Accepted Manuscript
Title: Synthesis of Water-Using Network with Central Reusable Storage in Batch Processes
Authors: Cheng-Liang Chen, Jui-Yuan Lee, Ju-Wei Tang, Ying-Jyuan Ciou
PII: S0098-1354(08)00192-0
DOI: doi:10.1016/j.compchemeng.2008.09.013
Reference: CACE 3696
To appear in: Computers and Chemical Engineering
Received date: 2-7-2007
Revised date: 18-9-2008
Accepted date: 19-9-2008
Please cite this article as: Chen, C.-L., Lee, J.-Y., Tang, J.-W., & Ciou, Y.-J., Synthesis of Water-Using Network with Central Reusable Storage in Batch Processes, Computers and Chemical Engineering (2008), doi:10.1016/j.compchemeng.2008.09.013
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Accepted Manuscript
submitted to Computers and Chemical Engineering (1st revision)
Synthesis of Water-Using Network with Central Reusable
Storage in Batch Processes
Cheng-Liang Chen,
∗Jui-Yuan Lee,
Ju-Wei Tang, Ying-Jyuan Ciou
Department of Chemical Engineering
National Taiwan University
Taipei 10617, Taiwan, ROC
September 18, 2008
Abstract
The starting and finishing times of water-using operations in batch plants are de-pendent on the production planning and scheduling. For maximizing the possibility of water reuse/recycle, or for minimizing the freshwater consumption, central stor-age facilities in a batch water-using system can collect the reusable water from the earlier operations and then supply the remaining water to the later operations. For a predefined production schedule, this paper presents a mathematical formulation for the synthesis of batch water-using networks with central storage tank(s). Superstruc-tures that incorporate all possible flow connections are built to facilitate the problem modeling. The design problem is formulated as a nonlinear program (NLP) if there is no limitation to water reuse/recycle, or a mixed-integer nonlinear program (MINLP) when forbidding water reuse between assigned water-using units and water recycle of certain water-using units is considered. A published literature example and some adaptation examples are provided to demonstrate the good expectation of the pro-posed, superstructure-based NLP and MINLP formulations.
As supported by the results, superstructures with the design scheme can effec-tively accomplish the objective of solving the design problem for minimum freshwa-ter consumption and offering the optimal network configuration.
Keywords: Batch process, Synthesis, Water-using network, Superstructure,
Nonlin-ear program (NLP), Mixed-integer nonlinNonlin-ear program (MINLP)
Accepted Manuscript
1
Introduction
Process integration is one of the most important strategies for increasing the overall utilization efficiency of resources and utilities. In contrast with the remarkable advances and extensive applications of heat integration techniques in process industry, only limited investigations on water minimization can be found. Furthermore, most of the methodolo-gies for water minimization in literature are concerning continuous processes. However, because batch operations have become the ordinary works in the production of specialty chemicals of high commercial value, the development of systematic approaches for water minimization in batch plants appeals to academics and practitioners.
It has been shown in literature that adequate arrangements of water reuse/recycle can considerably reduce the total freshwater consumption for water-using systems (Wang & Smith, 1994, 1995a, 1995b). Water reuse concerns the use of outlet water from one water-using unit in another unit, whereas water recycle refers to the use of outlet water from one water-using unit in the same unit. In contrast with their continuous counterpart where the time dimension is completely overridden, the major difficulty in water minimization for batch processes is the inherent time dependence on production planning and scheduling that the operating conditions and freshwater demands are variable with time. Even for a predefined production schedule, the opportunities of water recovery are still constrained by different starting and finishing times of water-using operations.
Wang and Smith (1995b) developed a graphical technique of time pinch analysis for wastewater minimization in batch processes. Both constraints of time and concentration driving force are taken into consideration in their work. In addition, water reuse from the earlier time intervals to the later also indicates the significance of storage facility. Follow-up works on graphical techniques were respectively presented by Foo et al. (2005) and Majozi et al. (2006). However, as pointed out by Majozi (2005a, 2005b), those graphical techniques are only applicable to processes characterized by single contaminants. Majozi (2005a, 2005b, 2006) developed a mathematical formulation to investigate the function of a central reusable water storage facility in bypassing the time limitation for wastewater
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minimization in batch plants. The formulation can not only be readily extended to multiple contaminant problems, but also used to consider the production scheduling, if necessary. Further studies about the optimal design of water reuse networks in batch processes are also found in literature (Almato et al., 1999; Li & Chang, 2006).
Although water reuse/recycle is effective to reduce the consumption of freshwater in process industries, and the existence of central storage facilities is capable of attenuating the time dependence of batch operations to enhance the opportunities of water recovery, the subject of forbidding water transfers from one water-using unit to another unit is not examined as yet. This paper aims to propose a mathematical formulation to deal with the cases that water reuse between assigned water-using units is forbidden. Three illustrative examples are provided to display the validness of the proposed formulation.
