ISSN 2434-1088

KURNS-EKR-5

アジア原子炉物理国際会議 2019 論文集

Proceedings ofReactor Physics Asia Conference 2019
(RPHA19)

開催日：2019 年 12 月 2 日および 3 日(December 2 and 3, 2019)

編集： 卞 哲浩、大岡靖典、辻田浩介、山中正朗
Edited by: Cheol Ho Pyeon, Yasunori Ohoka,
Kosuke Tsujita and Masao Yamanaka

京都大学複合原子力科学研究所
Institute for Integrated Radiation and Nuclear Science, Kyoto University

Preface
Establishing this RPHA (Reactor PHysics Asia) conference series was agreed by the
Reactor Physics Divisions (and/or mathematics and computation) of nuclear (or atomic
energy) societies of Korea, China and Japan to promote knowledge and information exchange
between the nuclear reactor physics related groups of the nuclear industry, academia, andresearch sectors in Asian countries. With the growing need for the safe and effective
utilization of nuclear energy in the Asian region which lacks abundant sources of cheap andclean energy, the outcomes of the theoretical, experimental, computational, and operational reactor
physics research and developments of the Asian reactor physics related groups could play an
instrumental role in the stable and expanded use of nuclear power. RPHAs, which will takeplace every two years by rotating the host among the three countries, could provide a very
efficient meeting ground for sharing technical information and encouraging mutual
collaboration. The regional closeness and the cultural similarities of the Asian countries
would be another important benefits of the RPHA meetings as well.
The RPHA19 (Reactor PHysics Asia Conference 2019) will be held at the Osaka
International House Foundation in Japan on December 2 and 3, 2019, organized by the
Reactor Physics Division of the Atomic Energy Society of Japan, the Reactor Physics &Mathematical Computational Division of the Korean Nuclear Society, and the MathematicalComputational Division of the Chinese Nuclear Society, and supported by the Kindai
University Atomic Energy Research Institute, and the Institute for Integrated Radiation and
Nuclear Science, Kyoto University.
Finally, we would like to give special thanks for their support and patience, by all
members of the Organizing committees in RPHA19, to hold this conference.

Cheol Ho Pyeon (KURNS)
Yasunori Ohoka (Nuclear Fuel Industries, Ltd.)
Kosuke Tsujita (Nuclear Engineering, Ltd.)
Masao Yamanaka (KURNS)

November 2019
Keywords:
Reactor physics, Mathematic and Computation, Nuclear datai

要旨
アジア原子炉物理国際会議 RPHA（Reactor PHysics Asia）は，アジア諸国の原子力産業や研究開発機関における原子炉物理学の情報交換を促進するため，韓国，中国，日本の原子力学会炉物理部会（及び計算科学部会）によって設立されたものであり，3 ヵ国間でホストを交代しながら 2 年毎に開催している原子炉物理学に特化した国際会議である。経済的でクリーンなエネルギー源に乏しいアジア地域において，原子力エネルギーの安全で効率的な利用に関するニーズは高まりに伴い，アジア諸国の原子炉物理学に基づく理論や実験，解析手法，
原子炉の運用手法といった研究開発は，原子力エネルギー利用の拡大に対し重要な役割を果たすことができる。RPHA は，そのようなニーズに応えるため，最新の研究成果を共有し，相互協力を促進するために有効な会議体であり，加えてアジア諸国の地理的な近さや文化的な類似性に関してもまた RPHA における重要な利点である。
2019 年 12 月 2 日および 3 日に大阪国際交流センターにおいて開催される RPHA19（Reactor
PHysics Asia 2019 Conference）は、日本原子力学会 炉物理部会，韓国原子力学会炉物理 計
算科学部会，中国原子力学会 計算科学部会が主催し，近畿大学 原子力研究所，京都大学 複合原子力科学研究所の後援を受けている。
最後に，RPHA19 組織委員会のメンバーに心より感謝申し上げます。

京都大学複合原子力科学研究所

卞 哲浩

原子燃料工業株式会社

大岡 靖典（株）原子力エンジニアリング

辻田 浩介

京都大学複合原子力科学研究所

山中 正朗

2019 年 11 月
キーワード：
炉物理，計算科学，核データ

iiContents
SS-01 Advanced Monte Carlo Modeling・・・・・・・・・・・・・・・・・・・・・・

1

SS-02 Source-Driven Subcritical System・・・・・・・・・・・・・・・・・・・・・・

18

SS-03 Nuclear Education and Training [Panel Discussion]
SS-04 Fukushima Challenge・・・・・・・・・・・・・・・・・・・・・・・・・・・

30

TS-01 Deterministic Transport Method and Applications・・・・・・・・・・・・・・・ 44
TS-02 Monte Carlo Method and Applications・・・・・・・・・・・・・・・・・・・・ 98
TS-03 Core Analysis Method・・・・・・・・・・・・・・・・・・・・・・・・・・・ 119
TS-04 Reactor Kinetics・・・・・・・・・・・・・・・・・・・・・・・・・・・・・

200

TS-05 Reactor Physics Experiments・・・・・・・・・・・・・・・・・・・・・・・・ 216
TS-06 Nuclear Data and Evaluation・・・・・・・・・・・・・・・・・・・・・・・・ 228
TS-07 Uncertainty Quantification and Data Assimilation・・・・・・・・・・・・・・・ 248
TS-08 Multiphysics Power Reactor Simulations・・・・・・・・・・・・・・・・・・・ 286

iii

目次

SS-01 高度モンテカルロ法・・・・・・・・・・・・・・・・・・・・・・・・・・

1

SS-02 中性子源駆動未臨界システム・・・・・・・・・・・・・・・・・・・・・・

13

SS-03 原子力教育・訓練（パネルディスカッション）
SS-04 福島原子力発電所事故に対する技術開発・・・・・・・・・・・・・・・・・

30

TS-01 決定論的輸送計算手法とその応用・・・・・・・・・・・・・・・・・・・・

44

TS-02 モンテカルロ法とその応用・・・・・・・・・・・・・・・・・・・・・・・

98

TS-03 原子炉炉心計算手法・・・・・・・・・・・・・・・・・・・・・・・・・・

119TS-04 原子炉炉心動特性・・・・・・・・・・・・・・・・・・・・・・・・・・・

200

TS-05 原子炉物理実験・・・・・・・・・・・・・・・・・・・・・・・・・・・・

216

TS-06 核データ評価・・・・・・・・・・・・・・・・・・・・・・・・・・・・・

228

TS-07 不確かさ評価，データ同化・・・・・・・・・・・・・・・・・・・・・・・

248

TS-08 マルチフィジクス解析・・・・・・・・・・・・・・・・・・・・・・・・・

286

iv

SS-01

Advanced Monte Carlo ModelingNo.

RPHA19-1058

Title

MCS Multi-group Cross Sections Generation for Fast Reactor
Analysis

Authors

Tung Dong Cao Nguyen, Hyunsuk Lee, Xianan Du, Vutheam Dos, Tuan
Quoc Tran, Deokjung Lee (Ulsan National Institute of Science andTechnology, Korea)
2

No.

RPHA19-1079

Title

A Systematic Way to Determine Neutron Generation Size in
Monte Carlo Simulation Accelerated by the CMFD

Authors

Inhyung Kim, Yonghee Kim (Korea Advanced Institute of Science andTechnology, Korea)
6

No.

RPHA19-1112

Title

Validation of Activation Analysis Method on RV/RVI of Kori
Unit 1 Using McCARD/ORIGEN2

Authors

Young In Kim, Sang Hoon Jang, Dong Hyuk Lee, Hyung Jin Shim (Seoul
National University, Korea)
10

No.

RPHA19-1062

Title

Uncertainty Quantification of Neutronics Characteristics in
Thermal Systems Using Random Sampling and Continuous
Energy Monte-Carlo Methods

Authors

Hiroya Oike, Ryoichi Kondo, Tomohiro Endo, Akio Yamamoto (Nagoya
University, Japan)
141Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

MCS Multi-group Cross Sections Generation for Fast Reactor Analysis
Tung Dong Cao Nguyen, Hyunsuk Lee, Xianan Du, Vutheam Dos, Tuan Quoc Tran and Deokjung Lee
Ulsan National Institute of Science and Technology (UNIST)
50 UNIST-gil, Ulsan 44919, Republic of Korea*
Corresponding author: deokjung@unist.ac.kr

Abstract
Current work introduces a brief methodology for multi-group (MG) cross sections(XSs) generation by Monte Carlo (MC) code MCS, which can be compatible withnodal diffusion code, PARCS. The applicability of the methodology is quantified
on the sodium fast reactor ABR-1000 design with a metallic fuel loaded. The MG
XSs generated by MCS with a 2D sub-assembly are well consistent with those of
SERPENT 2. Furthermore, the solutions of beginning-of-cycle steady-state MG
calculation of MCS/PARCS for a whole-core problem, including the core keff andpower profiles, are compared to those of the MCS MC code. Overall, the code-tocode comparison indicates a reasonable agreement between deterministic andstochastic codes, with the difference in keff less than 100 pcm and the root-meansquare error in assembly power less than 1.15%. Therefore, it is successfully
demonstrated the employment of the MCS MG XSs generation for PARCS is apromising system to accurately perform neutronic analyses for fast reactors.
Key Words: Monte Carlo, multi-group cross sections, fast reactor, MCS
1. Introduction

MCS is a 3D continuous-energy neutron-physics code forparticle transport based on the MC method, underdevelopment at UNIST since 2013 [1-2]. MCS canconduct criticality runs for reactivity calculations andfixed-source runs for shielding problems. MCS has been
designed from scratch since 2013 to conduct whole-core
criticality simulation with pin-wise depletion andthermal/hydraulic feedback. MCS neutron transport
capability is verified and validated against several
benchmark problems, including the BEAVRS benchmark,~300 cases from the International Criticality Safety
Benchmark Experimental Problem (ICSBEP) and the
Jordan Research and Training Reactor (JRTR).

In general, a Monte Carlo (MC) code or a deterministiccode is to be employed to simulate the nuclear reactor.
The main advantages of MC codes are in the accuracy ingeometry simulation and neutron interactions. However,those are counterweighted with the high computational
cost due to a complex and detail level of a certain model.
On the other hand, deterministic code’s capability is toprovide an adequately accurate result, which requires lesscomputational demand. Nevertheless, their disadvantage
is the simplicity in geometry and transport/diffusion
physics. Recently, the combination of both stochastic anddeterministic codes has become more and more attractivein the framework of establishing the basic viability of theadvanced fast reactor. This approach ensures thesuperposition of the benefits of these methods. The major
purpose of a code system is to achieve accurate evaluation
and analysis in a systematic manner for a wide variety offast reactor. Therefore, current research is focused on theuse of both MC and deterministic codes for 2D/3D fast
reactor simulation and analysis. In this work, thegeneration of the multi-group (MG) cross sections (XSs)
data by MC method is implemented in the in-house
UNIST MC code MCS. These XSs data is then converted
into the compatible database that is able to be used in
nodal diffusion code PARCS.

3.2 SERPENT 2
The SERPENT MC code [3] is a continuous energy MC
reactor physics burnup code with recent applications inradiation shielding, multi-physics, and fusion neutronics.
It is currently employed for reactor physics applications,
including homogenized group constant generation,
burnup calculations, the modeling of small research
reactor cores and multi-physics calculations. SERPENT
has been developed at VTT Technical Research Centre of
Finland since 2004 and the current development version,
SERPENT 2, has notably diversified the applications of
the code.

2. Computer Codes

3.3 PARCS

3.1 UNIST Monte Carlo code MCS

2

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

PARCS is a 3D reactor core simulator, which solves thesteady-state and time-dependent, MG neutron diffusion
and SP3 transport equations in square and hexagonal
geometries [4].

those of SERPENT 2. Then, the XS set for 24-groupenergy structure [7] is obtained by MCS and converted
into a compatible database that can be used in the nodal
diffusion code PARCS for the fast reactor 3D core
problem. The two energy structures are listed in Tables I
and II. The ENDF/B-VII.0 library is used, where the fueland other material temperature is set 900K and 600K,
respectively.

3. Multi-group Cross Sections Generation with
Monte Carlo Method
A general goal of the MC code is to estimate the average
neutron flux for a certain geometrical region and for anumber of energy ranges (energy group) [5]:g =1 Eg −1 (r , E )dEdV ,V VEg

Table I. 4-group Energy Structure.

No.
1
2
3
4

(1)

where

No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

V = volume for averaging flux region andg = group index.
From this, the specific reaction rates can be obtainedby multiplying flux estimates by an interest XS:
1 Eg −1
 x ( r , E )  (r , E )dEdV ,V VEg(2)

Division of these two quantities gives the MG XS fora certain reaction type x and energy group g: x, g =1 Eg −1
 x ( r , E )  (r , E )dEdV
V VEg

g

,

(3)

As discussed, the generated XSs are to be utilized in
MG nodal diffusion code, the required data are therefore
the total XS, the absorption XS, the fission and fission
production XS, fission spectrum, scattering matrix andtransport XS. It is noted that the Pn scattering matrix inthis study is only weighted by the scalar flux:

ls , g →g' =1 Eg ' ls ( r , E → E ')  (r, E )dEdV
V VEg

g

,

(4)

G

Lower E (MeV)

1.11090E-01
1.72230E-03
2.67030E-05
1.00000E-11

Upper E (MeV)
1.96403E+01
1.00000E+01
6.06531E+00
3.67879E+00
2.23130E+00
1.35335E+00
8.20850E-01
4.97871E-01
3.01974E-01
1.83156E-01
1.11090E-01
6.73795E-02
4.08677E-02
2.47875E-02
1.50344E-02
9.11882E-03
5.53084E-03
3.35463E-03
2.03468E-03
1.23410E-03
7.48518E-04
4.53999E-04
3.04325E-04
1.48625E-04

Lower E (MeV)
1.00000E+01
6.06531E+00
3.67879E+00
2.23130E+00
1.35335E+00
8.20850E-01
4.97871E-01
3.01974E-01
1.83156E-01
1.11090E-01
6.73795E-02
4.08677E-02
2.47875E-02
1.50344E-02
9.11882E-03
5.53084E-03
3.35463E-03
2.03468E-03
1.23410E-03
7.48518E-04
4.53999E-04
3.04325E-04
1.48625E-04
1.00001E-11

4.1 Multi-group cross section comparison between MCS
and SERPENT 2

where l = scattering order, which is also the lth order
Legendre polynomial coefficient.
The transport XS is estimated by using the outerscatter approximation:
tr , g = t , g −  1s , g → g ' ,1.41910E+01
1.11090E-01
1.72230E-03
2.67030E-05

Table II. 24-group Energy Structure.

 ( r, E ) = the space and energy dependent flux

Rx =Upper E (MeV)

Since the original aim of this work is to serve in the fast
reactor analysis and development, the fuel sub-assemblyof a typical sodium fast reactor, ABR-1000 [6], is
employed in this analysis. Fig. 1 illustrates the 2D layoutof the ABR-1000 sub-assembly [6]. Table III summarizes
the 4-group flux and XSs generated by MCS compared tothose of SERPENT 2. It is seen that MCS solution is wellagreed with SERPENT 2 solution, and it also notice thatthe considered error at group 4 is due to the poor
uncertainties as lack of neutrons in the small energy
ranges. Figs. 2 and 3 shown the P0 and P1 scattering andtheir uncertainties (in %) between MCS and SERPENT 2.
The index in the green row stands for the departure energy(5)g ' =1

where G = the total number of groups.
4. Numerical Results
First, the XS set for 4-group energy structure and for a 2D
assembly problem is generated by MCS and compared to3Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019group, while the vertical orange column indicates thearrival energy group. All the group-wise scattering XSs
by MCS are well consistent with those of SERPENT 2,
except group 4 as a result of the lack of neutron in thethermal energy range of a fast reactor assembly.

Fig. 2. P0 scattering matrix error, MCS vs. SERPENT 2.
Error (% )

1

2

3

4

1

-0.01

--

--

--

2

0.01

0.02

--

--

3

14.05

0.11

-0.44

--

4

-117.12-101.91214.67-103.86

SD (% )

1

2

3

4

1

0.02

--

--

--

2

0.09

0.05

--

--

3

23.63

0.19

1.26

--

4

115.765681.69

107.31

166.93Fig. 3. P1 scattering matrix error, MCS vs. SERPENT 2.
4.2 ARB-1000 analysis by MCS/PARCS
The radial core layout of the ABR-1000 is shown in Fig.
4 [6]. XSs for fuel sub-assemblies are generated using asingle 2D model of a fuel sub-assembly with reflective
boundary conditions, as in Fig. 1. All MG XSs for nonmultiplying regions are generated using 2D super-cellmodels as shown in Fig. 5 [7]. Those regions are locatedat the center of the model and surrounded by fuelassemblies to approximate the flux that those experiencein a core. The keff for 2D/3D core calculation at beginningof-cycle (BOC) all rod out (ARO) by MCS/PARCS andMCS is summarized in Table IV. The radial assemblywise power distributions by MCS/PARCS and MCS are
shown in Figs. 6-8. It is clearly seen that a great agreement
is achieved due to the difference in keff less than 100 pcm
and the root-mean-square (RMS) in power error is lessthan 1.15% and the maximum error is less than 2.5% forradial power and 4.73% for axial power.

Fig. 1. ABR-1000 fuel sub-assembly geometry.
Table III. Summary of 4-group Flux and XSs [cm-1].
Diff. vs.
SERPENT 2
Parameter
G
(%)
Value
SD (%)
ValueSD
1
1.18E+02
7.46E-03
0.22
0.01
2
6.53E+01
1.08E-02
0.21
0.01
Flux
3
1.12E+00
7.46E-02
0.29
0.11
4
3.03E-06
2.74E+01-11.31 45.63
1
2.08E-01
1.21E-03
0.00
0.00
2
3.02E-01
2.11E-03
0.02
0.00
Total
3
3.91E-01
6.40E-03
-0.01
0.02
4
5.51E-02
2.34E+01-93.11 24.38
1
4.39E-03
2.95E-03
0.02
0.01
2
6.93E-03
3.85E-03
0.02
0.01
Absa
3
2.67E-02
2.50E-02
-0.03
0.05
4
2.51E-02
3.90E+01-90.54 44.04
1
8.79E-03
5.06E-03
0.01
0.01
2
6.62E-03
2.05E-03
0.02
0.01
NSFb
3
2.13E-02
2.64E-02
-0.07
0.06
4
3.29E-02
5.92E+01-91.47 64.77
1
9.85E-01
1.90E-03
0.00
0.00
2
1.46E-02
1.28E-01
0.24
0.18
Chic
3
2.87E-05
3.04E+00
-0.09
4.30
4
5.01E-08
7.06E+01-44.60 97.40
1
1.71E-01
2.72E-03
0.01
0.00
2
2.89E-01
2.73E-03
0.02
0.00
Transport
3
3.87E-01
1.00E-02
-0.01
0.02
4
2.21E-02
2.24E+01-97.33 25.31
aAbsorption, bFission production, cFission spectrum
MCS

Error (% )

1

2

3

4

1

0.01

--

--

--

2

0.05

0.02

--

--

3

-2.65

0.14

-0.02

--

4

-62.01152.9826.76-95.27

SD (% )

1

2

3

4

1

0.00

--

--

--

2

0.02

0.00

--

--

3

3.07

0.09

0.03

--

4

62.96

75.01

51.58

28.60

Fig. 4. ABR-1000 radial core layout.

Fig. 5. Example for super-cell models.

4

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table IV. Summary of keff, MCS/PARCS vs. MCS.
MCS/
Diff.
Case
MCS
PARCS
(pcm)
2D core1.22120±0.00010
1.221299
3D core1.03011±0.00011
1.0310493

5. Conclusions
In this study, the feasibility of using the MCS MC codefor generating MG XSs at BOC for fast spectrum 2D/3D
simulation with the PARCS simulator is investigated. The
4-group XSs by MCS are compared with SERPENT 2
results, and a great agreement is achieved. The final stepis to perform steady-state analysis with PARCS using the
24-group XS data generated by MCS and conduct a
comparison in keff and power distributions. Results pointout a reasonable consistency, mainly explained by thedifference between deterministic and stochastic code. The
outcome of this study proves the MCS can be a promising
tool for MG XSs generation for fast reactor analysis. The
future work is focused on the application of MCS for
temperature dependent and coolant density dependent
MG XSs for thermal-hydraulics feedback for fast reactor.
Acknowledgements
This work was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea
government (MSIT). (No.NRF-2017M2A8A2018595)

Fig. 6. Normalized assembly radial power distribution
for 2D core, MCS/PARCS vs MCS.

References
1. Lee H et al., “MCS – A Monte Carlo Particle
Transport Code for Large-scale Power Reactor
Analysis,” Annuals of Nuclear Energy (2019). (Under
review)
2. Jang J et al., “Validation of UNIST Monte Carlo code
MCS for Criticality Safety Analysis of PWR Spent
Fuel Pool and Storage Cask,” Annuals of Nuclear
Energy, 114, 495-509 (2018).
3. Leppänen J et al., “The Serpent Monte Carlo Code:Status, Development and Applications in 2013,”
Annals of Nuclear Energy, 82, 142-150 (2015).
4. Downar T, “PARCS v2.7 U.S. NRC Core Neutronics
Simulator USER MANUAL,” U.S. NRC (2006).
5. Hoogenboom J et al., "Generation of Multi-group
Cross Sections and Scattering Matrices with theMonte Carlo Code MCNP5," Proc. Joint International
Topical Meeting on Mathematics & Computation andSupercomputing in Nuclear Applications, Monterey,
California, April 15-19, p.15-19 (2007).
6 Bernnat W et al., "Benchmark for Neutronic Analysis
of Sodium-Cooled Fast Reactor Cores with Various
Fuel Types and Core Sizes," Nuclear Energy Agency
Technical Report, NEA/NSC (2015).
7 Pereira G et al., "Applying the Serpent-DYN3D Code
Sequence for the Decay Heat Analysis of Metallic
Fuel Sodium Fast Reactor," Annals of Nuclear Energy,
125, 291-306 (2019).

Fig. 7. Normalized assembly radial power distribution
for 3D core, MCS/PARCS vs MCS.

Fig. 8. Normalized axial power distribution,
MCS/PARCS vs MCS.

5

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

A Systematic Way to Determine Neutron Generation Size in Monte Carlo Simulation
Accelerated by the CMFDa

Inhyung Kima and Yonghee Kima*
KAIST, Daehak-ro 291, Daejeon, Republic of Korea (34141)*
Corresponding author: yongheekim@kaist.ac.kr

Abstract
A modified particle ramp-up (m-PRUP) method was suggested and improved tosystematically determine the minimum generation size, optimum generation size,
and number of inactive cycles in a Monte Carlo simulation for an efficient andaccurate calculation. This paper describes the concept and algorithm for the mPRUP method in details. It provides a guideline for source convergence conditions.
It also explains how to determine the minimum generation size based on probability and statistics. The m-PRUP method was applied to a simple reactor problem tocharacterize its numerical performance compared to the standard MC calculation.
The convergence behavior was examined, and the multiplication factor, computing
time, and figure-of-merits (FOMs) were evaluated and compared for each method.
Key Words: m-PRUP method, Minimum generation size, Optimum generation size
convergence of the FSD and thus decrease the unnecessary computing time in the inactive cycle. After the roughconvergence is achieved, the number of histories increases. With new generation size, another stationary distribution is searched. This procedure continues until theconvergence of the FSD is achieved with the final generation size.

1. Introduction
In a Monte Carlo (MC) simulation, generation size, i.e.
the number of histories per cycle, is a significant factor
which can influence the reliability of the solution and theefficiency of the calculation. If generation size is huge,
the calculation can be inefficient due to a waste of time inunnecessary calculations during inactive cycles to determine a converged fission source distribution (FSD). On
the other hand, if the generation size is too small, the solution may contain a bias due to generation-correlation [1].
In addition, in the CMFD-assisted by MC calculation, it
may result in numerical instability and inconsistency inthe deterministic solution [2].
The modified particle ramp-up (m-PRUP) method
was proposed to give a guideline for an optimum generation size [3]. In this work, this method was further improved to suggest a minimum generation size which enhances the calculation efficiency and guarantees the numerical stability. Moreover, it can judge the convergenceof the FSD and systematically switch to the active MC
calculation. In this paper, the concept of the m-PRUP is
introduced in detail, and the method is applied to a benchmark problem to characterize its performance.
2. Modified Particle Ramp-up Method
2.1 Concept and algorithm
The m-PRUP method [3-4] is an algorithm for providing a minimum generation size, an optimum generation
size, and the number of inactive cycles in a systematicway. This method begins with a small number of histories.
The small generation size can quickly reach the roughFig. 1 Flowchart of the m-PRUP method
The algorithm of the m-PRUP method is shown in Fig.
1. In the figure, N 0 is the initial generation size, i is6

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019achieved depending on the initial generation size, or may
lessen the calculation efficiency. However, the stopping
criterion should be relatively tight for a reliable convergence of the FSD. In this regard, the stopping criterion ismore rigorous than the convergence criterion.

the cycle number, and N is the increment of the generation size. The convergence check is carried out by evaluating Shannon entropy [5]. If the difference of the entropy values is within the criterion, the method increases
the generation size (1st check), finds the optimal generation size (2nd check), and automatically switches to the active simulation (3rd check).
The entropy is estimated by taking the average of cycle-wise entropies such as
1 i
(1)
ei1 
 ek ,c k i c 1and
1 i c(2)ei0 
 ek ,c k i  2c 1

Table I. Convergence criteria for the m-PRUP method
Criterion
1
3C / N2
1C / N*C  0.03 and N is the current generation size

2.3 Minimum generation size
Initial generation size for the m-PRUP method is also
an important variable to enhance the calculation efficiency. If the initial generation size is pointlessly large, anamount of numerical effort would be wasted to determine
the converged FSD due to an inaccurate flat source data.
On the contrary, too small generation size can lead to numerical instability and inconsistency. Therefore, it is important to find the proper minimum generation size for them-PRUP method.
For a reactor core problem, zero flux or zero current
physically does not make sense. The given neutron particles per cycle should at least yield non-zero reactor parameters. In this respect, the probability of crossing allneighboring surfaces with N neutrons per node was estimated based on the Buffon’s needle problem [6].
According to the Buffon’s needle problem, the probability that a neutron traveling length l crosses any ofparallel lines each distance t apart can be obtained bywhere c is the cycle-accumulation length. If the relative
error of two quantities is within the first criterion such ase j1  e j0
 1 ,(3)
e j1
where j is the index for the generation size, and  1 isthe convergence criterion, then the generation size increases by the increment N . If the error of the entropies with two different generation sizes is within the second criterion such ase j1  e j11
 2,
(4)
e j1
where  2 is the stopping criterion, then the generation
size is determined and fixed. Last, if the difference of theentropies with the final (optimal) generation size is within
the last criterion such ase j1  e j0
 2 ,(5)
e j1

pcross 

2l 2 2 22l  t  1  sin 1 (t / l )
t t


(6)

A neutron track-length can be approximated by a crowflight distance (CFD) of the neutron which is defined as astraight root-mean-square distance from birth to death.
The CFD of the neutron is given bythen the simulation turns off the m-PRUP method andmoves into the active calculation.

l  6L,

2.2 Convergence criteria
Because the Shannon entropy is a statistical parameter,
it follows the intrinsic nature of stochastic parameters in
the MC simulation that the uncertainty decreases inversely proportional to the square root of the number ofhistories. Therefore, the convergence criteria are determined based on the function of the generation size.
The constant, C , for the criteria is derived by trial anderror based on numerous simulations. The inconsistentconclusion may be drawn depending on the constant C,
but many tests and simulations demonstrated that the adequate constant for the m-PRUP method was almost invariant regardless of the problems.
In the m-PRUP method, it is not necessary to accomplish
a strict convergence of the FSD in the first convergence
check. Too tight convergence condition may not be(7)

where L  D /  a is the diffusion length. The diffusion coefficient and absorption cross section can be obtained by the group condensation and spatial homogenization over the active core. The fast group constants could
draw the conservative conclusion, but the one-group constant also can yield sufficiently acceptable results. In thisanalysis, the one-group diffusion coefficient and absorption cross section were used.
The probability of crossing each surface (i.e. x, y, andz) can be estimated with consideration of radial and axial
node size. One neutron randomly distributed on a latticegrid can encounter total 8 cases including no surface
crossing (p), 3 cases with only one surface crossing (px,py, and pz), 3 cases with two surfaces crossing (pxy, pyz, andpzx), and crossing all surfaces simultaneously (pxyz).

7Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Therefore, the probability of crossing all surfaces with
one neutron can be calculated as

p1cross zero surface parameters at a probability less than 0.01%.
Therefore, in the PWR type reactor core problem, at least
4 to 5 neutrons per node are necessary for the efficient andreliable MC calculation.

pxyz

p  px  p y  pz  pxy  p yz  pzx  pxyz

 pxyzTable II. No. of neutrons per node depending on
the axial node size
Axial node size
No. of neutrons per node
5 cm
4.62
10 cm
4.73
20 cm
4.81

(8)

 pi  p j  pk
which is the only one case among 8 cases under the assumption that the probability of crossing each surface isindependent.
Next, the crossing probability with two neutrons canbe considered. Combination of two neutrons can make total of 82 cases. Among them, the cases which two neutrons
cross all surrounding surfaces are estimated as 27 cases.
2
pcross 2  ( p  pxyz  px  p yz  p y  pzx  pz  pxy

 pxy  p yz  p yz  pzx  pzx  p yx  px  pxyz
 p y  pxyz  pz  pxyz  pxy  p xyz  p yz  p xyz

The higher probability indicates that fewer neutrons
are required per node to make sure that the surface parameters are non-zero in the MC simulation. That is, when 5
cm of axial node size is used for a mesh grid, the numberof neutrons per node can be smaller. However, in terms of
the whole core simulation, the total number of nodes increases with decreasing the node size because of the many
axial division. As a result, the total number of neutrons
required for a whole core simulation will be higher withthe smaller axial node size.

(9)

 pzx  pxyz )  pxyz  pxyz .
In this manner, the probability of crossing all surfaces with N neutrons can be calculated. However, thenumber of cases and combinations exponentially increases with increasing the number of neutrons considered. Therefore, a simple computer code was developed
to numerically calculate the cases and probabilities for agiven condition. The computing time for the calculationof the cases and probability is almost negligible comparedto the MC simulation. It only requires less than 10 seconds
with the given data such as the reactor type, the problem
size, and so on.

3. Results and Discussion
3.1 Problem description
A simple reactor core problem was analyzed to verify
the numerical performance of the m-PRUP method in theMC simulation. The full core consists of 8 by 8 homogeneous uranium fuel assemblies surrounded by a reflector
as shown in Fig. 3. The radial assembly pitch size is 21.42
cm, and the height of the core is 64.26 cm. The core isaxially divided into three regions for the entropy calculation.

Fig. 3 Configuration of a reactor coreFig. 2 Probability of crossing all surfaces with N neutrons as a function of axial node sizes

3.2 Minimum generation size
Because each assembly contains 17 by 17 pin nodes,
the total number of pin node in this problem can be estimated to be 36,992 nodes. Therefore, according to the mPRUP method, the necessary minimum generation sizefor this problem is 177,931 by multiplying the total number of nodes and the necessary neutrons per node (4.81).
The MC simulation started with the 180,000 initial(minimum) histories, and the increment of the generation
size (ΔN) is also determined to be 180,000. The final (optimum) generation size and the number of inactive cycles

For a standard PWR type reactor core where the lowenriched uranium fuel is loaded and the pin pitch size is
1.26 cm, the probability of crossing all surfaces can becalculated as a function of the axial node size and thenumber of neutrons per node (Fig. 2). The probability dramatically increases with increasing the number of neutrons and decreasing the axial node size.
Using a fitting curve based on the data points, thenumber of neutrons per node can be derived to have non-8

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019are automatically obtained with the m-PRUP method.

4. Conclusions
The m-PRUP method was suggested and improved toprovide a guideline for systematically determining theminimum generation size, the optimum generation size,
and the number of inactive cycles in a MC calculation.
From the numerical analysis, the m-PRUP method
properly estimated the target variables and increased thecalculation efficiency by reducing the total number of histories necessary to determine the converged FSD. Thismethod can be applied to the CMFD-assisted MC simulation which often suffers from some divergence or bias depending on the mesh grid system, problem size, and thegeneration size etc. The m-PRUP method can stabilize thecalculation by the ramp-up technique and the algorithm togenerate the optimal generation size.

3.3 Convergence behavior
Fig. 4 compares the convergence behavior of the FSD
for the standard MC calculation and the m-PRUP method
application. The m-PRUP method begins with smaller
histories (i.e. 170,000) and allows a quick source convergence. When the difference of the entropies is smallenough, it increases the generation size and searches another stationary state with the new generation size. Afterseveral increments, the optimum generation size is finallyevaluated to be 540,000 histories per cycle and the simulation moves into the active calculation at cycle 47. On
the other hand, the standard MC calculation which uses
540,000 histories per cycle from the beginning shows theslow convergence in the FSD and reaches the stationary
level at around cycle 70.

Acknowledgements
This work was supported by the National Research
Foundation of Korea Grant funded by the Korean government NRF-2016R1A5A1013919.
References1.

2.

Fig. 4 Shannon entropy of the MC calculation
Table III demonstrates the numerical performance ofeach method through the FOM estimation. Both calculations show a good agreement of the multiplication factor
each other, and have a similar statistical error. However,
because the m-PRUP method uses much smaller histories
to obtain the converged source distribution, it can reduce
the numerical cost during the inactive cycle. As a result,
the total computing time is shortened, and thus the mPRUP gives the higher FOM than the standard MC calculation.

3.

4.

Table II. Comparison of reactor parameters
Parameters
Standard MC
m-PRUPkeff

1.38376

1.38376SD (pcm)

10.3

10.7

Inactive

48

13

Active

59

55

Total

106

68

14710

21401

Computing
time
(min)
FOM

5.

6.

9

Ely M. Gelbard, “Biases in Monte Carlo Eigenvalue
Calculations”, invited paper for the International
Symposium on Scientific Computing and Mathematical Modeling (IMACS), Bangalore, India, December
7-11, (1992).
YuGwon Jo & Nam Zin Cho, “Acceleration and Real
Variance Reduction in Continuous Energy Monte
Carlo Whole-Core Calculation via p-CMFD Feedback”, Nuclear Science and Engineering, Vol. 189(1), pp. 26-40, (2018).
Inhyung Kim & Yonghee Kim, “A Study on Deterministic Truncation of Monte Carlo Transport Calculation”, Reactor Physics (PHYSOR 2018), Cancun,
Mexico, April 22-26, (2018).
Amanda L. Lund Paul K. Romano, and Andrew R.
Siegel, "Accelerating Source Convergence in Monte
Carlo Criticality Calculations Using a Particle Rampup Technique," in International Conference onMathematics & Computational Methods Applied to
Nuclear Science & Engineering (M&C 2017), Jeju,
Korea, (2017).
Forrest B. Brown, "On the Use of Shannon Entropyof the Fission Distribution for Assessing Convergence of Monte Carlo Criticality Calculations," inReactor Physics (PHYSOR 2016), Vancouver, BC,
Canada, (2006).
Schroeder L., “Buffon’s Needle Problem: An Exciting Application of Many Mathematical Concepts”,
Mathematics Teacher, Vol. 67 (2), pp. 183-6, (1974).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Validation of Activation Analysis Method on RV/RVI of Kori Unit 1 Using
McCARD/ORIGEN2
Young In Kim, Sang Hoon Jang, Dong Hyuk Lee and Hyung Jin Shim*
Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea*
Corresponding author: shimhj@snu.ac.kr

Abstract
Activation analysis on Reactor Vessel and Reactor Vessel Internal of Kori Unit 1
is conducted for the planning of decommission. Unlike conventional activation
analysis methods that calculate radioactivity of radioactive isotopes
conservatively for convenience of calculation, this study pursued more accurate
calculation while keeping efficiency. Representative cores which can represent
the full cycle is selected and the calculation results are compared with the casewhich reflects full operating history. It is evaluated whether such a method isvalid by comparing the time plot of the specific activity.
Key Words: Activation Analysis, Kori Unit 1, McCARD, ORIGEN2, Sp
ecific Activity
1. Introduction

calculated by using McCARD, a Monte Carlo (MC)particle transport analysis code, and the ORIGEN2 code.
The radioactivity is compared with the result using themost precise way which traces the full cycle operating
history.

In the reactor decommissioning process, RV (Reactor
Vessel) and RVI (Reactor Vessel Internal) need to be
dismantled which are impossible to dispose of in their
original form. For the planning of the dismantling process,high-performance high-precision volumetric activation
analysis must precede which enables classifying of theradioactive waste level. Conventionally, activation
analysis is conducted in two-step procedure whichcalculates neutron flux and reaction cross sections at thedesired location by neutron transport code and calculates
radioactivity of RI(radioactive isotope) using depletion
code. The most precise way to get neutron flux is tracing
the full cycle operating history of the power plant andreflecting the fuel loading pattern of each cycle to thewhole core neutron transport analysis. But it takes a lot of
calculation time and is a burden to perform in this way
while getting neutron flux with a sufficiently small
statistical error. So several conventional methods choose
to get a conservative result rather than a precise result.
The radioactivity calculation using neutron flux obtained
from the last working core in the operating history gives
conservative results because neutron flux increases overtime in a cycle of operation. When calculating neutron
flux using neutron transport code, assuming low burnup
fuel assemblies are located on the periphery of the corealso gives conservative results due to maximized leakage
caused by the arrangement. Although these methods can
reduce the computational burden, they overestimate theradioactivity of RI.
This paper aims to devise activation method whichguarantees the accuracy of result with efficient calculation.
For the efficient activation analysis of Kori Unit1,
representative cores that can represent the full cycle
operating history are selected and radioactivity is

2. Modeling of Kori Unit 1

Fig. 1. Top view and side view of Kori Unit 1
The left side of Fig. 1 illustrates a top view of Kori Unit1.
It consists of 11×11 fuel assemblies, baffle, former,
thermal shield, reactor vessel, and insulation. Fuelenrichment, burnable poison rad and loading pattern wereconsidered for modeling of Kori Unit 1 core. The Rightside of Fig. 1 illustrates a side view of Kori Unit1. Various
guide tubes, instrument tubes, support plates were10

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

implemented in the geometry modeling. Leg nozzles andsimplified concrete structures are also modeled which are
not fully shown in the figure.
3. Methodology
To get the most accurate neutron flux and cross-sections
at the structures, several feedback effects were applied to
McCARD calculation. The calculation is performed in astate where the temperature gradient of fuel and coolant,boron concentration and xenon concentration areconverged at each burnup step by using the critical boron
concentration search module, the thermal-hydraulicfeedback module, and the equivalent xenon module.
Fig. 3. Calculation procedure of simple core tracking
method

3.1 Whole core tracking calculation method

The fuel composition of the MOC core was calculated
in two steps. First, the composition of a fuel assembly fordetailed burnup steps was calculated by McCARD
assembly-wise burnup calculation. Then, the compositionof the MOC core was acquired by interpolation according
to burnup. Activation calculation was performed using
McCARD / ORIGEN2 system which calculates neutron
flux and cross sections at the desired location with
McCARD and conducts depletion calculation with
ORIGEN2.

Fig. 2. Calculation procedure of whole core tracking
methodThe whole core tracking calculation method can produce
a reference solution to the activation analysis because it
is a method to simulate the lifetime of a reactor as it is.
The calculation of each cycle repeats neutron transport
calculation and depletion calculation using McCARD
according to burnup step given in nuclear data report andupdates the composition of fuel and structure. The
structural information for fuel assemblies and burnablepoison rods was also acquired from the nuclear design
report.
At the end of a cycle calculation, discharged fuel isreloaded according to the core loading pattern or new fuelis loaded. Then the whole core tracking calculation is
performed following the previous cycle.

Fig. 4. Neutron flux and conversion factor at baffle
Figure 4 illustrates neutron flux and conversion factor,
which is ratio of real flux to estimated flux from MC
calculation, calculated using a simple core tracking
method and whole core tracking method.
As a result of the comparison up to 15 cycles, neutron flux
and the conversion factor evaluated by the simple coretracking method were well followed by the results of thewhole core tracking method and the validity of the simple
core tracking method was verified.

3.2 Simple Core tracking calculation method
While the whole core tracking method can produce areference solution with no approximation, it requires longcomputing time and is challenging. Therefore, we carriedout activation analysis using a simple core tracking
method which assumes that the MOC(middle of the cycle)
core represents a cycle without performing burnup
calculation according to detailed burnup steps. So a totalof 32 cores was modeled for simple core tracking
calculation.

3.3 Representative core calculation method
Simple core tracking calculation is still a burdensome
method. Therefore, the final calculation is carried out by
the representative core calculation method which selects
the cores that can represent the whole cycle. Based on the11Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

results of neutron flux calculation using a simple coretracking method, cores that have neutron flux equivalent
to mean neutron flux of each section were selected as therepresentative core. This considers that the radioactivityof structure is proportional to fluence, the total amount of
neutrons irradiated on the structure over a given time.

inaccurate and actually be deduced to follow the result ofsimple core tracking calculation. TableⅠ shows the
specific activity of Fe-55, Ni-59, Co-60, Ni-63 at threedifferent regions after 10 years from when the reactor
operation stopped. The specific activity ratio of therepresentative core calculation method is alike a simple
core tracking method, and it also backs the claim that the
representative core calculation method is valid. It can bealso shown that the results of whole core tracking method
are about 10% higher than those of other methods and it
is explained as inaccuracy resulting from extrapolation.

Fig. 5. Decision of representative core with neutron flux
at baffle
As shown in Fig. 5, Results of simple core tracking
calculations show that the transition of neutron flux ishighly fluctuated in the anterior cycle and reaches
equilibrium state after a certain cycle. Therefore, selectionof representative core was performed by selecting twocores representing the transient cycle of 1 to 11 cycles anda core representing the equilibrium cycle of 12 to 32
cycles, each of which was determined as cycle 5 and cycle
29.

Fig. 6. Specific activity of Ni-63 at thermal shield

4. Calculation Results
Radioactivity of various radioactive isotope at RV/RVI
was calculated using three different methods. Whole coretracking calculation was performed up to cycle 14 and thevalue of remaining cycles are obtained by extrapolation.
Figures 6 and 7 show that the representative corecalculation method using MOC core eventually follows
Fig. 7. Specific activity of Co-60 at barrel
the result of a simple core tracking calculation. It can beinferred that the results obtained from extrapolation are
Table I. Comparison of the specific activity of radioactive isotopes at the barrel, thermal shield, a reactor vessel
RegionBarrelThermalShieldReactor Vessel

RI
Fe55
Ni59
Co60
Ni63
Fe55
Ni59
Co60
Ni63
Fe55
Ni59
Co60
Ni63

Specific Activity (Bq/g)
Whole core tracking Representative core Simple core tracking
Ratio (B/A) Ratio (C/A)
(A)
(B)
(C)
7.27E+07
6.15E+07
6.28E+07
0.85
0.86
9.65E+05
8.16E+05
8.58E+05
0.85
0.89
2.64E+08
2.21E+08
2.22E+08
0.84
0.84
1.25E+08
1.10E+08
1.13E+08
0.88
0.90
1.11E+07
9.99E+06
9.86E+06
0.90
0.89
1.47E+05
1.38E+05
1.37E+05
0.94
0.93
4.57E+07
4.20E+07
4.13E+07
0.92
0.90
1.87E+07
1.74E+07
1.73E+07
0.93
0.93
4.69E+05
3.77E+05
3.91E+05
0.80
0.83
2.90E+02
2.53E+02
2.64E+02
0.87
0.91
8.92E+04
7.93E+04
7.88E+04
0.89
0.88
3.68E+04
3.14E+04
3.27E+04
0.85
0.89
12

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

5. Conclusions

9.

Activation analysis of RV/RVI in Kori Unit1 is performed
using McCARD. Three different methods, whole coretracking calculation method, simple core tracking
calculation method, representative core calculation
method are applied to calculate specific activity of various
radioactive isotopes. For precise calculation, criticalboron concentration, thermal- hydraulic effect, xenon
concentration were considered at all of the methods. The
transition of the specific activity of isotope according totime was compared. The most simplified method,
representative core calculation method, showed similar
results to other methods and estimated to be sufficiently
valid. For the representative core calculation method, we
have to decide which core states can represent wholecycles and simple core tracking calculation should be
preceded. But it still holds its efficiency when we roughly
obtain neutron flux trend through cycles with relatively
small number of histories, and conduct main calculation
with selected representative core with a larger number ofhistories.
References
1.
2.

3.

4.

5.

6.

7.

8.

B. Roque et al. "International Comparison of a
Depletion Calculation Benchmark on Fuel
Cycle Issues", NEA/NSC/DOC(2013)1.
H. J. Shim, et al. "McCARD: Monte Carlo code
for advanced reactor design and analysis."
Nuclear Engineering and Technology 44.2
(2012): 161-176.
MJ Bell, “ORIGEN: the ORNL isotope
generation and depletion code”, No. ORNL—
4628. Oak Ridge National Lab., Tenn.(USA),1973.
DE Peplow, et al. "Monaco/MAVRIC:
Computational Resources for Radiation
Protection and Shielding in SCALE."
TRANSACTIONS-AMERICAN NUCLEAR
SOCIETY 95 (2006): 669.
Sanghwa Shin and Jooho Whang, "A Study on
Source Term Assessment of the Kori unit-1
Reactor Vessel", Journal of NUCLEARSCIENCE and TECHNOLOGY, Supplement 4,
p. 86-89, March, 2004.
Yong IL Kim, "Application of SCALE 6.1
MAVRIC Sequence for Activation Calculation
in Reactor Primary Shield Concrete",
Transactions of Korean Nuclear Society, Jeju,
Korea, 2014.
NUREG/CR-3474, J. C. Evans et al., "LongLived Activation Products in Reactor
Material,"
NUREG/CR-3474,
Pacific
Northwest Laboratory (1984).
Kontogeorgakos, D., F. Tzika, and I. E.
Stamatelatos, "Neutron Activation Study of the
GRR-1 Research Reactor Core Supporting
Plate," Nuclear Technology, 175[2], 435, 2011.

13

Ben Volmert, Elena Tamaseviciute, ManuelPantelias, Pavol Zvoncek and Beat Bitterli,"MCNP neutron streaming investigations fromthe reactor core to regions outside the reactor
pressure vessel for a Swiss PWR", Progress in
Nuclear Science and Technology, Vol 4, pp.
481-485, 2014.

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

UNCERTAINTY QUANTIFICATION OF NEUTRONICS CHARACTERISTICS IN
THERMAL SYSTEMS USING RANDOM SAMPLING AND CONTINUOUS ENERGYMONTE-CARLO METHODS
Hiroya OIKE*, Ryoichi KONDO, Tomohiro ENDO, and Akio YAMAMOTO
Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, 464-8603, Japan,*
Corresponding author: h-oike@fermi.energy.nagoya-u.ac.jp

Abstract
Uncertainty quantification of neutronic characteristics using the random sampling(RS) technique and a continuous energy Monte-Carlo code(MCNP) for thermal
neutron systems is conducted. The RS code of the ACE-format cross section is used,which is developed in Nagoya University using the modules of FRENDY. In thisstudy, all reactions in multi-group covariance data of SCALE 6.2.3 are considered.
Through perturbation of nuclear data and statistical processing of 𝑘eff obtained by
MCNP6.2, the uncertainty quantification of 𝑘eff is conducted. Comparison of thepresent result and the deterministic result obtained by the first-order perturbation
theory (TSUNAMI-1D) showed that the uncertainty quantification results in thermal
neutron systems using random sampling and MCNP are adequate.
Key Words: Uncertainty Quantification, Random Sampling, MCNP,FRENDY, TSUNAMI-1D, Thermal Neutron Systems
method. Kondo et al. developed an RS code for the ACE(A Compact ENDF) cross section files [3]. This code uses
the functional modules of the FRENDY [4] code, which
is a nuclear data processing system developed by Tada etal.
In the previous research[3], the RS method wasapplied to the Godiva core, which is a simple and baresphere of 235U, with MCNP, and its validity wasconfirmed through comparison with the deterministic
method(TSUNAMI-1D). However, the validity of the RS
method in a complicated thermal system with light
nuclides, moderators and reflectors has not been
confirmed yet.
The first purpose of this study is the investigation of
the validity of the RS method using the continuous energyMonte-Carlo method in the thermal systems containing
light nuclides.
The second purpose of this study is decomposition of
total uncertainties obtained by the RS method. Usually,
the RS method can quantify only total uncertainties even
if there are many sources of uncertainties. When
breakdown of uncertainties is necessary, the RS methodshould be repeatedly used for each source of uncertainty.
When the number of uncertainty sources is large, it will
be impractical. Therefore, possibility of uncertainty
decomposition using a set of random sampling
calculations considering all sources of uncertainty is
pursued in the present study [5].

1. Introduction
The aim of the present paper is the uncertainty
quantification of neutronics characteristics in thermal
systems using the random sampling and a continuous
energy Monte-Carlo methods. An analysis result of areactor core inevitably includes the uncertainty. To
improve the reliability of core analysis, uncertainty
quantification of prediction results is important [1]. The
uncertainties of predicted results mainly come fromanalysis methods and input parameters. The former isclassified into modeling approximations in a deterministic
method and statistical error of a probabilistic method.
Examples of the later source are fuel composition,
reactor/fuel geometry, and nuclear data. Recently, therandom sampling method has been used to evaluate theuncertainty of core characteristics in light water reactor
cores [2].
In the deterministic method, various approximations
are used in a core analysis such as discretization of space,angle, and energy to solve the Boltzmann transport
equation. By contrast, in the probabilistic method,especially in the continuous energy Monte-Carlo method,
fewer approximations are used. Therefore, theprobabilistic method is considered to be more accurate
than the deterministic method. In the previous study, theRS method has been used with a deterministic code andmulti-group cross section covariance of nuclear data [2].
However, from the viewpoint of accuracy, application of
the RS with a probabilistic method and continuous energy
cross section is desirable.
Perturbation of cross sections is necessary for the RS

14

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

2. Random Sampling-based Uncertainty
Quantification Method using MCNP

𝑇

(4)𝑉𝜎̃𝑘 = 𝑺𝑘𝜎̃ 𝑽𝜎̃,𝜎̃ 𝑺𝑘𝜎̃ ,where 𝜎̃ is specific nuclear data, 𝑉𝜎̃𝑘 is uncertainty due
to 𝜎̃ , which is extracted from total uncertainty; and𝐕𝜎̃𝑡𝑟𝑢𝑒is the multi-group covariance of nuclear data and,𝜎̃𝐕𝜎̃,𝜎̃ is a sample covariance matrix of nuclear dataobtained by the RS calculations. 𝐒𝜎𝑘̃ is a sensitivity
coefficients of neutronics parameter k with respect to 𝜎̃,and is calculated in Eq. (5).
𝐒𝜎𝑘̃ = 𝐕𝜎̃−1
(5)
̃,
,𝜎
̃ 𝐕𝑘,𝜎where 𝐕𝑘,𝜎 is a sample covariance vector between k and𝜎̃ . We attempt to decompose total uncertainty into
contributions from individual nuclides using Eq. (3) or (4)and Eq. (5).

This section describes outline of the uncertainty
quantification using the RS method and MCNP [2][3].
Firstly, multi-group relative variance-covariancematrix of nuclear data R is created. The diagonal elementsof this matrix contain variances for nuclear data and theoff-diagonal elements represent covariances among them.
Secondly, I sets of perturbation factors f, which are thefactors to make perturbations on cross sections, aregenerated. Note that dimension of f is (number of nuclide
× number of energy groups × number of reaction types).
The cross sections are assumed to obey a Gaussian
distribution. Using the matrix R and the multivariate
normal random numbers z in Eq. (1), the perturbed crosssections can be obtained considering the variance andcovariance of cross sections.
(1)
𝐟 = 𝐀𝐳 + 𝛍,where z is a vector of random number obeying thestandard normal distribution, 𝛍 is a vector whose
elements are 1, A is the matrix satisfying Eq. (2).
(2)
𝐑 = 𝐀𝐀𝑻 ,where T denotes a matrix transpose.
The perturbed cross sections are obtained bymultiplying f to the original(unperturbed) cross sections.
If negative value(s) is contained in the perturbation vectorf, this vector is discarded and regenerated. This treatment
may lead to positive bias on the result. However, negative
perturbation factors are generated for cross sections withlarge uncertainty, which would be cross sections of minor
nuclides. Generally, such nuclides have minor impact on
the results and our experiences in the previous studies
support this expectation.
Thirdly, the generated perturbation factors are
multiplied to continuous energy cross section data in theACE-format for MCNP to make perturbed cross sections.
The RS code for ACE-format cross sections was used
[3][4]. Note that the created matrix R is multi-group, thus
continuous energy cross sections are uniformly perturbed
within the energy range of a multi-group.
Fourthly, neutronics calculations using MCNP is
carried out using perturbed ACE-format cross sections [6].
Finally, I output parameters, specifically the effective
multiplication factors, are obtained and the relative
standard deviations are calculated in order to estimate theuncertainty of multiplication factor.

4. Calculation Conditions
In this study, multi-group relative variance-covariancematrix is taken from the 56 energy group covariance dataof SCALE6.2.3 (56groupcov7.1) [7].
ENDF/B-VII.1 in the ACE-format is used for
MCNP[8].
If the effective rank of covariance matrix 𝐕𝜎̃,𝜎̃ issmaller than that of matrix size, the singular value
decomposition and the low-rank approximation, or
pseudoinverse can be used to avoid a numerical issue tocalculate 𝐕𝜎̃−1,𝜎̃ . Note that the pseudoinverse is used in thepresent study [5].
A sphere composed of high enriched uranium
oxyfluoride solution in the ICSBEP handbook (HEUSOL-THERM-012) [9] is calculated in this study.
Number densities of the core are shown in Table I.
Hydrogen to Uranium-235 ratio is 1272. Radius of thecore and volume are 27.9244 cm and 91.21 liters,
respectively. The core is surrounded by 0.2 cm thickspherical 1100 aluminum tank and 15 cm thick waterreflector. The benchmark system is under room
temperature. The measured 𝑘eff is 0.9999 ± 0.0058.
All nuclides and reactions having the covariance data,
which are shown in Table II, were perturbed in this study.
Firstly, four sets of RS calculations were carried out
making the perturbation for 1H, 16O, 235U, and all nuclides.
The sample size is 100 for each set because the available
computational time was limited. Total number of neutron
histories was 50 million, i.e., numbers of neutron histories
per cycle, active cycles, and skip cycles are 10000, 5025,
and 25, respectively, for one sample.

3. Decomposition Method of Total Uncertainty

Table I. Compositions of HEU-SOL-THERM-012 coreThis section describes outline of the decomposition
method of total uncertainty into contribution from eachnuclide.
An estimation method of sensitivity coefficients ofneutronics parameters using the RS method was proposedin the previous study [5]. When the sensitivity coefficientsare available, the partial uncertainty owing to specificnuclear data can be calculated by Eq. (3) or Eq. (4).
𝑇𝑘
(3)𝑉𝜎̃𝑘 = 𝐒𝜎𝑘̃ 𝐕𝜎̃𝑡𝑟𝑢𝑒
,𝜎
̃ 𝐒𝜎
̃,

Elements
234
U
235
U
236
U
238
U
19
F
16
O
17
O
1
H

15

atoms/barn-cm
5.5393 × 10-7
5.2444 × 10-5
2.8022 × 10-7
2.9675 × 10-6
1.1249 × 10-4
3.34603 × 10-2
1.27197 × 10-5
6.6722 × 10-2

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table II. All Nuclides and reactions
considered in the calculation.
Reaction
Elastic(n, n′ )(n, 2n)Fission(n, γ)(n, p)(n, d)(n, t)(n,3 He)(n, α)(n, νtotal )Fission spectrum

Nuclide
1
H
16
O
17
O
19
F
27
Al
28
Si
29
Si
30
Si
55
Mn
63
Cu
65
Cu
234
U
235
U
236
U
238
U

(a)1H

Secondly, 2000 RS calculations are carried out todecompose total uncertainty into contribution from eachnuclide. In this case, the sample size is determined toreduce statistical error of estimated covariance matrix
𝐕𝑘,𝜎̃ . In these calculations, number of active cycles are
reduced to 525 to shorten computation time.

(b)235U

5. Numerical Results
Figure 1 shows the distributions of effective
multiplication factors obtained by RS for 1H, 16O, 235U,
and all nuclides.
Table III provides the comparison of the uncertanities
obtained by the present RS calculations and by thedeterministic method based on the first-order perturbation
theory(TSUNAMI-1D)[7]. The 95 % confidence interval
obtained by the bootstrap method with 10000 bootstrap
samples is also shown in Table III [10][11].
Table III indicates that the results obtained by the RS
and the deterministic method agree within the range ofstatistical error, which shows the validity of the present
method in the thermal neutron system.

(c)All nuclides
Fig. 1 The histogram of 100 𝑘eff in the RS method
for 3 cases. The number of bins in the histograms
is 20.
Table IV shows the decomposition results of total
uncertainty using Eq. (3) or (4) and Eq. (5) with 2000
samples and deterministic classification results of eachnuclide uncertainty.
The “Total all nuclide uncertainty” in Table IV was
calculated by Eq. (3) or Eq. (4) considering all nuclear
data. The “Sum up each nuclide uncertainty” wassummation of Eq. (3) or Eq. (4) for individual nuclides.
The covariance matrix obtained by the RS calculations isused for the results in 𝐕𝜎̃,𝜎̃ column calculated by Eq. (3)while the true covariance matrix 𝐕𝜎̃𝑡𝑟𝑢𝑒
is used for the,𝜎̃
results in 𝐕𝜎̃𝑡𝑟𝑢𝑒
columncalculated
by
Eq.
(4).
,𝜎
̃

Table III Comparison of relative 𝑘eff uncertainty
TSUNA
95% conf.
RS
MI-1D
interval
All nuclides
0.81
0.81
0.698-0.919
1
H
0.44
0.44
0.371-0.496
16
O
0.10
0.10
0.090-0.116
235
U
0.67
0.60
0.511-0.693

16Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

RS method. Comparison of the results by the RS method
and the deterministic method (TSUNAMI-1D) indicatesthe validity of present method. Decomposition of total
uncertainty is also tried and the results are promising; theuncertainties by the major contributors can be accurately
decomposed.
The present study clarifies the applicability of the RS
method coupled with accurate continuous energy MonteCarlo method. The present approach is flexible, accurate
and general, and can be used to obtain the reference valueof uncertainty.

Table IV Breakdown of uncertainty of 𝑘eff

Nuclide
Al
Cu
65
Cu
19
F
1
H
55
Mn
16
O
17
O
28
Si
29
Si
30
Si
234
U
235
U
236
U
238
U
Total ofall nuclide
uncertainties
Sum up ofeach nuclide
uncertainty
27

63

relative standard deviation(∆k/k%)
𝐕𝜎̃,𝜎̃

𝐕𝜎̃𝑡𝑟𝑢𝑒,𝜎̃

TSUNAMI
-1D

0.02
0.03
0.03
0.02
0.44
0.03
0.10
0.03
0.02
0.02
0.02
0.04
0.64
0.03
0.04

0.01
0.03
0.03
0.02
0.44
0.02
0.10
0.01
0.02
0.02
0.02
0.03
0.67
0.03
0.03

0.00
0.00
0.00
0.00
0.44
0.00
0.10
0.00
0.00
0.00
0.00
0.00
0.67
0.00
0.00

0.76

0.81

References1.

Cacuci DG, Sensitivity and Uncertainty Analysis:
Theory, Vol. 1, Chapman and Hall/CRC, Florida,
USA (2003).
2. Yamamoto A, Kinoshita K, Watanabe T, et al.,“Uncertainty Quantification of LWR Core
Characteristics Using Random Sampling Method,”
Nucl. Sci. Eng., 181, 2, 160 (2015).
3. Kondo R, Endo T, Yamamoto A, et al., ”
Implementation of Random Sampling for AceFormat Cross Sections using FRENDY andApplication to Uncertainty Reduction,” Proc. M&C
2019, Portland, OR, Aug.25-29, 2019, American
Nuclear Society (2019).
4. Tada K, Nagaya Y, Kunieda S, et al., “Development
and Verification of a New Nuclear Data Processing
System FRENDY,” J. Nucl. Sci. Technol., 54, 7,
806 (2017).
5. Chiba G, Kawamoto Y, Tsuji M, et al., “Estimation of
Neutronics Parameter Sensitivity to Nuclear Data in
Random Sampling-Based Uncertainty Quantification
Calculations," Ann. Nucl. Energy, 75, 395 (2015)
6. Werner CJ (Ed.), “MCNP User’s Manual Code
Version 6.2,” LA-UR-17-29981, Los Alamos
National Laboratory (2018).
7. Rearden BT, Jessee MA (Eds.), ”SCALE Code
System,” ORNL/TM-2005/39 Version 6.2.3, Oak
Ridge National Laboratory (2018).
8. Chadwick MB, Herman M, Obložinský P, et al.,“ENDF/B-VII.1 Nuclear Data for Science andTechnology: Cross Sections, Covariances, Fission
Product Yields and Decay Data,” Nucl. Data Sheets,
112, 12, 2887 (2011).
9. Pitts M, Rahnema F, Williamson TG, et al., “WaterReflected 91-Liter Sphere of Enriched Uranium
Oxyfluoride Solution,” NEA/NSC/DOC/(95)03/II,
HEU-SOL-THERM-012, OECD Nuclear Energy
Agency (2004).
10. Efron B, “Bootstrap Methods: Another Look at the
Jackknife,” Ann. Stat., 7, 1, 1 (1979).
11. Endo T, Yamamoto A, Watanabe T, “ConfidenceInterval Estimation by Bootstrap Method for
Uncertainty Quantification using Random Sampling
Method,” J. Nucl. Sci. Technol., 52,7, 993 (2015).

0.81
0.78

0.81

Table IV indicates that the decomposition results of
total uncertainty were appropriately estimated except forminor contributors. The statistical error of the MCNP in
this calculation conditions is about 0.025 %; thus
decomposition of minor contributors having small
uncertainty was not accurate.
Table IV also indicates that the results using 𝐕𝜎̃𝑡𝑟𝑢𝑒
,𝜎̃column are closer to the deterministic results than thoseusing 𝐕𝜎̃,𝜎̃ since 𝐕𝜎̃𝑡𝑟𝑢𝑒
utilizes the true covariance.
,𝜎̃
Even if the number of samples increases, 𝐕𝜎̃,𝜎̃ is notequal to 𝐕𝜎̃𝑡𝑟𝑢𝑒
because perturbation factor f has positive
,𝜎
̃
bias.
The values “Total of all nuclide uncertainties” and“Sum up of each nuclide uncertainty” using 𝐕𝜎̃𝑡𝑟𝑢𝑒,𝜎̃ areclose; thus correlation among nuclides is sufficiently
small. Therefore, “Total of all nuclide uncertainties” and“Sum up of each nuclide uncertainty” using 𝐕𝜎̃,𝜎̃ will becloser when the number of samples increases.
6. Conclusions
The uncertainty quantification of neutronics
characteristics in the thermal system is carried out using
the continuous energy Monte-Carlo code MCNP and the17

SS-02

Source-Driven Subcritical System

No.

RPHA19-1024

Title

Experimental Analyses of Spallation Neutrons by 100 MeV
Protons and Lead-Bismuth Target at Kyoto University
Criticality Assembly

Authors

Kota Morioka (Osaka Prefecture University College of Technology,
Japan), Masao Yamanaka (Kyoto University, Japan), Kimihiko Sugiura(Osaka Prefecture University College of Technology, Japan), Cheol Ho
Pyeon ( Kyoto University, Japan)
19

No.

RPHA19-1003

Title

Applicability of Extended Kalman Filter Technique toReactivity Monitoring at Kyoto University Criticality Assembly

Authors

Masao Yamanaka, Cheol Ho Pyeon (Kyoto University, Japan)
22

No.

RPHA19-1051

Title

Core Design of Accelerator Driven System with Reactivity
Control Method Using Thorium

Authors

Kentaro Nakamura, Hideki Saito, Naoto Aizawa, Tomohiko Iwasaki(Tohoku University, Japan)
26

No.

RPHA19-1081

Title

Neutron Correlation Analysis for Subcritical Reactor System
Driven by a Pulsed Spallation Neutron Source in KUCA

Authors

Kunihiro Nakajima, Atsushi Sakon, Sin-Ya Hohara, Kazuki Takahashi,
Tadafumi Sano (Kindai University, Japan), Masao Yamanaka, Cheol Ho
Pyeon (Kyoto University, Japan), Kengo Hashimoto (Kindai University,
Japan)

No.

RPHA19-1074

Title

A Correlation Characteristics of Spurious Counts of Fission
Counter Installed in Kyoto University Reactor for Reactor
Operation

Authors

Sin-Ya Hohara, Atsushi Sakon, Tadafumi Sano, Kunihiro Nakajima,
Kazuki Takahashi, Kengo Hashimoto (Kindai University, Japan)

18

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Experimental Analyses of Spallation Neutrons by 100 MeV Protons and Lead-Bismuth
Target at Kyoto University Criticality Assembly
Kota Moriokaa,*, Masao Yamanakab, Kimihiko Sugiuraa, Cheol Ho Pyeonb
Technological Systems, Mechanical Engineering Course, Osaka Prefecture University College of Technology, 26-12
Saiwai-cho, Neyagawa, Osaka 572-8572, Japan;bResearch Center for Safe Nuclear System, Institute for Integrated Radiation and Nuclear Science, Kyoto University,
Asashiro-nishi, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan;*Corresponding author: f18022@osaka-pct.ac.jpaAbstract
Irradiation experiments are carried out at the Kyoto University Criticality
Assembly aimed at revealing the accuracy of numerical calculations of spallation
reactions by 100 MeV protons and a lead-bismuth target. The neutron yield andthe spectrum of spallation neutrons are obtained by reaction rate analyses of analuminum, an indium and a bismuth foils. The experimental analyses are
performed by using PHITS3.10 together with a nuclear model INCL and several
neutron data libraries. The results show the agreement of the neutron yield andthe neutron spectrum within the relative difference of 10% and 20% between
measured and calculated reaction rates. Finally, the accuracy of calculations isrevealed through a series of the irradiation experiments.
Key Words: Spallation neutron, Reaction rate analysis, neutron spectrum,
Neutron yield, KUCA
reveal the accuracy of analyses of ADS experiments with
Pb-Bi target.
The purpose of this study was to examine the accuracyof spallation calculations on the neutron yield and theneutron spectrum through the comparison between
calculated and measured reaction rates.

1. Introduction
The accelerator-driven system (ADS) was proposed
aiming at a large amount transmutation of minor actinides
and long-lived fission products by operating subcritical
rector with spallation neutrons generated by high-energy
proton injections onto a heavy metal (lead-bismuth (PbBi)) target [1-2]. At the Kyoto University Critical
Assembly (KUCA), the basic research on feasibility of
ADS has been accumulated with the combined use of
subcritical cores and spallation neutrons generated by
100MeV protons, revealing that the neutronic
characteristics are varied by the spectrum of spallation
neutrons [3]. Furthermore, for the ADS design, the studyof the neutron source is emphasized quite important since
the power of the reactor could be affected by the neutron
yield that is the number of neutrons generated by injecting
a proton onto a target.
The spectrum analysis was conducted for spallation
neutrons generated by high-energy (150, 190 and 235
MeV) protons and tungsten target with reaction rates ofbismuth (Bi) [4-5]. In previous study [5], the spectrum of
spallation neutrons was correctly evaluated by Bi foils ingood agreement within the relative difference of 10%
between the calculation (MCNPX) and the experiment.
Also, the analyses have been conducted on reaction rates
in the research at the KUCA. Although the Pb-Bi target
was employed in the ADS experiments since the target isplanned to be used in actual ADS, the analyses ofspallation neutrons by Pb-Bi target are a remained to

2. Experimental Settings
Irradiation experiments were carried out at the beam
dump to obtain the reaction rate on the yield and thespectrum of the spallation neutrons. In case of themeasurement of the neutron yield, an aluminum (Al) andan indium foils (10×10×1 mm) were used for obtaining
reaction rates by protons and neutrons by attaching onto
the upper stream and the downstream of the Pb-Bi target(50 mm diam. and 18 mm thick; Fig. 1), respectively. In
case of the neutron spectrum, the Bi foil (50 mm diam.
and 2 mm thick) was attached on to the downstream of thetarget.
The proton beam was injected by an operation of
proton (FFAG) accelerator at pulsed frequency of 30 Hz,
and 300 pA and 1 nA during irradiation time of 2.5 and 4
hours in irradiation experiments of Al and In foils, and of
Bi foil, respectively.
Reaction rates of irradiated foils were measured by thesaturated activities D∞ (s-1·cm-3) deduced from γ-ray
measurements with the use of the highly-purified
germanium detector as follows:19

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019D 

 TC C  D  E 1  exp   Ti  exp   Tw   1 exp   TC   M,

4.0 [8] for neutrons (under 20 MeV). The transport
calculation was conducted with 5×105 total histories. The
proton spectrum injected into Bi foil indicated by monoenergy around 100 MeV and moderated spectrum, as
shown in Fig. 2. The combined spectrum was caused byscattering attributed to injecting 100 MeV protons onto
beam window (3 mm thick) before the Bi foil. By the
injection of the protons onto the Pb-Bi target, thespallation neutrons with wide range of spectrum weregenerated with two components: a most dominant
spectrum similar to that by fission reactions and a uniquehigh-energy spectrum ranging between 10 and 100 MeV.
The reaction rate analysis was conducted by multiplying
calculated neutron or proton spectra and target crosssections with PHITS3.10 code. Here, ENDF/B-VI.8 [9],JENDL/D-99 [10] and JENDL/HE-2007 [11] were used
for reaction rates of 27Al(p, n+3p)24Na, 115In(n, n’)115mIn
and 209Bi(n, xn)210-xBi (n = 4 to 8) reactions, as shown in
Fig.3.

(1)

where λ indicates the decay constant, Tc the counting time,
C the count rate, εD the detection efficiency, εE theemission rate, Ti the irradiation time, Tw the waiting time
before the start of the decay measurement, ρ the density,and M the mass of the foil, respectively. Here, thefundamental characteristics of irradiation foils used inexperiment are shown in Table I.
Lead-bismuth target(50 mm diam. and 18 mm)

Beam window(3 mm thick)Proton beam

100
Spallation neutrons
10-1

Fig. 1

BiBifoilfoil(50mm(50 mm diam.
×3 mm)
diam.×2mm)
or
In foil (10×10×1 mm)

Flux [A. U.]

Al foil (10×10×1 mm)

Schematic of irradiation experiments.

4.486 h

10-3

10-5
10-3

10-2

10-1

101

27

24

115

115m

206

Bi

209

205

Bi(n, 6n)204Bi

209

Al(p, n+3p) Na(22.56 MeV)
209
Bi(n, 5n)15.31 d703.5 (0.31)(29.63 MeV)
209
Bi(n, 6n)11.22 h374.8 (0.82)
670.7 (0.11)(38.08 MeV)
209
Bi(n, 7n)

11.76 h(45.34 MeV)
209
Bi(n, 8n)1.71 hCross section [barn]Bi(n, 4n)803.1 (0.99)

101

102

Energy [MeV]209

6.243 d100

Fig. 2 Spectra of injected protons on to Bi foil andgenerated spallation neutrons.

336.2 (0.37)(0.34 MeV)
209
Bi(n, 4n)(54.24 MeV)

10-2

10-4

Table I. Specification of measured reactions and γ-ray.
Reaction,
Half
Energy [keV](Thresholdlife(Emission rate [-])
energy)27
Al (p, n+3p)
15.00 h1368.6 (1.00)(3.25MeV)
115
In(n, n’)Injected protons

209

100

In(n, n')Bi(n, 5n)InBi

Bi(n, 7n)203Bi

209

Bi(n, 8n)202Bi

10-1

10-2
10-1

100

101

102

Energy [MeV]

822.7 (0.45)
1679.6 (0.09)
1847.3 (0.12)
422.1 (0.84)
657.5 (0.02)
960.7 (0.99)

Fig. 3 Cross sections of 27Al(p, n+3p)24Na, 115In(n,n’)115mIn and 209Bi(n, xn)210-xBi (x = 4 to 8) reactions.
3.1 Neutron yield
The neutron yield (number of generated neutrons by the
injection of a proton) was investigated through thecomparison of reaction rates by 27Al(p, n+3p)24Na and
115
In(n, n’)115mIn reactions for measuring the number ofprotons and neutrons respectively, at the location of PbBi target.
Measured 115In(n, n’)115mIn reaction rates were

3. Analyses of Spallation Neutrons
Experimental analyses were performed by PHITS 3.10
code [6] with cross sections by the nuclear model INCL
[7] for proton and neutrons (above 20 MeV) and JENDL-

20

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

normalized by the measured 27Al(p, n+3p)24Na ones so as
to represent the neutron yield in the experiment. Here,115
In(n, n’)115mIn indicates constant cross section ranging
between 2.5 and 10 MeV (Fig. 3), predicting to play therole of a neutron detector for dominant neutron spectrumof spallation neutrons (Fig. 2). The experimental analyses
agreed with the measured neutron yield within a relative
difference of 13%, as show in Table II, indicating thevalidity of the calculated spallation neutron spectrum andthe cross section of 27Al(p, n+3p)24Na reaction.

4. Conclusions
Irradiation experiments were carried out so as to examine
the prediction accuracy of the spectrum and the yield of
spallation neutrons with 100 MeV proton accelerator and
the Pb-Bi target. The experimental analyses of reaction
rates were numerically analyzed by PHITS3.10 together
with INCL and nuclear data libraries.
Calculated reaction rate of the neutron yield comparedwith measured one within the relative difference of 13%.
Furthermore, the reaction rates on spallation neutron
spectrum indicated agreement within 20% except for
209
Bi(n, 7n)206Bi reaction, demonstrating the validity ofspallation neutron calculations and nuclear data libraries.
For comprehensive evaluation of spallation calculations,various nuclear data libraries are examined inexperimental analyses of the neutron spectrum and theneutron yield to evaluate accuracy of the series of reaction
rate analysis as the future work. Also, the calculation
model is planned to be elaborated for the beam spot.

Table II. Comparison between the calculation and theexperiment in terms of neutron yield.
Neutron /Calculation
Experiment
C/E
proton115In(n,n’)
115mIn/Al (p, n+3p)24Na27

0.210±0.002

0.185±0.004

1.13±0.03

3.2 Neutron spectrum

References

The irradiation experiment was carried out on the
spectrum evaluation of spallation neutrons with Bi foil.
Measured reaction rates were normalized by 209Bi(n,4n)206Bi reaction rate for organizing reaction rates interms of neutron spectrum. Here, 209Bi(n, xn)210-xBi crosssections cover neutron energy ranging between 20 and
100 MeV for the high-energy component of the spallation
neutron spectrum in Fig. 2.
The results of experimental analyses indicatedagreement between measured and calculated reaction
rates within a relative difference of 20% except for 209Bi(n,7n)206Bi cross section, as shown in Table III. A
discrepancy of 209Bi(n, 7n)206Bi reaction was attributed to
the nuclear cross section since the cross section of 209Bi(n,xn)210-xBi reaction overlaps in very narrow energy and no
drastic variation was observed in the spectrum of thespallation neutrons ranging between 20 to 100 MeV.
Further investigation was considered necessary forspectrum estimation by comparison between various
cross sections for protons in a future work. Additionally,
elaboration of the beam spot modeling needs to beconsidered since cylindrical spot was approximately usedin this study.

1.

Rubbia C, et. al., “A Conceptual Design of a Fast
Neutron Operated High Power Energy Amplifier,”
CERN/AT95-44 ET, Switzerland (1995).4.
2. Sugawara T, et. al., “Research and Development
Activities for Accelerator-Driven System in JAEA,”
Prog. Nucl. Energy, 106, 27 (2018).
3. Pyeon CH, et. al., “Neutron Characteristics of Solid
Targets in Accelerator-Driven System at Kyoto
University Critical Assembly,” Nucl. Technol., 192,
181 (2015).
4. Kim E, et. al., “Measurement of Neutron Spallation
Cross Sections of 12C and 209Bi in the 20- to 150-MeV
Energy Range,” Nucl. Sci. Eng., 129, 209 (1998).
5. Pyeon CH, et. al., “Reaction Rate Analysis of Nuclear
Spallation Reactions Generated by 150, 190, and 235
MeV Protons,” J. Nucl. Sci. Technol., 47, 1090 (2010).
6. Sato T, et. al., “Features of Particle and Heavy Ion
Transport code System (PHITS) version 3.02,” J.
Nucl. Sci. Technol. 55, 684 (2018).
7. Boudard A, et. al., “New Potentialities of the Liège
Intranuclear Cascade Model for Reactions Induced by
Nucleons and Light Charged Particles,” PhysicalReview C87, 014606 (2013).
8. Shibata K, et. al., “JENDL-4.0: A New Libraryfor Nuclear Science and Technology,” J. Nucl. S
ci. Technol. 48, 1 (2011)..
9. “Evaluated Nuclear Data File,” ENDF/B-VI,
National Nuclear Data Center, Brookhaven National
Laboratory (1990).
10. Kobayashi K, et. al., “JENDL Dosimetry File 99(JENDL/D-99),” JAERI 1344, Japan Atomic Energy
Agency (2002).
11. Takada H, et. al., “Validation of JENDL High-Energy
File Through Analyses of Spallation Experiments at
Incident Proton Energies from 0.5 to 2.83 GeV,” J.
Nucl. Sci. Technol., 46, 589 (2009).

Table III. Reaction rates and comparison between thecalculation and the experiment in terms of neutron
spectrum
Reaction
Measured reaction
C/E
rate [1/cm3/s]
209

Bi(n, 4n)(8.22±0.09)×104-

209

Bi(n, 5n)(4.29±0.42)×104

0.84±0.09

209

Bi(n, 6n)(2.23±0.02)×104

1.20±0.03

209

Bi(n, 7n)(1.42±0.04)×104

1.44±0.06

209

Bi(n, 8n)

(3.54±0.09)×103

1.01±0.05

21

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Applicability of Extended Kalman Filter Technique to Reactivity Monitoringat Kyoto University Criticality AssemblyaMasao Yamanakaa, , Cheol Ho Pyeona
Research Center for Safe Nuclear System, Institute for Integrated Radiation and Nuclear Science, Kyoto University,
Asashiro-nishi, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan;*
Corresponding author: m-yamanaka@rri.kyoto-u.ac.jp

Abstract
Transient experiments with external neutron source are carried out at the Kyoto
University Criticality Assembly with the variation of subcriticality, so as toinvestigate applicability of the extended Kalman filter (EKF) technique to on-line
monitoring of subcriticality. The external neutron source is generated by an
injection of 100 MeV protons onto a lead-bismuth target, observed inherent
instability (beam swing and intense oscillation) in beam current. In the EKF
technique, two different state space models are prepared for an estimation of only
reactivity and a simultaneous estimation of reactivity and beam current. The resultof subcriticality estimation is revealed accurate for reference subcriticality by
MCNP6.2, indicating the applicability of EKF technique to reactivity monitoring.
Key Words: On-line monitoring of subcriticality, Extended Kalman filter,
Transient experiment, Spallation neutron source, KUCA
1. Introduction

2. Extended Kalman Filter

The Kalman filter technique has been adopted for
accurately evaluating reactivity variation by combined
use of measured data and theoretical modeling in aframework of data assimilation [1]. The reactivity
variation could be directly recognized by the response ofneutron detectors. The reactivity variation is easily
observed by the filtering technique even when thedetector count is very poor. For accurate subcriticality
monitoring, the particle filter was introduced as a strong
tool so as to take into account the non-linearity in the state
space model based on the one-point kinetic equation [2].
Among the filtering techniques, the extended Kalmanfilter (EKF) treats 1st-order approximation of thenonlinearity with slight calculation cost.
The EKF was applied to the subcriticality monitor
only in rod drop transient in critical experiments andsubcritical experiments with a stable neutron source [3-4].
Here, subject was remained to accurately estimate thesubcriticality when varying the neutron source intensity,
such as beam trip and restart behavior in an acceleratordriven system. Then, two different state space models
were attempted to examine the capability for single
parameter (reactivity) estimation and simultaneous
estimation of reactivity and neutron source intensity.
The purpose of this study was to investigate
applicability of the EKF technique to the subcriticality
monitor even with unstable neutron source.

The EKF was proposed to be applicable to indirectly
estimating parameters of the non-linear system from the
Kalman filter. On the basis of the one-point kinetic
equation, the state space model and the observation
equation were prepared as follows:x  k 1  f  x  k    b v  k  ,(1)y  k   h  x  k   w  k  ,(2)where x is state space variable, y observation matrix, f amatrix expressing the state space, b a vector distributing
system noise, v system noise, h a the observability matrix,
and w observation noise. The non-linearity wasconsidered by derivative A and cT, as follows:Ak  cT  k  f  x  k  xh  x  k  
x

,

(3)

,

(4)



x  x k x  x k 

where 𝑥̂ − is a priori estimate. The priori estimate isevaluated with the use of state estimate 𝑥̂ in previous
time step, as follows: Corresponding author, E-mail: m-yamanaka@rri.kyoto-u.ac.jp

22Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019x  k   f x  k  1

 .

were initiated at stable state first. Then, initial values inx(0) were set by solving the one-point kinetic equation instable state with measured neutron count at time = 0 s.
In case of simultaneous (reactivity and beam intensity)
estimation, observed current and differential of thecurrent in time step were appended to Eqs. (10), (11) and(12), as follows:(5)

The priori error covariance matrix P - is evaluated, as
follows:
P   k   A  k  P  k 1 AT  k    v 2b bT(6),g   k   P   k  C  k   C T  k  P   k  C  k    w2  .
1







(7)

(14)

6

   k    effn  k    i Ci  k    S  k   
 n  k   T 

i 1





  eff ,1C1  k   T n  k   1 C1  k  









  eff ,2

C
k
T
n
k

C
k


  2 2  



2 
 





  eff ,3C3  k   T n  k   3 C3  k  







f  x  k     eff ,4 ,C4  k   T n  k   4 C4  k  









  eff ,5C5  k   T n  k   5 C5  k  









  eff ,6C6  k   T n  k   6 C6  k   (15)  k S  k     k T k 



 .

(8)

Finally, for next time step, the posteriori error covariancematrix is prepared, as follows:y k   n S  ,t

The state estimate is evaluated with the use of observation
results and the priori state estimate by the most likelihood
parameter g, as follows:


(13)

t

where P is a posteriori error covariance matrix. The
Kalman gain g is determined, as follows:x k   x k   g k  y k   h x k 

x  k    n C1 C2 C3 C4 C5 C6  S   ,P k   Ι  g k   CT k  P  k  .

(9)In this study, simultaneous estimation of reactivity andneutron source intensity was attempted besides single
estimation of reactivity. In case of single (reactivity)
estimation model, the x, y and f were expressed asfollows:

where S is beam current and ω slope in the variation ofbeam current.

x  k    n C1 C2 C3 C4 C5 C6   ,t(10)

3. Experimental Settingsy  k    n ,(11)Transient experiments were carried out by combining
subcritical cores and an external neutron source generated
by an injection of 100 MeV protons on to a lead-bismuth(Pb-Bi) target. The subcritical cores were constituted bylead-loaded fuel rods “f” (central region: highly-enricheduranium (HEU) plate and lead plate; both sides: HEU
plate and polyethylene moderator) and normal fuel rod “F”(HEU plate and polyethylene moderator) as shown in Figs.
1 and 2. The core spectrum in the central region of fuelrod “f” was approximated to that of a fast rector byloading lead plates. Furthermore, the core spectrum in thenormal fuel region had relative hard spectrum by H/Uratio of approximately 50 in polyethylene-moderated
cores at KUCA.
The proton beam was generated by an operation of
proton (FFAG) accelerator at pulsed frequency of 20 Hz,
and varied beam current by the accelerator instability:
swing and rapid oscillation. The spallation neutrons weregenerated with Pb-Bi target placed at [A, 15; Figs. 1 and2], and its intensity of the neutron source was differed by
the variation of the beam current (approximately10 to 70
pA).
In transient experiments, control rods and safety rods
were inserted during the variation of beam current for twot6   k    effn  k    i Ci  k    I  k   
 n  k   T 

i 1





  eff ,1C1  k   T n  k   1 C1  k  









  eff ,2
C2  k   T n  k   2 C2  k  


 





  eff ,3C3  k   T n  k   3 C3  k  f  x  k   

 ,







eff ,4
C4  k   T n  k   4 C4  k  


 





  eff ,5C5  k   T n  k   5 C5  k  









  eff ,6C6  k   T n  k   6 C6  k   (12)
 




k





where n is the neutron count, ρ reactivity, βeff, i effective
delayed neutron fraction of i-th group, λi delayed neutron
decay constant of i-th group, Ci density of delayed neutron
precursor, ε source efficiency (determined by initialsubcriticality and neutron count), and I observed beam
current source T the time step. The transient experiments

23

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019different subcritical cores so as to investigate theapplicability of EKF technique to the monitoring in a wide
range of subcriticality and variation of the neutron source
intensity. Here, the subcriticality was numericallyobtained by using MCNP6.2 together with ENDF/B-VII.1,
as shown in Table I.
Time evolution of neutron signals was recoded for a
BF3 detector placed at [Q, 5; Figs. 1 and 2] to prevent thevariation of detector efficiency and neutron source
efficiency during transient experiments. In addition toneutron signals, the time series data on the proton beam
current were obtained for the observable parameter in the
EKF. The time resolution in on-line monitoring was 1 s.
1

2

3

4

5

6

7

Gr

A

8

Proton beam transient experiments were carried out fordifferent type of variation in the beam current: swing andintense oscillation involving short beam trip. With the useof these transient experiments, the applicability of thereactivity monitor was revealed by comparing theaccuracy between different modeling for the state spacewith and without estimation of the source intensity as
described in Sec. 2.
4.1 Swing of proton beam current
Beam swing transient experiment were conducted by
gradually varying the beam current. The neutron countrate was varied as shown in Fig. 3 by the beam transient
as shown in Fig. 4. The beam transient actually involved
spiked beam trip and restart, however, the focus wasplaced on the global variation of the beam swing.
The subcriticality was also varied during the beamswing by an actuator-driven insertion of control andsafety rods through Cases I-1 to I-4 (Table I) between
1000 and 1900 s. From the count rate distribution (Fig. 3),
reactivity variation could be recognized only at 1200 s and1800 s because of beam current increase (Fig. 4).
However, the emphasis should be placed where reactivity
monitoring correctly indicated subcriticality variationcompared with reference subcriticality during the beamswing. Furthermore, the accuracy was revealed that thesimultaneous estimation is better than the single(reactivity) estimation in the beam swing transient.

9

Gr

B

p

p

p

p

p

p

p

p

p

D

p

p

p

p

p

p

p

p

p

E

p

p

p

p

p

p

p

p

p

F

p

p

p

p

p

p

p

p

p

G

p

C3

p

F

F

F

p

S5

p

H

p

p

F

F

F

F

F

p

p

I

p

p

F

F

F

F

F

p

p

J

p

S4

F

f

f

f

F

C1

p

K

p

p

F

f

f

f

F

p

p

L

p

p

F

f

f

f

F

p

p

M

p

C2

F

F

F

F

F

S6

p

N

p

p

p

p

p

p

p

p

p

P

p

p

p

p

p

p

p

p

p

Q

p

p

p

p

p

p

p

p

p

BF3 detector

4. Transient Analyses

F

Normal fuel rodfLead-loaded fuel rod

P

New polyethylene reflector

C

Control rod

S

Safety rod

Polyethylene reflector

Gr

Graphite rod

Pb-Bi target

Fig. 1 Subcritical core (HEU: 3960 plates) for beamswing transient experiment.
2

3

4

5

6

7

8

9

p

p

p

p

p

p

p

p

p

G

p

C3

F

F

p

F

F

S5

p

H

p

p

F

F

F

F

F

p

p

I

p

p

F

F

F

F

F

p

p

J

p

S4

F

f

f

f

F

C1

p

K

p

p

F

f

f

f

F

p

p

L

p

p

F

f

f

f

F

p

p

M

p

C2

F

F

F

F

F

S6

p

N

p

p

p

p

p

p

p

p

p

200

Neutron count rate [1/s]1Fig. 2 Subcritical core (HEU: 4080 plates) for beam
oscillation transient experiment.

I-4

3960
3960

C1
C1, C2, C3

II-1

4080

II-2

4080

C1

5296

II-3

4080

C1, C2, C3

6314

500

1000

1500

2000

Fig. 3 Measured neutron count rate and its estimation
result by simultaneous estimation model in beam swing
transient experiment.
70
Experiment

6790
9010

Expeirent

Time [s]5796

C1, C2, C3
S4, S5, S6
-

50

0

Proton beam current [pA]

I-3

3960

100

Estimate by EKF

Table I. Condition of transient experiments.
Case
Fuel
Inserted
Subcriticality
platescontrol rod
[pcm]
I-1
3960
4633
I-2

150

4160

Estimate by EKF

60

50

40

30
0

500

1000

1500

2000

Time [s]Fig. 4 Measured beam current and its estimation result
by simultaneous estimation model in beam swing
transient experiment.

24

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

current and discontinuous trend, and was considered
unnecessary to estimate by EKF technique. In othermethod, time resolution (1s) should be increased to follow
the current variation.

-4000

Reactivity [pcm]Case I-1-6000
Case I-2
Case I-3
EKF technique-8000

Case II-1-4000

Single estimation of reactivity-10000

0

500

1000

Reactivity [pcm]Simultaneous estimation
Case I-4
1500

2000

Time [s]Fig. 5 Comparison of reactivity estimations in beamswing transient experiment.

Case II-2

-5000

-6000

EKF technique
Simultaneous estimation
Single estimation of reactivity-7000

4.2 Oscillation of proton beam current

0

300

600

Case II-3
900

1200

Time [s]Fig. 8 Comparison of reactivity estimations in beam
oscillation transient experiment.

The beam oscillation transient experiment was conducted,
when the accelerator was unstably resulted in intense
oscillation of count rate (Fig. 6), by insertion of control
and safety rods during 650 s and 1000 s in Fig. 7 throughCases II-1 to II-3 (Table I).
For unrecognizable subcriticality variation from thecount rate distribution (Fig. 6), subcriticality monitoring
was performed by the EKF technique with simultaneous
estimation and single (reactivity) estimation (Fig. 8). The
result of simultaneous estimation was revealed difficult tofollow the intense current oscillation as shown in Fig. 7,
indicating difficult to deduce correct subcriticality caused
by inaccurate estimation of the beam current.

5. Conclusions
Transient experiments with external neutron source andsubcriticality were carried out at KUCA so as toinvestigate applicability of EKF technique tosubcriticality monitoring. Among two types of transient
experiments, the simultaneous estimation of reactivity
and beam intensity was very accurate for beam swing
transient even in deep subcriticality (9%dk/k) bycomparing the reference by MCNP6.2. In case of
intense beam oscillation, no simultaneous estimation was
worked to follow rapid variation of the current.
Conversely, the single (reactivity) estimation indicatedaccurate subcriticality variation during unstable neutron
source. For simultaneously estimating beam current, time
resolution is considered to be increased.

300

Neutron count rate [1/s]Expeirment
250

Estimate by EKF

200
150
100

Acknowledgements

50

0

300

600

900

The authors are grateful to Prof. T. Endo of Nagoya
University for his advice in the extended Kalman filter.

1200

Time [s]Fig. 6 Measured neutron count rate and its estimation
result by simultaneous estimation model in beam
oscillation transient experiment.

References
1. Venerus JC, et. al., “Estimation of the Dynamic
Reactivity Using Digital Kalman Filtering,” Nucl. Sci.
Eng., 40, 199 (1970).
2. Ikeda T, et. al., “Estimation of Subcriticality Using
Particle Filter Method,” Trans. Am. Nucl. Soc., 118,
851 (2018).
3. Shimazu Y, “Qualitative Performance Comparison of
Reactivity Estimation between the Extended Kalman
Filter technique and the inverse point kinetic method,”Ann. Nucl. Energy, 66, 161 (2014).
4. Yamanaka M, et. al., “Subcriticality Estimation by
Extended Kalman Filter Technique in Transient
Experiment with External Neutron Source at Kyoto
University Critical Assembly,” Eur. Phys. J. Plus
(2019) [to be submitted].

Proton beam current [pA]90
80

Experiment70

Estimate by EKF

60
50
40
30
20
10
0

300

600

900

1200

Time [s]Fig. 7 Measured beam current and its estimation result
by simultaneous estimation model in beam oscillationtransient experiment.
Conversely, single (reactivity) estimation was revealed
applicable even for intensely unstable neutron source. The
correct estimation was attributed to accurately measured

25

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Core design of accelerator driven system with reactivity control method using thorium
Kentaro Nakamura, Hideki Saito, Naoto Aizawa, Tomohiko Iwasaki
Tohoku University, 6-6-01-2, Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan*
Corresponding author: kentaro_nakamura062@neutron.qse.tohoku.ac.jp

Abstract
Accelerator-driven system (ADS) is planned to be operated at stable thermal
power by increasing the beam current to compensate the fuel burnup, but theburden of core components increases with the beam current increase. The researchon the reactivity control method using burnable poison for ADS has been
conducted to minimize the increment of beam current. However, the burnablepoison-loaded core has a potential to reach criticality by the unexpected
withdrawal of burnable poison. The present study works on the reactivity control
method with thorium to take advantage of its characteristics as fertile. Three typesof thorium-loaded core are discussed from the aspect of the core homogeneity to
study the effective reactivity control: homogeneous core, heterogeneous core andheterogeneous fuel assembly-loaded core. The homogeneous core is composed ofsingle fuel assembly and all fuel pins are the mixture of original ADS fuel((MA+Pu)N) and thorium. The heterogeneous core is consisted of two fuelassemblies: MA-bearing fuel assembly and thorium-only fuel assembly. The
heterogeneous fuel assembly-loaded core is composed of single fuel assembly andthe assembly is composed of MA-bearing fuel pins and thorium fuel pins. The core
designs are performed by changing the amount of thorium for each core model.
The heterogeneous fuel assembly-loaded core which the assembly contains 85
thorium pins out of the 391 pins shows the best reactivity control among the threecore models, and 97% decrease of the burnup reactivity compared to that oforiginal ADS is achieved.
Key Words: Core design, ADS, reactivity control
current has been performed by the introduction of
burnable poison of gadolinium to increase the reactivity
at the end of the operation cycle in the past studies. The
various cores with gadolinium were designed with using
various moderators of zirconium hydride, zirconium
deuteride and beryllium oxide [1-3]. A series of core
designs revealed that the burnup reactivity loss wasdecreased up to 0.75 %Δk/k with the use of gadolinium
hydride as a result of optimizations whereas the loss in thereference core (no poison-loaded core) was 4.3 %Δk/k.
However, the gadolinium-loaded core had the inherent
characteristic that the unexpected withdrawal of burnablepoison leads to reach criticality because these coressuppressed the excess reactivity by the poison at the
beginning of cycle.
In the present study, the core design with reactivity
control method which can eliminate the potential for
criticality was conducted by the use of thorium todecrease the potential excess reactivity and takeadvantage of its characteristics as fertile. The corecharacteristics were also calculated to evaluate theinfluence of the use of thorium. In Section 2, the reference
core model was introduced, and the method of the core

1. Introduction
Accelerator-Driven System (ADS) has been studied totransmute Minor Actinides (MA) and long-lived fission
products in a high level radioactive waste effectively forthe mitigation of the burdens related to the disposal of thewaste. ADS consists of a high-intensity proton accelerator,
a spallation target, and a subcritical core. The thermal
power of ADS is maintained at a constant level bychanging the beam current in a normal operation
condition, and the beam current increases as the operation
progresses to compensate the fuel burnup. However, theincrease of beam current leads to grow the burden ofreactor components. The large damage is considered tooccur in the beam window, which is the important
division wall between the accelerator beam duct and thespallation target and is in very severe conditions such as
an external pressure, a heat generation by the proton beam
and an irradiation damage. Therefore, it is desired that theextra load on the beam window is avoided and theincrement of the beam current from the initial state isminimized.
The core design to decrease the maximum beam26

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

design was described. Section 3 presented the result ofburnup reactivity performance and the core characteristics.
Finally, the present study was summarized in Section 4

2.2 Core design of Thorium-loaded core
Thorium-loaded core was designed on the basis of thereference ADS core. For the design of thorium-loaded
core, the following design requirements were set:

2. Analysis condition
2.1 Reference core designThe reference core design employed in this studies was
JAEA-proposed 800 MWt lead-bismuth cooled ADS core[4-5]. Figure 1 shows the schematic view of the core, andthe basic specifications are presented in Table 1. The fuelis composed of the mixed plutonium-minor actinide
nitride ((Pu + MA)N) and the inert matrix of zirconium
nitride (ZrN). The coolant and the spallation target is leadbismuth eutectic. The proton beam accelerator is assumed
the linac and its beam energy is 1.5 GeV. The thermal
output is maintained at 800 MWt by adjusting the proton
beam current (about 10 -20 mA in the reference design)
throughout the operation cycle.







The chemical form of the thorium was assumed
the nitride (ThN).
The structure of fuel assembly was the same as
the reference design.
One type of fuel assembly and thorium-onlyassembly were employed in the core.
The number of fuel pins was conserved.
No moderators (such as hydride) in the core
The initial keff was set to 0.97 by changing theratio of ZrN in the fuel.

There are many ways to load thorium in the core, andthree types of the core were designed in this study fromthe aspect of the core homogeneity to study the effective
reactivity control. Each loading pattern is shown in Figs.
2-4. The core which was composed of single fuelassembly and whose fuel was the homogeneous mixtureof (Pu + MA + Th)N and ZrN (hereafter called
homogeneous core) is shown in Fig, 2. The ratio of ZrN
was decreased to compare with the reference core, andThN was added instead of ZrN. The ratio of thorium waschanged from 10 to 50 %. The heterogeneous core in Fig.
3 was consisted of two fuel assemblies: MA-bearing fuel((Pu + MA)N + ZrN) assembly and thorium-only (ThN)
assembly. The heterogeneous core introduced thoriumonly assemblies in addition to the MA-bearing fuelassembly, and therefore, the core size was enlarged toconserve the number of fuel pins. The number of thoriumonly assembly was changed from 12 to 48 in the processof the optimizations. The heterogeneous fuel assemblyloaded core is composed of the single fuel assembly, andthe assembly is composed of MA-bearing fuel ((Pu +MA)N + ZrN) pins and thorium (ThN) pins. The numberof pins in the assembly was 391, and the number ofthorium pin was changed from 48 to 144. The chemical
form of thorium in all core designs was assumed thenitride as listed in the design requirements since thehomogeneous core should adopt the nitride to consider the
manufacture process of the original fuel.
The analysis of the designed core was performed withthe use of the PHITS [6], MVP [7] and MVP-BURN [8].
The external neutron source was calculated by using
PHITS which carried out the transport calculations until
the spallation neutrons generated by 1.5 GeV protons
were below 20 MeV. The MVP calculations were
performed on the basis of the external neutron source dataobtained by PHITS. The nuclear data library employed in
MVP was the JENDL-4.0 [9], and the burnup calculation
was based on the chain data including actinides and 50
fission products. The calculation was performed with thetime interval of 200 days in the burnup calculations, andthe total history at each time step was set to 1,000,000 in
the fixed-source calculation of MVP.

Table I. Basic specification of reference ADS design
Plant
Thermal Power
800 MWt
initial keff
0.97
Fuel
(Pu + MA)N+ZrN
Lead-Bismuth
Coolant and Target
eutectic
Coolant inlet
300 °C
temperature
Coolant velocity
2.0 m/sFuelAssembly pitch
233.9 mm
Flat to Flat
232.9 mm
Active fuel length
1000 mmPu ratio in
36.2 vol%(Pu + MA)NZrN ratio of
67.1 vol%reference core fuelAccelerator Accelerator type
linac
Proton energy
1.5GeV

Fig. 1. Reference ADS core design27Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

heterogeneous fuel assembly-loaded core, the minimum
burnup reactivity of 0.15 %Δk/k was obtained when 85 of
the 391 fuel pins in the fuel assembly are replaced withthorium pin, and that showed the most effective burnup
reactivity control among the three cores.
From the aspect of core performances, the amount oftransmutation and the radial peaking factor deteriorated inany of the cores compared to that in the reference core
design as described in Table II. The peaking factor waslarge enough even in the reference core but it wasworsened in the all thorium-loaded core design. The
mitigation of the peaking factor was also the issue in thethorium-loaded core design.

Fig. 2. Homogeneous core (type1)

Fig. 3. Heterogeneous core (type2)
Fig. 5. Time variation of keff with three types of thoriumloaded ADS coreFig. 4. Heterogeneous fuel assembly-loaded core (type3)
3. Analysis result
Figure 5 shows the time variation of the keff with the
optimal core designs in each pattern of thorium-loaded
core. In the homogeneous core, the burnup reactivity
tended to decrease as the ratio of thorium in the fuelincreased. When the ratio was 10%, the burnup reactivity
was 2.49 %Δk/k, and the keff showed almost flat curvethorough the cycle when the ratio of thorium was 25 %.
The keff exceeded 0.97 of the design requirement over 30 %of the ratio of thorium. As for the heterogeneous core, theeffect of the burnup reactivity control depended on thelocation of thorium assembly, and the core with thethorium assembly at the second innermost fuel region
from the target made most effective reactivity control.
The optimal case of 18 assemblies at second innermost
fuel region showed the minimum burnup reactivity loss of
0.70 %Δk/k in the heterogeneous core. In the28

Table II. The summary of burnup reactivity and corecharacteristics in each pattern of thorium-loaded core
BurnupTransmuta Radial
reactivity tion
peakingloss
a m o u n t factor*[%Δk/k] [kg]Homogeneous core 0.31
447.6
2.17
Heterogeneous core 0.70
438.2
2.31
Heterogeneous fuel 0.15
454.3
2.33
assembly-loaded core
Reference core
4.32
484.7
1.89
* The ratio of the maximum radial neutron flux tothe average radial neutron flux
4. Summary
The reactivity control method using thorium for ADS is
studied. Three types of thorium-loaded core are discussed
from the aspect of the core homogeneity to study theeffective reactivity control. The heterogeneous fuelassembly-loaded core which the assembly has 85 thorium
pins out of the 391 pins shows the most effective
reactivity control among the three core models, and 97%
decrease of the burnup reactivity compared to the original
ADS is achieved.

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

References
1.

Aizawa N, et al., “Study of Reactivity Control
Method by the Application of Gadolinium Hydride to
Accelerator-Driven System,” J. Nucl. Sci. Technol.,53, 2, 240, (2016)
2. Yamaguchi H, et al., “Study of Reactivity Control Method by Metallic Deuteride Burnable Poison
for Accelerator Driven System,” Proc. RPHA15,
Jeju, Korea, Sep. 16-18, 2015, Korean Nuclear
Society (2015) (on website)
3. Aizawa N, et al., “Study of Reactivity Control
Method for Acceleratory-Driven System by the
Use of Burnable Poison Oxide,” Proc. TCADS-3,
Mito, Japan, Sep. 6-9, 2016, OECD/NEA (2017)
4. Tsujimoto K, et al., “Accelerator-driven system fortransmutation of highlevel waste,” Progress in
Nuclear Energy, 37, 339, (2000)
5. Tsujimoto T, et al., “Neutronics design for leadbismuth cooled accelerator-driven system fortransmutation of minor actinide,” J. Nucl. Sci.
Technol., 41, 1, 21, (2004)
6. Sato T, et al., “Particle and Heavy Ion Transport Code
System PHITS, Version 2.52”, J. Nucl. Sci. Technol.
50, 9, 913 (2013)
7. Nagaya Y, et al., “MVP/GMVP II: general purpose
Monte Carlo codes for neutron and photon transport
calculations based on continuous energy andmultigroup methods,” JAERI 1348, Tokaimura(Japan): Japan Atomic Energy ResearchInstitute (2005).
8. Okumura K, “Validation of a continuous-energyMonte Carlo burn-up code MVP-BURN and its
application to analysis of post irradiation experiment,”
J. Nucl. Sci. Technol., 37, 2, 128, (2000)
9. Shibata K, et. al., “JENDL-4.0: a new library for
nuclear science and engineering,” Journal of Nuclear
Science and Technology, 48, 1, 1, (2011)29

SS-04

Fukushima Challenge

No.

RPHA19-1080

Title

Peak Power Characteristics of Postulated Criticality in Fuel
DebrisAuthors

Yuichi Yamane (Japan Atomic Energy Agency, Japan)
31

No.

RPHA19-1085

Title

Development of Integral Kinetic Model with Delayed Neutrons
Effect for Criticality Accident Analysis of Fukushisma Daiichi
NPP Fuel Debris

Authors

Hiroki Takezawa (Tokyo City University, Japan), Delgersaikhan Tuya
(Japan Atomic Energy Agency, Japan), Toru Obara (Tokyo Institute ofTechnology, Japan)
34

No.

RPHA19-1090

Title

Radiation Dose by Criticality Accidents of Fuel Debris in Water

Authors

Kodai Fukuda, Jun Nishiyama, Toru Obara (Tokyo Institute of technology,
Japan)
36

No.

RPHA19-1091

Title

Development of Criticality Safety Evaluation Method Based on
the Actual Dynamic Behavior of the Fuel Debris in Water

Authors

Takeshi Muramoto, Jun Nishiyama, Toru Obara (Tokyo Institute ofTechnology, Japan)
40

30

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Peak Power Characteristics of Postulated Criticality in Fuel Debris
Yuichi Yamane
Japan Atomic Energy Agency, 2-4 Shirakata, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan*
Corresponding author: yamane.yuichi@jaea.go.jp

Abstract
For safe operation at the retrieval of fuel debris in Fukushima Daiichi Nuclear
Power Plant, inadvertent criticality should be considered and the prediction of its
peak power is important for the evaluation of the shielding between the fission
source and operators, while measured data or analytical values in past criticality
accidents are quite rare. Characteristics of the peak power under a postulated
criticality accident condition have been investigated. One-point kinetics basis
analysis of a postulated criticality in the fuel debris in comparison with that of JCO
criticality accident showed its dependency on the parameters such as excess
reactivity, delayed neutron fraction, neutron generation time, etc. In conclusion,
the difference of the peak power values is mainly from heat capacity, and the key
factor is the size of fuel debris at criticality.
Key Words: Criticality accident, Fuel debris, Fukushima Daiichi NPP,
Peak power, One-point kinetics
1. Introduction

Equations derived from one-point kinetics equation
were used with the analysis parameters of fuel debris in
Fukushima Daiichi NPP and JCO criticality accident. It isreported that numerical calculation by using one-point
kinetics code AGNES [5] reproduced power profiles
which showed good agreements with TRACY
experimental data [6]. That indicates one-point kinetics
describes the power profile very well. Therefore, ameaningful result is expected from the analysis based on
the one-point kinetics.

For safe operation at the retrieval of fuel debris in
Fukushima Daiichi Nuclear Power Plant (NPP),
inadvertent criticality should be considered. The retrieval
work should be planned to be conducted in a subcriticalstate, while the countermeasure and evacuation procedureagainst inadvertent criticality should be ready, because theproperty of the fuel debris such as the composition, inner
structure and outmost geometry is not clear.
For the operator’s safety, the prediction of its peak
power is important for the evaluation of the shielding
between the fission source and operators. Exposure todirect radiation of the neutron and gamma-ray from thefissions is the dominant factor of the operator’s dose. The
shielding between them and the fuel debris reduces theirdose. The effect of shielding should be evaluated before
starting the work.
Data of the peak power in past criticality accidents arevery little. Many critical accidents and estimated numberof the fissions are reported [1], but the peak power is not.
A paper reports an estimated power profile at the JCO
criticality accident [2]. The importance of such data has
been not recognized enough and the characteristics of thepeak powers have been not often discussed.
In this study, the characteristics of the peak power
under a postulated criticality accident condition have been
investigated, in order to find the important factors whichshould be considered in the prediction and estimation ofpeak power for better evaluation of shielding. Semiquantitative analysis was done because it is known thatthe non-stochastic effect of direct radiation depends on
the order of dose [3,4].

2. Parameters of Fuel Debris and JCO Accident
For the purpose of highlighting the difference between
solid and solution systems, the calculation conditions of
the fuel debris [7] and JCO accident [8] were compared.
From many calculations done for those, a typical case foreach was chosen for comparison. A case of fresh UO2 fuelwas chosen for fuel debris for simplicity.
In the study, it was assumed that 3.0$ of excess
reactivity was instantaneously inserted. When a portion ofdebris is being removed, the operation must be slowly
done to keep the reactivity insertion rate very small, andthe fission power reaches the 1st peak by temperature
feedback reactivity soon after the excess reactivity
exceeds 1$. For the excess reactivity close to 1$, there isno simple expression of the peak power and more than
3$ would give rise to boiling of the fuel solution in the
JCO condition. For simplicity of the study, 3.0$ waschosen as the representative of other lower excess
reactivity cases.
Some parameters were in the same order to each otheras shown in Table I, in other words, those values were31

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

similar to each other from the point of semi-quantitative
view. The nominal value of the temperature feedback
reactivity of fuel debris linearly increased as the increasein temperature and was smaller than that of the solution
system as shown in Fig. 1.
The differences were seen in the 2nd order coefficientof temperature feedback reactivity and reciprocal heat
capacity. The 2nd order coefficient of temperature
feedback reactivity was much smaller than the 1 st and it
was neglected in the following calculation for the fueldebris condition. The value of the reciprocal heat capacityof fuel debris was an order of magnitude smaller than thatof the solution system and the ratio of them was 7.6 (=3.7
×10-5/4.9×10-6), i.e. the ratio of heat capacities was 0.13(=1/7.6).

reactivity temperature feedback coefficient, K, reciprocalheat capacity, 𝓵, neutron generation time.
Another denoted by Ps was derived by Yamane fornon-linear temperature feedback system [12],𝟑𝟐(𝝆𝟎 − 𝜷 + 𝜶𝟐 𝑨𝟐 )𝟐 ×𝐏𝒔 =𝟑√𝜶𝟐 𝑲𝓵
𝟏𝟑

𝟐
𝟐𝟑
𝜶 𝟐 𝑨𝟐𝟏
𝜶 𝟐 𝑨𝟐[𝟏 − ()+()] , (2)𝟐 𝝆 𝟎 − 𝜷 + 𝜶 𝟐 𝑨𝟐
𝟐 𝝆 𝟎 − 𝜷 + 𝜶 𝟐 𝑨𝟐

where

𝜶𝟏
(3)
,
𝟐𝜶𝟐
and 𝜶𝟏 and 𝜶𝟐 were the nominal values of the 1st and2nd order coefficients of reactivity temperature feedback.
Equation (1) was applied to the fuel debris because the
2nd order coefficient was much smaller than the 1st andwas ignored. Equation (2) was applied to the solution
system because its 2nd coefficient was not too small to beignored compared to the 1st. In the calculation by using
Eq.(1), the value of 𝜶 was set as 1.2×10-4 .
𝐀=

Table I. A typical value of analysis parameterssolutionparameter
fuel debris
system-2
2.1×10
2.4×10-2
excess reactivity (-)
𝝆𝟎(3.0$)(3.0$)
neutron-5𝓵
5.4×10
3.0×10-5
generation time (s)effective delayed
𝜷6.9×10-3
7.9×10-3
neutron fraction (-)
𝜶𝟏
1streactivity-1.2×10-4-2.3×10-4
temperature (1/K) (𝜶)feedback
2nd
𝜶𝟐
4.6×10-8-1.4×10-6
coefficient (1/K2)
reciprocal heat
𝑲
3.7×10-5
4.9×10-6
capacity (K/J)

4. Comparison and Result
Eqs. (1) and (2), show that peak power is proportional
to the inverse of neutron generation time and the inverseof reciprocal heat capacity, i.e. heat capacity.
The calculated peak power by using those two
equations and values shown in Table I is shown in Table
II, in which the peak power of fuel debris is an order ofmagnitude smaller than that of the solution system. The
ratio of them is 0.14 (=4.2×108/3.0×109) and that is very
close to the ratio of their heat capacities, 0.13.
Table II. Calculated peak power
solutioncasefuel debris
systempeak power (W)

4.2×1083.0×109

That implies the difference in heat capacity is the maincause of the difference in peak power.
Specific heat 𝑪𝒑(𝒔𝒐𝒍) (𝑱⁄(𝒈𝑲)) of uranyl nitratesolution is expressed as a function of uranium
concentration [13] as,𝑪𝒑(𝒔𝒐𝒍) = 𝟒. 𝟏𝟖𝟔(𝟎. 𝟗𝟗𝟖 − 𝟗. 𝟔𝟑𝟎 × 𝟏𝟎−𝟒 × 𝑪𝒖
(4)
−𝟒. 𝟖𝟓𝟎 × 𝟏𝟎−𝟐 × 𝑪𝑵 )where Cu is uranium concentration (g/Lit) and CN isacidity (mol/Lit) and 𝑪𝒑(𝒔𝒐𝒍) is in the order of 𝑶(𝟏𝟎𝟎 )for 𝑪𝒖 = 𝟑𝟕𝟎 (g/Lit) and 𝑪𝑵 = 𝟎. 𝟓 (mol/Lit).
Specific heat of uranium di-oxide and SUS are in theorder of 𝑶(𝟏𝟎−𝟏 ) and smaller than that of aqueous
solutions as shown in table III.

Fig. 1. Temperature feedback reactivity calculated withthe values from Table I.
3. Equations Used for Analysis
For the analysis, two equations of the peak power for𝝆𝟎 ≫ 𝜷 derived from the one-point kinetics equation
were used instead of numerical calculation code for clear
visibility.
One was the equation derived by Nordheim and Fuchs
for linear temperature feedback system [9-11],(𝝆𝟎 − 𝜷)𝟐
(1)
𝐏𝒅 =
,
𝟐𝜶𝑲𝓵
where Pd was the peak power, 𝝆𝟎 , excess reactivity, ,effective delayed neutron fraction, , nominal value ofTable III. Specific heat of UO2 and SUS

32

Material

UO2 [14]

SUS304 [15]Specific heat(J/(gK))

0.24 (300K)
0.29 (600K)

0.50 (293K)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019And a homogeneous mixture of such solids has the sameorder of magnitude of their specific heat. Therefore, if thesize of the debris at criticality and that of a solution system
are close to each other, the difference in their specific heatmust be the main cause of the difference in their heat
capacity.
For a single homogeneously mixed fuel debris withsimilar parameter values to that of UO2 fuel, the order of
𝜷 and 𝓵 are expected to be 𝑶(𝟏𝟎−𝟑 ) and 𝑶(𝟏𝟎−𝟓 ),respectively, and that of 𝜶 is 𝑶(𝟏𝟎−𝟒 )[7].
From the consideration above, it is reasonable for the
first rough estimation that the peak power in a postulated
criticality of the fuel debris is in one order of magnitude
lower than that of the solution system like the JCO
criticality accident and that an extremely larger peak
power than that we have ever experienced from thecriticality accidents like the JCO will not appear in an
advertent criticality if the operation is cautiously
conducted so that the reactivity insertion rate is very low.
The effect of surrounding fuel debris is the next issue
to be taken into account, while it may not significantly
change the order of magnitude of the peak power because
the total reactivity of each is negative.
The concerned heat capacity, however, depends on themass or volume under criticality condition and the degreeof mixture of the materials. Therefore, the range of thesize is important and a sample of the fuel debris would
provide the information useful to estimate its size undercriticality condition.

5. Nakajima K, Yamane Y, Miyoshi Y, “A kinetics codefor criticality accident analysis of fissile solution
systems: AGNES2,” JAERI-Data/Code 2002-004,
Japan Atomic Research Institute (2002).
6. Miyoshi Y, Yamane Y, Okubo K, Reverdy L, Grivot
P, Konishi H, Mitake S, Liem PH, “Inter-code
comparison exercise for criticality excursion analysis:benchmarks phase I: pulse mode experiments withuranyl nitrate solution using the TRACY and SILENE
experimental facilities,” Report No.06285, OECD
NEA, (2009).
7. Yamane Y, Numata Y, Tonoike K, “Exploratory
Investigation for Estimation of Fuel Debris Criticality
Risk,” Proc. ICNC2019, Paris, France, September 1520, 2019, OECD/NEA (2019) (CD-ROM).
8. Yamane Y, Nakajima K, Yamamoto T, Nakamura T,
Miyoshi Y, “Analysis of TRACY Experiment andJCO Criticality Accident by Using AGNES Code,”
JAERI-Conf 2000-012, Japan Atomic ResearchInstitute, (2000) [in Japanese].
9. Fuchs K, “Efficiency for very slow assembly,” LA596, Los Alamos National Laboratory (1946).
10. Nordheim LW, “Pile kinetics,” MDDC-35, Oak Ridge
National Laboratory (1946).
11. Hetrick DL. Dynamics of nuclear reactors, p.164,
American Nuclear Society, La Grange Park, Illinois(1993).
12. Yamane Y, “Improvement in estimation of maximum
first peak power based on non-linear temperature
feedback reactivity in criticality accident withinstantaneous reactivity insertion," J. Nucl. Sci.
Technol. 52, 11, 1425 (2015).
13. Sugikawa S, Umeda M, Izawa N, Yamagami S, Ota
K, Hosoya N, “Performance analysis of uranium
evaporator by evaporator simulation code (I),” Proc.
the Atomic Energy Society of Japan Spring Meeting
1997, Tokyo, Japan, Mar 24-26, 1997, Atomic Energy
Society of Japan (1997) [in Japanese].
14. Yamane Y, Sakai M, Abe H, Yamamoto T, Okuno H,
Miyoshi Y, “Material Data for Criticality Transient
Phenomena Analysis of MOX Powder System -Dataof MOX and Zinc Stearate Powders-,” JAEAData/Code 2006-021, Japan Atomic Energy Agency,(2006) [in Japanese],
15. Peckner D, Bernstein IM, “Handbook of StainlessSteel,” McGraw-Hill Book Company, New York,USA (1977).

5. Conclusion
For the first rough estimation, the peak power in a
postulated criticality of the fuel debris is in one order ofmagnitude lower than that of the solution system like theJCO criticality accident. The difference of the peak power
value of the fuel debris from the solution system is mainly
from its low heat capacity and the key factor for the safetyof the operators is the size of the fuel debris undercriticality condition.
References
1. McLaughlin TP, Monahan SP, Pruvost NL, Frolov
VV, Ryazanov BG, Sviridov VI, “A Review ofCriticality Accidents,” LA-13638, Los Alamos
National Laboratory (2008).
2. K. Tonoike, T. Nakamura, Y. Yamane, Y. Miyoshi,“Power profile evaluation of the JCO precipitation
vessel based on the record of the γ-ray monitor,” Nucl
Technol, 143, 364 (2003).
3. Rechet P, Andrews GA, Auxier JA, Dousset M,
Duncan KP, Stevens DJ, Strambi E, Wald N, “Theprinciples and general procedures for handlingemergency and accidental exposures of workers,”
ICRP Publication 28, ICRP (1978).
4. Upton AC, Brown JM, Casarett GW, Field SB,
Lincoln TA,Withers HR, “Nonstochastic effects ofionizing radiation,” ICRP Publication 41, ICRP
(1984).

33

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Development of Integral Kinetic Model with Delayed Neutrons Effect for
Criticality Accident Analysis of Fukushisma Daiichi NPP Fuel Debris
Hiroki Takezawaa, Delgersaikhan Tuyab and Toru Obarac,
Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo, Japanb
Japan Atomic Energy Agency, 2-4 Shirakata, Tokai-mura, Ibaraki, Japanc
Tokyo Institute of Technology, 2-12-1-N1-19 Ookayama, Meguro-ku, Tokyo, Japan*
Corresponding author: takezawa@tcu.ac.jpaAbstract
Multi-region integral kinetic code MIK with a capability of handling delayed neutrons
effect has been under development for criticality accident analysis of Fukushisma
Daiichi NPP defueling activities. Cumulative secondary-fission distribution functionof delayed neutrons that is essential for MIK code was calculated for Godiva core
without delay of emission by modifying continuous energy neutron transport Monte
Carlo calculation code MVP3.0. Calculation result was consistent with experimental
delayed neutron fractions.

Key Words: Space-dependent kinetic analysis, Integral Kinetic Model,
Criticality accident, Fuel debris.
1. Introduction

2. Method

Fuel debris of Fukushima Daiichi NPP will remain
uncertain in composition and spatial distribution. Hence,
it is necessary to establish counter measures for
anticipated re-criticality accident in addition to avoiding
re-criticality associated with defueling activities. Onepoint kinetic model (PKM) is generally used for criticality
accident analysis. However, the model is not applicable tosome fuel debris systems that consists of fragmented fueldebris with water surrounded by re-solidified fuel debris
bed, i.e. the combination of thermal and fast spectrum
regions as shown in Fig. 1 as an example.
On the other hand, integral kinetic model (IKM) [1]and transient fission matrix approach (TFM) [2] arespace-dependent kinetic models and are applicable to socalled coupled reactors including such a fuel debris
system. IKM and TFM are the same in mathematicalfoundation. Multi-region integral kinetic code MIK [3]that has been developed on IKM has more general
feedback treatment than TFM. However, MIK code has
been developed only for super prompt-criticality transient
analysis. MIK code with a capability of handling delayed
neutrons effect has been under development. This papershares a part of the development progress.

IKM calculates fission rate in region i at the present timet 𝑁 𝑡 [fissions/sec] by taking all past fission
contributions in region j into account using Eq. (1):Cooling water

𝑁 𝑡

∑

𝛼

𝜏

𝛼

𝜏

𝑁 𝑡 𝑑𝑡

(1),

where
/
𝛼
𝜏 : probability density function of secondaryfissions in region i triggered by prompt neutrons ordelayed neutrons originates from a source fission inregion j with time difference 𝜏 ≡ 𝑡 𝑡′ [secondaryfissions@i/source fission@j/sec].
Eq. (1) is numerically solved simply by forward
calculation method using the cumulative secondary[cumulative
fission distribution function 𝐶 𝜏secondary-fissions@i/source fission@j] defined asfollows:𝐶 𝜏𝛼𝜏′ 𝑑𝜏′(2).

Continuous energy neutron transport Monte Carlocalculation code MVP3.0[4] was modified to calculate
𝐶 𝜏 of delayed neutrons without delay of emission.
Delay of emission is included in the forward calculation
module in MIK code. In this paper, 𝐶 𝜏 of delayed
neutrons without delay of emission was calculated for
Godiva core [5]. The calculation was performed by aneigenvalue calculation of MVP3.0 with nuclear datalibrary JENDL-4.0 [6]. Calculation conditions are

Fragmented fuel debris

Re-solidified fuel debris bed

Fig. 1. An example of fuel debris system.

34

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

summarized in Table I and II.

𝐶 𝜏 distribution functions of prompt and delayed
neutrons will be verified in comparison of MIK code
calculation results with Godiva super-prompt transient
experimental data considering delayed neutrons effects.
Subsequent development progress will be shared in thepresentation.

Table I. MVP3.0 calculation conditions
Number of history per batch
500,000
Total batches (skip)
100 (50)
Table II. Godiva core specification
Core division
2 (Inner/Outer)
Inner (ID=1) 5.0 cm
RadiusOuter (ID=2) 8.85 cm
Mass density
18.71 g/cm3
U234 1.0 wt%
Composition
U235 93.7 wt%
U238 5.3 wt%
Temperature
293 K

References
1. Gulevich A, “Methods for Calculating Coupled
Reactor Systems,” At. Enegy, 97, 803 (2004).
2. Laureau A, “Transient Fission Matrix: Kinetic
Calculation and Kinetic Parameters βeff and Λeff,” Ann.
Nucl. Energy, 85, 1035 (2015).
3. Tuya D, “Development of Monte Carlo Neutron
Transport Method-Based Supercritical Transient
Code with Time-Dependent Feedback Capability,”
Nucl. Sci. Eng., 193, 481 (2019).
4. Nagaya Y, “MVP/GMVP Version 3: General Purpose
Monte Carlo Codes for Neutron and Photon Transport
Calculations Based on Continuous Energy andMultigroup Methods,” JAEA-Data/Code 2016-018,
Japan Atomic Energy Agency (2016).
5. Peterson R. E., “Lady Godiva: An Unreflected
Uranium-235 Critical Assembly,” LA-1614, Los
Alamos Scientific Laboratory (1953).
6. Shibata K, “JENDL-4.0: A New Library for Nuclear
Science and Engineering,” J. Nucl. Sci. Technol., 48,
1 (2011).

3. Result𝐶 𝜏 of delayed neutrons𝐶 𝜏 of prompt neutrons
Same unit as left

Cumulative secondary-fission distribution functions𝐶 𝜏 of prompt (dashed lines) and delayed neutrons(solid lines) for Godiva core are shown in Fig. 2.
Difference in neutron spectrum between prompt anddelayed neutrons has been considered, but delay ofemission for delayed neutrons has not included in thecalculation. Horizontal axis shows time between anemission of a source prompt/delayed neutron and asecondary-fission. It is shown that both prompt andsec to 1.0
delayed neutrons fission in 1.0 10
10 sec after their emission, that is consistent to prompt
neutron lifetime. 𝐶 𝜏 of prompt neutrons are
relatively larger than 𝐶 𝜏 of delayed neutrons. Thisdifference was checked by confirming delayed neutron
fractions between experimental [5] and calculated values
were consistent each other.

Fig. 2. An example of cumulative secondary-fission
distribution functions for Godiva core.
4. Conclusions
MIK code with a capability of handling delayed
neutrons effect has been under development. Cumulativeof
secondary-fission distribution function 𝐶 𝜏delayed neutrons was calculated without considering
delay of emission by modifying continuous energy
neutron transport Monte Carlo calculation code MVP3.0.

35

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Radiation Dose by Criticality Accidents of Fuel Debris in Water
Kodai Fukuda, Jun Nishiyama and Toru Obara*
Laboratory for Advanced Nuclear Energy, Institute of Innovative Research, Tokyo Institute of Technology,
2-12-1-N1-19 Ookayama, Meguro-ku, Tokyo 152-8550, Japan*
Corresponding author: tobara@lane.iir.titech.ac.jp

Abstract
Removal of the fuel debris at Fukushima Dai-ichi Nuclear Power Station (1F-NPS)
must be started after the evaluation of the consequences of possible criticality
accidents. Especially, a prediction of radiation dose by such accidents can be
crucial for the safety of machines, workers and public. In this research work, aradiation dose by possible super prompt criticality accidents was predicted in a
simple fuel debris system. Particularly, we focused on the relationship between thewater level surrounding the fuel debris and the radiation dose during thesupercritical condition. This is according to that the water level may influence on
both the reactivity and the shielding of radiation. For that, space-dependent kinetic
analysis and radiation transport analysis were connected. For the space-dependent
kinetic analysis, the MIK code, which is a unique methodology based on Monte
Carlo neutron transport calculation, was used in order to consider special
characteristics of fuel debris systems. Radiation transport analysis was performed
using PHITS code. The obtained results indicated that the radiation dose during
the supercritical condition can be largest when a part of fuel debris is exposed tothe air, even though the total energy release is smaller compared with systems
which have higher water level. These results will be useful when the water level
will be determined during the removal of fuel debris.
Key Words: Criticality safety, Space-dependent kinetic analysis, Fuel debris,
Decommissioning of Fukushima Dai-ichi Nuclear Power Station
effective measures to protect workers. Specifically,evaluation of radiation doses resulting from criticality
accidents of fuel debris is important in this context.
In the case of criticality accidents, the water level
surrounding the fuel debris is thought to give an impact
on the radiation dose around the position of the criticality.
This is due to that the water level affects both thereactivity and the shielding of radiation. If the fuel debris
is partially in water, a higher water level will give higher
reactivity. However, if the fuel debris is completely
submerged in water, the radiation shielding effect will besignificant. This may be an important phenomenon forpredicting the radiation dose in the case of such accidents.
However, no study about this relationship has not been
conducted.
Therefore, the aim of this work was to make it clearthat the relationship between the water level and radiationdose when fuel debris becomes prompt supercritical inwater. A simple assumptive fuel debris system wasconsidered for that. In the system, supercritical kinetic
analysis was performed for different water level using the
MIK code [2], which is a space-dependent kinetic code
based on Monte Carlo neutron transport calculations. The
detailed information of the MIK code is described in the

1. Introduction
The Great East Japan Earthquake brought about a severe
accident at Fukushima Dai-ichi Nuclear Power Station(1F-NPS) in 2011. As a result, a part of the reactor coremelted down and fuel debris was thought to be formed [1].
To reduce long-term risk, a removal of the fuel debris is
one of the most crucial operation in the decommissioningof 1F-NPS.
During the removal of fuel debris, water will be usedbecause it has ability to shield operators and machinesfrom radiation emitted from fission products and preventthe diffusion of radioactive dust. In that case, criticality
accidents may happen because water can be a moderator
for neutrons. Therefore, it is necessary to prepare
measures to prevent such accidents in advance. However,
it is possible to happen the criticality accidents
unintentionally even if the measures are adequate. If thetransient of accidents is slow, measures can be taken evenafter the detection of the criticality. However, in the caseof prompt super criticality accidents, there is no time totake actions when the criticality is detected. Therefore,
before the removal of fuel debris, the effect of such
accidents should be evaluated in advance to think of36

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

later section. The region-dependent number of fissions
which were obtained by the MIK code was used for dose
evaluation.

shape was chosen for the first step because of its
simplicity.
The characteristics of the system are shown in Table 1.
In the system, particles of fuel debris that assemble into asphere are submerged in water. The composition of thefuel debris is unknown, therefore, the fuel debris was
assumed to be UO2 for simplicity. The UO2 was set tohave ordinary density and enrichment. The unit wascomposed of 0.1-cm-radius spherical particles. The
volume-packing fraction in the unit was 0.6, representing
random packing fraction. The radius of the unit was set tobe that the effective multiplication factor was 1 when thehalf of the unit was submerged. The thickness of the water
was 30 cm, representing an infinite reflector size. The
bottom of the unit was considered the zero water level in
the analysis. Different systems were simulated at different
water levels ranging from 50 cm to 150 cm.
The systems were assumed to be at a critical state of
low power, 1W. Step-wise positive reactivity was inserted
by increasing the water level. The situations were
simulated when the water level became a specific value in
an instant. It indicates that different water levels resulted
in different reactivities.

2. Methodology
It is necessary to perform kinetic analyses and estimate
the number of fissions in the case of supercriticalconditions in order to predict possible radiation doses.
The characteristics of fuel debris can be different indifferent locations. In specific, compositions, shapes,
quantities, and submerged conditions can be different indifferent positions. These difference may bring about anon-negligible space-dependent difference in terms ofneutronics. In such system, methods based on diffusion
theory, for example, traditional point-reactor kinetic
model (PKM) [3] may be out of the applicable range.
Hence, space-dependent kinetic analysis methods basedon neutron transport theory should be used for such fueldebris systems instead.
In this work, the Multi-region Integral Kinetic Code(MIK) was applied. The MIK code makes it possible toperform space-dependent kinetic analyses based on
neutron transport theory with reasonable calculation time.
The basis of the MIK code is the Integral Kinetic
Model (IKM) [4]. Fundamentally, the IKM describes thefission rate at time t in region i for a system consisting of
n fissile regions as:𝑡𝑝

𝑁𝑖 (𝑡) = ∑𝑛𝑗=1 (∫−∞ 𝛼𝑖𝑗 (𝑡 − 𝑡 ′ )𝑁𝑗 (𝑡 ′ )𝑑𝑡 ′ +𝑡𝑑 (𝑡− 𝑡 ′ )𝑁𝑗 (𝑡 ′ )𝑑𝑡 ′ ),∫−∞ 𝛼𝑖𝑗(1)where 𝑁𝑖 (𝑡) is the total fission rate at time t in region i(fissions in i/sec), and 𝛼𝑖𝑗 is the secondary fission rate inregion i provided by the first fission in region j with a timedifference 𝜏 (fissions in i/sec /fission in j ). Superscripts
p and d show that prompt neutrons and delayed neutrons
cause secondary fissions, respectively. The MIK code
calculates 𝑁𝑖 (𝑡) discretely using output data of theneutron transport calculation with the continuous-energyMonte Carlo Code MVP 2.0[5].
Detailed information about the MIK code is given
elsewhere [2].
In addition, dose evaluation was performed by the
PHITS code [6] (ver. 2.86) which is a radiation transport
calculation code based on the Monte Carlo method.
As an input data, the number of fissions obtained by
space-dependent kinetic analysis using the MIK code wasused.

Fig. 1. Cross-sectional view of system (not to scale).
Table 1. Characteristics of the system
Particle composition
UO2
3
Density of UO2, g/cm
10.97
Enrichment of 235U. wt%
2.95
Particle radius, cm
0.1
Volume packing fraction
0.6
Unit radius, cm
49.6
Water level, cm
50-1503. Analysis conditions

3.2 Other analysis conditions

3.1 Simple spherical fuel debris system

For neutron transport calculations, sixty batches (20skipped batches + 40 active batches) with 500,000
histories/batch were used with the JENDL-4.0 nuclear
data library [7]. Delayed neutrons were ignored in thecalculations because the target of this work was prompt
supercritical conditions. The fuel debris unit was divided

The analysis was performed in the spherical fuel debris
system shown in Fig. 1. This system was considered
because spherical system is the simplest of all systems interms of geometry. In reality, it is difficult for the fueldebris to make a spherical shape when thedecommissioning is conducted. However, the spherical37

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

into 2 fissile regions when part of the unit was exposedto the air. It was treated as 1 fissile region when theentire unit was submerged. In the analysis, the Doppler
feedback effect was considered.
The radiolysis gas and the heat transfer from the fueldebris to water were ignored. In fact, the instantevaporation of water and the production of radiolysis gas
are both expected as results of the supercritical condition.
However, both phenomena have a negative feedback
effect. This means that the release of the power is lower
when the phenomena are considered. Therefore, theapproximation disregarding these factors can be regarded
a conservative analysis condition.
For radiation transport calculation, one million
histories were used with the JENDL-4.0 as well. The
source locations are equivalent to fission regions whichwere considered in the space-dependent kinetic analysis.
Neutrons and photons were used as source particles. A
prompt neutron specrum [8] and prompt gamma ray
specrum [9], which were from fissions induced bythermal neutrons, were used. They were tallied at the
point 3 m above the center of the unit. The energy rangeof neutrons was between 10-3 and 103 MeV in the analysis.
Simiraly, that of gamma rays was between 10-10 and 20
MeV in the analysis. The effective dose conversion factor
under the antero-posterior irradiation condition [10] wasset to convert fluence to dose.

brought on by a rise of temperature of the fuel. It should
be noted that the heat transfer between the fuel debris andwater was not included. Thus, the temperature of the fueldebris was kept high even after the power declined andthe system became subcritical. Therefore, only the first
spike was obtained and later spikes could not be obtained.
However, the aim of this study can be satisfied byobserving the first spike. This is due to that the released
energy during the first spike is dominant for the total
released energy during the prompt supercritical condition
according to criticality experiments [11]. Therefore, thislimitation may be acceptable in this work.

4. Results and discussion

Fig. 3. Change of fission rate (water level = 52.1 cm).

The relationship between the water level and the prompt
multiplication factor is shown in Fig.2. It should be notedthat the dotted line shown in the figure indicates the waterlevel of 99.2 cm, which is the same as the diameter of theunit. According to Fig.2, the prompt multiplication factor
increased as the water level increased. This was due to theincrease of fissile regions which was submerged andreacted with thermal neutrons.

By the time-integration of the pulse such as shown in
Fig. 3, the number of total fissions in each region were
calculated. Those values were used in dose evaluation as
input values. Fig.4 shows the relationship between theradiation dose during supercritical condition and thewater level. The result shows that the maximum dose wasfound at approximately 95 cm of water level. Therefore,
in this system, the maximum dose during supercriticalcondition was expected in a system in which part of thefuel debris was exposed to the air.
Those results indicate that the reactivity is higher
when water level is higher. However, the dose is higher ina system when water level is lower. Therefore, when theremoval of fuel debris is performed at a lower water level,
more attentions should be paid in the case of criticality
accidents, although the total energy release may besmaller. On the other hand, a lower radiation dose can be
anticipated when the water level is higher, even though
the energy release and a probability of criticality may belarger.

