Targeting for multiple resources in batch processes

Chemical Engineering Science 104 (2013) 1081–1089

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Targeting for multiple resources in batch processes Nitin Dutt Chaturvedi, Santanu Bandyopadhyay n Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

H I G H L I G H T S








Methodology is developed to minimize operating cost using multiple resources in batch processes. Proposed algebraic methodology is based on the concept of prioritized cost of every resource. Prioritized cost of a resource in any time interval depends on the highest pinch quality of all the subsequent intervals. Mathematical results are proved rigorously to gurantee optimum solution. Proposed methodology is applicable to fixed-scheduled, fixed-flow batch process.

art ic l e i nf o

a b s t r a c t

Article history: Received 20 March 2013 Received in revised form 4 August 2013 Accepted 15 October 2013 Available online 22 October 2013

Operating cost of a batch process can be reduced using multiple resources. In this paper, a methodology is proposed to target multiple resources in order to minimize the operating cost of the overall batch process. The proposed methodology is based on the concept of prioritized cost, originally developed for continuous processes. Prioritized cost of a resource depends on the pinch quality, quality of the resource and its cost. The concept of prioritized cost is extended to optimize the operating cost of a batch process. It is proved that to minimize the operating cost a batch process, prioritized cost of a resource in any time interval may not necessarily depends on the pinch quality of that interval. Prioritized cost of a resource in any time interval depends on the highest pinch quality of all the subsequent intervals, including itself. This important result is proved mathematically. The proposed methodology is applicable to fixedscheduled, fixed-flow batch processes involving single quality. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Batch Mathematical modeling Optimization Systems engineering Process integration Multiple resources

1. Introduction Natural resources such as energy, hydrogen, freshwater, cooling water, raw materials, etc. are important elements for any chemical process industry. Conservation and efficient management of these natural resources are important for market competitiveness and overall sustainability. In recent years, various methodologies have been proposed to minimize the resource requirement for continuous processes: limiting composite curve (Wang and Smith, 1994), evolutionary tables (Sorin and Bedard, 1999) water surplus diagram (Hallale, 2002) material recovery pinch diagram (independently by El-Halwagi et al., 2003 as well as Prakash and Shenoy, 2005) source composite curve (Bandyopadhyay et al., 2006) water cascade analysis (Foo et al., 2006b), etc. A graphical method to conserve freshwater in a semi-continuous process was proposed by Wang and Smith (1995b). Majozi et al. (2006) presented a

n

Corresponding author. Tel.: þ 91 22 25767894; fax: þ 91 22 25726875. E-mail address: [email protected] (S. Bandyopadhyay).

0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.10.018

similar methodology to conserve freshwater in pure batch processes. These methodologies are applicable only to mass transfer based operations. Foo et al. (2005) extended applicability of water cascade analysis to cyclic batch process to include non-mass transfer based operations. Liu et al. (2007) developed a timedependent concentration interval analysis (CIA) method to solve the batch water-using system involving both mass and non-mass transfer based operations. Chen and Lee (2008) proposed a graphical technique to deal with hybrid system comprising of both truly batch and semi-continuous operations. Kim (2011) proposed a methodology for semi-continuous batch processes with fixed load operations. The upper as well as the lower bounds on freshwater requirement are set prior to the actual design of the allocation networks (Kim, 2011). Recently, Chaturvedi and Bandyopadhyay (2012) proved that the overall minimum resource requirement for a single batch operation can be determined through sequential transfer of waste from one time interval to the next time interval. On the other hand, the minimum resource requirement can be targeted directly for a cyclic batch operation after collapsing all the time intervals into a single interval. Various

