Mass exchanger network synthesis using genetic-alopex algorithms

Mass exchanger network synthesis using genetic-alopex algorithms

Process SystemsEngineering2003 B. Chen and A.W.Westerberg(editors) 9 2003 Publishedby ElsevierScienceB.V. 1310 Mass Exchanger Network Synthesis Usin...

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Process SystemsEngineering2003 B. Chen and A.W.Westerberg(editors) 9 2003 Publishedby ElsevierScienceB.V.

1310

Mass Exchanger Network Synthesis Using Genetic-Alopex Algorithms Shaojun Li a, Yongrong Yang b, Feng Qian a aInstitute of Automation, East China University of Science and Technology, Shanghai 200237, China bSchool of Chemical and Materials Engineering, Zhejiang University, Hangzhou 310027, China

Abstract In order to achieve the cost-effective targets and reduce the passive environmental influence of chemical processes, mass exchanger network synthesis (MENs) is proposed to reduce the waste to an acceptable level at the cheapest cost. Now pinch analysis is the primary method to synthesize mass exchanger networks. But the capital cost could not be considered in the network analysis or it is difficult to be considered in the synthesizing process. Pointed to the deficiency of pinch analysis method, the genetic-alopex (algorithm of pattern extraction) algorithm is brought forward to synthesize the mass exchanger networks by solving a superstructure mathematical model. The model considers not only operating cost but also capital cost. By solving the mathematical model using genetic-alopex algorithm, we can get the optimal network with the least total cost. The examples computed by geneticalopex algorithm show that the method is effective.

Keywords mass exchanger network, synthesis, genetic algorithm 1 INTRODUCTION Mass exchange includes the unit operations of absorption, adsorption, liquid-liquid extraction, desorption, and stripping. The operating utilities for these operations are mass separating agents (MSA) such as adsorbents, ion-exchange resins, or solvents, just as steam and cooling water in heat exchange. The design of these separation systems involves choosing the appropriate MSAs and determining their optimal flow rates to achieve the specified recovery of the key component. Separation system synthesis has been pursued through a variety of approaches. In the late 1980s, E1-Halwagi and Manousiouthakis [1] introduced the concept of mass exchanger networks (MEN) synthesis and first used the pinch technology to optimize the mass exchanger network. They showed how specifying a minimum concentration difference locates the mass transfer pinch, which is the thermodynamic bottleneck for mass transfer between process streams. Their work resulted in targets for the minimum amount of MSAs. The pinch divides the problem into two separate regions, each of which is in mass balance with its MSA. No mass should be transferred across the pinch if the minimum MSA target is to be met in design. Later E1-Halwagi and Manousiouthakis [2] presented an automated synthesis procedure. This procedure first used linear programming to determine the pinch points and minimum lean utility targets. Mixed integer linear programming was then used to synthesize all possible networks featuring the minimum number of units. The main shortcoming of this procedure is that the capital cost and operating cost could not be considered simultaneously. Papalexandri et al [5] recognized the shortcoming and applied MINLP to MENs problem. They did not use the pinch division and attempted to minimize the total annual cost by optimizing a network superstruture containing many mass exchanger alternatives. However, the methods for solving a MINLP problem often fail to achieve satisfactory results. For a non-convex mathematical form and complicated superstructure, it is necessary to develop a more effective method to solve this kind of problem.

