Continuous-Time Formulation for the Synthesis of Water-Using

7818

Ind. Eng. Chem. Res. 2008, 47, 7818–7832

Continuous-Time Formulation for the Synthesis of Water-Using Networks in Batch Plants Cheng-Liang Chen,* Chia-Yuan Chang, and Jui-Yuan Lee Department of Chemical Engineering National Taiwan UniVersity Taipei 10617, Taiwan, Republic of China

Water minimization is conducted by exploiting all possibilities of water reuse and recycle to reduce the freshwater consumption, as well as the wastewater generation. Because the starting and finishing times of batch water-using tasks are dependent on the production schedule as the inherent time dependence in batch processes, storage facilities are commonly equipped for the temporary storage of reusable water to partially bypass the time limitation. With a fixed production schedule, this paper presents a mathematical formulation for the synthesis of water-using networks in batch plants. Superstructures that incorporate all possible flow connections are built for modeling the batch water system. The proposed formulation is based on a continuoustime representation where different design objectives have been considered for an applicable network configuration. The design problems for the minimization of freshwater consumption, storage capacity, and the amount of connecting flows are formulated as nonlinear programs (NLPs), whereas the design problem for minimizing the number of connections will be a mixed-integer nonlinear program (MINLP). Representative examples from literature are provided to demonstrate the effectiveness of proposed formulation. Furthermore, the application of a fictitious contaminant is also developed, to address the forbidden match between assigned water-using tasks. 1. Introduction With the rapid industrial growth, many practitioners and savants realized the tremendous consumption of resources and utilities. For this reason, process integration has received more and more attention over the past decades and it is believed that the overall utilization efficiency can be increased by the optimal network design. In addition to heat integration, water minimization is one of the most popular issues to be studied in such a research field. An applicable strategy of water conservation not only reduces the operating cost but also reduces the environmental impact. However, the methodologies on water minimization are mainly concerned with continuous processes,1-3 because such processes are in the majority and usually have larger capacities. Because batch processes have become the ordinary works in the production of specialty chemicals of high commercial value, it is also necessary to develop systematic approaches for water minimization in batch plants. In contrast with their continuous counterpart, batch processes are more complicated to design, because both the operating conditions and resource demands are variable with time. Wang and Smith4 presented a systematic procedure with new graphical representations to target the minimum wastewater for a given set of batch water-using processes, in which time is treated as the primary constraint and concentration driving force is treated as the secondary constraint. This approach is applicable to semicontinuous operations, because water reuse is allowed between operations with overlapping durations. More recently, Majozi et al.5 developed a similar technique for freshwater and wastewater minimization in completely batch operations, where no reuse potential could be realized during the course of the process. Foo et al.6 proposed the application of water cascade analysis for the synthesis of maximum water recovery network in batch processes, and the authors also proved that the targeting technique from continuous processes can be validly extended to batch processes. The major drawback of graphical techniques * To whom correspondence should be addressed. Tel.: 886-223636194. Fax: 886-2-23623040. E-mail address: [email protected].

is the limitation in dimensionality, which makes it difficult to examine the multiple contaminant problem. In contrast, the ability to handle the multidimensionality of multiple contaminant problems with relative ease is exactly the advantage of mathematical techniques. Almato et al.7 presented a methodology for water-use optimization in batch process industries and a mathematical model was formulated to describe the water-reuse system through storage tanks. Furthermore, the connections between available tanks and the water streams are determined under different criteria, such as freshwater demand, water cost, utility demand of water streams, and water-reuse network cost. Majozi8-10 published a series of works for water minimization in multipurpose batch processes based on a continuous-time formulation for equipment scheduling. The author also proposed a two-stage algorithm to minimize the reusable water storage capacity while minimizing the amount of freshwater requirement and wastewater generation. Li and Chang11 proposed a general mathematical programming model for the design of discontinuous water-reuse systems. The flow and concentration equalization options are also incorporated into the superstructure to address the practical needs in wastewater treatment. Cheng and Chang12 developed a general mathematical model in which three optimization problems, including the batch schedule, the water-reuse network, and the wastewater-treatment network, are incorporated for generating the integrated water networks in batch processes. Shoaib et al.13 introduced a threestage hierarchical approach for the synthesis of cost-effective batch water networks, where the key assumption is that all water reuse/recycle between water sources and sinks is conducted between two consecutive batches of operation via water storage to avoid scheduling problems. Gouws and Majozi14 presented a mathematical technique for the minimization of multiplecontaminant wastewater where there are multiple storage vessels. This technique was extended to include operations where wastewater produced in one batch is reusable as feed for subsequent batches of the same product, thereby allowing an almost zero-effluent fashion.

10.1021/ie800573r CCC: $40.75  2008 American Chemical Society Published on Web 09/23/2008

Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008 7819

Because of the inherent time dependence of batch processes, for most mathematical approaches, the time representation is the key point of problem formulation. Generally speaking, there are two types of time representations to be employed: discrete time representations and continuous time representations. The former is conceptually simpler but entails a notably larger number of variables (especially binary variables), whereas the latter would complicate the formulation but reduces the number of variables effectively. Therefore, the mathematical formulation presented by Cheng and Chang12 may experience higher computation intensity when applied to large-scale problems, because their formulation is based on a discrete time model. On the other hand, the works developed by Majozi8-10 and Gouws and Majozi14 are based on a continuous time model, which renders their formulation amenable to large-scale problems as encountered in practice. Unfortunately, the drawback of their works is that only short-term problems had been discussed: the periodic problem was not addressed. This paper presents a mathematical approach for the synthesis of batch water-using networks, in which the scheduling framework is based on the continuous Resource-Task-Network (RTN) formulation.15,16 To concentrate the attention on the investigation of different scenarios in batch water systems, the problem has been simplified with the redefinition of binary variables to corresponding binary parameters under a fixed production schedule. Although it has been well-discussed that the presence of storage facilities can dramatically reduce the freshwater consumption in industrial plants, there is no systematic research on the subject of forbidden match between assigned water-using tasks. However, sometimes such practical constraints are essential to be considered because of some probable pollution or operational problems. Therefore, this work also aims to develop an effective technique to address the forbidden match. Illustrative examples from the literature are provided to demonstrate the adequacy of proposed formulation. 2. Problem Statement The synthesis problem of batch water-using networks addressed in this paper can be briefly stated as follows. Givens are a set of water-using tasks i ∈ I to be executed in their respective processing units for the removal of a set of transferable contaminants c ∈ C with known mass loads M(load) ; a set ic of water sources w ∈ W with specific concentrations Cwc to be purchased for utilization and some of them supply freshwater w ∈ F ⊆ W; a set of storage tanks s ∈ S to be equipped for the temporary storage of reusable water. The production schedule is assumed to be fixed; that is to say, the starting and finishing times of each task are specified a priori and assigned to corresponding time points t ∈ u. Furthermore, the maximum permissible inlet and outlet concentrations for each task (C(in) ic,max (out) and Cic,max ) are also defined as the constraints on concentration driving force. The objective is to determine the time-dependent operating strategy, which targets the minimum freshwater requirement, and the solution includes the overall freshwater consumption, required capacities of storage tanks and the network configuration. Note that the investment costs will not be tackled in this paper, although the comprehensive of economic objectives into optimization is part of the motivation for applying the mathematical programming approach. In contrast to the noticeable tradeoff between the utility consumption and the capital cost for heat exchanger areas in heat integration, the capital cost for processing units in water system is irrelevant to the freshwater consumption and mostly fixed. While there still some elements,

Figure 1. Superstructure of water-using tasks.

