In the framework of the OECD/NEA project on Benchmark for Uncertainty Analysis in Modeling (UAM) for Design, Operation, and Safety Analysis of LWRs, several approaches and codes are being used to deal with the exercises proposed in Phase I, “Specifications and Support Data for Neutronics Cases.” At UPM, our research group treats these exercises with sensitivity calculations and the “sandwich formula” to propagate cross-section uncertainties. Two different codes are employed to calculate the sensitivity coefficients of
As stated in the Introduction of [
For these calculations, the main source of uncertainty taken into account is the cross section uncertainties which are propagated throughout the different simulation levels.
There are mainly two different approaches to propagate uncertainties: The first one is based on a Monte Carlo approach where a large amount of calculations are performed sampling the problem parameters as random variables, and then carrying out a statistical analysis; the second one relies on sensitivity coefficients and the “sandwich formula.” The latter approach is the one employed in this work.
The way of obtaining the sensitivity coefficients of the response functions depends on which code is used. In this case, two different techniques are used: the Adjoint-Weighted Technique by SCALE-6.1 [
This work is aimed to present how the uncertainty quantification is carried out using the sensitivity approach and how the sensitivity coefficients are calculated with SCALE-6.1 and MCNPX-2.7e. Afterwards, in the framework of Exercise I-2, both codes are used to perform the uncertainty quantification on the
The uncertainty quantification based on sensitivity coefficients relies on the “sandwich formula” obtained with the propagation of moments, as presented in [
Being
Then, taking the parameters of the system as random variables,
Because usually the sensitivity coefficients are calculated as relative values,
The two codes which perform the criticality calculations in this paper use different methods for calculating the sensitivity coefficients necessary to carry out the uncertainty quantification: SCALE-6.1 uses the Adjoint-Weighted Technique. MCNPX-2.7e uses the Differential Operator Technique.
The Adjoint-Weighted Technique is used in SCALE-6.1 inside the TSUNAMI sequence, and the theory applied is stated in the SAMS module manual [
The implicit term,
In any TSUNAMI sequence, the forward and adjoint transport problems are solved in order to calculate the neutron flux and its adjoint using XSDRNPM module for TSUNAMI-1D, NEWT module for TSUNAMI-2D, and KENO-V.a or KENO-VI for TSUNAMI-3D. Then, the SAMS module is applied for calculating the sensitivity coefficients for every energy-group reaction cross section. The sensitivity to the average number of neutrons per fission,
The Differential Operator Technique is applied in MCNPX-2.7e to calculate the change
For the sensitivity coefficient, only the first derivative is required. Using the first term provided by the PERT card (using the keyword
Exercise I-2 [
The specifications of the fuel assembly are given in [
Geometrical model implemented in MCNPX-2.7e for
Unrodded
Rodded
The criticality calculations for the
For the criticality calculations with sensitivity analysis and uncertainty quantification of SCALE-6.1, the TSUNAMI-2D sequence is chosen. There, the NEWT module [
For the criticality calculations of MCNPX-2.7e, the KCODE card [
After preparing the inputs for the criticality calculations, PERT cards are added in order to calculate the sensitivity coefficients of
Reaction cross sections perturbed using PERT cards in the MCNPX-2.7e calculations.
Reaction | Isotopes |
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To calculate the same sensitivity coefficients as SCALE-6.1, the keyword that sets to which reaction cross section is assigned the perturbation on the
Once the sensitivity coefficients are calculated, the “SCALE Nuclear Data Covariance Library” is processed by the VIEWCVX code (provided as a module of ERRORJ code [
The
Unrodded | Rodded | |||
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HZP | HFP | HZP | HFP | |
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SCALE-6.1 | 1.41227 | 1.39802 | 1.07160 | 1.05834 |
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Difference | 383 pcm | 457 pcm | 211 pcm | 295 pcm |
The uncertainty results obtained by SCALE-6.1 and MCNPX-2.7e are presented in the tables: for the unrodded fuel assembly at HZP (Table
Comparison of the uncertainty contribution to
HZP
Reaction | Reaction | MCNP-2.7e | SCALE-6.1 | Ratio | ||
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0.23145 | 0.24582 | 0.942 |
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0.19674 | 0.19970 | 0.985 |
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0.10733 | 0.10895 | 0.985 |
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0.09078 | 0.09455 | 0.960 |
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0.08368 | 0.08500 | 0.985 |
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0.01363 | 0.01366 | 0.998 |
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0.00645 | 0.00683 | 0.945 |
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0.00476 | 0.00488 | 0.977 |
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0.00246 | 0.00257 | 0.957 |
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0.00129 | 0.00128 | 1.006 |
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−0.00476 | −0.00504 | 0.944 |
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Total | 0.34577 | 0.35940 | 0.962 | |||
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— | 0.26937 | |
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— | 0.08531 | |
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Total SCALE | 0.46441 | 0.774 |
HFP
Reaction | Reaction | MCNP-2.7e | SCALE-6.1 | Ratio | ||
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0.23793 | 0.25293 | 0.941 |
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0.19738 | 0.20025 | 0.986 |
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0.10697 | 0.10860 | 0.985 |
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0.09808 | 0.09962 | 0.985 |
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0.08333 | 0.08470 | 0.984 |
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0.01634 | 0.00943 | 1.732 |
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0.01407 | 0.01412 | 0.997 |
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0.00653 | 0.00693 | 0.943 |
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0.00480 | 0.00496 | 0.967 |
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0.00259 | 0.00272 | 0.953 |
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0.00135 | 0.00132 | 1.023 |
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Total | 0.35331 | 0.36607 | 0.965 | |||
