平成 27 年度 修士論文

Study on Uncertainty Quantification of Decay Heat in Light Water Reactor
Laboratory of Nuclear Reactor Engineering
Takanori Kajihara

北海道大学大学院 工学院
エネルギー環境システム専攻

Study on Uncertainty Quantification of Decay Heatin Light Water Reactor
byTakanori Kajihara
Submitted to the Division of Energy and Environmental Systems
in partial fulfillment of the requirements for the degree ofMaster of Engineering in the field of Energy and Environmental Systems
at theHokkaido University
March 2016

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Division of Energy and Environmental Systems
February 1, 2016
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Go Chiba
Associate Professor
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tadashi Narabayashi
Professor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michitsugu Mori
Professor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tamotsu Kozaki
Professor

Contents
1 Introduction

9

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2

Fission Pulse and Finite Irradiation Experiments . . . . . . . . . . . . . . 11

1.3

Motivation of Uncertainty Quantification of Decay Heat . . . . . . . . . . 11

2 Theory and Calculation System
2.1

2.2

13

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1

Uncertainty quantification . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2

𝜒2 analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Calculation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2

CBZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3

Burn-up Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Decay Heat Uncertainty in the Case of Fission Pulse and Finite Irradiation Experiments

17

3.1

Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2

Calculation Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3

Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1

Results of Uncertainty Quantification . . . . . . . . . . . . . . . . . 18

3.3.2

Experimental cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.3

Comparison between Numerical Results with Experimental Data . 28

3.3.4

Statistical Analyses for the Comparison . . . . . . . . . . . . . . . 37
1

4 Decay Heat Uncertainty in the Case of Actual Reactors

45

4.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2

Calculation Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Decay Heat Uncertainty Regarding Thorium Fuel

57

5.1

Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2

Calculation Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Conclusion

63

2

List of Figures
3-1 Decay heat uncertainty of U-235 thermal neutron pulse fission using the
JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3-2 Decay heat uncertainty of Pu-239 thermal neutron pulse fission using the
JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3-3 Covariance matrix of U-235 thermal neutron pulse fission using the JENDL. 20
3-4 Decay heat uncertainty of U-235 fast neutron pulse fission using the JENDL. 21
3-5 Decay heat uncertainty of Pu-239 fast neutron pulse fission using the JENDL. 21
3-6 Decay heat uncertainty of U-235 thermal neutron pulse fission using the
ENDF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3-7 Decay heat uncertainty of U-235 thermal neutron pulse fission using the
JEFF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3-8 Beta component of decay heat uncertainty of U-235 thermal neutron pulse
fission using the JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3-9 Gamma component of decay heat uncertainty of U-235 thermal neutron
pulse fission using the JENDL. . . . . . . . . . . . . . . . . . . . . . . . . 24
3-10 Decay heat uncertainty of U-238 thermal neutron pulse fission using the
JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3-11 Decay heat uncertainty of Pu-241 thermal neutron pulse fission using the
JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3-12 Decay heat uncertainty of Np-237 thermal neutron pulse fission using the
JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3-13 Decay heat uncertainty of U-235 thermal neutron with finite irradiation
using the JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3

3-14 Decay heat uncertainty of Pu-239 thermal neutron with finite irradiation
using the JENDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3-15 Comparison between numerical results with experimental data in the caseof beta decay heat of U-235 thermal neutron. . . . . . . . . . . . . . . . . 29
3-16 Comparison between numerical results with experimental data in the caseof gamma decay heat of U-235 thermal neutron. . . . . . . . . . . . . . . . 29
3-17 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron. . . . . . . . . . . . . . . . . 30
3-18 Comparison between numerical results with experimental data in the caseof beta decay heat of U-235 fast neutron. . . . . . . . . . . . . . . . . . . . 30
3-19 Comparison between numerical results with experimental data in the caseof gamma decay heat of U-235 fast neutron. . . . . . . . . . . . . . . . . . 31
3-20 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 fast neutron. . . . . . . . . . . . . . . . . . . 31
3-21 Comparison between numerical results with experimental data in the caseof beta decay heat of Pu-239 thermal neutron. . . . . . . . . . . . . . . . . 32
3-22 Comparison between numerical results with experimental data in the caseof gamma decay heat of Pu-239 thermal neutron. . . . . . . . . . . . . . . 33
3-23 Comparison between numerical results with experimental data in the caseof total decay heat of Pu-239 thermal neutron. . . . . . . . . . . . . . . . 33
3-24 Comparison between numerical results with experimental data in the caseof beta decay heat of Pu-239 fast neutron. . . . . . . . . . . . . . . . . . . 34
3-25 Comparison between numerical results with experimental data in the caseof gamma decay heat of Pu-239 fast neutron. . . . . . . . . . . . . . . . . 35
3-26 Comparison between numerical results with experimental data in the caseof total decay heat of Pu-239 fast neutron. . . . . . . . . . . . . . . . . . . 35
3-27 Comparison between numerical results with experimental data in the caseof beta decay heat of Np-237 fast neutron. . . . . . . . . . . . . . . . . . . 36
3-28 Comparison between numerical results with experimental data in the caseof gamma decay heat of Np-237 fast neutron. . . . . . . . . . . . . . . . . 36
4

3-29 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 10 seconds irradiation.

37

3-30 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 100 seconds irradiation. 38
3-31 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 1,0000 seconds irradiation. 38
3-32 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 10,000 seconds irradiation. 39
3-33 Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 100,000 seconds irradiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3-34 𝜒2 /𝑛 values for all experimental data and numerical results set in the caseof fission pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3-35 𝜒2 /𝑛 values for all experimental data and numerical results set in the caseof fission pulse (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3-36 𝜒2 /𝑛 values for all experimental data and numerical results set in the caseof finite irradiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3-37 𝜒2 /𝑛 values for set of experimental data and nuclides. . . . . . . . . . . . 43
3-38 𝜒2 /𝑛 values for set of experimental data and nuclides. . . . . . . . . . . . 44
4-1 Comparison between numerical results with experimental data for CLAB
experiment of Rhinghals-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4-2 Comparison between numerical results with experimental data for GEMorris experiment of Point Beach 2 and San Onofre 1. . . . . . . . . . . . 54
5-1 Th-U-235 decay heat uncertainty in 20GWd/t burn-up. . . . . . . . . . . . 59
5-2 Th-U-233 decay heat uncertainty in 20GWd/t burn-up. . . . . . . . . . . . 59
5-3 Th-Pu decay heat uncertainty in 20GWd/t burn-up. . . . . . . . . . . . . 59
5-4 Th-U-235 decay heat uncertainty in 20GWd/t burn-up. . . . . . . . . . . . 60
5-5 Th-U-233 decay heat uncertainty in 20GWd/t burn-up. . . . . . . . . . . . 60
5-6 Th-Pu decay heat uncertainty in 20GWd/t burn-up. . . . . . . . . . . . . 61
5

5-7 Th-U-235 decay heat uncertainty in 60GWd/t burn-up. . . . . . . . . . . . 61
5-8 Th-U-233 decay heat uncertainty in 60GWd/t burn-up. . . . . . . . . . . . 61

6

List of Tables
4.1

Fuel data of CLAB experiment in Ringhals-2 . . . . . . . . . . . . . . . . 47

4.2

Assembly design data of CLAB experiment in Ringhals-2

4.3

Burn-up history of CLAB experiment in Ringhals-2 . . . . . . . . . . . . . 48

4.4

Burn-up history of CLAB experiment in Ringhals-2(Cont’) . . . . . . . . . 49

4.5

Fuel data of GE=Morris experiment in Point Beach 2 and San Onofre 1 . 50

4.6Assembly design data of GE-Morris experiment in Point Beach 2 and San. . . . . . . . . 47

Onofre 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.7

Burn-up history of GE-Morris experiment in Point Beach 2 . . . . . . . . 51

4.8

Burn-up history of GE-Morris experiment in San Onofre 1

4.9

Measured decay heat results for Rhingals-2 PWR reactor . . . . . . . . . . 52

. . . . . . . . 51

4.10 Calculated decay heat results with uncertainty for Rhingals-2 PWR reactor 53
4.11 Measured decay heat results for Point Beach 2 and San Onofre 1 PWR
reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.12 Calculated decay heat results with uncertainty for Point Beach 2 and San
Onofre 1 PWR reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1

Initial fuel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7

8

Chapter 1

Introduction
The decay heat, which is heat released by fission products and actinides from reactor
operation, is important reactor physics quantities for operation and safety analyses ofnuclear power plants and nuclear waste disposal process. It is significant to predict decay
heat accurately to ascertain the credibility of safety system at nuclear power plants
and relevant nuclear facilities such as reprocessing units and disposal sites. For thesafety analyses and reactor design, the uncertainty quantification of decay heat is alsoessential to rationalize the systems. Nuclear data uncertainty quantification have been
continuously conducted roughly[1] since the early development of the nuclear facilities.
The quantification of uncertainties is essential method to measure quantitatively ourimperfect knowledge of the true values of reactor physics parameters. For practical
purposes, quantitative uncertainty information of decay heat is necessary to support ourdecision making.
Chapter 2 describes the calculation system of the decay heat with uncertainty andthe statistical ways to evaluate the validity of numerical results.
Chapter 3 describes the result of decay heat with uncertainty in the case of special
experiments: fission pulse and finite irradiation experiment.
Chapter 4 describes the result of decay heat with uncertainty in the case of actual
reactor system focusing on Pressurized Water Reactor (PWR).
Chapter 5 describes the result of decay heat with uncertainty in the case of thorium
fuel loaded PWR system.
91.1

