Ind. Eng. Chem. Res. 1993,32, 26362646
2636
Multiobjective Optimization Approach to Sensitivity Analysis: Waste Treatment Costs in Discrete Process Synthesis and Optimization Problems Amy R. Ciric' and Sebastien G. Huchette Department of Chemical Engineering, University of Cincinnati, Cincinnati, Ohio 45221-01 71
This paper presents an approach for determining the sensitivity of the maximum net profits of a chemical process to changes in the waste treatment cost when discrete terms must be optimized. The approach employs a basic relationship between the sensitivity problem and a multiobjective optimization problem that maximizes profiles and minimizes waste. It is shown that all solutions of the original process optimization problem must lie in the concave portions of the solution set of the multiobjective problem and that a simple transform exists between the multiobjective optimization problem and the sensitivity problem. The proposed approach uses a modified form of the outer approximation method t o identify discretely different regions of the multiobjective solution set. The complete solution set is generated with a sequential approximation algorithm. The approach is illustrated with a case study of the production of ethylene glycol from ethylene oxide and water. 1. Introduction
The costs of waste treatment and disposal are becoming increasingly important in the design and operation of chemical manufacturingprocesses. Estimating these costs can be difficult, as both direct costs (treatment, facilities, tipping fees, etc.) are rapidly increasing and indirect costs (liability, paperwork,public relations, etc.) are significant but hard to quantify. Rising waste treatment and disposal costs create an economic incentive to reduce waste productionat its source within a chemical process. There are some basic strategies for accomplishing this. For instance, the flow rate of a bleed or purge stream can be reduced by lowering the purge fraction, using a higher purity feedstock, or adding an additional separation unit to the recycle or purge stream. In addition, production of an unwanted reaction byproduct can be reduced by using a different reaction path, employing a different catalyst, modifying the reactor operating conditions, changing the feed ratio, or recycling byproducts back to the reactor so that they accumulate to equilibrium levels. Lastly, solvent wastes can be reduced by recovering and recycling the spent solvent, replacing the system with a solventless process, or replacing the solvent with a less toxic or more easily recovered solvent. Notice that these strategies will often require modifying the size or operating conditions of several units within a chemical process: for example, a smaller purge fraction may lead to larger recycle flows, larger equipment sizes, and higher operating costs. Clearly, source reduction of waste streams may require additional capital investment and/or higher operating costs, and the amount of additional investment or operating costs may increase with waste reduction. Finding the optimal design requires trading off treatment costs against profitability, and since waste treatment and disposal costs are often uncertain, this tradeoff can be difficult to make. The effect of this uncertainty upon the net profit of a chemical process can be captured by a sensitivity analysis of the changes in the maximum net profit in response to changes in the waste treatment costs. Figure 1shows the sensitivity of net profits np = p - aw to waste treatment costs a for a typical source reduction project. Here, p is
* Author to whom correspondence should be addressed. Electronic mail address:
[email protected].
m
Treatment Cost Figure 1. Net profita versus treatment costa in a chemical process optimization problem.
the profit before waste treatment and w is the waste flow rate. Notice that a t low treatment costs the net profit falls off rapidly; as the treatment cost rises, the net profile levels off. This happens became the slope of the sensitivity curve is equal to the waste flow rate. At low treatment costs, a high level flow rate of waste is economically desirable, but it makes the net profits very sensitive to increases in the treatment costs. As treatment costs increase, waste production levels drop, and the sensitivity of the net profit to waste treatment costs decreases. Traditional sensitivity methods (Edgar and Himmelblau, 1988) use derivatives to determine the localized response of an objective to changes in a parameter. In a waste minimization project, this involves computing the derivative of the net profit with respect to waste treatment costs, and then approximating the sensitivity curve with a linear expansion. Since the derivative of the net profits
OSSS-5SS5/93/2632-2636$04.00/0 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2637 with respect to the treatment costs is simply the waste flow rate, this analysis is equivalent to assuming that the waste production level remains constant as treatment costa increase. Consequently, the relationship between treatment costs and waste production levels is not captured, and the net profit is underestimated. The global sensitivity could be captured accurately by solving the process optimization problem for a range of waste treatment costs. However, this exhaustive approach is inefficient, as the computational effort is scattered across the entire range of treatment costs, rather than in the region where waste production levels are changing. Ciric and Jia (1992) presented an alternative approach for determining the sensitivity of net profits to waste treatment costs for continuous optimization problems. They formulated the sensitivity problem as a parametric optimization problem