A Mathematical Model for Water Network Synthesis Involving Mixed

Article pubs.acs.org/IECR

A Mathematical Model for Water Network Synthesis Involving Mixed Batch and Continuous Units Jui-Yuan Lee, Cheng-Liang Chen,* and Chun-Yen Lin Department of Chemical Engineering, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, 10617 Taiwan ABSTRACT: This paper presents a mathematical model for the synthesis of water networks for systems consisting of process units of different operation modes, namely, truly batch, semicontinuous, and continuous units. By treating a continuous process as a special case of a semicontinuous process, the original problem becomes to synthesize a batch water network comprised of truly batch and semicontinuous units operated cyclically with a fixed schedule. The model is formulated as a mixed-integer nonlinear program based on a unit-tank superstructure including all possible network interconnections. Three modified literature examples are used to illustrate the proposed approach, with both in-plant and interplant water integration analyzed.

1. INTRODUCTION Water is a key resource widely used in the process industry as a raw material, heat-transfer medium, mass-separating agent, etc. Common uses of water include steam stripping, liquid−liquid extraction, and a variety of washing operations. With rapid industrial growth, the increasing water demand leads to not only more fresh water consumed but also more wastewater generated and has, consequently, caused many environmental and economic problems, such as worldwide water pollution and rising costs of fresh water and effluent treatment. The latter is partly due to the (predicted) scarcities of industrial water and ever-stricter discharge regulations. As well as increased public awareness toward environmental sustainability, the abovementioned factors are among those calling for efficient and responsible utilization of water in industry. To make efficient use of water in the process industry, water recovery through the synthesis of water networks (WNs) has been commonly accepted as an effective means, with reuse, recycle, and regeneration being options for the reduction of industrial fresh-water intake and wastewater discharge.1 Over the past decades, numerous research works on WN synthesis have been reported both for continuous and for batch processes based on process integration techniques. Most of these works, along with the history of investigations and development (on WNs), are analyzed in the review papers of Bagajewicz,2 Foo,3 Jeżowski,4 and Gouws et al.5 Among the various water minimization methodologies, such as insight-based pinch analysis and mathematical optimization approaches, the latter is very useful in dealing with complex systems (e.g., multiple contaminants, cost considerations, topological constraints, limited piping connections, etc.) and often preferred when a systematic strategy leading to an optimum solution is required. Apart from in-plant water recovery, opportunities for interplant water integration may be explored to achieve further recovery in the case of industrial complexes with multiple plants or processes, where water-using operations are usually grouped in different geographical locations. The first work addressing this issue was reported by Olesen and Polley6 using a pinchbased approach. Several related works using mathematical techniques were later published.7−10 © 2013 American Chemical Society

Note that most WN works were developed for process systems containing either continuous11−13 or batch water-using units.14−21 However, there exist processes with a mixture of continuous and batch operations, for example, breweries, sugar mills, and tire-production plants. This calls for the need to develop a new model handling such processes, which is the subject of this work. In the following sections, a formal problem statement is first given, with the fundamental concepts for model formulation discussed next. The mathematical model is then presented, and its application is demonstrated using three illustrative examples.

2. PROBLEM STATEMENT The problem addressed in this paper can be formally stated as follows: Given is a set of water-using units i ∈ 0 , which consists of batch and continuous units (i ∈ 0 c). The former may operate in truly batch (i ∈ 0 b) or semicontinuous mode (i ∈ 0 sc). In addition, it is assumed in this work that batch operations are carried out cyclically with a fixed schedule. These units require water to remove a set of contaminants c ∈ * with fixed mass loads from the process materials. Available for service are a set of fresh-water sources w ∈ > with different qualities. Effluents from the units are sent to a set of wastewater disposal systems d ∈ + for final discharge or can be reused or recycled for reducing fresh-water usage and wastewater generation. To facilitate water allocation and recovery, a set of storage tanks s ∈ : for temporary water storage may be used. The objective of this work is to synthesize an optimal WN that achieves the minimum fresh-water consumption while satisfying all process constraints. A mathematical model is therefore developed for locating the maximum extent of water recovery and determining the corresponding WN configuration, the water-storage profile(s), and the required storage capacity. Special Issue: PSE-2012 Received: Revised: Accepted: Published: 7047

September 17, 2012 January 3, 2013 January 4, 2013 January 4, 2013 dx.doi.org/10.1021/ie302521v | Ind. Eng. Chem. Res. 2013, 52, 7047−7055

Industrial & Engineering Chemistry Research

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3. FUNDAMENTAL CONCEPTS In this section, the fundamental concepts on which the mathematical model is based are presented, namely, those relating to water-using unit types, the integration of units operating in different modes, and the representation of time. 3.1. Operation Modes of Water-Using Units. Figure 1 illustrates three common operation modes for water-using

Figure 2. Continuous-time representation over a cycle.

represented by the same index and set t ∈ ; . Note that time points are used to describe the water use of truly batch units and time intervals for semicontinuous units.

