DOI: 10.3303/CET23103135 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 07 May 2023; Revised: 06 June 2023; Accepted: 11 July 2023 
Please cite this article as: Ibrić N., Adams II T.A., Gundersen T., 2023, Synthesis of Heat-Integrated Water Networks with Exergo-economic 
Criteria, Chemical Engineering Transactions, 103, 805-810  DOI:10.3303/CET23103135 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 103, 2023 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Panos Seferlis, Yee Van Fan, Athanasios Papadopoulos 

Copyright © 2023, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-02-1; ISSN 2283-9216 

Synthesis of Heat-Integrated Water Networks with Exergo-

economic Criteria 

Nidret Ibrić*, Thomas A Adams II, Truls Gundersen 

Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim, Norway  

nidret.ibric@ntnu.no 

This paper considers the exergy analysis of Heat-Integrated Water Networks (HIWNs) rarely addressed in the 

literature. Two objective functions were analyzed, with the first minimizing the total annualized cost (TAC) and 

the second minimizing the exergy losses of the network. The exergetic objective function minimizes exergy 

losses due to non-isothermal mixing, heat exchange, and exergy losses due to friction. The HIWN was optimized 

under a constant pressure assumption. However, exergy losses due to pressure drops are considered to 

balance network complexity. The proposed nonlinear programming (NLP) model and iterative solution strategy 

showed that by using exergetic criteria as an objective function, good solutions could be obtained compared to 

the solution of the mixed integer nonlinear programming (MINLP) model minimizing the TAC. The advantage is 

that exergy-based model is an NLP, and cost data for the utilities and equipment are not required.  

1. Introduction 

Global water and energy consumption continuously increase due to increasing industrial, domestic and 

agricultural demands. This increasing demand adds to emissions of greenhouse gases and pollution of existing 

freshwater resources. Accordingly, there is a need to conserve water and energy, which is a main priority for 

many industries. Water is a natural resource essential to process industries such as chemical, pharmaceutical, 

food, and others. In energy production industries, it is critical for hydropower production and as cooling water in 

thermal or nuclear power plants. Process water is used for various purposes, such as cleaning, washing, 

scrubbing, and absorption, as well as water in the final products from food and pharmaceutical industries. Water 

and energy are closely related, and reducing freshwater consumption reduces energy for heating and cooling. 

This can be achieved systematically using Pinch Analysis (PA) (Savulescu et al., 2005) and Mathematical 

Programming (MP) (Bagajewicz et al., 2002). In PA methods, water and energy targeting is performed 

separately from water and Heat Exchanger Network (HEN) design using tools such as two-dimensional grid 

diagrams (Savulescu et al., 2005), temperature versus concentration diagrams (Martínez-Patiño et al., 2011), 

or water energy balance diagrams (Leewongtanawit and Kim, 2009). Thus, obtaining a design with minimum 

cost is difficult because a good trade-off between freshwater, utility consumption, and HEN investment is not 

achieved with these approaches. Targeting models for simultaneous water and energy minimization (Yang and 

Grossmann, 2013) can also help find theoretical limits to resource consumption but do not estimate the total 

cost of the network. This issue was addressed in (Nemet et al., 2022) with a targeting model to find minimum 

freshwater and utility consumption that can find promising matches for heat exchange and provide a 

corresponding cost estimate for the heat exchangers. This is the state of the art for targeting based on PA 

methods, but its usefulness is moreso in generating theoretical water and energy targets and bounds on 

performance rather than practical and real solutions. MP methods aim at more practical, real solutions. For 

example, Ahmetović and Kravanja (2012) used a sequential approach to first synthesize a Water Network (WN) 

under the assumption of constant temperatures using the global optimization solver BARON. In the second step, 

the HEN is synthesized using a stage-wise superstructure (Yee and Grossmann, 1990). Kermani et al. (2019) 

solved three sub-problems sequentially and iteratively to generate a set of potential solutions implementing 

integer cut constraints. The first problem minimizes operating costs by selecting potential thermal matches, and 

the second problem minimizes the number of thermal matches given the Heat Recovery Approach Temperature 

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(HRAT). In the final step, the optimum HEN design with minimum investment cost is determined by solving a 

nonlinear programming (NLP) model. In an alternative, simultaneous approach, freshwater and utility 

consumption are minimized simultaneously with the investment cost for heat exchangers optimizing the 

exchanger's minimum approach temperature. This can be achieved using multistep approaches consisting of 

targeting and design steps (Ibrić et al., 2014) or a single-step simultaneous approach (Ibrić et al., 2022) 

simultaneously combining WN with the HEN. The downside is that the complexity of MP problems grows rapidly 

with the scale of the HIWN. It can quickly become intractable to find global solutions or even local ones. Our 

goal is to develop methods with the scalability of PA and yet attain the rigorous, practical solutions of MP 

approaches. 

