## Two-Flux Approximation

* Following from: *Differential approximations

The simplest approximation for one-dimensional radiative transfer problems in a plane-parallel layer of absorbing and scattering medium is based on the assumption of isotropic radiation intensity over the forward and backward hemispheres. This model was proposed independently by Schuster (1905) and Schwarzschild (1906), and it is also called the Schuster-Schwarzschild approximation. In this article, we follow the majority of the publications and use the name “two-flux approximation” for this model.

The two-flux approximation is usually used for isotropically scattering media. Of course, one can also consider the case of the transport scattering function. The radiation intensity is expressed in the form

(1) |

where *z* is the coordinate across the layer, 0 *< z < d*, and θ is the angle measured from the *z* axis. In the transport approximation, the one-dimensional radiative transfer equation (RTE) can be written as follows:

(2) |

In two-flux approximation, the integration over the hemispheres leads to the following coupled ordinary differential equations instead of Eq. (2):

(3a) |

(3b) |

or

(4a) |

(4b) |

where

(5) |

Obviously, spectral radiation energy density *I*_{λ}^{0} = *g*_{0} and spectral radiation flux *q*_{λ} = -*h*_{0}/2. Equations (5) and (6) can be written as follows:

(6a) |

(6b) |

where *D*_{λ} = 1/(4β_{λ}^{tr}). One can see that the two-flux approximation leads to the typical relations of the diffusion approximation. The boundary conditions for the second-order differential [Eq. (8)] depend on the physical problem statement. Some variants of these conditions will be considered below. In all cases, the solution to the resulting boundary-value problem can be easily obtained even for variable radiative properties of the medium. Obviously, the two-flux approximation can also be employed in one-dimensional problems with axial or spherical symmetry (Rekin, 1978). The equations of two-flux approximation can be formally derived for the arbitrary scattering function, and the resulting formulation will include some characteristics of the scattering function (Brewster and Tien, 1982). However, it does not mean that the complex scattering function can be adequately taken into account in this approach.

In earlier studies by Kubelka and Munk (Kubelka and Munk, 1931; Kubelka, 1948), the phenomenological two-flux model (irrespective of the RTE) has been proposed for analyzing the reflectance and transmittance of paint layers or other scattering coatings. This approach, known as the Kubelka-Munk theory, is almost universally used in the paint industry. The equations of this approximation are usually written in the form:

(7a) |

(7b) |

where *q*^{∓} = *I*_{0}^{∓}/2 are the radiation fluxes in the backward and forward hemispheres, and *K*_{λ} and *S*_{λ} are the coefficients of absorption and backscattering, respectively. One can see that Eqs. (10) and (11) coincide with Eq. (3) in the case of the medium’s own negligible radiation. It is sufficient to assume that *K*_{λ} = 2α_{λ} and *S*_{λ} = σ_{λ}^{tr}. The specifics of paint coatings become important when taking into account the light refraction at the coating/air interface (which is the so-called Saunderson correction) (Saunderson, 1942) and possible illumination of the interface by a collimated external radiation. The latter is described by a modified approach known as the four-flux model. According to the four-flux model, two additional collimated components of the radiation are introduced in the forward and backward directions. Both the two-flux (Kubelka-Munk) and four-flux models are widely used in characterizing various paints and thin composite coatings in the visible and infrared spectral ranges. The reader can refer to the optical literature for details on these models and their present-day applications (Fukshansky and Kazarinova, 1980; Hecht, 1983; Maheu et al., 1984; Maheu and Gouesbet, 1986; Nobbs, 1985; Latimer and Noh, 1987; Mandelis and Grossman, 1992; Prishivalko, 1996; Orel et al., 1997; Orel and Gunde, 2001; Vargas and Niklasson, 1997a,b; Vargas et al., 1998, 2000; Vargas, 1998, 1999; Gunde and Orel, 2000; Rozé et al., 2001; Yang, 2003; Limare et al., 2003; Yang and Kruse, 2004; Levinson et al., 2005; Liu et al., 2005; Sokoletsky, 2005; Yang et al., 2005; Erdström, 2007; Hébert et al., 2007; Murphy, 2007). Note that the physical interpretation and accuracy of the Kubelka-Munk theory are also discussed in a recent paper by Kokhanovsky (2007). Additional information on engineering applications of the traditional two-flux method (the Schuster-Schwarzschild approximation) and the use of the same idea for multi-dimensional problems can be found in other works (Siegel and Spuckler, 1994; Spuckler and Siegel, 1996; Dembele et al., 2000; Dubroca and Klar, 2002; Gusarov and Kruth, 2005; Ripoll and Wray, 2005; Dombrovsky et al., 2005, 2006, 2007; Jensen et al., 2007).

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