4036
Ind. Eng. Chem. Res. 1998, 37, 4036-4048
Optimal Design of Distributed Wastewater Treatment Networks B. Galan† and I. E. Grossmann*,‡ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 and Department of Chemistry, ETSII, Universidad de Cantabria Santander, 39005 Spain
This paper deals with the optimum design of a distributed wastewater network where multicomponent streams are considered that are to be processed by units for reducing the concentration of several contaminants. The proposed model gives rise to a nonconvex nonlinear problem which often exhibits local minima and causes convergence difficulties. A search procedure is proposed in this paper that is based on the successive solution of a relaxed linear model and the original nonconvex nonlinear problem. Several examples are presented to illustrate that the proposed method often yields global or near global optimum solutions. The model is also extended for selecting different treatment technologies and for handling membrane separation modules. 1. Introduction Wastewater is treated before release to remove contaminants and to meet environmental regulations. The practice in the past has been to operate a centralized industrial wastewater treatment plant rather than to provide individual treatment for each production unit. Centralized treatment implies that all the process streams are collected and treated in a common facility. There are generally three stages of wastewater treatment: (1) primary treatment that uses physical operations to remove free oil or suspended solids, (2) secondary treatment for removal of dissolved contaminants through chemical or biological action, and (3) tertiary treatment for removal of residual contaminants (Allen and Rosselot, 1997). In distributed wastewater treatment, streams are either treated separately or only partially mixed which reduces the flow rate to be processed when compared to centralized wastewater treatment systems. This in turn reduces investment because the capital cost of most treatment operations is proportional to the total flow of wastewater and the operating cost for treatment increases with decreasing concentration for a given mass contaminant (see McLaughin et al., 1992). This suggests that the design of effluent treatment systems should segregate the streams for treatment and only combine them if it is appropriate. Technologies for the separation of contaminants in these systems include membrane technologies such as reverse osmosis, ultrafiltration, and electodialysis (Metcalf and Eddy, Inc., 1991; Freeman and Harris, 1995). The increasing importance of environmental regulations motivates the need of systematic design tools for waste treatment systems (e.g., see Mishra et al., 1974). In the past decade, a number of studies on wastewater treatment design have been reported. Takama et al. (1980) developed an optimization strategy and applied it in a petroleum refinery. These authors transformed the model into a series of problems without inequality constraints by employing a penalty function and solving them with the “complex” method. The optimal recycle/ reuse networks for waste reduction has been studied * To whom correspondence should be addressed. † Universidad de Cantabria. ‡ Carnegie Mellon University.
recently through the concept of mass-exchange networks (MENs), making use of pinch analysis. Different cases have been covered: single pollutant (El-Halwagi and Manousiouthakis, 1989), multiple pollutants (Gupta and Manousiouthakis, 1994; El-Halawagi, 1997) and one pollutant in liquid and gaseous wastes (El-Halwagi et al., 1996), and applications of reverse osmosis in (ElHalwagi, 1992). Papalexandri et al. (1994) have considered for these problems simultaneous optimization models in contrast to the sequential design strategies used by the other authors. Wang and Smith (1994) and Kuo and Smith (1997) approach the general problem for the design of the final disposal wastewater network using graphical representations and techniques on superstructures of alternative designs. Alva-Argaez et al. (1998) have proposed a solution approach based on a recursive MILP to optimize the Wang and Smith (1994) model. Optimality with this method is not guaranteed. In this paper we will present an NLP and MINLP model for the superstructures presented by Wang and Smith (1994) for the design of distributed wastewater treatment plants. Since these models involve bilinearities in the constraints that often cause convergence failures in NLP algorithms, or that lead to suboptimal local solutions, an effective heuristic procedure for global optimization is presented. An important observation of this paper is that the design of the wastewater treatment network can be related to the design of separation networks. In wastewater treatment the separation of the toxic components is the main goal. The synthesis of multicomponent process networks has been addressed in the literature, for instance, by Mahalec and Motard (1977), who developed heuristic techniques to generate network configurations. More recently, Quesada and Grossmann (1995) developed a global optimization method for handling the nonconvexities that arise due to the bilinear terms in the mass-balance equations of these process networks. The basic procedure in this method consists of a spatial branch and bound algorithm that relies on the use of linear underestimators of the bilinear terms. To tackle more general nonconvex NLP problems, a number of algorithms have been proposed, and a recent review can be found in Grossmann (1996). While significant progress has been made in global optimization methods, they often require significant computational effort. On the other hand, these methods
S0888-5885(98)00133-X CCC: $15.00 © 1998 American Chemical Society Published on Web 09/04/1998
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4037
Figure 2. Mixer model. Figure 1. Superstructure for three streams and three components.
often find the global optimum from the initial relaxation and spend most of the time in the branch and bound search to verify optimality of the initial solution. In this paper instead of applying a rigorous deterministic global optimization algorithm, a search procedure is proposed that is based on a relaxation used in the global optimization of bilinear problems. Numerical results show that the proposed method yields the global optimum in many cases. If a rigorous global optimization is applied, the proposed method can be used to predict a very good upper bound, which reduces the computational effort of these methods. The paper also presents models for selecting treatment technologies and for handling membrane modules such as nondispersive solvent extraction. 2. Problem Statement The problem of synthesizing a distributed wastewater network can be stated as follows: Given is a set of process liquid streams, with known flow rates, that contain certain pollutants with known concentrations. Given is also a set of technologies for the removal of each pollutant. The goal of this problem is to identify the interconnections of the technologies, and their corresponding flow rates and compositions that will meet the discharge composition regulations for each pollutant at minimum total cost. Two important aspects to be considered in this problem are the possibility of mixing streams to reach feasible levels of concentration for the processing of any of the technologies and to adjust the final concentration of the contaminants to the limit allowed by regulations. In the first part of this paper we will assume that the technologies are described by simple linear models with constant removal ratios for each component. For this case we will present a heuristic global optimization procedure that yields the global optimum in many cases. In the second part of the paper we introduce the possibility of selecting technologies by using binary variables, and in the third part we extend the proposed method to models for describing the use of membrane technologies through nondispersive solvent extraction. The first step for determining the least-cost wastewater treatment is to develop a general superstructure of the network (see Wang and Smith, 1994; Kuo and Smith, 1997). That superstructure includes splitters (S), mixers (M), treatment units (T), and their interconnections. Figure 1 shows the network superstructure for the determination of the optimal wastewater treatment network for the case of three process streams and three treatments units. The three initial streams are divided by initial splitters (S1, S2, and S3) into streams that are directed to mixers at the inlets of the treatment
units (M1, M2, and M3) and at the discharge point (M4). After the treatment units, splitters (S4, S5, and S6) are placed to direct the treated stream to another treatment process or to final discharge. Note that this superstructure includes all the alternatives: (i) an initial stream that can be discharged into a sewerage directly without any treatment, (ii) a stream or part of it can be processed with one or more treatment units (X, XX, and XXX), (iii) a stream or part of it that can be reprocessed by another technology. 3. NLP Model As explained in the previous section, the process network consists of splitters, mixers, and treatment units. Initially, we will assume that the treatment units are fixed and can be modeled so that the output concentration of the contaminants can be expressed as a linear relation of the inlet concentrations and that the total flow rate of the inlet streams do not change during the treatment. The NLP formulation for optimizing process networks involving mixers, splitters, and linear process units will be based on the use of individual contaminants and total flows. The major difficulty in optimizing this type of model arise from the bilinearities in the equations of the splitter units. These bilinear terms may give rise to local optima, and when flows take values of zero, numerical singularities arise that often cause failures in the NLP algorithms for finding feasible points (see Quesada and Grossmann, 1995). The optimization problem that is shown below consists of selecting the individual flows and the total flows of the streams in order to optimize an objective function given by total flow processed by the units. The NLP model is given as follows: Objective Function. Each treatment unit (TU) has associated an inlet stream i. For simplicity we consider the objective function expressed in terms of total inlet flow of the streams i ∈ TU (later in the paper in section 8 we will consider cost of units):
φ)
Fi ∑ i∈TU
(1)
Mixer Units. A mixer k ∈ MU consists of a set of inlet streams i that are specified in the index set Mk and an outlet stream k as shown in Figure 2. The total mass balance for a mixer k is given by eq 2 and the mass balance for each contaminant j in that mixer is given by eq 3:
Fk)
F i, ∑ i∈M
∀ k ∈ MU
(2)
k
f kj )
f ij, ∑ i∈M k
∀ j, ∀ k ∈ MU
(3)
4038 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
f ij ) β kj f kj ,
Figure 3. Splitter model.
