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These lines are shown in Fig. 2.96. Note that the 7-line passes through the intersection of the two component balance lines.

step 4 Step off the stages. Start off at the point (xD,xD>. Move horizontally to the left until you meet the equilibrium curve. The point of intersection with the equilibrium curve represents the vapor and liquid compositions of stage 1 vj. Then move vertically down to the point (xl,ya), which is located on the component balance line [Eq. (2.9)]. Move horizontally to the left until you meet the equilibrium curve at point (x2>y2). Continue stepping off stages until reaching the bottom composition^. The number of times the equilibrium curve is met is the number of stages. Note that the optimum feed point is where the component balance lines intersect. The number of stages above the intersection point is the number of rectifying stages. The number of stages below is the number of stripping stages.

This procedure is illustrated in Fig. 2.9c. For this example, just over 9 stages are required; 10 stages will ensure that the separation is achieved. The best feed point is stage 6, giving 5 rectifying and 5 stripping stages.

If the condenser were a partial condenser, the condenser would have been stage 1. In this case, the number of stages in the column would have been reduced from 10 to 9.

2.2.5 Optimum feed stage and pinching

In Sec. 2.2.4, it was assumed that the feed enters the column at the optimum feed stage, which is located at the intersection of the component balance lines. At that point, the construction was switched from the rectifying section component balance line to the stripping section component balance line.

This switch could have been made earlier or later, depending on the location of the actual feed point. Figure 2.10a shows a switch taking place earlier, because the column feed point is located between stages 4 and 5 (compared to stage 5 and 6 in Sec. 2.2,4). Figure 2.106 shows a switch taking place later, because the column feed point is located between stage 8 and 9. In either case, more stages are required (11 and 12 stages in Fig. 2.10a and 6, respectively, compared to 10 stages in Fig. 2.9c).

The reason for the greater number of stages is that steps become smaller as the component balance line moves closer to the equilibrium curve, and therefore more steps are required. The optimum feed point is therefore achieved when the "active" component balance line is as far as possible from the equilibrium curve.

Pinching. As the component balance line approaches the equilibrium curve, the steps become smaller. An infinite number of stages is required to reach the intersection of the component balance line and the equilibrium curve. This intersection is termed the pinch point. The bottom pinch point is (0.22, 0.4) in Fig. 2.106, and the top pinch point

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