Dynamic Water Network Topology Optimization of Batch Processes

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Dynamic Water Network Topology Optimization of Batch Processes Elena-Lacramioara Dogaru* and Vasile Lavric* Chemical Engineering Department, University Politehnica of Bucharest, RO-011061, Polizu 1-7, Bucharest ABSTRACT: A water network (WN) for a batch process could be seen as a dynamic structure which changes its topology at fixed time intervals delimited by events. During each time interval, the WN could be abstracted into an oriented graph, using a secondary ranking criterion, apart from schedule. The optimization strategy implies ordering the batch units twofold: by time, according to the predefined task schedule of each water-using unit (WU), and by maximum allowable outlet concentration of contaminants, the latter being consistent with the principle of driving force equipartition across the process. This paper focuses upon two aspects: (a) the optimization of the WN topology particular to each period, in close correlation to the storage tank behavior and (b) the dynamics of this optimized topology, as the batch process runs sequentially toward its completion. The objective is to minimize the fresh water consumption by raising the internal wastewater reuse between water-using units or from the storage tank (ST). The aforementioned optimization of the mathematical model, consisting of a system of differential algebraic equations with restrictions, is carried out using genetic algorithms, which generate the internal flows appropriate for each time window structure. A synthetic network of six WUs, three contaminants, one fresh water source, and one ST is presented and optimized under various scenarios, and the obtained results are analyzed and discussed.

1. INTRODUCTION Water is an important resource and because it is used for multiple purposes (reactant, thermal agent, solvent etc.) a large quantity is consumed in industry yearly.1 The contaminated water is sent to water treatment networks or is discharged in the environment, if the environmental regulations are met. More and more environmental and economic constraints together with rising the sustainability consciousness1,2 lead to a higher interest for the effective use of industrial water, due to its limited availability with respect to the ever increasing demand.1,3 To minimize both freshwater and energy consumption, several techniques appeared (internal water reuse, process scheduling, localized regeneration, retrofit, etc.) with a high degree of generality and promising results.4,7 Batch processes are used considerably for the manufacture of small volume products that have relatively high value8 (pharmaceutical products, inks), and also in food industry (margarine production, creams, etc.). Their main characteristics are great versatility and flexibility,3 allowing the use of the same equipment for the production of different chemicals having similar recipes.9 A water system associated to a batch process consists in the network linking different types of water using operations, each of them having different water requirements, qualitative and quantitative, depending on their task (washing, heating, extracting, etc.). Therefore, different quantities or flow rates of water are discharged from each of these water-using operations (WUOs), with different qualities according to their particular tasks. The WUOs could be lumped into two broad categories. (1) Fixed load, based on mass transfer operations, where water is the carrier of the contaminant load (e.g., washing, extraction): the inlet and outlet flow rates are assumed to be equal, and the contaminant load together with inlet and outlet concentration restrictions for r 2010 American Chemical Society

each operation are known. (2) Fixed flow rate: Water is either part of the chemical process (reactant/product), or thermal agent; the inlet and outlet flow rates are specified and in some cases the inlet and outlet streams have specific concentration values (e.g., maximum allowable inlet/outlet concentrations). Even though much of the research is focused on the optimization of continuous water and wastewater processes10-14 nowadays the discontinuous processes gained increasing attention.8,9 One of the reasons for which the optimization of batch water network (WN) is only in the early stage is that it is more difficult to directly apply the techniques derived from water minimization of the continuous networks, to the batch processes as the latter are time dependent, not only with respect to the water usage, but also to the network architecture.15 In continuous processes, the feasibility of water reuse within the units’ network is controlled by the concentration constraints. This means that water exiting from a continuous unit i, cannot be further used in other units, unless the contaminants’ concentrations of the outlet stream (for unit i) meet the inlet restrictions of the other units, concerning the pollutant concentrations. If not, sometimes it is possible to dilute the reused water streams with some fresh water, until these restrictions are met.15 Still, these makeup flows should be chosen such that the fresh water consumption should be kept as low as possible.16,17 For the batch processes, the time dimension becomes the major issue for water reuse, in addition to the concentration constraints. The former refers to the fact that water exiting from Special Issue: Water Network Synthesis Received: February 5, 2010 Accepted: June 10, 2010 Revised: May 28, 2010 Published: August 06, 2010 3636

dx.doi.org/10.1021/ie100284t | Ind. Eng. Chem. Res. 2011, 50, 3636–3652

Industrial & Engineering Chemistry Research the generic batch unit i, even fulfilling the concentration constraints, cannot be further used in the rest of the units, unless they share the same time-window (when the WUs are semicontinuous with respect to water) or they start immediately after this generic unit.15,18 This underlines the importance of process scheduling in the optimization of batch water-using units.19-21 The time constraint was partially outweighed by the use of storage tanks (STs). These are reservoirs that collect the water used in one or several batch units, for it to be reused by the WUs starting later (if the concentration restrictions allow this). However, the issue still standing is the limited capacity22-25 of these containers together with their number; when one of the pollutants has damaging effects upon several units, which otherwise could reuse some water from the ST, at least two STs should be used, one for the streams with the incriminated impurity and the other for the rest of the streams.3 The inlet concentrations restrictions are partially outweighed as well by the use of regeneration units, but this aspect gained attention only in the past few years.4,26,27 The main water minimization techniques could be classified in two broad categories: (1) insight based—graphical and algebraic and (2) mathematical model optimization methods. Insight-Based Methods. Many insight-based methodologies were developed for optimizing WNs.28-30 Their main advantage is that the established targets (e.g., fresh feed flow rate) are identified before the detailed network design.3 But, on the other hand, these methods are limited to single contaminant systems,5 their generalization to networks dealing with multiple pollutants being difficult. Another drawback consists in scaling problems in the case of very different units’ compositions or flow rates, which can distort the representation.3 Wang and Smith29 introduced the graphical technique of time pinch analysis for the minimization of wastewater in batch processes. Both constraints, concentration and time, are considered simultaneously. The authors divided the operations on concentration intervals, and tried to reuse water from a lower concentration interval to a higher one. Fresh feed is used when water is not available for reuse. The major drawback of this approach is that all WUOs are formulated as mass transfer processes, and therefore the approach is not suitable for fixed flow operations. Also, the technique proved to be applicable only to semibatch rather than strictly batch operations, as water reuse is allowed between operations with overlapping functional time intervals. The fixed flow rate operations were first approached by Foo et al.30 They developed an algebraic technique, water cascade analysis, for targeting the minimum water flows for a batch process. The proposed method separated the cases of networks with or without storage tanks, and did not permit water sources having different impurity concentration to mix; therefore too many storage tanks would have been necessary for a higher number of different concentration water sources. The approach is suitable for repeated batch processes, but not for single ones with storage tanks. To overcome the limitations of Wang and Smith, Majozi et al.22 proposed a graphical technique for water minimization in batch processes, that is, where water reuse or recycle can only be effected either at the beginning or at the end of the process.22 This technique considers both concentration and time as constraints and proved the possibility to use off-duty WUs as STs. The user can choose between concentration and time as primary constraint. Reversing the priority order has proven not to have

