# Uncertainty Analysis of Light Water Reactor Fuel Lattices

Authors: C. Arenas; R. Bratton; F. Reventos; K. Ivanov
Journal: Science and Technology of Nuclear Installations
 (2013)

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## Abstract

The study explored the calculation of uncertainty based on available cross-section covariance data and computational tool on fuel lattice levels, which included pin cell and the fuel assembly models.  Uncertainty variations due to temperatures changes and different fuel compositions are the main focus of this analysis. Selected assemblies and unit pin cells were analyzed according to the OECD LWR UAM benchmark specifications. Criticality and uncertainty analysis were performed using TSUNAMI-2D sequence in SCALE 6.1. It was found that uncertainties increase with increasing temperature, whilekinf decreases. This increase in the uncertainty is due to the increase in sensitivity of the largest contributing reaction of uncertainty, namely, the neutron capture reaction 238U(n, γ) due to the Doppler broadening. In addition, three types (UOX, MOX, and UOX-Gd2O3) of fuel material compositions were analyzed. A remarkable increase in uncertainty in kinf was observed for the case of MOX fuel.  The increase in uncertainty of kinf in MOX fuel was nearly twice the corresponding value in UOX fuel. The neutron-nuclide reaction of 238U, mainly inelastic scattering (n, n′), contributed the most to the uncertainties in the MOX fuel, shifting the neutron spectrum to higher energy compared to the UOX fuel.


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## Body

## 1. Introduction

The demand for the best estimate calculations in nuclear reactor core modeling and design has increased in recent years. Uncertainty analysis has been highlighted as an important part of the design and safety analysis of modern nuclear reactors. The modeling aspects of uncertainty analysis and sensitivity analysis are to be further developed and validated on scientific grounds in support of their performance. The Organization for Economic Cooperation and Development (OECD)/Nuclear Energy Agency (NEA) initiated the Benchmark for Uncertainty Analysis in Modeling, Design, Operation, and Safety Analysis of Light Water Reactor (OECD LWR UAM benchmark). The general objective of the benchmark is to propagate the uncertainty through complex coupled multiphysics and multiscale simulations. The benchmark is divided into three phases with Phase I highlighting the uncertainty propagation in neutronics calculations, while Phases II and III are focused on uncertainty analysis of reactor core and reactor system, respectively.In Phase I of the OECD LWR UAM benchmark, the exercises are divided into three parts: cell physics (Exercise I), lattice physics (Exercise II), and core physics (Exercise III) [1]. This paper will discuss Exercises I and II.


## 2. Uncertainty Calculations

In general, uncertainty is calculated based on covariance matrix and weighting factor coefficients [2]:
(1)Δ2=α[cov]αT,
where α is the weighting factor matrix, αT is the transpose of the weighting factor matrix, [cov] is the covariance matrix, and Δ2 is the uncertainty.In order to obtain the uncertainty of the response of interest, which may be the critical eigenvalue, the reactivity difference between two reactor states, or the ratio of reactions rates, sensitivity coefficients (S) are used as weighting factors to the covariance matrix
(2)Δkeff2=S[cov]ST.


### 2.1. Covariance Matrix of Nuclear Data

The evaluation of nuclear data induced uncertainty is possible by the use of nuclear cross-section variance and covariance data. By including the uncertainty or covariance information, the analyst can propagate cross-section data uncertainties through sensitivity studies to the final calculated quantities of interest. The covariance data files provide the estimated variance for the individual data as well as any correlation that may exist. In principle, the covariance matrices can be now self-shielded in the same way as the cross-sections, although in practice this is rarely done. The impact of this treatment on the obtained covariance matrices and their dependence on energy group structure needs to be studied. The SCALE 6.1/TSUNAMI-2D [3] sequence uses a completely different approach to address the impact of self-shielding in sensitivity and uncertainty analysis. Rather than modifying the covariance data, the sensitivity coefficients are modified to include the “implicit effects” of perturbations in the group cross-sections caused by perturbations in self-shielding. Implicit effects account for impact of resonance self-shielding on sensitivity coefficients and uncertainty evaluations. This allows the use of unshielded covariance data. Treatment of implicit effects is a standard part of the TSUNAMI-2D analysis and has been shown to be a significant sensitivity component in some cases.The SCALE 6.1 covariance library data corresponds also to 44-group relative uncertainties assembled from a variety of sources, including evaluations from ENDF/B-VII, ENDF/B-VI, JENDL-3.3, and more than 300 approximated uncertainties from a collaborative project performed by Brookhaven National Laboratory (BNL), Los Alamos National Laboratory (LANL), and Oak Ridge National Laboratory (ORNL).It is assumed that the same relative (rather than absolute) uncertainties can be applied to all cross-section libraries, even if these are not strictly consistent with the nuclear data evaluations. In addition, the assumption that there are no covariance correlations between energy groups is applied [1].


