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Ind. Eng. Chem. Res. 2005, 44, 588-599
Automated Design of Total Water Systems M. Gunaratnam, A. Alva-Arga´ ez, A. Kokossis, J.-K. Kim, and R. Smith* Department of Process Integration, UMIST, Manchester M60 1QD, U.K.
An automated method for the design of total water systems is developed in this paper. This approach considers simultaneously the optimal distribution of water to satisfy process demands and optimal treatment of effluent streams. Treatment can be for discharge to the environment or for regeneration of wastewater. The cases of regeneration reuse and regeneration recycle can be distinguished in the approach. It combines engineering insights with mathematical programming tools based on a superstructure model that results in a mixed-integer nonlinear programming problem. The approach features a fast and robust solution strategy. Complex tradeoffs involving operating, as well as capital, costs and other practical constraints have been included. In particular, piping and sewer costs, which are a major element in the capital cost of such systems, can be included. Network complexity is controlled by specifying the minimum permissible flow rates in the network, the maximum number of streams allowed at mixing junctions, and the inclusion of piping costs in the problem formulation. In addition to being able to solve the problem of total water system design, the approach is capable of designing water-using systems and effluent treatment systems when considered individually. Case studies are used to demonstrate the method. 1. Introduction
2. Previous Work
Escalating freshwater and effluent treatment costs are creating increased incentives to use an integrated approach for designing complete water networks. Such a methodology, which integrates water-using operations and water-treating systems within a single network, is termed the total water system design. The network can facilitate simultaneously the optimal distribution of water resources to satisfy process demands and optimal treatment of effluent to comply with environmental regulations at minimum total cost. Treatment can be for discharge to the environment or for regeneration of wastewater. The implementation of water reuse or recycle options and related regeneration opportunities can improve the efficiency of the network configurations. However, there are many cases when it is important (and it is sometimes crucial) for the design to not feature recycling. For example, in a food processing environment, reuse is often acceptable, but recycling must not be featured because of the potential to build up contamination from the growth of microorganisms in the recycle. In other cases, for example, traces of byproducts of corrosion can contaminate catalysts in the recycle. The design of a water network depends on the type of process operations and its requirements. Accurate design and implementation require details related to the capacity of operations and treatment systems, freshwater quality and quantity required, contaminants present, possibilities of reusing water, and quality and quantity of wastewater streams. The design procedure consists of a nonlinear optimization problem having a mixture of continuous and discrete decision variables. The presence of many interacting variables, constraints, and nonlinear terms increases the difficulties associated with solving the problem.
Takama and co-workers1 concluded that it was possible to reduce the large quantities of freshwater used and wastewater generated by industrial processes by considering the entire water network as a total system. The authors used mathematical programming to optimize a superstructure of the system and remove irrelevant and uneconomic connections. The model was transformed into a series of problems without inequality constraints by employing a penalty function and was solved using their “complex” method. However, the proposed superstructure was restricted and incorporated a centralized treatment system. More recently, the research effort in the water system design has explored the options of reusing water and wastewater in processes to facilitate improved water network designs. When both the water-using operations and effluent treatment systems have been examined, most of the investigations considered the two systems in isolation. Wang and Smith2 considered the problem of water minimization by maximum reuse using a graphical approach for targeting and a manual approach to design. Doyle and Smith3 proposed to solve the multicomponent version of the problem from the optimization of a superstructure by combining linear programming (LP) and nonlinear programming (NLP) optimization in an iterative procedure. Wang and Smith4 and Kuo and Smith5 developed manual approaches to the design of distributed effluent treatment systems. Galan and Grossman6 introduced an approach to the design of distributed effluent treatment systems based on the optimization of a superstructure. Rather than use treatment processes for discharge, they can be used for the regeneration of wastewater for further reuse or recycling. Regeneration reduces the volume of freshwater and wastewater and takes up part of the final effluent treatment load. Wang and Smith4
* To whom correspondence should be addressed. Tel.: +44 (0)161 200 4382. Fax: +44 (0)161 236 7439. E-mail:
[email protected].
10.1021/ie040092r CCC: $30.25 © 2005 American Chemical Society Published on Web 12/30/2004
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and Kuo and Smith7 explored graphical methods for regeneration of wastewater. The distinction was introduced between regeneration reuse and regeneration recycling. In regeneration reuse, the same water should only be allowed to enter the same operation once. In regeneration recycling, the same water is allowed to enter the same operation many times. It is important to distinguish between the two cases because regeneration recycling can in some cases create problems through the buildup of contaminants in the recycle not removed by the regeneration process. For total water systems, Kuo and Smith8 developed graphical techniques to explore the interactions between water reuse and effluent treatment system design. Later, Alva-Arga´ez9 introduced an approach to the total water system design based on the optimization of a superstructure representation formulated as a mixedinteger nonlinear programming (MINLP) problem. The problem was decomposed into two related subproblems. A feasible solution was found by solving the two relaxed problems sequentially within an iterative procedure. It involved a relaxation technique similar to that used by Doyle and Smith.3 However, in this case an augmented penalty function was employed to ensure feasibility. It was acknowledged that at times the procedure resulted in a sequence of infeasible problems. Benko et al.10 and Huang et al.11 presented MINLP or NLP models to solve the same problem. Jo¨dicke et al.12 presented a mixed-integer linear programming (MILP) model in which the reusability of water is exploited from the concept of the connectivity matrix. Ullmer et al.13 developed a MINLP optimization framework for water systems that incorporated heuristic rules. Although the research effort in this area has increasingly focused on mathematical programming methods, the solutions will be assisted by exploiting insights from conceptual approaches in order to simplify solution procedures. 3. Research Objective This paper describes the development of an automated strategy for designing complete industrial water networks. The approach considers simultaneously the optimal distribution of water, as well as the treatment of wastewater streams, such that process constraints, water demands, and environmental discharge limits are met at minimum cost. The water treatment can, in principle, be for discharge to the environment for regeneration of wastewater to allow further reuse or recycling of wastewater. Further, for the case of regeneration, the method should be capable of distinguishing between regeneration reuse and regeneration recycling. The present work is based on the reformulation of the model introduced by Alva-Arga´ez9 to overcome previous drawbacks with the aim of developing a more robust strategy. 4. Design Problem A total water system consists of a number of waterusing and water-treating systems. Each water-using operation requires a different quality and quantity of water to meet its process needs. Multiple sources of freshwater might be available with different qualities and costs. Effluent streams from water-using operations can be sent to treatment systems and treated water discharged to the environment or reused or recycled.
