Advertisement

Optimal Synthesis of Batch Water Networks Using Dynamic Programming

  • Zhiwei Li
  • Thokozani Majozi
Original Research Paper
  • 79 Downloads

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 processes 

Introduction

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:

A set of batch processes m with specific process data, including
  1. (i).

    the start and end time of each process,

     
  2. (ii).

    the limiting water quantity and quality of each process,

     
  3. (iii).

    the maximum storage capacity for each material,

     
  4. (iv).

    the time horizon of interest, and

     
  5. (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.

In this work, the states for every stage are freshwater consumption, the quantity of stored water, and wastewater generated. In each stage, a decision could be made on whether to use freshwater and/or reuse wastewater or regenerated water. No matter what stage the system is in, one or more decisions must be made in order for the analysis to proceed to the next stage. These decisions may depend on either precious stage or state or both. When a decision is made, a return cost or reward is obtained and the system undergoes a transformation to the next stage. The return cost is determined by a known single-valued function of the input state. In this work, the return cost mentioned in the following sections refers to the freshwater consumption. The entire process can be divided into N stages (1, 2, …, k, …, N) based on the start and end time of the operations. The relationship between stage and time is displayed in Fig. 1, where tsm and tem represent the start time and end time of operation m. It is noted that the problem was solved using the backward procedure of DP (Roberts 1964). The stages are thus presented in reverse order.
Fig. 1

The relationship between time and stage

Suppose that in kth stage there are Ik water sources and Jk water sinks, resulting in M(k) possible states which are numbered 1, 2, …., M(k). In other words, there are M(k) possibilities pertaining to the allocation of freshwater to water sinks in kth stage. The potential states in each stage are represented by the circles, as shown in Fig. 2. Let \( {F}_{j,k}^{Fw} \) denote the allocation state of freshwater to sink j in kth stage of the process. In Fig. 2, \( {F}_{j,k}^{Fw} \) is represented by the arc, indicating the chosen path from the state of previous stage to the state of next stage. If \( {F}_{j,k}^{Fw} \) is the state chosen in the kth stage, a solution to the whole problem may be denoted by the sequence \( \left({F}_{j,1}^{Fw},{F}_{j,2}^{Fw},\dots, {F}_{j,k}^{Fw},\dots, {F}_{j,N}^{Fw}\right) \). \( f\left({F}_{j,1}^{Fw},{F}_{j,2}^{Fw},\dots, {F}_{j,k}^{Fw},\dots, {F}_{j,N}^{Fw}\right) \) denotes the return cost associated with the sequence \( \left({F}_{j,1}^{Fw},{F}_{j,2}^{Fw},\dots, {F}_{j,k}^{Fw},\dots, {F}_{j,N}^{Fw}\right) \). The objective is to minimize \( f\left({F}_{j,1}^{Fw},{F}_{j,2}^{Fw},\dots, {F}_{j,k}^{Fw},\dots, {F}_{j,N}^{Fw}\right) \) subject to any constraints which are placed on the choice of the states in each stage, and these constraints may depend on the states entered in stages 1, 2, …, N-1. In this work, the return cost function is set as the minimum freshwater consumption over the kth stage, which equals the minimum freshwater consumption of all the water sinks in the kth stage (\( \sum \limits_{j=1}^{J_k}{F}_{j,k}^{Fw} \)) and return cost results (\( {f}_{k-1}\left({F}_{j,k-1}^{Fw}\right) \)) from the stage (k-1). This work aims to find the path with the minimum result from stage N to stage 1.
Fig. 2

Potential states in each stage

The objective of the problem is defined as:
$$ {f}_k\left({F}_{j,k}^{Fw}\right)=\mathrm{minimum}\ \mathrm{freshwater}\ \mathrm{consumed}\ \mathrm{over}\ \mathrm{the}\ k\ \mathrm{stages} $$
where,
$$ {f}_k\left({F}_{j,k}^{Fw}\right)=\left\{\begin{array}{c}\mathrm{Freshwater}:\kern1.00em \begin{array}{cc}& \end{array}\sum \limits_{j=1}^{J_k}{F}_{j,k}^{Fw}+{f}_{k-1}\left({F}_{j,k-1}^{Fw}\right)\\ {}\mathrm{Reuse}/\mathrm{recycle}:\begin{array}{cccc}\begin{array}{cccc}& & & \end{array}& & & \end{array}{f}_{k-1}\left({F}_{j,k-1}^{Fw}\right)\end{array}\right\},k=1,2,3,\dots .N $$
(1)
$$ {f}_0\left({F}_{j,0}^{Fw}\right)=0 $$
(2)

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.

When determining water target in each stage, two main categories of water network synthesis exist, i.e., fixed flowrate and fixed-mass load problems. For different types of problems, the resulting freshwater consumption may be different. Because the number of variables presented in the constraint equations is more than the number of equations, it is difficult to determine a sole solution of these equations for the system. Therefore, when attempting to identify the minimum freshwater consumption, a set of rules is generated in order to appropriately select from the available water resource and achieve the optimal design for water-using network.
  1. (1)

    When reuse water and regenerated water are available at the same time, reuse water should be used preferentially;

     
  2. (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. (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. (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. (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