Water level : 52.1 cm

Fission rate [fissions/s]

2.0x1020

Prompt multiplication factor

1.5x1020

1.06

1.04

1.02

1.00
50 60 70 80 90 100 110 120 130 140 150
Water level [cm]Fig. 2. Prompt multiplication factor and water level.
Fig. 3 shows the time-dependent fission rate
calculated by the MIK code in which the water level was
52.1 cm as an example. As shown in Fig. 3, a pulsatile
fission rate was obtained. It is observed that an increaseof power caused by the insertion of positive reactivity anda decrease of power due to the Doppler feedback effect38

kp=1.0042

1.0x1020
5.0x1019
0.0
0.00

99.2 cm

Debris in air
Debris in water

0.05

0.10

0.15 0.20
time [s]

0.25

0.30

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

MVP/GMVP II,” JAERI-1348 (2005).
6. T. Sato et al., “Features of Particle and Heavy Ion
Transport code System (PHITS) version 3.02,” J.
Nucl. Sci. Technol, 55, 6, 684 (2018).
7. K. Shibata et al., “JENDL-4.0: A New Library for
Nuclear Science and Engineering,” J. Nucl. Sci.
Technol, 48, 1, 1 (2011).
8. Atomic Energy Society of Japan, “Neutron shielding
design handbook,” (1993).
9. T. E. Valentine, “MCNP-DSP USERS MANUAL,”Oak Ridge, TN (United States) (2001).
10. N. Petoussi-Henss et al., “Conversion Coefficients forRadiological Protection Quantities for External
Radiation Exposures,” Ann. ICRP, 40, 2, C. H.
Clement, Ed., 1 (2010).
11. Y. Yamane et al., “Transient Characteristics Observedin TRACY Supercritical Experiments,” in
ICNC2003: Proceedings of the seventh international
conference on nuclear criticality safety, Challenges
in the pursuit of global nuclear criticality safety(2003).

99.2 cm
1.2x107

Neutron
γ-ray
Total

7

Dose [mSv]1.0x108.0x106
6.0x106
4.0x106
2.0x106
0.0
50 60 70 80 90 100 110 120 130 140 150
Water level [cm]Fig. 4. Radiation dose during supercritical condition.
5. Conclusions
In order to make it clear that the relationship between thewater level and radiation dose when fuel debris becomes
prompt supercritical in water, connected analysis was
performed and discussed. For space-dependent kinetic
analysis, the MIK code, which is unique methodology
based on Monte Carlo neutron transport calculation, wasused. The PHITS code was utilized for radiation transport
calculation with the results of space-dependent kinetic
analysis as an input data.
The results mainly showed that the dose caused bysupercritical condition is the largest when a part of the fueldebris is exposed to the air. Thus the dose in those systems
can be greater than that in systems which the all of fueldebris is completely submerged in water.
Hence, the radiation dose caused by criticality
accidents may be decreased if the removal of fuel debris
is conducted with a higher water level at 1F-NPS.
However, in that case, the possibility of an accident andthe total energy release may be higher. These results anddiscussions will contribute to the consideration about thewater level during the removal of fuel debris at 1F-NPS.
References
1. Tokyo Electric Power Company Holdings,“Evaluation of the situation of cores andcontainment vessels of Fukushima Daiichi Nuclear
Power Station Units-1 to 3 and examination into
unsolved issues in the accident progression,”
Progress Report No.1 (2013).
2. D. Tuya et al.,” Improved Approach to Multiregion
Supercritical Transient Analysis Based on the
Integral Kinetic Model and Monte Carlo Method,”
Nucl. Sci. Eng., 188, 1, 33 (2017).
3. J. J. Dudestadt and L. J. Hamilton, Nuclear reactor
analysis, John Wiley and Sons Inc (1976).
4. A. V. Gulevich and O. F. Kukharchuk, “Methods forcalculating coupled reactor systems,” At. Energy, 97,
6, 803 (2004).
5. Y. Nagaya, “General purpose Monte Carlo Codes for
Neutron and Photon Transport Calculations based on
Continuous Energy and Multigroup Methods:

39

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Development of Criticality Safety Evaluation Method
Based on the Actual Dynamic Behavior of the Fuel Debris in Watera

Takeshi Muramotoa, Jun Nishiyamaa, Toru Obaraa*,
Laboratory for Advanced Nuclear Energy, Institute of Innovative Research, Tokyo Institute of Technology,
2-12-1-N1-19, Ookayama, Meguro-ku, Tokyo 152-8550, Japan*
Corresponding author: tobara@lane.iir.titech.ac.jp

Abstract
In the decommissioning of the Tokyo Electric Power Company’s Fukushima
Daiichi nuclear power station, criticality safety assessment taking into account the
dynamic behavior of fuel debris in water is important for the safety of workers.
The purpose of this study is to show that it is possible to evaluate criticality safety
using a computational system combining moving particle semi-implicit method(MPS) and Monte Carlo neutron transport calculation code (MVP). Experiments
and calculations were first performed to validate the accuracy of the MPS
calculations. We then combined MPS and MVP and calculated the change in
effective multiplication factor at each time during fuel debris falling in water. The
results showed that it is possible to evaluate criticality safety taking into accountthe actual dynamic behavior of fuel debris in water using the computational system.
Key Words: Criticality Safety, Fuel Debris, Moving Particle Semi-Implic
it Method, Monte Carlo Neutron Transport Calculation
Eulerian method use calculation grids. The Lagrangian
method is suitable for solving physical phenomena likefree surfaces and large deformations compared with the
Eulerian method such as finite difference method, finite
volume method, and finite element method. The moving
particle semi-implicit (MPS) method [3] is one of the
Lagrangian methods. It can calculate multiphase flows
like the dynamics behavior of debris in water and various
complex boundary conditions. Thus, in this study weselected the MPS method using Particleworks [4]simulation software for computer-aided engineering(CAE) to calculate three-dimensional (3-D) solid-fluid
multiphase flows. In addition, NVIDIA Tesla k40 which
is one of the Graphics Processing Unit (GPU) was used as
the calculator.

1. Introduction
In the decommissioning of the Tokyo Electric Power
Company’s Fukushima Daiichi nuclear power station, thefuel debris retrieval work is going to start within a few
years based on the Japanese government’s mid- and longterm roadmap [1]. It is important to remove the fuel debris
in order to reduce radiation risk. At present, it is estimatedthat the fuel debris is not critical. However, it may bepossible to become critical if the fuel debris shape or
cooling water volume changes with the fuel debris
removal work. Thus, it is necessary to establish an
analysis method that can perform criticality evaluation
with higher accuracy for the protection of workers andpublic health in the decommissioning.
The conventional criticality assessment methods canevaluate only under conservative and static conditions
without considering the behavior of the fuel debris inwater. [2]. Thus, no study has yet been able to evaluate
criticality with the actual dynamic behavior of fuel debris
in water. The purpose of this study is to show that it ispossible to evaluate criticality safety using the calculation
system combining computational fluid dynamics (CFD)
and Monte Carlo neutron transport calculation code.

2.2 Validation of MPS simulation
2.2.1 Experiment of falling plate in water
In the behavior of fuel debris in water, the force fromgravity and fluid is applied to the fuel debris. In order toconfirm if these forces can be properly calculated by the
MPS method, validation experiments of falling plates inwater were performed. Because the plate shape is easily
affected by the resistance from water, it is suitable forverification experiments. The plate materials were
stainless steel (SS) (8.25 g/cm3) and aluminum (2.78
g/cm3), high- and low-density materials respectively.
Although studies on the physical properties of the fueldebris were in progress, the exact density andcomposition have not been clarified yet. Therefore we

2. Numerical simulation for fuel debris movement
2.1 CFD simulation method
The CFD calculation method is divided into theEulerian method and Lagrangian method. The Lagrangian
method uses calculation particles as opposed to the40Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019carried out verification experiments using these different
density substances. The plates were dropped into roomtemperature water by sliding down a 30-degree, 150-mm
high acrylic slope. The movement of the plate wascaptured using a high-speed camera with a frame rate of
120 fps from the water tank until it came to rest. The
calculation conditions are shown in Table 1. The discrete
particles were 2 mm in diameter, and there were 460,068
total particles. The validation method was a comparisonof the experimental and simulation results for the centerof the gravity position of the plate as it fell in water. The
experimental measurements were calculated using image
data processing of the pictures filmed by the camera. Eachexperiment was conducted 5 times, and the standard
division (σ) of the measurements was calculated as theexperimental uncertainty.
Snapshots of experimental and simulation results areshown in Figure. 1. The plate fell down in water withswinging due to water resistance. The aluminum plate fellin water with more complicated behavior compared to thestainless steel plate. Qualitatively, the MPS method has

been able to simulate the complex falling behavior in bothof experiments. In the case of stainless steel, thesimulation results were within the 1.96 σ of theexperimental results. On the other hand, in the case ofaluminum, the simulation results were almost within the
1.96 σ of experimental results but deviation between
experiment and simulation results increased with time. It
indicates that the calculation accuracy may be not well as
the falling time becomes longer when calculating a lightmaterial by the MPS method. As results, the MPS method
was shown to be capable of calculating gravity and thefluid resistance force well except for a long time falling
phenomenon of light materials.
2.2.2 Experiment of sedimentation of bolts in water
In order to confirm whether the interaction between
the objects can be calculated properly by MPS method,
we carried out experiments of sedimentation of bolts
which is hexagon head bolts (nominal diameter, 6 mm;
length of head, 4 mm; nominal length, 10 mm) to simulate
the debris. The experimental apparent is as follows. A
concrete board formed the bottom of the water tank tosimulate the environment in a reactor. The water surface
was 100 mm above the concrete board. A polyvinyl
chloride (PVC) pipe, 60 mm in diameter and filled with500 bolts was dropped from 150 mm into the water bypulling up the stopper. The bolts were made of stainlesssteel (7.90 g/cm3) and aluminum (2.62 g/cm3). The
calculation conditions are shown in Table 2. The diameterof discrete particles was 2 mm, and there were a total of
1,110,050 particles. The validation method compared theHeight (H), Width (W), and Depth (D) of thesedimentation shape between the experimental andsimulation results. Experimental measurements wereconducted using a ruler. Each experiment was repeated 5
times, and the standard division of the measurements wascalculated as the experimental uncertainty.

Table. 1 Simulation conditions of falling plate.
Water density

1,000 kg/m3

Water kinetic viscosity

1.0×10-6 m2/s

Stainless steel density

8,250 kg/m3

Aluminum density

2,780 kg/m3

Discrete particle diameter2.0×10-3 m

Number of particles

460,068

Time step

1.0×10-4 s

Table. 2 Simulation conditions of sedimentation of bolts.

Figure. 1 Comparison between experimental (left side) and simulation (right side) results of falling pla
te in water. (a) Stainless steel, (b) Aluminum.

41

Water density

1,000 kg/m3

Water kinetic viscosity

1.0×10-6 m2/s

Stainless steel density

7,900 kg/m3

Aluminum density

2,620 kg/m3

Friction coefficient of stainless steel

0.387

Friction coefficient of aluminum

1.09

Spring constants

0.05

Damping coefficient

0.05

Discrete particle diameter2.0×10-3 m

Number of particles

125,288

Time step

1.0×10-4 s

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Figure 2 contains snapshots of the experimental andsimulation results. The simulation results for themeasured values were within 1.96σ of the experimental
results, except for D in the case of aluminum. In theexperiment using aluminum bolts, this slight difference
was made because the mountain collapsed in the D
direction and this slight difference is smaller than thelength of one bolt. Therefore we conclude that we canreasonably simulate sedimentation process of debris using
the MPS method. We verified the simulation accuracy of
the MPS method by comparing with the experiments. As
a result, it clarified that MPS method can be simulated
within the constant fluctuation of experimental result.

3.2 Criticality evaluation in the fuel debris fall in water
accident during removal process
The purpose of this section is clarified to that it ispossible to evaluate criticality with the actual dynamic
behavior of fuel debris in water by developing acalculation system. We calculated criticality evaluation in
the fuel debris fall in water accident during removal
process. As a specific situation, it is assumed that thecollected fuel debris falls to the bottom of the pedestal
where the cooling water is accumulated during the fueldebris removal. A calculation system is shown in Figure.
3. Here, 1,728 debris cubes, 50-mm per side were dropped
from the funnel in water whose surface is 1 m high fromthe bottom of the pedestal. In addition, we calculated twocases to fall in water from two different funnels withdifferent outlet diameters (φ1m or 0.2 m). It makes
possible to evaluate the impact on the criticality by thedifference in the way of falling of the fuel debris in water.
The composition of the fuel debris and conditions iscurrently unknown and so we used UO2 (5wt%) as thecomposition of the fuel debris. The diameter of discrete
particles was 25 mm, and there was a total of 636,142
particles. In neutron transport calculation, the history
number is 2,000,000 and the number of batches is 80 with
40 discard batches. After the calculation, the effective
multiplication factor keff during the fuel debris
sedimentation in water was evaluated. The criticality
calculation is normal eigenvalue calculation considering
delayed neutrons and prompt neutrons.

Figure. 2 Experimental (left side) and simulation (right side) results of sedimentation of 500 bolts inwater. (a) Stainless steel, (b) Aluminum.
3. Analysis of criticality as fuel debris falls in water
3.1 Algorithm coupling the MPS simulation and Monte
Carlo neutron transport calculation
We used the neutron transport calculation with Monte
Carlo code MVP2.0 [5] and the JENDL - 4.0 [6] nuclear
data library. In the calculation by MVP code, it is possible
to create analysis geometry as input data by using theposition of the object. In the calculation system that
couples the MPS method with MVP, we first performed
fluid analysis by MPS. The analysis results, such as 3-D
position information and speed of all discretized particles
was output as CSV data for each calculation step. Eachparticle had an identification (ID). Using this ID, we could
track specific particles. Therefore, we could search andextract the ID of the particle discretizing the solid at eachtime. Using this 3-D position information, we could create
MVP input data. We basically focus on evaluation of thecriticality in the subcritical state and how to ensure safety.
So both of thermal feedback and delayed neutron are notconsidered in the system at this stage.

Figure. 3 The fuel debris falls to the bottom of thepedestal where the cooling water is accumulated
during the fuel debris removal in case of the funnel
outlet diameter (a) φ1m, (b) φ0.2m.

42Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The time change of the effective multiplication factor
keff is shown in Figure. 4. In case of the funnel outlet
diameter φ1m, the fuel debris starts to sink in water about
0.3 seconds from the start of the fall from air. At the sametime, keff rapidly increases because of reactivity increase.

the error bar is not displayed because it is too small. The
uncertainty by MPS method is not evaluated.
keff reaches a maximum just before the debris clump hits
the pedestal bottom. Then in the process of deposition, keffis reduced by the effect of the whole fuel debris shape
becoming flat and the neutron leakage becoming large.
On the other hand, In case of the funnel outlet diameter
φ0.2m, the fuel debris falls little by little because theoutlet of the funnel is narrow and so the increase in keff isslow. Furthermore, the maximum value of keff is smaller
than in another case. These results indicate that it may be
more likely to lead to a criticality accident if the collectedfuel debris falls more rapidly than gradually.
These calculations are to clarify that it is possible toevaluate criticality with the actual dynamic behavior offuel debris in water by developing a calculation systemthat combines MPS and MVP. Also the obtained facts was
first revealed quantitatively by our developed computing
system and it is an important result when considering therational removal working method in the future.

Table. 3 Calculaion condition.
MPS
3Water density

1,000 kg/m

Water kinetic viscosity

1.0×10 m /s

Cube density

10,950 kg/m

Friction coefficient
Spring constants
Damping coefficient

0.5
0.8
0.8

Discrete particle diameter2.5×10 m

Number of particles

636,142

Time step

5.0×10 s

-6

2

3

-2

4. Conclusions-4

In this study, we clarified that it is possible to evaluate
criticality with the actual dynamic behavior of fuel debris
in water by developing a calculation system that combinescomputational fluid dynamic and the Monte Carloneutron transport calculation code.
In the future works, we will clarify what kind of theremoval work conditions are safe and reasonable from theviewpoint of working cost and time in decommissioningof the TEPCO's Fukushima Daiichi nuclear power stationby using our developed calculation system.

MVP
Number of history

2,000,000

Number of batches

80 with 40 discard
batches

References
1. Inter-Ministerial Council for Contaminated Water
and Decommissioning Issues. Mid-And-Long-Term
roadmap towards the decommissioning of TEPCO’s
Fukushima Daiichi nuclear power station (2017). (InJapanese).
2. Hiroki Takezawa and Toru Obara., Passive measurefor preventing recriticality of fuel debris dust fordefueling at Fukushima Daiichi nuclear power station,
Journal of Nuclear Science and Technology, Vol. 53
No. 12 pp. 1960-1967 (2016).
3. S. Koshizuka, Y. Oka, “Moving-particle semiimplicit method for fragmentation of incompressible
fluid”, Nuclear Sci. Eng. 123, 421-434 (1996).
4. Prometech Software「Particleworks」,
http://www.prometech.co.jp/particleworks.html.
5. Yasunobu Nagaya et al. MVP/GMVP II: General
purpose Monte Carlo codes for neutron and photontransport calculations based on continuous energy andmultigroup method. Technical Report JAERI 1348,
Japan Atomic Energy Research Institute (2005).
6. Shibata Keiichi et al. JENDL-4.0: A new library for
nuclear science and engineering. Journal of Nuclear
Science and Technology, 48 (2011).

Figure. 4 Time change of effective multiplication factor
in case of the funnel outlet diameter (a) φ1m, (b) φ0.2m.
The uncertainty of the effective multiplication factor wasless than 0.05% by the Monte Carlo method. Therefore,43

TS-01

Deterministic Transport Methods and Applications

No.

RPHA19-1032

Title

BEAVRS Benchmark Simulation by Using NECP-Bamboo

Authors

Jiewei Yang, Hongchun Wu, Yunzhao Li (Xi'an Jiaotong University,
China)
47

No.

RPHA19-1056

Title

Optimization and Verification of PWR Pin-by-pin
Management Calculation Code NECP-Bamboo2.0

Authors

Sicheng Wang, Liangzhi Cao, Yunzhao Li, Wen Yang (Xi'an Jiaotong

Fuel

University, China)
51
No.

RPHA19-1018

Title

Application of cosKERNEL Code for NPP Shielding Calculation

Authors

Wen Song, Yeshuai Sun, Shuo Li, Xing Wang, Hui Yu, Yixue Chen (State
Power Investment Corporation Research Institute , China)
55

No.

RPHA19-1034

Title

GPU/CPU Concurrent Heterogeneous Parallel MOC Calculation
with Asynchronous Communication Scheme

Authors

Liang Liang, Peitao Song, Qian Zhang, Zhijian Zhang, Qiang Zhao
(Harbin Engineering University, China)
58

No.

RPHA19-1019

Title

Impact of Diffusion Coefficient and Correction Term on theConvergence of CMFD Acceleration for MOC

Authors

Yoshiki Oshima, Tomohiro Endo, Akio Yamamoto (Nagoya University,
Japan)
62

No.

RPHA19-1073

Title

Application of Improved Tone’s Method in STREAM for Fast
Reactor Analysis

Authors

Xianan Du, Sooyoung Choi, Jiwon Choe, Woonghee Lee, Deokjung Lee(Ulsan National Institute of Science and Technology, Korea)
66

44

No.

RPHA19-1004

Title

Application of the SuPer-Homogenization Method in Fast
Reactor Analysis System SARAX

Authors

Linfang Wei, Youqi Zheng, Bowen Xiao, Longfei Xu, Hongchun Wu (Xi'an
Jiaotong University, China)
70

No.

RPHA19-1047

Title

Progress on Multi-physics Calculations of Nuclear Reactor
Cores with the IGA Method

Authors

WilFred G. van Rooijen (University of Fukui, Japan)
74

No.

RPHA19-1052

Title

Research on a Method for Self-Shielding Calculations Based on
IGA-Method

Authors

Matthias Nezondet, WilFred G. van Rooijen (University of Fukui, Japan)
78

No.

RPHA19-1010

Title

A Deep Learning Model for Solving the Eigenvalue of the
Diffusion Problem of 2-D Reactor Core

Authors

Qian Zhang (Harbin Engineering University, China)

No.

RPHA19-1029

Title

Study on Effective Cross Section of Pu Spot in MOX Fuel

Authors

Tatsuya Kawano, Satoshi Takeda, Takanori Kitada (Osaka University,
Japan), Yasunori Ohoka, Shogo Matsuoka, Hiroaki Nagano (Nuclear Fuel
Industries, Ltd., Japan)
82

No.

RPHA19-1031

Title

Modification of MOC for Considering Double Heterogeneity
due to Pu Spots in MOX Fuel

Authors

Akifumi Ogawa, Hikaru Yamaguchi, Satoshi Takeda, Takanori Kitada(Osaka University, Japan), Yasunori Ohoka, Shogo Matsuoka, Hiroaki
Nagano (Nuclear Fuel Industries, Ltd., Japan)
86

45

No.

RPHA19-1027

Title

Reactivity-Equivalent
PhysicalTransformationfor
Homogenization Analysis of Double-Heterogeneous Fuel and
Burnable Poison

Authors

Lei Lou, Yao Dong, Chai Xiaoming, Yu Yingrui, Peng Xingjie (Nuclear
Power Institute of China, China)
90

No.

RPHA19-1095

Title

Prediction Analysis of 243 Am/ 235 U Fission Rate Ratio at KUCA
Solid Moderator Core

Authors

Tadafumi Sano (Kindai University, Japan), Jun-ichi Hori, Yoshiyuki
Takahashi (Kyoto University, Japan)
94

46

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

BEAVRS Benchmark Simulation by Using NECP-Bamboo
Jiewei Yang, Hongchun Wu and Yunzhao Li
Xi’an Jiaotong University: 28 West Xianning Road, Xi’an, Shaanxi 710049, China* Yunzhao Li: yunzhao@mail.xjtu.edu.cn

Abstract
NECP-Bamboo, a PWR-core fuel management calculation system developed by
NECP lab from Xi’an Jiaotong University, is based on improved two-step method.
By using the BEAVRS benchmark released by MIT of America based on theWatts-Bar nuclear power plant measurements, it is verified and validated in thispaper. Results of this paper cover critical boron concentrations, detector signal,
reactivity coefficient and control rod worth. The average absolute value of error
between calculated and measured value is 10ppm, the maximum error of detector
signals in BOC is 4.346%, and the maximum error of control rod wroth is -4%.
Key Words: PWR, BEAVRS, NECP-Bamboo1.0, Verification
1. Introduction

2. Theoretical Models

For large commercial PWR-core, the two-step method isstill usually used in the simulation with depletioncalculation, thermal-hydraulics and critical searchcalculation of large PWR.

2.1 The lattice heterogeneous calculation
With the autonomous multi-group cross section datalibrary and depletion data library from the ENDF/VII.0 .
The assembly modular Method of Characteristic(MOC) is
employed to solve the multi-group neutron transport
equation in the entire lattice directly[3]. It can handle
complex geometry including pin-cell (i.e., pellet, clad,and moderator), IFBA surrounding the fuel rod, water gap
surrounding the fuel assembly, baffle or barrel, et al.
Besides that, to decrease the cost of calculation, 1/4 or 1/8
modular can be described.

Firstly, the two-dimensional fine-group lattice neutrontransport calculation is carried out to obtain the few-groupconstants for multiple states of each assembly withreflective boundary condition. The few-group constants
can then be parameterized into the function of the stateparameters such as assembly burnup, boron concentration,
fuel and moderator temperatures. Those typical stateparameters are defined to describe the lattice state andshould cover the boundary of state during the entire core
operation process. The function relationships are stored in
the so-called few-group constants data library. After that,three-dimensional whole core neutron diffusion
calculation coupling thermal-hydraulics calculation,
depletion calculation and critical search calculation will
be carried out. Based on the two-step method, the PWR
core nuclear design code system NECP-Bamboo [1-2] hasbeen developed, including the two-dimensional latticecalculation code Bamboo-Lattice, three-dimensional
whole-core calculation code Bamboo-Core and three
dimensional whole core transient simulation code
Bamboo-Transient.

Before the transport calculation, the subgroup method is
employed to eliminate the limitation in assembly
geometry and to guarantee both accuracy and efficiency
in resonance calculation.
2.2 The cross section model
Considering the fact that it is impossible to take all states
during reactor operation into consideration in latticecalculation, it is necessary to provide a continuousfunction relationship between few-group constants andstate parameters for core calculation. It is the few-groupconstants parameterization or cross section model.

This paper introduces basic theoretical models and mainly
introduces the modeling of BEAVRS benchmark and thesolution by NECP-Bamboo code system. The main reason
for choosing this benchmark is that this benchmark has
measured value and this benchmark is widely used in lotsof code including deterministic and probabilistic code forthe verification and validation.

There are lots of ways to achieve the few-group constants
parameterization. The selection of state parameters type,the fitting form of state parameters, and the order in eachstate parameters will influence the accuracy. A generalcode [4] was developed to achieve the few-groupconstants parameterization. In this code, there is nolimitation on the type and number of few-group constants,
the type and number of state parameters, and the47Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

polynomial order of the fitting function
A recommended parameterization form is given as default
option in Bamboo-Core as following:  Bu, Tm, Tf , Cb   f  Bu , Cb   f  Bu , Tm, Tf  (1)During the active reactor core fuel cycle operation
simulation, the critical state with the effective
multiplication factor of the core equal to unity has to befound by changing the concentration of the boron in the
coolant or the control rod position, named as criticalsearch. For the second generation reactor, soluble boron
is more commonly used to control reactivity. Taking theboron concentration as an example.

where Bu, Cb, Tm, Tf represent node-averaged depletion(MWd/Tu), soluble boron concentration (ppm),
moderator temperature (K) and effective fuel temperature(K). The Σ can stand for macroscopic cross section (m-1),
microscopic cross section (barn), discontinuity factor andpin-power form factor, which means those variables use
the same function.

The linear interpolation is obtained to estimate the critical boron concentration (CBC):

2.3 The Variational Nodal Diffusion Method

CB2

CB1

Instead of using the transverse integration technique, the
Variational Nodal Method (VNM) directly expand thevolumetric flux and surface partial currents by using
three- and two-dimensional spatial orthogonal
polynomials respectively to transform the continuous
neutron diffusion equation about neutron flux into a linear
algebra system in terms of the expansion moments [5].

CBi 1

CBi

where, keff ,iCBi

represents the effective multiplication

The challenge of modeling the first cycle is how to deal
with the complex power history. In this study, a fine power
history modeling is carried out to describe the true power
history and make the minimum information loss,
guarantee both accuracy and efficiency.

dz

(3)

As can be seen in Fig. 1, the black line is the real power
history with 573 days data. It is complex because of thefrequent shutdown and power drop. In order to describe
those power history, typical days are chosen to comparedwith measured value. In those days, the core condition isexactly the same as real condition, including the power
level, control rod position, inlet coolant temperature, coremacroscopic depletion and et al. Between those typicaldays, extra time points were used to describe the powerchange and guarantee the depletion of typical days is sameas real depletion.

where w is the coolant mass flow (kg/s), h is the coolant
enthalpy (J/kg), q is the linear power in coolant (w/m).
After that, with known coolant temperature, the fuel rod
effective temperature can be obtained by solving the one
dimensional heat conduction model in rod geometry. The
radial heat conduction from the fuel pellet to the gap andeven to the clad, and coolant can be described by thefollowing equation:
1  
T 
(4)
k T  r
q 0
r r 

CBi 1

Besides those material and geometry description, thebenchmark also provides the hot zero power measurement,
measured data obtained during the reactor operation andthe operational power history.

For a single phase flow under the above conditions, thecoolant temperature is calculated by the following masscontinuity and energy conservation equation:
dw
(2)
0
q

keff ,iThe BEAVRS benchmark [ 7 ] is proposed by theComputational Reactor Physics Group of MIT, providing
detailed measured value and information for modeling.
The reactor comes from a Westinghouse four-loop PWR
with 3411MWth. The core is loaded with 193 fuelassemblies with 17×17 pin arrangement and 3
enrichments in the first cycle including 1.6, 2.4 and 3.1
wt.%. High leakage loading pattern is employed.

Ignoring the axial pressure drop and coolant mass
exchange between adjacent channels, the one dimensional
mass and energy conservation equations can be solved
from the core bottom to the top sequentially.

dzkeff ,i 1(5)

3. BEAVRS Benchmark

2.4 Thermal-hydraulics Modeld  hw Guess
1 keff ,ifactor corresponding to boron concentration CBi (ppm),
i is number of the critical boron concentration search,
Guess is initial prediction boron worth (ppm-1).

Compared with nodal methods based on integration, on
one hand, the homogeneous pin-power or flux can bedirectly obtained by combining the expansion moments
and the corresponding basis functions. On the other hand,the control rod cusping effect can be fully eliminated byusing heterogeneous VNM [6].

and1 keff ,1r 

where k(T) is the heat conductivity (w·cm-1) at
temperature T, q is the volumetric heat source (w·cm-3).
2.4 Critical Search Model48

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig. 2 Radial detector response distribution and relative
error between Bamboo-Core1.0 and Measured value for
HZP condition

meaured data
simulation data

Power

80

Table II Control rod worth and error between BambooCore1.0 and Measured value
Control rod
NECPMeasured Error/bank state
Bamboo
value
pcm
D in
804
788
16
C with D in
1219
1203
16
B with D C in
1196
1171
25
A with D, C, B
552
548
4
in
SE with D, C,
460
461
-1
B, A inSD with D, C,
741
772
-31
B, A, SE in
SC with D, C,
1118
1099
19
B, A, SE, SD in

60

40

20

0
0

100

200

300

400

500

Time/Day

Fig. 1 power history of the first cycle (black) and thepower follow approximation (red)Finally, the number of calculation time points is 61, muchless than the original 573.
4.Simulation Results

1.6

4.1 Hot Zero Power Simulation
The results in hot zero power condition include criticalboron concentration, control rod worth and detector
signals.

Detector signal

1.2

Table I shows the result of critical boron concentration.
The results of NECP-Bamboo agree well with themeasured values. Fig. 2 shows the results of detector
response. The maximum error from measured value is4.346% while the Root-Mean-Square (RMS) error is
1.89%. Table II provides the control rod worth in hot zero
power condition. The maximum error is -31pcm, while
the maximum relative error is -4%. Fig. 3 shows theresults of axial detector response and the results of NECPBamboo agree well with the measured values.

0.728
0.704
-3.242
1.242
1.249
0.531
0.984
0.972
-1.1891.212
1.207-0.421
0.968
0.979
1.105
1.138
1.147
0.808
0.897
0.900
0.301
1.065
1.068
0.300

0.584
0.575
-1.627

1.307
1.364
4.346

1.249
1.263
1.129
0.968
0.979
1.105
1.143
1.149
0.542
0.94
0.930
-1.1060.958
0.952-0.605
1.196
1.214
1.480
1.343
1.346
0.186

1.212
1.207
-0.4210.974
0.959-1.540
1.147
1.149
0.174

0.702
0.719
2.379
0.852
0.864
1.397
1.196
1.214
1.480
1.307
1.364
4.346
0.984
0.972
-1.189
1.168
1.186
1.558
0.935
0.926
-0.941

1.242
1.249
0.531
0.873
0.850
-2.5891.264
1.242-1.772

0.6

0.2
0.0

0

50

100

150

200

250

300

350

Axial Location/cm

Fig. 3 Axial detector response distribution and relative
error between Bamboo-Core1.0 and Measured value for
HZP condition
4.2 Hot Full Power Simulation
Hot full power calculations are performed for the firstcycle at 3411 MWth with all rods out of core (ARO). It
can be seen that the results of NECP-Bamboo show good
agreement with the reference in Fig. 4. The maximum
error is 22ppm, while the average absolute error is 7ppm.

Measured Value
NECP-Bamboo1.0
Relative Error/%
RMS/%
1.89
0.702
0.719
2.379
0.958
0.952
-0.6050.8

700

Boron letdown curve in BEAVRS
NECP-Bamboo1.0

600

Boron concentration

0.815
0.782
-4.0740.873
0.850-2.589
1.168
1.186
1.558
0.974
0.959
-1.540
1.143
1.149
0.542
0.897
0.900
0.301
1.011
1.020
0.880
0.779
0.779
0.026

1.0

0.4

Table I critical boron concentration results
NECPMeasured
Error/ppm
Bamboovalue
CBC975ppm
979ppm
-4ppm
0.778
0.751
-3.4581.264
1.242-1.7720.935
0.926-0.941
1.147
1.149
0.174
0.94
0.930
-1.106
1.065
1.068
0.300
0.779
0.779
0.026

Measured Data
NECP-Bamboo 1.0

1.4

0.584
0.575
-1.6270.728
0.704-3.2420.815
0.782-4.0740.778
0.751-3.458

40
30

500

20
10

400

0
300

-10
-20

200

-30

100
0

-40
0

2000

4000

6000

8000

Burnup/MWD/tU

49

50

10000

12000

14000

-50

Error/ppm

100Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig. 4 Comparison of boron letdown curve with provided
data in BEAVRS

By using the BEAVRS benchmark, NECP-Bamboo code
system is verified and validated in this paper. The
calculation includes HZP condition, HFP condition andpower history simulation. The results of NECP-Bamboo
give good accuracy within the criteria in the standard at
major core properties such as critical boron
concentrations, control rod worth and detector response.
Besides, NECP-Bamboo can give pin power distribution,
flux distribution, temperature distribution and et al.

At BOL of HFP condition, the pin power also can be given
by NECP-Bamboo. The guide tubes, fuel rods, andabsorbers can be recognized in Fig. 5.

Acknowledgements
This work is financially supported by the Project
11775171 supported by National Natural Science
Foundation of China.
References
1.
Fig. 5 Pin power of quarter core at BOL
4.3 Power History Simulation2.

Different with HFP condition, the power history andcontrol rod move are considered. The simplified power
history is shown in Fig. 1 as red line. The measured data
is from the detector signals folder rather than BEAVRS
text.

3.

4.

50
40

Measured CBC
NECP-Bamboo1.0

30

800

5.

20
10

600

0
400

-10

Error/ppm

Boron Concentration/ppm

1000

6.

-20

200

-30
-40

0
0

100

200

300

400

500

600

7.

-50

Days

Fig. 6 Comparison of boron letdown curve with provided
measured value.
Considering the complex of power history within the first
500 days, the results of NECP-Bamboo show great
agreement with measured value in Fig. 6. The maximum
error is less than 20ppm and the average absolute error is7ppm. After 500 days, the error is greater than that before
500 days, but the maximum error is -42ppm, within thecriteria. It is necessary to add more time points insimulation to reduce the error after 500 days.
6. Conclusions

50

Y. Li, B. Zhang, et al., “Development andVerification of PWR-Core Fuel Management
Calculation Code System NECP-Bamboo: Part I
Bamboo-Lattice,” Nuclear Engineering and Design,
335, pp.432-440 (2018).
Yang W, Wu HC, Li YZ, et al. “Development andverification of PWR: Core nuclear design code
system NECP-Bamboo, Part II: Bamboo-Core,”
Accepted by Nuclear Engineering and Design.
C. Tian, Y. Zheng, et al., “Neutron transport study
based on assembly modular ray tracing MOC
method,” Nuclear Power Engineering, VOL 36, pp.
157-161, June 15, (2015).
S. Gao. “PWR few-group constant parameterization
methodology study and code development,” Master
Thesis, Xi’an, China: Xi’an Jiaotong Univ. (2015).
Y. Li. “Advanced reactor core neutronics
computational algorithms based on the variational
nodal and nodal SP3 Methods,” PhD Thesis, Xi’an,
China: Xi’an Jiaotong Univ. (2012).
Li Y, Liang B, Wu H, et al. “Heterogeneous
Discontinuity Factor Treatment in Variational Nodal
Method,” Annals of Nuclear Energy(Online; DOI:
10.1016/j.anucene.2018.12.021)
N. HORELIK, B. HERMAN, M. ELLIS, et al.
“Benchmark for Evaluation and Validation of
Reactor Simulations (BEAVRS), Release rev.2.0.2”,
MIT Computational Reactor Physics Group (2018).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Optimization and Verification of PWR Pin-by-pin Fuel Management
Calculation Code NECP-Bamboo2.0
Sicheng Wang, Liangzhi Cao, Yunzhao Li* and Wen Yang
Xi’an Jiaotong University: 28 West Xianning Road, Xi’an, Shaanxi 710049, P. R. China*
yunzhao@mail.xjtu.edu.cn

Abstract
As an improved two-step calculation scheme for PWR-core fuel management, pinby-pin calculation can improve the simulation accuracy by eliminating
approximations including assembly homogenization and pin-power reconstruction.
Accordingly, the NECP-Bamboo2.0 code has been developed by using theapproaches of two-dimensional whole-assembly direct neutron transport
calculation, seven-group pin-cell homogenization and three-dimensional wholecore pin-by-pin multi-physics coupling iteration. Different from the traditional
assembly-homogenization based two-step scheme, the refinement in spacerequires not only higher-order angular discretization, such as from diffusion to SP3,
but also the increased number of energy-group, such as from 2 to 7. Unfortunately,
both the storage of the few-group constants and the computing time would be thenincreased dramatically. Thus, in this paper, an energy group structure optimization
has been done to provide a suitable 4-group structure condensed from the 69-group
WIMS library. The improved code has been verified by using a 2D multi-assembly
neutron transport calculation and a 3D whole-core power-history following
simulation.
Key Words: PWR, Pin-by-pin, Energy-group optimization, NECP-Bamboo
Pin-cell is usually optically thin, making the pin-cell
homogenized few-group constants more dependent on its
environment [4], especially when the target pin locates at
the assembly boundary or is adjacent to some special pincell like strong absorbers. Therefore, the two-groupstructure is insufficient to represent the coupling in pinby-pin simulation. An increased number of energy groups,
such as seven or even more, is widely used to capture thespectrum interference effect [5-6].

1. Introduction
There are two steps in the pin-by-pin calculation scheme,
including two-dimensional lattice high-order neutrontransport calculation [1] and three-dimensional wholecore low-order neutron transport calculation. The former
provides a series of discrete relationships between the
few-group constants of each pin-cell and its stateparameters such as burnup, fuel temperature, moderator
temperature and et al. After obtaining the continuous
relationship between few-group constants and stateparameters by parameterization in the form of few-groupconstants data library, multi-physics coupled iteration canbe carried out in the second step, including neutrontransport, thermal-hydraulics, fuel depletion, critical-state
searching and et al. Different from the traditional two-step
scheme with assembly homogenization, lattice is only
homogenized within each pin-cell to fully eliminate thepin-power reconstruction to provide the pin-power profilefor safety analysis. Based on this scheme, Nuclear
Engineering Computational Physics (NECP) laboratory
in Xi’an Jiaotong University has developed a new PWRcore pin-by-pin fuel management system named NECPBamboo2.0, including the lattice calculation code
Bamboo-Lattice2.0 [2] and the steady-state pin-by-pin
core calculation code Bamboo-Core2.0. [3]However, more energy groups would increase the amountof equations needed to be solved in the whole-corecalculation dramatically, and then deteriorate thecomputational efficiency. Besides, compared with thetwo-group structure, the increase in storage caused by theseven-group structure is about triple. In order to keep the
efficiency and to reduce the storage requirement, a
suitable few-group structure, which is less than sevengroups and can guarantee an acceptable accuracy at thesame time, would be very meaningful. This is what would
be introduced in this paper.
The rest of this paper is organized as following. The
theoretical models employed in the NECP-Bamboo2.0
code are described in Section 2, including the energygroup structure optimization method and a suitable fourgroup structure condensed from the 69-group WIMS

51

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019library. In Section 3, verification of the improved code isintroduced by using two typical calculations bycomparing with one-step calculation or engineering
measurements. Finally, Section 4 summarizes theconclusions.

normalized one. When energy spectrums were obtained
from different calculation models, there would always
exist adjacent fine-group intervals wheresingleenv

variesgently. For those adjacent fine-group intervals, thefollowing equalities can approximately hold:(2)
msingle c   env,h

2. Theoretical Models

m ,hIn this section, the main models in NECP-Bamboo2.0 are
introduced in the sequence of actual calculation scheme
as following: 2D whole-assembly heterogeneous neutrontransport, four-group pin-cell homogenization, Fewgroup cross-section parameterization and 3D whole-core
multi-physics coupling iteration.

hom , single
x,i , g     V=,  Vhg mihet
x , m, hhg mi

2.1 2D whole-assembly heterogeneous neutron transporthom , single
x,i , gsinglem, hsinglem, hmhom , env
x,i , gm     V=,  Vhg mihet
x , m, h

env

m ,h

env

m

m ,h

hg mim     V    c    V=  V c    V
hg mihet
x , m, hhg misinglem, hsinglem, hmmhg mihet
x , m, h

env

m ,h

env

hg mim ,hmhom , env
x,i , gm

(3)where superscripts single and env stand for singleassembly and whole-core models respectively. Eq. (3)shows the fact that in those fine-group intervals which
could be then condensed into a single few group, thesingle-assembly model and the whole-core model can
provide the same few-group constants. Therefore, to findcertain adjacent fine-group intervals where fine-group

Other than the legacy equivalence theory, the subgroupmethod is employed in Bamboo-Lattice 2.0 to eliminate
the limitations in lattice geometry and to guarantee bothaccuracy and efficiency. Besides, to solve the twodimensional steady-state multi-group neutron transport
equation, an assembly-modular Method of Characteristics(MOC) is employed in Bamboo-Lattice 2.0. Since the
MOC method has a capability of processing complex
geometry model, the code can module two-dimensional
single or multiple assembly, while each assembly can besurrounded by a thin water gap and be constructed by acertain number of rectangular pins. In addition, each pin
can be constructed by straight lines and circle curves.

singlevaries gently could be a guiding principle toenv

determine the few-group boundary.

In this paper, the difference between the energy spectrums
obtained from single-assembly model and whole-coremodel is analyzed. An optimized four-group energy
structure is proposed in Table I finally.

2.2 Four-group pin-cell homogenization
Either the generalized equivalent theory or the superhomogenization technique can be utilized to obtain the
few-group cross-sections after the lattice heterogeneous
calculation with the reflective boundary condition [4].
The definition of the homogenized few-group crosssection in the multi-group calculation is applied asfollowing: het
x , m , h  m , h  Vmhg mihom
 x,i , g =(1)m,h  Vm

Table I. The upper bounds of energy group structures
1
2
3
4
5
6
7

hg mi

7-group(eV)
1.000E+07
5.000E+05
9.118E+03
4.000E+00
6.250E-01
1.400E-01
5.800E-02

4-group(eV)
1.000E+07
9.118E+03
4.000E+00
6.250E-01

2-group(eV)
1.000E+07
6.250E-01

2.3 Few-group cross-section parameterization

Theoretically, if the fine-group energy spectrum used forhomogenization was obtained from whole-core highorder heterogeneous calculation, few-group low-ordercore calculation would be expected to provide consistent
results with heterogeneous calculation. Unfortunately,
considering the computational cost, pin-by-pin
calculation scheme can only get the homogenized fewgroup parameters by using the energy spectrum obtained
from single-assembly calculation. Obviously, the energy
spectrum obtained by the single-assembly model withreflective boundary differs from the spectrum from thewhole-core calculation due to the so-called environmental
effects.

NECP-Bamboo2.0 adopts the least-squares polynomial
fitting approach. A recommended few-group constants
parameterization function form is given as following: Bu , Tm, Tf , Cb  f  Bu , Cb   f Bu , Tm, Tf
(4)









where Bu, Cb,Tf ,Tm represent pin-averaged burnup(GWd / tU ) , boron concentration ( PPM ) , moderator
temperature ( K ) and effective fuel temperature
respectively

 K

 can stand for the macroscopic

cross section, microscopic cross section, effective yieldfor fission products (Iodine, Xenon, Promethium,
Samarium) and discontinuity factors.

It should be noted that the spectrum used in Eq. (1) is a52

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

distribution is shown in Fig. 3.
Table Ⅱ. Eigenvalue results of multi-assembly problems

2.4 3D whole-core multi-physics coupling iterationreference
(69G)
k-eff

NECP-Bamboo2.0 proposes a multi-physics couplingstrategy for pin-by-pin core calculation, which is a tightcoupled iterative process. Firstly, the cell homogenizedparameters generated by the Bamboo-Lattice2.0 code are
fitted and then the fission source iteration is performed ateach burnup point. At each outer iterative process, few
mesh acceleration calculation, SP3 multi-groupcalculation, updated fission source and keff, criticalsearchcalculation,
thermal-hydraulicfeedback
calculation, burnup calculation, and cross-section
calculation are performed in sequence. Among them,critical search calculation, thermal-hydraulic calculation
and burnup calculation are performed at certain outer
iterative interval numbers Nc, Nt, and Nd respectively
cross-section calculation must be performed wheneverany of the three calculations above is performed.

2G

case1 1.00388

1.00568

case2 1.18499
case3

1.03943

case4 0.89868

4G


/pcm

k-eff

7G



k-eff

/pcm

180

1.00473

1.1857071

1.04566623

0.90100

232


/pcm

85

1.00519131

1.1851920

1.1851314

1.04139196

1.04078135

0.8992759

0.8990335

Table Ⅲ. Storage Requirement and Calculation Time ofMulti-Assembly Problems
2G
Storage/Mb

Storage
/Mb

6.31

case1
case2

1.35

case3

3. Verification

4G
Time
/s
9.97
16.30

Storage
/Mb

8.00
2.58

13.88

case4

7G
Time
/s
10.92
25.26

Time
/s
17.88

4.78

16.79

37.70
55.81
51.13

3.1 Verification of the new four-group structure
In order to verify the optimized four-group structure
applied in the pin-by-pin calculation, four multi-assembly
problems with reflective boundary conditions have been
calculated. Six single assembly with different enrichment
or different numbers of burnable absorbers and control
rods are shown in Fig. 1. The configurations of the
problems are shown in Fig. 2, and the reference solution
was provided by 2D one-step transport calculation.

UOX-1

UOX-1-BA

UOX-1-CR

UOX-2

UOX-2-BA

UOX-2-CR

Fig. 1. Geometries of assemblies

case-1case-2case-3case-4
Fig. 2. Configuration of multi-assembly problems

2G

4Ga) case-1

7G

2G

4G
b) case-2

7G

2G

4G
c) case-3

7G

2G

4G
d) case-4

7G

Fig. 3. Relative pin-power bias

As in Table II, better results could be obtained from thefour- and seven-group structures compared with thetraditional two-group structure, while keff biases of the
four- and seven-group are both less than 200 pcm, which
satisfies the error requirements. Moreover, Table Ⅲ
shows that both computing time and storage requirementof four-group structure is almost half of those of sevengroup structure. Besides, the reference pin power

3.2 Verification of NECP-Bamboo2.0
With the new four-group structure, a 3D whole-core
power-history following simulation for a commercialpressurized water reactor currently in operation was
carried out to compare with the 13 active measurements.
There are 157 assemblies with different numbers of53

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019burnable absorbers and control rods in the core. Fig. 4
shows the critical boron concentrations, while Fig. 5-7
show the assembly power distributions of BOC, MOC
and EOC. What needs to be pointed out is that a simplified
operating history was employed in the calculation, while
the measured values comes from the actual and complex
operating history.
0
Meas.
Calc.
Error

1000

-5
-10
-15

800

-20
600

-25
-30

400

Error (ppm)

Critical boron concentration (ppm)

1200

0.765
0.753
-1.48%
0.925
0.909-1.68%
1.139
1.149
0.88%
1.073
1.073
0.02%
1.173
1.186
1.13%
1.040
1.042
0.21%
0.997
1.004
0.70%
0.974
0.982
0.81%

-35

Meas.
Calc.
(C-M)/M
0.911
0.901-1.08%
1.097
1.096-0.08%
1.062
1.061-0.08%
1.175
1.187
1.03%
1.052
1.054
0.22%
1.147
1.160
1.17%
1.040
1.042
0.20%

0.657
0.647
0.00%
1.041
1.039
-0.18%
1.130
1.132
0.19%
1.062
1.061
-0.07%
1.175
1.187
1.03%
1.069
1.070
0.13%
1.173
1.186
1.12%

0.728
0.719
-1.18%
0.991
0.984-0.63%
1.130
1.132
0.19%
1.062
1.061
-0.08%
1.169
1.182
1.19%
1.073
1.073
0.02%

0.728
0.719
-1.18%
1.041
1.039-0.19%
1.097
1.096-0.08%
1.037
1.031-0.59%
1.139
1.148
0.87%

0.657
0.647
-1.64%
0.911
0.901-1.08%
1.038
1.034-0.32%
0.925
0.909-1.68%

Fig. 7. Assembly power distribution of EOC

-40

200

0.630
0.620
-1.65%
1.038
1.035-0.31%
1.037
1.031-0.58%
1.169
1.182
1.19%
1.069
1.070
0.13%
1.147
1.160
1.17%
1.014
1.018
0.45%
0.997
1.004
0.70%

0.630
0.620
-1.66%
0.765
0.753-1.48%-45
0

4. Conclusions

-50
0

1

2

3

4

5

6

7

8

9

10

11

12

Burnup depth (GWd/tU)

NECP-Bamboo2.0 developed by NECP in Xi’an Jiaotong
University was improved by optimizing a four-groupstructure and verified by using a two-dimensional wholecore problem and a three-dimensional whole-core powerhistory following simulation. Compared with the original
seven-group structure, about 60% of computing time canabout of 50% of storage requirement can be saved.

Fig. 4. Letdown curve of the cycle
0.807
0.814
0.87%
0.857
0.846
-1.29%
1.095
1.103
0.80%
1.110
1.105
-0.46%
1.199
1.211
1.03%
0.963
0.970
0.69%
1.033
1.046
1.24%
1.061
1.077
1.53%

0.642
0.653
1.62%
1.008
1.014
0.64%
1.050
1.039
-1.12%
1.130
1.141
0.90%
1.008
1.006
-0.17%
1.058
1.076
1.73%
0.949
0.959
1.11%
1.033
1.046
1.24%

0.724
0.712
-1.57%
0.844
0.828-1.84%
1.095
1.098
0.33%
1.117
1.112
-0.49%
1.250
1.269
1.53%
1.101
1.113
1.05%
1.114
1.128
1.26%
1.113
1.127
1.26%

0.588
0.579
-1.45%
0.955
0.948-0.72%
1.026
1.010-1.48%
1.172
1.184
1.01%
1.111
1.116
0.45%
1.190
1.218
2.36%
1.087
1.101
1.36%
1.114
1.128
1.26%

Meas.
Calc.
(C-M)/M
0.997
0.996-0.08%
1.063
1.057-0.63%
1.019
1.007-1.22%
1.211
1.214
0.23%
0.993
0.991
-0.16%
1.058
1.076
1.72%
0.963
0.970
0.68%

0.701
0.701
0.00%
1.042
1.035
-0.63%
1.198
1.184-1.14%
1.061
1.047-1.27%
1.211
1.214
0.23%
1.008
1.006
-0.18%
1.199
1.211
1.03%

0.797
0.796
-0.13%
1.075
1.059-1.56%
1.198
1.184-1.14%
1.019
1.007-1.22%
1.130
1.140
0.88%
1.110
1.105
-0.48%

Acknowledgements
0.797
0.796
-0.13%
1.042
1.035-0.64%
1.063
1.056-0.64%
1.050
1.038-1.14%
1.095
1.103
0.79%

0.701
0.701
0.06%
0.997
0.996
-0.09%
1.008
1.014
0.62%
0.857
0.846
-1.32%

Fig. 5. Assembly power distribution of BOC

This work was supported by National Natural Science
Foundation of China (No. 11775171 and No.11735011).
References
1.

0.642
0.653
1.59%
0.807
0.813
0.84%

2.

Meas.
Calc.
(C-M)/M
0.887
0.875-1.37%
1.041
1.032-0.85%
1.062
1.055-0.69%
1.237
1.253
1.31%
1.096
1.112
1.46%
1.190
1.218
2.36%
1.101
1.113
1.05%

0.631
0.620
0.00%
0.981
0.972
-0.92%
1.139
1.135-0.43%
1.093
1.090-0.35%
1.237
1.253
1.31%
1.111
1.116
0.45%
1.250
1.269
1.53%

3.
0.714
0.706
-1.10%
0.985
0.973-1.22%
1.139
1.135-0.43%
1.062
1.055-0.69%
1.172
1.183
1.00%
1.117
1.112
-0.50%

0.714
0.706-1.10%
0.981
0.971-0.93%
1.041
1.032-0.85%
1.026
1.010-1.49%
1.095
1.098
0.28%

4.
0.631
0.620
-1.62%
0.887
0.875-1.37%
0.955
0.948-0.73%
0.844
0.828-1.86%

Fig. 6. Assembly power distribution of MOC

5.
0.588
0.579
-1.45%
0.724
0.712-1.57%

6.

54

C. Tian, Y. Zheng, et al. Neutron transport study
based on assembly modular ray tracing MOC
method (in Chinese), Nuclear Power Engineering,
36:157-161, 2015.
Y. Li, B. Zhang, et al. Development and verificationof PWR-core fuel management calculation code
system NECP-Bamboo: Part I Bamboo-Lattice,
Nuclear Engineering and Design, 335:432-440,
2018.
W. Yang. Study on PWR Three-dimensional Wholecore Pin-by-pin Physical Calculation Method, Xi’an
Jiaotong University, 2018.
B. Zhang, H. Wu, et al. Evaluation of Pin-Cell
Homogenization Techniques for PWR Pin-by-Pin
Calculation, Nuclear Science and Engineering,
186(2):134-146, 2016.
M. Tatsumia, Y. Ohokaa, et al. Verification of the
AEGIS/SCOPE2 in-core fuel management system,
International Conference on the Physics of Reactors,
Casino-Kursaal Conference Center, Interlaken,
Switzerland, September 14-19, 2008.
C. Beckert and U. Grundmann, Development andverification of a nodal approach for solving themultigroup SP3 equations, Annals of Nuclear Energy
35:75–86, 2008.

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Application of cosKERNEL code for NPP Shielding Calculation
Wen Song, Yeshuai Sun, Shuo Li, Xing Wang, Hui Yu, Yixue Chen
Nuclear Power Software Development Center, SPICRI, Beijing, China, 102209
National Energy Key Laboratory of Nuclear Power Software*
Corresponding author: songwen@spic.com.cn

Abstract
As the basis of shielding design, radiation dose calculation is to ensure thatradiation level is lower enough to satisfy a safe dose for relevant personnel. The
cosKERNEL is a code to compute the dose rate of nuclear power plant system andequipment based on the point kernel integral method with considering of thevarious geometries and materials to make sure that the code can meet therequirements. In the paper, four examples are designed and calculated for different
situation to prove that the code can be used for shielding calculation analysis of
the nuclear power plant.
Key Words: Radiation Dose, cosKERNEL, Point Kernel Integral Methodgeometric space and improve the computation speed of
optical distance by ray tracing means. The cosKERNEL
code can calculate the photon flux and dose rate at themulti-dose point position of the multi-radiation source.
The calculation model of dose rate is as follows.

1. Introduction
Radionuclides produced during reactor operation can be
carried by coolant and flow into the system and equipment,
it can cause radiation exposure to nuclear power plant
workers. The calculation of radiation dose for system andequipment can ensure radiation dose safety for staff andnearby residents. The point kernel integral method is amethod to calculate and process the radiation shielding
problem.
The State Power Investment Corporation ResearchInstitute Nuclear Power Software Development Center ofChina has developed a code package for design andanalysis of Core and System INtegrated Engine (named
COSINE) [1]. The radiation and shielding code system
cosSHIELD [2] is a member of COSINE code package,
which contains a Monte Carlo method code cosRMC, adiscrete ordinate method code cosSN, a point kernel
integral method code cosKERNEL and a radiation source
term code cosSOURCE. This paper mainly describes thecosKERNEL code.
The cosKERNEL code is widely used for radiation
shielding of the main and auxiliary circuit of nuclear
power plants. It supports the definition of point source,
surface source and body source. The calculated geometricspace can be formed by the combination of different basic
body. It can be used for containment building, nuclear
auxiliary building and other shielding calculation.
This paper mainly introduces the function ofcosKERNEL code, and the model calculation results are
displayed.

Di  DFi  V

S r ,i  e  L
4  r 2 B  E , L   dv ,

(1)

where
Sr,i= gamma source strength of source point r at energy
group i, the formula is as shown in equation (2);L= optical distance, the formula is as shown inequation (3);r= distance between the source and the dose point;
B(E,L)= build-up factor, the formula is as shown inequation (4);DFi= flux dose conversion factor;
Di= dose rate of energy group i.
2.1 Source Information
The input mode of source term information can beselected as nuclide activity or gamma source strength.
Multiple sources can be computed simultaneously. The
discrete source strength of energy group i is shown below.

S r ,i  S r  f i ,

(2)

where
Sr= gamma source strength of source point r;
fi= photon energy spectrum for each energy group.

2. Main Function

2.2 Optical DistanceThe cosKERNEL uses point kernel integral method tosimulate the penetrating behavior of gamma rays in

The optical distance is an important part of the calculation.

55

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The cosKERNEL uses the ray tracing method to enhance
the computing efficiency and the computational formula
indicated below.
NL   n Sn ,Example(3)

Mode II

n 1

where
μn= cross section of area n;△Sn= pass through distance of the ray at area n.
2.3 Build-up Factors
The cosKERNEL can calculate photon flux rate and dose
rate of many dose point at the same time. It covers ten
kinds of materials for the build-up factors numeration.
Goldstein Wilkins (GW) method [3] is used in the code tocompute the build-up factors and the formula is as follows.
3

Model III5

B  E , L    Cij  L  Es ji 0 j 0 heavy material 
Es  1/ E ,  light material 

Es  E ,

,

Model
IV

(4)

Description
Source: cylinder, 10cm (radius), 50cm
(height), water (1.0g/cm3), the distance
between each source is 7cm.
Shield: cylinder, 1cm (radius), stainlesssteel (7.86 g/cm3); cuboid, 20cm concrete(2.24 g/cm3). The closest cylinder is 5cm
away from the shield of the cuboid.
Dose Point: 2-A, 2-B, 2-C, 2-D, 30 cm
from the cuboid shielding surface.
Source: cuboid, 50×20×80 (cm), water(0.976 g/cm3).
Shield: The one that is closer to the sourceis stainless steel(7.86 g/cm3) with 1cm
thickness, the other is concrete (2.24
g/cm3) with 20cm thickness.
Dose Point: 3-A, 30cm from the shielding
surface.
Source: sphere, 25cm (radius), water (0.8g/cm3).
Shield: cuboid, 20cm concrete (2.24
g/cm3).
Dose Point: 4-A, 30cm from the shielding
surface.

The information of source strength and flux doseconversion factors are shown in table II. All diagrams inthis section are top views and schematic. The air density
is 1.29E-03 g/cm3.

where
Cij= parameters related to the source energy andmaterial;
L= optical distance;
E= source energy.

Table II. Part of Source Information
Average
SourceFlux Dose Conversion
EnergyStrength
Factors [6](MeV)(1/cm3s)(10-12Svcm-2)
0.31
2.50E+03
1.319E+00
0.65
5.98E+04
2.893E+00
1.10
6.77E+04
4.666E+00
1.45
7.52E+04
5.877E+00
2.08
8.46E+03
7.791E+00
2.25
2.78E+01
8.218E+00
3.01
3.11E+01
1.012E+01
Total Source Strength
2.41E+05(MeV/cm3s)

3. Calculation Results
3.1 Examples description
This part shows four examples and their parameters areshown in table I. The comparison code used in this paper
is cosRMC.
cosRMC is a member of COSINE code package. It isa Monte Carlo particle transport code jointly developedby State Power Investment Corporation Research Institute
Nuclear Power Software Development Center of China(National Energy Key Laboratory of Nuclear Power
Software) [4] and Tsinghua University [5]. cosRMC now
has such functions as criticality calculation, fixed-source
calculation, burnup calculation and kinetics simulations.

The geometric model of four examples are shown in fig.
1-fig. 4.

Table I. Parameters of four examples
ExampleDescription
Source: cylinder, 9cm (radius), 50cm
(height), mixed with water (0.657 g/cm3)and stainless steel (0.976 g/cm3).
Shield: cylinder, 1cm stainless steel (7.8g/cm3); cuboid, 20cm concrete (2.24
Model I
g/cm3), 30cm away from the center of thesource.
Dose Point: 1-A: 5 cm from the cylinder
shielding surface.1-B: 30 cm from the
cuboid shielding surface.

Z
Y

1-A

X

Fig. 1. Geometric model of model I.

56

1-B

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Dose
Point

2-A

2-D
2-C

Z

3-A

2-D

Y

4-A

X

Dose Rate (mSv/h)cosRMCcosKERNEL
1.44E-01
1.72E-01(0.76%)
6.12E-01
7.22E-01(0.60%)
3.41E-01
3.38E-01(0.75%)

Relative
Deviation
19.15%
17.85%-0.89%

From table III we can see that the relative deviation
between cosKERNEL and cosRMC is less than 28%. The
comparison results show that the cosKERNEL calculation
results are reasonable and reliable.

2-B

Fig. 2. Geometric model of model II.

4. Conclusions
This paper describes the point kernel integral codecosKERNEL in COSINE code package for plantshielding design.
The code supports the combination modeling ofvarious geometric objects and it can calculate build-up
factors of various materials. The results display that thecode can be used for shielding calculation and analysis ofreactor systems and equipment.

Y

Z

3-A

X

Fig. 3. Geometric model of model III.

Acknowledgements

Z

Y

This work was supported by National Science andTechnology Major Project of Large Advanced PWR andHTGR Nuclear Power Plant (2017ZX06004-002).
4-A

ReferencesX

1. YIXUE CHEN, HUI YU, “Development Status ofReactor Physics Codes in COSINE Project,” Proc.
The 18th Pacific Basin Nuclear Conference, Busan,
Korea, March 18-23, 2012, Pacific Nuclear Council(2012).
2. Yeshuai Sun, Wen Song, et al, “Preliminary
Application of Radiation and Shielding Code System
of COSINE Code Package,” PHYTRA4, Marrakech,
Morocco, September 17-19, 2018, Moroccan
Association for Nuclear Engineering and Reactor
Technology (2018) (CD-ROM).
3. Goldstein H, Wilkins J E. “Calculations of thepenetration of gamma rays,” International Conferenceon E-business & Inform (1954).
4. Yao Qin, Hui Yu, et al, “Verification of Monte Carlocode cosRMC based on VERA core physics
benchmark,” High Power Laser and Particle beams,
29(12), (2017).
5. Wang K , Li Z , She D , et al. “RMC – A Monte Carlocode for reactor core analysis,” Annals of Nuclear
Energy, 82:121-129, (2015).
6. International Commission on Radiological Protection.
Data for use in protection against external radiation: areport of a Task Group of Committee 3 of theInternational Commission on Radiological Protection[M]. Pergamon (1987).

Fig. 4. Geometric model of model IV.
3.2 Calculation Results
The calculation results of the above four examples arelisted in table III. The parenthesis contents shown insecond column represents relative statistical error ofcosRMC. The relative deviation means (cosKERNEL cosRMC) / cosRMC×100%.
Dose
Point
1-A
1-B
2-A
2-B
2-C

Table III. Calculation results
Dose Rate (mSv/h)
Relative
Deviation
cosRMCcosKERNEL
5.29E+00
5.71E+00
7.86%(0.08%)
6.11E-02
7.80E-02
27.72%(0.76%)
3.20E-01
3.81E-01
19.46%(0.63%)
3.18E-01
3.81E-01
19.81%(0.64%)
1.45E-01
1.72E-01
18.23%(0.76%)

57

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

GPU/CPU Concurrent Heterogeneous Parallel MOC Calculation with Asynchronous
Communication SchemeaLiang Lianga,, Peitao Songa, Qian Zhanga, Zhijian Zhanga, Qiang Zhaoa
Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, College of Nuclear Science andTechnology, Harbin Engineering University, Harbin, Heilongjiang 150001, China*
Corresponding author: liangliang_ls@hrbeu.edu.cn

Abstract
Recently, method of characteristics (MOC) has been widely developed as the mostpromise method to process the whole-core transport calculation. Meanwhile,
GPU/CPU heterogeneous parallel calculation has been widely used and great
performance has been achieved. In this research, GPU/CPU concurrent
heterogeneous parallel MOC calculation is implemented to exploit all thecomputational resource in the heterogeneous high performance computer (HPC)
while the asynchronous communication scheme is introduced in to improve theparallel efficiency. In order to accomplish this scheme, both MPI, OpenMP, CUDA
protocols are introduced. The spatial domain decomposition (SDD) techniqueprovides the coarse-grained parallelism with the MPI protocol while the finegrained parallelism is exploited through OpenMP (in CPU calculated domain) and
CUDA (in GPU calculated domain) based on the ray parallelization. Numericalresults indicate that both the concurrent heterogeneous parallel calculation and theasynchronous communication scheme are effective to improve the performance ofthe parallel MOC calculation. Moreover, the CPUs/GPUs heterogeneous clusters
significantly outperform the CPUs clusters, which makes the large-scale wholecore transport calculation is more practicable.
Key Words: Whole-core Transport, GPU/CPU Concurrent Heterogeneous
Parallel, MOC, Spatial Domain Decomposition, Asynchronous commu
nicationparallel spatial domain decomposition (SDD) [9] MOC
calculation is implemented. The SDD technique provides
the coarse-grained parallelism with the MPI protocol
while the fine-grained parallelism is exploited through
OpenMP (in CPU calculated domain) and CUDA (in GPU
calculated domain) based on the ray parallelization.
Moreover, the asynchronous communication scheme
is applied to reduce the communication time. The
overlapping asynchronous communication between GPU
and CPU has been introduced into this research to overlap
the CPU communication and the GPU transport
calculation. The asynchronous data copy between GPU
and CPU are processed to further improve the parallel
performance.
Numerical results indicate that the GPU/CPU
concurrent heterogeneous calculation is able to exploit all
the computational resource in the heterogeneous nodes
while the asynchronous communication scheme
significantly improve the parallel efficiency.

1. Introduction
Recently whole-core transport calculation is widely
developed in reactor physics calculation to obtain highfidelity simulation. Most of the research is based on the
MOC method [1], such as CRX [2], DeCART [3],nTRACER [4], and NECP-X [5]. CPU-based large-scaleparallel algorithms have been widely introduced in anddeployed on the supercomputer [6]. However, the wholecore high-fidelity simulation is still time consuming, evenusing the newest CPU-based supercomputers.
Heterogeneous parallelization is one of the mostpromising approaches to improve the computational
performance of the modern HPC systems. With thedramatically enhanced performance, the heterogeneous
hybrid parallel MOC method is practicable to accomplish
the whole-core transport calculations. Choi et al.
proposed a technique to accelerate MOC calculation withanisotropic scattering source on GPUs [7]. In our former
research, a 2D MOC parallel calculation based on GPU is
implemented and a performance analysis model isintroduced to guide the performance optimization [8].
In this paper, GPU/CPU concurrent heterogeneous

2. Concurrent Parallel Heterogeneous MOC
Calculation

58

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019assigned to the corresponding CPU or GPU. As
mentioned above, the ray parallelization is performed by
OpenMP protocol in CPU and CUDA protocol in GPU.
The concurrent MPI + OpenMP/CUDA programming
model is employed in this implementation to perform theconcurrent CPU/GPU heterogeneous parallelization. As
illustrated in Fig. 1, this is a heterogeneous cluster whichcontains 𝑁 compute nodes connected by high-speed
interconnection fabrics. Each heterogeneous nodecontains 𝑁𝑐 CPU-cores and 𝑁𝑔 GPUs, and commonly
𝑁𝑐 is assumed to be larger than 𝑁𝑔 . Based on thishardware model, two levels of parallelism are utilized toaddressing the MOC application onto heterogeneous HPC
cluster.

s3

s2

rank 4s4

1 G
  s, g  gg  (r)4 g 1

Node n2

 g is the fission spectrum.

ray k

GPU
2

GPU
3

CUDA

ray k
ray k

/ group 1/ group 2

GPU thread t+G
ray k / group G
GPU thread t+G+1 ray k+1/ group 1
GPU thread t+G+2 ray k+1/ group 2

OpenMP

CPU thread mray (n+0)+0*n_thread / group 1~Gray (n+0)+1*n_thread / group 1~Gray (n+0)+2*n_thread / group 1~G
…

GPU
1

GPU thread t
GPU thread t+1

keff is the eigenvalue.

ray n

ray ID < total number of rays
CPU thread m+1
ray (n+1)+0*n_thread / group 1~Gray (n+1)+1*n_thread / group 1~Gray (n+1)+2*n_thread / group 1~G
…

GPU
0

Where;

5

rank 5s5CPU0SpatialSomainDecomposition
(MPI)

g G
Qg (r )=
 f , g g  (r)4 keff g 1


s5

…

r.

s1

ray ID < total number of rays
CPU thread m+2…

the position r .
Qg (r ) is the total source of group g in the position

s4 s0

rank 3s3

m is the discrete direction.
g is the energy group.
 m , g (r ) is the angular flux of the direction m andenergy group g in the position r .
 t , g (r ) is the transport cross section of group g inrank 2s2

 , ,  is the direction cosine.

…

Where;

rank 1
s1

The basic theory of MOC method is shown as below. The
single group neutron transport equation is as Eq. (1) andthe MOC formed neutron transport equation is as Eq. (2).
 m, g (r ) m, g (r ) m, g (r )



(1)
x
y
z
 t , g (r ) m, g (r )  Qg (r)rank 0s0

2.1 MOC Basic Theory

Fig. 1. MPI + OpenMP/CUDA Programming Modelwith Concurrent SDD and Ray Parallelization.

 f , g  is the fission neutron number of group g  s , g  g is the scatter cross section from g  to g

3. Asynchronous Communication Schemeg  is the scalar flux of group g  .
 t , g (r ) m, g (s)  Qg (r )dsWith the flat source and the flat cross section
approximation, the analytical solution of Eq. (2) can beeasily obtained like Eq. (3).
Q
 s
 s
(3)
 m, g (s)   min, g e t ,g  g (1  e t ,g )t , g

The communication time in GPU/CPU heterogeneous
parallel calculation contains two parts: 1) thecommunication between each CPU process, 2) the data
copy between GPU and CPU. In order to reduce thecommunication impact, the overlapping asynchronous
communication scheme is implemented. The basic idea isthat CPU processes the communication meanwhile GPU
performs the calculation.

2.2 Implementation of GPU/CPU Concurrent
Heterogeneous Parallel MOC Calculation

3.1 Overlapping asynchronous communication between
CPU

SDD technique is important for the simulations of thewhole-core transport calculation on the HPC cluster. The
spatially decomposed subdomain is managed by a MPI[10] process, which will be calculated by GPU or
OpenMP [11] multi-threads. The independence of thoserays provides the possibility and the flexibility of massive
ray parallelization. As a result, the ray parallelization
involves parallelizing the loop over all rays in all
discretized directions. The ray parallelization is
implemented using a shared memory model (e.g. OpenMP,
CUDA [12]).
According to the computational ability, the wholeproblem is divided into several subdomains, which is

The overlapping asynchronous communication between
CPU is accomplished by starting process communication
between CPU immediately after initializing the GPU. The
MPI communication and the MOC sweep on GPU (CPU)are simultaneously performed. This scheme achieves theoverlap between communication and the MOC sweep,
which will reduce the overall runtime. The negative
impact is the iteration degradation, which is negligible. In
general, the overlapping asynchronous communication
scheme will hide the communication time while theiteration degradation is negligible.

d m , g ( s )(2)

3.2 Asynchronous data-copy between GPU and CPU

59

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019suggest that GPU/CPU concurrent calculation is able toexploit all the computational resource in the heterogeneous
node, which can bring 14.5% performance improvement
when using 5 nodes. With the increase of the nodes, thisvalue increases slightly.
Moreover,
MPI+CUDA+OpenMP+Asynchronous
means that the asynchronous communication scheme isalso applied and 21.8% improvement is observed whenusing 5 nodes. Since the asynchronous communication
scheme significantly reduce the communication time, theperformance improvement increases with the number ofthe nodes arising.

According to the former research, the communication costof the data-copy between GPU and CPU also have large
impact on the performance of the parallel efficiency of theheterogeneous application. To address this issue, theasynchronous data-copy between GPU and CPU is
implemented like shown in Fig. 2, In the original way, the
GPU/CPU data-copy, GPU sweep, CPU communication
are executed serially while in the overlapping
asynchronous
communication
scheme,thecommunication and the computation are executed
concurrently, which will reduce lots of runtime on each
iteration. In this research this new scheme is implemented
and the numerical results demonstrate that the great
improvement is obtained.

Fig. 2. the Asynchronous Communication Scheme

Fig. 4. Performance Comparison of Each Parallelization

4. Numerical Results

With the increase of the number of the calculation
nodes, all the parallel efficiency decrease. The parallel
efficiency of the parallelization without asynchronous
communication scheme decreases dramatically. According
to Fig. 5, the asynchronous communication scheme
contributes remarkable improvement. The overall parallel
efficiency is 95.8% when using 5 nodes to calculate while
the parallelization without asynchronous communication
scheme is only 84.5%.

The TianHe-ES supercomputer (TH-ES for short) serves
as the computational platform which is deployed in the
National Supercomputing Center in Tianjin. The
computing subsystem of TH-ES is constructed with 40
heterogeneous nodes, each of which is configured with
two Intel Xeon CPU E5-2690 v4 CPUs and two Tesla K80
GPUs.
In order to evaluate the performance of the newheterogeneous algorithm, a series of runs are performed fora 2D fictitious whole-core problem. This problem consistsof 100 UO2 and MOX fuel assemblies from the C5G7 2D[13] benchmark aligned in a chessboard pattern in Fig. 3.

Vacuum B.C.

Vacuum B.C.

Vacuum B.C.

UO2
Assembly

Vacuum B.C.
MOX
Assembly

Reflector

Fig. 3. The Mock Whole-core Problem.

Fig. 5. Parallel Efficiency with and without
Asynchronous Communication Scheme

Fig. 4 shows the runtime of each parallelization withdifferent number of heterogeneous nodes while Fig. 5
shows the parallel efficiency. In Fig. 4, MPI+CUDA
means that CPUs do not perform the MOC sweep while the
MPI+CUDA+OpenMP means both CPU and GPU
perform the MOC sweep concurrently. Numerical results

The same C5G7 based fictitious 2D whole-core
problem is also performed in the CPU nodes in TH-ES
HPC system. Fig. 6 illustrates the performance comparisonof the CPUs/GPUs heterogeneous nodes and CPUs nodes.
The MPI parallelization is performed with 5,10, 15, and 20

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

CPUs nodes. As shown in Fig. 6, The CPUs/GPUs
heterogeneous nodes significantly outperform the CPUs
nodes, and the computing capability of one CPUs/GPUs
heterogeneous node is equal to five CPUs nodes in our
implementation.

2.

3.

4.

5.
6.

Fig. 6 Performance Comparison of the CPU/GPU
Heterogeneous Node and the CPU Nodes
5. Conclusion

7.

In this paper, GPU/CPU concurrent heterogeneous
parallel SDD MOC calculation is implemented and theasynchronous communication scheme is applied to reducethe communication time. A 2D fictitious whole-core
problem is calculated to analyze the performance of the
new implementation.
Numerical results show that both the concurrent
heterogeneous calculation and the asynchronous
communication scheme are effective to reduce the overallruntime. Overall parallel performance improvement isabout 21.8% when using 5 heterogeneous nodes.
Numerical results also indicate that the computing
capability of the CPUs/GPUs heterogeneous node isbasically five times faster than the CPUs node in our
implementation.
This work performs remarkable improvement, whichmake the large-scale whole-core transport calculation
more practicable. The implementation and performance
analysis of the 3D whole-core transport calculation on
heterogeneous cluster will be the future focus.

8.

9.

10.

11.

Acknowledgements

12.

This work is supported by the fund provided by the
Science and Technology on Reactor System Design
Technology Laboratory, the Heilongjiang Province
Science Foundation for Youths [QC2018003], andResearch on Key Technology of Numerical Reactor
Engineering [J121217001]. This work is also supportedby the project from the Nuclear Power Technology
Innovation Center.

13.

References

1.

Askew JR, 1972. “A Characteristics Formulation

61of the Neutron Transport Equation in ComplicatedGeometries”. AEEW-M, 1108.
Cho NZ, Lee GS, Park CJ. “Partial current-based
CMFD acceleration of the 2D/1D fusion method
for 3D whole-core transport calculations,” Trans.
Am. Nucl. Soc., 88, 594-594 (2003).
Joo HG, Cho JY, Kim KS, et al., “Methods andperformance of a three dimensional whole-coretransport
code
DeCART,”International
conference on physics of reactors, Chicago, Illinois,
USA, Apr 25-29 (2004).
Jung YS, Shim CB, Lim CH, et al., “Practicalnumerical reactor employing direct whole coreneutrontransport
andsubchannel
thermal/hydraulic solvers,” Ann. Nucl. Energy, 62,
357–374 (2013).
Chen J, Liu ZY, Zhao C, et al., “A new high-fidelity
neutronics code NECP-X,” Ann. Nucl. Energy, 116,
417–428, (2018).
Ryu M, Jung YS, Cho HH, et al., “Solution of the
BEAVRS benchmark using the nTRACER direct
whole core calculation code,” Journal of Nuclear
Science and Technology, 52(7-8), 961-969, (2015).
Choi N, Kang J, Joo HG, “Massively parallel
method of characteristics neutron transport
calculation with anisotropic scattering treatment
on GPUs,” Proceeding of International conferenceon high performance computing in Asia PacificRegion, Chiyoda, Tokyo, Japan, January 28-31.
(2018).
Song PT, Zhang ZJ, Liang L, et al.,
“Implementation and performance analysis of themassively parallel method of characteristics basedon GPU,” Ann. Nucl. Energy, 131, 257-272, (2019).
Kosaka S, Saji E, “Transport theory calculation forheterogeneous multi-assembly problem bycharacteristics method with direct path linking
technique.” J. Nucl. Sci. Technol, 37, 1015-1023.
(2000).
Mathematics and Computer Science, The Message
Passing
Interface(MPI)
standard,
http://www.mcs.anl.gov/research/projects/mpi/,(2019).
The OpenMP API specification for parallel
programming, https://www.openmp.org/, (2019).
CUDA toolkit documentation, CUDA C
programming
guide,
https://docs.
nvidia.com/cuda/cuda-c-programmingguide/index.html, (2019).
Lewis EE, Palmiotti G, Taiwo TA, et al.,“Benchmark Specifications for Deterministic
MOX Fuel Assembly Transport Calculations
without Spatial Homogenization,” Organization
for Economic Co-operation and Development’s
Nuclear Energy Agency, (2003).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Impact of Diffusion Coefficient and Correction Term on the Convergence of CMFD
Acceleration for MOCaYoshiki Oshimaa,, Tomohiro Endoa and Akio Yamamotoa
Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, 464-8603, Japan*
Corresponding author: y-oshima@fermi.energy.nagoya-u.ac.jp

Abstract
The Coarse Mesh Finite Difference (CMFD) acceleration is widely used forneutron transport calculations, e.g., for the Method of Characteristics (MOC).
However, numerical instability is observed for optically thick meshes. In theprevious studies, the diffusion coefficient used in CMFD was corrected to improvethe convergence. The present study tries to clarify the impact of diffusion
coefficient and magnitude of correction term, which are used in CMFD, on theconvergence performance of CMFD. The results indicated that the correction term
became smaller when the modified diffusion coefficient, which improved thestability of CMFD, was used. However, when the diffusion coefficients were
further adjusted to make correction terms smaller, further improvement of theconvergence was not observed. These results imply that the limitation of thecurrent finite difference approximation formula used for CMFD acceleration.
Key Words: CMFD acceleration, diffusion coefficient, correction term,
convergence, MOC
transport calculation and diffusion calculation become
equivalent [7]. Their effectiveness is evaluated by thelinearized Fourier analysis.
These conventional studies on the stability of CMFD
generally focused on the diffusion coefficient used in thefinite difference approximation formula. Therefore, themagnitude of the correction term in CMFD has not been
explicitly discussed. CMFD uses the finite difference
approximation formula based on the diffusion theory.
When the correction term is large, it deviates from theoriginal finite difference approximation formula.
Therefore, the magnitude of the correction term may
have an impact on the convergence performance.
This study confirms the magnitude of the correction
term for various diffusion coefficients. Impact of thecorrection term on the convergence property is also
discussed.

1. Introduction
The neutron transport method such as the Method of
Characteristics (MOC) requires longer computation time
for a fuel assembly or a core analysis when noacceleration method is applied. Convergence
performance of MOC depends on its scattering ratio,
which is the ratio of scattering cross section to total crosssection, and eigenvalue dominance ratio. Especially in alight water reactor analysis, an efficient convergenceacceleration method is crucial and the Coarse Mesh
Finite Difference (CMFD) acceleration method is
generally used because scattering ratio and eigenvalue
dominance ratio are close to 1 in a light water reactor
analysis [1–3].
In CMFD, diffusion calculations are performed using
the modified finite difference diffusion approximation incoarse meshes. Generally, since the net neutron current
obtained by MOC is not consistent to that by a finite
difference diffusion approximation, a correction term isused to reproduce the net current by MOC. Though the
CMFD method is very efficient, the convergence issue isobserved for optically thick meshes [4].
Various studies have been carried out to improve thestability of CMFD and it is known that a correction of the
diffusion coefficient is a remedy [5]. Various methods
have been proposed e.g., multiplying the diffusion
coefficient by a constant [4], the effective diffusion
coefficient [6], adding a constant to the diffusion
coefficient [5] and the diffusion coefficient where

2. Theory
Let us consider a two-node problem shown in Fig. 1

Fig. 1. 𝜙, 𝐷, 𝐽 in coarse meshes.

62

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

In CMFD, the scalar fluxes and the net neutron current incoarse meshes have the following relation:𝐹𝐷̂𝑖+1/2
𝐽𝑖+1/2 = −𝐷
(𝜙𝑖+1 − 𝜙𝑖 )(1)
𝐶𝑂𝑅
̃𝑖+1/2(𝜙𝑖+1 + 𝜙𝑖 ),+ 𝐷where 𝐽 is the net neutron current, 𝜙 is the scalar flux,
𝐹𝐷̃ 𝐶𝑂𝑅 is the correction factor, 𝐷̂𝑖+1/2
𝐷is given by:
2𝐷
𝐷
𝑖
𝑖+1
𝐹𝐷
̂𝑖+1/2(2)
𝐷=,
𝐷𝑖 Δ𝑖+1 + 𝐷𝑖+1 Δ𝑖where 𝐷 is the diffusion coefficient, Δ is the coarse
mesh size, 𝑖 is an index of the coarse mesh, 𝑖 + 1/2
indicate the mesh interface between 𝑖 and 𝑖 + 1.
𝑀𝑂𝐶
Using 𝜙𝑖𝑀𝑂𝐶 , 𝐽𝑖+1/2
calculated by MOC and Eq. (1),𝐶𝑂𝑅̃
𝐷𝑖+1/2 is written as:
𝑀𝑂𝐶
𝐹𝐷
𝑀𝑂𝐶
̂𝑖+1/2
𝐽𝑖+1/2+𝐷(𝜙𝑖+1− 𝜙𝑖𝑀𝑂𝐶 )𝐶𝑂𝑅
̃𝑖+1/2
𝐷=.
(3)
𝜙 𝑀𝑂𝐶 + 𝜙 𝑀𝑂𝐶

The GENESIS code is used for MOC calculations [9].
The calculation conditions are as follows: ray trace widthis 0.1 cm, the number of azimuthal angles is 48 for 2π,
the number of polar angles 4 for π (TY-quadrature), thenumber of energy group is 7, the number of azimuthal
divisions in a pin-cell is 16, the number of radialdivisions in a pin-cell is 6, the number of maximum inner
iteration is 2, the number of maximum outer iteration is50, convergence criteria for 𝑘𝑒𝑓𝑓 and 𝜙 of MOC are
5.0 × 10−6 and 1.0 × 10−5 , convergence criteria for𝑘𝑒𝑓𝑓 and 𝜙 of CMFD are 5.0 × 10−7 and 1.0 ×
10−6 . The GENESIS code can set CMFD meshregardless of the input geometry. In this analysis, twodifferent mesh sizes for CMFD are used: 1 × 1 cell and5 × 5 cells for a CMFD mesh, i.e., mesh sizes are 1.26
cm and 6.30 cm, respectively.

In this study, diffusion coefficient 𝐷𝑖 of thefollowing four definitions are used:(a) Conventional diffusion coefficient:
1
𝐷𝑖𝐶𝑜𝑛𝑣 =,(4)3Σ𝑡𝑟,𝑖where Σ𝑡𝑟 is macroscopic transport cross section.

4. Results and Discussions

𝑖+1

𝑖

4-1. Results of mesh size 1.26 cm
Figure 3 shows the correlation between the converged
neutron currents estimated by MOC and that by finite
difference approximation using 𝐷𝑖𝐶𝑜𝑛𝑣 and 𝐷𝑖𝐴𝐺𝐷 for
all CMFD cells. The horizontal and vertical axes of Fig.
3 correspond to the first and negative of the second termson the right side of Eq. (3), respectively. In the vicinityof the red dotted line, these terms cancel out each other
𝐶𝑂𝑅
𝐶𝑂𝑅
̃𝑖+1/2
̃𝑖+1/2and then 𝐷≈ 0 . The histogram of |𝐷| isshown in Fig. 4.
𝐶𝑂𝑅
̃𝑖+1/2
Figures 3 and 4 show that the magnitudes of 𝐷
are small for both diffusion coefficients. Moreover, MOC
using either diffusion coefficient converged. These
results suggest that the finite difference approximation
formula of Eq. (1) is adequate for optically thin meshes.

(b) Artificial grid diffusion coefficient:Δ(5)
𝐷𝑖𝐴𝐺𝐷 = 𝐷𝑖𝐶𝑜𝑛𝑣 + 𝑖 .
4
The convergence of CMFD is equivalent to that of pCMFD when AGD is used [8].
(c) Diffusion coefficient multiplied by α:𝐷𝑖∗ = 𝛼𝐷𝑖𝐶𝑜𝑛𝑣 ,(6)𝐶𝑂𝑅
̃𝑖+1/2where 𝛼 is determined to minimize |𝐷| asdiscussed in Section 4.
(d) Diffusion coefficient added by energy groupdependent 𝛼𝑔 Δ𝑖 :𝐷𝑖+ = 𝐷𝑖𝐶𝑜𝑛𝑣 + 𝛼𝑔 Δ𝑖 ,(7)
where 𝑔 is an index of the energy group. The 𝛼𝑔 isdetermined in each energy group.
3. Calculation Conditions
The KAIST benchmark problem 2A shown in Fig. 2 isused for verification calculations. The fuel assemblies
consist of arrays of 17 × 17 fuel rods. The symbols(Ⅰ)–(V) represent UOX-1, UOX-2(CR), UOX-2(BA16 ),MOX-1, and MOX-1(BA8 ), respectively.
Vacuum boundary

Fig. 3. Relation between the neutron currents obtainedby MOC and finite difference diffusion approximation( mesh size 1.26 cm ).

Ⅰ

Ⅰ

Ⅳ

Ⅱ

Ⅰ

Ⅱ

Ⅴ

Ⅱ

Ⅰ

Ⅲ

Ⅱ

Ⅳ

Ⅰ

BaffleVacuum boundary

Reflective boundary

Reflector

Reflective boundary

Fig. 2. KAIST benchmark problem 2A.

63

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The above results suggest that the magnitude of𝐶𝑂𝑅
̃𝑖+1/2𝐷
becomes smaller when 𝐷𝑖𝐴𝐺𝐷 , which shows noconvergence issue, is used. Figure 5 suggests that further
corrections on the diffusion coefficient are possible to𝐶𝑂𝑅̃𝑖+1/2
make the magnitude of 𝐷
smaller. Therefore, the∗diffusion coefficient 𝐷𝑖 , which is calculated by Eq. (6),is considered. The correction value 𝛼, which is energy
group independent value, is determined to minimize
𝐶𝑂𝑅̃𝑖+1/2overall |𝐷| . Furthermore, Fig. 7 shows energy
𝐶𝑂𝑅̃𝑖+1/2group dependent 𝐷
in Fig. 5 for 𝐷𝑖𝐴𝐺𝐷 . Figure 7
suggests that corrections on the diffusion coefficient
would be energy group dependent to further minimize
𝐶𝑂𝑅̃𝑖+1/2|𝐷| . Therefore, in addition to 𝐷𝑖∗ , 𝐷𝑖+ , which has
the group dependent correction term, is considered.

𝐶𝑂𝑅̃𝑖+1/2
Fig. 4. The histogram of |𝐷| for mesh size 1.26 cm.

4-2. Results of mesh size 6.30 cm
Figures 5 and 6 show the results for mesh size 6.30 cm inthe same manner as Figs. 3 and 4. The magnitudes of𝐶𝑂𝑅
̃𝑖+1/2𝐷using either diffusion coefficient become larger
than that for mesh size 1.26 cm. However, when 𝐷𝑖𝐴𝐺𝐷
𝐶𝑂𝑅
̃𝑖+1/2is used, the magnitude of 𝐷
becomes smaller in𝐶𝑜𝑛𝑣comparison to 𝐷𝑖. Note that the MOC calculations
using 𝐷𝑖𝐶𝑜𝑛𝑣 and 𝐷𝑖𝐴𝐺𝐷 diverges and converges,
respectively.

Fig. 7. Relation between the neutron currents obtainedby MOC and finite difference diffusion approximation( 𝐷𝑖𝐴𝐺𝐷 , group dependent, mesh size 6.30 cm ).
The correction values 𝛼 and 𝛼𝑔 are determined by trial
𝐶𝑂𝑅̃𝑖+1/2and error to minimize |𝐷| . As a result, 𝛼 = 7.5
and 𝛼𝑔 = {2.0,1.0, 0.7, 0.5, 0.3, 0.2, 0.1} are used.
𝐶𝑂𝑅̃𝑖+1/2
Figures 8 and 9 show the magnitudes of 𝐷
using
𝐶𝑜𝑛𝑣
𝐴𝐺𝐷
∗
+
𝐷𝑖
, 𝐷𝑖 , 𝐷𝑖 and 𝐷𝑖 .

Fig. 5. Relation between the neutron currents obtainedby MOC and finite difference diffusion approximation( mesh size 6.30 cm ).

Fig. 8. Relation between the neutron currents obtainedby MOC and finite difference diffusion approximation( various diffusion coefficients, mesh size 6.30 cm ).

𝐶𝑂𝑅
̃𝑖+1/2
Fig. 6. The histogram of |𝐷| for mesh size 6.30 cm.

64

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

5. Conclusions
In this study, impact of the diffusion coefficient and thecorrection term in the CMFD acceleration method were
investigated to improve the convergence of CMFD. The
magnitude of the correction term is small and the validityof the finite difference formula is confirmed for opticallythin meshes. Contrary, when the optical mesh size
becomes large, the correction term becomes larger andthe convergence becomes worse. However, when themodified diffusion coefficient (AGD) which improves
the stability of CMFD is used, the correction term
becomes smaller and the calculation converges.
Although the diffusion coefficient was further corrected
to make the correction term smaller, the improvement of
the convergence was not observed. These results suggestthat the finite difference formula currently used in CMFD
has the limitation and the improvement of the formula
including reduction and prolongation would be necessary
to improve the convergence performance.

𝐶𝑂𝑅̃𝑖+1/2
Fig. 9. The histogram of |𝐷| using various
diffusion coefficients.

The convergence performance for 𝐷𝑖𝐴𝐺𝐷 , 𝐷𝑖∗ and 𝐷𝑖+ isshown in Figs. 10 and 11. Note that MOC using 𝐷𝑖𝐶𝑜𝑛𝑣
diverges. Figures 10 and 11 indicate that no convergence
improvement is observed for 𝐷𝑖∗ and 𝐷𝑖+ . Namely, even
if the correction term is further minimized, more
improvement of convergence is not achieved. Thissuggests that the limitation of the current finite difference
formula and prolongation process, in which scalar flux
distribution inside a CMFD mesh is considered.

References
1. Smith K, “Nodal Method Storage Reduction by Nonlinear Iteration,” Trans. Am. Nucl. Soc., 44, 265–266(1983).
2. Smith K, Rhodes JD, “Full-core, 2-D, LWR Core
Calculations with CASMO-4E,” Proc. PHYSOR 2002,
Seoul, South Korea, Oct. 7–10 (2002) (CD-ROM).
3. Joo HG, Cho JY, Kim KS, et al., “Methods andPerformance of a Three-dimensional Whole-CoreTransport Code DeCART,” Proc. PHYSOR2004,
Chicago, Illinois, USA, Apr. 25–29 (2004) (CD-ROM).
4. Yamamoto A, “Generalized Coarse Mesh Rebalance
Method for Acceleration of Neutron Transport
Calculations,” Nucl. Sci. Eng., 151, 274–282 (2005).
5. Jarrett M, Kochunas B, Zhu A, et al., “Analysis of
Stabilization Techniques for CMFD Acceleration of
Neutron Transport Problems,” Nucl. Sci. Eng., 184,
208–227 (2016).
6. Larsen E, “Infinite Medium Solutions to the Transport
Equation, Sn Discretization Schemes, and the
Diffusion Approximation,” Proc. M&C + SNA, Salt
Lake City, UT, USA, Sep. 9–13 (2001) (CD-ROM).
7. Yamamoto A, Endo T, Giho A, “Transport Consistent
Diffusion Coefficient for CMFD Acceleration andComparison of Convergence Properties,” J. Nucl. Sci.
Technol.,
DOI:10.1080/00223131.2019.1618405(2019).
8. Cho NZ, Lee GS, Park CJ, “Partial Current-based
CMFD Acceleration of the 2D/1D Fusion Method for
3D Whole-core Transport Calculations,” Trans. Am.
Nucl. Soc., 88, 594–596 (2003).
9. Yamamoto A, Giho A, Kato Y, et al., “GENESIS: aThree-dimensional Heterogeneous Transport Solver
based on the Legendre Polynomial Expansion ofAngular Flux Method,” Nucl. Sci. Eng., 186, 1–22
(2017).

1.00E+03

φ residual error

AGD
7.5
EG

1.00E+00

1.00E-03

1.00E-06

0

5

10 15 20 25 30 35 40 45 50
Outer iteration

Fig. 10. The residual errors of 𝜙.
1.00E+00

keff residual error

AGD
1.00E-02

7.5∗

1.00E-04

EG+

1.00E-06

1.00E-08
1.00E-10

0

5 10 15 20 25 30 35 40 45 50
Outer iteration

Fig. 11. The residual errors of 𝑘𝑒𝑓𝑓 .

65Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Application of Improved Tone’s Method in STREAM for Fast Reactor Analysis
Xianan Du, Sooyoung Choi, Jiwon Choe, Woonghee Lee, and Deokjung Lee
Ulsan National Institute of Science and Technology (UNIST)
50 UNIST-gil, Ulsan 44919, Republic of Korea
Corresponding author: deokjung@unist.ac.kr*

Abstract
In order to perform reactor core analysis with high accuracy, appropriate resonancetreatment method plays a very important role within the whole calculation
procedures. On the other hand, the fast reactor designs are the best choices to solve
future energy problem. Therefore, the in-house code STREAM, which is initiallydeveloped for light water reactor calculation, is decided to upgrade for fast reactor
analysis. Current work implements the Improved Tone’s Method into the
STREAM code to perform 2-D pin-cell/assembly calculation. Several verification
tests show the accuracy of new STREAM code, with less than 50 pcm differenceon keff value.
Key Words: STREAM; Improved Tone’s Method; Fast Reactor Analysis
1. Introduction

where

In the previous work[1], the STREAM code was coupledwith TULIP[2] code to perform fast reactor analysis. In
that work, the Tone’s method was used to calculate selfshielded cross sections. Due to the limitation of TULIP
code, only 1-D geometry can be dealt with during theresonance calculation. An equivalent 1-D cylindergeometry should be used to generate self-shielded crosssection for the 2-D assembly problem. This way works
only if the assembly has the natural ring-wise characters.
Therefore, it is necessary to have the 2-D resonancecalculation capability in the STREAM code.

isotope r and region i.
The escape cross section is calculated by using
background cross section:

e, g ,r ,i = Nr ,i t0, g ,r ,i − k r t , g ,k ,i0

the nuclide density.
To determine the background cross sections, the collision
probabilities and total cross sections can be used: t0,g ,r ,i =

j

k r

ji , gji , gt , g ,k , j

r, j

j

(4)

j

where Pij,g is the collision probability from region i toregion j of group g. V is the volume.
The Eq. (4) comes from the original Tone’s method[6]. In
the 1-D problem, the collision probabilities are easy toobtain. However, it is quite time consumption to getcollision probabilities in the 2-D or complex geometry.

To start with, the self-shielded cross sections of isotope r
in the region i can be expressed as:

According to the improved Tone’s method, thebackground cross sections can be alternatively obtainedwith two fixed-source equations: g ,r ,i =   r ,i ( E )r ,i ( E )dE /  r ,i ( E )dE (1)Eg  1, g ,r (r, ) + t , g (r ) 1, g ,r (r, ) =  k  r t , g ,k (r)Based on the narrow resonance approximation, theneutron flux is approximately calculated by the total crosssection and escape cross sections:

1
1
E t ,i ( E )+e, g ,r ,i P  V
P N Vj

2. Methodology

r ,i ( E ) (3)where  t , g ,r ,i is the background cross section and N is

Compared with subgroup method or hyperfine-groupmethod, the improved Tone’s method attracts lots ofattention in the recent decade due to its enough accuracy
and efficiency[3-5]. In this paper, the improved Tone’s
method is implemented into the STREAM code.
Considering the existed nuclear data library of TULIP
code, these two codes are merged to perform 2-D fast
reactor assembly calculation.

Eg

e, g ,r ,i is escape cross sections for group g,  2, g ,r (r, ) + t , g (r ) 2, g ,r (r, ) = N r (r)(5)

And the background cross sections are determined byusing the solution of Eq.(5):

(2)

66

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 20190t , g , r ,idV  d = dV   d 
Vi

4

Vi

4

cylinder geometry. Figures from Figure 3 to Figure 8
show the microscopic total cross sections and its
difference of each nuclide. For a pin-cell problem, it does
not have too much different between 1-D and 2-D
resonance calculation. Therefore, for the generated crosssections set, the biggest relative differences are around
2.5%, and RMS of relative differences of each nuclide are
0.02%(U-5), 0.13%(U-8), 0.09%(Pu-9), 0.04%(Cr-52),
0.13%(Fe-56), 0.03%(Na-23).

(r, )

= 1, g ,r ,i (6)2, g ,r ,i2, g , r (r , )

1, g , r

Figure 1 shows the flowchart of new STREAM code withimproved Tone’s method.

Figure 3 Microscopic total cross section of U-235 and its
difference

Figure 1 Flowchart of STREAM code using improved Tone’s
method

3. Verification Tests
3.1 Simplified Pin-cell Problem
For the initial test, a simplified rectangular fast reactor
pin-cell calculation has been performed. Figure 2 shows
the geometry information and Table 1 shows the
specification of the problem.

Figure 4 Microscopic total cross section of U-238 and its
difference

Figure 2 Simplified pin-cell problem
Table 1 Geometry and composition of simplified pin-cell
problem
RegionRadius/Pitch,
cm

Fuel

0.3236

Cladding

0.3857

Coolant

0.95186

Isotope
U235
U238
Pu239
Cr52
Fe56
Na23

Nuclide Density,
1024 at/cm3
3.08220E-05
1.82440E-02
2.81470E-03
8.68256E-03
6.39008E-02
2.22720E-02

Figure 5 Microscopic total cross section of Pu-239 and its
difference

The kinf values are summarized in the Table 2. For thereference, the in-house MCS[7] Monte Carlo calculation
was performed. In the calculation, P0, P1, and out-flowtransport correction are used to take the anisotropic
scattering into account. In the TULIP/STREAM

At first, the 1968-group self-shielded cross sections arecompared between TULIP and STREAM code. In TULIP
code, the cross sections are prepared by 1-D equivalent67

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019calculation, the 1968-group cross sections are prepared by
TULIP code and STREAM performs the calculation withmacroscopic cross sections.

STREAM directly calculation show good agreementcompared with MCS reference solution. The relative
differences are less than 30 pcm.
Based on the Eq.(6), the accuracy of cross sections
depends on the accuracy of solution of two fixed-source
problems. In order to see what happened when the loose
convergence criteria is used, the STREAM calculations
with different criteria (fsp-eps) were performed. The
results are shown in Table 3. It indicates that the loose
convergence criteria can be used without losing too muchaccuracy.
Table 3 Summary of kinf values of pin-cell problem withdifferent convergence criteria
Code
MCS

Figure 6 Microscopic total cross section of Cr-52 and its
difference

STREAM

fsp-eps
1e-4
1e-3
5e-3
1e-2

Rel.diff, pcm

kinf
1.47705 ± 0.0005
1.47768
1.477641.47753
1.47762

28
27
22
26

3.2 Fast Reactor Assembly Problem
In this section, two sets of fast reactor assembly problem
have been calculated. The specifications of first threecases are shown in Figure 9. The sodium is used as thecoolant with metallic fuel pellet. The fuel pins withdifferent enrichment are loaded in one assembly. The
composition of each material is from MET-1000 core
design, which is the 1000 MWth medium-size metallic
benchmark core consisting of U-Zr fuel, HT-9 structure
and, Sodium coolant [8].

Figure 7 Microscopic total cross section of Fe-56 and its
difference

Case
Case 1 1

Case
Case
3 3

Case
Case 22

Figure 8 Microscopic total cross section of Na-23 and its
difference
Table 2 Summary of kinf values of pin-cell problem
Code
MCS
TULIP
STREAMSTREAM

PN Order

kinf

P0
TR
P1
P0
TR
P1

1.47705 ± 0.0005
1.476481.47638
1.47646
1.47768
1.47758
1.47766

Rel.diff,pcm

Low enrichment metallic fuelHT9 claddingHigh enrichment metallic fuelSodium coolant

Figure 9 Specification of assembly problem, Case1~Case3

The second set of Case 4 and Case 5 is selected based on
the PASCAR core inner/outer assembly. The specificationof the problem can be found in the reference [1].

-26
-31
-27
28
24
27

As shown in the table, both TULIP/STREAM and68

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

reasonable accuracy. The loose convergence criteria canbe used while solving two fixed-source equations.
Compared with 1-D resonance calculation, the new
STREAM code with 2-D resonance calculation obtains
more accurate results. In the future, more verifications
need to be done, especially for the hexagonal assembly.
Currently, there is no optimization in the resonancecalculation. Therefore, the resonance calculation costs thesame CPU times compared with MOC transport
calculation. For example, Case 3 in Table 4, the resonancecalculation took 903 seconds while MOC transport
calculation took 806 seconds. In the future, several
optimizations will be implemented for resonancecalculation, such as OpenMP parallel calculation,
selection of resonance nuclide.

Figure 10 Specification of assembly problem, Case4 and Case5

Table 4 shows the kinf results. In the calculation, the outflow transport correction was applied for each case. For
each case, the STREAM results still have good agreementcompared with references. The largest relative differenceis 34 pcm with negative sign. The difference does notchange a lot when the problem has more fuel pins.

Acknowledgements

Table 4 Summary of kinf values of assembly problem
Case

MCS

STREAM12
3
4
5

1.38880 ± 0.00010
1.35624 ± 0.00010
1.37392 ± 0.00010
1.15252 ± 0.00008
1.32603 ± 0.00009

1.38916
1.35676
1.37448
1.15245
1.32543

This work was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea
government (MSIT). (No.NRF-2017M2A8A2018595)

Rel.diff,
pcm
19
28
30
-5
-34

References
1. X.N. Du, et al., Application of TULIP/STREAM codein 2-D Fast Reactor Core High-Fidelity Neutronic
Analysis, Nucl. Eng. Tech, Accepted for publication,
2019. https://doi.org/10.1016/j.net.2019.06.010.

In the previous study, the 1-D equivalent cylindergeometry was used for rectangular assembly self-shieldedcross sections’ generation. For the Case 2 and Case 3, it ishard to use that geometry for the resonance calculation.
To show the difference between 1-D and 2-D resonancetreatment, the Case 4 was calculated again with TULIP 1D resonance calculation module. The results are
summarized in the Table 5. When the cross sections wereprepared with 1-D equivalent geometry, the final kinf value
has -143 pcm difference compared with reference, whichis acceptable. Compared with 1-D resonance calculation,
the 2-D resonance calculation improves the final kinf value
around 138 pcm. That is to say, the assembly calculation
with 2-D resonance treatment is necessary from theviewpoint of eigenvalue.

2. X.N. Du, et al., A hybrid method to generate fewgroup cross sections for fast reactor analysis, J. Nucl.
Sci. Technol., 2018, 55(8):931-944.
3. C.H. Lee, et al., An Improved Resonance Selfshielding Method for Heterogeneous Fast Reactor
Assembly and Core Calculations, M&C 2013, Sun
Valley, Idaho, May 5-9, 2013.
4. H. Yu, T. ENDO, A. YAMAMOTO, ResonanceCalculation for Large and Complicated Geometry
Using Tone’s Method by Incorporating the Method of
Characteristics, Journal of Nuclear Science andTechnology, 2011, 48(3): 330-336.

Table 5 Summary of kinf values of assembly problem withdifferent resonance calculation module

5. L. Mao, et al., A new Tone's method in APOLLO3®
and its application to fast and thermal reactor
calculations, Nucl. Eng. Tech, 2017, 49(6): 1269-1286.

Code
MCS
TULIP/STREAM
STREAM

6. T. Tone, A numerical study of heterogeneity effects infast reactor critical assemblies, J. Nucl. Sci. Technol.,
1975, 12(8): 467-481.

kinf
1.15252 ± 0.00008
1.15062
1.15245

Rel.diff, pcm-143-5

7. J.R. Jang, et al., Validation of UNIST Monte Carlo
Code MCS for Criticality Safety Analysis of PWR
Spent Fuel Pool and Storage Case, Annals of Nuclear
Energy, 2018, 114:495-509.

4. Discussions and Conclusions
Previously, the STREAM code is coupled with TULIP
code for the fast reactor analysis. One limitation of thecoupled code system is the resonance calculation needs tobe done with 1-D equivalent geometry. In order toovercome this limitation, the improved Tone’s method isimplemented into the STREAM code in this study.

8. OECD/NEA. Benchmark for neutronic analysis of
sodium-cooled fast reactor cores with various fueltypes and core sizes. France: OECD/NEA; 2016. (no.
NEA/NSC/R(2015)9).

For the verifications, one pin-cell problem and several
assembly problems have been calculated. Based on the
numerical results, the improved Tone’s method is
successfully implemented into the STREAM code with69

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Application of the SuPer-Homogenization Method in Fast Reactor Analysis System SARAX
Linfang Wei, Youqi Zheng*, Bowen Xiao, Longfei Xu, Hongchun Wu
Xi’an Jiaotong University, No 28 Xianning West Road, Xi’an 710049, Shaanxi, China*
Corresponding author: yqzheng@mail.xjtu.edu.cn

Abstract
For better estimation of few-group cross-section considering heterogeneous effectduring subassembly homogenization in fast reactor analysis, especially in themulti-region system, the traditional spatial SuPer-Homogenization (SPH) method
was implemented in the few-group cross-section generation module, named
TULIP, in the SARAX system to preserve the reaction rate conservation during
spatial homogenization. Recently, the Energy-collapsing SuPer-Homogenization(ESPH) method was introduced in the new TULIP module to obtain themultigroup-fewgroup equivalence during group condensation. Using ESPH
method, the infinite multiplication factor and the region-wise reaction rate ofsubassembly calculation were strictly conserved before and after energy
collapsing. Besides, this paper also provides another way to generate few-groupcross-section by using a whole core spectrum when condensing, in which the
ESPH method was also used.
Key Words: fast reactor, few-group cross-section, heterogeneous effect, SuPerHomogenization, SARAX
1. Introduction

applied to the few-group cross-section generation module
TULIP [9] in the SARAX system to preserve themultigroup - fewgroup equivalence. With this method, theinfinite multiplication factor and the region-wise reaction
rate conservation were strictly preserved. In order toaccount for the neutron leakage between different
material regions, high order scattering matrices were also
corrected using the ESPH method.

Heterogeneous effect in fast reactor has become more andmore significant along with the newly designed
subassemblies nowadays [1-2]. For subassembly
homogenization in fast reactor analysis, especially in themulti-region system, it is always essential to preserve thereaction rate conservation during the homogenization
procedure. In early years, the SuPer-Homogenization(SPH) method [3] was applied to light water reactor(LWR) calculations, to preserve the reaction rates
obtained from the subgroup calculations whentransforming a multiband problem into a simple
multigroup problem in heterogeneous cases by Hébert [4].
Nearly identical to the SPH method, the multigroupequivalence method was applied to the fine structure
method in APOLLO3 code [5] without explicitly defining
the SPH factors. This method is used to preserve thereference reaction rates obtained from the heterogeneoushomogeneous equivalence, which has been involved in
the SARAX system [6].
In addition to this, researches show that during thegeneration of few-group cross-section, energy collapsing(or group condensing) procedure does not always
preserve the reaction rates obtained from the multigroupcalculations and few-group calculations in the multiregion system, which has been studied in the LWR
calculations [7-8]. The same problem also occurs in fast
reactor calculation. Therefore, the Energy-collapsing
SuPer-Homogenization (ESPH) method was recently

Different from the traditional two-step homogenization
method in fast reactor analysis, another few-group crosssection generation method is proposed in this paper
coupling with the newly-developed three-dimensional
discrete ordinate transport code HYDRA [10]. By using awhole core spectrum provided by the SN code whencondensing, with the following ESPH correction, theeffective multiplication factor and the reaction rates ofrespective materials in the reactor core are designed to bepreserved.
2. Methodology
2.1 Subassembly homogenization with SPH method

1
70

The heterogeneous effect in resonance self-shielding was
taken into account by region-wise escape cross-sections
in original TULIP module. The one-dimensional transport
solver was based on collision probability method (CPM)for slab and cylindrical geometries. For heterogeneous
calculation models, the homogenization and groupcondensation processes were separated into two steps.

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The first step was the spatial homogenization, with spatial
SPH correction to proceed, in which the heterogeneoushomogeneous equivalence was made. The second step
was the energy collapsing, with ESPH correction toproceed, in which the reaction rate conservation should bealso kept to ensure the multigroup-fewgroup equivalence.
The energy collapsing procedure was done by TULIP
without whole core spectrum calculation, or done by SN
transport calculation code HYDRA. By computing two or
three-dimensional spectrum of reactor core withapproximately real model, HYDRA was expected togenerate more accurate few-group constants considering
neutron leakage between different types of subassembly
precisely. Both flow paths were ended by a new methodof ESPH correction, as shown in Figure 1.
0D UFG Transport
Calculation

2.2.1. Energy collapsing using TULIP
Numerical results of fast reactor have shown that after
group condensing, the kinf results of few-group transport
calculation was to some extent differ from that ofmultigroup transport calculation before condensing. The
ever-present deviation indicated that energy collapsing
would probably introduce inaccuracy of homogeneous
parameters, particularly in the case that final
homogenized region number larger than one, as in theheterogeneous super-subassembly models. This was
because of the directly energy collapsing leading to nonconservation of reaction rates.
In order to enforce the reaction rate conservation while
doing energy collapsing, the technique of ESPH was used
in TULIP. Different from SPH, ESPH was applied within
a broad group containing tens of fine groups. Detailed
description can be found in the reference [12].

1D UFG Transport
Calculation

2.2.2. Energy collapsing using HYDRA

SpatialHomogenization

In order to consider the neutron leakage between different
material regions in the whole core analysis, the SN code
HYDRA was used to provide the high order flux momentfor group condensing. Recently, the group condensing
function of HYDRA has been supplemented to generate
few-group cross-sections, which are provided for core
analysis in SARAX. However, there also remains the nonconservation of reaction rate before and after energy
collapsing, thus the ESPH method was also applied. For
different material regions in core calculation, the energy
collapsing procedure (also the fine group structure being
1968 with the corresponding broad group structure being33) can be represented as:

SPH Correction

2D/3D Spectrum
Calculation?

Y

Energy Collapsing(Using TULIP)

2D/3D SN Core
Calculation

Energy Collapsing(Using HYDRA)

ESPH Correction

Fig. 1. The new computational flow chart of subassembly
homogenization.

 m,bg ,bg33m,bg =g = g1 , g1968(4)

 m , gm , g

The ESPH factor m, g of material m in group g was defined

In the cases that final homogenized region number is morethan one during spatial homogenization, e.g., ahomogenization case of three regions with the first two
regions merging into one, it is a typical multi-region casewhere the SPH method is used. According to the reaction
rate conservation, it can be represented as:(1)
1, g1, gV1 + 2, g2, gV2 =  gg (V1 + V2 )

as:

g2

m, g =

g = g1

m, g

(5)

m,bg

Thus the few-group cross-section of material m wascalculated as:Using the method proposed by Hébert [11], SPH factor
was introduced in TULIP which is represented as: V + V(2)
 g = 1, g 1 2, g 2
g (V1 + V2 )In this work, the SPH factors were calculated for all
homogenized regions.
Thus the corrected cross-section is represented as:
*g =  g  g =

g2



g2

*m,bg =  m ,bg  m , g = 

g = g1
g2

g2

m, g m, gg = g1



m, g



g = g1

m, g

(6)

m,bg

To ensure the convergence of iteration between m, g and*m,bg , the constraint to m, g was set as:(7)
0.8  m, g  1.2
If the calculated ESPH factor exceeded the limit, theboundary values were used. The convergence criterion of
ESPH factor was

1, g1, gV1 +  2, g2, gV2 1, gV1 + 2, gV2 1, g1, gV1 +  2, g2, gV2
=1, gV1 + 2, gV2
g (V1 + V2 )g (V1 + V2 )(3)
The conservation of reaction rate is always strictly
enforced by iteration of  g and *g .

G

 [(
g =1

2.2 Energy collapsing with ESPH methodm, g− m , g ) / m , g ]2G271

 1e−3(8)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The ESPH factors were updated by the iteration of Eq. (5),Eq. (6) and few-group PN-consistent transport equation.
For high order scattering matrices correction, the evenorder and odd-order matrices were treated separately. The
negative flux moments were replaced by their absolutevalues while condensing. The difference between ESPH
method involved in HYDRA and in TULIP was that, theintegrated flux normalization was managed according to
the whole core spectrum in the ESPH correction involvedin HYDRA. And because of the number of ESPH factors
multiplied by material number, it took more iteration
times for the ESPH convergence.

from -1749 pcm to 5 pcm by using the spatial SPH
correction after homogenization. And in the second
homogenization procedure of TULIP, the reference ofenergy collapsing was from the homogeneous multigrouptransport calculation. It is also presented that the error
introduced by energy collapsing is reduced from -1450
pcm to -19 pcm by using ESPH correction. The kinf resultof homogeneous few-group transport calculation agrees
well with that of multigroup transport calculation.
0.990

SPH
ESPH

0.988
0.986

3. Numerical Results
k-infinity

0.984

3.1 Results of SPH and ESPH method in TULIP
In this section, the SPH and the ESPH method arerespectively tested in separate procedures of TULIP using
the flow path without 2D/3D spectrum calculation aspresented in Figure 1. The verification test is aheterogeneous one-dimensional super-subassemblymodel constructed by inner control subassembly andouter fuel subassembly as designed in Figure 2.

Referen
ce

0.978

0.974
0.972
0.970
0

Spatial0.9882homogenizati 0.98822a 0.97143b
7conEnergy0.98827c 0.97431d—collapsinga Reference.
b Homogenized, w/o SPH, w/o condensation.
c Homogenized, w/ SPH, w/o condensation.
d Homogenized, w/ SPH, condensed, w/o ESPH.
e Homogenized, w/ SPH, condensed, w/ ESPH.

2

3

4

5

6

7

8

9

10

11

12

13

14

Fig. 3. kinf convergence with SPH/ESPH iterations.
Figure 3 shows the convergence of kinf with SPH/ESPH
iterations. Compared to SPH iterations, it takes fewer
iterations for ESPH to obtain a converged kinf result of
few-group transport calculation to the same convergencecriterion.
In fact, the deviation of kinf value is according to thedeviation of reaction rate estimation. Application of SPH
and ESPH correction respectively amends thehomogenized cross section after spatial homogenization
and after energy collapsing to more reasonable values.
Therefore, the reaction rate conservation is virtuallyrealized with these methods.
Table 2. Normalized total reaction rate using ESPH.

w/ Correction

SPH

1

Iterations

Table 1. kinf results of super-subassembly calculation
Procedure

0.980

0.976

Fig. 2. Schematics of control subassembly [13] andequivalent super-subassembly model.
w/oCorrection0.982

Absorber
regionFuelregionESPH—
0.98808e

Referen
ce

w/o
ESPH

Error/%w/
ESPH

Error’/%

0.118070.123144.29

0.11813

0.05

0.88193

0.87686-0.57

0.88187-0.01

Table 2 presents that energy collapsing without ESPH
correction will cause overestimation of the total reaction
rate in absorber region, which leads to kinf smaller than thevalue it should be. After ESPH correction, the normalized
reaction rates in both two regions are very closed to thereference.

In this case, the final homogenized region number was 2
including one control subassembly region and one fuelsubassembly region. The reference of spatialhomogenization was from the heterogeneous multigrouptransport calculation. It is presented in Table 1 that theerror introduced by spatial homogenization is eliminated

3.2 Results of ESPH method in HYDRA

3
72

In this section, the ESPH method is tested using
OECD/NEA SFR benchmarks [13], calculated by threedimensional SN transport calculation code HYDRA, with
S4P1 approximation. The homogenized multigroup cross-Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019section was provided by TULIP. During the ESPH
iterations, the averaged flux of each few group was kept,
and the ESPH factors were calculated for all the materials.

utility of ESPH method during energy collapsing using
the whole core spectrum. The numerical results indicatethat with the new ESPH correction method, the accuracyof few-group cross-section generation in SARAX system
has been significantly improved.

Table 3. keff of SFR benchmarks using HYDRA.
Bench
mark
MET1000
MOX3600
MOX1000

1968groupreference

33group
w/o
ESPH

Error
/
pcm

33-group
w/ ESPH

Error
’/pc
m

1.03026

1.02498-528

1.03001-25

1.01151

1.00904-247

1.01032-119

1.01791

1.01339-452

1.01774-17

References
1.

B. Faure, P. Archier, J-F. Vidal, et al., “Neutronic
calculation of an axially heterogeneous ASTRID fuelassembly with APOLLO3: Analysis of biases andforeseen improvements,” Ann. Nucl. Energy. 115, p.
88–104 (2018).
2. M.G. Jarrett, E.R. Shemon, C.H. Lee,
“Heterogeneous multigroup cross sections for fast
reactor calculations with MC2-3/Proteus,” Proc.
PHYSOR 2016, Sun Valley, ID, May 1-5 (2016).
3. A. Kavenoky. The SPH Homogenization Method,
Proc. Specialists’ Mtg. Homogenization Methods inReactor Physics, International Atomic Energy
Agency, Lugano, Switzerland, November 13-15,
1978, IAEA-TECDOC-231. International Atomic
Energy Agency, p. 181 (1978).
4. A. Hébert, “Advances in the development of a
subgroup method for the self-shielding of resonant
isotopes in arbitrary geometries,” Nucl. Sci. Eng.
126(3), p. 245–263 (1997).
5. D. Schneider, et al., “APOLLO3: CEA/DEN
deterministic multi-purpose code for reactor physics
analysis,” Proc. PHYSOR 2016, Sun Valley, ID, May
1-5 (2016).
6. Y. Zheng, X. Du, Z. Xu, et al., “SARAX: A new codefor fast reactor analysis part I: Methods,” Nucl. Eng.
Des. 340, p. 421–430 (2018).
7. N. Sugimura and A. Yamamoto, “Resonance
Treatment Based on Ultra-fine-group Spectrum
Calculation in the AEGIS Code,” J. Nucl. Sci.
Technol. 44(7), p. 958–966 (2007).
8. H. Park, H.G. Joo, “Practical resolution of angle
dependency of multigroup resonance cross sections
using parametrized spectral superhomogenization
factors,” Nucl. Eng. Technol. 49(6), p. 1287–1300(2017).
9. X. Du, L. Cao, Y. Zheng, et al., “A hybrid method togenerate few-group cross sections for fast reactor
analysis,” J. Nucl. Sci. Technol. 55(8), p. 931–944(2018).
10. L. Xu, L. Cao, Y. Zheng, et al., “Development of anew parallel SN code for neutron-photon transport
calculation in 3-D cylindrical geometry,” Progress in
Nuclear Energy. 94, p. 1–21 (2017).
11. A. Hébert, “Applied Reactor Physics”, Canada,
Presses Internationales Polytechnique, p. 225 (2009).
12. L. Wei, Y. Zheng, H. Wu, “Improvement of fewgroup cross-section generation in fast reactor
analysis system SARAX,” Ann. Nucl. Energy. 132, p.
149–160 (2019).
13. OECD/NEA, “Benchmark for neutronic analysis of
sodium-cooled fast reactor cores with various fueltypes and core sizes,” NEA/NSC/R (2015)9.

ESPH Factor

Table 3 shows that the keff deviations between multigroup(1968) and few-group (33) calculation are significantly
reduced with ESPH method. As the ESPH iteration times
becomes larger, the error of few-group keff becomes
smaller and finally leads to a converged value of corecalculation. The remaining deviation between 1968-groupcalculation and converged 33-group calculation is due tothe neutron current non-conservation during ESPH
iterations for vacuum boundary conditions.

1.040
1.035
1.030
1.025
1.020
1.015
1.010
1.005
1.000
0.995
0.990
0.985
0.980
0.975
0.970
0.965
0.960
0.955
0.950

group=11
group=22

0

10

20

30

40

50

60

70

80

90

ESPH Iterations

Fig. 4. ESPH factors of material 1.
Figure 4 shows the ESPH factor convergence of fuelmaterial 1 of MET-1000 benchmark in 33-group transport
calculation done by HYDRA. In different energy groups,
the ESPH factor variation tendency can be opposite, but
eventually converges to a fixed value.
4. Conclusions
This paper described the SuPer-Homogenization method
implemented in the fast reactor analysis system SARAX.
For subassembly calculation, the kinf result was correctedto the true value, and more accurate homogenized fewgroup cross-sections were generated with this method.
For the three-dimensional whole core calculation, thefew-group keff result was much closer to the multigroupone, and the accuracy of normalized total reaction rates offuel materials was improved, which were benefit from the473Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Progress on Multi-physics Calculations of Nuclear Reactor Cores with the IGA method
W.F.G. van Rooijena,
Research Institute of Nuclear Engineering, University of Fukui〒914-0055 Fukui-ken Tsuruga-shi Kanawa-chō 1-3-33, JAPAN*
Corresponding author: rooijen@u-fukui.ac.jpaAbstract
A method is presented to solve the SN neutron transport equation with IsoGeometric Analysis (IGA). This method allows to solve the neutron transport
equation on arbitrary geometrical domains. In the current work, a computer code
has been developed to solve the multi-group SN transport equations with IGA. The
theory is being expanded to include diffusion theory and thermal conduction in a
first approach to multiphysics calculations. The current computer code is actively
being developed. Accuracy is good but calculation time is a weak point.
Key Words: Iso-Geometric Analysis, SN Neutron Transport Theory, Numerical Simulation, Diffusion Theory
geometrical shapes as a (set of) mathematical equations.
One theory for such a representation is based on so-called
NURBS (Non-Uniform Rational B-Splines, [2]). In
NURBS theory, one starts with a set of basis functions
𝑁𝑖,𝑝 (𝑢) of degree p and = 1, … , 𝐼 , defined on aparameter space 0 ≤ 𝑢 ≤ 1 . For each basis function
𝑁𝑖,𝑝 (𝑢) there is a corresponding point 𝑷𝑖 in 3D space,called a “control point”, and a weight, 𝜔𝑖 . A curve isdefined as

1. Introduction
The steady-state neutron transport equation (Boltzmanequation) is defined over three independent variable, i.e.
space, direction of neutron movement, and neutron energy.
In computer simulations, each of these independent
variables must be discretized. Each discretizationnecessarily implies an approximation and becomes asource of uncertainty (error) in the calculation.
Furthermore, in the particular case of the space variable,
the conventional discretization methods are often limited
to rather simple shapes, such as cylinders, sphere, andplanes. In the current work, the so-called Iso-GeometricAnalysis (IGA) is applied for the discretization of space.
The IGA method allows to solve the neutron transport
equation in arbitrary geometry without meshing or otherdiscretization. With the IGA method, it is possible toanalyze geometrical shapes which are (slightly) deformed,
for example, due to thermal expansion. A calculation code
has been developed to solve the multi-group SN transport
equation as well as the neutron diffusion equation with the
IGA method. In this manuscript theoretical background isgiven as well as some results of initial calculations.
In Section 2 an overview is given of the theory behind the
IGA method, and in Section 3 the theory is given to apply
IGA to the SN neutron transport equation. In Section 4
detailed results are given, as well as an extension todiffusion theory and multiphysics calculations. Section 5gives conclusions.

𝐂(u) =∑𝐼𝑖=0 𝜔𝑖 𝑁𝑖,𝑝 (𝑢)𝑷𝑖∑𝐼𝑗=0 𝜔𝑗 𝑁𝑗,𝑝 (𝑢)(1)Thus, as the parameter u goes from 0 to 1, the curve 𝐂(u)“interpolates” the control points resulting in a 3D curve.
By the proper selection of the control points and weights,
any desired curve can be defined. The theory can also be
extended to 2D surfaces and 3D volumes.
The theory of NURBS also provides the following: A recursive definition of the basis functions which iseasy to program and numerically stable.
 It is possible to determine the tangent vectors in eachparameter direction and thus is it possible to determine
the normal vector to a surface at any point.
 The NURBS basis functions have some features
which make them attractive for application in the
Finite Element Method (FEM), details in Ref. [1].
The Finite Element Method (FEM) is a generalmathematical theory to solve Partial Differential
Equations (PDE), such as the neutron transport equation.
The PDE is defined on some geometrical domain 𝛤 withcorresponding boundary conditions. The basic approachof IGA is to use NURBS theory and the NURBS basis
functions to define the geometrical domain 𝛤, and thenalso use those same NURBS basis functions in the FEM

2. NURBS and Iso-Geometric Analysis
The introduction of IGA here is very limited. For moredetails, see reference [1]. For numerical analysis ofarbitrary geometrical shapes, such as in Computer-Aided
Design (CAD) systems, it is needed to represent arbitrary74

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019solution of the PDE. Thus, the PDE can be solved on any
arbitrary geometrical domain.

expanded into a set of basis functions, i.e. 𝜓(𝒓) =∑𝐼𝑖=1 𝛼𝑖 𝐵𝑖 (𝒓) , and to select a special weight function
η(𝒓) = Bj (𝒓):3. IGA, SN transport, and diffusion theory𝐼̂𝑘 ⋅ ∫
𝛀In the following, 2D geometry is assumed but the theory
applies to 3D geometry as well. The simplest form of thetransport equation is the following:̂ ⋅ 𝛁𝜓 𝑔 (𝒓, 𝛀̂ ) + Σ𝑡g 𝜓 𝑔 = 𝑄𝑔 (𝒓, 𝛀
̂)
𝛀

𝜕Γ+

𝑖=1

Γ

(2)

𝑖=1

Γ

̂ 𝑘 ⋅ ∫ 𝐵𝑗 (𝒓)𝜓𝑖𝑛 𝒏= ∫ 𝐵𝑗 (𝒓)𝑄𝑘 𝑑Γ − 𝛀̂ 𝑑𝑙 (4)𝜕Γ−ΓIn the IGA method the NURBS basis functions which
define the geometrical domain 𝛤 are used also to expand
the neutron flux. The basis functions are defined on theparameter space (𝑢, 𝑣) , thus the easiest option is toperform all the integrations of Eq. (4) in the parameterdomain. This requires a coordinate transform, but that isprecisely defined by the NURBS definition of the patch
𝛤; the NURBS theory also defines the normal vectors on
the edges of the domain so that it becomes possible todetermine the incoming and outgoing boundaries.
Eq. (4) is one equation, but there are 𝐼 expansion
coefficients for the flux, thus one needs 𝐼 equations. Thisis achieved using the weight function: by selecting 𝐵𝑗 ,𝑗 = 1, … , 𝐼 one obtains 𝐼 equations. The entire problem
can now be written in the form of a matrix multiplication,and the solution can be obtained by inversion of thematrix, for which many methods are available.
In the case of diffusion theory, the equation to solve is:

Γ

g

(5)

−𝛁 ⋅ 𝐷𝛁𝜙 𝑔 (𝒓) + Σ𝑡 𝜙 𝑔 = 𝑄𝑔 (𝒓)̂ 𝑘 ⋅ ∫ 𝜂𝜓𝑖𝑛 𝒏̂ 𝑑𝑙 (3)= ∫ 𝜂𝑄𝑘 𝑑Γ − 𝛀Γ𝐼⋅ ∫ 𝛁Bj (𝒓) ∑ 𝛼𝑖 𝐵𝑖 (𝒓) 𝑑Γ + ∫ 𝐵𝑗 (𝒓)Σ𝑡 ∑ 𝛼𝑖 𝐵𝑖 (𝒓) 𝑑Γ̂ k ⋅ ∫ 𝛁𝜂𝜓𝑑Γ + ∫ 𝜂Σ𝑡 𝜓𝑑Γ̂ 𝑘 ⋅ ∫ 𝜂𝜓𝒏̂ 𝑑𝑙 − 𝛀
𝛀
Γ

𝑖=1
𝐼

Where 𝒓 indicates space, 𝑔 indicates the energy group,̂ indicates neutron direction, Σ𝑡g is the total cross𝛀
section and 𝑄𝑔 is the neutron source; the equation isdefined on a geometrical domain 𝛤, called a “patch”. In
general, a complete geometry is built-up of several
patches in a so-called “multipatch geometry”. In the SN̂ ismethod, the variable of neutron direction 𝛀̂ 𝑘 and Eq. (2) is solved for eachdiscretized into a set 𝛀̂ 𝑘 in the set. To obtain the form of the equation for use
𝛀with the FEM, multiply the equation with an (arbitrary,
but non-zero) weight function η, integrate over the patch
𝛤, and use the Gauss divergence theorem; furthermore,since neutrons travel in a straight line, the boundary of thedomain can be divided into two parts, 𝜕Γ − where
neutrons enter the domain (incoming boundary), and𝜕Γ + where neutrons leave the domain (outgoingboundary). See also Fig. 1:𝜕Γ+̂k𝐵𝑗 (𝒓) ∑ 𝛼𝑖 𝐵𝑖 (𝒓) 𝒏̂ 𝑑𝑙 − 𝛀𝜕Γ−Similar to the transport equation, multiply by a weightfunction and integrate over the patch. In the case ofdiffusion theory, one cannot make a distinction between
incoming and outgoing boundaries, instead, the boundary
is divided into 3 parts: (1) boundary with Dirichletconditions 𝜕Γ𝐷 (flux is known on the boundary); (2)boundary with Neumann conditions 𝜕Γ𝑁 (derivative of
the flux is known on the boundary) and (3), boundary with
Robin conditions 𝜕Γ𝑅 (a superposition of Dirichlet andNeumann conditions). On 𝜕Γ𝐷 , the flux is already known,
thus that part of the boundary must not be included in thecalculation. Assume that the only other boundary
conditions are either perfect reflection (zero-derivative
boundary condition) or perfect vacuum (i.e. 𝛁𝜙 ⋅ 𝒏̂=−ϕ/dex , where dex is the extrapolation distance; thisboundary condition is a special case of the Robinboundary condition). As before, expand the flux into basis
functions 𝜙(𝒓) = ∑𝐼𝑖=1 𝛼𝑖 𝐵𝑖 (𝒓) and choose the weightfunction as η(𝒓) = Bj (𝒓) to find:(a)(b)
Fig. 1. (a) Geometrical domain 𝛤 for transport theory:̂ , incoming boundary ∂𝛤 − ,neutron direction 𝛀
outgoing boundary ∂𝛤 + and normal vector 𝒏̂ areindicated. (b) Geometrical domain for diffusion
calculations, with Dirichlet, Neumann, and Robinboundaries indicated as ∂𝛤𝐷 , ∂𝛤𝑁 and ∂𝛤𝑅𝐼𝐼

𝐷 ∑ 𝛼𝑖 ∫ ∇𝐵𝑗 ⋅ ∇𝐵𝑖 𝑑Γ + 𝛽𝐷 ∑ 𝛼𝑖 ∫ 𝐵𝑗 𝐵𝑖 𝑑𝑙

On the incoming boundary ∂Γ − the flux is known(boundary condition), and on the outgoing boundary∂Γ + it is assumed that the flux is the same as the patch
flux. The following step is to assume that the flux can be

𝑖=1

Γ

𝑖=1

𝐼

∂ΓR

+ Σ𝑡 ∑ 𝛼𝑖 ∫ 𝐵𝑗 𝐵𝑖 𝑑Γ = ∫ 𝐵𝑗 𝑄𝑑Γ (6)
𝑖=1

75

Γ

Γ

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

defined with each of the four pins; the IGA mesh is shownin Fig. 2, and the results are shown in Table I.

where 𝛽 is a term due to the Robin-boundary conditions.
As in the case of transport theory, choose the basis
functions Bj with j = 1, … , I to make 𝐼 equations for
𝐼 unknowns. Again, all integrals are evaluated in the
NURBS parameter domain, and the matrix inversion canbe performed with any of the standard numerical
techniques.
From a theoretical perspective, diffusion theory is inferior
to transport theory. There are two reasons to include
diffusion theory:
1. Diffusion theory can be used as a preconditioner fortransport theory, i.e. diffusion theory can be used tofind an approximate solution of the scalar flux which
can be used as a starting guess for the transport theory
solution
2. The neutron diffusion equation is the same as theequation for heat conduction, thus diffusion theory
provides a way to include the calculation oftemperature.

(a)(b)
Fig. 2. Example of the 4 pin geometry used for 2D
calculations with the IGA code. (a): the IGA multipatch
geometry, (b) the allocation of materials.
Table I: Results for 2D PWR pin cells of the C5G7
benchmark, and our own 4-pin model from Fig. 2.
NEWT
IGA
UO2
1.323135
1.323086
MOX 4.3%
1.132954
1.130616
MOX 7.0%
1.1569781.153789
MOX 8.7%
1.170037
1.166476
Four pin1.18736
1.18544

3.1 Some implementation details
There a few very important differences between thesolutions for diffusion theory and transport theory. Firstof all, in diffusion theory the solution is calculated for all
patches in the geometry at once, while for transport theory
the solution proceeds on a patch-by-patch basis. In
diffusion theory, the boundary condition on the boundary
between patches is that the flux is continuous. This is
implemented in practice by “merging” the basis functions
across the boundary. Effectively, from the point of viewof the calculation, the individual patches are “glued
together” to form one big patch. This has an effect on thepatch matrices in the calculation. The patch matrices intransport theory are quite dense, i.e. most of the elementsof the matrix are not zero, whereas the matrices indiffusion theory can be very sparse. In the present work,for transport theory all matrices are stored as dense
matrices and inverted with standard routines from BLAS[3] and LAPACK [4], whereas the matrices for diffusion
theory are stored as sparse matrices and the inversion is
performed with a special method for sparse matrices(SPARSKIT [5]).

Initial calculations showed that the calculation time of the
IGA code was excessively long. To increase performance,
an optimization of the patch order was implemented. Aŝ 𝑘 determines theillustrated in Fig. 3, the direction 𝛀“natural” order of the solution for the patches. The
problem is thus to determine how neutrons “flow” from̂ 𝑘 . This is really a
one patch to the next as a function of 𝛀
problem of graph theory, and the GRAFPACK software[10] was used to find optimal solutions. The improvementof the calculation time is shown in Table II: calculation
time was decreased by 60% - 75%.