Figure 4. Treatment unit.
where Fi and Fk are the total flow of the streams i and k, f ij is the flow of the contaminant j in stream i, and f kj is the flow of the contaminant j in stream k. It is important to note that both equations are linear and that the outlet flows for the final mixer are constrained to the desired discharge specifications. Splitter Units. A splitter k ∈ SU consists of an inlet stream k and a set of outlet streams i specified in the index set Sk, (Figure 3). The total mass balance for a splitter is given by the following linear equation:
∑ F i ) F k,
∀ k ∈ SU
(4)
i∈Sk
while the mass balance for each contaminant j is given by the linear equation
f ij ) f kj , ∑ i∈S
∀ k ∈ SU
(5)
k
It is also necessary to enforce the condition that the streams leaving the splitter have the same ratios in flow for each contaminant. Therefore, the split fraction, ζ, is introduced as an additional variable that represents the fraction of the inlet flow k that goes to the outlet of stream i:
f ij ) ζikf kj ,
∀ j, ∀ i ∈ Sk, ∀ k ∈ SU
(6)
0 eζik e 1
(7)
where
ζik ) 1, ∑ i∈S
∀ k ∈ SU
(8)
k
That split fraction is also used for distributing the total flow: i
F )
ζikF k,
∀ i ∈ Sk, ∀ k ∈ SU
(9)
Equations 6 and 9 are nonlinear and give rise to the nonconvexities of this model. Treatment Units. Each treatment unit k ∈ TU consists of an inlet stream k and an outlet stream i, as is shown in Figure 4. It is assumed that the individual flow of the outlet stream i can be expressed as a linear relation of the individual flow of the inlet stream k. As the concentration is low, it is also assumed that the total flow of the outlet stream does not change in the treatment unit.
F i ) F k,
∀ k ∈ TU
(10)
∀ j, ∀ k ∈ TU
(11)
where βkj is the removal ratio of the unit k for the component j. The model given by eqs 1-11 corresponds to an NLP model since eqs 6 and 9 are nonlinear. Since these equations involve bilinear terms, standard algorithms may lead to a local optimum. Global optimization techniques can be applied to obtain the global optimum. However, in this paper instead of applying a rigorous method (e.g., see Grossmann, 1996) we propose a heuristic procedure based on the solution of two models: the NLP model defined by eqs 1-11 and a relaxed LP model of the original nonconvex NLP model. 4. LP Model The relaxed LP model proposed in this paper incorporates linear bounding information in order to replace the nonconvex bilinear equations (6) and (9). The key feature of bounding is its ability to provide a rigorous lower bound to the global optimum. Several bounding techniques based on convex underestimators have been reported in the literature. McCormick (1976) introduced a general method for constructing convex/concave envelopes for factorable functions. Sherali and Alameddine (1992) developed the formulation and linearization technique for developing a tight LP to bound bilinear programs. The LP relaxation that is proposed is similar to the one employed by Quesada and Grossmann (1995) who embedded it in a branch and bound procedure to obtain the global optimum. The procedure for obtaining the LP underestimators consists of determining the bounds of the variables involved in the nonconvex terms, total flows, and individual flows and then applying these bounds to the underestimators in the LP subproblem. Following this procedure, the nonconvex equations (6) and (9) can be replaced by the linear inequalities
( (
)
i iL k kL iL f kj ζik g f kL j ζk + ζk f j - f j ζk k i kU i iU k iU f j ζk g f j ζk + ζk f j - f kU j ζk i iL k kU iL f kj ζik e f kU j ζk + ζk f j - f j ζk i iU k kL iU f kj ζik e f kL j ζk + ζk f j - f j ζk ∀ j, ∀ i ∈ Sk, ∀ k ∈ SU (12)
)
k kL iL F kζik g F kLζik + ζiL ζk k F - F iU k k i kU i kU iU F ζk g F ζk + ζk F - F ζk k kU iL F kζik e F kUζik + ζiL ζk k F - F iU k k i kL i F ζk e F ζk + ζk F - F kLζiU k ∀ i ∈ Sk, ∀ k ∈ SU (13)
where FiLand FiU are lower and upper values of the total kU are the lower and upper flow rate F and f kL j and f j values of the individual flow rate f ij. To tighten the lower bound predicted by the relaxation, additional constraints which are nonredundant in the linear relaxation of the problem are generated. The new equations relate the individual (f ij) and the total flows (Fi) through the concentration of each contaminant j in the stream i (cij). For those streams located after the initial splitters, (S1, S2, and S3 in Figure 1), the new equations are linear since the
concentration of the contaminant is the initial concentration. For all other streams, the new equations are nonlinear because the concentration of the contaminant is unknown. These bounding constraints can be defined by the following equations:
(
)
∀ j, ∀ i ∈ initial SU
f ij ) F icij,
(14)
i iL iL f ij g F iLcij + ciL j F - F cj i iU iU f ij g F iUcij + ciU cj j F - F iU i iU iL i iL i f j e F cj + cj F - F cj i iL iU f ij e F iLcij + ciU j F - F cj ∀ j, ∀ i ∉ initial SU (15)
where ciU and ciL are lower and upper values of the concentration of the contaminants. Taking into consideration eqs 12-15, the LP model is finally defined by the following set of equations. Objective Functions (O). Two types of objective functions will be considered. (1) Minimizing the flow rate processed by each treatment unit:
∀ i ∈ TU
φ ) F i,
(16)
(2) Minimizing the total flow rate processed by all units:
φ)
Fi ∑ i∈TU
(17)
Mixer Equations.
