Benchmark solutions of radiative integral transfer equations for three-dimensional emitting–absorbing and scattering media bounded by gray walls

Abstract

Three-dimensional dimensionless radiative integral transfer equations (RITEs) for a cubic emitting–absorbing and isotropically scattering homogeneous medium of constant properties bounded by gray walls are solved using the method of “product integration”. The resultant system of linear equations for the incident energy is solved iteratively. Evaluation of the accuracy of the numerical solution is achieved by using the computer codes to make predictions for an idealized enclosure in which the exact analytical solution is possible. Comparison of the analytical and numerical values of medium temperatures and heat fluxes has shown that our method is logically correct and has good accuracy. Four benchmark problems for participating medium subjected to various combinations of internally uniform/non-uniform source terms are solved. To make the benchmark problems more complex and more general, different scattering albedo (ω=0.1, 0.5, 0.9), different wall emissivity (ε=0.1, 0.5, 0.9) and different optical thickness (τ=0.1, 1.0, 5.0) are considered. The solutions for the temperature of the medium and the heat flux components are given in tabular form and they may serve as standard values to assess the accuracy of other methods more conveniently.