Nonclassical Particle Transport in 1-D Random Periodic Media

We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path-length $s$), and models particle transport taking place in homogenized random media in which a particle's distance-to-collision is not exponentially distributed. To solve the nonclassical equation one needs to know the $s$-dependent ensemble-averaged total cross section, $\Sigma_t(\mu,s)$, or its corresponding path-length distribution function, $p(\mu,s)$. We consider a 1-D spatially periodic system consisting of alternating solid and void layers, randomly placed in the $x$-axis. We obtain an analytical expression for $p(\mu,s)$ and use this result to compute the corresponding $\Sigma_t(\mu,s)$. Then, we proceed to numerically solve the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions $\mu=\pm 1$. To assess the accuracy of these solutions, we produce "benchmark" results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely-used atomic mix model in most problems.