In this article, we consider the well-known simple approximation
for radiative transfer in scattering media. The spherical harmonics method was
developed by Jeans (1917) in his work on radiative transfer in stars. Further
description of the method, as it applies to radiative transfer, was given by
Kourganoff (1952, 1963). Application of the spherical harmonics method to
neutron transport problems was considered by Davison (1957) and Murray (1957).
With the use of the spherical harmonics method, the spectral radiation
intensity is presented in a series of spherical functions. In the first
approximation of this method, *P*_{1}, which is the usual version
of the diffusion approximation, the following linear dependence is assumed:

Note that Eq. (3) is obtained for the arbitrary scattering function but it is the same as that for the transport approximation. This means that the *P*_{1} approximation is insensitive to the details of the scattering function, and the asymmetry factor of scattering μ_{λ} is the only characteristic of the scattering anisotropy taken into account in this approach. The combination of Eq. (3) with the radiation balance equation

Hereafter, condition (7) will be called the Pomraning boundary condition. Both variants of the boundary condition will be considered below in analysis of *P*_{1} approximation accuracy. It should be noted that the approximation, which is equivalent to *P*_{1} for one-dimensional problems, was developed independently by Milne (1930) and Eddington (1959), and this model is also known as the Milne–Eddington approximation.

The spherical harmonics approximation is used in analysis of thermal radiation from rocket plumes (Baudoux et al., 2001), in calculations of neutron transport in nuclear engineering (Ackroyd et al., 1999; Ziver et al., 2005), and in complex atmospheric problems (Nakajima and Tanaka, 1988; Evans, 1993, 1998; Trasi et al., 2004) including in image formation by observing some discrete light sources through clouds (Zardecki et al., 1986). In the last case, even *P*_{1} may give fairly accurate predictions for highly scattering optically dense media. Some additional references concerning the use of the diffusion approximation in solving multi-dimensional problems including present-day biological and medical studies will be given in the article Diffusion Approximation in Multi-Dimensional
Problems.

It should be noted that
spherical harmonics and the simplest diffusion
approximation are considered as a present-day tool for unsteady problems,
including the case when a pulsed laser beam illuminates a highly scattering
medium (Olson et al., 2000; Aydin et al., 2005). In the latter case, the
diffusion approximation can be employed to the radiation scattered from the
laser beam. Some modifications of spherical harmonics and the *P*_{1}
approximation as it applies to unsteady problems can be found in the studies done
by Morel (2000) and Frank et al. (2007).

#### REFERENCES

Abramson, M. N. and Lisin, F. N., Approximate solution of transfer equation in radiating, absorbing, and scattering layer, *Ind. Heat Eng.*, vol. **7**, no. 1, pp. 25–30 (in Russian), 1985.

Ackroyd, R. T., de Oliveira, C. R. E., Zolfaghari, A., and Goddard, A. J. H., On a rigorous resolution of the transport equation into a system of diffusion–like equations, *Prog. Nucl. Energy*, vol. **35**, no. 1, pp. 1–64, 1999.

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., Nekrasov, V. P., and Trofimov, V. P., Modeling radiant–heat–transfer problems in media of nonplane geometry, *J. Eng. Phys. Thermophys.*, vol. **36**, no. 2, pp. 148–158, 1979.

Atalay, M. A., *P*_{N} solutions of radiative heat transfer in a slab with reflective boundaries, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **101**, no. 1, pp. 100–108, 2006.

Aydin, E. D., Katsimichas, S., and de Oliveira, C. R. E., Time–dependent diffusion and transport calculations using a finite–element–spherical harmonics method, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **95**, no. 3, pp. 349–363, 2005.

Barichello, L. B. and Siewert, C. E., On the equivalence between the discrete ordinates and the spherical harmonics methods in radiative transfer, *Nucl. Sci. Eng.*, vol. **130**, no. 1, pp. 79–84, 1998.

Baudoux, P. E., Roblin, A., and Chervet, P., New approach for radiative–transfer computations in axisymmetric scattering hot media, *J. Thermophys. Heat Transfer*, vol. 15, no. 3, pp. 317–325, 2001.