4. Results
Application of the IGA method requires that one has adescription of the geometrical domain in the form of
NURBS. Currently the GeoPDEs package [6] is used. The
GeoPDEs package is an IGA package for Matlab [7] andGNU Octave [8]. GeoPDEs has options to define 2D and3D NURBS geometries.

̂1
Fig. 3. A multipatch geometry and 2 directions 𝛀̂̂and 𝛀2 . For 𝛀1 , the solution should start in patch 1
̂ 2 the order should be exactly
and finish in 6, for 𝛀
opposite.

4.1 Two-dimensional neutron transport
Calculations were first performed on PWR pin cell
models, for which the C5G7 benchmark [9] was used.
Calculations were performed for the four fuel types
included in the benchmark, i.e. a UOX pin and three MOX
pins (4.3%, 7.0%, 8.7%). Also a test geometry was

4.2 Three-dimensional neutron transport
Preliminary calculations were performed for 3D
geometries as well. In particular, the Jezebel and Godiva

76Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

5. Conclusions and future outlook
Table II. PWR pin cell, UOX fuel, calculation time (inseconds) before and after optimization of the sweep order.
GRAFPACK S2
S8
S16
No
52.15
456.5
3170
Yes
14.51
178.9
1163

The theory for the solution of the neutron transport
equation with multigroup SN theory and the IGA method
was successfully derived and a computer code has beendeveloped. The IGA method makes it possible to solve theneutron transport equation on arbitrary geometry, butthere are a few aspects which require optimization, suchas the sweeping order over the patches and even thesweeping order inside a patch. The current code includes
anisotropic scatter and shows good accuracy but
calculation time is clearly a problem. A diffusion versionof the code was also developed and showed good accuracy.

benchmarks were analyzed (anisotropic scatter isimportant in these benchmarks), as well as the Takeda-1
benchmark and a 3D system inspired by the Takeda-2
benchmark [11]. Results are given in Table III.
Table III: Results for 3D transport calculations with the
IGA code, compared to 1D SN transport (XSDRNPM),
and Monte Carlo (KENO-VI)
Jezebel-1
Jezebel-2
Godiva-1
XSDRNPM 1.08383
1.07335
1.11013
S4P0
IGA
1.072421.06244
1.10373S4P0
XSDRNPM 0.99380
0.982221.02022
S4P5
IGA
0.980930.96987
1.01232
S4P5
Takeda benchmark, model 1: KUCA. S8
Benchmark IGA
KENO-VI
CR up
0.97660.978800.97750
CR down
0.96220.96361
0.96258
CR worth
0.01540.015180.01492
Takeda benchmark model 2 inspired. S8IGA, 2nd IGA,
4th KENO-VI
degreedegreeCR up
0.99147
0.99161
0.99042
CR down
0.95577
0.956470.95544
CR worth
0.03570.035140.03498

Future developments are focusing on the following
aspects:
 The multi-group theory requires self-shielded groupcross sections. Research is currently ongoing todevelop the necessary theory and numerical tools(submitted for presentation at the RPHA19 meeting) Multiphysics calculation taking into account heat
conduction.
 The IGA method has many parameters, such as thedegree of the basis functions, the support of the basis
functions, etc. The optimal combination of theseparameters is to be determined.
 At present, at least two reasons for the long calculation
times have been found and remedies are being
investigated.
References
1. J. Austin Cottrell, Thomas J.R. Hughes, and Yuri
Bazilevs, Isogeometric Analysis – Toward integrationof CAD and FEA, Wiley (2009)
2. Les Piegl and Wayne Tiller, The NURBS Book,
Monographs in visual communication, Springer (1995)
3. BLAS, Basic Linear Algebra Subprograms, v3.8.0,
http://www.netlib.org/blas/
4. LAPACK, Linear Algebra PACKage, v3.8.0,
http://performance.netlib.org/lapack/
5. Y. Saad, “SPARKSIT, a basic tool kit for sparse matrix
computations,
Version2”,
https://wwwusers.cs.umn.edu/~saad/software/SPARSKIT/
6. R. Vázquez, “A new design for the implementation of
isogeometric analysis in Octave and Matlab:
GeoPDEs 3.0”, IMATI REPORT Series No. 16-02,
CNR IMATI (2016)
7. https://www.mathworks.com/products/matlab.html
8. https://www.gnu.org/software/octave/
9. Benchmark on Deterministic Transport Calculations
Without Spatial Homogenisation, A 2-D/3-D MOX
FuelAssembly
Benchmark,OECD,
NAE/NSC/DOC(2003)16
10. GRAFPACK
Graph
Computations,
https://people.sc.fsu.edu/~jburkardt/f_src/grafpack/gr
afpack.html
11. Toshikazu Takeda and Hideaki Ikeda, “3-D neutrontransport benchmarks”, NEACRP-L-330, OECDNEA (1991)

4.3 Diffusion theory.
The diffusion calculation is presently under development.
At present, only one result can be presented, see Table IV.
The UOX pin cell from the C5G7 benchmark wasanalyzed with diffusion theory and SN transport theory. As
can be seen, in comparison with S4 calculations, thereduction of calculation time is about 2 orders ofmagnitude.
Table IV: Comparison of SN calculations and diffusion
theory for a UOX PWR pin cell.
Time [s]k∞Diffusion
1.32599
4.40
S41.32314466
S81.324071603
Code development is currently ongoing for diffusion
theory, 3D calculations, and the calculation of heat
conduction. Our final contribution will contain more
results in these areas.

77Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Research on a Method for Self-Shielding Calculations Based on IGA-Method*Matthias Nezondeta , W.F.G Van Rooijenba University of Fukui, Graduate School of Engineering,

3-9-1 Bunkyo, Fukui-shi, Fukui 910-8507, Japanb University of Fukui, Research Institute of Nuclear Engineering,

1-3-33 Kanawa-cho, Tsuruga-shi, Fukui 914-0055, Japan*E-mail : nmatth@u-fukui.ac.jp

Abstract
An innovative calculational method for neutron transport is described. This code uses a particularmethod to treat the spatial domain. This code allows to perform nuclear reactor physics evaluations
for any arbitrary geometrical domains compared to conventional calculational methods that use commonly only ideal shapes (cylinders, spheres, plates,...) to describe the spatial domain. This new method
is based on the Iso-Geometric Analysis. To treat the energy variable we use the Multi-group method,
so it is essential to make self-shielding calculations to take into account the resonance phenomenon ofcross sections to determine correctly and accurately the behavior of the core and the safety parameters.
The self-shielding method that we use is based on a subgroup method developed by P.Ribon, known asmethod of the moments. This method replace the punctual micro cross sections by a density of probability that can be described by a quadrature set named probability table. It is also relevant to present
this probability tables and the method to create them. From now, we have calculated our probability
table and we will show their validity by making a comparison of the cross section calculated with theprobability table and with an other method.
Key Words: IGA-Method, Self-shielding, Cross sections, Sub-group method, Resonances, Probability table.

1. Introduction

2. Iso-Geometric Analysis

In conventional calculational methods for neutron transport, the spatial domain is often described based on ”ideal”
shapes. However, a nuclear reactor is not an ideal shape;for example, thermal expansion and/or mechanical stressdeform the shape of the reactor. The effects of such spatial deformations can be large, but in conventional calculation methods such effects are often ignored or taken into
account with correction factors. In recent years, in our lab
we are developing a numerical method based on so-called
Iso-Geometric Analysis (IGA) to simulate the transport ofneutrons through arbitrary geometrical domains, which allows to perform simulations using the actual shape of thespatial domain [1]. In numerical simulation of nuclear reactor physics one problem occurs, the solution of the transport
equation is linked to the treatment of the energy of the neutrons. The interaction between neutrons and atomic nuclei
has so-called resonances, where the reaction rate can varyover several orders of magnitude in a narrow range of energy. The presence of resonances gives rise to the so-called
self-shielding effect.

In numerical simulations of nuclear reactor physics, thetreatment of the spatial domain and the energy of the particles are directly related to each other. In recent years, amethod to simulate the transport of neutrons through arbitrary spatial domains has been developed in our laboratory.
To take into account every kind of shapes, our numericalcode is based on the IGA which was firstly used in CAD
design tools.
Previously in CAD software, the geometric shapes weredelimited by curved surfaces described by NURBS (NonUniform Rational Basis Splines), function defined piecewise by polynomials. NURBS is a mathematical model
used for generating and representing curves and surfaces.
A NURBS curve is defined by its order, a set of weighted
control points, and a knot vector. The order defines the number of nearby control points that influence any given pointon the curve and set the order of the polynomials that represent the curves, the control points determine the shape ofthe curve by adjusting the weight of each point and the knot
vector is a sequence of parameter values that determines78

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

where and how the control points affect the curve. Thismethod allows to represent every kind of curves and makes
complex geometries.
On the other hands for finite element method (FEM), thegeometric volumes are meshed, divided into polyhedrons(hexahedrons, cubes, tetrahedrons, prisms, . . . ). The way todescribe the geometric shapes and the geometric volumes
are different, so it is necessary to go through a conversion
step: the meshing. The meshing of the geometrical domain
is an approximation which has an impact on the accuracyof the calculation. To circumvent the issues related to themeshing, the IGA method has been improved [2, 3].

discrete ordinates Sn theory that we use in our code basedon IGA to resolve the transport equation.
4. Sub-group method and probability tables
4.1 Generalization of the sub-group method
The term of sub-group method was introduced in the 70s
by Nikolaev [4]. All the different sub-group methods use
the probability density of the microscopic cross section of
the resonant isotopes. Indeed, the microscopic cross section
can be defined by a function Π(σ ) so that Π(σ )dσ is theprobability for the microscopic cross section of the resonant
isotope to have a value between σ and σ + dσ . This function Π(σ ), which is called probability density, is a positive
function and this integral is equal to 1. A representation of
the probability density function is shown on Fig.1.
With the definition of the probability density function it
is possible to replace every Riemann integral with a σ dependent integrand by a Lebesgue integral :3. Self-shielding calculations
A basic topic in nuclear reactor analysis is the problem ofself-shielding, which is due to the very fine energy structureof the nuclear cross sections. An adequate treatment of selfshielding is essential for the correct prediction of the safetyparameters of a nuclear reactor (Doppler coefficient, void
coefficient, etc). Self-shielding calculations commonly take
into account only a few standard geometric shapes (plates,cylinders, spheres). Thus for IGA, it becomes necessary to
calculate self-shielding in arbitrary geometry.
To resolve the transport equation of neutrons, it is often
necessary to discretize the energy by using a certain number of groups (a dozen or hundreds groups). In that case, weneed to take into account the resonant behavior of the crosssections because some group include many resonances thathas an impact on the determination of the group cross sections. The goal of the self-shielding model is to calculate thegmulti-group self-shielded cross section σρ,ifor any reaction
ρ, in any group g and geometric zone i :

Zug

min(σ
Z )

f [σ (u)]du =ug−1

Π(σ ) f (σ )dσ(2)max(σ )

If we look at Fig.1, the function Π(σ ) can be approximated by a series of δ -functions centered at discrete values
σk and characterized by discrete weights wk (the weights are
positive and their sum is equal to 1) that represent the probability of the micro cross section to be equal to σk . Eachvalue represents a discrete level that can be assimilated to agroup inside the group g, hence the term of sub-group.
K

Rug
g
σρ,i
=

1
∆ug

σρ,i (u)Φi (u)du

ug−1

Rug

Φi (u)duhσρ,i Φi ig=
hΦi igΠ(σ ) = ∑ δ (σ − σk )wk

(3)

k=1

(1)
We substitute the Eq.(2) by the Eq.(3) and we obtain thefollowing discretization :

ug−1

where
u = ln(E0 /E) = lethargy
σρ,i (u) = microscopic cross section
Φi (u) = neutron flux

1
∆ug

Zug

K

f [σ (u)]du = ∑ wk f (σk )
ug−1

(4)

k=1

If we consider a heterogeneous system described in the Snformalism, where the direction of movement is discretized
into a set of directions Ω̂n and where a resonant isotope N ∗is in a non-resonant background N + , the transport equation
is given by :There are different methods for self-shielding calculations. The main ones are : method of equivalence, Stammlers method, Sub-group methods.
A lot of methods require a background cross section
which is a function of the geometry. However most of thesemethods use simple shapes, but we need to use a methodthat does not rely on the assumption of simple geometrical
domains. For our simulation code, we have decided to use
the sub-group method because this method can be used forevery geometric domain. Moreover, this method is compatible with any flux solution technique, more precisely with theΩ̂n ∇Φg (r, Ω̂n ) + [N ∗ σt (u) + N + σt+ ]Φg (r, u, Ω̂n )= Qg (r, Ω̂n )(5)where Qg (r, Ω̂n ) is the source of neutrons that can be due
to scattering neutrons, fission neutrons or neutrons from external source.
79

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig.1. Representation of the probability density of the function (x + 1)sin(x)In the case we have a calculation code that is based on the
Sn theory, it is possible to resolve the Eq.(5) and obtain theflux that will be a function of:Φg (r, Ω̂n ) = f [Qg (r, Ω̂n ), σt (u)]Using negative and positive moments allow to conserve a
correct reaction rate at low and high values of the resonantcross sections. In the original Ribons method, they were no
obligation for the choice of the different values of ”l”. The
moments method is based on the Stieljes series and the Padeapproximant. Considering the Stieljes function F(z) of thetotal cross-section moments, F(z) is defined as :(6)

Since the flux is a quantity which depends on σt (u),we
can apply the sub-group method that we presented above
and obtain the self-shielded cross section :
Rug

max(σ
Z )

F(z) =

Kσρ,i (u)Φi (u)du∑ σρ,k fk [Qg (r, Ω̂n ), σt,k ]
ug−1
k=1
g
σρ,i ==KRug
Φi (u)du∑ fk [Qg (r, Ω̂n ), σt,k ]
k=1
ug−1

min(σ )

K(zσ )1−K
dσ = ∑ zl Ml + O(zK+1 ) (10)
1 − zσl=1−K

Then we use the Pade approximant to describe F(z) :
K−1

(7)

∑ ai zi
i=0
zl Ml−K+1 = K−1
l=0∑ b j z j + zK
j=0

2K−1

zK−1 F(z) = ∑4.2 Moments method and mathematical probability tables
There are several subgroup methods. Their differences areabout how to create the probability tables, which is the term
to describe a quadrature set that represents a probability
density. This term was introduced by Levitt [5]. In ourcase we will use a subgroup method that use mathematicalprobability tables. This method is a moments method which
is based on Ribons method and more precisely on the Ribons extended self-shielding model proposed by Hebert andCoste. The moment approach allows to calculate the basepoints σk and the weights wk simultaneously [6, 7, 8].
The first step of the method is to calculate the positive andnegative moments Ml which are computed from the Riemann integrals of the microscopic cross-sections :
1
Ml =
∆ug

Zug

σ (u)l duThen by identification, we obtain the following linear
system :





MK
MK−1 · · ·M1
b0
M0MK−1 MK−2 · · · M−1 M0   b1 


=
−
 ..



 .. 
..
..
..
..
 .




.
.
.
.
. 
M1

M0

···

M2−K

bK−1

M1−K

(12)
Resolving the Eq.(12) gives the coefficients bk . Thanks
to the property of the Pade approximant of a Stieljes series,
the base point of the probability table σk are the roots of thefollowing polynomial :K(8)∏ (z − σk ) = 1 + bK−1 z + · · · + b1 zK−1 + b0 zK

ug−1

(13)

k=1

The second step is to calculate the corresponding weights
wk . They are chosen to preserve the moments of the total
cross section :A K-order probability table is defined to preserve the 2K
selected moments of Eq. (5) :K∑ wk (σk )l = Ml , 1 − K ≤ l ≤ K

(11)

K

Ml = ∑ wk σkl

(9)

k=1

k=1

80

(14)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

In our approach we have chosen to use a different set
for the values of ”l” compare to the set used in the Ribons extended method. The work of G. Chiba and H. Unesaki proved that preserving some values of the moments for−1 ≤ l ≤ 0 improve the accuracy of the probability tables
description [9]. Eq.(15) gives the following linear system :

 
1
1
···
1
w1
σ −(K−1)− 1 σ −(K−1)− 1 · · · σ −(K−1)− 1   w2 
 1
 
K
2

  .. 
..
..
..
..

 . 
.
.
.
.
−1
−1
−1
wK
σ1
σ2
···
σK


1
M−(K−1)− 1 


(15)
=

..


.
M−1

of dilution. As we can see, the results are good because theerrors are less than 0.06703%.
Tab. 1. Errors between the two different cross section
Dilution (b) Avg err (%) Max err (%)
0
4.523e-2
6.703e-21001.352e-32.319e-2
1.0e+4
1.219e-5
9.084e-5
1.0e+101.046e-56.6566e-5
5. Conclusion
We have started to improve this neutron transport code
based on IGA method to take into account self-shielding
effect. At the moment, we have found the self-shielding
model, calculated and validated our probability table.
The next step is to code the sub-group method for ”Sn”
and perform calculations for a simple case. Then implement
the method in our neutron calculation code based on IGA.

Resolving this linear system gives us the weights wk . The
set {σk , wk ; k = 1, K} is the probability table of the microscopic cross sections for the group g.

References
4.3 Verification of the Probability Table[1] W.F.G Rooijen and R. Horita. Three-dimensional ”Sn”
neutron transport calculations with iso-geometric analysis (IGA). Marrakech, Morocco, 17-19 September
2018. Proc. Int. Conf. PHYTRA4.

Before using our probability table to perform calculation,
we need to verify them by making a comparison between
multi-group cross section calculated using the probability
table and multi-group cross section using the Bondarenko
self-shielding method.
For our example we calculated the multi-group fission
cross section of Pu239 with the two methods for different
values of dilution. On the Fig.2 we can see these cross sections.

[2] Piegl L. and Tiller W. The NURBS Book. Springer,
1997.
[3] Hughues T.J.R. Austin Cottrell J. and Bazilevs Y. Isogeometric Analysis : toward integration of CAD and FEA.
Wiley, 2012.
[4] Nikolaev M.N. Comments on the Probability Table
Method. Nucl. Sci. Eng., 61(286), 1976.
[5] Levitt L.B. The probability table metod for treating
unresolved neutron resonances in monte carlo calculations. Nucl. Sci. Eng., 49(450), 1972.
[6] Cullen D.E. Application of the Probability Table
Method to Multigroup Calculations of Neutron Transport. Nucl. Sci. Eng., 55(387), 1974.
[7] Ribon P. and Maillard J-M. Les Tables de Probabilite. Application au Traitement des Sections Efficaces
pour la Neutronique. Technical report, Commisseriat
l’Energie Atomique, 1986.

Fig.2. Multi-group fission cross section for Pu239[8] Hebert A. The Ribon Extended Self-Shielding Model.
Nucl. Sci. Eng., 151(1-24), 2005.

To see the real differences between the different type ofcross section we also calculated the relative change. On thefollowing table we can see the average and maximum error
between the two different cross section for different values[9] Chiba. G and Unesaki H. Improvement of MomentBased Probability Table for Resonance Self-Shielding
Calculation. Annals of Nuclear Energy, 33(1141-1146),
2006.
81

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Study on Effective Cross Section of Pu Spot in MOX Fuel
Tatsuya Kawanoa,, Satoshi Takedaa, Takanori Kitadaa, Yasunori Ohokab, Shogo Matsuokab and Hiroaki Naganobab

Osaka University, 1 Yamadaoka 1-chome Suita city Osaka Japan

Nuclear Fuel Industories, Ltd., 950 Asashiro-Nishi 1-chome Kumatori-cho Sennan-gun Osaka Japan*
Corresponding author: t-kawano@ne.see.eng.osaka-u.ac.jp

Abstract
It has been confirmed that Pu spot observes in MOX fuel pellet. In the study basedon the deterministic method, it is necessary to calculate the appropriate effective
cross section of Pu spot to evaluate the effect of Pu spot. Since the self-shielding
effect of resonance varies depending on the particle size and the position in a fuelpellet, the effective cross section should be appropriately taken into account.
Therefore, in this study, the influence of changing the particle position and particle
size on the effective cross section of Pu spot was evaluated. The numerical results
show that, in the case of calculation in the 107 group, the influence of changing
the particle position and particle size on the effective cross section of Pu spot ismainly due to the effect of the resonance between 2 and 3 eV of Pu-242.
Furthermore, it was also found the influence of changing the particle size is larger
than that of the particle position.
Key Words: Pu spot, self-shielding, resonance, effective cross section, reaction
rate
were arranged at the center and the edge in the fuel pellet.
In addition, very large particle (400 μm) and small particle(50 μm) were used for the investigation of the effect of theparticle size on the effective cross section. The reaction
rate was calculated by GMVP [4] which is a Generalpurpose Monte Carlo code for neutron and photontransport calculation based on the multi-group method,
and the influence of the reaction rate due to the change ofeffective cross section was confirmed. Furthermore,
based on the result, it was confirmed which nuclide of Pu
is mainly attributable to the effect. Figure 1 shows
calculation geometry. Table I shows temperatures in eacharea. Table II shows the Pu enrichment of the matrix andPu spot, the U-235 enrichment. Table III shows Pu isotope
ratio in MOX fuel. Table IV shows calculation condition
on GMVP. Figure 2 shows the calculation procedure. As
shown in Fig. 2, effective cross sections in each area (Puspot, matrix, cladding, moderator) were calculated by
SRAC2006, then reaction rate was calculated by GMVP.

1. Introduction
It has been confirmed that Pu spot observes in MOX fuelproduction process. Recently, the importance of the
double-heterogeneous model considering Pu spot for
MOX fuel analysis has been emphasized [1], and it isnecessary to calculate the effective cross section of Pu
spot to evaluate the effect of Pu spot in the double-heterogeneous model. These effective cross sections need
to be calculated appropriately because self-shielding
effect of resonance varies depending on the particle
size and particle position in a fuel pellet. Therefore, inthis study, the particle position and particle size werearranged to investigate the largest impact of changing
them on the effective cross section, and the influenceof changing the particle position and particle size on
the effective cross section of Pu spot was evaluated.
2. Calculation Condition
In this study, in order to evaluate the effective crosssection of particles in MOX fuel, the calculation of 107
groups for the pin cell model of a pressurized water
reactor (PWR) is performed by integrated nuclear
calculation code SRAC2006 [2]. The nuclear data libraryused in this study is JENDL-4.0 (Japanese evaluatednuclear data library) [3]. In order to evaluate the effect ofthe position on the effective cross section, the particles

82Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

3. Result and Discussion
3.1 Influence of changing the particle position on theeffective cross section

Based on the calculation conditions given in Figs. 1, 2,
and Tables I-III, the effective fission cross section and theeffective capture cross section were calculated for theparticle at the center and the edge in the fuel pellet.
Figures 3 and 4 show the effective macroscopic fission
and capture cross section. The relative differences shownin Figs. 3-5 were calculated by the comparison of theeffective cross section at the edge in the fuel.

Fig. 1. Calculation geometry based on PWR pin-cell
model
Table I. Temperatures in Each Area
Fuel
Cladding
Moderator
temperature (K)

900

600

600

Table II. PuO2 Content and U-235 Enrichment
MatrixPu spot
PuO2 content
1 wt %
100 wt %(PuO2 / (PuO2+UO2))
U-235 enrichment
0.2 wt %(U-235 / U)
Table III. Pu Isotope Ratio (wt %)Pu-2385
Pu-23945
Pu-24010
Pu-24125
Pu-242
15

Fig. 3. Effect of position on effective macroscopic
fission cross section

Fig. 4. Effect of position on effective macroscopic
capture cross section
As shown in Figs. 3 and 4, the relative difference wasmainly observed in the resonance energy region, and thelargest relative difference was observed at resonancebetween 2 and 3 eV in the effective macroscopic capture
cross section. Furthermore, based on the result, it was
confirmed which nuclide of Pu is mainly attributable tothe effect. The numerical results show that Pu242 ismainly attributable to the largest relative difference.
Figure 5 shows effective macroscopic capture crosssection of Pu-242. As shown in Fig. 5, it was found thatthe relative difference between 2 and 3 eV in effective
cross section is due to Pu-242.

Fig. 2. Calculation procedure
Table IV. Calculation Condition of GMVP
Number of histories per batch
100000Total number of batch
2050
Number of skip batch
50

83

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

As shown in Figs. 6 and 7, the relative difference wasmainly observed in the resonance energy region, and thelargest relative difference was observed at resonancebetween 2 and 3 eV in the effective macroscopic capture
cross section. Furthermore, based on the result, it was
confirmed which nuclide of Pu is mainly attributable tothe effect. The numerical results show that Pu242 ismainly attributable to the largest relative difference.
Figure 8 shows the effective macroscopic capture crosssection of Pu-242.

Fig. 5. Effect on effective macroscopic capture crosssection of Pu-242 by change of the position
3.2 Influence of changing the particle size on theeffective cross section

Based on the calculation conditions given Figs. 1, 2, andTables I-III, effective fission cross section and effective
capture cross section were calculated for large particle(400 μm) and small particle (50 μm). Figures 6 and 7
show the effective macroscopic fission and capture crosssection, respectively. The relative differences shown in
Figs. 6-8 were calculated by the comparison of theeffective cross section of small particle (50 μm).

Fig. 8. Effect of particle size on effective macroscopic
capture cross section of Pu-242
As shown in Fig. 8, it was found that the relative
difference between 2 and 3 eV in effective cross sectionis due to Pu-242. Furthermore, it was also found that thechange in particle size has a larger effect on the effective
cross section than the change in position.

3.3 Influence of changing the particle size on thereaction rate
As mentioned in section 3.2, it was also found that thechange in particle size has a larger effect on the effective
cross section than the change in position. Therefore, thedependency of reaction rate on change particle size wasdescribed in this section. Based on the calculation
conditions given Figs. 1, 2, and Tables I-IV , t h emacroscopic fission reaction rate and the macroscopic
capture reaction rate were calculated by GMVP. Figures
9 and 10 show macroscopic fission reaction rate andmacroscopic capture reaction rate, respectively. Figure 11
shows macroscopic capture reaction rate of Pu-242. The
relative differences shown in Figs. 9-11 were calculatedby the comparison of the effective cross section of small
particle (50 μm). The reaction rates shown in Figs. 9-11
were calculated as reaction rates per production rate.

Fig. 6. Effect of particle size on effective macroscopic
fission cross section

Fig. 7. Effect of particle size on effective macroscopic
capture cross section84Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table V. Energy Group Structure between 2 and 3 eV
Energy group
Upper energy
Lower energy
number(eV)(eV)
61
3.05902.382062
2.38201.8550
4. Conclusion
In this study, in the case of calculation in the 107 group,
the influence of changing the particle position andparticle size on the effective cross section of the Pu spot
is mainly due to the effect of the resonance between 2 and
3 eV of Pu-242. It was also found that the change inparticle size has a larger effect on the effective crosssection than the change in position. Furthermore, it isconsidered that the effect of the resonance between 2 and
3 eV of Pu-242 on the effective cross section can be
decreased by increasing the division of energy range
between 2 and 3 eV.

Fig. 9. Effect of particle size on macroscopic fission
reaction rate

References1.

2.
Fig. 10. Effect of particle size on macroscopic capture
reaction rate
3.

4.

Fig. 11. Effect of particle size on macroscopic capture
reaction rate of Pu-242

3.4 Discussion about energy group structure
As shown in Table V, in the 107-group structure, energy
range between 2 and 3 eV is divided into about 2 groups.
It is considered that the effect of the resonance of Pu-242
between 2 and 3 eV on the effective cross section can be
decreased by increasing the division of energy range
between 2 and 3 eV.

85

Reactor Integral Test Working Group, JENDL
Committee, “Compilation of the Data Book on Light
Water Reactor Benchmark to Develop the Next
Version of JENDL– Utilization of Criticality Data in
ICSBEP and IRPhEP Open Databases,” Japan
Atomic Energy Agency (2017).
Okumura K, Kugo T, Kaneko K and Tsuchihashi K,“SRAC2006: A Comprehensive Neutronics
Calculation Code System,” Japan Atomic Energy
Agency (2007).
Shibata K, Iwamoto O, Nakagawa T, Iwamoto N,
Ichihara A, Kunieda S, Chiba S, Furutaka K, Otuka
N, Ohsawa T, Murata T, Matsunobu H, Zukeran A,
Kamada S, and Katakura J, "JENDL-4.0: A New
Library for Nuclear Science and Engineering," J.
Nucl. Sci. Technol., 48, 1, 1 (2011).
Nagaya Y, Okumura K, Sakurai T and Mori T,“MVP/GMVP Version 3: General Purpose Monte
Carlo Codes for Neutron and Photon Transport
Calculations Based on Continuous Energy andMultigroup Methods,” Japan Atomic Energy Agency(2017).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Modification of MOC for Considering Double Heterogeneity due to Pu Spots in MOX Fuel
Akifumi Ogawaa*, Hikaru Yamaguchia, Satoshi Takedaa, Takanori Kitadaa,
Yasunori Ohokab, Shogo Matsuokab and Hiroaki NaganobaOsaka University, 1 Yamadaoka 1-chome Suita city Osaka Japanb
Nuclear Fuel Industries, Ltd., 950 Asashiro-Nishi 1-chome Kumatori-cho Sennan-gun Osaka Japan*
Corresponding author: a-ogawa@ne.see.eng.osaka-u.ac.jp

Abstract
The heterogeneity of Mixed Oxide (MOX) fuel due to Plutonium spots has
impacts on criticality. In this study, a double heterogeneous model was introduced
into OpenMOC by the model derived by R. Sanchez. The calculation result ofmodified OpenMOC was compared with that of GMVP. In the verification, thecalculations were performed in pin-cell model. From the calculation results, thereactivity effect obtained by modified OpenMOC agrees with the results of GMVP
within 0.1%Δk/kk′. Since the discrepancy comes from the difference between thelattice model used for GMVP and double heterogeneity model used for modified
OpenMOC, it is confirmed that OpenMOC is properly modified for considering
double heterogeneity due to Pu Spots.
Key Words: Pu-spot, MOC, GMVP, Double-Heterogeneity
1. Introduction
Plutonium (Pu) is produced through the operation of
nuclear power plants. It is used as Mixed Oxide (MOX)
fuel, but Pu spots exist in the Mixed Oxide (MOX) fuelpellets. It was found that the heterogeneity due to Pu spots
has impacts on criticality in some systems [1]. The
heterogeneous effects have been analyzed by Monte Carlocalculation. It can evaluate nuclear characteristics indetail, but Monte Carlo calculation results include thestatistical error and the calculation cost is high comparedto the deterministic method. Method Of Characteristics(MOC) is one of the deterministic method and the results
don’t include the statistical error. The double
heterogeneous model for MOC has been derived by R.
Sanchez [2-4], and Han Gyu Joo has derived analytic
solution [5]. This model has been developed for the fuelcompact of high-temperature gas-cooled reactor forconsidering. In this study, OpenMOC [6] is modified byapplying the model in order to consider the heterogeneity
due to Pu spots. As the verification, the results of modified
OpenMOC is compared with that of GMVP.

Fig. 1. Calculation flow of the original code

2. Modification of OpenMOC
The calculation flow in the original OpenMOC code andthe modified OpenMOC code are shown in Figs.1 and 2,
respectively. Table I shows the newly added parameters
for modification.

Fig. 2. Calculation flow of modified code86Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table I. Newly Added Parameters for Modification
𝜙0 Neutron flux in matrix
Neutron flux in Pu spots
𝜙𝑖
𝜙′Average neutron flux
Cross section of matrix
Σ0
Cross section of Pu spots
Σ𝑖
Homogenized cross sectionΣEscape probability for neutrons generated in Pu𝑔𝐸𝑖
spots to escape from the Pu spots without any
collision
Transmission probability of neutron entering𝑔𝑇𝑖through surface of Pu spots
Reducedescape probability which corresponds𝑔𝐸̂𝑖
to reduced cross section (Σ𝑖 – Σ)
𝑔
𝑇̂
𝑖

𝑆0
𝑆𝑖
𝜑𝑎𝑠
𝐿𝑖
𝑝0
𝑝𝑖
Σ̃

4𝜋

𝑆𝑖 =

𝑘𝑒𝑓𝑓
𝜒

Source on matrix
Source on Pu spots
Asymptotic flux
Pu spots diameter
Volume fraction of matrix
Volume fraction of Pu spots
Adjusted total cross section defined as
1 𝑝𝑖
𝑔
(Σ0 +
(1 − 𝑇𝑖 ))
̅

+

0

𝐿𝑖

2

𝑦
𝑔
−(Σ𝑖 −Σ′)𝑦𝐸̂(1 − ( ) ) 𝑑𝑦,
𝑖 =∫𝑒
𝐿𝑖
0

𝐿𝑖

𝑔
𝑇̂
𝑖 = ∫𝑒02−(Σ −Σ′)𝑦
𝑖

𝐿𝑖 2

𝑦 𝑑𝑦.

𝑔

(8)

𝐸𝑖 𝑆𝑖 ).

0)(1 − 𝑒 −Σ x )𝐸̂
∆𝜑𝑖 = (𝜑𝑎𝑠 − 𝜑𝑖𝑛
𝑖 .

(11)

0
𝜑𝑖𝑛

𝑔

is in-coming angular flux.

(1)

+

4π ′
𝑔
𝑔
(𝛴 𝜑𝑎𝑠 𝐸𝑖 + (1 − 𝐸𝑖 )𝑆𝑖
𝛴′

𝑃
1
∑
∑ 𝜔𝑚(𝑘) 𝜔𝑝 𝜔𝑘 sin 𝜃𝑘 ∆𝜑𝑖 ),
𝐴𝑛
𝑘∈𝐴𝑛
𝑝=1

𝜙′ =

(2)

𝑦 𝑑𝑦,

𝑝𝑖
𝑝0

(7)

(10)

𝜙𝑖 =

𝐿𝑖

𝐿𝑖 2

1

𝛴

νΣ𝑖𝐹 𝜙𝑖 ),v) Calculations of neutron fluxes 𝜙0 , 𝜙𝑖 , 𝑎𝑛𝑑 𝜙 ′The neutron fluxes 𝜙 ′ , 𝜙0 , and 𝜙𝑖 are calculated by
the following formulas by using 𝑆0 , 𝑆𝑖 , 𝜑𝑎𝑠 , 𝛥𝜑 ′ , 𝛥𝜑0 ,and 𝛥𝜑𝑖 :4π
𝜙0 = ′ (𝛴 ′ 𝜑𝑎𝑠
𝛴
(12)
𝑃
1
+
∑
∑ 𝜔𝑚(𝑘) 𝜔𝑝 𝜔𝑘 sin 𝜃𝑘 ∆𝜑0 ),
𝐴𝑛
𝑘∈𝐴𝑛
𝑝=1

i) Calculations of newly added parameters𝑔𝑔𝑔
̂𝑔
The parameters 𝐸𝑖 , 𝑇𝑖 , 𝐸̂𝑖 , and 𝑇𝑖 are calculated by
the following formulas:

2

𝜒𝑖
𝑘𝑒𝑓𝑓

0
)(1 − 𝑒 −Σ x ),∆𝜑0 = (𝜑𝑎𝑠 − 𝜑𝑖𝑛where,The modifications of OpenMOC from the original tomodified code are shown here.

𝐿𝑖(Σ𝑆,𝑖 𝜙𝑖 +′Adjusted total cross section defined as𝑔(𝑝0 Σ0 + 𝑝𝑖 Σ𝑖 𝐸̂
𝑖 )Effective multiplication factor
Fission neutron energy spectrum𝑔𝑇𝑖 = ∫ 𝑒 −Σ𝑖 𝑦

1
4𝜋

′

Adjusted total cross section defined as
1 𝑝𝑖
𝑔
(Σ0 +
(1 − 𝑇̂
𝑖 ))
̅

0

(5)

iv) Calculations of angular neutron flux differences
𝛥𝜑 ′ , 𝛥𝜑0 and 𝛥𝜑𝑖𝛥𝜑 ′ , 𝛥𝜑0 , and 𝛥𝜑𝑖 are obtained by the following
formulas using φas :
Σ̂
′
0
(9)
)(1 − 𝑒 −Σ x ) ′ ,∆𝜑 ′ = (𝜑𝑎𝑠 − 𝜑𝑖𝑛𝛴Transmission probability which corresponds toreduced cross section (Σ𝑖 – Σ)1𝑦 2𝑔𝐸𝑖 = ∫ 𝑒 −Σ𝑖 𝑦 × (1 − ( ) )𝑑𝑦,
𝐿𝑖
𝑙𝑖̅

.

𝑘𝑒𝑓𝑓

𝜑𝑎𝑠 = ̃ (𝑆0 +

𝑝0 𝑙𝑖

Σ̂

𝛴 ((𝑝0 𝑣𝛴0𝐹 𝜙0 + 𝑝𝑖 𝑣𝛴𝑖𝐹 𝜙𝑖 ) × 𝑉) × 𝑘𝑒𝑓𝑓iii) Calculations of sources S0 and Si
S0 , Si , and φas are calculated by the following formulas:1𝜒𝑆0 = (Σ𝑆,0 𝜙0 + 0 νΣ0𝐹 𝜙0 ),
(6)

𝑝0 𝑙𝑖

Σ′

𝜙

𝜙=

4π ′
(𝛴 𝜑𝑎𝑠𝛴′𝑃1∑∑ 𝜔𝑚(𝑘) 𝜔𝑝 𝜔𝑘 sin 𝜃𝑘 ∆𝜑 ′ ).
𝐴𝑛
𝑘∈𝐴𝑛
𝑝=1

(13)

(14)

3. Code verification
3.1 Calculation condition(3)

This section shows the calculation condition of modified
OpenMOC and the reference calculation obtained by
GMVP [7]. The effective cross section was obtained by
the collision probability method (PIJ) of SRAC2006 [8].
107 energy groups structure is used. The fast neutronenergy region is divided into 61 groups, and the thermal
neutron energy region is divided into 46 groups. The fueltemperature is 900K and moderator temperature is 600K.
JENDL-4.0 is used [9]. The verification was performedby changing the diameter of Pu spots. In the verification,(4)ii) Normalizations of neutron flux 𝜙 ′ , 𝜙0 , and 𝜙𝑖
𝜙 ′ , 𝜙0 and 𝜙𝑖 are normalized based on the following
formula:87

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

the Pu spots volume fraction is about 1% and the Pu
enrichment of Pu spots is 100wt%. U-235 enrichment of
the matrix is 0.2wt%. These calculations were performed
under the conditions shown in Tables II and III. The total
number of histories in GMVP calculation was 20 million
excluding skipped histories.
Table II. Calculation Conditions on Modified OpenMOC

Fig. 4. Overview of lattice model for 100μm Pu spots

Table III. Calculation Conditions on GMVP

Fig. 5. Overview of lattice model for 400μm Pu spots
Table IV. Pu Spots Conditions
Modified OpenMOC and GMVP calculations were
performed in the pin-cell model. The geometry of pin-cellmodel is shown in Fig.3. In the GMVP calculation, Pu
spots were treated by lattice model which assumes thespherical Pu spots is regularly arranged in the matrix.
Note that the volume fraction of the Pu spots is not
rigorously 1% in lattice model since the number of Pu
spots in lattice model is integer value. Figures 4 and 5
show the overview of the lattice model for 100μm Pu
spots and 400μm Pu spots, respectively. Table IV shows
the diameter of the Pu spots, the number of Pu spots, andvolume fraction of the Pu spots. Table V shows isotopic
ratio of Pu.

Table V. Isotopic Ratio of Pu (wt%)

The heterogeneity reactivity effect is defined as thereactivity which is obtained by changing from thehomogeneous model to heterogeneous model. The effectis defined as:ℎ𝑒𝑡𝑒𝑟𝑜𝑔𝑒𝑛𝑒𝑖𝑡𝑦 𝑟𝑒𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑒𝑓𝑓𝑒𝑐𝑡=1𝑘(ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 𝑚𝑜𝑑𝑒𝑙)−

1𝑘′(ℎ𝑒𝑡𝑒𝑟𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 𝑚𝑜𝑑𝑒𝑙)

,

(15)

3.2 calculation results
Table VI and VII show the calculation result of modified
OpenMOC and GMVP, respectively. Figure 6 shows thecomparison of the heterogeneity reactivity effect between
modified OpenMOC and GMVP.

Fig. 3. Geometry of Pin-cell model88

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4. Conclusion

Table VI. Result of Modified OpenMOC
homogenous model
heterogeneous model

Pu spotsdiameter-infinite-multiplication heterogeneity
factorreactivity effect1.185485-100μm

1.184218-0.090%200μm

1.182571-0.208%400μm

1.178735-0.483%In this study, OpenMOC was modified in order to treat the
double heterogeneity due to the Pu spots in MOX fuel.
The modified OpenMOC was verified by the comparison
with the calculation results obtained by GMVP. The
results show that the heterogeneity reactivity effects
obtained by modified OpenMOC and GMVP agree within
0.1%Δk/kk′ each other. The discrepancy is assumed tocome mainly from the difference between the latticemodel and random model, it is confirmed that OpenMOC
is properly modified for considering double heterogeneity
due to Pu spots.

Table VII. Result of GMVP

homogenous model
heterogeneous model

Pu spotsdiameter-infinite-multiplication
factor1.18689

statisticalerror (1σ)
0.009%

heterogeneity
error(1σ)
reactivity effect
-

100μm

1.18633

0.009%-0.040%0.011%

200μm

1.18509

0.008%-0.128%0.011%

400μm

1.18109

0.008%-0.414%0.010%

References
1.

Reactor Integral Test Working Group, JENDL
Committee “Compilation of the Data Book on Light
Water Reactor Benchmark,” Japan Atomic Energy
Agency (2017).
2. Sanchez R, Pomraning GC, “A Statistical Analysisof the Double Heterogeneity Problem,” Ann. Nucl.
Energy., 18, 7, 371 (1991).
3. Sanchez R, Masiello E, “Treatment of the double
heterogeneity with the method of characteristics,”
PHYSOR 2002, October 7-10, (2002).
4. Sanchez R, “Renormalized treatment of the double
heterogeneity with the method of characteristics,”
PHYSOR 2004, April 25-29, (2004).
5. Joo HG, Park TK, “Derivation of Analytic Solution
and MOC Calculation Procedure for Double
Heterogeneity
Treatment,”
Seoul
National
University. SNURPL-SR-001 (2007).
6. William B, et.al “The Open MOC Method of
Characteristics neutral particle transport code,” Ann.
Nucl. Energy., 68, 43 (2014).
7. Nagaya Y, Okumura K, Sakurai T, Mori T,“MVP/GMVP Version 3: General Purpose Monte
Carlo Codes for Neutron and Photon Transport
Calculations Based on Continuous Energy andMultigroup Methods,” JAEA-Data/Code 2016-018
Japan Atomic Energy Agency (2017).
8. Okumura K, Kugo T, Kaneko K, Tsuchihashi K,“SRAC2006: A Comprehensive Neutronics
Calculation Code System,” Japan Atomic Energy
Research Institute (2007).
9. Shibata K, Iwamoto O, Nakagawa T, Iwamoto N,
Ichihara A, Kunieda S, Chiba S, Furutaka K, Otuka
N, Ohsawa T, Murata T, Matsunobu H, Zukeran A,
Kamada S, and Katakura J: “JENDL-4.0: A New
Library for Nuclear Science and Engineering,” J.
Nucl. Sci. Technol., 48 1, 1 (2011).
10. Yamamoto T, Sakai T, and Iwahashi D, “Effect of
Pu-rich agglomerates in MOX fuel on reactivity
analysis of light water reactor MOX core physics
experiments,” J. Nucl. Sci. Technol., 55, 4, 438(2018).

Fig. 6. Heterogeneity reactivity effect against Pu spots
diameter
In both results of modified OpenMOC and GMVP, theheterogeneity reactivity effect is negatively increased
with the increase of the Pu spots diameter. The magnitudeof the heterogeneity reactivity effect obtained by modified
OpenMOC is larger than the results of GMVP within
0.1 %Δk/kk′ . The research by T. Yamamoto [10] also
shows the magnitude of the heterogeneity effect obtainedby the lattice model is smaller than that obtained by the
random model. Although the statistical error of theheterogeneity effect is not negligible, the difference in theheterogeneity effect between the lattice model andrandom model is estimated about or less than the
0.05 %Δk/kk′ . Therefore, it is considered that thedifference in the heterogeneity effect between the resultsof GMVP and that of modified OpenMOC mainly comesfrom the difference between the lattice model and random
model. From the calculation results, it is confirmed that
OpenMOC is properly modified for considering double
heterogeneity due to Pu spots in MOX Fuel. As the future
research, more verification changing the Pu spots volume
fractions is planned.

89Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Reactivity-Equivalent Physical Transformation for Homogenization Analysis of DoubleHeterogeneous Fuel and Burnable Poisona

Lou Leia, Yao Donga, Chai Xiaominga, Yu Yingruia and Peng Xingjiea,∗
National Key Laboratory of Science and Technology on Reactor System Design Technology, Nuclear Power Instituteof China, No.328, Section 1, Changshun Avenue, Shuangliu, Chengdu, Sichuan, China*
Corresponding author: loulei.necp@qq.com

Abstract
The traditional volumetric homogenization method on fuel and burnable poison
with double heterogeneity will bring a certain calculation deviation. In the paper,from the DH(Double Heterogeneous) inequality criterion, the relationshipbetween the calculation deviation of volumetric homogenization method andvarious factors on the dispersed particle-type fuel and burnable poison are
analyzed. It is pointed out that for high enrichment or large size fuel particles andall types of burnable poisons, the traditional volumetric homogenization method
will bring a certain calculation deviation. And then RPT(Reactivity-Equivalent
Physical Transformation) method was used to treat dispersed particle-type fueland burnable poison, the results show that the RPT method can be used to treat
the dispersed particle-type fuel and burnable poison with relatively not too large
absorption cross section which have double heterogeneity, and the calculation
accuracy is similar to that of the grain model.
Key Words: dispersed particle-type fuel burnable poison
geneity RPT method

double-hetero

with radius of 100μm are dispersed randomly inzirconium matrix, the distribution is shown in Fig. 1.

1. Introduction
Dispersed particle-type fuel[1] has been widely usedin FCM(Fully Ceramic Micro-encapsulated Fuel) fueland HTGRs(High Temperature Gas-cooled Reactors )fuel, and dispersed particle-type burnable poison has
been more and more concerned because of its lastin
g and adjusting reactivity control ability. But the sp
ace self-shielding effect of the dispersed particle-typ
e fuel and burnable poison will bring double hetero
geneity which is the heterogeneity between dispersed
fuel and matrix and the heterogeneity of fuel pin, cl
ad and moderate in the cell level. The space self-sh
ielding effect will make neutron flux in the center of fuel lower than that in the outer layer of the parti
cle and that in the matrix, so the traditional pressuri
zed water reactor calculation program with the tradit
ional volumetric homogenization method will bring acertain calculation deviation. It’s the challenge of double-heterogeneity for the calculation of dispersed pa
rticle-type fuel and burnable poison.

Side view

Top view

Fig. 1. dispersed positions of 1000 fuel particles wit
h the radius of 100μm
In order to compare the accuracy of the VHM(Volumetric Homogenization Model) and the GM(Grain
Model), the following calculation case was construct
ed. With the parameters of Table 1, results of the
VHM and the GM are compared in Fig. 2.
Table 1 Parameters of calculation caseParameter
Lattice pitch, cm
Pin radius, cm
Clad thickness, cm
Gas thickness between clad and pin, cm
Pin height, cm

2. Homogenization Model
In order to better describe the random distribution ofdispersed particles in the matrix, the Monte Carlo
program RMC[2] was used to model the dispersed
particles. For example, when 1000 UO2 fuel particles

90

Value
1.0
0.6
0.1
0.00080.6

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Dispersed particle radius, μm
Dispersed particle number, number
UO2 fuel enrichment, wt%

100
1000
20.0

(a) different diameter of fuel particles

Fig. 2. Comparison of kinf as function of burnup
between the VHM and the GM
It can be observed from Fig. 2 that at the beginning of
burnup kinf of the VHM is larger than that of the GM, and as the burnup increases, relative deviation of the kinf be
tween the VHM and the GM is basically maintained at around 4‰~5‰. This relative deviation may increase withcertain factors of the particle, and it may not meet the design requirements when the accuracy requirements arehigh. So it’s necessary to analyze the deviation of the double heterogeneous dispersed fuel and study the appropri
ate method for neutronic calculation.

(b) different fuel enrichment
Fig. 3. Comparison of kinf as function of burnup bet
ween the VHM and the GM with different diameterof fuel particles and different fuel enrichment

3. Deviation Analyze of Volumetric Homogenization
Model

It can be observed from F i g . 3 that the reactivity
deviation between the VHM and the GM is lower than5‰ when the fuel enrichment is 20‰ and the diameterof fuel particles is less than 100μm. And the reactivity
deviation is growing larger as the diameter of fuelparticles or the fuel enrichment increases.
According the above results, the diameter of fuel particles and the fuel enrichment are the main influencing factors of the reactivity deviation. When the fuel enrichment or the diameter of fuel particles is relatively large, the trad
itional volumetric homogenization method will bring rel
atively large reactivity deviation, so it’s necessary to study the computational model that can be applied to dispers
ed fuel and burnable poison.

The traditional volumetric homogenization method
underestimates the space self-shielding of the particles,
and it’s the cause of the deviation. There is a DH
inequality
criterion| Σ matrix − Σ grain | ⋅d grain > ε , which can be used

to determine if the space self-shielding of the particles
need to be considered. In the above DH inequality
criterion,

Σ matrix and Σ grain are the macro crosssection of the matrix and grain, d grain is the diameterof the particle, and ε is the limit of optical length
generally set to be 0.1 which means the flux deviation
limit between the particles and the matrix is 10%.
According the above DH inequality criterion, the ma
in influencing factors of the space self-shielding effe
ct are the diameter of fuel particles and the fuel enrichment. So in the following the reactivity deviatio
n between the VHM and GM will be analyzed.

4. RPT Model
4.1 RPT method introduction
For columnar and spherical geometry, the process of RP
T[3]( Reactivity-equivalent Physical Transformation)
method can be found in Fig. 4. The first step is tocompress all fuel particles in a smaller fuel zone, and thesecond step is to homogenize the fuel zone with volume
weight. After the above two steps the traditional PWR p

91

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019rogram can be used to calculate the homogenized system. In RPT method, the diameter of the compressed fuel zo
ne can be determined by ensuring the kinf of system equal
to the reference value which is obtained by a high fideli
ty deterministic program or a Monte Carlo program.

(c) 235U nuclear density of fuel enrichment 20%
Fig. 4. The RPT Process
4.2 Numerical results
In order to get the radius of the compressed fuel zone in
PRT method, an external script program was made tosearch the radius in which the kinf of the system will beequal to that of RMC reference result, and then theburnup calculation was made to analyze the kinf ofsystem and other parameters change with burnup.
(d) 235U nuclear density of fuel enrichment 90%
Fig. 5. kinf and 235U nuclide density as function of b
urnup in cells with different enrichments and differe
nt models
It can be observed from Fig. 5 that for dispersed fuelparticles deviation of kinf and 235U nuclide density after
RPT method keeps very small in the burnup life, whichmeans RPT method can be used to calculate theneutronic parameters of dispersed fuel particles withdouble heterogeneity.
Because of the space self-shielding effect particle typeburnable poison also need RPT method to treat the
double heterogeneity. The reactivity deviation ofdifferent models as function of burnup shows in Fig. 6.

(a) kinf of fuel enrichment 20%(b) kinf of fuel enrichment 90%(a) AG92

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019(f) Gd2O3(b) HF

Fig. 6. kinf as function of burnup in cells with differ
ent burnable poison and different models
5. Conclusions
In this article, the calculation deviation of volumetrichomogenization method and various factors on thedispersed particle-type fuel and burnable poison isanalyzed. The research results shows that the enrichmentof fuel particles, the size of fuel and burnable poison
particles and the type of burnable poison are the main
factors of double heterogeneity calculation. When thesefactors are relatively large, it will bring ineligible
deviation. And then RPT method in the calculation ofdispersed particle-type fuel and burnable poison showsthat the RPT method can be used in the neutronic
calculation of dispersed particle-type fuel and burnablepoisons and the calculation deviation can be controlled
within an acceptable range. For dispersed particles ofburnable poison with large absorb cross section it’snecessary to find a new method to solve the double
heterogeneity.

(c) Dy2O3

Acknowledgements
This work is supported by the National Natural Scie
nce Foundation of China (Approved number Nos.117
0051016).
(d) Eu2O3

References
[1]

[2]

[3]

(e) B4C

93

IAEA-TECDOC-1797, Accident Tolerant Fuel
Concepts for Light Water Reactor, Proceedings of aTechnical Meeting held at Oak Ridge National
Labratories, United States of America, 1316,October 2014.
Shichang Liu, Ding She, Jin-gang Liang and Kan
Wang ， Random Geometry Capability in RMC
Code and Applications to Stochastic Media
Analysis，RPHA15,Jeju, Korea, Sept.16-18, 2015.
Kim, Y., Kim, K., & Noh, J. Reactivity-Equivalent
Physical Transformation for Homogenization of
Double-Heterogeneous Fuels, Transactions of
Korean Nuclear Society Autumn Meeting, Busan,
Korea, October 27-28, 2005.

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Prediction Analysis of 243Am/235U Fission Rate Ratio at KUCA Solid Moderator Core
Tadafumi Sanoa,*, Jun-ichi Horib and Yoshiyuki TakahashibaKindai University Atomic Energy Research Institute,3-4-1, Kowakae, Higashi-Osaka, Osaka, 577-8502, Japanb
Institute for Integrated Radiation and Nuclear Science, Kyoto University,
2-1010, Asashiro-nishi, Kumatori-cho, Sennan-gun, 590-0494, Japan*
Corresponding author: t-sano@kindai.ac.jp

Abstract
We are planning new measurements of 243Am/235U fission rate ratio by using a backto-back (BTB) fission chamber at two KUCA solid moderator core with difference
neutron spectrums. The one core is named “EE1” which is one of typical experimental
core at KUCA. Another core which named “EEG14” is newly construct for the present
study. The “EEG14” core has hard neutron spectrum than EE1. In this study, prediction
calculations of the criticalities and the fission rate ratio in the two KUCA cores. As theresults, the keff of the EE1 is 1.00454±0.0139 % and that of the EEG14 is
1.00386±0.0144 % by using MVP3.0 with JENDL-4.0. The results of the fission rate
ratio by using the JENDL-4.0 in the EE1 is 1.70E-2±1.86 %, the EEG14 is 5.75E-2.
In addition, only the 243Am cross sections were taken from JENDL-4.0, JENDL-3.3
and ENDF/B-VII.0 but the other nuclides were done from JENDL-4.0, to examine an
impact of the difference of 243Am cross section among these nuclear libraries to thefission rate ratio. As a result, the fission rate ratio with JENDL4.0 was about 10%

larger than that of other libraries.

Key Words: Minor Actinide, Fission Rate Ratio, Integral Experiment,
KUCA, Back-to-back fission chamber
fission cross section of 243Am is very small compare to thecapture cross section. Then, in order to enhance theintegral data of 243Am fission cross section, we areplanning new measurements of 243Am/235U fission rate
ratio by using the BTB fission chamber at the KUCA solid
moderator core. The experiments will be performed withtwo cores which are soft and hard neutron spectrum cores.
Then, in this study, prediction calculations of criticalities,
neutron spectrum and 243Am/235U fission rate ratio in thetwo cores which are soft and hard spectrum cores arecarried out.
The calculational models are explained at section II. In
section III, the results of prediction calculations is
described.

1. Introduction
Accurate nuclear data of minor actinides (MAs) haveimportant contribution for developments of nuclidetransmutation technique, design of fuel cycle and disposalof high level radioactive waste (HLW). In order to verify
evaluated nuclear data of MAs, a lot of integral
experiments at various critical assemblies and research
reactors were carried out. The measured integral
neutronics characteristics were criticality, reactivity,
sample reactivity worth, reaction rate distribution,
reaction rate ratio and so on. The measurements of MAs(237Np, 241Am, 243Am, 245Cm and 246Cm) reaction rate orreaction rate ratio were performed to obtain thermal
neutron capture cross sections at various critical assembly
and research reactor [1-3]. On the other hand, in order toperform the integral evaluation for the nuclear datalibraries, 237Np/235U and 241Am/235U fission rate ratios byback-to-back (BTB) fission chamber and 237Np/197Au
capture rate ratios has been obtained in the Kyoto
University Critical Assembly (KUCA) [4-6].
243
Am is an important nuclide for the nuclear design andthe design of HLW disposal, because 243Am is a fertile
nuclide of high-order minor actinides such as Cm. So, a
lot of integral data for 243Am capture cross section.
However, there are few integral data of 243Am fission
cross section in thermal neutron system, because the2. Calculational model
2.1 BTB fission chamber
Figure 1 shows a sample foil (235U or 243Am) withbacking plate and a BTB fission chamber. The backing
plate is made with titanium. The BTB fission chamber andthe electrode is aluminum alloy (Al5052). The connector
is made with cupper. A diameter of the foil is 20 mm. The
backing plate has 30 mm diameter and 0.1 mm thickness.
The distance from the sample deposit to electrode is 14mm.

94Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Sam p le Dep osit A

spectrum shown in Fig. 3. The EEG14 is newly set up forpresent study. The neutron spectrum in the core is more
hardly than EE1. Then, the sensitivity of resonance fission
in the core will be larger than EE1. The fuel element (F)is consisted of 41 unit cells and sandwiched by 2”
bdellium and 16” polyethylene reflector. The unit cell has
two 1/16” EU plates and one graphite plate of 1/4”
thickness. The unit cell of driver fuel has one 1/16” EU
plate and two 1/8” polyethylene plates. The driver fuelelement (D) is consisted of 43 unit cells and sandwichedby 2” bdellium and 16” polyethylene reflector. The
EEG14 is consisted of 24 fuels and 26 Driver fuels. In
addition, the core are surrounded by graphite reflectors.
The BTB fission chamber is set into “F-7”.

Sam p le Dep osit B
Cham b er (alum inum )Connector

φ 30.0m m

φ 35.0m m

φ 40.0m m

φ 49.0m m

Electrode (anode)

2.0m m1

2.5m m

18.5m m21.0m m

C
D

C3

E
F

2.2 Core Configuration

3

4

5

6

7

8

9

S4

G
H
I

In KUCA, fuel and moderator plates in fuel element
were set in a 1.5mm thickness aluminum sheath and allmaterial plates have nominal cross section of 2” square.
In this study, two cores with different neutron spectrum
will be constructed. The one core is named EE1. The EE1
is one of typical experimental core at KUCA. The coreconfiguration and the fuel element configuration are
shown in Fig. 2. The fuel element is consisted of 60 unit
cells and sandwiched by an upper and a lower
polyethylene reflector. The upper and lower polyethylene
reflector has 18” length. The unit cell has two enricheduranium (EU) plates of 1/16” thickness, one polyethylene
plates of 1/8” thickness. The core is consist of 29 fuelelement and one void tube with the BTB fission chamber.
2

3

B

Fig. 1. BTB fission chamber.

1

2

5

6

7

8

9

10 11 12 13

G

G

G

G

G

G

G

J

F

C2

G

G

D

D

D

D

D

G

G

G

G

D

F

F

F

F

F

D

G

G

D

D

F

F

F

F

F

D

D

G

G

D

D

F

F

B

F

F

D

D

G C1

G

D

D

F

F

F

F

F

D

D

G

G

D

F

F

F

F

F

D

G

G

G

G

D

D

D

D

D

G

G

G

G

G

G

G

G

G

G

G

S5

Fuel element（F）

F

FUEL

D

Driver FUEL

Polyethylene reflector

S6

K

G

Graphite reflectorr

B

Voild tube with BTB

1/16"EU+1/16"EU+1/4"Graphite

Al ２”
PE ref. 2"Be
Graphite2"BePE ref.

２”Graphite

41 unit cells
Driver fuel (D）Al

２”
PE ref.
Graphite

1/16"EU+1/8"PE+1/8"

Be refBe ref PE ref.

２”
Graphite

43 unit cells

Fig. 3. Configurations of EEG14 core, Fuel element (F)and Driver fuel element (D).
(C : Control rod, S : Safety rod)

10 11 12 13

A

4

A

FUEL

B
C
D

C3

E
F

S4

G
H

C2

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

B

F

F

F

F

F

F

F

F

F

F

F

F

3. Prediction Analysis and Discussion

Polyethylene reflector
S5

C1

B

3.1 Criticality and Neutron Spectrum

Voild tube with BTB

In this sub-section, the criticalities and the neutron
spectrum in the BTB fission chambers were calculated by
the use of MVP3 [7] with JENDL-4.0 [8]. In particular、
the criticality and the neutron spectrum of EEG14 isimportant. In the calculations, 300M neutron histories are
generated to suppress the statistical error of keff to lessthan 0.03 % (1σ) because the typical experimental error
for the keff is estimated to be about 0.03 %dk/k [9]. As theresults, the keff of the EE1 (Fig.2) is 1.00454±0.0139 %and that of the EEG14 (Fig.3) is 1.00386±0.0144 %. As aresult, it was confirmed that both cores reached criticality.
Figure 4 shows the neutron spectrum in the BTB fission
chamber. Here, a spectrum index is defined as follows;

S6

I
J
K

Fuel element（F）
Al ２”
Graphite

1/16"EU+1/16"EU+1/8"PEPE ref.

PE ref.

２”Graphite

60 unit cells

Fig. 2. Configurations of EE1 core and Fuel element (F).
(C : Control rod, S : Safety rod)
Another core is named EEG14 with hard neutron

95

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019
1.0 𝑒𝑉

Spectrum Index =𝑑𝐸∫1𝐸−5 𝑒𝑉 𝜙 𝐸
20 𝑀𝑒𝑉𝑑𝐸 .

∫1𝐸−5 𝑒𝑉 𝜙 𝐸

respectively. 235U fission rate in the EE1 are found to be
dominant over the thermal region shown in Fig.7. In the
EEG14, 235U fission rate in the thermal energy region isdecreased by hard neutron spectrum.

(1)

The spectrum index at the position of the BTB fission
chamber in EE1 core is 0.19. In the EEG15, the spectrum
index at the position of the BTB fission chamber is 0.02.

1.E+0

Reaction rate ratio (A.U)

1.E-1

EE1

1.E-2
1.E-3
1.E-4
1.E-5
1.E-6

EE1 (U235)
EE1 J33

1.E-7
1.E-8
1.E-4

1.E-2

1.E+0
1.E+2
Energy (eV)EE1 J40
EE1 B70
1.E+4

1.E+6

Fig. 4. Energy wise contribution of 243Am and 235U
fission rate by MVP3 in EE1.

Fig. 3. Neutron spectrum in BTB fission chamber.
3.2 Fission Rate Ratio

1.E+0
1.E-1
Reaction rate ratio (A.U)

The prediction analysis of the fission rate ratio arecarried out by using MVP3 with JENDL-4.0, where only
the 243Am cross sections were taken from JENDL-4.0,
JENDL-3.3 [10] and ENDF/B-VII.0 [11] but the othernuclides were done from JENDL-4.0, to examine an
impact of the difference of 243Am cross section among
these nuclear libraries to the fission rate ratio. In thecalculations, 1G neutron histories are generated tosuppress the statistical error of fission rate ratio to lessthan 2.0 % (1σ) because the typical experimental error for
the fission rate ratio is estimated to be about 2.0-2.3 % [9].
The results of the calculated fission rate ratio are shownin Table.1. From table 1, the fission rate ratio in EEG14
are about 3 times larger than EE1 because of thedifference of neutron spectrum. Using the 243Am crosssection in JENDL-3.3 and ENDF/B-VII.0 from JENDL4.0, the fission rate ratio in the EE1 is reduced about 3 %.
In the EEG14, the reduction is 9 to 10 %.

EEG14

1.E-2
1.E-3
1.E-4
1.E-5
1.E-6
EEG14 (U235)
EEG14 (Am243) J33

1.E-7
1.E-8
1.E-4

1.E-2

1.E+0
1.E+2
Energy (eV)

EEG14 (Am243) J40
EEG14 (Am243) B70
1.E+4

1.E+6

Fig. 5. Energy wise contribution of 243Am and 235U
fission rate by MVP3 in EEG14.
5. Conclusion

Table 1. Calculated 243Am/235U fission rate ratio
EE1
EEG14
JENDL-4.0
1.70E-2±
5.75E-2±
1.86 %
1.93 %JENDL-3.3
1.66E-2±
5.23E-2±
1.88 %
1.94 %
ENDF/B-VII.0
1.65E-2±
5.20E-2±
1.86 %
1.94 %For the purpose of enhancing the integral data to
improve the accuracy of Am243 fission cross section, weare planning new measurements of 243Am/235U fission rate
ratio by using the BTB fission chamber at the KUCA solid
moderator core. The experimental cores were two cores
which are soft and hard neutron spectrum cores. In thisstudy, the prediction calculation of criticalities, neutron
spectrum and 243Am/235U fission rate ratios were carriedout.
1. Criticality and neutron spectrum
The multiplication factor of EE1 was 1.00454±0.01
39 % and that of the EEG14 is 1.00386±0.0144 %..
As the results, it was confirmed that both cores reached criticality. The spectrum index of EE1 was 0.

Figure 4 and 5 shows the energy distribution of thefission rate ratio in the EE1 and the EEG14. From the bothof figures, the 243Am fission rate has a high sensitivity ina region of 2 MeV. In addition, the fission rate has
sensitivities at resonance of 1.356 eV and 1.744 eV,

96

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 201919, and EEG14 was 0.02.
2. Fission rate ratio
The prediction calculation of 243Am/235U fission rate
ratios in the EE1 and EEG14 were carried out. The ratiousing by the JENDL-4.0 were 1.70E-2 ±1.86 % in theEE1 and the EEG14 was 5.75E-2±1.93 %.
Only the 243Am cross sections were taken from JENDL4.0, JENDL-3.3 and ENDF/B-VII.0 but the other nuclides
were done from JENDL-4.0, to examine an impact of thedifference of 243Am cross section among these nuclear
libraries to the fission rate ratio.
From the above results, it was found that 243Am/235U
fission rate ratio depending on the neutron spectrum
can be measured in KUCA solid moderator core.

9.

Unesaki H, “Study on Verification of Nuclear Data
for Nuclear Design of Next Generation Thermal
Reactors,” dissertation, Kyoto University (2001).
10. Shibata K, et al., “Japanese Evaluated Nuclear Data
Library Version 3 Revision-3: JENDL-3.3,” J. Nucl.
Sci. Technol. 39, 1125 (2006).
11. Chadwick MB, et al., “ENDF/B-VII.0: Next
Generation Evaluated Nuclear Data Library for
Nuclear Science and Technology,” Nucl. Data
Sheets. 107, 2931 (2006).

Acknowledgements
The present work was performed as a joint research
program of the KUCA at the Institute for Integrated
Radiation and Nuclear Science, Kyoto University andsupported by Kansai Genshiryoku Kondankai.
References
1.

2.
3.

4.

5.

6.

7.

8.

Kobayashi K, et al., “Measurements of Thermal
Neutron Cross Section and Resonance Integral for
237Np (n, γ) 238Np Reaction,” J. Nucl. Sci. andTechnol., 31, 1239 (1994).
Harada H, et al., “Measurement of Effective Capture
Cross Section of 238Np for Thermal Neutrons,” J.
Nucl. Sci. and Technol., 41, 1 (2004).
Ohta M, et al., “Measurement of Effective Capture
Cross Section of Americium-243 for Thermal
Neutrons,” J. Nucl. Sci. and Technol., 43, 1441
(2006).
Iwasaki T, et al., “Measurement and Analysis of
Capture Reaction Rate of 237Np in Various Thermal
Neutron Fields by Critical Assembly and Heavy
Water Thermal Neutron Facility of Kyoto
University,” Nucl. Sci. Eng., 136, 321 (2000).
Unesai H, et al., “Measurement 237Np Fission Rate
Ratio Relative to 235U Fission Rate in Cores with
Various Thermal Neutron Spectrum at the Kyoto
University Critical Assembly,” J. Nucl. Sci. Technol.,37, 627 (2000).
Unesaki H, et al., “Measurement and Analysis of241
Am Fission Rate Ratio Relative to 235U Fission
Rate in Thermal Neutron Systems Using Kyoto
University Critical Assembly,” J. Nucl. Sci. Technol.,38, 600 (2001).
Nagaya Y, et al., “MVP/GMVP Version3: General
purpose Monte Carlo codes for neutron and photontransport calculations based on continuous energy
and multigroup methods,” JAEA-Data/Code 2016018, JAEA (2016).
Shibata K, et al., “JENDL-4.0: A New Library for
Nuclear Science and Engineering,” J Nucl Sci
Technol., 48, 1 (2011).

97TS-02Monte Carlo Methods and Applications

No.

RPHA19-1107

Title

Effect of Neutron Absorber on Production of Plutonium238 for
BWR Assembly

Authors

Hiroo Ohuchi, Satoshi Takeda, Takanori Kitada (Osaka University, Japan)
99

No.

RPHA19-1022

Title

Study on The Optimization of Component Design of SinglePass Super-Critical Water-Cooled Fast Reactor

Authors

Liping Huang, Zijing Liu (University of South China, China), Xiang Wang
(Harbin Engineering University, China)
103

No.

RPHA19-1043

Title

Application of Direct Simulation Method for Neutron SpaceTime Kinetics Based on RMC

Authors

Jian Lijun, Shang Xiaotong, Guo Xiaoyu, Wang Kan (Tsinghua University,
China)
108

No.

RPHA19-1057

Title

Core Design of a Molten Salt Reactor with Chloride Fuel

Authors

Masahiro Watanabe, WilFred G. van Rooijen (University of Fukui, Japan)
112

No.

RPHA19-1088

Title

Monte Carlo Based Analysis for CANDLE Burning Reactor

Authors

Hoang Hai Nguyen, Jun Nishiyama, Toru Obara (Tokyo Institute ofTechnology, Japan)
11698Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Effect of Neutron Absorber on Production of Plutonium238 for BWR Assembly
Hiroo Ohuchia,, Satoshi Takedaa and Takanori Kitadaa
Osaka University, Yamadaoka2-1, Suita, Osaka, Japan*
Corresponding author: h-ohuchi@ne.see.eng.osaka-u.ac.jpaAbstract
The burnup calculations were performed for BWR assembly by using 162Dy and
Gd in order to evaluate the effect of the neutron absorber on the production of
238Pu. From the results, it was found that 238Pu production is decreased by theaddition of 162Dy, and is increased by the addition of Gd. By evaluating the capture
rate which is related to 238Pu production, it is confirmed that 162Dy decreases 238Pu
production by reducing the capture rate of 236U, and Gd increases 238Pu production
by increasing the capture rates of 235U, 236U, and 237Np.
Key Words: 238Pu, 162Dy, Gd, BWR Assembly, Neutron capture reaction
1. Introduction

2. Calculation Condition

Transuranic (TRU) nuclides produced by the use of
nuclear reactor have high radiotoxicity for a long time. In
order to reduce the radiotoxicity, the fast reactor and theaccelerator driven systems have been studied. These
innovative transmutation systems are very effective to
reduce the radiotoxicity of the spent fuel. On the other
hand, for the early reduction of radiotoxicity, a concept of
FORSETI (Fuel cycle Of light water Reactor with Surplus
Enriched uranium for TRU-production Inhibition) has
been proposed[1]. In a feasibility study of FORSETI, Wadaet al. showed that the potential radiotoxicity of the spent
fuel is reduced by increasing the enrichment of uranium,
and 238Pu and 241Pu have high potential radiotoxicity
through 100 years after discharge. In addition, the highenriched uranium fuel requires the burnable poison
because of its high reactivity[2].
In the previous study[2], in order to decrease
radiotoxicity of the spent fuel, neutron absorbers wereadded to the fuels. For BWR pin cell model, neutron
absorbers were added to 10 wt% uranium fuels, and 162Dy
turned out to be effective for reducing the production of238
Pu. It was found that the capture rate of 236U is mainly
occurred around 5.5 eV and the capture rate around 5.5
eV is reduced by adding 162Dy, and 162Dy reduces thecapture rate of 236U by their resonance around 5.5 eV.
238
Pu is produced in the burnup process 235U  236U 
237
U  237Np  238Np  238Pu[3], therefore reducing theneutron capture reaction of 236U results in reducing the
production of 238Pu.
In this study, the burnup calculation is performed for
BWR assembly to evaluate the effect of the neutron
absorber on the production of 238Pu. Gd is generally used
for the neutron absorber. Therefore, in addition to 162Dy,
Gd is used for BWR assembly.

For BWR assembly, burnup calculations were performedby two patterns. The first pattern is 9x9 assembly in which
neutron absorbers are added to all fuel rods. The second
pattern is 9x9 assembly in which neutron absorbers areadded to several fuel rods. In the case of 162Dy, burnup
calculations are performed at both patterns. In the case ofGd, the calculations are performed at the former pattern.
The calculation model is shown in Figures 1 and 2. 9x9
assembly has two water rods around the center ofassembly.

Fig. 1. 9x9 BWR assembly (neutron absorbers are added
to all fuel rods)
1
2
4

3

Fig. 2. 9x9 BWR assembly (neutron absorbers are added
to several fuel rods)99

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

For the production of 238Pu in each case, theproduction of 238Pu in case 2 is increased by 23.1 %, theproduction of 238Pu in case 3 is decreased by 0.12 %, andthe production of 238Pu in case 4 is decreased by 38.2 %.
In Figure 2, selected four fuel rods were shown. Fuelrod number 1 is the farthest from the fuel rods withneutron absorbers. Fuel rod number 2 is next to the fuelrods with neutron absorbers. Fuel rod number 3 issurrounded by the 4 fuel rods with neutron absorbers. Fuelrod number 4 is the rod with neutron absorbers.
Figures 4-7 show the relationship between the burnup
and the production of 238Pu.
Number density of 238 Pu [1024 /cm3 ]Figure 1 shows ‘green’ fuel rods are 10 wt% uranium
fuels without or with 162Dy. Figure 2 shows ‘Red’ fuelrods are 10 wt% uranium fuels, and ‘yellow’ fuel rods are
10 wt% uranium fuels with neutron absorbers. The
enrichment of uranium fuel rod was 10 wt%. The burnup
calculations were performed in 4 cases. 10 wt% uranium
fuels (case 1) in Figure 1. Gd-bearing 10 wt% uranium
fuels (case 2) in Figure 2. 162Dy-bearing 10 wt% uranium
fuels at several positions (case 3) in Figure 2. 162Dybearing 10 wt% uranium fuels at all positions (case 4) inFigure 1.
Burnup calculation was performed by MVP-BURN[4],and nuclear library was JENDL-4.0[5]. In the burnup
calculation, the number of histories was 200,000 at eachburnup step. The fuel region was annularly divided into
10 regions, and the volume of each subdivided region was
the same. The number of burnup step was 40, andmaximum burnup was set to 45.0 GWd/t.
3. Calculation Results and Discussion
Figure 3 shows the result of setting infinite multiplication
factor at 22.5 GWd/t in order to determine the enrichmentof adding neutron absorbers.

7.0E-06
6.0E-06

5.0E-06
4.0E-06
3.0E-06
2.0E-06
1.0E-06
0.0E+00

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

Burnup [GWd/t]

1.5case 1case 2case 3

1.4

Fig. 4. Number density of 238Pu at the fuel rod number 1
Number density of 238 Pu [1024 /cm3 ]Infinite multiple factor

1.3
1.2
1.1
1
0.9
0.8

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5 45.0
Burnup [GWd/t]
4wt% uranium fuelscase 2case 3case 44.0E-06
3.5E-06
3.0E-06

2.5E-06
2.0E-06
1.5E-06
1.0E-06
5.0E-07
0.0E+00

0.0

10.0

20.0

Fig. 3. Burnup dependence of multiplication factor
case 1

The amount of 162Dy or Gd is adjusted so that theinfinite multiplication factor for 10 wt% uranium fuels
becomes equal to that for 4 wt% uranium fuels. The
enrichment of case 2 was 12.65 %, that of case 3 was
18.65 %, and that of case 4 was 2.27 %. These values ofthe enrichment was set at 22.5 GWd/t. Table I. shows theaverage number density of 238Pu at the 45.0 GWd/t andthe relative difference from case 1.

10wt% Uranium fuels（case 1）
Gd-bearing 10wt% Uranium fuels（case 2）
162Dy-bearing 10wt% Uranium fuelsat several positions（case 3）
162Dy-bearing 10wt% Uranium fuels

at all positions（case 3）

Number density of 238 Pu [1024 /cm3 ]Relative difference[1024/cm3][%]

2.062×10-4－

2.538×10-4

23.1%

2.059×10-4-0.12%

1.273×10-4-38.2%

40.0

50.0case 2case 3Fig. 5. Number density of 238Pu at the fuel rod number 2

Table I. Average Number Density of 238Pu and Relative
Difference
Average production

30.0

Burnup [GWd/t]

2.5E-06
2.0E-06
1.5E-06
1.0E-06

5.0E-07
0.0E+00

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Burnup [GWd/t]case 1case 2case 3Fig. 6. Number density of 238Pu at the fuel rod number 3

100Number Density of 238 Pu[1024 /cm3 ]Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019
2.5E-06

65.8

54.8

50.1

48.1

47.7

48.5

50.6

55.2

65.3

54.8

42.8

38.5

37.4

37.8

38.2

39.3

43.5

55.2

50.1

38.5

35.6

37.3

40.9

40.5

37.6

39.3

50.6

48.1

37.4

37.3

44.4

40.5

38.2

48.5

47.7

37.8

40.9

40.9

37.8

47.7

48.5

38.2

40.5

44.4

37.3

37.4

48.1

50.6

39.3

37.6

40.5

40.9

37.3

35.6

38.5

50.1

55.2

43.5

39.3

38.2

37.8

37.4

38.5

42.8

54.8

65.3

55.2

50.6

48.5

47.7

48.1

50.1

54.8

65.8

2.0E-06
1.5E-06
1.0E-06
5.0E-07
0.0E+00

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

Burnup[GWd/t]case 1case 2case 3Fig. 7. Number density of 238Pu at the fuel rod number 4
As shown in Figures 4-5, for the production of Pu
on burnup, the tendency of increasing the production of
238
Pu is almost the same. The addition of 162Dy decreases
the production of 238Pu in case 3. In the fuel rod number
4, 238Pu production obtained by case 2 is greatly increased.
Next, to investigate the difference of position where162
Dy is added, the variation in burnup in case 3 and thatin case 4 are compared. Figures 8 and 9 show burnup
distributions at end of cycle. The unit of numerical value
in Figures 8 and 9 is GWd/t. The standard deviations of
the pin-wise burnup in Case 1, 2, 3, and 4 are respectively
7.14, 12.75, 9.91, and 7.75. The production of 238Pu
increases exponentially at high burnup. Therefore, even if
the average burnup in assembly is same, 238Pu production
becomes large when the standard deviations of the pinwise burnup is high. The standard deviations of the pinwise burnup in case 3 is larger than that in case 4.
Therefore, compared with case 4, the production of 238Pu
becomes large in case 3 from the difference of burnup
distribution.
238

67.7

57.4

51.9

49.9

49.5

49.9

52.0

57.4

44.3

33.7

37.6

38.4

38.0

34.2

44.4

57.4

51.9

33.7

34.0

31.6

39.7

34.0

35.7

34.2

52.0

49.9

37.6

31.6

42.9

34.0

38.0

49.9

49.5

38.4

39.7

39.7

38.4

49.5

38.0

34.0

42.9

31.6

37.6

In case 2, as shown in Fig.7, the production of 238Pu isdifferent greatly between Gd-bearing and 162Dy-bearing
at the fuel rod number 4. To investigate the difference of238
Pu production between case 2 and 3, energy
distribution in terms of neutron spectrum is investigated.
the neutron capture reaction of 235U, 236U, and 237Np are
evaluated at the fuel rod number 4. The capture rates are
summarized in Table II. Neutron capture rates of 235U,
236
U, and 237Np are normalized by fission rate at each fuelrod. Table II shows the statistical error of neutron capture
rates of 235U, 236U and 237Np at the fuel rod number 4.
Table II. The Statistical Error of Neutron Capture Rates
at the Fuel Rod Number 4
Neutron Capture Rates of 235U

6.79 %

Neutron Capture Rates of 236U

43.3 %

Neutron Capture Rates of 237Np

9.71 %

67.9

57.4

49.9

Fig. 9. Burnup distribution at 45 GWd/t in case 4

Table III shows neutron capture rates of 235U, 236U,
and 237Np at fuel rod number 4.
Table III. Neutron Capture Rates of 235U, 236U, and 237Np
at the Fuel Rod Number 4

49.9

52.0

34.2

35.7

34.0

39.7

31.6

34.0

33.7

51.9

57.4

44.4

34.2

38.0

38.4

37.6

33.7

44.3

57.4

67.9

57.4

52.0

49.9

49.5

49.9

51.9

57.4

67.7

Neutron Capture

Neutron Capture

Neutron Captureof 235Uof 236Uof 237Np

case 17.00×10-2

4.35×10-29

1.47×10-28

case 2

8.91×10-2

1.39×10-28

3.03×10-28

case 3

7.44×10-2

1.76×10-29

1.89×10-28

case2/case1

1.27

3.20

2.06

case3/case1

1.06

0.40

1.29

The relationship between neutron energy and neutron
capture rate is shown in Figures10-12.

Fig. 8. Burnup distribution at 45 GWd/t in case 3

101

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4. Summary

8.E-03
7.E-03In the burnup calculation, 238Pu production is evaluatedfor BWR assembly by adding Gd or 162Dy to fuel rods.
Compared with 10 wt% uranium fuels without neutron
absorber, it was found that the addition of 162Dy decreases
238
Pu production and that of Gd increases 238Pu production.
The addition of Gd increases 238Pu production since Gd
increases the capture rates of 235U, 236U, and 237Np.

Neutron Capture Rate

6.E-03
5.E-03

4.E-03
3.E-03
2.E-03
1.E-030.E+00
1.E-05

1.E-03

1.E-01

1.E+01

1.E+03

1.E+05

References

1.E+07

Neutron Energy [eV]case 1case 2

1.

case 3

Fig.10. The energy distribution of neutron capture rate of235
U at the fuel rod number 42.

1.E-28

Neutron Capture Rate

1.E-288.E-293.

6.E-294.E-292.E-290.E+00
1.E-054.
1.E-03

1.E-01

1.E+01

1.E+03

1.E+05

1.E+07

Neutron Energy [eV]case 1case 2case 3

Fig.11. The energy distribution of neutron capture rate of236
U at the fuel rod number 4

5.

5.E-29Neutron Capture Rate

4.E-293.E-292.E-29

1.E-290.E+00
1.E-05

1.E-03

1.E-01

1.E+01

1.E+03

1.E+05

1.E+07

Nautron Energy [eV]case 1case 2case 3

Fig.12. The energy distribution of neutron capture rate of237
Np at the fuel rod number 4
In case 2, Figure 10 shows the neutron capture rate of
U is large in the energy region over 1 eV. Figure11
shows the capture cross section of Gd has the large
neutron capture rate of 236U in the energy region around
5.5 eV. Figure 12 shows that of Gd has that of 237Np in theenergy region over 1 eV. As a result, the reason for large
capture rate of 235U, 236U, and 237Np is the effect ofincreasing the proportion of neutrons more than 1 eV byadding Gd to fuel rods.
235

102

S. Wada, R. Kimura, S. Sakurai, K. Hiraiwa, K.
Yoshioka, T. Sugita, “Reduced-radiotoxicityproduction system by increasing initial and residual
fuel enrichment,” Chengdu, China, Reactor Physics
Asia (2017).
Takeda S, Kitada T. Study on suppression of TRU
production by adding poison materials. Paper
presented at: Pacific Basin Nuclear Conference(2018).
Satoshi Takeda, Takanori Kitada, Feasibility study
on burnable absorber for FORSETI to reduce
productions of Pu238 and Pu241, Journal of Nuclear
Science and Technology, published online (DOI:
10.1080/00223131.2019.1659876) (2019).
Nagaya Y, Okumura K, Sakurai T, Mori T,“MVP/GMVP Version 3: General Purpose Monte
Carlo Codes for Neutron and Photon Transport
Calculations Based on Continuous Energy andMultigroup
Methods,” JAEA-Data/Code 2016018 Japan Atomic Energy Agency (2017).
Shibata K, Iwamoto O, Nakagawa T, Iwamoto N,
Ichihara A, Kunieda S, Chiba S, Furutaka K, Otuka
N, Ohsawa T, Murata T, Matsunobu H, Zukeran A,
Kamada S, and Katakura J, “JENDL-4.0: A New
Library for Nuclear Science and Engineering,” J.
Nucl. Sci. Technol., 48 1, 1 (2011).

Study on the Optimization of Assembly Design of Single-Pass Supercritical Water-Cooled Fast Reactor
Liping HUANGa,b, Zijing LIUa,∗ and Xiang WANGb,*a

Department of Nuclear Science and Technology, University of South China, 421001, Hengyang, China
College of Nuclear Science and Technology, Harbin Engineering University, 150001, Harbin, China*
Zijing LIU: liuzijing1123@163.com, Xiang WANG: xiang.wang@hrbeu.edu.cnbAbstract
Supercritical water-cooled reactor (SCWR) is one of six types of Gen-IV
reactors. Compared with conventional water-cooled reactor, current designs haveshown a wider range of neutron spectrum and their corresponding core structure.
This study, using MCNP4B to establish the neutron physics model, focused on thesensitivity of solid moderator, cladding material and fuel axial enrichment on theassemblies’ characteristics in the fast spectrum single-pass SCWR (SCWRsf). It isfound that reduced mass fraction of Pu isotope in the fuel brings more negative
void reactivity coefficient, and adopting reasonable axial Pu enrichment
stratification of fuel rods can improve the effective multiplication factor and reduce
the axial power peaking factor. The reasonable thickness of solid moderator ZrH1.7
layer can meet the negative void coefficient and flat the axial power distribution of
the core.
Key Words: Single-pass, SCWR, neutron physics; fuel assemblies
1. Introduction
SCWR is the only water-cooled reactor among six
Gen-IV nuclear energy systems. According to the featuresof reactor type and neutron energy spectrum, SCWR isclassified in different ways[1]. For fast spectrum SCWR, it
can be realized by supercritical water with low density
and weak moderation under high temperature. Its
advantages include high fuel efficiency, high power
density and combustible actinides. However, the maintechnical challenges like the positive temperature
coefficient, the complex design of the reactor core andfuel assemblies are also critical[2]. Besides, it is moredifficult to select core-cladding materials.
SCWR (Fig. 1) has some designs in common with
BWR and PWR, while the differences between is as wellsignificant[3]. In the design of SCWR, the main circulation
pump provides the driving pressure head, which enables
the fluid to enter the reactor core through the main
feed-water line. After nuclear heating, it becomes"supercritical steam" with high temperature and pressure(no phase transition), which enters the turbine through themain steam line. After that, the steam is further cooled in
the condenser to liquid water, which returns to the main
pump inlet and close the direct circulation loop.
At present, few countries have conducted researcheson the fast spectrum SCWR. The most mature design isthe one[4] has both two-pass (Fig. 2, a, b) and single-pass(Fig. 2, c) flow. Compared to two-pass design, thesingle-pass design simplifies the core structure. However,
because the coolant density varies greatly along the axial
direction, the core has extremely uneven axial power
distribution and fluctuation during the cycle. This paper
also carries out sensitivity analysis for this point.

Fig. 1. Schematic diagram of SCWR system.

Fig. 2. Coolant flow schemes of Super FR.
2. Modelling of SCWRsf
2.1 Core Design Criteria
The design criteria of SCWRsf (SCWR with single-passand fast spectrum) were adopted by previous studies [5]. (1)core power is 1000 MW. (2) the average core outlet
temperature is above 500 ℃ to ensure the thermal
efficiency of 44%. (3) the overall volume power densityof the core is more than 100 W/cm3 (including gas
chamber and radial reflector). (4) in normal operation, the1103

maximum cladding surface temperature is lower than
650℃. (5) negative void reactivity is ensured. (6) undercold and hot operating conditions, the balance cycle andthe new reactor core both have a sufficient shutdown
margin of 1%.

In. dia. of CR guide tube
CR absorber/follower

13.8 mm
B4C/
SUS304-

No. of ZrH
0/270/19.64 mm
rods/thickness
No. of SS rods
0210/480
top/bottom
ZrH/SS dia.
6.5 mmChannel box thickness
3.0 mm*Pu isotopic 238/239/240/241/242 composition in wt%:
0.4/51.3/37.8/6.5/4.0.

2.2 Core Assembly design
The SCWRsf core is divided into two zones, namely, theseed and the blanket zone, filled with seed (SA) andblanket assemblies (BA), respectively. The design of the
SA (Fig. 3) is similar to that of traditional fast reactor
assembly, while that of the BA is slightly different as its
function is to breed fertile material and to reduce the void
coefficient (VC) of the core. Unlike SAs, BAs use a thicklayer of ZrH (Fig. 4) rods as a solid moderator. ZrH
moderated fast neutron produced from SAs, while fertile
nuclides in BAs increase neutron absorption to provide
negative void reactivity. Since ZrH fuel rods cause fuelpeak between adjacent assemblies, the decorated
stainless-steel rods (SS rods, Fig. 5) made of SUS304 arearranged to ease the peak. The entire core structure isshown in Fig. 6. The purple areas on the periphery
represent water, the yellow areas represent the MOX fuelrods in SAs with six purple control rod guides evenly
distributed in the middle, the dark-blue areas represent SS
rods in BAs, the light-blue areas represent ZrH rods inBAs, and the pink areas represent the UO2 fuel rods in
BAs.
Some specific parameters of the assembly are shownin Table Ⅰ, which refers to the design of professor Oka[6].
SAs use MOX fuel in the rod, while in BAs the upper partof the rod is depleted uranium, where 238U absorbs
neutrons and converts into 239Pu. From the safety
perspective, due to void effect, the resonance absorptionof 238U of neutrons from SAs makes VC more negative.
The lower part uses MOX fuel to flat the power change.
The active zone of the assembly is divided into 10
sections in the axial direction. Different coolant densities
are taken for upward flow positions of the coolant (Fig. 7,
inlet 310℃, outlet 501.2℃). Two water reflection layersof 40 cm thick are located at both ends of the assembly,
represent the chambers of the pressure vessel.

Fig. 3. Cross-sectional view of 1/4 SA.

Fig. 4. Top cross-sectional view of 1/4 BA.

Fig.5. Bottom cross-sectional view of 1/4 BA.

Table I. Design parameters of SA and BA
Assembly type
SA
BA
Top: UO2/1.8 m
MOX
Types of fuelBottom: MOX*/(2.4m)
0.6 m
Fuel rod dia.
5.5 mmFuel pellet dia.
4.65 mmCladding mat/thickness
SUS304/0.4 mm
Gap clearance
0.07 mm
Fuel rod pinch
6.545 mm
Number of fuel rods
978547
Assembly pinch
21.76 cm
Active length
240 cmGas plenum length
40 cm
Gas plenum location
Upper
No. of CR guide tube
70Fig. 6. In-core configuration in single-pass core design.

Fig. 7. Axial coolant density distributions in SA and BA.

2104

3. Results

Fig. 10. keff with solid moderator layer thickness.

3.1 Steady-state assessment
The calculations are performed by MCNP4B with nuclear
date library ENDF-b602[7]. The keff is 1.03933±0.00055
and the residual reactivity is 0.0378, which enables thecore to breed as shown in the pink areas of Fig. 6.
The change of reactivity with the void fraction (ϕ,Fig. 8) shows that reactivity fluctuates significantly with
the increase of ϕ. When the void fraction reaches 40%,
the VC changes from negative to positive.
The void fraction is changed by changing the densityof the coolant. If the coolant density decreases, the void
fraction will increase. The most negative VC it can reachis -11.323 pcm/% as shown in Fig. 8, which results innegative feedback effect and thus maintains stable
operation.
For a single assembly, the radial length is extremely
small compared with the axial length, and the fuel rod
arrangement is relatively dense, so the axial powerchange is not obvious. Therefore, this paper only focusedon the axial power change. Axial power distributions ofSA and BA (Fig. 9) are extremely uneven, and the power
peak (PP) appears in the lower part of the assembly. Thisis because that the fission by Pu causes larger power
output in the MOX fuel at the bottom. This helps toincrease the coolant outlet temperature and helps tomitigate the later power change of BAs. During theoperation, the PP shifts from bottom upwards in BAs. The
power peaking factor (PPF) of SAs is 1.77 and that of
BAs is 4.84, which is large and needs to be improved. Fig.
10 assesses the keff with ZrH thickness.

Fig. 11. Reactivity with ϕ, solid moderator thicknesses.

Fig. 12. Axial power of SA with moderator thicknesses.

Fig. 13. Axial power of BA with moderator thicknesses.
3.2 Sensitivity of solid moderator thickness
The ZrH1.7 solid moderator layer in BAs is applied due toits good thermal stability, strong moderation, hightemperature and radiation resistance. It is also easy to be
manufactured[6]. The reactivity (Fig. 11) and the keff (Fig.
14) change with ϕ, solid moderator thicknesses arestudied. Axial power distributions of assemblies are in Fig.
12, 13.
Though the axial PPF is the smallest with a thicknessof 13.09 mm, its VC which is more likely to be positive
results in worse reactor safety. When the ZrH thickness
increases, the axial PPF increases. Therefore, considering
keff, VC and axial PPF, 19.635mm was selected as the
optimal moderator layer thickness in this paper to ensurea negative VC and a small axial PPF with the keff greaterthan 1.

Fig. 8. Reactivity with ϕ.

Fig. 9. Axial power distributions in SAs and BAs.

Fig. 14. keff, VC and axial PPF with ZrH thickness
3.3 Sensitivity of cladding materials
Supercritical water is irradiated and decomposed in the3105core during the operation. The cladding materials shouldwithstand large pressure and ultra-high temperature, resist
strong corrosion and little hydrogen embrittlement. It isalso necessary to have a small neutron absorption crosssection as possible and to resist radiation damage. T91
and Zr-4 are selected to compare with SUS304 to study
the impact of these materials on keff, VC and axial PPF.
The Composition of SUS304, T91, Zr-4 are shownin Table Ⅱ~Ⅳ. The results are plotted in Fig. 15~18. It isclear that Zr-4 and T91 have larger keff , VCs and axial
PPFs of SA. Therefore, SUS304 is more suitable as the
cladding material here.

Fig. 18. keff, VC, axial PPF with cladding materials.
3.4 Optimization of axial enrichment of fuel rods
Based on the results, the axial power density distinguishes,
and the PP appears at the bottom. To flat the axial power,
Pu enrichment in SAs is stratified according to the axial
PP position. The enrichment should be set to ensure
negative VC and smaller axial PPF, and keff should begreater than 1. The enrichment of Pu was originallydivided into two layers. On the basis of the original layer,
the enrichment was divided into four layers, which
reduced the enrichment at large power level, increased theenrichment at small power level. (Tab. V, average for23.365%). The ZrH is 19.635 mm thick, and the cladding
is SUS304.
The keff is 1.02298±0.00055, and the VC is -30.742
pcm/%. The reactivity with the ϕ is shown in Fig. 19
while the axial power distribution in Fig. 20. The
comparison of parameters before and after optimization isshown in Table Ⅵ. The axial power distribution of the
improved SAs is more uniform. The reactivity is slightly
reduced after optimization, because the average Pu
enrichment of fuel after optimization (23.365%) isslightly lower than before (23.4%). The VC is morenegative, so the safety is improved.

Table Ⅱ. Composition of SUS304 (mass fraction)
C
Si
Cr
Mn
Ni
P
S
0.06
0.48 18.25 1.45
8.16 0.024 0.005
Table Ⅲ. Composition of T91 (mass fraction)
C
Si
Nb
P
Mn
0.100
0.028
0.090
0.009
0.450
N
V
Mo
Cr
Ni
0.038
0.200
0.900
8.370
0.210
Table Ⅳ. Composition of Zr-4 (mass fraction)
C
Cr
Fe
Sn
Ni
0.002
0.100
0.200
1.500
0.007

Table Ⅴ. Pu enrichment at axial positions of SAs
Axial position (cm)
Enrichment (%)-120~-60
19.57-60~-2421.28-24~2425.48
24~12027.16

Fig. 15. Reactivity with ϕ and cladding materials.

Table Ⅵ. Comparison before and after optimization
Parameters
BeforeAfter
Reactivity (Keff)
0.038 (1.03933)
0.023 (1.02298)
VC (pcm/%)-11.323-30.742axial PPF of SA
1.770
1.427
axial PPF of BA
4.840
2.080

Fig.16. Axial power distribution of SA with materials.

Fig. 17. Axial power distribution of BA with materials.
Fig. 19. Reactivity with ϕ after stratification

4106

Fig. 20. Axial power distribution of assemblies after
stratification
4. Summary
This paper refers to given assembly design of SCWRsf.
Based on the given condition, a series of sensitivity
calculations are carried out.
The optimal moderator layer thickness is 19.635 mm,
when the VC is negative and the axial PPF is small. With
the increase of the thickness of moderator layer, the keffincreases, the VC tends to be negative, but the axial PPF
increases.
The best cladding material is SUS304, which has
better corrosion resistance and negative void reactivity insupercritical water compared with T91 and Zr-4.
By stratifying the axial enrichment of SAs, arelatively flat distribution of the axial power of theassembly can be achieved, and the axial PPF can be
reduced to some extent.
References

1. Gabriel O. O., Alade A. M., Chad-Umoren Y. E.,“Science and Technology of Supercritical Water
Cooled Reactors: Review and Status,” Journal ofEnergy Technologies & Policy, 15, 4, p. 201, (2013).
2. Mukohara T., Koshizuka S. I., Oka Y., “Core design ofa high-temperature fast reactor cooled by supercriticallight water,” Annals of Nuclear Energy, 26, 16, p. 1423,(1999).
3. Oka Y., Koshizuku S., “Once-through Cycle Light
Water Cooled Reactor Concept,” J. Nucl. Sci.
Technol., 38, 12, p. 1081, (2001).
4. Oka Y., Morooka S., Reactor Design and Safety, p.
21, Springer, Tokyo, Japan, (2014).
5. Oka Y., Super Light Water Reactors and Super Fast
Reactors, p. 12, Springer, Tokyo, Japan, (2014).
6. Liu Q., Oka Y., “Single pass core design for a Super
Fast Reactor Annals Nuclear Energy,” Annals of
Nuclear Energy, 80, 10, p. 451, (2015).
7. X-5 Monte Carlo Team. “Mcnp4B general Monte
Carlo N-particle transport code,” LA-CP-03-0245,
Los Alamos National Laboratory (2003).

5
107

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Application of Direct Simulation Method for Neutron Space-Time Kinetics Based on RMCaJian Lijuna, Shang Xiaotonga, Guo Xiaoyua, Wang Kana
Department of engineering physics, Tsinghua university, Beijing*
Corresponding author: wangkan@mail.tsinghua.edu.cn

Abstract
This paper studies the direct simulation method of neutron space-time kinetics,
based on the RMC developed by Reactor Engineering Analysis Lab(REAL),
Department of Engineering Physics, Tsinghua University. A comb sampling
method that adjusts the neutron history numbers is developed to control thecomputational cost of the Monte Carlo algorithm, and is verified by the TRACYr61 case. A method based on precursor forced decay and roulette is developed toreduce the variance of precursor simulation, and is verified by a single group point
reactor model. The simulation process of absorption reaction and fission reaction
was modified to smoothen the number of neutrons in each counting time box.
Furthermore, after the critical calculation module is added, the code is verified by
the case 16A. The results show that the simulation of the code reaches goodprecision.
Key Words: Neutron space-time kinetics; direct simulation; RMC; comb
sampling method; precursor forced decay
1. Introduction

2.1 comb sampling method

The numerical calculation methods of reactor can bedivided into deterministic method and Monte Carlo
method. In transient analysis of nuclear reactor, somepeople have studied the deterministic method to solve.
But limited by the previous computational conditions,most of them used simplified transport model, such as
diffusion theory [1]. Later, the method of using neutrontransport equation was gradually developed, but in thiscase, the phase space needed to be stored after thegeometric structure was very large. At present, an
improved quasi-static method [2] is widely adopted,
which holds that transient state is composed of the steadystate at each moment, and the quasi-static method andMonte Carlo method are combined in the calculation.
However, this method also has some limitations, such as
kinetic parameters needed for time-related parts [3-4].
Direct simulation method is another idea of simulatingtransient problems. This method directly simulates
transient problems through Monte Carlo and obtains arandom process with exactly the same average result as
the described problem. In principle, the accuracy can becontrolled by statistical methods [5]. However, theconsumption of computer resources is relatively large, butwith the development of computers, the disadvantage
becomes smaller. The work of this paper is based on the
RMC [6] fixed source neutron start-up mode [7], which
has the computational framework of direct simulation
method.

To control the change of neutron quantity in the transient
process, comb sampling method [8-9] is added at theinitial moment of each time interval.
The main idea of the comb method is to adjust theweight of neutrons to change the neutron quantity while
keeping the total weight of neutrons unchanged. It is
assumed that there are K particles before the execution of
the comb method program, and the number of target
particles after adjustment is M. Imagine that the original
K particles are arranged in a row according to their weight
widths, and the total length of this long row is the total
weight W of K particles
W = ∑𝐾
𝑘=1 𝑤𝑘(1)

The average weight of the original K particles is
𝑊
𝑤𝑎𝑣 =
(2)
𝑀
It is considered that the weights of M particles after the
comb method are equal. Therefore, the original K
particles are rearranged into M particles.
The position of the first comb tooth is 𝜉𝑤𝑎𝑣 , and 𝜉 is
random number between 0 and 1. The other comb teeth
are arranged at equal intervals 𝑤𝑎𝑣 with the previous one,so that the position of each comb tooth can be obtained𝑊𝑊
𝑡𝑚 = 𝜉 + (𝑚 − 1)
𝑚 = 1, ⋯ , 𝑀(3)𝑀
𝑀
Each particle on the tooth is reserved with weight 𝑤𝑎𝑣
and continue simulation.
It is unbiased after the comb method, which can beproved mathematically. For a particle with the original

2. Theoretical Model108

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019adjusted.
The program uses implicit capture to deal withabsorption reactions, rather than the direct simulation
before. After sampling the type of reaction nucleus,
implicit capture directly adjusts the weight of particle 𝑤𝑛
to 𝑤𝑛′
𝜎𝑤𝑛′ = (1 − 𝑎 ) × 𝑤𝑛
(14)weight of 𝑤𝑘 , the probability that a comb tooth falls
within this weight range is𝑃ℎ𝑖𝑡,𝑘 =

𝑤𝑘
𝑤𝑎𝑣

(4)

The expectation that this particle can be retained aft
er the comb method is𝐸𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑒𝑑 = 𝑃ℎ𝑖𝑡,𝑘 𝑤𝑎𝑣 = 𝑤𝑘
(5)

𝜎𝑡

The program is inefficient when the weight is low, sowhen the new neutron weight 𝑤𝑛′ is less than a certain
threshold, a roulette process like equation (13) is carriedout. In the program, the roulette lower threshold is set as
𝑤𝑚𝑖𝑛 = 0.2 × 𝑤
̅(15)
While 𝑤
̅ is the average neutron weight at the beginningof the initial time.
As for fission reactions, after sampling the number offission neutrons in the original procedure, there is only
one secondary neutron, whose weight is the number offission neutrons sampled multiplies by the weight of theneutron before fission.
Correspondingly, when the neutron weight isextremely large, it is also divided by roulette. The upper
threshold of roulette is set as
𝑤𝑚𝑖𝑛 = 4 × 𝑤
̅(16)
While 𝑤
̅ was introduced earlier.

2.2 precursor forced decay
Compared with life-time prompt, the decay time ofprecursor can be very long, so some precursor will notdecay in the calculative time interval, which will enlarge
large variance. To solve this problem, the program uses
the method of precursor forced decay, so that all precursor
come into play. It is considered that each precursor isforced to yield a delayed neutron in each interval [10], so
as to influent the neutron transport process, and theunbiasedness is kept by adjusting weight.
Setting the time interval small enough, it is thatassumed that precursor yield uniformly at each timeinterval t1 to t1 + 𝛥𝑡, the probability of decay is
1𝑝̅ (𝑡) =(6)Δ𝑡
To maintain the unbiased result, the weight of delayed
neutrons is supposed to be exponential decay. The initial
weight of precursor is set as w𝑐 , and the weight of decay
in the time interval is𝑝(𝑡)𝑤𝑑 (𝑡) = 𝑤𝑐
= 𝑤𝑐 Δ𝑡 ∑𝑖 𝑓𝑑𝑖 𝜆𝑖 𝑒 −𝜆𝑖(𝑡−𝑡0)(7)

3. Numerical Verification
We used TRACY-r61 of continuous energy to verify thecorrectness of the comb method and single group point
reactor model to verify that the program of the precursor
forced decay and modification of nuclear reaction
simulation is correct. After adding the critical calculation
module, we tested the 16A transient benchmark of onedimensional two groups. The total weight of neutrons isselected as the output index, since it corresponds toneutron flux of the system.

𝑝̅ (𝑡)

𝑓𝑑𝑖 =

𝛽𝑖
𝛽

(8)

𝑡1 < t ≤ t1 + 𝛥𝑡 , 𝛽𝑖 is the precursor share of precur
sor group i and 𝜆𝑖 is the decay probability of precu
rsor group i.
Average weight of delayed neutrons 𝑤𝑑,𝑎𝑣 in thistime interval is
1 t +𝛥𝑡
𝑤𝑑,𝑎𝑣 = ∫t 1 𝑤𝑑 (𝑡)𝑑𝑡(9)Δ𝑡 1
𝑤𝑑,𝑎𝑣 = 𝑤𝑐 ∑𝑖 𝑓𝑑𝑖 (𝑒 −𝜆𝑖(𝑡𝑖 −𝑡0) − 𝑒 −𝜆𝑖 (t1+𝛥𝑡−𝑡0) ) (10)Similarly, the weight of the precursor 𝑤𝑡𝑖𝑚𝑒𝑑 also decreases exponentially𝑡𝑤𝑡𝑖𝑚𝑒𝑑 = 𝑤𝑐 (1 − ∫𝑡 ∑𝑖 𝑓𝑑𝑖 𝜆𝑖 𝑒 −𝜆𝑖 (𝑡−𝑡0) 𝑑𝑡) (11)

3.1 TRACY-r61 neutronics benchmark
We test the comb method with r61 of TRACY experiment[11]. There is a uranyl nitrate solution in the shape of ahollow cylinder as fuel, surrounded by a cladding with airon the outside. With the result of the original program asa reference, the subcritical and instantaneous supercriticalconditions are adopted for testing. The test items include
three cases: no control of particle number, comb controlof particle number to 4000 neutrons and comb control ofparticle number to 10000 neutrons. The test results are as
follows0

𝑤𝑡𝑖𝑚𝑒𝑑 = 𝑤𝑐 ∑𝑖 𝑓𝑑𝑖 𝑒 −𝜆𝑖(𝑡−𝑡0)(12)
To cancel the tracing of the inefficient precursor, theroulette is executed when the weight of the precursor
𝑤𝑡𝑖𝑚𝑒𝑑 is less than the threshold 𝑤𝑠𝑢𝑟𝑣 . The new weight𝑤𝑛𝑒𝑤 can be summarized as0𝜉 < 𝑤𝑡𝑖𝑚𝑒𝑑 /𝑤𝑠𝑢𝑟𝑣
𝑤𝑡𝑖𝑚𝑒𝑑 = {(13)
𝑤𝑠𝑢𝑟𝑣 𝜉 > 𝑤𝑡𝑖𝑚𝑒𝑑 /𝑤𝑠𝑢𝑟𝑣
𝜉 is random number between 0 and 1.
2.3 adjustment of nuclear reaction simulation
The comb method regulates the quantity of neutrons
totally and the comb method can only be used at the
beginning or end of each time interval. However, due tothe characteristics of the direct simulation method, thenumber of neutrons may vary greatly in each time interval.
That’s why the nuclear reaction process has also been109

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the procedure still maintains a certain amount ofcalculation. In prompt supercritical, under the premise ofsimilar calculation results, the calculation time of the
program is greatly shortened, indicating that the comb
method balances the variance and calculation amountwell. It is concluded that the comb method program
makes it feasible to simulate the transient process directly.

Subcriticality
100000No control of particle
numberParticle number of 4000
neutrons
Particle number of 10000
neutrons

Total neutron weight

10000
1000

3.2 a single group point reactor model

100

After the nuclear force decay and the modification of thenuclear reaction simulation, we use the simplest one
group point reactor model to test. Taking the results of thepoint reactor program as reference, only the precursor
forced decay and the precursor forced decay withmodified nuclear reaction were tested in subcritical state.
The controlled number of the comb method was 10000.
The results are as follows.

10

1

0

0.005

0.01

0.015

0.02

Time(s)Fig. 1. Calculation results of r61 in subcritical state.
Table I. Single-core time of r61 in subcritical stateParticle
No control
Particlenumber ofof particle
number of
10000
number4000 neutrons
neutrons
1.67min

10000

Total neutron weight

1.55min3.02min

Prompt supercritical
1E+09
100000000

Total neutron weight

10000000

1000

100

point reactor
Precursor forced decay
Adjustment of nuclear reaction

10
1

0

0.00005

0.00010.00015

Time(s)Fig. 3. Calculation results of a single-group point reactor
model in subcritical state.

1000000
100000

10000
1000
100
10
1

Criticality

1000000

0.01

No control of particle
numberParticle number of4000 neutrons
Particle number of
10000 neutrons

From the calculation results, we can see that after theforced decay of precursor, the calculation results arecorrect except that the volatility is relatively high. The
volatility was reduced after adjusting for the nuclear
reaction simulation.

0.02

3.3 16A neutronics benchmark

0.03

0.04

Time(s)

The transient process is considered to start from steadystate. After adding the critical calculation module, thedistribution of neutron and precursor in the initial steadystate is obtained through the critical calculation, and thenthe subsequent simulation is carried out. Transient
benchmark 16A [12] of one-dimensional two-group wastested in the critical condition(16A-0) and the prompt
supercritical condition(16A-2). This model consists ofseven uniform plates, which adjust the state of the system
by changing the density of the fuel. The test results are as
follows

Fig. 2. Calculation results of r61 in prompt supercritical.
Table II. Single-core time of r61 in prompt supercritical
Particle
No control
Particlenumber ofof particle
number of
10000
number4000 neutrons
neutrons
Memory3.54min
8.29mincorruption
In subcritical state, the fluctuation of the total neutron
weight decreases significantly after the addition of the
comb method, which indicates that with the comb method,110Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

1000000

Total neutron weight

References

Criticality

1. Boer, B., et al. “Validation of the DALTONTHERMIX code system with transient analyses of the
HTR-10 and application to the PBMR”, Nuclear
technology, 170.2, 306-321 (2010).
2. Kaplan, E. L. “Monte Carlo methods for equilibrium
solutions in neutron multiplication”, UCRL-5275-T,
Lawrence Radiation Lab (1958).
3. Carta, M., et al. “Calculation of the effective delayed
neutron fraction by deterministic and Monte Carlo
methods”, Science and Technology of Nuclear
Installations, 2011 (2011).
4. Verboomen B., et al. “Monte Carlo calculation of theeffective neutron generation time”, Annals of nuclear
energy, 33.10, 911-916 (2006).
5. Sjenitzer B. L., “The dynamic Monte Carlo method
for transient analysis of nuclear reactors”, Delft
University of Technology (2013).
6. Wang K, et al. “Analysis of BEAVRS two-cycle
benchmark using RMC based on full core detailed
model”, Progress in Nuclear Energy, 98, 301-312(2017).
7. Shang X. T., et al. “Burst waiting time simulation of
CALIBAN pulse reactor with RMC”, ANS
Transactions, (2017).
8. Booth T E. “A weight (charge) conserving
importance-weighted comb for Monte Carlo”. LAUR-96-0051; CONF-960415-10. Los Alamos
National Lab., NM (United States) (1996).
9. Kuilman F, et al. “Implementing the combing methodin the dynamic Monte Carlo”. Bachelor End Thesis,
Faculteit Technische Natuurwetenschappen (2012).
10. Legrady D, et al. “Scouting the feasibility of Monte
Carlo reactor dynamics simulations”. Proc.
PHYSOR-2008, Interlaken, American Nuclear
Society (2008).
11. Miyoshi, et al. “Inter-code Comparison Exercise for
Criticality Excursion Analysis”, Benchmark Phase I:
Pulse Mode Experiments with Uranyl Nitrate
Solution in the TRACY and SILENE Facilities, NEA
6285 (2009).
12. Edgar C. A., “An adaptive Runge-Kutta-Fehlberg
method for time-dependent discrete ordinatetransport”, Georgia Institute of Technology (2015).

100000

10000

Reference value

1000

Calculated value

100
10
1

0

0.00050.001

0.0015Time(s)Fig. 4. Calculation results of 16A-0 in critical state.

Prompt supercritical

10000000
1000000

Total neutron weight

100000
10000

Reference value

1000

Calculated value

100
10
1

0

0.00050.001

Time(s)Fig. 5. Calculation results of 16A-0 in prompt
supercritical.

0.0015

It can be seen from the calculation results that the test
results of critical and prompt supercritical agree well.
4. Conclusions
In this paper, based on RMC, we study several methodsof direct simulation method for neutron space-time
kinetics. Through the comb method and the modificationof the nuclear reaction simulation, the fluctuation of thenumber of neutrons against time in the actual
simulation is decreased. The forced decay method isadopted for the precursor, and the critical calculation
module is added to obtain the initial steady state
conditions. Finally, we used TRACY-r61, single group
point reactor model and 16A to verify the correctness of
the improved program. The results show that thealgorithm can get the correct results and reduce thevariance of the calculated results.
Acknowledgements
The work in this paper is supported by Science andTechnology on Reactor System Design Technology
Laboratory in China.

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Osaka, Japan, Dec. 2-3, 2019

Core Design of a Molten Salt Reactor with Chloride Fuel
Masahiro Watanabe a, W.F.G. van Rooijenb
Graduate school of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui city, Fukui 910-8507 Japanb
Research Institute of Nuclear Engineering, University of Fukui, 1-3-33, Kanawa, Tsuruga city, Fukui 914-0055,
Japan*
Corresponding author: jn180101@u-fukui.ac.jpaAbstract
The disposal method of spent fuel is a problem all over the world. Minor actinide (MA)that has a long half-life and is the cause of the problem, it is considered that the methodof MA reduction in the reactor due to depletion. In this study, basic reactor analysis
for a simple cylindrical type of a small chloride molten salt reactor aimed at MA
reduction is carried out using CSAS6 and TRITON in SCALE6. Criticality calculationand depletion calculation are performed, and the shape of the reactor and the changes
in the K-effective due to the composition of the fuel salt, and the radio isotopes due tofuel depletion have been confirmed.
Key Words: Molten salt reactor, Core design, Chloride fuel, Depletion calculation
used for the reflector. The fuel operating temperature is
500℃. The composition of the plutonium was calculatedwith TRITON in SCALE6 on the basis of a PWR fuelassembly model. The assumption was that the fresh fuelwas 4.5% enriched UO2. The composition of theplutonium is shown in Table Ⅰ, using the plutonium weightratio in the spent fuel after 4 cycles in pressurized water
reactor (PWR) operation. Fig. 1 shows the system of thereactor in this study. The fuel is represented in orange
color and the reflector is blue color. It is considered thatthe core power density should be about 100 MWth/m3.
Thus, for a power of 50 MWth, the core volume must be
0.5 m3. The core radius should be 43 cm and the coreheight 86 cm. The density of PuCl3-KCl-LiCl[3] iscalculated from equation 1 to 4. In molten salt reactor, thefuel flows and the temperature distribution is generated in
the reactor. In this study, it was assumed that thetemperature distribution was uniform, and ENDF/B-Ⅶ.1
is used.

1. Introduction
Molten salt reactors were studied using fluoride at the Oak
Ridge Laboratory in the United States from 1964 to 1969.
At present, a reactor using molten chloride salt is
considered, and since it is possible to make the fuelconcentration higher than fluoride eutectic salt, it isconsidered to work as a fast reactor. It is expected to beusable for transmutation of MA. The aim of this study is
to design a small reactor with a maximum thermal powerof 50MWth. Therefore, the reactor is designed as small as
possible. The K-effective changes depending on thereactor system, dimensions, fuel temperature, and thecomposition and concentration of fuel and structural
material. In this study, K-eff was calculated by changing
fuel fraction, reflector thickness and fuel temperature.
Furthermore, the depletion calculation was done changing
the natural abundance of Li and Cl, and the concentrationof Li-6 and Cl-35. In depletion calculation, the abundanceof Li-6 and Cl-35 with large neutron absorption crosssection in the fuel is changed, and the amount of H-3
produced and the Cl-36 with a long half-life are evaluated.

𝜌𝐾𝐶𝑙 = 2.1359 − 5.831 × 10−4 (273.15 + 𝑇),(1)

2. Parameter study of molten salt reactor withchloride fuel𝜌𝐿𝑖𝐶𝑙 = 1.8842 − 4.328 × 10−4 (273.15 + 𝑇),(2)

In this study, when deciding the reactor system, K-eff
and neutron calculation are evaluated using CSAS6 which
is a module of SCALE6[1]. Besides depletion calculationis performed using the determined fuel radius andreflector thickness, and the influence of the abundance
ratio of Li-6 and Cl-35 on the amount of H-3 and Cl-36
produced and the K-effective is confirmed. PuCl3-KClLiCl is used for the fuel salt, the mole% of fuel salt isdetermined from the ternary phase diagram[2], considering
the melting point and the concentration of Pu. SUS316 is𝜌𝑃𝑢𝐶𝑙3 = 6.6252 − 1.522 × 10−3 (273.15 + 𝑇),

(3)

1

(4)

𝜌𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑𝑤= ∑( 𝑖 ),𝜌𝑖

where
𝑇 ∶ Temperature (℃).
𝑤𝑖 : Weight fraction of molten salt.
𝜌𝑖 : Density of molten salt (g/cm3).

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Osaka, Japan, Dec. 2-3, 2019
1.50
1.40

Table Ⅰ. Plutonium abundance ratio in fuel salt
Plutonium abundance (%)

Pu-238

1.0

Pu-23967.0

Pu-240

17.0

Pu-241

12.0

Pu-242

3.0

1.30
1.20
1.10
K-eff

Nuclide

1.00
0.90
0.80
0.70

PuCl3 10mol%

PuCl3 15mol%

0.60

PuCl3 17mol%

PuCl3 20mol%

0.50

PuCl3 25mol%

PuCl3 30mol%

0.40

20

25

30
35
40
Reflector thickness(cm)4550

Fig. 2. The K-eff by different molar fraction of fueland Reflector thickness.
Table Ⅱ. Calculation conditions of Fuel, reflector andgeometry

Fig. 1. Core geometry on this study (Fuel is orange
color, Reflector is blue color, there are reflectors at the
top and bottom of the core).

Parameters

Parameters

Fuel

Composition

Reflector
Geometry
Boundary

2.1 Core design by fuel fraction, reflector thickness andfuel temperature

TemperatureDensity

PuCl3-KCl-LiCl(17-50-33 mole %)500℃
2.6222g/cm3 (500℃)

Composition
Density
type
type

Stainless steel 316
8.0300g/cm3
CylinderVacuum2.1.2 The K-eff by fuel temperature

In this section, the change in K-eff due to the fraction offuel in the eutectic salt and reflector thickness. From theresults, the thickness of the reflector and fraction of fuelare determined, and the change in K-eff is calculated dueto the change in fuel temperature.

The K-eff is calculated by changing fuel temperature from
500 ℃ to 750℃ every 50℃. Fig. 3 shows the K-eff bychanging fuel temperature.
It can be seen that K-eff decreases almost linearly withtemperature rise. It decreases approximately 0.008 every
50℃. The statistical error in KENO-6 is about 0.001.
From the result of Fig.3, temperature reactivity coefficientis approximately -16×10-5 [Δk/k/℃].

2.1.1 The k-eff by fraction of PuCl3 and reflector
thickness
In this study, k-eff was evaluated with CSAS6 (KENO6), and it was determined fuel fraction and reflector
thickness. The fuel fraction was increased from 10% andthe reflector thickness was increased from 25 cm to 45 cm.
Boundary condition is set as vacuum. In the KENO-6
calculations, the statistical error was smaller than 0.001.
It can be seen that k-eff varies greatly depending on thefuel fraction, and k-eff does not change when the reflector
thickness is 40 cm or more.
Since K-eff does not change greatly when the reflector
is 40cm and 45cm, it is considered that the infinite
thickness is at 40cm.
From the results of Fig. 2, the fuel fraction for whichk-eff is about 1.1 was determined to be 17 mol%, thereflector thickness was determined to be 40 cm, and theparameters in this analysis are shown in Table Ⅱ.

1.120
1.110

K-eff

1.100
1.090
1.080
1.070
1.060

450 500 550 600 650 700 750 800
Fuel temperature(℃)

Fig. 3. The K-eff by fuel temperature

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2.2 Depletion calculation

1.20E-05

Li-6 has large cross section of (n,T) reaction that absorbs
neutrons and emits H-3. Released H-3 is considered toaffect the operation of the reactor due to gaseous H-3 asvoid reactivity and the released H-3 chemically reacts
with chlorine to form hydrogen chloride. Hydrogenchloride may corrode the reactor vessel.
Cl-35 absorbs neutrons and Cl-36 is produced. Cl-36 hasa long half-life that is approximately 301,000 years. The
period at the time of landfill disposal may be long. Fig. 4
shows (n,γ) cross section of Cl-35 and Cl-37, it can beseen that the neutron absorption cross section of Cl-35 islarger than that of Cl-37. Therefore, it is necessary toconsider the concentration of Li-6 and Cl-35.
Depletion calculations are performed with TRITON.
The operation cycle is set as follows: the operation period
is about 10 months, and the inspection period is 2 months,
and calculated 4 cycles. It shows the cycle in Table Ⅲ.

300
300
300
300

60
60
60
60

Atoms density (atoms/cm3)

8.00E-06
6.00E-06
4.00E-06
2.00E-06
0.00E+00

0

200 400 600 800 1000 1200 1400 1600
Time (day)Fig.5. Produced Light elements by depletion.
2.2.2 Effect of concentration of Li-6

Table Ⅲ. Cycle condition of depletion calculation
Operation
Inspection
Burnup
Cycle
Time
time
(MWd/tHM)
(day)
(day)
1
2
3
4

H-1
He-4
H-3
Ar-39
K-40
Cl-36

1.00E-05

The concentration of Li-6 is changed to 0.01, 0.1, 1.0,
2.0, 4.0, 6.0, 7.59%, and calculated the influence of
depletion of Li-6. The concentration of Cl-35 is naturalabundance in this calculation.
Fig. 6 shows changes of the K-eff by concentration of
Li-6. There is no significant change in K-eff at 0.01, and0.1%. However there is a difference of K-eff about 0.044
between concentration of 0.1% and natural abundance. In
each cases, K-eff decreases approximately 0.042 at theend of 4 cycles due to depletion.
Fig. 7 shows the result of changes of mass of H-3 byconcentration of Li-6. It can be seen that as theconcentration of Li-6 decreases, the production rate ofH-3 decreases.

5595
11813
18030
24247

1.390
1.380
1.370
1.360
1.350
K-eff

Fig. 4. (n,γ) cross section of Cl-35 and Cl-37(Cl-35 is redcolor, Cl-37 is blue color)
2.2.1 Depletion calculation of natural abundance of Li
and Cl

1.340
1.330
1.320
Li-6 0.01%
Li-6 0.1%
Li-6 1.0%
Li-6 2.0%
Li-6 4.0%
Li-6 6.0%
Li-6 7.59%(Natural)

1.310

Fig 5 shows produced Light elements by depletion. H-1,
He-4, H-3, Ar-39, K-40 and Cl-36 are produced bydepletion and activation reaction. H-1 is produced byproton emission reaction such as (n,p) reaction. He-4 isconsidered to be emitted of alpha decay of Pu and FP. Ar39 is considered to be a product of (n,p) reaction of K-39.
In the light elements, hydrogen is the most abundant, so
there is a concern that it forms hydrogen chloride in thereactor and accelerates the corrosion of the structural
material.

1.300
1.290
1.280

0

200

400

600
800
Time(day)

1000

1200

1400

Fig. 6. K-eff by different concentration of Li-6
(Concentration of Cl-35 is natural abundance).

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1.00E-04

2.50E-05

Li-6 7.59%(Natural)
Li-6 6.0%
Li-6 4.0%
Li-6 2.0%
Li-6 1.0%
Li-6 0.1%
Li-6 0.01%

1.50E-05

8.00E-05
Mass of Cl-36 (g/cm3)

Mass of H-3 (g/cm3)

2.00E-05

1.00E-05

0

0

200 400 600 800 1000 1200 1400 1600
Time (day)Fig.9. Mass of Cl-36 by different concentration of Cl-35.

2.2.3 Effect of concentration of Cl-35

3. Conclusions

In this simulation, the concentration of Cl-35 is changed
to 0.01, 0.1, 1.0, 10, 20, 40, 60, 75.76% and calculated theinfluence of depletion of Cl-35. The concentration of Li6 is natural abundance in this calculation. Fig. 8 showschanges of the K-eff by concentration of Cl-35. There isno significant change in K-eff at 0.01 and 0.1%.
Comparing the case of 0.1% with the case of 75.76%,
there is a difference of K-eff about 0.086. In each cases,
K-eff decreases approximately 0.043 at the end of 4 cycles
due to fuel depletion.
Fig. 9 shows the result of changes of mass of Cl-36 bydifferent concentration of Cl-35. It can be seen that the
produced amount of Cl-36 is almost the same value fromthe result of case of 0.01% to 1.0 %. From Fig. 9 it appears
that there is not a large difference in the production of Cl36, going from natural chlorine to enriched chlorine.
However, the long-term effect on the radioactive waste
due to Cl-36 has not yet been investigated.

It was found that effect of the fuel fraction andconcentration of isotope in nuclear fuel at molten salt fast
reactor with chloride salt.
In order to keep the value of K-eff above 1.00, it isnecessary to keep the fuel fraction 15mol% or more, andinfinite thickness of the reflector was found to be 40cm.
It was confirmed that the negative reactivity was added by
the increase of the chloride fuel molten salt temperature.
Temperature reactivity coefficient is approximately-16×10-5 [Δk/k/℃].
In depletion calculation, noble gases such as He and Ar
are released by depletion and radioactivation. They canalso affect the nuclear characteristics as void reactivity.
Hydrogen and tritium react with chloride in the reactor.
And it forms hydrogen chloride, which may corrode
structural materials.
The changes in the concentration of Li-6 and Cl-35 havelittle effects on the amount of produced H-3 and Cl-36
during operation in 4 years. However, the concentrationsof Li-6 and Cl-35 have a significant influence for K-eff,and need to be considered sufficiently.

1.430
1.410

References

1.390
1.370

1. B. T. Rearden and M. A. Jessee, Eds., SCALE Code
System, ORNL/TM-2005/39, Version 6.2.3,Oak
Ridge National Laboratory, Oak Ridge, Tennessee(2018)
2. Wentao Zhou, “Thermodynamic evaluation of LiClKCl-PuCl3 system”, Journal of Alloys andCompounds 695 (2017) 2306-2313.
3. Robert D. Mariani, “Modeled salt density for nuclear
material estimation in the treatment of spent nuclear
fuel”, Journal of Nuclear Materials 404 (2010) 25-32.

1.350
K-eff

4.00E-05

0.00E+00

200 400 600 800 1000 1200 1400 1600
Time (day)Fig.7. Mass of H-3 by different concentration of Li-6.

1.330
1.310

Cl-35 0.01%
Cl-35 0.1%
Cl-35 1.0%
Cl-35 10.0%
Cl-35 20.0%
Cl-35 40.0%
Cl-35 60.0%
Cl-35 75.76%(Natural)

1.290
1.270
1.250
1.230

6.00E-05

2.00E-05

5.00E-06

0.00E+00

Cl-35 75.76%(Natural)
Cl-35 60.0%
Cl-35 40.0%
Cl-35 20.0%
Cl-35 10.0%
Cl-35 1.0%
Cl-35 0.1%
Cl-35 0.01%

0

200

400

600

800

1000 1200 1400

Time(day)Fig. 8. The K-eff by different concentration of Cl-35(Concentration of Li-6 is natural abundance).

115

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Monte Carlo Based Analysis for CANDLE Burning Reactor
Hoang Hai Nguyen, Jun Nishiyama and Toru Obara
Laboratory for Advanced Nuclear Energy, Institute of Innovative Research, Tokyo Institute of Technology, 2-12-1 N119, Ookayama, Meguro-ku, Tokyo, 152-8550 Japan*
Corresponding author: tobara@lane.iir.titech.ac.jp

Abstract
This study presents a Monte Carlo based analysis for CANDLE burning reactor.
In this procedure, neutronic calculation was performed by MVP code withJENDL-4.0 library and burnup calculation was performed by MVP-BURN codewith the detailed burnup chain. To simulate the fuel movement process, anauxiliary program was developed by Python language. To demonstrate thefeasibility of this procedure, a trial calculation was performed for a reference core
design. The results showed that this Monte Carlo based procedure was developed
successfully.
Key Words: CANDLE, Monte Carlo, MVP, MVP-BURN
process, an auxiliary code was developed by Python
language.
By repeating this procedure, the core equilibrium
condition can be achieved based on the convergence ofthe effective multiplication factor keff, neutron flux, power
density and nuclide density distribution.

1. Introduction
CANDLE (Constant Axial Shape of Neutron Flux,
Nuclide Densities and Power Profile During Life ofEnergy Production) burning concept was proposed by
Sekimoto et al. [1]. Most of the studies of CANDLE
reactor were performed by deterministic method. The
most advantage of deterministic method is computational
time. However, in deterministic method, several
approximations must be used to obtained the groupconstants for several burnup stages. Therefore, it isdifficult to improve the accuracy of results. On the other
hand, in Monte Carlo method, the accuracy of results canbe improved by increasing the number of neutron
histories.
The purpose of this study is to develop a procedure to
analyze the CANDLE burning by Monte Carlo based
method to overcome the limitation of deterministic
method.

2.2 Reference core design
To demonstrate the feasibility of this procedure, a trial
calculation was performed for a reference core design. In
fuel pin design, the innermost layer is the cylindricalshaped fuel pellet. In this study, metallic uranium with thezirconium content of 10 % is used as fuel. The next layer
in lead bismuth eutectic (LBE) bond. The ODS cladding
layer covers the fuel and bond. The outermost layer is
LBE coolant with hexagonal shape.
Table I. Core Design Parameters
Design parameters
Value
Thermal power [MWt]200Core height [cm]220Core radius [cm]145
Reflector thickness [cm]50
Fuel compositionU – 10 wt% Zr
Bond material
LBE
Cladding material
ODS
Coolant material
LBE
Reflector material
LBE
Fuel pin pitch [cm]1.08
Cladding thickness [cm]0.06
Bond thickness [cm]0.06
Fuel pellet radius [cm]0.39

2. Methodology
2.1 Procedure of CANDLE burning analysis by Monte
Carlo method
The procedure of CANDLE burning analysis consists ofthe following steps. Firstly, the neutron transport
calculation is performed by MVP code [2] with JENDL4.0 nuclear data library. The microscopic reaction rates,
obtained by neutron transport calculation, are transferred
into MVP-BURN code [3] to solve the burnup calculation.
After one fuel movement cycle (FMC), the fuel shufflingprocess is applied. In this process, spent fuel zones are
removed from the core top and fresh fuel zones are loaded
into core bottom. After this process, the new axial nuclide
density distribution is established. To stimulate thisThe core design (figure 1 and table I) consists of 127 fuel116Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019assemblies and each assembly is made from 397 fuel pins.
In axial direction, the core is divided into 55 zones. In theinitial core, the first 12 zones from the top to bottom areloaded with enriched uranium and the other zones areloaded with natural uranium.

a) Beginning of cycle

Fig. 1. Reference core layoutb) Middle of cycle

3. Results
To make the maximum uncertainty of calculated results
less than 0.01%, 100,000 neutron histories per batch and150 calculated batches were decided. In the burnup
calculation, several calculations were performed withdifferent burnup steps. These results show that with theburnup step less than 1250 days, there is no significant
difference between results. Therefore, 1250 days was
decided as the proper burnup step.

FMC-25
FMC-28

FMC-26
FMC-29Neutron flux
(neutrons/source/volume)

2.E-01c) End of cycle

FMC-27

Fig. 3. Neutron flux profile in one fuel movement cycle.
In equilibrium condition, the keff at the beginning of cycle
and at the end of cycle are 1.0172 and 1.0176, respectively.
Figure 2 shows the axial neutron flux profile for thedifferent FMCs. It shows that the flux profile is converged
to the equilibrium condition.
Figure 3 shows the flux profile from the beginning to
the end of one FMC. In radial direction, the neutron flux
at the core center reaches the highest value and decreases
to core edge. In axial direction, the highest region ofneutron flux is around the middle height. From the
beginning to the end of one FMC, the flux moves fromthe top to bottom without changing the shape.
With the convergence of neutronic characteristics,
such as neutron flux, to the equilibrium condition and the

2.E-01
1.E-01
5.E-020.E+00040

80 120 160
Axial position (cm)200

Fig. 2. Axial neutron flux distribution at each fuelmovement cycle.

117Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

movement of neutron flux from the top to the bottom of
the core without changing the shape, it can be concluded
that the CANDLE burning strategy has been developed
successfully.
It is difficult to make the exactly similar geometry forcomparison between deterministic method and Monte
Carlo method. In deterministic method, the reactor core
was simplified as cylindrical geometry. On the other hand,in Monte Carlo method, the hexagonal fuel pins werearranged in the hexagonal lattice to form the fuelassembly and the fuel assemblies were arranged in the
hexagonal lattice to establish the reactor core
However, the accuracy of Monte Carlo method has been
widely accepted. Therefore, with the enough number ofneutron histories, it can be expected that Monte Carlo
method provide the better accuracy of results. In addition,
for further detailed analysis, instead of deterministic
method, Monte Carlo method can provide more
information, such as the neutron leakage from the core.
4. Conclusions
In this study, a procedure for CANDLE burning analysis
based on the Monte Carlo method was proposed to obtain
the high accuracy of result. In this procedure, MVP codeand MVP- BURN code were applied for neutron transport
calculation and burnup calculation. To simulate the fuelmovement process, an auxiliary code was developed by
Python language. The results show that by repeating thefuel movement process, the equilibrium condition wasachieved successfully. This means this Monte Carlo based
procedure has been developed successfully.

References
1. SEKIMOTO H, RYU K, and YOSHIMURA Y,“CANDLE: The New Burnup Strategy”, Nuclear
Science and Engineering, 139, p. 306 (2001).
2. NAGAYA Y et al., “MVP/GMVP II: General
Purpose Monte Carlo Codes for Neutron and Photon
Transport Calculations Based on Continuous Energy
and Multigroup Methods”, JAERI 1348, Japan
Atomic Energy Research Institute (2005).
3. OKUMURA K et al., “MVP-BURN: Burn-up
Calculation Code Using a Continuous-energy Monte
Carlo Code MVP”, Draft Report for JAEAData/Code, Japan Atomic Energy Agency (2005).

118

TS-03

Core Analysis Methods

No.

RPHA19-1067

Title

Nuclear Waste Inventory Calculations from Fuel Cycle with Fast
Reactors by a Reactor Physics Code System CBZ

Authors

Yu Kobayashi, Go Chiba (Hokkaido University, Japan)
123

No.

RPHA19-1059

Title

Verification of FRBurner Module of CBZ Code System Based on
OECD/NEA Benchmark Report

Authors

Junshuang Fan, Go Chiba (Hokkaido University, Japan)
127

No.