1311 In this paper, we bring forward using a Genetic-Alopex algorithm to solve this type of problem and a mathematical model is proposed for addressing the mass exchanger network synthesis problem. The model doesrd] depend on the minimum composition difference, pinch location and sub-network partition, while the capital and operating costs are considered simultaneously during the solution process. 2 GENETIC-ALOPEX ALGORITHMS [3] Genetic algorithm (GA) is a stochastic optimization method based on the principles of natural selection. It has been applied successfully to real-world problems and exhibited, in many cases, better search efficiency compared with traditional optimization algorithms. Genetic algorithm has many advantages, such as that it does not need the mathematical information, it does not require assumption for the problem continuity, differentiability or convexity and is inherently suitable for use on parallel computers. But standard GA has the weak capacity of climbing hill and it gets into the local minimum easily. The algorithm of pattern extraction (Alopex) was proposed in 1974 and it is enlightened by the influence of a previous change of the independent variables with respect to the objective function. Alopex also utilizes noise to get out of local optima, which improves hill-climbing capacity to some extent. In order to solve a large-scale optimisation problem and make use of the advantages of GA and Alopex, the hybrid Genetic-Alopex algorithm is proposed. Alopex operation is conducted respectively after the ordinary operations of crossover and mutation of GA. Namely, new individuals are provided with certain noises to enhance their capacity of hill climbing and escaping local optima easily. Usually, after certain generations of GA, the approximate optima occupy most of the population, making genetic algorithm difficult to cast off local optima. While concerning the high similitude between the individuals, the son generations from parents with crossover and mutation have few differences, which result in slow convergence and low speed of computing. In order to ensure global convergence, the diversity of individuals is desired in the solution space thus avoid the loss of effective genes. By introducing Alopex into GA, the differences between son and parent generations will be obtained because Alopex can generate certain noises during the son generation, ensuring the diversity of the population. That introducing the alopex into genetic algorithm can lead to an enhancement of the algorithm's hill-climbing capacity, while converging to the global optimum and accelerating the speed of convergence to some extent. 3 THE MINLP MODEL OF NON-SPLITTING MASS EXCHANGER NETWORK A typical mass exchanger network can be described as [4]: Given a number of rich streams, and a number of lean steams (mass separating agents), it is desired to synthesize a costeffective network of mass exchangers that can preferentially transfer certain species from the rich streams to the MSAs. Given also are the flow rate of each rich stream, Gi, its supply (inlet) composition, yi s, and its target (outlet) composition, yi t. In addition, the supply and target compositions, xjs and xjt, are given for each MSA. The mass transfer equilibrium relations are also given for each MSA. The flow rate of each MSA is unknown and is to be determined as part of the synthesis task. The candidate MSAs (lean streams) can be classified as either process MSAs or external MSAs. The process MSAs already exist on the plant site and can be used at a low cost (often virtually free). The flow rate of each process MSA, Lj, is bounded by its availability in the plant and may not exceed a value of Ljc. On the other hand, the external MSAs can be purchased from the market and their flow rates are to be determined by economic considerations. The objective is minimization of the annual cost. In this paper we only consider the instance of non-splitting MEN. The basic idea of

1312 synthesizing mass exchanger network is to set several match phases, in every phase each potential match between a rich stream and a lean stream corresponds to a potential mass exchanger. For a MEN system with NR rich streams and Np process lean streams, assuming the same match between a rich stream and a process lean stream isn't over twice, there are 2NRNp mass exchangers at most, which can exist in different match sequences. Each rich stream matches with process lean stream in a sequence from higher concentration to lower concentration. Rich stream matches with external lean stream only once at the end of the lowest concentration. The matching sequence and mass of exchanging are optimized simultaneous in the synthesizing process. Without loss of generality, the optimizing process utilizes the following assumption: 1) The mass flow rate of each stream remains constant throughout the network. 2) The equilibrium does not depend on the other solutes. 3) The mass exchangers are considered of the counter-current type. 4) No mass exchange between rich streams is allowed. 5) The network is operated under constant pressure. The mass exchanger network synthesis problem can be formulated as a MINLP problem. The optimal network configuration, MSA consumption and equipment design parameters (such as stage numbers) are determined simultaneously. The objective function is the minimization of the total annualized cost, which includes the annualized operating cost, equation (1), and annualized capital cost, equation (2). Nt,

NE

~Cp,•

+ECej•

i=1

(1)

.

j=l

N R Np N K

NE

y' (y] y] (a o. x l/ijk x NTok ) + E (ao x ~j x NT~j )) i=1 j=l k=l

(2)

j=l

The constrains include the mass balances of each rich and lean stream, the mass balance at each mass exchanger, constraints forbidding a match and so on. 1. The mass balance of each rich stream Np N x

NE

(rich stream i)

(3)

(process lean stream j )

(4)

( external lean stream 1)

(5)

Z Z (Y~k - YO~ + Z (Yi~ - y o ) = y~ _ yO j=l k=l

/=1

2. The mass balance of each lean stream NR NK o

-

x o

-

x,

~

/ ~ k=l

(xi~ - x~) = x~ - x/

3. ~Iass balance at each mass exchanger I Gi (Ytk - Yij~ ) = L i (xij~ - xuk )

(6)

4. Mass exchange feasibility constraints I

O

.

