Superstructure Optimization in Heat Exchanger Network (HEN

Therefore, the major agents used in the architecture of such problems .... the objective function is calculated directly without the second run stage...
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Ind. Eng. Chem. Res. 2010, 49, 4731–4737

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Superstructure Optimization in Heat Exchanger Network (HEN) Synthesis Using Modular Simulators and a Genetic Algorithm Framework Roghayeh Lotfi* Sharif Energy Research Institute (SERI), Sharif UniVersity of Technology, P.O. Box 11365-8639, Tehran, Iran

Ramin B. Boozarjomehry Chemical & Petroleum Engineering Department, Sharif UniVersity of Technology, P.O. Box 11365-9465, Tehran, Iran

Heat exchanger network synthesis (HENS) is one of the most efficient process integration tools to save energy in chemical plants. In this work, a new optimization framework is proposed for the synthesis of HENS, based on a genetic algorithm (GA) coupled with a commercial process simulator through the ActiveX capability of the simulator. The use of GA provides a robust search in complex and nonconvex spaces of mathematical problems, while the use of a simulator facilitates the formulation of rigorous models for different alternatives. To include the most common heat exchanger structures in the model, a promising superstructure has been used. Allowing nonisothermal mixing of streams in the new simulation-based approach leads to the discovery of truly optimal network configurations. The performance of the proposed approach is demonstrated using some case studies, and the obtained solutions are compared with those available in the literature. 1. Introduction Heat exchanger network synthesis (HENS) is one of the most important fields in chemical process synthesis, and it has received much attention over the past four decades. Its significance can be attributed to the large savings achievable in terms of energy costs. So high energy prices motivate industry to apply heat integration in their facilities intensively and make the problem remain open-ended.1-4 In general, HENS can be solved by either sequential or simultaneous approaches. Simultaneous methods can themselves be classified as either a superstructure-based framework5,6 or a nonsuperstructure framework.7-9 In this paper, the former framework has been implemented through a new approach. On the other hand, from the optimization point of view, simultaneous HENS methods are often categorized as mixedinteger nonlinear programming (MINLP) models and can be solved by either gradient-based or stochastic methods. Furman and Sahinidis illustrated that solving HENS are NP-hard of strong sense.10 This limits the usefulness of gradient-based optimization techniques such as Outer Approximation and Generalized Benders Decomposition, because their computation time increases exponentially with problem size.11 Moreover, these methods usually suffer from getting trapped in local minima and a convergence of these methods may be achieved only if a set of appropriate initial estimates for the decision variables were provided. Therefore, stochastic methods such as SimulatedAnnealing,12 TabuSearch,13 andGeneticAlgorithm7,14,15 can be important approaches for tackling large-scale problems like HEN synthesis. Among these methods, genetic algorithms (GAs) are less sensitive to arbitrary initial guesses, because they keep track of a population of potential solution. In particular, genetic algorithms (GAs) work very well on mixed (continuous and integer) combinatorial problems. A few articles have reported the use of Genetic Algorithms to HEN synthesis, but these works have often involved the use * To whom correspondence should be addressed. Tel.: 98-2166036096. E-mail: [email protected].

of GAs just for one of the parametric or structural optimization. For instance, Lewin7 and Androulakis et al.16 used GA for structural optimization, whereas, in the studies of Stair and Fraga,17 GA performs a parametric optimization. Although in a recent work of Dipama et al.,18 GA has been used for both the structural and parametric optimization of the HEN problem; however, this work does not support stream splitting. In typical HEN synthesis problems, all design conditions are assumed as a constant value and the effect of temperature and pressure on physical properties of streams is neglected. This consideration can lead to solutions very far from the industrial point of view. Therefore, a proposal for a truly complete formulation of the HENS problems without any simplifying assumptions is still needed. The main novelty of this paper is to achieve this goal by using a commercial simulator. Although some works in the literature have proposed the use of a stochasticoptimizationmethodcoupledwithmodularsimulators,19-21 however, to the authors’ knowledge, the approach has not yet been implemented to HENS problems. Process simulators include a variety of highly efficient thermodynamic models and design models that make it possible to evaluate different flowsheets and modeling options in an easy way. The simulator has the capability of including real-life constraints such as pressure drop and fouling, which are essential factors in HENS. The use of a simulator also can help to ease the formulation of the synthesis problem regarding to the blackbox model concept. Furthermore, the new approach can facilitate the incorporation of HENS concepts with other aspects of the process synthesis, such as distillation sequences and reactor network synthesis, to lead to simultaneous synthesis of the total process flowsheet. The work is organized as follows. The next section contains a brief overview of genetic algorithms. In section 3, the implementation of the model is described. Section 4 contains the results of case studies.

