David P. Griesheimer

Numerical Methods in Coupled Monte Carlo and Thermal-Hydraulic Calculations

Large-scale reactor calculations with Monte Carlo (MC), including nonlinear feedback effects, have become a reality in the course of the last decade. In particular, implementations of coupled MC and thermal-hydraulic (T-H) calculations have been separately developed by many different groups. Numerous MC codes have been coupled to a variety of T-H codes (system level, subchannel, and computational fluid dynamics). In this work we review the numerical methods that have been used to solve the coupled MC–T-H problem with a particular focus on the formulation of the nonlinear problem, convergence criteria, and relaxation schemes used to ensure stability of the iterative process. We use a simple pressurized water reactor pin cell problem to numerically investigate the stability of commonly used schemes and which problem parameters influence the stability—or lack thereof. We also examine the role that the running strategy used in the MC calculation plays in the convergence of the coupled calculation. Results indicate that the instability in fixed-point iterations is driven by the Doppler feedback effect and that underrelaxation can be used to restore stability. We also observed that a form of underrelaxation could be achieved by performing the coupled iterations without converging the MC fission source each iteration. By performing many iterations of few histories, we observed rapid convergence to the coupled MC–T-H solution in a relatively small number of batches. Numerical results also showed that the presence of instability in the fixed-point iteration is independent of the stochastic noise in the MC simulation.

Reactivity Effects of Spatial Homogenization of Thermal Feedback Regions in Monte Carlo Reactor Calculations

An integrated thermal-hydraulic feedback module has previously been developed for the Monte Carlo transport solver MC21. The module incorporates a flexible input format that allows the user to describe heat transfer and coolant flow paths within the geometric model at any level of spatial detail desired. The effect that the varying levels of spatial homogenization of thermal regions has on the accuracy of the Monte Carlo simulations is examined in this study. Six thermal feedback mappings are constructed from the same geometric model of the Calvert Cliffs core. The spatial homogenization of the thermal regions is varied, giving each scheme a different level of detail, and the adequacy of the spatial homogenization is determined based on the eigenvalue produced by each Monte Carlo calculation. The purpose of these numerical experiments is to determine the level of detail necessary to accurately capture the thermal feedback effect on reactivity. Several different core models are considered: axial flow only, axial and lateral flow, asymmetry due to control rod insertion, and fuel heating (temperature-dependent cross sections). The thermal results generated by the MC21 thermal feedback module are consistent with expectations. Based on the numerical experiments conducted, it is concluded that the amount of spatial detail necessary to accurately capture the feedback effect on reactivity is relatively small. Homogenization at the assembly level for the Calvert Cliffs pressurized water reactor model results in a power defect similar to that calculated with individual pin cells modeled as explicit thermal regions.