Optimal Synthesis of Batch Water Networks Using Dynamic Programming
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Abstract
Water minimization in the process industry is becoming increasingly important due to increasingly stringent environmental legislation, especially for batch plants. This work proposes a dynamic programming (DP) method for the optimal design of water-using networks in batch plants. DP is a powerful framework for dealing with a large spectrum of multistage decision-making problems and has been applied in numerous chemical engineering problems. The proposed methodology is explained as follows. Firstly, based on the start time and end time of each operation, the whole process is divided into N stages. Secondly, the water requirement of water-using units in each stage is satisfied and the state of stored water and wastewater generation is determined. The backward procedure of DP is used to solve the DP problems. The target of freshwater consumption of the process and the optimal design of the water network are obtained simultaneously. In order to display the versatility of the proposed approach, four examples from literature are considered. Example 1 is a completely batch process with a fixed flowrate problem. Example 2 is a hybrid batch water system comprising various modes of operations and operating patterns. Example 3 is a fixed-mass load problem with a regeneration unit, while example 4 considers the batch water network design with multiple contaminants. The results obtained in this work were comparable with the results from literature, implying that it can be applicable to both mass transfer-based and non-mass transfer-based batch water networks.
Keywords
Dynamic programming Water integration Network design Batch processesIntroduction
Conservation and improvements in water resource management have been a global challenge (Bagatin et al. 2014). Water cascading utilization and approaches that aim to further increase water efficiency can be considered guiding principles for sustainable water resource management. The stringent regulation of environmental emissions has been driving the industry and academia to find ways of minimizing freshwater consumption and wastewater generation. Reduction of freshwater consumption could result in higher profitability and less adverse impacts on the environment. Although continuous processes have the advantage of manufacturing on a large scale in chemical industry, numerous chemical products still retain batch processing as their primary method of manufacture, such as pharmaceutical, agricultural, and food products. Batch processes have been widely used in the chemical industry due to their suitability for the production of small volume, high value-added products, as well as their capability of adjusting to rapid market changes. Therefore, it would be favorable to develop a systematic approach to design optimal water networks for batch processes.
Unlike continuous processes, batch operations do not only have to obey the concentration constraints, but also consider time feasibility issues. For example, wastewater generated by a process can be recovered if it obeys the inlet concentration constraint of the receiving unit and if the receiving unit is operated during or after the wastewater is generated. In the past decades, much effort has been placed on the optimal design of water-using networks in batch processes. An overview of the developments and methodologies proposed for batch water networks was presented by Gouws et al. (2010). These methodologies can roughly be divided into insight-based and mathematical techniques. Insight-based techniques for batch water networks share their roots with their continuous counterparts. However, apart from concentration constraints, time constraints should be taken into account during the targeting of batch processes. Hence, batch processes tend to be more complex because of the existence of scheduling issues.
Mathematical programming techniques offer a general modeling framework in the synthesis, optimization, and planning of batch chemical processes (Majozi 2010). Two main approaches arise when considering water integration for batch processes. One approach focuses on the minimization of water within a predefined schedule where timing of operations is stipulated a priori. This particular approach bears many similarities to that of insight-based techniques for batch processes. The other approach involves minimization of water where the start and finishing times are not known beforehand. In this approach, only duration is specified and the optimal schedule that achieves the minimum water targets is determined (Gouws et al. 2010).
The earliest contribution, such as Almató et al. (1997) and Kim and Smith (2004), utilized storage tanks to override the time gap between the finishing time of discharging task and starting time of the receiving task. In both these contributions, each storage tank is dedicated to a single reuse opportunity, resulting in a higher capital cost. Majozi (2005) circumvented the use of central storage tank to exploit opportunities of water reuse/recycle. In the subsequent work, a two-stage approach is proposed to minimize the freshwater consumption and the capacity of reusable water storage tank (Majozi 2006). In addition to minimization of freshwater consumption and storage capacity, Chen et al. (2008) considered minimizing the number of connections by formulating a mixed integer nonlinear (MINLP) program. The results from these papers indicate that freshwater can be substantially reduced through water reuse/recycle. However, wastewater treatment has not been adequately considered in water integration of batch processes. Cheng and Chang (2007) developed a general MINLP model to synthesize water networks in batch processes by optimizing batch schedules, water-reuse subsystems, and wastewater treatment subsystems simultaneously. In practice, there exist processes with a mixture of continuous and batch operations, for example, breweries, sugar mills, and tire-production plants. Lee et al. (2013) considered the synthesis of water networks for systems consisting of truly batch, semi-continuous, and continuous units. Furthermore, in their subsequent work, inter-plant water network was synthesized in process units operated in mixed continuous and batch modes (Lee et al. 2014).
