Optimal design of plate-fin heat exchangers by a hybrid evolutionary algorithm

International Communications in Heat and Mass Transfer 39 (2012) 258–263

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Optimal design of plate-fin heat exchangers by a hybrid evolutionary algorithm☆ M. Yousefi a,⁎, R. Enayatifar b, A.N. Darus a a b

Department of thermo-fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia Faculty of computer science, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia

a r t i c l e

i n f o

Available online 9 December 2011 Keywords: Plate-fin heat exchanger Particle Swarm Optimization Genetic Algorithm Hybrid algorithm Optimization

a b s t r a c t This study explores the first application of a Genetic Algorithm hybrid with Particle Swarm Optimization (GAHPSO) for design optimization of a plate-fin heat exchanger. A total number of seven design parameters are considered as the optimization variables and the constraints are handled by penalty function method. The effectiveness and accuracy of the proposed algorithm is demonstrated through an illustrative example. Comparing the results with the corresponding results using GA and PSO reveals that the GAHPSO can converge to optimum solution with higher accuracy. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Because of their adaptability to a wide range of applications, high compactness and relatively good heat transfer efficiency, plate fin heat exchangers (PFHEs) are widely used in different aspects of industry such as automobile, chemical and petrochemical processes, cryogenics and aerospace. The design of a PFHE is a complex task based on trialand-error process in which geometrical and operational parameters are selected to satisfy specified requirements such as outlet temperature, heat duty and pressure drop. Moreover, optimization based on the desired objective should always be taken into consideration. According to the literature, the common objectives in heat exchanger design are associated with minimizing capital cost and original cost. Practically, a higher velocity yields to higher heat transfer coefficient which consequently leads to smaller heat transfer area and lower capital cost. It should be noticed that, however, higher velocity results in higher pressure drop and power consumption, too. Therefore, before the optimal design is performing, the objective function should be considered based on the requirements. In most cases a compromise between the capital cost and power cost should be achieved by the design parameters. Many works have been devoted to the optimization of heat exchangers using traditional mathematical methods[1–5], In addition, recently, GAs, as stochastic global search algorithms, have been widely implemented in design and optimization of compact heat exchangers[6–17] since they have been proved to be very effective tools in finding near optimal solutions without having information of the derivatives. Particle Swarm Optimization (PSO), a new evolutionary based technique, has been recently

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: yousefi[email protected] (M. Yousefi). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.11.011

introduced by Kennedy and Eberhart [18] and has shown its effectiveness in design of CHEs [19,20]. Similar to GAs, PSO starts with an initial population of the possible solutions. Each solution is called a ‘Chromosome’ in GA and a ‘Particle’ in PSO where on the contrary to the former new solutions are not created from the parents within the evolution process. In PSO, any individual just tries to evolve its social behavior and move towards destination. Since PSO and GA are both working with a random initial population of solutions, Lu and Juang [21] combined their searching abilities of these two methods and proposed a new search method called, hybrid GA with PSO (HGAPSO). They successfully applied HGAPSO in design of a fuzzy controller. In this work, it is desired to see the feasibility of this newly introduced metaheuristic algorithm in optimization of plate fin heat exchangers.

2. Thermal modeling A schematic of a typical cross-flow plate fin heat exchanger with offset strip fin can be seen in Fig. 1. In the analysis, for the sake of simplicity, the variation of physical property of fluids with temperature is neglected where both fluids are considered to be ideal gases. Other assumptions are as follows. 1– Number of fin layers for the cold side (Nb) is assumed to be one more than the hot side (Na). It is a conventional way in design of heat exchangers in order to avoid heat waste to the ambient. 2– Heat exchanger is working under steady state condition. 3– Heat transfer coefficient and the area distribution are assumed to be uniform and constant. 4– The thermal resistance of walls is neglected. 5– Since the influence of fouling is negligibly small for a gas-to-gas heat exchanger, it is neglected.