that treated waste treatment costs as variable parameters. This problem was shown to be related to a multiobjective optimization problem that sought to simultaneously maximize profits before waste treatment and minimize waste production. This multiobjective problem was solved for the noninferior solution set, defined as the set of solutions where the profit cannot be increased without simultaneously increasing waste production. Ciric and Jia (1992) showed that (a) the optimal solutions of the sensitivity problem would lie in the convex hull of the noninferior curve and (b) a simple transform existed between the noninferior set and the sensitivity curve. These properties were used to develop a sequential approximation method that placed increasingly tight bounds upon the convex hull of the noninferior curve, yielding an efficient method for determining the sensitivity of the net profit to the waste production levels. This approach addressed the continuous optimization problems that arise when the basic structure of a chemical process flowsheet is fixed (Le., all equipment and the connections between them have been selected). Consequently, that work is primarily an analysis tool, rather than a synthesis tool. This paper extends the work of Ciric and Jia (1992) to process optimization problems that contain discrete variables, such as those that arise in synthesis. A three-step approach is proposed that (a) identifies the discretely different regions of the noninferior solution set using a novel solution algorithm, (b) finds the noninferior curve using a sequential approximation method, and (c) uses the basic relationship between sensitivity and multiobjective optimization to transform the noninferior solution set into a sensitivity plot. A brief overview of process synthesis by mixed integer nonlinear programming (MINLP) and multiobjective optimization will be presented, followed by a development of the sensitivity algorithm. The method is illustrated with a case study. 2. Process Synthesis with Mixed Integer Nonlinear Programming
Before explaininghow to use multiobjective optimization to determine economic sensitivity, it is convenient to briefly review mixed integer nonlinear programming approaches to process synthesis. The basic idea is to pose a superstructure that contains several attractive flowsheets embedded within it. Integer variables are introduced to represent the existence of discrete processing units and continuous variables to represent temperatures, pressures, flow rates, etc. A model of the superstructure is constructed from material and energy balances, thermodynamic relationships, product specifications, and equipment sizing correlations. This
model is used as the constraint set of a mixed integer nonlinear optimization problem that seeks to minimize the net cost; solvingthis MINLP extracts the economically optimal flowsheet from the alternatives embedded within the superstructure. In short, MINLP approaches to chemical process synthesis find the optimal flowsheet by solving problems of the form
subject to
h(x) -Ay = 0 g(x) - By I0
w = W(X,Y) Here, p is the profit before waste treatment, x is the vector of continuous process variables, y is the vector of integer variables, w is the vector of waste streams, a is the vector of waste treatment costs, and the constraints are material and energy balances. When the vector a is fixed, (Pl) can be solved for the optimal chemical process flowsheet. There are two common approaches to solving MINLP’s. Generalized Benders decomposition (Geoffrion,1974)uses a Lagrangian relaxation method to decompose the original MINLP into a primal NLP (nonlinear program) and a master MILP (mixed integer linear program). The master MILP contains a collection of constraints estimating the Lagrange function. This estimate is created from the values of the continuous variables and Lagrange multipliers at the optimum solution of one of the NLP’s. The solution of the master MILP yields a set of integer variables (Le., a flowsheet) and a lower bound on the true optimum, while the solution of the NLP optimizes a single flowsheet and provides (a) an upper bound on the true optimum and (b) information for updating the master problem. Iterating between the primal and master subproblems gives increasingly tighter bounds on the final solution, and ultimately converges to the true optimum. The key idea of the outer approximation method (Duran and Grossmann, 1986) is that linear overestimators exist for all nonlinear functions in the MINLP. Replacing these nonlinear functions with linear estimations leads to a mixed integer linear programming problem (MILP) that can be easily solved by branch-and-bound techniques. The solution of this approximate MILP is an estimation of the true optimum of the original MINLP; adding additional linear overestimators improves this estimate. In the outer approximation method, this MILP serves as a master problem that identifys attractive integer combinations and provides a lower bound on the final optimum. A primal problem is obtained by fixing the integers in the original MINLP; the solution of the primal problem yields a lower bound on the final optimum and information for strategicallyadding new linear estimators to the master problem. Iterating between primal and master subproblems leads to the optimal solution of the original MINLP. Both of these approaches are iterative, and generate an integer combination (i.e., a flowsheet) and optimize a related NLP at each iteration. The efficiency of both methods arises from two sources: (a) a single integer combination is evaluated no more than once and (b) the optimum solution of the MINLP can be found without examining every feasible integer combination.
