Exact Solutions of the Two-Dimensional Cell Problem

Abstract

A method has been developed that provides analytic solutions for two-dimensional cell problems for the neutron transport equation. This is made possible by assuming an infinite, repeating lattice of rectangular regions. The solution is effected by means of a finite Fourier transform, the periodicity of which is related to the size of the unit cell. In order to drive the flux, we assume that the cell is composed of two regions: an inner circular region and the remaining exterior part. Different sources are placed in each region thereby leading to a situation rather like the conventional reactor cell problem but with no spatial variation of the cross sections. The method is illustrated by two examples: the Levermore-Pomraning equations and the two-group equations. In the former case, we have obtained the stochastically averaged flux within the cell and also the Pomraning χ-function. In addition, we have calculated the ratio of the spatially averaged flux in the outer region to that in the inner circular region, i.e., the disadvantage factor. Fluxes and disadvantage factors are also obtained for the two-group equations, and the rate of convergence is shown. These results are exact transport theory solutions and are offered as benchmarks for checking transport theory codes. The calculations are also repeated using diffusion theory. The SPN method, which we show to be exact for our problem, is used to demonstrate the rate of convergence of the PN method for two-dimensional cell problems.