Simultaneous Optimization of NonIsothermal Design of Water Networks with Regeneration and Recycling
 44 Downloads
Abstract
Most of the water network design strategies considering regeneration and recycling are using the assumptions that the processes are isothermal and steady state. However, almost all of the available processes have temperature change from the inlets to the outlets of the units. Moreover, to maintain the process units’ temperature under the specified limit, additional heat exchangers are usually required. The traditional approach of designing water network and heat exchanger network are usually made separately. It is reasonable to expect that a simultaneous design procedure of water network and heat exchanger network that considers process temperature change may result in a better design option. In this work, systematic design of nonisothermal water networks with regeneration and recycling is proposed. Superstructure model with nonlinear programming (NLP) model is developed accordingly to incorporate all possible networks. Regeneration units and heat exchangers are introduced to achieve the specified contaminant requirements and temperature limits of each process unit. The superstructure model is enhanced with regeneration unit model by considering the removal ratio of contaminants. As opposed to the traditional model, the novel heat exchanger model in this work is constructed separately for the heating side and the cooling side to increase further flexibility to the original model. The overall model is optimized simultaneously to achieve feasible heat exchange network and optimal configuration. The total annual cost (TAC) and global equivalent cost (GEC) are considered as the objective function to represent the economic benefits of the given case study. Several case studies are provided to demonstrate the effectiveness of the proposed design strategy.
Keywords
Nonisothermal water networks NLP WAHEN Superstructure Mathematical programmingNomenclature
Subscript
 \( {\displaystyle \begin{array}{l}{P}_1,{P}_2,{P}_3,\dots \in P\\ {}{P}_1,{P}_2,{P}_3,\dots \in {P}^{\hbox{'}}\\ {}{P}_1,{P}_2,{P}_3,\dots \in {P}^{\hbox{'}\hbox{'}}\end{array}} \)
P ≠ P^{'}and P^{'} ≠ P^{''} (Process)
 \( {\displaystyle \begin{array}{l}{R}_1,{R}_2,{R}_3,\dots \in R\ \\ {}{R}_1,{R}_2,{R}_3,\dots \in {R}^{\hbox{'}}\\ {}{R}_1,{R}_2,{R}_3,\dots \in {R}^{\hbox{'}\hbox{'}}\end{array}} \)
R ≠ R^{'}and R^{'} ≠ R^{''} (Regeneration)
 \( {\displaystyle \begin{array}{l}{Eh}_1,{Eh}_2,{Eh}_3,\dots \in Eh\\ {}{Eh}_1,{Eh}_2,{Eh}_3,\dots \in {Eh}^{\hbox{'}}\\ {}{Eh}_1,{Eh}_2,{Eh}_3,\dots \in {Eh}^{\hbox{'}\hbox{'}}\end{array}} \)
Eh ≠ Eh^{'}and Eh^{'} ≠ Eh^{''} (Exchanger heating side)
 \( {\displaystyle \begin{array}{l}{Ec}_1,{Ec}_2,{Ec}_3,\dots \in Eh\\ {}{Ec}_1,{Ec}_2,{Ec}_3,\dots \in {Eh}^{\hbox{'}}\\ {}{Ec}_1,{Ec}_2,{Ec}_3,\dots \in {Eh}^{\hbox{'}\hbox{'}}\end{array}} \)
Ec ≠ Ec^{'}and Ec^{'} ≠ Ec^{''} (Exchanger cooling side)
 C_{1}, C_{2}, C_{3}, … ∈ C
Component
 Un ∈ P, R, Eh, Ec
Units
 Hu_{1}, Hu_{2}, Hu_{3}, … ∈ Hu
Hot utility
 Cu_{1}, Cu_{2}, Cu_{3}, … ∈ Cu
Cold utility
Superscript
 in
Input condition
 out
Output condition
 f
Fresh water
 w
Waste water
Parameter
 M_{C,P}
Amount of contaminant C generated by the process P (g/h)
 \( {C}_{\max C,P}^{in} \)
Limiting inlet concentration C of process P (ppm)
 \( {C}_{\max C,P}^{out} \)
Limiting outlet concentration C of process P (ppm)
 PR_{C,R}
Removal ratio of contaminant C in regeneration R
 PHD_{P}
Heat duty of process P (kW)
 RHD_{R}
Heat duty of regeneration R (kW)
 T^{f}
Temperature of freshwater inlet
 \( {T}_{\max P}^{in} \)
Maximum for limit inlet temperature of process P (°C)
 \( {T}_{\min P}^{in} \)
Minimum for limit inlet temperature of process P (°C)
 \( {T}_{\max P}^{out} \)
Maximum for limit outlet temperature of process P (°C)
 \( {T}_{\min P}^{out} \)
Minimum for limit outlet temperature of process P (°C)
 \( {T}_{\max \mathrm{R}}^{in} \)
Maximum limit for inlet temperature of regeneration R (°C)
 \( {T}_{\min \mathrm{R}}^{in} \)
Minimum limit for inlet temperature of regeneration R (°C)
 \( {T}_{\max \mathrm{R}}^{out} \)
Maximum limit for outlet temperature of regeneration R (°C)
 \( {T}_{\min \mathrm{R}}^{out} \)
Minimum limit for outlet temperature of regeneration R (°C)
 \( {C}_{\max C}^w \)
Maximum limit for concentration of contaminant C in waste flow (ppm)
 \( {T}_{\max T}^w \)
Maximum limit for temperature of waste flow (°C)
 C_{Fw}
Cost of fresh water
 C_{Hu,Un}
Cost of hot utility
 C_{Cu,Un}
Cost of cold utility
 \( {T}_{Hu, Un}^{in} \)
Inlet temperature of hot utility
 \( {T}_{Cu, Un}^{in} \)
Inlet temperature of cold utility
 \( {T}_{Hu, Un}^{out} \)
Outlet temperature of cold utility
 Cp
Heat capacity of water
 C_{fixed}
Fixed charges for heat exchangers and utility units
 C_{area}
Area cost coefficient for heat exchangers and utility units
 U_{Eh,Ec,hu,cu}
Overall heat transfer coefficients for hot stream, cold stream, and heat exchanger
 H
Hours of plant operation
 T^{f}
Inlet temperature of fresh water
 \( {T}_{\max T}^w \)
Maximum limit for temperature of wastewater stream
 α
Regeneration cost factor
 β
Wastewater cost factor
 γ
Minimum temperature difference of the heat exchangers’ inlet and outlet
 λ
Exponent parameter for area cost
 N_{Eh,Ec}
Number of heat exchanger
 N_{Hu,Un}
Number of hot utility
 N_{Cu,Un}
Number of cold utility
Positive Variable
 \( {F}_P^f \)