Fk)
F i, ∑ i∈M
∀ k ∈ MU
(18)
k
f kj )
∑ f ij,
∀ j, ∀ k ∈ MU
(19)
i∈Mk
Splitter Equations.
F i ) F k, ∀ k ∈ SU ∑ i∈S
(20)
k
(
f ij ) f kj , ∑ i∈S k
∀ k ∈ SU
(21)
)
ki
∀ k ∈ SU
)
k kL iL F i g FkLζik + ζiL ζk k F - F i kU i kU iU iU k F g F ζk + ζk F - F ζk k kU iL F i e FkUζik + ζiL ζk k F - F i kL i kL iU iU k F e F ζk + ζk F - F ζk ∀ i ∈ Sk, ∀ k ∈ SU (24)
Treatment Unit Equations.
f ij ) βkj f kj ,
{{
∀ j, ∀ k ∈ TU
(25)
Linear Bounding Constraints.
f ij ) ficij i iL iL f ij g F iLcij + ciL j F - F cj i iU iU f ij g F iUcij + ciU cj j F - F iL i i iU i iU iL f j e F cj + cj F -F cj i iL iU f ij e F iLcij + ciU j F - F cj
}
∀ i ∈ initial SU ∀ i ∉ initial SU
}
∀ j (26)
where i iU i iU iL f iL e F i e F iU, ciL j e f j e f j , F j e cj e cj
The motivation for using the objective functions in (16) for each treatment unit is to be able to generate multiple starting points in addition to the one that is obtained with the original objective in (17). Note that the motivation in using (16) is to find flow conditions where the load for each treatment unit is the smallest. The LP model given by 16-26 can be reliably and efficiently solved to optimality using existing algorithms. The solution of this LP model with the objective function in (17) will predict a lower bound to the objective function of the original NLP model. Furthermore, the values of the optimal variables of the LP will be used as a starting point for the solution of the NLP model. The nonconvex model can be solved by application of a standard NLP algorithm. However, there is no guarantee that the global optimum will be obtained. Since the objective functions, (16) and (17), will be used in the LP model, different initial points for the NLP model will be obtained. Each of these starting points can be used to solve the original nonconvex model. If the bounds of the variables in (22), (24), and (26) are tight, good initial starting points for the NLP can be expected from the LP that in most cases will lead to a feasible local solution of the NLP. The detailed procedure to search for the global optimum is described in the following section. 5. Search Procedure
i iL k kL iL f ij g f kL j ζk + ζk f j - f j ζk i kU i iU k iU f j g f j ζk + ζk f j - f kU j ζk i iL k kU iL f ij e f kU j ζk + ζk f j - f j ζk i iU k kL iU f kj ζik g f kL j ζk + ζk f j - f j ζk ∀ j, ∀ i ∈ Sk, ∀ k ∈ SU (22)
ζik ) 1, ∑ i∈S
(
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4039
(23)
The goal of this method is to obtain a good upper bound of the global optimum of the NLP model. The method is based on the successive solution of the LP and NLP models described in the previous sections. The procedure uses the LP model to generate good initial points for the NLP model. Since NLP algorithms do not guarantee the global optimum for nonconvex models, different objective functions are used in the LP model to generate different starting points. Finally, among all the results, the best objective function is selected. The proposed procedure consists of the following four steps.
4040 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
Figure 6. Optimal solution for example 1. Table 3. Wastewater Stream Data for Example 2 Figure 5. Superstructure for example 1. Table 1. Wastewater Stream Data for Example 1 stream number
flow rate (t/h)
contaminants
Ci (ppm)
1
40
2
40
A B A B
100 20 15 200
Table 2. Treatment Process Data for Example 1 process number X XX
contaminants
removal ratio (%)
A B A B
95 0 0 97.6
Step 1. Solve the NLP model given by eqs 2-11 using the objective function (1) and the upper bound of the variables as starting points. If a local optimum is obtained, it corresponds to an upper bound of the objective function. Step 2. For each treatment unit i ∈ TU, solve the LP model given by eqs 18-26 using the objective function (16). Solve the NLP model (1)-(11) using as the starting point the optimum value of variables from the LP problems. This step is repeated for each treatment unit i ∈ TU. Step 3. Solve the LP model given by eqs 18-26 using the objective function (17). The solution of this problem yields a rigorous lower bound to the global optimum. Solve the NLP model (1)-(11) using as the starting point the optimum values of the variables from this LP problem. Step 4. Select among the NLP solutions in steps 1-3 the one with the lowest objective function value. An estimate of the error is given by the difference of the upper and lower bounds. 6. Examples The procedure proposed in previous sections is used for solving several examples that have been reported in the literature. In all the examples GAMS has been used as the equation modeling system, GAMS/OSL as the LP solver and GAMS/MINOS as the NLP solver. The computer that was used is an HP9000/C110. Example 1. The first example illustrates a case of two streams, each containing two contaminants A and B (Wang and Smith, 1994). The data for the two effluent streams are given in Table 1 and the data for the two wastewater treatments with the removal ratios and concentrations constraints are given in Table 2. For both treatment processes, costs are optimized by minimizing the flow rate treated and the environmental discharge limit is 10 ppm for A and B. The superstructure for this problem is presented in Figure 5. The NLP model is given by eqs 1-11 and the LP is given by eqs 16-26. The results obtained at each step of the search procedure are as follows:
stream number
flow rate (t/h)
Ci (ppm)
1 2 3
40 30 20
400 100 30
Step 1. Solve NLP model with objective φ ) F3 + F4. Local optimum ) infeasible. Step 2. (a) (i) Solve LP model with objective φ ) F3. Optimum ) 40. (ii) Solve NLP model with objective φ ) F3 + F4. Local optimum ) 110.95. (b) (i) Solve LP model with objective φ ) F4. Optimum ) 40.98. (ii) Solve NLP model with objective φ ) F3 + F4. Local optimum ) 89.83. Step 3. (i) Solve LP model with objective φ ) F3 + F4. Lower bound ) 80.8. (ii) Solve NLP model with objective φ ) F3 + F4. Local optimum ) 110.95. Step 4. Select the best result for the NLP model. Upper bound ) 89.83. As can be seen, solving the NLP model without any special provisions (step 1) leads to an infeasible solution. The second and third steps solve the LP and the NLP model sequentially. It is observed that using better initial points in the NLP model than those in step 1 leads to feasible solutions. Two different local solutions are found: 89.83 and 110.95. Also, in step 3 the lower bound of 80.9 is determined. Finally, step 4 selects the best result: 89.83. In this example we can see that the proposed procedure yields essentially the same solution as the one presented by Wang and Smith, whose solution has an objective function of 90. The network obtained is shown in Figure 6. Note that splitters 1 and 2 do not divide the initial streams. Stream 1 goes to treatment