Simultaneous Optimization of Heat-Integrated Water Allocation

Article pubs.acs.org/IECR

Simultaneous Optimization of Heat-Integrated Water Allocation Networks Using the Mathematical Model with Equilibrium Constraints Strategy Li Zhou, Zuwei Liao,* Jingdai Wang, Binbo Jiang, Yongrong Yang, and Huanjun Yu State Key Laboratory of Chemical Engineering, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, P. R. China

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S Supporting Information *

ABSTRACT: A general optimization model is proposed for the simultaneous optimization of integrated water allocation and heat exchanger networks (WAHEN). An improved superstructure and mathematical model for the WAHEN problem is presented. Complementary formulations are applied to model the discrete decisions, and a mathematical model with equilibrium constraints (MPEC) is formulated. Both the MINLP model and the MPEC model are developed and solved for the problem. The effectiveness of the proposed model is illustrated with two case studies. A comparison between the results obtained by the proposed model and those obtained from the model in the literature is carried out. Case studies show that the proposed simultaneous methodology performs well and is efficient.

1. INTRODUCTION Water and energy are indispensable resources for the normal functioning of process industries. Large amounts of water are consumed in petroleum refineries,1 breweries,2,3 and paper mills4 by washing, liquid−liquid extraction, absorption, sterilization, and so on. Energy is considered as one of the dominant contributors to the daily operation cost of the process industries. To achieve better economic performances, water and energy integration are considered important for the process industry, which has long been the research focus of the process system engineering community. Over the past few decades, numerous research works have been carried out for the synthesis of water networks (WN)5−7 and heat exchanger networks (HEN).8 The synthesis techniques addressing the efficient utilization of water resources can be classified into two categories, optimization-based mathematical programming method and insight-based pinch analysis technology. For more comprehensive information, see the thorough review works by Jezowski5 and Foo.7 The primal scientific work focusing on HEN synthesis dates back to the middle 1960s.9,10 Since then, this area has received considerable attentions and an excellent review has been contributed by Furman and Sahinidis.8 Quality and temperature are two important aspects for the water utilization. In the process industry, water is not only consumed to remove the desired/undesired materials but also used for heating and cooling. Strong interaction exists between the WN and the HEN, as water carries both the exchanged matter and energy during its utilization. Despite the strong interaction, the WN and the HEN was not investigated jointly until the middle 1990s. In 1995, Huang and Edgar11 introduced the concept of combined exchange network (CEN) for mass and heat exchanger networks and proposed a three-step knowledge-based synthesis approach. Later on, a great number of conceptual tools12−18 were developed. Savulescu19 introduced a two-dimensional grid diagram for the grassroots design © 2015 American Chemical Society

of a water and heat exchanger network. Systems with no water reuse20 and with maximum water reuse21 were both considered in their work. Also, other insight-based methods, such as source-demand energy composite curves,22 graphical thermodynamic rules,23 a heat surplus diagram,24 and a water energy balance diagram25 were presented. Superstructure-based mathematical programming techniques were also widely investigated, which can be classified into two categories: sequential approaches and simultaneous methods. The sequential approaches26−29 were investigated first. Bagajewicz et al.29 presented a mathematical method for energy efficient water network design based on necessary conditions of optimality30 and a heat trans-shipment model. Sequential optimization steps including minimum freshwater targeting, minimum utility targeting, heat exchanger network design and stream merging were proposed in their work. Feng et al.31 analyzed the energy performance of different water networks and concluded that reducing the number of temperature local fluctuations was beneficial for the energy performance. Based on the analysis, they proposed a new mathematical model for the water network design taking energy performance into consideration, followed by the synthesis of HEN. Liao et al.32 introduced a two-step systematic optimizing method. In their model, a trans-shipment model was adopted to target the number of heat exchange matches, on the basis of which a modified stage-wise superstructure27 was performed. Another two-step methodology was later proposed by Boix et al.33 In the first step, the freshwater consumption, energy consumption, and the number of interconnections and heat exchangers were taken into consideration, and an MILP model was solved. In the second step, an MINLP procedure was carried out to Received: Revised: Accepted: Published: 3355

May 14, 2014 December 29, 2014 March 12, 2015 March 12, 2015 DOI: 10.1021/ie501960e Ind. Eng. Chem. Res. 2015, 54, 3355−3366

Article

Industrial & Engineering Chemistry Research

Figure 1. Superstructure for the synthesis of heat integrated water network.

