# Procedure 2 Optimum Economic Design Of Conventional And Complex Distillation Columns

Conventional Distillation Columns

In the formulation of this problem, it is supposed that in addition to the usual specifications, two purity specifications (bl/dl 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 /c2, and the corresponding values of the operating variables Lx/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 is to be minimized subject to the limits kiL <kx< kw k2L <k2< k2V ^

The capital and operating costs are denoted by Cc and Ca, 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 ctD + 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.

### Initial Search

In the first initial search, the objective function, given by Eq. (9-1) is searched over the independent variables {ku k2} at some arbitrarily selected value of Ll/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 {ku Lt /D} with k2 and D being dependent. Note k2 = N - 2, and D depends upon the fixed value of N.

The final search is initiated by finding exact solutions at perturbations of the vertices {ku k2, D, L^/D} found by the second initial search. Then the economic objective function given by Eq. (9-15) is searched over these variables by use of the complex method of Box. To demonstrate the application of this procedure, Example 9-2 is presented in Table 9-7. The results of the initial and final searches are presented in Table 9-8.

### Table 9-7 Statement of Example 9-2

Design a conventional distillation column to process the feed stream whose composition and thermal condition are stated in Table 9-1. The following separation is to be performed at the least possible cost per mole of distillate product.

V<*i ^ V>,/d,)u = 0.1248 bJdh>(bJdh)L=: 7.201

Use the design equations and cost data given in App. 9-4. Other design data and costs needed are as foliows:

Overall heat transfer coefficient of condenser = 250 Btu/(h ft °F). Cooling water at 55°F is available at a cost of S0.295/M gal. Saturated steam at 490°F is available S2.50/MM Btu.

Table 9-8 Initial and final searches for Example 9-2

First initial search (at a fixed reflux ratio LJD = 4)

Initial simplex

 Variables