Pinch point curves

Figure 12.15 The locus of tangents to the residue curves gives the pinch point curve.

The rectifying or stripping section of a column must operate somewhere between total reflux and minimum reflux conditions. The range of feasible operation of a column section can thus be defined for a given product composition. It can be seen in Figure 12.19 that these section profiles are bounded for a stage column by the distillation line and the pinch point curve. As noted previously, the pinch point curve provides a minimum reflux boundary for both staged and packed columns,... 

Also shown in Figure 12.19 is the residue curve projected from the same product composition. The area enclosed within the residue curve and the pinch point curve thus provides the feasible compositions that can be obtained by a packed column section from a given product composition. For any given product composition, the operation leaf of feasible operation for a column section can be defined by plotting the distillation line (or residue curve) and the pinch point curve1314. The column section must operate somewhere between the total and minimum reflux conditions. 

Figure 12.19 The operation leaf is bounded by the distillation line or residue curve and the pinch point curve. (From Castillo F, Thong DY-C, Towler GP, 1998, Ind Eng Chem Res, 37 987 reproduced by permission of the American Chemical Society).

Figure 12.25 The difference point allows the pinch point curves for the middle section operation leaf to be constructed. 

Bottom Product B with a straight line joining the Distillate D and Entrainer Feed E, as shown in Figure 12.24. Pinch point curves for the middle section can now be constructed by drawing tangents to the residue curves from the difference point (net overhead product). This is shown in Figure 12.25 for the system ethanol-water-ethylene glycol. The area bounded by the pinch point curves defines the middle section operation leaf. 

There is a point P that resides on both pinch point curves in Fig. 52. We can make the following argument concerning P ... 

By con.struction, P is simultaneously on both pinch point curves. 

As it is on the pinch point curve for D, the residue curve passing through P is tangent to the line from P to D. 

Fig. 53. Topology of the point P, the intersection point for a liquid distillate and bottom product pinch point curves.

If the feed is Fi, a point lying between D and P, then the pinch curve through D lies entirely to one side of the point F and curves back to cross at P. The corresponding distillation curve lies even farther away to the same side (the pinch point curve turns more sharply than the distillation curve). The reachable region for D excludes all points between D and P. Thus the points between F and P are unreachable by the top part of the column. As D could move to be coincident with F, points beyond F are not excluded. 

The region between F and P is a function only of F, the feed to the column, as we can find P by finding the pinch point curve for F from the arguments above. We have subtly turned our attention from D and B to F. Thus, we can map out this unreachable region by knowing only the column feed. 

Next, pick a bottoms product, B. that lies on this straight line above the inflection point. The pinch point curve emanating from B will move to the left initially because of the curvature of the local residue curves. It will move left until it encounters point a which, by construction, is a pinch point for B. It cannot cross this residue curve, however, because any residue curve an infinitesimal bit to its left can have no pinch point with B. On the other hand, a curve just to the right will have two pinch points with B, one just before a and one just after. The pinch point trajectory thus reflects off this residue curve. After encountering point a, it heads to the lower right and ultimately to the chloroform node. 

Finally, we note with a similar set of arguments that no product in the left-hand-side region can have a pinch point curve that crosses the distillation boundary, thereby justifying the statement appearing in the literature that one cannot cross such boundaries from the convex side. 

One line we have to think more carefully about is the pinch point line as we move to other dimensions. Each point on the line is the end point of a operating line for a distillation column operating with a fixed distillate (or bottoms) product, varying parametrically with the reflux ratio we use to define that operating line. Thus, a pinch point curve emanating from a fixed distillate composition is a sequence of points. It remains a line. It may bifurcate, as we have shown above, but it remains a line. 

The S shape of the residue curves gives us another pinch point curve for A. It starts at the azeotrope and ends at the node for A. Above this curve, movement is again in the wrong direction. Thus, we have only the region between these two curves in which we can correctly operate the section of the column between the two feeds. 

What does it mean to have two pinch point curves for A To operate as we wish, we need to keep the composition of the liquid on the trays between these two curves for the section of the column between the feeds. We shall now show that one of the curves dictates a minimum reflux ratio and the other a maximum reflux ratio that must be used to operate the column for a fixed solvent ratio. We shall then show that there is a minimum solvent ratio we can use—one where the pinch point trajectories just touch. We will also show that larger ratios move the pinch curves apart, making it easier to effect the separation. 

FUjURE 3.19 Movement of TTs for a constant relative volatility system with a = [5,1,2] for a set X4 = [0.2, 0.3] and vaiying reflux along the pinch point curve. 

FIGURE 13 Regions of unique pinch point curve bdiavior the diffmnt placements of 5 (a) S pair 1, (b) 8 pair 2, and (c) 8 pair 3. 

Notice that there are six possibilities of positive and negative net flow combinations, each producing unique pinch point curves, but their pinch point curve behavior suggests that these six different flow possibilities can be divided into three pairs because of the continuation of these curves. A pair essentially consists of two opposite flow patterns. These three pairs are imp ative for the design of this simplified system, since a mass balance around the thermally coupled junction in Figure 7.2 gives... 

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