10.1021/ie901215w  2010 American Chemical Society Published on Web 04/21/2010

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preference for using binary encoding is typically derived from the schema theory of genetic algorithms, which analyzes genetic algorithms according to their expected schema sampling behavior. Also, binary representation is a robust method for MINLP problems, because every integer variable in the model can be handled efficiently by using just a set of bits. Because the proposed algorithm improves the probability of the selection of the topmost individual in the mating pool, it will converge in fewer generations. The termination criterion of the algorithm is based on the satisfaction of a predetermined similarity ratio between chromosomes of the current generation. 3. Implementing the Model

Figure 1. Flowchart of Genetic Algorithm (GA).

2. Genetic Algorithm Genetic algorithms (GAs) are among the most widely used stochastic optimization techniques and represent a promising alternative to gradient-based techniques for certain classes of problems, such as complex and nonconvex spaces optimization problems, especially those characterized by mixed continuousdiscrete variables.22 In a GA, a population of solutions represented in chromosal-like fashion is manipulated similar to the process of natural evolution found in nature. The algorithm requires the following operators: (a) population initialization, (b) an eValuation procedure, (c) a selection procedure, and (d) crossoVer and mutation operators. Each of these operators usually depends on the application domain in which the GA is applied. Figure 1 presents the general flowchart of the GA. Generally, two main approaches are used to implement the GA: binary encoding and decimal encoding. In the former encoding, each chromosome consists of a binary string in which each gene can take a value of 0 or 1, while in the latter approach, each chromosome consists of a decimal string and each gene can take a decimal number. Although sometimes the decimal GA algorithm may perform better (i.e., converge faster) for a specific group of problems, it degrades the generality of the GA and its resemblance to what happens in nature.23 The strong

Figure 2. Schematic of the superstructure presented by Yee and Grosssmann.6

The proposed work uses a commercial simulator to simulate the structures suggested by the optimizer. These simulators provide the detailed economical evaluation of a process flowsheet; however, the inclusion of different structures in an internal optimization step is not possible without varying the structures by hand. This means that, to automate the synthesis of the network, it is necessary to use an interface to connect the optimizer and simulator. Therefore, the major agents used in the architecture of such problems consist of user, optimizer, interface, and simulator. In the present work, the user interacts with the GA that defines the parameters of the algorithm, constraints, and termination criteria, with the interface, to set up the control variables and with the simulator, to select the thermodynamic model and type of calculation. 3.1. The Superstructure Selection. In a superstructure-based approach of HENS problems, a framework called the HEN superstructure is postulated to formulate the synthesis problem as an optimization problem. This superstructure embeds several alternative HEN designs. The two famous stream superstructures proposed in the literature: one was devised by Ciric and Floudas5 and the other was reported by Yee and Grossmann.6 While the former embeds almost all different alternative structures, it involves more nonlinear heat balance constraints and, therefore, is more difficult to solve. On the other hand, the superstructure proposed by Yee and Grossmann,6 because of its stagewise nature, allows both series and parallel decoupling of the heat exchangers (see Figure 2). Although the model cannot generate some HEN structures, Yee and Grossmann6 illustrated that good HEN structures can be obtained. The main drawback of this superstructure is the isothermal mixing assumption, which is made to avoid nonlinear

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Figure 3. Sample chromosome for optimization problem.