4. MODEL FORMULATION With all continuous units being treated as semicontinuous ones, the original problem becomes the synthesis of a WN for truly batch and semicontinuous units. A mathematical model is then developed to address the remaining problem. As an extension of the formulation of Chen et al.,21 this model consists mainly of mass-balance equations and is based on a superstructure including all feasible network connections between water-using units (Figure 3a,b) and storage tanks (Figure 4). Notation used in the formulation is given in the Nomenclature section.

Figure 1. Types of water-using operations.

units. While a continuous unit operates uninterruptedly for a long duration (e.g., 8000 h/year) spanning many operation cycles, truly batch and semicontinuous units are scheduled to operate within certain periods of time (as short as a few hours or days) and often in a cyclic manner. In addition, the operating period of either a truly batch or semicontinuous unit is always shorter than the batch cycle time. Regarding water usage, a truly batch unit takes in water at the start and discharges wastewater at the end of its operation. A typical example for this kind of operation is the batch reaction for agrochemical production in which water is used as the reaction solvent and for product washing. By contrast, water intake and discharge for continuous and semicontinuous units take place steadily during the course of the operation. Typical examples for such operations include the various extraction and washing processes in the chemical industry. 3.2. Integration of Batch and Continuous Units. For WN synthesis involving units of different operation modes, the main challenge would be to integrate these units. In this work, it is proposed to treat a continuous operation as a special case of a semicontinuous operation existing over the whole batch cycle time, as shown in Figure 1. This approach is based on the assumption that changes in the water supply are acceptable even for continuous water-using operations. Because water intake and discharge for truly batch units take place at time points and for semicontinuous units in time intervals, direct water transfers between truly batch and semicontinuous units are not happening. Therefore, water integration between these two types of units can only be carried out indirectly via waterstorage tanks. 3.3. Representation of Time for Cyclic Operation. The structure of water minimization formulations for batch processes is largely dictated by the treatment of time. In most formulations, the time horizon of interest, or the cycle time (H), is divided into several time intervals, with events taking place only at the interval boundaries. In this work, a continuous representation of time is employed in which a cycle is divided into T time intervals that are not necessarily of equal duration. As shown in Figure 2, the interval boundaries (time points) in each cycle are numbered from t = 1 to T + 1, with the latter coinciding with the start of the next cycle, i.e., t = 1. Using this time representation, both time intervals and points can be

Figure 3. Schematics of water-using units: (a) truly batch and (b) semicontinuous units.

4.1. Mass Balance for Truly Batch Units. Figure 3a shows a schematic diagram of a truly batch unit i ∈ 0 b. Its inlet water may come from other truly batch units i′ (∈0 b), storage tanks s, or fresh-water sources w, and the outlet water may be sent to other truly batch units i′, storage tanks s, or wastewater disposal systems d. Equations 1 and 2 describe the inlet and

Figure 4. Schematic of a storage tank. 7048

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outlet water flow balances for unit i ∈ 0 b at time point t, respectively. By assuming truly batch units to operate without water losses or gains, the overall water flow balance for unit i ∈ 0 b is quite simple, as given in eq 3. Note that ; SE i is the set of time point pair(s) corresponding to the start and end times of the operation of unit i ∈ 0 b. qitin =

∑

qi ′ it +

i ′∈ 0b

∑ qsit + ∑

s∈:

w∈>

∑ i ′∈ 0b

qii ′ t +

s∈:

d∈+

(3)

The lower and upper bounds for the inlet and outlet water flows of truly batch units are given by eqs 4 and 5

Q iLYitE

≤

≤

YSit

≤

qitout

Q iUY itS

≤

b

∀i∈ 0, t ∈ ;

Q iUYitE

b

∀i∈ 0, t ∈ ;

in f iin c ict ̅ =

(4) (5)

i ′∈ 0

b

qi ′ it ciout ′ ct +

∑ qsit csctout + ∑

s∈:

w∈>

cictin ≤ Cicin,maxY itS cictout

≤

Cicout,maxYitE

+

∑ d∈+

fidt

b

∀ c ∈ *, i ∈ 0 , t ∈ ;

(12)

i ′∈ 0

sc

fi ′ it c i̅ out ′ ct +

∑ fsit c sct̅ out + ∑

s∈:

w∈>

fwit Cwc (13)

∀ c ∈ *, i ∈ 0 sc, t ∈ ;

(14) (15)

∀ c ∈ *, i ∈ 0 sc, t ∈ ; (16)

