Calculation of Direct Exchange Areas for Nonuniform Zones Using a Reduced Integration Scheme

Abstract

In the zonal method, considerable computational resources are needed to calculate the direct exchange areas (DEA) among the isothermal zones due to integrals with up to six dimensions, while strong singularities occur in the integrands when two zones are adjacent or overlaping (self-irradiation). A special transformation of variables to reduce a double integral into several single integrals is discussed in this paper. This technique was originally presented by Erkku (1959) for calculation of DEA using a uniform zone system in a cylindrical enclosure. However, nonuniform zones are needed for applications with large thermal gradients. Thus we extended this technique to calculate the DEA for non-uniform zones in an axisymmetrical cylinder system. A six-fold reduction in computational time was observed in calculating DEA compared with cases without a variable transformation. It is shown that accuracy and efficiency of estimation of radiation heat flux is improved when using a nonuniform zone system. Reasonable accuracy of all DEA are calculated without resorting to the conservative equations. Results compared well with analytical solutions and numerical results of previous researchers. This technique can be readily extended to rectangular enclosures with similar reduction in computation time expected.