RPHA19-1092

Title

Development of a Steady-State Core Analysis Code System for
VVERs

Authors

Lulin Yu, Guohua Chen, Jiadong Bao, Lingfeng Wei (NuStar Nuclear
Power Technology Co., Ltd., China), Zejun Liao (Nuclear PowerOperations Research Institute, China), Shaohong Zhang (NuStar Nuclear
Power Technology Co., Ltd., China)
131

No.

RPHA19-1012

Title

BEAVRS Benchmark Evaluations with Studsvik CMS5 CodePackage

Authors

Tamer Bahadir (Studsvik Scandpower, Inc., United States), Masatoshi
Yamasaki (Studsvik Japan, Ltd., Japan)
135

No.

RPHA19-1008

Title

An Improved Chebyshev Rational Approximation Method Baseon Order Reduction of Krylov Subspace Method

Authors

Yuying Hu, Qi Luo, Dong Yao, Yingrui Yu, Hongkuan Liao, Bingyan Zhou
(Nuclear Power Institute of China, China)
139

No.

RPHA19-1069

Title

Improvement of Optimally-Weighted Predictor-Corrector
Method for Nuclear Fuel Burnup Calculations

Authors

Jumpei Sasuga, Go Chiba (Hokkaido University, Japan), Yasunori Ohoka,
Kento Yamamoto, Hiroaki Nagano (Nuclear Fuel Industries, Ltd., Japan)
143

119

No.

RPHA19-1075

Title

Robustness of the GPS Functions in Pinwise Neutronics
Analysis of PWRs

Authors

Hwanyeal Yu, Yonghee Kim (Korea Advanced Institute of Science andTechnology, Korea)
147

No.

RPHA19-1101

Title

APEC-corrected NEM Analysis of the VERA Core

Authors

Seongdong Jang (Korea Advanced Institute of Science and Technology,
Korea), Kyunghoon Lee (Korea Atomic Energy Research Institute, Korea),
Yonghee Kim (Korea Advanced Institute of Science and Technology,
Korea)
151

No.

RPHA19-1064

Title

Discontinuity Factor; A Discontinuity Condition for Angular
Flux?

Authors

Akio Yamamoto, Tomohiro Endo (Nagoya University, Japan)
155

No.

RPHA19-1013

Title

Calculation of Higher Eigen-modes of the Forward and Adjoint
Neutron Diffusion Equations Using IRAM Algorithm Based on
Domain Decomposition

Authors

Wenbin Wu, Yingrui Yu, Qi Luo, Dong Yao, Qing Li, Xiaoming Chai
(Nuclear Power Institute of China, China)
158

No.

RPHA19-1070

Title

Performance of the RSE (Resonance calculation using energy
Spectral Expansion) Method for Heterogeneous Pin-cell
Geometry

Authors

Ryoichi Kondo, Tomohiro Endo, Akio Yamamoto (Nagoya University,
Japan), Satoshi Takeda (Osaka University, Japan), Hiroki Koike, Kazuya
Yamaji, Daisuke Sato (Mitsubishi Heavy Industries, Ltd., Japan)
162

120

No.

RPHA19-1020

Title

Application of Singular Value Decomposition and Low Rank
Approximation for Compression of Macroscopic andMicroscopic Cross Section Table for Core Calculations

Authors

Masato Yamamoto, Tomohiro Endo, Akio Yamamoto (Nagoya University,
Japan)
166

No.

RPHA19-1044

Title

Compression of Multi-Physics Simulation Output Data Using
Principle Component Analysis

Authors

Alexey Cherezov, Jinsu Park, Hanjoo Kim (Ulsan National Institute of
Science and Technology, Korea), Nur Syazwani Mohd Ali (University ofTechnology, Malaysia, Malaysia), Deokjung Lee (Ulsan National Instituteof Science and Technology, Korea)
170

No.

RPHA19-1103

Title

Refinement of Convolutional Neural Network for Neutronic
Design Parameter Prediction of a Loading Pattern

Authors

Hyunbin Jang (Pusan National University, Korea), Ho Cheol Shin (Korea
Hydro and Nuclear Power Central Research Institute, Korea), Hyun Chul
Lee (Pusan National University, Korea)
175

No.

RPHA19-1048

Title

Experimental Validation for the Pin-wise Isotope Prediction
Methodology

Authors

Minjae Lee, Taeyang Noh, Myung Hyun Kim (Kyung Hee University,
Korea)
179

No.

RPHA19-1036

Title

The
NeutronStreaming
Correction
MethodHomogenization of Pebble-bed HTGR Reflector

Authors

Yutong Wen, Ding She, Lei Shi, Jing Zhao (Tsinghua University, China)

in

the

183

121

No.

RPHA19-1039

Title

Application of Neutron Balance Analysis in Enhancing theTemperature Reactivity Feedback of Gas-cooled Fast Reactor

Authors

Ce Zhang, Youqi Zheng, Linfang Wei ( Xi’an Jiaotong University , China)
187

No.

RPHA19-1040

Title

Innovative Design Concepts of Burnable Poison Rods for PWR

Authors

Aiman Dandi, Myung Hyun Kim (Kyung Hee University, Korea)
191

No.

RPHA19-1087

Title

Thermal Hydraulic Analysis for a Breed and Burn Core with
Spiral Fuel Shuffling

Authors

Kazuki Kuwagaki, Jun Nishiyama, Toru Obara (Tokyo Insutitute ofTechnology, Japan)

No.

RPHA19-1017

Title

Research on Core Design of Pebble Bed Advanced High
Temperature Reactor

Authors

Wang Lianjie, Sun Wei, Xia Bangyang (Nuclear Power Institute of China,
China), Zou Yang, Yan Rui (Chinese Academy of Sciences, China)
196

122

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Nuclear Waste Inventory Calculations from Fuel Cycle with Fast Reactors
by a Reactor Physics Code System CBZYu Kobayashia.*, Go Chibaa
Hokkaido University, Kita-13 Nishi-8, Sapporo, Japan*
Corresponding author: esaking5884@eis.hokudai.ac.jpaAbstract
We investigate the effect of minor actinide recycling on nuclear wastes from fast
reactors by using an updated version of a FRBurner module developed for fast
reactor whole-core burn-up calculations in a code system CBZ.
The update of CBZ enables calculation of inventory of transuranic wastes
composed of hulls and end-pieces, so calculations of decay heat of high-level
wastes and transuranic wastes become possible. Decay heat of cobalt-60 fromhulls and end-pieces is as large as decay heat of fission products. It is turned outthat minor actinide recycling conditions and cooling time changes are notimportant for decay heat of transuranic wastes in comparison with that of highlevel wastes.
Key Words: fast reactor, fuel cycle, MA conversion, TRU wastes
1. Introduction

amount of wastes, but the reduction amount would be
dependent on these uncertainties. Therefore, we have toknow how much the fuel cycle with fast reactors including
uncertainties affects the amount of the nuclear wastes.
In order to carry out these calculations, we have
implemented a new capability into a FRBurner module ina reactor physics code system CBZ. In the present work,some calculations are conducted with various conditions
to demonstrate the effectiveness of the new capability. In
the present work, the number of recycled MA nuclides isvaried according to 4 cases for HLW first. All MA
nuclides are recycled in case 1 and fewer MA nuclides arerecycled in cases 2, 3 and 4 in sequence. Second,sensitivity study about recycling (separation) efficiencyof MA nuclides in spent fuel reprocessing is carried out.
Finally, the cooling time from the fuel discharging to
reprocessing is changed for TRU wastes calculations.
Four-, 10-, 25- and 50-year cooling are considered.

When we utilize nuclear energy, we cannot avoid
management of nuclear wastes. Nuclear wastes have to be
managed for hundreds or thousands of years because of
their decay heat (DH) and potential radiotoxicity. In orderto properly and efficiently manage nuclear wastes,
accurate evaluations of the inventory of radioactive
nuclides such as fission products (FP) and minor actinide(MA) nuclides are necessary.
Some MA nuclides continue to emit DH for long
period, so they increase the number of the waste units andrequired area of disposal sites because heat production of
a waste unit is restricted. In addition, MA nuclides havepotential radiotoxicity. From these reasons, reduction of
the amount of MA nuclides is important.
MA nuclides can be converted to short-lived
radioactive nuclides through burn-up in fast reactors.
Thus, the nuclear fuel cycle with fast reactors is expected
to reduce the area of disposal sites of the high-level wastes(HLW). However, the transuranic (TRU) wastes from thenuclear fuel cycle with fast reactors are generated but are
not often considered. In this paper, the wastes only fromhulls and end-pieces of fuel assemblies are called TRU
wastes. Hulls and end-pieces of fuel assemblies include
Co-59 as impurity and it is converted to the radioactive
nuclide, Co-60, by capturing neutron during reactor
operation [1]. The volume of TRU wastes is not small, so
it is necessary to consider TRU wastes also when
discussing about the nuclear wastes.
Moreover, the nuclear fuel cycle with fast rectors have
many uncertainties, such as cooling time before
reprocessing, operation time of fast reactors and so on.
The fuel cycle with fast reactors is expected to reduce the

2. Implementation of Nuclear Waste Inventory
Calculation Capability to CBZ/FRBurner
The FRBurner module has been developed to perform
whole-core burn-up calculations for fast reactors in CBZ.
On fuel assembly calculations, homogeneous or onedimensional heterogeneous model can be adopted, andassembly-homogeneous (or homogenized) multi-groupcross sections are used in subsequent whole-corecalculations based on the transport or diffusion theory. A
reactor core is modeled to the two-dimensional cylinder.
More detailed information about FRBuener and results of
its verification calculation are presented in a different
paper in this conference [2]. In the present study,
homogeneous assembly model and diffusion theory123Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

module are used.
Generally, in fast reactor burn-up calculations, fission
product nuclides are treated as simple pseudo nuclides,and explicit representations of important fission productnuclides in burn-up calculations are not introduced. On
inventory calculations in nuclear waste, one-point fuelburn-up codes like ORIGEN are generally utilized. This
procedure can roughly estimate inventory of nuclear
wastes with reasonable computation time, but highaccuracy cannot be expected because neutron flux energy
spectrum is significantly dependent on spatial position ina fast reactor core and it is quite difficult to define unique
one-group cross sections required for one-point burn-up
calculations. To overcome this difficulty, a detailed burnup chain consisting of 197 fission product nuclides canbe used in whole-core burn-up calculations in FRBurner.
In addition to that, activation calculations of Co-59
included in SUS cladding as an impurity material, which
is important to quantify DH in TRU wastes, are also
possible.
Whole-core burn-up calculations with FRBurner can
provide nuclides inventory information included indischarged fuels, and using these information, we cancalculate characteristics of HLW and TRU wastes. Decay
heat of HLW and TRU wastes can be calculated bycooling calculation capability of a Burner module of CBZ,
which uses rigorous burn-up chain model consisting of all
the fission product nuclides defined in JENDL fission
products yield data file-2011 (JENDL/FPY-2011) and Co60.

nuclides (Cm, Am and Np) remain in wastes in case 4.
3.1 Decay heat of high-level waste
The total DH in HLW of each case is shown in Fig. 1.
Reduction of the number of recycled MA nuclides
increases DH of the wastes. DH of cases 3 and 4 are same,and this means that the contribution of Np recycling to
reduction of wastes is negligible.
Component-wise DH of cases 1 and 2 are shown in
Figs. 2 and 3. It represents the effect of recycling of Cm.
In case 1, all MA nuclides are recycled, so DH of FP is
identical with the total DH of wastes. In case 2, DH ofheavy metals (HM) appears and be dominant around 300
years after reprocessing.
According to Fig. 3, at a time of reprocessing, DH of
FP is dominant, but some years later, contribution of HM
becomes large. Actually, the number of canisters is notaffected by DH of HM since it is determined by DH of FP,
but the interval of neighboring canisters at disposal site isdependent on DH of HM. That’s why MA nuclides
recycling reduce the area of disposal site.

3. Sample Calculation
The target reactor core is a large MOX-fuel high-internalconversion representative core provided in the final reporton phase 2 of feasibility study on commercialized fast
reactor cycle systems from Japan Atomic Energy Agency[3]. This is called JSFR-1500 and is a 1500 MWe sodiumcooled reactor. It’s designed to achieve long operation
term and high Pu generation ratio. This JSFR-1500 has
two different core concepts, the transition-phase core andthe equilibrium-phase core, and we consider the latter inthe present calculation.
Decay heats of HLW and TRU wastes are shown in
the following sub-sections.
In both calculations, burn-up calculations of JSFR1500 are carried out with FRBurner and nuclide number
densities of discharged fuel per one batch at an
equilibrium cycle are obtained. Then 4-year cooling
calculations are carried out with Burner, and nuclides
number densities in HLW and TRU wastes generated
from spent fuel reprocessing are calculated. Separation
efficiency of MA nuclides is assumed 100%; all therecycled MA nuclides are included to the nuclear fuel andno recycled MA remains in HLW wastes. Time-dependentradioactivity and DH after reprocessing are calculated.
In spent fuel reprocessing, four cases with different
conditions about recycled MA nuclides are considered; allMA nuclides are recycled in case 1, Cm remains in wastes
in case 2, Cm and Am remain in case 3 and All MA

Fig. 1. The total decay heat of HLW

Fig. 2. Component-wise decay heat of HLW in case 1

124

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Because HM remain in other cases when separation ratiois 100%, the contribution of 0.01% of HM remaining in
HLW is not significant. That’s why the change appears
only in case 1. Figure 6 shows DH of each nuclide ofcase 1 with 99.9% separation ratio. Decay heat of Cm and
Pu become larger than that of FP when 400 years passed
and soon DH of Pu becomes dominant.

Fig. 3. Component-wise decay heat of HLW in case 2
Figure 4 shows element-wise DH in case 2. According
to this figure, DH of Pu becomes larger than the others. It
means that if Cm remains in HLW, the contribution of Pu
becomes larger.

Fig. 6. Element-wise decay heat of HLW in case 1(separation ratio is 99.9%)
3.2 TRU Wastes
Since detailed information about reprocessing of spent
fuel from fast reactors is not available, it is assumed thatcondition of TRU waste generation in reprocessing in thepresent study is same as that of light water reactors. A few
FP and HM in spent fuel attach to TRU wastes, and theattachment rate of FP and HM are assumed 0.3% and0.2%, respectively [4]. Heavy metals are recycled under
the same conditions adopted in sub-section 3.1.
Decay heat of Co-60 from hulls and end-pieces has
become possible to be calculated by the updated
FRBurner module. Decay heat of Co-60 is large as wellas other FP and HM, so its contribution to volume ofwastes is large.
Figure 7 shows DH of four cases. Few differencesamong four cases exist. When the number of recycled MA
nuclides increases, the amount of MA nuclides in spent
fuel increases, so DH of case 1 is the largest.

Fig. 4. Element-wise decay heat of HLW in case2
Next, let’s see the effect of MA separation ratiochange in spent fuel reprocessing. Total DH of HLW withseparation rate of 99.9% is shown in Fig. 5. In thiscalculation, 0.1% of recycled MA is included in HLW.

Fig. 5. The total decay heat of HLW(separation ratio is 99.9%)
When separation ratio is 99.9%, only DH of case 1
increases from that with separation ratio of 100%.

Fig. 7. The total decay heat of TRU wastes

125Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Figure 8 shows the component-wise DH. At first, theDH of Co-60 is as large as DH of FP, and that’s why
considering Co-60 in TRU wastes are important. Whentens of years passed, the DH of Co-60 decreases rapidly
in comparison with the DH of FP.

core burn-up calculation in a code system CBZ. Fourcases with different reprocessing conditions are
considered for HLW and TRU wastes; no MA nuclides
remains, Cm remains, Cm and Am remain and all MA
nuclides remain in the HLW. By a comparison between
the above two cases, the effect of Cm recycling has
become clear. Curium recycling reduces DH around 100
years after reprocessing. If Cm remains in the wastes, thecontribution of Pu is large.
The effect of changing separation ratio has been
turned out. When separation ratio is not 100% but 99.9%,
the DH of the case that all MA nuclides are recycled has
increased to about ten times because FP do not initiallyexist in case 1, so the contribution of 0.1% of MA nuclides
is large.
Update of CBZ has enabled calculation of inventoryof TRU wastes. Decay heat of Co-60 from hulls and endpieces is as large as DH of FP. Thus, TRU wastes should
be considered when we use nuclear energy and manage
the wastes from it. Minor actinide nuclides recycling
conditions and cooling time changes are not important for
DH of TRU wastes by comparison to it of HLW.
A calculation of DH of HLW and TRU wastes havebecome possible now. As a future study, we will consider
impacts of uncertainties in various conditions on nuclear
waste from nuclear fuel cycle with fast reactors.

Fig. 8. Component-wise decay heat of TRU wastes incase 1
Finally, DH with various cooling time in case 2 isshown in Fig. 9. Here, cooling time means the time fromfuel discharging to reprocessing. Four cases are calculated:4-, 10-, 25- and 50-year cooling. The DH curves start at ayear after reprocessing.

Acknowledgements
Some of the results shown in the present paper are basedon the research program conducted by the Radioactive
Waste Management Funding and Research Center(RWMC), entitled “Study on the effects of advanced
nuclear fuel cycle technology to the geological disposal
concept”. The authors are grateful to Dr. Hidekazu
Asano and Dr. Masahiro Kawakubo of RWMC for theirsupports on the present work.
References
1.

Hirano F, et al., "Thermal impact of geologicaldisposal of hull and end piece wastes resulting fromhigh-burn-up operation of LWR and introduction of
MOX fuels into LWR,” J. Nucl. Sci. Technol., 46,443 (2009).
2. Fan J, Chiba G, “Verification of FRBurnermodule of
CBZ code system based on OECD/NEA benchmark
report,” this conference.
3. Ohki S, et al., “FBR core concepts in the “FaCT”
project in Japan,” Proc. PHYSOR 2008, Interlaken,
Switzerland, Sep. 14–19, 2008, (2008) (CD-ROM).
4. Hirano F, et al., “Burning of MOX fuels in LWRs; fuelhistory effects on thermal properties of hull and end
piece wastes and the repository performance” J. Nucl.
Sci. Technol., 49, 310 (2012).

Fig. 9. The total decay heat of TRU wastes in case 2 withdifferent cooling timeLong cooling time increases DH of HLW after the longperiod disposal because Pu-241 is converted to Am-241
before reprocessing. In contrast, there are few changes in
DH of TRU wastes because Pu-241 is not removed from
TRU wastes when reprocessing.
The above results suggest that recycling conditions of
MA nuclides and expanding cooling time are negligible
when considering TRU wastes
4. Conclusion
We have investigated the effect of MA nuclides
recycling on nuclear wastes from fast reactors by using
the FRBurner module developed for fast reactors whole-126

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Verification of FRBurner Module of CBZ Code System
Based on OECD/NEA Benchmark Report
Junshuang Fana,*, Go Chibaa
Hokkaido University, Kita 13, Nishi 8, Sapporo, Japan*
Corresponding author: fanjsh@eng.hokudai.ac.jpaAbstract
We verify the accuracy of our tools about sodium-cooled fast reactor
calculation in this work by comparing results with the OECD/NEA benchmark
report. Besides, we analyze the influences about transport theory, fuel assembly
heterogeneity, and PN-SN condition on transport calculation on global
parameters of core calculation. Maximum bias of keff, βeff, Δρvoid, and ΔρDoppler
are less than 0.80%, 3.2%, 8.9% and 9.0% respectively. Bias comparison
shows our tool could provide reasonable results on sodium-cooled fast reactor
cores with various types of fuel and sizes. Maximum value of transport theory
correction effect for all cores on keff, βeff, non-leakage and leakage componentof Δρvoid, Δρvoid, and ΔρDoppler are around than 1.21%, 0.2%, 0.7%, 7.2%, 8.1%
and 0.6% respectively. Transport correction effect indicates that it makes large
differences in keff and Δρvoid calculations. Maximum heterogeneous correction
effects of these parameters are around 0.67%, 0.6%, 10.1%, 0.6%, 13.4% and8.9% respectively, which indicates it has large influence on keff and reactivity
calculation. Independence between heterogeneous lattice model and transport
theory calculation is observed. Besides, comparison between P1S4 and P3S8
shows negligible differences. Particularly, relative large transport correction
effect occurs at keff and the leakage component of Δρvoid for MET-1000. Thisphenomenon is caused by a fact that neutron spectrum of the MET-1000 core isharder than the other cores. We also find transport theory correction effect willchange with core size, fuel type and the number of energy groups.
Key Words : fast reactor, burn-up calculation, verification, correction effect
1. Introductionkeff, effective delayed neutron fraction βeff, Δρvoid,
Doppler reactivity ΔρDoppler are compared withbenchmark report. Particularly, we put different
model/theory into account. We are able to quantify theseeffects consistently in our code system by doing thiswork.

We are going to develop our original fast reactor core
concept in the near future and new module of our code
system for fast reactor calculation has been developed.
Consequently we choose one benchmark problem of
OECD/NEA [1] as our reference to verify whether thisnew module could fit with various fuel types and coresizes or not.
This benchmark report provides four core concepts.
Two of them are large size cores at 3600 MWe(MOX-3600, CAR-3600) and 2 of them are middle sizecores at 1000 MWe (MOX-1000, MET-1000). Core
name indicates the fuel type of each core (“CAR” means
carbide fuel). Various institutions such as JAEA, ANL,
CEA and so on, participated in this benchmark work andgive their results based on different libraries, calculation
theory, model, and energy group structure. Thisbenchmark report indicates several points related to thiswork: (1) nuclear data library used has a large impact on
the effective neutron multiplication factor keff; (2) celland control rod models (heterogeneous vs.
homogeneous) employed for lattice and whole corecalculations also impact on keff, sodium void reactivity
Δρvoid, and control rod worth.

2. Numerical Tools and Methods
A FRBurner module has been developed to perform
whole-core burn-up calculations for fast reactors in CBZ,
which is a general-purpose reactor physics code systemdeveloped by our laboratory.
On fuel assembly calculations, homogeneous or
one-dimensional heterogeneous model can be adopted,
andassembly-homogeneous(or
homogenized)
multi-group cross sections are used in subsequent
whole-core calculations. In resonance self-shielding
calculations, a 280-group library generated from theevaluated nuclear data files is used in the present work.
A group structure of this 280-group library is generated
from the 70-group structure adopted in the JAERI fast
set-3 (JFS-3) [2], and each energy group in JFS-3 isdivided into four groups with equal lethargy width. For
heterogeneous model calculations, Tone’s method [3] is127Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019used to calculate background cross sections considering
heterogeneity effect. Neutron transport calculations inheterogeneous assembly model are carried out with thecollision probability method.
A reactor core is modeled to a two-dimensional
cylinder. Neutron multiplication factor and neutron flux
spatial and energy distributions are calculated by the
diffusion or transport theory. In diffusion theory
calculations, a finite volume method-based module,
PLOS, is used. In transport theory calculations, a SNRZ
module based on the diamond-differencing spatial
discretization
anddiscrete-ordinate
angulardiscretization schemes is used. In SNRZ calculations,arbitrary number of the Legendre expansions foranisotropic scattering cross sections can be used. In
nuclear fuel burn-up calculations, fission productnuclides are treated as simple pseudo nuclides. Sodium
void and Doppler reactivity calculations and effective
delayed neutron fraction calculations can be carried out
also by the diffusion or transport theory, and reactivity
is calculated by the perturbation theory.
In this benchmark, nuclides number density
information at the beginning of cycle at the equilibrium
state are provided for each of the four cores. With thesenumber densities at BOC, users of this benchmark
problem should perform nuclear fuel burn-up
calculations to obtain number densities at the end ofcycle (EOC). Hence comparisons at EOC should also
include burn-up effects.

Fig. 1-b. Biases between CBZ and reference at EOC
In the CBZ calculations, we use JEFF-3.1.1-based
library , the same as CEA-1. In Fig. 1-a and Fig. 1-b,
biases on keff is multiplied by 10 for clear comparison.
For MET-1000, maximum biases of theseparameters are around 0.52%, -3.2%, -8.8% and -6.5%
respectively. Bias of EOC is also consistent with that of
BOC.
For MOX-1000, maximum biases of theseparameters are around 0.80%, -3.0%, -6.0% and -8.9%
respectively. Bias of EOC is also consistent with that of
BOC.
For MOX-3600, maximum biases of theseparameters are around 0.32%, -2.5%, -8.9% and -5.5%
respectively. Comparison of βeff at EOC is not available
because this reference has no data of βeff at EOC.
For CAR-3600, maximum biases of CBZ arearound 0.40%, -1.9%, -2.4% and -8.6%. Consistency
between BOC and EOC is also obtained.
The above comparisons show us that CBZ could
give reasonable results about burn-up calculations of
sodium-cooled fast reactor with various types of fuel(metallic, MOX, carbide fuels) and different sizes. The
maximum biases of our tool are less than 0.8% in keff,
4% in βeff, 10% in two reactivities. The remaining bias
would come from our 2-dimensional model in wholecore calculation procedure whereas the reference applies
3-dimensional model.

3. Bias Comparison
Different reference values are provided by the
benchmark report but only four departments give resultsof all the fore cores result. To keep the consistency ofreference value we choose results provided as CEA-1(heterogeneous model, transport theory,
JEFF-3.1.1-based library) as our reference. Figure 1-a
and Figure 1-b show the biases between CBZ systemand the references under heterogeneous lattice modelwith transport theory (P3-S8), which are consistentconditions with references.

4. Correction Effect
After the verification work is finished, we figure out theeffect of heterogeneous lattice model and transport
theory which are compared with homogeneous latticemodel and diffusion theory.
Transport theory effect is compared underhomogeneous and heterogeneous lattice models, as
shown in Figure 2. Heterogeneous lattice model effect iscompared under diffusion and transport theory (P3-S8),
as shown in Figure 3. Effects on keff is multiplied by 10
for clear comparison for the both figures. Figure 2 andFigure 3 only show two correction effects at BOC
because the effects at EOC are almost the same as BOC.
In all figures, Δρ(N-Lk) and Δρ(Lk) mean non-leakage

Fig. 1-a. Biases between CBZ and reference at BOC

128

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

component and leakage component of sodium void
reactivity.

calculation.
Next, heterogeneous correction effect shows
relatively large influence on non-leakage component ofΔρvoid and ΔρDoppler. The maximum value on non-leakage
component is around 10% which is indicated in the
MOX-3600 case, and around 8.6% on Doppler reactivity
for all the cores. The reason of large negative changes
on non-leakage component of Δρvoid is yield component
increasing with the decreasing in absorption andscattering components. Among them, yield component
is negative effect, absorption and scattering components
are positive effect. The dominant reason for negative
reactivity increasing of yield component is homogenizedmacroscopic fission cross section decreasing. This isbecause in heterogeneous model, neutron flux leveloutside fuel region increases and the weight of neutron
flux inside fuel region decreases when sodium void
occurs, and macroscopic cross section is calculated bythis weight of neutron. Finally, the net effect on
non-leakage component of Δρvoid is negative.

Fig. 2. Transport theory correction effect at BOC
Firstly, as shown in Figure 2, transport theory causes
large increasing on keff, about 0.5%~1.2%. Reason ofthis increasing on keff is that diffusion theory
overestimates neutron leakage. As a consequence,transport theory gives larger value of keff.
There almost no influences on βeff and ΔρDoppler.
As for Δρvoid, we decompose it into non-leakage
component and leakage components. Non-leakage
component includes yield, absorption, scattering and(n,2n) fractions. We can find that transport theory leads
to a slight increasing on non-leakage component, but it
has large negative effect on leakage component. The
reason of negative effect on Δρ(N-Lk) is that neutron
leakage is overestimated with diffusion theory.
Consequently, transport theory corrects this error andleakage component of sodium void reactivity will
decrease with transport theory calculation.
Particularly, we could easily find that correction
effect on keff and leakage component for the MET-1000
core is larger than the other cores. This is because
metallic fuel core has harder neutron spectrum than theothers. Another interesting thing is that we could
observe obvious difference between large size core andmiddle size core. This phenomenon tells us transport
theory calculation is important for small size systems
because small size systems have larger neutron leakage.
Here we have to emphasize one important thing:transport theory correction effect will change withenergy structure, i.e., transport theory effect incoarse-energy group and that in fine-energy group are
different with each other, especially for keff. Thus resultof Figure 2 is only suitable for 280-group libraryFig. 3. Heterogeneous lattice model correction effect at
BOC
The reason of large increasing on ΔρDoppler inheterogeneous model is because the background crosssection in heterogeneous model is smaller than it inhomogeneous model. Since Doppler reactivity is mostly
caused by U-238 resonance absorption and thisabsorption is related to background cross section.
Regarding keff, 0.47%~0.64% differences caused byheterogeneous effect is found. This is because energy
self-shielding and spatial self-shielding effects. On thecontrary, heterogeneous effect is not considerable for βeff.
Maximum value is around 0.7% which occurs in

129

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

MOX-3600 case.
Thirdly, we can easily observe that both effects show
independence, i.e., transport effect under homogeneousmodel is consistent with it under heterogeneous model;
heterogeneous effect under diffusion theory is consistent
with it under transport theory.

all four cores on keff, βeff, non-leakage Δρvoid, leakage
Δρvoid, Δρvoid, and ΔρDoppler are around 1.21%, 0.2%,
0.7%, 7.2%, 8.1% and 0.6% respectively. This result
indicates that it makes large difference on keff and Δρvoid.
Maximum value of heterogeneous correction effect for
all four cores on keff, βeff, non-leakage Δρvoid, leakage
Δρvoid, Δρvoid, and ΔρDoppler are around 0.67%, 0.6%,
10.1%, 0.6%, 13.4% and 8.9% respectively, and thisresult indicates it has large influence on keff and Δρvoid
calculation. Besides, we have found that these two
effects are independent with each other. Furthermore,
we have observed transport theory correction effectrelates to core size and fuel type.
Thirdly, PN-SN effect has been quantified, which is
not estimated by benchmark report. Maximum
difference between P3-S8 and P1-S4 of all four cores on
keff, βeff, Δρvoid, ΔρDoppler are around -0.12%, -0.02%,
0.32% and -0.1% respectively. The PN-SN effect shows
negligible differences which means P1-S4 is better thanP3-S8 due to P3-S8 is time-consuming.
Consequently, these three comparisons indicates weshould apply heterogeneous lattice model with transport
theory (P1-S4) calculation in our future work with CBZcode system.

5. PN-SN Effect
In transport theory we use PN order and SN order tocontrol the degree of approximation of transport theory
calculation. PN is the maximum order of Legendre
polynomials for anisotropic scattering cross section
expansion. SN is the order of discrete points in angle. In
the present work, we use the level symmetric quadraturesets.
We compared the differences between P1-S4 andP3-S8, and the result is shown in Figure 4. Differencesof keff are multiplied by 10 for clear comparison. Figure
4 only shows PN-SN effect at BOC because it is almost
the same as EOC.
We could find that P3-S8 makes little difference toP1-S4. This means P1-S4 is enough for transport theory
calculation because P3-S8 is quite time-consuming.

References
1. “Benchmark for Neutronic Analysis of Sodium
cooled Fast Reactor Cores with Various Fuel
Types and Core Sizes”, NEA/NSC/R(2015)9, OECD
Nuclear Energy Agency (2016).
1. Takano H, Ishiguro Y, “Production and Benchmark
Tests of Fast Reactor Group Constant Set JFS-3-J2,”
JAERI-M 82-135, Japan Atomic Energy ResearchInstitute (1978).
2. Tone T, “A Numerical Study of Heterogeneity
Effects in Fast Reactor Critical Assemblies,” J. Nucl.
Sci. Technol., 12, 467 (1975).

Fig. 4. PN-SN effect at BOC
6.

Conclusion

In this work, we have verified the accuracy of our tools
about fast reactor burn-up calculation. Totally, bias of all
four cores for keff, βeff, Δρvoid, ΔρDoppler are less than
0.80%, 3.2%, 8.9% and 9.0% respectively. This means
new module for CBZ could provide acceptable result on
sodium-cooled fast reactor with various types of fueland sizes.
Secondly, we have quantified transport andheterogeneous correction effects for each core.
Maximum value of transport theory correction effect for130

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka,Japan,Dec. 2-3, 2019

Development of aSteady-State Core Analysis Code System for VVER
Lulin Yua,*, Guohua Chena, Jiadong Baoa, Lingfeng Weia, Zejun Liaob, Shanghong Zhangaa

NuStar Nuclear Power Technology Co., Ltd., Suite 1101,1698Yishan Road, Shanghai, China.
b
Nuclear Power Operations Research Institute, Hangzhou, Zhejiang, China*

Corresponding author:lulinyu@nustarnuclear.com

Abstract
Recently, a variant of NuStar's core analysis code system for western type PWR,
has been developed in NuStar to satisfy the need of VVER applications. It is
composed of NuStar's latest version of lattice code, ROBIN2, the variant of
NuStar's 3D core code for hexagonal geometry, EGRET-H, and othercomponents for cross section representation, high productivity and good user
experience. This paper outlines the technical features of this new code system
ORIENT-H, especially the features that are introduced recently. Verification andvalidation results of the code system are also briefly introduced.
Key Words: Core Analysis, VVER, ORIENT-H, Codearea of reactor physics method and computation but also
existing in the area of computer science and software
engineering. It is a fruit of joint efforts by NuStar's
reactor physicists and computer software engineers. The
neutronics kernel of the system is composed of a 2D

1. Introduction
As the major vendor-independent core analysis codedeveloper and provider in mainland China, NuStar has
been dedicating itself to satisfy the need coming fromthe ever-growing domestic nuclear power industry.
Currently, NuStar's ROBIN/EGRET code system[1,2],
which was designed for western type PWR, is being
adopted by 11 operating units for reactor core operation
support, independent reload design audit and advanced
reactor physics test applications. To further serve thedomestic clients, including utilities who are operating
the Russian type VVER units, NuStar is independently
developing a code system for VVER, since so far there is
no domestically-developed vendor-independent code
applicable to the operating 4 VVER-1000 units and theother 2 VVER-1200 units expected to be in operation inyear 2026 and 2027.
Considering
the
fact
that
the
original
ROBIN/EGRET code system had been extensively
verified and validated before it was licensed for
production use, and its performance has been repeatedly
recognized during its about 50 reactor-years of on-site
application,
theauthors
adopt
as
many
modules/subroutines as possible from it whendeveloping its variant for VVER. Therefore, in this paper,
only these technical features/functions that are newly
developed for VVER applications are emphasized.
Section 2 outlines the key component codes for thesystem, while Section 3 introduces the verification andvalidation work so far has been performed and presents
the preliminary results. Section 4 concludes the paper.

lattice code ROBIN2 and a 3D core code EGRET-H.
2.1 Lattice code ROBIN2
ROBIN2 is the latest version of NuStar’s lattice code.
Compared with its predecessor, one of the noticeable
improvements of ROBIN2 is its applicability tocomplicated lattice geometries. In previous generation of
ROBIN, there are only two types of geometry elements(i.e the box and the pin cell geometry) pre-defined in thecodes, which limits the codes only applicable to typicalwestern type PWR geometries and a small portion of
BWR geometries. While for ROBIN2, since the idea ofpre-defined geometry element is abandoned and the Rfunction method[3] is employed instead, its geometry
applicability to complex fuel lattices significantly
enhanced. As an example, Fig.1 illustrates three types oflattice geometries that ROBIN2 is applicable to.

Fig.1 ROBIN2 applies both to regular and irregularlattice geometries.
Besides the above-mention enhancement in geometry
applicability, there are also other enhancements adopted
in ROBIN2 either to further improve the code's
computational efficiency for transport calculation or to

2. Component Code Development
The ORIENT-H system is developed by extensively
employing today's technology not only existing in the131Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka,Japan,Dec. 2-3, 2019improve its overall performance for industrial
applications. For instance, while one set of space andangle discretization parameters are adopted for all theenergy groups to perform MOC transport calculations inprevious generation of ROBIN, discretization parameterschange in ROBIN2 for different energy ranges as theneutron mean free path changes noticeably between fastneutron and thermal neutron.
In short, the main functionalities and features of
ROBIN2 can be summarized as below:
1) Adopting R-function method to handle irregular
geometry.
2) Rational approximation coupled with enhanced
neutron current method[4] as its main resonancecalculation method.
3) Pre-tabulated resonance interference factor table toaccount for the resonance interference effects
among multiple resonance nuclides.
4) MOC [5,6] for transport calculation.
5) Assembly-based modular ray generation[7] andcyclic ray tracing for both square and hexagonal
geometry.
6) Cell-based CMFD acceleration for both regular andirregular lattice geometry.
7) Smart azimuthal angle discretization scheme[8].
8) Different space and angle discretization scheme fordifferent energy ranges.
9) Linear rate (LR) depletion method for normal fuelpins.
10) Logarithmic linear reaction rate (LLR) depletion
method [9] for gadolinium-bearing fuel pins.

Fig. 2 Conformal mapping of a hexagon to arectangle.
By applying this mapping, the original two-dimensionalgroup-wise neutron diffusion equation defined incomplex z-plane (as given in below),22
 D ( 2  2 ) ( x, y )   r ( x, y )  Q ( x, y ) , (2)xyis now defined in complex w-plane and has a new form
as follows,2 2
D( 2  2 )(u, v) r g2 (u, v)(u, v)  g2 (u, v)Q (u, v) . (3)u vCompared against Eq.(2), the new form of Eq.(3) is very
similar. The only difference is that the new form ofequation has a non-uniform cross section within therectangular node.
2.2.2 Semi-Analytical Nodal Solution
The method employed in EGRET to solve the 3D nodal
diffusion problem is the so-called semi-analytical nodal
method (SANM). By following the common practice oftransverse integration technique and then approximately
expanding the 1D nodal fluxes by a combination ofanalytical and polynomial base functions, one may solve
the original 3D nodal diffusion problem iteratively.
Enormous practical applications demonstrate that themethod possesses an outstanding overall performance.

2.23D Core Code EGRET-H
EGRET[10] is NuStar's 3D core code for steady-statecriticality and depletion calculation for western type
PWR. As mentioned earlier, this code, as a key
component for NuStar's core analysis code system, has
so far been extensively verified and validated. With thiscode at hand, the key task to develop a variant for VVER
is therefore to develop a 3D neutron diffusion problem
solver that applies to hexagonal geometry.

As for the hexagonal geometry case, as illustrated in
Fig.2, for each radial direction perpendicular to a pair ofsurfaces, the hexagonal node can be mapped into a
corresponding rectangle. Then by applying thetransverse integration technique to Eq.(3) within therectangle, one may obtain the following 1D equation,
which is still quite similar to the form of the 1D equation
solved by SANM in EGRET.
2
 D 2 u (u )   r g 2 (u )u (u )  g 2 (u )Qu (u )  Lu (u ) , (4)
u
where

To reuse modules/subroutines of EGRET as many as
possible and also to minimize the code verification andvalidation efforts, the authors choose to employ Chao's
conformal mapping method[11]to develop this diffusion
problem solver.
2.2.1 Conformal Mapping
According to the conformal mapping theory, as
illustrated in Fig.2, by introducing the Schwarzchristoffel mapping, a hexagon in complex z-plane canbe mapped to a rectangle in another complex w-plane
and the corresponding mapping function g(u,v)is definedas,dz
g (u, v) .
dw

1 av / 2 (u, v)dv
av  av /2
1 az /2 (u, v)    ( x, y, z )dzaz  az /2u (u ) 

g 2 (u )  (1)av / 2

 av / 2132

g 2 (u , v ) (u , v)dv /  (u )

(5)
(6)
(7)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka,Japan,Dec. 2-3, 2019

and tools from modern web development, to provide
convenient interaction scheme without the need of
installing a client application, i.e. a user can model areactor, define a core loading scheme, specify acalculation task, and view the corresponding results withthe help of a web browser.

Lu (u )  LRu (u)  g 2 (u ) LZu (u) (8)LRu (u) 

1 av /2
2
 D 2  (u, v)dvav  av /2v(9)

1 az /22
D
 (r )dzaz  az /2
z 2

(10)

LZu (u ) 

3. Code Verification and Validation
Unlike the case of rectangle node, where only three
coupled 1D equations are obtained after the applicationof transverse integration, one now obtains a coupled four
integrated 1D equations, i.e. one from axial direction andthe other three from radial directions perpendicular to

Currently, verification and validation tests are being
extensively performed for each component code and the
ORIENT-H system as a whole. For verification purpose,emphases are directed to these modules that are newly
developed or modified for VVER applications. Results
obtained are compared either with internationallyrecognized benchmark results or with results obtained byother codes that are already verified or validated. Whilefor validation purpose, as the first attempt, thecalculation results for a VVER-1000 reactor arecompared against the measurement data previously
published in a proposal for an international benchmark
problem.[13]three sets of surfaces of a hexagon. By some minor
adaptation, the original EGRET SANM kernel can beused to solve this set of coupled 1D equations.
2.2.3 CIAMA technique
One noticeable feature of EGRET is that it adopts theinnovative Channel-wise Intrinsic Axial Mesh
Adaptation (CIAMA) technique[12] to practicallyeliminate the long-standing non-physical control rod
cusping issue for the modern coarse-mesh nodal
methods. Another advantage of this technique is that one
may also explicitly represent the local effect caused by
the fuel grid directly in the whole-core coarse-meshnodal calculation. EGRET-H continues to employ thisinnovative technique unique to EGRET.

3.1 Validation Results for VVER-1000 Benchmark
The comparison of the critical boron concentrations forthe first cycle is given in Fig. 3. While the corresponding
comparison for axial offset (AO) is given in Fig.4. Fromthese comparisons, one may notice that for most of thecore depletion history, the ORIENT-H prediction results
agree well with the measurement data. However, there
are also some statuses that noticeably large discrepancies
are observed. It is no doubt that the whole system andthe problem modeling are still needed to be further tested,
especially for these modules to perform TH feedback
and to generate the homogenized parameters for thereactor baffle, the authors incline to the view that suchlarge discrepancies are very likely to be caused by theinconformity of the core conditions between calculationand measurement. As shown in Fig.5, the operation
history for the cycle is quite unstable, especially for the
BOC and EOC period. Since any change in the reactor
power and the insertion position of the control rod will
inevitably result in the corresponding changes of thetotal xenon poison and its spatial distribution, and suchchanges always lag behind, the core conditions will
never reach stable in the days after each change happens.
Considering the fact that the measurement data is justprovided once for a day, it is quite apprehensible thatlarge discrepancies are apt to see for these days, sincecalculations are actually performed for core states thatare noticeably different from these where themeasurement is performed. The authors incline tobelieve that this fact could also account for the similar
phenomenon that people see in results from other code
systems[13].

2.2.4 Intranode Representation
To capture the burnup gradient effect within a hexagonalnode, the spatial variation of cross sections inside ahexagon is represented in EGRET-H. The diffusion
coefficients are assumed to have negligible burnup
gradient effect, while other cross sections are tracked
with seven values per node, i.e. the values on the six
hexagon surfaces and the node-averaged value. In
SANM calculation, the effect of heterogeneous crosssection is considered as an extra contribution to theneutron source.
2.3 System Integration
As an engineer-oriented software package, ORIENT-H isdesigned to encapsulate the process of the codes within
two-steps framework, and provide a user-friendly
interface, with a relatively gentle learning curve.
On the backend, besides the aforementioned latticecode ROBIN2 and 3D core nodal code EGRET-H,
ORIENT-H relies on other components, including a
database OASIS for storing modeling and calculation
information, a management platform NYMPH for
manipulating files and driving calculations, a crosssection representation tool IDYLL for cross-section table
pre-generation. With such an architecture, it simplifies
the users’ work and releases them from writing textual
input card, which is tedious and prone error.
On the frontend, ORIENT-H adopts the techniques133Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka,Japan,Dec. 2-3, 2019

to Dr. Yung-An Chao for his kind help and invaluable
discussions during this study.
Validation against the VVER-1000 benchmark was
mainly undertaken by Mr. Lingfeng Wei during his
internship in the company, who was then a graduate
student of the Sino-French Institute for Nuclear Energy
and Technology of Sun Yat-sen University.
References
1. Zhang SH et al., "Validation of NuStar's PWR Core
Analysis System", Proc PHYSOR2014, Kyoto, Japan,
Sept. 28-Oct. 3, 2014, American Nuclear Society(2014).
2. Zhang SH, "Current Status of NuStar’s PWR Core
Analysis Code System", Proc RPHA15, Jeju, Korea,
Sept. 16-18, 2015.
3. Altiparmakov DV, Belicev P, "An Efficient Study of
the R-function Method Applied as Solid Modeler forMonte Carlo Calculations". Progress in Nuclear
Energy, 24, p77 (1990).
4. Yamamoto A, "Evaluation of Background Cross
Section for Heterogeneous and ComplicatedGeometry by the Enhanced Neutron Current
Method". J. Nuclear Science and Technology, v45,
n12, p1287 (2008).
5. Askew R, "A Characteristics Formulation of the
NeutronTransport
Equation
inComplicatedGeometries." AEEW-M 1108 (1972).
6. Goldberg L et al., "The Method of Characteristics inGeneral Geometry". Trans. Am. Nucl. Soc., 73, 173(1995).
7. Cho JY et al., "Whole Core Transport Calculation
Employing Hexagonal Modular Ray Tracing and
CMFD Formulation". J Nuclear Science Technology,
45, p.740-751(2008).
8. Yamamoto A et al., "Derivation of Optimum Polar
Angle Quadrature Set for the Method of
Characteristics Based on Approximation Error for
the Bickley Function". J Nuclear Science
Technology, 44, 129 (2007).
9. Carpenter DC et al., "The Log Linear Rate ConstantPower Depletion Method". Proc PHYSOR 2010,
Pittsburgh, Pennsylvania, May 9 - 14, 2010.
10. Zhang, SH, "The Multigroup Neutronics Model of
NuStar’s 3D Core Code EGRET", Proc
PHYSOR2014, Kyoto, Japan, Sept. 28-Oct. 3, 2014,
American Nuclear Society (2014).
11. ChaoYA et al., "Conformal Mapping and Hexagonal
Nodal Methods". Nuclear Science and Engineering,
121, 202-225 (1995).
12. Lu D et al., "A Three-dimensional Nodal Methodwith Channel-wise Intrinsic Axial Mesh Adaptation",
Annals of Nuclear Energy, 79, 152-157, 2015.
13. Lötsch T et al., "Corrections and Additions to the
Proposal of a Benchmark for Core Burnup
Calculations for a VVER-1000 Reactor", Proc AER
2010, Hanasaari, Espoo, Finland, Sept. 20–24, 2010.

Fig. 3 Critical boron concentration of cycle 1.

Fig. 4 Axial offset of cycle 1.

Fig. 5 Operation history of cycle 1.
4. Conclusions
A new variant of NuStar's core analysis code system hasrecently been developed for VVER application. It ismainly composed of NuStar's latest version of latticecode, ROBIN2 and the variant of NuStar's 3D core codefor hexagonal geometry, EGRET-H. The first validation
results obtained for the initial cycle of a VVER-1000
reactor has demonstrated that the development of thesystem is successful. Extensive validation against
multiple units and multiple cycles are currently underway, and the results will be published in the near future.

Acknowledgements
The authors would like to express their sincere gratitude134Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

BEAVRS Benchmark Evaluations with Studsvik CMS5 Code Package
Tamer Bahadira,*, Masatoshi Yamasakib
Studsvik Scandpower, Inc, 10 Langley Rd, Suite 201, Newton, MA 02459, USAb
Studsvik Japan, Ltd, Rinku Gate Tower Bldg 1409, 1 Rinku Oraikita, Izumisano-city, Osaka, 598-0048 Japan*
Corresponding author: tamer.bahadir@studsvik.comaAbstract
The MIT BEAVRS benchmark problem is evaluated using Studsvik’s state-of-theart in-core fuel management code package CMS5, which includes CASMO5 and
SIMULATE5. The CMS5 calculated results for low-power physics tests (hot zero
power critical boron, control rod worth and isothermal temperature coefficients)and full power operation (boron letdown and flux map reaction rate distributions)are compared to plant measured data provided in the benchmark specification.
The CMS5 calculations are repeated with three different evaluated nuclear datalibraries: ENDF/B-VII.1, ENDF/B-VIII.0 and JENDL-4.0. The CMS5 model
predicts HZP critical boron concentration for all-rods-out conditions within 10
ppm for both Cycle-1and Cycle-2; the control rod worth is predicted with adifference of 0.7% ± 3.8%, where the maximum difference is less than 9%. For
the core follow calculations at the hot full power condition, the average differencein predicting the critical boron concentration is less than 20 ppm. In addition, theradial and nodal reaction rate distributions are predicted with a mean difference ofabout 1.6 percent and 3.8 percent, respectively. No significant difference isobserved in predicting measured plant parameters with different nuclear datalibraries.
Key Words: CMS5, CASMO5, SIMULATE5, BEAVRS benchmark
1. Introduction

power and at-power conditions. The benchmark was setup
primarily for the verification and validation of highfidelity tools that have coupled neutron transport,
thermal-hydraulics, and fuel isotopic depletion models.
Public availability of such detailed data makes this avaluable tool for validating any in-core fuel management
code package. This work presents the results of such
validation effort undertaken with the CMS5 code package.

Studsvik’s in-core fuel management code package
CMS/CMS5, which includes lattice physics code
CASMO4/5 [1] and 3D nodal code SIMULATE3/5 [2],has been widely used in the industry for LWR reactor
analysis for more than 35 years. The initial lattice physics
code, CASMO-3, and 3D nodal code, SIMULATE-3,
developed in early 1980s were replaced with the state-ofthe-art CASMO5 and SIMULATE5 codes in mid 2000s
by taking advantage of today’s modern computerresources to implement advanced physics and numericalmodels, as well as employing new nuclear data libraries.
The CMS/CMS5 code package has been applied toapproximately 200 reactors, analyzing more than athousand cycles. It is proven to be accurate for typical
LWR reactors. The CMS5 code package has recently been
approved by the US NRC for generic application to
Pressurized Water Reactors in the US [3].
The MIT BEAVRS reactor as defined in reference [4]contains information for the initial cycle 1 and thereloaded cycle 2 of a typical 4-loop Westinghouse reactor.
The benchmark contains not only the detailed descriptionof the reactor core, its fuel and other essential components,
but also provides detailed measured data at hot-zero-

2. CASMO5 and SIMULATE5 Overview
CASMO5 is a two-dimensional characteristics-based
neutron and gamma transport theory lattice physics codewith depletion capability. CASMO5 generates crosssections and discontinuity factors for both square lattice
BWR/PWR and hexagonal lattice, VVER, diffusion
theory 3-D nodal core analysis. CASMO5 includes many
advancements both in physics modeling and capabilities,
including updated neutron data libraries containing morethan one thousand unique nuclides and materials,
extended depletion chains solved with Chebyshev
Rational Approximation Method (CRAM), refined 586
energy-group structure, quadratic gadolinium depletion,
Characteristics-based Dancoff, and enhanced geometry
modeling. Ref. [1] presents the details of CASMO5135

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

capabilities and features.
SIMULATE5 is a three-dimensional multi-group
analytical nodal diffusion code for both square andhexagonal geometries. It solves either the diffusion or
optionally SP3 equation, which may become importantfor
MOX
cores.
It
employsahybridmacroscopic/microscopic depletion model that tracks
approximately 60 isotopes per node. It was designed fromits inception to handle heterogeneities of modern fuelassemblies, both in the radial and axial directions, byusing on-the-fly axial homogenization and a radial submesh model. The embedded fuel temperature calculation
is part of the new detailed thermal/hydraulic module. Ref.
[2] presents the details of SIMULATE5 features.

factors and pin form factors, compared to conventional
methods, which use either a full- or quarter-assemblytransport calculation run with periodic boundary
conditions.
The cross-section library for the top and bottom
reflectors are generated from conventional onedimensional fuel and reflector CASMO5 calculations.
The top and bottom reflector geometries account for allmaterial compositions within the first 35 cm above the top
and below the bottom of active fuel. The improved radialreflector model is employed for this benchmark problem[5]: Nine position-dependent radial reflector segments are
generated from a multi-assembly CASMO5 calculation,
as depicted for an octant core in Fig. 1.

3. BEAVRS Benchmark Problem
The MIT BEAVRS benchmark provides a detailed coreand fuel description of the initial two cycles of a 4-loop
Westinghouse reactor:
Cycle 1 contains 3 different fuel types with respect toU-235 enrichment (1.6%, 2.4% and 3.1.%) and 5 different
configurations of PYREX® boron glass BP (Burnable
Poison/Absorber pins) loaded.
Cycle 2 contains 129 shuffled assemblies from EOC1, two different fresh fuel types with respect to U-235
enrichment and three different configurations of
PYREX® boron glass BP (Burnable Poison/Absorberpins). For all shuffled fuel from cycle 1, the PYREX®
boron glass BP has been withdrawn in cycle 2.

Fig. 1. CASMO5 multi-assembly geometry for generating
position dependent radial reflector data.

4. CMS5 BEAVRS Core Model

CASMO5 calculations are performed using threedifferent evaluated nuclear data libraries: ENDF/B-VII.1(E7R1), ENDF/B-VIII.0 (E8R0) and JENDL-4.0 (J4).
The E7R1 library is the default CASMO5 library.
CASMO5 E8R0 library has been released recently in May
2019. The use of J4, Japanese Evaluated Neutron Data
Library, is mostly limited to Japan.
All CASMO5 calculations are performed using thedefault calculational model options, using 19 energy
groups in the 2D transport solution for fuel and 95 energy
groups for reflector segments.

The core model of the BEAVRS benchmark is set up withthe conventional two-step procedure: using cross-sections
generated from 2D CASMO5 calculations in 3D
SIMULATE5 calculations.
4.1 CASMO5 Fuel and Reflector Models
CASMO5 2D single assembly calculations, either inoctant or full assembly geometry, are set up for all axial
variation of the fuel within the active length. The
variations in the axial regions include fuel with andwithout Burnable Poison (BP) or control rod as well as
guide tubes with different dimensions, generally referred
to as “DASHPOT” and “No DASHPOT” regions. The
spacers in the active fuel zone are modeled as anadditional delta cross-section dependency, generated fromCASMO5 branch calculations, and are included in the
SIMULATE5 library.
The initial cycle of the BEAVRS core contains
asymmetric fuel assemblies with 6 and 15 BP pins loadedon core edges to flatten the power distribution. The CMS5
model makes use of a new model for building asymmetricassemblies by extracting the quarter assembly nodal data,in the linking code level, from the solution of the fullassembly CASMO5 calculations. The new model better
captures the radial heterogeneity of the fuel assembly,
both in nodal cross-sections/assembly discontinuity

4.2 SIMULATE5 Core Model
SIMULATE5 core follow calculations are run in full coreusing 4 radial nodes per assembly and 24 uniformly
spaced axial nodes. The diffusion calculations are
performed in the default 4 energy groups. The sub-meshmodel divides each quarter assembly into 5x5 sub-meshes.
The microscopic depletion model tracks approximately
60 isotopes per node, and 5 actinides per sub-mesh, where
the later captures intra-assembly burnup shape. The flux
map evaluations are performed utilizing the benchmark
provided, pre-processed (aligned and calibrated),measured reaction rate data files.
For the core follow calculations, the operating
conditions are taken from the benchmark specification.
The coolant inlet temperature of 560 K is assumed for the
hot zero power calculations of both cycles.

136Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
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5. Benchmark results

SC
SB
SA
Total

5.1 Zero-Power Physics Testing
The BEAVRS benchmark provides various zero-power
physics test measurements: critical boron concentration,
control rod bank worth, reactivity coefficients for cycle 1
and 2, and also one flux map taken in cycle 1 at 0.71%
power, which is not included in this work. SIMULATE5
accuracy is presented in Table I.
Meas.
(ppm)

ARO
D In
C, D In
A,B,C,D In
A,B,C,D,SE,
SD,SC In

975
902
810
686
508

CMS5 - Measured
E7R1
E8R0
J4
Cycle 1
-10
4
-4
4
18
9
1
14
7
-7
4
-1
-15
-6
-9

ARO
C In

1405
1273

Cycle 2
-9
26

-10
25

Rod
Position

-7
29

D
C
B
A
SE
SD
SC

788
1203
1171
548
461
772
1099

D
C
B
A
SE
SD

426
1014
716
420
438
305

(1 - CMS5/Meas)%
E7R1
E8R0
J4
Cycle1
-1.5
-1.9
0.9
1.5
0.3
-0.2
3.2
6.0
5.9
8.5
0.6
0.5
-0.6
-1.0
Cycle 2
5.3
5.3
-1.8
-2.2
-7.0
-5.4
-2.4
-4.5
-3.3
-4.3
6.2
7.3

6.4
-1.6
4.7
5.3

ITC
pcm/°FARO
D In
C, D In

-1.75
-2.75
-8.01

ARO

1.71

CMS5-Measured
E7R1
E8R0
J4
Cycle 1
-0.43
0.01
-0.35
-0.63
-0.17
-0.55
-0.03
0.44
0.05
Cycle 2
0.20
0.30
0.05

5.2 Hot Full Power (HFP) Boron Predictions
The core follow calculations are performed with acondensed number of state points compared to theoperational logs provided in the benchmark. Sufficientdetail is included such that the operating history (power,
inlet temperature and control rod bank position) is
accurately represented. The condensed Cycle 1 and Cycle
2 depletion decks contain 66 and 45 state points,
respectively. Figures 2 and 3 present the measured and
CMS5 predicted critical boron concentrations. The CMS5
predictions with different libraries are indistinguishable inthese plots.
Except the initial 150 EFPD of Cycle 1, the CMS5
reactivity predictions are in good agreement vs the plantmeasurement. There is a notable difference (Calc - Meas)of around -33 ppm initially in the cycle (from around 25125 EFPD). This coincides with a period of operation
rarely at full power. One possible explanation for thedifference is that the measured boron concertation was
adjusted to nominal 100% power and ARO condition, as
this is a common practice in some utilities. Unfortunately,
no direct information on the conditions of thesemeasurements is found in Ref [4].
In the exposure interval from 50-300 EFPD,
comparing calculated boron with CMS5 E7R1 libraryversus the measured boron gives an average bias of -20
ppm. At the end of the cycle, the agreement is good andhas an average difference of -16 ppm. In Cycle 2, theaverage bias between CMS5 E7R1 model prediction andthe measurement is 22 ppm.
In general, all three nuclear data libraries yield similar
reactivity and critical boron predictions at power. The
higher reactivity observed with the E8R0 library at Cycle
1 startup quickly burns out after the first 100 EFPD
operation. Overall, compared to the E7R1 library, theE8R0 library results in 3 and 6 ppm lower boron in Cycles

Table II. Control Rod Bank Worth (pcm)
Rod Worth
(pcm)

6.4
-1.6
4.7
5.3

Table III. Isothermal Temperature Coefficient (pcm/°F)

The CMS5 critical boron concentration predictions for
all rod configurations are in good agreement with themeasurements. In Cycle 1 startup for all rods out (ARO)
position, the CMS5 predictions are within 10 ppm against
the measurement. In general, the E8R0 library givesslightly higher, 5-15 ppm, critical boron concentrationcompared to E7R1 and J4 libraries. The differences
between the libraries disappear with Cycle 2 predictions
where the maximum error is 10 ppm underprediction.
Table II shows the comparisons between
SIMULATE5 control rod bank worth predictions and themeasured values for a number of different bankconfigurations. The maximum error in Cycle 1 isapproximately 8% encountered for the shutdown bank-SE.
The E8R0 library results have slightly larger errors
compared to the other two libraries. No inner or outerbank relation could be noted in the results presented.
Rod
Position

5.2
-2.0
3.0
5.3

Isothermal temperature coefficient (pcm/°F) predictions
are given in Table III. The ITC predictions in both cycles
are in excellent agreement vs the measurement. Note thatthe cycle 2 measured ITC is corrected to +1.71 pcm/°F [6],not -1.71 pcm/°F as reported in Ref. [4]. This would becorrected in newer revisions of the benchmark
specification.

Table I. Critical Boron Concentration at Zero PowerRod Position

307
781
326
4733

-1.3
0.6
1.6
1.8
4.6
0.6
-0.5
5.3
-2.3
-5.6
-4.2
-4.0
7.3

137

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

1 and 2 respectively; and the J4 library results in 6 ppm
lower boron for the same cycles.

library.
Power

800

Critical Boron Concentration (ppm)

700

Cycle 1
All
>95%
Cycle 2
All
>95%

E7R1
E8R0

600

J4
500

Measured

400

200
100
0
50

100

150

200

250

300

350

Cycle Exposure (EFPD)

Fig. 2. Cycle 1 core follow boron.

E7R1

Critical Boron Concentration (ppm)

800

E8R0

700

J4R0
Measured

600
500
400
300
200
100
0
100

150

200

250

300

Cycle Exposure (EFPD)

Fig. 3. Cycle 2 core follow boron.
5.3 Flux Map Comparisons

Reaction Rate RMS (%)

9Radial RMS

8

Nodal RMS

Cycle 1

Cycle 2

5
4
3
2
1
0
0

100

2.46
2.02

2.66
2.17

2.42
1.99

4.51
3.69

4.65
3.77

4.49
3.69

14
11

1.42
1.19

1.42
1.18

1.46
1.24

3.97
3.96

3.98
3.96

4.00
4.00

References
1. J. Rhodes, K. Smith and D. Lee, "CASMO5
Development and Applications," Proc. in PHYSOR
2006, Vancouver (2006).
2. T. Bahadir and S.-Ö. Lindahl, "Studsvik’s Next
Generation Nodal Code SIMULATE5," Proc. in
Advances in Nuclear Fuel Management IV, Hilton
Head Island, South Carolina, USA (2009).
3. B. Haugh, "Generic Application of the Studsvik
Scandpower Core Management System to
Pressurized Water Reactors," Studsvik Scandpower,
Inc., SSP-14/P01-028-TR-NP-A (2017).
4. N. Horelik and B. Herman, “MIT Benchmark for
Evaluation and Validation of Reactor Simulations(BEAVRS)”, MIT Computational Reactor Physics
Group, Rev. 2.0.2. (2018).
5. T. Bahadir, "Improved PWR Radial Reflector
Modeling with SIMULATE5," Proc. in Advances in
Nuclear Fuel Management V, Hilton Head Island,South Carolina, USA (2015)..
6. K. Smith, Personal e-mail communication, Sept 10,
2019.

10

6

23
11

The authors acknowledge the former Studsvik employee
Magnus Kruners for his work constructing the initial
BEAVRS core model with CMS5.

The 2D assembly and 3D nodal comparisons of the preprocessed measured and calculated detector reaction rates
at the measured assembly locations are given in Fig.4 forCycles 1 and 2 with the E7R1 model. The Cycle 1 flux
map at low power on Day 7 (at 4% power) is excluded
from comparisons. The results presented include all flux
maps taken at low power as well as at hot full power. Table
IV presents the flux map statistics, with three nuclear datalibraries, for all flux maps and those taken above 95%
power, separately.

7

Nodal RMS (%)
E7R1
E8R0
J4

The MIT BEAVRS benchmark is evaluated with
Studsvik’s CMS5 code package using three different
nuclear data libraries. The CMS5 calculated coreparameters at zero and full power are in good agreementwith the plant measurements: The boron predictions arewithin 30 ppm at zero power and less than 20 ppm at
hot full power condition; the control rod worth ispredicted with 0.7 ± 3.8% accuracy; the radial andnodal reaction rate distributions are predicted with amean difference of about 1.6% and 3.8%, respectively.
No significant difference is observed in predicting
measured plant parameters with different nuclear datalibraries.
Acknowledgment

900

50

Assembly RMS (%)
E7R1
E8R0
J4

6. Conclusions

1000

0

# of
Maps

The noted RMS values, combined for 22 flux maps
with power > 95%, show excellent predictions with the
2D radial below 2.1% and the 3D nodal below 4.0% for
HFP conditions. The flux map statistics in general areinsensitive (less than 0.2% difference) to the choice of thenuclear data library.

300

0Table IV. Flux Map Statistical Summary

200
300
400
Cumulative Exposure (EFPD)

500

600

Fig. 4. Flux map assembly and nodal RMS with the E7R1

138

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

An Improved Chebyshev Rational Approximation Method Base on Order Reduction of
Krylov Subspace Method
Yuying HUa, Qi LUOb*, Dong YAOa, Yingrui Yua, Hongkuan LIAOa, Bingyan ZhouaaScience and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China,
Chengdu, China
Nuclear Power Institute of China, Chengdu, Chinab*

Corresponding author: 18200315798@sina.cn

Abstract
Burnup is an important part of physical calculation in reactors. The accuracy of results
deeply influences critical physical characteristics of reactor such as the cycle length,
reactivity and power distribution. Algorithm of solving burnup equations directly relates tothe accuracy and efficiency of burnup calculation. In this study, an improved Chebyshev
Rational Approximation Method, Combined algorithm, has been used to solve burnup
equations. According to verification and analysis, comparing to CRAM, Combinedalgorithm has higher efficiency in large matrices. For the little burnup benchmark, therelative error could reach to 10-5, worse than CRAM in calculation accuracy, but it basicallymeets the calculation requirements.

Key Words: Burnup calculation, Combined algorithm, Improved CRAM,
Benchmark verification
2. Combined Algorithm

1. Introduction
With the development of computer science and theimprovement of nuclear technology, physical calculationof high accuracy and efficiency in reactors becomes more
and more important. Using one-step method to complete
physical design in cores becomes a trend. Thus, burnup
algorithm needs to be improved. Nowadays, major
methods to solve burnup equations contain Taylor series
expansion, Chebyshev Rational Approximation Method(CRAM), Transmutation Trajectory Analysis (TTA) andothers.
There are thousands of nuclides in the database, andthe lives and cross section of nuclides differ greatly. Thus,
compressed database has been used to improve
calculation efficiency. The main idea of Krylov Subspacemethod is reducing the order of matrices to improve
calculation efficiency[6]. The biggest character of CRAM
is the computational stability[7]. Besides, results canalways be carried out with CRAM even if the matrix is
ill-conditioned. In this study, the idea of order reduction
in Krylov Subspace has been used before solve burnup
equations through CRAM, so that it improves thecalculation efficiency while keeping the accuracy on an
acceptable level.

2.1 Order reduction in Krylov Subspace Method
In this method[1], the vector norm ‖𝑵‖2 was gained
from initial nuclides density 𝑵 , and the firstorthonormal basis 𝒗1 was built. According 𝒗1 andburnup matrix 𝑨𝑛×𝑛 , orthonormal basis matrix
𝑽 = [𝒗1 , 𝒗2 , ⋯ , 𝒗𝑚 ] could be built. With matrix 𝑽 and𝑨𝑛×𝑛 , matrix 𝑯𝑚×𝑚 of reduced order could be gained.
Thus, the initial exponential matrix calculation 𝑵𝑒 𝑨𝑡
has been changed to calculate ‖𝑵‖2 𝑽𝑚 𝑒 −𝑯𝑚 𝑒1 ,reaching the goal of reducing matrix scale andimproving efficiency.
Re-orthogonalization Arnoldi method is used togenerate the matrix of reduced-order, and orthonormal
basis matrix 𝑽 is generated by modified Gram-Schmidt
orthogonalization algorithm. The details are shown as
below.
(1) Calculating vector norm ‖𝑵‖2 according initial
nuclides density 𝑵;(2) 𝒗1 = 𝑵/‖𝑵‖2 , 𝑽i,1 = 𝒗1 ;(3) Gaining orthonormal basis matrix 𝑽:
for 𝑗 = 1: 𝑚𝒗j [k] = 𝑽k,j𝒘 = 𝑨 × 𝒗j
for 𝑖 = 1: 𝑗𝒗i [k] = 𝑽k,i

139

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Series Expansion was used to calculate matrix
exponential in Krylov Subspace Method. The results areshown as below.
When calculating random matrices, the relative errorof CRAM is about 10-12 and the same with Combined
Algorithm, while the relative error of Krylov Subspace
Method being 10-15. However, when calculating
ill-conditioned matrices such as Hilbert matrix, theaccuracy of Krylov Subspace Method is quite low, andthe max relative error reaches to 13%. As for CRAM,
the relative error is still 10-12, with 10-11 of Combined
Algorithm. These two methods remain a level of highaccuracy.
From Fig. 1, Combined Algorithm has highefficiency than Krylov Subspace Method whencalculating in the same sub-steps. Compared with
CRAM, Combined Algorithm behaves better with highorder matrices even 20-substeps calculation.

𝑯𝒕i,j = 𝐰 × 𝒗i𝒘 = 𝒘 − 𝑯𝒕i,j × 𝒗i

end
for 𝑖 = 1: 𝑗s = 𝒘 × 𝒗i𝒘 = 𝒘 − s × 𝒗i
end
𝑽k,j+1 = 𝒘k ⁄‖𝒘‖2end(4) 𝑯m×m = 𝑽T × 𝑯𝒕 × 𝑽.
Through this method, original nth order matrix A
has been translated to mth order matrix H . In theverification later, m = n/4.
2.2 Solve equations by Chebyshev Rational
Approximation Method
After reducing burnup matrix order with Krylov
subspace, new exponential matrix would be solved by
CRAM[4], shown as follows.
⁄

𝑵𝑛 = α0 𝑵0 + 2Re[∑kj=12 αj (𝑯t − θj I)−1 ]𝑵0(1)

k is the order of CRAM, and the convergenceaccuracy of CRAM is 9.3−k , calculation timeincreasing with k enlarging. α and θ are given
parameters, depending on the calculation order. In theverification later, k = 16.
3. Verification Analysis
The advantage of Krylov subspace method isimproving efficiency through reducing matrix order.
Efficiency of exponential matrix calculation could beheld on a high level by using CRAM when calculating
ill-conditioned matrix. Combined Algorithm combines
the advantages of two methods to improve calculation
efficiency with high accuracy. In this study, large scale
matrices and IAEA little burnup benchmark has beenverified.

Fig. 1 Calculation time of random matrices
3.2 Verification of IAEA benchmark
In the burnup chains of this benchmark[3], there are
34 nuclides containing neutron capture, decay andgeneration of some important fission products. The
burnup chains are shown in Fig. 2 Burnup time is 50
days, the initial density of 235U and 238U are 7.4003×10-5
g-atom and 6.9360×10-3 g-atom. The results are shownin Table I. The calculation results of TTA[2] have beenused as reference value because there is no annularclosed burnup chain.
From Table I, when the number of sub-steps is small,the accuracy of Combined Algorithm is not very high.
However, the accuracy of 20-substeps calculation could
reach to 10-3, while 50-substeps reaching to 10-6.

3.1 Verification of large scale matrices
Compared with Krylov Subspace Method and
CRAM, Combined Algorithm has better behavior in
efficiency of large scale matrices calculation. In thisstudy, different orders of matrices are generated with
Matlab code[5], and verification has been done withthese three methods.
Matrices orders range from 50~1500th. When using
Krylov Subspace Method and Combined Algorithm,
there is 1-substep, 5-substeps, 10-substeps and20-substeps calculation in a single burnup step. Taylor

140Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig. 2 Burnup chains of IAEA benchmark
Society for Industrial and Applied Mathematics
Review, 45, 1 (2003).
2. Jerzy C, “General solution of Bateman equations fornuclear transmutations,” Annals of Nuclear Energy,
33 (2006).
3. M. G, “Neutronic Depletion Benchmark Problems,”
Trans. Am. Nucl. Soc, 23 (1976).
4. Pusa M., “Rational Approximations to the Matrix
Exponential in Burnup Calculations,” Nuclear
Science and Engineering, 169, 2 (2011).
5. Z Z, “Mastering MATLAB R2011a,” Beijing
University of Aeronautics and Astronautics Press,
Beijing(2011).
6. Akio Y, Masahiro T, Naoki S. “Numerical Solutionof Stiff Burnup Equation with Short Half Lived
Nuclides by the Krylov Subspace Method,” Journalof Nuclear Science and Technology, 44, 2(2007).
7. Songqian T, Feng X, Yi T, Shuping W. “Analysis ofComplex Nuclide Chain Equation Calculation Based on
Different Method,” Nuclear Power Engineering, 37,
1(2016).

4. Conclusions
According to the verification of large scale randommatrices and IAEA benchmark, Combined Algorithm
has higher efficiency than CRAM because of reducing
orders before solve burnup equations. Some information
is lost due to reducing the matrix order, so calculation
accuracy of Combined Algorithm is lower than CRAM.
However, calculation accuracy could be improved bychoosing suitable burnup step. In the further work,
algorithm optimization would become a key point, andaccuracy of burnup could be improved by matrix
reconstruction and other methods.
Reference
1. Cleve M., “Nineteen Dubious Way to Compute theExponential of a Matrix, Twenty-five Years Later,”141

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table I. Results of CRAM and Combined Algorithm
nuclide

TTA

CRAM

Relative

Combined(20)Relative

Combined(50)Relativeg-atomg-atomerror

g-atomerror

g-atom

error

234

0

0

0

0

0

0

0

235

5.833942×10-5

5.833939×10-5-5.77×10-7

5.833882×10-5-1.02×10-5

5.833942×10-5-4.41×10-10236

2.860583×10-6

2.860583×10-6-1.58×10-7

2.860563×10-6-6.93×10-6

2.860583×10-6

1.88×10-9237

2.061026×10-8

2.061026×10-8-1.01×10-7

2.061034×10-8

3.92×10-6

2.061026×10-8

3.05×10-9238

6.919236×10-3

6.919233×10-3-4.88×10-7

6.919169×10-3-9.75×10-6

6.919236×10-3-1.56×10-10

239

U

7.183703×10-9

7.183700×10-9-4.88×10-7

7.183617×10-9-1.20×10-5

7.183703×10-9-1.96×10-10

237

Np

4.508447×10-8

4.508447×10-8-1.57×10-8

4.508197×10-8-5.55×10-5

4.508447×10-8-3.89×10-8

238

3.248943×10-10

3.248943×10-10-6.01×10-9

3.248732×10-10-6.50×10-5

3.248943×10-10-4.51×10-8

239

1.029451×10-6

1.029451×10-6-4.56×10-7

1.029441×10-6-9.80×10-6

1.029451×10-6-2.30×10-102401.322940×10-11

1.322940×10-11-4.56×10-7

1.326491×10-11

2.68×10-3

1.322940×10-11-4.12×10-9

238

1.472831×10-9

1.472831×10-9

3.08×10-8

1.471592×10-9-8.41×10-4

1.472831×10-9-5.83×10-7

239

1.057469×10-5

1.057469×10-5-1.49×10-7

1.057456×10-5-1.18×10-5

1.057469×10-5

5.57×10-10240

9.959276×10-7

9.959276×10-7-5.41×10-8

9.959928×10-7

6.54×10-5

9.959277×10-7

3.10×10-8241

3.342055×10-7

3.342055×10-7

8.61×10-10

3.341756×10-7-8.93×10-5

3.342055×10-7-2.83×10-8242

1.636514×10-8

1.636514×10-8

3.48×10-8

1.634770×10-8-1.07×10-3

1.636512×10-8-7.34×10-7

243

Pu

1.362797×10-11

1.362797×10-11

3.56×10-8

1.289188×10-11-5.40×10-2

1.362796×10-11-7.50×10-7

241

Am

5.864330×10-10

5.864330×10-10

3.34×10-8

5.859058×10-10-8.99×10-4

5.864327×10-10-5.72×10-7

242

6.172483×10-12

6.172483×10-12

3.52×10-8

6.166452×10-12-9.77×10-4

6.172479×10-12-6.25×10-7

242m

Am

5.048715×10-12

5.048715×10-12

5.05×10-8

5.038860×10-12-1.95×10-3

5.048708×10-12-1.42×10-6

243

Am

4.568499×10-10

4.568500×10-10

5.78×10-8

4.561127×10-10-1.61×10-3

4.568486×10-10-2.85×10-6244

6.377320×10-14

6.377321×10-14

5.31×10-8

6.238801×10-14-2.17×10-2

6.377297×10-14-3.58×10-6242

5.499801×10-11

5.499801×10-11

4.99×10-8

5.481788×10-11-3.28×10-3

5.499785×10-11-2.92×10-6

243

1.153872×10-13

1.153871×10-13-2.30×10-7

1.146019×10-13-6.81×10-3

1.153863×10-13-7.75×10-6244

2.068299×10-11

2.068299×10-11

5.47×10-8

2.053931×10-11-6.95×10-3

2.068283×10-11-7.69×10-6

245

Cm

2.434189×10-13

2.434188×10-13-3.04×10-7

2.405039×10-13-1.20×10-2

2.434152×10-13-1.50×10-5135I8.827566×10-9

8.827563×10-9-4.23×10-7

8.824156×10-9-3.86×10-4

8.827566×10-9-3.94×10-10

135

Xe

9.147625×10-10

9.147621×10-10-4.24×10-7

9.144409×10-10-3.52×10-4

9.147625×10-10-4.48×10-10

147

Nd

1.211571×10-7

1.211571×10-7-2.84×10-7

1.211490×10-7-6.72×10-5

1.211571×10-7-1.16×10-8

147

Pm

2.018147×10-7

2.018146×10-7-9.31×10-8

2.018217×10-7

3.48×10-5

2.018147×10-7

1.10×10-8148

4.570362×10-9

4.570362×10-9-7.32×10-8

4.570592×10-9

5.03×10-5

4.570363×10-9

1.43×10-8

148m

Pm

3.867448×10-9

3.867448×10-9-7.42×10-8

3.867638×10-9

4.92×10-5

3.867448×10-9

1.41×10-8

149Pm

1.996828×10-8

1.996827×10-8-3.08×10-7

1.996855×10-8

1.39×10-5

1.996828×10-8

5.46×10-9

149

Sm

1.197771×10-8

1.197770×10-8-2.92×10-7

1.197792×10-8

1.78×10-5

1.197771×10-8

6.23×10-9FP

1.452279×10-5

1.452279×10-5-1.39×10-7

1.452262×10-5-1.14×10-5

1.452279×10-5-2.73×10-9

U
U
U
U
U

Np
Np
Np
Pu
Pu
Pu
Pu
Pu

Am

Am
Cm
Cm
Cm

Pm

142

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Improvement of Optimally-Weighted Predictor-Corrector Method
for Nuclear Fuel Burnup Calculations
Jumpei Sasugaa,* , Go Chibaa, Yasunori Ohokab, Kento Yamamotob and Hiroaki Naganoba
Hokkaido University, Kita 13 Nishi 8, Sapporo, Hokkaido, Japanb
Nuclear Fuel Industries, Ltd., 950 Asashiro-Nishi 1-chome Kumatori-cho, Sennan-gun, Osaka, Japan*
Corresponding author: qazxcvbnm@eis.hokudai.ac.jp

Abstract
A new algorithm to determine a proper weight in the optimally-weighted
predictor-corrector (OWPC) method is proposed. This concept is based on
problem simplification in which the proper weight can be easily obtained.
Verification calculations are performed for three different PWR fuel assemblies
including burnable absorber. The new OWPC method shows better performance
than the original PC method and the projected PC method.
Key Words: burnup calculation, burnable poison, predictor-corrector
method
1. Introduction

to as 𝑁𝑐 . This is referred to as a corrector calculation.
With those two NDs, 𝑁𝑝 and 𝑁𝑐 , we finally calculate
ND at the end of time step N as follows:

Generally, fuel assemblies of Light Water Reactors(LWRs) contain burnable absorbers such as gadolinium(Gd). Since impact of this burnable absorber on reactor
physics property drastically changes even during small
time period, accurate numerical scheme for time
discretization is required. The predictor-corrector (PC)method [1] has been well known as one of the accurate
schemes, but further improvement of the numerical
scheme has been attempted and proposed by many
researchers so far [2-4]. Okumura and Chiba haveproposed improved methods based on the PC method: a
Weighted Predictor-Corrector (WPC) method which
introduces weight to the PC method and an OptimallyWeighted Predictor-Corrector (OWPC) method which
introduces an algorithm to obtain proper weightautomatically in WPC [5]. During our recent works,
however, we have found that the original OWPC method
does not work well in certain conditions. In addition, there
still be a free parameter which should be determined byusers in the original OWPC method. Under thesecircumstances, we propose a new OWPC method which
does not contain any users-defining parameters and isapplicable for various condition.

𝑁=

𝑁𝑝 + 𝑁𝑐.
2(1)

We have another choise to calculate N as follows [5]:ln𝑁𝑝 + ln𝑁𝑐
𝑁 = exp (
).
2

(2)

This is based on simple averaging of reaction rate andlinear dependence of reaction rate with time is implicitly
assumed in this definition.
In case of calculation for ND of neutron absorbers,
there is a systematic error when we use the PC method
since the following two approximations are introduced in
the PC method.
One is the approximation about reaction rates in thecorrector calculation 𝑅𝑐 . This is calculated from 𝑁𝑝 , but
this is not the true value of ND at the end of time step.
This ND, 𝑁𝑝 , is calculated from the predictor
calculations in which reaction rates are assumed constantduring the time step. Reaction rate of neutron absorbing
materials should increase during time step because ofneutron energy spectrum softening, so ND at the end oftime step should be smaller than 𝑁𝑝 and reaction rate at
the end of time step should be larger than 𝑅𝑐 .
Another one is the approximation introduced at the
final step in the procedure. As mentioned above,
averaging of final ND using Eq. (2) is based on theassumption that neutron absorber reaction rate changes
linearly with time.
In order to improve accuracy of the PC method,

2. Theory
First, let us consider the original PC method. We use the
PC method in burnup calculations for neutron absorbing
nuclides. In the PC method, we calculate number density(ND) at the end of time step 𝑁𝑝 using the reaction rate at
the beginning of time step 𝑅𝑝 . This procedure is called apredictor calculation. Next, we obtain the reaction rate at
the end of time step 𝑅𝑐 from 𝑁𝑝 . Then, using 𝑅𝑐 , weagain calculate ND at the end of the step, and it is denoted143Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Okumura and Chiba have proposed new advanced
methods based on the original PC method: the WPC andOWPC methods [5]. Instead of Eq. (2), the following
equation is used in the WPC method:𝑁 = exp{(1 − 𝜔)ln𝑁𝑝 + 𝜔ln𝑁𝑐 } .

MulticellBurner in a deterministic reactor physics code
system CBZ which is under development at Hokkaido
University. 107-group self-shielded cross sections arecalculated based on the advanced Bondarenko modelfrom a 107-group CBZLIB generated from JENDL-4.0[6]. Lattice effect is taken into account by the Dancoff
factor method. Neutron transport equation of multicell
problems is solved by a module MEC based on themethod of characteristics. Scattering anisotropy is taken
into account by the P0 transport approximation. With
neutron flux and self-shielded cross sections, fuel burnup
calculations are carried out. A simplified burnup chain
consisting of 138 fission products is used and burnup
equation is solved by the matrix exponential method. The
matrix exponential is numerically calculated by the minimax polynomial approximation method. This nuclear fuelburnup calculation capability of CBZ has been wellverified in the previous work [5].

(3)

We introduce a weight ω to obtain more accurate ND at
the end of time step than the PC method. In the originalOWPC method, an algorithm to determine ω has been
proposed, but recently we have found that it does not work
well in certain conditions, so we propose a new algorithm
to determine ω for the OWPC method as follows.
Let us consider that ND of neutron absorbing
materials such as Gd-155 and -157 changes mainly due
to absorbing neutrons. In our study, we use the following
simplified model. This model assumes that ND of theseneutron absorbing nuclides changes as the following
simple differential equation:𝑑𝑁(𝑡)= −𝑅(𝑡)𝑁(𝑡).
𝑑𝑡

3.2 Test problem
Numerical calculations are curried out for three PWR fuelassemblies. One of them is shown in Fig.1 called as the
Okumura problem [5] in the present paper. This contains
burnable poison rods composed of 9.6 wt% Gd 2O3 and
UO2 whose uranium-235 enrichment is 2.6 wt%.
Geometrical configuration of this burnable poison rod issame as that of the UO2 rod. In addition, 25 guide tubes,
in which water is filled inside of cladding whose inner andouter radii are 0.569 cm and 0.61 cm, are also located.
Total thermal power of 5.75523×10-2 MW per unit heightin cm is assumed. Temperature of fuel, cladding andmoderator are 900, 600, 591 K, respectably. The other two
PWR fuel assemblies are the 2O and 2P problems in the
VERA depletion benchmark model [7]. Gadolinium
concentration of a gadolinium pins in these assemblies is
5 wt%.

(4)

It means that ND of this nuclide decreases by only neutron
absorption. Yamamoto has numerically presented thatreaction rate of Gd-155 and -157 decrease linearly with
ND of themselves [3]. Since we calculate 𝑁𝑝 and 𝑅𝑐
during the procedure of the PC method, we can determine𝑑𝑅from (𝑁0 , 𝑅𝑝 ) and (𝑁𝑝 , 𝑅𝑐 ) where 𝑁0 is the ND
𝑑𝑁
at the beginning of time step. By assuming a linear
dependence of 𝑅 on 𝑁 , we can easily obtain accurate
numerical result of ND at the end of time step, 𝑁𝑟 , in thissimple model represented by Eq. (4) with fine time meshdivisions. From 𝑁𝑟 , we can obtain corresponding
reaction rate 𝑅𝑟 as follows:
𝑁𝑟 = 𝑁0 exp(−𝑅𝑟 ∆𝑡) .

(5)

At the same time, calculation based on the PC method is
carried out in the same simple model, and the reaction rate𝑅̃𝑐 is calculated. Using 𝑅𝑟 , 𝑅𝑝 and 𝑅̃𝑐 , a proper
weight ω for the OWPC method can be defined asω=(𝑅𝑟 − 𝑅𝑝 ).
(𝑅̃𝑐 − 𝑅𝑝 )(6)

Using this 𝜔 , we can obtain ND at end of the time stepin the original problem from Eq. (3). In the present
paper, this method is referred to as the OWPC method.
Note that the concept of this new OWPC method isquite similar to the quadratic depletion model proposedby Lee. [4]Fig. 1. Geometric configuration of the Okumura problem

3. Test Problems and Numerical Procedure

We compare infinite neutron multiplication factor kinf
to test our method. References of kinf are calculated using
the conventional PC method with 0.055 GWd/t burnup

3.1 Brief description of numerical methods of CBZ
All the calculations are carried out with a module named

144

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

step. The PC method is generally considered effective
during Gd depletion, so we adopt our proposed method
𝑁 −𝑁
only in the condition of 0 𝑝 > 0.01.
𝑁0

3.3 Numerical procedure
We calculate ND by the OWPC method as the following:
1. Carry out a one-step burnup calculation with theoriginal PC method and obtain the parameters about
nuclide 𝑖 (𝑖=155,157): 𝑁0𝑖 , 𝑅𝑝𝑖 , 𝑁𝑝𝑖 and 𝑅𝑐𝑖 .
2. We assume relation between microscopic absorbing
reaction rate and ND of themselves linearly in eachtime step and

𝑑𝑅 𝑖
𝑑𝑁𝑖

Fig. 2. Errors of kinf in the VERA/2O problem

is obtained as follows:𝑑𝑅𝑖 𝑅𝑝𝑖 − 𝑅𝑐𝑖=.
𝑑𝑁 𝑖 𝑁0𝑖 − 𝑁𝑝𝑖

(7)
𝑑𝑅 𝑖

3. With the parameters, 𝑁0𝑖 , 𝑅𝑝𝑖 ,and ∆𝑡 , a𝑑𝑁𝑖
simplified model presented as Eq. (4) is prepared and
𝑁𝑟𝑖 is numerically calculated. In addition, we
calculate 𝑅̃𝑐𝑖 in the simplified model also.
4. From Eq. (6), a proper weight ω is calculated, andthen 𝑁 𝑖 is obtained from Eq. (3).

Fig. 3. Errors of kinf in the VERA/2P problem

4. Numerical Results
Differences in kinf obtained by the PC method and the
OWPC method against the references are shown in Figs.
2 to 4. The OWPC method shows better performance thanthe original PC method and Projected Predictor-Corrector
method (PPC method) whose time step width is equal tothat of the OWPC method. The PPC method modifies
reaction rate at the end of time step. However, the otherapproximation remains in the PPC method. On the other
hand, our OWPC method does not introduce theseapproximations. Therefore, the PPC method performs
better than the PC method but inferior to the OWPC
method. Notably, the OWPC method shows good
performance in the 2O and 2P fuel assemblies.
Gadolinium concentration of these two assemblies is lessthan that of the Okumura problem. When the large amountof Gd is included, thermal neutron absorption is enhanced,
and neutron flux energy spectrum becomes hard. Insuch cases, contributions of other reaction except thethermal neutron absorption become large. In the OWPC
method, we assume that ND of Gd changes mainly due toneutron absorption. That is why the OWPC method shows
poor performance in the Okumura problem in comparison
with the 2O and 2P fuel assemblies.

Fig. 4. Errors of kinf in the Okumura problem
We assume that the microscopic reaction rate is
proportional to the ND for each isotope: Gd-155 reaction
rate with ND of Gd-155 and Gd-157 reaction rate with
ND of Gd-157. This is because ND of Gd influences
neutron spectrum. Since neutron absorbing cross sectionof Gd-157 is larger than that of Gd-155, Gd-157 burns out
earlier than Gd-155. A comparison of macroscopic
absorption reaction rates is shown in Fig. 5. This figuresuggests that Gd-155 absorption becomes dominant after
the burnout of Gd-157. Therefore, we should use
correlation between microscopic reaction rate of Gd-157
and ND of Gd-155 in such situations. Based on thisobservation, we can say that during the period when Gd155 absorption is dominant, we should use correlation145

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

between ND of Gd-155 and microscopic reaction rate of
Gd-157. We judge a timing when the isotope which burnup mainly changes by comparing depletion quantity ofthese two isotopes. In the case that ND of Gd-157
decreases more than that of Gd-155, we use originalcorrelation of the OWPC method. On the other hand, inthe case that Gd-155 decreases more than Gd-157, insteadof𝑑𝑅 157𝑑𝑁157

5. Conclusion
A new algorithm to determine a proper weight in the
optimally-weighted predictor-corrector method has been
proposed.
This concept is based on problem
simplification in which proper weight can be easily
obtained. Verification calculation has been performed forthree PWR fuel assemblies including Gd2O3. Better
performance of the OWPC method over the original PC
method and the PPC method has been demonstrated. In
addition, we have examined which correlation is better toimprove calculation accuracy.

, we use the correlation as follows:
𝑅𝑝157 − 𝑅𝑐157
𝑑𝑅157
=
.
𝑑𝑁 155 𝑁0155 − 𝑁𝑝155(9)

We call this method in which we change correlation as an
OWPC-MIX method.

References
1.
2.

3.

4.

Fig. 5. Macroscopic reaction rates of gadolinium isotopes
Result of the OWPC-MIX method in the VERA/2O
problem is shown in Fig. 6. Using the OWPC-MIX
method, the calculation accuracy is not affected, andsimilar results are obtained in the other two assemblies.
The isotope effect of the correlation between ND andreaction rate is not important in these problems.

5.

6.

7.

Fig. 6. Errors of kinf in the VERA/2O problem

146

Stamm’er RJJ, et al., Methods of Steady-State
Reactor Physics in Nuclear Design, Academic Press(1983).
Yamamoto M, “Coarse Time-Step Integration
Method for Burnup Calculation of LWR Lattice
Containing Gadoninium-Poisoned Rods,” J. Nucl.
Sci. Technol., 22, 1 (1985).
Yamamoto A, Tatsumi M, Sugimura N, “Projected
Predictor-Corrector Method for Capability for LWR
Fuel Assembly in Reactor Physics Burnup
Calculation,” Nucl. Sci. Eng., 163, 144 (2009).
Lee DJ, Rhodes J, Smith K, “Quadratic Depletion
Model for Gadolinium Isotopes in CASMO-5,” Proc.
Int. Conf. Adv. Nucl. Fuel Management, ANFM 2009,
Hilton Head Island, SC, April 12-15, 2009 (2009).
Okumura S, Chiba G, “Development of Nuclear Fuel
Depletion Calculation for LWR Fuel Assembly in
Reactor Physics Code System CBZ,” Proc. of
Reactor Physics Asia 2017, RPHA17, Chengdu,
China, August 24-25, 2017 (2017).
Shibata K, Iwamoto O, Nakagawa T, Iwamoto N,
Ichihara A, Kunieda S, Chiba S, Furukawa K, Otuka
N, Ohsawa T, Murata T, Matsunobu H, Zukaran A,
Kameda S, Katakura J, “JENDL-4.0: A new libraryfor nuclear science and technology,” J. Nucl. Sci.
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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Robustness of the GPS Functions in Pinwise Neutronics Analysis of PWRsa

Hwanyeal Yua and Yonghee Kima,
Dept. of Nuclear and Quantum Eng., KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, Korea, 34141*
Corresponding author: yongheekim@kaist.ac.kr

Abstract
The generalized equivalence theory (GET) plus superhomogenization (SPH)[GET Plus SPH (GPS)] method, which is a new leakage correction method has
been investigated to demonstrate its applicability to PWR cores with different
conditions. In the GPS method, the pin-wise cross section (XS)-dependent SPH
factors are parameterized as a function of normalized leakage information, i.e.,current-to-flux ratio (CFR). In this paper, the given GPS functions, which are predetermined at BOC HFP condition have been applied to two different conditions(HZP and CZP). Both UOX-loaded and partially MOX-loaded small PWR cores
are analyzed using given HFP GPS functions to investigate the temperaturedependency of the GPS method. The two-dimensional method of characteristics(MOC)-based DeCART2D code is used for both lattice and reference corecalculation in this work. The GPS method is implemented to an in-house nodal
expansion method (NEM)-based pin-wise diffusion code with the hybrid CMFD(HCMFD) algorithm. Based on the comprehensive result of this work, it isconcluded that the GPS method has minor temperature dependency.
Key Words: Pin-by-pin core analysis, Leakage correction, GPS methodcores and lattice calculations. The GPS method isimplemented to an in-house pin-wise diffusion solver
with the hybrid CMFD (HCMFD) algorithm [6].

1. Introduction
The conventional two-step core analysis is based on thehomogenized parameters obtained by the heterogeneoustransport calculation with the all-reflective boundary
condition. Consequently, the homogenized groupconstants (GCs) are always subjected to an inherent
spatial homogenization error due to the loss of
information and the unphysical boundary condition.
Recently, a new leakage correction method, called
GET plus SPH (GPS) method was proposed to overcome
this unavoidable error of conventional two-step core
analysis. As a combination of generalized equivalence
theory (GET) [1] and super-homogenization method(SPH) [2], the modified SPH concept was introduced tocorrect the pin-wise cross section (XS) of conventional
GET-based two-step core analysis. The feasibility of the
GPS method was demonstrated in the previous studies[3,4].
In this paper, the applicability of the GPS method todifferent temperature conditions (including densitychange of coolant) is investigated. Both UOX-loaded andpartially MOX-loaded small PWR cores with threedifferent conditions were considered: Hot Full Power(HFP), Hot Zero Power (HZP), Cold Zero Power (CZP).
These small PWR cores were analysis with given GPS
functions, which are pre-determined at HFP condition(hereafter called HFP-GPS function) in previous studies[3,4]. A 2-D method of characteristics (MOC) basedlattice code, DeCART2D [5] was used for the reference

2. The GPS Methodologies
In the GPS method, a modified SPH factor, called XSdependent SPH factor is introduced to correct the pinwise XSs of the conventional two-step core analyses ref  corrected ,() )SPH  SPH   , g   , g
 ,gg

SA, gSA, gwhere g is the group index,  denotes reaction type,SAref, g is the reference pin-wise XS,  , g is the pin-wise
XS obtained from the 2-D lattice calculation, and SPHg isthe standard SPH factor.
The XS-dependent SPH factor is parameterized as afunction of pin-wise leakage information, current-to-flux
ratio (CFR), defined as in Eq. (2):
CFR g 

J ,
s

g

s
g

(2)

where g is the group index, s is the surface index, J gs isthe outward net current on a surface, and g is the pinwise volume-averaged flux.
In the GPS method, the change of both XS-dependent
SPH factor and CFR are defined as:(3)
SPH ,g  SPH ,g  SPHSA,g ,147

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

CFRg  CFRg  CFRgSA ,where

SPHSA, gand

CFRgSA

300 K () .0 g/cm3). In case of HZP condition, temperatures
were set to be 555 K (0.7603 g/cm3 for coolant density)
for all with an assumption of conventional temperature
difference (30 K) between core inlet and outlet
temperature.
To investigate the temperature dependency of the GPS
method, the HFP-GPS functions from previous studies[3,4] were used to analyze small PWR cores of HZP and
CZP condition. To focus on temperature dependency of
the GPS function only, the pin-wise two-group constants
used for each condition are determined fromcorresponding condition. Similar with the pin-wise groupconstants, the SPHSA, g and CFRgSA in Eqs. (3) and (4) are(4)

are the pin-wise XS-

dependent SPH factor and CFR from the infinite latticecalculation, respectively.
The changes in the pin-wise XS-dependent SPH
factors and their initial values from the lattice calculation
are functionalized by the change of pin-wise two-group
CFR for reaction  and group g (‘F’ for fast group and‘T’ for thermal group)SPH , F  a1, ,F CFRF  a2, , F CFRT  a3, , F , (5)SPH ,T  a1, ,T CFRF  a2, ,T CFRT  a3, ,T ,(6)

where a1, , g , a2, , g , and a3, , g are coefficients.

also set to be values from corresponding conditions.

3. Numerical Results and Discussions

3.1 UOX-loaded small PWR

To investigate temperature dependency of the GPS
method, a small PWR [3] in Fig. 1 and a partially MOXloaded PWR (KAIST 1A benchmark) [4,7] in Fig. 2 wereconsidered. The FA designs are same as the one of Ref. 3
and 4. For the consistency, the baffle-reflector regions arealso treated with pin-wise GCs. The pin-wise GCs ofbaffle-reflector are determined by same processes with
Ref. 3.

Table 1 shows summaries of an UOX-loaded small
PWR with different conditions. It is clearly shown that, in
HZP and CZP conditions, the GPS correction by HFPGPS functions can provides similar accuracy in terms ofeigenvalue and pin-power as shown in Table 1 and Figs.
3 to 5. It is understandable since the change of XSdependent SPH factor is parameterized with the change of
CFR information and temperature effect is already taken
into account by the pin-wise group constants. Therefore,
the temperature effect on GPS functions is minor.
Table ) . Summaries of UOX-loaded small PWR
Condition

k-eff

∆𝛒
[pcm]

HFP condition
1.112455
1.113307
68.81
1.112681
18.28
HZP condition
Ref. DeCARD2D
1.123510
Standard two-step*
1.124351
66.55
GPS
1.123737
17.97
CZP condition
Ref. DeCARD2D
1.139892
Standard two-step*
1.140636
57.24
GPS
1.140155
20.21*
GET-based two-step core analysis
Ref. DeCARD2D
Standard two-step*GPS

Fig. ) . Configuration of the small PWR benchmark

Pin-power% errorMax (RMS)-2.95 (0.82)
1.64 (0.21)-2.98 (0.87)
1.38 (0.22)
3.02 (0.95)
1.70 (0.33)

Fig. 2. Configuration of the KAIST ) A benchmark
The HFP and CZP conditions of both small PWRs are
the same as the one of Ref 7. For HZP condition, theaverage temperatures of coolant, cladding, and fuel are
570 K (0.7295 g/cm3), 630 K, and 900 K, respectively.
For CZP condition, all temperatures are assumed to be

Fig. 3. FA-wise maximum and RMS pin-power %error
distribution (HFP condition)148

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig. 4. FA-wise maximum and RMS pin-power %error
distribution (HZP condition)

Fig. 6. FA-wise maximum and RMS pin-power %error
distribution (HFP condition)

Fig. 5. FA-wise maximum and RMS pin-power %error
distribution (CZP condition)

Fig. 7. FA-wise maximum and RMS pin-power %error
distribution (HZP condition)

3.2 Rodded UOX-loaded small PWR
A rodded UOX-loaded small PWR was also
considered. In this study, only two rodded FAs are locatedin inner core regions as shown in Fig. 1. As shown in
Table 2 and Figs 6 to 8, the GPS correction by HFP-GPS
functions effectively improved the accuracy of the pinby-pin core analyses with different conditions. Although
the maximum error in CZP condition is still significant,the normalized pin-power of corresponding pin isrelatively low, 0.115.
Table 2. Summaries of rodded UOX-loaded small PWR
Condition
Ref. DeCARD2D
Standard two-stepGPS
Ref. DeCARD2D
Standard two-stepGPS
Ref. DeCARD2D
Standard two-step
GPS

k-eff

∆𝛒
[pcm]

HFP condition
1.014952
1.016220122.971.014905-4.57
HZP condition
1.025144
1.026401
119.451.025096-4.56
CZP condition
1.042166
1.043337
107.671.042374
19.14

Fig. 8. FA-wise maximum and RMS pin-power %error
distribution (CZP condition)

Pin-power% errorMax (RMS)

3.3 Partially MOX-loaded small PWR-4.15 (1.14)-1.57 (0.59)

Table 3 shows summaries of a partially MOX-loaded
small PWR with different conditions. Figures 9 to 11
show the FA-wise maximum and RMS pin-power %error
distribution of each condition. As expected, even if
different GPS functions are applied in terms oftemperature condition, the GPS corrections improveaccuracy of pin-by-pin core analysis.

-4.21 (1.20)-1.89 (0.66)-4.98 (1.29)-3.36 (0.67)

149Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019
Table 3. Summaries of partially MOX-loaded PWR
Condition
Ref. DeCARD2D
Standard two-stepGPS
Ref. DeCARD2D
Standard two-stepGPS
Ref. DeCARD2D
Standard two-step
GPS

k-eff

∆𝛒
[pcm]

HFP condition
1.116271
1.118000
138.571.116599
26.30
HZP condition
1.127773
1.129454
132.001.128095
25.32
CZP condition
1.140499
1.142178
128.92
1.141020
40.03

distribution (CZP condition)

Pin-power% errorMax (RMS)

It is noted that, in rodded and partially MOX-loaded
cores there are slightly higher errors of CZP condition
compared with HZP and HFP condition. It is expectedthat it is caused by huge temperature difference anddifficulty of problem itself. Fortunately, the impact isminor and the accuracy is acceptable.

3.94 (1.15)-1.78 (0.34)
3.89 (1.19)-1.90 (0.36)

4. Conclusions
In this paper, the applicability of the GPS functions todifferent temperature conditions was investigated. BothUOX-loaded and partially MOX-loaded small PWR coreswith three different conditions, HFP, HZP, and CZP wereanalyzed with given GPS functions obtained from HFP
conditions. Based on the implementation of given GPS
correction on the different temperature conditions, it canbe concluded that the GPS method has minor temperaturedependency since the change of XS-dependent SPH
factor is parameterized by the change of CFR informationin the GPS functionalization. The temperature effect will
be considered by the pin-wise group constants from THcoupling.

4.13 (1.22)-2.18 (0.43)

Acknowledgements
This work was supported by the National Research
Foundation of Korea (NRF) Grant funded by the Korean
Government (MSIP) (NRF-20) 6R) A5A) 0) 39) 9)

Fig. 9. FA-wise maximum and RMS pin-power %error
distribution (HFP condition)

References
1. K. S. Smith, “Assembly Homogenization Techniquesfor Light Water Reactor Analysis,” Progress in
Nuclear Energy, 17, 303, 1986.
2. A. Hebert, “A Consistent Technique for the Pin-byPin Homogenization of a Pressurized Water Reactor
Assembly”, Nucl. Sci. Eng., 113, 227, 1993.
3. H. Yu, W. Kim, and Y. Kim, “A Leakage Correction
with SPH Factors for Two-group Constants in GETbased Pin-by-Pin Reactor Analyses”, Annals of
Nuclear Energy, Vol. 129, p. 30-50, July, 2019.
4. H. Yu, J. Kim, and Y. Kim, “Pinwise Diffusion
Solution of Partially MOX-loaded PWRs with the
GPS (GET PLUS SPH) Method”, Nuclear Scienceand Engineering, online published, June, 2019
5. J. Y. Cho, DeCART2D v) .) User’s Manual,
KAERI/UM-40/2016, 2016
6. S. Song, H. Yu, and Y. Kim, “One-node and two-node
hybrid coarse-mesh finite difference algorithm for
efficient pin-by-pin core calculation”, Nuclear
Engineering and Technology, Vol. 50, Issue 3, 327339, 2018
7. N. Z. Cho, “Benchmark Problem ) A: MOX FuelLoaded Small PWR Core (MOX Fuel with Zoning)”,
KAIST/NuratT, http://nurapt.kaist.ac.kr/benchmark,June 23 (2000)

Fig. )0 . FA-wise maximum and RMS pin-power %error
distribution (HZP condition)

Fig. )) . FA-wise maximum and RMS pin-power %error

150Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

APEC-corrected NEM Analysis of the VERA Core
Seongdong Janga, Kyunghoon Leeb and Yonghee Kima,∗
Korea Advanced Institute of Science and Technology, 291, Daehak-ro, Yuseong-gu, Daejeon, 34141, Korea
b
Korea Atomic Energy Research Institute, 111, Daedeok-daero 989beon-gil, Yuseong-gu, Daejeon, 34057, Korea*
Corresponding author: yongheekim@kaist.ac.kraAbstract
The limitation of the conventional two-step method for the modern reactor physics
analysis is quite obvious due to the lack of leakage information. To overcome its
genetic limitation caused by unphysical boundary condition at the single assembly
lattice calculation, the albedo-corrected parameterized equivalence constants(APEC) method has been proposed. The main idea of this APEC method is toconstruct linear APEC functions taking into account normalized leakage
information and then use it during NEM calculation. The APEC functions were
set up in terms of the current-to-flux ratio, spectral index and flux ratio to consider
leakage information. The purpose of this study is to verify the impact of the APECcorrected NEM analysis by solving 2-D VERA core benchmarked problem at the
0 MWD/MTU burnup. All of the single assembly lattice and color-set wascalculated by DeCART2D code and NEM nodal analysis with APEC method was
conducted by in-house NEM nodal code. The results showed that the solution of
the APEC-corrected NEM analysis can be improved in terms of the multiplication
factor and relative power error. It is concluded that the APEC method is a one of
the improved methodology to assist the conventional two-step method.
Key Words: APEC Method, VERA Core, In-situ Leakage Correction
1. Introduction

2. APEC Leakage-corrected Two-step Method

The conventional two-step method based on thegeneralized equivalence theory (GET) is the one of thecorner-stone in the modern reactor physics analysis. The
main idea of the GET is that the equivalence can beapplied into the homogeneous diffusion nodal analysis bypreserving the reaction rate of the heterogeneous transport
analysis via homogenized group constants (HGCs) anddiscontinuity factors (DFs). The actual application to theconventional two-step method is through the single
assembly lattice calculation to obtain the HGCs, which
are also called flux weighted constants (FWCs), andassembly DFs (ADFs). However, some errors of themultiplication factor and power distribution are occurred
due to its limit by neighboring effects so the neighbor- andspectrum-irrelevant leakage correction are mainly used.
Recently, the albedo-corrected parameterized
equivalence constants (APEC) method was developed tocorrect all of the lattice-based HGCs and DFs during
nodal analysis. It was showed that the HGCs and DFs canbe updated as an in-situ correction method during thenodal analysis through the APEC functions taking into
account normalized leakage parameters such as a currentto-flux ratio (CFR) and flux ratio (FR). In this study, theeffect of the APEC-corrected NEM analysis is discussed
by solving 2-D VERA core.

The main principle of the APEC leakage-corrected
method is that the HGCs and DFs are corrected by APEC
functions which are predetermined by the color-set
calculation. It is possible to obtain all of the coefficientsof the APEC functions, which can consider the actualleakage information based on their lattice-wise HGCs andADFs, by multiple linear regression using the results of
the additional color-set model. In this study, to focus on
the effect of the APEC functions for the fuel assembly(FA), the reference HGCs and DFs, which were obtained
from the whole core transport calculation, were used forthe reflector region. All of the single assembly lattice andcolor-set model was calculated by DeCART2D code andthe homogeneous diffusion NEM nodal calculation basedon APEC method was analyzed by in-house NEM nodal
code.
The philosophy of the APEC methodology is based onconventional two-step method as follows: 1) obtaining thesingle assembly-wise HGCs and ADFs by latticecalculation, 2) constructing APEC function bydetermining all of coefficients in terms of leakage
parameter via additional color-set calculation including
the results of the lattice calculation and 3) applying the insitu correction of the HGCs and DFs through the updated
leakage parameters in the iteration of nodal calculation.

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Osaka, Japan, Dec. 2-3, 2019

3. Numerical Results

2.1 APEC functions for the HGCs
To do the in-situ correction of the HGCs during NEM
nodal calculation, the APEC cross section (XS) functions
are set up in terms of node- and group-wise CFRs. As
shown in Eq. (1), the HGCs based on the single assembly
HGCs can be corrected through the updated △XS terms.
The APEC XS functions are defined as △XS for eachenergy group as shown in Eq. (2) and (3), respectively.

Σ x , g = Σ SA
x , g + ∆Σ x , g=∆Σ x , F ax , F

s
F

(1),
s
T

∑J +b ∑J +c
∑ φ
∑ φ
x, F

s

F

s

x, F

s
T

s

(2),

T

2

∑ J + b  ∑ J  + c
 φ ∑ φ∑ =∆Σ x ,T ax ,TTo verify the APEC-corrected NEM analysis, 2-D VERA
core was analyzed as a benchmark problem. The core andFA configuration are as shown in Fig. 1.

s
T

x ,T

T

s

x ,T

(3),

T

where, Σ SA : single assembly HGC,
F : fast group, T :thermal group,savg
J : surface current, φ s: surface flux, φ :average flux,

= 0 for interior FA.
cx , G ≠ 0 for peripheral FA
Note that the constant terms are designated depending on
the position of the FA. Using the node-wise CFRparameters calculated by color-set calculation, thecoefficients can be determined. Then, these coefficientscan do the in-situ update of the △XS by node-wise CFRs
determined by NEM calculation during its iteration.
2.2 APEC functions for the DFs

In case of the APEC DF functions, improved model isconsidered as follows:=
DFg ADFg + ∆DFg
Avg

ag ,1
∆DF
=
g

φg

s

φ g

+ ag ,2

s
g

J
∑J +c
+ ag ,3
g
s
s
φ
φ
s
g

g

∑

Fig. 1. 2-D VERA core and fuel assembly configuration.

(4),

The color-set was calculated based on the list of them andtheir models as shown in Table I and Fig.2, respectively.

(5).

Table I. List of Color-set Calculationsg

It was demonstrated that surface-wise leakage parameters,such as FR and CFR, can improve the accuracy of the DF
even for partially MOX-loaded PWRs. The generalized
procedure to obtain all of the coefficients of APEC DF
functions is as follows: 1) classification by FA types, 2)(optional) classification by surface directions, 3) (optional)
filtering by adjacent FA types, and 4) generation of APEC
DF functions. In this study, optional surface-wise APEC
DF functions are considered for the peripheral FA which
is mark by (*) with the name of the FA type as shown in
Fig. 3.

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019
A0
1.29
-2.07
-0.78
-0.22
0.22

B2
1.13
-4.20
-0.40
-0.11
-0.51
A0
1.26
-1.90
-0.67
-0.22
0.32

Fig. 2. Color-set models for the APEC XS and DF.
In this study, the sensitivity test of the APEC DF constantterm in Eq. (5) was conducted. Only for the APEC DF
functions, they were categorized depending on theexistence of the constant term as follows: 1) APEC-a (w/oconstant) and 2) APEC-b (w/ constant).
Table Ⅱ. Results for VERA 2-D Core
RMS
∆ρ
XS
DF
keff
(%)
(pcm)
DeCART2D
1.162661
FWC
ADF
1.161181 -109.59 1.988
APEC
ADF
1.162309 -26.04 0.456
APEC APEC-a 1.162415 -18.20 0.121
APEC APEC-b 1.162667
0.43
0.306

A0
1.26
-1.67
-0.64
-0.19
0.25
B3
1.06
-4.58
-0.57
-0.09
-0.20
A0
1.26
-1.09
-0.56
-0.13
0.45

B2
1.14
-3.33
-0.29
-0.01
-0.45
A0
1.28
-0.93
-0.59
-0.10
0.37
B2
1.14
-2.68
-0.20
0.05
-0.31
A0
1.27
0.32
-0.43
0.00
0.68

FA Type
DeCART2D
FWC / ADF (%)
APEC XS/ ADF (%)
APEC XS/ APEC-a DF (%)
APEC XS/ APEC-b DF (%)

Max.
(%)

A0
1.27
-0.48
-0.57
-0.06
0.38
B2
1.13
-2.48
-0.13
0.11
-0.35
A0
1.27
0.27
-0.49
-0.03
0.31
B2
1.08
-0.93
0.09
0.18
-0.43
B0
1.22
2.29
-0.05
0.42
0.01

B2
1.06
-2.16
-0.05
0.00
-0.45
A0
1.21
0.33
-0.37
-0.03
0.26
B1
1.12
-0.68
0.19
0.06
-0.09
A0
1.11
2.12
0.05
-0.07
0.32
B3
0.76
1.22
0.79
0.14
-0.62
C2*
0.71
4.87
2.09
0.43
0.13

A0
1.03
0.81
-0.12
-0.15
0.24
C4
0.95
-1.73
0.11
0.07
-0.20
A0
1.03
2.02
0.12
0.04
0.32
C3*
0.89
1.97
1.03
0.03
-0.06
C0*
0.77
4.99
1.15
0.03
-0.59
C0*
0.50
5.95
1.17
-0.12
-0.26

C2
0.68
0.52
0.06
-0.47
-0.34
C0*
0.78
1.97
-0.15
-0.03
-0.09
C1*
0.68
2.62
0.82
0.06
0.44
C0*
0.55
4.23
0.85
0.00
0.22

Fig. 3. Power distribution of VERA 2-D core

The reference power distribution of the VERA core, theresults of the conventional two-step method and theresults of the in-situ APEC correction were calculated asshown in Fig. 3. The results showed that the relative errorof the power distribution from the conventional two-stepmethod can be reduced to around zero by applying APEC
correction. Specifically, the maximum and minimum
relative error of the conventional two-step method, (5.95,-4.58), could be reduced to (2.09, -0.78) respectively,
when only APEC XS correction was applied. In case ofin-situ APEC XS and DF correction, these maximum andminimum values could be improved to (0.43, -0.47) by
APEC-a (w/o constant), and (0.68, -0.62) by APEC-b (w/constant).

5.954
2.091
0.473
0.677

The results showed that APEC XS and APEC DF
functions can improve the results in terms of themultiplication factor and relative power error as shown in
Table Ⅱ. Although the results of the conventional two-stepmethod using FWC XS and ADF could be significantly
improved by applying only in-situ APEC XS update withconventional ADF, it was still not good enough toovercome the limit of the conventional two-step method.
It was noted that introduced in-situ APEC DF correction
can enhance the results when in-situ APEC XS correctionis also applied simultaneously. It can be interpreted that
this APEC DF correction can overcome the limit which isoccurred when APEC XS correction is only used.
Note that the constant term in the APEC DF functions
may affect to the multiplication factor because it isintegrated parameter of the whole core. Depending on
their power distribution, the multiplication factor may besensitive due to the error cancellation. Therefore, it might
be hard to say which APEC DF function is better than theother in point of view of the multiplication factor. In termsof the relative power error, both of the APEC DF function
can improve the results compared to those of theconventional two-step method. Regarding the sensitivityof the constant term in the APEC DF function, the resultsof the APEC-a (w/o constant) are better than those of
APEC-b (w/ constant) in terms of the relative power error.

Table Ⅲ. Relative RMS Error of the HGCs
RMS Error
APEC XS /APEC XS /FWC
APEC-a DF
APEC-b DF(%)
1.45E-01
6.60E-02
6.59E-02
D1
4.85E-03
1.06E-03
1.30E-03
Σ𝑎𝑎1
υΣ𝑓𝑓1
1.78E-03
6.22E-04
7.77E-04
2.05E-02
3.31E-03
4.06E-03
Σ𝑠𝑠1→2
D2
Σ𝑎𝑎2
υΣ𝑓𝑓2

Σ𝑠𝑠2→12.73E-01
5.79E-02
1.21E-01
6.53E-04

9.23E-03
1.61E-03
3.08E-03
1.76E-05

2.68E-02
4.63E-03
1.04E-02
5.75E-05

The results of Table Ⅲ showed the comparison of HGCs
RMS error based on those of reference. The FWC

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019indicates the HGCs calculated by lattice calculation and
the APEC XS means the converged HGCs in in-situ
APEC corrected nodal calculation. It is noted that the
HGCs could be converged different values depending on
the APEC DF function because the leakage parameters
might be differently updated depending on their APEC
function.
The results showed that all of the HGCs converged by
APEC correction could be improved compared to those of
conventional two-step method.

CASL-U-2012-0131-004, Oak Ridge National
Laboratory (2014).
6. S. Jang, and Y. Kim “Improved Discontinuity Factor
Modeling for the in-situ APEC Leakage Correction in
Nodal Analysis,” Transactions of ANS Winter,
November 17-21, Washington, D.C., 2019

Table Ⅳ. Relative RMS Error of the DFs
RMS Error (%)
ADF
APEC-a APEC-b
Fast Group
Thermal Group

0.794
1.821

0.369
0.665

0.462
1.245

Similar to Table Ⅲ, the relative RMS error of the DFs
compared to the reference are arranged in Table Ⅳ. The
results showed that the converged APEC DF values are
improved compared to ADFs which are determined bylattice calculation. Especially for APEC-a (w/o constant)function, the thermal group DFs were significantly
improved compared to those of APEC-b (w/ constant)function.
4. Conclusions
The APEC-corrected NEM analysis has been successfully
verified by solving the 2-D VERA core benchmark
problem. It is demonstrated that the in-situ APEC leakage
correction method can improve the results in terms of themultiplication factor and relative power error, when bothof the HGCs and DFs are corrected simultaneously.
Acknowledgements
This work was supported by the National Research
Foundation of Korea (NRF) Grant funded by the Korean
Government (MSIP) (NRF-2016R1A5A1013919)
References
1.

2.

3.

4.
5.

K. Lee, W. Kim and Y. Kim, “Improved Nodal
Equivalence with Leakage-corrected Cross Sections
and Discontinuity Factors for PWR Depletion
Analysis,” Nucl. Eng. Tech., 51, 5, (2019).
W. Kim, W. Heo and Y. Kim, “Improvement of
Nodal Accuracy by Using Albedo-Corrected
Parameterized Equivalence Constants,” Nucl. Sci.
Eng., 188, 207 (2017).
W. Kim, K. Lee and Y. Kim, “Functionalization of
the Discontinuity Factor in the Albedo-corrected
Parameterized Equivalence Constants (APEC)
Method,” Nucl. Sci. Eng., 192, 1 (2018).
J. Y. Cho, et al., “DeCART2D v1.0 Methodology
Manual,” KAERI/TR-5283/2013, Korea Atomic
Energy Research Institute (2013).
A. T. Godfrey, “VERA Core Physics Benchmark
Progression Problem Specifications, Revision 4,”154Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Discontinuity Factor: A Discontinuity Condition for Angular Flux?
Akio Yamamotoa, Tomohiro Endoa
Nagoya Unversity, Furo-cho, Chikusa-ku, Nagoya, Japan*
Corresponding author: a-yamamoto@energy.nagoya-u.ac.jpaAbstract
The discontinuity factor (DF) has been considered as a discontinuity condition forscalar flux, partial current, or angular flux. However, DF can be also considered
as a partial reflective condition at a region interface and is consistent with therefractive index used in optics. The unphysical interpretation regarding to DF, e.g.,loss or production of neutrons at an interface, can be avoided through this new
insight.
Key Words: Discontinuity factor, scalar flux, partial current, angular flux,
refractive index
1. Introduction

surfaces, respectively.
By applying the relationship between the partial
currents and scalar flux in the diffusion theory, thefollowing relation for partial currents are obtained at theinterface (Fig.1):
𝑓 𝑓
2𝑓
𝐽
𝐽
𝐽 ,
𝑓 𝑓
𝑓 𝑓
(2)
𝑓
𝑓
2𝑓
𝐽
𝐽
𝐽 ,
𝑓 𝑓
𝑓 𝑓
where
𝐽 and 𝐽 : incoming and outgoing partial currents at
the left region surface,
𝐽 and 𝐽 : incoming and outgoing partial currents at
the right region surface.
𝑓𝑓

The discontinuity factor (DF) is a well-known concept in
the reactor analysis field and is widely used [1]-[3]. The
use of DF greatly reduces the homogenization error in theadvanced nodal method and is a key technology of its
success.
The application area of DF has become wider and it iscurrently applied not only to reduce homogenization
errors but also to reduce spatial and angular discretization
errors in diffusion and transport calculations.
Though there is no doubt on its usefulness, its concept
is somewhat strange since application of DF makes partial
current or angular flux discontinuous, i.e., “stream” ofneutrons becomes discontinuous. We usually accept thisinconsistency by excusing that the scalar flux/partial
current/angular fluxes appeared in homogenized (orenergy collapsed) calculations are not “physical”
quantities thus they can be discontinuous at an interface.
In the present summary, another interpretation of DF
that can avoid the above inconsistency is described [4].

𝐽
𝐽
𝐽

𝐽

𝐽

𝐽

2. Theory
Left region

There are various ways to apply DF. In this summary, thecontinuous condition is applied to the odd-parity angular
fluxes (or flux moments) while the discontinuous
condition is allowed for even-parity angular fluxes (orflux moments).
Firstly, let us consider the diffusion theory. In this case,the following interface condition is used:
𝑓𝜙
𝑓𝜙 ,
(1)
𝐽
𝐽 ,
where
𝑓 and 𝑓 : DFs for the left and the right region surfaces,
respectively,
𝜙 and 𝜙 : scalar fluxes at the left and the right region
surfaces, respectively,
and 𝐽: net currents at the left and the right region
𝐽

Right region

Fig 1. Relationship among partial and net currents
Equation (2) indicates that the partial currents are
discontinuous at the interface while the net neutron
current is continuous:
(3)
𝐽
𝐽
𝐽
𝐽
𝐽
𝐽 .
Next, we consider the transport equation in the Pnform. In this case, the discussion for the diffusion theory
can be extended straightforward manner, i.e., the scalar
flux and the net current are considered as the even and odd
angular flux moments, respectively.
Thirdly, the transport equation for even-parity angular
flux is considered. In this case, the following interface

155Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019condition is applied:𝑓 𝛀 𝜓 𝛀𝑓 𝛀 𝜓 𝛀 ,
(4)
𝜓 𝛀 ,
𝜓 𝛀
where
𝑓 𝛀 and 𝑓 𝛀 : DFs for even-parity angular flux atthe left and the right surfaces for direction 𝛀 , and𝛀 ,𝑓 𝛀
𝛀 .
𝑓𝑓𝑓 𝛀𝜓 𝛀 and 𝜓 𝛀 : even-parity angular fluxes at the left
and the right surfaces for direction 𝛀,𝜓 𝛀 and 𝜓 𝛀 : odd-parity angular fluxes at the left
and the right surfaces for direction 𝛀.
The even and odd parity angular fluxes are defined by:
𝜓
𝛀
𝜓 𝛀
,
𝜓 𝛀
2
𝜓
𝛀
𝜓 𝛀
𝜓 𝛀
,
2
(5)
𝜓
𝛀
𝜓 𝛀
𝜓 𝛀
,
2
𝜓
𝛀
𝜓 𝛀
𝜓 𝛀
,
2
where 𝜓 𝛀 and 𝜓 𝛀 are the angular flux at the left
and the right surfaces for direction 𝛀, respectively.
By inserting Eq.(5) into Eq.(4), the following relations
are obtained:

Reflection coefficient for angular flux:

𝑓 𝛀
𝑓 𝛀
𝛀

𝑓 𝛀
𝜓
𝑓 𝛀

2𝑓 𝛀
𝜓
𝑓 𝛀
𝑓 𝛀
𝑓 𝛀
𝑓 𝛀

𝛀 ,

(6)

𝛀
𝑓 𝛀
𝜓 𝛀 .
𝑓 𝛀

3. Discussion
By applying the discontinuity condition for the scalar fluxand even-parity angular flux, the partial current and theangular flux become discontinuous as shown in Eqs.(2)and (6) except for DFs for the left and right regions are
identical. The partial current and angular flux represent a
stream of neutrons for a particular direction. Therefore,
Eqs. (2) and (6) indicate that the stream of neutrons is
discontinuous at an interface when DFs of the left andright regions are different. In this case, neutrons look like
disappeared or produced at an interface. Apparently,
neutrons are not disappeared or produced at the interface
thus the above interpretation is unphysical.
However, if Eqs. (2) and (6) are carefully observed,
they can be also considered as the transmission and thereflection of partial current or angular flux at an interface.
For example, the transmission and the reflection
coefficients for the partial current or angular flux goingfrom left to right are expressed as:Transmission coefficient for partial current:
Reflection coefficient for partial current:

𝑓
𝑓

2𝑓

𝑓

𝑓
𝑓
.
𝑓

2𝑓 𝛀
,
𝑓 𝛀
𝑓 𝛀
𝑓 𝛀
𝑓 𝛀

𝑓 𝛀
.
𝑓 𝛀

(8)

Summation of the transmission and the reflection
coefficients are 1 as derived from Eqs.(7) and (8).
The above interpretation (the transmission andreflection) is more natural and intuitive. Namely, whendifferent regions having different DFs are adjacent,stream of neutrons (partial current or angular flux) arepartially reflected at the interface. On the other hand,when the adjacent regions have identical DF, no partialreflection occurs and all neutrons are transmitted toadjacent region. It should be noted that the reflection
considered in this discussion is different from physicalone. In physical reflection, flight direction of a neutronchanges at the surface of reflection. However, in thereflection described in Eq.(6), a neutron entering aninterface reflected to the opposite of entering direction
regardless of the entering angle to the interface.
It is worthwhile to see the relation with the optics. In
the case of light, the amplitude of the transmitted andreflected lights are described by:
2𝑛
𝐴
𝐴,
𝑛
𝑛
(9)
𝑛
𝑛
𝐴
𝐴,
𝑛
𝑛
where
𝑛 and 𝑛 : refractive indexes for the left and the rightmaterials,
𝐴 , 𝐴 , and 𝐴 : amplitude of entering, transmitted, andreflected lights.
Interestingly, there is a clear relationship between
Eqs.(8) and (9). The DF in Eq.(8) correspond to therefractive indexes in optics in Eq. (9). The light entering
to different material is partly reflected and the rest istransmitted. No reflection occurs when the same materials(or materials having the identical refractive indexes) areadjacent. This is an excellent analogy for the neutrontransport with DF. When different materials are adjacent,part of the entering neutrons are “reflected” and the rests
are transmitted. For example, Eq.(2) indicates that 𝐽 isa summation of the transmitted partial current from theleft region and the reflected partial current from the rightregion. As a result, we see the discontinuity of partial
current or the angular flux, which is an analogy of the
behavior of light.
𝑓 , the coefficients for the “reflection”
When 𝑓𝑓 / 𝑓 𝑓and the transmission term
term 𝑓
2𝑓 / 𝑓 𝑓 in Eq.(2) become negative and greaterthan 1, respectively. In this case, the transmitted partial
current or the angular flux is “amplified”. In order topreserve the neutron balance at the interface, the reflectedpartial current or the angular flux becomes negative.
The DF has been considered as a discontinuous
condition for scalar flux, partial current, or angular flux.
However, when the present discussion is taken into
account, the DF can be considered as the partial reflective
boundary condition at the material interface. In the most

2𝑓 𝛀
𝜓 𝛀
𝑓 𝛀
𝑓 𝛀

𝜓 𝛀

𝜓

Transmittion coefficient for angular flux:

,

(7)

156

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

simple and coarse case, a partial reflective boundary
condition for partial current is considered that reproduce
the reference reaction rate in a region. As a more detailcondition, a partial reflective boundary condition can beconsidered for face average angular flux or angular flux
on a ray trace. Spatial and angular resolution of partialreflective boundary condition can be adjusted by the
purpose. The advantage of the present approach is that theneutron balance at interfaces is always preserved.
In summary, no unphysical interpretation is necessary
by adopting this concept and it can be applied to diffusion
or transport theory with flexibility.
4. Conclusions
In the present summary, a new interpretation of DF is
described. The DF can be understood through the analogy
of optics, i.e., partial reflection and transmission of lightat a material interface. By adopting this concept, an
unphysical interpretation that requires artificial loss andproduction of neutrons at an interface is no morenecessary.
Acknowledgments
The author sincerely thanks Dr. Yung-An Chao for his
advice and insightful comments on this topic.
References
1. Koebke K., “A New Approach to Homogenization
and Group Condensation,” Proceedings of IAEA
Technical Committee Meeting on Homogenization
Methods in Reactor Physics, Lugano, Switzerland,(1978).
2. Smith K. S., “Assembly Homogenization Techniquesfor Light Water Reactor Analysis,” Prog. Nucl.
Energy, 17, 303 (1986).
3. Sanchez R., “Assembly Homogenization Techniques
for Core Calculations,” Prog. Nucl. Energy, 51, 14(2009).
4. Yamamoto A, Endo T, “A New Interpretation of
Discontinuity Factor,” Nucl. Sci. Eng., (2019).
https://doi.org/10.1080/00295639.2019.1579514

157

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Calculation of Higher Eigen-modes of the Forward and Adjoint Neutron Diffusion
Equations Using IRAM Algorithm Based on Domain Decomposition
Wenbin Wua, Yingrui Yua, Qi Luoa, Dong Yaoa, Qing Lia, and Xiaoming Chaia
Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China
NO. 328 Section 1 Changshun Ave, Shuangliu District, Chengdu, Sichuan Province, China 610213*
Corresponding author: wenbinwu@126.com

aAbstract
The higher eigen-modes of the forward and adjoint neutron diffusion equations
are widely used in several application perspectives for engineering purpose, suchas the power distribution reconstruction, the stability analysis, and generalized
perturbation theory (GPT) based sensitivity and uncertainty analysis. In thisstudy, the higher eigen-modes of both the forward and adjoint neutron diffusion
equations are calculated using implicitly restarted Arnoldi method (IRAM) basedon domain decomposition parallelism and the multi-group multi-domain coupled
PGMRES algorithm. A multi-dimensional LWR diffusion problem is solved forverification. Numerical results demonstrate that the forward equation and the
adjoint equation have the same eigenvalue spectrum, and the obtained higher
order eigen-vectors follow the orthogonality relationship. The numerical results
agree well with the anticipated theory analysis.
Key Words: Neutron diffusion equation, Forward and adjoint equations,
IRAM algorithm, PGMRES algorithm, Higher eigen-modes, Orthogonality.
domain coupled PGMRES algorithm.

1. Introduction
The neutron diffusion equation is usually represented asa generalized eigenvalue problem, of which thefundamental solution is the effective multiplication
factor of a fission system, i.e. 𝑘𝑒𝑓𝑓 , and the flux
distribution. For this diffusion eigenvalue system, thehigher eigen-modes could be computed for theory
analysis and some engineering applications in the
aspects of the power distribution reconstruction [1], thetransient analysis [2], the stability analysis [3], and theloading pattern optimization [4].

2. Methodologies
2.1 The forward and adjoint diffusion equations
The neutron diffusion equation is derived based on the
Fick’s law, which introduces a heuristic relation between
the neutron current and the gradient of the neutron flux[8]. The multi-group form of the forward neutron
diffusion equation is−𝛁 ∙ 𝐷𝑔 (𝒓)∇𝜙𝑔 (𝒓) + Σ𝑟𝑔 (𝒓)𝜙𝑔 (𝒓) = 𝑄𝑔 (𝒓),The adjoint (in contrary to forward) neutron diffusion
equation is also a generalized eigenvalue problem, of
which the fundamental solution, known as the adjoint
neutron flux, has the physical meaning of neutron
importance. The adjoint neutron flux is widely used in
the perturbation theory [5] and the sensitivity anduncertainty (SU) analysis. Similar as the forward neutron
diffusion equation, the higher eigen-modes of the adjoint
equation could also be computed, which have potential
value in high-order perturbation theory and generalized
SU analysis [6].

(1)

where
𝑄𝑔 (𝒓) = 𝑄𝑠𝑔 (𝒓) + 𝑄𝑓𝑔 (𝒓),𝐺

𝑄𝑠𝑔 (𝒓) = ∑ ℎ=1 Σ𝑔←ℎ (𝒓)𝜙ℎ (𝒓),ℎ≠𝑔

𝑄𝑓𝑔 (𝒓) =𝐺𝜒𝑔 (𝒓)∑ 𝜐Σ𝑓ℎ (𝒓)𝜙ℎ (𝒓).
𝑘ℎ=1

The symbols have their usual physical meanings
consistent with the definitions in literature [8]. The static
neutron diffusion equation is a generalized eigenvalue
problem, of which the fundamental solution corresponds
to the keff and the neutron flux.

In this study, the IRAM [7] algorithm is employed forthe calculation of the higher eigen-modes of both theforward and adjoint neutron diffusion equations based on
domain decomposition and the multi-group multi-

158

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

discretized diffusion equation. By moving the scattering
source term to the left side, the diffusion equation is rearranged as

The multi-group form of the adjoint neutron diffusion
equation is represented as−𝛁 ∙ 𝐷𝑔 (𝒓)∇𝜙𝑔∗ (𝒓) + Σ𝑟𝑔 (𝒓)𝜙𝑔∗ (𝒓) = 𝑄𝑔∗ (𝒓),

(2)

𝑳𝝓 =

where

1
𝑭𝝓 𝑜𝑟 𝑳−𝟏 𝑭𝝓 = 𝑘𝝓,
𝑘

(4)

where

𝜙𝑔∗ (𝒓) = adjoint flux,

𝝓 = scalar flux vector,
𝑳 = loss operator,
𝑭 = multiplicative operator.

∗
∗ (𝒓)
(𝒓),
𝑄𝑔∗ (𝒓) = 𝑄𝑠𝑔+ 𝑄𝑓𝑔
𝐺Generally, the diffusion equation is computed with thepower iteration method in the form of inner-outer
iteration. In this study, IRAM algorithm serves as the
eigenvalue solver and the multi-group multi-domain
coupled PGMRES algorithm serves as the linear solver.
These two algorithms will be described in detail in thefollowing sections.

∗ (𝒓)
𝑄𝑠𝑔
= ∑ ℎ=1 Σℎ←𝑔 (𝒓)𝜙ℎ∗ (𝒓),
ℎ≠𝑔

∗
(𝒓) =
𝑄𝑓𝑔

𝐺
𝜐Σ𝑓𝑔 (𝒓)∑ 𝜒ℎ (𝒓)𝜙ℎ∗ (𝒓).
𝑘ℎ=1

As we can see from equations (1)(2), the adjoint form isstraightforwardly obtained by permuting primary andsecondary group indices of the forward form. The
adjoint equation could be solved in the same methodwith the forward equation by properly manipulating theenergy groups.

2.3 The multi-group multi-domain coupled PGMRES
algorithm
For either the power iteration method or the IRAM
algorithm, it is necessary to solve the linear system
𝑳𝒙 = 𝑭𝝓 also known as the fixed source problem (FSP).
There exist many advanced algorithms widely used in
the reactor physics community, such as the GMRES
algorithm [9].

2.2 Discretization of the neutron diffusion equations
It is usually impossible to analytically solve the diffusion
equations. Hence, discretization methods such as thefinite difference and the finite element are employed totransform the algebraic operator into a matrix operator inorder to solve numerically.

GMRES is one of the famous Krylov sub-space methods(in contrary to the stationary iterative methods), which
employs orthogonalization based Arnoldi process. For
the Arnoldi process, several orthogonalization
techniques such as classical Gram-Schmidt, modified
Gram-Schmidt, and Householder transformations could
be adopted. All these techniques need the operation of
matrix-vector multiplication 𝒚 = 𝑳𝒙 . GMRES
algorithm is implemented in many excellent numericallibraries, such as Intel MKL and PETSc, which provide
convenient reverse communication interface. Only thematrix-vector multiplication operator should be provided
to the GMRES interface. In this study, domain
decomposition parallelism is applied. Hence, PGMRES,
the parallel version of the GMRES algorithm is used.