improve the results obtained from the first step. When applying sequential approaches, the problem was divided into several subproblems and solved systematically. Though the problem can be solved easier in this way, the interconnections between the synthesis of WN and HEN cannot be fully explored, which may lead to suboptimal results. Consequently, simultaneous approaches34−37 were investigated. Leewongtanawit and Kim38 proposed a simultaneous design methodology that combined the trans-shipment model with a hyperstructure model. Their model was solved by an iterative procedure. Dong et al.39 presented a modified state-space representation to capture richer structural characteristics of the combined network. An interactive iteration method was also proposed for the solving procedure. Bogataj and Bagajewicz40 modified the stage-wise superstructure41 for allowing cross stream mixing and splitting in the HEN. Ahmetovic and Kravanja42 also modified the stagewise superstructure with cross stream mixing and splitting arranged outside the HEN. Recently, Yang and Grossmann43 proposed a water targeting model for the simultaneous optimization of WAHEN. The HEN superstructure proposed by Yee and Grossmann41 was frequently adopted by researchers in both the sequential and simultaneous synthesizing methods. The main difference between these modified stage-wise superstructures is the identification of the hot and cold streams. Different water network structures feature different hot and cold streams, resulting in HENs with different economic performance. Therefore, trade off exists between WN and HEN. To investigate the trade off, Bogataj and Bagajewicz40 labeled the

hot and cold streams by presolving a WN model. Chen et al.44 and Ahmetovic and Kravanja42 introduced mathematical formulations using binary variables to identify hot and cold streams. By assuming that the temperature and flow rates of the streams changed monotonously, Liao et al.32 identified the hot and cold streams in a general way without any binary variables. In the proposed manner of stream identification, special possibilities in stream splitting and mixing are involved in the HEN superstructure. In this paper, we adopt the stream identification method proposed by Liao et al.,32 and extend the original sequential approach to a simultaneous method. An improved superstructure and mathematical model for the combined water network and heat exchanger network is presented. The stagewise superstructure32,41 is incorporated for the HEN design without any previous targeting procedure. Possible network configurations featuring stream splitting and mixing as well as isothermal and nonisothermal mixing are included. The temperatures of the water streams in the HEN are treated as optimizing variables. Furthermore, complementary formulations are applied to model the binary decisions in the model. The MINLP model is reformulated and solved as an MPEC model. Two case studies are presented to indicate the efficiency of the proposed method in solving the simultaneous optimization problem of WAHEN. A comparison between the MINLP model and the MPEC model is carried out. 3356

DOI: 10.1021/ie501960e Ind. Eng. Chem. Res. 2015, 54, 3355−3366

Article


Figure 2. Stage-wise superstructure for heat exchange network.

Figure 3. Illustration of the hot and cold stream definition.

2. PROBLEM STATEMENT The WAHEN problem addressed in this article can be stated as follows. Given is a set of process units with a certain mass load of contaminant to be removed, by consuming water of preferable quality and temperature. Available resources include freshwater with certain temperature and hot and cold utility for heating and cooling task. The effluent is required to be discharged at a specific temperature. It is desired to build up a mathematical programming model to determine the network configuration which satisfies the units operation requirements while minimize the total cost. The proposed method is based on the following assumptions: 1) single contaminant; 2) continuously steady state operation; 3) isothermal operation; 4) mass transferred to the water stream are negligible comparing to the stream flow rate.

requirement, freshwater is preheated in the HEN, or it can be directly sent to the WN and mixed with other hot streams. Water generated at the outlet of each operation is identified as hot or cold streams. The identification process will be discussed later in this section. The hot and cold streams can be sent back to the WN for reuse before or after heat exchange in the HEN. It should be noted that freshwater is not allowed to be directly discharged as wastewater as this is a waste of freshwater. Both direct and indirect heat exchange are considered in the model. Inside the HEN, direct heat exchange only occurs between the hot streams or the cold streams. In other words, cold streams can only blend with cold streams, and hot streams can only mix with hot streams. Hot stream and cold stream blending can only happen outside the HEN. Labeling of the potential hot and cold water streams is addressed as follows. Identifying the potential hot and cold water streams from the possible connections is a key issue when synthesizing a WAHEN simultaneously. So far, two types of techniques have been reported to tackle this issue. One way is to apply mathematical formulations to determine the binary variables that indicate the stream identities.42,44 The other way is to prespecify the streams without introducing extra binary variables and equations.32 The use of binary variables increases the complexity of the problem as well as the computational cost. In this work, the prespecifying method is employed to label the potential hot and cold streams. It is also assumed that the temperatures of the streams inside the HEN increase/decrease monotonously.

