Resource allocation network for segregated targeting problems with

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Resource allocation network for segregated targeting problems with dedicated sources Sheetal Jain, and Santanu Bandyopadhyay Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03329 • Publication Date (Web): 20 Oct 2017 Downloaded from http://pubs.acs.org on October 24, 2017

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Industrial & Engineering Chemistry Research

Resource allocation network for segregated targeting problems with dedicated sources

Sheetal Jain

and

Santanu Bandyopadhyay*

Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

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

ACS Paragon Plus Environment


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ABSTRACT

Resource allocation networks (RAN) are efficiently optimized using the techniques of Pinch Analysis. Primarily, these RANs consist of a set of source streams and a set of demands. However, in the past a special type of RAN, called segregated targeting problems, is identified which consists of multiple sets of demands called zones. The resource and cost optimality of such problems are determined by using the sequential targeting and the concept of prioritized cost. In this paper, segregated targeting problems are extended to include dedicated sources in each zone. In segregated targeting problems, sources are shared by all zones. However, in the extended version of the problem, there are some dedicated sources specific to a zone in which they are present and are not shared with other zones due to proximity, safety, operability, etc. The primary objective of this paper is to develop an algorithm, based on the principles of Pinch Analysis, to determine the minimum resource requirement for a segregated targeting problem with dedicated sources. Using rigorous mathematical arguments, a non-dimensional number is determined that dictates the allocation of an internal source to a zone for overall resource optimality. As a part of the proposed algorithm, we have rigorously analyzed and characterized pinch jump due to addition and removal of flows from a RAN. The applicability of the proposed algorithm is demonstrated through three examples from diverse domains: cooling water network, carbon constrained energy sector planning and water allocation network.


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1. INTRODUCTION Conservation of resources is the priority for any process industry. Industries make several efforts to minimize their resource requirement by using more efficient technology, reusing and recycling the process streams in a resource allocation network (RAN). Pinch Analysis has proved itself as one of the promising techniques in optimizing RANs. RAN primarily consists of a set of source streams and a set of demands. These sources and demands are characterized by two parameters, quantity and quality. The resource is minimized in a RAN by utilizing the maximum source streams to meet the demand quality constraints. Pinch Analysis has evolved itself to determine the optimal resource requirement for many RANs. Pinch Analysis was initially developed as a tool for energy integration in heat exchanger network.1 Over the years, it has proved itself as a well-established technique for conservation of resources in RANs. Pinch Analysis has helped in exploiting the underlying structure of many resource conservation problems and it has marked its applicability in various domains. For example, due to the structural similarity in heat and mass transfer processes, the application of Pinch Analysis is extended to mass exchange networks.2 Within the framework of mass exchanger networks, water networks3 and hydrogen networks4 gained utmost importance. Pinch Analysis has also been applied to aggregate production planning,5 property based material recovery problems,6 energy sector planning,7 emergy transformity based planning,8 cogeneration analysis for site utility systems,9 design and optimization of isolated energy systems,10 integrate bioehtanol as an in-process material in biorefineries,11 production planning in small medium industries,12 design optimal bioethanol networks with purification for integrated biorefineries,13 planning of unconventional gas field development,14 industrial safety risk and environmental management,15 greenhouse gas emission reduction in municipal solid waste,16 biochar based



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carbon management networks,17 etc. It has extended from heat recovery Pinch Analysis1 to material Pinch Analysis for global steel flows,18 strategies for reduction of greenhouse emission,19 selection of energy conservation projects through Financial Pinch Analysis,20 Power Pinch Analysis for to size renewable energy systems with uncertainties,21 segmented Pinch Analysis for environmental risk management,22 etc. In addition to these single objective problems, Krishna Priya and Bandyopadhyay23 have successfully applied Pinch Analysis to multiple objectives problems involving power system planning. Tan et al.24 elaborated many such novel applications of Pinch Analysis in diverse domains. Recently, Basu et al.25 applied the principles of Pinch Analysis to determine gaps in health delivery systems. The application of Pinch Analysis in different domains is possible because of various mathematical and graphical solution procedures reported in literature. Wang and Smith3 developed limiting composite curve for wastewater minimization. Savelski and Bagajewicz26 introduced necessary conditions for optimality of water using networks. A graphical approach based on the concepts of water surplus diagram27 was introduced for targeting the minimum freshwater requirement. Banyopadhyay28 had developed an algorithmic procedure to reduce waste generation and the targeting philosophy is represented by source composite curve.29 All these cases describe problems where minimizing energy, freshwater, hydrogen, etc. are involved. However, apart from targeting and minimizing resources and energy, Pinch Analysis has been successfully applied for minimization of capital and operating costs of the system.30 Shenoy and Banyopadhyay31 introduced the concept of prioritized cost to determine the cost optimal solution in RANs involving multiple resources. Prioritized cost shows the trade-off between the cost of one resource and its potential to replace another costlier resource to minimize the overall cost of the system. Bandyopadhyay and Desai32 performed the cost optimal energy sector planning by using the concepts of Pinch Analysis.


