SESSION 4

RADIATIVE HEAT TRANSFER MODELS

Chairman: N. Selçuk

THERMAL RADIATION PROPERTIES OF DISPERSED MEDIA: THEORETICAL PREDICTION AND EXPERIMENTAL CHARACTERIZATION

Jean-François 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 semi-transparent media which are either porous materials or media containing particulates which play a key-role in the radiative transfer mechanisms. Some examples are: fluidized and packed beds, combustors, catalytic reactors, surface pigmented coatings, soot and fly-ash, 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 pseudo-continuum 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

THE THIN LIMIT OF RADIATION-CONDUCTION INTERACTION IN A SEMI-TRANSPARENT 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, 790-600, Korea

ABSTRACT. The thin limit of radiation-conduction interaction in a semi-transparent 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 non-grayness factor. An alternative definition of conduction-radiation 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 two-flux 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 Newton-Raphson method. The results show that the radiative heat flux, as well as the radiative conductivity calculated from the two-flux model, is approx. 15 % in error while a P-1 equation gives very accurate results compared to higher order spherical harmonics approximations. The computational costs of low-order spherical harmonics are acceptable, although not as low as when using the two-flux model. The costs of using very high approximation orders such as P-21 are considerable. However, the P-l and P-3 approximations do not model the angular distribution of the intensity at the walls as precisely as do the P-11 and P-21 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 well-posed direct problem. The reconstruction of the scattering phase function from the symbol is an ill-posed 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 Wick-Chandrasekhar-Gauss 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.