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any effect on the water target. It is recommended though to choose time as the primary constraint for problems involving a smaller number of concentration intervals, and the other way around for problems having a smaller number of time intervals. The approach is applicable to batch processes having single contaminant streams, which is a common drawback of graphical methods. Liu et al.28 proposed a time-dependent concentration interval analysis (CIA) technique (algebraic insight based) for targeting both the single and repeated discontinuous WU systems involving the two types of WUOs: fixed load and fixed flow rates. The objective was to minimize the freshwater consumption and wastewater generation through maximization of the available water resources in designing the WN for batch process and reusing water between operations within the same time interval. Also for fixed flow problems, but this time focusing on batch water networks design, Chen and Lee25 presented a graphical technique. Besides the design aspects, the method also ensures finding a minimum ST capacity. Mathematical Modeling-Minimizing Fresh Water Consumption. For more complex systems, the mathematical models represent the methodology of choice, since they are suitable for multiple contaminants problems.5 Another advantage is the flexibility and adaptability of the included performance indicator, for example, the objective function which could be plain fresh water consumption or more elaborated cost functions.15,31-33 The mathematical techniques are used as a synthesis tool,32,34,35 and they usually depend on a pre-established process schedule, although there are approaches for simultaneous optimization of the schedule and the WN topology.5,9,15,22,36-38 But the use of mathematical models does not provide the same perception as insight-based methods. Furthermore, when the mathematical formulation is represented by nonconvex nonlinear problems (NLP) or by mixed-integer nonlinear problems (MINLP), obtaining the global solution may involve great efforts.5 Starting from a given production plan, Almato et al. used a mathematical optimization approach to synthesize the batch water network.32 The connections between the available STs and the WUs had been determined using different objectives: freshwater demand, water cost, utility demand of water streams, and water reuse network costs. The water reuse system had been modeled, simulated, and optimized using a superstructure, including all connections between the units and the ST. Majozi5 presented a continuous-time mathematical method for multipurpose batch plants which is used for freshwater and wastewater minimization, with and without a central reusable water storage facility. This goal was achieved through the exploitation of recycle and reuse opportunities. The study assumed a predefined schedule, with specified beginning and ending times, which become later optimization variables. For the case without ST, the WUs’ outlet concentrations or the contaminant mass load are fixed, and the mathematical formulation is represented as a mixed integer linear problem (MILP). But this representation changes to MINLP if either the outlet concentration varies within an interval or the ST is used, and therefore the global optimality cannot be guaranteed for complex problems. In a later paper,22 Majozi presented a methodology aimed to minimize both the reusable water storage capacity and the amount of freshwater requirement and wastewater generation. These objectives are contradictory, because when increasing the reusable water storage capacity it allows a reduction in freshwater use, due to increased water reuse. On the other hand, reducing 3637

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Industrial & Engineering Chemistry Research the reusable water storage capacity implies the increase of freshwater use, as a result of reduced water reuse. A two-stage solution procedure was proposed in order to handle the conflicting objectives. The mathematical model is of a MINLP type and nonconvex, while the methodology is applicable to single contaminant batch processes only. A mathematical model for designing the optimal buffer system for the equalization of flow rates and contaminant concentrations of its output was developed by Chang and Li,9 targeting the integration of batch WUs networks equipped with buffer tanks. Shoaib et al.4 presented an approach for synthesizing batch WWNs, having multiple objectives. Because of the complexity, the proposed approach has a three-stage hierarchy: minimization of freshwater, minimization of the number of storage tanks, and the minimization of interconnections between units and storage tanks. Insights gained from the water pinch analysis are also incorporated, in order to further reduce water consumptions through water regeneration units. Recently, Chen et al.39 presented a mathematical formulation for the synthesis of batch WWNs with central STs, starting from a predefined production schedule. They use superstructures including all possible flow connections and the design was formulated as a NLP, when there was no limitation of water reuse/ recycle, or a MINLP, when forbidding water reuse between the assigned WUs and water recycle of certain WUs. Traditionally, the tasks of optimizing batch schedules, water reuse, and wastewater treatment subsystems were performed individually.15 Chang and Cheng were the first who developed a procedure to incorporate the three components into a broader mode.18 To overcome the shortcomings of the aforementioned study, Zhou et al.15 modified the state-space superstructure to formulate a MINLP model for simultaneous optimization of batch process schedules and single- or multicontaminant water allocation system. The advantage of this approach is that not only all possible connections between operations and/or equipments can be easily incorporated, but the batch concept as well. In addition, the network complexity can always be reduced by introducing additional costs of splitters and mixers, and all examples provided proved that it is also an effective way for the simultaneous overall designs and complexity evaluation. By interactively applying perturbations, based on the sensitivity analysis in this strategy, the global optimum can almost always be identified in all case studies presented.15 There is a recent trend to combine both techniques to benefit from all their advantages.40 The current paper presents a new approach for the WN dynamic topology optimization for batch processes with the associated WUs experiencing a continuous throughput of water flow rate. The optimization is carried out considering that there is only one ST available, which acts as both sink and contaminated water source. The mathematical model, grouping steady state and dynamic mass balance equations, is of the DAE type and includes also the usual concentration restrictions associated to each WU belonging to the network. An objective function aimed to minimize the fresh water consumption is superposed to this DAE system, and the optimal dynamic topology is found, for the overlapping WUs on each time interval, using GA41-44 which generates the sequential optimal internal flows eventually. This new approach is suited for batch processes having a pre-established schedule and continuous water flow throughput, although the schedule optimization could be implemented as well. For the moment, our methodology does not work for pure batch

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processes, including here the repeated batch ones, where both the WU and the water utility are fed at the beginning of the stage then discharged at the end. If the repeated batch processes means the use of water as utility in two or several sequential stages, the present approach applies simply considering these stages as separate but consecutive time-windows. To illustrate the advantages of this technique, a batch process using six WUs, three contaminants and, one ST is presented, optimized, and analyzed. Several scenarios are then proposed and thoroughly discussed with respect to this original case.