Flowrate from fresh water to process P (t/h)
 \( {F}_R^f \)
Flowrate from fresh water to regeneration R (t/h)
 \( {F}_{Eh}^f \)
Flowrate from fresh water to exchanger heating side Eh (t/h)
 \( {F}_{Ec}^f \)
Flowrate from fresh water to exchanger cooling side Ec (t/h)
 F_{P→P'}
Flowrate from process P to process P’ (t/h)
 F_{R→R'}
Flowrate from regeneration R to regeneration R’ (t/h)
 F_{Eh→Eh'}
Flowrate from exchanger heating side Eh to exchanger heating side Eh’ (t/h)
 F_{Ec→Ec'}
Flowrate from exchanger cooling side Ec to exchanger cooling side Ec’ (t/h)
 F_{P→R}
Flowrate from process P to regeneration R (t/h)
 F_{P→Eh}
Flowrate from process P to exchanger heating side Eh (t/h)
 F_{P→Ec}
Flowrate from process P to exchanger cooling side Ec (t/h)
 F_{R→P}
Flowrate from regeneration R to process P (t/h)
 F_{R→Eh}
Flowrate from regeneration R to exchanger heating side Eh (t/h)
 F_{R→Ec}
Flowrate from regeneration R to exchanger cooling side Ec (t/h)
 F_{Eh→P}
Flowrate from exchanger heating side Eh to process P (t/h)
 F_{Eh→R}
Flowrate from exchanger heating side Eh to regeneration R (t/h)
 F_{Eh→Ec}
Flowrate from exchanger heating side Eh to exchanger cooling side Ec (t/h)
 F_{Ec→P}
Flowrate from exchanger cooling side Ec to process P (t/h)
 F_{Ec→R}
Flowrate from exchanger cooling side Ec to regeneration R (t/h)
 F_{Ec→Eh}
Flowrate from exchanger cooling side Ec to exchanger heating side Eh (t/h)
 \( {F}_P^w \)
Flowrate from process P to discharge (t/h)
 \( {F}_R^w \)
Flowrate from regeneration R to discharge (t/h)
 \( {F}_{Eh}^w \)
Flowrate from exchanger heating side Eh to discharge (t/h)
 \( {F}_{Ec}^w \)
Flowrate from exchanger cooling side Ec to discharge (t/h)
 \( {C}_{C,P}^{in} \)
Inlet concentration C of process P (ppm)
 \( {C}_{C,P}^{out} \)
Outlet concentration C of process P (ppm)
 \( {C}_{C,R}^{in} \)
Inlet concentration C of regeneration R (ppm)
 \( {C}_{C,R}^{out} \)
Outlet concentration C of regeneration R (ppm)
 C_{C,Eh}
Inlet concentration C of exchanger heating side Eh (ppm)
 C_{C,Ec}
Inlet concentration C of exchanger cooling side Ec (ppm)
 \( {C}_C^w \)
Concentration C of wastewater (ppm)
 \( {T}_P^{in} \)
Inlet temperature of process P (°C)
 \( {T}_P^{out} \)
Outlet temperature of process P (°C)
 \( {T}_R^{in} \)
Inlet temperature of regeneration R (°C)
 \( {T}_R^{out} \)
outlet temperature of regeneration R (°C)
 \( {T}_{Eh}^{in} \)
Inlet temperature of exchanger heating side Eh (°C)
 \( {T}_{Eh}^{out} \)
outlet temperature of exchanger heating side Eh (°C)
 \( {T}_{Ec}^{in} \)
Inlet temperature of exchanger cooling side Ec (°C)
 \( {T}_{Ec}^{out} \)
Outlet temperature of exchanger cooling side Ec (°C)
 T^{w}
Temperature of wastewater (°C)
 \( {Q}_P^{in} \)
Inlet heat duty of process P inlet using hot utility (ton.°C.C_{P}/h)
 \( {Q}_R^{in} \)
Inlet heat duty of regeneration R using hot utility (ton.°C.C_{P}/h)
 \( {Q}_{Eh}^{in} \)
Inlet heat duty of exchanger heating side Eh using hot utility (ton.°C.C_{P}/h)
 \( {Q}_{Ec}^{in} \)
Inlet heat quality of exchanger cooling side Ec using hot utility (ton.°C.C_{P}/h)
 \( {T}_{Hu, Un}^j \)
Temperature for unit before heated by hot utility
 \( {T}_{Hu, Un}^k \)
Temperature for unit after heated by hot utility
 \( {T}_{Cu, Un}^j \)
Temperature for unit before cooled by cold utility
 \( {T}_{Cu, Un}^k \)
Temperature for unit after cooled by cold utility
 ΔT_{Eh,Ec}
Temperature difference of the mixing exchangers’ inlet and outlet
 ΔT_{Hu,Un}
Temperature difference of the hot utility’ inlet and outlet
 ΔT_{Cu,Un}
Temperature difference of the cold utility’ inlet and outlet
 Q_{Eh,En}
The exchanged energy by the heat exchanger
 A_{Eh,Ec}
Heat transfer area of mixing heat exchanger
 A_{Hu,Un}
Heat transfer area of hot utility
 A_{Cu,Un}
Heat transfer area of cold utility
Negative Variable
 \( {D}_P^{in} \)
Removal heat duty in process P inlet using cooling utility (ton.°C.C_{P}/h)
 \( {D}_R^{in} \)
Removal heat duty in regeneration R inlet using cooling utility (ton.°C.C_{P}/h)
 \( {D}_{Eh}^{in} \)
Removal heat duty in exchanger heating side Eh inlet using cooling utility (ton.°C.C_{P}/h)
 \( {D}_{Ec}^{in} \)
Removal heat duty in exchanger cooling side Ec inlet using cooling utility (ton.°C.C_{P}/h)
 \( {D}_w^{in} \)
Removal heat in exchanger cooling side Ec inlet using cooling utility (ton.°C.C_{P}/h)
Free Variable
 C^{annual}
Annual cost (M$)
Introduction
In process industries, water plays an important role to remove process contaminants and to maintain process operating temperature. Under the regulated environmental laws and the everincreasing freshwater price, water allocation in process industries is given higher attention in recent years. Contaminants in process water usually were taken away through regeneration units, and the water is recycled back into the process or discharged into the environment. Depending on the source, water can be classified as fresh water, regeneration water, and wastewater. The freshwater is assumed to be zero contaminants. The freshwater will have an increase in contaminants concentration after it is used in a process. The water can be discharged as wastewater or can be regenerated in water treatment units to remove the contaminants. Note that the allowed contaminant concentration of each process unit is different. Some of the process units may allow fresh water intake, while the rest of the process units allows regeneration water intake. Finally, the