4.3. Mass Balance for Storage Tanks. Figure 4 shows a schematic diagram of a storage tank s. Its inlet water may come from truly batch units i ∈ 0 b at time points or semicontinuous units i ∈ 0 sc in time intervals; the outlet water of tank s may be sent to units i ∈ 0 b or 0 sc. Equations 17 and 18 describe the water flow balances between tank s and truly batch units at time point t. Note that if there are inlet and outlet water flows for a tank at the same time point, the inlet flow is assumed to occur before the outlet flow. The water flow rate balances between tank s and semicontinuous units in time interval t are given by eqs 19 and 20. qstin =

(7)

∀ c ∈ * , i ∈ 0 b, t ∈ ;

∑

out out,max op c ict Y it ̅ ≤ Cic

(6)

; SE i

(11)

∀ i ∈ 0 sc

in in,max op c ict Y it ̅ ≤ Cic

qitincictin + Mic = qitout c out ′ ict ′ ∀ c ∈ *, i ∈ 0 , (t , t ′) ∈

∑ fist

s∈:

∀ c ∈ *, i ∈ 0 sc, t ∈ ; iop

qwit Cwc

∀ c ∈ * , i ∈ 0 b, t ∈ ;

b

fii ′ t +

in out out f iin c ict ̅ + M̅ ic = f i c ict ̅

where and are binary parameters indicating the start and end times of truly batch operations, respectively. These two constraints ensure a reasonable amount of water entering and leaving a truly batch unit and that no water enters or leaves if it is not time to begin or cease operation. Apart from the water flow balances, contaminant balances for truly batch units are also considered. For unit i ∈ 0 b, eq 6 defines the inlet balance at time point t and eq 7 the overall balance. The maximum inlet and outlet concentrations for a truly batch unit are specified by eqs 8 and 9. Note that these two constraints force the inlet and outlet concentrations of unit i ∈ 0 b to be zero when there is no water intake or discharge.

∑

(10)

∀ c ∈ *, i ∈ 0 sc, t ∈ ;

YEit

qitincictin =

fwit

w∈>

Contaminant balances for semicontinuous units are also considered. For unit i ∈ 0 sc, eq 13 defines the inlet balance in time interval t and eq 14 the overall balance, which is performed for the operating periods. The maximum inlet and outlet concentrations for a semicontinuous unit are specified by eqs 15 and 16. These two constraints, similar to eqs 8 and 9, force the inlet and outlet concentrations of unit i ∈ 0 sc to be zero when it is not in operation.

(2)

∀ i ∈ 0b, (t , t ′) ∈ ; SE i

qitin

i ′∈ 0 sc

f iin = f iout

qidt

∀i∈ 0, t ∈ ;

Q iLY itS

∑

+

s∈:

∀ i ∈ 0 sc, t ∈ ;

b

qitin = qitout ′

∑

f iout Y itop =

(1)

∑ qist + ∑

∑ fsit

fi ′ it +

i ′∈ 0 sc

∀ i ∈ 0 sc, t ∈ ;

qwit

∀ i ∈ 0 b, t ∈ ; qitout =

∑

f iin Y itop =

∑ qist

qstout =

(8) (9)

f stin =

4.2. Mass Balance for Semicontinuous Units. Figure 3b shows a schematic diagram of a semicontinuous unit i ∈ 0 sc. Its inlet water may come from other semicontinuous units i′ (∈ 0 sc), storage tanks s, or fresh-water sources w, while the outlet water may be sent to other semicontinuous units i′, storage tanks s, or wastewater disposal systems d. Equations 10 and 11 describe the inlet and outlet water flow rate balances for unit i ∈ 0 sc in time interval t, respectively. Note that Yop it is a binary parameter indicating whether unit i ∈ 0 sc operates in time interval t. With the same assumption made for truly batch units, the overall water flow rate balance for unit i ∈ 0 sc is as simple as that in eq 12.

∀ s ∈ :, t ∈ ; (17)

i ∈ 0b

∑ qsit

∀ s ∈ :, t ∈ ;

∑

∀ s ∈ :, t ∈ ;

(18)

i ∈ 0b

i ∈ 0 sc

f stout =

fist

∑ i ∈ 0 sc

fsit

(19)

∀ s ∈ :, t ∈ ; (20)

Equation 21 defines the overall water balance for tank s: the amount of water stored in tank s at a time point (t) is equal to that at the previous time point (t − 1) adjusted by the inlet and outlet water flows during time interval t − 1 (i.e., the interval between time points t − 1 and t) and those at time point t. Note that eq 21 is applicable at time points t > 1. The overall water balance for tank s at the first time point (t = 1) is given by eq 22, where T denotes the last time interval of a cycle. 7049