Figure 1. Design problem of a total water system.
The distinction should be made between regeneration reuse and regeneration recycling. Expressions describing the performance of the treatment systems are necessary. At the conceptual design stage, these are normally specified by simply an outlet concentration or a removal ratio. The two most significant elements in the capital cost of water-using networks are the capital cost of treatment and that for pipework and sewers to transport the water and wastewater.9 The cost of pipework and sewers will be included in the optimization. This requires the geographic distances between sources and sinks to be specified. However, it should be noted that these distances should not be the shortest distances between two points but should consider the practical issues of routing pipes along pipe racks, pipe trenches, and pipe bridges. Similarly, routing new sewers should take into account the practical restrictions for their routing. For retrofit, existing pipes and sewers are given zero capital cost. It is also necessary to specify environmental discharge levels. Figure 1 illustrates the design problem. 5. Combined Superstructure Approach A superstructure approach that considers the waterusing and water-treating subsystems simultaneously has been suggested by Alva-Arga´ez.9 Figure 2 illustrates such a superstructure through a representation consisting of two water sources, two water-using operations, two effluent discharge streams, and one effluent treatment process. This superstructure includes all of the interactions and possible connections between the waterusing and water-treating subsystems, as well as those between the process operations, freshwater sources, and wastewater discharges. Furthermore, water reuse and recycle options and the related regeneration options are also incorporated. Complex tradeoffs that arise through the minimization of water demands and effluent treatment can be explored through this approach. Polley and Polley14 discussed the tradeoffs between water consumption, network complexity, and capital cost. As a result of such tradeoffs, the final network design often uses more than the minimum quantity of water required. The distinction between treatment and regeneration systems also becomes irrelevant. A given water reuse scheme defines the performance of a specific effluent treatment system. The principal features of the combined superstructure representation are as follows: (i) each freshwater stream that enters the network is either sent directly or split into smaller streams before being sent to the water-using operations; (ii) each operation is preceded by a mixer, which is fed by freshwater and reuse streams originating from the outlets of other water-
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Figure 2. Combined superstructure representation.
using operations, as well as treated streams from the outlets of treatment processes; (iii) each water-using operation and treatment system is followed by a splitter that feeds the final mixer before discharge, as well as other water-using operations and treatment processes; (iv) effluent streams from each process are mixed at a final discharge mixer to ensure compliance with the environmental discharge limits. 6. Design Methodology The automated design methodology is based on the optimization of a combined superstructure representation that considers all of the possible design options for a complete water network. This representation is mathematically modeled, and its formulation results in a MINLP problem, which is solved by utilizing an optimization strategy. Physical insights gained through conceptual approaches2,4,5,7,815 and aspects of mathematical programming provide the foundation to the development of the solution strategy for solving the design problem. Relevant constraints are added to the mathematical formulations to enable the solution strategy to locate a feasible design configuration based on the available data and limitations. This results in a nonconvex nonlinear optimization problem. A two-stage optimization is used involving MILP in the first stage to initialize the problem. For this first stage, a decomposition strategy divides the problem into MILP and LP problems. This strategy is based on the physical nature of the problem, whereby depending on preset water demands, uneconomic features or connections are removed sequentially. The approach exploits the rewards of water reuse or recycle options, and the associated regeneration schemes, to explore different design outlines, as well as to locate an optimal design. In the second stage, the design is fine-tuned using MINLP. The synthesis objective examines related cost tradeoffs between freshwater consumption, effluent treatment, and investment costs. Therefore, it is plausible to obtain designs that feature cost-efficient schemes rather than those that present minimum freshwater consumption options. The capital cost includes piping costs. The approximate length of the pipe can be specified for each possible connection, together with the materials of construction. The automated methodology also allows control of network complexities, as well as the exploration of the interactions within the system.
The nonlinear terms in the formulation are due to bilinear terms that appear in the mass balance equations involving the mixers and splitters within the superstructure representation, as well as in the cost functions involving power terms. Engineering insights gained through conceptual approaches allow projection of the bilinear terms by following a recursive procedure. Binary variables are introduced to account for the existence or nonexistence of different structural features. The variables are useful for specifying many practical constraints such as geographical constraints, flow-rate restrictions, and forbidden or compulsory matches, which address issues related to network complexity. By contrast, the continuous variables determine the optimal values of continuous design and operating parameters, such as water flow rates, as well as pollutant concentrations. Assumptions that have been made are as follows: (i) water-using operations are described by maximum and minimum concentration levels at the inlet and outlet, as well as in terms of either pollutant mass load or the limiting water flow rate;2 (ii) the number of water-using operations and water-treating systems is fixed; (iii) the mass load picked up in each water-using operation is constant (the formulation is readily adapted to problems in which the outlet concentration is fixed; see work by Doyle and Smith3); (iv) the water flow rate through all of the operations is constant (but a water loss can be allowed at the inlet or a water gain at the outlet from an operation or treatment process); (v) treatment units are expressed through fixed pollutant removal ratios (the formulation is readily adapted to a fixed outlet concentration); (vi) the cost of effluent treatment is a function of the effluent flow rate; (vii) the influence of thermal and pressure effects is negligible. 7. Problem Formulation A set of constraints formulated on the basis of the illustration of the design problem is included in the design and optimization model. The constraints are devised using continuous and integer variables. The sets are defined as follows: Sets C ) {c|c is a contaminant present in the water}, c ) 1, 2, ..., NC S ) {s|s is a freshwater source available}, s ) 1, 2, ..., NS E ) {e|e is a wastewater discharge point}, e ) 1, 2, ..., NE