There are other contributions that are worthy of mention in water minimization in batch plants using a flexible schedule. For example, Adekola and Majozi (2011) addressed the problem of simultaneous production scheduling and wastewater minimization by employing a wastewater regenerator to further reduce freshwater consumption. Similarly, Chen et al. (2011) proposed a mathematical model for simultaneous scheduling and water minimization in multipurpose batch plants. However, in their work, the scheduling framework is based on the Resource-Task Network representation. Chaturvedi and Bandyopadhyay (2014) developed a multiple objective formulation to simultaneously target the minimization of fresh water requirement and the maximization of production in a batch process. Lee and Foo (2017) proposed an integrated technique of simultaneous targeting and scheduling in batch-processing plants by combining pinch-based automated targeting model and production scheduling model based on state-task network representation.
The seminal work of insight-based approaches for the optimal design of batch water networks was proposed by Wang and Smith (1995). This approach treated time as the primary constraint and concentration feasibility as a secondary constraint. However, the authors only considered situations where batch processes consume or produce water continuously during their operation in a semi-continuous manner. Based on this observation, Majozi et al. (2006) proposed a graphical technique for wastewater minimization in completely batch operations. Firstly, it took the time dimension as the primary constraint and concentration as the secondary constraint. Next, the priority of constraints was reversed so as to demonstrate the effect of the targeting procedure of the proposed approach on the final design. Chen and Lee (2008) presented a graphical technique to deal with a hybrid batch water system comprising fixed-mass load and fixed flowrate problems. They also considered completely batch and semi-continuous operations. Kim (2011) proposed a two-stage graphical approach to minimize freshwater consumption and wastewater generation for discontinuous water systems. In the first step, the lower and upper bound targeting of the water systems are obtained by pinch analysis. In the second step, the optimal water network is designed to achieve the lower bound target. Chaturvedi et al. (2016) provided a simplified conceptual approach of dealing with scheduling problems of a batch water network with multiple water resources and analyzed the effect of multiple water resources on a batch water network schedule. Furthermore, some algebraic methods have been developed to perform water reuse/recycle for batch process. Foo et al. (2005) developed a time-dependent water cascade analysis for the synthesis of a maximum water recovery network for a batch process. Similarly, a time-dependent concentration interval analysis method was proposed by Liu et al. (2007) to solve the problems associated with synthesis of discontinuous or batch water-using systems involving both non-mass transfer-based and mass transfer-based operations. Foo et al. (2012) proposed a systematic procedure to perform targeting and design of a batch resource conservation network involving material regeneration and waste treatment. A new formulation was proposed by Oliver et al. (2008), which combines water pinch technology with mathematical modeling to design the water network for batch processes. The formulation is capable of satisfying the minimum water target while determining the minimum number of water storage tanks. Foo (2010) proposed an automated targeting technique to determine the minimum resource and waste targets for batch process integration problems. Although the technique is formulated as a mathematical optimization model, the fact that it is built on the insight-based pinch analysis technique enables the minimum resource targets to be identified prior to detailed design. Li et al. (2013) developed some simple design heuristics to manually generate batch water networks in systems with multiple contaminants.
In the aforementioned methods, the design of batch water networks is divided into many intervals based on the starting and finishing time of each operation. In each interval, the designer would identify freshwater consumption and the quantity of reuse water from earlier processes. It can be formulating multistage decision processes. Dynamic programming (DP) is a powerful formal framework for dealing with a large spectrum of multistage decision-making problems (Bellman 1957). In fact, DP provides a decomposition strategy for large optimization problems. It has been applied in numerous multistage decision processes, for example, the allocation of feed in a multi-reactor system, equipment replacement, tray-by-tray calculations in distillation and absorption as well as multiple flash tank separation (Roberts 1964). El-Halwagi et al. (2003) employed DP techniques to derive the mathematical conditions and characteristics of an optimal solution strategy for material reuse/recycle networks. In the work of Diban et al. (2016), DP was used to obtain an optimum replanting policy that achieves minimum carbon emissions over a finite time horizon for commercial agriculture plantations.