2
Problem Statement
The problem of synthesizing batch water-using network in this paper can be briefly
stated as follows. Givens are a set of water-using units i∈ I in which some transferable
contaminants c∈ C with fixed mass loads Mic(load)require to be removed and the maximum
permissible inlet/outlet concentrations for each unit, Cic,max(in) /Cic,max(out) , are also specified; a
set of water sources w ∈ W with specific concentrations Cwcfor utilization, where some
of them supply freshwater w ∈ F ⊆ W; a set of storage tanks s ∈ S for the temporary
storage of used water to enhance water recovery. Furthermore, the production scheduling is assumed to be known and the starting and finishing times of water-using operation in
each unit have been assigned to their corresponding time points p∈ P. The objective is to
determine the time-dependent operating strategy which targets the minimum freshwater consumption. The solution of problem provides the minimized freshwater consumption, resultant network configuration and accumulation of used water in storage tanks. It has to be mentioned that the investment costs will not be addressed in this work, even though the adaptability to include economic factors among design criteria is part of the motivation for applying mathematical techniques. To emphasize the impact of available storage tanks on the reduction of freshwater, the investment costs are temporarily left out from the problem
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Water-using unit i M q S q Q q q q q q q C C C C C C M F ro m w ater-using unit i' T o w ater-using unit i' F ro m sto rage tank s F ro m w ater so urc e w T o sto rage tank s T o d isc h arge p o int d (a) Storage tan k s M q q S ! "#! q$ q&% q $ q'% C ( "# ! C ( ! C$( ! ! C'%( Q F rom w ater- u s i n g u n i t i F rom s torage tan k s' T o s torage tan k s' T o w ater- u s i n g u n i t i (b)Figure 1: Superstructures for (a) water-using unit and (b) storage tank.
formulation and this issue will be tackled in the future work.
3
Superstructures and Problem Formulation
In order to formulate the synthesis problem, generic superstructures for water-using unit and storage tank are constructed to incorporate all possible flow connections within the batch water system, as shown in Fig.s1(a) and (b) respectively. The problem formula-tion presented in the following secformula-tions mainly involves water and contaminant balances around each water-using unit and storage tank. All required indices, sets, parameters and variables are marked on Fig. 1 and listed in the table of Nomenclature.
3.1
Water balance around water-using units
The overall water balances toward water-using units are formulated on the basis of the superstructure in Fig. 1(a), including the inlet flow from sources, the outlet flow to sinks
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and the relation between inlet and outlet flows. For dealing with the time dependence of
batch process, two sets of binary parameters, y(st.)ip and yip(fin.), are predefined to indicate the
starting and finishing times of all operations in water-using units. yip(st.) = 1 means that the
operation in unit i is starting at time point p, while y(fin.)ip = 1 signifies that the operation
in unit i is finishing at time point p.
qip(in) = X ∀i0∈I qi0ip+ X ∀s∈S qsip+ X ∀w∈W qwip ∀i ∈ I, p ∈ P (1) qip(out) = X ∀d∈D qidp+ X ∀i0∈I qii0p+ X ∀s∈S qisp ∀i ∈ I, p ∈ P (2) QLiyip(st.) ≤ qip(in) ≤ QU i y (st.) ip ∀i ∈ I, p ∈ P (3) QL iy (fin.) ip ≤ q (out) ip ≤ QUi y (fin.) ip ∀i ∈ I, p ∈ P (4)
qip(in)− qi,p+∆(out) i − y
(in) ip Q
(loss)
i = 0 ∀i ∈ I, p ∈ P (5)
Equations (1) and (2) are water balances around mixing point M and splitting point S of unit i at time point p. Equations (3) and (4) give lower and upper bounds for the inlet and outlet flows of unit i at time point p. The amount of water entering and leaving unit i at time point p will be bounded in a reasonable range if the operation is about to start or finished at that time point. On the other hand, those two equations also ensure that no water flows in and out when the time point is neither a starting nor finishing time for the operation. Equation (5) is the water balance throughout unit i, where the amount of water
loss during operation is assumed to be a known constant or function. Note that∆i is the
distance between time points for the starting and finishing of the operation in unit i, which reflects a certain duration to achieve the expected effect.
3.2
Contaminant balance around water-using units
As stated in the following, Eq. (6) is the contaminant balance around the mixing point before unit i at time point p. Equation (7) is the contaminant balance throughout unit i, in which the concentration of water loss is assumed to be a known parameter. Equation (8) gives the maximum permissible inlet and outlet concentrations of contaminant c for unit i as the constraints on concentration driving force. Besides, it should be realized that the
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outlet flow is always prior to the inlet flow when a water-using operation is finishing and starting immediately at the same time point.
qip(in)Cicp(in) = X
∀i0∈I qi0ipCi(out)0cp + X ∀s∈S qsipCscp(out) + X ∀w∈W qwipCwc (6) ∀c ∈ C, i ∈ I, p ∈ P q(in)ip Cicp(in)− qi,p+∆(out) iC
(out) ic,p+∆i + y (in) ip (M (load) ic − Q (loss) i C (loss) ic ) = 0 (7) ∀c ∈ C, i ∈ I, p ∈ P
Cicp(in) ≤ Cic,max(in) ; Cicp(out) ≤ Cic,max(out) ∀c ∈ C, i ∈ I, p ∈ P (8)
3.3
Water balance around storage tanks
Based on the superstructure in Fig.1(b), water balances for storage tanks are similar to those for water-using units, as stated below. Equations (9) and (10) are the water balances around the mixing and splitting points of storage tank s at time point p. Equations (11) and (12) provide the water balance throughout storage tank s to calculate the amount of
remaining water therein at time point p. Note that Z(cyc.)is a predefined binary parameter
to describe two operating modes. Z(cyc.) = 0 denotes the single operation where no initial
storage is assumed, and Z(cyc.) = 1 indicates the cyclic operation where the initial storage
is equivalent to the remaining water at the last time point P for the previous batch cycle. Equation (13) gives an upper bound for the capacity of storage tank s.