concentration of the contaminant in wastewater is restricted under the environmental laws. The issue to optimally allocate water utilization in process units has been considered in water allocation network (WAN) studies. Wang and Smith (1994) have first reported the design of WAN with multiplecontaminants for minimum freshwater consumption. Ideally, the cost factor of water network should not only be the freshwater consumption. Therefore, the minimum freshwater target may not represent the comprehensive economic benefits of WAN. Feng et al. (2008) have developed multiplecontaminants WAN design using mixed integer nonlinear programming (MINLP) model to determine the optimal minimum fresh water, regeneration water, process contaminant load and connection number. The WAN optimization alone is a complex task, but it is important to consider the heat utilization since most of the process industries are consuming energy and energy conservation is essential to save costs.
Heat exchanger network (HEN) design has been studied extensively for reducing energy consumption. Pinch analysis is the most extensive method used in HEN design. Linnhoff and Hindmarsh (1983) proposed the concept which evaluates the optimum ∆T_{min} to obtain the pinch point and the minimum energy requirement (MER) for hot and cold utilities. Mathematical programming has also been used on HEN design. Yee and Grossmann (1990) have proposed a simultaneous process and HEN synthesis. Similar to the WAN problem, the HEN model can be constructed with nonlinear programming (NLP) or MINLP model to optimize the economic benefit of HEN.
Note that water network with regeneration and recycling while maintaining process units’ operation temperature is critical in the process industries. In response to this issue, it is necessary to integrate WAN model and HEN model and combine them into water allocation heat exchanger network (WAHEN). Recently, the WAHEN design has been recognized as an active research field in process system engineering. The review by Ahmetović et al. (2015) has covered most of the studies on nonisothermal water network synthesis. There are two main optimization methods for WAHEN design, twostep method and simultaneous method. Boix et al. (2012) proposed twostep method to optimize the WAHEN design. The first step is to optimize WAN by mixed integer linear programming (MILP) model, then optimize HEN by MINLP model to solve water and energy allocation problem with four criteria, i.e., freshwater consumption, energy consumption, number of interconnections, and number of heat exchangers. DeLeón Almaraz et al. (2016) also solved the WAHEN design problem with twostep method. The first step is to optimize WAN by minimizing the global equivalent cost (GEC). The second step of optimizing the HEN problem uses the pinch analysis and mathematical programming algorithm (Yee and Grossmann 1990) where annual cost results from both optimization methods are then compared. DeLeón Almaraz et al. (2016) showed that the annual cost results from the mathematical programming algorithm method are lower than that of with pinch analysis method. The HEN superstructure in proposed by Yee and Grossmann (1990) has been proven to be effective, but there is lack of flexibility which can be found in water network where the operating fluids are allowed to be mixed from different streams. Bogataj and Bagajewicz (2008) proposed a simultaneous method to optimize the WAHEN design with MINLP model. The HEN superstructure model is developed from that of Yee and Grossmann’s (1990) where additional mixers and splitters model are introduced to enhance the configuration flexibility. The addition of mixers and splitters model increased the constraints number and consequently increase the model complexity. Furthermore, there are several works have been published using simultaneous method (Ahmetović and Kravanja 2013) and (Yan et al. 2016).
In this work, heat exchanger model with mixing stream capability is introduced into the superstructure model developed by DeLeón Almaraz et al. (2016). The mixing heat exchanger is constructed separately for the heating side and the cooling side. From the original model of DeLeón Almaraz et al. (2016), the superstructure is enhanced with four units (process units, regeneration units, mixing exchangers heating side, and mixing exchangers cooling side). The novelty of this work is that now the heat exchangers for WAHEN have the capability of mixing the water streams which open new structure possibilities that have not been explored before. At the same time, that different input and output temperatures are integrated into each process and each regeneration unit and that the change of phase of some streams in the HEN can be made possible in the design algorithm. The WAHEN problem in this work is formulated as NLP problem where the objective function of total annual cost is minimized. Isothermal and nonisothermal case studies will be demonstrated to show the effectiveness of the new mixing heat exchanger model.
The rest of this paper is organized as follows: Section 2 is dedicated to the general formulation of WAHEN and its superstructure. Section 3 describes the general mathematical model of the proposed design strategy. The case study and optimization results are discussed in Section 4. Finally, conclusions and perspectives are given in the last section.
WAHEN General Formulation