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259

Nomenclature A, AHT Aff C Cr C1 C2 Dh f f(X) g(X) G h H j l L m n Na, Nb NTU Pi Pg pm Pr Q rand() Rand() Re R1 t T U V X

heat exchanger surface area (m 2) free flow area (m 2) heat capacity rate (W/K) Cmin/Cmax Cognition factor social collaboration factor hydraulic diameter (m) friction factor objective function constraint function mass flow velocity (kg/m 2s) convective heat transfer coefficient(W/m2K) height of fin (m) Colburn factor lance length of the fin (m) heat exchanger length (m) mass flow rate (kg/s) fin frequency (fins per meter) number of fin layers for fluid a and b number of transfer units Particle's best position best particle in the current generation mutation probability Prandtl number rate of heat transfer (W) Random function Random function Reynolds number penalty parameter fin thickness (m) temperature °C overall heat transfer coefficient Particle velocity Particle position

Fig. 1. (a) Schematic representation of cross-flow plate-fin heat exchanger, and (b) detailed view of offset-strip fin.

In the mentioned equation, Cr = Cmin/Cmax. Neglecting the thermal resistance of the walls and fouling factors, NTU is calculated as follows. 1 1 1 ¼ þ UA ðhAÞa ðhAÞb

ð2Þ

UA Cmin

ð3Þ

NTU ¼ Greek symbols ε effectiveness μ viscosity (N/m2s) ρ density (kg/m3) () penalty function ΔP Pressure drop (N/m2) ω Inertia weight

Heat transfer coefficient is calculated from j Colburn factor. −23

h ¼ j:G:Cp:Pr

:

ð3Þ

m In this formula, G ¼ Aff , where Aff is free flow cross-sectional area which is calculated considering the geometrical details in Fig. 2.

Subscripts a,b fluid a and b i,j variable number max maximum min minimum

Aff a ¼ ðHa −ta Þð1−na ta ÞLb Na

ð5Þ

Aff b ¼ ðHb −tb Þð1−nb tb ÞLa Nb

ð6Þ

Heat transfer area for both sides can be calculated similarly. Aa ¼ La Lb Na ½1 þ 2na ðHa−ta Þ

ð7Þ

Ab ¼ La Lb Na ½1 þ 2nb ðHb −tb Þ

ð8Þ

Then, total heat transfer area is given by: In the present work, since the outlet temperature of the fluids is not specified the ε-NTU method is used for rating performance of the heat exchanger in the optimization process. The effectiveness of cross-flow heat exchanger, for both fluids unmixed is proposed as [22].

AHT ¼ Aa þ Ab : Heat transfer rate is calculated as follows.   Q ¼ ε C min Ta;1 −Tb;1

ð9Þ

ð10Þ

Frictional pressure drop in both sides is given by: 
 n h i o 1 0:22 0:78 NTU ε ¼ 1− exp −1 exp −Cr:NTU Cr

ð1Þ

ΔPa ¼

2f a La G2a ρa Dh;a

ð11Þ

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M. Yousefi et al. / International Communications in Heat and Mass Transfer 39 (2012) 258–263

mutation. More details on GA and its application in heat transfer problems can be seen in [24]. In PSO the process begins with generation of a random population of possible solutions, each of them called a ‘Particle’. Throughout the process, each particle, i, having the current position of Xi in the search space, flies with the velocity of Vi towards the solution while keep track of its best position, Pi, in all previous cycles. Moreover, the position of the best particle (a particle with the best fitness) in the current cycle is represented by Pg. The velocity of each particle in each cycle is calculated as follow. New Vi ¼ ω   current  Vi þ c1  randðÞ  ðPi −Xi Þ þ c2  RandðÞ  Pg −Xi

ð16Þ

Subject to: Vmax ≥ V1 ≥ − Vmax And the new position becomes: Fig. 2. Flow of the hybrid algorithm.