In this study, the mesh centered finite difference method
is employed to discretize the neutron diffusion equation,
as depicted in Fig. 1. The balance relation of the
diffusion equation is represented by the neutron currentsof all six surfaces as
𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

𝐽𝜐+,𝑔 − 𝐽𝜐−,𝑔∑𝜐𝜖(𝑥,𝑦,𝑧)Δ𝜐 𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

+ Σ𝑟,𝑔 𝜙𝑔

𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

− 𝑄𝑠𝑔 = 𝑄𝑓𝑔 ,

(3)

where
𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

𝑖+1,𝑗,𝑘𝐽𝑥+,𝑔 = 𝐷𝑒𝑓𝑓,𝑥+,𝑔 (𝜙𝑔

𝑖,𝑗,𝑘

− 𝜙𝑔

𝑖,𝑗,𝑘

𝑖,𝑗,𝑘

𝐷𝑒𝑓𝑓,𝑥+,𝑔 =

−2𝐷𝑔

𝑖,𝑗,𝑘

Δ𝑥 𝑖+1,𝑗,𝑘 𝐷𝑔),

In this study, the PGMRES algorithm is from the PETSc
library [10]. PETSc supports the matrix-free methods,
which require no explicit storage of the matrix. However,users should define routines representing operations of
the implicit matrix on a vector.

𝑖+1,𝑗,𝑘𝐷𝑔𝑖+1,𝑗,𝑘+ Δ𝑥 𝑖,𝑗,𝑘 𝐷𝑔.

Specific to the discretized multi-group and domaindecomposed diffusion equation, the input vector of themultiplication operation includes the fluxes of all groups
in all sub-domains, and the results vector is obtained bysubstitute the fluxes to the left hand side of equation (3).
Each sub-domain is assigned to a MPI process, which
stores only the fluxes of meshes within the individual

Fig. 1 Illustration of mesh centered finite difference
discretization method
Equation (3) is the fundamental equation of the159Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

sub-domain. Since the multiplication operation needs
fluxes of neighbors, message passing is applied for
fluxes transferring, which happens primarily in themultiplication operation.

〈𝑳∗ 𝝓∗ , 𝝓〉 =1 1
( − ∗ ) 〈𝑭𝝓, 𝝓∗ 〉 = 𝟎.
𝑘 𝑘

IRAM is an efficient and robust Krylov subspace
iterative method to calculate part or all of the higher
eigen-modes simultaneously. As with most Krylov
subspace iterative methods, only the matrix-vector
multiplication operator should be provided to the IRAM
interface to complete the Arnoldi process. For equation(4), the matrix-vector multiplication is written as

Inversion of a large matrix is often unacceptable. Hence,
it’s transformed into the FSP (6), which is solved by themulti-group multi-domain coupled PGMRES algorithm
proposed above.

The IRAM implementation in the ARPACK library is
widely used. Similarly as the PETSc library, the
ARPACK library wrapped in the SLEPc library [11]provides a reverse communication interface. It simplifies
the IRAM implementation into practical applications
because the IRAM solver can simply be wrapped aroundthe existing FSP module without explicitly constructing
the matrix 𝑳−𝟏 𝑭.

The mesh size employed for the finite difference
discretization is 1.25cm×1.25cm×2.5cm in the 𝑥𝑦𝑧
directions, leading to about 3 millions of meshes. The
stopping criteria are set to be 10-5 for PGMRES and 10-6
for IRAM. The platform where ParaFiDi runs is a clusterwith Intel SandBridge E5-2670 CPU linked by Infinite
band network. 320 CPU cores are used, and the total
elapsed time is around 1100 seconds for 500 orders of
eigen-modes.

The forward and adjoint forms of the diffusion equation
are represented as

1 ∗ ∗
𝑭𝝓.
𝑘∗

(8)

3.2 Accuracy of the fundamental solution

By multiplying equation (7) with 𝝓∗ and equation (8)with 𝝓, and integrating both equations over phase space,equation (7)(8) are re-written as〈𝑳𝝓, 𝝓∗ 〉 =1〈𝑭𝝓, 𝝓∗ 〉,𝑘(13)

A multi-dimensional LWR diffusion problem [12] issolved to demonstrate the capability of ParaFiDi for the
calculation of higher eigen-modes of forward and adjoint
neutron diffusion equations. It is a 3D 2-group 1/4
symmetric diffusion problem with the size of
170cm×170cm×380cm. Details of this problem isoriginally from reference [12].

2.5 The theory analysis of forward and adjoint higher
eigen-modes𝑳∗ 𝝓 ∗ =and 𝑘𝑚 = 𝑘𝑛∗ ; 〈𝑭𝝓𝑚 , 𝝓𝑛∗ 〉 ≠ 𝟎 𝑓𝑜𝑟 𝑚 = 𝑛.

3.1 The benchmark description and computing
parameters

(6)

(7)

(12)

In this study, two Krylov subspace iterative methods are
employed: IRAM for eigenvalue problems along with
PGMRES for linear systems. The PGMRES
implementation is from the PETSc library, and the
IRAM implementation is from the SLEPc library, which
wraps around the ARPACK and PARPACK libraries andmakes use of PETSc data structures. According to themethodologies above, the ParaFiDi (Parallel Finite
Difference) code based on domain decomposition isdeveloped using MPI (Message Passing Interface).

where 𝝓 is the input vector and 𝒚 is the result vector.

1
𝑭𝝓,
𝑘

〈𝑭𝝓𝑚 , 𝝓𝑛∗ 〉 = 𝟎 𝑓𝑜𝑟 𝑚 ≠ 𝑛,

3. Numerical results

(5)

𝑳𝝓 =

(11)

Therefore, the orthogonality relationship for the forward
and adjoint higher eigen-modes is expressed as

2.4 The application of IRAM for higher eigen-modes

𝑳𝒚 = 𝑭𝝓,(10)

Applying the definition of adjoint operators andsubtracting equation (9) from (10) results in

The fluxes of all groups in all sub-domains are solved
simultaneously by the PGMRES algorithm. Thus, this iscalled the multi-group multi-domain coupled PGMRES
algorithm.

𝒚 = 𝑳−𝟏 𝑭𝝓,

1 ∗ ∗〈𝑭 𝝓 , 𝝓〉.
𝑘∗Since the fundamental solution corresponds to the keffand neutron flux, it’s compared with the benchmark
results. keff by ParaFiDi is 1.02904, which shows 1 pcm
error with the reference. The maximum error of
assembly power is -0.94% located at the outer part of thecore. This implies the fundamental solution has good
accuracy.

(9)

160

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the IRAM and PGMRES algorithms, and making use of
these higher eigen-modes in GPT based SU analysis.

3.3 The eigenvalue spectrum
500 order eigenvalues of both the forward and adjoint
equations along with their relative differences are plottedin Fig. 2, which shows that the relative differences arewithin 5 pcm. This implies the forward and adjoint
equation have the same spectrum, which is consistent
with the adjoint theory.

Acknowledgements
This work is supported by the Exploring Grant of
Nuclear Power Institute of China under the file No.
JJXM-TSJJ-15-12-01.
References
1 C. Wang, H. Wu, L. Cao, et al. "On-Line
Reconstruction of in-Core Power Distribution by
Harmonics Expansion Method," Nucl Eng Des.
241(8), pp. 3042-3050 (2011).
2 Z. R. de Lima, F. C. Da Silva, A. C. M. Alvim. "AModal Multidimensional Kinetics Method Using
Pseudo-Harmonics," Ann Nucl Energy. 36(6), pp.
752-759 (2009).
3 F. Zinzani, C. Demazière, C. Sunde. "Calculation of
the Eigenfunctions of the Two-Group Neutron
Diffusion Equation and Application to Modal
Decomposition of Bwr Instabilities," Ann Nucl
Energy. 35(11), pp. 2109-2125 (2008).
4 X. Zhaodong. Study On the Improvement of LowOrder Harmonies Expansion Method. Beijing:
China Institute of Atomic EnergyNuclear Scienceand Engineering, 2007.
5 E. E. Lewls, J. R. W. F. Mlller. Computational
Methods of Neutron Transport. La Grange Park,
lliinois USA: American Nuclear Society, 1993.
6 S. Carney, F. Brown, B. Kiedrowski, et al. "Theory
and Applications of the Fission Matrix Method forContinuous-Energy Monte Carlo," Ann Nucl Energy.
73, pp. 423-431 (2014).
7 Y. Saad. Numerical Methods for Large Eigenvalue
Problems. Halsted Press, 1992.
8 A. Hébert. Multigroup Neutron Transport and
Diffusion Computations Handbook of Nuclear
Engineering. Springer, 2010:874.
9 Y. Saad. Iterative Methods for Sparse Linear
Systems. Society for Industrial and Applied
Mathematics, 2003.
10 S. Balay, J. Brown, K. Buschelman, et al. ”Petsc
Users Manual (Revision 3.4)”Argonne National
Laboratory（2013）.
11 C. Campos, J. E. Roman, E. Romero, et al. ”Slepc
Users Manual: Scalable Library for Eigenvalue
Problem Computations (Revision 3.4)”Universidad
Politecnica de Valencia, Spain（2013）.
12 A. C. Center. ”Benchmark Problem Book”,
ANL-7416. Computational Benchmark Problems
Committee of the Mathematics and Computation
Division of the American Nuclear Society（1977）.

Fig. 2 Eigenvalues of both the forward and adjoint
equations along with their relative differences
3.4 The orthogonality relationship
The angle of a forward eigenvector and an adjoint
eigenvector is defined as𝜃𝑚𝑛 = 𝑐𝑜𝑠 −1〈𝑭𝝓𝑚 , 𝝓𝑛∗ 〉, 𝑓𝑜𝑟 𝑚 ≠ 𝑛.
‖𝝓𝑚 ‖‖𝝓𝑛∗ ‖(14)

The results of 𝑀𝐴𝑋, 𝐴𝑉𝐺, 𝑅𝑀𝑆 are separately 1.25,
6.25×10-5, 8.30×10-6 in degree (90 means absolute
orthogonality), which implies the orthogonality
relationship for the forward and adjoint higher eigenvectors are well satisfied, being consistent with thetheory.
6. Conclusions
In this study, the higher eigen-modes of the forward andadjoint neutron diffusion equations are calculated using
IRAM along with the multi-group multi-domain coupled
PGMRES algorithm based on domain decomposition.
The ParaFiDi code is developed accordingly and a multidimensional LWR diffusion problem is solved to verify
ParaFiDi. Numerical results demonstrate that thefundamental solution agrees well with the reference
results, the forward and adjoint equation have the same
spectrum, and the forward and adjoint higher
eigenvectors follow the orthogonality relationship. The
ParaFiDi results are consistent with the anticipatedtheory analysis.
Future work will include employing pre-conditioner for161

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Performance of the RSE (Resonance calculation using energy Spectral Expansion) Methodfor Heterogeneous Pin-cell Geometry
Ryoichi Kondoa,, Tomohiro Endoa, Akio Yamamotoa, Satoshi Takedab,
Hiroki Koikec, Kazuya Yamajic and Daisuke Satoca
Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi, 464-8603, Japanb
Osaka University, 2-1 Yamadaoka, Suita-shi, Osaka, Japan, 565-087cMitsubishi Heavy Industries, Ltd., 1-1-1 Wadasaki-Cho, Hyogo-ku, Kobe, Japan, 652-8585*
Corresponding author: r-kondo@fermi.energy.nagoya-u.ac.jp

Abstract
A Resonance calculation using energy Spectral Expansion (RSE) method has been
applied to a heterogeneous pin cell geometry and its validity is confirmed throughcomparison with the ultra-fine group calculation. The RSE method utilizes
expansion of angular flux by orthogonal basis on continuous energy. The singularvalue decomposition (SVD) and the low-rank approximation (LRA) are used toconstruct the orthogonal basis from ultra-fine group spectrum calculations in thehomogeneous geometry. The calculation results show the validity of the present
method.
Key Words: Resonance calculation, effective cross section, reduced ordermodel, low-rank approximation, singular value decomposition
1. Introduction

2. Theory

Various resonance calculation methods have beendeveloped so far, e.g., the equivalence, the sub-group, andthe ultra-fine group methods [1]. These methods haveadvantages and disadvantages from the viewpoints ofcomputational efficiency. Since the most accurate method
is the ultra-fine group method, it would be suitable forhigh-fidelity core simulations. However, its application toa large geometry is still limited from the viewpoint ofcomputational time.
The RSE method has been recently proposed by theauthors as an alternative resonance calculation method [2].
In this approach, continuous (pointwise) neutron energy
spectra in heterogeneous regions are expanded byorthogonal basis on continuous energy. The singular value
decomposition (SVD) and the low-rank approximation(LRA) [3][4] are used to construct orthogonal basis on
continuous energy from the calculation results of ultrafine group spectrum in homogeneous geometry for typicalmaterials and background cross sections.
In principle, the proposed method has several
advantages over conventional methods. For example,
accuracy is not dominated by the number of resonances ina group and the resonance interference can be explicitly
considered.
In the present summary, the RSE method is applied toa heterogeneous geometry and its accuracy is verified
through comparisons with the reference results obtainedby the ultra-fine group calculation in a heterogeneousgeometry.

The transport equation assuming isotropic scattering andneutron sources is:𝛀 ∙ ∇𝛹(𝐫, 𝛀, 𝐸) + Σ𝑡 (𝐫, 𝐸)𝛹(𝐫, 𝛀, 𝐸)
1=
𝑄(𝐫, 𝐸).
4𝜋(1)𝛀 ∙ ∇𝛹(𝐫, 𝛀, 𝐸𝑔 ) + Σ𝑡 (𝐫, 𝐸𝑔 )𝛹(𝐫, 𝛀, 𝐸𝑔 )1=
𝑄(𝐫, 𝐸𝑔 ).
4𝜋(2)Consider a multi-group energy range (energy range
for multi-group 𝑔, 1 ≤ 𝑔 ≤ G):

Angular flux, scalar flux, and neutron source are
expanded by the orthogonal bases on energy:𝑁𝛹(𝐫, 𝛀, 𝐸𝑔 ) = ∑ 𝑓𝑖,𝑔 (𝐸𝑔 )𝛹𝑖,𝑔 (𝐫, 𝛀),

(3)

𝑖=1
𝑁

𝜙(𝐫, 𝐸𝑔 ) = ∑ 𝑓𝑖,𝑔 (𝐸𝑔 )𝜙𝑖,𝑔 (𝐫) ,

(4)

𝑖=1
𝑁

𝑄(𝐫, 𝐸𝑔 ) = ∑ 𝑓𝑖,𝑔 (𝐸𝑔 )𝑄𝑖,𝑔 (𝒓) ,

(5)

𝑖=1

where the orthogonal property of the bases is:∫ 𝑓𝑖,𝑔 (𝐸𝑔 )𝑓𝑗,𝑔 (𝐸𝑔 )𝑑𝐸𝑔 = δ𝑖𝑗 .
Δ𝐸𝑔

(6)

By substituting Eqs.(3)–(5) into Eq.(2), andmultiplying 𝑓𝑛,𝑔 (𝐸𝑔 ) to the both sides of Eq.(2) andperforming energy integration for Δ𝐸𝑔 , we have:162

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compared with the reference obtained in Step (8).

𝑁𝛀 ∙ ∇𝛹𝑛,𝑔 (𝐫, 𝛀) + ∑ Σ𝑡,𝑔,𝑛,𝑖 (𝐫)𝛹𝑖,𝑔 (𝐫, 𝛀)
𝑖=1

(7)

Σ𝑡,𝑔,𝑛,𝑖 (𝐫) = ∫ 𝑓𝑛,𝑔 (𝐸𝑔 )Σ𝑡 (𝐫, 𝐸𝑔 )𝑓𝑖,𝑔 (𝐸𝑔 )𝑑𝐸𝑔 .

(8)

1
=
𝑄 (𝐫),
4𝜋 𝑛,𝑔

where
Δ𝐸𝑔

The four cases shown in Table I are considered. Notethat these four cases are chosen to investigate the impact
of NR approximation on orthogonal basis and slowing
down source. For example, the NR approximation is
generally used for slowing down source calculation in the
sub-group method. A pin cell geometry shown in Fig.1 isused in the present study. The fuel pellet is divided into
two annular regions in order to verify the variation ofneutron spectra and effective cross sections due to thespatial self-shielding effect. Number densities of fuel andcoolant are summarized in Table II and the temperature of
all material is 600 K.

In the RSE method, Eq.(7) is solved with the crosssections defined by Eq.(8). Note that the neutron source
in Eq.(7) is directly obtained by an ultra-fine groupcalculation in heterogeneous geometry in this study.
Treatment of the scattering source will be one of the futuretasks.
One of the key points of the RSE method is theselection of orthogonal basis. In principle, any complete
orthogonal basis can be used for energy spectra [5]–[9].
However, if an inappropriate basis is used, the number of
expansions to achieve sufficient accuracy will increase,which makes the present method inefficient.
In order to make efficient orthogonal basis that well
reproduces various neutron spectra in heterogeneous
regions, SVD, LRA, and ultra-fine group calculations inhomogeneous geometry with various background crosssections are used in this study. Namely, neutron spectra ineach region in a heterogeneous geometry are expressed by
the orthogonal basis obtained in a homogeneous geometry.
This process is called reduced order modeling (ROM)[10].

Table I. Calculation cases
Spectra used for
Slowing down
Caseorthogonal basis
source in Eq.(7)
1
SLD1
SLD2
2
SLD1
NR2
3
NR1
SLD2
4
NR1
NR2
SLD1: Obtained by ultra-fine group slowing down
calculation in homogeneous geometry
SLD2: Obtained by ultra-fine group slowing down
calculation in heterogeneous geometry
NR1: Obtained by ultra-fine group calculation inhomogeneousgeometry
using
the
NR
approximation
NR2: Obtained by the NR approximation

3. Calculations

Moderator

The following calculation flow is used in the present
study:(1) Perform ultra-fine group slowing down calculations inhomogeneous geometry with various background crosssections. The slowing down calculations are carried out
for materials (e.g., UO2) appeared in the heterogeneousgeometry. Background cross sections are chosen to
cover from the fully shielded (very small backgroundcross section) to the fully diluted (very large
background cross section) conditions.
(2) Repeat Step (1) for materials used in the calculationgeometry.
(3) Collect ultra-fine group spectra obtained by Step (2)and perform SVD and LRA for each multi-group.
(4) Generate Σ𝑡,𝑔,𝑛,𝑖 using Eq.(8).
(5) Perform MOC transport calculation using Eq.(7). Note
that, the direct numerical solution using the matrix
exponential is adopted in this study.
(6) Once 𝛹𝑖,𝑔 is obtained in Step (5), ultra-fine group
angular flux is reconstructed by Eq.(3).
(7) Effective microscopic cross sections for each energy
group are generated by angular flux obtained in Step(6) and continuous energy microscopic cross sections.
(8) The ultra-fine group slowing down calculation iscarried out in the heterogeneous geometry using theconventional MOC to obtain the reference solution.
(9) Effective cross section obtained in Step (7) isFuel0.98 cm
Region 1
Region 2

1.26 cm

Fig. 1. Pin cell geometry
Table II. Number densities
Material
Fuel (UO2)Moderator(H2O)

Nuclide
U235
U238
O16
H1
O16

Density[1/barn/cm]
1.1e-32.1e-24.4e-24.7e-22.35e-2

For the pointwise cross sections used in the present
method, the ACE formatted cross sections, which are
generated by the FRENDY code [11] using JENDL-4.0[12], is applied. The GENESIS code [13] is used as the
MOC transport solver for the ultra-fine group calculation163

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Error of effective total cross section [%]in the pin cell geometry with the following calculation
conditions:-Number of polar angles: 6 for π using the TY quadrature
set [14]-Number of azimuthal angles: 32 for 2π-Ray trace width: 0.05 cm-Convergence criterion for scalar flux: 10−5
The energy group structure for ultra-fine groupcalculation is shown in Table III. The XMAS 172 groupstructure is adopted for multi-group.
Table III. Energy group structure for ultra-fine groupcalculation
Number of divisions
Upper energy(Equal division forboundary [eV]lethargy)
20,000,000
10,00052,47556,0009118.812,0004307.412,000961.128,000
130.07
12,0000.32242
10,000Lower energy boundary: 0.00001 eVError of effective total cross section [%]4. Results

10.0
8.0
6.0
4.0
2.0
0.0
Case 4
Case 3
Case 2
Case 1

-6.0
-8.0
-10.0
1.0E+01.0E+1
1.0E+2
Energy [eV]

Case 4
Case 3
Case 2
Case 1

6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
1.0E+01.0E+1
1.0E+2
Energy [eV]

1.0E+3

10.0
8.0
6.0
4.0
2.0
0.0
-2.0

Case 4
Case 3
Case 2
Case 1

-4.0
-6.0
-8.0
-10.0
1.0E+01.0E+1
1.0E+2
Energy [eV]

1.0E+3Fig. 4. Error of U235 effective total cross section in 172
XMAS energy group structure (Region 2, outerregion of pellet).
Error of effective total cross section [%]Error of effective total cross section [%]Relative errors of the effective microscopic total crosssections of U235 and U238 in the energy range including
major resolved resonance (1 eV – 1000 eV) are shown in
Figs. 2-5. The relative error is defined by (Present –Reference)/Reference. The number of bases is fixed to 11
in these figures to observe the effects of the NR
approximation in orthogonal bases and neutron source on
the calculation results.

-4.0

8.0

Fig. 3. Error of U238 effective total cross section in 172
XMAS energy group structure (Region 1, inner
region of pellet).

A maximum number of the orthogonal bases is 11 on
the present calculation since the background crosssections for UO2 and H2O are 1010-101 barn (10 points)
and 1010 barn (1 point), respectively.

-2.0

10.0

10.0
8.0

Case 4
Case 3
Case 2
Case 1

6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
1.0E+01.0E+1
1.0E+2
Energy [eV]

1.0E+3Fig. 5. Error of U238 effective total cross section in 172
XMAS energy group structure (Region 2, outerregion of pellet).

1.0E+3Figures 2-5 indicate that the RSE method for Case 1,
in which NR approximation is not used, well reproduces
the reference effective cross section. When the NR
approximation is applied to the neutron source, theFig. 2. Error of U235 effective total cross section in 172
XMAS energy group structure (Region 1, inner
region of pellet).

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Error of effective total cross section [%]accuracy of the RSE method decreases. This result
suggests that consideration of fine energy dependence of
scattering neutron source would be necessary.
Next, the impact of the number of bases used in the
RSE method is discussed. The relation between thenumber of orthogonal bases and the accuracy of theeffective cross section for group 88 in the XMAS 172
group structure, in which a large resonance of U238 exists,
is investigated. Figure 6 indicates that errors are almost
converged to small value at five orthogonal bases. The
values of error in Region 1 are slightly larger than thosein Region 2. The slight differences come from theapproximation in the present study, i.e., the neutron
spectra in heterogeneous geometry is expanded by thosein homogeneous geometry.

2. Yamamoto A et al., “A Resonance Calculation
Method using Energy Expansion Bases based on a
Reduced Order Model,” Proc. M&C2019, Portland,
Oregon, Aug.25–29, (2019).
3. Matsushita M, Endo T, Yamamoto A, Kitao T,“Development of Reduced Order Model of Severe
Accident Analysis Code for Probabilistic Safety
Margin Analysis,” Proc. PHYSOR2018, Cancun,
Mexico, Apr. 22–Apr. 26, 2018 (2018). [USB].
4. Reed R, Roberts J, “Enhancements to the Discrete
Generalized Multigroup Method,” Trans. Am. Nucl.
Soc., 120, 475 (2019).
5. Rahnema F, Douglass S, Forget B, “Generalized
Energy Condensation Theory,” Nucl. Sci. Eng., 160,
41 (2008).
6. Zhu L, Forget B, “A Discrete Generalized Multigroup
Energy Expansion Theory,” Nucl. Sci. Eng., 166, 239(2010).
7. Le Tellier R, Fournier D, Ruggieri JM, “A WaveletBased Finite Element Method for the Self-Shielding
Issue in Neutron Transport.” Nucl. Sci. Eng., 163, 34(2009).
8. Yang W, Wu H, Zheng Y, Cao L, “Application of
Wavelets Scaling Function Expansion Method in
Resonance Self-Shielding Calculation,” Ann. Nucl.
Energy, 37, 653 (2010).
9. Rooijen W, “Feasibility of Wavelet Expansion
Method to Treat the Energy Variable,” Proc.
PHYSOR2012, Knoxville, TN, Apr. 15–20, 2012(2012). [USB-DRIVE]
10. Abdel-Khalik HS, Feasibility Study of an Integrated
Framework for Characterization of Uncertainties
with Application to CANDU Steady State andTransient Reactor Physics Simulation, Canadian
Nuclear Safety Commission, Canada (2015).
11. Tada K, Nagaya Y, Kunieda S, Suyama K, Fukahori
T, “Development and Verification of A New Nuclear
Data Processing System FRENDY,” J. Nucl. Sci.
Technol., 54, 806 (2017).
12. Shibata K et al., “JENDL-4.0: A New Library for
Nuclear Science and Engineering,” J. Nucl. Sci.
Technol. 48, 1–30 (2011).
13. Yamamoto A, Giho A, Kato Y, Endo T, “GENESIS –A Three-dimensional Heterogeneous Transport
Solver based on the Legendre Polynomial Expansionof Angular Flux Method,” Nucl. Sci. Eng., 186, 1(2017).
14. Yamamoto A, Tabuchi M, Sugimura N, Ushio T, Mori
M, “Derivation of Optimum Polar Angle Quadrature
Set for the Method of Characteristics Based on
Approximation Error of the Bickley Function,” J.
Nucl. Sci. Technol., 44, 129 (2007).

10.0
8.0
6.0
4.0
2.0
0.0
-2.0

U235, Region 1
U238, Region 1
U235, Region 2
U238, Region 2

-4.0
-6.0
-8.0
-10.0

1

2

3

4
5
6
7
8
9
Number of orthogonal bases

10

11

Fig. 6. Comparison of effective total cross section forvarious number of orthogonal bases in group 88.
5. Conclusions
The Resonance calculation based on energy Spectral
Expansion (RSE) is applied to a pin cell geometry and its
accuracy is verified through comparison with the resultsby the ultra-fine group MOC calculation. Impact of thegeneration method of the orthogonal bases and theneutron source treatment is quantified through thebenchmark calculation.
Though the results obtained by the present study is
promising, the following issues will be addressed: 1)
treatment of scattering source, 2) generation method of
orthogonal basis, 3) application to more complicatedconditions including multi-cell geometry with different
fuel compositions/temperatures 4) efficient numericalcalculations for matrix exponential 5) computational costof the proposed method.
References
1. Cacuci DG ed., Handbook of Nuclear Engineering,
Chap.9, Lattice Physics Computations, Springer,
ISBN 978-0-387-98150-5, (2010).

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Application of Singular Value Decomposition and Low Rank Approximation for
Compression of Macroscopic and Microscopic Cross Section Table for Core Calculations
Masato Yamamotoa*, Tomohiro Endoa, Akio Yamamotoa
Nagoya University: Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan*
Corresponding author: m-yamamoto@fermi.energy.nagoya-u.ac.jpaAbstract
The singular value decomposition and the low-rank approximation are applied toreduce size of macroscopic and microscopic cross sections of heterogeneousmaterial regions obtained from a typical lattice physics calculation consideringvarious state points including burnup and branches. The impact of the compression
is investigated by calculating the effective multiplication factor and scalar flux ofheterogeneous regions using the reference (original) and compressed crosssections. In this study, approximately 99% reduction in cross section table size isachieve while suppressing the error of the effective multiplication factor due tocompression of cross section below 0.1%.
Key Words: macroscopic cross section, microscopic cross section,
dimensionality reduction, singular value decomposition, low-rankapproximation, heterogeneous assembly calculation, heterogeneous corecalculationconsidered and they are compressed by the singular value
decomposition and the low-rank approximation. The impactof the compression on the effective multiplication factor andscalar flux is investigated in single assembly geometry using
the compressed and the original cross section data.

1. Introduction
In the current core analysis, an assembly calculation iscarried out and then assembly average, pin average, ormaterial average multi-group macroscopic and microscopic
cross sections are edited. These multi-group cross sections
are generally pre-tabulated and later used for successive coresimulations.
In the today’s state-of-art core analysis, space, andenergy resolutions are increased, and such simulations
naturally require spatially and energetically detailed
macroscopic and microscopic cross sections. Therefore, thedata size of these cross sections also increases as the spatial
or energetic resolution becomes higher.
Spatial and energetic resolutions of core analysis tend
to continuously increase. Thus the development of an
efficient reduction method of a cross section table size isdesirable. In the previous study [1], the dimensionality
reduction technique using the singular value decomposition
and the low-rank approximation is used for cell-averaged
macroscopic cross sections.
Though the dimensionality reduction technique wasapplied to multi-group macroscopic cross sections ofhomogenized pin-cell in the previous study, a high fidelity
core analysis will require macroscopic and microscopic
cross sections for heterogeneous material regions. In addition,
the data size of microscopic cross sections is proportional toas the number of nuclides considered in a core calculation.
Therefore, the reduction of not only macroscopic crosssections but also microscopic cross sections in each material
region is necessary. In this study, 16-group macroscopic
and microscopic cross sections in heterogeneous material
regions obtained by a typical lattice physics calculation are

2. Theory for Cross Section Compression
A single assembly calculation using a typical casematrix is performed to generate multi-group macroscopic
and microscopic cross sections for heterogeneousmaterial regions. The cross sections in the fuel region are
considered in this study, and the cross section matrix 𝐀 isgenerated. Note that moderator and cladding crosssections are not discussed in the present paper since theirvariations are not very large.
Multi-group macroscopic
and microscopic cross sections(Σ𝑠, 𝑔→𝑔′ , 𝜎𝑎,𝑔 , 𝜎𝑓,𝑔 , 𝜈𝜎𝑓,𝑔 , 𝜎𝑐,𝑔 for each nuclide)
State points
× Number of fuelregions

cross
sectionmatrixFig. 1. Generated cross section matrix.
In the matrix 𝐀 , the cross sections for each fuel166

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019𝑢11
𝐔 =( ⋮𝑢𝑚1
𝑣11
𝐕′ = ( ⋮𝑣𝑛1region at each burnup, fuel temperature, moderator
temperature, boron concentration, and void fraction arearranged as rows. Then, if a pellet region in a fuel rod isannularly divided, the cross sections of the divided
regions are arranged in different rows. When the numberof state points and fuel cells are I and J, respectively, andthe fuel region in a pellet is divided to K regions, there are
I × J × K rows in the matrix. The types of cross sections
considered in this study are macroscopic scattering crosssection Σ𝑠, 𝑔→𝑔′ and microscopic cross sections
𝜎𝑎,𝑔 , 𝜎𝑓,𝑔 , 𝜈𝜎𝑓,𝑔 , 𝜎𝑐,𝑔 for each nuclides, which arearranged as columns. When the number of energy groups
and the number of nuclides are N and M, respectively,
there are N2 + 4NM columns in the matrix (the numberof Σ𝑠, 𝑔→𝑔′ is N2, and that of microscopic cross sections is
4 reaction types × N groups × M nuclides = 4NM).
The matrix 𝐀 has 𝑚 (= I × J × K) rows and 𝑛 (= N2+ 4NM) columns.
The singular value decomposition [2] is applied to
𝐀 (here 𝑚 > 𝑛 is assumed) as shown in Eqs. (1) – (4).
𝐀 = 𝐔𝚺𝐕 𝐓
𝑆1 0 ⋯ 0
0 𝑆2⋮⋮ 0 ⋱ 0
𝚺= ⋮
⋮
0 𝑆𝑛
⋮
⋮
⋮
0
⋮
⋮
⋮
⋮
(0 0 0 0)
𝑢11 ⋯ 𝑢1𝑚
⋱
⋮ )
𝐔=( ⋮
𝑢𝑚1 ⋯ 𝑢𝑚𝑚
𝑣11 ⋯ 𝑣1𝑛
⋱
⋮ )
𝐕=( ⋮
𝑣𝑛1 ⋯ 𝑣𝑛𝑛

′

⋯ 𝑢1𝑘
⋱
⋮ )
⋯ 𝑢𝑚𝑘
⋯ 𝑣1𝑘
⋱
⋮ )
⋯ 𝑣𝑛𝑘

(6)
(7)

Storage of 𝚺 ′ is not necessary when it is multiplied to
the left (or right) singular vector as shown Eq. (8).
𝑆1 𝑢11 ⋯ 𝑆𝑘 𝑢1𝑘
⋱
⋮ )
(8)
𝐔′𝚺′ = ( ⋮𝑆1 𝑢𝑚1 ⋯ 𝑆𝑘 𝑢𝑚𝑘
By multiplying 𝐔 ′ , 𝚺 ′ and 𝐕 ′ , a reconstructed crosssection matrix 𝐀′ is obtained as shown Eq. (9).
𝐓
(9)𝐀′ = 𝐔 ′ 𝚺 ′ 𝐕 ′The number of the singular values adopted in 𝚺 ′ canbe changed from 1 to 𝑛, and 𝐀′ can be calculated in eachcase. Apparently, a smaller number of the adopted singularvalues results in the smaller size of the cross section set while
the reconstruction accuracy decreases.
Macroscopic cross sections except for the scattering
are reconstructed by Eq. (10).

(1)Σ𝑥 = ∑ 𝑁𝑖 𝜎𝑥,𝑖 ,

(10)

𝑖

where
𝑁𝑖 : nuclide number density of nuclide 𝑖,
𝜎𝑥,𝑖 : microscopic cross section of nuclide 𝑖 of reaction
𝑥. Note that energy group index is omitted.
The computing time for the reconstruction is
negligibly short compared to that of an assembly or corecalculation.

(2)

(3)

3. Calculation Condition(4)

In this study, a heterogeneous single assembly
calculation was performed using AEGIS [3] for the 17 ×
17 PWR fuel assembly (235U enrichment: 4.8 wt %)shown in Fig. 2 considering the burnup and branch points.

𝐔 of Eq. (3) and 𝐕 of Eq. (4) are unitary matrixes
composed by the left and the right singular vectors,
respectively. The diagonal components of 𝚺 in Eq. (2) are
the singular values. By singular value decomposition, 𝑛singular values of 𝑆1 , ⋯ , 𝑆𝑛 are obtained. The low-rankapproximation is performed by neglecting the small singularvalues (𝑆𝑘+1 , ⋯ , 𝑆𝑛 ) as shown in Eq. (5).
𝑆1 0 ⋯ ⋯ 0
0 ⋱⋮⋮ 0 𝑆𝑘
⋮
0 ⋱ 0
𝚺′ =
⋮
0 𝑆𝑛
(5)
⋮
0
⋮
⋮
⋮
⋮
⋮
(0 0 0 0 0)
𝑆1 ⋯ 0≈(⋮ ⋱ ⋮)
0 ⋯ 𝑆𝑘As the result, the left singular vector 𝐔 and the rightsingular vector 𝐕 can also be approximated as Eqs. (6) and(7).

UO2 Rod (235U: 4.8 wt%)
Water Rod (G/T, I/T)℄℄

Fig. 2. 17×17 UO2 fuel assembly for PWR.
Then, 189 state points for various burnup, fueltemperatures,moderator
temperatures,
boron

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concentrations, and void fractions whose ranges areshown in Table I, are considered. The 16-group
microscopic cross sections, nuclide number density foreach nuclide, and macroscopic scattering cross section areedited for each material region in fuel rods, guidethimbles (G/T), or an instrumentation thimble (I/T). The
material regions of a fuel cell are shown in Fig. 3 and thegeometry is shown in Table II. Note that the total numberof cells is 45 (39 fuel rods and 6 water rods) consideringoctant symmetry.
Range of state point parameters

Parameters

4. Numerical Results

Value (or State)

Burnup [MWd/kg]
Temperature of Fuel [K]Temperature of Moderator [K]Boron Concentration [ppm]Void Fraction [%]Control Rod

The singular values obtained by the singular value
decomposition are shown in Fig. 4.

0.1 – 70556 or 900
556 – 596
0 – 2400
0 – 80No

Magnitude of singular value

Table I.

singular values. The reconstructed macroscopic crosssection Σ𝑥 required for a heterogeneous assembly
analysis is calculated by Eq. (10) using the nuclide
number densities calculated by AEGIS and thereconstructed microscopic cross sections. Transport
calculations in heterogeneous assembly geometry were
performed by GENESIS [5] using the reconstructed andthe original macroscopic cross sections to estimate theerror of 𝑘𝑒𝑓𝑓 and scalar flux due to compression of thecross sections.

UO2 fuel pellet
Gap
Cladding

1.0E+08
1.0E+05

1.0E+02
1.0E-01

1.0E-04
1.0E-07
1.0E-10Moderator
Background Mesh

100

1000

Order of singular value

10000

The maximum error of 𝑘𝑒𝑓𝑓 is defined by the
maximum relative error among all state points. The
maximum and the root mean square (RMS) errors of 𝑘𝑒𝑓𝑓
are shown in Fig. 5. Note that error of scalar fluxes will beshown in the presentation.

Table II. Geometry of a square cell (cell pitch: 1.26 cm)

Radius [cm]0.36630.3663 – 0.4095
0.4095 – 0.4180
0.4180 – 0.4750
0.4750 – 0.5647
0.5647 – Boundary

1.0E+02keff error (%)

Materials

10

Fig. 4. The magnitude of singular values.

Fig. 3. Material regions in a fuel cell.

Fuel Pellet (inner)
Fuel Pellet (outer)
Gap
Cladding
Moderator (inner)
Moderator (outer)

1

Among the cross sections edited for each material
region, we focused on those of fuel regions to generate thecross section matrix. There are 221 nuclides in a fuelregion and the number of the cross sections for a region is
14400 (number of Σ𝑠,𝑔→𝑔′ is 16 × 16 =256, and thoseof 𝜎𝑎,𝑔 , 𝜎𝑓,𝑔 , 𝜈𝜎𝑓,𝑔 , 𝜎𝑐,𝑔 are 4 × 16 × 221 = 14144,
thus 256 + 14144 = 14400). The fuel pellet region isdivided into two and cross sections for these regions arearranged in different rows. Therefore, the generated
matrix has 14742 rows (= 39 cells × 2 regions × 189
state points) and 14400 columns, and has 14400 singularvalues.
Singular value decomposition was performed forthis matrix using the numpy.linalg.svd function of the
Python’s numerical calculation library NumPy [4]. The
singular values were truncated from 14400 using the lowrank approximation and the cross section matrix wasreconstructed using the reduced singular vectors and1.0E+01

Maximum

1.0E+00

RMS

1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05

10

100

1000

Order of singular values

Fig. 5. The maximum and RMS errors of 𝑘𝑒𝑓𝑓 using thereconstructed cross sections(from 10 to 1000 singular values).
RMS error of 𝑘𝑒𝑓𝑓 is calculated by Eq. (11).

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 20192.

∆𝑘𝑒𝑓𝑓,𝑅𝑀𝑆
𝑘𝑒𝑓𝑓,𝑟𝑒𝑐𝑜𝑛𝑠𝑡,𝑖 − 𝑘𝑒𝑓𝑓,𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙,𝑖 2
𝑠𝑡𝑎𝑡𝑒
(11)
∑𝑁
(
)
𝑖
𝑘𝑒𝑓𝑓,𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙,𝑖√=𝑁𝑠𝑡𝑎𝑡𝑒where
𝑁𝑠𝑡𝑎𝑡𝑒 : number of state points,
𝑘𝑒𝑓𝑓,𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙,𝑖 : 𝑘𝑒𝑓𝑓 in the 𝑖-th state point calculated fromthe original cross sections,𝑘𝑒𝑓𝑓,𝑟𝑒𝑐𝑜𝑛𝑠𝑡,𝑖 : 𝑘𝑒𝑓𝑓 in the 𝑖-th state point calculated fromthe reconstructed cross sections.
In Fig. 5, the maximum error of 𝑘𝑒𝑓𝑓 isapproximately 0.1% when the number of adopted
singular values is 100. When 100 singular values are
adopted, 14742 × 14400 cross section matrix iscompressed into 14742 × 100 left singular vector(multiplied by singular values) and 14400 × 100 rightsingular vector, which means 98.6% reduction of theoriginal

cross

section14742 × 100 + 14400 ×100
14742 × 14400

data

3.

4.
5.

(= 1 −).

5. Conclusions
Reduction of cross section table size including
macroscopic and microscopic cross sections isinvestigated by applying the singular value
decomposition and the low-rank approximation. Amongthe material-wise cross sections obtained by a latticephysics calculation, those of fuel pellet are considered inthis study. The impact of dimensionality reduction isinvestigated with a comparison of the assembly
calculations using the reconstructed and original crosssections. In this study, approximately 99% reduction incross section table size is achieved while suppressing theerror of 𝑘𝑒𝑓𝑓 below 0.1%.
Application of the present method to other lattice
types is one of the future studies. Furthermore,development of the tabulation method using
dimensionality reduced cross sections is considered fromthe viewpoint of the applicability of the present method
for actual core calculations.
Acknowledgments
The authors sincerely thanks to Dr. Masato Tabuchiof Nuclear Engineering Ltd. for his technical support. The
cross section data used in this study were obtained during
the internship in NEL.
References
1.

Yamamoto M, Endo T, Yamamoto A, “Reduction of
Cross Section Table Size for Core Analysis using
Dimensionality Reduction Technique,” Trans. Am.
Nucl. Soc., 119, 1226 (2018) (USB).

169

Golub G, Van Loan C, Matrix Computations 3rd ed.,Chapter 2, Johns Hopkins University Press,
Baltimore, Maryland (1996).
Yamamoto A, Endo T, Tabuchi M, et al., “AEGIS: An
Advanced Lattice Physics Code for Light Water
Reactor Analyses,” Nucl. Eng. Technol., 42, 5, 500(2010).
“Scipy.org,” https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.svd.html(access: 2019/06/07).
Yamamoto A, Giho A, Kato Y, et al., “GENESIS: A
Three-Dimensional Heterogeneous Transport Solver
Based on the Legendre Polynomial Expansion ofAngular Flux Method,” Nucl. Sci. Eng., 186, 1, 1
(2017).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Compression of the Multi-Physics Simulation Output Data
Using Principle Component AnalysisaAlexey Cherezova, Jinsu Parka, Hanjoo Kima, Nur Syazwani Mohd Alib and Deokjng Leea,Ulsan National Institute of Science and Technology, 50 UNIST-gil, Eonyang-eup, Ulju-gun, Ulsanb
University of Technology, Malaysia, Skudai, Johor, 81310*
Corresponding author: deokjung@unist.ac.kr

Abstract
High-fidelity multi-physics reactor core simulation includes the pin-by-pin
neutronics, thermal-hydraulics, thermal-mechanics and fuel performancephysical models. Every run of a high-fidelity code consumes a lot ofcomputation time, memory and usually requires an expansive multi-corecomputational cluster. All output data is to be saved for further analysis if one
does not want to repeat the expansive calculations. The memory amount of highfidelity output data can be too large, e.g. in the case of uncertainty analysis.
Therefore, the data compression algorithms are needed for reduction of thememory without accuracy loose. In this paper we suggest such algorithm basedon Principle Component Analysis (PCA) adopted for multi-physics data.
Key Words: Multiphysics, Compression, PCA, RIA
1. Introduction
The multi-physics code has been developed at UNISTCore Lab. for steady-state and transient simulation of
PWR cores. It consists of three program modules forcalculation of: 1) reactor core neutron field and isotope
number density; 2) coolant thermal-hydraulics; 3) fuelrod thermal-mechanics and fuel performance. Below we
describe each of these modules.
RAST-K [1] is a neutron diffusion nodal code with
pin-power reconstruction designed for the steady-state
and transport calculation of the pressurized water reactor
cores with rectangular assemblies. The code utilizes thetwo-group cross-section library calculated by a latticecode.
CTF [2] is a thermal-hydraulic code that uses thesubchannel modeling approach for light water reactor
simulation. The code uses a two-fluid, three-fieldmodeling approach and was originally developed by
Pacific Northwest Laboratory.
FRAPTRAN [3,4] is a fuel performance code
designed to perform the transient and hypotheticalaccidents simulation of light-water reactor fuel rods. The
phenomena modeled by the code include heat
conduction, fuel and cladding mechanical deformation,
cladding oxidation, fission gas release and fuel rod gas
pressure.
FRAPCON is a fuel performance code designed to
calculate the steady-state response of light-water reactor
fuel rods during long term burnup. The numericalmethods and modeled phenomena of the code are similar
with FRAPTRAN. Both codes were developed by
Pacific Northwest National Laboratory and are used for

The continuous evolution of reactor core numericalmethods and computer technologies has approached thehigh-fidelity neutron transport simulation with pin-bypin resolution [1,2]. The further improvement of thenuclear reactor simulation requires the tight coupling of
neutronics, coolant thermal-hydraulics, fuel rod thermalmechanics and other physical models.
This work is devoted to the coupled multi-physics
code designed for multi-physics and hypotheticalaccident analysis of pressurized water reactor (PWR)
cores. The code provides the pin-by-pin nodal diffusion
neutronics, coolant thermal-hydraulic, one-dimensional
thermal-mechanics of pellet and cladding, fuelperformance. The code is considered to be used for PWR
uncertainty analysis. However, every single run of thecode produces large amount of output data. In order toreduce the memory size the data compression algorithms
based on Principle Component Analysis (PCA) andsparse matrix representation has been developed.
The paper is organized as follow. Section 2 is devoted
to the multi-physics code physical models and couplingalgorithm description. Section 3.1 briefly explains the
PCA method. In section 3.2 we discuss the residual
matrix and sparse matrix representation approaches.
Section 3.3 is about the compression rate analysis. In
section 3.4 we present the step-by-step description of thecompression algorithm. Section 4 presents thecalculation results.
2. Multi-Physics Code170

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Osaka, Japan, Dec. 2-3, 2019n · Vm2 = trace [ (I - P) × M ].

certification of the light-water core designs.
In order to use the FRAPCON and FRAPTRAN fuelperformance codes for the time integration with a tightcoupling, the Fuel Rod Analysis Program Interface
FRAPI has been developed. FRAPI runs a fuelperformance code and updates a fuel rod state at the eachtime step with respect to the data transferred by an
external code. Also FRAPI restarts the previous fuel rod
state when within time step iteration is repeated.

The trace of a matrix is the sum of its eigenvalues.
Projector I - P is orthogonal, therefore spectrum of
matrix (I - P) × M is a subset of the spectrum of matrix
M. In order to minimize the variance matrix P should be
a projector on any eigensubspace of M. Therefore, basis
vectors of Y called principle components are theeigenvectors of matrix M and the minimal variance isthe sum of eigenvalues σi of M, i.e.

3. Methodology
n · Vm2 = σm+1 + σm+2 + … + σn.
3.1 Principle Component Analysis
Obviously, the principle components should belong to
the largest eigenvalues of covariance matrix M. Solutionof the eigenvalue problem is conducted by the Singular
Value Decomposition (SVD) algorithms. It is worth tonote that PCA is efficient for the high correlated data.
Otherwise the truncation error, especially in the
maximum norm, is huge.

The data compression method under consideration isbased on the Principle Component Analysis (PCA) [5].
PCA is the statistical method that processes the linear
correlations of a vector set and produces the optimalbasis vectors such that the variance of truncation error isminimal. This concept is well-known and it is widelyused for many applications of reactor physics [6-11].
Therefore, here we explain only the main idea of themethod.
Let's consider (n, N)-matrix X having n rows and N
columns. In our application n is a number of time steps
and N is a total number of spatial nodes. The total
number of spatial nodes is equal to a number of fuel rods
by a number of axial nodes. We assume that n << N andthat the mean vector of rows of X is equal to zero.
Let’s approximate data X by an orthonormal set of mbasis vectors. The basis vectors forms (m,N)-matrix
which we denote as Y. The truncation error is thedifference between original vectors X and a linear
combination of basis vectors, i.e.

3.2 Residual Matrix and Sparse Representation
The ouput data of multiphysics simulation contains
many variables distributed in time and space, such as
linear power, fuel temperature, cladding stress and strain
etc. Although these variables can be processed as a large
data set, we begin our research from the independent
compression of each variable and looking forward amore general approach in future works. In additional,
this approach is beneficial from the computational pointof view since the covariance matrix rank is not too large
and computational cost of SVD algorithm is reasonable.
Consider multi-physics variable u and (n, N)-array U
which contains the corresponding numerical simulation
data. One can turn U into a centered data set X andcalculate m principle components such that the resulting
truncation error satisfies to a given tolerance. However,these direct approach does not show enough well results
due to few features of the considered distributions:
1) presence of large variations over time and space;2) weak spatial correlation of some locations.
The first problem is fixed by a proper normalizationof data U. Introduce (n,)-vector A which is the sum of N
columns of data U. Vector A depends on the time stepand represents the magnitude of variable u,

X – C × Y.
Matrix C is a (n,m)-matrix of approximation
coefficients. Here and further the symbol × denotes thematrix dot product, while · denotes the tensor product.
The minimum value of the truncation error is observed if
vectors C × Y are the orthogonal projection of vectors X
onto the linear subspace formed by Y. Therefore, theapproximation coefficients are
C = X × YT.

Ai = Ui1 + … + UiN

Now let’s consider the variance of the truncation error
and find the optimum basis vectors of Y. Remember,that the basis vectors are normalized such that YYT = I.
The variance Vm depends on a number m of basis
functions and it is calculated as follow(1)Along with the magnitude vector we introduce the(N,)-vector B that catches the average shape of variable
u over a time,
Bj = (U1j / A1 + … + Unj / An) / n

n · Vm2 = trace [ (X – C × Y)T × (X – C × Y) ](2)

Finally, we normalize and center the given data set U
as follow
Xij = Uij / (δ + Ai Bj ) - 1.
(3)

We denote I is an identity matrix, P = YT×Y is a
projection matrix and M = XT×X is a covariance matrix.
All these matrices have the shape (N,N). Then theprevious equation is transformed in the form

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The absolute tolerance δ is used here in order to preventthe division by zero. One can easily confirm that X1j + …
+ Xnj = 0 with the accuracy δ.
As we noted above, PCA is able to approximate wellonly the high-correlated data. However, for example,
some locations in a power distribution can be notcorrelated due to the different position of control rods.
Therefore, along with PCA another approach is needed
for catch and save the uncorrelated part. Here we use theresidual matrix Z which is the simple difference between
the original data set X and its PCA approximation. Thus
the approximation formula of original data U is
U = A × B × (I + C × Y + Z)

number of principle components M < min(n,N). Initial
number of principle components m = 0. Then algorithm
makes the next steps:
1. Calculate the magnitude vector A
2. Calculate the average shape function B
3. Normalize and center original data U
4. Calculate first M principle components by PCA
5. Calculate approximation coefficients C
6. Calculate residual matrix Z for m principle
components
7. Eliminate the close to zero elements from the residual
matrix according to the rule |zij| < ε and count the numberof remaining non-zero elements K
8. Estimate memory size Sm by formula 5. If Sm-1 < Sm go
to step 6 with m := m + 1, otherwise exit from the cycle.
The algorithm output are vectors A and B, dense
matrices Y and C, sparse residual matrix Z.

(4)It should be noted, that this representation is accurate if
there is not any assumptions about matrix Z. However,
we assume the most elements of the residual matrix areclose to zero. Therefore, the small elements of Z arefiltered in order to the essential remaining part can be
stored in the sparse format.
3.3 Compression Rate
A compression algorithm is designed to minimize thememory costs of data. Let us evaluate the memory costs
required according to formula 4. Assume that the
memory of a float number is 8 bytes and the memory of
an integer number is 4 bytes. Then the memory size ofthe original data U is
SU = 8 · n · N .
As mentioned above, residual matrix Z is represented in
the sparse format. If the number of non-zero elements is
K << n · N then 16 · K bytes is needed to store the threearrays for represent the residual matrix in the coordinate
format: a) data array (8·K bytes); b) column coordinates(4·K bytes) and c) row coordinates (4·K bytes). The total
memory size of approximation 4 is
Sm = 8 · (m · N + n · m + n + N + 2 · Km).

Fig.1. Flowchart of the data compression algorithm(5)

4. Numerical Results

The more principle components m is employed, the lessnumber Km of non-zero elements is needed in Z.
However, because m and K have the different multipliersin formula 5, there is an optimal number of principle
components m* for which Sm takes a minimal value.
For further numerical estimations we define thecompression rate as a ratio between the memory of theoriginal and compressed data,

The compression algorithm has been tested for the
output data file obtained by simulation of the control rod
ejection accident in pressurized water reactor core OPR1000. The fresh core of first cycle is considered. The
initial total power is 1 MW.
OPR-1000 core consists of 177 square assemblies
arranged as depicted in Fig.2. Each assembly contains 5
guide tubes and 236 fuel rods placed in 16x16 lattice.
Sizes of a fuel assembly and a fuel rod are shown in
Fig.3. Number of axial layers in the fuel performanceand thermal-hydraulic models is set to 10. At the initialstate all control rods of 1 – 6 groups are located at thebottom position while the groups 7, 8 and 9 are fully
ejected out of the core. From 100 ms to 200 ms thecontrol rods are gradually ejected from the core until theupper limit.

R = 100% Sm* / SU
3.4 Compression Algorithm
The compression algorithm consists of few steps
depicted in Fig.1. The input arguments are data U,
relative tolerance ε, absolute tolerance δ and a maximum

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For demonstration of the compression algorithm we
present the case with three ejected control rods placed as
depicted in Fig.2. In this case the inserted reactivity is
about 1.2 $. It causes a large power pulse suppressed byincrease of the fuel temperature and Doppler effect.
Power distribution at the pick of the pulse shown in
Fig.4. The simulation is performed using time step 10 ms
and total number of made time steps is 50. Therefore, theshape of the data to be compressed is n = 50 rows and N= 16·16·236·10 columns. We demand the compression ofdata with a small loose of accuracy withing tolerance δ =10-3.
In the considered example the compression rate oflinear power distribution is about 30% only. That is
because the power strictly depends on the control rod
position and many components are needed for an
accurate approximation in the region from 100 and 200
ms. However, the other parameters are approximatedmuch better that is shown in Table I. Mostly one or two
components is enough for achieve the accuracy 0.1%,
while the fraction of non-zero elements in the residual
matrix is about 1%.
The compression algorithm has been applied forvarious configurations of ejected control rods, initial
power levels, number of axial nodes and time step sizes.
Some of the compression results are presented in Table
II. Refining of the axial and time mesh does not lead tothe principle component increase. However, the size ofthe principle components depends on the number ofaxial nodes. Therefore, the compression rate drops twice
for the twice decreased time step size, while keeps thesame if the number of axial steps increases.

Fig.3. Fuel assembly and fuel rod geometry and sizes.

Fig.4. Linear power pulse (W/cm) and shape function at
the middle height of reactor core
TABLE I. Compression Rate of Multiphysics Parameters
Parameter
UnitsmR, Max. rel.
% error, %Cladding temperature

K

1

4

0.1

Cladding hoop stress

MPa

1

4

0.2

Pellet hoop strain

%

2

6

0.0

Coolant temperature

K

1

4

0.0

Gap thickness

μmm

1

4

0.1

Fuel enthalpy

cal/g

1

4

0.1

W/(K·m )

1

6

0.2

Linear power

W/cm

13

30

1.3

Pellet temp.

K

2

6

0.1

Gap HTC

Fig.2. Control rods groups of OPR-1000 reactor core.
Ejecting groups are colored by orange.

2#

TABLE II. Compression Rate ofMultiphysics Simulation Output Data FilesNz Δtt,Size, GBR, %ms
Original
Compressed

1

5

5

4.3

0.14

3

2

5

10

2.2

0.14

6

3

10

5

8.7

0.26

3

4

10

10

4.4

0.29

6

6. Conclusions

173Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The multiphysics pin-by-pin code for steady-state
and transient simulation has been developed at UNIST.
The multiphysics code is based on the coolant thermalhydraulic code CTF, fuel performance code FRAPTRAN
and neutron nodal diffusion code RAST-K. The code issupposed to be used for uncertainty analysis of PWR
using sampling approach. The result of the sampling islarge number of output data files that have to be stored in
hardware for further processing. These files require a lotof memory, therefore the storage and propercompression of the multiphysics data becomes theimportant problem.
The developed data compression algorithm isdesigned to decrease the memory costs of transient
multiphysics output data. The algorithm is based on
decomposition of data on high and low correlated parts.
The high correlated part is approximated by PCA, while
the low correlated part is saved in a sparse format.
The compression algorithm has been applied to thecontrol rods ejection accident of OPR-1000 coresimulated by the developed multiphysics code. It has
been shown that the data can be compressed up to 3-6 %of initial size of data. The relative error of thecompressed data is less than 1%.
Although the developed algorithm demonstrates
enough well results, the further improvement issuggested:
1. Use one set of principle components which iscommon for all multi-physics variables in order toimprove the compression rate
2. Take into account the correlations of assembly
shapes for further compression rate improvement
3. Develop a recurrent algorithm for on-the-fly datacompression in order to save a memory during
calculation

4. M.J. Thurgood et al. COBRA-TF Development.
In 8th Water Reactor Safety Information Meeting,1980.
5. A. Gorban et al. Principle Manifolds for Data
Visualization and Dimension Reduction. SpringerVerlag Berlin Heidelberg, 2008
6. A. SARTORI, A. CAMMI, and L. LUZZI, “A MultiPhysics Reduced Order Model for the Analysis oftheLead Fast Reactor Single Channel,”Annals of
NuclearEnergy,87, 198–208 (2016).
7. Buchan, A., Calloo, A., Goffin, M., Dargaville, S.,
Fang, F., Pain, C., Navon, I., 2015. A POD reducedorder model for resolving angular direction in
neutron/photon transport problems. J. Comput. Phys.
296, 138.
8. Pasetto, D., Ferronato, M., Putti, M., 2017. A reducedorder model-based preconditioner for the efficientsolution of transient diffusion equations. Int. J.
Numer. Meth. Eng. 109, 1159.
9. Semenov, A., Shchukin, N., Ryabov, N., 2007. Usingof the reduction order model for the reconstruction of
the power distribution by the lateral ionizationchambers. J. Nucl. Eng. 4, 47
10. A. YANKOV, “Analysis of Reactor Simulation
Using Surrogate Model,” Ph.D. thesis, University ofMichigan (2015).
11. M. Matsushita, T. Endo, A. Yamamoto, T. Kitao,"Development of Reduced Order Model of Severe
Accident Analysis Code for Probabilistic Safety
Margin Analysis," Proc. PHYSOR2018, Cancun,
Mexico, Apr. 22- Apr. 26, 2018, (2018).

Acknowledgements
This research was partially supported by the project(L17S018000) by Korea Hydro & Nuclear Power Co.
Ltd. This work was partially supported by the National
Research Foundation of Korea (NRF) grant funded by
the Korea government (MSIT) (No. NRF2019R1C1C1010063).
References
1. F. Khoshahva, M. Park et al. Vanadium, rhodium,
silver and cobalt self-powered neutron detector
calculations by RAST-K v2.0. Annals of Nuclear
Energy, vol. 111, pp. 644-659, 2018
2. K.J. Geelhood, W.G. Luscher, and J.M. Cuta.
FRAPTRAN-2.0: A Computer Code for theTransient Analysis of Oxide Fuel Rods. PNNL19400, Vol. 1, rev.1, Pacifc Northwest National
Laboratory, Richland, WA, 2016
3. K.J. Geelhood and W.G. Luscher. FRAPTRAN-2.0:
Intenral Assessment. PNNL-19400, Vol. 2 Rev. 2,
Pacifc Northwest National Laboratory, Richland,WA, 2016

174

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Refinement of Convolutional Neural Network for Neutronic Design Parameter Prediction of
a Loading PatternaHyunbin Janga, Ho Cheol Shinb, Hyun Chul Leea*
School of Mechanical Engineering, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan,
46241, KoreabCore and Fuel Analysis Group, Korea Hydro and Nuclear Power Central Research Institute (KHNP-CRI), Daejeon,
Korea*
Corresponding author: hyunchul.lee@pusan.ac.kr

Abstract
The computational cost of numerical analysis codes for optimizing the loading
pattern is expensive. Therefore, in the previous study, the convolutional neuralnetwork was selected as the deep learning algorithm to replace the numericalanalysis codes in order to quickly find the optimal loading pattern. Furthermore,in this study, we improved the prediction model of peaking factor and cycle
length by applying internal structure modification, the normalization andregularization. The improved model predicts the reactor design parameters wellwithin a small error range compared with the reference numerical code.
Key Words: Loading pattern optimization, Deep learning, Convolutional
neural network, Reactor design parameter
1. Introduction

factor and the cycle length because it can be used forlearning by extracting more complex and nonlinear
features using multiple hidden layers.

Optimization for the loading pattern of the reactor
core is to find the most economical loading pattern
among all the various cases that satisfy the safety limit.
To obtain an optimal loading pattern, all loading patterns
must be computed as numerical analysis codes. However,despite improved computer performance, it takes a lot ofcomputation time to find optimized loading patterns. If
the reactor design parameters can be predicted with low
computational cost, the optimized loading pattern can befound faster than the numerical methods.
In the previous study, a comparison of prediction
performance between Convolutional Neural Network(CNN) and Deep Neural Network (DNN) was performed[1]. As a result of the comparison, CNN algorithm
showed higher performance in both the peaking factor
and the cycle length.
In this study, internal structure modification of the
CNN algorithm was performed to improve prediction
accuracy. Also, normalization and regularization wereapplied to the peaking factor and the cycle length
prediction algorithm. Furthermore, sensitivity tests basedon the depth of the prediction models were performed tofurther improve the prediction accuracy.

2.1 Automatic generation of training data
A system for generating training data was establishedto obtain a large amount of training data. The training
data was automatically generated by the system using
the STREAM / RAST-K 2.0 [2-4] code system andupdated with the big data. The STEAM code is a neutrontransport analysis code for Light Water Reactor (LWR)
core calculation and developed at Ulsan National
Institute of Science and Technology (UNIST) and the
RAST-K 2.0 code is a diffusion nodal code for
Pressurized Water Reactor (PWR) core analysis
developed by UNIST.
2.2 Selection of deep learning algorithm
The CNN algorithm was adopted to preserve spatial
information and to consider the effects of surrounding
assemblies. The comparison of prediction performance
between the DNN which is the basic algorithm of deeplearning and the CNN was performed. As a result of thecomparison, the CNN showed better performance thanthe DNN, but it was necessary to improve prediction
performance. Table Ⅰ and Table Ⅱ summarize thepeaking factor and the cycle length prediction errors of
the DNN and CNN.

2. Review of Previous Works
The development of an artificial neural network(ANN) algorithm to find the optimal loading pattern was
performed to replace the numerical analysis codes. A
deep learning algorithm was used to predict the peaking

175Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table I. Prediction error of the peaking factor using
DNN and CNN
Prediction error
Algorithm
RMS (%)Max (%)
DNN
9.28
60.91
CNN
1.74
13.81

regularization method that prevents overfitting resulting
from large weighting parameters. The new loss function
𝐿′ is implemented by adding a regularization term to thegeneral loss function L.
𝜆𝑚
𝐿′ = 𝐿 + 2𝑚 ∑𝑖=1
𝜔𝑖2 ,

(2)

Where,
𝜆 = regularization parameter,𝜔 = weighting parameter.

Table II. Prediction error of the cycle length using DNN
and CNN
Prediction error
Algorithm
RMS (%)Max (%)
DNN
1.06
3.98
CNN
1.07
2.12

In the gradient descent parameter update, using the l2
regularization reduces all weight values.
3.2 Internal structure modification of CNN
In order to increase the prediction performance of thepeaking factor, learning was performed by increasing theweight layer depth like VGG NET [6]. Also, the max
pooling layer was removed to prevent loss of assembly
information. Therefore, the training data size is fixed to
8x8. Fig. 2 shows the internal structure of the prediction
models, where model 1 is the configuration used in theprevious study and model 2 is the modified
configuration through sensitivity test in this study.

3. Improvement of Prediction Accuracy
In the previous study, the CNN was adopted as thedeep learning method to replace numerical analysis
codes in loading pattern optimization. Furthermore, inthis study, internal structural modification of the CNN,
normalization and regularization were applied to
improve prediction performance [5].
3.1 Normalization and regularization
Normalization of input data is a technique ofmachine learning used when the scale of input data isdifferent. The normalization means normalizing the data
dimensions to the same scale. Applying thenormalization leads to more stable and faster learning. In
this study, Min-Max scaling is applied and normalized
data can be obtained as:𝑥−𝑥

𝑥𝑛𝑜𝑟𝑚 = 𝑥𝑑𝑎𝑡𝑎 − 𝑥 𝑚𝑖𝑛 ,
𝑚𝑎𝑥

𝑚𝑖𝑛

(1)

Where,
𝑥𝑑𝑎𝑡𝑎 = original value,
𝑥𝑚𝑎𝑥 = maximum value in data,𝑥𝑚𝑖𝑚 = minimum value in data.
The input data size is 8x8 of the quarter core size andthe input parameters are fuel enrichment, BP fraction,
number of BP pins, and assembly burnup. Fig. 1 shows aexample of input data.

Fig. 2. Internal structures of the prediction models
4. Results and Discussion
In this section, the prediction performance of the
improved model is evaluated. The normalization and theL2 regularization were applied to both the peaking factor
and the cycle length prediction models. Since the cycle
length prediction model showed sufficient prediction
accuracy, the sensitivity test through internal structure
modification was performed only for the peaking factor
prediction model. The overall structure of the prediction
model is shown in Fig. 2. In addition, 3x3 convolution
filter, about 42,000 training data, and 6,000 test data
were used equally in all models. The prediction models
were implemented using the TensorFlow library [7].

Fig. 1. Example of input data
Overfitting refers to the problem of poor
performance in test data because it is learned too closely
fit to limited training data. In this study, the L2
regularization was applied to improve the performanceby preventing overfitting. It is the most common

176Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4.1 Training Result of Peaking Factor
In this content, the performance of the improved
peaking factor prediction model is evaluated. The
performance of model 1 and 2 using normalization andL2 regularization (M1NR and M2NR) are comparedwith the results of the previous study (M1). Fig. 3 shows
the process of the prediction error convergenceaccording to the training step using test data. Fig. 4~6
show the training results of peaking factor using M1,
M1NR, and M2NR.
Fig. 6. Training result of peaking factor using M2NR
As the results of training using the M1, 98.7% of thetest data is predicted within 5% error range. With M1NR,
98.5% of the test data is predicted within 4% error range,
and with M2NR, 98.3% of the test data is predictedwithin 3% error range. The results show that the M2NR
has the best performance in peaking factor prediction.
Table Ⅲ summarizes the peaking factor prediction errors.
Table Ⅲ. Prediction error of peaking factor
Prediction error
Model
RMS (%)Max (%)
M1
1.74
13.81
M1NR
1.42
8.41
M2NR
1.14
6.92

Fig. 3. Convergence process for peaking factor
prediction

4.2 Training Result of Cycle Length
In this content, the performance of M1, M1NR and
M2NR in cycle length prediction were compared. Fig. 7
shows the process of the prediction error convergenceaccording to the training step using test data. Fig. 8~10
show the training results of the cycle length using M1,
M1NR and M2NR.
Fig. 4. Training result of peaking factor using M1

Fig. 7. Convergence process for cycle length prediction
Fig. 5. Training result of peaking factor using M1NR

177

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

5. Conclusions
In this study, we successfully applied the internal
structure modification, the normalization and theregularization as a way to improve the prediction
performance. As a result of the model improvement, theprediction performance was improved in both thepeaking factor and the cycle length. In particular, thecycle length prediction model has sufficient prediction
accuracy, while the peaking factor prediction model stillneeds improvement.
Therefore, the peaking factor prediction model needs
to improve the performance through additional
sensitivity test for the weight layer depth and the numberof filters. Furthermore, we will also develop theprediction models of other parameters such as power
distribution and MTC as future work.

Fig. 8. Training result of cycle length using M1

Acknowledgements
This research was supported by the project(Development of Artificial Intelligence Technology for
Multi-Cycle Core Design Based on Machine Learning)
by Korea Hydro & Nuclear Power Co. Ltd.
References
1. H. Jang, et al, “Prediction of Pressurized Water
Reactor Core Design Parameters Using Artificial
Neural Network for Loading Pattern Optimization,”
Korea Nuclear Society Jeju, Korea, May 22-24(2019).
2. S. Choi, et al, “Resonance Treatment using PinBased Pointwise Energy Slowing-Down Method,”
Journal of Computational Physics, 330, pp.134(2017).
3. J. Choe, et al, “Verification and validation of
STREAM/RAST-K for PWR analysis,” Nuclear
Engineering and Technology, 51, 2, pp.356 (2019).
4. F. Khoshahval, et al, “Vanadium, rhodium, silver
and cobalt
self-powered neutron detector
calculations by RAST-K v2.0,” Annals of Nuclear
Energy, 111, pp.644 (2018).
5. Karpathy, et al, "Cs231n convolutional neuralnetworks for visual recognition," Neural networks, 1(2016).
6. K. Simonyan, et al, “Very deep convolutional
networks for large-scale image recognition,” arXiv
preprint arXiv:1409.1556.
7. M. Abadi, et al., “TensorFlow: A System for Largescale Machine Learning,” Proceeding of the 12th
USENIX conference on Operating Systems Design
and Implementation, Savannah, Nov. 2-4, pp.265,
GA (2016).

Fig. 9. Training result of cycle length using M1NR

Fig. 10. Training result of cycle length using M2NR
As the results of training using the M1, 96.9% of thetest data is predicted within 1.5% error range. With
M1NR, 95.6% of the test data is predicted within 0.5%
error range, and with M2NR, 99.0% of the test data ispredicted within 0.5% error range. The results show that
the M2NR has the higher performance in cycle length
prediction. Table Ⅳ summarizes the cycle length
prediction errors.
Table Ⅳ. Prediction error of cycle length
Prediction error
Model
RMS (%)Max (%)
M1
1.07
2.12
M1NR
0.26
1.44
M2NR
0.18
1.96

178

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Experimental Validation for the Pin-wise Isotope Prediction Methodology
Minjae Lee, Taeyang Noh and Myung Hyun Kim
Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do, Republic of Korea, 17104*
Corresponding author: mhkim@khu.ac.kr

Abstract
Detail and reliable isotope assessment of spent nuclear fuels is an essential task
for criticality analysis, shielding design, decay heat cooling analysis and others in
the application for storage, transport and disposal. Because of time limitation anddata processing issues, material assay has been evaluated in the unit of assembly.
A mathematical methodology to predict pin-wise isotope inventory was developed
and used for the reutilization of spent fuel pins in PWR. In this paper, validationof this method is done by comparing with experimental data published by
OECD/NEA. Validation was carried out by comparing with experimental data forthree reactor cases. Because quite a few data are available from experiment,
calculation results were compared only for a few fuel pins in an assembly. For themost of major isotopes which has large impact, it was found that error to theexperimental data is less than 5%. However, in case of minor isotopes with small
number density and low importance to criticality, error exceeds over 5%, while
they are within an error band of 30%. Importance of each isotope to the criticality
was measured by an index of importance; as ratio of isotope contribution to total
absorption or fission rates. It was found that all isotopes showing large error over
5% have very low importance compared with major isotopes. In case of short halflife fission products, calculation was overestimated because of fast radioactive
decay and unknown cooling time of spent fuels. Overall estimation results arereasonably good within an acceptable range.
Key Words: Validation, Pin-wise, Isotope prediction, Experimental dataassemblies. Second step is the production of database as
many tables of isotope number densities vs. pin burnup.
Last stage is to match the pin information to the proper
table and interpolate to assess the fuel pin composition.
We may establish many different model options
depending on the level of depth and width of treatment.
In this study, validation of mathematical model has
been tried. A numerical benchmark may be the easiest
way to be achieved in a near future. Monte Carlosimulation for all spent fuel pins in a full-sized reactor
model cannot be solved for all kinds of isotopes and for along period of fuel depletion history. A simplified
numerical benchmark problem can be developed as a
future work scope. In the other stream, experimental
assessment is available for a limited number of isotopes.
Recently experimental assessment results for a few fuelpins from many PWR and BWR cores were published.
OECD/NEA organized an international collaborationteam to assess the spent fuel compositions and published
two documents [3, 4]. In this paper, simulation ofassembly-wise depletion was performed and predictedisotopic compositions were compared with OECD/NEA
experimental data.

1. Introduction
Management of spent nuclear fuels has been studied inorder to reduce safety margin related to many applications
such as reprocessing, interim storage, cask transportationand final disposal. Technical assessments for criticality
analysis, shielding design, decay heat evaluations needtime-dependent data of detail isotopic concentrations inall spent fuel pins. However, it has been out of scope ofcore designer to treat all kinds of isotopes in all fuel pins.
Modern core design methodologies based on nodal
methods gave big benefits in saving of computationaltime and memory. However, detail pin-wise information
has been lost during book keeping for spent fuelmanagement. Therefore, available data for all spent fuels
are not complete. Most of isotopic inventory data aremanaged as assembly-averaged data and number ofisotopes traced are limited to a certain number.
A mathematical methodology to predict pin-wiseisotope inventory was developed and used for thereutilization of spent fuel pins in PWR [1, 2]. Pin-wiseisotope prediction is done by three stages. First stage is topredict pin-wise burnup distribution by simulating corecalculation with historical information about spent fuel179

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table I. OECD/NEA Measurement Data List

2. Analysis method

Reactor

2.1 Pin-wise isotope prediction methodology

Assembly

Pin
SF95

Pin-wise isotope prediction methodology consists ofthree stages. Figure 1 shows the pin-wise isotope
prediction methodology procedure. First stage is topredict the pin-wise burnup distribution through corecalculation and reconstruction techniques. Core
calculation was done by DeCART2D and MASTER code
package which was developed by Korea Atomic Energy
Research Institute (KAERI) [5, 6]. Group constant andform function library is produced by DeCART2D and
HELIOS 47group library. Then, core calculation is doneby MASTER with group constant and form function
library, and pin-wise burnup distribution was forecasted
by using reconstruction method.
Second stage is production of isotope number density
tables as a function of burnup through assembly
calculation by using DeCART2D. In this stage, groupingof fuel pins inside of an assembly is an essential work to
reduce the number of tables. Isotope number density
tables are classified along with fuel pin types and loaded
place within assembly. Fuel pins in the center of assemblyand near the water hole have a characteristic to deplete
faster than other pins while fuel pins located in edge of
assembly fuel pin is depleted slowly. Through this way,
isotope number density tables between 3 and 6 is
produced in accordance with assembly loading pattern.
Last Step is to predict isotope composition by usualinterpolation technique using prepared group of tables
and predicted burnup information.

NT3G23
SF96

Takahama
Unit 3

NT3G24

SF97

D047

MKP
109

D101

MLA
098

G13

N13

CalvertCliffsUnit 1C5

Sample
Name
2
3
4
5
2
3
4
5
2
3
4
5
6
CC
LL
P
BB
JJ
P
91E07
89G01

Burnup(GWd/tU)
24.35
35.42
36.69
30.4
16.44
28.2
28.91
24.19
30.73
42.16
47.03
47.25
40.79
37.12
27.35
44.34
26.62
18.68
33.17
52.43421.465

CoolingTime (d)
841

1051

1358

4656
1824
No data

89G03
28.71789G08
30.172No data
F4
89G10
38.496
O13 89G05
25.137* Cooling time means the days from shutdown to the measured day
Ohi
Unit 1&2

17G

Because detailed core configuration is not described, corecalculation is not able. Instead of output of corecalculation, measured burnup is used. The burnup
calculated by the MASTER code is assumed to be sameto the burnup in Table I. Finally, number density is
calculated by the DeCART2D code. Cooling time is notconsidered in this step, because the number of isotopes are
not enough in spite of complex decay chain.
2.3 Evaluation procedure
Results were checked by using the ratio of calculated
value to experimental data (i.e., C/E ratio). In this study,
acceptable error was defined to 5% for major isotopes and
30% for insignificant isotopes which have negligible
amount or affect negligible in terms of criticality. Amountof isotope is evaluated by number density and impact on
criticality is assessed by cross section. Impact is evaluatedby ratio of isotope contribution to total absorption, whichis named "importance” in this paper. The importance isdefined as the followings and they are evaluated by
MCNP6 where ENDF-VII.1 is used as nuclear librarydata. In order to suggest trustworthy results, relative error
was set under 50pcm. Also, cycle is divided into 50
inactive cycle and 100 active cycle that each cycle has
50000 neutron histories.

Fig. 1. Pin-wise isotope prediction methodology scheme
2.2 OECD/NEA experimental data
In this study, OECD/NEA experimental data is used forvalidating pin-wise isotope prediction methodology.
These experimental data are managed by OECD/NEA forthe purpose of burnup credit license and many types ofreactor are included in this package. Since methodology
was developed only for Pressurized Water Reactor (PWR)
cores, selection of data set for comparison was limited to
PWR cores.

Importance
𝑝𝑖𝑛

Σ𝑓,𝑁𝑖
𝑝𝑖𝑛

=

Σ𝑎,𝑡𝑜𝑡𝑎𝑙
𝑝𝑖𝑛

Σ𝑓,𝑁𝑖
𝑝𝑖𝑛

(Σ𝑎,𝑡𝑜𝑡𝑎𝑙

180, (𝑓𝑜𝑟 𝑓𝑖𝑠𝑠𝑖𝑙𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙), (𝑓𝑜𝑟 𝑓𝑒𝑟𝑡𝑖𝑙𝑒 𝑎𝑛𝑑 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table III. Fuel Pin Number Density – 50GWd/tU

3. Results

Isotope

3.1 Takahama Unit 3

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Results of takahama unit 3 is shown in Fig. 2 and Fig.
3. Isotope distribution along error band is listed in Table
II. Error under 5% means to predict suitably. Evaluation
for number density and importance is conducted forisotope above 5% error. In below figures, the boundaries
of 5% and 30% errors are represented by the red line andblue line, respectively.

U238
U235
Pu239
U236
Cs137
Pu240
Nd144
Pu241
Nd143
Nd146
Nd145
Nd148
Pu242
Np237
Ce144
Sm150
Ru106
Nd150
Sm148
Cs134

NumberDensity
2.1430E-02
1.8755E-04
1.4265E-04
1.3013E-04
7.0714E-05
6.5680E-05
5.8388E-05
4.1129E-05
4.0688E-05
3.8602E-05
3.5367E-05
2.0039E-05
1.9741E-05
1.5468E-05
1.5419E-05
1.5346E-05
1.1559E-05
9.6805E-06
8.3964E-06
8.3225E-06

Isotope
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38

Pu238
Sm152
U235
Am243
Sm147
Cm244
Sm154
Eu154
Nd142
Am241
Sb125
Sm151
Cm242
Sm149
Cm245
Am242mCm243
Cm246

NumberDensity
7.4531E-06
5.0241E-06
4.8103E-06
4.2682E-06
3.8790E-06
1.9880E-06
1.9858E-06
1.5008E-06
1.3353E-06
1.2836E-06
8.1856E-07
5.7244E-07
5.6097E-07
1.4852E-07
1.3374E-07
2.9900E-08
1.9222E-08
1.4228E-08

Table IV. Importance for Criticality – 50GWd/tU
IsotopeFig. 2. C/E ratio – Takahama Unit 3 (actinide)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Fig. 3. C/E ratio – Takahama Unit 3 (fission product)
Table II. Isotope List according to Relative Error
Relative Error
Isotope235
U, 236U, 238U, 239Pu, 240Pu, 241Pu,
242
Pu, 143Nd, 145Nd, 146Nd, 148Nd,
Under 5%150
Nd, 149Sm, 150Sm, 151Sm, 152Sm,
154
Sm, 137Cs
234
U, 238Pu, 241Am, 242mAm, 243Am,
From 5% to244
Cm, 245Cm, 142Nd, 144Nd, 148Sm,
30%
154
Eu, 134Cs, 144Ce
237
Nd, 242Cm, 243Cm, 147Sm, 106Ru,
Above 30%
125
Sb

U238
Pu239
U235
Pu240
Pu241
Nd143
U236
Pu242
Np237
Sm149
Sm152
Sm151
Eu154
Nd145
Am243
Pu238
Sm150
Am241
Cs134
Sm147

NumberDensity
2.6333E-01
1.5645E-01
8.1254E-02
7.6283E-02
5.0495E-02
1.1440E-02
9.6123E-03
6.6572E-03
6.2807E-03
6.2133E-03
5.1523E-03
4.3474E-03
3.8636E-03
3.6388E-03
3.0888E-03
2.2487E-03
2.1295E-03
1.6682E-03
1.2551E-03
1.1605E-03

Isotope
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38

U234
Cm244
Nd144
Cm245
Am242m
Nd148
Sm148
Nd146
Nd150
Cm242
Sm154
Cs137
Ce144
Nd142
Sb125
Cm243
Ru106
Cm246

NumberDensity
1.0486E-03
4.5129E-04
2.7590E-04
2.3868E-04
2.2667E-04
1.4737E-04
1.3824E-04
9.6285E-05
6.5580E-05
4.5848E-05
3.8387E-05
2.6230E-05
2.4606E-05
2.3695E-05
2.1928E-05
2.1789E-05
1.2698E-05
7.9081E-07

3.2 Calvert Cliffs Unit 1 and Ohi Unit 1&2
Calvert Cliffs Unit 1 and Ohi Unit 1&2 also apply same
method for evaluation. Results of Calvert Cliffs and Ohi
is shown in Fig. 4, 5 and 6. 237Nd, 242Cm and 243Cm ismatched well in Calvert Cliffs and Ohi reactor while error
is large in Takahama reactor.
However, some isotopes are overestimated. These
isotopes are listed in Table V. In case of 241Pu, it isoverestimated a little bit for Calvert Cliffs Unit 1. But thisisotope is estimated very accurately in other cases. There
are no reasons to be overestimated, without decay-out. We
guess that workers had experiment with too long coolingtime in the Calvert Cliffs reactors, therefore short-lived
isotopes were decayed out. Half-life of 241Pu and 155Eu are
14.3 years and 4.75 years, respectively. On the other hand,calculated value are results at the time of discharged point
without any cooling time because there are no data of

Evaluation of number density was performed on
50GWd/tU burnup. Number density is shown in Table III.
Except for 144Nd and 144Ce, isotopes have small number
density. Therefore, these isotopes are accepted up to 30%
error. However, there are much number density for 144Nd
and 144Ce, so importance evaluation has to be done. Table
IV shows importance for criticality. While 144Nd and 144Ce
have a large number of number density, importance is low
because of low XS. Therefore, 30% error is permitted for144
Nd and 144Ce.

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

cooling time in the referred documents. Five isotopes may
be overestimated for the same reasons in case of Ohi Unit
1&2. Half-life of 154Eu, 125Sb, 134Cs, 106Ru and 144Ce are
8.6 years, 2.76 years, 2.07 years, 371.8 days and 284.9
days, respectively. These are the shortest value in case of
Ohi Unit 1&2.

4. Conclusions
In this paper, validation of pin-wise isotope prediction
was tried by comparing with OECD/NEA experimental
data. Twenty-five assay sample in three reactors wereused for validation. Objective error for major isotopes was
set as 5% and 30% error was permitted for minor isotopes
with small amount and low importance to criticality.
Isotope’s amount is evaluated by number density andimportance is assessed by ratio of isotope’s reaction rate
to fuel pin total reaction rate. Based on these results, mostisotopes satisfy the objective error. However, there were
overestimated isotopes which did not satisfied theobjective error. These isotopes have relatively short halflife. It can cause the radioactive decay during cooling timeand it will be studied in the future.
Acknowledgements

Fig. 4. C/E ratio – Calvert Cliffs Unit 1 (actinide)This work was supported by Korea Hydro & Nuclear
Power Company, LTD (Project No. 2017-Tech-12).
References
1. Aung T.D. and M.H. Kim, “Investigation into Fuel
Pin Reshuffling Options in PWR In-core FuelManagement for Enhancement of Efficient Use of
Nuclear Fuel”, Nucl. Eng. Des., 273, 332, (2014).
2. Z. Zafar, Y.S. Park and M.H. Kim, “Evaluation of thereutilization of used nuclear fuel in a PWR core
without reprocessing”, Nucl. Eng. Tech., 2, 51, 345355, (2018).
3. ALEJANO.C, et al., Spent Nuclear Fuel Assay Datafor Isotopic Validation, NEA/NSC/WPNCS/DOC(2011)5, Nuclear Science Committee, Working partyon Nuclear Criticality Safety (WPNCS), Expert
Group on Assay Data of Spent Nuclear Fuel
(EGADSNF), OECD/NEA, Jun. (2011).
4. P.Ortego, et al., Evaluation Guide for the EvaluatedSpent Fuel Nuclear Fuel Assay Data Base,
NEA/NSC/R(2015)8, Nuclear Science Committee,
Working party on Nuclear Criticality Safety(WPNCS), Expert Group on Assay Data of Spent
Nuclear Fuel (EGADSNF), OECD/NEA, Feb. (2016).
5. B.O. Cho, H.G. Joo, J.Y. Cho and S.Q. Zee,
“MASTER-3.0: Multi-purpose Analyzer for Staticand Transient Effects of Reactors”, KAERI/TR2061/2002, Korea Atomic Energy Research Institute,(2002).
6. J.Y. Cho, J.C. Lee, K.H. Lee, S.Y. Park, H.J. Park,
H.Y. Kim, K.Y. Kim, H.J. Jung, J.S. Song, S.Q. Zee,
C.K. Jo and H.C. Lee, “DeCART2D v1.0 User`s
Manual”, KAERI/TR-5116/2013, Korea Atomic
Energy Research Institute, (2013).

Fig. 5. C/E ratio – Calvert Cliffs Unit 1 (fission product)

Fig. 6. C/E ratio – Ohi Unit 1&2 (all isotopes)
Table V. Overestimated Isotopes (C/E ratio)

Calvert Cliffs Unit 1
MKP109 MKP109 MKP109 MLA098 MLA098 MLA098
Isotope
CC
LL
P
BB
JJ
P
Pu241 1.347
1.372
1.323
1.391
1.380
1.413
Eu155 2.618
2.607
2.422
Ohi Unit 1&2
N13
C5
C5
F4
F4
O13
Isotope
91E07
89G01 89G03 89G10 89G08 89G05
Eu154 1.727
1.545
1.635
1.887
1.894
1.604
Sb125 8.020
7.497
7.100
8.758
8.876
7.600
Cs134 5.419
5.128
4.864
5.485
5.280
4.897
Ru106 36.036 31.488 28.657 34.144 31.255 28.589
Ce144 109.996 88.012 86.157 94.216 81.858 81.806

182

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The Neutron Streaming Correction Method in the Homogenization of Pebble-bed
HTGR Reflector
Yutong Wen*, Ding She, Lei Shi and Jing Zhao

Institute of Nuclear And New Energy Technology, Tsinghua University
Tsinghua University, Haidian, Beijing, China

wenyt17@mails.tsinghua.edu.cn

Abstract
In pebble bed high temperature gas-cooled reactors (HTGRs), the graphitereflector generally contains void regions including control rod channels, absorberball channels and coolant channels. However, reactor physics design of HTGR
based on deterministic codes usually models the reactor by two dimensional r-z
coordinate system, which means the regions must be homogenized in azimuthal
direction. PANGU is a modern computer code for pebble-bed HTGR physics
analyses and fuel cycle simulations. This paper demonstrates a new method, forthe homogenization of the reflector, implemented in PANGU code. The new
method, taking into account the neutron streaming effect, can improve theaccuracy of neutronics calculation.
Key Words: HTGR, PANGU code, void regions, neutron streaming,
homogenization
1. Introduction

2. Neutron streaming effect correction in HTGRIn HTGR physics analysis deterministic codesuch as VSOP [1] and PANGU [2], the reactor isusually modelled in two dimensional r-z coordinate
system, assuming that the model is azimuthally
uniform. While, in high temperature gas-cooled
reactors (HTRGs), the graphite reflector generally
contains certain void regions, such as control rod
channels, absorber ball channels and coolant channels,
which have azimuthal dependence. It is required thatthe regions containing void channels must be
homogenized in the azimuthal direction. The mostconvenient method is to treat the regions by volumetrichomogenization. However, because of the neutron
streaming effect of voided region, the volumetrichomogenization will bring about remarkable deviation.
In this paper, a new method for homogenization is
introduced. In this method, the equivalent diffusion
coefficient will be calculated to take into account theneutron streaming effect. The method has been
implemented in PANGU code.
The remainder of the paper is organized as follows.
In Section 2, the general correction method for neutron
streaming in HTGR has been introduced. The new
method is described in Section 3. In Section 4, thenumerical tests of HTR-10 and HTR-PM aredemonstrated, and the conclusion remarks are provided
in Section 5.

In the regions, which contains void space,neutrons can stream through the void space, increasing
the migration length. The simple volumetrichomogenization can lead to the underestimation of theneutron transport, resulting in less neutron leakage outof the region [3]. In pebble bed HTGR, neutron
streaming effect is remarkable in pebble bed. To treat
the streaming effect in pebble bed, the general method
is to calculate the spatially homogenized diffusion
coefficient, using streaming effect correction factor of
diffusion coefficient, as is proposed by Behrens (1949)[4]−

Dhom = Cstr Dvol

(1)

Dhom is the homogenized diffusion coefficient. The

formula of C str can be referred in Reference [4]. Dvol isthe volume-averaged diffusion coefficient.

Dvol =

1
3 tr ,vol(2)

Where  tr,vol is the volume-averaged transport crosssection.
Lieberoth etc (1980) proposed a new formula for183

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019the correction factor C str , on basis of Behrens’ work,
which can be referred in Reference [5] The formula isof general validity in various HTGR physics analysis
codes [6][7]. The HTGR graphite reflector also
contains void regions. However, the spatial distribution
and geometry shape of the void regions, as well as theneutron’s anisotropy are all distinctly different fromthat in pebble bed. Therefore the correction factor of
pebble bed can not be applied to the reflector. But thestreaming correction of pebble bed provides the idea
that, the streaming effect can be taken into account by
adjusting the spatially homogenized diffusion
coefficient.

Fig. 1. The transmission model of the reflector

Where L is the side length of the square. f is thevoid fraction. R is the radius of the void circle. The
model assumes that a parallel neutron beam transverses
the square. The penetrating probability p can becalculated by:

3. Description of the method

The homogenized diffusion
calculated by the following:
Dhom =

Where

−

 tr

1
−

3  trcoefficient

is

2 L2
(5)
du  e −tr l (u )L 0And l (u ) is the length of the neutron trajectory inreflector material.
L − 2 R 2 − u 2uRl (u ) = Lu = R(3)

p=is the spatially homogenized transport−

cross-section. The simplest way to obtain  tr is byvolume-averaging as eq (2). However, In general, it iscalculated by following some equivalence principles.
Reference [8] suggests a method to obtain
homogenized cross-sections, which approximates theneutron’s transmission behavior, from a simplified
physics model. This method follows the equivalenceprinciple that the transmission probability of the region
is preserved. This method is designed for the spatialhomogenization of fuel grains in HTGR fuel elements.
The idea of the method is applied in this work to−
calculate the  tr .
In the HTGR reflector, there are various voidchannels, which vary distinctly in geometry shape. For
the sake of simplicity and convenience, a simplified
transmission model is established. As is shown in
Figure 1. The void region is supposed to be located at
the center of a square in circle form (or infinite length
cylinder), surrounded by the reflector material. The
area of the circle is the mean value of the crosssectional areas of all void channels, and the side lengthof the square is determined by the following:
L=


f

R

(6)

The integration in Eq (5) is solved numerically.
Then, it is assumed that the square is filled withhomogeneous media. Importantly, this work assumes
that the homogenization procedure preserves thepenetration probability. The penetration probability of
the homogenized square is−p = e−tr L
(7)

(4)

Fig. 2. The spatial homogenization, by preserving thepenetration probability

To preserve the penetration probability, theequivalent transport cross section is calculated as:−

 tr = −

184

ln p
L

(8)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019by transmission model, to obtain the homogenized−transport cross sections  tr . The radius of the void
circle R in Eq(4) is determined by the mean value ofthe cross sectional areas of all void channels in theannular zone.
HTR-PM is a modular pebble bed HTGR
demonstration power plant, approved as part of the
National Major Science and Technology Project inChina. The distribution of the void channels in HTRPM is similar with that in HTR-10, and the design
parameters can be found in Reference [11].

The homogenized diffusion coefficient is calculated by
Eq (3).
4. Numerical Validation

4.1 Description of Test Cases
The method for homogenized diffusion
coefficient has been implemented in PANGU code. In
order to validate the method for homogenized diffusion
coefficient calculation, The calculation models of
HTR-10 and HTR-PM are tested. The calculation
results of keff is compared with that by MCNP.
The HTR-10 is a test modular pebble bed HTGR,
designed and constructed by Tsinghua University’s
Institute of Nuclear Energy Technology (INET) . The
horizontal cross sectional view of HTR-10 is shown in
Fig.3 (a). As is shown in Fig 3 (b), the annular regions
containing void channels are homogenized in theazimuthal direction, for the 2D physics calculation by
PANGU.

4.2. Calculation Results and Analysis
4.2.1 HTR-10
Due to the fact that PANGU is a deterministiccode, quite different from Monte Carlo code inneutronics calculation. It is necessary to evaluate theinherent difference between the two codes. The
deviation caused by the void channels must be
removed at first. Hence, the annular zones containing
void regions are firstly treated by volumetrichomogenization, as is shown in Fig 3 (b). The 2Dmodel is calculated by MCNP and PANGU. The resultsof the keff are given in Table I. It can be concluded that,without the void regions, the neutronics calculation
results of PANGU code agrees well with the MCNP
code.
Table I. Calculation results of HTR-10 (without voidchannels) by PANGU and MCNP
code
MCNP
PANGU

(a) The 3D high-fidelity model of HTR-10

keff
1.22225
1.22119

Error(pcm)-106

The calculation results of HTR-10 are
demonstrated in Tab II. It can be found that, without
correction, the keff calculated by PANGU code ishigher than that by MCNP code by more than 1000
pcm. While, after the streaming correction, thedeviation decreases remarkably.

(b) The simplified 2D model of HTR-10 [9]Table II. Calculation results of keff for HTR-10 modelby PANGU and MCNP

Fig. 3. The horizontal cross sectional view of HTR-10

There are two annular zones requiring
homogenization. The inner annular zone contains 13
control rods channels and the 7 absorber ball channels.
And the outer annular zone contains 20 coolant flow
channels. The detailed design parameters of HTR-10
can be found in Reference [10].
The two annular zones are separately calculated

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

code

keff

MCNP(3D
highfidelity)
PANGU(2D,withoutcorrection)PANGU(2D)

1.21071

improve the accuracy of PANGU 2D neutronics
calculation.

Error(pcm)

References
1.221191048

1.21026-45

1. E. TEUCHERT et al., “V.S.O.P. (’94) Computer Code
System for Reactor Physics and Fuel Cycle Simulation,”
Juel-2897, Institut für Sicherheitsforschung und
Reaktortechnik (1994).
2. Ding She et,PANGU code for pebble-bed HTGR
reactor physics and fuel cycle simulations,Annals of
Nuclear Energy,Volume 126,2019
3. Auwerda G J , Kloosterman J L , Lathouwers D , et al.
Effects of random pebble distribution on themultiplication factor in HTR pebble bed reactors[J].
Annals of Nuclear Energy, 2010, 37(8):1056-1066.
4. Behrens D J . The Effect of Holes in a Reacting
Material on the Passage of Neutrons[J]. Proceedings of
the Physical Society, 2002, 62(10):607.
5. Lieberoth J , Stojadinovi? A . Neutron Streaming in
Pebble Beds[J]. Nuclear Science and Engineering, 1980,
76(3):336-344.
6. Gougar H D . The Application of the PEBBED CodeSuite to the PBMR-400 Coupled Code Benchmark - FY
2006 Annual Report[J]. Office of Scientific & Technical
Information Technical Reports, 2006.
7. Bernnat W , Feltes W . Models for reactor physics
calculations for HTR pebble bed modular reactors[J].
Nuclear Engineering and Design, 2003, 222(2-3):331-347.
8. She D , Liu Z , Shi L . An Equivalent Homogenization
Method for Treating the Stochastic Media[J]. Nuclear
Science & Engineering the Journal of the American
Nuclear Society, 2017, 185(2):351-360.
9. Terry W K , Kim S S , Montierth L M , et al. Evaluationof the HTR-10 Reactor as a Benchmark for Physics CodeQA[J]. Office of Scientific & Technical Information
Technical Reports, 2006.
10. Evaluation of high temperature gas cooled reactor
performance: Benchmark analysis related to initial testingof the HTTR and HTR-10, IAEA-TECDOC-1382,
International Atomic Energy Agency, Vienna 2003.
11. Zhang Z , Wu Z , Sun Y , et al. Design aspects of theChinese modular high-temperature gas-cooled reactor
HTR-PM[J]. Nuclear Engineering and Design, 2006,
236(5-6):485-490.

4.2.2 HTR-PM
Table III demonstrates the test results of keff for
HTR-PM. Similar with the results of HTR-10, thestreaming correction significantly reduces thedifference between the calculation results of the MCNP
3D model and PANGU 2D model.
Table III. Calculation results of keff for HTR-PM
model by PANGU and MCNP
code

keff

MCNP(3D
highfidelity)
PANGU(2D,without
correction)PANGU(2D)

1.24337

Error(pcm)1.254981161

1.24575238

4.2.3 Analysis
From the results of HTR-10 and HTR-PM, it canbe found that the results of calculation by PANGU,
with streaming correction, agrees well with that by
MCNP, which illustrates that streaming effect can betaken into account by calculating the homogenizeddiffusion coefficient, thus reducing the deviation
caused by the homogenization from 3D model to 2Dmodel.
5.Conclusion
In this work, a method is proposed to treat theneutron streaming effect in the homogenization of
HTGR reflector, which contains void channels, bycalculating the equivalent homogenized diffusion
coefficients. In this method, a simplified physics modelis proposed to approximate the regions containing voidchannels. The homogenization follows the rule that, theneutron penetrating probability should be preserved.
The method has been implemented in PANGU code.
The numerical results of a simplified HTGR model
agree well with Monte Carlo 3D high-fidelity solutions.
It can be concluded that the method effectively

186

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Application of Neutron Balance Analysis in Enhancing the Temperature Reactivity
Feedback of Gas-cooled Fast Reactora

Ce Zhanga, Youqi Zhenga,* and Linfang Weia
School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China*
Corresponding author: yqzheng@mail.xjtu.edu.cn

Abstract
In recent years, more gas-cooled fast reactor are designed for various
applications. However, for all kinds of gas coolant, the temperature reactivity
feedback is generally very small, which will cause safety problem in thetransients. Out of this consideration, the optimization methods for enhancing thenegative feedback are investigated. A series of gas-cooled fast reactor core
designs are calculated by the SARAX (System for Advanced Reactor Analysis at
Xi’an Jiaotong University) code system. To further study the influence factors oftemperature reactivity feedback, the neutron balance method is applied.
Reactivity decomposition based on neutron balance method is available for thequantitative analysis of reactivity change. On the basis of neutron balance, thecoolant expansion reactivity can be decomposed into the contribution fromreaction-types, nuclides, or energy-groups. The decomposition results support
that the optimization measures are effective.
Key Words: reactivity feedback, neutron balance, core design, GFR
1. Introductionchange of temperature. The difference of coolant
temperature between the inlet and outlet of PWR core ismostly 30℃, which leads to small overall temperaturerise of the core material. However, this difference of gas
cooled fast reactor core is 150~200℃. In addition to the
feedback brought by fuel Doppler Effect and coolant
expansion effect, the expansion effect of core material
should also be considered.
The Doppler Effect mainly comes from the change ofneutron absorption capacity of heavy nuclides in the core.
When the temperature increases, the height of theresonance peak of the nuclide decreases and the width
becomes wider, and the total resonance absorption
capacity will become stronger, which will cause theabsorption cross section of the neutron to become larger
and the total reactivity of the system to become smaller.
The coolant expansion feedback mainly comes fromthe change of neutron slowing ability caused by thechange of coolant density and the influence of thedecrease of density on the neutron leakage of reactor.
The specific contribution of gas coolant expansion
feedback has not been studied quantitatively. It can referto two main reasons for the density change of water in
PWR. First, the absorption of neutrons by the coolant
decreases when the density changes, while the thermal
neutron utilization coefficient of fuel increases, leading
to the increase of reactivity. Second, as the densitychanges, the coolant slow down less neutrons, which
increases the number of epithermal neutrons, and thus
the resonance absorption of the fuel increases, making
the reactivity decrease. Therefore, only appropriate

Since GFR was listed as one of the six generation IV
reactors, the application and performance of gas coolant
in nuclear reactors have been developed greatly. The
Helium, the He-Xe mixed gas and so on have been
investigated a lot.
However, all kinds of gas-cooled fast reactors havepotential problems in the transients. Because of its small
density and cross-sections, the coolant provides little
reactivity feedback, while it has much weaker heattransfer capacity than water or liquid metal. Thus, it’srequired to take a special design to enhance thetemperature feedback of the reactor to improve thesafety performance.
This paper describes the works on enhancing thetemperature reactivity feedback of a medium size gascooled fast reactor. Two methods are tried. One is to use
the moderator material to adjust the neutron spectrum,
which aims to enhance the Doppler feedback. The otheris to enhance the leakage of the core, which aims to
make the coolant expansion feedback larger. To guidethe tests, the neutron balance method is applied in theanalysis, which quantitatively supports the enhancing
approaches.
2. Reactivity temperature feedback analysis
The feedback of PWR mainly comes from Doppler
Effect of fuel and the feedback of the influence of
coolant density on the neutron slowing ability with187

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

water-uranium ratio can make the coolant void feedback
negative [1]. Different from PWR, the reaction crosssection of gas coolant with neutrons is much smaller
than that of water, and the absorption of neutrons ismuch smaller. The influence of the neutron moderationand leakage in the reactor core needs to be further
analyzed in combination with specific reactor cores.
The expansion effect of structure materials is
considered by referring to sodium-cooled fast reactor,
which is mainly divided into axial expansion of fuel,radial expansion of cladding and assembly wall, andexpansion of grid plate under the core. The feedback
effect of axial expansion of fuel mainly comes from thereduction of reactivity caused by the decrease of nuclear
density of fuel and the increase of reactivity caused byaxial elongation of fuel. The radial expansion feedback
effect of cladding and component wall is mainly due tothe reduction of coolant caused by volume expansion of
cladding and component wall, which further influences
the reactivity. The influence of expansion of the lower
grid plate on reactivity is increased by increasing the
spacing of components and increasing coolant and coreleakage.
In our research, the reactivity temperature feedback
is composed of Doppler Effect, coolant expansion
reactivity and the expansion effect of structural materials.
The structure materials used in the nuclear reactor are alltested by engineering application. It’s not wise to
optimize the reactivity feedback in terms of materials
selection. In consequence, our optimization focuses on
the Doppler Effect and coolant expansion reactivity.

     r,E, dEd  dV    r,E   r,E,  dEd  dV
1
 f  r,E'   r,E',  '  dE' d  ' dV
keff 

The transport solver of SARAX uses the SN node
method based on arbitrary triangular mesh. Calculation
mesh can be shown like Fig. 1

Fig. 1. Triangular calculation mesh.
The single-group steady-state neutron transport
equation in m-th discrete direction in the mesh is:2 xm xm  2um um  2vm vm m       Q
h
m
z

m
z

surfaces and z axis,  x ,  x and  x are the threesurfaces flux,  z  and  z  are the nodal top andbottom surface flux, hz is the nodal height,  is thenodal average flux.  t is total cross section and Q isthe neutron generation term.
Integrate Eq. (3) over space and energy like Eq. (2).
It can lead to the discrete neutron balance equation:M  w  2   2   2  dVm 1M   wm
m 1

m
x

m

m

m
x

m
u

m
u

m
v

m
z

M

m
z

m 1

z



t

(1)

m
v

  dV    w   dVh   r,E,      r,E   r,E,  

m

A

m

(4)

1 M
 f wm m dV
keff  m 1

Where wm is the weight coefficient in each discrete

eff

s

(3)

Where m is the number of energy group,  x , u , v
and  are the quadrature for the three triangularIt can be known from the above qualitative analysis that
we can enhance the Doppler Effect by softening neutronenergy spectrum and reduce coolant expansion reactivity
by increasing core neutron leakage. But the traditional
way can only provide qualitative judgment. Neutronbalance method is able to help analyze reactivity
feedback of core design quantitatively.
The neutron balance method can be used for detailed
reactivity decompositions in terms of different reactiontypes, nuclides or energy groups [2]. Reference to themethod of decompositions of void reactivity in thesodium-cooled fast reactor, we take a similar way toanalyze the Doppler Effect and the coolant expansion ofgas-cooled fast reactor core.
In the core, the steady-state neutron transport
equation can be shown as below:  

mt

z

3. Neutron balance method

( E )
 f  r,E'   r,E', '  dE' d  '
4 k

(2)

A

M

direction, and meets that  wm  1 .

 r; E', '  E,   r,E', '  dE' d  'm 1

The Eq. (4) is the form of neutron balance used in
SARAX. The first integral term represents radial leakage( Lr ), the second is axial leakage ( La ), the third is

If Eq. (1) is integrated over space and energy in a
specific region, we can obtain neutron balance equation:

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019absorption ( A , it can be decomposed into capture C
and fission F ). And the right of Eq. (4) is neutron
production ( P ). We simplify this equation and replacekeff into reactivity  :  1

LC  FP

8.60 cm(5)

When the reactivity changes, it’s approximately
believed that the neutron reaction rates are independentof each other, and that the change of reactivity is linearwith the change of the disturbance quantity of eachreaction rate. We can get that:            L   C   F    P

L

C

F
 




 P 
L ' L C ' C F ' F P ' P




(6)
P
P
P
keff P

Fig.2. Fuel assembly design diagram.
4.2 Core layout

  L  C   F   P

The core layout is shown in Fig. 3. In order to obtainlarge neuron leakage rate, it adopts a complex geometry
layout. The active region of inner fuel assemblies isdivided into three layers by two blanket layer axially.
And blanket assemblies are also arranged radially.

CU  235 CU  238


P
P



Inner fuelControl-2

Steel reflector

BlanketShut-down

Axial
reflectorGas
plenum(7)

CC 12 CO 16

(8)
P
P

200

20

C 

Gas space

Activecore

20

4. Core design and numerical result130 G 1  G

Control-130  g1  g 2 Outer fuel30

Eq. (6) is the result of decomposition reactivity
based on neutron balance method. The changes ofreactivity are decomposed into different reaction type.
According Eq. (4), when we integrate the reaction rates
by singe energy or nuclide, it can be decomposed into
every energy group or every nuclide. In consideration ofthat neutron leakage is weakly correlated with nuclide,and that fission reaction and neutron production can’t be
optimized by choosing nuclides, we only take thedecomposition into capture reaction of each nuclide.
These can be expressed as Eq. (7) and Eq. (8).

1030

A medium size gas-cooled fast reactor core is analyzed
in this paper. Some special designs are proposed toenhance the temperature reactivity feedback. There are
two main optimization measures. One is that ZrH1.7
material layer is added in the process of assembly design.
The other is increasing the neutron leakage by special
core layout.

BlanketZrH1.7
B4C
absorberFig. 3. Layout of gas-cooled core design
The depleted uranium blanket is expected to absorb
neutrons to increase leakage rate, and then enhance thecoolant expansion reactivity [4]. The radial gas space isdesigned to improve the radial leakage [5]. When thetemperature of core increases, the density of the gas inthis space decreases. So that more neutrons can run outof the active core.

4.1 Fuel assembly design
The fuel assembly design is shown in Fig. 2. The flat-toflat distance is 8.50 cm, thickness of assembly wall is
0.2 cm, and there is coolant in a thickness of 0.1 cm
between assemblies. So the center distance of assemblies
is 8.60 cm. And in the fuel region, there are 6 rings or 91
rods in total. There is a ZrH1.7 layer thick outside thefuel region. It can slow down the neutrons, enhancing
the Doppler Effect [3].

4.3 Numerical results of the design
The isothermal coefficient is calculated by SARAX in a
critical condition. The coolant expansion reactivity is
obtained when coolant density decreased by 1%. The

189

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Decompositon in energy-group /pcm

Doppler Effect is calculated at 600K and 900K of fueltemperature. The structure expansions reactivity changes
are all calculated in the condition of expansion 1%. Thatthe respective reactivity divides by the corresponding
temperature will reach the isothermal coefficient.
As we can see in Table I given below, the comparison
between a normal gas-cooled core and optimized one.
The optimization measures greatly enhanced Doppler
Effect, while coolant expansion reactivity and thestructure expansion gets smaller. The ZrH1.7 material
slows down neutrons, that decreases the neutron freepath, enhancing the Doppler Effect as well as letting theleakage down. The blanket in the core absorb neutrons
and increase the leakage on the opposite. The sum
reactivity feedback is enhanced by the optimization
measures.

0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2

0

5

10

15

20

25

30

Group

Fig. 4 Coolant expansion reactivity decomposition inenergy group

Table I. Isothermal coefficient of the core
NormalOptimization
Reactivity feedback/pcm·K-1/pcm·K-1
Coolant expansion-1.884-1.072
Doppler-0.219-2.478
Fuel axial expansion -0.378-0.168
Cladding and wall
0.098
0.006
radial expansion
Grid expansion-1.066-0.394Sum-3.449-4.11

5. Conclusions
This paper describes the work on the enhancing thetemperature reactivity feedback of a medium size gascooled fast reactor. This paper quantitatively analyzes
the physical reasons, develops reactivity decomposition
based on neutron balance method, and calculates thenumerical results. The results show that the two
optimization measures are effective to enhance thetemperature reactivity feedback, which are supported by
the results in the neutron balance analysis.

The coolant expansion reactivity decomposition inreaction-type and nuclide are given in Table II and Table
III, in energy-group shown in Fig. 4. As it can be seen,
the main components are radial leakage and capture
reaction. Table III lists a part of nuclides contribution on
the capture. It mainly consists of the U-238 and structurematerials element such as Fe, Ni and so on. As for theenergy-group decomposition, it’s basically negative inall group and closely related to neutron flux.

Acknowledgements
This work is supported partly by the National Natural
Science Foundation of China (Approved Number:
11775170 and 11735011).
References
1. Wu HC, Cao Lz, Zheng YQ, et al, Nuclear reactor
physics, 104-107, Atomic Energy Press, Beijing,(2014).
2. Sun K, Krepel J, Mikityuk K, et al, “Void reactivity
decomposition for the Sodium-cooled Fast Reactor
in equilibrium fuel cycle,” Annals of Nuclear Energy,(38), 1645-1657, (2011).
3. Yoo J, “Three-Dimensional Core Design of Large
Scale Supercritical Light Water-Cooled Fast
Reactor,” The University of Tokyo, Tokyo, (2006).
4. Beck T, Blanc V, Escleine JM, et al, “Conceptual
design of ASTRID fuel sub-assemblies,” Nuclear
Engineering and Design, (315), 51-60, (2017).
5. Michael AP, “Thermal hydraulics design of a
2400MWth Direct supercritical CO2-cooled fast
reactor,” Thesis of Massachusetts Institute ofTechnology, (2006).

Table II. Coolant expansion reactivity decomposition inreaction-type
Reaction-types
Value/pcm
Radial leakage-2.743
Axial leakage-0.902
Capture-2.809Fission-0.448
Production-0.833Sum-6.842
Table II. Capture reaction decomposition in nuclide
NuclidesValue/pcm
U-235
1.227
U-238
-1.044Fe-0.498Ni-0.427...
...
Sum-2.809190

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Innovative Design Concepts of Burnable Poison Rods for PWR
Aiman Dandi and Myung Hyun Kim
Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do, Korea, 17104*mhkim@khu.ac.kr

Abstract
The Burnable Poisons (BPs) are the main players to hold-down the initial excess
reactivity and to control the power peaking for Pressurized Water Reactors(PWRs). Even though all conventional BPs have some drawbacks, they satisfy thedesign requirements for 18-month cycle PWR. On the other hand, theseconventional BPs cannot meet the design requirements for a longer operation cycle
PWR cores. In this paper two BP design concepts are presented: Matryoshka Doll
Burnable Poison (MDBP) for Westinghouse (WH) fuel assembly and Burnablepoison Attached to Guide tube (BAG) for Combustion Engineering (CE) fuelassembly. The MDBP consists of two tubes inside each other and covered withclad. The outer tube contains of natural boron and the inner tube contains oferbium as absorber materials. While, the BAG is a 4 thin wires that contain of
natural boron as absorber material and attached to the guide tubes in the outercorners. The assembly calculations of the two new designs were performed by
DeCART2D code and compared with the conventional BPs results. These results
show that the MDBP can hold-down the initial excess reactivity as much as Gad
and stronger than any other conventional BPs. Even though BAG+Erbia casecontains much lower Erbia pins than in Erbia case, both cases reduce the initialexcess reactivity to the same level. The new designs can control the excess
reactivity much longer than any other conventional BPs. Also, they show
reasonable values of power peaking factor and moderator temperature coefficient(MTC).
Key Words: Burnable Poison, PWR, Innovative Design Concept
1. Introduction

2. Designing Tools and Reference Design

The Burnable Poison (BP) is the main feature to maintain
operating Pressurized Water Reactor (PWR) core within
the design constraints of average excess reactivity level,
power peaking and moderator temperature coefficient(MTC). There are different conventional BP designs thatare compatible with 18-month cycle PWR regardless eachBP design flaws. These flaws become more effective
when operate PWR longer than 18 months due to therequirement of large number of conventional BP pins tohold-down the unavoidable higher excess reactivity into
appropriate level. This leads to a failure to meet the design
constraints of PWR core operation [1, 2]. Based on that,inventing a new BP design with good properties thatallows PWR core to operate longer cycle than 18 months
within the design constraints is essential.
In the presented study, an attempt was done to invent
two new BP designs for both standard fuel assembly types:
Westinghouse (WH) fuel assembly and Combustion
Engineering (CE) fuel assembly. Only assembly
calculations were performed by DeCART-2D code [3].
The obtained results were compared with theconventional BP results in terms of k-infinite letdown
curve, power peaking and MTC.

In this study, DeCART-2D code is used to perform theassembly calculations. The reference designs used hereare 17×17 WH and 16×16 CE fuel assembly types asshown in Figs. 1 and 2. Table I shows design parametersof the two reference designs [4,5]. The direct increase
method of fuel enrichment is adopted in order to simulate
a long cycle operation PWR core [6,7]. Consequently,
only fuel enrichment is increased to 6.96w/o for normal
fuel rod and 4.10w/o for zoning fuel rod with keeping theother design parameters as the same as in both reference
designs.
3. Innovative Design Concepts
To design BP suitable for long operation cycle, someconditions should be considered. These conditions are:the BP should be able to hold-down the initial excess
reactivity to an adequate level; the consumption rate ofabsorber material should be as flat as possible; and at EndOf-Cycle (EOC), the residual reactivity penalty should be
as small as possible [8].

191Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

mixed homogeneously with 40w/o of Er2O3 and the outertube made of Al2O3-B4C. This design is loaded in the GTs
without control rod. The design structure can be changed
to three tubes or more and use different absorber material
with different concentration, depend on the required
performance. For example the expected performance of
this design is to have a good ability to hold-down theinitial excess reactivity, the burnup rate of absorbermaterial is slow and to induce the MTC toward morenegative value due to the resonant behavior of erbium atthe epithermal range (~0.5 eV) [9], which is suitable forlonger operation cycle and high burnup fuel as targeted inthis study.
Fig. 1. Standard WH fuel assembly type.

Fig. 3. MDBP design for WH fuel assembly type.
BAG design is based on a simple idea; in the outside
corners of each GT there are some spaces that canaccommodate small amount of absorber material [10].
Therefore, in this study Al2O3-B4C made as a thin wire(radius ≈ 0.235 cm) and welded at the outside of the GTat the free spaces as shown in Figs. 4 and 5. Similar to
MDBP design, the absorber material in BAG design also
can be changed to any absorber material with any
concentrations. This design has small amount of absorbermartial which is not enough to reduce the initial excess
reactivity to the needed level by itself, but due to its goodcharacteristics, it can give support to any conventional BP
design and improve their performance to be suitable for
PWR core design with any operation cycle, i.e., 18
months operation cycle or longer.

Fig. 2. Standard CE fuel assembly type.
Table I. WH and CE Fuel Assembly Design Parameters
Design Parameter
WH
CE
Fuel Rod Array
17×17
16×16
Number of Fuel Rods
264
236
Active Fuel Length
365.76381
Number of Guide Tube (GT)24
4
Number of Instrumentation
1
1
tube
Fuel Assembly Length (cm)
406.3
452.8
Fuel Assembly Pitch (cm)21.5040 20.7772
Fuel Rod Length (cm)
388.1
409.4
Cell Pitch (cm)
1.260
1.285
Fuel Diameter (cm)0.81920.8192
Cladding material
ZIRLO
ZIRLO
Cladding I.D. (cm)
0.83570.8357
Cladding O.D. (cm)
0.95
0.95
GT material
ZIRLO
ZIRLO
GT I.D. (cm)
1.008
2.286
GT O.D. (cm)1.224
2.4892The two BP design concepts are: ‘Matryoshka Doll
Burnable Poison’ (MDBP) for WH fuel assembly and
Burnable poison Attached to Guide tube (BAG) for CE
fuel assembly. Figure 3 shows the structure of MDBP
which comprises of two tubes inside each other andcovered with clad. The inner tube consists of ZIRLO

Fig. 4. Standard CE fuel assembly type with BAG design.

192

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
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96IFBA cases have the lowest residual reactivity penalty
which is less than 300 pcm for each.

Fig. 5. BAG design for CE fuel assembly type.
MDBP and BAG designs are considered as a discrete BP
which they are not mixed within the fuel thus does not
displace the fissile material, and does not reduce thethermal conductivity of the fuel. Furthermore, these
designs are easy to manufacture, during manufacturing noneed for a radiological regulated facility and during therefueling time they can be removed from the core to avoid
the effect of any undestroyed high absorption crosssection isotopes on the next cycles.
In the next section, MDBP and BAG designs arecompared with the conventional BP designs: Wet Annular
Burnable Absorber (WABA), Gadolinia (Gad), Erbia andIntegral Fuel Burnable Absorber (IFBA) in terms of kinfinite, power peaking and MTC. Also, a combined caseof BAG+Erbia is added to the comparison as an example
to show the benefit of combining BAG design with any of
the conventional BP design. In WH design, each case of
WABA, Gad and MDBP consist of 24 pins. Moreover,
Erbia and IFBA consist of 96 pins loaded in the fuelassembly. In CE design, BAG and Gad cases consist of 20
pins of BP. Further, IFBA and Erbia cases consist of 112
and 132 pins of BP respectively. On the other hand, thecombination of BAG and Erbia case consists of 20 wiresof BAG and 64 pins of Erbia. In both designs, 8.0w/o of
Gd2O3 is mixed with 2.5w/o of UO2 and 2.0w/o of Er2O3
is mixed with fully enriched UO2. Lastly, the boron in
WABA, MDBP and BAG cases is natural boron withconcentration of 10.96w/o.

Fig. 6. Comparison of excess reactivity for WH fuelassembly type.
Fig. 7 shows the k-infinite versus burnup (MWD/kgHM)
for CE fuel assembly type of No BP, Erbia, Gad, IFBA,
BAG and BAG+Erbia cases. At the BOL all cases havealmost the same effect on holding-down the initial excess
reactivity except the BAG case has the lowest value which
is about 6,404 pcm. This is due to the fact that BAG
design can contain only small amount of absorbermaterial in few locations. However, this value is morethan half of the other cases values which means the
combination of BAG with any conventional BP designs
can reduce the number of this conventional BP pins byhalf to achieve the same effect on holding-down the initialexcess reactivity. For instance, the combination of 20
wires of BAG with 64 pins of Erbia can hold-down theinitial excess reactivity as the same as 132 pins of Erbia.
This reduction of Erbia pins can be reflected positively on
the residual reactivity penalty. At the burnup step of 21
MWD/kgHM, all cases lost their ability to hold-down theexcess reactivity except Erbia and BAG+Erbia cases
which can hold-down the excess reactivity effectively
much longer. The residual reactivity penalty at the EOLof BAG+Erbia case is 468 pcm which is much lower thanthe values of Gad and Erbia cases by about 2,066 pcm and798 pcm respectively. The other cases of IFBA and BAG
have the lowest value of this penalty among all cases.

4. Calculation Results
Fig. 6 shows the k-infinite versus burnup (MWD/kgHM)
for WH fuel assembly type of No BP, 24WABA, 24Gad,
96Erbia, 96IFBA and 24MDBP cases. At the BeginningOf-Life (BOL) 24MDBP and 24Gad cases have thestrongest ability among all cases to hold-down the initialexcess reactivity by 12,500 pcm and 12,890 pcm
respectively. At the burnup step of 25 MWD/kgHM all thecases lost their ability to hold-down the excess reactivity
except 24MDBP case which it can hold-down the excess
reactivity effectively until the burnup step of 36
MWD/kgHM. Since one of the tubes in MDBP designcontains Erbia, thus some of undestroyed high neutron
absorption cross section isotopes cause high residual
reactivity penalty at End-Of-Life (EOL) [9] which isabout 1,162 pcm. This value is similar to the value of
Erbia case but it is much lower than the value of Gad caseby about 1,638 pcm. On the other hand, 24WABA andFig. 7. Comparison of excess reactivity for CE fuelassembly type.
Fig. 8 illustrates the power peaking factor in 2-D
assembly geometry versus burnup (MWD/kgHM) of the193Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019same cases for WH fuel assembly type. At BOL the power
peaking factor of 24MDBP case is higher than all cases
except 24Gad case. Around the burnup step of 25
MWD/kgHM the values of all cases become very close toeach other except 24Gad case which is much higher.

Fig. 10. Comparison of MTC for WH fuel assembly type.
Fig. 11 demonstrates the MTC (pcm/C°) versus burnup(MWD/kgHM) of the same cases for WH fuel assembly
type. At the BOL Erbia and BAG+Erbia cases have themost negative MTC values among all cases with thevalues of -42.28 pcm/C° and -37.18 pcm/C° respectively.
Around the burnup step of 50 MWD/kgHM, the values ofthese two cases become almost the same values until the
EOL.
Worth mentioning, for the power peaking factor and
MTC values any design satisfy the design limitations is
acceptable. The values of power peaking factor and MTC
no need to be very low.

Fig. 8. Comparison of power peaking factor in 2-D
assembly geometry for WH fuel assembly type.
Fig. 9 demonstrates the power peaking factor in 2-D
assembly geometry versus burnup (MWD/kgHM) of thesame cases for CE fuel assembly type. Gad case has thehighest power peaking factor among all cases throughout
the burnup. At the BOL the power peaking factor of BAG
case is little bit higher than the other cases. Around theburnup step of 13 MWD/kgHM, the values of all cases
except Gad case become very close to each other until the
EOL. Generally, the highest power peaking factor of Gad
cases in both fuel assembly types is due to the low content
of U235 with high concentration of gadolinium in Gad pins
[9].

Fig. 11. Comparison of MTC for CE fuel assembly type.
5. Conclusions
Matryoshka Doll Burnable Poison (MDBP) and Burnablepoison Attached to Guide tube (BAG) are new designconcepts that can be optimized to be suitable for any cycle
operation length for PWR core.
DeCART2D code was used in this study as a design
tool to perform assembly calculations for WH and CE fuelassembly types. The obtained results of these two new BP
concepts were compared with the results of conventional
BP designs regarding k-infinite, MTC and power peaking
factor.
24MDBP and BAG+Erbia cases reduced the initialexcess reactivity by about 43%. Also, they have the ability
to hold-down the excess reactivity for a longer time thanany other cases. Both cases provide a very good MTC
values due to the presence of erbium in both designs.
BAG+Erbia case provides good power peaking factor andlow residual reactivity penalty. On the other hand, theFig. 9. Comparison of power peaking factor in 2-D
assembly geometry for CE fuel assembly type.
Fig. 10 shows the MTC (pcm/C°) versus burnup(MWD/kgHM) of the same cases for WH fuel assembly
type. At the BOL, 24MDBP and 96Erbia cases havealmost the same MTC values which are the most negativevalues among all cases. At the burnup step of 20
MWD/kgHM until the EOL, the values of 24MDBP casebecome more negative than the values of 96Erbia case.

194Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
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power peaking factor and residual reactivity penalty
values of 24MDBP case are reasonable.
Although 24MDBP and BAG+Erbia cases in thisstudy are designed to be suitable for long operating cycle
and high burnup fuel, both designs have a degree of
freedom to be adjusted to satisfy any cycle length by using
appropriate absorber material.
This study is prelaminar study therefore moreinvestigation is needed.
References[1] J. Secker, J. Brown, Westinghouse PWR Burnable
Absorber Evolution and Usage, in: 2010 ANS WinterMeeting, Las Vegas, NV, USA, Nov. 7-11, (2010).
[2] Jeffery A. Brown, Ho Q. Lam, Hybrid IFBA Gad
Assembly Designs for Long PWR Cycles, in: 2017 Water
Reactor Fuel Performance Meeting, Ramada Plaza Jeju,
Jeju Island, Korea, Sep. 10-14, (2017).
[3] J.Y. Cho, J.C. Lee, K.H. Lee, S.Y. Park, H.J. Park,
H.Y. Kim, K.Y. Kim, H.J. Jung, J.S. Song, S.Q. Zee, C.K.
Jo, H.C. Lee, DeCART2D v1.0 User’s Manual,
KAERI/TR-5116/2013, Korea Atomic Energy ResearchInstitute (2013).
[4] Young Kim Jang, Kyeong Lak Jeon. (2010). Inreactor Irradiation Performances of Advanced Fuels,
ACE7™, for Westinghouse Type Nuclear Power Plants,
Transactions of the Korean Society of MechanicalEngineers Spring Meeting, 5, pp. 343-344, (2010).
[5] Kyu-Tae Kim, Young-Ki Jang, Jae-Ik Kim, InReactor Performance of an Advanced PWR Fuel, PLUS7,
for OPR1000s in Korea, Journal of Nuclear Science andTechnology, 45:8 (2008) 836-849.
[6] Aiman Dandi, MinJae Lee, Myung Hyun Kim, Soon
Ki Kim, Sang Rin Shon, Combination of Burnable Poison
Pins for 24 months Cycle PWR Reload Core, in: 2018
Transactions of the Korean Nuclear Society Spring
Meeting, Jeju, Korea, May 17-18, (2018).
[7] MinJae Lee, Aiman Dandi, Myung Hyun Kim, Soon
Ki Kim, Sang Rin Shon, The Combinational Use of
Burnable Poison Pins for 24 Months Cycle PWR, in: 2018
Transactions of the American Nuclear Society Summer
Meeting, Philadelphia, Pennsylvania, USA, June 17–21,(2018).
[8] J. Choe, H.C. Shin, D. Lee, New burnable absorber for
long-cycle low boron operation of PWRs, Ann. Nucl.
Energy 88, pp. 272-279, (2016).
[9] F. Franceschini, B. Petrovic, Fuel with advavced
burnable absorbers design for the IRIS reactor core:
combined erbia and IFBA, Ann. Nucl. Energy 36,
pp.1201-1207, (2009).
[10] H. Yu, M.-S. Yahya, Y. Kim, A Reduced-Boron
OPR1000 Core Based on the BigT Burnable Absorber,
Nuclear Engineering and Technology (2016).

195

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Research on Core Design of Pebble Bed Advanced High Temperature Reactor
Wang Lianjiea, Sun Weia, Xia Bangyanga, Zou Yangb, Yan Ruib
Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China,
Chengdu, China;b
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai, 201800, China *aAbstract
We find that evacuating molten salt and injecting poison into coolant both can beused as the auxiliary system of the second shutdown system in Pebble Bed
Advanced High Temperature Reactor (PB-AHTR). Compared with injecting
poison into coolant, the effect of evacuating molten salt on the core is smaller
and more beneficial to the engineering realization. Compared to one time
charging scheme, batch charging scheme can ensure that a small core excess
reactivity in whole life-time which is easier to be controlled, but more complex
to operation. In one time charging scheme, the second shutdown system gains
enough fast shutdown margin by increasing the number of second control rod,instead of reducing the stack height of core activity. In this study, A PB-AHTR
core with burn-up life-time of 100 equivalent full power days (EFPDs) is
proposed. The shutdown margins of both the first shutdown system and thesecond shutdown system meet the design requirements.
Key Words: Pebble Bed Advanced High Temperature Reactor, Second s
hutdown system, Evacuating molten salt, Core design
1. Introduction

2. Calculation CodeIn 2002, based on the consideration of sustainability,
economy and safety, reliability and nuclear nonproliferation, the International Symposium on Fourth
Generation Nuclear Reactors screened out six fourth
generation nuclear reactors from more than 100 types ofreactors [1]: Molten Salt Reactor (MSR), Gas-cooled Fast
Reactor (GFR), Lead-cooled Fast Reactor (LFR),
Sodium-cooled Fast Reactor (SFR), Super Critical
Water-cooled Reactor (SCWR), Very-High Temperature
Reactor (VHTR). The pebble bed advanced hightemperature reactor (PB-AHTR) is a combination ofmolten salt reactor and very-high temperature reactor. It
is an advanced high temperature reactor using molten
salt as coolant. It has five characteristics: molten saltcooling, coated granular fuel and passive cooling safetysystem, supercritical water energy circulation systemand conventional island design. Due to the inheritanceof many advantages and technical foundations of theexisting cores, the PB-AHTR fully meets therequirements of the fourth generation nuclear reactor
core. Moreover, the commercial application of the PBAHTR is also highly feasible under the conditions ofcurrent technical foundation.
Studies on the core design of the PB-AHTR havebeen carried out. The key parameters of core design are studied, and then the preferred design concept of
the PB-AHTR core is proposed.

This study uses the MCNP code [2] and the ORIGEN
code [3]. The MCNP code is a computer code developedby the Los Alamos National Laboratory for transport
calculation based on the Monte Carlo method. It canhandle the movement of neutrons, protons and photons
in various media and give corresponding results. It has
been widely used in various aspects such as reactor
physics and radiation protection.
The ORIGEN code is a code system developed by
the Oak Ridge National Laboratory for radionuclide
ignition, decay calculation, and radioactive material
processing. The ORIGEN code is primarily used for fuelconsumption calculations, as well as flux-to-power
conversion calculations. ORIGEN provides various
databases for different reactor types, different fuelcombinations and different fuel burn depths, which canbe selected by users according to actual needs [3].
A large number of open literatures have applied
MCNP code and the ORIGEN code to investigate thedesign of pebble bed reactors [4-6].
3. Research on key parameters of core design
3.1 Overall parameters of core
In this paper, the thermal power of the reactor isdesigned as 10 MWt, the coolant is 2LiF-BeF2 molten

196Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

salt, and the spherical fuel element is undertaken as sameas the high temperature gas cooled reactor used. It
contains TRISO coated particulate fuel and graphite base.
The diameter of the fuel sphere is 6.0 cm, the fuel is UO2
with 235U enrichment of 17.0 wt%, and the uranium
loading amount of per fuel sphere is 7.0 g.

cking Equivalence
3.3 Research on the design of the second shutdown
systemThe fuel ball of pebble bed advanced hightemperature reactor core is arranged in a crossarrangement, hence the control rod can only be arrangedin the outer reflection layer of the active area that greatly
reduces the value of control rod. Due to the limitation of
peripheral position, only 16 B4C control rods arearranged so far, and the total value of these 16 control
rods is also small. If 16 control rods are divided into twoparts, one for the first shutdown system and another forthe second shutdown system, while meeting the safe
margin, the initial residual reactivity of the core is small,that seriously affects the core life. In order to increase
the options and diversity of the scheme, it considers to
conduct an analysis of other possible alternative
auxiliary shutdown methods for the second shutdown
system for design reference, including: evacuation ofmolten salt and addition of soluble poisons.
(1) Evacuate Molten Salt
In the calculation and analysis, the core active zone
is loaded to the lower end plane of the cylinder. In orderto prevent the pebble bed from gradually sinking andbeing out of shape when the molten salt is emptied, thegraphite ball (diameter is 6 cm) is loaded into the fuelpebbles located at the bottom of active zone Of the 16
control rods, 2 (2 and 9 in Figure 1a) control rods serveas the second shutdown system control rods for fast
shutdown.
Table I shows the Keff values after inserting thesecond shutdown rods at different burn-up and normal
cold state equilibrium Xenon (the first set of control rods
is at the critical rod position). As can be seen from Table
I, the maximum Keff value is 0.99728 after the core isinserted into the second set of shutdown rods. It can beseen that when the first set of control rods is stuck in the
power operation critical rod position, the second set of
shutdown rods can bring the reactor into the cold
shutdown state. As the xenon is reduced, the Keff valueof the core at the cold state without xenon is greater than
1 (Table I). In order to avoid the reactor core returning to
criticality, it is necessary to start the molten salt systembefore the xenon poison disappears.

The pebble bed advanced high temperature reactor
core consists of a core active zone and a graphitereflective layer (Fig. 1). The upper part of the active area
is placed a fuel pebble, the lower part settles a graphite
pebble, the remaining space is filled with molten salt,
and the gap between the pebbles forms a random
irregular flow path for the molten salt coolant. The
coolant takes heat generated by fission away frombottom to top. As a neutron reflecting material, thegraphite reflective layer constructs an active area of theupper and lower circular table and the middle column,
and accommodates special channels such as neutron
source and shutdown system. The positions of 16 control
rods are shown in Fig. 1. The core active zone issurrounded by the core’s graphite member forming as
upper and lower circular tables and a middle cylindricalregion. The cylinder height is 180.0 cm.

a. Core Cross Section

b. Core Longitudinal Section

Fig.1 Schematic Diagram of PB-AHTR Core
3.2 Calculation Model
The fuel pebble TRISO particle geometry model uses
the same structure as the actual TRISO grain which mea
ns a fuel core with addition of four coating layers. The T
RISO particles are filled in the fuel sphere using a simpl
e cubic equivalent simulation of TRISO particles [4-5] (Fig. 2).
According to the literature research, the duty cyc
le of the fuel sphere in the core is about 60 % [68] that can be modeled by body-centered cubic repeating cell [6]. In this paper, a 60% duty cycle body
Table I Keff value of core after inserting second sets of-centered cubic repeating cell model is used, and th
shutdown rods
e graphite pebbles at the bottom of the core are arr
Keff Value
Keff Vaulue
anged in the same way.
Burn-up
EFPD(Cold State With
Equilibrium Xenon)(Cold State Without
Xenon)

0.0

0.99728(0.00041*)

0.99728(0.00041)

2.0

0.99325(0.00040)

1.01754 (0.00040)a) TRISO b) TRISO Rule Packing c) Profile of Fuel Sphere

100.0

0.98822(0.00039)

1.01369(0.00043)

Fig.2 Schematic Diagram of TRISO Particle and Pa

150.0

0.98729(0.00043)

1.01199(0.00041)

197Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Note: * is the statistical error of the Monte Carlo code

two methods: ①reducing the height of the core active
zone; ②while meeting the requirements of the first
shutdown system, distributing some control rods of thefirst set of shutdown systems to the second set of
shutdown systems.
(1) Reducing the height of the core active zone
Table IV shows the Keff value of the core under vari
ous operating conditions when the core loading height of
the active zone is 150 cm. It can be seen from Table IVthat even if the height of the core active zone is lowered,in case of the second set of shutdown systems only maintains two control rods, the Keff value will be 0.99212 andthe margin is small, too. The reason is mainly because of the second set of shutdown system control rods are inse
rted at the critical rod position of the first set of shutdow
n system control rods to achieve rapid shutdown, and tot
al value of these two rods is basically unchanged. Even if the height of active area is reduced, the shutdown marg
in of the second shutdown system is also small.

