Evolution of Resource Allocation Networks - Industrial & Engineering

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Ind. Eng. Chem. Res. 2009, 48, 7152–7167

Evolution of Resource Allocation Networks Ashish K. Das Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

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

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

It is often desirable to evolve a resource allocation network (RAN) into a simpler one for improved controllability, operability, and ease of implementation. In this paper, a methodology, based on the concept of graph theory and process integration, is proposed to evolve a preliminary RAN into a simpler one involving fewer interconnections (matches) by incurring the minimum resource penalty. The proposed methodology is essentially based on four evolution strategies: loop breaking without violation, two-loop relaxation, loop breaking with path relaxation, and direct path elimination. The procedure is applied to various RANs such as water allocation, hydrogen allocation, and material allocation in a reuse network. The evolved networks are compared with results obtained from a new mathematical optimization formulation involving mixed integer linear programming. 1. Introduction Process industries require significant amounts of different natural resources such as energy, freshwater, cooling water, hydrogen, and raw materials. Because of the scarcity of these natural resources and greater awareness toward environment sustainability, efficient resource utilization and waste reduction have become important issues in the process industries. Moreover, considerable increases in the costs of natural resources as well as stringent environmental legislation have forced process designers to optimize resource allocation networks (RANs). Optimization of RANs implies minimization of the resource requirements, the wastes generation, and the operating costs. One of the effective ways for minimizing resource requirement and waste generation is to maximize in-plant reuse-recycle of the resource. Typically, maximum reuse of the resource complicates the allocation network, and it may often be beneficial to simplify the overall network to improve its controllability, operational flexibility, and ease of implementation. In some cases, such as water and hydrogen allocation networks, a simplified network may decrease capital investment through reduction in piping and/or operation costs associated with pumping and compression. 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. Tools of process integration have been successfully applied in analyzing heat exchanger networks,1-4 utility systems,5-10 mass exchanger networks,11,12 * To whom correspondence should be addressed. Tel.: +91-2225767894. Fax: +91-22-25726875. E-mail: [email protected].

water networks,13-17 distillation columns,18-22 fired heaters,23,24 production planning,25,26 renewable energy systems,27-32 energy sector planning with carbon constraint,33 and isolated power systems.34 These methods are either based on the conceptual approach of pinch analysis or based on mathematical modeling and optimization. The conceptual approaches give physical insight into the problem through graphical representations and tabular calculations, while the mathematical approaches are effective in optimizing large-scale systems with complex constraints. Conceptual approaches based on pinch analysis are used for targeting the minimum resource requirement and the minimum waste generation before actually designing the allocation network. This allows a process designer to evaluate the benefits that can be achieved prior to the detailed design of the RAN and thus gives the designer more flexibility in decision making. Various targeting techniques have been proposed such as problem table algorithm for heat exchange networks35 and mass exchange networks;11,36 limiting water profile13 and internal water main37 for water networks; surplus diagram,38 cascade analysis,39 and composite table algorithm40 for both water and hydrogen networks; invariant rectifying-stripping curves18 for distillation columns; and material recovery pinch diagram41,14 for material recycle-reuse. Recently, Bandyopadhyay et al.16 have introduced an approach based on the source composite curve for simultaneously targeting the minimum freshwater requirement and the distributed effluent treatment system. Bandyopadhyay42 has generalized the concept to reduce waste generation for a variety of applications. Shenoy and Bandyopadhyay43 have addressed the resource allocation problem for multiple resources. Apart from targeting, various design techniques have also been proposed to achieve the target. Prakash and Shenoy14 proposed an algorithmic method, known as the nearest neighbors algorithm (NNA), for designing RANs. It may

10.1021/ie9003392 CCC: $40.75  2009 American Chemical Society Published on Web 07/02/2009

Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009

be noted that the resource conservation problems have multiple allocation networks satisfying the minimum resource requirement.44 The primary objective of different methodologies proposed in the literature is to target and design optimum RANs irrespective of the number of interconnections (matches). For a simple plant involving only a few processes, the design and implementation of the optimum RAN may be simple. However, for a large plant involving many processes, the design of the optimum resource network may be complex, and its implementation may be difficult. It may be desirable to evolve the RAN into a simpler one for improved implementation, controllability, operability, and flexibility. A RAN may be evolved by reducing the number of interconnections. Similar to the evolution of heat exchanger networks,45 the interconnections in a RAN may be eliminated by loop breaking. Wang and Smith13 have identified the possibilities of loop breaking for water allocation networks without proposing a detailed methodology. Prakash and Shenoy15 have proposed a conceptual methodology to evolve water allocation networks for fixed flow rate problems through a source-shift technique. However, the procedure based on the source-shift technique is iterative in nature and based on trialand-error. Ng and Foo46 have improved the source-shift technique to reduce the iterations involved. Ng and Foo46 have also proposed the concept of water path analysis (WPA) to systematically reduce the number of interconnections at the cost of freshwater penalty. However, WPA does not always evolve to the best network in terms of freshwater penalty as will be shown later. Bagajewicz47 presented a mathematical model to determine the network with the least capital cost. Bagajewicz47 aimed at fixing the bound, the designer may be willing to accept, on the freshwater penalty and then searched for the network with the least capital cost. As pointed out by Harkisanka,48 this can be misleading as a much improved network may be feasible in terms of lower capital cost at a slightly higher freshwater penalty or in terms of a much lower freshwater penalty at a slightly higher capital cost. Thus, the main drawback of this approach is the fixing of the penalty bounds a priori. Lee and Grossmann49 also proposed a mathematical model with the objective function being the minimization of the freshwater as well as the piping cost of each interconnection. However, all the mathematical approaches discussed above generate a new network with a lesser number of interconnections, which may or may not bear any resemblance to the preliminary RAN. The objective of this paper is to propose a unified and rigorous methodology that can systematically evolve the preliminary RAN. 2. Mathematical Formulation and Problem Statement Evolution of a RAN to get a simplified network is an important step since a network satisfying the minimum resource target may be complex with many recycles and reuse. The evolution of a preliminary RAN may be mathematically stated as follows. In a process, a set of Ns internal sources (streams) and a set of Nd internal demands (units) are given. Each source produces a flow Fsi with a given quality qsi. Each demand accepts a flow Fdj with a quality that has to be less than a predetermined maximum limit qdj. There is an external source called the resource with a quality qrs, and there is an external demand called the waste without any maximum quality limit. There is no flow limitation associated with either resource or waste. Flows are denoted by non-negative real numbers. Quality is defined as a real number with inverse scale such that a higher numerical

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42

value indicates inferiority. In other words, a source with a higher numerical value of quality is inferior to another source with a lower numerical value of quality. A preliminary RAN with an allocation of flows from sources (including resource) to demands (including waste) is also given. An allocation of a positive flow from a source to a demand is called an interconnection. The objective of this work is to develop a methodology that will systematically evolve a preliminary RAN into another allocation network with fewer numbers of interconnections while incurring minimum penalty for resource requirement. The product of quality with flow is defined as quality load (Q). Before developing an appropriate mathematical formulation for the above problem, conservation equations for flows and quality loads are defined.42 Whenever two streams with flows F1 and F2 and qualities q1 and q2, respectively, are mixed (or added), it produces a mixed stream with flow, F3, and quality, q3. Flows and quality loads are said to be conserved as they obey the following relationships: F1 + F 2 ) F3

