Targeting for Multiple Resources - Industrial & Engineering Chemistry

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Ind. Eng. Chem. Res. 2007, 46, 3698-3708

Targeting for Multiple Resources Uday V. Shenoy Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India

Santanu Bandyopadhyay* Energy Systems Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India

The source composite curve is useful in simultaneously setting targets for the minimum freshwater requirement and minimum effluent treatment. The source composite curve also helps in reducing waste production for a variety of applications. In this paper, a methodology is developed to target multiple resources through the source composite curve in order to minimize the operating cost of the overall process. It is observed that it is not necessary to maximize the usage of the resource with the minimum unit cost. Hence, a prioritized cost for each resource is devised to select appropriate resources that minimize the operating cost subject to the availability of the resources. 1. Introduction Techniques of process integration are primarily used for process design (both grassroots and retrofits) with special emphasis on the efficient utilization of resources and the reduction of environmental pollution. Process integration is a system-oriented approach to industrial process design with an objective of sustainable development. Pinch analysis has established itself as a tool for analyzing and developing efficient processes through process integration. Pinch analysis, which began as a thermodynamics-based approach to energy conservation,1 has evolved over the years to become a powerful tool for process integration and resource optimization.2-4 Pinch analysis has been fruitfully used in analyzing heat exchanger networks,3 utility systems,5 mass exchanger networks,6,7 water networks,8-11 distillation columns,12-16 production planning,17,18 renewable energy systems,19,20 etc. Pinch analysis recognizes the importance of setting targets before design. This allows different process design objectives to be screened prior to the detailed design of the process. Pinch analysis also provides the process designer with graphical representation tools for easier visualization and more control over the decision-making processes. Bandyopadhyay et al.11 have introduced an approach based on the source composite curve for simultaneously targeting the minimum freshwater requirement and the distributed effluent treatment system. Bandyopadhyay21 has generalized the concept of the source composite curve to reduce waste production for a variety of applications by applying the unified algorithmic procedure, called the waste targeting algorithm (WTA), to water management, hydrogen management, and material reuse/recycle in a metal degreasing process. Multiple resources may be used to reduce the operating cost of processes. For instance, multiple hot and cold utilities in heat exchanger networks may be selected through profile matching on the grand composite curve.1 Furthermore, the optimum loads of multiple hot and cold utilities may be determined by considering the cost tradeoffs in energy and capital using the cheapest utility principle.5 The principles to select external mass separating agents for synthesizing mass exchange networks have been presented by Fraser et al.22 They have observed that, to * To whom correspondence should be addressed. Tel.: +91-2225767894. Fax: +91-22-25726875. E-mail: [email protected].

minimize the operating cost, it is not necessary to maximize the usage of the external mass separating agent with the minimum unit cost (i.e., cost per unit mass). The methodology based on the composite table algorithm (CTA) for water and hydrogen networks23 has been extended to target multiple resources24 but without considering their cost. In this paper, a methodology is developed to target multiple resources in order to minimize the operating cost of the overall process. The methodology is general enough to be applied to various network flow problems such as water, hydrogen, and material recycle. In a manner similar to Fraser et al.,22 it is observed that it is not necessary to maximize the usage of the resource with the minimum unit cost (i.e., cost per unit mass). Therefore, a prioritized cost for each resource is developed to select appropriate resources that minimize the operating cost subject to the availability of the resources. 2. Problem Statement and Mathematical Formulation The general problem of targeting multiple resources using the source composite curve may be mathematically stated as follows. In a process, a set of Ns internal sources (streams) is given. Each source produces a flow Fsi with a given quality qsi. A set of Nd internal demands (units) is also given. Each demand accepts a flow Fdj with a quality that has to be less than a predetermined maximum limit qdj. There is a set of Nr external sources, and these external sources are called resources. Each resource is available with a quality qrk and a cost per unit flow crk. The availability of each resource is limited to a specified maximum Frk,max. There is an external demand, called waste, without any maximum quality limit or any flow limitation. Flows are denoted by a non-negative real number. Quality is defined by a non-negative real number with a higher numerical value indicating more contamination and, therefore, greater inferiority.21 In other words, a source with a higher numerical value of quality is more contaminated or inferior to another source with a lower numerical value. The objective of this work is to develop an algorithmic procedure with a graphical representation that will identify an optimum strategy for integrating sources and demands to minimize operating cost. A network representing the above problem is shown in Figure 1. Before developing an appropriate mathematical formulation for the above problem, the conservation equations for flows and

10.1021/ie070055a CCC: $37.00 © 2007 American Chemical Society Published on Web 04/27/2007

Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3699 Nr N d

R)

∑∑

Nr

fkj )

k)1 j)1

∑Frk

(8)

k)1

Taking the summation over all internal sources and demands on eqs 3 and 4, the following overall flow balance across the process can be established. Nr N d

Ns

Nd

Ns

Fsi ) ∑Fdj + ∑ fiw ∑∑ fkj + ∑ i)1 j)1 i)1

(9)

k)1 j)1

Using eqs 7 and 8, the overall flow balance across the process in eq 9 can be simplified as Nr N d

R)

Figure 1. Network representing a generalized multiple resource optimization problem.

qualities may be defined.21 Note that two streams with flows F1 and F2 and qualities q1 and q2, respectively, may be mixed to produce a stream with flow F3 and quality q3. The product of quality with flow may be defined as quality load Q. Then, flows and quality loads are said to be conserved because the following relationships are satisfied.