3. SUPERSTRUCTURE FOR THE WAHEN The proposed mathematical programming model for the WAHEN problem is formulated based on the superstructure given in Figure 1 and Figure 2. The superstructure consists of two coupled networks: water network and heat exchanger network interconnected by hot and cold water streams. Figure 1 gives the superstructure for the WN and its interconnection with the HEN, while Figure 2 represents the detailed stage-wise superstructure41 for the HEN. Within the superstructure, the freshwater is provided at a temperature lower than any of the units’ operation temperature. To satisfy the temperature 3357


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Industrial & Engineering Chemistry Research Based on the starting and ending temperatures, three types of streams can be identified. Considering a water stream from source i (Ti) to sink j (Tj), it could be any of three types of streams. (1) Hot water stream (red line). When Ti > Tj, the water stream is labeled as hot stream with the corresponding parameter denoting its identity hoti,j be set to 1. (2) Cold water stream (blue line). When Ti < Tj, the stream is labeled as cold stream with hoti,j be set to −1. (3) Ordinary stream. When Ti = Tj, the stream is an ordinary stream, for example the recycling stream of an operation. The corresponding parameter hoti,j is set to be 0. The ordinary streams only appear inside the water allocation network as they require no heating or cooling. Beside the recycling stream, water reuses between units with the same operation temperature are also recognized as ordinary streams. Though these streams are not shown in the superstructure, they are considered in the proposed model. It should be stressed that, in the superstructure, the blue and red lines representing the cold and hot streams only exist when the corresponding cold and hot streams exist. For example, only a hot stream exists at the outlet of the unit with the highest operation temperature. In this case, only the red lines exist in the final superstructure. As is shown in Figure 3, an operation is possibly connected with two hot streams and two cold streams. At the inlet, the mixing hot water stream coming from other sources is denoted as hot demand stream ld, while the mixing cold water stream coming from other sources is denoted as cold demand stream md. At the outlet, the water stream sent to other demand points as hot stream is denoted as hot source stream ls, while the water stream sent to other demand points as cold stream is denoted as cold source stream ms. For an operation with the highest operation temperature, only a cold demand stream at the inlet and a hot source stream at the outlet are involved. Similarly, for an operation with the lowest operation temperature, only a hot demand stream at the inlet and a cold source stream at the outlet exist. Wastewater is required to be discharged at a specific temperature which is lower than any of the operation temperatures, thus the discharging streams are all hot streams. The calculation of the stream flow rates will be described in the following section.

minTAC = H op ∑ CiFWFii + C hu

∑

+ C cu

Qc l d +

d

l ∈ HD

+

∑

∑ ∑ ∑ C fixedxl ,m,k l∈H m∈C k∈K

∑ ∑ ∑C

AlB, m , k +

∑

C

A mBd , HU +

m ∈ CD

+

∑

C

hu

∑

C fixedxc l d

d

l ∈ HD

Area

A lBd , CU

l d ∈ HD

CFW i ,

C fixedxhmd

md ∈ CD

Area

d

∑

Area

l∈H m∈C k∈K

+

Qhmd

md ∈ CD

i∈w

(1)

cu

where C , and C represent the unit price for fresh water, hot utility, and cold utility, Cfixed and CArea are the fixed charge and area cost coefficient of heat exchangers. xl,m,k, xhmd, xcld represent the existence of heat exchanger, heater and cooler, respectively. The heat exchange area needed is determined by the amount of heat exchanged and the temperature differences between the hot and cold streams. The temperature difference between the hot and cold streams is calculated by the approximation model proposed by Chen.45 Al , m , k =

Q l ,m,k 1/3 ⎡ (Δt 1 + Δt 2 ) ⎤ Ul , m⎢(Δtl1, m , k ·Δtl2, m , k) · l ,m,k 2 l ,m,k ⎥ ⎦ ⎣

∀l ∈ H , m ∈ C , k ∈ K

(2)

A md , HU =

Qhmd 1/3 in ⎡ (Δt d + (THU − Tj)) ⎤ in Umd , HU ⎢Δt md , HU ·(THU − Tj) · m ,HU 2 ⎥⎦ ⎣

∀ md ∈ CD , j ∈ p ̂ , md = j A l d , CU =

(3)

Qc l d 1/3 in ⎡ (Δt d + (Tj − TCU )) ⎤ in Ul d , CU ⎢Δt l d , CU ·(Tj − TCU ) · l ,CU 2 ⎥ ⎦ ⎣

∀l d ∈ HD , j ∈ p ̂ , l d = j

(4)

As the stream temperatures in the HEN are variables, the temperature driving forces for heat transfer are considered as optimization variables. Constraints for the temperature driving force are given as below: Δtl1, m , k ≤ (thil , k − tcom , k ) + Ω(1 − xl , m , k) ∀ l ∈ H, m ∈ C, k ∈ K

4. MATHEMATICAL FORMULATION

(5)

Δtl2, m , k ≤ (thol , k − tcim , k) + Ω(1 − xl , m , k)

4.1. Review of Previous Two-Step Model. Objective Function. The objective is to design a water allocation network with minimal network cost. The network cost considered includes the operation cost and the investment cost. The operation cost of the system consists of the cost for freshwater, hot utility, and cold utility. The investment cost of the network considered includes the investment for heat exchangers, heaters, and coolers. For the cost estimation of heat exchangers, the annualized capital cost model for the conventional shelland-tube heat exchanger39 is adopted.

∀ l ∈ H, m ∈ C, k ∈ K

(6)

The heat amount exchanged between hot stream l and cold stream m at interval k Ql,m,k is determined by eq 7 and 8.