Background

Since the evaluation of decay heat is very important for the various areas of nuclear engineering: normal operation and accidental transient of nuclear reactors, the processing,transportation, and store of spent fuel or nuclear waste, scientists and engineers haveeagerly conducted the measurements and calculation of decay heat. The measurementof decay heat in the nuclear reactor was initiated at the same time as the launch of firstnuclear reactor during 1940s, but these measurements are much less accurate because
detectors are not sensitive to the calorific value. The first decay heat calculation followed
these measurements in 1948 by Way and Winger[2]. Then, Perkins and King conducted
the first summation calculation using 123 fission products’ decay data generated by thermal neutron fission of U-235 in 1958[3]. However, at that time, the accuracy of summation
calculation did not show good agreement with experimental data because the amountof nuclear data is not sufficient to reproduce decay heat. For the accurate calculationof decay heat, the accurate nuclear data for each nuclide is required. For the accurate
calculation, integral measurements have been undertaken to obtain short cooling time
decay heat. Nuclear models have been also applied to short half-lived nuclides[4]. These
accumulated information relevant to decay heat is included in the decay and fission yields
data of the evaluated nuclear data libraries with sub libraries: JENDL-4.0[5] ENDF/BVII.1[6], and JEFF-3.1.1.[7]. These nuclear data contain uncertainty of nuclear data.
Hence, the uncertainty of decay heat is also focused on for more accurate and rational
evaluation.
The uncertainty quantification of decay heat has been conducted through several
countries. Since one of major sources of decay heat uncertainty is nuclear data uncertainty, researchers have investigated the propagation of uncertainty. A research inlatest years has quantified the uncertainty of decay heat in the system of fission pulse
experiments[8]. This research calculated the uncertainties through analytical sensitivity
calculation with recent nuclear data. However this analysis cannot be adapted to actual
reactors, and there still remain some approximations.
10

1.2

Fission Pulse and Finite Irradiation Experiments

For the measurements of decay heat, fission pulse and finite irradiation experiments havebeen conducted. The decay heat value of these experiments can identify decay heat
contribution in each nuclide. In the system of these experiments, the thermal neutron
or fast neutron irradiations are conducted using accelerators or research reactors. The
target is pure heavy metal nuclide such as U-235 and Pu-239. In the case of fission pulse
experiment, short time of irradiation such as a few seconds or at most about 20 seconds
is conducted. Then, the irradiation time is normalized as one fission per one second.
the finite irradiation is continuous irradiation from around a few seconds to 105 seconds.
This experiment is also normalized as one fission per second during irradiation. After theirradiation, irradiated nuclide is measured for the magnitude of beta and gamma energy
by beta and gamma detectors.

1.3

Motivation of Uncertainty Quantification of Decay Heat

The motivations of current research are to identify which uncertainties of nuclear data
are influential to the uncertainty of decay heat in experimental and actual systems, andto assure more accurate decay heat uncertainty for practical purposes such as designusing validated, accurate numerical calculation system. The identification of decay heat
uncertainty source can be led to the prospective reduction of uncertainty by additional
experiments. The uncertainty quantification of current light water reactor and thorium
fuel reactor is conducted through the sensitivity analyses of burn-up. More accurate
evaluation of numerical value for decay heat is essential to the rational design of reactor
system. Thanks to the latest evaluated nuclear data library and approximation-free numerical calculation, the reliability of decay heat and decay heat uncertainty is improved.
Even though there is a research of decay heat uncertainty analysis with latest evaluatednuclear data[8], the accuracy of uncertainty is not so well optimized, and there is no quantitative analyses for the validation process. This research conducts accurate uncertainty
calculation and adopt quantitative analyses for the comparison between numerical results
and experimental data, which can realize the best suited uncertainty quantification of11decay heat with credibility.

12Chapter 2Theory and Calculation System
2.1

Theory2.1.1

Uncertainty quantification

The decay heat and the decay heat uncertainty are calculated through the following
process.
The decay heat at time 𝑡, 𝐷𝐻(𝑡), is calculated by summing the nuclide-wise decay
heats by a following equation:
𝐷𝐻(𝑡) =

∑︁𝜆𝑖 𝑁𝑖 (𝑡)𝐸𝑖 ,

(2.1)

𝑖=1

where 𝜆𝑖 is decay constant, 𝑁𝑖 is number density, and 𝐸𝑖 is mean decay energy of nuclide𝑖. Number densities are obtained by solving a following burn-up equation:𝑑𝑁 (𝑡)= 𝐴(𝑡)𝑁 (𝑡),
𝑑𝑡

(2.2)

where 𝐴 and 𝑁 are a burn-up matrix and a nuclide number density vector respectively.
Equation 2.2 is solved by a new numerical method of the matrix exponential, a MiniMax Polynomial Approximation (MMPA) method[9]. Then, the sensitivity coefficientsof decay heat 𝐷𝐻(𝑡) to five nuclear data components are calculated. These components
are decay energy, independent fission yield, half-life, neutron cross-section, and branching13

ratio. Following equations are for calculation of sensitivity of decay heat.
(1) Decay energy
𝐷𝐻
𝑆𝐸
=
𝑖

𝜕𝐷𝐻 𝐸𝑖𝐸𝑖 𝜆𝑖 𝑁𝑖·=,𝜕𝐸𝑖 𝐷𝐻𝐷𝐻(2.3)(2) Independent fission yield
𝑦𝑖
𝜕𝐷𝐻
·
𝑆𝑦𝐷𝐻
=
=
𝑖
𝜕𝑦𝑖 𝐷𝐻(︃)︃∫︀ 𝑃
𝐸𝑖 𝜆𝑖 0 𝑁 * ∆𝐴𝑁 𝑑𝑡
𝑦𝑖
,
∆𝑦𝑖
𝐷𝐻

(2.4)

(3) Half-life
𝐸𝑖 𝜆𝑖 𝑛𝑖𝜕𝐷𝐻 𝜆𝑖
·
=
+
𝑆𝜆𝐷𝐻
=
𝑖
𝜕𝜆𝑖 𝐷𝐻
𝐷𝐻

(︃

)︃
∫︀ 𝑃
𝐸𝑖 𝜆𝑖 0 𝑁 * ∆𝐴𝑁 𝑑𝑡
𝜆𝑖
,
∆𝜆𝑖
𝐷𝐻

(2.5)

(4)Branching ratio
𝐷𝐻
𝑆𝐵𝑅
=
𝑖,𝑗

𝜕𝐷𝐻 𝐵𝑅𝑖,𝑗·=𝜕𝐵𝑅𝑖,𝑗 𝐷𝐻

(︃

)︃
∫︀ 𝑃
𝐸𝑖 𝜆𝑖 0 𝑁 * ∆𝐴𝑁 𝑑𝑡 𝐵𝑅𝑖,𝑗,∆𝐵𝑅𝑖,𝑗𝐷𝐻(2.6)

where 𝑁 * is an adjoint number density vector, and 𝑃 is end time of burn-up. The
uncertainty of decay heat 𝑣 is derived through the following uncertainty propagation
equation:𝑣 = 𝐺𝑇 𝑉 𝐺,

(2.7)

where 𝐺 is sensitivity coefficient vector and 𝑉 is variance-covariance matrix of nuclear
data. Note that 𝑉 is relative covariance.

2.1.2

𝜒2 analysis

For quantitative comparison between numerical results and experimental data, we calculate 𝜒2 by a following equation.

𝜒2 = (𝐶 − 𝐸)𝑡 (𝑉𝑒 + 𝑉𝑚 )−1 (𝐶 − 𝐸),(2.8)

where 𝐶 and 𝐸 are calculated results vector and experimental data vector respectively. 𝑉𝑒 and 𝑉𝑚 are variance-covariance matrix for experimental data and calculated14

results. Note that 𝑉 is absolute covariance on the contrary to Eq. 2.7. Through thispaper, the correlation coefficient of 𝑉𝑒 is set 0.7. We deal with 𝑛 as degree of freedom,which is corresponding to the number of experimental data. When the 𝜒2 /𝑛 is less thanunity, the compared targets is explained as good agreements.

2.2

Calculation System

2.2.1

Outline

The burn-up calculation and sensitivity calculation are conducted with CBZ[10, 11], ageneral-purpose deterministic reactor physics calculation code. For the burn-up calculation, there are two modules: Burner and Cooler. Burner is pin cell calculation system,
which simulates the simplified actual reactor core. Burner is implemented to simulate
actual light water reactor. Cooler offers spatially one-point system. It does not consider
the effect of all neutron reactions of fission products, but realizes high speed calculation.
Hence, Cooler is used for the simulation of fission and finite irradiation experiments.
Generalized perturbation theory is adopted to the sensitivity calculation[12].