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 591 U ) {u|u is a process operation within the water network}, u ) 1, 2, ..., NU WU ) {wu|wu is a water-using operation within the water network}, wu ) 1, 2, ..., NWU TU ) {tu|tu is a treatment unit within the water network}, tu ) 1, 2, ..., NTU WU . TU ) U Decision Variables (i) Continuous variables associated with flow rates w Fs,u ) freshwater flow from a freshwater source s ∈ S to operation u ∈ U Ftu ) total flow through operation u ∈ U out ) flow from operation u ∈ U to discharge point e ∈ E Fu,e ua Fu,ua ) flow from operation u ∈ U to operation ua ∈ U
(ii) Contaminant concentration and mass flow in process operations out Cc,u ) concentration c ∈ C in streams leaving operation u ∈U in Mc,u ) mass flow entering operation u ∈ U out Mc,u ) mass flow leaving operation u ∈ U loss Mc,u ) mass flow loss from operation u ∈ U gain Mc,u ) mass flow gain in operation u ∈ U
(iii) Cross-sectional area of the pipes connecting different operations fw As,u ) pipe connections between freshwater source s ∈ S and operation u ∈ U ua Au,ua ) pipe connections between operation u ∈ U and operation ua ∈ U out ) pipe connections between operation u ∈ U and Au,e discharge point e ∈ E
(iv) Cost terms in the objective function Costfw s ) cost of freshwater supply s ∈ S Costtu u ) cost of water treatment tu ∈ TU fw,pipe Costs,u ) piping cost from freshwater source s ∈ S to operation u ∈ U ua,pipe Costu,ua ) piping cost from operation u ∈ U to operation ua ∈ U out,pipe ) piping cost from operation u ∈ U to discharge Costu,e point e ∈ E Ocost ) total annualized cost (v) Binary variables related to existence and/or nonexistence of connections fw Bs,u ) stream from freshwater source s ∈ S to operation u ∈U ua Bu,ua ) stream from operation u ∈ U to operation ua ∈ U out Bu,e ) stream from operation u ∈ U to discharge point e ∈ E G1 Bu,ua ) operation u ∈ U belongs to group 1 with respect to treatment tu ∈ TU G2 Bu,ua ) operation u ∈ U belongs to group 2 with respect to treatment tu ∈ TU
Parameters (i) Concentration bounds in,max Cc,u ) maximum inlet concentration c ∈ C to operation u∈U out,max Cc,u ) maximum outlet concentration c ∈ C from operation u ∈ U
(ii) Distances between water sources and sinks fw ) distance between freshwater source s ∈ S and ds,u operation u ∈ U ua du,ua ) distance between operation u ∈ U and operation ua ∈ U out du,e ) distance between operation u ∈ U and discharge point e ∈ E
(iii) Flow velocities within the pipe connections fw vs,u ) velocity in pipes between freshwater source s ∈ S and operation u ∈ U ua vu,ua ) velocity in pipes between operation u ∈ U and operation ua ∈ U out vu,e ) velocity in pipes between operation u ∈ U and discharge point e ∈ E
(iv) Regression parameters for piping costs fw as,u ) freshwater flow from source s ∈ S to operation u ∈ U fw bs,u ) freshwater flow from source s ∈ S to operation u ∈ U ua ) flow from operation u ∈ U to operation ua ∈ U au,ua ua bu,ua ) flow from operation u ∈ U to operation ua ∈ U out ) flow through operation u ∈ U to discharge point e ∈ au,e E out au,e ) flow through operation u ∈ U to discharge point e ∈ E
(v) Other parameters fw Cc,s ) concentration of contaminant c ∈ C in freshwater source s ∈ S ) environmental discharge limit on contaminant c ∈ Cenv c C loss Cc,u ) concentration c ∈ C at which the loss occurs in operation u ∈ U ) water flow-rate loss from operation u ∈ U Floss u ml Lc,u ) limiting mass load of contaminant c ∈ C from operation wu ∈ WU RRc,u ) removal ratio of contaminant c ∈ C in treatment tu ∈ TU (0 e RRc,u < 1) con,k Cc,u ) fixed concentration c in streams leaving operation u∈U fw,k ) fixed concentration in freshwater source s ∈ S to Cs,u operation u ∈ U fua,k Cu,ua ) fixed concentration in streams from operation u ∈ U to operation ua ∈ U COfw s ) cost per unit volume of freshwater from source s ∈ S ) scalar value for the maximum number of sources NSmax u into operation u∈ U Ft,max ) maximum water flow rate in the operation u ∈ U u ) minimum water flow rate in the operation u ∈ U Ft,min u
Equations and Constraints. The problem involves a nonlinear objective function, together with linear and nonlinear constraints. It consists of mass balances around the operations, mixers, and splitters present in the superstructure. The objective function determines the appropriate design solution to the problem. Expressions related to capacity constraints, environmental limits, design equations, cost calculations, and logical statements are also included. The constraints and equations must reflect the total mass and component balances that constitute the process constraints, as well as design specifications, restrictions, feasibility constraints, and logical constraints. A schematic representation of a basic operation and its formulation is illustrated in Figure 3.
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specified small matches. These constraints may require additional freshwater as a penalty for the network simplifications. More sophisticated constraints involving binary variables can also be introduced such as threshold flow-rate constraints or a maximum number of inlet connections for an operation throughout the network.
(i) Upper and lower bounds on the flow rates:
Figure 3. Schematic representation of a basic operation.
(A) Balances around Operations, Mixers, and Splitters.
(i) Overall balance around the entire water system: w out - ∑ ∑ Fu,e ) ∑ Floss ∑ ∑ Fs,u u s∈S u∈S e∈E u∈U u∈U
∑
ua Fua,u
ua∈U
+
∑
ua Fu,ua s∈S
-
∑
out Fu,e
)
(1)
Floss u
(2)
u∈U
(ii) Contaminant mass balance for each operation:
∑
ua (Fua,u
out Cc,ua )
out Cc,u [
∑
fw Cc,s )
-
in Mc,u
)0
(3)
w ua out Fs,u + ∑ Fua,u - Floss ∑ u ] - Mc,u ) 0 s∈S ua∈U
(4)
ua∈U
+
w (Fs,u
s∈S
in out ml loss Mc,u - Mc,u + Lc,u - Floss Cc,u )0 u
∀ u ∈ WU (5)
in loss out (1 - RRc,u)Mc,u - Floss Cc,u - Mc,u ) 0 ∀ u ∈ TU u (6)
(i) Constraints of water flow rate: Ft,min e Ftu u
(7)
g Ftu Ft,max u
(8)
where w ua Fs,u + ∑ Fua,u - Floss ∑ u ] ) 0 s∈S ua∈U
(ii) Constraints on the quality and quantity of water: in in,max t Mc,u e Cc,u Fu
(9)
Equation 9 is significant in order to enforce water quality specifications for each process operation and ensure that a suitable quality of water is supplied. Water losses (Floss u ) have been assumed to be constant and known. It has been assumed that these flow losses only occur at the inlet of the process operation before the water enters the operations. (iii) Constraint on the environmental discharge limit of contaminants:
∑
u∈U
out (Cc,u
out Fu,e )
e
Cenv c
∑
out Fu,e
(11)
w fw - Ls,uBs,u g0 Fs,u
(12)
out out out - Uu,e Bu,e e 0 Fu,e
(13)
out out out - Lu,e Bu,e g 0 Fu,e
(14)
ua ua ua - Uu,ua Bu,ua e0 Fu,ua
(15)
ua ua ua - Lu,ua Bu,ua g0 Fu,ua
(16)
out ua where Ufw s,u, Uu,e , and Uu,ua ) upper bounds for the out ua water flow in the connection and Lfw s,u, Lu,e , and Lu,ua ) scalar defining the minimum allowable flow in the connection. Constraints (11)-(16) are introduced to correlate the binary variables with the continuous variables in order to represent the upper and lower limits to the water flow rates. Restricting water flow rates to be above a minimum threshold is a significant feature in controlling network complexity.
