McCabe-Thiele method minimum stages

The feed, at a flow rate of 100 kmol/h, is sent as saturated vapor to the distillation column. The column is equipped with a partial condenser with a vapor product, and a reboiler. For a solvent rate of 500 kmol/h, it is required to determine the required number of equilibrium stages and the optimum feed location for a reflux ratio of 1.5 times the minimum. The McCabe-Thiele method may be used on a solvent-free basis. 

FIGURE 5.3-9 Minimum stages, McCabe-Thiele method. 

Total reflux corresponds to a situation where there is no feed, distillate, or bottoms and where the minimum number of stages is required to achieve a desired separation. It will be recalled that in the McCabe-Thiele method this situation corresponded to having the operating line coincide with the y = x line. 

According to the McCabe-Thiele method, the system for separation is considered on a quasi-binary basis. In this approach, it must be possible to neglect the influence of the solvent, which is acceptable if the phase boundary lines (solubilities) do not change much with concentration during separation. In this case, the number of theoretical stages, the minimal reflux (ratio), the minimum number of theoretical stages, and their mutual dependence can be determined. 

Fig. 2.3.4-2 Mass balance of an extraction (a) and application of the McCabe-Thiele method (b) for determining the theoretical number of stages N and the minimum amount of solvent...

Equilibrium data must be converted to ratio units, Y vs. X. These values can be greater than 1.0, since Y = y/(l - y) and X = x/(l - x). The Y = X line has no significance in absorption. As usual the stages are counted at the equilibrium curve. A minimum S/G ratio can be defined as shown in Figure 12-9. If the system is not isothermal, the operating line will not be affected, but the equilibrium line will be. Then the McCabe-Thiele method must be modified to include changing equilibrium curves. 

For a given rectification problem the minimum reflux ratio v ,jn and the minimum number of separation stages may be determined by the McCabe-Thiele method or calculated by Eqs. (2-74) and (2-77). With these, the required number of theoretical stages for any chosen reflux ratio V is found from the Gilliland diagram. The best economic operating range of rectification columns is... 

Minimum reflux, by definition, is that condition whereby a zone of an infinite number of stages exists immediately on each side of the feed location. In the McCabe-Thiele method of graphical representation and calculation, both operating lines intersect with the locus of the feed partitioning, all at a point on the fC-value plot (or equilibrium curve). It is therefore impossible to calculate away from this point of intersection, in either direction. 

The operation of a binary distillation has two important limiting cases minimum and maximum stages for a given separation. Minimum stages occur when the column is operated with total reflux (i.e., no top product is withdrawn). This situation, when applied to the McCabe-Thiele method, uses the y = x line as the mass balance for both rectifying and stripping. The situation is depicted in Figure 12-11. 

Figure 12-11. Total reflux and minimum stages with McCabe-Thiele method. (Reproduced with permission from reference 18. Copyright 1997, American Chemical Society.)...

If graphical methods are used for binary systems, the minimum reflux ratio may be estimated directly. For McCabe-Thiele, it can be determined from the minimum slope of the upper operating line, which can be found graphically and often represents coincidence with the intersection of the line with the equilibrium curve, as shown in Fig. 5.3-11. Unusual curvature of the equilibrium relationship can limit the minimum reflux ratio to a higher value as also shown in Fig. 5.3-11. In stepping off stages, starting at the top, a... 

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