SESSION 4
RADIATIVE HEAT TRANSFER MODELS
Chairman: N. Selçuk
THERMAL RADIATION PROPERTIES OF DISPERSED MEDIA: THEORETICAL PREDICTION AND EXPERIMENTAL CHARACTERIZATION
JeanFrançois SACADURA Institut National des Sciences Appliquées Centre de Thermique de Lyon (CETHIL) Bâtiment 404  F69621 VILLEURBANNE CEDEX, FRANCE
ABSTRACT
Thermal radiation is a predominant mode of energy transfer in many engineering systems. A
wide variety of these involve semitransparent media which are either porous materials or media
containing particulates which play a keyrole in the radiative transfer mechanisms. Some examples
are: fluidized and packed beds, combustors, catalytic reactors, surface pigmented coatings, soot and
flyash, sprayed fluids and a variety of insulating materials like fibers, foams, porous and reticulated
ceramics, microspheres, multilayered particles.
Modern design of these systems requires an accurate modeling of the radiative heat transfer.
This is commonly done by solving the Radiative Transfer Equation (RTE) combined to the energy
conservation equation (and other equations for fluid media that may furthermore be reactive). The
RTE is written in terms of spectral intensity of the radiation propagating in a given direction
:
This equation involves the material spectral radiative properties which are the refractive index
(implicitly contained in the blackbody intensity expression , in the boundary conditions and in
the wavelength definition), the absorption coefficient , the scattering coefficient and the
scattering phase function . Instead of and , one may alternatively use the extinction
coefficient and the albedo . These properties are those of a
pseudocontinuum medium equivalent, in terms of radiative transport, to the real dispersed material under
consideration.
An alternative way of treating the radiative transfer in dispersed media is to use discontinuous
modeling.
Anyway in addition to accurate RTE solution techniques in continuous or discontinuous
formulations which currently are available through a variety of methods, radiation heat transfer
modeling also requires a good knowledge of the material optical and radiative properties. This
remains an important source of uncertainty. Therefore this survey will be focused on the
determination through theoretical prediction or experiments of the radiative properties of dispersed
media. As an extensive review of radiative transfer in dispersed media has been performed by
Viskanta and Mengüç in 1989, the current survey will be concentrated on developments which were
published in the nineties. Nevertheless a limited number of prior relevant references are also recalled
as a background to this presentation.
According to the particle size and the particle separation distance compared to wavelength, the
particle shape and the optical properties of the particle material and of the background medium,
different ways of radiation property modeling may be employed. Since particulate media concerned
by most engineering applications cover a wide category of shapes, sizes and refractive indexes of
particle and dispersion medium, the way of modeling the properties migth be the basis of the
organization of the paper. Instead we preferred to start the survey with a brief background on
radiation property modeling in which the focus is put on the question of independent/dependent
scattering. Afterwards the theoretical prediction of radiative properties is scanned according to
several families of particles : spherical particles, fibers, foams and reticulated ceramics, soot and
agregates.. Then experimental work aiming to the property characterization is reported, with a
special attention to inverse techniques. Finally some concluding remarks are made, trying to point
out some needs for future work.
REFERENCE
Viskanta, R., and Mengüç, M.P., 1989, Radiative transfer in dispersed media, Appl. Mech. Rev. 42, No. 9, 241259
THE THIN LIMIT OF RADIATIONCONDUCTION INTERACTION IN A SEMITRANSPARENT SLAB WITH A CONVECTIVE BOUNDARY
Seppo A. Korpela* and Hoyoung Kim** * Department of Mechanical Engineering The Ohio State University, Columbus, Ohio 43210 USA ** Steel Process Division Research Institute of Industrial Science and Technology, Pohang, 790600, Korea
ABSTRACT. The thin limit of radiationconduction interaction in a semitransparent slab
with convective boundary is examined in the differential approximation by using perturbation
methods. The spectral properties are accounted for by mean coefficients and a nongrayness
factor. An alternative definition of conductionradiation parameter is used in order to make
the optical thickness alone govern the opacity of the medium.
