Non-Negativity, Non-Singularity and Energy Conservation of the Zonal Method with Non-Reciprocal Exchange Factors
This paper proves that the zonal balances of Hottel's method, written in power variables for a first-interaction exchange factor matrix $\mathbf{F}$, have a unique and non-negative solution for any combination of prescribed temperatures and prescribed source terms in which at least one element with prescribed temperature absorbs radiation, and that they conserve energy as an algebraic identity. The results hold for media ranging from transparent to absorbing, emitting and scattering, and for boundaries ranging from absorbing to reflecting. They require only that $\mathbf{F}$ be non-negative, row stochastic and irreducible: reciprocity is not needed, so the guarantees hold for Monte Carlo exchange factors as traced, without smoothing, which the symmetric notation of the zonal method does not make apparent. Both the non-singularity and the non-negativity follow from a single discrete minimum principle, and both have been machine-checked in Lean 4. To this end the balances are written as one linear system in the total radiant powers, $\mathbf{M}\mathbf{j} = \mathbf{h}$, assembled row by row from the prescribed temperatures and source terms, with a system matrix that inherits the sparsity of $\mathbf{F}$; the total exchange areas of Noble's matrix formulation are in general dense, and the two agree symbolically and numerically to machine precision. Validation is provided symbolically against the textbook closed-form solutions for infinite parallel plates and concentric cylinders, numerically against the results of Crosbie and Schrenker for pure and partial scattering, against an exact symmetry identity in a three-dimensional enclosure, and against the diffusion approximation in an optically thick problem with about $10^6$ elements, solved in about one minute.