Yijk > moxok + bo, I

(higher concentration)

(7)

(8) 5. Constraints for forbidding matches (0-1 variables) If the match of a rich stream i and a lean stream j is forbidden, then Vijk=0, otherwise yo.~ > m~ixiik + bij .

( l o w e r concentration)

Wijk=l.

4 APPLICATION OF G E N E T I C - A L O P E X A L G O R I T H M S F O R MEN SYNTHESIS If a match between the same rich and the same process lean streams can occur twice, for a MEN system with NR rich streams and Np process lean streams, there are at most 2NRNp mass exchangers, which can exist in a different match sequence. The match sequences are

1313 represented by a one-dimension array S and the mass exchange loads are represented by another array F. Then the individual of GAA can be represented as {S[1 ], S[2], "", S[2NRNp], S[I+2NRNp], " " , S[2NRNp +NRNE]} and {F[S[1]], F[S[2]], " - ' , F[S[2NRNp]], F[S[1 +2NRNp]], "", F[S[2NRNp +NRNE]]}. The array S is ascertained by definition as follows" The serial numbers of first rich stream matching with other process lean streams in the k th match level are: (k- 1)NRNp + 1, --', (k- 1)NRNp+Np, respectively, The serial numbers of second rich stream matching with other process lean streams in the k th match level are: (k- 1)NRNp+ Np+ 1, ".', (k- 1)NRNp+2Np, respectively, ooo

The serial numbers of NR rich stream matching with other process lean streams in the k th match level are: (k- 1)NRNp+Np(NR- 1)+ 1, "', kNRNp, respectively. The matching sequences of exchangers between rich and external lean streams are assumed under the same assumption above and their serial number are kNRNp +1, "", kNRNp+ NRNE, respectively. The first generation solution is engendered as follows: Initially, a feasible value for flow rate of every lean stream and a sequence S[1 ], S[2], "", S[kNRNp] are generated stochastically. Thus the serial numbers of rich and lean streams can be obtained according to the definition above. According to the sequence of array S, the inlet concentration of the rich stream and the outlet concentration of lean stream, as well as the flow rates of streams, the maximum quantity of mass exchanged between the rich and the lean streams can be computed. Then the feasible value of the first matching can be stochastically chosen between 0 and maximum quantity of mass exchanged. The other variables can be determined using a similar method. The column stage numbers and the capital cost of every match can be computed according the matching sequences and loads. Finally, the total annualized cost is ascertained. Every individual has different total annualized cost. According to the cost, the fitness of every individual can be computed. Thus the genetic-Alopex algorithms can be used to solve the problem of mass exchange according to the genetic operator, crossover and mutation. Crossover: the variable array S and variable array F make crossover respectively according to the individual crossover probability. The real variable F carries out a continuous crossover according to the genetic-Alopex crossover operator. Because the match between a rich and a lean stream in one matching level can exist only once, the individual after the crossover still asks for the match only once. We chose the part-matching crossover for the part of array S. Mutation: the variable array S and variable array F make mutation respectively according to the mutating probability. Integer variable S carries out mutation following the form of reversing mutation and location mutation. The real variables array F carries out continuous mutation according to the genetic-Alopex mutation operator. After crossover and mutation, the individual may become infeasible according to the constraints, and it should be adjusted to satisfy the constraints. The adjustment of F is according to the match sequence array S and the concentrations of correlative streams. 5 EXAMPLES Now we choose two examples to show the effective of the method, and the biggest match times between a rich and a lean stream is twice in the model. Example 1 comes from the literature [1,4]. E1-Halwagi [1] and Hallale [4] used this example to illustrate their MENs synthesizing procedure. The objective of the network is to remove hydrogen sulphide from two rich streams. These are a coke-oven gas stream and the tail gas from a Claus unit. Two MSAs are utilized in this process. The cheapest is an aqueous ammonia stream, a process MSA, which is supplemented, as needed, by chilled methanol, which is an external MSA. The

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Flow(kg/s) 0.9

/

?0.06981 00066

0.0003

?

