Following from: Computational models for radiative transfer in disperse systems, The radiative transfer equation
Leading to: Differential approximations
One of the following methods is commonly used in approximation of scattering function: expansion in a series of the Legendre functions or description of the main maxima of the scattering function by several delta functions with some weight coefficients. Some combinations of these two approaches are also employed (Dombrovsky, 1996a; Modest, 2003). The simplest approximations of each method are well known. If one is restricted to two terms in expansion on Legendre functions, the result is the following linear-anisotropic approximation:
where μλ is the asymmetry factor of scattering:
The linear-anisotropic approximation loses any physical sense with μλ > 1/3; i.e., at large scattering anisotropy, since Φλ(μ0) becomes negative when μ0 < -1/(3μλ). The other drawback of linear approximation (1) should also be noted: the linear dependence on the cosine of the scattering angle is too far from the typical scattering functions of disperse systems (Dombrovsky, 1996a). Note that the linear-anisotropic approximation has been widely used in early studies to estimate the role of anisotropic scattering in model radiative transfer problems (Özişik, 1973).
If one takes into account only the forward and backward scattering, presenting the scattering function as a linear combination of Dirac delta functions δ(1 + μ0) and δ(1 - μ0), the simplest “back-scattering” model is derived. This approximation, in which the integral term in the radiative transfer equation (RTE) disappears, gives good results in some one-dimensional problems (Belov, 1982) but it cannot be applied to the scattering description in multi-dimensional problems for inhomogeneous and nonisothermal disperse systems.
The well-known transport approximation appears to be a highly successful method (Davison, 1957; Dombrovsky, 1996a,b). According to this approximation, the scattering function is replaced by a sum of the isotropic component and the term describing the peak of forward scattering:
With the use of transport (tr) approximation, the RTE can be written in the same way as that for isotropic scattering; i.e., with Φλ ≡ 1:
where the “transport” scattering and extinction coefficients are defined as follows:
The transport approximation has been widely used in neutron transport and radiative transfer calculations for many years. The quality of this approach has been analyzed in earlier studies by Pomraning (1965), Bell et al. (1967), Potter (1970), and Crosbie and Davidson (1985). In the case of strong forward and backward scattering, one can introduce an additional delta function in the backward direction (Sjöstrand, 2001). According to Williams (1966) this scattering function was used already by Fermi.
One should also remember the simplest version of the delta-Eddington approximation (Joseph et al., 1976; Wiscombe, 1977; Modest and Azad, 1980; Davies, 1980; Azad and Modest, 1981; McKellar and Box, 1981; Mengüç and Viskanta, 1983; Truelove, 1984), which presents the scattering function as a combination of the linear function and delta function in the forward direction. The delta-Eddington approximation can also be completed by description of backward scattering in accordance with the backscattering model as was done by Modest and Azad (1980). In a more complex version of the delta-Eddington approximation, the remaining part of the scattering function (in addition to the forward scattering peak) is expressed as a truncated Legendre series (McKellar and Box, 1981; Crosbie and Davidson, 1985). An alternative approach for the one-dimensional azimuthally symmetric problem was suggested by Berdnik and Loiko (1999). This approach is based on the separation of the delta anisotropy in the azimuth-averaged scattering function.
The following analytical approximation for scattering functions suggested by Henyey and Greenstein (1941) is widely used in radiative transfer problems:
This simple relation enables one to describe anisotropic scattering including the case of strong forward scattering typical of particles of a size much greater than the radiation wavelength. It was proven by Pomraning (1988) that one should not choose parameter μλ from the condition of the best fitting of the real scattering function. Contrary, the value of μλ must be equal to the real value of the asymmetry factor of scattering. The latter choice minimizes the error in the RTE solution. Both van de Hulst (1968) and Hansen (1969) have shown that function (6) gives very accurate results for radiative transfer by multiple scattering of light in cloudy and hazy planetary atmospheres. Of course, the Henyey-Greenstein function is not the only smooth approximation of the scattering functions at an arbitrary asymmetry factor. For example, in the treatment of light scattering by blood, two other approximations of the same type have been suggested by Pedersen et al. (1976). The Henyey-Greenstein function also shows significant discrepancies compared with the Mie theory in the forward-peak range when asymmetry factor of scattering μλ > 0.8 (van de Hulst, 1957, 1981). The high values of μλ are typical of biomedical media. In this case, a new analytical function proposed by Liu (1994) is employed in radiative transfer calculations. In some specific cases of disperse systems, such as polydisperse large spherical particles, one can suggest more accurate analytical approximations for the scattering function as was done by Caldas and Semião (2001).
