The Analytical Discrete Ordinates Method for a One-Dimensional Model of Neutral Particle Transport in Ducts

The analytical discrete ordinates (ADO) method is used to develop an approximate, but accurate, solution to a one-dimensional model of neutral particle transport in ducts proposed originally by Prinja and Pomraning. The implementation of the ADO method is facilitated by a variable transformation that is used to rewrite the Prinja-Pomraning equation in a form very similar to that of the Bhatnagar-Gross-Krook model equation in rarefied gas dynamics. Techniques of linear algebra are used to find an analytical solution for the linear system that has to be solved for the superposition coefficients of the ADO method in the case of a semi-infinite duct. Numerical results for the reflection and transmission probabilities that illustrate the capability of the method are tabulated for semi-infinite and finite ducts of circular cross section and two types of particle incidence: isotropic incidence and incidence described by the Dirac delta distribution. It is concluded that the ADO method can achieve a desired precision in the reflection and transmission probabilities with a much lower quadrature order than previously used numerical implementations of the discrete ordinates method and consequently is much more efficient.