For the freshwater scenario, the water requirements of sinks are satisfied with freshwater, reusable wastewater, and/or regenerated water. The freshwater consumption in the kth stage can be obtained by Eqs. (3) and (4). The priority of water utilization is based on the aforementioned rules. The procedure is explained as follows: Firstly, the process is to use a mixture of reuse water and freshwater to meet the water requirement. If the quantity of reuse water in the same stage is insufficient, the deficit water is supplied with regenerated water. After all the states of freshwater consumption are determined, the minimum freshwater consumption is chosen as the optimal state in this stage.
$$ \sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind}+{F}_{j,k}^{\operatorname{Re}g}+{F}_{j,k}^{Fw}={F}_{j,k} $$
(3)
$$ {\displaystyle \begin{array}{c}\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d{c}_{i,j,k}^{d,l}+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind}{c}_{i,j,k}^{ind,l}+{F}_{j,k}^{\mathrm{Reg}}{c}_{\operatorname{Re}g}^l+{F}_{j,k}^{Fw}{c}_{Fw}^l={F}_{j,k}{c}_{j,k}^{\max, in,l}\\ {}\forall i\in {I}_k,\forall j\in {J}_k,\forall l\in L,k=1,2,\dots, N\end{array}} $$
(4)
For the reuse/recycle scenario, the specific water requirement for water sink j in stage k can be obtained by Eqs. (5) and (6). In the equations, \( {F}_{i,j,k}^d \) and \( {F}_{i,j,k}^{ind} \) represent the direct reuse water and indirect reuse water, respectively. Based on the aforementioned rule (2), direct reuse water should be selected preferentially. If the quantity of direct reuse water is not enough to meet the water demand, the deficit water is supplemented by indirect reuse water. Furthermore, if the quantity of indirect reuse water is still deficient or not available, the remaining water is supplied by regenerated water.
$$ \sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind}+{F}_{j,k}^{\mathrm{Reg}}={F}_{j,k} $$
(5)
$$ {\displaystyle \begin{array}{c}\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d{c}_{i,j,k}^{d,l}+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind}{c}_{i,j,k}^{ind,l}+{F}_{j,k}^{\mathrm{Reg}}{c}_{\operatorname{Re}g}^l={F}_{j,k}{c}_{j,k}^{\max, in,l}\\ {}\forall i\in {I}_k,\forall j\in {J}_k,\forall l\in L,k=1,2,\dots, N\end{array}} $$
(6)

Fixed-Mass Load Problems

For fixed-mass load problems, the supplied water quantity can be less than the limiting water requirement and the inlet concentration can also be less than the maximum allowable inlet concentration. In order to obtain an optimal result of freshwater consumption, it is prudent to use the nonlinear programming (NLP) to determine the results of freshwater consumption and reuse water from previous processes. The NLP model is explained as follows: The objective function is to minimize the freshwater consumption, as shown in Eq. (7). Equation (8) states that the removal of mass load must be greater than or equal to the limiting mass load if the outlet concentration reaches its limiting data. Equation (9) represents the calculation of inlet concentration. Equation (10) ensures that the actual removal of mass load is equal to limiting mass load. The quantity constraint is expressed in Eq. (11). Equation (12) indicates that the actual inlet and outlet concentration should be limited by the limiting data.
$$ \min {F}_{j,k}^{Fw} $$
(7)
$$ \left(\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{in d}+{F}_{j,k}^{Fw}+{F}_{j,k}^{\mathrm{Reg}}\right)\left({c}_{j,k}^{\max, out,l}-{c}_{j,k}^{in,l}\right)\ge {M}_{j,k}^l $$
(8)
$$ {c}_{j,k}^{in,l}=\frac{\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d{c}_{i,j,k}^{d,l}+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{in d}{c}_{i,j,k}^{in d,l}+{F}_{j,k}^{\mathrm{Reg}}{c}_{\operatorname{Re}g}^l+{F}_{j,k}^{Fw}{c}_{Fw}^l}{\sum \limits_{i=1}^I{F}_{i,j,k}^d+\sum \limits_{i=1}^I{F}_{i,j,k}^{in d}+{F}_{j,k}^{\mathrm{Reg}}+{F}_{j,k}^{Fw}} $$
(9)
$$ \left(\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{in d}+{F}_{j,k}^{Fw}+{F}_{j,k}^{\operatorname{Re}g}\right)\left({c}_{j,k}^{out,l}-{c}_{j,k}^{in,l}\right)={M}_{j,k}^l $$
(10)
$$ \sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind}+{F}_{j,k}^{Fw}+{F}_{j,k}^{\operatorname{Re}g}\le {F}_{j,k} $$
(11)
$$ {\displaystyle \begin{array}{c}{c}_{j,k}^{in,l}\le {c}_{j,k}^{\max, in,l},{c}_{j,k}^{out,l}\le {c}_{j,k}^{\max, out,l}\\ {}\forall i\in {I}_k,\forall j\in {J}_k,\forall l\in L,k=1,2,\dots, N\end{array}} $$
(12)
For the reuse/recycle scenario, water sinks are only fed by the available wastewater from storage tanks, or wastewater which is produced in the same stage, or regenerated water. The NLP model is expressed by Eqs. (13)(18). The difference between this scenario and the freshwater scenario is that the freshwater term is removed in all of the constraints. The objective function expresses the minimization of the sum of reuse water and the regenerated water. It applies individually to each stage.
$$ \min \sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind} $$
(13)
$$ \left(\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{in d}+{F}_{j,k}^{\mathrm{Reg}}\right)\left({c}_{j,k}^{\max, out,l}-{c}_{j,k}^{in,l}\right)\ge {M}_{j,k}^l $$
(14)
$$ {c}_{j,k}^{in,l}=\frac{\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d{c}_{i,j,k}^{d,l}+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{in d}{c}_{i,j,k}^{in d,l}+{F}_{j,k}^{\mathrm{Reg}}{c}_{\operatorname{Re}g}^l}{\sum \limits_{i=1}^I{F}_{i,j,k}^d+\sum \limits_{i=1}^I{F}_{i,j,k}^{in d}+{F}_{j,k}^{\mathrm{Reg}}} $$
(15)
$$ \left(\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{in d}+{F}_{j,k}^{\operatorname{Re}g}\right)\left({c}_{j,k}^{out,l}-{c}_{j,k}^{in,l}\right)={M}_{j,k}^l $$
(16)
$$ \sum \limits_{i=1}^{I_k}{F}_{i,j,k}^d+\sum \limits_{i=1}^{I_k}{F}_{i,j,k}^{ind}+{F}_{j,k}^{\operatorname{Re}g}\le {F}_{j,k} $$
(17)
$$ {\displaystyle \begin{array}{c}{c}_{j,k}^{in,l}\le {c}_{j,k}^{\max, in,l},{c}_{j,k}^{out,l}\le {c}_{j,k}^{\max, out,l}\\ {}\forall i\in {I}_k,\forall j\in {J}_k,\forall l\in L,k=1,2,\dots, N\end{array}} $$
(18)