2.2.2

CBZ

A general-purpose reactor physics code system, CBZ, is a calculation code system for
reproducing physical phenomena of neutron and gamma-ray transport in and aroundthe reactor, and interactions of neutrons with atoms or nuclides. For example, fission
chain reaction, burnup of nuclear fuel, shielding of radiation are typically simulated
through the system. It is being developed by nuclear reactor engineering laboratory at
Hokkaido University since April 2012. This code is based on CBG which was developed
by Japan Atomic Energy Agency (JAEA). The program code is written by C++ andall information related to the reactor physics calculation (geometry of a core or fuelassemblies, the medium consisting of core, calculation method, convergence condition,etc.) are defined as Class of C++. Some solvers for solving transportation equation anddiffusion equation of neutron and gamma-ray are implemented on CBZ. These solvers
integrate multiple, related classes. The exchanges of data, such as neutron cross-section
15

data, are conducted through instances of the class. So, this code can realize a wide
variety of reactor physics calculation easily by combining multiple solvers.

2.2.3

Burn-up Calculation

Number density of several nuclides in nuclear fuel changes as reactor operates. This
calculation is realized by solving burn-up equation. CBZ offers Burnup class module forsolving the equation. CBZ adopts matrix exponential method as numerical solution ofburn-up equation. It solves matrix exponential by Mini-Max Polynomial Approximation(MMPA)[9] or Chebyshev rational approximation method[13] (these methods can calculate the burn-up chain including short half-lived nuclides which are not able to solveby Krylov subspace method[14]). Burn-up Chain Generator module for making burn-up
chain is implemented. We can make arbitral burn-up chains which consist of nuclides
chosen by users using evaluated decay data file and fission yield data file. Often, weuses the chain consisting of the most detailed chain of SRAC code (193 FP nuclides)
plus four additional nuclides. This chain can evaluate decay heat and radioactivity in areactor after a year of shutdown. Burn-up class calculates number density after burn-up
by inputting nuclides number density and burnup period. Neutron flux information is
given by neutron transportation or diffusion solver.
Burner is implemented as a tool for burn-up calculation about pin cell system. Burner
calculates square and hexagonal systems consisting of three regions of fuel, cladding, andcoolant. Neutron flux calculation is conducted by PJI module based on the collision
probability method. Burner can calculate both in operation and cooling after the operation of a reactor. Therefore, Burner can calculate number density, radioactivity, anddecay heat of every nuclide depending on burn-up (cooling) period. A method which calculates sensitivity to nuclear data regarding nuclear property (number density, effective
multiplication factor, reactivity) after burn-up is implemented[11, 15]. For the special
experimental system such as fission pulse and finite irradiation, the Cooler module canbe used. This module can offer spatially one-point calculation.

16

Chapter 3

Decay Heat Uncertainty in the Caseof Fission Pulse and Finite
Irradiation Experiments
3.1

Background and Motivation

The fission pulse and finite irradiation experiments are parts of decay heat measurement
methods from the perspective of spectroscopic category. In these experiments, the gamma
and beta radiation spectra of individual nuclides are measured by appropriate detectors
for each energy range. For the validation of the decay heat calculation, the comparison
between numerical results and experimental data have been continuously conducted allover the world. The accomplishments of these attempts were successful to such extent
that numerical results of decay heat can be used for the evaluation of safety regulation
with conservative uncertainty evaluation. However, there was only a comparative analysis
with uncertainties of calculation results[16], which is a qualitative manner.
This chapter shows the uncertainty of decay heat in the case of fission pulse and finite
irradiation experiments, and compares numerical results with representative experimental
results with quantitative evaluation of agreements.
17

3.2

Calculation Condition

The uncertainty quantifications are conducted in the case of fission pulse and finite
irradiation experiments. Regarding the fission pulse experiment, U-233, U-235, U-238,
Np-237, Pu-239, Pu-241, and Th-232 are selected as irradiated nuclides. Thermal andfast neutrons are considered for each case. The cooling time after irradiation is from10−1 to 105 seconds. This calculation is simulated by the Cooler module. For thefinite irradiation experiment, U-235 and Pu-239 thermal neutron fissions are calculated
between 10 and 105 seconds irradiation time. In the same way, the cooling time after
irradiation is from 10−1 to 105 seconds. In this study, yield and decay data are quoted
from three evaluated nuclear data libraries: JENDL/FPD-2011 and FPY-2011[17], which
are defined JENDL-2011 on this chapter, ENDF/B-VII.1[6], and JEFF-3.1.1.[7].

3.3Results and Discussions

3.3.1

Results of Uncertainty Quantification

The calculated uncertainties of pulse fission decay heat by thermal neutron on JENDL2011 are shown in Figures 3-1 and 3-2 for U-235 and Pu-239 respectively. Thesefigures also show four components of uncertainty: decay energy, fission yield, half-life,and branching ratio. Around 10−1 to 100 seconds after instantaneous pulse fission, theuncertainties are about 10% and gradually decreases along the cooling time. Throughthe colling time, the dominant uncertainty source is decay energy in the short term.
The contribution of half-life and branching ratio is extremely small. Very early part ofcooling time shows relatively high uncertainty because short half-lived nuclides typicallycontain large uncertainty. Around 100 seconds after pulse fission, there is a small peak of
uncertainty, which is formed by a few nuclides: Cs-140, Sr-94, and Rb-90. The difference
between U-235 and Pu-239 is not so large, and both of them show similar trends. As
for the impact of yield uncertainty, the case of Pu-239 is slightly larger. This a bit large
yield uncertainty contributes the small difference of total uncertainty between U-235
and Pu-239. Figure 3-3 is a correlation matrix in the case shown in Fig. 3-1. This
figure explains that there is strong positive correlation between nearby cooling times. It
18

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-1: Decay heat uncertainty of U-235 thermal neutron pulse fission using the
JENDL.

also shows there is no negative correlation through the whole cooling time. This factshows that the source of uncertainty have similar trend through the region with strong
correlation.
Figures 3-4 and 3-5 show nearly same condition of decay heat uncertainty following
pulse fission, but the energy spectrum of neutron is fast neutron. The trends of riseand fall through the cooling time is almost same between U-235 and Pu-239, but theuncertainty of early part is slightly large.
Figures 3-6 and 3-7 show the decay heat uncertainty following thermal neutron
pulse fission using ENDF/B-VII.1 and JEFF-3.1.1. respectively. The result of ENDF isvery similar to that of JENDL, however, there is a small peak at 105 seconds in the caseof ENDF. This uncertainty is induced by fission yield. The dominant effect is a chain of
atomic mass 151. On the other hand, the results of JEFF are a bit smaller than JENDL
and ENDF cases. The maximum uncertainty of JEFF is about 8 % instead of 10 % in
the other cases. The trend of JEFF is similar to other two evaluated nuclear data files,
and eventually it reaches about 1%.
Figures 3-8 and 3-9 show the beta and gamma components of decay heat uncer19

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-2: Decay heat uncertainty of Pu-239 thermal neutron pulse fission using the
JENDL.

-1

1.0

10

Cooling time [s]

100
0.5
1

10

2

0.0

10

3

10

-0.5
4

10

105
10-1 100

101

102

103

104

105

-1.0

Cooling time [s]Figure 3-3: Covariance matrix of U-235 thermal neutron pulse fission using the JENDL.

20

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-4: Decay heat uncertainty of U-235 fast neutron pulse fission using the JENDL.

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
-1
10

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

5

10

Figure 3-5: Decay heat uncertainty of Pu-239 fast neutron pulse fission using the JENDL.

21

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-6: Decay heat uncertainty of U-235 thermal neutron pulse fission using the
ENDF.

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
-1
10

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

5

10

Figure 3-7: Decay heat uncertainty of U-235 thermal neutron pulse fission using the
JEFF.

22

TotalEnergyYield
HalfLife
BranchingRatio

14

Uncertainty [%]

12
10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-8: Beta component of decay heat uncertainty of U-235 thermal neutron pulse
fission using the JENDL.

tainty following thermal neutron pulse fission using JENDL. In this case, the uncertainty
induced by decay energy is large in whole. The other nuclear data components are notdifferent from total uncertainty.
Figures 3-10 - 3-12 show fission pulse decay heat uncertainty of other nuclides:
U-238, Pu-241, and Np237. The energy of irradiated neutron is thermal. The decay heat
uncertainty of U-238 is about 9.5 % and decline slope through cooling time is gentle. The
contribution of fission yield is slightly larger than that of U-235. The peak uncertaintyof Pu-241 is 11 %, which is the largest values among the representative nuclides. As
for Pu-241, the high contribution of Half life before one seconds of cooling time is also
distinctive. In the case of Np-237, the trend of uncertainty is similar to the trend of
Pu-239, and the maximum peak uncertainty is about 10 %.

23

TotalEnergyYield
HalfLife
BranchingRatio

14

Uncertainty [%]

12
10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-9: Gamma component of decay heat uncertainty of U-235 thermal neutron pulse
fission using the JENDL.

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
-1
10

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

5

10

Figure 3-10: Decay heat uncertainty of U-238 thermal neutron pulse fission using the
JENDL.