To do this, we note that most objective functions in MP approaches are the minimum total annualized cost 

(TAC). These require cost estimates for resources and equipment that are sometimes unreliable and are almost 

always nonlinear and sometimes even discrete. It is a major contributor to the computational difficulty of finding 

the optimal solution. For this reason, a few papers studied the use of exergy as a useful thermodynamic metric 

that aids in design. For example, Dong et al. (2014) presented a simultaneous optimization approach of an 

integrated heat, mass, and pressure exchange networks and applied it to a modified HIWN problem considering 

pressure exchange networks. Exergo-economic optimization was performed to minimize economic cost per unit 

exergy of the final water flows. Meramo and González-Delgado (2023) analyzed heat-integrated water 

regeneration networks with different objectives: minimizing TAC, minimizing exergy losses, and a two-

dimensional approach in searching for a solution with a good trade-off between the first two scenarios. Both 

recognized the value of exergy but still required the use of TAC and retained the same scalability challenges 

associated with the use of nonlinear TAC. We propose using exergetic objective functions without the use of 

TAC in an MP approach for synthesizing a HIWN. The objective function minimizes exergy losses by introducing 

exergy losses due to friction to balance network complexity with thermo-mechanical exergy losses of the overall 

system. The proposed objective function does not rely on cost data for resources and equipment and provides 

an alternative approach to solving HIWN problems which solves much more quickly and leads to very similar 

rigorous solutions to the more computationally intensive TAC-based MP approaches. 

2. Problem statement 

Figure 1 shows a representation of the synthesis problem of the HIWN. Water is supplied to process water-

using units (p ∈ P) with the freshwater sources s ∈ S, and wastewater is available for reuse. Process units 

operate at a specified temperature and given maximum inlet contaminants k ∈ K concentration. Water can be 

directly used within water-using units or heated/cooled within a heat exchanger network (HEN). The water 

network (WN) and heat exchanger network (HEN) are interconnected, enabling heat integration of water 

streams by direct heat exchange. Within the HEN, indirect heat exchange opportunities are enabled within heat 

exchangers (e ∈ E), heaters (h ∈ H), and coolers (c ∈ C). The hot utility is low-pressure steam, and the cold utility 

is cooling water. Wastewater is discharged into the environment at a specified temperature. The objective is to 

find the optimum HIWN design by minimizing economic (TAC) or exergetic (exergy losses) criteria.  

 

Figure 1: Problem representation of the HIWNs 

3. Superstructure and modeling  

In this work, a recently published paper (Ibrić et al., 2022) is extended regarding superstructure design and 

mathematical modeling to include an exergy analysis of HIWN. The superstructure proposed by Ibrić et al. 

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(2022) consisted of a single set of heat exchange systems (e ∈ E) consisting of heat exchanger, a heater and a 

cooler. The coolers and heaters are present in the superstructure, even if not required for specific problems. 

The superstructure has been modified to define separate sets for heat exchangers (e ∈ E), heaters (h ∈ H), and 

coolers (c ∈ C). This enables more control over the superstructure design and facilitating the solution to the 

problem.  

3.1 Objective function 

In this paper, two objective functions were analyzed. The objective economic function given by Eq(1) minimizes 

TAC, including operating costs for the freshwater, utilities, pumping for overcoming pressure drop per unit length 

of pipes, and annualized investment costs for the heat exchangers. 

min ∑FSsCsH

s

+∑ q
h
CHU

h

+∑ q
c
CCU

c

+ CelH∑Npump + af∑(zexca + bAexc
n )

exc

 (1) 

The exergetic objective function minimizes thermo-mechanical exergy losses related to non-isothermal mixing 

and heat exchange and exergy losses due to friction, as given by Eq(2).  

min∑FSses
s

+∑mh

(HU)
∆eHU

h

+∑mc
(CU)

∆eCU
c

- FWWeWW + ∑Eloss,p (2) 

The classic TAC economic objective function considers both the continual costs of operation (the terms without 

the underbar) balanced with the one-time costs of capital (the terms with the underbar), with their relative 

influence weighted by the annualization factor af. The capital costs term factors in both nonlinear costs 

associated with heat exchanger size, but also costs of “existence” such that there is a cost penalty associated 

with the discrete number of exchangers (and the number of network connections) at any size.  

