It is relatively common in separations (especially when we look
at **cascades** of separation units) to examine graphical
approaches to solutions.

The graphical solutions to be used always include an
**operating line** and an **equilibrium curve**

The **equilibrium curve** is a plot of the composition of a
component in one phase versus the composition of the *same*
component in the other phase.

A simple way to obtain the equilibrium curve is by using the relative volatility, which you may recall is defined as (for a binary system)

$\alpha_{AB} = \frac{K_A}{K_B} = \frac{y_A/x_A}{y_B/x_B} = \frac{y_A(1-x_A)}{(1-y_A)x_A}$

We can rearrange this to give:

$y = \frac{\alpha_{AB}x}{1+(\alpha_{AB}-1)x}$

The **operating line** is a line representing the material
balance equations.

There are several options of operating line format, but one form can be obtained, by rearranging the component mass balance for y:

y = -(L/V)x + (F/V)z

we then take the total material balance and we divide both sides by V to give:

(L/V) = (F-V)/V = (1-V/F)/(V/F)

combining these two equations gives:

y = [(V/F-1)/(V/F)]x + (F/V)z

If we denote the fraction of the feed that is vaporized (i.e., V/F) as f = V/F, we get the operating line as:

$y = \frac{f-1}{f}x+\frac{z}{f}$

Plotting these two lines yields the composition of both the exiting vapor (y) and liquid (x) as the intersection of the lines.

The slope and intercept of the operating line are typically used
to plot it in the first place. The dotted line along the diagonal
is the y=x line and is sometimes used to facilitate plotting the
operating line (i.e., set y=x in the feed line equation in order to
find the location where the feed line crosses the dotted line).
*It has no physical significance in this application*.

Derive and plot the operating equation for a binary flash distillation on a y-x diagram