Operating Lines

There are three operating lines used in the McCabe-Thiele Method. Each of these are straight lines, and therefore can be expressed by the standard equation of a line. Two of these lines are operating lines for the two sections of a distillation column, above and below the feed plate, and are derived from material balances. The third equation relates feed conditions.

The portion of the distillation column above the feed is called the rectifying section. If it is assumed that the heat of mixing is negligible and that the difference in the molar latent heats of vaporization of the binary system also differs by a negligible amount, then the observed relationship between any two passing streams is simplified to:
          
     y = mole fraction of more volatile component in the vapor
     x = mole fraction of more volatile component in the liquid
     x_D = mole fraction of more volatile component in the distillate
     D = Distillate product flow (mol/t)
     L = total flow rate of the liquid stream in the rectifying section (mol/t)
     V = total flow rate of the vapor stream in the rectifying section (mol/t)

This is known as the operating line for the rectifying section, or the upper operating line, in short. Remembering that the x-y diagram is a plot of vapor (V) vs. liquid (L), it can be seen that the operating line is a simple y=mx+b equation. Therefore, with the point (x_D,y₁) and the slope (L/V), this operating line can be plotted.

What are passing streams?

Each tray can be simplified to a system with two entering streams and two exiting streams. Vapor enters the stage from the bottom and leaves from the top, and liquid enters from the top and leaves from the bottom. This process is modeled in the following diagram.

As streams (2) and (3) enter, they mix in the stage. It is assumed that full contact is made, and the exiting streams are in equilibrium. The term "passing streams" refers to either set (1) and (2) or (3) and (4), where the streams are depicted as passing each other as they either enter or exit the stage.

 

Alternative Equation

There is a second equation which can describe the upper operating line, the derivation of which will not be discussed. However, often this equation is more useful when exact flows are not known. The following equation describes the upper operating line in terms of reflux ratio.

The portion of the distillation column below the feed is called the stripping section. With the previously stated assumptions, the observed relationship between any two passing streams is simplified to:
          
     y = mole fraction of more volatile component in the vapor
     x = mole fraction of more volatile component in the liquid
     x_B = mole fraction of more volatile component in the bottom product
     B = Bottom product flow (mol/t)
     L_bar = total flow rate of the liquid stream in the rectifying section (mol/t)
     V_bar = total flow rate of the vapor stream in the rectifying section (mol/t)

This is known as the operating line for the stripping section, or the lower operating line, in short. Again, it can be seen that this is a simple line equation with the slope (L_bar/ V_bar). With the slope and point where x_D meets the diagonal, the lower operating line can be plotted.

A third useful equation is the Feed-Line Equation, commonly known as the q-line. The equation for the q-line is:
          
     q = mole fraction of saturated liquid in feed stream
     x = mole fraction of any component in the liquid
     z_F = mole fraction of any component in the feed
On the McCabe-Thiele diagram, the q-line is useful because it represents the intersection of the upper and lower operating lines. The stage that crosses the q-line is the optimum feed plate. If the slope of the q-line is not directly known, it may be helpful to look at the feed plate enthalpic conditions. Two cases in particular are useful.

If either condition is met, the q-line can immediately be drawn, as seen in the example on the right.

Below is an example system with two possible feed conditions, saturated liquid and saturated vapor. Notice how the slopes of the upper and lower operating lines do not change, but the number of necessary stages changes due to feed conditions. While viewing the graphs, notice the differences:

Saturated Vapor

Through the use of and energy balance around the feed plate, an alternative form of the q-line equation can be found. This is an energy flow diagram of the feed plate:

Without derivation, this is the resulting equation:

     H_V = enthalpy per mol of vapor
     h_l = enthalpy per mol of liquid
     h_F = enthalpy per mol of feed

Both H_V and h_l are assumed independent of composition.