Table II Keff value of core after evacuating molten salt
(cold state without xenon)
Burn-up/EFPD

Keff Value

0.0

0.84777(0.00043)

2.0

0.88951(0.00045)

100.0

0.91448(0.00046)

150.0

0.92906(0.00042)

After the core has been shut down by the second setof shutdown rods, Table II gives the Keff value of thecore after the molten salt is emptied at the cold state
without xenon in the core at different burn-up states. It
can be seen from Table III that after the core is shut
Table IV Keff value of core under different working
down by the second set of shutdown rods, the molten
conditions when the core height is 150 cm and thesalt evacuation system can ensure that reactor
second shutdown system has two control rods
maintaining a cold shutdown state before the core
Conditions
Keff Value
returns to critical due to the xenon poison consumption.
Withdraw All Control Rods
1.05974（0.00039）(2)Add Soluble Poison
Two rods of the second set of
The injection of poison into coolant is also a
shutdown system（The position iscommon means of shutdown, so that a neutron poison
2, 9 in Figure 1）are both insered
0.99212（0.00040）
salt which is soluble and compatible with the molten salt
（Cold State With Equilibrium
can be injected as a shutdown means. The calculation
Xenon）
conditions are the same as venting molten salt.
All control rods of the first set ofTable III shows the Keff value of the core when shutdown system are all inserted
0.95400（0.00041）
injected into the Li-6 solution at the cold state without（Cold State）
xenon in different burn-up. It can be seen from Table III
that after the core is shut down by the second set of(2) Increase the number of control rods in the second set
shutdown rods, the injection of Li-6 solution can ensureof shutdown systems
that the reactor maintains a cold shutdown state before
The Keff values of the cores under various working
the core returns to critical due to xenon poison
conditions when the core loading height of the active
consumption.
zone is 150, 160 and 165 cm are calculated. When thethree rods of the second set of shutdown systems are all
Table III Keff value of core after injecting Li-6
inserted, the shutdown margin meets the requirements
Burn-up/EFPD
Condition 1①
Condition 2②(Keff < 0.99 when the second set of control rods isinserted).
00.96833（0.00042） 0.95085（0.00040）
4. Preferred core design20.98709（0.00041） 0.97038（0.00041）
100

0.98322（0.00041）

0.96507（0.00042）

150

0.98131（0.00041）

0.96363（0.00039）

Compared with the core injecting poisonous molten
salt, the method of evacuating the molten salt is morepractical, because after the core is injected with poison,
the core will be contaminated, and the entire reactor is
scrapped. In order to meet the requirements of the first
set and the second set of shutdown systems at the sametime, the final core loading height of the active zone is
160 cm, and the second set of shutdown system has 3rods(the position is 2, 7, 10 in Figure 1) all inserted.

Note: ①Li-6 accounts for 0.05% of the mass of thecoolant; ②Li-6 accounts for 0.08% of the mass of thecoolant.
3.3 Research on core fuel loadage design
When the core active zone is loaded to the lower end
plane of the cylinder, 2 rods of the second set is inserted
while the first set of control rods at the critical rod
position. It can be seen from Table 2 that the core’s Keffof the initial is 0.99728 whilst the quick shutdown
margin is small. In order to solve this problem, there are

4.1 numerical result
At the beginning of the life whilst the core is at coldstate and control rods are all withdrawn, Keff value is
1.07762, and will be 0.96603 while control rods are all198

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019studies on the set of second shutdown systems and corefuel loading of PB-AHTR. It has been found that venting
molten salt and injecting poison into the coolant can beused as an auxiliary system for the second shutdown
system of the core. However, the effect of evacuating
molten salt on the core is smaller than that of the core
injected with poisonous molten salt and more practical.
It is impossible to make the second shutdown systemhave sufficient rapid shutdown margin by reducing theheight of the core active zone. Unlikely, it is possible to
do so by increasing the number of control rods in thesecond shutdown system. Based on the above research,
the optimal core design scheme of 10 MW PB-AHTR is
proposed. The scheme has a burn-up life reaching 100
EFPD. The first shutdown system has a cold shutdown
depth of -3516 pcm whilst the second shutdown system
has a cold shutdown depth of -10223 pcm. This design
has a sufficient cold shutdown depth to meet design
requirements.

inserted; The cold shutdown depth is -3516 pcm (1pcm=10-5).
Table V shows the Keff values when the second setof shutdown rods are inserted under different burn-up
while the core is at cold state with equilibrium xenon(the first set of control rods are at the critical rod
position). It can be seen from Table V that the maximum
Keff value is 0.98438 after the second set of shutdown
system is inserted. It can be seen that when the control
rods of the first set of shutdown system are stuck at thepower operation critical rod position, the control rods ofthe second set of shutdown system can bring the reactor
into the cold shutdown state.
After inserting the control rods of the second set of
shutdown system to achieve shutdown, in order to avoid
the reactor core returning to criticality, it is necessary tostart the molten salt evacuation system before the xenon
poison disappears. Table VI shows the Keff value of thecore by evacuating molten salt at cold state without
xenon in different burn-up. It can be seen from Table VI
that after shutdown by inserting the control rods of thesecond shutdown system, the molten salt evacuation
system can ensure that the reactor maintains a cold
shutdown state if it is started before the reactor return tocriticality for xenon poison reducing. Among which, themaximum Keff value after evacuating molten salt indifferent burn-up is 0.90725 (corresponding to the cold
shutdown depth is -1023 pcm), which meets the design
requirements.

References
1. “US DOE Nuclear Energy Research Advisory
Committee and the Generation IV International
Forum,” A Technology Roadmap for Generation IV
Nuclear Energy Systems, GIF-002-00 (2002).
2. J.F. Briesmeister, “MCNP – A General Monte Carlo
N-Particle Transport Code,” Los Alamos National
Laboratory (1997).
3. A.G. Croff, “A User’s Manual for ORIGEN2
Computer Code,”USA：Oak Ridge National
Laboratory (1980).
4. M.Q. Huda, T. Obara, “Development and testing ofanalytical models for the pebble bed type HTRs,”
Annals of Nuclear Energy, 35 (2008).
5. Volkan Seker，Uner Colak, “HTR-10 full core firstcriticality analysis with MCNP,” Nuclear
Engineering and Design, 222(2003).
6. G.J.Auwerda, J.L.Kloosterman, D. Lathouwers,
T.H.J.J. van der Hagen. “Effects of random pebble
distribution on the multiplication factor in HTR
pebble bed reactors,” Annals of Nuclear Energy,
37(2010).
7. Yixiang Gan, Marc Kamlaha, Jorg Reimanna,“Computer simulation of packing structure in pebble
beds,” Fusion Engineering and Design, 85(2010).
8. Joerg Reimanna, Ali Abou-Senaa, Rainer Nippenb,
Paul Tafforeauc, “Pebble bed packing in prismatic
containers,” Fusion Engineering and Design,
88(2013).

Table V Keff value of core when the second control rods
inserting (cold state with equilibrium xenon)
Burn-up/EFPD

Keff Value

0.0

0.98438(0.00042)

2.0

0.97900(0.00040)

50.0

0.97628(0.00040)

100.0

0.97469(0.00040)

Table VI Keff value of Core after evacuating Molten Salt
(Cold State without Xenon)
Burn-up/EFPD

Keff Value

0.0

0.84305(0.00044)

2.0

0.88201(0.00045)

50.0

0.89643(0.00042)

100.0

0.90725(0.00045)
5. Conclusion

This paper investigates and conducts a series of199

TS-04

Reactor Kinetics

No.

RPHA19-1065

Title

C5G7-TD Benchmark Analysis Using Multigrid Amplitude
Function Method

Authors

Kosuke Tsujita (Nuclear Engineering, Ltd., Japan), Tomohiro Endo, Akio
Yamamoto (Nagoya University, Japan)
201

No.

RPHA19-1083

Title

FAST (Floating Absorber for Safety at Transient) for the
Improved Safety of MOX-loaded Sodium-cooled Fast Reactors

Authors

Chihyung Kim, Yonghee Kim (Korea Advanced Institute of Science andTechnology, Korea)
205

No.

RPHA19-1102

Title

A Comparative Study on the IQS (Improved Quasi-Static) and
PCQS (Predictor Corrector Qusai-Static) Methods

Authors

Taesuk Oh, Yunseok Chung, Yonghee Kim (Korea Advanced Institute of
Science and Technology, Korea)
209

No.

RPHA19-1106

Title

Point Kinetics Analysis for Source-jerk Experiment at AGN201K

Authors

Seulki Lim, Hyung Jin Shim (Seoul National University, Korea), MyungHyun Kim (Kyung Hee University, Korea)
213

200

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

C5G7-TD Benchmark Analysis using Multigrid Amplitude Function Methoda

Kosuke TSUJITAa*, Tomohiro ENDOb, and Akio YAMAMOTOb
Nuclear Engineering, Ltd., 1-3-7 Tosabori, Nishi-ku, Osaka, Japanb
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan*
Corresponding author: ktsujita@neltd.co.jp

Abstract
C5G7-TD 2D benchmark analysis is performed using the linear source
approximation and the multigrid amplitude function (MAF) method. Through thesensitivity analysis for neutron flight direction, spatial mesh division, and time
step size, spatially and temporally fine kinetic calculations are carried out for theTD1 perturbation condition of the C5G7-TD benchmark problem. The initialeigenvalue and the core power transitions are compared with that of MCNP and
MPACT codes respectively and they are in good agreement.
Key Words: method of characteristics, multigrid amplitude function method,linear source approximation, C5G7-TD benchmark problem
1. Introduction
𝑄𝑔 =Time-dependent transport calculations without
spatial homogenization are one of the most challenging
topics since the computational cost of the method ofcharacteristics (MOC) [1] in large heterogeneousgeometry is still expensive. Thus, various efficient
numerical methods for solving the time-dependenttransport equation is developed to improve thecomputational efficiency of the time-dependent transport
calculations. Especially, the multigrid amplitude function(MAF) method [2] archives high computational
efficiency with the minimum degradation of theaccuracy by using the coarse mesh and fine time step
diffusion calculation. However, since the flat source
assumption is employed in the previous study [3], a large
number of flux regions are required in strongly
heterogeneous geometry to obtain fine calculation results,
e.g., the C5G7-TD benchmark problem [4].
In this study, C5G7-TD 2D benchmark analysis using
the MAF method with the linear source approximation[5] is carried out to provide spatially and temporally finesolution. In the present summary, the theoretical basis of
the MAF method with the linear source approximation is
described in section 2. The calculation results of the TD1
perturbation condition are described in section 3. Finallyconcluding remarks are summarized in section 4.

(2)
(3)

where,
Ω : the unit vector in the direction of flying neutron,𝜓 : angular flux,
𝜙 : scalar flux,𝜒 : fission spectrum,Σ : cross section,
𝐶 : precursor density,
𝜆 : decay constant,𝛽 : delayed neutron fraction, v : neutron velocity,g, j, p, and d are the subscript for neutron energy group,
delayed precursor group, prompt and delayed neutrons,
respectively. An approximation for the temporal
derivative of the angular flux is necessary to perform thekinetic calculation using MOC within practical memory
requirement. Since the previous study revealed that theisotropic approximation for the temporal derivative of
the angular flux has a negligible impact for the accuracy[6], thus Eq.(1) is approximated as:
1 1 𝜕𝜙𝑔= −Ω ⋅ ∇𝜓𝑔 − Σ𝑡,𝑔 𝜓𝑔 + 𝑄𝑔 ,4𝜋 v𝑔 𝜕𝑡(4)

𝜕𝜓𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡)
1 1 𝜕𝜙𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡)=−
4𝜋 v𝑔,𝑟𝜕𝑡𝜕𝑠−Σ𝑡,𝑔,𝑟 𝜓𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡) + 𝑄𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡) ,(5)

Discretizing the neutron flight direction and space for Eq.
(4), the balance equation along a neutron path line is
described as:

2. Multigrid Amplitude Function Method withLinear Source Approximation

where,: coordinate along the neutron flight direction.
𝑠𝑚 : neutron flight direction,𝑘 : sequential number of the path line,: segment,𝑟 : flux region,𝑖and Fig. 1 shows their definitions.

In this section, the theoretical basis of the MAF methodwith the linear source approximation is described. The
time-dependent transport equations are written as:
1 𝜕𝜓𝑔= −Ω ⋅ ∇𝜓𝑔 − Σ𝑡,𝑔 𝜓𝑔 + 𝑄𝑔 ,v𝑔 𝜕𝑡

𝜒𝑝,𝑔 (1 − 𝛽) ∑ 𝜈Σ𝑓,𝑔´ 𝜙𝑔´1𝑔´,
4𝜋 + ∑ Σ𝑠,𝑔´→𝑔 𝜙𝑔´ + ∑ 𝜒𝑑,𝑗,𝑔 𝜆𝑗 𝐶𝑗
{
}
𝑔´
𝑗
𝜕𝐶𝑗
= 𝛽𝑗 ∑ 𝜈Σ𝑓,𝑔´ 𝜙𝑔´ −𝜆𝑗 𝐶𝑗 ,
𝜕𝑡
𝑔´

(1)

201

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019k-th
path line

1 1 𝜑𝑔,𝑟 (𝑡𝑛+1 ) − 𝜑𝑔,𝑟 (𝑡𝑛 )4𝜋 v𝑔,𝑟Δ𝑡𝑠ℎ𝑎𝑝𝑒
𝜙𝑔,𝑟 (𝑡𝑛+1 ) 𝜔𝑔,𝐼 (𝑡𝑛+1 )𝑅𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡𝑛+1 ) −
4𝜋
v𝑔,𝑟
=
,
𝑃𝑔,𝐼 (𝑡𝑛+1 )r-thflux regionsi-th segment

where 𝑛 and Δ𝑡𝑠ℎ𝑎𝑝𝑒 denote time step and time step
size for the shape function, respectively. Since the shapefunction weakly depends on time, a coarse time step canbe used for the temporal discretization. Multiplying the𝑃𝑔,𝐼 (𝑡𝑛+1 ) for both sides, the balance equation for the
angular flux and thus the scalar flux is obtained as:

m-th direction

Fig. 1. Flight direction, path line, segment, flux region
In the present study, the linear source approximation [5]is employed to reduce the spatial discretization error, i.e.,the scalar flux is described as:
𝑥 (𝑡)𝜙𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡) = 𝜙𝑔,𝑟 (𝑡) + 𝑥𝑚,𝑘,𝑖 (𝑠)𝜙𝑔,𝑟(6)𝑦+ 𝑦𝑚,𝑘,𝑖 (𝑠)𝜙𝑔,𝑟 (𝑡),where, 𝑥𝑚,𝑘,𝑖 (𝑠) and 𝑦𝑚,𝑘,𝑖 (𝑠) are the local coordinates𝜕𝜓𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡𝑛+1 )+ Σ𝑡,𝑔,𝑟 𝜓𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡𝑛+1 )𝜕𝑠
1 𝜙𝑔,𝑟 (𝑡𝑛 ) 𝑃𝑔,𝐼 (𝑡𝑛+1 )= 𝑄𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡𝑛+1 ) +4𝜋 v𝑔,𝑟 Δ𝑡𝑠ℎ𝑎𝑝𝑒 𝑃𝑔,𝐼 (𝑡𝑛 )𝜙𝑔,𝑟 (𝑡𝑛+1 )1−(𝜔𝑔,𝐼 (𝑡𝑛+1 ) +).
4𝜋v𝑔,𝑟Δ𝑡𝑠ℎ𝑎𝑝𝑒whose origin is the geometrical centroid of the flux
region [5], and thus the neutron source is described as alinear function of the track distance. The 1st order𝑦𝑥, represent
coefficients of the scalar flux, 𝜙𝑔,𝑟and 𝜙𝑔,𝑟
the local shape of the scalar flux distribution inside theflux region and their temporal derivative are considered
to be small. Therefore, the 0th order coefficient of thescalar flux is factorized as a product of the shape andamplitude functions:
𝜙𝑔,𝑟 (𝑡) ≡ 𝜑𝑔,𝑟 (𝑡)𝑃𝑔,𝐼 (𝑡) ,(7)where 𝜑𝑔,𝑟 (𝑡) denotes the shape function that represents
the strong and weak dependences for space and time,
respectively. On the other hand, the amplitude function
𝑃𝑔,𝐼 (𝑡), which is defined as Eq. (8), represents the weak
and strong dependences for time and space, respectively.
𝑃𝑔,𝐼 (𝑡) ≡ ∑𝑟∈𝐼𝜙𝑔,𝑟 (𝑡)𝑉,v𝑔,𝑟 𝑟𝐶𝑀(𝑡)
1 𝑑𝜙𝑔,𝐼𝐶𝑀 (𝑡)𝜙 𝐶𝑀= −𝐿𝑔,𝐼 (𝑡) − Σ𝑟,𝑔,𝐼
𝑔,𝐼 (𝑡)
𝐶𝑀
𝑑𝑡
v𝑔,𝐼
𝐶𝑀
+ 𝑄𝑔,𝐼
(𝑡),
𝐶𝑀
𝐶𝑀
(𝑡)
𝐽𝑔,𝐼+1/2 (𝑡) − 𝐽𝑔,𝐼−1/2
𝐿𝑔,𝐼 (𝑡) = ∑
,
Δℎ𝐼

(8)

(16)

where 𝐽𝑔𝐶𝑀 and Δℎ𝐼 represent the neutron current at thecoarse mesh surface and the width of the coarse mesh inx- or y-direction, respectively. To obtain the consistentneutron current between the TCMFD and MOC
calculations, the correction term for the neutron current
is employed in the TCMFD calculation [3]. Employing
the theta method, Eq. (15) is written as
𝐶𝑀 (𝑡𝐶𝑀𝐿𝑔,𝐼 (𝑡) + Σ̂𝑟,𝑔,𝐼
𝑁+1 )𝜙𝑔,𝐼 (𝑡𝑁+1 )𝐶𝑀 (𝑡 )
𝜙𝑔,𝐼 𝑁1 − 𝜃 𝐶𝑀(17)
𝐶𝑀 (𝑡= 𝑄𝑔,𝐼+
𝑅𝑔 (𝑡𝑁 ),𝑁+1 ) +
𝐶𝑀
𝜃
𝜃v𝑔,𝐼
Δ𝑡𝑎𝑚𝑝
1
𝐶𝑀 (𝑡
𝐶𝑀
Σ̂𝑟,𝑔,𝐼,(18)
𝑁+1 ) = Σ𝑟,𝑔,𝐼 (𝑡𝑁+1 ) +
𝐶𝑀
𝜃v𝑔,𝐼
Δ𝑡𝑎𝑚𝑝
𝐶𝑀 (𝑡)𝜙 𝐶𝑀 (𝑡)𝐶𝑀 (𝑡),
𝑅𝑔𝐶𝑀 (𝑡) = −𝐿𝑔,𝐼 (𝑡) − Σ𝑟,𝑔,𝐼+ 𝑄𝑔,𝐼(19)
𝑔,𝐼
where 𝜃, 𝑁, and Δ𝑡𝑎𝑚𝑝 denote the weighting parameter

dynamic frequency defined as:(10)

Substituting Eq. (9) into Eq. (5) and solving it for the
temporal derivative of the shape function,
1 1 𝑑𝜑𝑔,𝑟 (𝑡)4𝜋 v𝑔,𝑟𝑑𝑡
1
1 𝜔𝑔,𝐼 (𝑡)(𝑠, 𝑡) −=(𝑅
𝜙𝑔,𝑟 (𝑡)) ,𝑃𝑔,𝐼 (𝑡) 𝑔,𝑚,𝑘,𝑖
4𝜋 v𝑔,𝑟𝜕𝜓𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡)
𝑅𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡) = −− Σ𝑡,𝑔,𝑟 𝜓𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡)𝜕𝑠+ 𝑄𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡) .

(15)

𝑥,𝑦

𝜕𝜙𝑔,𝑚,𝑘,𝑖 (𝑠, 𝑡) 𝑑𝜙𝑔,𝑟 (𝑡)
≈
𝜕𝑡
𝑑𝑡
(9)
𝑑𝜑𝑔,𝑟 (𝑡)=𝑃𝑔,𝐼 (𝑡) + 𝜙𝑔,𝑟 (𝑡)𝜔𝑔,𝐼 (𝑡) ,𝑑𝑡
where 𝜔𝑔,𝐼 (𝑡) is the space- and energy-dependent
1 𝑑𝑃𝑔,𝐼 (𝑡).
𝑃𝑔,𝐼 (𝑡) 𝑑𝑡(14)

In the MAF method, the fine temporal dependence iscaptured in Eq. (14) by the amplitude function and the
dynamic frequency. In order to calculate them, the timedependent coarse mesh finite difference (TCMFD)
method [2] is employed in the present study.
In the TCMFD method, both sides of Eq. (4) areintegrated for total solid angle and spatially averaged
within each coarse mesh 𝐼. The balance equation is
written as:

where 𝐼 and 𝑉𝑟 represent the coarse mesh which
includes the flux region r and the volume of the flux
region, respectively. Using the Eq. (7) and assuming the
temporal derivative of 1st order coefficients of the scalar
flux as zero, the temporal derivative of the scalar flux isapproximated as:

𝜔𝑔,𝐼 (𝑡) ≡(13)

in the theta method, the time step, and the time step sizefor the TCMFD calculation, respectively.
Note that Eq. (17) should be solved with finer time
step size than that of shape function from t = tn to t = tn+1
because the amplitude function and the dynamic
frequency are calculated by the scalar flux in the coarse
mesh as follows:

(11)

(12)

𝑃𝑔,𝐼 (𝑡) = ∑Employing the fully implicit method for the temporal
derivative of the shape function,𝑟∈𝐼

𝜔𝑔,𝐼 (𝑡) =

202

𝐶𝑀
𝜙𝑔,𝐼
(𝑡)
𝜙𝑔,𝑟 (𝑡)𝑉𝑟 =𝐶𝑀 𝑉𝐼 ,
v𝑔,𝑟
v𝑔,𝐼

𝐶𝑀
(𝑡)
1 𝑑𝑃𝑔,𝐼 (𝑡)1 𝑑𝜙𝑔,𝐼= 𝐶𝑀,
𝑃𝑔,𝐼 (𝑡) 𝑑𝑡
𝑑𝑡
𝜙𝑔,𝐼 (𝑡)

(20)
(21)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The TCMFD calculation and the transport calculation
using MOC are iterated until the neutron flux at t = tn+1
in fine geometry sufficiently converges.

Table II shows the initial eigenvalue, maximum andminimum pin power for each test case. In Table II, coreaveraged pin power is normalized as 1.0.

3. C5G7-TD Benchmark Analysis

Table II. Sensitivity for azimuthal angle division, ray
separation, and flux region division

The C5G7-TD benchmark problem [4] consists of
two UO2 and MOX PWR heterogeneous fuel assemblies
surrounded by water reflectors. The cross sections forseven energy groups and the delayed neutron parametersfor 8 precursor groups specified in the benchmark
problem are used in the present calculations.

Case
Nazim
Δpath
FR

3.1 Sensitivity analysis for the initial condition
To confirm sufficient divisions for neutron flight
directions and spatial meshes, a sensitivity analysis forthe azimuthal angle division, ray separation, and flux
region division is carried out. Table I shows the casematrix of the present sensitivity analysis. Figure 2 shows
the flux region structure used in the present calculations.

Ref.
128
3
0.01
Fig. 2(a)

*N
azim

:

Npol

:

Δpath
FR

:
:

Case 1128
3
0.01
Fig. 2(b)Case 2128
3
0.01
Fig. 2(c)Case 364
30.01
Fig. 2(b)1.26 cm

1.26 cm

10x10 meshes

Case 4128
3
0.02
Fig. 2(b)1.26 cm

2.500
0.232
-

2.500
0.232
0.007%

2.500
0.232
0.030%

2.501
0.232
0.010%

2.499
0.232
0.050%:::(+2pcm)(+7pcm)(-9pcm)(+17pcm)

Maximum pin power (-),
Minimum pin power (-),
Root mean square difference with respect to theresference solution (Case : Ref.).

Ref.
128
3
0.01
Fig. 2(b)1000 msec
1 msecCase 1128
3
0.01
Fig. 2(b)1000 msec
10 msec

Case 2128
3
0.01
Fig. 2(b)1000 msec
100 msec

The reference core power transition, which is shown in
Fig. 3, is calculated with Δtamp = 1 msec.

cylindrical meshesof 0.6 cm+
8 azimuthal division

1.26 cm

1.26 cm

Pmax
Pmin
RMS

Case
Nazim
Npol
Δpath
FR
ΔtshapeΔtampcylindrical meshesof 0.1~0.5cm+
8 azimuthal division

1.2

Core power (-)

3x3 meshes
1.26 cm

Case 4
128
0.02
Fig. 2(b)1.18669

Table III. Calculation conditions for the time step

Reflector cell
Other cells(b) Material boundary + thin cylinder mesh formoderator + octant division, 3x3 meshes for reflector

Reflector cell

Case 364
0.01
Fig. 2(b)1.18638

3.2 Sensitivity analysis for the time step size
A sensitivity analysis for the time step size is carriedout for the TD1 perturbation condition of the C5G7-TD
benchmark problem [4] using the MAF method. Table
III shows the case matrix of the present sensitivity
analysis. The weighting parameter θ in the theta method
was 1.0 for all calculations (the fully implicit method).

Reflector cell
Other cells(a) Fine mesh structure for reference calculation

3x3 meshes

Case 2
128
0.01
Fig. 2(c)1.18657

As shown in Table II, small discretization error remains
in the case of the azimuthal angle division = 64 and theray separation = 0.02 cm. The flux region division is
sufficiently fine in the case of Fig. 2(b). The present
result is also in good agreement with the MCNP5’s
initial keff solution, which is provided as 1.18646±0.07%.

azimuthal angle division for 2π using cyclic
quadrature set,polar angle division for π/2 using TabuchiYamamoto optimum quadrature set,ray separation (cm),flux region division.

20x20 meshes

Case 1
128
0.01
Fig. 2(b)1.18651

Pmax
Pmin
RMS

Table I. Calculation condition for the neutron flight
directions and spatial meshes
Case
Nazim
Npol
Δpath
FR

keff

Ref.
128
0.01
Fig. 2(a)1.18649

4 azimuthal division

1.0

TD 1-1
TD 1-2
TD 1-3
TD 1-4
TD 1-5

0.8
0.6
0.4

Other cells

0(c) Material boundary + quarter division,
3x3 meshes for reflectors
Fig. 2. Flux region divisions

1

2

3

4 5 6
Time (sec)

7

8

9

10

Fig. 3. Reference core power transition for TD1
perturbation203

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019agreement, where the relative difference is calculated as:(𝑃𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑠𝑡𝑢𝑑𝑦 − 𝑃𝑀𝑃𝐴𝐶𝑇 )/𝑃𝑀𝑃𝐴𝐶𝑇 × 100 (%) .
(23)

The accuracy of the core power transitions with respect
to the reference solution is evaluated for Δtamp = 10 msec
and Δtamp = 100 msec, respectively. Figure 4 shows therelative difference of the core power, which is calculated
as:

Relative difference (%)(𝑃 − 𝑃𝑟𝑒𝑓 )/𝑃𝑟𝑒𝑓 × 100 (%) ,(𝑃: 𝑐𝑜𝑟𝑒 𝑝𝑜𝑤𝑒𝑟, 𝑟𝑒𝑓: 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛).

1.0

(22)

Relative difference (%)

0.10
0.05
0.00

0.5

TD 1-3

0.0
-0.5

0
-0.05

TD 1-1
TD 1-4

-0.10
1

2

3

TD 1-2
TD 1-54 5 6Time (sec)

TD 1-3
7

8

9 10

0.00

-0.10
0

1

2

3

TD 1-2
TD 1-54 5 6Time (sec)

TD 1-378

9 10(b) Δtamp = 100 msec
Fig. 4. Relative difference of the core power comparing
to the reference solution

TD 1-3

TD 1-4

TD 1-51.00000 1.00000 1.00000 1.00000
0.73048 0.95381 0.97044 0.68931
0.55530 0.90639 0.93938 0.50412
0.66388 0.93674 0.95945 0.61909
0.88531 0.97872 0.98654 0.87035
0.90237 0.98180 0.98846 0.88975
0.91104 0.98339 0.98946 0.89955
0.92115 0.98525 0.99062 0.91098
0.93205 0.98723 0.99186 0.92329
0.93868 0.98844 0.99261 0.93078
0.94336 0.98928 0.99312 0.93607* calculation results of Δtamp = 1 msec

1.00000
0.666290.477130.59455
0.862020.88275
0.893180.90534
0.918430.926390.93202

7

8

9

10

1. Halsall MJ, CACTUS, A Characteristics Solution to
the Neutron Transport Equation in ComplicatedGeometries, UKAEA, Winfrith (1980).
2. Ban Y, Endo T, Yamamoto A, “A Unified Approachfor Numerical Calculation of Space-dependent Kinetic
Equation,” J. Nucl. Sci. Technol. 49, 496 (2012).
3. Tsujita K, Endo T, Yamamoto A, “Application of theMultigrid Amplitude Function Method for Timedependent Transport Equation using MOC,” Proc.
M&C 2013, Sun Valley, Idaho, May 5–9, 2013,
American Nuclear Society (2013) (CD-ROM).
4. Boyarinov VF, Fomichenko PA, Hou JJ, et al.,“Deterministic Time-dependent Neutron Transport
Benchmark without Spatial Homogenization (C5G7TD),” OECD Nuclear Energy Agency (2017).
5. Ferrer R, Rhodes J, “Linear Source Approximation in
CASMO5,” Proc. PHYSOR 2012, Knoxville,
Tennessee, Apr. 15–20, 2012, American Nuclear
Society (2012) (CD-ROM).
6. Tsujita K, Endo T, Yamamoto A, et al., “Higher Order
Treatment on Temporal Derivative of Angular Flux forTime-dependent MOC,” Proc. M&C 2013, Sun Valley,
Idaho, USA, May 5–9 (2013) (CD-ROM).
7. Shen Q, Wang Y, Jabaay D, et al, “Transient analysis
of C5G7-TD benchmark with MPACT,” Ann. Nucl.
Energy, 125, 107 (2019).

Table IV. Summary of the core power transition*
TD 1-24 5 6Time (sec)

References

As shown in Fig. 4, the temporal discretization error issufficiently reduced at Δtamp = 10 msec for all test
problems. Table IV shows a summary of the core powertransitions in the present study.

TD 1-13

Analysis of the C5G7-TD 2D benchmark is carriedout by MOC using the linear source approximation and
the MAF method. In the present summary, sensitivity
analyses for neutron flight directions, spatial meshstructures, and the time step size are carried out andspatially and temporally fine transitions are obtained.
The initial eigenvalue and the core power transitions forTD1 perturbation conditions were compared with thoseof the MCNP and MPACT code, respectively. They arein good agreement. The more detailed pin-wisecomparison will be carried out in the future study.

0.05

TD 1-1
TD 1-42

5. Conclusions

0.10

-0.05

1

Fig. 5. Comparison of the core power transition with the
MPACT code(a) Δtamp = 10 msec

Time
(sec)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4.0
6.0
8.0
10.0

TD 1-2
TD 1-5-1.0

0

Relative difference (%)

TD 1-1
TD 1-4

3.3 Verification
To verify the present calculation results, acomparison of the core power transition with the othercalculation code is carried out. Figure 5 shows therelative difference of the core power between the present
study and the MPACT code [7] and they are in good204

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

FAST (Floating Absorber for Safety at Transient) for the Improved Safety of MOX-loaded
Sodium-cooled Fast Reactors
Chihyung Kim and Yonghee Kim*
Dept. of Nuclear & Quantum Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea*
Corresponding author: yongheekim@kaist.ac.kr

Abstract
This paper presents passive safety device called FAST (Floating Absorber for
Safety at Transient) for the improved inherent safety of sodium-cooled fast
reactors (SFR). FAST is designed to cause negative reactivity feedback against
coolant temperature rise compensating the positive reactivity induced by positive
coolant temperature reactivity feedback. This study investigates the transient
response of FAST in MOX-loaded SFR considering three ATWS (Anticipated
Transient Without Scram) scenarios, which are ULOF (Unprotected Loss of Flow),
ULOHS (Unprotected Loss of Heat Sink) and UTOP (Unprotected Transient
Overpower). The ATWS simulations are performed with an in-house computercode taking into account all possible reactivity feedback components including the
FAST movement and core expansion coefficients, and EOL (End-Of-Life)
condition of the core with most positive CTC (Coolant Temperature Coefficient)
is considered for the conservative analysis. Temperatures of core components andreactor power are considered as main factors of interest and it is confirmed thatconsequence of ULOF, ULOHS and UTOP in a MOX-loaded SFR core can benoticeably mitigated by the application of FAST device.
Key Words: SFR, passive safety device, FAST, coolant temperature coeff
icient, ATWS
1. Introduction
Floating absorber for safety at transient (FAST) is apassive safety device for sodium-cooled fast reactors,
which inserts negative reactivity in case of coolant
temperature rise or coolant voiding [1]. Previous study
showed that FAST in metallic fuel-loaded breed-and-burn
sodium-cooled fast reactor (B&BR) core effectively
works to compensate the coolant temperature feedback
induced positive reactivity by inserting the negative
reactivity in case of coolant temperature rise [2]. However,feasibility of FAST in metallic fuel-loaded B&BR withmoving active core cannot represent the feasibility of
FAST in oxide fuel-loaded SFR or burner-type SFR. In
this regard, this study investigates the feasibility of FAST
in MOX-loaded burner-type SFR core. The response of
FAST at transient in advanced burner reactor (ABR)
developed by ANL (Argonne National Laboratory) [3] isanalyzed. In-house thermal hydraulics coupled point
kinetics based transient code are used for the analysis.

Fig. 1. Concept of FAST
The inside of the FAST is filled with coolant and theneutron absorber module is located in the coolant. The
axial position of the absorber module is determined by thebalance of buoyancy and gravity. There are several holes
at the top and bottom of the FAST pin to allow the inflow
and outflow of the coolant. The absorber module consistsof the absorber and void canister to adjust the magnitudeof buoyancy force. The absorber part is B4C enclosed in a

2. Description of the FAST
FAST apparently has the same geometry as the fuel rod,but it does not contain the fuel inside as shown in Fig 1.

205

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

SiC/SiC composite cladding and void part is filled with
noble air. SiC/SiC composite is helium permeable so thatthe helium produced by (n,α) reaction of B-10 can bevented out through the cladding to release the internalpressure [4]. It is noteworthy that void part and absorberpart are not attached to each other to increase the freedomof drop path in case of fuel pin or assembly bowing. One
can also consider the separation of absorber part ofabsorber module in several pieces.
3. Reference Core: Advanced Burner Reactor
Fig. 2. Axial power profile of reference core1000 MWth advanced burner reactor (ABR) coredeveloped by ANL (Argonne National Laboratory) [3] ischosen as a reference core for the analysis of FAST
behavior during the transient. Reactivity feedback
coefficients of the reference core listed in Table I are used
for the reactivity feedback during the transient.

4.2 FAST movement model
Assuming that the fluid is incompressible, irrotationaland fully developed, velocity field of coolant surrounding
the absorber module (Vcoolant) in FAST pin can be derived
by Navier-Stokes equation in cylindrical coordinate [6].
d ( Vcoolant (r ))(r )  dp
d   dV r   r  coolant  (4)Table I. Reactivity feedback coefficients of ABR
Reactivity feedback coefficients
Value
Fuel temperature (pcm/K)-0.372
Coolant temperature (pcm/K)
0.496
Axial expansion (pcm/K)-0.930
Radial expansion (pcm/K)-0.155dt

dz

dr  

dr

B.C.: Vcoolant ( rFAST )  VFAST , Vcoolant (rpin )  0

The equations for forces acting on FAST are tabulated
in Table II.

TABLE II. Mathematical formulation of forces acting on FAST

4. Methodologies for FAST Behavior Analysis

Force type
Gravity

4.1 Heat transfer model

Buoyancy

EquationFg  FASTVolumeFAST g
Fb   coolant  z  gdV
V

Assuming radial temperature distribution of coolant isflat, heat transfer by the coolant flow can be modeled by
1-D time-dependent energy and mass conservation in theaxial direction:
TT c pcoolant coolant   c pcoolant vcoolant coolant  q '''
t
z
(1)
 coolant
 coolant
vcoolant vcoolant  coolant
0
t
z
z
(2)
In Eqs. (1) and (2), ρcoolant and Tcoolant are density andtemperature of the coolant, respectively, and vcoolant isvelocity of the coolant flow. cp,coolant means the specificheat of coolant and it is assumed to be a constant since it
is almost constant between 600 K and 900 K [5]. The heatsource term for each axial node in the fuel region is
calculated by considering the axial power distribution. A
chopped cosine shape is assumed for the axial power
distribution as shown in Fig. 2. It is assumed that thepower distribution does not change during the transient
since power distribution hardly changes in fast reactors.
Heat transfer in the fuel pin and FAST pin is treated asa radial 1-D heat conduction in cylindrical coordinate
since axial heat conduction in the fuel rod is negligible
compared to the radial heat conduction:
T 1   T 
(3)
cp

 kr
  q '''
t r r  dr 
Eq. (3) is solved by typical finite difference method
using coolant temperature as a boundary condition.

Drag
PressureFD   

dVcoolant
dAFAST _ sidedrFp  p  AFAST _ front

4.3 Point kinetics model
The standard point kinetics model is considered tosimulate the change of reactivity in accordance with thetemperature change of core components and insertion of
the FAST. The point kinetics equation is solved by thesimple finite difference method and the reactivity change
is calculated by the equation (5).
 (t )   0   f T f   c Tc   ex    FAST
(5)
4.4 Reference FAST design
Detailed design parameters of reference FAST used forthe transient analysis are listed in Table III.
Table III. Reference FAST design parameters
Design parameters
Value
Absorber height, cm
60
Void height, cm
20
Absorber module average density, g/cc
0.832
Absorber module radius, cm
0.20
FAST pin radius (inner/outer), cm
0.3215 / 0.3775
It should be noted this feasibility study considers
arbitrarily determined design parameters of reference206

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

FAST to be physically feasible. Maximum reactivity
worth of FAST can be controlled by adjusting theenrichment of B4C in absorber module and maximum
reactivity of 1$ is considered for this study. Analyses inRef. 2 showed that FAST reactivity worth of 1$ can beeasily achieved by installing 3~4 FAST pins in each
assembly. Position-wise reactivity worth of FAST
absorber module is shown in Fig. 3 and it is determinedassuming typical S-shape curve similar to that of thereactivity worth of control rods. At nominal state, void
part of absorber module is fully sank in active core region,
while absorber part is floating above the active core asshown in Fig. 3 to avoid the positive reactivity insertion
by void insertion at transient and to minimize thereactivity perturbation by FAST at nominal state.

is slightly lower than the melting point of typical UO2.
The coolant boiling point of 1,150 K at 1 bar is considered
to be a failure limit for the coolant in this work. However,
the transient simulations are not interrupted even whentemperatures of the core components reach the failure
limit due to difficulties and uncertainties in modelling thefuel melting and coolant boiling.
5.1 Unprotected loss of flow (ULOF)
Figure 4 compares the transition of core power andmaximum temperature of fuel and coolant with andwithout FAST. The power and the temperatures of fueland coolant decreased sharply in the presence of FAST.
One can note in Fig. 4 that FAST can prevent the coolant
boiling that happens in case without FAST and it is clearlyshown that FAST effectively mitigates the temperaturerise of core components during the ULOF.

Fig. 3. Position-wise reactivity worth of FAST
5. Transient Response with the FAST Device

Fig. 4. Time evolution of the core power in ULOF

Three representative anticipated transient without
scram (ATWS) scenarios are considered. All the transient
events are triggered in an initially steady state.
In the ULOF scenario, failure of all the coolant pumps
in the primary system is assumed. During the ULOF
transient, coolant mass flow rate decreases with pumphalving time of 5 seconds and 5% natural circulation isconsidered. Inlet coolant temperature of the core isassumed to remain unchanged during the ULOF.
In the ULOHS scenario, a complete loss of heatremoval capacity in intermediate heat exchangers (IHXs)
is considered and it is assumed that IHX heat removal
capacity decreases from 100% to 0.0% over 20 seconds.
It should be noted that the flow rate of the coolant in the
primary circuit does not change during the ULOHS.
The UTOP scenario in this study assumes 50 secondsof external reactivity insertion with a ramp rate of 0.02$/sec, while keeping the nominal coolant flow rate. Twoheat removal scenarios in the IHX are considered for thesimulation of UTOP: constant temperature drop in IHX
and constant core inlet coolant temperature. The constanttemperature drop in IHX scenario assumes that theamount of heat removal in IHX is always the same as thenominal full power. In the case of the constant inlet
temperature scenario, core inlet coolant temperature isalways kept the same as in the nominal condition,
regardless of core outlet temperature.
Failure limit of the fuel is assumed to be 3000 K, which

5.2 Unprotected loss of heat sink (ULOHS)
Figure 5 compares the transient progress of with andwithout FAST in ULOHS scenario.

Fig. 5. Time evolution of temperature in ULOHS
In the presence of FAST, power decrease at ULOHS ismore rapid and therefore, the temperature rise of the corecomponents is much less than that without FAST. The
reference core with FAST quickly shutdown and thetemperatures of core components quickly converge to theequilibrium during the ULOHS as shown in Fig 5.

207

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

5.3 Unprotected transient overpower (UTOP)

noticeable oscillation in case of external reactivity
insertion during the UTOP case, in spite of very
successful mitigation of power increase at the early stageof UTOP.
For more practical performance evaluation of FAST, it
is necessary to analyze the ATWS transients by using
accurate system code in the future.

Simulation results for the hypothetical UTOP case areshown in Figs. 6 and 7. In the absence of FAST, the powerof the core increases sharply due to positive external
reactivity insertion. On the other hand, the reactor powerat the initial stage of UTOP is effectively suppressed by
the FAST device and the core can survive the serious
UTOP transient much longer with FAST with any IHX
models.
However, oscillatory behavior of power andtemperature due to refloating of FAST is observed in casewith FAST. Although previous study showed that low
reactivity worth of FAST is better to minimize theoscillation during the UTOP [2], it is difficult to make any
concrete conclusions about the performances of FAST
due to the highly uncertain IHX models. Consequently, areliable reactor system model with a practical IHX modelis necessary for a more accurate simulation of the UTOP
scenario.

Acknowledgements
This work was supported by the National Research
Foundation of Korea Grant funded by the Korean
government NRF-2016R1A5A1013919.
References
1. Donny Hartanto, Chihyung Kim, and Yonghee Kim,
“A Comparative Physics Study for an Innovative
Sodium-Cooled Fast Reactor (iSFR),” International
Journal of Energy Research, Vol. 42, Issue 1, January(2018).
2. Chihyung Kim, “Passive Safety Device to Improve
Coolant Temperature Coefficient of Reactivity for
Sodium-cooled Fast Reactors,” PhD Dissertation,
KAIST (2019).
3. Cahalan, James E. et al. “Advanced Burner Reactor
1000MWth Reference Concept,” ANL-AFCI-202,
Argonne National Laboratory (2007).
4.
Y. Katoh, et al., “Mechanical Properties ofChemically Vapor-Infiltrated Silicon Carbide
Structural Composites with Thin Carbon Interphases
for Fusion and Advanced Fission Applications,”
Materials Transactions 46 (3), 527-535 (2005).
5. Fink, J. K. and L. Leibowitz., “Thermodynamic andtransport properties of sodium liquid and vapor,”
ANL/RE-95/2, Argonne National Laboratory (1995)
6. Khalil, M. F., Kassab, S. Z., Adam, I. G., & Samaha,
M. "Laminar flow in concentric annulus with amoving core," 12th International Water Technology
Conference, IWTC12 (2008).

Fig. 6. Temperature change during the UTOP withconstant temperature drop in IHX modelFig. 7. Temperature change during the UTOP withconstant inlet temperature IHX model
6. Conclusions
The transient analysis using a neutronics-thermal
hydraulics coupled model showed that the FAST device
can effectively prevent early failure of a MOX-fueled
SFR core during the hypothetical ULOF, ULOHS andUTOP scenarios. In the case of the ULOF and ULOHS
transients, a long-term safety is shown to be almost
achievable with the aid of FAST. On the other hand, it isconfirmed that the current FAST design is prone to a208

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

A Comparative Study on the IQS (Improved Quasi-Static)and PCQS (Predictor Corrector Quasi-Static) MethodsaTae-suk Oha, Yunseok Chunga, and Yonghee Kima,*
Dept. of Nuclear and Quantum Eng., KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, Korea, 34141*
Corresponding author: yongheekim@kaist.ac.kr

Abstract
In order to acquire the space-time solution of neutron balance equation within a tolerable computational burden, factorization of flux into the product of
two functions, which are namely ‘amplitude’ and ‘shape’ is widely utilized.
The shape function varies in a longer time-scale than the amplitude, whichmakes the implementation of different time steps viable. In this paper, IQS(Improved Quasi-Static) and PCQS (Predictor Corrector Quasi-Static) methods have been employed, which contains Point Kinetic Equations (PKE), to solvea transient problem in a 1D-1group slab reactor. The new method to harness
physical quantities of Kinetic Equation to attain the variation of amplitude is
implemented along with the conventional approach. It has been shown thatthe proposed method more aptly reflects the physical perturbation.
Key Words: Transient problem, IQS, PCQS, Point Kinetic Equations
1. Introduction𝑝(𝑡) and shape function 𝜓(𝒓, 𝐸, 𝑡).
f (r, E , t ) = p(t )y (r, E , t ) ,Being able to investigate the behavior of the reactor
subjected to various perturbations is important while
designing or performing safety analysis of the nuclear
power plants. However, directly solving the timedependent neutron balance equation to acquire space-time
solution imposes unaffordable computation burden. In
order to circumvent such a situation, the quasi-static (QS)
approach which factorizes the neutron flux had beendevised and is widely used [1, 2].
Recently, the QS method had been revisited that
illustrates the algorithms of Improved Quasi-Static (IQS)
and Predictor Corrector Quasi-Static (PCQS) methods [3].
Both methods were implemented to ascertain the responseof a one-dimensional slab reactor having localized
perturbation in the multiplication properties.
In this paper, both IQS and PCQS methods are briefly
introduced along with the key-concept of factorization
approach. The authors also propose an improved concept
to estimate the kinetic parameters for PCQS method
which alleviates errors originating from the factorization.
A numerical test on a one-dimensional one-group slab
reactor diffusion problem was performed for two different
perturbation scenarios (step-reactivity and ramp-up
reactivity insertions) to validate the applicability of theaforementioned methods.

(1)

which is not an approximation but instead requires
additional equation for being unique
1
v

1
v

y 0* , y := òòy 0* (r, E ) y (r, E , t ) drdE1
= c º y 0 , f0
v

.

(2)

*

Time-dependent neutron balance equation can be
expressed as
1 ¶f= (Fp - M )f (r, E, t) + S d [f (r, E, t)] + S ext (r, E, t),v ¶t,
S d [f (r, E, t)] = å lk c dk (E)C k (r, t )(3)kwhere f and Ck denote energy-time dependent neutron
flux and delayed neutron precursor concentrations. The
operator Fp - M denotes the time-dependent operator
that includes prompt fission, absorption, scattering, andstreaming. 𝐒𝐝 and 𝐒𝐞𝐱𝐭 indicate the contribution fromdelayed neutrons and external source respectively. Thebalance for precursor concentration is expressed as

2. Quasi-Static Approach of a Time-Dependent
Neutron Balance Equation¥¶Ck= -lk c dk (E)C k (r, t ) + ò dE ¢n dk S f (r, E ¢, t )f (r, E ¢, t ),.
¶t0

The quasi-static approach hinges on the factorizationof neutron flux into the product of amplitude function

k = the number of considered precursor groups

1
209

(4)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

For one-dimensional one-group problem, eqs (3) and(4) reduces into
1 ¶f
= (Fp - M )f (r, t) + S d [f (r, t)],v ¶t,S d [f (r, t)] = å lk C k (r, t )

(5)

¶Ck
= -lk Ck (r, t ) + n dk S f (r, t )f (r, t ) ,¶t(6)[1] Calculate EPK parameters (9) using current shapefunction 𝜓(𝒓, 𝑡).
[2] Solve EPK for micro-step 𝛿𝑡 until 𝑡 + ∆𝑡 to acquire
𝑝 and 𝑝̇ at 𝑡 + ∆𝑡.
[3] Calculate the shape function at 𝑡 + ∆𝑡 with eq (7).
;789:∗ , 5= ?@7<,[4] Calculate the normalized difference 𝜀5 =@and normalize the updated shape function to c.
[5] Repeat 1-4 until 𝜀5 becomes smaller than certain
value. Note that EPK parameters must be re-evaluatedusing the updated shape functionkwhile neglecting the contribution from external source.
Taking eq (1) into account, the balance equation can be
expressed as

2.2 Predictor Corrector Quasi-Static (PCQS)(7)

PCQS method directly solves the neutron balance
equation rather than employing eq (7) for macro-step ∆t.
After acquiring the updated flux and precursor
information, EPK parameters are evaluated and used forsolving eq (8) for micro-step 𝛿𝑡.

where 𝑝̇ denotes the time derivative of power. The
evolution of amplitude can be treated via exact-point
kinetic equation (EPK)[1] Solve eq (3) to acquire flux and precursor
concentration at 𝑡 + ∆t. Such solutions are denoted as
𝜙C(𝒓, 𝑡) and 𝐶EF (𝒓, 𝑡).

dp r (t ) - b (t )1=
p (t ) +å lkxk (t ) ,dt
L (t )
L0 k[2] Calculate

1
S d [ p(t ¢)y (r, E, t ¢)] =p (t ),p!
1 ¶y (r, E , t )y (r, E , t ) +¶tvpv(Fp - M )y (r, E , t ) +1
z = y 0* , y / cv

to obtain the shapeI

I

where each parameter can be calculated asfunction 𝜓(𝒓, Δ𝑡) ∶= J 𝜙C(𝒓, Δ𝑡) where 𝑐 ∶= < 𝜙/∗ , M 𝜙/ >[3] Evaluate the EPK parameters using eq (9) and solveeq (8) until 𝑡 + ∆t to obtain 𝑝(𝑡 + ∆𝑡).
[4] Correct the flux and precursor concentrations.
f (r, E , t + D) = p(t + Dt )y (r, E , t + Dt ) ,F (t ) = y 0* , Fy ,Ck (r, t + D t) =L (t ) = c+dx k
F (t )= -lkx k (t ) +b k (t ) p(t ) ,dt
F0

F (t )(8),b k (t ) =1
y 0* , Fdky ,F (t )x k (t ) =1
y 0* , Ck (r, t ) ,F0

1
r (t ) =y 0* ,[DF - DM ]y .
F (t )

1
1 + lk Dt

Ck (r, t)b k DtnS f f (r, E , t + Dt )dE ¢1 + lk Dt ò

.

(10)

3. Numerical Results and Discussion
To investigate the reliability of IQS and PCQS
methods, a one-dimensional one-group 1 [m] slab reactor
subjected to localized perturbation (ranging from 20 [cm]to 30 [cm]) in the absorption cross section was considered.
The perturbation was given at 𝑡 = 2.0 [𝑠] in two different
ways: Case (1) step-transition, and Case (2) ramp-up
transition for 1 [s]. The micro-step was fixed to 0.1 [ms]throughout the simulation. The detailed physicalinformation regarding the problem is depicted in Table 1(9)Notation 𝜓/∗ indicates the adjoint flux (shape) which isconventionally employed as a weighting factor.
From the reasonable conjecture that the shape of theflux varies much slower than its amplitude, different timesteps are implemented while numerically solving eqs (7)and (8) which is the key idea of QS approach. It is
worthwhile to underline that 𝑝 and 𝑝̇ from eq (8) issubsumed while deducing the shape 𝜓(𝒓, 𝑡) from eq (7),and the (updated) shape is then used to acquire EPKparameters by eq (9).

Table 1. Summary of Physical Information
1D 1Group Slab Reactor
ΣV
0.066 [1/cm]
𝜈ΣX
0.070 [#/cm]
0.9 [cm]𝐷
300,000 [cm/sec]𝑣/
Slab Length
100 [cm]
𝛿ΣV
[Σ
±5%
V

2.1 Improved Quasi-Static (IQS)
IQS method can be achieved by the following steps,
which requires iteration(s) to satisfy the convergence ofthe shape function.
2
210

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig. 1. Power and reactivity from IQS and PCQS forstep-transition.

Fig. 3. Power and reactivity from linear interpolation
PCQS for step-transition.

Fig. 2. Power and reactivity from IQS and PCQS for
ramp-up transition.

Fig. 4. Power and reactivity from linear interpolation
PCQS for ramp-up transition.

Calculated power and reactivity evolution via IQS andPCQS after the onset of transitions are shown in Fig. 1
and Fig. 2. For obtaining IQS solution, lenientconvergence criterion was imposed in order to minimize
the computation burden originating from the iterative
procedure. Such treatment renders the IQS method to beconsistent with the PCQS method in terms of noniterative attribute. It can be seen that reference solution,
which was obtained by solving eq (3) within a fine timestep, is preceded by PCQS solution whereas the IQS
solution tends to lag with respect to the reference for bothcases. Such behavior originates from the usage of updated
shape function for PCQS and initial shape for IQS while
deducing the EPK parameters. Furthermore, since theEPK parameters are assumed to be constant within amacro-step ∆𝑡, saw-tooth like evolution of power wasobserved.

Fig. 3 and 4 depicts the time evolution of power andreactivity after the perturbation estimated by PCQS (reddashed), PCQS with linear EPK interpolation (bluedashed dot), and PCQS with linear shape interpolation(cyan-dotted) after step and ramp-up transient
respectively.
There was no noticeable difference between the shapeinterpolated result and the conventional PCQS result
except for the foremost macro-step after the onset of steptransition as shown in Fig. 3. This implies that the shape
does not vary substantially except for severely abrupt
perturbation, i.e., step-transition. For ramp-up transient, it
can be seen that power and reactivity obtained from linear
interpolation of EPK adequately resembles the reference
solution since change in the absorption cross section
varies linearly.
3.2 Consideration of Different Macro-Steps

3.1 Linearization Approaches to PCQS method

The magnitude of macro-step contributes to bothcomputational burden and accuracy. For more realistic
problems, e.g., three-dimensional multigroup whole core
problem, solving the space-time dependent equation forshape (or flux) demands majority of the computational
resources than solving exact point kinetic equation.
Furthermore, since the variation of power within a macrostep is estimated through point kinetic equation, one canconclude that having larger macro-step is beneficial interms of computation time; however, a drawback from theperspective of accuracy.

The saw-tooth like evolution can induce a significant
error in the calculated power when the calculation is
performed with a large time-step. To overcome such aproblem, the authors liberated the constraint of having
constant EPK parameters while solving eq (8). Based on
the PCQS approach, after acquiring the updated EPKparameters which hinge on the updated shape, either EPK
parameters or shape can be linearly interpolated for eachmicro-step. Then the exact point kinetic equation can besolved from such parameters.
3
211Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table 2. Summary of computational time
Computational
Method∆𝑡 [s]
Time [s]Case 1. Step-Transition
Reference
0.0001141.445
IQS [w/o iteration]
0.1
2.454
PCQS
0.2
2.542
PCQS [EPK Linear]
0.2
2.670
PCQS [Shape Linear]
0.2
7.856
PCQS
0.1
2.593
PCQS [EPK Linear]
0.1
2.670
PCQS [Shape Linear]
0.1
7.749
PCQS
0.05
2.980
PCQS [EPK Linear]
0.05
3.081
PCQS [Shape Linear]
0.05
7.938

Fig. 5. Power evolution for step and ramp-up transient
for macro-step of 0.2 [s]Computational
Time [s]Case 2. Ramp-up Transition
Reference
0.0001147.542
IQS [w/o iteration]
0.1
2.433
PCQS
0.2
2.515
PCQS [EPK Linear]
0.2
2.613
PCQS [Shape Linear]
0.2
7.710
PCQS
0.1
2.595
PCQS [EPK Linear]
0.1
2.819
PCQS [Shape Linear]
0.1
7.985
PCQS
0.05
3.001
PCQS [EPK Linear]
0.05
3.074
PCQS [Shape Linear]
0.05
8.464
MethodFig. 6. Power evolution for step and ramp-up transient
for macro-step of 0.05 [s]Fig. 5 depicts the estimated power evolution for
macro-step of 0.2 [s] for both step and ramp-up transientcases. Compared to macro-step of 0.1 [s] results, which
are shown in Fig. 3 and 4, the calculated solution deviates
further from the reference. Fig. 6 shows the result
obtained for macro-step of 0.05 [s]. The saw-tooth likeevolution feature from the original PCQS and linearshape interpolated PCQS dwindles as macro-stepincreases which implies that having shorter macro-stepincreases the accuracy. Required computational time foreach case is tabulated in Table 2. Considering both thecomputational time and accuracy of the solution, one canconclude that PCQS with linear EPK interpolation is themost suitable method among the proposed methods.

∆𝑡 [s]

It has been shown that implementation of PCQS
method aided with interpolation of kinetic parameters
results in the most physically acceptable solution.
Consideration of higher order interpolation along with theadaptive time-step scheme will be pursued in the future.
Acknowledgements
This work was supported by the National Research
Foundation of Korea (NRF) Grant funded by the Korean
Government (MSIP) (NRF-2016R1A5A1013919)

4. Conclusions

References

In this paper, IQS and PCQS methods were
implemented to solve space-time dependent neutronbalance equation for one-dimensional one-group slab
reactor subjected to localized perturbation. Due to theusage of initial and updated shape while attaining EPKparameters, which do not vary during EPK calculations,the IQS tends to lag and PCQS tends to be ahead of thereference solution respectively. Furthermore, bothmethods become unreliable in terms of power when themacro-step increases.
To surmount such defects, the authors proposed a new
method to harness EPK parameters for each micro-step by
employing either linear interpolation of EPK parametersor the shape.

1. A. F. Henry & N. J. Curlee (1958), “Verification of a
Method for Treating Neutron Space-Time Problems”,
Nuclear Science and Engineering, 4:6, 727-744.
2. K. O. Ott & D. A. Meneley (1969), “Accuracy of the
Quasistatic Treatment of Spatial Reactor Kinetics”,
Nuclear Science and Engineering, 36:3, 402-411.
3. Sandura Dulla et al., “The quasi-static method
revisited”, Progress in Nuclear Energy, 50 (2008)
908-920

4212

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Point Kinetics Analysis for Source-jerk Experiment at AGN-201K
Seulki Lima, Hyung Jin Shima*, Myung-Hyun Kimb
a Nucl. Eng. Dept., Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea
b Nucl. Eng. Dept., Kyung Hee University 1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do
17104, Koreas*
Corresponding author: shimhj@snu.ac.kr

Abstract
Point kinetics analysis is a widely used for performing analyses of dynamic
phenomena. However, it is based on kinetics theory developed for critical reactor
studies, and thus its applicability to subcritical systems is subject to investigation.
In this paper, we conducted point kinetics analysis on source-jerk experiment at
AGN-201K reactor, Kyung Hee University. And the point kinetics analysis results
are compared with experiment data.
Key Words: point kinetics analysis, subcritical system, source-jerk
total 235U amount of 690g in the fuel disks and thecontrol rods.

1. Introduction
Point kinetics analysis is a widely used for performing
analyses of dynamic phenomena.
It has been extensively applied for the transient design
analysis of existing reactors and forms the basis of manytransient analysis computational codes. However, it isbased on kinetics theory developed for critical reactor
studies and thus its applicability to subcritical systems issubject to investigation. [1] Thus, in this paper, we
conducted point kinetics analysis on source-jerk
experiments at AGN-201K [2,3] reactor in subcriticalstate. The main objectives of this study is to confirm thechange of the detector signal according to the initial
location of the source, and to compare the result of thepoint kinetics analysis with the experimental result thatchanges subcriticality.

2.2 Source jerk experiment
The experiment goal is to measure the detector signal thatappears when removing the source from the subcritical
reactor that contains the neutron source. In thisexperiment, we used Ra-Be neutron source
Before the experiments, we conducted a task of finding
the critical state control rod position of this reactor and as
a result, this reactor’s critical state control rod position is
FR 15.56cm, CR 20.67cm, SR 23cm.
The experiment is conducted in two stage. First, put aneutron source into the subcritical state reactor and wait
150 seconds for the detector signal to stabilize. Second,
inserted neutron source quickly removes at 151 seconds
and detector signal measured up to 300 seconds. In thisexperiment, Helium-3 detector was used to measure thecount rate.
Figure 1. shows the planar configuration of AGN-201K
used in the experiment and Table I. shows details of theexperiments that we conducted.

2. Source jerk experiments
2.1 AGN-201K reactor
The experiment was conducted with AGN-201K reactor
at Kyung Hee University. The AGN-201K reactor has
been operated for research and education since 1982.
AGN-201K consists of polyethylene disks embedding
19.5 wt. % enriched UO2 particles, control rods, graphitereflector, lead and water shields. [4]The AGN-201K reactor has four control rods –twosafety rods (SR 1 and SR 2), one coarse rod (CR), and one
fine rod (FR) – which can be inserted by 16 cm from thebottom of the fuel disk 9 in a total flight length of 24cm
within vertical control element tubes. The reactor operator
can insert negative reactivity by withdrawing the control
rods. The composition of control rods is the same as thefuel disks. The thicknesses of its cladding and control
element tube made of aluminum are 0.25 cm and 0.25 cm,
respectively. The FR contains 2.5 g of 235U with the innerdiameter of 2 cm. Then AGN-201K becomes to have theFig. 1. Planar Configuration of experiment213

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Table I. Experiment conditions
Case

Control Rod Position
Fine
rod

1

10cm

2
3
4

Coarserod

0cm

Safetyrod

Sourceposition

Initialcount rate

Center

5,610

30cm
west

1,066

Center

4,378

30cm
west

753

Center

1,571

30cm
west

338

23cm

23cm
0cm

5

0cm

6

The signal after 150 seconds was measured every 5
seconds. It was showed that the count rate drops rapidly
as soon as the neutron source was removed.
3. Point kinetics analysis for source-jerk experiment
3.1 Point kinetics analysis for experiments.
To conduct point kinetics analysis for this experiments,
we should obtain initial reactivity and kinetics parameter.
To get initial reactivity and kinetics parameter, theMcCARD[6] eigenvalue calculations are performed with1,000,000 histories per cycle on 50 inactive and 400
active cycles using the continuous-energy cross section
libraries produced from ENDF/B-VII.1 and thermal
scattering libraries, for water, graphite, and polyethylene.
Table Ⅱ. shows the keff and initial reactivity that
calculated by McCARD. We get initial reactivity as therelative difference between experiment case and criticalstate. Table Ⅲ. shows kinetics parameters that calculatedby McCARD.

2.3 Experiment result
Figures 2.–4. show result of source jerk experiment.
6000

Table Ⅱ. McCARD calculation of keff and reactivity
Reactivity
keff (S/D)

Count Rate

5000
4000

(1/ kcritical – 1/ kexperiment)

Case1

3000

Case 1,2

Case2

2000
1000

Case 3,4

0
150

200

250

300

350

Case 5,6

Time(sec)

Fig. 2. Experiment result (Case 1,2)

CriticalstateCount Rate

5000
Case3

2000

Case4

1000
0
150

200

250

300

350

Fig. 3. Experiment result (Case 3,4)Count Rate

2000
1500
Case5
Case6

500
0
150

200

250

300

-1,074 pcm-3,326 pcm-

𝛽𝑖,𝑒𝑓𝑓 (S/D)

0.00757(0.00283)

0.00761(0.00278)

0.00760(0.00284)

𝛽1 (S/D)

0.00026(0.01454)

0.00028(0.01473)

0.00027(0.01575)

𝛽2 (S/D)

0.00139(0.00641)

0.00137(0.00657)

0.00138(0.00684)

𝛽3 (S/D)

0.00131(0.00660)

0.00133(0.00670)

0.00132(0.00669)

𝛽4 (S/D)

0.00291(0.00450)

0.00291(0.00455)

0.00290(0.00457)

𝛽5 (S/D)

0.00120(0.00727)

0.00122(0.00684)

0.00122(0.00726)

0.00050(0.01118)
0.00007(0.00069)

0.00051(0.01073)
0.00007(0.00064)

0.00051(0.01128)0.00008(0.00068)

Time(sec)

1000-720 pcm

Table Ⅲ. McCARD calculation of Kinetics Parameters
Kinetics
Case 1,2Case 3,4Case 5,6
parameter

4000
3000

1.00383(0.00004)
1.00028(0.00004)
0.97824(0.00004)
1.01114(0.00005)

𝛽6 (S/D)

350

Time(sec)�(S/D)

Fig. 4. Experiment result (Case 5,6)

214

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019in subcritical state, the point kinetics analyses for thesource jerk experiments are quite in accordance withexperiments data. The reason seems to be that theexperimental reactor subcriticality is slightly smaller than
1.0 and it is not deep enough.
For the future work, point kinetics analysis of thesource trip experiment will be carried at KUCA with deep
subcriticality condition.

3.2 Comparison between experiment and numerical result

Nomilized Count Rate

Figure 5.-7. show comparisons between point kinetics
analyses and experiments.
The figures showed that even when the experiment was
carried out by changing the position of the neutron sourcewith the same rod position, the dropping rate was not
significantly different. And it showed that the point
kinetics analyses are quite in accordance withexperiments that carried out in a subcritical state.

References
1. Eriksson M, Cahalan JE, Yang WS. On the
Performance of Point Kinetics for the Analysis of
Accelerator-Driven Systems. Nucl. Sci. Eng. (2005)
2. M. H. Kim, Research & Educational Reactor AGN201K, Reactor Research & Education Center, Kyung
Hee University, (2018).
3. M. H. Kim, Utilization of AGN-201K for Education
and Research in Korea. No. INIS-BE--11N0031.
(2011).
4. Kyoseong Song., McCARD Analysis for AGN-201K
Reactor Physics Experiments, Seoul National
University, Korea nuclear society (2018)
5. H. J. Shim et al., McCARD for Neutronics Design
and Analysis of Research Reactor Cores, Seoul
National University, Annals of Nuclear Energy, 82,
pp.45-53, (2015).

1.2
1
0.8

PKE
Case1
Case2

0.6
0.4
0.2
0
0

100

200

300

400

Time(sec)

Nomilized Count Rate

Fig. 5. Comparison between point kinetics analysis and experiment(Case 1,2)
1.2

1
PKE
Case3
Case4

0.8
0.6
0.4
0.2
0
0

100

200

300

400

Time(sec)

Nomilized Count Rate

Fig. 6. Comparison between point kinetics analysis and experiment(Case 3,4)
1.2
1
PKE
Case5
Case6

0.8
0.6

0.4
0.2
0
0

100

200

300

400

Time(sec)

Fig. 7. Comparison between point kinetics analysis and experiment(Case 5,6)
4. Conclusion
It was confirmed that there was no significant differencein the dropping rate seven when the source jerk
experiment was conducted with different neutron source
positions. And although this experiments were conducted215

TS-05

Reactor Physics Experiments

No.

RPHA19-1026

Title

A Subcriticality Measurement of AGN-201K Reactor Using the
Rossi-α Method

Authors

Sungho Moon, Jong Hoon Kim, Myung Hyun Kim, Ser Gi Hong (Kyung
Hee Univerisity, Korea)
217

No.

RPHA19-1035

Title

Mockup Critical Experiments Validation for SARCS

Authors

Mancang Li, Zhang Chen, Dong Yao (Nuclear Power Institute of China,
Chine)

No.

RPHA19-1076

Title

Feynman-α Method Experiment Using a Random Selection
Method for AGN-201K

Authors

Jong Hoon Kim, Sung Ho Moon, Ser Gi Hong, Myung Hyun Kim (Kyung
Hee University, Korea)
221

No.

RPHA19-1086

Title

A Verification of Delayed Neutron Dataset with Comparing
Theoretical and Experimental Delayed-to-Prompt Ratio Index
Defined in Feynman-α Analysis

Authors

Kazuki Takahashi, Kunihiro Nakajima, Atsushi Sakon, Tadafumi Sano,
Sin-Ya Hohara, Kengo Hashimoto (Kindai University, Japan)

No.

RPHA19-1050

Title

Education on Reactor Physics and Fuel
Experimental Facilities of the Nuclear Industry

Authors

Kenichi Yoshioka, Tsukasa Sugita, Satoshi Wada, Rei Kimura, Yu

Cycle

Using

Yamashita, Hironori Kumanomido, Tadaharu Masuyama, Masahide
Akiyama, Haruo Miyadera, Kouji Hiraiwa (Toshiba Energy Systems &Solutions, Japan)
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Osaka, Japan, Dec. 2-3, 2019

A Subcriticality Measurement of AGN-201K Reactor using the Rossi-α Method
Sungho Moon, Jong Hoon Kim, Myung Hyun Kim and Ser Gi Hong
Kyung Hee University, Department of Nuclear Engineering
1732 Deogyeong-daero, Giheung-gu, Yongin, Gyeonggi-do, 17104 - Korea*
Corresponding author: sergihong@khu.ac.kr

Abstract
A continuous subcriticality monitoring using ex-core detectors signal can provide
operational margin with respect to criticality of nuclear reactors. The Rossi-αmethod which is one of the most popular noise analysis methods estimates theprompt neutron decay constant(α) by using the probability representing autocorrelation of the detector signals in time and it can be used in estimating keff withkinetic parameters. In this study, subcriticality measurement using the Rossi-αmethod is performed at AGN-201K. The results show that the subcriticality forseveral control rod positions can be estimated within ~330 pcm Δk.
Key Words: Subcriticality, Rossi-α method, Noise analysis method, AGN-201K
1. Introduction
Most nuclear facilities are designed to have conservative
subcritical margin to accidentally uncontrolled neutron
multiplications. Therefore, an accurate real-timemeasurement of subcriticality can provide a helpful way
to guarantee the safe operation of nuclear facilities. Noiseanalysis methods have been studied for a long time forthis purpose. The most noise analysis methods have thesame basis that the properties of a subcritical system canbe determined by measuring fluctuations in the fission
chain processes. The fluctuations in fission chain
processes are dependent on the stochastic nature of chain
reactions. The Rossi-α method is one of the popular noise
analysis methods. In this work, subcriticality experimentis performed with the Rossi-α method for checking its
suitability at AGN-201K which is unique zero-power
research and training reactor in our country.

Fig. 1. Random branching process of fission neutron

p0 (t0 ) dt0  Fdt0 ,(1)

where F is the average fission rate. The probability ofdetecting correlated neutron detection event near time t1
and t2 (i.e., the probability that the neutrons generated
from a fission reaction at t0 are detected at t1+ dt1 and dt2
respectively) are given by

2. Theory and Method for Rossi-α Method
2.1 Review of Rossi-α method
An understanding of the distinction between correlated
neutron pairs which are derived from same neutron
ancestor and uncorrelated neutron pairs is essential to thecomprehension of how the Rossi-α method wasdeveloped. Fig. 1 provides a random branching process ofneutron in the view point of correlated and uncorrelated
neutron pairs. The left hand fission event represents therelationship of correlated fission neutrons between A and
B. On the other hand, the event C describes theuncorrelated event with the event A or B. The probability
that a neutron is counted at time t0=0 can be defined asfollows :p1 (t1 )dt1   v p f e  (t1 t0 ) dt1
p2 (t2 )dt2   (v p  1) f e  (t2 t0 ) dt2,(2)

where ε is the detector efficiency and νp is the average
number of prompt neutrons generated by one fission, υ isthe neutron velocity, and Σf is the macroscopic fission
cross section. It is noted that the ν term has been modified
to (ν-1) to account for the neutron lost at t1 to the fission
chain. Therefore, the total probability of detecting two
chain-related counts initiated by a fission at time t0 can be217Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019derived as follows :pc (t1 , t2 )dt1dt2  F

22.2 Subcriticality Measurement System (SMS)
In this study, a time-series data of neutron counts within afine unit gate time of 10sec is acquired using the
Subcriticality Measurement System (SMS) which wasdeveloped by Korea Electric Power Research Institute(KEPRI) for measuring the ex-core detector signal fromcommercial PWR to get the condition of large
subcriticality [4]. Since the neutron generation time (Λ)
of AGN-201K is estimated about 50~60 μsec, the shorter
gate time can acquire more detailed information for
estimating the α value [4]. Therefore, in this work, the unit
gate time is set to 10 μsec. Due to this measuring system,
the neutron counts in a few seconds provide a large
number of count data for a huge number of time bins. Fig.
2 shows the measured neutron counts during 0.1 seconds
containing 10,000-time bin data using 10 μsec unit gate time.

t1

 p(t )dt p(t )dt FdtD k p2
2(1  k p ) e1 ( t2 t1 )

1

2

2

0

dt1dt2,(3)where Dν is the Diven’s factor considering an average ofthe number of prompt neutrons emitted, kp is the prompt
critical, and Λ is the neutron generation time. The
probability of detecting two random uncorrelated events
which are not derived from same ancestor can be seen in
Eq. (4)pr (t1 , t2 )dt1dt2  F 2 2 dt1dt2 ,(4)

The final total probability of detecting two neutron counts
is obtained by summing the correlated and uncorrelated
event probabilities, which is given byp(t1 , t2 )dt1dt2
D k p2.
 F 2  F e (t2 t1 )  dt1dt2

2(1  k p )



(5)

It is shown that a correlation between two detecting points
decreases exponentially with the form of Eq. (5) as theirtime difference increases. This method can determine theprompt neutron decay constant (α) by considering theauto-correlation of the detector signals in time using Eq.
(6), which is supposed to be equivalent to Eq. (5)theoretically.

1 N kP ( )  P ( k t ) C (i )C (i  k ) ,N  k i 1Fig. 2. Time-series data of neutron counts by using SMS
for 0.1 sec
3. Description of AGN-201K and Subcritical States
3.1 AGN-201K(6)

AGN-201K which is a zero-power reactor for educationaland research purposes is polyethylene-moderated andgraphite-reflected reactor [5]. The fuel material is ahomogeneous mixture of UO2 and polyethylene. As
shown in Fig. 3, the fuel is comprised of 10 disks with
12.8 cm radius and 25 cm active core height [6]. Uranium
enrichment of the fuel is about 19.5 wt%. The active coreis surrounded by 25 cm thick graphite reflector followed
by a 10 cm thick lead gamma shield. For fast neutron
shielding, the outside of the core tank is filled with waterof ~47.5 cm thickness [6]. The control rods which havethe same composition as the fuel material consist of two
Safety Rods (SR), one Coarse Rod (Cr), one Fine Rod(FR). During operation, reactor power is controlled by CR
and FR [5]. In particular, external Ra-Be source located in
the left-upper beam port supplies neutrons with anintensity of 10 mCi. A He-3 ex-core detector is located inright-lower beam port.

where τ is the time interval between two detecting time
points (=t2-t1), N is the total number of time bins, and C isthe detector counts in a given time bin. By using thefollowing least square fitting of the Rossi probability
which is equivalent with Eq. (5) versus the gate time τ, αcan be obtained [1, 2] :Fitting Curve = A + Be ,(7)

After the determination of the prompt neutron decay
constant (α), the effective multiplication factor can becalculated using Eq. (8) derived from point kinetics
equation [3] as follows :keff 

1.
1    

(8)

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019considered was 25 million counts (i.e., 25,000,000τ, τ=10μsec). For curve fitting, the length of gate time was up to
0.05 sec (i.e., 5,000τ). Table II shows the results of
subcriticality measurements using Rossi-α method. The
α-PKE values in Table II represent the ideal value giving
the reference keff values using the given kinetic
parameters in Table I. In other words, the α-PKE value
can be obtained by Eq. (8). As shown in Table II, theRossi-α method using the whole data gives good
agreements in keff within 330 pcm Δk for all the cases.
Table II. The results of Rossi-α method
Condition
keff
α-PKE
k-est
α-est
0.99086
300.000.98764 358.95 a
SCR1b-322.4658.95
0.99889
159.920.99668 200.53
SCR2-220.6240.62
0.99939148.770.99737 186.03
SCR3-201.5737.26
0.99982
143.680.99811 175.53
SCR4-171.4031.86
1.00031135.56
0.99885 162.60
SCR5-145.92
27.04

Fig. 3. Axial configuration of AGN-201K modelled with
MCNP6

a
b

[(keff) – (k-est)] (pcm Δk)[(α-PKE) – (α-est)] (1/s)

Figs. 4 to 8 show the illustration of Rossi-α method withcurve fitting. It is noted that the more subcritical condition
has a larger dispersion as the time increase.

3.2 Subcritical states
We considered five different subcritical statescorresponding to five different control rod configurations
to apply the Rossi-α method. Table I describes the control
rod positions and the keff values and kinetic parameters
estimated with MCNP6 for the specified five subcriticalstates [7]. Estimation of subcriticality depends heavily on
the kinetic parameters such as effective delayed neutron
fraction (β) and neutron generation time (Λ). In order tominimize the statistical error of keff and kinetic parameters,
the MCNP6 criticality calculations are performed with
100 inactive and 5,000 active cycles of 100,000 histories,
which give the standard deviation of keff less than 3 pcm
with ENDF/B-VII.1 neutron cross-section library. As
shown in Table I, the SCR1 state has the highest
subcriticality of 1,236 pcm Δk.
4. Results and Discussion
A series of neutron counts was obtained for the fivesubcritical conditions by using SMS with a unit gate timeof 10 μsec during 4~5 minutes. The number of time bins

Fig. 4. Rossi-α fitting curve for SCR1

Table I. Calculated reference multiplication factor and kinetic parameters for considered control rod position
Inserted rod position (cm)σΛCondition
keff
β-eff
(pcm)
(μsec)
SR#1
SR#2
CR
FR
0.9876430.00755 55.89873
23.07
23.44
0
12.56
SCR1
0.996683
0.00761 54.55938
23.07
23.44
17.25
12.56
SCR2
0.997373
0.00746 54.27638
23.07
23.44
18.25
12.56
SCR3
0.998113
0.00757 53.91283
23.07
23.44
19.25
12.56
SCR4
0.9988530.00763 54.00546
23.07
23.44
20.25
12.56
SCR5

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Fig. 5. Rossi-α fitting curve for SCR2

Fig. 8. Rossi-α fitting curve for SCR5
5. Conclusions
In this work, Rossi-α method using auto-correlation of
detector signals was performed for subcriticality
measurements for AGN-201K. From the results ofapplication to the considered five subcritical states, it was
shown that the Rossi-α method estimated thesubcriticalities within 330 pcm Δk.
Acknowledgements
This work was supported by the National research
Foundation of Korea(NRF) grant funded by the Korean
government (MSIP : Ministry of Science, ICT and Future
Planning) (No.2017M2B2B1072806). We would like tothank for Dr. Eun Kee Lee and Mr. Ho Cheol Shin at CRI(Central Research Institute) of KHNP (Korea Hydro andNuclear Power) Company and Prof. Deokjung Lee at
UNIST (Ulsan National Institute of Science andTechnology).

Fig. 6. Rossi-α fitting curve for SCR3

References
1. Uhrig. R.E., Random noise techniques in nuclear
reactor systems, p. 60, The Ronald Press Company,
New York (1970).
2. Mckenzie G.E., “Modern Rossi Alpha Measurements,”
University of Illinois (2014).
3. Kong C, et al., “Stability improvement of noise
analysis method in the case of random noisecontamination for subcriticality measurements,”
Annals of Nuclear Energy., 71, p. 245-253 (2014).
4. Lee E.K., et al., “Experimental determination of
subcriticality at subcritical PWRs in Korea,” Annalsof Nuclear Energy., 37, p. 1601-1608 (2010).
5. Kim M.H., et al., “KHU AGN-201K technical
Specification,” Kyung Hee University (2014).
6. Gorham M., “Evaluation of the AGN-201 Reactor at
Idaho State University,” NEA/NSC/DOC(2006)1,
Idaho National Laboratory (2006).
7. Goorley J.T., et al., “MCNP6TM USER’S MANUAL
Version 1.0,” LA-CP-13-00634, Los Alamos
National Laboratory (2013).

Fig. 7. Rossi-α fitting curve for SCR4

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Feynman-α Method Experiment
Using a Random Selection Method for AGN-201K
Jong Hoon Kim, Sung Ho Moon, Ser Gi Hong and Myung Hyun Kim
Kyung Hee University, Department of Nuclear Engineering
1732 Deogyeong-daero, Giheung-gu, Yongin si Gyeonggi-do, Republic of Korea, 17104*
mhkim@khu.ac.kr

Abstract
In this study, Feynman-α experiment was conducted on AGN-201K research
reactor of Kyung Hee University, Korea. A sufficient number of data is required
to use the Feynman-α method. For this purpose, the experiment was conductedwith a device designed by Korea Electric Power Research Institute (KEPRI)
using 10 μsec as the gate time. To calculate the criticality, the time swap method
and the random selection method were used as the experimental data processing
method. The time-swap method is called “Bunching technique” and is one of thewidely used methods in the Feynman-α experiment. The Random selecting
method is a method in which several Y values are calculated at the same gate
time by randomly setting the start point of data selection from experimental data
and the interval between gate time and gate time. In order to compare theexperiment data, the k-value was calculated using MCNP6 and estimated throughthe control rod worth. The criticality, the delayed neutron fraction and neutron
generation time were calculated using the MCNP6 under the same conditions asin the experiment, and the criticality was estimated using the control rod worth
obtained from the compensation method by comparing the control rod position
between the critical state of experiment and each experiments. As a result, thedifferences of the criticality between estimation from rod worth, MCNP6 results
and the methods are within 300pcm.
Key Words: Noise analysis method, Feynman-α method, sub-criticality,
AGN-201K,
1.

Introduction

1.1 Purpose

1.2 Description of AGN-201K

In this study, we carried out noise analysis experimenton AGN-201K, a research reactor of Kyung Hee
University, Korea. Feynman-α method, widely used as
one of the noise analysis methods, was used in thisexperiment.
AGN-201K is a small reactor with a total height of 280
cm and a diameter of 198 cm. Because of its small size,
there is a lot of noise being measured on the detector. In
this respect, the Feynman-α method, one of the noise
analysis methods to be applied, is highly worthy ofexperimentation. Therefore, in this study, we applied the
Feynman-alpha method using the data obtained throughAGN-201K and compared it with the MCNP calculation
results.
From the data obtained by the experiment, the k value
can be calculated through the Feynman-α method. For thecomparison, we used the MCNP6 to calculate thecriticality, the delayed neutron fraction and neutron
generation time. We also used the estimated criticality
through the compensation method which is the control rod
worth measuring method for the comparison.

AGN-201K research reactor in Korea is very safe reactor
because of the limited excess reactivity and the strong
negative temperature feedback coefficient. The reactor
core tank is sealed with the aluminum of 2mm thickness
to keep the fission gas and its diameter is 32.2cm, heightis 76cm. The reactor core is surrounded by the reflector of
the high-density graphite with 20cm thickness. The
gamma shield surrounding the reflector is the lead of
10cm thickness.
The reactor tank is comprised of core tank, reflector,
lead shielding, control rod, Glory-Hole and Access Ports.
There is a movable thermal column on the top of thereactor tank and reactor tank which is surrounded by thelight waters (about 1,000 gallon) of 55cm thickness
except bottom part. AGN-201K has four control rods, one
fine control rod, one coarse control rod, and two safetycontrol rods.
There are a total of six detectors in the reactor. It has 3
He-3 detectors, 3 BF3 ionization chambers, and 1 Fission
chamber. Experiments were conducted using a He-3
detector, one of the portable detectors.

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2. Experiment Using Feynman- α method

selected data, a graph was drawn using the ratio of theaverage and the number of neutrons measured per gatetime. Using the right-hand side of Eq.(1), the appropriatealpha value was obtained by least squares fitting method.
We proceeded to calculate the criticality by substituting
the delayed neutron fraction and neutron regeneration
time in Eq (2).

2.1 Feynman-α method
Feynman-α method is a method of calculating thecriticality using the mean and variance ratio of thenumber of neutrons detected in regular intervals (Gatetime). The significant equation can be represented as [1] :̅̅̅𝑍 2 − 𝑍̅ 2𝑍̅2

𝜀𝐷𝑣 (1 − 𝛽)
1−𝑒
[1 −
2
(𝛽 − 𝜌)𝛼𝑡1 − 𝑒 −𝛼𝑡
= 1 + 𝑎 [1 −]= 1+𝑌𝛼𝑡≅ 1+

3. Data selection−𝛼𝑡

The data obtained from the experiment consists of abundle of neutron numbers measured within 10 μsec ofcontinuous time as mentioned above. In this study, we
proceeded in two ways to select the data from theexperimental data for using the Feynman-α method.
For the first time, data selection was performed using amethod called "Bunching-technique (Time swap)" [2, 3,
4]. This method is a widely known method for using the
Feynman- α method to obtain the number of data bundles
obtained from experimental data even in data selection ofdifferent conditions.

](1)

where Z is the count rate detected over time t, 𝐷𝑣 isDiven Factor, 𝛽 is the delayed neutron fraction and 𝜌
is the reactivity. The effective multiplication factor can becalculated as:
𝑘𝑒𝑓𝑓 =

1
1 − 𝛽 + 𝛼𝛬(2)where α is the prompt decay constant, Λ is the neutron
generation time. The effective multiplication factor can becalculated by substituting the prompt decay constantobtained from Eq. (1).
2.2 Description of the experiment process
Noise analysis is a method of determining the criticality
by analyzing the noise of detector signal under a subcritical state. Therefore, this has the advantage of not
requiring a lot of other equipment. Generally, when
applying the Feynman-alpha method, there should be
sufficient measurement data to ensure statisticalsignificance by increasing the gate time and sufficientnumber of measurements.
In order to obtain enough measurement data, Korea
Electric Power Research Institute (KEPRI) developed ameasuring device with a minimum gate time of 10 μsec.
In this study, sub-criticality measurement experiment was
performed using this measuring device [2].
The Feynman-alpha method experiment was conductedusing a He-3 detector while changing the position of theCoarse Rod. The initial state of the reactor was critical at
20.25 cm for the position of the Coarse Rod.
The experiment data were acquired by measuring theminimum gate time at 10 μsec for about 5 minutes. In theprocess of the experiment, the coarse control rod positions
for data measurements were changed to 19.5 cm, 18.5 cm,
17.5 cm, and 0 cm, and the data were collected for 5
minutes in the same process. The data format consists ofa set of neutrons counts, measured in 10 μsec
continuously, in time and neutron numbers. That is, thedata consists of about 30,000,000 bundles, assuming ameasurement time of about five minutes.
As a result, sufficient data were obtained to apply the
Feynman-alpha method. In addition, the obtained data canbe selectively used according to the purpose. In theFig. 1. Schematic view of the time-swap technique
The second method is a continuous random selecting
method. As shown in Fig. 2, this method sets the gate timeand randomly sets the interval time and the initial time
between the gate times to have several Y values at thesame gate time. The initial time is an arbitrary time from0 second to 1 second, and the interval time is from 0
second to 0.01 second. In this method, the Y value has atotal of 10 different values through different initial timeand interval time for each gate time, and the alpha value
is calculated using this values.

Fig. 2. Schematic view of Random selecting method
4. Results

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4.1 Criticality Estimation with Control rod calibration
curve

4.2 Criticality Calculation with MCNP6As shown in Figure 4, we calculated the criticality by
MCNP6 based on the structure of AGN-201K describedin Section 1. The keff values and kinetic parameters were
calculated according to the positions of the control rods ineach experiment and summarized in the Table 1 [6].

To compare the criticality calculated by the Feynmanα method, we use the data obtained from the control rod
worth measurement experiment to calculate the criticality
according to the control rod position in the above fiveexperiments. KHU Research Reactor
By design, the control rods are the same in configurationand only differ in size. Since the control rod worth is
known through the experiment, the coarse control rod is
estimated through the fine control rod in order to calculate
the criticality according to the reactor state. The finecontrol rod was measured using the compensation method,
one of the control rod worth measurement experiments.
Experiments were conducted using DDRCS (Direct
Digital Reactivity Computer System) to obtain the
integral rod worth of fine control rod. Integral rod worthof the coarse control rod was calculated using the control
rod design value (CR: 1250pcm, FR: 310pcm) [5]. Thiscan be seen in Figure 3 below.

Fig. 4. Axial configuration of AGN-201K modelled with
MCNP6
Fig. 3. Integral rod worth of Coarse rod and Fine rod
Table I. Calculated reference multiplication factor and kinetic parameters for considered control rod position
Experiment
Inserted rod position (cm)
Λ
σ
β-eff
keff
Order
(μsec)
(pcm)
SR#1
SR#2
CR
FR
23.07
23.44
20.25
12.56
0.00763
54.00546
3
10.99885
23.07
23.44
19.25
12.56
0.00757
53.91283
3
2
0.99811
23.07
23.44
18.25
12.56
0.0074654.27638
3
3
0.99737
23.07
23.44
17.25
12.56
0.0076154.55938
3
4
0.99668
23.07
23.44
0
12.56
0.00755
55.89873
350.98764
TableⅡ. Difference of the multiplication factor between Estimation value from Rod worth and Each method.
Order Estimation by Rod worth
Time swap method
Random selecting method
α-valuekeff (A)α-valuekeff (B) A-B (pcm)
α-valuekeff (C) A-C (pcm)
152.5770.99939
123.400
1.00097125.386
1.000861-158-147
163.374
0.99876137.271
1.00017138.2291.000122-141-135
178.534
0.99778147.5170.99945145.7700.999553-168-177
194.432
0.99701
156.4620.99907149.009
0.999484-206-247
333.3280.98904
278.0440.99207216.601
0.995465-303-642
TableⅢ. Difference of the multiplication factor between Calculated value using MCNP6 and Each method.

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Order
1
2
3
4
5

MCNP6
α-valuekeff (A)
162.601
0.99885
175.535
0.99811186.028
0.99737200.535
0.99668
358.9470.98764

Time swap method
α-valuekeff (B) A-B (pcm)
123.400
1.00097-212
137.271
1.00017-206
147.5170.99945-208
156.4620.99907-239
278.0440.99207-443

4.3 Comparison of Multiplication factor

Random selecting method
α-valuekeff (C) A-C (pcm)
125.386
1.00086-201
138.2291.00012-201
145.7700.99955-218
149.009
0.99948-280
216.601
0.99546-782

The Feynman- α experiment to measure sub-criticality
in the Research reactor AGN-201K of Kyung Hee
University, Korea was carried out.
The data was collected using a device with a minimum
gate time of 10 μsec, which was used by KEPRI, and thecode was programmed to select the desired data to obtainthe sub-criticality.
In order to verify the calculated criticality, we used thecontrol rod worth of the reactor to estimate the criticality
under the same experimental conditions. Also MCNP6 isused to verify the calculated multiplication factor
obtained by each method.
As a result, it can be confirmed in that the differences of
the criticality of the two methods were relatively small forforth experiment, but difference of criticality in the lastexperiment in which coarse rod (CR) is all out state waslarge. Since the experiments in the intermediate range
have not been carried out, further experiments will be
carried out.

We compared the criticality obtained from theexperiment using the Feynman- α method and Time swap
method between the estimated criticality using the control
rod and calculated criticality using MCNP6. TableⅡshows difference of the criticality between estimatedvalue from Rod worth and each method. TableⅢ showsdifference of the multiplication factor between calculated
value using MCNP6 and each method.
As a result of the comparison, the calculation errors of
the two methods are within the range of 300pcm until thecriticality reaches 0.98. However, we could see that thedifference of the criticality was large in the experiments
where the coarse control rod was all rod out state. Below
Figure 4,5 shows Feynman- α curve of Time swap andRandom selecting method.

Acknowledgements
This work was supported by the National Research
Foundation of Korea(NRF) grant funded by the Korean
government (Ministry of Science and ICT)(2017M2B2B1072803)
References1.
2.

Fig. 4. Feynman- α curve of Time swap method

3.

4.

5.
6.

Fig. 5. Feynman- α curve of Random selecting method
5. Conclusions

224

Juan Blazquez, Remarks on kinetics parameters of a
sub-critical reactor for nuclear waste incieraation,
NEA, 2001 pp826-829.
Lee E.K., et al., “Experimental determination of
subcriticality at subcritical PWRs in Korea,” Annalsof Nuclear Energy., 37, p. 1601-1608 (2010)
Misawa T., et al., “Measurement of Prompt Neutron
Decay Constant and Large Subcriticality by the
Feynman-α Method,” Nuclear Science andEngineering., 104, p. 53-65 (1990)
Okuda R., et al., “An improved Feynman-α analysis
with a moving-bunching technique,” Journal of
Nuclear Science and Technology., 53, 10, p. 16471652 (2016).
KHU AGN-201K Technical Specification”, Kyung
Hee University, 2014.
Goorley J.T., et al., “MCNP6TM USER’S MANUAL
Version 1.0,” LA-CP-13-00634, Los Alamos
National Laboratory (2013).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Education on Reactor Physics and Fuel Cycle using Experimental Facilitiesof the Nuclear Industry
Kenichi Yoshioka, Tsukasa Sugita, Satoshi Wada, Rei Kimura, Yu Yamashita, Hironori Kumanomido,
Tadaharu Masuyama, Masahide Akiyama, Haruo Miyadera and Kouji Hiraiwa
Toshiba Energy Systems & Solutions Corporation, 4-1 Ukishima-cho, Kawasaki-ku, Kawasaki, Japan*
Corresponding author: kenichi.yoshioka@toshiba.co.jp

Abstract
Education programs using experimental facilities of Toshiba Energy Systems &Solutions are being conducted for college students. The reactor physics & fuel cycle
education course is one of the programs. There are experimental items such as a
critical experiment with a critical assembly, a moderation experiment with aneutron source, gamma-ray measurement of a fuel rod, and a chemical separation
experiment employing ion exchange resin. Students study the basic theory of a
nuclear reaction, power plants, and the fuel cycle through these experimental items.
This education has been conducted for over 10 years. More than 400 students havereceived the education. Some of the students currently work in the nuclear industry.
Key Words: Reactor physics, fuel cycle, education, critical assembly,
moderation, gamma-ray measurement, chemical separation
1. Introduction

the basic theory of nuclear engineering can be covered in
lectures prior to the course.
This paper introduces the outline of the reactor
physics & fuel cycle education course.

Reactor physics is one of the most important educational
themes concerning the utilization of nuclear energy.
However, many students experience difficulty in studying
reactor physics because neutron transport theory iscomplex and hard to visualize. On the other hand, the fuelcycle is also important. In particular, the disposal ofradioactive waste is a critical issue with an important
bearing on the continuing use of nuclear energy. Many
students may not understand how to dispose ofradioactive waste safely and be skeptical about theutilization of nuclear energy.
Experiments can be effective for understanding
complex concepts. There are some experimental facilities
in the nuclear industry. These facilities are also attractive
for use in the education of college students who are
studying nuclear engineering.
Education programs using experimental facilities ofToshiba Energy Systems & Solutions are being
conducted for college students [1]-[2]. The reactor
physics & fuel cycle education course is one of the
programs.
The reactor physics & fuel cycle education course
consists of a critical experiment with a critical assembly,
a moderation experiment with a neutron source, gammaray measurement of a fuel rod, a chemical separation
experiment with ion exchange resin, etc.
Target students are technical college students,
undergraduate students and graduate students. The periodof the education course is about 1 week. Educational
items are adjusted in light of the level of the students’knowledge. For students unfamiliar with nuclear energy,

2. Critical ExperimentToshiba Nuclear Critical Assembly (NCA) is a slightly
enriched, uranium-fueled, light-water-moderated criticalassembly. NCA was utilized to validate an LWR nuclear
design methodology and to develop a new fuel designconcept [3]-[5]. Figure 1 shows the overview of the NCA
core. Operators directly handle fuel rods to set up an
experimental core because the activity of the fuel rods islow.
Figure 2 shows the concept of NCA. Reactivity iscontrolled by adjusting the water level. The maximum
power of NCA is 200 W. The critical experiment tomeasure critical mass is mostly conducted at less than 1
W. Figure 3 shows the vertical cross section of NCA. The
fuel rod of NCA consists of UO2 pellets of 10 mmdiameter and aluminum cladding. The active length of thefuel rod is 1500 mm.
Figure 4 is an example of a critical approach by astudent. Students also view the experimental core directly
as shown in Fig.4. Reactivity measurements with theperiod method and neutron flux distribution
measurements have been conducted for education.
The control panel of NCA shown in Fig.4 is used forthe education on safety systems in nuclear power plants,
such as “fail safe” or “interlock”.
NCA has already shut down. Although thisexperimental item is not conducted as of 2019, inverse

225

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019multiplication plots using historical data and education
using the control panel continues to be conducted.

with a Ge detector and gamma-ray spectrum.

3. Neutron Moderation and Fission Measurement
1840mm

The concept of moderation is important for understanding
fission and the chain reaction. A moderation and fission
experiment is conducted with a standard neutron source,
polyethylene blocks, a He-3 neutron detector, cadmium
sheets, and fuel rods. Figure 5 shows an example of anexperimental setup conceived and implemented by astudent. In particular, this experiment helps students who
are not majoring in nuclear engineering to understand thebasic theory of nuclear power.

AlAl

Al spacer
Al

Core
Core
tank
tank

UO2
1500mm(critical water level)

Active core hight2000mmWater level

Lower tieplate(Al)

120mm

70mm

Fig 1. NCA core

Fuel support plate
Structureal
Material
Structural material

40mm

Base
(Concrete, Iron)

Fig 3. Vertical cross section of NCA

Fig 4. Critical approach by a student
Fig 2. Concept of NCA
4. Gamma-ray Measurement of Fuel Rods
The NCA facility possesses a variety of fuel rods such asfuel rods with 1wt% to 4.9wt% enrichment and gadoliniabearing fuel rods. Gamma rays emitted from U-235 andU-238 in the fuel rods are measured with a Ge detector.
Students speculate on the difference of enrichment or
uranium inventory in the fuel rods in light of themeasurements. Even a slight amount of Cs-137 can bedetected. The gamma-ray intensity of Cs-137 is
proportional to total fission in a fuel rod. Students study
the concept of burnup and the difference of power
between a critical assembly and a commercial plant.
Figure 6 shows an example of fuel rod measurement

Fig 5. Moderation and fission measurement226

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig.8 Trend of the number of participants

Fig.6 Gamma-ray measurement of a fuel rod

7. Conclusions

5. Chemical Separation Experiment

Reactor physics education using experimental facilities ofToshiba Energy Systems & Solutions is being conducted.
Experimental items have contributed to the education ofcollege students. Reactor physics education is becoming
increasingly important for maintaining human resourcesin the field of nuclear energy.

The disposal of radioactive waste is one of the important
technical issues concerning the sustainable utilization of
nuclear energy. As well as studying reactor physics,
learning how to treat radioactive waste helps students
understand the concept of the nuclear fuel cycle.
Chemical separation is an essential technique for reducing
radioactive waste.
Unsealed radioactive isotopes of Co-60 and Cs-137
are used. These isotopes are separated by a chemical
separation technique employing an ion exchange resin.
Figure 7 shows an example of the chemical
separation experiment.

8. Future Plan
Although NCA has already shut down, we intend to
improve the education course using the equipment
available and to continue conducting these education
programs in order to develop human resource for the
nuclear industry.

6. Number of Participants

Acknowledgement

This reactor physics education has been conducted forover 10 years. Figure 8 shows the trend of the number ofparticipants. More than 400 students have received theeducation. Some students currently work in the nuclear
industry.

These education programs have been conducted as a
subsidy project of the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) of Japan.
1.

2.
3.
4.

5.
Fig.7 Chemical separation experiment

227

References
Hiraiwa K, et al., “Education programs for students
and graduate students with experimental facilities fornuclear energy in Toshiba,” PHYSOR 2014(Onlypresentation), Kyoto, Japan, Sep 28-Oct 3, 2014.
Ito R, et al., “Approach to Light-Water Reactor
Safety Education,” ICONE2017 (Only presentation),
Shanghai, China, July 2-6, 2017.
Kurosawa F, Sekiguchi Y and Yokoyama T, “Criticalexperiments in a BWR lattice using NAIG criticalassembly,” Toshiba review (1972).
Umano T, et al., “Critical experiments simulating theoperating conditions of PWRs in the Toshiba NCA
Facility,” Proc. ICAPP09, Tokyo, Japan, May 10-14,
2009.
Yoshioka K, et al., “Critical experiments for BWR
fuel assemblies with cluster of gadolinia rods,” Proc.
PHYSOR 2014, Kyoto, Japan, Sep 28-Oct 3, 2014.

TS-06

Nuclear Data and Evaluation

No.

RPHA19-1006

Title

Treatment of R-matrix Limited Formula in FRENDY

Authors

Kenichi Tada, Satoshi Kunieda (Japan Atomic Energy Agency, Japan)
229

No.

RPHA19-1030

Title

Validation of New Fission Yield by Analysis of Post-Irradiation
Examination

Authors

Atsuya Iso, Satoshi Takeda, Takanori Kitada (Osaka University, Japan)
233

No.

RPHA19-1042

Title

Continuous Thermal Neutron
Capability in RMC Code

Authors

Lei Zheng, Kan Wang, Wei Wang (Tsinghua University, China)

Scattering

Data

Processing

236
No.

RPHA19-1053

Title

Thermal Neutron Scattering Data Generation Function in the
Nuclear Data Processing Code NECP-Atlas

Authors

Yongqiang Tang, Tiejun Zu, Siyu Yi, Jialong Xu, Liangzhi Cao (Xi'an
Jiaotong University, China), Jin Yang (Hunan University of Technology,
China)
240

No.

RPHA19-1068

Title

Quantification of Effectiveness of Integral Data Using Active
Sub-Space in Nuclear Data Testing

Authors

Daichi Imazato, Go Chiba (Hokkaido University, Japan)
244

228

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Treatment of R-matrix Limited Formula in FRENDY
a

Kenichi Tadaa* and Satoshi Kuniedaa
Japan Atomic Energy Agency, Shirakata 2-4, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan*

Corresponding author: tada.kenichi@jaea.go.jp

Abstract
The R-matrix limited formula is formatted by the current nuclear data format andit is adopted some nuclei in the latest evaluated nuclear data library. Since theprocessing of the R-matrix limited formula is significantly different to the otherresonance formulae, it is difficult to treat this formula without large modificationof the nuclear data processing code. In this study, we implemented one of the Rmatrix code AMUR to treat this formula in FRENDY. The processing results of
FRENDY are compared to those of NJOY2016 to verify FRENDY. The
comparison results indicate that FRENDY appropriately treat the R-matrix limited
formula with similar computational time.
Key Words: FRENDY, AMUR, R-matrix limited formula, nuclear dataprocessing
1. Introduction

treat the total, elastic scattering, fission, and radiative
capture cross sections. The R-matrix limited formula cantreat other reactions and the differential cross-sections,
i.e., angular and energy distribution of the secondary
particle. Therefore, the large modification is required totreat the R-matrix limited formula in the nuclear dataprocessing code.
JAEA has developed AMUR (A Multi-channel Rmatrix code) in order to evaluate the cross-section andcovariance for the light-mass nuclei, e.g., 16O [8]. AMUR

Nuclear data processing is an important interface between
evaluated nuclear data library and neutronics transport
codes. The neutronics transport codes cannot directly treat
an evaluated nuclear data library and these codes requirea cross-section data library which is generated by the
nuclear data processing code as shown in Fig. 1.
Japan Atomic Energy Agency (JAEA) has developed
a new nuclear data processing code FRENDY (FRom
Evaluated Nuclear Data librarY to any application) andreleased FRENDY version 1 [1]. FRENDY version 1 cangenerate the cross-section library for a continuous energyMonte Carlo calculation codes. The many processes, e.g.,resonance
reconstruction,
Dopplerbroadening,
generation of the probability table, are required togenerate cross section library. We have been improved theprocessing method of FRENDY [2]. In this study, we
focused on the expansion of the resonance reconstruction.
The R-matrix limited formula is one of the resonanceformula for the resolved resonance region. The current
nuclear data format, i.e., the ENDF-6 format [3], prepares
five resonance formulae in the resolved resonance region.
The Single- and Multi- Level Breit Wigner (SLBW) [4]and the Reich-Moore (RM) resonance formulae [5] aremainly adopted in the current evaluated nuclear datalibrary. The R-matrix limited formula is more rigorous
formula than SLBW and RM formulae. Currently, thisformula is adopted in 35Cl, 40Ca, 54Fe, 57Fe, 63Cu, and 65Cufrom ENDF/B-VIII.0 [6] and 63Cu and 65Cu from JEFF3.3 [7].
The processing of the R-matrix limited formula issignificantly different to the other resonance formulae.
For example, the conventional resonance formulae only

Fig.1. Calculation flow from evaluated nuclear datalibrary to nuclear calculation.

229

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019is a multi-level and multi-channel R-matrix code based on
the Wigner and Eisenbud formalism [9] with RM
approximation [5] for the radiative neutron capturechannels Since AMUR can treat the cross-sections and thedifferential cross-sections, we implemented AMUR totreat the R-matrix limited formula in FRENDY.
This paper shows the overview of nuclear dataprocessing code FRENDY and verification of the Rmatrix limited formula calculation function using AMUR.
To verify FRENDY, we compared the processing resultsof FRENDY to those of NJOY2016 [10] which is theconventional nuclear data processing code.

and covariance for the light-mass nuclei [8]. Values of theresonance parameters are estimated from theexperimental cross-sections with the generalized leastsquare method. This code is also designed to take accountof the experimental resolution, renormalization and so on
to obtain reasonable resonance parameters as much as
possible.
AMUR is also prepared the cross-section calculation
function using the resonance parameter in the ENDF-6
format. We implemented this function to processing of theR-matrix limited formula in FRENDY.
ENDF‐6
formatGNDS
formatEndf6Parser/Writer

GndsParser/Writer

Endf6
Converter

Gnds
Converter

2. Overview of FRENDY and AMUR
2.1 Overview of FRENDY
FRENDY is developed in order to process JENDL [11]which is the evaluated nuclear data library developed in
JAEA [1]. The current version of FRENDY can generate
ACE (A Compact ENDF) formatted files for continuous
energy Monte Carlo calculation codes such as MCNP [12]and PHITS [13]. It uses the same processing method as
NJOY because the implementation of the conventional
method is an important step to develop the new code.
FRENDY is written in the object-oriented language
C++ so as to enhance maintainability, modularity,
portability and flexibility. FRENDY is developed not only
to process nuclear data libraries but also to implement the
FRENDY modules to other calculation codes. Users caneasily use many functions e.g., reading, writing, andprocessing the nuclear data library, in their own codes.
The system structure of FRENDY is shown in Fig. 2.
The modules with solid-lined shapes have been already
implemented, while the ones with dashed-lines shapes
have not been developed yet. In this study, we focus on
the improvement the “ResonanceReconstructor” module
to treat the R-matrix limited formula.
The current version of FRENDY only treats the
ENDF-6 format. Recently, the introduction of the new
nuclear data format GNDS (Generalized Nuclear Data
Structure) has been considered [14]. This format uses
XML format and it is quite different from the current
ENDF-6 format. The difference of the nuclear data format
has no large impact on the processing since FRENDY
converts the nuclear data format from each nuclear data
format to “NuclearDataObject”. FRENDY can treat sucha new nuclear data format when parser, writer andconverter modules are implemented.
FRENDY accepts two types of input formats. One is
the NJOY compatible format and the other is the original
input format [15]. The original input format requires only
a processing mode and the file name of an evaluatednuclear data library at minimum. Therefore, everyone canprocess the nuclear data library without expert knowledgeof the nuclear data processing.

Resonance
Reconstructor

NuclearData
ObjectDoppler
Broader

AceData
Generator

Gas
Production
CrossSection
Calculator

AceDataObject

ThermalScattering
DataProcessor
Unresolved
Resonance
DataProcessor

AceDataParser/Writer

Implemented
ACE
formatNot
Implemented

Fig. 2. The system structure of FRENDY.
3. Verification of FRENDY
The processing results of FRENDY are compared to thoseof NJOY2016 [10] to verify calculation of R-matrix
limited formula. The all nuclei which use R-matrix
limited formula in ENDF/B-VIII.0 and JEFF-3.3 are used
for comparison.
The processing results of FRENDY and NJOY2016
and its comparison for 35Cl from ENDF/B-VIII.0 are
shown in Figs. 3-7. As shown in Figs. 3-7, the differenceis only found in the discontinuity region where theboundary of the resolved and unresolved resonanceregions. Since the difference is only found in small energy
region and the maximum relative difference is less than afew percent, this difference has not impact on theneutronics calculations. As shown in Figs. 3-7, thedifference in the other region is so small. These results
indicate that the processing results of FRENDY show
good agreement with those of NJOY2016.
The R-matrix limited formula can treat not only total,

2.2 Overview of AMUR
AMUR is developed in order to evaluate the cross-section230

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019elastic scattering, fission, and radiative capture crosssections but also other reactions. 35Cl from ENDF/BVIII.0 has resonance parameter for (n, p0) cross section.
As shown in Fig. 6, the relative differences of the (n, p0)
cross section is similar to those of other reactions. This
result indicate that FRENDY appropriately treat the all
reaction prepared in the R-matrix limited formula.
35
Cl from ENDF/B-VIII.0 prepares the (n, p) crosssection. The (n, p) cross section is the total proton
production cross section, i.e., the sum of (n, pn) crosssection. The (n, p) cross section must be modified whenthe (n, p0) cross section is calculated. As shown in Fig. 7,
FRENDY appropriately treat the total proton production
cross section.
Other nuclei which use R-matrix limited formula in
ENDF/B-VIII.0 and JEFF-3.3 are also show similardifference. These results indicate that FRENDY
appropriately calculate the R-matrix limited formula.
The processing time of FRENDY is also compared tothat of NJOY2016. Table I shows the comparison of theprocessing time and the energy grid number. As shown in
Table I, the processing time of FRENDY is similar to thatof NJOY2016 in many nuclei. However, FRENDY
requires long computational time to process 40Ca and 57Fe
from ENDF/B-VIII.0. The energy grid number of
FRENDY is similar to that of NJOY. Therefore, thedifference of energy grid number is not cause of the longprocessing time. The improvement of the processing timeof these nuclei is now undergoing.

Fig. 5. The comparison of the radiative capture crosssection for 35Cl from ENDF/B-VIII.0.

Fig. 6. The comparison of the (n, p0) cross section for 35Cl
from ENDF/B-VIII.0.

Fig. 7. The comparison of the (n, p) cross section for 35Cl
from ENDF/B-VIII.0.
Table I. Comparison of the processing time and energy
grid number.
Calculation time

Fig. 3. The comparison of the total cross section for 35Cl
from ENDF/B-VIII.0.

FRENDY NJOY

F/NFRENDY NJOY

Cl

63.4

49.7

1.3

22,844 22,656

Ca

35
40

B80

F33

Fig. 4. The comparison of the elastic scattering crosssection for 35Cl from ENDF/B-VIII.0.

Energy grid

221.2

19.9

11.1

27,861 28,376

54

13.2

23.8

0.6

49,311 43,038

57

Fe

41.3

6.8

6.1

16,993 16,380

63

Cu

17.9

26.0

0.7

54,931 43,582

65

Cu

13.6

19.5

0.7

44,502 35,794

63

Cu

41.2

57.4

0.7

90,334 93,973

65

Cu

35.6

44.9

0.8

105,670 81,380

Fe

4. Conclusions and future works
The processing of R-matrix limited formula isimplemented in FRENDY. AMUR is used to process the
R-matrix limited formula.
The processing results of FRENDY are compared to231Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

those of NJOY2016 to verify FRENDY. The processing
results of FRENDY show good agreement with those of
NJOY2016. This result indicate that FRENDY
appropriately process the R-matrix limited formula. The
processing time of FRENDY is similar to that of NJOY in
many nuclei.
The current version of FRENDY only calculate thecross-sections. The R-matrix limited formula can treat not
only the cross-section but also the differential crosssections, i.e., angular and energy distribution of thesecondary particle. The calculation function of thedifferential cross-sections are developed in the near future.

14. Mattoon C. M., Beck B. R., Patel N. R., et al.,"Generalized Nuclear Data: A New Structure (withSupporting Infrastructure) for Handling Nuclear
Data," Nucl. Data Sheets, 113, 3145 (2012).
15. Tada K., Kunieda S., Nagaya Y., "Nuclear Data
Processing Code FRENDY Version 1," JAEAData/Code 2018-014, Japan Atomic Energy Agency(2019).

References
1. Tada K., Nagaya Y., Kunieda S., et al., "Development
and verification of a new nuclear data processing
system FRENDY," J. Nucl. Sci. Technol., 54, 806(2017).
2. Tada K., "Improvement of Probability Table
Generation Using Ladder Method for a New Nuclear
Data Processing System FRENDY," Proc. PHYSOR
2018, Cancun, Mexico, Apr. 22–26, 2018, American
Nuclear Society (2018).
3. Trkov A., Herman M., Brown, D. A., "ENDF-6
Formats
Manual,"BNL-203218-2018-INRE,
Brookhaven National Laboratory (2018).
4. Breit G., Wigner E., "Capture of Slow Neutrons,"
Phys. Rev., 49, 519 (1936).
5. Reich C. W., Moore M. S., "Multilevel Formula for
the Fission Process," Phys. Rev., 111, 929 (1958).
6. Brown D. A., Chadwick M. B., Capote R., et al.,"ENDF/B-VIII.0: The 8th Major Release of the
Nuclear Reaction Data Library with CIELO-project
Cross Sections, New Standards and Thermal
Scattering Data," Nucl. Data Sheets, 148, 1 (2018).
7. Cabellos .O, Alvarez-Velarde F., Angelone C. J., etal., "Benchmarking and Validation Activities within
JEFF Project," EPJ Web of Conf., 146, 06004 (2017).
8. Kunieda S., "Status of the R-matrix Code AMUR
toward a consistent cross-section evaluation andcovariance analysis for the light nuclei," EPJ Web ofConf., 146, 12029 (2017).
9. Wigner E. P., Eisenbud L., "Higher AngularMomenta and Long Range Interaction in Resonance
Reactions," Phys. Rev., 72, 29 (1947).
10. MacFarlane R. E. and Kahler A. C., "The NJOY
Nuclear data Processing System Version 2016," LAUR-17-20093, Los Alamos National Laboratory(2017).
11. Shibata K., Iwamoto O., Nakagawa T. et al., "JENDL4.0: A New Library for Nuclear Science andEngineering," J. Nucl. Sci. Technol., 48, 1 (2011).
12. X-5 Monte Carlo Team, "MCNP – A General Monte
Carlo N-Particle Transport Code, Version 5," LA-UR03-1987, Los Alamos National Laboratory (2003).
13. Sato T., Iwamoto Y., Hashimoto S., et al., "Featuresof Particle and Heavy Ion Transport code System(PHITS) version 3.02," J. Nucl. Sci. Technol., 55, 684(2018).

232

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Validation of new Fission Yield by Analysis of Post-Irradiation Examination
Atsuya Isoa,, Satoshi Takedaa and Takanori Kitadaa
Osaka University, Yamadaoka2-1, Suita, Osaka, Japan*
Corresponding author: a-iso@ne.see.eng.osaka-u.ac.jpaAbstract
Fission yield is an important physical quantity to evaluate the amount of fission
products after irradiation. Tokyo Institute of Technology (Tokyo Tech) newly
evaluated fission yield. It is necessary to confirm the validity of the fission yield
evaluated by Tokyo Tech because the fission yield is expected to be included in
the next Japanese Evaluated Nuclear Data Library (JENDL). In this study, theanalysis of post-irradiation examination using JENDL FP Fission Yields Data File2011 (FPY-2011) and new fission yield was performed. The integrated burnup
calculation code system SWAT4.0 was used for the analysis. BM5 sample of the
Swiss Beznau-1 PWR was used in this study. It was found that the C/E values
using new fission yield for Sm151, Sm152, and Eu151 are improved more than
10 %. The improvement of the C/E value of Sm151, Sm152 and Eu151 mainly
come from the updating fission yield of Pu241.
Key Words: Fission yield, FPY-2011, analysis of post-irradiation examination,
SWAT4.0

1. Introduction

core is Swiss Beznau-1 PWR. BM5 sample measured by

Fission yield is an important physical quantity to evaluate
fission products after irradiation. The fission yield used inJapanese Evaluated Nuclear Data Library (JENDL)
employs ENDF/B data. The fission yield has recently
been evaluated in Japan based on the various theories.
Tokyo Institute of Technology (Tokyo Tech) newly
evaluated fission yield [1]. It is necessary to confirm thevalidity of the new fission yield because this fission yield
is expected to be included in the next JENDL. In thisstudy, the analysis of Post-Irradiation Examination (PIE)using the fission yield evaluated by Tokyo Tech and
JENDL FP Fission Yields Data File 2011 (FPY-2011) [2]that is the latest fission yield data in Japan was performed
and validation of new fission yield was checked.

about 60 days, and the final burnup was about 56.6GWd/t
with the consideration of the cooling time between
irradiation cycles. Figure 1 shows the position of BM5

Studiecentrum voor Kernenergie-Centre d'étude del'Energie Nucléaire (SCK.CEN) was analyzed. The
sample was chosen because its measurement error isrelatively small among samples. BM5 sample wasobtained from M308 assembly and this assembly has adiagonal symmetry. Burnup periods were separated bysample in M308 assembly and Table Ⅰ summarizes
initial composition of BM5 sample. Table Ⅱ shows
calculation condition of MVP.

2. Calculation Condition
The burn-up calculation was performed by SWAT4.0 [3]using two data of FPY-2011 and the fission yield
evaluated by Tokyo Tech, and the calculation results iscompared with the experimental value. The continuous
energy Monte Carlo code MVP version 3 was used as aneutron transportation solver. JENDL-4.0 was used in the
MVP calculation. In the analysis, the fission yield of
ORLIBJ40 [4] was replaced by using FPY-2011 and thefission yield evaluated by Tokyo Tech. Fission yields ofTh232, U233, U235, U238, Pu238, Pu241, Cm245, andCf249 are used because ORLIBJ40 has fission yields ofthese 8 fissile nuclides. The PIE data of ARIANE
program was used [5]. In the ARIANE program, target233

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019
A
B
C
D
E
F
G
H
I
J
K
L
M
N
1

2

3

4

5

6

7

8

9 10 11 12 13 14

Guide Tube

Fig. 3. C/E values of fission products

Instrumentation Tube
MOX High Pu Content

The results of actinides and fission products for FPY-2011
and fission yield evaluated by Tokyo Tech are almost thesame as shown in figs 2 and 3. The results of the fission
products with mass number 90 to 137 are also nearly thesame for two fission yields. The C/E values using thefission yield evaluated by Tokyo Tech are improved over
10 % for Sm151, Sm152, and Eu151. These nuclides are
known for nuclides that those absorption cross sections
are not negligible and affect the criticality as described inreference [6]. It was found that the fission yields of
Sm151, Sm152, and Eu151 are improved in the fission
yield evaluated by Tokyo Tech compared to FPY-2011.

MOX Intermediate Pu Content
MOX Low Pu Content
BM5 sample

Fig. 1. Position of the sample obtained from the fuelassembly
Table I. Initial Composition of BM5 Sample

Nuclides
U234
U235
U238
Pu238
Pu239

Composition
(wt%)
0.002
0.218
94.2320.033
3.666

Nuclides
Pu240
Pu241
Pu242
Am241

Composition
(wt%)
1.287
0.364
0.147
0.051

4. Discussion
The reason for the improvement of the results of Sm151,
Sm152, and Eu151 is considered in this chapter. Table Ⅲ
shows the amounts of productions of Sm151, Sm152, andEu151 and the relative differences between the results of
FPY-2011 and the new fission yield evaluated by Tokyo
Tech.

Table Ⅱ. Calculation Condition of MVP
Number of History per Batch
5000
Total Number of Batch
220
Number of Skip Batch
20
3. Calculation Results

Table Ⅲ. Amounts of Production and Relative
Differences between two Fission Yields

Based on the calculation condition mentioned in previous
chapter, burn-up calculation was carried out for the fuelassembly. Figures 2 and 3 show C/E values of the amountof actinides and fission products. Error bar in figs 2 and 3shows experimental error and this confidence level is
95 %.

NuclidesFPY-2011① (g/tU)

Fission Yield evaluated by
Tokyo Tech ② (g/tU)

Relative
Differences(②/①-1)(%)

Sm151

23.60

27.15

15.1

Sm152
Eu151

177.82
0.63

198.27
0.72

11.5
15.1

Figure 4 shows relative differences in productions of
Sm151, Sm152, and Eu151 when the yield of U235, U238,
Pu239, or Pu241 is changed from FPY-2011 to new
fission yield. It was found that fission yield of Pu241
mainly affects the production of Sm151, Sm152, andEu151. As an example, the fission yield of Sm151 and its
parent nuclides produced from Pu241 was shown in Table
IV. The relative difference between the fission yields for
Ce151 is more than 700% and their differences affect the
improvement of the C/E value. Therefore, relative
differences shown in Table Ⅲ mainly come from thedifference of the fission yield of Pu241.
Fig. 2. C/E values of actinides

234

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019improvement of the C/E value of Sm151, Sm152, andEu151 is mainly due to the update of the fission yield of
Pu241.
References
1.
2.
3.
(a) Sm151

4.

5.
(b) Sm1526.

(c) Eu151
Fig.4. Relative differences in production of Sm151,
Sm152, and Eu151 when yield of U235, U238, Pu239,
or Pu241 is changed from FPY-2011 to new fission yield
Table IV. Comparison between FPY-2011 and fission
yield evaluated by Tokyo Tech
FPY-2011 ① (%)

Fission Yield evaluated by
Tokyo Tech ② (%)

La151

8.10E-03

1.09E-02

34.9

Ce151

3.63E-02

3.04E-01

737.5

Relative Differences (②/①-1)(%)

Pr151

5.15E-01

5.31E-01

3.2

Nd151

8.80E-02

6.68E-02

-24.1

Sm151

2.58E-07

1.34E-07-47.9

5. Conclusion
The results for FPY-2011 and fission yield evaluated by
Tokyo Institute of Technology are almost the same. As for
Sm151, Sm152, and, Eu151, compared to FPY-2011, it
was found that the C/E values are improved over 10 % byusing the fission yield evaluated by Tokyo Tech. The

235

Tsubakihara K et al. To be submitted to JNST; S.
Chiba, private communication.
Katakura J, “JENDL FP Decay Data File 2011 andFission Yields Data File 2011,” JAEA-Data/Code
2011-025, Japan Atomic Energy Agency (2011).
Kashima T, Suyama K and Takada T, “SWAT4.0 The Integrated Burnup Code System Driving
Continuous Energy Monte Carlo Codes MVP,MCNP and Deterministic Calculation Code SRAC,”
JAEA-Data/Code 2014-028, Japan Atomic Energy
Agency (2014).
Okumura K, Sugino K, Kojima K, Jin T, Okamoto T,
and Katakura J, “A Set of ORIGEN2 Cross Section
Libraries Based on JENDL-4.0: ORLIBJ40,” JAEAData/Code 2012-032, Japan Atomic Energy Agency(2012).
Primm R, “ARIANE International Programme Final
Report,” ORNL/SUB/97-XSV750-1, Oak Ridge
National Laboratory (2003).
Parks C, DeHart M, and Wagner J, “Review andPrioritization of Technical Issues Related to Burnup
Credit for LWR Fuel,” NUREG/CR-6665,
ORNL/TM-1999/303,
Oak
Ridge
National
Laboratory (2000).

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Continuous Thermal Neutron Scattering Data Processing Capability in RMC CodeZHENG Leia , WANG Kana , WANG Weiaa
Department of Engineering Physics, Tsinghua University, 1st Qinghuayuan, Haidian District,Beijing, China*
Corresponding author: zl17@mails.tsinghua.edu.cn*

Abstract
A new continuous representation of the ACE format thermal neutron scattering
data was released in 2007, this representation gives more accurate description forthe secondary neutron energies, but was not widely known. In this paper, thecontinuous representation was introduced and compared with the traditional
discrete one, the continuous thermal neutron scattering data processing capability
was also developed in RMC code. The capability was validated by a set ofbenchmarks, of the 63 testing cases, only 2 cases show difference a bit more than
3σ for the discrete representation and the continuous representation. Results showthat a change from the traditional discrete representation to the continuous one
does not produce noticeable changes for the eigenvalues when calculating criticalexperiments.
Key Words: Continuous Thermal Neutron Scattering Data, RMC,
Benchmarks, ENDF/B-Ⅷ.0
representation of the thermal neutron scattering data wasprovided by Los Alamos National Laboratory (LANL). It
is therefore necessary to use the more physical andrigorous thermal neutron scattering data representation
for the high fidelity neutronics calculation. This paper
introduces the new continuous representation briefly, theimplementation of the continuous thermal neutron
scattering data processing capability in Reactor Monte
Carlo code RMC [10], and the verification & validation
results.

1. Introduction
In the thermal energy range (<10eV), neutron scattering
events can occur resulting in a gain or loss in energy frominteraction with the target material. Meanwhile, theneutron scattering data is affected by the chemical bindingof the target to other nuclides in a material or crystalline
structure effects. All these effects complicate the physics
of the scattering process and the calculation of thescattering data. To provide high fidelity Monte Carloneutronics calculations for thermal reactors, A Compact
ENDF (ACE) format thermal neutron scattering dataprocessed through NJOY [1] is always used.
Since the thermal scattering data is complicated, large
amount of computer memory is need to store all thedetailed thermal neutron scattering information, a discrete
representation of the secondary neutron spectrum andangular distribution has been used since the early 1980s
due to the limited storage capabilities of computers [2].
The use of the discrete representation does not produce
noticeable effects in integral calculations such as keffeigenvalues [3], but can produce unphysical phenomena[4] and noticeable deficiencies for differential
calculations [5].
A new continuous ACE representation of the thermal
neutron scattering data was firstly created by Los Alamos
National Laboratory in 2007 and implemented in MCNP
in 2008 [6].This representation has been available online
since 2007, but was not widely known. In 2012 [7] and2014 [8], continuous ACE representation of thermal
neutron scattering data was released subsequently basedon ENDF/B-Ⅶ.0 and ENDF/B-Ⅶ.1 libraries. In thelasted ENDF/B-Ⅷ.0 [9], only the continuous ACE

2. Continuous Thermal Neutron Scattering Data
From the traditional discrete representation of thermal
neutron scattering data, the secondary neutron is limited
to discrete energies and angles, which is not the real casefor the scattering kernel. While the continuous
representation allows users to process the secondary
neutron energies in a continuous manner, with a memorystorage consumption increasing of ~10 times larger thanthe previous discrete one [7].
In order to obtain the continuous representation
thermal neutron scattering data, the aceth module in
NJOY has been modified to convert the secondary energy
distributions from thermr into probability density
functions (PDFs) and cumulative density functions (CDFs)
form, PDFs and CDFs are packed into the big inelasticarray of the ACE format thermal scattering data [3]. The
continuous representation contains a table of incident
energies, a table of inelastic cross sections, a table ofpointers to the start of each distributions, a table of thenumber of points in each distribution and the secondary
energy and angle distributions. Detailed formats and data236

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

blocks of the discrete and continuous representation for
the inelastic thermal scattering data are given in Fig. 1. It
can be noted that two extra data blocks as well as
PDFs/CDFs for secondary energy distributions are added
to the continuous representation.

elastic cross section, 1 is the random number.

1  inel inel   el

(1)(2) Search the location of the incident energy Ein in
the energy table, and calculate the energy fractionr . Ei and Ei 1 are the energy grids in theincident energy table.
Ei  Ein  Ei 1(2)
Ein  Ei  r  Ei 1  Ei (3) Chose the closest energy as the “incident energy”,
if r > 0.5, Ei is chosen as the “incident energy”,
otherwise, Ei 1 is chosen.
(4) Sample random number  2 , search CDFs todetermine the outgoing energy grid, cl ,k andcl ,k 1 are the CDFs corresponding to the incident
energy.
Fig. 1 Discrete (left) and continuous (right) represen
tation for inelastic thermal scattering data.

cl ,k  2  cl ,k 1(3)(5) Calculate the outgoing energy E and theinterpolation fraction ri for outgoing angle. El ,k'

To obtain the continuous ACE representation thermal
neutron scattering data, the weighting option (iwt flag) incard 9 of the ACER module should be set to 2 (tabulated)when using NJOY to process the thermal neutron
scattering data.

and El ,k 1 are the sampled outgoing energy grids
corresponding to  2 , pl ,k and pl ,k 1 are the PDFs .
pl ,k 1  pl ,k pl ,k |El ,k 1  El ,kpl ,k 1  pl ,kEl ,k 1  El ,k| pl2,k  2  2  cl ,k 
E '  El ,k 

3. Implementation of Continuous Thermal Neutron
Scattering Data Processing Capability(4)

E '  El ,k(5)
El ,k 1  El ,k 1(6) Sample random number  3 , and search theoutgoing angle grid. NIL is the inelastic
dimensioning parameter.
(6)j  int  NIL  1  3 ri 

In the thermal energy range, neutrons will scatter
coherently or incoherently with target atoms dependingon the material. Inelastic scattering is important for all
thermal materials, coherent elastic scattering is of great
importance for powdered crystalline materials, andincoherent elastic scattering is significant for partiallyordered materials such as polyethylene [11].Each thermalmaterial will have inelastic scattering and may have either
coherent or incoherent scattering, but not both.
The continuous thermal neutron scattering dataprocessing capability was developed in RMC code. For agiven incident energy, the treatment starts with thesampling of scattering type depending on the crosssections of elastic scattering and inelastic scattering, thenthe closest energy in the incident energy table is matched,
the cumulative distribution function data is searched forthe random sample, the PDFs and CDFs are used to
calculate the outgoing energy. Outgoing angle is thensampled and calculated to the outgoing energy. Next, theoutgoing energy is projected up or down to correspond to
the actual incident energy. The projection scheme is much
faster than the real interpolation of incident energies, but
still produces relatively accurate results since the incident
energy points are enough [3]. The calculation steps areshown as below:(1) Sample the scattering type for incident energy, inel and  el are the inelastic cross section and the(7) Calculate the outgoing angle out , l ,k , j is theoutgoing angle grid corresponding to theoutgoing energy.
(7)
out  l ,k , j  ri  (l ,k 1, j  l ,k , j )  d  (24 1)
d  min  d1 , d 2 

d1 

(8)

( l ,k , j  l ,k , j 1 )  ri   l ,k 1, j  l ,k 1, j 1    l ,k , j 1  l ,k , j  (9)( l ,k , j 1  l ,k , j )  ri   l ,k 1, j 1  l ,k 1, j    l ,k , j  l ,k , j 1  

(10)

d2 

2

2

(8) Project the outgoing energy, and calculate thesecondary energy. The down projection and up
projection schemes are given in Eq. (11) and (12).
 ' 2 Ein  Ei ', E  0.5Ei
E Ei
Eout   E '  E  E , E '  0.5E
inii ' 2 Ein  Ei 1 ', E  0.5 Ei 1
E 
Ei 1
Eout   E '  E  E , E '  0.5E
in
i 1
i 1


237

(11)
(12)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4. Verification and Validation
To evaluate the validity of the continuous thermal neutron
scattering capability and the effects of the discrete andcontinuous representation, a set of 63 benchmarks fromthe International Handbook of Evaluated Criticality
Safety Benchmark Experiments (ICSBEP) [12] wasselected. The suite is divided into five categories
depending on the major fissile material, the neutron
spectrum can be divided into fast spectrum, intermediate
spectrum and thermal spectrum according to its
contribution to fission rate. General information for thesuite is given in Table Ⅰ.

Case name
umf5-1

Thermalmaterial
Be

Experimen
t keff1.0000(30)Continuous keff
0.9975(1)Discretekeff
0.9976(1)umf5-2Be

1.0000(30)0.9977(2)0.9972(2)usi1-1H2O/poly

1.0000(83)0.9872(2)0.9872(2)ust1-1H2O/poly

1.0000(31)1.0011(1)1.0012(1)ust1-2H2O/poly

1.0005(33)1.0010(1)1.0009(1)ust1-3

H2O/poly

1.0006(33)1.0006(1)1.0007(1)ust1-4

Be/H2O

0.9998(33)1.0006(1)1.0006(1)ust1-5

H2O

0.9999(33)1.0001(1)1.0001(1)

ust-8

H2O

1.0006(29)1.0012(1)1.0011(1)Case namehmf4-1

Thermalmaterial
H2O

Experimen
t keff
1.0020(10)Continuous
keff
1.0056(2)Discretekeff
1.0052(2)hmf4-2H2O

0.9985(10)1.0014(2)1.0013(2)hmf9-2BeO

0.9992(15)0.9956(1)0.9953(1)hmf-11Poly

0.9989(15)0.9984(2)0.9982(1)hmf192
hmf202
hmf2611hmi6-1Grph

1.0000(28)1.0006(1)1.0006(1)Poly

1.0000(28)1.0004(1)1.0004(1)Poly

1.0000(38)1.0014(2)1.0018(2)Grph

0.9977(8)0.9989(2)0.9988(2)hmi6-2Grph

1.0001(8)1.0026(2)1.0024(2)hmi6-3Grph

1.0015(9)1.0043(2)1.0042(2)hmi6-4Grph

1.0016(8)1.0062(2)1.0060(2)hst13-1

H2O

1.0012(26)0.9982(1)0.9983(1)hst13-2

H2O

1.0007(36)0.9977(1)0.9978(1)hst13-3

H2O

1.0009(36)0.9946(1)0.9947(1)hst13-4

H2O

1.0003(36)0.9962(1)0.9962(1)

hst32

H2O

1.0015(26)0.9994(1)0.9994(1)Case name
imf4-2

Thermal m
aterialGrph

Experimen
t keff1.0000(30)Continuous keff1.0049(1)ict2-3H2O

1.0017(44)1.0063(2)

Case
name
lst2-1

Thermalmaterial
H2O

Experiment
keff
1.0004(26)Continuous
keff
0.9997(1)Discretekeff
0.9996(1)lst2-2

H2O

1.0003(32)0.9956(1)0.9954(1)lst7-14

H2O

0.9961(9)0.9955(1)0.9953(1)lst7-30

H2O

0.9973(9)0.9978(1)0.9977(1)lst7-32

H2O

0.9985(10)0.9965(1)0.9966(1)lst7-36

H2O

0.9988(11)1.0165(1)1.0167(1)lst6-49

H2O

0.9983(11)0.9977(1)0.9977(1)Table Ⅰ. General information for the validation suite.
Fissile mat
erial
233
U
HEU
IEU
LEU
Pu
Total

Fast
2
7
1
0
7
17

Number of benchmarks
intermediate
Thermal
1
6
4
5
0
1
0
19
1
9
6
40

Total
9
16
2
19
17
63

Thermal materials covers light water, 10%-porosity
graphite [13], beryllium oxide, polyethylene, andberyllium metal. The latest thermal neutron scattering
data library and neutron data library from ENDF/B-Ⅷ.0
were employed, version 2016.20 [14] of NJOY was used
to produce the ACE format neutron data and discrete
representation thermal neutron scattering data, thenumber of exit energy grids for the discrete representation
is 64. The continuous representation thermal scattering
data was taken from LANL.
The 63 benchmarks using the continuous
representation and discrete representation data were runwith RMC code, version 2.5.7. All cases were run in thecriticality mode, using 100 inactive cycles, 700 additional
active cycles, and 30000 neutrons per cycle. Uncertainties
of k-effective were less than 20pcm for all cases.
5. Results
Results of the calculated keff values are given in Table Ⅱ.
to Ⅵ. Keff values of the discrete representation and thecontinuous representation agreed within statistics formost cases: 40 of 63 cases showed difference not morethan 1σ, 49 of 63 cases showed difference not more than
2σ, with 9 cases differed by more than 3σ.
Detailed examination is conducted for cases: hmf9-2,
lct8-2, lct8-5, lct8-7, lct8-11, pmf21-1, pst9-3a, pst18-9.
It is observed that uncertainties of these cases are near but
a bit smaller than 15pcm, in the final comparsion, these
uncertainties are treated as 10pcm because of rounding,resulting in the overestimated deviation. 3 of these 7 cases
showed difference less than 2σ, 4 of these 7 cases differedby less than 3σ using the real uncertainties. Finally, only
2 of 63 cases differed by a bit more than 3σ using the real
uncertainties, which is not noticeable.
Table Ⅱ. 233U benchmark results

238

Δkeff
0.0001(1)0.0005(3)0.0000(3)0.0001(1)0.0001(1)0.0001(1)0.0000(1)
0.0000(1)
0.0001(1)

Table Ⅲ. HEU benchmark results

Δkeff
0.0004(3)0.0004(3)0.0003(1)0.0002(3)0.0000(1)
0.0000(1)
0.0004(3)0.0001(3)0.0002(3)0.0004(3)0.0002(3)0.0001(1)0.0001(1)0.0001(1)0.0000(1)
0.0000(1)

Table Ⅳ. IEU benchmark results

Discretekeff
1.0049(1)1.0068(2)Δkeff
0.0000(1)
0.0005(3)

Table Ⅴ. LEU benchmark results

Δkeff
0.0001(1)0.0002(1)0.0002(1)0.0001(1)0.0001(1)0.0002(1)0.0000Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019lct8-1H2O

1.0007(16)1.0026(1)1.0033(1)lct8-2

H2O

1.0007(16)1.0031(1)1.0034(1)lct8-5H2O

1.0007(16)1.0019(1)1.0023(1)lct8-7H2O

1.0007(16)1.0013(1)1.0018(1)lct8-8H2O

1.0007(16)1.0008(1)1.0015(1)lct8-11

H2O

1.0007(16)1.0034(1)1.0039(1)mct2-pnl30
mct2-pnl31
mct2-pnl32
mct2-pnl33
mct2-pnl34
mct2-pnl35

H2O

1.0024(60)1.0005(2)1.0005(2)H2O

1.0009(47)1.0015(2)1.0023(2)H2O

1.0042(31)1.0014(2)1.0013(2)H2O

1.0024(24)1.0058(2)1.0059(2)H2O

1.0038(25)1.0023(1)1.0024(2)H2O

1.0029(27)1.0054(1)1.0056(2)

Case
name
pmf11

Thermalmaterial
H2O

Experiment
keff
1.0000(10)Continuous
keff
1.0011(2)Discretekeff
1.0012(2)

pmf18

Be

1.0000(30)0.9985(1)0.9984(1)

pmf19

Be

0.9992(15)1.0001(1)0.9999(1)pmf211
pmf212
pmf232
pmf242
pci1

Be1.0000(26)1.0042(1)1.0038(1)BeO

1.0000(26)0.9929(1)0.9928(1)Grph

1.0000(20)0.9986(1)0.9986(1)Poly

1.0000(20)1.0014(1)1.0015(1)Grph

1.0000(11)1.0092(1)1.0093(1)pst9-3aH2O

1.0000(33)1.0129(1)1.0126(1)pst1116-1
pst1116-5
pst1118-1
pst1118-6
pst18-9

H2O

1.0000(52)1.0028(2)1.0028(2)H2O

1.0000(52)0.9993(2)0.9993(2)H2O

1.0000(52)0.9871(2)0.9873(2)H2O

1.0000(52)0.9930(2)0.9931(2)H2O

1.0000(34)0.9998(1)0.9995(1)pst21-1

H2O

1.0000(32)0.9981(2)0.9980(2)pst21-3

H2O

1.0000(65)1.0005(2)1.0001(2)

pst34-1

H2O

1.0000(62)0.9968(2)0.9962(2)Acknowledgements(1)0.0007(1)0.0003(1)0.0004(1)
0.0005(1)0.0007(1)0.0005(1)0.0000(3)0.0008(3)0.0001(3)0.0001(3)0.0001(3)0.0002(3)This work was supported by the Science Challenge
Project, No. TZ2018001.
References1.