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qst = qs , t − 1 + (f sin, t − 1 − f sout )Δt − 1 + qstin − qstout ,t−1 ∀ s ∈ :, t ∈ ;, t > 1

min ϕ1 =

x1∈Ω1

(33) ⎧ c in , ⎪ ictin ⎪ fi , ⎪ in x1 ≡ ⎨ qit , ⎪ ⎪ ⎪ ⎩

∀s∈ : (22)

To provide sufficient time for water to be well mixed within a tank, so that the outlet concentration from the tank can be a constant in any time interval, eqs 23 and 24 are introduced to forbid the inlet and outlet water flows from occurring in the same time interval, with binary variable yinst used to indicate the occurrence of an inlet water flow to tank s in time interval t. Equation 25 states that the amount of water leaving tank s in time interval t cannot exceed the amount of water stored at time point t. The capacity constraint for storage tanks is given by eqs 26 and 27. FsLystin ≤ f stin ≤ FsUystin

∀ s ∈ :, t ∈ ;

(23)

f stout ≤ FsU(1 − ystin )

∀ s ∈ :, t ∈ ;

(24)

∀ s ∈ :, t ∈ ;

(26)

∀s∈ :

(27)

min ϕ2 =

In addition to the water balances, contaminant balances for storage tanks are also considered. Equations 28 and 29 describe the inlet balances for tank s for contaminant flows from truly batch units at time point t and from semicontinuous units in time interval t, respectively. The overall contaminant balance for tank s is given by eqs 30 and 31. Because the inlet and outlet water flows of tank s cannot exist in the same time interval, the outlet concentration from tank s in time interval t will be equal to the concentration inside tank s at time point t, as given in eq 32. in qstincsct =

∑ qist cictout

x2 ∈ Ω 2

i∈0

sc

out fist c ict ̅

∀ c ∈ *, s ∈ :, t ∈ ; (29)

out qst csct = qs , t − 1cscout, t − 1 + (f sin, t − 1 c sc̅ in, t − 1 − f sout c ̅ out, t − 1)Δt − 1 , t − 1 sc in out + qstincsct − qstoutcsct

(30)

out in in out out in in qs1cscout 1 = qsT cscT + (f sT c scT ̅ − f sT c scT ̅ )ΔT + qs1 csc1

− qsout c out 1 sc1 out out c sct ̅ = csct

∀ c ∈ *, s ∈ :

∀ c ∈ *, s ∈ :, t ∈ ;

(37)

∑ ∑ ∑ w ∈ > i ∈ 0 sc t ∈ ;

⎫ ⎪ ⎬ fwit Δt = ϕ1*⎪ ⎭ (38)

Note that the second objective function (representing the total storage capacity) can be regarded as a linear proxy for the capital cost of storage tanks. However, it is sometimes more appropriate to use a capital cost function with both fixed and variable cost terms, especially when the number of tanks is also to be optimized. Similar sequential approaches are found in the literature. Majozi18 presented a two-stage algorithm for fresh-water and reusable water-storage minimization in batch plants. In the first stage, the fresh-water requirement is minimized; in the second stage, the amount of water stored over the time horizon is minimized subject to the earlier set fresh-water target. However, the objective of the second stage may fail to eliminate the peak of the storage profile, which determines the required tank size. In addition, Shoaib et al.19 proposed a hierarchical approach for the synthesis of batch WNs. This approach involves three stages in which the fresh-water flow, the number of tanks, and the number of connections are minimized sequentially.

∀ c ∈ *, s ∈ :, t ∈ ;

∀ c ∈ *, s ∈ :, t ∈ ;, t > 1

(36)

s∈:

⎧ Equations (1)−(32) ⎪ Ω 2 = ⎨x2 ∑ ∑ ∑ qwit + ⎪ ⎩ w ∈ > i ∈ 0b t ∈ ;

i ∈ 0b

∑

∑ qscap

x2 ≡ x1 ∪ {qscap , ∀ s ∈ :}

(28) in f stin c sct ̅ =

(35)

where x1 is a vector of the variables and Ω1 a feasible solution space defined by the constraints. Because of the presence of bilinear terms in the contaminant balance equations (eqs 6, 7, 13, 14, and 28−31) and the use of binary variables for storage tanks, the model is a mixed-integer nonlinear program (MINLP). Because there would be degenerate solutions with the same minimum fresh-water consumption, a second objective has to be imposed to obtain the preferred designs among the alternatives. The second objective function used in this work is to minimize the total storage capacity (eq 36), in which case the capacity of individual tanks (Qcap s ) in eqs 26 and 27 becomes a variable (rewritten as qscap) to be optimized. Additionally, the earlier determined minimum fresh-water consumption (ϕ1*) is added as a constraint. The resulting model is also an MINLP.

