Design of Optimal Water-Using Networks with Internal Water Mains

The most attractive networks, which have near-minimal freshwater consumption, relatively simple structures, and promising controllability, can be furt...
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Ind. Eng. Chem. Res. 2006, 45, 8413-8420

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Design of Optimal Water-Using Networks with Internal Water Mains Xuesong Zheng, Xiao Feng,* and Renjie Shen Department of Chemical Engineering, Xi’an Jiaotong UniVersity, Xi’an 710049, People’s Republic of China

Warren D. Seider Department of Chemical and Biomolecular Engineering, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104-6393

To provide more efficient usage of water resources and to minimize wastewater discharge, the design of water allocation networks has drawn increased attention over the past two decades. However, water system integration yields more structurally complex plants with greater interactions among the process units. Therefore, it is desirable to simplify the network structure with internal water mains, which help to attenuate the propagation of disturbances through the more complex water networks. In this paper, a general methodology for the design of optimal water-using networks with internal water mains is proposed using a new superstructure and a mixed-integer nonlinear programming (MINLP) strategy, given specifications of the mass load of the contaminants to be removed from each process unit and the lower and upper bounds on the contaminant concentrations. The mathematical program is formulated with constraints that bound the maximum number of outlet streams from each process unit. The most attractive networks, which have near-minimal freshwater consumption, relatively simple structures, and promising controllability, can be further designed to assess their operating and installation costs and controllability. This approach is applied to multicontaminant, waterusing systems and is used for the design of water-using networks with multiple internal water mains. 1. Introduction The mass-integration of water-using systems simultaneously reduces their freshwater consumption and wastewater discharge. This is gaining importance, especially in developing countries, where water resources are scarce and the need to protect the environment is becoming well-recognized. During the past 20 years, many new analysis and design techniques have been developed to improve water-using systems.1-10 Water-using networks generated either by using compositioninterval or graphical methods,5 often involving pinch compositions, or by mathematical programming methods10-12 are more highly integrated. For small water-using systems, the designs are simple, and the freshwater savings are satisfying. But for large water-using systems, the designs are more complex, and consequently, when the water flow rate or contaminant mass load for any process unit changes, the operations of many other process units are potentially affected. Consequently, it becomes important to control each process unit, using freshwater to counter contaminant concentration increases. Clearly, for large water-using systems, integrated water-using networks can have lower flexibility and be more difficult to control. The internal water mains introduced by Feng and Seider1 are intended to overcome the above limitation to some extent. From one perspective, the internal water mains can be regarded as buffers or mixing tanks, in which water streams are mixed before being reused in other process units. From another, each internal water main can be controlled easily using freshwater. When the concentrations of the contaminants in the internal water mains are controlled using freshwater, disturbances from the process units are more easily rejected, having less negative impact on process units downstream. Clearly, the holdups of the internal water mains influence the quality of the disturbance * Corresponding author. Tel.: 85-29-82668980. Fax: 86-2983237910. E-mail: [email protected].

rejection, with larger holdups being more effective but more expensive to installsan important consideration during the final design stages. Thus, compared with the networks generated by traditional methods (without internal water mains), networks with internal water mains are normally more convenient to operate and control and more flexible. It is noteworthy that internal water mains have been applied successfully for the water-system integration of industrial chemical plants.2 Thus far, the methodology for designing water-using networks with internal water mains has been developed for singlecontaminant, water-using systems1 or for multiple-contaminant, water-using systems with one internal water main.3 Both papers introduce algorithmic solution strategies, rather than mathematical programming strategies. Neither consider the usage of multiple internal water mains to enhance the water-savings effect. To generalize the design methodology for multiple-contaminant systems with multiple internal water mains, this paper presents a new mathematical programming technique, beginning with a general superstructure. Two case studies show the application of this methodology, with emphasis on the comparisons among water-using networks with different structures. Before proceeding, it is important to recognize that these strategies are intended for water-network synthesis in the early stages of process design, when the mass loads of the contaminants to be removed from each process unit are known and the lower and upper bounds for the water concentrations in each process unit are known. These strategies are designed to synthesize the most promising networks; that is, networks that have low freshwater consumption and wastewater production, low installation costs, and easy controllability. For the most promising networks, design teams are advised to design the equipment; that is, estimate equipment sizes and costs, as well as carry out controllability studies, designing the control systems for the process units.