F 1 + F2 ) F3

(1)

F1q1 + F2q2 ) F3q3

(2)

Now, let fij denote the flow transferred from source i to demand j. Similarly, let fkj and fiw represent the flow transferred from resource k to demand j and flow transferred from source i to waste, respectively. Thus, f denotes the flow in the interconnections shown by dashed arrows in Figure 1. Due to the flow conservation in eq 1, the flow balance for every internal source and for every internal demand may be written as follows.

fij + fiw ) Fsi ∑ j)1

∑

k)1

for every internal source i ∈ Ns (3)

Ns

fkj +

(10)

Ns Nd Fsi - ∑j)1 Fdj is a constant for a given process. where ∆ ) ∑i)1 The term ∆ signifies the overall flow loss/gain in the system. A positive ∆ signifies that there is a flow gain in the system, whereas a negative ∆ implies an overall flow loss. By definition, every demand accepts a flow Fdj with a quality that has to be less than a predetermined maximum limit qdj. Utilizing the quality load conservation in eq 2, the quality requirement for any internal demand may be mathematically expressed as

Nr

∑

k)1

Ns

fkjqrk +

fijqsi e Fdjqdj ∑ i)1

for every internal demand j ∈ Nd (11)

The operating cost Φ of the process may be expressed as Nr

Φ)

∑crkFrk

(12)

k)1

Nd

Nr

∑∑ fkj ) W - ∆

k)1 j)1

fij ) Fdj ∑ i)1

for every internal demand j ∈ Nd (4)

Additionally, the total requirement of resource k may be calculated to be Nd

fkj ) Frk ∑ j)1

for every resource k ∈ Nr

(5)

The availability of every resource is limited. This may be expressed mathematically as

Frk e Frk max for every resource k ∈ Nr

(6)

The total waste generation from the process can be expressed as Ns

W)

fiw ∑ i)1

(7)

Similarly, the total resource requirement may be calculated to be

The objective is to minimize Φ subject to the constraints given by eqs 3, 4, 6, and 11. As all the constraints as well as the objective function are linear, this is a linear programming problem. There are (NsNd + NrNd + Ns) flow variables with (Ns + Nd) equality constraints and (Nr + Nd) inequality constraints. Therefore, the degrees of freedom for the linear programming problem are (NsNd + NrNd - Nd). The above linear programming problem is solved below through an algorithmic procedure based on WTA. The proposed algorithmic procedure is associated with a graphical representation called the source composite curve. 3. Targeting Multiple Resources for Minimum Operating Cost The algebraic procedure to target minimum operating cost for a given problem with multiple resources is discussed in this section. The applications of this targeting technique are demonstrated in the subsequent section. The algorithm for generation of the source composite curve is discussed first. 3.1. Source Composite Curve and Waste Targeting Algorithm. Bandyopadhyay et al.11 have introduced the source composite curve for simultaneously targeting the distributed effluent treatment system and the minimum freshwater requirement. Recently, Bandyopadhyay21 has extended the concept of the source composite curve to reduce waste production for a variety of applications. The steps of the waste targeting

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Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007

Table 1. Formulas for Generating Source Composite Curve first column quality

second column net flows

third column cumul flows

fourth column quality load

fifth column cumul quality load

q1 q2 ... qm ... qn

F1 F2 ... Fm ... Fn

F1 F 1 + F2 ... m ∑l)1 Fl ... n ∑l)1 Fl ) ∆

Q1 ) 0 Q2 ) (q1 - q2) F1 ... m-1 Qm ) (qm-1 - qm)(∑l)1 Fl) ... n-1 Qn ) (qn-1 - qn)(∑l)1 Fl)

Q1 ) 0 Q1 + Q2 ) F1 (q1 - q2) ... m m-1 ∑l)1 Ql ) ∑l)1 Fl(ql - qm) ... n n-1 ∑l)1 Ql ) ∑l)1 Fl(ql - qn) ) QT

first row second row ... mth row ... nth (last) row

algorithm (WTA) and generation of the associated source composite curve are outlined below. The formulas for each step are given in Table 1. Step 1: The qualities of all internal sources, demands, and resources are tabulated in decreasing order in the first column. If the value of a particular quality occurs more than once, it is not repeated. Without loss of generality, it can be said that the quality for the mth row is denoted as qm (Table 1) such that

q1 > q2 > ‚‚‚ > qm > ‚‚‚ > qn

(13)

