2 Grey Matrix-based Methods
2.1 Introduction
The grey formulation of radiative transfer, where material properties are treated as wavelength-averaged, marks the historical and conceptual foundation of all matrix-based solution methods for the Radiative Transfer Equation (RTE). By neglecting spectral dependence while retaining spatial and directional effects, early researchers were able to capture the essential physics of emission, absorption, and reflection without the additional complexity of spectral variation. This simplification proved decisive: it allowed the energy exchange between discrete regions of a domain to be expressed as a system of linear equations, forming the basis for the first matrix representations of radiative transport.
The evolution of grey matrix-based methods can be traced through several complementary lines of development.
In thermal engineering research, Hottel and co-workers introduced the Zone Method (Hottel and Cohen 1958; Hottel and Sarofim 1967; Noble 1975), in which radiative interactions between finite surface and gas elements were formulated using integrals embodying the geometric and physical coupling between emitters and absorbers.
In parallel, atmospheric and astrophysical studies built on Chandrasekhar’s foundational work (Chandrasekhar 1950), developing matrix-operator and adding–doubling techniques for layered media and later, generalizing these ideas in numerically stable discrete-ordinate solvers such as DISORT (Stamnes et al. 1988), which handle both scattering and spectrally dependent conditions.
During the same period, advances in computational methods extended the discrete ordinates approach beyond layered systems, leading to general multidimensional formulations (Fiveland 1984, 1988) and related finite-volume methods (Raithby and Chui 1990; Chai, Lee, and Patankar 1994), which cast the RTE directly into sparse matrix form suitable for numerical solution.
Together, these developments established the conceptual link between radiative transfer and linear systems theory: radiative emission and absorption can be viewed as source and sink terms, while scattering and reflection act as coupling operators between elements of the domain.
The remainder of this chapter surveys the principal grey matrix-based approaches and highlights their common features, strengths, and limitations.
2.2 Exchange Factor Methods
Seeking to quantify combustion heat transfer, Hottel and co-workers (Hottel and Cohen 1958; Hottel and Sarofim 1967; Noble 1975) pioneered solutions to the RTE using a discretized integral approach. Fundamentally, this integral approach produces matrices of exchange areas, which describe the radiative connectivity of the discretized domain. The integrals for the exchange areas are:
Surface-surface: \overline{s_j s_k} = \frac{1}{\pi}\int_{A_j}\int_{A_k} \frac{\overline{t}(S_{j-k})\mathrm{cos}(\theta_j)\mathrm{cos}(\theta_k)}{S^2_{j-k}}dA_k dA_j \tag{2.1}
Surface-gas and gas-surface: \overline{s_k g_\gamma} = \overline{g_\gamma s_k} = \frac{\kappa}{\pi} \int_{V_\gamma}\int_{A_k}\frac{\overline{t}(S_{k-\gamma}) \mathrm{cos}(\theta_k)}{S^2_{k-\gamma}} dA_k dV_\gamma \tag{2.2}
Gas-gas: \overline{g_\gamma g_{\gamma*}} = \frac{\kappa^2}{\pi} \int_{V_\gamma}\int_{V_{\gamma*}} \frac{\overline{t}(S_{\gamma-\gamma*})}{S^2_{\gamma-\gamma*}}dV_{\gamma*} dV_\gamma \tag{2.3}
where \overline{t}=\mathrm{exp}(-\kappa S) is the transmissivity of the participating medium, \theta is the angle with the surface normal, \mathrm{cos}(\theta) of the emitter surface accounts for the diffuse Lambertian emission and \mathrm{cos}(\theta) of the absorbing surface accounts for the projected area effect, S is the distance from the point of emission to the point of absorption and \kappa is the absorption coefficient of the medium. These integrals, being analytically intractable, were initially, before the general advent of modern day digital computers, solved graphically and tabulated. Once the exchange areas have been determined for all pairs of emitters and absorbers in the system, subsequent solution of the RTE is achieved using linear algebra.