McCabe-Thiele diagrams

If we were to start with a liquid containing 0.5 mole fraction C3 (point A) and allow it to evaporate partially in a closed container to equilibrium conditions, then the vapour would have the composition at B. If we were to remove the vapom and condense it, it would then contain about 0.75 mole fraction of C3. The liquid left in the container would of course be substantially richer in C4. This is one theoretical tray. 

The method as shown assumes total reflux, i.e. no overhead product is drawn. The reflux ratio (R/D) is therefore infinite. It therefore gives us the minimum number of theoretical trays. 

The molar feed rate to the column is F. It has feed quality of q. This rather confusing term has nothing to do with the composition of the feed but is a measure of its enthalpy. Its definition is given by 

But perhaps an easier way of understanding it is thinking of it as the fraction of the feed which leaves the feed tray as liquid. Under normal conditions the liquid held on each tray is 

However, if the feed entering is liquid exactly at its bubble point, no heat exchange is necessary. Thus all of the liquid entering as feed will leave the feed tray as liquid and so q will be 1. Similarly, if the feed entering is vapour exactly at its dew point, all of the feed will leave the feed tray as vapour and q is 0. Values of q between 0 and 1 are possible. Under these circumstances the feed will be a mixture of vapour and liquid at the saturation temperature. In fact q is then a measure of the wetness of the mixture. 

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Fig. 25. McCabe-Thiele diagram, (a) Number of theoretical plates, 5 (b) number of actual plates, 8.

McCabe-Thiele diagrams for nonlinear and more practical systems with pertinent inequaUty constraints are illustrated in Figures 11 and 12. The convex isotherms are generally observed for 2eohtic adsorbents, particularly in hydrocarbon separation systems, whereas the concave isotherms are observed for ion-exchange resins used in sugar separations. 

Alternative approaches are to be found in the hterature. Derivations of the above equations are given in numerous texts (2,10—12), which also describe graphical or analytical solutions to the problem. Many of these have direct analogues in other separation processes such as distillation (qv) and hquid—hquid extraction, and use plots such as the McCabe-Thiele diagram or Ponchon-Savarit diagram. 

Fig. 3. A McCabe-Thiele diagram for a hypothetical square cascade section illustrating pinching. Terms are defined ia text.

Fig. 8. McCabe-Thiele diagram. Terms are defined in text.

FIG. 13-37 McCabe-Thiele diagrams for limiting cases, a) Minimum stages for a column operating at total reflux with no feeds or products, (h) Minimum reflux for a binary system of normal volatility. 

FIG. 13-38 McCabe-Thiele diagram for columns with and without an interre-hoiler and an intercondenser. 

For certain simplified cases it is possible to calculate directly the number of stages required to attain a desired product composition for a given set of feed conditions. For example, if equilibrium is attained in all stages and if the underflow mass rate is constant, both the equilibrium and operating lines on a modified McCabe-Thiele diagram are straight, and it is possible to calculate direc tly the number of ideal stages required to accommodate arw rational set of terminal flows and compositions (McCabe, Smith, and Harriott, op. cit.) ... 

Figure 14.1 The McCabe-Thiele diagram for the calculation of the number of theoretical stages required to separate two liquids to yield relatively pure products...

In addition to the previously mentioned shortcut equations, plotting a McCabe-Thiele diagram is also a very useful tool. The equation for the equilibrium X-Y diagram and plotting of the operating lines are described next. 

Kister shows how the McCabe-Thiele Diagram is an excellent tool for analyzing computer simulation results. It can be used to... 

Revised McCabe-Thiele Diagram (For Reiative Volatilities under 1.25)... 

The familiar steps for the theoretical stages on the McCabe-Thiele diagram are modified on the Ryan plot. The vertical portions remain vertical, but the horizontal portions become slanted with a slope of -1.00 (see Figure 2). 

Ryan, J. M., Ryan Consulting, Inc. (jamesryancon sulting juno.com) Replotting the McCabe-Thiele Diagram , Chemical Engineering, May, 2001, p. 109. 

Solution Constructed McCabe-Thiele diagram. Pinch found. (See Section 3 Fractionators Graphical Methods.)... 

On a y-x (McCabe-Thiele) diagram, this equation represents the operating line which extends between the points (y , x" ) and (yf , x") and has a slope of Lj/Gi, as shown in Fig. 2.5. Furthermore, each theoretical stage can be represented by a step between the operating line and the equilibrium line. Hence, NTP can be determined by stepping off stages between the two ends of the exchanger, as illustrated by Fig. 2.5. 

Fig. 5-8. McCabe-Thiele diagram for a

Dechman [109] illustrates a modification to the usual McCabe-Thiele diagram that assumes constant molal overflow in a diagram that recognizes unequal molal overflow. 

The system material balance from Treybal [129] using a heated kettle and distillation column following a McCabe-Thiele diagram, using reflux, but having only a batch (kettle) charge ... 

Tf = ratio of key components at intersection point of operating line (from a McCabe-Thiele diagram). 

Total reflux for a symmetric separation. Note, the term symmetric separation is used here to mean that on a McCabe-Thiele diagram, the liquid phase compositions of the overhead product and bottom product are roughly equidistant from 0.5. 

The situation for any plate n, with liquid composition x corresponding to an equilibrium vapour composition y , but with actual vapour composition y , is represented on a small section of the McCabe-Thiele diagram in Fig. 3.62. 

When concentrations of the more volatile component of either product is very low the steps on the McCabe-Thiele diagram become very small and difficult to plot. This problem can be overcome by replotting the top or bottom sections to a larger scale, or on log-log paper. In a log plot the operating line will not be straight and must be drawn by plotting points calculated using equations 11.21 and 11.22. This technique is described by Alieva (1962) and is illustrated in Example 11.2. 

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