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

This example is taken from Majozi et al. (2006). In their work, all operations consume water only at the beginning of operation and generate water only at the end, i.e., truly batch processes are considered. Additionally, water intake and discharge are assumed to take place instantly. The operations are based on a fixed water requirement. The start and end times of each operation are known a priori. In order to explain the method clearly, in each stage if an operation requires water, it is termed a water sink. If an operation generates water which becomes available for later reuse, it is termed a water source. The limiting data are summarized in Table 1.
Table 1

Limiting data for example 1

Process

Number

Sinks

Time (h)

cmax, in (kg salt/kg water)

Water (kg)

Stage

A wash

1

SK1

0

0

1000

7

B reaction

2

SK2

0

0.25

280

7

B wash

3

SK3

4

0.1

400

4

C reaction

4

SK4

2

0.25

280

6

C wash

5

SK5

6

0.1

400

2

Process

Number

Sources

Time (h)

cmax, out (kg salt/kg water)

Water (kg)

Stage

A wash

1

SR1

3

0.1

1000

5

B reaction

2

SR2

4

0.51

280

4

B wash

3

SR3

5.5

0.1

400

3

C reaction

4

SR4

6

0.51

280

2

C wash

5

SR5

7.5

0.1

400

1

The whole process is divided into seven stages, based on the start and end times of each batch operation. The start and end times of each operation and the corresponding stages are shown in Fig. 3. The time points 0 h and 2 h are set as stages 7 and 6, respectively. The end time point of the last operation, i.e., 7.5 h, is stage 1. During each stage, the designer would make the decision to choose the freshwater scenario or reuse/recycle scenario to meet the water requirements. It is worth noting that in this example, water could not be reused in adjacent stages directly due to the discrete time of each stage. For example, stage 3 is at 5.5 h, and stage 2 is at 6 h. It is impossible to reuse the water which is available in stage 3 to stage 2 directly without the presence of a storage tank.
Fig. 3

The relationship between stage and time of example 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

In this stage, water sinks SK1 and SK2 can only be fed with freshwater because there is no available water source for reuse. Hence, there is only a state of freshwater consumption in stage 7, with a quantity of 1280 kg of freshwater consumed in this stage, i.e., \( {F}_{SK_1,7}^{Fw} \) = 1000 kg and \( {F}_{SK_2,7}^{Fw} \) = 280 kg. As there are no alternative water sources, the quantity of stored water is zero. The selected state and return cost for stage 7 is shown below. This work does not consider the cost of consumed freshwater consumption. Therefore, the return cost for the problem is the extent of freshwater consumption.
$$ {\displaystyle \begin{array}{c}{S}_{St age7}=\left\{\sum \limits_j^{J_7}{F}_{j,7}^{Fw}=1,280 kg,{F}_{\mathrm{Tank}1}^{St}=0 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{\sum \limits_j^{J_7}{F}_{j,7}^{Fw}+{f}_6\left({F}_{j,6}^{Fw}\right)\right\}=\min \left\{1,280 kg+{f}_6\left({F}_{j,6}^{Fw}\right)\right\}\end{array}} $$

Stage 6

In stage 6, there is no water available for reuse and so the water sink SK4 is also fed by freshwater, i.e., 280 kg. Therefore, the chosen state of stage 6 is shown below. The recursive function of the problem could be updated by the state of freshwater consumption in stage 6. The return cost is the sum of the minimum freshwater consumption in stages 6 and 7 and the freshwater consumption over the remaining stages (stages 1–5).
$$ {\displaystyle \begin{array}{c}{S}_{St age6}=\left\{\sum \limits_j^{J_6}{F}_{j,6}^{Fw}=280 kg,{F}_{\mathrm{Tank}1}^{St}=0 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1,280 kg+\sum \limits_j^{J_6}{F}_{j,6}^{Fw}+{f}_5\left({F}_{j,5}^{Fw}\right)\right\}=\min \left\{1,560 kg+{f}_5\left({F}_{j,5}^{Fw}\right)\right\}\end{array}} $$

Stage 5

In stage 5, a water source SR1 exists. This means the state of freshwater consumption is zero, i.e., \( \sum \limits_j^{J_5}{F}_{j,5}^{Fw} \) = 0 kg. Water source SR1 is of a high enough quality, i.e., 0.1 kg salt per kilogram of water, that it could be sent to the storage facilities for indirect reuse in later stages. In theory, there are a great number of states of stored water in this stage, i.e., the quantity of wastewater sent to the storage tank ranges from 0 to 1000 kg. The available wastewater is 1000 kg. The quantity of stored water mainly depends on the water requirements of water sinks in later stages. Once the states of freshwater consumption in stages 2 and 4 are determined, the maximum required capacity of the storage tank can be identified. At this phase of the procedure, it is assumed that all the water from source SR1 is sent to stored tank, i.e., tank 1. Note that the quantity of stored water in stage 5 is not the final result. It will be updated if the later stage does not require the current quantity of stored water. Therefore, the selected state of stored water in stage 5 is determined, i.e., \( {F}_{\mathrm{Tank}1}^{St} \) = 1000 kg. The state of freshwater and stored water and return cost of the problem is expressed as follows.
$$ {\displaystyle \begin{array}{c}{S}_{St age5}=\left\{\sum \limits_j^{J_5}{F}_{j,5}^{Fw}=0 kg,{F}_{\mathrm{Tank}1}^{St}=1,000 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1,560 kg+\sum \limits_j^{J_5}{F}_{j,5}^{Fw}+{f}_4\left({F}_{j,4}^{Fw}\right)\right\}=\min \left\{1,560 kg+{f}_4\left({F}_{j,4}^{Fw}\right)\right\}\end{array}} $$

