Page 13 - gas transport in porous media

6
Neglecting gravity and rearranging
µ g Webb
∇P g =− u g
k g
Note that the Darcy velocity, u g , is not a physical velocity. Rather, it is a superfi-
cial velocity based on the entire cross section of the flow, not just the fluid flow
cross-section. The Darcy velocity is related to the pore velocity, V g , through the
porosity, φ,or
u g
V g =
φ
The gas-phase permeability, k g , is a proportionality constant that is usually experi-
2
mentallydeterminedwithunitsoflength . Thegas-phasepermeabilitymaybeslightly
different than the liquid-phase permeability due to the effects of the fluids. Values
2
of the liquid-phase permeability vary widely, from 10 −7 to 10 −9 m for clean gravel
2
down to 10 −18 to 10 −20 m for granite (Bear, 1979, pg. 68). The unit Darcy is often
2
used, which is equal to 0.987 × 10 −12 m .
Darcy’s law is applicable to low velocity flow, which is generally the case in porous
media flow, and to regions without boundary shear flow, such as away from walls.
When wall shear is important, the Brinkman extension can be used as discussed below.
For turbulent flow conditions, the Forchheimer equation is appropriate. In some
situations (e.g., Vafai and Tien, 1981), the Brinkman and Forchheimer equations
are both employed for a more complete momentum equation. For a more detailed
discussion of the various flow laws, see Nield and Bejan (1999), Kaviany (1995), or
Lage (1998).

2.1.2 Brinkman Extension
The Brinkman extension to Darcy’s law equation includes the effect of wall or
boundary shear on the flow velocity, or

µ g 2
∇P g =− u g +˜µ ∇ u g
k g
where gravity has been ignored for clarity. The first term on the RHS is immediately
recognizable as the Darcy expression, while the second term is a shear stress term
such as would be required by a boundary wall no-slip condition. The coefficient ˜µ is
an effective viscosity at the wall, which in general is not equal to the gas viscosity, µ g ,
as discussed by Nield and Bejan (1999). For many situations, the use of the boundary
shear term is not necessary. The effect is only significant in a region close to the
1/2
boundary whose thickness is of order of the square root of the gas permeability, k g
(assuming ˜µ = µ g ), so for most applications, the effect can be ignored.
The Brinkman equation is also often employed at the interface between a porous
media and a clear fluid, or a fluid with no porous media, in order to obtain continuity
of shear stress. This interfacial condition is discussed in more detail by Nield and
Bejan (1999) and Kaviany (1995).