In this work, DP is used to design a water network for batch processes. Although, the procedure is similar to the stepwise method of water network design. The difference is that in every stage, the optimal solution to the decomposed problem is chosen to form the overall solution. When transforming from a stage to the next stage, the overall solution is not optimal, it will return back to the previous stage to choose another solution to ensure finding the globally optimal solution for the original problem. The rest of the paper is organized as follows. Section “Problem Statement” describes the problem statement of this work. A description of the methodology employed in this work is explained in Section “Methodology”. Section “Illustrative examples” presents case studies reproduced from the literature. Finally, Section “Conclusions” concludes the paper and suggests prospects for further research.
Problem Statement
The problem addressed in this paper is formally stated as follows.
Given:
- (i).
the start and end time of each process,
- (ii).
the limiting water quantity and quality of each process,
- (iii).
the maximum storage capacity for each material,
- (iv).
the time horizon of interest, and
- (v).
general information of regeneration, e.g., fixed outlet concentration or remove ratio.
Determine the optimum water network that achieves minimum freshwater consumption and minimum storage capacity.
Methodology
Dynamic programming (DP) is generally used to reduce a complex problem with many variables into a series of optimization problems with one variable. Thus, the decomposed problems are comparatively easy to solve. In the analysis, DP is characterized fundamentally in terms of stages and states. A difficult problem can be divided into a series of consecutive stages. Each stage constitutes a new problem to be solved in order to find the optimal result. In each stage, the system can be described or characterized by a relatively small set of parameters called the state variables. The states correspond to the alternative decisions which could be made in this stage and those will often be a range of possible values for control variables.
Equation (1) represents the objection function of the problem when it is located in kth stage. In every stage, the designer has two options for satisfying the water demand of an operation, i.e., the freshwater scenario, reuse/recycle scenario. For the freshwater scenario, the water requirement of a sink is satisfied with freshwater in addition to any reused, recycled, and regenerated water. In this scenario, freshwater must be used to dilute wastewater in order to meet the water requirement. However, for the reuse/recycle scenario, no freshwater is required to satisfy the water demand. In other words, the available reusable wastewater or regenerated water is capable of achieving the target. Equation (2) is the objective function of the origin stage, setting as zero. It does not have physical meaning.
- (1)
When reuse water and regenerated water are available at the same time, reuse water should be used preferentially;
- (2)
In order to reduce the cost of stored water, if a water sink has a few alternative water sources, the water source which is generated latest should be reused preferentially;
- (3)
If the process chooses a freshwater scenario which requires some reuse water, the reuse water with the highest quality, i.e., lowest concentration, should be used preferentially, in as great a quantity as possible;
- (4)
If a process is fed solely by wastewater, i.e., no freshwater needed, wastewater with the contaminant concentration closest to the maximum inlet concentration of the process should be used preferentially, in as great a quantity as possible;
- (5)
If the water sink has a few available water sources with same concentration, the alternative water source with sufficient amount of reuse water should be used preferentially.
Fixed Flowrate Problems
Fixed-Mass Load Problems
For simplicity, in this work, only the states of freshwater and stored water are displayed in each stage. Once the freshwater consumption of each process is determined, the quantities of reused water and wastewater, as well the maximum capacities of storage tanks can be identified. The problem was solved using the backward procedure of DP (Roberts 1964). In the following section, four examples adopted from the literature are explored in detail using the above methodology.
Illustrative Examples
To illustrate the application of proposed approach to network design in batch processes, four examples extracted from literature are investigated. Example 1 is composed of fixed quantity operations, while example 2 is a hybrid system that includes different types of water-using operations. Examples 3 and 4 are fixed-mass load operations. Example 3 considers water network design with a regeneration scheme and example 4 focuses on water network design with multiple contaminants.
