Membranes - Introduction to Carbon Capture and Sequestration - page 288

can solve the differential equation. If we use as boundary conditions
ρ (0)

= ρ 0 and ρ (L) = ρ L , the concentration profi le is given by:

z

()

( )

ρ=ρ+ρ−ρ z ,

0 L 0

L

and the corresponding fl ux can now be obtained from Fick's equation:

D

( )

j =− ρ −ρ

L 0

L

If we assume that the adsorption in the membrane is in the Henry regime
(see Section 6.3) and assume perfect mixing on both sides of the mem-
brane, we can relate the concentrations at z

L to the pressure
of the gas on the retentate and permeate side (see Figure 7.2.2 ),
respectively:

= 0 and z =

ρ= Hp and ρ= Hp

0 R L P

Substitution of these Henry coeffi cients into the expression for the fl ux
gives:

DH

( )

j = p − RP p

L

This is an important result as we see that the fl ux through the mem-
brane depends on two material properties: the diffusion coeffi cient,
which expresses whether molecules diffuse fast or slowly in the mate-
rial, and the Henry coeffi cient, which expresses whether the molecules
adsorb or not. Compare with the following: cats can swim relatively
fast, yet the fl ux of cats through a river is very small as their affi nity with
water is so low.
Experimentally, it is often easier to measure the overall transport
through a material as a function of the concentration differences rather
than measuring the adsorption and diffusion separately. To use this infor-
mation, we defi ne two new quantities: the permeability P and perme-
ance P

. The permeability relates the fl ux to the concentration difference
and depends on the thickness of our membrane. Permeance P

′

′ is a

material property independent of the thickness:

P

L ′

( ) ( )

j = p − p = P p − p

RP RP