An approximate method for solving radiation and neutron transport problems in spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. This is done by assuming, a priori, a functional form for the stochastic flux in terms of the members of a random set function. For the case of a two-phase medium, two coupled integro-differential equations are obtained for the deterministic functions that arise and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those of the Levermore–Pomraning (LP) theory, but they offer an opportunity to deal with more general forms of stochastic processes. It is also shown that the coupling coefficient between the phase equations is directly proportional to the gradient of the autocorrelation function evaluated at the origin; a feature which has been noted in other fields in which random media occur. By making plausible assumptions about the functional form of the autocorrelation function, different forms of the transport equations can be obtained, according to the structure of the medium. For the one-dimensional case, we may show an exact correspondence with the LP equations. Discussions are given regarding the application of the method to three-dimensional problems for which we expect it to be a good approximation for the mean. We also note that the equations are applicable to realistic problems, such as grains embedded in a background matrix, and not restricted to slabs. Investigations into the variance have also been made and a simple approximation scheme developed which gives reasonable agreement with the simulation results of Adams et al. [Adams, M.L., Larsen, E.W., Pomraning, G.C., 1989. Benchmark results for particle transport in a binary Markov statistical medium. Journal of Quantitative Spectroscopy & Radiative Transfer, 42, 253].