Figure 4. Schematic of the network corresponding to binary variables of Figure 3.

constraints, and it might cause suboptimal networks. Although Bjork and Westerlund24 removed the assumption with convexification, because this technique will give rise to approximate problems, the precision of their method is under question. In the present work, the superstructure proposed by Yee and Grossmann6 has been used as the framework, and because of using a robust optimizer coupled with a black box model, the isothermal assumption of the formulation is thoroughly eliminated to obtain more-reliable networks. 3.2. The Algorithm. The HEN solution provides both the structure and operating parameters related to the internal (heat exchangers) and external (heaters and coolers, if required) networks. Thus, the solution includes a set of binary and continuous variables:

of each continuous variable. For a continuous decision variable such as xi, whose value is between ai and bi and its precision is pi, the number of genes can be calculated through the following equation: ni ) int

[

]

log(bi - ai)/pi +1 log 2

(3)

The total number of genes in a chromosome, specified to continuous variables (ng,c), is calculated by the following equation: ng,c )

∑n

i

(4)

i

The length of the chromosome (l) is equal to

{Zi,j,k, Qi,j,k, Xi,j,k, Yi,j,k} where the binary variable Zi,j,k denotes the possible match of internal network between hot stream i and cold stream j in stage k of the superstructure. Note that, considering the dimension of the variable, the use of three indices creates a four-dimensional matrix. The operating variables are presented with Qi,j,k, Xi,j,k, and Yi,j,k denoting heat exchanger duty and the hot and cold stream split ratios. For a problem of m hot streams and n cold streams and s stages, the number of binary variables (nz) and the number of continuous variables (nc) is given as nz ) m × n × s

(1)

nc ) 3 × m × n × s

(2)

To begin the procedure, the GA generates an initial population of 0-1 chromosomes randomly. The length of each chromosome is a predefined constant value for each problem. This value is calculated according to the assumption of existence of all possible matches between the streams in the superstructure. Therefore, the number of genes occupied by binary variables is equal to nz, while the number of genes occupied by continuous variables depends on the precision and the range of variation

l ) nz + ng,c

(5)

To specify the values of the variables, the above encoding should be decoded. The arrangement of the variables in the chromosome is one of the important challenges of the decoding stage. It has a significant effect on the performance of the GA solution. After several runs of the program, it concluded that the most efficient arrangement is to specify the first nz bits of the chromosome to the binary variables and the remaining part to the continuous variables. As it was mentioned previously, the variables are manipulated through four-dimensional matrices, while the GAs work with one-dimensional chromosomes. Therefore, it is necessary to convert the 1-D chromosome into a 4-D matrix by using the interface. In the next step, binary variables are mapped into specified heat exchangers of the superstructure. In the case of existence of a heat exchanger (i.e., nonzero binary value), the exchanger would be added to the flowsheet through the ActiveX capability of the simulator. As an example, consider a problem with two hot streams, two cold streams and two stages. This problem requires eight binary variables. If the binary bits of the chromosome represented in Figure 3 correspond to these variables, then, according to the decoding procedure used in

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Figure 5. Schematic of the algorithm of solution procedure. Table 1. Problem Data for Case Study Aa cost stream TIN (K) TOUT (K) Fcp (kW/K) h (kW/cm2 K) ($/kW per day) H1 H2 C1 C2 S1 W1

443 423 293 353 450 293

333 303 408 413 450 313

30 15 20 40

0.8 0.8 0.8 0.8 0.2 0.8

80 20

a Heat exchanger except heaters ($ per year), 4000 + 700a0.8. Heaters ($ per year), 4000 + 560a0.8.

