Convergence characteristics of temperature in radiation problems

Abstract

Numerical convergence of temperature in radiation problems with absorbing, emitting,and nonscattering media is examined. Iteration processes for the solution of blackbody intensity (or temperature) when using the pointwise or the control-volume energy equation are scrutinized and a sufficient condition for convergence is presented. An explicit and a few implicit methods for prescribed heat source distribution in one-, two-, and three-dimensional enclosures are devised. They are incorporated and tested with the discrete ordinates method, the discrete ordinates interpolation method [1], and the zone method, and comparison of the number of iterations for convergence as well as the solution accuracy is made. The implicit methods significantly reduce the number of iterations, and this is prominent for optically thick cases. However, they reduce the number of iterations marginally as the number of control volumes increases. A fully implicit scheme reduces the number of iterations more effectively than a semi-implicit one does. Also, taking finer (and thus more) control volumes increases the number of explicit iterations asymptotically, reaching all upper bound as a function of the total optical thickness and the convergence criterion.