### 2.2. Sensitivity Coefficients

For light water reactors, two components of sensitivity coefficient are needed. Explicit sensitivity represents the sensitivity of the calculatedkinf to components of the group-wise cross-section data. Implicit sensitivity addresses the impact of resonance self-shielding effect in a system. Together, the explicit and implicit sensitivity are combined to produce the total sensitivity:
(3)STotal=SExplicit+SImplicit.


### 2.3. Computational Tools

The SCALE 6.1/TSUNAMI-2D sequence is used to perform the study. First, the ENDF/B-VII.0 based 238-group microscopic cross-section data library is processed using BONAMIST and CENTRM/PMC. Next, forward and adjoint calculations are performed using NEWT, a 2D transport solver. Finally, sensitivity coefficients are calculated, and uncertainty data is generated by SAMS using the default covariance data library in 44 groups (44groupcov).The SCALE 6.1 sensitivity and uncertainty methodology is based on the first-order perturbation theory to calculate response sensitivity coefficients, which are then folded with nuclear data covariances to obtain the response uncertainty. TSUNAMI-2D applies the generalized perturbation theory (GPT) to generate uncertainties associated with the few-group assembly homogenized neutron cross-section data [2].


## 2.1. Covariance Matrix of Nuclear Data

The evaluation of nuclear data induced uncertainty is possible by the use of nuclear cross-section variance and covariance data. By including the uncertainty or covariance information, the analyst can propagate cross-section data uncertainties through sensitivity studies to the final calculated quantities of interest. The covariance data files provide the estimated variance for the individual data as well as any correlation that may exist. In principle, the covariance matrices can be now self-shielded in the same way as the cross-sections, although in practice this is rarely done. The impact of this treatment on the obtained covariance matrices and their dependence on energy group structure needs to be studied. The SCALE 6.1/TSUNAMI-2D [3] sequence uses a completely different approach to address the impact of self-shielding in sensitivity and uncertainty analysis. Rather than modifying the covariance data, the sensitivity coefficients are modified to include the “implicit effects” of perturbations in the group cross-sections caused by perturbations in self-shielding. Implicit effects account for impact of resonance self-shielding on sensitivity coefficients and uncertainty evaluations. This allows the use of unshielded covariance data. Treatment of implicit effects is a standard part of the TSUNAMI-2D analysis and has been shown to be a significant sensitivity component in some cases.The SCALE 6.1 covariance library data corresponds also to 44-group relative uncertainties assembled from a variety of sources, including evaluations from ENDF/B-VII, ENDF/B-VI, JENDL-3.3, and more than 300 approximated uncertainties from a collaborative project performed by Brookhaven National Laboratory (BNL), Los Alamos National Laboratory (LANL), and Oak Ridge National Laboratory (ORNL).It is assumed that the same relative (rather than absolute) uncertainties can be applied to all cross-section libraries, even if these are not strictly consistent with the nuclear data evaluations. In addition, the assumption that there are no covariance correlations between energy groups is applied [1].


## 2.2. Sensitivity Coefficients

For light water reactors, two components of sensitivity coefficient are needed. Explicit sensitivity represents the sensitivity of the calculatedkinf to components of the group-wise cross-section data. Implicit sensitivity addresses the impact of resonance self-shielding effect in a system. Together, the explicit and implicit sensitivity are combined to produce the total sensitivity:
(3)STotal=SExplicit+SImplicit.