Stage 4

A water sink SK3 and a water source SR2 exist in this stage. In order to meet the water requirement of SK3, there are two possible states of freshwater consumption. As restored water from stage 5 has a concentration of 0.1 kg salt per kilogram of water, which is equal to the inlet concentration of water sink SK3, the stored water can be reused by water sink SK3. Hence, the first possible state is that the indirect reuse water \( {F}_{Tank1,{SK}_3,4}^{ind} \) from tank 1 is 400 kg and the freshwater consumption is zero. The second possible state is that water sink SK3 is fed with freshwater, i.e., \( {F}_{SK_3,4}^{Fw} \) = 400 kg. Because the stored water from stage 5 is enough to supply the water sink in this stage, and the quality of water from source SR2 is inferior, i.e., 0.5 kg salt per kilogram of water, to that of the stored water, the water from SR2 is discharged as wastewater. The state of stored water in the tank 1 is thus 600 kg, i.e., 1000 kg (from stage 5) to 400 kg (to stage 4). The states of freshwater consumption for stage 4 are shown as follows.
$$ {\displaystyle \begin{array}{c}{S}_{St age4}^1=\left\{\sum \limits_j^{J_4}{F}_{j,4}^{Fw}=0 kg,{F}_{Tank1,{SK}_3,4}^{ind}=400 kg,{F}_{\mathrm{Tank}1}^{St}=600 kg\right\}\\ {}{S}_{St age4}^2=\left\{\sum \limits_j^{J_4}{F}_{j,4}^{Fw}=400 kg,{F}_{Tank1,{SK}_3,4}^{ind}=0 kg,{F}_{\mathrm{Tank}1}^{St}=1,000 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1,560 kg+\sum \limits_j^{J_4}{F}_{j,4}^{Fw}+{f}_3\left({F}_{j,3}^{Fw}\right)\right\}=\min \left\{1,560 kg+{f}_3\left({F}_{j,3}^{Fw}\right)\right\}\end{array}} $$

Stage 3

In stage 3, only a water source SR3 exists. Because SR3 has a lower concentration, i.e., 0.1 kg salt per kilogram water, which matches the maximum inlet concentration of water sink SK5 in stage 2, all of the available water from SR3can be stored in tank 2 for indirect reuse. The state of freshwater consumption \( \sum \limits_j^{J_3}{F}_{j,3}^{Fw} \) is zero for stage 3. Since in the later stages, stages 1 and 2, containing a single water sink with a demand of 400 kg, the water from water source SR3 can be stored in a separate tank (tank 2) for reuse. Therefore, the state of stored water in stage 3 is 400 kg, with a concentration of 0.1 kg salt per kilogram of water.
$$ {\displaystyle \begin{array}{c}{S}_{St age3}=\left\{\sum \limits_j^{J_3}{F}_{j,3}^{Fw}=0 kg,{F}_{\mathrm{Tank}1}^{St}=600 kg,{F}_{\mathrm{Tank}2}^{St}=400 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1560 kg+\sum \limits_j^{J_3}{F}_{j,3}^{Fw}+{f}_2\left({F}_{j,2}^{Fw}\right)\right\}=\min \left\{1,560 kg+{f}_2\left({F}_{j,2}^{Fw}\right)\right\}\end{array}} $$

Stage 2

Water sink SK5 and water source SR4 exist in stage 2. There are two possible states of freshwater consumption. In other words, water sink SK5 can be fed with the stored water from stage 3 instead of freshwater, \( {F}_{SK_5,2}^{Fw} \) = 0 kg, or it can be fed with only freshwater, \( {F}_{SK_5,2}^{Fw} \) = 400 kg. For the state of reuse water, the sink can use the water stored in tank 1 from stage 5 (\( {F}_{Tank1,{SK}_5,2}^{ind} \) = 400 kg) or the water stored in tank 2 from stage 3 (\( {F}_{Tank2,{SK}_5,2}^{ind} \) = 400 kg). In order to reduce the cost of storage, it is better to select the water which was stored at the time closest to stage 2, i.e., the water stored in tank 2. This is the implementation of rule 2 as mentioned in Section of Methodology. Due to the higher concentration (0.51 kg salt per kg water), all the water from water source (SR4) should be discharged as wastewater. The states of freshwater are presented as follows.
$$ {\displaystyle \begin{array}{c}{S}_{St age2}^1=\left\{\sum \limits_j^{J_2}{F}_{j,2}^{Fw}=0 kg,{F}_{Tank2,{SK}_5,2}^{ind}=400 kg,{F}_{\mathrm{Tank}1}^{St}=600 kg,{F}_{\mathrm{Tank}2}^{St}=0 kg\right\}\\ {}{S}_{St age2}^2=\left\{\sum \limits_j^{J_2}{F}_{j,2}^{Fw}=400 kg,{F}_{\mathrm{Tank}1}^{St}=1000 kg,{F}_{\mathrm{Tank}2}^{St}=0 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1,560 kg+\sum \limits_j^{J_2}{F}_{j,2}^{Fw}+{f}_1\left({F}_{j,1}^{Fw}\right)\right\}=\min \left\{1,560 kg+{f}_1\left({F}_{j,1}^{Fw}\right)\right\}\end{array}} $$

