Simultaneous Targeting and Scheduling for Batch Water Networks

Article pubs.acs.org/IECR

Simultaneous Targeting and Scheduling for Batch Water Networks Jui-Yuan Lee*,† and Dominic Chwan Yee Foo‡ †

Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, 1, Sec. 3, Zhongxiao E. Road, Taipei 10608, Taiwan, R.O.C. ‡ Department of Chemical and Environmental Engineering/Centre of Excellence for Green Technologies (CEGT), University of Nottingham Malaysia Campus, Broga Road, 43500 Semenyih, Selangor, Malaysia S Supporting Information *

ABSTRACT: Batch processes have received considerable attention from both industry and academia because of their flexibility and suitability to produce low volume, high value-added products (e.g., fine/specialty chemicals) and also as a result of the recent trend toward product-centered manufacturing globally. This paper presents an optimization technique for water network synthesis in batch processing plants. The mathematical formulation is a mixed integer linear program (MILP) based on the pinch-based automated targeting model (ATM) and a statetask network (STN)-based discrete-time scheduling model. Embedding the ATM in the scheduling framework enables simultaneous process scheduling and water minimization and the true minimum flow/cost targeting prior to detailed water network design. Two modified literature examples and an industrial case study on polyvinyl chloride (PVC) manufacturing are solved to demonstrate the application of the proposed MILP model. Results for the PVC case study show that significant water (50−56%) and cost savings (61−67%) can be achieved through optimal production scheduling and water recovery.

1. INTRODUCTION A batch process consists of steps operating discontinuously and, distinct from its continuous counterparts, delivers its product in discrete amounts. Thus, properties such as heat, mass, temperature, and concentration vary with time.1 The time dimension brings constraints that are not experienced in the design of continuous processes. On the other hand, batch processes are ideally suited for the production of low volume, high value-added products (e.g., pharmaceuticals, fine and specialty chemicals) and flexible in accommodating production changes. Furthermore, batch processes allow the use of multipurpose equipment for producing a variety of products from the same plant. Such flexibility and adaptability, however, make the design of batch processes a more complex problem, hence the need and constant development of effective scheduling and optimization techniques since the 1990s. Considerable amounts of wastewater are normally generated in batch plants during multipurpose equipment cleaning, which is mandatory for the changeover between two different products.2 The same happens when water is used as a solvent and discharged at the end of the process or when used in various extraction and washing operations. With ever stricter discharge regulations and increased environmental awareness, the rise in fresh water and effluent treatment costs has urged batch plants to make efficient use of water resources, in which case both fresh water consumption and wastewater generation can be reduced through reuse, recycling, and regeneration. The latter are three common options to be considered in wastewater minimization through water network synthesis.3 Much research © XXXX American Chemical Society

has been conducted in the past decades to address such problems for batch processes using various process integration techniques. Readers may refer to the review papers by Foo,4 Jeżowski,5 Gouws et al.,6 and Klemeš et al.7 for more details. As with those for continuous processes, the methodologies for synthesizing water networks in batch plants, or batch water networks (BWNs), can be classified as pinch analysis and mathematical optimization techniques.6 The former normally consists of two steps to identify rigorous water targets prior to detailed network design. Such approaches provide good insights into debottlenecking and process modification, although limited to a single contaminant and not readily applied to cases with practical constraints on network design or cost considerations. Mathematical approaches, by contrast, offer good flexibility to take into account additional process constraints, various cost elements, and multiple water quality indices for more complex problems. However, nonlinearity and nonconvexity to arise from contaminant mass balances may cause difficulties in problem solving, along with the disadvantage that global optimality cannot always be guaranteed. The first contribution to BWN synthesis based on pinch analysis was presented by Wang and Smith,8 who developed a systematic procedure to determine the minimum wastewater Received: Revised: Accepted: Published: A