Step 2: Net flows (i.e., algebraic sum of flows corresponding to a quality) are tabulated in the second column. Consider the flows corresponding to internal sources as positive and the flows corresponding to internal demands as negative. For the mth row, the net flow is denoted as Fm (Table 1). Step 3: Cumulative flows are tabulated in the third column. m The summation of the net flows for all previous rows (∑l)1 Fl) gives the cumulative flow for the mth row. The last entry in this column represents ∆ as defined in eq 10. Step 4: The fourth column comprises the quality load for each quality interval (Qm). The quality load for each quality interval may be calculated using the following formula.

[

Qm ) 0

for m )1 m-1

) (qm-1 - qm)(

Fl) ∑ l)1

for m > 1

]

(14)

The first entry in the fourth column is set to zero. For the remaining entries, the difference between the last two qualities is multiplied by the cumulative flow to calculate the quality load. Step 5: Cumulative quality loads are calculated by summing the quality loads for all previous rows ( ∑lem Ql) and tabulated in the fifth column. Using eq 14, the cumulative quality load for the mth row may be calculated from

[

m

Ql ) 0 ∑ l)1

for m ) 1

m-1

)

Fl(ql - qm) ∑ l)1

for m > 1

]

targeting multiple utilities on the grand composite. However, in the present case, resources represent nonpoint utilities. 3.2. Waste Line for the Purest Resource. The source composite curve helps in determining the waste generation.21 The resource requirement can be then determined from the overall flow balance on the system. To determine the waste line (Figure 2) corresponding to the purest resource (i.e., with the lowest value of quality), it may be assumed (without loss of generality) that k ) 1 represents the purest resource (i.e., qr1 < qrk for any k > 1). Then, any line with a negative slope on this quality q vs quality load Q diagram (Figure 2) and passing through the point (QT, qr1) represents a waste line for the purest resource. The equation for the waste line may be expressed as

Q ) QT - W(q - qr1)

(17)

From the above equation, note that the waste line has a negative slope equal to the reciprocal of the waste flow W. It should be noted that, at any quality, the waste line cannot pickup more quality load than what is available (as given by the source composite curve or the cumulative quality load available at any given quality). Therefore, the minimum waste can be targeted (as shown in Figure 2) by rotating the waste line upward with (QT, qr1) as the pivot point until it just touches the source composite curve. If the waste line passes through (QT, qr1) and ( ∑lem Ql, qm) on the quality q vs quality load Q diagram, the waste flow may be determined as m

QT W1m )

Ql ∑ l)1

qm - qr1

for qm > qr1

(18)

The maximum numerical value of the waste flow calculated using eq 18 determines the minimum waste flow for the purest resource based on the waste line that just touches the source composite. In continuation of the previous algorithmic steps,

(15)

The bottom entry in the fifth column signifies the total quality load QT thrown to the waste. n

QT )

∑ l)1

n-1

Ql )

Fl(ql - qn) ∑ l)1

(16)

Now, the cumulative quality load (fifth column) may be plotted against the quality (first column) to obtain the source composite curve. A schematic source composite is shown in Figure 2. It may be noted that the source composite curve is equivalent to the grand composite curve in heat exchanger networks, and targeting multiple resources is analogous to

Figure 2. Source composite curve and targeting for purest resource.

Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3701

Figure 4. Composite waste line with introduction of below pinch resource with reduced purest resource.

Figure 3. Composite waste line with introduction of (a) below pinch resource and (b) above pinch resource.

the following step may be performed to calculate the minimum waste flow for the purest resource. Step 6: The waste flow for the purest resource is calculated using eq 18 for rows such that qm > qr1 and is tabulated in the sixth column (not shown in Table 1, but illustrated through four examples later). The largest entry in this column is the target for the minimum waste flow corresponding to the purest resource. The largest entry may be denoted by W1. The quality that corresponds to the minimum waste flow is called the pinch quality (qp) and the point at which the waste line touches the source composite curve is known as the pinch point. In accordance with eq 10, the flow of the purest resource (that corresponds to the minimum waste flow) may be determined from

Fr1 ) W1 - ∆

(19)