A reasonable choice of the scattering function approximation depends on the specifics of the radiation transfer problem to be solved. The experience of this author has shown that transport approximation is sufficiently good in heat transfer problems when the integral characteristics of the radiation field, such as radiation flux and the heat generation rate, are calculated. For this reason, the transport approximation is widely used for solving thermal radiation problems. It was shown by Dombrovsky et al. (1991) that scattering treatment with the use of the transport approximation provides a high accuracy of calculations not only for thermal radiation flux but also for the radiation intensity of a scattering medium in different directions. This allows developing an approximate method that employs a simplified approach to determine the radiation energy density at the first stage of iterative solution (Bobcôo, 1967; Edwards and Bobcôo, 1967; Adzerikho and Nekrasov, 1975; Adzerikho et al., 1979). This method was used by Dombrovsky and Barkova (1986) in calculations of thermal radiation of two-phase exhaust jets in rocket engines. A similar approach was also employed in the interpretation of dusty-plasma diagnostics results (Vladimirov et al., 1988) and in determination of a laser beam scattering in a two-phase erosional plume (Kolpakov et al., 1990; Dombrovsky et al., 1991).
Recent studies by Dombrovsky et al. (2005, 2006, 2007) have shown that transport approximation is a good approach for calculating hemispherical transmittance and reflectance of a collimated beam by samples of various scattering materials. It makes this approach useful in identification of the spectral radiative properties of real disperse materials. Of course, the approach based on transport approximation is not universal and more accurate models for scattering functions should be used in some special cases.
Adzerikho, K. S. and Nekrasov, V. P., Luminescence characteristics of cylindrical and spherical light-scattering media, Int. J. Heat Mass Transfer, vol. 18, no. 10, pp. 1131-1138, 1975.
Adzerikho, K. S., Antsulevich, V. I., Lapko, Ya. K., and Nekrasov, V. P., Luminescence of two-phase inhomogeneous media of cylindrical geometry, Int. J. Heat Mass Transfer, vol. 22, no. 1, pp. 131-136, 1979.
Azad, F. H. and Modest, M. F., Evaluation of the radiative heat flux in absorbing, emitting and linear-anisotropically scattering cylindrical media, ASME J. Heat Transfer, vol. 103, no. 2, pp. 350-356, 1981.
Bell, G. I., Hansen, G. E., and Sandmeier, H. A., Multitable treatment of anisotropic scattering in SN multigroup transport calculations, Nucl. Sci. Eng., vol. 28, pp. 376-383, 1967.
Belov, G. Ya., Effectiveness of backward scattering model in the radiation transfer theory in media with axisymmetrical scattering functions, High Temp., vol. 20, no. 6, pp. 1102-1109 (in Russian), 1982.
Berdnik, V. V. and Loiko, V. A., Modelling of radiative transfer in disperse layers of a medium with a highly stretched phase function, J. Quant. Spectrosc. Radiat. Transf., vol. 61, no. 1, pp. 49-57, 1999.
Bobcôo, R. P., Directional emissivities from a two-dimensional, absorbing-scattering medium: The semi-infinite slab, ASME J. Heat Transfer, vol. 89, no. 4, pp. 313-320, 1967.
Caldas, M. and Semião, V., A new approximate phase function for isolated particles and polydispersions, J. Quant. Spectrosc. Radiat. Transf., vol. 68, no. 5, pp. 521-542, 2001.
Crosbie, A. L. and Davidson, G. W., Dirac-delta function approximation to the scattering phase function, J. Quant. Spectrosc. Radiat. Transf., vol. 33, no. 4, pp. 391-409, 1985.
Davies, R., Fast azimuthally dependent model of the reflection of solar radiation by plane-parallel clouds, Appl. Opt., vol. 19, no. 2, pp. 250-255, 1980.
Davison, B., Neutron Transport Theory, London: Oxford University Press, 1957.
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996a.
Dombrovsky, L .A., Approximate methods for calculating radiation heat transfer in dispersed systems, Therm. Eng., vol. 43, no. 3, pp. 235-243, 1996b.
Dombrovsky, L. A. and Barkova, L. G., Solving the two-dimensional problem of thermal-radiation transfer in an anisotropically scattering medium using the finite element method, High Temp., vol. 24, no. 4, pp. 585-592, 1986.
Dombrovsky, L. A., Kolpakov, A. V., and Surzhikov, S. T., Transport approximation in calculating the directed-radiation transfer in an anisotropically scattering erosional flare, High Temp., vol. 29, no. 6, pp. 954-960, 1991.
Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.
Dombrovsky, L., Randrianalisoa, J., and Baillis, D., Modified two-flux approximation for identification of radiative properties of absorbing and scattering media from directional-hemispherical measurements, J. Opt. Soc. Am. A, vol. 23, no. 1, pp. 91-98, 2006.
Dombrovsky, L. A., Tagne, H. K., Baillis, D., and Gremillard, L., Near-infrared radiative properties of porous zirconia ceramics, Infrared Phys. Technol., vol. 51, no. 1, pp. 44-53, 2007.
Edwards, R. N. and Bobcôo, R. P., Radiant heat transfer from isothermal dispersion with isotropic scattering, ASME J. Heat Transfer, vol. 89, no. 4, pp. 300-308, 1967.
Hansen, J. E., Exact and approximate solutions for multiple scattering by cloudy and hazy planetary atmospheres, J. Atmos. Sci., vol. 26, no. 3, pp. 478-487, 1969.
Henyey, L. G. and Greenstein, J. L., Diffuse radiation in the galaxy, Astrophys. J., vol. 93, no. 1, pp. 70-83, 1941.
Joseph, J. H., Wiscombe, W. J., and Weinman, J. A., The Delta-Eddington approximation for radiative flux transfer, J. Atmos. Sci., vol. 33, no. 12, pp. 2452-2459, 1976.
Kolpakov, A. V., Dombrovsky, L. A., and Surzhikov, S. T., Transfer of directed radiation in an absorbing and anisotropically scattering medium, High Temp., vol. 28, no. 5, pp. 753-757, 1990.
Liu, P., A new phase function approximating to Mie scattering for radiative transport equations, Phys. Med. Biol., vol. 39, no. 6, pp. 1025-1036, 1994.
McKellar, B. H. J. and Box, M. A., The scaling group of the radiative transfer equation, J. Atmos. Sci., vol. 38, no. 5, pp. 1063-1068, 1981.
Mengüç, M. P. and Viskanta, R., Comparison of radiative transfer approximations for a highly forward scattering planar medium, J. Quant. Spectrosc. Radiat. Transf., vol. 29, no. 5, pp. 381-394, 1983.
Modest, M. F., Radiative Heat Transfer, 2nd ed., New York: Academic, 2003.
Modest, M. F. and Azad, F. H., The influence and treatment of Mie-anisotropic scattering in radiative heat transfer, ASME J. Heat Transfer, vol. 102, no. 1, pp. 92-98, 1980.
Özişik, M. N., Radiative Transfer and Interaction with Conduction and Convection, New York: Wiley, 1973.
Pedersen, G. D., McCormick, N. J., and Reynolds, L. O., Transport calculations for light scattering in blood, Biophys. J., vol. 16, 199-207, 1976.
Pomraning, G. C., A flexible numerical approach to linear transport problems, Trans. Am.. Nucl. Soc., vol. 8, 488-489, 1965.
Pomraning, G. C., On the Henyey-Greenstein approximation to scattering phase function, J. Quant. Spectrosc. Radiat. Transf., vol. 39, no. 2, pp. 109-113, 1988.
Potter, J. F., The delta function approximation in radiative transfer theory, J. Atmos. Sci., vol. 27, no. 6, pp. 943-949, 1970.
Sjöstrand, N. G., On an alternative way to treat highly anisotropic scattering, Ann. Nucl. Energy, vol. 28, no. 4, pp. 351-355, 2001.
Truelove, J. S., The two-flux model for radiative transfer with strongly anisotropic scattering, Int. J. Heat Mass Transfer, vol. 27, no. 3, pp. 464-466, 1984.
van de Hulst, H. C., Light Scattering by Small Particles, New York: Wiley, 1957.
van de Hulst, H. C., Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers, J. Comput. Phys., vol. 3, no. 2, pp. 291-306, 1968.
van de Hulst, H. C., Light Scattering by Small Particles, New York: Dover, 1981.
Vladimirov, V. I., Gorshkov, Yu. A., and Krymasov, A. V., Effects of anisotropic scattering on dusty-plasma temperature measurements, High Temp., vol. 26, no. 2, pp. 266-272, 1988.
Williams, M. M. R., The Slowing Down and Thermalization of Neutrons, Amsterdam: North-Holland, 1966.
Wiscombe, W. J., The delta-M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions, J. Atmos. Sci., vol. 34, no. 9, pp. 1408-1422, 1977.