[0.0013 ~

0.1

'[00006 0'0006"2"189

000464 000085

0.053 0.004 0.00828 0.0005 6 trays 2 trays 5 trays 1 tray / . . . . . 0 003 @/0.000' ~0.0002

0.383

0.00114 0.000123 3 trays 2 trays Figure 1. Optimal network structure using GAA mass exchangers are sieve tray columns and it will be assumed that the capital cost of an exchanger is $4552 per year per equilibrium stage. In literature [4] the operating cost is $313261, the total number of stages is 25 and the capital cost is $113800. Figure 1 shows the network structure designed by Genetic-Alopex algorithms. The total number of stages is 19. In contrast with the literature [4], the flow rate of lean stream is higher and the operating cost is $324483. However, the total number of exchanger stages in the network is lower and the capital cost is $86488. The operating cost is higher than literature [4], but the capital cost is lower. By trading off between operating and capital cost, the total cost is $419071. This is a better network than that of the published literatures. Example 2 is also taken from literature [4] and involves the removal of phenol from two aqueous streams, R1 and R2 by solvent extraction. Two process MSAs are available: these are gas oil, S1, and lube oil, $2. The addition of phenol to these oil streams is actually beneficial for them and it is specified that the entire gas-oil stream should be used. The external MSA in this example is specified to be light oil, $3, with data taken from literature [4]. The annual operating time is assumed to be 8600h. The mass exchangers are sieve tray columns and it will again be assumed that the capital cost of an exchanger is $4552 per year per equilibrium stage. Literature [4] gained an optimal network structure using the pinch technology, whose flow rates of process lean streams and external lean stream are 5kg/s, 2.22kg/s and 0.704kg/s respectively. The annual operating cost of the network is $217960 and the total colunm stages are 28; the annual capital cost is $127456. Thus the total annual cost is $345416. The network structure gotten by using genetic-Alopex algorithms is shown in Figure 2. The annual operating cost is $205623, total stages are 28, the annual capital cost is $122904 and the total cost is $333079 per year. Flow-rate, kg/s ~

|

[email protected] [email protected]

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trays 2 trays 3 trays

3 trays 5 trays

['~

0015

I

5.00 |

0.010

2.245

~'00381 F)0.013 _

4 trays

1 tray

Figure2. The optimal network structure using GAA

0.664

1315 6 CONCLUSION In this paper a new optimization approach has been presented. The method doesn't depend on pinch technology and it can get the optimum structure considering the operating cost and the capital cost simultaneously. Two examples have been tackled show that the method is effective to synthesize mass exchanger networks. NOMENCLATURE a price of column tray per annual, $/a b intercept of equilibrium line C price of lean stream, $/kg F matching load array G flow rate of rich stream, kg/s k the match times between a rich stream and a lean one L flow rate of lean stream, kg/s m slope of equilibrium line NE number of external lean stream Np number of process lean stream NR number of rich stream NT number of column trays S matching sequence array V 0-1 variable denoting the column existing x composition of lean stream (mass fraction) y composition of rich stream (mass fraction) SUBSCRIPTS E external lean stream E i rich stream I j lean steam j k match level k P process lean stream P SUPERSCREIPTS I inlet O outlet s supply t target REFERENCES [1] Halwagi, M. M., Manousiouthakis, V. (1989), Synthesis of mass exchanger networks, AIChE J, 35(8), 1233-1244 [2] E1-Halwagi, M. M., & Manousiouthakis, V. (1990), Automatic synthesis of mass exchanger networks with single-component targets, Chemical Engineering Science, 45(9), 2813-2831 [3] Shaojun Li, Hui Wang and Pingjing Yao, Study of genetic algorithm-alopex for global optimization. Information and Control, 2000,29(4), 304-308,314 (in Chinese) [4] Hallale, N., & Fraser, D.M. (2000), Capital and total cost targets for mass exchanger networks, Computer and Chemical Engineering, 23(11/12), 1661-1699 [5] Papalexandri, K. P., Pistikopoulos, E. N., & Floudas, C. A. (1994), Mass exchange networks for waste minimisation: a simultaneous approach, Transactions IChemE, 72,279-294