Optimization of Water Systems with the Consideration of Pressure

ARTICLE pubs.acs.org/IECR

Optimization of Water Systems with the Consideration of Pressure Drop and Pumping Szu Wen Hung† and Jin-Kuk Kim*,‡ †

Centre for Process Integration, School of Chemical Engineering and Analytical Science, The University of Manchester, Manchester M13 9PL, U.K. ‡ Department of Chemical Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul, Republic of Korea 133-791 ABSTRACT: Novel automated design method has been developed to consider impacts of pressure drop and pumping arrangement in the water network in which water reuse is exploited to reduce freshwater consumption in process industries. A superstructurebased optimization framework has been developed to systematically consider all the key design issues simultaneously, including flow rate constraints, pressure-drop constraints, and other operational constraints, as well as to fully accommodate rigorous economic trade-off between freshwater cost, piping cost, and pumping cost. The proposed optimization study enables the identification of the optimal distribution of water within the network, together with the most appropriate location and capacity of pumps required for water systems. A robust and reliable solution strategy has been developed to effectively deal with computational difficulties associated with solving a mixed-integer nonlinear programming problem, with the aid of the effective application of physical insights gained from conceptual understanding of the design problem. The network complexity has been readily controlled by imposing design constraints in the optimization. Examples have been provided to prove the effectiveness of the design method developed in this paper and to demonstrate the importance of considering pressure drop constraints in practice.

1. INTRODUCTION There has been a growing perception of environmental protection and sustainable development in society, which raises public concern on the use of natural resources by industries. Water is a key nature resource, which has been heavily extracted and used in process industries as raw material, mass separating agent, or heat-transfer medium. On the other hand, costs related to freshwater supply and effluent disposal have been significantly increased, which requires urgent attention for achieving efficient use of water in process industries. Due to this economic and socioenvironmental pressure, academic and industrial communities have been actively engaged in the search of new opportunities and technologies for fundamentally improving the efficiency of industrial water use. One of the widely used methods in process industries is to exploit the reusability of water within a system boundary, based on the application of process integration concept, with which a systematic way of reusing water is identified. Water reuse can be coupled with regeneration facilities which remove contaminants of water streams before being reused or recycled within water systems. Wang and Smith1 proposed a holistic approach to minimize freshwater consumption by implementing water reuse, regeneration, and recycling and, consequently, minimize wastewater generation at the same time. The concept of reusing water is to take the effluent from water-using operations and to reuse it partly or totally in the other water-using operations, rather than discharge it directly. With the installation of water-treating operations, the intensity of water reusability within the water systems can be considerably increased at the expense of additional capital investment. How water reuse, regeneration, and recycling are introduced in the context of water networks depends on the operational r 2011 American Chemical Society

constrains of the water-using operations. For example, there are constraints on the maximum allowable inlet and outlet concentration based on design and operating factors, such as minimum mass-transfer driving force, maximum solubility, and corrosion restriction. These limiting conditions are imposed to ensure the feasibility of water reuse in the network. A wide range of design methods has been proposed to design the water networks, and those available methods are broadly classified into two categories: insight-based pinch analysis and mathematical optimization technique. For both categories, most of the studies carried out in the early days had treated water-using operations and water-treating operations as separated systems.13 There had been attempts to address the interactions between water-using systems and water-treating systems with a conceptual approach4 or an automated design method.5,6 Gunaratnam et al.5 presented an optimization framework for the integrated design of total water networks which combined water-using operations and water-treating operations as a single system and effectively considered complex design interactions existed between water reuse, regeneration, recycling, and distributed treatment in the total water systems. Further research has been carried out to exploit benefits gained from an integrated concept.79 From the viewpoint of water pinch analysis, various methods have been developed to target the minimum freshwater consumption.1,3,4,1013 The most widely used concept for targeting is to construct a limiting water composite curve and to manipulate it with a target water supply line.14 The limiting water Received: August 10, 2011 Accepted: November 22, 2011 Revised: November 7, 2011 Published: December 15, 2011 848

dx.doi.org/10.1021/ie201775y | Ind. Eng. Chem. Res. 2012, 51, 848–859

Industrial & Engineering Chemistry Research

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composite curve is plotted on the diagram with mass load against concentration, which captures the overall characteristics of whole water-using operations. For the network design, one of the widely used design methods is a “water main concept” proposed by Kuo and Smith,4 with which the available matches between water sources and sinks are identified through material balances around a water main with the aid of graphically represented conditions of water use. Because it is not straightforward to perform economic tradeoff among the costs of different elements in the network through a water pinch approach, optimization techniques have been widely adopted in the design of water systems.5,1518 With such automated design using optimization techniques, the costeffective network configuration can be obtained through rigorous economic trade-off between operating and capital costs, as well as screening different design options available in a water network. Another benefit from using optimization techniques is that it can be effectively made to incorporate engineering constraints to be avoided in the network design or to enforce preferred features into the water network. However, the optimization-based design method often requires considerable computational efforts when stochastic optimization is used, or to provide robust solution strategies when deterministic approach is employed to solve highly nonlinear optimization problems. The conceptual insights obtained from graphic-based water pinch analysis not only enhance the understanding of the design problem but also can be very useful in developing an effective solution strategy for the optimization. Various design issues had been addressed in the past to enhance the applicability of water network design methodologies; however, most of the studies carried out in industrial and academic communities had not included about how pressure drop should be considered in the water network and how the pumping of water within the network should be arranged. A conventional design practice is, first, to consider water reuse, regeneration, and recycling in the water network first and to determine the water network configuration. With the confirmed water network, the pressure drop is calculated and, correspondingly, the design of water pumping is considered. There had been an attempt to calculate the overall pressure drop of cooling water in the design of cooler networks, together with designing a cooling water network.19 Although the overall pressure drop of the cooling water network was calculated during the network optimization and was used as a design guideline for the optimization, it had not systematically considered impacts of the network configuration on pressure drop and the arrangement of pumping. Also, Kim and Smith’s work19 did not provide any information for how pressure drop influences the pumping of water and what is the best pumping arrangement for the network. A large number of structural options are available in the design of water networks when an integrated water network is desired to reduce freshwater consumption through water reuse. The resulting water network is complicated, and it is not straightforward to identify the most appropriate location of pump and its optimal size. It would be ideal to simultaneously consider pressure drop and pumping arrangement during the water network design. For most of the published works, the consideration of piping cost in the optimization of water networks had been reported,4,6 but design issues associated with pressure drop and pumping had not been thoroughly addressed. This paper focuses on the development of an automated design method which is able to simultaneously perform the

Figure 1. Superstructure representation.

calculation of pressure drop and design the pumping of water in the context of a water network. The developed optimization framework systematically carries out rigorous economic trade-off between freshwater cost, pipeline cost, and pumping cost, which provides the optimal distribution of water and the optimal location and capacity of the pump for the network. The current study has been limited to study of the design problem of water reuse in the network without considering water-treating operations.

