- Here we're gonna look at distillation of a binary mixture
in a batch distillation where we have a total condenser
which means that the vapor, so this V is a molar flow rate.
A vapor enters the condenser.
This is where we're removing heat.
So we form complete condensation to a liquid.
Some of the liquid becomes the distillate.
So this is a molar flow rate of the distillate.
The rest becomes a liquid flow rate that's returned
to the column.
So we have vapor flowing up, liquid flowing down
and we have a still here where we start
with some mole fraction of
and what we'll use the mole fractions
So I'm gonna use X and Y
and mole fractions of the more volatile component.
And in particular, if I write X in
and Y in it means the Mole fractions leaving that stage.
So for to liquid X three, X two, X one, Y three
Y two Y one, oh, this would be Y zero.
The distillate mole fraction is x d.
Well, that's the same mole fraction here.
And Y D has the same value as X D.
So Y three and Y D are the same.
Other term they're going to use is the reflux ratio.
So that's going to be the molar flow rate.
L liquid over the molar flow rate D, the amount
the distillate we collect.
So V course is L plus D, right?
V is the flow rate coming
into here and we split it into two straight.
So what we wanna do is do a mass balance
on this more volatile component
the one that we're using X and Y and let's
Let's do that.
What I'm gonna do is just pick lice that includes two
of the stages and write a balance
around that on this more volatile component.
And so balance how much flows in better equal
The amount that flows out well
in is Y one times the vapor, the the gas molar flow rate.
And then out is X two here, the liquid flow rate
plus the other stream leaving is the D stream X, D times D.
So I can write that.
Let me rearrange it.
So here's the rearrange version.
And in general I could write it this way.
So this would be n plus one.
In other words, I could pick to do the balance
around the exit and stage three, stage one.
And of course, in a system we could have more
than three stages.
So this is a general form of that balance.
Now, we often write this in terms of this reflux ratio
and I'm just gonna write the answer down that L over V if
We just do the rearrangement is R Over R plus one
and then D V is one over R plus one.
And that's just by rearranging that we get this relationship
and it's just sometimes easier to work
in terms this reflux ratio.
Well, this equation or this one same equation
is the operating line on a x Y diagram where
we want to determine the number of stages.
Let's look at that diagram.
So on the left I've shown the complete diagram.
This line then is this equation.
The operating line.
This curve is the equilibrium expression
for example, routes law
something that relates the mole fraction, the vapor phase
to the mole fraction, liquid phase, at equilibrium.
And then what we're showing is how we step off the stages
so we can determine what's the distillate
for this setup
What would be the distillate mole fraction that we collect
And what I've done is expand just part of that figure
but keep in mind that Y and X
and so for this particular example
the mole fraction in the still is 0.4 for the liquid.
And if we then look at this value 0.65 for the mole fraction
of vapor, that's the vapor above the liquid.
That's the in equilibrium
So, so that would be Y zero.
So if I now look at the point on the operating curve
this would be X one and Y zero.
Now where as that's when I do a mass balance
Those are the two terms that are gonna show
up in this equation.
And that's why I've drawn this line.
So this point is still X one
but it's Y one because X one and Y one are an equilibrium.
They're the mole fractions leaving.
This is Y one, X one and Y one and equilibrium.
So they're on the equilibrium line.
This point then would be X two
and Y one values here that we, again
when we did a mass balance substitute into this equation.
So an operating line, and we continue this X two and Y two
they're an equilibrium X three, Y three.
So this would be X of the distillate.
And, and so for a batch distillation, keep in mind this is
at an instant in time as we carry out the distillation
we're changing the mole fraction still.
So, so this point would move this curve
which is related to this would change as XD change
because XD changes as we change the composition and still
but this hopefully gives you a better idea of why we count
off stages the way we do in this batch distillation.