No heat loss.

Concentration and temperature are ideally mixed.

Steadystate operation.

If phase change occurred in the system, the latent heat would be considered.
WAHEN Mathematical Model
The following mathematical model is constructed according to the superstructure shown in Fig. 1. The mathematical model consists of each unit mass balances, energy balances, temperature and concentration upper or lower limits, and the physical limits of the heat exchangers.
To compute the economic benefit of the WAHEN, the objective function of the NLP formulation is defined in Eq. (39) which consist of two parts, the TAC (equipment costs) and the GEC (water costs). The TAC is presented as annual cost whereas the GEC is presented with cost per ton flowrate multiplied by the total annual operating time. Both terms TAC and GEC are on the same cost unit. Moreover, the objective function strictly follows the relevant literatures for fair comparison of economic benefit between different methods and the original case studies while simultaneously provide good insights on the energy utilization and the water consumption. On the other hand, it is possible to add other variables (i.e., pipe cost, binary variables, etc.) in the objective function with appropriate modification on the NLP model but these cost factors are case by case which could be considered in the future.
The number of mixing heat exchangers should be chosen at the beginning prior solving the problem. The WAHEN superstructure NLP model is built and solved accordingly with the chosen number of mixing heat exchangers. The results of WAHEN configuration with different mixing heat exchangers number are then collected and compared accordingly. The most appropriate number of mixing heat exchangers is reported as the optimal result. It is suggested to start with a low number of mixing heat exchangers to avoid a large number of units and interconnections which may reduce the overall operability of the WAHEN configuration.
Case Studies
To show the extensive applicability of the proposed model, two different systems, isothermal systems (simple case) and nonisothermal system (complex case), are carried out in this work for comparison and discussion.
In isothermal systems, two case studies by Savulescu et al. (2005) and Dong et al. 2008 are considered for optimization. Since the process operating temperatures are constant, the process heat duty, in this case, is ignored. Moreover, the possible number of interconnections is not too much. Hence, the use of binary variables is unnecessary.
Gunaratnam et al. (2005) reported their result on a nonisothermal system which is suitable for comparison purpose. DeLeón Almaraz et al. (2016) solved the problem by twostep method mathematical programming. The proposed model shown in the previous section has greater flexibility compared to the model by DeLeón Almaraz et al. (2016). Note that the proposed model complexity is also increased because of the simultaneous nature of the mathematical model. To reduce the model complexity, the heat transfer area in Eq. 32 is calculated after the NLP model has been optimized. Also, the use of binary variables is avoided to reduce computational time and therefore MINLP model is not chosen for this particular work. All of the computational works are carried out in a desktop server platform with Intel® Xeon® CPU E52620 v4 @ 2.10Ghz and 256 GB RAM. All of the WAHEN NLP models were solved with the most recent GAMS/BARON version 25.0.2.
Isothermal System
Case Study 1
Case study 1 process data (Savulescu et al. 2005)
Process unit  M _{ C,P } (g/h)  \( {C}_{\max C,P}^{in} \) (ppm)  \( {C}_{\max C,P}^{out} \) (ppm)  Operating temperature (°C) 

1  7200  0  100  40 
2  18,000  50  100  100 
3  108,000  50  800  75 
4  14,400  400  800  50 
Case study 1 cost and operating parameters
Parameter  

C _{Fw}  Cost of fresh water  0.375 $/T 
C _{ Hu,Un }  Cost of hot utility  377 $/(kW.a) 
C _{ Cu,Un }  Cost of cold utility  189 $/(kW.a) 
\( {T}_{Hu, Un}^{in} \)  Inlet temperature of hot utility  120 °C 
\( {T}_{Cu, Un}^{in} \)  Inlet temperature of cold utility  10 °C 
\( {T}_{Hu, Un}^{out} \)  Outlet temperature of cold utility  20 °C 
Cp  Heat capacity of water  4.2 kJ/kg.°C 
C _{ fixed }  Fixed charges for heat exchangers and utility units  8000$/a 
C _{ area }  Area cost coefficient for heat exchangers and utility units  1200 $/(m^{2}.a) 
λ  Exponent parameter for area cost  0.6 
U _{ Eh,Ec,Hu,Cu }  Overall heat transfer coefficients for hot stream, old stream, and heat exchanger  0.5 kW/(m^{2}.°C) 
H  Hours of plant operation  8000 H per year 
T ^{ f }  Inlet temperature of fresh water  20 °C 
\( {T}_{\mathrm{max}}^w \)  Maximum limit for temperature of wastewater stream  30 °C 
β  Cost factor for wastewater  0 
Case study 1 results comparison (Ahmetović et al. 2015)
Method proposed by the authors  Solution method  Annual fresh water cost ($/y)  Annual cold utility cost ($/y)  Annual hot utility cost ($/y)  Annual investment cost ($/y)  Total annual cost ($/y) 

Savulescu et al. (2005)  PA  972,000  91,665  1,607,905  369,236  3,040,806 
Hou et al. (2014)  PA  972,000  0  1,425,060  364,587  2,761,647 
Polley et al. (2010)  PA  972,000  0  1,425,060  344,905  2,741,965 
MartínezPatiño et al. (2012)  PA  972,000  0  1,425,060  341,919  2,738,979 
Dong et al. (2008)  MP  972,000  0  1,425,060  341,047  2,738,107 
Luo et al. (2012)  PA  972,000  0  1,425,060  320,782  2,717,842 
Bagajewicz et al. (2002)  MP  972,000  0  1,425,060  314,495  2,711,555 
Bogataj and Bagajewicz (2008)  MP  972,000  0  1,425,060  310,495  2,711,555 
Liu et al. (2015)  PA MP  972,000  0  1,425,060  310,342  2,707,402 
Leewongtanawit and Kim (2009)  PA  972,000  0  1,425,060  310,293  2,707,353 
Liao et al. (2011)  MP  972,000  0  1,425,060  304,644  2,701,704 
Ahmetović and Kravanja (2013)  MP  972,000  0  1,425,060  255,899  2,652,959 
Ibrić et al. (2013a)  MP  972,000  0  1,425,060  255,899  2,652,959 
Zhou et al. (2015)  MP  972,000  0  1,425,060  255,899  2,652,959 
This work  MP  972,000  0  1,061,060  418,977  2.452,037 
Case Study 2
Case study 2 process data (Dong et al. 2008)
Process unit  Contaminant  \( {C}_{\max C,P}^{in} \) (ppm)  \( {C}_{\max C,P}^{out} \) (ppm)  M_{ C,P } (g/h)  Operation temperature (°C) 

