P_{1} Approximation of the Spherical Harmonics Method
Following from: Differential approximations
Leading to: The simplest approximations of double spherical harmonics, Radiation of an isothermal plane-parallel layer, Radiative equilibrium in a plane-parallel layer, Radiation of a nonisothermal layer of scattering medium, An estimate of P_{1} approximation error for optically inhomogeneous media, Diffusion approximation in multi-dimensional radiative transfer problems
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:
(1) |
By multiplying the radiative transfer equation
(2) |
with , and integrating it over a solid angle, one can find that the spectral radiation flux is related to the spectral radiation energy density I_{λ}^{0} by the following equation:
(3) |
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
(4) |
leads to the following equation for the spectral radiation energy density:
(5) |
The Marshak boundary condition (Marshak, 1947) is usually employed in the P_{1} approximation. In the absence of walls and external sources of radiation, this condition has the following form (Case and Zweifel, 1967):
(6) |
where is the external normal to a boundary surface. According to Pomraning (1964) and Shokair and Pomraning (1981), we consider also the following corrected boundary condition:
(7) |
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.
We do not consider high-order approximations (P_{N}) of the spherical harmonics method in this article. The reader can refer to the available literature on this subject (Kofink, 1959; Bayazitoglu and Higenyi, 1979; Yücel and Bayazitoglu, 1983; Karp et al., 1980; Ou and Liou, 1982; Khan and Thomas, 2005; Atalay, 2006). Note that the P_{N} approximation is equivalent to the discrete ordinate method based on the corresponding order of the Gaussian quadrature for the integral term of the radiative transfer equation (RTE) (Barichello and Siewert, 1998).
The P_{1} approximation is a simplified approach, which is expected to be fairly good with absorbing and highly scattering media at large optical distances from boundaries or interfaces that have a strong variation of temperature and radiative characteristics of the medium. Nevertheless, the simplicity and clear physical sense of P_{1} have attracted the attention of many researchers who have suggested various modifications of this approach as applied to some specific radiative transfer problems. One can remember the modified differential approximation (MDA) by Olfe (1967, 1968, 1970) [see also Glatt and Olfe (1973) and an improvement of the MDA by Wu et al. (1987)] and the improved differential approximation (IDA) by Modest (1974, 1975, 1990). Both methods separate the radiation emitted by the walls from the thermal radiation of the medium. However, these methods are not as simple as the ordinary P_{1} and require the evaluation of some integral correction factors (Modest, 2003). Formal attempts to improve the P_{1} accuracy by modifying the boundary conditions were made by Liu et al. (1992a,b) and Su (2000). We will not consider the possible modifications of the boundary conditions for P_{1} and high-order approximations of the same method here. Some additional results on this subject can be found by Lii and Özişik (1973) and Chien and Wu (1991), and also by Modest (2003). In one-dimensional problems, the accuracy can also be improved by applying the P_{1} approximation separately to different solid angle ranges, as was done by Mengüç and Subramaniam (1992).
The most widely used approximations are the P_{1} and P_{3} approximations of the spherical harmonics method. However, high-order P_{N} approximations are also employed in solving one-dimensional problems. For example, Mengüç and Viskanta (1988) employed the P_{9} approximation to solve the RTE for a stratified fly-ash cloud near the walls of a pulverized-coal furnace to study the effect of spectral radiative properties on the blockage of radiation to the walls. High-order solutions, up to P_{11}, for gray medium between concentric spheres have been considered by Tong and Swathi (1987) and by Li and Tong (1990). For solving the atmospheric problems, Karp et al. (1980) employed the spherical harmonics approximation up to 99th order (P_{99}) with up to 100 terms of the scattering function and considering up to 15 homogeneous layers with any optical thickness. In solving one-dimensional radiative transfer problems, many researchers are not limited by diffusion approximation, but employ it only for one of the solution stages–such as at the initial approximation, as reported by Adzerikho and Nekrasov (1975) and Sutton and Özişik (1979), or at the first iterative step, as was done by Abramson and Lisin (1985). Mengüç and Viskanta (1985, 1986) limited their development to the P_{3} approximation but considered the three-dimensional problem in Cartesian coordinates and a two-dimensional axisymmetric problem. The diffusion approximation has been used in similarity analysis of radiative transfer problems in scattering media (Popov, 1980; McKellar and Box, 1981; Lee and Buckius, 1982; Kim and Lee, 1990; Liu et al., 1992a,b; Tagne and Baillis, 2005) and in obtaining analytical solutions for simple geometrical forms of a radiating volume (Popov, 1980; Adzerikho et al., 1979).
