McCabe-Thiele analysis for (almost) arbitrary reflux ratios is similar to total reflux, if slightly harder.

In this case the two operating equations (lines), an equilibrium expression (line) using the relative volatility, and a feed expression (line) do not simplify, and thus must be used in their normal forms:

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)

We first plot the equilibrium line on an x-y diagram. We then
plot the rectifying and stripping operating lines -- using the
x_{D} and x_{B} points on the y=x line as our first
points and the slopes L/V and L'/V', respectively.

The feed line will start at the x_{F} composition on the
y=x line and go to the intersection of the two operating lines. It
is possible that we might need to use this info rather than the
distillate and bottoms compositions and slopes. One option would be
to set the feed equation equal to one of the other operating
equations to analytically find the intersection point.

The steps in this case still go horizontally (left) for equilibrium, but now when we go vertically (down) for our material balance, we go to the rectifying operating line (prior to the intersection point) and down to the stripping operating line (after the intersection point):

It is possible that we will not match our distillate and bottoms
compositions both *exactly*. In this case, we say we need 2.9
stages, for example. In reality we obviously can only have an
integer number of stages (but would achieve a better separation
than expected).

If we think about it, we can see that there is a limit to how small our reflux ratio (L/D) can be. As L/D decreases, we also decrease our slope L/V (prove this to yourself with an overall material balance on the rectifying section). In this case, we might obtain a graph like the following:

Clearly we cannot operate the column in this way, as we would
need to move horizontally **beyond the equilibrium point**,
which is physically impossible. Instead there is a minimum value of
the reflux ratio that can be determined by finding the conditions
under which the intersection point *just* "pinches" the
equilibrium line:

The **minimum reflux ratio** is the ratio of L/D that leads
to the intersection of the rectifying and stripping operating lines
falling on the equilibrium curve (rather than inside it).

This condition, however, would require an infinite number of stages (can you see why?).

In reality, each stage will not quite achieve the equilibrium compositions (due to poor mixing, or finite contact times), therefore our predictions from a McCabe-Thiele analysis will be slightly low relative to the actual number of stages required.

Determine the number of stages required to achieve a separation at minimum or higher reflux ratios using the McCabe-Thiele method