P_{1}  A B C  0 0 0  100 80 60  10,800 8640 6480  100 
P_{2}  A B C  50 40 15  150 115 105  14,400 10,800 12,960  75 
P_{3}  A B C  50 50 30  125 80 130  5400 2160 7200  35 
Case study 2 cost and operating parameters
Parameter  

C _{Fw}  Cost of fresh water  0.45 $ per T 
C _{ Hu,Un }  Cost of hot utility  388 $/(kW.a) 
C _{ Cu,Un }  Cost of cold utility  189 $/(kW.a) 
\( {T}_{Hu, Un}^{in} \)  Inlet temperature of hot utility  150 °C 
\( {T}_{Cu, Un}^{in} \)  Inlet temperature of cold utility  10 °C 
\( {T}_{Hu, Un}^{out} \)  Outlet temperature of cold utility  20 °C 
Cp  Heat capacity of water  4.2 kJ/kg.°C 
C _{ fixed }  Fixed charges for heat exchangers and utility units  8000$/a 
C _{ area }  Area cost coefficient for heat exchangers and utility units  1200 $/(m^{2}.a) 
λ  Exponent parameter for area cost  0.6 
U _{ Eh,Ec,Hu,Cu }  Overall heat transfer coefficients for hot stream, cold stream, and heat exchanger  0.5 kW/(m^{2}.°C) 
H  Hours of plant operation  8000 H per year 
T ^{ f }  Inlet temperature of fresh water  80 °C 
\( {T}_{\mathrm{max}}^w \)  Maximum limit for temperature of wastewater stream  60 °C 
β  Cost factor for wastewater  0 
NonIsothermal System
Case study 3 process data (DeLeón Almaraz et al. 2016)
Process unit  Contaminant  \( {C}_{\max C,P}^{in} \) (ppm)  \( {C}_{\max C,P}^{out} \) (ppm)  M_{ C,P } (g/h)  \( {T}_{\max P}^{in} \) (°C)  \( {T}_{\min P}^{in} \) (°C)  \( {T}_{\max P}^{out} \) (°C)  \( {T}_{\min P}^{out} \) (°C)  PHD_{ P }(ton.°C.C_{P}/h) 

Stream stripping (P_{1})  HC H_{2}S SS  0 0 0  15 400 35  750 20,000 1750  290  180  1000*  20*  − 3500 
HDS1 (P_{2})  HC H_{2}S SS  20 300 45  120 12,500 180  3400 414,800 4590  1000*  20  35  20*  466 
Desalter (P_{3})  HC H_{2}S SS  120 20 200  220 45 9500  5600 1400 520,800  75  75  75  75  0 
VDU (P_{4})  HC H_{2}S SS  0 0 0  20 60 20  160 480 160  290  250  1000*  20*  − 1680 
HDS2 (P_{5})  HC H_{2}S SS  50 400 60  150 8000 120  800 60,800 480  1000*  20  40  20*  510 
Performance of the treatment units (DeLeón Almaraz et al. 2016)
Regeneration unit  Removal ratio  α value  \( {T}_{\max R}^{in} \) (°C)  \( {T}_{\min R}^{in} \) (°C)  \( {T}_{\max R}^{out} \) (°C)  \( {T}_{\min R}^{out} \) (°C)  RHD _{R} (ton.(°C).C_{P}/h)  

HC  H_{2}S  SS  
Steamstripping column (R_{1})  0  0.999  0  3.13  1000*  110  1000  0  − 4462 
Biological treatment (R_{2})  0.7  0.9  0.98  2.34  30  30  30  30  0 
API separator (R_{3})  0.95  0  0.5  0.89  35  35  35  35  0 
Case study 3 cost and operating parameters (DeLeón Almaraz et al. 2016)
Parameter  

C _{Fw}  Cost of fresh water  0.375 $ per T 
C _{ Hu,Un }  Cost of hot utility  377 $/(kW.a) 
C _{ Cu,Un }  Cost of cold utility  189 $/(kW.a) 
\( {T}_{Hu, Un}^{in} \)  Inlet temperature of hot utility  120 °C 
\( {T}_{Cu, Un}^{in} \)  Inlet temperature of cold utility  10 °C 
\( {T}_{Hu, Un}^{out} \)  Outlet temperature of cold utility  20 °C 
Cp  Heat capacity of water  4.2 kJ/kg.°C 
C _{ fixed }  Fixed charges for heat exchangers and utility units  8000$/a 
C _{ area }  Area cost coefficient for heat exchangers and utility units  1200 $/(m^{2}.a) 
λ  Exponent parameter for area cost  0.6 
U _{ Eh,Ec,Hu,Cu }  Overall heat transfer coefficients for hot stream, cold stream, and heat exchanger  0.5 kW/(m^{2}.°C) 
H  Hours of plant operation  8000 H per year 
T ^{ f }  Inlet temperature of fresh water  20 °C 
\( {T}_{\mathrm{max}}^w \)  Maximum limit for temperature of wastewater stream  30 °C 
β  Cost factor for wastewater  5.625 
For the operational purpose, it is favorable to have less unit number for any configuration due to the operation simplicity. In most of the design optimization with superstructure, users try to minimize capital and operating costs. There are occasions where the number of required units may be enormous, and the total capital costs are much lower than the operating costs. Although it is feasible, the operator of such configuration will have greater difficulties than to operate a configuration with less unit number. In the following case studies, the optimization campaign is started with an incremental number of allowed mixing heat exchangers. That way, the superstructure starts from small, and the structural possibility is increased in a controlled manner. The optimal configuration results from the work of DeLeón Almaraz et al. (2016) will be compared with the results of the proposed design method with increasing number of mixing heat exchangers.
Cost results for HEN design with mathematical programming approach (DeLeón Almaraz et al. 2016)
Ex. number  Match  Q (kW)  Area (m^{2})  Exchanger cost (M$)  Utility (M$) 