In the absence of any other resource, eq 19 represents the target for the minimum resource requirement and, hence, corresponds to the minimum operating cost. However, this may not be true if multiple resources are available. 3.3. Prioritization of Multiple Resources. The purest resource flow obtained using eq 19 represents the minimum resource requirement as well as corresponds to the minimum operating cost when no other resource is available. If another resource with a quality qr2 (such that qr1 < qr2) is introduced, then another waste line with the same flow is added to the original waste line. Such a composite waste line is shown in Figure 3. The initial part of the composite waste line may be plotted using eq 17 as the waste generated corresponds to the purest resource (i.e., W ) W1 ) Fr1 + ∆ for qr1 < q < qr2). As

the second resource is added with a quality qr2 (with qr1 < qr2), then another waste line with the same flow is added to the original waste line to generate the composite waste line. Equation 17 may be slightly modified to represent the composite waste line Q ) QT - W1(qr2 - qr1) - (W1 + Fr2)(q - qr2). Whenever qr2 < qp, the composite waste line deviates from the pinch point and lies entirely below the source composite curve without touching it (Figure 3a). This implies a possibility to reduce the purest resource by decreasing the corresponding waste flow. On the other hand, the introduction of a resource above the pinch (qr2 > qp) does not reduce the purest resource because the composite waste line still touches the source composite at the pinch point (Figure 3b). Therefore, it may be concluded that no resource should be introduced above pinch. This observation is equivalent to the rules of pinch analysis for heat exchanger network synthesis,1 i.e., hot utility should not be introduced below the pinch and cold utility should not be introduced above the pinch. Similar observations for placing side exchangers in distillation columns have been made by Bandyopadhyay et al.12 The introduction of another resource below the pinch reduces the requirement of the purest resource. However, it should be analyzed whether this leads to a reduction in operating cost. Consider the case where the waste generation is W1 for the purest resource (as in Figure 4). Then, the corresponding operating cost Φ1 may be calculated from

Φ1 ) cr1Fr1 ) cr1(W1 - ∆)

(20)

Without loss of generality, it may be assumed that another resource is available with a quality qr2 such that qp > qr2 > qr1. Assume that the waste generation for the purest resource is reduced by δW (Figure 4). To take care of the resource requirement, a second resource is introduced. Then, the minimum requirement of the second resource and the corresponding operating cost (Φ2) are given by

Fr2 )

W1(qp - qr1) - (W1 - δW)(qr2 - qr1) (qp - qr2)

-

(qp - qr1) (21) (W1 - δW) ) δW (qp - qr2) Φ2 ) cr1Fr1 + cr2Fr2 ) cr1(W1 - δW - ∆) + (qp - qr1) (22) cr2δW (qp - qr2)

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Table 2. Process Data for Example 1 quality (contaminant conc; ppm)

flow (water flow rate; t/h)

Internal Sources 100 100 800

S1 S2 S3

20 100 40

Internal Demands 0 50 50 80

D1 D2 D3 D4

20 100 40 10

Table 3. Resource Specifications for Example 1a quality (contaminant conc; ppm)

cost ($/t)

maximum flow (water flow rate; t/h)

0 25

cr1 cr2

∞ ∞

FW1 FW2

Figure 5. Source composite curve and optimized composite waste line for example 1.

a Actual cost data are not available. However, it may be assumed that cr1 . cr2.

In eq 24, QTk signifies the quality load removed by purer resources and is calculated from

The introduction of the second resource is advantageous if Φ2 < Φ1. Using eqs 20 and 22, this condition may be expressed as

cr2 (qp - qr2)

qrk

(24)

The largest entry in this column is the target for the minimum waste flow corresponding to the kth resource. The largest entry may be denoted by Wk.

k-1

QTk ) QT -

Wl(qrl+1 - qrl) ∑ l)1

(25)

The resource requirement for the kth resource may be then calculated from

[

Frk ) W1 - ∆ for k ) 1 ) Wk - Wk-1 for k > 1

]

(26)

The general equation for the composite waste line may be expressed as follows: k-1

Q ) QT -

Wl(qrl+1 - qrl) - Wk(q - qrk) ∑ l)1

for q > qrk (27)

It may occur that, due to the introduction of an intermediate resource, the pinch point 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. The resource requirement target obtained using eq 26 minimizes the operating cost without considering the maximum availability limitations for different resources. If the resource requirement for the kth resource obtained using eq 26 violates eq 6, the flow of the kth resource should be set to its maximum value (Frk ) Frk,max). After the kth resource is set to its maximum flow, it may be considered as another source and the previously

Table 4. Source Composite Curve and Targeting Multiple Water Resources for Example 1 quality contaminant conc (ppm)

net flows net flow rate (t/h)

cumul flows cumul flow rate (t/h)

quality load mass load (kg/h)

cumul quality load cumul mass load (kg/h)

waste flow for purest resource wastewater flow rate for FW1 only (t/h)

800 100 80 50 25 0

40 120 -10 -140 0 -20

40 160 150 10 10 -10

0 28 3.2 4.5 0.25 0.25

0 28 31.2 35.7 35.95 36.2

45.25 82 62.5 10 10 -

waste flow for first resource wastewater flow rate for FW1 (t/h)

waste flow for second resource wastewater flow rate for FW2 (t/h) 46.39 106 86.36 10