24

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

Figure 3-11: Decay heat uncertainty of Pu-241 thermal neutron pulse fission using the
JENDL.

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
-1
10

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

5

10

Figure 3-12: Decay heat uncertainty of Np-237 thermal neutron pulse fission using the
JENDL.

25

Figures 3-13 shows the series of calculated U-235 uncertainties using the JENDL2011 after 101 - 105 seconds finite irradiation. They indicates that longer finite irradiation
time makes the uncertainty of decay heat smaller compared to the case of fission pulse.
It is consistent with research of an early date [18, 19]. In a practical situation such asfuel pin in a light water reactor, the uncertainty of decay heat is around 3% around 10−1
seconds of cooling time. Figures 3-14 shows the series of calculated Pu-239 uncertainties
using the JENDL-2011 after 101 - 105 seconds finite irradiation[20, 21].
12

12
Total
EnergyYield
HalfLife
BranchingRatio

8
6
4
2
0
-1
10

Total
EnergyYield
HalfLife
BranchingRatio

10
Uncertainty [%]Uncertainty [%]

10

8
6
4
2

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

0
-1
10

5

10

0

10

(a) 10 seconds

3

4

10

4

10

10

5

12
Total
EnergyYield
HalfLife
BranchingRatio

8

Total
EnergyYield
HalfLife
BranchingRatio

10
Uncertainty [%]10
Uncertainty [%]

2

10
10
Cooling time [s](b) 100 seconds

12

6
4
2
0
-1
10

1

10

8
6
4
2

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

0
-1
10

5

10

0

10

(c) 1,000 seconds

1

10

2

3

10
10
Cooling time [s]105(d) 10,000 seconds

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

(e) 100,000 seconds

Figure 3-13: Decay heat uncertainty of U-235 thermal neutron with finite irradiation
using the JENDL.

26

12

12
Total
EnergyYield
HalfLife
BranchingRatio

8
6
4
2
0
-1
10

Total
EnergyYield
HalfLife
BranchingRatio

10
Uncertainty [%]Uncertainty [%]

10

8
6
4
2

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

0
-1
10

5

10

0

10

(a) 10 seconds

3

4

10

4

10

10

5

12
Total
EnergyYield
HalfLife
BranchingRatio

8

Total
EnergyYield
HalfLife
BranchingRatio

10
Uncertainty [%]10
Uncertainty [%]

2

10
10
Cooling time [s](b) 100 seconds

12

6
4
2
0
-1
10

1

10

8
6
4
2

0

10

1

10

2

3

10
10
Cooling time [s]

4

10

0
-1
10

5

10

0

10

(c) 1,000 seconds

1

10

2

3

10
10
Cooling time [s]105(d) 10,000 seconds

12
Total
EnergyYield
HalfLife
BranchingRatio

Uncertainty [%]

10
8
6
4
2
0
10-1

100

101

102
103
Cooling time [s]

104

105

(e) 100,000 seconds

Figure 3-14: Decay heat uncertainty of Pu-239 thermal neutron with finite irradiation
using the JENDL.

27

3.3.2

Experimental cases

Numerical results of decay heat with uncertainty are compared with experimental data.
Those data due to the fast system were taken at YAYOI[22, 23, 24, 25] experimental reactor at the University of Tokyo. Thermal system data were taken by or at Lowell[26], Oak
Ridge National Laboratory[27, 28, 29] (ORNL), Tobias[30], and Uppsala University[31].
ORNL, Lowell, and Uppsala offer the data of U-235 thermal neutron fission. YAYOI
and Lowell offer the data of U-238 fast neutron fission. ORNL and Lowell offer Pu-239
thermal neutron fission data. ORNL offers the data of Pu-241 thermal neutron fission.

3.3.3

Comparison between Numerical Results with Experimental Data

Here, numerical results calculated by CBZ using JENDL, ENDF, and JEFF and representative experimental data are compared. In the following figures, y-axis represents
the product of decay heat and time. The upper and lower limits of uncertainty range
belonging numerical results are shown as fine broken lines. Here, the uncertainty range
is defined as 1 𝜎. The cases of thermal and fast energy of U-235 and Pu-239, and fast
energy of Np-237 are raised as representative of fission pulse cases. As for finite irradiation, thermal energy of U-235 and Pu-239 between 10 - 100,000 seconds of irradiation
cases is covered.
Figures 3-15 - 3-17 show the comparison between numerical results with experimental data in the case of beta, gamma, and total decay heat of thermal pulse fission on
U-235. Lowell, ORNL, and Tobias data are plotted for beta and gamma energy cases.
ORNL and Tobias data are plotted for total cases. Regarding beta component and total decay heat, numerical results and experimental data are overlapping in the range of
uncertainty. In the case of gamma component, ORNL and Tobias experimental data isunderestimated in the range of 100 - 1,000 cooling time beyond uncertainty range.
Figures 3-18 - 3-20 show the comparison between numerical results with experimental data in the case of beta, gamma, and total decay heat of fast neutron fission pulse
on U-235. For all cases, YAYOI experimental data are compared with numerical results.
All of comparisons indicate all experimental data are within the range of uncertainties.
Figures 3-21 - 3-23 show the comparison between numerical results with experi28

Decay heat x t[MeV/fission]

0.9

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Lowell Exp.
ORNL Exp.
Tobias Exp.

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-15: Comparison between numerical results with experimental data in the caseof beta decay heat of U-235 thermal neutron.

Decay heat x t[MeV/fission]

0.9

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Lowell Exp.
ORNL Exp.
Tobias Exp.

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
100

101

102
103
Cooling time[s]

104

105

Figure 3-16: Comparison between numerical results with experimental data in the caseof gamma decay heat of U-235 thermal neutron.

29

U-235(thermal) Total power

Decay heat x t[MeV/fission]

1.6

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
ORNL Exp.
Tobias Exp.

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-17: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron.

1.0

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

Decay heat x t[MeV/fission]

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
100

101

102
103
Cooling time[s]

104

105

Figure 3-18: Comparison between numerical results with experimental data in the caseof beta decay heat of U-235 fast neutron.

30

Decay heat x t[MeV/fission]

0.8

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-19: Comparison between numerical results with experimental data in the caseof gamma decay heat of U-235 fast neutron.

Decay heat x t[MeV/fission]

1.6

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

1.4
1.2
1.0
0.8
0.6
0.4
0.2
100

101

102
103
Cooling time[s]

104

105

Figure 3-20: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 fast neutron.

31Decay heat x t[MeV/fission]

0.8

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Lowell Exp.
ORNL Exp.
Tobias Exp.

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-21: Comparison between numerical results with experimental data in the caseof beta decay heat of Pu-239 thermal neutron.

mental data in the case of beta, gamma, and total decay heat of thermal neutron fission
pulse on Pu-239. Lowell, ORNL, and Tobias data are plotted for beta and gamma energy
cases. ORNL and Tobias data are plotted for total cases. In the case of beta components, Tobias shows relatively high values around 10 seconds of cooling time, and Lowellshows a bit low values around 10,000 seconds. The gamma components of calculation
results are slightly overestimated even though numerical results and experimental data
are overlapped within the uncertainty region. The experimental data of Lowell is low inthe late cooling time. This trend is also observed in U-235 thermal neutron experimentof Lowell, which infers that there is some measurements problem. Here the calculation
result using JEFF file is low compared to using other two nuclear data. These results are
explained as pandemonium effect[32]. In the total decay heat, numerical results are a bit
underestimated around 1,000 seconds of cooling time, but it is in the range of uncertainty.
Figures 3-24 - 3-26 show the comparison between numerical results with experimental data in the case of beta, gamma, and total decay heat of fast neutron fission pulse on
Pu-239. YAYOI data is plotted on each case. Even though the case of gamma component
32

Decay heat x t[MeV/fission]

0.6
0.5
0.4
0.3
JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Lowell Exp.
ORNL Exp.
Tobias Exp.

0.2
0.1
0

10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-22: Comparison between numerical results with experimental data in the caseof gamma decay heat of Pu-239 thermal neutron.

Decay heat x t[MeV/fission]

1.4

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
ORNL Exp.
Tobias Exp.

1.2
1.0
0.8
0.6
0.4
0.2
100

101

102
103
Cooling time[s]

104

105

Figure 3-23: Comparison between numerical results with experimental data in the caseof total decay heat of Pu-239 thermal neutron.

