The Exchange Factor Method: An Alternative Basis for Zonal Analysis of Radiating Enclosures

Abstract

In zonal analysis of an enclosure containing a radiatively participating medium, the volume and bounding surface are divided into volume and area elements which are assumed to be isothermal. For a system of M elements consisting of N surfaces and Γ volume elements (M = N + Γ), M2 relationships (direct- or total-exchange areas of conventional zonal analysis) define all possible radiant interchange between element pairs. Due to reciprocity there are at most M(M + 1)/2 unique direct-exchange areas. The use of symmetry may greatly reduce the number of unique exchange areas. In the alternative form presented, exchange factors are defined such that a gray medium’s gas-to-gas exchange factors describe only the redistribution of radiative source terms. Only surface-to-surface and surface-to-gas (gas-to-surface) relationships are required to wholly define the system in radiative equilibrium. That is, a system in radiative equilibrium may be characterized by as many as Γ(Γ + 1)/2 fewer factors. These same exchange factors may be augmented by gas-to-gas exchange factors to analyze media not in radiative equilibrium. In an isotropically scattering, gray medium, exchange factors are functions of system geometry and total extinction only. The degree of extinction due to scattering does not affect exchange factors. Transformations between direct-exchange areas and exchange factors are developed.

Comparison between Kim, Wiscombe and least squares scattering phase function renormalization methods

“…Larsen and Howell [24], then Méchi et al [25] applied this method to normalize the direct exchange areas in zonal analysis which embody the enclosure's optical and geometric properties [26,27]. Ou and Wu [7] have performed this method for inverse radiation problems in conjunction with the LevenbergMarquardt algorithm.…”

On the use of flux limiters in the discrete ordinates method for 3D radiation calculations in absorbing and scattering media

“…Solutions to the same problem are also given by Fiveland [5] using a S 8 approximation for the step and diamond spatial discretizations, and by Larsen [47] using the zone method. For a fair evaluation of the exact solution the integral analytical expressions given by Crosbie and Schrenker [24] are solved numerically using Gauss-Seidel iterations on a 100 Â 100 Â 100 mesh to ensure mesh independence up to a 10 À8 tolerance.…”

New analytical methodologies for radiative heat transfer in enclosures based on matrix formalism and network analogy

“…The net radiation method (NRM) proposed by Hottel (1954) [2], which had been developed earlier by Poljak with some differences (1935) [3], was further approached by Gebhart (1961) [4] and Sparrow (1963) [5]. It represented the radiative transfer equations in discrete analog form for each surface and marked a valuable instrument of linear analysis of the radiative interactions surface-surface in enclosures that later (1958) evolved in surface-gas and gasgas specific for zone method (ZM) [6,7,8]. Matrix formalism began to be used and simplified the operation with the system of equations of radiative variables defined in their discrete analog form [9][10][11].…”