# U

The following lower bounds were selected: klL = k2L = k3L = 3. The following upper bounds were selected: klv = k2U = k3V = 200.

The upper and lower bounds on d and w are established by use of the results obtained by finding the solutions at total reflux at the maximum and minimum number of plates as described in App. 9-1. The lower bounds on d and w were taken to be a fraction of the smaller values found at total reflux. Similarly, the upper bounds on d and w were taken to be a fraction larger than their corresponding values at total reflux. For the examples presented, the multipliers of 0.9 and 1.1 were used to compute the lower and upper bounds, respectively.

The calculational procedure employed is the same as the one described for conventional distillation columns except for the difference in the search variables and the objective function. In the initial search, the independent or search variables are {kl9 k2. k3} and the dependent variables are {D, W} as shown in Table 9-6. To find the initial vertex, pick an arbitrary value of k3 lying between the upper and lower bounds as follows where r3 is a random number lying between 0 and 1. On the basis of this value of /c3, the values of kl and k2 are obtained in the following manner where rx and r2 are random numbers lying between 0 and 1. The corresponding values of the dependent variables were selected as described in App. 9-2. The objective function given by Eq. (9-12) is used in both the initial and final searches.

In the final search, the function 0 is searched over {kl9 k2, k3, D, W}. The final search is initiated by finding a solution to the equations for the complex column by an exact calculational procedure for a set of vertices which are obtained by use of a 10 fjfercent perturbation (by random numbers) about the final vertices {kt. k2, k3\$D, W} of the initial search. The remainder of the calculational procedure is analogous to that described for conventional columns.

Procedure 1 is readily generalized for problems involving other types of