OCR Text |
Show - 4 - (Beckermann et aI., 1988). The porous bed-gas system is assumed to be a gray medium capable or absorbing, emitting and scattering thermal radiation. The gas is considered to be transparent, because its opacity based on the mean distance between the particles in the bed is very small. Hence, radiative transrer is assumed to take place only between the particles comprising the solid bed and between the bed and the tubes. The radia~ive transrer is tw~dimensional and is modeled using a differential approximation (Vortmeyer, 1978; VlSkanta and Menguc, 1989). The absorption and scattering coefficients, single scattering albedo as well as the rorward and backward scattering rractions ror the porous bed are assumed to be gray. Little data are available ror the radiative properties; thererore, a detailed treatment or radiative transrer in the bed does not appear to be warranted at the present time. Model Equations The temperature or the porous solid (T s) is assumed to be different than that or the combustion products (T g). Heat transfer in the solid is coupled to that in the gas through the convection which is expressed in terms of the convective heat transfer coefficient between the solid and the gas (h) and the temperature difference between the solid bed and the gas (Ta - T,). The porous bed with the coolant pipes embedded in it is assumed to be tw~dimensional, i.e., the temperature changes in the z-direction (perpendicular to the plane or the figure) are negligible in comparison to the x- and ydirections. The energy equation ror the porous bed matrix matrix is o = (1-¢»ks V2T. - \l·i - ~(T. - T,) The conservation equations for the gas are; Mass: Momentum: Energy: -aaxu+ -O&=yv 0 o = : + l/V2u - (l/jK + C,tVK I u I}u o = -: + l/V2v - (l/jK + C,tVK I v I}v (1) (2) (3) (4) or, or, ~ p,cp,(u~ + v Oy ) = ¢>k, v-T, - ~(T, - Ta) + ¢>~8(y) (5) The boundary conditions are taken to be: |