We're going to talk about series resistances, and looking at fluxes of a component through
a membrane.
Now what we're considering here is that the component may have to diffuse through a boundary
layer on the feed side before getting to the membrane surface, and then it may have to
diffuse through a boundary layer again on the other side after going through the membrane.
For example we could have a liquid on one side of the membrane, and a liquid on the
other side of the membrane, and the idea is that these two liquids will have different
compositions, and the concentration of our component of interest in the liquid phase
is C1, and then as it diffuse through whatever component is not going through the membrane
as fast its concentration will decrease to C1 at the interface, and then there's a discontinuity
in general at the interface because at that point we're now looking at the concentration
inside the membrane itself, so then this composition would be C1is.
In other words the concentration of component i on side one, and then at the surface.
Then the component diffuses through the membrane with some diffusivity we'll call Da, and then
on the other side it's going to have C2is, and then there's an equilibrium constant that
relates the compositions at that point, and in the liquid phase on the downstream side,
and then there can be a boundary layer where it then diffuses down to a composition of
C2 in the permeate.
So the flux is in this direction, and that we're going to call NA, and we use a mass
transfer coefficient on each side to represent the proportionality constant between the driving
force, which is the concentration difference, and the flux of the component.
Now, let's look at the fluxes.
So the fluxes through all of the layers must be the same at steady state, so they must
be equal at steady state.
So we can write the flux in each zone, so on the feed side here, and the membrane here,
and then in the permeate side here.
So all of those fluxes must be equal, so the flux on the feed side is proportional to,
by a proportionality constant of KC1, the concentration in the bulk on the feed side
minus the concentration at the interface on the feed side.
Similarly, the flux in the liquid phase on the permeate side is proportional to, with
a proportionality constant of KC2, the concentration on the permeate side at the surface, minus
the concentration in the bulk of the permeate side, and then through the membrane we have
the dissolved phase concentrations, the proportionality constant there is the diffusivity over the
thickness of the membrane, and the concentration difference is C1is minus C2is.
At the interface the concentration in the membrane, C1is, we'll call that equal to K'
times C1i.
So this K' is an equilibrium constant representing how well that component dissolves into the
membrane.
Similarly, C2is maybe equal to K'C2i.
In general those K's can be different, although we're going to assume for this example that
they are the same.
OK, they are like a Henry's law constant, they represent how much is absorbed given
a fluid concentration, so in this case how much is absorbed on the surface, or absorbed
into the material, given a fluid concentration.
In the gas to gas mass transfer situation it looks very much the same, except that we
talk about partial pressures instead of concentrations in this case.
But other than that the idea is the same, we set the fluxes through each boundary layer
and through the membrane all equal to each other.