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Simultaneous synthesis of process water and heat exchanger networks Elvis Ahmetovi c a, b, Zdravko Kravanja b, * a b

Faculty of Technology, University of Tuzla, Univerzitetska 8, 75000 Tuzla, Bosnia and Herzegovina Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova ul. 17, 2000 Maribor, Slovenia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 September 2012 Received in revised form 3 February 2013 Accepted 17 February 2013 Available online 6 April 2013

This paper presents a novel superstructure and optimization model for the simultaneous synthesis of process water and heat exchanger networks. This superstructure combines the water network and heat exchanger network using interconnecting hot and cold streams. The water network has been extended for both direct and indirect heat exchanges. In addition, opportunities for heat integration between hot and cold streams, splitting and mixing of the freshwater and wastewater streams are incorporated within the superstructure. The proposed model is formulated as a non-convex MINLP (mixed-integer non-linear program), where the objective is to minimize the total annual costs of the network. A new convex hull formulation is presented for identifying the streams’ roles within the network. Three examples involving single and multiple contaminant problems are presented in order to illustrate the applicability and capabilities of the proposed superstructure and model. In all cases the resultant networks exhibit lower total annual costs, whilst the freshwater and utilities consumption are the same as reported in the literature. In addition, novel designs for heat-integrated process water networks with smaller or same number of heat exchangers are presented. 2013 Elsevier Ltd. All rights reserved.

Keywords: Simultaneous synthesis Heat-integrated water networks Superstructure MINLP (mixed-integer non-linear program) model

1. Introduction Process industries consume large amounts of water and energy for different purposes (washing operations, extraction, absorption, reaction, cooling, heating, etc.). Therefore, one of the main tasks of industry has been devoted to discovering the best ways to reduce water and energy consumptions, whilst satisfying strict environmental regulations. To achieve this important industrial goal, a stronger interaction between water and energy systems needs to be considered and explored. In order to systematically address this problem, a conceptual method can be used based on process insights and heuristic rules, pinch analysis and/or a mathematical programming approach based on superstructure optimization. For a deeper understanding of the above-mentioned systematic methods and tools for solving WN (water network) problems the _ reader is referred to the review papers by Bagajewicz [1], Jezowski [2,3], Foo [4], and the HEN (heat exchanger network) by Furman and Sahinidis [5], or books relating to systematic methods of

* Corresponding author. Tel.: þ386 2 22 94 481; fax: þ386 2 25 27 774. E-mail addresses: [email protected] (E. Ahmetovi c), [email protected] uni-mb.si, [email protected] (Z. Kravanja). 0360-5442/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.02.061

chemical process design [6], sustainability within the process industry [7], and sustainable design through process integration [8]. Over the last decade and a half, the synthesis and optimization of heat-integrated water networks have received considerable attention. The main objective was to simultaneously consider water re-use and heat exchange, and design a combined network (WN and HEN) with minimum freshwater, hot and cold utilities consumption, and minimum total annual cost. It should be mentioned that graphical methods and heuristic rules can provide good understanding of the interconnections within WN and HEN. However, they are of a sequential nature and thus cannot simultaneously consider strong interactions between WN and HEN, especially for multiple contaminant problems when minimizing the total annual cost. Apart from sequential procedure, in most of the published papers based on pinch analysis and mathematical programming several opportunities for heat integration between streams, freshwater and wastewater splitting and mixing within a network have not as yet been fully considered and explored. An ideal case would be to have a network superstructure which captures all possible network conﬁgurations including direct and indirect heat recovery, as well as freshwater and wastewater splitting and mixing allowing for an optimization model for ﬁnding the best network solution. However, such a model would be very complex and very difﬁcult to solve. In addition, it is worth

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

pointing out that within the combined network (WNeHEN) there are interconnecting hot and cold streams with unknown temperatures as well as ﬂowrates, thus making the synthesis problem more difﬁcult to solve especially when a large number of these streams exist within the network. Therefore, developing methodologies and approaches to deal with these problems is still a big challenge for the process systems engineering community. The objectives of this work are to present a brief literature review, a novel superstructure, and mathematical model for the simultaneous synthesis of process water and heat exchanger networks. As will be shown, the superstructure involves additional opportunities for heat integration and freshwater/wastewater splitting and mixing. Direct heat exchange by the mixing of streams and indirect heat exchange in heat exchangers is considered within the superstructure. The model is formulated as a MINLP where the objective is to minimize the total annual costs. A set of new constraints is formulated for identifying the interconnecting hot and cold streams between WN and HEN. The model is tested on single and multiple contaminant problems. The outline of the paper is as follows. Firstly, a brief literature review is given of those contributions related to the synthesis of heat-integrated water networks. Then the problem statement is deﬁned, and a description given of the superstructure and optimization model. Three case-studies considering single and multiple contaminant problems are solved for demonstrating the capabilities and effectiveness of the proposed model. Finally, the last section presents the general conclusions, and comments on future study. 2. Literature review This section presents a brief literature review focused on the synthesis of heat-integrated water networks. Generally speaking, from 1998 until late-to-mid 2009 less than 20 papers were published regarding this ﬁeld [3], whilst over the last couple of years the synthesis of heat-integrated water networks has been an active research area. Also, it will be one of the main directions for future research in order to produce water and energy efﬁcient and sustainable solutions. The ﬁrst works concerning the topic of heat integration within WN problems were from Savulescu and Smith [9], Savulescu et al. [10], and Bagajewicz et al. [11]. In the ﬁrst two studies, an insight-based approach was used, whilst in the latter one an optimization based approach was presented. Later, using the above-mentioned approaches to heat integration within WN problems, a more-active research area developed. Savulescu et al. [12] studied simultaneous energy and water minimization with no water re-use, and with maximum re-use of the water [13]. They explored the conceptual approach as well as direct and indirect heat-recovery opportunities, in order to identify a network design with minimum freshwater and energy consumption. In addition, they introduced a new grid representation, called the twodimensional grid diagram. Bogataj and Bagajewicz [14,15] proposed an approach for the simultaneous synthesis of energyefﬁcient WN using mathematical programming and superstructure optimization. They modiﬁed the HEN introduced by Yee et al. [16] for the case of mixing and splitting streams within HEN superstructure and combined with WN. The combined model was solved by a two-step solution strategy where in the ﬁrst step a WN (NLP (non-linear programming)) model was solved following by a solution of a combined WN þ HEN (MINLP) model during the second step. The WN model was solved in order to realize good initialization and to classify the streams into hot and cold. The proposed approach was used for solving single and multiple contaminant problems. The resulting network had a smaller number of heat exchangers and a smaller consumption of utilities than the network reported in Savulescu et al. [13]. Leewongtanawit

