In this screencast we are going to work through an example problem involving the separation
of two species, in this case acetone and ethanol using distillation. And we are going to use
the McCabe-Thiele method to do this. So I will give you the time to pause this and read
the question, so you understand what we are doing, and in other screencasts you can see
how these operating lines and methods are derived but in this screencast we are just
going to work through this problem.
Now the first thing you would want to do is collect
vapor liquid equilibrium data and plot the x vs y or liquid vs vapor mole fractions of
the more volatile species. So in this case we have pressure at 101.3 kPa. We have our mole
fraction of acetone in the liquid and the mole fraction of the acetone in the vapor
plotted on this xy diagram. I also have an x equals y line that I have thrown into the plot.
This will help us as we go through the McCabe-Thiele method. Let's go ahead and determine our feed
line. So our feed operating is defined as y, the vapor mole fraction of acetone, is
equal to the quality of the feed, q, over the quality minus 1, times the liquid mole
fraction minus zF to designate our feed composition and this is over our quality minus 1. So what
is our feed quality? The feed quality is an indication of how much of our feed is liquid
or vapor. And we could write this as a balance of our liquid flow rates in and out of our
feed plate over the amount that we feed. And so when we look at this we want to get an
idea of what our feed plate is. So I have drawn our feed, coming into our feed plate.
We know that we have a vapor flow rate leaving, a liquid flow rate coming in, the vapor
coming from the stripping section we designate Vbar and the liquid leaving the feed plate we designate Lbar.
And these flow rates are designated differently because we're not sure what the feed is. The
feed could be some form of a vapor or it could be some form of a liquid. Now in this case
we have a sub cooled liquid so we have no vapor that is coming in from our feed. And
more so, some of the vapor actually comes in and gets condensed to join that liquid leaving.
So we lose some of our vapor so we would expect that our V, our vapor flow rate
leaving the feed, is going to be less than our Vbar. So to determine this quantity, we
need to write a balance and substitute the values we have into this. So we know Lbar
is going to be equal to L plus our feed F, plus, for every six moles of our feed we condense
1 mole of our vapor. So we could say that it is also going to add 1/6 of our feed. We
can use this information to plug into q. This becomes Q is equal to F plus one sixth F over F and we
get a quality of 7/6. Plug that in here and we know that our feed composition is equimolar
so we have a 0.5 mole fraction for zF, and now we could write our operating line equation
for the feed. So our feed operating line ends up having the equation y equals 7x minus 3. And so
we could plot this on our xy diagram. So I have plotted my feed operating line, we can
start working on our rectifying operating line. So our rectifying line has the equation
y equals L over V, L being the liquid flow rate and V being the vapor flow rate in this section,
times x, the liquid mole fraction, plus D, the distillate flow rate over V times xd,
which is the liquid mole fraction of our distillate. The other equation you may see is in terms
of the reflux ratio, R, which then has the form, R over R plus 1 times x, plus 1 over R plus 1, times xd.
So depending on the information you are using, the reflux ratio or some ratio of the
flow rates you would determine this line. Now from the problem statement we are told
that we are operating L over V at 1.4 times L over Vmin. So we don't know what this minimum ratio
is but we do know how we could calculate that. Whenever we have a minimum ratio of liquid
being sent back into the distillation column we know that that minimum flow rate is going
to be correlated with needed an infinite amount of equilibrium stages to achieve the separation
that we are doing. And so we go back to our xy diagram and I am going to draw our 0.95
liquid mole fraction point. This is our xd, what we are looking to get from our separation.
So get an infinite number of stages we know that our operating line for both the feed
and the rectifying section have to intersect. If we draw this line through our highest point
of our feed equilibrium intersection there then we are going to get a pinch point at
that location and that pinch point is associated with needing an infinite amount of stages.
And I will show you what I mean by this so it will make sense as we start doing this
more. So now that I have these two points you would want to determine the slope of a
line through that. Now the reason that this is our L/V min is again recall that the equation
for our operating line in this section has the following form, and the slope is L over
V. So by drawing this line that is associated with infinite stages and minimum reflux we
could determine that now our L over V that we are actually operating at is 1.4 times
that value which gives me a new slope of 0.868. So I am going to use 0.87 and now I could
determine my operating line for the rectifying section as 0.87x plus D over V times our distillate
composition. So we know our distillate composition is 0.95. We don't know what D is. However
we do know that V is equal to the liquid going back into the column plus our distillate leaving
the column. Therefore if dividing everything by V, we could see that 1 is equal to L/V
plus D/V. Which allows us to substitute in 1 minus L/V in place for D/V. So when I do that
I get 0.13 times 0.95 is 0.12. So now we have our operating line for our rectifying section
that we could add to the McCabe-Theile plot. So we know the intercept is about 0.12, about
the closest I could draw on this. We are going to make a line that goes through these two
points, and that is our rectifying line. So the last line we need to add to our plot is
that for the stripping section. And now that we have two of our operating lines we know
that the stripping operating line must also coincide with an intersection at these points.
And we see an intersection between our feed line and our rectifying line here. The other
point we know is at the bottoms composition is 0.05 which falls on our xy line. So we
could just draw a line between these two points to determine our stripping line. So I like
cleaning up our plot, we don't need our rectifying line going all the way into our y axis. We
also don't need our feed line going past our intersection point. And that leaves us with
a cleaned up McCabe-Thiele diagram. So we have successfully plotted our operating lines
and in another screencast we will work through determining the equilibrium stages and the
molecular flow rates that we were asked to in the problem.