Some binary mixtures, such as water and ethanol, have a non-ideal behavior that is characterized by the presence of a positive azeotrope. This minimum-boiling azeotrope makes the preparation of anhydrous ethanol a challenging task that requires special techniques, such as extractive distillation and heteroazeotropic distillation using ethyleneglycol and benzene as entrainers, respectively.

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This Demonstration presents the McCabe-Thiele graphical method applied to the water-ethanol mixture. First, the rectifying (cyan) and stripping (magenta) operating lines are plotted using a reflux ratio equal to 1.5 times the minimum reflux ratio. These lines are derived by writing global and partial material balances around different sections of the column. The feed line (black) is also plotted. If the construction is correct, these three lines must intersect at the same point. For the separation to be feasible, you must choose values for the parameters' feed quality, bottom, feed and distillate compositions that cause the intersection point to fall below the equilibrium curve (red). The equilibrium curve is obtained using the van Laar model for the computation of the activity coefficients, which are used in the modified Raoult's law. The stages are stepped off using the equilibrium curve and the two operating lines in order to determine the number of equilibrium stages or the number of theoretical column plates. The points that allow the construction of the steps are alternatively on the equilibrium curve or on the operating line. If one point is on the equilibrium curve, it corresponds to two streams in equilibrium leaving a particular stage. When the point is on the operating line, it corresponds to two passing streams, one entering and one leaving the stage.

The horizontal and vertical axes are the liquid and vapor mole fractions. The red curve is the equilibrium curve. In fact, every liquid with mole fraction is in equilibrium with its vapor with mole fraction . The line segments allow a visualization of the steps, which are equilibrium stages.

One limitation of this approach is that it will not work when the feed is saturated vapor (feed quality ), because the slope of the feed line becomes infinite. The Demonstration displays the number of stages, which is the sum of the plates in the rectifying and stripping sections. The optimum feed plate is the stage where the operating lines intersect. If the user sets the distillate specification close to the azeotropic composition, then the number of plates becomes very large. Indeed, there is a severe tangent pinch (i.e., the operating line is almost tangent to the equilibrium curve).

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## Details

Using the van Laar model, the activity coefficient for component 1 is given by:

,

where and are the binary interaction parameters and is the liquid mole fraction of component .

Pure component vapor pressure, , is given by Antoine's equation:

, where is in mmHg, the temperature, , is expressed in °C, and , , and are Antoine's constants that depend on the pure component considered.

VLE data is computed using the modified Raoult's law, which is given by:

, where and are the liquid and vapor mole fractions of component , is the vapor pressure of component , its partial pressure, and finally is the activity coefficient of component *i*.