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Lee et al.33 identified a special class of RAN in carbon-constrained energy sector planning which consists of a set of sources and multiple sets of demands called zones. This type of problem is termed segregated targeting problem. In segregated targeting problems, individual resources are needed for different zones and typically these zone-specific resources are different in qualities. On the other hand, the internal sources are shared by all the zones. Bandyopadhyay et al.8 applied the concepts of Pinch Analysis to propose a decomposition algorithm for determining the resource optimal solution for segregated targeting problems. The decomposition algorithm works by arranging the zones in increasing order of resource quality and targeting them sequentially. Chandrayan and Bandyopadhyay34 introduced the economic aspect to the segregated targeted problem. Each resource is now characterized by two parameters: quality and cost. The objective of the problem is to minimize the total cost. The concept of prioritized cost has been applied to achieve the desired objective of cost optimality. In general, not all the sources can be shared between all the zones. Limitations such as transportation over long distance, flexible operation of certain zones, safety, etc. force some sources to restrict to certain demand zones and not shared by other ones. These problems are described in this paper as segregated targeting problem with dedicated sources that can be used only in a particular zone. Due to presence of dedicated sources, existing algorithms cannot be applied directly to solve these problems. The objective of the work is to determine the resource optimal solution for such problems using pinch analysis. In this paper, a novel non-dimensional number is derived which dictates the distribution of internal sources among different zones to achieve resource optimality. This paper is organized as follows. Next section deals with the formal statement of the problem and its mathematical formulation. Then, the problem is mathematically analyzed and effect of addition and subtraction of flow in a RAN on pinch point is discussed in detail. Then an algorithm is proposed for determining the resource optimal



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solution for segregated targeting problem with dedicated sources. Then, the applicability of the developed algorithm is demonstrated with the help of three illustrative examples from diverse domains which includes cooling water network, carbon constrained energy sector planning, and water allocation network.

2. PROBLEM STATEMENT AND MATHEMATICAL FORMULATION A general segregated targeting problem with dedicated internal sources can be physically described with the help of the superstructure shown in Fig. 1, which has multiple zones and a set of common internal sources. Each zone consists of a set of demands, a resource and a set of dedicated sources. Dedicated sources can be considered as sources which can transfer flow only to the demands of the zone in which they are present. That is, dedicated sources from one zone cannot transfer the flow to the demands of other zones. The objective of the problem is to minimize the total resources being utilized in the overall problem.

The mathematical definition of the above mentioned problem is as follows. A set of internal

sources is given. Each source ( = 1,2, … , ) produces a flow with quality . A set of multiple regions or zones = 1,2, … , is also given. Each zone consists of a set of dedicated sources, a resource and a set of demands. Each dedicated source = 1,2, … , of the

zone produces flow with quality . Resources have no flow limitations and each resource corresponding to zone has a quality associated with it. Each demand =

1,2, … , of zone accepts a flow with a maximum allowable quality of from

internal sources, dedicated sources and resource. Flows are denoted by non-negative real


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numbers. Qualities are denoted by real numbers and follow an inverse scale. That is, lower numerical value of quality indicates its superiority.15 The unused flow from the dedicated internal sources and common internal sources are thrown to an external demand called waste (Figure 1). In this case, waste does not have any flow and quality limitations as it is not directly discharged to the environment. Appropriate treatment or interception units can be installed before discharging the waste to the environment.

Let be the flow transferred from internal source to demand of zone . Similarly, is

the flow transferred from dedicated source to demand of the zone . Let be the flow from

internal source to the waste and be the flow from dedicated source of zone k to the

waste. Let be the flow transferred from the resource of zone to the demand of the same

zone. The flow constraint for each internal source is given by eq 2 and flow constraint equation for every dedicated source is given by eq 3. Eqs 4 and 5 represent the flow and quality load constraint for every demand.



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Resource = 1 "1 )

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Zone 1

Dedicated sources

= 1 [#$11 , #$11 ] [#$ 1 , #$ 1 ] = #$1 [#$#$ 1 1 , #$#$ 1 1 ]

= 1 [ 1 , 1 ]

Demands

= 1 [%11 , %11 ] [% 1 , % 1 ] = %1 [%% 1 1 , %% 1 1 ]

Internal source, Resource " ) Dedicated sources

= 1 [#$1 , #$1 ]

[ , ]

Internal source, [#$ , #$ ] = #$ [#$#$ , #$#$ ]

= [ ,

Internal source,

]

Resource & "&

Zone Demands

= 1 [%1 , %1 ] [% , % ] = % [%% , %% ]

Zone &

Dedicated sources

= 1 [#$1& , #$1& ] [#$ & , #$ & ] = #$& [#$#$ & & , #$#$ & & ]

Demands

= 1 [%1& , %1& ] [% & , % & ] = %& [% % & & , % % & & ]

Waste

Figure 1. Representation for segregated targeting problem with dedicated sources.


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The objective is to minimize the total resource requirement (eq 1). The overall optimization problem may be expressed as:

+ *+ ∑, Minimize ' = ∑,-

)

)

Subject to:

)/ *+ ∑,∑) ,- + = *+ ∑) ,- + =

(1) for every internal source ∈ {1,2, … , }

(2)

in zone ∈ {1,2, … , }

(3)

for every dedicated source ∈ {1,2, … , }

)34+ 5 ∑)

,- + ∑,- + =

for every demand ∈ {1,2, … , } in zone ∈ {1,2, … , }

(4)

5 34+ + ∑ , + ∑, ≤ for every demand ∈ {1,2, … , }

)

)

in zone ∈ {1,2, … , }

(5)

The objective function (eq 1) and the constraints given by eqs 2-5 are linear in nature. Hence, this is a linear programming problem. It is mathematically analyzed and solved using the concepts of Pinch Analysis.