September 23, 2016 December 27, 2016 January 20, 2017 January 20, 2017 DOI: 10.1021/acs.iecr.6b03714 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research for a given set of f ixed load semicontinuous water-using processes. This seminal work stimulated several follow-ups, including the works by Foo et al.,9 Majozi et al.,10 and Chen and Lee11 to address f ixed flow problems, which are commonly found in truly batch operations. These pinch-based techniques, although useful, are limited to single-contaminant cases and incapable of handling time as a variable. This implies that the scheduling of the operations has to be predefined. Both limitations can be overcome by using mathematical techniques. Early works in this regard focused more on water allocation and assumed the schedule to be fixed. Examples include the works of Almotó et al.,12 Kim and Smith,13 and Majozi.14 Later extensions have considered water regeneration,15,16 forbidden matches,17,18 property-based water quality indices,19,20 and interplant water integration.21,22 More recent works tend to treat time as an optimization variable, in which case the schedule is allowed to change. Majozi23 presented a mathematical technique for water minimization in multipurpose batch plants based on a continuous-time formulation for production scheduling. His work was then extended to address multiple contaminant streams24 and wastewater regeneration.25 Chaturvedi and Bandyopadhyay26 proposed a mathematical formulation to minimize the operating cost of a BWN by utilizing multiple fresh water resources. Their methodology incorporates variable process scheduling for a given production. The effect of multiple water resources on a BWN schedule was analyzed later by Chaturvedi et al.27 Apart from the pinch-based and mathematical approaches, there are also hybrid ones taking the advantages of both sides. A typical example is the automated targeting model (ATM), which was originally developed for mass exchange network synthesis for continuous processes by El-Halwagi and Manousiothakis.28 This technique was then extended to address various resource conservation network (RCN) problems, including concentrationbased,29,30 property-based,31,32 and interplant RCNs.33 Like mathematical techniques, the ATM has the adaptability to accommodate design constraints and cost trade-offs, while providing the same insight into RCN synthesis as conventional pinch analysis. For batch process integration, the ATM was further extended by Foo34 but limited to fixed-schedule cases, in which the water and storage targets identified may not be the true optimum. It should be noted that most research works dealing with simultaneous scheduling and water minimization focused on truly batch operations, with only limited works on semicontinuous processes. The latter can be more challenging because of the need to define time intervals where water streams may exist and direct water reuse/recycling can take place. Dong and co-workers35,36 developed superstructurebased mixed integer nonlinear programming (MINLP) models and hybrid optimization strategies integrating deterministic and stochastic search techniques to address BWN problems featuring semicontinuous processes. However, their design methods, even for a relatively small example, appear to be too complicated for engineers to apply. In this paper, the ATM is embedded within a scheduling framework to allow simultaneous targeting and process scheduling, thus enabling the determination of the true flow/cost targets for BWNs. This provides a simple yet effective method for the problem. In the following sections, a formal problem statement is first given. The mathematical formulation is presented next. Three case studies are then used to illustrate the proposed approach.

Finally, conclusions and prospects for future work are given at the end of the paper.

2. PROBLEM STATEMENT The problem addressed in this paper is formally stated as follows. Given: • Process scheduling data, including (i) the production recipe for each product, (ii) the available units and their capacities, (iii) the maximum storage capacity for each material, and (iv) the time horizon of interest. • A set of water-consuming units in the process, designated as process sinks, j ∈ J. Each sink requires a fixed water flow (Fj) with an impurity concentration no greater than the allowable maximum (Cj). • A set of water-generating units or streams, designated as process sources, i ∈ I. Each source is characterized by a fixed water flow (F i ) and a constant impurity concentration (Ci) and can be connected to the sinks for water reuse/recycling. The unutilized wastewater from sources will be sent for treatment before final discharge to the environment, which is considered an external sink. Note that the impurity concentration can be changed to some other quality index (e.g., stream properties in property integration34,37) for more general RCN problems. As process sources and sinks may occur in different time intervals, depending on the process schedule, water storage tanks may be used to facilitate water recovery by allowing water to be stored temporarily and reused later. When the available sources and water storage cannot supply enough water to the sinks, fresh water may be purchased from external source(s) to meet the sink requirements. The objective is to determine the optimum production schedule that achieves minimum fresh water consumption, minimum BWN cost, or maximum profit. 3. MODEL FORMULATION The overall model consists of two interconnected modules. One module focuses on the exploration of water reuse/recycle opportunities using the ATM, and the other on proper sequencing and scheduling based on a discrete-time formulation. These two modules are presented below separately. 3.1. Water Reuse/Recycle Module. For application in batch processes, the ATM developed by Foo34 is adopted. It should be noted that the ATM is a mathematical optimization approach based on the concept of pinch analysis, thereby enabling the minimum flow/cost targets to be identified prior to detailed network design. Figure 1 shows the ATM framework for direct reuse/ recycling in a BWN.34 Note that each of the flow ( f, δ) and load terms (ε) has index t to indicate the time interval in which it occurs. The presence of flow terms for water stored from earlier ( f t−1,t) and for later time intervals ( f t,t+1) allows the transfer of water across different time intervals. Note also that water storage is available at all concentration levels except the last one (k = K). However, the actual storage requirement is only determined through optimization. The ATM formulation is given as follows: δk , t = fFW, t |k = 1 + δk − 1, t |k > 1 + (∑ fi , t − i∈I

+ (ft − 1, t − ft , t + 1 )k B

∑ f j ,t )k j∈J

∀ k ∈ K, k < K , t ∈ T

(1)

DOI: 10.1021/acs.iecr.6b03714 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research qWS, t + 1, k = qWS, t , k − ft − 1, t , k + ft , t + 1, k ∀ k ∈ K, t ∈ T qWS, t , k ≤ qSC, k

∀t∈T

εk , t = εk − 1, t + δk − 1, t(Ck − Ck − 1)

(2)

∀ k ∈ K, t ∈ T (3)

εk = 1, t = 0

∀t∈T

(4)

