|
Open-Celled Materials.
Reticulated ceramic foams have many high-temperature
applications and the reticulated structure allows long radiation paths. Howell
et al. (1996) and Kamiuto (2008) review the radiative properties of commercially
available reticulated porous ceramic foams. These materials have an open pore
structure in a random dodecahedral matrix created by the interconnected struts
or webs that make up the foam, and it is difficult to specify a particle shape
or size that is representative of the material. Most researchers and
manufacturers specify an effective pore diameter, dpore, and
the scale of the structure is described by dpore and the
porosity. The porosity is typically about 85%, and pore sizes
range from approximately 25 to 2 pores per centimeter (ppcm) [65 to 8 pores per
inch (ppi)]. This is well outside the range where dependent scattering should be
of importance. The foam composition is usually specified by the manufacturer in
terms of a base material (silicon carbide, silicon nitride, mullite, cordierite,
alumina, and zirconia ) plus a stabilizing binder (e.g., magnesia and yttrium).
Properties are usually specified by the composition and the
manufacturer's specification of "pores per centimeter" or ppcm. However, the
actual pore diameter varies even among materials with the same specified ppcm
supplied by the same manufacturer. Thus, when using experimental property data,
significant error may be introduced unless reliable property data are available
on exactly the material being modeled.
When the porous material is treated as a homogeneous absorbing and
scattering medium, the effective absorption and scattering coefficients are
needed along with the scattering phase function. Hsu and Howell (1992) used a
two-flux radiation model to infer the effective radiative extinction coefficient
b from experimental
heat transfer measurements. They obtained the result as a function of pore size
for partially stabilized zirconia reticulated ceramic foam in the form of a data
correlation [0.3 mm < dpore < 1.5 mm]:
b = 1340-1540dpore+527dpore2
(B-9)
where b
is the mean (spectrally averaged for long wavelength radiation) extinction
coefficient in m-1 and dpore is the actual pore
diameter in mm. The data are for 290<T< 890 K, although no significant
temperature dependence was observed. Additionally, they
present a relation based on geometric optics that well predicted the trend of
the data for dpore > 0.6 mm:
b = (3/dpore)(1-e)
(B-10)
where e
is the porosity of the sample, which varied over a narrow range from 0.87 at
large pore diameters to 0.84 at the smallest diameters. The inversion method to
obtain these correlations assumed isotropic scattering. It was not possible to
determine independent values of albedo or scattering or absorption coefficients.
Hale and Bohn (1993) measured scattered radiation from a sample of
reticulated alumina that resulted from an incident laser beam at 488 nm. They
then matched Monte Carlo predictions of the scattered radiation calculated from
various values of extinction coefficient and scattering albedo, and chose the
values that best matched the experimental data for reticulated alumina samples
of 4, 8, 12 and 26 ppcm. They found that a scattering albedo of 0.999 and an
assumed isotropic scattering phase function reproduced the measured data for all
pore sizes.
Hendricks and Howell (1994, 1996) measured the normal spectral
transmittance and normal-hemispherical reflectance of three sample thicknesses
each of reticulated partially stabilized zirconia and silicon carbide at pore
sizes of 4, 8 and 26 ppcm for 400<l<5000 nm. They used an inverse
(Levenberg-Marquardt) discrete ordinates method to find the spectrally-dependent
absorption and scattering coefficients as well as the constants appropriate for
use in either the Henyey-Greenstein approximate phase function or a composite
isotropic/forward scattering phase function.
Integration over wavelength of the spectral results of Hendricks and
Howell provides mean extinction coefficient data that can be compared with those
of Hsu and Howell (1992). The Hale and Bohn (1993) data are
for the single laser-source wavelength of 488 nm. Hendricks and Howell (1996)
found that a modified geometrical optics relation fit the data for the
integrated extinction coefficient of both zirconia and silicon carbide. They
recommend the relation
b = (4.4/dpore)(1-e)
(B-11)
where b
is in m-1 and dpore is the actual pore diameter
in mm.
High-temperature experiments (1200-1400K) were performed by
Mital et al. (1996) on reticulated ceramic samples of mullite, silicon carbide,
cordierite, and yttria-zirconia-alumina. They reduced the data using a gray
two-flux approximation assuming isotropic scattering. Temperature dependence was
found to be small. They present the relations for absorption
and scattering coefficients for these materials as
When summed to obtain the extinction coefficient, the result
agrees with Eq. (B-10).
Fu et al. (1997) used a unit cell model to predict the extinction
coefficient and single scattering albedo of reticulated ceramics, and compared
their results with available data. Kamiuto and Matsushita
(1998) predicted the extinction coefficient of various open-celled structures
simulated by cubic unit-cells, based on the work of Kamiuto (1997). They
compared the predictions with experimental measurements obtained by an inversion
technique using emission data from a heated plane layer of porous material. They
assumed a Henyey-Greenstein phase function, and obtained good agreement between
experiment and prediction for the extinction coefficient and asymmetry factor
for Ni-Cr and cordierite porous plates.
Moura
et al. (1998) examined four generic experimental techniques that can be used to
generate data that can then be inverted to find the fundamental radiative
properties of porous materials. Loretz et al. (2006) compared the predictions of
various models for the radiative properties of metallic foams, and Coquard et
al. (2009) used a detailed cell model to predict the spectral transmittance and
reflectance of expanded polyethylene foams, and showed good agreement with
experiment. |
(B-12)
(B-13)