Figure 2. Superstructure of storage tanks.

such as the capital cost for storage tanks and the piping cost, do compromise the freshwater consumption in a way, they are just for a smaller part. On the other hand, the main points of this work are to synthesize applicable water-using networks with the minimum freshwater consumption and to analyze the impact of present storage tanks on freshwater reduction. For this reason, the investment costs are temporarily left out from the problem formulation and will be kept for future work. 3. Superstructures and Mathematical Formulation The mathematical formulation is presented in two parts for different types of operations: The first part addresses the single batch, and the second portion examines a cyclic batch. The modeling of batch water system mainly consists of the material balances around tasks (processing units) and storage tanks. Besides the balance equations, reasonable lower and upper bounds for connecting flows between processing units and storage tanks are also set to delimit an appropriate searching space and accelerate the convergence of optimization. All sets, parameters, and variables are listed in the Nomenclature section. 3.1. Formulation for Single Batch. “Single batch” means a certain time horizon of interest and some tasks are proceeding within it, which is analogous to a short-term problem in production scheduling. 3.1.1. Material Balances for Water-Using Tasks in Single Batch. The material balances for water-using tasks are based on the superstructure shown in Figure 1, which incorporates all possible flow connections. For the time dependence in j itt′ are predefined batch processes, a set of binary parameters N to indicate the beginning and the end of all tasks. Specifically, j itt′ ) 1, provided that task i is proceeding within the time N interval from time point t to t′. Note that the difference between time points t′ and t reflects the required duration of task i to achieve the desired effect (e.g., mass transfer or cleanliness of the vessel). Those binary parameters are quite significant, because they represent the distribution of tasks over the time horizon. Constraint 1 states the water balance around the mixing point M before the processing unit of task i: The inlet water to task i at time point t comprises the water directly reused from other task i′, the water reused from storage tank s, and the water supplied from water source w. Similarly, constraint 2 states the water balance around the splitting point S after the processing unit of task i: The outlet water from task i at time point t′ is either directly reused in other task i′, sent to storage tank s for

7820 Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008

later reuse, or discharged to end-of-pipe treatment d. Constraint 3 depicts the water balance throughout the processing unit of task i, where both amounts of water generation and loss are assumed to be known, and it is obvious that constraint 3 is effective only when task i is under operation during the time j itt′ ) 1. interval t to t′, i.e., N

∑f

∑f

∑f

∑f

fit(in) )

iit +

(out) fit )

s∈S

ii′t′ +

wit

∀ i ∈ I, t ∈ T

∑f

idt′

∀ i ∈ I, t ′ ∈ T (2)

(1)

w∈W

ist′ +

i′∈I

∑f

sit +

i∈I

s∈S

d∈D

(in) ) fit(in)Cict

(out) + Fi(gen) + Fi(loss))Nitt′ ) 0 (fit(in) - fit

Fmin i

∑N

∑N

t′∈T

t′∈T

t′>t

t′>t

(in) max itt′ e fit e Fi

itt′

∑N

∑N

t∈T

t∈T

t t), the amount of water leaving at time point t′ will be bounded in a reasonable range. Similarly, the j itt′ in constraint 5 must not greater than one, summation of N because task i can be only finished once at the same finishing point t′. On the other hand, these two constraints ensure that no water passes in and out if task i does not start at time point t nor finish at time point t′. Fmin i

straint 9 states the contaminant balance around the mixing point before the processing unit of task i at time point t. Constraint 10 depicts the contaminant balance throughout the processing unit of task i during the time interval t to t′, in which both concentrations of water generation and loss are assumed to be known parameters. Note that the contaminant balance around the splitting point after the processing unit of task i is omitted from the problem formulation, because all branches are the same concentration. Furthermore, constraints 11 and 12 give the maximum permissible inlet and outlet concentrations of contaminant c for task i, respectively.

∀ i, i ′ ∈ I, t ∈ T

(6)

∑f

(out) sitCsct +

s∈S

∀ c ∈ C, i ∈ I, t ∈ T

(11)

∀ c ∈ C, i ∈ I, t ′ ∈ T

(12)

(in) (in) Cict e Cic,max (out) (out) Cict e Cic,max

3.1.2. Material Balances for Storage Tanks in Single Batch. Based on the superstructure shown in Figure 2, the material balances for storage tanks are similar to those for waterusing tasks. Constraints 13 and 14 provide the water balances around the mixing and splitting points of storage tank s (that is, for the inlet and outlet flows of storage tank s) at time point t. Obviously, the inlet flow of a storage tank does not include any water from water source w and the outlet flow of a storage tank will not be discharged to end-of-pipe treatment d. This is because storage tanks are equipped to enhance the possibility of water reuse, and, therefore, only the reusable used water must be stored. Constraint 15 describes the amount of water remaining in storage tank s at time point t is equal to the water accumulated from previous time point t - 1 adjusted by the input and output at time point t. Note that the first term in the right-hand side of constraint 15 is present for t > 1, because it is assumed that a single batch starts with zero storage. Constraint 16 ensures that the amount of water entering storage tank s at time point t will not exceed the maximum capacity of storage tank s, and constraint 17 ensures a reasonable amount of water remaining in storage tank s at time point t.