X, where pollutant A is removed, and stream 2 goes to treatment XX, where pollutant B is removed. After treatment X, splitter 3 divides stream 5 into two streams 52 and 53. Stream 52 flows to mixer 2 to join stream 22 to eliminate contaminant B in treatment XX. Stream 6 is not divided by splitter 4 and finally streams 53 and 63 are collected in mixer 3 to be discharged. For the remaining examples we present a brief description and in the following section the results are summarized. Example 2. This wastewater treatment system is from Wang and Smith (1994) and involves only one contaminant. The streams, flow rates and the inlet concentration for three effluent streams are given in Table 3. The environmental discharge limit is 20 ppm and a treatment process with a removal ratio of 0.99 is used to carry out the treatment task. Example 3. The third example illustrates a case where a maximum concentration for the inlet streams is specified for the treatment units (Wang and Smith, 1994). The data for two wastewater streams containing one contaminant are given in Table 4. The environmental limit to the concentrations of the contaminant is 10 ppm. The removal ratios and concentration constraints for the two treatments available are given
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4041 Table 4. Wastewater Stream Data for Example 3
Table 10. Wastewater Stream Data for Example 6
stream number
flow rate (t/h)
Ci (ppm)
stream number
flow rate (t/h)
contaminants
Ci (ppm)
1 2
60 20
400 800
1
13.1
2
32.7
3
56.5
A B C A B C A B C
390 16780 25 10 110 100 250 400 350
Table 5. Treatment Process Data for Example 3 process number
removal ratio (%)
Cmax (ppm)
X XX
99 80
200 1000
Table 6. Wastewater Stream Data for Example 4 stream number
flow rate (t/h)
contaminants
Ci (ppm)
1
13.1
A B C A B C A B C
390 16780 25 10 110 100 25 40 35
2
32.7
3
56.5
Table 7. Treatment Process Data for Example 4
Table 11. Treatment Process Data for Example 6 removal ratio (%) process number X XX XXX
A
B
C
99.9 90 0
0 70 70
0 98 50
Table 12. Wastewater Stream Data for Example 7 stream number
flow rate (t/h)
contaminants
Ci (ppm)
1
20
2
15
3
5
A B C A B C A B C
600 500 500 400 200 100 200 1000 200
removal ratio (%) process number X XX XXX
A
B
C
99.9 90 0
0 90 95
0 97 20
Table 8. Wastewater Stream Data for Example 5 stream number
flow rate (t/h)
Ci (ppm)
1 2 3
20 30 50
800 400 200
Table 9. Treatment Process Data for Example 5 process number
removal ratio (%)
CI, max (ppm)
X XX
90 99
600 200
in Table 5 and in both cases cost is optimized for minimum flow rate treated. Example 4. Table 6 gives the data for three wastewater streams by Takama et al. (1980) and Wang and Smith (1994). The streams contain three contaminants and the environmental limits of the concentration of the three contaminants are CA,max ) 2 ppm, CB,max ) 2 ppm, and CC,max ) 5 ppm. Three treatment units are available and in all cases cost is optimized for minimum flow rate treated. The removal ratios are given in Table 7. Example 5. Tables 8 and 9 show the data for a set of three effluent streams and treatment methods. This problem has a single contaminant for which the discharge limit is 30 ppm. Example 6. Example 6 is similar to example 4, but several data were changed by Kuo and Smith (1997) in order to make them more realistic. Tables 10 and 11 present the data for the streams and for the treatments. The environmental limits of the concentration of the three contaminants have also been changed to CA, max ) 5 ppm, CB,max ) 20 ppm, and CC,max ) 100 ppm. Example 7. This problem addresses the optimal design of a distributed treatment system for a set of three waste streams with contaminants A, B, and C at different concentration levels. The streams, flow rates and initial contaminant concentrations are shown in Table 12. The maximum pollutant concentration allowed at the wastewater discharge is 100 ppm for each
Table 13. Treatment Process Data for Example 7 removal ratio (%) process number X XX XXX
A
B
C
90 0 0
0 99 0
0 0 80
contaminant. Each of the effluent treatment processes X, XX, and XXX can remove only one contaminant according to the removal ratios provided in Table 13. 7. Computational Results The results obtained for examples 1-7 are shown in Tables 14-16. The local solutions for wastewater networks using the NLP model and setting the variables to their upper bounds for the starting points are presented in the fourth column of Table 14. That column shows the results for the first step of the procedure described in section 5 and therefore gives the upper bound of the objective function. In Table 14 the dimensions of the NLP problems are also given in terms of the number of variables and constraints. The second part of Table 14 shows the solutions of the LP model (step 3) and the corresponding number of variables and constraints. The seventh column of the table shows the solution of the LP model that corresponds to the lower bound of the global optimum. Table 15 shows the detailed results for the heuristic procedure at section 5 involving the LP model with the different objective functions and NLP model (steps 2-4 of section 5). The top entry in columns 3-5 of Table 15 show the results of the second step when the objective functions in the LP model are the flow rates of the individual treatments. Column 6 shows the third step when the objective function of the LP model is the sum of the flow rate of all treatments. The bottom entry shows the results of the NLP model that was run with the same
4042 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Table 14. Solutions for the NLP and LP Models for Examples 1-7 first NLP
lower bound LP
variables
constraints
local solution
variables
constraints
optimal solution
38 32 53 166 65 166 169
78 38 55 176 69 176 179
infeasible 59.79 130.7 274.7 99.49 227.47 87.91
97 38 67 232 79 235 235
242 76 147 620 164 623 620
80.98 58.44 106.27 126.22 88.98 113.68 64.83
example 1 example 2 example 3 example 4 example 5 example 6 example 7
Table 15. Solutions for Steps 2-4 for Examples 1-7 treat. X
treat. XX
example 1 LP 40.0 NLP 110.95
40.98 89.83
treat. XXX
example 2 LP NLP example 3 LP 36.36 45.5 NLP 130.7 130.7 example 4 LP 3.3 NLP 269.4 example 5 LP NLP
28.1 99.49
61.3 229.7
example 7 LP NLP
27.7 80.7
14.1 80.7
80.8 110.95
89.8
90
58.4 59.8
59.8
60
106.27 130.7 130.7
153
7.01 126.2 274.7 229.7
4.0 99.49
example 6 LP 3.3 19.9 NLP 173.28 179.8
bestall upper reported treat. bound solution
88.98 99.49 9.9 173.8 21.2 80.7
113.6 215.6 64.83 85.14
229.7 99.49
221.8 99.5
173.28 176.6 80.77
80.77
Table 16. CPU Times for Examples 1-7a example 1 example 2 example 3 example 4 example 5 example 6 example 7 a
LP
NLP
total
0.12 0.05 0.07 0.26 0.09 0.26 0.25
0.15 0.03 0.05 1.48 0.10 0.25 0.32
0.27 0.08 0.12 1.74 0.19 0.51 0.57
HP 9000/C110 workstation.
