A CONCEPTUAL ANALYSIS FOR RESOURCE 

OPTIMIZATION IN VARIABLE TIME 

MULTIPURPOSE BATCH PLANTS 

by 

Zhiwei Li (1569114) 

 

A thesis submitted in partial fulfillment of the requirements for the degree of  

Doctor of Philosophy (Chemical Engineering)  

 

 

Submitted to 

School of Chemical and Metallurgical Engineering, Faculty of Engineering and the Built 

Environment, University of the Witwatersrand, Johannesburg, South Africa 

 

Supervisor: Prof. Thokozani Majozi 

February 2020 

 

 



Declaration 

I 

 

Declaration 

 

I, Zhiwei Li, with student number 1569114, declare in terms of Rule G27 that: 

 

1. I understand what plagiarism is and I am aware of the University policy in this regard. 

 

2. This thesis is my own original work. Where other people’s work has been used (either 

from a printed source, Internet or any other source), this has been properly 

acknowledged and referenced in accordance with departmental requirements. 

 

3. This thesis and all of its contents has not been used as a submission for any other degree 

or submitted at any other university. 

 

4. I have not used work previously produced by another student or any other person to 

hand in as my own. 

 

5. I have not allowed, and will not allow, anyone to copy my work with the intention of 

passing it off as his or her own work. 

 

 

____________________       

Signature 

         

 

06/02/2020 

______________________ 

Date 



Acknowledgements 

II 

 

Acknowledgements 

The journey to completing this thesis has a limit, but knowledge has none. I would like to thank all 

of my supports for navigating my way around and making the thesis possible. 

First, I would like to express my most profound appreciation to my supervisor, Prof. Thokozani 

Majozi, for the continuous support throughout the course of this study. Every time I got stuck in 

my research, his discussions gave me new ideas and inspired me to find the right directions, as 

described in the poem from the Southern Song Dynasty: “After endless mountains and rivers that 

leave doubt whether there is a path out, suddenly one encounters the shade of a willow, bright 

flowers and a lovely village.” I would like to thank him for his patience and time that he devoted 

in helping me to achieve the objectives of this thesis.  

Besides my supervisor, I would like to thank my colleagues at Sustainable Process Engineering 

group at University of the Witwatersrand. They acted as my sisters and brothers and always gave 

me their hands whenever I faced with difficulties.  

I would also like to thank my lovely wife, Dr Menglu Song. During my study period, she always 

gave me encouragement and supports when I was frustrated. I am also indebted to my son, Yunan 

Li. He brings so many joys to me in the revised process of my thesis. Special thanks also go to my 

parents and my sister for supporting me spiritually throughout my life.   

Last but not the least, I would also like to thank the National Research Foundation (NRF) of South 

Africa for funding this research.  



Publications 

III 

 

Publications 

Below is a list of journal and conference papers presented in this dissertation. 

Journal papers 

Li, Z. and Majozi, T., 2019. Optimal Design of Batch Water Network with a Flexible 

Scheduling Framework. Industrial & Engineering Chemistry Research, 58(22): 9500-9511.  

Li, Z. and Majozi, T., 2018. Optimal Synthesis of Batch Water Networks Using Dynamic 

Programming. Process Integration and Optimization for Sustainability, 2(4), 391-412. 

Conference papers 

Li, Z. and Majozi, T., 2018. Optimal Synthesis of Batch Water Networks with a Flexible 

Scheduling Framework Using Dynamic Programming. Computer Aided Chemical 

Engineering, 44, 205-210 

Li, Z. and Majozi, T., 2017. Dynamic Programming for Optimal Synthesis of Water Networks 

in Batch Processes. Computer Aided Chemical Engineering, 40, 919-924. 

Li, Z. and Majozi, T., 2017. Optimal Design of Batch Water Network with a Flexible 

Scheduling Framework. Chemical Engineering Transactions, 61, 139-144 

 

 



Synopsis 

IV 

 

Synopsis 

Batch processing has been embraced by numerous chemical industries, such as the food, 

beverage, specialty chemical, and pharmaceutical industries because of its flexibility of 

accommodating changes in demand and responding effectively to uncertainties in market and 

supply chains. On the other hand, wastewater generated by batch processes depends on the 

specific tasks. Thus, the methods of wastewater minimization developed for continuous 

processes do not apply to batch processes. Few studies based on insight-based techniques have 

been reported to address wastewater recovery issues for flexible batch processes. The flexible 

batch process refers to that in which the durations of batch operations are treated as variables, 

not parameters. Therefore, a conceptual analysis was performed in this work to address the 

wastewater minimization in batch plants with flexible scheduling. 

A dynamic programming approach was first proposed to design a batch water network based 

on predefined scheduling. Insight from these results could assist the designer in rescheduling 

operations to maximize water recovery through direct water reuse. However, the extent of 

performance improvement is limited because the rescheduling process only involves a few 

operations, not all of the water operations. Thus, the dynamic programming method was 

extended to flexible batch processes in which only durations of batch operations are given. The 

difference is that in the former method, the stage is determined by the timings of water 

operations, while for the latter approach, the stages are identified based on the inlet 

concentration of water operations. 

Although dynamic programing approach, as an algebraic method, could provide a graphical 

representation of the targeting process, it is difficult to find the optimal solution when faced 

with complex problems. A match ranking matrix approach was proposed to prioritize matches 

between water sources and sinks. Batch water network could be synthesized by choosing 

optimal matches between water sources and sinks. Besides, a hybrid method of combing water 

pinch analysis and match ranking matrix method was presented to design batch water network, 

which takes advantages of graphical representation of pinch analysis and insight for the design 

of batch water network from match ranking matrix method.  

Finally, the main conclusions and future work are summarized. These conceptual methods in 

this work could be used to address the wastewater minimization in flexible batch plants. 

 



Content 

V 

 

Content 

 

Declaration................................................................................................................................ I 

Acknowledgements ................................................................................................................. II 

Publications ........................................................................................................................... III 

Synopsis .................................................................................................................................. IV 

Content ..................................................................................................................................... V 

List of Figures ........................................................................................................................ IX 

List of Tables ........................................................................................................................ XII 

List of Acronyms ................................................................................................................ XIII 

INTRODUCTION.................................................................................................................... 1 

1.1 Background ...................................................................................................................... 1 

1.2 Motivation ........................................................................................................................ 3 

1.3 Problem statement ............................................................................................................ 3 

1.4 Research objectives .......................................................................................................... 4 

1.5 Scope of work .................................................................................................................. 4 

1.6 Thesis structure ................................................................................................................ 5 

References .............................................................................................................................. 6 

LITERATURE REVIEW ....................................................................................................... 7 

2.1 Operational philosophy .................................................................................................... 7 

2.2 Production scheduling ...................................................................................................... 8 

2.2.1 Schedule-graph (S-graph) ......................................................................................... 8 

2.2.2 State task network (STN) .......................................................................................... 9 

2.2.3 State sequence network (SSN) ................................................................................ 10 

2.2.4 Resource task network (RTN) ................................................................................ 10 



Content 

VI 

 

2.3 Synthesis of Water Networks ........................................................................................ 11 

2.3.1 Water reuse/recycle ................................................................................................. 11 

2.3.2 Regeneration reuse/recycle ..................................................................................... 12 

2.3.3 Total water network ................................................................................................ 13 

2.4 Water integration techniques for continuous processes ................................................. 14 

2.4.1 Insight-based techniques ......................................................................................... 15 

2.4.2 Algebraic methods .................................................................................................. 20 

2.4.3 Mathematical optimization-based approaches ........................................................ 22 

2.5 Water integration techniques for batch processes .......................................................... 25 

2.5.1 Insight-based techniques for batch processes ......................................................... 25 

2.5.2 Mathematical programming method for batch processes ....................................... 27 

2.6 Water Network Synthesis Challenges ............................................................................ 34 

2.6.1 Nonconvexity .......................................................................................................... 34 

2.6.2 Nonlinearity ............................................................................................................ 35 

2.6.3 Uncertainty .............................................................................................................. 36 

2.6.4 Future research direction for water integration ....................................................... 37 

2.7 Conclusions .................................................................................................................... 38 

References ............................................................................................................................ 39 

OPTIMAL SYNTHESIS OF BATCH WATER NETWORKS BASED ON DYNAMIC 

PROGRAMMING ................................................................................................................. 54 

3.1 Introduction .................................................................................................................... 54 

3.2 Problem statement .......................................................................................................... 58 

3.3 Methodology .................................................................................................................. 58 

3.4 Illustrative examples ...................................................................................................... 64 

3.4.1 Example 1 ............................................................................................................... 65 

3.4.2 Example 2 ............................................................................................................... 75 



Content 

VII 

 

3.4.3 Example 3 ............................................................................................................... 80 

3.4.4 Example 4 ............................................................................................................... 85 

3.5 Conclusions .................................................................................................................... 90 

Nomenclature ....................................................................................................................... 90 

References ............................................................................................................................ 91 

SYNTHESIS OF FLEXIBLE BATCH WATER NETWORKS USING DYNAMIC 

PROGRAMMING ................................................................................................................. 94 

4.1 Introduction .................................................................................................................... 94 

4.2 Problem statement .......................................................................................................... 97 

4.3 Methodology .................................................................................................................. 98 