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— | 0.26834 | |
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— | 0.08823 | |
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Total SCALE | 0.46984 | 0.779 |
Comparison of the uncertainty contribution to
HZP
Reaction | Reaction | MCNP-2.7e | SCALE-6.1 | Ratio | ||
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0.21670 | 0.23086 | 0.939 |
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0.17403 | 0.17602 | 0.989 |
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0.16286 | 0.16089 | 1.012 |
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0.11027 | 0.11189 | 0.985 |
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0.10708 | 0.10936 | 0.979 |
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0.07569 | 0.07831 | 0.967 |
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0.06786 | 0.06965 | 0.974 |
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0.03640 | 0.03627 | 1.003 |
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0.02242 | 0.02237 | 1.002 |
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0.00747 | 0.00817 | 0.915 |
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0.00744 | 0.00738 | 1.008 |
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0.00556 | 0.00584 | 0.952 |
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0.00322 | 0.00337 | 0.955 |
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0.00162 | 0.00154 | 1.050 |
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Total | 0.37315 | 0.38518 | 0.969 | |||
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— | 0.25594 | |
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— | 0.13360 | |
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Total SCALE | 0.49199 | 0.783 |
HFP
Reaction | Reaction | MCNP-2.7e | SCALE-6.1 | Ratio | ||
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0.22281 | 0.23783 | 0.937 |
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0.17426 | 0.17639 | 0.988 |
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0.16419 | 0.16909 | 0.971 |
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0.10985 | 0.11143 | 0.986 |
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0.10703 | 0.10925 | 0.980 |
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0.07689 | 0.07969 | 0.965 |
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0.06925 | 0.07100 | 0.975 |
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0.03707 | 0.03690 | 1.005 |
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0.02313 | 0.02310 | 1.001 |
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0.00770 | 0.00825 | 0.934 |
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0.00762 | 0.00748 | 1.019 |
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0.00569 | 0.00592 | 0.961 |
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0.00341 | 0.00356 | 0.958 |
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0.00166 | 0.00159 | 1.048 |
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Total | 0.37973 | 0.39380 | 0.964 | |||
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— | 0.25451 | |
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— | 0.13788 | |
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Total SCALE | 0.49966 | 0.788 |
There is a good agreement between MCNPX-2.7e and SCALE-6.1 results except when the
However, the
It is necessary to implement more PERT cards in MCNPX-2.7e, because only with the ones calculated, the total uncertainty in the
The differences between HZP and HFP are explained later through the comparison of the sensitivity profiles, because the variance-covariance matrices used in the “sandwich formula” do not change between cases.
The sensitivity profiles of 238U reaction cross sections calculated by MCNPX-2.7e and SCALE-6.1 are presented in Figures
Sensitivity profiles of 238U reaction cross sections calculated by MCNPX-2.7e and SCALE-6.1 for unrodded case at HZP.
Sensitivity profiles of 238U reaction cross sections calculated by MCNPX-2.7e and SCALE-6.1 for rodded case at HZP.
Sensitivity profiles of
Unrodded
Rodded
There is good agreement between all reaction cross sections except for
The temperature effect on the sensitivity profiles is analysed. Only for the unrodded case there is a noticeable change in the 238U
The effect of the control rods can be observed comparing Figures
Another source of difference, apart from the methodology used by each code to calculate the sensitivity coefficients, is that SCALE-6.1 provides the sensitivity profiles in 238 energy groups, while for MCNPX-2.7e the sensitivity profiles are calculated in 44 energy groups. Therefore, the energy group structure could lead to differences in between both codes.
SCALE-6.1 provides a useful result: the integrated sensitivity coefficients of every reaction involved in the criticality calculations. It is the sum of the sensitivity coefficients over all energy groups of the same reaction. This value reflects how sensitive is the
With the sensitivity profiles provided by MCNPX-2.7e, the integrated sensitivity values can be calculated. Because comparing the integrated sensitivity coefficients is equivalent to compare sensitivity profiles, only the SCALE-6.1 results are shown in this section.
Table
Largest integrated sensitivity coefficients of the reaction cross sections for the unrodded case at HZP and HFP sorted in descending order provided by SCALE-6.1.
HZP
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HFP
Reaction | Int.sen.coef. | |
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Largest integrated sensitivity coefficients of the reaction cross sections for the rodded case at HZP and HFP sorted in descending order provided by SCALE-6.1.
Rodded at HZP
Reaction | Int.sen.coef. | |
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Rodded at HFP
Reaction | Int.sen.coef. | |
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The propagation of cross section uncertainties in criticality calculations for a
The two approaches have been presented and compared through this exercise. The
In general, both approaches/codes are in good agreement, in spite of the differences in the
The inability of calculating the contribution due to
Thus, MCNPX-2.7e can deal with the uncertainty quantification problem as SCALE-6.1 does, but improvements should be done in the PERT card capabilities such as the proper calculation of the sensitivity coefficients of
The research leading to these results has received funding from specific collaborative agreement P110530207 between CSN and UPM in the area of “uncertainty propagation in nuclear criticality safety.” Also, this work is also partially supported by the Spanish Ministry of Education of Spain through the FPU Program for teaching and researching formation under Grant AP2009-1801 for the first author.