A Numerical Solution of Variable Porosity Effects on Natural Convection in a Packed-Sphere Cavity

The problem of natural convection in differentially heated vertical cavities filled with spherical particles saturated with Newtonian fluids is investigated numerically. The Brinkman–Darcy–Ergun equation is used as the momentum equation, and the wall effect on porosity variation is approximated by an exponential function. The effect of variable stagnant thermal conductivities is taken into consideration in the energy equation. The formulation of the problem shows that the flow and heat transfer characteristics depend on six dimensionless parameters, namely, the Rayleigh and Prandtl numbers of the fluid phase, the dimensionless particle diameter, the conductivity ratio of the two phases, the bulk porosity, and the aspect ratio of the cavity. The influences of these parameters on the heat transfer rate are thoroughly investigated. The predicted Nusselt numbers are compared with existing experimental results. It is found that the computed Nusselt numbers based on the present model compare the best with experimental data.