Let's discuss another form of one dimensional mass transfer via diffusion termed unimolecular
diffusion or UMD. We are going to start with a container filled with a liquid and say that
we have some stagnant fluid. In this case maybe air, which we will term as B, that this
fluid will diffuse through to get to the top of this container. And once it reaches the
top of the container, because of any overdraft, or current above the container, we are going
to assume that that fluid, we are going to term A, diffuses to the surface and then gets
carried away by this passing fluid. So we really just want to focus in on this area.
So let's give the system some coordinates. We are just looking at this film layer that
this fluid A is diffusing through. So if we call z our coordinate system for our vertical
direction, and we are going to look at some z for our surface and we will say z1 is the
top of our container. Again we are looking at A as it diffuses through B. So at the surface
we have some concentration we are going to say is yA, and at the top we know that yA
is just equal to zero. We are going to start again with our flux equation for one dimensional
mass transfer based on a fixed coordinate system, for a binary system of A and B that
we saw in our previous screencast where we derived this. And now we said the flux of
A in the z direction as equal to negative the concentration of our system times the
diffusivity times our gradient, in this case in terms of the mole fractions in the z direction.
And then we are going to include the term for our bulk flow mass transfer as such. So
here is our constitutive equation that we're going to start with and examining the system.
So if we look at the steady state mass transfer of A through a stagnant layer B, we are saying
that B has a negligible solubility in A, as well as that A and B do not react, then
we could reduce our system to where we have a flux of A, this NAz, but no flux of B, since
it is stagnant. That means we get to cross this term out. That leaves us with the following
equation and we can see mathematically that we do have that bulk flow term still and its
effect. But we want to rearrange this into a Fick's Law form and bring our flux term
on to one side. When we do this we are going to have one minus yA, the mole fraction A in the gas
stream in our denominator. So this term accounts for the bulk flow effect. If we assume that
no accumulation of A in our film space with time, that the concentration is constant, and the
diffusion coefficient is not a function of concentration, then we can integrate this to
look like the following. We are going to bring our dz term over to the left side, and since
our concentration and our diffusivity is constant we bring it out of our integral and we will
have dyA over one minus yA. Now importantly, if we do a quick material balance, we know that
whatever enters our film space, so remember our film space is pretty much everything from
the surface of the liquid to the top of the container. Anything that enters, all the A
that enters that film, must exit since we are assuming that there is no accumulation.
That means that in equals out. This means that our flux in must equal our flux out.
So our change in our flux with position must be equal to zero. Therefore, our flux is constant.
That means for this left side of our integral we could bring it out. Now we could evaluate
this from z1 to z2. And therefore we are going to have the mole fraction of A at position 1
and the mole fraction of A at position 2. Now we get our flux times z2 minus z1 is equal
to negative concentration, diffusivity, times the natural log of 1 minus yA1 over 1 minus
yA2. Since it is in the denominator we end up flipping them. And we can distribute this
negative sign as well as bring the z's to the right side, to get our final flux equation
to look like the following. Now z2 is not specified and we are just looking at it in
terms of z and the mole fraction yA but you just take these 2 subscripts out of our equation
and this would be our final form to characterize our flux as a function of position and concentration.
Now it may be of interest to plot the mole composition, yA, as a function of position,
z. We could just rewrite this equation looking at yA as a function of z and we get 1 minus,
1 minus yA1 times the exponential of our flux, times our film thickness over our concentration
and our diffusivity. Now the last thing I want to show you is that you may also
see this equation in a different form using what is known as the log mean. You may have
the subscript LM. So you may recall, the log mean of something, in this case I am going
to write log mean, LM is equal to our difference in compositions. So yA2 minus yA1 over the natural
log of yA2 over yA1. So we could take this to the case since we have 1 minus yA, the log mean
of 1 minus yA would have a very similar form, 1 minus yA2 minus 1 minus yA1 over the natural log of the ratio
of the two. And when we simplify this, this is just yA1 minus yA2 over the natural log
of this ratio. And if you recall, this natural log ratio is in our equation up here or at
least a form of it. If you look a little further where we had designated it z2 you see it right
there. And when we designated it with z2 you saw it as such. So, plugging this log mean
term in instead, we get the following expression and just to make it a little harder, sometimes
you even see this as yB, mole fraction of our other component. So hopefully that gives
you an idea of where these equations are derived from and in another screencast we will work
through an example using unimolecular diffusion to calculate our mass transfer of a species.