A Radiation Exchange Factor Transformation with Proven Convergence, Non-Negativity, and Energy Conservation

TL;DR

This paper introduces a matrix-based transformation for radiative transfer problems that guarantees convergence, non-negativity, and energy conservation, improving accuracy over classical methods and applicable to complex media.

Contribution

It develops a novel exchange factor transformation with proven convergence, non-negativity, and energy conservation, addressing discrepancies in classical formulations.

Findings

Method guarantees convergence, non-negativity, and energy conservation.

Validated against diffusion approximation and classical results.

Applicable to medium-scale and large-scale scattering problems.

Abstract

This paper presents a matrix-based exchange factor transformation for solving coupled mixed boundary condition radiative transfer problems on general domains. The method applies to participating media ranging from transparent to absorbing, emitting, and scattering, with boundaries ranging from absorbing to reflecting. Given a first-interaction exchange factor matrix $\mathbf{F}$, the transformation produces an absorption matrix $\mathbf{A}$ and a multiple reflection-scattering matrix $\mathbf{R}$ through a Neumann series that analytically traces all reflection-scattering paths to steady state. The paper establishes rigorous conditions under which the method guarantees convergence, non-negative radiation, and exact energy conservation to machine precision. A comparison with Noble's matrix formulation of Hottel's zonal method reveals a previously unidentified discrepancy in that classical approach; the proposed transformation eliminates this discrepancy. The method is validated against the diffusion approximation in the high-extinction limit and against results of Crosbie and Schrenker for pure and partial scattering cases. The method is applicable to medium-scale general reflecting-scattering problems and scales to large problems when negligible reflection-scattering and high extinction ensure matrix sparsity.