objective function: sum of the flow rates of all the treatment units. Column 7 of Table 15 shows the fourth step when the best solution of all local optima is selected (upper bound), and column 8 shows the best-reported solution in the literature (Wang and Smith, 1994; Kuo and Smith, 1997; Zamora and Grossmann, 1998). Table 14 indicates that by using the algorithm MINOS, the initial NLP model converged to a local minimum in all cases, except in example 1 in which there was a failure in the convergence. From Table 15 it can be seen that in example 2 the lower bound predicted by the LP model of step 3 is very close to the upper bound. In examples 1, 3, 5, and 7 the bounds are within 20% of the upper bound. For examples 4 and 6 the differences are larger. It can also be seen that it is only in example 4 that the solution obtained is worse than the best solution reported in the literature. This is because we considered only three treatment units, while four pieces of equipment were used by Wang and Smith (1997). If we use the fourth piece of equipment, the result obtained using the proposed method is 221.3. From the comparison of columns 7 and 8 in Table 15, it can be seen that the heuristic search procedure used for solving examples 1-7 gives very good results compared to solutions reported previously in the literature. Note that Zamora and Grossmann (1998) solved ex-
Figure 7. Superstructure for selection of technologies at each task unit.
ample 7 by applying a rigorous global optimization algorithm and found the global optimum of 80.77. It can be seen that for example 7 the upper bound is also 80.77. Therefore, for this example, the upper bound corresponds to the global optimum. As can be seen in Table 16, the problem that required the largest CPU time was only 1.74 s (HP9000/C110). We also applied our method to a much larger problem that is described in Appendix A, which involves 12 streams, 6 contaminants, and 5 units. The corresponding NLP has 958 variables and 999 constraints. The proposed procedure required only 5.5 s to solve this problem. A rigorous global optimization method is likely to take a much longer time. 8. Selection of Technologies In this section we introduce an extension which is frequently found in the wastewater treatment systems: two or more technologies can remove one or more pollutants with different costs (investment and/or operating), and one of the technologies has to be selected. The problem is formulated as a mixed-integer nonlinear program where the objective is to determine the equipment and the sequence of the streams. The binary variables are associated with the selection of the technologies and the continuous variables are assigned to total and individual flow rates, concentrations, and split fractions. We propose a heuristic method to find the optimum value of the variables similar to the method explained in section 5, but in this case binary variables are included. The first step is to develop a general superstructure for the new system. That superstructure includes splitters (S), mixers (M), tasks (Ta), and their interconnections. Task is the elimination of one or more pollutant by any physical, chemical, or biological treatment. Each task can be carried out by more than one treatment and one of them has to be selected. Figure 7 shows the network superstructure for the case of two initial streams and two tasks with two possible technologies each (TU). The second step is to obtain the model of the process. For the MILP and MINLP models, equations for mixer and splitters can be used without any changes, but the
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4043
objective functions and the treatment units have to be reformulated in order to introduce costs and the task definition, respectively. For the NLP model only changes in the objective function are needed. Objective Function. We consider the case where the total capital cost (CT) and total operating cost (OT) of the treatment processes are functions of the flow rates. Following the search procedure described before, two types of objective functions will be considered. Equation 27 considers the total cost of each treatment unit, and eq 28 considers the sum of the individual costs. i R
i
i
∀ i ∈ TAU
i
φ ) C (F ) + O F ,
∑
φ)
(Ci(F i)R + OiF i)
(28)
where 0 < R e 1. Equation 27 will be used in the MILP model to minimize the cost of each treatment unit and eq 28 will be used in the MILP, MINLP, and NLP models to minimize the total cost. As capital costs are expressed as a function of R where, 0 < R e 1, eqs 27 and 28 are nonlinear. These equations can be linearized using linear underestimators (Zamora and Grossmann, 1998) that are defined by eq 29:
(
)
L gi(xU i ) - gi(xi ) L xU i - xi
(xi - xLi ) e gi(xi) (29)
where gi(xi) is a concave function and gi(xiU) and gi(xiL) are the upper value and the lower value of the function. Therefore, eqs 27 and 28 can be replaced by the linear equations (30) and (31):
ˆ i + OiF i, φ ) CiF φ)
∀ i ∈ TAU
(CiF ˆ i + OiF i) ∑ i∈TAU
(30) (31)
where
ˆ iL)R + F ˆ i ) (F
(
)
(F iU)R - (F iL)R F
iU
-F
iL
(F i - F iL) (32)
Task units. Each task unit k ∈ TAU consists of treatment units (h ∈ TU) that are specified in the index set NE and one inlet stream k and one outlet stream i. It will be assumed that the treatment units can be linearly modeled and the total flow rate of the inlet streams does not change during the treatment. The model for the treatment units is given by eqs 33-37:
∑ f k,h j ,
∀ j, ∀ k ∈ TAU
(33)
f i,h ∑ j , h∈NE
∀ j, ∀ k ∈ TAU
(34)
f kj )
h∈NE
f ij )
k,h i f i,h j ) f j βj ,
iU
k,h f i,h j e fj Y ,
∀ h ∈ TU, ∀ j, ∀ k ∈ TAU ∀ h ∈ TU, ∀ j, ∀ k ∈ TAU
removal ratio (%) equipment
A
B
C
X
task
E1 E2
50 90
0 0
0 0
XX
E1 E2
0 0
90 99
0 0
XXX
E1 E2
0 0
0 0
60 80
Table 18. Treatment Process and Cost Data for Example 9
(27)
i∈TAU
gˆ i(xi) ) gˆ i(xLi ) +
Table 17. Treatment Process Data for Example 8
(35) (36)
costsa
removal ratio (%) equipment
A
B
C
investment ($)
X
E1 E2
99 99
0 0
0 0
16800F0.7 168000F0.7
1F 10F
XX
E1 E2
90 90
90 30
97 0
12600F0.7 126000F0.7
0.0067F 0.067F
XXX
E1 E2
0 40
95 50
20 0
4800F0.7 48000F0.7
task
operating ($/h)
0 0
a F is the flow rate treated given in t/h. Annual rate of return: 10%. Operating hours: 8600 (h/year).
