Ind. Eng. Chem. Res. 2004, 43, 2175-2191
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Superstructure Decomposition and Parametric Optimization Approach for the Synthesis of Distributed Wastewater Treatment Networks Rogelio Herna´ ndez-Sua´ rez,†,‡ Julia´ n Castellanos-Ferna´ ndez,‡ and Juan M. Zamora*,† Department of Process Engineering, Universidad Auto´ noma Metropolitana-Iztapalapa, 09340 Mexico City, Mexico, and Direccio´ n Ejecutiva Te´ cnica, Instituto Mexicano del Petro´ leo, 07730 Mexico City, Mexico
Simultaneous design techniques for the synthesis of distributed wastewater treatment networks rely on the solution of nonconvex mathematical models, which give rise to multiple suboptimal solutions and often cause standard local optimization techniques to fail. A superstructure decomposition and parametric optimization approach is presented in this paper for the synthesis of distributed wastewater treatment networks with no stream recycles or recirculations. Within the developed methodology, a typical complex network superstructure for simultaneous design is decomposed into a set of basic network superstructures, which partitions the design search space. The best treatment network design embedded in each of the basic network superstructures is determined by solving a set of linear programming problems that is generated from a structured nonconvex mathematical model by fixing a small number of key problem variables. Under the most generally accepted assumptions, the systematic exploration of the parametric space defined by the key problem variables renders, from the solution space spanned by the basic network superstructures, a most certainly globally optimal network design. Indeed, the best network design of a problem involving one treatment unit can be obtained by solving a single linear programming problem. The best designs of systems containing two and three treatment units can be obtained by carrying out the parametric optimization over only one and three key problem variables, respectively. The application of the proposed methodology is illustrated with the detailed solution of several design problems. 1. Introduction Water is a resource that is used intensively for many different purposes in industry. The ordinary contamination of water used in the processing industry generates great volumes of wastewaters that discharge conventional [e.g., biochemical oxygen demand (BOD), total suspended solids (TSS), pH, and oil and grease], nonconventional [e.g., ammonia, chemical oxygen demand (COD), chlorine, fluorides), and toxic (e.g., acrylonitrile, benzene, carbon tetrachloride, chloroform, phenol, lead, toluene) pollutants into different water disposal sites, with a corresponding impact on the environment. An effective effluent treatment system is essential and can significantly influence the performance of a processing industry. The main task of the effluent treatment system is to mitigate the impact of pollutant discharge over the environment. Although the typical approach used in industry is a sequential treatment arrangement in two or three stages, the so-called primary, secondary, and tertiary treatments,1,2 the specific contaminants present in effluent streams might require specific treatments. Also, the concentrations of toxic, conventional, and nonconventional pollutants might be very different in the various effluent points, especially in industrial complexes. Additionally, wastewater treatment costs depend heavily on the type of contaminants present in the effluent streams and on the amount of wastewater that is treated.3,4 * To whom correspondence should be addressed. E-mail:
[email protected]. † Universidad Auto´noma Metropolitana-Iztapalapa. ‡ Instituto Mexicano del Petro´leo.
The different costs between particular treatments, and the differences in the concentrations of contaminants in the effluent streams, call for sewage segregation and a decentralized treatment system so that the volumes to be treated can be reduced, particularly in the more expensive treatment processes. These and other significant advantages that segregated treatment of wastewater streams could have over the most common centralized treatment, in which large volumes of wastewater with low concentrations of contaminants have to be processed, have been stressed by a number of authors.5,6 Wastewater treatment costs are also influenced by discharge standards. These standards are the result of regulations based on the water quality of the receiving stream, and they include toxicity-based limits.6-11 The main trend in wastewater regulations is to set the most stringent limits based on the performance of the best available technologies, particularly for new plants that have the opportunity to install the most efficient processes to reduce discharges and stateof-the-art treatment technologies at the time of startup. The increasing severity in water quality specifications and the need for water conservation have brought new attention to the synthesis and optimization of distributed wastewater treatment systems and integrated water management networks.12-54 In 1989, El-Halwagi and Manousiouthakis55 introduced the problem of synthesizing mass exchange networks (MENs), in which certain species from a set of rich streams must be transferred to a set of lean streams. Paralleling concepts and techniques previously developed for the synthesis
10.1021/ie030389+ CCC: $27.50 © 2004 American Chemical Society Published on Web 03/31/2004