1  H1.C2.1  424  29.5  $ 0.035  – 
2  H1.C2.3  140  12.0  $ 0.014  – 
3  H1.C3.1  1500  90.8  $ 0.108  – 
4  H2.C1.2  894  66.5  $ 0.079  – 
5  H2.C1.3  317  30.6  $ 0.036  – 
6  H2.C4.1  869  27.2  $ 0.032  – 
7  H6.C1.3  707  67.8  $ 0.081  – 
8  H6.C5.1  1761  174.1  $ 0.208  – 
Total  6611  498.3  $ 0.598  $–  
Hot utility  
1  HUC1  7376  37.2  $ 0.044  $ 2.780 
2  HUC2  1575  12.9  $ 0.015  $ 0.593 
3  HUC3  1462  6.2  $ 0.007  $ 0.551 
4  HUC4  1202  5.0  $ 0.006  $ 0.453 
5  HUC5  814  3.1  $ 0.003  $ 0.306 
6  HUC6  289  0.9  $ 0.001  $ 0.109 
Total  12,719  65.3  $ 0.078  $ 4.794  
Cold utility  
1  CUH1  600  51.6  $ 0.061  $ 0.113 
2  CUH2  465  44.9  $ 0.053  $ 0.087 
3  CUH3  156  13.5  $ 0.016  $ 0.029 
4  CUH4  15  0.9  $ 0.001  $ 0.002 
5  CUH5  33  4.1  $ 0.005  $ 0.006 
6  CUH6  428  29.3  $ 0.035  $ 0.080 
7  CUH7  29  2.1  $ 0.002  $ 0.005 
8  CUH8  290  20.9  $ 0.025  $ 0.054 
Total  2017  167.3  $ 0.200  $ 0.381  
Fixed cost for 22 heat exchangers = M$ 0.176  
*Water cost = M$ 0.174 
General results with mathematical programming approach (DeLeón Almaraz et al. 2016)
Results  New equipment  Total Area (m^{2})  GEC (M$)  Utility cost (M$)  TAC (M$)  Annual cost (M$) 

Exchangers  8  498.40  2.2347  5.17  6.226  8.4607 
Heaters  6  65.25  
Coolers  8  167.35  
Total  22  731 
WAHEN Configuration with no Mixing Heat Exchanger
Unit parameter results of WAHEN configuration with no mixing heat exchanger
Unit  T^{ in } (°C)  T^{ out } (°C)  C^{ in } (ppm)  C^{ out } (ppm)  

HC  H_{2}S  SS  HC  H_{2}S  SS  
P_{1}  180  110  0  0  0  15  400  35 
P_{2}  21.28  35  8.09  300  45  108.09  12,500  180 
P_{3}  75  75  48.76  19.7  89.68  149.94  45  9500 
P_{4}  250  40  0  0  0  20  60  20 
P_{5}  20  30  4.96  12.98  60  36.34  2397.3  78.82 
R_{1}  110  62  50.08  4799.63  91.82  50.08  4.8  91.82 
R_{2}  30  30  149.63  51.43  9474.24  44.89  5.143  189.49 
R_{3}  35  35  47.85  4.95  133.68  2.39  4.95  66.84 
Cost results of WAHEN configuration with no mixing heat exchanger
Heat exchanger  Q (kW)  A^{ λ }(m^{2})  Exchanger cost (M$)  Utility (M$) 

Total hot utility  
Q_{P1}  9295.55  15.71  0.0189  3.5044 
Q_{P3}  449.09  5.77  0.0069  0.1693 
Q_{P4}  2137.98  8.05  0.0097  0.806 
Q_{R1}  4615.22  35.29  0.0423  1.7399 
Total  16,497.84  64.82  0.0778  6.2196 
Total cold utility  
D_{P2}  460.75  13.47  0.0162  0.0871 
D_{P5}  409  12.72  0.0153  0.0773 
D_{R2}  2295.85  23.2  0.0278  0.4339 
D_{R3}  2572.06  19  0.0228  0.4861 
D_{W}  283.36  8.31  0.01  0.0536 
Total  6021.02  76.7  0.0981  1.138 
Fixed cost for 9 heat exchangers = M$ 0.072 
WAHEN Configuration with One Mixing Heat Exchanger
Unit parameter results of WAHEN configuration with one mixing heat exchanger
Unit  T^{ in } (°C)  T^{ out } (°C)  C^{ in } (ppm)  C^{ out } (ppm)  

HC  H_{2}S  SS  HC  H_{2}S  SS  
P_{1}  180.2  110.26  0  0  0  15  400  35 
P_{2}  21.28  35  20  300  44.737  120  12,500  179.738 
P_{3}  75  75  87.597  8.191  123.447  188.42  33.397  9500 
P_{4}  250.16  40.16  0  0  0  20  60  20 
P_{5}  23.79  40  16.092  400  33.498  46.225  2690.09  51.578 
R_{1}  110  36.18  87.597  8191.32  123.447  87.597  8.191  123.447 
R_{2}  30  30  25.525  48.534  4946.14  7.658  4.853  98.923 
R_{3}  35  35  188.42  33.397  9500  9.421  33.397  4750 
Eh_{1}  100.59  30  15.69  353.103  32.931  15.69  353.103  32.931 
Ec_{1}  20  90.59  0  0  0  0  0  0 
Waste  303  –  11.169  5  100  –  –  – 
Cost results of WAHEN configuration with one mixing heat exchanger
Heat exchanger  Q (kW)  A^{ λ }(m^{2})  Exchanger cost (M$)  Utility (M$) 