(25)

∀ s ∈ :, t ∈ ;, t > 1

⎫ ⎪ ⎪ ⎪ qitout , qidt , qii ′ t , qist , qsit , qst , qstin , qstout , qwit , ystin ⎬ ⎪ ⎪ ∀ c ∈ *, d ∈ +, i , i′ ∈ 0, s ∈ :, t ⎪ ∈ ;, w ∈ > ⎭ (34)

in out in out in out cictout , c ict ̅ , c ict ̅ , csct , csct , c sct ̅ , c sct ̅ , out in out f i , fidt , fii ′ t , fist , fsit , f st , f st , fwit ,

Ω1 = {x1|Equations (1)−(32)}

qs , t − 1 + (f sin, t − 1 − f sout )Δt − 1 + qstin ≤ Q scap ,t−1

in out qsT + (f sT − f sT )ΔT + qsin1 ≤ Q scap

w ∈ > i ∈ 0 sc t ∈ ;

(21)

in out qs1 = qsT + (f sT − f sT )ΔT + qsin1 − qsout 1

f stout Δt ≤ qst

∑ ∑ ∑ qwit + ∑ ∑ ∑ fwit Δt

w ∈ > i ∈ 0b t ∈ ;

(31) (32)

5. ILLUSTRATIVE EXAMPLES Three literature examples are solved to illustrate the proposed approach. In these examples, the model is implemented in the

4.4. Objective Functions. For maximum water recovery, the objective function is to minimize the total fresh-water consumption for the water-using operations: 7050



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GAMS environment22 on a Core 2, 2.00 GHz processor with BARON as the MINLP solver. 5.1. Example 1: Integration of Truly Batch and Continuous Units. The first example is adapted from Majozi17 by adding a continuous unit (F) to the five truly batch units (A−E). Table 1 shows the operating data for these

Table 2. Operation Parameters for Example 1 time point/interval parameter

unit

t=1

t=2

t=3

t=4

t=5

t=6

YSit

A B C D E A B C D E F

1 1 0 0 0 0 0 0 0 1 1

0 0 0 1 0 0 0 0 0 0 1

0 0 0 0 0 1 0 0 0 0 1

0 0 1 0 0 0 1 0 0 0 1

0 0 0 0 0 0 0 1 0 0 1

0 0 0 0 1 0 0 0 1 0 1

Table 1. Operating Data for the Water-Using Operations for Example 1 YEit

limiting concentration (ppm) unit A B C D E F

limiting flow (t)

[300, 400] [300, 400]

time (h)

Cin,max ic

Cout,max ic

mass load

start

end

0 250 100 250 100 100

100 510 100 510 100 250

100 kg 72.8 kg 0 kg 72.8 kg 0 kg 25 kg/h

0 0 4 2 6 0

3 4 5.5 6 7.5 7.5

Yop it

water-using units, with the corresponding Gantt chart shown in Figure 5. The cycle time for repeated batch operation is 7.5 h. According to the scheduled start and end times of operation for units A−E, the cycle time is divided into six time intervals with six time points used, and the values of binary parameters YSit, YEit , and Yop it are set as in Table 2. Note that unit F is treated as a semicontinuous unit operating in all of the time intervals. In this example, a single uncontaminated fresh-water source (Cwc = 0) is available for use. Before considering water integration, in which case all waterusing units use fresh water without reuse/recycle, the minimum fresh-water requirement is calculated to be 2635.49 t per cycle. With two storage tanks available to facilitate water recovery, the model involves 486 constraints, 615 continuous variables, and 12 binary variables. It is solved in 4 CPU s with the minimum fresh-water consumption determined to be 1150 t per cycle. This corresponds to a 56.36% reduction in the fresh-water use compared to the situation without water recovery. Figure 6 shows the optimal WN configuration. In this arrangement, only units A and F use fresh water; both tanks are employed for water reuse, and there is no direct water transfer between units. Tank 1 (ST1) stores effluents from units A, C, and E for reuse in units C, E, and F, while tank 2 (ST2) allows effluents from units A and F to be reused in units B−D. Figure 7 shows the water-storage profiles of the tanks. The capacities required are 720 t for ST1 and 280 t for ST2.

Figure 6. WN configuration for example 1.

5.2. Example 2: Integration of Semicontinuous and Continuous Units. The second example is adapted from Kim and Smith15 by adding a continuous unit (U5) to the four semicontinuous units (U1−U4). Table 3 shows the limiting conditions and timing data for the water-using operations, with the corresponding Gantt chart shown in Figure 8. The cycle time for repeated batch operation is 5 h. According to the scheduled operating periods of units U1−U4, the cycle time is divided into four time intervals with four time points used, and the values of binary parameter Yop it are set as in Table 4. In this example, a single pure fresh-water source is available for use. Before carrying out water integration, the minimum freshwater requirement is calculated to be 480 t per cycle. With the

Figure 5. Gantt chart for repeated batch operation for example 1. 7051



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Figure 7. Water-storage profiles for example 1.

