Radiative Property Prediction 

If a simple geometry is representative of a porous material, it is possible to predict the equivalent radiative properties of a homogeneous medium using radiative transfer analysis based on surface-surface interactions.

 

Cell models 
Most early models of radiative properties in porous materials are based on simplified models of the structure. For example, random packings of spheres are assumed to have characteristics of regular packings. This clearly allows much simpler analysis because of geometric regularity. Usually, a unit cell of the regular packing is analyzed, and the characteristics of this cell are then assumed to be repeated throughout the porous medium. Radiation and conduction are assumed to occur in some parallel-series mode that is dependent on the cell structure. Cell models often predict agreement for a limited range of parameters, but fail outside of that range. Further, the radiative behavior of the cell components is usually assumed to be governed by diffuse reflections, and this is often far from the case. Finally, most early cell models did not recognize the importance of dependent scattering effects. Vortmeyer (1978) and Baillis and Sacadura (2000), review cell models, and compare their predictions with experiment and with each other. The models agree best with each other when the bed particles are radiatively black, as might be expected because radiative transfer for each model becomes less dependent on the surface reflective behavior.

 

Measured Properties of the Equivalent Homogeneous Material 
Because most treatments of radiative transfer in porous media rely on solution of the RTE, it is necessary to measure or predict the effective continuum radiative properties of the porous medium. Approaches to finding these properties can be separated into two classes; direct or indirect measurement of the required properties, or prediction of the properties from cell models and surface properties of the porous material. The properties are dependent on the particle scale, material and geometry of the particular porous material, and may be functions of temperature and wavelength. Depending on the criteria for particle size and clearance relative to important wavelengths, dependent scattering and absorption effects may also become important. The problem of adequately describing particle geometry is the most daunting.

 

Most measurements of radiative properties are made by inferring the detailed properties from measurements of transmission or reflection of radiation from the porous material. Inversion of these measurements using methods outlined in Chap. 8 is then used to find the best set of scattering coefficient, absorption coefficient and phase function that will predict the measured effects.

 

            Two problems are inherent in these measurements. First, inverse techniques are susceptible to large uncertainties in the inferred values that depend on the experimental uncertainty in the measured values, and there are questions of uniqueness as well (Chap. 8). Second, the model of radiative transfer that is used in the inversion may have particular assumptions embedded within it. For example, a simple phase function behavior (isotropic, linearly anisotropic, etc.) may be assumed to simplify the radiation model used in the inversion. If this is done, then the resulting values of inferred absorption and scattering coefficient will depend on that assumed type of phase function. It follows that the radiative model used in describing radiation in a porous material must use the same assumptions, or the absorbing and scattering coefficients will not be compatible. This may be overlooked when reported properties are taken from the literature and then using them in an incompatible model of radiative transfer. 

Dependent scattering 
For closely-spaced particles, fibers, or other bodies, the scattering and absorption of radiation is affected by near-field radiation interactions (Chapter 16). As the volume fraction occupied by particles increases (porosity decreases), these effects become more important. Yamada et al. (1986) present experiments for particles with dependent scattering.

 

             For spherical particles, Kaviany and Singh (1993) recommend that independent scattering can be assumed when the criterion

 

                                                           (B-4)

 

is met, where C is the interparticle clearance distance. This criterion can also be written as

 

                                                            (B-5)

 

where the clearance parameter xC = pC/l. If this criterion is not met, near-field effects must be considered. This result was derived for porosities typical of rhombohedral packing, e » 0.26, but dependent scattering in packed beds remains an important effect for bed porosities as high as 0.935, and the effect is most pronounced for opaque particles. Both the porosity requirement and the relation between C and l [Eqs. (B-4) or (B-5)] must be satisfied before the assumption of independent scattering can be used with confidence. Brewster and Tien (1982) provided the criterion for independent scattering as

 

                                         (B-6)

 

where the independent scattering region (i.e., deviation of more than five percent from independent scattering results) is demarcated when C/l = 0.5 is inserted into the equation [Tien (1988)]. This relation is based on far-field interference effects.