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methodologies for water minimization for continuous and batch processes have been reviewed by Foo (2009) and Gouws et al. (2010) respectively. Multiple resources may be utilized to minimize the operating cost of the process. It is important to note that methodologies proposed for single resource, cannot be applied directly for targeting multiple resources. Various methodologies have been proposed to optimize the use of multiple resources in continuous processes (Wang and Smith, 1995a; Huang et al., 1999; Almutlaq et al., 2005; Jezowski et al., 2006; Shenoy and Bandyopadhyay, 2007; Foo, 2007; Wałczyk et al., 2007; Leewongtanawit and Kim 2008; Liu et al., 2009; Deng and Feng, 2011). Wang and Smith, (1995a) presented a graphical methodology to target multiple resources and multiple contaminants WANs. The methodology is based on limiting composite curve (Wang and Smith, 1994). Almutlaq et al. (2005) proposed another graphical approach to target multiple resources WAN which was based on material recycle pinch diagram (El-Halwagi et al., 2003; Prakash and Shenoy, 2005). It may be noted that economic factors are not included directly in these two methodologies. Jezowski et al. (2006) proposed a linear programming model which accounts for multiple resources of different prices and concentrations along with availability limits. Shenoy and Bandyopadhyay (2007) introduced the concept of prioritized cost to target multiple resources to minimize the operating cost of the overall process. Foo (2007) presented a three-step procedure based on water cascade analysis (Manan et al., 2004) to calculate the minimum pure and impure fresh water sources requirements. Deng and Feng (2011) extended the concept of prioritized costs for property-based water networks with multiple water resources. Liu et al. (2009) proposed a simultaneous procedure for design and targeting WAN with multiple resources. Wałczyk et al. (2007) included multiple resources in their mathematical formulation while optimizing WANs. Leewongtanawit and Kim (2008) incorporated presence of multiple resources in optimizing heat integrated WANs. Recently, Chandrayan and Bandyopadhyay (2013) proposed a generalized methodology to target cost optimal allocation of resources in segregated targeting problems. The methodology adopts concept of prioritized costs to decompose targeting problem into multiple sub-problems. Due to presence of additional time direction in a batch process, these methodologies cannot be applied directly to design and synthesize resource allocation networks (RANs) for a batch process. In a mixed-integer nonlinear programming (MINLP) problem formulation, Li and Chang (2006) have included the possibility of presence of multiple fresh water sources. However, the methodology has not been demonstrated with any illustrative example. In this paper an algebraic methodology is proposed to target multiple resources in order to minimize the operating cost in a batch process. The concept of prioritized cost, originally introduced by Shenoy and Bandyopadhyay (2007) for optimizing multiple resources in a continuous process, is extended for targeting multiple resources in batch processes. The proposed methodology is applicable to fixed-scheduled, fixed-flow batch processes involving single quality. The proposed methodology is proved mathematically to guarantee optimality. Applicability of the methodology is demonstrated with various illustrative examples.

RNr Nr external resources (Crk, Frk,max)

Rk R1

Fsi, qsi SNs Si DNd

S1 Dj D2 Fdj, qdj D1

Time (h) Fig. 1. A schematic of batch process showing demand, sources, resources, etc.










to be less than a predetermined limit qdj. It may be noted that quality follows the inverse scale (Bandyopadhyay et al., 2006). A set of Ns internal sources is given. Each source generates a flow Fsi with a given quality qsi for a fixed interval of time. The flow from an internal source can be reused/recycled to any other internal demand that appears during or at some later time. A set of Nr external resources is also given. Each resource is available during the entire time horizon of the batch process with a quality qrk and a cost per unit flow Crk. The availability of each resource may be limited to a specified maximum of Frk,max. The objective is to develop an algorithm to determine the minimimum operating cost. It may be noted that the proposed objective is to minimize the operating cost and this does not include capital cost of any equipment, piping networks and storage units.

Entire time horizon of the batch process is sub-divided into several time intervals (say I1, I2, I3, …) such that all sources and/or demands must start or end at these time intervals. In other words, neither any demand nor any source is allowed to end or to start in between these time intervals. For a batch process, the overall resource requirement and equivalently the overall waste generation is the sum total of the resource requirements and the waste generations in individual intervals. n

Rk ¼ ∑ RI ik i¼1 n

W ¼ ∑ W Ii i¼1

ð1Þ

ð2Þ

2. Problem definition 3. Analysis of a single batch process The general problem of targeting multiple resources for a fixedscheduled, fixed-flow batch process (Fig. 1) may be given as follows:


A set of internal demands Nd is given. Each demand accepts a flow Fdj in a given fixed interval of time with a quality that has

In a single batch process, sources available in later time interval could not supply to the demands in some prior time intervals. Therefore, it can be concluded that for a single batch process the sources available in interval Ii can be used in an interval Il if and

N.D. Chaturvedi, S. Bandyopadhyay / Chemical Engineering Science 104 (2013) 1081–1089

only if ir l. Initially, batch process containing two intervals is analyzed. Subsequent to this, a batch process containing three intervals is analyzed. These results are generalized for any single batch process and an algorithm for targeting minimum operating cost employing multiple resources is proposed. Chaturvedi and Bandyopadhyay (2012) proposed a methodology for targeting batch RANs with single resource. It has been proved that for a single batch operation, targeting through sequential transfer of waste from one time interval to the next time interval always leads to the overall minimum resource requirement (Proposition 1, as reproduced below) and for cyclic batch all the time intervals may be collapsed as a single interval and the minimum resource requirement can be targeted directly (Proposition 2, as reproduced below). Proposition 1:. In a batch resource allocation network, targeting via sequential transfer of waste profile always leads to the overall minimum resource requirement. Proposition 2:. For targeting a cyclic batch process, all the time intervals may be collapsed as a single interval and the overall minimum resource requirement can be determined for that single interval. Chaturvedi and Bandyopadhyay (2012) observed that, whenever pinch quality of first interval is higher than that of second interval (qPI1 4qPI2), there is no change in overall waste generation due to transfer of waste from interval I1 to interval I2. However, there is a possibility of change of waste profile or redistribution of waste generation. Chaturvedi and Bandyopadhyay (2012) also proved following important result related to the pinch jump with addition and deletion of flows for any RAN: Proposition 3:. In any resource allocation network, addition of flow may lead to pinch jumping towards lower quality index and removal of flow may lead to pinch jumping towards higher quality index. These results are used in this work to prove various results related to multiple resources targeting in a batch process.

