1. Comparison of Sensitivity Analysis Techniques in Monte Carlo Codes for Multi-Region Criticality Calculations Bradley T. Rearden Douglas E. Peplow Oak Ridge National Laboratory

2. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 2 Sensitivity and Uncertainty Analysis Techniques New techniques recently developed at ORNL to determine applicability of critical experiments for criticality code validation. Use energy-dependent sensitivity coefficients and cross section covariance data to assess similarity of systems.

3. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 3 Sensitivity Methods in Monte Carlo Codes Differential Sampling –Calculates the derivatives of the response with respect to some parameter at the same time the response is calculated. –First, second, third, etc. and cross derivatives can be computed. –Available in MCNP and can be used for fixed source or criticality problems. –MCNP manual cautions: “The track length estimate of k eff in KCODE criticality calculations assumes the fundamental eigenvector (fission distribution) is unchanged in the perturbed configuration.” –This means that the source dependence on the parameter of interest is not taken into account by the differential sampling, which could lead to incorrect values of the derivatives if that source dependence is large.

4. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 4 Sensitivity Methods in Monte Carlo Codes Perturbation Theory –Independent forward and adjoint calculations –Group-wise fluxes are computed and folded with cross- section data to produce sensitivity coefficients. –Only valid for small perturbations with linear response. –Employed inSEN3 analysis sequence using KENO V.a in SCALE.

5. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 5 GODIVA Sphere 8.7414 cm bare HEU metal 18.74 g/cm 3 93.71% 235 U, and 5.27% 238 U, 1.02% 234 U ENDF/B-V data were used in both codes to ensure consistency

6. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 6 Sensitivity Calculations Sensitivity of k eff to fuel density was investigated. –Over entire system –Only in outer 2-cm thick shell Code generated sensitivity coefficients Direct Recalculation +/- 10% in MCNP +/- 5% in KENO V.a

7. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 7 Sensitivity Results Reasonable results for both codes for whole system. MCNP over-predicts outer shell by 40%.

8. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 8

9. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 9

10. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 10 Explanation Differential Sampling –When density of whole system is perturbed, uniform change in source distribution. –Only outer 2-cm thick shell is perturbed, non-uniform change in source distribution. Perturbation Theory –Forward fluxes are weighted with adjoint flux or “importance function”

11. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 11 Possible Solution for Differential Operator Fixed source problem where source is function of material density, accumulator stores sum of the relative derivatives of interaction probabilities, P ij To include effect of source that is proportional to density (S=c  ), start accumulator with derivative of source with respect to density

12. UT-BATTELLE U.S. Department of Energy Oak Ridge National Laboratory 12 Application to Criticality Problems Must be extended from generation to generation. Effect of each energy-dependent cross-section of interest would have to be taken into account. Would provide powerful tool for analysts with ability to predict non-linear response to perturbations. For now, perturbation theory provides adequate results for criticality applications.