10.1021/ie050911n CCC: $33.50 © 2006 American Chemical Society Published on Web 11/02/2005

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Figure 1. Schematic for water-using process unit.

Also, before proceeding, it is helpful to consider some aspects of process control in water networks. For individual process units (so-called mass exchangers) that remove contaminants with “reuse water”, when a contaminant concentration increases because of a disturbance, either the flow rate of the reuse water can be increased or freshwater can be introduced. The latter, although more effective, adversely affects the total freshwater consumption, and consequently, the former is commonly implemented. However, for large internal water mains, which act as buffers or mixers, contaminant concentration controllers respond to concentration increases with the addition of freshwater. Large internal water mains tend to absorb large concentration increases from upstream process units, thereby preventing their propagation throughout the network. The methods introduced herein are intended to synthesize networks that prevent this propagation before equipment design and controllability analysis take place. 2. Superstructure of Network with Internal Water Mains In this section, the steady-state model for the water-using process units is presented first. Then, the rules for positioning the internal water mains into water networks are discussed. Finally, a superstructure for mathematical programs to synthesize optimal water networks is introduced. 2.1. Model for Water-Using Process Units. To simplify the mass-integration problem in water-network synthesis, it is assumed that (1) each water-using process unit operates at steady state with a constant mass load of each contaminant to be removed and (2) only physical processes occur, not involving chemical reactions. The first is a typical assumption in the early stages of process design. The second acknowledges that, when reactions occur, for example, involving electrolytes, they must be accounted for in the selection of the contaminating species. Let P and C denote the sets of water-using process units and contaminants,

P ) {i | i is the index of water-using process units; i ) 1, ..., P} (1) C ) {s | s is the contaminant index; s ) 1, ..., S}

(2)

where P is the number of process units and S is the number of contaminants. A schematic of a water-using process unit is shown in Figure 1. Mass transfer of the contaminants is from the rich process stream to the water stream. Consequently, the concentrations in the rich process stream decrease as those in the water stream increase. For each water-using process unit i, the overall water balance, assuming no water loss, is out i∈P Fin i ) Fi

(3)

Figure 2. Water-using network with one internal water main. out where Fin i is the inlet flow rate and Fi is the outlet flow rate. Similarly, the species balances for contaminant, s, are

in out out Fin i ∈ P, s ∈ C i Ci,s + Mi,s ) Fi Ci,s

(4)

out where Cin i,s and Ci,s are the inlet and outlet concentrations of contaminant s and Mi,s is the mass load of contaminant s to be removed. In addition to the mass balances, concentration limits are necessary:

in,max i ∈ P, s ∈ C Cin i,s e Ci,s

(5)

out,max i ∈ P, s ∈ C Cout i,s e Ci,s

(6)

2.2. Structure of the Network with Internal Water Mains. All water networks involve two external water mains, that is, freshwater and wastewater mains. The internal water mains have intermediate contaminant concentrations, CWM m,s , m ) 1, ..., WM, above zero and less than those in the wastewater main, CWW s , s ) 1, ..., S, where WM is the number of internal water mains and m is the internal water main counter. They receive used water from process units having lower outlet concentraWM tions, that is, Cout i,s e Cm,s , i ∈ P, s ∈ C, m ∈ WM, where WM is the set of internal water mains. They provide water for process in units having higher inlet concentrations; that is, CWM m,s e Ci,s, i ∈ P, s ∈ C, m ∈ WM. In summary, water networks are comprised of a freshwater and wastewater main, internal water mains, and water-using process units. One or more internal water mains receive discharged water from some process units and provide used water for other process units. Each water-using process unit is connected to water mains only, receiving water from the freshwater main and one or more internal water mains and discharging water into the wastewater main or internal water mains. Because the process units are not connected to each other, concentration disturbances in any process unit are buffered by an internal water main and are not transmitted directly to the other process units. Figure 2 shows a typical water network with one internal water main. In Figure 2, the three bold vertical lines denote the freshwater, internal water, and wastewater mains, respectively. The contaminant concentrations in each main are annotated above the lines, with the total water flow rate to and/or from each main annotated below the lines. The numbered boxes represent the water-using process units, with the arrows representing their inlet and outlet streams.