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It has been proved by Chaturvedi and Bandyopadhyay (2012) that waste transfer from a higher pinch quality interval to lower pinch quality interval does not affect the resource requirement; however, it may change the waste profile (this possibility is included in next sub-section). Hence, for case 1, the overall operating cost remains undisturbed when wastes of interval I1 is transferred to interval I2 (as pinch quality of interval I1 is higher than the pinch quality of interval I2). Therefore, the minimum operating cost for the entire batch process is sum of the operating cost of individual intervals. Minimum operating cost of an interval can be targeted using the methodology proposed by Shenoy and Bandyopadhyay (2007). For case 2; let R1 and R2 are the two resources (without loss of generality, we can assume that qr1 rqr2). In this case, pinch quality of first interval is lower than that of the second interval, so waste transfer from the first interval to the second interval will affect the overall resource requirement. Introduction of the second resource will change the waste profile of the first interval and this will affect the overall resource requirements and hence, the overall operating cost. Let ‘δ’ amount of resource R2 is added in first interval. The change in waste (ΔWI1) and change in cost (ΔCI1) due to this perturbation in the first interval are determined to be:

ΔW I1 ¼ δ

qr2  qr1 qPI1  qr1 

ΔC I1 ¼ C r2 δ þ C r1 δ 1 

ð3Þ qr2  qr1 qPI1  qr1

 ð4Þ

Eqs. (3) and (4) can be derived based on the algebraic expression developed in Pillai and Bandyopadhyay (2007). Similarly, change in cost of second interval (ΔCI2) due to the extra waste transfer is expressed as    q q qr2 qr1 ΔC I2 ¼  C r1 δ PI1 r1 ð5Þ qPI2  qr1 qPI1 qr1 Addition of resource R2 in the first interval is beneficial if only if the overall change in operating cost (ΔC) is negative. Using Eqs. (4) and (5), this condition may be expressed as

3.1. Batch process containing two intervals

C r2 C r1 r qPI2  qr2 qPI2  qr1

Consider a batch process having only two intervals I1 and I2. Let qPI1 and qPI2 be the individual pinch qualities of these two intervals when solved using sequential transfer of waste profile (Proposition 1, as described in Chaturvedi and Bandyopadhyay, 2012) considering the purest resource only. There could be two possible cases according to pinch qualities of two intervals (Fig. 2).

Exactly the same expression can be obtained for cost optimal introduction of R2 in the second interval. The quantity C ri =ðqP  qri Þ in Eq. (6) is called the prioritized cost of a resource (Shenoy and Bandyopadhyay, 2007). Prioritized cost of a resource is proportional to its actual cost and inversely proportional to the difference between the pinch quality and the quality of the resource. It is interesting to note that Eq. (6) is independent of qPI1. This implies that cost prioritization of resources for both intervals I1 and I2 are governed according to pinch quality of interval I2. This observation can be summarized as follows:

Case 1:qPI1 Z qPI2 Case 2:qPI1 r qPI2

Pinch quality

I1

I2

Case 2

ð6Þ

Lemma 1:. Minimum operating cost of a batch process containing two intervals can be determined via cost prioritization of resources according to the pinch quality of last interval whenever the pinch quality of the last interval is greater than that of the first interval, otherwise cost prioritization of resources should be done independently.

Case 1, 2 3.2. Batch process containing three intervals

Case 1

Time Fig. 2. Schematic of batch process containing two time intervals showing various possible cases based on pinch quality.

For a batch process containing three time intervals, there could be six possible cases according to pinch qualities of individual time interval (Fig. 3). Case 1:qPI3 ZqPI2 ZqPI1 Case 2: qPI2 Z qPI3 Z qPI1

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I1

I2

I3

Pinch quality

Case 1, 5 Case 1, 2, 3 Case 2 Case 1-6 Case 6 Case 4, 5, 6 Case 3, 4 Time Fig. 3. Schematic of batch process containing three time intervals showing various possible cases based on pinch quality.

