Analytical and numerical study of natural convection heat transfer in a vertical porous annulus

A study is made of quasi-steady state natural convection in a vertical annular porous layer when the inner wall is heated by a constant heat flux and the other walls are maintained adiabatic. On the basis of the Darcy-Oberbec-Boussinesq equations, the problem is solved analytically in the limit of a long shallow annulus (A >> 1), where A is the aspect (height-to-gap width) ratio. The analytical solution for the flow and heat transfer, based on a parallel flow assumption, is derived in terms of the Darcy-Rayleigh number R and radius ratio K. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is conducted to assess the validity of the analytical results. It is demonstrated that the analytical solution predicts well the flow structure and the heat transfer for a wide range of R and K. In the boundary layer regime it is shown that the Nusselt number, based on the gap width of porous annulus, is Nu = R-2/5 [2K(2) InK - (K-2- 1)(2)]/[2(K - 1)(2) (K + 1)(6/5)].