4.4 Illustrative examples .................................................................................................... 105 

4.4.1 Example 1 ............................................................................................................. 105 

4.4.2 Example 2 ............................................................................................................. 113 

4.4.3 Example 3 ............................................................................................................. 120 

4.5 Conclusions .................................................................................................................. 123 

References .......................................................................................................................... 124 

DESIGN OF FLEXIBLE BATCH WATER NETWORK USING MATCH RANKING 

MATRIX METHOD ............................................................................................................ 127 

5.1. Introduction ................................................................................................................. 127 

5.2. Problem statement ....................................................................................................... 132 

5.3. Methodology ............................................................................................................... 133 

5.4. Illustrative examples ................................................................................................... 140 

5.4.1 Example 1 ............................................................................................................. 141 

5.4.2 Example 2 ............................................................................................................. 144 

5.4.3 Example 3 ............................................................................................................. 147 

5.5. Conclusions ................................................................................................................. 148 

Nomenclature ..................................................................................................................... 149 



Content 

VIII 

 

References .......................................................................................................................... 150 

SYNTHESIS OF FLEXIBLE BATCH WATER NETWORKS BASED ON PINCH 

ANALYSIS ........................................................................................................................... 153 

6.1 Introduction .................................................................................................................. 153 

6.2 Problem statement ........................................................................................................ 157 

6.3 Methodology ................................................................................................................ 157 

6.4 Case studies .................................................................................................................. 160 

6.4.1 Example 1 ............................................................................................................. 160 

6.4.2 Example 2 ............................................................................................................. 163 

6.4.3 Example 3 ............................................................................................................. 166 

6.4.4 Example 4 ............................................................................................................. 169 

6.5 Conclusions .................................................................................................................. 172 

References .......................................................................................................................... 172 

CONCLUSIONS AND RECOMMENDATIONS ............................................................. 176 

7.1 Conclusions .................................................................................................................. 176 

7.2 Recommendations ........................................................................................................ 178 

Appendix A ........................................................................................................................... 179 

Appendix B ........................................................................................................................... 185 

Appendix C ........................................................................................................................... 188 

 



List of Figures 

IX 

 

List of Figures 

Figure 1.1 The distribution of water scarcity area (Mekonnen and Hoekstra, 2016) ................ 1 

Figure 2.1 Batch water network with reuse scheme....………………………………………12 

Figure 2.2 Batch water network with direct water recycling scheme ...................................... 12 

Figure 2.3 Water network with regeneration reuse .................................................................. 13 

Figure 2.4 Water network with regeneration recycling ........................................................... 13 

Figure 2.5 Schematic of the total water network ..................................................................... 14 

Figure 3.1 The relationship between time and stages .............................................................. 59 

Figure 3.2 Potential states in each stage .................................................................................. 61 

Figure 3.3 The relationship between stages and time of Example 1 ....................................... 66 

Figure 3.4 The optimal state of Example 1 (first sequence) .................................................... 71 

Figure 3.5 The resultant network for Example 1 (first sequence) ........................................... 71 

Figure 3.6 The resultant network for Example 1 after rescheduling (first sequence) .............. 72 

Figure 3.7 The optimal state of Example 1 (cyclic operation) ................................................ 74 

Figure 3.8 Water network for Example 1 (cyclic operation) ................................................... 74 

Figure 3.9 Final network for Example 1 after rescheduling (cyclic operation) ....................... 75 

Figure 3.10 The relationship between time and stage for Example 2 ...................................... 76 

Figure 3.11 The optimal state of Example 2 (first sequence) .................................................. 76 

Figure 3.12 Resultant water network of Example 2 (first sequence) ....................................... 77 

Figure 3.13 Water network of Example 2 after rescheduling (first sequence) ........................ 78 

Figure 3.14 The optimal state of Example 2 (cyclic operation) .............................................. 78 

Figure 3.15 Resultant water network for Example 2 (cyclic state) .......................................... 79 

Figure 3.16 Water network for Example 2 after rescheduling operations (cyclic state) ......... 80 

Figure 3.17 The relationship between time and stage for Example 3 ...................................... 81 

Figure 3.18 The optimal states for Example 3 (reuse scenario) .............................................. 82 

Figure 3.19 Optimal network for Example 3 (reuse scenario)................................................. 83 



List of Figures 

X 

 

Figure 3.20 Water network for Example 3 after rescheduling operations (reuse scenario) .... 83 

Figure 3.21 The optimal states for Example 3 (regeneration scenario) ................................... 84 

Figure 3.22 Water network for Example 3 with regeneration unit .......................................... 85 

Figure 3.23 The optimal states for Example 4 with two tanks ................................................ 87 

Figure 3.24 The final batch water network of Example 4 with two tanks ............................... 87 

Figure 3.25 The optimal states for Example 4 with one tank .................................................. 89 

Figure 3.26 The final batch water network of Example 4 with one tank ................................. 89 

Figure 4.1 The relationship between the concentration of water sinks and stages .................. 99 

Figure 4.2 Potential state in each stage .................................................................................. 100 

Figure 4.3 The design procedure for batch water network .................................................... 105 

Figure 4.4 The identification of stages for Example 1 ........................................................... 106 

Figure 4.5 The schematic diagram of states in each stage for Example 1 ............................. 110 

Figure 4.6 Water network of Example 1 with reuse scenario ................................................ 111 

Figure 4.7 Water network of Example 1 with a regeneration unit......................................... 113 

Figure 4.8 The identification of stages for Example 2 ........................................................... 115 

Figure 4.9 The schematic diagram of states in Example 2 .................................................... 118 

Figure 4. 10 Water network for Example 2 (reuse scenario) ................................................. 119 

Figure 4.11 Water network for Example 2 (regeneration scenario) ...................................... 120 

Figure 4.12 The assignment of units for each stage ............................................................... 121 

Figure 4.13 The states of each stage for a single batch operation ......................................... 122 

Figure 4.14 Results of Example 3 with reuse scenario .......................................................... 123 

Figure 4.15 Results of Example 3 with regeneration scenario .............................................. 123 

Figure 5.1 Schematic representation of the problem ............................................................. 132 

Figure 5.2 Schematic representation of the approach ............................................................ 133 

Figure 5.3 Initial water network of Example 1 with reuse scenario ...................................... 140 

Figure 5.4 Resultant network of Example 1 (reuse/recycle scenario) ................................... 141 



List of Figures 

XI 

 

Figure 5.5 Evolved water network of Example 1 (reuse/recycle scenario) ........................... 142 

Figure 5.6 Water network for regeneration scenario of Example 1 ....................................... 143 

Figure 5.7 Evolved water network of Example 1 (regeneration scenario) ............................ 144 

Figure 5.8 Resultant water network for the reuse scenario of Example 2 ............................. 146 

Figure 5.9 Resultant water network of Example 2 (regeneration) ......................................... 147 

Figure 5.10 Resultant water network for Example 3 (First sequence) .................................. 148 

Figure 6.1 The design procedure of the proposed method ....................................................... 158 

Figure 6.2 The matrix for water sources and sinks ................................................................ 159 

Figure 6.3 The graphical representation of targeting for Example 1 ..................................... 161 

Figure 6.4 The matching matrix of Example 1 ...................................................................... 162 

Figure 6.5 Water network of Example 1 ................................................................................. 162 

Figure 6.6 Material recovery pinch diagram of Example 2 ..................................................... 164 

Figure 6.7 The matching matrix of Example 2 ...................................................................... 164 

Figure 6.8 The final water network for Example 2 (first sequence) ......................................... 165 

Figure 6.9 The final water network of the cyclic batch for Example 2 .................................... 166 

Figure 6.10 Graphical representation of the targeting process of Example 3 ........................ 167 

Figure 6.11 The matching matrix of Example 3 .................................................................... 167 

Figure 6.12 The water network of Example 3 ....................................................................... 168 

Figure 6.13 Graphical representation of targeting for Example 4 ......................................... 169 

Figure 6.14 The matching matrix of Example 4 .................................................................... 170 

Figure 6.15 The water network for Example 4 (single batch) ............................................... 171 

Figure 6.16 Gantt chart for Example 4 .................................................................................. 171 

 



List of Tables 

XII 

 

List of Tables 

Table 3.1 Limiting data for Example 1 .................................................................................... 65 

Table 3.2 Problem specification for Example 2 ....................................................................... 75 

Table 3.3 Limiting water data for the Example 3 .................................................................... 81 

Table 3.4 Specification of Example 3 with central buffer tank ............................................... 82 

Table 3.5 Specification of Example 3 with regeneration unit ................................................. 84 

Table 3.6 Limiting data for Example 4 .................................................................................... 86 

Table 3.7 Specification concentration of each operation ......................................................... 88 

Table 4.1 Limiting water data for the Example 1 .................................................................. 106 

Table 4.2 The limiting data of Example 2 ............................................................................. 114 

Table 4.3 The limiting data of Example 3 ............................................................................. 121 

Table 5.1 Limiting data for Example 1 .................................................................................. 134 

Table 5.2 The matching matrix for reuse scenario (unit: t) ................................................... 135 

Table 5.3 The ranking matrix for water sources and sinks .................................................... 138 

Table 5.4 The potentially optimal matches ............................................................................ 138 

Table 5.5 The initial potential optimal match between sources and sinks............................. 139 