Equation 1 describes the water cascade across all concentration levels for all time intervals. The net water flow from concentration level k in time interval t (δk,t) is the result of the water cascaded from the previous concentration level (δk−1,t) adjusted by the difference between the source and sink flows (∑i f i,t,k − ∑j f j,t,k), as well as the amounts of water taken from and sent to storage (f t−1,t,k − f t,t+1,k). Note that the water flow into the first concentration level (k = 1) in each time interval is defined as the fresh water flow (i.e., δk=0,t = f FW,t, as shown in Figure 1). On the other hand, the water flow to the last concentration level (δk=K−1,t) is identified as the wastewater flow, as given in eq 2. Next, eq 3 describes the impurity load cascade for all time intervalsthe residual impurity load to concentration level k (εk,t) is contributed by that cascaded to the previous concentration level (εk−1,t) and the impurity load within the concentration interval (k − 1, k). The latter is given by the product of the net water flow (δk−1,t) and the difference in concentration between the two adjacent levels (Ck − Ck−1). It should be noted that the impurity load cascade in each time interval starts with zero. This is shown in Figure 1 and stated in eq 4. The source and sink flow rates are given by eqs 5 and 6, respectively. fi , t , k = Fy i i , t Δt Yi , k f j , t , k = Fjyj , t Δt Yj , k

∀ i ∈ I, k ∈ K, t ∈ T ∀ j ∈ J, k ∈ K, t ∈ T

∀ k ∈ K, t ∈ T

(8)

3.2. Sequencing/Scheduling Module. The scheduling module adopted in this work is a modified state-task network (STN)-based discrete-time formulation,38,39 where the processing tasks to be scheduled are expressed using the state sequence network (SSN) representation.40 More specifically, a task is identified by one of its material inputs (i.e., the effective state) and the process unit where it takes place, without having to define a dedicated index and set for the tasks. This approach has proved to require fewer binary variables compared to models based on other network representations. On the other hand, the discrete representation of time is simple and readily applied to the integration of batch and semicontinuous processes with water streams occurring in time intervals, despite the increase in the model size with a larger number of constraints and variables. Equation 9 states that at most one task can be performed in unit u during time interval t. It should be noted that this constraint requires fixed and predefined processing times for all tasks to be scheduled. In addition, it is implicitly assumed that all tasks have to release the allocated processing unit upon completion; in other words, processing units are not to be used as temporary storage vessels.

Figure 1. ATM framework for direct reuse/recycle BWN with water storage.

δk = K − 1, t = fWW, t

(7)

t − τ(s in, u) + 1

∑

∑

s in, u ∈ S*in, u

t ′= t

y(s in, u , t ′) ≤ 1

∀ t ∈ T, u ∈ U (9)

where τ(sin,u) is the total operation time of the processing unit. Normally this operation time is equal to the task duration α(sin,u); however, when there is some associated task(s) to be performed in the same unit, for example, equipment washing, the operation time will be the sum of the task duration and the washing time. The amount of material starting to be processed in the task in unit u at time t (i.e., the beginning of time interval t) is bounded by the minimum and maximum capacities of the unit: Vuminy(s in, u , t ) ≤ m(s in, u , t ) ≤ Vumaxy(s in, u , t ) ∀ s in, u ∈ S*in, u , t ∈ T, u ∈ U

(10)

Note that eq 10 forces the batch size to be zero if y(sin,u,t) = 0 (i.e., the task not started at the beginning of time interval t). Equation 11 describes the mass balance for material storage. This constraint states that the amount of state s stored at time t is equal to that stored at time t − 1 adjusted by the difference of the amounts produced and consumed at time t. The initial amount of each state is assumed to be known, thus allowing the initial condition of all material inventories to be specified.

(5) (6)

In addition, eqs 7 and 8 are used for water storage sizing at each concentration level. Equation 7 states that at concentration level k, the amount of stored water available for the next time interval (qWS,t+1,k) is equal to that for the current time interval (qWS,t,k) adjusted by the amounts of water used ( f t−1,t,k) and stored (f t,t+1,k) in time interval t. The amount of water stored at any time cannot exceed the storage capacity (qSC,k), as stated in eq 8. In other words, the capacity required is determined by the peak of the water storage profile over the time horizon.

qs , t = qs , t − 1 + −

∑ s in, u ∈ Bcs

∑ s in, u ∈ Bsp

ρsp (s in, u)m(s in, u , t − α(s in, u))

ρsc (s in, u)m(s in, u , t )

∀ s ∈ S, t ∈ T (11)

Equation 12 states that the amount of state s stored at any time cannot exceed the maximum storage capacity. It should be C


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Industrial & Engineering Chemistry Research noted that dedicated storage units are assumed to be available for each state. qs , t ≤

Q smax

∀ s ∈ S, t ∈ T

RC =

s ∈ Sr s in, u ∈ Bcs t ∈ T

(12)

∑ fFW,t

t∈T

4. CASE STUDIES The application of the proposed approach is illustrated through three case studies, in which the model developed is implemented and solved in the GAMS environment41 on a Core i5-4310M, 2.70 GHz processor, utilizing CPLEX as the MILP solver. All solutions were found with reasonable processing time. 4.1. Case Study 1. The first case study is adapted from Kim and Smith,13 involving four semicontinuous water-using operations with a single contaminant. To evaluate the impact of process scheduling on water integration, only the duration, instead of start and end times, is specified for these operations. Table 1 shows the limiting conditions and timing data for this

Minimizing fresh water flow also leads to the minimum wastewater flow. A two-stage optimization approach may be followed, with a second objective to determine the minimum water storage capacity required (eq 14): min ϕSC =

∑ qSC,k

(14)

k∈K

Table 1. Limiting Water Data for Case Study 1

Note that an additional constraint given in eq 15 is needed in this case to limit the fresh water flow to the earlier determined minimum.

