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Procedure 1. Determination of the Minimum Number of Stages Required to Effect a Specified Separation at a Given Reflux Ratio for a Complex Distillation Column

For definiteness, consider the complex column shown in Fig. 9-2 which has a sidestream W withdrawn in addition to the distillate and bottoms. For a complex column, kt is equal to the number of plates above and inclusive of the plate from which the sidestream W is withdrawn, k2 is equal to the number of plates above and inclusive of the plate on which the feed F enters, and k3 denotes the total number of plates in the column, k3 = N — 2.

The withdrawal of this sidestream introduces two new variables which may be specified such as the stage number kt + 1 of the sidestream withdrawal, and its flow rate W. Instead of kx and W, any other two product specifications may be made such as {wr/dr, ws/ds}, {xWsi xWr\ and {Tw, xWr} as shown in Table 9-5. Components r and s consist of any two components of the feed which are selected in the order of increasing volatility; s is more volatile than r. In the following application of Box's complex method, it will be supposed that the additional specifications for the stream W are {ws/ds, wr/dr}. The complete set of additional purity specifications are taken to be

as shown by set 2 of item I of Table 9-5. At the specified reflux ratio Li/D, the total number of plates N — 2 is to be minimized subject to the purity constraints. When the purity constraints are included in the objective function, one obtains

Figure 9-2 A complex distillation column with one sidestream.

Table 9-5 Various sets of specifications for complex columns

I. Complex column with one sidestream (Fig. 9-2)

Additional specifications

Set

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