Case Case Case Case

3: 4: 5: 6:

qPI2 ZqPI1 ZqPI3 qPI1 ZqPI2 ZqPI3 qPI3 ZqPI1 ZqPI2 qPI1 ZqPI3 ZqPI2

For case 1, arguments, similar to the one for case 2 of batch process with two intervals, prove that pinch quality of third interval I3 governs the cost prioritization of various resources. For cases 2 and 3, pinch quality of second interval is the highest. Therefore, for first two intervals, cost prioritization of resources to be calculated based on the pinch quality of I2 (Lemma 1). As, qPI2 ZqPI3, waste transfer of from I2 to I3 does not affect the overall minimum cost (equivalent to case 1 of batch process with two intervals). To summarize, cost prioritizations for I1 and I2 are to be determined based on qPI2 and cost prioritizations for I3 are to be determined based on qPI3. In case 4, waste transfer from one interval to the next does not affect resource requirements as the pinch qualities are in descending order (Chaturvedi and Bandyopadhyay, 2012) and hence, cost prioritizations for each interval to be determined independently. In case 5, qPI3 is the highest; hence, waste generated from I1 and I2 can be utilized in I3. There are two ways waste can be transferred from I1 to I3: (a) waste of I1 is transferred directly to I3 (direct transfer) and (b) waste of I1 is first transferred to I2 and then to I3 (sequential transfer). It may be noted that qPI1 ZqPI2, waste from I1 cannot reduce the waste generation in I2. However, there is a possibility of waste profile change without any change in overall waste generation (Chaturvedi and Bandyopadhyay, 2012). To study the effect of waste profile change, let δI1 amount of flow is extra generated (qPI1) in I1, and gets distributed in two different sources, when transferred from I1 to I2. Let flow δu is generated at higher quality (qu) and δlo flow is generated at lower quality (qlo). Following equations can be written based on mass balances.

δI1 ¼ δu þ δlo

ð7Þ

qPI1 δI1 ¼ qu δu þ qlo δlo

ð8Þ

Change in cost between direct transfer (ΔCD) and sequential transfer (ΔCS) are calculated using expression of cost change developed in Eq. (4) as per various cases and their difference is expressed as follows. Detailed derivation of the following equation is not reported due to brevity.    ðq q Þðq  q Þδ C r2 C r1 ΔC D  ΔC S ¼  PT1 lo u PT3 T1  qu  qlo qPT3  qr2 qPT3 qr1 ð9Þ

From Eq. (9), it can be observed that reduction in cost for sequential transfer is higher than the reduction in cost for direct transfer. Therefore sequential transfer should always be preferred, which implies that for this case extra waste generated in first interval should be transferred to third interval via its redistribution. Similar to case 1, pinch quality of third interval I3 governs the cost prioritization in this case also. For case 6, qPI1 is the highest; hence, waste generated from I1 cannot be utilized in subsequent intervals. On the other hand, waste generated from I2 can be utilized in I3. However, waste of I1 can get redistributed in I2 (Chaturvedi and Bandyopadhyay, 2012) and may lead to reduction in overall operating cost (equivalent to case 5). Therefore, for case 6, I1 is solved for multiple resources and the waste is transferred to I2. For the remaining intervals, cost prioritizations to be calculated based on qPI3 (Lemma 1). The above discussion can be summarized as follows. In cases when pinch quality of either first or second interval is the highest, the problem may be decomposed in to two sub-problems: up to highest pinch quality interval and the rest. Cost prioritization of resources for these two sub-problems can be determined independently (applying Lemma 1) and the overall problem may be solved through sequential waste transfer. On the other hand, when pinch quality of the third interval is the highest, following lemma can be stated. Lemma 2:. For a batch process containing three time intervals, overall minimum operating cost can be determined via cost prioritization of resources according to the pinch quality of last interval whenever pinch quality of the last interval is the highest. Combining Lemmas 1 and 2, along with general observations made during proving Lemma 2, following generalized result can be stated. Theorem 1:. Cost prioritization of resources for an interval is to be determined based on the highest pinch quality of all the subsequent intervals, including itself. Proof: to prove above theorem, the principle of induction is applied. It may be noted that the theorem is true for batch processes involving two (Lemma 1) and three time intervals (Lemma 2). Let us assume that Theorem 1 is true for any batch process with n–1 time intervals. Let us consider a batch process with n time intervals and pinch quality of every interval is calculated using only the purest resource. There are two cases possible: (A) pinch quality of the last interval is highest, and (B) pinch quality of some other interval is the highest. For case (A), as the pinch quality of the last interval is the highest, arguments, similar to the one for case 2 of batch process with two intervals, prove that pinch quality of the last interval governs the cost prioritization of various resources. For case (B), the overall problem can be divided in to two sub-problems: one up to the interval of highest pinch quality and the remaining intervals. These two sub-problems can be solved independently and Theorem 1 is applicable as each one of them contains less than n  1 intervals (by assumption). This proves Theorem 1. & 4. Targeting algorithm Based on above theorem and lemmas, following algorithm is proposed to target the minimum operating cost for a batch process with a given schedule. Fig. 4 shows the flow chart of the proposed algorithm. 1. Entire time horizon of batch process is sub-divided into several time intervals (say I1, I2, I3…Im), such that all sources and/or demands must start or end at the end points of these time

N.D. Chaturvedi, S. Bandyopadhyay / Chemical Engineering Science 104 (2013) 1081–1089

intervals. In other words, neither any source nor any demand is allowed to end or start in between these time intervals (as defined by Chaturvedi and Bandyopadhyay, 2012).