2. DESIGN OF WATER NETWORKS The water systems discussed in this paper consists of waterusing operations in which water is contaminated through the mass transfer of contaminants from process streams. Several freshwater sources may be available for the water systems, while there is a sink to take the contaminated wastewater discharged from water-using operations. A mixer and a splitter may be required before and after each water-using operation. The mixer is used to blend all of the inlet streams from different freshwater sources and/or from different water-using operations, while the splitter facilitates the distribution of the exiting stream to other water-using operations or a sink for discharged water. The cost considered in the water-using network in this study includes operating costs related to freshwater and electricity costs for operating pumps and capital cost for pipeline and pump. A superstructure approach is used in this work that considers design interactions between pipeline arrangement and pump installation. Figure 1 illustrates a superstructure with one available water source and two water-using operations. All of the piping options for connecting among freshwater sources, water-using operations, and wastewater discharges, as well as all the potential options for installing pumps are incorporated in the superstructure. A mixed-integer nonlinear programming (MINLP) model based on this superstructure has been developed for the network design. For the simplification in the installation of pumps, when there are two streams or more to be pumped simultaneously, the single pump is used if these streams are designated to the same sink or originated from the same source. Otherwise, it is assumed 849

dx.doi.org/10.1021/ie201775y |Ind. Eng. Chem. Res. 2012, 51, 848–859


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that individual pumps are introduced for pumping each stream. For example, when a single pump is introduced for pumping two streams which are not originated from the same water source, the mixing is necessary for pumping these two streams with a single pump. Therefore, unnecessary extra mixing can be avoided due to the introduction of a pump in the water network. This is a reasonable and acceptable simplification because mixing between water streams should be selected such that freshwater consumption in the network is minimized, according to the limiting water conditions. This simplification is fully justified when the freshwater cost is compared to pumping cost from the optimization result of cases studies carried out in section 4 of the current paper. The dominance of freshwater cost in the overall water reuse network can be found from the previous work.5 Another reason to avoid extra mixing is that the required head elevation varies from pipeline to pipeline, and in some scenarios, extra mixing for sharing the pump causes a great loss in rhe pressure head. The superstructure introduced above includes all the possible stream connections, but it is not enough for calculating the pressure drop of the network. The calculation of pressure drop is determined not only by individual pipelines and water-using operations, but also by the configuration. It is necessary to introduce the concept of “critical path algorithm” used for the calculation of pressure drop in the network.19 The critical path algorithm is used to find the path with a maximum length among all the paths available from a starting point and a final point. When this algorithm is applied for finding the maximum pressure drop, the maximum pressure-drop path can be found by minimizing the pressure difference between a source node and a sink node, without violating the pressure drop constraints between intermediate nodes.19 This network design problem has been formulated as a MINLP problem, which is solved with an objective function to minimize total annualized cost. The key nonlinear terms in the formulation appeared in the contaminant balance equations around mixer, splitter, and the water-using operation, as well as in the equation representing the pumping cost, which caused computational difficulties in the optimization. To effectively deal with nonlinearity in the optimization model, a robust solution strategy of this MINLP problem has been also proposed by incorporating physical insights in the design of the water network and its conceptual understanding. For the developed model, operational constraints have been imposed to ensure the feasibility. Besides, the model is possible to control the network complexity and flow rate restriction by manipulating binary variables. There are some assumptions made in the network design as follows: (i) The number of water-using operations is fixed and its masstransfer behavior is specified by maximum inlet and outlet concentrations and the mass load to be removed. (ii) The mass load to be removed in each water-using operation is constant. (iii) The pressure at any point in the system is above the required net positive suction head of every pump. (iv) The pressure loss of each water-using operation is constant. (v) Pressure loss related to fitting, sudden change in the direction of the pipeline, valves, or measuring devices is neglected. (vi) All of the water-using operations and other equipment are located at the same gravity level. (vii) The suction loss of the pump is negligible.

Figure 2. Schematic representation of a water-using operation.

3. MATHEMATICAL FORMULATION The mathematical formulation includes all of the feasible connections based on the proposed superstructures. The first step for developing the mathematical model is to define sets, parameters, and variables representing the water systems. Model Equations and Constraints. The mathematical model to design the integrated water network and determine the allocation of pumps in the network involves a nonlinear objective function, together with linear and nonlinear constraints. The model comprises the flow rate balance equations, contaminant balance equations, equations to calculate pressure drop, operational constraints, and logical constraints. A schematic representation of the water-using operation is shown in Figure 2, in which inlet streams may come from freshwater sources and/or other water-using operations and outlet streams may be transported to other water-using operations and/or a discharged point directly. The formulated MINLP model is as follows: Balances around Mixer, Splitter, and Water-Using Operations flow rate balances around the mixing and splitting points : ua ¼ Fut ∑ Fs;uw þ ua∑∈ U Fua;u

s∈S

∑

ua ∈ U

ua Fu;ua þ

∑ Fu;eout ¼ Fut

e∈E

"u∈U

ð1Þ

"u∈U

ð2Þ

mass balances of each contaminant around the water-using operation:

∑ Fs,wu Cfwc, s þ ua∑∈ U Fua,ua uCout c, ua þ Mc, u

s∈S

¼ Fut Cout c, u

" u ∈ U, c ∈ C

ð3Þ

Operational Constraints on the Concentration of Inlet and Outlet Streams. There are limitations for the concentration levels for both inlet and outlet water for the water-using operation. ua t in;max Cout ∑ Fs;uw Cfwc;s þ ua∑∈ U Fua;u c;ua e Fu Cc;u

s∈S

" u ∈ U; c ∈ C

ð4Þ Cout c;u

e

Cout;max c;u

" u ∈ U; c ∈ C

ð5Þ

Logic Constraints for Streams in the Integrated Water Network. upper and lower bounds of the flow rate : w fw fw Fs;u e Us;u Bs;u

850

" s ∈ S; u ∈ U

ð6Þ


Industrial & Engineering Chemistry Research w fw Fs;u g Lfw s;u Bs;u

" s ∈ S; u ∈ U

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Equation 17 is taken to estimate the fanning friction factor.19 In this study, it has been assumed that the flow velocity is 2 m/s in all of the pipelines. Under this assumption, the diameter of the pipeline is a function of the flow rate. Equations 18 and 19 show how the diameter of the pipeline between the freshwater source and waterusing operation is calculated. The diameter of other pipelines is calculated in a similar way. constraints of “critical path algorithm” :

ð7Þ

ua ua Fu;ua e Uu;ua Bua u;ua

" u; ua ∈ U

ð8Þ

ua ua Fu;ua g Lua u;ua Bu;ua out out out e Uu;e Bu;e Fu;e out out out Fu;e g Lu;e Bu;e

