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 = x_{D}D + x_{B}B

and an energy balance:

$\Delta H = Q_R + Q_C$

where Q_{R} is positive and Q_{C} 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:

V_{j+1} = L_{j} + D

y_{j+1}V_{j+1} = x_{j}L_{j} +
x_{D}D

$\Delta H = Q_C$

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 x_{D} to x_{0}. For each of the
following stages, j+1, we use the previous balances (on j) to tell
us x_{j} composition of the incoming liquid stream.

Balances on the stripping section,

take the form:

V'_{f+1} = L'_{f} - B

y_{f+1}V'_{f+1} = x_{f}L'_{f}
- x_{B}B

$\Delta H = Q_R$

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 x_{B} to x_{6}. For each of the
following stages, f-1, we use the previous balances (on f) to tell
us x_{f} composition of the incoming liquid stream.

Write the mass and energy balances and equilibrium expressions for any stage in a column