# Theoretical number of plates for distillation column

1. Mar 5, 2015

### Maylis

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

With my solution, I think cold feed means $q = \infty$, which is seen from my xy diagram. I have some issue understanding the statement "for each mole of feed 0.15 mol of vapor is condensed at the feed plate". I assumed that to mean that after the top stream goes through the condenser, 0.15 moles are sent back to the column in the split with the distillate stream. However, it became clear that my mass balance is wrong, since there are more moles of methanol coming from the distillate than is being fed, which does not make sense. So I don't really know what I am supposed to do with this critical piece of information.

Last edited: Mar 5, 2015
2. Mar 5, 2015

### Staff: Mentor

3. Mar 11, 2015

### Maylis

Sorry for my late response, been very busy.

So I know the equation
$$q = \frac {\bar {L} - L}{F}$$

Where $\bar {L}$ is the liquid feed below the feed tray, $L$ is the feed above the feed tray, and $F$ is the feed. Now I am thinking of what the liquid feed below the feed tray, $\bar {L}$, should be in this instance. Well, I suppose it would be the liquid feed from the tray above, plus 0.15F. So $\bar {L} = L + 0.15F$. Well, then I plug into the equation for the q-line,

$$q = \frac {L + 0.15F - L}{F} = 0.15 < 1$$

However, the PDF says that $q > 1$ for a cold liquid feed, therefore my definition of $\bar {L}$ is incorrect. What can I be missing?

EDIT: I see from my textbook that it would be L + qF, hence L + 1.15F. Well, I am having a little difficulty understanding it then. Wouldn't some of the feed vaporize when it enters the column? It seems to mean that all the feed goes to the stripping section, and it also condenses some vapor rising from the plate below in addition to the feed and liquid coming down from the bottom tray.

Last edited: Mar 11, 2015
4. Mar 11, 2015

### Maylis

Apologies for the sideways image, but I am wondering geometrically how to get that fraction of a theoretical plate. The distance from plate 3 to $x_{B}$, the bottoms mole fraction, is 0.1. The distance from $x_{B}$ to the equilibrium line is 0.08. I suppose if it was split even 0.05 and 0.05, it would be 2.5 plates plus a reboiler, but I am unsure what ratio I should properly use.

Side note, the q-line doesn't seem to have mattered for solving part (a)

5. Mar 11, 2015

### Staff: Mentor

From the reference I recommended (which looks great), I get q = 1.15.

Chet

6. Mar 11, 2015

### Staff: Mentor

If you have a non-integer number of plates, you round up to the next largest integer.

To get the minimum number of plates, you are using total reflux, so no feed is being added. So, in this case, there is no q.

Chet

7. Mar 11, 2015

### Maylis

In the textbook, there are non-integer values for theoretical plates (for the real plates, that is when you round up). I know it sounds stupid, but we are supposed to report the answer as some fraction of a plate.

8. Mar 11, 2015

### Staff: Mentor

OK. Start at the bottom and step upward, then start at the top and step downward. Then take the average of the two numbers of plates. I'm sure the two numbers will come out very close to the same value. Anyhow, in the end, its a judgement call.

Chet