New position Xi ¼ current position Xi þ New Vi :

2

ΔPb ¼

2f b Lb Gb : ρb Dh;b

ð12Þ

There are many correlations for evaluation of Colburn factor j and Fanning factor f for offset strip fin. Eqs. (13) and (14) are the correlation presented by Manglik and Bergles [23] which is used in this work. −0:5403 ðαÞ−0:1541 ðδÞ0:1499 ðγÞ−0:0678 j ¼ 0:6522ðReÞ i0:1 h −5 1:34 0:504 ðδÞ0:456 ðγÞ−1:055  1 þ 5:26910 ðReÞ ðαÞ

f ¼ 9:6243ðReÞ−0:7422ðαÞ−0:1856 ðδÞ0:3053 ðγÞ−0:2659 h i −8 4:429 0:920 3:767 0:236 0:1 1 þ 7:669  10 ðReÞ ðαÞ ðδÞ ðγÞ

ð13Þ

ð14Þ

 ′ Where α ¼ hs , δ ¼ lft , γ ¼ st considering s ¼ 1 n −t and h ¼ H−t. Hydraulic diameter and Reynolds number are defined as below.

ð Þ

4shl

Dh ¼ 2 sl þ h′ þ th′ þ ts

ð15Þ

The above mentioned equations are valid for 120 b Re b 10 4, 0.134 α b 0.997, 0.012 b δ b 0.048 and 0.041 b γ b 0.121. These equations correlate j and f factors from experimental data within +20% accuracy in laminar, transition and turbulence flow regimes, therefore, there is no need to describe the flow regime for a specified operating condition and hence very useful in most practical applications. 3. GA and PSO GAs are based on the Darwinian idea of the survival of the fittest. A solution is called a ‘chromosome’ and consists of different genes any of them represent a variable of the possible solution. A population consists of an arbitrary number of chromosomes randomly created within the search space. The merit of each member is evaluated using a desired objective function. To mimic the survival of the fittest concept, best individuals, based on their merits, are selected to create the next generation of chromosomes. Usually a large number of generations are needed to arrive at a near-optimum solution. Basically GAs consist three main operators namely, selection, crossover and

ð17Þ

In the above formula, c1 and c2 are two positive numbers associated with cognition and social collaboration of a particle respectively (usually c1 = c2 = 2) where rand() and Rand() are uniformly distributed random functions in the range of [0,1]. ω is an inertia weight for controlling the effect of the previous velocity on the current one. This operator balances the role of the local search and the global search. Vmax and –Vmax are upper and lower bonds of the velocity. The computation of PSO is easy and using it in the GA does not implement huge computation task. 4. GA hybrid with PSO A schematic of the GAHPSO which is mainly consists of three main operators, enhancement, crossover and mutation is shown in Fig. 2. Enhancement: In this algorithm, after evaluating all the individuals by using the fitness function, the top-half of the population is selected as elites. However, as opposed to elite GAs, these individuals are not reproduced to the next generation directly not after being enhanced by PSO first. Using these enhanced elites as parents usually results in better-performing offsprings than those mated by original elites. Each member of the selected elites is considered a particle and the whole group a swarm in PSO process. In calculation of velocities and new positions from () and (), Pg is the best-performing particle among all the enhanced elite and offspring. Current Vi and pi are set to zero and x respectively in the case elite i is the offspring of the previous generation otherwise they record the current velocity of the particle and its best-performance position evolved so far. Half of the chromosomes (population) in the next generation are the enhanced elites while the others are produced by crossover operator. For the crossover operation, parents are only selected from the enhanced elites to increase the possibility of producing better performing individuals. In the proposed algorithm, tournament selection is used where two enhanced elite are selected at random and the better performing one is chosen as one of the parents while the other parent is selected by the same manner. Two offsprings are generated by performing two-point crossover where two side of the crossover (parents) are chosen arbitrarily Mutation: In the proposed algorithm, mutation happens during crossover. In the following simulations, a constant mutationprobability Pm = 0.1 is used.

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Table 1 Operating parameters selected for the case study. Parameters

Hot side

Cold side

Mass flow rate, m (kg/s) Inlet temperature (°C) Specific heat, Cp (J/kg K) Density, ρ (kg/m3) Dynamic viscosity, μ (Ns/m2) Prandtl number, Pr Maximum pressure drop, ΔP (N/m2)

1.66 900 1122 0.6296 401E-7 0.731 9.50

2 200 1073 0.9638 336E-7 0.694 8.00

5. Objective functions, design parameters and constraints For the case study A, total heat transfer area, AHT, is considered as the objective function and is which calculated using Eq. (9). In the case study B the total pressure drop is calculated as follows.

f ðxÞ ¼

ΔPa ΔPb þ ΔPa; max ΔPb; max

ð18Þ

Fig. 3. Evolution process of the objective of heat transfer area.

