Analytical benchmark for non-equilibrium radiation diffusion in finite size systems

TL;DR

This paper presents an analytical solution for non-equilibrium radiation diffusion in finite-sized planar and spherical systems, providing a benchmark for validating computational models in various applications.

Contribution

It introduces an analytical solution using Laplace transforms for non-equilibrium radiation diffusion in finite geometries, aiding code validation.

Findings

Analytical solutions match finite difference results.

Steady state energy densities are linear in slabs.

Non-linear dependence observed in spherical shells.

Abstract

Non-equilibrium radiation diffusion is an important mechanism of energy transport in Inertial Confinement Fusion, astrophysical plasmas, furnaces and heat exchangers. In this paper, an analytical solution to the non-equilibrium Marshak diffusion problem in a planar slab and spherical shell of finite thickness is presented. Using Laplace transform method, the radiation and material energy densities are obtained as a function of space and time. The variation in integrated energy densities and leakage currents are also studied. In order to linearize the radiation transport and material energy equation, the heat capacity is assumed to be proportional to the cube of the material temperature. The steady state energy densities show linear variation along the depth of the planar slab, whereas non-linear dependence is observed for the spherical shell. The analytical energy densities show good agreement with those obtained from finite difference method using small mesh width and time step. The benchmark results obtained in this work can be used to validate and verify non equilibrium radiation diffusion computer codes in both planar and spherical geometry.