∑ fFW,t

* ≤ ϕFW

limiting concentration (ppm)

(15)

t∈T

Alternatively, the objective function may be set to minimize the total annual cost (TAC) of a BWN, taking into account both operating and capital cost elements. As given in eq 16, the operating cost consists of fresh water and wastewater treatment costs, while the capital cost is contributed mainly by storage tanks. t∈T

∑ (CSTfixyST,k k∈K

var + CST qSC, k )

(16)

where N is the quotient of the annual operating time divided by the time horizon of interest (N = AOT/H), assuming repeated cycles. Equation 17 is used in conjunction with eq 16 for the calculation of fixed storage cost. qSC, k ≤ Q kmaxyST, k ∀ k ∈ K

unit

inlet

outlet

limiting flow rate (t/h)

duration (h)

limiting flow (t)

1 2 3 4

0 50 400 50

100 100 800 800

20 100 10 40

1 2.5 2 2

20 250 20 80

case study. It is assumed that operations 1 and 2, as well as operations 3 and 4, are sequential steps of the batch process. Furthermore, the overall process is assumed to operate in repeated batch cycles with the cycle time being 5 h. As demonstrated by Foo et al.,9 the inlet and outlet flows of fixed load operations can be represented as the water sinks and sources. Table 2 shows the corresponding sink and source data. In this instance, SK1-SK4 and SR1-SR4 are sinks and sources for operations 1−4, respectively. Note that the limiting water flow is simply the product of the limiting flow rate and the duration. Without water reuse/recycling, both the fresh water and wastewater flows are calculated to be 370 t/cycle.

min ϕTAC = N ∑ (C FWfFW, t + C WTfWW, t ) + AF

Table 2. Water Sink and Source Data for Case Study 1

(17)

Qmax k

where is the maximum water storage at concentration level k (determined by the availability of water sources). From the point of view of the profit, the objective function may be formulated as the difference between the product revenue (PR) and the costs of raw materials (RC) and water (WC), as given in eq 18. max ϕP = PR − RC − WC

sink SK1 SK2 SK3 SK4

∑ Price(s)qs ,t |t = T + 1

s ∈ Sp

limiting flow rate (t/h)

concentration (ppm)

duration (h)

limiting water flow (t)

0 50 400 50 concentration (ppm)

1 2.5 2 2 duration (h)

20 250 20 80 limiting water flow (t)

100 100 800 800

1 2.5 2 2

20 250 20 80

(18)

source

20 100 10 40 limiting flow rate (t/h)

(19)

SR1 SR2 SR3 SR4

20 100 10 40

where PR =

(21)

t∈T

With the presence of binary variables, the overall model is a mixed integer linear program (MILP), which can be readily solved to global optimality without major computational difficulties.

(13)

t∈T

(20)

WC = C FW ∑ fFW, t + C WT ∑ fWW, t

In addition to the targeting and scheduling formulations presented above, timing constraints (as those presented in the Supporting Information) are also needed to locate when the process water sources and sinks occur during the courses of the corresponding tasks/operations. The timing constraints thus connect the two basic modules by correlating binary variables y(sin,u, t) with yit and yjt. However, such constraints are casespecific and will only be discussed later in the case studies. 3.3. Objective Functions. For water resource conservation, the objective may be set to minimize the fresh water flow, as given in eq 13:

min ϕFW =

∑ ∑ ∑ Cost(s)ρsc (sin,u)m(sin,u , t )

D


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Industrial & Engineering Chemistry Research On the basis of the durations of the water-using operations and water sinks/sources, the length of the time intervals is chosen to be 0.5 h. The timing constraints for the sinks and sources are given in eqs S.1−S.4 in the Supporting Information. Two scenarios are analyzed in this case study. In scenario 1, the objective is to minimize fresh water consumption, while in scenario 2 it is to minimize water and storage cost. Solving the MILP model consisting of eqs 1−13 and S.1−S.4 for scenario 1 gives the minimum fresh water consumption of 185 t/cycle. The minimum water storage capacity required is then determined to be 2.86 t by solving the model consisting of eqs 1−12, 14, and 15 and S.1−S.4. Figure 2 shows the optimal

AF =

r(1 + r ) p (1 + r ) p − 1

(22)

where the interest rate (r) is taken to be 5% over a period of five years (p = 5).1,37 Solving the MILP model consisting of eqs 1−12, 16, and 17 and S.1−S.4 gives the minimum total cost of US$300,000/y. Figure 4 shows the optimal schedule and the corresponding

Figure 4. Optimal schedule and water network for scenario 2.

