Simultaneous Optimization of Non-Isothermal Design of Water Networks with Regeneration and Recycling

• Wen-Hao Cheng
Original Research Paper

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 non-isothermal water networks with regeneration and recycling is proposed. Superstructure model with non-linear 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

Non-isothermal water networks NLP WAHEN Superstructure Mathematical programming

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)

C1, C2, C3, … ∈ C

Component

Un ∈ P, R, Eh, Ec

Units

Hu1, Hu2, Hu3, … ∈ Hu

Hot utility

Cu1, Cu2, Cu3, … ∈ Cu

Cold utility

in

Input condition

out

Output condition

f

Fresh water

w

Waste water

Parameter

MC,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)

PRC,R

Removal ratio of contaminant C in regeneration R

PHDP

Heat duty of process P (kW)

RHDR

Heat duty of regeneration R (kW)

Tf

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)

CFw

Cost of fresh water

CHu,Un

Cost of hot utility

CCu,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

Cfixed

Fixed charges for heat exchangers and utility units

Carea

Area cost coefficient for heat exchangers and utility units

UEh,Ec,hu,cu

Overall heat transfer coefficients for hot stream, cold stream, and heat exchanger

H

Hours of plant operation

Tf

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

NEh,Ec

Number of heat exchanger

NHu,Un

Number of hot utility

NCu,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)

FP→P'

Flowrate from process P to process P’ (t/h)

FR→R'

Flowrate from regeneration R to regeneration R’ (t/h)

FEh→Eh'

Flowrate from exchanger heating side Eh to exchanger heating side Eh’ (t/h)

FEc→Ec'

Flowrate from exchanger cooling side Ec to exchanger cooling side Ec’ (t/h)

FP→R

Flowrate from process P to regeneration R (t/h)

FP→Eh

Flowrate from process P to exchanger heating side Eh (t/h)

FP→Ec

Flowrate from process P to exchanger cooling side Ec (t/h)

FR→P

Flowrate from regeneration R to process P (t/h)

FR→Eh

Flowrate from regeneration R to exchanger heating side Eh (t/h)

FR→Ec

Flowrate from regeneration R to exchanger cooling side Ec (t/h)

FEh→P

Flowrate from exchanger heating side Eh to process P (t/h)

FEh→R

Flowrate from exchanger heating side Eh to regeneration R (t/h)

FEh→Ec

Flowrate from exchanger heating side Eh to exchanger cooling side Ec (t/h)

FEc→P

Flowrate from exchanger cooling side Ec to process P (t/h)

FEc→R

Flowrate from exchanger cooling side Ec to regeneration R (t/h)

FEc→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)

CC,Eh

Inlet concentration C of exchanger heating side Eh (ppm)

CC,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)

Tw

Temperature of wastewater (°C)

$${Q}_P^{in}$$

Inlet heat duty of process P inlet using hot utility (ton.°C.CP/h)

$${Q}_R^{in}$$

Inlet heat duty of regeneration R using hot utility (ton.°C.CP/h)

$${Q}_{Eh}^{in}$$

Inlet heat duty of exchanger heating side Eh using hot utility (ton.°C.CP/h)

$${Q}_{Ec}^{in}$$

Inlet heat quality of exchanger cooling side Ec using hot utility (ton.°C.CP/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

ΔTEh,Ec

Temperature difference of the mixing exchangers’ inlet and outlet

ΔTHu,Un

Temperature difference of the hot utility’ inlet and outlet

ΔTCu,Un

Temperature difference of the cold utility’ inlet and outlet

QEh,En

The exchanged energy by the heat exchanger

AEh,Ec

Heat transfer area of mixing heat exchanger

AHu,Un

Heat transfer area of hot utility

ACu,Un

Heat transfer area of cold utility

Negative Variable

$${D}_P^{in}$$

Removal heat duty in process P inlet using cooling utility (ton.°C.CP/h)

$${D}_R^{in}$$

Removal heat duty in regeneration R inlet using cooling utility (ton.°C.CP/h)

$${D}_{Eh}^{in}$$

Removal heat duty in exchanger heating side Eh inlet using cooling utility (ton.°C.CP/h)

$${D}_{Ec}^{in}$$

Removal heat duty in exchanger cooling side Ec inlet using cooling utility (ton.°C.CP/h)

$${D}_w^{in}$$

Removal heat in exchanger cooling side Ec inlet using cooling utility (ton.°C.CP/h)