∑

Fhl , k (thil , k − thol , k )cp =

Q l ,m,k

∀ l ∈ H, k ∈ K

m∈C

(7)

Fcm , k(tcom , k − tcim , k)cp =

∑ Q l ,m,k

∀ m ∈ C, k ∈ K

l∈H

(8) 3358


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Industrial & Engineering Chemistry Research thil , k ≥ thol , k

∀ l ∈ H, k ∈ K

tcom , k ≥ tcim , k

∀ m ∈ C, k ∈ K

Each stream in the WN, except for the streams with the same starting and ending temperature, is identified as hot/cold stream based on its starting and ending temperature. Mass balance for the hot and cold streams at the inlet and outlet of each operation are formulated as

(9) (10)

When heat exchange occurs, new equipment is required. The logic constraints for heat exchangers are Q l , m , k − xl , m , kQ lU, m ≤ 0

∀ l ∈ H, m ∈ C, k ∈ K

∑ Fi ,j

Fll s =

(11)

hot i , j(1 + hot i , j) 2

j ∈ p̂

∀ md ∈ CD

Qhmd − xhmdUh ≤ 0

Qc l d − xc l dUc ≤ 0

∀ l d ∈ HD

echl = Fll(thinl − thoutl)cp

∀l∈H

eccm = Fmm(tcoutm − tcinm)cp

∀m∈C

∀ i ∈ p̅ , j ∈ p̂ , i = j

∑ Fi ,j

Fmmd = −∑ Fi , j

(15) (16)

trhl s , l d +

i∈p

∑ trck m ,m ,k = −Fi ,j s

(27)

hot i , j(1 − hot i , j)

d

k s

2

d

∀ m = i, m = j, k ∈ K

(28)

where trhls,ld and trcms,md denote the hot and cold water streams that bypass the HEN, while trhkls,ld,k and trckms,md,k stand for the hot and cold streams that were sent for heat exchange in the HEN. In the HEN, stream splitting and mixing are allowed in order to explore possibilities for heat recovery. Both the flow rates and temperatures of the streams are treated as optimization variables. The mathematical model for HEN is formulated as follows. Mass balance for each splitter and mixer. Hot stream splitter:

(20)

The necessary conditions of optimality presented by Savelski and Bagajewicz30 is adopted in order to cut down the complexity of the model. Mass balance for the contaminant and the constraint for maximum inlet concentration are defined by eq 21 and eq 22.

max max ∑ Fi ,j(cout, i − c in, j ) ≤ 0

2

d

∀ l = i, l = j, k ∈ K

(19)

∀ j ∈ p̂ Fhl s , k

(21)

i∈p

hot i , j(1 + hot i , j)

s d

k

∀ i ∈ p̅

max max ∑ Fi ,j(cout, i − cout, j) + Lj = 0

2

∑ trhk l ,l ,k = Fi ,j

s

∀ j ∈ p̂

∀ i = w , j = ww

∀ md = j , j ∈ p ̂

where Flls represents the total flow rate of hot source stream, and Flld stands for the total flow rate of hot demand stream. Similarly, Fmms denotes the total flow rate of cold source stream, and Fmmd represent the total flow rate of cold demand stream. 4.2. Novel Formulations for the Heat Exchanger Network. According to the superstructure in Figure 1, the hot and cold water streams can bypass or go through the HEN. This is one of the modifications of this work. In this way, more network structures can be included.

(14)

To minimize the freshwater consumption, freshwater is not allowed to be discharged directly. Fi , j = 0


(26)

trc ms , md +

j ∈ p̂

∀ ms = i , i ∈ p ̅

2

i∈p

(18)

∑ Fi ,j

(24)

(25)

The outlet streams can be reused or discharged. Mass balance at the outlet of each process unit is formulated as

Fii =

∀ l d = j , j ∈ p̂


j ∈ p̂

(17)

i∈p

2

Fmms = −∑ Fi , j

The inlet of an operation is a mixed flow of the streams coming from all the possible sources, including the freshwater and the reusing streams. Mass balance at the inlet of each process unit is given as Fjj =

hot i , j(1 + hot i , j)

i∈p

Mathematical Model for WN. The amount of the contaminant transferred to the water stream during the process is negligible compared to the water flow rate. The inlet and outlet mass balance for each process unit is given as Fii = Fjj

∑ Fi ,j

Fll d =

(13)

∀ l ∈ H, m ∈ C

(23)

(12)

where QUl,m is the upper bound for the heat exchange amount between l and m, while Uh and Uc represent the upper bound for heating and cooling. Equations 14−16 define the upper bound for heat exchange between two streams. For the MINLP model, xl,m,k, xhmd, and xcld are binary variables denoting the existence of heat exchangers. For the MPEC model, complementary formulations are employed for the modeling of these discrete decisions, and xl,m,k, xhmd, and xcld are positive variables. The complementary formulations will be fully described later in the next section. Q lU, m = min(echl , eccm)

