Effective optical depth as a means of simplifying radiation-conduction analysis

Abstract

Paper presented at the 5th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, South Africa, 1-4 July, 2007.Analysis of any heat transfer problem that includes thermal radiation with absorption and emission is always complicated by the “action-at-a-distance” nature of thermal radiation. This implies that the entire temperature field has to be determined simultaneously rather than focusing on a single location and the immediate neighborhood about that location as in a conduction-only problem. This paper introduces the concept of the effective optical depth (EOD) which limits the range about a location over which a solution is conducted rather than solving the entire temperature field at once. In this study, we investigate the error introduced in the solution of a planar, gray, radiation-conduction heat transfer problem with black and grey boundaries for a range of EOD/optical thickness ratio from 0.01 to 10 and conduction-radiation parameter from 0.01 to 10. In general, the accuracy of the predicted temperature profiles and total heat flux was within a few percentage points and was observed to be as high as 10% for small EOD/optical thickness ratios. Computational times using the finite element method were estimated to be 12.5% or less using the EOD as compared to solving the entire temperature field for each element. This savings in computational time may justify the small errors introduced by using the EOD approximation.