237

and Kim [17] studied the simultaneous optimization of a combined WN and HEN superstructure. They used modiﬁed WN [18], heat transhipment [19], and HEN superstructure [20] model. The overall MINLP problem was decomposed into two sub-problems MILP (mixed-integer linear programming) and NLP which were solved using an iterative procedure. The solution procedure was sequential, and their approach can be used for solving large-scale multiple contaminant problems. Dong et al. [21] modiﬁed the state-space superstructure [22] for simultaneously synthesizing WN and HEN. The problem was formulated as an MINLP. The feasible solutions were produced using randomly-generated initial estimations followed by improving the candidate solutions using perturbation techniques, and generating alternative network structures by shifting heat loads in loops along the utility paths. This approach can be used for single and multiple contaminant problems. Feng et al. [23] analyzed the reason why different WN structures have different energy performances for the same freshwater consumption. In their proposed model they used a variable to describe the temperature peak or vale, and introduced some new constraints. The problem was solved over two steps. The ﬁrst one was the solving of the standard WN problem followed by the synthesizing of HEN. They found that reducing the number of temperature ﬂuctuations along the WN subsystems was beneﬁcial for energy performance. Leewongtanawit and Kim [24] presented a graphical approach for the design of heat-integrated water networks. This approach was based on the Water and Energy Balance Diagram which is an extension of studies by Savulescu et al. [12,13]. The design interactions between WN and HEN were explored and energy-efﬁcient and cost-effective conﬁgurations for heat recovery were identiﬁed. Their approach was applied on a singlecontaminant WN problem. Xiao et al. [25] used holistic mathematical programming to formulate a MINLP for the heat-integrated WN problem. A hybrid optimization strategy based on stochastic and deterministic search techniques was used for solving the proposed model. They presented sequential and simultaneous solution procedures for single and multiple contaminants. Chen et al. [26] developed a MINLP model based on the superstructures of WN and HEN, and proposed a set of constraints to identify cold and hot streams in WN. This model was solved using a two-step sequential strategy. Polley et al. [27] developed a simple methodology based on a design insight for the designing of WN and HEN. They demonstrated that WN and HEN can be de-coupled and solved. The resulting network exhibited the minimum water and energy consumption, and provided simple structures for single-contaminant problems. Liao et al. [28] introduced a step-wise systematic procedure for the synthesis of heat-integrated water networks, and proposed a procedure for the identiﬁcation of hot and cold streams within WN, followed by targeting and design steps. During the targeting step, the identiﬁcation of the promising matches between hot and cold streams was performed, whilst in the design step, a stage-wise superstructure was used to deal with the features of mixing and splitting inside HEN. The problem was formulated as a MINLP, and single-contaminant WN problems were considered. Sahu and Bandyopadhyay [29] formulated the heat-integrated water network problem as linear programming in order to determine the minimum freshwater and utilities consumption. They analyzed both the isothermal and non-isothermal mixings of streams for single and multiple contaminant problems and proposed some theorems. Boix et al. [30] presented an approach based on mathematical programming to solve WN and HEN by considering several objectives such as freshwater consumption, energy consumption, interconnections number, and the number of heat exchangers. A two-step solution procedure was used for problem solving, and single-contaminant problems were considered. Martínez-Patiño et al. [31,32] studied the interactions between water

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and energy systems, and proposed a heuristic procedure for the synthesis of the heat-integrated water networks. This approach is based on the use of a temperature vs. concentration diagram. Yiqing et al. [33] studied the effect of non-isothermal mixing on water network’s energy performance and how the energy penalty can be eliminated or avoided. They used a problem table algorithm [6] for heat integration to analyze the energy performance of single-contaminant WN problem. They showed that heterogenous mixing of water streams can decrease the utility consumption of WN, which is not case for the homogenous mixing. According to the published literature, a small number of contributions have addressed the simultaneous synthesis of heat-integrated water networks using mathematical programming. However, only this approach can simultaneously explore trade-offs between freshwater cost, utility cost, and the investment costs for heat exchangers. This paper presents a new contribution and results in this ﬁeld using simultaneous synthesis and the mathematical programming approach based on superstructure optimization. 3. Problem statement The general formulation of the heat-integrated water network problem can be stated as follows. For a given a set of freshwater sources and a set of water-using units that require water of a certain quality and temperature, it is necessary to determine the interconnections, ﬂowrates, contaminant concentrations, and the temperatures of each stream within the network. The freshwater sources can be with different qualities and temperatures. The maximum inlet and outlet concentrations of contaminant, the contaminant load to be transferred to the water stream, and the operating temperatures for water-using units are speciﬁed. Single and multiple contaminants are considered, and the temperature given of the discharged wastewater. The standard assumptions are used as given in the literature. The number of water sources and the number of water-using units are speciﬁed, the water streams have constant heat capacities and heat-transfer coefﬁcients, the heat exchangers are counter-current, the network operates continuously, with one hot and one cold utility being available. The main goal was to simultaneously synthesize the heat-integrated water network with the minimum total annual cost and the optimal consumptions of the freshwater and utilities. 4. Superstructure Fig. 1 shows the development of a combined WN and HEN superstructure from the general representation (Fig. 1a) of a heatintegrated water network problem. The combined superstructure is proposed for systematically addressing the simultaneous synthesis of the heat-integrated process water networks, and exploring strong interactions between WN and HEN. The combined superstructure consists of a WN superstructure, HEN superstructure, and their complicated interconnecting streams, which can be hot, cold, or bypass streams (Fig. 1b). Within this superstructure, the freshwater stream can be preheated in HEN in order to satisfy the temperature of the process units in WN, or it can be directly introduced to WN as a bypass stream. The wastewater is generated after being used as freshwater in WN. The wastewater stream from WN can be sent to HEN for heat recovery or as a bypass stream mixed with other wastewater streams from HEN, and then discharged into the environment. Therefore, the freshwater can be regarded as a cold or bypass stream, whilst the wastewater as a hot or bypass stream. Additionally, the outlet water stream of one process unit can be re-used in another process unit. In addition to these streams, streams from mixers to process units or streams from process units to splitters can be hot, cold or bypass streams

with the temperatures being optimization variables (Fig. 1c). For most of these streams it is impossible to determine in advance whether they should be characterized as hot or cold streams. As shown in Fig. 1c, i.e. a stream from the mixer process unit (point M) can be sent to HEN for heat recovery as a cold or hot stream. It is worth pointing out that matches between these two streams are forbidden (see the ﬁrst and the ﬁfth matches from left to the right in Fig. 1c) in HEN and they can only exchange heat with other streams in HEN. After heat recovery in HEN the streams are returned to WN. In general, when WN consists of more process units, the number of cold and hot streams increase linearly, and the number of matches in HEN exponentially. In addition, if multiple contaminants are considered, the synthesis problem of the heat-integrated water network can be very complex and difﬁcult to solve. Accordingly, the development of methodologies and approaches for successfully dealing with these kinds of problems represents a big challenge for the scientiﬁc community. In order to develop a combined superstructure model, the recently proposed WN model [34] has now been extended for both direct and indirect heat exchanges, additional freshwater and wastewater splitting and mixing, and combined with the heat exchanger network model [16,35,36]. 5. Mathematical model The model of the superstructure for the simultaneous synthesis of a HIPWNs (heat-integrated process water networks) consists of mass balance equations for water and contaminants, and heat balance equations. In order to circumvent the problem with streams’ characterizations, each stream within the network is represented by a couple of alternative hot and cold streams, and described with its convex hull formulation for identifying its role in the HEN. The HIPWN model can be efﬁciently solved based on this formulation and connecting constraints. The proposed model is formulated as a non-convex MINLP (mixed-integer non-linear programming) problem with additional 0e1 variables included for the identiﬁcation of the streams’ roles in HEN. The non-linearities within the models appear in the mass and heat balance equations in the forms of bilinear terms (ﬂowrate multiplied by the concentration and the ﬂowrate multiplied by the temperature). The other non-convexities are the concave investment cost terms within the objective function. The objective is to simultaneously synthesize HIPWN with the minimal TAC (total annual cost). 5.1. Mathematical model of water network superstructure Fig. 2 presents a new general superstructure for WN including water splitting and mixing, as well as direct and indirect heat transfer between hot and cold streams. Based on this superstructure, the mathematical model has been developed and combined with HEN. 5.1.1. Initial splitters of freshwater The freshwater of an initial splitter SIs from freshwater source s ˛ SW can be sent to the mixer before each process unit and to the freshwater splitter SSp. The overall mass balance for the initial splitter is given by Eq. (1), and the ﬂowrate and temperature equality constraints by Eqs. (2)e(4).