∀ l s = i , i ∈ p̅

∀ j ∈ p̂ (22) 3359

⎧ Fl s − ∑ trh s d − ∑ trhk s d l ,l l ,l ,k ⎪ l l d ∈ HD l d ∈ HD ⎪ ⎪ ∀ l s ∈ HS , k = 1 =⎨ ⎪ Fhl s , k − 1 − ∑ trhk l s , l d , k ⎪ l d ∈ HD ⎪ s ⎩ ∀ l ∈ HS , k = 2, ..., K

(29)


Article

Industrial & Engineering Chemistry Research Hot stream mixer:

Fhl d , k

rqc md =

⎧ trhk l s , l d , k ∀ l d ∈ HD , k = 1 ⎪ s∑ ⎪ l ∈ HS = ⎨ Fh d ⎪ l , k − 1 + s∑ trhk l s , l d , k l ∈ HS ⎪ ⎪ ∀ l d ∈ HD , k = 2, ..., K ⎩

(30)

⎧ Fm s − ∑ trc s d − ∑ trck s d m ,m m ,m ,k ⎪ m md ∈ CD md ∈ CD ⎪ s ⎪ ∀ m ∈ CS , k = K =⎨ ⎪ Fc ms , k + 1 − ∑ trck ms , md , k ⎪ md ∈ CD ⎪ s ⎩ ∀ m ∈ CS , k = 1, ..., K − 1

(32)

tcims , k

⎧ s ∀ l s ∈ HS , k = 1 ⎪ thin l ⎨ =⎪ s ⎩ thol s , k − 1 ∀ l ∈ HS , k = 2, ..., K

(33)

⎧ s ∀ ms ∈ CS , k = K ⎪ tcin m =⎨ s ⎪ ⎩ tcoms , k + 1 ∀ m ∈ CS , k = 1, ..., K − 1

(34)

⎧ 1 Q l ,m,k > 0 ⎪ xl , m , k = ⎨ ⎪0 Q =0 l ,m,k ⎩

which can be transformed into complementary constraints as bellow.

Heat balance for each mixer:

Fhl d , k thil d , k

⎧ trhk l s , l d , kthinl s ∀ l d ∈ HD , k = 1 ⎪ s∑ ⎪ l ∈ HS = ⎨ Fh d tho d ⎪ l , k − 1 l , k − 1 + s∑ trhk l s , l d , kthil s , k l ∈ HS ⎪ ⎪ ∀ l d ∈ HD , k = 2, ..., K ⎩

xl , m , k = 0·xfl , m , k + 1·(1 − xfl , m , k ) ∀ l ∈ H , m ∈ c, k ∈ K

∀ l ∈ H , m ∈ c, k ∈ K 0 ≤ xfl , m , k ⊥λl1, m , k ≥ 0

Fc md , ktcimd , k

∑ s

l ∈ HS

trhl s , l dthinl s+

∑ Fhl ,kthol ,k d

d

(41)

∀ l ∈ H , m ∈ c, k ∈ K

0 ≤ 1 − xfl , m , k ⊥λl2, m , k ≥ 0

(42)

∀ l ∈ H , m ∈ c, k ∈ K (43)

where ⊥ is the complementarity operator compelling at least one of the complementing bounds to be active. By inspection it can be seen that if heat exchange occurred, that is, Ql,m,k > 0, the auxiliary variable λ1l,m,k will be assigned with a positive value by eq 41. With λ1l,m,k being positive, eq 42 will set the switching variable xf l,m,k to zero, which makes xl,m,k equal to 1, indicating that the heat exchanger exists. It should be stressed that, when heat exchange does not happen, xl,m,k can be assigned with any value in the range of 0 to 1. In this case, the minimization option will set xl,m,k to zero.

(36)

Heat balance at the inlet of each operation. The heat balances for the hot and cold streams are described as rqhl d =

(40)

(Q l , m , k − 0) − λl1, m , k + λl2, m , k = 0

(35)

⎧ trck ms , md , ktcinms ∀ md ∈ CD , k = K ⎪ ∑ s ⎪ m ∈ HS = ⎨ Fc d tco d trck ms , md , ktcims , k ⎪ m ,k+1 m ,k+1 + ∑ ms ∈ HS ⎪ ⎪ ∀ md ∈ CD , k = 1, ..., K − 1 ⎩

(39)

where Qhmd and Qcld are the required amount of hot and cold utility. MPEC Strategies for the Modeling of Discrete Decisions. For most of the models that were reported for the WAHEN design, binary variables were applied in modeling the discrete decisions. Though a mixed integer programming approach is the most general technique to handle logical disjunctions, the computational expense can be relatively high when dealing with large complex systems. Sometimes, an MINLP model may not even provide integer solution. In the present work, attempts are made to model the discrete decisions by applying complementary formulations. Complementarity is a relationship between variables where either one or both must be at its bound, which can be used to model disjunctive decisions instead of binary variables. The application of complementary formulations in chemical engineering was first introduced by Baumrucker et al.46 Later, it was introduced to the pipeline operation optimization problems47,48 to model the piecewise functions. The binary variables in WAHEN can also be treated as piecewise functions:

Heat balance for the splitters and mixers. Heat balance for each splitter: thil , k

(Qhmd − Qc l d) = Tj(Fll d + Fmmd ) cp

∀ l d ∈ HD , md ∈ CD , j ∈ p , l d = md = j

Cold stream mixer:

s

(38)

When the temperature of the inlet stream cannot meet the requirement, an additional heater or cooler is employed. The heat balance at the inlet of an operation is formulated as rqhl d + rqc md +

⎧ trck ms , md , k ∀ md ∈ CD , k = K ⎪ ∑ s ⎪ m ∈ CS = ⎨ Fc d trck ms , md , k ⎪ m ,k+1 + ∑ ms ∈ CS ⎪ ⎪ ∀ md ∈ CD , k = 1, ..., K − 1 ⎩

k=1

∀ m ∈ CD

(31)

Fc md , k

trc ms , mdtcinms+ ∑ Fc md , ktcomd , k

d

Cold stream splitter:

Fc ms , k

∑ ms ∈ CS

∀ l d ∈ HD

k=K

(37) 3360


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5. CASE STUDIES To illustrate the application of the proposed optimization method, two case studies considering both separated waste-

Additionally, MINLP models based on the same superstructure are also solved for the two examples considered in this work. DICOPT is used as the MINLP solver. 5.1. Case a: Uniform Wastewater Treating. The first example was first given by Savulescu and Smith,19 which was then widely studied by the research community.20,21,32,39,42 The problem consists of four water-using operations with a single contaminant. The operation data for the case study is provided as Supporting Information. Table 1 presents the cost and operating parameters. Freshwater is provided at 20 °C, while the wastewater is required to be discharged at 30 °C. The annual operating hours is 8000 h. The number of temperature intervals for this case is 2. The MPEC model for this example consists of 2014 constraints and 1861 continuous variables, while the MINLP model consists of 909 constraints, 1196 continuous variables, and 220 discrete variables. The optimal result from the MPEC model and the MINLP model are illustrated in Figure 4 and Figure 5, respectively. To compare the result obtained by this work and the result published in the literature, the costs and the required heat transfer area of the resultant network in the literature are recalculated by using the parameters and the method used in the present work. Detailed comparisons between the network generated by this study and the networks obtained by other approaches in the literature are given in Table 2 and Table 3. Table 2 shows the network configuration including freshwater consumption, hot and cold utility demand, and new equipment requirement as well as the total heat exchange area. Table 3 gives the annual network cost. Among all these resultant networks, the network obtained by the proposed one-step mathematical model is economically equal to or better than the literature results. Compared to the two step methods and the iterative methods, the proposed technique can further improve the performance of the heat exchanger network (lower equipment investment) while ensuring the optimality of the water network (simpler network

Table 1. Cost and Operating Parameters for the Case Studies parameter CFW i Chu Ccu TinHU TinCU Hop cp Cfixed CArea B Ul,m, Umd,HU, Uld,CU

0.375 $/t 377 $/(kW a) 189 $/(kW a) 120 °C 10 °C 8000 h 4.2 kJ/(kg °C) 8000 $ 1200 $/m2 0.6 0.5

water treating and uniform wastewater treating are presented in this section. The MPEC models are implemented in GAMS software49 (based on the PC specification: Intel D CPU 3.00 GHz, 4 GB RAM) using NLPEC as solver. The MPEC model is transformed into the NLP model and then solved by a standard NLP solver. In our case, CONOPT is set to be the active NLP solver. The reformulation options that were used for solving the MPEC model are listed below: (1) Reftype mult (specify the reformulation type: the inner product reformulations); (2) Slack positive (set the slack variables used in the reformulation process to positive value); (3) Constraint equality (set the inner products equal to zero). For more information, the readers are referred to the detailed NLPEC solver manual by Ferris.50

Figure 4. Optimal network design result for case a generated by the MPEC model. 3361


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Figure 5. Optimal network design result for case a generated by the MINLP model.

Table 2. Comparison of the Resultant Networks of Case a of Different Methods different methods

FFW(kg/s)

Qhot total (kW)

Qcold total (kW)

Savulescu et al. /graphical method Dong et al.39/interactive iteration mathematical method Liao et al.32/two-step mathematical method Ahmetovic et al.42/one-step mathematical method This paper/one-step mathematical method MINLP model MPEC model

90 90 90 90

8640 3780 3780 3780

0 0 0 0

3 4 3 2

90 90

3780 3780

0 0

2 heat exchangers, 1 heater 2 heat exchangers, 1 heater

21

a

new equipment heat heat heat heat

exchangers, exchangers, exchangers, exchangers,

1 1 1 1

heater heater heater heater

Atotal (m2)

Qtotal (kW)

3514.8a 4049.6 3819.5 3960.8

23100.2 22680 22008 22344

3993.0 3960.8

22361.7 22344

Note that the hot utility temperature is not given in the original paper, 150 °C is used for area calculation.