Macfarlane RE, and Kahler AC, “Methods for
Processing ENDF/B-VII with NJOY,” Nucl. Data
Sheets, 111, 12, 2739 (2010).
2. MacFarlane RE, “A discussion of the MCNP
Thermal Scattering Model,” (March 22 2005).
3. Pavlou AT, Brown FB, Martin WR, et al,“Comparison of discrete and continuous thermal
neutron scattering treatments in MCNP5,” Proc.
PHYSOR 2012, LaGrange Park, IL, April 15–20,
2012, American Nuclear Society (2012).
4. Conlin JL, Parsons DK, Brown FB, et al,“Continunous S(α,β) Capability in MCNP,” Proc.
ANS Annual Meeting 2012, Chicago, IL June 24–28,
2012.
5. Cullen DE, Hansen LF, Lent EM, et al, “Thermal
Scattering Law Data: Implementation and Testing
using the Monte Carlo neutron transport codes COG,
MCNP and TART,” UCRL-ID-153656, Los Alamos
National Laboratory (2003).
6. Booth TE, Brown FB, Bull JS, et al, “MCNP5 1.50
Release Notes,” LS-UR-08-2300, Los Alamos
National Laboratory (2008).
7. Parsons DK and Conlin JL, “Release of Continuous
Representation for S(α,β) ACE Data,” LA-UR-1200800, Los Alamos National Laboratory (2012).
8. Parsons DK and Conlin JL, “Release of Continuous
Representation for S(α,β) ACE Data,” LA-UR-1421878, Los Alamos National Laboratory (2014).
9. Brown DA, Chadwick MB, Capote R, et al,“ENDF/B-VIII.0: The 8th, Major Release of the
Nuclear Reaction Data Library with CIELO-project
Cross Sections, New Standards and Thermal
Scattering Data,” Nucl. Data Sheets, 148, 1 (2018).
10. Wang K, Li Z, She D, et al, "RMC – A Monte Carlocode for reactor core analysis." Ann. Nucl. Energy,
82, 121 (2015).
11. Trkov A, Herman M, Brown DA, et al, “ENDF-6
Formats Manual: Data Formats and Procedures for
the Evaluated Nuclear Data Files ENDF/B-VI,
ENDF/B-VII and ENDF/B-VIII”, BNL-2032182018-INRE, Cross Sections Evaluation Working
Group (2018).
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Benchmark Evaluation Project." NEA/NSC/DOC(95)03, OECD Nuclear Energy Agency (2011).
13. Parsons DK, “NJOY Processing of ENDF/B VIII.0
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Table Ⅵ. Pu benchmark results

Δkeff
0.0001(3)0.0001(1)0.0002(1)0.0004(1)
0.0001(1)0.0000(1)
0.0001(1)0.0001(1)0.0003(1)0.0000(1)
0.0000(1)
0.0002(3)0.0001(3)0.0003(1)0.0001(3)0.0004(3)0.0006(3)

6. Conclusions
The continuous ACE representation of the thermal
neutron scattering data was introduced in this paper, theprocessing capability was also implemented in theReactor Monte Carlo code RMC. Of the 63 testing cases,
only 2 cases show difference a bit larger than 3σ for thediscrete representation and the continuous representation.
Results show that a change from the traditional discrete
representation to the continuous one does not produce
noticeable changes for the eigenvalues when calculating
critical experiments.

239

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Thermal neutron scattering data generation functionin the nuclear data processing code NECP-Atlas
Yongqiang Tanga, Tiejun Zua,, Siyu Yia, Jialong Xua, Liangzhi Caoa and Jin Yang b
School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, Chinab
College of Urban and Environmental Sciences, Hunan University of Technology, ZhuZhou, Hunan 412007, China*
Corresponding author: tiejun@xjtu.edu.cn (T. Zu)aAbstract
A new module called sab_calc was implemented into NECP-Atlas, a nuclear dataprocessing code developed at Xi’an Jiaotong University in China, to evaluate thethermal neutron scattering law (TSL), which describes the thermal scattering frombound moderators. Coupling this new module and thermal scattering cross section
calculation module, NECP-Atlas can provide accurate thermal scattering crosssection for various moderators used in nuclear reactors. In this paper, ZrH2 wasselected as an example to illustrate the function to generate thermal scattering laws
and cross sections in NECP-Atlas. The Finite Difference Method (FDM) wasapplied to calculate crystal force constants, which is transferred to a latticedynamics code Phonopy to calculate phonon density of states (DOS), which is thebasis to evaluate TSL data. The five isotopes of Zr in ε-phase ZrH2 were treatedindividually, then a new ACE format library using the evaluated ZrH2 TSL data inthis work was generated. Six benchmarks from International Criticality Safety
Benchmark Evaluation Project (ICSBEP) with ZrHx (x=1.9) moderator were
calculated using the Monte Carlo code, and the numerical results show that εphase ZrH2 can provide more precious thermal scattering cross section than theapproximated ZrHx in ENDF/B VIII.0.
Key Words: NECP-Atlas, Thermal neutron scattering, First-principles
calculations, ENDF/B VIII.0
newly developed to establish a platform to carry out
deeper researches on nuclear data processing methods.
All the different types of TSL and free gas model can beprocessed by therm_clac module in NECP-Atlas, andpoint-wise cross sections can be transformed into ACE
library by ace_outp module as described by Zu et al. [2].
However, NECP-Atlas didn’t have the function to
calculate the TSL data based on the first-principle
calculations. In this work, a new module called sab_calc
was implemented in NECP-Atlas to provide accurate
thermal scattering data for the materials without TSL data
in ENDF.
In this work, the zirconium hydride (ZrH2) moderator
was selected as the example to illustrate the function to
generate thermal scattering data in NECP-Atlas. Six
benchmark experiments with the ZrH2 were analyzed andevaluated to verify the precision of the thermal neutron
scattering data. The calculated results were comparedwith the benchmark results and experimental results.

1. Introduction
The performance of thermal neutron driven reactors
physics calculating is limited by the accuracy of thethermal scattering cross sections. In thermal energy
region, the wavelengths of neutrons are similar to theinteratomic distances in matter, so the neutron scattering
behavior in the energy region of atomic thermal motion isfundamentally different from high energy regions. And
neutron energies are similar to those of excitations inmatter, therefore, interactions between neutrons andmaterials will excite or deexcite the system’s energy
states[1]. The TSL data is used to describe the thermal
scattering from bound moderators in thermal energy
region. The TSL of several materials are given in theevaluated nuclear data file (ENDF). For these materials,
the thermal scattering cross sections are evaluated directly
using the TSL data. While, for the materials without TSL
data, the free gas model is used to evaluate the thermal
scattering cross sections, which will introduce large errors.
Therefore, a numerical method is applied to calculate the
TSL data based on the first principle, such as the LERPR
module in NJOY.
The nuclear data processing code NECP-Atlas is

2. Theory of Thermal Scattering Cross Section
From Born scattering theory and Femi’s Golden rule, thedouble differential scattering cross section can be derived

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019as:
  ( E  E' ,  ) E

1
2 k BTE

 coh S ( ,  )   inc Ss ( ,  )    

whereE   E  2  E E
Ak BT



E  Ek BT
The first term in Eq. (1) is coherent scattering in which
neutron waves scattered from different particles interface
with each other. The second term is incoherent scattering,where there are no interference effects between waves
scattered by different particles. Using incoherent
approximation, the coherent and incoherent inelastic canbe considered as a whole part, and formulas are listed asfollow.
Inelastic scattering:E
 ( E  E' ,  )  b
S s ( ,  ) 
2k B T E b   inc   coh 

3.1 Crystal Structure of ZrH2
Through systematically investigation on the structure of
Zr hydrides ZrHx (x =0.5, 1.5, 2)[4], ZrH2 (ε-phase) is
Fluorite-like structured and crystallizes in the tetragonal
I4/mmm space group. Thus, I4/mmm (c/a=1.25) structure
is taken into consideration.
The structure of the ZrH2 unit cell is shown in Fig. 1
with the Zr atoms colored green and the H atoms colored
blue. The structure has the constants a=b=3.54186 Å andc= 4.45270 Å, c/a = 1.257, which agree well with theprevious calculation by Zhu et al[4]. a=b=3.538 Å,
c=4.406 Å, c/a=1.25.
The calculations of the structural properties in thisstudy were carried with the Vienna ab initio simulation
package [5,6]based on Finite Difference Method (FDM)with plane-wave basis sets and the Projector augmented
wave (PAW) pseudopotential. The generalized gradientapproximation (GGA) of Perdew-Burker-Ernzerhof (PBE)
functional has been applied to evaluate exchangecorrelation functions.

S s is the asymmetric form of the scattering law:1  i  t  ( t ) e e dt 
2 
where tˆ is time in units of  / k BT , and the second
exponential can be expanded as a Taylor series known asphonon expansion,Ss ( ,  ) a central-force lattice dynamics model [3]. Generally,
zirconium hydride has several lattice structures dependent
on the hydrogen content. In this work, the approximation
in ENDF/B thermal scattering library was removed, andthe pure face centered tetragonal ε-phase ZrH2 wasinvestigated. Also, the five isotopes of Zr were treatedindividually, because of their different atomic mass ratios
and bound scattering cross sections.

e  (t ) 


e



n


  1 
 ( )e   / 2 e  i t d   
2  sinh(  / 2)n 0 n ! As seen, the PDOS  (  ) is the important needed input
to calculate the scattering law.
Incoherent elastic scattering (the same in all directions): 1  e4 wE  ( E  E )  b  
2  2 wE 
where the Debye-Waller integral w is computed fromthe PDOS as ( )w
e   / 2 d   2k T  sinh(  / 2)
B
The above equations calculating the thermal scattering
law are newly implemented in NECP-Atlas as a newmodule called sab_calc.

Fig. 1. The ZrH2 unit cell

3. Results
In this work, ZrH2 (ε-phase) was selected as an example
material to explain the workflow of producing the thermal
scattering cross sections. Six benchmarks from the
ICSBEP handbook were used to verify the thermal crosssections.
In ENDF/B VIII.0, the crystal structure of ZrHx isapproximated to a face-centered cubic structure based on

Fig. 2 Ab initio calculated ZrH2 dispersion relations241

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

3.3 Verifications with ICSBEP benchmark calculation
HEU-COMP-MIXED-003 benchmark experiment[9],with highly enriched fuel (approximately 96% U-235),
zirconium hydride moderator and beryllium reflector, wascalculated using ACE format libraries prepared by NECPAtlas. The Beryllium, Beryllium Oxide and HinH2O
thermal scattering libraries were generated by NECPAtlas, using the PDOS and atoms site position provided
by ENDF/B-VIII.0.The atomic densities of the mainmaterials
10

Inelastic Scattering Cross section (b)Fig. 3 Calculated partial phonon DOS of ZrH2
Fig. 2 shows the dispersion relations along the highest
symmetry points of the first reduced Brillouin zone,
calculated by the Phonopy package[7]. Fig. 3 shows thepartial phonon frequency distributions for H in ZrH2 andZr in ZrH2. This is consistent with the work of Zhu et al[4].
3.2 Thermal Scattering Cross section

1

0.1
1E-5

Zr-90
Zr-91
Zr-92
Zr-94
Zr-96

1E-4

0.001

0.01

0.1

1

Energy (eV)In ENDF/B VIII.0, Zr in ZrHx was treated as a natural
nuclide. Generally, natural zirconium is formed of fiveisotopes: Zr-90, Zr-91, Zr-92, Zr-94 and Zr-96. Table I
shows abundance and neutron scattering cross sections ofzirconium and their isotopes. Besides the calculatedpartial phonon DOS, atomic mass ratios and atom
scattering cross sections are important inputs to calculate
TSLs using the sab_module in NECP-Atlas. Therefore, inthis work, there five isotopes were treated individually.
Fig. 4 shows inelastic scattering cross section of Zr-90,
Zr-91, Zr-92, Zr-94 and Zr-95 at 300K generated bytherm_calc module in NECP-Atlas. Fig. 5 shows elastic
scattering cross section of Zr-90, Zr-91, Zr-92, Zr-94 andZr-95 at 300K generated by therm_calc module in NECPAtlas. Fig. 6 shows scattering cross section of the crystal
models for H in ZrH2 at 300K generated by NECP-Atlas.
Fig.7 shows the differences of HinZrH2 thermal
scattering cross sections between ENDF/B-VIII.0 TSL
and NECP-Atlas TSL in this work. The inelastic
scattering cross section generated by NECP-Atlas TSL
shows higher values at energy rangy of 0.002~0.1eV.

Fig. 4 Inelastic scattering cross section of Zr-90, Zr-91,
Zr-92, Zr-94 and Zr-95 at 300K

Elastic Scattering Cross section (b)

10

Zr-90
Zr-91
Zr-92
Zr-94
Zr-96

8

6

4

2

0
1E-5

1E-4

0.001

0.01

0.1

1

Energy (eV)Fig. 5 Elastic scattering cross section of Zr-90, Zr-91, Zr92, Zr-94 and Zr-95 at 300K

Table I. abundance and neutron scattering cross sectionsof zirconium and their isotopes[8]Coh xsa Inc xsb Scatt xsc
abundance
/barn
/barn
barn
Zr
--6.44
0.02
6.46
Zr-90
51.45%
5.1
0
5.1
Zr-91
11.22%
9.5
0.15
9.7
Zr-92
17.15%
6.9
0
6.9
Zr-94
17.38%
8.4
0
8.4
Zr-96
2.80%
3.8
0
3.8
notes
a bound coherent scattering cross section
b bound incoherent scattering cross section
c total bound scattering cross section

Inelastic
Elastic
Total

Cross section (b)

100

10

1
1E-5

1E-4

0.001

0.01

0.1

1

Energy (eV)Fig. 6 Thermal scattering cross section of the crystal
models for H in ZrH2 at 300K

242

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019
Inelastic in this work
Elastic in this work
Total in this work
Inelastic from ENDF/B VIII.0
Elastic from ENDF/B VIII.0
Total from ENDF/B VIII.0

Cross section (b)100can be easily found in ICSBEP handbook. The ratio of H
and total Zr atomic density is about 1.9, which isconsistent with pre-calculated ε-phase zirconium hydride.
Table II shows experimental, benchmark and calculated
keff by MCNP using different libraries. Compared to the
ZrHx TSL from ENDF/B VIII.0, the evaluated ZrH2 TSL
in this work performed about 100 pcm closer toexperiment values, while TSL from ENDF/B VIII.0
shows a better results in benchmark hcm003_05. The lasttwo columns show that Zr in ZrH2 treated with as fiveisotopes performed a slight improvement compared to Zr
treated as a natural nuclide.

10

1
1E-5

1E-4

0.001

0.01

0.1

1

Energy (eV)Fig. 7 A comparison of HinZrH2 thermal scattering crosssections between the ENDF/B-VIII.0 TSL and NECPAtlas TSL in this work

Table II Experimental, benchmark and calculated keff using different libraries.

hcm003_01
hcm003_02
hcm003_03
hcm003_04
hcm003_05
hcm003_06

experiment

ENDF/B VIII.0-TSL
Zr-nata

Atlas-TSL
Zr-natb

Atlas-TSL
Zr-isoc

1.00000
1.00000
1.00000
1.00000
1.00000
1.00000

0.99834(8)0.99905(8)0.99898(8)0.99885(8)0.99989(8)0.99886(8)0.99974(8)1.00028(8)1.00011(9)1.00033(8)1.00114(8)1.00026(8)0.99980(8)1.00017(8)1.00028(8)1.00035(8)1.00109(8)1.00026(8)notes
a. Zr in ZrHx was treated as a natural nuclide. Zr TSL in ENDF/B VIII.0 was used.
b. Zr in ZrH2 was treated as a natural nuclide. Zr TSL generated in this work was used.
c. Zr in ZrH2 was treated as five isotopes. Zr TSL generated in this work was used.
3rd ed., Cambridge University Press (2012).
2. ZU T et al., “NECP-Atlas: A new nuclear dataprocessing code,” Ann. Nucl. Energy 123, 153,
Elsevier Ltd (2019).
3. MACFARLANE R, “New Thermal Neutron
Scattering Files for ENDF/B-VI Release 2”(1994).
4. ZHU X et al., “Structure and thermodynamic
properties of zirconium hydrides by structure
search method and first principles calculations,”
Comput. Mater. Sci. 150 April, 77, Elsevier(2018).
5. KRESSE G, FURTHMÜLLER J, “Efficiency of abinitio total energy calculations for metals andsemiconductors using a plane-wave basis set,”Comput. Mater. Sci. 6 1, 15, Elsevier (1996).
6. KRESSE G, FURTHMÜLLER J, “Efficient iterative
schemes for ab initio total-energy calculations
using a plane-wave basis set,” Phys. Rev. B 54
16, 11169, American Physical Society (1996).
7. TOGO A, OBA F, TANAKA I, “First-principles
calculations of the ferroelastic transition between
rutile-type and CaCl2-typ SiO2 at highpressures,” Phys. rev. B 78 13, 134106 (2008).
8. SEARS V, “Neutron scattering lengths and crosssections,” Neutron News 3, 26 (1992).
9. OECD-NEA, “International Criticality Safety
Benchmark Evaluation Project” (2014).

4. Conclusions
A new module sab_calc was implemented in NECPAtlas to generate thermal scattering library. Starting fromfirst-principles calculations, the complete generation
workflow of thermal scattering laws and cross sections
was investigated. ZrH2 (ε-phase) was selected as an
example material to validate the workflow of thermal
cross sections generation. Zr in ZrH2 was separated to fiveisotopes. The numerical results of benchmark experimentfrom ICSBEP handbook shows a higher accuracy thermal
scattering library of ZrH2 was developed. The reason isthat for zirconium hydride with the ratio of H and total Zr
atomic density around 2, the lattice structure should be
pure face centered tetragonal ε-phase. The workflow ofproviding thermal neutron scattering data based on
NECP-Atlas is workable, and more validation will bedone in the future.
Acknowledgements
This research was supported by National Natural Science
Foundation of China (No. 11605128)
References
1. BUCKNALL D, Introduction to the Theory of
Thermal Neutron Scattering, 3rd edn., by G.L.
Squires, 3rd ed., in Contemporary Physics 53 6,

243

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Quantification of Effectiveness of Integral Data
Using Active Sub-Space in Nuclear Data TestingaDaichi Imazatoa*, Go Chibaa
Hokkaido University, Kita-13 Nishi-8, Sapporo, Japan*
Corresponding author: daiti@eis.hokudai.ac.jp

Abstract
We develop a quantification method of effectiveness of a set of integral data forverification of nuclear data. This study uses the singular value decomposition, theorthogonal projection and the concept of active sub-space. By using the proposed
method, we can quantify the independency of a set of integral data and the possibilityof verification of individual nuclear data by a set of integral data. In this study, we
consider fictitious homogeneous spherical critical assemblies: bare assemblies andwater-reflected assemblies, and the proposed method is tested against theseassemblies. Through numerical calculations it is found that the proposed method is
promissing.
Key Words: nuclear data, integral data, sensitivities, singular value
decomposition, active sub-space, orthogonal projection
1. Introduction

them is 𝑛. In such a case, a matrix 𝐴𝑚×𝑛 consisting ofthese vectors is defined as

Nuclear data values are evaluated by using results oftheoretical calculations and nuclear data experiments, andintegral data experiments are used for verification andupdating of evaluated nuclear data. True values of nuclear
data can be theoretically obtained from integral data if thenumber of independent integral data is larger than thenumber of nuclear data. This is not realistic because thenumber of nuclear data is generally much larger than thenumber of integral data. However, there is a possibility
that some of nuclear data can be independently verified
by using integral data. Therefore, it is important toquantify the effectiveness of a set of integral data forverifying individual nuclear data. Also, various different
integral data can be obtained by changing theexperimental condition, but there should be dependencies
among the different integral data because of similarity ofexperiment condition, material composition, and so on.
Consequently, it is also important to quantify the
independency of each integral data in a set of integral data.
In this study, we propose a method to quantify how
much individual nuclear data can be verified with a set of
integral data, and how much each of integral data aredependent on each other. These are attained by using
mathematical techniques, such as the singular value
decomposition (SVD), the orthogonal projection [1] andthe concept of active sub-space (ASS) [2].

𝐴𝑚×𝑛 = (𝑎⃗1(1)

⋯ 𝑎⃗𝑛 ).

𝑎⃗2This matrix can be transformed by SVD as(2)

𝐴𝑚×𝑛 = 𝑈𝑚×𝑚 𝐷𝑚×𝑛 (𝑉𝑛×𝑛 )𝑇 ,where the superscript 𝑇 is for the matrix transposition.
When we assume 𝑛<𝑚, 𝐷𝑚×𝑛 can be presented as
𝜎1
0

𝐷𝑚×𝑛 =

0
𝜎2
0

0
0
𝜎3
0

⋮
0

⋯

⋮
(0

⋮
⋯

⋯
⋯
⋯
⋱
0

0
0
0
0
𝜎𝑠
⋯

⋮

⋮

⋮

⋯
⋯
⋯
0
⋱
0
⋮
⋯

0
0
0
0
0
,
0
𝜎𝑛
0
⋮
0)

(3)

where 𝜎𝑖 are singular values arranged by descendingorder. The matrices 𝑈𝑚×𝑚 and 𝑉𝑛×𝑛 are the unitarymatrices and presented as
𝑈𝑚×𝑚 = (𝑢⃗⃗1 𝑢⃗⃗2 ⋯ 𝑢⃗⃗𝑠 ⋯ 𝑢⃗⃗𝑚 ),(4)
𝑉𝑛×𝑛 = (𝑣⃗1 𝑣⃗2 ⋯ 𝑣⃗𝑠 ⋯ 𝑣⃗𝑛 ).
(5)Then, the sets of 𝑢⃗⃗𝑖 and 𝑣⃗𝑖 are the orthonormal basis
components of the m-dimensional and the n-dimensional
sub-spaces, respectively, so
𝑈𝑈 𝑇 = 𝑈 𝑇 𝑈 = 𝐼,

2. Theory and Numerical Procedure

𝑉𝑉 𝑇 = 𝑉 𝑇 𝑉 = 𝐼,

2.1 Singular value decomposition

(6)
(7)

where 𝐼 is the unit matrix.
Let us consider a 𝑚-dimensional vector 𝑟⃗, which is
described as a linear combination of 𝑎⃗𝑖 . This vector 𝑟⃗

Let us consider 𝑚-column vectors 𝑎⃗𝑖 and the number of244

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019can be expressed as

𝑃𝑈′ = 𝑈′𝑁×𝑁 (𝑈′𝑁×𝑁 )𝑇 ,′𝑟1𝑟2𝑟⃗ = ( ⋮ ) = (𝑎⃗1
𝑟𝑚

𝑟1
𝑟2 ′
𝑎⃗2 ⋯ 𝑎⃗𝑛 ) ( )⋮𝑟𝑛 ′= ∑ 𝑎⃗𝑖 𝑟𝑖 ′ = 𝐴𝑚×𝑛 ⃗⃗⃗⃗
𝑟 ′.

where 𝑈′𝑁×𝑁 is the orthonormal basis matrix whichspans the sub-space of the data set. Multiplying 𝑃𝑈′ to avector enables us to know how much this vector can bepresented by this sub-space. If this vector is a column unit
vector ⃗⃗⃗𝑒𝑗 in which the 𝑗-th entry is unity, the following
equation is obtained:(8)

If the matrix 𝐴𝑚×𝑛 is decomposed as Eq. (2), Eq. (8) canbe written as⃗⃗⃗⃗
𝑒𝑗′ = 𝑃𝑈′ ⃗⃗⃗.
𝑒𝑗𝑛𝑟⃗ = 𝑈𝑚×𝑚 𝐷𝑚×𝑛 (𝑉𝑛×𝑛 )𝑇 ⃗⃗⃗⃗
𝑟 ′ = ∑(𝜎𝑖 𝑣⃗𝑖𝑇 ⃗⃗⃗⃗𝑟 ′) 𝑢
⃗⃗𝑖 .
𝑖=1

𝑠

(10)

𝑖=1

2.3 Sensitivity

This shows that an arbitrary vector represented as a linear
combination 𝑎⃗𝑖 can be described by a linear combinationof 𝑢⃗⃗𝑖 ; orthogonal basis of a space spanned by 𝑎⃗𝑖 is 𝑢⃗⃗𝑖 .
In this study, the matrix 𝐴𝑚×𝑛 is constructed by a setof vectors, and SVD is adopted to obtain the orthonormal
basis vectors 𝑢⃗⃗𝑖 . These vectors are used for thesubsequent orthogonal projection and the quantification
methods of the effectiveness of a set of integral data.

We use sensitivities of integral data with respect tonuclear data. Sensitivities are impact of change in nuclear
data on integral data. A sensitivity 𝑆 is defined as𝑆=

Let us consider a vector 𝑎⃗ in a M-dimensional space, andthis vector is orthogonally projected to a N-dimensional(𝑁< 𝑀) sub-space.
In the case of 𝑀=3 and 𝑁=2, 𝑎⃗ is expressed as

Active sub-space (ASS) is a sub-space spanned by only
effective orthogonal basis components in an original
space spanned by all the orthogonal basis vectors. As an
example, let us consider a simple case that there are anumber of vectors in a three-dimensional space and ASS
spanned by these vectors. If those vectors almost exist on
the plane including the origin within the original threedimensional space, the dimension of this ASS is two.
In this study, we use ASS for the subsequent
quantification method for the independency of data set.

(11)

𝑏1
𝑢2 ⃗⃗⃗⃗⃗)⃗⃗⃗⃗⃗
𝑢3 ∙ (𝑏2 ) = 𝑈𝑏⃗⃗,
𝑏3

(12)

𝑏⃗⃗ = 𝑈 𝑇 𝑎⃗.

(13)

where 𝑈 is the unitary matrix. By multiplying by 𝑈 𝑇 , Eq.
(12) is transformed to the following equation:

2.5 Quantification method of independency of data set
Quantification process of independency of data set isdescribed below.
First, we perform SVD with a matrix which isconstructed by a set of vector data, and then obtain theorthonormal basis vectors. Second, for searching ASS, allvector data are orthogonally projected on sub-spacespanned by the limited number of basis vectors. When all
norms become enough close to unity, this sub-space canbe regarded as ASS, and the dimension of this ASS
corresponds to independency of the data set. In this study,
we regard the norm value is enough close to unity whenthe value is larger than 0.99.
When we adopt the above concept to nuclear data
testing with integral data, the matrix consists of sensitivity
column vectors of a set of integral data. The number ofthe sensitivity column vectors correspond to the number

An orthogonally-projected-two-dimensional vector 𝑐⃗ of𝑎⃗ is written as𝑐⃗ = 𝑏1 ⃗⃗⃗⃗⃗
𝑢1 + 𝑏2 ⃗⃗⃗⃗⃗,
𝑢2

(14)

and this equation can be transformed as follows:𝑇𝑇𝑐⃗ = (𝑢⃗⃗⃗⃗⃗1 ∙ 𝑎⃗)𝑢⃗⃗⃗⃗⃗1 + (𝑢⃗⃗⃗⃗⃗2 ∙ 𝑎⃗)𝑢⃗⃗⃗⃗⃗2 = 𝑈 ′ 𝑈 ′𝑇 𝑎⃗,(18)

2.4 Active Sub-Space

where ⃗⃗⃗⃗
𝑢𝑖 is the orthonormal basis of the M-dimensional
space. Equation (11) can be transformed to𝑎⃗ = (𝑢⃗⃗⃗⃗⃗1

𝜕 𝑝 ⁄𝑝,𝜕 𝜎 ⁄𝜎where 𝑝 is integral data, and 𝜎 is nuclear data. In thepresent study, sensitivity is presented as a column vector
whose size is the number of nuclear data.

2.2 Orthogonal projection𝑎⃗ = 𝑏1 ⃗⃗⃗⃗⃗
𝑢1 + 𝑏2 ⃗⃗⃗⃗⃗
𝑢2 + 𝑏3 ⃗⃗⃗⃗⃗,
𝑢3

(17)

The 2-norm of ⃗⃗⃗⃗
𝑒𝑗′ corresponds to how much the vector ⃗⃗⃗𝑒𝑗
is presented by the sub-space defined by 𝑈′𝑁×𝑁 .
This orthogonal projection is used for two purposes.
One is the subsequent quantification method of thedimension of ASS. The other is the quantification methodof the effectiveness of the set of integral data.

(9)

If the singular values 𝜎𝑖 (𝑖 > 𝑠) are extremely small, Eq.
(9) can be transformed to𝑟⃗ = ∑(𝜎𝑖 𝑣⃗𝑖𝑇 ⃗⃗⃗⃗𝑟 ′) 𝑢
⃗⃗𝑖 .

(16)

(15)

where 𝑈′ = (𝑢⃗⃗⃗⃗⃗1 ⃗⃗⃗⃗⃗)
𝑢2 This means that the orthogonal
projection can be performed by multiplying theorthonormal basis matrix of the sub-space to be projected,
and its transposition to the objective vector.
Based on the above, the orthogonal projection of theobjective data on the sub-space of the data set is expressedas245

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019of integral data. By using the orthonormal basis vectors
generated by SVD, we obtain the dimension of ASS of thesensitivities. The dimension means the independency ofthe set of integral data for verification of nuclear data.
2.6 Quantification method of the possibility of verificationof individual nuclear data by using the set of integral data
If a nuclear data vector exists in ASS spanned bysensitivity vectors, this nuclear data can be presented by
the set of sensitivity data which spans this ASS. Thismeans that this nuclear data can be independently verified
from the set of integral data. On the other hand, if anuclear data vector does not exist on the ASS, we regard
a norm of an orthogonally-projected vector of this nuclear
data vector to the ASS as possibility of independent
verification of this nuclear data.

Fig. 1 Norm of orthogonal projected vectors of 94
integral data on sub-spaces with different dimensions
Second, we investigate the influence on independency
by adding a new set of integral data. Additional integral
data are also fictitious spherical critical assemblies withwater reflector whose thickness is 20 cm. The assembly
size is also adjusted to make the multiplication factor
unity. As the same manner with the bare assemblies, 94
water-reflected assemblies are prepared by changing U235 enrichment from 7% to 100%.
We calculatesensitivities of these assemblies, and now the size of the
new matrix is 280×188.
The quantification method is adopted to this new
matrix. Figure 2 shows norms of orthogonally-projected
vectors of 188 integral data to sub-spaces. This resultshows that dimension of ASS becomes four. In otherwords, the independency of the set of integral data
increases from three to four by addition of a new set of
integral data to the original data set.

3. Numerical Result
3.1 Numerical method
In this study, sensitivities of integral data with respect tonuclear data are calculated with a reactor physics code
system CBZ, which is under development at Hokkaido
University. Multi-group cross sections are obtained
through resonance self-shielding calculations inhomogeneous infinite media, and forward and adjoint
angular neutron fluxes in spherical cores are calculatedwith a discrete-ordinate neutron transport solver SNR.
With these neutron fluxes, sensitivities of neutron
multiplication factors to reaction cross sections arecalculated by the first-order perturbation theory. In thisstudy, we use 280-energy group cross sections, andsensitivities of effective multiplication factor 𝑘eff areconsidered. As nuclear data, only fission cross section of
U-235 is considered.
3.2 Independency of two sets of integral data
First, let us consider a set of fictitious homogeneous barespherical critical assemblies which are composed of U235 and U-238. We prepare 94 different assemblies bychanging U-235 enrichment from 7% to 100%. The sizeof each critical assembly is adjusted so as to make themultiplication factor unity. We calculate sensitivities of allthese assemblies, and the size of a sensitivity matrix
becomes 280×94.
For searching the dimension of ASS of the set of these
integral data, the quantification method presented above
is adopted
Figure 1 shows norms of orthogonally-projected
vectors of each integral data on sub-spaces with different
dimensions. In the case of a three-dimensional sub-space,
all norms are larger than 0.99. This means that the numberof dimensions of ASS, that corresponds to the
independency of the set of integral data, is three in thisdata set.

Fig. 2 Norm of orthogonal projected vectors of 188
integral data on sub-spaces with different dimensions
Figure 3 shows the orthonormal basis vectors of the
four-dimensional ASS constructed from the 188 integral
data.

246

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019by the sensitivity vectors. This corresponds to thepossibility of independent verification of individual
nuclear data by using the set of integral data. Maximum
norms are almost 0.3 in two cases in which one uses only
bare assembly data and the other uses all the data. This
result suggests that individual nuclear data verification isdifficult by using these set of integral data.

Fig.3 Basis vectors of four-dimensional ASS constructedby 188 integral data
Then, we quantify which additional integral data
contribute to increase of independency. To do this we
perform orthogonal projection of each of additional 94
integral data to the three-dimension ASS constructed by
the original 94 integral data. When a norm is 0, thisadditional integral data is totally independent on the set oforiginal 94 integral data and this additional integral data
contributes to increase of the independency.
Figure 4 shows norms of orthogonally-projected
vectors of each of integral data vectors of the additional
94 assemblies. This figure shows that the additional
data with low enrichment relatively contribute to increaseof the independency, and that the most efficient additional
data is the assembly with enrichment of 10%. This isbecause this assembly has large fission reactions inthermal neutron energy.

Fig. 5 Norm of orthogonal projected vectors of nuclear
data vector on ASS
4. Conclusions
We have proposed the quantification method of theeffectiveness of the set of integral data for verification of
nuclear data by using SVD, the orthogonal projection andthe concept of ASS. The effectiveness of the set of integral
data means two things:⚫ The independency of the set of integral data
⚫ The possibility of verification of individual nuclear
data by the set of integral data
This study has treated two analysis conditions which
are bare spherical critical assembly and that one plus
spherical critical assembly with water reflector.
We could quantify above two things with two
conditions. Concretely the independencies of the sets of
94 integral data and 188 integral data are three and four,
respectively. Moreover, it is difficult to verify individual
nuclear data by using these sets of integral data.
References1.

Meyer C, Matrix Analysis and Applied Linear
Algebra, Society for Industrial and Applied
Mathematics, Philadelphia, (2000).
2. Abdel-Khalik HS, et al., “Efficient subspace methodsbased algorithms for performing sensitivity,
uncertainty, and adoptive simulation of large-scalecomputational models,” Nucl. Sci. Eng., 159, 256(2008).

Fig. 4 Norm of orthogonal projected vectors ofadditional 94 integral data on ASS spanned by theoriginal 94 integral data
3.3 Effectiveness of the set of integral data
For quantification of the possibility of verification ofindividual nuclear data by ASS of the set of integral data,
the proposed method is adopted.
Figure 5 shows norms of orthogonally-projected
vectors of each of nuclear data vector to the ASS spanned247TS-07

Uncertainty Quantification and Data Assimilation

No.

RPHA19-1049

Title

Estimated Criticality Lower-Limit Multiplication Factor
Considering Neutronic Similarity and Uncertainties ofEffective Multiplication Factor Using the Bootstrap Method (1)TheoryAuthors

Takuto Hayashi, Fuga Nishioka, Tomohiro Endo, Akio Yamamoto(Nagoya University, Japan)
250

No.

RPHA19-1061

Title

Estimated Criticality Lower-Limit Multiplication Factor
Considering Neutronic Similarity and Uncertainties ofEffective Multiplication Factor Using the Bootstrap Method (2)Application

Authors

Fuga Nishioka, Takuto Hayashi, Tomohiro Endo, Akio Yamamoto(Nagoya University, Japan)
254

No.

RPHA19-1055

Title

Covariance-Oriented
Uncertainty Analysis

Authors

Zhuojie Sui, Liangzhi Cao, Chenghui Wan (Xi’an Jiaotong University,

SampleTransformation

Methodfor

China)
258
No.

RPHA19-1002

Title

Effectiveness of Subcritical Measurement at Solid Moderated
KUCA Core for Reducing Nuclear Data-Induced Uncertainties
in Other Light Water Reactor Analysis

Authors

Tomohiro Endo, Akio Yamamoto (Nagoya University, Japan)
262

No.

RPHA19-1084

Title

Benchmark Study of Nuclear Processed Systems with NCA Data

Authors

Satoshi Wada, Kenichi Yoshida, Tsukasa Sugita (Toshiba Energy Systems& Solutions, Japan)
266

248

No.

RPHA19-1045

Title

Sensitivity Analysis for Generalized Response with RMC Code

Authors

Guanlin Shi, Conglong Jia, Quan Cheng, Kan Wang (Tsinghua University,
China)
270

No.

RPHA19-1009

Title

Uncertainty Analysis of Neutron Parameters for NESTOR

Authors

Hongkuan Liao, Qing Li, Yingrui Yu, Yuying Hu (Nuclear Power Instituteof China, China)
274

No.

RPHA19-1071

Title

Comparison of Methods of Generating Covariance Matrix of
Fission Yield

Authors

Keisuke Honta, Go Chiba (Hokkaido University, Japan)
278

No.

RPHA19-1015

Title

Application of Surrogate Modeling with Singular Value
Decomposition for Design Basis Accident Aiming Statistical
Safety Analysis

Authors

Masaki Matsushita, Tomohiro Endo, Akio Yamamoto (Nagoya University,
Japan)
282

249

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Estimated Criticality Lower-Limit Multiplication Factor Considering Neutronic Similarity
and Uncertainties of Effective Multiplication Factor Using the Bootstrap Method(1) Theorya

Takuto Hayashia,, Fuga Nishiokaa, Tomohiro Endoa, and Akio Yamamotoa
Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi, 464-8603, Japan*
Corresponding author: t-hayashi@fermi.energy.nagoya-u.ac.jp

Abstract
The present paper describes a calculation method for evaluation of the estimatedcriticality lower-limit multiplication factor (ECLLMF) considering neutronic
similarity and uncertainties of effective multiplication factor (𝑘eff ) using thebootstrap method. The normal distribution is assumed for a probability distribution
with respect to calculation biases of 𝑘eff to evaluate ECLLMF in the conventional
method. An estimation method of ECLLMF was already developed using thebootstrap method, where the assumption of the normal distribution is not necessary.
In this study, the calculation method for evaluation of ECLLMF using thebootstrap method is improved. The point of improvement is taking into accountneutronic similarity between a target system and the criticality benchmark
experiments, experimental uncertainties of benchmark data, and statisticaluncertainties of 𝑘eff by a continuous energy Monte Carlo code.
Key Words: estimated criticality lower-limit multiplication factor, bootstrap
method, neutronic similarity, experimental error, Monte Carlo code,
statistical error
1. Introduction

nuclear-data-induced uncertainties.
Hence, the present paper aims to develop a calculation
method for evaluation of the ECLLMF taking into
account (1) representative factor that incorporates thedegree of neutronic similarity between a target system anda critical benchmark experiment system, (2) experimental
uncertainties of benchmark data, and (3) statisticaluncertainties of calculated 𝑘eff by a continuous energyMonte Carlo code.
In section 2, the conventional calculation method of
ECLLMF, where the assumption of the normal
distribution is necessary, is briefly described. In addition,
the improved calculation method of ECLLMF using thebootstrap method is proposed. Then, a numerical analysis
is carried out for a bimodal distribution in order to verify
the proposed method in section 3. Finally, concluding
remarks are provided in section 4.

The estimated criticality lower-limit multiplication factor(ECLLMF) is the upper limit of the neutron multiplication
factor where a target system may be judged subcritical [1].
The ECLLMF is evaluated from (1) many benchmark
calculations using experimental data which areneutronically similar to a target system, and (2)statistically processing of calculated effective
multiplication factors ( 𝑘eff ) obtained by benchmark
calculations. However, the statistical calculation methodof the ECLLMF would be complicated and can be
improved especially when a probability distribution of𝑘eff does not obey the normal distribution because thenormal distribution is assumed in the conventional
evaluation process.
Therefore, in order to propose a methodology without
the assumption of the normal distribution, an evaluation
method of the ECLLMF was developed using thebootstrap method, where the assumption of the normal
distribution was not necessary [2]. In the previous study,
the following issues remained to be addressed:● Consideration of neutronic similarities between atarget system and the criticality benchmark
experiments in the evaluation process.
● Consideration of experimental uncertainties ofbenchmark data, statistical uncertainties of calculated𝑘eff by a continuous energy Monte Carlo code, and

2. Theory
2.1 Conventional method
The calculated effective multiplication factors (𝑘eff ) are
assumed to follow the normal distribution in theconventional method. Then, the conventional estimatedcriticality lower-limit multiplication factor (ECLLMF)
𝑘sub,t using the non-central t-distribution is calculated by
[1]:

250

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019𝑘sub,t = 𝑘̅ − 𝛼t 𝑠,
wherefactors 𝜎calc,1 , 𝜎calc,2 , … , 𝜎calc,𝑛 by the benchmark
calculations for 𝑛 criticality experiments.
2. The experimental and statistical uncertainties of 𝑘eff,𝑖are synthesized by:

(1)

𝑛

1
𝑘̅ = ∑ 𝑘eff,𝑖 ,
𝑛
𝑛

(2)

2
2
𝜎𝑖 = √𝜎bench,𝑖+ 𝜎calc,𝑖
.

𝑖=1

1
2
𝑠2 =
∑(𝑘eff,𝑖 − 𝑘̅ ) , (𝑛 ≥ 2).
(3)
𝑛−1
𝑖=1
Note that 𝑘̅ is the sample mean of 𝑘eff,1 , 𝑘eff,2 , … , 𝑘eff,𝑛obtained by the benchmark calculations for 𝑛 criticality
experiments; 𝛼t is a margin value, and 𝑠 is the sample
standard deviation of 𝑘eff,1 , 𝑘eff,2 , … , 𝑘eff,𝑛 .
The margin value of 𝛼t can be calculated using theproperties of the normal distribution and the non-central
t-distribution by setting 𝑝 (excess probability of criticalcondition) and 𝛾 (success probability of setting
ECLLMF). The value of 𝑝 and 𝛾 are set individually byan analyst, e.g., 𝑝 = 0.025 and 𝛾 = 0.975. When 𝑛 issufficiently large, for example, 𝛼t approaches 1.96 with
𝑝 = 0.025. The value of 𝛼t increases as 𝑛 becomes
smaller.

(4)

3. A multi-modal distribution 𝑝(𝑥) is defined by a
mixture of weighted 𝑛 normal distributions [5]:𝑛2𝑤𝑖(𝑥 − 𝑘eff,𝑖 )
𝑝(𝑥) = ∑exp (−),(5)2𝜎𝑖2√2𝜋𝜎𝑖2𝑖=1

where the weighing factor 𝑤𝑖 is determined by alinear model with respect to the representative factors
𝑐𝑘,𝑖 . The relationship between the representative factor
and weight factor is represented by:
𝑐𝑘,𝑖 − 𝑐𝑘,acc
𝑤𝑖 = max {0,},(6)𝑐𝑘,max − 𝑐𝑘,acc
where the subscript 𝑖 indicates the 𝑖 th criticalexperiment; 𝑐𝑘,max is the maximum value of 𝑐𝑘,𝑖 ; and𝑐𝑘,acc is the acceptable minimum value of 𝑐𝑘 . If therepresentative factors are not considered, 𝑤𝑖 = 1/𝑛,i.e., each of critical experiments has the same weight.
4. Calculate the lower 100 𝑝 percentile of the multimodal distribution expressed by Eq. (5). The lower
100𝑝 percentile is defined by the value of a randomvariable when the cumulative distribution function
matches. Hereafter, the percentile is called 𝑘eff,𝑝 . For
example, 𝑘eff,𝑝 can be evaluated using the Monte
Carlo integration.
5. Generate a new set of resamples 𝑘1∗𝑏 , 𝑘2∗𝑏 , … , 𝑘𝑛∗𝑏 by
the parametric bootstrap sampling based on the multimodal distribution, where the superscript ‘∗ 𝑏’ means
the 𝑏 th bootstrap sample. To consider the
representative factors in this bootstrap sampling, theweighting factor 𝑤𝑖 is used as the probability toselect for the corresponding normal distribution
𝒩(𝑘eff,𝑖 , 𝜎𝑖2 ).
6. Calculate the bootstrap sample mean 𝑘̂ ∗𝑏 and thebootstrap sample standard deviation 𝑠̂ ∗𝑏 for the 𝑏th
set of resamples, respectively.
̂ ∗ = (𝑘̂ ∗1 , 𝑘̂ ∗2 , … , 𝑘̂ ∗𝐵 )7. Obtain𝐤and𝐬̂ ∗ =∗1 ∗2∗𝐵(𝑠̂ , 𝑠̂ , … , 𝑠̂ ) by repeating the procedure 5–6 in
𝐵 times.
1
2
𝐵
8. 𝐤 sub,boot = (𝑘sub,boot, 𝑘sub,boot, … , 𝑘sub,boot)isobtained by:̂ ∗ − 𝛼boot 𝐬̂ ∗ ,(7)
𝐤 sub,boot = 𝐤

2.2 Proposed method
The bootstrap method is one of uncertainty evaluation
methods and is used for evaluation of the confidence
interval of the mean and the standard deviation of sample
data or other statistical values [3], [4].
The ECLLMF using the bootstrap method without theassumption of the normal distribution was alreadydeveloped [2]. In this method, the ECLLMF wasestimated by generating new sets of samples throughrandom sampling with replacement. From the viewpointof the skewness and the kurtosis of a probability
distribution of 𝑘eff , the previous study was confirmedthat the ECLLMF using the bootstrap method was morereasonable than 𝑘sub,t when a probability distribution of𝑘eff did not obey the normal distribution.
In this study, the calculation method for evaluation of
ECLLMF using the bootstrap method is improved. The
point of improvement is taking into account (1)representative factor (𝑐𝑘 ) that represents the degree ofneutronic similarity between a target system and a criticalbenchmark experiment system, (2) experimental
uncertainties of benchmark data ( 𝜎bench ), and (3)statistical uncertainties of calculated 𝑘eff (𝜎calc ) by acontinuous energy Monte Carlo code. The proposed
method has an advantage that the improved ECLLMF canbe equal to the previous ECLLMF using the bootstrap
method when all representative factors are the same anduncertainties of 𝑘eff can be regard as zero.
Calculation procedures of the ECLLMF using thebootstrap method 𝑘sub,boot are as follows:
1. Obtain
effective
multiplication
factors𝑘eff,1 , 𝑘eff,2 , … , 𝑘eff,𝑛 ,representative
factors
𝑐𝑘,1 , 𝑐𝑘,2 , … , 𝑐𝑘,𝑛 , experimental uncertainties ofbenchmark data 𝜎bench,1 , 𝜎bench,2 , … , 𝜎bench,𝑛 , andstatistical uncertainties of effective multiplication

where the margin value 𝛼boot is determined so that𝑘sub,boot,𝛾 reproduces 𝑘eff,𝑝 as presented in thefollowing procedure 9. Here, 𝑘sub,boot,𝛾 means thelower 100 𝛾 percentile of 𝐤 sub,boot in ascendingorder.
9. Solve 𝑓(𝛼boot ) = 0 using the bisection method [6],where the function 𝑓(𝛼boot ) is defined by:𝑓(𝛼boot ) = 𝑘sub,boot,𝛾 − 𝑘eff,𝑝 .

(8)

In the bisection method, the iteration is performed251

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019(9)𝑘sub,boot = 𝑘̅ − 𝛼boot 𝑠,

10where the parameters 𝑘̅ and 𝑠 are the mean and thestandard deviation of the multi-modal distribution
expressed by Eq. (5), respectively.

20

p(x)

30

40

until |𝑘sub,boot,𝛾 − 𝑘eff,𝑝 | < 𝜀 is satisfied. The
convergence criteria 𝜀 is 10-7 because the sufficiently
small 𝜀 should be set compared with the typicalstatistical errors (about 10-5) in a continuous energyMonte Carlo code.
10. The ECLLMF 𝑘sub,boot is finally obtained by:

0

3. Verification

0.97

A numerical analysis is carried out for the virtual 𝑘eff
which obey the bimodal distribution in order to verify thecalculation method of the ECLLMF considering
neutronic similarity and uncertainties of 𝑘eff using thebootstrap method. Then, the verification of the proposed
method is achieved by comparing the ECLLMF and theanalytical solution. In this verification, the value of 𝑝and 𝛾 are 0.025 and 0.975, respectively.
A bimodal distribution means a weighted mixture of
two normal distributions 𝒩(𝑘eff,1 , 𝜎12 ), 𝒩(𝑘eff,2 , 𝜎22 ). A
probability density function of the bimodal distribution is
obtained by Eq. (5). The mean and variance of thebimodal distribution 𝜇, 𝜎 2 are analytically obtained by:

𝑛

𝑖=1

𝑖=1

(11)

𝑖=1

where 𝑤𝑖 is the weighting factor.
The probability distribution of a bimodal distribution
with the calculation condition summarized in Table I isshown in Fig. 1.
Table I. Parameter of the bimodal distributionNo𝑘eff,𝑖
𝜎𝑖
𝑤𝑖
1
0.990
0.007
0.6
2
1.009
0.004
0.4
𝜇
0.9976𝜎20.0001224

1.01

1.02

1.03

When the sample size is sufficiently large, ECLLMF
approaches the lower 100 𝑝 percentile of the bimodal
distribution 𝑘eff,𝑝 , thus a comparison between 𝑘eff,𝑝 and𝑘sub,boot is carried out for verification. Using the Newton
method, 𝑘eff,𝑝 is calculated by the cumulative
distribution function of the bimodal distribution, thus thenumerical value of 𝑘eff,𝑝 is 0.977878. In addition, acomparison between a saturation value 𝛼∞ and 𝛼boot iscarried out. The value of 𝛼∞ can be analytically solvedby:𝜇 − 𝑘eff,𝑝
(12)
𝛼∞ =
= 1.78.
𝜎
By changing the sample size, the estimation results of𝑘sub,t and 𝑘sub,boot , margin values of 𝛼t and 𝛼boot ,and probabilities of ECLLMF exceeding 𝑘eff,𝑝 areshown in Fig. 2–4, respectively. In addition, probability
distributions of 100 virtual experimental value 𝑘eff ,𝑘sub,t , and 𝐤 sub,boot include a blue dashed vertical lineof 𝑘eff,𝑝 are shown in Fig. 5 as an example.
As shown in Fig. 2, the ECLLMF by 𝑘sub,t did not
approach 𝑘eff,𝑝 despite of the sufficiently large sample
size. The ECLLMF by 𝑘sub,boot was smaller than 𝑘eff,𝑝when the sample size was small, then it approached 𝑘eff,𝑝
as the sample size became larger.
The differences between 𝑘sub,t and 𝑘sub,boot comefrom the differences in the margin values shown in Fig. 3.
The conventional 𝛼t value approached 1.96, while
𝛼boot approached 1.78 in Fig. 3. Figure 4 indicates thatthe probability of ECLLMF exceeding 𝑘eff,𝑝 of theconventional method was not equal to the value of (1−𝛾)= 0.025. It means that the success probability of setting
ECLLMF was not correctly set as 𝛾 in the conventional
method. As can be seen from Fig. 5, the probability
distribution of 𝑘sub,t is shifted to the left direction (i.e.,the margin value of 𝛼t is too large). This is caused by theassumption of the normal distribution for the virtual
experimental value 𝑘eff in the conventional method. On
the contrary, the probability of ECLLMF exceeding 𝑘eff,𝑝of the proposed method was always equal to (1−𝛾) for
any sample size. It means that the more appropriate

𝑛
2

1.00

Fig. 1. Probability distribution of the bimodal distribution.

(10)

𝜎 2 = ∑ 𝑤𝑖 𝜎𝑖2 + ∑ 𝑤𝑖 (𝜇 − 𝑘eff,𝑖 ) ,

0.99

x

𝑛

𝜇 = ∑ 𝑤𝑖 𝑘eff,𝑖 ,0.98

𝜎0.0111This verification can be also regarded as the following
virtual conditions: calculated 𝑘eff of the total criticality
experiments obey the bimodal distribution; 60% of thetotal calculated 𝑘eff obey the normal distribution
𝒩 (0.990, 0.0072) and the others obey the normal
distribution 𝒩(1.009, 0.0042).

252

Probability distribution of 𝑘sub,t

ECLLMF can be determined by the proposed method sothat the success probability is equal to 𝛾.
Consequently, it was confirmed that the proposed
method was more reasonable than the conventional
method when a probability distribution of 𝑘eff obeyed
the bimodal distribution. Another companion paper of thisconference will report application results of estimation
ECLLMF for more complicated probability distributions
obtained by actual critical experiments.

𝐤 sub,bootProceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019𝑘eff,𝑝100 virtual multiplication factors

0.980

Value of ECLLMF

0.975

0.970

Fig. 5. Probability distributions of virtual experimental
value 𝑘eff , 𝑘sub,t , and 𝐤 sub,boot include 𝑘eff,𝑝 .

0.965

keff,p

0.960

0.950
1.E+00

4. Conclusions

ksub,t0.955

ksub,boot
1.E+01

1.E+02 1.E+03
Sample size

1.E+04In the present paper, the calculation method using thebootstrap method was improved to evaluate the ECLLMF.
The proposed ECLLMF was calculated taking into
account neutronic similarity, experimental uncertainties
of benchmark data, and statistical uncertainties of 𝑘eff
by a continuous energy Monte Carlo code. The numericalanalysis was carried out for the virtual 𝑘eff which obey
the bimodal distribution in order to verify the proposed
method. From the verification, it was confirmed that the
proposed method was more reasonable than theconventional method when a probability distribution of𝑘eff obeyed the bimodal distribution.
The followings are future researches:● Consideration of nuclear-data-induced uncertainties.
● Comparison of the proposed ECLLMF and a baseline
upper-subcritical-limit obtained by Whisper-1.1 [7].

1.E+05

Fig. 2. Estimation results of 𝑘sub,t and 𝑘sub,boot .
4.0

1.78

3.5

alpha_t

Value of alpha

3.0

alpha_boot

2.5
2.0
1.5

1.0
0.5
0.0
1.E+00

1.E+01

1.E+02 1.E+03
Sample size

1.E+04

References

1.E+05

1. “Nuclear Criticality Safety Handbook, 2; English
Translation,” JAERI-Review-2001-028 (2001).
2. Hayashi T, Endo T, and Yamamoto A, “Calculation
Method of Estimated Criticality Lower-Limit
Multiplication Factor Using the Bootstrap Method,”
Proc. M&C 2019, Portland, OR, August 25-29, 2019,
American Nuclear Society (2019).
3. Efron B, “Bootstrap Methods: Another Look at the
Jackknife,” Ann. Stat., 7, 1, pp. 1-26 (1979).
4. Efron B and Tibshirani RJ, An Introduction to the
Bootstrap, Chapman and Hall/CRC, Florida (1994).
5. Lindsay BG, Mixture Models: Theory, Geometry andApplications, 5, Institute of Mathematical Statistics,
Hayward, California (1995).
6. Burden RL and Faires JD, Numerical Analysis, 3rdedition, pp. 39-46, Brooks Cole, Boston, MA (2002).
7. Kiedrowski
BC
et
al.,
“Whisper:Sensitivity/Uncertainty-Based
Computational
Methods and Software for Determining Baseline
Upper Subcritical Limits,” Nucl. Sci. Eng., 181, 1, pp.
17-47 (2015).

Fig. 3. Margin values of 𝛼t and 𝛼boot .
Excess probability of ECLLMF

0.040

Conventional method

0.035

Proposed method

0.030
0.025
0.020
0.015
0.010

0.005
0.000

-0.005
1.E+00

1.E+01

1.E+02 1.E+03
Sample size

1.E+04

1.E+05

Fig. 4. Probabilities of ECLLMF exceeding 𝑘eff,𝑝 .

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Estimated Criticality Lower-Limit Multiplication Factor Considering Neutronic Similarity
and Uncertainties of Effective Multiplication Factor Using the Bootstrap Method(2) Application
Fuga Nishioka a, , Takuto Hayashia, Tomohiro Endo a, and Akio Yamamoto aaNagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, 464-8603, Japan* Corresponding author: f-nishioka@fermi.energy.nagoya-u.ac.jp

Abstract
The present paper describes the calculation results of the estimated criticality
lower-limit multiplication factor (ECLLMF) considering neutronic similarity anduncertainties of effective multiplication factor (𝑘eff) using the bootstrap method.
A unit cell of a UO2 sphere with a radius of 1.0 cm surrounded by a watermoderator is considered as a target system. The proposed ECLLMF was calculated
taking into account (1) representative factor that represents the degree of neutronic
similarity between a target system and a critical benchmark experiment system, (2)experimental uncertainties of benchmark data, and (3) statistical uncertainties ofcalculated 𝑘eff by a continuous energy Monte Carlo code. From the estimation
results, it was confirmed that the distribution of 𝑘eff and the ECLLMF change
depending on the consideration of (1)–(3).
Key Words: estimated criticality lower-limit multiplication factor, bootstrap
method, neutronic similarity, experimental error, Monte Carlo code,
statistical error
1. Introducti on

2. Calculation Conditions

In our recent research, an evaluation method of theestimated criticality lower-limit multiplication factor(ECLLMF) was newly developed using the bootstrap
method, where the assumption of the normal distribution
is not necessary. The motivation and background of thepresent study are described in the companion paper [1].
The present paper describes the calculation results ofthe ECLLMF considering neutronic similarity anduncertainties of effective multiplication factor (𝑘eff ) using
the bootstrap method. A unit cell of a UO2 sphere with aradius of 1.0 cm surrounded by a water moderator, wheremoderation ratio is almost optimum, is considered as atarget system. The proposed ECLLMF is calculated
taking into account (1) representative factor ( 𝑐𝑘 ) that
represents the degree of neutronic similarity between atarget system and a critical benchmark experiment system,(2) experimental uncertainties (𝜎bench ) of benchmark data,and (3) statistical uncertainties (𝜎calc ) of calculated 𝑘eff
by a continuous energy Monte Carlo code.
In section 2, the calculation conditions of numericalanalysis are described. In section 3, estimation results of
ECLLMF using the conventional method and the
proposed method are described. Finally, concluding
remarks are provided in section 4. Detail calculation
procedures are described in the companion paper [1].

2.1 Calculation geometry and compositions
The numerical analysis was carried out for a unit cell of a
UO2 sphere with a radius of 1.0 cm surrounded by a 0.5
cm thickness of water moderator with the white boundary
condition. Here, the moderation ratio is almost optimum.
The calculation geometry model is shown in Fig. 1.
Compositions of fuel and moderator are shown in Table I.
Water moderator

UO2 fuel
White boundary

1.0 cm

0.5 cmFig. 1. Cell geometry and boundary condition.

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Table I. Compositions of fuel (5 wt.%, 293 K) andmoderator. (0.9970 g/cm3 , 293 K) [2]Atomic number density
Element(×1024 atoms/cm3 )235 U
1.1757×10-3
238
Fuel
U
2.2057×10-2
O
4.6465×10-2
H
6.6658×10-2
Moderator
O
3.3329×10-2

Table II. Calculation conditions for MCNP6.2.
Nuclear data library
ENDF/B-VII.1
Number of neutrons/cycle 100,000
Number of active cycles
500
Number of inactive cycles 100
1.0

Values of weight factor

0.9

2.2 Selection of criticality benchmark experiment data
Whisper-1.1 [3] was used for benchmark calculations anda selection of critical experiment data contained in theInternational Criticality Safety Benchmark Evaluation
Project (ICSBEP) Handbook [4]. The present study
utilizes the calculated 𝑘eff values selected by Whisper1.1.
Effective multiplication factors 𝑘eff,1 , 𝑘eff,2 , … , 𝑘eff,𝑛obtained by the benchmark calculations for 𝑛 criticalexperiments were corrected by subtracting the bias fromthe calculated 𝑘eff value. Here, the bias is defined bysubtracting unity from the experimental 𝑘eff value.
For the target system in Fig. 1, the calculation
conditions of MCNP6.2 [5] used in Whisper-1.1 areshown in Table II. The relationship between the
representative factor ( 𝑐𝑘 ) and weight factor ( 𝑤 ) isrepresented by Eq. (1):𝑐𝑘,𝑖 − 𝑐𝑘,acc},
𝑤𝑖 = max {0,(1)𝑐𝑘,max − 𝑐𝑘,acc
where the subscript 𝑖 indicates the 𝑖 th criticalexperiment; 𝑐𝑘,max is the maximu m value of 𝑐𝑘,𝑖 ; and𝑐𝑘,acc is the permissible minimum value of 𝑐𝑘 . The valueof 𝑐𝑘,acc is 0.6026 for the target system as shown in Fig.
1. The critical experiments whose representative factor
𝑐𝑘,𝑖 is larger than the value of 𝑐𝑘,acc are selected by
Whisper-1.1. For the critical experiments selected by
Whisper-1.1, the weight factors 𝑤𝑖 are summarized inFig. 2. The total number of critical experiments ( 𝑛 )selected by Whisper-1.1 is 85. When the representative
factors are not considered, 𝑤𝑖 = 1/𝑛.
The experimental uncertainties (𝜎bench ) of benchmark
data, and the statistical uncertainties (𝜎calc ) of calculated𝑘eff are obtained from the Whisper-1.1 database. These
uncertainties of 𝑘eff are synthesized by Eq. (2):22𝜎𝑖 = √𝜎bench,𝑖+ 𝜎calc,𝑖
.

0.7
0.6
0.5
0.4
0.3
0.2
0.1

leu-comp-therm-008
leu-comp-therm-011
leu-comp-therm-005
leu-comp-therm-060
leu-comp-therm-017
leu-comp-therm-010
heu-sol-therm-042
u233-comp-therm-004
leu-sol-therm-020
leu-sol-therm-002
heu-sol-therm-032
leu-sol-therm-021
leu-comp-therm-001
leu-comp-therm-028
leu-sol-therm-004
leu-comp-therm-007
leu-sol-therm-007

0.0

Report ID in ICSBEP

Fig. 2. Weight factors for each critical experiment selectedby Whisper-1.1.
3. Estimation Results of ECLLMF
The conventional and proposed methods used the valuesof 𝑝 = 0.025 (excess probability of critical condition),𝛾 = 0.975 (success probability of setting ECLLMF) [6].
The total number of the bootstrap procedures (𝐵) is 106 .
The number of bootstrap procedure (𝐵) is equivalent to
the number of the bootstrap samples. The values of 𝑘sub ,tand 𝑘sub,boot mean ECLLMFs(= 𝑘̅ − 𝛼𝑠) estimated by
the conventional method based on the non-central tdistribution and by the proposed method using thebootstrap method, respectively. Note that 𝐤 sub,boot in
Figs. 3–6 which are described later means bootstrap
distribution, not ECLLMF.
Values of 𝑘̅ , 𝑠 , 𝑘eff,𝑝 , 𝛼 and ECLLMF aresummarized in Table III. Note that 𝑘̅ is the sample meanof 𝑝(𝑥) obtained by the benchmark calculations for 𝑛
critical experiment; 𝛼 is a margin value, and 𝑠 is the
sample standard deviation of 𝑝 (𝑥) . The value of 𝑘eff,𝑝means the lower 100𝑝 percentile of 𝑝(𝑥) in ascendingorder.
For the following four cases, the random sampling
results of probability density distributions for 𝑘eff and𝑘sub,boot are shown in Figs. 3-6, respectively:
 Case 1: no considering representative factor 𝑐𝑘 ,without uncertainties 𝜎bench and 𝜎calc (Fig. 3)
 Case 2: considering 𝑐𝑘 , without 𝜎bench and 𝜎calc(Fig. 4)(2)

For the parametric bootstrap method in the evaluationof ECLLMF, the probability distribution of 𝑘eff isexpressed by the following multi-modal distribution
𝑝 (𝑥) , which is defined by a mixture of weighted 𝑛normal distributions:𝑛2(𝑥 − 𝑘eff,𝑖 )
𝑤𝑖).
𝑝(𝑥) = ∑exp (−(3)2𝜎𝑖2√2𝜋𝜎 2
𝑖 =1

0.8

𝑖

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Case 3: no considering 𝑐𝑘 , with 𝜎bench and 𝜎calc(Fig. 5)Case 4: considering 𝑐𝑘 , with 𝜎bench and 𝜎calc(Fig. 6)
Each of Figs. 3–6 includes a red solid vertical line of𝑘sub,t and a blue dashed vertical line of 𝑘eff,𝑝 .
The estimated value of 𝑘sub ,boot in Case 1 is slightly
smaller than conventional 𝑘sub ,t by 0.00032 [ ∆𝑘] . Thisdifference is caused by the difference of 𝛼 between Case
1 and the conventional method, i.e., the probability
density distribution of 𝑘eff does not necessarily obey thenormal distribution. Here, an extension of Shapiro andWilk’s W test [7], which is a normality test, was carriedout for a probability density distribution of 𝑘eff and theobtained p-value was 2.99 × 10-7 . Since the p-value issmaller than 0.05, it was confirmed that the probability
density distribution of 𝑘eff did not obey the normal
distribution.
The estimated value of 𝑘sub ,boot in Case 2 is slightly
smaller than that in Case 1 by 0.00023 [ ∆𝑘] . When 𝑐𝑘 is
considered, 𝑘eff with a small weight factor is rarely
selected as a bootstrap sample. It is considered that the
total number of effective bootstrap samples is reduced.
Therefore, the value of 𝛼 increased and the value of𝑘sub,boot decreased in Case 2.
The estimated value of 𝑘sub ,boot in Case 3 is smaller
than that in Case 1 by 0.01185 [ ∆𝑘] . By considering
𝜎bench and 𝜎calc , the probability distribution of 𝑘effchanged, and the value of 𝑠 increased. Therefore, thevalues of 𝑘eff,𝑝 and 𝑘sub ,boot decreased in Case 3.
The estimated value of 𝑘sub ,boot in Case 4 is smaller
than that in Case 1 by 0.01008 [ ∆𝑘] , but larger than thatin Case 3 by 0.00177 [ ∆𝑘] . Since 𝑘̅ and 𝑠 are almost
the same in Cases 3 and 4, it is considered that thedifference in 𝑘sub ,boot between Cases 3 and 4 is causedby the changes in the probability distribution shape (e.g.,
kurtosis and skewness) of 𝑘eff considering 𝑐𝑘 , 𝜎benchand 𝜎calc .
Fig. 3. Probability density distributions of random
sampling results of 𝑘eff , 𝐤 sub,boot and 𝑘sub ,t , 𝑘eff,𝑝 .
(Case 1)

Fig. 4. Probability density distributions of random
sampling results of 𝑘eff , 𝐤 sub,boot and 𝑘sub ,t , 𝑘eff,𝑝 .
(Case 2)

Table III. Summary of 𝑘̅ , 𝑠, 𝑘eff,𝑝 , 𝛼 and ECLLMFs.
Conventional
Estimator
Case 1Case 2
method0.999430.999430.99959𝑘̅
0.002430.002420.00249𝑠𝑘eff,𝑝0.995190.995192.37942.52192.6146𝛼ECLLMF
0.99364
0.99332
0.99309
Estimator𝑘̅𝑠𝑘eff,𝑝𝛼
ECLLMF

Case 30.999430.00389
0.99170
4.6156
0.98147

Case 40.99959
0.00386
0.99258
4.23380.98324256

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Osaka, Japan, Dec. 2-3, 2019

The future research is the evaluation of the ECLLM F
for the molten core-concrete interaction (MCCI) products.
References
1. Hayashi T, Endo T, and Yamamoto A, “Calculation
Method of Estimated Criticality Lower-Limit
Multiplication Factor Using the Bootstrap Method,”
Proc. M&C 2019, Portland, Oregon, August 25–29,
2019, American Nuclear Society (2019).
2. Izawa K et al., “Infinite Multiplication Factor of LowEnriched UO2 -Concrete System,” J. Nucl. Sci.
Technol., 49(11), pp. 1043–1047 (2012).
3. Kiedrowski BC et al., “Whisper: Sensitivity/
Uncertainty-Based Computational Methods andSoftware for Determining Baseline Upper Subcritical
Limits,” Nucl. Sci. Eng., 181(1), pp. 17–47 (2015).
4. “International Handbook of Evaluated Criticalit y
Safety Benchmark Experiments,” https://www.oecdnea.org/science/wpncs/icsbep/handbook.html (2018).
5. “MCNP User’s Manual – Code Version 6.2,”
https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/l
a-ur-17-29981.pdf (2017).
6. “Nuclear Criticality Safety Handbook, 2; English
Translation,”
http://dx.doi.org/10.11484/JAERI Review-2001-028 (2001).
7. Royston JP, “An Extension of Shapiro and Wilk’s W
Test for Normality to Large Samples,” Appl. Statist.,
31(2), pp. 115–124 (1982).

Fig. 5. Probability density distributions of random
sampling results of 𝑘eff , 𝐤 sub,boot and 𝑘sub ,t , 𝑘eff,𝑝 .
(Case 3)Fig. 6. Probability density distributions of random
sampling results of 𝑘eff , 𝐤 sub,boot and 𝑘sub,t , 𝑘eff ,𝑝 .
(Case 4)
4. Conclusions
The present paper aimed to calculate the ECLLM F
considering neutronic similarity, and experimental andstatistical uncertainties of effective multiplication factor( 𝑘eff ) using the bootstrap method. In this study, theproposed method was applied to a unit cell of a UO2
sphere with a water moderator to confirm the validity. As
a result, it was confirmed that the probability density
distribution of 𝑘eff and the ECLLMF change dependingon the consideration of the neutronic similarity, and theexperimental and errors. Thus, the proposed method
enables to evaluate the ECLLMF taking into account theprobability distribution shape of 𝑘eff , which changes
according to the neutronic similarity and the experiment alcalculation errors.

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Covariance-Oriented Sample Transformation Method for Uncertainty Analysis
Zhuojie Suia, Liangzhi Caoa*, Chenghui Wanaa

School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China*
Corresponding author: caolz@mail.xjtu.edu.cn (Cao L.).

Abstract
In this paper, the method of Covariance-Oriented Sample Transformation(COST) has been proposed in the uncertainty analysis for reactor-physics
modeling and simulation. As verification and application, the COST
method has been applied in the nuclear-data uncertainty analysis for the
TMI-1 pin-cell, propagating the nuclear-data uncertainties to the eigenvalueof the pin-cell. From the numerical result comparisons, it can be observedthat the consistent uncertainty analysis results can be provided by COST
with very small sample size, compared with the conventional sampling
method with very huge sample size.
Key Words: Uncertainty analysis; COST; Minimum Sample Size

where XS is a sample set of the input parameter X; ZS is asample set from the standard normal distribution Z ~ NNV(0, I). XS and ZS have the same dimension NV×NS in
which NS is the number of samples for each parameter of
X and Z. A is the transformation matrix with thedimension of NV×NV. V represents the extended matrixof μ with the dimension of NV×NS.
For the determination of matrix A, this paper
presents a new method which is different from theconventional ones.
Firstly, based on the covariance information of thenuclear data in the evaluation nuclear database, therelative input-parameter covariance matrix Σ of theconsidered multi-group cross-section characterized canbe constructed. Therefore, the joint probability density
function (PDF) of the input parameters is determined asfollows:

1. Introduction
The statistical sampling method is a widely capable
technology for the Uncertainty Quantification (UQ) [1] of
the reactor-physics modeling and simulation (M&S),
especially for the non-linear systems. However, in thosesystems with multidimensional input parameters, theconventional sampling methods[2] always have a hugedemand for sample size and are not able to propagate theuncertainty of input parameters completely, resulting incorresponding computational challenge and accuracy
loss. Therefore, it is an issue waiting to be solved that
how to notably reduce and determine the sample size ofthe input parameters on the basis of ensuring theconsistent UQ results with those under infinite sample
size and infinitesimal statistical fluctuations. In thiscontext, through combining the most-basic ideas of theconventional sampling method and the Deterministic
Sampling (DS) method [3-4], the Covariance-Oriented
Sample Transformation (COST) has been proposed togenerate multivariate normal distribution samples for
uncertainty analysis.
2. Theory and Method
The input parameter uncertainties in the reactor-physics
M&S are usually assumed to obey the normal
distribution. Assume that the input parameter is X withthe dimension of NV, the mean vector of X is μ=[μ1, μ2,...,μNV]T and the input parameter covariance matrix is Σ,
then X ~ NNV (μ, Σ). The samples from general normal
distribution are usually generated by multiplying thesamples from the standard normal distribution by thetransformation matrix, as shown in Eq.(1):X S  AZ S  V ；NSV  , , ,  gX 

1
2

NV / 2T 1
exp    X     1  X     (2)2where X = [σ1,σ2,…,σNV]T, represents the input
parameter vectors consisting of multi-group crosssections of all nuclides, reaction channels and energy
groups that need to be analyzed, in which NV is thedimension of it; μ is the mean vector of the input
parameter. Then the eigenvalue decomposition of thematrix is used to diagonalize Σ. The eigenvalues andcorresponding eigenvectors are arranged in descendingorder. The number of non-zero eigenvalues, i.e. the rankof Σ is counted which can be denoted by r, and here, r is
the minimum sample sizes required for convergence: = U  U T
(3)
Secondly, give a standard normal distribution
population with the dimension equaling NV, which can

(1)

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Osaka, Japan, Dec. 2-3, 2019

be denoted as NNV (0, I). Given any sample size not lessthan r, the standard normal distribution is sampled byuser-specified sampling technique to generate sample
space characterized as ZS. It should be noted that the
sample size should be big enough to fully describe theone-dimensional standard normal distribution when r is
too small.
Then the users should calculate the sample
covariance matrix of ZS which can be characterized as I*,
and the eigenvalue decomposition of the matrix is used todiagonalize I*. The number of non-zero eigenvalues, i.e.
the rank of I* is counted and can be denoted by k :I * = P I* PT(4)sample size.
Table Ⅱ shows the results of the sample verificationin an arbitrary random numerical experiment based on
235
U-σ(n,γ) covariance matrix whose eigenvalue vector is
λΣ. Since the rank of the input parameter covariancematrix is 56, COST requires 56 samples, while theconventional sampling method is also given the same
sample size for comparison. The top 20 largest
eigenvalues are listed here while COST-Var(XPS ) and
LHS-Var( XPS ) represent the variance of sample
projection on corresponding eigenvectors by COST andconventional sampling method using LHS, respectively.
From comparison shown in the table, the following
conclusions can be drawn: the values of λΣ and
COST-Var(XPS ) are completely equal, hence the COST
sample covariance matrix can fully map back to thetarget input parameter covariance matrix Σ. This means
the samples can carry all the information of the target
covariance matrix which describes the uncertainty of
input parameters. Furthermore, this phenomenon alsomeans the uncertainty-calculation samples are generated
correctly by COST method. However, in theconventional sampling method, the eigenvalue vector ofΣ cannot be reconstructed through the sample, whichmeans the sample loses part of the uncertainty
information of the given input parameter.

Note that there is a condition k ≥ r that needs to besatisfied. If k<r, repeat this step to reproduce ZS until thecondition is met.
Thirdly, matrix L and matrix R are determined by
the following formula：(5)
L  Pk 1/I *2k ；R  U k 1/k2
Then find an invertible matrix F which cantransform L into the row simplest matrix. Meanwhile, asquare matrix S of NV×NV is constructed based onS   R O  , whose order is equal to the dimension of

the input parameter.
Finally, based on matrix S and F, the transformation
matrix A can be obtained by A  SF for linear
transformation. Like the last step of the conventional
sampling method, samples XS are generated using Eq.(1)to propagate the input parameter uncertainties completely.
Then the users can continue the next step of uncertainty
analysis based on the sampling method.

3.2 Samples for uncertainty analysis
Figure 1 shows the performance of COST samples
in the uncertainty analysis of TMI-1 pin at HFP
conditions. The considered response is k∞. In the figure,the starting point of COST curve is the minimum sample
size point required by COST. The perturbation quantityof Direct Numerical Perturbation (DNP)[6] is set to 1%.
LHS represents the conventional sampling method using
Latin hypercube sampling technique. LHS-10000
represents the results of conventional sampling method(LHS) using 10000 samples as the rough reference value,
although there are still some statistical fluctuations. It
should be noted that all the points in Figure 1 are froman arbitrary random numerical experiment.
According to Figure 1, the following conclusions
can be drawn:(1) With the increase of the sample size, theuncertainty results curves of the response with respect tothe sample size based on the COST method keep almost
flat and the fluctuation range of them is very small,while the LHS method has notable statisticalfluctuations and the curves fluctuate greatly.
(2) The minimum sample size can be determinedby COST according to the input parameter covariancematrix in advance. Therefore, the computational cost of
the sample-size test in conventional sampling method
can be avoided.
(3) The consistent UQ results can be provided by
COST with the minimal sample size, compared with theconventional sampling method with very huge sample
size: Overall, the difference between COST results andLHS-10000 results is small in all cases, which are

3. Verification and Application
In order to verify the COST method in uncertainty
analysis, the uncertainty analysis has been performed to
the TMI-1 pin-cell, proposed by UAM[1]. In the process,the UNICORN code[5] has been applied to propagate thenuclear-data uncertainties to the eigenvalue of the TMI-1
pin-cell.
3.1 Sample characteristics and verification
For intuitive understanding the characteristics of samples
generated by COST compared with those generated byconventional sampling method, two methods are used tosample the two-dimensional standard normal distribution
N2 ( 0 , I2 ). The covariance matrix and distribution of thegenerated samples from an arbitrary random numericalexperiment are shown in Table I.
It can be found that when the sample size increases
from 30 to 20000, the sample covariance matrix
generated by the conventional sampling method (i.e.
LHS) is more likely to approximate the unit matrix I2,
but there are still differences which can be the sourcesof errors in the subsequent uncertainty propagation. For
COST, on the other hand, the sample covariance matrix
can converge to the unit matrix I2 at an initial smaller259

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019within acceptable range. It is worth noting that theresults of LHS-10000 are only taken as the roughreference value; In the first six cases, the results of
COST and DNP are very close, even completely
consistent. This means, the COST method can achieve
high precision reconstruction of the covariance of input
parameters through linear transformation, which will

give a very good approximation of the propagateduncertainty. However, in the latter two cases, thedifferences between deterministic method and sampling
method are obvious, especially for 238U-σ(n,elas), σ(n,inel),σ(n,2n), σ(n,f), σ(n,γ), σ(n,ν), the difference can account for
24.2% of COST and 23.6% of LHS -10000 results.

Table I. Sample distribution and sample covariance matrix of the two method
Conventional sampling method (LHS)

Sample size

Sample covariance matrix

COST

Sample distribution

Sample covariance matrix

300.9410 0.2899 0.2899 1.0902 

1 0 
0 1 



200

0.9921 0.1781
0.1781 1.0072


1 0 
0 1 


2000

 1.0000 -0.0412
 -0.0412 1.0023 

1 0 
0 1 



20000

1.0000 0.0140 
0.0140 0.9999


1 0 
0 1 
Sample distribution

Table Ⅱ. Sample verification results: Top 20 largest eigenvalues and variance of sample projection on corresponding
eigenvectors of 235U-σ(n,γ) input-parameter covariance matrix
Index

λΣ

COST-Var(𝑿𝑷𝑺 )LHS-Var(𝑿𝑷𝑺 )

Index

λΣ

COST-Var(𝑿𝑷𝑺 )LHS-Var(𝑿𝑷𝑺 )1

1.189932053

1.189932053

1.350837908

110.001060731

0.001060731

0.000823979

20.601402038

0.601402038

0.709092319

120.000515916

0.000515916

0.000395084

30.066016454

0.066016454

0.055161019

130.000490640

0.000490640

0.000558967

40.033705620

0.033705620

0.031554108

140.000415083

0.000415083

0.000421806

50.009126260

0.009126260

0.011340763

150.000361361

0.000361361

0.000483240

60.004452515

0.004452515

0.005338535

160.000317896

0.000317896

0.000402570

70.004093384

0.004093384

0.003433930

170.000214704

0.000214704

0.000131445

80.002369551

0.002369551

0.002732174

180.000209785

0.000209785

0.000195423

90.001746378

0.001746378

0.001893451

190.000195123

0.000195123

0.000228734

100.001236687

0.001236687

0.000899516

200.000188628

0.000188628

0.000207613(a)U-  (n, )235(b) 1H-  (n,elas ) ,  (n, )260(c) 90Zr-  (n,elas ) ,  (n,inel) ,  (n, )Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019(d) 16O-  (n,elas ) ,  (n,inel) ,  (n, ) ,  (n, )(e) 235U-  (n,elas ) ,  (n,inel) ,  (n,2n) ,  (n,f ) ,  (n, ) ,  (n, )(f) 238U-  (n,elas ) ,  (n,inel) ,  (n,2n) ,  (n,f ) ,  (n, ) ,  (n, )Fig. 1. Uncertainty analysis results of TMI-1-pin at HFP condition
analysis has been proposed. By investigating theperformance of samples generated by COST in theuncertainty analysis of UAM-TMI-1, the consistentuncertainty analysis results can be provided by COST
with very small sample size, compared with theconventional sampling method with very huge sample
size. It is clear that there are still some areas for COST
method deserve further research and improvement. For
example, if the users want to strictly guarantee theindependence between nuclides, when many nuclides
are considered, the number of computations will
inevitably become very large.
Acknowledgements
This work is supported by the National Natural Science
Foundation of China (Grant No. 11735011)

Fig. 2. Comparison of uncertainty results of different
methods when the 6 reaction channels σ(n,elas), σ(n,inel),σ(n,2n), σ(n,f), σ(n,γ), σ(n,ν) of 238U are perturbed separately

References
1. Ivanov K, Avramova M, Kamerow S, et al.
Benchmarks for Uncertainty Analysis in Modeling(UAM) for the Design, Operation and Safety
Analysis of LWRs. Volume I: Specification andSupport Data for the Neutronics Cases (Phase I)[R].
OECD Nuclear Energy Agency, (2007).
2. Rochman D, Leray O, Hursin M, et al. Nuclear Data
Uncertainties for Typical LWR Fuel Assemblies anda Simple Reactor Core[J]. Nuclear Data Sheets,
139:1-76, (2017).
3. Rahman S, Karanki D R, Epiney A, et al. Deterministic
sampling for propagating epistemic and aleatory
uncertainty in dynamic event tree analysis[J].
Reliability Engineering & System Safety, 175:62-78,(2018).
4. Sahlberg A., Ensemble for Deterministic Samplingwith positive weights: Uncertainty quantification
withdeterministicallychosensamples[D].
UPPSALA University, (2016).
5. Wan C, Cao L, Wu H, et al. Total sensitivity anduncertainty analysis for LWR pin-cells withimproved UNICORN code[J]. Annals of Nuclear
Energy, 99:301-310, (2017).
6. Ball M R, Novog D R Luxat J C. Analysis of
implicit and explicit lattice sensitivities using
DRAGON[J]. Nuclear Engineering and Design,
265(Complete):1-12, (2013).

In order to find the source of the difference, eachreaction channel was analyzed separately, and the results
are shown in Figure 2. In the figure, LHS represents theconventional sampling method, and the sample size used
is 10000. Because the ranks of covariance matrices ofσ(n,inel) and σ(n,2n)are 12 and 2 respectively, which are too
small to accurately describe the standard normal
distribution, the sample size of them is set to 30 in
COST. Therefore, for σ(n,elas), σ(n,inel), σ(n,2n), σ(n,f), σ(n,γ),σ(n,ν), the sample size determined by COST is 51, 30, 30,
69, 51, 37 in turn. The perturbation quantity of DNP is
set to 1%.
As can be seen from Figure 2, the uncertainty
analysis results of the partial reaction channels obtainedby the three methods are consistent. However, thecumulative uncertainty results of 238U by DNP is smaller
than the two sampling methods. This phenomenonshows that in DNP, although the global input parameter
covariance matrix of the 6 reaction channels is used in
calculating the uncertainty of k∞ caused by 238U when
considering the correlation between reaction channels,
the accuracy of the results is reduced by the low-orderapproximation of deterministic method.
4. Conclusion
In this paper, the COST method of generating
multivariate normal distribution samples in uncertainty

261

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Effectiveness of Subcritical Measurement at Solid Moderated KUCA Core for Reducing
Nuclear Data-Induced Uncertainties in Other Light Water Reactor AnalysisaTomohiro Endoa,, Akio Yamamotoa
Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, 464-8603, Japan*
Corresponding author: t-endo@energy.nagoya-u.ac.jp

Abstract
This paper aims to clarify that the subcritical measurement at the solid moderated
KUCA core is useful to reduce nuclide data-induced uncertainties of the effective
neutron multiplication factors 𝑘eff in other light water moderated systems. For
this purpose, we pay attention to the prompt neutron decay constant 𝛼, which wasmeasured by the Feynman-𝛼 method with the moving block bootstrap method forthe deep subcritical KUCA core under the shutdown state without any external
neutron source. The sensitivity analyses of 𝛼 for the subcritical KUCA core andof 𝑘eff for various light water moderated critical cores were carried out based on
the first-order perturbation theory. Consequently, it is demonstrated that thenuclear data-induced uncertainties of 𝑘eff can be reduced by the bias factor
method using the measurement value of 𝛼, if the target 𝑘eff value has a strong
correlation with the measured 𝛼 value in the subcritical KUCA core.
Key Words: effective neutron multiplication factor, uncertainty, KUCA,
subcritical measurement, prompt neutron decay constant, data assimilation
it was demonstrated that the DA technique using theprompt neutron decay constant 𝛼 measured at KUCA
solid moderated core under the shutdown state can reduce
the nuclear data-induced uncertainty of 𝑘eff under thecritical state. Namely, the effectiveness was confirmed forthe just same core.
The purpose of this study is to clarify whether the
subcritical measurement value of 𝛼 in the polyethylene
moderated core of KUCA is also effective for thecriticality analysis for other light water-moderated
reactors.

1. Introduction
This study presents how effective the subcriticalmeasurement at the solid moderated core of Kyoto
University Criticality Assembly (KUCA) is to reduce thenuclear data-induced uncertainties of effective neutron
multiplication factors 𝑘eff for the light water reactor
analysis.
In the field of the reactor physics, 𝑘eff is an important
core-characteristics-parameter to quantify the criticality.
Using the numerical analysis such as the deterministic
method or the Monte Carlo method, 𝑘eff can be
numerically predicted with uncertainties due to thefollowing factors: (1) uncertainties of input parameters(e.g., nuclear data, nuclide compositions, and geometries),and (2) errors of analytical modeling methods (e.g.,
spatial and angular discretization, and energy collapsingin the deterministic method, or statistical error of theMonte Carlo method. Thus, Best Estimate Plus
Uncertainty (BEPU) is an important subject to assure thequality of numerically predicted results. The data
assimilation (DA) technique using actual measurement
results (e.g., the cross section adjustment method [1], andthe bias factor method [2]) is effectively utilized to
improve the prediction accuracy and precision ofnumerical results of 𝑘eff .
Recently, to effectively utilize reactor-physicsexperiment facilities, authors have been investigating DA
using the subcritical measurement [3], instead of
conventional critical experiments. In the previous study,

2. Subcritical Experiment and Sensitivity Analysis of
𝜶 measured at KUCA
The zero-power reactor noise experiment was conductedin the A-core (A3/8” p36EU-NU) at KUCA under the
shutdown state (𝑘eff ≈ 0.937) [4, 5]. Figure 1 shows the
top view of the experimental core and the side view of theloaded fuel assembly. The core averaged 235U enrichment
was ~5.4wt%, which was adjusted by the unit fuel cell(one highly enriched uranium-aluminum alloy plate, one
natural uranium plate, and three polyethylene plates). The
deep subcriticality was achieved by fully inserting all thesix control- and safety-rods and by unloading 3×3 fueland reflector assemblies.

262

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

p

p

p

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p

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1

p

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p

p

p

p

p

p

p

p

p

p

p

F

F

F

F

F

S5

p

p

p

F

F

F

F

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p

p

p

p

p

S4

F

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F

F

F

C1

p

p

p

p

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F

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p

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2

p

p

C2

F

F

F

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F

S6

p

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p

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F

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3

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(a) Top view of core

partial fuel assemblyp

new polyethylene
reflector
airsafety &control rods
3He detector

51.1 cmC3

fuel assembly

×36 polyethylene

p
p

F

20

43.6 cmp
4

3 3 assemblies
are withdrawn

51.4 cmppolyethylenep(b) Side view offuel assembly F

Fig. 1 Subcritical core geometry (A3/8” p36EU-NU).

(a) Top viewb) Side viewFig. 3 Calculation geometry (A3/8” p36EU-NU).

The Feynman-𝛼 experiment was carried out during~933.5 min with only inherent neutron sources, such as
spontaneous fission of 238U and (α,n) reaction of 27Al due
to α-decays of uranium isotopes. In other words, there is
no external neutron source (e.g., Am-Be and Cf source).
To increase the detection efficiency and mitigate thehigher-order mode components in the Feynman- 𝛼method, all the time-series data using four 3He detectors
were summed for the Feynman- 𝛼 method. Using themoving block bootstrap method [4, 5], the measured valueof 𝛼 was obtained as 1618.5±6.5 (1/s) as shown in Fig. 2.
In the bootstrap method, the sample size of bootstrap
resamples was 1000 and the bootstrap correlation of 𝑌(𝑇)between different neutron counting gate widths 𝑇 weretaken into account in the non-linear least square fitting
procedure. Figure 2 indicates that the bootstrap histogramof 𝛼 was well approximated by the normal distribution.

3. Sensitivity Analysis of 𝒌𝐞𝐟𝐟 for Light Water
Moderated CoresFirst, using the Whisper1.1 [12], some criticalexperiments were picked up, according to the strength of
nuclear data-induced correlation (or representative factor)
between 𝑘eff values of the KUCA subcritical core andvarious ICSBEP experiments [13]. Consequently, thefollowing five light water moderated critical experiments
were selected as shown in Fig. 4:
 LEU-COMP-THERM-007, Case 2
 LEU-COMP-THERM-039, Case 5
 LEU-COMP-THERM-002, Case 3
 LEU-COMP-THERM-009, Case 4
 LEU-COMP-THERM-010, Case 3

LEU-COMP-THERM-007, Case 2

LEU-COMP-THERM-039, Case 5

LEU-COMP-THERM-009, Case 4

LEU-COMP-THERM-010, Case 3

LEU-COMP-THERM-002, Case 3

Fig. 2 Bootstrap histogram of 𝛼.

Fig. 4 Top view of selected ICSBEP benchmarks.

The sensitivity analysis (SA) of 𝛼 with respect to the
nuclear data was conducted based on the first-order
perturbation theory, as proposed in the previous study [6].
For SA, the 56 neutron-energy group SN calculations with
P2 anisotropic scattering using PARTISN [7]. The
effective microscopic cross sections were evaluated by
the ultra-fine energy group resonance calculation using
SCALE6.2.3/CENTRM [8] with the ENDF/B-VII.1
library [9]. The calculation geometry was spatiallydiscretized as shown in Fig. 3. The vacuum condition wasused for all boundary because of the asymmetry in theexperimental cores. The total number of discretizedneutron-flight-directions was 384 (points/4π) by an
improved variant of EON quadrature [10,11]. The
numerical result of 𝛼 was 1631.2 (1/s), which agreed
well with the measurement value by comparison with
nuclear data-induced uncertainty 10.0 (% Δ𝛼 ⁄𝛼 ).

Furthermore, to investigate 𝑘eff in the case of acommercial scale light water reactor, a preliminary
numerical analysis of the BEAVRS Cycle 1 HZPbenchmark was carried out based on the information
specified in the document ver. 2.02 [14]. The ACE
formatted libraries at the HZP condition (565.5K) were
produced using the FRENDY [15].
Unlike in the case of SA of 𝛼 using the deterministiccodes, the SA of 𝑘eff for these light water moderated
cores was accomplished by the continuous energy Monte
Carlo code MCNP 6.2 [16] with the ENDF-B/VII.1
library [9]. The relative sensitivity coefficients of 𝑘effwith respect to the nuclear data (elastic+S(α,β), total
inelastic, fission, (n,2n), (n,γ), (n,p), (n,d), (n,t), (n,3He),(n,α), total fission 𝜈 , total fission 𝜒 ) were evaluatedusing the iterated fission probability method. For the
ICSBEP benchmarks, the total number of neutron263

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

histories was 109, i.e., neutron history per cycle=100000,
active cycle=10000, and skip cycle=100. For the
BEAVRS benchmark of which the dominance ratio is
close to unity because of the large core geometry, the total
number of neutron histories was the same but neutron
history per cycle=1000000, active cycle=1000, and skip
cycle=500.

respectively. Based on the previous DA study [4], Vexp,𝛼and Vmodel,𝛼 were determined to be 1.6 × 10−5 and2.0 × 10−4 , respectively.
Table I. Results of the bias factor method using 𝛼.
A posteriori
A priori 𝑘eff
-uncertainty
𝑘eff 𝑐𝛼,𝑘eff
Benchuncertainty𝑇
mark√𝐒𝑘eff ,𝜎 𝚺𝜎 𝐒𝑘eff ,𝜎(-)√V𝑘′ eff (%dk/k)(%dk/k)leu-comptherm-0.866
0.809
0.417
007-002leu-comptherm-0.861
0.802
0.420
039-005
leu-comptherm-0.852
0.765
0.412
002-003leu-comptherm-0.847
0.740
0.404
009-004
leu-comptherm-0.844
0.707
0.389
010-003beavrscycle1-0.254
0.565
0.546
hzp

4. Bias factor method using 𝜶
As discussed in the previous study [2], the nuclear datainduced correlation between the measurement andnumerical values is a key parameter to reduce theuncertainties of numerically predicted results. Thus, using
the sandwich formula with both the relative sensitivity
coefficient matrices of 𝛼 and 𝑘eff (𝐒𝛼,𝜎 and 𝐒𝑘eff,𝜎 ) as
presented in Secs. 2 and 3, the correlation between 𝛼 and𝑘eff can be evaluated as follows:
𝐒𝛼,𝜎 𝚺𝜎 𝐒𝑘𝑇eff,𝜎𝑐𝛼,𝑘eff ≡
,
(1)
𝑇
𝑇 ×𝐒
√𝐒𝛼,𝜎 𝚺𝜎 𝐒𝛼,𝜎𝑘eff ,𝜎 𝚺𝜎 𝐒𝑘eff ,𝜎where 𝚺𝜎 is the relative variance-covariance matrix ofnuclear data. In this study, the 56 group SCALE
covariance data (scale.rev08.56groupcov7.1) was used as𝚺𝜎 [8]. Note that, in this study, 𝐒𝛼,𝜎 and 𝐒𝑘eff ,𝜎 were
calculated by the deterministic and the Monte Carlo
methods, respectively; thus, the estimated correlation by
Eq. (1) has an error owing to the difference in theanalytical modeling method.
Table I summarizes the nuclear data-induced
correlation between (1) the 𝛼 value for the polyethylene
moderated KUCA core and (2) the value for each of lightwater moderated benchmark problems. From Table I, theselected ICSBEP benchmarks have the strong correlation
with the 𝛼 value measured at the subcritical KUCA core,
although the geometry of unit fuel cell (pin or plate-typecell) and the fuel forms (UO2 or uranium-aluminum alloy)
are different. The reason is that the relative variation in
the sensitivity coefficients of 𝛼 and 𝑘eff with respect tothe neutron energy are the almost the same each other inthese cases, as presented in Fig. 5. On the other hand, theenergy dependencies of the 𝑘eff -sensitivity coefficientsin the BEAVRS benchmark are different compared withother cases because the energy spectrum of neutron flux
is different owing to the difference in the core temperatureand the existence of boron in the borated water.
For supplemental information, the bias factor method
using the measurement value of 𝛼 was applied to check
how small the nuclear data-induced uncertainty of 𝑘eff
can be reduced. By this DA, a priori nuclear data-induced
variance of 𝑘eff ( 𝐒𝑘eff,𝜎 𝚺𝜎 𝐒𝑘𝑇eff,𝜎 ) is updated into a
posteriori variance (V𝑘′ eff ) as follows:V𝑘′ eff ≈ 𝐒𝑘eff ,𝜎 𝚺𝜎 𝐒𝑘𝑇eff ,𝜎 −𝑇𝐒𝑘eff,𝜎 𝚺𝜎 𝐒𝛼,𝜎
𝐒𝛼,𝜎 𝚺𝜎 𝐒𝑘𝑇eff,𝜎
𝑇 +V
𝐒𝛼,𝜎 𝚺𝜎 𝐒𝛼,𝜎exp,𝛼 + Vmodel,𝛼2= {1 −

𝑇
𝐒𝛼,𝜎 𝚺𝜎 𝐒𝛼,𝜎(𝑐𝛼,𝑘eff )} 𝐒𝑘eff ,𝜎 𝚺𝜎 𝐒𝑘𝑇eff,𝜎 ,𝑇 +V𝐒𝛼,𝜎 𝚺𝜎 𝐒𝛼,𝜎exp,𝛼 + Vmodel,𝛼Table I also provides nuclear data-induced
uncertainties of 𝑘eff before and after the bias factor
method, i.e., √𝐒𝑘eff ,𝜎 𝚺𝜎 𝐒𝑘𝑇eff,𝜎 and √V𝑘′ eff , respectively.
As expected from Eq. (2), a posteriori uncertainty of 𝑘eff
becomes smaller, as the absolute value of the correlation|𝑐𝛼,𝑘eff | approaches to unity. Consequently, it wasdemonstrated that the stronger correlation results between
𝛼 and 𝑘eff result in the more efficient reduction of 𝑘eff uncertainty.
5. Conclusions
As clarified in this study, the subcritical measurement of
𝛼 measured at the solid moderated KUCA core is usefulto reduce nuclear data-induced uncertainties of 𝑘eff inother light water moderated cores, if the correlation
between 𝛼 and 𝑘eff is strong.
In this investigation, the sensitivity coefficients of 𝛼
and 𝑘eff were estimated by different numerical methods(i.e., the deterministic and the Monte Carlo methods,
respectively) because no existing continuous energyMonte Carlo code was available to carry out SA of 𝛼.
This difference may degrade the correlation between 𝛼
and 𝑘eff , and the subsequent uncertainty reduction by DA.
Thus, further research and development of numericalmethodology of SA of a using the continuous Monte Carlocode are important to address this problem.

(2)

where Vexp,𝛼 and Vmodel,𝛼 are variances owing to theexperimental errors and the analytical modeling errors,264

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4. Endo T, Yamamoto A, “Comparison of Theoretical
Formulae and Bootstrap Method for Statistical Error
Estimation of Feynman-α Method,” Ann. Nucl.
Energy, 124, 606 (2019).
5. Endo T, Yamamoto A, Yamanaka M, et al.,“Experimental Validation of Unique Combination
Numbers for Third- and Fourth-order Neutron
Correlation Factors of Zero-power Reactor Noise,” J.
Nucl. Sci. Technol., 56, 4, 322 (2019).
6. Endo T, Yamamoto A, “Sensitivity Analysis ofPrompt Neutron Decay Constant using Perturbation
Theory,” J. Nucl. Sci. Technol., 55, 11, 1245 (2018).
7. Alcouffe RE, Baker RS, Dahl JA, et al., “PARTISN:A Time-dependent, Parallel Neutral Particle Transport
Code System,” LA-UR-08-07258 (2008).
8. Rearden BT, Jessee MA (Eds.), “SCALE Code
System,” ORNL/TM-2005/39 Version 6.2.3 (2018).
9. Chadwick MB, Herman M, Obložinský P, et al.,“ENDF/B-VII.1 Nuclear Data for Science andTechnology: Cross Sections, Covariances, Fission
Product Yields and Decay Data,” Nucl. Data Sheets,
112, 12, 2887 (2011).
10. Endo T, Yamamoto A, “Development of New Solid
Angle Quadrature Sets to Satisfy Even- and OddMoment Conditions,” J. Nucl. Sci. Technol., 44, 10,1249 (2007).
11. Endo T, Watanabe K, Chiba G, et al., “Nuclear Datainduced Uncertainty Quantification of Prompt
Neutron Decay Constant based on Perturbation
Theory for ADS Experiments at KUCA,” J. Nucl. Sci.
Technol., DOI: 10.1080/00223131.2019.1647893
12. Kiedrowski BC, Brown FB, Conlinet JL, et al.,“Whisper:Sensitivity/Uncertainty-Based
Computational Methods and Software for
Determining Baseline Upper Subcritical Limits,” Nucl.
Sci. Eng., 181, 1, 17 (2015).
13. International Handbook of Evaluated Criticality
Safety Benchmark Experiments, OECD-NEA, Paris,
France, https://www.oecdnea.org/science/wpncs/icsbep/handbook.html (2018).
14. Horelik N, Herman B, Ellis M, et al., “MITBenchmark for Evaluation and Validation of Reactor
Simulations: Release ver. 2.02” MIT Computational
Reactor Physics Group (2018).
15. Tada K, Nagaya Y, Kunieda S, et al., “Development
and Verification of a New Nuclear Data Processing
System FRENDY,” J. Nucl. Sci. Technol., 54, 7, 806(2017).
16. Werner CJ (Ed.), “MCNP User’s Manual Code
Version 6.2,” LA-UR-17-29981 (2017).

(a) 235U: prompt 𝜒p for 𝛼, and total 𝜒 for 𝑘eff(b) 1H: (n,γ)
Fig. 5 Comparison of sensitivity coefficients. The
sensitivity coefficients are normalized by
dividing the maximal or minimal value.
Acknowledgements
This work has been carried out in part under the Visiting
Researcher’s Program of the Research Reactor Institute,
Kyoto University. The authors are grateful to all thetechnical staff of KUCA for their assistance during theexperiment. This work was supported by the Japan
Society for the Promotion of Science Grant-in-Aid for
Scientific Research (C) [Grant Number 19K05328].
References
1. Yokoyama K, Kitada T, “Generalized Formulation ofExtended Cross-section Adjustment Method based on
Minimum Variance Unbiased Linear Estimation,” J.
Nucl. Sci. Technol., 56, 1, 87 (2019).
2. Endo T, Watanabe T, Yamamoto A, “Bias Factor
Method using Random Sampling Technique,” J. Nucl.
Sci. Technol., 53, 10, 1494 (2016).
3. Endo T, Yamamoto A, “Data Assimilation using
Subcritical Measurement of Prompt Neutron Decay
Constant,” Proc. M&C2019, Portland, Oregon, USA,
August 25–29, 2019, American Nuclear Society(2019).

265

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Benchmark study of nuclear processed systems with NCA dataaSatoshi Wadaa, Kenichi Yoshiokaa and Tsukasa Sugitaa,
Toshiba Energy Systems & Solutions Corporation, 4-1 Ukishima-cho, Kawasaki-ku, Kawasaki 210-0862 Japan*
Corresponding author: satoshi9.wada@toshiba.co.jp

Abstract
Critical experiment data of Toshiba Nuclear Critical Assembly were used tobenchmark for FRENDY and NJOY codes, both with three neutron libraries:
JENDL-4, ENDF/B-VII.1, and ENDF/B-VIII.0. The results obtained by using
ACE-libraries processed by FRENDY and NJOY were in good agreements withthe each other.
Key Words: FRENDY, NCA, Benchmark study, NJOY
1. Introduction

2.2 Experimental cases

It is important to accumulate benchmarks that can bereferred in the reactor physics studies i.e. testing new fuels,
designing new cores, and code developments. Toshiba
Nuclear Critical Assembly (NCA) started its operation in
1963, therefore, there are vast data available on the criticalexperiments. Those can be used as reference data when
testing new fuel assembly concepts, void effect, or
gadolinia effect [1]. Recently, a novel nuclear processed
system, FRENDY, was developed by Japan Atomic
Energy Agency (JAEA) [2]. FRENDY is expected to theevaluated nuclear data library JENDL. In addition, thelatest JENDL and/or other evaluated nuclear data libraries
will be able to use by FRENDY treating the GNDS [3]format and the MVP library. In this paper, two nuclear
processed systems, FRENDY and NJOY [4-6] were
evaluated by using the NCA critical-experiment data.

The experimental conditions are summarized in Table I,
and the core configurations are shown in Fig. 3. Theseexperiments simulated fuel assemblies of the boilingwater reactor (BWR): effects of the void fraction and thegadolinia were simulated in the experiments.
3. Critical Analysis
3.1 Calculation conditions
Calculations were carried out by using MCNP 6.2 code [7]with ACE libraries processed by FRENDY [2] and NJOY
[4-6] with JENDL-4.0 [8], ENDF/B-VII.1 [9], and
ENDF/B-VIII.0 [10]. In the current paper, the following
conditions were used in the calculation: 20,000 particle
histories for each batch; total batch of 6,050; 50 skip batch.
The thermal scattering data is used only for hydrogen in
H2O with the following libraries;
1) ENDF/B-VI.8 thermal scattering data and JENDL-4.0
neutron library,2) ENDF/B-VII.1 thermal scattering data and ENDF/BVII.1 neutron library,3) ENDF/B-VIII.0 thermal scattering data and ENDF/BVIII.0 neutron library.

2. Critical Experiments
2.1 Toshiba Nuclear Critical Assembly
Critical experiments were conducted at the Toshiba
Nuclear Critical Assembly (NCA). The main purpose of
NCA is to develop and test new core and fuel assembly
concepts and to develop new nuclear codes. The NCA isa light-water-moderated critical assembly that is allowed
to operate with uranium-dioxide (UO2) fuel rods ofmultiple enrichments. Also, three types of gadoliniabearing UO2 rods are available for the critical experiments.
Fig. 1 shows the vertical cross-section of the NCA-coretank. The criticality of the NCA core can be adjusted by
the water level. For designing light-water reactor (LWR)
cores, void fractions and water densities are the key issues,
therefore, void tubes are used in order to simulate hot 0%
and 40% void conditions in the NCA as shown in Fig. 2.
The cladding of fuel rods and void tubes are made ofaluminum alloy.

3.2 Calculation results
The calculation results are shown in Fig. 4. In the caseswith higher void fraction, the neutron multiplication
factor (k-effective) tended to overestimate theexperimental data. The results obtained by JENDL-4.0
were smaller than that obtained by using the ENDF
libraries. The FRENDY and NJOY results agreed within
0.1 %dk. The effect of the gadolinia-bearing rod wassmaller than that of the void fraction.

266

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4. Conclusions
The current paper evaluated two nuclear processed
systems, FRENDY and NJOY, by using MCNP6.2 andcritical experiment data obtained at the NCA. These
calculation results are in good agreement with the ACEformat libraries processed by FRENDY and NJOY. The
detailed evaluation results and discussions are planned topresent in the conference.
Acknowledgements
The authors would like to thank Dr. K. Tada for kind
support on the calculation with the FRENDY code.
References
Yoshioka K, et al., “CRITICAL EXPERIMENTS
FOR BWR FUEL ASSEMBLIES WITH CLUSTEROF GADLINIA RODS,” Proc. PHYSOR 2014,
Kyoto, Japan, Sep. 28 – Oct. 3, 2014, Japan Nuclear
Energy Society (2014) (CD-ROM).
2. Tada K, et al., “Development and verification of a
new nuclear data processing system FRENDY,” J.
Nucl. Sci. Technol., 54, pp.806-817 (2017).
3. C. M. Mattoon, et. al, “Generalized Nuclear Data:a New Structure (with Supporting Infrastructure) for
Handling Nuclear Data,” Nuclear Data Sheet, 113,
pp. 3145-3171 (2012).
4. MacFarlane R. E., et al., “The NJOY Nuclear
Data Processing System, Version 91,” LAUR-12740-M (1994).
5. MacFarlane R. E., et al., “The NJOY Nuclear
Data Processing System, Version 2012,” LAUR-12-27079 (2012).
6. MacFarlane R. E., et al., “The NJOY Nuclear
Data Processing System, Version 2016,” LAUR-17-200093-M (2016).
7. Werner J., et al, “MCNP6.2 Release Notes,” LAUR18-20808 (2018).
8. Shibata K, et al., “JENDL-4.0: A New Libraryfor Nuclear Science and Engineering,” J. Nucl.
Sci. Technol., 48, pp.1-30 (2011).
9. Chadwick M. B., et al., “ENDF/B-VII.1 nuclear
data for science and technology: Cross
section, covariances, fission product yields anddecay data,” Nuclear Data Sheets, 112(12),pp.2887-2996 (2011).
10. Brown D. A., et al., “ENDF/B-VIII.0: The 8th
Major Release of the Nuclear Reaction Data
Library with CIELO-project Cross Sections, New
Standards and Thermal Scattering Data,” Nuclear
Data Sheets, 148, pp.1-142 (2018).
1.

Fig. 1. Vertical cross section of NCA

Fig. 2. Geometry of fuel rod and void tube

267

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Case ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22

Void fraction
Cold
Hot 0% Void
Hot 40% VoidCold
Hot 0% Void
Hot 40% Void
Hot 0% Void
Hot 40% VoidCold
Hot 0% Void
Hot 40% Void
Hot 40% VoidCold
Hot 0% Void
Hot 40% VoidCold
Hot 0% Void
Hot 40% Void
Hot 40% VoidCold
Hot 0% Void
Hot 40% Void

Table I. Summary of test cases
Gadolinia condition
No gadolinia rod
No gadolinia rod
No gadolinia rod
With 5wt% gadolinia rod
With 5wt% gadolinia rod
With 5wt% gadolinia rod
With 5wt% gadolinia rod
With 5wt% gadolinia rod
With 10wt% gadolinia rod
With 10wt% gadolinia rod
With 10wt% gadolinia rod
With 5wt% gadolinia rod
With 10wt% gadolinia rod
With 10wt% gadolinia rod
With 10wt% gadolinia rod
With 5, 15wt% gadolinia rod
With 5, 15wt% gadolinia rod
With 5, 15wt% gadolinia rod
With 5, 15wt% gadolinia rod
With 10wt% gadolinia rod
With 10wt% gadolinia rod
With 10wt% gadolinia rod

Fig. 3. Core configuration

268

Critical water level (mm)1,137.9907.4
1,019.3
949.6
1,071.5
951.8
877.5
968.9
966.1
874.1
1,182.4
925.7
838.4
935.9
1,079.9
1,026.1
1,176.9
1,060.0
969.8
963.8
1,055.1
1,195.8

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019A) k-effectiveB) comparison of k-effective between using the NJOY and the FRENDY
Fig. 4. Calculation results

269

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Sensitivity analysis for generalized response with RMC Code
Guanlin Shi*, Conglong Jia, Quan Cheng, and Kan Wang
Department of Engineering Physics
Tsinghua University, Beijing, China, 100084,*
shigl16@mails.tsinghua.edu.cn

Abstract
In this paper the generalized sensitivity analysis capability based on GEneralized
Adjoint Response in Monte Carlo (GEAR-MC) method and direct perturbation (DP)method have been developed in the Reactor Monte Carlo Code RMC. The GEARMC method calculates the generalized sensitivity coefficient based on generalized
perturbation theory. Whereas, the direct perturbation method is based on thedefinition of sensitivity coefficient. The newly developeded methods were verified
by comparing the results of DP and GEAR-MC methods with the results of already
implemented collision history-based method through Godiva and PWR pin cellbenchmark problems. Results show that, in general, DP and GEAR-MC methods
agree well with collision history-based method.
Key Words: Monte Carlo, generalized sensitivity analysis, RMC
perturbation for the nuclear data cross section x

1. Introduction

2.2 GEAR-MC method
GEAR-MC method is put forward to perform sensitivity
analysis for the response function in the form of ratio of
two reaction rates, which is expressed as

2. TheoryR

2.1 Direct perturbation method

R R ,(1)
x x
where x is the cross section that need to be analyzed.
Using central differences the following formula can beobtainedx R + -R  ,(2)
S xR 
R x  x

where x and x(3) is the neutron flux, 1 and  2 are any

kinds of nuclear data.

is integration over somephase space.
The change of generalized response function to the
nuclear data x is given by

S xR 



1 ,  ,2 , 

where

The sensitivity coefficient of response R is defined asrespectively. And R and R are the corresponding
results.
In practice, a multi-group structure perturbation factor
library is prepared in advance. If we want 1% increase inelastic cross section then the data saved in perturbation
factor library is 1.01. In transport canculation, theperturbation is introduced by multiplying the perturbation
factor to the cross section. After perturbation, the elastic
cross section will increase  s . In order to keep balance,
the total cross section will also increase s .

In past few years, sensitivity and uncertainty analysis has
become a hot spot in reactor physics and several tools
with sensitivity analysis capability are developed. For
example, the TSUNAMI-3D [1] code of SCALE code
package, McCARD [2], MONK [3], SERPENT2 [4],Open MC [5], MCNP6 [6] and Reactor Monte Carlo code
(RMC) [7]. But only part of them have the ability toperform sensitivity analysis of generalized response
function. In previous work, the collision history-based
method has been implemented in RMC to computegeneralized sensitivity coefficients. In this work, theGEAR-MC method put forward by C. Perfetti et al [8] andthe direct perturbation method are implemented in RMC.
And, the results computed by the GEAR-MC and direct
perturbation methods have been compared with thatcomputed by the collision history-based method.

R 

R
R 
x
x .
x
 x

(4)

So, the generalized sensitivity coefficient can be
expressed as

represent the positive and negative270

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

 R R R x R  x .
S xR 

x x
x R  x R

Assume the neutron source is point source, which can be
expressed as(5)

S  S0 ( P  P0 ) .

The first term on the right hand side of Eq. (5) is known
as the direct term, it represents the influence of theperturbation of the nuclear data on the response function.
And with a little mathematical manipulation, the directterm can be expressed as
R x

x R

d1
d 2
x
x
dx
dx

1
2

.

Substitute Eq. (15) into Eq. (14), the expression for thegeneralized adjoint flux at phase space P0 can be obtained.
* ( P0 ) 



1 x
2 x
x
x
.

1
2

( P0  r )In order to solve the indirect effect terms, The Boltzmann
equation and the generalized adjoint equation will be used.
The Boltzmann equation and generalized adjoint equation
can be expressed as Eq. (8) and Eq. (9) respectively

(A* 

1
* (r ) M ( s  r )  1* F1  *2 F2 +*3 F3 +...+0 .
k(9)where superscript * denotes conjugation of symbols.
Through perturbing the Boltzmann equation, Eq. (8) canbe obtained(kA  M)  ( M  k A)   kA .

 (kA  M)   ( M  k A)   k  A ,
*

*

(10)

To verify the RMC capability of calculating generalized
sensitivity coefficients, the following linear response
function is selected.

(11)

 
R
 

(12)

Combine Eq. (11) and Eq. (12). We can derive that theindirect effect terms can be expressed asR  x1 dM dA x* ( ) .
 x R
k dx dx(13)1
* S  Q*  * M .
k

(14)

238

f
235

f

U

(r , E )(r , E, )dEdrd U(r , E )(r , E, )dEdrd 

.

(19)

The response is calculated in Godiva and UAM TMIPWR pin cell problems. The results of three methods arecompared for Godiva problem. And only the results of
GEAR-MC method and collision history method arecompared for UAM TMI PWR pin cell problem.

and1
 (A*  M* )* = Q* .
k(18)

3. Results

Multiply Eq. (9) by  and Eq. (10) by * , and thentake the inner product respectively, we can get*(17)

It can be solved by using the method similar to iterated
fission probability (IFP) method. But the difference is thatin the IFP method only in asymptotic population theimportance will be calculated whereas in GEAR-MC
method the importance is calculated in both latent andasymptotic generation.

(8)

1 * *
M ) =Q* ,k ( P0  r ) 2 ( P0  r ) .
1 R
 1

R 
1
2 

It can be solved by using Contributon-Linked eigenvalue
sensitivity/Uncertainty estimation via Track length
importance Characterization (CLUTCH) method.
The second term on the right hand side is known as intergenerational term and it can be expressed as(7)kA  M ,

1
1 R
1 * 1
 ( P0  r ) (r ) M ( P0  r ) . (16)
S0
R 
S0
k

The generalized adjoint flux includes two terms. The firstterm on the right hand side is called intra-generationalterm and it can be expressed as(6)

The second term on the right hand side of Eq. (4) is called
indirect effect terms. It represents the impact of
perturbation of nuclear data on the neutron flux andinfluences the response function further more. And it canbe expressed asR  x x R(15)

3.1 Godiva benchmark
The energy-resolved F28/F25 sensitivity coefficients withregard to U-235 inelastic cross section, fission crosssection, disappearance cross section and total crosssection are presented in Fig. 1. In general, GEAR-MC
method and collision history method agree well with eachother.

In order to solve the generalized adjoint flux, the GEARMC method is developed by C. Perfetti.
Multiply Eq. (10) by * and Eq. (9) by  andintegrating over some phase space, respectively, we canobtain271

Sensitivity coefficient per Unit Lethargy

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

0.0

-0.1

-0.2

-0.3

GEAR-MC elastic
GEAR-MC n,n'
GEAR-MC fission
GEAR-MC n,gamma
Collision history elastic
Collision history n,n'Collision history fission
Collision history n,gamma-0.4
10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107

Energy (ev)

Fig 1. Sensitivity coefficients of 235U

Fig 3. Sensitivity coefficients of 235U

Some energy integrated sensitivity coefficients of 235U arepresented in Table I. As can be seen from the Table I, threemethods agree well with each other and the GEAR-MC
method could obtain lower variance than the collisionhistory-based method. From Fig.1, it is easy to notice that
the F28/F25 are more sensitive to the fission cross sectionof 235U. This is mainly because the contribution from thedirect term in Eq. (6).
3.2 UAM TMI PWR pin cell benchmark
Fig. 2 through Fig. 5 show the energy-resolved F28/F25
sensitivity coefficients with regard to 1H n,gamma crosssection, 235U fission cross section and n,gamma crosssection, 238U inelastic cross section, fission cross section
and 16O elastic cross section respectively. From thesefigures we can also find that GEAR-MC method agree
well with collision history method.

Fig 4. Sensitivity coefficients of 238U

Fig 5. Sensitivity coefficients of 16O

Fig 2. Sensitivity coefficients of 1H

Some energy integrated sensitivity coefficients of this pin
cell problem are presented in Table II. As can be seen fromthe Table II, relative differences between GEAR-MC
method and collision history-based method are within 4%,
which means GEAR-MC method generally agrees with272

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

collision history-based method.
4. Conclusions

3.

In this work the capability of computing generalized
sensitivity coefficients based on the GEAR-MC methodand direct perturbation method have been implemented in
RMC. The Godiva benchmark and the UAM TMI PWR
pin cell benchmark are used to verify the newly
implemented methods. The results calculated by these
two methods agree well with previously implemented
collision history method..

4.

5.
6.

Acknowledgements
This research is partially supported by China ScholarshipCouncil, National Natural Science Foundation of China(GrantNo: 11475098).

7.

References
1.

2.

Perfetti C M, Rearden B T, Martin W R. “SCALE
Continuous-Energy
Eigenvalue
Sensitivity
Coefficient Calculations.” Nucl. Sci. Eng. 182,
pp.332–353, (2016).
Shim H J, Kim C H. “Adjoint sensitivity anduncertainty analyses in Monte Carlo forward

8.

calculations.” J. Nucl. Sci. Technol. 05, pp. 1453–
1461(2011).
Baker C, Smith P N, Mason R, et al. “Calculating
Uncertainty on k-effective with Monk 10.” ICNC
2015, Charlotte, NC, September 13-17, (2015).
Aufiero, M., Bidaud, A., Hursin, M., et al. “A
collision
history-based
approachtosensitivity/perturbation
calculations
inthecontinuous energy Monte Carlo code SERPENT.”
Annals of Nuclear Energy. 85, pp. 245-258(2015).
Peng, Xingjie, et al. “Development of continuousenergy sensitivity analysis capability in OpenMC.”
Annals of Nuclear Energy 110, pp. 362–383(2017).
Kiedrowski, Brian C., F. B. Brown, and P. P. H.
Wilson. “Adjoint-Weighted Tallies for k-Eigenvalue
Calculations with Continuous-Energy Monte Carlo.”
Nuclear Science & Engineering the Journal of theAmerican Nuclear Society 168(3), pp. 226-241(2011).
Qiu, Y., Aufiero, M., Wang, K., Fratoni, M.
“Development of sensitivity analysis capabilities ofgeneralized responses to nuclear data in Monte Carlocode RMC.” Annals of Nuclear Energy 97, pp. 142152(2016).
Perfetti, Christopher M., and B. T. Rearden.
“Development of a Generalized Perturbation Theory
Method for Sensitivity Analysis Using ContinuousEnergy Monte Carlo Methods.” Nuclear Science &Engineering the Journal of the American Nuclear
Society 182(3), pp. 354-368(2016).

Table I. Energy integrated sensitivity coefficients of 235U
Measurement

DP

GEAR-MC

Relativestandard
deviation

Collision
history

Relativestandard
deviation
1.02E-02

GERA-MCCollisionhistory
Relativedifference (%)-0.14

DPCollisionhistory
Relativedifference (%)
3.40

elastic-7.60E-02-7.34E-02

7.59E-03-7.35E-02

inelastic-3.33E-01-3.44E-01

9.92E-04-3.36E-01

1.22E-03

2.38

-0.89

fission-8.52E-01-8.62E-01

3.54E-04-8.64E-01

3.58E-04

-0.23

-1.39

capture3.90E-02

3.57E-02

8.50E-04

3.60E-02

2.68E-03

-0.83

8.33

Table II. Energy integrated sensitivity coefficients of different nuclide
NuclideReaction

GEAR-MC

92235

fissioncaptureinelastic
captureelastic
capture-2.80E-01
1.66E-01-2.26E-01
2.09E-01-7.51E-02
4.17E-02

92238
8016
1001

Relativestandard
deviation
7.56E-04
3.53E-04
7.84E-04
4.01E-04
9.56E-03
3.89E-04

273

Collision history-2.76E-01
1.63E-01-2.19E-01
2.05E-01-7.60E-02
4.10E-02

Relativestandard
deviation
1.27E-03
8.30E-04
1.21E-03
7.94E-04
1.26E-02
1.60E-03

Relativedifference (%)
1.45
1.84
3.20
1.95
-1.18
1.71

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Uncertainty analysis of neutron parameters for NESTOR
Hongkuan LIAO, Qing LI, Yingrui YU*, Yuying HU
Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu,
China*
Corresponding author: yuyr78@163.com

Abstract
Due to the complexity of the reactor system, many approximations are used in the nuclear
design and calculations inevitably. The accuracy of the nuclear design software is closely
related to the safety of the reactor design and operation. Besides, improving the accuracy
is an effective way to excavating the economy of nuclear power plants. In this study, On
the basis of the random sampling statistical analysis (RSSA) method, the uncertainty ofcontrol rod worth and boron concentration were obtained by comparing the measured
values and the theoretical values modeled by NESTOR. The results indicate that the
RSSA is feasible in the uncertainty analysis of nuclear design. As a result, the final
computational uncertainty of the worth of control rod is ±7.447% and the final
computational uncertainty of the boron concentration is ±49.71 ppm of NESTOR
software.

Key Words: NESTOR, Uncertainty Analysis, RSSAsp

Nomenclature
Average of samples
Ensemble averagespSample standard deviation

Nt

Total number of samples
T distribution

2

X95/95

 p,95%

 p,95%P
BURNUPCB
EFPD
ppm

Chi square distribution
Under the limit of 95%
confidence coefficient and 95%
probability
Ensemble average of 95%
confidence coefficient
Total standard deviation of 95%

pcm
ρ

CNP650Confidence level
Probability level
The PWR designed by China,
consisting
of
121
fuel
assemblies, power rating of1930MW[3].
A nuclear design software
package designed by CNNC[4].

Burnup unit
All Rod Clusters Out
Main regulating rod group D all
in the core, while others all outof the core
Relative power of the core
Burnup in the core
Boron Concentration
Equivalent Full Power Day
10-6（weight ratio）, using insoluble boron concentration
10-5(Δk/k)
The worth of control rod
1. Introduction

Uncertainty always exists in the engineering design
such as the uncertainty of engineering system and theuncertainty of the environment[1]. So the uncertainty
represents the credibility and trust of the system.
The reactor design is a typical and complex
engineering design, and its uncertainty is very importantto the thermal hydraulics analysis.
NESTOR nuclear design software package can beused to analyze full range of nuclear design contents. In
this study, as important neutron parameters, the worth ofthe control rod and the boron concentration are chosen tobe evaluated.

confidence coefficient
Owen factor[2]NESTOR

Din

Total standard deviation

H  NP 

MWd/tU
ARO

1274

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

H(Np) is Owen factor. It is decided by the size of thesample space, which can be obtained by table look-up or
calculation with the equation, as below .

CORCA software is the core software of NESTOR,
which is a 2D finite difference software. The theoreticalvalues of the worth of control rod and the boron
concentration are provided by CORCA, and theexperimental measurements are used for the value
comparison. After evaluating the applicability of RSSA bystatistical analysis of the result, the uncertainty of thecontrol rod worth and the boron concentration of
NESTOR were given. The theoretical value of the control
rod is obtained by simulating the ARO core state and thefull insertion of the tested control rod group into the corestate. The theoretical value of boron concentration is
obtained by simulating the physical test state of thenuclear power plant once a month, and the consideredstate parameters include core power, control rod position,
temperature and pressure.
2. Method and Code

 1.3528  N P  1
1  1  1 
 NP 1  NP
H  N P   1.64485 
1.35281
NP 1
2.2 Nonparametric Statistics
The nonparametric statistics is to test the generalcharacteristic to obtain the population tolerance limit
under certain confidence and certain probability.
The process of quantifying the uncertainty of input
parameters by the nonparametric statistics is shown in
Fig. 1.

Sensitivity and uncertainty analysis method could be
divided into two categories which are the statisticalanalysis and the deterministic calculation. The mostwidely used statistical analysis method is the sampling
method. In this study, we have quantified the influence ofimportant plant state parameters with the parametricstatistics method and the nonparametric statistics
method[2].











 x ，x ， x

2
n 1 
  X 1 
 1




  X 2 
 x ，x 2， x n 2 
input  operation matrix  1
 





  X  N
 x1，x 2， x n N  Y1 Y 
code cacultion
 output 2 


Y 
N



2.1 Parametric Statistics

Fig.1 the Process of Nonparametric Analysis

Normal distribution confidence limit method is atypical parametric statistics method. First of all, all the
distribution characteristics of the input parameters should
be confirmed. There are 3 steps in the method.
 Distribution test,
 Estimating the population mean and standard
deviation,
 Analyzing the 95/95 limits.
Firstly, the distribution of samples would be tested byrelationship among the mean and standard deviation ofsamples and the population would be presented as below
if the test was passed.
s   p( N  1) s 2(1)~ t N 1
~  2 N 1

This method was presented by Wilks in 1941[4, 5].
Firstly, the important input parameters and theirdistribution should be determined, with sampling theseparameters and obtaining the operation matrix. All thesamples would generate the result of the output parameter
after the code calculation. Then the tolerance limit could
be obtained by sorting the output parameter.
For the single tolerance limit, the confidence and theprobability should satisfy the condition described below.
(5)  1  N
For the bilateral tolerance limit, the confidence andthe probability should satisfy the condition as Equation(6).
(6)
  1   N  N 1     N-1

Secondly, the population mean and standard deviation
could be presented by the sample mean and standard
deviation shown as Equation (2) and Equation (3).
 s  t /2  N  1  s / N  ＜p＜ s  t /2  N  1  s / N  (2)3. Uncertainty Analysis

the test method such as

s



N




 s   N  12

 2  N  12

fitting testing[3]. The

 p2



   N  1
   N  1

Based on the given confidence level βand probability
level γ, the required number of samples N can be
determined through the inequalities (5) and (6).

2＜ p 2＜ s2

In this section, the worth of control rod and the boron
concentration are been analyzed. The theoretical model ofplant was established by NESTOR software according tothe operating status data of the nuclear power plant suchas power, burnup, and boron concentration.

(3)
Thirdly, the 95/95 limit would be obtained as below.
(4)
X95/95   p，95%  H  NP   p，95%
/2

1 /2

2

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Osaka, Japan, Dec. 2-3, 2019

Samples are needed to be determined by sampling for
RSSA method, but in this study, as the measured value ofcontrol rods coming from the start-up physicalexperiments of nuclear power plants, the measured boron
concentration comes from the periodic physical test of thenuclear power plant once a month during the operation ofplants which are obtained by chemical titration. It can beconcluded that the calculation of control rod value andboron concentration is closely related to the uncertainty ofreactivity calculation, but not to the specific state of thereactor core. Therefore, in this study, all the measured
data is directly used to instead of sampling to obtain corestatus.
For RSSA method, relative or absolute deviation can
be chosen as the object of uncertainty analysis according
to the characteristics of the study, which will not affect theconclusions. In this study, the unit of control rod is pcm.
As the value of each group of rods varies greatly, relative
variation is used, (pcal-pmes)/pmes. As the concentration ofboron varies with burnup, the absolute variation is used ,CBcal-CBmes.

3
4
5
6
7

(-3%,-1%] 0.208947(-1%,1%] 0.260961(1%,3%] 0.212480(3%,5%] 0.112772(5%，∞) 0.049194

total(-∞，∞)
According to

20.6858
25.8351
21.0356
11.16454.8702

1

22 1.31
25 -0.84
19 -2.04
15 3.84
3 -1.87

0.083500.02700
0.19698
1.31769
0.7182099

4.2731 2 fitting testing[3], as it is shown in

Table 1, pi is the theoretical probability in the span, and n
is the total number of samples. ni is the number ofsamples in actual range, and  ni  n *pi 2  n *pi means the divergence of each value interval. The totalk  ni  n *pi   n *pi  is 4.2731, smaller
2
2than   k  r  1  0.05  4   9.4877 . The degree ofdivergence2i1

difference presents that the result meets normal
distribution.
Thus, the bilateral tolerance interval of  p is [0.651%, 0.729%], and the tolerance interval of  p is

3.1 The worth of control rod[0.0263, 0.0349]. The Owen factor H(Np) is 1.92499.
The upper limit under the condition of 95%
confidence coefficient and 95% probability of the worthof control rod is 7.447%, and the lower limit is -7.369%.
The uncertainty of the worth of control rod is±7.447%.

In order to ensure the conservativeness of the analysis
results, there are totally 99 measured values of unit 1 to 4
of Qinshan II Nuclear Power Plant, unit 1 and unit 2 ofDaya Bay Nuclear Power Plant are selected. And the 99
theoretical values of the measured states were obtained by
NESTOR software. Then, the uncertainty of control rod
worth was provided by comparing the theoretical values
and measured values. Figure 2 presents the deviation of
the theoretical values and measured values.

3.2 The boron concentration
Similar to the method of analyzing control rod worth,
there are totally 112 measured values of unit 1 to 4 of
Qinshan II Nuclear Power Plant. Unit 1 and unit 2 ofDaya Bay Nuclear Power Plant are selected to analysis
the uncertainty of the boron concentration calculated by
NESTOR software. And the uncertainty of the boron
concentration was provided by comparing the theoreticalvalues and measured values of the 112 measured states.
Figure 3 presents the deviation between the theoreticalvalues and measured values.

Fig. 2 Deviation of the Worth of the Control Rod

Table 1 Calculation data of 
i
1
2

2

fitting testing
nin*pi(nin*pi)2/n*pi(-∞，-5%] 0.046593 4.6127 2 -2.61(-5%,-3%] 0.109052 10.7961 13 2.20

1.479850.44988

Span

pi

n*pi

ni

Fig. 3 Deviation of the Boron Concentration

3276

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Osaka, Japan, Dec. 2-3, 2019

The degree of difference presents that the result meetsnormal distribution. Thus, the bilateral tolerance intervalof  p is [11.48 ppm, 17.70 ppm], and the toleranceinterval of  p is [12.89, 16.79]. The Owen factor H(Np)is 1.906487.
The upper limit under the condition of 95%
confidence coefficient and 95% probability of the boron
concentration is 49.71 ppm, and the lower limit is -20.60
ppm. The uncertainty of the boron concentration is±49.71 ppm.
4. CONCLUSION
The computational uncertainty of important neutron
parameters is very important to evaluate the economy andsafety of nuclear power plants. In this study, theuncertainty of control rod worth and the boron
concentration are analyzed. Finally the computational
uncertainty of the worth of control rod is ±7.447% and thecomputational uncertainty of the boron concentration is±49.71 ppm of NESTOR software.
It is proved that RSSA method is applicable to theuncertainty analysis of the neutron parameters; it can beapplied to the uncertainty analysis of other neutron
parameters such as moderator temperature coefficient andboron differential value in the future.
References
1.

2.

3.
4.
5.

Wen Yao, Xiaoqian Chen, Wencai Luo, Michel van
Tooren, Jian Guo, “Review of uncertainty-based
multidisciplinary design optimization methods foraerospace vehicles,” Progress in Aerospace
Sciences, 47(2011).
Glaeser H, “RS Method for uncertainty andsensitivity evaluation of code results andapplications , “cience and Technology of Nuclear
Installations, 2008.
Dovore, Jay L, Probability and Statistics forEngineering and Sciences, The Thomsom
Corporation, 2004.
Wilks S S, Determination of sample sizes for setting
tolerance limits[J], Heat Nuclear
Technology,1995,112:355-358.
Wilks S S, Statistical prediction with specialreference to the problem of tolenrance limits[J], The
annals of mathematical statistics, 1942:400-409.