2638 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
The key difference between approaches is that the master problem in Bender’s decomposition approximates the dual region of the original MINLP, while the master in the outer approximation method is obtained by directly overestimating the original constraint set. As a consequence, the constraint set of the master problem in Bender’s decomposition will change dramatically with the objective function, while the constraint set of the master problem for the outer approximation method remains largely unchanged. This flexibility makes the outer approximation method ideal for multiobjective optimization and sensitivity studies where the objective function is likely to change dramatically.
D
Profit
3. Multiobjective Optimization
A multiobjective formulation of the chemical process synthesis problem seeks to simultaneously maximize the profits before waste treatment and minimize the waste production: Waste
Figure 2. Noninferior solution set of an optimizationproblem with discretely different feasible regions.
subject to
h(x)- Ay
0
g(x) - By I 0
Multiobjective optimization is an attractive systems analysis approach to problems with several noncommesurate objectives,and has awidespread range of applications. The basic solution methods have been recently reviewed by Haimes and Li (1988); engineering applications by Goicoecheaet al. (1980),chemical engineering applications by Clark and Westerberg (19831, and theory by Sawargi et al. (1985). A key feature of multiobjective optimization is this: there is rarely a single solution that simultaneously optimizes all objectives. Instead, there is a set of noninferiorsolutions where one objective cannot be improved except at the expense of another. This set is readily visualized by projecting all feasible solutions to the constraint set of (Ml) onto thep-w plane (Figure 2). Notice that this figure shows discretely different feasibility regions: this reflects the discrete elements of the optimization problem. Point A in Figure 2 is an inferior solution to the optimization problem: moving in direction u leads to higher profits and less waste. Point B is also an inferior solution, as changing to a discretely different solution set would yield a range of solutions with higher profits and lower costs than B. Lastly, Point C is also an inferior solution, as moving in direction v leads to a higher profits and less waste. Conversely, point F is a noninferior solution of (Ml): one cannot increase profits without producing more waste, or reduce waste without decreasing profits. The set of curve segments D-E and F-G comprise the set of noninferior solutions: at each point on these curve segments, one cannot increase profits without producing more waste, or lower waste production without reducing profits. Although multiobjective optimization has been extensively researched for continuous optimization problems, relatively little work has been done for multiobjective
integer programming problems. Zionts (1981) presented an interactive approach for choosing among explicitly identified alternatives. Marcotte and Soland (1986) presented a modified branch-and-bound technique, while Villareal and Karwan (1981) presented a method based upon dynamic programming. Solution methods for multiobjective mixed integer linear programming problems have been presented by Bitran (1977,1979)and Klein and Hannan (1982). More recently, Esawaran et al. (1989) presented a parametric decomposition method for generating the discrete noninferior set. This paper presents a novel method for identifying the noninferior set in a discrete multiobjective optimization problem. The method is based upon a modification of the outer approximation method, and is discussed in more detail below. 4. Relationship between Multiobjective Optimization and Sensitivity
A relationship between the multiobjective optimization problem (Ml) and the sensitivity problem (Pl) can be established through the following set of theorems. Theorem One. The solution of the sensitivityproblem for an a I 0 is a noninferior solution of ( M l ) . Proof. The net profit np increases with increasing p and decreasing w. At every inferior solution (p, w) of (Ml), there is at least one point @’, w’) where p‘ 1 p and w‘ I w; at this point, np’ I np. Thus, the optimum solution of (Pl) cannot lie at an inferior point of (Ml). Theorem Two. At the optimum solution of (Pl),6Pl 6w = a.
Proof. It follows from theorem one that the optimum of (Pl) can be found by searching on the noninferior set only: max P(w) - ciw
(Pa
Here, P(w) is a functional expression of p along the noninferior set.