" u; ua ∈ U

ð9Þ

" u ∈ U; e ∈ E

ð10Þ

" u ∈ U; e ∈ E

ð11Þ

Equations 611 specify the upper and lower bounds of water streams. fw ua out Us,u, Uu,ua, and Uu,e are defined as upper bounds of corresponding fw ua , Lu,ua, and Lout streams, while Ls,u u,e represent lower bounds. The lower bounds given eliminate the possibility of the existence of a stream with a very small flow rate which is not economical or practical.

pump Puin Ps þ ΔPspump þ ΔPs;u

þ LV ð1 Bfw s;u Þ g ΔPs;u

"u∈U

þ LV ð1 Bua u;ua Þ g ΔPu;ua

ð12Þ

"u∈U

þ LV ð1 Bout u;e Þ g ΔPu;e

w fw ¼ Afw Fs;u s;u vs;u sffiffiffiffiffiffiffiffiffi w 4Fs;u Ds;u ¼ fw πvs;u

ð21Þ

" u ∈ U; e ∈ E

ð22Þ

According to the “critical path algorithm”, the overall pressure drop of water networks cannot be calculated until the network configuration is known. To address the problem without knowing the network configuration in advance, eq 20, eq 21, and eq 22 for the connections between each mixing and splitting node are formulated with the aid of binary variables which are introduced with a sufficiently large value to consider whether this constraint is activated or not. In the model considering the allocation of pumps, the pumping term, which represents the pressure elevation of each pumping node (the available place to put the pump between the mixing node and splitting node), is included. With reference to eq 20 as an example, when the connection between water source s and water-using operation u exists (Bfw s,u = 1), the pressure difference between the splitting node of the freshwater source and the mixing node of the water-using operation need to be larger than the pressure drop caused by the friction in between. If the pressure difference between the splitting node and mixing node is not large enough, the pump is introduced to compensate for the pressure drop. For the connection between a freshwater source and a waterusing operation, there are two possible places to install a pump. The first option is to pump the water at the freshwater source before the water is distributed. Another choice is to pump the water at the beginning of the connection between the freshwater source and the water-using operation. The pressure elevated from these two pumps and ΔPpump in eq 20, respectively. For corresponds to ΔPpump s s,u illustration, three possible pumps can be installed between a freshwater source and water-using operation schematically shown in and Figure 1. The first two possible locations are related to ΔPpump s in eq 20, and ΔPin,pump in eq 13 indicates the third location ΔPpump s,u u in front of water-using operations. Equation 21 is used to calculate the pressure drop related to the connection between different waterusing operations, such that the pressure increased by the pump around the water-using operation is considered between the inlet node of operation ua and the outlet node of operation u. Similarly, eq 2 is introduced for calculating the pressure drop associated with the connection between water-using operations and the discharge point .

ð13Þ

The pressure drop of the water-using operation, ΔPu, has been assumed to be constant and preknown in eq 13 in this study. However, the detailed calculation of pressure can be readily added in this framework, as long as the detailed unit operation of water-using operations and its pressure drop model is available, and the formulation for the pressure drop calculation is reasonindicates the pressure elevation of the ably simple. ΔPin,pump u pump installed in the front of water-using operations. pressure drop in pipes : ! ! fw ds;u Fvs;u 2 ΔPs;u ¼ 4f0 2 Ds;u ! ! ua du;ua Fvu;ua 2 ΔPu;ua ¼ 4f0 2 Du;ua ! ! out du;e Fvu;e 2 ΔPu;e ¼ 4f0 2 Du;e ! 0:046 f0 ¼ Re0:2

" u; ua ∈ U

pump Pe Puout þ ΔPuout;pump þ ΔPu;e

To control the complexity of the water network, eq 12 can be introduced to restrict the total number of streams allowed to the inlet of water-using operations. NSmax u represents the maximum allowable number of inlet streams. The formulations given so far are based on the superstructure of Figure 1 and the critical path algorithm for the calculation of pressure drop. Pressure Calculation around Water-Using Operation Puin þ Uuin, pump ΔPu ¼ Puout

ð20Þ

in pump Pua þ ΔPuout;pump þ ΔPu;ua Puout

maximum number of inlet streams:

∑ Buaua, u þ s ∑∈ S Bfws, u e NSmax u ua ∈ U

" s ∈ S; u ∈ S

ð14Þ

ð15Þ

ð16Þ ð17Þ ð18Þ ð19Þ

The pressure drop in a pipe depends on the fanning friction factor, the density of the flow, the flow velocity in the pipeline, the diameter of the pipeline, and the length of the pipeline. The general formulation to calculate the pressure drop in the pipe is shown in eqs 1416.

upper and lower bound for pressure elevation of pump: ΔPspump e Uspump Bpump s 851

"s∈S

ð23Þ


Industrial & Engineering Chemistry Research ΔPspump g Lpump Bpump s s

"s∈S

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ð24Þ

pump pump ΔPs,pump u e Us, u Bs, u

" s ∈ S, u ∈ U

ð25Þ

pump pump ΔPs,pump u g Ls, u Bs, u

" s ∈ S, u ∈ U

ð26Þ

pump pump ΔPu,pump ua e Uu, ua Bu, ua

" u, ua ∈ U

ð27Þ

ΔPu,pump ua

g

ΔPuin, pump

pump Lpump u, ua Bu, ua

e

" u, ua ∈ U

Piping cost relates to the cross-sectional area and the distance of the pipeline. Note that the relationship between cross-sectional area and the flow rate has been specified in eq 18. Pumping Cost. Pumping cost includes the capital cost of the pump and its operating cost. Capital cost depends on the power required to drive the pump. The way to calculate the cost of the pump at the freshwater source is demonstrated in eqs 40, 41, and 42. The costs associated with other pumps are calculated in a similar manner. costspump;cap ¼ rðQs Þw

ð28Þ

pump Uuin, pump Bin, u

"u∈U

ð29Þ

pump in, pump ΔPuin, pump g Lin, Bu u

"u∈U

ð30Þ

pump ΔPuout, pump e Uuout, pump Bout, u

"u∈U

ð31Þ

pump out, pump Bu ΔPuout, pump g Lout, u

"u∈U

ð32Þ

pump pump ΔPu,pump e e Uu, e Bu, e

" u ∈ U, e ∈ E

ð33Þ

pump pump ΔPu,pump e g Lu, e Bu, e

" u ∈ U, e ∈ E

ð34Þ

costspump;op

ð

∑ costfws Þ þ ðs ∑∈ S u ∑∈ U costfw,s, u pipe pipe pipe þ ∑ ∑ costua, þ ∑ ∑ costout, ÞAF u, ua u, e u ∈ U ua ∈ U e ∈ Eu ∈ U cap cap þ ∑ ∑ costpump, þ ð ∑ costpump, s s, u s∈S s ∈ Su ∈ U cap pump, cap þ ∑ ∑ costpump, þ ∑ costin, u, ua u u ∈ U ua ∈ U u∈U pump, cap cap þ ∑ costout, þ ∑ ∑ costpump, ÞAF u u, e u∈U u∈Ue∈E op op þ ∑ ∑ costpump, þ ð ∑ costpump, s s, u s∈S s ∈ Su ∈ U op pump, op þ ∑ ∑ costpump, þ ∑ costin, u, ua u u∈U s ∈ S ua ∈ U pump, op op þ ∑ costout, þ ∑ ∑ costpump, Þ ð35Þ u u, e u∈U u∈Ue∈E

cap ≈ hQs þ gBpump costpump, s s

Piping Cost.