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of heat exchanger networks (e.g., Linnhoff and Hindmarch,56 Linnhoff57), El-Halwagi and Manousiouthakis defined composite curves to represent the mass exchange operations and to determine targets for the maximum extent of mass exchange among process streams and minimum usage of external lean streams. Several limitations of the MEN approach when applied to the synthesis of freshwater and wastewater minimization methods have been pointed out in the literature (Dhole et al.,19 El-Halwagi and Spriggs,58 Huang et al.,32 El-Halwagi et al.53). Nevertheless, the use of the key ideas behind the composite curves introduced by ElHalwagi and Manousiouthakis gave rise to a variation of MENs known as water pinch, which has been in development by several authors, starting from the initial work by Wang and Smith.16 The water pinch approach now includes several methodologies for the synthesis of both water networks and distributed effluent treatment networks. For a summary of advances in the systematic solution of these two problems, the reader is referred to the review by Bagajewicz.41 In particular, the water pinch approach for the synthesis of distributed effluent treatment systems (Wang and Smith,16 Kuo and Smith21) generates distributed effluent treatment designs in several stages in a sequential manner. A network configuration is produced and evolved in several design steps that require the determination of targets for the minimum effluent treatment flow rates through the treatment plants available for the removal of contaminants. The targeting task is performed graphically with the aid of composite curves that relate the concentration of each contaminant in the wastewater streams with the corresponding mass load of contaminant that needs to be removed and operating lines that describe the removal of contaminants in the treatment plants. Using the composite curves for the individual contaminants and the determined flow rate targets, the design methodology for distributed wastewater treatment networks produces independent treatment subnetworks for the removal of each of the contaminants present in the wastewater streams. By merging the resulting treatment subnetworks, an initial total network design is obtained. This initial total network design is then evolved and optimized to determine the final network design. The design technique proposed by Wang and Smith16 and Kuo and Smith21 is very useful and insightful; nevertheless, several of its drawbacks have to be pointed out. Because of the graphical nature of the targeting step, this technique is able to manage only simple design constraints. Also, the minimum total flow rate target determined in the solution of multiple-unit, multicontaminant problems is not rigorous. Furthermore, to have a clean generation and merging of subnetworks in distributed systems involving two or more treatment units capable of removing a given contaminant, decisions have to be made with respect to the mass load distribution of the contaminant and the sequencing of treatment units. Alternatively, the sequencing of units is given, and maximum inter-unit concentration levels for key contaminants must be assumed for the design technique to be effective. These assumptions might drive the design procedure toward suboptimal network designs. Simultaneous mathematical programming approaches for the design of distributed effluent treatment systems, or for the more general synthesis problem in which the
design of water networks and the design of distributed effluent treatment systems are merged into a single integrated water management problem, have also been proposed in the literature. In their pioneering work, Takama et al.12 addressed the optimal allocation of water in a petroleum refinery within a nonconvex NLP (nonlinear programming) framework based on a network superstructure that embeds a high connectivity for wastewater reuse and all possible treatment process arrangements for the wastewater streams. Takama et al. utilized the complex method to develop optimal network designs. In 1998, Galan and Grossmann26 developed nonconvex NLP and MINLP (mixed-integer nonlinear programming) models for the design of distributed wastewater treatment plants utilizing the network superstructure presented by Wang and Smith.16 On the basis of the solution of a set of relaxed LP (linear programming) models, which provide initialization points for the solution of the original NLP problem, these authors proposed a multistart heuristic procedure for the global optimization of the developed nonconvex model. Alva-Arga´ez et al.27 proposed an integrated methodology for the design of industrial water systems that incorporates insights from the water pinch approach within a mathematical programming modeling framework. The network superstructure utilized by Alva-Arga´ez et al. includes all possibilities for water reuse, regeneration, recycling, and treatment. A feasible solution of the associated nonconvex MINLP model is obtained by decomposing the original MINLP model into a sequence of relaxed MILP (mixed-integer linear programming) problems, in which the objective function is augmented with an increasing penalty term that pursues a reduction of the problem infeasibilities. In 1999, Huang et al.32 utilized an NLP approach for the synthesis of integrated water usage and distributed wastewater treatment networks. In their design approach, which is similar to the one presented by Galan and Grossmann26 for the design of distributed wastewater treatment networks, a modified version of the superstructure proposed by Takama et al.12 is extended by including the possibility of multiple water sources and sinks, water losses, and repeated water treatment units. The resulting NLP model in the work by Huang et al.32 is solved using initial feasible points generated through the water pinch methodology or by solving a nonlinear system of equations that result from fixing, in the developed NLP model, several key design variables at “reasonable” levels. In 2000, Benko et al.37 proposed the cover-and-eliminate NLP approach for the synthesis of integrated water management networks. In their approach, Benko et al. cover all candidate system alternatives by including them in a network superstructure. Inferior network designs are then eliminated from consideration in a recursive manner by interpreting zero or near-zero results from the optimal solution of the NLP program and dropping the corresponding units or streams from the superstructure before the solution of a new NLP problem is carried out. The superstructure optimization process stops when no more structural features are eliminated. In spite of the contributions described above, the development of globally optimal designs for distributed wastewater treatment networks or integrated water management networks remains a challenging problem. The basic idea behind the simultaneous synthesis approaches is fundamentally the same: the optimiza-