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2639 b At the optimum of (P2), 6PI6w = CY.I t should be noted A that P is not necessarily smooth; consequently, the slope a is computed in terms of differences rather than derivatives. Theorem Three. All solutions of the sensitiuity problem ( P l ) lies at either an extreme point or a concaue portion of the noninferior solution set. Proof. For every point @, w) on the noninferior curve, there is some value of CY = 6pI6w. For small perturbations about this point, np = p - CYW= npo + ( 1 / 2 ) 6 ~ ~ ( 6 ~ P I 6 ~ ~ ) 6 w . Notice that at all points except the extreme points, every value 6w is feasible. Consequently, if @, w ) is not an Profit extreme point, it must satisfy one of two conditions: a. There is no direction 6w such that ~ w ~ ( ~ ~ P I ~ w ~ ) ~ w 1 0. In this case, the noninferior curve P is concave at @, w ) (Avriel, 19761, and @, w ) is a local maximum of np = p - aw. Note that the global maximum of np = p - CYW B will lie on the convex hull of the noninferior set. b. There is at least one direction 6w such that 8wT(b2Pl 6w2)6w10. In this case, P(w) is not concave at w (Avriel, 1976). Moving in the direction 6w increases np; moving in this direction will lead either to a new point where Spl 6w = a or to an extreme point. Waste These theorems establish a simple relationship between Figure 3. Direct application of the sequential approximation the solution of the sensitivity problem (Pl) and the approach to discrete optimization problems. multobjective optimization problem (MU. In particular, once the concave portions of the noninferior solution set 1992). Lastly, the concave portions of the curve are of (Ml) have been identified, the solution of the sensitivity transformed into sensitivity data by plotting np = p problem can be obtained by plotting from the transfor(6pI6w)w versus (Y = 6pIGw. mation max np(a) = p - (6pI6w)w versus CY = 6pI6w. 5.1. Overview of MOMINLP Solution Technique. These mathematical properties also provide a method The concave portions of a tradeoff curve or noninferior for bounding the noninferior set. From theorem three, it set can be readily bounded by linear under- and overis evident that only the extreme points (i.e., max p , min estimators. Ciric and Jia (1992) used this to develop a w ) and the concave portions of the noninferior curve are sequential approximation method for pollution prevention relevant for a sensitivity analysis. It should be noted that optimization problems, and applied it to problems that a line connecting two points on a concave curve will could be modelled as continuous nonlinear programming underestimate the curve, and a line tangent to the curve (NLP) optimization problems. This approach involved will overestimate the noninferior set (Avriel, 1976). Consolving a sequence of nonlinear programming problems; sequently, under- and overestimators for the relevant the solution of each problem yielded a point on the tradeoff portions of the noninferior set always exist. curve and provided improved upper and lower bounds on These properties lead to an efficient approach for the rest of the curve. determining the sensitivity of the maximum net profit to One approach to the MOMINLP problem presented in changes in the waste treatment cost. this paper is to use the method of Ciric and Jia (1992) directly. One would begin by solving an MINLP that 5. Overview of Solution Approach maximized profit (point A on Figure 3) and an MINLP that minimized waste production (point B on Figure 3). In a pollution prevention or source reduction design The noninferior set of solutions would then be bounded problem, waste treatment and disposal costa are ambiguous within the convex hull formed from the tangents at A and and difficult to determine, and a parametric sensitivity B and from the line connecting A to B. A third point in analysis must be performed. The previous section of this the noninferior set is found by setting CY equal to the slope paper established a fundamental relationship between this of the underestimator formed by the line connecting A sensitivity analysis and a related multiobjective optimiand B. Solving the MINLP optimization problem with zation problem. It was established from first principles this value of CY would yield a third point on the noninferior that the sensitivity analysis is equivalent to finding the curve (point C). The line segments AC and BC form an concave portions of the tradeoff curve or noninferior set improved underestimator, and the tangent at C provides of a multiobjective optimization problem that seeks to a new overestimator. This procedure would be repeated maximize profits and minimize wastes simultaneously. until the area of each hull fell below some previously When discrete process alternatives are present, this leads specified tolerance. to a multiobjective mixed integer nonlinear programming This approach to the MOMINLP would require solving problem. a mixed integer nonlinear programming (MINLP) problem The proposed approach for determining the sensitivity at each iteration of the sequential approximation method. of a pollution prevention design problem to changes in Such an approach would be inefficient, as (a) it would not the waste treatment cost is as follows. First, the design use the considerable information generated during each problem is modeled as a multiobjective mixed integer iteration of the MINLP algorithms, and (b) there is no nonlinear programming problem (MOMINLP). Second, guarantee that an integer combination of variables will the discrete portions of the MOMINLP are identified using not be evaluated several times during the overall procedure. a modified outer approximation approach, discussed below. An alternative approach that avoids these inefficiencies The continuous portions of the curve are identified with is pursued in this paper. The basic idea is to first identify the sequential approximation approach (Ciric and Jia,
2640 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
Profit
Profit
Waste
Waste
Figure 4. Identifying discrete noninferior regions by elimination.