Table 3. Limiting Water Data for Example 2

Figure 9. WN configuration for example 2.

limiting concentration (ppm)

time (h)

unit

Cin,max ic

Cout,max ic

mass load (kg/h)

start

end

U1 U2 U3 U4 U5

0 50 50 400 200

100 100 800 800 400

2 5 30 4 20

0 1 3 1 0

1 3.5 5 3 5

Figure 10. Water-storage profile for example 2.

previous two, with the units in example 1 constituting the first plant (plant 1) and those in example 2 the second plant (plant 2). However, plant 1 is operated in overlapping batches, which decreases the cycle time from 7.5 to 5 h. According to the Gantt charts for both plants shown in Figure 11, the common Figure 8. Gantt chart for repeated batch operation for example 2.

Table 4. Operation Parameters for Example 2 time interval parameter

unit

t=1

t=2

t=3

t=4

Yop it

U1 U2 U3 U4 U5

1 0 0 0 1

0 1 0 1 1

0 1 1 0 1

0 0 1 0 1

use of one storage tank, the model involves 160 constraints, 255 continuous variables, and 4 binary variables. It is solved in 1 CPU s with the minimum fresh-water consumption determined to be 356.25 t per cycle. This corresponds to a 25.78% reduction in the fresh-water use compared to the case without water recovery. Figure 9 shows the optimal WN configuration. Note that most water reuse is carried out directly between units; only part of the effluent from U2 is stored for reuse in U5. Figure 10 shows the water-storage profile of the tank. The capacity required is 47.5 t. 5.3. Example 3: Interplant Integration of All Three Types of Units. To demonstrate the applicability of the proposed model for WN synthesis involving truly batch, semicontinuous, and continuous units, a combined two-plant example is considered. This example is a combination of the

Figure 11. Gantt charts for example 3.

cycle time of 5 h is divided into eight time intervals with eight time points defined. Both in-plant and interplant integration scenarios are analyzed in this example. Prior to exploration of the water reuse/recycle opportunities, the minimum fresh-water requirements in plants 1 and 2 are 7052



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Table 5. Operation Parameters for Example 3 time point/interval parameter

unit

t=1

t=2

t=3

t=4

t=5

t=6

t=7

t=8

YSit

A B C D E A B C D E F U1 U2 U3 U4 U5

1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1

0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1

0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1

0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1

0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1

0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1

0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1

0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1

YEit

Yop it

Figure 12. Interplant WN configuration for example 3.

Figure 13. Water flow rate profiles between continuous units and tanks.

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consumption and network complexity. There are also cases with the trade-off between the water recovery potential and the requirement for storage or regeneration systems to be analyzed.24 Hence, a more comprehensive formulation would be required to address the conflicting objectives in WN synthesis.

calculated to be 2385.49 and 480 t per cycle, respectively, with a total of 2865.49. This is taken as the base case. In-plant water integration is first considered for both plants. In the presence of two storage tanks, the minimum fresh-water consumption of plant 1 is found to be 1000 t per cycle. Note that optimizing water utilization for plant 1 alone needs only seven time intervals and seven time points. The corresponding model has 565 constraints, 717 continuous variables, and 14 binary variables and is solved in 3 CPU s. For plant 2, the minimum fresh-water consumption is the same as that in example 2, i.e. 356.25 t per cycle. The total of 1356.25 t per cycle corresponds to a 52.67% reduction in the fresh-water use compared to the base case. Detailed results for water integration in plant 1 are not shown for brevity. Interplant water integration is then considered. Table 5 shows the values of binary parameters (YSit, YEit , and Yop it ) for this case. With the use of two centralized storage tanks, the overall model involves 869 constraints, 1429 continuous variables, and 16 binary variables. The minimum fresh-water consumption of 1185 t per cycle is obtained in 18 CPU s. This result corresponds to a 58.65% reduction in the fresh-water use compared to the base case and shows an almost further 6% reduction compared to the in-plant integration case. Figure 12 shows the optimal interplant WN configuration, with detailed flow rate profiles between continuous units and tanks shown in Figure 13. Note that in plant 1 the fresh-water intake is more than the wastewater discharge, while a contrary situation is found in plant 2. This indicates a net cross-plant water flow of 372.4 t per cycle from plant 1 to 2. Figure 14 shows the waterstorage profiles of the tanks. The required capacities of ST1 and ST2 are 1148 and 245.19 t, respectively.

