An evaluation of approximate methods for calculating anisotropic flux escape and transmission probabilities for cylindrical annular regions

Abstract

In the interface current method for solving the integral transport equation, most of the computer time is spent in the precalculation of various region transmission, escape and collision probabilities. Therefore, any reduction in computer time used to calculate these probabilities will directly affect the efficiency of this method. The normal methods such as the Trapezoidal rule or Gauss-quadrature formula for calculating the various probabilities require the evaluation of many Bickley functions per integral, which is time consuming. This paper discusses efficient methods for calculating the various probabilities corresponding to anisotropic terms of the angular flux expansion at region interfaces for cylindrical annular geometry. The paper first discusses the application of Bonalumi's Pseudo-Linear (P-L) approximation and the second-order correction to it which requires the evaluation of one and two Bickley functions, respectively, per integral. The accuracy of the second-order approximation is of the order of 0.1% for all the probabilities except the transmission probability from outer to outer surface ( P ρv ∞) of the region (where ρ and v denote the order of anisotropy of the flux entering and leaving the outer boundary of the annular region). The paper then discusses a Gauss-quadrature formula with exponential weighting and its modified form which is suitable for calculating P ρv ∞ efficiently.