∑f

fst(in) )

∑f

ist +

s′st

i∈I

∀ s ∈ S, t ∈ T

(13)

∀ s ∈ S, t ∈ T

(14)

s′∈S s′*s

fst(out) )

∑f

sit +

∑f

ss′t

s′∈S s′*s

∑N

itt′

∀ i, i ′ ∈ I, t ∈ T

(7)

t′∈T t′>t

Similarly, constraint 8 gives an upper bound for the water discharged from task i to end-of-pipe treatment d at time point t′. fidt′ e Fmax i

∀ c ∈ C, i ∈ I,t ∈ T (9)

∀ c ∈ C, i ∈ I, t, t ′ ∈ T, t < t′ (10)

Mic(loan))Nitt′ ) 0

t′1 + fst(in) - fst(out)

∀ s ∈ S, t ∈ T

(15)

fst(in) e Fsmax

∀ s ∈ S, t ∈ T

(16)

Fst e Fsmax

∀ s ∈ S, t ∈ T

(17)

Constraint 18 gives an upper bound for the water stored from task i to storage tank s at time point t, and constraint 19 gives an upper bound for the water reused from storage tank s to task i at time point t.

t∈T t1 + fst Csct - fsc Csct

(in) (in) e Csc,max Csct

∑N

(in) max itt′ e fit e Fi

fst(out) )

∑f

sit +

i∈I

∑f

ss′t

s′∈S s′*s


|

|

Fst ) Fs,T-1 t)1 + Fs,T-1 t>1 + fst(in) - fst(out)

∀ s ∈ S, t ∈ T* (37)

fst(in) e Fsmax

∀ s ∈ S, t ∈ T*

(38)

Fst e Fsmax

∀ s ∈ S, t ∈ T*

(39)

Constraint 40 gives an upper bound for the water stored from task i to storage tank s at time point t, and constraint 41 gives an upper bound for the water reused from storage tank s to task i at time point t. fist e Fmax i

∑ Nj

∀ i ∈ I, s ∈ S, t ∈ T*

(40)

∑ Nj

∀ i ∈ I, s ∈ S, t ∈ T*

(41)

it′t

t′∈T*

fsit e Fsmax

itt′

t′∈T*

Constraints 42 and 43 provide the contaminant balances around storage tank s at time point t, and constraint 44 gives the maximum permissible inlet concentration of contaminant c for storage tank s. (in) ) fst(in)Csct

∑f

(out) istCict +

i∈I

∑f

(out) s′stCsct

s′∈S s′*s

∀ c ∈ C, s ∈ S, t ∈ T* (42)

|

|

(out) (out) (out) (in) (in) ) Fs,T-1Csc,T-1 FstCsct t)1 + Fs,t-1Csc,t-1 t>1 + fsc Csct (out) fst(out)Csct ∀ c ∈ C, s ∈ S, t ∈ T*(43)

∀ c ∈ C, s ∈ S, t ∈ T*

(in) (in) e Csc,max Csct

min J1† )

x∈Ω1†

{ |∑ ∑ ∑

Ω2† ) Ω1† ∩ x′

P3 :

(for single batch) (47)

Ω1,cyclic ) {x|constraints 23-44}

(for cyclic batch) (48)

Although solving P1 is useful to locate the minimum freshwater consumption, it usually results in multiple solutions for network configuration, because the alternatives are relatively broad when merely considering the freshwater usage. For a preferable design, one can minimize the capacity of storage tanks as the secondary objective in another design problem (P2) under the minimum freshwater consumption J/1† obtained in P1. Here, the required capacity of each storage tank Qs is considered to be a design variable and the searching space is contracted to Ω2† with the additional constraints.

∑Q

s

s∈S

† ∈ {single,cyclic}

(49)

ist +

i∈I s∈S t∈T

∑∑∑f

† ∈ {single,cyclic} (52)

ss′t

s∈S s′∈S t∈T


∀ d ∈ D, i ∈ I, t ∈ T

(54)

∀ i, i ′ ∈ I, t ∈ T

(55)

fist (fist - fisL) g 0

∀ i ∈ I, s ∈ S, t ∈ T

(56)

fsit (fsit - fsiL) g 0

∀ i ∈ I, s ∈ S, t ∈ T

(57)

∀ s, s ′ ∈ S, t ∈ T

(58)

∀ i ∈ I, t ∈ T, w ∈ W

(59)

fidt (fidt - fidL) g 0 L fii′t (fii′t - fii )g0

L fss′t (fss′t - fss )g0

Ω1,single ) {x|constraints 1-22}

∑∑∑f

∀ s ∈ S, t ∈ T }

(in) (out) (in) (out) fit(out), fst(in), fst(out), Fst ;Cict , Cict , Csct , Csct ;

min J2† )

ii′t +

i∈I i′∈I t∈T

Ω3† ) Ω2† ∩ {x|Fst e Qs∗ ;

x ≡ {fidt, fii′t, fist, fsit, fss′t, fwit, fit(in),

x′∈Ω2†

∑∑∑f

sit +

(45)

P2 :

min J3† )

x∈Ω3†

∑∑∑f

w∈F i∈I t∈T

∀c ∈ C,d ∈ D, i, i ′ ∈ I, s ∈ S, t ∈ T, w ∈ W } (46)


The solution of P2 actually gives a more-suitable design. Nevertheless, there are still numerous implementations of the pipeline for the minimized freshwater consumption and storage capacity, especially in some large-scale problems with complex connections. To further reduce the alternatives and improve the network configuration, there are two options for the tertiary objective. Considering that the piping cost is roughly proportional to the throughput of piping system, the first option is to minimize the amount of connecting flows, as stated in P3. Note that the summation of freshwater consumed and wastewater discharged over the time horizon is omitted from the objective function because both of them have been determined in P1 already. Here, the capacity of each storage tank is limited by its minimum requirement Q/s obtained in P2. Besides, constraints 54-59 (shown as follows) could be appended to the problem formulation for the elimination of some uneconomically small flows, if necessary. Those constraints are used to ensure that the flow in the connection is either zero or larger than a given lower bound.

† ∈ {single,cyclic}

wit

}

∀ s ∈ S, t ∈ T

s∈S i∈I t∈T

∑ ∑∑f

(50)

fwit e J∗1†; Fst e Qs,

w∈F i∈I t∈T

(44)

3.3. Objective Functions. The primary objective of this work is to minimize the freshwater consumption on a batch water system, and the design problem (P1) is formulated as a nonlinear program (NLP), because of the bilinear terms that emerge from contaminant balances. As stated in the following, x denotes the vector of all design variables and Ω1† denotes the feasible searching space that is delimited by all constraints. P1:

x′ ≡ x ∪ {Qs ; ∀ s ∈ S }

L fwit (fwit - fwi )g0

The solution of P3 can remove the unnecessary flows in connections and lower the expenditure of piping system conceptually, but such a design problem is not so explicit to influence on the number of connections in a water-using network. Sometimes, the network complexity becomes a critical issue to be considered, and, therefore, the second option of a tertiary objective is to minimize the number of connections, as stated in P4. Note that the design problem has become a mixedinteger nonlinear program (MINLP), because of the introduction Table 1. Problem Specification for Examples 1 and 2 Cic,max (kg salt/kg water) [Fimin,Fimax] (in) task (kg) Cic,max A

[0,1000]

0

B C D E

[0,280] [300,400] [0,280] [300,400]