Yk,h ) 1, ∑ h∈NE
∀ k ∈ TAU
(37)
where Yk,h are the binary variables for the piece of equipment h of the task unit k and f iU j is the maximum value of the f ij. It is observed that eqs 33-37 are linear equations. The procedure to obtain an upper bound for the global optimum is as follows. Step 1. Solve the MINLP model given by eqs 2-10 and 33-37 using the objective function in (27) and the upper bound of the variables as starting points. The local optimum obtained corresponds to an upper bound of the objective function. Step 2. For each task unit i ∈ TAU, solve the MILP model given by (18)-(24), (26), and (33)-(37) and using the objective function described by eq 30. Then, solve the NLP model in (2)-(11) with the objective function in (28), using the equipment selected by the MILP model and the optimal values of the continuos variables from the MILP problems as a starting point. This step has to be repeated for each i ∈TAU. Step 3. Solve the MILP model given by (18)-(24), (26), and (33)-(37) using the objective function in (31). The solution to this model yields a lower bound. After that, solve the NLP model in (2)-(11) with the objective function in (28), using the equipment selected by the MILP model and as the starting point the optimal values of the continuous variables from this MILP problem. Step 4. Select the final solution of the NLP model with the lowest value of the objective function. This procedure was applied in examples 8-10. In example 8, it is assumed that the cost is proportional to the flow and therefore the objective functions are given by eqs 1, 16, or 17. In examples 9 and 10, the cost was given by an explicit function shown in Tables 18 and 20 and the objective functions are given by eqs 27-32. In all the examples GAMS has been used as the equation modeling system, GAMS/DICOPT as the MINLP solver, GAMS/OSL as the MILP solver, and GAMS/MINOS as the NLP solver.
4044 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Table 19. Wastewater Stream Data for Example 10 stream number
flow rate (t/h)
contaminants
Ci (ppm)
1
20
2
15
3
5
A B C A B C A B C
1100 300 400 300 700 1500 500 100 600
Figure 8. Optimal solution for example 10.
Table 20. Treatment Process and Cost Data for Example 10 costsa
removal ratio (%) equipment
A
B
C
investment ($)
X
E1 E2 E3
90 50 0
0 70 80
40 0 0
3480F0.7 469F0.7 26F0.7
0 10F F
XX
E1 E2 E3
0 0 50
90 99 99
0 0 80
726F0.7 1260F0.7 5000F0.7
0.0089F 0.018F 5.8F
XXX
E1 E2 E3
80 0 0
0 0 0
60 80 40
320F0.7 58F0.7 10F0.7
6F 15F F
task
operating ($/h)
a F is the flow rate treated given in t/h. Annual rate of return: 10%. Operating hours: 8600 (h/year).
As in previous examples the results are shown in two tables: Tables 21 and 22. Table 22 also indicates which piece of equipment was finally selected for each task. It is shown that the use of an MINLP leads to an infeasible solution in example 9. However, the proposed procedure leads to feasible solutions in all examples. The solutions obtained for examples 8 and 9 can be compared with reported solutions and in both cases the solutions are equal. It is important to note that 80.77 is the global optimum for example 8. Figure 8 shows the optimal solution for example 10 and also indicates the equipment selected. The total time required for solving each problem on a HP9000/C110 workstation were 0.67, 0.89, and 0.92 s. 9. Nondispersive Solvent Extraction
Example 8. In this example data from example 7 were used for initial and final concentration, and flow rates of the streams. Removal ratios of the treatment units are given in Table 17. It can be seen that the removal ratios are lower for the piece of equipment E1 than for E2, and at the same time the removal ratios of E1 fit the removal ratios for example 7. This situation should lead to an upper bound equal to the upper bound in example 7. It is also important to note that the global optimum of example 7 was found by Zamora and Grossmann (1998). Example 9. The data for concentrations and streams are from example 4. In this case the cost correlations were used for the treatments units. Also we consider several technologies with different removal ratios and different costs as shown in Table 18. Example 10. Data of the streams and concentration, of the contaminants are shown in Table 19. In this case three technologies are available in each task with different removal ratios and different costs (Table 20).
Separation processes constitute an important part of the waste minimization procedure. Until this section we have simplified the process model by considering that the output concentrations of the contaminants can be expressed as a linear relation of the inlet concentrations. However, not all technologies follow this approximation. Membrane technologies are included among them. Over the past decade, membrane separation processes have gained a growing level of application. Among all the membrane separation technologies, nondispersive solvent extraction was selected to extend the model and search procedure proposed in section 5. In this technology the outlet concentration depends on the inlet concentration of the pollutant in the streams and on the flow rates. Solvent extraction processes using hollow fiber modules have been used for the removal and/or concentration of different metallic elements, anion or organic compounds. NDSX overcomes most of the shortcomings of conventional liquid extraction systems: backmixing, emulsion generation, flooding limitations on indepen-
Table 21. Solutions for the MINLP and MILP Models for Examples 8-10 first MINLP example 8 example 9 example 10
lower bound MILP
cont. variables
binary variables
constraint
local solution
cont. variables
binary variables
constraint
optimum
243 243 283
6 6 9
259 259 270
92.07 infeasible 1.99 106
315 315 342
6 6 9
688 688 712
64.83 6.5 105 1.6 105
Table 22. Solutions for the NLP and MILP Models for Example 8-10 treat. XX
treat. XXX
MILP NLP
27.7 88.24
treat. X
14.14 80.77
21.25 93.79
example 9
MILP NLP
1.1 × 105 infeasible
2.2 × 105 4.6 × 105
example 10
MILP NLP
0 2.0 × 106
1.7 × 103 infeasible
example 8
840 4.6 ×105 0 1.69 × 106
all treat. 64.83 80.77 6.5 × 105 9.15 × 105 1.6 × 105 infeasible
upper bound/equipment selected 80.77 E2-E2-E2 4.6 ×105 E1-E1-E1 1.69 × 106 E1-E3-E3
best-reported 80.77
4.6 ×105
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4045 Table 23. Value of He and Co for Example 12
Figure 9. Setup of nondispersive solvent extraction system.
dent phase flow rate variations, and requirements of density differences (Galan et al., 1994). Nondispersive contactors also have several advantages, namely (i) a very large interfacial area without direct mixing of the aqueous and organic phases, (ii) capability of treating dilute solutions, (iii) reduction of the solvent losses, and (iv) reduction in the equipment volume and space. In hollow fiber contactors, aqueous and organic solutions flow continuously: one of the solutions through the lumen of the fiber and the other through the shell side. Both phases get into contact through the pores of the fiber wall. Using hydrophobic membranes, the phase dispersion can be avoided by applying differential static pressure in the aqueous phase. The performance of the NDSX for the separation/ reuse of the components includes three steps: (1) Reduction of the concentration of the pollutant present in the aqueous phase during the extraction processes to the level required to be discharged or to be reused as freshwater. (2) Regeneration of the organic phase in the stripping module, minimizing the losses of the extractant and making it possible to work with it in a closed loop. (3) Increase the concentration of the pollutant during the stripping processes into an aqueous strip phase to be recycled to the industrial processes. To carry out simultaneously the three objectives, two different hollow fiber modules were used. In one module the extraction is performed and in the second one stripping takes place. A schematic diagram of the setup is shown in the Figure 9.
