Evolutionary Design Methodology for Resource Allocation Networks

ARTICLE pubs.acs.org/IECR

Evolutionary Design Methodology for Resource Allocation Networks with Multiple Impurities Guilian Liu,*,† Mingyuan Tang,‡ Xiao Feng,† and Chunxi Lu† † ‡

College of Chemical Engineering, China University of Petroleum, Beijing 102249, People's Republic of China Department of Chemical Engineering, Xi’an Jiaotong University, Xi’an 710049, People's Republic of China ABSTRACT: An evolutionary method is proposed for designing resource allocation networks with multiple impurities. With every match possibility considered, the maximum match flow rate matrix (M matrix), potential match flow rate matrix (P matrix), and optimal match flow rate matrix (O matrix) can be determined in turn. Both the complementary advantage of source streams and the supplement possibilities are considered in the design procedure. Furthermore, the minimum flow rate rule and maximum flow rate rule are proposed to guide the design. The proposed method can be applied to resource networks with multiple impurities, no matter whether the utility contains impurities. The application of this method is illustrated by two case studies, and the results show that this method can successfully identify the minimum utility consumptions and optimal networks.

1. INTRODUCTION Along with the growth of the world population and economic development, the consumption of resources such as natural gas, crude oil, and water are increasing rapidly. Because of the scarcity of these resources and greater awareness toward environment sustainability, efficient resource utilization and waste reduction have become important issues in the process industries.1 Process integration is a holistic approach to process design, retrofitting, and operation and has been proved to be a promising tool for reducing both fresh resource consumption and waste generation.2-7 Numerous tools of process integration have been developed and applied for optimizing resource networks. These methods are either based on the conceptual approach of pinch analysis or based on mathematical modeling and optimization.1 Pinch analysis, which began as a thermodynamic-based approach to energy conservation,8,9 has evolved over the years to become a powerful tool for process integration and resource optimization. 2,3 In 1989, El-Halwagi and Manousiouthakis extended the concept of pinch analysis to mass exchange networks.10 Since then, numerous methods have been proposed for different material networks. With all water-using processes taken as mass transfer operations, Wang and Smith11 developed a two-stage water pinch approach which utilizes a limiting composite curve to locate the minimum fresh water and wastewater flow rate systems. Later, Dhole et al. 12 proposed a new water source and demand composite curve methodology which can further reduce the fresh water consumption by mixing and bypassing. However, it was found that, if the correct stream mixing cannot be identified, the utility target identified by this method might be larger than the true minimum value.13 Sorin and Bedard14 developed the evolutionary table for targeting the minimum utility. In principle, this method is equivalent to Dhole’s composite curve method. Hence, the solution of this method also depends on whether the correct mixing can be found. r 2011 American Chemical Society

The surplus diagram, which was originally proposed for targeting the minimum utility consumption of hydrogen networks,15 was extended to water networks by Hallale.16 In this method, only when the initial utility consumption is specified can the water surplus diagram be constructed. Hence, iterative calculation is required. El-Halwagi et al.17 developed a noniterative graphical technique. In this method, source and sink composite curves are plotted on a load vs flow rate diagram. The minimum utility consumption and the pinch point can be identified by shifting the composite curves. Later, Prakash and Shenoy18 developed a similar noniterative graphical approach for water targeting along with network synthesis tools. On the basis of the pinch principles, Bandyopadhyay et al.19,20 developed another graphical representation, source composite curves. Manan et al.21 developed a noniterative method, the water cascade analysis technique. This method is equivalent to the water surplus diagram,16 with its iterative calculation steps eliminated. Later, this method was extended to the optimization of systems with single and multiple impure feeds.22,23 On the basis of the composite curves proposed by El-Halwagi et al.,17 Almutlaq et al.24,25 proposed an alternative algebraic tool to synthesize resource allocation networks. Furthermore, Towler et al.26 proposed a method based on value composite curves to analyze the distribution of hydrogen streams in oil refining. Their method can be used to set targets for hydrogen recovery from process off-gases and for hydrogen production in a hydrogen plant. Later, hydrogen surplus diagrams,15,27 material recovery pinch diagrams,17,28,29 limiting composite curves,30 and cascade analysis31 were also developed for utility gas networks. Although the minimum utility consumption can be obtained from the above-mentioned pinch-based Received: May 3, 2010 Accepted: December 21, 2010 Revised: December 14, 2010 Published: January 26, 2011 2959

dx.doi.org/10.1021/ie101010m | Ind. Eng. Chem. Res. 2011, 50, 2959–2970

Industrial & Engineering Chemistry Research targeting methods, many graphical methods are cumbersome and algebraic approaches require multiple repetitive steps. Also, most of these methods can only be used for systems with single impurities. Few graphical methods can be used for material networks with multiple impurities; the calculation process is very cumbersome as a large amount of data and graph transformation are needed. Superstructure-based mathematical optimization approaches can overcome the disadvantage of pinch-based targeting methods and have received much attention. For the optimal design of water-using networks with a single contaminant, Savelski and Bagajewicz32 studied the necessary conditions of optimality and proposed a MILP formulation and a noniterative algorithmic procedure.33,34 For multiple contaminant water networks, different design methods are developed, such as the design methodology with a single internal water main35 and step by step linear programming design method.36 On the other hand, mathematical optimization techniques are developed for hydrogen networks, and the use of purification techniques in enhancing hydrogen recovery is also considered.37-39 Global optimization methods40-42 with all feasible design alternatives incorporated can be used to design material networks with multiple impurities. Furthermore, there are some design methods with different characteristics, including the symmetric fuzzy linear programming procedure43 for systems with imprecise data, the automated design method44 with the treatment of effluent streams considered, and the mathematically rigorous methodology 45 combined with the simplicity of the pinch analysis. Compared with the pinch-based methods, mathematical optimization methods can be easily used in networks with multiple impurities. However, for this kind of system, the superstructure model is a complex MINLP model, and it is difficult to integrate the experience and judgment of the engineers into the solution process. Therefore, the solution and the convergence of the method are relatively difficult. In this paper we present an evolutionary integration method for resource allocation networks with multiple impurities. The match possibilities of every pair of sink and source streams will be analyzed according to all impurities. This method can be applied to target the minimum utility consumption and the optimal network simultaneously. Its applicability and implementation will be illustrated by two case studies.

