Membrane Gas Separation

5.4
Knudsen Diffusion 
Knudsen diffusion [20,30,32,37 – 40] depending on gas pressure and mean free path in the 
gas phase applies to pores between 10 Å and 500 Å in size; however, there are examples 
in the literature where it was observed for much larger pores [41] . In this region, the mean 
free path of molecules in gas phase  
λ
  is much larger than the pore diameter d . It is common 
to use the so - called Knudsen number K
n
=  
λ
  / d to characterize the regime of permeation 
through pores. When K
n
<< 1, viscous (Poiseuille) fl ow is realized. The condition for 
Knudsen diffusion is K
n
>> 1. An intermediate regime is realized when K
n
≈
1. 
The Knudsen diffusion coeffi cient can be expressed in the following form
D
d
u
K
=
3
τ
(5.12)

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90
Membrane Gas Separation
where  
τ
  is the pore tortuosity and u¯ is the average molecular speed. This expression shows 
that the separation outcome should depend on the differences in molecular speed (or 
molecular mass). The average molecular speed u¯ is calculated using the Maxwell speed 
distribution,
u
RT
m
=
8
π
(5.13)
where m is the molecular mass and hence the diffusion coeffi cient can be presented as
D
d
RT
m
K
1 2
3
=
(
)(
)
τ
π
8
(5.14)
For the fl ux in the Knudsen regime the following equation holds [42,43] :
J
n d
pD
RTL
=
π
2
4
Δ
K
(5.15)
where n is the surface concentration of the pores with diameter d ,
Δ
p is the pressure drop 
across the membrane and L is the membrane thickness. After substituting Equation (5.14)
into Equation (5.15) , one has the following expressions for the fl ux J and permeability 
coeffi cient P :
J
n
d
p
L
mRT
=
(
)
(
)
π
τ
1 2
3
1 2
6
2
Δ
(5.16)
P
n
d
mRT
=
(
)
(
)
π
τ
1 2
3
1 2
6
2
(5.17)
Two important conclusions can be made from analysis of Equations (5.16) and (5.17) . 
First, selectivity of separation in Knudsen regime is characterized by the ratio 
 
α
 
ij
= ( M
j
/ M
i
) 
1/2
. It means that membranes where Knudsen diffusion predominates are 
poorly selective. For example, separation factor of separation of O 
2
/N 
2
pair is 1.07. The 
highest gas separation selectivity can be observed in separation of lightest and heaviest 
gases, e.g. hydrogen and butane: in this case  
α
  = 5.4. Another unusual feature of Knudsen 
diffusion is that increases in temperature result in slight decreases of the fl ux and perme-
ability coeffi cient, as J and P depend on temperature as T
1/2
. Numerous confi rmation of 
this dependence can be found in the literature.