Table 5.6 The limiting water data of Example 2 ................................................................... 145 

Table 5.7 The limiting data of Example 3 ............................................................................. 147 

Table 6.1 Limiting water data for Example 1 .......................................................................... 160 

Table 6.2 Limiting water data for Example 2 .......................................................................... 163 

Table 6.3 Limiting water data for Example 3 .......................................................................... 166 

Table 6.4 Limiting water data for Example 4 .......................................................................... 169 

 



List of Acronyms 

XIII 

 

List of Acronyms  

CIS Common intermediate storage 

DP Dynamic programming 

FIS Finite intermediate storage 

HQR Higher quality region 

LP Linear Programming 

LQR Lower quality region 

MEN Mass Exchange Network 

MINLP Mixed integer nonlinear programming 

MILP Mixed integer linear programming 

NLP Non-linear programming   

NIS  No intermediate storage 

RTN Resource task network 

S-graph Schedule-graph 

SK Water sink 

SR Water source 

SSN State sequence network 

STN State task network 

TWN Total water network 

UIS  Unlimited intermediate storage 

WSD    Water source diagram 

 

 

 



Introduction                                                                             │Chapter 1 

1 

 

1 
INTRODUCTION 

1.1 Background 

Water scarcity is becoming a threat to sustainable development all over the world due to 

increasing water demands and the effects of the degradation of water quality caused by human 

activities. According to the report, about four billion people are suffering from water scarcity 

at least one month of the year, as shown in Figure 1.1. About half a billion people all over the 

world suffer from severe water scarcity all year round (Mekonnen and Hoekstra, 2016). South 

Africa is a water-scarce country in which only 13% of the land is arable, and the population 

experiences significant disparities in access to clean water (Gulati et al., 2013). Cape Town is 

the second-most populous city in South Africa. In 2018, the government of the city estimated 

that it could become the first major city in the world to have no water supply (Maxmen, 2018). 

Finally, the real “Day Zero” did not happen because of the coming of the rainy season. 

Although it was a false alarm, it has affected the lives of residents and the regular operation of 

industries due to water shortage. It also indicates that water crisis is not just a concept perceived 

in mind, but a reality.  

 

Figure 1.1 The distribution of water scarcity area (Mekonnen and Hoekstra, 2016) 



Introduction                                                                             │Chapter 1 

2 

 

The increasing climate variability and uncertainties caused by global warming are contributing 

to reducing the sustainability of water resources. Furthermore, the water scarcity will have 

significant impacts on the food provision and energy supply due to the water-energy-food 

nexus. It is projected that annual precipitation in southern Africa will be decreased by as much 

as 20% by the 2080s (Conway et al., 2015). According to results of the evaluation of water-

energy-food nexus for BRICS countries, i.e., Brazil, the Russian Federation, India, China, and 

South Africa, the relationship between carbon dioxide emissions and economic growth for 

Brazil, India, and South Africa can confirm the Environmental Kuznets curve hypothesis 

(Panayotou, 1993). These countries should construct an adequate water resource management 

to enhance agricultural productivity (Ozturk, 2015). The chemical industry is the largest of the 

manufacturing industries in the world and is central to the economic growth of South Africa. 

It contributes around 5% to the GDP of South Africa and nearly 25% of manufacturing sales 

in 2016/17 (SA Year Book 2016/17). Though the chemical industry provides a common 

commodity for the people, it also consumes huge quantities of energy and water and discharges 

wastewater to the environment. Also, stringent regulations and social pressures have created 

the need for improving the efficiency of water utilization in the chemical industry and 

preventing the deterioration of the water system via wastewater minimization.  

In general, according to the products, two different processing modes are used in the chemical 

industry, i.e., continuous and batch processes (Rippin, 1993). For continuous processes, raw 

materials and products are operated continuously at known flowrates. The states of the plant 

and operating conditions for the continuous processes are not changing along the time horizon. 

Therefore, continuous processes are usually used to produce typical products on a large scale, 

such as petroleum refineries, natural gas processing plants. However, in batch processes, 

materials are treated in finite quantities. Raw materials are feeding the vessel and then react for 

a given duration until physical properties of the production is achieved. Finally, the production 

is then discharged to the storage tank for delivery. Thus, batch processes are widely used to 

process low volume, high value-added products, such as agrochemical, pharmaceuticals, 

specialty chemicals, and cosmetics. In general, new products are initially manufactured in batch 

processing mode and then evolved to a continuous process because of upgrading technologies 

and market acceptance. Hence, this work focuses on wastewater recovery for batch process. 

There are two categories of batch processes, i.e., multiproduct and multipurpose batch process. 

In the multiproduct batch plant, all of the batches are handled in the same way, and processing 

sequences of batches in each unit are identical. In multipurpose batch plants, on the other hand, 



Introduction                                                                             │Chapter 1 

3 

 

the same unit could process different products and the processing paths of the batches may be 

different. For multipurpose batch processes, various tasks can be performed in the same unit, 

resulting in the problem of how to allocate the unit optimally to the tasks. Thus, most of the 

research focuses on multipurpose batch process as compared to multiproduct batch processes, 

and the industry pays more attention to multipurpose batch processes. In this work, conceptual 

analysis is performed to maximize water recovery for multipurpose batch processes by 

rescheduling the batch operations. 

1.2 Motivation 

The first motivation behind this work is that in the chemical industry, most industrial plants are 

processing through continuous operations because it is suitable to produce products on a large 

scale. Batch processing is also adopted in the chemical industry due to its ability to adjust to 

changing market conditions and capability of producing high value-added and small quantity 

products. In general, consumers prefer tailor-made products, rather than the generic products. 

This trend provides an opportunity for expanding the batch processing model in the industry. 

In batch processes, freshwater is usually used to wash the units after production. Thus, the 

composition of the contaminant in wastewater is complex and changing all the time. Also, the 

discrete nature of batch operations makes wastewater treatment uneconomical as compared to 

the continuous process. Industrial activities not only consume freshwater resources but also 

discharge wastewater into the environment. This results in the fact that water ecosystem is 

unable to maintain its natural, original state. Furthermore, stringent legislation on industrial 

wastewater has led to an increase in effluent treatment costs. Thus, development of a systematic 

tool to maximize water recovery, reduce freshwater intake and to minimize wastewater 

discharge from industrial plants is necessary. 

The final motivation for this work is that most work on the synthesis of batch water networks 

focuses on predefined scheduling. In batch processes, the recipe of production is always 

changing to adjust to the demand of the market, resulting in a change in the water network. 

Therefore, it is required to develop a systematic method of considering water recovery and 

process scheduling simultaneously.  

1.3 Problem statement 

The problem to be addressed by this research can be stated as follows. 

 



Introduction                                                                             │Chapter 1 

4 

 

Given: 

⚫ Production scheduling data, such as mean processing time, and the relationship between 

different productions, i.e. recipe, 

⚫ Limiting water data, including water quantity and water quality constraints of 

operations,  

⚫ the maximum capacity of the storage vessel, 

⚫ the performance of the regeneration units, such as removal ratio or outlet concentration 

and 

⚫ Time horizon of interest 

The objective is to find an optimal production schedule that achieves the minimum amount of 

freshwater consumption by employing water reuse/recycling and regeneration scheme. 

1.4 Research objectives 

The objectives of this research are summarized as follows. It is to present a general conceptual 

approach that could be used to optimize water utilization in variable time multipurpose batch 

plants. In the analysis, variable time refers to the scenario that the duration of each batch 

process is given, rather than the starting and ending times of operations. Consequently, starting 

and ending times become optimization variables, hence the name ‘variable time’. Freshwater 

could be minimized through water reuse/recycling and regeneration by rescheduling the 

process operations. Most of the work in published literature in the area of water optimization 

in multipurpose batch plants tends to be steeped in mathematical modeling, with very limited 

conceptual analysis. This results in models that are very difficult to solve at best and almost 

impossible to solve at worst. The main contribution, therefore, is the development of models 

that are solvable in real-time or reasonable time. 

1.5 Scope of work 

The main objective of this research is to develop a general framework for the synthesis of the 

water network in a flexible batch plant. This work proposes a dynamic programming approach 

for the design of a batch water network based on predefined scheduling. Insights from these 

results could assist in rescheduling operations to maximize water recovery through direct water 

reuse. However, the improvement of targets is limited because rescheduling only involves a 

few operations and, as a consequence, inherently entails fewer degrees of freedom compared 



Introduction                                                                             │Chapter 1 

5 

 

to a complete scheduling problem. Thus, the dynamic programming method is also extended 

to flexible batch processes in which only duration of operations is given. Although dynamic 

programing approach as an algebraic method could provide a graphical representation of the 

results, insights from the analysis are sometimes not obvious, depending on the complexity of 

the problem. A match ranking matrix approach is then proposed to prioritize matches between 

water sources and sinks. Based on the ranking of matches, batch water networks could be 

synthesized by considering the time nature of batch processes. Besides, a hybrid method of 

combing water pinch analysis and match ranking matrix is presented to design batch water 

networks. This hybrid method takes advantages of graphical representation and conceptual 

nature of pinch analysis. Finally, the main conclusions and future work are summarized. 

1.6 Thesis structure 

The outline of this thesis is arranged as below. 

