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Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem
Annals of Nuclear Energy xxx (xxxx) xxx
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Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem Chenghui Wan a,⇑, Zhuojie Sui a, Bo Wang a, Liangzhi Cao a, Zhouyu Liu a, Jason Hou b a b
School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695, USA
a r t i c l e
i n f o
Article history: Received 10 August 2019 Received in revised form 1 October 2019 Accepted 14 October 2019 Available online xxxx Keywords: Uncertainty analysis Nuclear-data uncertainties C5G7 benchmark problem Transient simulation
a b s t r a c t In this study, uncertainty analysis has been performed to the time-dependent transient simulation of the C5G7-TD benchmark problem, propagating the nuclear-data uncertainties to the key parameters of interest. The detailed material compositions and geometries of C5G7-TD have been applied for the time-dependent transient simulation. For uncertainty analysis of the transient simulation, multigroup cross-section covariance library has been generated based on ENDF/B-VII.1 using the NJOY code. Our home-developed uncertainty-analysis code named UNICORN has been utilized for the uncertainty analysis, using the statistical sampling method. For the steady-state and transient simulations, our home-developed high-fidelity neutronics code NECP-X has been utilized. The relative uncertainties of the fuel-assembly normalized power and pin-wise power have been quantified as function of time. The numerical results show that the maximum relative uncertainties for the assembly normalized power can up to be about 2.14% and the value for the pin-wise power distributions can be about 3.54%. Ó 2019 Published by Elsevier Ltd.
1. Introduction It has been an increasing demand for the best estimate predictions to be provided with their confidence bounds in many domains, including the nuclear research, industry, safety and regulation (Ivanov et al., 2013). Motivated by UAM expect group, many research works have been focused on the uncertainty analysis of the LWR system at steady-state modeling and simulations (Yankov et al., 2012; Zhou et al., 2016; Pusa and Isotalo, 2017). However, few research of uncertainty analysis was performed to the transient related modeling and simulation, which is significant for the safety analysis. With the great efforts made by OECD/NEA, a series of space–time neutron kinetics benchmark exercises, dubbed the C5G7-TD benchmark, have been proposed and developed (Boyarinov et al, 2016; Hou et al, 2017), based on the wellstudied steady-state neutron transport C5G7 benchmark. These benchmark problems can be generally divided into the 2dimensional (2D) and 3-dimensional (3D) transient problems. The 2D problem includes four different transient exercises, in which the control-rod cluster moves and the moderator density changes with various rate and magnitude. The 3D problem adopts the 3D configuration of the C5G7 core, and two different exercises
⇑ Corresponding author.
are defined for the control-rod insertion/withdrawal and the moderator density change with various rate and magnitude. Moreover, both the few-group homogenization constants and detailed material compositions of the materials have been provided for these C5G7-TD benchmark problems. A number of transient simulation results of the C5G7-TD benchmark problem have been reported (Wang et al., 2018; Shen et al., 2019; Ryu et al., 2017) and the comparative analysis results were summarized (Ogujiuba et al., 2019). However, all the activities related to the current benchmark have been focused on the transient modeling and simulation, no study was performed to the uncertainty analysis, which is the objective of the last phase of this benchmark. Therefore, the uncertainty analysis has been focused on in this study, aimed at propagating the nuclear-data uncertainties to the key parameters of interest in the transient simulations for the C5G7 benchmark. Through the study implemented in this study, the uncertainty-analysis research can be expanded to the transient simulation, which would provide the preliminary observations for the safety analysis. In order to propagate the nuclear-data uncertainties through the reactor-physics modeling and simulation, the deterministic method (Pusa, 2012; Jessee and Dehart, 2009) and statistical sampling method (Herranz et al., 2008; Williams et al., 2012; Zwermann et al., 2013) have been proposed and widely applied. For the deterministic method, the uncertainties of the interested
E-mail address: [email protected] (C. Wan). https://doi.org/10.1016/j.anucene.2019.107122 0306-4549/Ó 2019 Published by Elsevier Ltd.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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parameters are quantified using the ‘‘sandwich rule”, in which the nuclear-data covariance is combined with the sensitivity coefficients of the parameters with respect to the nuclear data determined by the sensitivity analysis. For the statistical sampling method, the uncertainties of the interested parameters are quantified with the parameters’ samples, which are determined through the reactor-physics simulations with corresponding nuclear-data samples generated with the nuclear-data covariance. Comparing these two methodologies indicate that the first-order approximation exists in the deterministic method, as only the first-order sensitivity coefficients are applied in the uncertainty quantification. Therefore, the statistical sampling method is considered more appropriate to perform the uncertainty analysis for the transient simulations, and hence selected in this study to propagate the nuclear-data uncertainties to the key parameters of interest. In this context, the uncertainty analysis has been implemented to the transient simulations of the C5G7-TD benchmark problem, quantifying the uncertainties of the key parameters of interest during the transient process due to the nuclear-data uncertainties. For the uncertainty analysis, the statistical sampling method implemented in our home-developed code named UNICORN (Wan et al., 2015, 2017, 2019) has been applied. The multigroup crosssection covariance libraries, which contains the nuclear-data uncertainties, were generated by NJOY (MacFarlane et al., 2016) based on the ENDF/B-VII.1 library in this study. In order to compatible with the multigroup microscopic cross-section covariance libraries, the microscopic description of the C5G7-TD benchmark was utilized in the transient modeling and simulation. Moreover, our home-developed high-fidelity neutronics code named as NECP-X was used for the transient modeling and simulation for the C5G7 benchmark. As both the multigroup and continueenergy cross-section library were required in the NECP-X simulations (Chen et al., 2018; Liu et al., 2018a; Zhao et al., 2018; Liu et al., 2018b; Wang et al., 2018), the continue-energy crosssection perturbation model has been developed in UNICORN to generate the cross-section samples based on the corresponding covariance libraries. Using the approach described above, the nuclear-data uncertainties can be propagated to the key parameters of interest during the transient process for the C5G7-TD benchmark problem. As the numerical results, the relative uncertainties of the assembly normalized power and also pin-wise power distributions have been quantified during the transient process. The paper is structured as follow. The brief introduction to the methodology for the transient uncertainty analysis and detailed description of the continue-energy cross-section perturbation model are provided in Section 2. Section 3 shows the detailed numerical solutions and corresponding analysis results. Summaries and conclusions are given in Section 4. 2. Flowchart and methods for the transient uncertainty analysis In this section, the uncertainty analysis method for the transient simulation using the high-fidelity NECP-X code established in UNICORN is briefly explained. Due to the specific requirement that the continue-energy cross-section library is applied in NECP-X for the resonance shielding calculation, corresponding continue-energy cross-section model has been developed in UNICORN. 2.1. Uncertainty analysis method for transient simulation As a short introduction, the NECP-X (Chen et al., 2018; Liu et al., 2018a; Zhao et al., 2018; Liu et al., 2018b; Wang et al., 2018) is a new high-fidelity neutronics code developed by NECP (Nuclear Engineering Computational Physics) Laboratory at Xi’an Jiaotong