2. PROBLEM STATEMENT In a material network with m sources (including the hydrogen utility), n sinks, and nc impurities, each source can be discharged or reused/recycled by matching with any sink, no matter whether the sources with lower impurity concentrations are used up. Source SRi (i = 1, 2, ..., m) has flow rate FSRi and is characterized by a constant concentration of impurity k (k = 1, 2, ..., nc), CSRi,k. For sink streams, there are limitations on both material and/or impurity concentrations. Once the balance of all impurities is defined, the balance for the material is determined correspondingly. Therefore, only impurities will be considered in this work. Sinks can accept sources via reuse/ recycle. Each sink SKj (j = 1, 2, ..., n) requires flow rate F SK j and can accept an average inlet impurity concentration from the source that is lower than its maximum allowable impurity

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concentration, Cmax SK j ,k : CSKj , k e Cmax SK j , k

ðj ¼ 1, 2; :::; n; k ¼ 1, 2; :::; ncÞ

ð1Þ

where CSKj,k is the inlet concentration of sink SKj. Hydrogen utility is a fresh (external) source purchased from other factories or specially produced and may contain some impurities. The objective of the methodology is to identify the material allocation network with minimum utility consumption, minimum waste discharge, and maximum material reuse.

3. THEORETICAL DEVELOPMENT AND MATHEMATICAL FORMULATION For each sink or source, there are two fundamental parameters: flow rate and impurity concentration. The product of these two parameters is defined as the impurity load. For source SR i and sink SK j , the loads of impurity k, L SR i,k and L SK j,k , can be calculated according to eqs 2a and 2b, respectively. LSKj , k ¼ FSKj CSKj , k

ðk ¼ 1, 2, :::, ncÞ

ð2aÞ

LSR i , k ¼ FSR i CSR i , k

ðk ¼ 1, 2, :::, ncÞ

ð2bÞ

When the flow rate of a sink is fixed, the constraint shown by eq 1 can be rewritten in terms of the impurity load: LSKj , k e Lmax SK j , k

ðj ¼ 1, 2, :::, n; k ¼ 1, 2, :::, ncÞ

max Lmax SK j , k ¼ FSK j CSK j , k

ðk ¼ 1, 2, :::, ncÞ

ð3aÞ ð3bÞ

To maximize the reuse of the source and minimize the utility consumption, two important criteria need to be fulfilled: (1) the sum of the match flow rates between sink SKj and all sources (including the utility) should equal the required flow rate of the corresponding sink, FSKj; (2) the total impurity load of SKj, LSKj,k (k = 1, 2, ..., nc), should satisfy eqs 3a and 3b. Hence, an evolutionary method can be developed. 3.1. Maximum Match Flow Rate Matrix (M Matrix). Generally, not all impurity loads of a match can reach their upper bounds at the same time. When the load of one impurity reaches its upper bound, the match flow rate cannot increase anymore, although the loads of the other impurities are lower than their upper bounds. Therefore, between source SRi and sink SKj, the match flow rate at which the load of one impurity will reach its upper bound while those of other impurities are less than their upper bounds is defined as their maximum match flow rate. At the maximum match flow rate between source SRi and sink SKj, the impurity load of the match should satisfy eqs 3a and 3b. On this basis, the maximum match flow rate between each pair of sink and source streams can be identified by the following two steps: (1) Calculate the maximum possible match flow rate between source SRi and sink SKj, MPi,j,k (k = 1, 2, ..., nc), according to different impurities by the following equation: MPi, j, k ¼

Lmax SK j , k CSR i , k

ði ¼ 1, 2; :::; m; j ¼ 1, 2; :::; nÞ

ð4Þ

where CSRi,k is the concentration of impurity k in source SRi and max represents the upper bound of the impurity load in sink SKj. LSK j,k 2960

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Industrial & Engineering Chemistry Research

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Table 1. Relevant Sink and Source Data of a Hydrogen Network

Table 2. Maximum Possible Match Flow Rates of the System Shown in Table 1

impurity concentration

SK1

(ppm) flow rate (t/h)

source stream utility (fresh hydrogen)

HC

C

5.67

8.16

8.50

56.00

0.90

62.22

SR3 0.00 0.00 0.00

3.09 226.67

1.61

30.55 24.89

11.79

SR3

56

220

45

950

concentration (ppm) H2S

C

SK1

45

0

0

0

SK2

34

20

300

45

SK3

56

120

20

200

HC denotes the light hydrocarbons.

Equation 4 can only be applied to systems with pure utility. If the utility stream also contains impurities, the maximum possible match flow rate can be calculated according to the following equation: Cmax SK j , k - CSR 0 , k CSR i , k - CSR 0 , k

ð5Þ

where CSR0,k is the concentration of impurity k in the utility stream. (2) Identify the maximum match flow rate, Mi,j. Different maximum possible match flow rates can be obtained according to different impurities. On the basis of the fact that no impurity load can exceed the corresponding upper bound, the minimal flow rate rule is proposed to identify Mi,j: The maximum match flow rate between source SRi and sink SKj, Mi,j, is the minimum one among all MPi,j,k (k = 1, 2, ..., nc) values, i.e. Mi, j ¼ minðMPi, j, k Þ

H2S

SR2 0.00 0.00 0.00

SR1 0.00 0.00 0.00 45.33

35 180

MPi, j, k ¼ FSKj

HC

0

upper bound of impurity

a

C

0 400 1250

HC

H2S

0

C

15 120

flow rate (t/h)

HC

H2S

45 34

sink stream

C

SK3

HCa

SR1 SR2

a

H2S

SK2

ðk ¼ 1, 2; :::; ncÞ

ð6Þ

At the maximum match flow rate, the impurity whose load reaches its upper bound is defined as the key impurity of the match and is represented by KIi,j. In case two or more maximum possible match flow rates equal the maximum match flow rate, the loads of the corresponding impurities will reach their upper bounds simultaneously at the maximum match flow rate. Hence, all of these impurities should be taken as the key impurities of the match. The maximum match flow rates between all sources and sinks can be illustrated by a matrix, as shown by eq 7. In this work, it is termed the maximum match flow rate matrix (M matrix).