6. CONCLUSIONS In this paper, with a continuous process treated as a special case of a semicontinuous process, a mathematical model has been developed for the synthesis of WNs for truly batch and semicontinuous units in cyclic operation with a fixed schedule. Three examples adapted from literature were solved to illustrate the proposed approach. The results show that significant reductions in fresh-water consumption as well as wastewater generation can be achieved through in-plant and interplant water integration. Future work will focus on other cases of WN synthesis with more different types of units, e.g., truly batch/ semicontinuous water sources and sinks and their various combinations, along with the development of suitable case studies. The incorporation of regeneration processes and a scheduling framework into the model also remains as the subject of future work. It should be noted that having a flexible schedule can further reduce the need for storage.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +886-2-33663039. Fax: +886-2-23623040. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the National Science Council for supporting this research under Grants NSC101-3113-E-002-004 and NSC99-2221-E-002-187-MY3.

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NOMENCLATURE

Indices and Sets

c ∈ * = contaminants d ∈ + = wastewater disposal systems i ∈ 0 = water-using units i ∈ 0 b ⊂ 0 = truly batch units i ∈ 0 sc ⊂ 0 = semicontinuous units i ∈ 0 c ⊂ 0 sc = continuous units s ∈ : = storage tanks t ∈ ; = time points/intervals (t, t′) ∈ ; SE i = start and end time points for the operation of truly batch unit i t ∈ ; op i = operating periods of semicontinuous unit i w ∈ > = fresh-water sources

Figure 14. Water-storage profiles for example 3.

Parameters

5.4. Discussion. All of the minimum fresh-water consumptions reported in examples 1−3 agree with the water targets obtained using the insight-based targeting methods (such as the time-dependent water cascade analysis technique23) and have proven to be globally optimal. However, because of the MINLP formulation, global optimality for the minimization of storage capacity cannot be guaranteed. Instead of taking water reduction as the overriding concern, the minimum fresh-water constraint may sometimes be relaxed to further simplify the WN for the trade-off between water

Cin,max = maximum inlet concentration of contaminant c for ic unit i Cout,max = maximum outlet concentration of contaminant c ic for unit i Cwc = concentration of contaminant c in fresh-water source w FLs = lower bound for the inlet water flow rate to tank s FUs = upper bound for the inlet/outlet water flow rate of tank s Mic = mass load of contaminant c in unit i ∈ 0 b 7054



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M̅ ic = mass load of contaminant c in unit i ∈ 0 sc QLi = lower bound for the inlet/outlet water flow of unit i ∈ 0b QUi = upper bound for the inlet/outlet water flow of unit i ∈ 0b Qcap s = storage capacity of tank s YEit = binary parameter indicating whether truly batch unit i ceases to operate at time point t Yop it = binary parameter indicating whether semicontinuous unit i operates in time interval t YSit = binary parameter indicating whether truly batch unit i begins to operate at time point t Δt = length of time interval t