Figure 5. Candidate regions of the noninferior set.
a set of candidates for the discrete regions of the noninferior set through a screening process. The actual curve of each candidate region is not identified in the initial screen. Instead, continuous portions of the noninferior curve are bounded with upper and lower approximators. The overall curve is found by applying Ciric and Jia’s sequential approximation method to each candidate region of the noninferior set. The candidate solution regions are identified using a modified form of the outer approximation method. An MILP master problem is constructed by linearizing all nonlinear equations. This MILP is solved to find an integer combination that maximizes the profit in the linearized model (region 1 in Figure 4). The full nonlinear optimization problem is solved in two primal steps, yielding the maximum profit (point A) and minimum waste (point B) for the fixed integer combination identified by the master problem. This information is used to (a) generate an underestimator (line AB), (b) eliminate a patently region in the p , w space (hatched region), and (c) bound the noninferior curve associated with this particular discrete region (triangle ABU). The approximating MILP is updated by adding (a) additional linear estimations of the full nonlinear problem, (b) an integer cut excluding the particular integer combination, and (c) a set of constraints defining the underestimator of the noninferior set. The updated MILP is solved for a new combination of integer variables (region 2). The procedure is repeated, adding and deleting candidate regions as necessary, until the master problem is infeasible. The product is a set of candidate noninferior regions (Figure 5). Once the discrete noninferior regions have been identified, the full noninferior curve is found by applying the sequential approximation method to each discrete region. In the last step, this information is transformed to give the sensitivity of the maximum net profits to variations in the treatment cost.
a number of advantages over conventional sensitivity analysis techniques, such as a linear sensitivity analysis or an exhaustive search. A linear sensitivity analysis approximates the max np vs a curve with a line whose slope equals the derivative of the curve at some nominal value of a. It is easy to show that this derivative of this curve is equal to the waste production. Thus, a linear sensitivity analysis implicitly assumes that the waste production level is constant, and will consistently underestimate the maximum net profit. In contrast, the proposed approach accurately identifies the sensitivity curve. An exhaustive search would involve finding the maximum net profit for a number of different values of a. Solving a mixed integer nonlinear programming problem at each value of a is inefficient, since the same inferior flowsheet may be visited computationally several times. Moreover,an exhaustive search scatters the computational effort over the entire range of the waste treatment costs a,rather than focusing the effort in the region where the waste treatment levels (and hence the slope) is changing. In contrast, the proposed approach visits each discrete flowsheet exactly once, and the computational effort is naturally focused upon regions of the sensitivity curve where the slope is changing. It should be noted that many MINLP’s for process synthesis are nonconvex. This can complicate the technique, since a Taylor series expansion of a nonconvex function may slice off part of the feasible region, which may in turn prevent the master problem from identifying all noninferior integer solutions. This problem may be overcome by using a combined penalty-function outerapproximation method for the MINLP (Viswanathan and Grossmann, 1990). The nonconvexities may also cause multiple optima in a primal problem. This type of problem will be detected in the sequential approximation stage, since it will lead to crossed under- and overestimators.
6. Discussion of the Proposed Approach
7. Solution Algorithm
The proposed approach is an efficient method for determining the sensitivity of the maximum net profit to changes in the waste treatment cost. This approach has
The solution algorithm can be stated formally as follows: Step 1. Select an initial combination of integer variables.
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2641
Step 2. Solve the NLP for the maximum profit and then for the minimum waste. Step 3. Using the information obtained from the primal subproblems, (a) Generate linear overestimators of the nonlinear functions and append them to the master subproblem. (b) Check the feasibility of the new master problem and correct any errors. (c) Add an integer cut that excludes the previous integer combination. (d) Add appropriate constraints for the linear underestimator of the noninferior curve. Step 4. Solve the modified master problem, maximizing profit. If the master problem is feasible, return to step 2. Step 5. Generate the noninferior curve for each candidate region identified in steps 1-4 using the sequential approximation method. Eliminate any inferior regions, and substitute any convexregions of the overall noninferior set with a line connecting the endpoints of the convex regions. Step 6. Transform the noninferior curve into the sensitivity curve using the relationships np=p-Ew
9
8
7
6
5
P 4
3
2
This algorithm will be used to determine the sensitivity of net profits to waste treatment costs with two examples: a simple problem that illustrates the approach and a case study exploring the production of ethylene glycol. Each example was modeled using GAMS (Brooke et al., 19881, with MINOS5.1 used to solve nonlinear programming problems and ZOOM used to solve mixed integer linear problems.