Example 1
Limiting data for example 1
Process | Number | Sinks | Time (h) | c^{max, in} (kg salt/kg water) | Water (kg) | Stage |
A wash | 1 | SK_{1} | 0 | 0 | 1000 | 7 |
B reaction | 2 | SK_{2} | 0 | 0.25 | 280 | 7 |
B wash | 3 | SK_{3} | 4 | 0.1 | 400 | 4 |
C reaction | 4 | SK_{4} | 2 | 0.25 | 280 | 6 |
C wash | 5 | SK_{5} | 6 | 0.1 | 400 | 2 |
Process | Number | Sources | Time (h) | c^{max, out} (kg salt/kg water) | Water (kg) | Stage |
A wash | 1 | SR_{1} | 3 | 0.1 | 1000 | 5 |
B reaction | 2 | SR_{2} | 4 | 0.51 | 280 | 4 |
B wash | 3 | SR_{3} | 5.5 | 0.1 | 400 | 3 |
C reaction | 4 | SR_{4} | 6 | 0.51 | 280 | 2 |
C wash | 5 | SR_{5} | 7.5 | 0.1 | 400 | 1 |
In the synthesis of batch water networks for completely batch operations, first sequence and cyclic-state targeting are essential. The first sequence targeting focuses on a single batch over a relatively short time horizon of interest. The cyclic-state targeting occurs over an extended time horizon when more than one batch has to be produced in multistage operations. In the following subsection, the first sequence and cyclic-state scenarios are considered.
First Sequence Scenario
When the freshwater consumption of all the stages has been determined, the final quantities of stored water and generated wastewater can be calculated from the water balances of each operation. Furthermore, the maximum capacity of the storage vessel can also be determined. Hence, for simplicity, only the states where freshwater is consumed and initial quantity of stored water are displayed in this work. In every stage, the freshwater and reused water, for freshwater scenario, can be determined from Eqs. (3) and (4). For the reuse/recycle scenario, the quantity of reused water is determined by Eqs. (5) and (6). The detailed processes are described as follows.
Stage 7
Stage 6
Stage 5
Stage 4
Stage 3
Stage 2
Stage 1
Cyclic Operation Scenario
In industry, more than one batch is operated for certain operation in order to achieve the required capacity or meet a required product demand. Therefore, consecutive batches are repeated with the same schedule in an extended time horizon. This kind of operating mode is termed cyclic, which is the equivalent of steady state in continuous operations. In cyclic batch operations, processes can use surplus water from previous batches through the use of storage tanks. This means that later water sources in previous batches can be used to supply the earlier water sinks in later batches.
Based on the foregoing analysis of the single batch case, insight on the potential reuse of water between water sinks and sources can be obtained. For example, 600 kg of high-quality water from SR_{1} at 0.1 kg salt/kg water in stage 5 and 400 kg of the same quality water from SR_{5} in stage 1 are discharged as wastewater in the first sequence scenario. These two water sources could instead be stored in a tank for reuse in the next batch case. In order to reduce the freshwater consumption as much as possible, water from these two water sources should be sent to storage tank for future reuse in possible sinks. Hence, the maximum amount of stored water in tank 1 (\( {F}_{\mathrm{Tank}1}^{St} \)) is 1000 kg, i.e., 600 kg from SR_{1} and 400 kg from SR_{5}. It should be noted that only the stages which are different from the first sequence are presented in detail and the repeated stages are omitted.
Stage 7
Stage 6
Example 2
Problem specification for example 2
Sinks | Time (h) | C^{max,in} (kg salt/kg water) | Water (t) | Sources | Time (h) | C^{max,out} (kg salt/kg water) | Water (t) |
---|---|---|---|---|---|---|---|
SK_{1} | 0.0–1.0 | 0.0 | 20 | SR_{1} | 4.0–5.0 | 0.2 | 20 |
SK_{2} | 0.0–0.5 | 0.25 | 16 | SR_{2} | 4.5–5.0 | 0.5 | 16 |
SK_{3} | 5.0–6.5 | 0.1 | 15 | SR_{3} | 5.0–6.5 | 0.1 | 15 |
SK_{4} | 2.0–2.5 | 0.25 | 24 | SR_{4} | 6.5–7.0 | 0.4 | 24 |
SK_{5} | 7.0–8.5 | 0.1 | 15 | SR_{5} | 7.0–8.5 | 0.12 | 15 |
First Sequence Scenario
Cyclic-State Scenario
Example 3
In the previous examples, the problems are mainly based on fixed flowrate assumption. Although in example 2, some operations are fixed-mass load problems, it is not enough to prove the proposed method can be applied to fixed-mass load problems. In this subsection, a literature example adopted from Liu et al. (2009) with fixed-mass load problem is illustrated to explain the method. Two scenarios are taken into account, i.e., reuse scenario with central buffer tank and water reuse scenario with central buffer tank and regeneration.