Total mixing heat exchanger  
Eh_{1}Ec_{1}  4776.86  61.39  0.0737  – 
Total hot utility  
Q_{P1}  5327.16  12.4  0.0149  2.009 
Q_{P3}  2520.1  13.94  0.0167  0.95 
Q_{P4}  1393.8  6.93  0.0083  0.5254 
Q_{R1}  5128.95  30.88  0.037  1.9334 
Total  14,370.01  64.15  0.077  5.4178 
Total cold utility  
D_{P2}  362.81  12.45  0.0149  0.0686 
D_{P5}  286.66  9.52  0.0114  0.0542 
D_{R2}  558.53  11.62  0.0139  0.1056 
D_{R3}  2357.49  18.03  0.0216  0.4456 
D_{W}  19.2  1.79  0.0021  0.0036 
Total  3584.69  53.41  0.0641  0.6776 
Fixed cost for 10 heat exchangers = M$ 0.08 
WAHEN Configuration with Two Mixing Heat Exchangers
Unit parameter results of WAHEN configuration with two mixing heat exchangers
Unit  T^{ in }(^{°}C)  T^{ out }(^{°}C)  C^{ in } (ppm)  C^{ out } (ppm)  

HC  H_{2}S  SS  HC  H_{2}S  SS  
P_{1}  180  110  0  0  0  15  400  35 
P_{2}  20  33.72  20  300  45  120  12500  180 
P_{3}  75  75  83.65  19.78  118.75  184.52  45  9500 
P_{4}  250  40  0  0  0  20  60  20 
P_{5}  20  40  15.29  400  34.69  46.66  2784.31  53.52 
R_{1}  110  34.51  88.84  8372.39  126.27  88.84  8.37  126.27 
R_{2}  30  30  42.05  50  5000  12.31  5  100 
R_{3}  35  35  184.52  45  9500  9.23  45  4750 
Eh_{1}  107.81  30  184.52  45  9500  184.52  45  9500 
Eh_{2}  107.81  30  15.26  400.13  34.66  15.26  400.13  34.66 
Ec_{1}  20  88.83  0  0  0  0  0  0 
Ec_{2}  20  97.81  96.2  9347.14  138.96  96.2  9347.14  138.96 
Waste  303    12.31  5  100       
Cost results of WAHEN configuration with two mixing heat exchangers
Heat exchanger  Q (kW)  A^{ λ }(m^{2})  Exchanger cost (M$)  Utility (M$) 

Total mixing heat exchanger  
Eh_{1}Ec_{1}  4550.74  48.71  0.0584  – 
Eh_{2}Ec_{2}  4550.74  54.5  0.0654  – 
Total  9101.48  103.21  0.1238  – 
Total hot utility  
Q_{P1}  5296.8  12.31  0.0148  1.997 
Q_{P3}  2484.77  13.84  0.0167  0.9368 
Q_{P4}  1585.95  7.35  0.0089  0.5979 
Q_{R1}  1427.03  20.59  0.0247  0.538 
Q_{Eh1}  1918.93  20.48  0.0246  0.7234 
Total  12,713.48  74.57  0.0897  4.7931 
Total cold utility  
D_{P2}  449.63  14.3  0.0172  0.085 
D_{P5}  296.34  11.58  0.0139  0.056 
D_{R2}  302.54  8.6  0.0103  0.0572 
D_{Ec2}  921.63  19.85  0.0238  0.1742 
Total  1970.14  54.33  0.0652  0.3724 
Fixed cost for 11 heat exchangers = M$ 0.088 
WAHEN Configuration with More Mixing Heat Exchangers
The results comparison of this work with that of the reference
Number of heat exchanger  Number of utility heat exchanger  Cost of utility (M$)  TAC (M$)  GEC (M$)  Annual cost (M$)  Number of interconnections 