(1)

F1q1 + F2q2 ) F3q3

(2)

Let fij denote the flow transferred from source i to demand j. In other words, fij denotes an interconnection or a match between source i and demand j. Furthermore, let yij be the binary variable associated with this match: yij is 1 if fij is positive; otherwise, yij is 0. Similarly, frs,j represents the flow from the resource to the demand j, and fiw represents the flow from the source i to the waste. Associated binary variables are represented as yrs,j and yiw, respectively. As a result of flow conservation (1), the flow balance for every internal source and every internal demand may be written as follows. Nd

∑f

ij

+ fiw ) Fsi

for every internal source i

(3)

j)1

Ns

frs,j +

∑f

ij

) Fdj

for every internal demand j

i)1

(4) By definition, every demand accepts a flow Fdj with a quality that has to be less than a predetermined maximum limit qdj. Utilizing quality load conservation eq 2, the quality load requirement for any internal demand may be mathematically expressed as: Ns

frs,jqrs +

∑f q

ij si

e Fdjqdj

for every internal demand j

i)1

(5) The binary variables associated with every interconnection can be determined from the following relationships: frs,j - Umaxyrs,j e 0

(6)

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fij - Umaxyij e 0

(7)

fiw - Umaxyiw(i) e 0

(8)

frs,j - Uminyrs,j g 0

(9)

fij - Uminyij g 0

(10)

fiw - Uminyiw(i) g 0

(11)

Table 1. Limiting Process Data for Example 1 sources

max

min

U and U are positive constants with very high and very low values. These constants along with the above equations make sure that the value of a binary variable is zero or unity whenever the associate continuous variable is either zero or positive. Then the total number of interconnections (I) in any given network is given by Nd

I)

∑y

rs,j

j)1

Nd

+

Ns

∑ ∑y j)1 i)1

Ns

ij

+

∑y

iw

(12)

i)1

Total resource requirement can be expressed as: Nd

R)

∑f

rs,j

(13)

j)1

The objective is to minimize R subject to a given value of I and the constraints given by eqs 3-11. Evolution of a given RAN aims at minimizing resource requirement R subject to the constraints in eqs 3-11 such that number of interconnections is at least one less than that originally. This problem can be now solved as a mixed integer linear programming (MILP) formulation. In this paper, a methodology based on four evolution strategies is proposed, and the resource requirement at every stage is compared with the result obtained by solving the MILP problem. 3. Representing Resource Allocation Networks (RANs) A RAN consists of sources (including resource) and demands (including waste). An interconnection or a match connects a source with a demand. A RAN may be represented as a bipartite graph. Often, when there are a large number of processes, the graphical representation in the form of a process flow diagram is tedious to design and understand because of the complex and large number of recycles and reuse. Additionally, in the language of graph theory, a bipartite graph with higher number of nodes is not planar. That means a bipartite graph with higher number of nodes cannot be drawn without their edges crossing. This leads to overlapping of different interconnections in case of a RAN. The process flow diagram of a RAN can be conveniently and compactly represented as a matching matrix. Both these representations are equivalent to each other. The matching matrix representation was first introduced by Prakash and Shenoy14 for water allocation networks. In a matching matrix, the sources appear as rows while demands appear as columns, and they are arranged in increasing order of quality.15 Flow transferred (fij) from source i to demand j is represented as the (i, j) element of the matching matrix. The sum of all the flows in a particular row equals the total flow available from the particular source, while the sum of all the flows in a particular column equals the total flow required for the particular demand. The matching matrix representation is useful during network evolution because the matches to be

S1 S2 S3 S4

demands

flow rate F (t/h)

quality q (ppm)

50 100 70 60

50 100 150 250

D1 D2 D3 D4

flow rate F (t/h)

quality q (ppm)

50 100 80 70

20 50 100 200

removed and incorporated can be quickly identified. A small modification is done in the matching matrix representation proposed by Prakash and Shenoy.15 Two rows at the bottom and one column at the right are added. The modified matching matrix representation helps in quickly checking the flow and quality constraints. The total number of nonzero elements in a matching matrix represents the total number of interconnections or matches. The matching matrix and the process flow diagram are dual of each other since one representation can be developed using the other. It may be noted that the matching matrix is the adjacency matrix of the process flow diagram. The modified matching matrix representation for a RAN is illustrated through an example. Let us consider the fixed flow rate example 1 for water management consisting of four sources and four demands. The process data50 for this example are given in Table 1. Here, quality is defined in terms of the contaminant concentration in ppm, and the water flow rate in t/h is considered as flow. The resource is freshwater with qrs ) 0 ppm, while the waste is wastewater in this case. On the basis of various targeting algorithms, the minimum freshwater requirement is determined to be 70 t/h and the corresponding wastewater flow rate is 50 t/h. One of the many possible water allocation networks, generated based on NNA, is shown in Figure 1. The first column of the matching matrix represents all the sources with their flow rate and contaminant concentration in brackets. The first row indicates all the demands with flow rate as well as the maximum contaminant concentration in brackets. Each element represents the allocation of water from a source to a demand. The last two rows represent the actual flow rate and contaminant quality for every demand. Similarly, the last column represents the total flow rate available from each source. As stated earlier, the last two rows and the last column are added to the original matching matrix to provide a quick check for the flow and quality constraints during evolution. It may also be noted that the pinch division can be illustrated on the matching matrix. For example 1, the pinch concentration is 150 ppm. The upper rectangular box (shaded in Figure 1) involves sources below the pinch and demands above the pinch, and similarly, the lower rectangular box (shaded in Figure 1) involves sources above the pinch and demands below the pinch. These elements represent cross-pinch flow transfer. To meet

Figure 1. Representing one of the many possible water allocation networks satisfying the minimum requirement of freshwater for example 1 through matching matrix. A loop and a path are also illustrated on the matching matrix.