Water Reuse Scenario with Central Buffer Tank
Limiting water data for the example 3
Operation | Quantity/t | C^{in}/(μg/g) | C^{out}/(μg/g) | T_{in}/h | T_{out}/h | M/kg |
---|---|---|---|---|---|---|
A | 50 | 0 | 400 | 0 | 2 | 20 |
B | 30 | 100 | 400 | 3 | 4 | 9 |
C | 10 | 200 | 500 | 3 | 6 | 3 |
D | 24 | 350 | 600 | 3.5 | 7.5 | 6 |
E | 40 | 450 | 700 | 6 | 8.5 | 10 |
The total freshwater consumption is identified as 80.5 t, indicating that the configuration of water reuse between different operations is also determined. Because operations A and B have the same outlet concentration and wastewater from these two operations has a high quality, it is assumed that all the water discharged is sent to the tank during the targeting process. When the freshwater target is determined, there is surplus wastewater stored in the tank. Hence, it is necessary to check the amount of water sent to the tank in order to employ a minimum capacity of tank.
Specifications of example 3 with central buffer tank
Operation | Quantity/t | C^{in}/(μg/g) | C^{out}/(μg/g) | M (kg) |
---|---|---|---|---|
A | 50 | 0 | 400 | 20 |
B | 22.5 | 0 | 400 | 9 |
C | 10 | 200 | 500 | 3 |
D | 24 | 350 | 600 | 6 |
E | 36.67 | 427.27 | 700 | 10 |
Water Reuse Scenario with Central Buffer Tank and Regeneration
Specifications of example 3 with a wastewater regeneration unit
Operation | Quantity/t | C^{in}/(μg/g) | C^{out}/(μg/g) | M (kg) |
---|---|---|---|---|
A | 50 | 0 | 400 | 20 |
B | 25.469 | 46.63 | 400 | 9 |
C | 10 | 200 | 500 | 3 |
D | 21.094 | 315.55 | 600 | 6 |
E | 16.667 | 100 | 700 | 10 |
Example 4
Limiting data for example 4
Operation | Contaminant | C^{max, in} (ppm) | C^{max, out} (ppm) | Mass load (kg) | Water (t) | Start time (h) | End time (h) |
---|---|---|---|---|---|---|---|
1 | C1 | 0 | 20 | 4 | 200 | 0 | 0.5 |
C2 | 0 | 400 | 80 | ||||
C3 | 0 | 50 | 10 | ||||
2 | C1 | 50 | 100 | 15 | 300 | 1 | 2 |
C2 | 200 | 1000 | 240 | ||||
C3 | 50 | 12,000 | 3585 | ||||
3 | C1 | 10 | 200 | 28.5 | 150 | 2 | 3.5 |
C2 | 50 | 100 | 7.5 | ||||
C3 | 300 | 1200 | 135 | ||||
4 | C1 | 30 | 75 | 9 | 200 | 1 | 2 |
C2 | 100 | 200 | 20 | ||||
C3 | 200 | 1000 | 160 | ||||
5 | C1 | 150 | 300 | 15 | 100 | 4 | 4.5 |
C2 | 200 | 1000 | 80 | ||||
C3 | 350 | 1200 | 85 | ||||
6 | C1 | 0 | 150 | 22.5 | 150 | 5.5 | 6.5 |
C2 | 0 | 300 | 45 | ||||
C3 | 50 | 2500 | 367.5 | ||||
7 | C1 | 100 | 200 | 5 | 50 | 8 | 10 |
C2 | 150 | 1500 | 67.5 | ||||
C3 | 220 | 1000 | 39 |
Specific concentration of each operation for example 4 with two tanks
Operation | Contaminant | C^{in} /(ppm) | C^{out} /(ppm) | Mass load/(kg) | Water (t) |
---|---|---|---|---|---|
1 | C1 | 0 | 20 | 4 | 200 |
C2 | 0 | 400 | 80 | ||
C3 | 0 | 50 | 10 | ||
2 | C1 | 10 | 60 | 15 | 299.37 |
C2 | 198 | 1000 | 240 | ||
C3 | 25 | 12,000 | 3585 | ||
3 | C1 | 10 | 200 | 28.5 | 150 |
C2 | 33.6 | 83.6 | 7.5 | ||
C3 | 168 | 1068 | 135 | ||
4 | C1 | 4 | 59.52 | 9 | 161.54 |
C2 | 76 | 200 | 20 | ||
C3 | 10 | 1000 | 160 | ||
5 | C1 | 21 | 181 | 15 | 93.58 |
C2 | 145 | 1000 | 80 | ||
C3 | 292 | 1200 | 85 | ||
6 | C1 | 0 | 150 | 22.5 | 150 |
C2 | 0 | 300 | 45 | ||
C3 | 0 | 2450 | 367.5 | ||
7 | C1 | 9 | 118 | 5 | 45.92 |
C2 | 30 | 1500 | 67.5 | ||
C3 | 151 | 1000 | 39 |
However, the water network obtained in the scenario with two tanks could be evolved to have one storage tank. As shown in Fig. 19, wastewater generated from operation 1 at 0.5 h is sent to tank for future reuse. An amount of 179.19 t wastewater is then reused by operations 2 and 4 at 1 h and the remaining, i.e., 20.81 t, is reused at 4 h. The tank is empty from 4 h to the end time horizon of interest. It is possible to merge these two tanks into a single tank.