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tion of an NLP or MINLP model associated with a network superstructure removes its redundant features and produces a final network design. These simultaneous design techniques allow for the automation of the design procedure, but because of the nonconvex nature of the mathematical models involved, suboptimal solutions or nonoptimal stationary points might be obtained when standard (local) optimization techniques are utilized. Moreover, linearizations of the nonconvex constraints of feasible design problems can define infeasible regions or produce indefinite Hessian matrices that often cause the failure of standard local optimization techniques.59 Other theoretical and practical issues related to the complex problem of determining a globally optimal solution for nonconvex optimization models can be found, for instance, in Horst and Pardalos,60 Horst and Tuy,61 Floudas et al.,62 and Tawarmalani and Sahinidis.63 The use of stochastic or deterministic global optimization techniques for the synthesis of distributed wastewater treatment networks, or integrated water management networks, is a relatively new matter that has emerged in only a few recent works. In 2001, Tsai and Chang46 extended the design approach utilized by Huang et al.32 for the synthesis of integrated water usage and distributed wastewater treatment networks by furnishing the network superstructure with a set of fictitious mixer units that perform no stream transformation but expand the design search space by providing additional stream mixing and splitting nodes in the network superstructure. Tsai and Chang obtained interesting results through the implementation of a genetic algorithm (e.g., Holland,64 Goldberg,65 Mitchell66) for the solution of the NLP model associated with their network superstructure. The main results include the retrofit of a water usage and treatment network in a refinery, involving three water sources, three water sinks, seven water-using operations with three contaminants, and two water treatment units. The application of rigorous global optimization techniques for the determination of the best structural and operational features of distributed wastewater treatment networks, or water networks, has been limited to the solution of small design problems that involve few streams and few contaminants. The deterministic global optimization techniques applied to this kind of problem include the branch-and-contract algorithm by Zamora and Grossmann67,68 and the recently developed global optimization algorithm for the solution of nonconvex generalized disjunctive programming problems by Lee and Grossmann.69 A superstructure decomposition and parametric optimization approach is presented in this paper to search for globally optimal designs in the synthesis of distributed wastewater treatment networks. A formal statement of the problem considered is provided in the next section. The main concept behind the proposed methodology consists of partitioning the design search region spanned by a typical complex network superstructure for simultaneous design. The partition of the design search region is accomplished by decomposing the complex network superstructure into a set of basic network superstructures that lack stream recycles and recirculations. These ideas and a structured nonconvex mathematical model for the optimization of basic network superstructures are presented in section 3, along with a model reduction scheme to generate linear
Figure 1. (a) Design of distributed wastewater treatment system. (b) Superstructure.
programming models, which are utilized within a solution procedure devised to conduct a systematic search for globally optimal network designs in the solution space spanned by the basic network superstructures. The use and advantages of the proposed methodology, as well as the derived mathematical models, are illustrated in section 4 with the detailed solution of four design problems, including the synthesis of distributed effluent treatment systems involving up to five contaminants, seven effluent streams, and three treatment units. It is shown that the proposed design methodology not only helps in determining globally optimal designs, but also allows for the robust solution of the design problem under a variety of different scenarios, enabling in some cases the identification of design regions over which a given network topology prevails over other alternative ones. Finally, in section 5, some conclusions are drawn. 2. Problem Statement The problem of distributed effluent treatment system design addressed in this paper can be stated as follows (see Figure 1). Given are a set of effluent streams I with different flow rates Si, i ∈ I, carrying a set of contaminants J with contaminant concentrations Ci,j, i ∈ I, j ∈ J. The concentrations of all of the contaminants in the effluent streams must be reduced in a set of treatment U , before the plants, K, to meet environmental limits, cj,e final discharge. Each of the available treatment plants is able to remove, to some extent, a subset of the contaminants present in the effluent streams. A matrix of constant removal ratios, Rj,k, j ∈ J, k ∈ K, specifies the mass fractions of the contaminants that the available treatment units are able to remove. It is assumed that the treatment processes operate with no loss of wastewater and that the capital and operating costs are proportional to the treated flow rate. The design task under consideration consists of determining the topology and operating conditions of the wastewater treatment network that will achieve the required removal of contaminants at minimum cost. Additional design constraints might include, for instance, the enforcement of total, partial, or no treatment at all for a subset of the
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Figure 3. Basic network superstructures involving two treatment units.