(ii) Maximum number of sources to feed each operation:
(B) Availability and Capacity Constraints.
Ftu - [
w fw fw Fs,u - Us,u Bs,u e0
(10)
u∈U
Constraint (10) guarantees that environmental discharge limits are met. (C) Logic Constraints. Uneconomically small flow rates between operations can be eliminated by the addition of supplementary constraints that remove
ua fw Bua,u + ∑Bs,u e NSmax ∑ u u∈U s∈S
(17)
in constraint (17) represents the The parameter NSmax u maximum number of water sources, which includes both freshwater and reuse water, that can be fed to an operation. Such a restriction provides an additional constraint in the design of water networks and is a significant parameter in controlling network complexity. (iii) Elimination of regeneration recycling: ua G1 Bu,tu - Bu,tu e0
(18)
ua G2 Btu,u - Bu,tu e0
(19)
G2 G1 ) 1 - Bu,tu Bu,tu
(20)
G2 G2 ua + Bua,tu ) g Bu,ua 2 - (Bu,tu
(21)
G1 G2 + ∑Bu,tu ) NOP - 1 ∑u Bu,tu u
(22)
where NOP ) number of operations in the network. The elimination of the water recycle stream connections from the superstructure representation during the optimization is a complex task because the identity of the connecting water streams is undefined when the location of a mixer or splitter is determined within the network. The design of a total water system involves the above complex features and requirements because the abandonment of these characteristics has a signifi-
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cant impact on the design configurations to meeting practical designs. A grouping strategy initially introduced by Kuo17 and Kuo and Smith8 has been developed to ensure the elimination of recycle streams within the network. The principal consideration is to separate all of the operations into two groups, groups 1 and 2. Two different sources of water are considered for supplying the water required by the water-using operations, namely, a freshwater source and a regenerated water supply. The operations in group 1 are defined as those operations that feed a treatment system directly, while operations in group 2 are defined as those being fed by that particular treatment system. This grouping scheme needs to be performed on every treatment system present in the network. Eliminating the possibilities of water reuse options from group 2 operations to those in group 1 eliminates water recycle streams in the network. Based on the grouping strategy, eq 18 enables any operation that is feeding the treatment systems to be assigned to group 1, while eq 19 assigns to group 2 any operation that is receiving regenerated water from the treatment systems. Constraint (20) is required to ensure that all of the operations present within the network belong either to group 1 or group 2 and not both. To remove all of the reuse connections from the operations in group 1 to those in group 2 and eliminate the recycle schemes between the operations in both the groups, eq 21 is introduced. Finally, eq 22 ensures that all of the water-using operations are assigned to either group 1 or group 2. Hence, upon decomposition of the regeneration problem, there are essentially two water sources available, freshwater and regenerated water, as well as two subgroups of water users; one is fed by freshwater and the other by regenerated water.
(iv) Elimination of direct recycling: ua ua Bu,ua + Bua,u e1
(23)
(D) Objective Function. The objective function represents the total annualized cost of the complete water network, taking into account costs associated with freshwater supply, water, and wastewater treatment as well as piping and sewers to link the entire network. (i) Freshwater Supply Cost. The cost of freshwater supply is expressed as follows:
Costfw s )
w COfw ∑ s Fs,u u∈U
(24)
w fw fw Fs,u ) As,u vs,u
(26)
The above expression relates piping from a freshwater source to a water-using operation. The pipe crosssectional area can be varied to consider various data or problem needs. The velocity through the pipe is generally assumed to be between 1 and 2 m‚s-1. Similarly, corresponding equations relating piping linking treatment systems and water-using operations, as well as these process operations, to the final discharge point are also generated. Thus, the piping cost is given by: fw,pipe fw fw fw fw fw ) [(as,u As,u ) + (bs,u Bs,u )]ds,u Costs,u
(27)
fw The values of afw s,u and bs,u represent the interrelated variable costs, as well as fixed costs. Both values depend on the materials of construction employed for the pipe fabrication. In the same way, the piping costs for the other connections can be estimated. (iv) Overall Objective Function. Taking into account the above cost functions and equations, the objective function is to minimize the overall annualized cost, which includes the costs associated with freshwater supply, water treatment, piping, and sewer costs:
fw,pipe Costfw + ∑ s ) + (∑ ∑ Costs,u s∈S s∈S u∈U ua,pipe out,pipe Costu,ua + ∑ ∑ Costu,e ) + ( ∑ Costtu ∑ ∑ u) u∈U ua∈U e∈E u∈U t∈TU
Ocost ) (
(28) The design and optimization of a complete water network is a MINLP problem. The nonlinear terms are due to the presence of expressions with bilinear features present in mass balance equations and power terms representing the cost functions that appear in the objective function. These terms often cause convergence failures in NLP algorithms or lead to suboptimal local solutions. 8. Solution Strategy
(ii) Water and Wastewater Treatment Cost. Effluent treatment cost is based on the assumption that its cost is proportional to the total flow of wastewater requiring treatment. As such, it is estimated as follows: tu t βu Costtu u ) COu Fu
source and sink it is connecting rather than the shortest distance. Pipes will normally follow pipe racks, pipe bridges, and pipe trenches. The cost of sewers can also be related to the flow rate of effluent, although this is less straightforward. As with piping, sewers will tend to follow directions other than the shortest distance between a source and sink. The pipe cross-sectional area is determined as follows:
(25)
(iii) Piping and Sewer Cost. The costs related to the piping are a function of the cross-sectional area of the pipes and velocity through the pipe. The approximate length of the pipe can be specified for each possible connection, together with the materials of construction of the pipe. Note that the materials of construction required might depend on the direction of the flow between two operations. The approximate length should take into account the route between the
Standard solvers have a limited potential for solving such design problems. Therefore, a more explicit solution strategy that is able to utilize the physical insights of such systems is necessary. The initial formulation of the design problem results in a MINLP scheme, which is here decomposed into two stages. The first stage consists of a MILP-LP formulation that is solved in an iterative manner to provide an initial starting point. The solution from this initialization stage is refined in the second stage to a final solution in a MINLP scheme. A systematic decomposition method is presented that exploits the concepts from the graphical approaches. The decomposition follows successive projections on the concentration space to enforce the feasibility of the relaxed constraints. The availability of a feasible starting point or an effective initialization stage can assist