water reuse/recycle network for scenario 2. It can be seen that water storage is eliminated at a slight increase of 1.35% in fresh water consumption, as compared to the results for scenario 1. This indicates the trade-off between water and storage costs. The computational results for both scenarios are summarized in Table S1 in the Supporting Information. Note that the targets for the predefined schedule are taken from the previous works9,13 and can also be obtained using the proposed model. 4.2. Case Study 2. The second case study, adapted from Kondili et al.,38 considers a multipurpose batch facility producing two products from three raw materials. Figure 5 shows the STN representation of the production process, which involves five tasks. The recipe is as follows. i. Heating: Feed A (s1) is heated for 1 h. ii. Reaction 1: 50% feed B (s2) reacts with 50% feed C (s3), lasting for 2 h, to form intermediate BC (s6). iii. Reaction 2: 40% hot A (s5) reacts with 60% intermediate BC (s6), lasting for 2 h, to form intermediate AB (s8; 60%) and product 1 (s7; 40%). iv. Reaction 3: 20% feed C (s4) reacts with 80% intermediate AB (s8), lasting for 1 h to form impure E (s9). v. Separation: Impure E (s9) is distilled to separate pure product 2 (s10; 90%, after 1 h) and pure intermediate AB (s8; 10%, after 2 h). The latter is then recycled. Tables 3 and 4 present the data required for process scheduling. It is further assumed in this work that the reactors need to be cleaned at the end of each reaction to remove impurities and ensure product integrity. The data pertaining to the washing operations are shown in Table 5. Note that the washing time is included in the total operation time of the processing unit. In addition, the costs of fresh water (contaminant-free) and wastewater treatment are taken to be $2/t and $3/t, respectively. On the basis of the durations of the processing tasks and washing operations, the length of the time intervals is chosen to be 0.5 h. The timing constraints for the water sources and sinks are given in the Supporting Information (see eqs S.5−S.8). The objective for this case study is to maximize the profit given in eq 18. Solving the MILP model consisting of eqs 1−12 and 18 and S.5−S.8 over a 12-h time horizon (cycle time) yields the results

Figure 2. Optimal schedule and water network for scenario 1.

schedule and the corresponding water reuse/recycle network for scenario 1. This result shows a 50% reduction in fresh water usage compared to the base case without water integration. Although no further water saving is observed when compared with the results for the predefined schedule (see Figure 3), in

Figure 3. Optimal water network for the predefined schedule.

which case the minimum fresh water consumption and the minimum water storage capacity were determined to be 185 t/ cycle and 30 t respectively,9,13 the optimized schedule results in a significant reduction of more than 90% in the capacity of water storage. It should be noted that, with repeated batch operation, the minimum water targets can always be achieved regardless of the schedule.9 However, taking into account process scheduling can maximize direct water reuse and minimize the need for water storage. For scenario 2, it is assumed that the process operates for 8000 h, or 1600 batch cycles annually (N = 8000/5 = 1600). The fresh water cost is taken to be US$1/t, while the fixed and variable cost coefficients for water storage are US$20300 and US$235/t, respectively.34 It is further assumed that the piping cost has a small impact on the overall water network design and may be omitted in the objective function. The annualization factor is calculated to be 0.231 using the following equation:42 E


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Industrial & Engineering Chemistry Research

Figure 5. STN representation for case study 2.

6a and 7 show the optimal production schedule for these two cases. It can be seen that a tank with a required capacity of 45 t

Table 3. Processing Data for Case Study 2 task heating reaction 1 reaction 2 reaction 3 separation

unit heater reactor reactor reactor reactor reactor reactor still

duration (h)

capacity limits (kg)

1 2 2 2 2 1 1 1,2

10−100 10−80 10−50 10−80 10−50 10−80 10−50 10−200

1 2 1 2 1 2

Table 4. Material Data for Case Study 2a

a

state

storage capacity (kg)

initial inventory (kg)

cost/price ($/kg)

s1 s2 s3 s4 s5 s6 s7 s8 s9 s10

UL UL UL UL 100 150 UL 200 100 UL

AA AA AA AA 0 0 0 0 0 0

1 1 1 1 10 10

Figure 6. (a) Optimal production schedule (numbers in task boxes indicating the batch size in kg) and (b) water storage profile for case study 2.

UL, unlimited; AA, available as/when required.

Table 5. Limiting Water Data for Case Study 2 impurity concentration (ppm) task reaction 1 reaction 2 reaction 3

unit reactor reactor reactor reactor reactor reactor

1 2 1 2 1 2

washing time (h)

inlet/ sink

outlet/ source

0.5 0.5 0.5 0.5 0.5 0.5

100 100 0 0 100 100

400 400 200 200 200 200

limiting flow (t) 10 8 10 8 10 8

Figure 7. Optimal production schedule for case study 2 without water storage (numbers in task boxes indicating the batch size in kg).

to store 100 ppm water (see Figure 6b) gives a further 6.6% savings in fresh water cost/consumption. In addition, water reuse/recycling achieves a 19.9−24.6% improvement in the profit when compared with the case without water integration. The latter is taken as the base case, identified by replacing the

in Table S2 in the Supporting Information. The maximum profit is determined to be $1511/cycle in the presence of water storage or $1455/cycle in the absence of water storage. Figures F


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dryer to remove moisture before the PVC product is sent for storage. In this case study, water recovery is to be conducted through reuse/recycling for overall water reduction. The water sinks and sources originate from the PVC manufacturing process (reactor feed, cleaning, and decanter discharge), the utility section (boiler and cooling tower makeup), and miscellaneous usage (e.g., floor cleaning etc.), where the suspended solid content is the main impurity of concern for water reuse/recycling. Note that the four decanters with the same characteristics are treated as two water sources, one for each downstream processing train. It should be noted that, due to piping constraints, feedwater can only be charged to one reactor at a time. The same goes for the blowdown operation after the reaction; only one reactor can discharge the effluent to the blowdown vessel at a time. Therefore, both the feeding and blowdown operations of each reactor are to be scheduled in sequence.43 By merging the parallel identical reactors and taking the blowdown vessel as finite intermediate storage in process scheduling, while focusing on the identified water sinks and sources, the PVC manufacturing process may be simplified as shown in Figure 10a, together with the operation timings

ATM with the calculation of the total sink (fresh water) and source (wastewater) flows. Note, however, that the water costs in the base case are not much higher than those in the cases with water integration because of fewer reactions performed, as shown in Figure 8. This indicates a trade-off between increased product revenue and reduced water cost savings.