3. MATHEMATICAL ANALYSIS Let us assume that there are two zones with resource qualities, - and 7 . Let both the zones be

targeted individually by their respective dedicated internal sources. Three possibilities may arise after targeting is performed: Case 1: Both the zones have pinch points.

Case 2: One zone has the pinch point whereas the other zone does not have it (i.e., threshold problem). Case 3: None of the zones have pinch points (i.e., both are threshold problems).



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In all the three cases, additional reduction in resource requirement can be achieved by supplying flow from the common internal sources. However, the maximum reduction in overall resource requirement depends on the distribution of flow from internal sources to both the zones.

3.1. Condition of optimality for case 1

Consider case 1, when both the zones have the pinch points. Let us assume that 8- and 87

are the pinch points in zone 1 and zone 2 respectively. The resource requirement for these zones is given by eq 6, as derived by Bandyopadhyay35: ' = ∑

=99 ? )*+ =9 >9*:+? ,95G ?

(10)

(11)

Whenever D ≥ 8 , the quantity − D / − in eq 10 is positive. Combining with

the fact that '8 ≥ ' for all si, it can easily be shown that ' S 8 ≥ '′ . Therefore, if ∆ is added in the above pinch region (D ≥ 8 ) then the pinch point does not change. This mathematical analysis proves lemma 2.

Lemma 2: The pinch point of a zone remains unchanged when the flow is added in the above pinch region.



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However, if ∆ is added in the below pinch region, the pinch point cannot jump in the above

pinch region. The argument for this lies in the fact that whenever ≥ 8 , ' S 8 ≥ '′ (as 95B >95G 95B >9@5

≥

=9< >95G ? =9< >9@5 ?

)for all such i’s as can be seen from eq 10 and 11.

This discussion proves lemma 3.

Lemma 3: The pinch point of the zone can jump only in the below pinch region when the flow is added in the below pinch region. Eq 10 and 11 are equated for all i’s (eq 12) to determine the maximum value of ∆ that can be

added in the below pinch portion of a zone (D < 8 without changing the pinch point of that zone.

='8 − ' ? 9 5B >9

9 >9@5 =9< >9@5 ?

@5 =9< >95B ?

if D <

∆ = V =9< >9@5 ? ='8 − ' ? =9 >9 ? if D ≥

5G

9 =9 >9 ? D < 8 5G @5 < 5B ∆ = V 95B >9@5 ='8 − ' ? 9 >9 D ≥ 8 9 >9@5 =9< >9@5 ? 5B

(13)

5G

Lemma 7: The maximum amount of flow that can be extracted from a zone without changing the pinch point of that zone is the minimum of all ∆ ’s given by eq 13.

3.3. Condition of optimality for other cases Case 2 considers that one zone has the pinch point while the other one is a threshold problem, that is, without a pinch point. Without loss of generality, let us assume that zone 1 has a pinch point and zone 2 does not have a pinch point. In such cases, a dedicated pseudo source is introduced in zone 2 at a very high quality such that the pinch point for zone 2 lies on that pseudo source. Once both the zones have the pinch point, the methodology developed in case 1 is followed for further resource reduction. At the end, the elimination of remaining pseudo source is performed by utilizing the leftover internal sources and resource. It may be noted that the pseudo source does not exist in reality, it is only assumed to make the problem pinched and hence, at the end of the problem solution, it should be completely removed.



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Similarly, for case 3 where none of the zones have pinch points, dedicated pseudo sources are introduced in both the zones to create the pinch points. Once, both the zones get pinched, the solution procedure is same as that for case 2. Again, the pseudo sources are removed at the end of the solution procedure. Using these results, an algorithm is developed to determine the resource optimal solution for segregated problems with dedicated sources.

4. PROPOSED ALGORITHM The algorithm developed for determining the resource optimal solution for segregated targeting problem with dedicated sources is shown with the help of a flowchart in Figure 3. The steps of the proposed algorithm are as follows: Step 1: Solve each zone individually to determine its resource requirement without using the internal sources. It can be done by using any of the established Pinch Analysis based techniques like source composite curve, limiting composite curve, material recovery pinch diagram, etc. Step 2: After targeting all the zones with their respective dedicated internal sources, check whether all the zones have the pinch points or not. If all the zones have the pinch points, then go to step 4. Else, go to step 3. Step 3: Introduce dedicated pseudo source at a very high quality in the zones which do not have pinch points. Flow is added from these pseudo sources to the respective zones until these pseudo sources become the pinch points. Once all the zones have the pinch point, go to step 4.


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Step 4: Calculate the Benefit number (eq 9) for all the zones for all the internal sources. If an internal source has zero or negative values of Benefit number at all the zones, then the flow from such internal sources is neglected.

Step 5: Transfer X (eq 15) amount of flow from the internal source ( ≠ 0) to that zone which corresponds to the highest value of Benefit number.

X = min [ , ∆]

where ∆ is calculated by following Lemma 4.

(15)

After each addition, update = − X.

If < ∆, then repeat Step 5 with the subsequent value of J , until all the internal

sources gets exhausted, then go to Step 6.

If ≥ ∆, then the pinch point jumps from L to L′ . Update L = L′ and repeat Steps 4 and 5.