In the M matrix, each row corresponds to a source and each column corresponds to a sink. Mi,j represents the maximum

25.50 43.71 448.00

2.80 320.00

match flow rate between source SRi and sink SK j . If M i,j equals 0, source SR i cannot be matched with sink SKj . Otherwise, there is the match possibility between source SR i and sink SK j . For easy understanding, an example is used to illustrate the calculation procedure. In a hydrogen network, there are three sources (excluding the utility), three sinks, and three impurities (HC, H2S, and C), as shown in Table 1. Here, HC represents the light hydrocarbons and C represents CO and CO2. For each pair of sink and source streams, the maximum possible match flow rates, MPi,j,HC, MPi,j,H2S, and MPi,j,C, can be calculated according to eq 4, as shown in Table 2. In the first three columns of Table 2, all elements are equal to 0; therefore, no source can be matched with sink SK1, and the maximum match flow rate between sink SK1 and each source is 0. However, for the match between source SR1 and sink SK2, the maximum possible match flow rates calculated according to the three impurities are different. According to the minimal flow rate rule, it can be identified that the maximum match flow rate of this match equals the maximum possible match flow rate calculated according to H2S, i.e. M1, 2 ¼ MP1, 2, H2 S ð8Þ At the maximum match flow rate, the inlet concentration of H2S will reach its upper bound. Hence, H2S is the key impurity of the match. Similarly, the maximum match flow rates and key impurities can be identified for all the other matches, and the following M matrix can be obtained (note that the key impurities are given in parentheses):

3.2. Potential Match Flow Rate Matrix (P Matrix). In the M matrix, there may be more than one source that can be matched with the same sink. However, they cannot be matched with this sink simultaneously at their maximum match flow rates, as the impurity loads will exceed their upper bounds. To achieve the maximum reuse, the optimal one among them needs to be identified, and it is termed the potentially optimal source. To determine the potentially optimal source, the potential match flow rate is proposed. If source SRi is the potentially optimal match source of sink SKj, its potential match flow rate, represented by Pi,j, will not be 0; however, for the sources which are not the potentially optimal match source of SKj, their corresponding potential match flow rates will all be 0. In the identification process of the potentially optimal source, the complementary advantage between different sources should be considered. 2961



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3.2.1. Complementary Advantage. In this section, an example is used to explain the concept of complementary advantage. For sink SKj, both sources SRq and SRp (p 6¼ q) can be matched with it, and their corresponding maximum match flow rates and key impurities are Mq,j, KIq,j and Mp,j, KIp,j, respectively. When they are mixed, the flow rate and impurity concentration of the mixture can be calculated according to the following mass balance equations:

3.2.3. Sources Have Complementary Advantage. For two sources, SRq and SRp (p 6¼ q) with complementary advantage, when they are matched with sink SKj, their mixture can be matched with this sink at a larger match flow rate. For the match between the mixture and SKj, when both key impurities KIq,j and KIp,j, reach their upper bounds simultaneously, the match flow rate will reach the maximum value, and the corresponding mix flow rate ratio between these two sources will be optimal. In this case, the two key impurities satisfy eqs 15 and 16.

Mp0 , j CSR p , KIq, j þ Mq0 , j CSR q , KIq, j ¼ FSR p - q CSR p - q , KIq, j

ð10Þ

Mp0 , j CSR p , KIp, j þ Mq0 , j CSR q , KIp, j ¼ FSR p - q CSR p - q , KIq, j

ð11Þ

Mp0 , j CSR p , KIq, j þ Mq0 , j CSR q , KIq, j ¼ FSK j CSKj , KIq, j

ð15Þ

ð12Þ

Mp0 , j CSR p , KIp, j þ Mq0 , j CSR q , KIp, j ¼ FSKj CSKj , KIp, j

ð16Þ

Mp, j0 þ Mq, j0 ¼ FSR p - q

where M0q,j and M0p,j are the mix flow rates of sources SRq and SRp, respectively, FSRp-q is the flow rate of the mixture, and CSRp-q,KIp,j and CSRp-q,KIq,j represent the concentrations of the two impurities in the mixture, respectively. If KIq,j 6¼ KIp,j, CSRp,KIq,j < CSRq,KIq,j, and CSRq,KIp,j < CSRp,KIp,j, the concentrations of the two key impurities in their mixture satisfy the following relations: CSR p , KIq, j < CSR p - q , KIq, j < CSR q , KIq, j