qsp(in) = X ∀i∈I qisp+ X ∀s0∈S qs0sp ∀p ∈ P, s ∈ S (9) qsp(out) = X ∀i∈I qsip+ X ∀s0∈S qss0p ∀p ∈ P, s ∈ S (10) Qsp = Qs,p−1+ q(in)sp − q (out) sp ∀p ∈ P ∗ , s∈ S (11) Qsp = Z(cyc.)QsP + q(in)sp − q (out) sp ∀p = 1, s ∈ S (12) Qsp ≤ QU s ∀p ∈ P, s ∈ S (13)
3.4
Contaminant balance around storage tanks
The contaminant balances for storage tanks are also formulated as follows. Here, it is assumed that the inlet flow is prior to the outlet flow when there are water flows entering
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and leaving a storage tank at the same time point. Equation (14) is the contaminant balance around the mixing point of storage tank s at time point p. Moreover, Equations (15) and (16) are the contaminant balance throughout the storage tank s at time point p.
qsp(in)Cscp(in) = X
∀i∈I
qispCicp(out) +
X
∀s0∈S
qs0spCs(out)0cp (14)
∀c ∈ C, p ∈ P, s ∈ S QspCscp(out) = Qs,p−1Csc,p−1(out) + q(in)sp C
(in) scp− q (out) sp C (out) scp (15) ∀c ∈ C, p ∈ P∗, s∈ S
QspCscp(out)= Z(cyc.)QsPCscP(out)+ q (in)
sp Cscp(in)− qsp(out)Cscp(out) (16)
∀c ∈ C, p = 1, s ∈ S
3.5
Objective functions
The main objective in this work is to minimize the freshwater consumption on water-using operations, and the design problem is formulated as the following nonlinear program
(NLP). Note that x1is the vector of all design variables, andΩ1 is the feasible searching
space delimited by all design constraints.
P1: min x1∈Ω1 J1 = X ∀w∈F X ∀i∈I X ∀p∈P qwip (17) x1 ≡
qidp, qii0p, qisp, qsip, qwip, q(in)ip , qip(out)
q(in)
sp , qsp(out), Qsp, Cicp(in), C (out)
icp , Cscp(in), Cscp(out)
∀c ∈ C, d ∈ D, i, i0 ∈ I, p ∈ P, s ∈ S, w ∈ W (18) Ω1 = {x1| Eqs. (1) − (16)} (19)
Because design problem P1 may result in multiple solutions in terms of network con-figuration, one can minimize the overall capacity of storage tanks as the secondary objec-tive to obtain a preferable design, as stated in the following NLP. Note that the maximum
capacity of each storage tank Qmax
s is considered as a design variable at this moment, and
the searching space is contracted toΩ01 with the additional constraint, for maintaining the
minimum consumption of freshwater.
P2: min x0 1∈Ω 0 1 J2 = X ∀s∈S Qmaxs (20)
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x0 1 ≡ x1∪ {Qmaxs , ∀s ∈ S} (21) Ω01 = Ω1∩ x0 1 X ∀w∈F X ∀i∈I X ∀p∈P qwip ≤ J ∗ 1 Qsp ≤ Qmaxs , ∀p ∈ P, s ∈ S (22)4
Illustrative Examples
Three examples are present to illustrate the validness of the proposed formulation for synthesizing batch water-using networks. All mathematical programs are solved by using the General Algebraic Modeling System (GAMS, Brooke et al., 2003) in a Core 2, 2 GHz processor. Moreover, the solver selected for NLP is SNOPT.
4.1
Example 1: five single-stage operations
The first example is taken from Majozi (2005a, 2005b and 2006) regarding the execu-tion of five water-using operaexecu-tions in an agrochemical manufacturing facility: freshwater is introduced to form an aqueous phase for the liquid-liquid extraction to remove sodium chloride from the organic phase. The basic processing data for each operation, including lower and upper bounds for the inlet and outlet water flows, maximum permissible inlet and outlet concentrations, starting and finishing times and contaminant mass load, are all given in Table 1. According to the known schedule, Fig.2 shows the corresponding Gantt chart of this example. Note that the constant interval between any two time points adopted
in the problem formulation is0.5-h, and therefore, sixteen time points are required over
a7.5-h time horizon. Also note that, for this example, the five water-using operations are
included into a single batch operation.
For a start, the mode of single operation without a storage tank is considered. As the resultant network configuration shown in Fig. 3(a), every unit has used freshwater with
a total consumption of1767.84 kg per batch. Part of the outlet water from unit B (D) is
reused in unit C (E) after the dilution with freshwater. Nevertheless, the outlet water from unit A is not reused in other units but directly discharged because of the time limitation, although it is at a qualified concentration for all other units. An existing storage tank can temporarily store the outlet water of unit A and then supply it to units C and E for reuse,
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Table 1: Processing data for examples 1 and 2.
