Numerical solution of the radiative-transfer equation for an absorbing, emitting, and scattering medium with a complex 3-D geometry

Abstract

A method is proposed for numerically solving the integro-differential, radiative-transfer equation with the use of its piecewise-analytic solutions obtained by the discrete-ordinate method and grids constructed by the finiteelement method. The advantages of the method proposed and some results of calculation of the radiativetransfer characteristics for one-, two-, and three-dimensional problems are discussed.Radiative energy transfer is of crucial importance in many natural and technical processes of energy exchange. This pertains equally to high-temperature processes (combustion of organic fuels, thermal treatment of metals, hightemperature synthesis and pyrolysis in chemical technologies, etc.) where the radiative energy transfer accounts for 90% or more of the total energy exchange (see, for example, [1][2][3][4]) and to the processes occurring at lower atmospheric temperatures [5,6]. It is known that an exact estimation of the characteristics of heat and mass transfer in technological processes allows one to obtain a significant economical effect, i.e., to increase the quality and functional characteristics of products and decrease their cost, as well as to make for good environmental conditions and to conserve material, energy, and manpower resources. For example, the temperature fields of many technological processes (several tens of them are described in [1,2]) occurring at temperatures from 125 to 1600 o C should be calculated with an accuracy of ,1-2 o C.Radiative heat exchange plays a dominant role in the total heat exchange in high-temperature processes in gaseous media. The accuracy of estimation of the temperature fields of such media is primarily dependent on the correctness of calculation of the radiative-transfer characteristics. This is also very important for optimization of the heating of steel products having a different geometry in ring furnaces with a moving bottom in which the working temperatures can reach 1200 o C.Mathematical Model. It is difficult to calculate the characteristics of radiative heat transfer in selectively emitting, absorbing, and scattering media because, in this case, it is necessary to take into account the multiple processes of reradiation on solid particles, the selectivity of the radiation of gas components, and the temperature inhomogeneity and complex configuration of the radiating volume. The correctness of estimation of the radiative-heat-exchange characteristics depends, to a large extent, on the correctness of solution of the radiative-transfer equation [7][8][9]. In the case of a local thermodynamic equilibrium, this equation defines the law of conservation of radiant energy in the process of its propagation in an absorbing, emitting, and scattering medium: l⋅∇I λ (r, l) + [χ λ (r) + σ λ (r)] I λ (r, l) = χ λ (r) B λ (T (r)) + σ λ (r) 4π ∫ 4π p λ (r, l, l ′ ) I λ (r, l ′ ) dΩ ′ .(1)The boundary conditions for Eq. (1) are determined by the radiation and reflection processes occurring on the boundary surfaces of the medium and can be written, in the general cas...

Calculation of the Characteristics of Transfer of Thermal Radiation in the Workspace of an Annular Furnace

“…Based on the procedure presented in this work and in [18], we developed a computer program for computation of the characteristics of radiation transfer in the workspace of an annular furnace. Using this computer program, we investigated the influence of the furnace-arch height and the emissivity factor of the lining on the maximum specific output of the furnace at a constant temperature along the furnace channel.…”

“…In [18], we reviewed modern methods of solution of the equation of radiation transfer. As has already been noted, it takes much computer time to solve this equation in the annular-furnace space with an accuracy acceptable for practice.…”

“…To calculate the values of Q res and J we used the method described in [18]. The investigations were carried out for the range of optical thicknesses τ [0.01 − 10].…”