0.25 0.1 0.25 0.1

(out) Cic,max

0.1 (example 1) 0.2 (example 2) 0.51 0.1 0.51 0.1

ti(s) (h) ti(f) (h) Mic(load) (kg) 0

3

0 4 2 6

4 5.5 6 7.5

100 72.8 0 72.8 0


of binary variables to identify the existence of connections. The following expressions (constraints 62-67) are used to correlate the binary variables with the continuous variables, where Z//′t ) 1 (//′ ∈{id, ii′, is, si, ss′, wi}) indicates the existence of connecting flow from / to /′ with the amount positioned between a pair of specified lower and upper bounds. P4 :

∑ ∑ ∑Z

min J4† )

idt +

x′′∈Ω4†

∑∑∑Z

i∈I d∈D t∈T

ist +

i∈I s∈S t∈T

∑∑∑Z

∑∑∑Z

ii′t +


sit +

s∈S i∈I t∈T

∑ ∑∑Z

wit

∑∑∑Z

ss′t +



w∈F i∈I t∈T

x ′′ ≡ x ∪ {Zidt, Zii′t, Zist, Zsit, Zss′t, Zwit ; ∀i, i ′ ∈ I, s, s ′ ∈ S, t ∈ T } (61) ∀ d ∈ D, i ∈ I, t ∈ T

(62)

∀ i, i ′ ∈ I, t ∈ T

(63)

fisL Zist e fist e fisUZist

∀ i ∈ I, s ∈ S, t ∈ T

(64)

fsiL Zsit e fsit e fsiUZsit

∀ i ∈ I, s ∈ S, t ∈ T

(65)

∀ s, s ′ ∈ S, t ∈ T

(66)

∀ i ∈ I, t ∈ T, w ∈ W

(67)

fLidZidt e fidt e fUidZidt L U fii Zii′t e fii′t e fii Zii′t

L U fss Zss′t e fss′t e fss Zss′t

fLwiZwit e fwit e fUwiZwit

|

Ω4† ) Ω2† ∩ {x″ constraints 62-67; Fst e Qs∗, ∀ s ∈ S, t ∈ T

}


4. Illustrative Examples Three examples are performed to illustrate the adequacy of proposed formulation for the synthesis of batch water-using networks. To solve the mathematical programs, a high-level language, General Algebraic Modeling System (GAMS),18 is used as the optimization platform in a Core 2 2.00 GHz processor. The solver selected for NLP/MINLP is BARON, which is advertised as a superior solver to obtain the global optimum.19 4.1. Example 1: Agrochemical Manufacturing Facility. The first example is taken from Majozi et al.5 and Majozi8-10 regarding the utilization of freshwater as the aqueous phase for the liquid-liquid extraction to remove sodium chloride (NaCl) in the production of agrochemicals. All operating data for waterusing tasks are given in Table 1, including lower and upper bounds for water passing through the processing units, maximum permissible concentrations for inlet and outlet streams, starting and finishing times, and contaminant mass loads. It can be observed that there is no contaminant mass to be removed in either task C and task E, because most of them are removed with the reaction solvents in tasks B and D, respectively. However, as stated by Majozi et al.,5 those secondary washing steps (tasks C and E) should not be discarded as they constitute a quality control precaution in case of unforeseen process problems. Figure 3 shows the corresponding Gantt chart for this example, where seven time points are required over a 7.5-h time horizon. Before the exploration of water reuse/recycle, the base case shown in Figure 4a indicates that 1885.49 kg of freshwater will be consumed. To start with, the scenario of single batch without a storage tank is considered and the resultant network is shown in Figure 4b. The freshwater consumption is 1767.84 kg, which corresponds to a 6.24% reduction, compared to the base case. As shown in the figure, some of water used from both tasks B and

Figure 3. Gantt chart for the production schedule of examples 1 and 2.

D is diluted with freshwater and then supplied to tasks C and E, respectively, for reuse. Note that the outlet water from task A is directly discharged as the possibility of reuse is obviated by the time limitation. From the Gantt chart, it is obvious that none of the task commences when task A finishes after the third hour. Subsequently, the scenario of single batch with one storage tank is considered and the maximum capacity of the storage tank (Fsmax) is given as 800 kg. As shown in Figure 4c, the presence of a storage tank provides temporary storage of the reusable water, to partially bypass the time limitation and creates more possibilities for water reuse. Accordingly, the freshwater consumption is decreased from 1767.84 to 1285.49 kg, which corresponds to a 31.82% reduction in freshwater demand. Besides, the peak of the storage profile signifies that the required capacity of storage tank is 300 kg. Finally, the scenario of cyclic batch with one storage tank (Fsmax ) 800) is considered, and the resultant network is shown in Figure 4d. The freshwater consumption can be further decreased from 1285.49 kg to 1000 kg (i.e., a 45.37% reduction per batch under cyclic operation), but the required capacity of the storage tank has a slight increase, from 300 kg to 355.12 kg. The model statistics of example 1 are summarized in Table 2, including problem structure, the number of constraints, the number of continuous variables and discrete variables, CPU time, and objective values for all scenarios. Note that the superscript number indicates the corresponding design problem (for instance, “1” in the table represents P1). For the purpose of comparison, the results of Majozi8-10 and Majozi et al.5 are also presented in Table 2. First, design problems on the minimization of freshwater consumption, storage capacity, and the amount of connecting flows (i.e., P1, P2, and P3) are all formulated as nonlinear programs, because the production schedule is known a priori and there is no need of binary variables for scheduling. However, when taking the minimum number of connections as the design objective, binary variables are still required to identify the existence of connection. Thus, P4 will be a MINLP. Second, for rather simple scenarios (such as single batch without and with one storage tank), the solutions of P1 or P2 have already exhibited the optimal network configurations, which is the reason why only the statistics of P1 and P2 are to be reported. Third, the same network configurations can be obtained when solving P3 and P4 for the scenario of cyclic batch with one storage tank; in other words, the design with the minimum amount of connecting flows can also exhibit the least number of connections (under minimized freshwater consumption and storage capacity) in this instance. Nevertheless, the design with the least number of connections


Figure 4. Resultant networks for example 1: (a) base case, (b) single batch without a storage tank, (c) single batch with one storage tank, and (d) cyclic batch with one storage tank. Table 2. Model Statistics for Example 1a Single Batch no storage tank

Cyclic Batch one storage tank

one storage tank

Results of Proposed NLP/MINLP Formulation problem structure number of constraints number of continuous variables number of discrete variables CPU time (s) objective value freshwater usage (kg) storage capacity (kg)