the minimum resource target, there should not be any crosspinch flow transfer.44 4. Elements of a Resource Allocation Network (RAN): Loops and Paths Prior to developing different evolution strategies, the basic elements in a RAN, such as loops and paths, and their properties are discussed in this section. 4.1. Loops. A loop in a RAN is defined by a series of connected matches such that it starts and ends at the same node (either a demand or a source). Loops in the water allocation network of example 1 are illustrated in Figure 1. The loop 2 (in Figure 1) consists of the following interconnections: FWD2, FW-D3, S2-D3, and S2-D2. Some of the properties of a loop are identified below. (i) Every Loop Contains an Even Number of Matches. This property can easily be proved as follows. Without loss of generality, let us start from a source node. A match from that source will lead to a demand. To complete the loop, we have to come back to the same source. Therefore, another match is required to connect a demand to a source. Every odd match from a source node ends in a demand node, and always an even number of matches are required to come back to a source. Hence a loop always contains an even number of matches. It may be noted that this observation is true for any bipartite graph. (ii) There Exists No Loop with Only Two Matches. Since parallel matches can always be combined, it may be easily proved that every loop contains four or more number of matches. Note that in heat exchange networks (HENs) and mass exchange networks (MENs), parallel matches can exist, and thus a loop with only two exchangers can exist between only two streams. (iii) Loops Can Be Categorized into Two: Above-Pinch and Below-Pinch. The RAN is divided into two: an abovepinch region and a below-pinch region. It has been proved that there exist no cross-pinch interconnections when the minimum resource is utilized.44 The loop 2, illustrated in Figure 1, is a below-pinch one, as all the participating sources and demand are below the pinch. However, it may be noted that as the resource requirement increases during evolution, cross-pinch loops may be formed. In contrast, cross-pinch loops may exist in HENs, and utility penalties may be incurred when these loops are broken. (iv) Flow is Conserved in a Loop. If δ amount of flow is added to every alternate match and δ amount of flow is reduced from the remaining ones, flows from every participating source node and flows to every participating demand node will remain constant. Therefore, if the flows in a loop are perturbed as described above, flow is conserved; however, quality constraints (eq 5) for some of the demands may be violated. (v) Matches in a Loop Can Be Partitioned into Two Groups. It may be noted from the previous property that matches in a loop can be partitioned into two groups of every alternate connection. During flow perturbation as described earlier, flows through every match in one group increases by an equal amount while flows through every match in the other group get reduced by the same amount. (vi) Loops May Be Broken and Number of Matches Reduced through Flow Perturbation. In any graph, the number of interconnections and nodes always obey Euler’s theorem. Similar to HENs, every loop in a RAN accounts for an additional match. Therefore, the number of interconnections can be reduced by breaking a loop. By perturbing the flow in a loop, flow through an interconnection in either group can be made zero and thereby the number of matches can be reduced.

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As mentioned earlier, while breaking a loop through flow perturbation, the quality constraints (eq 5) for some of the demands may be violated. Let δ be the flow perturbation in the loop, then penalty (in terms of quality load) for a demand j is given by P ) δ(qdjC - qdj)

(14)

where qdjC is the actual quality of demand j obtained after loop breaking. For the purpose of evolution of RANs, we generalize the definition of a loop to include matches with zero flows. We call such loops with one or more zero-flow matches artificial loops. For such an artificial loop, all properties remain valid. However, care should be taken during flow perturbation to ensure that the flow through any match should not be negative. 4.2. Paths. A path in a RAN is defined by a series of connected matches such that it starts from a resource and ends at the waste. One of the paths for the water allocation example is also highlighted in Figure 1. The path consists of the following interconnections: FW-D3, S3-D3, and S3-WW. Some of the properties of a path are identified below. (i) Every Path Contains an Odd Number of Matches. This property can be proved as follows. Every match from a resource will lead to a demand, and every match to the waste will come through a source. To complete a path, we have to connect the demand to the source through only odd matches. Therefore, every path contains an odd number of matches. (ii) There Exists No Path with a Single Match. The only path that can contain only a single interconnection is the direct path between a resource and the waste. However, such a path is not possible.44 Therefore, every path must contain at least three matches. (iii) Every Path Must Be a Cross-Pinch Path. Since a path involves a resource and the waste and they lie on different sides of the pinch, every path must be cross-pinched. (iv) Flows Are Conserved for Every Internal Source and Demand. Similar to a loop, δ amount of flow is added to every alternate match and δ amount of flow is reduced from the remaining ones. Then, flows from every participating internal source and flows to every participating internal demand will remain the same. Therefore, if the flows over a path are perturbed as described, flows are conserved for every internal source and demand. However, because of overall conservation of flows, flow from the resource and flow to the waste will be changed by the same amount. It may be noted that if flow is perturbed as described, the quality constraints (eq 5) for some of the demands may be violated. (v) Matches in a Path Can Be Partitioned into Two Groups. From the previous property, it is clear that matches in a path can be partitioned into two groups of alternate connections. One group includes both the resource-demand and the source-waste matches, while the other group involves only demand-source matches. During flow perturbation as described earlier, flows through every match in one group increase by an equal amount while flows through every match in the other group reduce by the same amount. (vi) Quality Constraints May Be Restored and Number of Matches Reduced through Path Relaxation. Loop breaking may give rise to a violation of a demand. The violation can be restored within its quality limits by increasing the resource requirement on identifying a path that involves the demand in question. Similarly, an interconnection may be eliminated through appropriate flow perturbation in any path. It should be noted that the amount of extra resource added should equal the

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amount of extra waste generated due to overall conservation of the flow. This method of restoring the quality constraint of a demand or reducing the number of matches is referred to as path relaxation. For the purpose of evolution of RANs, we generalize the definition of path to include matches with zero flows. We call such generalized paths with one or more zero-flow matches as artificial paths. For such an artificial path, all properties remain valid. As before, care should be taken during flow perturbation to ascertain that the flow through any match should not be negative. 5. Evolution of Resource Allocation Networks (RANs) For evolution of a RAN, the following four basic strategies are proposed: (i) loop breaking without violation; (ii) two-loop relaxation; (iii) loop breaking with path relaxation; and (iv) direct path relaxation. Out of these strategies, both loop breaking without violation and two-loop relaxation result in zero penalty for the resource requirement as these procedures do not involve any flow transfer across the pinch. Loop breaking without violation is a singlestep procedure because no quality load constraint for any demand will be violated. On the other hand, two-loop relaxation is a two-step procedure since the violated quality constraint for a demand is restored by flow perturbation through another loop involving the particular demand. Therefore, loop breaking without violation is preferred ahead of two-loop relaxation strategy. The last two strategies, that is, loop breaking with path relaxation as well as direct path relaxation, involve path relaxation and thereby a resource penalty. Thus, to meet our objective, these two strategies are applied after loop breaking without violation and two-loop relaxation strategies. Loop breaking with path relaxation is a two-step procedure where the violated quality constraint for a demand is restored through path relaxation. On the other hand, direct path relaxation is a single-step procedure. Depending on the topology of the RAN, loop breaking with path relaxation may give less, more, or the same penalty as compared to the direct path relaxation procedure. Therefore, depending on the resource penalty, either strategy may be adopted first. The criterion for selecting the appropriate strategy is discussed later. Development of these strategies along with different criteria for appropriate application to optimal evolution of a preliminary RAN is discussed in this section. 5.1. Loop Breaking without Violation. Loop breaking without violation can be achieved in either of the following two cases: first, when there are no constraints on the quality load for some demands; and second, when the combined quality of the flows supplied to a demand is lower than its maximum allowable value. The first case arises when a loop is formed involving waste, while the second case arises often at the later stage of the evolution when the resource penalty relaxes the quality load constraints for some demands (i.e., eq 5 becomes a strict inequality). In both these cases, the demand can accept more flows from a dirtier source in exchange of flows from a cleaner one. In the former case, it is always possible to reduce one match through appropriate flow perturbation while in the latter case, the amount of flow perturbation is bounded by both the flow and quality constraints. Therefore, it may or may not be possible to reduce a match. Let the demand Dk receive a combined quality of qdkC which is less than the maximum allowable limit of qdk. The maximum flow that can be exchanged without violating any demand is given by

x)