Start Division into time intervals Targeting with purest resource

Identification of time interval (Ihigh) having highest pinch quality (qPhigh). Prioritization of resources for the time intervals prior to Ihigh according to pinch quality of Ihigh (qPhigh).

Targeting of resources according to their controlling pinch quality for the time intervals up to Ihigh Identification of highest pinch quality interval in remaining set of intervals Is last interval reached?

No

Yes Total resource requirement= Sum of individual resource requirements of each interval

End Fig. 4. Flow chart for targeting operating cost with multiple resources.

Table 1 Limiting water data for example 1. Source

Concentration (ppm)

Duration (h)

Flow (t)

S1 S2 S3 S4

12 15 25 10

0.0–1.0 1.0–2.0 2.0–3.0 3.0–4.0

7 9 20 10

Demand

Concentration (ppm)

Duration (h)

Flow (t)

D1 D2 D3 D4 D5

5 11 12 15 5

0.0–1.0 0.0–2.0 1.0–2.0 2.0–3.0 3.0–4.0

2 10 5 10 5

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2. Calculate pinch quality (qPIi) of each interval using the purest resource following sequential transfer of waste profile (Chaturvedi and Bandyopadhyay, 2012). It may be noted that the pinch quality for each interval may be calculated using any of the proposed methodologies: limiting composite curve (Wang and Smith, 1994), evolutionary tables (Sorin and Bedard, 1999), water surplus diagram (Hallale, 2002), material recovery pinch diagram (El-Halwagi et al., 2003; Prakash and Shenoy, 2005), source composite curve (Bandyopadhyay et al., 2006), water cascade analysis (Foo et al., 2006b), etc. 3. Identify the time interval with highest pinch quality (say Ihigh). Determine the prioritized cost of resources for the intervals prior to Ihigh according to the highest pinch quality (qPhigh). For minimum operating cost, the acceptable resources will form a sequence such that their qualities are in increasing order while their prioritized costs are in decreasing order. This may be called prioritized sequence. 4. Introduce the resources according to prioritized sequence using the methodology proposed by Shenoy and Bandyopadhyay (2007) for this set of intervals. It should be noted that, due to the introduction of an intermediate resource, the pinch point of Ihigh may jumps to a lower quality. In such a case, the prioritized cost for the new resource has to be calculated based on the new pinch quality. If the prioritized cost is still less than that of other purer resources, the algorithm may be continued with the new pinch point. However, if the prioritized cost increases due to the pinch jump, then the waste flow should be adjusted such that the waste composite line passes through both the original pinch point and the new pinch point (Shenoy and Bandyopadhyay, 2007). 5. Consider the next set of intervals and repeat steps 3 and 4 till last interval. Total resource requirement is sum individual resource requirement of each interval (Eq. (1)). It has been proved that the targeting procedure for a cyclic batch process is equivalent to that of a continuous process (Foo et al., 2005; Chaturvedi and Bandyopadhyay, 2012). Therefore, targeting multiple resources for a cyclic batch process can be carried out applying the procedure proposed by Shenoy and Bandyopadhyay (2007) after collapsing all the time intervals into single time interval.

5. Illustrative examples

5.1. Illustrative example 1

Table 2 Resource data for example 1.

FW1 FW2 FW3

The proposed algorithm may be applied to minimize operating cost in different batch RANs. Applicability of the proposed algorithm is demonstrated through various illustrative examples.

Quality (contaminant conc; ppm)

Cost ($/t)

0 10 20

45 25 15

The limiting data for this example are given in Table 1 and resource specifications are listed in Table 2. The four sources and four demands of this example can be categorized in four time intervals (see Table 3). Initially, targeting is carried out using purest resource (FW1) applying sequential transfer of waste profile. The calculated

Table 3 Various time intervals for examle 1. I1(0–1.0 h)

I2(1.0–2.0 h)

I3(2.0–3.0 h)

I4(3.0–4.0 h)

Conc. (ppm)

Flow (t)

Conc. (ppm)

Flow (t)

Conc. (ppm)

Flow (t)

Conc. (ppm)

Flow (t)