## 2.3. Computational Tools

The SCALE 6.1/TSUNAMI-2D sequence is used to perform the study. First, the ENDF/B-VII.0 based 238-group microscopic cross-section data library is processed using BONAMIST and CENTRM/PMC. Next, forward and adjoint calculations are performed using NEWT, a 2D transport solver. Finally, sensitivity coefficients are calculated, and uncertainty data is generated by SAMS using the default covariance data library in 44 groups (44groupcov).The SCALE 6.1 sensitivity and uncertainty methodology is based on the first-order perturbation theory to calculate response sensitivity coefficients, which are then folded with nuclear data covariances to obtain the response uncertainty. TSUNAMI-2D applies the generalized perturbation theory (GPT) to generate uncertainties associated with the few-group assembly homogenized neutron cross-section data [2].


## 3. Temperature Sensitivity Study

The study begins with specification of fuel pin cells of three light water reactor types and two critical experiments provided by the OECD LWR UAM benchmark within Exercise I-1 [4]. This exercise evaluates the multigroup microscopic cross-section uncertainties associated with multigroup cross-section libraries used as an input in lattice physics codes. The details of the specifications are available in [1]. kinf and propagated cross-section uncertainties obtained for different pin cell test problems are shown in Table 1.Table 1
Uncertainty inkinf in LWR unit cells.
Fuel
Operating conditions
k
inf
Uncertainty inkinf (%  Δk/k)
Largest  uncertainty-contributing  reaction
BWR
HZP
1.3382
0.52
238U(n,γ)
HFP (40% void)
1.2208
0.62
238U(n,γ)
PWR
HZP
1.4206
0.48
238U(n,γ)
HFP
1.4017
0.49
238U(n,γ)
VVER
HZP
1.3448
0.51
238U(n,γ)
HFP
1.3270
0.52
238U(n,γ)
KRITZ 2.1
HZP
1.2323
0.59
238U(n,γ)
HFP
1.1837
0.63
238U(n,γ)
KRITZ 2.13
HZP
1.2642
0.55
238U(n,γ)
HFP
1.2329
0.58
238U(n,γ)Observations of the results showed thatkinf reduces as the temperature rises. This is due to the effect of Doppler broadening and the reduction in moderation due to the changing moderator density. The uncertainty in kinf seemed to increase with increasing temperature. The largest contributor to the uncertainty is due to 238U(n,γ) reaction. The majority of neutron capture in 238U occurs at intermediate energy, and thus as the temperature increases, the neutron spectrum shifts into the epithermal range. Figure 1 shows such a shift of the neutron spectrum into the epithermal range when changing from Hot Zero Power (HZP) to Hot Full Power (HFP) conditions.Figure 1
The neutron flux of PWR unit cell at two different operating conditions.The resonance absorption due to Doppler broadening is reflected in the calculation of the sensitivity, mainly the implicit sensitivity. This implicit sensitivity accounts for the self-shielding effect. Figure2 illustrates the relative change in sensitivity of 238U(n,γ) in each unit cell analyzed between HZP and HFP conditions.Figure 2
Relative change in the sensitivity of238U(n,γ).In addition, three fuel assemblies of three light water reactor types provided by the OECD LWR UAM benchmark were analyzed. The details of the specifications are readily available [1]. The kinf and uncertainties obtained are shown in Table 2.Table 2