Stage 1

Stage 1 is the last stage of the problem and consists of a single water source SR5. Therefore, the state of freshwater consumption is zero. Water from SR5 is discharged as wastewater.
$$ {\displaystyle \begin{array}{c}{S}_{St age1}=\left\{\sum \limits_j^{J_1}{F}_{j,1}^{Fw}=0 kg,{F}_{Tank1}^{St}=600 kg,{F}_{Tank2}^{St}=0 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1,560 kg+\sum \limits_j^{J_1}{F}_{j,1}^{Fw}\right\}=\left\{1,560 kg+0 kg\right\}=1,560 kg\end{array}} $$
At this point, it can be concluded that the total freshwater consumption is 1560 kg. Two tanks with capacity of 1000 kg and 400 kg, respectively, are required to store wastewater for reuse. However, in stage 1, tank 1 is not empty but contains 600 kg reuse water. This implies that only 400 kg of water from SR1 in stage 5 should be sent to tank 1 and the remaining 600 kg should be discharged as wastewater. As a result, the stored water from stage 5 is completely consumed in stage 4; tank 1 is empty again in stage 3. Hence, only one tank is required for this problem. In other words, water from SR3 in stage 3 can be send to tank 1. The maximum required capacity of the tank is 400 kg. The chosen states for the problem are represented in solid circle, as shown in Fig. 4. The arc represents the chosen path from the state of previous stage to the state of next stage. The solid arcs indicate the path from the optimal state in previous stage to the optimal state in the adjacent stage, while the dashed arc is the alternative state. The values along each arc are the state of freshwater consumption from the present stage to the adjacent stage. Therefore, the sum of all the values along with the solid arc is the target of freshwater consumption, i.e., 1560 kg.
Fig. 4

The states of example 1 (first sequence)

The final network of water flows of operations is shown in Fig. 5. The freshwater consumption and wastewater generation of example 1 (first sequence) are both 1560 kg. The targets identified in this work match with those reported by Majozi et al. (2006).
Fig. 5

The resultant network for example 1 (first sequence)

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 SR1 at 0.1 kg salt/kg water in stage 5 and 400 kg of the same quality water from SR5 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 SR1 and 400 kg from SR5. 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

Since water sink SK1 has a maximum inlet concentration of 0 kg salt per kilogram of water, it should be fed with freshwater. The inlet concentration of SK2 is 0.25 kg salt per kilogram of water, which implies that it can be supplied with stored water from previous batch cycles, which is of a concentration of 0.1 kg salt per kilogram of water. Hence, the state of freshwater consumption in stage 7 is determined as \( {F}_{SK_1,7}^{Fw} \) = 1000 kg. It should be noted that the state of stored water at this point is an initial result and that will be updated based on the specific consumption of stored water.
$$ {\displaystyle \begin{array}{c}{S}_{St age7}=\left\{\sum \limits_j^{J_7}{F}_{j,7}^{Fw}=1,000 kg,{F}_{\mathrm{Tank}1}^{St}=720 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{\sum \limits_j^{J_7}{F}_{j,7}^{Fw}+{f}_6\left({F}_{j,6}^{Fw}\right)\right\}=\min \left\{1,000 kg+{f}_6\left({F}_{j,6}^{Fw}\right)\right\}\end{array}} $$

Stage 6

In the first sequence scenario, there is no available water for reuse in this stage and the water sink SK4 was still fed by freshwater, i.e., 280 kg. However, for cyclic-state scenario, the water stored in the tank is of sufficient quantity and quality to supply SK4. Hence, the state of freshwater consumption is zero.
$$ {\displaystyle \begin{array}{c}S{}_{St age6}=\left\{\sum \limits_j^{J_6}{F}_{j,6}^{Fw}=0 kg,{F}_{\mathrm{Tank}1}^{St}=440 kg\right\}\\ {}{f}_7\left({F}_{j,7}^{Fw}\right)=\min \left\{1,000 kg+\sum \limits_j^{J_6}{F}_{j,6}^{Fw}+{f}_5\left({F}_{j,5}^{Fw}\right)\right\}=\min \left\{1,000 kg+{f}_5\left({F}_{j,5}^{Fw}\right)\right\}\end{array}} $$
As the remaining stages are the same as the first sequence scenario as described in the previous subsection, it is not necessary to repeat this information here. The chosen states for this scenario are shown in Fig. 6. The values along each arc are the state of freshwater consumption from the present stage to the adjacent stage. Therefore, the sum of all the values along with the solid arc is determined as the target of freshwater consumption, i.e., 1000 kg. The same method can be used to identify the target of wastewater generation.
Fig. 6

The states of example 1 (cyclic operation)

Both the freshwater demand and wastewater generation for the cyclic operation are 1000 kg. This result is identical to the results of Majozi et al. (2006). Compared to the first sequence scenario, the difference here is the storage tank which provides enough water for SK2 (stage 7) and SK4 (stage 6). As a result, the maximum capacity of the storage tank is 560 kg, which equals to Majozi et al. (2006). The water reuse network is shown in Fig. 7.
Fig. 7

Water network for example 1 (cyclic operation)

Example 2

In practice feeding and discharging, the system is not accomplished instantly. The operations have specific processing times during input and output periods. The limiting water data for example 2 are shown in Table 2, where operations 1, 2, and 4 are formulated as fixed-mass load operations and operations 3 and 5 are treated as fixed flowrate operations (Chen and Lee 2008; Foo 2012). Although water charging and discharging are not accomplished rapidly, the proposed method can also be applied to this example. This is because the process of decision-making is an instant action and it does not last for the whole input and output periods. It has no impact on the target of freshwater consumed in this example. Therefore, those operations can still be treated as a completely batch model.
Table 2

Problem specification for example 2

Sinks

Time (h)

Cmax,in (kg salt/kg water)

Water (t)

Sources

Time (h)

Cmax,out (kg salt/kg water)

Water (t)

SK1

0.0–1.0

0.0

20

SR1

4.0–5.0

0.2

20

SK2

0.0–0.5

0.25

16

SR2

4.5–5.0

0.5

16

SK3

5.0–6.5

0.1

15

SR3

5.0–6.5

0.1

15

SK4

2.0–2.5

0.25

24

SR4

6.5–7.0

0.4

24

SK5

7.0–8.5

0.1

15

SR5

7.0–8.5

0.12

15

The stage is set based on the starting point of feeding period and discharging period, as shown in Fig. 8. For fixed flowrate operations, the freshwater and wastewater are determined by Eqs. (3)–(6). For fixed-mass load operations, the freshwater and wastewater are determined by Eqs. (7)–(18). In this example, first sequence and cyclic-state batch process are also introduced.
Fig. 8