University. The pseudo-resonance-nuclide subgroup method is applied for the resonance treatment. For the 3D neutronics simulation, the 2D/1D method is utilized to solve the neutron-transport equation. The CRAM method is applied for the depletion calculation and the sub-channel method is utilized for the thermal-hydraulics calculation. The matrix-free CMFD method and multi-level parallel strategy are applied for high efficiency. The NECP-X code is capable of performing both the steady-state and transient simulations for the pressurized water reactors (PWR). The UNICORN code (Wan et al., 2015, 2017, 2019) is also developed by NECP Laboratory for the sensitivity and uncertainty analysis. The direct numerical perturbation method is applied for the sensitivity analysis and the statistical sampling method is implemented for the uncertainty analysis. Based on the sensitivity and uncertainty analysis, the UNICORN code has the capabilities of the similarity analysis (Zheng et al., 2019) and nuclear-data adjustment (Wan et al., 2019). Based on the multigroup cross-section covariance libraries, the relative perturbation factors of the multigroup cross sections, referred to as cross section perturbation factors, can be generated, which contains the uncertainty information of the nuclear data. Therefore, it is the vital step that how to generate the samples of the microscopic cross-section libraries based on the samples of the cross-section perturbation factors. It should be noted that the pseudo-resonance-nuclide subgroup method (Liu et al., 2018) implemented in NECP-X for the resonance shielding calculation requires both the multigroup and continue-energy cross-section library. The multigroup cross-section perturbation model has been developed and implemented in UNICORN previously (Wan et al., 2015). In this study, a newly perturbation model was developed to generate samples for the continue-energy cross-section library, as will be introduced in Section 2.2. It should be noted that the newly perturbation model is applied for the continue-energy cross sections within the resonance energy ranges, while for perturbing the cross sections, the multigroup cross-section perturbation model has been applied to the multigroup cross sections for the fast and thermal groups. After the samples of both the multigroup and continue-energy cross-section libraries having been generated, the NECP-X code was executed once for each sample. In this process, corresponding samples for the key parameters of interest for the transient simulations can be determined. Finally, the statistical analysis can be done using all the samples of outputs to quantify the uncertainties and corresponding correlations for all system response of interest. Applying the process described above, the nuclear-data uncertainties can be propagated into the transient simulations, quantifying the uncertainties of the key parameters of interest. The flowchart of uncertainty analysis for the transient simulation based on NECP-X implemented in the UNICORN code has been shown in Fig. 1. 2.2. Continue-energy Cross-section perturbation model For better understanding the context, some detailed introduction towards the pseudo-resonant-nuclide subgroup method are given in this section. First, a pseudo resonant nuclide is defined as the mixture of all the resonant nuclides in each fuel rod. The 0-dimentional (0-D) neutron slowing-down equations are solved over a range of dilution cross sections by the hyper-finer energy group method (~1 M energy groups) to obtain the cross-section tables of the pseudo resonant nuclide. The hyper-finer cross sections are generated with the continue-energy cross-section library. Second, the physical subgroup probability tables are generated based on the pseudo-resonant nuclides and the intermediate resonance approximation. Third, a set of equivalent 1-dimental (1-D) cylinder problem are generated, by preserving the Dancoff
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 1. Flowchart for the transient uncertainty analysis based on NECP-X.
correlation factors, which are calculated by the neutron current method within the whole simulation ranges. Finally, the subgroup equations are solved for the 1-D cylindrical problems to get the effective self-shielding cross sections. According to the resonance-treatment method implemented in NECP-X, both the multigroup and continue-energy cross-section libraries are required for the high-fidelity neutronics modeling and simulations. There are 258 nuclides in the library, including 75 resonant nuclides. The multigroup cross-section libraries were generated for all nuclides and the continue-energy cross-section libraries were generated for all resonant nuclides. The conventional 69-group energy-group structure has been applied for the multigroup cross-section libraries, with the resonance-energy range extended to be 0.625 eV to 24.78 keV (13th–45th group). For the continue-energy cross-section libraries, the whole energy range has been covered and temperature interval is set to be 50 K from 300 K to 1800 K. The hyperfine energy grids are utilized, hence the cross sections between any two adjacent energy points can be interpolated linearly. Moreover, only the cross-section information for the rf, relas and rt are saved in the continueenergy cross-section libraries. Based on the relative covariance matrices for the cross sections, the samples for the relative perturbation factors can be generated. For any sample, the relative perturbation factors can be characterized as d = [d1, d2,. . ., dM]T, in which M presents the total number of the nuclides, cross-section types and groups. Detailed description towards the multigroup cross-section perturbation has been explained in our previous, and the following context will be focused on how to perturb (or sample) the continue-energy cross-section library with the relative perturbation factors. With the relative perturbation factor dm (m = 1, 2,. . ., M) for the gthgroup energy range, the cross-section sampling process can be treated as perturbing the cross sections within corresponding energy range, which can be characterized as:
r0x ðE; TÞ ¼ ð1:0 þ dm Þrx ðE; TÞ; Eg 6 E < Eg1
ð1Þ
where rx(E,T) and r’ x(E,T) present the initial and perturbed (or sampled) cross sections for the reaction , and the Eg and Eg-1 are the energy boundaries of the group g within the multigroup energy-group structure. In the continue-energy cross-section library applied in NECP-X, only the cross-section data for the rf, relas and rt are saved. Using the saved cross sections, the total absorption cross section can be calculated:
ra ðE; TÞ ¼ rt ðE; TÞ relas ðE; TÞ rf ðE; TÞ
ð2Þ
where ra presents the total absorption cross section, which can be re-characterized in detailed as:
ra ðE; TÞ ¼ rinel ðE; TÞ þ rc ðE; TÞ þ
X x
rðn;xnÞ ðE; TÞ þ
X
rðn;PÞ ðE; TÞ
P
ð3Þ where rinel is the inelastic cross section and rc is the capture cross section. The r(n,xn) includes the cross section (n,2n), (n,3n), (n,4n) etc. The r(n,P) presents the other cross-section types with particle productions, including the (n,P) for proton produce, (n,D) for deuterium produce, (n,T) for tritium produce, (n,a) for alpha produce etc. In the UNICORN code, relative-perturbation samples need to be generated for all the basic cross sections, including the relas, rinel, rf, rc, r(n,xn), r(n,P) etc. The multigroup cross-section perturbation model was implemented for sampling the multigroup crosssection library (Wan et al., 2015). As only the cross-section data for the rf, relas, rt and ra can be provided by the continueenergy cross-section library applied in NECP-X, it is not available to generate the samples for the basic cross-section types presented in the right-hand side of Eq. (3). Therefore, when relative perturbations are applied to these basic cross-section types, corresponding multigroup cross-section data were used to determine the relative perturbations, which would be applied to the saved rf, relas, ra and rt. Different approaches should be taken based on the reaction types. When the samples are generated for the rf and relas, which are saved in the continue-energy cross-section library, the relative perturbations will be applied to the cross-section data directly. This case is straightforward and the implementation can be done as shown in Eq. (1). When the relative perturbations are applied to these basic cross sections contained in ra, corresponding multigroup cross-section data are applied to determine the relative perturbations for ra:
r0a;g ðTÞ ¼ ra;g ðTÞ þ
X
dx;g rx;g ðTÞ
ð4Þ
x2a
da;g ðTÞ ¼
r0a;g ðTÞ 1:0 ra;g ðTÞ
ð5Þ
where da,g is the relative perturbations added to the absorption cross section within the gth-group energy ranges. The corresponding perturbed or sampled continue-energy absorption cross section within group g can be determined as:
