Details

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 two-parameter Margules model:

,

,

where and are the dimensionless Margules parameters for a methanol-water mixture.

The operating line for the rectifying section from stages 1 to is found from a material balance:

,

written in slope-intercept 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 slope-intercept 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 slope-intercept 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 step-by-step 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.

References

[1] J. D. Seader, E. J. Henley and D. K. Roper, Separation Process Principles: Chemical and Biochemical Operations, 3rd ed., Hoboken, NJ: Wiley, 2011.

[2] LearnChemE. McCabe–Thiele Graphical Method Example Part 1 [Video]. (Feb 2, 2017) www.youtube.com/watch?v=Cv4KjY2BJTA.

[3] LearnChemE. McCabe–Thiele Graphical Method Example Part 2 [Video]. (Feb 2, 2017) www.youtube.com/watch?v=eIJk5uXmBRc.

[4] LearnChemE. McCabe–Thiele: Stepping Off Stages [Video]. (Feb 2, 2017) www.youtube.com/watch?v=rlg-ptQMAsg.