Precise Derivations of Radiative Properties of Porous Media Using Renewal Theory

Abstract: This work uses the mathematical machinery of Renewal/Ruin (surplus risk) theory to derive preliminary explicit estimations for the radiative properties of dilute and disperse porous media otherwise only computable accurately with Monte Carlo Ray Tracing (MCRT) simulations. Although random walk and Levy processes have been extensively used for modeling diffuse processes in various transport problems and porous media modeling, relevance to radiation heat transfer is scarce, as opposed to other problems such as probe diffusion and permeability modeling. Furthermore, closed form derivations that lead to tangible variance reduction in MCRT are widely missing. The particular angle of surplus risk theory provides a richer apparatus to derive directly related quantities. To the best of the authors' knowledge, the current work is the only work relating the surplus risk theory derivations to explicit computations of ray tracing results in porous media. The paper contains mathematical derivations of the radiation heat transfer estimates using the extracted machinery along with proofs and numerical validation using MCRT.