Assemblykinf values and associated uncertainties.
Fuel
Operating conditions
k
inf
Uncertainty inkinf (%  Δk/k)
Largest uncertainty-contributing reaction
BWR
HZP
1.1116
0.50
238U(n,γ)
HFP (40% void)
1.0779
0.56
238U(n,γ)
PWR
HZP
1.4130
0.46
23U(n,γ)
HFP
1.3968
0.47
238U(n,γ)
VVER
HZP
1.3164
0.47
238U(n,γ)
HFP
1.3115
0.47
238U(n,γ)Similar to the fuel pin models, in fuel assembly models, the uncertainty inkinf increases with increasing temperature, while kinf decreases [5]. As explained earlier, this effect arises because of the increase in sensitivity of the largest contributor of uncertainty, the 238U(n,γ) reaction. The majority of neutron capture in 238U occurs at intermediate energy, and thus as the temperature increases, the spectrum shifts to the epithermal range (i.e., becomes harder). Likewise, 40% void exhibit larger uncertainty in kinf also due to a harder flux spectrum.The fuel assembly analysis (as part of Exercise I-2) includes the propagation of multigroup cross-section uncertainties (multigroup covariance matrix) to two-group homogenized cross-section uncertainties (two-group covariance matrix). The two-group cross-section uncertainties are obtained using the SCALE-6.0 44-group covariance matrix as input to the TSUNAMI-2D sequence with GPT in SCALE 6.1. The obtained results are shown in Table3.Table 3
Two-group cross-section uncertainty in LWR fuel assembly.
Response cross-section
Uncertainty (%ΔR/R)
Uncertainty (%ΔR/R)
Uncertainty (%ΔR/R)
BWR
PWR
VVER
HZP
HFP
HZP
HFP
HZP
HFP
∑
t
1
0.84
0.91
0.87
0.88
0.81
0.82
∑
t
2
0.13
0.15
0.14
0.14
0.12
0.12
D
1
0.84
0.91
0.87
0.88
0.81
0.82
D
2
0.13
0.15
0.14
0.14
0.12
0.12
∑
a
1
0.78
0.83
0.86
0.87
0.81
0.82
∑
a
2
0.20
0.22
0.22
0.22
0.21
0.21
∑
f
1
0.68
0.72
0.36
0.36
0.47
0.47
∑
f
2
0.32
0.32
0.32
0.32
0.32
0.32
∑
11
0.84
0.91
0.87
0.87
0.81
0.81
∑
1
2
1.10
1.22
1.20
1.21
1.03
1.03
∑
2
1
0.27
0.34
0.30
0.33
0.26
0.29
∑
22
0.13
0.16
0.15
0.15
0.13
0.13One can define nine-dimensional response vectorR=[Σa1,Σa2,Σf1,Σf2,νΣf1,νΣf2,D1,D2,Σ12] for two-group assembly homogenized cross-sections and obtain a corresponding covariance matrix in which the diagonal elements are the relative standard deviations, while off-diagonal elements are the correlation coefficients. The covariance matrices for the BWR and PWR cases are displayed in Figures 3, 4, 5, and 6.Figure 3
BWR HZP covariance matrix.Figure 4
BWR HFP covariance matrix.Figure 5
TMI HZP covariance matrix.Figure 6
TMI HFP covariance matrix.The obtained results for different LWR types and cases indicate the following tendencies.(a)
Group 1 (fast) cross-section uncertainty is ~2-3 times larger than Group 2 (thermal) cross-sections uncertainty.(b)
Uncertainty contributions:
(i)
a major contributor to Group 1 (fast) cross-section uncertainties is U-238 inelastic scattering;(ii)
U-238 inelastic scattering uncertainty is quite large;(iii)
40% void (and higher) exhibit larger uncertainty ink∞ due to harder flux spectrum.(c)
Uncertainty (correlation) contribution:
(i)
U-238 inelastic scattering uncertainty is quite large and dominates correlation coefficient.