Bayazitoglu, Y. and Higenyi, J. Higher–order differential equations of radiative transfer: *P*_{3} approximation, *AIAA J.*, vol. **17**, no. 4, pp. 424–431, 1979.

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

Chien, H. H. and Wu, C. Y., Modified differential approximation for radiative transfer in a scattering medium with specularly reflecting boundaries, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **46**, no. 5, pp. 439–446, 1991.

Davison, B., *Neutron Transport Theory*, London: Oxford University Press, 1957.

Derby, J. J., Brandon, S., and Salinger, A. G., The diffusion and *P*1 approximations for modeling buoyant flow of an optically thick fluid, *Int. J. Heat Mass Transfer*, vol. **41**, no. 11, pp. 1405–1415, 1998.

Dixon, C. M., Yan, J. D., and Fang, M. T. C., A comparison of three radiation models for the calculation of nozzle arcs, *J. Phys. D: Appl. Phys.*, vol. **37**, no. 23, pp. 3309–3318, 2004.

Dombrovsky, L. A., A theoretical investigation of heat transfer by radiation under conditions of two-phase flow in a supersonic nozzle, *High Temp.*, vol. **34**, no. 2, pp. 255–262, 1996a.

Dombrovsky, L. A. Quartz-fiber thermal insulation: Infrared radiative properties and calculation of radiative-conductive heat transfer, *ASME J. Heat Transfer*, vol. **118**, no. 2, pp. 408–414, 1996b.

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

Evans, K. F., Two-dimensional radiative transfer in cloudy atmospheres: The spherical harmonic spatial grid method, *J. Atmos. Sci.*, vol. **50**, no. 18, pp. 3111–3124, 1993.

Evans, K. F., The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer, *J. Atmos. Sci.*, vol. **55**, no. 3, pp. 429–446, 1998.

Frank, M., Klar, A., Larsen, E. W., and Yasuda, S., Time–dependent simplified *P*_{N} approximation to the equations of radiative transfer, *J. Comput. Phys.*, vol. **226**, no. 2, pp. 2289–2305, 2007.

Freton, P., Gonzalez, J. J., Gleizes, A., Peyret, F. C., Caillibotte, G., and Delzenne, M., Numerical and experimental study of the plasma cutting torch, *J. Phys. D: Appl. Phys.*, vol. **35**, no. 2, pp. 115–131, 2002.

Glatt, L. and Olfe, D. B., Radiative equilibrium of a gray medium in a rectangular enclosure, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **13**, no. 9, pp. 881–895, 1973.

Hartung, L. C. and Hassan, H. A., Radiation transport around axisymmetric blunt body vehicles using a modified differential approximation, *J. Thermophys. Heat Transfer*, vol. **7**, no. 2, pp. 220–227, 1993.

Jeans, J. H., The equations of radiative transfer of energy, *Mon. Not. R. Astron. Soc.*, vol. **78**, no. 1, pp. 28–36, 1917.

Karp, A. H., Greenstadt, J., and Fillmore, J. A., Radiative transfer through an arbitrarily thick, scattering atmosphere, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **24**, no. 5, pp. 391–406, 1980.

Khan, T. and Thomas, A., Comparison of *P*_{N} or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices, *Opt. Commun.*, vol. **255**, no. 1–3, pp. 130–166, 2005.

Kim, T. K. and Lee, H. S., Scaled isotropic results for two-dimensional anisotropic scattering media, *ASME J. Heat Transfer*, vol. **112**, no. 3, pp. 721–727, 1990.

Klason, T., Bai, X. S., Bahador, M., Nilsson, T. K., and Sundén, B., Investigation of radiative heat transfer in fixed bed biomass furnaces, *Fuel*, vol. **87**, no. 10–11, pp. 2141–2153, 2008.

Kofink, W., Complete spherical harmonics solution of the Boltzmann equation for neutron transport in homogeneous media with cylindrical geometry, *Nucl. Sci. Eng.*, vol. **6**, pp. 473–486, 1959.

Kourganoff, V. *Basic Methods in Transfer Problems*, Oxford, UK: Clarendon, 1952.

