In this screen-cast we are going to look at operating a batch distillation unit with a
constant reflux ratio.
In the problem statement we are given the total number of equilibrium stages in the
system (2), we are also given a reflux ratio of 3, and then we are posed with the scenario
that at the distillate composition of 60 mole % we know how much is in the reboiler in terms
of number of moles.
And then we are asked to determine the amount of liquid left in the reboiler when the instantaneous
distillate composition is a later value of 50 mole %. So let's take a second to think
about what this distillation unit looks like.
Early, simple, distillation units consist of just the pot still with a condenser to
get the distillate product out.
In this case we are not looking at that, we are actually thinking about taking this basic
reboiler system and adding an additional column to the top of that.
So we have reflux returning to the column and we have additional equilibrium stages
associated with that.
And in this operation we have the Rayleigh equation to relate the compositions and the
amounts of the material in the still at any given time.
So the overall plan is to first find a relationship between the distillate and the boiler compositions.
And we are going to do that by constructing McCabe-Thiele Diagrams.
And then we are going to take that and plug it into a numerical integration of the Rayleigh
equation.
And for this situation we are going to use the Simpon's rule to do that numerical integration.
So first, in order to get started with the McCabe-Thiele diagrams we are going to first
find vapor-liquid equilibrium data.
And we have some provided here for the ethanol water system.
And then on that diagram we are going to zoom into the region of interest and also plot
our operating line.
Our operating line for this case is going to be bound by the point x is equal to y is
equal to our distillate composition of 0.5.
We also know we have a reflux ration of 3 and from that we calculate a slope of 3/4
for our operating line.
When we plot that operating line we can then step off two stages and determine, when we
have a distillate composition of 0.5, which is our end point on our distillation operation,
we know that the composition in the reboiler is 4% ethanol.
So now we have a relationship between xd and xw at the end of our process.
Now we need to repeat this to relate xd and xw both at the beginning of the process and
also one point directly in the middle.
We are going to talk about why we chose those points a little later when we get into the
details of the Simpson rule.
So if we pick the point in the middle and start evaluating the relationship between
xd and xw we repeat the same process.
Since our reflux ratio is constant we still have the same slope of an operating line.
But now, since we have a new distillate composition of 55 mole % ethanol we now have translated
that line further up that x=y reference line.
So now we step off two stages and see that when we have a distillate composition of 55
mole % ethanol, we have a reboiler concentration or an xw of 6% ethanol.
Repeat the process one more time and now we are evaluating it at the beginning of the
distillation process where we have a distillate composition of 60 mole % ethanol and we see
that is related to a reboiler composition of 9% ethanol.
Now we just simply tabulate the results we have generated so far.
And I've shown that here with one column indicating our reboiler compositions, xw, and then one
column indicating our distillate compositions, xd.
So now that we've related the distillate and the reboiler compositions, now we want to
return to our original problem statement using the Rayleigh equation.
And we want to take this equation and we want to evaluate this side of the equation, the
right side of the equation, using the Simpson's rule which is a numerical integration technique.
Now the Simpson's rule just takes a look at the function that is inside the integral and
evaluates that function at three different points, which would be the initial point,
the final point, and a point directly in between.
For our case we actually have a function of 1/xd-xw.
So now we simply have to evaluate this f(x) at three different point in the distillation
operation.
So let's go back to exel to do this.
And what we see here, is you can see I've created a new column in excel that simply
takes 1 divided by the distillate composition minus the reboiler composition, our xw, and
calculates this for each of our three points.
Once we have that, we then create a new cell, which I'm calling the Simpson's Rule cell
here.
And basically I've entered the formula for a Simpson's rule, which is our final reboiler
composition minus the initial reboiler composition, divided by 6, plus our initial state, plus
4* our halfway point, plus the final state.
And when we evaluate that point we know have an approximation for the right hand side of
the Rayleigh equation.
And we can then just do the simple math using our initial charge to the reboiler and calculate
the final number of moles in that reboiler.
I have done that here in excel, where the final amount of material in the reboiler is
related to the initial number of moles in the reboiler times e^(-sum that we determined
by the Simpson's rule).
And when we evaluate that we see that the final number of moles in the reboiler is 677
moles.