Fdk(qdk - qdkC) ∆qL

(15)

where ∆qL is the quality difference of the loop chosen. Case (a). If x g δ (δ being the flows through an interconnection that has to be eliminated from the loop), then it is possible to get rid of at least one match without paying any penalty on the resource requirement. Case (b). If x < δ, then removing the match will violate the corresponding demand, and hence, loop breaking without violation is not applicable. Three independent loops for example 1 with 12 matches are identified in Figure 1. In Figure 1, Loops 1 and 2 involves demands that are already accepting maximum quality load. These loops cannot be broken without quality load violation. However, loop 3 (S3-D4, S3-WW, S4-WW, and S4-D4) is the waste-process loop involving wastewater. As there is no quality load restriction on waste, this loop can be broken without violation. There are two matches involving wastewater with the smallest flow rate, S3-WW and S4-WW, both having a flow rate of 25 t/h. By a flow perturbation of 25 t/h, either of these matches can be eliminated. If the S4-WW match is removed, 60 t/h of flow from S4 and 10 t/h of flow from S3 will be supplied to D4. This will violate the quality load constraint of D4. On the other hand, the S3-WW match can be safely be eliminated as a cleaner source of 25 t/h at 150 ppm (from S3) is shifted from waste to demand D4 while an equivalent amount is shifted from demand D4 to waste. It may be noted that the wastewater quality changes from 200 ppm to 250 ppm after eliminating S3-WW match. The resultant network with 11 matches is shown in Figure 2. 5.2. Two-Loop Relaxation. The strategy of two-loop relaxation involves two loops with two demands. When the flow is perturbed in a primary loop involving two demands, flow from one demand is transferred to the other. In the process, the quality constraint for the demand that is accepting flow from a dirtier source will be violated whereas the demand that accepts flow from a cleaner source will be relaxed. Therefore, the second demand can accept flow from another dirtier source in exchange of flow from a cleaner one. Thus, flow can be perturbed in a secondary loop involving these two demands to restore the quality constraints. In Figure 3, the schematic diagram of both the primary and the secondary loops for two-loop relaxation are shown. There are three sources Si, Sj, and Sk with quality qsi, qsj, and qsk, respectively, such that qsi < qsj < qsk. The primary loop involves Sj and Sk, while the secondary loop involves Si and Sj. Both loops involve demands Dm and Dn. Let y amount of flow be perturbed in the primary loop and g(y) amount of flow be perturbed in the secondary loop. The modified flows through every match are shown in Figure 3. It has already been proved that the flow is always conserved in a loop. However, the same cannot be claimed for conservation of quality load. Simultaneous flow perturbations in both the loops will not result in violation, provided the following relation holds: g1(y) ) y

(

qsk - qsj qsj - qsi

)

(16)

It may be noted that the participating interconnections can be partitioned into two sets: one involving - signs (viz., Si-Dm, Sj-Dn, and Sk-Dm) and the other involving + signs (viz., Si-Dn, Sj-Dm, and Sk-Dn). Now, any of the three elements in a particular set can be eliminated using an appropriate value of y. To eliminate one match from the negative set, either g1(y) )


Figure 2. Evolved network containing 11 matches for example 1 after eliminating the waste-process loop.

Figure 3. Perturbing primary and secondary loops in a RAN for two-loop relaxation strategy.

Fim or g1(y) + y ) Fjn or y ) Fkm. Therefore, the appropriate value of y may be obtained as follows:

( (

y ) min Fim

) (

) )

qsj - qsi qsj - qsi , Fjn , Fkm qsk - qsj qsk - qsi

(17)

The choice of y as the minimum value from the set ensures that flows through none of the interconnections will be negative. The match corresponding to the minimum from the above will be eliminated. Similarly, to eliminate one match from the positive set, the appropriate value of y may be obtained as follows:

( (

y ) max -Fin

) (

) )

eliminated while one match will be introduced in the negative set. Thus, the overall number of interconnections can be reduced from 11 to 10. The new network is shown in Figure 4. Two more candidates from Figure 4 may be considered for two-loop relaxation. One involves sources FW, S1, and S2 with demands D1 and D2. The other candidate involves sources FW, S1, and S3 along with demands D1 and D2. It may be noted that both of these candidates contain artificial matches. Although application of the two-loop relaxation strategy does not reduce number of interconnections, topologically different networks are obtained. These networks are shown in Figure 5. 5.3. Loop Breaking with Path Relaxation. Breaking a loop through flow perturbation may violate the quality constraint for one of the participating demands. The quality constraint for that demand may be restored through path relaxation. The minimum amount of additional resource required to restore the quality constraint for the violating demand may be determined as follows. Suppose, after loop breaking, a demand Dk, having a maximum allowable quality of qdk, receives flows from different sources at a quality qdkC such that the quality constraint is violated, that is, qdkC > qdk. Let a path be chosen to restore the quality constraint such that the sources SR and ST be the participating sources with respect to demand Dk (Figure 6). Further, let P be the required penalty in resource flows. Hence to make qdkC equals to qdk, penalty P must be added from the cleaner source (SR in Figure 6) while the same amount should be reduced from the dirtier source (ST in Figure 6). Then, the overall quality load balance for the demand may be written as F1kqs1 + ... + (FRk + P)qsR + ... + (FTk - P)qsT + ... + FNkqsN ) Fdkqdk (19) Rearranging the above equation, penalty P can be calculated directly: N

(18)

It may be noted that if two or all the three values within the brackets in eqs 17 or 18 are same, it is possible to reduce a larger number of matches through the two-loop relaxation procedure. The same method can be applied with artificial matches (i.e., matches involving zero flow). Care should be taken as flow through any element cannot be negative. However, if more artificial elements are present, then they have to be in the same set. Two-loop relaxations will either simply reduce the number of interconnections or give a structurally different network with same number of interconnections. The latter case may arise when any of the loops involves an artificial element. If no artificial element is involved in either of the loops, then it is always possible to get rid of at least one match. The modified network shown in Figure 2 does not involve any candidate for two-loop relaxation without artificial elements. One of the possible candidates with an artificial element involves sources FW, S2, and S3 with demands D2 and D3. It may be noted that flow through the match S3-D2 is zero. Applying eqs 17 and 18, it may be noted that y is either a minimum of {70, 130/3, 0} or a maximum of {-10, -70/3, -10}. Therefore, by choosing y ) -10, two matches from the positive set can be