NLP1 7111 3861

NLP1,2 845,1 8462 491,1 4922

0.33 s1 1767.841 1767.84 kg

0.44 s,1 0.34 s2 1285.49,1 3002 1285.49 kg 300 kg

NLP,1,2,3 MINLP4 725,1 726,2,3 12664 426,1,4 427,2 4613 3154 1.05 s,1 1.44 s,2 1.45 s,3 2.16 s4 1000,1 355.12,2 1610.24,3 114 1000 kg 355.12 kg

Results of Majozi8-10 and Majozi et al.5 problem structure number of constraints number of continuous variables number of discrete variables CPU time objective value freshwater usage storage capacity a

MILP 1320 546 120 1.61 s 1767.84 1767.84 kg

MINLP 5534 1217 280 309.41 s,M1 1.51 sM2 1285.49 s,M1 600 sM2 1285.49 kg 300 kg

1000 kgb 560 kgb

Superscripts denote the design problem (P1, P2, P3, P4). b Graphical result.

does not necessarily exhibit the minimum amount of connecting flows. As shown in Figure 5a and b, there are other two acceptable alternatives that would be found in P4 (besides the resultant design in Figure 4d). Each of them has the least number of connections (11) but a larger amount of connecting flows (3732.68 kg/3910.24 kg), compared to that of the resultant design (3610.24 kg). Therefore, one can further minimize the

amount of connecting flows under the least number of connections to obtain a more-precise solution from P4, as stated in P3′. Here, Ncon is the total number of connections within a water/ using network and J4† is the least number of connections obtained in P4. Fourth, both the number of constraints and the number of continuous variables entailed in the scenario of cyclic batch with one storage tank are smaller than those in the scenario


Figure 5. Candidate networks of dsign problem P4 for example 1, cyclic batch with one storage tank: (a) 11 connections with 3732.68 kg of connecting flows and (b) 11 connections with 3910.24 kg of connecting flows.

of a single batch with one storage tank, because of the fewer time points involved. P3 ′ :

min J3′† )

x′′′∈Ω3′†

∑∑∑f

ii′t +


∑∑∑f

ist +

i∈I s∈S t∈T

∑∑∑f

sit +

∑∑∑f


ss′t

s∈S i∈I t∈T


x ′′′ ≡ x ′′ ∪ {Ncon} con

N

)

∑ ∑ ∑Z

idt +

i∈I d∈D t∈T

∑∑∑Z

ii′t +


∑∑∑Z

sit +

s∈S i∈I t∈T

|

∑∑∑Z

∑∑∑Z

ist +

i∈I s∈S t∈T

ss′t +


Ω3′† ) Ω4† ∩ {x ′′′ Ncon e J∗4†}

(70)

∑ ∑∑Z

wit

(71)

w∈F i∈I t∈T


It is obvious from Table 2 that the proposed NLP/MINLP formulation in this paper can provide agreeable solutions to those of Majozi8-10 and Majozi et al.5 with satisfactory computation efficiency. Note that, in Majozi’s formulation,8-10 binary variables are used to correlate the starting/ending of water-using tasks, as well as the connections between processing units and the storage tank with the time dimension. Therefore, those binary variables are required for example 1 although it is actually a fixed schedule problem with given starting and ending times for each water-using task. Also note that the storage capacity for the cyclic batch reported by Majozi et al.5 is 560 kg, because the water requirement for tasks B and D and tasks C and E are set at 280 and 400 kg, respectively, during their targeting procedure. 4.2. Example 2: Modified Case with Higher Outlet Concentration. The second example is a modification of example 1 used to generate a relatively complex case. As shown in Table 1, the maximum outlet concentration of task A is altered to 0.2 kg salt/kg water, and the other parameters remain unchanged. Because of the change in outlet concentration, one can imagine that the water used from task A will become lesser in amount but higher in contamination. Moreover, the emergence of an extra concentration level induces greater complexity in the water-reuse system. Following the precedent of example 1, there are three different scenarios to be considered and compared with the base case.

Without considering the possibility of water reuse/recycle, Figure 6a shows the base case of example 2 in which 1385.49 kg of freshwater will be consumed. For the scenario of single batch without a storage tank, Figure 6b shows that the freshwater consumption is 1267.84 kg, which corresponds to an 8.49% reduction, in comparison to the base case. As mentioned previously, only a small amount of water savings can be achieved, because the possibility of water reuse is quite limited by different starting and finishing times of individual tasks. For the scenario of single batch with one storage tank (Fmax ) 800), s Figure 6c shows that the freshwater consumption is decreased from 1267.84 kg to 935.49 kg and the reduction of freshwater is improved from 8.49% to 32.84% with a required capacity of 300 kg to the storage tank. With regard to the scenario of cyclic batch with one storage tank (Fsmax ) 800), Figure 6d shows that the freshwater consumption is further decreased from 935.49 kg to 617.58 kg (i.e., a 55.42% reduction per batch), but the required storage tank capacity experiences a large increase, from 300 kg to 800 kg. However, the result shown in Figure 6d is merely the solution under a limiting storage capacity; in other words, such a design has not achieved the maximum reduction of freshwater and there is still room for improvement in regard to water recovery. Although the water used from task A is fit for both tasks B and D, storing the outlet water from task A renders the concentration of water in storage higher than the tolerance of task C. For this reason, some additional freshwater is always required for task C, to dilute the water from the storage tank before reuse. Considering that the degradation of water storage may increase the freshwater demand, one workable way is to add another storage tank, to avoid such circumstances. Figure 6e shows the resultant network for cyclic batch with two storage tanks, in which the freshwater consumption is decreased from 617.58 kg to 500 kg (i.e., a 63.91% reduction per batch) by handling reusable water of different concentrations separately. As shown in the figure, tank 1 is dedicated to the water used from task C and tank 2 is reserved for the water used from task A. Moreover, the required capacities for tanks 1 and 2 are 300 and 469.68 kg, respectively. Because the close confinement to the inlet concentration of task A signifies that at least 500 kg of freshwater is required for operation, the achievement of maximum water recovery has been ensured.


Figure 6. Resultant networks for example 2: (a) base case, (b) single batch without a storage tank, (c) single batch with one storage tank, (d) cyclic batch with one storage tank, and (e) cyclic batch with two storage tanks.

Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008 7827 a

Table 3. Model Statistics for Example 2

Single Batch no storage tank problem structure number of constraints number of continuous variables number of discrete variables CPU time objective value freshwater usage (kg) storage capacity (kg) a

one storage tank

NLP1 7111 3861

NLP1,2 845,1 8462 491,1 4922

0.33 s1 1267.841 1267.84

0.84,1 0.942 935.49,1 3002 935.49 300

Cyclic Batch one storage tank

two storage tanks

NLP,1,2,3 MINLP4 725,1 726,2,3 12664 426,1,4 427,2 4613 3154 4.81,1 24.17,2 3.06,3 7.664 617.58,1 800,2 1452.47,3 124 617.58 800

NLP,1,2,3 MINLP4 839,1 840,2,3 15004 516,1,4 518,2 5613 3854 475.64,1 548.48,2 76.39,3 37.674 500,1 769.68,2 1839.35,3 114 500 300, 469.68

Superscripts denote the design problem (P1, P2, P3, P4).

