Page 14 - gas transport in porous media

Chapter 2: Gas Transport Mechanisms
7
2.1.3
Forchheimer Extension
At low pore velocities, Darcy’s law works quite well. However, as the pore velocities
increase, the flow becomes turbulent, the flow resistance becomes non-linear, and the
Forchheimerequationismoreappropriate. FromJosephetal. (1982), theForchheimer
equation is
µ g −1/2
∇P g =− u g − c F k ρ g u g u g

g
k g
where c F is a constant and gravity has been ignored. The first term on the RHS is
again immediately recognizable as Darcy’s law. The second term on the RHS is a non-
linear flow resistance term. According to Nield and Bejan (1999), the above equation
is based on the work of Dupuit (1863) and Forchheimer (1901) as modified by Ward
(1964). The value of c F is approximately 0.55 based on the work of Ward (1964).
However, later work indicates that c F is a function of the porous medium and can
be as low as 0.1 for foam metal fibers as summarized by Nield and Bejan (1999). In
addition, Beavers et al. (1973) showed that bounding walls can change the value of
c F significantly.
The above equation can be rearranged in terms of a permeability-based
Reynolds number, where the characteristic dimension is the square root of the gas
permeability, or
1/2
ρ g u g k g
Re k =
µ g
The Forchheimer equation can be rearranged in terms of the value of c F and the
Reynolds number, or

1
∇P g ∝ + c F
Re k
According to Nield and Bejan (1999), the transition from Darcy’s law (c F = 0.) to
the above Forchheimer equation occurs in the permeability-based Reynolds number
range of 1 to 10. Note that this transition is based on liquid flow through an isothermal
liquid-saturated porous medium, not an all-gas system. At low Reynolds numbers,
Darcy’s law is recovered (c F 1/Re k ). As the Reynolds number increases, the pres-
sure drop increases above that predicted by Darcy’s law. For further details, see the
discussion in Nield and Bejan (1999).
More recently, porous media approaches have been developed that include a two-
equation turbulence model similar to that used in clear fluid computational fluid
dynamics codes as exemplified by Masuoka and Takatsu (1996), Antohe and Lage
(1997), and Getachew et al. (2000).

2.1.4 Low Permeability Effects
Gas advection through porous media can be idealized as flow through numerous
capillary tubes. For large capillary tubes, the gas molecular mean free path is much