The equilibrium curve is calculated using the modified Raoult's law:
where
and
are the liquid and vapor mole fractions (
for methanol,
for water),
,
,
is total pressure (bar), and
is the saturation pressure (bar), which is calculated using the Antoine equation:
where
is temperature (K) and
,
and
are constants.
The activity coefficients
are calculated using the twoparameter Margules model:
,
,
where
and
are the dimensionless Margules parameters for a methanolwater mixture.
The operating line for the rectifying section from stages 1 to
is found from a material balance:
,
written in slopeintercept form as
,
where
and
are the liquid and vapor molar flow rates in the rectifying section,
is the distillate molar flow rate, and
is the mole fraction of methanol in the distillate stream.
Under the assumption of constant molar overflow (CMO), the reflux ratio
is the same everywhere. Because
, the rectifying section's operating line can be written in terms of the reflux ratio:
The operating line for the stripping section from stages
to
is found from a material balance:
,
which, when written in slopeintercept form, is:
,
where
and
are the liquid and vapor molar flow rates in the stripping section,
is the bottoms molar flow rate, and
is the mole fraction of methanol in the bottoms stream.
The boilup ratio
is constant everywhere. Combining
with a material balance around the partial reboiler,
, the operating line for the stripping section can be written in terms of the boilup ratio:
.
The feed operating line (
line) can also be used to locate the stripping operating line, because all three operating lines share an intersection point. The parameter
is defined as the ratio of the increase in molar reflux rate across the feed stage to the molar feed rate:
,
where
is the feed molar flow rate. The
line can be derived from the following mass balance:
.
By combining an overall material balance around the column
(where
is the mole fraction of methanol in the feed), a material balance around the feed stage
, and the definition of
, the
line can be written in slopeintercept form as
Once the operating lines and compositions are plotted on the

diagram, the number of equilibrium stages needed to achieve a bottoms composition of
is determined. Start at the top of the column at
. On each stage, the liquid and vapor streams are in VLE, so the stream has a liquid composition
and a vapor composition
. A vertical line (purple) is drawn from the equilibrium curve at
to the

line at
. This process is repeated until the bottoms composition
is reached. Typically the number of stages drawn is not a whole number. The partial reboiler is considered an equilibrium stage. View the screencast videos [2, 3] for a stepbystep explanation of the McCabe–Thiele method. The screencast video [4] explains how material balances and VLE data are used to determine the number of equilibrium stages.
[1] J. D. Seader, E. J. Henley and D. K. Roper,
Separation Process Principles: Chemical and Biochemical Operations, 3rd ed., Hoboken, NJ: Wiley, 2011.