Vector radiative transfer equation

M. 1. Mishchenko (2002). Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles a microphysical derivation from statistical electromagnetics, Opt. 41,7114-7134. 

It is important to realize that the solution of the radiative transfer equation is required only to obtain the divergence of the radiative flux vector that is a total quantity (i.e., inte-... 

The DO-FV model [ 17] solves the radiative transfer equation (RTF) as a field equation for a finite number of discrete solid angles each associated with a vector direction s fixed in the global Cartesian system. The procedure involves the solution of as many transport equations as there are solid angles. 

Equation (5-121) specifically includes those zones which may not have a direct view of the refractory. When Qr = 0, the refractory surface is said to be in radiative equilibrium with the entire enclosure. Equation (5-121) is indeterminate if = 0. If ,. = 0, rdoes indeed exist and may be evaluated with use of the statement Er = Hr = Wr. It transpires, however, that I, is independent of Erfor all 0 < , < 1. Moreover, since Wr = Hr when Q, = 0, for all 0 < ty < I, the value specified for , is irrelevant to radiative transfer in the entire enclosure. In particular it follows that if Qr = 0, then the vectors W, H, and Q for the entire enclosure are also independent of all 0 < e, < 1.0. A surface zone for which e, = 0 is termed a perfect diffuse mirror. A perfect diffuse mirror is thus also an adiabatic surface zone. The matrix method automatically deals with all options for flux and adiabatic refractory surfaces. 

In addition to Eq. (34) (that is exact), the construction of macroscopic models consisting of a closed set of equations for the moments of the distribution function (or the intensity), usually requires to formulate approximations. In fluid mechanics, this approximation leads, for example, to the Navier-Stokes equation. The most common approximate macroscopic radiative models describe radiative transfer with heat-like equations (eg, see the Rosseland approximation and the PI approximation). Among them, the PI approximation leads to Pick s equation of the flux density vector Jr (see Eq. (73) substituting Pick s equation into Eq. (34) yields the following heat-like equation for the irradiance G (in the absence of a source term) ... 

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