Unstructured Grids and the Multigroup Neutron Diffusion Equation

Abstract

The neutron diffusion equation is often used to perform core-level neutronic calculations. It consists of a set of second-order partial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved by discretizing the neutron leakage term using a structured grid. This work introduces the alternatives that unstructured grids can provide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities they offer. It is by understanding the basic mathematics that lie beneath the equations that model real physical systems; better technical decisions can be made. It is in this spirit that this paper is written, giving a first introduction to the basic concepts which can be incorporated into core-level neutron flux computations. A simple two-dimensional homogeneous circular reactor is solved using a coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements method. Also, the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries, meshing algorithms, basic geometric entities, discretization schemes, and characteristic grid lengths, giving even more insight into the peculiarities that arise when solving the neutron diffusion equation using unstructured grids.

Assembly Discontinuity Factors for the Neutron Diffusion Equation discretized with the Finite Volume Method. Application to BWR

“…conditions most commonly used are the zero flux and the reflective flux, which are shown in Equations (6,7), respectively. The neutron current continuity is shown in Equation 8, for the adjacent cells i and l. The heterogeneous neutron flux condition is calculated in Equation (9), for the adjacent cells i and l, by using the ADFs definition, which is expressed in Equation (10).…”

“…In contrast, the Finite Volume Method (FVM) is easily applied to unstructured meshes. In addition, the application of the FVM to the NDE is feasible, as discussed by Bernal et al [5,6] and Theler [7].…”

Response Surface Methodology Control Rod Position Optimization of a Pressurized Water Reactor Core Considering Both High Safety and Low Energy Dissipation

“…For simplification, the Neutron Diffusion Equation (NDE) is also used. These equations can be solved by the Finite Different Method (FDM) [5][6][7], the Finite Volume Method (FVM) [8][9][10][11], the Finite Element Method (FEM) [10,[12][13][14], etc. Normally, the neutron flux density distribution matches the fission heat source distribution.…”

Development of a 3D Modal Neutron Code with the Finite Volume Method for the Diffusion and Discrete Ordinates Transport Equations. Application to Nuclear Safety Analyses

“…In the last years, a number of works has been published, which use Krylov-Schur methods to calculate multiple eigenvalues of the λ-eigenvalue problem, of the Neutron Diffusion Equation (Bernal et al 2017a, Vidal-Ferrandiz et al 2014, Bernal et al 2018, Theler 2013, and Carreño et al 2017. Actually, one can find a great analysis of the application of Krylov-Schur methods for the calculation of different kinds of eigenvalue problems of the Neutron Diffusion Equation in Carreño et al 2017.…”