ð13aÞ

CSR q , KIp, j < CSR p - q , KIp, j < CSR p , KIp, j

ð13bÞ

From eqs 13a and 13b, it can be seen that the concentrations of both key impurities can be reduced by mixing. Because of this, the maximum match flow rate between the mixture and sink SKj will be larger than both Mq,j and Mp,j, and the corresponding key impurity can be either impurity KIq,j or impurity KIp,j, or even both. In general, when two sources, SRq and SRp, satisfy two conditions [(1) when they are matched with the same sink, SKj, respectively, the corresponding key impurities, KIq,j and KIp,j, are different and (2) CSRp,KIq,j < CSRq,KIq,j, CSRq,KIp,j < CSRp,KIp,j], their mixture can be matched with SKj at a flow rate larger than both Mq,j and Mp,j. This is called the complementary advantage of sources. According to whether the sources have complementary advantage, the corresponding identification procedures of the potential match flow rate can be proposed. 3.2.2. Sources Do Not Have Complementary Advantage. If the sources that can be matched with sink SKj do not have complementary advantage, matching their mixture with sink SKj cannot result in better reuse and hence does not need to be considered. In this case, the potential match flow rate of this sink can be identified according to Mi,j (i = 1, 2, ..., m). To maximize reuse, the maximal flow rate rule is proposed for identifying the potential match flow rate: the source with the maximum Mi,j (i = 1, 2, ..., m) can be taken as the potentially optimal match source, and the corresponding potential match flow rate can be set as its maximum match flow rate, i.e., Pi,j = Mi,j; however, the potential match flow rate between sink SKj and other sources is set as 0, i.e. ( Mi, j if Mi, j ¼ maxðM1, j , :::, Mi, j , :::, Mm, j Þ Pi, j ¼ 0 if Mi, j 6¼ maxðM1, j , :::, Mi, j , :::, Mm, j Þ ð14Þ

According to eqs 15 and 16, eqs 17 and 18 can be deduced. CSR q , KIp, j CSK j , KIq, j - CSR q , KIq, j CSKj , KIp, j Mp0 , j ¼ FSKj ð17Þ CSR p , KIq, j CSR q , KIp, j - CSR q , KIq, j CSR p , KIp, j Mq0 , j ¼ FSK j

CSR p , KIq, j CSK j , KIp, j - CSR p , KIp, j CSK j , KIq, j CSR p , KIq, j CSR q , KIp, j - CSR q , KIq, j CSR p , KIp, j

ð18Þ

The maximum match flow rate between the mixture and SKj, Mq-p,j, equals the sum of M0q,j and M0p,j, i.e. Mq - p, j ¼ Mq0 , j þ Mp0 , j

ð19Þ

It should be noted that, although Mq-p,j is larger than both Mq,j and Mp,j, it may be less than the other maximum match flow rates, Mi,j (i = 1, 2, ..., m; i 6¼ q; i 6¼ p). To determine the potential match flow rate of sink SKj, it is necessary to compare Mq-p,j with all the other Mi,j (i = 1, 2, ..., m; i 6¼ q; i 6¼ p) values. To achieve this, it is necessary to take the mixture of sources with complementary advantage as a pseudosource stream. With all maximum match flow rates between such types of pseudosources and corresponding sinks calculated, the potential match flow rate of SKj, Pi,j, can be identified according the following procedure. (1) Identify the M matrix. (2) For each sink, identify its maximum match flow rate with the mixture of sources which have complementary advantage according to the following steps. (a) Identify the sources with complementary advantage and set their maximum match flow rates as 0. (b) Take the mixture of these sources as a pseudosource, set its maximum match flow rate with all sinks (except SKj) as 0, and add a corresponding row in the M matrix. (c) Calculate the maximum match flow rate between the pseudosource and SKj according to eqs 17-19. (d) Identify the potential match flow rate of this sink according to eq 14. (3) For the sink whose potential match flow rate with the pseudosource is nonzero, set the potential match flow rate of the sources which can be mixed into the pseudosource as the corresponding mix flow rates. Equations 17 and 18 can only be applied to the case where two sources have complementary advantage. In a system with nc impurities, the number of sources with complementary advantage might be more, and the maximum number is equal to the number of impurities, nc. No matter how many sources have 2962


Industrial & Engineering Chemistry Research complementary advantage, the match flow rate between their mixture and the corresponding sink reaches the maximum value when all key impurities reach their upper bounds simultaneously. Hence, the mass balance equations of these impurities, which are similar to eqs 15 and 16, can be written. In addition, the equations which can be directly used to calculate the mix flow rate of each source can also be deduced from mass balance equations. However, they are much more complex, as more sources and more impurities are involved. For example, if three sources (SRp, SRq, SRr) have complementary advantage when they are matched with sink SKj, their key impurities, KIp,j, KIq,j, and KIr,j, are different. The maximum match flow rate between the mixture of these sources and SKj can be calculated according to the following equations: ð20Þ Mp - q - r, j ¼ Mp0 , j þ Mq0 , j þ M0r, j Mp0 , j

c1 b2 d3 þ b1 d2 c3 þ d1 c2 b3 - b1 c2 d3 - d1 b2 c3 - c1 d2 b3 ¼ c1 b2 a3 þ b1 a2 c3 þ a1 c2 b3 - b1 c2 a3 - a1 b2 c3 - c1 a2 b3 ð21Þ

Mq0 , j ¼

M0r, j ¼

a1 c2 d3 þ c1 d2 a3 þ d1 a2 c3 - c1 a2 d3 - d1 c2 a3 - a1 d2 c3 a1 c2 b3 þ c1 b2 a3 þ b1 a2 c3 - c1 a2 b3 - b1 c2 a3 - a1 b2 c3 ð22Þ a1 b2 d3 þ b1 d2 a3 þ d1 a2 b3 - b1 a2 d3 - d1 b2 a3 - a1 d2 b3 a1 b2 c3 þ b1 c2 a3 þ c1 a2 b3 - b1 a2 c3 - c1 b2 a3 - a1 c2 b3 ð23Þ

where Mp-q-r,j is the maximum match flow rate between the mixture and SKj, M0p,j, M0q,j, and M0r,j are the mix flow rates of SRp, SRq, and SRr, respectively, a, b, and c represent the impurity concentration in sources SRp, SRq, and SRr, respectively, d represents the impurity upper bound of sink SKj, and the subscript 1, 2, and 3 represent key impurities KIq,j, KIp,j, and KIr,j, respectively. Therefore, when more than three sources have complementary advantage, instead of deducing the complex equations similar to eqs 21-23, it is recommended to directly solve the mass balance equations by software. For example, the mass balance equations can be directly solved by Lingo or Microsoft Office Excel without a complex programming process. With all Pi,j (i = 1, 2, ..., m; j = 1, 2, ..., n) values determined, the following P matrix can be obtained:

Take the system shown in Table 1 as an example. From its M matrix, it can be seen that no source can be matched with sink SK1. Therefore, the potential match flow rates between the three sources and SK1 are all 0. On the other hand, all three sources can be matched with sinks SK2 and SK3, and their complementary advantage needs to be analyzed. First, the complementary advantage of sources SR1 and SR2 is analyzed. From the M matrix, it can be seen that, when these two sources are matched with sink SK2, their key impurities are H2S and HC, respectively. However, the concentrations of these two

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Table 3. Complementary Possibility of the Sources Matched with SK2 comparison of source pair SR1 and SR2 SR2 and SR3 SR1 and SR3 a

key

the key impurity

complementary

impurity

concentrations

advantagea N

SR1 (H2S)

CSR1,H2S < CSR2,H2S

SR2 (HC)

CSR1,HC < CSR2,HC

SR2 (HC)

CSR2,HC < CSR3,HC

SR3 (C)

CSR2,C < CSR3,C

SR1 (H2S)

CSR1,H2S < CSR3,H2S

SR3 (C)

CSR1,C < CSR3,C

N Y

Key: N, the pair of sources do not have complementary advantage; Y, the pair of sources do have complementary advantage.

key impurities in SR1 are both less than those in SR2, as shown in Table 3. Therefore, SR1 and SR2 do not have complementary advantage. Similarly, SR2 and SR3 do not have complementary advantage either, and the detailed information is shown in Table 3. When SR1 and SR3 are matched with sink SK2, their key impurities are H2S and C (CO and CO2), respectively, and CSR1,H2S > CSR3,H2S and CSR1,C < CSR3,C. Therefore, SR1 and SR3 have complementary advantage and can be mixed to match sink SK2. In addition, the maximum match flow rate between their mixture and SK2 can be calculated according to eqs 17-19. The results show that, when SR1 and SR3 are mixed at flow rates 25.42 and 0.67 t/h, respectively, their mixture has the maximum match flow rate (25.99 t/h) with SK2. Using the same method, it can be found that SR1 and SR3 also have complementary advantage when they are matched with SK3, and the maximum match flow rate between their mixture and SK3 is 13.21 t/h, which is mixed by 1.48 t/h SR1 and 11.73 t/h SR3. With all the potential match flow rates identified, the following potential match flow rate matrix (P matrix) can be obtained:

3.3. Optimal Match Flow Rate Matrix (O Matrix). In the calculation of the P matrix, the flow rates of the sources and the sinks are not considered. However, their values also affect the match between sink and source streams. For example, if the potential match flow rate of a match, Pi,j, is larger than the flow rate of the corresponding source, SRi, i.e., Pi,j > FSRi, SRi cannot match SKj at flow rate Pi,j. Therefore, it is necessary to compare Pi,j with the flow rates of SRi and SKj and identify their feasible match flow rate, which is termed the optimal match flow rate in this work. However, the optimal match flow rate cannot be identified in one step; two intermediate variables, which are termed the potentially optimal match flow rate and optimal supplement flow rate, need be determined first. 3.3.1. Identification of POi,j. Since a source might be matched with several sinks, its distribution between sinks needs to be considered and is affected by its flow rate. Hence, the identification ofP POi,j will be discussed in the following two n situations: (a) j = 1P i,j e FSRi . In row i, if the sum of all elements is not larger than the flow rate of source SR i, i.e., P n j = 1 P i,j e FSRi , source SR i can satisfy the requirement of all 2963



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sinks with corresponding potential match flow rates. In this case, POi,j can be determined according to the required flow rate of sink SKj, FSKj: ( Pi, j if Pi, j < FSKj ð26Þ POi, j ¼ FSKj if Pi, j > FSKj P (b) nj = 1Pi,j > FSRi. If the flow rate of Psource SRi is smaller than the sum of all elements in row i, i.e., nj = 1Pi,j > FSRi, this source cannot supply all sinks with flow rate Pi,j. Therefore, it is necessary to determine its optimal distribution. First, the required flow rate of each sink, FSKj, is considered. If Pi,j is larger than the sink’s required flow rate, FSKj, the flow rate supplied by SRi to this sink can be reduced: Pi, j ¼ FSKj

Pn

ð27Þ

Thus, j = 1Pi,j can be reduced. If it is less than FSRi, the potentially optimal match flow rate between source SRi and sink SKj, POi,j, can be determined according to eq 26. Otherwise, if P n j=1Pi,j is still larger than FSRi, this source cannot supply all sinks with corresponding potential match flow rates, and its optimal distribution should be determined. To identify the optimal distribution of source SRi between different sinks, an important parameter, resource-reusing ratio (Ri,j), is proposed. When sink SKj is matched with SRi, the ratio between their potential match flow rate and the flow rate of SKj reflects the source reuse ratio of sink SKj and is termed the resource-reusing ratio (Ri,j). Its analytical definition can be written as eq 28. Ri, j ¼

Pi, j FSKj

ð28Þ

From the deduction process of Pi,j, it can be obtained that Pi, j ¼

0 0 LSK FSKj CSK j , KIi, j j , KIi, j ¼ CSR i , KIi, j CSR i , KIi, j

CSKj , KIi, j0 CSR i , KIi, j

ð30Þ

From eq 30, it can be seen that the sink with a smaller Ri,j has a lower CSKj,KIi,j and can be matched with fewer sources, while the sink with a higher Ri,j has a higher relative key impurity concentration and can be matched with more sources. To guarantee the sinks with smaller Ri,j values can be matched by sources with lower impurity concentrations, it is necessary to first fulfill them by source SRi. Therefore, the sinks can be sorted in the increasing order of their resource-reusing ratio (Ri,j), and the sources can be distributed among them according to this order. The sink with a lower resource-reusing ratio can be matched with source SRi at the corresponding potential match flow rate, i.e. POi, j ¼ Pi, j