33

Decay heat x t[MeV/fission]

0.7

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

0.6
0.5
0.4
0.3
0.2
0.1
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-24: Comparison between numerical results with experimental data in the caseof beta decay heat of Pu-239 fast neutron.

show some parts of underestimate and overestimate, all cases are good agreements.
Figures 3-27 and 3-28 show the comparison between numerical results with experimental data in the case of beta and gamma decay heat of fast neutron fission pulse on
Np-237. For the beta component, experimental data and numerical results are completely
different between 100 and 1,000 seconds of cooling time. Since these YAYOI experiments
showed large underestimate even in the case of U-235, which can be well measured ingeneral, some experimental errors are inferred. On the contrary, the case of gamma component shows good agreement. There are some scattering points in experimental data.
The cause of them is owing to hardness for adjusting raw values in this cooling time
region. There is a significant effect of (n,𝛾) reaction.
Figures 3-29 - 3-33 show the series of comparison between numerical results withexperimental data in the case of decay heat of thermal neutron finite irradiation on U-235
in the irradiation time from 10 to 100,000 seconds. The data of Tobias is adopted for thecomparison. Across the board, calculated results and experimental data show very good
agreements through the cooling time. At 1,000 seconds of cooling time in the case of34

Decay heat x t[MeV/fission]

0.6
0.5
0.4
0.3
0.2
0.1
0
10

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.
1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-25: Comparison between numerical results with experimental data in the caseof gamma decay heat of Pu-239 fast neutron.

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

Decay heat x t[MeV/fission]

1.2
1.0
0.8
0.6
0.4
0.2
100

101

102
103
Cooling time[s]

104

105

Figure 3-26: Comparison between numerical results with experimental data in the caseof total decay heat of Pu-239 fast neutron.

35

1.0

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

Decay heat x t[MeV/fission]

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-27: Comparison between numerical results with experimental data in the caseof beta decay heat of Np-237 fast neutron.

Decay heat x t[MeV/fission]

0.9

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
YAYOI Exp.

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
100

101

102
103
Cooling time[s]

104

105

Figure 3-28: Comparison between numerical results with experimental data in the caseof gamma decay heat of Np-237 fast neutron.

36

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Tobias Exp.

14.0
Decay heat x t[MeV]

12.0
10.0
8.0
6.0
4.0
2.0
0

10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-29: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 10 seconds irradiation.

10,000 and 100,000 seconds of irradiation, numerical results are slightly overestimated.
In the case of 100,000 second irradiation time, numerical result of JEFF match better
than the others clearly.
In broad terms, numerical results and experimental data are well overlapped in therange of uncertainty. In the next subsection, more quantitative comparison will be conducted.

3.3.4

Statistical Analyses for the Comparison

The 𝜒2 /𝑛 values are calculated for the all representative experimental data using JENDL,
ENDF, and JEFF as shown in Figure 3-34. In addition to experimental data which
former subsection raised, some experimental data of YAYOI, Lowell, and ORNL are
added for U-233, U-238, and Pu-241. Figure 3-35 is a figure eliminating the case of
U-235-Lowell-g, U-238-Lowell-g, and Pu-239-Lowell-g for easy-to-see.
Since the values are less than about 1.4 for the most cases in the fission pulse, numerical results match well with experiments. However, some data using JEFF represent37120.0Decay heat x t[MeV]100.0

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Tobias Exp.

80.0
60.0
40.0
20.0
0.0
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-30: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 100 seconds irradiation.

800.0

Decay heat x t[MeV]

700.0
600.0

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Tobias Exp.

500.0
400.0
300.0
200.0
100.0
0.0
100

101

102
103
Cooling time[s]

104

105

Figure 3-31: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 1,0000 seconds irradiation.

38

4000.0

Decay heat x t[MeV]

3500.03000.0JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Tobias Exp.

2500.02000.01500.01000.0
500.0
0.0
0
10

1

10

2

3

10
10
Cooling time[s]

4

10

5

10

Figure 3-32: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 10,000 seconds irradiation.

16000.0

Decay heat x t[MeV]

14000.012000.0

JENDL-2011
ENDF/B-VII.1
JEFF-3.1.1
Tobias Exp.

10000.0
8000.06000.04000.02000.0
0.0
100

101

102
103
Cooling time[s]

104

105

Figure 3-33: Comparison between numerical results with experimental data in the caseof total decay heat of U-235 thermal neutron with 100,000 seconds irradiation.

39

0

40
Pu241-T-ORNL-b
Pu241-T-ORNL-g
Pu241-T-ORNL-t

70

Pu239-T-Lowell-b
Pu239-T-Lowell-g
Pu239-T-ORNL-b
Pu239-T-ORNL-g
Pu239-T-ORNL-t
Pu239-T-Tobias-b
Pu239-T-Tobias-g
Pu239-T-Tobias-t
Pu239-F-Yayoi-b
Pu239-F-Yayoi-g
Pu239-F-Yayoi-t

U238-T-Lowell-b
U238-T-Lowell-g
U238-T-Yayoi-b
U238-T-Yayoi-g
U238-T-Yayoi-t

U235-T-Lowell-b
U235-T-Lowell-g
U235-T-ORNL-b
U235-T-ORNL-g
U235-T-ORNL-t
U235-T-Tobias-b
U235-T-Tobias-g
U235-T-Tobias-t
U235-F-Yayoi-b
U235-F-Yayoi-g
U235-F-Yayoi-t

U233-F-Yayoi-b
U233-F-Yayoi-g
U233-F-Yayoi-tχ2 / n
80

JENDL
ENDF
JEFF

60

50

40

30

20

10

Figure 3-34: 𝜒2 /𝑛 values for all experimental data and numerical results set in the caseof fission pulse.

041
Pu241-T-ORNL-b
Pu241-T-ORNL-g
Pu241-T-ORNL-t

6

Pu239-T-Lowell-b
Pu239-T-ORNL-b
Pu239-T-ORNL-g
Pu239-T-ORNL-t
Pu239-T-Tobias-b
Pu239-T-Tobias-g
Pu239-T-Tobias-t
Pu239-F-Yayoi-b
Pu239-F-Yayoi-g
Pu239-F-Yayoi-t

U238-T-Lowell-b
U238-T-Yayoi-b
U238-T-Yayoi-g
U238-T-Yayoi-t

U235-T-Lowell-b
U235-T-ORNL-b
U235-T-ORNL-g
U235-T-ORNL-t
U235-T-Tobias-b
U235-T-Tobias-g
U235-T-Tobias-t
U235-F-Yayoi-b
U235-F-Yayoi-g
U235-F-Yayoi-t

U233-F-Yayoi-b
U233-F-Yayoi-g
U233-F-Yayoi-tχ2 / n
7

JENDL
ENDF
JEFF

5

4

3

2

1

Figure 3-35: 𝜒2 /𝑛 values for all experimental data and numerical results set in the caseof fission pulse (2).

2.5
JENDL
ENDF
JEFF

χ2 / n

2.0
1.5
1.0

Pu239-T-Tobias-1E+5

Pu239-T-Tobias-1E+4

Pu239-T-Tobias-1E+3

Pu239-T-Tobias-1E+1

U235-T-Tobias-1E+5

U235-T-Tobias-1E+4

U235-T-Tobias-1E+3

U235-T-Tobias-1E+2

U235-T-Tobias-1E+1

0.0

Pu239-T-Tobias-1E+2

0.5

Figure 3-36: 𝜒2 /𝑛 values for all experimental data and numerical results set in the caseof finite irradiation.

impossibly high values because JEFF does not consider the pandemonium effect. In
addition, some experimental data such as the gamma component of Lowell show highvalue. They are caused by large discrepancy around 10,000 seconds cooling time. Few
numbers of experimental data are also affect higher 𝜒2 /𝑛 value. About from 2 to 6 isobtained as 𝜒2 /𝑛 values. From the definition of 𝜒2 , these values represent that there arewrong in the experimental data and/or numerical results as for such cases. To be more
precise, the value more than 2.0 is statistically impossible.
In the case of finite irradiation as shown in Figure 3-36, most of comparisons are in
the range of good fit. However, JENDL and JEFF represent higher value for U-235 andPu-239 respectively compared to ENDF. Even though uncertainty of numerical results
are low, it can be said that numerical calculation reproduce the data of finite irradiation
experiments well.
Figure 3-37 and 3-38 are 𝜒2 /𝑛 values by set of nuclides and experimental cases.
Here, the experimental correlation coefficient among different experiments are set 0. This42

JENDL
JEFF
ENDF

40.0
35.0
30.0

χ2 / n

25.0
20.0
15.0
10.0

all

Ir-Pu239

Ir-U235

Pulse

Yayoi

Tobias

ORNL

Lowell0.0

Irradiation

5.0

Figure 3-37: 𝜒2 /𝑛 values for set of experimental data and nuclides.

results clarify that the ORNL data show low 𝜒2 /𝑛 values, and Lowell and Tobias havelarge discrepancy in some data. Even though total 𝜒2 /𝑛 becomes large because of theexistence of deviant values in Lowell and Tobias, it is infered that basically experimental
data and calculation results show good agreement.
The uncertainties of fission products’ decay heat have been quantified in the case offission pulse and finite irradiation experiments. These analyses have been carried out by
the decay data and fission yield data from the latest JENDL, JEFF, and ENDF files.
Decay energy is dominant component of decay heat uncertainty. In the case of finite
irradiation, the uncertainty is gradually decrease along the increase of irradiation time.
From the value of 𝜒2 /𝑛, calculation results can be considered accurate for many fission
pulse experiments.
43

10.0

JENDL
JEFF
ENDF

8.0

χ2 / n

6.0

4.0

Figure 3-38: 𝜒2 /𝑛 values for set of experimental data and nuclides.