12 11 5

7 5 2

15 12 11

9 5 5

25 15

20  10

10 5

10 5

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resource requirement, operating cost and pinch quality for each interval are shown in Table 4. The operating cost is calculated to be $408.6. The pinch quality interval I3 is highest; prioritized cost of resources for intervals I1, I2 and I3 should be governed according pinch quality of I3 (Theorem 1). Prioritized cost for I4 to be calculated independently. The prioritized costs of the resources FW1, FW2 and FW3 are calculated (prioritized cost ¼C ri =ðqP  qri Þ) to be 1.8, 1.67 and 3$/t ppm, respectively for the first three intervals. As the prioritized cost of FW3 is higher than that of FW2, FW3 cannot substitute FW2. Therefore, for the first three intervals, only FW2 and FW1 should be used. For the last interval, FW1 is the only resource that can be used. For time interval I1 the resource requirement of FW1 and FW2 calculated (Table 5) using source composite method (Shenoy and Bandyopadhyay, 2007) to be 1 and 3.5 t respectively and a 4.5 t waste is generated at 12 ppm. It may be noted that in this paper, source composite method is used. However, the proposed methodology is generic in nature and any established methodology can be used for multiple resources targeting for a particular interval. Similarly, with transferring waste profile sequentially, resource requirements of intervals I2 and I3 are calculated (not shown due to brevity). As there are no other resources that can be utilized in I4, minimum FW1 requirement remains same as reported in Tables 4, i.e., 2.5 t. Results are summarized in Table 6. The minimum operating cost is $389, a 20.5% reduction from using only the purest resource. One of the possible water allocation networks to achieve this target is shown in Fig. 5. It may be noted in Fig. 5, storage tanks are required to carry forward a source from one time interval to a later time interval. For cyclic batch resource requirements can be calculated by collapsing all the intervals into a single interval (Table 7). The resource requirement for FW1 and FW2 are calculated to be 3.5 t and 1.76 t, respectively. The minimum operating cost is calculated to be $201.67.

5.2. Illustrative example 2 The limiting water data for this example are given in Table 8 (Chaturvedi and Bandyopadhyay, 2012). In this example, there are five sources and five demands which can be categorized into four time intervals (Chaturvedi and Bandyopadhyay, 2012). Two external resources are assumed to be available for this example (Table 9). Table 4 Interval wise cost requirement for example 1 using purest resource.

Pinch quality (ppm) Resource requirement (t), FW1

I1

I2

I3

I4

12 1.58

15 2

25 3

10 2.5

Total operating cost ($)

The operating cost is calculated to be $4800, when targeting is carried out using only the purest resource (R1), applying sequential transfer of waste profile. It may be observed that pinch quality (at 60 ppm) of the last interval is highest, and hence, prioritized costs of resources for all intervals are governed by the pinch quality of the last interval I4. Accordingly, prioritized cost R1 and R2, are calculated to be 1.34 and 0.4$/t ppm respectively. As prioritized cost of R2 is less than prioritized cost of R1 therefore R2 is introduced from I1 to I4 following sequential transfer of waste profile to substitute R1 (detailed calculations are not shown for brevity). It should be noted that due to addition of resource R2 pinch quality of fourth interval pinch quality jumps to a lower pinch quality of 20ppm. However, at new pinch point prioritized cost of R2 (2$/t ppm) is still less than that of other R1 (4$/t ppm), the algorithm may be continued with the new pinch point. Overall cost for resource supply is calculated to be $3968 and resource requirements of R1 and R2 are calculated to be 40 t and 38.4 t respectively. A reduction of 17.3% is observed from the base case. Fig. 6 shows one of the possible water allocation networks to achieve this target. For cyclic batch resource requirements for R1 and R2 are calculated to be 40 t and 35 t respectively. The cost of resource supply for this case is $3900. 5.3. Illustrative example 3 The limiting water data for the example are given in Table 10 and resource data are listed in Table 11. The total operating cost is calculated to be $657.67 using purest resource (R1), applying sequential transfer of waste profile. The pinch quality for each interval is shown in Table 12. It may be observed from Table 12 that pinch quality I2 is highest; prioritized cost of resources for intervals I1 and I2 should be governed according pinch quality of I2. For remaining set of intervals, I3 and I4 pinch quality of I4 is highest; prioritized cost for I3 and I4 should be governed according to pinch quality of I4 (Theorem 1). The prioritized cost of FW1, FW2 and FW3 are calculated to be 1$/t ppm, 0.82$/t ppm and 0.78 $/t ppm for first two intervals. As prioritized cost of FW2 is less than that of FW1: FW2 is introduced to substitute FW1 in intervals I1 and I2. It should be noted that introduction of FW2 to substitute FW1 does not affect the pinch quality of I2. Furthermore, prioritized cost of FW3 is less than that of FW2; hence FW3 is introduced to substitute FW2. However, substitution of FW2 by FW3 after a limit changes pinch quality of I2 and at new pinch (i.e., 20 ppm) prioritized cost of FW3 is no longer less than prioritized Table 6 Interval wise cost requirement for example 1.