Free Variable

Cannual

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 ever-increasing 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 multiple-contaminants 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 multiple-contaminants WAN design using mixed integer non-linear 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 ∆Tmin 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 non-linear 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 non-isothermal water network synthesis. There are two main optimization methods for WAHEN design, two-step method and simultaneous method. Boix et al. (2012) proposed two-step 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. De-León Almaraz et al. (2016) also solved the WAHEN design problem with two-step 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. De-León 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 De-León Almaraz et al. (2016). The mixing heat exchanger is constructed separately for the heating side and the cooling side. From the original model of De-León 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 non-isothermal 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 The WAHEN model is constructed with the superstructure as shown in Fig. 1. From a given number of process units (P), regeneration units (R), mixing heat exchanger heating side (Eh), and mixing exchanger cooling side (Ec), all the possible connections between them may exist, except stream connections from the same units. The inlet water in a processing unit can be freshwater, used water coming from other processes and/or recycled water coming from a regeneration unit. Each process unit has contaminants mass loads that may increase contaminants concentration in the inlet water. Several contaminants can exist in the system. The flow rate, concentration, and temperature of each stream would be presented in streamlines. Inlet or outlet flow rate, contaminant concentrations, and temperature are imposed by the user and will constitute bounds for the optimization problem. The water output from process units may be directly discharged and/or distributed to all other units. Similarly, for a regeneration unit, the input water may come from either process units, other regeneration units or heat exchangers. The regeneration units have been given processing capacity for specific contaminants. Streams are allowed to be heated or cooled by external utilities before entering process units to satisfy the inlet temperature of the corresponding process units. In this work, the unit inlets and outlets are specified with the required water quality and temperature. The process units are specified with mass load and heat duty. The regeneration units are specified with removal ratio and heat duty. The mixing heat exchangers are allowed to exchange heat from its heating side and its cooling side. Some assumptions are made as follows: • No heat loss. • Concentration and temperature are ideally mixed. • Steady-state 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. The mass balance for process units (P) is shown in Eq. (1). $${\displaystyle \begin{array}{l}{F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\\ {}={F}_P^w+\sum \limits_P{F}_{P\to P\hbox{'}\hbox{'}}+\sum \limits_P{F}_{P\to R}+\sum \limits_P{F}_{P\to Eh}+\sum \limits_P{F}_{P\to Ec}\end{array}}$$ (1) The mass balance for regeneration units (R) is shown in Eq. (2). $${\displaystyle \begin{array}{l}{F}_R^f+\sum \limits_P{F}_{P\to R}+\sum \limits_{R\hbox{'}}{F}_{R\hbox{'}\to R}+\sum \limits_{Eh}{F}_{Eh\to R}+\sum \limits_{Ec}{F}_{Ec\to R}\\ {}={F}_R^w+\sum \limits_R{F}_{R\to P}+\sum \limits_R{F}_{R\to R\hbox{'}\hbox{'}}+\sum \limits_R{F}_{R\to Eh}+\sum \limits_R{F}_{R\to Ec}\end{array}}$$ (2) The mass balance for mixing heat exchangers heating side (Eh) is shown in Eq. (3). $${\displaystyle \begin{array}{l}{F}_{Eh}^f+\sum \limits_{\mathrm{P}}{F}_{P\hbox{'}\to Eh}+\sum \limits_R{F}_{R\to Eh}+\sum \limits_{Eh\hbox{'}}{F}_{Eh\hbox{'}\to Eh}+\sum \limits_{Ec}{F}_{Ec\to Eh}\\ {}={F}_{Eh}^w+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Eh}{F}_{Eh\to R}+\sum \limits_{Eh}{F}_{Eh\to Eh\hbox{'}\hbox{'}}+\sum \limits_{Eh}{F}_{Eh\to Ec}\end{array}}$$ (3) The mass balance for mixing heat exchangers cooling side (Ec) is shown in Eq. (4). $${\displaystyle \begin{array}{l}{F}_{Ec}^f+\sum \limits_{\mathrm{P}}{F}_{P\hbox{'}\to Ec}+\sum \limits_R{F}_{R\to Ec}+\sum \limits_{Eh}{F}_{Eh\to Ec}+\sum \limits_{Ec\hbox{'}}{F}_{Ec\hbox{'}\to Ec}\\ {}={F}_{Ec}^w+\sum \limits_{Ec}{F}_{Ec\to P}+\sum \limits_{Ec}{F}_{Ec\to R}+\sum \limits_{Ec}{F}_{Ec\to Eh}+\sum \limits_{Ec}{F}_{Ec\to Ec\hbox{'}\hbox{'}}\end{array}}$$ (4) The contaminant (C) balance for process units (P) inlet is formulated in Eq. (5). $${\displaystyle \begin{array}{l}\sum \limits_{P\hbox{'}}\left({F}_{P\hbox{'}\to P}\times {C}_{C,P\hbox{'}}^{out}\right)+\sum \limits_R\left({F}_{R\to P}\times {C}_{C,R}^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh\to P}\times {C}_{C, Eh}\right)+\sum \limits_{Ec}\left({F}_{Ec\to P}\times {C}_{C, Ec}\right)\\ {}=\left({F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\right)\times {C}_{C,P}^{in}\end{array}}$$ (5) The contaminant (C) balance for regeneration units (R) inlet is formulated in Eq. (6). $${\displaystyle \begin{array}{l}\sum \limits_P\left({F}_{P\to R}\times {C}_{C,P}^{out}\right)+\sum \limits_{R\hbox{'}}\left({F}_{R\hbox{'}\to R}\times {C}_{C,R\hbox{'}}^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh\to R}\times {C}_{C, Eh}\right)+\sum \limits_{Ec}\left({F}_{Ec\to R}\times {C}_{C, Ec}\right)\\ {}=\left({F}_R^f+\sum \limits_P{F}_{P\to R}+\sum \limits_{R\hbox{'}}{F}_{R\hbox{'}\to R}+\sum \limits_{Eh}{F}_{Eh\to R}+\sum \limits_{Ec}{F}_{Ec\to R}\right)\times {C}_{C,R}^{in}\end{array}}$$ (6) The contaminant (C) balance for mixing heat exchangers heating side (Eh) is formulated in Eq. (7). $${\displaystyle \begin{array}{l}\sum \limits_P\left({F}_{P\to Eh}\times {C}_{C,P}^{out}\right)+\sum \limits_R\left({F}_{R\to Eh}\times {C}_{C,R}^{out}\right)+\sum \limits_{Eh\hbox{'}}\left({F}_{Eh\hbox{'}\to Eh}\times {C}_{C, Eh\hbox{'}}\right)+\sum \limits_{Ec}\left({F}_{Ec\to Eh}\times {C}_{C, Ec}\right)\\ {}=\left({F}_{Eh}^f+\sum \limits_{\mathrm{P}}{F}_{P\hbox{'}\to Eh}+\sum \limits_R{F}_{R\to Eh}+\sum \limits_{Eh\hbox{'}}{F}_{Eh\hbox{'}\to Eh}+\sum \limits_{Ec}{F}_{Ec\to Eh}\right)\times {C}_{C, Eh}^{in}\end{array}}$$ (7) The contaminant (C) balance for mixing heat exchangers cooling side (Ec) is formulated in Eq. (8). $${\displaystyle \begin{array}{l}\sum \limits_P\left({F}_{P\to Ec}\times {C}_{C,P}^{out}\right)+\sum \limits_R\left({F}_{R\to Ec}\times {C}_{C,R}^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh\to Ec}\times {C}_{C, Eh}\right)+\sum \limits_{Ec\hbox{'}}\left({F}_{Ec\hbox{'}\to Ec}\times {C}_{C, Ec\hbox{'}}\right)\\ {}=\left({F}_{Ec}^f+\sum \limits_{\mathrm{P}}{F}_{P\hbox{'}\to Ec}+\sum \limits_R{F}_{R\to Ec}+\sum \limits_{Eh}{F}_{Eh\to Ec}+\sum \limits_{Ec\hbox{'}}{F}_{Ec\hbox{'}\to Ec}\right)\times {C}_{C, Ec}^{in}\end{array}}$$ (8) The contaminant (C) balance for process units (P) outlet is formulated in Eq. (9). $${\displaystyle \begin{array}{l}\left({F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\right)\times {C}_{C,P}^{in}+{M}_{C,P}\\ {}=\left({F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\right)\times {C}_{C,P}^{out}\end{array}}$$ (9) The removal ratio (RR) balance for regeneration units (R) is formulated in Eq. (10). $${RR}_{C,R}=\frac{C_{C,R}^{out}}{C_{C,R}^{in}-{C}_{C,R}^{out}}$$ (10) The contaminant (C) balance for wastewater outlet is formulated in Eq. (11). $${\displaystyle \begin{array}{l}\sum \limits_P\left({F}_P^w\times {C}_{C,P}^{out}\right)+\sum \limits_R\left({F}_R^w\times {C}_{C,R}^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh}^w\times {C}_{C, Eh}\right)+\sum \limits_{Ec}\left({F}_{Ec}^w\times {C}_{C, Ec}\right)\\ {}=\left(\sum \limits_P{F}_P^w+\sum \limits_R{F}_R^w+\sum \limits_{Eh}{F}_{Eh}^w+\sum \limits_{Ec}{F}_{Ec}^w\right)\times {C}_C^w\end{array}}$$ (11) The energy balance for process units (P) inlet is formulated in Eq. (12). $${\displaystyle \begin{array}{l}\frac{Q_P^{in}+{D}_P^{in}}{Cp}+\left({F}_P^f\times {T}^f\right)+\sum \limits_{P\hbox{'}}\left({F}_{P\hbox{'}\to P}\times {T}_{P\hbox{'}}^{out}\right)+\sum \limits_{\mathrm{R}}\left({F}_{R\to P}\times {T}_R^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh\to P}\times {T}_{Eh}^{out}\right)\\ {}+\sum \limits_{Ec}\left({F}_{Ec\to P}\times {T}_{Ec}^{out}\right)=\left({F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\right)\times {T}_P^{in}\end{array}}$$ (12) The energy balance for regeneration units (R) inlet is formulated in Eq. (13). $${\displaystyle \begin{array}{l}\frac{Q_R^{in}+{D}_R^{in}}{Cp}+\left({F}_R^f\times {T}^f\right)+\sum \limits_P\left({F}_{P\to R}\times {T}_P^{out}\right)+\sum \limits_{\mathrm{R}\hbox{'}}\left({F}_{R\hbox{'}\to R}\times {T}_{R\hbox{'}}^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh\to R}\times {T}_{Eh}^{out}\right)\\ {}+\sum \limits_{Ec}\left({F}_{Ec\to R}\times {T}_{Ec}^{out}\right)=\left({F}_R^f+\sum \limits_P{F}_{P\to R}+\sum \limits_{R\hbox{'}}{F}_{R\hbox{'}\to R}+\sum \limits_{Eh}{F}_{Eh\to R}+\sum \limits_{Ec}{F}_{Ec\to R}\right)\times {T}_R^{in}\end{array}}$$ (13) The energy balance for mixing heat exchangers heating side (Eh) inlet is formulated in Eq. (14). $${\displaystyle \begin{array}{l}\frac{Q_{Eh}^{in}+{D}_{Eh}^{in}}{Cp}+\left({F}_{Eh}^f\times {T}^f\right)+\sum \limits_P\left({F}_{P\to Eh}\times {T}_P^{out}\right)+\sum \limits_{\mathrm{R}}\left({F}_{R\to Eh}\times {T}_R^{out}\right)\\ {}+\sum \limits_{Eh\hbox{'}}\left({F}_{Eh\hbox{'}\to Eh}\times {T}_{Eh\hbox{'}}^{out}\right)+\sum \limits_{Ec}\left({F}_{Ec\to Eh}\times {T}_{Ec}^{out}\right)\\ {}=\left({F}_{Eh}^f+\sum \limits_{\mathrm{P}}{F}_{P\hbox{'}\to Eh}+\sum \limits_R{F}_{R\to Eh}+\sum \limits_{Eh\hbox{'}}{F}_{Eh\hbox{'}\to Eh}+\sum \limits_{Ec}{F}_{Ec\to Eh}\right)\times {T}_{Eh}^{in}\end{array}}$$ (14) The energy balance for mixing heat exchangers cooling side (Ec) inlet is formulated in Eq. (15). $${\displaystyle \begin{array}{l}\frac{Q_{Ec}^{in}+{D}_{Ec}^{in}}{Cp}+\left({F}_{Ec}^f\times {T}^f\right)+\sum \limits_P\left({F}_{P\to Ec}\times {T}_P^{out}\right)+\sum \limits_{\mathrm{R}}\left({F}_{R\to Ec}\times {T}_R^{out}\right)\\ {}+\sum \limits_{Eh}\left({F}_{Eh\to Ec}\times {T}_{Eh}^{out}\right)+\sum \limits_{Ec\hbox{'}}\left({F}_{Ec\hbox{'}\to Ec}\times {T}_{Ec\hbox{'}}^{out}\right)\\ {}=\left({F}_{Ec}^f+\sum \limits_{\mathrm{P}}{F}_{P\hbox{'}\to Ec}+\sum \limits_R{F}_{R\to Ec}+\sum \limits_{Eh}{F}_{Eh\to Ec}+\sum \limits_{Ec\hbox{'}}{F}_{Ec\hbox{'}\to Ec}\right)\times {T}_{Ec}^{in}\end{array}}$$ (15) The energy balance for process units (P) outlet is formulated in Eq. (16). $${\displaystyle \begin{array}{l}\left({F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\right)\times {T}_P^{in}+\raisebox{1ex}{{PHD}_P}\!\left/ \!\raisebox{-1ex}{ Cp}\right.\\ {}=\left({F}_P^f+\sum \limits_{\mathrm{P}\hbox{'}}{F}_{P\hbox{'}\to P}+\sum \limits_R{F}_{R\to P}+\sum \limits_{Eh}{F}_{Eh\to P}+\sum \limits_{Ec}{F}_{Ec\to P}\right)\times {T}_P^{out}\end{array}}$$ (16) The energy balance for regeneration units (R) outlet is formulated in Eq. (17). $${\displaystyle \begin{array}{l}\left({F}_R^f+\sum \limits_{\mathrm{P}}{F}_{P\to R}+\sum \limits_{R\hbox{'}}{F}_{R\hbox{'}\to R}+\sum \limits_{Eh}{F}_{Eh\to R}+\sum \limits_{Ec}{F}_{Ec\to R}\right)\times {T}_R^{in}+\raisebox{1ex}{{RHD}_P}\!\left/ \!\raisebox{-1ex}{ Cp}\right.\\ {}=\left({F}_R^f+\sum \limits_{\mathrm{P}}{F}_{P\to R}+\sum \limits_{R\hbox{'}}{F}_{R\hbox{'}\to R}+\sum \limits_{Eh}{F}_{Eh\to R}+\sum \limits_{Ec}{F}_{Ec\to R}\right)\times {T}_R^{out}\end{array}}$$ (17) The heat transfer energy balance between mixing heat exchangers heating side (Eh) and cooling side (Ec) is shown in Eq. (18). $${\displaystyle \begin{array}{l}\left({F}_{Eh}^f+\sum \limits_P{F}_{P\to Eh}+\sum \limits_R{F}_{R\to Eh}+\sum \limits_{Eh\hbox{'}}{F}_{Eh\hbox{'}\to Eh}+\sum \limits_{Ec}{F}_{Ec\to Eh}\right)\times \left({T}_{Eh}^{in}-{T}_{Eh}^{out}\right)\\ {}=\left({F}_{Ec}^f+\sum \limits_P{F}_{P\to Ec}+\sum \limits_R{F}_{R\to Ec}+\sum \limits_{Eh}{F}_{Eh\to Ec}+\sum \limits_{Ec\hbox{'}}{F}_{Ec\hbox{'}\to Ec}\right)\times \left({T}_{Ec}^{out}-{T}_{Ec}^{in}\right)\end{array}}$$ (18) The energy balance for wastewater outlet is shown in Eq. (19). $${\displaystyle \begin{array}{l}{\mathrm{D}}^{w, in}+\sum \limits_P\left({F}_P^w\times {T}_P^{out}\right)+\sum \limits_R\left({F}_R^w\times {T}_R^{out}\right)+\sum \limits_{Eh}\left({F}_{Eh}^w\times {T}_{Eh}\right)+\sum \limits_{Ec}\left({F}_{Ec}^w\times {T}_{Ec}\right)\\ {}=\left(\sum \limits_P{F}_P^w+\sum \limits_R{F}_R^w+\sum \limits_{Eh}{F}_{Eh}^w+\sum \limits_{Ec}{F}_{Ec}^w\right)\times {T}^w\end{array}}$$ (19) The inlet and outlet contaminant concentration upper limit for process units (P) is defined in Eq. (20) and Eq. (21). $${C}_{C,P}^{in}\le {C}_{\max\ C,\mathrm{P}}^{in}$$ (20) $${C}_{C,P}^{out}\le {C}_{\max\ C,\mathrm{P}}^{out}$$ (21) The contaminant concentration upper limit for wastewater is defined in Eq. (22). $${C}_C^w\le {C}_{\max\ C}^w$$ (22) The inlet temperature upper and lower limit for process units (P) is defined in Eq. (23). $${T}_{\min\ P}^{in}\le {T}_P^{in}\le {T}_{\max\ P}^{in}$$ (23) The outlet temperature upper and lower limit for process units (P) is defined in Eq. (24). $${T}_{\min\ P}^{out}\le {T}_P^{out}\le {T}_{\max\ P}^{out}$$ (24) The inlet temperature upper and lower limit for regeneration units (R) is defined in Eq. (25). $${T}_{\min\ \mathrm{R}}^{in}\le {T}_{\mathrm{R}}^{in}\le {T}_{\max\ \mathrm{R}}^{in}$$ (25) The outlet temperature upper and lower limit for regeneration units (R) is defined in Eq. (26). $${T}_{\min\ \mathrm{R}}^{out}\le {T}_{\mathrm{R}}^{out}\le {T}_{\max\ \mathrm{R}}^{out}$$ (26) The temperature upper limit for wastewater is defined in Eq. (27). $${T}^w\le {T}_{\mathrm{max}}^w$$ (27) The physical limit for mixing heat exchangers heating side (Eh) and cooling side (Ec) is shown in Eq. (28) and Eq. (29). $${T}_{Eh}^{in}>{T}_{Eh}^{out}$$ (28) $${T}_{Ec}^{in}<{T}_{Ec}^{out}$$ (29) Eq. (30) and Eq. (31) are formulated for temperature difference of the mixing exchangers’ inlet and outlet. Eq. (33)–(35) are formulated for temperature difference of the utility heat exchangers’ inlet and outlet. Tj,Hu,Un is the inlet temperature for operating unit before heated by hot utility; Tk,Hu,Un is the inlet temperature for operating unit after heated by hot utility; Tj,Cu,Un is the inlet temperature for operating unit before cooled by cold utility; Tk,Cu,Un is the inlet temperature for operating unit after cooled by cold utility. The parameter γ is the minimum temperature difference of the heat exchangers’ inlet and outlet. $$\Delta {T}_{1, Eh, Ec}={T}_{Eh}^{out}-{T}_{Ec}^{in}\ge \gamma$$ (30) $$\Delta {T}_{2, Eh, Ec}={T}_{Eh}^{out}-{T}_{Ec}^{in}\ge \gamma$$ (31) $$\Delta {T}_{1, Hu, Un}={T}_{Hu, Un}^{in}-{T}_{j, Hu, Un}\ge \gamma$$ (32) $$\Delta {T}_{2, Hu, Un}={T}_{Hu, Un}^{in}-{T}_{k, Hu, Un}\ge \gamma$$ (33) $$\Delta {T}_{1, Cu, Un}={T}_{j, Cu, Un}-{T}_{Cu, Un}^{out}\ge \gamma$$ (34) $$\Delta {T}_{2, Cu, Un}={T}_{k, Cu, Un}-{T}_{Cu, Un}^{in}\ge \gamma$$ (35) The heat transfer area of mixing heat exchangers, hot utility, and cold utility is calculated by Eq. (36)–(38) (Chen 1987). QEh,Ec is the total exchanged heat of the corresponding mixing heat exchanger; AEh,Ec , AHu,Un, ACu,Un are the heat transfer area of the mixing heat exchangers, hot utility, and cold utility heat exchangers. For non-isothermal systems (complex cases), Eqs. (36)–(38) are calculated subsequently after the overall NLP model has been computed in order to reduce the model complexity and avoid the difficulties to find reliable solutions. It is arguable that this approach might affect the objective function due to the different total area with the same number of exchangers with each calculation. Therefore, the model has to be calculated several times to ensure reliable solutions. On the other hand, the heat transfer area model can be simultaneously computed in isothermal systems (simple cases) due to its simpler mathematical model structure. $${A}_{Eh, Ec}=\frac{Q_{Eh, Ec}}{U{\left[\left(\varDelta {T}_{1, Eh, Ec}\times \varDelta {T}_{2, Eh, Ec}\times \frac{\varDelta {T}_{1, Eh, Ec}+\varDelta {T}_{2, Eh, Ec}}{2}\right)\right]}^{\raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{3}\right.}}$$ (36) $${A}_{Hu, Un}=\frac{Q_{P,R, Eh, Ec}^{in}}{U{\left[\left(\Delta {T}_{1, Hu, Un}\times \Delta {T}_{2, Hu, Un}\times \frac{\Delta {T}_{1, Hu, Un}+\Delta {T}_{2, Hu, Un}}{2}\right)\right]}^{\raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{3}\right.}}$$ (37) $${A}_{Cu, Un}=\frac{D_{P,R, Eh, Ec}^{in}}{U{\left[\left(\Delta {T}_{1, Cu, Un}\times \Delta {T}_{2, Cu, Un}\times \frac{\Delta {T}_{1, Cu, Un}+\Delta {T}_{2, Cu, Un}}{2}\right)\right]}^{\raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{3}\right.}}$$ (38) 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. Both total annual cost (TAC) and global equivalent cost (GEC) will be minimized to find the lowest cost of the overall WAHEN configuration. The TAC breakdown is shown in Eq. 40, where CHu,Un is the cost of hot utility; CCu,Un is the cost of cold utility; $${A_{Eh, Ec}^{\lambda}}^{\ast }{C}_{area}$$, $${A_{Hu, Un}^{\lambda}}^{\ast }{C}_{area}$$ and $${A_{Cu, Un}^{\lambda}}^{\ast }{C}_{area}$$ are heat transfer area multiplied by the unit cost of heat transfer area; λ is the exponent parameter for area cost; NEh, EcCfixed, NHu, UnCfixed and NCu, UnCfixed are the number of heat exchanger multiplied by the fixed cost of the heat exchanger. The GEC is also broken down in Eq. 41 where FF is the total freshwater flow rate; αFR is regeneration water cost multiplied by its factor; βFW is the total wastewater flow rate multiplied by its factor. α is regeneration cost factor; β is wastewater cost factor. α and β are equipment specific, hence corresponding values are quoted case by case. $$Obj=\mathit{\min}\left( TAC+ GEC\right)$$ (39) $${\displaystyle \begin{array}{l} TAC=\left(\sum \limits_P{Q}_P^{in}+\sum \limits_R{Q}_R^{in}+\sum \limits_{Eh}{Q}_{Eh}^{in}+\sum \limits_{Ec}{Q}_{Ec}^{in}\right)\ast {C}_{Hu, Un}\\ {}+\left(\sum \limits_P{D}_P^{in}+\sum \limits_R{D}_R^{in}+\sum \limits_{Eh}{D}_{Eh}^{in}+\sum \limits_{Ec}{D}_{Ec}^{in}\right)\ast \left(-{C}_{Cu, Un}\right)\\ {}+\sum \left({A_{Eh, Ec}^{\lambda}}^{\ast }{C}_{area}\right)+\left({N_{Eh, Ec}}^{\ast }{C}_{fixed}\right)\\ {}+\sum \left({A_{Hu, Un}^{\lambda}}^{\ast }{C}_{area}\right)+\left({N_{Hu, Un}}^{\ast }{C}_{fixed}\right)\\ {}+\sum \left({A_{Cu, Un}^{\lambda}}^{\ast }{C}_{area}\right)+\left({N_{Cu, Un}}^{\ast }{C}_{fixed}\right)\end{array}}$$ (40) $${\displaystyle \begin{array}{l} GEC=\Big(\sum \limits_P{F}_P^f+\sum \limits_R{F}_R^f+\sum \limits_{Eh}{F}_{Eh}^f+\sum \limits_{Ec}{F}_{Ec}^f\\ {}+\sum \limits_P\alpha {F}_{P\to R}+\sum \limits_{R\hbox{'}}\alpha {F}_{R\hbox{'}\to R}+\sum \limits_{Eh}\alpha {F}_{Eh\to R}+\sum \limits_{Ec}\alpha {F}_{Ec\to R}\\ {}+\sum \limits_P\beta {F}_p^W+\sum \limits_R\beta {F}_R^W+\sum \limits_{Eh}\beta {F}_{Eh}^W+\sum \limits_{Ec}\beta {F}_{Ec}^W\Big)\ast {H}^{\ast }{C}_{Fw}\end{array}}$$ (41) 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 non-isothermal 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 non-isothermal system which is suitable for comparison purpose. De-León Almaraz et al. (2016) solved the problem by two-step method mathematical programming. The proposed model shown in the previous section has greater flexibility compared to the model by De-León 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 E5-2620 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 by Savulescu et al. (2005) is considered for water-using operations with a single contaminant. The operation data are shown in Table 1. Table 2 presents the cost and operating parameters. Case study 1 has been discussed and addressed in many published works regarding the synthesis of WAHEN design. Table 3 shows the comparison the results of the case study 1 (Ahmetović et al. 2015). Table 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 Table 2 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$/a C area Area cost coefficient for heat exchangers and utility units 1200$/(m2.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/(m2.°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

