Effects of heterogeneity in forced convection in a porous medium: parallel plate channel or circular duct

The effects of variation (in the transverse direction) of permeability and thermal conductivity, on fully developed forced convection in a parallel plate channel or circular duct filled with a saturated porous medium, is investigated analytically on the basis of a Darcy or Dupuit–Forchheimer model. It is shown that the Dupuit–Forchheimer problem reduces to the Darcy problem with a changed permeability variation. The cases of isoflux and isothermal boundaries are treated in turn. The bulk of the results pertain to a two-step variation, but the case of a weak continuous variation is also considered. The results for the parallel plate geometry and for the circular duct geometry are qualitatively similar. The replacement of isoflux boundaries by isothermal boundaries leads to a reduction of Nusselt number but otherwise there is little change in the pattern. The results demonstrate that the effect of permeability variation is that an above average permeability near the walls leads to an increase in Nusselt number, and this is explained in terms of variation in the curvature of the temperature profile. The effect of conductivity variation is more complex; there are two opposing effects and the Nusselt number is not always a monotonic function of the conductivity variation.