COMPARISON OF RADIATIVE HEAT TRANSFER MODELS IN MINERAL WOOL AT ROOM TEMPERATURE
F.M.B. Andersen*, S. Dyrbol** * Department of Energy Engineering Technical University of Denmark, 2800 Lyngby. Denmark. ** Rockwool International A/S and Department of Buildings and Energy Technical University of Denmark, 2800 Lyngby. Denmark.
ABSTRACT. Heat transport in fibrous insulation is modelled, emphasis being placed on radiative
heat transfer. The model further includes conductive heat transport in the gaseous and fibrous
phases. The case is a planar layer of fibrous insulation with a heated and a cooled plate on the two
sides as in a standard test apparatus for measuring the apparent thermal conductivity. The
absorption and scattering coefficients are calculated using the Mie theory, a measured statistical
fibre diameter distribution, and the orientation of the fibres. The radiative heat transfer is modelled
using two models: the twoflux model and the spherical harmonics method with arbitrary
approximation order. The system of governing equations is discretized using finite differences and
the system of algebraic equations is solved by the NewtonRaphson method. The results show that
the radiative heat flux, as well as the radiative conductivity calculated from the twoflux model, is
approx. 15 % in error while a P1 equation gives very accurate results compared to higher order
spherical harmonics approximations. The computational costs of loworder spherical harmonics are
acceptable, although not as low as when using the twoflux model. The costs of using very high
approximation orders such as P21 are considerable. However, the Pl and P3 approximations do
not model the angular distribution of the intensity at the walls as precisely as do the P11 and P21
approximations.
RADIATIVE HEAT TRANSFER IN DISPERSED MEDIA: NEW PHENOMENOLOGICAL APPROACH
Andrei V. Galaktionov Institute for High Temperatures Russian Academy of Sciences, 13/19 Izhorskaya, Moscow, 127412, Russia
ABSTRACT. This paper proposes a new phenomenological approach for the radiative heat transfer
description in dispersed particulate, porous, and cellular media capable of absorbing, emitting, and
scattering thermal radiation. The approach does not use any assumptions about the medium. It is free
of geometrical optics restrictions and useful in the medium with significant dependent absorption and
scattering effects. The approach is formulated in terms of the symbol (i.e. the Fourier transformation
of Green's function) of the operator of radiative heat transfer. General features of the symbol are
deduced from the thermodynamics laws, Curie's principle, and symmetry considerations. The symbol
on the real axis is an even, real, positive, bounded, and steadily increasing function, which vanishes at
the origin of coordinates. Singularities of the symbol belong to imaginary axis.
Correspondence between the approach and classical radiation transfer equation is established. The
symbol calculation problem is a wellposed direct problem. The reconstruction of the scattering
phase function from the symbol is an illposed inverse problem. It is shown that several widely used
models of radiation transfer, namely, limits of optically thick and thin medium, spherical harmonics
technique, and WickChandrasekharGauss discrete ordinate technique are the particular cases of
proposed approach.
Possible realization and applications of the approach are discussed. It is a powerful tool for
comparison between known models of radiative heat transfer and for development of new approximate
methods for practical calculations. Numerical techniques may be directly analyzed as well.
Two possible ways of symbol measurements are discussed. Traditional inverse problems, which are
based on measurements of the reflectivity and transmissivity of a sample, determine the situation of
symbol singularities on the imaginary axis of complex plane. New temperature wave technique for
optical properties measurements determines the symbol near the diagonal line of complex plane.
Proposed approach is preferable if structure of examined medium is unknown or it is too difficult to
propose an adequate model of radiative heat transfer. This situation is typical for such materials as
microsphere insulation, conglomerated soot particles, deposited soot, packed and fluidized beds.
