Page 20 - gas transport in porous media

Chapter 2: Gas Transport Mechanisms
diffuse reflection equal to unity
1/2 13
2 8RT
D iK,capillary = r capillary
3 πm i
This equation can be written as
1/2
T
D iK,capillary = 97.0 r capillary
m i
2
where D iK,capillary is in m /s, T is in K, m is the molecular weight, and r capillary is
in m. While this equation may be appropriate for flow in well-defined capillaries, it
is not directly useful for porous media applications. However, there is an alternative
way to determine the Knudsen diffusion coefficient that includes the complexity of
the porous media.
The equation for the molecular flux from Knudsen diffusion can be rewritten in
the same form as Darcy’s law. From this expression, and Klinkenberg’s formula,
the Knudsen diffusion coefficient, D iK , can be related to the Klinkenberg factor as
follows (Thorstenson and Pollock, 1989a, eqn 60)

k g b i
D iK =
µ i
As discussed in the advection part of this chapter, there are a number of correla-
tions for the Klinkenberg coefficient, b i , as a function of the porous medium, the fluid,
and the temperature. The Klinkenberg coefficient can be used in the above equation
to evaluate the Knudsen diffusion coefficient for a porous medium. The Klinken-
berg coefficient implicitly takes into account the structure of the porous medium as
reflected through the permeability. The modifications to the Klinkenberg factor due
to the gas (molecular weight and viscosity) and the temperature should be used as
discussed earlier in Section 2.1.4.
As the permeability of the medium gets even lower, the pore dimensions approach
those of a single molecule. At this point, the flow mechanisms change, and con-
figurational diffusion (Cunningham and Williams, 1980) becomes important. In
configurational diffusion, the size of the molecules is comparable to the pore dimen-
sions, and the molecular configuration becomes important (e.g., Xiao and Wei,
1992a, b). Membrane diffusion occurs at even smaller pore sizes where the chemical
characteristics of the molecules are important (Cunningham and Williams, 1980).
Cunningham and Williams (1980) suggest that configurational diffusion may be
encountered when the pore sizes are less than about 10 Å. Note that the molecular
size can be characterized by the Lennard-Jones length constant, σ, which varies from
about 2.5 to 7.5 Å as given by BSL (1960, pg. 744). Assuming a porosity of about
10%, and calculating the tortuosity by the Millinton and Quirk relationship given
2
earlier, the effective Knudsen diffusion coefficient will be about 10 −9 m /s. Using
the Jones and Owens (1980) correlation for the Klinkenberg coefficient, the perme-
2
ability is about 10 −21 m . This prediction of the transition from Knudsen diffusion to