Chapter 2 gives a comprehensive survey of the literature related to this work, including the 

topic about production scheduling, water minimization of continuous and batch processes, as 

well as simultaneous optimization of production scheduling and water utilization. Literature is 

also given on the different techniques of water minimization problems, such as insight-based 

methods and mathematical model-based optimization methods.  

In Chapter 3, a dynamic programming method is used to design an optimal batch water network 

with a predefined schedule. Four examples adapted from literature are demonstrated to 

illustrate the proposed method. Furthermore, rescheduling is then performed to maximize direct 

reuse and to reduce water storage. 

Chapter 4 shows the extension of Chapter 3. It introduces how to use dynamic programming 

to design optimal batch water network with flexible production scheduling.  

In Chapter 5, a matrix matching method is developed to design the optimal batch water network 

with flexible production scheduling framework.  

Pinch analysis is widely recognized as an efficient method of water recovery in continuous 

processes. In Chapter 6, conventional pinch analysis is combined with match ranking matrix 

method to design optimal batch water network with flexible production scheduling. 

Chapter 7 gives conclusions and recommendations for future work. 



Introduction                                                                             │Chapter 1 

6 

 

References 

Conway, D., van Garderen, E.A., Deryng, D., Dorling, S., Krueger, T., Landman, W., 

Lankford, B., Lebek, K., Osborn, T., Ringler, C., Thurlow, J., Zhu, T., Dalin, C., 2015. Climate 

and southern Africa's water–energy–food nexus. Nature Climate Change 5, 837. 

Gulati, M., Jacobs, I., Jooste, A., Naidoo, D. and Fakir, S., 2013. The water–energy–food 

security nexus: Challenges and opportunities for food security in South Africa. Aquatic 

Procedia, 1, 150-164. 

Majozi, T., Seid, E.R., Lee, J.-Y., 2017. Understanding Batch Chemical Processes: Modelling 

and Case Studies. CRC Press, Taylor & Francis Group, Boca Raton, Florida, United States. 

Maxmen, A., 2018. As Cape Town water crisis deepens, scientists prepare for ‘Day Zero’. 

Nature 554, 13-14. 

Mekonnen, M.M., Hoekstra, A.Y., 2016. Four billion people facing severe water scarcity. 

Science Advances 2(2), e1500323. 

Ozturk, I., 2015. Sustainability in the food-energy-water nexus: Evidence from BRICS (Brazil, 

the Russian Federation, India, China, and South Africa) countries. Energy 93, 999-1010. 

Panayotou, T., 1993. Empirical Tests and Policy Analysis of Environmental Degradation at 

Different Stages of Economic Development, working paper WP238. International Labor 

Office, Geneva, Switzerland. 

Rippin, D.W.T., 1993. Batch process systems engineering: A retrospective and prospective 

review. Computers & Chemical Engineering 17, S1-S13. 

SA Year book 2016/17, Accessed at 

www.gcis.gov.za/sites/default/files/docs/resourcecentre/yearbook/SAYearbook2016-17.pdf. 

 

http://www.gcis.gov.za/sites/default/files/docs/resourcecentre/yearbook/SAYearbook2016-17.pdf


Literature review                                                                         │Chapter 2 

7 

 

2  
LITERATURE REVIEW 

In this chapter, a literature review is summarized to explain the progress of production 

scheduling and water minimization techniques in chemical processes. In Section 2.1, the 

operational philosophy for batch processes is introduced. Section 2.2 describes the techniques 

of production scheduling. In Section 2.3, it has briefly introduced the synthesis of water 

network and different water reuse schemes. Since the existing insight-based methods for the 

synthesis of water network mainly focus on continuous processes, water integration techniques 

for continuous and batch processes are reviewed in Sections 2.4 and 2.5, respectively. Section 

2.6 summarizes the challenges of synthesis of water networks and future research. 

2.1 Operational philosophy 

The flexibility in batch processes arises mainly because of the availability of intermediate 

storage between processing stages in a batch plant. According to the nature of intermediate 

storage, there are four main operational philosophies, i.e., unlimited intermediate storage (UIS), 

finite intermediate storage (FIS), no intermediate storage (NIS), and common intermediate 

storage (CIS) (Majozi, 2010; Romero et al., 2004).  

In general, a batch chemical process is broadly categorized into a single stage or multistage 

processes. For a single stage process, each stage may consist of a single unit or multiple parallel 

units based on the structure of processing networks. For the multistage process, it can be 

classified into multiproduct and multipurpose batch plants (Majozi et al., 2017; Reklaitis et al., 

1996). Short-term scheduling in batch processes addresses the problems of allocating 

production resources to meet the product demand and managing the material transfer from one 

operation to another in a coordinated time. In chemical batch plants, short-term scheduling is 

similar to the general job-shop problems in operations research. However, there are some 

differences between these two activities. In chemical batch plants, additional constraints, such 

as the storage policy, the physical property of materials, should be considered in the production 

scheduling (Sanmartí et al. 2002). On the other hand, multiproduct batch processes follow the 



Literature review                                                                         │Chapter 2 

8 

 

same sequence of unit operations. However, each batch does not have to produce the same 

product. The processing time of tasks corresponding to different products could be different. 

In contrast to multiproduct plants, multipurpose batch production does not require common 

processing sequences between the products (Majozi, 2010; Reklaitis et al., 1996). 

2.2 Production scheduling 

Batch processes are a widely accepted procedure in the chemical industry because of the 

flexibility in changing production recipe and the suitability of producing small-quantity 

products. The competitive pressure calls for an improved design of batch plants and better use 

of existing facilities. An effective production schedule must optimally allocate limited 

resources to tasks achieving the goal of either maximizing throughput or minimizing the total 

cost. There have been reported four scheduling methods for chemical processes in literature, 

i.e., Schedule-graph (S-graph), State task network (STN), State sequence network (SSN), 

Resource task network (RTN). 

2.2.1 Schedule-graph (S-graph)  

Most of the research in production scheduling focuses on developing general mathematical 

formulations. However, this general-purpose approach has limitations in its application to 

large-scale industrial problems in terms of computational time involved (Mokashi and 

Kokossis, 2002). Therefore, a few researches tried to explore the opportunities of using graph 

theory to solve scheduling problems.  

Wu et al. (1999) proposed a graph-theoretic approach to determine job-shop scheduling 

problems. The results obtained by this graphical approach are more robust as compared to the 

existing mathematical methods. Mokashi and Kokossis (2002) proposed an efficient algorithm 

called the maximum order tree to address the product distribution problem. This graph-based 

method can reduce the CPU time dramatically compared to conventional methods without 

compromising on the quality of the solution. These graph-theoretic approaches can be used to 

solve scheduling problems. However, it is only applicable to job-shop problems with UIS 

policy.  

In the chemical industry, production scheduling is different from that in job-shop problems. 

For example, there are many storage policies due to the characteristics of intermediate 

materials. Sanmartí et al. (1998; 2002) proposed the S-graph method to deal with scheduling 

problems with NIS policy. In the S-graph method, the graph nodes are the tasks of operations, 



Literature review                                                                         │Chapter 2 

9 

 

while the arcs connecting different nodes are the precedence relationship of tasks. Both NIS 

and UIS policies can be represented by choosing the appropriate precedence relationships. 

Besides, other algorithms, such as branch-and-bound method can be easily incorporated into 

the S-graph to solve the problem.  

The most widely utilized mathematical programming methods for tackling the scheduling 

problem are expensive in terms of computational time. Because S-graph is flexible in 

considering various production structures and having advantages of graph theory, it has been 

used to perform production scheduling in the batch plant. For example, Holczinger et al. (2002) 

applied the S-graph to solve the scheduling problem in multipurpose batch processes. An 

acceleration algorithm was developed to increase computational efficiency to find the optimal 

solutions for significant scheduling problems. Hegyháti and Friedler (2011) listed several 

combinatorial algorithms that can be combined with S-graph to address the scheduling problem 

of batch processes, which can be used to solve practical problems with additional requirements. 

Romero et al. (2004) modified the existing S-graph representation by introducing a new type 

of node for representing CIS policy. A set of rationales is defined to accelerate the solution 

strategy by reducing the searching space. In the work of Majozi and Friedler (2006), S-graph 

was used to optimize the production schedule for a multipurpose batch plant by maximizing 

the throughput. A guided search algorithm is proposed to ensure that the results are the global 

optimum. However, their paper only considered the non-intermediate storage policy. Hegyháti 

et al. (2009) considered scheduling problems with complicated recipes based on S-graph. NIS, 

ZW, and CIS policies are taken into account in their paper. Compared to mathematical 

programming, S-graph can avoid the occurrences of cross-transfer, which can lead to a real 

optimum (Hegyháti et al., 2009). Furthermore, S-graph can deal with production scheduling 

with uncertainty. Laínez et al. (2010) considered the uncertainties caused by exogenous factors 

in the scheduling process for a batch process based on the S-graph approach.  