10


Figure 6. Water allocation network with two water resources for example 1 (The values show flow rate in tons per hour with contaminant concentrations in part per million within braces). Table 5. Resource Specifications for Example 2a quality (contaminant conc; ppm)

cost ($/t)

maximum flow (water flow rate; t/h)

0 25 60

cr1 cr2 cr3

∞ ∞ 100

FW1 FW2 FW3

a Actual cost data are not available. However, it may be assumed that cr1 . cr2 . cr3.

described steps may be repeated to obtain the target requirements of other resources. In the following section, the application of the source composite curve for targeting multiple resources is illustrated with various examples. 4. Illustrative Examples 4.1. Example 1 with Two Water Resources. The first example is taken from the domain of water management. The data for this example25 are given in Tables 2 and 3. For this example, the actual water flow rate in tons per hour is considered to be the flow and contaminant concentration in parts per million is considered to be the quality. Water flow rate and contaminant load are conserved due to conservation of mass, and they satisfy eqs 1 and 2. In this example, there are three internal sources and four internal demands. Two water sources (qr1 ) 0 ppm and qr2 ) 25 ppm) exist as resources without any availability limitation, and wastewater is the waste. It may be noted that the costs for different resources are not available. However, with respect to the original example,25 it may be assumed that cr1 . cr2. The objective is to minimize the purest water (with no contaminant) requirement satisfying all the process constraints. The numerical values calculated by applying the proposed algorithm are shown in Table 4.

For this problem, ∆ ) -10 t/h (last entry in the third column) which suggests that there is an overall water loss of 10 t/h. The last entry of the fifth column suggests that 36.2 kg/h of contaminant load is thrown to the wastewater (QT ) 36.2 kg/ h). The source composite curve for this water management example is shown in Figure 5. The sixth column in Table 4 represents the target for purest water resource (i.e., FW1). It identifies the pinch quality as 100 ppm and the minimum wastewater flow rate as 82 t/h. Applying eq 19, the purest water requirement is obtained as 92 t/h. This suggests that, in the absence of FW2, a minimum of 92 t/h of freshwater is required for this example. In the absence of cost data (as mentioned earlier), it may be assumed that the prioritized cost for the first resource is more than that of the second. This assumption is consistent with the original problem definition. This leads to the sequential minimization of the purer resource. The results of step 7 of the proposed algorithm are shown in the seventh and eighth columns of Table 4. The wastewater flow rate due to FW1 is targeted to be 10 t/h, and the wastewater flow rate due to combined FW1 and FW2 is 106 t/h. Applying eq 26, the targets for the two resources are obtained as 20 t/h for FW1 and 96 t/h for FW2. These results are identical to those reported by Wang and Smith.25 A water allocation network satisfying these targets is shown in Figure 6. It may be noted that this network is different from the one presented in the work of Wang and Smith.25 This network is obtained using the nearest neighbor algorithm proposed by Prakash and Shenoy.9 It may also be noted that though the targets for the two resources are unique, the network associated with these targets is not unique. This leads to an opportunity for further study, analysis, and structural optimization of different networks depending on various objective functions such as piping cost, pressure drop, controllability, etc. 4.2. Example 2 with Three Water Resources. The process data for this example25 are the same as those given in Table 2. In this example, there are three water resources (qr1 ) 0 ppm, qr2 ) 25 ppm, and qr3 ) 60 ppm), and the corresponding data24 are given in Table 5. It should be noted that the availability of the third resource (FW3) is limited to 100 t/h. Cost data for the different resources are not available, and it may be assumed that cr1 . cr2 . cr3. The numerical values calculated by applying the proposed algorithm are shown in Table 6. As the process data remain unchanged, the total water loss (∆ ) -10 t/h) and the total contaminant load thrown to the wastewater (QT ) 36.2 kg/h) also remain unchanged. As in the previous example, the freshwater target is obtained as 92 t/h from eq 19 based on the minimum wastewater flow rate of 82 t/h (as per the sixth column in Table 6) corresponding to a pinch quality of 100 ppm. In the absence of cost data, it may be assumed that the prioritized cost for the first resource is more than that of the second and the prioritized cost for the second resource is more than that of the third. The results of

Table 6. Source Composite Curve and Targeting Multiple Water Resources for Example 2

quality contaminant conc (ppm)





800 100 80 60 50 25 0

40 120 -10 0 -140 0 -20

40 160 150 150 10 10 -10

0 28 3.2 3 1.5 0.25 0.25

0 28 31.2 34.2 35.7 35.95 36.2

waste flow for purest resource wastewater flow rate for FW1 only (t/h) 45.25 82 62.5 33.33 10 10


waste flow for second resource wastewater flow rate for FW2 (t/h)

waste flow for third resource wastewater flow rate for FW3 (t/h) 46.22 155 150

50 10 10

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Figure 7. Source composite curve and composite waste line for example 2.