ln(HeCA - C0) ) ln(HeCA0 - C0) -
Fi
(38)
where CA0 is the inlet concentration of the contaminant and CA is the outlet concentration, at is the surface area of the hollow fiber module, NM is the number of modules, Km is the membrane transport coefficient, He is the distribution constant of the pollutant between the organic phase and the aqueous phase, and C0 is the concentration of the contaminant in the organic phase (Ortiz et al., 1996). In the simplified case, where
C
treat. X:
He C0
1900 200
1700 200
0 0
treat. XX:
He C0
0 0
1700 200
1900 200
treat. XXX:
He C0
1700 200
0 0
1500 200
atKmHeNM , Fi ∀ j, ∀ k ∈ TU (39)
ln(Hef ij - C0F i) ) ln(Hef kj - C0F k) -
Equation 39 can be arranged and expressed in exponential terms:
(
Hef ij - C0F i ) exp -
)
atKmHeNM F
i
(Hef kj - C0F k),
∀ j, ∀ k ∈ TU (40)
Therefore, the NLP model for NDSX is finally described by eqs 1-10 and by eq 40. LP Model The bilinear term of the right-hand side of eq 40 can be linearized following the procedure given in section 4, and the exponential term can be linearized using the underestimators defined by eq 29. Applying this linearization technique to the exponential term of eq 40, the following equation is obtained:
(
gˆ (F i) ) exp -
(
(
exp -
atKmHeNM
B
extraction and back-extraction are carried out at the same rate, we can assume that C0 remains constant. In this paper we model the extraction step since we only take into consideration the separation of the toxic component, but we do not cover its reuse. Equation 11 from the NLP model proposed in section 3 is replaced by the following equation:
Nonconvex NLP Model The NLP model for NDSX makes use of the previous NLP model except for eq 11. For NDSX the outlet concentration depends on the inlet concentration of the pollutant in the streams and on the flow rate. However, as in the previous model, the flow rate of the inlet stream is assumed not to change during the treatment since the concentration of the pollutants is low. A short-cut model of the NDSX is used in this paper. The equation for the NDSX treatment is as follows:
A
)
atKmHeNM
F iL atKmHeNM F
iU
)
+
(
- exp -
)
atKmHeNM F iL
F iU - F iL
)
×
(F i - F iL) (41)
Finally, eq 43 is formulated for the bilinear terms in eq 40:
(
(Hef kj - C0F k) ) Mkj
(Hef ij - C0F i) g gˆ (F iL)Mkj + MkL ˆ (F i) - gˆ (F iL)Mj j g
)
(42) kL
(Hef ij - C0F i) g gˆ (F iU)Mkj + MkU ˆ (F i) - gˆ (F iU)Mj j g (Hef ij - C0F i) e gˆ (F iU)Mkj + MkL ˆ (F i) - gˆ (F iU)Mj j g
kU
kL
kU
(Hef ij - C0F i) e gˆ (F iL)Mkj + MkU ˆ (F i) - gˆ (F iL)Mj j g
∀ j, ∀ k ∈ TU (43)
Therefore, the LP model for NDSX will be given by eqs 15-24, 26, and 41-43. The search procedure for obtaining an upper bound for the NDSX models is identical to the procedure
4046 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
Figure 10. Optimal solution for example 11. Table 24. Solutions for the NLP and LP Models for NDSX Examples first NLP example 11 example 12
variables
constraints
187 205
197 215
lower bound LP local solution 66.06 infeasible
variables
constraints
optimum
256 277
623 659
65.5 137.77
Table 25. Solutions for Steps 2-4 for NDSX Examples treat. X example 11 example 12
treat. XX
LP NLP
25.5 infeasible
19.99 71.81
LP NLP
0.0 infeasible
0.0 infeasible
explained in section 5. This method was applied in examples 11 and 12, and the results are shown in Tables 24 and 25. The modules considered for the NDSX application in examples 11 and 12 are based on extra-flow contactors built by Hoechst with catalog number 5PCM-106. The contactors have 135 m2 of membrane area consisting of 216 500 hollow fiber membranes (Celgard ×10). Each fiber has 240-µm inner diameter and 30-µm thickness. Even though the surface area is very high, the dimension of the cartridge module is quite small: 71-cm length and 25.4-cm external diameter. The material of the fibers is polypropylene and the material of the cartridge is 304 SS. Since the membrane is hydrophobic, the pressure in the aqueous phase has to be slightly higher than that in the organic phase. Example 11. For this case the data of example 7 were used, but the treatment technology is NDSX. The values of Km, He, C0, and NF, were taken from the literature (Ortiz et al., 1996) and some of them were modified for the inlet concentration of this example. The final values are: Km ) 2.2 × 10-8, He ) 900, C0 ) 200, and NM ) 15. Example 12. Data of example 6 were used, but the concentration of contaminant B in the stream 2 is reduced to 148 ppm since NDSX is mainly used for low concentrations of the contaminants. The values of Km, and NM are the same as those in the previous example and the data of He and C0 are shown in Table 23. Table 24 shows the local solution for the NLP model (step 1) and the number of variables and constraints involved in these problems. The optimal solution and number of constraints and variables for the LP model are also presented in Table 24. The local optima for the NLP model in step 1 involves two different solutions. In example 11, a local optimum that is very close to the LP optimum is obtained. In example 12 no feasible
treat. XXX
all treat.