2.2.2 State task network (STN) 

Kondili et al. (1993) proposed the STN representation to capture the characteristics of 

production scheduling in batch plants. In the representation, two types of nodes are defined, 

i.e., task nodes that represent individual batch operations and state nodes that indicate raw 

material, intermediate and final products. The binary variables in the formulation imply 

whether the tasks are implemented in a specific piece of equipment at a specific event point or 

not. This representation can consider the general production operation, involving stream 

mixing and splitting, material reuse and recycling, storage policies, and so on. The scheduling 



Literature review                                                                         │Chapter 2 

10 

 

problem is formulated as a mixed-integer linear programming (MILP) model because of the 

deployment of binary variables. However, in the work of Kondili et al. (1993), a discrete-time 

representation was employed where the time horizon is partitioned into many intervals with 

equal duration. It usually leads to a large model and results in sub-optimal solutions (Floudas 

and Lin, 2004). Bose and Bhattacharya (2009) proposed a discrete-time STN model to deal 

with the scheduling in multi-stage multi-product continuous facilities. Because discrete-time 

representation usually leads to a large model, various specific techniques were developed by 

Shah et al. (1993) to enhance computational efficiency.  

Although significant improvements have been made on the discrete-time model, the 

assumption of constant processing times force operations occurring at the start or end of a 

specific time interval, preventing its full application in the industry. Therefore, the continuous-

time STN formulation was developed to overcome the limitations of discrete-time models. 

Compared with general discrete-time STN formulations, continuous-time STN model is faster 

in CPU time (Maravelias and Grossmann, 2003a, 2003b, 2006).  

2.2.3 State sequence network (SSN) 

SSN was proposed by Majozi and Zhu (2001) to perform short-term scheduling for batch plants 

in order to reduce the model size. In the representation, only state nodes are required because 

the tasks and units can be eliminated without any loss of accuracy. Thus, the SSN based model 

requires fewer binary variables (Majozi, 2010). Based on SSN representation, resource 

allocation model was incorporated into the overall plant model to exploit resource conservation 

and optimal scheduling (Zhu and Majozi, 2001). Because the SSN representation has fewer 

binary variables, it has been used as a cornerstone for the optimization of batch process 

scheduling in wastewater minimization, and heat integration (Majozi, 2010; Majozi, 2017). 

2.2.4 Resource task network (RTN) 

To describe all resources in a unified manner, Pantelides (1994) proposed a unified 

mathematical formulation to represent production scheduling for batch plants. It is to model a 

mixed production facility as a collection of facilities performing a variety of tasks which 

consume some resources and generate others. The resources can be feeds, intermediates, 

products, utilities, equipment, storage facilities, workforce, etc.  

Zhang and Sargent (1996) extended concepts of the RTN to a mixed production facility and 

determined the optimal operating conditions. In the work of Castro et al. (2003), discrete- and 

continuous-time were used to determine optimal cyclic operations for a batch plant based on 



Literature review                                                                         │Chapter 2 

11 

 

RTN. Results show that discrete-time formulation is more accessible to solve to optimality than 

the continuous-time counterpart. Chen and Chang (2009) modified the work of Castro et al. 

(2003) and proposed an integrated formulation for multipurpose batch processes, which 

considers production scheduling and heat recovery problems. Furthermore, Chen et al. (2011) 

presented an integrated model for multipurpose batch plants by simultaneously considering 

scheduling and water recovery based on RTN approach. Castro et al. (2013) provide a generic 

modeling framework for production scheduling under energy constraints based on RTN. Castro 

and Grossmann (2014) addressed the modeling of crude oil operations in refineries based on 

RTN representation. 

2.3 Synthesis of Water Networks  

Rudd (1968) proposed a system principle to instruct the chemical engineer how to combine 

different processes to achieve the assigned task. This concept is the initial definition of process 

synthesis. This definition has evolved over the past few decades. The commonly accepted 

definition of process synthesis is discrete decision-making activities of choosing some 

available elements to obtain an optimal solution for a given design problem from a systematic 

perspective (El-Halwagi, 1997). There are two crucial areas in process synthesis, i.e., mass 

exchanger network synthesis (mass integration) and heat exchanger network synthesis (heat 

integration). Within the framework of mass integration, water integration serves as a special 

case. This review focuses on water integration for the continuous and batch processes. 

In the synthesis of water networks, there are three subclasses: (1) water reuse/recycle; (2) water 

regeneration; (3) total water network synthesis. In the total water network, water reuse 

subsystem, regeneration unit, and wastewater treatment unit are integrated from a systematic 

perspective. The description of these subclasses is described below. 

2.3.1 Water reuse/recycle 

Figure 2.1 depicts a water reuse network for the process industry. It allows water to be directly 

or indirect reused in other processes where the contaminant levels of reusable water are within 

the process limits. In Figure 2.1 (a), the reusable water is directly reused in other processes, 

while in Figure 2.1 (b) the reusable water is first sent to the storage and then reused by other 

processes. Indirect reuse is a typical case in batch processes because of the time constraint. In 

a water reuse network, reusable water can be blended with freshwater or reusable water from 

other processes to meet the contaminant levels of water-receiving processes. 



Literature review                                                                         │Chapter 2 

12 

 

Process unit 1 Process unit 2
Freshwater 

To waste

Process unit 1 Process unit 2
Freshwater 

To waste

(b) Indirect reuse

(a) Direct reuse 

 

Figure 2.1 Batch water network with reuse scheme 

Process unit 1 Process unit 2
Freshwater To waste

(a) Water recycling for continuous processes

Process unit 1 Process unit 2
Freshwater To waste

(b) Water recycling for batch processes

Tank

 

Figure 2.2 Batch water network with direct water recycling scheme 

Figure 2.2 shows the water recycling scenario where reusable water is reused in the same 

process. In the direct recycling scenario, wastewater from the process is partially treated and 

then sent back to the system. Freshwater is usually used to mix this reusable water to meet the 

contaminant limits of water-receiving processes. Because water recycling can bring in 

contaminant accumulation and demand more freshwater, this scheme is thus not used in 

industrial activities. Figure 2.2(a) represents the water recycling scenario for continuous 

processes, while in Figure 2.2(b) the reusable water is first sent to the tank and then mixed with 

freshwater to feed the water-using operation. 

2.3.2 Regeneration reuse/recycle 

Figure 2.3 presents the regeneration reuse scenario where wastewater is treated to remove the 

contaminants and then reused in other processes (Wang and Smith, 1994a). The regenerated 

water might be directly reused in other operations or mixed with wastewater from other 

processes and freshwater to feed water-using operations. In regeneration reuse scenario, the 

regenerated water is not allowed to reenter the processes where the regeneration water comes 

from to prevent the accumulation of harmful contaminants in the activities of water cascade 



Literature review                                                                         │Chapter 2 

13 

 

utilization. However, sometimes the regenerated water can reenter the process where this water 

is previously reused before regeneration; this refers to a different scenario of wastewater 

minimization, as shown in Figure 2.4.  

Process unit 1

Process unit 2

Freshwater 

To waste

Regeneration unit

 

Figure 2.3 Water network with regeneration reuse 

Regeneration recycling refers to the situation where wastewater is partially treated and recycled 

in process units where it was generated (Wang and Smith, 1994a). Although regeneration can, 

to some extent, reduce wastewater generation, it might not be allowed in practice due to the 

accumulation of contaminants. In practice, two or more methods are combined to achieve the 

target of wastewater minimization. Although the description above explains how to reduce 

freshwater requirement, systematic approaches are still required to determine the target. The 

water integration techniques of combining the schemes above are introduced in the following 

section. 

Process unit 1

Process unit 2

Freshwater 

To waste

Regeneration unit

 

Figure 2.4 Water network with regeneration recycling  

2.3.3 Total water network 

In previous subsections, direct reuse/recycling and regeneration reuse/recycling are introduced. 

Here, the concept of total water network (TWN) is introduced by integrating direct 

reuse/recycling and regeneration reuse/recycling in the same water system. Kuo and Smith 

(1998a) proposed the concept of TWN for fixed load problems. As shown in Figure 2.5, the 

TWN includes three subsystems, i.e., direct reuse/recycle, regeneration, and waste treatment. 



Literature review                                                                         │Chapter 2 

14 

 

Because these water schemes consider water utilization from different views, it is necessary to 

treat them as an overall framework. If the potential of water recovery is maximized through 

water reuse/recycle scheme, employing a regeneration unit for partial purification of the 

wastewater could lead to further reduction of freshwater consumption. In some cases, the 

industrial plants utilize freshwater to dilute the wastewater to meet the emission standard. 

Because of the increasing cost of freshwater and stringent regulation on the environment, 

wastewater treatments are employed to treat the effluent to protect the environment and reduce 

freshwater consumption. 

Water-using processes

Regeneration Wastewater treatment

Freshwater

Direct reuse/recycle

Regeneration 

reuse/recycle

 

Figure 2.5 Schematic of the total water network 

2.4 Water integration techniques for continuous processes 

Mass integration, as a branch of process integration, provides a systematic methodology for 

optimizing the mass flow within a process based on setting performance targets by splitting, 

mixing different streams and species (El-Halwagi, 1998; Klemeš et al., 2013). El-Halwagi and 

Manousiouthakis (1989) proposed a graphically based approach to maximize mass exchange 

between different mass transfer operations, such as absorbers, extractors. Water integration is 

a specific application of mass integration in water systems. Many methods for the synthesis of 

water networks have been reported in the literature. These approaches can be divided into two 

groups, i.e., insight-based techniques and mathematical programming methods. 