Table 3

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 ($/y) Savulescu et al. (2005) PA 972,000 91,665 1,607,905 369,236 3,040,806 Hou et al. (2014) PA 972,000 0 1,425,060 364,587 2,761,647 Polley et al. (2010) PA 972,000 0 1,425,060 344,905 2,741,965 Martínez-Patiño et al. (2012) PA 972,000 0 1,425,060 341,919 2,738,979 Dong et al. (2008) MP 972,000 0 1,425,060 341,047 2,738,107 Luo et al. (2012) PA 972,000 0 1,425,060 320,782 2,717,842 Bagajewicz et al. (2002) MP 972,000 0 1,425,060 314,495 2,711,555 Bogataj and Bagajewicz (2008) MP 972,000 0 1,425,060 310,495 2,711,555 Liu et al. (2015) PA- MP 972,000 0 1,425,060 310,342 2,707,402 Leewongtanawit and Kim (2009) PA 972,000 0 1,425,060 310,293 2,707,353 Liao et al. (2011) MP 972,000 0 1,425,060 304,644 2,701,704 Ahmetović and Kravanja (2013) MP 972,000 0 1,425,060 255,899 2,652,959 Ibrić et al. (2013a) MP 972,000 0 1,425,060 255,899 2,652,959 Zhou et al. (2015) MP 972,000 0 1,425,060 255,899 2,652,959 This work MP 972,000 0 1,061,060 418,977 2.452,037 Using the proposed model, the NLP model consists of 145 continuous variables and 87 equations. The total computational time required is 1428 s. Figure 2 shows the optimal configuration. In case study 1, only fresh water cost is considered for the GEC. As shown in Table 3, the annual cost of this work is 2,452,037$/y, which is lower than the previous works’ annual costs. It is worth to mention that the wastewater temperature is defined as a variable, not as a constant. The resulting temperature of wastewater is 27 °C after optimization, which is different to that of the references (30 °C). It is reasonable that by having lower wastewater temperature, the hot utility demand can also be reduced by 2814.57 kW. Moreover, the wastewater temperature lower than 30 °Cis not harmful to the environment.

Case Study 2

Case study 2 is including three processes and specifically consists of three contaminants system (Dong et al. 2008). The process operating conditions are shown in Table 4. Yan et al. (2016) have reported their best solution where the freshwater flowrate of the optimal configuration is 252 ton/h, the hot and cold utility consumption rates are 172 and 6052 kW, and CPU time is 20 s, respectively. One heat exchanger, one heater and two coolers are used in the reported optimal structure (Table 5).
Table 4

Case study 2 process data (Dong et al. 2008)

Process unit

Contaminant

$${C}_{\max C,P}^{in}$$ (ppm)

$${C}_{\max C,P}^{out}$$ (ppm)

MC,P (g/h)

Operation temperature

(°C)