44

all

Ir-Pu239

Ir-U235

Pulse

Yayoi

ORNL

0.0

Irradiation

2.0

Chapter 4

Decay Heat Uncertainty in the Caseof Actual Reactors
4.1

Background

For the validation of numerical results calculated by CBZ, it is essential to compare withdecay heat data obtained by actual reactors. It is ideal to measure the decay heat generated from the fuels in reactors. However, it is very tough to measure accurate decay
heat in the actual operating reactors. For this reason, the accurate measurements ofdecay heat have been conducted by measuring spent fuel assemblies, which are storedin fuel storage facilities. Spent fuel is located into the special vessel with water, andthe temperature in the vessel is measured by calorimeter. There are several measurements of decay heat using spent fuels. General Electric (GE) conducted measurements
using a pool calorimeter at the GE Morris Operations (GE-Morris) spent fuel storage
facility in Illinois. As a recent calorimeter measurements of spent fuel assemblies, theSwedish central interim storage facility for spent fuel (CLAB)[33] performed the decay
heat measurements. This facility is located in Oskarshamn, Sweden.
In the case of GE-Morris, Decay heat was measured for fuel assemblies which is in
San Onofre 1, Point Beach, and Turkey Point in the case of PWRs. Regarding BWR,
the Dresden, Cooper and Monticello power plants’ fuel assemblies was measured. In
total, there are 65 assemblies at the GE-Morris. The calorimeter is located underwater45

in the spent fuel storage bay. To reduce heat loss to the pool, there is a urethane foam
insulation which separate two concentric carbon steel pipes. The length of calorimeter is
4.6 m. The systematic error is estimated to be about 2% for a thermal output of 700W,
then it increases to 4% in the range of 200W. The random error is about 4.3%.
In the project of CLAB, SKB (Swedish nuclear fuel and wastes management Co.)established a large amount of database of decay heat measurements including 39 BWR
assemblies and 34 PWR assemblies. They focused the fuels from Barseb¨𝑎ck, Ringhals,
Forsmark, and Oskarsham nuclear power plants. The design of the calorimeter at the
CLAB experiment is 4.9 m long and composed of two concentric pipes separated bypolyurethane foam to provide insulation between the interior vessel and the pool water.
The measured temperature is converted to the estimated decay heat using calibration
curve, which is evaluated by the preliminary experiment using electric heated assembly.
The effect of gamma ray which escapes the calorimeter system is measured by five gamma
monitors around vessel. the information of gamma monitors are used for correcting thecalorimeter data without gamma energy. The accuracy for the calorimeter is about 2% at
the 95 % confidence level. There is the other random error induced by the measurements.
For example, when the output is 50W in the system, error is 9%. For output about 300W,
the error is about 2.7%. At that time, there is direct proportional relationship between
error and heat output.
In this study, the 15 × 15 PWR fuels obtained from Ringhals-2 and 14 × 14 PWR
fuels obtained from San Onofre 1 and Point Beach 2 are adopted for the comparative
candidates of decay heat. These fuel assemblies do not have the pin of burnable poison.
Hence, the calculation system of CBZ can simulate burn-up process most accurately.
That is why this assemblies set is selected as a candidate.

4.2

Calculation Condition

The uncertainty quantifications are conducted in the case of PWR actual reactors. The
calculation condition is set for emulating fuel composition, assembly geometry, and burnup history. These conditions are set as shown in Tables 4.1, 4.2, and 4.3 for CLAB
measurements.
46

Table 4.1: Fuel data of CLAB experiment in Ringhals-2
Assembly ID

Type

C01
C12
C20
D27
D38
E38
E40
F14
F21
F25
F32
G11
G23
I09
I24
I25

1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2

Fuel density(g/cm3 )
10.23
10.18
10.21
10.10
10.13
10.12
10.13
10.18
10.17
10.21
10.20
10.18
10.18
10.21
10.03
10.11

U-238 (g)weight441,485
439,496
440,486
418,326
419,898
419,527
420,157
422,232
421,803423,107422,824
422,077
421,947
423,149
415,641
418,995

U-235 (g)weight14,107
14,043
14,075
14,066
14,11913,86913,89013,953
13,93913,982
13,972
13,906
13,981
14,007
13,75913,870U-236 (g)weight
69
69
69
69
69
69
69
69
69
69
69
69
69
69
69
69

U-234 (g)weight
128
128
128
128
128
128
128
128
128
128
128
128
128
128
128
128

Table 4.2: Assembly design data of CLAB experiment in Ringhals-2
Design data
Fuel rod pitch(cm)Fuel rod diameter(cm)Fuel clad thickness(cm)Fuel pellet diameter(cm)Fuel clad material

47

Type1
1.43
1.072
0.0618
0.929
Zr4

Type2
1.43
1.075
0.07250.911
Zr4

Table 4.3: Burn-up history of CLAB experiment in Ringhals-2
Cycle
Startup date
Shutdown date
Operating days
Downtime days
Assembly ID
C01
C12
C20
D27
D38
E38
E40
F14
F21
F25
F32
G11
G23
I09
I24
I25

1
6/19/1974
4/13/1977
1029
85
11,247
11,247
11,2472
3
4
5
7/7/1977 5/26/1978 6/25/1979 6/18/1980
3/31/1978 4/3/1979
4/1/1980
4/4/1981
267
312
281
290
56
83
78
80
Assembly burn-up(MWd/MTU)
20,65028,21936,68820,56527,95536,385
20,62428,0789,510
22,39931,666
6,367
15,69823,05631,757
7,568
16,02625,905
7,705
14,95425,6095,069
15,824
4,767
11,0848,307
19,056
10,5536,890

48

6
6/23/1981
5/6/1982
317
84

39,40333,97334,33925,722
21,13027,372
21,162
17,312
10,268

Table 4.4: Burn-up history of CLAB experiment in Ringhals-2(Cont’)
Cycle
Startup date
Shutdown date
Operating days
Downtime days
Assembly ID
C01
C12
C20
D27
D38
E38
E40
F14
F21
F25
F32
G11
G23
I09
I24
I25

7
7/29/1982
4/28/1983
273
91

8
9
10
11
7/28/1983 7/12/1984 6/14/1985 7/2/1986
4/13/1984 4/4/1985 4/30/1986 4/25/1987
260
266
320
297
90
71
63
54
Assembly burn-up(MWd/MTU)

12
6/18/1987
5/12/1988
329—

35,72039,67634,00929,38535,352
29,55325,18020,3038,245
5,207

36,27332,12327,921
6,727
17,212
10,19835,46335,633
15,677
26,35620,001

49

37,31443,94350,962

24,742
34,29428,99932,31040,18836,859

Also, the fuel composition, assembly geometry, and burn-up history are set as shownin Tables 4.5, 4.6, 4.7, and 4.8 for GE-Morris measurements.
Table 4.5: Fuel data of GE=Morris experiment in Point Beach 2 and San Onofre 1
ReactorAssembly ID

Point Beach 2
Point Beach 2
Point Beach 2
Point Beach 2
Point Beach 2
Point Beach 2
San Onofre 1
San Onofre 1
San Onofre 1
San Onofre 1
San Onofre 1
San Onofre 1
San Onofre 1
San Onofre 1

C52
C56
C64
C66
C67
C68
C01
C16
C19
C20
D01
D46
E18
F04

Fuel density(g/cm3 )
9.467
9.467
9.467
9.467
9.467
9.467
10.03710.03710.03710.03710.03710.03710.03710.037

U-238
wt%
3.397
3.397
3.397
3.397
3.397
3.397
3.865
3.865
3.865
3.865
4.005
4.005
4.005
3.996

U-235
wt%
96.55796.55796.55796.55796.55796.55796.08996.08996.08996.08995.94995.94995.94995.958

U-236
wt%
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03

U-234
wt%
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016
0.016

Table 4.6: Assembly design data of GE-Morris experiment in Point Beach 2 and San
Onofre 1
Design data
Fuel rod pitch(cm)Fuel rod diameter(cm)Fuel clad thickness(cm)Fuel pellet diameter(cm)Fuel clad material

Point Beach 2
1.412
1.07190.06180.9290
Zr4

50

San Onofre 1
1.4122
1.07190.04190.9741
Zr4

Table 4.7: Burn-up history of GE-Morris experiment in Point Beach 2
Cycle
Startup date
Shutdown date
Operating days
Downtime days
Assembly ID
C52
C56
C64
C66
C67
C68

1A
1B
2
3
8/1/1972
5/1/1973
12/20/1974 3/29/1976
5/1/1973 10/16/1974 2/26/1976
3/3/1977
273
533
433
339
0
65
32
–
Assembly burn-up(MWd/MTU)
677
10,801
23,117
31,914
1,033
16,47529,35638,917
1,061
16,92029,76439,384732
11,66824,92435,433
1,041
16,60029,401
38,946817
13,034
13,03437,059

Table 4.8: Burn-up history of GE-Morris experiment in San Onofre 1
Cycle
Startup date
Shutdown date
Operating days
Downtime days
Assembly ID
C01
C16
C19
C20
D01
D46
E18
F04