Total

9.08 408.6

Resource requirement, FW1 (t) Resource requirement, FW2 (t) Cost ($)

I1

I2

I3

I4

Total

1 3.5 132.5

0 4.3 107.5

0 1.47 36.75

2.5 0 113

389

Table 5 Resource requirement calculation for time interval I1 for example 1. Contaminant conc. (ppm)

Net flow (t)

Cum. flow (t)

Quality load mass load (kg)

Cum. mass load (kg)

Waste flow for purest resource FW1 only (t)

12 11 10 5 0

7 5 0 2 0

7 2 2 0 0

0 7 2 10 0

0 7 9 19 19

1.583333 1.090909 1 0

Waste flow for first resource FW1 (t)

Waste flow for second resource FW2 (t) 4.5 2

1

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Table 9 Resource data for example 2.

Tank 2 Tank 1

R1 R2

7

Quality (contaminant conc; ppm)

Cost ($/t)

0 10

80 20

2 Tank 2

S3 0.73 S2

Tank 1

D4

0.27 2

3

S2

D3

5

S4

2.5

D2

4.5

D2

S4

S1

Fresh water 2

1

D1

D1

1.7

S3

D5

1

1

2

4

35

R2

Time (h)

D3

40

R1

Fig. 5. Water allocation network for example 1.

0

1

D5

2

3

4

Time (h)

Table 7 Resource requirement calculation for cyclic batch (example 1).

Fig. 6. Water allocation network for example 2.

Contaminant conc. (ppm)

Net flow (t)

Cum. flow (t)

Mass load (kg)

Waste flow Cum. mass load for FW1 only (t) (kg)

Waste flow for FW2 (t)

25 15 12 11 10 10 5 0

20 1 2  10 10 0 7 0

20 19 21 11 21 21 14 14

0 200 57 21 11 0 105 70

0 200 257 278 289 289 394 464

19.27 17.8 16 11

18.56 17.6 17.25 16.90 17.5 17.5 14

2.5

84.9 S5

3.4

3

10.9

D4

2.5

0

87.5

9.1

2.5 Fresh water 1

12.5

S1

2.5

Table 8 Limiting water data for example 2. Source

Flow rate (t/h)

Concentration (ppm)

Duration (h)

Flow (t)

S1 S2 S3 S4 S5

20 12.5 90 15 100

37.5 60 30 40 20

0.0–1.0 1.0–4.0 1.0–2.0 2.0–3.0 2.0–3.0

20 37.5 90 15 100

Demand

Flow rate (t/h)

Concentration (ppm)

Duration (h)

Flow (t)

D1 D2 D3 D4 D5

12.5 100 35 100 20

30 33.75 10 22.5 0

0.0–1.0 1.0–2.0 3.0–4.0 2.0–3.0 2.0–4.0

12.5 100 35 100 40

cost of FW2. So FW3 is introduced till pinch quality does not changes and FW3 is cheaper than FW2. Overall fresh water requirements for FW1, FW2 and FW3 are calculated to be 1.43 t, 5.17 t and 2.4 t respectively for the two intervals. The operating cost till second interval is $249.

Table 10 Limiting water data for example 3. Source

Concentration (ppm)

Duration (h)

Flow (t)

S1 S2 S3 S4

20 40 10 30

0.0–1.0 1.0–2.0 2.0–3.0 3.0–4.0

7 8 8 10

Demand

Concentration (ppm)

Duration (h)

Flow (t)

D1 D2 D3 D4 D5

12 5 20 6 12

0.0–1.0 0.0–1.0 1.0–2.0 2.0–4.0 3.0–4.0

5 5 6 12 8

Table 11 Resource specifications for example 3.

FW1 FW2 FW3

Quality (contaminant conc; ppm)

Cost ($/t)

0 7 12

40 27 22

For next set of intervals, I3 and I4, pinch quality of fourth interval controls the prioritization. The prioritized cost of fresh water FW1, FW2 and FW3 are calculated to be 1.33, 1.17 and 1.22 $/t ppm. As the prioritized cost of FW3 is higher than that of FW2, FW3 cannot substitute FW2. Therefore, only FW2 and FW1 should be used. Resource requirements are calculated to be 10.08 t for FW2 and 1.81 t for FW1. Table 13 shows results for this example. The minimum operating cost is $578.7. A reduction of 12% in cost

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Table 12 Intervalwise pinch quality for example 3.

Table 14 Limiting water data for example 4.

Time interval

Pinch quality (ppm)

I1

20 40 10 30

I2 I3 I4

Class I–II water (S1) Class III–IV water (S2) Back grinding (D1) Marking (D2)

Flow rate (gal/ min)

r (kΩ cm)

Duration (h)

Flow (gal  106)

650 600 1000 700

14,000 12,000 16,000 10,000

2.0–3.0 2.0–3.0 0.0–2.0 1.0–3.0

2.34 2.16 7.2 5.04

Table 13 Interval wise cost requirement for example 3.