Article

REFERENCES

(1) Wang, Y. P.; Smith, R. Wastewater minimisation. Chem. Eng. Sci. 1994, 49 (7), 981−1006. (2) Bagajewicz, M. A review of recent design procedures for water networks in refineries and process plants. Comput. Chem. Eng. 2000, 24 (9−10), 2093−2113. (3) Foo, D. C. Y. State-of-the-art review of pinch analysis techniques for water network synthesis. Ind. Eng. Chem. Res. 2009, 48 (11), 5125− 5159. (4) Jeżowski, J. Review of water network design methods with literature annotations. Ind. Eng. Chem. Res. 2010, 49 (10), 4475−4516. (5) Gouws, J. F.; Majozi, T.; Foo, D. C. Y.; Chen, C.-L.; Lee, J.-Y. Water minimization techniques for batch processes. Ind. Eng. Chem. Res. 2010, 49 (19), 8877−8893. (6) Olesen, S. G.; Polley, G. T. Dealing with plant geography and piping constraints in water network design. Process Saf. Environ. Prot. 1996, 74 (4), 273−276. (7) Chew, I. M. L.; Tan, R.; Ng, D. K. S.; Foo, D. C. Y.; Majozi, T.; Gouws, J. Synthesis of direct and indirect interplant water network. Ind. Eng. Chem. Res. 2008, 47 (23), 9485−9496. (8) Lovelady, E. M.; El-Halwagi, M. M. Design and integration of eco-industrial parks for managing water resources. Environ. Prog. Sustain. Energy 2009, 28 (2), 265−272. (9) Chen, C.-L.; Hung, S.-W.; Lee, J.-Y. Design of inter-plant water network with central and decentralized water mains. Comput. Chem. Eng. 2010, 34 (9), 1522−1531. (10) Rubio-Castro, E.; Ponce-Ortega, J. M.; Nápoles-Rivera, F.; ElHalwagi, M. M.; Serna-González, M.; Jiménez-Gutiérrez, A. Water integration of eco-industrial parks using a global optimization approach. Ind. Eng. Chem. Res. 2010, 49 (20), 9945−9960. (11) Gunaratnam, M.; Alva-Argáez, A.; Kokossis, A.; Kim, J.-K.; Smith, R. Automated design of total water systems. Ind. Eng. Chem. Res. 2005, 44 (3), 588−599. (12) Karuppiah, R.; Grossmann, I. E. Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng. 2006, 30 (4), 650−673. (13) Ahmetović, E.; Grossmann, I. E. Global superstructure optimization for the design of integrated process water networks. AIChE J. 2011, 57 (2), 434−457. (14) Almató, M.; Espñna, A.; Puigjaner, L. Optimisation of water use in batch process industries. Comput. Chem. Eng. 1999, 23 (10), 1427− 1437. (15) Kim, J.-K.; Smith, R. Automated design of discontinuous water systems. Process Saf. Environ. Prot. 2004, 82 (3), 238−248. (16) Li, B.-H.; Chang, C.-T. A mathematical programming model for discontinuous water-reuse system design. Ind. Eng. Chem. Res. 2006, 45 (14), 5027−5036. (17) Majozi, T. Wastewater minimisation using central reusable water storage in batch plants. Comput. Chem. Eng. 2005, 29 (7), 1631−1646. (18) Majozi, T. Storage design for maximum wastewater reuse in multipurpose batch plants. Ind. Eng. Chem. Res. 2006, 45 (17), 5936− 5943. (19) Shoaib, A. M.; Aly, S. M.; Awad, M. E.; Foo, D. C. Y.; ElHalwagi, M. M. A hierarchical approach for the synthesis of batch water network. Comput. Chem. Eng. 2008, 32 (3), 530−539. (20) 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 (20), 7818−7832. (21) 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 (1), 267−276. (22) Rosenthal, R. E. GAMSA User’s Guide; GAMS Development Corp.: Washington, DC, 2008. (23) Foo, D. C. Y.; Manan, Z. A.; Tan, Y. L. Synthesis of maximum water recovery network for batch process systems. J. Clean. Prod. 2005, 13 (15), 1381−1394. (24) Nun, S.; Foo, D. C. Y.; Tan, R. R.; Razon, L. F.; Egashira, R. Fuzzy automated targeting for trade-off analysis in batch water networks. Asia-Pac. J. Chem. Eng. 2011, 6 (3), 537−551.

Variables

cinict = inlet concentration of contaminant c to unit i ∈ 0 b at time point t b cout ict = outlet concentration of contaminant c from unit i ∈ 0 at time point t cini̅ ct = inlet concentration of contaminant c to unit i ∈ 0 sc in time interval t sc cout i̅ ct = outlet concentration of contaminant c from unit i ∈ 0 in time interval t cinsct = inlet concentration of contaminant c to tank s at time point t cout sct = outlet concentration of contaminant c from tank s at time point t cins̅ ct = inlet concentration of contaminant c to tank s in time interval t cout s̅ ct = outlet concentration of contaminant c from tank s in time interval t f ini = inlet water flow rate to unit i ∈ 0 sc in time interval t sc f out i = outlet water flow rate from unit i ∈ 0 in time interval t f idt = water flow rate from unit i ∈ 0 sc to disposal system d in time interval t f ii′t = water flow rate from unit i ∈ 0 sc to unit i′ ∈ 0 sc in time interval t f ist = water flow rate from unit i ∈ 0 sc to tank s in time interval t fsit = water flow rate from tank s to unit i ∈ 0 sc in time interval t finst = inlet water flow rate to tank s in time interval t fout st = outlet water flow rate from tank s in time interval t f wit = water flow rate from fresh-water source w to unit i ∈ 0 sc in time interval t qinit = inlet water flow to unit i ∈ 0 b at time point t b qout it = outlet water flow from unit i ∈ 0 at time point t qidt = water flow from unit i ∈ 0 b to disposal system d at time point t qii’t = water flow from unit i ∈ 0 b to unit i′ ∈ 0 b at time point t qist = water flow from unit i ∈ 0 b to tank s at time point t qsit = water flow from tank s to unit i ∈ 0 b at time point t qst = amount of water stored in tank s at time point t qinst = inlet water flow to tank s at time point t qout st = outlet water flow from tank s at time point t qwit = water flow from fresh-water source w to unit i ∈ 0 b at time point t yinst = binary variable indicating whether there is inlet water flow to tank s in time interval t 7055


A Mathematical Model for Water Network Synthesis Involving Mixed

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