ΔPs;u ¼

fw 0:04F0:8 μ0:2 us;u 2:4 π0:6 ds;u w Þ1:4 ðFs;u

ΔPu;ua ¼ ð36Þ

fw;pipe fw fw fw fw ¼ ½ðafw costs;u s;u As;u Þ þ ðbs;u Bs;u Þds;u

ð37Þ

ua;pipe ua ua ua ua costu;ua ¼ ½ðaua u;ua Au;ua Þ þ ðbu;ua Bu;ua Þdu;ua

ð38Þ

out;pipe out out out out costu;e ¼ ½ðaout u;e Au;e Þ þ ðbu;e Bu;e Þdu;e

ð39Þ

ð42Þ

ð43Þ

(2) The linearization of equations related to the calculation of a pressure drop in the pipes (eq 14, eq 15 and eq 16). At first, eqs 17 and 19 are substituted into eqs 1416, as given in eqs 44, 45, and 46, respectively.

Fresh Water Cost ψs Fs,wu ∑ ∑ s ∈ Su ∈ U

3600F

ð41Þ

Qs is the power required for the pump. With the assumptions given in the previous section, the electricity required is calculated as in eq 42, which relies on the water flow rate to be pumped and the pressure to be elevated through hthe pump. Solution Strategy. Highly nonlinear terms included in the model present a major difficulty to solve the optimization problem directly by using a standard solver, and a solution strategy to cope with the nonlinear equations is necessary. It is essential to provide a good starting point for the highly nonlinear MINLP problem, for example, using a simpler optimization model or some initial points obtained from conceptual insights when a conventional optimization solver is applied. Because the optimization framework developed in this work is highly nonlinear, it is not straightforward to provide a starting point. Therefore, the decomposition of the original problem into subproblems is considered, with which solutions can be obtained in confidence without compromising the quality of the optimal solution. In the proposed model, there are four sets of nonlinear equations: the mass balance equations of contaminants, the equations to calculate the pressure drop in pipes, the equations to calculate the capital cost of the pump, and the equations to calculate the supply power of the pump. The proposed solution strategy includes the following four elements: (1) The equations to calculate the capital cost of the pump are linearized by the regression in the format of eq 43. There is a small degree of compromise in the accuracy related to the calculation of pump capital cost, due to the introduction of linearization.

s∈S

costfw s ¼ hour

∑

u∈U

Qs ηs ¼

Equations 2334 give the upper and lower bounds for the pump pressure elevation of the pump. ΔUpump , ΔUpump s s,u , ΔUu,ua , in,pump out,pump pump , ΔUu , and ΔUu,e are the upper bounds while ΔUu pump in,pump , Lpump , Lout,pump , and Lpump represent the Lpump s s,u , Lu,ua , Lu u u,e lower bounds. Binary variables which indicate the existence of the pump are incorporated into these equations. Objective Function. The objective of this optimization model is set to minimize total annualized cost. The objective function is represented in terms of the cost of freshwater, the cost of pipelines, the capital cost of the pump, and the operating cost of the pump. obj function ¼ ð

¼ kQs hour w Fs;u ÞΔPspump

ð40Þ

ΔPu;e ¼

ua 0:04F0:8 μ0:2 uu;ua 2:4 π0:6 du;ua

ua Þ1:4 ðFu;ua out 0:04F0:8 μ0:2 uu;e 2:4 π0:6 du;e ua Þ1:4 ðFu;e

ð44Þ ð45Þ ð46Þ

Note that eqs 4446 are univariate equations of the corresponding flow rate. However, in this case, they cannot be fit properly by only one linear equation. Therefore, SOS2 technique has been adopted to approximate these equations by dividing 852



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them into several linear segments. For example, eq 44 can be approximated by NL

∑ ΔPs,q u λwq q¼1

ΔPs, u ¼

are likely to be close to the maximum outlet concentration allowed, in order to reduce freshwater consumption of individual operations. Fixing outlet concentration in the model leads to an infeasible solution because not all of the contaminants are required to be at the maximum outlet concentration. Iteration mechanism has been introduced to regain the feasibility of the model by relaxing eq 3 to eq 56 with the introduction of slack variables. From subproblem P1, the water flow rate is optimized with fixed outlet concentration. The water flow rate obtained from P1 is now fixed in the subproblem P2, and outlet concentration is optimized with minimizing slack variables. The value of Cout,k c,ua is updated through iteration until convergence criteria are met.

ð47Þ

together with w Fs;u ¼

NL

∑ Fs;uw;q λwq q¼1

∑q λwq ¼ 1

ð48Þ ð49Þ

ΔPqs,u, the pressure drop in the pipe between the freshwater source s ∈ S and the water-using operation u ∈ U, is obtained w from eq 44 when the flow rate is Fw,q s,u . λq is a set of SOS2 variables. Equation 44 is replaced by eqs 47, 48, and 49. Equations 45 and 46 are also substituted in the same way, as shown in eqs 50 and 53. ΔPu;ua ¼

NL

∑

q¼1

q ΔPu;ua λwq

NL

ua;q ua λq ∑ Fu;ua

ua ¼ Fu;ua

¼ Fut Cout c, u

ΔPu;e ¼

NL

∑ ΔPu;eq λoutq q¼1 NL

out;q out Fu;e λq ∑ q¼1

∑q λoutq ¼ 1

" u ∈ U, c ∈ C

ð56Þ

The decomposition strategy had been coupled with the linearization and the substitution together, which completes the overall solution strategy for the optimization. The reformulated objective function for P1 is given as

ð50Þ ð51Þ

obj function ¼ ð

q¼1

∑q λuaq ¼ 1

out ¼ Fu;e

k ∑ Fs,wu Cfwc, s þ ua∑∈ U Fua,ua u Cout, c, ua þ Mc, u þ sc, u

s∈S

ð52Þ

∑∈ costfws Þ þ ð ∑∈ u ∑∈ U costfw,s, Upipe

out, pipe Þ ∑∈ ua∑∈ U costua,ua, pipe ua þ ∑ ∑ costu, e ∈ u∈U þ f ∑ ½vΔPspump ð ∑ Fs,wu Þ þ zBpump s ∈ u∈U w pump þ ∑ ∑ ½vΔPs,pump ua Fs, u þ zBs, u s ∈ Su ∈ U ua pump þ ∑ ∑ ½vΔPu,pump ua Fu, ua þ zBs, ua u ∈ U ua ∈ U pump þ ∑ ½vΔPuin, pump Fut þ zBin, u ∈ pump þ ∑ ½vΔPuout, pump Fut þ zBout, u ∈ out pump þ ∑ ∑ ½vΔPu,pump e Fu, e þ zBu, e g u∈Ue∈E þ ½ ∑ αvΔPspump ð ∑ Fs,wu Þ hour ∈ u∈U w þ ∑ ∑ αΔPs,pump u Fs, u hour s ∈ Su ∈ U ua þ ∑ ∑ αΔPu,pump ua Fu, ua hour u ∈ U ua ∈ U þ ∑ αΔPuin, pump Fut hour þ ∑ αΔPuout, pump Fut hour ∈ ∈