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Figure 3. Superstructure for the water-using network with internal water mains.

One or more internal water mains can be utilized. In general, as internal water mains are added to the water-using network, the freshwater consumption is reduced together with the wastewater discharged. Normally, with just two or three internal water mains, the amount of freshwater consumed approaches the minimum for a set of process units. However, the network structure becomes more complex, and consequently, during the design process, the number of internal water mains must be balanced against the freshwater consumption and the treatment capacity for wastewater. Factors such as cost, controllability, and flexibility must be taken into account. 2.3. Superstructure for Water-Using Network with Internal Water Mains. The superstructure in Figure 3, which uses constructs similar to those introduced by Quesada and Grossmann13 for process networks in general, includes all of the potential design configurations for general water-using networks having internal water mains. In this figure, the following notation applies: 1. FW denotes the freshwater main. The arrows originating from it denote freshwater streams sent to water-using process units. 2. M denotes a mixing point for streams before they enter water-using process units, Pi, internal water mains, or the wastewater main. Arrows pointing to the mixing points before the process units represent streams from the freshwater main and internal water mains, and arrows pointing to the mixing points before the internal water mains, WMi, denote streams from water-using process units as well as excessive water discharged from the previous internal water main. 3. S denotes splitting points for the streams from the process units or the internal water mains. Arrows spread from the splitting points represent the outlet streams from process units sent to the internal water mains or streams to the process units from the internal water mains. 4. P denotes water-using process units. The arrows originating from them denote the water discharged to the internal water mains or the wastewater main. Each process unit discharges its used water through the splitter, S, to one or more water mains (either one or more of the internal water mains or the wastewater main). 5. WW denotes the wastewater main. Arrows pointing to it represent the wastewater streams discharged from water-using process units and the excess water discharged from the last internal water main. 3. Optimal Design Procedure for Water Networks with Internal Water Mains Before the mathematical program for the optimal design of the water network is formulated, it is necessary to specify the number of internal water mains, WM. This specification has a major impact on the structure of the optimal network and the freshwater consumption. As the number of internal water mains

Figure 4. Optimal design procedure.

increases, the freshwater consumption approaches the minimum obtained using the traditional water-using network (without internal water mains). However, a large number of internal water mains complicates the mathematical program and its solution algorithm and, more importantly, yields a complex network structure with increased capital costs. For this reason, an iterative design procedure is introduced beginning with one internal water main. The minimum freshwater consumption, FFW min, is computed, and if unacceptable, a second internal water main is added. Additional internal water mains are included, one at a time, until the calculated minimum freshwater consumption is satisfactory. Note that, in the early design stages, it is sufficient to qualitatively consider the tradeoffs in network complexity, controllability, and flexibility, as the freshwater consumption is decreased. As mentioned in the Introduction and Section 2.2, for the most promising water networks, equipment sizes and costs can be estimated and quantitative profitability and controllability studies can be undertaken. This design procedure is summarized in Figure 4. Note that considerations in setting the maximum number of outlet streams for each process unit, NSmax i , i ) 1, ..., P, are discussed in Section 3.2 and in the case studies in Section 4.

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Figure 5. Nomenclature for process units and internal water mains.

