Numerical solution of Rosseland model for transient thermal radiation in non-grey optically thick media using enriched basis functions

Abstract

Heat radiation in optically thick non-grey media can be well approximated with the Rosseland model which is a class of nonlinear diffusion equations with convective boundary conditions. The optical spectrum is divided into a set of finite bands with constant absorption coefficients but with variable Planckian diffusion coefficients. This simplification reduces the computational costs significantly compared to solving a full radiative heat transfer model. Therefore, the model is very popular for industrial and engineering applications. However, the opaque nature of the media often results in thermal boundary layers that requires highly refined meshes, to be recovered numerically. Such meshes can significantly hinder the performance of numerical methods. In this work we explore for the first time using enriched basis functions for the model in order to avoid using refined meshes. In particular, we discuss the finite element method when using basis functions enriched with a combination of exponential and hyperbolic functions. We show that the enrichment can resolve thermal boundary layers on coarse meshes and with few elements. Comparisons to the standard finite element method for thermal radiation in non-grey optically thick media with multi-frequency bands show the efficiency of the approach. Although we mainly study the enriched basis functions in glass cooling applications the substantial saving in the computational requirements makes the approach highly relevant to a large number of engineering applications that involve solving the Rosseland model.