Step 6: Check whether the pseudo source is still utilized in the network. If it is not utilized, then the algorithm terminates else eliminate the pseudo source still utilized in the system by using the neglected internal sources and the resource. The solution obtained by this algorithm for segregated targeting problem with dedicated sources is resource optimal. The applicability of the proposed algorithm is demonstrated in the next section.



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START Target each zone individually All the zones have pinch points? Yes

No

Calculate B for all the zones at all internal sources

Introduce pseudo source at a very high quality

Check, if there exist internal sources with negative or zero value of B for all zones? Yes

No

Neglect those internal sources Any internal source left?

Choose the highest value of B Yes

Determine its corresponding ith internal source and kth zone

No Pseudo source is still in use? No

Transfer α = min(Fsi, ∆) from that ith source (Fsi ≠ 0) to that kth zone

Yes Use neglected internal sources to eliminate it

Pseudo source is not in use?

Update Fsi = Fsi - α

Fsi = 0 for all that ith internal source

Pseudo source is still in use? Use resource to eliminate it

No

Fsi = 0 for all internal source i?

Pinch point jumps Update the new pinch point

Yes

STOP

Figure 3. Algorithm for resource optimality of segregated targeting problems with dedicated sources.


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5. ILLUSTRATIVE EXAMPLES The applicability of the proposed algorithm is demonstrated by using three different examples from diverse domains: cooling water network, carbon constrained energy sector planning and water allocation network. Structural similarities of these illustrative examples in terms of flow and quality variables are listed in Table 1 and the overall superstructure of these problems is shown in Figure 1.

Table 1. Stream quality and flow for three illustrative examples. Illustrative Example

Quality

Flow

Cooling water network

Temperature (°C)

Heat

capacity

flow

rate

(kW/°C) Carbon constrained sector planning

energy CO2 emission factor (t/TJ)

Water allocation network

Concentration (ppm)

Energy (TJ)

Water flowrate (t/h)

5.1. Example 1: Cooling Water Network Cooling towers are used to reject heat to the atmosphere. The basic data for this example have been taken from Majozi and Moodley.36 In the original problem, there are three cooling towers (T1, T2 and T3) supplying cooling water to three independent zones. For each independent zone, there are demands, sources and a specified cooling tower. The cooling water supplied by T1, T2 and T3 are 65.75 kW/°C, 34.8 kW/°C and 15 kW/°C. Each cooling tower has a specified capacity. It is assumed that four additional cooling water heat exchangers are commissioned. This leads to four sources of cooling water (S1, S2, S3 and S4) and four demands of cooling water (D4, D7, D8 and D10). Due to geographic proximity, these additional demands are placed



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in three zones (D4 is placed in zone 1; D7 and D8 are placed in zone 2; and D10 is placed in zone 3). However, four sources cannot be placed in those zones due to capacity limitation as after introducing these additional heat exchangers, the cooling water requirement from T1 increased considerably (80.75 kW/°C). Therefore, it is decided to distribute these sources in such a way to minimize the overall cooling water flow. This leads to a problem of segregated targeting with dedicated sources. The heat capacity for the water is assumed to be constant throughout the network and it is also assumed that the supply temperature of each cooling tower is fixed. The overall data for this example are given in Table S1 (Supporting information). The proposed algorithm is applied to determine the resource optimal solution for this problem. According to Step 1 of the algorithm, all the zones are solved individually without using the internal sources. In this example, the targeting is performed by using limiting composite curve.3 It is observed that the pinch points for these zones are at 45 °C, 55 °C and 60 °C, as shown in Figure 4. The overall cooling water requirement is 140.16 kW/°C (80.75 kW/°C in zone 1, 39.41 kW/°C in zone 2 and 20 kW/°C in zone 3). It may be noted that the cooling water requirement is usually expressed as heat capacity flow rate (i.e., the product of mass flow rate with specific heat). Values of Benefit number for all internal sources at different zones are calculated and shown in Table 2. The internal sources S3 and S4 have negative values of Benefit number for all the zones, and therefore, these internal sources are neglected for overall integration and are utilized for balancing the flow in different cooling towers as shown in Figure 5.


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65

pinch point 60

60

Zone 3 55

Zone 1

pinch point Zone 2

55

50

pinch point

quality (°C)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


45

45

Limiting composite curve 40

minimum resource line

35

30

25 22 20 0

500

1000

1500

2000

2500

cumulative quality load (kW)

Figure 4. Limiting composite curves for all three zones of example 1. As can be seen from Table 2, the highest value of Benefit number is obtained at internal source

S1 for zone 3. According to Step 5, X amount of flow is transferred from S1 to zone 3. Applying

eq 15, the maximum value of X is calculated to be 10 kW/°C (the minimum of 10 kW/°C, flow available in S1 and 48 kW/°C, flow required for pinch jump). S1 gets exhausted and the overall

resource requirement is reduced to 136.41 kW/°C (80.75 kW/°C in zone 1, 39.41 kW/°C in zone 2 and 16.25 kW/°C in zone 3). The next highest value of Benefit number is 0.25 and this corresponds to S2 at zone 3. S2 has a flow of 10 kW/°C and a flow of 31.67 kW/°C leads to pinch jump. Therefore, only 10 kW/°C of S2 is supplied to zone 3. After this flow addition, all the internal sources get exhausted and the algorithm terminates. The overall cooling water requirement reduced to 133.91 kW/°C (80.75 kW/°C in zone 1, 39.41 kW/°C in zone 2 and 13.75



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kW/°C in zone 3). This result is verified by mathematical optimization technique. The cooling water allocation network for this example is given in Figure 5. It is to be noted that the network shown in Figure 5 in not unique and many such networks can be designed for a given pinch target.