FWs ¼ FWout þ s

X

FIPs;p

cs˛SW

(1)

p˛PU

¼ FSSin FWout s p

cs˛SW; cp˛PU; p ¼ 1

(2)

TFWs ¼ TSSin p

cs˛SW; cp˛PU; p ¼ 1

(3)

TFWs ¼ TIPs;p

cs˛SW; cp˛PU

(4)

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

Fig. 1. Development of a combined WN and HEN superstructure.

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E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

240

Fig. 2. Extended WN superstructure including direct and indirect heat transfer and additional opportunities for freshwater/wastewater splitting and mixing.

5.1.2. Freshwater splitters The freshwater from the initial splitter SIs can be sent to HEN in order to heat and satisfy the temperature constraints of the process units. In addition, the outlet stream of freshwater from HEN can be split and mixed with the cold stream of freshwater from the initial splitter or re-used stream of water from the process units. The mass balance for the freshwater splitters is given by Eqs. (5)e(7), the contaminant concentration equality constraint by Eq. (8), and the temperature constraints by Eqs. (9) and (10). out FSSin p ¼ FSSp þ

FSSin p

¼

X

X

FSSMp;p0

cp; p0 ˛PU; psjPUj

cp; p ˛PU; p ¼ jPUj

FSSMp0 ;p ,xSSout p0 ;c þ

þ

X

FIP,xWin s;c

s˛SW

X

FPp0 ;p ,xSPUout p0 ;c

p0 ˛PU psp0 ;Rp ¼ 0

þ

X

FPp0 ;p ,xSPUout p0 ;c

cp˛PU; cc˛CC

0

p ˛PU Rp ¼ 1

(12) mix ¼ FPUin p ,Tp

0

X p0 ˛PU

(5)

p0

FSSMp;p0

in FPUin p ,xPUp;c ¼

(6)

p0

X p0 ˛PU

þ

FSSMp0 ;p ,TSSout p0 þ X

X

FIP,TIPs;p

s˛SW

FPp0 ;p ,TSPUout p0

0

p ˛PU

FSSout p

¼

FSSin pþ1

cp˛PU; psjPUj

¼ TSSin TSSout p pþ1

(8)

cp˛PU; psjPUj

(9)

cp˛PU

(10)

5.1.3. Mixer process units The outlet stream of the mixer process unit MPUp consists of a set of inlet streams from the initial splitter, the splitter process unit, and the freshwater splitter. This stream is directed to the process unit, and can be a hot, cold or bypass stream. The mass balance for the mixer is given by Eq. (11), the mass balance for each contaminant c by Eq. (12) and the heat balance by Eq. (13).

FPUin p ¼

X

FSSMp0 ;p þ

p0 ˛PU

þ

X p0 ˛PU Rp ¼ 1

X

FIPs;p þ

s˛SW

X

FPp0 ;p

cp˛PU

FPp0 ;p ,TSPUout p0

cp˛PU

(13)

Rp ¼ 1

5.1.4. Process units The process unit PUp consists of an inlet stream FPUin p from the from the process mixer process unit and an outlet stream FPUout p unit. The water ﬂowrate through the process units is assumed to be a continuous variable. The outlet stream from PUp is directed to the splitter process unit, and can be a hot, cold or bypass stream. The mass balance is given by Eq. (14), the mass balance for each contaminant c by Eq. (15), and the equality contaminant concentration by Eq. (16). out FPUin p ¼ FPUp

cp˛PU

(14)

in out out FPUin p ,xPUp;c þ LPUp;c ¼ FPUp ,xPUp;c cp˛PU; cc˛CC

(15)

in xPUout p;c ¼ xSPUp;c

(16)

cp˛PU; cc˛CC

p0 ˛PU psp0 ;Rp ¼ 0

FPp0 ;p

X

þ

p0 ˛PU

cs˛SW; cp0 ˛PU; cc˛CC

out xWin s;c ¼ xSSp0 ;c

out TSSin p TSSp

(7)

psp0 ;Rp ¼ 0

(11)

5.1.5. Splitter process units The splitter process unit SPUp consists of an inlet stream from the process unit, and a set of outlet streams directed to the ﬁnal mixer MF, the mixer process unit MPUp, and the wastewater mixer

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

MMFp. The mass balance for the splitter process unit is given by Eq. (17) and the equality contaminant concentration by Eq. (18).

X

FPUout ¼ p

p0 ˛PU

p0 ˛PU

out out FPOp0 ,xSPUout ,xc ; p ¼ 1;cc˛CC p0 ;c ¼ F

(28) out FMMout p ,TMMp þ

psp0 ;Rp ¼ 0

cp˛PU

(17)

cp˛PU; cc˛CC

(18)

FPp;p0

out FMMout p ,xMMp;c þ

FPp;p0

p0 ˛PU

X

þ

X

FSMMp;p0 þ FPOp þ

241

X

p0 ˛PU

X

FPOp0 ,TSPUout p0

p0 ˛PU

¼ F out ,T out ;

cp˛PU; p ¼ 1

(29)

Rp ¼ 1

in xSPUout p;c ¼ xSPUp;c

5.1.6. Wastewater mixers The wastewater mixer MMFp consists of a set of inlet streams from the splitter process unit. The outlet stream of the wastewater mixer is sent to HEN for heat integration. In addition, the outlet stream of wastewater from HEN can be mixed with wastewater streams from the process units. The mass balance for the wastewater mixer is given by Eqs. (19)e(21), the mass balance equation for each contaminant c by Eqs. (22) and (23), the heat balance by Eqs. (24) and (25), and the temperature constraint by Eq. (26).