Table 2. Problem Data for Case Study Ba cost stream TIN (K) TOUT (K) Fcp (kW/K) h (kW/cm2 K) ($/kW per day) H1 H2 H3 H4 H5 C1 S1 W1 a

500 480 460 380 380 290 700 300

320 380 360 360 320 660 700 320

6 4 6 20 12 18

1 1 1 1 1 1 1 1

140 10

Heat exchanger ($ per year), 4000 + 700a0.8.

this work, four heat exchangers highlighted in Figure 4 would be added to the flowsheet. The remaining part of the chromosome corresponds to the continuous variables. If a binary variable were equal to zero, then the corresponding continuous variables of that exchanger would not be used in the solution procedure and the related genes can be easily set to zero. To utilize the genetic operators in the most efficient way, it would be better to lead such genes to the last part of the chromosome as the part of bits specified with “Non-Used Continuous Variables” in Figure 4. Therefore, in the case of happening crossover or mutation operators in this

section of chromosome, these operators can be directed to other sections by the use of a random function. After decoding the chromosome, the simulator receives the values of variables through the interface. Then, the network is simulated, and after the run, the outlet temperatures of the streams are checked. If these temperatures do not satisfy the specified target values of the problem (i.e., the hot streams should be hotter than the target temperature and cold streams be colder than the target temperature), an external network should be constructed that includes the required heaters and coolers. The heat duties of hot and cold utility streams are determined through the overall energy balance for each stream. After the run of the new network, the objective function is calculated using the areas of internal and external exchangers and the heat duties of utility exchangers. The objective function is defined as the annual cost for the network, which involves the combination of the utility cost, the fixed charges for exchangers, and the area cost for each exchanger. Using this objective function, one can rank various networks based on the issues corresponding to both design and operational concerns. This is due the fact that, for a network that consists of exchangers whose capital costs are low, the operational cost (mainly consists of utility cost) would be high and vice versa. Hence, there is a tradeoff between operational cost and capital cost. Hence, minimizing the linear combination of these costs would results in an optimum HEN. Note that, in the case of the satisfaction of target temperatures just by the internal network, the objective function is calculated directly without the second run stage. In the final step, the objective function value is returned to the GA, which develops other networks through the use of different operators. This process continues until the termination criterion is satisfied. The optimal network and the corresponding

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Figure 6. Final optimal network of case study A.

Figure 7. Final optimal network of case study B.

variables then are reported to the user. Figure 5 shows a brief flowchart of the solution procedure. To handle the constraints in the model, penalty functions have been used. For simulations ending with warnings or errors (infeasible designs), a large penalty is defined, whereas for successful simulations that cannot achieve the desired specifications, a smaller penalty is used. 4. Case Studies The performance of the algorithm is evaluated with several HENS from the literature: two small-scale and two midscale. The examples are organized according to size, such that the smallest problem is the first one and the largest problem is last. The objective is to minimize the annualized cost expressed as the sum of the utility costs, fixed charges for each heat exchanger, and an area-based cost for each heat exchanger. All calculations were performed using a Pentium IV CPU (1.8 GHz) with 1 GB RAM. In all case studies, the following parameters and units were used: temperature (T, K); heat capacity flow rate (Fcp, kW/K); convective heat transfer (h, kW/(cm2 K)); and cost ($/kW per year). 4.1. Case Study A. This is a well-known case study that has been taken from the work of Yee and Grossmann6 and involves two cold and two hot streams. Table 1 shows the problem data, as well as exchanger cost equations. By selecting the number of stages to be equal to 2, the number of binary and continuous variables would be equal to 8 and 16. The final

network is presented in Figure 6. This network consists of four process heat exchangers and one cooler. The total annual cost is $75 890, while the reported annual cost in the reference paper is $80 274. 4.2. Case Study B. This example is taken from Lewin14 and involves five hot streams and one cold stream. Its problem data, as well as exchanger cost equations, are presented in Table 2. As shown in this table, the corresponding temperature for the hot utility (i.e., steam) is 700 K, but in the steam table, there is no such saturated steam. This problem shows that, although in many approaches, the heat exchanger network synthesis is solved just using mathematical models and the process insights are neglected, in practice, the problem cannot be solved using Table 3. Problem Data for Case Study Ca cost stream TIN (K) TOUT (K) Fcp (kW/K) h (kW/cm2 K) ($/kW per day) H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 S1 W1 a

423 522 544 500 472 355 366 311 333 389 509 311

366 411 422 339 339 450 478 494 433 495 509 355

8.79 10.55 12.56 14.77 17.73 17.28 13.90 8.44 7.62 6.08

Heat exchanger ($ per year), 145.63a0.6.