Where ϕ is a penalty function defined as, 2

ϕððXÞÞ ¼ R1:ðgðXÞ : In this work, the optimization is targeting two single-objective functions separately. The first is the minimum heat transfer area which is mainly associated with the capital cost of the heat exchanger and the other is minimum total pressure drop that represents the operating cost. In summary the optimization problem at hand is a large-scale, combinatorial problem which deals with both continuous and discrete variables. In the present optimization problem, GAHPSO is used for a constrained minimization. The problem can generally be stated as follows. Minimise f ðxÞ

X ¼ ½x1 …xk :

ð19Þ

Where constraints are stated as gj ðXÞ≤0; j ¼ 1; …; m

ð20Þ

and xi; min ≤xi ≤xi; max ;

i ¼ 1; …; k:

ð21Þ

To handle the constraints in the optimization algorithm, a static penalty function approach is applied. A penalty value is added to the objective function which converts the unconstrained problem to a constrained one.   m Minimise f ðxÞ þ ∑j¼1 ϕ g j ðXÞ

ð22Þ

R1 is the penalty parameter which comparing to the f(x) have a relatively large value. 6. Illustrative example An illustrative example from the work of Shah [25] is considered to clarify the application of the mentioned algorithm. A gas-to-air single pass cross-flow heat exchanger having heat duty of 1069.8 kW is needed to be designed for the minimum heat transfer area and total pressure drop separately. Maximum flow length and no-flow length of the exchanger is 1 m and 1.5 m respectively. Gas and air inlet temperatures are 900 °C and 200 °C respectively and gas and air mass flow rates are 1.66 kg/s and 2.00 kg/s respectively. Pressure drops are set to be limited to 9.50 kPa and 8.00 kPa. The gas and air inlet pressures are 160 kPa and 200 kPa absolute. The offset strip fin surface is used on the gas and air sides. Operating conditions are listed in Table 1. In this study, a total number of seven parameters namely, hot flow length (La), cold flow length (Lb),number of hot side layers (Na),fin frequency (n), fin thickness (t),fin height (H) and fin strip length (lf) are considered as optimization variables. All variables except number of hot side layers are continuous. Thickness of the plate; tp is considered to be constant at 0.5 mm and is not to be optimized. The variation ranges of the variables are shown in Table 2. Additional constraint is implementing to ensure that a minimum required heat transfer is achieved. Table 3 Preliminary design and GAHPSO results for minimum heat transfer area.

Subject to; Design variables

Table 2 Variation range of design parameters. Parameter

Min

Max

Hot flow length (La) (m) Cold flow length (Lb) (m) Fin height (H) (mm) Fin thickness (t) (mm) Fin frequency (n) (m− 1) Fin offset length (lf) (mm) Number of hot side layers (Na)

0.1 0.1 2 0.1 100 1 1

1 1 10 0.2 1000 10 200

ð24Þ

Constrained Condition Objective

Variables

Preliminary design

Optimal design

Side a length (La) (m) Side b length (Lb) (m) Fin height (H) (mm) Fin thickness (t) (mm) Fin frequency (n) (m− 1) Lance length (lfa) (mm) Number of hot side layers (Na) ΔPa at hot side (kPa) ΔPb at clod side (kPa) No-flow length, LC (m) Heat transfer area (m2)

0.3 0.3 2.49 0.10 782 3.2 167

0.21 0.23 5.9 0.10 1000 2.1 91

9.34 6.90 1 142.75

9.50 8.00 1.18 112.69

262

M. Yousefi et al. / International Communications in Heat and Mass Transfer 39 (2012) 258–263 Table 5 Comparison of results from GAHPSO and GA methods.