∑F q

ik si

P)

qsj - qsi qsj - qsi , -Fjm , -Fkn qsk - qsj qsk - qsi

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- Fdkqdk

i)1

∆qsP

with ∆qsP * 0

(20)

where ∆qsP () qsT - qsR) denotes the quality difference of the participating sources of the path chosen to restore the quality constraint of the violating demand. From eq 20, it may be noted that to relax the quality constraint of a violating demand through path relaxation, a path must be identified through the violating demand such that it must contain two participating sources with different qualities. Furthermore, the penalty (P) is inversely proportional to ∆qsP. Therefore, the penalty can be minimized through selecting a path such that ∆qsP is as high as possible. When the combined quality of the flows supplied to a demand is lower than its maximum allowable value, it may be possible to break the loop without violation. Otherwise, the minimum penalty required can be calculated using eq 20. Now, we will separately analyze the case when the demands are already receiving flows with the maximum allowable quality. This condition normally arises at the start of the evolution as various design methodologies such as NNA ensure that all the demands have their quality equal to the maximum allowable value.14 Now, if a demand is receiving flows at the maximum allowable quality, then the quality load balance equation is as follows: F1kqs1 + ... + FRkqsR + ... + FTkqsT + ... + FNkqsN ) Fdkqdk (21)

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Figure 4. Evolved network containing 10 matches for example 1 obtained using two-loop relaxation procedure.

Figure 5. Topologically different networks containing 10 matches for example 1 are obtained using two-loop relaxation procedure: (a) two-loop relaxation on sources FW, S1, and S2 with demands D1 and D2 and (b) two-loop relaxation on sources FW, S1, and S3 along with demands D1 and D2.

Let δ be the flow perturbation in a loop (Figure 6) which leads to the violation of a participating demand. On the basis of the value of δ, the quality constraint for either of the demands may be violated. If δ is positive, then the quality constraint for demand Dk will be violated as flow from a cleaner source (SR) is replaced by flow from a dirtier source (ST). On the other hand, for a negative δ, the quality constraint for demand Dm will be violated. Without loss of generality, δ is assumed to be positive. F1kqs1 + ... + (FRk - δ)qsR + ... + (FTk + δ)qsT + ... + FNkqsN ) Fdkqdk + δ(qsT - qsR) (22) Substituting eq 22 into eq 20, we get P)

δ∆qsL ∆qsP

(23) Figure 6. Representing a flow perturbation.


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Figure 7. Evolved network containing 9 matches for example 1 obtained using loop breaking with path relaxation procedure.

where ∆qsL () qsT - qsR) denotes the quality difference of the participating sources of the loop where flow perturbation will lead to the violation of the demand. Let us now define ξ as the ratio of the quality difference between the participating sources of the loop to be broken to that between the participating sources of the path to be chosen for restoring the quality load violation. ξ)

∆qsL ∆qsP

(24)

Whenever ξ < 1, that is, the loop involves participating sources with lesser quality difference compared to that of the participating sources of the path, eq 23 suggests that the penalty will be less than the amount of flow perturbed to break the loop. On the other hand, whenever ξ g 1, eq 23 suggests that the penalty will be greater than or equal to the amount of flow perturbed to break the loop. In such a case, it is preferred to go for direct path relaxation strategy as the penalty involved in a direct path relaxation is always equal to the flow perturbed to reduce a match. Three different water allocation networks for example 1 with 10 matches are shown in Figures 4 and 5. The water allocation network shown in Figure 4 contains a loop involving the following interconnections: FW-D1, FW-D2, S1-D2, and S1D1. If the match S1-D1 is eliminated by perturbing 20 t/h of flow, D2 which is already receiving flows at its maximum allowable quality limit will get violated. It can be restored by relaxing through the path FW-D2, S3-D2, S3-D4, S4-D4, and S4-WW. Applying eq 23, the penalty is calculated to be 6.67 () 20 × 50/150) t/h. It may be noted that the match FW-D1 (and simultaneously S1-D2) cannot be eliminated as a path to restore the quality constraint for demand D1 does not exist. It should also be noted that the penalty of 6.67 t/h may not be the smallest possible. To find the smallest penalty other water allocation networks, shown in Figure 5, should also be examined. The water allocation network shown in Figure 5b contains a loop involving the following interconnections: FW-D1, FWD2, S3-D2, and S3-D1. If the S3-D2 interconnection is removed, the penalty is calculated to be 3.33 () 3.33 × 150/150) t/h. On the other hand, a penalty of 6.67 () 6.67 × 150/150) t/h is incurred if the match S3-D1 is removed. Similarly, the loop containing the matches FW-D1, FW-D2, S2-D2, and S2-D1 in Figure 5a can be broken. It may be observed that the match S2-D2 cannot be removed as it will leave no path for relaxation. The penalty for removing the match S2-D1 is 6.67 () 10 × 100/150) t/h. Therefore, the minimum penalty of 3.33 t/h is

incurred when the match S3-D2 in Figure 5b is eliminated. The new network with 9 matches is shown in Figure 7. It may be noted that for evolution of HENs, a “smallest heat load” heuristic is typically used related to a loop breaking. It suggests that the match with the smallest heat load in a loop is broken and the temperature penalty is usually relaxed using path relaxation. An equivalent heuristic can be developed for evolution of RAN. It has been proved earlier that matches in a loop in a RAN can be partitioned into two groups. Smallest flow match (equivalent to the smallest heat load in HEN) in either group may be broken. Therefore, instead of the overall smallest, the smallest from either group may be broken. However, this generalized heuristic is applicable to HEN also. 5.4. Direct Path Relaxation. As discussed earlier, interconnections in a path can be partitioned into two groups of alternate connections: one includes both the resource and the waste, while the other involves only demand-source matches. A match with the minimum flow in the group that does not include the resource and the waste can also be eliminated directly through flow perturbation. In direct path relaxation, the penalty incurred exactly equals the flow of the match eliminated. Direct path relaxation is applied when no loop exists but a path exists, or if the penalty is greater than or equal to the amount of flow perturbed to break the loop as discussed earlier. Ng and Foo46 have also proposed this approach. However, as pointed out earlier, this may incur higher penalties than other strategies. Figure 7 shows the water allocation network for example 1 with 9 matches. This network does not contain any loop, but there still exists a path (FW-D1, S3-D1, S3-D4, S4-D4, and S4-WW). Interconnections in this path can be partitioned into two groups. One group contains FW-D1, S3-D4, and S4-WW, while the other group contains S3-D1 and S4-D4. Note that the flow through both the matches S3-D1 and S4-D4 is the same. Therefore, both these matches can be eliminated directly through flow perturbation by incurring a penalty of 6.67 t/h. The final evolved network with 7 matches is shown in Figure 8. 6. Illustrative Examples Applications of the proposed evolution methodology to different examples are illustrated in this section. 6.1. Example 1 of Water Allocation Network. The process data50 for this water allocation example are given in Table 1. The minimum freshwater requirement is determined to be 70t/h and the corresponding wastewater flow rate is 50t/h. Figure 1 represents a network with 12 matches. As discussed earlier, subsequent networks with reduced number of matches are shown in Figures 2, 4, 5, 7, and 8. It may be noted that the network

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Figure 8. Evolved network containing 7 matches for example 1 obtained using the direct path relaxation procedure.

