In this screen cast we are going to work through the approach to a mass transfer problem, specifically
one in which we have a reaction occurring at a surface.
So here we have a thin film of metal which we are calling species C and it is being deposited
onto a silicon wafer by the instantaneous reaction that is given below.
You want to come up with an equation which could be solved to predict the moe fraction
of A as a function of Z.
And if you look at our reaction we have 2A in a gaseous state and it is fluxing through
this gas layer down to the surface, along with B, also in a gaseous state.
And these two react to form C and 4D.
And D is a gas.
And it is fluxing away from the surface.
So we have said that the surface is our Z=0 and the top of our gas layer is some delta
for our thickness and our mole fraction of A is some known concentration of A which we
are designating as yA0.
Based on this we can see that since we have a flux of both A and B through the gas layer
to the surface and a flux of D away from the surface, we are looking for an equation for
the flux of A to give us our concentration of A.
We are going to look at both the diffusive and convective terms to characterize this
flux.
So we will start by writing out our flux equations.
And we will start with the continuity equation for species A.
We have dCa over dt plus the divergence of the flux of A, or del dot NA.
And this is equal to our reaction term RA.
So the question is, is this a steady state process?
Do we have anything changing with time?
Now at the startup, or shutdown of this process we would have unsteady state concentrations
and thus the concentration of A would be a function of time, however if we are assuming
that this process has begun and there is no limit to A and B being feed to the surface,
C is a thin film such that the gas layer thickness is not changing sufficiently we could assume
that this is steady state and cross out this first term.
Now for our divergence term, we are examining the divergence term in the z direction only.
We are going to write this as dNA as a function of z, so dNA/dz.
Now we set this equal tour action term- however our reaction term is since we do not have
a reaction occurring within the diffusional direction of A.
This is important, the reaction term, this RA is on a per volume basis within the control
volume.
And in this case the control volume is this space, this gas layer which we are looking
at.
And so since the reaction isn't occurring within that control volume where we are examining
the flux this reaction term is zero.
So when we go through these terms what we are left with is that the flux of A in the
z direction is not changing.
So from here we will use Fick's law to describe the flux of A in terms of the diffusion of
A and the bulk flow of A. so when we write out Fick's law for the flux of A in the z
direction.
We get negative concentration times the diffusivity of A through B dAy/dz.
This is the diffusive part, the bulk flow, or convective part is the mole fraction yA
times the flux of A plus the flux of B plus the flux of D. Now we don't considerC because
that is a solid layer and that is not fluxing.
Now our first simplification to this is going to be to write the fluxes in terms of the
flux of A. Now let's look at B and A.
We know that 2A react per each B molecule.
So since we have twice as much A reacting with one B we need the flux of A to be twice
as fast as the flux of B. So this means that the flux of B is going o be equal to half
the flux of A. When we compare A and D we see that for every 2 flux of A into the surface
we have 4 D fluxing away.
Thus the flux of D with respect to A will be twice that of A and in the opposite direction.
So 2NA is equal to -ND.
So we are going to write the flux of ND in the z direction as -2NA.
Now we could plug these into our fick's law equation and simplify this and you could see
that because this is non-zero, we do have a bulk flow component, and therefore do have
both diffusive flux and a convective flux of A through this gas layer.
So what we are going to do is bring this term over to the left side and we are going to
group the flux of A in the z direction together and now we can bring this term back over and
we have a relationship with the flux of A in the z directions.
Now this isn't quite what we are looking for so we are going to combine the continuity
and Fick's law to come up with the following relationship.
We know the differential of the flux of A in the Z direction with respect to z is equal
to 0.
So we are going to want to take Fick's Law and we are going to want to take the derivative
of it with respect to z.
So to approach this, and combining them we can take a first look at the continuity equation
and since the flux of A with respect to z is equal to 0 we can say that the flux of
A in the z direction must be constant and we can call this NAz0.
So now we know that Fick's law the flux of z (he means A) must be equal to NAz0 at every
position of Z.
So we can set this equal to -cDAB(dyA/dz) all over (1+(yA/z)).
Now if we are going to solve for yA as a function of z we need to rearrange terms and integrate
using boundary conditions you know.
Now the boundary conditions are that at z=0 we have an instantaneous reaction so yA is
going to be equal to 0, whereas z=delta we know yA is equal to yA0.
So if our end goal is to get an equation for yA as a function of z we first need to solve
for NAz0.
And when we solve for NAz0 we use the boundary conditions.
So what we are going to do is separate our variables and write this as NAz0, dz, evaluate
this from 0, z=0 to z=delta, and set this equal to -cDAB since these are constants times
the integral from 0 to yA0.
And this is going to by dyA over 1+ (yA/z).
So now evaluating the left side we get NAz0 times delta and evaluating the right side
we get 2 times the natural log of (1+ yA/z).
So now to get an equation for NAz0 we just divide by delta and we get the following equation
for NAz0.
So lastly we could put this back into our equation to solve for y as a function of z.
Now if we set these equal, simplify and rearrange terms you should get the following.
However, this time we are evaluating from 0 to z and therefor 0 to yA.
So solving these integrals and simplifying we get the following relationship where we
could figure out the concentration or mole fraction yA as a function of z.