Procedures for solving several types of optimization problems commonly encountered in the design and operation of conventional and complex distillation columns are presented. The continued increase in energy costs for operating distillation columns has created the need for rapid calculational procedures both for the design of new distillation columns and for the selection of optimum operating conditions for existing columns. Problems of the following types are solved: (1) determination of the minimum number of stages required to effect a specified separation at a given reflux ratio, (2) optimum economic design of a distillation column, and (3) minimization of the reflux ratio for an existing column by determination of the optimum feed plate location. These problems are solved by use of a modified form of the search procedure called the complex method, which was proposed by Box.2 The primary modification consists of including the constraints in the objective function which reduces the time required to solve a problem by a factor of 1/10 or less. Furthermore, the difficulty of finding feasible solutions (solutions which satisfy the constraints) throughout the domain of the search variables is thereby eliminated. The solution for which the objective function is a minimum is either a feasible solution or is in the near neighborhood of a feasible solution.

After the near neighborhood of the optimum solution has been located, the equations describing the distillation column are solved exactly. These exact solutions may be found by use of any algorithm which one might have available. Thus, this adaptation of the complex method makes it possible to solve a large variety of optimization problems by use of the calculational procedure which is the most efficient one for the particular system under consideration.

One of the first efforts to solve design problems by use of an optimization procedure was made by Srygley and Holland19 who made use of the Hooke and Jeeves13 search procedure. The proposed procedure was limited to the determination of the minimum number of stages required to effect a specified separation at a given reflux ratio. Sargent and Gaminibandara16 considered the more general problem of the optimum configuration of columns needed to effect a specified separation. More recently, an iterative procedure based on the Naphtali and Sandholm14 formulation of the Newton-Raphson method was proposed by Ricker and Grens15 for the minimization of the number of stages required to effect a specified separation at a given reflux ratio.

The calculational procedures are presented first for conventional distillation columns and then for complex distillation columns. The conventional distillation column is completely determined by fixing the following variables: (1) the complete definition of the feed (total flow rate, composition, and thermal condition), (2) the column pressure (or the pressure at one point in the column, say in the accumulator), (3) the type of condenser, (4) ku the number of plates above and including the feed plate, (5) k2, the total number of plates, and (6) two other specifications which are usually taken to be the reflux ratio and the distillate rate {Li/D, D} or two product specifications such as {6/M, ^/iM,}, {Xdi> xBh}> {Td, 7^}, or combinations of these. The subscript / is used to denote the light key and the subscript h is used to denote the heavy key. In all of the optimization problems considered herein, the variables listed in items (1), (2), and (3) are always fixed. For convenience these variables are referred to collectively as the "usual specifications." The remaining four variables, ku k2, and two other specifications such as LJD and D are called "additional specifications."

Since all of the variables included in the usual specifications are commonly fixed, only the four variables classified as additional specifications need to be fixed in order to completely determine the column. These four variables may be picked from the set {ku k2,LJD, D, bjd^ bh/dh}. Instead of bljdl and bh/dh9 any other pair of specifications such as those enumerated above may be selected. Since integral numbers of plates are to be used, the specified values for both bxldx and bh/dh cannot necessarily be made exactly. However, better separations but not poorer separations of these components would be acceptable, which is expressed in the form of constraints where the subscripts U and I denote the upper and lower bounds, that is, the optimum design and operation of distillation columns 301

largest and smallest values of bt/di and bh/dh, respectively, which are acceptable.

The light- and heavy-key components are usually but not necessarily adjacent in volatility. The feed may contain components having volatilities lying between those of the keys. The specification of the ratios bl/dl and bh/dh is equivalent to specifying the molar flow rates or fractional recoveries of the keys in the distillate D and the bottoms B. A material balance on the light key gives

Thus and the fractional recovery is given by d, i

Expressions of the same form as those shown above are obtained for the heavy key by replacing the subscript / by the subscript h. For definiteness in the formulation of the optimization problem, the two product specifications are taken to be bl/dl and bh/dh (or and dh).

Let the total number of stages be denoted by N which includes the reboiler and the condenser, partial or total. The number of plates is then equal to N — 2 which is also equal to k2 for a conventional column. The upper and lower bounds on are denoted by kiV and /c1L, respectively. Similarly, the upper and lower bounds on the total number of plates are denoted by k2U and k2L, respectively. For example, values of k2 greater than 50 or 100 are seldom encountered, and the designer might well take kiv = k2U = 100 (or 200). The lower bounds might well be taken to be k1L = k2L = 1.

The proposed application of the complex method makes it possible to use any existing calculational procedure for solving the equations for a distillation column exactly. However, since repetitive calculations are involved in the search technique, computer time can be conserved by use of the calculational procedure which is most efficient for a given type of column. In a series of papers,4,5'8'9 comparisons of the computer times required to solve a wide variety of numerical examples by use of the 0 method and formulations of the Newton-Raphson method are presented. The following conclusions were reached. (1) For distillation-type columns in the service of separating ideal or near ideal solutions, the 6 method is more efficient than the Newton-Raphson formulations; (2) for absorber-type columns in the service of separating ideal or near ideal solutions, the formulation of the Newton-Raphson method in terms of IN independent variables (where N is equal to the number of stages) is recommended; (3) for columns of all types in the service of separated highly nonideal solutions, the Almost Band Algorithm presented in Chap. 5 is recommended.8

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