The exergetic objective function is structured analogously; the continual thermo-mechanical exergy losses from 

operation (the terms without the underbar) are compared against an term representing the exergetic losses 

associated with network complexity (the terms with the underbar). The thermo-mechanical exergy losses are 

calculated as exergy inlet to the system with fresh water and utilities minus the exergy outlet with the wastewater 

stream, assuming constant system pressure. These thermo-mechanical  exergy losses inherently include losses 

for non-isothermal mixing and exergy losses in heat exchangers, heaters, and coolers. Deng et al. (2023) 

already showed that the exergy of utilities could be used as a proxy for the utility value and the exergy losses 

can be used as a direct analogous substitute for utility cost in the exergy objective function. However, if only the 

thermo-mechanical exergy losses due to non-isothermal mixing and heat exchange are minimized, solutions 

with high capital investment and many network connections would be obtained, which is undesirable. Therefore, 

in this work, the exergy loss due to friction is introduced (underbar), to balance the network complexity. Friction 

losses make a good predictor of the number of network connections (more connections require more pipes). 

The term assumes a constant pressure loss per unit pipe length for simplicity. This paper estimates the thermo-

mechanical exergy of water streams by fitting data from the literature (Deng et al., 2023). 

3.2 Solution strategy 

The mathematical programming model with the objective function minimizing the TAC given by Eq (1) (OBJ-1) 

is an MINLP model solved by the SBB solver using Conopt4 as an NLP sub-solver. The second mathematical 

programming model, with the objective function given by Eq (2) (OBJ-2), is an NLP model solved with the 

Conopt4 as an NLP solver. Models are solved by using the General Algebraic Modeling System (GAMS, 2021), 

with a single-step iterative solution strategy proposed by Ibrić et al. (2022). This strategy provides random 

initialization to the optimization variables between lower and upper bounds generated based on initial data.  

4. Example 

The proposed model is demonstrated with a typical literature example (Bogataj and Bagajewicz, 2008) with data 

for the process water-using units given in Table 1. Freshwater is available at 20 °C and 0 ppm contaminants, 

and wastewater is discharged into the environment at 30 °C. Other network design data and cost parameters 

are taken from the literature (Ibrić et al., 2022). 

Optimization problem with economic objective function (OBJ-1) was solved for the investment annualization 

factor (af) equal to 1. Exergy losses for the optimal network design were calculated after the optimization. In 

addition, the same problem was solved with the exergetic objective function (OBJ-2). The TAC of the optimal 

design chosen was calculated after the exergetic optimization. Ultimately, the solutions obtained by the exergy 

objective function are almost, but not exactly, identical to the solutions obtained by the economic one. The 

optimum network design for the objective function minimizing TAC is given in Figure 2. The network exhibits 

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minimum freshwater consumption of 25 kg/s and a hot utility consumption of 1,050 kW, with a network design 

including one heat exchanger and one heater. The heat exchanger recovers 5,530 kW of heat from the 

wastewater stream to preheat freshwater from 20 °C to 82 °C. The external hot utility is required to deliver 

freshwater at a maximum operating temperature of 100 °C. Exergy losses of non-isothermal mixing and heat 

exchange are 87.97 and 160.65 kW. The exergy losses due to friction are 58.56 kW per unit length of pipes. 

Total exergy losses for the optimal design are 307.19 kW, and the TAC of the network is 846,927 $/y. 