X

FMMin p þ

cp˛PU; psjPUj

FSMMp0 ;p ¼ FMMout p

out FMMin p ¼ FMMpþ1

cp˛PU; p ¼ jPUj

cp˛PU; psjPUj X

out FMMout pþ1 ,xMMpþ1;c þ

p0 ˛PU out ¼ FMMout p ,xMMp;c

p0 ˛PU

(20)

(21)

FSMMp0 ;p ,xSPUout p0 ;c

cp˛PU; psjPUj

out out FSMMp0 ;p ,xSPUout p0 ;c ¼ FMMp ,xMMp;c

(22) cp˛PU; p ¼ jPUj (23)

out FMMout pþ1 ,TMMpþ1 þ

X

(30)

X s˛SW

FWs ,xWin s;c þ

X

LPUp;c ¼ F out ,xout c ;

cc˛CC

(31)

p˛PU

in ¼ FMMout p ,TMMp

cp˛PU; psjPUj

in FSMMp0 ;p ,TSPUout ¼ FMMout p ,TMMp p0

(24) cp˛PU; p ¼ jPUj (25)

cp˛PU

5.2.1. Streams from mixer to process unit Fig. 3 shows the superstructure of water streams (hot, cold, and bypass) after the mixer process unit. The inlet water streams to the mixer process unit can be different temperatures and involved in direct heat transfer by mixing. The temperature of the mixed stream of water can be: Tpmix > TPUin p (hot stream), Tpmix ¼ TPUin p (bypass stream).

p0 ˛PU

in TMMout p TMMp

In WN there are certain streams i.e. after the mixer process units or after process units, the temperatures of which are optimization variables and thus unknown. For these streams it is impossible to determine whether they should be characterized in HEN as hot or cold streams ahead of the optimization. In order to circumvent this problem when identifying the hot and cold streams, a new convex hull formulation is presented in this section for the identiﬁcation of streams’ roles in the HEN.

Tpmix < TPUin p (cold stream) or

FSMMp0 ;p ,TSPUout p0

p0 ˛PU

X

FWs ¼ F out

s˛SW

(19)

p0 ˛PU

X

X

5.2. Convex hull formulation for the identiﬁcation of streams for HEN

FSMMp0 ;p ¼ FMMout p

p0 ˛PU

X

5.1.8. Total mass balance and contaminant mass balance for the network The total mass balance for the network is given by Eq. (30) and the total contaminant mass balance by Eq. (31).

(26)

According to this, the temperature of the mixed stream is optimization variable and the mathematical formulation is required for identifying the hot and cold streams as follows. The inequality binary constraint for the selection of hot or cold streams is given by Eq. (32) and the temperature constraints by Eqs. (33)e(37). Note, if a stream is unselected, then both binary variables are set to zero and the stream is sent as a bypass from the mixer to the process unit (from MPUp to PUp in Fig. 3).

5.1.7. Final mixer The ﬁnal mixer MF consists of a set of inlet streams from the splitter process unit, and the outlet stream from the ﬁrst wastewater mixer. The mass balance for the ﬁnal mixer is given by Eq. (27), the mass balance equation for each contaminant c by Eq. (28) and the heat balance by Eq. (29).

FMMout p þ

X p0 ˛PU

FPOp0 ¼ F out

(27) Fig. 3. Superstructure of streams (hot, cold, and bypass) after the mixer process unit.

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

242 hp

cp

yp þ yp 1;

p˛PU

(32)

cp hp Tpmix ¼ T hot;in ; þ Tjcold;in þ TPUin p , 1 yp yp i

(33)

p˛PU; i˛HP; j˛CP; i ¼ p; j ¼ p hp

Tihot;in TPUin;max ,yp ; p

p˛PU; i˛HP; i ¼ p

(34)

5.3. Mathematical model of a heat exchanger network superstructure The presented WN model is combined with the HEN model introduced by Yee et al. [16]. In WN and HEN models, the ﬂowrates and temperatures of the streams are treated as optimization variables, so that trade-offs can be obtained between freshwater cost, utility cost, and the investment cost for heat exchangers. The mathematical model of HEN is given by Eqs. (44)e(63), as follows.

Tihot;in TPUin p ,yp ;

hp

p˛PU; i˛HP; i ¼ p

(35)

5.3.1. Overall heat balance for each stream

cp

p˛PU; j˛CP; j ¼ p

(36)

ðthini thouti Þ,fhi ¼

Tjcold;in TPUin p ,yp ; Tjcold;in TFWmin ,ycp p ; s

X X

qi;j;k þ qcui

i˛HP

(44)

j˛CP

(45)

k˛ST j˛CP

p˛PU; j˛CP; j ¼ p

(37)

X X tcoutj tcinj ,fcj ¼ qi;j;k þ qhuj ; k˛ST i˛HP

5.2.2. Streams from process unit to splitter Fig. 4 shows the superstructure of streams (hot, cold, and bypass) after the process unit. The temperature of the outlet stream of the splitter process unit SPUp can be:

5.3.2. Heat balance at each stream

X qi;j;k ; thi;k thi;kþ1 ,fhi ¼

in TSPUout p > TPUp (hot stream),

X qi;j;k ; tcj;k tcjþ1 ,fcj ¼

in TSPUout p < TPUp (cold stream) or in TSPUmix out ¼ TPUp (bypass stream).

p˛PU

(38)

cps

TSPUout ¼ T ihot;out þ Tjcold;out þ TPUout p , 1 yp p p˛PU; i˛HP; j˛CP;

hps Tihot;out TPUout p ,yp ;

cps

j˛CP; k˛ST

(47)

;

5.3.3. Assignment of superstructure inlet temperatures

thini ¼ thi;1 ;

i˛HP

tcinj ¼ tcj;NOKþ1 ;

(48)

j˛CP

(49)

5.3.4. Feasibilities of temperatures

thi;k thi;kþ1 ;

i˛HP; k˛ST

(50)

(39)

tcj;k tcj;kþ1 ;

j˛CP; k˛ST

(51)

(40)

thouti thi;NOKþ1 ;

i ¼ p þ jPUj; j ¼ p þ jPUj

p˛PU; i˛HP; i ¼ p þ jPUj

,yp ; Tjcold;out TPUout;max p cps Tjcold;out TPUout p ,yp ;

hps

yp

p˛PU; i˛HP; i ¼ p þ jPUj

Tihot;out TFWmin ,yhps p ; s

(46)

i˛HP

The mathematical formulation for identifying the hot and cold streams after the process unit is given, as follows. The inequality binary constraint for the selection of hot or cold streams is given by Eq. (38) and the temperature constraints by Eqs. (39)e(43). Note again, if a stream is unselected, then both binary variables are set to zero and the stream is sent as a bypass from the process unit to the splitter (from PUp to SPUp in Fig. 4).

yhps þ ycps p p 1;

i˛HP; k˛ST

j˛CP

p˛PU; j˛CP; j ¼ p þ jPUj

p˛PU; j˛CP; j ¼ p þ jPUj

(41) (42) (43)

tcoutj tcj;1 ;

i˛HP; k˛ST

(52)

j˛CP; k˛ST

(53)

5.3.5. Hot and cold utility load

thi;NOKþ1 thouti ,fhi ¼ qcui ; tcoutj tci;1 ,fcj ¼ qhuj ;

i˛HP

j˛CP

(54) (55)

5.3.6. Logic constraints

Fig. 4. Superstructure of streams (hot, cold, and bypass) after the process unit.

