A Radiation Exchange Factor Formulation with Proven Non-Negativity and Unconditional Energy Conservation

Abstract

This paper presents a matrix formulation of the scalar laws of radiative transfer. The method applies to coupled mixed boundary condition problems on general domains. Participating media can range from transparent to absorbing, emitting, and scattering, with boundaries ranging from absorbing to reflecting. Given a non-dimensional first-interaction exchange factor matrix $\mathbf{F}$, the formulation partitions $\mathbf{F}$ into a single-step absorption matrix and a single-step reflection-scattering matrix via Hadamard products with a column-constant matrix of reflection-scattering coefficients. The resulting linear system encodes the radiative energy balance for arbitrary combinations of prescribed temperatures and prescribed source terms, with a proven non-singularity result for the mixed-boundary system. The method is shown to admit a unique non-negative solution for non-negative source terms whenever the maximum reflection-scattering coefficient is strictly less than unity, with unconditional energy conservation to machine precision. Validation is provided symbolically against the textbook closed-form solutions for infinite parallel plates and concentric cylinders, and numerically against the diffusion approximation in the high-extinction limit and against the results of Crosbie and Schrenker for pure and partial scattering cases. A comparison with Noble's matrix formulation of Hottel's zonal method reveals a discrepancy in that classical approach, not previously identified to the author's knowledge; the proposed formulation avoids this discrepancy. The method requires a single linear solve whose sparsity inherits from that of $\mathbf{F}$, making it applicable to medium-scale dense problems and to large-scale sparse problems with high extinction.