*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*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).

_{N}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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