Table 1: Data for the process units 

Process unit Contaminant mass 

load (g/h) 

Maximum inlet contaminant 

concentration (ppm) 

Maximum outlet contaminant 

concentration (ppm) 

Temperature 

(°C) 

P1 2,000 0 100 40 

P2 5,000 50 100 100 

P3 30,000 50 800 75 

P4 4,000 400 800 50 

 

Figure 2: Optimal network design with economic criteria  

Table 2: Comparison of the results with the literature 

Reference Freshwater 

consumption 

(kg/s) 

Hot utility 

consumption 

(kW) 

HEN 

investment 

($/y) 

TAC ($/y) TAC including 

pumping ($/y) 

Bogataj and Bagajewicz (2008) 25 1,050 146,748 812,598 - 

Ibrić et al. (2014) 25 1,050 134,227 800,077 - 

Yan et al. (2016) 25 1,050 131,525 797,375 - 

Ibrić et al. (2022) 25 1,050 131,527 797,377 - 

This paper OBJ-1 25 1,050 134,227 800,077 846,927 

This paper OBJ-2 25 1,050 134,795 800,645 847,008 

 

Figure 3 shows the optimum network design with the exergetic objective function. The solution is selected from 

the set shown in Figure 4 obtained using an iterative solution strategy. Figure 4 shows the TAC for each solution 

calculated after the exergetic optimization that corresponds to the minimum exergy losses. The solution with 

minimum exergy losses can be easily identified. However, alternative local solutions with different network 

designs can also be analyzed. The optimum network exhibits the same freshwater and utility consumption as 

the network with minimum TAC. The network design includes the same number of heat exchangers and heaters 

and a similar network design. The network exhibits the same number of non-isothermal mixing points (five) and 

one less splitting point (four) compared to the network shown in Figure 2. Exergy losses of non-isothermal mixing 

808



and heat exchange are 88.12 and 160.50 kW. In addition, the exergy losses due to friction are slightly reduced 

from 58.56 to 57.95 kW. The TAC of the optimal design with minimum exergy losses is 847,008 $/y, which is 

comparable with the objective of minimizing the TAC of the network. Table 2 shows a comparison of the results 

with those from the literature. TAC, including pumping cost were not compared as those are not optimized in 

other studies. However, the design with minimum TAC is the same as in previous studies (Ibrić et al., 2014), 

and given electricity cost 0.1 $/(kWh), the network would give the same TAC by including pumping. 

 

Figure 3: Optimal network design with exergetic criteria 

 

Figure 4: Set of solutions obtained for the objective function minimizing exergy losses 

5. Conclusions 

This paper presented an alternative approach for synthesizing Heat Integrated Water Networks (HIWNs) with 

an exergetic objective function. The objective function of the nonlinear programming (NLP) model minimizes 

thermo-mechanical exergy losses of the system and exergy related to frictional pressure drop to balance 

network complexity. The NLP model requires less computational effort to be solved compared to the mixed 

integer nonlinear programming (MINLP) model with economic objective function and finds designs essentially 

equal in quality to those of the cost-based approaches. The solution obtained with exergetic objective function 

is within 0.04 % of the best solution in the literature when minimizing total annualized cost. In addition, the 

exergetic objective function is not reliant on the cost of utilities and equipment. The model can be modified to 

include wastewater regeneration units for distributed wastewater treatment. In addition, the non-constant 

pressure of water streams can be considered by including pressure change networks.  

Nomenclature

a – fixed cost coefficient, $ 

Aexc – heat exchanger area, m2 

af – annualization factor for the investment, - 

b – area cost coefficient, $/m2  

CCU – cost of cold utility, $/(kW y) 

Cel – cost of electricity, $/(kW h) 

CHU – cost of hot utility, $/(kW y) 

Cs – cost of freshwater, $/kg    

∆eCU – exergy change of cold utility, kJ/kg 

∆eHU – exergy change of hot utility, kJ/kg 

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es – exergy of freshwater, kJ/kg 

eww – exergy of wastewater, kg/kg 

Eloss,mix – exergy loss of mixing, kW 

Eloss,exc – exergy loss of heat exchangers, kW 

Eloss,p – exergy loss due to friction, kW 

FSs – freshwater consumption, kg/s 

FWW – wastewater flowrate, kg/s 

H – plant operating hours, h/y 

mc
(CU)

 – mass flowrate of cold utility, kg/s 

mh

(HU)
– mass flowrate of hot utility, kg/s  

Npump – energy, kW 

n – heat exchanger cost exponent, - 

q
c
 – cold utility consumption, kW 

q
h
 – hot utility consumption, kW 

zexc – binary variable representing heat 

exchangers  

 

Acknowledgments 

This publication is funded by the HighEFF - Centre for an Energy Efficient and Competitive Industry for the 

Future. The authors gratefully acknowledge the financial support from the Research Council of Norway and user 

partners of HighEFF, an 8-year Research Centre under the FME scheme (Centre for Environment-friendly 

Energy Research, 257632).  

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