5.2. Case b: Separated Wastewater Treating. This example is an extension of the first case. The operation data for each water-using unit and the cost parameters are the same. The only difference is that the wastewater streams are required to be treated separately. In other words, more wastewater disposal units are involved. Consequently, the number of potential water streams is larger than that of the first case, which makes the problem more complex. The number of temperature intervals for this case is 3. Case b was also solved by our previous two-step synthesis method32 and a heat-integrated water network with four heat exchangers and one heater was obtained. Applying the proposed method to this case, a much simpler network design with less number of heat exchangers is achieved. The MPEC model consists of 4742 equations and 4371 variables. The model is solved in GAMS platform. The optimal network structure generated by this study is illustrated in Figure 6. An identical network design is also obtained by the MINLP model. To compare the results obtained by this study with the results given by Liao et al.32 The required heat exchange area and the corresponding cost obtained by Liao et al.32 are recalculated by

Table 3. Comparison of the Resultant Network Costs of Case a of Different Methods different methods

freshwater cost ($/a)

hot utility cost ($/a)

972000 3257280 Savulescu et al.21/ graphical method 39 972000 1425060 Dong et al. / interactive iteration mathematical method 972000 1425060 Liao et al.32/two-step mathematical method Ahmetovic et al.42/one972000 1425060 step mathematical method This paper/one-step mathematical method MINLP mode1 972000 1425060 MPEC model 972000 1425060

investment cost ($/a)

total cost ($/a)

306885.3

4536165

341047.4

2738107

311926.8

2708987

255878.9

2652939

256778.6 255878.9

2653839 2652939

with the same freshwater and utility consumption). The resultant optimal network of the proposed model can fairly match the best network design reported until now.42 3362


Article


Figure 6. Optimal network design result of case b.

Table 4. Comparison of the Resultant Networks of Case b of Different Methods different methods

FFW (kg/s)

Qhot total (kW)

Qcold total (kW)

new equipment

Atotal (m2)

Qtotal (kW)

Liao et al. /two-step mathematical method this paper/one-step mathematical method

90 90

3780 3780

0 0

4 heat exchangers, 1 heater 3 heat exchangers, 1 heater

5530.6 5086.4

25102.1 24600

32

6. CONCLUSIONS

Table 5. Comparison of the Resultant Network Costs of Case b of Different Methods different methods Liao et al.32/two-step mathematical method this paper/one-step mathematical method

freshwater cost ($/a)

hot utility cost ($/a)

investment cost ($/a)

97200

1425060

375003

2772063

97200

1425060

336692.6

2733753

In this paper, we extent our previous two-step approach to a one-step methodology addressing the simultaneous synthesis of WAHEN. The superstructure and the mathematical model are improved, thus more possible network structure are included. Direct and indirect heat exchanges are fully explored. Additionally, complementary formulations are also applied in modeling discrete decisions instead of binary variables. Both the MINLP model and the MPEC model are developed for the problem optimization. The capabilities and efficiency of the proposed method are illustrated with two case studies taken from the literature. It is clearly indicated that the network designs achieved by the proposed method are economically equal to or better than the literature results. The MPEC strategy shows good performance and high efficiency in the case studies. Compared to the previous sequential method, the proposed method can generate better results with less effort. Compared to another simultaneous method published in the literature (for example, the model proposed by Ahmetovic and Kravanja42), our method offers an alternative comparable way to synthesize WAHEN simultaneously.

total cost ($/a)

using the parameters used in this work. Comparison of the network configuration and the corresponding cost are given in Table 4 and Table 5, respectively. Table 6 gives the comparison of the model status and the corresponding computational efforts between the proposed model and the model in the literature. It can be seen from the table that, for the same case study, the scale of the MPEC model is larger than the MINLP model. This is because, when applying the complementary formulations, three more variables are introduced (xf l,m,k, λ1l,m,k and λ2l,m,k), and four extra equations are added (eqs 40− 43) for the calculation of one discrete decision (xl,m,k). However, binary variables are excluded in this way and the model can be solved more efficiently.

