Resolution of the Generalized Eigenvalue Problem in the Neutron Diffusion Equation Discretized by the Finite Volume Method

Álvaro Bernal,Damián Ginestar,Gumersindo Verdú,Rafael Miró

doi:10.1155/2014/913043

Álvaro Bernal, Damián Ginestar + Show 2 more

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https://doi.org/10.1155/2014/913043

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Abstract

Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Element Method (FEM) and the Finite Volume Method (FVM) are easily applied to unstructured meshes. On the one hand, the FEM can be accurate for smoothly varying functions. On the other hand, the FVM is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm implemented in the ARB Partial Differential Equations solver has been used to discretize the neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library.

Highlights

The neutron diffusion equation is used to calculate the neutron flux distribution, which is one of the most important variables in a Nuclear Power Reactor (NPR)
The use of the neutron diffusion equation is justified by the lower computational time and relatively low heterogeneity of commercial NPR
The homogeneous reactors check the discretization of the equations without taking into account the use of the global diffusion coefficient and the heterogeneous reactors check the different approaches of the global diffusion coefficient developed in this study

Summary

Introduction

The neutron diffusion equation is used to calculate the neutron flux distribution, which is one of the most important variables in a Nuclear Power Reactor (NPR). This equation is a simplification of the neutron transport equation using Fick’s Law, as discussed by Stacey [1]. The calculation of several eigenvalues and eigenvectors is important for different applications as the modal analysis of nuclear reactors and BWR instabilities analysis, as discussed elsewhere [2, 3]

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