Figure 2. Typical network superstructures for the design of distributed wastewater treatment systems.
wastewater streams; the specification of minimum and maximum flow rates through particular treatment units; the specification of maximum concentrations of contaminants at the inlets of particular treatment units; and the specification of target concentrations of contaminants at the outlets of particular treatment units. 3. Proposed Design Approach 3.1. Superstructure Decomposition. Typical network superstructures for the solution of the problem stated above within a simultaneous mathematical modeling framework are presented in Figure 2 for a design problem involving three treatment units.21,26,27,32,46,67 The network superstructures shown in Figure 2 include the treatment units, stream splitters at the inlet of the system and outlet of the treatment units, mixers at the inlets of treatment units and at the outlet of the treatment system, and a rich connectivity that embeds all possible treatment arrangements and stream recirculations, many possible combinations of effluent streams being fed to the treatment units, and effluent stream bypasses to the discharge point. Although these superstructures are versatile and comprehensive, their optimization requires mathematical models that include nonconvex bilinear terms, involving, for instance, products of stream flow rates and concentrations in mass balances for contaminants at mixers27,67 or products of contaminant flow rates and split fractions in mass balances for contaminants at splitters.26 Therefore, the selection of a particular network superstructure to develop a distributed wastewater treatment network design should take into consideration the complexity of the associated mathematical model. Most likely, a greater number of different topological features incor-
porated in the superstructure leads to a mathematical model with higher complexity. Furthermore, although a structural feature such as the stream recycles shown in Figure 2b might be necessary in some particular cases to make the operation of a treatment system feasible or to lower the concentration of contaminants at the inlet of a treatment unit, its inclusion should be carefully evaluated. Otherwise, a stream recycle might constitute an unnecessary feature whose only practical consequence is to complicate the mathematical representation of the superstructure. With the purpose of conducting a systematic search to determine globally optimal designs in the synthesis of distributed wastewater treatment networks, a superstructure decomposition approach is utilized in this paper. Within the proposed approach, a typical network superstructure involving n treatment units is decomposed into a set of n! basic network superstructures that are individually optimized to obtain a set of alternative network designs from which a final network design is selected. The basic network superstructures present, as main topological features, different treatment unit arrangements and all effluent and treated streams flowing in a downstream direction. Figure 3 presents the two basic network superstructures that are obtained by decomposing a typical network superstructure involving two treatment units. An important trade-off is implied in the decomposition scheme. The optimization of the full superstructure design problem is exchanged for the optimization of a set of less complicated problems based on basic network superstructures. Also, the complexity of the network superstructure for the solution of a design problem is reduced at the expense of excluding portions of the solution space that include network designs that feature stream recirculations and recycles. Because wastewater reuse is not a concern in this work, this last consideration constitutes a reasonable simplifying assumption. 3.2. Modeling Framework. The superstructure decomposition scheme simplifies the design problem by partitioning and excluding some portions of the solution
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space. In addition, the lack of backward connectivity in a basic network superstructure decreases the mathematical complexity of the subproblems that need to be addressed. Nevertheless, these methodological elements do not suffice to prevent the use of nonconvex, complicating terms and variables in the associated mathematical models for the synthesis of distributed wastewater treatment networks. Another step toward the reduction of the number of complicating terms and variables is accomplished through the development of the following structured mathematical model for the optimization of a basic network superstructure involving n treatment units. Model BNS-n Indices i ) effluent stream j ) stream contaminant k, l ) treatment unit e ) discharge point Sets I ) {i : i is an effluent stream} J ) {j : j is a contaminant} K ) {k : k is a treatment unit} Parameters Ci,j ) concentration of contaminant in effluent stream (ppm) U ) maximum concentration of contaminant alcj,e lowed at discharge point (ppm) CCk ) capital cost coefficient COk ) operating cost coefficient Rj,k ) contaminant removal ratio Si ) effluent stream flow rate (t/h) Fe ) total effluent flow rate at discharge point (t/h)
Fe )
Si ∑ i∈I
t)
(1)
Model Constraints Mass balances for wastewater at the initial network splitters
i∈I
(2)
Mass balances for wastewater at the mixers preceding treatment units
∑ i∈I
fi,k +
∑ l∈K lk
Mass balance for wastewater at the final network mixer
fi,e + ∑ Rk,etk ) Fe ∑ i∈I k∈K
(5)
Mass balances and removal of contaminants in the treatment units
∑ i∈I
fi,kCi,j +
∑ l∈K
103∆mj,l
Rl,k(1 - Rj,l)
l
