Solution of neutron diffusion equation in 2d polar (r,θ) coordinates using Nodal Integral Method

The nodal methods are significantly more accurate than the traditional methods such as finite difference method (FDM), finite element method (FEM) etc. However, these methods can be used only for the nodes of only a few limited shapes such as rectangular (in 2D) or cuboidal (in 3D). In this paper the approach for solving neutron diffusion equation in 2-d cylindrical polar geometry (r,θ) using nodal method is discussed. The analytic nodal method using transverse integration process is used to solve the 2-d neutron diffusion equation in polar coordinate. The problem involved with the transverse integration in θ-direction has been resolved by approximating the average of products by product of averages. This approximation leads to three different formulations of the scheme. A detailed study of the numerical error for source problems having analytical solutions is carried out. Using this analysis it has been shown that the error is second order for these problems irrespective of the formulation used. In addition to that the methodology is used to solve criticality problems for which analytical or benchmark solutions are available. The solution of the source problem shows that method maintains its accuracy and order even with approximations made to deal with transverse integration in θ-direction. The comparison of eigenvalues obtained with current methodology with those obtained analytically or available as benchmark show that the method is capable of accurately predicting the eigenvalues.