8. Examples 8.1. Example 1. The first example is a simple problem designed to illustrate the approach to multiobjective mixed integer nonlinear programming. The process optimization problem is given by max - a w WUlY
subject to
OSpllO OIw15
The discrete noninferior set was identified after three iterations of the modified outer approximation algorithm. The candidate noninferior regions are development are shown in Figure 6. The actual tradeoff curve was identified to within f0.1 after four iterations of the sequential
1
I
1
I
2
I
3
I
4
l
5
e
W Figure 6. Example 1: candidate noninferior regions.
approximation method. The entire procedure required less than 5 s of CPU time on a Sparc 2 workstation computer. The concave portions of the final tradeoff curve is shown in Figure 7. Notice that the curve has discrete transitions from one integer solution to another at waste production levels of 1.7, 3, 3.3, and 4.6. The dashed lines in Figure 7 span the convex portions of the noninferior set that are created by the overlapping feasibility regions of the discretely different solutions. The upper dashed line has a slope of 0.85, while the lower line has a slope of 3.2. The sensitivity curve is shown in Figure 8. Note that the sensitivity curve is continuous, but has nonsmooth points at a = 0.85 and at a = 3.2. At these values of a, the tangent of the noninferior curve lies along one of the dashed lines that span a locally convex region. A t these points, there are two values of w that maximize np = p aw. On the sensitivity curve, this translates into a point with more than one tangent line, giving a nonsmooth point. It should be noted that identifying these transition points is particularly important in pollution prevention problems, since they indicate the minimum waste treatment cost that will provide the economic incentive for waste reduction by significant process modifications. 8.2. Example 2. Production of Ethylene Glycol. Ethylene glycol is manufactured by hydrating ethylene oxide: C,H40 + H,O C2H60, (3) Waste materials are produced when the glycol reacts +
2642 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 water
9
Evaporator Train oxide
8
L -
/-I
7
Lb6
I -t
k
byproducts
Figure 9. Example 2: ethylene glycol process.
P
5
,/
I
4
c)
2
1
k
I
cY3=l + I
2
1
I
3
4
I
)
5
W Figure 7. Example 1: concave portions of the noninferior set.
"A 0
6
4-
2-
&I
II
a z
0-
4
-2 0 -4
1
2
3
4
Alpha Figure 8. Example 1: sensitivity curve.
further with the oxide to form diglycols, triglycols, and higher glycols:
--
C2H40+ C2H602 C4HI0O3
(4)
(5) The reaction is typically performed in a liquid-phase mixture at 20 atm and 200 "C. All reactions are irreversible C2H40
I
Figure 10. Example 2: evaporator train superstructure. 1
I
I
k I
/
3
+ C4H1003
'BH14'4
and have second-order kinetics. It was assumed that the activation energy of all reactions was the same, and that the oxide reactions with heavy glycols had the same reaction constants as the oxide reaction with ethylene glycol. This allows the diglycols, triglycols, and heavy glycols to be lumped together as a single byproduct. In addition, it is assumed that all the ethylene oxide is consumed in the reactor. These assumptions lead to a closed-form algebraic expression of the outlet concentrations of water, glycol, and byproduct in the reactor effluent stream in terms of the inlet concentrations of oxide and water. These expressions are derived in Appendix A. The conventional flowsheet for the production of ethylene glycol from ethylene oxide and water is shown in Figure 9. Water and oxide are fed to a reactor at approximately 20 atm and 200 "C. Excess water is used so as to minimize the production of byproducts. The reactor effluent is passed through an evaporator train that removes most of the excess water, with the remainder distilled off in a distillation column. The excess water is recycled to the reactor, while the ethylene glycol is distilled from the heavy glycol wastes in a second distillation column. The amount of heavy glycol waste produced in the process is determined by the amount of excess water fed to the reactor: the higher the waterloxide ratio, the better the selectivity. An inherent tradeoff exists between waste production and profitability, as increasing the selectivity by raising the watertoxide ratio leads to lower raw material costs, but requires more energy and capital equipment for recovering the unconsumed water. The evaporator train was modeled with the superstructure shown in Figure 10. The reactor effluent is heated, passed through a valve, and then flashed. The bottoms product is split into two streams: one stream passes to the distillation column; the other is passed through a valve,
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2643
4 evavoraors
0
__
I
I
I
UIO
1000
lux)
2000
Waste (Kgmoldyear) Figure 11. Example 2 noninferior curve.