Unit [Qmin i , Qmaxi ] C (in) ic,max C (out) ic,max t (st.) ip t (fin.) ip M (load) ic (kg) (kg salt/kg water) (h) (h) (kg) A [0, 1000] 0 0.1 0 3 (Example 1) 100 4.5 (Example 2) B [0, 280] 0.25 0.51 0 4 72.8 C [300, 400] 0.1 0.1 4 5.5 0 D [0, 280] 0.25 0.51 2 6 72.8 E [300, 400] 0.1 0.1 6 7.5 0 0 1 2 3 4 6 7 T i m e ( h )5 A B C D E
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A B C D E 0 2 3 4 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 42. 7 4 1 42. 7 4 241 . 1 8 241 . 1 8 1 7 67 . 8 4 1 7 67 . 8 4 1 000 300 300 8 3. 9 2 8 3. 9 2 5 8 . 8 2 5 8 . 8 2 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) 300 300 (a) A B C D E 0 2 3 4 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 42. 7 4 1 42. 7 4 1 28 5 . 49 1 28 5 . 49 7 00 300 1 42. 7 4 1 42. 7 4 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) 300 300 300 300 300 300 3 4 5 . 5 6 T i m e ( h ) T a n k A B C D E 0 2 3 4 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 000 644. 8 8 300 1 7 7 . 5 6 1 7 7 . 5 6 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) T a n k 300 35 5 . 1 2 300 300 1 7 7 . 5 6 1 7 7 . 5 6 1 7 7 . 5 6 35 5 . 1 2 3 4 5 . 5 6 T i m e ( h ) 5 5 . 1 2 5 5 . 1 2 35 5 . 1 2 35 5 . 1 2 35 5 . 1 2 0 0 2 7 . 5 (b) (c)Figure 3: Resultant networks of example 1: (a) single operation without a storage tank, (b) single operation with one storage tank, and (c) cyclic operation with one storage tank.
as shown in Fig. 3(b). Consequently, the freshwater consumption for single operation is
decreased from1767.84 kg to 1285.49 kg (per batch) with the existence of storage tank.
Moreover, the required capacity of storage tank is300 kg, and the storage profile is also
shown in Fig.3(b). Because of the fact that the outlet water of units C and E can be stored and reused by units B and D in the next batch operation, the mode of cyclic operation will
further reduce the freshwater consumption from1285.49 kg to 1000 kg (per batch), with
a slight increase of storage capacity from300 kg to 355.12 kg, as shown in Fig. 3(c). All
computation results, including problem structure, the number of variables, the number of constraints and CPU time, are summarized in Table 2.
4.2
Example 2: modified case with longer operating time
The second example is a modification of example 1 where the operating time of unit
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Table 2: Computation results of examples 1 and 2.
Operating mode Single op. Cyclic op.
No. of storage tanks 0 1
Problem structure NLP
No. of constraints 881 977 977
Continuous variables 881 1121 1121
Ex. 1 Freshwater (kg) 1767.84 1285.49 1000
Storage cap. (kg) 0 300 355.12
CPU time (sec) 0.17 0.20 0.22
Ex. 2 Freshwater (kg) 1767.84 1526.67 1000
Storage cap. (kg) 0 300 655.12
CPU time (sec) 0.19 0.20 0.22
operation without and with one storage tank are1767.84 kg and 1526.67 kg (per batch),
respectively, while the freshwater consumption for cyclic operation with one storage tank
is1000 kg per batch. The resultant network configurations of this example are similar to
those of the previous one, as shown in Fig.s4(a)-(c). Furthermore, the required capacities
of storage tanks for single and cyclic operations are300 kg and 655.122 kg, respectively.
All computation results of example 2 are also summarized in Table 2.
4.3
Forbidden water reuse/recycle
It is quite common that the water reuse between certain water-using units is forbidden for antipollution. In other words, the feasibility of water reuse from one water-using unit to another is dependent on not only the concentration constraint but also the operational consideration. Until now, only a few works in literature have addressed forbidden matches for continuous processes using mathematical formulation (Bagajewicz & Savelski, 2001; Gunaratnam et al., 2005), but there is no parallel information for batch processes.
Review the previous two examples and suppose that the water reuse between units C and E is forbidden, and what’s more, the water recycle of units C and E is also prohibited. Accordingly, some of the resultant networks should be changed to satisfy the operational constraints. As observed in both Fig.s 3(b) and (c), the original designs of example 1 for single and cyclic operations each with one storage tank, the reuse of water from unit C to unit E is undesired. Besides, as the original design of example 2 for cyclic operation with
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A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 42. 7 4 1 42. 7 4 241 . 1 8 241 . 1 8 1 7 67 . 8 4 1 7 67 . 8 4 1 000 3 00 3 00 8 3 . 9 2 8 3 . 9 2 5 8 . 8 2 5 8 . 8 2 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) 3 00 3 00 (a) A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 42. 7 4 1 5 26. 67 1 5 26. 67 7 00 3 00 1 42. 7 4 8 3 . 9 2 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) T a n k 3 00 3 00 3 00 3 00 4. 5 6 T i m e ( h ) 241 . 1 8 5 8 . 8 2 3 00 1 42. 7 4 A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 000 644. 8 8 3 00 1 7 7 . 5 6 1 7 7 . 5 6 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) T a n k 3 00 3 5 5 . 1 2 3 00 3 00 1 7 7 . 5 6 1 7 7 . 5 6 3 00 4. 5 4 5 . 5 6 T i m e ( h ) 47 7 . 5 6 0 2 7 . 5 65 5 . 1 2 65 5 . 1 2 65 5 . 1 2 3 5 5 . 1 2 0 3 5 5 . 1 2 (b) (c)Figure 4: Resultant networks of example 2: (a) single operation without a storage tank, (b) single operation with one storage tank, and (c) cyclic operation with one storage tank.