Kourganoff, V. *Basic Methods in Transfer Problems*, New York: Dover, 1963.

Lee, H. and Buckius, R. O., Scaling anisotropic scattering in radiation heat transfer for a planar medium, *ASME J. Heat Transfer*, vol. **104**, no. 1, pp. 68–75, 1982.

Li, W. and Tong, T. W., Radiative heat transfer in isothermal spherical media, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **43**, no. 3, pp. 239–251, 1990.

Lii, C. C. and Özişik, M. N., Hemispherical reflectivity and transmissivity of an absorbing, isotropically scattering slab with a reflecting boundary, *Int. J. Heat Mass Transfer*, vol. **16**, no. 3, pp. 685–690, 1973.

Liu, F., Swithenbank, J., and Garbett, E. S., The boundary condition of the *P*_{N} approximation used to solve the radiative transfer equation, *Int. J. Heat Mass Transfer*, vol. **35**, no. 10, pp. 2043–2052, 1992a.

Liu, F., Garbett, E. S., and Swithenbank, J., Effects of anisotropic scattering on radiative heat transfer using the *P*1 approximation, *Int. J. Heat Mass Transfer*, vol. **35**, no. 10, pp. 2491–2499, 1992b.

Marakis, J. G., Papapavlou, C., and Kakaras, E., A parametric study of radiative heat transfer in pulverised coal furnaces, *Int. J. Heat Mass Transfer*, vol. **43**, no. 16, pp. 2961–2971, 2000.

Marshak, R. E., Note on the spherical harmonics method as applied to the Milne problem for a sphere, *Phys. Rev.*, vol. **71**, no. 7, pp. 443–446, 1947.

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., Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous, anisotropically scattering media, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **33**, no. 6, pp. 533–549, 1985.

Mengüç, M. P. and Viskanta, R., Radiative transfer in axisymmetric finite cylindrical enclosures, *ASME J. Heat Transfer*, vol. **108**, no. 2, pp. 271–276, 1986.

Mengüç, M. P. and Viskanta, R., Effect of fly–ash particles on spectral and total radiation blockage, *Combust. Sci. Technol.*, vol. **60**, no. 1–3, pp. 97–115, 1988.

Mengüç, M. P. and Subramaniam, S., Radiative transfer through an inhomogeneous fly–ash cloud: Effect of temperature and wavelength dependent optical properties, *Numer. Heat Transfer*, Part A, vol. **21**, no. 3, pp. 261–273, 1992.

Milne, F. A., Thermodynamics of the Stars, in *Handbuch der Astrophysik*, Berlin: Springer, pp. 65–255, 1930.

Modest, M. F., Two-dimensional radiative equilibrium of a gray medium in a plane layer bounded by gray non-isothermal walls, *ASME J. Heat Transfer*, vol. **96**, no. 4, pp. 483–488, 1974.

Modest, M. F., Radiative equilibrium in a rectangular enclosure bounded by gray walls, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **15**, no. 1, pp. 445–461, 1975.

Modest, M. F., The improved differential approximation for radiative transfer in multi–dimensional media, ASME *J. Heat Transfer*, vol. **112**, no. 3, pp. 819–821, 1990.

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

Morel, J. E., Diffusion-limit asymptotics of the transport equation, the *P*_{1/3} equations, and two flux–limited diffusion theories, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **65**, no. 5, pp. 769–778, 2000.

Murray, R. L., *Nuclear Reactor Physics*, Engelwood Cliffs, NJ: Prentice Hall, 1957.

Nakajima, T. and Tanaka, M., Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncated approximation, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **40**, no. 1, pp. 51–69, 1988.

Olfe, D. B., A modification of the differential approximation for radiative transfer, *AIAA J.*, vol. **5**, no. 4, pp. 638–643, 1967.

Olfe, D. B., Application of a modified differential approximation to radiative transfer in a gray medium between concentric spheres and cylinders, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **8**, no. 3, pp. 899–907, 1968.

Olfe, D. B., Radiative equilibrium of a gray medium bounded by nonisothermal walls, in *Prog. Astronaut. Aeronaut.*, vol. **23**, pp. 295–317, 1970.