then exchanges heat with the stream from the first evaporator, and is flashed. The superstructure contains 10 evaporators: evaporator chains containing up to 10 evaporators can be extracted from this superstructure. Any remaining water is removed from the glycol in a finishing column. In the last step of the process, ethylene glycol is separated from the heavy waste in a second distillation column. Both of these columns were modeled with shortcut simulation methods. The MINLP model for this process is given in Appendix B. The discrete noninferior solution set was estimated after four iterations of the modified outer approximation method. The procedure required approximately500 CPU s on a Sparc 2 workstation. The actual noninferior curve was identified in nine iterations of the sequential approximation method,and required approximately 900CPU s on a Sparc 2 workstation computer. The modified outer approximation method identified flowsheets with two to four evaporators as potentially noninferior solutions. The noninferior curves of these four flowsheets were found using the sequential approximation method. Although the initial bound on the noninferior curve of these solutions overlapped, the actual curves do not intersect. Consequently, of the four candidate noninferior solutions identified by the modified outer approximation method, only the four-evaporator flowsheet proved to be noninferior. The true noninferior solution set is shown in Figure 11. The sensitivityof net profits to fluctuations in treatment costs was obtained by plotting np = p - (bp/bw)wversus a = bp/bw. This curve is shown in Figure 12. 9. Conclusions
The rising cost of waste treatment makes process synthesis and optimization an uncertain task. This paper
4000 I
-
-3000
0
2
4
6
8
1 0 1 2 1 4
Alpha Figure 12. Example 2: ethylene glycol process, sensitivity c w e . a, four evaporators.
addressed this problem and presented a method for determining the sensitivity of net profits to variations in waste treatment costa in process synthesis problems that require optimizing a discrete set of variables. This sensitivity was readily obtained from the concave portions of the noninferior set of a multiobjective optimization problem that sought to simultaneouslymaximize profits and minimize wastes. The noninferior set was found with a two-step procedure that first identified the discretely different regions of the noninferior curve with a modified outer approximation method and then identified the full curve with a sequential approximation method. The approach was derived from a fundamental relationship between sensitivity analysis and multiobjective
2644 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
optimization. Consequently, the approach is general and can be applied to any waste reduction problem featuring discrete alternatives. It should be noted that the approach can be used to determine the sensitivity to other terms in the objective, such as raw material and/or product prices. The technique was applied to a case study of the production of ethylene glycol from ethylene oxide and water.
Appendix A. Ethylene Glycol Reactor Model The ethylene glycol reactor is assumed to be an adiabatic plug flow reactor. In addition, it is assumed that (a) all ethylene oxide fed to the reactor is consumed; (b) the activation energy is the same for all reactions; (c) the formation of diglycol and higher glycols have the same reaction rate constant, and this constant is X times larger than the constant for ethylene glycol formation; (d) all reactions are second order; and (e) there is no glycol or byproducts in the reactor feed. These assumptions lead to the following set of differential equations that describe the concentration profile along the plug flow reactor:
r--: -
kcOcw- Xkcocg
Here, co is the oxide concentration, cw is the water concentration, cg is the ethylene glycol concentration, cgi is the concentration of higher glycols,and r is the residence time. This problem can be simplified by defining m
cb
=C
cgi r=2
as the byproduct concentration, and defining y as the fraction of unconverted water, cw/cwo.Substituting these terms into (6)-(9)and dividing the resulting differential equations by (7) yields the followingsimplified description of the reactor, modeled in terms of the water concentration:
H, O/Oxide brio
Figure 13. Example 2 experimental data (McKetta, 1984) vs modeling predictions for ethylene glycol production.
fraction of unconverted water, and the parameter X:
coo - R = (1- X)(1- y) - X In y
Equations 13 and 14 can be used to find the selectivity of ethylene glycol: (16) The parameter X was found to be 4.6 by fitting a single data point (S = 0.82, R = 0.1) to (15) and (16), yielding the following model for the reactor performance: =
( 5 (y ) - y4.6) 3.6
cw
Equations 11 and 12 can be solved to give
(15)
cwo
= CWY
-coo- - -3.6(1 - y) - 4.6 In y two
(19) (20)
This model provides an excellent fit to published sensitivity data (Figure 13). c g = (&)Q-Y? Substituting these relationships into (10) and noting that there is no oxide in the reactor leads to the following relationship between the feed ratio of oxide to water, the
Appendix B. Mathematical Formulation of Ethylene Glycol Process Optimization Problem The ethylene glycol case study was modeled as the following mixed integer nonlinear programming problem:
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2645 Pe+l= P v
subject to p X lo4 = 3.585D2, - 1.23F10- E L 2 5 X 10"Ye
+
e
2.63
10"ZFIi,,] -0.01 - 0.041FL - 0.0225 - 3 X
X
I
1041D1 - 9.7
- 8.6 X 103(R1 + 1 ) Z D l i
10%
X
FVi,e = Fi,e+l FHX,,, = FT,,, HHX,,, IX,FTi,, = HVic HTi,, - HHX,,, Distillation Columns.
