Membranes - Principles of Chemical Separations with Environmental Applications - page 242

where P i is the pressure drop of component i across the membrane and M i is the

molecular weight of component i . The flux through a microporous membrane of thickness
l is given by:

GS P i

l

J i = √ 2 , (9.13)

π M i RT

where G is a geometric factor that takes into account tortuosity and porosity. Note that
the flux is independent of the average pressure as long as the pressure is in the Knudsen
diffusion regime.
An equimolar mixture of feed gas, diffusin g across a membrane in the Knudsen diffusion
regime, will have a separation factor

α ij = M j /

M i when the permeate side is at vacuum.
Otherwise, the separation factor will be smaller. The narrow pore-size distributions and the
small pores of ceramic and glass membranes allow separation due to Knudsen diffusion
(for the appropriate pressure range) by preferential diffusion of the lighter component
through the membrane. In composite membranes, the thin permselective layer can be in
the Knudsen diffusion regime and thus be responsible for all the separation. The support
layers, with their larger-diameter pores, are usually in the viscous-flow regime.
Separation by Knudsen diffusion has some limitations because only the lighter compo-
nent can be preferentially removed. The best separation in the Knudsen diffusion regime
is thus obtained for H 2 . When the molecular weight difference between components to
be separated is small, an economical separation probably cannot be obtained by Knudsen
diffusion.

9.7.2 Surface diffusion

A process that can occur in parallel with Knudsen diffusion is surface diffusion
(Figure 9.2(b)). A gas can chemisorb or physisorb on the pore walls and migrate along
the surface. Surface diffusion increases the permeability of the more strongly adsorbed
components in a diffusing mixture while simultaneously reducing the permeability of the
gas diffusing components by decreasing the effective pore diameter. Thus, this diffusion
mechanism is more important for membranes with small pores. For example, the number
of molecules in a monolayer on the wall of a 5-nm-diameter pore at 0.1 MPa can be
over 200 times larger than the number of molecules in the gas phase. As the temperature
increases, species desorb from the surface, surface diffusion becomes less important, and
Knudsen diffusion predominates. When surface diffusion occurs, competitive adsorption
must also be considered.
The equation describing surface flow is

J s =− ρ (1 − ε ) D s µ s d q / d l (9.14)

where J s is the surface diffusion flux, ρ is the true density of the adsorbed layer, D s is the

d l is the surface
concentration gradient. Surface diffusion must usually be determined experimentally.

surface diffusion coefficient, µ s is the tortuosity of the surface, and d q /