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the solution strategy in locating an optimal solution without great difficulty. However, it is a challenging requirement and often depends on the problem formulation. Previous conceptual studies indicate that at least one of the contaminants must reach a maximum outlet concentration level in order to minimize freshwater use. Furthermore, if the pollutant removal ratio in effluent treatment systems is fixed, this ensures that one contaminant is treated to a maximum load for a minimum treatment flow rate. The present solution strategy involves two initial assumptions. First, it assumes initially that all of the streams exiting the water-using operations have attained their maximum allowable outlet concentration levels. Upon maximization of the difference between the outlet and inlet contaminant concentrations, the water flow rate is reduced to a minimum because the mass load through an operation is fixed in this case. The other initial assumption is that the treatment systems are performing perfectly. The concentration of the streams leaving the treatment systems is set to zero, which is the minimum outlet concentration for a perfect treatment system. These assumptions are able to secure an initial conservative starting point. However, the assumption that contaminants reach their maximum values in water use may not always hold because of the interactions between the water-using operations, and a perfect treatment system does not exist. Therefore, these assumptions may contribute to highly nonoptimal initial points. The present formulation results in a nonconvex nonlinear design problem that exhibits local minima and/or has convergence difficulties, rendering the application of standard NLP techniques ineffective. These conditions arise because of the presence of the bilinear terms in the mass balance equations, which are comprised of products of the two continuous variables of flow rate and concentration, which are both unknown. Setting the outlet concentration levels of the process operations to their maximum values removes these terms. A fixed structure of the network is obtained based on the above scheme. This aids in the elimination of uneconomic connections and options. Nevertheless, it is important to note that the feasible network obtained might still have many unnecessary interconnections. In addition, flow rates having zero values introduce numerical difficulties in the procedure to determine feasible solutions. The current approach differs from that of AlvaArga´ez,9 which used an augmented objective function with penalty terms to ensure feasibility. The weight associated with the penalty term in each iteration was increased until the procedure converged to a feasible solution. The present methodology avoids the use of augmented objective functions with penalty terms within the solution strategy. Instead, slack and surplus variables known as mass loss and mass gain terms have been included in the contaminant mass balance constraints. The sum of the mass loss and mass gain terms is to be reduced to zero to guarantee that the variations or violations in the mass balances are removed. This ensures that contaminant concentrations are within the set limits and prevents these concentration levels from exceeding allowable limits. This condition acts as a convergence criterion.
A feasibility criterion is necessary to simultaneously exclude certain subregions from further consideration while refining the search in other subregions. This decisive factor ensures that the search is confined to a smaller solution space, thereby accelerating the convergence of the algorithm. It can also ensure that the search does not progress too far into the infeasible region and prevent premature termination in a feasible, but far from an optimal, solution. MILP Formulation. The problem involves the design constraints (5) and (6) that are essential for formulating the network configuration constraints relaxed in the form in out ml loss gain loss Mc,u - Mc,u + Lc,u - Floss Cc,u + Mc,u - Mc,u )0 u (29) in loss out gain (1 - RRc,u)Mc,u - Floss Cc,u - Mc,u - Mc,u ) 0 (30) u
Constraints (3) and (4) are projected onto the contaminant concentration solution space with respect to the outlet concentration, Cout c,u , where k indicates the iteration number. ua con,k w fw in (Fua,u Cc,ua ) + ∑(Fs,u Cc,s ) - Mc,u )0 ∑ u∈U s∈S con,k [ Cc,ua
w ua out + ∑ Fua,u - Floss ∑Fs,u u ] - Mc,u ) 0 ua∈U
(31) (32)
s∈S
The MILP problem consists of eqs 1, 2, 7-9, and 1123 from the original problem formulation, as well as the above-mentioned constraints of eqs 29-32. In this problem, the concentrations of the streams leaving the water-using operations are fixed to the specified maximum outlet concentrations, while the flow rates are varied in order to satisfy these constraints. Flow rates corresponding to the fixed concentrations are then obtained. When the contaminant concentrations are maintained, the problem is rendered linear, thereby removing the nonlinear terms. Therefore, the uncoupling of the bilinear terms (flow times concentration) with the introduction of the mass flows and relaxation strategy allows the search to be performed in the space defined by the convex feasible region in the MILP model. The mass loss and mass gain terms that represent the slack and surplus variables respectively in eqs 29 and 30 account for the mass load variations in the process operations. These variables take values depending on whether the removal ratio is adequate to achieve the projected outlet contaminant concentration in relation to the performance of the treatment systems. The objective function in this scheme is to minimize the total cost, consists of linear equations, and assumes the form fw,pipe + ∑Costfws ) + (s∈S ∑ u∈U ∑ Costs,u ua,pipe out,pipe + ∑ ∑ Costu,e ) + ( ∑ Costtu ∑ ∑ Costu,ua u) u∈U ua∈U e∈E u∈U t∈TU
cost )( Omilp
s∈S
(33) The freshwater cost is assumed to be a linear function of the freshwater flow rate. The same applies to the terms describing the piping cost. The nonlinear concave cost functions corresponding to the effluent treatment systems can be linearized, together with a fixed charge
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 595
for treatment. The linearization procedure can be performed over a range of effluent flow rates. LP Formulation. The problem includes the relaxed mass balance equations around treatment operations, eqs 29 and 30. Constraints (3) and (4) are projected onto ua , Fout the flow-rate solution space with respect to Fu,ua u,e , w and Fs,u, where the iteration number is indicated by k. ua,k out w,k fw in Cc,ua ) + ∑(Fs,u Cc,s) - Mc,u )0 ∑ (Fua,u s∈S
(34)
ua,k out w,k + ∑ Fua,u - Floss ∑Fs,u u ] - Mc,u ) 0 ua∈U
(35)
ua∈U
out [ Cc,u
s∈S
The LP model consists of eqs 29 and 30, as well as constraints (33) and (35). The outlet concentrations (Cout c,u ) are varied in order to satisfy the flow-rate requirements. These values are determined by solving a set of simultaneous linear equations. However, because of the conservative assumption that all treatment systems perform perfectly and all outlet concentration levels are fixed to the maximum outlet values, an inconsistent system of equations may arise. Hence, the optimal objective value for the LP problem may be greater than zero. The LP problem is solved against an objective that minimizes the slack and surplus variables (mass loss and mass gain terms) and assumes the form:
M
terms
)
∑c ∑u
loss Mc,u
+
∑c ∑u
gain Mc,u
(36)
