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B.  RADIATIVE TRANSFER IN POROUS AND DISPERSED MEDIA

Radiative transfer in porous and dispersed media is reviewed in Tien and Drolen (1987), Tien (1988), Dombrovsky (1996), Baillis-Doermann and Sacadura (1998), Howell (2000), and Kamiuto (1999, 2008). These references provide extensive bibliographies. Many early models neglected the effect of anisotropic and/or dependent scattering that have been found important in some applications. If the porous structure is well defined, newer models have moved toward numerical and statistical modeling approaches that incorporate the detailed porous structure and its thermal properties. Considering radiative transfer among the structural elements provides detailed radiative transfer results, but does not yield simplified engineering heat transfer correlations. If the structural elements are translucent, refraction at the solid-fluid interfaces must be considered as well as the external and internal reflections that occur at these interfaces (see Sec. 17-5.3). Almost always, the structural elements are so closely spaced that dependent scattering occurs (Chapters 15, 16). Independent scattering has been shown to fail for systems with low porosity and for packed beds [Singh and Kaviany (1994), Coquard and Biallis (2004)], and deviations from independent scattering theory have been found at porosities as high as 0.935.

A less detailed engineering approach has been to assume that the porous material can be treated as a continuum. This depends on the minimum porous bed or medium dimension, L, relative to the particle or pore diameter, D_p or D_pore, and the particle size parameter ξ = πD_P/λ. For most practical systems the porous material is many pore or particle diameters in extent (L/D_pore or L/D_p > ~10), and the pores or particles are large relative to the wavelengths of the important radiative energy (ξ > ~ 5). In this case, the porous medium may be treated as continuous, and the effective radiative properties are measured by averaging over the pore structure. Traditional radiative transfer analysis methods for translucent media can then be applied without using detailed element-to-element modeling. When temperature gradients are large and/or the thermal conductivity of the elements is small, the assumption of isothermal elements may not be accurate [Singh and Kaviany (1994); Robin et al. (2006)]. Combustion in some liquid- and gas-fueled porous burners occurs within the porous structure, producing large temperature gradients; temperatures within the porous medium can increase by over 1000 K within a few pore diameters.

The energy equation for porous media analysis depends on the complexity of the particular problem to be solved. To determine the energy transfer between the porous solid and a flowing fluid, energy equations are written for both the solid and the fluid, with a convective transfer term providing the coupling between the two equations. This approach permits calculation of the differing bed and fluid temperatures. If the fluid is transparent (no absorption or scattering) the radiative flux divergence is only in the equation for the absorbing and radiating solid structure of the porous material. In this case, the energy equations (in one dimension) become, for the fluid and solid phases,

                        (B-1)

                                        (B-2)

The h_v is the volumetric convective heat transfer coefficient, and  is the volumetric energy source in the fluid from combustion or other internal energy sources. The properties are effective properties that depend on the structure of the solid and the flow configuration. If multiple species are present, as for combustion, additional terms for species diffusion must be included in the fluid-phase equation. The coupling of (B-1) and (B-2) through the convective transfer terms shows that radiation affects both the solid and fluid temperature distributions even when a radiation term is only in the solid energy equation.

If there is no fluid within the porous medium, a single energy equation is used for the temperature distribution of the solid structure. Then Eqs. (B-1) and (B-2) reduce to the steady form of (10-3) with no viscous dissipation. Alternatively, if the fluid flow rate is sufficiently large, the volumetric heat transfer coefficient becomes large and the local fluid and solid temperatures become essentially equal; then Eqs. (B-1) and (B-2) combine to give

                                    (B-3)

In this equation, k_eff is the effective thermal conductivity of the fluid-saturated porous medium, and the other properties are corrected for the porosity and are on a per unit of total volume basis.

Most analyses of radiative transfer in porous media rely on solving the radiative transfer equation (RTE) and it is necessary to measure or predict the effective continuum radiative properties of the porous medium. This can be done by direct or indirect measurements, or by predicting the properties using models of the geometrical structure and surface properties of the porous structural material. Most radiative property measurements are made by inferring the detailed properties from measurement of radiative transmission or reflection by the porous material. Measured and predicted properties of various packed beds are discussed in Howell (2000), and the properties of open-cell foam insulation are studied in Baillis et al. (2000). Haussener et al. (2009) use computerized tomography to determine the physical characteristics of a packed bed of CaCO₃ particles, and then use forward Monte Carlo to determine the spectral scattering and absorption coefficients as well as the spectral phase function. Although the results by Haussener et al. were independent of the reflectivity characteristics of the system boundaries (specular or diffuse), they were strongly dependent on the assumed reflectivity characteristics of the particles themselves. Lalich et al. (2009) examine the radiative properties of nanoporous matrices composed of silica particles. 

Solutions are difficult when the radiative mean free path is of the order of the overall bed dimensions. Simplifying assumptions cannot be made, such as an optically thick medium, and in this case a complete solution of the RTE may be required. The two-flux method is often used when one-dimensional radiative transfer is assumed; however, if the particles in the bed are absorbing, this method does not give satisfactory results [Singh and Kaviany (1994)]. If the solid structure of the porous medium has a defined shape it is possible to use conventional surface-to-surface radiative interchange analysis. This is used in Antoniak et al. (1996) and Palmer et al. (1996) with a Monte Carlo cell-by-cell analysis to simulate radiative transfer among arrays of geometric shapes. Some research has tried to define an effective radiative conductivity that can be combined with heat conduction. For use in optically thick systems with porosity between 0.4 and 0.5, composed of opaque solid spherical particles with surface emissivity ε_r and thermal conductivity of the solid k_s, the radiative conductivity can be approximated by [Singh and Kaviany (1991, 1994), Kaviany (1991)]:

            (B-4)

where . This is reasonably accurate for both diffuse and specular reflecting particles (the constants in (B-4) from Singh and Kaviany (1994) are specifically for diffuse surfaces), and is weakly dependent on bed porosity. Coquard and Baillis (2004) treat beds of specular and diffuse opaque spheres by Monte Carlo, and find good agreement with the correlations of Singh and Kaviany.