Figure 7. Gantt chart for the production schedule of example 3. Table 4. Concentration Data for Example 3 Cic,max (ppm) contaminant

(in) Cic,max

(out) Cic,max

Mic(load) (g)

limiting flow (ton)

Task 1 c1 c2 c3

5 0 0

15 0 0

c1 c2 c3

0 50 0

0 100 0

c1 c2 c3

120 200 200

220 450 9500

675 0 0

67.5

Task 2 0 2500 0

50

5600 14000 520800

56

Task 3

The model statistics of example 2 for all scenarios are summarized in Table 3. First, it is similar to the circumstance in example 1, in that the optimal network configurations have been obtained already by solving P1 or P2 for simple scenarios such as single batch without a storage tank and with one storage tank. Second, for the scenario of cyclic batch with one storage tank, all design problems (i.e., P1, P2, P3, and P4) will lead to the same network configuration, because there is only one way

Figure 8. Base case of example 3.

to achieve the better reduction in freshwater consumption with limited storage facilities. Third, the same network configuration can be obtained when solving P3 and P4 for the scenario of cyclic batch with two storage tanks; however, of course, other acceptable alternatives would be also found in P4. 4.3. Example 3: Multicontaminant Case. The third example is adapted from Gouws and Majozi,14 regarding the work involving the execution of three repeated tasks in their respective processing units for the production of specific products. Figure 7 shows the Gantt chart for the production schedule, in which tasks 1, 2, and 3 are repeated for four, three, and five times (with durations of 2, 2.5, and 1.5 h, respectively), and, furthermore, 11 time points are required through an 8-h time horizon. The concentration data for all water-using tasks are given in Table 4, including contaminant species, maximum permissible concentrations for inlet and outlet streams, contaminant mass loads, and limiting flows. It can be observed that the water used for task 1 contains a single contaminant c1, and the water used for task 2 contains another contaminant c2, whereas the water used for task 3 contains three contaminants (namely c1, c2, and c3). To enhance the possibility of water reuse/recycle, for this instance, two storage tanks are considered for the collection of reusable water; one is dedicated to the outlet water from task 1, and the other is present solely for the outlet


Figure 9. Resultant networks of example 3 for single batch with two storage tanks obtained from design problems (a) P3 and (b) P4.

water from task 2. Note that the foregoing conditions can be easily fitted by setting the permissible concentrations for contaminants into storage tanks. The maximum capacity of each storage tank is given as 200 tons. In addition to the production schedule and the concentration data, it is quite important to note that the objective of Gouws and Majozi14 for the original problem is not to minimize the freshwater consumption but to maximize the profit, where the profit is the difference between the revenue of products and the combined cost of raw materials and treatment for effluent. Such

an objective has a tendency to maximize the throughput of processing units, because the yield of products is proportional to the amount of water being processed. Therefore, in this example, the amounts of inlet and outlet water for each waterusing task are set at the limiting flow to keep to the original intention. Before the exploration for the possibility of water reuse/ recycle, Figure 8 shows the base case of example 3 in which 700 tons of freshwater will be consumed. Considering the scenario of single batch with two storage tanks, the freshwater


Figure 10. Resultant networks of example 3 for cyclic batch with two storage tanks obtained from design problems (a) P3 and (b) P4.

consumption is deceased from 700 tons to 397.88 tons (i.e., a 43.16% reduction of freshwater). In addition, the required capacities are 45 tons for tank 1 and 47.54 tons for tank 2. Two resultant networks obtained from design problems P3 and P4 are respectively shown in Figure 9a and b. Overall, the difference between them is not significant; the first one has 37 connections with 1234 tons of connecting flows, and the second one has 36 connections with 1235.18 tons of connecting flows. In contrast with the result of Gouws and Majozi (461.7 tons of

freshwater consumption),14 the proposed formulation provides more-complex network configurations for better achievement, with regard to water minimization. After the discussion on single batch, the scenario of cyclic batch with two storage tanks is also considered. The freshwater consumption is further decreased from 397.88 tons to 275.26 tons (i.e., a 60.68% reduction per batch under cyclic operation). Moreover, the required capacities are 42.79 tons for tank 1 and 50 tons for tank 2. Two resembling networks obtained from design problems P3 and

7830 Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008 Table 5. Model Statistics for Example 3 design problem P1

design problem P2

design problem P3

design problem P4

Single Batch, 2 Storage Tanks problem structure number of constraints number of continuous variables number of discrete variables CPU time objective value

NLP 1281 760

NLP 1282 762

NLP 1282 760

3.81 s 397.88

17.67 s 92.54

4.08 s 438.23

MINLP 1877 760 297 78.72 s 36

Cyclic Batch, 2 Storage Tanks problem structure number of constraints number of continuous variables number of discrete variables CPU time objective value

NLP 1169 694

NLP 1170 696

NLP 1170 715

4.37 s 275.26

4.39 s 92.79

4.69 s 596.83

Table 6. Related Parameters on Forbidden Reuse between Tasks C and E in Single Batch Cic,max (kg salt/kg water) min

max

[Fi ,Fi

] (kg)

contaminant

(in) Cic,max

(out) Cic,max

Mic(load) (kg)

Task A [0,1000]

NaCl fic-C

[0,280]

NaCl fic-C

[300,400]

NaCl fic-C

[0,280]

NaCl fic-C

0 1

0.1 1

100 0

Task B 0.25 1

0.51 1

72.8 0

0.1 1

0 30

0.51 1

72.8 0

0.1 0

0 0

Task C 0.1 1 Task D 0.25 1 Task E [300,400]

NaCl fic-C

0.1 0

Table 7. Related Parameters on Forbidden Reuse between Tasks C and E in Cyclic Batch Cin,max (kg salt/kg water) [Fimin,Fimax] (kg)

contaminant

[0,1000]

NaCl fic-C fic-E

[0,280]

NaCl fic-C fic-E

[300,400]

NaCl fic-C fic-E

[0,280]

NaCl fic-C fic-E

[300,400]

NaCl fic-C fic-E

(in) Cin,max

(out) Cin,max

Mic(load) (kg)

Task A 0 1 1

0.1 1 1

100 0 0

Task B 0.25 1 1

0.51 1 1

72.8 0 0

0.1 1 0

0 30 0

0.51 1 1

72.8 0 0

0.1 0 1

0 0 30

Task B 0.1 1 0 Task D 0.25 1 1 Task E 0.1 0 1

P4 are respectively shown in Figure 10a and b as the resultant designs, where the first one has 35 connections with 1147.34 tons of connecting flows and the second one has 34 connections