Figure 9. Intermediate network containing 8 matches for example 1. Table 3. Process Data for Example 2

Table 2. Comparison of Results for Example 1 freshwater flow rate (t/h)

sources

this work number of matches

Prakash and Shenoy15

Ng and Foo46

evolution strategy

MILP formulation

12 11 10 9 8 7

70 70 70 74 80 80

70 70 70

70 70 70 73.33 76 80

70 70 70 73.33 76 80

80 100

shown in Figure 7 contains 9 matches, while the network shown in Figure 8 contains 7 interconnections. A network containing 8 matches can easily be constructed from Figure 7 using the loop breaking with path relaxation strategy. Consider the artificial loop S3-D1, S3-D4, S4-D4, and S4-D1. Since the matches S3-D1 and S4-D4 have the same flow, both of them can be eliminated by flow perturbation while a new match (i.e., S4-D1) is added. This results in a quality constraint violation for demand D1, which can be restored by relaxing through the path FW-D1, S4-D1, and S4-WW. Applying eq 23, the additional penalty is calculated to be 2.67 () 6.67 × 100/250) t/h. The intermediate network with 8 matches is shown in Figure 9. The final network (Figure 8) may be obtained from the network shown in Figure 9 through direct path relaxation. The same example has been solved by Prakash and Shenoy15 and Ng and Foo46 through different evolution techniques. Table 2 compares the results obtained in this paper with those reported in the literature as well as with the minimum penalty obtained through the mathematical optimization technique. It may be noted that the results obtained through the proposed methodology are identical to those obtained by mathematical optimization based on the MILP formulation in Section 2. The results are

S1 S2 S3 S4 S5

demands

flow rate F (t/h)

quality q (ppm)

120 80 140 80 195

100 140 180 230 250

D1 D2 D3 D4 D5 D6

flow rate F (t/h)

quality q (ppm)

120 80 80 140 80 195

0 50 50 140 170 240

better than those reported by Prakash and Shenoy15 using the source-shift technique and those reported by Ng and Foo46 using the improved source-shift algorithm and water path analysis. 6.2. Example 2 of Water Allocation Network. To illustrate the application of the proposed evolution procedure, another example related to a water allocation network consisting of six demands and five sources is considered. The process data51 are shown in Table 3. Similar to the previous example, the actual water flow rate in t/h is considered as flow, while the contaminant concentration in ppm is considered as quality. The resource is freshwater with a quality of 0 ppm, and the waste is wastewater. The minimum water flow rates are targeted to be 200 t/h for freshwater and 120 t/h for wastewater, with two pinch qualities identified at 100 and 180 ppm. One of the various possible water allocation networks, obtained using NNA, is shown in Figure 10. This initial network contains 15 interconnections. The loop, consisting of S3-D6, S3-WW, S5-WW, and S5D6, involves waste. The match S5-WW can be eliminated without any violation. The subsequent network containing 14 interconnections is not shown for brevity. Application of the two-loop relaxation involving sources S1, S2, and S3 with demands D4 and D5 produces a topologically different network


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Figure 10. Initial water allocation network containing 15 matches for example 2.

Figure 11. Topologically different water allocation network containing 14 matches for example 2, obtained using two-loop relaxation.

Figure 12. AIntermediate suboptimal network containing 14 matches for example 2.

as shown in Figure 11. To obtain a network with 13 interconnections incurring the least penalty, the network shown in Figure 11 is evolved through an intermediate nonoptimal network. First, the match S1-D3 in the artificial loop with elements S1-D3, S1-D4, S2-D4, and S2-D3 is eliminated. The quality constraint for demand D3 is violated, and it can be restored through path relaxation with a penalty of 11.43 t/h. The resultant intermediate network is shown in Figure 12. It may be noted that both the demands D4 and D5 receive flows with qualities that are less than the allowable maximum. Therefore, it is possible to break the loop with elements S2-D4, S2-D5, S3-D5, and S3-D4 without any violation. The match S2-D5 cannot be eliminated without violation. However, the match S2-D4 can be eliminated without any violation. The resultant network with 13 interconnections is shown in Figure 13. Interconnections S2-D3 and S3-D5 have the same flow, and both of them are eliminated simultaneously by perturbing 28.6 t/h of flow through the artificial loop, involving the intercon-

nections S2-D3, S2-D5, S3-D5, and S3-D3. Demand D3 which is already receiving flows at its maximum allowable quality limit is violated, and it is restored by relaxing through the path FWD3, S3-D3, S3-D6, S5-D6, and S5-WW. Applying eq 23, the additional penalty is calculated to be 6.35 t/h. The water allocation network with 12 interconnections is shown in Figure 14. Now, the interconnection S3-D3 can be eliminated through the direct path relaxation of the path FW-D3, S3-D3, S3-D6, S5-D6, and S5-WW with an additional penalty of 22.2 t/h. The resultant network is not shown for brevity. Similarly, the interconnection S5-D6 can be eliminated through the direct path relaxation of the path FW-D2, S1-D2, S1-D4, S4-D4, S4-D6, S5-D6, and S5-WW with an additional penalty of 35 t/h. The final network comprising 10 matches with a total penalty of 75 t/h is shown in Figure 15. This is the network with the minimum number of interconnections, since no more loops and paths remain in the network.

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Figure 13. Optimal water allocation network containing 13 matches for example 2.

Figure 14. Water allocation network containing 12 matches for example 2.

Figure 15. Water allocation network containing the minimum interconnections for example 2. Table 4. Comparison of Results for Example 2 freshwater flow rate (t/h) this work number of matches Ng and Foo46 evolution strategy MILP formulation 15 14 13 12 11 10

200 230 240 275

200 200 211.4 217.8 240 275

200 200 211.4 217.8 240 275

Ng and Foo46 have also considered the same example for evolution using water path analysis. Table 4 compares the results obtained here with those reported by Ng and Foo46 and those obtained using mathematical optimization. It may be noted that the proposed methodology produces optimal networks at every stage of evolution and is better than the existing evolution techniques. Moreover, existing techniques such as the source-

shift technique and water path analysis may not evolve to the global minimum number of interconnections. The minimum number of interconnections produced using the water path analysis of Ng and Foo46 is 11, while it is possible to reach a water allocation network with 10 interconnections. 6.3. Example 3 of Hydrogen Allocation Network. Hydrogen is an important and expensive resource in oil refining and petrochemicals processing. For continuous reduction of the allowed sulfur content in fuels, to hydrogenate aromatics and olefins, for upgrading large hydrocarbons into lighter hydrocarbons, and other such uses, hydrogen is required for operations such as hydro-treating and hydro-cracking. Additionally, regulations on low-aromatics gasoline have reduced some of the sources of hydrogen previously available to refineries. To illustrate the methodology developed in the previous section, an example of a hydrogen allocation network is considered. The process data52 for this example are given in Table 5. There are

Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009 a

Table 5. Process Data for Example 3 sources

demands

flow rate purity quality F (mol/s) y (%) q (%) SRU CRU HCU DHT NHT CNHT a

623.8 415.8 1801.9 346.5 138.6 457.4

93 80 75 73 75 70

7 20 25 27 25 30

flow rate purity quality F (mol/s) y (%) q (%) HCU DHT NHT CNHT

2495 554.4 180.2 720.7

80.61 77.57 78.85 75.14

19.39 22.43 21.15 24.86

Imported hydrogen with a purity of 95% (q ) 5%) is the resource.

six internal sources of hydrogen: catalytic reformer (CRU), steam reformer (SRU), hydro-cracker (HCU), straight-run naphtha hydro-treater (NHT), cracked naphtha hydro-treater (CNHT), and diesel hydro-treater (DHT). There are four internal demands: hydro-cracker (HCU), straight-run naphtha hydrotreater (NHT), cracked naphtha hydro-treater (CNHT), and diesel hydro-treater (DHT). There is a nearby hydrogen facility, and the imported hydrogen from this facility is the resource. Excess hydrogen, which is the waste for this example, is fed to the fuel system. For this example, the molar flow rate of hydrogen in moles per second (mol/s) has been considered as the flow, and the quality is defined as q ) (100 - y), where y is the percent purity of hydrogen. It is targeted that 268.8 mol/s has to be imported from the nearby hydrogen facility, and the pinch quality is 30% (corresponding to 70% purity). A hydrogen allocation network, satisfying the targets, is generated using the NNA (Figure 16). The preliminary hydrogen allocation network, shown in Figure 16, contains three independent loops: (i) IMPORT-HCU, IMPORT-NHT, CRU-NHT, and CRU-HCU, (ii) IMPORTHCU, IMPORT-DHT, HCU-DHT, and HCU-HCU, and (iii) CRU-NHT, CRU-CNHT, CNHT-CNHT, and CNHT-NHT. It may be noted that none of these loops involve waste, and hence, loop breaking without violation is not applicable at this stage. Applications of two-loop relaxation for different sources and demands produce topologically different networks, and the number of interconnections cannot be reduced. The CRU-NHT interconnection in loop 3 can be eliminated, and the violating demand NHT can be restored through path relaxation with a penalty of 0.61 mol/s. The loop comprising matches IMPORT-HCU, IMPORTDHT, HCU-DHT, and HCU-HCU cannot be broken as eliminating any match from this loop will leave no path for relaxation. Another loop is considered for loop breaking. The new loop consists of the following interconnections: IMPORT-HCU,

Figure 16. Hydrogen allocation network containing 14 matches for example 3.

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IMPORT-NHT, CNHT-NHT, CNHT-CNHT, CRU-CNHT, and CRU-HCU. Elimination of the match CNHT-NHT through flow perturbation results in violation of the quality constraints for two demands, HCU and CNHT. On the other hand, the quality of the flows accepted by NHT is lower than its maximum allowable limit. Therefore, the match IMPORT-NHT is removed in the artificial loop IMPORT-HCU, IMPORT-NHT, SRUNHT, and SRU-HCU. Now the quality constraint for the demand CNHT is restored by relaxing the path IMPORT-HCU, CRUHCU, CRU-CNHT, CNHT-CNHT, and CNHT-FUEL. The final network with 12 interconnections and total penalty of 4.62 mol/s is shown in Figure 17. Similar to the previous stage, no natural loop can be broken as eliminating any match from any natural loop leaves no path to restore the violating demand. Now, the matches SRU-NHT, NHT-NHT, and DHT-HCU are removed respectively from the following artificial loops: (i) SRU-HCU, SRU-NHT, CRU-NHT, and CRU-HCU, (ii) CRU-NHT, CRU-CNHT, NHT-CNHT, and NHT-NHT, and (iii) DHT-HCU, DHT-CNHT, CNHT-CNHT, and CNHT-HCU. These result in an intermediate suboptimal network with 13 matches and a violating demand, as shown in Figure 18. The quality constraint for the demand HCU and the optimal hydrogen allocation network with 11 matches can be obtained by perturbing 3.76 mol/s of flow through the path IMPORT-HCU, CRU-HCU, CRU-CNHT, CNHT-CNHT, and CNHT-FUEL and by perturbing 15.274 mol/s of flow through the path IMPORT-HCU, CNHT-HCU, and CNHT-FUEL. The resultant network is shown in Figure 19. Evolution of the network containing 11 interconnections, shown in Figure 19, to a hydrogen allocation network containing 10 matches with the lowest penalty goes through many intermediate suboptimal networks. The following evolution steps may be taken to evolve to a network with 10 interconnections: (i) Perform two-loop relaxation involving demands HCU and DHT along with the sources IMPORT, HCU, and CNHT. The resultant network contains 11 interconnections. (ii) Eliminate CNHT-FUEL match from the loop DHT-CNHT, DHT-FUEL, CNHT-FUEL, and CNHT-CNHT. The resultant network contains 12 matches. (iii) Eliminate HCU-DHT match from the loop HCU-DHT, HCU-CNHT, NHT-CNHT, and NHT-DHT. The resultant network contains 13 matches. (iv) Perform two-loop relaxation involving demands CNHT and DHT along with the sources IMPORT, NHT, and CNHT. The resultant network contains 13 interconnections. (v) Perform two-loop relaxation involving demands CNHT and DHT along with the sources IMPORT, CRU, and CNHT. The resultant network contains

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Figure 17. Evolved hydrogen allocation network containing 12 matches.

Figure 18. Intermediate suboptimal hydrogen allocation network containing 13 matches and a violating demand.

Figure 19. Hydrogen allocation network containing 11 matches.

13 interconnections. (vi) Eliminate CNHT-DHT match from the loop DHT-DHT, DHT-CNHT, CNHT-CNHT, and CNHT-DHT. The resultant network contains 13 matches, and the quality constraint for demand CNHT is violated. Eliminate the DHTCNHT match from the path IMPORT-CNHT, DHT-CNHT, and DHT-FUEL. The resultant network contains 12 matches, and the quality constraint for demand CNHT is restored. (vii) Eliminate the DHT-DHT match from the path IMPORT-DHT, DHT-DHT, and DHT-FUEL. The resultant network contains 11 matches. (vii) Eliminate both IMPORT-DHT and CRU-NHT matches by perturbing the artificial loop IMPORT-NHT,

IMPORT-DHT, CRU-DHT and CRU-NHT. The resultant network contains 10 matches, as shown in Figure 20. It may be noted that these evolution steps are not unique. A different set of evolution steps is illustrated by Das.53 The IMPORT-CNHT match can further be eliminated from the loop IMPORT-HCU, IMPORT-CNHT, HCU-CNHT, and HCU-HCU. The CNHT-CNHT match is eliminated by perturbing the path IMPORT-HCU, HCU-HCU, HCU-CNHT, CNHTCNHT, and CNHT-FUEL. Finally, the violating demand CNHT is restored by removing the SRU-HCU match from the artificial loop containing SRU-HCU, SRU-CNHT, HCU-CNHT, and


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Figure 20. Hydrogen allocation network containing 10 matches.