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one storage tank shown in Fig. 4(c), the water reuse from unit C to unit E, from unit E to unit C, and the water recycle of unit C are all undesired.
4.3.1 Logical constraints for forbidding water reuse/recycle
To achieve practical designs without the forbidden water reuse/recycle, some relevant binary variables and constraints must be added into the problem formulation. Firstly, a set
of binary variables Zii0p, Zispand Zsipare introduced to indicate the existence of “direct”
connections. The following expressions, Eqs. (23)-(25), are used to correlate the binary
variables with the continuous variables. For instance, Zii0p = 1 indicates the existence of
connecting flow from unit i to another unit i0
at time point p, with the amount positioned between a pair of reasonable lower and upper bounds.
QL
ii0Zii0p ≤ qii0p ≤ QUii0Zii0p ∀i, i
0
∈ I, p ∈ P (23)
QLisZisp ≤ qisp ≤ QUisZisp ∀i ∈ I, p ∈ P, s ∈ S (24)
QL
siZsip ≤ qsip ≤ QUsiZsip ∀i ∈ I, p ∈ P, s ∈ S (25)
In addition to the direct connections, the existence of “indirect” connections has to be identified because of the fact that water transfers between water-using units and/or storage tanks will be also in a circuitous way. For example, the outlet water from one water-using unit to another may pass through other units or storage tanks on the way. For this reason,
another set of binary variables Xii0p, Xispand Xsipare further introduced to include such
indirect connections: Xii0p = 1 represents that unit i
0
has received the water from unit i at
time point p in direct or indirect ways; Xisp = 1 denotes that storage tank s has received
the water from unit i at time point p in direct or indirect ways, whereas Xsip= 1 signifies
that unit i has received the water from storage tank s at time point p in direct or indirect ways. In order to make an adaptation of the current situation as well as for the purpose of simplification, only one storage tank is considered in the design system, for the moment.
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Logical relations between direct and indirect connections are presented as follows.
Xii0p ≥ Zii0p ∀i, i 0 ∈ I, p ∈ P (26) Xii0p ≥ Xii00,p−∆i00Zi00i0p ∀i, i 0 , i00 ∈ I, p ∈ P, p > ∆i00 (27) Xii0p ≤ Zii0p+ X ∀i00∈I (Xii00,p−∆i00Zi00i0p) p>∆i00 ∀i, i0 ∈ I, p ∈ P (28)
Equations (26) and (27) state that unit i0 will be recognized as receiving the water from
unit i at time point p if there exist direct flow from unit i or indirect flow through unit i00
.
Equation (28) ensures that no water from unit i has been received by unit i0 at time point
p if there is neither direct nor indirect connection. It is important to note that any possible
paths of water transfers included in Xii0pare not involved the storage tank.
Xisp≥ Zisp ∀i ∈ I, p ∈ P, s ∈ S (29)
Xisp≥ Xii0,p−∆i0Zi0sp ∀i, i 0 ∈ I, p ∈ P, p > ∆i0, s∈ S (30) Xisp≤ Zisp+ X ∀i0∈I (Xii0,p−∆i0Zi0sp) p>∆i0 ∀i ∈ I, p ∈ P, s ∈ S (31)
Equations (29) and (30) state that storage tank s will be recognized to receive the water
from unit i at time point p if there exist direct flow from unit i or indirect flow via unit i0.
Note that the circuitous way from one water-using unit to storage tank is simply passing through other units for the situation of single storage tank. Equation (31) ensures that no water from unit i has been received by storage tank s at time point p providing there exists no direct nor indirect connection.
Xsip≥ Zsip ∀i ∈ I, p ∈ P, s ∈ S (32)
Xsip≥ Xsi0,p−∆i0Zi0ip ∀i, i 0 ∈ I, p ∈ P, p > ∆i0, s∈ S (33) Xsip≤ Zsip+ X ∀i0∈I (Xsi0,p−∆i0Zi0ip) p>∆i0 ∀i ∈ I, p ∈ P, s ∈ S (34)
Similarly, Eqs. (32) and (33) express that unit i will be recognized to receive the water from storage tank s at time point p if there exists direct flow from storage tank s or indirect
flow via unit i0. Equation (34) ensures that no water from storage tank s has been received
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Apart from the identification of all flow connections, the condition inside the storage
tank is also considerable and binary variables Y(occ.)
sp and Espp0are defined accordingly. As
presented in Eq. (35), Ysp(occ.)= 1 denotes that storage tank s keeps a finite storage at time
point p, on the other hand, Y(occ.)
sp = 0 reflects that the storage tank is empty.