þ

ð53Þ ð54Þ ð55Þ

(3) Equation 42 is substituted in eqs 40 and 41. Then, eqs 40 and 41 are substituted into the corresponding terms of the objective function. By these substitutions, the nonlinear terms to calculate the power required for pumping no longer appear in the constraints of the model. All of them are given in the objective function. This decreases the computational difficulty to solve the optimization model. (4) With the linearization of equations above, the only remaining set of nonlinear constraints in the model is the mass balance equations of contaminants. The nonlinear terms come from the multiplication of outlet concentration and flow rate. A concept to decompose the MINLP problem into MILP and LP subproblems had been developed by Gunaratnam et al.,5 which had been extended for proposing a new solution strategy in this study. The assumption is made such that the value of the outlet concentration is fixed, which helps to eliminate the nonlinear terms in the mass balance equations. With this strategy, the MINLP problem is decomposed into two subproblems, P1 (MINLP model) and P2 (LP model). In P1, the projection of the nonlinear constraints onto the concentration domain is made and the maximum outlet concentrations are assumed as initial guesses for the outlet concentrations for all of the contaminants out,k out,max = Cc,u ). The selection of these starting points for (Cc,u concentration variables is reasonable and effective, as outlet concentrations of water-using operations in the optimal solution

þ

out ∑ ∑ αΔPu,pump e Fu, e hour þ ω ∑ ∑ sc, u ∈ c∈C

u∈Ue∈E

ð57Þ

where ω is a weight factor for infeasibilities. In eq 57, the following parameters are used. v ¼ 0:000 14;

z ¼ 3401:1;

α ¼ 3:55 108

The subproblem P1 is formulated as an MINLP problem which is subject to constraints eqs 1 and 2, eqs 411, eq 13, eqs 2034, eqs 3639, and eqs 4756 with an objective function of eq 57. With the exception of the objective function, all of the constraints in this problem are linear. Subproblem P2 is the LP model which includes eq 59 with projection on to the flow rate domain. The outlet concentration of water-using operation becomes variable. 853



ARTICLE k ua/ t/ w,k w/ Step 2: Fw/ s,u , Fua,u, and Fu are forwarded to (LP) (Fs,u = Fs,u ; ua/ t,k t/ out/ Fua,u; Fu = Fu ), and outlet concentration, Cc,u , is obtained from solving (LP)k. out out,max , Step 3. The value of Cout c,u is checked. If Cc,u is greater than Cc,u out out,max Cc,u is adjusted to be Cc,u .

The objective function of P2 is set as follows: obj function ¼

∑∈ c ∑∈ C sc, u

ð58Þ

Subproblem P2 is subject to

∑ Fs,w,uk Cfwc, s þ ua∑∈ U Fua,ua, ukCout c, ua þ Mc, u þ sc, u

s∈S

¼ Fut, k Cout c, u

" u ∈ U, c ∈ C

Table 5. Results for Example 1

ð59Þ

cost

ua,k t,k Fw,k s,u , Fua,u, and Fu are fixed flow rates obtained from P1 in iteration k. New values for outlet concentration are forwarded to in the next subproblem P1 and are to be used for Cout,k+1 c,ua iteration. Note that the new outlet concentration may violate out is, then, set as the constraint stated in eq 5 in subproblem P1. Cc,u out,max out out,max when Cc,u g C0,u . Cc,u Overall, the iterative procedure proposed in this study is summarized as below. (MINLP)k and (LP)k are the formulation of (MINLP) and (LP) at iteration k, respectively. Step 1: In the first iteration (k = 1), the outlet concentration of the stream leaving the water-using operations is fixed with its out,k out,max = Cc,u ) in (MINLP).1 maximum allowable values (Cc,u ua/ The corresponding optimal flow rate values are Fw/ s,u , Fua,u, t/ and Fu .

overall cost

freshwater

piping

pumping

scenario 1

$1 489 190

$1 422 225

$55 574

$11 391

scenario 2

$1 195 905

$1 137 780

$47 785

$10 340

scenario 3

$1 180 067

$1 137 780

$36 081

$6 206

scenario 4

$1 180 366

$1 137 780

$36 081

$6 505

Table 1. Operating Data for Water-Using Operation in Example 1 operation

max inlet

max. outlet

mass load

concn (ppm)

concn (ppm)

(g/h) 2 000

O1

0

100

O2

50

100

5 000

O3

50

800

30 000

O4

400

800

4 000

Figure 3. Network configuration for example 1 in scenario 1.

Table 2. Distance Matrix (m) for Example 1 O1

O2

O3

O4

discharge

S1

50

70

210

370

400

O1

0

30

80

90

270

O2

30

0

65

35

320

O3 O4

80 90

65 35

0 80

80 0

95 75

Table 3. Pressure Drops for Water-Using Operations in Example 1 operation

pressure drop (Pa)

operation

pressure drop (Pa)

O1

500

O3

100

O2

1000

O4

700


Table 4. Computational Data for Example 1 scenario 1

no. of continuous variables no. of binary variables overall computational timea (s) a

scenario 2

scenario 3

scenario 4

MINLP

LP

MINLP

LP

MINLP

LP

MINLP

LP

180 24

9

180 24

9

225 53

9

225 53

9

0.016

0.016

0.016

0.016

Intel(R) Core2 Duo CPU T8300 @ 2.40 GHz. 854



ARTICLE

Table 7. Distance Matrix (m) for Example 2 O1

Table 6. Operating Data for Water-Using Operation in Example 2 max inlet

max outlet

mass

concn (ppm)

concn (ppm)

load (g/h)

operations

contaminant

O1

A

0

50

1 250

B C

0 0

100 50

2 500 1 250

A

0

100

7 000

B

0

300

21 000 42 000

O4

O5

O6

O7

O4

O5

O6

O7

discharge

S1

50

63

70

87

75

37

62

200

O1

0

125

175

200

225

150

75

150

O2

125

0

15

50

100

125

150

60

O3

175

15

0

50

25

125

150

50

O4

200

50

50

0

75

200

175

50

O5

225

100

25

75

0

250

150

100

O6

150

125

125

200

250

0

50

112

O6

75

150

150

175

150

50

0

125

operation

O3

O3

Table 8. Pressure Drops for Water-Using Operations in Example 2


O2

O2

C

0

600

A

20

150

4 550

B

50

400

12 250

C

50

800

26 250

A B

50 110

600 450

22 000 13 600

C

200

700

20 000

A

20

500

3 840

B

100

650

4 400

C

200

400

1 600

A

500

1100

30 000

B

300

3500

160 000

C A

600 0

2500 15

95 000 675

B

0

400

18 000

C

0

35

1 575

pressure drop (Pa)

operation

pressure drop (Pa)