3.1. Basic Mathematical Program. Before examining the basic mathematical program, consider Figure 5, which summarizes the nomenclature. Here, FFW is the flow rate of i is the flow rate of wastewater freshwater to process unit i, FWW i from process unit i, FWM m,i is the flow rate of water from internal water main m to process unit i, Fi,m is the flow rate of water discharged by process unit i to internal water main m, and WM Fm-1,m is the excess flow rate of water through internal water main m-1, which is sent to internal water main m (to avoid accumulation in internal water main m-1). To determine the optimal design, given specifications for the mass loads to be removed from the rich streams in each process unit, Mi,s, and the maximum inlet and outlet concentrations for in,max out,max and Ci,s , the following the contaminant species, Ci,s nonlinear program (NLP) can be solved for the unknown flow rates and concentrations at the minimum total freshwater flow rate, FFW:

NLP min FFW )

FFW ∑ i i∈P

(7)

S. T.: 1. Overall mass balance (no water losses) for process unit i. FFW + i

WM Fm,i ) FWW + ∑ Fi,m ∑ i m∈WM m∈WM

i∈P

(8)

2. Species balances for the inlet mixing point before process unit i.

∑

WM WM Fm,i Cm,s ) (FFW + i

m∈WM

∑

WM in Fm,i )Ci,s i ∈ P, s ∈ C (9)

m∈WM

3. Species balances for mass transfer in process unit i (derived using eq 3). WM WM WM out Fm,i Cm,s + Mi,s ) (FWW + ∑ Fi,m )Ci,s ∑ i m∈WM m∈WM

i ∈ P, s ∈ C (10) 4. Bounds on inlet and outlet contaminant concentrations in process unit i. in in,max e Ci,s i ∈ P, s ∈ C Ci,s

(11)

out out,max e Ci,s i ∈ P, s ∈ C Ci,s

(12)

5. Overall water mass balance for internal water main m. Fi,m + ∑ i∈P

WM Fm-1,m

)

∑ j∈P

WM Fm,j

+

WM Fm,m+1

m ∈ WM (13)

6. Species mass balances for internal water main m. out WM WM WM WM WM Fi,mCi,s + Fm-1,m Cm-1,s ) (∑Fm,j + Fm,m+1 )Cm,s ∑ i∈P j∈P

m ∈ WM, s ∈ C (14) 7. Internal water main concentration constraints. WM WM Cm,s e Cm+1,s m ∈ WM

8. Nonnegativity constraints for all variables.

(15)

Note that the model has bilinear equality constraints with a linear objective function. When solving for the unknown flow rates and concentrations and the minimum freshwater consumption, FFW min, the following constraints impose practical restrictions on the solution space. 3.2. Structural Constraints. In the NLP, the number of outlet streams, NS, from the process units is unbounded. However, because the outlet streams are sent to internal water mains or the wastewater main, NS in the NLP solution cannot exceed the number of internal water mains plus one. Clearly, as NS increases, the network becomes more complex. Hence, when computing solutions, especially the initial solutions, it helps to enforce an upper bound, NSmax. Furthermore, because many streams in the superstructure do not exist in the optimal solution, having zero flow rates, it is helpful to introduce binary variables into the mathematical program, thereby converting the NLP into the following mixed-integer nonlinear program (MINLP):

MINLP min FFW )

FFW ∑ i i∈P

(7)

S. T.: Constraints in eqs 8-15 and Fi,m - yi,mU e 0 i ∈ P, m ∈ WM

(16)

- yWW Ue0 i∈P FWW i i

(17)

yi,m + yWW e NSmax ∑ i i m∈WM

i∈P

(18)

Here, the specification for U in the constraints in eqs 16 and 17 must be sufficiently large. Furthermore, when process unit i discharges water into internal water main m, Fi,m > 0 and, consequently, yi,m ) 1. Similarly, when process unit i discharges > 0 and yWW ) 1. water into the wastewater main, FWW i i Otherwise, the binary variables are zero. Finally, the inequality in eq 18 ensures that the upper bound, NSmax, is enforced. Initially, the MINLP is solved with NSmax ) 1, giving a relatively simple network and, normally, a high freshwater consumption. This solution usually gives a good initialization to determine subsequent solutions with NSmax g 2. Note that, in some cases, the initial solution has an acceptably low freshwater flow rate. As mentioned in the Introduction and Section 2.2, for the potentially attractive solutions, equipment sizes and costs can be estimated and controllability studies can be undertaken to identify the most attractive water networks. It is also possible to place lower or upper bounds on the total flow rate of water associated with the process units; for example,

∑

Fi,m + FWW e FUi i

(19)

m∈WM

where FUi is the bounding flow rate for process unit i. In addition, constraints can be added to achieve desired network features. In the next section, the case studies illustrate the usage of this methodology to obtain various solutions for two waterusing systems. These solutions were computed using the commercial software package, LINGO. A successive linear programming solver was selected. Using a branch-and-bound strategy, an initial result was obtained, that is, FFW 1 . Then, a FW second solution was obtained using the constraint, FFW 2 e F1 .