qi;j;k U,zi;j;k 0;

i˛HP; j˛CP; k˛ST

(56)

qcui U,zcui 0;

i˛HP

(57)

qhuj U,zhuj 0;

j˛HP

(58)

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

5.3.7. Logic constraints for temperature differences

Dti;j;k thi;k tcj;k þ G, 1 zi;j;k ;

i˛HP; j˛CP; k˛ST

Dti;j;kþ1 thi;kþ1 tcj;kþ1 þ G, 1 zi;j;k ; i˛HP;j˛CP;k˛ST

243

fcj ¼ FPUout p ,Cp ;

cp˛PU; j˛CP; j ¼ p þ jPUj

(59)

tcinj ¼ TPUout p ;

(60)

cps tcoutj ¼ Tjcold;out þ TPUout ; p , 1 yp

cp˛PU; j˛CP; j ¼ p þ jPUj cp˛PU; j˛HP;

j ¼ p þ jPUj

Dti;CU thi;NOKþ1 TOUTCU þ G,ð1 zcui Þ;

Dtj;HU TOUTHU tcj;1 þ G, 1 zhuj ; Dti;j;k ; Dti;CU ; Dtj;HU EMAT;

i˛HP

(73) (74)

(75)

(61)

j˛CP

(62)

i˛HP; j˛CP; k˛ST

(63)

5.4.3. Connecting equations for HEN for freshwater streams between freshwater splitters

fcj ¼ FSSin p ,Cp ;

cp˛PU; j˛CP; j ¼ p þ 2,jPUj

Note that all the temperature driving forces within exchangers, coolers, and heaters are optimization variables. Logical constraint Eq. (63) puts the EMAT (exchanger minimum allowable temperature) as a lower bound on them. Note also that Eqs. (59)e(62) deﬁne positive driving forces as differences between the temperatures of hot and cold streams when matches are selected (binary variable z ¼ 1) or make this relation also redundant for rejected matches (z ¼ 0) for those cases when actual differences between streams’ temperatures can be negative.

5.4.4. Connecting equations for HEN for wastewater streams between wastewater mixers

5.4. Connecting equations between WN and HEN

fhi ¼ FMMout p ,Cp ;

Based on ﬂowrates, temperatures, and binary variables from WN, connecting equations, Eqs. (64)e(89), relating to the heat capacity ﬂowrates of hot and cold streams and their corresponding inlet and outlet temperatures can be deﬁned in order to simultaneously solve the combined model of HIPWN minimizing the TAC of the network.

tcinj ¼ TSSin p ;

cp˛PU; j˛CP; j ¼ p þ 2,jPUj

tcoutj ¼ TSSout p ;

thini ¼ TMMin p ; thouti ¼ TMMout p ;

cp˛PU; j˛CP; j ¼ p þ 2,jPUj

cp˛PU; i˛HP; i ¼ p þ 2,jPUj cp˛PU; i˛HP; i ¼ p þ 2,jPUj cp˛PU; i˛HP; i ¼ p þ 2,jPUj

(76) (77) (78)

(79) (80) (81)

5.4.5. Logical constraints for connecting streams in WN in HEN 5.4.1. Connecting equations for HEN for streams from mixer to process unit

fhi ¼

FPUin p ,Cp ;

cp˛PU; i˛HP; i ¼ p

hp thini ¼ Tihot;in þ TPUin ; p , 1 yp

cp˛PU; i˛HP; i ¼ p

zi;j;k yhp p ; (64) (65)

p˛PU; i˛HP; j ¼ CP; k˛ST; i ¼ p

(82)

cp

p˛PU; i˛HP; j ¼ CP; k˛ST; j ¼ p

(83)

zcui yp ;

hp

p˛PU; i˛HP; i ¼ p

(84)

p˛PU; j˛CP; j ¼ p

(85)

zi;j;k yp ;

thouti ¼ TPUin p ;

cp˛PU; i˛HP; i ¼ p

(66)

zhuj ycp p ;

fcj ¼ FPUin p ,Cp ;

cp˛PU; j˛CP; j ¼ p

(67)

zi;j;k yp ;

p˛PU; i˛HP; j ¼ CP; k˛ST; i ¼ p þ jPUj

(86)

(68)

zi;j;k ycps p ;

p˛PU; i˛HP; j ¼ CP; k˛ST; j ¼ p þ jPUj

(87)

(69)

zcui yp ;

hps

p˛PU; i˛HP; i ¼ p þ jPUj

(88)

cps

p˛PU; j˛CP; j ¼ p þ jPUj

(89)

cp tcinj ¼ Tjcold;in þ TPUin p , 1 yp ; tcoutj ¼ TPUin p ;

cp˛PU; j˛CP; j ¼ p

cp˛PU; j˛CP; j ¼ p

5.4.2. Connecting equations for HEN for streams from process unit to splitter

fhi ¼

FPUout p ,Cp ;

thini ¼ TPUout p ;

cp˛PU; i˛HP; i ¼ p þ jPUj cp˛PU; i˛HP; i ¼ p þ jPUj

(70) (71)

hps thouti ¼ Tihot;out þ TPUout ; cp˛PU; i˛HP; i ¼ p þ jPUj p , 1 yp (72)

hps

zhuj yp ;

5.4.6. Objective function of the proposed model The objective function of the model, given by Eq. (90) and formulated to minimize TAC, consists of cost terms for freshwater, utilities, and the investment for the heat exchangers. The cost terms for the investment in heat exchangers consist of a ﬁxed charge for exchanger units and an area cost for heat exchangers, and these terms are calculated as given in Eq. (90). In order to make a comparison between ours and others’ solutions, the annualized capitalcost model for a conventional shell-and-tube heat exchanger is

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

244

taken from Dong et al. [21], and given as 8000zi;j þ 1200A0:6 i;j where the heat-transfer area Ai,j is in m2. The annual interest rate is set at 10% and the plant is assumed to be operated continuously for 8000 h a year.

TAC ¼ H,

X

FWs ,CFWs þ

s˛SW

þ

X X X X

CFi;j ,zi;j;k þ

þ

X

CHU ,qhuj

j˛CP

X

CFi;CU ,zcui

i˛HP

CFj;HU ,zhuj þ

j˛CP

X

CCU ,qcui þ

i˛HP

i˛HP j˛CP k˛ST

þ

X

X X X

B

i;j Ci;j ,Ai;j;k

i˛HP j˛CP k˛ST

Bi;CU Ci;CU ,Ai;CU

þ

i˛HP

X

B

j;HU Cj;HU ,Aj;HU

j˛CP

(90)

The A (areas), LMTD (temperature differences), and U (overall heat-transfer coefﬁcients) of the exchangers, coolers, and heaters are given by Eqs. (91)e(99).

