Radiative transfer theory

According to the ground rules laid down at the beginning of this book, multiple scattering is excluded from consideration. But it is not always prudent to pretend that multiple scattering does not exist. Fortunately, it is almost trivial—the mathematical apparatus of radiative transfer theory is unnecessary—to extend our treatment of scattering and circular polarization to multiple scattering media, and in this instance it is worth the small amount of effort required to do so. 

Slurry photocatalytic reactors are characterized by the presence of catalyst particles with sizes in the micrometer and submicrometer range. Such small particles produce in general a fair amoimt of radiation scattering, as will be discussed later. Radiative transfer theory (Mahan, 2002 Modest, 2003) provides the tools to analyze and evaluate the absorption of radiation inside a scattering/absorbing medium. 

Radiative transfer models can be very complicated and their solution cumbersome, so it is always desirable to have simplified tools for this analysis. Very simple and intuitive models can be developed in some cases to study radiation transfer inside photocatalytic reactors. However, it must be pointed out that these models must be well grounded in radiative transfer theory, in order to provide results that are consistent and of lasting engineering interest. Here, we present the basic concepts and discuss some approximations and methods that have been used. 

Besides the approximate Equation (35), another more fundamental equation from radiative transfer theory also relates Ga with qA (see, for instance. Modest, 2003, pp. 312-314)... 

R.W. Preisendorfer (1961). Application of radiative transfer theory to light measurements in the sea. Union geod. Geophys. Inst. Monogr., 10,11-30, cited in [3]. 

Radiative transfer theory to calculate the upscatter fraction p as a function of particle size and solar zenith angle 0o was developed by Wiscombe and Grams (1976). Figure 24.5 gives p as a function of p0 = cos 0q and particle radius. For Sun at the horizon... 

EIGENVALUE SHIFTING - A NEW ANALYTICAL-COMPUTATIONAL METHOD IN RADIATIVE TRANSFER THEORY... 

The exact solution to Eq. (14) is well known from the standard radiative transfer theory (cf. [1] or [6]). In particular, the intensity of emergent radiation can be expressed in terms of Chandrasekhar s //-function in the form... 

It shottld be mentioned that many of the various computational, analytical or semi-analytical standard approaches of the radiative transfer theory can be applied also to the transformed radiative transfer equation Eq. (42). In particrrlar, this is tme also for principles of invariance (cf., e g., [1] or [5]), which can be applied to derive the well-known nonlinear integral eqrration also for the albedo matrix Rc (ji, /jq ),... 

The optical measmements of diffuse reflectance are dependent on the composition of the system. Several theoretical models have been proposed for diffuse reflectance, which are based on the radiative transfer theory, and all models consider that the incident hght is scattered by particles within the medium. The most widely used theory in photometric sensors is the Kubelka-Munk theory, in which it is assumed that the scattering layer is infinitively thick, which may, in practice, be the case with the chemical transducers utilized in photometric sensors. The absolute value of the reflectance R is related to the absorption coefficient K and the scattering coefficient S by the equation... 

Similarly, the calculations of Qae are carried out based on radiative transfer theory and the known absorptivity of atmospheric constituents and the vertical distribution of temperature. The radiatively active gases are H2O, CO2, O3, methane (CH4), nitrous oxide (N2O), carbon monoxide (CO), and chlorofluorocarbons (CFCs). Since both upward and downward fluxes are involved in the infrared radiation calculation, the handling of cloud effects in determining Qae becomes more complicated than the calculation of Qas-... 

A discussion of applications of these concepts to neutron transport theory can be found in the papers [4 5 6 7 8 9]. For the interested reader we also call attention to the applications to radiative transfer theory [1 10 11], to wave propagation, [12 13 14], to random walk and multiple scattering [15 16], and to problems with moving boundaries [9]. 

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