FSR i -

n X j¼1

POi, j < Pi, g

In this case, POi,g can be set as FSRi POi, g ¼ FSR i -

ð31Þ

ð32Þ

Pn

n X j¼1

j = 1POi,j,

i.e. ð33Þ

POi, j

3.3.2. Supplement Source and Maximum Supplement Flow Rate. With all PO i,j values identified, there might exist the case that, for sink SKj , PO i,j < P i,j. If the deficit in the flow rate is supplemented by utility, the inlet concentrations of all impurities will be less than the corresponding limitations of sink SK j, and the utility consumption cannot be minimized. To avoid this, when all POi,j (i = 1, 2, ..., m; j = 1, 2, ..., n) values are identified, another source, SRi , which is unused or not used up can be applied to supplement the requirement of sink SK j . In a given system, possibly several sources can be used to supplement the same sink. To maximize the resource reuse, the source which can supplement the sink with maximum flow rate should be selected as the optimal supplement source, and the corresponding supplement flow rate is taken as the optimal supplement flow rate (OSi,j). The identification procedure of the maximum supplement flow rate is similar to that of the maximum match flow rate, as shown by the following steps. Step 1: Calculate the maximum load of impurity k (ΔMk) that each source can supplement to sink SKj. For source SRi which is not used up, the maximum impurity load (ΔMk) it can supplement to sink SKj equals the deficit amount of the sink, as shown by eq 34.

ð29Þ

where LSKj,KIi,j0 is the key impurity load that can be supplied by SRi and CSKj,KIi,j0 is the key impurity concentration of SKj that can be fulfilled by SRi. If Pi,j = Mi,j 6¼ 0, then CSKj,KIi,j0 = CSKj,KIi,jmax and LSKj,Kli,j0 = LSKj,KIi,jmax; if Pi,j < Mi,j, then CSKj,KIi,j0 < CSKj,KIi,jmax and LSKj,KIi,j0 < LSKj,KIi,jmax. According to eqs 28 and 29, eq 30 can be deduced. Ri, j ¼

P Since nj = 1Pi,j > FP SRi, when it is time to set the value of POi,g for sink SKg, FSRi - nj = 1POi,j might be less than Pi,g, i.e.

ΔMk ¼ FSK j Cmax SK j , k -

m X i¼1

POi, j CSR i , k

ð34Þ

Step 2: Calculate the potential maximum supplement flow rate according to different impurities by the following equation: PMSi, j, k ¼

ΔMk CSR i , k

ðk ¼ 1, 2; :::; ncÞ

ð35Þ

where PMSi,j,k is the possible maximum supplement match flow rate of source SRi determined according to impurity k. This equation can only be used when the utility is pure. If there are some impurities in the utility, eq 36 can be used to calculate PMSi,j,k. PMSi, j, k ¼

ΔMk - CSR 0 , k ðFSKj -

m P i¼1

POi, j Þ

CSR i , k - CSR 0 , k ðk ¼ 1, 2; :::; ncÞ

ð36Þ

Step 3: Identify the maximum supplement flow rate. Here, the minimal flow rate rule can also be used: The maximum supplement flow rate (MSi,j) between source SRi and sink SKj is the minimum one among all PMSi,j,k (k = 1, 2, ..., nc) values. In the identification of the maximum supplement flow rate, there is another constraint, the remaining flow rate of source 2964



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Figure 1. Optimal network of the system shown in Table 1 (FSR1 = 45 t/h).

Figure 2. Optimal network of the system shown in Table 1 (FSR1 = 26 t/h).

P SRi (FSRi - nj = 1POi,j), which is the upper bound source SRi can supplement SKj. Therefore, MSi,j can be identified according to the following equation: MSi, j ¼ minðPMSi, j, k , FSR i -

n X j¼1

ðk ¼ 1, 2; :::; ncÞ

POi, j Þ ð37Þ

Step 4: Identify the optimal supplement source and optimal supplement flow rate, OSi,j. Maybe more than one source can be used to supplement the requirement of sink SKj. To maximize the reuse of the source stream and minimize the utility consumption, the selection of the optimal supplement source and optimal supplement flow rate, OSi,j, can be classified as the following two situations. P (1) If all supplement flow rates are less than FSKj - m i = 1POi,j, then the source with the maximum supplement flow rate can be selected as the optimal supplement source, and the optimal supplement flow rate, OSi,j, is set as ( MSi, j if MSi, j ¼ maxðMS1, j , :::, MSi, j , :::, MSm, j Þ OSi, j ¼ 0 if MSi, j 6¼ maxðMS1, j , :::, MSi, j , :::, MSm, j Þ ð38Þ (2) If the supplement flow rates of some sources are larger P than FSKj - m i = 1PO i,j, then among these streams, the one with the minimum supplement flow rate can be selected as the optimal supplement source, and the corresponding optimal

P supplement flow rate, OSi,j, is set as FSKj - m i = 1 POi,j, while for other sources, their optimal supplement flow rates are all set as 0, i.e. 8 m X > > POi, j > FSKj > > > i¼1 < > > > > > > : 0 if

OSi, j ¼ if

MSi, j ¼ minðMSi, j , MSi, j ∈½MSi, j jMSi, j > FSKj -

m X i¼1

POi, j , i∈½1, mÞ

MSi, j 6¼ maxðMS1, j , :::, MSi, j , :::, MSm, j Þ

ð39Þ Following the above steps, the optimal supplement flow rate of all pairs of sinks and sources can be identified. 3.3.3. Optimal Match Flow Rate. With all potentially optimal match flow rates and the optimal supplement flow rates determined, the optimal match flow rate, Oi,j, can be calculated according to the following equation: Oi, j ¼ POi, j þ OSi, j

ð40Þ

Then the optimal match flow rate matrix (O matrix) can be constructed with all optimal match flow rates written in a matrix. In each column of the O matrix, the sum of all elements will not be larger than the flow rate of the corresponding sink, i.e. m X i¼1

2965

Oi, j e FSKj

ðj ¼ 1, 2; :::; nÞ

ð41Þ



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Figure 3. Flowchart of the design procedure.