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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r0a ðE; TÞ ¼ ð1:0 þ da;g Þra ðE; TÞ; Eg 6 E < Eg1
ð6Þ
After the cross-section perturbations or sampling, the consistent rule should be applied to balance the total cross section and the components, which can be characterized as:
r0t ðE; TÞ ¼ r0a ðE; TÞ þ r0elas ðE; TÞ þ r0f ðE; TÞ
ð7Þ
Through the continue-energy cross-section perturbation model, the samples for the continue-energy cross-section libraries can be generated and provided to the NECP-X code for performing simulations and generating corresponding samples for the key parameters of interest. 3. Numerical results and analysis
Table 2 Isotopic distributions of the materials. Nuclides
Concentrations/1024 at./cm3 MOX 4.3%
MOX 7.0%
MOX 8.7%
235
U 238 U 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 16 O
5.0000E-5 2.2100E-2 1.5000E-5 5.8000E-4 2.4000E-4 9.8000E-5 5.4000E-5 1.3000E-5 4.6300E-2
5.0000E-5 2.2100E-2 2.4000E-5 9.3000E-4 3.9000E-4 1.5200E-4 8.4000E-5 2.0000E-5 4.6300E-2
5.0000E-5 2.2100E-2 3.0000E-5 1.1600E-3 4.9000E-4 1.9000E-4 1.0500E-4 2.5000E-5 4.6300E-2
Material
Nuclides
Concentrations/1024 at./cm3
UO2
235
U U 16 O
8.6500E-4 2.2250E-2 4.6220E-2
1
H O B 11 B
6.7000E-2 3.3500E-2 5.9667E-6 2.1833E-5
90
Zr Zr 92 Zr 94 Zr 96 Zr
2.2124E-2 4.8246E-3 7.3745E-3 7.4734E-3 1.2040E-3
238
In this section, the numerical results of the transient problem with insertion and withdraw of bank 1 for C5G7 will be provided and analyzed. First, the detailed information of the C5G7 microscopic benchmark will be given, based on which the steady-state and transient simulations. Second, the uncertainty analysis has been performed within the framework of the high-fidelity modeling and simulation, to quantify the uncertainties of the key parameters during the transient simulation, due to the nuclear-data uncertainties.
Moderator
16 10
Zirconium
91
3.1. Steady-state and transient modeling and simulation for C5G7
Aluminum
27
6.0000E-2
Detailed C5G7-TD benchmark specifications were used in the problem setup of the transient simulation, including the geometries, configuration and material compositions. The detailed information for the geometries and materials is shown in Table 1. The pin-cells in C5G7 are the square lattice with cylindrical fuel rods. The detailed isotopic distributions of these loaded materials are listed in Table 2. The void is fulfilled with the nuclide He-4 with the atomic concentrations of 1.60535E-4 atom/barncm. The configurations of the pin-cells and assemblies are identical to the C5G7 benchmark specifications, as shown in Figs. 2 and 3. Based on the essential information about the geometry information for the pin-cells and assemblies and isotopic concentrations for all the loaded materials, the NECP-X code is applied to execute the simulation for the transient exercise of C5G7 benchmark from the microscopic cross-section library. It should be noted that the 2D model has been applied for the transient simulation. When the control rods insert in or withdraw out of the core, the material compositions of the guide tubes are changed, using the volumes of the moderator and Ag-In-Cd as weights to generate the mixture compositions. And the volumes of the moderator and Ag-In-Cd are calculated in advance, according to the insertion fraction of the control rods. The transient exercise analyzed in this study is the process to insert and then withdraw the bank 1, which is at the position of the UO2-loaded fuel assembly in middle of the core. At beginning,
Ag-In-Cd
107
2.2711E-2 2.2711E-2 8.0008E-3 2.7241E-3
Table 1 Pin-cell geometries and materials. Pin-cell
Material
External radius/cm
Fuel cells
Fuel Void Zirconium Void Aluminum Clad Moderator (square lattice pitch)
0.4095 0.4180 0.4750 0.4800 0.5400 1.26
Guide-tube cells
Moderator Aluminum Clad Moderator (square lattice pitch)
0.3400 0.5400 1.26
Al
Ag Ag In 113 Cd 109
115
the insertion fraction of bank 1 in the TD0-1 exercise proposed in C5G7 transient benchmark has been selected and tested. However, it was observed that the negative reactivity was very huge (about 16$) and the fuel-assembly power would decrease to be almost zero in very short time. Therefore, an appropriate insertion fraction (about 1%) has been determined to be applied for the transient modeling and simulation, as shown in Fig. 4. In the transient exercise, the bank 1 is inserted with the insertion fraction of 1% at the initial time t = 0.0 s and the position of bank 1 is hold until the time t = 1.0 s; at time t = 1.0 s, the bank 1 is withdraw out of the core. In this study, the 2D model has been applied to perform the transient simulation for the transient exercise. During the transient process, the thermal-hydraulics feedback is not considered, and the temperatures of the fuel and moderator remain at the hot conditions with the room temperature. Therefore, two different steady states have been modeled and simulated to determine the reactivity introduced in the core with the insertion of bank 1, including the state that the bank 1 fully withdraw and 1% inserted in the core. The keff of these two steady states and corresponding reactivity introduced by the bank insertion can be determined, as shown in Table 3. Applying the NECP-X code, the transient process has been modeled. For the transient modeling and simulation, the time step is set to be 0.05 s in the range of 0.0–2.0 s and 0.1 s between 2.0 and 3.0 s. The numerical results for the normalized power of the fuel assembly during the transient process are as shown in Fig. 5. During the time 0.0 s–1.0 s, the normalized fuel-assembly powers decrease with the time, as the bank 1 was inserted into the core at t = 0.0 s; at t = 1.0 s, the bank 1 was withdraw out of the core, and corresponding fuel-assembly powers increase with the time. These numerical results of the transient characteristic are reasonable and within what are expected.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 2. the fuel-assembly configuration for the C5G7 benchmark problem.
Fig. 3. the control-assembly configuration for the C5G7 benchmark problem.
Fig. 4. Controlrod movement in the transient exercise.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Table 3 The keff and reactivity introduced by bank-1 insertion of C5G7. keff with bank-1 withdraw
keff with bank-1 insertion
Reactivity introduced
1.202867
1.146980
4050.8pcm
Table 4 The isotopes and corresponding cross-section types in uncertainty analysis. Isotopes
Cross sections
235
r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c), v r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c), v r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,c), r(n,a) r(n,elas), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c)
U U Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 16 O 1 H 90 Zr 91 Zr 92 Zr 94 Zr 109 Ag 238 238
3.2. Uncertainty analysis results for transient simulation Based on the transient model above, the uncertainty analysis has been performed. The multigroup cross-section covariance library containing the uncertainty and correlation information was generated based on the ENDF/B-VII.1 library using the NJOY code. For the uncertainty quantification, the UNICORN code has been applied using the statistical sampling method. The Latin Hypercube Sampling (LHS) method has been utilized to generate 200 relative perturbation factors and corresponding multigroup and continue-energy cross-section library. In the uncertainty analysis, the analyzed isotopes and corresponding cross sections are listed as shown in Table 4. It should be noted that for the material Ag-In-Cd, the isotopes of In and Cd have no covariance data in ENDF/B-VII.1 and only 109Ag has the covariance data. Therefore, on the isotope 109Ag and corresponding cross-section types have been analyzed for the control-rod material. For the absorption material in the moderator, the B-10 of boron, very extremely large
data exists in the covariance of r(n,inel) in ENDF/B-VII.1, which is not appropriate considered in uncertainty analysis. The uncertainty analysis has been conducted focusing on the fuel-assembly and pin-wize power during the transient process. The normalized fuel-assembly power and corresponding relative uncertainties are shown in Figs. 6 and 7, respectively. From the numerical results, several observations can be found. For the
Fig. 5. The normalized fuel-assembly powers during transient process.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 6. Normalized assembly powers and corresponding uncertainties.