P1

A

B

C

0

0

0

100

80

60

10,800

8640

6480

100

P2

A

B

C

50

40

15

150

115

105

14,400

10,800

12,960

75

P3

A

B

C

50

50

30

125

80

130

5400

2160

7200

35

Table 5

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 $/(m2.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/(m2.°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 The NLP model of the proposed simultaneous strategy consists of 104 continuous variables and 102 equations. The total computational time required is 300 s. The optimal configuration of the proposed model is shown in Fig. 3. The total freshwater flowrate is 252 ton/h, whereas the hot utility and cold utility consumption are 172.19 and 6052.65 kW. The optimal configuration was similar to that of Yan et al. (2016), but the CPU time (300 s) is longer than the previously reported work. This case study shows that the proposed model is capable of computing optimal configuration at the same level with the previous work for isothermal system although requiring more computational time thanks to the enhanced flexibility of the proposed model that allows simultaneous optimization for both isothermal and non-isothermal operations. Non-Isothermal System A case study 3 for a simplified petroleum refinery (Gunaratnam et al. 2005) is considered and analyzed to design the WAHEN. The process unit (P1: stream stripping, P2: hydrodesulfurization (HDS-1), P3: desalter, P4: vacuum distillation unit (VDU) and P5: HDS-2), and three regeneration units (R1: steam-stripping column, R2: biological treatment unit, R3: API separator) with three species of contaminants (C1: hydrocarbon (HC), C2: hydrogen sulfide (H2S), C3: suspended solids (SS)). Tables 6, 7, and 8 are presented to show the required parameters for the corresponding WAHEN design (De-León Almaraz et al. 2016). Note that the hot utility temperature is set to be 120 °C. Therefore, the upper limit for the temperature of the streams heated by the hot utility is 110 °C. With the same manner as the hot utility, the temperature of streams cooled by cold utility lower limit is 20 °C. Moreover, process units P1 and P4 require over 180 °C and 250 °Cof inlet temperatures. Therefore, the temperature of hot utility for P1 and P4 is assumed to be at 300 °C. Consequently, the maximum temperature for streams that are heated by the 300 °C hot utility is 290 °C. Table 6 Case study 3 process data (De-León Almaraz et al. 2016) Process unit Contaminant $${C}_{\max C,P}^{in}$$ (ppm) $${C}_{\max C,P}^{out}$$ (ppm) MC,P (g/h) $${T}_{\max P}^{in}$$ (°C) $${T}_{\min P}^{in}$$ (°C) $${T}_{\max P}^{out}$$ (°C) $${T}_{\min P}^{out}$$ (°C) PHDP(ton.°C.CP/h) Stream stripping (P1) HC H2S SS 0 0 0 15 400 35 750 20,000 1750 290 180 1000* 20* − 3500 HDS-1 (P2) HC H2S SS 20 300 45 120 12,500 180 3400 414,800 4590 1000* 20 35 20* 466 Desalter (P3) HC H2S SS 120 20 200 220 45 9500 5600 1400 520,800 75 75 75 75 0 VDU (P4) HC H2S SS 0 0 0 20 60 20 160 480 160 290 250 1000* 20* − 1680 HDS-2 (P5) HC H2S SS 50 400 60 150 8000 120 800 60,800 480 1000* 20 40 20* 510 *The value of the original data is not available; therefore, the value is set in this work to suit the corresponding mathematical formulation Table 7 Performance of the treatment units (De-León 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).CP/h) HC H2S SS Steam-stripping column (R1) 0 0.999 0 3.13 1000* 110 1000 0 − 4462 Biological treatment (R2) 0.7 0.9 0.98 2.34 30 30 30 30 0 API separator (R3) 0.95 0 0.5 0.89 35 35 35 35 0 *The value of the original data is not available; therefore, the value is set in this work to suit the corresponding mathematical formulation Table 8 Case study 3 cost and operating parameters (De-León 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$/a C area Area cost coefficient for heat exchangers and utility units 1200$/(m2.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/(m2.°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

In this case, phase changes occur at process units P1 (10 bar, LVS = 2013 kJ/kg) and P4 (40 bar, LVS = 1713 kJ/kg). The latent heat is a local problem, which only occurs in the particular process units, and it is independent of the process temperature. Therefore, the latent heat could be calculated separately after the optimization campaign is completed. It is done similarly as in De-León Almaraz et al. (2016) work. The latent heat formula is shown in Eq. (36).
$$F\ast LVs=\mathrm{the}\ \mathrm{latent}\ \mathrm{heat}\ \mathrm{for}\ \mathrm{process}\ \left(\mathrm{kW}\right)$$
(36)

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 De-León Almaraz et al. (2016) will be compared with the results of the proposed design method with increasing number of mixing heat exchangers.

For comparison purpose, the results from De-León Almaraz et al. (2016) are shown as a reference (Fig. 4). The optimal WAN configuration was obtained from the first step of the MINLP model by De-León Almaraz et al. (2016) with the GEC of 744.9 (t per h). The second optimization step incorporating the HEN model to obtain the WAHEN configuration is shown in Fig. 5. There are eight heat exchangers, six hot utilities, and eight cold utilities required for the optimal configuration with mathematical programming method. Tables 9 and 10 show the corresponding overall heat exchanger data, GEC, TAC, and the annual costs.
Table 9

Cost results for HEN design with mathematical programming approach (De-León Almaraz et al. 2016)

Ex. number

Match

Q (kW)

Area (m2)

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

HU-C1

7376

37.2

$0.044$ 2.780

2

HU-C2

1575

12.9

$0.015$ 0.593

3

HU-C3

1462

6.2

$0.007$ 0.551

4

HU-C4

1202

5.0

$0.006$ 0.453

5

HU-C5

814

3.1

$0.003$ 0.306

6

HU-C6

289

0.9

$0.001$ 0.109

Total

12,719

65.3

$0.078$ 4.794

Cold utility

1

CU-H1

600

51.6

$0.061$ 0.113

2

CU-H2

465

44.9

$0.053$ 0.087

3

CU-H3

156

13.5

$0.016$ 0.029

4

CU-H4

15

0.9

$0.001$ 0.002

5

CU-H5

33

4.1

$0.005$ 0.006

6

CU-H6

428

29.3

$0.035$ 0.080

7

CU-H7

29

2.1

$0.002$ 0.005

8

CU-H8

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

*The water cost of $M 0.174 is ignored in this work, therefore the water cost is deducted from the reference’s TAC and the annual cost Table 10 General results with mathematical programming approach (De-León Almaraz et al. 2016) Results New equipment Total Area (m2) GEC (M$)

Utility cost (M$) TAC (M$)

Annual cost (M$) Exchangers 8 498.40 2.2347 5.17 6.226 8.4607 Heaters 6 65.25 Coolers 8 167.35 Total 22 731 WAHEN Configuration with no Mixing Heat Exchanger As a base case, the WAHEN is optimized using the proposed model without any mixing heat exchanger. The hot and cold utilities are allowed to provide or take heat from the processing units to satisfy the required inlet and outlet temperatures. The NLP model consists 175 continuous variables, 160 equations and the computational time required is 17 s. The optimal configuration is shown in Fig. 6. Note that there are two processing units with phase change, i.e., P1 and P4. Since the water flow rate of both units is the same to the results of De-León Almaraz et al. (2016), the required latent heats are necessarily the same, 28,000 kW for P1 and 3800 kW for P4. The freshwater demand is higher than that of the reference; hence, the GEC is also higher (M$ 3.0107) than that of the reference GEC (M$2.2347). In this model, no mixing heat exchanger is allowed; therefore, the hot and cold utility play unique role to satisfy the unit temperature limits. As a consequence, the utility cost is much higher in this case (M$ 7.3576) than that of the reference utility cost (M$5.175). The units’ temperature and concentration level are shown in Table 11, where the value of the parameters is all within the specified limits. The required number of utility heat exchangers and utility loads is shown in Table 12. Unit P1 is the primary consumer of the hot utility as the inlet requires a high temperature (180 °C) and high freshwater flow rate (50 t/h). The annual cost of this configuration is found to be M$ 10.6010, which is much higher than that of the reference cost (M$8.4607). Table 11 Unit parameter results of WAHEN configuration with no mixing heat exchanger Unit Tin (°C) Tout (°C) Cin (ppm) Cout (ppm) HC H2S SS HC H2S SS P1 180 110 0 0 0 15 400 35 P2 21.28 35 8.09 300 45 108.09 12,500 180 P3 75 75 48.76 19.7 89.68 149.94 45 9500 P4 250 40 0 0 0 20 60 20 P5 20 30 4.96 12.98 60 36.34 2397.3 78.82 R1 110 62 50.08 4799.63 91.82 50.08 4.8 91.82 R2 30 30 149.63 51.43 9474.24 44.89 5.143 189.49 R3 35 35 47.85 4.95 133.68 2.39 4.95 66.84 Table 12 Cost results of WAHEN configuration with no mixing heat exchanger Heat exchanger Q (kW) Aλ(m2) Exchanger cost (M$)