0.852 0.852 0.852 0.852 0.852 0.852 0.852 0.852 0.852 0.852 1.136 0.852

37.64 18.12

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Figure 8. Final optimal network of case study C.

just mathematical formulations. Nevertheless, to synthesis this problem, it is assumed that, initially, the hot stream is cooled to the saturation temperature, and at this temperature, it exchanges heat with cold streams. If, for the sake of simplicity, the number of stages were assumed to be 2, there would be 10 and 30 binary and continuous variables. The GA converges to a total annual cost of $375 100, while Lewin has reported $573 205. It is obvious that the comparison between these numbers is not logical, since the cold stream C1 cannot be heated to 660 K. The final network with corresponding areas of exchangers is presented in Figure 7. 4.3. Case Study C. This case study is also taken from Lewin,14 and it involves five cold streams and five hot streams. Table 3 shows the problem data, as well as exchanger cost equations. By selecting the number of stages equal to 3, the number of binary and continuous variables would be equal to 75 and 225. The final network is presented in Figure 8. This network consists of eight process heat exchangers and three coolers. The total annual cost is $44 477, while the reported annual cost in the reference paper is $43 452. There was an interesting point in running the code for this case study; i.e., after running the GA several times, no feasible solution was obtained, so a migration operator was used. This result is not far from expectations, because it is obvious that the efficiency of superstructure-based methods decreases by the increase of problem size. For example, in this case study, the number of variables is equal to 300 and the corresponding chromosome length is 1662 and migration operator should be used to give the solution. A comparison of the case study data is given in Table 4. 5. Conclusion This paper has presented a novel approach for the synthesis of heat exchanger networks, relying on a genetic algorithm (GA) to perform both the structural and parametric optimizations. The optimizer determines the final exchanger network and computes

Table 4. Comparison Data for the Case Studies Annual Cost ($)a

Number of Variables case study

binary

continuous

chromosome length

from previous works

from this work

A B C

8 10 75

16 30 225

128 128 1662

80274 573205 43452

75890 375100 44477

a

Here, the annual cost is considered the onbjective function.

the optimum stream split flows and heat exchanger duties, for a given heat exchanger network (HEN) structure. The newly developed optimization environment has the ability of treating arbitrary flowsheets, including structure and parametrization of the system in question simultaneously in the optimization procedure. The simulations and the cost calculations exploit the complete process modeling accuracy without the necessity of simplifications due to restrictions imposed by the optimization method. This seems to be the major reason why the proposed framework leads to the networks whose performances are better than those reported by other methods. The given results demonstrate the suitability of GAs for HEN optimization. Although the proposed procedure greatly improved the performance of the synthesis, the computational requirements are still a major task when using superstructures. Therefore, it might be beneficial for future works to use a nonsuperstructurebased framework coupled with the suggested algorithm. Work is underway on the nonsuperstructure-based methods, considering practical issues such as pressure drop and fouling. Literature Cited (1) Gundersen, T.; Naess, L. The synthesis of cost optimal heat exchanger networks. Comput. Chem. Eng. 1988, 12 (6), 503–530. (2) Furman, K. C.; Sahinidis, N. V. A critical review and annotated bibliography for heat exchanger network synthesis in 20th century. Ind. Eng. Chem. Res. 2002, 41, 2335–2370. (3) Jezowski, J. Heat exchanger network grassroot and retrofit design. The review of the state-of-the art: Part I. Heat exchanger network targeting and insight based methods of synthesis. Hung. J. Ind. Chem. 1994, 22, 279– 294.

Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010 (4) Jezowski, J. Heat exchanger network grassroot and retrofit design. The review of the state-of-the art: Part II. Heat exchanger network synthesis by mathematical methods and approaches for retrofit design. Hung. J. Ind. Chem. 1994, 22, 295–308. (5) Ciric, A. R.; Floudas, C. A. Heat exchanger network synthesis without de-composition. Comput. Chem. Eng. 1991, 15 (6), 385–396. (6) Yee, T. F.; Grossmann, I. E. Simultaneous optimization models for heat integration. 2. Heat exchanger network synthesis. Comput. Chem. Eng. 1990, 14 (10), 1165–1184. (7) Lewin, D. R.; Wang, H.; Shalev, O. A generalized method for HEN synthesis using stochastic optimization. 2. General framework and MER optimal synthesis. Comput. Chem. Eng. 1998, 22 (10), 1503–1513. (8) Chakraborty, S.; Ghosh, P. Heat exchanger network synthesis: The possibility of randomization. Chem. Eng. J. 1999, 72, 209–216. (9) Errico, M.; Maccioni, S.; Tola, G.; Zuddas, P. A deterministic algorithm for the synthesis of maximum energy recovery heat exchanger network. Comput. Chem. Eng. 2007, 31, 773–781. (10) Furman, K. C.; Sahinidis, N. V. Computational complexity of heat exchanger network synthesis. Comput. Chem. Eng. 2001, 25, 1371–1390. (11) Floudas, C. A. Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications; Oxford University Press: New York, 1995. (12) Dolan, W. B.; Cummings, P. T.; Levan, M. D. Process optimization via simulated annealing: Application to the network design. AIChE J. 1988, 35 (5), 725–736. (13) Lin, B.; Miller, D. C. Solving heat exchanger network synthesis problems with Tabu Search. Comput. Chem. Eng. 2004, 28, 2287–2306. (14) Lewin, D. R. A generalized method for HEN synthesis using stochastic optimization. 2. The synthesis of cost optimal networks. Comput. Chem. Eng. 1998, 22 (10), 1387–1405. (15) Ravagnani, M. A. S. S.; Silva, A. P.; Arroyo, P. A.; Constantino, A. A. Heat exchanger network synthesis and optimization using genetic algorithms. Appl. Therm. Eng. 2005, 25, 1003–1017.

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(16) Androulakis, I. P.; Venkatasubramanian, V. A genetic algorithm framework for process design and optimization. Comput. Chem. Eng. 1990, 15 (4), 217–228. (17) Stair, C.; Fraga, E. S. Optimization of unit operating conditions for heat integrated processes using genetic algorithms. In Proceedings of the 1995 Institute of Chemical Engineering Research EVent, 1995; Vol. 1, pp 95-97. (18) Dipama, J.; Teyssedou, A.; Sorin, M. Synthesis of heat exchanger networks using genetic algorithms. Appl. Therm. Eng. 2008, 28 (14-15), 1763–1773. (19) Gross, B.; Roosen, P. Total process optimization in chemical engineering with evolutionary algorithms. Comput. Chem. Eng. 1998, 22, 229–236. (20) Leboreiro, J.; Acevedo, J. Process synthesis and design of distillation sequences using modular simulators: A genetic algorithm framework. Comput. Chem. Eng. 2004, 28, 1223–1236. (21) Jang, W.-H.; Hahn, J.; Hall, K. R. Genetic/quadratic search algorithm for plant economic optimizations using a process simulator. Comput. Chem. Eng. 2005, 30, 285–294. (22) Rao, S. S. Optimization: Theory and Applications.: Wiley Eastern, Ltd.: New Delhi, 1985. (23) Boozarjomehry, R. B.; Massoori, M. Which method is better for the kinetic modeling: Decimal encoded or Binary Genetic Algorithm? Chem. Eng. J. 2007, 130, 29–37. (24) Bjork, K. M.; Westerlund, T. Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption. Comput. Chem. Eng. 2002, 26, 1581–1593.

ReceiVed for reView March 31, 2009 ReVised manuscript receiVed March 15, 2010 Accepted April 5, 2010 IE901215W