Case study A Case study B

Parameters

GAHPSO

GA

PSO

Heat transfer area (m2) CPU time (s) Total pressure drop CPU time (s)

112.7 3.32 0.056 3.25

132.5 4.41 0.070 4.35

123.5 3.57 0.063 3.49

Both hot-flow and cold-flow length are increased to their maximum limits while the number of fin layers and fin frequency are decreased. Fig. 4. Evolution process of the objective of total pressure drop.

7. Results and discussion 7.1. Minimum heat transfer area (case study A) For the prescribed heat duty and allowable pressure drop, the optimization problem is finding the design variables that minimize heat transfer area which is mainly associated with the capital cost of the heat exchanger. Fig. 3 demonstrates the iteration process of GAHPSO method. A significant decrease in the target function is seen in the beginning of the evolution process (first 10 decades).After certain decades (more than 50) the changes in the fitness function become relatively small. Finally the minimum heat transfer area of the PFHE is found to be 112.69 m 2 after 200 generations. Table 3 shows the PFHE preliminary design and the optimum configuration found by GAHPSO. A considerable reduction (21%) in the heat transfer area is presented by the optimization method comparing to the preliminary design. It is seen that in the minimum heat transfer area design the fin frequency is increased from 782 to 1000 and reached to its maximum limit while the fin strip length is decreased from 3.2 mm to 2.1 mm. The pressure drop on both sides, as predicted, is close to their maximum allowable amount, so the operating cost associated with the prime movers of the fluids will rise.

7.2. Minimum total pressure drop (case study B) Fig. 4 shows the evolution of the desired objective. A noteworthy decline in the target function is noticed in the beginning of the evolution process (first 10 decades). After certain decades (more than 30) the fitness function shows no changes. The final result is 0.056 which is considerably smaller than the 1.846 preliminary designs. Table 4 demonstrates the results of the optimization process.

7.3. A Comparison among the proposed algorithm, GA and PSO The efficiency of the GAHPSO algorithm in comparison to the GA and PSO algorithms is investigated in terms of computational time and accuracy. The results are demonstrated in Table 5. For the GA and PSO algorithms the population and iteration number are set similar to the hybrid algorithm. All algorithms are programmed in MATLAB® and run on an Intel® Core™ i5 CPU. The mentioned CPU time is the average of 10 executions of the computer code. The above mentioned results indicate that the proposed hybrid of GA and PSO can converge to the optimum results in both cases with higher accuracy in less computational time. In the case study A the result obtained by GAHPSO is 17.5% and 9.5% better than the results of GA and PSO respectively. The same trend can be noticed in the case study B where the hybrid algorithm presents result which is 25.0% and 12.5% better than the results of GA and PSO respectively. In both case studies the computational time of the hybrid algorithm is less than the original GA and PSO. 8. Conclusions This study presents the successful application of the GA hybrid with PSO for the optimal design of plate-fin heat exchangers. The GAHPSO concept, which is a combination of GA and PSO, is simple and easily can be implemented in engineering applications. Moreover, the combination of two different algorithms increase the diversity of the solutions and the probability of getting trapped in the local optima, which is the main disadvantage of PSO, is decreased. This features increase the applicability of this algorithm in thermal engineering problems which are contain a large number of discreet and continues variables and a large amount of discontinuity. A total number of seven design parameters are considered as the optimization variables and the constraints are handled by adding a penalty function to the fitness function. The results both on the total heat transfer area and total pressure drop show the great performance of the proposed algorithm over the preliminarily design and also GA and PSO. Acknowledgment

Table 4 Optimized results for minimum pressure drop.

Design variables

Constrained Condition Objective

This work was financed by institutional scholarship provided by University Teknology Malaysia and the ministry of higher education.

Variables

Optimized results

Side a length (La) (m) Side b length (Lb) (m) Fin height (H) (mm) Fin thickness (t) (mm) Fin frequency (n) (m− 1) Lance length (lfa) (mm) Number of hot side layers (Na) ΔPa at hot side (kPa) ΔPb at clod side (kPa) No-flow length, LC (m) Total pressure drop

1.00 1.00 10 0.1 241 10 71 0.29 0.21 1.5 0.056

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