Table 2. The values of Benefit number (B(m,k)) for different internal sources at different zones for example 1.

Internal Sources S1 (45°C)

S2 (50°C)

S3 (65°C)

S4 (70°C)

Zone 1 (p1 = 45°C)

0

-0.25

-1

-1.25

Zone 2 (p2 = 55°C)

0.303

0.1515

-0.303

-0.455

Zone 3 (p3 = 60°C)

0.375

0.25

-0.125

-0.25


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80.75 kW/°C 10 kW/°C

T1

30 kW/°C

25°C

D1

OP1

DS1

3.75 kW/°C

11.25 kW/°C 3.75 kW/°C

D2

32 kW/°C

D3

OP2

DS2

15 kW/°C

DS3

32 kW/°C

80.75 kW/°C OP3

15 kW/°C 15 kW/°C

D4

NOP1

S4

30 kW/°C

39.41 kW/°C 4.35 kW/°C

D5

OP4

DS4

9.41 kW/°C

T2 15.65 kW/°C

22°C 30.46 kW/°C

D6

OP5

DS5

16.24 kW/°C

39.41 kW/°C

5.65 kW/°C 4.35 kW/°C

D7

NOP2

S2

10 kW/°C

9.16 kW/°C 10.59 kW/°C 0.25 kW/°C

D8

NOP3

20 kW/°C

S3

13.75 kW/°C

T3 16.25 kW/°C

20°C 11.75 kW/°C

D9

DS6

OP6

23.75 kW/°C

2 kW/°C

13.75 kW/°C 2 kW/°C

D10

NOP4

S1

8 kW/°C

Figure 5. Allocation of cooling water for example 1.


10 kW/°C


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5.2. Example 2: Carbon-constrained Energy Sector Planning The data for this example are given in Table S2 and are adopted from Lee et al.33 They considered two sectors of energy consumption, namely transportation sector and industrial sector. Coal, Oil, and natural gas are considered as internal sources for both the sectors and low carbon fuels such as biodiesel in transportation sector and hydropower in industrial sector are used as resources for these sectors.33 Biodiesel has a cost of 0.031 $/MJ and hydropower has a cost of 0.028 $/MJ as described by Chandrayan and Bandyopadhyay.34 However, in recent years, the economy of transportation sector is shifting from coal to petroleum products and the consumption of coal in transportation sector is decreasing gradually. Therefore, it is assumed that coal serves as a dedicated source for industrial sector in this problem. The proposed algorithm is applied to determine the resource optimal solution for this problem. By following Step 1 of the algorithm, both the sectors are solved individually using the methodology of material recovery pinch diagram.37 It is observed that coal serves as a pinch source for industrial sector with the CO2 emission factor of 105 t/TJ (Figure S1, supporting information). However, transportation sector does not have a pinch point. The overall energy requirement obtained is 332.19 × 104 TJ (184 × 104 TJ for transportation sector and 148.19 × 104 TJ for industrial sector) and the total operating cost is 98.53 billion $ (57.04 billion $ for transportation sector and 41.49 billion $ for industrial sector). A dedicated pseudo source is introduced in transportation sector which has the CO2 emission factor of 200 t/TJ. It may be noted that Pseudo source is introduced in such a way that it has the highest CO2 emission factor among all the sources and demands. 25.31 × 104 TJ of energy is added from this pseudo source to make it as a pinch point in transportation sector and it reduced the overall energy requirement to 306.88 × 104 TJ (158.69 × 104 TJ for transportation sector and 148.19 × 104 TJ for industrial sector) and the total operating cost is reduced to 90.68 billion $


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(49.19 billion $ for transportation sector and 41.49 billion $ for industrial sector). The values of Benefit number are calculated at both the internal sources for different sectors (Table3 ). Benefit number is the highest corresponding to natural gas at transportation sector. According to Step 5, 80 × 104 TJ (minimum of 80 × 104 TJ, energy available in natural gas and 120.62 × 104 TJ, energy required for pinch jump) of energy is transferred from natural gas to transportation sector. After this flow addition, the energy utilized from pseudo source is reduced to 8.52 × 104 TJ and the overall energy requirement is reduced to 243.67 × 104 TJ (95.48 × 104 TJ for transportation sector and 148.19 × 104 TJ for industrial sector). The total operating cost is reduced to 71.09 billion $ (29.6 billion $ for transportation sector and 41.49 billion $ for industrial sector).

Table 3. The values of Benefit number (B(m,k)) for different internal sources at different sectors for example 2. Internal Sources Natural Gas (55 t/TJ)

Oil (75 t/TJ)

Transportation sector (p1 = 200 t/TJ)

0.79

0.68

Industrial Sector (p2 = 105 t/TJ)

0.48

0.29

Transportation sector (p1 = 75 t/TJ)

0.34

0

0.48

0.29

Industrial Sector (p2 = 105 t/TJ)

Remarks

Pinch point of transportation sector jumps from pseudo source to Oil when 26.74 × 104 TJ of energy is transferred from oil to transportation sector.

