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, D,
and the particle size parameter _{pore}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 > ~10), and the pores or
particles are large relative to the wavelengths of the important radiative
energy (_{p}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
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,
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 CaCO
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 k, the
radiative conductivity can be approximated by [Singh and Kaviany (1991, 1994),
Kaviany (1991)]:_{s}
(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. |