FIG. 13-33 Typical construction for a sidestream showing the intersection of the two operating lines with the q line and with the x - y diagonal. (a) Liquid sidestream near the top of the column. (b) Vapor sidestream near the bottom of the column.
Pressures can be specified at any level below the safe working pressure of the column. The condenser pressure will be set at 275.8 kPa (40 psia), and all pressure drops within the column will be neglected. The equilibrium curve in Fig. 13-35 represents data at that pressure. All heat leaks will be assumed to be zero. The feed composition is 40 mole percent of the more volatile component 1, and the feed rate is 0.126 (kgmol)/s [1000 (lbmol)/h] of saturated liquid (q = 1). The feed-stage location is fixed at stage 4 and the total number of stages at eight.
Answers are desired to the following two questions. First, what bottom-product composition xB will the column produce under these specifications? Second, what will be the top vapor rate VN in this operation, and will it exceed the maximum vapor-rate capacity for this column, which is assumed to be 0.252 (kgmol)/s [2000 (lbmol)/h] at the top-tray conditions?
The solution is started by using Eq. (13-25) to convert the external-reflux ratio of 4.5 to an internal-reflux ratio of L/V = 0.818. The xD = 0.95 value is then located on the diagonal, and the upper operating line is drawn as shown in Fig. 13-35.
If the xB value were known, the bottom operating line could be immediately drawn from the xB value on the diagonal up to its required intersection point with the upper operating line on the feed q line. In this problem, since the number of stages is fixed, the xB which gives a lower operating line that will require exactly eight stages must be found by trial and error. An xB value is assumed, and the resulting lower operating line is drawn. The stages can be stepped off by starting from either xB or xD; xB was used in this case.
Note that the lower operating line is used until the fourth stage is passed, at which time the construction switches to the upper operating line. This is necessary because the vapor and liquid streams passing each other between the fourth and fifth stages must fall on the upper line.
The xB that requires exactly eight equilibrium stages is x1 = 0.026. An overall component balance gives D = 0.051 (kg mol)/s [405 (lbmol)/h]. Then,
which exceeds the column capacity of 0.252 (kg mol)/s [2007 (lbmol)/h]. This means that the column cannot provide an overhead-product yield of 40.5 percent at 95 percent purity. Either the purity specification must be reduced, or we must be satisfied with a lower yield. If the xD = 0.95 specification is retained, the reflux rate must be reduced. This will cause the upper operating line to pivot upward around its fixed point of x = 0.95 on the diagonal. The new intersection of the upper line with the q line will lie closer to the equilibrium curve. The xB value must then move upward along the diagonal because the eight stages will not "reach" as far as formerly. The higher xB concentration will reduce the recovery of component 1 in the 95 percent overhead product.
Another entire column with a partially vaporized feed, a liquid-sidestream rate equal to D and withdrawn from the second stage from the top, and a total condenser is shown in Fig. 13-36. The specified concentrations are xF = 0.40, xB = 0.05, and xD = 0.95. The specified L/V ratio in the top section is 0.818. These specifications permit the top operating line to be located and the two top stages stepped off to determine the liquid-sidestream composition xS = 0.746. The operating line below the sidestream must intersect the diagonal at the "blend" of the sidestream and the overhead stream. Since S was specified to be equal to D in rate, the intersection point is x = (1.0)(0.746) + (1.0)(0.95) = 0848 1.0 + 1.0 . This point plus the point of intersection of the two operating lines on the sidestream q line (vertical at xS = 0.746) permits the location of the middle operating line. (The slope of the middle operating line could
also have been used.) The lower operating line must run from the specified xB value on the diagonal to the required point of intersection on the feed q line. The stages are stepped off from the top down in this case. The sixth stage from the top is the feed stage, and a total of about 11.4 stages is required to reach the specified xB = 0.05.
Fractional equilibrium stages have meaning. The 11.4 will be divided by a tray efficiency, and the rounding to an integral number of actual trays should be done after that division. For example, if the average tray efficiency for the process being modeled in Fig. 13-36 were 80 percent, then the number of actual trays required would be 11.4/0.8 = 14.3, which would be rounded to 15.
Feed-Stage Location The optimum feed-stage location is that location which, with a given set of other operating specifications, will result in the widest separation between xD and xB for a given number of stages. Or, if the number of stages is not specified, the optimum feed location is the one that requires the lowest number of stages to accomplish a specified separation between xD and xB. Either of these criteria will always be satisfied if the operating line farthest from the equilibrium curve is used in each step as in Fig. 13-35.
It can be seen from Fig. 13-35 that the optimum feed location would have been the fifth tray for that operation. If a new column were being designed, that would have been the designer's choice. However, when an existing column is being modeled, the feed stage on the diagram should correspond as closely as possible to the actual feed tray in the column. It can be seen that a badly mislocated feed (a feed that requires one to remain with an operating line until it closely approaches the equilibrium curve) can be very wasteful insofar as the effectiveness of the stages is concerned.
Minimum Stages A column operating at total reflux is diagramed in Fig. 13-37a. Enough material has been charged to the column to fill the reboiler, the trays, and the overhead condensate drum to their working levels. The column is then operated with no feed and with all the condensed overhead stream returned as reflux (LN+1 = VN and D = 0). Also all the liquid reaching the reboiler is vaporized and returned to the column as vapor. Since F, D, and B are all zero, Ln +1 = Vn at all points in the column. With a slope of unity (L/V = 1.0), the operating line must coincide with the diagonal throughout the col umn. Total-reflux operation gives the minimum number of stages required to effect a specified separation between xB and xD.
Minimum Reflux The minimum-reflux ratio is defined as that ratio which if decreased by an infinitesimal amount would require an infinite number of stages to accomplish a specified separation between two components. The concept has meaning only if a separation between two components is specified and the number of stages is not specified. Figure 13-37£> illustrates the minimum-reflux condition. As the reflux ratio is reduced, the two operating lines swing upward, pivoting around the specified xB and xD values, until one or both touch the equilibrium curve. For equilibrium curves shaped like the one shown, the contact occurs at the feed q line. Often an equilibrium curve will dip down closer to the diagonal at higher concentrations. In such cases, the upper operating line may make contact before its intersection point on the q line reaches the equilibrium curve. Wherever the contact appears, the intersection of the operating line with the equilibrium curve produces a pinch point which contains a very large number of stages, and a zone of constant composition is formed.
Intermediate Reboilers and Condensers A distillation column of the type shown in Fig. 13-2a, operating with an interreboiler and an intercondenser in addition to a reboiler and a condenser, is diagramed with the solid lines in Fig. 13-38. The dashed lines correspond to simple distillation with only a bottoms reboiler and an overhead condenser. Total boiling and condensing heat loads are the same for both columns. As shown by Kayihan [Am. Inst. Chem. Eng. J. Symp. Ser. 76, 192, 1 (1980)], the addition of interreboilers and intercondensers increases thermodynamic efficiency but requires additional stages, as is clear from the positions of the operating lines in Fig. 13-38.
Optimum Reflux Ratio The general effect of the operating reflux ratio on fixed costs, operating costs, and the sum of these is shown in Fig. 13-39. In ordinary situations, the minimum on the total-cost curve will generally occur at an operating reflux ratio of from 1.1 to 1.5 times the minimum R = LN+i/D value, with the lower value corresponding to a value of the relative volatility close to 1.
Difficult Separations Some binary separations may pose special problems because of extreme purity requirements for one or both products or because of a relative volatility close to 1. The y-x diagram
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