The next highest value of Benefit number corresponds to oil at transportation sector. Oil has an available energy of 100 × 104 TJ and the energy required for pinch jump is 26.74 × 104 TJ.



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Therefore, 26.74 × 104 TJ of energy is transferred from oil to transportation sector and the overall energy requirement is reduced to 225.46 × 104 TJ (77.26 × 104 TJ for transportation sector and 148.19 × 104 TJ for industrial sector) and the overall operating cost is reduced to 65.44 billion $ (23.95 billion $ for transportation sector and 41.49 billion $ for industrial sector). Pseudo source is completely eliminated after this flow addition and oil serves as a pinch source for transportation sector. The values of Benefit number are calculated with the updated pinch point (Table 3). As the natural gas is already exhausted, the next highest value of Benefit number is 0.29, which corresponds to oil at industrial sector. The energy available in oil is 73.27 × 104 TJ and the energy required for pinch jump is 94.93 × 104 TJ. Therefore, 73.27 × 104 TJ of energy is transferred from oil to industrial sector and the overall energy requirement is reduced to 204.52 × 104 TJ (77.26 × 104 TJ for transportation sector and 127.26 × 104 TJ for industrial sector). The total operating cost is reduced to 59.58 billion $ (23.95 billion $ from transportation sector and 35.63 billion $ from industrial sector). As all the internal sources get exhausted, the algorithm terminates. One of many possible energy allocation networks for this example is given in Figure 6. The overall energy requirement obtained by following the proposed algorithm is same as the energy requirement obtained by Lee et al.33 where coal is not considered as a dedicated source. It shows that without changing the resource requirement of the overall problem, a benefit on economic aspect (elimination of transport channel from coal to transportation sector) may be achieved by considering coal as a dedicated source. However, this is a special case where dedicating a source to a particular zone does not lead to change in overall resource requirement and it may not hold true in general. It may also be noted that the total operating cost obtained here might not be the minimum operating cost for this problem.


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Biodiesel

(16.5 t/TJ)

Hydropower 1272.58 PJ

772.65 PJ

Transportation Sector

Industrial Sector

Demands 140.26 439.48

Demands 259.74

960

T1 (30 t/TJ, 400 PJ) T2 (40 t/TJ, 720 PJ)

220.26 T3 (50 t/TJ, 720 PJ)

(0 t/TJ)

I1 (30 t/TJ, 1600 PJ)

280.52

270.67

232.39

41.91

Dedicated Source 640 92.65

I2 (40 t/TJ, 480 PJ)

Coal (105 t/TJ, 5000 PJ) 154.77 PJ

116.68 I3 (50 t/TJ, 80 PJ) 38.09

800 PJ

Natural Gas (55 t/TJ, 800 PJ) 4845.23 PJ

267.35 PJ

Oil (75 t/TJ, 1000 PJ)

732.65 PJ

Unutilised Energy

Figure 6. Allocation of energy for example 2.

5.3. Example 3: Water Allocation Network The data for this example are adopted from Bandyopadhyay et al.8 In this example interplant water integration is performed in two zones. Zone 1 has freshwater supply at 0 ppm (1.5 $/t) and zone 2 has freshwater supply at 10 ppm (1$/t). It is assumed that DS1 and DS2 serve as dedicated source for zone 1 and DS3 and DS4 serve as dedicated sources for zone 2. The data for this example is given in Table S3.



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The proposed algorithm is applied to determine the resource optimal solution for this problem. According to Step 1 of the algorithm, both the zones are solved individually using source composite curve.28 It was observed that zone 1 has a pinch point at 800 ppm (Figure S2) and zone 2 does not have a pinch point. The overall freshwater requirement obtained is 326.25 t/h (156.25 t/h is required for zone 1 and 170 t/h is required for zone 2) and the total operating cost is 404.37 $/h (234.37 $/h for zone 1 and 170 $/h for zone 2). A pseudo source at 1000 ppm is introduced in zone 2 and 0.81 t/h of flow is added from this pseudo source such that it becomes the pinch point for zone 2. The values of Benefit number are calculated at each internal source for both the zones as shown in Table S4. The highest value of Benefit number is 0.959, which corresponds to S1 at zone 2. By following step 5, 20 t/h (minimum of 50 t/h, flow available with S1 and 20 t/h, flow needed for pinch jump) is added to zone 2 from S1 and the pinch point of zone 2 jumps at DS4. The overall freshwater requirement is reduced to 306.25 t/h (156.25 t/h is required for zone 1 and 150 t/h is required for zone 2) and the total operating cost is reduced to 384.37 $/h (234.37 $/h for zone 1 and 150 $/h for zone 2). The flow utilized from pseudo source is completely eliminated after this flow addition. The values of Benefit number for different internal sources are calculated again with the updated pinch point as shown in Table S4. The next highest value of Benefit number is 0.938 which corresponds to S1 at zone 1. The flow available with S1 is 30 t/h and the minimum flow required for pinch jump is 145.33 t/h. Therefore, 30 t/h is transferred from S1 to zone 1 and the overall freshwater requirement is reduced to 278.13 t/h (128.13 t/h is required in zone 1 and 150 t/h is required in zone 2). The total operating cost is reduced to 342.19 $/h (192.19 $/h for zone 1 and 150 $/h for zone 2).