2. PROBLEM STATEMENT There is a discontinuous process with a predefined schedule and an associated dynamic WN formed of N batch WUs with respect to the raw material transformation, but continuous with respect to the water/wastewater flow throughput, dealing with K contaminants, and one ST having infinite capacity; later within this paper, the constraint of finite capacity of the ST will be dealt with. The purpose of optimization is to search for the network topology particular to each time window of the schedule together with the associated internal flows which ensure the minimum fresh water consumption by increasing the internal reuse of water streams. The internal reused flows can originate from both the ST and the active WUs, the latter ordered according to some appropriate ranking criterion, which will be presented in the next section. The optimization of the aforementioned objective function superposed to the pending DAE mathematical model of the WUs is carried out using GA as implemented in Matlab. 3. PHYSICAL MODEL A discontinuous WN could be seen as a dynamic structure which changes its configuration after the occurrence of an event, defined as the start or the end of overlapping or sequential WUs/ WUOs. During each fixed time interval delimited by two consecutive events, the WN topology remains unchanged, but the water streams could change at least with respect to their composition. Since during a time interval the WN can be seen as stationary with respect to the topology, at least, the techniques developed for steady-state continuous optimization could be applied.41-44 Still, care should be taken when considering the possible variations of flows and/or compositions, due to the everchanging composition of the ST collecting outputs from the active WUs. During each time interval, the WN could be abstracted to an oriented graph.41,44 The nodes represent the WUs and the ST, while the arches represent the unidirectional flow connections. The batch units are ordered not only by time, according to the task schedule of each WU, but a second ranking by maximum critical outlet concentration is applied, consistent with the principle of driving force equipartition across the process. Figure 1 should provide the reader a clear image about the result obtained when applying the two ordering criteria. The time schedule covered by the N WUs can be divided into at most 2N - 1 time intervals, limited by two events. As an example the first time interval (I) is fixed by the beginning of operation A and the beginning of operations B and F, while the second time window (II) is delimited by the latter events and the beginning of operations C and D (Figure 1). At a conceptual level, the graph mentioned before exists virtually, linking all the overlapping WUs. The connections between the units, including the ST, are symbolized by arches. These encode virtual pipes through which the flow has only one direction. 3638


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Table 1. The WUs Which Could Be Tagged Forbidden Unit during the Batch Process from Figure 1 I

time interval a

forbidden unit

II

III

IV

V

VI

VII

B

D

D

A

C

E

a

Not always will the WU ending a sequence in a time window operate as a forbidden unit, but only when its exit stream attains at least one of the exit concentrations limits.

Figure 1. Diagram of batch units ordered by schedule time windows and by Cmax out . The time schedule covered by the N batch WUs (horizontal lines) can be divided into at most 2N - 1 time intervals, limited by two consecutive events. An event is defined as the moment a WU begins or ends its operating regime (vertical dotted lines). The second ranking criterion is the maximum allowable exit concentration (from the least pollutant, unit F in the second interval, to the most pollutant, unit B in the same time window, i.e., the exit flow of F unit could be reused in A and B units, while the exit of A unit can be reused in B unit only).

For example, there could be no virtual link between WUs B and E from Figure 1, since they do not share any time-window. On the contrary, WU E should have virtual links to WUs A, C, D, and F, since they share at least one period and WU E is the first node of the oriented graph. During the optimization procedure, the virtual connections for which there is no flow assigned during any time interval vanish. The rest of them become physical connections which could be active or not after each event, as the water passes through them or not, according to the results of the optimization algorithm. The complexity of the topology changes with each event, because the WUs connections are different for each defined interval. When the units belonging to the same period are sorted in ascending order, according to their maximum possible outlet concentration, the water reuse between units in cascade should boost. This way, the hope is that the ST will be kept as a last resort side/supplemental source. To avoid excessive contamination of the ST, the output of the last unit in the graph/network (the WU with the highest maximum output concentration from all the overlapping WUs) could be sent directly to treatment, bypassing the ST, if the exit water reaches at least one of its output limits in the current time interval. Such an unit is tagged forbidden unit (see Table 1 for the possible forbidden units during the batch process from Figure 1). The supply for any WU can be obtained mixing streams from the followings sources: (i) fresh water—alone, in the special cases when no reuse is possible, or as makeup water, when necessary (ii) partially contaminated water from previous WUs (see Figure 1); (iii) contaminated water from ST, which collects the excess water from all the WUs that overlap during the current time interval, except when the forbidden unit concept applies. The outlet water from a WU can have three destinations: (i) the other WUs situated after the current one in the graph, as feed (see Figure 1), the outlet stream can be split according to the operational needs; (ii) the ST; (iii) the treatment network, when this WU is operated as forbidden unit. 3.1. Simplifying Assumptions. The fresh water feed is considered contaminant free (CFk = 0, "k ∈ K). Every WU may send water either directly to following WUs common to the same time-window (if the concentration constraint allows this after the use of some makeup fresh water, eventually) or to the ST. The last WU in the network, corresponding to a particular

time window, can even send heavily contaminated water directly to wastewater treatment, bypassing the ST, when it becomes forbidden unit. Conceptually, the stream heading toward treatment does not represent a separate unknown; although its destination changed, its origin remains the same. In the case of finite capacity ST, water from the ST is sent to the treatment network only when overflowing; the concentrations’ limit is kept at lower levels through the forbidden unit concept. Every ST with finite capacity has an upper limit, which triggers the withdrawal of water to be sent to treatment, and a lower limit, which stops this water withdrawal. This volume change will affect the time profile of the ST composition and, thus, the possibility of water reuse from the tank. The contaminant load for each water-using unit, Δmki, is timeindependent, and so are the internal and supply flows Xji, Xij, Sin i (see Figure 2). It should be emphasized that, even in the case where the contaminant load would change during the time window, the approach is still applicable using the maximum value for this load during the working period with the drawback that the fresh water consumption would be slightly overestimated. Starting from the aforementioned abstraction and these assumptions, an upper triangular matrix containing the unknowns of the WN can be built, having as elements the internal flows, Xji and Xij, together with the supplies from the ST, Sin i , characteristic to each generic unit i (see Figure 2). The elements on the column i, Xji, j < i, represent the streams that may enter the WU i, in the time interval τn, reused internally from the previous WUs, while the elements on the line i, Xij, j > i, are streams being reused from the WU i to the rest of the WUs, ordered according to the ranking criterion of ascending maximum output concentrations. The streams Sin i are uploaded from the ST, to diminish the fresh water consumption. When there are no more reusing possibilities, the remaining flow from each WU is directed to the ST as Sout i .

4. MATHEMATICAL MODEL For each time interval defined by two successive events, the WN is considered stationary with respect to its topology, but dynamic with respect to water composition. When the inlet stream of a WU (a mix of the internally reused streams from the previous units and the stream coming from the ST) has higher concentrations than the inlet restrictions allow, fresh water should be used as makeup; this fresh water stream can be adjusted in time, if the tank concentrations become too high and the inlet/outlet WU’s restrictions are breached. This increase will be balanced by the stream heading toward the ST which will be augmented accordingly. The mathematical model describing the WN consists of (a) steady state overall and partial mass balance equations for the WUs operating in the particular time-window network topology and (b) dynamic overall and partial mass balance equations for the ST. 3639



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Figure 2. The upper triangular matrix with all the possible reuse flows in a concentration ordered series corresponding to a time interval.

Figure 3. Schematic representation of flows around generic WU i.

