Membranes - Introduction to Carbon Capture and Sequestration - page 286

We have the feed gas containing different components, and this feed
splits into two parts: the permeate , which is the part that goes through
the membrane, and the retentate , which is the part that does not go
through the membrane. The idea is that if there is a difference in the ease
with which components in the gas pass through the membrane, these
components will get separated between the permeate and the retentate.
For example, for the permeate gas mixture to be enriched in CO 2 , the
CO 2 molecules must fl ow more easily through the membrane than the
other fl ue gas components (i.e., N 2 ); the retentate will consequently be
depleted of CO 2.

Permeation and permeability

Before we consider an actual separation, let us look in more detail at a
pure component. Suppose we have a feed with a given pressure, p R ,
and on the other side of the membrane we maintain a pressure, p P
(see Figure 7.2.2 ). Because of this pressure difference, the molecules will
fl ow through the material. If we continuously remove the permeate and
maintain a constant fl ow of the feed, the system will reach a steady state.
We would like to know the fl ux of molecules (mol/s) through our mem-
brane and examine how this fl ux depends on the properties of our mate-
rial. We can do some simple experiments with our membrane, such as
change the area A and the thickness L . How will this change the fl ux?
What is the density of gas molecules across the membrane?
We address these questions by looking at a small volume of the
material and applying a mass balance over this control volume (see
Figure 7.2.3 ), similar to what we have done in our calculations for diffu-
sion limitation (Section 5.4) and the breakthrough curves (Section 6.3).
The change of the concentration, , in our control volume is equal to the

difference of the fl ux, j , entering and leaving our volume. In differential
form, this reads:

ρ

()

dj z

d
dt

ρ

=−

dz

The fl ux is related to the concentration gradient and the diffusion coeffi -
cient (Fick's law):

D d ρ

d

j =−