Graphical Design of a Distillation Column
Start with the the feed of the previous illustration, consisting of 50 mol % n-pentane and 50 mol % n-heptane, and determine how many equilibrium stages are needed to recover 95 percent of the n-pentane in the feed in a stream that contains 98 mol %n-pentane. At the top of the column, 1 mole of product will be withdrawn for every 9 moles that are returned to the column, and the distillation column will operate at 1.013 bar pressure.
Figure 10.1-10 shows the constant pressure x-y plot for this system. For the example considered here, q = L/D = 9/1 = 9. Consequently, the rectification section operating line starts at x = 0.98 and has a slope of q/(q + 1) = 9/(9 + 1) = 0.9. The stripping section operating line starts at x = 0.0485 and has a slope of (q + F/D)/(q + 1) = (9 + 100/48.47)/(9 + 1) = (9 + 2.063)/10 = 1.106. Both of these operating lines are drawn on the x-y diagram. Note that these operating lines cross at the feed composition.
The rectification operating line relates the composition of the vapor entering an equilibrium stage (from the stage below) to the liquid leaving that stage, while the equilibrium line relates the composition of the equilibrium liquid leaving the stage to the equilibrium vapor also leaving that stage and entering the one above it (or the condenser, if it is the top stage). Thus, starting with the vapor composition at the top stage, which is equal to the liquid distillate concentration (that is, the point x_{D}=y_{1}), a horizontal line is drawn until it intersects the equilibrium curve. This intersection gives the liquid of composition x_{1} in equilibrium with y_{1}. Now drawing a vertical line from this point until it intersects the stripping section operating line gives the vapor composition y_{2} entering stage 1. Another horizontal line then gives the liquid composition x_{2} in equilibrium with this vapor, and so on. This graphical stage-to-stage construction is repeated until a liquid composition equal to or less than the feed composition is reached. The optimal stage at which to inject feed is the one that has a liquid composition closest to that of the feed. Therefore, once a liquid stage composition is less than that of the feed, we identify that as the feed stage and continue the graphical construction using the stripping (lower) operating line until a composition equal to or less than the desired bottoms composition is reached.
This graphical construction is shown in Fig. 10.1-10. From this figure, we see as before that the desired separation can be achieved in a total of four equilibrium stages, one of which is the reboiler, and that the feed should be on the second stage from the top of the column. The temperature on each tray, the condenser, and the reboiler can now be determined from the T-x-y diagram for this mixture using the liquid composition in each of these locations.
Comments
Note the difference between this method of calculation and the one used in the previous illustration. There we did vapor-liquid equilibrium calculations only for the conditions needed, and then solved the mass balance equations analytically. In this illustration we first had to do vaporliquid equilibrium calculations for all compositions (to construct the x-y diagram), and then for this binary mixture we were able to do all further calculations graphically. As shown in the following discussion, this makes it easier to consider other reflux ratios than the one used in this illustration.
The choice of reflux ratio q = L/D, that is, the ratio of the liquid returned from the condenser to distillate, was chosen arbitrarily to be 9 in this example. Suppose instead a value of 0.5 had been chosen. Then the rectification section operating line would still start at x = 0.98 but now have a slope of q/(q + 1) = 0.5/(0.5 + 1) = 0.333. The stripping section operating line would again start at x = 0.0485 and have a slope of (q+F/D)/(q+1) = (0.5+100/48.47)/(0.5+1)= 2.563/1.5 = 1.709. The graphical stage-to-stage calculation for this case is shown below. In this case, we need a total of six equilibrium stages,with the feed on the third stage. Thus by decreasing the reflux ratio, we need two additional stages in the distillation column, which increases the capital cost. However, by decreasing the reflux ratio, for a fixed amount of product, less vapor and liquid are circulating through the column so a column of smaller diameter can be used, which decreases the capital cost. More important, less liquid is being vaporized in the reboiler and less vapor is being condensed in the condenser, resulting in a very significant decrease in utility costs. So by decreasing the reflux ratio, the cost of separation can be greatly reduced.
However, the reflux ratio cannot be reduced indefinitely. There is a value of the reflux ratio below which it is no longer possible to achieve the desired separation. This is the value of q that results in the simultaneous intersection of two operating lines and the equilibrium line. For the system being considered, this limiting reflux ratio is q = 0.237. At this reflux ratio the desired separation cannot be achieved since, in stepping of stages, it is not possible to get beyond the intersection point of the two operating lines. This point of intersection between the equilibrium and operating lines is referred to as a pinch point and is shown in Figure 10.1-11. For any reflux ratio greater than this minimum value, the separation is possible. As a (very) rough rule of thumb, a reflux ratio that is 20 percent greater than theminimum required for the separationmay be close to the economic optimum between equipment and operating costs.
Another limiting case is the minimum number of stages required for a separation, which occurs when the reflux ratio is infinity, so that all the condensed vapor is returned to the column, there is no overhead or distillate product, and also no bottoms product and therefore no feed to the column. In this case, from Eq. 10.1-12, y_{ i +1}=x_{ i } \text { and from Eq. 10.1-14, } x_{ I +1}=y_{ I }, so that the upper and lower section operating lines are coincident with the x=y \text { or } 45^{\circ} line. For the case here, from Fig. 10.1-12, this results in slightly more than three equilibrium stages as the minimum number required to obtain the desired separation. [Using an Aspen Plus^R a simulation using the distillation shortcut method DSTWU is available on Wiley website for this book in the folder Aspen Illustration>Chapter 10.1>10.1-7.]