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The next highest value of Benefit number is 0.875 corresponding to S2 at zone 1. It is calculated that 60.71 t/h (minimum of 220 t/h, flow available in S2 and 60.71 t/h, flow required for pinch jump) of flow is added from S2 to zone 1 and the pinch point of zone 1 jumps at S2. The freshwater requirement is reduced to 225 t/h (75 t/h is required in zone 1 and 150 t/h is required in zone 2) and the total operating cost is reduced to 262.5 $/h (112.5 $/h for zone 1 and 150 $/h for zone 2). The values of Benefit number for different internal sources are calculated again with the

updated pinch point (Table S4). The highest value of J K, is 0.833, corresponding to S1 at

zone 2. The flow from S1 which was added in zone 1 in previous iterations (30 t/h) is extracted

out from there and added to zone 2 (minimum of 30 t/h, flow extracted from zone 1 and 135t/h, flow required for pinch jump). To maintain the pinch point of zone 1 at S2, 15 t/h (eq 12) of flow is added from S2 to zone 1. After this flow addition, the overall freshwater requirement is reduced to 215 t/h (90 t/h is required in zone 1 and 125 t/h is required in zone 2) and the total operating cost is reduced to 260 $/h (135 $/h for zone 1 and 125 $/h for zone 2). The next highest value of Benefit number is 0.625 which corresponds to S2 at zone 2. The flow available from S2 is 144.29 t/h and the flow required from pinch jump is 53.33 t/h. Therefore, 53.33 t/h of flow is added from S2 to zone 2. After this flow addition, the pinch point of zone 2 jumps at DS3 and the overall freshwater requirement is reduced to 181.67 t/h (90 t/h is required in zone 1 and 91.67 t/h is required in for zone 2). The overall operating cost is reduced to 226.67 $/h (135 $/h for zone 1 and 91.67 $/h for zone 2). The values of Benefit number are again calculated for all the internal sources with the updated pinch point (Table S4). The highest value of Benefit number is 0.357 (as flow from S1 is already exhausted in zone 2), which corresponds to S2 at zone 2. The flow transferred from S2 to zone 2 is 54.44 t/h (minimum of 144.29 t/h, flow available in S2 and 54.44 t/h, flow required for pinch



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jump). The pinch point of zone 2 jumps at S2 and the overall freshwater requirement is reduced to 162.22 t/h (90 t/h is required for zone 1 and 72.22 t/h is required for zone 2). The overall operating cost is reduced to 207.22 $/h (135 $/h for zone 1 and 72.22 $/h for zone 2). The values of Benefit number are calculated again with the updated pinch point (Table S4). According to the step 4 of the algorithm, the flow from S2 is neglected as it has 0 values of Benefit number for both the zones. The algorithm terminates as all the internal sources get exhausted. In this example, the results obtained by the proposed algorithm for resource optimality matches with the result obtained by Bandyopadhyay et al.8 However, it is noted that the total operating cost obtained here is not the minimum operating cost for this problem. Two different flow allocation networks for this example are given by Figures 7a and 7b.

Resource 1

(0 ppm)

Resource 2 (10 ppm)

90 t/h

72.2 t/h

Zone 1 Dedicated Sources

Zone 2 Demands

DS1 (800 ppm, 40 t/h)

Demands

20

37.5

50

34.7

D1 (0 ppm, 20 t/h) 50

DS2 (800 ppm, 10 t/h)

20

D2(50 ppm, 100 t/h)

D5 (20 ppm, 50 t/h)

DS3 (150 ppm, 70 t/h)

37.5 D6(50 ppm, 100 t/h)

20

DS4 (250 ppm, 60 t/h)

27.8

D3(50 ppm, 40 t/h)

D7(100 ppm, 80 t/h)

D4(400 ppm, 10 t/h)

D8(200 ppm, 70 t/h) 35

80 5.7 4.3

35

S1 (50 ppm, 50 t/h) 75.7 t/h

10 t/h

Dedicated Sources

12.5

35.7 t/h

S2 (100 ppm, 220 t/h)

50 t/h

107.8 t/h

36.5 t/h

Waste

Figure 7(a). Allocation of water flows (t/h) for example 3.


35 t/h

25 t/h

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Resource 1

(0 ppm)

Resource 2 (10 ppm)

90 t/h

72.2 t/h

Zone 1 Dedicated Sources

Zone 2 Demands

DS1 (800 ppm, 40 t/h)

Demands

20

44.4

50

27.8

D1 (0 ppm, 20 t/h) 50

DS2 (800 ppm, 10 t/h)

20 5.7 2.3

D2(50 ppm, 100 t/h)

D5 (20 ppm, 50 t/h) D6(50 ppm, 100 t/h)

20

DS3 (150 ppm, 70 t/h)

50

DS4 (250 ppm, 60 t/h)

22.2

D3(50 ppm, 40 t/h)

D7(100 ppm, 80 t/h)

D4(400 ppm, 10 t/h)

D8(200 ppm, 70 t/h) 35

80

2

35

S1 (50 ppm, 50 t/h) 75.7 t/h

8 t/h

Dedicated Sources

5.6

37.7 t/h

S2 (100 ppm, 220 t/h)

50 t/h

107.8 t/h

36.6 t/h

35 t/h

25 t/h

Waste

Figure 7(b). Alternate allocation of water flows (t/h) for example 3.

