Pareto Approach in Designing Optimal Semicontinuous Water Networks

Article pubs.acs.org/IECR

Pareto Approach in Designing Optimal Semicontinuous Water Networks Elena-Lăcrămioara Dogaru and Vasile Lavric* University Politehnica of Bucharest, Chemical Engineering Department RO-011061, Polizu 1-7, Bucharest ABSTRACT: Pareto optimization approach was used in searching for the particular successions of topologies and operating conditions for a semicontinuous water network (batch with respect to the raw materials) that minimize both the freshwater consumption and the investment and operating costs. The semicontinuous wastewater network (SWN) formed of N water-using units (WUs), each handling at most K contaminants, and one storage tank (ST) is optimized on time intervals by increasing the wastewater reuse opportunities. A single freshwater source is available and later a regeneration unit (RU) is added to the network as a fine-tuning tool in the Pareto optimization, and further its influence is discussed. The WUs are ordered simultaneously by schedule and by maximum allowable outlet concentrations of contaminants. Thus, the rules concerning the internal reuse between the overlapping WUs correspond to an oriented graph: no recycle is allowed against the ordered sequence. The differential-algebraic equations model consisting in a set of differential equations for the ST and a nonlinear system mass balance equations with restrictions for the overlapping WUs is optimized using the Matlab built in functions for both the genetic algorithms (GA) and the RK-type integrator. The topology of the SWN changes for each time interval, its optimal structure depending upon the minimum freshwater consumption and the costs. A complex synthetic case study was subjected to optimization, and several scenarios are analyzed.

1. INTRODUCTION Many efforts have been directed to the development of new and effective methods for reducing resource utilization (water, energy). Omnipresent in industry, water is consumed in large quantities yearly in continuous or batch processes, either as raw material (reactant), utility such as a transfer agent (cooling, heating), carrier (extraction, washing), etc. Consequently, large amounts of wastewater, which have been subjected to treatment so that the environmental regulations are met, are discharged into the environment. When used as raw material, water consumption can be curbed by modifying the technological process. To mitigate the level of the water consumed as utility, three main methods are envisaged: reuse, recycle, and regeneration.1 The large volumes of wastewater produced by continuous processes represent one of the reasons why these processes were the first subjected to optimization. The methods used for water minimization can be categorized into insight-based (mainly pinch analysis and derived techniques) and mathematical modeling approaches. There are three reviews concerning the published studies presenting the methodologies used on wastewater minimization in continuous processes.1−3 In the past decade, the studies on batch water networks (BWNs) were intensified, one reason being the increasing pressure of the market for small volume products with relatively high value (pharmaceuticals, flavors, etc.).4−9 An excellent review regarding the state of the art of water networks, including here the BWNs, is due to Jeszowski.3 The methodologies on wastewater minimization used for continuous processes were extended and adjusted for BWNs.10,11 The most obvious difference between the continuous and batch processes is the discreteness of tasks for the latter. While for continuous processes water can be reused/recycled as long as © 2012 American Chemical Society

the inlet concentration of contaminants in water is below the maximum allowable limit of the recipient WU, in batch processes an additional constraint is involved: time. Therefore, even though the concentration constraints are observed, water cannot be reused in another WU of the network unless either they operate simultaneously, or this other unit starts operating immediately after the former one has ceased operating. The time constraint was partially overcome by using STs, which are reservoirs collecting the used water, which might be reused when necessary. From the operating conditions point of view, the BWNs can be classified as a true (completely) batch12 (TBWNs) and as semicontinuous (SWNs). In the TBWNs, the inlet and outlet flows happen at the beginning and the end of the technological processing time. However, in the SWNs, waterseen as utilityflows throughout the WUs continuously, although the raw material is processed in batches. Wang and Smith10 were the first to extend an insight based technique from continuous to batch water networks. Their study addressed SWNs, the approach being suitable for fixed load problems, where the inlet and outlet flow rates are assumed to be equal. Concentration intervals are built based on a predefined schedule so that the water is cascaded from the lowest to the highest concentration interval. The excess water is stored for reuse from one time interval to another. When reuse is not possible, freshwater is used. After freshwater and storage requirements are determined, the network is built. Among the disadvantages, the procedure is not fitted for repeated batches Received: Revised: Accepted: Published: 6116

October 28, 2011 February 26, 2012 April 5, 2012 April 5, 2012 dx.doi.org/10.1021/ie2024728 | Ind. Eng. Chem. Res. 2012, 51, 6116−6136

Industrial & Engineering Chemistry Research

Article

technique, time-dependent water cascade analysis, in order to further reduce the fresh and wastewater flows. Tokos and Pintarič17 extended the work of Kim and Smith23 by adding regeneration, with the possibility to operate the network either as true batch or semicontinuously. The semicontinuous regenerated water stream can be used once it is needed, unlike the batch regenerated water, which can be reused only after the regeneration is complete. Liu et al.20 proposed a mathematical-based approach for designing an optimal BWN with one central RU, during multiple repeated batch cycles. To ensure the continuous operation of the RU, two STs are included in the network: one receiving contaminated water from the batch units and sending it to other batch units for reuse and/or to regeneration, and the other one collecting the regenerated water and sending it to reuse. Although the methods of choice for optimization are of a deterministic nature,3 the use of evolutionary algorithms in BWN is steadily increasing.24−26 Still, their usage is far less frequent than in the case of continuous water networks.3 Among the stochastic methods, genetic algorithms are the most popular for water network design.27−30 Tsai and Chang31 were the first to use them to optimize a WN. Another method of choice for optimizing water networks is the adaptive random search, due to its simpler application.24,32 Zhou et al.22 were the first to use a stochastic based genetic algorithm to overcome the inherent nonlinearity of the batch water network problem. Dogaru and Lavric27 used the ranking of the overlapping WUs during a time interval as a method for reducing the freshwater requirements of a SWN with a defined schedule; the wastewater cascades from WUs with lower maximum allowable outlet concentrations to WUs with higher outlet restrictions. The method applies to a SWN with fixed loads and defined schedule, and can be easily extended to processes with water losses and/or gains. The optimized topology is dynamic in nature, changing from one time interval to another, as the overlapping WUs change as well. Moreover, because this method uses a two-level optimization strategy, it permits the simultaneous optimization of the dynamic network topology and the schedule.28 The present study extends the work of Dogaru and Lavric.27,28 The purpose of optimization is to find the best successive topologies, together with the appropriate internal flows for the reused wastewater, which ensure two minima simultaneously: freshwater consumption and investments and operating costs. The optimization is performed using the Pareto approach with a vector-objective function: freshwater consumption and total investment and operating costs of the network pipes, designed for optimum diameters. The method was tested on a case study composed of six WUs, three contaminants, one ST, and one optional RU. Several scenarios are discussed: to use or not the RU and working with a ST of limited/unlimited capacity.

and also it cannot be applied to processes with water losses and/or gains because of the assumed inlet and outlet flow rates equality. From the same category of insight-based approaches, several algebraic techniques were proposed.13−15 Based on cascading water from lower to higher concentration intervals, these methods (water cascade analysis,13 time dependent concentration interval analysis14) apply also to fixed flow problems (when the inlet/outlet flow rates are specified), semicontinuous and true batch processes; they are suited for both single (with or without storage) and repeated batch processes.16 When it comes to complex problems (e.g., multiple contaminants) the insight-based techniques are no longer efficient, and the mathematical modeling approach proves to be more appropriate. Generally, these techniques involve a superstructure, a mathematical model, based upon the total and partial mass balances, which represent the constraints and one or more objective functions. Even when the BWN is optimized with respect to the freshwater consumption, some of the WUs are not optimally operated; neither their inlet concentration constraints, nor their outlet ones are attained by the water streams flowing through them, but because of the interconnectivity between the WUs, a further decrease in freshwater feed for these units is not possible, since the rest of the WUs work optimally. Local regeneration of the streams associated with these nonoptimal WUs could render them optimal. Therefore, another way to minimize the freshwater consumption is to introduce RUs at the water network level, followed by reuse/recycle of the regenerated water streams wherever is needed. Water regeneration implies the partial treatment of contaminated water streams to overcome a certain limit of contamination, in order to bring them to a level acceptable for further internal reuse. The regeneration alternative was analyzed for batch processes too, but the number of studies addressing it together with wastewater treatment is low.17−20 Gouws et al.16 present an outline of the water minimization techniques and future directions on this matter. Cheng and Chang19 used simultaneous water minimization with scheduling; batch schedules, water-reuse, and watertreatment subsystems were integrated in the same mathematical formulation. The method is based on a superstructure built considering a set of rules: reuse/recycle is possible only via buffer tanks and no direct connections are allowed between WUs. The constraints are represented by mass balances and network structure specificities.19,21 There are two main disadvantages: the models rapidly become intractable, due to the excessive number of unknowns, and the loss of accuracy with respect to scheduling when long time intervals are involved. To overcome these limitations, Zhou et al.22 modified the state-space superstructure to formulate a mixed integer nonlinear programming (MINLP) model for simultaneous optimization of batch process schedules and a single- or multicontaminant water allocation system. The main advantage is that the global optimality is reached almost always in the presented case studies. A hierarchical approach with multiple objectives for synthesizing batch wastewater networks was developed by Shoaib et al.18 Owing to the complexity of the problem, the minimization is done in three stages: freshwater, number of STs, and interconnections between WUs and STs. The placement of RUs in the BWN is assessed using a hybrid

2. PROBLEM STATEMENT Consider a discontinuous (DC) process with respect to the raw material transformation, for which the schedule is given. This means that the working time for the N belonging WUs is known, together with the time they start operating from the beginning of the DC process; each WU is continuous with respect to the water/wastewater flow throughput (seen here as utility), and could release in water up to K contaminants. The schedule is defined as the succession of time intervals delimited 6117

dx.doi.org/10.1021/ie2024728 | Ind. Eng. Chem. Res. 2012, 51, 6116−6136


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the active WUs, the ST, and the RU (when active), while the arches constitute the unidirectional flow connections.27 The graph has a dynamic structure, modifying its configuration whenever an event occurs. An event represents the starting or the ending moment of a WU’s operating period; two consecutive events define a time interval (Figure 2). Since

by the start and/or the end of at least one WU and represents the time line for processing the raw materials into products. For each time interval, the overlapping WUs form a SWN receiving water from the freshwater source and, when possible, from any other supplemental supply. When passing to the next time interval of the schedule, the topology of the new SWN changes, according to the different WUs working together and the associated operating conditions. One ST of unlimited capacity and one RU are also part of the network; the ST with limited capacity represents a particular case. The purpose of this study is to search for the dynamics of the water/wastewater network topology during the whole schedule, with the associated wastewater internal reuse (treating the ST as a simultaneous sink and source) and the appropriate level of local regeneration that would increase this reuse, the goal being simultaneous minimization of the freshwater quantity and the investment and operating costs. Since the only difference between topologies, at the whole batch scale, is given by the network of pipes, the costs associated to the WUs being constant, we safely assumed that the investments and operating costs are given by the pipes belonging to the network, which should have optimum diameters, and by the flows through these pipes.

Figure 2. The Gantt diagram with the WUs ranked according to their schedule (primary ranking) and outlet maximum allowed concentration (secondary ranking). The third time interval, as delimited by two successive events, is active (see the correspondence with Figure 1).

3. PHYSICAL MODEL The physical model largely presented in Dogaru and Lavric27 is maintained; one single RU is added to the network, receiving wastewater from ST and distributing regenerated water to the WUs when necessary (Figure 1). For the sake of completeness, the abstraction of the SWN corresponding to a time interval is shortly described. 3.1. Model Description. The SWN, such as depicted in Figure1, is abstracted as an oriented graph: its nodes represent

the topology of the SWN does not change during a time interval and the water/wastewater flows continuously throughout the overlapping WUs, the use of some techniques from continuous WN optimization27 is straightforward. The Gantt diagram of the WUs (Figure 2) shows the use of a second ranking criterion, besides time: maximum critical outlet concentration of contaminants27 (the overlapping WUs over a time-interval are ordered from the lowest to the highest maximum critical outlet concentration of contaminants). In this respect, wastewater might be reused from WUs with a lower outlet maximum allowed concentration, to other WUs having a higher value for this concentration, all of them sharing the same time interval. As a matter of fact, four ranking criteria were tried to order the overlapping WUs from their lowest to their highest value: maximum critical inlet concentration of contaminants (A), maximum critical outlet concentration of contaminants (B), maximum contaminants load (C), and maximum freshwater for each WU, but this latter criterion gave the same outcome as (C). The results obtained after performing the optimization using each of the four criteria are presented in Figure 3 and Figure 4. Although using criteria A resulted in lower total costs (Figure 4), the second criterion usage permitted a significant decrease in freshwater consumption (Figure 3) against the other two; since the primary concern should be freshwater consumption minimization (a strategic environmental objective), the maximum critical outlet concentration of contaminants was kept as the second ranking criterion. The ST operates as a sink and, possibly, as a source. It receives wastewater which cannot be directly reused in other units due to either its concentrations, or time constraints, or surplus wastewater. At the same time, the ST represents a supplemental source of contaminated water; it is worth mentioning that the wastewater concentration in the ST is ever-changing, at every moment, due to its dynamicseach time interval is characterized by different concentrations for the

Figure 1. Schematic representation of the flows around the generic WU i, together with its positioning in the cascaded SWN corresponding to a standard time interval. For the sake of simplicity, the time interval index is omitted from the notations. It should be emphasized that each time interval of the schedule has its own SWN formed by the overlapping (active) WUs; the full line WUs are active, the dashed line WUs are inactive. Within appropriate conditions, RU starts working, not necessarily at the beginning of the current time interval, replacing ST as supply (dotted arrows). When the forbidden unit concept27 is in use, the heaviest polluted stream coming from the last WU in the ranked network sends its exit wastewater flow to the treatment section, instead of the ST (the dashed arrow). 6118



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with cleaner water. Therefore, the freshwater consumption is diminished, but at the expense of increasing the operating costs. The RU, which could have limited or unlimited capacity, can be of two types: fixed output or fixed ratio. In the former case, the wastewater is partially decontaminated until the concentration of pollutants drops to some predefined level. In the latter case, the outlet concentrations of the RU stay in the same ratio with respect to the inlet concentrations. When the concentration in the ST reaches the inlet threshold values for all the contaminants, the RU starts operating (see Figure 1), distributing treated water for reuse in the active WUs. From this moment, the ST remains the ultimate sink for the WUs, but becomes a source only for the RU. If the concentrations of contaminants in the ST fall below the thresholds, the RU ceases operating. A disadvantage can arise, namely the discontinuous operation of the RU. However, judging from the results after multiple runs, the RU works continuously once started, as the regeneration changes the quality of the WUs inlet water and not their output concentrations, responsible for the accumulation of pollutants into the ST. There is another way of placing the RU in the network, namely each WU directly feeding the RU, but there are several drawbacks associated to this arrangement. First, the RU would be subjected to strong fluctuations of flow rates and concentrations from time-interval to time-interval. Second, the regenerated flow rate needed by a WU would be the same, irrespective if it comes from ST or as exit of a group of WUs, in both cases the wastewater being processed by the RU. This implies that the RU effort would be higher in the latter case (higher inlet concentrations from the WUs) than in the former, since the RU’s exit is the same. Third, there are issues related to costs and operability: linking the WUs directly to the RU would at least double the number of pipes as the ones connected to the ST cannot be excluded due to the time constraint for batch processes. More, if the exit flow from WUs is higher than the treated water flow needed for the next units, a supplemental regulatory system should be provided for each WU, ensuring the proper split of the WUs exit: a fraction for RU and the rest for ST, different for each time-interval. On the contrary, since the ST works as a buffer too, feeding the RU from the ST in order to dampen these fluctuations in both concentration and flow at the RU entrance seemed a better choice. The water sources and sinks for the generic WUi are presented in Table 1.

Figure 3. Freshwater consumption obtained after optimization of the WN of the case study (section 6) using as second ranking criteria the maximum critical inlet concentration of contaminants (A), maximum critical outlet concentration of contaminants (B), and contaminants load (C).

Figure 4. Total cost obtained after optimization of the WN of the case study (section 6) using as second ranking criteria the maximum critical inlet concentration of contaminants (A), maximum critical outlet concentration of contaminants (B), and contaminants load (C).

incoming wastewater, which mixes with the water already present in the ST. Supplementary, the water volume in the ST changes as well, due to the imbalance between the inlet and the withdrawal flows, the latter being for internal reuse in SWN. For the purpose of keeping the contaminants’ concentration at an acceptable level, thus diminishing the freshwater needed to possibly dilute the reused water from the ST, the concept of forbidden unit was introduced.27 This term designates the last active WU in the time interval network, for which wastewater reaches the outlet restriction for at least one of the contaminants. In this situation, the wastewater is sent directly to the treatment network (TN), bypassing the ST (see Figure 1, the dashed arrow marked Wi, Ck,i). When a batch is finished, with respect to the raw materials processing, the ST is emptied, the accumulated wastewater being sent to the TN (see Figure 1, the dashed arrow marked WST, Ck,ST). The RU is placed after the ST, its task being to partially remove the pollutants from the flow withdrawn from the ST (see Figure 1). The RU starts working when the contaminants concentrations overcome some threshold levels, characteristic to each of the K pollutants. Wastewater enters the RU if, and only if, all contaminants have concentrations greater than their corresponding thresholds. This restriction prevents the regeneration of slightly polluted wastewaters, which could be reused as such in the SWN, thus avoiding an increase in the operating costs. In this way the operating WUs are provided

Table 1. Water Sources and Sinks for a Generic WUi supply source freshwater WUja (j = 1, i − 1) ST RU

generic unit

sinks

→ WUi →

WUjb, j = i + 1, N ST/TUc

a

Predecessor WUs in the modified Gantt diagram. bSuccessor WUs in the modified Gantt diagram. cWhen forbidden unit concept applies.

3.2. Simplifying Assumptions. The simplifying assumptions cover the WUs, ST,27 and also the RU. • Freshwater is free of contaminants: CFk = 0, ∀k = 1, K. • When the ST has limited volume (1) an upper mass/ volume limit is imposed; when reached, water is withdrawn and sent to TN at a given flow rate and (2) 6119



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Partial mass balance

water is withdrawn from the ST until some previously set lower limit is reached. • The contaminant load for each WU, Δṁ k,i, k = 1, K; i = 1, N, is constant over the entire operating period of the WU. • The internal and supply flows (Xj,i, j = 1, i − 1; Xi,j, j = i + 1, N; Sini , i = 1, N) are assumed constant over each time interval. • The regeneration capacity of the working RU is unlimited, its outlet concentrations being guaranteed, irrespective of the inlet flow, as long as all of the corresponding concentrations are greater than their thresholds (for the present study, these thresholds correspond to the minimum outlet restrictions for each contaminant over the whole WUs); (1) Fixed outlet. The water exiting the RU has imposed concentrations, suitably chosen; for the present study, these outlet concentrations correspond to the minimum inlet restrictions for each contaminant over the whole WUs, except for the WU fed with freshwater. (2) Fixed ratio. The water exiting the RU has the concentrations diminished with a given percentage, suitably chosen, with respect to the inlet concentrations; for the present study, a value of 90% removal for each contaminant was arbitrarily chosen.

i−1 j=1

Fi +

j=1

j=i+1

j=i+1

where R,out ⎧ , if RU is active ⎪ Ck CkP = ⎨ ⎪ S otherwise ⎩ Ck ,

It should be emphasized that when the concept of forbidden unit applies and the implied conditions are met, Sout becomes Wi, i without changing either its flow rate or its composition, but only the destination, TN instead of ST The RU is seen as a bypass on the pipe going from ST to the distribution system, toward the active WUs. The flow rates Sini and Ri from Figure 1 are mutually exclusive; the WUi is fed with reused water from ST, directly, or after regeneration. This is why the generic flow Pi was introduced, after the junction between the exit of the RU and the pipe going from ST to the distribution system. The concentrations of this generic flow are either those corresponding to the ST, or the RU exit, when active. The acknowledgment of this mutual exclusiveness of Sini and Ri flow rates prevents the use of binary variables to distinguish which one of the connections should be used in the mathematical model and the solving algorithm. Each WU is constrained by the maximum inlet and outlet concentrations of contaminants. The eqs 3 and 4 that follow mean that the concentration of each pollutant should cope, simultaneously, with both inlet and outlet constraints. The optimum situation is to have both equalities satisfied in eqs 3 and 4 for at least one of the contaminants, as this would translate into the right amount of partially contaminated water spent and no excess of freshwater. (b) Meeting WUi inlet restrictions. i−1

Ckin, i

=

P ∑ j = 1 (Xj , iCk , j) + PC i k i−1

Fi + ∑ j = 1 Xj , i + Pi

≤ Ckin,max ,i (3)

(c) Meeting WUi outlet restrictions. i−1

Ckout ,i

=

P ∑ j = 1 (Xj , iCk , j) + PC i k + Δṁ k , i N*

∑ j = i + 1 Xi , j + Siout

≤ Ckout,max ,i (4)

The ST receives wastewater concomitantly from the overlapping WUs during a time-interval; thus, wastewater accumulates in the ST and, consequently, the contaminants composition changes during this time interval. The overall and partial dynamic mass balance equations for the ST capture these changes, considering the latter as a perfectly mixed vessel.27 Total mass balance over the ST

N*

∑ Xj ,i + Pi + Δṁ i = ∑

Xi , j + Siout)Ck , i (2)

4. MATHEMATICAL MODEL Water originating from several sources (freshwater supply, predecessor WU(s), ST/RU) feeds the generic WU i , incorporates contaminants, and then it is directed to the following sinks (successor WU(s), ST/TN). The stream of wastewater entering the ST changes the contamination level of the stored water. Part of the ST content is outputted to be reused in the WN, diluted with freshwater, where necessary. When the concentration of the stored water reaches the thresholds for the RU to start, the exit stream of the ST is sent to regeneration. Then, the regenerated water is distributed to the WUs for reuse. The mathematical model, consisting of total and partial mass balances for each WU and ST, is the same as in Dogaru and Lavric;27 supplementary, the mathematical model is completed with the overall and partial mass balance equations for the RU. (a) Mass balances over the generic WUi with 1≤ i ≤ N*, and a typical time interval (for brevity, the index of the time interval is disregarded): Overall mass balance i−1

N*

P ∑ (Xj ,iCk ,j) + PC i k + Δṁ k , i = ( ∑

Xi , j + Siout (1)

where ⎧ R if RU is active ⎪ i, Pi = ⎨ ⎪ in ⎩ Si , otherwise

dm = dt

Although there is this common practice of neglecting Δṁ i = ∑kK= 1Δṁ k,i contribution to the total mass balance, it is better to include it and not only from a mathematical point of view. When dealing with many units operating over large timeintervals, the accumulation of the overall mass of contaminants could have significant values, thus affecting the ST mass balance and influencing the contaminated water reuse from the ST.

N*

∑ (Siout − Siin)

(5)

i=1

Partial mass balance over the ST d (mCkS) = dt 6120

N*

N*

∑ (SioutCk ,i) −

∑ (SiinCkS)

i=1

i=1

(6)



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turbulent

Differentiating eq 6 and using eq 5 into the result, we obtain the equations of time change of the concentrations of the contaminants:

⎡ 6.04 × 10−4Dnq 2.84ρ0.84μ0.16K (1 + J )H ⎤1/(4.84 + n) r ij y c ⎥ Dij(u) = ⎢ ⎢ ⎥ n + u XEK (1 ) F ⎦ ⎣

N*

∑ Siout(Ck , i−CkS) dCkS = i=1 dt m

(7)

laminar ⎡ 0.1628Dnq 2μ K (1 + J )H ⎤1/(4 + n) r ij c y ⎥ Dij(u) = ⎢ ⎢ ⎥ n (1 u ) XEK + F ⎣ ⎦

Equation 7 clearly shows that even if Ck,i would be constant during a time interval, the accumulation of mass m in the ST renders its contaminants’ concentration time-dependent. (d) Dual-objective function. In the present study, the Pareto approach using a dual-objective function is used in order to reach the sequence of the optimum topologies. The first objective is the same as in Dogaru and Lavric,27 namely freshwater consumption, Gw, eq 8. The minimization of freshwater consumption exploits the internal wastewater reuse opportunities, including here ST as well, regardless of the network complexity. N

Gw =

(10)

(11)

The dimensionless total cost of the network, Γ, is given by the topology of the network, which is represented by the grid of active pipes (nonzero throughput flows):

Q

∑ ∑ (Fiq·τq) i=1 q=1

(8)

where Fiq represents the fresh water flow feeding WUi during the time interval τq. The second component is the cost-based objective function, which takes into consideration the on-field geometry of the network, through the distances between WUs themselves, WUs and both freshwater source and ST. The investment and operating costs of the pipes’ network will be the only ones included in the second objective function. The computations address merely the active pipes, meaning the pipes throughout which there is a flow in the time interval under consideration, since the WUs’ associated costs remain unchanged irrespective of the water network topology and operating conditions. Minimizing this cost-based objective function we obtain the optimum topology of a specified SWN with respect to costs. Moreover, in order to consolidate the optimality of the solution, every pipe of the wastewater system should have an optimum economic diameter, computed such as to minimize the friction losses due to the fluid velocity.33 To keep the computations as simple as possible, still preserving the main characteristics of the problem at hand, no adjacent costs were included in the objective function, like effluent disposal charges based on volume and/or contaminant loading, or on-site treatment costs to upgrade the water quality of the effluent for reuse.29,34 Considering one year as the time basis, the unit length costs associated to a pipe having optimum diameter33 is

The denominator in eq 12 is computed using the largest diameter value as resulted from applying eq 9 with the highest freshwater inflow for all the pipes of the WWN, regardless if they are active or not (no active pipe means no water flowing through it). The result is a dimensionless total cost of the active pipes with values lower than unity even when all the pipes of the network are active, since many flows will be lower than the highest values.The dimensionless dual-objective function eq 13 is obtained considering the vector whose components are the freshwater consumption eq 8 and the total cost of the active pipes eq 12: ⎧Gw ⎫ fob = ⎨ ⎬ ⎩Γ⎭

5. SOLUTION ALGORITHM The Pareto optimization approach seeks the simultaneous minimization of this dual-objective function; the solution is not trivial since the complete model and the objective function are fully nonlinear. The optimization procedure is performed using the GA function gamultiobj which implements in Matlab (MathWorks, Inc., USA) the Pareto approach. The procedure follows the details given in Dogaru and Lavric27 though some modifications aimed to incorporate regeneration effects and the costs computation were done. The internal flows are used as genes, defining a chromosome. The restrictions are dealt with during the population generation when the individuals outside the feasible domain given by the restrictions, that is, eq 3 and eq 4, are eliminated. The number of individuals and the number of generations can be changed manually by means of a friendly user interface, the actual values used in this study being 200 for both parameters. The individuals are interbreeding according to their selection frequency, using one-point crossover method, and then mutation is applied to arbitrarily selected ones. For mutation we use the constraint-dependent default which chooses an adaptive feasible mutation rate. The fitness functions are evaluated separately for each member of the population. Moreover, an in-house user-friendly interface was built which offers an easy access for the following alternatives to

Cij*(u) = {[C*(u)]pumping + [C*(u)]pipe }ij ⎡ χq aρβμγK (1 + J )H ⎤ y ij ⎥ ′ =⎢ + B δ ⎢ ⎥ ( ) D u ij ⎣ ⎦ pumping ⎡ ⎛ Dij(u) ⎞ ⎤ ⎟⎟KF⎥ + ⎢(1 + u)X ⎜⎜ ⎢⎣ ⎝ Dr ⎠ ⎥⎦ pipe

(13)

(9)

The flow regime, as implied by the flow rate u, and the value of the Fanning friction factor determine the exponents of the pumping term in eq 9. Dij(u) stands for the optimum economic diameter of the pipe linking units i and j, while Dr is the reference diameter.29,33,34 The formula for computing Dij(u) depends upon the flow regime: 6121



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be tested: limited/unlimited ST, considering/disregarding the forbidden unit concept, applying/denying regeneration, choosing one or two objective functions. In the case of limited ST, the upper and lower limits shall be specified, together with the actual withdrawal flow rate. For the RU, the choice to be made is between fixed output and fixed ratio. The software proposes the inlet thresholds and the output concentrations, according to the aforesaid criteria, but the user has the possibility to change these values at will. For the cost objective function, the distances between the WUs should be provided. The intrinsic dynamic of the ST, as resulted from the integration of the DAE system formed by the eqs 1−7, affects the operating conditions of all the overlapping WUs which need wastewater reused from ST. Therefore, the solving procedure starts, at the beginning of each time-interval, with the search for the optimum topology of the overlapping WUs. Then, as the DAE system is integrated, the violation of the critical inlet/outlet concentrations, due to the possible increase of contaminants’ concentration in ST, is checked at every integration time step. If this is the case, the freshwater flow for the WU whose inlet/outlet critical limits are violated is slightly increased. The complete algorithm used to solve the DAE system, while adjusting the freshwater intake and the output directed to ST, can be found in Dogaru and Lavric.27

Figure 5. The diagram of the WUs ordered by both schedule and maximum allowable outlet concentrations; WUs 1 and 3 are inlet-free of contaminants, so they must be placed at the beginning of the graph (reused wastewater flows from WUs on the bottom to the WUs on the top of the diagram).

others, from WU4 to the ones above it in Figure 5, but no water can be reused from WU2 to WU4 or WU6. The optimization performed on each time interval (Table 3) using the dual-objective function eq 13 searches for the best topology which minimizes the freshwater and the costs.

6. RESULTS AND DISCUSSION The present methodology aims to search for the optimum topology that ensures maximization of contaminated water reuse to minimize the freshwater intake and at the same time to keep total costs at minimum. No losses are accounted for, to keep the model as simple as possible, but they can be easily incorporated. The present paper analyzes the optimization results obtained on a synthetic case study: six WUs dealing with three contaminants, having one ST of unlimited capacity. The data defining this semicontinuous network are presented in Table 2.

Table 3. Time Intervalsa on Which Optimization Is Performed time chronology of events associated interval no. with overlapping WUs (h) I II III IV V VI VII VIII IX X

Table 2. Operating and Restriction Data for the WUs together with their schedule WU1a

a

Δṁ 1i Δṁ 2i Δṁ 3i

0.35 0.25 0.35

Cin,max 1i Cin,max 2i Cin,max 3i Cout,max 1i Cout,max 2i Cout,max 3i

0 0 0 35 45 55

tbeg

5

δt

45

WU2

WU3a

Loads, kg/h 0.15 0.55 0.35 0.45 0.25 0.15 Restrictions, ppm 15 15 20 35 25 0 70 75 115 95 110 85 Beginning, h 10 0 Duration, h 70 25

WU4

WU5

WU6

0.45 0.15 0.45

0.25 0.65 0.65

0.65 0.55 0.85

25 45 45 105 85 100

45 35 55 90 105 120

35 20 25 85 110 95

35

20

0

65

55

70

a

0−5 5−10 10−20 20−25 25−35 35−50 50−70 70−75 75−80 80−100

active WUs WU3,WU6 WU1, WU3,WU6, WU1, WU3,WU6, WU2 WU1, WU3,WU6, WU2, WU5 WU1, WU6, WU2, WU5 WU1, WU4, WU6, WU2, WU5 WU4, WU6, WU2, WU5 WU4, WU2, WU5 WU4, WU2 WU4

The meaning of the time interval is described in section 3.1.

The objective functions belonging to eq 13 are dichotomic: less freshwater means more wastewater internal reuse, at the expense of more pipes and higher flows, both increasing the investment and operating costs. Because of this antagonistic nature of the dual-objective function components, a Pareto Front (PF) containing a set of equally optimal points is obtained for each time interval (see Appendix 4). Assuming the minimum number of points in the PF for each time interval, that is, 2, a number of 210 (1024) runs have to be done to cover the whole optimality spectra associated to this basic SWN, all of them being of equal optimality. As the number of points belonging to the PF increases, so does the number of runs, rendering the problem unsolvable in a decent amount of time. On the basis of the authors’ experience showing that the points from the PF differentiate themselves seldom through their topology and often through their operating conditions, a less time-consuming strategy was adopted. Once the PF is obtained for the current time interval, the points with values lower than 1 for the first objective function are selected. Afterward, the point

Inlet free of contaminants WU.

The WUs are ordered using the schedule and the outlet maximum allowable concentration as ranking criteria (Figure 5). To make things clearer: Water can be reused from WU1 to all the 6122



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situated in the middle of this region of the PF30 is chosen as starting point for the next time interval, and the optimization continues. In Appendix 4, the PFs are represented for each time interval. For the first time interval, all the points are displayed, and the middle point chosen according to the aforementioned strategy is highlighted. For the rest of the time intervals only the points with values lower than 1 for the first objective function are shown. 6.1. Unlimited ST. The base case scenario, as presented in Table 2, was optimized considering that the ST has unlimited capacity; this assertion simply means that, since the optimization is done for design purposes, the necessary volume of the ST will thus be found. If the volume is too large, implying high investments and environmental issues, an analysis will be done to establish what volume of the ST would be reasonable, and how limiting the capacity of the ST would affect the design of the base case (see the section ST with Limited Capacity).27 The result of the optimization represents a series of optimal SWNs formed by the overlapping WUs (Table 3), one for each time interval covering the schedule.27 The following figures present the results of the Pareto optimization for the whole batch, using summative quantities: The global freshwater consumption (Figure 6), the ST capacity (Figure 7), and the

Figure 7. Comparison of the ST capacity for single and doubleobjective optimization.

Figure 8. Wastewater reused from ST for single and double-objective optimization.

done considering the dual-objective function (FOB1&2), the freshwater consumption rises to 1813.65 tonnes, which means 6.22% more as compared to the FOB1 case (Figure 6). Still, the freshwater consumption, in this later case, gets reduced by 13.85% compared to the unoptimized network. The smaller increase in freshwater consumption, compared to the FOB1 case, is the result of the dichotomic nature of the two components of the objective functionless freshwater consumed means more wastewater reused from ST (see Figure 8, FOB1 and FOB1&2), which means more pressure losses for the internal active pipes’ network. The effect should be corroborated with the pipes’ standardization (their diameter does not vary continuously), meaning that there is a rather large interval of flow rates for which the optimum diameter has the same value (see Appendixes 2 and 3 for such flow rates rather different but giving the same optimum diameter). Consequently, there is a particular value of the flow from which the optimum diameter of the pipe jumps to the next standardized value where it remains for another certain range of flows. Thus, the investment does not change, while the friction losses stabilize (see Tudor and Lavric30 for the same discussion, but for continuous water/wastewater networks). The FOB2 minimization has a more complex effect upon the SWN topology and its operating conditionsit diminishes the internal network complexity and favors the large flows against the small ones; thus, the freshwater consumption increases to 4713 tonnes, meaning 176% higher than in the FOB1 case (Figure 6). This result confirms the tendency of the optimization algorithm to search for lower investment and operating costs, increasing the flow rate throughputs, thus decreasing the friction factor (see Appendixes 2 and 3 for a comparison between flows and optimum diameters). And this

Figure 6. Comparison of freshwater consumption for single and dualobjective optimization (maximum freshwater corresponds to the lack of internal wastewater reuse either between WUs or from the ST).

overall contaminated water reused from ST (Figure 8). These values are shown against the results of the optimal design for the same conditions but using, separately, only one of the two objectives. The results are compared to the ones obtained when regeneration is used (see the section Influence of Regeneration). The results presented in Figures 6−8 show both the dichotomic nature of the functions used in the Pareto approach to the design of the base case and the compromise obtained when doing it. Using freshwater as the only objective function (FOB1), the lowest value for its consumption among all scenarios was found, 1707.41 tonnes (see Figure 6), which is lower than the freshwater consumed by the unoptimized network, 2105.25 tonnes (Figure 6, Maximum Freshwater). When the design is 6123



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is done increasing both freshwater consumption and wastewater reused from ST (Figure 8). This large increase of the former should be related to the lack of pricing the freshwater, its quantification being done through its flow rate only and no upper threshold being used during optimization. There are two main reasons to prefer the results of the Pareto approach instead of the FOB2. The first is the decreased ST capacity, which should accommodate the overall freshwater entered the system during a batch. Pareto approach gave a ST capacity which represents 38.5% of the ST capacity obtained using FOB2, thus a large cut in the investment costs. The second is related to the treatment costs associated to the contaminated water collected in the ST, which are higher in the case of FOB2 usage, since the amount of the processed wastewater in the treatment system is 2.6 times higher than in the FOB1&2 case and the contaminants’ concentrations are significantly lower, as presented in Figures 9 and 10.

Figure 10. Concentration profiles in the ST for the three contaminants during the entire batch−double-objective optimization.

When FOB1 is used for optimization, the mean availability is higher than zero for WU1 on time intervals II, III, and IV, and for WU2 on time intervals VIII and IX. The higher amount of freshwater used when FOB2 is optimized dilutes the concentrations of contaminants and the units WU1, WU2, WU5 are not working optimally anymore, their availability becoming nonzero. The compromise is realized by the use of the Pareto approach, when only WU1 departed from optimality. When minimizing FOB1, the low freshwater consumption raises the final contaminant concentrations in the ST (Figure 11).

Figure 9. Concentration profiles in the ST for the three contaminants during the entire batch−costs minimization only.

The concept of availability, as introduced by Iancu et al.,35 is used to assess the optimality of the results. The WU’s availability represents the mean overall mass transfer pseudodriving force a unit still has for the contaminant k, defined as the logarithmic mean of concentration differences at the entrance and at the exit of each water-using unit:

Αk , i

⎧ 0, if [(C in,max − C in ) = 0 or k ,i k ,i ⎪ ⎪ (Ckout,max − Ckout ,i , i ) = 0] ⎪ ⎪ = ⎨ in,max − Ckin, i) − (Ckout,max − Ckout ⎪ (Ck , i ,i ,i ) , ⎪ in ⎞ ⎛ Ckin,max − Ck , i ,i ⎪ ⎜ ⎟ ln ⎪ − Ckout ⎝ Ckout,max ,i ⎠ ,i ⎩

Figure 11. Concentration profiles in the ST for the three contaminants during the entire batch−freshwater minimization only.

Therefore less contaminated water is used from the ST, since this would mean diluting it with freshwater to cope with the inlet restrictions of the overlapping WUs. When FOB2 is minimized, the large quantities of freshwater (see Figure 6) are used to diminish the concentrations of contaminants for the reused wastewater in every WU. FOB2 favors the internal reuse, at the expense of the availability, which is higher, hence WUs are not working optimally. The tendency is to simplify the topology by eliminating the connections to ST (see Appendix 2). Likewise the contaminants concentrations in the ST are the lowest (Figure 9), hence WU5 is fed with wastewater from ST only, except one interval (VIII) when freshwater is also used, canceling some internal connections (see Appendix 1 and Appendix 2 for comparison, intervals V and VIII). A comparison of the costs of the optimized versions of the SWN, resulted when FOB1&2 and FOB2 are used as objective functions, is presented in Table 4.

otherwise

(14)

The overlapping WUs’ availabilities could be used to define the mean availability of the network with respect to the contaminant k: N*

Αk =

∑i = 1 Αk , i N*

(15)

As the mean availability approaches zero, the WUs belonging to the same network are closer to optimality, meaning that their inlet and/or outlet concentrations are nearer their restrictions. A WN works optimally with respect to the contaminant k when Ak = 0, meaning there is no driving force left for the mass transfer in any WU. 6124



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Table 4. Total Costs Associated to the Optimized SWN, for each Time Interval, When Using Either the Active Pipes Investment and Operating Costs Only or the Dual-Objective Optimization. The Costs Are Expressed in Arbitrary Monetary Units (m.u.) FOB2 ×10−4 FOB1&2 ×10−4

I

II

III

IV

V

VI

VII

VIII

IX

X

4.961 5.101

8.441 7.987

16.028 8.518

17.758 15.406

11.496 11.899

15.274 13.839

11.561 10.370

9.252 6.863

6.068 3.525

2.618 2.616

Figure 12. Regenerated flow profiles during the entire batch for freshwater minimization only and double-objective optimization.

6.3. Influence of Regeneration. A further improvement in wastewater reuse for any type of water network is obtained by placing a local RU within the network, hence reducing the freshwater consumption. The wastewater coming from ST and having the concentrations of all contaminants higher than the imposed RU inlet limit is first processed by this unit; the concentrations of the pollutants are decreased to 90% of their entering values, provided that these latter are higher than the RU’s inlet restrictions. The regenerated stream is then distributed to the overlapping WUs according to their needs. As expected, the influence of regeneration is significant with respect to both freshwater consumption reduction and topology. When FOB1 only is minimized, the freshwater consumption decreases to 59.16% (Figure 6), while the whole reused wastewater, 1270.7645 tonnes (a spectacular increase of 2221.61%, see Figure 8), is subject to regeneration (Figure 12). The Pareto approach lowers the level of wastewater reused to 1102.336 tonnes (though a very nice increase of 1784.26%, see Figure 8), from which 1055.418 tonnes are regenerated. At the same time, the freshwater consumption drops by 28.69% (Figure 6). When FOB2 only is the used objective function, no wastewater reused from the ST is subject to regeneration, since the concentration of the contaminants is lower than the RU inlet threshold and no restrictions were imposed in using freshwater to dilute the contaminants concentration at the WUs inlet. With respect to network dynamics, the connections set between the operating WUs and ST, when the latter was used as supplemental supply, are replaced by the connections between WUs and the RU, when the latter starts functioning, and the connection between ST and RU, which is the same,

When Pareto approach is used, the costs associated with the investments and network operation are 16.75% lower than the value obtained when only FOB2 was minimized. In the latter optimization, the strategy used for cost minimization is reducing the connections to the ST, thus reducing the investment costs, and this can be achieved by increasing the internal flows of reused wastewater and freshwater flows. This leads to an increase in the pipe diameters and therefore to higher costs. The Pareto optimization succeeds in obtaining a value below the optimum one resulted by optimizing costs only, because also with freshwater consumption reduction as an optimization objective, it chooses to eliminate some internal pipes rather than the connections to the ST. 6.2. ST with Limited Capacity. If the investment costs associated to a large ST are important, a way of avoiding them is to analyze the consequences of using a ST with limited capacity, emptied from time to time when some upper threshold value is reached. Although this upper limit and the withdrawal flow could be subject to optimization themselves, after several trials a suitable value for this threshold appeared to be 700 m3 (tonnes). When this limit is reached, the wastewater within is sent to treatment with a flow rate of 100 tonnes/h until a lower, safety limit for the volume is reached, 200 m3 left in the ST (see the discussions in Dogaru and Lavric27). When the optimization is performed with respect to costs only, WU5 and WU4 receive wastewater from the ST on several time intervals. Limiting the capacity of the ST leads to higher contaminant concentrations in the ST, and as a result the quantity of wastewater reused from ST is diminished. 6125



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irrespective of the network topology, can thus be disregarded. Even if the internal flows have a high level of contaminants, given by the WUs outlet restrictions with which they still cope, the regenerated water is used to dilute them before entering any WU. If needed, some freshwater is added, but the priority is granted to the regenerated stream.

Table 6. The Economic Optimum Diameters Minimizing the Investments and Operating Costsa

Table 5. The WUs Operating Only with Wastewater on Certain Time Intervals when the RU Is Active (DualObjective Optimization) water using unit

time intervals when freshwater flow is zero

WU2 WU4 WU5 WU6

V, VIII VI, X V, VI, VII, VIII II, V, VII

It is worth mentioning that when using regeneration, several WUs are not fed with freshwater during some time intervals (see Table 5). They are fed with internally reused water (intensified reuse) and with regenerated water. When RU was not working, WU4 only received wastewater from the ST during the last time interval, X (see Appendix 3). When RU is present, the regenerated streams feed all the WUs (except WU1 and WU3 which requires freshwater only, see Table 2) after RU starts operating. As an observation, the investments cost of the RU is not considered, since, like in the case of the ST, it does not modify when the topologies and operating conditions of the SWN change. A RU, especially one capable of treating all the contaminants in the same placethe present casewill involve high investments, operating costs, and maintenance. What change for the RU, when the operating conditions and the topology of the network vary, are the operating costs, which depend upon both the flow throughput and the amount of pollutants to be removed. But the flows are accounted for when computing the optimum diameter. So, we believe that not taking into consideration the costs associated with regeneration does not introduce an important bias in the present analysis. 6.4. Analysis of the Optimal Networks. The result of the optimization process, irrespective of whether it involved a Pareto approach or a single objective function minimization, either FOB1 or FOB2, is a succession of topologies, together with their associated operating conditions, namely the flows. According to these flows, the economic optimum diameters (ODs) were computed, minimizing the investments and operating costs, as presented in Table 6. Since WU1 and WU3 are inlet free of contaminants units, they accept only freshwater, whose flows remain the same irrespective of the time intervals; though, the WU1 freshwater flow is increased when WU3 becomes inactive. The rest of the WUs could accept reused wastewater, diluted eventually with freshwater, whose flow will change from time interval to time interval. For example, WU4 should have three ODs for the freshwater feeding pipe during its working period: 0.0231 m (1.391 t/h), 0.0394 m (4.499 t/h and 4.770 t/h) and 0.0398 m (5.256 t/h). Since endowing WU4 with three feeding pipes is an economic nonsense, one diameter should be kept for all time intervals. To pick up the right diameter, the flow regime should be taken into consideration; in the present case study the flow regime is turbulent for all streams. Consequently, the

a

Values computed for each time interval, as the result of the Pareto approach, for the freshwater (rows to the left of WUs) and reused wastewater (rows to the right of WUs); when the results were the same, for different time intervals, a single optimum diameter is shown. The rightmost ODs correspond to the flows sent to ST.

OD 0.0398 m will be kept, since it corresponds to the maximum flow throughput, any lower flow meaning less friction thus less pumping energy, which is the case for the other two flows. The same judgment applies for the rest of the WUs, highest ODs being chosen, as shown in Table 6. The flows sent to ST, from each WU, are, in almost all cases, in turbulent regime; therefore the highest ODs have to be chosen as well. On the other hand, when the flow regime is laminar the smallest ODs are preferred, since an increase of the flow lowers the friction factor and, thus, the operating costs. The influence of the Pareto approach compared to the optimization performed only with respect to costs (FOB2) is perceived at all levels: freshwater consumption (Figure 6), costs (Table 4), and topology (Appendix 2 and Appendix 3). The topology of the network optimized with respect to costs only is simplified (Appendix 2) as compared to the one resulted after applying the Pareto approach optimization. The connections with the ST are suppressed for WU1 on time intervals I, V, and VI, for WU3 on time interval IV, for WU4 on intervals VI and VII and for WU6 on intervals IV, V, VI. The effects are visible on the internal flows directed to the other active WUs. They rise and require the use of larger diameter pipes The finally selected optimum diameters resulted from the minimization of the investment and operational costs are presented in Table 7. It should be noted that, with few exceptions, there are notable differences between FOB2 optimization and the Pareto approach. For instance, the optimum diameter found for the stream directed from WU1 to WU5 is higher when performing FOB2 optimization only. This higher value corresponds to the time interval IV (Appendix 2) when three pipes directed to ST are suppressed 6126



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Table 7. The Finally Selected Economic Optimum Diameters, Minimizing the Investments and Operating Costsa

a

Values computed for each time interval, as the result of the optimization with respect to costs only, for the freshwater (rows to the left of WUs) and reused wastewater (rows to the right of WUs). The rightmost ODs correspond to the flows sent to ST. The bolded values are identical to the ones selected for the double-objective optimization (Table 6).

(including the one from WU1 to the ST); therefore three pipe connections present in the Pareto approach are eliminated in order to save investment and operational costs. Although, this action should result in lower costs, the effect is completely the opposite because the internal flow rates increase together with their diameters and thus with the costs (Table 4). As an additional observation, with FOB2 the energy savings increase (less pumping is performed since the pipes are eliminated) at the expense of an increased freshwater consumption. As a general remark concerning the comparison of the two optimizations using costs in the objective function (FOB2 against FOB1&2), when the target is limited to minimizing the costs only, the connections to the ST are suppressed. The network’s structure is simpler as it involves less pipe connections, but more freshwater is used (Figure 6). Therefore, an increased flow rate of fresh feed requires large diameter pipes (flow regime is turbulent when discussing freshwater inlet and wastewater stream directed to ST) that minimize the friction and hence the pumping/ operational costs. This analysis of the optimized networks’ ODs shows that, no matter how good the results are for each time interval and how fine-tuned the subnetworks are to the peculiarities of a given time interval, when using multiobjective optimization to seek for the best dynamic topology of the semicontinuous water network, there is no global optimality, at the scale of both each time interval and batch time scale. This suboptimal behavior is not related to the choice of the optimization algorithm, which may or may not guarantee global optimality for a time interval, but to the choice of both the point on the PF used to continue the search for the optimal topology on the next time interval and the pipes OD. To not render the optimization problem intractable with respect to the CPU time, one has to pick-up only one point from the optimally equivalent solutions offered by the PF for one time interval, to continue the optimization problem in the next time interval. Even if a scalar objective function would be used (avoiding the set of equally optimal solutions), when the working time of two WUs operating simultaneously covers more than one time-interval, it is more likely that the reused flow from one of the WUs to the next would change, and accordingly, the pipe’s optimum diameter would change too. This means that only one of the two pipes’ diameters should be kept, which renders suboptimal the other time slice. Consequently, even when using an algorithm guaranteeing global optimality for each time-interval, the network would

behave suboptimally due to, on one hand, the need to choose only one point from the PF to continue optimization of the subnetwork in the next time interval, and, on the other, to the aforementioned choice of only one diameter from two or several optimum diameters for the same WU.

7. CONCLUSIONS The present study extends the work of Dogaru and Lavric27 providing a RU to the semicontinuous WN and using a dualobjective function in the Pareto approach to search for the particular network topology (seen here as a succession of subnetworks, optimized for each time interval the schedule of the batch is divided into) which should ensure minimum freshwater consumption and minimum investment and operating costs. The integration of the DAE system is done with function ode15s, while the optimization is performed using GA function gamultiobj both built-in in Matlab. The optimization is performed on time intervals, for which a PF being obtained (see Appendix 4), due to the dichotomic nature of the components of the objective function, which cannot be concomitantly minimized. To not end-up optimizing too many subnetworks, a consequence of moving from the current Pareto front to the next Pareto fronts that resulted from each point of the former one (all the points are of equal optimality), the approach is to choose the topology and operating conditions corresponding to the middle point of the region containing the points with values lower than 1 for the first objective function on the PF, making a compromise between both freshwater consumption and pipe network cost. The methodology was tested on a case study composed of six WUs, three contaminants, one ST having infinite/finite capacity and one RU. The Pareto approach succeeds in finding a series of topologies for each time interval, together with the optimum internal flows for the reused wastewater. According to these flows, the economic optimum diameters (ODs) are computed, minimizing the investments and operating costs. The freshwater consumption is higher than the one used by the unoptimized network, but also the costs associated with the investments and network operation are 16.75% lower than the value obtained when only FOB2 was minimized. In the case of costs optimization only, the strategy involves reducing the number of connections to the ST, determining in this manner a reduction in the investment costs. As a result though, the internal flows and their corresponding diameters rise, increasing 6127



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the total cost. The Pareto optimization having also freshwater consumption reduction as an optimization objective chooses to eliminate some internal pipes rather than the connections to the ST, obtaining a lower total cost than when optimization by FOB2 only occurs. Less freshwater consumed results in less wastewater reused from ST (see Figure 8, FOB1 and FOB1&2), and these both imply smaller flow rates throughout the pipes. In the applied dual-optimization, minimizing the freshwater quantity prevented the excessive usage of wastewater from increasing the flow rates through pipes (increased friction), thus rising the operating costs. The standardization of the pipes (their diameter does not vary continuously) participates to the costs augmentation. As two WUs of the network are connected through a pipe having a standardized diameter, the variation of flows on time intervals requires more energy to minimize the friction. Therefore the pumping costs increase. Hence, a substantial mitigation of the operating costs is generated by a small increase in both freshwater and wastewater reused from ST. Although a high investment, the RU influences significantly the freshwater consumption as it provides the possibility of reusing an increased amount of wastewater (see Figures 6−10). Also the investment cost is reduced when pipe connections feeding some WUs with freshwater are suppressed. When a WU working time spreads over several time intervals, it is normal to have different ODs for the same connection of this WU with another unit (may it be another WU or the ST), provided different flows link these two units. Since there cannot be more than one pipe connecting two units, the final optimum diameter is chosen depending on the flow regime: for turbulent flow, it is the highest OD from the set, while for laminar flow, it is the lowest. Taking this final selection of the pipes’ ODs into account together with the choice of the operating conditions from the PF as starting point for the optimization of the next time interval, the resulted whole network will be suboptimal on the entire batch, no matter how good the results are for each time interval and how well the subnetworks accommodated the restrictions for every time interval. A solution for bringing the suboptimal network closer to optimality would be to quantify the impact of the selection of the final ODs upon the energy consumption at the semicontinuous network level and decide which OD should be kept for a given WU based upon this impact, and this will be further investigated.

Figure 1A. Network topology on time interval I. Single optimizationminimization of freshwater consumption only (FOB1).

Figure 2A. Network topology on time interval II. Single optimization−minimization of freshwater consumption only (FOB1).

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APPENDIX 1 Figures 1A−10A as described in the text and in the figure captions.

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APPENDIX 2 Figures 1B−10B as described in the text and in the figure captions.

■

APPENDIX 3

Figures 1C−10C as described in the text and in the figure captions.

■

APPENDIX 4 Figures 1D−9D as described in the text and in the figure captions.

Figure 3A. Network topology on time interval III. Single optimization−minimization of freshwater consumption only (FOB1). 6128



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Figure 4A. Network topology on time interval IV. Single optimization−minimization of freshwater consumption only (FOB1).

Figure 7A. Network topology on time interval VII. Single optimization−minimization of freshwater consumption only (FOB1).

Figure 5A. Network topology on time interval V. Single optimization−minimization of freshwater consumption only (FOB1).

Figure 8A. Network topology on time interval VIII. Single optimization−minimization of freshwater consumption only (FOB1).

Figure 6A. Network topology on time interval VI. Single optimization−minimization of freshwater consumption only (FOB1).

Figure 9A. Network topology on time interval IX. Single optimization−minimization of freshwater consumption only (FOB1). 6129



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Figure 10A. Network topology on time interval X. Single optimization−minimization of freshwater consumption only (FOB1).

Figure 3B. Network topology on time interval III. Single optimization−minimization of investment and operational costs only (FOB2).

Figure 1B. Network topology on time interval I. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 4B. Network topology on time interval IV. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 2B. Network topology on time interval II. Single optimization−minimization of investment and operational costs only (FOB2).

Figure 5B. Network topology on time interval V. Single optimization− minimization of investment and operational costs only (FOB2). 6130



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Figure 6B. Network topology on time interval VI. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 9B. Network topology on time interval IX. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 7B. Network topology on time interval VII. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 10B. Network topology on time interval X. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 8B. Network topology on time interval VIII. Single optimization− minimization of investment and operational costs only (FOB2).

Figure 1C. Network topology on time interval I. Dual optimization− minimization (FOB1&2). 6131



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Figure 2C. Network topology on time interval II. Dual optimization− minimization (FOB1&2).

Figure 5C. Network topology on time interval V. Dual optimization− minimization (FOB1&2).

Figure 6C. Network topology on time interval VI. Dual optimization− minimization (FOB1&2).

Figure 3C. Network topology on time interval III. Dual optimization− minimization (FOB1&2).

Figure 4C. Network topology on time interval IV. Dual optimization− minimization (FOB1&2).

Figure 7C. Network topology on time interval VII. Dual optimization−minimization (FOB1&2). 6132



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Figure 8C. Network topology on time interval VIII. Dual optimization−minimization (FOB1&2).

Figure 1D. Pareto front corresponding to time interval I.

Figure 9C. Network topology on time interval IX. Dual optimization− minimization (FOB1&2).

Figure 2D. Pareto front corresponding to time interval II.

Figure 10C. Network topology on time interval X. Dual optimization−minimization (FOB1&2).

Figure 3D. Pareto front corresponding to time interval III. 6133



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Figure 4D. Pareto front corresponding to time interval IV.

Figure 7D. Pareto front corresponding to time interval VII.

Figure 5D. Pareto front corresponding to time interval V.

Figure 8D. Pareto front corresponding to time interval VIII.

Figure 6D. Pareto front corresponding to time interval VI.

Figure 9D. Pareto front corresponding to time interval IX. 6134



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List of Abbreviations

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Elena-Lăcrămiora Dogaru gratefully acknowledges the financial support of the Sectorial Operational Programme Human Resources Development 2007−2013 of the Romanian Ministry of Labor, Family and Social Protection through the Financial Agreement POSDRU/88/1.5/S/61178. Professor Vasile Lavric and Elena-Lăcrămiora Dogaru acknowledge the financial support of UEFISCSU Project no. 663/19.01.2009.

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NOTATIONS Ck,j, Ck,i = concentration of contaminant k exiting unit j, respectively i, ppm CFk = concentration of contaminant k in feed, ppm CPk = CR,out when the regeneration is active, or CPk = CSk k otherwise CSk = concentration of contaminant k in ST, ppm CR,out = concentration of contaminant k in the regenerated k stream, ppm Cink,i = concentration of contaminant k in the inlet stream of unit i, ppm Cout k,i = concentration of contaminant k in the outlet stream of unit i, ppm in,max Ck,i = maximum allowable inlet concentration of contaminant k for unit i, ppm out,max Ck,i = maximum allowable outlet concentration of contaminant k for unit i, ppm Dr = reference diameter Di,j = optimum economic diameter of the pipe linking the WU i and j Δṁ k,i = load of contaminant k for water-using unit i, kg/h Δṁ i = total load of contaminant provided by each unit, kg/h F = Fanning friction factor Fi = necessary flow of fresh feed, t/h Fk,i = freshwater flow for contaminant k, used in unit i, t/h Fiq = freshwater flow feeding WUi in the time interval τq Gw = global fresh water quantity, t Γ = total cost, m.u. (monetary units) i, j = indexes for batch WUOs K = number of contaminants k = index for contaminants Li = water losses for unit i, t/h m = mass of wastewater in the storage tank, t N = number of overall semicontinuous WUs N* = number of overlapping WUs in a time interval Q = number of time intervals Pi = flow rate of wastewater coming from the ST and supplying unit i, if regeneration is inactive; or flow rate of regenerated water feeding the WU if regeneration is active Ri = flow rate of regenerated water feeding the WU Sini = flow coming from the ST and supplying unit i, t/h = flow directed from unit i to ST, t/h Sout i t = time, h τq = time interval Wi = stream of unit i directed to treatment network, t/h Xj,i, Xi,j = internal flow from unit j to unit i, respectively from unit i to unit j, t/h

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BWN = batch water network CPU = central processing unit DC = discontinuous DAE = differential algebraic equations FOB1 = objective function 1: freshwater consumption FOB2 = objective function 2: investment and operational costs FOB1&2 = double objective function GA = genetic algorithms MINLP = mixed-integer nonlinear programs PF = pareto front RU = regeneration unit ST = storage tank SWN = semicontinuous water network TN = treatment network TBWN = truly batch water network TU = treatment unit WN = water network WU = water-using unit

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