In order to determine the number of stages necessary for a given separation (i.e., known feed composition, bottoms composition, and distillate composition), we could analytically solve stage-wise balance equations starting from the top (with the rectifying equations) and switching over to the stripping equations, once we reach the feed composition.

A simpler technique, however, involves a graphical approach. We already have two operating equations (lines), an equilibrium expression (line) using the relative volatility, and a feed expression (line).

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

RECTIFYING LINE: y = xL/V + x_{D}(1-L/V)

STRIPPING LINE: y = xL'/V' - x_{B}(L'/V'-1)

FEED LINE: y = xq/(q-1) - z/(q-1)

The **McCabe-Thiele method** is a graphical separations
procedure that exploits the fact that a separation stage involves
interaction between the mass balance and vapor-liquid
equilibrium.

The simplest example of a McCabe-Thiele analysis involves the special case of no material being removed from the top or the bottom, so that the L=V=L'=V' and both the stripping and rectifying lines reduce to:

y = x

This operating condition corresponds to a reflux ratio (L/D) and boil-up ratio (V/B) of infinity. The feed line is useless in this case.

This condition leads to a plot that involves the equilibrium curve and a 45 degree operating line

If we start at the desired distillate composition x_{D},
and recalling that we need to use a total condensor if D=0, we note
that the vapor composition y_{j} coming off the top stage
of the column is the same as x_{D} (as is the liquid
composition returing to the column x_{0}). The 45 degree
line, therefore, shows what the composition y_{j} actually
is.

The composition of the liquid in equilibrium with that vapor
(i.e., the one with composition x_{j}) can be determined by
looking vertically at the x value of the curve at that point
y_{j}.

Our material balance (operating line) tells us the relationship between the current vapor composition and the composition of the next vapor stage. We, therefore, "perform" our material balance by first moving vertically (down) to the operating line (in this case the y=x line) then moving horizontally (left) to the equilibrium line, to tell us the y composition of the next lower stage. In order to solve the remainder of the problem, we continue with stage-wise stepping until we reach the bottoms composition:

Each stair-step represents a stage within the column. This example, therefore, involved 3 stages.

Determine the number of stages required to achieve a separation at total reflux using the McCabe-Thiele method