For a packed bed of spherical particles with an average bed temperature near 500K, Wien’s displacement law  gives the wavelength at the peak of the blackbody spectrum as lmax = C3/T = 2898(mm K)/500K = 5.8 mm. This wavelength splits the energy in the blackbody spectrum such that 25 percent of the radiant power is at shorter wavelengths, and the remainder is at longer wavelengths. This provides a reasonable criterion for an approximate determination of whether dependent scattering will be important. If the bed is closely packed so that the clearance distance can be taken near zero, then the particle diameter below which dependent scattering becomes important according to Eq. (B-4) is dp=5lmax = 29 mm. At the same temperature, Eq. (B-6) with e = 0.26 gives the minimum particle diameter for assuming dependent scattering as dp = 30.6 mm. Lower temperature beds will have correspondingly greater values of lmax and particle diameters in the dependent scattering region will be larger.

 

Kaviany and Singh (1993) also present a relation for a scaling factor that gives the relation between transmission through a packed bed of opaque particles with independent scattering to the result assuming dependent scattering, and show that this result is only dependent on the bed porosity (i.e., independent of particle radiative emissivity).

 

Closely-spaced parallel cylinders are of interest in applications such as in-situ curing by an infrared source during filament winding and the determination of nanomaterial properties. Lee (1994) and Chern et al. (1995c) determined the radiative properties of arrays of parallel cylinders subject to normal irradiation and far-field dependent effects. They show the ranges of cylinder volume fraction and size parameter where these effects become important for various values of the fiber effective refractive index. Lee and Grzesik (1995) later extended the analysis to coated and uncoated fibers subject to obliquely incident radiation.  Chern et al. (1995b) show that dependent scattering effects for parallel cylinders are important in treating radiative transfer within filament-wound epoxy/glass-fiber composites because of the large fiber volume fraction and small size parameter of the fibers typical of these systems. For fiber layers, the interaction among wavelength, refractive index, fiber diameter, and fiber spacing is complex, and no simple criterion is available for deciding whether dependent scattering will be important. However, for x (based on cylinder diameter) > 2, the dependent scattering is relatively independent of the refractive index of the fibers and the size parameter, and dependent scattering is present for porosity < 0.925. Yamada and Kurosaki (2000) examined the characteristics of fibers with large size parameter.

 

Beds of spheres. 

Chen and Churchill (1963) measured the transmission of radiation through an isothermal bed of randomly packed equal diameter spheres, as well as beds of cylinders and irregular grains. They included materials of glass, aluminum oxide, steel and silicon carbide. They used these data to determine the bed effective scattering and absorption cross-sections based on a two-flux model. Kamiuto et al. (1990) measured the angularly-dependent reflectance of planar beds of glass and alumina spheres contained between glass plates due to normally incident radiation. An inversion technique was used to find the extinction coefficient, albedo, and Henyey-Greenstein asymmetry factor.  Kudo et al. (1991a) measured the transmittance through a randomly packed bed of polished ball bearings of uniform diameter, and observed significant transmittance through the bed in regions near the duct surfaces, while the transmittance near the bed centerline decreased rapidly with increasing bed thickness.

 

            A porous medium composed of regularly or randomly packed equal-diameter spheres has a well-determined size parameter, x = pdp/l based on the diameter dp the spheres. The absorption and scattering characteristics for such a medium are discussed in classic texts (Bohren and Huffman, 1983). If the spheres are close-packed and have small diameters relative to the important wavelengths, then the potential for near-field effects should be checked by using Eqs. (B-4) or (B-5).

 

            Ray tracing and Monte Carlo techniques have successfully predicted radiation transfer through regular and random arrays of particles and fibers. Yang et al. (1983) derived the statistical attributes of a randomly packed bed of identical cold (non-emitting) spheres (distribution of number of contact points between spheres and the conditional distributions of contact angles for spheres with a given number of contact points.) Distributions were generated from a zero-momentum random packing model.  The upper layer of spheres in a randomly packed container is not level in either real or simulated packed beds. Thus, defining the bed thickness is somewhat arbitrary, especially for beds that are only a few sphere diameters in depth. Wavy container boundaries were assumed so that the radial dependence of packing characteristics present in rigid containers could be ignored. Yang et al. used the bed packing distributions to construct a ray-tracing algorithm for specularly reflecting spheres, and computed the fraction of diffuse incident radiation transmitted through the bed. No dependent scattering effects were considered. Comparisons with the experimental results of Chen and Churchill (1963) showed similar trends of transmittance vs. bed thickness, but not exact agreement.

 

            When particle sizes are large such that geometric optics is applicable, Saatdjian (1987) provides a model for the extinction behavior of uniformly dispersed spheres, and shows that an extinction coefficient can be predicted that conforms to exponential attenuation in the limit of small spheres. A porosity range of 0.70 < e < 1.0 is covered by the theory. This work is applicable to cases where the particles are separated such as in particle-laden flows or in radiating-droplet space radiators where only radiation transfer governs interparticle energy transfer.

 

            Tien (1988) discusses all of the above references and others in detail, and shows the comparisons of the various models with the experimental data for transmission through a bed of spheres generated by Chen and Churchill (1963).

 

            Kudo et al. (1991a) used a model similar to that of Yang for determining transmittance, and added further experimental results for comparison. They simulated random packing by perturbing regular packing within a bounded region and then removing those spheres that overlapped the boundary after perturbation. This produced a low-density region near the boundary as observed in real packed beds. This was found to introduce a considerable increase in bed radiative transmissivity. Both specularly and diffusely reflecting spheres were modeled.

Singh and Kaviany (1991, 1994) and Kaviany and Singh (1993) extended the Monte Carlo analysis of packed beds of spheres to consider semitransparent and emitting spheres, and in particular to show that dependent scattering effects are quite important even for quite large bed porosities. Kaviany and Singh (1993) pointed out that the results of the earlier analyses by Yang and Kudo did not account for the non-diffuse nature of the incident radiation in the Chen and Churchill experiments, and thus tended to underpredict the experimental bed transmittance.