O1

800

O5

700

O2

1500

O6

1500

O3

1000

O7

1500

O4

2000

4. CASE STUDY Two examples are given in this paper to illustrate the proposed approach, and four scenarios are discussed in each example for comparison. Scenario 1: An objective function is to minimize freshwater consumption. The water network is designed without water reuse and recycling and optimization of pumping systems. Scenario 2: An objective function is to minimize freshwater consumption. The reuse and recycling of water is allowed, while pumping systems are not optimized. Scenario 3. Total annualized cost is minimized while water recycling is allowed. Pumping systems optimization is carried out, and all of the available locations of pumps identified in the node superstructure are fully considered. Scenario 4. Minimum total annualized cost is sought when water recycle is allowed, but water can only be pumped together at the beginning of the network. Because water reuse and recycling is forbidden, the upper bounds of water reuse and recycling streams are set to be zero for scenario 1. When the optimization of pumping systems is not considered in the model, for example, scenarios 1 and 2, eqs 2034 are not included in the optimization accordingly. The results from scenario 1 serve as a base case for comparison with other scenarios. It is worth mentioning that the pumping cost in scenarios 1 and 2 is calculated under the assumption that all of the pipelines and water-using operations have their own pump and the pressure head elevation of the pumps is the same as the pressure drop of the water-using operations/pipelines attached to the pumps. The results from scenario 3 include water network design information, including optimal distribution of water, as well as optimal location of pump and its capacity simultaneously. In scenario 4, the restriction on the location of the pump has been implemented by setting the upper bound of the pressure eleva= Upump tion of the nonexisting pump to be zero (Upump s,u u,ua = in,pump out,pump pump = Uu = Uu,e = 0). Uu For both examples, it is assumed that a single uncontaminated freshwater source is available at the cost of $1.47/ton. It has been assumed that the annual operating hour is 8600 h, and an

Step 4. The new set of concentration values replaces the concentration values in the MINLP problem in the next out,k+1 out/ = Cc,u . The corresponding flow iteration (k = k + 1); Cc,u rate value is obtained from solving (MINLP)k+1. The iterative procedure continues until the sum of slack variables is small enough, which indicates that all of the constraints in the model are not violated. It should be noted that decomposition method developed by Gunaratnam et al.5 requires additionally an MINLP optimization after successive iterations between MILP and LP subproblems, while the developed decomposition method is based on iterative optimization of MINLP and LP subproblems, which effectively deals with nonlinearity imposed from the simultaneous optimization of water network and pumping systems. 855



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Table 9. Computational Data for Example 2 scenario 2 scenario 1

scenario 2

MINLP LP MINLP LP no. of continuous variables

470

no. of binary variables

63

overall computational timea (s) a

0.048

43

470

scenario 3

(simplified network) MINLP

LP

470

43

43

63

MINLP LP 565

63

0.047

scenario 3

43

141 0.047

0.079

scenario 4

(simplified network) MINLP 565

LP 43

141

scenario 4 MINLP LP 565 141

0.046

.094

43

(simplified network) MINLP

LP

565

43

141 0.062

Intel(R) Core2 Duo CPU T8300 @ 2.40 GHz.

Table 10. Results for Example 2 cost overall cost

freshwater

piping

pumping

scenario 1

$3 379 842

$3 323 256

$42 849

$13 737

scenario 2

$2 408 159

$2 321 005

$69 629

$17 525

scenario 2 (simplified network)

$2 482 935

$2 405 676

$61 403

$15 856

scenario 3

$2 398 693

$2 325 293

$62 588

$10 812

scenario 3 (simplified network) scenario 4

$2 475 222 $2 424 324

$2 406 721 $2 351 159

$58 058 $55 905

$10 443 $17 260

scenario 4 (simplified network)

$2 501 019

$2 427 960

$52 937

$20 122

annualized factor is 0.1. The flow velocity in all the pipes has been assumed to be 2 m/s, and the parameters for piping cost are ua out fw ua out afw s,u = au,ua = au,e = 3603.4; bs,u = bu,ua = bu,e = 124.6 (the chemical engineering index is based on 389.1, which is then updated with 666.0.).5 Carbon steel has been chosen as the material of construction. Besides, the pump in this example has been assumed as a centrifugal pump. The parameters for the capital cost of pumps are given by r = 1.97 103 and w = 0.35 (the chemical engineering index is based on 435.8, which is then updated with 666.0.), and 10% of the purchased cost is added for installation cost.14 After the linear regression, h and g in eq 43 are obtained (h = 431.69; g = 3401.1). Moreover, the parameters for the operating cost of the pump is given by k = 0.1085 which relates to the price of electricity. The optimization platform employed in this work is GAMS (general algebraic modeling system).20 In this study, CPLEX is chosen as a solver for the LP problem, and DICOPT is used as a solver for the MINLP problem with CONOPT as an NLP solver and CPLEX as an MIP solver. Example 1. The first example is a single-contaminant problem which is taken from Argaez.21 Table 1 shows the operating data for the water-using operations, which include the maximum inlet and outlet concentrations and the mass load. Table 2 presents the distance matrix for example 1. The pressure drops of water-using operations are given in Table 3. Note that this simple example can be solved without iterative solution strategies in all scenarios. The reason is that this example is a single-contaminant problem in which the outlet concentrations of that single contaminant tends to reach their maximum allowable concentrations in the design of optimal network structure. In this example, the assigned out,k out,max = Cc,u ) which have values of outlet concentrations (Cc,u been given in the first step of solution strategy coincidently meet the requirement in the optimal network. Therefore, the slack variables become zero, and no further iteration is necessary. The computational data and results for example 1 are shown in Tables 4 and 5, respectively.