Ai;j;k ¼

qi;j;k ; Ui;j ,LMTDi;j;k

Ai;CU ¼

qi;CU ; Ui;CU ,LMTDi;CU

i˛HP

(92)

Aj;HU ¼

qj;HU ; Uj;HU ,LMTDj;HU

j˛CP

(93)

i˛HP; j˛CP; k˛ST

(91)

2

LMTDi;j;k

31=3 Dti;j;k þ Dti;j;kþ1 6 7 ¼ 4 Dti;j;k ,Dti;j;kþ1 , 5 ; 2

(94)

i˛HP; j˛CP; k˛ST

LMTDi;CU ¼

"

Dti;CU ,ðTOUTi TINCU Þ ,

#1=3

Dti;CU þ ðTOUTi TINCU Þ 2

"

i˛HP

(95)

Dtj;HU , TINHU TOUTj ,

LMTDj;HU ¼

;

Dtj;HU þ TINHU TOUTj

#1=3

2 1 1 1 ¼ þ ; Ui;j hi hj

i˛HP; j˛CP

;

j˛CP

(96)

(97)

HRAT (heat recovery approach temperature), as all temperature driving forces are now considered as optimization variables. 6. Solution strategy The proposed model of HIPWN is formulated as a non-convex MINLP (mixed-integer non-linear programming) problem with additional 0e1 variables included for the identiﬁcation of cold and hot streams. Due to its non-convex and highly non-linear nature, the MINLP problem is very difﬁcult to solve especially if the desire is to explore all interactions within the combined network. However, by using the proposed model we managed to solve the combined HIPWN whilst simultaneously minimizing the total annual costs. It is worth mentioning, that the problem can be solved using two solution approaches. The ﬁrst one is by directly solving the combined network as one system. In the second approach, WN is solved ﬁrst, where the objective is to minimize freshwater consumption and provide a good initial point. After that the overall HIPWN is solved. The problem is modeled in GAMS (General Algebraic Modeling System) [37] and for model solving different solvers from GAMS library were used, like BARON (Branch-AndReduce Optimization Navigator) and SBB (Simple Branch-andBound). In the second approach BARON was used for WN and SBB for the combined HIPWN. To illustrate the capabilities of the proposed model and solution strategy, we solved three examples including single and multiple contaminants. In all examples, the resulted networks exhibited lower total annual costs compared to the reported results in the literature, because the superstructure captured additional opportunities for heat integration. 7. Application examples This section presents three examples involving single and multiple contaminant problems to illustrate the applicability and capabilities of the proposed optimization model. The examples were implemented in GAMS [37] and solved on a PC (personal computer) machine (2.67 GHz, 8 GB RAM (random-access memory)), with reasonable computation time (in less than 70 CPUs). In the ﬁrst and second examples, the model consisted of 1031 constraints, 928 continuous variables, and 176 discrete variables. The model of the third example consisted of 1075 constraints, 970 continuous variables, and 176 discrete variables. The cost and operating parameters given in Table 1 [21] were used in all examples. Examples have been solved by several authors using different models and solution strategies based on pinch analysis or mathematical programming. However, in all cases, networks obtained using the approach presented in this paper had lower total annual costs compared to results in the literature. In addition, the resulting Table 1 Cost and operating parameters for Examples 1e3. Parameter

1

1 1 ¼ þ ; hi hCU

i˛HP

(98)

1 1 1 ¼ þ ; Uj;HU hj hHU

j˛CP

(99)

Ui;CU

This model enables the obtaining of an appropriate trade-off between utility consumption, water usage, and investment. In cases with temperature pinches it, in principle, allows for a reduction in energy consumption when compared to existing models, since the heat integration applied is performed simultaneously and the proposed model does not rely on the assumption of a ﬁxed

Freshwater cost Cooling utility (cooling water) cost Heating utility (low pressure steam, 120 C) cost Fixed charge for heat exchangers Area cost coefﬁcient for heat exchangers Cost exponent for exchangers Overall heat-transfer coefﬁcient (individual heat-transfer coefﬁcients for streams and utilities were assumed to be 1 kW(m2 C) Working hours of plant per year Temperature of freshwater Temperature of wastewater The inlet and outlet temperature of cooling water Speciﬁc heat capacity of water

0.375 $/t 189 $/(kW a) 377 $/(kW a) 8000 $ 1200 $/m2 0.6 0.5 kW/(m2 C)

8000 h 20 C 30 C 10 C and 20 C 4.2 kJ/(kg C)

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

networks had simpler designs with the same or a smaller number of heat exchangers. In order to compare the presented model’s results with the results published in the literature, data from Table 1 [21] was used to recalculate costs (freshwater cost, hot utility cost, cold utility cost, investment cost, and ﬁnally the total annual cost) for all cases taken from the literature. 7.1. Example 1 Example 1 is a well-known case study given by Savulescu and Smith [9] and Savulescu et al. [12,13], and involves WN with four processes and a single contaminant. Data for the processes, contaminants, and temperatures are presented in Table 2. This problem Table 2 Water-using operation data. Process number

Contaminant load (g/s)

Maximum inlet concentration (ppm)

Maximum outlet concentration (ppm)

Temperature ( C)

1 2 3 4

2 5 30 4

0 50 50 400

100 100 800 800

40 100 75 50

245

has been studied by several authors [9,11e13,15,21,24,25,27,28,31]. In all the reported networks freshwater consumption was the same (90 kg/s), whilst in Savulescu et al. [13] both the hot utility consumption (4265 kW) and the cold utility consumption (485 kW) were greater than in other cases (3780 kW and 0 kW). In addition, the reported networks had different numbers of heat exchangers (see Table 3). The superstructure for this example is given in Fig. 5 and consists of all possible connections between the splitting points of sequentially heating-up freshwater, process units, and the mixing points of sequentially cooling discharged wastewater. The heat transfer between the hot and cold streams in the superstructure is possible through direct heat transfer (mixing of water streams), and indirect heat transfer (through the heat exchanger area). It is worth pointing out that allowing for additional water splitting and mixing increases the possibilities for direct heat transfer. Consequently, networks can be obtained with a reduced number of heat exchangers (see Tables 3 and 4). In Case c, i.e. when one heat exchanger and one heater are selected within a network with the a total annual cost of 2,668,035 $/a, the heat-transfer area of the exchanger is very large (about 5938 m2) and impractical because the shell-and-tube heat exchangers usually have heat-transfer areas [38,39] of up 4000 m2 or 4647 m2. However, it may be

Table 3 The results for the freshwater and utilities consumption, the total heat ﬂow exchanged in the heat exchangers, and the total area of the heat exchangers. Method proposed by the authors

FW (kg/s)

HU (kW)

CU (kW)

Equipment requirements (/)

Qtotal (kW)

Atotal (m2)

Savulescu et al. [13] Xiao et al. [25] Polley et al. [27] Case a Case b Case c Martínez-Patiño et al. [31] Dong et al. [21] Bagajewicz et al. [11] Bogataj and Bagajewicz [15] Leewongtanawit and Kim [24] Liao et al. [28] Case a Case b This paper Case a Case b Case c

90 90

4265 3780

485 0

3 Heat exchangers, 1 heater, 1 cooler 3 Heat exchangers, 2 heaters

23,585 26,040a

4549 4690

90 90 90 90 90 90 90 90

3780 3780 3780 3780 3780 3780 3780 3780

0 0 0 0 0 0 0 0

3 3 4 3 4 3 3 3

26,040 23,940 23,100 26,040 22,680 22,008 22,008 22,260

4690 4258 4088 4722 4050 3782 3782 3776

90 90

3780 3780

0 0

3 Heat exchangers, 1 heater 4 Heat exchangers, 1 heater

22,008 25,102

3633 5531

90 90 90

3780 3780 3780

0 0 0

3 Heat exchangers, 1 heater 2 Heat exchangers, 1 heater 1 Heat exchanger, 1 heater

22,400 22,344 23,688

3424 3961 5938

a

Heat Heat Heat Heat Heat Heat Heat Heat

exchangers, exchangers, exchangers, exchangers, exchangers, exchangers, exchangers, exchangers,

2 3 4 1 1 1 1 1

heaters heaters heaters heater heater heater heater heater

Note that in Xiao et al. [25] the total amount of heat ﬂow should be 26,040 kW instead of 22,480 kW (see Fig. 11 in Xiao et al. [25]).