4277

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Comparison of Methods of Generating Covariance Matrix of Fission Yield
Keisuke Hontaa , Go Chibaa
Hokkaido University, Sapporo, Japan,*
Corresponding author: keisukehon@eis.hokudai.ac.jpaAbstract
The modern evaluation method of generating covariance matrix of independent
fission yields based on the generalized least square method is introduced, anduncertainties in nuclides number densities after nuclear fuel burnup arecalculated using the generated covariance matrix and burnup sensitivities. In thiswork, calculations are performed using the pincell system as a preliminary study.
Comparison with results based on the simple covariance generation method ismade, and uncertainties of fission product nuclides number densities after
burnup varies depending on the nuclides by switching the methods of generating
covariance matrix of fission yields.
Key Words: fission yield, uncertainty calculation, correlation
1. Introduction

By the GLS method, original nuclear data 𝑻𝟎 is adjusted
̂ , and the covariance matrix of the original nuclear
to 𝑻
̂ . In Eqs (1) and (2), 𝑹(𝑻𝟎 ) isdata 𝑽𝝈 is updated to 𝑽
integral data predicted by using 𝑻𝟎 , 𝑮 represents̃ is measured
sensitivity of 𝑹(𝑻𝟎 ) to 𝑻𝟎 , and 𝑹
̃ and 𝑹(𝑻𝟎 )integrated data. The covariance matrix of 𝑹
is represented as 𝑽𝒆𝒎 .
In the application of the GLS method to fission yield
covariance matrix generation, independent fission yields
are adjusted by using several physical constraint
conditions.

In evaluated nuclear data files, only variance is provided
as uncertainty information on fission yield. In our
research group, works on uncertainty propagation fromfission yield to reactor physics parameters have been
conducted [1], and fission yield covariance has been
generated by a method adopted in the previous work by
Katakura [2]. In the present paper, this method is referred
to as Katakura’s method. Katakura’s method canapproximately consider the fission yield covarianceamong fission product (FP) nuclides belonging to thesame mass chain. In recent years, a new evaluation
method for fission yield covariance matrices have been
proposed by Fiorito et al [3]. This method is based on the
Generalized Least Square (GLS) method. Under thiscircumstance, we introduce this new evaluation method,and compare results of uncertainty quantification with
Katakura’s method. Comparisons are made about FP
nuclides number density uncertainty induced by fission
yield uncertainties.

2.2 GLS updating technique
The fission yield covariance matrix adjustment procedureusing the GLS method which has five steps is shown in
Fig. 1.

2. Theory
2.1 Basic equation of GLS
The GLS method is a method of revising the original input
data by using new observable data dependent on the input
data. In the field of nuclear reactor physics, it is used toadjust nuclear data using integral data. Basic equations
used to adjust the nuclear data are shown in Eqs. (1) and(2) [4].
̂ = 𝑻𝟎 + 𝑽𝝈 𝑮𝑻 (𝑽𝒆𝒎 + 𝑮𝑽𝝈 𝑮𝑻 )−𝟏 (𝑹̃ − 𝑹(𝑻𝟎 )),𝑻(1)

̂ = 𝑽𝝈 − 𝑽𝝈 𝑮𝑻 (𝑽𝒆𝒎 + 𝑮𝑽𝝈 𝑮𝑻 )−𝟏 𝑮𝑽𝝈 .
𝑽(2)Fig. 1. Updating scheme used to adjust mean values andcovariance matrix of fission yields

278

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

In the following, each of the five steps is described indetail. Note that we do not consider independent fission
yields of light charged particles in the present study.

yields of nuclides with large mass number is fixed. The
sensitivity vector G in this case is 0 for nuclides withsmall mass number and 1 for nuclides with large mass
number.

(A) Condition on chain yields
The first step is about a constraint of chain yields. A chain
yield 𝐶ℎ𝑖 represents the total yield for a given decay
chain and is evaluated after both prompt and delayed
neutron emissions, and 𝑖 is a mass number of a stable
nuclide at the end of this decay chain. A vector of chain
yields 𝑪𝒉 can be represented by independent fission
yield vector 𝒀 and a matrix 𝑫 as follows:
𝑪𝒉 = 𝑫𝒀.

For the diagonal component of 𝑽𝒆𝒎 for the condition(A), variance of the cumulative fission yields in
JENDL/FPY-2011 are used. Note that non-diagonalcomponents of 𝑽𝒆𝒎 are 0 since no correlation isassumed among different chain yields. For the conditions(B) to (E), 𝑽𝒆𝒎 are set as 0.01 % since these conditions
should be strictly preserved.
̃ = 𝑹(𝑻𝟎 ) , so
In all the conditions, we assume 𝑹
independent fission yield itself is not updated and only
covariance matrix is updated in the present calculation.
On the other hand, it is possible to give a condition that,for example, Eq. (7) becomes 2.0, but in such a case, it is
not guaranteed that 𝑻 becomes identical to 𝑻𝟎 .

(3)

In the present study, the 𝑫 matrix is created as follows.
A square matrix corresponding to the number of nuclides
whose size is 1,400 is prepared, and its diagonalcomponent is 1 for stable nuclides and 0 for other nuclides.
In the non-diagonal components of this matrix, the decaybranching ratio from nuclide 𝑖 to nuclide 𝑗 given by the
nuclear data is put to the (𝑗, 𝑖) element of this matrix. After
that, the matrix is multiplied iteratively. When there is nodifference between before and after the multiplication, thematrix is regarded as the 𝑫 matrix.
In this condition, a constraint of chain yields is posed.
Sensitivity vector 𝑮 in the adjustment equation is the 𝑫
matrix itself.

3. Numerical Test
Independent fission yields of 23 fissile nuclides shown in
Table I are considered in the present study. All the data aretaken from JENDL/FPY-2011. Covariance matrix of thesefission yield data are updated, and then using the updated
covariance matrix, uncertainties of FP nuclides number
densities after burnup in UO2 fuel with a uranium
enrichment of 4.1 wt% and MOX fuel are calculated. The
burnup is 45 GWD/t, and the number of target FP nuclides
is 92.
All the calculations are carried out with a reactor
physics code system CBZ, which is under development at
Hokkaido University. Nuclear data-induced uncertainties
of nuclides number densities after burnup are quantified
by the so-called sandwich formula with sensitivities of
nuclides number densities with respect to nuclear data.
These sensitivities are calculated by a Burner module forpin-cell burnup calculations. In the Burner module thedepletion perturbation theory considering spatial
dependence is implemented [5]. Fuel burnup calculations
are carried out with 107-group cross sections and adetailed burnup chain consisting of 1,400 fission productnuclides which are defined in JENDL/FPY-2011.
Sensitivities of number densities of concerned 92 nuclides
to half-lives, decay branching ratios and fission yields ofall the fission product nuclides are calculated by the
Burner module. For the sensitivity analysis of burnup
calculation, the implicit effect has not been considered.

(B) Condition on atomic number∑ 𝑍𝑖 𝑌𝑖 = 𝐶𝑜𝑛𝑠𝑡.

(4)𝑖

𝑍𝑖 and 𝑌𝑖 are the charge value and independent fission
yield of fission product nuclide 𝑖 , so this equationsuggests that the number of protons is fixed. In thesensitivity vector G, the entry becomes charge value ofthe corresponding nuclide.
(C) Condition on mass number∑ 𝐴𝑖 𝑌𝑖 = 𝐶𝑜𝑛𝑠𝑡.

(5)𝒊

𝐴𝑖 is the mass number of fission product nuclide 𝑖 , so
this equation suggests that the number of nucleons is fixed.
In the sensitivity vector G, the entry is the mass numberof the corresponding nuclide.
(D) Condition on fission yield (1)∑ 𝑌𝑖 = 𝐶𝑜𝑛𝑠𝑡.

(6)

Th-232
U-235
Pu-238Am-241
Cm-244𝑖This equation suggests that the sum of independent fission
yields is fixed. All the entries of the sensitivity vector G
are unity.

∑

𝑌(𝐴𝑖 ) = 𝐶𝑜𝑛𝑠𝑡.

U-232
U-237
Pu-240
Am-243
Cm-246

U-233
U-238
Pu-241
Cm-242

U-234
Np-237
Pu-242
Cm-243(With underline: fast fission Others: thermal fission)(E) Condition on fission yield (2)𝐴𝑐𝑛 −𝝂̅𝑝 (𝐸)𝐴𝑖 ≥2Table I. Fissile materials

Pa-231
U-236
Pu-239Am-242m
Cm-245(7)This equation suggests that the sum of independent fission279

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

4. Numerical Result

between the GLS method and Katakura’s method. As Fig.
2, forty-six (half of 92) nuclides are shown. According to
Fig. 3, when using the GLS method, it can be seen that theuncertainty tends to increase. Furthermore when it
increases, the difference is up to about 12 % in the case of
MOX fuel. Also, looking at the difference due to fuel, thetendency of increase or decrease in uncertainty does notchange, but there is a difference in degree.

4.1 Changes in uncertainty due to correlation
Although the independent fission yield should beconsidered for correlation in nature, it is necessary toquantify the correlation in some way because it is not
given in evaluated nuclear data files. Figure 2 shows FP
nuclides number density uncertainty for two cases when
the correlation of independent fission yield covariance istaken into account and not taken into account in the caseof UO2 fuel at 45 GWD/t. Correlation is considered by
Katakura’s method. Forty-six (half of 92) nuclides areshown here due to page limitations. It is possible toconfirm that the uncertainty decreases when considering
the correlation among all the target FP nuclides [3].

Fig. 3. Uncertainty difference due to adjustment method
at 45 GWD/t
4.3 Impact assessment of each step in the GLS method
Figure 4 shows the contribution to FP nuclides number
density uncertainty for every five steps in the GLS methodin U-235 thermal fission case. Although the reduction
effect of each step differs depending on the nuclide, it canbe seen that the reduction effect of (A) chain yield
condition is the largest. As shown in the reference [3], it
is dominant that the influence of the (A) chain yield
condition to update the covariance matrix.

Fig. 2. FP nuclides number density uncertainty of UO2 at
45 GWD/t
4.2 Uncertainty difference due to covariance matrix
generating method
The difference in the uncertainty of the FP nuclides
number density after burnup using the conventional
Katakura’s method and the GLS method is shown in Fig.
3. The values in Fig. 3 are the results of difference280Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019(b) The GLS method case

4.4 Comparison of correlation coefficient by covariancematrix generation method

5. Conclusion
The new evaluation method of covariance matrix ofindependent fission yields has been introduced.
Furthermore, the burnup calculation has been performed
using the result given by the new method. As a result ofcomparing the evaluation methods, the uncertainty of FP
nuclides number density after burnup have varied
depending on the nuclides. In particular, when increasing,some have exceeded 10 % or more. In this work,
calculations have been performed using the pincell
system as an initial study, but in the future, it will benecessary to perform calculations close to reality, such as
calculation of fuel assemblies, for evaluation.

Fig. 4. Uncertainty reduction effect per step at U-235
thermal fission

The target is Ce-144, which has a relatively large
difference between the two methods in Fig. 3, and thenuclide is important nuclide in the passive gamma method.
Focusing on the sensitivity to fission yield of the nuclides
number density after burnup, it is possible to grasp the
difference due to the adjustment method by confirming
the correlation coefficient matrix of the independent
fission yields of the nuclides having relatively larger
sensitivity. Table II describes the names of nuclides witha sensitivity of 1.0 % or more for Ce-144 nuclides in U235 thermal fission yield. In Ce-144, all sensitive
nuclides have the same mass, and it becomes Cs-144 →Acknowledgements
This work was supported by the Nuclear Regulation
Authority (NRA). Authors thanks to Dr. Toshihisa
Yamamoto and Dr. Tatsuya Fujita for their helpful adviceson this work.

Ba-144 → La-144 → Ce-144 by a series of 𝛽 − decay.
The adjusted correlation coefficient matrix of nuclides
with these sensitivities is shown in Table II. Table III (a)and (b) show the result of the conventional Katakura’s
method and the GLS method, respectively. The GLS
method weakens all negative correlations other than thecorrelation coefficient of each nuclide itself as comparedwith Katakura’s method. In particular, the correlation
coefficient between Ba-144 and La-144 is -0.805 in
Katakura's method and -0.199 in the GLS method. This
indicates that the negative correlation is significantly
weakened.

References
1. Nihira S, Chiba G, "Uncertainty Quantification of
Nuclides Number Densities after Fuel Depletion by
Random Sampling Method," Proc. Reactor Physics
Asia 2017 Conference, RPHA2017, Chengdu, China,
Aug. 24-25, 2017 (2017).
2. Katakura J, “JENDL FP decay data file 2011 andfission yields data file 2011.” JAEA-Data/Code
2011-025, Japan Atomic Energy Agency, (2012).
3. Fiorito L, et al., "Generation of Fission Yield
Covariances to Correct Discrepancies in the Nuclear
Data Libraries." Ann. Nucl. Energy, 88, 12 (2016).
4. Dragt, JB, et al., "Methods of adjustment and error
evaluation of neutron capture cross sections;application to fission product nuclides." Nucl. Sci.
Eng. 62, 117 (1977).
5. Chiba G, et al., "Development of a fuel depletion
sensitivity calculation module for multi-cell
problems in a deterministic reactor physics code
system CBZ." Ann. Nucl. Energy, 96, 313 (2016).

Table II. Sensitivity of certain nuclides fission yieldto Ce-144 fission yield in U-235 thermal fission
FP
Sensitivity

Cs-1440.0384Ba-1440.3770La-1440.1010

Table III. Specific correlation coefficient matrix of
U-235 fission yield after adjustment for Ce-144(a) Katakura’s method case281

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Application of Surrogate Modeling with Singular Value Decomposition for Design Basis
Accident aiming Statistical Safety Analysisa

Masaki Matsushitaa,*, Tomohiro Endoa and Akio Yamamotoa
Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, 464-8603, Japan*
Corresponding author: m-matsushita@fermi.energy.nagoya-u.ac.jp

Abstract
This study aims to reduce the computational cost by an accident analysis code (e.g.,
RELAP5/SCDAPSIM, MAAP, MELCORE) for statistical safety analysis. In
order to reduce the computational cost, this study applies the reduced ordermodeling (ROM) with the singular value decomposition and the low-rankapproximation to time-series data obtained by a safety analysis code. Applying the
ROM to the time-series data of RELAP5/SCDAPSIM, statistical safety analysis
is carried out by the ROM instead of many RELAP5/SCDAPSIM calculations.
This paper applies the ROM to the load rejection accident with an open failure of
the turbine bypass valve for BWR, which is categorized as a design basis accident.
The result of statistical safety analysis by ROM is almost consistent with that by
RELAP5/SCDAPSIM. Therefore, statistical safety analysis by ROM is revealed
applicable with reduction of calculations by RELAP5/SCDAPSIM from 1,000 to
135 in this study.
Key Words: Statistical Safety Analysis, Singular Value Decomposition,
Reduced Order Modeling, RELAP5/SCDAPSIM, Design Basis Accident,
Load Rejection of BWR
severe accident, to evaluate core damage frequency witha small computational cost. However, there are many
accident scenarios in nuclear power plants. Therefore, it
is desirable that the applicability of the proposed ROM
method will be confirmed for various accident scenarios.
The purpose of this study is to apply the proposed
ROM theory to a design basis accident (the load rejection
with an open failure of the turbine bypass valve in a
BWR) and to show the applicability of ROM for thestatistical safety analysis.

1. Introduction
This study presents the application of a surrogate modelfor the load rejection with an open failure of the turbine
bypass valve, which is categorized as a design basis
accident for BWR aiming statistical safety analysis.
After the severe accident at Fukushima Daiichi
nuclear power plant in March 2011, the safety for nuclear
power plant has been reaffirmed as the highest priority. In
order to evaluate the safety, statistical safety analysis iscarried out. In the statistical safety analysis, theuncertainty should be evaluated by calculations of a safety
analysis code (SA-code). In general, random sampling
can be used for the estimation of uncertainty. However,enormous number of samples is necessary for accurate
prediction of the uncertainty, resulting in increase of thecomputational cost. Therefore, development of thesurrogate model, which can rapidly reproduces the resultsof a SA-code, is desirable to evaluate the uncertainty
through random sampling.
The reduced order modeling (ROM) with the singularvalue decomposition (SVD) and the low-rankapproximation (LRA) was developed in the previous
study in order to reduce the computational cost for
probabilistic safety margin analysis using a severe
accident analysis code [1]–[2]. In the previous study, theROM method is applied to the station black out with total
loss of feedwater capabilities, which is categorized as a

2. Outline of ROM theory
This section briefly explains the concept of ROM with
SVD and the application procedures of the ROM to theresults of a SA-code.
2.1 Concept of ROM with SVD and LRA
The technique of SVD is widely used in the engineering
field (e.g., image compression and so on). SVD candecompose a matrix A, which consists of row vectors𝑎⃗ (1 ≤ 𝑛 ≤ 𝑚), into three matrices as:𝑎⃗𝑎⃗𝐀== 𝐔𝚺 𝐕,(1)⋮𝑎⃗where 𝐔 = (𝑢⃗ , 𝑢⃗ , … , 𝑢⃗ ) , 𝚺 = 𝑑𝑖𝑎𝑔(𝜎 , 𝜎 , … , 𝜎 )282

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019and𝐕 = (𝑣⃗ , 𝑣⃗ , … , 𝑣⃗ ) contain the left singular
vectors, the singular values, and the right singular vectors,
respectively (𝑚 ≤ 𝑡). By using these matrices with LRA,𝑎⃗ in the matrix A is expressed as:
𝑎⃗ =

𝑢 𝜎

𝜷=𝐔𝚺
= 𝛽⃗ 𝛽⃗ ⋯ 𝛽⃗ ⋯ 𝛽⃗
𝑢 𝜎
𝑢 𝜎
⋯ 𝑢 𝜎
(4)
𝑢 𝜎
𝑢 𝜎
⋯ 𝑢 𝜎
=
,
⋮
⋮
⋱
⋮
𝑢 𝜎 𝑢 𝜎 ⋯ 𝑢 𝜎
Each element in the expansion coefficient 𝛽⃗ isrepresented by a function of input data used for training.
5. For “𝑝” input parameter, which is not included in input
data for training, the expansion coefficients 𝑢 𝜎 canbe calculated by the function obtained in Step 4.
6. The time-series result for “𝑝” input parameter can bepredicted by the expansion coefficients obtained in
Step 5 and the orthogonal basis 𝑣⃗ as:

𝑣⃗

(2)
≈

𝑢 𝜎

𝑣⃗ ,

where k is the number of singular values considering in
ROM (𝑚 ≫ 𝑘), which is determined by a criterion (e.g.,
the cumulative contribution ratio or the scree plot [3]);𝑢 and 𝜎 is the the n-th element of 𝑖-th left singular
vector and the i-th singular value, respectively. The above
description also holds true for 𝑚 > 𝑡.
From Eq. (2), any 𝑎⃗ in the matrix A can be
expanded by linear combinations of representative rightsingular vectors. In other words, SVD and LRA cangenerate the dominant vectors, which cover overallcharacteristics of the matrix A. This property contributes
modeling of the time-series results obtained by a SA-code
[1].

𝑝⃗ ≈

𝑢𝑝𝑖 𝜎𝑖 𝑣⃗ ,(5)

3. Calculation Conditions
3.1 Safety analysis code and plant model
The RELAP5/SCDAPSIM code and the plant model of
Lagna Verde Unit 1 and 2 (rated power: 1931MWth),which is BWR-5 designed by General Electric company,
in Mexico are used.

2.2 Application of ROM to accident analysis results
A safety analysis code considers many input parameters
due to the sophisticated modeling of physical phenomenathat occurs in the accidents. Each input parameter shouldbe ranked in the view of sensitivity for the accident inorder to clarify the effect of input parameters. Therefore,
preliminary sensitivity analyses of input parameters aredesirable before construction of ROM. The number ofsampling should be increased as the sensitivity of the
input parameter becomes larger in order to accurately
capture the effects of the input parameter for the analysis
result.
The application procedures of the present method are
as follows:
1. Sample m input parameters and perform m calculations
by a SA-code The input parameters and the results areused to construct ROM.
2. Construct matrix A as shown in Eq. (1). Numbers ofrows and columns are “the number of samples (m)” and“the number of time-steps in an analysis result (t)”,respectively.
3. Apply LRA to matrix A after SVD. Then matrix A is
decomposed as:(3)
𝐀 ≈ 𝐔′𝚺′ 𝐕′,
where 𝐔′ = (𝑢⃗ , 𝑢⃗ , … , 𝑢⃗ ), 𝚺′ = 𝑑𝑖𝑎𝑔(𝜎 , 𝜎 , … , 𝜎 )and 𝐕 = (𝑣⃗ , 𝑣⃗ , … , 𝑣⃗ ). The m simulation results
are expanded using the right singular vectors obtainedby SVD. (Eq. (2))
4. Set functions from correlation among the expansion
coefficients 𝛽⃗ and input training data, where 𝛽⃗ iscalculated by the left singular vectors and the singularvalues:

3.2 Accident scenario
The accident scenario is the load rejection with an open
failure of the turbine bypass valve. At 𝑡 = 0 s, plant is astable condition. However, the load rejection of turbine
generator occurs at 𝑡 = 10 s due to its failure or themismatch between demand and supply, and so on. Thenthe turbine governor valve rapidly closes to protect
turbine followed by the failure of turbine bypass valve toopen. Therefore, the primary pressure rapidly increases
resulting collapse of void in the core. Thus, plant power
rapidly increases due to the void reactivity insertion.
When the neutron flux exceeds 120% (relative value torated power), the SCRAM signal is automaticallydispatched. The accident is successfully terminated due tothe negative Doppler feedback and the large negative
reactivity by SCRAM.
In the above scenario, MCPR (Minimum Critical
Power Ratio) is an important parameter in order to judge
the integrity of the fuel. However, RELAP5/SCDAPSIM
cannot evaluate MCPR due to the limitation of modeling.
When power increases by reactivity insertion, themaximum heat flux in the core will become closer to thecritical heat flux and MCPR has correlation with core
power. From this consideration, the relative core power isused instead of MCPR in this study.
The total simulation time for the load rejection
accident scenario is 30 seconds. The number of the time
steps is 30 because the time step size of one second isapplied to edit the results. The time-step size in
RELAP5/SCDAPSIM calculations is automaticallyvaried from 10 to 0.1 seconds.

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Osaka, Japan, Dec. 2-3, 2019

3.3 Perturbed input parameters

3.4 Procedures of ROM construction

Among many input parameters that have asignificant/considerable effect in the load rejection
accident, the uncertainty of reactivity parameters related
to neutronic calculation (void, Doppler, and SCRAM
reactivity) are considered due to their impact on the
results. In this study, the uncertainty (1𝜎 ) of all input
parameters is assumed as 10%, where it is known that theactual uncertainty would be around approximately 10 %.
These reactivities are perturbed as Figures 1–3. The
values of reactivities are based on those of typical BWRs
and the perturbations are performed by multiplying a
perturbation factor to reactivity. For ROM construction,
the perturbation range of input parameters is ±20% tocover the uncertainty of input parameters within 2σ.

The actual calculation procedures of ROM construction
are as follows:
1. Perform sensitivity analysis of input parameters for the
result obtained by RELAP5/SCDAPSIM (Eq.(6)). The
sensitivity becomes smaller in the order of the void,
SCRAM, and Doppler reactivity.
∆𝑚𝑎𝑥{𝑓(𝑥)} 𝑚𝑎𝑥{𝑓(𝑏𝑎𝑠𝑒 + σ )} − 𝑚𝑎𝑥{𝑓(𝑏𝑎𝑠𝑒)}
≈
,
∆𝑥
σ

(6)

where 𝑥 is the input parameter (i.e., reactivity), 𝑓
is a function of RELAP5/SCDAPSIM, and σ is theuncertainty of 𝑥 .
2. The range 2σ of input parameters are divided into
equal width based on Step 1. The void, SCRAM, andDoppler reactivity are divided into nine, five, and three,respectively.
3. By using 135 (= 9 × 5 × 3) combinations of input
parameters,
perform
calculations
of
RELAP5/SCDAPSIM.
4. The input parameters and results are used to construct
ROM.
(Section
2.2.)
Note
that
the
LinearNDInterpolator function of the Scipy package[4], which is the linear interpolation method, is used toestimate the expansion coefficient in ROM.
5. The number of 𝑘, which is the number of truncatedbases in LRA, is determined by the scree plot, which isa criterion in the principal component analysis. In thisanalysis, 𝑘 = 27 is used for ROM.

Fig. 1. The perturbation of void reactivity.

3.5 Procedures of statistical safety analysis
In order to validate the statistical safety analysis by ROM,the uncertainties of maximum power during the transient
calculated by RELAP5/SCDAPSIM and ROM arecompared. The procedures are described as follows:
1. Perform 1,000 times random sampling of input
parameters. When the sampled values exceed the rangeof 2σ, resampling is performed since the range of input
parameters covered by ROM is within 2σ and theextrapolation by ROM has low accuracy.
2. By using the 1,000 input parameters, ROM and
RELAP5/SCDAPSIM calculations are carried out.
Note that the identical input parameters are given for
ROM and RELAP5/SCDAPSIM.
3. The average and the standard deviation of the ROM and
RELAP5/SCDAPSIM
resultsareestimated.
Furthermore, the Pearson’s correlation coefficient
between ROM and RELAP5/SCDAPSIM results arealso evaluated.

Fig. 2. The perturbation of Doppler reactivity.

4. Results
Figure 4 shows the time-series data of core power in thebase case, in which no uncertainty is considered for all
reactivities. Figure 5 shows the histogram of the
maximum power on output training data. ROM isconstructed by using these 135 input and output training
data.
Figure 6 and Table I show the comparison ofFig. 3. The perturbation of SCRAM reactivity.

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

uncertainty
estimated
by
ROM
and
RELAP5/SCDAPSIM. From Fig. 6, the maximum power
estimated by ROM can approximately reproduce that by
the RELAP5/SCDAPSIM. From Table I, ROM also canreproduce the average and the standard deviation obtainedby many RELAP5/SCDAPSIM calculations. ROM
results well reproduce those of RELAP5/SCDAPSIM.
However, there is a small difference between ROM
and RELAP5/SCDAPSIM in Fig. 6 though identical input
parameters are used. There are two possible reasons forthe difference: 1. the lack of input training data to learn
the behavior of the expansion coefficients and 2. the errorof linear interpolation to estimate the expansion
coefficients.
As a result, the reduction of computational cost by
ROM will be possible. In this paper, the number ofcalculations by RELAP5/SCDAPSIM decreases from
1,000 to 135.

ROM and RELAP5/SCDAPSIM using 1,000 samples.
Table I. Comparison of average and standard deviation of
maximum power during the transient. The correlation
coefficient is also shown.

Average [%]Standard deviation [%]Correlation coefficient

ROM
229.1
18.8

RELAP
229.8
18.9
0.999

6. Conclusions
This study aims to reduce the computational cost by aSA-code for statistical safety analysis. In order to reducethe computational cost, the ROM is developed with SVD
and LRA for time-series data obtained by a SA-code.
In this paper, ROM is constructed to estimate themaximum power in the load rejection accident for a BWR
with an open failure of the turbine bypass valve, which iscategorized as a design basis accident. The input
parameters related to feedback and reactivity, whichstrongly affects the core power, is considered for thesource of uncertainty. The reduction of computational
cost by ROM will be possible since the maximum core
power calculated by ROM is consistent with those by
RELAP5/SCDAPSIM.
The following issues are considered as future
problems; consideration of actual uncertainties in random
sampling, consideration of additional input parameters for
random sampling, and application to the other accident
scenarios.

Fig. 4. Time-series data of power ratio in the base case.

Acknowledgments
The RELAP/SCDAPSIM used in this study has been
introduced by the funding of Nuclear Regulatory
Authority (NRA) for the educational program of nuclear
regulation and nuclear safety.
References
1. Matsushita M, Endo T, Yamamoto A, Kitao T,“Development of Reduced Order Model of Severe
Accident Code for Probabilistic Safety Margin
Analysis,” Proc. PHYSOR 2018, Cancun, Mexico,
April 22–28, 2018, American Nuclear Society (2018)(USB).
2. Matsushita M, Endo T, Yamamoto A, “Surrogate
Model of Severe Accident Analysis Code for SBO
Aiming Probabilistic Safety Margin Analysis,” Trans.
Am. Nucl. Soc., 119, p. 900, American Nuclear
Society (2018).
3. Cattell RB, “The Scree Test for the Number ofFactors,” Multivariate behavioral research, 1, 2, p.
245, (1966).
4. “Scipy.org,” https://docs.scipy.org/doc/scipy/referen
ce/generated/scipy.interpolate.LinearNDInterpolator.
html?highlight=linearnd#scipy.interpolate.LinearND
Interpolator (access: 2019/5/21).

Fig. 5. The histogram of the maximum power for output
training data (135 samples).

Fig. 6. Comparison of the maximum power between285TS-08Multiphysics Power Reactor Simulations

No.

RPHA19-1037

Title

Preliminary Performance Assessment of the GPU Acceleration
Module in a Pinwise Core Thermal Hydraulics Code ESCOT

Authors

Kyung Min Kim, Namjae Choi, Jaejin Lee, Han Gyu Joo (Seoul National
University, Korea)
288

No.

RPHA19-1078

Title

BEAVRS Benchmark Analysis by a Whole Core Transport CodenTER

Authors

Hojin Park, Jin Young Cho (Korea Atomic Energy Research Institute,
Korea)
292

No.

RPHA19-1097

Title

Initial Assessment of Anderson Acceleration on Pinwise
Coupled Neutronics/Thermal-Hydraulics Code nTER/ESCOT

Authors

Jaejin Lee (Seoul National University, Korea), Jin Young Cho (Korea
Atomic Energy Research Institute, Korea), Han Gyu Joo (Seoul National
University, Korea)
296

No.

RPHA19-1108

Title

Pin-by-Pin Multi-physics Analysis and Evaluation of the
Critical Heat Flux (CHF) in a PWR Core

Authors

Jaeha Kim, Khurrum Saleem Chaudri, Yonghee Kim (Korea Advanced
Institute of Science and Technology, Korea)
300

No.

RPHA19-1093

Title

Functional Expansion Tallies in Monte Carlo High Fidelity LWR
Analysis

Authors

Bamidele Ebiwonjumi, Hyunsuk Lee, Peng Zhang, Deokjung Lee (Ulsan
National Institute of Science and Technology, Korea)
304

286

No.

RPHA19-1099

Title

Hybrid Parallelism of Internal Coupling Method between
Monte Carlo Code RMC and Sub-channel Thermal-Hydraulic
Code CTF

Authors

Kaiwen Li (Tsinghua University, China), Shichang Liu (North China
Electric Power University, China), Juanjuan Guo, Kan Wang (Tsinghua
University, China)
308

No.

RPHA19-1041

Title

NECP-X and CTF Solutions to VERA Benchmark

Authors

Bo Wang, Zhouyu Liu, Liangzhi Cao (Xi'an Jiaotong University, China)
312

287

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Preliminary Performance Assessment of the GPU Acceleration Module in a Pinwise Core
Thermal Hydraulics Code ESCOT
Kyung Min Kim, Namjae Choi, Jaejin Lee, and Han Gyu Joo,
Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea*
Corresponding author: skstr3182@snu.ac.kr

Abstract
Even though there have been significant advances in CPU computing power,
whole-core multi-physics calculations still remain impractical for the industrial
applications. Furthermore, as the development of the CPU technology is being
challenged nowadays, heterogeneous computing is emerging as an alternative.
This work suggests a strategy to accelerate pin-wise sub-channel T/H calculations
on GPUs, which involves the solutions of linear systems, the manipulation of theequation of state, and the setup of the linear systems for the various componentsof the governing equations. A group of GPU acceleration methods were
implemented in a drift-flux model based pin-wise sub-channel T/H code, ESCOT,
which is being developed at Seoul National University. The performance results
obtained with 4 GPUs for a OPR1000 quarter core appear comparable with theparallel calculation employing 104 CPU cores. It is demonstrated that evenconsumer-grade commercial GPUs can be used as an effective means in the subchannel T/H calculation.
Key Words:
drift flux model, GPU, pressure equation, equation of state, BiCGSTAB
1. Introduction

be carried out. Especially, this paper focuses on
optimizing the pressure equation solution, the evaluationof IAPWS-IF97 [8] steam table formulations, and the
setup of linear system which had been the major
challenges.

As the CPU processing power had increased rapidly overthe past decades, numerical solutions with higher
resolutions and fewer approximations became more
affordable and employable. In the reactor physics field, it
had enabled whole-core multi-physics calculations by
combining a high-fidelity neutronics code and asubchannel thermal-hydraulics (T/H) code. Current
mainstream codes, which are CPU-based, include
DeCART [1] coupled with MATRA [2], MPACT [3]coupled with COBRA-TF (CTF) [4], and nTRACER [5]coupled with ESCOT [6].
However, as the CPU technology faces the limitations
due to the power consumption issue, researchers are now
paying more attention to deploying heterogeneouscomputing powered by Graphics Processing Units (GPU)to achieve even higher performance. And so far it has
succeeded in many computational physics fields. In
keeping with this, nTRACER had also established a plan
to offload entire core follow processes onto GPU. So far
the neutronics solver has shown significant improvementsin performance over the traditional CPU-based approach[7].
Therefore, as the next step, this research aims at
extending the GPU acceleration capability of nTRACER
to actual operational conditions that include T/H
feedbacks. In order to achieve this goal, the GPU
acceleration of the pin-wise core T/H code ESCOT is to

2. Scope of Research
Even though there exist efficient algorithms for massive
parallelization of whole-core calculations on many-coresystems, such systems are not yet practically affordable.
Therefore, we continue to focus on planar decomposition
strategy which, we think, is still an effective approach forpractical whole-core calculations. Instead, we introduce
GPUs to overcome the limitation of computing resourcesin a cost-effective way.
We expect that a moderate-sized cluster, whichcontains a dozen of nodes with multiple commercial
GPUs in each node, will be affordable to most researchand user groups, and we are focusing on developing
algorithms that are most efficient for such small systems.
Currently, we are building the Soochiro 4 cluster in SNU,
which is a representative form of our oriented system:
Table I. Specifications of the Soochiro 4 cluster# of Nodes
6
2 × Intel Xeon E5-2630 v4
CPU
20 Cores, 2.4 GHz (Boost)GPU
4 × NVIDIA GeForce GTX 1080

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Osaka, Japan, Dec. 2-3, 2019

MemoryInterconnect

Meanwhile, GPUs can exert their maximum
performance when employing single precision units.
BiCGSTAB algorithm, however, contains a couple ofinner products and matrix-vector multiplications in each
iteration, and single precision suffers from round-off
errors in those reduction operations due to the illconditioned linear system. Therefore, several methods
and preconditioners were investigated to replace orenhance BiCGSTAB, and solve the equation through theiterative refinement [7], but all of them failed or showed
extreme performance degradation as follows:

8 × 16GB DDR4 RAM
Infiniband (56 Gbps)

To maximize the performance of multi-physics wholecore calculation in such small systems, tandem approach
is favorable as in Fig. 1. Neutronics calculation requires
binding each plane to a GPU, which will occupy most of
the available nodes. If the T/H calculation can be carriedout as fast as the neutronics iteration using the remaining
resources, the T/H time can be fully hidden behind theneutronics time by the tandem manner.

Table II. Solutions tested using the iterative refinement
SolverPreconditioner Convergence
ILU0
Converged
BiCGSTABSPAI [11]Diverged
Converged
AAJ [12]
Not feasible
AMG [13]
Diverged
BiCGSTAB(l) [14]Diverged
IBiCGSTAB [15]Diverged

Node 9⁝⁝nTRACER⁝⁝
Node 2
Node 1ESCOT

As in Table II, all the attempts failed to converge
except AAJ and BiCGSTAB with ILU0 preconditioner.
However, the ILU0 preconditioner cannot be parallelized,
because of the forward and backward substitutions. In
addition, AAJ takes too much time to fully converge. So,
it was determined that AAJ and BiCGSTAB with the
ILU0 preconditioner are not practical to apply.
As a consequence of the investigation, it was
confirmed that double precision approach to the pressure
equation is indispensable. That being so, it became a main
objective to optimize the double-precision BiCGSTAB sothat the algorithm becomes the best-suited on GPU,
instead of implementing new solvers or preconditioners.
As follows are the conclusions of the optimization
research:

Node 0Fig. 1. Topology for the multi-physics simulation intandem manner.
Therefore, this research targets to accelerate ESCOT codeusing only a single node to minimize the overhead fromthe T/H calculation by employing the tandem scheme.
3. Optimization Process
In this paper, only brief introductions and optimization
strategies for the major bottleneck operations arepresented. Full physical backgrounds and implementation
schemes of ESCOT are explained in [6].

(1) ELLPACK matrix format is much better thanCompressed Sparse Row (CSR) format for the highly
structured pressure equation matrix.
(2) BiCGSTAB with Jacobi (diagonal) preconditioner,
which can be applied in parallel, outperforms
BiCGSTAB without a preconditioner.
(3) All the linear algebra routines operating for thepressure matrix on GPUs are memory-bounded, so
double precision calculations are only twice slower
than single precision counterparts.

3.1 Pressure Equation
SIMPLEC [9] algorithm incorporates specific enthalpy,
void fraction, and pressure to construct a linear system ata certain calculation node (I,J). Of these three scalarquantities, the pressure linear system, which is solved
every outermost iteration, has the form: n +1n +1
1 +  bnb  PI , J −  bnb Pnb = s3 ,nb
 nb 

(1)

In the new BiCGSTAB solver, collective operations
like inner products and l2-norm are taken from CUBLAS,
and the ELLPACK format matrix-vector multiplicationand element-wise vector operations are implemented bymanually tuned kernels.

wherebnb = coefficients for each pressure elements
PIn, J+1 = next time pressure of the node (I,J)
s3 = source element.

3.2 Equation of StateThe coefficient matrix is septa-diagonal. Its diagonaldominance is very weak because the off-diagonal terms
are much greater than unity. The CPU-based ESCOT
employs BiCGSTAB in the PETSc [10] library by default.

ESCOT sets the mixture velocity, void fraction,
specific enthalpy, and pressure as primary variables.
Among these, mixture velocity and pressure are289

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019calculated from solving each linear system, and the rest
are solved for each calculation node. On the other hands,
the other variables are designated as the secondary ones,
which are determined by the equation of state (EOS) using
the IAPWS-IF97 steam table. IAPWS-IF97 classifies thesteam regions according to pressure and temperature,described in Fig. 2.

Runtime and accuracy are compared between the original
CPU-based and GPU-accelerated ESCOT for OPR1000
quarter core problem. In common, BiCGSTAB with theJacobi preconditioner is used as the linear solver. The
examinations were carried out on the Soochiro 4 cluster,and the calculation conditions are summarized in Table III.
Table III. Calculation conditions
VersionCPU
GPU# of calculation
370,208 (11569 × 32)
nodesIntel FortranPGI FortranCompiler
17.0.4
18.10
JacobiJacobipreconditioned preconditioned
Linear solver
BiCGSTABBiCGSTAB(PETSc)(Manual)# of processors
52 cores
4 GPUs/ devices
104 cores

Fig. 2. Steam regions specified in IAPWS-IF97.

As previously mentioned, the GPU-accelerated code
was run using only a single node while CPU-based oneused up all the available CPU cores. The results are
summarized in Table IV and Table V. The three main
bottlenecks are listed, and the rest are omitted.

In real implementation, property functions require not
only the base properties, but also the steam region of theproperties. The region 1 and 2 are typical subcooled liquidand superheated vapor, but the region 3 is between theregion 1 and 2, most of which is in the critical state andso algorithms are much more complex. The quantities in
the region 1 and 2 consist of a set of multiple functions,in which dimensionless Gibbs free energy is calculated
from expansions of polynomials, so it is one of the major
calculation burdens.
In fact, since the EOS calculation is independent
between meshes, coalesced access can be achieved, so it
is suitable for the parallel execution scheme of GPUs.
Moreover, IAPWS formulation for the steam table is acompute-intensive task, because it contains expansion ofhigh order polynomials, which is also proper for GPUs.
Although the steam table requires steam region of eachcalculation node, ESCOT utilizes and requires both liquidand vapor quantities and specifies the steam region, andtherefore it is obvious that branch divergence barely
occurs in the EOS procedure.

Table IV. Time summary of 52 cores and 4 GPUs
Calculation
CPUGPU
SpeedupLinear system setup
8.15s
3.82s
2.1
Equation of State
18.60s
5.93s
3.1
Solving the pressure
24.32s32.30s0.8
equation
Total
73.76s53.70s
1.4
Table V. Time summary of 104 cores and 4 GPUs
Calculation
CPUGPU
SpeedupLinear system setup
4.21s
3.82s
1.1
Equation of State
9.78s
5.93s
1.7
Solving the pressure
15.77s32.30s0.5
equation
Total
46.35s53.70s
0.9

3.3 Linear System Setup

It can be observed that the linear system setup and theEOS procedures are fully parallelized so that they show
good speed-up. Therefore, the GPU version outperforms
52 cores and is comparable with 104 cores.
However, in terms of solving the pressure equation,
the GPU version could not exceed the CPU-based codeeven in 52 cores. This is due to cache utilization andmemory bandwidth, which are the most notabledifferences between the two architectures. For themassive parallelization based on CPUs, the matrix is split
into a number of small matrices, all of which can beloaded on CPU cache. For example, since the pressure
matrix of the OPR1000 problem occupies 20 MiB
memory, 52 and 104 cores can split it into 0.4 MiB and
0.2 MiB per processor, respectively, and these magnitudes
are sufficiently small to be loaded on cache. However, theparallel strategy based on a couple of GPUs cannot get thedata reside in cache, because size of the data is too big.

In the ESCOT algorithm, there are 4 linear systems thatare set up every outer iteration, which are the axial andthe radial momentum equations, the pressure equation andthe scalar equation. For realistic problems, the number ofthe outer iterations is a hundred or over. Therefore, it is asevere burden unless hundreds of cores are employed interms of the domain decomposition.
The linear system setup procedures, alike the EOS, areinherently independent between the meshes. However, incontrast to the EOS, they need much larger data than the
EOS routines. Therefore, prefetch of global memory toeach thread is done in the first stage and after that,
registers are exploited mostly in the calculation to
maximize flops.
4. Results and Discussion

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Osaka, Japan, Dec. 2-3, 2019

Acknowledgements

And in terms of bandwidth, multiple CPU cores are much
faster than several GPUs, since all the data reside in cachefor CPUs. Although each of cache bandwidth is lower
than GPU global memory bandwidth, the memory is
loaded through tens of cores for CPU, while only 4 GPUs
load the data.
Next, the accuracy of the GPU-accelerated version
was examined. Although not explained here, the iterativerefinement strategy is employed in solving the axial
momentum equation. However, it does not affect thesolution as shown in Fig. 3 below. The figure shows therelative difference of outlet density in upper triangularpart and outlet temperature in lower triangular part
between solutions from CPU single core and 4 GPUs,
respectively. It can be observed that magnitude of therelative difference is only 10-5% order at best.

This research is supported by National Research Foundation of
Korea (NRF) Grant No. 2016M3C4A795263 (Realization of
Massive Parallel High Fidelity Virtual Reactor).

References[1] J. Y. Cho, S. Yuk, “Massive Parallel Computation for an
Efficient Whole Core Transport Calculation,” Transactions of
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[2] H. Kwon et al., “Validation of a Subchannel Analysis Code
MATRA Version 1.1,” KAERI/TR-5581/2014, Korea Atomic
Energy Research Institute, 2014.
[3] B. Kochunas, T. Downar, Z. Liu, “Parallel 3-D Method of
Characteristics in MPACT,” International Conference onMathematics and Computational Methods Applied to Nuclear
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[4] R. Salko and M. Avramova, "COBRA-TF Subchannel
Thermal-Hydraulics Code (CTF) Theory Manual-Revision 0,"
The Pennsylvania State University, 2015.
[5] Y. S. Jung et al., "Practical numerical reactor employing
direct whole core neutron transport and subchannel
thermal/hydraulic solvers," Annals of Nuclear Energy, vol. 62,
pp. 357-374, 2013.
[6] J. Lee, A. Facchini., H. G. Joo, “Development of a Drift-Flux
Model Based Core Thermal-Hydraulics Code for Efficient
High-Fidelity Multiphysics Calculation,” Nuclear Engineeringand Technology(2019), doi: 10.1016/j.net.2019.04.002[7] N. Choi, J. Kang, H. G. Joo, “Preliminary Performance
Assessment of GPU Acceleration Module in nTRACER,”
Transactions of the Korean Nuclear Society Autumn Meeting,Yeosu, Korea, October 25 - 26.
[8] J. R. Cooper, “Revised Release on the IAPWS Industrial
Formulation 1997 for the Thermodynamic Properties of Waterand Steam,” International Association for the Properties ofWater and Steam Publication, 2012.
[9] J. P. Van Doormaal, G. D. Raithby, “Enhancement of the
SIMPLE Method for Predicting Incompressible Fluids Flow,”
Numerical Heat Transfer, vol. no. 2, pp. 147-163, 1984.
[10] PETSc Team, "PETSc Users Manual," ANL-95/11 - Rev.
3.10, Argonne National Laboratory, 2018.
[11] M. J. Grote, T. Huckle, “Parallel Preconditioning withSparse Approximate Inverses,” SIAM J. Sci. Compute., 18 (3)(1997), pp. 838-853.
[12] P. Pratapa, P. Suryanarayana, J. Pask, “Anderson
Acceleration of the Jacobi Iterative Method: An Efficient
Alternative to Krylov Methods for Large, Sparse Linear
Systems,” Journal of Computational Physics, vol. 306, pp. 43–
54 (2016).
[13] R. Gandham, K. Esler, Y. Zhang, “A GPU Accelerated
Aggregation Algebraic Multigrid Method,” Computers &Mathematics with Applications, vol. 68, Issue 10, pp. 1151 –
1160, 2014.
[14] G. L. G. Sleijpen, D. R. Fokkema, “BiCGSTAB(L) for
Linear Equations Involving Unsymmetric Matrices with
Complex Spectrum,” Electronic Transactions on Numerical
Analysis, vol. 1, pp. 11 – 32, 1993.
[15] L .T. Yang, R. P. Brent, “The Improved BiCGSTAB
Method for Large and Sparse Unsymmetric Linear Systems on
Parallel Distributed Memory Architectures,” Proceedings ofthe Fifth International Conference on Algorithms andArchitectures for Parallel Processing, 2002, pp. 324 - 328.

Fig. 3. Relative differences of outlet density andtemperature
5. Conclusion
GPU acceleration capability has been implemented in
ESCOT. The modules are still under development, so
there exist yet under-optimized routines, such as the linear
solver. Apart from this, the other main bottlenecks are
sufficiently accelerated so that only a single computing
node outperforms the many-core CPU performance. Also,
the accuracy of the solutions from commercial GPUs
were verified.
Still, the poor performance of the manually written
BiCGSTAB solver should be enhanced to achieve theultimate goal of the tandem coupling with the GPUaccelerated nTRACER. There are several ways to resolve
it: 1) devising a new solver more proper for GPUs, 2)
investigation of preconditioners which can be generated
and applied in a parallel way, and significantly reduce thenumber of BiCGSTAB iterations. In order to carry out
these tasks efficiently, the PETSc library will be used,which provides a variety of solvers and preconditioners
for the GPU systems. Finishing the tasks will make itavailable to accomplish a whole-core steady-statecalculation including T/H feedbacks within 5 minutes ona moderated-sized GPU cluster.

291

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

BEAVRS Benchmark Analysis by a Whole Core Transport Code nTERa

Ho Jin Park a,*, Jin Young Choa
Korea Atomic Energy Research Institute, 111, Daedeok-daero 989beon-gil, Daejeon, 34057, Korea*
Corresponding author: parkhj@kaeri.re.kr

Abstract
The BEAVRS benchmark problem was solved by a newly developed whole coretransport code, nTER (Neutron Transport Evaluator for Reactor) to verify its
code systems through its core follow calculations. The nTER results for control
rod worth, radial detector signal, and boron letdown curve during two cycles agree
well with the measured data. Therefore, it is concluded that the nTER code is welldeveloped in the terms of the solution accuracy for the high-fidelity nuclear
parameter evaluation.
Key Words: nTER, BEAVRS, Verification and Validation, Benchmark
1. Introduction

0.05 cm, 2 polar angles of 90°, and 8 azimuthal angles of
90° were used for nTER MOC calculations. All nTER
calculations were conducted with the newly-generated
ENDF/B-VII.1 based PV05 47-group cross section
library and the sub-group method.

A new transport code, nTER (Neutron Transport
Evaluator for Reactor) [1], for direct whole corecalculations was developed by Korea Atomic Energy
Research Institute (KAERI) with the cooperation of
Korea Hydro and Nuclear Power Central ResearchInstitute (KHNP CRI). The nTER code uses radial methodof characteristics (MOC) transport and the axial P3
coupled method to obtain high-fidelity transport solutions.
Moreover, the nTER code is equipped with up-to-date
methodologies such as a radial MOC with anisotropic
scattering source, high-performance computation basedon node-wise MPI (Message Passing Interface)parallelization. The detailed methodologies andfunctionality of the nTER code are described in [1].
Recently, the various benchmark calculations such as
C5G7 and VERA benchmark have been applied to
examine the performance of the nTER code. The
Benchmark for Evaluation and Validation of Reactor
Simulations (BEAVRS) [2] was proposed by the
Massachusetts Institute of Technology (MIT)computational reactor physics group, and this provides
detailed specifications and various measurement datasuch as control rod bank worth, radial detector signal,
isothermal temperature coefficients, and critical boron
concentration (CBC) during two cycles. The BEAVRS
solutions have been reported by various reactor physics
codes such as VERA [3], nTRACER [4], SCALE/PARCS[5], MCS [6], and DeCART [7]. For this study, as anextension of the verification and validation (V&V) of thenTER code, a whole-core benchmark analysis for the
BEAVRS benchmark was performed.

2.1 Hot Zero Power Results by nTER
Table I compares the kinf‘s of two codes for fourteen 2D
BEAVRS fuel assemblies (FAs), which consist of the
BEAVRS cycle 1 and cycle 2 core. The root mean square(RMS) differences of the kinf between the nTER and
McCARD [8] reference solutions is about 63 pcm.
Table I. kinf of BEAVRS 2D FA problems
kinf for 2D FA
Enri.
No.
w/o%
BA* McCARD
nTER Diff (pcm)
1.6
2.4

3.1

3.2
3.4

0
0
12
16
0
16
20
6
15
0
4
8
12
0

0.99646
1.13932
1.01575
0.977421.22135
1.06547
1.02975
1.16459
1.08032
1.230421.18869
1.14807
1.10825
1.24916

0.99749
1.140431.01537
0.97669
1.22263
1.06529
1.02945
1.165201.08043
1.23179
1.18950
1.14848
1.108391.25057

103
85
-37
-77
86
-16
-29
45
9
17
90
57
31
11

*BA=burnable absorber

2. BEAVRS Benchmark Results by nTER

Table II compares the control rod bank worth results bynTER for hot zero power (HZP) condition with themeasured values. The benchmark sheets provide theIn the BEAVRS benchmark calculations, a ray-spacing of292

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019in detector measurements, and control rod bank D
positions for detailed calculations at cycle 1 and cycle 2.
Power History (Cycle 1)

Percent Rated Power [%]Table II. Total and individual control rod bank worth
Control rod bank worth (pcm)
Case
MeasurednTER
Diff. (%)(M)(N)
(M-N)/M
D in
788
784
0.5%
C with D in
1203
1256
-4.4%
B with D, C in
1171
1255
-7.2%
A with D,C,B in
548
508
7.3%
SE with All R in
461
467
-1.3%
Total
4171
4271
-2.4%
D

C

B

A

0.78
0.75
-3.3%
1.01
1.00
-1.0%

1.07
1.05
-1.3%
0.90
0.88
-2.2%
1.14
1.13
-0.5%

0.94
0.91
-3.1%
1.14
1.14
-0.3%
0.97
0.96
-0.8%
1.25
1.25
0.4%

1.15
1.15
0.0%
0.97
0.95
-2.6%
1.21
1.20
-0.6%

0.94
0.93
-1.0%
1.17
1.20
2.7%
0.98
0.97
-1.5%
1.31
1.37
4.8%
1.20
1.20
0.6%
0.85
0.85
-0.2%

1.26
1.28
0.9%
0.87
0.86
-1.1%
1.24
1.28
2.9%

0.78
0.78
-0.3%
0.82
0.81
-0.6%
0.73
0.72
-0.5%
0.58
0.59
1.3%

10

11

1.34
1.33
-0.8%

Measured
12

nTER
Diff

13

RMS diff.
Max.

1.06

1.82%
4.84%

0.94

120

40

80

20

40
Power History
Detector Measurements
Bank D Position (Step)

0

100

200

300

400

500

0

600

Calendar Days from BOC(a) cycle 1
Power History (Cycle 2)

228 (ARO)

100

200

80

160

60

120

40

80

20

Power History
Detector Measurements
Bank D Position (Step)

0
0

50

100

150

200

250

40

0
350

300

Calendar Days from BOC(b) cycle 2

0.96
0.96
0.0%
0.70
0.71
1.0%

Fig. 2. Power history and control rod bank positions
Figures 3 and 4 show the boron letdown curve calculated
by 100% HFP condition. For cycle 1 and 2, the RMS
errors of critical boron concentration are 8 and 4 ppm.
During two cycles, it was observed that the CBC resultsfrom nTER agree reasonably well with the measurements.
Figures 5 and 6 show the boron letdown curve calculatedby the detailed calculation with variable parameters in
detector measurements.

Fig. 1. Comparison of assembly-wise tilt-corrected
detector signal.
Table III. Isothermal temperature coefficients
Isothermal temperature coefficients(pcm/F°)
Case
Measured
nTER
Diff.
(M)
(N)
(M-N)
ARO
-1.75
-3.11
1.36
D in
-2.75
-4.65
1.90
C with D in
-8.01
-9.55
1.54

1000

100

BEAVRS Boron Letdown Curve (Cycle 1)- 100% Power900

2.2 Cycle 1 and Cycle 2 Results by nTER
The BEAVRS benchmark provides two different types ofboron letdown curves during cycle 1 and cycle 2
operations. One is based on the 100% hot full power (HFP)
conditions while the other is based on the detailed
calculations with variable parameters in detector
measurements based on power history data. Figure 2
presents the 24 hour-averaged power history, time points

293

800

75
50

700
600

25

500

0

400

-25

300
200

-50

Measurements (M)nTER (N)Difference (M-N)

100
0
-100

Difference (ppm)

E

160

60

Percent Rated Power [%]

9

0.91

F

80

0

Boron Concentration (ppm)

8

G

200

Bank D position (step)H

228 (ARO)

100

Bank D position (step)

regulating rod and shutdown bank worth for eachconfiguration. The calculated individual control rod bankworth agrees well with the measured one within 7.3%.
Figure 1 shows the tilt-corrected FA-wise detector signal
results from nTER. The relative RMS and maximum
errors from C11 FA, which is the large tilt position, are
1.8% and 4.8%. Table III shows the isothermal
temperature coefficients (ITC). The maximum error of
ITC is 1.90 pcm/F°

0

2

4

6

-75

8

10

12

14

-100

Burnup (MWd/kgU)

Fig. 3. Boron letdown curve for cycle 1 (100% Power)Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

1000

8

75
50

800

25

600

0
-25

400

0

2

4

1.309

9

G
1.042
1.045
0.3%
1.338
1.356
1.3%

10

11

-75

6

8

Burnup (MWd/kgU)

10

12

13

RMS diff.
Max.

H
8

9

12

13

25

600

0
400

Measurements (M)nTER (N)Difference (M-N)-75

8Burnup (MWd/kgU)

10

12

14

-100

Fig. 5. Boron letdown curve for cycle 1 (detector
measurements)

1000

50

800

25
0

600

-25
400

0

-50

Measurements (M)nTER (N)Difference (M-N)

200

0

2

4

-75

6

8

10

12

0.983
3.8%
0.651
0.651
0.0%

0.716
0.720
0.6%
0.512
0.523
2.1%

F
1.042
1.059
1.6%
1.069
1.085
1.5%
1.219
1.234
1.2%

MeasurednTER
Diff
RMS diff.
Max.

E
1.053
1.057
0.4%
1.184
1.201
1.4%
1.079
1.074
-0.4%
1.317
1.308
-0.7%

D
1.124
1.132
0.7%
1.088
1.081
-0.6%
1.259
1.269
0.8%
1.063

1.216

1.10%
2.77%

C
1.028
1.016
-1.2%
1.153
1.069
1.044
-2.4%
1.226
1.219
-0.6%
1.070
1.051
-1.8%
0.884
0.874
-1.2%

B
1.001
0.985
-1.6%
1.021
1.022
0.1%
1.052
1.035
-1.6%
0.961

A
0.762
0.772
1.3%
0.789
0.791
0.3%
0.712
0.705
-0.9%
0.552
0.536
-2.8%

0.791
0.789
-0.3%
0.533
0.526
-1.1%

For convenient neutronics and thermal/hydraulics (T/H)
coupling calculations, ESCOT [10] (Efficient Simulatorof Core Thermal-Hydraulics) code was built in the nTER
code. ESCOT is a drift-flux model (DFM) based pinresolved T/H analysis with conducting equations for twofluid analyses by adopting the mixture velocity. In thesame manner as the nTER stand-alone (SA) calculations,the CBC calculations on the detailed calculations asshown in Fig. 5 was conducted by the nTER/ESCOT
coupled (CO) calculations. To decrease the possibility ofdivergence or oscillations, under-relaxation scheme wasadopted in the iterations between nTER and ESCOT as
following equations:75

Difference (ppm)

Boron Concentration (ppm)

BEAVRS Boron Letdown Curve (Cycle 2)- Detector Measurements

1.143

3. BEAVRS Benchmark Results by nTER/ESCOT

100
1200

0.772

Figure 7 compares the calculated assembly-wise detector
signals with the measurements at the time points in
detector measurements. The relative RMS and maximum
errors at 1.50 MWd/kgU (169 day) of cycle 1 are 1.9%
and 3.8% and those at 10.43 MWd/kgU (296 day) of cycle
2 are 1.1% and 2.8%. The error of the nTER-calculated
detector signals seems not to fluctuate with the burnup.

-50

6

1.247
3.3%

Fig. 7. Assembly-wise detector signal at various burnup
step.

-25

4

1.044

A
0.648
0.651
0.4%
0.667
0.667
-0.1%
0.588
0.601
2.1%
0.485
0.484
-0.1%

(b) 10.43 MWd/kgU (cycle 2)

50

2

B
1.159
1.189
2.6%
0.791
0.784
-0.9%
1.119
1.148
2.6%

1.93%
3.79%

11

75

800

1.243

G
1.031
1.032
0.1%
1.035
1.047
1.1%

10

Difference (ppm)

Boron Concentration (ppm)

BEAVRS Boron Letdown Curve (Cycle 1)- Detector Measurements

0

C
0.977
0.980
0.3%
1.224
1.251
2.2%
0.952
0.967
1.6%

(a) 1.50 MWd/kgU (cycle 1)

100

0

D
1.294
1.327
2.6%
1.069
1.069
0.0%
1.278
1.311
2.6%

-100

For cycles 1 and 2, the RMS errors are 20 ppm. It is notedthat the error at 12.34 MWd/kgU burnup (cycle 1)
exceeds 50 ppm limits, which is well known as the typicalacceptance criteria. However, the error of CBC at thesame burnup point from CASMO/Simulate-3 calculations
from the BEAVRS uncertainty quantification report is 46
ppm [9]. In the comparison of the PARCS results [5], it
was the only point higher than 50 ppm. Therefore, a large
error may come from the uncertainty of measurement, andthe approximation of geometric model.

200

E
1.124
1.112
-1.1%
1.364
1.382
1.3%
1.094
1.105
1.0%
1.321
1.355
2.5%

Measured
nTER
Diff

12

Fig. 4. Boron letdown curve for cycle 2 (100% Power)

1000

F
1.356
1.376
1.5%
1.121
1.118
-0.3%
1.362
1.393
2.3%

-50

Measurements (M)nTER (N)Difference (M-N)

200

0

H

100

BEAVRS Boron Letdown Curve (Cycle 2)- 100% Power

Difference (ppm)

Boron Concentration (ppm)

1200

-100

Burnup (MWd/kgU)

Fig. 6. Boron letdown curve for cycle 2 (detector
measurements)

294

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

…(1)
…(2)

P( k )  (1   )  P( k 1)    f N ( P( k 1) ,T ( k 1) ),

T

(k )

 f E (P( k 1),T( k 1)

code as the V&V tasks. It is noted that the nTER results
for control rod bank worth, ITC, radial detector signal,
and CBC during two cycles agree well with the measured
data. Therefore, it is concluded that the nTER code is welldeveloped in terms of solution accuracy for high-fidelity
nuclear parameter evaluation. In addition, the multiphysics calculations by nTER/ESCOT code coupling
were successfully performed in the same manner. The
considerable difference between SA and CO results were
not observed in CBC and detector signal.

).

where P(k) and T(k) are the pin-power distributions andtemperatures at k-th iteration step and fN(P(k-1),T(k-1)) andfE(P(k-1),T(k-1)) indicate the nTER and ESCOT calculations
using the pin-power distributions and temperatures at theprevious iteration, respectively. Noted that the underrelaxation factor,  , was set as 0.7 for this study.
20

BEAVRS Boron Letdown Curve (Cycle 1)- Detector Measurements

800

10
5

600

0
400

-5

-10

Measurements
nTER Stand-Alone (SA)nTER/ESCOT Coupled (CO)Difference (SA-CO)

200

0
0

2

4

6

8

Burnup (MWd/kgU)

This study was supported by KHNP CRI (Korean Hydro
and Nuclear Power Central Research Institute).

15

Difference (ppm)

1000

Boron Concentration (ppm)

Acknowledgements

-15

10

12

-20

14

Fig. 8. Comparison of CBCs by SA and CO calculations(detector measurements)

8

9

H
0.923
0.935
1.2%

G
0.763
0.764
0.2%
1.013
1.014
0.0%

10

F
1.063
1.063
-0.1%
0.886
0.886
-0.1%
1.143
1.143
-0.1%

11

12

13

SA
CO
Diff
RMS diff.
Max.

E
0.919
0.918
-0.1%
1.149
1.148
-0.1%
0.966
0.966
0.0%
1.258
1.260
0.2%

D
1.155
1.153
-0.2%
0.954
0.952
-0.1%
1.209
1.209
0.0%
1.057
1.061
0.3%
1.322
1.338
1.2%

0.39%
1.23%

C
0.928
0.926
-0.2%
1.203
1.201
-0.2%
0.969
0.968
-0.1%
1.366
1.368
0.1%
1.194
1.197
0.3%
0.841
0.843
0.2%

B
1.275
1.271
-0.3%
0.861
0.859
-0.2%
1.275
1.272
-0.2%
0.933
0.933
-0.1%
0.949
0.949
0.1%
0.701
0.702
0.1%

A
0.771
0.769
-0.3%
0.805
0.803
-0.3%
0.720
0.718
-0.2%
0.587
0.586
-0.2%

Fig. 9. Comparison of detector-signals by SA and CO
calculations at 0 MWd/kgU (detector measurements)
Figures 8 and 9 compare the CBC and detector signal by
SA calculations with those by CO calculations. The RMS
and maximum difference of CBC between SA and CO are
1 ppm and 5 ppm. Noted that the difference between
temperatures by nTER-equipped simple 1D T/H
calculations and ESCOT elaborate drift-flux calculations
is not largely affected in the behavior of CBC anddetector-signal.
4. Conclusions
In this study, BEAVRS benchmark analyses were
performed using the newly developed nTER transport

295

References
1. J. Y. Cho et al., “Performance of a Whole Core
Transport Code, nTER,” the 6th International
Conference on Nuclear and Renewable Energy
Resources (NURER2018), Ramada Plaza Jeju, Korea,
30 Sep. ~ 03 Oct. (2018).
2. N. Horelik et al., “Benchmark for Evaluation andValidationofReactor
Simulations,”MIT
Computational Reactor Physics Group, rev03, 29
August, (2018).
3. B. S. Collins et al., “Simulation of the BEAVRSBenchmark using VERA,” International Conference onMathematics & Computational Method Applied to
Nuclear Science and Engineering (M&C 2017), Jeju,
Korea, 16~20 Apr., (2017).
4. M. Ryu et al, “Solutions of the BEAVRS benchmark
using the nTRACER direct whole core calculation code,”
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5. P. Darnowski et al, “Analysis of the BEAVR PWRBenchmark
using
SCALE
and
PARCS,”NUKLEONIKA, 64(3), 87, (2019).
6.Jiankai Yu et al, “Application of MCS/CTF/FRAPCON
Multi-physics Coupling Code System in BEAVRSCycle 1 Depletion Simulation,” Transactions of the
Korean Nuclear Society Autumn Meeting, Yeosu, Korea,
October 25-26, (2018).
7. H. J. Park et al., “BEAVRS Benchmark Analysis by
DeCART Standalone and DeCART/MATRA MultiPhysics Code Coupling Calculations” M&C 2019,
Portland, OR, August 25-29, (2019).
8. H. J. Shim et al., “McCARD: Monte Carlo Code forAdvanced Reactor Design and Analysis,” Nucl. Eng.
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9. S. Kumar et al., “Integral Full Core Multi-Physics PWR
Benchmark with Measured Data,” NEUP 14-6741,(2018).
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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Initial Assessment of Anderson Acceleration on Pinwise Coupled Neutronics/ThermalHydraulics Code nTER/ESCOTJaejin Leea, Jin Young Chob and Han Gyu Jooa,
Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Republic of Korea
b
Korea Atomic Energy Research Institute, 111 Daedeok-daero 989Beon-gil, Yuseong-gu, Daejeon, 34057, Republic of
Korea
aCorresponding author: joohan@snu.ac.kr*

Abstract
In order to improve the convergence behavior of the fixed-point iteration (Picard
iteration) for neutronics/thermal-hydraulics coupled problems, Anderson
acceleration is implemented in a pin-wise whole core analysis code nTER/ESCOT.
The fixed-point map of Anderson acceleration is established for serially coupledwhole core transport code and subchannel code. The performance of Anderson
acceleration is examined with single assembly problems having low and highboron concentrations. The Anderson scheme shows the comparable convergence
behavior to that of the optimum fixed-point iteration with under-relaxation factor.
Key Words: Multiphysics, Fixed-point iteration, Anderson acceleration,
Nonlinear problem, nTER, ESCOT
1. Introduction
Pin-resolved whole core multiphysics calculations
become more practical by the help of advanced
computing resources and numerical methodologies.
Especially, coupling a neutronics code and a thermalhydraulics (T/H) code in pin-level is popular because theirsolutions have a strong inter-dependency in the Neutron
Transport Equation[1]. For the scheme of coupling
between physics, the fixed-point (or Picard) iteration (FPI)
is widely used because of its ease of implementation.
However, some convergence issues of the fixed-point
iteration are raised when the physics are tightly coupled.
Recently, the Jacobian-free Newton-Krylov (JFNK)[2]
Methods have been suggested to resolve that problem, but
calculating the residual of each system might require abunch of modifications of codes. Moreover, a good
preconditioner is essential for the excellent performanceof JFNK methods, and it is opposite of the Jacobian-freefeature.
For those reason, we apply Anderson acceleration(AA)[3] as an alternative coupling scheme to the fixedpoint iteration. Anderson acceleration is a method which
accelerates the convergence rate of the fixed-point
iteration by utilizing secant information of previous
iterations. The advantage of Anderson acceleration is thatno extra information is required than the fixed-point
iteration, so the physics codes can be treated as blackboxes. In this study, the deterministic transport codenTER[4] embedded with the detailed two-phasesubchannel T/H module ESCOT[5] is used as a test bedof Anderson acceleration. Firstly, the nTER/ESCOT
multiphysics platform is introduced briefly. Secondly, the296

implementation of Anderson acceleration will bedemonstrated. Lastly, the performance of Anderson
acceleration compared to the fixed-point iteration will bepresented with sample problems.
2. Coupling Scheme of nTER/ESCOTnTER (Neutron Transport Evaluator for Reacotor) is adeterministics transport code under development by
Korea Atomic Energy Research Institution (KAERI) withthe cooperation of Korea Hydro and Nuclear PowerCentral Research Institute (KHNP-CRI). The 2D/1D
method with the radial MOC and axial PN solution isapplied to nTER, and solutions are accelerated by the
CMFD method. nTER is highly parallelized with MPI, as
its domain can be decomposed by assembly-wise in aradial direction and plane-wise in an axial direction.
Although a built-in simplified T/H module was
implemented in nTER, a detailed two-phase subchannel
T/H code, ESCOT (Efficient Simulator of Core Thermalhydraulics), is recently integrated in nTER for the highfidelity analysis. ESCOT is a subchannel code developed
by Seoul National University (SNU). The four equations
drift-flux model is adopted for two-phase calculations,and numerical solutions are obtained by applying SemiImplicit Method for Pressure Linked Equation (SIMPLE)
algorithm. ESCOT is also MPI-based parallelized withthe same domain decomposition scheme of nTER.
nTER and ESCOT are serially coupled where nTER isa main code and ESCOT takes charge of the T/H moduleof nTER. At each iteration of T/H feedback calculations,nTER provides pin-wise power distribution (q) to ESCOT.
ESCOT performs its own calculation and returns T/H

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019solutions such as coolant densities (ρc), coolant
temperatures (Tc), and fuel temperatures (Tf). nTER
resumes the calculation after updating group constants.
The flowchart of nTER/ESCOT is illustrated in Fig. 1.

Based on the above definition, the algorithm of Anderson
acceleration can be expressed as Table II.
Table II. Algorithm of Anderson acceleration
Given x0 and m≥1,
Set x1=G(x0).
for k=1,2,....
Set mk = min{m,k}.
Compute G(xk) and let fk = G(xk)-xk.
Set Fk  [ f k  mk ,..., f k ]Determine (k ) [ 0( k ) ,...,  m( kk ) ]T that solves

 min T Fk  2 [0 ,..., mk ].
 mk
s.t.   i  1
 i  0m

Set xk 1    i( k ) G xk  mk i .
end

Here, m is the storage depth which determines how many
histories of previous iterations will be stored, so
Anderson-m denotes the algorithm of Table II having
specific m. The idea of Anderson acceleration is todetermine the solution at (k+1)-th iteration as a linear
combination of G where the coefficients make the normof residual vectors the minimum value. The least-squares
problem can be solved by QR decomposition. The detailderivation and algorithm are described in [6,7]Fig 1. Flowchart of nTER/ESCOTThe above procedure is a typical fixed-point iteration
with the Gauss-Seidel scheme as shown in Table I. The
solution function f and g can be corresponded to nTER
and ESCOT solution algorithm respectively. The solution
vector x which is solutions of nTER and inputs of ESCOT
refers power profile q. Likewise, the solution vector yconsists of ρc, Tc, and Tf.

3.2 Construction of the fixed-point map

Table I. Fixed-point iteration with Gauss-Seidel scheme
Given x0, y0
for k=0,1,....
Solve x( k 1)  f (x( k ) , y ( k ) )In order to implement Anderson acceleration to thenTER/ESCOT code, the fixed-point map G has to beconstructed for the system[8]. Firstly, the solutionfunction of nTER is defined as below:

Solve y ( k 1)  g (x( k 1) , y ( k ) )endx(Nn 1)  g N (x N , Tc , c , T f )3. Implementation of Anderson Acceleration

3.1 The Anderson acceleration method
In many scientific applications, a system of nonlinear
equation can be generally expressed as below:
n



n

.

q ( n 1)   E , N (x(Nn 1) )

(1)

n



n

.

(4)

where subscript E denotes ESCOT, so [E,N] implies thatthe solutions are transferred from nTER to ESCOT. The
solution function of ESCOT can be defined as samemanner:

The eq. (1) can be rearranged to an equivalent a fixedpoint problem as follows:x  G(x), G :(3)where subscript N means nTER, and superscript (n+1)
denotes the next iteration while the notation for thecurrent iteration is omitted. The solution vector xN
contains group scalar fluxes and eigenvalue. The pin
powers can be calculated by multiplying the scalar fluxes
and kappa-fission cross sections, so that process can be
expressed with a transfer function τ:

In this chapter, the algorithm of Anderson acceleration ispresented, and the implementation process in the code is
explained.

F (x)  0, F :

i 0

(2)

x(En 1)  g E (xE , q( n 1) ) .

297

(5)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

The solution vector xE contains main variables of T/Hsolutions such as velocities, pressure, and water
properties at each subchannel. Extracting essential T/Hsolutions from the xE can be expressed with followedtransfer functions:Tc( n1)   N , E ,Tc (x(En1) )(6)

Tf( n1)   N , E ,Tf (x(En1) )(7)

c(n1)   N , E ,  (x(En1) ) .

(8)cFig. 3 shows that how the axial power profiles arechanged as the fixed-point iteration goes. Because coolant
with low boron concentration has a strong negative
Moderator Temperature Coefficient (MTC), the axial
power shape is down-shifted and more vigorous changes
are observed.

Therefore, the fixed-point map G can be constructed as: E , N g N Tc , c , T f   q ( n 1) q   ( n 1) ( n 1)  
 
T f   N , E ,Tc g E  q
 Tf


.
 ( n 1)   G    ( n 1)
T
 
 Tc

 c   N , E ,Tc g E  q  ( n 1)  c   c t N , E ,Tc g E  q ( n 1)  











Fig. 3. Axial power profile change by fixed-point iteration(9)

For the fixed-point iteration, the under-relaxation
factors are applied for fuel temperatures as below:
T f  Tnew  1    Told .

The eq. (9) can be further simplified by eliminating q,
because it only depends on ρc, Tc, and Tf. Finally, thefollowing fixed-point map is derived as:

  
  
  

(11)

In case of Anderson acceleration, storage depth of 1 and
2 are applied. The norm 2 of fission source pseudo errorof MOC is monitored to investigate the convergence rate,and it is illustrated in Fig. 4 and Fig. 5. The convergencecriterion is set to 10-5.
The major results can be found in Fig. 4, Fig5, Table
III, and Table IV. The eigenvalues are agreed within 2 pcm
difference. The optimum under-relaxation factor of FPI isdifferent by 1.0 and 0.7 for 1,000 ppm and 0 ppm caserespectively, so it can be considered that the optimum
under-relaxation factor of the FPI would be varied by thecharacteristics of a problem. Anderson-1 shows
comparable performance with that of the optimum FPI inboth cases, while Anderson-2 does only for 1,000 ppm
case.

  
   .(10)   N , E ,Tc g E  E , N g N  Tc ,  c , T f  T f( n 1)  Tf   ( n 1)   Tc
  G  Tc    N , E ,Tc g E  E , N g N Tc , c , T f  ( n 1)    c 
 c  t
 N , E ,Tc g E  E , N g N Tc , c , T f 

4. Performance Examination with Single Assembly
Problems
The performance of Anderson acceleration is examinedwith single assembly problems. The configuration of theassembly is a typical PWR 16x16 rod array as shown in
Fig. 2. The axial computational nodes are constructed by
diving 4.4 m height into 26 non-uniform planes. Typical
Hot Full Power (HFP) conditions of PWRs are applied as
15.5 MPa outlet pressure, 296 ℃ inlet temperature, and15.9 MWth of power. Two different boron concentrations,0 ppm and 1,000 ppm, are used to simulate different
feedback effects.

Fig. 2. Material configuration of a single assembly
problemFig. 4. Norm 2 of fission source errors by FPI and AA of
Boron 1,000 ppm case298

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Acknowledgements
This work was supported by Korea Atomic Energy
Research Institution (KAERI) and funded by Korea
Hydro and Nuclear Power Central Research Institute(KHNP CRI)
References
1. Kochunas R., et al., “VERA Core Simulation
Methodology for Pressurized Water Reactor Cycle
Depletion,” Nucl. Sci. Eng., 185 (2017)
2. Knoll D.A. and Keyes D.E., “Jacobian-free NewtonKrylov Methods: a Survey of Approaches andApplications,” J. Comput. Phys., 193, 357-397 (2004)
3. Anderson D.G., “Iterative Procedures for Non-linear
Integral Equations,” J. ACM 12, 547, 547-560 (1965)
4. Cho J.Y., et al., “Performance of a Whole Core
Transport Code, nTER,” NURER2018, Jeju, Korea,
Sep. 30 – Oct. 03 (2018)
5. Lee J., et al., “Development of a drift-flux model
based core thermal-hydraulics code for efficient highfidelitymultiphysics calculation,” Nucl. Eng. and Tech,https://doi.org/10.1016/j.net.2019.04.002 (2019)
6. An H., et al., “Anderson Acceleration and Application
to the Three-Temperature Energy Equations,” J.
Comput. Phys., 347, 1-19 (2017)
7. Walker H.F., “Anderson Acceleration: Algorithms and
Implementations”, Research Report, MS-6-15-50,
Worcester Polytechnic Institute Mathematical
Sciences Department (2011).
8. Toth A., et al. “Local Improvement Results forAnderson Acceleration with Inaccurate Function
Evalutions,” SIAM J. Sci. Comput., 39(5), 47-65 (2017)

Fig. 5. Norm 2 of fission source errors by FPI and AA of
Boron 0 ppm case
Table III. Result comparison of FPI and AA of boron
1,000 ppm case


k-eff
FPI w=1.0
FPI w=0.7
FPI w=0.5
FPI w=0.3
AA-1
AA-2

(pcm)

0.964700.964700.964700.96469
0.964710.96470

Ref.
0
0
1
-1
0

Rel. Pin Power
diff.
(Max/RMS, %)Ref.
0.14/0.06
0.14/0.05
0.21/0.10
0.10/0.02
0.10/0.02

# of
FPI
6
7
10
16
6
6

Table IV. Result comparison of FPI and AA of boron
0 ppm case


k-eff
FPI w=1.0
FPI w=0.7
FPI w=0.5
FPI w=0.3
AA-1
AA-2

(pcm)

1.09060
1.09059
1.090621.090621.09059
1.09061

Ref.
1
-2
-2
1
-1

Rel. Pin Power
diff.
(Max/RMS, %)Ref.
0.45/0.24
0.48/0.32
0.72/0.46
0.89/0.58
0.43/0.23

# of
FPI
10
7
10
16
6
10

6. Conclusions
In order to improve the convergence behavior of fixedpoint iteration in neutronics-T/H coupled calculations,
Anderson acceleration is implemented to nTER/ESCOT.
Through the performance examinations in two different
boron concentrations, it was figured out that Anderson
acceleration can follow the optimum performance of theunder-relaxed FPI, so we can expect the improved
convergence behavior of nonlinear iterations by Anderson
acceleration without heuristic scanning.
However, the deteriorated performance of higher
order Anderson acceleration is a remaining issue, so it will
be studied in the future. Moreover, the applicability ofAnderson acceleration will be widely investigated forcore size problems and depletion calculations.

299

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Pin-by-Pin Multi-physics Analysis and Evaluation of the Critical Heat Flux (CHF) in a PWRCoreJaeha Kim, Khurrum Saleem Chaudri, and Yonghee Kim∗
Department of Nuclear and Quantum Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, 34141, Republic of Korea*
Corresponding author: yongheekim@kaist.ac.kr

Abstract
A diffusion-based pin-by-pin neutronics and sub-channel thermal hydraulics
coupled code system has been developed. The pin-by-pin neutronics calculations
are performed using HCMFD algorithm while the sub-channel thermal hydraulics
solution is provided by the START code. It is shown that a steady-state pin-by-pin
coupling analysis can be performed within a reasonably short time with the code
system. With the sub-channel T/H coupling, the radial power peaking factor in aconverged solution becomes 1.26 as compared to 1.46 for uncoupled case.
Similarly, the Critical Heat Flux Ratio (CHFR) gives a value of 1.78 for converged
coupled solution as compared to 1.62 for non-coupled solution, hence showing
more margin of safety.
Key Words: HCMFD algorithm, START code, Neutronics-T/H coupling,
Pin-by-pin diffusion analysis, Sub-channel T/H analysis
1. Introduction

To enable an efficient parallel computing, the wholedomain is decomposed into subdomains, the fuelassemblies in a LWR.
The global balance is governed by solving the global
eigenvalue problem with the one-node CMFD method,
represented by Eq. (1).

In view of the safety and operational margin of nuclear
reactors, accurate pin-wise power distribution is essential
and a lot of efforts are being made for more accurate 3-D
pin-power information. However, the massive
computational loads for a high-fidelity solution is always
a major obstacle.
In our research group, the Hybrid Coarse-Mesh Finite
Difference (HCMFD) algorithm [1] and the GET Plus
SPH (GPS) methods [2] are suggested to carry out pinby-pin neutronics calculations accurately in a short time.
Similarly, the START code [3] provides capability to carryout sub-channel based thermal hydraulics calculations forlarge LWR core in a relatively short time. In this study,
HCMFD algorithm based code has been coupled with the
START code to perform whole-core pin-by-pin
neutronics/thermal-hydraulics coupled calculations for alarge LWR core.
Critical Heat Flux Ratio (CHFR), ratio of Critical Heat
Flux (CHF) to local heat flux, is a very important safetyparameter for reactor design and safety. Conservativevalue estimation for CHFR is done based on hot channel
values. Effect of coupled calculation on CHFR prediction
is studied in this work.

globalglobalAcmfd
Φ cmfd=1 global global
Fcmfd Φ cmfd ,k(1)

global: global node-averaged fluxes,
Φ cmfdglobalAcmfd : global system matrix,
global: global fission source operator,
Fcmfd
k : multiplication factor.

Then the global balance information is transferred to each
subdomain in terms of the boundary conditions, the fixedfission source and incoming currents on the boundary.
The local fixed source problems, represented by Eq.
(2), are solved by the conventional two-node CMFD
method based on NEM, with the given boundary
conditions. The solutions of the local problems are thenhomogenized for the following global calculation.
local, i 1, 2,...N ,M ilocal=φilocal S=i

2. Methodology(2)

φilocal : i -th node local fluxes,

2.1 HCMFD Algorithm

M ilocal : i -th node local system matrix,
Silocal : i -th node local fixed source.

In the HCMFD algorithm, two CMFD (Coarse-Mesh
Finite Difference) methods are nonlinearly coupled with
the nodal expansion method (NEM) applied on pin-level.

In this HCMFD framework, the whole-core pin-by-300Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

pin analysis can be performed very effectively by anefficient parallel computing in a local-global non-linear
iterative scheme as shown in Fig. 1. More details of thework can be found in the reference [1].

been integrated into the HCMFD code as a T/H module.
The main program and START module continuouslyinterchange required data in each neutronics and T/Hcalculation, and do feedbacks for a converged multiphysics solution. Detailed list of data interchanged is
introduced in Table II.
Table II: Flow of data in coupled analysis
Data flow
List of data
HCMFD → START
Pin-by-pin linear power.
Coolant temperature,
START → HCMFD
Coolant density,
Fuel temperature.
Meanwhile, a module for fuel temperature analysis
has been added. It performs a 1-D cylindrical fueltemperature analysis using the rod-centered coolant
temperature and corresponding surface heat transfer
coefficient. The effective fuel pin temperature is simply
calculated by using a 30/70 weighted average of fuelcenterline and fuel surface temperature [4]. The obtained
pin-by-pin fuel temperature data is then transferred to
HCMFD for the T/H feedback. The cross-section
feedback is done by Eq. (3) with the cross-section
derivatives prepared in advance.

Fig. 1: Schematic diagram of the HCMFD algorithm
2.2 START code
The START (Steady and Transient Analyzer for Reactor
Thermal Hydraulics) code is an in-house developed code
to perform sub-channel thermal hydraulics analyses for
LWRs. Special emphasis is on fast execution of the code
to perform coupled neutronics/thermal hydraulics
analysis in a reasonable time. OpenMP parallelization isapplied to several parts of the code.
The START code is based on homogeneous two-phasemodel. Basic conservation equations (mass, momentum
and energy), based on sub-channel formulation, aresolved using marching algorithm. Newton-Raphson
iterations determine pressure drop for axial and radialpressure drop used in axial and lateral momentum
equations. Time-dependent solution is based on an
implicit scheme. The START code solution has beenvalidated against PSBT benchmark. More details of thework can be found in the reference [3]. Various
correlations and model used in the START code are given
in Table I.
CHFR is determined using local conditions. EPRI
correlation is used for calculation of CHF value. Relevantcorrection factors such as grid spacer correction factor
and effect of non-uniform axial heat flux are also included
in calculations.

Σ(T f , Tc , Dc ) =Σ ref +

∂Σ∂Σ
∆ Tf +
∆Tc
∂Tc
∂ Tf

+

∂Σ
∂ 2Σ
∆Dc +
(∆Dc ) 2
∂Dc
∂Dc 2

(3)

The coupled analysis is initiated by execution of
HCMFD code with constant values for fuel temperature,
coolant temperature, and coolant density followed by T/Hcalculations. After every START run, convergence ischecked for coolant temperature, density, fueltemperature, and linear power. Iterations continue until
the convergence criteria is met. Flow chart for coupledcalculations is shown in Fig. 2.
Start

HCMFD
Calculation
Linear Power Density

Table I: Correlations and models used in START code
Parameter
Correlation
Two-phase friction multiplier
Armand Correlation
Grid spacer pressure drop
K. Rehme Model
CHF Prediction
EPRI Correlation
HTC (Single phase/subcooled Dittus-Boelter/and saturated nucleate boiling) Dittus+Thom

NO

START
Calculation
Coolant Temperature & Density, Fuel Temperature
Convergence
YES

2.3 HCMFD/START Coupled Analysis

Stop

As the START code is written in a modular form, it has

Fig. 2: Flow chart for coupled calculations301

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

3. Numerical Results

HCMFD. Initially, a URF value of 0.5 is applied. This
helps dampening the oscillations in power profile andachieve convergence in less number of iterations, as
shown by previous studies [6]. The variation in linear
power density (radial average value) for the 19 axial
nodes during coupled process is shown in Fig. 5.
It is seen that after fourth iteration, the values are quite
close to each other showing convergence of solution is
being approached. In Fig. 5, it can be seen that 1st iteration
profile is perfectly chopped cosine type, which we expect
due to use of constant T/H parameters, as compared tobottom-skewed profile obtained after convergence of
coupled solution.

In this work, all calculations are performed with 40 coreson Intel Xeon Gold 6148 CPUs (2.40 GHz). To assess thecomputational performance of the coupled code system,
an arbitrary OPR-1000(Optimized Power Reactor,
1000MW)-size core was analyzed. The fuel loading
pattern with 6 types of assemblies (A0, B0, B1, B2, C0,C1) is shown in Fig. 3. A total of 177 fuel assemblies of
16 by 16 fuel pins is present in the core. The core has anactive height of 3.81 m which is divided into 19 axial
segments. The assembly geometry is shown in Fig. 4, andthe pin-wise cross-sections are generated by pin-cell
homogenization in each assembly lattice model using a 2D transport code, DeCART2D [5] varying thetemperature conditions. The baffle surrounding the fuelassemblies is also appropriately considered in the coremodel. The pin-wise discontinuity factors are applied
over the whole core.

1
2
3
4
5
6
7
8
9
10

Linear Power Density (W/m)

30000
25000
20000
15000
10000
5000
0

0

2

4

6

8

10

12

14

16

18

20

Node

Fig. 5: Linear power density variation during iterations
Convergence behavior of various parameters with
iteration number is shown in Fig. 6.
It is observed that reactivity (pcm), shows a value of lessthan 5 pcm at 9th iteration. For converged solution i.e. 10th
iteration, a reactivity difference of less than 2 pcm isobtained. The accumulated variation in k-eff is -771 pcm
from the initial condition.
Critical Heat Flux Ratio (CHFR) shows an increase invalue with coupled iterations, hence greater margin ofsafety. Initial value of CHFR was 1.62 that converges to avalue of 1.78 at 10th iteration. Due to bottom-skewed
converged power profile, location of CHFR also shifts
accordingly.
1.144
k-eff
Coolant Density
Fuel Temperature
1.142
Linear Power
Coolant Temperature

80
70

Fig. 4: Assembly geometry and pin-cell homogenization

60

Relative Error (%)

For convergence criteria, it is observed that maximum
linear power error shows slowest convergence. Especiallythe points near the bottom or top of reactor core, where
the power density is small, show larger deviations inrelative error. For a maximum linear power error of lessthan 0.1%, maximum deviation in coolant temperature,density, and fuel temperature is already below 0.1%. So
from all the nodes, maximum linear power error needs tobe below 0.1% in order for coupled calculations to stop.
For speedy convergence and avoiding oscillations inthe solutions, under relaxation factor (URF) is applied
when thermal hydraulics parameters are transferred to

1.140

50

1.138

40
30

k-eff

Fig. 3: Fuel loading pattern

1.136

20
1.134

10

1.132

0
1

2

3

4

5

6

Iteration #

7

8

9

10

--

Fig. 6: Convergence behavior of various key parameters302Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Radial power shape for converged solution is shownin Fig. 7. The converged power profile becomes more
evenly spread in the core rather than just peaking at thecenter, and then falling off quickly for peripheral
assemblies. Radial assembly peaking factor values of 1.43
and 1.35 are obtained for initial core and converged corerespectively.

4. Conclusions
Whole-core pin-by-pin coupled neutronics/thermal
hydraulics simulation for an OPR-1000 core has been
carried out using in-house codes. Pin-by-pin neutronics
calculations are carried out using HCMFD algorithm
while sub-channel thermal hydraulics solution is carriedout using the START code. The simulations have shownthat within a very reasonable time (~3.5 minutes), with asmall commercial cluster system, pin-by-pin scale
coupled simulations are possible. Coupled solution effectcan be seen in radial power profile which is showing aradial power peaking value of 1.35 for converged core ascompared to 1.46 for initial core. Similarly, CHFR valueof 1.78 is obtained for converged coupled case ascompared to 1.62 for standalone calculation. Clearlybottom-skewed axial power profile as compared toassumed chopped cosine also shows need and importanceof coupled calculations for design and safety analyses.
The location of CHF also shifted from E8 to F10 for initialand converged solution respectively.

Power Peaking Factor
15

1.350

14
13
1.080

12
11
10

0.810098
70.54006
540.2700

3
2
1

0.000

A B

C D E

F G H

I

J

Acknowledgements

K L M N O

Fig. 7: Assembly-wise radial power peaking profile

This work was supported by the National Research
Foundation of Korea (NRF) Grant funded by the Korean
Government (MSIP) (NRF-2016R1A5A1013919).

Calculation time for the simulation is of particularinterest. Both the codes, HCMFD and the START code,
have computational times less than a minute for wholecore calculation. In order to further decrease thecomputational time, part of thermal hydraulic solution(flow rates and pressure drops) from previous iteration
was used as an initial guess for current iteration. This
caused a significant decrease in computational time oncethe solution started to approach convergence.
Individual code timing in each iteration, reported inTable III, confirms this. The overall calculation time forcoupled codes to produce a converged solution turn out tobe 203 seconds for 10 coupled iterations. Both HCMFDand START show very good parallel efficiency of values
above 70%. Calculation time for fuel temperature analysis
module is negligible (approx. 3.0 sec for 10 iterations)
showing very good parallel performance.

References
1. J. Kim and Y. Kim, “Development of 3-D HCMFD
algorithm for efficient pin-by-pin reactor analysis,”Ann. Nucl. Energy, vol. 127, pp. 87–98, May (2019)
2. H. Yu, J. Kim, and Y. Kim, “Pinwise Diffusion
Solution of Partially MOX-Loaded PWRs with the
GPS (GET PLUS SPH) Method,” Nucl. Sci. and Eng.,
June (2019).
3. K. S. Chaudri, J. Kim, and Y. Kim, “Development andvalidation of a fast sub-channel code for LWR multiphysics analyses,” Nucl. Eng. Technol., February(2019).
4. T. Kozlowski and T. J. Downar, “PWR MOX/UO2Core Transient Benchmark Final Report,”
NEA/NSC/DOC, p. 20 (2006)
5. “DeCART 2D v1.0 User's Manual,” KAERI/TR5116/2013, Korea Atomic Energy Research Institute(2013)
6. C. L. Waata, “Coupled neutronics thermal hydraulics
analysis of a high-performance light-water reactor
fuel assembly” (2006)

Table III: Computing time
Iteration
#
1
2
3
4
5
6
7
8
9
10
Total (sec)

HCMFD
START
Parallel
Parallel
CPU time
efficiency
efficiency
(sec)
(%)
(%)
8.642
71.5
15.74387.1
15.70674.8
13.59689.9
13.90875.3
11.610
93.2
9.950
75.2
10.83894.7
10.71274.9
10.499
95.6
9.748
75.0
10.385
95.7
7.580
75.4
10.331
95.9
3.092
75.6
10.311
95.9
4.543
74.8
10.293
95.9
2.971
75.5
10.27696.0
203.460

CPU time
(sec)

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Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Functional Expansion Tallies in Monte Carlo High Fidelity LWR Analysis
Bamidele Ebiwonjumi, Hyunsuk Lee, Peng Zhang, Deokjung Lee
Department of Nuclear Engineering, Ulsan National Institute of Science and Technology, 50 UNIST-gil, Eonyangeup, Ulju-gun, Ulsan, 44919, Korea*
Corresponding author: deokjung@unist.ac.kr

Abstract
Functional expansion tallies (FETs) are implemented in the Monte Carlo code
MCS to improve the fidelity of multi-physics (MP) solution of light water reactor(LWR) problems. Neutronics (N) parameters (spatial flux or power) are expanded
in a set of orthogonal basis functions (Legendre and/or Zernike polynomials) andthe coefficients of the functional expansions are determined from the Monte Carlo(MC) simulations. With the coefficients, the required distribution is reconstructed
continuously in space. Moreover, thermal-hydraulic (TH) quantities (fueltemperature, coolant density) obtained from the TH calculations of MCS are
integrated into polynomial functions and their coefficients are calculated. Coupled
MP simulation results are presented in this paper using a three-dimensional (3D)PWR fuel pin. FETs coefficients of power are transferred to the TH solver. Thenthe power distribution is reconstructed for the TH calculations. Subsequently, thecoefficients of fuel temperature/coolant density are returned as feedback, and thefuel temperature/coolant density profiles are reconstructed for the next neutronic
cycle. The results obtained with the coefficients demonstrates promise of the FETs
for high fidelity MP coupled calculations.
Key Words: MCS, functional expansion tallies, Legendre polynomials, Z
ernike polynomials, Multiphysics, high fidelity
1. Introduction

depletion, on-the-fly doppler broadening of cross-sections,
S(α, β) data and doppler broadening rejection correction.
MCS employs the conventional MC tally bins to estimate
neutronic parameters using the collision and track-length
estimators. MCS does not contain FET capabilities andare currently being implemented as detector tallies and for
MP calculations.

MC methods have the advantage to utilize continuousenergy cross-sections, and model arbitrary complex
geometry with very few approximations. However, MC
methods are well suited for the estimation of integral
quantities and to determine quantities of interest in a given
volume or point. To estimate a distribution (spatial flux or
power), the geometry of the problem is discretized, andthe integrated quantity is determined in each bin. The
fidelity of this discrete representation is usually improvedby using finer bins. However, small bin size means that
small number of neutrons contribute to the tally bin, anda large bin score standard deviation. Large number ofparticle histories are thus needed to obtain finer
distributions and decrease the bin-wise standard deviation.
Large particle histories also imply increased run time.
One of the solution to the aforementioned problem is touse functional expansion tallies (FETs) [1].

3. Functional Expansion Tallies
FETs enables the continuous representation of spatial
distributions. The discrete representation is limited to thezeroth moment. FETs produce high order moments basedon a set of orthogonal basis functions, which can be used
to reconstruct the spatial distribution. A given distribution
can be expressed as a functional expansion according to
Eq. (1)
𝐹(𝜉) = ∑∞𝑛=0 𝑎𝑛 𝑘𝑛 𝑓(𝜉)(1)where 𝜉 denote the spatial variable, {𝑓(𝜉)}∞0 forms acomplete set of orthogonal basis functions, 𝑘𝑛 is thenormalization constant, 𝑎𝑛 represent the expansion
coefficients given by Eq. (2), in which 𝜌(𝜉) is aweighting function.

2. Monte Carlo Code MCS
MCS is a 3D continuous-energy MC code underdevelopment at the Ulsan National Institute of Scienceand Technology (UNIST), for the purpose of highfidelity large scale power reactor analysis [2]. MP
coupling in MCS allows the modeling of thermalhydraulics and fuel behavior feedback [3], with fuel𝑎𝑛 = ∫ 𝐹(𝜉)𝑓(𝜉)𝜌(𝜉) 𝑑𝜉

304

(2)

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

simulation. With a collision estimator, the coefficients oftallied quantity 𝐹(𝜉) are shown in Eq. (10).

3.1 Legendre polynomials
Legendre polynomials can be used to express the axial
distribution of flux, power, fuel temperature and coolant
density in a fuel pin. Legendre polynomials are defined in
Eq. (3) [4]
𝑃𝑛+1 (𝜉) =(2𝑛+1)𝜉𝑃𝑛 (𝜉)−𝑛𝑃𝑛−1 (𝜉)
𝑛+1

1

2

(3)

(4)

with 𝛿𝑖𝑗 = 1 for 𝑖 = 𝑗 and 𝛿𝑖𝑗 = 0 otherwise. The
normalization constant is given by Eq. (5) and theweighting function for Legendre polynomials is 1.
𝑘𝑛 =

2𝑛+1

(6)

where n and m are the order of the radial and azimuthal
polynomials respectively. The radial polynomial isdefined in Eq. (7). Zernike polynomials are orthogonal in[0, 1]. The orthogonality relations are shown in Eq. (8a)and (8b) for 𝑚 ≠ 0 and 𝑚 = 0 respectively. The
normalization constants are presented in Eq. (9). Only theradial functions corresponding to 𝑚 = 0 are consideredin this work.
𝑛−𝑚2
𝑅𝑛𝑚 (𝑟) = ∑𝑠=0(−1)𝑠 (𝑛−𝑠)!
𝑛+𝑚
𝑛−𝑚
−2]![
−𝑠]!
2
2

𝑠![

𝑟 𝑛−2𝑠

(7)

1

2𝜋

𝜋

(8a)

1

2𝜋

𝜋

(8b)

𝑚′ (𝑟,𝜃)𝑟𝑑𝑟𝑑𝜃 =𝛿 𝛿∫0 ∫0 𝑍𝑛𝑚 (𝑟, 𝜃)𝑍𝑛′
2𝑛+2 𝑛,𝑛′ 𝑚,𝑚′𝑚′ (𝑟,𝜃)𝑟𝑑𝑟𝑑𝜃 =𝛿 𝛿∫0 ∫0 𝑍𝑛𝑚 (𝑟, 𝜃)𝑍𝑛′
𝑛+1 𝑛,𝑛′ 𝑚,𝑚′2𝑛+2𝜋𝑘𝑛𝑚 = { 𝑛+1
𝜋

, 𝑚≠0
, 𝑚=0

𝑓𝑛 (𝜉𝑖 )

(10)

−1

𝑟′
𝑅

(11)
(12)

4. Numerical Tests

Zernike polynomials can be used to represent the radial
distributions on unit disks. They have been used in optics
and are defined in Eq. (6) as the product of radial andazimuthal polynomials [5]. Application of Zernike
polynomials to FETs can be found in [6-7].
𝑚>0
𝑚<0
𝑚=0

𝑧−𝑧𝑚𝑖𝑛𝑧𝑚𝑎𝑥 −𝑧𝑚𝑖𝑛𝑟=3.2 Zernike polynomials

𝑍𝑛𝑚 (𝑟, 𝜃) = 𝑅𝑛𝑚 (𝑟) 𝑐𝑜𝑠(𝑚𝜃)𝑍𝑛𝑚 (𝑟, 𝜃) = 𝑅𝑛𝑚 (𝑟) 𝑠𝑖𝑛(𝑚𝜃)𝑍𝑛𝑚 (𝑟, 𝜃) = 𝑅𝑛𝑚 (𝑟)𝛴𝑡 (𝜉𝑖 )

𝜉=2

(5)

2

𝑤𝑖 𝐹(𝜉𝑖 )where 𝑛 is the nth coefficient of the expansion, 𝑁 is thenumber of histories, 𝑤𝑖 is the weight of particle 𝑖 prior
to collision, 𝛴𝑡 is the total macroscopic cross section,and 𝑓(𝜉) represents the Legendre and/or Zernike
polynomials evaluated at the collision point. By applying
the appropriate normalization constant, the continuous
distribution can be reconstructed according to Eq. (1). It
should be noted that the particle position must betransformed to the Legendre and/or Zernike space asshown in Eq. (11) and Eq. (12), where the problem
geometry spans the axial and radial regions, 𝑧𝑚𝑖𝑛 ≤ 𝑧 ≤𝑧𝑚𝑎𝑥 and 0 ≤ 𝑟 ′ ≤ 𝑅, respectively.

with 𝑃0 (𝜉) = 1 and 𝑃1 (𝜉) = 𝜉 and n is the polynomial
order. Legendre polynomials are orthogonal in [-1, +1]
and the orthogonality relation is shown in Eq. (4).
∫−1 𝑃𝑖 (𝜉)𝑃𝑗 (𝜉)𝑑𝜉 = 2𝑖+1 𝛿𝑖𝑗

1𝑁

𝑎𝑛 = ∑𝑁𝑖=1(9)3.1.1 Evaluation of expansion coefficients
The expansion coefficients in Eq. (2) are an integral
quantity. They can be estimated in a Monte Carlo305

The FETs are tested on a 3D fuel pin to verify the
implementation. The pin-cell test employ reflective
boundary conditions in the x, y directions and blackboundary conditions in the z direction. The pin-cell is a
3.0 wt.% 235U, 400cm long fuel pin with 0.41195 cm
radius. All materials are 293.6 K and moderator density is
0.7 g/cm3. 200 inactive cycles, 200 active cycles and
500,000 neutron histories were simulated. Fig. 1 shows
the continuous axial power distribution in the fuel pin
reconstructed with 10th order Legendre polynomials andcompared to the discrete representation tallied in 24 axial
nodes. FETs are also collected using the convolution of
10th order Legendre and Zernike Polynomials. The
continuous power distribution in the r-z direction of thefuel pin is presented in Fig. 2. Power in the right-hand side
is higher than in the left-hand side because the right-handside is nearer to the boundary of the fuel pin where there
is increased fission power due to the fuel’s closeness tothe moderator.
With the same pin-cell, coupled MP simulation was
performed with FET coefficients. Coefficients of power
are transferred to the TH solver and the power distribution
is reconstructed for the TH calculations. The fueltemperature and coolant density are obtained respectively
by solving the heat conduction and mass/energy
conservation equations with the reconstructed power
distribution. Moreover, the TH results are then integrated
into Legendre and Zernike polynomial functions
according to Eq. (13) with coefficients determined from
Eq. (14). Subsequently, coefficients of fuel temperatureand coolant density are returned as feedback. Then thefuel temperature and coolant density profiles are thenreconstructed for the next neutronic cycle. An example ofthe fuel temperature reconstructed from coefficients isshown in Fig. 2. MP simulation results based on theProceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019coefficients are presented in Fig. 3. The temperature anddensity distributions are compared in order to verify the
MP results based on coefficients with those obtained withthediscretepower/temperature/coolant
density
distributions. Fig 3 illustrates the comparison of the power,
average fuel temperature, maximum fuel temperature,
coolant density and temperature distribution respectively.
The power distribution has been normalized to the total
system power. The MCS/FET results are reconstructedfrom polynomial coefficients. The MCS/TH1D results arethose obtained with the discrete representations of power,
fuel temperatures and coolant densities. The axial
MCS/FET
fueltemperatureandcoolant
density/temperature distributions used during thesimulation are reconstructed using 10th order Legendre
polynomial coefficients and the radial fuel temperature isreconstructed with 4th order Zernike polynomial
coefficients at all axial levels. The k-eff of MCS/FET is 8
pcm less compared to MCS/TH1D (see Table I), and thisdifference is within one standard deviation of thequadratically combined k-eff uncertainty.

2. Lee H, “Development of a New Monte Carlo Code forLarge-Scale Power Reactor Analysis,” (PhD thesis),
Ulsan National Institute of Science and Technology,
Ulsan (2019).
3. Yu J, Lee H, Lemaire H, et al., “MCS based
Neutronics/Thermal-hydraulics/Fuel-performance
Coupling with CTF and FRAPCON,” Computer
Physics Communications, 238, 1 (2019).
4. Abramowitz M, Stegun IA, Handbook of Mathematical
Functions:withFormulas,Graphs,andMathematical Tables, Courier Corporation (1964).
5. Wang JY, Silva DE, “Wave-front Interpretation with
Zernike Polynomials,” Applied Optics, 19, 1510
(1980).
6. Ellis M, “Methods for Including Multiphysics
Feedback into Monte Carlo Reactor Physics
Calculations,” (PhD thesis), Massachusetts Instituteof Technology (2017).
7. Wendt B,
Kerby L, et al., “Advancement offunctional expansion capabilities: Implementationand optimization in Serpent 2,” Nucl. Eng. Des, 334,
138 (2018).

𝑚 𝑚𝑛
𝐹(𝑟, 𝜃, 𝑧) = ∑𝐿𝑙=0 ∑𝑁
𝑛=0 ∑𝑚=−𝑛 𝑐𝑛,𝑙 𝑍𝑛 (𝑟, 𝜃)𝑃𝑙 (𝑧) (13)
1

1

2𝜋

𝑚
𝑐𝑛,𝑙
= ∫−1 𝑑𝑧 ∫0 𝑑𝑟 ∫0 𝑑𝜃 𝐹(𝑟, 𝜃, 𝑧)𝑍𝑛𝑚 (𝑟, 𝜃)𝑃𝑙 (𝑧) (14)

6. Conclusions
FETs are presented in this paper as an alternative
approach to improve the fidelity of MC LWR analysis.
Spatial distributions of N-TH parameters are expanded ina set of orthogonal basis functions including Legendre
and Zernike polynomials. The coefficients of thefunctional expansions are evaluated and then used toreconstruct the distribution continuously during a MP
simulation. Good agreement is obtained when comparedto the discrete case which demonstrates that the FETs canbe applied in MP coupled calculations. The resultspresented in this paper are preliminary and foundational
to the future work which includes neutron transport
through materials of continuously varying properties,
application of the FETs to MP calculations of fuelassembly and ultimately large scale commercial PWRs.

Fig. 1. Comparison of axial power distribution in a fuelpin.

Acknowledgements
This work was partially supported by the National
Research Foundation of Korea (NRF) grant funded by the
Korea
government(MSIT).
(No.
NRF2019M2D2A1A03058371). This research was partially
supported by the project(L18S015000) by Korea Hydro& Nuclear Power Co. Ltd..
References
1.

Griesheimer DP, Martin WR, Holloway JP,“Estimation of Flux Distributions with Monte Carlo
Functional Expansion Tallies,” Rad. Prot. Dos., 115,
428 (2005).

Fig. 2. Continuous power/fuel temperature distribution.

306

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Fig. 3. Comparison of axial distribution of power, fuel temperature, coolant density and temperature.
Table I. Comparison of the Fuel Pin k-eff Values
MCS/TH1D
1.10022 ± 7.52317E-05
MCS/FET
1.10014 ± 7.78896E-05∆𝑘 (pcm)-8.00000E-05 ± 1.08290E-04

307

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

Hybrid Parallelism of Internal Coupling Method between Monte Carlo Code RMC and Subchannel Thermal-Hydraulic Code CTF
Kaiwen Lia, Shichang Liub, Juanjuan Guoa and Kan Wanga,∗
Department of Engineering Physics, Tsinghua University, 100084 Beijing, CHINAb
School of Nuclear Science and Engineering, North China Electric Power University, 102206, Beijing, CHINA*
Corresponding author: wangkan@mail.tsinghua.edu.cnaAbstract
Internal coupling interface for the CTF Thermal-Hydraulics code developed in the
RMC Monte Carlo code has been implemented for multi-physics simulations on
nuclear reactors. Fixed-point iteration scheme is utilized to take the ThermalHydraulic feedback into account during the neutronic transport calculation in
RMC. This paper introduces the scheme of internal coupling, the novel hybrid
parallelism method and the validation results based on a simple single fuel pin test
case and a checkerboard 2-by-2 assemblies case. The results demonstrated theaccuracy and parallel capability of the coupling system.
Key Words: RMC, CTF, Internal Coupling, Hybrid Parallelism
1. Introduction

Physics at Tsinghua University, Beijing, China[12]. Onthe-fly treatment of temperature dependent cross section,
including improved Gauss-Hermite Quadrature Method
in resolved resonance regions(RRR), on-the-fly
probability table interpolation in unresolved resonanceregions(URR) and on-the-fly sab interpolation in thermalenergy regions, has been well developed[13]. Moreover,
RMC has become the first Monte Carlo code that
accomplished the fully two-cycle simulation of BEAVRSbenchmark with COBRA-EN[14].
RMC has already been successfully coupled with
CFD code CFX in 2012[15] and sub-channel thermal
hydraulic code COBRA-EN in 2015[16]. In 2017, Guo’s
work has implemented the coupling between RMC and
CTF with a hybrid coupling scheme[17].
COBRA-TF (CTF), Coolant Boiling in Rod ArraysTwo Fluid, is a sub-channel thermal-hydraulic code
designed specially for Light Water Reactor (LWR) vessel
and core analysis. CTF is developed and improved byReactor Dynamics and Fuel Modeling Group (RDFMG)
in North Carolina State University (NCSU)[3].
Hybrid coupling method has been successfully
applied to the simulation of VERA Benchmark Problem
6 to 9. However, to fully make use of the capability ofboth RMC and CTF, internal coupling method has to be
developed.
Compared to hybrid coupling with CTF, internal
coupling method has some advantages listed below:• Data transferred between RMC and CTF by memory,which may avoid the truncation and save the disk I/Otime.
• Precise 3D power distribution is served as the heatsource for CTF, instead of the reconstruction from 2Dradial distribution and 1D axial distribution read from
CTF input files. Therefore, for some situations where

Multi-scale and multi-physics high-fidelity analysis onlarge PWRs and other reactors has become a significant
issue nowadays. With years of efforts all around the world,
including the rapid development of high performancecomputing, it is more and more practical to conduct fullcore pin-by-pin multi-physics simulation.
Therefore, many attempts to couple neutronics
transport codes with thermal hydraulics codes have beentaken over the world. The MOC code MPACT[1] in
CASL[2] project has been coupled with CTF[3] to build
a multi-physics coupling system based on deterministic
neutron transport method[4]. This coupling system has
been successfully applied to analysis on large commercial
LWRs to obtain pin-wise distributions of neutronicparameters and thermal hydraulic parameters. Besides,
another deterministic code nTRACER[5] has been
coupled with CTF and MATRA[6].
For Monte Carlo method, efforts have been made to
reduce the time and memory consumption and these years,
coupling systems based on Monte Carlo method are being
established around the world. MCNP[7] developed by
LANL has been coupled with FLICA[8] by KIT in 2011,
Serpent 2 has been coupled with SubChanFlow by KIT in
2015[9], MC21 has been coupled with CTF in 2017[10],and MCS developed by UNIST has been coupled with
CTF in 2018[11].
In the aspect of Neutronics-Thermal-Hydraulics
coupling, REAL (Reactor Engineering Analysis
Laboratory) group in Tsinghua University, Beijing, China,
has made a lot of progress based on the state-of-art Monte
Carlo code RMC. RMC is a continuous-energy Reactor
Monte Carlo neutron, photon and electron transport codedeveloped by REAL group in Department of Engineering308Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019

the power distribution is far from symmetric, internal
coupling will give more precise description andconsequently more accurate results.
• RMC and CTF are regarded as two functions or
modules rather than two programs in the internal
coupling system, thus on some batch platforms like
Tianhe-2 and Sunway-Taihu Light HPC Clusters, theneutronic and TH calculation is integrated into one job
rather than several separated jobs, which may reduce
the queueing time.

Equ. (5)(6) can be solved by a neutron transport codeand Equ. (7) can be handled by a thermal hydraulic code.
The initial temperature distribution 𝑇𝑇 and number
density distribution 𝑁𝑁 are set even practically.
The data transferred between those two modules are
power distribution 𝜓𝜓, temperature distribution 𝑇𝑇 (bothfuel temperature and moderator temperature distribution
included) and number density distribution 𝑁𝑁 (especiallythe number density distribution of the coolant). Fig. 1
below demonstrates the data transfer in Picard iteration
process.

The preprocessing procedure of CTF is required only
once in the internal coupling method, while several runs
are essential in hybrid coupling method.
2. Methodology
2.1 Coupling Scheme
When the TH feedback is not considered, the neutrontransport equation can be interpreted as Equ. (1).
1
(1)
𝐿𝐿𝐿𝐿 + 𝑅𝑅𝑅𝑅 = 𝐹𝐹𝐹𝐹,
𝑘𝑘
where 𝐿𝐿 is the loss operator, 𝑅𝑅 is the removal operator,
𝐹𝐹 is the fission operator and 𝑘𝑘 is the eigenvalue, that is,
the multiplication factor of the reactor.
In Equ. (1), operator 𝐿𝐿, 𝑅𝑅, 𝐹𝐹 are all functions ofmacro cross sections, which vary according to thetemperature distribution 𝑇𝑇(𝑟𝑟⃗) and number density
distribution 𝑁𝑁(𝑟𝑟⃗) . Therefore, from the multi-physics
aspect, the neutron transport equation evolves to Equ. (2)below.
1(2)
𝐿𝐿(𝑇𝑇, 𝑁𝑁)𝜙𝜙 + 𝑅𝑅(𝑇𝑇, 𝑁𝑁)𝜙𝜙 = 𝐹𝐹(𝑇𝑇, 𝑁𝑁)𝜙𝜙,𝑘𝑘
The temperature distribution 𝑇𝑇(𝑟𝑟⃗) and number
density distribution 𝑁𝑁(𝑟𝑟⃗) can be obtained from thermal
hydraulic calculation. In Equ. (3), 𝜓𝜓 is the power
distribution, and TH(𝜓𝜓) represents the thermal hydraulicequations operator when the power distribution is 𝜓𝜓. Equ.
(4) is the flux to power distribution process.
𝑇𝑇𝑇𝑇(𝜓𝜓)(𝑇𝑇, 𝑁𝑁) = 0,
(3)
(4)
𝑃𝑃(𝜙𝜙) = 𝜓𝜓,In the coupling system, Equ. (2)(3)(4) are coupled
together. There are different methods to handle thoseequations. For example, some regard the TH feedback as
a perturbation while others may recommend to use
Jacobian or Jacobian-Free Newton Krylov method. The
most straightforward and practical method, however, isPicard iteration method. In this work, Picard iteration
method is chosen, and the iteration process can beclarified as below:
𝐿𝐿�𝑇𝑇 (𝑛𝑛) , 𝑁𝑁 (𝑛𝑛) �𝜙𝜙 (𝑛𝑛) + 𝑅𝑅�𝑇𝑇 (𝑛𝑛) , 𝑁𝑁 (𝑛𝑛) �𝜙𝜙 (𝑛𝑛)1
(5)
= (𝑛𝑛) 𝐹𝐹�𝑇𝑇 (𝑛𝑛) , 𝑁𝑁 (𝑛𝑛) �𝜙𝜙 (𝑛𝑛) ,𝑘𝑘(6)𝑃𝑃�𝜙𝜙 (𝑛𝑛) � = 𝜓𝜓 (𝑛𝑛) ,(𝑛𝑛)(7)𝑇𝑇𝑇𝑇�𝜓𝜓 ��𝑇𝑇 (𝑛𝑛+1) , 𝑁𝑁 (𝑛𝑛+1) � = 0,

Fig. 1. Schematic diagram of coupling method: datatransfer between neutronic transport code RMC and subchannel code CTF.
From the experience, the influence of TH feedback on
eigenvalue k eff is no more than 1000pcm, typically
200pcm for the full core, thus the single physics solution
obtained from the first iteration is not too far from thecoupling one. Due to the negative reactivity coefficient of
TH feedback and the local convergence of Picard iteration
method, we may get a pretty good approximation after
only a few iterations.
2.2 Convergence Criteria
When the solution is closer to the actual one, the change
of TH feedback will become smaller and smaller, thus dueto the continuity of the equations, the flux calculated fromthe neutronics code will be similar to the previous one.
Therefore, the convergence criteria of Picard iteration is
set to the 2-order norm of the relative change of flux
distribution 𝜙𝜙.
𝜙𝜙 (𝑛𝑛+1) − 𝜙𝜙 (𝑛𝑛)∥ < 𝜖𝜖,(8)∥
𝜙𝜙 (𝑛𝑛)𝜖𝜖 can be set at some figure of the same order of thestandard deviation of neutron flux 𝜙𝜙 . In addition, toterminate from too many iterations, a maximum of
iteration number can also be defined.
2.3 MPI/OpenMP Hybrid Parallelism
To distribute the heavy work to thousands or even moreCPUs, MPI mechanism is used to communicate between
those processes. However, the memory stack for eachprocess is isolated and when burnup is involved, all thematerial data should be copied to each process, which
may exceed the limit of physical resources. Therefore,

where 𝑛𝑛 = 0,1,2, … .

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shared memory mechanism is utilized and the OpenMP
library is utilized. Currently, RMC has accomplished the
MPI/OpenMP hybrid parallelism and in this work, thismechanism is extended to the coupling module.
To include CTF into the MPI/OpenMP hybrid
parallelism scheme, some detailed analysis and the usageof MPI sub-communicator and group are needed – someof the MPI processes are reorganized as a subcommunicator to handle the CTF work while others
waiting for the TH feedback.
When OpenMP is also involved, a more complicatedbut innovative method is taken. For instance, in Fig. 2, themaximum number of process is 3 due to the memory limit,
but the number of assemblies is 4. As there is no OpenMP
support in CTF, the number of MPI processes required byparallel CTF is 4, the same as the number of assemblies.
In this innovative method, 4 processes are created but toreduce the memory usage, Proc. 3 will not cooperate in
the calculation of RMC, thus less memory is consumed
by this process.

Comparison between hybrid coupling and internal
coupling is performed to verify the correctness as the
hybrid coupling method has been validated by many cases
before.
The single pin has a length of 400 cm and is axiallydivided into 20 sections in the CTF model. The radius of
the fuel pellet is 0.41195 cm and the inner and outer radiusof the cladding is 0.41875 and 0.47585 cm. The pitch of
the pin is 1.26 cm.
In the neutron transport calculation, 200 inactive and
200 active cycles with 50,000 histories per cycle are
simulated respectively in hybrid coupling and internal
coupling.
Table I. Eigenvalue results from different cases.
Case
keff
std
w/o TH feedback
1.130519
0.000246
Hybrid coupling
1.120219
0.000244
Internal coupling
1.120033
0.000241
Table I gives the eigenvalues and their standard
deviation calculated in three cases. Significant
overestimation of eigenvalue as large as 1000pcm can beobserved without TH feedback, while the difference
between hybrid and internal coupling methods is only
18.6pcm, less than the standard deviation, which verifies
the correctness of the internal coupling between RMC andCTF.
3.2 Checkerboard 2-by-2 assemblies caseThis checkerboard case is proposed in Yu’s work todemonstrate the capability of MCS and CTF coupling[11]. The simulation results from RMC/CTF internal
coupling method are compared with those fromMCS/CTF coupling system to validate our couplingsystem.
The four assemblies in this case are the same except
for the enrichment – the top left and bottom right has anenrichment of 1.30 w%, while the other two of 2.36%.

Fig. 2. An example for demonstrating the innovative
hybrid coupling method for internal coupling between
RMC and CTF.
2.4 Cross Compilation
Cross compilation is required, for RMC is written in C++
while CTF in Fortran. Therefore, techniques like “extern“C”” in C++ and “name bind” in Fortran are used. The
cross compilation treatment is successfully tested bothunder GNU and Intel (R) compilers.
3. Simulation Results and V&V
Comparisons between hybrid coupling and internal
coupling on some test cases are performed to verify thecorrectness of the implementation of internal coupling.
And to validate the coupling system, the 2-by-2
assemblies case proposed in Yu’s work[11] is used tocompare RMC/CTF coupling and MCS/CTF coupling.
As not all parameters are described in [11]. , there may beslight difference between results calculated by MCS/CTF
and here calculated by RMC/CTF.

Fig.3. The radial profile of the 3D checkerboard 2-by-2
assemblies case[11].
Fig.3 shows the geometry of the case. The blue blocks
contain guide tubes.
100 inactive and 100 active cycles are simulated andthe number of neutrons per cycle is 500,000. In this

3.1 Single Fuel Pin Case
A typical PWR fuel pin case is used to demonstrate thecapability of internal coupling between RMC and CTF.

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summary, the comparison of eigenvalue and radial power
distribution are shown below in Table II and Fig. 4.

Subchannel Thermal-Hydraulics Code (CTF) Theory
Manual–Revision 0. Consortium for Advanced
Simulation of Light Water Reactors, CASL-U-20150054, 2015.
4. Kochunas, B., et al., Demonstration of Neutronics
Coupled to Thermal-Hydraulics for a Full-Core
Problem
using
COBRA-TF/MPACT.
CASL
Document CASL-U-2014-0051-000, 2014.
5. Ryu, M., et al., Solution of the BEAVRS benchmark
using the nTRACER direct whole core calculation
code. Journal of Nuclear Science and Technology,
2015. 52(7-8): p. 961--969.
6. Yoo, Y.-J., D.-H. Hwang, and Others, Development ofa subchannel analysis code MATRA applicable to
PWRs and ALWRs. Nuclear Engineering andTechnology, 1999. 31(3): p. 314--327.
7. Briesmeister, J.F., MCNP-A General Monte Carlo NParticle Transport Code, Version4A. LA-12625, 1993.
8. Aniel, S., et al. FLICA4: Status of numerical andphysical models and overview of applications. 2005.
9. Daeubler, M., et al., High-fidelity coupled Monte
Carlo neutron transport and thermal-hydraulicsimulations using Serpent 2/SUBCHANFLOW.
Annals of Nuclear Energy, 2015. 83: p. 352-375.
10. Kelly Iii, D.J., et al., MC21/CTF and VERA
multiphysics solutions to VERA core physics
benchmark progression problems 6 and 7. Nuclear
Engineering and Technology, 2017. 49(6): p. 1326-1338.
11. Yu, J., et al., Preliminary coupling of the
Thermal/Hydraulic solvers in the Monte Carlo code
MCS for practical LWR analysis. Annals of Nuclear
Energy, 2018. 118: p. 317--335.
12. Wang, K., et al. RMC-A Monte Carlo code for reactor
core analysis. 2014. EDP Sciences.
13. Liu, S., et al., Development of on-the-fly temperaturedependent cross-sections treatment in RMC code.
Annals of Nuclear Energy, 2016. 94: p. 144-149.
14. Wang, K., et al., Analysis of BEAVRS two-cycle
benchmark using RMC based on full core detailed
model. Progress in Nuclear Energy, 2017. 98: p. 301312.
15. Li, L.-S., H.-M. Yuan, and K. Wang, Coupling of
RMC and CFX for analysis of pebble bed-advanced
high temperature reactor core. Nuclear Engineeringand Design, 2012. 250: p. 385-391.
16. Liu, S.C., et al. Study of neutronics and thermalhydraulics coupling with RMC COBRA-EN system.
2015.
17. Guo, J., et al., Coupled neutronics/thermal-hydraulics
analysis of a full PWR core using RMC and CTF.
Annals of Nuclear Energy, 2017. 109: p. 327-336.

Table II. Eigenvalue results from different couplingsystems (w/ cross flow). To keep the number of digits thesame, results from RMC/CTF are rounded from 6 digits
to 5.
Case
keff
std
MCS/CTF
1.12095
0.00008
RMC/CTF
1.12104
0.00006

Fig. 4. Radial power distribution from RMC/CTF
coupling.
Compared with the results from MCS/CTF, the differenceof eigenvalue is only 9pcm, and the radial power
distributions of the two coupling systems are almost thesame, which validates the internal coupling method
implemented between RMC and CTF.
4. Conclusion
Internal coupling between the RMC neutronics code andthe CTF sub-channel thermal-hydraulic code has beensuccessfully implemented and MPI and OpenMP hybrid
parallelism is supported. Verification and validation forthis internal coupling system is firmly guaranteed by theconsistence with hybrid coupling and the MCS/CTF
coupling system, which enables the coupling system to becompetitive for high-fidelity full-core pin-by-pin multiphysics simulation.
Reference
1. Kochunas, B., et al. Overview of development anddesign of MPACT: Michigan parallel characteristics
transport code. 2013.
2. Turinsky, P.J. and D.B. Kothe, Modeling andsimulation challenges pursued by the Consortium for
Advanced Simulation of Light Water Reactors(CASL). Journal of Computational Physics, 2016. 313:
p. 367--376.
3. Salko, R.K. and M.N. Avramova, COBRA-TF

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NECP-X and CTF Solutions to VERA Benchmarka

Bo Wanga, *, Zhouyu Liua and Liangzhi Caoa
Xi’an Jiaotong University, No.28 Xianning West Road, Xi’an, Shaanxi, 710049, P.R.China*
Corresponding author: 756958413@qq.com

Abstract
An internal coupling scheme was proposed for the coupling between high-fidelity
neutronics code NECP-X and sub-channel thermal hydraulic code CTF. CTF was
involved in NECP-X as a shared library object and driven by NECP-X, NECP-X
and CTF have their communicator, respectively. Coupling data was transferred via
code memory rather than files. NECP-X/CTF was applied to VERA benchmark.
The results were compared with other high-fidelity coupling systems. The
numerical results shown that internal coupling method is an effective way forreactor core simulation and is accurate enough.
Key Words: NECP-X; CTF; coupling calculation; VERA Benchmark
1. Introduction

calculation are used; a pseudo-resonant-nuclide subgroupmethod is implemented; there are four methods foranisotropic scattering: the P0 isotropic approximation, theoutflow transport approximation, the inflow transport
approximation and the high order scattering method; thehybrid transport method based on 2D MOC and 1D SN isadopted in NECP-X, multi-group CMFD acceleration andmulti-level parallel method have been extended to thetransport calculation; a closed channel T/H model withoutcross flow based on pin level has been developed andcoupled into NECP-X; the transient calculation anddepletion based on the predictor-corrector quasi-static
method also are implemented in NECP-X.

With the growth of computing capability, the coupling of
high-fidelity neutronics and sub-channel thermal
hydraulic becomes more and more popular worldwide.
Coupling data is passed through computer memory ininternal coupling method, there are higher transform
efficiency and more complete iteration data, but usually
two codes need to be modified to meet the requirements
for internal coupling. Usually an interface transferring rod
power from neutronics code to thermal hydraulic code
and an interface transferring thermal hydraulic variablefrom thermal hydraulic code are needed.
In this paper, NECP-X and CTF are combined with theinternal coupling approach. First CTF is compiled to ashared library using CMake tool, then the shared libraryis linked in NECP-X using CMake tool when compiling
NECP-X. Note that the preprocessor of CTF is used togenerate the input files of CTF, which is designed forpressurized water reactor in a quick and less error-prone
manner.
This paper is organized as follows: in Section 2, NECPX, CTF are introduced in brief, respectively; in Section 3,
the coupling method is described, including coupling
approach, data transform method and mesh mapping; thecoupling results of VERA benchmark are listed andcompared in Section 4; finally the conclusions and future
work are presented in Section 5.

2.2 Sub-channel thermal hydraulic code CTF
COBRA-TF is a code developed originally in 1980 by
Pacific Northwest Laboratory under sponsorship of the
NRC, began as a T/H rod-bundle analysis code. The codeuses a two-fluid modeling approach with consideration
for three separate, independent flow fields; fluid film,
vapor, and liquid droplets. Both sub-channel and 3D
Cartesian forms of 9 conservation equations are available
for PWR modeling. COBRA-TF includes a wind range ofT/H models crucial to accurate LWR safety analysis
including, but not limited to, flow regime dependent twophase heat transfer, inter-phase heat transfer and drag,droplet breakup, and quench-front tracking.
CTF is an improved version of COBRA-TF developed
and maintained by RDFMG at PSU and NCSU [1]. There
are three symmetry models in CTF, including quarter-core
mirror symmetry, quarter-core rotational symmetry andeighth-core symmetry. The domain decomposition
method has been realized in CTF with assembly as thesmallest unit based on MPI.

2. Coupling codes
2.1 High-fidelity neutronics code NECP-X
NECP-X is a new high-fidelity neutronics code developedby NECP laboratory at Xi’an Jiaotong University. The 69group cross sections library for transport calculation andcontinuous energy cross sections library for resonance3. Coupling method

312Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
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In this section, the internal coupling approach is describedin Section 3.1, the radial mesh mapping is shown in
Section 3.2, the parallel coupling and call logic are
discussed in Section 3.3.

3.3 Parallel coupling and call logic
The minimum spatial decomposition unit in NECP-X isquarter assembly, while the minimum unit in CTF isassembly, so actually the number of cores needed forneutronics calculation is large than the number for T/H
calculation. The whole flow chart is given in Fig. 4 toillustrate the details of coupling process.

3.1 The coupling approach
CMake, gfortran and BLAS/LAPACK libraries areneeded prior the CTF building, PETSc and MPICH
libraries are necessary if the parallel option is on. To
couplewithotherneutronics
codes,a
CTF_Coupling_Interface has been developed. When
processing the building, firstly this interface is built as ashared library object via modifying the CTF CMakeLists
file, then the shared library object is included and linked
in NECP-X. All data is passed through the interface incoupling.

Start

Read NECP-X and CTF input

Pin power

CTF preprocessor

Set CTF rod power
NECP-X and CTF
initialization
CTF solve

Mapping
Get T/H data
NECP-X solve

3.2 The mesh mapping

Update NECP-X temperature

Fig. 1 shows the neutronics and thermal hydraulic model
to a quarter core. The pin power generated by NECP-X isflat source region level, but pin power transferred to CTF
is homogenized in a fuel pin and the power profile inside
the fuel rod is constant as Fig. 2 shown. Fig. 3 shows theT/H variable exchange, CTF can solve the 3D heat
conduction equations to obtain fine temperature
distribution inside a fuel rod, but just the volumeaveraged fuel temperature is used in NECP-X now.

No

Fuel temperature andcoolant temperature anddensity

Converged
Yes
End

Fig. 4. Flow chart of coupling calculation.
4. Numerical results
4.1 VERA 6
VERA #6 is a PWR fuel assembly case at HFP conditions[2]. The axial model of NECP-X is as Fig. 5 shown. The
convergence criteria is listed in Table 1.
518.937cm

Upper Reflector
Upper Core Plate
Top Nozzle
Upper Gap
Upper End Plug

PlenumEnd Grid

468.937cm
461.337cm
452.510cm
448.711cm

432.711cm

Intermediate Grid

Fig. 1. Model of quarter core.

393.108cm

Active FuelIntermediate Grid

132.108cm

End Grid

70.82cm
66.951cm

Lower End Plug
Lower Gap

Bottom Nozzle
Lower Core Plate

61.050cm
55.000cm
50.000cm

Lower Reflector
0.000cm

Fig. 2. Pin power transform between NECP-X and CTF.
Fig. 5. Axial model of NECP-X.
Table 1. Convergence criteria
Parameters
Value
Keff
1 pcm
Fuel rod temperature
1 K
Cladding temperature
1 K
Coolant temperature
1 K

Fig. 3 Thermal hydraulic variable transform.

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Osaka, Japan, Dec. 2-3, 2019

The keff results comparison [3] with other high-fidel
ity codes are listed in Table 2.

Fig. 8. Averaged fuel temperature at 25th layer.

Table 2. Keff Comparison
Code
Value
DifferenceMC21/CTF
1.16424±0.00003 0
MPACT/CTF
1.16361-63
NECP-X/CTF
1.16289-135
Fig. 6 shows the axial integral power comparison with
MPACT/CTF, the outlet temperature is compared in Fig.
7, the maximum fuel rod temperature is shown in Fig. 8,
last some typical channels and typical fuel rods results arecompared in Fig. 9 and Fig. 10.
Fig. 9. Comparison of 2 channels.

Fig. 6. Comparison of axial power

Fig. 10. Comparison of 3 fuel rod.
4.2 VERA 7
VERA 7 is run a full power and nominal flow condition.
The core is at beginning of life and hot full power
conditions, including nominal power and flow. The
geometry is identical with VERA 5 except for the positionof Bank D. The soluble boron concentration is calculated
to keep the core criterial.
The criterial boron concentration results are compared inTable 3, the radial assembly normalized power iscompared in Fig. 11.

Fig. 7. Outlet temperature comparison.

Code
MC21/CTF
RMC/CTF
NECP-X/CTF

314

CBC (ppm)
854.5
848
850.5

Difference (ppm)-6.5-4

Proceedings of the Reactor Physics Asia 2019 (RPHA19) Conference
Osaka, Japan, Dec. 2-3, 2019H
1.11798 1.1020-1.41881.0299
9 1.0178-1.1706
1.1148
10 1.1024-1.1130
1.0556
11 1.0465-0.8610
1.1566
12 1.1481-0.73171.0537
13 1.0517-0.1933
1.0492
14 1.0524
0.30380.756315 0.7653
1.1931G
1.0302
1.0178-1.1994
1.1081
1.0948-1.19900.98250.9728-0.9906
1.14711.1372-0.86111.0793
1.0738-0.5065
1.1557
1.1535-0.1915
1.01281.0186
0.57560.85650.8682
1.3625F
1.1156
1.1024-1.18390.98280.9728-1.0208
1.13101.1203-0.94441.07401.0667-0.6772
1.18501.1792-0.49121.12191.1226
0.05971.0565
1.06130.45720.76530.77571.3614E
1.0564
1.0465-0.9361
1.1475
1.1372-0.8957
1.07401.0667-0.6772
1.17981.1727-0.5977
1.07771.0750-0.2472
1.11931.12000.0586
0.9876
0.99610.8656
0.63430.64201.2215D
1.15711.1481-0.7746
1.07951.0738-0.5249
1.1844
1.1792-0.4408
1.0774
1.0750-0.2194
1.23771.23890.10010.86540.86830.34040.89230.90151.0305C
1.0531
1.0517-0.1365
1.15491.1535-0.1224
1.12101.12260.14001.11841.12000.13910.86500.86830.38680.86560.87320.88060.60770.61240.7786B
1.04871.05240.35161.01191.0186
0.66501.0552
1.06130.58090.98650.9961
0.97810.89140.90151.13250.60730.61240.8450A
0.75580.76531.26000.85580.8682
1.44540.76460.77571.4542
0.63350.64201.3493

MC21/CTF
NECP-X/CTF
Relative difference (%)

Fig. 13. The coolant density distribution.
6. Conclusions
An internal coupling approach is applied to high-fidelity
neutronics code NECP-X and sub-channel TH code CTF.
All coupling data is transferred via computer memory, andthe different parallel communicators are used for different
codes. The coupling system is used into VERA N/TH
coupling benchmark and numerical results show that the
coupling system can simulate the PWR and works well.
Acknowledgements

Fig. 11. The radial assembly normalized power

This work was supported by National Natural Science
Foundation of China No. 11605130, 11605128, 11775169.

The core coolant temperature distribution is shown in Fig.
12 and the outlet coolant density and coolant density
distribution are shown in Fig. 13.

References[1] Salko, Robert & Wysocki, Aaron & Avramova,
M. & Toptan, Aysenur & Porter, N. & Blyth, T. &Dances, C. & Gomez, A. & Jernigan, C. & Kelly, J.
(2016). CTF Theory Manual.
[2] GODFREY A T. “VERA core physics benchmar
k progression problem specifications,” Oak Ridge N
ational Laboratory, CASL-U-2012-0131-004, 2013.
[3] KELLY III D J, et al. “MC21/CTF and VERA
multiphysics solutions to VERA core physics benchmark progression problems 6 and 7,” Nuclear Engin
eering and Technology, 49, pp. 1326-1338(2017).

Fig. 12. The coolant temperature distribution.

315

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