The flow rates from the previous stage are fixed and corresponding outlet concentrations are determined through the LP problem as the sum of mass loss and gain terms are minimized to zero. The new outlet concentrations are then fixed again in the following MILP model, and corresponding flow rates are determined. This procedure is continued until the convergence criterion is satisfied. The feasible solution is then refined further by passing it to a MINLP algorithm. Outline of Solution Strategy. The overall iterative solution procedure is explained below. Let (MILP)k and (LP)k be the formulation of (MILP) and (LP) at iteration k. The algorithm involves the following: Step 1. In the first iteration (k ) 1), the MILP ) problem (MILP)k is initially solved by setting Ccon,k c,u cost to obtain O and optimal flow-rate values for Cout,max c,u milp out* ua* the network design, Fw* s,u, Fu,e , and Fu,ua, where * indicates that the corresponding value is optimal. Step 2. The flow rates are fixed as follows: Fw,k s,u ) w* ua,k ua* Fs,u, Fout,k ) Fout* u,e u,e , and Fu,ua ) Fu,ua, when solving the LP problem (LP)k to obtain a new vector of outlet concentration levels, Cout* c,u . Step 3. Concentration levels are set as follows for the k+1 ) Cout* next iteration (k ) k + 1), Cout,k+1 c,u c,u . The (MILP) problem is solved to obtain new values of the flow rates in the network, and the iterative procedure continues until convergence. The convergence criterion is the reduction of the objective function of the LP problem to zero as:
Mterms ) 0 ((tolerance limit - near zero) All of the mass loss and mass gain terms should be minimized to zero, indicating that the associated equations, as well as constraints, are not violated. If the
Figure 4. Solution strategy.
value of Mterms is reduced to zero or if it is within the tolerance limit, then proceed with step 4. Otherwise, return to step 2. Step 4. The MINLP scheme is solved with the initial starting point available from the initialization stage (MILP-LP), whereby all of the structural features are fixed according to the earlier scheme. The approach solves the MILP-LP problems iteratively until the convergence criterion is satisfied. The solution from the MINLP refining stage results in the optimal structure of the complete water network, taking into account the operating flow rates and contaminant concentrations in each connection stream. In the event that the initial assumption of a perfect treatment performance defines an infeasible point for the original MINLP problem, the outlet concentration levels from the treatment systems can be fixed to the environmental discharge limit. This assists the initialization procedure in providing a more feasible starting point when the existing procedure is unable to cope with the problem requirements. The complete solution strategy is illustrated in Figure 4. 9. Case Studies The applicability of the methodology for designing total water networks will now be demonstrated through case studies. The optimization platform employed in this work is GAMS (General Algebraic Modeling System16), which is a modeling system where optimization models can be specified in equation form and their solutions can be obtained with different solvers. The solvers employed were OSL for LP as well as MILP and DICOPT for MINLP problems, and the case study was performed on a Pentium 3, 400 MHz PC. 9.1. Case Study 1. Consider the water network design problem for a simplified petroleum refinery.17 The problem consists of five water-using operations and
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Table 1. Water-Using Operations for Case Study 1 operation
description
operation
description
O1 O2 O3
steam stripping HDS-1 desalter
O4 O5
VDU HDS-2
three treatment units with three contaminants. The objective is to design a complete water network. Table 1 identifies the water-using subsystems. The three contaminants are hydrocarbon, hydrogen sulfide, and suspended solids. The performance of the treatment units is defined in terms of removal ratios, which have been specified for different contaminants in Table 2. Table 3 presents the operating data for the waterusing operations and contaminants. The quality and quantity of water required by the water-using operations are given in this table. The inlet and outlet contaminant concentrations, together with the limiting mass load of each operation, are also presented. The capital and operating costs are a function of the total treatment flow rate entering a treatment unit, Fttu, and are as follows:
Table 2. Performance of the Treatment Units for Case Study 1 removal ratio operation
HC
H2S
SS
T1 T2 T3
0.00 0.70 0.95
0.999 0.90 0.00
0.00 0.98 0.50
Table 3. Operating Data for the Water-Using Operations for Case Study 1 operation
contaminant
in,max Cc,u (ppm)
out,max Cc,u (ppm)
ml Lc,u (g/h)
O1
HC H2S SS HC H2S SS HC H2S SS HC H2S SS HC H2S SS
0 0 0 20 300 45 120 20 200 0 0 0 50 400 60
15 400 35 120 12500 180 220 45 9500 20 60 20 150 8000 120
750 20000 1750 3400 414800 4590 5600 1400 520800 160 480 160 800 60800 480
O2 O3 O4 O5
(i) T1: Steam-stripping column 0.7 t Capital cost: CostTU T1 ($) ) 16800Ftu (t/h)
Table 4. Distance Matrix (m) for Case Study 1
Operating cost: Cost ($/h) )
Fttu
(t/h)
(ii) T2: Biological treatment unit t 0.7 Capital cost: CostTU T2 ($) ) 12600Ftu (t/h)
Operating cost: Cost ($/h) ) 0.0067Fttu (t/h)
S1 O1 O2 O3 O4 O5 T1 T2 T3
O1
O2
O3
O4
O5
T1
T2
T3
discharge
30 0 30 80 150 400 90 150 200
25 30 0 60 100 165 100 150 150
70 80 60 0 50 75 120 90 350
50 150 100 50 0 150 250 170 400
90 400 165 75 150 0 300 120 200
200 90 100 120 250 300 0 125 80
500 150 150 90 170 120 125 0 35
600 200 150 350 400 200 80 35 0
2000 1200 1000 800 650 300 250 100 100
(iii) T3: API separator 0.7 t Capital cost: CostTU T3 ($) ) 4800Ftu (t/h)
Operating cost: Cost ($/h) ) 0 The cost for freshwater supply S1 is $0.2/t. There is only one freshwater source in this problem. It is also assumed that the plant site operates for 8600 h annually with an annualization factor of 0.1. The environmental limits for the three contaminants are as follows: (i) HC, 20 ppm; (ii) H2S, 5 ppm; (iii) SS, 100 ppm. Piping costs have also been included in the design strategy. Table 4 presents the distances linking all of the operations within the network, as well as those between the various operations and discharge points. A flow velocity of 1 m/s has been assumed in all of the pipes throughout the network. The material of construction has been assumed to be carbon steel. The parameters of the costs for the piping work are as follows: fw ua out ) au,ua ) au,e ) 3603.4 as,u fw ua out ) bu,ua ) bu,e ) 124.6 bs,u
The Chemical Engineering Equipment (CE) index is based on the December 1997 value, which is 389.1. The problem was solved by taking into account three different scenarios: (i) network design allowing recy-
cling, (ii) network design without recycling, (iii) network design without recycling and including piping costs. Scenario i. The problem was solved by allowing recycling without taking into account piping costs. Therefore, the relevant equations and constraints (18)(22) that ensure recycling is avoided were initially omitted. The objective function for this scenario was minimization of freshwater and wastewater treatment costs, initially excluding piping costs. The present approach solves the problem within five iterations. To formulate the problem, the MILP model required 511 continuous and 75 binary variables, while the LP model needed 118 continuous variables. The current methodology is capable of locating a feasible starting point without violating any constraints during its initialization process. Like the previous method, the initial point was further improved by an MINLP algorithm, which resulted in a design configuration that requires an annual total expenditure of $578217. As mentioned earlier, the option to allow water recycling may not be desirable because of concerns over contaminants building up to unacceptable levels. Scenario ii. Recycling was then not allowed, and therefore the relevant constraints were included. The objective function for this scenario involves the minimization of freshwater and wastewater treatment costs. It also excludes the cost of piping. However, the equations that maintain the grouping of operations were introduced to avoid direct and indirect recycling of water.