4.1. Mass Balance over WUs. The generic WU i is represented in Figure 3. Starting from the above scheme the overall mass balance for the water-using units can be written as in eq 1. Therefore the total inlet flow which may be composed of fresh feed (used as makeup water), Fi, of streams coming from the units with a lower outlet concentration, Xji, or from the ST, Sin i , and of the total flow of contaminant provided by each unit, Δmi, as given by eq 2, should be balanced by the total outlet flow: streams sent directly to other units having a higher outlet concentration, Xij, losses, Li, and flow sent to the ST, Sout i , or in the case when i is operated as forbidden unit—flow sent to the treatment network, Wi Sout i . In the partial mass balance eq 3, the fresh water supply is assumed to be contaminant free; therefore, its flow contribution is disregarded in the inlet but taken into consideration in the outlet streams. Because of the oriented nature of the graph, only the preceding WUs can feed the current one, which could be a supplemental source to the next WUs in the network. To that is added the contribution of the ST and the total flow of contaminants withdrawn from unit i. i-1

Fi þ

∑

j¼1

The optimum operating condition for WU i is when at least one of its inlet or outlet (better both) restrictions are met. Although the ST is in a dynamic regime, all the water-using units are steady state, except for their inlet concentrations coming from the aforementioned ST; when the situation imposes, the fresh water flow will suffer incremental adjustments so that the unit’s restrictions are not violated. 4.2. Meeting Inlet Restriction. Equation 4 formulates the condition for optimal operation of unit i, regarding the inlet streams. The least quantity of fresh water is used when the total inlet flow reaches at least one of the maximum allowable inlet concentrations. From eq 4, the freshwater flow for each contaminant eq 5, necessary to reach its respective maximum allowable inlet concentration, is computed. Therefore, in order to find the value for the freshwater flow satisfying simultaneously every contaminant restriction, the maximum of Fin* ki should be retained eq 6. i-1

i-1

N

Xji þ Sin i

þ Δmi ¼

∑

N

in, max ∑ ðXji Ckj Þ þ Sini Csk ¼ ðFkiin þ j ¼∑i þ 1Xij þ Sout i ÞCki j¼1

j¼i þ 1

Xij þ Li þ Sout i

ð1Þ

Fkiin ¼

ð4Þ

N

in, max s out ∑ ðXji Ckj Þ þ Sin i Ck - ð ∑ Xij þ Si ÞCki j¼1 j¼i þ 1 in, max

Cki

ð5Þ

K

Δmi ¼ i-1

∑ Δmki k¼1

ð2Þ

Fiin ¼ max ðFkiin Þ k

N

∑ ðXji Ckj Þ þ Sini Csk þ Δmki ¼ ðFi þ j ¼∑i þ 1Xij þ Li þ Sout i ÞCki j¼1 ð3Þ

ð6Þ

4.3. Meeting Outlet Restriction. When the unit operates under minimum fresh water consumption, at least one of the outlet concentrations should equal its corresponding restriction, as in eq 7. Reasoning the same way as for inlet, the minimum 3640



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Figure 4. Schematic representations of the ST and its inlet and outlet flows

fresh water needed to obey all outlet restrictions is given by (9). i-1

∑ ðXjiCkj Þ þ Sini Csk þ Δmki

j¼1

N

¼ ðFkiout þ

out, max ∑ Xij þ Li þ Sout i ÞCki j¼i þ 1

i-1

N

out, max ∑ ðXjiCkjÞ þ Sini Csk þ Δmki - ðj ¼∑i þ 1Xij þ Li þ Sout i ÞCki

j¼1

out, max

Cki

ð8Þ

Fiout ¼ max ðFkiout Þ

ð9Þ

k

Finally, in order for the unit to respect both restrictions, the WU should be fed with the higher of the two flows, eq 10. This represents the minimum freshwater flow for the WU i. Fi ¼ maxðFiin , Fiout Þ

ð10Þ

4.4. Mass Balance over the ST. The infinite ST continuously receives contaminated water from the WUs and sends water back to them, if needed (see Figure 4). As discussed in the simplifying assumptions section, there could be situations in which WU i could be operated as forbidden unit, meaning that the stream Wi changes destination, heading to the treatment facility, Sout i instead of the ST. The stream sent to treatment, Wi, could appear also when the tank becomes filled because of its limited capacity and should be emptied to a lower level. Equation 11 stands for the total mass balance over the ST. The change in volume is due to the imbalance between the inlet (Sout i ) and outlet (Sin i ) streams that come and are sent respectively to the batch units.

dm dV ¼F ¼ dt dt

N

in ∑ ðSout i - Si Þ i¼1

ð11Þ

The partial mass balance over the ST is presented in eq 12. The ST is assumed to be perfectly mixed and therefore, at every moment t, there is a bulk concentration of the tank content. d ðmCsk Þ ¼ dt

N

N

in s ∑ ðSout i Cki Þ - ∑ ðSi Ck Þ i¼1 i¼1

ð12Þ

Taking into account the mass variation described by eq 11, the final relationship which describes the time change in concentrations of the tank reads N

dCsk dt

¼

s ∑ Sout i ðCki - Ck Þ i¼1

m

N

ð7Þ

Fkiout

¼

5. SOLVING ALGORITHM The problem at hand is to save as much fresh water from being used at the expense of wastewater internal reuse between units and/or supply water from the ST. This is a complex optimization task, involving a dynamic network topology, which changes from time interval to time interval, due to the discontinuous nature of the process with respect to the raw material processing. The simplest objective function to be minimized, in order to attain this task, is the global fresh water quantity used during the batch process:

ð13Þ

minðGw Þ ¼ min½

Q

∑ ∑ ðFiq τq Þ i ¼ 1q ¼ 1

ð14Þ

where Fiq represents the fresh water flow feeding WU i during the time interval τq. The general procedure to find this optimum value is to integrate the DAE system, knowing that initially there is no water in the ST, and minimize the fresh water consumption on each time interval delimited by two consecutive events. Since, from the point of view of water consumption, the WUs operating within one time interval are fully independent from the WUs operating within the next time-window, optimizing them implicitly means getting the batch process optimized. It is worth noting that sorting the units in ascending order of their maximum output concentrations during each time interval and using the forbidden unit concept preserve the ST from being overpolluted which would render its content harder to be used as supply water. The solving algorithm is described by the following steps: 1. Integrate the system for the first time interval, disregarding the contribution of the ST, but reusing as much water as possible, when several WUs are overlapping; for this, use GA to find the optimal topology together with the subjacent internal flows, Xij, i = 1, N, j = i þ 1, N. 2. At the beginning of each remaining time intervals, use GA to find the subjacent internal flows, Xij, i = 1, N, j = i þ 1, N and streams coming as supply of WUs from the tank, Sin i , i = 1, N. They will give the optimal topology—an actual pipe exists for every nonzero flow. During each time window, the flows resulting from applying GA are kept constant. 3. With the flows from step 2, compute the actual state of the streams Fi (how much makeup water, if any, should be together with the concentrations Cout used) and Sout i ki for each active WU i, ensuring the proper compliance with the restrictions. Water is sent directly to the water treatment network and not to ST if the last unit in the series (polluting the most) gets operated as a forbidden unit. 4. Integrate one step forward the DAE system to obtain the corresponding mass and concentrations in the ST. If the end of the current time window was not reached, continue with step 3. 5. Change the current time interval and repeat the procedure starting from step 2, until the end of the batch process. 6. Compute the optimum global fresh water quantity needed summing up the quantities obtained in each time-window. The core of the optimization procedure is the GA, as implemented in Matlab. For each time interval, the number of unknowns exceeds the number of mass balance equations with the internally reused flows, Xij, and flows coming from the ST, Sin i . Each of these unknown flows is set as a gene and the ensemble of 3641



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Table 2. Operating and Restriction Data for the Batch Process Units and Their Use Schedule According to the Technological Specificationsa Unit 1b

Unit 2

Unit 3b

Unit 4

Unit 5

Unit 6

Δm1i, kg/h

0.35

0.15

0.55

0.45

0.25

0.65

Δm2i, kg/h

0.25

0.35

0.45

0.15

0.65

0.55

Δm3i, kg/h

0.35

0.25

0.15

0.45

0.65

0.85

, ppm Cin,max 1i

0

15

15

25

45

35

, ppm Cin,max 2i , ppm Cin,max 3i

0 0

20 25

35 0

45 45

35 55

20 25

, ppm Cout,max 1i

35

70

75

105

90

85

, ppm Cout,max 2i

45

115

95

85

105

110

, ppm Cout,max 3i

55

110

85

100

120

95

tbeg, h

5

10

0

15

10

0

tend, h

40

70

15

80

45

60

Unit i

a

Run without any internal water reuse and no supply from the ST; the fresh water consumption of this batch process is 1688.28 t for the entire 80 h schedule. b Inlet free of contaminant unit.

genes forms the chromosome—direct storage was chosen as the encoding procedure, to decrease the CPU time, on one hand, and to cope easier with the associated restrictions, on the other. GA starts by generating a population of chromosomes for which the mathematical model is solved and the objective function is computed. Then, four genetic operators are used to improve the population: selection (the parents producing better offspring are chosen randomly, but with a probability proportional to their fitness, among the best ranked individuals), crossover (recombine genes of selected pairs of individuals using one-point method), mutation (randomly change one gene in chromosomes with a very small probability; uniformity and premature convergence onto a local optimum are thus prevented) and cloning (duplicate the individuals from the previous to the next generations). One of the novelties of our network optimization algorithm is the seamless time adjustment of the fresh water flow Fi for each WU active in a given time-window when its inlet/outlet constraints are about to be broken, as a consequence of the increase in the ST contaminant concentrations. If this is the case, Fi is gradually increased to dilute the inlet stream until the concentrations in question match the WU’s constraints. Consequently, Sout i has to be increased too. This safeguard mechanism is implemented in steps 3 and 4. If applicable, the most restrictive units (free of inlet contaminant) are not fed from the ST, but with fresh water only. Case Study. The proposed algorithm is applied to a synthetic test case with a batch process lasting 80 h and involving a total of six units and three contaminants. The data for the batch process are presented in Table 2. Then, various design strategies are tested, particular to either physical constraints, like the limited capacity of the ST, or to operating constraints, like a possible relaxed policy regarding the starting time for some WUs, which could vary between some limits. An opportunity time-window is defined as the time interval during which the WU could start without affecting the overall performance of the batch process, with respect to its commercial outputs, providing that it finishes before a critical moment.

6. RESULTS AND DISCUSSION The results of the optimization procedure will be presented and discussed assuming first that the ST has infinite capacity and

the batch process is scheduled according to Table 2, which will be denoted from now on as “original schedule”. To simplify the analysis, the possible losses are disregarded. Then maintaining the same relaxed conditions on ST, we will present the implications upon the topology and operating conditions of allowing two of the WUs to have opportunity time-windows. WUs 1 and 6 are permitted to start later than in the original schedule, after 5 h for WU1 and 10 h for WU6. Then, the same runs are repeated imposing a finite volume for the ST (700 m3). When reaching this limit, a flow of 100 m3/h is withdrawn from the ST and directed to the treatment facility, until a lower limit (200 m3) is attained. The values we used are the result of some previous runs. It has to be mentioned that these limits could be subject of optimization themselves, since the investments are proportional to the volume, the operating costs are proportional to the withdrawal flow, while the water reuse will depend upon the concentration of the contaminants accumulating in the ST; the lower the volume, the smaller the capacity of the ST to act like a buffer for the high concentration streams. 6.1. Original Schedule and Unlimited Capacity of the Storage Tank. According to the principle of driving force equipartition, the units are ordered as in Figure 5. In conformity to the schedule given by the events (start and end of each WU, in process hours), there are eight time intervals delimited as follows: I (0 ÷ 5), II (5 ÷ 10), III (10 ÷ 15), IV (15 ÷ 40), V (40 ÷ 45), VI (45 ÷ 60), VII (60 ÷ 70), and VIII (70 ÷ 80). WUs 3 and 6 are the first to start, and the WU 4 is the heaviest pollutant. Every WU can send water only to the WUs sharing the same time window and situated above it (see Figure 5 for details; note that WUs 1 and 3 shall have inlet free of contaminants). When there are no restrictions related to the storage capacity of the tank, the volume/mass of the contaminated water increases faster or slower, depending upon the reuse rate of this alternative supply source, as presented in Figure 6 for the original schedule (see Figure 5a for the corresponding ranking of the WUs with respect to their maximum output concentration constraints). The time profile of the mass of contaminated water is obtained solving the mathematical model given by the DAE system. The mass in the ST never decreases, due to the imbalance between inlet and outlet. Less water is sent as supply to the WUs, compared to the flow received from them, since there are always positive makeup fresh water flows designated to dilute the water reused. Figure 6 could be divided into four main time periods, according to the behavior of the mass of contaminated water from the ST: during the first (0 ÷ 10), the level in the tank rises slowly, through the second (10 ÷ 40) the mass increase has a greater slope, in the third (40 ÷ 60) the slope diminishes, and ultimately during the fourth (60 ÷ 80) the slope becomes even smaller. In the first two time intervals covered by the first period, the number of WUs increases (two for the first time interval, three for the second), so more fresh water is needed (see Figure 7a for details regarding the fresh water distribution in time and between WUs). Nevertheless, the increase in the ST mass is not so sharp, because of the small mass loads of the WUs scheduled for these periods. The outlet unit flow profiles dependency upon time mimics that of fresh water (see Figure 7c) since the assumption used when solving the mathematical model and optimizing the internal flows for each time interval was that the water supply flows from the tank, Sin i , are kept constant, while the fresh water consumption could be adjusted according to steps 3 and 4 of the aforementioned solving algorithm. Although there are only two and three, 3642



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Figure 5. The diagram of discontinuous units ordered by both schedule and the maximum allowable outlet concentrations; note the time intervals created by events—the internal currents flow from lower values of maximum exit concentrations to higher values, except for the WUs 1 and 3 which are inlet free of contaminants (dash-dot lines): (a) original schedule, (b) postponed WU1 schedule, (c) postponed WU6 schedule, (d) postponed WU1 and W6 schedule.

Figure 6. Time mass profile of the storage tank, for the batch process. Original schedule: mSin = 191.47 t, mSout = 1561.38 t, m = 1369.74 t. Postponed WU1 schedule: mSin = 286.91 t, mSout = 1681.74 t, m = 1394.66 t. Postponed WU6 schedule: mSin = 337.16 t, mSout = 1742.34 t, m = 1404.46 t. Postponed both WU1 and W6 schedule: mSin = 204.95 t, mSout = 1584.78 t, m = 1379.46 t.

respectively, WUs active in this first period (0 ÷ 10), there is substantial internal water reuse (Figure 8a,b). Moreover, in the second time interval (5 ÷ 10) of this period the ST acts as both

sink and source, contributing to diminishing the fresh water consumption of WU6, which also receives partially contaminated water from WUs 1 and 3 (Figure 8b). It should be noted that here 3643



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Figure 7. Time profiles for the water flows during the batch process for the original schedule: (a) fresh water flow profile to each unit inlet. (b) supply water flow profile from storage tank to each unit inlet; (c) wastewater flow profile to the storage tank from each unit outlet.

there was an important decrease in fresh water flow, more than 55% as compared with the same unit in Figure 8a. During the fourth time interval (15 ÷ 40), the inlet concentration for WUs 5 and 6 would exceed the maximum allowable concentration for contaminants 2 and 3, because the supply water from the tank, although keeping the same flow, has an ever increasing contaminant concentration due to the continuous accumulation of pollutants in the ST (see Figure 9, original schedule). To balance this and keep the water flowing into WUs 5 and 6 within the working limits, more fresh water feed is added quasi-continuously (see also Figure 7a, the increase of the fresh water flow when needed during this time interval). Therefore, the water mass increase in the ST is more rapid than for the previous period of time. The consequence of this policy of slowly increasing the fresh water flow when needed is that the ST concentration of contaminants 2 and 3, although increasing, remains at such levels which ensure its reuse in the feed for WUs 5 and 6. During this second period (10 ÷ 40) the network topology changes significantly, since all but one WU (the fourth in the time interval 10 ÷ 15, the third in the time interval 15 ÷ 40, respectively) become active, as depicted in Figure 8 panels c and d. Both networks are complex, with many internal water flows cascaded from the less pollutant to the most pollutant WUs (see also Figure 5 for the ranking according to the maximum allowable output concentration). Some of the internal pipes are active in both topologies (WU: 1f5, 1f5, and 6f5) but with the flows modified according to the deactivation/ activation of WUs 3 and 4, respectively. Once the next period (40 ÷ 60) begins, and the subnetwork changes (Figure 8e,f), the fresh feed changes too to new flows (Figure 7a), which remain quasi-constant along the subsequent

intervals, according to each unit’s constraints, so the mass in ST still rises, but slower. The topology changes since WUs 1 and 5 cease working during the two consecutive time intervals, 40 ÷ 45 and 45 ÷ 60. Although the internal flow is still present (see the continuity of links WU: 4f6 and 6f2), the use of the ST as supply increases significantly, lowering the fresh water consumption. This is the characteristic of the last period (60 ÷ 80), too, when the supply from the ST rises from 20% of the fresh water to more than 40% when only WU4 remains active (Figure 8g,h). The pipe 4f2 remains active until WU2 ends its task. With the analysis of the dynamics of the network topology, as resulting from Figure 8a-h, the benefit of ranking the overlapping units according to their maximum allowable exit concentrations should be emphasized. This ranking ensures stable links between the same units all along their working period, thus lowering the investments in pipes and fittings. Another characteristic of the ST is given by the time profile of the pollutants concentration (Figure 9). Tracking them is important because their actual values are responsible for the flows the network uses to minimize the fresh-water consumption, particular to each time interval. For the present process schedule, the contaminants concentration has a different behavior. At the beginning, they have a constant value corresponding to the first time interval, when the ST only receives water from WU 3 and 6. The concentration of contaminant 1 is constant over the first 5 h, while decreasing over the next 10 h. This decrease is explained by the larger flow of fresh feed introduced in this latter period as the number of units per interval is higher. After that, the interplay between the imbalance of the inlet and outlet flows of the ST, the reuse of water from one unit to another, and the supply from the ST result in an almost 3644



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Figure 8. The dynamics of the optimized network topology with respect to the time intervals I (a), II (b), III (c), IV (d), V (e), VI (f), VII (g), and VIII (h) of the original schedule; the arrows represent the arches of the graph with the corresponding flows, as resulted from optimization.

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Figure 9. Concentration profiles in the storage tank for the three contaminants during the entire working time.

constant concentration of this contaminant for the rest of the batch working time (see Figures 7a and 9). The concentrations of the second and third contaminants behave similarly during the period 15 ÷ 50; after a small decrease, due to the fresh water flow used for the WU1, they increase following a saturation trend, characteristic to the mass transfer processes, which are first order in nature. This is the effect of the decrease of fresh water consumption due to both internal reuse and supply from the ST, with respect to the value it should have lacking any feed with partially polluted water. One can notice in Figure 7a how Fi is continuously increased for WUs 4, 5, and 6 (more obvious for the latter) along the aforementioned period, in order to adjust the concentrations of the inlet stream to the inlet/outlet restrictions, about to be violated due to the increasing concentration of the second and third contaminants in the ST. This slow adaptation of the inlet fresh water flow contributes to the asymptotic behavior of both concentrations. During the rest of the schedule, 50 ÷ 80, the fresh water is used at such flows as to maintain constant the concentration of the first and third contaminant, at the expense of decreasing the concentration of the second. As can be seen from Figure 9, there is a lot of internal water reuse, together with significant supplies from the ST. The overall dynamics of the network’s architecture, changing its configuration from time interval to time interval, according to the schedule, reveals some characteristics (Figure 8). It is worth mentioning that, unlike its continuous network counterpart, where the optimization heavily prunes the unnecessary links and all the remaining connections are active, a discontinuous network has always two states for a physical pipe connecting two WUs: active or inactive. The switch between these states happens when the two WUs share the same time window and the water coming from one of the units could be reused, with or without makeup fresh water, in the other. Each WU receives fresh feed, additionally to the internal flows of contaminated water (coming from less pollutant units or from the ST); the use of the latter is prone to minimize the objective function which is the overall fresh water consumption. But, pending of the state of the WU, the physical pipe could be active or not (see Figure 8, WU1, for example). Another example is represented by the physical link between WU1 and ST, active for

Table 3. Operating Time Intervals for WUs 1 and 6, Possessing or Not an Opportunity Time-Window water using units 1 case original schedule postponed starting/ending time

tbeg, h

6 tend, h

tbeg, h

tend, h

5

40

0

60

10

45

10

70

a period of only 5 h, when there were not many WUs to receive the surplus of used water from WU1. After this period, as WUs 2 and 5 become active, they can reuse water from WU1 and the pipe linking the latter with the ST switches to inactive. The highest flow rate of fresh feed is used for the unit with the lowest minimum outlet concentration, since they cannot reuse much water; but they can send water to the following units for reprocess (in our case, Figure 5a, the ones situated above it). It can be observed that the trend is to keep the flow at a constant value along the intervals. There are also some exceptions, like WUs 5 and 6 within the third and fourth time intervals (10 ÷ 40). A continuous increase can be noticed, due to an inlet concentration that tends to exceed the maximum allowable one (see Figure 7a). 6.2. Opportunity Time-Window Effect upon Water Network with Unlimited Capacity of the Storage Tank. From the point of view of the commercialized products of the batch process at hand, WUs 1 and 6 have a less restricted relative starting time; they have opportunity time-windows during which they can start, without affecting the quality or the quantity of the end products (see Table 3 for details). From the perspective of fresh water consumption though, there could be some major implications; if not concerning the overall quantity which has to be used, at least regarding the dynamics of the WN and fresh water flow distribution. The opportunity time-window effect will be accounted for considering three possibilities: WUs 1 and 6 could be postponed independently or simultaneously. When the events defining a WU change, the number of time intervals changes too (see Figure 5 panels b, c, and d for these new periods associated with WU1, WU6, or both starting later). 3646



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Figure 10. a: Time profiles for the fresh water inlet flows for the opportunity time-window simulations: (a) postponed WU1 schedule, (b) postponed WU6 schedule, (c) postponed WU1 and WU6 schedule.

In this respect the flows of fresh feed and the reused water coming from ST on each interval will differ from the original schedule, as they depend on the number of WUs operating simultaneously. The concentration profiles for the ST will change as well, determining variations in Fi and, as a result of mass balance, in Sout i . The time mass profiles are significantly different, compared to the original schedule, only when WU6 has a postponed start (see Figure 6, WU1 postponed, WU6 postponed, and WU1 and WU6 postponed). Apart from a smaller delay as compared to WU6 (5 h against 10 h), WU1 inlet shall be free of contaminant, thus its impact upon the water reuse strategy is lower—the corresponding profile (Figure 6, WU1 postponed) stays closest to the original schedule profile (see the charts of flows repartition in Figures 7a and 9a). But, since WU1 accepts fresh water only as feed, its later start establishes slightly lower concentrations for the contaminants in the ST (Figure 9). Postponing the start of WU6 influences more the time profile of the mass accumulated in the ST, which is at lower values than in the original schedule case, until 65 h. Quite interesting, the mass profile increase does not change its slope when WU6 starts working (10 h), although its need of fresh water is quite important (Figure 10b), being adjusted continuously, until 40 h, to avoid breaching inlet/outlet restrictions. This happens because there is an important supply water from the ST to WU6 and even more to WU2 (data not shown), possible due to the low contaminant concentrations (Figure 9)—WUs 1 and 3 work alone during the first 10 h (Figure 7c), both being WUs with inlet free of contaminants. During the last half of its working period, the supply water from the ST used by WU6 increased from 2.22 t/h to 3.1 t/h (data not shown, but see, for comparison, Figure 7b) slightly increasing too

its fresh water consumption (Figure 10b). Ultimately, this led to the highest value of the final mass in the ST (Figure 6) but the smallest contaminant concentrations (Figure 9). The most dramatic change, with respect to the ST mass profile, at least for the first 70 h, is when both WU1 and WU6 start later, at the same time as WUs 2 and 5 (Figure 7d). This is because there is a complementary effect, at least regarding the contaminant concentration profiles and the supply water from the ST. Contrary to the previous case, the concentrations of the first two contaminants are not dropping when WUs 1 and 6 start, meaning the fresh water consumption is kept at lower values. Then, due to the better internal reuse, which keep the fresh water flows at lower values (Figure 10c), the concentrations start increasing until they become closest to the original schedule case (Figure 9). Still, moving the heaviest pollutant WU toward the end (WU6) determined a slightly higher mass in the ST, eventually (Figure 6). With respect to the active links between WUs throughout the schedule, there are no fundamental changes, meaning that postponing WUs 1 and/or 6 does not affect the investments. The flows are changing—fresh water, supply from the ST, and internal reuse—but not spectacularly (see the legend of Figure 6 and Figure 10). The highest difference is around 34 t (original schedule vs postponed WU6 schedule), which represents less than 2.5% of the pumped water. Although modifying the start of WUs 1 and 6 does influence the dynamics of WN with respect to contaminants concentrations and ST mass profile, in the end the variations are within the limits of losses (an upper limit of 5% is considered, in many situations, acceptable). 6.3. The Effect of the Forbidden Unit Concept upon the Water Network Topology. When the forbidden unit concept is applied, the wastewater exiting any final WU and having at least 3647



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Figure 11. Time profiles when the batch process (original schedule) is operated according to the forbidden unit concept: (a) mass of the storage tank (m = 1193.32 t); (b) postponed WU1 schedule; (c) supply water flow distribution (mSin = 256.85 t); (d) WUs exit flow distribution (mSout = 1625.72 t).

Figure 12. Time profile for the mass when the storage tank has finite volume (Vmax = 700 m3, Vmin = 200 m3, and FV = 100 m3/h). Original Schedule: mSin = 144.37 t, mSout = 1517.97 t. Postponed WU1 schedule: mSin = 235.9 t, mSout = 1632.97 t. Postponed WU6 schedule: mSin = 247.83 t, mSout = 1655.68 t. Postponed both WU1& W6 schedule: mSin = 213.5 t/h, mSout = 15920.18 t.

one of the contaminant concentrations at the constraint level of the unit will be redirected toward treatment, instead of the ST, thus maintaining the latter concentrations at a lower level. As expected, these changes in flow distribution and contaminant level modify the dynamics of the ST, thus increasing the supply

water distribution which curbs the fresh water consumption (Figure 11). When several WUs bypass the ST, since they are operated as forbidden units during some time intervals of the original schedule, the final mass content of the ST is smaller (Figure 11) while the concentration of the contaminants are a 3648



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Figure 13. Time profiles for the fresh water inlet flows for the opportunity time-window simulations when the storage tank has finite volume: (a) original schedule; (b) WU6 postponed; (c) postponed WU6 schedule; (d) postponed WU1 and WU6 schedule.

Table 4. Comparative Results for Fresh Water Consumption for the Cases Investigateda case investigated

ST with infinite capacity

ST with finite capacity (min 200 m3, max 700 m3)

mass of fresh water used, t

savings, t

fresh water only, no internal reuse, no ST

1866.28

forbidden unit original schedule original schedule

1359.87 1369.74

506.41 496.54

postponed WU1 schedule

1394.66

471.62


1404.46

461.82

postponed WU1 and WU6 schedule

1379.46

486.82

original schedule

1373.6

492.68


1397.1

469.18


1407.9

458.38

postponed WU1 and WU6 schedule

1378.7

487.58

a

Original schedule with forbidden unit concept applied shows maximum savings; limited storage capacity with both WUs postponed (1 and 6) ensures higher water savings than infinite capacity; the same schedule is applied.

little bit lower (data not shown) compared to Figure 9. Analyzing the fresh water flow distribution from Figure 11b against the one from Figure 7a, one can observe that the most affected are WUs 2, 4, and 6, due to the significant change in the supply profile with wastewater from the ST (Figure 7b and 11c). 6.4. The Effect of the Storage Tank’S Limited Capacity upon the Water Network Topology. When the ST is not infinite, there are three characteristics which have to be decided: its capacity, meaning how much wastewater could be stored prior to start the withdrawal pump which feeds the treatment section, its inferior limit, meaning how much wastewater should remain in the ST when the withdrawal pump stops working and the actual withdrawal flow which is directed toward treatment. Although these characteristics could be a matter of optimization

on their own, for the present study they were set, after some preliminary runs, to 700 tons for the capacity, 200 tons for the inferior limit and, respectively 100 tons/h for the withdrawal flow. Figure 12 shows the mass time profile in ST for the time schedule corresponding to the original schedule. Although the ST was filled in the shortest time for the original schedule, the second withdrawal started the last, due to the decrease in the fresh water consumption for the last part of the batch process (see, also, the discussions in section 6.1). This means that, on average, the ST has more water in it during the batch process than in the opportunity time-window cases, thus keeping the contaminant concentrations lower, favoring the use of ST as secondary supply. This is the reason why in the case of postponed 3649


Industrial & Engineering Chemistry Research WU1 and WU6 schedules the mass of the supply water from ST increased from 204.95 to 213.5 t, leading to an actual decrease of the fresh water used (see Figures 6 and 12). The repercussions of the cyclic withdrawal of wastewater from the storage tank are against the concentration time profiles of the pollutants which should affect the use of ST as secondary water supply. Because of the water level decrease in the ST, the contaminant species concentration will be affected more, depending upon the timewindow and the associated fresh water consumption (Figure 13), since the loads of WUs remain the same. The change in profiles consists in a rise in concentration for contaminants 2 and 3 until a maximum is attained, higher than for the infinite ST, while the concentration of the contaminant 1 reaches a minimum value around 45 h, then starts increasing. Also, the same decrease in the tank’s mass slows down the decline of concentrations for contaminants 2 and 3 until the end of the batch process (data not shown). No topology changes are observed against the original schedule irrespective of the opportunity time-window, only adjustments in the fresh water flows, affecting also the WUs outlet flows toward ST (not shown).

7. CONCLUSIONS The ever-tightening environmental regulations and the scarcity of water require developing proper water management methodologies with special emphasis on industrial water conservation. Although, in the literature, the related publications in this area are almost all concerned with the continuous processes, more and more intense studies have been made on discontinuous processes. The new paradigm introduced in this paper permits the dynamic optimization of a batch process formed of several WUs scheduled according to some technological prescriptions. For each time interval, as designated by two successive events, the WN is considered as pseudostationary. This means that the structure remains the same but, due to the ever changing compositions of the water in the tank, the fresh feed flow is established such that the inlet and outlet restrictions are to be observed, and accordingly the wastewater flow directed to the tank is to be consistent to the mass balance. GAs are used as optimization procedures and several runs were performed with different starting points in order to obtain the best solution, although care should be take in designating this as the optimal solution. For complex NLP problems and when using evolutionary algorithms, as is the present case, finding the true optimum solution is not guaranteed. Also, one important limitation of the present method is that it is applicable only for semibatch, fixed load processes. The new paradigm was tested against a case study having six discontinuous WUs, three contaminants, and one ST. It was proven that with the use of the latter together with the two ranking criteria (time and maximum outlet concentration) and the concept of internal reuse, the fresh water consumption can be reduced, as compared to the case when only fresh feed is used, as illustrated in Table 4. It is essential to observe the influence of the opportunity timewindow. Although, the topology is not affected, the choice regarding whether to postpone or not a WU influences water savings (Table 4). Therefore, depending on the objective, it represents an important aspect to be considered when establishing the process schedule.

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The forbidden unit concept should be of great help in curbing the fresh water consumption, although this could mean an increase in the investment costs, due to the supplemental pipes needed to connect the WUs that could be operated as forbidden units directly with the main collector of the treatment facility. Furthermore, the ST capacity, the limit of discharge and the withdrawal flow, presently chosen after some preliminary runs, could alter significantly the fresh water consumption. However, as mentioned, these parameters could be a matter of optimization on their own. Another issue to be further investigated is the extent at which the water left within the pipes, after the batch WU had finished its operating regime, influences the freshwater consumption when the batch process starts again—the length of the pipes introduces a time delay in the DAE system. Although the wastewater quantities remaining may not be significant, the concentrations may have an important contribution considering the inlet and outlet restrictions. This aspect will make the subject of another paper. This paper offers an engineering approach on the optimization of WN, considering fresh water minimization only. Additionally, a cost analysis should be made, where costs for pipes, storage facility, pumps, etc. are to be taken into account. After reviewing the economical aspects, a choice can be made between lower production costs and more fresh water savings, taking into consideration that environmental regulations are increasingly firm. Consequently, there should always be a balance between sustainability and profits.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]; [email protected].

’ ACKNOWLEDGMENT Prof. Vasile Lavric gratefully acknowledges the financial support of UEFISCSU Project No. 663/19.01.2009. ’ NOTATIONS Ckj, Cki = concentration of contaminant k exiting unit j, respectively i, ppm CFk = concentration of contaminant k in feed, ppm Csk = concentration of contaminant k in ST, ppm = maximum allowable inlet concentration for unit i, ppm Cin,max i Cout ki = outlet concentration of contaminant k for unit i, ppm Cmax out = maximum allowable outlet concentration, ppm = maximum allowable inlet concentration of contaminant Cin,max ki k for unit i, ppm = maximum allowable outlet concentration of contamiCout,max ki nant k for unit i, ppm Δmki = load of contaminant k for water-using unit i, kg/h Δmi = total flow of contaminant provided by each unit, kg/h Fi = necessary flow of fresh feed, t/h Fiq = fresh water flow into water-using unit i during the time interval τq, t/h Fin* ki = freshwater flow for contaminant k, necessary to reach its respective maximum allowable inlet concentration of unit i, t/h Fout* ki = freshwater flow for contaminant k, necessary to reach its respective maximum allowable outlet concentration of unit i, t/h 3650


Industrial & Engineering Chemistry Research Fin i = freshwater flow fulfilling every contaminant inlet restriction, t/h Fout i =freshwater flow fulfilling every contaminant outlet restriction, t/h Gw = global fresh water quantity, t i, j = indexes for batch WUOs K = number of contaminants k = index for contaminants L = losses of water, t/h m = mass of wastewater in the storage tank, t N = number of batch WUOs Q = number of time intervals q = index for time intervals F = water density kg/m3 Sin i = flow coming from the ST and supplying unit i, t/h mSin = total mass of the wastewater used as supply from the ST, t = flow directed from unit i to ST, t/h Sout i mSout = total mass of the wastewater which fed the ST, t t = time, h tbeg = beginning time of operation/process, h tend = ending time of operation/process, h τn, τq = time intervals, h V = volume of storage tank, m3 Wi = stream of unit i directed to treatment network, t/h Wsi = stream of the ST directed to treatment network, t/h Xji, Xij = internal flow from unit j to unit i, respectively from unit i to unit j, t/h Abbreviations

CIA = concentration interval analysis CPU = central processing unit DAE = differential algebraic equations GAs = genetic algorithms MILP = mixed integer linear problem MINLP = mixed-integer nonlinear problems NLP = nonconvex nonlinear problems ST = storage tank WN = water network WU = water-using unit WUO = water-using operation

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