Scenario 1: Freshwater is supplied in a parallel configuration, as shown in Figure 3. In Figure 3, the numbers without an underline represent the flow rate of each stream (unit, tons/h), and numbers with underline mean the pressure head elevation of the pump (unit, kPa). Scenario 2: The resultant network configuration is shown in Figure 4. Compared with the design in scenario 1, it shows 19.7% reduction in total annualized cost. Scenario 3: From the optimization, it has been identified to install two pumps in the networks, which is a much simpler pumping arrangement compared to scenario 2, as shown in Figure 5. Scenario 4: In this example, although the restriction on the location of the pump is imposed, the optimal water distribution is the same as that of scenario 3, while the pumping cost in this scenario is increased from $6206 to $6505. Example 2. A multiple-contaminant water network consisting of seven water-using operations and three types of contaminants is considered, which had been adapted from Chen et al.6 The operating data for water-using operations are presented in Table 6. Table 7 shows the distance matrix for example 2. The pressure drops for water-using operations are given in Table 8. Also, the optimization has been carried out with four scenarios. For all scenarios, one iteration between subproblems P1 and P2 is required until the optimal solution is achieved. The computational data and results for example 2 are shown in Tables 9 and 10, respectively. Scenario 1: The network configuration with one-through use of water is illustrated in Figure 6. The expenditure of freshwater, pipelines, capital, and operating cost of the pump are $3 323 256, $42 849, and $13 737, respectively. The total annual cost is determined to be $3 379 842. Scenario 2: As shown in Figure 7, the network design is very complicated. To control the complexity of the network, the additional constraint eq 12 is introduced, and the maximum number of inlet streams for a single water-using operation is set to be 2. A simplified network configuration is provided in Figure 8. 856



ARTICLE

Figure 6. Network configuration for example 2 in scenario 1. Figure 9. Network configuration for example 2 in scenario 3.



= 2) constraint on the maximum number of inlet streams (NSmax u is obtained, the total annual cost increases by 3.2%, compared with the network shown in Figure 9. The rise in total annualized cost mainly results from the increase in freshwater intake. Scenario 4: The optimal configuration is given in Figure 10. In this example, freshwater cost has a dominant impact on the network design. When Figures 9 and 10 are compared, the consideration of economic trade-off between freshwater cost, piping cost, and pumping cost in scenario 3 simplifies indirectly the design complexity of water networks. The simplification of network configuration is often required from operational purpose, as the large number of streams coming into the same operation may not be favored, due to potential difficulties in process control or operability. As shown in Figures 9 and 10, three different sources of water need to be merged and be supplied to water-using operation 3. This operational constraint can be readily added into the optimization model by setting the = 2), with maximum number of inlet streams as 2 (i.e., NSmax u which this complex junction of water streams can be avoided at the expense of increase in overall cost.

Figure 8. Simplified network configuration for example 2 in scenario 2.

Note that the piping cost reduces from $69 629 to $61 403, and pumping costs including capital cost and operating cost reduces from $17 525 to $15 856. However, the total annual cost increases from $2 408 159 to $2 482 935 due to the growth in freshwater consumption. Scenario 3: Figure 9 shows the optimal design network considering water distribution, capacity of pumps, and location of pumps at the same time. When the simplified network by imposing the

5. CONCLUSION A new automated method for the design of water networks has been developed for minimization of the total annualized cost 857



ARTICLE

Aua u,ua = cross-sectional area of a pipe connecting operation u ∈ U and operation ua ∈ U out = cross-sectional area of a pipe connecting operation u ∈ U Au,e and discharging point e ∈ E

which consists of freshwater cost, piping cost, and pumping cost. The introduction of pressure-drop constraints in the proposed model improves the network design through consideration of the optimal allocation and capacity of the pump. The economic tradeoffs between freshwater cost, piping cost, and pumping cost are carried out simultaneously, while through the manipulation of binary variables in the optimization model, the complexity of water networks can be effectively controlled, and other practical constraints (e.g., forbidden location for a pump) can be readily added. A robust and practical approach for optimizing the developed MINLP model has been proposed. The solution strategy includes the linearization of nonlinear constraints, the manipulation of equations to keep all the constraints in the model linear, except in the objective function, and an iterative scheme between MINLP and LP subproblems. The new automated design method not only considers technoeconomic impacts of water reuse, piping, but also the arrangement of water pumping in the network. Various pumping options are considered in the case studies presented in this paper, which provides valuable understanding on the implication of pressure drop in the water network design. In the future, the pump design problem addressed in this paper will be further extended in the total water systems in which both water-using operations and water-treating operations can be considered as a whole. It is because there are strong and complex design interactions between water-using operations and water-treating operations.4,5,13,22,23, which should be dealt with in an integrated manner.

Variables for Pressure Calculation

Pin u = pressure at the mixing point of operation u ∈ U Pout u = pressure at the splitting point of operation u ∈ U Pe = pressure at the discharge point e ∈ E ΔPs,u = pressure drop in pipe between freshwater source s ∈ S and operation u ∈ U ΔPu,ua = pressure drop in pipe between operation u ∈ U and operation ua ∈ U ΔPu,e = pressure drop in pipe between operation u ∈ U and discharge point e ∈ E = pressure elevation of the pump at freshwater source s ∈ S ΔPpump s = pressure elevation of the pump between water source s ∈ ΔPpump s,u S and operation u ∈ U = pressure elevation of the pump between operation u ∈ U ΔPpump u and operation ua ∈ U = pressure elevation of the pump in the front of operation ΔPin,pump u u∈U = pressure elevation of the pump at the end of operation ΔPout,pump u u∈U = pressure elevation of the pump between operation u ∈ U ΔPpump u,e and discharge point e ∈ E Variables for Cost Calculation

costfw s = cost of freshwater s ∈ S = piping cost of the connection between freshwater costfw,pipe s,u s ∈ S to operation u ∈ U = piping cost of the connection between operation costua,pipe u,ua u ∈ U to operation ua ∈ U = piping cost of the connection between operation costout,pipe u,e u ∈ U to discharge point e ∈ E = capital cost of pump at freshwater source s ∈ S costpump,cap s = capital cost of pump between supply water source costpump,cap s,u s ∈ S and operation u ∈ U = capital cost of pump between operation u ∈ U and costpump,cap u,ua operation ua ∈ U = capital cost of pump in the front of operation u ∈ U costin,pump,cap u = capital cost of pump at the end of operation u ∈ U costout,pump,cap u = capital cost of pump between operation u ∈ U and costpump,cap u,e discharge point e ∈ E = operating cost of pump at supply water source s ∈ S costpump,op s = operating cost of pump between supply water source costpump,op s,u s ∈ S and operation u ∈ U = operating cost of pump between operation u ∈ U and costpump,op u,ua operation ua ∈ U = operating cost of pump in the front of operation u ∈ U costin,pump,op u = operating cost of pump at the end of operation u ∈ U costout,pump,op u = operating cost of pump between operation u ∈ U and costpump,op u,e discharge point e ∈ E

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel.: +82 2 2220 2331.

