In the Monte Carlo (MC) burnup analyses, the uncertainty of a tally estimate at a burnup step may be induced from four sources: the statistical uncertainty caused by a finite number of simulations, the nuclear covariance data, uncertainties of number densities, and cross-correlations between the nuclear data and the number densities. In this paper, the uncertainties of
Monte Carlo (MC) burnup analysis codes [
The uncertainty quantification of a nuclear parameter, such as
The direct stochastic sampling methods can produce output distributions from a number of MC calculations each with different input data set sampled. This approach is easy to implement by running existing MC neutronics analysis codes with different input data sets but at the expense of high computational costs. This approach includes the XSUSA/SCALE [
In this paper, we perform the McCARD uncertainty propagation analysis for a PWR burnup pin-cell benchmark, one of the OECD benchmarks for uncertainty analysis modeling (UAM) for design, operation, and safety analysis of LWRs [
The MC depletion calculations consist of the successive MC transport analyses with updating the material compositions. Microscopic reaction rates are estimated at every beginning of a burnup step by the MC transport calculations. They are then used to solve the depletion equation to update isotopic number densities at the end of the burnup step. Thus, the uncertainties of the MC estimates on reaction rates due to the statistical and nuclear data and number density uncertainties cause those of the updated number densities. With the progress of the stepwise MC burnup calculations, the MC reaction rate uncertainties of a burnup step propagate to the number density uncertainties of the burnup step and to those of the following burnup steps. Figure
Uncertainty propagation in the MC burnup analysis.
In the McCARD uncertainty propagation formulation [
In the McCARD uncertainty propagation analysis, the partial derivatives in (
In exactly the same way as above for the variance of
The PWR burnup pin-cell benchmark problem in Phase I of the OECD LWR UAM benchmarks [
The McCARD analyses are conducted with the continuous-energy cross-section libraries processed by NJOY [
For the fresh burnup state of the TMI-1 pin-cell problem, the
Table
Comparison of
Covariance data | ENDF/B-VII.1 |
SCALE6.1/ | |
---|---|---|---|
RSD due to 235U (%) |
|
0.604 | 0.264 |
( |
0.216 | 0.211 | |
( |
0.075 | 0.076 | |
( |
0.081 | 0.075 | |
( |
0.001 | 0.002 | |
| |||
RSD due to 238U (%) |
|
0.071 | 0.070 |
( |
0.294 | 0.263 | |
( |
0.016 | 0.015 | |
( |
0.104 | 0.105 | |
| |||
Total | 0.729 | 0.463 |
The MC burnup uncertainty propagation analyses are conducted by using the covariance data of 10 isotopes—235U, 238U, 239Pu, 240Pu, 241Pu, 242Pu, 241Am, 242mAm, 243Am, and 244Cm. The McCARD eigenvalue calculations are performed on 100 active cycles with 10,000 histories per cycle. Table
Burnup (MWd/kgU) |
|
RSD (%) | |
---|---|---|---|
ENDF/B-VII.1 44G Cov. | SCALE6.1/ | ||
0.00 | 1.41701 | 0.731 | 0.469 |
0.10 | 1.39073 | 0.729 | 0.475 |
0.20 | 1.38327 | 0.726 | 0.465 |
0.50 | 1.36993 | 0.718 | 0.462 |
1.00 | 1.35687 | 0.714 | 0.455 |
2.00 | 1.34292 | 0.697 | 0.455 |
4.00 | 1.31499 | 0.666 | 0.446 |
6.00 | 1.28805 | 0.637 | 0.444 |
8.00 | 1.26320 | 0.608 | 0.448 |
10.00 | 1.23924 | 0.588 | 0.452 |
12.00 | 1.21683 | 0.569 | 0.464 |
14.00 | 1.19584 | 0.557 | 0.465 |
16.00 | 1.17646 | 0.532 | 0.478 |
18.00 | 1.15745 | 0.517 | 0.488 |
20.00 | 1.13972 | 0.499 | 0.492 |
30.00 | 1.05605 | 0.449 | 0.530 |
40.00 | 0.98051 | 0.397 | 0.579 |
50.00 | 0.91289 | 0.403 | 0.633 |
60.00 | 0.85671 | 0.411 | 0.682 |
Table
RSDs (%) of one-group reaction rates with ENDF/B-VII.1 and SCALE6.1/COVA covariance data
Cov. data | Burnup (MWd/kgU) |
235U ( |
235U ( |
238U ( |
238U ( |
---|---|---|---|---|---|
ENDF/B-VII.1 | 0 | 1.35 | 0.53 | 0.94 | 3.97 |
10 | 1.45 | 0.69 | 0.87 | 3.97 | |
20 | 1.55 | 0.88 | 0.83 | 3.85 | |
30 | 1.62 | 1.06 | 0.79 | 3.88 | |
40 | 1.73 | 1.28 | 0.76 | 3.79 | |
50 | 1.79 | 1.43 | 0.67 | 3.80 | |
60 | 1.88 | 1.60 | 0.70 | 3.71 | |
| |||||
SCALE 6.1 |
0 | 1.38 | 0.51 | 0.82 | 3.77 |
10 | 1.46 | 0.61 | 0.81 | 3.89 | |
20 | 1.55 | 0.79 | 0.75 | 3.85 | |
30 | 1.58 | 0.90 | 0.73 | 3.77 | |
40 | 1.63 | 1.04 | 0.68 | 3.83 | |
50 | 1.69 | 1.18 | 0.64 | 3.81 | |
60 | 1.75 | 1.29 | 0.66 | 3.79 |
RSDs (%) of number densities with ENDF/B-VII.1 and SCALE6.1/COVA covariance data
Cov. data | Bunrup or time | 235U | 239Pu | 240Pu | 241Pu | 242Pu |
---|---|---|---|---|---|---|
ENDF/B-VII.1 | 0 MWd/kgU | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
10 MWd/kgU | 0.14 | 0.99 | 1.37 | 1.38 | 2.72 | |
30 MWd/kgU | 0.58 | 1.47 | 1.78 | 1.31 | 2.47 | |
50 MWd/kgU | 1.33 | 1.95 | 2.22 | 1.77 | 2.51 | |
Shutdown | 1.95 | 2.16 | 2.47 | 2.06 | 2.60 | |
1 year cooling | 1.95 | 2.13 | 2.47 | 2.16 | 2.60 | |
100 years cooling | 1.95 | 2.13 | 2.34 | 2.06 | 2.60 | |
| ||||||
SCALE 6.1 |
0 MWd/kgU | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
10 MWd/kgU | 0.09 | 0.80 | 1.17 | 1.16 | 1.57 | |
30 MWd/kgU | 0.43 | 1.11 | 1.46 | 0.97 | 0.42 | |
50 MWd/kgU | 1.04 | 1.43 | 1.80 | 1.25 | 0.17 | |
Shutdown | 1.53 | 1.58 | 2.00 | 1.47 | 0.23 | |
1 year cooling | 1.53 | 1.56 | 1.99 | 1.54 | 0.23 | |
100 years cooling | 1.52 | 1.56 | 1.90 | 1.47 | 0.23 |
The McCARD uncertainty propagation analyses with different covariance data files have been performed for the TMI-1 burnup pin-cell problem in Phase I of the OECD LWR UAM benchmarks. The numerical results show that the uncertainty behavior over burnup strongly depends on the nuclear covariance data.
This work was performed through the contract with Korea Institute of Nuclear Safety as part of its research project “Development of Licensing Technologies for Very High Temperature Reactor” which is funded by Ministry of Education and Science of Korea.