Figure 9-1 A conventional distillation column.

minimize the total number of stages N. The objective function 0 to be minimized is then

subject to the constraints bdi, < (b,/d,)v bjdh > (bJdh)L (9-5)

and the limits klL — ^ klU L — ^2 ^ k2u DL<D <DV (9-6)

The lower bound on the distillate rate D is

where d, and dh are computed by use of the specified values of and bh/dh and the component-material balances; see Eq. (9-2). The upper bound on D is, of course, equal to the total feed rate F, that is,

For definiteness, take kl(J = k2U = 200, and ku = k2l = 2.

The problem may be solved by use of the complex method of Box2 which is based on the reduction of a simplex of solutions which satisfy the constraints. Finding such solutions can prove to be difficult, thereby requiring an excessive amount of computer time. To avoid this difficulty, the well-known technique of including the constraints in the objective function was employed. Several forms of the objective function including the one proposed by Srygley and Holland19 were investigated. The best of these consisted of including the constraints as an additive term as follows

where

The factors fx and f2 make it possible to place different weights on the number of plates (N — 2) and the constraints. In the initial search, fx and f2 were taken equal to a fraction or a multiple of the average of the simplex values of their respective numerators, and in the final search,/! was taken equal to a fraction or a multiple of the average of the simplex values of its respective numerator and f2 was set equal to unity. This choice off2 tended to drive the search either into the neighborhood or the region of feasible solutions.

Use of the square of the terms shown in Eq. (9-9) tended to reduce the number of trials required to minimize the function 0. In the initial search, Eq. (9-9) was used as stated, and in the final search it was used as stated for st > 1 and sh > 1. For either st < 1 or sh < 1, the term sz or sh was replaced by either In st or In sh. Use of this variable form of the function places a penalty on making separations which are better than those specified and tends to drive the objective function toward equality constraints.

In summary then, the function O given by Eq. (9-9) is to be searched for its minimum value over the variables {ku k2, D}. The search is carried out by use of the following procedure.

In the initial search, approximate solutions are obtained to the equations describing a distillation column; see App. 9-2. This procedure makes use of component-material balances and equilibrium relationships, and it reflects fairly accurately the effect of varying /cx and k2 on the separations {bjdi, bh/dh}. In the initial search, the minimum value of O is determined for the specified value of the reflux ratio. Also in the initial search, the total distillate rate D is taken to be dependent on N, and it is estimated as described in App. 9-2. Thus, the function O is searched over only two variables, kx and k2, as follows:

Step 1. Determine the feasibility of the purity specifications (the constraints) at N = Nv at total reflux. Use the Fenske equation as described in App. 9-1. If it is impossible to satisfy the constraints at total reflux, calculations are ceased. If the specified separations cannot be made at total reflux, they cannot be made, of course, at any operating reflux less than total. If the separations are feasible (all constraints are satisfied) at N = NL at total reflux, go to step 2.

Step 2. Search the objective function 0 over kt and k2 by use of a modified version of the complex method by Box2 which is described in App. 9-3. Commence by finding five solutions which define the initial simplex. To find the initial solution, pick arbitrary values of kx and k2 lying between the upper and lower bounds as follows k\ = k2 (9-10)

where r1 and r2 are random numbers lying between 0 and 1. First k2 is found by use of Eq. (9-10), and then kx is found by use of Eq. (9-10). For this set of values of kt and k2, take Z^ to be equal to the values predicted by the total reflux models as described in App. 9-2. For each of the remaining solutions, the values of kt and k2 are selected by use of random numbers as described above and the corresponding values of D are selected as described in App. 9-2.

Step 3. Search the function O by use of the complex method (see App. 9-3) over the variables kx and k2 with the value of D being determined by the choice of kt and k2 as described in App. 9-2.

t fw

In the final search, exact solutions of the equations describing the distillation column are used. The objective function is searched over the variables which are most conveniently fixed in the particular calculational procedure used to solve the equations for the distillation column. If the 0 method is used, the search variables are taken to be D, ki9 and k2. On the other hand, if the 2N Newton-Raphson method is used, the search variables are taken to be VN/B, ku and k2.

In the final search, the function O is searched over the variables D, ku and k2. The final search is initiated by finding a solution by an exact calculational procedure for each vertex Z), kl9 and k2 determined by the initial search. After the function 0 has been evaluated at each vertex, the complex method of Box is employed as described in App. 9-3.

In order to demonstrate the use of this procedure, Example 9-1 is presented. The statement of this example is given in Tables 9-1 and 9-2. Results of the initial and final searches are presented in Table 9-3, and the final solution appears in Table 9-4. The term44 iteration " which appears in Table 9-3 is used to

Component no. |
Component |
Feed composition xi |

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