Literature review                                                                         │Chapter 2 

15 

 

2.4.1 Insight-based techniques 

2.4.1.1 Graphical methods 

Graphical methods provide a systematic view of determining the performance target of the 

system without detailed design. Besides, the physical insights provided by the graphical 

representation could assist chemical engineers to implement the detailed design in practice, 

while mathematical programming only provides the results without giving more details of the 

computation processes. The development of graphical methods has two stages. In the first 

stage, the development mainly focuses on the composite curve of Wang and Smith (1994a) to 

cope with fixed mass load problems. In the second period, many techniques are proposed to 

deal with the design of the water network. These methods are no longer limited to fixed load 

problems, but fixed load and fixed flowrate problems.  

Wang and Smith (1994a) proposed the water pinch analysis to minimize wastewater generation 

in process industries. In their method, concentration is analogous to temperature, while the 

mass load is analogous to enthalpy. The limiting composite curve is constructed by combing 

all the segments of the processes on concentration vs. mass load diagram. A water supply line 

is drawn from the origin and rotates in an anticlockwise direction until it touches the limiting 

composite curve. The target of minimum freshwater flowrate is determined by the reciprocal 

of the slope of the water supply line (Wang and Smith, 1994a). The bottleneck of water 

recovery for the system can be determined from pinch point. However, the limitation is that 

the assumption of mass transfer is not always applicable to some non-mass transfer processes. 

Many processes, such as reactors, boilers, and cooling towers, do not have mass transfer 

phenomena. Therefore, another method was developed by Dhole et al. (1996) to determine the 

freshwater targets, which consists of supply composite and demand composite curves. 

Although their method can identify the target graphically, the results depend on the mixing of 

different sources, which could lead to inaccurate solutions. The limitation is that there exist 

many local pinch points in the approach of Dhole et al. (1996). This observation could obscure 

the physical meaning of the pinch point and cannot provides precise information to assist the 

designer in designing an optimal water network. In the method of Dhole et al. (1996), the 

operations of water splitting, and mixing are not considered, which could reduce the potential 

of water recovery. A Load Table method was developed by Olesen and Polley (1997), in which 

the penalty of freshwater caused by splitting can be determined. Furthermore, Olesen and 

Polley (1996) considered water transfer between different plants based on the Load Table. 



Literature review                                                                         │Chapter 2 

16 

 

Sorin and Bédard (1999) proposed a two-stage approach to design water reuse networks. In the 

first step of their method, the target and the pinch point can be determined using the 

Evolutionary Table. In the second step, some special rules are defined to guide the mixing 

operation of water sources above and below pinch point in the network design process. 

In order to avoid confusing representation of water operations, Castro et al. (1999) presented a 

simultaneous targeting-design algorithm called water source diagram (WSD) to attain 

minimum flowrate. However, this method is unable to determine the targets in regeneration 

scenario separately. Gómez et al. (2001) proposed a concentration grid method for the synthesis 

of water-using networks. The concentration grid is similar to WSD in Castro et al. (1999). But 

it represents the water sources in a different manner. Gomes et al. (2007) made some 

improvements for the WSD method, and thus, the modified WSD is more general for 

minimizing wastewater in process industries. Because WSD offers a dynamic and flexible 

framework for water management, Ulson de Souza et al. (2009) extended the WSD method to 

a petroleum refinery. The scenarios of maximum water reuse, regenerative end-of-pipe 

treatment, and differentiated regeneration were studied. Ulson de Souza et al. (2010) modified 

the WSD method and applied to a textile plant to reduce water requirement through water reuse 

scheme. A reduction of as much as 64% of water can be reduced compared to the scenario 

without water reuse. Gomes et al. (2013), and Calixto et al. (2015) also applied the WSD 

method to multiple contaminants systems. Besides, WSD can be applied to the hybrid systems 

with both mass transfer and non-mass transfer operations, as well as water gain/loss operations 

(da Silva Francisco et al., 2015; Francisco et al., 2018). 

By the analogy between the transfer of mass and heat, El-Halwagi and Manousiouthakis (1989) 

developed a systematic procedure for the synthesis of cost-effective mass-exchange networks 

(MENs). First, pinch points are identified based on the thermodynamic analysis, which limits 

the extent of mass exchange between the rich and the lean streams. Next, the network which is 

designed based on the insights from the first step is evolved to be practical by considering more 

constraints. In their subsequent paper, an automatic approach was developed to generate cost-

effective MENs (El-Halwagi and Manousiouthakis, 1990). The automatic method is based on 

a two-stage procedure. First, thermodynamic constraints are formulated as an LP problem. 

Next, binary variables are employed to choose the suitable match between the rich and lean 

streams to achieve the targets in the first step and to minimize the number of mass-exchange 

units.  



Literature review                                                                         │Chapter 2 

17 

 

In industrial processes, not all water-using operations involve mass transfer phenomena. In 

some cases, some operations only have water intake or waste discharge. Therefore, the problem 

of water usage and discharge can be presented by a source and a sink. Although Dhole et al. 

(1996) employed water demand and supply composite curves, pinch points defined in their 

method cannot represent the bottleneck of the problem. Some pinch points could be eliminated 

by mixing some water streams. To overcome these limitations, a material recovery pinch 

diagram was developed by El-Halwagi et al. (2003) to target the minimum flowrate of the fresh 

resource. It provides a systematic, and graphical method for the targeting and can yield the 

results without tedious iterations. Some important information, such as the maximum of 

resource recovery, and the number of unused materials, can be attained from the diagram. The 

same method was proposed separately by Prakash and Shenoy (2005b) almost at the same time. 

Prakash and Shenoy (2005b) also proposed a new design procedure for the synthesis of water 

networks based on nearest neighbors principle. For a specific water demand, the water sources 

to be chosen to feed this demand should be the nearest available neighbors in terms of 

concentration. A concept of source shifts was introduced by Prakash and Shenoy (2005a) to 

yield simpler networks. It could assist the designer in evolving these alternative networks 

systematically with fewer connections at the expense of a freshwater penalty.  

Dunn and Wenzel (2001) introduced a source-sink mapping diagram to determine the water 

reuse and recycling network. It is convenient to design a simple water network using the source-

sink mapping diagram, while for complex processes, the design of the piping network is 

complicated. Feng and Seider (2001) employed the water main to simplify the piping system. 

Also, internal water main was employed when designing multi-contaminant water network for 

water reuse scheme (Wang et al., 2003) and regeneration recycling scheme (Cao et al., 2004). 

These methods mentioned above provide a systematic view for targeting the minimum 

freshwater requirement based on the limiting composite curve. Except for the graphical 

representation of Wang and Smith (1994a), different graphical techniques were also proposed. 

Bandyopadhyay et al. (2006) developed a source composite diagram to minimize wastewater 

generation for water systems with fixed-mass load and fixed-flowrate operations. In addition 

to the target of freshwater, some critical factors, such as the minimum effluent flowrate for 

treatment, the minimum contaminant removal ratio, and the minimum number of treatment 

units can be determined. In the work of Bandyopadhyay et al. (2006), graphical representation 

of the method, as well as its analytical algorithms, were presented in detail to explain how to 

attain these targets. This graphical approach has been applied to water management, hydrogen 



Literature review                                                                         │Chapter 2 

18 

 

management (Bandyopadhyay, 2006). Also, Bandyopadhyay and Cormos (2008) applied the 

source composite diagram to water systems with a regeneration unit. However, this graphical 

approach has the same limitation as to other graphical methods. It only applies to water systems 

with a single contaminant. Shenoy and Bandyopadhyay (2007) extended the method to a water-

using system which was supplied by multiple resources. They found that maximizing the 

utilization of resources with the least unit cost does not lead to the optimal water network with 

the minimum operating cost.  

Except for these graphical methods for water allocation network, some approaches for 

wastewater treatment have also been developed. Wang and Smith (1994b) proposed a graphical 

approach for the synthesis of a distributed wastewater treatment network. If cost decreases 

continuously for decreasing flowrate, the proposed method can determine the minimum cost 

required to be treated wastewater. A treatment process network can be designed to achieve the 

flowrate targets with the minimum total treatment costs according to the design rules. For 

example, streams having a concentration higher than pinch point are fully treated. Those having 

the concentration equal to pinch point are partially treated, and partially bypass treatment and 

those starting below the pinch completely bypass treatment. The limitation of this work is that 

sometimes the minimum flowrate does not lead to the minimum treatment costs. This is 

because the relationship between wastewater flowrate and treatment cost is not linear. Also, 

the design for multiple treatment processes was not addressed clearly. Furthermore, it is 

assumed that there is no water loss for the flowrate in the treatment process. This assumption 

is not true for some wastewater treatment processes, such as the membrane technologies.  

Hence, Kuo and Smith (1997) considered the synthesis of effluent treatment networks with 

multiple treatment processes. In the targeting processes, some treatment lines are allowed to 

cross the composite effluent curve on an individual basis, while the composite treatment curve 

must not cross the composite effluent curve. They also extended the method to retrofit cases. 

However, for a water system with multiple contaminants, their method cannot guarantee that 

the minimum flowrate is obtained because the designed network is achieved by iterative 

procedures of targeting and network design. In their subsequent paper, Kuo and Smith (1998a) 

introduced a new method to explore the regeneration opportunities. In their method, two 

different groups were defined, i.e., one group only requires freshwater, and the other requires 

freshwater and regenerated water. A few rules are set to migrate the water transfer between 

these two groups. The target of freshwater consumption, regenerated water, as well the number 

of regeneration unit, can be determined.  



