McCabe-Thiele operating diagrams

To test the separation of the three actinides (Th, U and Pu) from each other, a modified Thorex solvent extraction flow sheet using 30% tributylphosphate (TBP) has been developed. Considerable work has already been reported in the literature for separations in Th-U systems ( 3, 4), but inclusion of Pu in the fuel cycle adds additional complexities (5). The flow sheet adopted is that in Figure 1, which also shows the relative flows (FL) of the inlet streams and concentrations (M or mol/L) of the major components. The flow ratios and acidities for each contactor were initially derived by constructing McCabe-Thiele operating diagrams based on unpublished distribution measurements made in our laboratories (6). 

A numerical solution of the SMB process in the nonlinear case, using the mixing cell method and applying the McCabe-Thiele operation diagram, was described by Ching et al. [114,115]. 

Figure 7.13 Removal of a single component contaminant from a feed continuous countercurrent adsorption system and McCabe-Thiele operating diagram (source Ruthven and Ching 1989).

Data could be obtained from such a graph to construct a McCabe-Thiele type diagram. Tie line points on the binodal curve would provide data for an equilibrium curve, and data could be obtained to construct an operating line (from lines drawn from 0 through the binodal curve). 

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. 

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.) ... 

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. 

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. 

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

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. 

The catalytic esterification of ethanol and acetic acid to ethyl acetate and water has been taken as a representative example to emphasize the potential advantages of the application of membrane technology compared with conventional distillation [48], see Fig. 13.6. From the McCabe-Thiele diagram for the separation of ethanol-water mixtures it follows that pervaporation can reach high water selectivities at the azeotropic point in contrast to the distillation process. Considering the economic evaluation of membrane-assisted esterifications compared with the conventional distillation technique, a decrease of 75% in energy input and 50% lower investment and operation costs can be calculated. The characteristics of the membrane and the module design mainly determine the investment costs of membrane processes, whereas the operational costs are influenced by the hfetime of the membranes. 

The graphical construction of an extraction isotherm, an operating line, and the stepwise evaluation of the number of stages in this manner is known as a McCabe-Thiele diagram. Flistorically, it found great application in a variety of mass transfer operations, from gas adsorption through distillation to solvent extraction. Flowever, the advent of modern computational techniques has made it largely redundant, as it is often easier and certainly more accurate to calculate the cascade directly. 

Figure 13.7. Features of McCabe-Thiele diagrams for constant molal overflow, (a) Operating line equations and construction and minimum reflux construction, (b) Orientations of -lines, with slope = ql q — 1), for various thermal conditions of the feed, (c) Minimum trays, total reflux, (d) Operating trays and reflux, (e) Minimum reflux determined by point of contact nearest xD.

Figure 13.9. Combined McCabe-Thiele and Merkel enthalpy-concentration diagrams for binary distillation with heat balances, (a) Showing key lines and location of representative points on the operating lines, (b) Completed construction showing determination of the number of trays by stepping off between the equilibrium and operating lines.

Example 7.1 The next part of the McCabe-Thiele diagram is the operating line. To generate the operating line, the problem for Fig. 7.2 must be stated. The following conditions are given. Find the required solvent rate and number of theoretical equilibrium stages to meet the treated water specification of 1.0% acid remainder. Benzene will be the solvent. Also determine what solvent inlet purity is required to meet this water specification. A step-by-... 

Figure 7.4 Operating line data in McCabe-Thiele diagram.

Figure 13 shows a McCabe-Thiele diagram, which can be used when the mixture to be distilled consists of only two components or can be represented by two components. Starting at the required overhead product composition x/), an upper-section operating line is drawn hav-... 

---