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.
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