Fig. 7. Relative uncertainty of the normalized fuel-assembly powers.
central assembly, the relative uncertainty of the normalized power decreases with the time after the control-rod insertion and increase immediately following the control-rod withdraw. It then
approaches to an asymptotic value afterwards For other fuel assemblies which do not witness the control-rod movements, the trend of the relative uncertainty of the normalized assembly
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 8. the pin-wise powers and corresponding relative uncertainties/% at t = 0.0 s.
powers is opposite: it increases over the time with the bank insertion and decreases with the bank withdraw before approach the asymptotic value. In order to explain the observed phenomenon mentioned above, the pin-wise power distributions are analyzed thereafter. The relative uncertainties of the pin-wise power at t = 0.0 s, 0.05 s, 1.0 s and 1.05 s are shown in Figs. 8–11 respectively, and the following observations can be provided. Before the bank 1 insertion, the pin-wise powers at #1 fuel assembly are the largest ones and those in #4 fuel assembly are the smallest. After the bank 1 inserted at the time t = 0.05 s, the pin-wise power decreases across the core and so does the fuel-assembly power. By comparing the power distributions between t = 0.0 s and t = 0.05 s, the power of Assembly #1 decreases from 0.48 to 0.318, with the decreased ratio is about 33.7%; while for the others fuel assemblies, the decreased ratios are respectively 31.5% for Assembly #2 and #3 and 30.5% for Assembly #4. Through the assembly-power normalization, the relative powers of Assembly #2, #3 and #4 increases due to the larger power decrease in Assembly #1. In this case, the relative contributions of the fission cross sections to the fuelassembly powers in #2, #3 and #4 fuel assemblies becomes larger after the bank 1 insertion. Therefore, after the bank 1 insertion, the relative uncertainties of the fuel-assembly power for #2, #3 and #4 assemblies increases lightly while #1 decreases lightly with the
time. The process above becomes opposite after the bank 1 withdraw when t = 1.0 s and t = 1.05 s. When the bank 1 withdraw out of the core, the fuel-assembly power for #1 assembly increases immediately and becomes dominant, hence the relative contribution of the fission cross sections for #1 assembly becomes dominant and the relative uncertainty of the power in #1 assembly increases after bank 1 withdraw.Fig. 9. From the numerical values, for the transient process defined in the study, the uncertainty-analysis results can be concluded as following. The maximum relative uncertainty for the normalized assembly powers can be about 2.14% and the minimum value is about 0.57%. For the pin-wise power distributions, the maximum relative uncertainty is about 3.54%. 4. Conclusions In this study, uncertainty analysis has been implemented to the transient simulation for the C5G7 benchmark problem, propagating the nuclear-data uncertainties to the key parameters of interest. The detailed description of the geometries and material compositions of C5G7 has been utilized for the transient simulation and corresponding uncertainty analysis. Our homedeveloped high-fidelity neutronics code name NECP-X has been utilized to perform the transient modeling and simulation. The
Fig. 9. the pin-wise powers and corresponding relative uncertainties/% at t = 0.05 s.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 10. the pin-wise powers and corresponding relative uncertainties/% at t = 1.0 s.
Fig. 11. the pin-wise powers and corresponding relative uncertainties/% at t = 1.05 s.
uncertainty analysis for the transient simulation has been implemented with our home-developed UNICORN code based on the statistical sampling method. In order to sample for the continueenergy cross-section library utilized in NECP-X, the perturbation model for the continue-energy cross sections has been developed in this study. Based on the above analysis code and method, the uncertainty analysis has been implemented to the transient simulation of C5G7 benchmark with the 2D model, in which the control-rod insertion and withdraw is modeled by mixing the moderator and control-rod absorber with the volume. The transient process is defined that the control rods loaded in the middle assembly are inserted in the core with the fraction of 1% and then withdraw out of the core after 1.0 s. For the uncertainty analysis, the crosssection covariance library is generated based on ENDF/B-VII.1 using the NJOY code. The relative uncertainties of the normalized fuel-assembly power and pin-wise power have been quantified as the function of transient time. It can be observed that the maximum relative uncertainty for the normalized fuel-assembly power and pin-wise power can reach about 2.14% and 3.54%. As function of the transient time, the relative uncertainty of the normalized fuel-assembly power for the assembly with control-rod movement decrease with the control-rod insertion and then increases with the control-rod withdraw, while the relative uncertainty of the
other assemblies have the opposite trending varying with the transient time. In this study, considering the huge computational cost for the transient simulation and corresponding uncertainty analysis, only one single case with the 2D model and the power distributions have been implemented for the C5G7-TD benchmark. In our further researches, the 3D model and more complex transient process proposed in the C5G7-TD benchmark problem will be focused on, and more key parameters of interest will be studied. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is supported by the Fundamental Research Funds for the Central University (Grant No. xzy012019026), the National Natural Science Foundation of China (Grant No. 11735011) and the China Postdoctoral Science Foundation (Grant No. 2018M643668).
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.anucene.2019.107122.
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Pusa, M., 2012. Incorporating sensitivity and uncertainty analysis to a lattice physics code with application to CASMO-4. Ann. Nucl. Energy 40, 153–162. Pusa, M., Isotalo, A., 2017. Uncertainty analysis of assembly and core-level calculations with application to CASMO-4E and SIMULATE-3. Ann. Nucl. Energy 104, 124–131. Ryu, M., Joo G. nTRACER whole core transport solutions to C5G7-TD benchmark. 2017, M&C2017, Jeju, Korea. Shen, Q., Wang, Y., Jabaay, D., et al., 2019. Transient analysis of C5G7-TD benchmark with MPACT. Ann. Nucl. Energy 125, 107–120. Williams, M., Wiarda, D., Smith, H., et al. April 15–20, 2012. Development of a Statistical Sampling Method for Uncertainty Analysis with SCALE. PHYSOR 2012, Tennessee, USA. Wan, C., Cao, L., Wu, H., et al., 2015. Code development for eigenvalue total sensitivity analysis and total uncertainty analysis. Ann. Nucl. Energy 85, 788– 797. Wan, C., Cao, L., Wu, H., Shen, W., 2017. Total sensitivity and uncertainty analysis for LWR pin-cells with improved UNICORN code. Ann. Nucl. Energy 99, 301–310. Wan, C., Cao, L., Wu, H., 2019a. Uncertainty quantification for the first-cycle modeling and simulations of the BEAVRS benchmark problem. Ann. Nucl. Energy 133, 378–386. Wan, C., Qiao, L., Zheng, Y., et al., 2019b. Validation of SARAX for the China Fast Reactor with the extrapolated experimental data. Ann. Nucl. Energy 127, 188– 195. Wang, B., Liu, Z., Chen, J., et al., 2018. A modified predictor-corrector quasi-static method in NECP-X for reactor transient analysis based on the 2D/1D transport method. Prog. Nucl. Energy 108, 122–135. Yankov, A., Klein, M., Jessee, M., et al., 2012. Comparison of XSUSA and ‘‘Two-Step” Approaches for Full-Core Uncertainty Quantification. PHYSOR 2012, Knoxville, Tennessee, USA. Zhao, C., Liu, Z., Liang, L., et al., 2018. Improved leakage splitting method for the 2D/ 1D transport calculation. Prog. Nucl. Energy 105, 202–210. Zheng, Y., Qiao, L., Wan, C., et al., 2019. Sensitivity analysis and similarity evaluation of sodium-cooled fast reactor in large size. Ann. Nucl. Energy 125, 283–290. Zhou, H., Jessee, M., Ward, A., et al., 2016. Uncertainty Quantification for Full-Core Steady-State PWR Core Simulation with SCALE and PARCS. PHYSOR2016, Sun Valley, ID. Zwermann, W., Gallner, L., Klein, M., et al., 2013. Status of XSUSA for sampling based nuclear data uncertainty and sensitivity Analysis. EDP Sci. 03003.