1
1/27/1968
10/2/1970
979
49
7,689
9,077
11,75014,4182
3
4
5
11/20/1970 2/24/1972 6/30/1973 4/23/1975
1/1/1972
6/2/1973 3/14/1975 9/3/1976
407
464
622
526
54
28
40
186
Assembly burn-up(MWd/MTU)
16,20526,54018,452
28,462
21,37230,42623,41032,363
9,738
20,592
31,393
10,02921,05532,318
8,068
19,47032,357
6,883
19,73351

6
4/4/1977
9/15/1978
529–

30,429

4.3

Results and Discussion

The calculation of decay heat and decay heat uncertainty is conducted for 16 assembliesof the Rhinghals-2 PWR reactor. Table 4.9 shows the results of measurement decay heat.
Because several assembly cases are measured more than once in different dates, such cases
are simulated as different cases. Figure 4.10 show the results of calculated decay heatand uncertainty. The comparison between numerical results and experimental data isshown in Table 4-1. It clearly represents that the difference between numerical resultswith experimental data are within uncertainty region. In the quantitative analysis, thevalue of 𝜒2 /𝑛 is calculated. When we consider the correlation of experiment in each case
0.7, 𝜒2 /𝑛 is 0.31. Hence, the calculation results and experimental data match well withuncertainty and error. In addition, the accuracy of calculation system is validated.
Table 4.9: Measured decay heat results for Rhingals-2 PWR reactor
Assembly
C01
C12
C20
C20
C20
C20
D27
D38
E38
E38
E40
F14
F21
F25
F32
G11
G23
I09
I24
I25

Measurementdate
6/10/2004
4/6/2004
4/14/2004
4/15/2004
4/16/2004
4/23/2004
4/26/2004
4/5/2004
3/30/2004
3/31/2004
6/14/2004
6/18/2004
6/23/2004
6/21/2004
5/28/2004
5/24/2004
5/18/2004
5/17/2004
5/26/2004
4/13/2004

Decay
Time
8468
8403
6950
6951
6952
6959
7669
8005
7999
8000
8075
7722
7376
7725
5860
6990
6984
5849
6601
6198

Calorimeter
decay heat(W)
406.3
401.6
406.1
416.4
419.2
425.9
444.6
431.6
366.9
364.9
372.2
372.5
410.2
386.7
675.4
405.7
409.6
494.5
398.2
434

Gamma
Escape(W)
9.4
8.7
9.7
9.6
9.7
9.8
11.4
10.8
9.4
9.4
9
9.3
10.7
10
16.6
10.6
11.1
13.4
11.8
11.8

Total
decay heat(W)
415.8
410.3
415.8
426
428.9
435.7
456.1
442.3
376.3
374.3
381.3
381.8
420.9
396.7
692
416.4
420.6
507.9
410.1
445.8

As cases of shorter cooling time of measured decay heat, decay heat and decay heat52

Table 4.10: Calculated decay heat results with uncertainty for Rhingals-2 PWR reactor
Assembly
C01
C12
C20
C20
C20
C20
D27
D38
E38
E38
E40
F14
F21
F25
F32
G11
G23
I09
I24
I25

Calculated decay heat(W)
417.3
412.6
428.7
428.7
428.6
428.5
449.5
439.9
370.0
370.0
374.0
379.1
414.1
397.3
692.7
409.1
415.3
511.8
398.1
446.5

Uncertainty(%)
2.76
2.76
2.70
2.70
2.70
2.70
2.80
2.82
2.82
2.82
2.82
2.83
2.83
2.82
3.05
2.83
2.85
2.87
2.86
2.87

uncertainty is calculated for 14 assemblies of PWR reactor in the GE-Morris experiments.
Table 4.11 shows the results of measurement decay heat. Figure 4.12 show the results ofcalculated decay heat and uncertainty. The comparison between numerical results andexperimental data is shown in Table 4-2. Most of cases have gaps within the range of
uncertainties and errors, but some have relatively large discrepancy. Since the calculations results are underestimated in general, the more accurate calculation setting may
be able to improve the reproducibility. In the quantitative analysis, the value of 𝜒2 /𝑛 is
calculated using 0.7 of the correlation of experiments. Since 𝜒2 /𝑛 is 0.39, the calculation
results and experimental data match well with uncertainty and error.
The decay heat of actual reactors is simulated. When the uncertainty of numericalresults are considered, the accuracy of numerical results are quantitatively supported.

53

700

Exp.
Cal.

Decay heat[W]

600
500
400
300
200

I25

I24

I09

G23

G11

F32

F25

F21

F14

E40

E38-2

E38-1

D38

D27

C20-4

C20-3

C20-2

C12

C01

0

C20-1

100

Figure 4-1: Comparison between numerical results with experimental data for CLAB
experiment of Rhinghals-2.

1200

Exp.
Cal.

Decay heat[W]

1000
800
600
400
200

F04

E18

D46

D01

C20

C19

C16

C01

C68

C67

C66

C64

C56

C52

0

Figure 4-2: Comparison between numerical results with experimental data for GE-Morris
experiment of Point Beach 2 and San Onofre 1.

54

Table 4.11: Measured decay heat results for Point Beach 2 and San Onofre 1 PWR
reactor
ReactorAssembly

Point Beach 2
Point Beach 2
Point Beach 2
Point Beach 2
Point Beach 2
Point Beach 2
San Onofle 1
San Onofle 1
San Onofle 1
San Onofle 1
San Onofle 1
San Onofle 1
San Onofle 1
San Onofle 1

C52
C56
C64
C66
C67
C68
D01
D16
C19
C20
D01
D46
E18
F04

Measurementdate
8/23/1981
8/241981
8/23/1981
8/20/1981
8/19/1981
8/20/1981
8/30/1981
8/31/1981
8/30/1981
8/29/1981
8/27/1981
8/29/1981
8/28/1981
8/28/1981

Decay
Time
1635
1634
1633
1630
1629
1630
3011
3012
3011
3011
2358
2360
1794
1078

Total
decay heat(W)
724.0
921.0
931.0
846.0
934.0
874.0
359.0
384.0
418.0
456.0
499.0
510.0
635.0
934.0

Table 4.12: Calculated decay heat results with uncertainty for Point Beach 2 and San
Onofre 1 PWR reactor
Assembly
C52
C56
C64
C66
C67
C68
D01
D16
C19
C20
D01
D46
E18
F04

Calculated decay heat(W)
686.6
881.6
895.4
800.1
883.6
841.3
335.4
362.8
390.5
419.6
457.0
474.2
590.7
909.9

55

Uncertainty(%)
1.99
1.99
1.99
1.95
1.98
1.96
2.72
2.70
2.68
2.66
2.50
2.49
2.13
1.49

56

Chapter 5

Decay Heat Uncertainty Regarding
Thorium Fuel
5.1

Background and Motivation

Recently, thorium has been much acknowledged as an important resource for nuclear fuel.
However, the nuclear characteristics of thorium fuels are not so well unraveled compared
to conventional fuels. Vast evaluation of reactor physics quantities is necessary to realize
thorium fueled reactors. There are some previous works of reactor physics simulation ofThorium Fuel Cycles(ThFCs) for the case of heavy water reactors in Canada[34, 35, 36].
In the European Union, ThFCs of light water reactors have been mainly studied in
France. There are some studies of pressurized water reactor neutronic characteristics
with thorium fuel[37, 38, 39]. In Japan, a laboratory in Hokkaido University investigated
the feasibility of advanced converter reactor with thorium fuel in BWR in 2010[40, 41].
Also, there is a ThFCs research of decay heat[42], which is one of the important physicalquantities for both accident and nuclear waste analyses.
However, compared to the conventional fuel simulation, the decay heat uncertainty
induced by nuclear data can be greater in the case of thorium fuel because it has toconsider the uncertainty of thorium-series nuclides. Hence, uncertainty quantificationof decay heat for thorium fuel is essential for high credibility. The motivation of thischapter is to clarify the decay heat uncertainty through the comparison between Th and57

conventional fuels, and identify the causal nuclides inducing high uncertainty of decay
heat. It can ascertain the feasibility of the system of ThFCs.

5.2

Calculation Condition

This study calculates decay heat and decay heat uncertainty for three compositions of Th
fuel in PWR. These fuels are Th-235 U[43], Th-Pu[44], and Th-233 U[45] whose components
are shown in Table 5.1. For the simplification, U-234 and U-236 are not considered asfuel composition. The burn-up of fuel is set 20 GWd/t and 60 GWd/t. Linear power
density during irradiation is 170 W/cm. At that time, detail burn-up history is notconsidered. The fuel depletion calculation and sensitivity calculation are conducted inpin cell system. To be more precise, Burner module in CBZ[10] is used. We adopt
JENDL-4.0, JENDL/FPD-2011, and JENDL/FPY-2011[17] as nuclear data, and also
use JEFF-3.1.1[7] for complimenting the FPs, which are not included in JENDL series.
Table 5.1: Initial fuel composition

NuclideTh-232
U-233
U-235
U-238
Pu-238
Pu-239Pu-240Pu-241
Pu-242
Am-241

5.3

UO2
4.1
95.9
-

composition ratio[wt%]
MOX Th-235 U Th-Pu Th-233 U
75
95.5
96.25
3.75
0.18
4.875
89.82
20.125
0.2
0.045
5.5
2.79
2.5
1.08
0.9
0.36
0.6
0.025
0.3
-

Results and Discussion

First, we simulate the case of short cooling time (0.1 - 100,000 seconds) after irradiation.
Figures 5-1, 5-2, and 5-3 show relative uncertainties of Th-235 U, Th-233 U, and ThPu fuels in 20GWd/t burn-up respectively. Left side figures represent five components
58

of uncertainty: cross section, decay energy, fission yield, half-life, and branching ratio.
Right side figures represent comparison between Th fuels and conventional fuels: UO2
and MOX fuels.

