Derivation of Analytic Solution and MOC Calculation Procedure for Double Heterogeneity Treatment

Abstract

The Sanchez-Pomraning method to resolve the double heterogeneity problem in the MOC transport calculation is described in detail. This method is founded on the collision probability method concepts which involves the collision and escape probabilities, and formally derived by using the statistical treatment of the neutron balance equation along a path. The statistical approach bring the concept of the chord and segment length distributions in the formulation of the integral equation for the grain surface and matrix fluxes, which later turns into collision and escape probabilities. For the analytic solution, a boundary layer of a grain thickness, in which no grain is present, is assumed within each flat source region. The analytic solution of the coupled integral equation which involves the convolution integral is derived first by Laplace transform, but finally by substitution. This solution introduces an effective cross section which represents the homogenized mixture of the matrix and grain. With observation that the resulting analytic solution for the matrix is the same as the MOC solution for the homogenized medium, the equivalent source is constructed so that the MOC calculation can be performed for the homogenized mixture. The assumption of the boundary layer causes, however, a problem in the neutron conservation which should be corrected by renormalization. This method requires very little modifications to the existing MOC code to implement the double heterogeneity treatment. Starting from the very basic collision probability relation, the exhaustive derivation and explanation of the all the solution and terms needed to establish the MOC calculation sequence with the double heterogeneity treatment are provided for complete understanding of the reader who might not have sufficient background on this subject.