MINLP 1710 694 297 28.84 s 34

with 1150.48 tons of connecting flows. The model statistics of example 3 are summarized in Table 5. 4.4. Practical Constraints on Design: Forbidden Matches. So far, the most emphasis is placed on the achievement of water recovery, and, as a consequence, water-using networks are synthesized by exploiting all possibilities of water reuse/recycle. However, some practical constraints on network configuration are still more important to be taken into consideration, especially the forbidden match between assigned waterusing tasks for the prevention of probable pollution or operational problems. By and large, such constraints have a significant effect on the resultant design, in terms of both freshwater consumption and storage capacity. The elimination of forbidden matches from a practical design is really not easy, because the connections between processing units and storage tanks are undefined before the network synthesis. Although the existence of connecting flows can be identified by introducing binary variables, as stated in section 3.3, it merely represents the “direct connections”. As the case stands, the transportation of water flows between processing units and storage tanks also will occur in a circuitous way. For example, only the connections from the task to the storage tank and the storage tank to the task are apparently seen when the outlet water from one task has been stored in a storage tank and then reused in another task; nevertheless, such an “indirect connection” between two tasks actually does exist. Therefore, the use of a large amount of logical constraints and binary variables to analyze the overall situation and capture the route of concerned flows seems to be unavoidable. To address the forbidden matches without burdening the computation efficiency, an effective technique is proposed from another point of view, which is readily understandable and solvable. When the used water of a water-using task is forbidden to be reused in another task, it is reasonable to assume that the water used from the former must contain an unfavorable component to the latter. Furthermore, a set of fictitious contaminants c ∈C fic⊂C are then invented to visualize the presence of such unfavorable components. One then can distinguish the water used for a specific task from others by embedding a certain amount of fictitious contaminant in the task to change its outlet concentration, and, therefore, all undesired matches can be well-eliminated by setting the related parameters appropriately. The application of fictitious contaminants will be elaborated in the following paragraphs with the further study of example 1. Review the first example for an agrochemical manufacturing facility, in which three scenarios have been discussed, with respective reductions in freshwater demand of 6.24%, 31.82%,


Figure 11. Resultant networks for the further study of example 1: (a) single batch and (b) cyclic batch with one storage tank and forbidden reuse between tasks C and E. Table 8. Model Statistics for the Further Study of Example 1 Single Batch, 1 Storage Tank

Cyclic Batch, 1 Storage Tank

design problem P1 design problem P2 design problem P1 design problem P2 design problem P3 design problem P4 problem structure number of constraints number of continuous variables number of discrete variables CPU time objective value

NLP 983 575

NLP 984 576

NLP 963 570

NLP 964 571

NLP 964 605

0.64 s 1285.49

0.45 s 600

3.78 s 1000

4.94 s 655.12

4.67 s 1732.68

and 45.37%. Suppose that the water reuse between tasks C and E is forbidden, that is to say, those tasks are not allowed to reuse outlet water from each other. Accordingly, some of the results for example 1 should be revised to meet a practical design. As shown in Figure 4c, the original design for single batch with one storage tank, it is undesired that the outlet water from task C be reused in task E through the storage tank. As shown in Figure 4d, the original design for cyclic batch with one storage tank, it is also undesired that the outlet water from task E be reused in task A and then be reused in task C through the storage tank. In single batch, the outlet water from task E is absolutely impossible to be reused in task C, because the finishing time of task E is later than the starting time of task C. Therefore, the only concern in this instance is to take precautions against the water reuse from task C to task E. For one thing, a fictitious contaminant (fic-C) is embedded in task C as an additional contaminant, where the mass load can be arbitrarily assigned, provided that it does not influence the water requirement of task C. For the next, the maximum permissible concentration of fic-C for task E is set at zero, to prohibit any water used from task C, which contains fic-C. In regard to the other tasks (A, B, C, and D), the maximum permissible concentrations for fic-C are all set to a large enough value to ignore the presence of fic-C. Table 6 gives an example of how to set the related parameters, and the problem is then expanded to two contaminants with the introduction of fic-C. Finally, the forbidden reuse between tasks C and E can be eliminated after solving the new-set problem. All computation results, including resultant network and model statistics, are shown in Figure 11a and Table 8, respectively. Fortunately, the freshwater consumption is still 1285.49 kg without penalty; however, the required capacity of storage tank doubles (i.e., 600 kg).

MINLP 1594 610 315 7.09 s 11

In cyclic batch, not only would the water reuse from task C to task E happen, but it is also possible for the outlet water from task E to be reused in task C. To prohibit the water reuse between tasks C and E in both directions (namely, from task C to task E and from task E to task C), two fictitious contaminants are then embedded in tasks C and E, respectively. All related parameters have been set as given in Table 7, and the forbidden reuse between tasks C and E can be eliminated after solving the new-set, a three-contaminants problem. The resultant network and model statistics are shown in Figure 11b and Table 8, respectively. It is similar to the previous case, in that the practical constraints do not cause any penalty for freshwater demand but do enlarge the requirement for storage capacity, and, as a consequence, the freshwater consumption is 1000 kg and the required capacity of storage tank is increased from 355.12 kg to 655.12 kg. 5. Conclusions On the basis of a fixed production schedule, this paper has presented a mathematical formulation for the synthesis of waterusing networks in batch plants. With the redefinition of binary variables to corresponding binary parameters, design problems for minimizing the freshwater consumption, the required capacity of storage tank(s) and the amount of connecting flows can be formulated as nonlinear programs (NLPs). When taking the least number of connections as the design objective, binary variables will be required to identify the existence of connection and the design problem becomes a mixed-integer nonlinear program (MINLP). Applicable network configurations with minimum freshwater consumption are then obtained through solving sequential design problems. Three representative examples are provided for different purposes and it has been shown that the adequate