Figure 8. Portion of the source composite curve and optimized composite waste line for the modified example 2.

Table 7. Modified Process Data for Example 2 to Incorporate Resource Availability quality (contaminant conc; ppm) FW3 S1 S2 S3 D1 D2 D3 D4

flow (water flow rate; t/h)

Internal Sources 60 100 100 800

100 20 100 40

Internal Demands 0 50 50 80

20 100 40 10

step 7 of the proposed algorithm are shown in subsequent columns of Table 6. The wastewater flow rate due to FW1 is targeted to be 10 t/h, the wastewater flow rate due to combined FW1 and FW2 is 50 t/h, and wastewater flow rate due to all three sources put together (FW1, FW2, and FW3) is 155 t/h. Applying eq 26, the targets for different resources may be summarized as 20 t/h of FW1, 40 t/h of FW2, and 105 t/h of FW3. These results are identical to those reported by Gupta.24 The source composite curve and the composite waste line are shown in Figure 7. It may be noted that the requirement for the third resource is more than the maximum available limit of 100 t/h. Now, to solve the problem with the availability constraint, FW3 is considered as an internal source with 100 t/h of flow at 60 ppm. The modified data for this problem are given in Table 7. It may be noted that the third resource is no longer available (because it is treated as an internal source stream) and the modified resource data are identical to those given in Table 3. On applying the proposed algorithm for this modified problem, the calculations in Table 8 show ∆ ) 90 t/h (last entry in the third column). This suggests that there is an overall water gain of 90 t/h (which is expected because of the overall water loss of 10 t/h in the original problem and the extra source FW3 of

Figure 9. Water allocation network with three water resources for example 2 (The values show flow rate in tons per hour with contaminant concentrations in parts per million within braces).

100 t/h in the modified problem). The last entry in the fifth column of Table 8 suggests that 42.2 kg/h of contaminant load is thrown to the wastewater (QT ) 42.2 kg/h), which is also consistent with the previous result because an extra contaminant load of 6 kg/h (i.e., 100 t/h × 60 ppm) is introduced in the modified problem. The sixth column identifies the minimum wastewater flow rate to be 142 t/h and the pinch quality as 100 ppm. Then, on applying eq 19, the freshwater target is obtained as 52 t/h. It suggests that, in the absence of FW2, a minimum of 52 t/h of FW1 and 100 t/h of FW3 are required for the original example. The results of step 7 of the algorithm are shown in the remaining two columns of Table 8. The wastewater flow rate due to FW1 is targeted to be 110 t/h, whereas the wastewater flow rate due to combined FW1 and FW2 is 152.67 t/h. Applying eq 26, the targets may be calculated as 20 t/h for FW1

Table 8. Source Composite Curve and Targeting Multiple Water Resources for the Modified Example 2 to Incorporate Resource Availability quality contaminant conc (ppm)





800 100 80 60 50 25 0

40 120 -10 100 -140 0 -20

40 160 150 250 110 110 90

0 28 3.2 3 2.5 2.75 2.75

0 28 31.2 34.2 36.7 39.45 42.2

waste flow for purest resource wastewater flow rate for FW1 only (t/h) 52.75 142 137.5 133.33 110 110


waste flow for second resource wastewater flow rate for FW2 (t/h) 50.9 152.67 150 150 110

110

Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3705 Table 9. Process Data for Example 3 purity (fraction) CRU HCU NHT CNHT DHT

0.80 0.75 0.75 0.70 0.73

HCU NHT CNHT DHT

0.8061 0.7885 0.7514 0.7757

quality (impurity in fraction)

flow (hydrogen flow rate; mol/s)

Internal Sources 0.20 0.25 0.25 0.3 0.27

415.8 1801.9 138.6 457.4 346.5

Internal Demands 0.1939 0.2115 0.2486 0.2243

2495 180.2 720.7 554.4

Table 10. Resource Specifications for Example 3a

IMPORT SRU

purity (fraction)

quality (impurity in fraction)

cost ($/mol)

max flow (hydrogen flow rate; mol/s)

0.95 0.93

0.05 0.07

cr1 cr2

∞ 623.8

Figure 10. Source composite curve and composite waste line for example 3.

a Actual cost data are not available. However, it may be assumed that cr1 . cr2.

and 42.67 t/h for FW2. The source composite curve and the optimized composite waste line for the modified example are shown in Figure 8. Therefore, the targets for the three resources may be summarized as 20 t/h of FW1, 42.67 t/h of FW2, and 100 t/h of FW3. These results agree with those reported by Gupta.24 A water allocation network satisfying these targets is shown in Figure 9. It may be noted that this network is different from the one presented by Gupta.24 4.3. Example 3 with Two Hydrogen Resources. This example is taken from the domain of hydrogen management. The data for this refinery example26 are given in Tables 9 and 10. For this example, the hydrogen volumetric flow rate in moles per second is considered as the flow. Considering an ideal gas, the volumetric flow balance satisfies eq 1. In this problem, the impurity concentration is defined as q ) 1 - y (where y is the hydrogen purity fraction) and it is considered as the quality. Volume conservation for hydrogen may be written as

