Analytical Discrete Ordinates Solution for a 1D Model of Particle Transport in Ducts that Includes Wall Migration

The analytical discrete ordinates (ADO) method is used to develop a solution to a one-dimensional model of particle transport in ducts that includes wall migration. Particle reemission from the wall is described by a nonlocal, exponential displacement kernel. Since the governing transport equation of the model is not directly amenable to a solution by the ADO method, an alternative transport equation is derived first. For an approximation based on a half-range quadrature of order , the ADO solution of the alternative equation becomes available once techniques of linear algebra are used to solve a quadratic eigenproblem of order for the eigenvalues and eigenvectors. The solution is expressed as a superposition of 4N modes, which are constructed from 2N positive/negative pairs of separation constants (the reciprocals of the square roots of the eigenvalues) and associated eigenvectors. Compatibility conditions that the solution must satisfy in order to also solve the governing equation of the model result in a reduction of the number of relevant modes to 2N + 2, just two in excess of the number of modes in the solution of the problem without wall migration. Highly accurate numerical results for the reflection and transmission probabilities are reported for isotropic and monodirectional incidence.