The relationship between time and stage for example 2

First Sequence Scenario

The same procedure is repeated in this scenario. The problem can be divided into eight stages based on the charging or discharging time. The detailed calculation processes are omitted for simplicity. The chosen states for this scenario are shown in Fig. 9. In order to easily understand which process has the water requirement or requires to discharge the wastewater, the limiting data of water sinks and sources are given in this example rather than the inlet and out concentration. During the calculation process, the water requirement at a specific stage is determined by mass load and the inlet and outlet concentration if it is formulated as a fixed-mass load process. For example, operation 2 has a limiting quantity of 16 t. But it is formulated as a fixed-mass load process. The freshwater consumption is thus determined as 8 t, rather than the limiting water flow of 16 t. The solid arcs represent the optimal path from stage 8 to stage 1. Therefore, the sum of all the values along with the solid arc is determined as the target of freshwater consumption, i.e., 44.5 t. It is easy to identify the target of wastewater generation.
Fig. 9

The states of example 2 (first sequence)

As a result, both the amount of freshwater and wastewater for first sequence of example 2 are 44.5 t. The final network is shown in Fig. 10. The targets identified in this scenario could match with those reported by Chen and Lee (2008). It is noted that, in this work, effluent of operation 3 is not allowed to feed itself. Hence, the reference of the freshwater consumption in Chen and Lee (2008) is set as 44.5 t, not 39.5 t, which is the scenario of effluent of operation 3 feeding itself without any treatment.
Fig. 10

Resultant water network of example 2 (first sequence)

Cyclic-State Scenario

Based on the single batch case, some insight could be obtained. For example, 12.5 t of water for operation 1 and 15 t of water for operation 5 are discharged as wastewater. However, operations 1 and 5 have an outlet concentration of 0.2 kg salt per kilogram of water and 0.12 kg salt per kilogram of water, respectively. The water from operations 1 and 5 could be sent to tanks 1 and 2, respectively, and stored for future reuse. The stored water in tanks 1 and 2 before the cyclic operation is 20 t and 15 t. The same procedure is similar to the previous subsections. The chosen states for this scenario are shown in Fig. 11. Therefore, the target of freshwater consumption is determined as 26.42 t.
Fig. 11

The states of example 2 (cyclic operation)

Since the quantity of freshwater consumed in this scenario is determined, the wastewater can be identified based water balance of each operation. In this scenario, both the amount of freshwater and wastewater for the cyclic operation are 26.42 t, as shown in Fig. 12. The targets identified in this scenario are less than those reported by Chen and Lee (2008) (27.5 t). This is because, in this work, all the wastewater from operation 1 is sent to storage tank, while in the work of Chen and Lee (2008), 1.88 t of water is discharged as wastewater. Furthermore, the freshwater consumption in this scenario is also less than that (31.021 t) of Liu et al. (2009). However, in this work, two storage tanks are used, while in Liu et al. (2009), only one tank is involved. But the results of this work have a less capacity of storage facility (20 t and 15 t) compared with those (16.22 t and 15 t) of Chen and Lee (2008). Noting that in this work, effluent of operation 3 is not allowed to feed itself. Hence, the result of this work can be comparable with the target of 27.5 t in Chen and Lee (2008), where the effluent of operation 3 is not allowed to feed itself without any treatment.
Fig. 12

Resultant water network for example 2 (cyclic-state)

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

This scenario deals with the integration of a water-using system that consists of fixed-mass load operations. The limiting water data are given in Table 3. For fixed-mass load problems, it is improper to divide the operation into water sources and sinks. This is because the water quantity of operation is dependent on inlet and outlet concentration. The detail process is similar to the previous examples. For simplicity of the analysis, only the state of freshwater and stored water is shown in detail. The quantity of freshwater and wastewater is identified by the mass load and the concentration difference of each operation, i.e., Eqs. (7)–(12) or Eqs. (13)–(18). Based on the starting and finishing times, this batch processes comprise of eight stages, as shown in Fig. 13. The details of targeting processes are also omitted for simplicity.
Table 3

Limiting water data for the example 3

Operation

Quantity/t

Cin/(μg/g)

Cout/(μg/g)

Tin/h

Tout/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

Fig. 13

The relationship between time and stage for example 3

The chosen states for this scenario are shown in Fig. 14. The target of freshwater consumption is therefore determined as 80.5 t.
Fig. 14

The states for example 3 (reuse scenario)

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.

It is proper to identify the stored water in each stage using backward approach. In other words, the stored water could be identified from stage 1 to stage 8. This is because operations A and B finish at an early time with a lower concentration and the capacity of storage tank is mainly determined by the amount of wastewater from operations A and B. It is easy to determine the capacity of tank based on the water requirement. For example, in stage 3, an amount of 26.67 t water is required from tank 1 to supply process E: that water should be stored in tank 1 prior to stage 3. In stage 4, wastewater generated from process B is 22.5 t. The deficit quantity of 4.17 t, i.e., 26.67–22.5 t, should be sourced from the previous stages. However, operation A is the only source of stored water. The maximum capacity of tank 1 is 30.17 t, which is the sum of indirect water at stage 6 (5 t) and stage 5 (21 t) and the deficit water for stage 4 (4.17 t). The final results of water network are shown in Fig. 15. The actual inlet and outlet concentrations and corresponding mass load for each operation are summarized in Table 4. The result of freshwater consumption matches with that of literature (Liu et al. 2009). However, the capacity (30.17 t) of tank in this work is less than the literature example (41.302 t). The difference is that, in this work, all the wastewater discharged from operation B is sent to the tank, while in Liu et al. (2009), an amount of 7.37 t water is discharged as wastewater.
Fig. 15