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Figure 5. Minimum total annualized cost design for scenario iii in case study 1.
The solution procedure converged in three iterations in comparison with the four stages required for the previous scenario. The MILP model required 511 continuous and 83 binary variables, while the LP model needed 118 continuous variables. The solutions from the optimization of the MINLP algorithm resulted in a design configuration that requires an annual total expenditure of $654245. Scenario iii. So far, piping costs have not been taken into account. The final scenario involves regeneration reuse in which piping costs were taken into account. All relevant data involving piping distances, pipe velocity, and material of construction of the pipes were included. The objective function of this scenario involved the minimization of the total annualized costs involving freshwater, wastewater treatment, and piping costs. Once the piping cost information is brought into the design procedure, the need for a simultaneous approach becomes more evident. The design obtained from this simultaneous approach minimizes the total annualized cost of the system. The equations that maintain the grouping of operations are also introduced to avoid indirect recycling. The solution strategy converged in four iterations. The MILP model required 511 continuous and 83 binary variables, while the LP model needed 118 continuous variables. The solution from the optimization resulted in a design configuration that requires an annual total expenditure of $683369. The design configuration for scenario iii is illustrated in Figure 5. The total cost of the design is also some 5% better than that obtained from a conceptual approach by Kuo17 if piping costs are left out, as Kuo17 did not consider piping costs. As mentioned earlier, no recycling appears in the network, but the regeneration reuse scheme is prominent. The network design introduces a different structure that minimizes simultaneously the piping costs. It can be concluded that decomposing the problem and making arbitrary decisions as to the manner in which the treatment units operate can result in the omission of certain cost-efficient options. The new approach is also capable of introducing constraints restricting flow rates to a minimum (e.g., a minimum flow rate in the network of 5 t/h). Furthermore, it is also able to provide designs with limits on the number of streams entering a mixing junction (e.g.,
Table 5. Limiting Water Data for Case Study 2 operation
in,max Cc,u (ppm)
out,max Cc,u (ppm)
ml Lc,u (g/h)
limiting flow rate Fml u (t/h)
O1 O2 O3 O4
0 50 50 400
100 100 800 800
2000 5000 30000 4000
20 100 40 10
no more than two streams entering a mixing junction). Such constraints, together with the inclusion of piping costs, give the designer a degree of control over network complexity. In general, the inclusion of piping costs alone leads to simpler network configurations. Without piping costs, there is no incentive for the optimization to simplify the network because the objective is to purely minimize operating costs. Inclusion of minimum threshold flow rates and constraints on mixing junction complexity adds additional control over the network complexity. 9.2. Case Study 2. The automated methodology developed for designing total water systems can also be employed for designing individual water systems. The applicability of the methodology for designing a waterusing subsystem is now addressed, and a water-treating subsystem will be addressed in case study 3. Consider a water network problem consisting of four water-using operations and a single contaminant, which has been taken from Wang and Smith.2 The objective is to find a network design that features minimum operating cost. This translates directly into the case for a design with minimum freshwater consumption. Table 5 shows the limiting water profiles as well as concentration data. A single, uncontaminated freshwater source is available (Cfw c,s ) 0). The freshwater cost is 0.75 $/t, and the annual operating hours have been assumed to be 8600 h with an annualization factor of 0.1. The base case of the problem is a network design without water reuse features (once-through use of water), which has been obtained from water pinch analysis. The minimum freshwater flow is found to be 112.5 t/h with an annual operating cost of $725625. Upon application of the current methodology, the problem was solved within a single iteration. The minimum flow rate consumed, as illustrated in the design, is only 90 t/h with an annual operating cost of only $580500. A 20% savings in terms
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Figure 6. Design with minimum freshwater cost for case study 2. Table 6. Limiting Water Data for Case Study 3
operation
contaminant
in,max Cc,u (ppm)
out,max Cc,u (ppm)
load Mc,u (g/h)
O1
H2S oil SS H2S oil SS H2S oil SS
0 0 0 0 0 0 0 0 0
390 10 25 16780 110 40 25 100 35
5109 131 327.5 548706 3597 1308 1412.5 5650 1977.5
O2 O3
limiting flow rate ml Fc,u (t/h) 13.1 32.7 56.5
Table 7. Removal Ratios of the Treatment Units for Case Study 3 treatment unit
RR(H2S)
RR(oil)
RR(SS)
T1 T2 T3 (T3E)
0.999 0.90 0
0 0.90 0.95
0 0.97 0.20
Table 8. Distance Matrix (m) for Case Study 3 distance (m) O1 O2 O3 T1 T2 T3 T3E discharge
O1
O2
O3
T1
T2
T3
T3E
discharge
0 0 0 30 150 50 100 170
0 0 0 40 25 80 50 90
0 0 0 35 120 40 100 100
30 40 35 0 80 30 60 35
150 125 120 80 0 65 140 55
50 80 40 30 65 0 15 40
100 50 100 60 140 15 0 65
170 90 100 35 55 40 65 0
of the water consumption was achieved, saving $145125. The network design is shown in Figure 6. 9.3. Case Study 3. Consider a water network problem consisting of three effluent streams and four treatment units with three contaminants, which has been taken from Wang and Smith.4 The objective is to find a network design that features minimum treatment cost by taking into account piping cost (i.e., no water reuse is considered). The three contaminants are hydrogen sulfide, oil, and suspended solids. Table 6 presents the limiting water profile data of the three effluent streams originating from three operations and shows the inlet and outlet contaminant concentrations together with the mass load of each operation. The four treatment units available are shown in Table 7, which includes a steam stripper (T1), an activated sludge system (T2), and two API separators (T3 and T3E). The distances between effluent streams and treatment units, as well as between treatment units and discharge points, are provided in Table 8. A flow velocity of 1 m/s is assumed for all pipes made from carbon steel. Other cost parameters are the same as those of case study 1, except that an annualization factor of 0.4 is
Figure 7. Minimum total annualized cost design for case study 3.