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Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem Chenghui Wan a,⇑, Zhuojie Sui a, Bo Wang a, Liangzhi Cao a, Zhouyu Liu a, Jason Hou b a b
School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695, USA
a r t i c l e
i n f o
Article history: Received 10 August 2019 Received in revised form 1 October 2019 Accepted 14 October 2019 Available online xxxx Keywords: Uncertainty analysis Nuclear-data uncertainties C5G7 benchmark problem Transient simulation
a b s t r a c t In this study, uncertainty analysis has been performed to the time-dependent transient simulation of the C5G7-TD benchmark problem, propagating the nuclear-data uncertainties to the key parameters of interest. The detailed material compositions and geometries of C5G7-TD have been applied for the time-dependent transient simulation. For uncertainty analysis of the transient simulation, multigroup cross-section covariance library has been generated based on ENDF/B-VII.1 using the NJOY code. Our home-developed uncertainty-analysis code named UNICORN has been utilized for the uncertainty analysis, using the statistical sampling method. For the steady-state and transient simulations, our home-developed high-fidelity neutronics code NECP-X has been utilized. The relative uncertainties of the fuel-assembly normalized power and pin-wise power have been quantified as function of time. The numerical results show that the maximum relative uncertainties for the assembly normalized power can up to be about 2.14% and the value for the pin-wise power distributions can be about 3.54%. Ó 2019 Published by Elsevier Ltd.
1. Introduction It has been an increasing demand for the best estimate predictions to be provided with their confidence bounds in many domains, including the nuclear research, industry, safety and regulation (Ivanov et al., 2013). Motivated by UAM expect group, many research works have been focused on the uncertainty analysis of the LWR system at steady-state modeling and simulations (Yankov et al., 2012; Zhou et al., 2016; Pusa and Isotalo, 2017). However, few research of uncertainty analysis was performed to the transient related modeling and simulation, which is significant for the safety analysis. With the great efforts made by OECD/NEA, a series of space–time neutron kinetics benchmark exercises, dubbed the C5G7-TD benchmark, have been proposed and developed (Boyarinov et al, 2016; Hou et al, 2017), based on the wellstudied steady-state neutron transport C5G7 benchmark. These benchmark problems can be generally divided into the 2dimensional (2D) and 3-dimensional (3D) transient problems. The 2D problem includes four different transient exercises, in which the control-rod cluster moves and the moderator density changes with various rate and magnitude. The 3D problem adopts the 3D configuration of the C5G7 core, and two different exercises
⇑ Corresponding author.
are defined for the control-rod insertion/withdrawal and the moderator density change with various rate and magnitude. Moreover, both the few-group homogenization constants and detailed material compositions of the materials have been provided for these C5G7-TD benchmark problems. A number of transient simulation results of the C5G7-TD benchmark problem have been reported (Wang et al., 2018; Shen et al., 2019; Ryu et al., 2017) and the comparative analysis results were summarized (Ogujiuba et al., 2019). However, all the activities related to the current benchmark have been focused on the transient modeling and simulation, no study was performed to the uncertainty analysis, which is the objective of the last phase of this benchmark. Therefore, the uncertainty analysis has been focused on in this study, aimed at propagating the nuclear-data uncertainties to the key parameters of interest in the transient simulations for the C5G7 benchmark. Through the study implemented in this study, the uncertainty-analysis research can be expanded to the transient simulation, which would provide the preliminary observations for the safety analysis. In order to propagate the nuclear-data uncertainties through the reactor-physics modeling and simulation, the deterministic method (Pusa, 2012; Jessee and Dehart, 2009) and statistical sampling method (Herranz et al., 2008; Williams et al., 2012; Zwermann et al., 2013) have been proposed and widely applied. For the deterministic method, the uncertainties of the interested
E-mail address: [email protected] (C. Wan). https://doi.org/10.1016/j.anucene.2019.107122 0306-4549/Ó 2019 Published by Elsevier Ltd.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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parameters are quantified using the ‘‘sandwich rule”, in which the nuclear-data covariance is combined with the sensitivity coefficients of the parameters with respect to the nuclear data determined by the sensitivity analysis. For the statistical sampling method, the uncertainties of the interested parameters are quantified with the parameters’ samples, which are determined through the reactor-physics simulations with corresponding nuclear-data samples generated with the nuclear-data covariance. Comparing these two methodologies indicate that the first-order approximation exists in the deterministic method, as only the first-order sensitivity coefficients are applied in the uncertainty quantification. Therefore, the statistical sampling method is considered more appropriate to perform the uncertainty analysis for the transient simulations, and hence selected in this study to propagate the nuclear-data uncertainties to the key parameters of interest. In this context, the uncertainty analysis has been implemented to the transient simulations of the C5G7-TD benchmark problem, quantifying the uncertainties of the key parameters of interest during the transient process due to the nuclear-data uncertainties. For the uncertainty analysis, the statistical sampling method implemented in our home-developed code named UNICORN (Wan et al., 2015, 2017, 2019) has been applied. The multigroup crosssection covariance libraries, which contains the nuclear-data uncertainties, were generated by NJOY (MacFarlane et al., 2016) based on the ENDF/B-VII.1 library in this study. In order to compatible with the multigroup microscopic cross-section covariance libraries, the microscopic description of the C5G7-TD benchmark was utilized in the transient modeling and simulation. Moreover, our home-developed high-fidelity neutronics code named as NECP-X was used for the transient modeling and simulation for the C5G7 benchmark. As both the multigroup and continueenergy cross-section library were required in the NECP-X simulations (Chen et al., 2018; Liu et al., 2018a; Zhao et al., 2018; Liu et al., 2018b; Wang et al., 2018), the continue-energy crosssection perturbation model has been developed in UNICORN to generate the cross-section samples based on the corresponding covariance libraries. Using the approach described above, the nuclear-data uncertainties can be propagated to the key parameters of interest during the transient process for the C5G7-TD benchmark problem. As the numerical results, the relative uncertainties of the assembly normalized power and also pin-wise power distributions have been quantified during the transient process. The paper is structured as follow. The brief introduction to the methodology for the transient uncertainty analysis and detailed description of the continue-energy cross-section perturbation model are provided in Section 2. Section 3 shows the detailed numerical solutions and corresponding analysis results. Summaries and conclusions are given in Section 4. 2. Flowchart and methods for the transient uncertainty analysis In this section, the uncertainty analysis method for the transient simulation using the high-fidelity NECP-X code established in UNICORN is briefly explained. Due to the specific requirement that the continue-energy cross-section library is applied in NECP-X for the resonance shielding calculation, corresponding continue-energy cross-section model has been developed in UNICORN. 2.1. Uncertainty analysis method for transient simulation As a short introduction, the NECP-X (Chen et al., 2018; Liu et al., 2018a; Zhao et al., 2018; Liu et al., 2018b; Wang et al., 2018) is a new high-fidelity neutronics code developed by NECP (Nuclear Engineering Computational Physics) Laboratory at Xi’an Jiaotong