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Figure 6. Optimal network with one internal water main for case study 1. Table 1. Limiting Water Data for Case Study 1 process i

Cin,max , ppm i

Cout,max , ppm i

Mi, g/h

1 2 3 4 5 6 7 8 9 10

25 25 25 50 50 400 200 0 50 150

80 90 200 100 800 800 600 100 300 300

2 000 2 880 4 000 3 000 30 000 5 000 2 000 1 000 20 000 6 500

This procedure was repeated until no solution was obtained. Finally, a global solver was used to obtain the solution. Note that this algorithm does not guarantee a globally optimal solution. However, for the first case study that follows, as the number of water mains is increased, the solutions approach the globally optimal solution without internal water mains computed by others.

Figure 7. Optimal network with two internal water mains for case study 1.

Figure 8. Minimum freshwater consumption for networks with different structures.

4. Case Studies 4.1. Single-Contaminant System. This system, involving 10 process units, was taken from Bagajewicz et al.,12 with the limiting water data for the process units shown in Table 1. The MINLP was formulated with WM ) 1, NSmax ) 1, and U ) 3 000. It contains 83 constraints (32 equations and 51 inequalities, 21 of which are nonlinear) and 91 variables. It gives the optimal solution, FFW min ) 178.86 t/h, with the integrated network shown in Figure 6. Note that its freshwater consumption is 12.59 t/h greater than the minimum freshwater consumption, 166.27 t/h (computed by Bagajewicz et al.12), for networks with the traditional structure, without internal water mains, shown in Figure 9. When the number of internal water mains was increased to 2, the minimum freshwater consumption was decreased to 169.04 t/h, 9.82 t/h less than the first design, with the associated network shown in Figure 7. Table 2 and Figure 8 show the minimum freshwater consumption as a function of the number of internal water mains and the number of streams leaving the process units. Note that the variation with NS decreases as the number of internal water mains increases.

Figure 9. Optimal network with traditional structure for case study 1.

The traditional network, without internal water mains, designed by Bagajewicz et al.,12 is shown in Figure 9, with the water allocation details presented in Table 3. Three key parameters are introduced to compare the networks having different structures in the early stages of process design, before equipment sizes and costs are estimated and control structures studied: (1) freshwater consumption, which directly affects the operating cost; (2) the number of connections among the process units and the water mains, which increases with

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Table 2. Minimum Freshwater Consumption for Networks with Different Structures NS FFWmin

1

2

3

4

WM 1 WM 2 WM 3

178.86 169.04 166.93

175.87 168.35 166.27

168.35 166.27

166.27

Table 3. Water Allocation Details for Figure 9 process unit i

Fi,j, t/h

Cin, ppm

Cout, ppm

FFWi, t/h

1 2 3 4

0 F1,2 ) 13.8462 F2,3 ) 6.3492 F1,4 ) 11.1538 F2,4 ) 23.4188 F4,5 ) 9.51428 F8,5 ) 10 F9,5 ) 0.16191 F9,6 ) 16.6667 F3,7 ) 1.19047 F4,7 ) 1.90476 F9,7 ) 1.90476 0 F2,9 ) 14.5397 F4,9 ) 26.9143 F3,10 ) 21.6667 F4,10 ) 21.6668

0 25 25 50

80 90 200 100

25 30.4615 16.5079 25.4273

50

800

20.3238

300 200

800 600

0 0

0 50

100 300

10 38.546

150

300

0

5 6 7 8 9 10

FFW min )

