ABSTRACT
Many are the applications of interest for inverse radiative transfer problems, in several fields, as cited, for example, in Sarvari and Mansouri (2004), Tito et al. (2004), Ren et al. (2006), Sanchez and McCormick (2008), Gryn (1995), Moura et al. (1998), Chaloub and Campos Velho (2003), Klose and Hielscher (2002), Liu et al. (2008), Özis¸ik and Orlande (2000), and Siewert (2002). Particularly, in Chapter 21, inverse thermal radiation problems are discussed. In this context, a fundamental issue required for several methodologies is a good solution for the direct problem, in this case, a solution for the radiative transfer equation (Chandrasekhar 1960, Özis¸ik 1973, Modest 1993). The analysis of radiative transport is somewhat more complex than heat transfer by
conduction or convection within a medium (Modest 1993). In fact, the temperature is not the main unknown in the balance equation, but the radiation intensity, which is function of space variables and also direction of the particles. Still, radiative properties may depend, along with direction, on the wavelength. Due to the complexity of the original mathematical model associated with radiative
transfer processes (an integro-differential equation where the unknown distribution depends on seven independent variables), several studies have been devoted to the challenge of developing accurate solutions adequate to different geometries. In regard to
Case (1960), who derived an exact solution for simpler transport models. Although its use is restricted to very simple physical models, it provided important theoretical information, which was very helpful in the development of the preliminar numerical approaches in this field. Nowadays, in general, two ways may be followed: the probabilistic approach and the deterministic approach. In the deterministic approach, one searches for exact solutions of approximated forms of the original equation. In this context, two classical methodologies associated with the solution of radiation problems are very well known and should be mentioned here: the spherical harmonics method and the discrete ordinates method. The fundamental idea involved in the spherical harmonics method (Davison 1957), also
referred to as PNmethod, is the approximation (expansion) of the angular dependence in the unknown function, as the radiation intensity, in terms of spherical harmonics functions, or, simply, Legendre polynomials. More recent developments improved important features of the PN solution, particularly making it more efficient from the computational point of view (Benassi et al. 1983, 1984). The generalization of such approach to the treatment of multidimensional problems and more complex geometries may be a very hard task, if possible. Still, it is always important to emphasize that the spherical harmonics approach provides a solution for the moments of the transport equation instead of the equation itself. The development of the discrete ordinates method in the solution of the radiative transfer
equation may be mostly associated with Chandrasekhar’s work (Chandrasekhar 1960), although it seems to be already proposed in Wick’s work (Wick 1943). The fundamental idea in the discrete ordinates method is the use of a quadrature scheme to deal with the integral termof the radiative transfer equation, such that the original problem is transformed into a system of differential equations. Under certain restrictions on the quadrature scheme and boundary conditions, it may be shown that the discrete ordinates method is equivalent to the spherical harmonicsmethod (Barichello and Siewert 1998). As extension of the original version of themethod, over the years, the discrete ordinatesmethod has been combinedwith finite-difference techniques (Fiveland 1984, Lewis and Miller 1984), when the spatial dependence of the problem is treated numerically, and multidimensional quadrature schemes have been developed (Lewis and Miller 1984) as well. In this chapter, we focus our attention on the solution of thermal radiation problems
based on a more recent analytical version of the discrete ordinates method: the ADO method (Barichello and Siewert 1999a, Barichello and Siewert 2002). Differently of the Chandrasekhar’s approach, the ADO approach (i) does not depend on any special properties of the quadrature scheme, (ii) has the separation constants defined as eigenvalues of a matrix instead of roots of a characteristic equation, and (iii) defines a scaling to avoid positive exponentials that cause overflows in numerical calculations. In addition, the ADO formulation leads to eigenvalue systems of reduced order, in comparison with standard discrete ordinates calculations, which results in computational gain. These features have made possible the development of concise and accurate solutions for a wide class of problems, including, with respect to radiative transfer applications, models that consider polarization effects (Barichello and Siewert 1999b) and Fresnel boundary conditions (Garcia et al. 2008), for example. Here, as an introductory study, simple models will be used for developing the ADO
solution, in order to provide a basic scheme for establishing a computational procedure, which may be, however, useful as benchmark case when solving more complex problems with numerical tools. In addition, taking into account that solutions are obtained in a closed form, this formalism may represent important computational gain if used in association with the solution of inverse problems (Barichello and Siewert 1997, Siewert 2002).
definitions to formulate the general model of interest. In Section 15.3, we focus our attention in the specific formulation for one-dimensional (plane-parallel) geometry. In Section 15.4, we describe the problem (gray, anisotropic medium) to which we develop the ADO solution in Section 15.5. Continuing, we deal with the simplest model, the isotropic case, in Section 15.6, to show that, in this case, explicit solutions can be found. In Section 15.7, we discuss computational aspects and list numerical results for a test case. Finally, in Section 15.8, we add some general and concluding remarks.
