Procedure 2. Optimum Economic Design of Conventional and Complex Distillation Columns

9-2 PROCEDURE 2. OPTIMUM ECONOMIC DESIGN OF CONVENTIONAL AND COMPLEX DISTILLATION COLUMNS 

In the formulation of this problem, it is supposed that in addition to the usual specifications, two purity specifications (bt/dt and bh/dh) are made, and that it is required to find the feed-plate location [plate number (kt + 1)], the total number of plates k2, and the corresponding values of the operating variables Li/D and D that minimize the operating costs and capital costs per mole of the most valuable product (D or B). Since only two of the four additional variables required to define the column are specified, many solutions may be obtained by making different choices for two of the four remaining variables. The best choice for the remaining variables has been made when the objective function is minimized. 

When the purity constraints are included in the objective function, the problem to be solved may be stated as follows. The economic objective function 

The capital and operating costs are denoted by Cc and C , respectively, and R is the fractional return on investment. If the bottom product B is more valuable than the top product D, then D is replaced by B in the objective function O given by Eq. (9-15). Alternatively, D could be replaced by the sum cxD + c2B, where Cj and c2 are the relative values of the distillate and bottoms. The formulation of the first term or the economic part of the objective function O is described in App. 9-4. Again the weight factors were selected as described in procedure 1. Also, the constraints s, (i = h, /) were replaced by (In s,) for values of s, less than unity. The objective function given by Eq. (9-15) was minimized by use of the following calculational procedure. 

In the first initial search, the objective function, given by Eq. (9-1) is searched over the independent variables kXl k2 at some arbitrarily selected value of Lx/D in the same manner as described in procedure 1. In this first initial search, D is taken as a dependent variable. After the minimum number of stages N has been found by this search, a second initial search is made with the total number of stages fixed at this value of N. In the second initial search, the economic objective function given by Eq. (9-15) is searched over kx, Lx /D with k2 and D being dependent. Note k2 = N - 2, and D depends upon the fixed value of N. 

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