166.2665

the complexity of the water-using network, directly affecting the capital cost; and (3) the maximum number of process units downstream from a process unit. The latter provides an approximate measure of the number of process units through which a large disturbance, which cannot be attenuated by the control systems for the process units, can propagate. With large internal water mains, however, the disturbances are attenuated, and consequently, process units downstream of internal water mains are not included. Table 4 lists these parameters for the three networks. A brief discussion for each network follows. 1. Network without water-reuse. The process units in this network utilize freshwater directly, with no water-reuse. Therefore, the largest amount of freshwater is consumed, the largest amount of wastewater is discharged, and the largest operating cost is incurred. However, because there are no connections between the process units, this network is the simplest to construct, having the lowest installation costs. Moreover, because the process units are independent of each other, no propagation of disturbances occurs. 2. Optimal network with traditional structure (no internal water mains). Because of the maximum water-reuse, these networks consume the least freshwater and discharge the least wastewater. Accordingly, they have the lowest operating cost. However, they have the most connections between the process units, the most complex piping networks, and, consequently, the largest installation costs. Furthermore, because of the tight integration, unattenuated disturbances can propagate among

many process units. For example, in Figure 9, when the concentration of contaminants in process unit 1 increases, because of a disturbance not controlled properly by its control system, it is propagated to process units 2 and 4 directly and, subsequently, to process units 3, 5-7, 9, and 10 indirectly. As can be seen, this corresponds to the maximum number of process units downstream from a process unit, which is eight for this design. 3. Optimal network with one internal water main. The freshwater consumption for this network is far less than that for the network without water-reuse and slightly greater than that for the optimal network with the traditional structure. Therefore, it has a relatively low operating cost. Furthermore, because of its relatively simple structure, it has a relatively low installation cost. Moreover, because of the internal water main, unattenuated disturbances are effectively isolated from the other process units. For example, when the concentration in process unit 1 increases beyond the set point of its controller, the concentration in the internal water main increases, but it is less because of its large holdup. Note that, normally, this disturbance can be rejected by the control system for the internal water main, which introduces freshwater as needed. Because no process units are connected directly with each other, the maximum number of process units downstream affected by this disturbance is zero. At this stage in design, this potential improvement in controllability is very attractive. For this reason, design teams are advised to estimate the equipment sizes and costs and to design and simulate the control system for this network. 4. Optimal network with two internal water mains. Because of the second internal water main, the freshwater consumption is reduced and is just slightly higher than that for the traditional structure. This reduction is balanced by an increase in installation costs. The number of units downstream remains zero. In summary, at this stage in process design, the networks with internal water mains are potentially attractive, providing low freshwater consumption with fairly simple structures and promising controllability. On the basis of these attractive features, design teams are advised to carry out more detailed design studies. 4.2. Multi-Contaminant System. This system, involving 7 process units and 3 contaminants, was taken from Wang et al.,3 with the limiting water data for the process units shown in Table 5. The MINLP was formulated with WM ) 1, NSmax ) 1, and U ) 3 000. It contains 116 constraints (53 equations and 63 inequalities, 45 of which are nonlinear) and 89 variables. It gives the optimal solution, FFW min ) 156.85 t/h, with the integrated network shown in Figure 11. Note that its freshwater consumption is 3.55 t/h less than the minimum freshwater consumption, 160.4 t/h, for the network designed by Wang et al., shown in Figure 10. Furthermore, because the minimum freshwater consumption using a traditional network is 139.7 t/h (computed by Wang et al.), a second internal water main is considered for the network, using NSmax ) 2. For the revised MINLP, there

Table 4. Comparison of Networks for Case Study 1 freshwater consumption, t/h

no. of connections

maximum no. of process units downstream

without water-reuse

252.42

20

0

traditional structure

166.27

27

8

one internal water main

178.86

23

0

two internal water mains

169.04

24

0

structure of optimal network

features of network the simplest structure no connections between process units the most complex structure most connections between process units fairly simple structure no connections between process units fairly simple structure no connections between process units

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Figure 11. Optimal network with one internal water main for case study 2. Figure 10. Optimal network with one internal water main for case study 2 (designed by Wang et al.3). Table 5. Limiting Water Data for Case Study 2 process

contaminant

, Cin,max i ppm

Cout,max , i ppm

limiting flow rate, t/h

Mi,s g/h

1

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

0 0 0 0 0 0 20 50 50 50 110 200 20 100 200 500 300 600 150 700 800

50 100 50 100 300 600 150 400 800 600 450 700 500 650 400 1 100 3 500 2 500 900 4 500 3 000