For each sink, the deficit in the flow rate can be supplemented by utility. The supplement flow rate, FSKjU, can be calculated by the following equation: U FSK ¼ FSKj j

m X i¼1

Oi , j

ðj ¼ 1, 2; :::; nÞ

ð42Þ

can be seen that the potentially optimal match flow rate between source SRi (i = 1, 2, 3) and sink SKj (j = 1, 2, 3) equals the corresponding Pi,j. Thus, the optimal match flow rate matrix (O matrix) can be determined, and it is the same as the P matrix:

Hence, the minimum utility consumption of the system, FU, can be calculated, i.e. FU ¼

n X j¼1

U FSK j

ð43Þ

In the P matrix of the system shown in Table 1, the sum of the elements in each row is less than the flow rate of the corresponding source. According to the above discussion, it

From the O matrix, it can be seen that source SR 1 can be matched with sinks SK 2 and SK 3 at flow rates 25.42 and1.48 t/h, respectively. Source SR3 can be matched with sinks SK2 and SK 3 at flow rates 0.67 and 11.73 t/h, respectively. SK1 needs to be entirely supplied by utility. The minimum utility 2966


Industrial & Engineering Chemistry Research consumption of this system is 95.70 t/h, and the optimal network with minimum utility consumption is shown in Figure 1. If the flow rate of source P SR1 is 26 t/h instead of 45 t/h, the P matrix will not change. 3j = 1P1,j (26.90 t/h) will be larger than the flow rate of SR1, and it is necessary to distribute SR1 between sinks. When matched with SR1, the resource-reusing ratio of SK2, R1,2, is 0.75, while that of sink SK3 is 0.03. Therefore, source SR1 can first supply SK3 at a flow rate of 1.48 t/h, and the remaining 24.52 t/h can be totally supplied to SK2, i.e., PO1,3 = 1.48 t/h and PO1,2 = 24.52 t/h. All POi,j values are shown in the following matrix:

Since PO1,2 is less than P1,2 (25.42 t/h), another source needs to be used to supplement the requirement of sink SK2. According to the procedure introduced above, it can be found that SR3 is the optimal supplement source, and the corresponding supplement flow rate is 0.04 t/h. For the other sinks, no supplement source is needed. Therefore, OS3,2 = 0.04, while all other OSi,j values can be set as 0. All the optimal supplement flow rates are shown in the following matrix:

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4. CASE STUDIES 4.1. Case 1. A hydrogen network containing three sources and three sinks will be studied. For all sources, their flow rates and impurity concentrations are fixed. While for each sink, there is an upper bound for impurity H2S, a lower bound for hydrogen, and no limitations for other impurities. Therefore, all impurities except H2S can be taken as an inert impurity, A. To consider hydrogen and H2S simultaneously, the lower bound limitation on the inlet concentration of hydrogen can be transferred into the upper bound limitation on the inlet concentration of inert impurity A, as shown in Table 4. Following the design procedure and the minimal flow rate rule, the M matrix of this system can be obtained:

In the third column of the M matrix, the key impurities for all three possible matches are the same. Therefore, the potential match source of sink SK3 is SR2, and the potential match flow rate is 260.35 mol/s. However, in the second column corresponding to sink SK2, the key impurities are different. In addition, it can be identified that SR1 and SR2 have complementary advantage. According to eqs 17 and 18, the mix flow Table 4. Data of the Sink and Source Streams of Case 1

According to the potentially optimal match flow rate and the optimal supplement flow rate, the following O matrix can be obtained. Comparing the optimal match flow rate matrix shown in eqs 44 and 45, it can be seen that the difference between these two matrices is the optimal match flow rates of sink SK2. The optimal network with minimum utility consumption is shown in Figure 2.

impurity concentration (mol %) flow rate (mol/s)

H2S 0

0

SR1 SR2

280 350

0 0.21

9.2 6.81

SR3

104

1.03

12.34

source stream utility

A

upper bound of the impurity concentration (mol %)

3.4. Complete Identification Procedure. The complete

procedure for determining the minimum utility consumption and the corresponding optimal network is illustrated in Figure 3. This method not only can be used when the utility is pure, but also can be used when there are impurities in the utility streams.

sink stream

flow rate (mol/s)

SK1

122

0

7.19

SK2

318

0.1

8.57

SK3

197

1

9

H2S

A

Figure 4. Optimal network of case 1 with minimum utility consumption. 2967



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rates between SR1 and SR2 can be calculated, which are 184.13 and 151.43 mol/s, respectively. Then, the following P matrix can be constructed:

In the P matrix, there are two nonzero values in the first row. They are both less than the required flow rate of the corresponding Table 5. Relevant Data of Case 2 impurity concentration (ppm) source stream

flow rate (t/h)

utility SR1

1

2

3

25

0 50

0 100

0 50

SR2

70

100

300

600

SR3

35

150

400

800

SR4

40

600

450

700

SR5

8

500

650

400

SR6

50

1100

3500

2500

SR7

30

900

4500

3000

sinks, and their sum is smaller than the flow rate of source SR1. In the second row, there are two nonzero values, and their sum is larger than the flow rate of source SR2. Since P2,3 is larger than the required flow rate of SK3, P2,3 can be reset as FSK3, i.e., P2,3 = FSK2 = 197 mol/s. Hence, the sum of nonzero values in the second row is reduced to 348.43 mol/s, less than the required flow rate of SR2. However, since the sum of the second column is larger than the required flow rate of SR2, the match flow rate between this source and the two sinks SK1 and SK2 should be adjusted until their sum equals the flow rate of sink SK2. It should be noted that either the match flow rate between SR1 and SK1 or that between SR2 and SK1, or even both, can be adjusted. Although the resulting optimal network will be different, the total utility consumption will not be affected. If PO1,1 = P1,1, PO1,2 = P1,2, and PO1,3 = P1,3, PO2,2 can be set as FSK2 - PO1,2, 133.87 mol/s, PO1,3 can be set as 0, and PO2,3 can be set as 197 mol/s. With all potentially optimal match flow rates identified, it can be found that no sink needs to be supplemented, as either their impurity loads or their flow rates reach the corresponding limitations. Therefore, the optimal match flow rate between each pair of streams equals their potentially optimal match flow rate. The resulting O matrix is as follows:

upper bound of the impurity concentration (ppm) sink stream

flow rate (t/h)