QLsYsp(occ.) ≤ Qsp ≤ QUsY (occ.)
sp ∀p ∈ P, s ∈ S (35)
Furthermore, Eqs. (36) and (37) are used to examine the internal condition of storage tank s during a particular time period. It is obviously that storage tank s will be recognized
to have been emptied during the time period from p to p0 (Espp0 = 1) if there is no storage
remaining in storage tank s for at least one time point p00∈ [p, p0
]. Espp0 ≥ 1 − Ysp(occ.)00 ∀p, p 0 , p00 ∈ P, p ≤ p00 ≤ p0 , s∈ S (36) Espp0 ≤ X p00∈[p,p0] (1 − Ysp(occ.)00 ) ∀p, p 0 ∈ P, s ∈ S (37)
On the basis of the above binary variables and logical expressions, the forbidden water reuse/recycle can be avoided by imposing the following constraints. It is clear as depicted
in Eq. (38) that Xii0pis forced to be zero to prohibit the water transfer from unit i to unit
i0
directly or via other units. Note thatIforbid is an assigned set of forbidden unit pairs, in
which both forbidden water reuse (i6= i0
) and recycle (i= i0
) are covered,∀i, i0
∈ I.
Xii0p = 0 ∀(i, i
0
) ∈ Iforbid, p∈ P (38)
Equations (39)-(41) are used in another case for water transfers involving the storage tank, where the receivability of water from storage tank s to unit i is examined. Definitely,
for the prohibition of water transfer from unit i to unit i0
, unit i0
cannot receive any water from storage tank s after unit i supplying water to the storage tank, until the water stored in storage tank s has been emptied by other units. In other words, as described in Eq. (39),
if storage tank s has received the water from unit i at an earlier time point p0 (Xisp0 = 1)
and keeps a certain amount of storage all the time during the time period from p0 to p− 1
(Esp0,p−1 = 0), unit i0 cannot receive any water from storage tank s at time point p (Xsi0p
Accepted Manuscript
the water from unit i at a later time point p0in the time scale and keeps the storage without
being emptied, past the current batch and continued to an earlier time point p− 1 in next
batch (Esp0P = 0 and Es1,p−1= 0), unit i0 cannot receive the water from storage tank s at
time point p. Equation (41) is specially for the first time point p= 1, where the condition
in the previous batch cycle is referred when cyclic operation.
Xsi0p ≤ 1 − Xisp0 + Esp0,p−1 (39) ∀(i, i0 ) ∈ Iforbid, p, p0 ∈ P, p > p0 , s ∈ S Xsi0p ≤ 1 − (Xisp0− Esp0P − Es1,p−1)Z(cyc.) (40) ∀(i, i0 ) ∈ Iforbid, p, p0 ∈ P∗ , p≤ p0 , s∈ S Xsi0p ≤ 1 − (Xisp0− Esp0P)Z(cyc.) (41) ∀(i, i0) ∈ Iforbid, p= 1, p0 ∈ P, s ∈ S
4.3.2 Practical design excluding forbidden water reuse/recycle
Revisit examples 1 and 2 with the additional binary variables and logical constraints. For the same objective to minimize the consumption of freshwater, the design problem is formulated as the following mixed-integer nonlinear program (MINLP).
P3: min x2∈Ω2 J3 = X ∀w∈F X ∀i∈I X ∀p∈P qwip (42) x2 ≡ x1∪ Zii0p, Zisp, Zsip, Xii0p
Xisp, Xsip, Ysp(occ.), Espp0
∀i, i0 ∈ I, p, p0 ∈ P, s ∈ S (43) Ω2 = Ω1∩ {x2| Eqs. (23) − (41)} (44)
To reduce multiple solutions for a more applicable network configuration, the minimiza-tion of storage capacity is further considered.
P4: min x0 2∈Ω 0 2 J4 = X ∀s∈S Qmaxs (45) x0 2 ≡ x2∪ {Qmaxs , ∀s ∈ S} (46) Ω02 = Ω2∩ x0 2 X ∀w∈F X ∀i∈I X ∀p∈P qwip ≤ J ∗ 3 Qsp ≤ Qmaxs , ∀p ∈ P, s ∈ S (47)
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A B C D E 0 2 3 4 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 42. 7 4 1 42. 7 4 1 28 5 . 49 1 28 5 . 49 400 300 1 42. 7 4 1 42. 7 4 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) 300 600 300 300 300 600 3 4 6 T i m e ( h ) T a n k A B C D E 0 2 3 4 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 000 344. 8 8 300 1 7 7 . 5 6 1 7 7 . 5 6 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) T a n k 300 65 5 . 1 2 300 300 1 7 7 . 5 6 1 7 7 . 5 6 1 7 7 . 5 6 35 5 . 1 2 3 4 6 T i m e ( h ) 5 5 . 1 2 65 5 . 1 2 35 5 . 1 2 35 5 . 1 2 0 0 2 7 . 5 (a) (b) A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 000 44. 8 8 3 00 1 7 7 . 5 6 1 7 7 . 5 6 ( 0. 1 ) ( 0. 1 ) ( 0. 1 ) ( 0. 5 1 ) ( 0. 5 1 ) T i m e ( h ) T a n k 3 00 9 5 5 . 1 2 3 00 3 00 1 7 7 . 5 6 1 7 7 . 5 6 3 00 4. 5 4 6 T i m e ( h ) 47 7 . 5 6 0 2 7 . 5 65 5 . 1 2 9 5 5 . 1 265 5 . 1 2 0 (c)Figure 5: Revised networks after forbidding water reuse/recycle in units C and E: (a) single and (b) cyclic operation with one storage tank for example 1, and (c) cyclic operation with one storage tank for example 2.
