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, while
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) [
In general, uncertainty is calculated based on covariance matrix and weighting factor coefficients [
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 (
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 [
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 [
For light water reactors, two components of sensitivity coefficient are needed. Explicit sensitivity represents the sensitivity of the calculated
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 [
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 [
Uncertainty in
Fuel | Operating conditions |
|
Uncertainty in |
Largest uncertainty-contributing reaction |
---|---|---|---|---|
BWR | HZP | 1.3382 | 0.52 |
238U( |
HFP (40% void) | 1.2208 | 0.62 |
238U( |
|
PWR | HZP | 1.4206 | 0.48 |
238U( |
HFP | 1.4017 | 0.49 |
238U( |
|
VVER | HZP | 1.3448 | 0.51 |
238U( |
HFP | 1.3270 | 0.52 |
238U( |
|
KRITZ 2.1 | HZP | 1.2323 | 0.59 |
238U( |
HFP | 1.1837 | 0.63 |
238U( |
|
KRITZ 2.13 | HZP | 1.2642 | 0.55 |
238U( |
HFP | 1.2329 | 0.58 |
238U( |
Observations of the results showed that
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. Figure
Relative change in the sensitivity of 238U(
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 [
Assembly
Fuel | Operating conditions |
|
Uncertainty in |
Largest uncertainty-contributing reaction |
---|---|---|---|---|
BWR | HZP | 1.1116 | 0.50 |
238U( |
HFP (40% void) | 1.0779 | 0.56 |
238U( |
|
PWR | HZP | 1.4130 | 0.46 |
23U( |
HFP | 1.3968 | 0.47 |
238U( |
|
VVER | HZP | 1.3164 | 0.47 |
238U( |
HFP | 1.3115 | 0.47 |
238U( |
Similar to the fuel pin models, in fuel assembly models, the uncertainty in
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 Table
Two-group cross-section uncertainty in LWR fuel assembly.
Response cross-section | Uncertainty (% |
Uncertainty (% |
Uncertainty (% |
|||
---|---|---|---|---|---|---|
BWR | PWR | VVER | ||||
HZP | HFP | HZP | HFP | HZP | HFP | |
|
0.84 | 0.91 | 0.87 | 0.88 | 0.81 | 0.82 |
|
0.13 | 0.15 | 0.14 | 0.14 | 0.12 | 0.12 |
|
0.84 | 0.91 | 0.87 | 0.88 | 0.81 | 0.82 |
|
0.13 | 0.15 | 0.14 | 0.14 | 0.12 | 0.12 |
|
0.78 | 0.83 | 0.86 | 0.87 | 0.81 | 0.82 |
|
0.20 | 0.22 | 0.22 | 0.22 | 0.21 | 0.21 |
|
0.68 | 0.72 | 0.36 | 0.36 | 0.47 | 0.47 |
|
0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 |
|
0.84 | 0.91 | 0.87 | 0.87 | 0.81 | 0.81 |
|
1.10 | 1.22 | 1.20 | 1.21 | 1.03 | 1.03 |
|
0.27 | 0.34 | 0.30 | 0.33 | 0.26 | 0.29 |
|
0.13 | 0.16 | 0.15 | 0.15 | 0.13 | 0.13 |
One can define nine-dimensional response vector
BWR HZP covariance matrix.
BWR HFP covariance matrix.
TMI HZP covariance matrix.
TMI HFP covariance matrix.
The obtained results for different LWR types and cases indicate the following tendencies. Group 1 (fast) cross-section uncertainty is ~2-3 times larger than Group 2 (thermal) cross-sections uncertainty. Uncertainty contributions: a major contributor to Group 1 (fast) cross-section uncertainties is U-238 inelastic scattering; U-238 inelastic scattering uncertainty is quite large; 40% void (and higher) exhibit larger uncertainty in Uncertainty (correlation) contribution: U-238 inelastic scattering uncertainty is quite large and dominates correlation coefficient.