F1y1 + F2y2 ) F3y3

(28)

On subtracting eq 28 from eq 1, eq 2 may be obtained. It implies that the quality defined above is conserved as in eq 2. In this example, there are five internal sources and four internal demands. There are two hydrogen sources (qr1 ) 0.05 and qr2 ) 0.07) available as resources, and the waste is the fuel system. The cost data for different resources are not available, and it may be assumed that cr1 . cr2. On applying the proposed algorithm, the results obtained are shown in Table 11. For this problem, ∆ ) -790.1 mol/s suggesting an overall hydrogen loss. The last entry in the fifth column suggests that 13.2 mol/s

Figure 11. Source composite curve and optimized waste line for modified example 3.

of impurity has to be thrown to the fuel system (QT ) 13.2 mol/s). The sixth column in Table 11 represents the target for the purest hydrogen resource (i.e., import). It identifies the pinch quality as 0.3 and, in the absence of hydrogen available from SRU, the hydrogen requirement through import may be obtained as 842.7 mol/s on applying eq 19. The results of step 7 of the algorithm are shown in subsequent columns of Table 11. On applying eq 26, the target obtained is 916 mol/s of hydrogen required from SRU with no import necessary. The source composite curve and the composite waste line for example 3 are shown in Figure 10. However, it may be noted that the hydrogen requirement is more than the maximum available limit of 623.8 mol/s from SRU. Therefore, SRU is considered as an internal source with 623.8 mol/s of flow at a quality of 0.07. The problem now simplifies to a single resource optimization problem, and Table 12 shows the results calculated on applying the proposed algorithm. For this modified problem, the value

Table 11. Source Composite Curve and Targeting Multiple Hydrogen Resources for Example 3 quality impurity (fraction)

net flows net flow rate (mol/s)

cumul flows cumul flow rate (mol/s)

quality load impurity load (mol/s)

cumul quality load cumul impurity load (mol/s)

0.3 0.27 0.25 0.2486 0.2243 0.2115 0.2 0.1939 0.07 0.05

457.4 346.5 1940.5 -720.7 -554.4 -180.2 415.8 -2495 0 0

457.4 803.9 2744.4 2023.7 1469.3 1289.1 1704.9 -790.1 -790.1 -790.1

0 13.7 16.1 3.8 49.2 18.8 14.8 10.4 -97.9 -15.8

0 13.7 29.8 33.6 82.8 101.6 116.4 126.8 28.9 13.2

waste flow for purest resource fuel flow rate for import only (mol/s) 52.6 -2.6 -83.2 -103.2 -399.7 -547.8 -688.6 -790.1 -790.1

waste flow for first resource fuel flow rate for import (mol/s)

waste flow for second resource fuel flow rate for SRU (mol/s) 125.9 76.2 -4.7 -26.2 -349.1 -513.6 -673.0 -790.1

-790.1

3706


Table 12. Source Composite Curve and Targeting Hydrogen Resource for Modified Example 3 quality impurity (fraction)

net flows net flow rate (mol/s)

cumul flows cumul flow rate (mol/s)

quality load impurity load (mol/s)

cumul quality load cumul impurity load (mol/s)

0.3 0.27 0.25 0.2486 0.2243 0.2115 0.2 0.1939 0.07 0.05

457.4 346.5 1940.5 -720.7 -554.4 -180.2 415.8 -2495 623.8 0

457.4 803.9 2744.4 2023.7 1469.3 1289.1 1704.9 -790.1 -166.3 -166.3

0 13.7 16.1 3.8 49.2 18.8 14.8 10.4 -97.9 -3.3

0 13.7 29.8 33.6 82.8 101.6 116.4 126.8 28.9 25.6

waste flow for purest resource fuel flow rate for import only (mol/s) 102.5 54.1 -20.8 -40.3 -328.1 -470.6 -605.5 -703.4 -166.3

Table 13. Process Data for Example 4 quality (contaminant; fraction)

flow (material flow rate; t/h)

W1 W2 W3

Internal Sources 0.12 0.2 0.25

20 70 100

G1 G2 G3

Internal Demands 0.1 0.15 0.2

100 20 60

Table 14. Resource Specifications and Respective Prioritized Costs for Example 4 quality (contaminant; fraction)

cost ($/t)

max flow (material flow rate; t/h)

prioritized cost ($/t)