Literature review                                                                         │Chapter 2 

19 

 

Kuo and Smith (1998b) proposed a graphical method which considers water-using subsystem 

and wastewater treatment subsystems as a whole. Their method offers insights to quantify the 

trade-off between water minimization and treatment systems. Wang and Smith (1994a) pointed 

out that the regeneration process must be put in the right place. Otherwise, it leads to an increase 

in the flowrate of regeneration.  

In general, the regeneration concentration is located at the pinch point. If not, the complexity 

increases, making targeting more complex. Feng et al. (2007) analyzed the effect of different 

regeneration concentration on the targets of freshwater and regenerated flowrate. In their 

method, sequential optimization strategy was adopted to obtain targets of freshwater 

consumption, minimum regenerated water flowrate, and optimal generation concentration. 

First, the outlet concentration of the regeneration unit is fixed. Next, a sensitivity analysis was 

carried out to evaluate the effect of the outlet concentration on these targets. Based on this 

method, Bai et al. (2007) considered the total regeneration and partial regeneration scenarios. 

Furthermore, an improved mass problem table was introduced to identify targets of freshwater, 

regeneration water, and regeneration concentration.  

Tan and Manan (2006) considered the retrofit of regeneration water networks. First, 

optimization schemes were chosen based on the insight provided by the targeting process. Next, 

an economic analysis of these process changes was performed to find the optimal solution with 

the minimum total cost. In their subsequent work, this method was applied to retrofit of water 

networks with regeneration for a papermaking process (Tan et al., 2007). In the work of Wang 

and Smith (1994a), mass load decomposition, and concentration decomposition were used to 

determine freshwater requirements of water-using processes. Bai et al. (2010) summarized 

some decomposition rules and combined with the mathematical model to optimize water 

system with regeneration reuse. Although process decomposition leads to the reduction of 

freshwater to some extent, it complicates network structure and results in infeasible solutions 

about economic concerns. 

In water pinch analysis, the composite curve is first drawn. Next, the water supply line with 

the steepest slope is chosen as the desired objective. Finally, the target of freshwater 

requirement is determined by reversing the slope of the desired water supply line. The targeting 

process is partially automated. Some efforts have been made to implement pinch analysis 

automatically. For example, Jacob et al. (2002) formulated the pinch analysis as an LP model. 

The targeting process is fully automated and can cope with significant size problems. It is 

helpful for the application of pinch analysis in practice. On the other hand, most of the paper 



Literature review                                                                         │Chapter 2 

20 

 

based on graphical techniques focused on the utility targets, rather than costs. It thus does not 

necessarily lead to the minimum capital cost for the network with the minimum freshwater 

consumption.  

A graphical method was proposed by Hallale and Fraser (1998) for targeting the minimum cost 

before detailed design. In their method, the minimum number of trays for a network of 

absorption columns was initially determined. Next, the capital cost compatible with this target 

was accounted for through the trade-off analysis between operating and capital costs. Sujak et 

al. (2017) presented a cost-effective method to design optimal water networks that consider 

economic factors while exploring all of the water minimization options. The advantage of this 

method lies in the simultaneous consideration of water management at all hierarchical levels 

and the economic analysis of water minimization schemes that achieve the minimum target of 

freshwater to attain a maximum net profit. 

2.4.2 Algebraic methods  

Graphical methods, such as composite curves are helpful for chemical engineers because they 

provide physical insights for the overall potentials of resource recovery. However, the targeting 

process will be cumbersome if the size of the problem is significant. Hence, numerical targeting 

tools have been proposed because of the advantages of graphical methods in the view of 

accuracy and speed (Manan et al., 2004).  

The composite curves presented by Wang and Smith (1994a) can find the pinch point and 

determine the minimum freshwater requirement before detailed design. However, in their 

method, only one pinch point was defined. In the paper of Dhole et al. (1996), many local pinch 

points were defined. This observation could obscure the designers and complicate the problem 

because the multiple pinch point can confuse the designer, which is the global pinch point. To 

rectify the shortcoming of these methods, a two-step procedure was proposed by Sorin and 

Bédard (1999). The Evolutional Table was used to identify these targets for the problem. 

However, Evolutionary Table failed to find the true pinch point if more than one global pinch 

points existed. In the work of Hallale (2002), water surplus diagram was proposed for the 

targeting process. This method is also tedious and time-consuming to plot as compared to the 

other forms of composite curves. Thus, a numerical equivalent form, i.e., water cascade 

analysis, was developed to simplify the targeting process (Manan et al., 2004). A similar 

cascade representation based on the load interval diagram was proposed by Almutlaq and El-

Halwagi (2007) to identify rigorous targets for water-using systems. Aly et al. (2005) modified 



Literature review                                                                         │Chapter 2 

21 

 

the load interval diagram and developed the load problem table to obtain freshwater 

requirement of water systems.  

In water-using systems, water quality is usually characterized by the contaminant 

concentration. In practice, except for the concentration, the property is also used to represent 

the quality of streams, such as density, electric resistivity, viscosity. A property-based algebraic 

technique was developed by Foo et al. (2006) to identify the targets for material reuse network. 

Hortua et al. (2013) proposed an integrated approach for simultaneous optimization of property 

and mass integration. However, in their work, only water reuse and recycling were taken into 

account, while water regeneration was not considered, which may further reduce the total cost. 

An Extended Composite Table Algorithm was developed by Parand et al. (2014) to determine 

the targets of the total water network. A fixed post-regeneration concentration was set to obtain 

freshwater and regeneration flowrate. The trade-off analysis between freshwater and 

regeneration flowrate was analyzed by relaxing post-regeneration concentration. 

When designing the water networks, the first step is to determine the targets by pinch analysis 

or algebraic methods. The designer then utilizes the insights from the targeting process to 

design the networks. In this process, a feasible network is attained by removing the loops 

created by the matching procedure. However, the loop breaking depends on the problem size 

and the experience of the designer. When the number of operations becomes more prominent, 

and the loops overlap, it is difficult to break all of the loops. Furthermore, sometimes the loop 

breaking could bring unrealistic splitting of operations.  

Hence, Savelski and Bagajewicz (2000) introduced the necessary conditions of optimality for 

assisting the designer in checking the feasibility of matches. These conditions can be used as 

heuristic rules for the design of water allocation problems. Based on these rules, an LP model 

was developed by Bagajewicz and Savelski (2001) to determine the target, and a MILP 

formulation was proposed to choose the optimal matches. Also, Bagajewicz et al. (2000) 

incorporated the necessary conditions into the model to design the multiple-contaminant water 

network. Teles et al. (2008) developed a sequential approach for multiple-contaminant water-

using networks. In their method, a superstructure was established to account for all possible 

connections among different water-using operations. Then, the water requirement of each 

operation is met in sequence from the first to the last element. Although this approach can 

transform NLP problems into a succession of LP problems, the interaction between different 

operations was not considered. Pan et al. (2012) presented an iterative method for the synthesis 

of optimal water networks with regeneration recycling. This method is capable of designing 



Literature review                                                                         │Chapter 2 

22 

 

networks with known outlet concentrations and given removal ratios. Kim (2012) proposed a 

combined approach to assess different water schemes. They pointed out that costs associated 

wastewater effluent treatment are the key determinant for choosing the regeneration unit. 

2.4.3 Mathematical optimization-based approaches  

Although graphical and algebraic methods are powerful and straightforward, their inability to 

account for multiple contaminants is a common drawback which limits its application. Hence, 

mathematical programming has been used to cope with the synthesis of water network 

(Bagajewicz, 2000; Foo, 2012; Klemeš, 2012, 2013). Mathematical programming follows a 

general superstructure that represents all possibilities of connections for the system under 

consideration. The principle of the superstructure-based methods is fundamentally the same. 

The NLP or MINLP model formulated for the superstructure is optimized by removing its 

redundant constraints and then generate a final result. Thus, the desired design is automatically 

obtained (Hernández-Suárez et al., 2004).  

The seminal work of addressing water allocation issues based on the superstructure was 

proposed by Takama et al. (1980), in which the superstructure was constructed to explore 

opportunities for possible water reuse and regeneration. Then, the model was solved to attain 

the optimal water network. Alva-Argáez et al. (1998) combined the insight provided by water 

pinch analysis with mathematical programming tools for the design of industrial water systems. 

A penalty form was added in the objective function to pursue a reduction of infeasibilities. The 

MINLP model was solved by decomposing into a sequence of relaxed MILP problems. In their 

subsequent work, a MILP formulation was presented to screen water network for multiple 

contaminants problem (Alva-Argáez et al., 1999). Huang et al. (1999) modified the model 

developed by Takama et al. (1980) to consider water recovery and wastewater treatment. Some 

useful features in realistic applications were considered, for example, multiple sources and 

sinks, water losses. In the solution process, the results obtained by water pinch analysis were 

set as the initial points, and some variables were bounded at a reasonable level.  