Figure 8. Optimal production schedule for case study 2 without water integration (base case; numbers in task boxes indicating the batch size in kg).

Since the case study has been significantly modified from the original38 to include the water integration aspect, the results are no longer comparable. Also, the results obtained in this case study are not comparable with those in the previous works24−27,35,36 because of different sets of water-using operations assumed. However, the purpose of this case study is to demonstrate the use of the proposed model for multipurpose batch plants and to show the impact of water integration in addition to process scheduling. 4.3. Case Study 3. The third case study is adapted from Chan et al.43 Figure 9 shows the process flow diagram for a polyvinyl chloride (PVC) plant, which operates in a mixedbatch and semicontinuous mode.37,43 Fresh vinyl chloride monomer (VCM) molecules are mixed with the recycled VCM and fed to 10 parallel batch polymerization reactors, where a large amount of water is used as a reaction carrier. Upon completion of the polymerization reaction, the reactor effluent in slurry form (containing PVC resin and water) is sent to a blowdown vessel, which acts as a buffer for two parallel, semicontinuously operated downstream processing trains. The reactor slurry is then sent to the steam stripper to recover unconverted VCM. The stripped VCM, together with the VCM emitted from the reactors and the blowdown vessel, is sent to the VCM recovery system. On the other hand, the stripped slurry enters two parallel decanters for water removal, and the

Figure 10. (a) Simplified process flowsheet and (b) operation timings.

Figure 9. Process flow diagram for a PVC manufacturing plant. G


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wastewater treatment, and the use of an 85.17-t capacity tank to store reusable water at 50 ppm. Figure 11 shows the optimal production schedule, and Figure 12 the corresponding water

(Figure 10b). As indicated by the latter, the separation can only start after the reaction is complete. In addition, the total capacity of the polymerization reactors is assumed to be twice the throughput of the downstream processing trains, hence, one reaction to be followed by two batches of separation. The limiting water data for the sinks and sources are shown in Table 6. Table 6. Limiting Water Data for Case Study 3 sink

operation

SK1 SK2 SK3

reactor feed line flushing boiler makeup 1 boiler makeup 2 reactor cleaning cooling tower flushing floor cleaning 1 floor cleaning 2

SK4 SK5 SK6 SK7 SK8 SK9

flow rate (kg/h)

concentration (ppm)

duration (h)

water flow (kg)

30000 3125 6600

10 10 10

3.5 5 0.5

105000 15625 3300

6600

10

0.5

3300

1200

20

5

6000

6300

50

15

94500

2000 3000

200 500

5 0.5

10000 1500

3000

500

0.5

1500

Figure 12. Water storage profile.

source

operation

flow rate (kg/h)

concentration (ppm)

duration (h)

water flow (kg)

SR1 SR2

decanter 1 decanter 2

6831.25 6831.25

50 50

10 10

68312.5 68312.5

storage profile. It is worth noting that the fresh water and wastewater flows determined for minimum TAC are identical to those in the minimum water solution obtained by solving eq 13 subject to eqs 1−12 and S.9-S.12. This is because water costs are dominating in this particular case. The foregoing results assume the PVC plant operates in repeated batch cycles, where water may be reused between successive cycles. In some cases, however, water storage tanks would need regular emptying and cleaning to prevent the buildup of contamination. This can be reflected in the model by setting the amount of water stored (i.e., qWS,t,k and f t,t+1,k) to zero at certain points in time. Assuming that the water storage tank needs to be emptied and cleaned once a cycle (in which case the amount of water stored at the beginning of a cycle qWS,1,k and the amount of water transferred between two successive cycles f192,1,k are set to zero), resolving the same MILP model (eq 16 subject to eqs 1−12 and 17 and S.9-S.12) yields the minimum TAC of US $73771, with 59548.35 t/y of fresh water consumption, 7706.55 t/y of wastewater treatment, and the use of a 33.66-t capacity tank to store reusable water at 50 ppm. Note that the need for tank emptying at the end of each cycle limits the water recovery potential; no water can be stored and reused in the beginning of the next cycle, hence, increased water flows and TAC, despite a much reduced water storage capacity (−60.5%). Figure 13 shows the optimal production schedule and Figure 14 the corresponding water storage profile. It can be seen that the schedule shown in Figure 13 is not much different from that in Figure 11. This is because in both cases the operations are scheduled for maximum water recovery and minimum TAC for the given production demand. The computational results for this case study are summarized in Table S3 in the Supporting Information. An overall comparison shows that significant reductions in fresh water