Fig. 5. Superstructure of HIPWN for Example 1.

246

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

Table 4 The results of the freshwater and utilities costs, the investment cost for heat exchangers and the total annual cost. Method proposed by the authors

FWC ($/a)

CUC ($/a)

HUC ($/a)

IC ($/a)

TAC ($/a)

Savulescu et al. [13] Xiao et al. [25] Polley et al. [27] Case a Case b Case c Martínez-Patiño et al. [31] Dong et al. [21] Bagajewicz et al. [11] Bogataj and Bagajewicz [15] Leewongtanawit and Kim [24] Liao et al. [28] Case a Case b This paper Case a Case b Case c

972,000 972,000

91,665 0

1,607,905 1,425,060

369,236 364,587

3,040,806 2,761,647

972,000 972,000 972,000 972,000 972,000 972,000 972,000 972,000

0 0 0 0 0 0 0 0

1,425,060 1,425,060 1,425,060 1,425,060 1,425,060 1,425,060 1,425,060 1,425,060

364,587 344,905 379,349 341,919 341,047 314,495 314,495 310,293

2,761,647 2,741,965 2,776,409 2,738,979 2,738,107 2,711,555 2,711,555 2,707,353

972,000 972,000

0 0

1,425,060 1,425,060

304,644 375,039

2,701,704 2,772,099

972,000 972,000 972,000

0 0 0

1,425,060 1,425,060 1,425,060

293,165 255,899 270,975

2,690,225 2,652,959 2,668,035

possible to install multiple units, or use plate and frame exchangers which are more compact, have lower cost and more ﬂexibility (extra plates can be added) compared to shell-and-tube heat exchangers. Compact heat exchangers have up to ﬁve times higher heat-transfer efﬁciency than shell-and-tube heat exchangers [40]. However, it should be mentioned that the largest plate-shell-type heat exchangers can have heat exchange surfaces of up to 10,500 m2 per single unit and have already been successfully used for energy saving within industries. Tables 3 and 4 show the results from this case study as obtained by using the approaches in the literature and the proposed approach given in this paper. The results for FW (freshwater) and utilities (HU (hot utility) and CU (cold utility)) consumptions, the Qtotal (total heat ﬂow) exchanged within the heat exchangers, and the Atotal (total area) for all exchangers, are summarized in Table 3,

and the corresponding annual costs for freshwater, utilities and investment in Table 4. It is interesting to note that, in all cases, better network designs (with lower TAC and simpler design) were obtained with the proposed method in this paper compared to the published results in the literature. All the resulting networks exhibited the same freshwater consumption (90 kg/s) and hot (3780 kW) and cold utility (0 kW) consumption as reported in the literature but TAC, (i.e. for Case a: 2,690,225 $/a, or Case b: 2,652,959 $/a) were lower due to smaller investment costs for heat exchangers. Accordingly, Fig. 6 presents a novel network design for this case study with the minimal TAC 2,652,959 $/a, involving three exchangers (two heat exchangers and one heater). The selected matches for indirect heat integration are between the inlet cold and outlet hot streams of the process unit with the highest temperature (100 C), and the cold freshwater and the hot wastewater streams.

Fig. 6. A novel network design with a smaller number of heat exchangers for Example 1.

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

In addition, direct heat recovery between water streams of different temperatures is selected within the network (i.e. see mixing points of PU1, PU2, PU3 as well as the mixing points of the wastewater streams). 7.2. Example 2 In this problem the contaminant loads are scaled-down by a factor of 3.6, whilst all other data are the same as given in Example 1. The network superstructure for this problem is the same as given in Example 1. Bogataj and Bagajewicz [15] solved this problem using the two-step solution strategy and obtained HIPWN with two heat exchangers and one heater. In their solution the interconnections between water-using process units were the same as in the network for the ﬁrst example (three heat exchangers and one heater) but the topology of HEN was changed due to economy

of scale. In order to compare their results with the results in this paper, using the cost and operating parameters from Table 1, the freshwater cost was recalculated, together with the hot utility cost, investment cost for heat exchangers, and TAC for the network design given in their paper (see Table 5). We solved this problem and obtained a simpler network with only one heat exchanger and one heater (see Fig. 7). The selected match for heat integration within the network is between the freshwater cold stream and the wastewater hot stream. The total heat duty and total area of heat exchangers are 6580 kW and 1725 m2. Due to economy of scale and additional opportunities for heat integration within the network, as well as the water splitting and mixing in the presented solution, the investment costs for the heat exchanger (134,227 $/a) and TAC of the network (800,077 $/a) were somewhat lowered compared to the result in Bogataj and Bagajewicz [15] (see Table 5). 7.3. Example 3

Table 5 Results for Example 2.

Heat exchangers cost ($/a) Freshwater cost ($/a) Hot utility cost ($/a) Total annual cost ($/a)

247

Recalculated results from Bogataj and Bagajewicz [15]

This paper (see Fig. 7)

146,748 270,000 395,850 812,598

134,227 270,000 395,850 800,077

This problem is an extension of the second one, but now considers multiple contaminants (A, B, C). The data for the problem (Table 6) are taken from Bogataj and Bagajewicz [15] where the HIPWN with freshwater consumption (95.527 t/h z 26.535 kg/s), three heat exchangers and one heater is given. Their results were recalculated using data from Table 1 in order to compare with the results obtained using the proposed model (see Table 7). The

Fig. 7. Optimal design of HIPWN with a smaller number of heat exchangers for Example 2.

