Effect of solid conductivity on radiative heat transfer in packed beds

THE SOLUTION of the radiative heat transfer problem in porous media has received considerable attention for a number of years (e.g.. Vortmeyer [I], Tien and Drolen [2], ’ nd Kaviany and Singh [3]). The medium may be considere d as a continuum or as a discrete collection of particles, depending on whether the packing lies in the dependent or independent scattering/absorption range. Independent scattering/absorption is said to occur when the interaction between the radiation and a particle is not influenced by the presence of the neighboring particles. Dependent scattering can be divided into a far-field interference influencing the scattering characteristics of the medium and a near-field multiple scattering within a representative elementary volume in which the scattering and absorption characteristics of the particles are affected. Singh and Kaviany [4] show that the limits of independent scattering are a minimum value of porosity (E z 0.95) and a minimum value of C/j.. The average interparticle clearance distance C for a rhombohedral packing is given by C/d = 0.905/[(1 -&)I ’ I]. Since ati practical packed beds and most of the fluidized beds have a porosity lower than this independent limit. the scattering/absorption will generally lie in the dependent range. This implies that the radiative properties of the bed cannot be predicted from the properties of a single particle by the theory of independent scattering/absorption. Then the continuum approach to the packed bed internal radiation becomes difficult unless the properties are determined either experimentally or from discrete models (Singh and Kaviany [5]). In the dependent range, ray tracing has been successfully used to solve the internal, bed radiative heat transfer (Chen and Tien [6] ; Yang er al. [7]). Singh and Kaviany [5] have extended this approach to include absorbing and emitting as well as transparent and semi-transparent particles. One limitation of this approach is that the spherical particle diameter has to be much larger than the wavelength of radiation, i.e. the particles must lie in the geometric optic range. However, this is generally satisfied in most packed-bed applications. The second limitation has been that the problem was solved by assuming that the solid conductivity is either very large (as compared with radiant conductivity) or is very small. The difference between the predicted values of the radiant conductivity from these two asymptotes can be quite large (as much as fivefold). This latter limitation provides the motivation for the solution of the generalized problem, i.e. the radiative heat transfer through a packed bed of absorbing-emitting-scattering spheres with an arbitrary solid conductivity. For optically thick media, the concept of radiant conductivity (Vortmeyer [I]) k, has been used and through this an exchange factor Fhas been introduced which depends on the particle properties (Tien and Drolen [2]). For opaque particles and for porosites characteristic of packed beds, the use of radiant conductivity is generally valid.