Coupled or combined radiation and conduction heat transfer takes place in heated semitransparent media that have a spectral range of partial transparency (frequency range where the value of absorption coefficient k is approximately in the interval 0.01 < k < 100 cm^{−1}) and where radiation is an appreciable part of the total energy flux.

The presence of a semitransparency range in the spectrum of thermal radiation is typical for dielectrics and semiconductors in condensed phases and for multiatomic gases with asymmetrical molecules.

Combined radiation and conduction heat transfer is of great practical importance for some semitransparent material manufacturing and heat treatment processes carried out at high temperatures: melting, moulding and fritting of glasses, growth of dielectric and semiconductor single crystals, sintering of ceramics, drawing out of fibers and light guides.

Also, combined radiation and conduction heat transfer is important where semitransparent materials are used at high-temperature conditions (ceramics and fiber thermal protection systems of reusable space vehicles entering the Earth’s atmosphere; thermal insulation of high-temperature furnaces and other industrial high-temperature equipment; intense radiation sources; solar volumetric receivers; radiative converters, packed beds of powders of oxides and other materials during remelting in solar and arc image furnaces; ceramic materials subjected to cutting and other processing by intense laser radiation).

The purpose of solving problems of combined radiation and conduction heat transfer is the calculation of temperature distribution and energy fluxes in the volume of semitransparent medium and on its boundaries. Side by side with this, inverse problems of combined radiation and conduction heat transfer are important, especially for determining true thermal conductivity of materials.

The theory of combined radiation and conduction heat transfer in media capable of absorbing radiation, as well as scattering radiation on various heterogeneities (pores, insertions of other phases), is based usually on simultaneous solution of the radiation transfer equation and of the energy conditions.

The formulation of radiative transfer equation for combined radiation and conduction heat transfer does not differ from the formulation for radiation transfer only.

The integro-differential energy conservation equation describing an energy balance on an elemental volume of medium has the form:

where is the total heat flux vector at point M:

and (M, t) is total radiation flux vector:

where Ω is the solid angle and ν the frequency. In Eq. (3), the integration over frequency ν for calculation of vector is fulfilled over the frequency range Q, corresponding to the region of media semitransparency.

It follows from Eq. (1) that for calculating temperature distribution over the volume of the semitransparent medium, it is necessary to know the radiation intensity distribution. Also, from the radiation transfer equation and its boundary conditions, it follows that for calculating radiation intensity distribution the temperature distribution must be known for all points inside the medium and on its boundaries. This is how the combination of heat transfer by radiation and conduction is manifested mathematically.

The simultaneous analytical solution of the radiation transfer and energy conservation equations is impossible and numerical solutions present a very difficult problem. The solution of the radiation transfer equation demands a large volume of calculations as radiation intensity is a function of point coordinates, direction and frequency of radiation. During combined radiation and conduction transfer calculations, the radiation transfer equation must be solved iteratively. A general solution of combined radiation problems for the general three – dimensional case considering the frequency, direction and temperature dependence of optical properties and temperature dependence of thermophysical properties has not so far been published. To decrease the computational effort, combined radiation and conduction heat transfer problems are usually simplified by considering one-dimensional cases, as a rule a plane layer. Combined radiation and conduction heat transfer problems in a plane layer have been solved successfully for nonscattering media. The computer codes developed give the possibility of calculating transient temperature distributions, considering the frequency and temperature dependence of the optical properties of the medium and its boundaries' optical properties.

For plane layers of scattering media, the problem is more difficult, since the radiation transfer equation includes not only the absorption coefficient k and refractive index n, but also the scattering coefficient β and phase function of scattering P — the latter depending on the frequency ν and the directions of incident and scattered radiation. Therefore, some simplifications and approximations have to be used for solving the radiation transfer equation.

In addition to the mathematical difficulties, the lack of data on frequency dependence of k, β and P is a significant obstacle for analysis of combined radiation and conduction heat transfer. Obtaining these characteristics by calculations using Mie theory is possible only for media with perfect spherical form and for cylindrical scatterers when the clearance between scatterers is bigger than the wavelength and the properties n and k of scatterer substance and surrounding medium are known. In practice this is very rarely possible. In most cases, the values k, β and P must be measured experimentally, but this is very difficult. Therefore, most calculations of combined radiation and conduction heat transfer use model "grey" media, where the scattering is assumed to be wavelength – independent and isotropic.

Instead of using the strict radiation transfer equation, approximations may be applied for decreasing the many parameters describing the optical properties. The two-flux approximation uses four parameters: the averaged absorption coefficient, the scattering coefficient, the back-scattered fraction factor and the refractive index. The two-flux *Kubelka-Munk* approximation is used for describing radiation transfer in a plane layer of highly-scattering medium. The parameters are the refractive index n and two constants, K and S, defining absorption and scattering. The *radiation diffusion approximation* may be used not only for one-dimensional transfer problems but also for two-and three-dimensional cases. It uses three parameters: the refractive index n, the absorption coefficient k and the radiation diffusion coefficient D. The optical properties K and S or k and D may be considered as the new phenomenological parameters evaluated from experiments. Here, they may be used for describing combined radiation and conduction heat transfer even if the radiation transfer equation is not applicable (e. g., for media with dense packing of scatterers).

The radiation thermal conductivity approximation is often used for describing combined radiation and conduction heat transfer in scattering media and is sometimes employed for two- and three-dimensional problems in nonscattering media. This approximation is applicable in the interior of an optically-thick medium when the influence of radiation at the boundaries disappears beyond the region adjacent to the surface. In this case, the combined radiation and conduction heat transfer calculation reduces to solving the Fourier thermal conductivity equation, where thermal conductivity is a temperature-dependent "effective" thermal conductivity λ_{eff} measured experimentally. It is a sum of "true" (conductive) thermal conductivity λ_{c} and "radiation" thermal conductivity λ_{R}:

If the radiation transfer equation is applicable and optical properties of medium are known, the radiation thermal conductivity may be calculated using the formula:

where μ, is the mean cosine of the scattering angle and I^{p}(T) is the intensity of equilibrium (Planckian) radiation at temperature T; the integration is performed over the frequency range Q corresponding to the region of medium semitransparency.

To use experimentally-measured values of λ_{eff}, local radiation equilibrium must apply for every point inside the medium. When the medium is highly scattering but its absorption coefficient is small (for example, the absorption coefficient of silica glass fiber thermal insulation is equal to k 10^{−3} cm^{−1} or less), local radiation equilibrium may be disturbed even though the mean free-path of radiation is within the limits, from a few to hundreds of microns, and thickness of the layer is in centimeters. Here, combined radiation and conduction heat transfer calculations using the radiation thermal conductivity approximation may lead to errors and it is preferable to use the diffusion approximation.

#### REFERENCES

Ozisik, M. N. (1973) *Radiative Transfer and Integration with Conduction and Convection*. New York: Werbel and Peck.

Seigel, R. and Howell, Y. R. (1992) *Thermal Radiation Heat Transfer*, third edition, Washington D.C.: Hemisphere. DOI: 10.1016/0927-0248(94)90242-9

Sparrow, E. M. and Cess, R. D. (1978) *Radiation Heat Transfer*, Belmont: Wadsworth.

Viskanta, R. and Anderson, E. E. (1975) *Heat Transfer in Semitransparent Solids: Advances in Heat Transfer*, New York: Academic Press, 317-441.