Table 6 Data for Example 3. Process number

1 2 3 4

Contaminant load (g/s)

Maximum inlet concentration (ppm)

Maximum outlet concentration (ppm)

A

B

C

A

B

C

A

B

C

2 5 30 4

1 0 4 22

3 15 0 17

0 50 100 400

15 100 100 380

0 30 100 250

100 100 800 800

100 200 750 800

100 250 600 800

Temperature ( C)

40 100 75 50

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

248 Table 7 Results for Example 3.

Heat exchangers cost ($/a) Freshwater cost ($/a) Hot utility cost ($/a) Total annual cost ($/a)

Recalculated results from Bogataj and Bagajewicz [15]

This paper (see Fig. 8)

199,861 286,581 420,624 907,066

190,890 286,579 420,157 897,626

costs and a reduced number of heat exchangers compared to the results given in the literature, due to the incorporated additional opportunities for heat integration within the superstructure. Future research will be directed towards an extension of the superstructure by allowing for both isothermal and non-isothermal heat transfer between the process-to-process streams in order to explore some additional opportunities for heat integration, e.g. multiple utility conﬁgurations. In addition, the effect of direct and indirect heat

Fig. 8. Optimal design of HIPWN with three heat exchangers and one heater for Example 3.

obtained network design given in Fig. 8 involves three heat exchangers and one heater which is the same number of heat exchangers as given in Bogataj and Bagajewicz [15]. However, the TAC of the network in our case (897,626 $/a) was lower than the TAC (907,066 $/a) of Bogataj and Bagajewicz [15] mainly due to the smaller investment cost for heat exchangers. It is interesting to note that in the proposed network design the inlet and outlet streams of PU4 are hot streams. Additionally, the inlet and outlet streams of the process units PU1 and PU4 play an important role within the network’s heat integration.

transfer, and freshwater/wastewater splitting and mixing on heat recovery within the network would be interesting to explore. Acknowledgments The authors are grateful to the Scholarship scheme for academic exchange between the EU and Western Balkan countries for a JoinEU-SEE postdoctoral fellowship. Financial support is also gratefully acknowledged from the Slovenian Research Agency (Program No. P2-0032), and the Federal Ministry of Education and Science, Bosnia and Herzegovina.

8. Conclusions Nomenclature This paper addressed the simultaneous synthesis of heatintegrated process water networks. A new combined superstructure for process water and heat exchanger networks has been presented involving additional opportunities for direct and indirect heat transfer and freshwater and wastewater splitting and mixing. These opportunities have been systematically incorporated into the mathematical model, which is formulated as a MINLP problem. Also, a new mathematical formulation has been proposed for identifying hot and cold streams (streams from mixers to process units, and streams from process units to splitters) in HEN. Example problems have clearly indicated that the proposed method can be successfully used for single and multiple contaminant water network problems. In addition, the model can produce an efﬁcient and better design of heat-integrated process water networks with lower total annual

Indices c i j k p s

contaminant hot process stream cold process stream index for stage and temperature location process unit freshwater source

Sets CC CP

contaminants cold process streams

E. Ahmetovic, Z. Kravanja / Energy 57 (2013) 236e250

HP PU ST SW

hot process streams process units stages in the HEN superstructure freshwater sources

TSSin p TSSout p TSPUout p

Parameters B exponent for area cost C area cost coefﬁcient, $/m2 CCU per unit cost for cold utility, $/(W a) CF ﬁxed charge for exchangers, $ cost of freshwater from source s, $/kg CFWs per unit cost for hot utility, $/(W a) CHU heat capacity of water, J/(kg K) Cp EMAT minimum approach temperature, K H hours of plant operation per annum, h h individual heat-transfer coefﬁcients, W/(m2 K) load of contaminant c in process unit p, kg/s LPUp,c local recycle around process unit p (Rp ¼ 0 does not exist, Rp Rp ¼ 1 if exists) temperature of freshwater source s, K TFWs temperature at inlet of process unit p, K TPUin p temperature at outlet of process unit p, K TPUout p TIN inlet temperature of utility stream, K TOUT outlet temperature of utility stream, K temperature of freshwater stream from source s to mixer TIPs,p process unit, K temperature of outlet stream from ﬁnal mixer, K Tout minimum temperature of freshwater source s, K TFWmin s U overall heat-transfer coefﬁcient, W/(m2 K) maximum concentration of contaminant c in inlet stream xPUin;max p:c to process unit, ppm maximum concentration of contaminant c in outlet xPUout;max p;c stream from process unit, ppm concentration of contaminant c in freshwater in outlet xSSout p0 ;c stream from splitter, ppm concentration of contaminant c in freshwater source s, ppm xWin s;c G upper bound for temperature difference U upper bound for heat exchange

Continuous variables fc, fh heat capacity ﬂowrate of cold and hot stream, W/K mass ﬂowrate of water stream from freshwater source s to FIPs,p process unit p, kg/s mass ﬂowrate of inlet stream to wastewater mixer, kg/s FMMin p mass ﬂowrate of outlet stream from wastewater mixer, FMMout p kg/s mass ﬂowrate of inlet freshwater stream to heat FSSin p exchange, kg/s mass ﬂowrate of outlet freshwater stream from heat FSSout p exchanger, kg/s FSSMp;p0 mass ﬂowrate of freshwater stream from splitter to mixer process unit, kg/s mass ﬂowrate of water stream from process unit p0 to FPp0 ;p process unit p, kg/s mass ﬂowrate of water stream from process unit p’ to ﬁnal FPOp0 mixer, kg/s mass ﬂowrate of inlet water stream to process unit p, kg/s FPUin p mass ﬂowrate of outlet water stream from process unit p, FPUout p kg/s mass ﬂowrate of outlet wastewater stream from ﬁnal Fout mixer, kg/s mass ﬂowrate of water for freshwater source s, kg/s FWs tc, th temperature of cold and hot stream, K

Tpmix TMMin p TMMout p thi,k tcj,k Dti,j,k

Dti,CU Dtj,HU qi,j,k qcui qhuj xout c xMMout p;c xPUin p;c xPUout p;c xSPUin p;c xSPUout p;c

249

temperature of inlet freshwater stream to heat exchanger, K temperature of outlet freshwater stream from heat exchanger, K temperature of outlet water stream from splitter process unit, K temperature of outlet stream from mixer process unit, K temperature of inlet wastewater stream to wastewater mixer, K temperature of outlet wastewater stream from wastewater mixer, K temperature of hot stream i at hot end of stage k, K temperature of cold stream j at hot end of stage k, K temperature approach for the match (i, j) at temperature location k, K temperature approach for the match of hot stream i and cold utility, K temperature approach for the match of hot utility and cold stream j, K heat ﬂow exchanged between hot process stream i and cold process stream j in stage k, W heat ﬂow exchanged between hot process stream i and cold utility, W heat ﬂow exchanged between hot utility and cold process stream j, W concentration of contaminant c in discharge stream to the environment, ppm concentration of contaminant c in outlet stream from wastewater mixer, ppm concentration of contaminant c in inlet stream to process unit p, ppm concentration of contaminant c in outlet stream from process unit p, ppm concentration of contaminant c in inlet stream to splitter process unit p, ppm concentration of contaminant c in outlet stream from splitter process unit, ppm

Binary variables hp existence of hot stream from mixer to process unit yp existence of cold stream from mixer to process unit ycp p hps

yp

existence of hot stream from process unit to splitter

ycps p zi,j,k zcui zhuj

existence existence existence existence

of of of of

cold stream from process unit to splitter match (i, j) in stage k match between cold utility and hot stream i match between hot utility and cold stream j

Subscripts, superscripts, abbreviations CU cold utility CUC cold utility cost FWC freshwater cost HEN heat exchanger network HIPWN heat-integrated process water network HU hot utility HUC hot utility cost IC investment cost in inlet stream max maximum min minimum MINLP mixed-integer non-linear programming out outlet stream TAC total annual cost WN water network

250

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