Conclusions
A DP approach is used to identify the optimal water-using policy that will achieve minimum freshwater consumption and wastewater generation, as well as the minimum capacity of storage facility. Four examples from literature are used to demonstrate the applicability of the proposed approach. The results obtained in this work match well with literature (Chen and Lee 2008; Kim and Smith 2004; Majozi 2010; Majozi et al. 2006). Furthermore, the capacity of storage facility obtained in this work is less than that in published literature. The advantage of DP approach is used to decompose a complex problem with many variables into a series of problems comparatively easy to solve. For example, in example 4, it considers water integration of batch processes with multiple contaminants. It is difficult to determine the globally optimal results using mathematical programming. This approach is simple and straightforward for the design of batch water networks. Future work should focus on exploring opportunities of water reuse/recycle and/or regeneration in batch processes with flexible product scheduling. Furthermore, it could be used to design batch water network design with multiple contaminants in the framework of flexible scheduling. Also, other types of regeneration unit should be employed to purify the wastewater for further reuse.
Notes
Nomenclature
k the kth stage
I_{k} a set of available water sources in stage k
J_{k} a set of available water sources in stage k
m the number of operations
L a set of contaminants l
ts_{m} the start time of operation m
te_{m} the end time of operation m
\( {F}_{j,k}^{Fw} \) the quantity of freshwater consumed by water sink j in stage k
\( {F}_{Tanki}^{St} \) the quantity of water stored in tank i
\( {F}_{i,j,k}^d \) the quantity of directly reused water from water source i to water sink j in stage k
\( {F}_{i,j,k}^{ind} \) the quantity of indirectly reused water from water source i to water sink j in stage k
\( {F}_{i,k}^{Ww} \) the quantity of wastewater from water source i in stage k
\( {F}_{j,k}^{\mathrm{Reg}} \) the quantity of regenerated water for water sink j in stage k
\( {F}_{i,k}^{st} \) the quantity of stored water from water source i in stage k
F_{j, k} the quantity of water for sink j in stage k
F_{i, k} the quantity of water for source i in stage k
\( {c}_{i,j,k}^{d,l} \) the concentration of contaminant l direct reuse water from water source i to water sink j in stage k
c_{i, j, kind, l} the concentration of contaminant l of indirect reuse water from water source i to water sink j in stage k
\( {c}_{j,\mathrm{k}}^{\max, in,l} \) the maximum inlet concentration of contaminant l of water sink j in stage k
\( {c}_{j,\mathrm{k}}^{\max, \mathrm{out},l} \) the maximum inlet concentration of contaminant l of water sink j in stage k
\( {c}_{j,\mathrm{k}}^{in,l} \) inlet concentration of contaminant l of water sink j in stage k
\( {c}_{Fw}^l \) the concentration of contaminant l of freshwater
\( {c}_{\operatorname{Re}g}^l \) the concentration of contaminant l of regenerated water
\( {M}_{j,k}^l \) the mass load of contaminant l of water sink in stage k
Funding Information
The authors thank the National Research Foundation (NRF) of South Africa for funding this work under the NRF/DST Chair in Sustainable Process Engineering at the University of the Witwatersrand, Johannesburg.
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