In the short cooling time cases, the uncertainties of all Th fuels do
4
Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

3
2

Uncertainty [%]Uncertainty [%]

4

1
0
10-1

100

101
102
103
Cooling time [s]104Th-U-235
UO2

3
2
1
0
10-1

105

(a) uncertainty component

100

101
102
103
Cooling time [s]

104

105

(b) comparison with UO2 fuelFigure 5-1: Th-U-235 decay heat uncertainty in 20GWd/t burn-up.

4
Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

3
2

Uncertainty [%]Uncertainty [%]

4

1
0
10-1

100

101
102
103
Cooling time [s]104Th-U-233
UO2

3
2
1
0
10-1

105

(a) uncertainty component

100

101
102
103
Cooling time [s]

104

105

(b) comparison with UO2 fuelFigure 5-2: Th-U-233 decay heat uncertainty in 20GWd/t burn-up.

4
Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

3
2

Uncertainty [%]Uncertainty [%]

4

1
0
10-1

100

101
102
103
Cooling time [s]

104

Th-Pu
MOX

3
2
1
0
10-1

105

(a) uncertainty component

100

101
102
103
Cooling time [s]

104

105

(b) comparison with MOX fuelFigure 5-3: Th-Pu decay heat uncertainty in 20GWd/t burn-up.
not change significantly till about 1,000 seconds of cooling time when we compare themwith that of UO2 and MOX fuels. The result of Th-235 U case is similar to those of
Th-233 U case, but the contribution of yield is slightly larger in the case of Th-233 U fuel59

owing to the uncertainty induced by U-233. In the comparison between Pu-containing
fuels, The uncertainty of Th fuel is even slightly small through the early a few seconds.
It represents the accumulation of TRU nuclides’ yield uncertainty outweighs Th yield
uncertainty. Dominant source of increasing uncertainty is cross-section of a few nuclides.
Around 10,000 seconds, capture cross-section of Th-232 is main source of uncertainty
according to a burn-up sensitivity analysis.
Next, we calculate the case of long cooling time (0.1 - 1,000 years). Figures 54, 5-5, and 5-6 show relative uncertainties of Th-235 U, Th-233 U, and Th-Pu fuels in
20GWd/t burn-up respectively. Left side figures represent components and right side
figures represent comparison with conventional fuels.

3

Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

4
Uncertainty [%]Uncertainty [%]

4

2
1
0
10-1

100

101
Cooling time [yr]102Th-U-235
UO2

3
2
1
0
10-1

103

In the longer cooling time scale,(a) uncertainty component

100

101
Cooling time [yr]

102

103

(b) comparison with UO2 fuelFigure 5-4: Th-U-235 decay heat uncertainty in 20GWd/t burn-up.

3

Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

4
Uncertainty [%]Uncertainty [%]

4

2
1
0
10-1

100

101
Cooling time [yr]

102

3
2
1
0
10-1

103

(a) uncertainty component

Th-U-233
UO2

100

101
Cooling time [yr]

102

103

(b) comparison with UO2 fuelFigure 5-5: Th-U-233 decay heat uncertainty in 20GWd/t burn-up.
the uncertainties are greater at the early part and around 1,000 years of cooling time.
In both areas, the influence of uncertainties induced by cross section is significant. The
contributed nuclides through early part are Th-232 in all cases. In the case of Th233 U, uncertainties after 100 years are large, which is the influence of Th-232.

60

On the3

Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

2
1
0
10-1

100

Th-Pu
MOX

4
Uncertainty [%]Uncertainty [%]4

101
Cooling time [yr]

102

3
2
1
0
10-1

103

(a) uncertainty component

100

101
Cooling time [yr]

102

103

(b) comparison with MOX fuelFigure 5-6: Th-Pu decay heat uncertainty in 20GWd/t burn-up.
other hand, the uncertainty of Th-Pu fuel is slightly greater compared with a MOX fuel.
These gaps are induced by the capture and fission cross-sections of Th-232 at 10 years
and capture cross-section of Th-232 at 300 years. These results also clearly articulate
the effect of U-233 is not significant even in the case of Th-233 U.
Figures 5-7 and 5-8 show relative uncertainty of Th-235 U and Th-233 U in the burnup of 60GWd/t. In the case of high burn-up fuel, 60GWd/t, the gap between Th-fuels

3

Total
CrossSection
EnergyYield
HalfLife
BranchingRatio

4
Uncertainty [%]Uncertainty [%]

4

2
1
0
10-1

100

101
Cooling time [yr]

102

3
2
1
0
10-1

103

Th-U-235
UO2(a) uncertainty component

100

101
Cooling time [yr]

102

103

(b) comparison with UO2 fuel

11
Total
10
CrossSection
9
EnergyYield
8
HalfLife
7 BranchingRatio
6
5
4
3
2
1
0
10-1
100

Uncertainty [%]Uncertainty [%]Figure 5-7: Th-U-235 decay heat uncertainty in 60GWd/t burn-up.

101
Cooling time [yr]

102

103

(a) uncertainty component

11
10
9
8
7
6
5
4
3
2
1
0
10-1

Th-U-233
UO2

100

101
Cooling time [yr]

102

103

(b) comparison with UO2 fuelFigure 5-8: Th-U-233 decay heat uncertainty in 60GWd/t burn-up.
and conventional fuels becomes much larger. The uncertainty of Th-235 U fuel reaches
61about 4% at 1,000 year, and that of Th-233 U fuel reaches about 10% at 1,000 year.
The main factors with high contribution to the uncertainty are capture cross-sectionof Th-232. The contribution of the capture and fission cross-section of U-233 is only
5% of total cross-section-induced uncertainty. It indicates that obtaining the additional
data of capture cross-section of Th-232 can reduce the decay heat uncertainty in ThFCs
significantly.
Through the uncertainty quantification of decay heat regarding thorium fuels, thedifference between Th fuel and conventional fuel has been clarified. It is relatively highfrom 100,000 seconds to 0.4 years and after 100 years. When the burn-up is small, there
is not so obvious shift as for Th fuels, however, some Th fuel cases with the large burn-up
such as 60GWd/t have shown the tremendous impact of the decay energy uncertainty
especially in the case of Th-233 U fuel. We have also identified that desirable nuclear data
to reduce the decay heat uncertainty regarding thorium fuels are capture cross-section of
Th-232[46].

62Chapter 6

Conclusion
Chapter 2 has explained the theory and calculation system for simulating the decay heatand the uncertainty of decay heat. Cooler module has been developed for calculating
numerous cases of fission pulse and finite irradiation. For the quantitative analysesof agreements between numerical results with uncertainty and experimental data witherrors, the value of 𝜒2 /𝑛 has been adapted.
Chapter 3 has shown the numerical results of decay heat uncertainty in the case of fission pulse and finite irradiation experiments. The calculation has been conducted Cooler
module. The dominant uncertainty component of nuclear data is decay energy throughthe cooling time for all representative irradiated nuclides. In the case of finite irradiation experiments, the uncertainty of decay heat is gradually decreasing in proportion as
the increase of irradiation time before cooling. The comparison with experimental dataclarify that numerical results and experimental data shows good agreement in almost
all cases. The quantitative analyses using 𝜒2 /𝑛 have inferred some experimental datacontain mistakes of measurements.
Chapter 4 has shown the numerical results of decay heat with uncertainty in the caseof actual PWR reactors. The calculated results are compared with spent fuel measurements data from CLAB. The difference between numerical results and experimental data
are within uncertainty range, which support the validity of calculated decay heat withhigh accuracy.
Chapter 5 has shown the simulation in the case of thorium fuel. The comparison
63between conventional fuel and thorium fuel has revealed that the uncertainty of thorium
fuel is relatively high from 100,000 seconds to 0.4 years and after100 years. If the burn-up
becomes larger and dominant composition shift into U-233, the uncertainty increasing
especially in long cooling time span. The source of uncertainty unique to thorium fuel ismainly capture cross section of Th-232.
Through the research conducted above, calculated decay heat is validated and uncertainty has been quantified more accurately by numerical calculation. Also, the uncertainty sources of decay heat is identified.

64

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68Acknowledgment
I am very grateful for the excellent guidance and support of my research advisors Professor
Go Chiba and Professor Tadashi Narabayashi through out this research and my own
personal academic career. I would also like to thank Ex Professor Masashi Tsuji for his
great support last year. In addition, I would also like to thank everyone of graduate
and undergraduate students in the laboratory. I would like to acknowledge the sincere
support by Mr. Hirai and Ms. Katsuyama.

69