In the work of Jödicke et al. ( 2001), except for the general costs, such as pumping costs, 

purchase cost of freshwater, and cost of wastewater treatment, costs of splitting and merging 

of pipes were also considered in the objective function. Hernández-Suárez et al. (2004) 

decomposed the complex superstructure into a series of basic superstructures. The advantage 

is that the problem is easy to solve. However, the reduction of complexity for the problem is 

accompanied by excluding some alternative solutions that may have stream recirculation and 



Literature review                                                                         │Chapter 2 

23 

 

recycle. Chang et al. (2009) proposed a systematic procedure to improve the operational 

flexibility of water networks. Two scenarios were considered. One is to relax the allowed 

minimum freshwater requirement at normal conditions, and the other is to deploy auxiliary 

pipelines to eliminate existing pipelines. Results indicate bottleneck of the problem is the point 

where the concentration of the mixture of freshwater and reuse water approaches their upper 

bound, while the maximum inlet and outlet concentration reach their lower bound. This finding 

can give the designer information on how to carry out process changes in the retrofitting of the 

existing network.  

In a recent paper, Jiang and Chang (2013) proposed an iterative strategy to solve the NLP 

model to enhance the flexibility index of the water network. Karuppiah and Grossmann (2008) 

considered the issues of the effects of uncertainties on the water network. The uncertainties in 

the problem include the amount of mass load and parameters of treatment units. Lagrangean 

decomposition for global optimization was used to solve the proposed model. In the process, 

the nonconvex problem was approximated by piecewise linear estimators, and tight lower 

bounds were thus attained by solving the relaxed problem.  

In the above literature, water reuse/recycle scenario is mostly considered. The synthesis of total 

water networks is rarely addressed because of complexity (Bagajewicz, 2000). Two factors 

mainly affect complexity. One is the bilinear terms in the mass balance equations in the form 

of water flowrate multiplying water concentration. The other is the cost terms in the objective 

function, which cannot ensure the solution is the global optimum.  

To eliminate nonconvex terms in mass balances, it is usually not allowed to mix streams before 

regeneration. Gabriel and El-Halwagi (2005) developed a simultaneous synthesis procedure 

for the design of material reuse and regeneration networks. The modeling of regeneration units 

was taken outside the optimization model, and several simplifying assumptions were proposed 

to transform the original model into a linear program. Nápoles-Rivera et al. ( 2012) extended 

the model to a multi-component case. The linear relationships obtained can ensure global 

optimality for the problem. Ahmetović and Grossmann (2011) developed a general 

superstructure for the synthesis of total water networks. Some additional connections were 

considered in their work, such as multiple sources with different quality, and hybrid operating 

units (mass transfer and non-mass transfer operations). Furthermore, pumping cost was also 

considered in the objective function, and tight bounds on the variables were added as 

constraints to solve the model. Boix et al. (2011) comprehensively considered the design of 

multi-contaminant water networks from the aspects of resource conservation, network 



Literature review                                                                         │Chapter 2 

24 

 

complexity, and economic feasibility. A multi-criteria decision-making procedure was 

provided for the practitioner to choose a proper network.  

Lovelady et al. (2009) presented a reverse model by combining the effluent of industrial 

processes with the watershed of the water ecosystem. The maximum allowable concentrations 

for the contaminants emitted from the industrial plants were determined based on the material 

flow analysis. These concentrations were incorporated into the water recovery model to 

generate the optimal design of a water network and to select a plant location. However, the 

limitation is that it does not optimize water recovery issues for different plants simultaneously. 

Lira-Barragán et al. (2011a) considered the location of a new plant based on an MINLP model 

by investigating the impacts of effluents on the surrounding watershed. The objective function 

is to minimize total annual cost, including wastewater treatment costs, and operating costs of 

the industrial plan, such as transportation of materials, land cost, and labor fees. However, 

wastewater recovery was not considered in the model. To overcome these limitations, Lira-

Barragán et al. (2011b) presented an integrated model combining water recovery models for 

the new plant with environmental modeling of the watershed system. A few alternative 

locations for the new plant were considered in their work. Compared with the previous work 

(Lira-Barragán et al., 2011a), the proposed formulation can obtain more reductions in the total 

cost due to the consideration of mass integration inside the plant.  

In the past, most papers focused on the development of complex models for water reuse and 

water regeneration for intra- and inter-plant water integration. The complexity of treatment 

processes and wastewater composition is ignored. Because the design of wastewater treatment 

plants not only needs to achieve the goal of contaminants removal but also is constrained by 

practical engineering constraints. Environmental impacts are also not considered from the 

perspective of the life cycle. Besides, the interactions between industrial plants, residence, and 

agriculture are not considered because in the past water shortage problem is not severe.  

In the work of Quaglia et al. (2014), a practical framework was presented to address water 

recovery issues in the industry. Wastewater engineering knowledge and optimization method 

were incorporated into a computer-aided environment to find the optimal solution. It makes the 

method more applicable to real industrial problems. Sotelo-Pichardo et al. (2014) tried to 

design a water network with the minimum cost from the life cycle of the project. Some 

uncertainties such as water demand and environmental regulations are considered.  



Literature review                                                                         │Chapter 2 

25 

 

García-Montoya et al. (2015) proposed a model to achieve water reuse in residence through 

water reuse, recycling, and regenerating, as well as the rainwater recovery activities. The 

method applies to the scenario where water demands are changing through different hours of 

the day and the situation in which water requirement of different seasons of the year are not 

equal. Rubio-Castro et al. (2016) proposed a multi-period MINLP model for water reuse in the 

agriculture sector. Although different scenarios and conditions were considered in the proposed 

optimization formulation, some important parameters, such as evapotranspiration, the 

precipitated water, were set at a fixed value or ignored, which can affect the results. 

2.5 Water integration techniques for batch processes 

The time dimension introduces an additional constraint when considering water recovery 

opportunities in batch plants. During water recovery activities, the constraints of water quality 

and quantity, as well as the time feasibility of these activities should be considered 

simultaneously. The water integration techniques for batch processes are similar to continuous 

processes. These approaches can also be classified as insight-based approaches and 

mathematical methods (Gouws et al., 2008). 

2.5.1 Insight-based techniques for batch processes 

Wang and Smith (1995) first extended the water pinch analysis developed for continuous 

processes to batch processes. The approach divides the water-using processes into many 

concentration intervals and time intervals based on the parameters of water-using operations. 

Next, time is treated as the primary constraint, while concentration feasibility is considered as 

a secondary constraint. In other words, for each concentration subinterval, water is reused from 

one time subinterval to the next, and the freshwater requirement for this concentration interval 

is determined. The sum of freshwater consumption across all the concentration intervals is the 

target for the batch process. However, in their work, streams are splitting so that water can 

enter the operation during the course of the process, which is not allowed in completely batch 

plants. Furthermore, it is assumed that the operations are fixed mass transfer units.  

To overcome these limitations, Majozi et al. (2006) reversed the priority of the constraints for 

water recovery activities. First, the sequence of constraints is the same as Wang and Smith 

(1995). Subsequently, time is treated as a primary constraint and concentration as a secondary 

constraint to investigate the effect of the priority on results. First sequence and cyclic operations 

were considered in their method. In the work of Chen and Lee (2008), a different representation 

of the targeting process was presented on a concentration vs. quantity diagram. This approach 



Literature review                                                                         │Chapter 2 

26 

 

can be applied to a hybrid system comprising various operations (fixed load and fixed flowrate) 

and operating models (completely batch and semi-continuous). Some useful concepts and 

principles were adopted to evolve the initial network and to ensure the network meets the 

practical constraints. Subsequently, this graphical approach was extended fixed flow rate 

problems. The freshwater consumption was identified in the first step, and then some 

constraints were used to reduce the network complexity by removing some connections (Chen 

and Lee, 2010). Kim (2011) addressed water reuse issues for discontinuous water systems. 

First, water upper and lower bound targets for each time interval were obtained based on the 

water composite curve of Wang and Smith (1994a). Next, the water network was designed to 

achieve the lower bound target within and beyond the particular time interval, and the 

requirements of the storage tank were then obtained when water reuse between different time 

intervals occurs. Chaturvedi (2017) proposed a graphical technique to perform energy 

management for batch water allocation network. However, in the proposed method, 

opportunities for water reuse were not considered. 

Foo et al. (2005) proposed a two-stage approach for the design of batch water networks. First, 

water cascade analysis (Manan et al., 2004) developed for continuous processes was extended 

to batch processes. For each time interval, water was cascaded from high-quality to low-quality 

streams. The sum of freshwater consumption for all the time intervals is the target of the 

problem. Next, the network was configured to achieve the targets in the first step. This method 

could be used to deal with water recovery issues for both mass transfer and non-mass transfer 

operations. A similar method was also proposed by Liu et al. (2007) to solve the water recovery 

problems for discontinuous or batch water systems. This method can obtain a different 

configuration with fewer storage tanks compared with Wang and Smith (1995). However, the 

network design and targeting process are performed simultaneously.  

Chaturvedi and Bandyopadhyay (2012) proved that in a single batch process, the target of 

freshwater consumed in a batch plant could be achieved through the sequential transfer of 

wastewater from one time interval to the next. For cyclic operations, the targeting process is 

the same as continuous processes by aggregating all the time intervals into a single interval. 

Chaturvedi and Bandyopadhyay (2013) addressed the targeting of multiple freshwater 

resources with different quality in batch processes. The concept of prioritized cost developed 

by Shenoy and Bandyopadh