The first step in doing solving distillation column problems is to examine the overall system (we have done this already):
We can clearly write two material balances:
F = B + D
zF = xDD + xBB
and an energy balance:
$\Delta H = Q_R + Q_C$
where QR is positive and QC is negative.
A critical component of solving distillation problem is knowing whether the condenser/reboiler are partial or total condensers/reboilers. Total condensers/reboilers mean that the compositions leaving the condenser/reboiler are the same for both streams, while partial condensers/reboilers mean the compositions leaving them are in equilibrium
Depending on what other information is given in the problem, these equations may need to be augmented by balances around the condenser or reboiler (if we know the reflux ratio L0/D or the boilup ratio V6/B).
In order to do design of distillation columns, we need to know the compositions at each stage (so we can determine how many to build). Doing a balance on a single stage within a distillation column, or even a group of stages, is much like doing a balance on a single flash drum.
For reasons that will become clear in the next section, we typically write our balance equations separately for the region above and below the feed stage.
The enriching section or rectifying section of the column are the stages (and condenser) above the feed stage.
The stripping section of the column are the stages (and reboiler) below the feed stage.
Balances on the rectifying section,
take the form:
Vj+1 = Lj + D
yj+1Vj+1 = xjLj + xDD
$\Delta H = Q_C$
This form of the balances works even for j=0, the condenser. Knowing the reflux ratio (L0/D) is very useful for that balance set.
If we start our procedure at the condenser, we know that determining whether the condensation is partial or total will allow us to relate xD to x0. For each of the following stages, j+1, we use the previous balances (on j) to tell us xj composition of the incoming liquid stream.
Balances on the stripping section,
take the form:
V'f+1 = L'f - B
yf+1V'f+1 = xfL'f - xBB
$\Delta H = Q_R$
This form of the balances works equally well for the reboiler. Knowing the boilup ratio (Vn/B) is very useful for that balance set.
If we start our procedure at the reboiler, we know that determining whether the vaporization is partial or total will allow us to relate xB to x6. For each of the following stages, f-1, we use the previous balances (on f) to tell us xf composition of the incoming liquid stream.
Write the mass and energy balances and equilibrium expressions for any stage in a column