arrangement of reusable water leads to considerable reduction of freshwater. The first example is taken from Majozi et al.5 and Majozi8-10 for testing and verifying the effectiveness of proposed formulation. After a little modification, the second example is used to discuss the impact of multiple storage tanks on water reuse/ recycle, where the reduction in freshwater demand is further increased to almost 64% for the help of additional storage tank. The third example is adapted from Gouws and Majozi14 and the proposed formulation is extended to handle more complicated cases within a multiple contaminant environment. In consequence, network configurations with higher network complexity but better reductions in freshwater demand have been obtained. Finally, the concept of fictitious contaminant is offered to address the forbidden match. Though such practical constraints do not cause any penalty for freshwater demand in the further investigation of the first example, they have enlarged the requirement of storage capacity. However, as mentioned by Majozi8 and Gouws and Majozi,14 it is really an oversimplification to assume that the production schedule of overall plant is known a priori, moreover, the water targets identified under this circumstance cannot be regarded as the absolute minimum but rather the minimum for that specific schedule. Therefore, the conjunction of network synthesis and scheduling will be the subject of future work. Acknowledgment Financial supports of the Ministry of Economic Affairs (under Grant No. 96-EC-17-A-09-S1-019) and the National Science Council of ROC (under Grant No. NSC96-2221-E-002-151MY3) are appreciated. Nomenclature Indices and Sets c ∈ C ) {1,..., C}, transferrable contaminants d ∈ D ) {1,..., D}, discharge to end-of-pipe treatment i ∈ I ) {1,..., I}, batch water-using tasks s ∈ S ) {1,..., S}, storage tanks t ∈ u ) {1,..., T}, time points t ∈ u* ) {1,..., T - 1}, time points without the last one w ∈ W ) {1,..., W}, water sources Parameters (in) Cic,max ) maximum inlet concentration of contaminant c for task i (out) Cic,max ) maximum outlet concentration of contaminant c for task i C(in) sc,max ) maximum inlet concentration of contaminant c for storage tank s Fimin ) lower bound for the inlet and outlet flows of task i Fimax ) upper bound for the inlet and outlet flows of task i Fsmax ) maximum capacity of storage tank s Mic(load) ) mass load of contaminant c to be removed in task i j itt′ ) ∈{0, 1}, ) 1 denotes task i proceeding in the time interval N t to t′ PositiVe Variables (in) Cict ) inlet concentration of contaminant c of task i at time point t (out) Cict ) outlet concentration of contaminant c of task i at time point t (in) Csct ) inlet concentration of contaminant c of storage tank s at time point t (out) Csct ) outlet concentration of contaminant c of storage tank s at time point t fidt ) amount of water from task i to end-of-pipe treatment d at time point t

fii′t ) amount of water from task i to other task i′ at time point t fist ) amount of water from task i to storage tank s at time point t fit(in) ) amount of inlet water to task i at time point t fit(out) ) amount of outlet water from task i at time point t fsit ) amount of water from storage tank s to task i at time point t Fst ) amount of water remaining in storage tank s at time point t fst(in) ) amount of inlet water to storage tank s at time point t fst(out) ) amount of outlet water from storage tank s at time point t fwit ) amount of water from water source w to task i at time point t Binary Variables Zidt ) 1 denotes the existence Zii′t ) 1 denotes the existence Zist ) 1 denotes the existence Zsit ) 1 denotes the existence Zss′t ) 1 denotes the existence Zwit ) 1 denotes the existence

of of of of of of

flow from i to d at time point flow from i to i′ at time point flow from i to s at time point flow from s to i at time point flow from s to s′ at time point flow from w to i at time point

t t t t t t

Literature Cited (1) Wang, Y. P.; Smith, R. Wastewater minimization. Chem. Eng. Sci. 1994, 49, 981. (2) Bagajewicz, M. A Review of Recent Design Procedures for Water Networks in Refineries and Process Plants. Comput. Chem. Eng. 2000, 24, 2093. (3) Yoo, C. K.; Lee, T. Y.; Kim, J.; Moon, I.; Jung, J. H.; Han, C.; Oh, J. M.; Lee, I. B. Integrated Water Resource Management through Water Reuse Network Design for Clean Production Technology: State of the Art. Korean J. Chem. Eng. 2007, 24, 567. (4) Wang, Y. P.; Smith, R. Time pinch analysis. Trans. Inst. Chem. Eng. 1995, 73a, 905. (5) Majozi, T.; Brouckaert, C. J.; Buckley, C. A. A Graphical Technique for Wastewater Minimization in Batch Processes. J. EnViron. Manage. 2006, 78, 317. (6) Foo, C. Y.; Manan, Z. A.; Tan, Y. L. Synthesis of maximum water recovery network for batch process systems. J. Clean. Prod. 2005, 13, 1381. (7) Almato, M.; Espuna, A.; Puigjaner, L. Optimisation of Water Use in Batch Process Industries. Comput. Chem. Eng. 1999, 23, 1427. (8) Majozi, T. Wastewater Minimisation Using Central Reusable Water Storage in Batch Plants. Comput. Chem. Eng. 2005, 29, 1631. (9) Majozi, T. An Effective Technique for Wastewater Minimisation in Batch Processes. J. Clean. Prod. 2005, 13, 1374. (10) Majozi, T. Storage Design for Maximum Wastewater Reuse in Multipurpose Batch Plants. Ind. Eng. Chem. Res. 2006, 45, 5936. (11) Li, B. H.; Chang, C. T. A Mathematical Programming Model for Discontinuous Water-Reuse System. Ind. Eng. Chem. Res. 2006, 45, 5027. (12) Cheng, K. F.; Chang, C. T. Integrated Water Network Design for Batch Processes. Ind. Eng. Chem. Res. 2007, 46, 1241. (13) Shoaib, A. M.; Aly, S. A.; Awad, M. E.; Foo, C. Y.; El-Halwagi, M. M. A Hierarchical Approach for the Synthesis of Batch Water Network. Comput. Chem. Eng. 2008, 32, 530. (14) Gouws, J. F.; Majozi, T. Impact of Multiple Storage in Wastewater Minimization for Multicomponent Batch Plants: Toward Zero Effluent. Ind. Eng. Chem. Res. 2008, 47, 369. (15) Castro, P. M.; Barbosa-Povoa, A. P.; Matos, H. A. Optimal Periodic Scheduling of Batch Plants Using RTN-Based Discrete and ContinuousTime Formulations. Ind. Eng. Chem. Res. 2003, 42, 3346. (16) Castro, P. M.; Barbosa-Povoa, A. P.; Matos, H. A.; Novais, A. Q. Simple Continuous-Time Formulation for Short-Term Scheduling of Batch and Continuous Processes. Ind. Eng. Chem. Res. 2004, 43, 105. (17) Shah, N.; Pantelides, C. C.; Sargent, R. W. H. Optimal Periodic Scheduling of Multipurpose Batch Plants. Ann. Oper. Res. 1993, 42, 193. (18) Brooke, A.; Kendrick, D., Meeraus, A.; Raman, R.; Rosenthal, R. E. GAMS: A User’s Guide; Scientific Press: Redwood City, CA, 2003. (19) GAMS: The SolVer Manuals; GAMS Development Corporation: Washington, DC, 2003.

ReceiVed for reView April 10, 2008 ReVised manuscript receiVed July 3, 2008 Accepted July 16, 2008 IE800573R

Continuous-Time Formulation for the Synthesis of Water-Using

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