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

Abstract

a b s t r a c tNon-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 functions 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 and the opacity to be independent of 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.

Asymptotic-Preserving Discrete Schemes for Non-Equilibrium Radiation Diffusion Problem in Spherical and Cylindrical Symmetrical Geometries

“…In the studies on linear diffusion and transport problems [23][24][25][26] or simpler nonlinear heat conduction problems [27], one can use classical strategy such as Laplace (Fourier) transformation for temporal (spatial) variant, or adopt some special function to obtain exact solution. But for scaled non-equilibrium radiation diffusion problems, it is rather difficult to construct exact solutions due to the strong nonlinearity and the scaling parameter.…”