0.05 0.15 0.24

7 3 1

∞ ∞ ∞

35 30 100

R1 R2 R3

of ∆ changes from -790.1 to -166.3 mol/s (because of the additional internal source flow of 623.8 mol/s). The minimum requirement of hydrogen through import is obtained as 268.8 mol/s, i.e., 102.5 - (-166.3) as per eq 19. The source composite curve and the optimized waste line for modified example 3 are shown in Figure 11. Therefore, the final targets may be summarized as 623.8 mol/s of hydrogen from SRU with an additional import of 268.8 mol/ s. These results are the same as those reported by Alves and Towler.26 A hydrogen allocation network satisfying these targets is shown in Figure 12. 4.4. Example 4 with Three Material Resources. This example is taken from the domain of material management. The data for this example27 are given in Tables 13 and 14. Material flow rate in tons per hour is considered to be the flow, and the contaminant fraction represents the quality. Flow and quality load are conserved due to mass conservation law. The numerical values calculated by applying the proposed algorithm are shown in Table 15. For this problem, ∆ ) 10 t/h suggesting an overall material gain. On the basis of the purest resource, the pinch quality is identified to be qp ) 0.25. Then, using eq 23, the

Figure 12. Hydrogen allocation network with two hydrogen resources for example 3 (The values show flow rate in moles per second with hydrogen purity in the fraction within braces).

prioritized costs for all resources are tabulated in the last column of Table 14. It may be noted that the prioritized cost for the third resource is more than that of the purest resource. Therefore, it is beneficial to use the first two resources only. Applying eq 26, the targets may be summarized as 44 t/h for resource R1 and 56 t/h for resource R2. It should be noted that the pinch point jumps to qp ) 0.20 for the second resource. The prioritized cost for resource R2 based on this new pinch point may be calculated to be $60/t, and this is more than that for resource R1. Therefore, for optimum operating cost, the utilization of resource R2 should be such that points (0, 0.25) and (5, 0.20) hold the pinch. The slope of the waste line passing through these points is 100 t/h. This line passes through the point (10, 0.15). The slope of the waste line (corresponding to resource R1) passing through the points (10, 0.15) and (15.9, 0.05) is 59 t/h.

Table 15. Source Composite Curve and Targeting Multiple Resources for Example 4 quality contaminant (fraction)



quality load contaminant load (kg/h)

cumul quality load cumul contaminant load (kg/h)

0.25 0.24 0.2 0.15 0.12 0.1 0.05

100 0 10 -20 20 -100 0

100 100 110 90 110 10 10

0 1 4 5.5 2.7 2.2 0.5

0 1 5 10.5 13.2 15.4 15.9

waste flow for purest resource waste flow rate for R1 only (t/h) 79.5 78.4 72.7 54 38.6 10

waste flow for first resource waste flow rate for R1 (t/h)

waste flow for second resource waste flow rate for R2 (t/h) 105 105.6 110

54 38.6 10


proposed algorithm is applicable for systems with a single quality. Current research is directed toward developing appropriate algorithms for systems with multiple qualities. Nomenclature

Figure 13. Source composite curve and composite waste lines for example 4.

c ) cost per unit flow F ) total flow f ) flow N ) cardinality of a set q ) quality Q ) quality load R ) total waste requirement W ) total waste generation y ) purity of hydrogen ∆ ) overall flow loss/gain in the system Φ ) operating cost of the process Subscripts d ) demand max ) maximum r ) resource s ) source T ) total w ) waste Literature Cited

Figure 14. Material allocation network with two resources for example 4 (The values show flow rate in tons per hour with contaminant in fraction within braces).

Therefore, the final targets may be summarized using eq 26 as (59 - 10) t/h, i.e., 49 t/h for resource R1 and (100 - 59) t/h, i.e., 41 t/h for resource R2, which corresponds to a minimum operating cost of $466/h. These results are the same as those reported by Almutlaq27 using mathematical programming. It should be noted that the minimization of the costliest resource need not always reduce the overall operating cost of the process. Similar observations were also made by Fraser et al.22 The source composite curve and the optimized composite waste line for this example are shown in Figure 13. A material allocation network satisfying these targets is shown in Figure 14. 5. Conclusions A simple conceptual methodology based on pinch analysis for targeting multiple resources has been proposed in this paper. The methodology has been presented in graphical form (as the source composite curve) as well as an algorithm and has been demonstrated through a variety of examples. The methodology is general enough to be applied to problems related to water management and hydrogen management, as well as material management through appropriate reuse and recycle. It has been observed that it is not necessary to maximize the usage of the resource with the minimum unit cost (i.e., cost per unit mass). A prioritized cost for each resource has been developed to select appropriate resources that minimize operating cost subject to the availability of the resources. It may be noted that the

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ReceiVed for reView January 10, 2007 ReVised manuscript receiVed March 24, 2007 Accepted March 27, 2007 IE070055A

Targeting for Multiple Resources - Industrial & Engineering Chemistry

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