Table 6. Comparison of the Problem Sizes and Computational Effort of Different Methods different methods case a

case b

Ahmetovic et al.42 Bogataj M et al.40 this paper this paper

MPEC MINLP MPEC MINLP

constraints

continuous variables

binary variables

1031

928 749 1861 1196 4371 2772

176 115

2014 909 4742 2077 3363

220 530

CPU time/s

1.259 5.054 1.332 31.187


Article


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Lj = mass load of water using process, g·s−1 Tj = operation temperature of operation j, °C TinHU = temperature of the hot utility, °C TinCU = temperature of the cold utility, °C thinl = initial temperature of hot stream l, °C tcinm = initial temperature of cold stream m, °C thoutl = expected ending temperature of hot stream l, °C tcoutm = expected ending temperature of cold stream m, °C Ω = large constant Uh = upper bound for heat exchanged between cold stream and hot utility Uc = upper bound for heat exchanged between hot stream and cold utility Hop = hours of plant operation per annum, h QUl,m = upper bound for heat exchanged between l and m, KW

ASSOCIATED CONTENT

S Supporting Information *

The operation data for the case studies is provided. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +86-571-87951227. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The financial support provided by the National Natural Science Foundation of China (21261130583), the Specialized Research Fund for the Doctoral Program of Higher Education (20110101120019), the Zhejiang Provincial Natural Science Foundation of China (LY14B060007), and the National Basic Research Program of China (2012CB720500) are gratefully acknowledged.

Positive Variables

Al,m,k = area of heat exchange, m2 Amd,HU = area of heater, m2 Ald,CU = area of cooler, m2 Fii = inlet water flow rate of process i, kg·s−1 Fjj = outlet water flow rate of process j, kg·s−1 Fi,j = water flow rate form source i to sink j, kg·s−1 Fhls,k = water flow rate of hot source stream ls in interval k, kg·s−1 Fhld,k = water flow rate of hot demand stream ld in interval k, kg·s−1 Fcms,k = water flow rate of cold source stream ms in interval k, kg·s−1 Fcmd,k = water flow rate of cold source stream md in interval k, kg·s−1 trhls,ld, trcms,md = hot/cold water directly from source ls/md to sink ld/md without heat exchange trhkls,ld,k, trckms,md,k = water flow rate transferred from hot/cold source ls/ms to sink ld/md in interval k, kg·s−1 thil,k, thol,k = inlet and outlet temperature of hot stream l at interval k, °C tcim,k, tcom,k = inlet and outlet temperature of cold stream m at interval k, °C Ql,m,k = heat exchanged between hot stream l and cold stream m at interval k, KW Qhmd = heat exchanged between cold stream md and hot utility, KW Qcld = heat exchanged between hot stream ld and cold utility, KW echl = maximal amount of heat that allowed to be recovered from hot stream l, KW eccm = maximual amount of heat that allowed to be transferred to cold stream m, KW rqhld = residual energy of hot demand stream ld leaving heat exchanger network, KW rqcmd = residual energy of cold demand stream md leaving heat exchanger network, KW rqrj = energy of the recycle stream of operation unit j, KW Δ1l,m,k, Δ2l,m,k = temperature difference between stream l and m which involving heat exchange at interval k xl,m,k = positive variable denoting the existence of heat exchanger between stream l and m at interval k xhmd = positive variable denoting the existence of heater xcld = positive variable denoting the existence of cooler xf l,m,k, λ1l,m,k, λ2l,m,k = auxiliary variables

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ABBREVIATIONS CEN = combined exchange network HEN = heat exchanger networks MPEC = mathematical programming with equilibrium constraints WAHEN = water allocation and heat exchanger networks WN = water networks

Nomenclature Sets

p = set of water using process p̅ = set of water sources, ={p, w} p̂ = set of water sinks, ={p, ww} w = freshwater ww = wastewater K = temperature interval HF = set of hot water flow, HF = {(i,j)/i ∈ p,j̅ ∈ p̂,Ti > Tj} CF = set of cold water flow, CF ={(i,j)/i ∈ p̅,j ∈ p̂,Ti < Tj} H = set of hot water stream, H ={l/l ∈ HS ∪ HD} C = set of cold water stream, C = {m/m ∈ CS ∪ CD} HS = set of hot source water streams, HS = {ls/ls = i,i ∈ HF} HD = set of hot demand water streams, HD = {ld/ld = j,j ∈ HF} CS = set of cold source water streams, CS = {ms/ms = i,i ∈ CF} CD = set of cold demand water streams, CD = {md/md = j,j ∈ CF} Parameters

Bl,m = exponent parameter for area cost cmax in,j = the maximum inlet contaminant concentration of process j, ppm cmax out,i = the maximum outlet contaminant concentration of process i, ppm CFW = cost of freshwater, $/t i Chu = cost of hot utility, $/(kW a) Ccu = cost of cold utility, $/(kW a) Area Cfixed l,m , Cl,m = fixed charge and area cost coefficient for heat exchangers, $, $/m2 cp = heat capacity of water, kJ/(kg °C) hoti,j = parameter indicating the stream states in the heat exchanger network

Subscript

i = water source 3364


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j = water sink l,ls,ld = hot stream, hot source stream, hot demand stream m,ms,md = cold stream, cold source stream, cold demand stream k = index for temperature interval, k = 1,2,...,K

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