Figure 21. Hydrogen allocation network containing 9 matches. Table 6. Comparison of Results for Example 3 hydrogen flow rate (mol/s) number of matches

evolution strategy

MILP formulation

14 13 12 11 10 9

268.8 269.43 273.44 292.47 512.8 970.2

268.8 269.43 273.44 292.47 512.8 970.2

Table 7. Process Data for Example 4 sources

demands

flow rate quality F (kg/s) q (atm1.44) C1 (condensate I) C2 (condensate II)

4 3

13.199 3.741

flow rate quality F (mol/s) q (atm1.44) D (degreaser) A (absorber)

5 2

4.865 7.362

HCU-HCU. The resultant network containing 9 matches is shown in Figure 21. The network shown in Figure 21 has the minimum number of interconnections; however, it incurs a high penalty of 701.34 mol/s. Such a simple network may not be implemented in real practice. However, the designer has the flexibility of choosing any one of the intermediate hydrogen allocation networks. Alves and Towler52 had reported an existing network containing 15 matches with a penalty of 277.2 mol/s. On the basis of the results tabulated in Table 6, it may be noted that the designer can choose a network with fewer interconnections and a similar

penalty or a network with similar number of interconnections and a lower hydrogen penalty. 6.4. Example 4 of Material Reuse Network. Properties such as pH, density, viscosity, volatility, reflectivity, vapor pressure, and solubility can be used as quality instead of concentration and purity. These properties are not conserved directly. However, they follow certain blending or mixing rules, and an appropriately defined quality based on the mixing rule is conserved.42,44 El-Halwagi and co-workers have developed tools to synthesize and analyze material reuse networks.54-58 To illustrate the applicability of the proposed methodology, an example of a metal degreasing process57 is considered. A fresh organic solvent is used in the degreaser of the reactive thermal degreasing process to remove grease from metal. The solvent is regenerated and reused in the degreaser. Fresh solvent is also used in an absorber column to arrest the light gases from the off-gas produced in the regeneration section before sending it to the flare. Condensates produced in two condensers are sent to the waste disposal unit. For this example, the mass flow rate in kg/s has been considered as the flow. The primary property of the solvent that is considered for reuse and recycle is Reid vapor pressure (RVP), an important property in characterizing volatility of the solvent.57 RVP as such is not conserved, but it follows the following blending rule:57 F1(RVP)11.44 + F2(RVP)21.44 ) F3(RVP)31.44

(25)

Therefore, in this problem, the quality is defined as q ) RVP1.44. The fresh organic solvent (S) is a resource with quality qrs )

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Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009 Table 8. Comparison of Results for Example 4 hydrogen flow rate (mol/s)

Figure 22. Material allocation network containing 6 matches.

Figure 23. Evolved material allocation network containing 5 matches.

Figure 24. Evolved material allocation network containing 4 matches.

2.713 atm1.44, and the condensate discharged to the waste disposal unit is the waste (W). Table 7 shows the process data for the metal degreasing process.57 The resource target based on a targeting algorithm is found to be 2.38 kg/s, and the corresponding pinch quality is 13.199 atm1.44 (corresponding to RVP of 6 atm). One of the material reuse networks achieving the minimum target is shown in Figure 22, represented in the form of a matching matrix. The same network has also been reported in Kazantzi and El-Halwagi.57 Applying two-loop relaxation involving demands D and A along with the sources S, C2, and C1, another network with 6 matches can be obtained, and the same network is reported by Bandyopadhyay.42 This network (not shown for brevity) contains a loop: C2-D, C2-A, C1-A, and C1-D. The C1-A match may be eliminated, and the quality constraint for demand D may be restored by relaxing the path S-D, C1-D, and C1-W. The evolved network with 5 interconnections and having a penalty of 0.7 kg/s is shown in Figure 23. At this stage, all loops are eliminated, and further reduction in number of interconnections is only possible through direct path elimination. The final network containing 4 interconnections is shown in Figure 24. This is the network with the minimum number of interconnections because there is no loop or path remaining. The results for different numbers of interconnections are tabulated in Table 8. 7. Conclusion For a simple plant involving only a few processes, design and implementation of the optimum RAN, obtained through different methodologies proposed in the literature, is simple. However, for a large plant involving many processes, imple-

number of matches

evolution strategy

MILP formulation

6 5 4

2.38 3.08 4

2.38 3.08 4

mentation of a complex resource network may be difficult. For improved implementation, controllability, operability, and flexibility, it may be desirable to evolve the preliminary RAN. Similar to the evolution of heat exchanger networks,45 the interconnections in a RAN may be eliminated using different evolution strategies. In this paper, different evolution strategies, based on the appropriate understanding of graph theory, have been proposed to evolve a preliminary network. The proposed evolution strategies are loop breaking without violation, two-loop relaxation, loop breaking with path relaxation, and the direct path relaxation. The proposed methodology has been applied to different resource allocation problems such as water management, hydrogen management, and material reuse networks. The complexity in the evolution of a RAN involves searching for the optimum through a discrete space. Through examples 2 and 3, it has been demonstrated that the evolution from an optimal configuration to another one with fewer interconnections may proceed through suboptimal intermediate networks. The proposed methodology is primarily based on the conceptual understanding of the process, and hence it gives more insight to the designer during evolution. The proposed methodology utilizes different evolution strategies at different stages of evolution to ensure minimum resource penalty. The results at each stage have been validated with optimum values obtained through mathematical optimization from a new MILP formulation. The mathematical optimization proposed in this paper is different from the ones reported in the literature. It may be noted that it is possible to enumerate different modifications and search for the optimum. A similar methodology is followed in mixedinteger optimization technique proposed in the paper. However, such an enumeration based methodology is time-consuming for RAN with a large number of interconnections. The proposed methodology gives physical insight during each stage of evolution, and it has been observed that the proposed methodology produces the lowest penalty for a given number of interconnections. The results obtained through the proposed methodology are superior to those reported earlier in the literature. The proposed methodology is applicable to fixed flow rate problems with a single quality. Current research is directed toward networks with multiple qualities. Literature Cited (1) Shenoy, U. V. Heat Exchanger Network Synthesis: Process Optimization by Energy and Resource Analysis; Gulf Publishing Company: Houston, 1995. (2) Smith, R. Chemical Process Design and Integration; John Wiley: West Sussex, 2005. (3) Kemp, I. C. Pinch analysis and process integration: A user guide on process integration for the efficient use of energy; Elsevier: New York, 2007. (4) El-Halwagi, M. M. Process Integration; Elsevier Academic Press: Amsterdam, 2006. (5) Hui, C. W.; Ahmad, S. Total site integration using the utility system. Comput. Chem. Eng. 1994, 18 (8), 729–742. (6) Raissi, K. Total site integration. Ph.D. Thesis, UMIST, Manchester, U.K., 1994.

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ReceiVed for reView March 1, 2009 ReVised manuscript receiVed April 18, 2009 Accepted June 5, 2009 IE9003392

Evolution of Resource Allocation Networks - Industrial & Engineering

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