University. The pseudo-resonance-nuclide subgroup method is applied for the resonance treatment. For the 3D neutronics simulation, the 2D/1D method is utilized to solve the neutron-transport equation. The CRAM method is applied for the depletion calculation and the sub-channel method is utilized for the thermal-hydraulics calculation. The matrix-free CMFD method and multi-level parallel strategy are applied for high efficiency. The NECP-X code is capable of performing both the steady-state and transient simulations for the pressurized water reactors (PWR). The UNICORN code (Wan et al., 2015, 2017, 2019) is also developed by NECP Laboratory for the sensitivity and uncertainty analysis. The direct numerical perturbation method is applied for the sensitivity analysis and the statistical sampling method is implemented for the uncertainty analysis. Based on the sensitivity and uncertainty analysis, the UNICORN code has the capabilities of the similarity analysis (Zheng et al., 2019) and nuclear-data adjustment (Wan et al., 2019). Based on the multigroup cross-section covariance libraries, the relative perturbation factors of the multigroup cross sections, referred to as cross section perturbation factors, can be generated, which contains the uncertainty information of the nuclear data. Therefore, it is the vital step that how to generate the samples of the microscopic cross-section libraries based on the samples of the cross-section perturbation factors. It should be noted that the pseudo-resonance-nuclide subgroup method (Liu et al., 2018) implemented in NECP-X for the resonance shielding calculation requires both the multigroup and continue-energy cross-section library. The multigroup cross-section perturbation model has been developed and implemented in UNICORN previously (Wan et al., 2015). In this study, a newly perturbation model was developed to generate samples for the continue-energy cross-section library, as will be introduced in Section 2.2. It should be noted that the newly perturbation model is applied for the continue-energy cross sections within the resonance energy ranges, while for perturbing the cross sections, the multigroup cross-section perturbation model has been applied to the multigroup cross sections for the fast and thermal groups. After the samples of both the multigroup and continue-energy cross-section libraries having been generated, the NECP-X code was executed once for each sample. In this process, corresponding samples for the key parameters of interest for the transient simulations can be determined. Finally, the statistical analysis can be done using all the samples of outputs to quantify the uncertainties and corresponding correlations for all system response of interest. Applying the process described above, the nuclear-data uncertainties can be propagated into the transient simulations, quantifying the uncertainties of the key parameters of interest. The flowchart of uncertainty analysis for the transient simulation based on NECP-X implemented in the UNICORN code has been shown in Fig. 1. 2.2. Continue-energy Cross-section perturbation model For better understanding the context, some detailed introduction towards the pseudo-resonant-nuclide subgroup method are given in this section. First, a pseudo resonant nuclide is defined as the mixture of all the resonant nuclides in each fuel rod. The 0-dimentional (0-D) neutron slowing-down equations are solved over a range of dilution cross sections by the hyper-finer energy group method (~1 M energy groups) to obtain the cross-section tables of the pseudo resonant nuclide. The hyper-finer cross sections are generated with the continue-energy cross-section library. Second, the physical subgroup probability tables are generated based on the pseudo-resonant nuclides and the intermediate resonance approximation. Third, a set of equivalent 1-dimental (1-D) cylinder problem are generated, by preserving the Dancoff
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 1. Flowchart for the transient uncertainty analysis based on NECP-X.
correlation factors, which are calculated by the neutron current method within the whole simulation ranges. Finally, the subgroup equations are solved for the 1-D cylindrical problems to get the effective self-shielding cross sections. According to the resonance-treatment method implemented in NECP-X, both the multigroup and continue-energy cross-section libraries are required for the high-fidelity neutronics modeling and simulations. There are 258 nuclides in the library, including 75 resonant nuclides. The multigroup cross-section libraries were generated for all nuclides and the continue-energy cross-section libraries were generated for all resonant nuclides. The conventional 69-group energy-group structure has been applied for the multigroup cross-section libraries, with the resonance-energy range extended to be 0.625 eV to 24.78 keV (13th–45th group). For the continue-energy cross-section libraries, the whole energy range has been covered and temperature interval is set to be 50 K from 300 K to 1800 K. The hyperfine energy grids are utilized, hence the cross sections between any two adjacent energy points can be interpolated linearly. Moreover, only the cross-section information for the rf, relas and rt are saved in the continueenergy cross-section libraries. Based on the relative covariance matrices for the cross sections, the samples for the relative perturbation factors can be generated. For any sample, the relative perturbation factors can be characterized as d = [d1, d2,. . ., dM]T, in which M presents the total number of the nuclides, cross-section types and groups. Detailed description towards the multigroup cross-section perturbation has been explained in our previous, and the following context will be focused on how to perturb (or sample) the continue-energy cross-section library with the relative perturbation factors. With the relative perturbation factor dm (m = 1, 2,. . ., M) for the gthgroup energy range, the cross-section sampling process can be treated as perturbing the cross sections within corresponding energy range, which can be characterized as:
r0x ðE; TÞ ¼ ð1:0 þ dm Þrx ðE; TÞ; Eg 6 E < Eg1
ð1Þ
where rx(E,T) and r’ x(E,T) present the initial and perturbed (or sampled) cross sections for the reaction , and the Eg and Eg-1 are the energy boundaries of the group g within the multigroup energy-group structure. In the continue-energy cross-section library applied in NECP-X, only the cross-section data for the rf, relas and rt are saved. Using the saved cross sections, the total absorption cross section can be calculated:
ra ðE; TÞ ¼ rt ðE; TÞ relas ðE; TÞ rf ðE; TÞ
ð2Þ
where ra presents the total absorption cross section, which can be re-characterized in detailed as:
ra ðE; TÞ ¼ rinel ðE; TÞ þ rc ðE; TÞ þ
X x
rðn;xnÞ ðE; TÞ þ
X
rðn;PÞ ðE; TÞ
P
