## Radiative transfer equation

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## Computational Models for Radiative Transfer in Disperse Systems

This introductory material begins a set of articles on computational models employed in radiative transfer calculations for disperse systems. We will not consider here the nature and basic laws of thermal radiation because this general knowledge is given in the well-known textbooks by Sparrow and Cess (1978), Siegel and Howell (2002), and Modest (2003). Nevertheless, before proceeding to mathematical formulation of radiation transfer problems for scattering media, it is reasonable to recall some of the definitions of the main physical quantities.

The radiation energy in wavelength interval (λ, λ + dλ), passing per time dt in solid angle d near direction through area dσ located at point and oriented perpendicular to , is equal to Iλ(,) dλ dt dσ d, where function Iλ(,) is the spectral intensity of radiation. This function is the most general characteristic of the radiation field in the case of unpolarized (randomly polarized) radiation. The polarization of electromagnetic waves is usually not important in the problems concerning thermal radiation, and it is sufficient to use the scalar function Iλ(,) instead of the Stokes parameters (Born and Wolf, 1999). The details concerning description of polarized radiation can be found in the classic book by Chandrasekhar (1950, 1960) and in the monographs by van de Hulst (1957, 1981) and Bohren and Huffman (1983), which are very close to the problems under consideration and should be included by a reader in a short list of the most important handbooks. The spectral intensity of equilibrium (the so-called “black body”) thermal radiation of an isothermal medium is given by the Planck’s function: Iλ = Bλ(T). The black-body radiation is homogeneous and isotropic; i.e., independent of both coordinate and direction . For radiating medium, a deviation of the function Iλ(,) from the intensity of equilibrium radiation at local temperature T() is described by the radiative transfer equation.

Absorption and scattering of radiation in a medium are described by spectral coefficients αλ and σλ, respectively, by the extinction coefficient βλ = αλ + σλ and by the scattering function Φλ(', ), which is also called the scattering phase function or indicatrix of scattering. The latter function presents the angular intensity distribution for the radiation scattered by a small (elementary) volume of the medium by one act of scattering. The scattering function satisfies the normalizing condition:

 (1)

Note that coefficients αλ, σλ, and βλ are also referred to as the medium elementary volume. It is assumed that the absorption and scattering characteristics of a small element of the medium can be determined on the basis of the so-called far-field single-scattering approximation. This assumption, which is also known as the independent scattering approximation, is correct in many applied problems concerning rarefied disperse systems when positions of single particles are random and uncorrelated and the average distances between the neighboring particles are greater than the particle size and radiation wavelength. The physical sense of this assumption has been considered in some detail by Mishchenko et al. (2004).

Note that the above definitions of the absorption and scattering characteristics of a medium correspond to the continuous model of the radiation propagation in the medium. It is a widely used approach, which appears to be fairly good in many practical problems concerning thermal radiation in disperse systems. However, there are some specific cases, such as particulate debris beds or large-scale cellular structures, when the classical continuum theory may not be appropriate and the discrete transfer models are physically more adequate to the real processes (Vortmeyer, 1978; Viskanta and Mengüç, 1989).

#### REFERENCES

Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.

Born, M. and Wolf, E., Principles of Optics, 7th ed. (expanded), New York: Cambridge University Press, 1999.

Chandrasekhar, S., Radiative Transfer, Oxford, UK: Oxford University Press, 1950.

Chandrasekhar, S., Radiative Transfer, New York: Dover, 1960.

Mishchenko, M. I., Hovenier, J. W., and Mackowski, D. W., Single scattering by a small volume element, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 71-87, 2004.

Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic, 2003.

Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., New York: Taylor & Francis, 2002.

Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, New York: McGraw-Hill, 1978.

van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.

van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.

Viskanta, R. and Mengüç, M. P., Radiative transfer in dispersed media, Appl. Mech. Rev., vol. 42, no. 9, pp. 241-259, 1989.

Vortmeyer, D., Radiation in packed solids, Proc. of 6th International Heat Transfer Conference, Toronto, vol. 6, pp. 525-539, 1978.

#### References

1. Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.
2. Born, M. and Wolf, E., Principles of Optics, 7th ed. (expanded), New York: Cambridge University Press, 1999.
3. Chandrasekhar, S., Radiative Transfer, Oxford, UK: Oxford University Press, 1950.
4. Chandrasekhar, S., Radiative Transfer, New York: Dover, 1960.
5. Mishchenko, M. I., Hovenier, J. W., and Mackowski, D. W., Single scattering by a small volume element, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 71-87, 2004.
6. Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic, 2003.
7. Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., New York: Taylor & Francis, 2002.
8. Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, New York: McGraw-Hill, 1978.
9. van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.
10. van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.
11. Viskanta, R. and Mengüç, M. P., Radiative transfer in dispersed media, Appl. Mech. Rev., vol. 42, no. 9, pp. 241-259, 1989.
12. Vortmeyer, D., Radiation in packed solids, Proc. of 6th International Heat Transfer Conference, Toronto, vol. 6, pp. 525-539, 1978.

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Computational methods for radiative transfer in disperse systems

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Computational methods for radiative transfer in disperse systems in Fundamentals
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