Utility (M$) Total hot utility QP1 9295.55 15.71 0.0189 3.5044 QP3 449.09 5.77 0.0069 0.1693 QP4 2137.98 8.05 0.0097 0.806 QR1 4615.22 35.29 0.0423 1.7399 Total 16,497.84 64.82 0.0778 6.2196 Total cold utility DP2 460.75 13.47 0.0162 0.0871 DP5 409 12.72 0.0153 0.0773 DR2 2295.85 23.2 0.0278 0.4339 DR3 2572.06 19 0.0228 0.4861 DW 283.36 8.31 0.01 0.0536 Total 6021.02 76.7 0.0981 1.138 Fixed cost for 9 heat exchangers = M$ 0.072

WAHEN Configuration with One Mixing Heat Exchanger

In this case study, one mixing heat exchanger is allowed in the superstructure model. Solving the NLP model which involves 216 continuous variables, 164 equations. The computational time is 3106 s. The optimal configuration is shown in Fig. 7. Tables 13 and 14 show the units’ temperature and concentration level and the required number of utility exchangers and utility loads, respectively. In the optimal configuration, there are 4776.86 kW of heat is exchanged within the mixing heat exchanger. With one mixing heat exchanger added, the superstructure model becomes more flexible. The optimal configuration flowrate maximizes the heat exchanger effectiveness while satisfying the temperature and concentration limits of the processing unit. The annual cost (M$8.5807) is now lower than that without mixing exchanger although it is still higher than that of the reference cost (M$ 8.4607). Note that by adding one mixing exchanger, the utility loads have been reduced quite significantly (M$6.0954) from that of the base case (M$ 7.304).
Table 13

Unit parameter results of WAHEN configuration with one mixing heat exchanger

Unit

Tin (°C)

Tout (°C)

Cin (ppm)

Cout (ppm)

HC

H2S

SS

HC

H2S

SS

P1

180.2

110.26

0

0

0

15

400

35

P2

21.28

35

20

300

44.737

120

12,500

179.738

P3

75

75

87.597

8.191

123.447

188.42

33.397

9500

P4

250.16

40.16

0

0

0

20

60

20

P5

23.79

40

16.092

400

33.498

46.225

2690.09

51.578

R1

110

36.18

87.597

8191.32

123.447

87.597

8.191

123.447

R2

30

30

25.525

48.534

4946.14

7.658

4.853

98.923

R3

35

35

188.42

33.397

9500

9.421

33.397

4750

Eh1

100.59

30

15.69

353.103

32.931

15.69

353.103

32.931

Ec1

20

90.59

0

0

0

0

0

0

Waste

303

11.169

5

100

Table 14

Cost results of WAHEN configuration with one mixing heat exchanger

Heat exchanger

Q (kW)

Aλ(m2)

Exchanger cost (M$) Utility (M$)

Total mixing heat exchanger

Eh1-Ec1

4776.86

61.39

0.0737

Total hot utility

QP1

5327.16

12.4

0.0149

2.009

QP3

2520.1

13.94

0.0167

0.95

QP4

1393.8

6.93

0.0083

0.5254

QR1

5128.95

30.88

0.037

1.9334

Total

14,370.01

64.15

0.077

5.4178

Total cold utility

DP2

362.81

12.45

0.0149

0.0686

DP5

286.66

9.52

0.0114

0.0542

DR2

558.53

11.62

0.0139

0.1056

DR3

2357.49

18.03

0.0216

0.4456

DW

19.2

1.79

0.0021

0.0036

Total

3584.69

53.41

0.0641

0.6776

Fixed cost for 10 heat exchangers = M$0.08 WAHEN Configuration with Two Mixing Heat Exchangers In this case study, two mixing heat exchangers are allowed in the superstructure model. The NLP model consists 278 continuous variables, 187 equations. The corresponding total computational time is 137,486 s. The optimal configuration is shown in Fig. 8. Tables 15 and 16 show the units’ temperature and concentration level and the required number of utility exchangers and utility loads, respectively. In the optimal configuration, there is now more heat exchanged in the mixing exchangers with total 9101.48 kW where shared equally on both mixing heat exchangers. Two mixing heat exchangers allow more possibility into the superstructure model and add more mixing opportunity of streams from the process units. Note that the annual cost (M$ 7.795) is now even lower than the previous case and below the reference cost (M$8.4607). Finally, although the stream number of units’ inlet is increased for configuration with one or two mixing heat exchangers, the total flow rate is close to the reference case. Table 15 Unit parameter results of WAHEN configuration with two mixing heat exchangers Unit Tin(°C) Tout(°C) Cin (ppm) Cout (ppm) HC H2S SS HC H2S SS P1 180 110 0 0 0 15 400 35 P2 20 33.72 20 300 45 120 12500 180 P3 75 75 83.65 19.78 118.75 184.52 45 9500 P4 250 40 0 0 0 20 60 20 P5 20 40 15.29 400 34.69 46.66 2784.31 53.52 R1 110 34.51 88.84 8372.39 126.27 88.84 8.37 126.27 R2 30 30 42.05 50 5000 12.31 5 100 R3 35 35 184.52 45 9500 9.23 45 4750 Eh1 107.81 30 184.52 45 9500 184.52 45 9500 Eh2 107.81 30 15.26 400.13 34.66 15.26 400.13 34.66 Ec1 20 88.83 0 0 0 0 0 0 Ec2 20 97.81 96.2 9347.14 138.96 96.2 9347.14 138.96 Waste 303 - 12.31 5 100 - - - Table 16 Cost results of WAHEN configuration with two mixing heat exchangers Heat exchanger Q (kW) Aλ(m2) Exchanger cost (M$)