A set of forbidden unit pairsIforbid = {(C, E), (E, C), (C, C), (E, E)} is assigned here for
the cases of single and cyclic operations in example 1, and cyclic operation in example 2. As the revised networks of example 1 shown in Fig.s 5(a) and (b), the outlet water from unit C is discharged instead of entering the storage tank. Besides, the revised network for example 2 in Fig. 5(c) shows that the outlet water from both units C and E is discharged rather than being stored. The freshwater consumptions for those cases are kept the same (1285.49 kg, 1000 kg and 1000 kg) at the expense of the enlarged storage capacities (600
kg,655.12 kg and 955.12 kg). The solver selected for MINLP is SBB, and all computation
Accepted Manuscript
Table 3: Computation results of cases with forbidden reuse/recycle.
Case Example 1 Example 2
Operating mode Single op. Cyclic op. Cyclic op.
Problem structure MINLP
No. of constraints 6294 7230 7125
Continuous variables 1121 1121 1121
Discrete variables 1392 1392 1392
Freshwater (kg) 1285.49 1000 1000
Storage cap. (kg) 600 655.12 955.12
CPU time (sec) 2.14 5.25 6.06
Table 4: Processing data for example 3.
Unit [Qmin i , Qmaxi ] C (in) ic,max C (out) ic,max t (st.) ip t (fin.) ip M (load) ic (kg) (kg salt/kg water) (h) (h) (kg) A [0, 1000] (0, 0, 0) (0.1, 0.1, 0.1) 0 4.5 (100, 0, 50) B [0, 364] (0.25, 0.1, 0.1) (0.51, 0.2, 0.2) 0 4 (72.8, 36.4, 0) C [300, 400] (0.1, 0.1, 0.1) (0.1, 0.1, 0.1) 4 5.5 (0, 0, 0) D [0, 364] (0.25, 0.1, 0.1) (0.51, 0.2, 0.2) 2 6 (72.8, 36.4, 0) E [300, 400] (0.1, 0.1, 0.1) (0.1, 0.1, 0.1) 6 7.5 (0, 0, 0)
4.4
Example 3: multiple contaminant case
The third example is an adaptation of example 2 for a multiple contaminant environ-ment as described in Table 4. The forbidden set of unit pairs for this example is defined as Iforbid = {(C, C), (E, E)}. The resultant network configurations and computation results
are presented in Fig.s6(a)-(d) and Table 5, respectively. It is worthy to note that although the operational constraints in terms of forbidden water recycles do not cause any penalty for the freshwater consumption and required storage capacity, the storage policy needs to be changed to meet a practical design.
5
Conclusions
Operations of water-using units in batch plants are almost time dependent. Neverthe-less, such time dependence can be partially bypassed by equipping central storage facili-ties, to temporarily collect the reusable water from the earlier operations and then supply that water for the later operations. With a predefined production schedule, this paper has
Accepted Manuscript
A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 8 2 1 8 2 225 225 1 8 1 4 1 8 1 4 1 000 3 00 3 00 1 07 1 07 7 5 7 5 (0.4,0.2,0) T i m e ( h ) 3 00 3 00 (0.4,0.2,0) (0.1 ,0,0.05 ) (0.1 ,0.05 ,0) (0.1 ,0.05 ,0) A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 8 2 1 5 8 9 1 5 8 9 7 00 3 00 1 8 2 1 07 T i m e ( h ) T a n k 3 00 3 00 3 00 3 00 4. 5 6 T i m e ( h ) 225 7 5 3 00 1 8 2 (0.1,0,0.05) (0.1,0,0.05) (0.4 ,0.2 ,0) (0.4 ,0.2 ,0) (0.1,0.05,0) (a) (b) A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 000 63 6 3 00 1 8 2 1 8 2 (0.1,0,0.05) (0.5,0.2 ,0.05) T i m e ( h ) T a n k 3 00 3 64 3 00 3 00 1 8 2 1 8 2 3 00 4. 5 4 5 . 5 6 T i m e ( h ) 48 2 0 2 7 . 5 664 664 664 3 64 0 3 64 (0.5,0.2 ,0.05) (0.1,0,0.05) (0.1,0,0.05) A B C D E 0 2 44. 5 5 . 5 6 7 . 5 Freshwater D i sc harg e 1 000 1 000 3 3 6 3 00 1 8 2 1 8 2 (0.1,0,0.05) (0.5,0.2 ,0.05) T i m e ( h ) T a n k 3 00 664 3 00 3 00 1 8 2 1 8 2 3 00 4. 5 4 5 . 5 6 T i m e ( h ) 48 2 0 2 7 . 5 664 664 664 3 64 0 (0.5,0.2 ,0.05) (0.1,0,0.05) (0.1,0,0.05) (c) (d)Figure 6: Resultant networks of example 3: (a) single operation without a storage tank, (b) single operation with one storage tank, (c) cyclic operation with one storage tank, and (d) cyclic operation with one storage tank after forbidding water recycle in units C and E.
Table 5: Computation results of example 3.