On the basis of the durations of the operations and water sources and sinks, the length of time intervals is chosen to be 0.5 h. The timing constraints for the water sources and sinks are given in the Supporting Information (see eqs S.9−S.12). It is assumed that the production demand requires four batches of polymerization reaction to be processed in the cycle time of 96 h (192 time intervals). The cost data used in this case study are as follows:34 • Fresh water cost = US$1/t • Wastewater treatment cost = US$1/t • Fixed storage cost = US$20,300 • Variable storage cost = US$235/t • Annualization factor = 0.231, calculated using eq 22 • Plant operation = 7982 h, or 83 cycles annually It is further assumed that the piping cost is small and may be neglected. Solving the optimization objective in eq 16 subject to constraints in eqs 1−12 and 17 and S.9−S.12 gives the minimum TAC of US$62428. This is concomitant with 52479.24 t/y of fresh water consumption, 637.44 t/y of

Figure 11. Optimal production schedule for case study 3. H


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Figure 13. Optimal production schedule for case study 3 (with tank emptying).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b03714. Timing constraints and computational results for case studies 1−3 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +886-2-27712171 ext. 2524. Fax: +886-2-27317117. Email: [email protected] (J.-Y. Lee).

Figure 14. Water storage profile (with tank emptying).

ORCID

Jui-Yuan Lee: 0000-0002-5879-3926 Dominic Chwan Yee Foo: 0000-0002-8185-255X consumption and TAC (more than 50% and 60%, respectively) can be achieved by simultaneous scheduling and water integration. Although the case study has been modified from its original43 to include further degrees of freedom in process scheduling, hence, no direct comparability of the results, it is still much the same as in the original solution that water integration can almost (99%) eliminate wastewater generation.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was funded by the Ministry of Science and Technology of Taiwan, R.O.C. (Project Nos. MOST104-2218E-027-009 and MOST105-2221-E-027-127).

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5. CONCLUSION A mathematical optimization technique for simultaneous process scheduling and water minimization in batch processes has been developed in this paper. The overall formulation incorporates the ATM of Foo34 and a simple discrete-time scheduling model, applying to fixed flow problems with multiple water sources and sinks. The main novelty of this newly proposed approach is that scheduling is now incorporated into the targeting concept of process integration. Prior to this no such work had been reported. With the incorporation of scheduling, the truly global optimal resource targets can be identified prior to detailed network design. Thus, it is a major breakthrough in batch process integration research. In addition, the proposed approach applies to both multiproduct and multipurpose batch plants. Three case studies were solved to demonstrate its application. The results indicate the trade-off between water and storage costs and show that significant water/cost savings can be achieved through water recovery. It should be noted that a major advantage of the proposed model is its linearity, hence, guaranteed global optimality. Although the present work is limited to problems with single contaminants, there are industrial cases where the proposed MILP model is applicable, for example, the pulp and paper, painting, steel, and semiconductor industries.44 Future work will consider more detailed batch process design and variable processing time/stream flow rates (as a function of batch size). Cases with multiple contaminants/quality indices may also be addressed.

NOMENCLATURE

Indices and sets

i ∈ I = process sources j ∈ J = process sinks k ∈ K = concentration levels s ∈ S = states s ∈ Sp = product states s ∈ Sr = raw material states sin,u ∈ Bcs = tasks consuming state s sin,u ∈ Bps = tasks producing state s sin,u ∈ Sin,u * = effective states representing tasks t ∈ T = time intervals u ∈ U = processing units

Parameters

α(sin,u) = constant processing time of the task AF = annualization factor CFW = unit cost of fresh water Cfix ST = fixed cost coefficient for water storage tanks Cvar. ST = variable cost coefficient for water storage tanks CWT = unit cost for wastewater treatment Ck = concentration at level k Δt = length of time interval t Fi = flow rate of source i Fj = flow rate of sink j N = number of batches/cycles/repeating periods p = number of years r = fractional interest rate per year

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Industrial & Engineering Chemistry Research ρcs (sin,u) = fraction of state s in the input material consumed by the task ρps (sin,u) = fraction of state s in the output material produced by the task Qmax = maximum capacity of water storage at concentration k level k Qmax = maximum storage capacity for state s s Vmax = maximum capacity of unit u u Vmin = minimum capacity of unit u u Yi,k = binary parameter indicating at which concentration level the source is located Yj,k = binary parameter indicating at which concentration level the sink is located

