A Robust Second-Order Multiple Balance Method and Alpha-Weighted Multigroup Constants for Time-Dependent Nuclear Reactor Simulations

Abstract

Time-dependent nuclear reactor simulations are essential in improving the safety, effectiveness, and efficiency of nuclear reactor designs, experiments, and operations. This thesis proposes, implements, and tests two new methods designed to improve two different aspects of time-dependent reactor simulation: (1) Multiple Balance Time-Discretization (MBTD), a robust second-order accurate time-stepping method, an alternative to the highly reliable Backward Euler (BE); and (2) α-Weighted Multigroup Constants (α-MGXS), an alternative formulation of multigroup constants that offers advantages over the traditionally-used k-Weighted Multigroup Constants (k-MGXS) for time-dependent neutron transport simulations.

Despite being only first-order accurate, BE has been the primary time-discretization method in reactor simulations due to its simplicity and robustness (unconditionally stable and free of spurious oscillations). The Multiple Balance method [Morel & Larsen 1990] was originally introduced as a spatial discretization for neutron transport methods. We show that its application to time-discretization (MBTD) yields a method that is not only robust like BE but also second-order accurate. MBTD consists of solving two coupled balance equations at each time step. In this thesis, three general strategies for solving these coupled equations are explored. MBTD adaptations are made for (1) the finite difference method (FDM) applied to the neutron diffusion equation and for (2) several techniques for the neutron transport equation, including Source Iteration (SI), applied to the Diamond-Difference (SN-DD) and Method of Characteristics (MOC). By exploiting the results of Fourier convergence analysis, an effective Diffusion Synthetic Acceleration (DSA) method for MBTD-SI is developed. Four representative kinetic problems are devised to test and assess the relative efficiency of MBTD versus BE. It is found that MBTD is about 2, 2.5, and 3 times computationally more expensive than BE for neutron diffusion with FDM, neutron transport DSA with SN-DD, and MOC, respectively, given the same uniform time-step size. However, due to its higher-order accuracy, MBTD is generally more efficient than BE: a larger time step can be used to achieve a certain accuracy. Finally, a similar trend is observed in a neutronics/thermal-hydraulics tight-coupling multi-physics application, where MBTD is more efficient than BE for reasonably accurate simulations (relative error less than ~10%).

Multigroup neutron transport methods remain as essential tools for reactor simulations, but their accuracy can only be as good as their multigroup constants (MGXS). Estimation of MGXS is traditionally based on the solution of the k-eigenvalue neutron transport calculation. However, the k-eigenfunction is not physically representative for systems that are far from critical, which is the case in many reactor transient simulations. Representing the asymptotic behavior of time-dependent transport problems, the α-eigenfunction may be a better alternative for the calculation of MGXS. In this thesis, physics-preserving MGXS for time-stepping methods are derived. A review of α-eigenvalue iteration methods is presented. A relaxed α-k Iteration developed to simulate the fundamental α-mode is implemented in the open-source Monte Carlo code OpenMC and verified with several benchmark problems. Results from four kinetics problems simulating absorber injection and removal to initially-critical infinite-medium fast and thermal systems emphasize that the fundamental α-eigenfunction—as a multigroup constant weighting spectrum—offers physical characteristics that make it advantageous (in producing accurate solutions) over the typically used fundamental k-eigenfunction.