This is the second in a series of using the Hunter-Nash method for the liquid-liquid
extraction (LLE) where we have a ternary system. And in the first screencast we looked at the
simple two stage system. And then we used the ternary phase diagram, we fixed the flow rate
of the feed and the solvent and desired extract concentration and then we showed how to determine
the mixing point, and I won't repeat that here. And then we showed how to determine
the operating point P. These are all based on mass balances. So what I want to do now
is once we have this operating point, how do we determine the number of stages in this
method. So let me take the same diagram, but now, what I have done is add in tie lines
into the two phase region. I remove the lines that helped us determine the mixing point
and the operating point so it is less complicated. And what I have left is the solvent feed,
the final raffinate that we calculated from mass balances for the final extract and what
the feed is. So the first thing I am going to do in this diagram is I am going to change
this R2 to R sub n. Because when I did the initial ways of determining the mixing and
operating point I did an example with just 2 stages, but this is going to take more than
2 stages, so let's determine the number of stages. So let's look at the case where there
are 4 stages. And I am doing that because I have already solved the problem and that
is approximately the number of stages it is going to take. And then we have mass balances,
again derived in the first screencast, that said the flow rate, E1 minus F, this difference,
the net flow out here must be the net flow in here, otherwise we are not at steady state.
And then we go down each stage and say the same thing. That the net flows must be equal.
I can then rearrange and I am going to do this because these mass balances are going
to allow us to draw lines connecting, for example E1, P and F, we have already done
that, but now E2 related to R1, E3 related to R2. So these are all mass balances. It
is important to keep that in mind. The other important information is equilibrium phases,
so if I look at stage 2, E2 leaving, must be in equilibrium with R2. Likewise, E1 must
be in equilibrium with R1 and likewise for the other stages. And this is important because
we are going to use this in the Hunter-Nash method to determine the number of stages.
So let's go back now and look at this ternary diagram and say, what happens, E1, leaving
stage 1, is in equilibrium with R1. So I am going to look at this diagram and R1 must
be in equilibrium with E1 which means on the tie line. So this point then corresponds to
R1 along this tie line we have equilibrium. We now can look at our mass balances. And
so here it says that P plus R1 is E2, in other words you get the composition E2 by summing
P and R1. And that means if I draw a line between R1 and P the intersection is going
to give me the value of E2. So let's do that. So I have drawn the orange lie, and this point
here then corresponds to E2, from our mass balance. And then we are going to do the same
thing again, E2 is going to be in equilibrium with R2. So I am going to draw a tie line.
I could certainly do it more accurately, but for this demonstration I am going to approximate
the tie line. So I have drawn the tie line and that means this must be R2, because remember
E2 if we go back and look at our system, E2 must be in equilibrium with R2 leaving stage
2. And therefore, we now have an equation between R2 and E3. So this means we are going to draw
another line between P and R2. I have drawn the line between P and R2 and this means this
point now is E3. Keep in mind that actually trying to draw these lines on a tablet is
not really as easy to do as on a piece of paper and so it may not look quite as nice as it would if
we do it on a piece of paper. Well now E3, must be in equilibrium with R3. So that means
along this line here, this tie line we have R3. Well we are going to do the same thing
again to find E4. So I drew the line between P and R3 and that means that this point is
E4. And we can follow along and so R4 is close to our calculated value overall mass balances,
from when we originally did this in the first screencast. So it looks like, to a pretty good
approximation, it says it is slightly less than 4 stages of course the accuracy of this depends
very much on how good I am at drawing these lines, but the idea is what we are trying
to get across here, and we can see that this point, of course, that is in equilibrium with
E4 is pretty close to our original Rn value. And so we conclude that for this system, we
take 4 stages to do the liquid-liquid extraction. So this hopefully gives you a good idea of
how to apply the Hunter-Nash method to liquid-liquid extraction when you have a ternary
system that we are applying it to.