How to Use the McCabe-Thiele Method for Fractional Distillation

This Demonstration shows how to use the McCabe–Thiele method to determine the number of equilibrium stages needed to separate a binary mixture (methanol and water) in a distillation column.
First, select "plot points" to show the location of the feed, distillate and bottoms compositions on the - diagram. The methanol feed mole fraction is , the desired methanol mole fraction in the distillate is , and the desired methanol mole fraction in the bottoms is . Next, select "draw operating lines" to set the feed quality with a slider and plot the operating lines for the feed, rectifying and stripping sections. The diagram on the right shows a representation of part of the distillation column and explains the equations used for the operating lines.
Then, select "count stages" and use the slider to step off stages and draw lines for each stage; the diagram on the right highlights the corresponding location on the column. The partial reboiler is considered an equilibrium stage. The green horizontal lines on the - diagram result from a material balance between stages. The purple vertical lines correspond to vapor-liquid equilibrium (VLE) on each stage. Finally, select "optimal feed location" to determine where the feed should enter the column; the feed should enter above the stage where the rectifying and stripping sections meet. Mouse over the stage numbers on the - diagram to see which section the numbers correspond to.

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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.
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