(10) Majozi, T.; Brouckaert, C. J.; Buckley, C. A. A Graphical Technique for Wastewater Minimisation in Batch Processes. J. Environ. Manage. 2006, 78, 317−329. (11) Chen, C.-L.; Lee, J.-Y. A Graphical Technique for the Design of Water-Using Networks in Batch Processes. Chem. Eng. Sci. 2008, 63, 3740−3754. (12) Almató, M.; Espuña, A.; Puigjaner, L. Optimisation of Water Use in Batch Process Industries. Comput. Chem. Eng. 1999, 23, 1427− 1437. (13) Kim, J.-K.; Smith, R. Automated Design of Discontinuous Water Systems. Process Saf. Environ. Prot. 2004, 82, 238−248. (14) Majozi, T. An Effective Technique for Wastewater Minimisation in Batch Processes. J. Cleaner Prod. 2005, 13, 1374−1380. (15) Li, B. H.; Chang, C. T. A Mathematical Programming Model for Discontinuous Water-Reuse System Design. Ind. Eng. Chem. Res. 2006, 45, 5027−5036. (16) Liu, Y.; Li, G.; Wang, L.; Zhang, J.; Shams, K. Optimal Design of an Integrated Discontinuous Water-Using Network Coordinating with a Central Regeneration Unit. Ind. Eng. Chem. Res. 2009, 48, 10924− 10940. (17) Chen, C.-L.; Chang, C.-Y.; Lee, J.-Y. Continuous-Time Formulation for the Synthesis of Water-Using Networks in Batch Plants. Ind. Eng. Chem. Res. 2008, 47, 7818−7832. (18) Chen, C.-L.; Lee, J.-Y.; Tang, J.-W.; Ciou, Y.-J. Synthesis of Water-Using Network with Central Reusable Storage in Batch Processes. Comput. Chem. Eng. 2009, 33, 267−276. (19) Ng, D. K. S.; Foo, D. C. Y.; Rabie, A.; El-Halwagi, M. M. Simultaneous Synthesis of Property-Based Water Reuse/Recycle and Interception Networks for Batch Processes. AIChE J. 2008, 54, 2624− 2632. (20) Chen, C.-L.; Lee, J.-Y.; Ng, D. K. S.; Foo, D. C. Y. A Unified Model of Property Integration for Batch and Continuous Processes. AIChE J. 2010, 56, 1845−1858. (21) Lee, J.-Y.; Chen, C.-L.; Lin, C.-Y. A Mathematical Model for Water Network Synthesis Involving Mixed Batch and Continuous Units. Ind. Eng. Chem. Res. 2013, 52, 7047−7055. (22) Lee, J.-Y.; Chen, C.-L.; Lin, C.-Y.; Foo, D. C. Y. A Two-Stage Approach for the Synthesis of Inter-Plant Water Networks Involving Continuous and Batch Units. Chem. Eng. Res. Des. 2014, 92, 941−953. (23) Majozi, T. Wastewater Minimisation Using Central Reusable Storage in Batch Plants. Comput. Chem. Eng. 2005, 29, 1631−1646. (24) Majozi, T.; Gouws, J. F. A Mathematical Optimisation Approach for Wastewater Minimisation in Multipurpose Batch Plants: Multiple Contaminants. Comput. Chem. Eng. 2009, 33, 1826−1840. (25) Adekola, O.; Majozi, T. Wastewater Minimization in Multipurpose Batch Plants with a Regeneration Unit: Multiple Contaminants. Comput. Chem. Eng. 2011, 35, 2824−2836. (26) Chaturvedi, N. D.; Bandyopadhyay, S. Optimization of Multiple Freshwater Resources in a Flexible-Schedule Batch Water Network. Ind. Eng. Chem. Res. 2014, 53, 5996−6005. (27) Chaturvedi, N. D.; Manan, Z. A.; Wan Alwi, S. R.; Bandyopadhyay, S. Effect of Multiple Water Resources in a FlexibleSchedule Batch Water Network. J. Cleaner Prod. 2016, 125, 245−252. (28) El-Halwagi, M. M.; Manousiouthakis, V. Automatic Synthesis of Mass-Exchange Networks with Single Component Targets. Chem. Eng. Sci. 1990, 45, 2813−2831. (29) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Automated Targeting Technique for Single-Impurity Resource Conservation Networks. Part 1: Direct Reuse/Recycle. Ind. Eng. Chem. Res. 2009, 48, 7637−7646. (30) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Automated Targeting Technique for Single-Impurity Resource Conservation Networks. Part 2: Single-Pass and Partitioning Waste-Interception Systems. Ind. Eng. Chem. Res. 2009, 48, 7647−7661. (31) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R.; Pau, C. H.; Tan, Y. L. Automated Targeting for Conventional and Bilateral Property-Based Resource Conservation Network. Chem. Eng. J. 2009, 149, 87−101. (32) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R.; El-Halwagi, M. M. Automated Targeting Technique for Concentration- and Property-

Variables

δk,t = residual water flow at concentration level k in time interval t εk,t = residual impurity load at concentration level k in time interval t f i,t,k = flow of source i in time interval t at concentration level k f j,t,k = flow of sink j in time interval t at concentration level k f FW,t = flow of fresh water in time interval t f WW,t = flow of wastewater in time interval t f t,t+1,k = amount of water stored at concentration level k and transferred from time interval t to t + 1 m(sin,u,t) = amount of material used for the task at the beginning of time interval t qSC,k = water storage capacity at concentration level k qs,t = amount of state s stored at the beginning of time interval t qWS,t,k = amount of water stored at concentration level k for time interval t yi,t = binary variable indicating if source i exists in time interval t yj,t = binary variable indicating if sink j exists in time interval t yST,k = binary variable indicating the use of water storage at concentration level k y(sin,u,t) = binary variable indicating if the task starts at the beginning of time interval t

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Simultaneous Targeting and Scheduling for Batch Water Networks

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