used. The environmental limits for the three contaminants are as follows: H2S, 2 ppm; oil, 2 ppm; suspended solids, 5 ppm. The solution reported by Wang and Smith4 is assumed to be the base case design configuration. The total annualized cost of the treatment units is $684280. If costs related to the pipe work are added, the total annualized cost increases to $756566. The solutions from the optimization of the MINLP algorithm using the current solution strategy resulted in a design configuration with an annualized cost of $705156, which is about 7% lower than that from water pinch analysis. This new design uses only three treatment units and is simpler as well as more cost-effective. The design configuration is illustrated in Figure 7. 10. Conclusions The automated approach developed in this paper presents improved (cost-effective) network configurations compared with those of previous approaches, while maintaining control over the complexities associated with network connections. The introduction of additional constraints through the application of binary variables also generates alternative designs. Many practical restrictions, for instance, compulsory or forbidden connections, piping costs, minimum or maximum allowable flow rates, as well as geographical, control, and safety constraints, can be included. The operating and capital costs are optimized simultaneously, allowing for the exploration of cost tradeoffs in the system. The method can handle large and complex problems without great difficulty because of the ability of the LP and MILP codes to handle a large number of variables and constraints. The approach is able to explore the synergies between the water-using and water-treating subsystems to ensure their efficient integration. It is also important to note that the methodology is capable of designing both water-using operations and water-treating systems when considered individually. Although the present design methodology provides a robust technique, it does not necessarily provide the global optimum. This automated method should be viewed as a decision support tool for screening alternatives, which can then be engineered and evaluated for implementation. The method has been tested on various examples. It presents a significant improvement in the efficiency of the solution strategy and the quality of designs. Literature Cited (1) Takama, N.; Kuriyama, T.; Shiroko, K.; Umeda, T. Optimal Water Allocation in a Petroleum Refinery. Comput. Chem. Eng. 1980, 4, 251.
Ind. Eng. Chem. Res., Vol. 44, No. 3, 2005 599 (2) Wang, Y.; Smith, R. Wastewater Minimisation. Chem. Eng. Sci. 1994, 49, 981. (3) Doyle, S.; Smith, R. Targeting Water Reuse with Multiple Contaminants. Trans. Inst. Chem. Eng. 1997, 75, Part B, 181. (4) Wang, Y.; Smith, R. Design of Distributed Effluent Treatment Systems. Chem. Eng. Sci. 1994, 49, 3127. (5) Kuo, W.; Smith, R. Effluent Treatment System Design. Chem. Eng. Sci. 1997, 52, 4273. (6) Galan, B.; Grossman, I. Optimal Design of Distributed Wastewater Treatment Networks. Ind. Eng. Chem. Res. 1998, 37, 4036. (7) Kuo, W.; Smith, R. Design of Water-Using Systems Involving Regeneration. Trans. Inst. Chem. Eng. 1998, 76, Part B, 94. (8) Kuo, W.; Smith, R. Designing for the Interactions Between Water Use and Effluent Treatment. Trans. Inst. Chem. Eng. 1998, 76, Part A, 287. (9) Alva-Arga´ez, A. Integrated Design of Water Systems. Ph.D. Thesis, UMIST, Manchester, U.K., 1999. (10) Benko, N.; Rev, E.; Szitkai, Z.; Fonyo, Z. Optimal Water Use and Treatment Allocation. Comput. Chem. Eng. 1999, 23, S589. (11) Huang, C.; Chang, C.; Ling, H.; Chang, C. A Mathematical Programming Model for Water Usage and Treatment Network Design. Ind. Eng. Chem. Res. 1999, 38, 2666.
(12) Jo¨dicke, G.; Fischer, U.; Hungerbu¨hler, K. Wastewater Reuse: A New Approach to Screen for Designs with Minimal Total Costs. Comput. Chem. Eng. 2001, 25, 203. (13) Ullmer, C.; Kunde, N.; Lassahn, A.; Gruhn, G.; Schulz, K. WADO: Water Design Optimization-Methodology and Software for the Synthesis of Process Water Systems. J. Clean. Prod. 2003, in press. (14) Polley, G.; Polley, H. Design Better Water Networkss Environmental Protection. Chem. Eng. Prog. 2000, Feb, 47. (15) Wang, Y.; Smith, R. Wastewater Minimisation with Flowrate Constraints. Trans. Inst. Chem. Eng. 1995, 73, Part A, 889. (16) Brooke, A.; Kendrick, D.; Meeraus, A. GAMSsA Users’ Guide; Scientific Press: Redwood City, CA, 1988. (17) Kuo, W. A Combined Approach to Water Minimisation and Effluent Treatment System Design. Ph.D. Thesis, UMIST, Manchester, U.K., 1996.
Received for review March 23, 2004 Revised manuscript received October 6, 2004 Accepted October 11, 2004 IE040092R