Optimal network for example 3 (reuse scenario)

Table 4

Specifications of example 3 with central buffer tank

Operation

Quantity/t

Cin/(μg/g)

Cout/(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

In some cases, the potential of water reuse in batch processes could be exhausted. It might therefore be required to employ the regeneration to purify wastewater for reuse. The outlet concentration of the central regeneration unit in this example is set to be 100 μg/g. The flowrate for regeneration is 11.875 t/h. As aforementioned, the whole batch process is divided into eight stages. The details of targeting processes are also omitted. The optimal states for the problem are shown by the solid circles in Fig. 16.
Fig. 16

The states for example 3 (regeneration scenario)

The total freshwater of this scenario is 68.594 t. The wastewater is 25.886 t. At the end of this batch process, the remaining regenerated water in the tank is 42.708 t. The capacity of tanks is 50 t and 42.708 t. The actual inlet and outlet concentrations are shown in Table 5. The final water network is shown in Fig. 17. Compared with Liu et al. (2009), this work has the same target of freshwater consumption. It should be noted that this work has a higher quantity of wastewater than that of Liu et al. (2009), i.e., 16.316 t. This is because, in Liu et al. (2009), a quantity of 9.569 t wastewater is stored in the storage tank at the end of operation. However, the network obtained in this wok has less connection. For example, in Liu et al. (2009), water requirement of process C is satisfied by three sources, i.e., freshwater, regenerated water, and indirect reuse water from process A. In this work, process C is fed with freshwater and indirect reuse water from process A.
Table 5

Specifications of example 3 with a wastewater regeneration unit

Operation

Quantity/t

Cin/(μg/g)

Cout/(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

Fig. 17

Water network for example 3 with a wastewater regeneration unit

Example 4

In industry, it is a common occurrence that wastewater is contaminated with multiple contaminants. Most of wastewater minimization methodologies have been proposed to deal with single contaminant systems. This restricts the application of the methodologies to a small range of problems. Therefore, in this example, DP is used to design batch water network with multiple contaminants. The example is extracted from the literature (Kim and Smith 2004; Majozi 2010). The example involves seven water-using operations with three contaminants presented in the system. This example is assumed to be a fixed-mass load problem. The limiting data for the example are given in Table 6.
Table 6

Limiting data for example 4

Operation

Contaminant

Cmax, in (ppm)

Cmax, 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

The whole batch process is divided into 11 stages through starting and ending time as demonstrated earlier in this paper. The same procedure in the previous subsections is applied to determine the optimal freshwater consumption in every stage. As this example considers multiple contaminants, in every stage, a nonlinear model with minimum freshwater as the objective function is constructed based on Eqs. (7)–(12) or Eqs. (13)–(18). The GRG (generalized reduced gradient) solver in Microsoft Excel is chosen to solve sets of nonlinear equations. The nonlinear functions are input as cells and equation residuals are minimized. The freshwater consumption and reuse water in every stage is then solved using algebraic methods. The calculation method is similar to the previous examples and is thus not explained in detail. The optimal states for the problem are represented in solid circles, as shown in Fig. 18.
Fig. 18

The states for example 4 with two tanks

The total freshwater consumption for the scenario with two tanks is 842.04 t. The results of example 4 with two tanks are shown in Fig. 19. The specific concentrations are shown in Table 7. It is worth noting that Kim and Smith (2004) included a second water source, other than freshwater. Consequently, the results of this work cannot be readily compared to the results of Kim and Smith (2004).
Fig. 19

The final batch water network of example 4 with two tanks

Table 7

Specific concentration of each operation for example 4 with two tanks

Operation

Contaminant

Cin /(ppm)

Cout /(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.

The states for example 4 with one tank are shown in Fig. 20. The chosen states are represented in solid circles. The resultant water network is shown in Fig. 21. The freshwater consumption is slightly increased, determined as 842.6 t. This is because, the mixing of two different water sources results in a degradation of water quality. Compared with Fig. 19, the difference is the quantity of freshwater consumption and reuse water of operations 5 and 7. In the scenario of one tank, the inlet and outlet concentrations of operations 5 are 19, 119, 263 ppm and 184, 1000, 1200 ppm, respectively. For operation 7, the inlet and outlet concentrations are 13, 81, 180 ppm and 118, 1500, 1000 ppm, respectively. They are less than the limiting concentration of operations 5 and 7. However, the result of example 4 with one tank is less than the result obtained by the exact model of Majozi (2010), i.e., 865.8 t, and is greater than the result obtained by the relaxed model of Majozi (2010), i.e., 769.3 t. Compared with the base scenario without reuse/recycle scheme, i.e., 1076.3 t, the freshwater consumption of the scenario with one tank could be reduced by 21.7%. The advantage of DP method is illuminated in the calculation process. For example, when considering formulating the model of water integration of batch processes with multiple contaminants, it is difficult to determine which contaminant reaching the maximum outlet concentration. If the designer set more than one contaminant reaching the maximum outlet concentration, it is difficult to solve the model. However, in each stage of DP method, the complex problem is decomposed into some small problems so that it can be solved easily. If a few stages compete with the same water sources, it can also obtain the optimal results for these stages by trial and error method, such as operation 5 in the stage 6 and operation 7 in the stage 2.
Fig. 20

The states for example 4 with one tank

Fig. 21

The final batch water network of example 4 with one 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

Ik a set of available water sources in stage k

Jk a set of available water sources in stage k

m the number of operations

L a set of contaminants l

tsm the start time of operation m

tem 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

Fj, k the quantity of water for sink j in stage k

Fi, 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

ci, 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.

