Improved Inverse Method for Radiative Characteristics
of Closed-Cell Absorbing Porous Media
Jaona Randrianalisoa
*
and Dominique Baillis
†
Centre de Thermique de Lyon, Institut National des Sciences Appliquées de Lyon,
69621 Villeurbanne Cedex, France
and
Laurent Pilon
‡
University of California, Los Angeles, Los Angeles, California 90095-1597
DOI: 10.2514/1.16684
Radiative characteristics such as the extinction coefficient, the scattering albedo, and the scattering phase function
of fused quartz containing closed cells are determined by using an inverse method based on theoretical and
experimental bidirectional transmittances. The theoretical transmittances are obtained by solving the radiative
transfer equation with the discrete ordinate method. Improvements have been made over previously reported
experimental determination of porous fused quartz radiative characteristics by using a more accurate phase function
and an adaptive quadrature to compute more precisely the intensities in the measurement directions. In addition, a
two-step inverse method to compute accurately and simultaneously the radiative parameters has been developed.
The results are shown to be independent of samples thickness. Exhaustive comparison between experimental
measurements of hemispherical transmittance and reflectance and computational results using the retrieved
radiative characteristics shows good agreement. The retrieved absorption coefficient of porous fused quartz appears
to be more realistic than that reported in our earlier publication.
Nomenclature
a = bubble radius, m
b = corrective factor used in Eq. (10)
c
ij
= matrix elements of the sensitivity coefficients J
e = sample thickness, m
f
1
, f
2
= spectral weights of the Henyey–Greenstein
phase function
HG
g = spectral asymmetry factor
g
1
, g
2
= spectral parameters of the Henyey–Greenstein
phase function
HG
I = spectral radiation intensity, W m
2
sr
1
J = matrix of the sensitivity coefficients
k = volumetric absorption coefficient, m
1
Mb = quadrature order of the discrete ordinate method
m = fused quartz refractive index
Nb = number of measurement directions
n = number of unknown parameters including !, ,
f
1
, g
1
, and/or g
2
p = unknown parameter such as !, , f
1
, g
1
, or g
2
Q = ratio of the measured scattered to the incident
radiation fluxes
r = interface reflectivity
S = minimization function
T = spectral transmittance or reflectance, sr
1
T = average spectral transmittance or reflectance,
sr
1
w = angular weight of the discrete ordinate method
w
0
= angular weight of the two Gaussian quadratures
associated to the experimental directions
x = bubble size parameter
y = spatial coordinate along the sample thickness, m
= angle between incident radiation and
measurement directions, rad
= volumetric extinction coefficient, m
1
= relaxation factor used in Eq. (5)
= divergence angle of the incident radiation, rad
= solid angle, sr
= Kronecker delta function
"
0
, "
1
, "
2
, "
3
= coefficients of the third order polynomial
estimating T
sca
in Eq. (17)
= cosine of the angle
= scattering angle defined in Eq. (21), rad
= angle between incident radiation direction and
radiation inside the porous medium, rad
= fused quartz absorption index
= radiation wavelength, m
= cosine of the angle
i
= weight associated to measurement in the
direction i 1 to Nb
= random number defined between 0 and 1
= dimensionless sensitivity coefficient
= standard deviation
0
= optical thickness
= spectral phase function
’ = azimuthal angle, rad
= experimental error, %
! = volumetric scattering albedo
Superscripts
= refers to hemispherical transmittance
= refers to hemispherical reflectance
Subscripts
bulk = refers to the continuous phase (quartz)
coll = refers to collimated radiation
Received 17 March 2005; revision received 26 October 2005; accepted for
publication 26 October 2005. Copyright © 2005 by the American Institute of
Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper
may be made for personal or internal use, on condition that the copier pay the
$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood
Drive, Danvers, MA 01923; include the code $10.00 in correspondence with
the CCC.
*
Ph.D. Student, CETHIL UMR CNRS 5008, Domaine Scientifique de la
Doua, INSA de Lyon, Bâtiment Sadi Carnot, 9 rue de la Physique; jaona.
randrianalisoa@insa-lyon.fr.
†
Assistant Professor, CETHIL UMR CNRS 5008, Domaine Scientifique
de la Doua, INSA de Lyon, Bâtiment Sadi Carnot, 9 rue de la Physique;
dominique.baillis@insa-lyon.fr.
‡
Assistant Professor, Mechanical and Aerospace Engineering Department,
37-132 Engineering IV, Box 951597; pilon@seas.ucla.edu.
JOURNAL OF THERMOPHYSICS AND HEAT TRANSFER
Vol. 20, No. 4, October–December 2006
871