Differential approximations

DOI: 10.1615/thermopedia.000117

Differential approximations

Leonid A. Dombrovsky

Following from: Computational models for radiative transfer in disperse systems, The radiative transfer equation, Transport approximation

Leading to: Two-flux approximation, P1 approximation of the spherical harmonics method, The simplest approximations of double spherical harmonics, Diffusion approximation in multi-dimensional radiative transfer problems

The complete mathematical formulation of radiative transfer problems is very complicated even for the simplest approximation of the scattering function. The main difficulty is an angular dependence of the radiation intensity. At the same time, this angular dependence appears to be rather simple in many important applied problems. It enables one to use this property of solution to derive simple, but fairly accurate, differential approximations.

The differential approximations have a long history. This is reflected in their well-known names: the Eddington method, the Schwarzschild-Schuster method, etc. The progress in computer engineering and numerical methods for boundary-value problems makes it possible to obtain more accurate solutions. Nevertheless, simple and physically clear differential approximations are widely used at present for solving the radiative transfer problems in scattering media, particularly in combined heat transfer problems (Öziik, 1973; Viskanta, 1982, 2005; Rubtsov, 1984; Dombrovsky, 1996). All of the differential approximations for the radiative transfer equation (RTE) are based on simple assumptions concerning the angular dependence of spectral radiation intensity Iλ. These assumptions enable us to deal with a limited number of functions Iλi() instead of function Iλ(,) and turn to the system of the ordinary differential equations by integration of the RTE. The same result can be obtained if the integral term in the equation


is expressed in the form of the Gaussian quadrature (Case and Zweifel, 1967). Particularly, through the use of the discrete ordinate method (DOM) one can derive the same system of the ordinary differential equations as those obtained by expansion of Iλ on the spherical functions. Differential approximations are suitable for calculation of radiative transfer at arbitrary optical depth, but their possibilities when accounting for real scattering functions are very limited. For example, the mathematical formulations in the first approximation of the spherical harmonics method for the linear scattering function


and transport scattering function


are identical.

The simplest differential approximations, brought together in this article as was done by Adrianov (1972) and Dombrovsky (1996), by the general term “diffusion approximation,” give the following representation of the spectral radiation flux:


and differ only by expression for radiation diffusion coefficient Dλ. Sometimes, the term “diffusion approximation” is related only to the case when Dλ = 1/(3βλtr) (Zeldovich and Raizer, 1966, 1967, 2002), which corresponds to the Eddington approximation. It will be shown below that Eq. (4) can be derived based on some assumptions concerning the angular dependence of radiation intensity. Substituting Eq. (4) into the radiation balance equation


we obtain the nonhomogeneous modified Helmholtz equation for the spectral radiation energy density:


For the internal region of the optically thick volume, one can use the equilibrium radiation intensity in Eq. (4) and the radiation diffusion coefficient from the Eddington approximation (Eddington, 1959):


Integration of Eq. (7) over the spectrum yields



Here kr is the so-called radiative conductivity and βRtr is the Rosseland mean transport extinction coefficient. Equations (8) and (9) are called the radiative conduction approximations or the Rosseland approximation (Rosseland, 1936). This approximate model is sometimes called the Rosseland diffusion approximation, but one should avoid confusion between this model and the above-defined diffusion approximation [Eq. (4)]. Obviously, the radiative conduction (Rosseland, 1936) approximation is applicable only inside optically dense media at large optical distances from the boundaries and from regions with strong variation of temperature and medium properties.


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Case, K. M. and Zweifel, P. F., Linear Transport Theory, Reading, MA: Addison-Wesley, 1967.

Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.

Eddington, A. S., The Internal Constitution of the Stars, New York: Dover, 1959.

Öziik, M. N., Radiative Transfer and Interaction with Conduction and Convection, New York: Wiley, 1973.

Rosseland, S., Theoretical Astrophysics: Atomic Theory and the Analysis of Stellar Atmospheres and Envelopes, Oxford, UK: Clarendon, 1936.

Rubtsov, N. A., Radiation Heat Transfer in Continuous Media, Novosibirsk: Nauka (in Russian), 1984.

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