References

  1. Adekola O, Majozi T (2011) Wastewater minimization in multipurpose batch plants with a regeneration unit: multiple contaminants. Comput Chem Eng 35:2824–2836CrossRefGoogle Scholar
  2. Almató M, Sanmartí E, Espuña A, Puigjaner L (1997) Rationalizing the water use in the batch process industry. Comput Chem Eng 21:S971–S976CrossRefGoogle Scholar
  3. Bagatin R, Klemeš JJ, Reverberi AP, Huisingh D (2014) Conservation and improvements in water resource management: a global challenge. J Clean Prod 77:1–9CrossRefGoogle Scholar
  4. Bellman R (1957) Dynamic programming. Princeton University Press, Princeton, New JerseyzbMATHGoogle Scholar
  5. Chaturvedi ND, Bandyopadhyay S (2014) Simultaneously targeting for the minimum water requirement and the maximum production in a batch process. J Clean Prod 77:105–115CrossRefGoogle Scholar
  6. Chaturvedi ND, Manan ZA, Wan Alwi SR, Bandyopadhyay S (2016) Effect of multiple water resources in a flexible-schedule batch water network. J Clean Prod 125:245–252CrossRefGoogle Scholar
  7. Chen C-L, Lee J-Y (2008) A graphical technique for the design of water-using networks in batch processes. Chem Eng Sci 63:3740–3754CrossRefGoogle Scholar
  8. Chen C-L, Chang C-Y, Lee J-Y (2008) Continuous-time formulation for the synthesis of water-using networks in batch plants. Ind Eng Chem Res 47:7818–7832CrossRefGoogle Scholar
  9. Chen C-L, Chang C-Y, Lee J-Y (2011) Resource-task network approach to simultaneous scheduling and water minimization of batch plants. Ind Eng Chem Res 50(7):3660–3674Google Scholar
  10. Cheng K-F, Chang C-T (2007) Integrated water network designs for batch processes. Ind Eng Chem Res 46:1241–1253CrossRefGoogle Scholar
  11. Diban P, Abdul Aziz MK, Foo DCY, Jia X, Li Z, Tan RR (2016) Optimal biomass plantation replanting policy using dynamic programming. J Clean Prod 126:409–418CrossRefGoogle Scholar
  12. El-Halwagi MM, Gabriel F, Harell D (2003) Rigorous graphical targeting for resource conservation via material recycle/reuse networks. Ind Eng Chem Res 42:4319–4328CrossRefGoogle Scholar
  13. Foo DCY (2010) Automated targeting technique for batch process integration. Ind Eng Chem Res 49(20):9899–9916Google Scholar
  14. Foo DCY (2012) Process integration for resource conservation. CRC Press, Boca Raton, FloridaGoogle Scholar
  15. Foo DCY, Lee J-Y, Ng DKS, Chen C-L (2012) Targeting and design for batch regeneration and total networks. Clean Techn Environ Policy 15:579–590CrossRefGoogle Scholar
  16. Foo DCY, Manan ZA, Tan YL (2005) Synthesis of maximum water recovery network for batch process systems. J Clean Prod 13:1381–1394CrossRefGoogle Scholar
  17. Gouws JF, Majozi T, Foo DCY, Chen C-L, Lee J-Y (2010) Water minimization techniques for batch processes. Ind Eng Chem Res 49:8877–8893CrossRefGoogle Scholar
  18. Kim J-K (2011) Design of discontinuous water-using systems with a graphical method. Chem Eng J 172:799–810CrossRefGoogle Scholar
  19. Kim JK, Smith R (2004) Automated design of discontinuous water systems process. Saf Environ 82:238–248Google Scholar
  20. Lee J-Y, Chen C-L, Lin C-Y (2013) A mathematical model for water network synthesis involving mixed batch and continuous units. Ind Eng Chem Res 52:7047–7055CrossRefGoogle Scholar
  21. Lee J-Y, Chen C-L, Lin C-Y, DCY F (2014) A two-stage approach for the synthesis of inter-plant water networks involving continuous and batch units. Chem Eng Res Des 92:941–953CrossRefGoogle Scholar
  22. Lee J-Y, Foo DCY (2017) Simultaneous targeting and scheduling for batch water networks. Ind Eng Chem Res 56:1559–1569CrossRefGoogle Scholar
  23. Li B-H, Liang Y-K, Chang C-T (2013) Manual design strategies for multicontaminant water-using networks in batch processes. Ind Eng Chem Res 52:1970–1981CrossRefGoogle Scholar
  24. Liu Y, Li G, Wang L, Zhang J, Shams K (2009) Optimal design of an integrated discontinuous water-using network coordinating with a central continuous regeneration unit. Ind Eng Chem Res 48:10924–10940CrossRefGoogle Scholar
  25. Liu Y, Yuan X, Luo Y (2007) Synthesis of water utilization system using concentration interval analysis method (II) discontinuous process. Chinese J Chem Eng 15:369–375CrossRefGoogle Scholar
  26. Majozi T (2005) Wastewater minimisation using central reusable water storage in batch plants. Comput Chem Eng 29:1631–1646CrossRefGoogle Scholar
  27. Majozi T (2006) Storage design for maximum wastewater reuse in multipurpose batch plants. Ind Eng Chem Res 45:5936–5943CrossRefGoogle Scholar
  28. Majozi T (2010) Batch chemical process integration: analysis, synthesis and optimization. Springer, HeidelbergGoogle Scholar
  29. Majozi T, Brouckaert CJ, Buckley CA (2006) A graphical technique for wastewater minimisation in batch processes. J Environ Manag 78:317–329CrossRefGoogle Scholar
  30. Oliver P, Rodríguez R, Udaquiola S (2008) Water use optimization in batch process industries. Part 1: design of the water network. J Clean Prod 16:1275–1286CrossRefGoogle Scholar
  31. Roberts SM (1964) Dynamic programming in chemical engineering and process control. Academic Press, LondonGoogle Scholar
  32. Wang Y, Smith R (1995) Time pinch analysis. Chem Eng Res Des 73:905–914Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.School of Chemical and Metallurgical EngineeringUniversity of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations