Ind. Eng. Chem. Res. 2010, 49, 8877–8893
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Water Minimization Techniques for Batch Processes Jacques F. Gouws,† Thokozani Majozi,*,† Dominic Chwan Yee Foo,‡ Cheng-Liang Chen,§ and Jui-Yuan Lee§ Department of Chemical Engineering, UniVersity of Pretoria, Lynnwood Road, Pretoria 0002, South Africa, Department of Chemical and EnVironmental Engineering, UniVersity of Nottingham Malaysia, Broga Road, 43500 Semenyih, Selangor, Malaysia, and Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China
Water minimization in the process industry is becoming increasingly important as environmental legislation becomes increasingly stringent and the awareness of the impact of industrial activities on the environment increases. Much work has been done on water minimization in continuous processes as evidenced by the detailed reviews of Bagajewicz (2000) and Foo (2009). Although water minimization for batch processes (batch water network in short) has been ignored in the past, it is steadily gaining more attention in research. An overview of the developments and methodologies proposed for batch water network is presented. The methodologies for water minimization can roughly be divided into insight-based and mathematical techniques. The former always consists of a two-step approach (targeting and design) in synthesizing a batch water network that features the minimum freshwater and wastewater flows for a given production schedule. The mathematical techniques, on the other hand, may be categorized into two subsectors, that is, with and without scheduling consideration. In this review, various water minimization methodologies are discussed and comparisons are made among them. When necessary, they are illustrated through examples. 1. Introduction The effect of industrial activity on the environment has been gaining more attention in the past few decades. The effects of industry are becoming more apparent, and general awareness of the public of these effects is increasing. This has led to the tightening of controls on the polluting effects of industry and more stringent allowable levels of pollution. Industry first responded to the required pollutant level by considering endof-pipe treatment methods. Traditionally, wastewater produced is collected and sent to the waste treatment facility before it is dispensed with. The treatment methods can be broadly divided into physical, chemical, and biological treatments as well as advanced waste treatment. These treatment methods ensure the effluent water meets the required discharge levels before being discharged to the environment. The above-mentioned methods can account for significant financial investment in a plant, as the cost of the treatment units is dependent on the volume of wastewater that requires treatment. Furthermore, in certain instances the level of treatment required is quite substantial. Hence, it would be favorable if the volume of water that requires treatment were reduced. Industry, therefore, has started looking at methods to reduce the volume of wastewater produced. This has resulted in much research into developing various in-plant water recovery methods, thereby reducing the overall volume of wastewater produced. In most instances processing units do not require freshwater to achieve the desired quality of the final product. These units can accept water with certain levels of contaminants present. Hence, implementing water minimization exercises means a dual reduction of freshwater and wastewater flow rates. * To whom correspondence should be addressed. Tel: +27 12 420 4130. Fax: +27 12 362 5173. E-mail:
[email protected] (J.F.G.);
[email protected](T.M.);
[email protected](D.C.Y.F.);
[email protected] (C.-L.C.);
[email protected] (J.-Y.L.). † University of Pretoria. ‡ University of Nottingham Malaysia. § National Taiwan University.
The methods developed for water minimization in continuous processes can roughly be divided into insight-based and mathematical techniques. The former1–14 are mainly based on pinch analysis techniques that were originally developed for heat15 and mass integration.16,17 On the other hand, the mathematical methods18–28 have their roots in optimization framework. Detailed reviews on methodologies for wastewater minimization in continuous processes (mainly on pinch analysis techniques) are given by Bagajewicz29 and Foo.30 It should be mentioned that most of the early research works on water minimization were developed for continuous processes. The development of water minimization techniques for batch processes (or batch water network) has always come second as compared to its counterpart in continuous processes. There are numerous reasons for this. The main reason is that continuous processes produce much larger volumes of wastewater as compared to batch processes, which has been the main driving force for industrial practitioners in implementing water recovery. Besides, techniques for batch water network are generally more complex than those for continuous processes due to the existence of time dimension apart from contaminant content. This point will be further illustrated later. However, recent development has seen the shift of research focus toward batch processing in general. In particular, development of batch water network techniques has also gained attention in the past decade. Several reasons may be tied to this trend. First, batch processes have always been poorly designed due to the lack of generic design techniques, as most conceptual design approaches are mainly dedicated to continuous processes.31–34 Second, batch processes have gained more attention in recent years due to the rise of various low-volume and highvalue-added products in the market, for example, pharmaceuticals and specialty chemicals. This has encouraged the development of various systematic design techniques to cater for this development. Third, wastewater produced from batch processes generally has higher toxicity than wastewater generated from continuous processes. Good examples of such wastewater
10.1021/ie100130a 2010 American Chemical Society Published on Web 08/23/2010
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the common operating conditions for various types of batch water-using processes. This is shown in the next section. 2. Operating Condition of Batch Water-Using Processes
Figure 1. Illustration of time and concentration constraints in a batch process.36
streams are those produced from agrochemical and pharmaceutical facilities. Wastewater produced by agrochemical facilities is often contaminated with cyanide, arsenic, and organic solvents, which are all highly toxic, whereas pharmaceutical wastewater often contains chemicals found in specialty facial creams, lotions, and hair care products, which are generally of very high toxicity. Consequently, the development of waste minimization strategies for batch processes has gained good attention in the past decade. The major difference between water minimization in batch and continuous processes lies mainly in the discreteness of tasks in the former. Consequently, units are not always active during the time horizon of interest. This means that water is not always required or produced during a time period. In a continuous process, the main constraint for wastewater recovery is the impurity concentration. However, as discussed earlier, the time dimension is another important constraint that needs to be simultaneously addressed in a batch water network. Figure 1 demonstrates exactly how concentration and time restrict the recovery of water sources. As shown, wastewater generated by a process can be recovered only if it obeys the inlet concentration constraint of the receiving unit and only if the receiving unit is operating during or after wastewater generation. However, wastewater produced at a later time cannot be used by a unit operating earlier in the time horizon. For instance, in Figure 1, the wastewater generated by operation B can be recovered to operation C because operation C has a higher inlet concentration than the concentration of wastewater from operation B. Moreover, operation C starts operating after operation B. On the other hand, wastewater produced by operation D cannot readily be recovered in operation A, even though the concentration constraint is met. This is due to the fact that operation D starts operating at a later time as compared to operation A. Similar to the case in continuous processes, the methodologies developed for a batch water network can be roughly divided into insight-based and mathematical techniques. Each of these techniques will be discussed in the subsequent sections of this review. Note that from the perspective of process integration, direct reuse refers to effluent from a process unit that enters to another process unit, whereas recycle refers to the re-entering of effluent to the process unit in which it is produced.1 On the other hand, indirect reuse/recycle refers to the use of a storage vessel in recovering the process effluent. However, before the wastewater minimization methodologies are discussed, it is essential to have a good understanding of
In this section, common operating conditions for batch waterusing processes are discussed. As shown in Figure 2, batch water-using processes may be broadly classified into truly batch or semicontinuous operations. For the former case, such as process 1 in Figure 2a, water is charged to the operation within a duration (t1 - t2) before the operation is carried out, whereas wastewater is discharged from the operation at a later duration (t3 - t4). A typical example for this kind of operation is the batch polymerization reaction in which water is used as a carrier. In this case, water is charged before the polymerization reaction takes place and is discharged at the end of the reaction. On the other hand, a semicontinuous operation charges freshwater and discharge wastewater during the course of the operation (t5 t6), such as what happens in process 2 in Figure 2b. Typical examples for this kind of operation include the various washing processes in the chemical industry. The washing process is normally carried out within the course of the operation and may be assumed as a conventional steady-state continuous operation if its dynamic behavior is insignificant. However, it operates for only a certain duration, which is always shorter than the cycle time of the complete batch operation. Note that for both types of processes there may be cases with or without water losses and/or gains. In some special cases, processes 1 and 2 in Figure 2 may experience situations with the existence of freshwater intake (Figure 3a,c) or wastewater discharge (Figure 3b,d) alone. A typical example for the former cases is the batch reactor in which water is used as raw material. As a result of the reaction, water is consumed completely (being a limiting reactant). Note that the reactor (process 1) does produce some product (not shown in the figures) with, however, insignificant water content. Thus, it may be assumed that no water discharge is worth recovering from the discharge point of this process. On the other hand, for cases in Figure 3b,d, water is formed as a product/byproduct in the reactor (process 2). A typical example for this operation is fermentation processes, by which significant water is produced. Note that because no water intake is needed for these kinds of processes, it is not represented on the diagram where water recovery is of concern. Note that in handling the common types of batch processes, one may utilize the approaches developed for either the fixed load or fixed flow problems. For the former, water is utilized as mass separating agent to partially remove a fixed amount of contaminant from the contaminant-rich stream. For this type of problem, water-using processes have always been treated as a unit with uniform/equal inlet (freshwater) and outlet (wastewater) flows. The approach works fine when there exist a small number of processes with water losses and/or gains. However, with the presence of many processes with water losses and/or gains, most researchers have adopted the fixed flow type approaches, in which the main concern is the water flow requirement of the water-using processes. For this latter approach, the water-using processes are segregated into sink (freshwater intake) and source (wastewater discharge), respectively, when analysis is carried out. Note that for the fixed flow type approaches, the flow requirement of the sink and the flow generation of the source are kept constant. The review by Foo30 has discussed the use of both approaches in detail. Note that
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Figure 2. Types of batch water-using processes: (a) truly batch; (b) semicontinuous operations.
Figure 3. Special cases of (a, b) truly batch and (c, d) semicontinuous processes.
for a single contaminant case, the two types of problems are compatible. However, this is not the case for multiple contaminants.30 3. Insight-Based Techniques for Water Minimization in Batch Processes Insight-based techniques for batch water network share their roots with their counterparts in continuous processes. However, apart from concentration constraint, time constraint is built into the various targeting tools for batch processes. It is worth mentioning that the insight-based techniques can further be classified as graphical and algebraic targeting techniques, resembling that in the continuous processes. Note that different insight-based techniques have been developed for both fixed load and fixed flow rate problems (see the characteristics of these problems in the review paper of Foo30). Note also that the major assumption in all insight-based approaches is that the operating procedure of the water-using processes is known a priori. With the known operating schedule, one may determine the minimum water flows within the time intervals of interest for a batch process. These various insight-based techniques are reviewed in the following sections. 3.1. Graphical Targeting Technique of Wang and Smith.35 The first extension of insight-based approach for batch water network was proposed by Wang and Smith.35 This seminal work on the insight-based approach was extended on the basis of their limiting composite curve for continuous processes1 and hence is based on the model of the fixed load problems. To consider concentration and time constraints simultaneously, the procedure
first segregates the water-using operations into concentration intervals, so that water may be cascaded from the lowest to the highest concentration intervals. The amount of water required for each process (∆G) within a concentration interval (∆C) is calculated using eq 1 ∆G )
∆m ) f∆t ∆C
(1)
where ∆m, f, and ∆t represent the mass load, water flow rate, and the duration of the operation, respectively. In the method by Wang and Smith,35 for a certain concentration interval, the amount of water flow (∆G) is plotted versus time for each operation. Each of these individual segments is then combined to give a composite curve for that concentration interval of interest. Once this has been done, the targeting within the concentration interval can be performed. The targeting procedure is best described using Figure 4, where two composite curves are shown. The first composite curve represents the processes that operate within the concentration interval of interest, and the second represents the available water flow from a previous concentration interval. The gradient of the individual segments of both curves varies throughout the time horizon due to different operations that exist within each time interval. If one considers the first time interval, direct reuse can take place because there is water available for reuse (from a lower concentration interval). The total amount of water needed is ∆G1. However, because an amount of ∆G3 is available for reuse, there exists a surplus of water (∆G3 - ∆G1) in this interval. This water surplus can be stored for reuse/recycle in subsequent
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Figure 4. Flow targeting technique of Wang and Smith,35 for selected concentration interval.
time intervals. The targeting procedure proceeds through all of the time intervals in the same concentration interval, before moving to subsequent concentration intervals, until the final one has been reached. In cases when no water is available for reuse/recycle, freshwater is used. Once the freshwater and the storage requirements are determined, the batch water network is then constructed. Note that the main objective for the work of Wang and Smith35 is on the flow targeting technique rather than network design. Hence, no systematic design procedure was proposed. The batch water network was constructed merely from the insights gained in the targeting stage. There are a few drawbacks with the targeting method of Wang and Smith.35 First, the technique assumes that each water-using process is operating in a semicontinuous mode during its operation. Hence, it does not cater to truly batch operations (Figure 2a). For the former, water is charged at the beginning of a water-using operation and discharged only upon the completion of its operation. Due to the water intake and discharge that occur at different time intervals, the approach of Wang and Smith35 does not apply to these processes. Second, the network design algorithm is not well developed because a simple case study was used in which network can be easily observed from the targeting stage.35 Third, the approach was not being explored for repeated-batch processes, which are commonly found in industrial practice. Finally, the targeting technique does not apply to processes with water losses/gains (as described in Figure 3), because the water-using processes are assumed to have uniform/equal inlet and outlet flows. 3.2. Graphical Targeting Technique of Majozi et al.36 Majozi et al.36 developed another graphical targeting method based on that developed by Wang and Smith.35 However, there are some fundamental differences between the two techniques. First, the water-using operations considered in the work of Majozi et al.36 are truly batch processes (Figure 2a); that is, water intake and discharge are found at the beginning and end of the water-using operation, respectively, which is not the case in the work of Wang and Smith.35 Second, Wang and Smith35 assume that the inlet and outlet concentrations of each operation to vary, but being lower than their respective maximum values, and hence the amount of water used by a water-using process may vary. Majozi et al.,36 however, fix the amount of water used by the water-using process, hence allowing its inlet and outlet concentrations to vary lower than their maximum values. It is worth noting that the similarity between the two methods is that the work of Majozi et al.36 is also developed for the fixed load problem.
Figure 5. Flow targeting technique of Majozi et al.36 with time as the main constraint.
Figure 6. Flow targeting technique of Majozi et al.36 with concentration as the main constraint.
Majozi et al.36 proposed two ways in which the targeting steps can be carried out, that is, either time can be taken as the primary constraint and concentration as the secondary constraint or vice versa. Both approaches are briefly reviewed next. The targeting approach with time as the main constraint is similar to that proposed by Wang and Smith.35 Each water-using process is plotted on a water demand versus time diagram, as is the available water source from the previous concentration interval. An example of this is given in Figure 5. As mentioned earlier, Majozi et al.36 assumed a fixed amount of water requirement for the water-using processes. This can be seen from the water demand profile, which consists of the horizontal segments in all time intervals. Another alternative procedure for the targeting technique is to treat concentration as the main constraint. In this case, a concentration versus water demand diagram for each process is drawn for each separate time interval. Different from the previous approach, each process remains a separate segment, in which the composite curve is not constructed. Water at the end of a time interval is stored for later use. However, note that the stored water will only be used provided that it meets the concentration constraint of the water-using processes. An example of this targeting step is given in Figure 6. For instance, wastewater from the completed operation 1 may be reused in operation 3, because the concentration of the former matches the maximum limit of the latter. The reuse of wastewater from operation 1 to operation 3 is dependent on when the former completes its operation. If operation 1 finishes at the start of operation 3, then wastewater from operation 1 can be directly
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Table 1. Targeting Technique of Foo et al.,
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reused in operation 3. In cases when operation 1 completes its process at the earlier time, water storage is needed for its wastewater for indirect reuse in operation 3. Majozi et al.36 also proposed a way to reduce the required storage for reusable water by using idle processing vessels for reusable water storage. This is done by drawing an inherent storage availability diagram (ISAD), in which each operation is plotted on a capacity versus time diagram. From the diagram, the available capacity for wastewater storage in each unit can be determined. In a nutshell, ISAD is used as a tool to assess the extent of idleness of processing units so as to determine their availability for storage. Similar to the case of Wang and Smith,35 the targeting technique is limited for water-using processes with uniform/ equal flow for water intake and discharge. Hence, it does not handle water-using processes with water losses or gains (Figure 3). 3.3. Algebraic Targeting and Network Design Techniques of Foo et al.37 The first work based on the fixed flow problem was reported by Foo et al.37 The authors proposed a two-stage targeting and network design approach for batch water network, following the conventional practice of pinch analysis techniques (e.g., heat and mass integration). For flow targeting, an algebraic method known as the time-dependent water cascade analysis (WCA) technique was developed to locate the minimum water flows in a batch process. The method was essentially a combination of two earlier developed targeting tools for water minimization in continuous processes, that is, WCA,7,38 as well as the mass exchange network synthesis for batch processes.39,40 As with previous insight-based techniques, the time horizon is divided into a number of time intervals in which targeting can take place. The time-dependent WCA technique is next illustrated. Table 1 shows a generic cascade table on how the targeting procedure is carried out. All water sources and sinks are first located at their respective impurity concentration levels (Ck) within the time interval (t) in which they exist. Within each time interval, the flow of the water sink (Fj) is then deducted from that of the source (Fi) at their respective concentration levels to yield the net flow rate, (ΣiFi - ΣjFj)k. The net flow surplus/deficit is cascaded down the concentration levels to yield the cumulative flow rate (FC, k). The load in column 7 (∆mk) is obtained from the product of FC,k and the difference across two concentration levels (Ck+1 - Ck). Cascading the load down the concentration levels yields the cumulative load (cum. ∆mk). Flow at the first level of the FC,k column corresponds to the minimum freshwater flow (FFW) of the network, whereas that at the final level represents the wastewater (FWW) generated from the
network. Note that the minimum flow of freshwater is determined from the infeasible cascade, but it is omitted here for brevity. The targeting procedure also differentiates between networks with and without a water storage system. For the latter, the targeting procedure is carried out in each time interval, and the total freshwater flows are summed across all time intervals. For networks with a water storage system (repeated-batch processes), a slight modification is needed for the targeting procedure. An overall flow targeting is first carried out for the network because repeated-batch processes with a storage system resemble a continuous process. The targeted freshwater flow is then allocated accordingly on the basis of the needs of the individual time intervals. Wastewater flow and storage capacity targets are next determined. This type of analysis is similar to that proposed by Majozi et al.,36 who considered concentration as the main constraint, in which each time interval is analyzed separately. Because the targeting technique is algebraic in nature, it does not require any drawing to be performed and, hence, is free from the common limitations of graphical plots, such as inaccuracy and time consumption. Because the approach is developed for the fixed flow problems, it treats water-using processes as sinks (where reuse/ recycle is possible) and sources (units that produce effluent) separately. In the context of batch processes, the sink and source represent the fixed amount of water flow needed and produced by a water-using process. Hence, this approach handles both truly batch and semicontinuous processes equally well, as well as processes with water losses and gains. There are several limitations with this targeting approach though. First, the targeting approach does not work for singlebatch processes with storage systems when there are multiple sinks and sources of different concentration levels, even though it works well for repeated batches. The reason for this is that, mixing of water sources of different qualities is allowed when water is stored at an earlier time interval for use at a later time. Hence, it is not obvious which water source(s) should be stored to achieve minimum water flows for the overall network. However, the technique works efficiently for simple batch processes. Second, the work assumes that a single water storage tank is used for storing water sources of the same quality for future reuse. In other words, no mixing is allowed for sources of different concentrations. This will lead to an unnecessarily large number of storage vessels when the number of water sources with different concentrations increases. However, it is worthy of mention that the identification of storage in the method is more straightforward than that presented by Wang and Smith.35
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Table 2. Targeting Technique of Liu et al.,42 Time-Dependent CIT t)1 Ck
ΣiFi
∆mk
Fi
∆m1
Fi
∆m2
C1 C2
t)2
cum ∆mk
FFW,k
cum ∆m1
FFW,1
cum ∆m2
ΣiFi
∆mk
Fi
∆m1
Fi
∆m2
FFW,2
cum ∆mk
FFW,k
cum ∆m1
FFW,1
cum ∆m2
FFW,2
Ck
Besides, the authors formalized the design procedure for the batch water network, which has been treated implicitly by earlier researchers.35,36 In principle, the water sinks and sources are segregated in their individual time intervals before they are matched to achieve the minimum flow targets. Note that this principle holds for all network design procedures proposed for the continuous processes,8,17,41 so long as the unit of water flow is used. A time-water network diagram (not shown) is also proposed to present the batch water network.37 3.4. Algebraic Targeting Technique of Liu et al.42 Another algebraic targeting technique called the time-dependent concentration interVal analysis (CIA) was developed by Liu et al.42 This technique stems from their earlier work done in continuous processes,43 which was extended to handle the fixed flow problems from the earlier version that was dedicated for the fixed load problems.44 Hence, the time-dependent CIA handles the fixed flow problems equally well (even though it has not been shown in their work). The basic principle of the time-dependent CIA is similar to that of the time-independent WCA,37 in which water sinks and sources are segregated in their respective time intervals before the targeting step is carried out. The procedure for this targeting technique is briefly described next, using the concentration interval table (CIT) in Table 2. As shown, the impurity concentrations (Ck) are first arranged in descending order, and water flow (Fi) is then located within the respective intervals. Impurity load (∆mk) in each concentration interval is then calculated from the product of the water flow with the concentration difference across the interval. The impurity load is then cascaded upward throughout the concentration intervals (cum. ∆mk). The minimum freshwater flow is then determined by dividing the cum ∆mk value by its corresponding concentration level. The targeting procedure is then carried out for all time intervals to locate the overall water flows. Different operation modes of the batch water network were analyzed, that is, single-batch operation with and without a water storage system as well as repeated-batch operation (with water storage). Finally, note that the technique has not been demonstrated for its applicability in truly batch processes, because only semicontinuous processes were analyzed.42 However, because the targeting technique treats water sinks and sources similarly to the time-independent WCA,37 it applies to truly batch processes equally well. This is demonstrated with an example in a later section. 3.5. Network Design Technique of Chen and Lee.45 Recently, Chen and Lee45 developed the first graphical technique for the design of a batch water network for fixed flow problems. Apart from the network design procedure, the technique also locates the minimum storage capacity as well as simplifies the preliminary network complexity. This is accomplished through a quantity-time diagram (Figure 7), in which operations are represented as water sinks and sources on the positive and negative regions of the quantity-axis, respectively. The timeaxis guides the designer to meet a certain water sink only with water sources that occur earlier in time, whereas the quantityaxis indicates the total water flow needed (by the sink) or
Figure 7. Quantity-time diagram of Chen and Lee45 (values in parentheses indicate impurity concentration in mass ratio). Table 3. Limiting Water Data for Example 1 concentrations (ppm)
time (h)
process
flow rate (t/h)
Cin,max
Cout,max
ts
tf
1 2 3
100 80 50
100 0 100
400 200 200
0.5 0 0.5
1.5 0.5 1.0
possessed (for the source). The procedure of utilizing the quantity-time diagram is briefly illustrated as follows. As shown in Figure 7, water sinks 1 (D1), 2 (D2), and 4 (D4) exist before sources 1 (S1), 2 (S2) and 4 (S4), whereas sink 3 (D3) exists between S2 and S4. D1, D2, and D4 are satisfied by freshwater, because they exist before any water sources. In contrast, one may use S1 or S2 to fulfill D3 that exist after them. To maximize water recovery, a source of a lower concentration is used (i.e., S1 with 0.2 impurity mass ratio), by diluting with freshwater to fulfill the flow requirement of D3. Upon reuse, flow of S1 is reduced. The unutilized portion of S1 will emit as wastewater along with S2 and S4. The main strength of this approach is that it guarantees the minimum water flows to be achieved even without prior knowledge of the flow targets (via targeting). In principle, the design procedure fulfills the necessary condition of optimality given by Savelski and Bagajewicz46 and the nearest-neighbor algorithm.8 All three modes of batch operation (single batch with and without storage; repeated batch with storage) are analyzed. Note also that because this is a network design technique, it allows the exploration of compulsory and forbidden matches between the water sinks and sources. 3.6. Illustrative Examples. To compare the various insightbased techniques, two examples are analyzed. Example 1 consists of a semicontinuous process originally presented by Wang and Smith.35 This example involves three water-using processes that operate within a specific duration. The schedule and limiting data are given in Table 3. The method developed by Majozi et al.36 is not considered in the comparison because it was developed for truly batch processes. The result of example 1 obtained via various techniques is given in Table 4. In essence, because the example is rather simple, there is no real difference in the minimum water flows. Because this is a fixed load case study, the wastewater flow is the same as the freshwater flow as identified in Table 4.
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Table 4. Result of Example 1 Obtained via Various Insight-Based Techniques Wang and Chen and Smith35 Foo et al.37 Liu et al.42 Lee45 freshwater target for single-batch process without storage (t) freshwater target for single-batch process with storage (t) number of storage vessels for single-batch process size of storage required for single-batch process (t) freshwater target for repeated-batch process with storage (t) number of storage vessels for repeated-batch process size of storage required for repeated-batch process (t)
NA
121.25
121.25
121.25
102.5
102.5
102.5
102.5
1
1
1
1 37.5
37.5
37.5
37.5
NA
102.5
102.5
102.5
NA
1
1
1
NA
37.5
37.5
37.5
Table 5. Limiting Water Data for Example 2
process
freshwater flow (kg)
1 2 3 4 5
1000 280 400 280 400
concentrations (kg of salt/kg of water)
time (h)
Cin,max
Cout,max
ts
tf
0 0.25 0.1 0.25 0.1
0.1 0.51 0.1 0.51 0.1
0 0 4 2 6
3 4 5.5 6 7.5
Table 6. Result of Example 2 Obtained via Various Insight-Based Techniques Foo et al.37 freshwater target for single-batch process without storage (t) freshwater target for single-batch process with storage (t) number of storage vessels for single-batch process size of storage required for single-batch process (t) freshwater target for repeated-batch process with storage (t) number of storage vessels for repeated-batch process size of storage required for repeated-batch process (t)
Majozi Chen et al.36 Liu et al.42 and Lee45
2052.30
2052.30
2052.30
2052.30
1560
1560
1560
1560
1
1
1
1
400
400
400
400
1000
1000
1000
1000
1
1
1
1
560
560
560
560
Example 2 is taken from Majozi et al.36 and involves five truly batch water-using processes in an agrochemical production. The schedule and limiting data are given in Table 5. A few remarks are worth mentioning in this example. First, even though the scheduled start and end times for the processes look similar to those in example 1, the water-using processes are actually operated in truly batch mode; that is, water is charged before the operation starts and can be discharged only after its operation is completed. In other words, no water intake and discharge is observed during the operation of the processes. Second, note that the limiting inlet and outlet concentrations of both processes 3 and 5 are the same, because these processes do not generate an impurity load when water is utilized. Table 6 shows the result of example 2 obtained via various insight-based techniques. The method of Wang and Smith35 is not considered in the comparison because it was developed for semicontinuous processes. 3.7. Overall Comparisons and Comments on Various Insight-Based Techniques. Table 7 shows the overall comparison for the five different insight-based techniques. As shown, since the approaches of Wang and Smith35 as well as Majozi et al.36 are developed for the fixed load problem, they are restricted
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in solving semicontinuous or truly batch processes, respectively. On the other hand, none of the approaches developed for the fixed flow problem are bound with this limitation (see columns 2-4). The final three columns list the modes of operation that the individual approaches may use to determine the water flow. Insight-based techniques for batch processes are useful in situations when the operation under investigation has a production schedule that is fixed or rarely changes. Because it provides good insights into the synthesis problem, it thus assists in the identification of bottleneck operations. These operations can be adjusted to allow for greater water recovery opportunities. In addition, the insight-based techniques do not require knowledge of mathematical optimization to find a feasible solution to a problem. However, insight-based techniques have several disadvantages. First, similar to their counterpart in continuous processes, they are restricted to single-contaminant problems and do not handle multiple-contaminant problems efficiently. Second, as will be demonstrated at a later point, minimizing water by assuming a fixed production schedule will not necessarily give the absolute minimum freshwater consumption. A change in the production schedule will possibly allow for greater water recovery. Third, complex process constraints, such as compulsory and forbidden matches or maximum number of pipe connections, can be handled only in the design step of the insight-based techniques, but not in the targeting step. Hence, the minimum flow situation is no longer guaranteed. Fourth, the method could be very complex for large problems with many concentration and/or time intervals. 4. Mathematical Techniques for Water Minimization in Batch Processes All mathematical-based techniques for batch water network follow the same general structure, in which their formulations comprise the various constraints that describe the water and contaminant balances across the water-using processes, process operation, capacity, and time dimension. The main difference between the various methodologies is in the treatment of the time dimension of the batch processes. Two main groups arise: The first group minimizes water flows within a predefined schedule, which implies that time is treated as a parameter, similar to that of insight-based techniques. The second group minimizes water flows and determines the schedule that achieves the minimum water flow targets. Consequently, time is treated as a variable in this latter category. 4.1. Water Minimization Techniques within a Fixed Schedule. Water minimization within a predefined schedule has been seen as a simpler problem to solve than that when the schedule is to be determined as part of the solution algorithm. Furthermore, due to the fact that batch scheduling methodologies have only reached a mature stage with reasonable problem sizes and duration, the batch water network has hence been classically solved with a predefined schedule. The methods that fall into this category are those proposed by Almato´ et al.,47,48 Kim and Smith,49 Majozi,50 Li and Chang,51 Shoaib et al.,52 Rabie and El-Halwagi,53 Ng et al.,54 and Chen et al.55–57 Most mathematically based models have a few characteristics in common. First, almost all methods are based on a superstructural approach, which is a representation of all possible solutions for the process under investigation. Second, each method has an objective function. The objective functions vary and are dependent on the way in which the problem is addressed and the nature of the mathematical model. However, the solution
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Table 7. Summary of Various Insight-Based Techniques fixed load/flow types fixed load Wang and Smith35 Majozi et al.36 Foo et al.37 Liu et al.42 Chen and Lee45 a
fixed flow
X X
types of batch processes semicontinuous
operating mode of batch processes
truly batch
single batch without storage
single batch with storage
repeated batches with storage
X X Xa X
Xa X X Xa
X X Xa X X
X X X X
X X X X
X X X
Not reported in the original work, but can be done.
strategy for these approaches varies. Each of these methods will now be discussed in the following section. 4.1.1. Superstructure. The purpose of a superstructure is to represent all of the water-using processes in a unified manner while considering all possible interconnections between the various processes. In essence, the superstructure should represent all possible solutions. The usage of a superstructure allows each process to be represented by a common set of constraints in which the optimization can take place. Most often, the superstructure embeds the assumption made in a model and dictates strategy in solving it. There are different ways of representing superstructures. In a broader perspective, one may classify them on the basis of the use of an intermediate storage tank. When a storage tank is used to reuse/recycle water among the waterusing processes, it is termed indirect integration. In contrast, direct integration refers to a batch water network without an intermediate storage tank. In the following section, the superstructure for indirect integration is first discussed. The seminal work of mathematical-based batch water network synthesis is reported by Almato´ et al.,47,48 who made use of a superstructure of the indirect integration scheme. In this model, all water-using processes are connected to their respective intermediate storage tanks for water recovery. In other words, no direct reuse/recycle options were included in the superstructure. Similar superstructures for indirect integration schemes were also proposed by Li and Chang,51 Shoaib et al.,52 and Rabie and El-Halwagi,53 in which water recovery was only possible through storage vessels. It is worth noting that in the work of Li and Chang51 the stored water may be reused/recycled by sinks of the same batch cycle. However, in the cases of Shoaib et al.52 and Rabie and El-Halwagi,53 the stored water may only be reused/recycled by sinks in the next batch cycle. Note further that the works of Almato´ et al.47,48 were developed for the fixed load problem, whereas that of the rest are based on the fixed flow problem (except that of Li and Chang,51 which is for both fixed load and fixed flow problems). An example of these superstructure representations is shown in Figure 8. As shown, all possible connections are embedded between water sources and sinks (via storage tanks). On the other hand, Majozi50 proposed a superstructure that considers only the direct integration scheme. Because the method does not consider intermediate storage for water recovery, the superstructure depicts only direct reuse/recycle connections between water-using operations as shown in Figure 9. Later works in batch water network synthesis generally consider both direct and indirect integration schemes simultaneously, because it is more generic. This includes the works of Kim and Smith,49 Ng et al.,54 and Tokos and Pintaricˇ58 as well as those by Chen et al.55–57 The superstructure presented by Kim and Smith49 is based on the fixed load model. As shown in the superstructure in Figure 10, water-using operations within the same time interval have direct reuse/recycle connections among them. Besides, storage tanks (indirect integration) are used among water-using operations of different time intervals.
Figure 8. Superstructure of Shoaib et al.52
Figure 9. Superstructure used by Majozi50 showing an individual unit.
Figure 10. Superstructure of Kim and Smith.49
An extension work based on the method of Kim and Smith49 is proposed by Tokos and Pintaricˇ.58 The additional feature of this work is the incorporation of regeneration processes that are operated in both batch and semicontinuous modes (see superstructure in Figure 11). The regenerated source from a semicontinuous unit may be reused in other processes immediately (when regeneration is in operation). In contrast, the regenerated source from a batch treatment unit can be reused only when its operation is fully completed. The model will also schedule the use for these regeneration units. The main limitation
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Figure 11. Superstructure of Tokos and Pintaricˇ.58
Figure 12. Superstructure of Ng et al.54
of this work is that the model is restricted to a single-batch process with a water storage tank. It has not been demonstrated for a single-batch process without a storage tank or for repeated batches. On the other hand, the superstructure of Ng et al.54 takes the form of source-storage-interception-storage-sink, as shown in Figure 12. Even though the approach is developed for a property network, it works well for a concentration-based batch water network. Note that when water-using processes are found in the same time interval, the storage tanks are omitted. On the other hand, the superstructure of Chen et al.55,56 takes the same form as that of Majozi50 (Figure 9), however, with the use of storage tanks for indirect water recovery. Because the superstructures used in the two works55,56 are quite similar, with just a slight difference in the consideration of water losses/gains, only the one of Chen et al.56 is shown in Figure 13. Besides, the work of Chen et al.57 was developed for the fixed flow problem. Their superstructure consists of the schematic representation for all process elements and has a similar form to that in Figure 13. Following the establishment of the superstructure, mass balance constraints that account for the movement of mass among the various water-using processes are next described. Note that the superstructures have a strong influence on the structure of the mass balance constraints. 4.1.2. Mass Balance Constraints. The structures of the mass balance constraints considered in each method are different, due to the various natures of the superstructures in each method.
Figure 13. Superstructure of Chen et al.56
Furthermore, the handling of time plays an important role in the derivation of the mass balance constraints. In some instances, time is overridden by assuming that all water reuse/recycle between sources and sinks is carried out between two consecutive batches of operation via water storage,52–54 and in others the mass balances are done over the duration of the operation of a unit.47–49,57 Besides, the structure of the mass balance constraints is also dependent on whether the model is described as the fixed load or fixed flow model. For the former, it is implicitly assumed that water is charged to and withdrawn from a unit’s operation uniformly, whereas the assumption is relaxed in the fixed flow model (see detailed discussion of these models in the work of Foo30). The mass balance constraints derived by Almato´ et al.47,48 are continuous in nature and take the form of a fixed load model. This is because the concentration of the stream is given as a function of time. These are presented in two forms, that is, integral or difference forms, as shown in eqs 2 and 3 respectively. It must be noted that the method presented by Almato´ et al.47,48 is based on the flow rate in each unit being fixed. The inlet and outlet concentrations of a unit are, however, not fixed. In eqs 2 and 3, Cs (t) is the contaminant concentration of stream s at any time t, Rs,s′ is a parameter relating stream s to s′. This could be interpreted as a stoichiometric factor. qs (t) is the flow rate of stream s at any time t, ms,s′ (t) is the
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contaminant mass variation between streams s and s′ at any time t′, tis′ and tfs′ are the initial and final times of stream s′, respectively. Finally, ∆Cs,s′ is the contaminant concentration variation between streams s and s′.
∑ R ( ∫ q (t)C (t) dt + ∫ q (t)C (t) dt + ∫ m S
Cs′(t) )
t
τ
s,s′
s)1
tis
s
s
tis′
s′
tis
s,s′(t)
dt
∫ q (t) dt + ∫ q (t) dt t
τ
|∑
)
τ
s′
tis
s
tis′
s′
S
∀s,s′
Rs,s′ > 0,
qs′ > 0,
τ ) min(t,tfs)
tis′ e t e tfs′,
s)1
(2)
S
Cs'(t) )
∑R
s,s′(Cs(t) + ∆Cs,s′),
s)1
∀s,s′, t
qs′ > 0,
|
S
∑R
s,s'
> 0,
s)1
tis′ e t e tfs′
(3)
One of the defining characteristics of the method by Almato´ et al.47,48 is that the authors define a variable which accounts for the fraction of a stream that is either sent to a tank from a process or from a tank to a process. There are various constraints derived to define each fraction. This is an important decision variable of the model, which is determined during the optimization. Puigjaner et al.59 implemented the method developed by Almato´ et al.47,48 in a prototype software package. A user interface was designed whereby all of the required input data could be entered and saved in a database. The data are then used to model the production process for a given demand and production sequence. The software package includes a scheduling module and a water minimization module. The scheduling module is used to provide the initial schedule for the water minimization problem. The solution from the water minimization problem is displayed graphically in the form of a Gantt chart, stream chart, tank-to-stream assignment diagram, and various diagrams showing water targets, tank levels, and concentrations. This is currently one of the only such packages described in the literature. It is interesting to note that the constraints proposed by later researchers49–52,58 are relatively simpler than those proposed by Almato´ et al.,47,48 in which no integral or difference forms are found. For most cases, each balance is done within each operational time interval (except from the works of Shoaib et al.,52 Rabie and El-Halwagi,53 and Ng et al.54). Note that the work of Kim and Smith49 is developed for semicontinuous type operations. Note also that water from storage is not explicitly taken into account in the mass balances. Instead, the mass balances are formulated as direct/indirect reuse between two operations not being expressly differentiated. An extra constraint is formulated, which assigns a storage tank to reuse streams where the water is produced at a time earlier from when it is PP is the binary variable used. This is given in eq 4, where Y n,n′ showing water reuse from operation n to operation n′, Y nST is the binary variable showing the existence of a storage tank to operation n, and T n′S and T nE are the starting and ending times of operations n′ and n, respectively. PP Yn,n′ - YST n e 0
S ∀n , where Tn′ g TEn
(4)
In the formulation proposed by Majozi,50 the time horizon is represented by a number of time points, and each variable within the formulation is being defined at each time point. In the formulation, a unit will start processing at a time point p and finish processing at time point p + 1. It is assumed that water
is used at time point p and produced at time point p + 1, that is, a truly batch operation. Therefore, an inlet water and contaminant balance is done at time point p and an outlet water balance is done at time point p + 1. The water and contaminant balances over a unit, however, comprise the inlet variables at time point p - 1 and outlet variables at time point p. The contaminant balance over a unit is presented in eq 5. In eq 5, mp(sout,j,p) is the amount of state sout produced by unit j at time point p, Cout(j,p) is the outlet concentration from unit j at time point p, mu(sin,j,p - 1) is the amount of state sin,j used at time point p - 1, Cin(j,p - 1) is the inlet concentration into unit j at time point p - 1, M(j) is the mass load of contaminant in unit j, and y(sin,j,p - 1) is the binary variable associated with usage of state sin,j at time point p - 1. Worthy of note is the fact that the meaning of state as used in this context dovetails with that encountered in state-task-network (STN), that is, it refers to any stream in the network. mp(sout,j,p)Cout(j,p) ) mu(sin,j, p - 1)Cin(j, p - 1) + M(j)y(sin,j, p - 1) ∀j ∈ J, sin,j ∈ Sin,j, sout,j ∈ Sout,j, p ∈ P, p > p1
(5)
As described above, the method of Majozi50 assumes that water is used or produced at a time point. This is similar to the methods presented by Chen et al.,55,56 but different from those presented by Almato´ et al.,47,48 Kim and Smith,49 Li and Chang,51 and Tokos and Pintaricˇ,58 in which water is constantly flowing during a unit’s operation. The mass balance constraints formulated by later researchers51–54,57 take a similar form based on the fixed flow problem. In this case, water and contaminant balances are done around each water sink and source. Li and Chang51 proposed mass balances for four sets of units, namely, water sources, water users, buffer tanks, and water sinks. The mass balances for each are different. The water and contaminant balances for sinks and sources are derived in such a way that they hold for each time interval within the time horizon. This is shown in eq 6 for a water source. For water users and buffer tanks, the water and contaminant balances follow the same way in substance. However, the mass balances over a water user are derived independent of the time intervals. The water going to and from each water user is summed over the subset of time intervals in which they occur. This is shown in eq 7 for a water balance source represents the flow rate over a water user. In eqs 6 and 7, f sa,t of freshwater from source sa during time interval t, fssa,mx,t is the flow rate in the split branch from source sa to mixing node in out and f u,t′ are the mx, DT is the duration of a time interval, f u,t inlet and outlet flow rates of water user u during time interval t, and DVu represents the water loss in the operation of water user u. source fsa,t )
∑
fssa,mx,t
∀sa ∈ SA, t ∈ T
(6)
mx∈MX
DT
∑
t∈TCu
in fu,t ) DT
∑
out fu,t′ + DVu
∀u ∈ U
(7)
t'∈TDu
Later works by Shoaib et al.,52 Rabie and El-Halwagi,53 and Ng et al.54 are built on the same format. Because the main assumption of the work is that water from sources in an earlier batch can be reused/recycled only in a later batch via indirect integration (exceptional for Ng et al.,54 who utilized an inspection to remove the excessive storage tank to allow for direct integration), the work is limited to cyclic operation.
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Typical water balances for source and sink are presented in eqs 8 and 9, respectively. In eq 8, FSRi is the total flow of source i, FSRi,STk is the flow from source i to storage tank k, and FWW,SRi is the wastewater flow from source i. In eq 9 FSKj is the flow demand for sink j, FSTk,SKj is the flow from storage tank k to sink j, and FFW,SKj is the freshwater flow to sink j. FSRi )
∑F
FSKj )
∑F
STk
STk
SRi,STk
STk,SKj
+ FWW,SRi
SRi ) 1,2, ..., Ni
(8)
+ FFW,SKj
SKj ) 1,2, ..., Nj
(9)
Equation 10 ensures that the mixture of the individual sources (including those from storage water, each with concentration CSTk) for the sink should satisfy the impurity constraints of the latter (CSKj).
∑F STk
STk,SKjCSTk
e FSKjCSKj
STk ) 1,2, ..., Nk
(10)
4.1.3. Network Structure Considerations. Apart from mass balance constraints, network structural constraints are also considered by various authors. Almato´ et al.48 do not specifically formulate constraints that restrict the number of storage tanks or number of pipelines. Kim and Smith49 propose a range of constraints to limit the numbers of inlet and outlet streams for each operation and control the compulsory or unacceptable matches. Li and Chang51 propose a number of constraints that simplify the structural complexity of the resulting reuse network. Structural considerations of the work by Shoaib et al.52 and Rabie and El-Halwagi53 are mainly on the number of storage tanks and piping connections. Majozi,50 however, does not consider any structural constraints. Chen et al.55 used the concept of fictitious contaminant to systematically address forbidden matches, especially for the case of forbidden reuse and/or recycle between assigned water-using tasks to prevent probable pollution or operational problems. Chen et al.56 also tackled the subject of forbidden water transfers between assigned water-using units by including related logical constraints into the mathematical model. Generally, once all of the constraints are formulated, the objective function must be defined. 4.1.4. Objective Function. The objective function used by each method is different, and in most methods the objective function can take on numerous forms depending on the problem and the emphasis of certain aspects in each problem. In most cases, they correspond to the minimization of costs (e.g., operating, fixed, and/or total costs), water flows, storage capacity, network connections, etc. The methods by Almato´ et al.,47,48 Kim and Smith,49 Li and Chang,51 and Rabie and ElHalwagi53 present at least one objective function that minimizes the overall cost of the water recovery system. The overall cost accounts for the costs of the water reuse network, freshwater, and intermediate storage tanks. Majozi50 does not present any such objective function because structural considerations are not taken into account in the method. Almato´ et al.48 present four different objective functions. The first one is the minimization of freshwater demand. The second one is the minimization of water cost, which is the sum of the freshwater cost and wastewater treatment cost. The third one is the minimization of water and utility costs, and the final one is the minimization of overall cost. The works of both Kim and Smith49 and Tokos and Pintaricˇ58 use only one objective function, the minimization of overall cost, which is the
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summation of the freshwater cost, storage tank cost, piping network cost, and regeneration cost (found only in Tokos and Pintaricˇ58). Majozi50 considers an objective function for the minimization of freshwater or effluent. Li and Chang51 present an objective function that minimizes the sum of annual freshwater cost, annualized installation costs, and annual treatment costs. The methods presented by both Shoaib et al.52 and Rabie and El-Halwagi53 have multiple objectives, which are solved in three steps. In the first step the objective function is the minimization of freshwater. The objective function in the second step is the minimization of the number of storage tanks. The two methods differentiate in the final step. The third objective function for the work of Shoaib et al.52 is the minimization of the number of interconnections between water sources, tanks, and sinks, whereas that of Rabie and El-Halwagi53 is to minimize the total annual cost (comprising fixed costs for the water storage vessels and the operational costs for freshwater use). Chen et al.55 considered a series of design objectives that included the minimum freshwater consumption, minimum capacity of storage tanks, and minimum amount of connecting flows or least number of connections. Chen et al.56 considered minimizing the freshwater consumption and then the capacity of storage tanks. Due to the fact that different models tend to adopt different objective functions, the comparison of solutions obtained is difficult. Furthermore, even if exactly the same objective function is used, the solution achieved using each method might be different as a result of various assumptions made in the formulation of each model. The final common characteristic of each of the methodologies is the solution procedure. Each method has a different solution procedure. 4.1.5. Solution Procedure. The solution procedure for each method varies significantly. This is as a result of the manner in which the nonlinearities are addressed and the multiple objectives in some formulations. Almato´ et al.47 use a two-step solution method. The first step involves a heuristic, which finds an initial tank-to-stream assignment. This provides an initial solution for the second step, which is an optimization step by which an NLP is solved. In subsequent work, Almato´ et al.48 consider not only the above procedure but also a simulated annealing solution procedure. In both situations the global optimality of solutions is not guaranteed. The solution strategy employed by Kim and Smith49 is very different from that adopted by the previous methods. The authors note that the concentration variables in the bilinear terms are likely to be close to their maximum levels in the final solution. An MILP-LP decomposition of the original MINLP problem is proposed to find an initial solution for the MINLP. The MINLP is converted to an MILP by fixing the outlet concentrations, hence eliminating the bilinear terms. The solution to the MILP gives the connections between the various units and tanks; this information and the values of the flow rates are then substituted into the original MINLP to give an LP. The values of the concentrations determined by the LP are then used in the MILP. The MILP and LP iterate until the required convergence criteria are met. The solution from this is then used as an initial solution for the MINLP. The formulation presented by Majozi50 is an MINLP, due to bilinear terms in the contaminant mass balances. To deal with this, an optimization criterion is used to fix concentrations at their respective maxima. Indeed, the fixing of concentrations
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at their respective maxima is a necessary condition, albeit not sufficient, for minimum water use in any network (Savelski and Bagajewicz46). The resulting nonlinearity that takes on the form of a continuous variable and a binary variable is then linearized exactly using Glover transformation.60 Li and Chang51 proposed a simple solution method. To find an optimal solution, the problem is solved multiple times with randomly selected initial points. As mentioned earlier, the method proposed takes into consideration water sources, water users, buffer tanks, and water sinks. Hence, the superstructure used is built using a number of rules that dictate which operations can be connected and which operations cannot be connected. Any reuse that occurs is done exclusively via storage vessels, which are also depicted in the superstructure. The authors discretize the time horizon into a number of time slots of equal length. This is a major drawback because this technique will lead to a larger number of binary variables and, hence, longer CPU times. Furthermore, there is little flexibility in the resulting reuse network because the cost was minimized on the basis of a water target for a fixed schedule. Shoaib et al.52 proposed a three-step algorithm in solving the optimization problems. In the first step, an LP model is solved to determine the minimum flow for the batch water network that ensures a globally optimal solution. However, in the second and third steps, the minimization of storage capacity and network connections leads to an MINLP problem. Shoaib et al.52 employed a linear relaxation technique proposed by McCormick.61 The solution to the resulting MILP provides a lower bound to the problem, and if this lower bound satisfies the original constraints, then it is seen as the global solution to the original problem. The method proposed by Rabie and El-Halwagi53 also comprises three steps in model solving. In the first step the resulting model is an NLP, and the resulting model in the second step is a MINLP. The solution algorithm proposed in the third step involves the iterative solving of the models proposed in the first two steps. In each iteration, the number of storage vessels is decreased by two (because it is assumed that a number of tank pairs are in service for storing water) until there are zero storage vessels. At this point one can select the solution with the lowest total annual cost from all of the iterations. The design problems in Chen et al.55 were formulated as NLPs in the steps of minimizing the freshwater consumption, capacity of storage tanks, and amount of connecting flows. When taking the least number of connections into account, binary variables are necessary to identify the existence of connections and the problem becomes an MINLP. However, no special solution strategy was proposed, and the design problems are directly solved by a superior solver. In Chen et al.,56 the design problem was formulated as an NLP if there are no constraints on water reuse/recycle or as an MINLP when forbidden water reuse/recycle between designated units is considered. The design problem is solved by conventional solvers with feasible initial solutions. The main limitation of the methods described above is that they are based on the optimal schedule being known a priori. The methods are therefore more suited to situations in which the schedule rarely changes. Furthermore, due to the fact that all of the methods synthesize a batch water network based on given schedules, the flexibility of the network is reduced. The various methods attempt to account for some variation by including storage tanks. Apart from the common limitation mentioned above, each method has its specific limitations. The method proposed by
Shoaib et al.52 is limited to plants operating in a cyclic manner, as is the method by Rabie and El-Halwagi.53 Hence, they do not handle cases with single-batch operation (even though this is rare in industrial practice). The solution method proposed by Kim and Smith49 can result in long solution times due to the solution strategy. As aforementioned, the main drawback of the method proposed by Li and Chang51 is the particularly large resulting models, due to the even discretization of the time horizon. The methods do, however, have the advantage of being able to deal with complex process-related constraints, which is an inherent advantage of mathematical formulations. It must be mentioned that the methods by Kim and Smith,49 Li and Chang,51 Rabie and El-Halwagi,53 and Chen et al.55,56 deal with the possibility of multiple contaminants in the system. Kim and Smith,49 Rabie and El-Halwagi,53 and Chen et al.55,56 define a distinct set of contaminants that accounts for all of the contaminants present in the system. Li and Chang51 define a similar set. However, the authors define this as a contaminant index. Almato´ et al.47,48 and Majozi50 do not deal with multiple contaminants, whereas Shoaib et al.52 consider only a key contaminant. 4.2. Simultaneous Water Minimization and Scheduling. The second group of mathematical optimization techniques for batch water network synthesis is discussed in this section. The formulations determine the minimum water flows and/or network cost and the corresponding production schedule. Hence, time is treated as a variable in this category. Determining the optimal schedule that will achieve the minimum water target poses many difficulties. First, methodologies that determine the minimum flow and/or cost and the corresponding schedule for the network must be based on an underlying scheduling framework. Much work in the past three decades has gone into formulating rigorous scheduling methodologies that are robust with low solution times. Early methodologies for batch water networks were based on scheduling methodologies, which were difficult to solve due to the resulting size and complexity of the formulation. Second, the resulting size of a water minimization formulation can take on large proportions. This results in extremely long solution times. Finally, determining a new batch water network for each schedule has some practical limitations due to excessively long solution times that can affect negatively production timelines. However, the attractiveness of these methods lies in the fact that the global minimum water flow and/or cost can be determined for a variety of operational requirements. Furthermore, these methods can easily accommodate the situation when the schedule is known and still determine the corresponding minimum water flow and/or cost solution. In the following subsections, these approaches are reviewed on the basis of different research groups. 4.2.1. Method of Majozi and Coworkers. Using the scheduling technique developed by Majozi and Zhu62 as a foundation, Majozi63 proposed a method for batch water network synthesis. This forms the basis for a number of subsequent works.64–67 The scheduling method proposed by Majozi and Zhu62 is based on the time horizon being represented by a number of time points, the position of which is unknown. The scheduling method is further based on a novel representation of a process, namely, the state sequence network (SSN). The SSN representation is different from the previous state task network (STN) in that the SSN represents only states, and tasks are implicitly included in the representation. The scheduling formulation,
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therefore, has a low number of binary variables due to the SSN representation and the use of a concept called effective states. The method of Majozi63 is derived for four scenarios. The first scenario assumes that the outlet concentration of each unit is fixed at its maximum value. This means that during the optimization the flow rate through each unit is free to vary below a defined upper limit. The second scenario assumes that the flow rate through each unit is fixed, and the inlet and outlet concentrations are free to vary during the optimization within predefined maximum values. The final two scenarios are the same as scenarios one and two but with a central reusable storage vessel being included in both scenarios, respectively. The method is based on a superstructure which, for scenarios one and two, is similar to that presented by Majozi50 given in Figure 9. The superstructure for scenarios three and four, where there is a central reusable water storage vessel, is very much the same as the previous, except a storage vessel and the corresponding connections are also included. In the third and fourth scenarios, both direct and indirect reuses between the various units are considered. Indirect recovery through the storage vessel will occur only when the time at which water is available for reuse/recycle and the time at which it is required by a unit are different. This is different from other methods, by which reuse can exclusively take place through a storage vessel.47,48,51,52 The main advantage of the method lies in the fact that the production schedule is determined together with the minimum water targets. This means that both scheduling and water minimization comprise the same modeling framework, which allows time to be treated as a variable rather than a parameter. Another advantage of the method is that the batch water network is determined during the optimization. The limitation of this early approach is that it is limited to single contaminant. The method was further extended to determine the optimum size of reusable water storage64 as well addressing systems that are characterized by multiple contaminants and multiple reusable storage units.66,68 In the work for multiple contaminant systems,66 water-using processes can roughly be divided into three groups, depending on the profile of contaminants present in the water streams. In the first group, different wastewater streams contain single but different contaminants. In the second group, each wastewater stream contains the same set of contaminants. The third group is a mixture of the first and second groups, that is, there are wastewater streams with single but different contaminants as well as wastewater streams with multiple contaminants. Each of these groups requires different storage options. The effect of the multiple storage vessels and multiple contaminants is shown in eq 11, which is an inlet contaminant balance into a unit. In this constraint, cin(j,c,p) is the inlet concentration into unit j of contaminant c at time point p, fu(j,p) is the amount of water into unit j at time point p, fr(j′,j,p) is the amount of water recycled between unit j′ and unit j at time point p, and cout(j′,c,p) is the outlet concentration of the water from unit j′ of contaminant c at time point p. Furthermore, fsout(u,j,p) is the amount of water going to unit j from storage vessel u at time point p and csout(u,c,p) is the concentration of contaminant c of the water inside a storage tank u at time point p. cin(j,c,p)fu(j,p) )
∑ f (j′,j,p)c
out(j′,c,p)
r
+
j′
∑ fs
out(u,j,p)csout(u,c,p)
u
∀j,j′ ∈ J, p ∈ P, c ∈ C, u ∈ U
(11)
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Due to bilinear terms arising from eq 11, the overall model is inherently nonlinear and nonconvex, hence the failure to ascertain global optimality. To reduce computational intensity, some of these nonlinearities were linearized using the relaxation-linearization method proposed by Quesada and Grossman.69 Operations in which water reuse/recycle could be exploited with the ultimate goal of near-zero effluent generation have been considered by Gouws and Majozi.65–67 Their method is based on a unique type of operation whereby wastewater generated is reused as raw material for a subsequent batch of product. In this type of operation wastewater is generally generated from cleaning operations and consequently contains residue of the product produced in the unit prior to the cleaning task. In addition to significant reduction in effluent, water reuse/recycle in this fashion allows for the recovery of valuable product residue. 4.2.2. Method of Cheng and Chang.70 Another work that considers simultaneous water minimization with scheduling problem is reported by Cheng and Chang.70 The optimization of the batch schedule, water reuse/recycle, and wastewater treatment network design are done in an integrated form. This allows one to optimize all of the water network elements while taking into account the interactions among them. The scheduling framework used by Cheng and Chang70 is based on the discrete representation of the time horizon. In this representation the time horizon is divided into a number of time slots of equal duration. At the start and end of each time slot, a time point is defined. Three binary variables are defined in the scheduling module, with the first indicating whether a unit starts operating at a time point, the second whether a unit is processing at a time point, and the third if the unit stops operating at a time point. Allocation constraints ensure that a unit is starting, operating, or ending its operation at a given time point. The objective function for the scheduling module is a function of profitability, given as the difference between the revenue from product and the cost of raw material. Note that the batch water network design module presented by Cheng and Chang70 is very similar to that of Li and Chang.51 As with the latter, a number of sets is introduced for sinks, sources, water users (units that both consume and produce water), buffer tanks, equipment, tasks to be performed, and pollution indices accounting for all of the contaminants in the system. As with all mathematical methods for batch water network, the method proposed by Cheng and Chang70 is based on a superstructure. Due to the wide variety of water sources and sinks considered, the superstructure is developed following a number of rules. These rules ensure that the correct sources, sinks, water users, and buffer tanks are connected to each other. A mixing node or splitting node is placed at each sink, source, water user, and buffer tank (similar to that in Figure 8). Water reuse/recycle is only allowed through buffer tanks, and hence there are no direct connections between the units. The constraints considered by Cheng and Chang70 in the water reuse module can be divided into two rough groups, namely, mass balance constraints and network structure constraints. The first group, the mass balance constraints, comprise water and contaminant balances at the splitting and mixing nodes, respectively. The contaminant balances at each node are done for each pollution index. Equation 12 is a water balance over a buffer tank, and eq 13 is the contaminant balance over the same buffer tank. In eq 12,Vb,t is the volume in and of water in buffer tank b at the end of time interval t, f b,t out f b,t are the inlet and outlet flow rates of buffer tank b during
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Figure 15. Batch water network synthesized using the method of Kim and Smith.49
Figure 14. Superstructure of Li et al.72
time interval t, respectively, and DT is the length of the time in out and c b,k,t represent the values of interval. In eq 13, c b,k,t pollution index k in the input and output streams of buffer tank b at the end of time interval t, respectively. in out Vb,t ) Vb,t-1 + (fb,t - fb,t )DT
b ∈ B,k ∈ K,t ∈ T
(12) out in in out out Vb,tcout b,k,t = Vb,t-1cb,k,t-1 + (fb,tcb,k,t - fb,t cb,k,t)DT b ∈ B,k ∈ K,t ∈ T
(13)
The second group of constraints considered are constraints dealing with the structure of the batch water network. These constraints are identical to those proposed by Li and Chang,51 which have been discussed earlier. To optimize the schedule and the batch water network simultaneously, Cheng and Chang70 assume that the amount of water consumed or generated is proportional to the amount of material produced. The authors then combine the two objective functions, the objectives from the scheduling module and water network module, by subtracting the water recovery objective function from the scheduling objective function. Cheng and Chang70 also consider wastewater treatment design in their method, which, however, will not be discussed here. The resulting models from the usage of the method developed by Cheng and Chang70 could reach large proportions due to the discrete representation of the time horizon. Furthermore, the accuracy of the method is dependent on the length of the individual time intervals. Long time intervals might lead to smaller problems, but be less accurate with regard to scheduling, whereas shorter time intervals might be more accurate, but the problems could be intractable. 4.2.3. Method of Dong and Co-workers. One of the latest works on simultaneous water minimization and scheduling is proposed by Dong and co-workers.71,72 The model formulation by Zhou et al.71 is based on the continuous-time representation with an improved state-space superstructure. Besides, a hybrid optimization strategy was developed for the resulting MINLP problem. Li et al.72 later developed another simultaneous water minimization and scheduling framework on a so-called state-time-space superstructure (Figure 14), which was based on the formulation proposed by Ierapetritou and Floudas.73 However, they use a discrete time formulation, which yields a nonconvex MINLP that is solved using a two-step procedure. In the first step, a scheduling problem that fixes the sequence of tasks is solved in the form of an MILP. Once the sequence is fixed, the detailed MINLP problem is solved using an
evolutionary algorithm, which is a stochastic optimization technique. The use of a combined deterministic and stochastic approach renders this contribution unique in the context of batch water network synthesis problem. However, the authors also acknowledge the computational difficulties associated with a problem of this nature. The methods that consider simultaneous water minimization and scheduling all have the major advantage of being able to identify the absolute minimum water flows. However, in certain cases the resulting model can become extremely large, rendering the problem intractable. Example 1 is revisited to compare the solution using methods of the fixed schedule and that with process scheduling. 4.3. Example 1 (Revisited). To demonstrate the advantage of a simultaneous water minimization and scheduling method, example 1 is reconsidered here, with its limiting data given in Table 3. The results of three methodologies will be compared. The first two methods stem from the application of the fixed schedule approaches by Kim and Smith49 and Majozi.50 However, the results are different because they treat water-using processes as semicontinuous and truly batch processes, respectively. In contrast, the third method is based on the simultaneous water minimization and scheduling type of Majozi.63 Using the method derived by Kim and Smith,49 a minimum freshwater flow of 102.5 t is identified. As can be seen from Figure 15, a storage tank with a capacity of 37.5 t is required to achieve the target. The storage vessel is required because the water-using processes are operated as semicontinuous mode, where water is continuously fed and discharged during the processes operation. These results are identical to those determined by the insight-based techniques (see Table 4). If one were to utilize the method of Majozi,50 the freshwater target identified is 107.5 t, with the resulting network shown in Figure 16. As can be seen from the figure, there is only one opportunity for water recovery, which is between processes 2 and 1. It must be noted that because this method is based on truly batch processes, effluent from a unit is available for reuse/ recycle only at the end of a process. On the other hand, applying the method of Majozi63 and discarding the given schedule in Table 3, a minimum freshwater flow of 102.5 t is found, with the resulting network as shown in Figure 17. Even though the water flow is similar to that in Figure 15, no storage vessel is required to achieve the minimum flow in this case. As can be seen from Figure 17, process 3 (P3) starts at an earlier time. This allows the wastewater produced from P3 to be reused in process 1. Comparing the results identified by the three methods reveals that the solution provided by the method proposed by Majozi50
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49
Figure 16. Batch water network synthesized using the method of Majozi.50
and Smith, the wastewater produced by process 3 can be reused by process 1 because this method is based on semicontinuous water-using processes. The freshwater target identified by the method proposed by Majozi63 is the same as that identified by Kim and Smith,49 which is 102.5 t. The solution obtained by the method of Majozi63 demonstrates a very important fact that only if a schedule is free to change can one find the absolute minimum freshwater flow for the network. In the solution provided by the method by Majozi,63 process 3 has started operating at the beginning of the time horizon, thereby allowing the reuse of the wastewater generated by process 3 in process 1. Furthermore, no storage was required to achieve the freshwater target. The above example demonstrates the difference one might get when using a method in which the schedule is fixed and one in which it is flexible. The use of a fixed schedule method or flexible schedule method is, however, dependent on the type of problem under consideration, the operating policies, and the main focus of the plant. 4.4. Overall Comparisons and Comments on Various Mathematical Optimization Techniques. Similar to the case of insight-based techniques, the various mathematical approaches may also be categorized on the basis of the model of water-using operations that they developed the approach, that is, fixed load or flow model. Besides, one may also identify if the developed model is suitable for a semicontinuous or truly batch processes (or both). As mentioned earlier, because the approaches developed for the fixed flow model take a sink/source representation, it works well for both semicontinuous and truly batch processes. A summary of the comparison is found in Table 8. 5. Hybrid Graphical and Mathematical Methodologies
Figure 17. Batch water network synthesized using the method of Majozi.63
gives the highest freshwater flow, that is, 107.5 t. This is, however, due to the fact that the method is based on truly batch processes with fixed schedule type operations. Even with the addition of storage, the freshwater target would not be reduced. This is because the wastewater generated by process 3 is only available during the operation of process 1 and can therefore not be reused. In the solution provided by the method of Kim
Oliver et al.74 derived a hybrid graphical and mathematical approach for batch water network. Oliver et al.74 first treat the batch process as a fully continuous process. Water pinch analysis1 is used to determine the lower bound of the freshwater target and possible reuse/recycle between the various units. To achieve a realistic batch water network, a number a storage tanks are used to overcome timing difficulties. A mathematical model is derived that determines the batch water network using the storage vessels. The mathematical model is derived around a schedule determined a priori. The constraints presented in the mathematical model are in essence more suited to semicontinuous, rather than truly batch processes.
Table 8. Summary of Various Mathematical Techniques fixed load/flow types fixed load
fixed flow
Almato´ et al.47,48 Kim and Smith49 Majozi50 Li and Chang51 Shoaib et al.52 Rabie and El-Halwagi53 Ng et al.54 Tokos and Pintaricˇ58 Chen et al.55,56 Chen et al.57
X X X X
X
Majozi63,64 Gouws and Majozi65–67 Cheng and Chang70 Zhou et al.;71 Li et al.72
X X X X
X X X X
X X X
types of batch processes semicontinuous
truly batch
operating mode of batch processes single batch without storage
single batch with storage
X X X X
X X X
Water Minimization with Fixed Schedule X X X X X X X X X X X X X X X
Simultaneous Water Minimization and Scheduling X X X X X X X X
X X X X X X
repeated batches with storage
X X X X X X X
X X
X X X X
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Apart from methodologies that minimize freshwater in batch processes, there also exist methodologies that minimize waste generated from a batch process. The waste considered by these methodologies consists not only of wastewater but byproduct waste generated from setting up of units for a specific task, etc. 6. Future Directions for Research Synthesis of batch water network will remain an active area of research in the near future. There is still much room for development in both insight-based and mathematical optimization approaches, particularly in the following areas. Incorporation of Source Interception Unit(s). Source interception is commonly used to purify water for further reuse/ recycle (more commonly known as regeneration) and/or for waste treatment prior to final discharge. Significant development has been reported for continuous processes for both insightbased (see the review of Foo30) and mathematical optimization techniques. However, most works in batch water network are still focusing on water reuse/recycle, with several recent contributions that reported regeneration and waste treatment.52,58,70–72 It is expected that approaches developed for continuous processes12–14,23,41,75,76 will be extended into batch processes in the near future. Simultaneous Energy and Water Reduction. Significant works were observed in this area for continuous processes;77–81 however, they have yet to be seen for batch processes. The extension of these approaches into batch processes is foreseen in the near future. Stochastic Techniques. Most mathematical optimization techniques developed thus far are based on deterministic approaches. It is foreseen that stochastic optimization techniques will soon be developed to synthesize a batch water network, particularly in addressing various MINLP or scheduling problems. The recent work of Zhou et al.,71 who used a stochasticbased genetic algorithm to address the inherent nonlinearity of the batch water network problem, is the first of this kind. More works are expected in the near future. Simultaneous Scheduling and Water Minimization in Multiple-Contaminant Multipurpose Batch Plants. As evinced by the foregoing review, most of the work in batch water network has focused on single- and multiple-contaminant systems with a predefined schedule. In situations when scheduling and water minimization are combined within a single framework, production recipes have tended to be serial or multiproduct in nature. The problem that entails multiple contaminants and complex recipes which exhibit multipurpose structure has largely been avoided within a combined water minimization and scheduling framework. Needless to mention, a problem of this nature is of utmost complexity in both modeling and consequent pursuit for optimum solutions. Nonetheless, research in this area still commands attention, because this situation is very commonly encountered in practice. 7. Conclusions Water minimization for batch processes remains a complex problem to be solved. Various methods have been proposed, each with various merits and disadvantages. Due to the large scope of the problem and the various interpretations of a batch processes, methodologies vary significantly. The choice of method to solve a certain problem is very much dependent on the nature of the problem and the operating policy of the process. It is, however, apparent that the insight-based approach (based on pinch analysis) is less flexible than the mathematically based
counterparts. However, they have proven to be valuable in providing good insights to the problem. The methodologies reviewed do, however, achieve their objective of minimizing the amount of freshwater and wastewater for a process. The optimality of the solution is not always guaranteed, because most formulations are highly nonconvex. Furthermore, the practical applicability of the solutions is somewhat limited, due to controllability and constraints on the physical water network. Even though the problems reported in the literature are based on industrial processes, they are always being simplified for ease of model solving. More realistic industrial cases should be developed in order for more practical and reliable models to be developed in the future. Currently, scheduling methodologies being developed are extremely robust in nature and their usage as an underlying formulation for batch water network is promising in the near future. Literature Cited (1) Wang, Y. P.; Smith, R. Wastewater minimization. Chem. Eng. Sci. 1994, 49, 981–1006. (2) Dhole, V. R.; Ramchandi, N.; Tainsh, R. A.; Wasilewski, M. Make your process water pay for itself. Chem. Eng. 1996, 103, 100–103. (3) Olesen, S. G.; Polley, S. G. A simple methodology for the design of water networks handling single contaminants. Chem. Eng. Res. Des. 1997, 75, 420–426. (4) Kuo, W. C. J.; Smith, R. Design of water using system involving regeneration. Process Saf. EnViron. Prot. 1998, 76, 94–114. (5) Hallale, N. A new graphical targeting method for water minimization. AdV. EnViron. Res. 2002, 6, 377–390. (6) El-Halwagi, M. M.; Gabriel, F.; Harell, D. Rigorous graphical targeting for resource conservation via material recycle/reuse networks. Ind. Eng. Chem. Res. 2003, 42, 4319–4328. (7) Manan, Z. A.; Foo, C. Y.; Tan, Y. L. Targeting the minimum water flow rate using water cascade analysis technique. AIChE J. 2004, 50, 3169–3183. (8) Prakash, R.; Shenoy, U. V. Targeting and design of water networks for fixed flowrate and fixed contaminant load operations. Chem. Eng. Sci. 2005, 60, 255–268. (9) Bandyopadhyay, S.; Ghanekar, M. D.; Pillai, H. K. Process water management. Ind. Eng. Chem. Res. 2006, 45, 5287–5297. (10) Bai, J.; Feng, X.; Deng, C. Graphical based optimization of singlecontaminant regeneration reuse water systems. Chem. Eng. Res. Des. 2007, 85, 1178–1187. (11) Feng, X.; Bai, J.; Zheng, X. On the use of graphical method to determine the targets of single-contaminant regeneration recycling water systems. Chem. Eng. Sci. 2007, 62, 2127–2138. (12) Ng, D. K. S.; Foo, D. C. Y.; Tan, Y. L.; Tan, R. R. Ultimate flow rate targeting with regeneration placement. Chem. Eng. Res. Des. 2007, 85, 1253–1267. (13) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Targeting for total water network. 1. Waste stream identification. Ind. Eng. Chem. Res. 2007, 46, 9107–9113. (14) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Targeting for total water network. 2. Waste treatment targeting and interactions with water system elements. Ind. Eng. Chem. Res. 2007, 46, 9114–9125. (15) Linnhoff, B.; Townsend, D. W.; Boland, D.; Hewitt, G. F.; Thomas, B. E. A.; Guy, A. R.; and Marshall, R. H. A User Guide on Process Integration for the Efficient Use of Energy. Rugby: IChemE, 1982. (16) El-Halwagi, M. M.; Manousiouthakis, V. Synthesis of mass exchange networks. AIChE J. 1989, 35, 1233–1244. (17) El-Halwagi, M. M. Pollution PreVention through Process Integration: Systematic Design Tools; Academic Press: San Diego, CA, 1997. (18) Takama, N.; Kuriyama, T.; Shiroko, K.; Umeda, T. Optimal water allocation in a petroleum refinery. Comput. Chem. Eng. 1980, 4, 251–258. (19) Doyle, S. J.; Smith, R. Targeting water reuse with multiple contaminants. Process Saf. EnViron. Prot. 1997, 75, 181–189. (20) Alva-Arga´ez, A.; Kokossis, A. C.; Smith, R. Wastewater minimization of industrial systems using an integrated approach. Comput. Chem. Eng. 1998, 22, s741–s744. (21) Jo¨dicke, G.; Fischer, U.; Hungerbu¨hler, K. Wastewater reuse: a new approach to screen for designs with minimal total costs. Comput. Chem. Eng. 2001, 25, 203–215. (22) Bagajewicz, M.; Savelski, M. On the use of linear models for the design of water utilization systems in process plants with a single contaminant. Chem. Eng. Res. Des. 2001, 79, 600–610.
Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 (23) Gabriel, F. B.; El-Halwagi, M. M. Simultaneous synthesis of waste interception and material reuse networks: problem reformulation for global optimization. EnViron. Prog. 2005, 24, 171–180. (24) Lavric, V.; Iancu, P.; Ples¸u, V. Cost-based design of wastewater network optimal topology. Resour. ConserV. Recycl. 2007, 50, 186–201. (25) Hul, S.; Ng, D. K. S.; Tan, R. R.; Chiang, C. L.; Foo, D. C. Y. Crisp and fuzzy optimization approaches for water network retrofit. Chem. Prod. Process Model. 2007, 2, Art. 6. (26) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Automated targeting technique for single-component resource conservation networks. Part 1: Direct reuse/recycle. Ind. Eng. Chem. Res. 2009, 48, 7637–7646. (27) Ng, D. K. S.; Foo, D. C. Y.; Tan, R. R. Automated targeting technique for single-component resource conservation networks. Part 2: Single pass and partitioning waste interception systems. Ind. Eng. Chem. Res. 2009, 48, 7647–7661. (28) Tan, R. R.; Ng, D. K. S.; Foo, D. C. Y.; Aviso, K. B. A superstructure model for the synthesis of single-contaminant water networks with partitioning regenerators. Process Saf. EnViron. Prot. 2009, 87, 197–205. (29) Bagajewicz, M. A review of recent design procedures for water network refineries and process plants. Comput. Chem. Eng. 2000, 24, 2093–2113. (30) Foo, D. C. Y. A state-of-the-art review of pinch analysis techniques for water network synthesis. Ind. Eng. Chem. Res. 2009, 48, 5125–5159. (31) Douglas, J. Conceptual Design of Chemical Processes; McGrawHill: New York, 1988. (32) Smith, R. Chemical Process Design; McGraw-Hill: New York, 1995. (33) Smith, R. Chemical Process Design and Integration; Wiley: New York, 2005. (34) Seider, W. D.; Seader, J. D.; Lewin, D. R. Product and Process Design Principles: Synthesis, Analysis, and EValuation; Wiley: New York, 2003. (35) Wang, Y. P.; Smith, R. Time pinch analysis. Chem. Eng. Res. Des. 1995, 73, 905–913. (36) Majozi, T.; Brouckaert, C. J.; Buckley, C. A. A graphical technique for wastewater minimization in batch processes. J. EnViron. Manage. 2006, 78, 317–329. (37) Foo, D. C. Y.; Manan, Z. A.; Tan, Y. L. Synthesis of maximum water recovery network for batch process systems. J. Clean. Prod. 2005, 13, 1381–1394. (38) Foo, D. C. Y.; Manan, Z. A.; Tan, Y. L. Use cascade analysis to optimize water networks. Chem. Eng. Prog. 2006, 102, 45–52. (39) Foo, C. Y.; Manan, Z. A.; Yunus, R. M.; Aziz, R. A. Synthesis of mass exchange network for batch processessPart I: Utility targeting. Chem. Eng. Sci. 2004, 59, 1009–1026. (40) Foo, C. Y.; Manan, Z. A.; Yunus, R. M.; Aziz, R. A. Synthesis of mass exchange network for batch processessPart II: Minimum units target and batch network design. Chem. Eng. Sci. 2005, 60, 1349–1362. (41) Kuo, W. C. J.; Smith, R. Effluent treatment system design. Chem. Eng. Sci. 1997, 52, 4273–4290. (42) Liu, Y. J.; Yuan, X. G.; Luo, Y. Q. Synthesis of water utilisation system using concentration interval analysis method (II). Discontinuous process. Chin. J. Chem. Eng. 2007, 15, 369–375. (43) Liu, Y. J.; Yuan, X. G.; Luo, Y. Q. Synthesis of water utilisation system using concentration interval analysis method (I). Non-mass-transferbased operation. Chin. J. Chem. Eng. 2007, 15, 361–368. (44) Castro, P.; Matos, H.; Fernandes, M. C.; Nunes, C. P. Improvements for mass-exchange networks design. Chem. Eng. Sci. 1999, 54, 1649–1665. (45) Chen, C.-L.; Lee, J.-Y. A graphical technique for the design of waterusing networks in batch processes. Chem. Eng. Sci. 2008, 63, 3740–3754. (46) Savelski, M.; Bagajewicz, M. J. On the optimality conditions of water utilization systems in process plants with single contaminants. Chem. Eng. Sci. 2000, 55, 5035–5048. (47) Almato´, M.; Sanmartı´, E.; Espun´a, A.; Puigjaner, L. Rationalizing the water use in the batch process industry. Comput. Chem. Eng. 1997, 21, s971–s976. (48) Almato´, M.; Espun´a, A; Puigjaner, L. Optimisation of water use in batch process industries. Comput. Chem. Eng. 1999, 23, 1427–1437. (49) Kim, J.-K.; Smith, R. Automated design of discontinuous water systems. Process Saf. EnViron. Prot. 2004, 82, 238–248. (50) Majozi, T. An effective technique for wastewater minimization in batch processes. J. Clean. Prod. 2005, 13, 1374–1380. (51) Li, B. H.; Chang, C. T. A mathematical programming model for discontinuous water-reuse system design. Ind. Eng. Chem. Res. 2006, 45, 5027–5036. (52) Shoaib, A. M.; Aly, S. M.; Awad, M. E.; Foo, D. C. Y.; El-Halwagi, M. M. A hierarchical approach for the synthesis of batch water network. Comput. Chem. Eng. 2008, 32, 530–539. (53) Rabie, A. H.; El-Halwagi, M. M. Synthesis and scheduling of optimal batch water-recycle networks. Chin. J. Chem. Eng. 2008, 16, 474–479.
8893
(54) Ng, D. K. S.; Foo, D. C. Y.; Rabie, A.; El-Halwagi, M. M. Simultaneous synthesis of property-based water reuse/recycle and interception networks for batch processes. AIChE J. 2008, 54, 2624–2632. (55) Chen, C.-L.; Chang, C.-Y.; Lee, J.-Y. Continuous-time formulation for the synthesis of water-using networks in batch plants. Ind. Eng. Chem. Res. 2008, 47, 7818–7832. (56) Chen, C.-L.; Lee, J.-Y.; Tang, J.-W.; Ciou, Y.-J. Synthesis of waterusing network with central reusable storage in batch processes. Comput. Chem. Eng. 2009, 33, 267–276. (57) Chen, C.-L.; Lee, J.-Y.; Ng, D. K. S.; Foo, D. C. Y. A unified model of property integration for batch and continuous processes. AIChE J. 2010, 56, 1845–1858. (58) Tokos, H.; Pintaricˇ, Z. N. Synthesis of batch water network for a brewery plant. J. Clean. Prod. 2009, 17, 1465–1479. (59) Puigjaner, L.; Espun˜a, A.; Almato´, M. A software tool for helping in decision-making about water management in batch process industries. Waste Manange. 2000, 20, 645–649. (60) Glover, F. Improved linear integer programming formulations of nonlinear problems. Manage. Sci. 1975, 22, 445–460. (61) McCormick, G. P. Computability of global solutions to factorable non-convex programs, part I. Convex underestimating problems. Math. Prog. 1976, 10, 147–175. (62) Majozi, T.; Zhu, X. X. A novel continuous-time MILP formulation for multipurpose batch plants. 1. Short-term scheduling. Ind. Eng. Chem. Res. 2001, 40, 5935–5949. (63) Majozi, T. Wastewater minimization using central reusable storage in batch plants. Comput. Chem. Eng. 2005, 29, 1631–1646. (64) Majozi, T. Storage design for maximum wastewater reuse in multipurpose batch plants. Ind. Eng. Chem. Res. 2006, 45, 5936–5943. (65) Gouws, J. F.; Majozi, T. Effective scheduling technique for zeroeffluent multipurpose batch plants. ReV. Chim. (Bucharest, Rom.) 2007, 58, 415–418. (66) Gouws, J. F.; Majozi, T. Impact of multiple storage in wastewater minimization for multi-contaminant batch plants: towards zero effluent. Ind. Chem. Eng. Res. 2008, 47, 369–379. (67) Gouws, J. F.; Majozi, T. A mathematical technique for the design of zero-effluent batch processes. Water SA 2008, 34, 291–295. (68) Majozi, T.; Gouws, J. A mathematical optimization approach for wastewater minimization in multiple contaminant batch plants. Comput. Chem. Eng. 2009, 33, 1826–1840. (69) Quesada, I.; Grossmann, I. E. Global optimization of bilinear process networks with multicompnent flows. Comput. Chem. Eng. 1995, 19, 1219–1242. (70) Cheng, K.-F.; Chang, C.-T. Integrated water network designs for batch processes. Ind. Eng. Chem. Res. 2007, 46, 1241–1253. (71) Zhou, R.-J.; Li, L.-J.; Wu, X.; Dong, H.-G. Simultaneous optimization of batch process schedules and water-allocation network. Comput. Chem. Eng. 2009, 33, 1153–1168. (72) Li, L.-J.; Zhou, R.-J.; Dong, H.-G. State-time-space superstructurebased MINLP formulation for batch water-allocation network design. Ind. Eng. Chem. Res. 2010, 49, 236–251. (73) Ierapetritou, M. G.; Floudas, C. A. Effective continuous-time formulation for short-term scheduling. 1. Multipurpose batch processes. Ind. Eng. Chem. Res. 1998, 37, 4341–4359. (74) Oliver, P.; Rodrı´guez, R.; Udaquiola, S. Water use optimization in batch processes industries. Part 1: Design of water network. J. Clean. Prod. 2008, 16, 1275–1286. (75) Galan, B.; Grossmann, I. E. Optimal design of distributed wastewater treatment networks. Ind. Eng. Chem. Res. 1998, 37, 4036–4048. (76) Bandyopadhyay, S. Targeting minimum waste treatment flow rate. Chem. Eng. J. 2009, 152, 367–375. (77) Bagajewicz, M. J.; Rodera, H.; Savelski, M. J. Energy efficient water utilization systems in process plants. Comput. Chem. Eng. 2002, 26, 59–79. (78) Savulescu, L. E.; Sorin, M.; Smith, R. Direct and indirect heat transfer in water network systems. Appl. Therm. Eng. 2002, 22, 981–988. (79) Savulescu, L.; Kim, J.-K.; Smith, R. Studies on simultaneous energy and water minimizationsPart I. Systems with no water re-use. Chem. Eng. Sci. 2005, 60, 3279–3290. (80) Savulescu, L.; Kim, J.-K.; Smith, R. Studies on simultaneous energy and water minimizationsPart II. Systems with maximum re-use. Chem. Eng. Sci. 2005, 60, 3291–3308. (81) Leewongtanawit, B.; Kim, J.-K. Improving energy recovery for water minimization. Energy 2009, 34, 880–893.
ReceiVed for reView January 20, 2010 ReVised manuscript receiVed July 7, 2010 Accepted July 20, 2010 IE100130A