ð3Þ where rinel is the inelastic cross section and rc is the capture cross section. The r(n,xn) includes the cross section (n,2n), (n,3n), (n,4n) etc. The r(n,P) presents the other cross-section types with particle productions, including the (n,P) for proton produce, (n,D) for deuterium produce, (n,T) for tritium produce, (n,a) for alpha produce etc. In the UNICORN code, relative-perturbation samples need to be generated for all the basic cross sections, including the relas, rinel, rf, rc, r(n,xn), r(n,P) etc. The multigroup cross-section perturbation model was implemented for sampling the multigroup crosssection library (Wan et al., 2015). As only the cross-section data for the rf, relas, rt and ra can be provided by the continueenergy cross-section library applied in NECP-X, it is not available to generate the samples for the basic cross-section types presented in the right-hand side of Eq. (3). Therefore, when relative perturbations are applied to these basic cross-section types, corresponding multigroup cross-section data were used to determine the relative perturbations, which would be applied to the saved rf, relas, ra and rt. Different approaches should be taken based on the reaction types. When the samples are generated for the rf and relas, which are saved in the continue-energy cross-section library, the relative perturbations will be applied to the cross-section data directly. This case is straightforward and the implementation can be done as shown in Eq. (1). When the relative perturbations are applied to these basic cross sections contained in ra, corresponding multigroup cross-section data are applied to determine the relative perturbations for ra:
r0a;g ðTÞ ¼ ra;g ðTÞ þ
X
dx;g rx;g ðTÞ
ð4Þ
x2a
da;g ðTÞ ¼
r0a;g ðTÞ 1:0 ra;g ðTÞ
ð5Þ
where da,g is the relative perturbations added to the absorption cross section within the gth-group energy ranges. The corresponding perturbed or sampled continue-energy absorption cross section within group g can be determined as:
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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r0a ðE; TÞ ¼ ð1:0 þ da;g Þra ðE; TÞ; Eg 6 E < Eg1
ð6Þ
After the cross-section perturbations or sampling, the consistent rule should be applied to balance the total cross section and the components, which can be characterized as:
r0t ðE; TÞ ¼ r0a ðE; TÞ þ r0elas ðE; TÞ þ r0f ðE; TÞ
ð7Þ
Through the continue-energy cross-section perturbation model, the samples for the continue-energy cross-section libraries can be generated and provided to the NECP-X code for performing simulations and generating corresponding samples for the key parameters of interest. 3. Numerical results and analysis
Table 2 Isotopic distributions of the materials. Nuclides
Concentrations/1024 at./cm3 MOX 4.3%
MOX 7.0%
MOX 8.7%
235
U 238 U 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 16 O
5.0000E-5 2.2100E-2 1.5000E-5 5.8000E-4 2.4000E-4 9.8000E-5 5.4000E-5 1.3000E-5 4.6300E-2
5.0000E-5 2.2100E-2 2.4000E-5 9.3000E-4 3.9000E-4 1.5200E-4 8.4000E-5 2.0000E-5 4.6300E-2
5.0000E-5 2.2100E-2 3.0000E-5 1.1600E-3 4.9000E-4 1.9000E-4 1.0500E-4 2.5000E-5 4.6300E-2
Material
Nuclides
Concentrations/1024 at./cm3
UO2
235
U U 16 O
8.6500E-4 2.2250E-2 4.6220E-2
1
H O B 11 B
6.7000E-2 3.3500E-2 5.9667E-6 2.1833E-5
90
Zr Zr 92 Zr 94 Zr 96 Zr
2.2124E-2 4.8246E-3 7.3745E-3 7.4734E-3 1.2040E-3
238
In this section, the numerical results of the transient problem with insertion and withdraw of bank 1 for C5G7 will be provided and analyzed. First, the detailed information of the C5G7 microscopic benchmark will be given, based on which the steady-state and transient simulations. Second, the uncertainty analysis has been performed within the framework of the high-fidelity modeling and simulation, to quantify the uncertainties of the key parameters during the transient simulation, due to the nuclear-data uncertainties.
Moderator
16 10
Zirconium
91
3.1. Steady-state and transient modeling and simulation for C5G7
Aluminum
27
6.0000E-2
Detailed C5G7-TD benchmark specifications were used in the problem setup of the transient simulation, including the geometries, configuration and material compositions. The detailed information for the geometries and materials is shown in Table 1. The pin-cells in C5G7 are the square lattice with cylindrical fuel rods. The detailed isotopic distributions of these loaded materials are listed in Table 2. The void is fulfilled with the nuclide He-4 with the atomic concentrations of 1.60535E-4 atom/barncm. The configurations of the pin-cells and assemblies are identical to the C5G7 benchmark specifications, as shown in Figs. 2 and 3. Based on the essential information about the geometry information for the pin-cells and assemblies and isotopic concentrations for all the loaded materials, the NECP-X code is applied to execute the simulation for the transient exercise of C5G7 benchmark from the microscopic cross-section library. It should be noted that the 2D model has been applied for the transient simulation. When the control rods insert in or withdraw out of the core, the material compositions of the guide tubes are changed, using the volumes of the moderator and Ag-In-Cd as weights to generate the mixture compositions. And the volumes of the moderator and Ag-In-Cd are calculated in advance, according to the insertion fraction of the control rods. The transient exercise analyzed in this study is the process to insert and then withdraw the bank 1, which is at the position of the UO2-loaded fuel assembly in middle of the core. At beginning,
Ag-In-Cd
107
2.2711E-2 2.2711E-2 8.0008E-3 2.7241E-3
Table 1 Pin-cell geometries and materials. Pin-cell
Material
External radius/cm
Fuel cells
Fuel Void Zirconium Void Aluminum Clad Moderator (square lattice pitch)
0.4095 0.4180 0.4750 0.4800 0.5400 1.26
Guide-tube cells
Moderator Aluminum Clad Moderator (square lattice pitch)
0.3400 0.5400 1.26
Al
Ag Ag In 113 Cd 109
115
the insertion fraction of bank 1 in the TD0-1 exercise proposed in C5G7 transient benchmark has been selected and tested. However, it was observed that the negative reactivity was very huge (about 16$) and the fuel-assembly power would decrease to be almost zero in very short time. Therefore, an appropriate insertion fraction (about 1%) has been determined to be applied for the transient modeling and simulation, as shown in Fig. 4. In the transient exercise, the bank 1 is inserted with the insertion fraction of 1% at the initial time t = 0.0 s and the position of bank 1 is hold until the time t = 1.0 s; at time t = 1.0 s, the bank 1 is withdraw out of the core. In this study, the 2D model has been applied to perform the transient simulation for the transient exercise. During the transient process, the thermal-hydraulics feedback is not considered, and the temperatures of the fuel and moderator remain at the hot conditions with the room temperature. Therefore, two different steady states have been modeled and simulated to determine the reactivity introduced in the core with the insertion of bank 1, including the state that the bank 1 fully withdraw and 1% inserted in the core. The keff of these two steady states and corresponding reactivity introduced by the bank insertion can be determined, as shown in Table 3. Applying the NECP-X code, the transient process has been modeled. For the transient modeling and simulation, the time step is set to be 0.05 s in the range of 0.0–2.0 s and 0.1 s between 2.0 and 3.0 s. The numerical results for the normalized power of the fuel assembly during the transient process are as shown in Fig. 5. During the time 0.0 s–1.0 s, the normalized fuel-assembly powers decrease with the time, as the bank 1 was inserted into the core at t = 0.0 s; at t = 1.0 s, the bank 1 was withdraw out of the core, and corresponding fuel-assembly powers increase with the time. These numerical results of the transient characteristic are reasonable and within what are expected.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 2. the fuel-assembly configuration for the C5G7 benchmark problem.