FW3 FW2 FW1

Requirement (t)

Operating cost ($)

2.39 15.25 3.24

95.6 411.75 71.28

Fresh water 1 Fresh water 2

578.7

Tank 1

10,000 12,000 13,000 14,000 18,000

S2 D3

0.64

6

1

2.4

Cost ($/106 gal)

Maximum flow (gal  106)

18,000 13,000

40 10

1 3

r 1/r Flow Cum flow (kΩ cm) (kΩ  1 cm  1) (gal  106) (gal  106)

S4

7.36

0.0001 8.33E 05 7.69E  05 7.14E  05 5.56E  05

 2.52 2.16 0.66 2.34

 2.52  0.36 0.3 2.64 2.64

Cum. load (kΩ cm gal  106)

Waste flow (gal  106)

0  4.20E  05  2.31E  06 1.65E  06 4.19E  05

0  4.20E  05  4.43E  05  4.27E 05  7.55E  07

D5

D1

1.6

r (kΩ cm)

Table 16 Resource requirement calculation for interval I3 (example 4).

Tank 2

S1

Table 15 Resource specifications for example 4.

S3 9.37

FW 1

D2

D4

1.98 1.43

0.66

0.64

1.9

0.4

D2

S1 2.16

3.57

FW 2 FW 3

0

1

2

3

S2

4

4.86

Time (h) Fig. 7. Water allocation network for example 3.

may be observed against using only the purest resource. One of the possible water allocation networks to achieve this target is shown in Fig. 7. For cyclic batch minimum operating cost is calculated to be $464.42. The resource requirements for FW1, FW2 and FW3 are calculated to be 4.57 t, 10.4 t and 0 t respectively. 5.4. Illustrative example 4 Apart from concentration, various other properties such as density, reflectivity, pH, solubility, conductivity, viscosity, etc. are also important for designing RAN. Proposed algorithm can be also applied to batch processes where such properties play an important role. A water minimization example in a semiconductor process is considered here to illustrate the applicability of the proposed algorithm. In this example, the resistivity (r) is the property in concern for water reuse/recycle As resistivity constitutes an index of the ionic content of aqueous streams and it follows the following mixing rule (Foo et al., 2006a). 1 x ¼∑ i r i ri

D1

FW1

ð10Þ

2.34

FW2

0

1

2 Time (h)

3

Fig. 8. Water allocation network for example 4.

The limiting water data for the example are given in Table 14 (Chaturvedi and Bandyopadhyay, 2012). Two external resources are assumed to be available to supply the requirements (Table 15). Source and demands can be categorized in three intervals. The operating cost using only FW1 (purest resource) is calculated to be $388.8. For multiple resources, proposed methodology is applied and operating cost is calculated to be $300.4, a 22.7% reduction may be observed. It should be noted that in last time interval FW2 exhausts and only 0:66  106 gal is available for this interval. To solve the problem with the availability constraint, FW2 is considered as an internal source with 0:66  106 gal of flow at 13000 kΩ cm (Table 16). Targeting for third interval is carried out using the same procedure. Fig. 8 shows possible network to achieve this target. For cyclic batch, the operating cost is calculated to be $222.8.

N.D. Chaturvedi, S. Bandyopadhyay / Chemical Engineering Science 104 (2013) 1081–1089

6. Conclusions A conceptual methodology based on pinch analysis for targeting multiple resources for batch process has been proposed in this paper. The methodology is based on rigorous mathematical proofs and therefore, guarantees the optimum results. The mathematical proofs take care of both the quality and time (i.e., schedule) constraints. It has been proved that to calculate minimum cost prioritization of resources for an interval is not always calculated using its own pinch quality rather it is to be determined based on the highest pinch quality of all the subsequent intervals, including itself. The concept of prioritized cost, developed by Shenoy and Bandyopadhyay (2007), is extended targeting multiple resources in batch processes. It may be noted that the proposed algorithm is applicable for batch process with a single quality. Future research activities are directed towards developing algorithms for batch process with multiple qualities. Nomenclature C Crk Fdj Frk,max Fsi FW Ii Nd Nr Ns q qdj qsi qPIi R RIik r W Ii W

¼cost ($) ¼cost per unit flow of resource Rk ($/t) ¼flow requirement by a jth internal demand (t) ¼specified maximum availability of a resource Rk (t) ¼flow generated by a ith internal source (t) ¼ fresh water ¼ith time interval ¼number of internal demands ¼number of external resources ¼number of internal sources ¼quality ¼maximum allowable quality of jth internal demand ¼quality of ith internal source ¼pinch quality of time interval Ii ¼resource ¼resource requirement of kth resource in interval Ii (t) ¼resistivity (kΩ cm) ¼waste generated in interval Ii (t) ¼waste generation (t)

Greek letters Δ

δ

¼difference ¼change

Subscripts D lo P S u high

¼direct ¼lower ¼pinch ¼sequential ¼upper ¼highest

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