SK1

25

SK2

70

0

0

0

SK3

35

20

50

50

SK4

40

50

110

200

SK5

8

20

100

200

SK6

50

500

300

600

SK7

30

150

700

800

1 0

2

3 0

0

It can be determined that the minimum fresh hydrogen (utility) consumption is 26.65 mol/s; the same result can be obtained by the linear programming method. The optimal hydrogen distribution network with minimum utility consumption is shown in Figure 4. 4.2. Case 2. This case is taken from Wang et al.35 The water system involves seven sources and seven sinks containing three impurities, and the relevant information is shown in Table 5.

Figure 5. Optimal network of case 2 with minimum utility consumption. 2968


Industrial & Engineering Chemistry Research According to the design procedure introduced in Figure 3, the M matrix and O matrix can be obtained:

Note that the key impurities are given in parentheses.

It can be calculated that the minimum fresh water (utility) consumption is 140.93 t/h, the same as that calculated by Li and Fei36 using the step by step linear programming and less than that found by Wang et al. (160.4 t/h)35 with one internal water main. The optimal water network with minimum utility consumption is shown in Figure 5.

5. CONCLUSIONS In this paper, an evolutionary method is developed for minimizing the utility consumption of resource networks with multiple impurities. The method contains three main stages: (stage 1) identification of the maximum match flow rate matrix (M matrix); (stage 2) identification of the potential match flow rate matrix (P matrix); (stage 3) identification of the optimal match flow rate matrix (O matrix). To minimize the utility consumption and maximize the resource reuse, the complementary advantage of source streams is considered in the procedure of identifying the P matrix, and the optimal supplement source and optimal supplement flow rate are determined in the procedure of identifying the O matrix. This method can identify the minimum utility consumption and the corresponding optimal network structure simultaneously and can be applied in resource networks no matter whether the utility is pure. The proposed method overcomes the drawback of the pinch technique and can be easily understood and coded into software. Also, the engineers’ experience and judgment can be easily integrated into the design procedure. Two case studies show that this method can successfully find the minimum utility consumptions and optimal networks. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ86(0) 29-8266 8980. Fax: þ86(0) 29-8323 7910. E-mail: [email protected].

’ ACKNOWLEDGMENT Financial support provided by the National Natural Science Foundation of China under Grants 20876124 and 20936004 is gratefully acknowledged.

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’ NOMENCLATURE a = impurity concentration in source SRp b = impurity concentration in source SRq c = impurity concentration in source SRr CSKj,k = inlet concentration of impurity k in sink SKj max CSK = maximum allowable inlet concentration of impurity k j,k in sink SKj 0 CSK = key impurity concentration of SKj that can be fulfilled j,KIi,j by SRi CSR0,k = concentration of impurity k in the utility stream CSRi,k = concentration of impurity k in source SRi CSRp,KIp,j = concentration of KIp,j in SRp CSRp,KIq,j = concentration of KIq,j in SRp CSRp-q,KIp,j = concentration of KIp,j in the mixture of SRp and SRq CSRp-q,KIq,j = concentration of KIq,j in the mixture of SRp and SRq CSRq,KIp,j = concentration of KIp,j in SRq CSRq,KIq,j = concentration of KIq,j in SRq d = impurity upper bound of sink SKj FU = minimum utility consumption of the system FUSKj = utility consumed by sink SKj FSKj = required flow rate of sink SKj FSRi = flow rate of source SRi FSRp-q = flow rate of the mixture mixed by SRp and SRq i = source i j = sink j k = impurity k KIi,j = key impurity of the match between SRi and SKj KIp,j = key impurity of the match between SRp and SKj KIq,j = key impurity of the match between SRq and SKj KIr,j = key impurity of the match between SRr and SKj LSKj,k = load of impurity k in sink SKj LSRi,k = load of impurity k in source SRi Lmax SRi,k = maximum allowable load of impurity k in sink SKj L0SKj,KIi,j = key impurity load that can be supplied by SRi to SKj m = total number of sources MPi,j,k = maximum possible match flow rate between source SRi and sink SKj calculated according to impurity k Mi,j = maximum match flow rate between SRi and SKj Mp,j = maximum match flow rate between SRp and SKj Mp-q-r,j = maximum match flow rate between the mixture (SRp, SRq, and SRr) and SKj M0p,j = mix flow rate of source SRp Mq,j = maximum match flow rate between SRq and SKj Mq-p,j = maximum match flow rate between the mixture (SRq and SRp) and SKj M0q,j = mix flow rate of source SRq Mr,j = maximum match flow rate between SRr and SKj M0r,j = mix flow rate of source SRr MSi,j = maximum supplement flow rate between SRi and SKj n = total number of sinks nc = total number of impurities Oi,j = optimal match flow rate between SRi and SKj OSi,j = optimal supplement flow rate between SRi and SKj Pi,j = potential match flow rate between SRi and SKj PMSi,j,k = potential maximum supplement flow rate of source SRi to sink SKj calculated according to impurity k POi,j = potentially optimal match flow rate between SRi and SKj Ri,j = resource-reusing ratio of SKj when matched with SRi SKj = sink j SRi = source i SRp = source p SRq = source q 2969


Industrial & Engineering Chemistry Research SRr = source r ΔMk = maximum load of impurity k each source can supplement to sink SKj Subscripts

1, 2, 3 = key impurities KIq,j, KIp,j, and KIr,j, respectively, in eqs 21-23

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Evolutionary Design Methodology for Resource Allocation Networks

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