Utility (M$) Total mixing heat exchanger Eh1-Ec1 4550.74 48.71 0.0584 Eh2-Ec2 4550.74 54.5 0.0654 Total 9101.48 103.21 0.1238 Total hot utility QP1 5296.8 12.31 0.0148 1.997 QP3 2484.77 13.84 0.0167 0.9368 QP4 1585.95 7.35 0.0089 0.5979 QR1 1427.03 20.59 0.0247 0.538 QEh1 1918.93 20.48 0.0246 0.7234 Total 12,713.48 74.57 0.0897 4.7931 Total cold utility DP2 449.63 14.3 0.0172 0.085 DP5 296.34 11.58 0.0139 0.056 DR2 302.54 8.6 0.0103 0.0572 DEc2 921.63 19.85 0.0238 0.1742 Total 1970.14 54.33 0.0652 0.3724 Fixed cost for 11 heat exchangers = M$ 0.088

WAHEN Configuration with More Mixing Heat Exchangers

Intuitively, adding more mixing heat exchangers will result in more economical annual costs. It is noteworthy that the model complexity will increase if there are more mixing heat exchangers added into consideration. Although the model can provide more solution space and better results, the impact is not as significant as the previous ones. In Table 17, the results of WAHEN with different mixing heat exchangers are summarized and compared with the results by De-León Almaraz et al. (2016). By increasing the number of mixing heat exchangers to 3 and 4, the utility costs can be saved by some with the tradeoff of increasing number of interconnections. Although the annual costs with a higher number of mixing heat exchangers are mostly lower than that of the reference, it is not favorable to add more mixing exchangers. With additional mixing exchangers, the operability issues, i.e., larger network, more interconnections, etc., become more prevalent. Furthermore, all the GECs in this work are slightly higher than that of the reference’s GEC. It is arguable that the proposed model provides more flexibility, in which more water is utilized for heat exchanging agent, and TAC and number of interconnections are decreased as the tradeoff. Overall, the results with less TAC that includes operating costs and with less number of interconnections will be favorable and advantageous for non-isothermal WAHEN operation.
Table 17

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 De-León 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%)

It is precisely shown that with two mixing heat exchangers, the annual cost can be reduced by 7.87% and with the advantage of fewer interconnections (25) and less utility heat exchangers (9) than that of the reference values. Figure 9 shows the annual cost comparison of different WAHEN configuration and the reference. Adding more mixing heat exchangers over two units do not significantly bring more economic benefits to the WAHEN configuration. Instead, it may burden the operation of the overall network.

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 non-isothermal systems, the proposed design strategy has been implemented in a case study with five process units, three regeneration units, and muti-contaminants. The effectiveness of the simultaneous optimization of the non-isothermal 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 106-2221-E-005-095-, 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 Chung-Hsing University 10617003G 新進教師經費補助.

References

1. Ahmetović E, Kravanja Z (2013) Simultaneous synthesis of process water and heat exchanger networks. Energy 57:236–250
2. Ahmetović E, Ibrića N, Kravanja Z, Grossmann IE (2015) Water and energy integration: a comprehensive literature review of non-isothermal water network synthesis. Comput Chem Eng 82:144–171
3. Bagajewicz M, Rodera H, Savelski M (2002) Energy efficient water utilization systems in process plants. Comput Chem Eng 26:59–79
4. Bogataj M, Bagajewicz MJ (2008) Synthesis of non-isothermal heat integrated water networks in chemical processes. Comput Chem Eng 32:310–3142
5. Boix M, Montastruc L, Pibouleau L, Azzaro-Pantel C, Domenech S (2012) Minimizing water and energy consumptions in water and heat exchange networks. Appl Therm Eng 36:442–455
6. Chen JJJ (1987) Comments on improvements on a replacement for the logarithmic mean. Chem Eng Sci 42:2488–2489
7. De-León Almaraz S, Boix M, Montastruc L, Azzaro-Pantel C, Liao Z, Domenech S (2016) Design of a water allocation and energy network for multi-contaminant problems using multi-objective optimization. Process Saf Environ Prot 103:348–364
8. Dong HG, Lin CY, Chang CT (2008) Simultaneous optimization approach for integrated water-allocation and heat-exchange networks. Chem Eng Sci 63:3664–3678
9. Feng X, Bai J, Wang H, Zheng X (2008) Grass-roots design of regeneration recycling water networks. Comput Chem Eng 32:1892–1907
10. Gunaratnam M, Alva-Argáez A, Kokossis A, Kim JK, Smith R (2005) Automated design of total water systems. Ind Eng Chem Res 44:588–599
11. Hou Y, Wang J, Chen Z, Li X, Zhang J (2014) Simultaneous integration of water and energy on conceptual methodology for both single- and multi-contaminant problems. Chem Eng Sci 117:436–444
12. Ibrić N, Ahmetović E, Kravanja Z (2013) A two-step solution strategy for the synthesis of pinched and threshold heat-integrated process water networks. Chem Eng Trans 35:43–48Google Scholar
13. Leewongtanawit B, Kim JK (2009) Improving energy recovery for water minimisation. Energy 34:880–893
14. Liao Z, Rong G, Wang J, Yang Y (2011) Systematic optimization of heat-integrated water allocation networks. Ind Eng Chem Res 50:6713–6727
15. Linnhoff B, Hindmarsh E (1983) The pinch design method for heat exchanger networks. Chem Eng Sci 38:745–763
16. Liu Y, Luo Y, Yuan X (2015) Simultaneous integration of water and energy in heat-integrated water allocation networks. AICHE J 61:2202–2214
17. Luo Y, Luo X, Yuan X (2012) Thermodynamic analysis of homogeneous non-isothermal mixing influence on the water-using networks’ energy target. Comput Aided Chem Eng 31:420–424
18. Martínez-Patiño J, Picón-Núñez M, Serra LM, Verda V (2012) Systematic approach for the synthesis of water and energy networks. Appl Therm Eng 48:458–464
19. Polley GT, Picón-Núñez M, López-Maciel JDJ (2010) Design of water and heat recovery networks for the simultaneous minimisation of water and energy consumption. Appl Threm Eng 30:2290–2299
20. Savulescu L, Kim JK, Smith R (2005) Studies on simultaneous energy and water minimisation—part II: systems with maximum re-use of water. Chem Eng Sci 60:3291–3308
21. Wang YP, Smith R (1994) Wastewater minimisation. Chem Eng Sci 49:981–1006
22. Yan F, Wu H, Li W, Zhang J (2016) Simultaneous optimization of heat-integrated water networks by a nonlinear program. Chem Eng Sci 140:76–89
23. Yee TF, Grossmann IE (1990) Simultaneous optimization models for heat integration—II. Heat exchanger network synthesis. Comput Chem Eng 14:1165–1184
24. Zhou L, Liao Z, Wang J, Jiang B, Yang Y, Yu H (2015) Simultaneous optimization of heat-integrated water allocation networks using the mathematical model with equilibrium constraints strategy. Ind Eng Chem Res 54:3355–3366