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Chapter 10: Natural Convection Gas Transport in Porous Media
steady case, when the wall temperature was a quadratic function along the surface, it
was shown by that a steady free convective boundary-layer flow solution existed if and
only if the quadratic wall temperature was a convex function. This was also confirmed
for the unsteady case for a concave parabolic wall temperature prescription. Rees and
Vafai (1999) investigated boundary and inertial effects for natural convection from
a horizontal plate embedded in a porous medium. The governing equations were
studied using both numerical and asymptotic methods.

10.3 NATURAL CONVECTION FROM A VERTICAL SURFACE

Buoyancy-induced flow along a vertical plate embedded in a porous medium is a fun-
damental problem. This problem has received considerable attention due to possible
applications in industries involving heat exchanger design, petroleum production,
filtration, chemical catalytic reactors, and nuclear waste repositories.
The problem of natural convection about a heated impermeable surface embedded
in fluid-saturated porous media was investigated by Cheng and Minkowycz (1977).
They obtained similarity solutions, which were based on Darcy’s law and boundary
layer approximations. To extend the range of applicability of the boundary layer
analysis to relatively lower modified Rayleigh numbers, Cheng and Hsu (1984) and
Joshi and Gebhart (1984) examined higher order effects such as the entrainment from
the edge of the boundary layer, the axial heat conduction, and the normal pressure
gradients using the method of matched asymptotic expansions. The Brinkman model
was used for the theoretical study of boundary effects for a natural convection in a
porous medium adjacent to a semi-infinite vertical plate with a power law variation
of wall temperature by Hsu and Cheng (1985).
The heat-transfer rate from an isothermal vertical plate placed next to saturated
high-permeability porous media was studied experimentally and analytically by
Kaviany and Mittal (1987). The medium was polyurethane foam saturated with air.
An integral method was applied in predicting the heat-transfer rate by including
non-Darcy effects, which was expected to be significant at high permeabilities and
high Rayleigh numbers. The agreement between experimental and predicted results
was good except when the permeability was relatively low and the Peclet number was
very small. Although some of the permeabilities encountered were large due to exper-
imental constraints, Rayleigh numbers were not very high, resulting in insignificant
non-Darcian effects.
Hongetal. (1987)examinedanalyticallytheeffectsofnon-Darcianandnonuniform
permeability conditions on natural convection from a vertical plate embedded within
a porous medium. The non-Darcian effects, which include the no-slip and inertia
effects, decrease the flow and heat transfer rate, while the nonhomogeneity effect
enhances the heat transfer. For packed spheres, in particular, the nonhomogeneity in
permeability due to the packing of spheres near the solid wall results in a strong flow-
channeling effect that significantly increases the heat transfer. The effect of transverse
thermal dispersion was also examined.