Radiation transfer equation

Note that using these assumptions does not make the mathematical solution of the problem significantly more difficult, for again, qr—and hence radiation transfer—is a known constant and enters only in the boundary condition. The differential equations describing the processes are not altered. 

According to our assumption the incident radiation transfers part of its momentum to the atom and electon without change of direction and we have, therefore, the equations ... 

The intensity of radiation, /, decreases along its path s due to absorption (scattering in plasma can be neglected) and increases because of spontaneous and stimulated emission. The radiation transfer equation in quasi-equilibrium plasma can then be presented as... 

Y. R. Sivathanu, J. P. Gore A Discrete Probability Function Method for the Equation of Radiative Transfer, J. Quant. Spectrosc. Radiat. Transfer 49(3), 269-280 (1993). 

Thus, three optical parameters need to be estimated to solve the radiation transfer equation ... 

Solving the radiation transfer equation is then the challenge of explaining the behavior of light in turbid media. By defining assumptions (that are beyond the scope of this chapter), the radiation transfer equation can be simplified and solved by different approaches ... 

Schuster solution [24] It consists in summarizing the radiation transfer by forward and reverse light flux with respect to the incident light. After derivation, the following equation is obtained ... 

Kubelka-Munk solution [25] While the Schuster solution is for one particle, Kubelka-Munk generalized the absorption and scattering phenomena to the whole sample (K and S) and solved the radiation transfer equation in a different way than Schuster. More information about the derivation is provided in Griffiths and Dahm [23]. The solution is... 

Finally, ofher solutions to the radiation transfer equation have been proposed using simulation to calculate the absorption and scattering coefficients. An example of simulation framework has been proposed by Wang et al. [26]. 

The change in intensity of a beam of radiation of selected wavelength in a path length dz within the medium is given by the radiation transfer equation, Eq. 2... 

Numerous researchers have developed their own simplified solutions to the radiation transfer equation. The first solution were Schuster s equations (3), in which, for simplification, the radiation field was divided into two opposing radiation fluxes (+z and -z directions). The radiation flux in the +z direction, perpendicular to the plane, is represented by /, and the radiation flux in the -z direction, resulting from scattering, is represented by J. The same approximation was used by Kubelka and Munk in their equations, in the exponential (4) as well as in the hyperbolic solution (5). In the exponential solution by Kubelka-Munk, a flat layer of thickness z, which scatters and absorbs radiation, is irradiated in the -z direction with monochromatic diffuse radiation of flux I. In an infinitesimal layer of thickness dz, the radiation fluxes are going in the + direction J and in the -direction I. The average absorption in layer on path length dz is named K the scattering coefficient is S. Two fundamental equations follow directly ... 

Much work that has been done on two-flux treatment of the diffuse reflection of radiation has evolved from a general radiation transfer equation. In simple terms, a radiation transfer equation can be written as... 

An equation such as this describes the change in intensity, dl, of a beam of radiation of a given wavelength in a sample, the density of which is p and for which the pathlength is ds. k corresponds to the attenuation coefficient for the total radiation loss whether that loss is due to scattering or absorption. The general form of the radiation transfer equation that is used in the derivation of most phenomenological theories considers only plane-parallel layers of particles within the sample and can be written as... 

Variables Used in the Development of Kubelka s Simplified Solution to the Radiation Transfer Equation... 

Because the simplified solufion obfained by Kubelka is a two-constant equation and therefore experimentally testable, and because so many other workers derivations are derivable from Kubelka and Munk s work, their solution is the most widely accepted, tested and used. Other workers have derived solutions to the radiation transfer equation that are more complicated than these two-constant formulas. For example, a third constant has been added to account for different fractions of forward and back scattering [36]. Ryde [37,38] included four constants since a difference in the scattering between incident light and internally diffused light is assumed, while Duntley [39] developed a model with eight constants, as a difference between both the absorption and scattering coefficients due to incident and internally diffused radiation was assumed. However, none of these theories is readily applicable in practice, and therefore the treatment of Kubelka is most often applied. 

Another two-constant approach, based on a discrete ordinate approximation of the radiation transfer equation [17,40] was recently applied to describe the diffuse reflectance in the NIR [41,42]. In this... 

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