Reference by DeLeón Almaraz et al. (2016)  
8  14  5.175  6.226  2.2347  8.4607  29 
This work  
0  9  7.3576 (+ 42.18%)  7.5994 (+ 22.06%)  3.0107 (+ 34.73%)  10.6101 (+ 25.4%)  15 (− 48.58%) 
1  9  6.0954 (+ 17.79%)  6.3165 (− 1.43%)  2.2642 (+ 1.32%)  8.58 (+ 1.41%)  18 (− 37.93%) 
2  9  5.1655 (− 0.18%)  5.532 (− 11.15%)  2.263 (+ 1.27%)  7.795 (− 7.87%)  25 (− 13.8%) 
3  11  5.0632 (− 2.16%)  5.5655 (− 10.61%)  2.4563 (+ 9.92%)  8.0218 (− 5.19%)  30 (+ 3.45%) 
4  13  5.0542 (− 2.37%)  5.5414 (− 11%)  2.2749 (+ 1.8%)  7.8163 (− 7.62%)  32 (+ 10.34%) 
Conclusions
In this work, the mixing heat exchanger heating side and cooling side models are constructed within a WAHEN superstructure model to increase further flexibility to the original model. Simultaneous optimization of WAHEN design has been successfully implemented using NLP model. The number of mixing heat exchangers should be incrementally increased, and the appropriate number is then chosen with consideration on the total annual cost and the total number of interconnections.
In isothermal systems, the case study 1 optimal results were lower than references, because the wastewater temperature is allowed to be lower than 30 . The results show that the wastewater temperature should be defined a variable to increase the flexibility of the model. In case study 2, the optimal result is similar to the reference, and it shows the potential of using the proposed model to achieve similar performance to the other design strategies to address simple isothermal WAHEN system design.
In nonisothermal systems, the proposed design strategy has been implemented in a case study with five process units, three regeneration units, and muticontaminants. The effectiveness of the simultaneous optimization of the nonisothermal design of water networks with regeneration and recycling has been demonstrated with better annual costs and efficient configuration. It is expected that this work will be implemented in process industries in the coming future.
In the future, the heat transfer area variables and latent heat variables will be integrated into the mathematical model. Integer variables may be considered for the existence of the mixing heat exchangers for automatic computation to determine the optimal number of mixing heat exchangers with the tradeoff of longer computational time.
Notes
Acknowledgments
The authors wish to thank the anonymous reviewers for the useful comments that helped improve the readability of this work. The research was partially supported by the Ministry of Science and Technology, Taiwan under contract no. MOST 1062221E005095, the 105科技部補助大專校院延攬特殊優秀人才 award, and a grant from National ChungHsing University 10617003G 新進教師經費補助.
References
 Ahmetović E, Kravanja Z (2013) Simultaneous synthesis of process water and heat exchanger networks. Energy 57:236–250CrossRefGoogle Scholar
 Ahmetović E, Ibrića N, Kravanja Z, Grossmann IE (2015) Water and energy integration: a comprehensive literature review of nonisothermal water network synthesis. Comput Chem Eng 82:144–171CrossRefGoogle Scholar
 Bagajewicz M, Rodera H, Savelski M (2002) Energy efficient water utilization systems in process plants. Comput Chem Eng 26:59–79CrossRefGoogle Scholar
 Bogataj M, Bagajewicz MJ (2008) Synthesis of nonisothermal heat integrated water networks in chemical processes. Comput Chem Eng 32:310–3142CrossRefGoogle Scholar
 Boix M, Montastruc L, Pibouleau L, AzzaroPantel C, Domenech S (2012) Minimizing water and energy consumptions in water and heat exchange networks. Appl Therm Eng 36:442–455CrossRefGoogle Scholar
 Chen JJJ (1987) Comments on improvements on a replacement for the logarithmic mean. Chem Eng Sci 42:2488–2489CrossRefGoogle Scholar
 DeLeón Almaraz S, Boix M, Montastruc L, AzzaroPantel C, Liao Z, Domenech S (2016) Design of a water allocation and energy network for multicontaminant problems using multiobjective optimization. Process Saf Environ Prot 103:348–364CrossRefGoogle Scholar
 Dong HG, Lin CY, Chang CT (2008) Simultaneous optimization approach for integrated waterallocation and heatexchange networks. Chem Eng Sci 63:3664–3678CrossRefGoogle Scholar
 Feng X, Bai J, Wang H, Zheng X (2008) Grassroots design of regeneration recycling water networks. Comput Chem Eng 32:1892–1907CrossRefGoogle Scholar
 Gunaratnam M, AlvaArgáez A, Kokossis A, Kim JK, Smith R (2005) Automated design of total water systems. Ind Eng Chem Res 44:588–599CrossRefGoogle Scholar
 Hou Y, Wang J, Chen Z, Li X, Zhang J (2014) Simultaneous integration of water and energy on conceptual methodology for both single and multicontaminant problems. Chem Eng Sci 117:436–444CrossRefGoogle Scholar
 Ibrić N, Ahmetović E, Kravanja Z (2013) A twostep solution strategy for the synthesis of pinched and threshold heatintegrated process water networks. Chem Eng Trans 35:43–48Google Scholar
 Leewongtanawit B, Kim JK (2009) Improving energy recovery for water minimisation. Energy 34:880–893CrossRefGoogle Scholar
 Liao Z, Rong G, Wang J, Yang Y (2011) Systematic optimization of heatintegrated water allocation networks. Ind Eng Chem Res 50:6713–6727CrossRefGoogle Scholar
 Linnhoff B, Hindmarsh E (1983) The pinch design method for heat exchanger networks. Chem Eng Sci 38:745–763CrossRefGoogle Scholar
 Liu Y, Luo Y, Yuan X (2015) Simultaneous integration of water and energy in heatintegrated water allocation networks. AICHE J 61:2202–2214CrossRefGoogle Scholar
 Luo Y, Luo X, Yuan X (2012) Thermodynamic analysis of homogeneous nonisothermal mixing influence on the waterusing networks’ energy target. Comput Aided Chem Eng 31:420–424CrossRefGoogle Scholar
 MartínezPatiño J, PicónNúñez M, Serra LM, Verda V (2012) Systematic approach for the synthesis of water and energy networks. Appl Therm Eng 48:458–464CrossRefGoogle Scholar
 Polley GT, PicónNúñez M, LópezMaciel JDJ (2010) Design of water and heat recovery networks for the simultaneous minimisation of water and energy consumption. Appl Threm Eng 30:2290–2299CrossRefGoogle Scholar
 Savulescu L, Kim JK, Smith R (2005) Studies on simultaneous energy and water minimisation—part II: systems with maximum reuse of water. Chem Eng Sci 60:3291–3308CrossRefGoogle Scholar
 Wang YP, Smith R (1994) Wastewater minimisation. Chem Eng Sci 49:981–1006CrossRefGoogle Scholar
 Yan F, Wu H, Li W, Zhang J (2016) Simultaneous optimization of heatintegrated water networks by a nonlinear program. Chem Eng Sci 140:76–89CrossRefGoogle Scholar
 Yee TF, Grossmann IE (1990) Simultaneous optimization models for heat integration—II. Heat exchanger network synthesis. Comput Chem Eng 14:1165–1184CrossRefGoogle Scholar
 Zhou L, Liao Z, Wang J, Jiang B, Yang Y, Yu H (2015) Simultaneous optimization of heatintegrated water allocation networks using the mathematical model with equilibrium constraints strategy. Ind Eng Chem Res 54:3355–3366CrossRefGoogle Scholar