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 [
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
Uncertainty in
Fuel | Compositions |
|
Uncertainty in |
Largest uncertainty-contributing reaction |
---|---|---|---|---|
MOX | 9.8% 239Pu | 1.0921 | 0.94 |
238U( |
6.5% 239Pu | 1.0540 | 0.97 |
238U( |
|
3.7% 239Pu | 1.0115 | 0.99 |
238U( |
|
| ||||
UOX | 4.2% 235U | 1.2431 | 0.51 |
238U( |
3.2% 235U | 1.1741 | 0.54 |
238U( |
|
2.1% 235U | 1.0490 | 0.59 |
238U( |
|
| ||||
UOX-Gd2O3 | 2.2% 235U | 0.2166 | 1.79 |
238U( |
1.9% 235U | 0.1997 | 1.94 |
238U( |
For each group of the fuel cells, several factors influence the changes in the uncertainty in
For each unit cell, the calculated
For the MOX fuel cells, the amount of 238U 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
However, the uncertainties of the MOX fuel cells were nearly twice than that of UOX fuel. The presence of 239Pu plays an important role in the increase in uncertainty. Figure
Plot of 239Pu absorption cross-sections compared to 235U absorption cross-sections.
As a consequence, from the fact that more neutrons are absorbed by 239Pu than 235U, more neutrons are produced by fission due to 239Pu. Table
Macroscopic fission cross-sections in MOX fuel cells.
Fuel | Composition |
|
235U |
239Pu |
---|---|---|---|---|
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 | 5.88 |
LWR neutronics parameters [
Parameter | 235U | 239Pu |
---|---|---|
Average |
2.4 | 2.9 |
Average |
2.0 | 1.9 |
Average |
280 barns | 790 barns |
Neutrons produced by 239Pu, 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
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 the 238U. The resonance absorption of 238U that occurs at higher neutron energy is very large.
The absorption of thermal neutrons by 239Pu 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(
The plot of inelastic cross-section of 238U as a function of energy.
The neutron flux spectra of three unit cell compositions.
The sensitivity profiles were compared in Figure
The sensitivity of 238U(
The covariance matrix of 238U(
The 2D plot of 238U(
The 2D plot of 238U(
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: type 1 (UOX type 2 (UOX type 3 (UOX type 4 (MOX).
The details of the specifications are readily available [
Uncertainty in
Fuel | Compositions |
|
Uncertainty in |
Largest uncertainty-contributing reaction |
---|---|---|---|---|
GEN III | Type 1 | 1.2501 | 0.49 |
238U( |
Type 2 | 1.1228 | 0.49 |
238U( |
|
Type 3 | 0.9564 | 0.53 |
238U( |
|
Type 4 | 1.0700 | 0.97 |
238U( |
Results show that
Additionally, the two-group cross-section uncertainties are presented in Table
Two-group cross-section uncertainty (%
Response cross-section | GEN III type 1 | GEN III type 2 | GEN III type 3 | GEN III type 4 |
---|---|---|---|---|
|
0.90 | 0.90 | 0.90 | 0.97 |
|
0.14 | 0.14 | 0.14 | 0.14 |
|
0.90 | 0.90 | 0.90 | 0.97 |
|
0.14 | 0.14 | 0.14 | 0.14 |
|
0.89 | 0.89 | 0.89 | 1.00 |
|
0.21 | 0.19 | 0.19 | 0.24 |
|
0.37 | 0.37 | 0.50 | 0.44 |
|
0.32 | 0.32 | 0.33 | 0.62 |
|
0.90 | 0.90 | 0.90 | 0.97 |
|
1.25 | 1.25 | 1.24 | 1.47 |
|
0.33 | 0.32 | 0.31 | 0.37 |
|
0.15 | 0.15 | 0.15 | 0.16 |
These results show, as expected, a larger uncertainty for the MOX fuel assembly (type 4) than for the UOX assemblies (types 1 through 3).
Figure
The neutron flux spectra of GEN-III LWR fuel assembly.
Sensitivity profiles of 238U(
The sensitivity of 238U(
Sensitivity studies have been performed using SCALE [ increasing temperature leads to increasing uncertainty in decreasing 238U in fuel composition leads to decreasing uncertainty in 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.
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 [
The authors would like to acknowledge the two Ph.D. fellowship grants, one at the Technical University of Catalonia (sponsored by the Spanish Nuclear Security Council) and the other at the Pennsylvania State University (sponsored by the US Nuclear Regulatory Commission), which supported the graduate students performing the studies reported in this paper.