Page 186 - gas transport in porous media

Khanafer and Vafai
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walls. In addition, natural convection may take place around horizontal and vertical
cylinders as well as around spheres. Natural convection may take place around other
immersed bodies such as cubes and spheroids.
Unlike the external natural convection boundary layer that is caused by the heat
transfer interaction between a single surface and a large reservoir, internal natural
convection in enclosures is a result of the complex interaction between the fluid and
all the surfaces that confine it.
In convection heat transfer, Grashof number replaces the Reynolds number in the
convection correlation equations. Grashof number is defined as the ratio between
the buoyancy force and the viscous force. In free convection, buoyancy driven flow
dominates the flow inertia and therefore the Nusselt number is a function of Grashof
and Prandtl numbers. However, Reynolds number is significant if there is an external
flow. In many situations, it is advantageous to combine the Grashof number and the
Prandtl number to define a new parameter called Rayleigh number (Ra = Gr Pr).
The significance of the Rayleigh number is to characterize the laminar to turbulent
transition of a natural convection boundary layer flow.
The boundary conditions for convection heat transfer problems include no slip
boundary conditions on the solid surfaces and either temperature or heat flux on the
walls. In forced convection, the free stream farfield velocity is assigned along with
the ambient temperature while temperature or heat flux may be applied on the solid
surfaces. For the case of natural convection, either temperature or heat flux can also
be applied on the solid walls while usually stagnant, isothermal infinite reservoir
boundary conditions are applied for the far field conditions.
Fundamental studies associated with natural convection gas transport in porous
media have increased substantially over the past decades because of the impor-
tance of porous media in diverse technological and industrial applications such as
migration of pollutants, storage of nuclear waste, oil recovery enhancement, thermal
insulation, electronics cooling and packed bed chemical reactors. Natural convec-
tion gas transport in porous media has been widely studied and well documented in
the literature both experimentally and numerically. Major developments have been
made in modeling natural convection gas transport in porous media including several
important physical aspects. Some of the studies have used what is now commonly
known as Brinkman-Forchheimer-extended Darcy or the generalized model. Signif-
icant advances have been made in developing the momentum equation that governs
the fluid flow in porous media starting from Darcy’s law to the generalized model.
Darcy’s law revealed proportionality between the velocity and the applied pressure
difference for low speed flow in an unbounded porous medium. As such Darcy’s law
does not account for inertial effects or no-slip condition at the wall. To account for
the solid boundary, Brinkman’s equation which also known as Brinkman’s extension
of Darcy’s law was developed. Brinkman’s equation incorporated two viscous terms.
The first is the typical Darcy term and the second is similar to the Laplacian term.
Darcy’s law is linear in the Darcy velocity, which holds for a sufficiently small veloc-
ity. At higher velocities, inertial effects become appreciable causing an increase in
the form drag. Forchheimer’s equation was developed as an extension to the Darcy’s