Currently, many commercial CFD codes have the P_{1} approximation as an optional solution technique for radiation calculations. The engineering calculation of heat transfer in combustion is one of the most well-known applications of P_{1} and P_{3} approximations (Viskanta and Mengüç, 1987; Onda, 1995; Dombrovsky, 1996a; Marakis et al., 2000; Viskanta, 2005; Klason et al., 2008). The P_{1} application is widely used in solving various combined heat transfer problems because this approximation usually gives a good estimate of the radiation energy density in scattering media. As examples of the use of the diffusion approximation in modeling of radiative–conductive heat transfer in thermal insulation, one should refer to the studies by Petrov (1993) and Dombrovsky (1996b). In the case of buoyant flow of an optically thick fluid, it was shown by Derby et al. (1998) that the P_{1} approximation yields surprisingly accurate results compared with the solutions obtained from the rigorous treatment. The same approach is employed in modeling radiative transfer in arcs and discharges (Freton et al., 2002; Dixon et al., 2004; Segur et al., 2006), and even in calculations of the radiation field around blunt-body vehicles in the atmosphere (Hartung and Hassan, 1993).
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).
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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.
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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. 25â€“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 diffusionâ€“like equations, Prog. Nucl. Energy, vol. 35, no. 1, pp. 1â€“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. 1131â€“1138, 1975.
- Adzerikho, K. S., Antsulevich, V. I., Nekrasov, V. P., and Trofimov, V. P., Modeling radiantâ€“heatâ€“transfer problems in media of nonplane geometry, J. Eng. Phys. Thermophys., vol. 36, no. 2, pp. 148â€“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. 100â€“108, 2006.
- Aydin, E. D., Katsimichas, S., and de Oliveira, C. R. E., Timeâ€“dependent diffusion and transport calculations using a finiteâ€“elementâ€“spherical harmonics method, J. Quant. Spectrosc. Radiat. Transf., vol. 95, no. 3, pp. 349â€“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. 79â€“84, 1998.
- Baudoux, P. E., Roblin, A., and Chervet, P., New approach for radiativeâ€“transfer computations in axisymmetric scattering hot media, J. Thermophys. Heat Transfer, vol. 15, no. 3, pp. 317â€“325, 2001.
- Bayazitoglu, Y. and Higenyi, J. Higherâ€“order differential equations of radiative transfer: P_{3} approximation, AIAA J., vol. 17, no. 4, pp. 424â€“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. 439â€“446, 1991.
- Davison, B., Neutron Transport Theory, London: Oxford University Press, 1957.
- Derby, J. J., Brandon, S., and Salinger, A. G., The diffusion and P1 approximations for modeling buoyant flow of an optically thick fluid, Int. J. Heat Mass Transfer, vol. 41, no. 11, pp. 1405â€“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. 3309â€“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. 255â€“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. 408â€“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. 3111â€“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. 429â€“446, 1998.
- Frank, M., Klar, A., Larsen, E. W., and Yasuda, S., Timeâ€“dependent simplified P_{N} approximation to the equations of radiative transfer, J. Comput. Phys., vol. 226, no. 2, pp. 2289â€“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. 115â€“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. 881â€“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. 220â€“227, 1993.
- Jeans, J. H., The equations of radiative transfer of energy, Mon. Not. R. Astron. Soc., vol. 78, no. 1, pp. 28â€“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. 391â€“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. 1â€“3, pp. 130â€“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. 721â€“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. 10â€“11, pp. 2141â€“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. 473â€“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. 68â€“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. 239â€“251, 1990.
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