## 4. Composition Sensitivity Study

Selected fuel pin cells and four types of fuel assemblies from a representative Generation III LWR (GEN-III) specification were analyzed for the purpose of comparing effect of the compositions on the uncertainty calculations. The specifications of the GEN-III unit cells and fuel assemblies are readily available [1].Three types of unit cells were analyzed at Hot Full Power; these include MOX, UOX, and UOX with Gd2O3. The multiplication factors and their uncertainties are presented in Table 4.Table 4
Uncertainty inkinf in GEN-III unit cells.
Fuel
Compositions
k
inf
Uncertainty inkinf (%  Δk/k)
Largest uncertainty-contributing reaction
MOX
9.8%239Pu
1.0921
0.94
238U(n,n′)
6.5%239Pu
1.0540
0.97
238U(n,n′)
3.7%239Pu
1.0115
0.99
238U(n,n′)
UOX
4.2%235U
1.2431
0.51
238U(n,γ)
3.2%235U
1.1741
0.54
238U(n,γ)
2.1%235U
1.0490
0.59
238U(n,γ)
UOX-Gd2O3
2.2%235U
0.2166
1.79
238U(n,n′)
1.9%235U
0.1997
1.94
238U(n,n′)For each group of the fuel cells, several factors influence the changes in the uncertainty inkinf, and each will be examined separately.For each unit cell, the calculatedkinf increases with increasing enrichment of fissile material, while the uncertainty in kinf decreases.For the MOX fuel cells, the amount of238U is reduced as the amount of 239Pu, the fissile material, is increased. The reduction in the amount of 238U means that there is less neutrons absorption by 238U nuclides. This is later found to be the most important nuclide contributor to uncertainty in kinf in the unit cell. Thus, with less absorption by 238U nuclides, the uncertainty is reduced when comparing only the changes in the enrichment of 239Pu in MOX fuel cell.However, the uncertainties of the MOX fuel cells were nearly twice than that of UOX fuel. The presence of239Pu plays an important role in the increase in uncertainty. Figure 7 shows that the neutron absorption by 239Pu outcompeted the neutron absorption by 235U.Figure 7
Plot of239Pu absorption cross-sections compared to 235U absorption cross-sections.As a consequence, from the fact that more neutrons are absorbed by239Pu than 235U, more neutrons are produced by fission due to 239Pu. Table 5 shows the dependency of fission macroscopic cross-sections to the composition of the fuel, and Table 6 displays the comparison of the neutrons production parameters of each nuclide.Table 5
Macroscopic fission cross-sections in MOX fuel cells.
Fuel
Composition
k
inf
235U ∑f (1/cm)
239Pu ∑f (1/cm)
MOX
3.7%  Pu329
1.0115
6.70
13.74
MOX
6.5%  Pu329
1.0540
4.65
8.40
MOX
9.8% Pu329
1.0921
3.69
5.88Table 6
LWR neutronics parameters [6, Table 3–1 page 88].
Parameter
235U
239Pu
Averageν
2.4
2.9
Averageη
2.0
1.9
Averageσfission
280 barns
790 barnsNeutrons produced by239Pu, in general, will have higher energy than that of 235U. In this case, the neutron spectrum is harder because more neutrons with higher energies are produced by 239Pu, the dominant fission nuclide.On the other hand, for UOX fuel cell with Gd2O3 added, similar finding (hardening of the neutron spectrum) occurred but due to a very different mechanism. The uncertainties in kinf are nearly triple of the fuel cell with similar 235U enrichment. This is due to the fact that gadolinium is a major thermal neutron absorber. The significant reduction in the thermal neutrons shifted the fission reactions toward neutrons with higher energy. Once again, the neutron spectrum is harder than that of UOX fuel cell without Gd2O3 added.The uncertainties in fuel cells with harder neutron spectrum seemed to be higher than fuel cells with softer neutron spectrum. This is due to the absorption of the238U. The resonance absorption of 238U that occurs at higher neutron energy is very large.The absorption of thermal neutrons by239Pu dominates the fission process. Since fission in 239Pu produces more fast neutrons, the neutron spectrum becomes harder. The harder the spectrum, the higher the 238U(n,n′) reaction rates. This effect can be seen in Figure 8, which depicts the cross-section of 238U(n,n′) reaction. When Gd2O3 is added, the largest nuclide reaction cross-section contributor to the uncertainty in kinf changed from 238U(n,γ) to 238U(n,n′). These changes are due to the presence of gadolinium, mainly a thermal neutron absorber, causing significant reductions in thermal neutron populations. The neutron flux spectrum is harder, and it can be seen in Figure 9.Figure 8
The plot of inelastic cross-section of238U as a function of energy.Figure 9
