In this screencast we are going to derive the constitutive equations that we are going
to use to help us solve systems in which we have mass transfer through diffusion. And so let us
start with the following idea: Say we have a tank of some fluid and with any good system
we should have some kind of coordinates. The only coordinates we are going to care about
in this is going to be z, which we are going to say is positive in the up direction. Now
let's say at some time, zero, we drop in some other species, some other liquid, and it covers
the surface of our tank. And so at some later time we notice that a lot of these particles
of this other species has diffused down in our container. At a later time t2 we see that particles have
moved farther down in the tank, but not, maybe, as many. And then at some time t3, we notice
that more of this fluid has moved closer to the bottom. Eventually if we leave this here
long enough, more and more of this fluid at the top will start moving down and we will
notice that we don't really notice that there's anything moving anymore and we have reached some kind
of equilibrium state. So from this we saw that we have some kind of mass transfer from
a high concentration to low concentration. We also notice that the net mass transfer
is equal to zero when the concentration is uniform. And we also get an idea that the
transfer rate is proportional to the area which it is transferring through. In this
case the cross sectional area. The larger the area the higher the transfer rate. But
we could also imagine that it wouldn't be very convenient if all we did was talk about
mass transfer for a system that is stationary. What if we looked at a system that was flowing
and all of a sudden at time equals zero we drop in some species into our flowing system. Maybe
at some later time this has started to spread out, at some later time, the spread has become
even bigger, and so forth, until we get a very large spread that initially started out
as a very small droplet. Do those same rules we had before still apply. Well not really-
but kind of. If we focus on the system with respect to it moving, then why not. If we
say the coordinate system is the same at each location as it moves and that is all we want
to define our movement from that point then it does kind of obey those statements that
we had before. We are going to come up with a way to characterize this mass transfer for
a moving system. And then we will relate that back to a stationary system because that is
kind of what we are going to care about more often than not. So recall back from momentum
transfer and heat transfer. It followed the same pattern. It followed the pattern that
we had a flux of something that was proportional to some gradient. And had some kind of resistance
to this flux by some property of our system. So for instance, momentum flux. We said that
the momentum flux, the shear stress was equal to the viscosity of the fluid times our velocity
gradient. This was Newton's law of viscosity. So we have our flux of momentum proportional
to a gradient, in this case a velocity gradient, that is resisted by the viscosity of that
fluid. Our heat flux, q, was equal to negative k dt/dx. This was Fourier's Law of heat conduction
where this was our heat flux, this was the thermal conductivity of our material and this
is our temperature gradient. So what is to believe that mass flux wouldn't follow the
same form? So if we were to look at a molar flux given a binary mixture of A and B. Fick
came up with his law of diffusion which followed the same pattern, saying that the molar flux
of A in the direction of z was proportional to a concentration gradient. We add a negative
sign because we know it goes from a high concentration to a low concentration and then our property
that helps dictate this is our diffusivity of A in B, this is also called a diffusion
coefficient. Now it is important to note that this is based on our moving system. It characterizes
the mass transfer due to ordinary diffusion which is just random molecular movement. When
our total concentration is constant this gets brought out of the differential and we write
it in terms of a mole fraction instead. So that is a common form we might see it. It
also could be rewritten in terms of a mass flux which we denote by a lowercase j and
this would be in terms of density. Still some diffusivity of A in B. And then in terms
of our mass fraction, w here. But this didn't quite cover mass transfer in all circumstances.
As we mentioned this was with respect to the molar average velocity of the mixture. This
explained the ordinary molecular diffusion but what if we had mass transfer due to our
bulk flow. What if we wanted to look at this system and instead try to characterize
the mass transfer that occurs from point A to point B. This is where we are going to
start deriving this. So let's look at the molar average velocity of our mixture, I will
designate as vm. We could say that this is the sum of the velocities of each component times
its concentration over the total concentration. And we know that the sum of the velocities
times the concentrations is going to equal the total net molar flux of both A and B,
which we could write as the flux of A plus the molar flux of B. To prove that again,
let's take a look at va ca. What are the units on this? We have a velocity, so length times
time, multiplied by a concentration, we are going to say a mole per length cubed. This
is going to give us a mole per length squared, per time. Which is the units for flux. Hence
why we have a flux on top. Similarly if we look at an individual species velocity with
respect to a stationary coordinate we are going to write the vi. We could say it is the flux of i over
the concentration of i. We also know that the mole fraction xi is equal to the concentration
of i over the total concentration. This means we could expand our molar average velocity of our mixture
as such, where we have the velocity of A times the concentration of A, plus the velocity
of B times the concentration of B, all over the total concentration. This, based on this
relationship is going to give us xa, the mole fraction of A times the velocity of A plus
the mole fraction of B times the velocity of B. So now if we want to separate out the
velocity due to the species diffusion in the mixture which again we relate back to that Ji, the
flux we are going to write vi d to indicate the diffusive velocity of the
species. We know this is going to be the actual velocity of the species minus our molar average
velocity of the mixture. So now we are isolating out that diffusive velocity and we could write
it in terms of our flux of i over our concentration of i. So very similar to what we had before,
where we said vi times ci was equal to our net flux of i. In this case, vi d times ci is going
to give us our molar flux with respect to our moving system. Again to characterize a
net flux because that is what we are really looking for, based on systems that have net
flow, both bulk and diffusive mass transfer. We want to put this in terms of a stationary
coordinate system. So let's rewrite this in terms of our stationary coordinate system
so vi will be equal to the velocity due to the diffusive mass transfer as well as our bulk
flow, in this case the average molar velocity of our mixture. We know vi is just Ni over
ci so plug that in over here. Our diffusive velocity is our molar flux, Ji over ci and
the molar average velocity, vm, we wrote as the combination of the fluxes of A and B
over the concentration. So then we can rearrange this equation a little bit by multiplying
through by the individual concentration ci, and then we could plug in what we know. In this
case we are going to look at the flux of A, so the diffusive flux of A through B in terms
of the mole fraction xa plus this case xa which is our ca over c times our total flux.
And we could write the same thing for our flux of B. Notice where I have substituted
B for A. We also know that the flux is the moles per area per time. So we could rewrite
our molar flux Na in terms of our total molar flow, little na over our area. So these two
equations become pretty important when we start looking at characterizing mass transfer
and there are a couple specific cases that we are going to start with, that of unimolecular
diffusion and equal molar counterdiffusion. And those are things we can find in other
screencasts so please check those out.