+
FCl, = XNF,,,
a
e
FC2, = Bl,
w =zB2, I
F1 = z F C l i , e I
Reactor Model. cg =
F2 = ZFCP,,,
(g)-
I
(y y4.7
FCl, = B1, + Dl, FC2, = B2, D2, FCl, xl, = F1 FC2, x2, = F2
+
cw = CWY co,,/c,, = -3.6(1- y) - 4.6 In y
FI, = cwoQ FI, = cOoQ FO, = ciQ FO, = 0
l;]
R2 = 1.2[ x2, + x -2 x2,(ag
V 1 = (R1+ 1 ) ZI D 1 , C H O i = Z[FI,CP,(TF - 27311 - HRX a
I
HO, = FO,CP,(TF - 273)
#
w
= HOi i # w H b , l = HOW+ aa
-- HL,e - HLw,e Xe
F T , e = FIw,eQe FTi,, = FIi,e- FBi,,
ID1 = 0.189V1°.5 ID2 = 0.239V2°.5 Integer Relationships. FIi,, - VU, I0 FBi,, - VU, I0 FTi,, - VU, I0 FVic - VU, I0 FHXi,, - VU, I0 NFi,, - U(Ye- Ye+,) I0
= FI,,eCP,(Te - 273)
Y e + l I Ye
HTic = HI,,, - HBi,, HT,,, = FT,,,[CP,(T,
Yl = 1
- 273) + A,]
Literature Cited
BW
T,= A, - In P,- c w X, = 45.079(1.73 - 0.00267T,)0*38
FB,,, = FVi,, + NF,,, P, IPV, TV, = HVi,,
BW
A,
- In PV, - cw
FBi,eCP,(TV,- 273)
+1)zD2, I
i
HOWI[CP,(TF - 273) + 32.821F0, Evaporator Trains.
Qe
V2 = (R2
i
#
w
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2646 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Duran, M. A.; Grossmann, I. E. An Outer Approximation Algorithm for A Class of Mixed Integer Nonlinear Programs. Math. Program. 1986,36,307-339. Edgar, T. F.; Himmelblau,D. M. Optimization ofchemicalhcesses; McGraw-Hill: New York, 1988. Eswaran, P. K.; Ravindran, A.; Moskowitz, H. Algorithms for Nonlinear Integer Bicriterion Problems. J. Optim. Theory Appl. 1989,63,261-279. Geoffrion, A. M. Generalized Benders Decomposition. J. Optim. Theory Appl. 1974,10,237-260. Goicoechea,A.; Hauseu, D. R.; Duckstien, L. MultiobjectiueDecision Analysis with Engineering and Business Applications; Wiley: New York, 1982. Haimes, Y. Y.; Li, D. Hierarchal Multiobjective Analysis for Large Scale Systems: Review and Current Status. Automatica 1988, 24,1361-1383. Klein, D.; Hannan, E. An Algorithm for the Multiple Objective Integer Linear Programming Problem. Eur. J. Oper. Res. 1982,9,37& 385. Marcotte, 0.; Soland,R. An Interactive Branch-and-Bound Algorithm for Multiple Criteria Optimization. Manage. Sci. 1986,32,61-75.
McKetta, J. J. Encyclopedia of Chemical Processing and Design; Marcel Dekker: New York, 1984. Sawaragi, Y.; Nakagama, H.; Tanino, T. Theory of Multiobjectiue Optimization; Acadcemic Press: Orlando, FL, 1985. Villareal, B.; Karwan, M. H. Multicriteria Integer Programming: A (Hybrid) Dynamic Programming Recursive Approach. Math. Program. 1981,21,204-221. Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990,7,769-782. Zionta, S.A Multiple Criteria Method for Choosing Among Discrete Alternatives. Eur. J. Oper. Res. 1981,7, 143-147. Received for review March 25, 1993 Revised manuscript received July 19, 1993 Accepted July 27, 1993.
Abstract published in Advance ACS Abstracts, October 1, 1993. @