The neutron flux spectra of three unit cell compositions.The sensitivity profiles were compared in Figure10. It was found that the unit cell containing gadolinium is very sensitive to the 238U(n,n′) reaction, and this large sensitivity contributes to increase in uncertainty.Figure 10
The sensitivity of238U(n,n′) as function of energy.The covariance matrix of238U(n,n′) is presented in Figure 11, while the covariance matrix of 238U(n,γ) is shown in Figure 12. It should be noted that at high energy, the coefficients of relative covariance of 238U(n,n′) cross-section reach 30% of the standard deviation. Thus, for a reactor with a harder spectrum, it produces a large contribution to the uncertainty of kinf.Figure 11
The 2D plot of238U(n,n′) covariance by energy group.Figure 12
The 2D plot of238U(n,γ) covariance by energy group.Four types of fuel assembly from a representative Generation III LWR specification (as part of Exercise I-2) were analyzed at Hot Full Power condition:(i)
type 1 (UOX4.2%235U);(ii)
type 2 (UOX4.2%235U + UO2Gd2O32.2%235U);(iii)
type 3 (UOX3.2%235U + UO2Gd2O31.9%235U);(iv)
type 4 (MOX).The details of the specifications are readily available [1]. The values calculated for kinf and their uncertainties are presented in Table 7.Table 7
Uncertainty inkinf in GEN-III fuel assembly.
Fuel
Compositions
k
inf
Uncertainty inkinf (%  Δk/k)
Largest uncertainty-contributing reaction
GEN III
Type 1
1.2501
0.49
238U(n,γ)
Type 2
1.1228
0.49
238U(n,γ)
Type 3
0.9564
0.53
238U(n,γ)
Type 4
1.0700
0.97
238U(n,n′)Results show thatkinf increases with increasing enrichment of fissile material, while the uncertainty in kinf decreases. Unlike the effect we found earlier for UOX fuel pin cell with Gd2O3 added, in this case the presence of Gd2O3 does not lead to a significant increase in uncertainty because of the homogenization process. For the MOX fuel assembly, there is a remarkable increase in uncertainty in kinf, and this result is similar to that observed in the corresponding pin cell analysis.Additionally, the two-group cross-section uncertainties are presented in Table8.Table 8
Two-group cross-section uncertainty (%ΔR/R) in GEN-III fuel assembly.
Response cross-section
GEN III type 1
GEN III type 2
GEN III type 3
GEN III type 4
∑
t
1
0.90
0.90
0.90
0.97
∑
t
2
0.14
0.14
0.14
0.14
D
1
0.90
0.90
0.90
0.97
D
2
0.14
0.14
0.14
0.14
∑
a
1
0.89
0.89
0.89
1.00
∑
a
2
0.21
0.19
0.19
0.24
∑
f
1
0.37
0.37
0.50
0.44
∑
f
2
0.32
0.32
0.33
0.62
∑
11
0.90
0.90
0.90
0.97
∑
12
1.25
1.25
1.24
1.47
∑
21
0.33
0.32
0.31
0.37
∑
22
0.15
0.15
0.15
0.16These results show, as expected, a larger uncertainty for the MOX fuel assembly (type 4) than for the UOX assemblies (types 1 through 3).Figure13 depicts the neutron flux of fuel assemblies from a representative GEN-III LWR at HFP condition. There is a significant reduction in the thermal neutrons for the MOX fuel assembly. The harder the spectrum, the higher the 238U(n,n′) reaction rates.Figure 13
The neutron flux spectra of GEN-III LWR fuel assembly.Sensitivity profiles of238U(n,n′) were compared in Figure 14. Fuel assembly containing Pu is very sensitive to 238U(n,n′) reaction, and its contribution to uncertainty is noticeable.Figure 14
The sensitivity of238U(n,n′) as function of energy.


## 5. Conclusions

Sensitivity studies have been performed using SCALE [7] to investigate the effect of temperature and material composition on cross-section uncertainty propagation at two levels—pin cell models (within the framework of Exercise I-1) and assembly models (within the framework of Exercise I-2). It was found that the uncertainty in kinf is dominated by neutron interactions with 238U in both studies with varying temperatures and compositions. The following tendencies have been observed:(a)
increasing temperature leads to increasing uncertainty inkinf;(b)
decreasing238U in fuel composition leads to decreasing uncertainty in kinf;(c)
major contributor to uncertainty is affected by the neutron spectrum.The fast group cross-section uncertainties are much larger than the thermal cross-section uncertainties due to the larger role of 238U.


## 6. Future Work

The future studies will be focused on Exercise I-3—propagation of few-group cross-section uncertainties to the core steady state stand-alone neutronics calculations using statistical methodology similar to the one reported in [8].



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