Fig. 3. the control-assembly configuration for the C5G7 benchmark problem.
Fig. 4. Controlrod movement in the transient exercise.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Table 3 The keff and reactivity introduced by bank-1 insertion of C5G7. keff with bank-1 withdraw
keff with bank-1 insertion
Reactivity introduced
1.202867
1.146980
4050.8pcm
Table 4 The isotopes and corresponding cross-section types in uncertainty analysis. Isotopes
Cross sections
235
r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c), v r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c), v r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,f), r(n,c) r(n,elas), r(n,inel), r(n,c), r(n,a) r(n,elas), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c) r(n,elas), r(n,inel), r(n,2n), r(n,c)
U U Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 16 O 1 H 90 Zr 91 Zr 92 Zr 94 Zr 109 Ag 238 238
3.2. Uncertainty analysis results for transient simulation Based on the transient model above, the uncertainty analysis has been performed. The multigroup cross-section covariance library containing the uncertainty and correlation information was generated based on the ENDF/B-VII.1 library using the NJOY code. For the uncertainty quantification, the UNICORN code has been applied using the statistical sampling method. The Latin Hypercube Sampling (LHS) method has been utilized to generate 200 relative perturbation factors and corresponding multigroup and continue-energy cross-section library. In the uncertainty analysis, the analyzed isotopes and corresponding cross sections are listed as shown in Table 4. It should be noted that for the material Ag-In-Cd, the isotopes of In and Cd have no covariance data in ENDF/B-VII.1 and only 109Ag has the covariance data. Therefore, on the isotope 109Ag and corresponding cross-section types have been analyzed for the control-rod material. For the absorption material in the moderator, the B-10 of boron, very extremely large
data exists in the covariance of r(n,inel) in ENDF/B-VII.1, which is not appropriate considered in uncertainty analysis. The uncertainty analysis has been conducted focusing on the fuel-assembly and pin-wize power during the transient process. The normalized fuel-assembly power and corresponding relative uncertainties are shown in Figs. 6 and 7, respectively. From the numerical results, several observations can be found. For the
Fig. 5. The normalized fuel-assembly powers during transient process.
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Fig. 6. Normalized assembly powers and corresponding uncertainties.
Fig. 7. Relative uncertainty of the normalized fuel-assembly powers.
central assembly, the relative uncertainty of the normalized power decreases with the time after the control-rod insertion and increase immediately following the control-rod withdraw. It then
approaches to an asymptotic value afterwards For other fuel assemblies which do not witness the control-rod movements, the trend of the relative uncertainty of the normalized assembly
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 8. the pin-wise powers and corresponding relative uncertainties/% at t = 0.0 s.
powers is opposite: it increases over the time with the bank insertion and decreases with the bank withdraw before approach the asymptotic value. In order to explain the observed phenomenon mentioned above, the pin-wise power distributions are analyzed thereafter. The relative uncertainties of the pin-wise power at t = 0.0 s, 0.05 s, 1.0 s and 1.05 s are shown in Figs. 8–11 respectively, and the following observations can be provided. Before the bank 1 insertion, the pin-wise powers at #1 fuel assembly are the largest ones and those in #4 fuel assembly are the smallest. After the bank 1 inserted at the time t = 0.05 s, the pin-wise power decreases across the core and so does the fuel-assembly power. By comparing the power distributions between t = 0.0 s and t = 0.05 s, the power of Assembly #1 decreases from 0.48 to 0.318, with the decreased ratio is about 33.7%; while for the others fuel assemblies, the decreased ratios are respectively 31.5% for Assembly #2 and #3 and 30.5% for Assembly #4. Through the assembly-power normalization, the relative powers of Assembly #2, #3 and #4 increases due to the larger power decrease in Assembly #1. In this case, the relative contributions of the fission cross sections to the fuelassembly powers in #2, #3 and #4 fuel assemblies becomes larger after the bank 1 insertion. Therefore, after the bank 1 insertion, the relative uncertainties of the fuel-assembly power for #2, #3 and #4 assemblies increases lightly while #1 decreases lightly with the
time. The process above becomes opposite after the bank 1 withdraw when t = 1.0 s and t = 1.05 s. When the bank 1 withdraw out of the core, the fuel-assembly power for #1 assembly increases immediately and becomes dominant, hence the relative contribution of the fission cross sections for #1 assembly becomes dominant and the relative uncertainty of the power in #1 assembly increases after bank 1 withdraw.Fig. 9. From the numerical values, for the transient process defined in the study, the uncertainty-analysis results can be concluded as following. The maximum relative uncertainty for the normalized assembly powers can be about 2.14% and the minimum value is about 0.57%. For the pin-wise power distributions, the maximum relative uncertainty is about 3.54%. 4. Conclusions In this study, uncertainty analysis has been implemented to the transient simulation for the C5G7 benchmark problem, propagating the nuclear-data uncertainties to the key parameters of interest. The detailed description of the geometries and material compositions of C5G7 has been utilized for the transient simulation and corresponding uncertainty analysis. Our homedeveloped high-fidelity neutronics code name NECP-X has been utilized to perform the transient modeling and simulation. The
Fig. 9. the pin-wise powers and corresponding relative uncertainties/% at t = 0.05 s.
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Fig. 10. the pin-wise powers and corresponding relative uncertainties/% at t = 1.0 s.
Fig. 11. the pin-wise powers and corresponding relative uncertainties/% at t = 1.05 s.
uncertainty analysis for the transient simulation has been implemented with our home-developed UNICORN code based on the statistical sampling method. In order to sample for the continueenergy cross-section library utilized in NECP-X, the perturbation model for the continue-energy cross sections has been developed in this study. Based on the above analysis code and method, the uncertainty analysis has been implemented to the transient simulation of C5G7 benchmark with the 2D model, in which the control-rod insertion and withdraw is modeled by mixing the moderator and control-rod absorber with the volume. The transient process is defined that the control rods loaded in the middle assembly are inserted in the core with the fraction of 1% and then withdraw out of the core after 1.0 s. For the uncertainty analysis, the crosssection covariance library is generated based on ENDF/B-VII.1 using the NJOY code. The relative uncertainties of the normalized fuel-assembly power and pin-wise power have been quantified as the function of transient time. It can be observed that the maximum relative uncertainty for the normalized fuel-assembly power and pin-wise power can reach about 2.14% and 3.54%. As function of the transient time, the relative uncertainty of the normalized fuel-assembly power for the assembly with control-rod movement decrease with the control-rod insertion and then increases with the control-rod withdraw, while the relative uncertainty of the
other assemblies have the opposite trending varying with the transient time. In this study, considering the huge computational cost for the transient simulation and corresponding uncertainty analysis, only one single case with the 2D model and the power distributions have been implemented for the C5G7-TD benchmark. In our further researches, the 3D model and more complex transient process proposed in the C5G7-TD benchmark problem will be focused on, and more key parameters of interest will be studied. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is supported by the Fundamental Research Funds for the Central University (Grant No. xzy012019026), the National Natural Science Foundation of China (Grant No. 11735011) and the China Postdoctoral Science Foundation (Grant No. 2018M643668).
Please cite this article as: C. Wan, Z. Sui, B. Wang et al., Nuclear-data uncertainty propagation in transient simulation for the C5G7-TD benchmark problem, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107122
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Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.anucene.2019.107122.
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