McCabe diagram

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.

Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liqmd-phase flow rate is assumed to be constant from tray to tray in each sec tiou of the column between addition (feed) and withdrawal (produc t) points. If the liquid rate is constant, the vapor rate must also be constant. 

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.)... 

The design of a distillation column is based on information derived from the VLE diagram describing the mixtures to be separated. The vapor-liquid equilibrium characteristics are indicated by the characteristic shapes of the equilibrium curves. This is what determines the number of stages, and hence the number of trays needed for a separation. Although column designs are often proprietary, the classical method of McCabe-Thiele for binary columns is instructive on the principles of design. 

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. 

Figure 8-2. Continuous fractionation of binary mixtures McCabe-Thieie Diagram with total condenser.

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 ... 

Figure 8-35. Batch distiilation constant reflux ratio after McCabe-Thieie diagram. Revised/adapted and used by pennission, Schweitzer, PA Handbook of Separation Techniques for Chemical Engineers, McGraw-Hiil Book Co. (1979) aiso reprinted by special permission, Chem. Eng. Jan. 23 (1961), p. 134., 1961 by McGraw-Hili, Inc., New York.

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

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