’ ACKNOWLEDGMENT We express our gratitude to the Process Integration Research Consortium (PIRC) at the University of Manchester for funding the research. J.-K.K. also acknowledges that this work is the outcome of a Manpower Development Program for Energy supported by the Ministry of Knowledge and Economy (MKE). ’ SETS, PARAMETERS, AND VARIABLES Set

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 discharge point},e = 1, 2, ..., Ne U = {u|u is a water-using operation in the water network}, i = 1, 2, ..., Nu Variables for Water Flow

Fws,u = freshwater flow from water source s∈S to operation u∈U Ftu = total water flow through the operation u∈U out = water flow from operation u∈U to discharge point e∈E Fc,u ua = water flow from operation u∈U to operation ua ∈ U Fc,u

Binary Variables

Bfw s,u = binary variable for stream from water source s ∈ S to operation u ∈ U Bua u,ua = binary variable for stream from operation u ∈ U to operation ua ∈ U out = binary variable for stream from operation u ∈ U to Bu,e discharge point e ∈ E = binary variable for pump at water source s ∈ S Bpump s

Variable for Concentration out Cc,u = concentration c ∈ C of stream leaving operation u ∈ U

Variables Associated with Cross-Sectional Area of Pipeline

Afw s,u = cross-sectional area of a pipe connecting freshwater source s ∈ S and operation u ∈ U 858



ARTICLE

Bpump = binary variable for pump between water source s ∈ S and s,u operation u ∈ U Bpump u,ua = binary variable for pump between operation u ∈ U and operation ua ∈ U = binary variable for pump in the front of operation Bin,pump u u∈U = binary variable for pump at the end of operation u ∈ U Bout,pump u = binary variable for pump between operation u ∈ U and Bpump u,e environment e ∈ E

(2) Castro, P.; Matos, H.; Fernandes, M. C.; Nunes, C. P. (1999) Improvements for mass-exchange networks design. Chem. Eng. Sci. 1999, 54 (11), 1649–1665. (3) El-Halwagi, M. M.; Gabriel, F.; Harell, D. Rigorous graphical targeting for resource conservation via material recycle/reuse networks. Ind. Eng. Chem. Res. 2003, 42, 4319–4328. (4) Kuo, W. C. J.; Smith, R. Designing for the interactions between water-use and effluent treatment. Chem. Eng. Res. Des. 1998, 76 (A3), 287–301. (5) Gunaratnam, M.; Alva-Argaez, A.; Kokossis, A.; Kim, J. K.; Smith, R. Automated design of total water systems. Ind. Eng. Chem. Res. 2005, 44 (3), 588–599. (6) Chen, C. L.; Hung, S. W.; Lee, J. Y. Design of inter-plant water network with central and decentralized water mains. Comput. Chem. Eng. 2010, 34 (9), 1522–1531. (7) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Targeting for total water network. 1. Waste stream identification. Ind. Eng. Chem. Res. 2007, 46, 9107–9113. (8) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Targeting for total water network. 2. Waste treatment targeting and interactions with water system elements. Ind. Eng. Chem. Res. 2007, 46, 9114–9125. (9) Bandyopadhyay, S.; Cormos, C. C. Water management in process industries incorporating regeneration and recycle through a single treatment unit. Ind. Eng. Chem. Res. 2008, 47, 1111–1119. (10) Hallale, N. A new graphical targeting method for water minimisation. Adv. Environ. Res. 2002, 6 (3), 377–390. (11) Prakash, R.; Shenoy, U. V. Targeting and design of water networks for fixed flowrate and fixed contaminant load operations. Chem. Eng. Sci. 2005, 60 (1), 255–268. (12) Foo, D. C. Y.; Manan, Z. A.; Tan, Y. L. Use cascade analysis to optimize water networks. Chem. Eng. Progress 2006, 102 (7), 45–52. (13) Foo, D. C. Y. State-of-the-art review of pinch analysis techniques for water network synthesis. Ind. Eng. Chem. Res. 2009, 48, 5125– 5159. (14) Smith, R. Chemical Process Design and Integration; Wiley: West Sussex, U.K., 2005. (15) Huang, C. H.; Chang, C. T.; Ling, H. C.; Chang, C. C. A mathematical programming model for water usage and treatment network design. Ind. Eng. Chem. Res. 1999, 38, 2666–2679. (16) Feng, X.; Seider, W. D. New structure and design methodology for water networks. Ind. Eng. Chem. Res. 2001, 40, 6140–6146. (17) Zheng, X. S.; Feng, X.; Shen, R. J.; Seider, W. D. Design of optimal water-using networks with internal water mains. Ind. Eng. Chem. Res. 2006, 45, 8413–8420. (18) Poplewski, G.; Walczyk, K.; Jezowski, J. Optimization-based method for calculating water networks with user specified characteristics. Chem. Eng. Res. Des. 2010, 88 (1A), 109–120. (19) Kim, J. K.; Smith, R. Automated retrofit design of cooling-water systems. AIChE J. 2003, 49 (7), 1712–1730. (20) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R.; GAMS: A User's Guide; GAMS Development: Washington, DC, 2005. (21) Argaez, A. A. Integrated design of water system. Ph.D. Thesis, University of Manchester Institute of Science and Technology, Manchester, U.K., 1999. (22) Klemes, J.; Friedler, F.; Bulatov, I.; Varbanov, P.; Sustainability in the Process Industry—Integration and Optimization; McGraw Hill: New York, 2011. (23) Li, B. H.; Chang, C. T. Evolution of water-using networks with multiple contaminants. Chem. Eng. Trans. 2011, 25, 599–604.

Parameters for Concentration Bounds in,max Cc,u = maximum inlet concentration of contaminant c ∈ C to operation u ∈ U out,max = maximum outlet concentration of contaminant c ∈ C Cc,u from operation u ∈ U

Parameters for Distances between Water Sources and Sinks

dfw s,u = distance between water source s ∈ S and operation u ∈ U dua u,ua = distance between operation u ∈ U and operation ua ∈ U out = distance between operation u ∈ U and discharge point e ∈ E du,e Parameters for the Flow Velocity in the Pipeline

vfw s,u = velocity in pipes between water source s ∈ S and operation u∈U vua u,ua = velocity in pipes between operation u ∈ U and operation ua ∈ U vout u,e = velocity in pipes between operation u ∈ U and discharge point e ∈ E Regression Parameters for Piping Costs

afw s,u = cost parameters “a” of freshwater flow from source s ∈ S to operation u ∈ U bfw s,u = cost parameters “b” freshwater flow from source s ∈ S to operation u ∈ U aua u,ua = cost parameters “a” of flow from operation u ∈ U to operation ua ∈ U bua u,ua = cost parameters “b” of flow from operation u ∈ U to operation ua ∈ U aout u,e = cost parameters “a” of flow from operation u ∈ U to discharge point e ∈ E bout u,e = cost parameters “b” of flow from operation u ∈ U to discharge point e ∈ E Other Parameters fw Cc,s = concentration of contaminant c ∈ C in water source s ∈ S out,k = outlet concentration of contaminant c ∈ C in stream Cc,u leaving operation u ∈ U in iteration k Hour = annual operating hour Mc,u = mass load of contaminant c ∈ C of water using operation u∈U = the maximum number of inlet streams entering operation Nmax u u∈U Ps = pressure at freshwater source s ∈ S ΔPu = pressure drop of operation u ∈ U ψs = unit cost of freshwater from water source s ∈ S η = efficiency of pump F = density μ = viscosity

’ REFERENCES (1) Wang, Y. P.; Smith, R. Wastewater minimisation. Chem. Eng. Sci. 1994, 49 (7), 981–1006. 859


Optimization of Water Systems with the Consideration of Pressure

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