Operating mode Single op. Cyclic op.
No. of storage tanks 0 1
Forbidden recycle No Yes
Problem structure NLP MINLP
No. of constraints 1521 1681 1681 7285
Continuous variables 1201 1505 1505 1505
Discrete variables − − − 1392
Freshwater (kg) 1814 1589 1000 1000
Storage cap. (kg) 0 300 664 664
Accepted Manuscript
presented a mathematical formulation for synthesizing batch water-using networks with central storage tank(s). Superstructures incorporating all flow connections are constructed to assist the problem formulation. The minimum freshwater consumption is targeted, and both operating modes (single and cyclic operations) are included. The design problems is formulated as a nonlinear program (NLP) if there is no constraint on water reuse and recy-cle, or a mixed-integer nonlinear program (MINLP) when forbidding water reuse/recycle is taken into consideration. Representative examples adopted and adapted from literature are provided to demonstrate the good expectation of the proposed, superstructure-based NLP and MINLP formulations. The logical expressions developed in this work are appli-cable to the cases with single storage tank. More possibilities of flow connections should be comprehended when considering the cases with multiple storage tanks.
Acknowledgment
Financial supports of the Ministry of Economic Affairs (under grand 96-EC-17-A-09-S1-019) and the National Science Council of ROC (under grand NSC96-2221-E-002-151-MY3) are appreciated.
Nomenclature
Indices and Sets
c∈ C = {1, ..., C}, contaminant species
d∈ D = {1, ..., D}, discharge points
i∈ I = {1, ..., I}, water-using units
p∈ P = {1, ..., P }, time points
p∈ P∗
= {2, ..., P }, time points except the first one
s∈ S = {1, ..., S}, storage tanks
Accepted Manuscript
Parameters
Cic,max(in) maximum inlet concentration of contaminant c of unit i Cic,max(out) maximum outlet concentration of contaminant c of unit i
Cwc concentration of contaminant c in water source w
∆i distance between time points of start/finish of the operation in unit i
Mic(load) mass load of contaminant c to be removed in unit i
QL
i lower bound for the water requirement of unit i
QU
i upper bound for the water requirement of unit i
QL
s lower bound for the capacity of storage tank s
QU
s upper bound for the capacity of storage tank s
QL
∗∗0 lower bound for the connecting flow from∗ to ∗
0
,∗∗0
∈ {ii0
, is, si}
QU
∗∗0 upper bound for the connecting flow from∗ to ∗
0
,∗∗0
∈ {ii0
, is, si} y(st.)ip ∈ {0, 1}, = 1 denotes unit i starting the operation at time point p y(fin.)ip ∈ {0, 1}, = 1 denotes unit i finishing the operation at time point p Z(cyc.) ∈ {0, 1}, = 1 denotes cyclic operation
Binary variables
Espp0 = 1 means storage tank s was emptied during the time period p to p0
Xii0p = 1 means unit i0 received the water from unit i at time point p
Xisp = 1 means storage tank s received the water from unit i at time point p
Xsip = 1 means unit i received the water from storage tank s at time point p
Y(occ.)
sp = 1 means storage tank s keeps a finite storage at time point p
Zii0p = 1 means existing flow from unit i to unit i0at time point p
Zisp = 1 means existing flow from unit i to storage tank s at time point p
Zsip = 1 means existing flow from storage tank s to unit i at time point p
Positive variables
Cicp(in) inlet concentration of contaminant c to unit i at time point p
Cicp(out) outlet concentration of contaminant c from unit i at time point p
C(in)
scp inlet concentration of contaminant c to storage tank s at time point p
C(out)
scp outlet concentration of contaminant c from storage tank s at time point p
qidp water flow from unit i to discharge point d at time point p
qii0p water flow from unit i to another unit i0 at time point p
qisp water flow from unit i to storage tank s at time point p
qsip water flow from storage tank s to unit i at time point p
qwip water flow from water source w to unit i at time point p
qip(in) inlet water flow to unit i at time point p
qip(out) outlet water flow from unit i at time point p
Qmax
s maximum capacity of storage tank s
Qsp amount of remaining water in storage tank s at time point p
q(in)
sp inlet water flow to storage tank s at time point p
Accepted Manuscript
References
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Captions of Figures and Tables
Figure 1 Superstructures for (a) water-using unit and (b) storage tank. Figure 2 Gantt chart for the production schedule of example 1.Figure 3 Resultant networks of example 1: (a) single operation without a storage tank, (b) single operation with one storage tank, and (c) cyclic operation with one storage tank.
Figure 4 Resultant networks of example 2: (a) single operation without a storage tank, (b) single operation with one storage tank, and (c) cyclic operation with one storage tank.
Figure 5 Revised networks after forbidding water reuse/recycle in units C and E: (a) single and (b) cyclic operation with one storage tank for example 1, and (c) cyclic opera-tion with one storage tank for example 2.
Figure 6 Resultant networks of example 3: (a) single operation without a storage tank, (b) single operation with one storage tank, (c) cyclic operation with one storage tank, and (d) cyclic operation with one storage tank after forbidding water recycle in units C and E.
Table 1 Processing data for examples 1 and 2. Table 2 Computation results of examples 1 and 2.
Table 3 Computation results of cases with forbidden reuse/recycle. Table 4 Processing data for example 3.
