For a vast range of engineering applications, radiation is a substantial method of heat transfer. Especially, in high temperature equipment like furnaces, boilers, gas turbine combustors, and nuclear reactors, where the combustion generating luminous flames includes combustion gases and other particles. Where the scattering is mostly anisotropic, the particles emit, absorb and scatter radiant energy. Therefore, the necessity for analysis of radiation heat transfer leads to an increase demand for developing well-designed radiation models, applicable to arbitrary shaped multi-dimensional geometries and capable of treating anisotropic characteristics in participating media. In recent decades, many researchers have tried to calculate the radiation transfer equation (RTE) in multidimensional complicated geometries. Considering computation costs and precision, three most adopted methods could be recalled the discrete transfer, the discrete ordinates and the finite-volume methods. The first description of the discrete transfer method (DTM) is presented by Lockwood and Shah [
1] and applied later to complex geometries by the cell-blocking process according to Cartesian coordinates [
2], and nonorthogonal grid systems [
3]. Chai et al. [
4] used the DOM with the blocked-off-region procedure. Fiveland and Jesse [
5] have accomplished a formulation of the discrete ordinates method (DOM) using finite element associated with curvilinear grids. The finite volume method (FVM) was adapted with various procedures that treated the problem of irregular geometries. The FVM was developed with nonorthogonal coordinate systems [
6], the blocked-off-region [
7,
8,
9], and the spatial-multiblock [
10] procedures. Coelho et al. [
11] modelled the radiation heat transfer in enclosures including blocks of narrow thicknesses by the above-mentioned methods. Guedri et al. [
12] investigated the impacts of baffles on radiation heat transfer in the 2-D and 3-D complicated geometries. The authors examined two different schemes: The STEP and CLAM schemes. Furthermore, they treated the effect of change of the absorption and albedo coefficients on the temperature profiles as well as net radiation heat flux distributions in a 3-D biomass pyrolysis reactor. The similar study is done by Abbassi et al. [
13] in a 2-D complex geometry. They examined the baffles shadow and soot volume fraction impacts on the temperature profiles and radiation heat flux. In all previous works, the problem of anisotropic scattering is processed in a simple way. Mengüç and Viskanta [
14] analyzed the radiation exchanges in a 3-D rectangular enclosure housing radiatively participating mixture of gases and anisotropic scattering particles applying the first and third-order spherical harmonics approximation. The delta-Eddington model is employed to define the scattering phase function. Kim and Lee [
15,
16] studied the impact of the anisotropic scattering in a 2-D rectangular enclosure using the S-N discrete ordinates scheme. The scattering phase function is expanded in a series of Legendre polynomials. Results indicated that the phase function anisotropy has a vital importance in the radiation heat transfer while the non-symmetric boundary conditions are considered in the problem. Farmer and Howell [
17] used the Monte Carlo approach to anticipate the radiation heat transfer within general inhomogeneous media that represents both highly spectral and anisotropic scattering behaviors. Guo and Maruyama [
18] investigated the scaled isotropic scattering radiation in a 3-D inhomogeneous medium by the radiation element method. In a 2-D rectangular enclosure, Trivic et al. [
19,
20,
21] coupled the FVM to calculate the radiation transfer equation employing Mie equations for evaluating the scattering phase function. This model can be applied to any given particle parameters regardless the previous designed analytical expressions for the scattering function. It is also adopted for the radiation heat transfer in a 3-D geometry with grey and non-grey anisotropically scattering media. In this work, the authors used two different numerical schemes to solve the radiation transfer equation: (i) The finite volume method with a highly refined angular discretization and (ii) the combination of zone method and the Monte Carlo statistical simulation method, and for the non-gray gases, they used the Smith’s weighted sum of gray gases model (WSGGM) for a hypothetical gas represented by 5 gray gases. Outcomes from the two methods are found in an acceptable agreement. Hunter and Guo [
22] presented a technique for the normalization of the phase function to meet the scattered energy conservation constraint and thus, minimize the numerical errors generated by the discretization of the integral numerical term in the RTE. This technique is simpler and applied to the 3-D FVM to enhance radiation calculation precision and efficiency in anisotropically scattering media. The outcomes matched well to those achieved by the Monte Carlo and discrete-ordinates methods for a cubic enclosure confining a highly-anisotropic scattering mediumproblem. The authors achieved an acceptable compromise between the FVM results by applying the normalization technique, and the two different methods by little effect on computational efficiency. Guedri et al. [
23] formulated and applied the FTn Finite Volume Method (FTn FVM) for transient radiation in a 3-D absorbing, emitting, and anisotropically scattering medium. The outcomes illustrated that FTn FVM decreases mostly the ray impacts and its accuracy is higher than the standard FVM. Moreover, FTn FVM has shorter convergence time compared to the standard FVM for all cases using both STEP and CLAM schemes.
The aim of this study is to understand the anisotropic scattering phenomenon impact on the radiation heat transfer in a 3-D complex geometry. The enclosure is a pilot plant for biomass pyrolysis reactor consisted of two pyrolysis chambers and a heat recuperator. The FVM is used in relation to the blocked-off-region process and Mie equations to evaluate the scattering phase function (SPF). The heat recuperator includes a gray, absorbing–emitting, and anisotropically scattering medium. Solid particles of many different coals and fly ash are formed during the pyrolysis process. The main advantage of the proposed procedure is that it can be readily applied to radiating particles of any types within the Mie scattering approximation limits, given the geometrical and optical particle parameters like the complicated index of refraction, wavelength of the incident radiation and particle mean diameter.