Olson, G. L., Auer, L. H., and Hall, M. L., Diffusion, *P*_{1}, and other approximate forms of radiation transport, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **64**, no. 6, pp. 619–634, 2000.

Onda, K., Prediction of scattering effect by ash polydispersion on spectral emission from coal-fired MHD combustion gas, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **53**, no. 4, pp. 381–395, 1995.

Ou, S. Ch. S. and Liou, K. N., Generalization of the spherical harmonics method to radiative transfer in multi-dimensional space, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **28**, no. 4, pp. 271–288, 1982.

Petrov, V. A., Complex approach to a radiative–conductive heat transfer problem in scattering semitransparent materials using a diffusive approximation as the base, *J. Eng. Phys. Thermophys.*, vol. **64**, no. 6, pp. 583–589, 1993.

Pomraning, G. C., Variational boundary conditions for the spherical harmonics approximation to the neutron transport equation, *Ann. Phys.*, vol. **27**, no. 2, pp. 193–215, 1964.

Popov, Yu. A., Radiation from scattering volumes having simple geometries, *High Temp.*, vol. **18**, no. 3, pp. 450–453, 1980.

Ségur, P., Bourdon, A., Marode, E., Bessieres, D., and Paillol, J. H., The use of an improved Eddington approximation to facilitate the calculation of photoionization in steamer discharges, *Plasma Sources Sci. Technol.*, vol. **15**, no. 4, pp. 648–660, 2006.

Shokair, I. R. and Pomraning, G. C., Boundary conditions for differential approximations, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **25**, no. 4, pp. 325–337, 1981.

Su, B., More on boundary conditions for differential approximations, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **64**, no. 4, pp. 409–419, 2000.

Sutton, W. H. and Özişik, M. N., An iterative solution for anisotropic radiative transfer in a slab, ASME J. Heat Transfer, vol. 101, no. 4, pp. 695–698, 1979.

Tagne, H. T. K. and Baillis, D., Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence, *J. Quant. Spectrosc. Radiat. Transf.*, vol. **93**, no. 1-3, pp. 103–113, 2005.

Tong, T. W. and Swathi, P. S., Radiative heat transfer in emitting–absorbing–scattering spherical media, *J. Thermophys. Heat Transfer*, vol. **1**, no. 2, pp. 162–170, 1987.

Trasi, N. S., de Oliveira, C. R. E., and Haigh, J. D., A finite element–spherical harmonics model for radiative transfer in inhomogeneous clouds: Part I. The EVENT model, *Atmos. Res.*, vol. **72**, no. 1–4, pp. 197–221, 2004.

Viskanta, R., *Radiative Transfer in Combustion Systems: Fundamentals and Applications*, New York: Begell House, 2005.

Viskanta, R. and Mengüç, M. P., Radiation heat transfer in combustion systems, *Prog. Energy Combust. Sci.*, vol. **13**, no. 2, pp. 97–160, 1987.

Wu, C. Y., Sutton, W. H., and Love, T. J., Successive improvement of the modified differential approximation in radiative transfer, *J. Thermophys. Heat Transfer*, vol. **1**, no. 4, pp. 296–300, 1987.

Yücel, A. and Bayazitoglu, Y., *P*_{N} approximation for radiative heat transfer in a nongray medium, *AIAA J.*, vol. **21**, no. 8, pp. 1196–1203, 1983.

Zardecki, A., Gerstl, S. A. W., and DeKinder, R. E., Jr., Two- and three-dimensional radiative transfer in the diffusion approximation, *Appl. Opt.*, vol. **25**, no. 19, pp. 3508–3514, 1986.

Ziver, A. K., Shahdatullan, M. S., Eaton, M. D., de Oliveira, C. R. E., Umpleby, A. P., Pain, C. C., and Goddard, A. J. H., Finite element spherical harmonics (*P*_{N}) solutions of the three-dimensional Takeda benchmark problems, *Ann. Nucl. Energy*, vol. **32**, no. 9, pp. 925–948, 2005.