Membrane Gas Separation

5.9.3
Predictions of Gas Separation 
In this section predictions are made using the model to demonstrate the different transport 
behaviours with varying pore size and temperature. Figure 5.11 predicts the permeability 
P as a function of pore size d for the different transport mechanisms, described earlier, 
with Equation (5.21) as the defi nition for W . For activated diffusion
Δ
E
a
= | W | and for 
surface diffusion
Δ
E
S
= aq = aW . The results represent the permeability within a single 
pore. In reality there will be a distribution of pore sizes, and therefore the transition 
Pore size (Å)
P
er
meability (arbitrar
y units)
0
5
10
15
20
25
30
-2
-1
0
1
2
10
10
10
10
10
10
3
Activation diffusion
Surface diffusion
Knudsen diffusion
Parallel transport
Figure 5.11 Model prediction of normalized permeability P as a function of pore size 
(distance between surface nuclei, d ). Modes of transport are indicated

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Modelling Gas Separation in Porous Membranes 
103
between activated diffusion and the other modes of transport will usually be smooth. Each 
mode of transport is scaled arbitrarily such that the trends may be clearly seen and there-
fore the magnitude is insignifi cant. The permeability for activated diffusion is a sharply 
increasing function of pore size as the energy barrier changes dramatically with pore size.
As explained earlier, the permeability for surface diffusion is dominated by the surface 
concentration and therefore the model predicts a peak at which the heat of adsorption is 
maximized. The permeability for Knudsen diffusion increases as a cubic function of pore 
size (see Equations 5.16 and 5.17 ) since the diffusivity depends linearly on the pore size 
d and the concentration depends on the volume of the pore d
2
(assuming cylindrical pores). 
The parallel transport model assumes that surface diffusion and Knudsen diffusion are 
occurring simultaneously such that the total permeability is given by Equation (5.18) . 
This model is explained in further detail earlier in this chapter and has been used by 
various groups [27,49,50] . Parallel transport is initially dominated by surface diffusion 
within the smaller pores where the surface concentration is high while the mode of 
Knudsen diffusion dominates within the larger pores. 
Permeability varies with temperature for each transport mechanism as demonstrated in 
Figure 5.12 . The permeability for activation diffusion is the only increasing function with 
respect to increasing temperature. When gases are in the mode of surface diffusion, the 
surface concentration decreases more than the increase in surface mobility, resulting 
in an overall decrease in permeability with increasing temperature. Knudsen diffusion 
displays decreasing permeability with increasing temperature as the concentration loss 
dominates the increase in diffusivity. As temperature increases, the surface diffusion part 
Temperature (K)
P
er
meability (arbitrar
y units)
100
150
200
250
300
350
400
10
-1
10
0
10
1
10
2
10
3
Activation diffusion
Surface diffusion
Knudsen diffusion
Parallel transport
Resistance in Series transport
Figure 5.12 Model prediction of permeability as a function of temperature. Modes of 
transport are indicated for the following pore sizes: activated diffusion ( d = 6.8 Å ), surface 
diffusion ( d = 10 Å ), Knudsen diffusion ( d = 10 Å ), parallel transport ( d = 10 Å ), and 
resistance in series transport ( d
small
= 6.8 Å , d
large
= 10 Å , x
K
= 0.8)

---

104
Membrane Gas Separation
of the parallel transport model has less infl uence causing the permeability to tend towards 
a Knudsen - type transport at high temperatures. The resistance in series transport model, 
detailed earlier in this chapter, assumes that the diffusing molecules travel through pores 
in the mode of Knudsen diffusion while occasionally encountering constrictions where 
activation diffusion occurs. The total permeability is therefore expressed by Equation 
(5.20) , where x
K
is the fraction of the pore length where Knudsen diffusion occurs. As 
shown in Figure 5.12 , the total permeability predicted by the resistance in series model 
behaves mostly in accordance with the mode of activated diffusion even for a small frac-
tion of constrictions (1 – x
K
).
In the interest of gas separation, the model prediction for CO 
2
/CH 
4
selectivity versus 
CO 
2
permeability has been calculated with varying pore size and temperature, and the 
results are shown in Figures 5.13 and 5.14 , respectively. Equation (5.21) is used for W
cyl
with the parameter values taken from Table 5.2 . Since the resistance in series transport 
behaves like activated diffusion, the predictions for activation diffusion have been omitted 
from the plot. The selectivity is high for small pores with surface diffusion as the transport 
mechanism where permeability is dominated by the concentration component for which 
CO 
2
forms denser surface layers than CH 
4
. As pores become larger, the enthalpy of 
adsorption results in a maximum CO 
2
permeability, followed by a decrease in the enthalpy 
of adsorption leading to a surface concentration loss. In this case, a single pore is 
considered and therefore the surface concentration eventually increases with increasing 
pore size according to the surface area of the cylindrical pore with the density of both 
gases tending toward that upon a fl at surface. The Knudsen diffusion selectivity favours 
CO
2
Permeability (arbitrary units)
Se
le
ct
iv
it
y
C
O
2
/C
H
4
10
0
10
1
10
2
10
0
10
1
Surface diffusion
Knudsen diffusion
Parallel transport
Resistance transport
Figure 5.13 Model predictions of CO 
2
 /CH 
4
selectivity versus CO 
2
permeability for 
varying pore size d . Arrows indicate the direction of increasing pore size. The pore size 
range, 7.22
>
d
>
30 Å , was chosen for all the modes of transport, apart from the 
resistance in series transport where the constriction size varied while the large pore size 
remained constant ( d
small
= 5 – 7 Å , d
large
= 10 Å , x
K
= 0.99, Equation 5.20 )

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Modelling Gas Separation in Porous Membranes 
105
CH 
4
because of its lighter mass resulting in a higher molecular velocity and does not 
change with pore size. Parallel transport follows the same trend as surface diffusion in 
small pores and tends toward Knudsen behaviour as the pore sizes increase. Finally, the 
resistance in series transport model predicts a decrease in selectivity as the permeability 
of CH 
4
increases more rapidly than for CO 
2
with increasing pore size.
As seen in Figure 5.14 , the selectivity is predicted to slightly increase with increasing 
temperature when gases are in the mode of surface diffusion. This is due to the larger 
adsorption energy that CH 
4
experiences over CO 
2
for this particular pore size. Knudsen 
selectivity does not depend on temperature. Parallel transport is dominated by the surface 
diffusion component at low temperatures and gradually becomes more dependent on the 
Knudsen diffusion component at high temperatures. Note that the trends will be different 
depending on the pore size. For example, in the case of d = 8 Å , the selectivity is predicted 
to decrease as the temperature increases. The resistance in series transport demonstrates 
an increase in selectivity as the CO 
2
permeability increases more than the CH 
4
permeabil-
ity with increasing temperature, as a consequence of the lower energy barrier that CO 
2
experiences for this particular pore size ( d = 6.5 Å ).