CONCLUSIONS In this paper, a Pinch Analysis based methodology is developed for minimizing the resource consumption in a segregated targeting problem which includes dedicated sources. These sources are dedicated to the zones in which they are present and are not shared with the other zones. They are included in a RAN to accommodate various industrial constraints such as geographical proximity, flexible operations, safety, etc. The rigorously developed methodology for resource optimality identifies a non-dimensional number (J K, ) or Benefit number, which dictates the distribution of flows in these problems. However, the concept of Benefit number is not restricted

to segregated targeting problems with dedicated source and can be extended to accommodate RANs involving multiple zones.



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In this paper, characteristics of pinch jump are also analyzed in detail. The effects of addition and subtraction of flows, from various zones on pinch points are exhaustively studied and mathematically proved. The analysis accurately expresses the flow required for the pinch jump and the location of next pinch point. The applicability of the proposed algorithm, is demonstrated with the help of three different examples from diverse domains which includes cooling water network, carbon constrained energy sector planning and water allocation network. However, as seen from example 3, the algorithm may become iterative and complex when many zones are involved. Additionally, operating cost for these systems can further be minimized by appropriately allocating resources to different zones. The future research is focused on representing the solution graphically so as to get an insight into the problem and determining the cost optimal solution for such problems.


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Nomenclature

Parameters Fsi

Flow of ith internal source (kW/°C, TJ, t/h)

Fdjk

Flow of jth demand of zone k (kW/°C, TJ, t/h)

FDSlk

Flow of lth dedicated source present in zone k (kW/°C, TJ, t/h)

qDSlk

Quality of lth dedicated source present in zone k (°C, t/TJ, ppm)

qdj

Quality of jth demand (°C, t/TJ, ppm)

qrk

Quality of resource at zone k (°C, t/TJ, ppm)

qsi

Quality of ith source (°C, t/TJ, ppm)

qsm

Quality of mth internal source (°C, t/TJ, ppm)

Variables B

Benefit number

fijk

Flow transferred from ith source to jth demand in zone k (kW/°C, TJ, t/h)

fiw

Flow transferred from ith source to waste (kW/°C, TJ, t/h)

frjk

Flow transferred from resource to jth demand of zone k (kW/°C, TJ, t/h)

qpk

Quality of pinch point of zone k (°C, t/TJ, ppm)

R

Resource (kW/°C, TJ, t/h)

Rsi

Resource required in a zone if ith source is the pinch source (kW/°C, TJ, t/h)

Rp

Actual Resource required in a zone (kW/°C, TJ, t/h)

xljk

flow transferred from lth dedicated source to jth demand in zone k (kW/°C, TJ, t/h)

xlkw

flow transferred from lth dedicated source of zone k to waste (kW/°C, TJ, t/h)

W

Waste (kW/°C, TJ, t/h)



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C

∆ α

Page 34 of 43

Flow transferred from internal source to different zones (kW/°C, TJ, t/h) Maximum flow transferred from internal source to change the pinch point (kW/°C, TJ, t/h) Flow transferred from internal source to the zone corresponding to highest value of B (kW/°C, TJ, t/h)

Subscripts DS

Dedicated source

d

Demand

i,m

Source index

j

Demand index

l

Dedicated source index

k

Resource index

max

maximum

min

minimum

p

Pinch point index

r

Resource indices

s

Source

w

Waste

1,2,..., N

indices


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Supporting Information This information is available free of charge via the Internet at http://pubs.acs.org/. Flow and temperature data for example 1 (Table S1), Flow and quality data for example 2 (Table S2), Flow and quality data for example 3 (Table S3), The values of Benefit number (B(m,k)) for different internal sources at different zones for example 3 (Table S4), Material recovery pinch diagram for industrial sector (Figure S1), and Source composite curve for zone 1 of example 3 (Figure S2).


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List of Figure Captions Figure 1

Representation for segregated targeting problem with dedicated sources.

Figure 2

Comparison of reduction in resource flow to determine the correct distribution of internal sources in both the zones.

Figure 3

Algorithm for resource optimality of segregated targeting problems with dedicated sources.

Figure 4

Limiting composite curves for all three zones of example 1.

Figure 5

Allocation of cooling water for example 1.

Figure 6

Allocation of energy for example 2.

Figure 7

(a) Allocation of water flows for example 3. (b) Alternate allocation of water flows for example 3.



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List of Table Captions Table 1. Stream quality and flow for three illustrative examples. Table 2

The values of Benefit number (B(m,k)) for different internal sources at different zones for example 1.

Table 3

The values of Benefit number (B(m,k)) for different internal sources at different sectors for example 2.


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Graphical Abstract Resource 1 "1 ) Dedicated sources

= 1 [#$11 , #$11 ]

Internal source, = 1 [ 1 , 1 ]

[#$ 1 , #$1 ] l = NDS1 [#$#$ 1 1 , #$#$ 1 1 ]

Internal source, [ , ]

Resource 2 "2 ) Dedicated sources

= 1 [#$12 , #$12 ]

Internal source, = [ , ]

[#$2 , #$2 ] l = NDS2 [#$#$ 2 2 , #$#$ 2 2 ]

Waste


Zone 1 Demands = 1 [%11 , %11 ] [% 1 , % 1 ] j = Nd1 [% % 1 1 , %% 1 1 ]

Zone 2 Demands = 1 [%12 , %12 ] [% 2 , % 2 ] j = Nd2 [% %2 2 , % % 2 2 ]

Resource allocation network for segregated targeting problems with

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