upper bound
17.46 66.06
65.50 66.06
66.06
0.0 195.62
137.7 195.39
195.39
solution is obtained. The local solution in example 11 is so close to the global LP solution that most probably the value of 66.06 is a global solution. Figure 10 shows the optimal design for this example. Table 25 shows the results for steps 2-4. Column 7 shows the best solution among all local optima for the different objective functions that were tested (steps 1-3). Note that for example 11 the best solution is equal to the solution obtained in step 1. These two examples show that the heuristic procedure described in section 5 can also be used with technologies where the output concentration of the contaminants is expressed as a nonlinear relation of the inlet concentrations. 10. Conclusions This paper has addressed the optimum design of distributed wastewater networks where multicomponent streams are considered. A heuristic search procedure was proposed to find a good upper bound of the global optimum. This bound could be used in a rigorous global optimization algorithm. The basic idea of the procedure is to solve a relaxed LP (MILP) model of the original nonconvex model and to use this solution as a starting point of the NLP problem. Different objective functions are used in the relaxed problem, which leads to different optima, and therefore to different starting points for the NLP problem, which may lead to different local optimum solutions. If several solutions are obtained, the best one is selected as the upper bound of the global optimum. The case where two or more technologies can remove one or more pollutants with different costs (investment and/or operating) has also been addressed. Finally, the procedure was also applied to membrane separation modules given by nondispersive solvent
Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4047 Table A1. Treatment Process Data for Large Example removal ratio (%) process number
A
B
C
D
E
F
X XX XXX XXXX XXXXX
99 0 0 0 0
0 99 0 0 0
0 0 99 0 0
0 0 0 99 0
0 0 0 90 99
0 0 0 0 99
Table A2. Wastewater Process Data for Large Example Ci of the contaminants
a
stream number
A
B
C
D
E
F
flow rate (t/h)
1 2 3 4 5 6 7 8 9 10 11 12
1100 40 200 60 400 0 610 370 290 0 10 300
500 0 220 510 170 0 310 120 350 0 50 10
500 100 200 500 100 0 500 100 200 200 1 20
200 300 500 200 300 500 2000 300 500 500 60 150
800 910 150 780 900 140 830 950 0 100 20 120
100 200 0 100 0 0 0 300 0 0 30 270
19 7 8 6 17 12 2 24 15 1 33 6
Maximum concentration of each contaminant at the final stream is 100 ppm (Tables A3 and A4).
Table A3. Solutions of Initial NLP and Lower Bound LP Model for Large Example first NLP
lower bound LP
variables
constraints
local solution
variables
constraints
optimal solution
958
999
infeasible
1228
2799
137.4
appendix
Table A4. Solutions for Steps 2-4 for Large Example appendix
LP NLP
treat. X
treat. XX
treat. XXX
treat. XXXX
treat. XXXXX
28.94
25.94
16.67
20.24
0
192.72
201.30
191.61
191.79
198.9
extraction (NDSX) units. In this technology the concentration of the pollutants after the treatment units depends on the inlet concentration of the pollutant and on the flow rate, resulting in a nonlinear expression. Acknowledgment This work has been supported by MEC under project PF 97 and by the Computer Aided Process Design Consortium at Carnegie Mellon. Appendix A We present in this appendix a substantially larger problem for optimizing a wastewater treatment system involving 12 streams with 6 contaminants that are to be treated in 5 units. The data and results are shown in Tables A1 and A2. Maximum concentration of each contaminant at the final stream is 100 ppm (Tables A3 and A4). Literature Cited Allen, D. A.; Rosselot, K. S. Pollution Prevention for Chemical Processes; John Wiley and Sons: New York, 1997. Alva-Argaez, A. A.; Kokossis, C.; Smith, R. Wastewater Minimisation of Industrial System Using an Integrated Approach. Comput. Chem. Eng. 1998, 22 (Suppl.), 5741-5744. El-Halwagi, M. M. Synthesis of Reverse-Osmoses Networks for Waste Reduction. J. 1992, 38 (8), 1185-1197. El-Halwagi, M. M. Pollution Prevention through Process Integration; Academic Press: San Diego, 1997.
all treat.
upper bound
137.4 infeasible
191.61
El-Halwagi, M. M.; Manousiouthakis, V. Synthesis of Mass Exchange Networks. AIChE J. 1989, 35, 1233-1244. El-Halwagi, M. M.; Hamad, A. A.; Garrison, G. W.; Synthesis of Waste Interception and Allocation Networks. AIChE J. 1996, 42 (11), 3087-3101. Freeman, H. M.; Harris, E. F. Hazardous Waste Remediation. Innovative Treatment technologies; Technomic Publishing Co.: Lancaster, PA, 1995. Galan, B.; Alonso, A. I.; Urtiaga, A. M.; Irabien, A.; Ortiz, M. I. Cleanup of plating effluents using hollow fiber modules. 4th Conf. Environ. Sci. Technol. 1994, 2, 59-68. Grossmann, I. E., Ed. Global Optimization in Engineering Design; Kluwer Academic Publishers: Amsterdam, The Netherlands, 1996. Gupta, A.; Manousiouthakis, V. Waste Reduction through Multicomponent Mass Exchange Network Synthesis. Comput. Chem. Eng. 1994, 18 (Suppl), S585-S590. Kuo, W. C. J.; Smith, R. Effluent Treatment Systems Design. Chem. Eng. Sci. 1997, 52 (23), 4273-4290. Mahalec, V.; Motard, R. L. Evolutionary Search for an Optimal Limiting Process Flowsheet. Comput. Chem. Eng. 1977, 13, 149. McCormick, G. P. Computability of Global Solutions to Factorable Nonconvex ProgramssPart Isconvex Underestimating Problems. Math. Prog. 1976, 10, 146-175. McLaughlin, L. A.; McLaughlin, H. S.; Groff, K. A. Develop an Effective Wastewater Treatment Strategy. Chem. Eng. Prog. 1992, Sept, 34-42. Metcalf and Eddy, Inc. Wastewater Engineering. Treatment, Disposal and Reuse, 3rd ed.; McGraw-Hill: New York, 1991. Mishra, P. N.; Fan, L. T.; Erickson, L. E. Application of Mathematical Optimization Techniques in Computer Aided Design of Wastewater Treatment. AIChE Symp. Ser. 1974, 145 (17), 136-153.
4048 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Ortiz, M. I.; Galan, B.; Irabien, J. A. Kinetic Analysis of the Simultaneous Nondispersive Extraction and Back-Extraction of Chromium (VI). Ind. Eng. Chem. Res. 1996, 35, 13691377. Papalexandri, K. P.; Pistikopoulos, E. N.; Floudas, C. A. Mass Exchange Networks for Waste Minimization: A Simultaneous Approach. Trans. I.Chem.E. Part A 1994, 72, 279-294. Quesada, I.; Grossmann, I. E. Global Optimization of Bilinear Process Networks with Multicomponent Flows. Comput. Chem. Eng. 1995, 19 (12), 1219-1242. Sherali, H. D.; Alameddine, A. A New Reformulation Linearization Technique for Bilinear Programming Problems. J. Global Optim. 1992, 2, 379-410.
Takama, N.; Kuriyama, Y.; Shiroko, K.; Umeda, T. Optimal Allocation in Petroleum Refinery. Comput. Chem. Eng. 1980, 4, 251-258. Wang, Y. P.; Smith, R. Design of Distributed Effluent Treatment Systems. Chem. Eng. Sci. 1994, 49 (18), 3127-3145. Zamora, J. M.; Grossmann, I. E. Continuous Global Optimization of Structured Process System Models. To appear in Comput. Chem. Eng. 1998.
Received for review March 2, 1998 Revised manuscript received July 7, 1998 Accepted July 8, 1998 IE980133H