25

1 250 2 500 1 250 7 000 21 000 42 000 4 550 12 250 26 250 22 000 13 600 20 000 3 840 4 400 1 600 30 000 160 000 95 000 22 500 114 000 66 000

2 3 4 5 6 7

70 35 40 8 50 30

are 130 constraints (57 equations and 73 inequalities, 48 of which are nonlinear) and 115 variables. It gives the optimal solution, FFW min ) 143.53 t/h, with the integrated network shown in Figure 12. This freshwater consumption is 13.32 t/h lower than that for the network with one internal water main, designed herein. To compare these designs, see Table 6, which lists the key parameters. When comparing the networks in Figures 10 and 11, the MINLP herein provides a more water-efficient network. Furthermore, the design in Figure 12 shows that, by relaxing the structural limits, considerable freshwater is saved, but with a significant increase in network complexity. 5. Conclusions Using the superstructure introduced herein, the general methodology for the design of optimal water-using networks

Figure 12. Optimal network with two internal water mains for case study 2.

with internal water mains proposed herein is found to be effective in the early stages of process design. The methodology permits experimentation with the number of internal water mains and the number of outlet streams from each process unit. Given specifications for the mass loads of contaminants to be removed in each process unit, and the minimum and maximum concentration possible in each process unit, water networks are designed having low freshwater consumption, with relatively simple structures, and potentially good controllability. Reliable solutions of the mathematical programs introduced herein are obtained using a successive linear programming algorithm in the LINGO software package.

Table 6. Comparison of Networks for Case Study 2 structure of optimal network

freshwater consumption, t/h

no. of connections

max no. of process units downstream

one internal water main (designed by Wang et al.) one internal water main (designed herein) two internal water mains (designed herein)

160.4 156.85 143.53

17 16 20

0 0 0

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Acknowledgment

Superscripts

Financial support provided by the National Natural Science Foundation of China under Grant No. 20376066 and Grant No. 20436040 and by the Major State Basic Research Development Program under Grant No. 2003CB214500 is gratefully acknowledged by the first three authors.

FW ) freshwater in ) inlet stream max ) maximum out ) outlet stream P ) process unit U ) upper bound WM ) internal water main WW ) wastewater main

Nomenclature C ) set of contaminants in water-using system C ) number of contaminants in water-using system in ) inlet concentration of contaminant s to process unit i, Ci,s ppm in,max Ci,s ) maximum inlet concentration of contaminant s in process unit i, ppm out Ci,s ) outlet concentration of contaminant s from process unit i, ppm out,max Ci,s ) maximum outlet concentration of contaminant s from process unit i, ppm WM ) concentration of contaminant s in internal water main Cm,s m, ppm Fi,m ) water flow rate from process unit i to internal water main m, t/h FFW ) total freshwater flow rate, t/h ) freshwater flow rate to process unit i, t/h FFW i WM ) water flow rate to process unit i from internal water Fm,i main m, t/h WM Fm-1,m ) water flow rate from internal water main m-1 to internal water main m, t/h ) wastewater flow rate from process unit i, t/h FWW i Mi,s ) mass load of contaminant s to be removed in process unit i, g/h NSi ) number of outlet streams for process unit i ) maximum number of outlet streams for process unit i NSmax i P ) set of water-using process units P ) number of water-using process units U ) flow rate constant for the inequalities in eqs 16 and 17, t/h WM ) set of internal water mains WM ) number of internal water mains yi,m ) binary variable ) 1 when stream between process unit i and internal water main m exists; otherwise ) 0 yWW ) binary variable ) 1 when stream between process unit i i and wastewater main exists; otherwise ) 0 Subscript s i ) process unit counter m ) internal water main counter min ) minimum s ) contaminant species counter

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ReceiVed for reView August 5, 2005 ReVised manuscript receiVed October 3, 2005 Accepted October 7, 2005 IE050911N