Configuration parameters design and optimization for plate-fin heat exchangers with serrated fin by multi-objective genetic algorithm

Energy Conversion and Management 117 (2016) 482–489

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Configuration parameters design and optimization for plate-fin heat exchangers with serrated fin by multi-objective genetic algorithm Jian Wen a, Huizhu Yang a, Xin Tong a, Ke Li a, Simin Wang b,⇑, Yanzhong Li a a b

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 2 November 2015 Received in revised form 22 February 2016 Accepted 17 March 2016 Available online 24 March 2016 Keywords: Plate-fin heat exchanger Serrated fin Genetic algorithm Kriging response surface Optimization

a b s t r a c t The effect of fin design parameters on the performance of plate-fin heat exchanges was investigated in the paper, in which an improved algorithm combing a Kriging response surface and multi-objective genetic algorithm was used. An ideal gas was adopted as the working fluid and e-NTU method was utilized to determine the heat transfer and pressure drop. The fin height h, fin space s, fin thickness t and interrupted length l of serrated fin and channel inlet Reynolds number are firstly optimized, while the j factor, f factor and JF factor are optimization goals. The results show that when the inlet channel flow is in the laminar flow (Re < 1000), it is beneficial to trade off the j factor and f factor. Furthermore, the total heat flow rate, total annual cost and number of entropy production units of plate-fin heat exchanges are optimized with the specified mass flow rate under given space by multi-objective optimization. Results obtained from the first two objectives and three objectives show that the fin design parameters are very similar except that the latter interrupted length is smaller slightly. A comparison between the results obtained by the previous approaches and the proposed algorithm shows that under the same effectiveness, the annual cost of the proposed algorithm is about 10% lower than the previous ones and it is faster to be converged. Therefore, this study demonstrated that the proposed algorithm is able to optimize the fin design parameter of serrated fin and the obtained results are beneficial to guide the design of plate-fin heat exchanges. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Heat exchangers are used to transfer thermal energy between two or more media and be known as one of the most essential equipment in almost every industrial plant, including power engineering, aerospace, electronics, automobile, petroleum refineries, cryogenic and chemical industries, etc. Plate-fin heat exchanges (PFHEs) are widely used in gas–gas applications for their high thermal effectiveness, high compactness, low weight, ease of handling multiple streams (up to 20 kinds of media) and low cost. They weigh 95% less than comparable conventional shell-and-tube heat exchangers and provide 1000–2500 square meters of heat transfer per cubic meter of exchanger volume [1]. A typical PFHE main includes nozzle, heater, distributor fin, cover plate, side bar, corrugated fins and cap sheet. However, the most important component of PFHEs is core, which is built by cover plates and layers of fins in a sandwich construction. Depending on the diverse applications, there are many types of fins such as corrugated, louver, perforated, ⇑ Corresponding author. E-mail address: [email protected] (S. Wang). http://dx.doi.org/10.1016/j.enconman.2016.03.047 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.

serrated strip and pin fins [2]. While among the many enhanced fin constructions, rectangular serrated fin is widely used. This type of fin is characterized by a high degree of surface compactness, high reliability and substantial heat transfer enhancement due to the boundary layer re-starting at the uninterrupted channels. However, there is, on the other hand, an associated increment in a large pressure drop consequently which leads to higher operational costs. Indeed, it has become necessary to find a trade-off between the heat transfer enhancement and the pressure drop increment. Unfortunately, the conventional design approaches of PFHEs are difficult to optimize the fin configurations owing to the complexity of PFHEs and the disable design approaches that mainly based on empirical chosen, checking computations with trial-and-error and the results obtained by predecessors. The application of genetic algorithm (GA) in optimization of PFHEs has shown the effectiveness and robust in the few decades. Ozkol et al. [3] employed a GA-based approach for design of PFHEs as an alternative to the previous trial-and-error based approaches and the numerical results indicated that the proposed approach was superior to its previous ones. Xie et al. [4] used the GA to achieve minimum total weight and total annual cost of a cross-flow PFHE with

J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489

483

Nomenclature a Achan Acov Af CA CInv COpe CTotal D f h j kel l L N Ns NTU Nu Pr Q Re

heat transfer coefficient, W m2 K1 heat transfer area in one channel, m2 heat transfer area in cover plate, m2 annual cost coefficient cost per unit surface area, $ m2 annual cost of investment, $ year1 annual cost of operation, $ year1 total annual cost, $ year1 hydraulic diameter of fin channel, m friction factor height of serrated fin, m colburn factor price of electrical energy, $ kW h1 interrupted length of serrated fin, m length of heat exchanger, m fin channel number number of entropy production units number of transfer units Nussle number Prandtl number total rate of heat transfer, W Reynolds number

given constrained condition. Tugrul Ogulata et al. [5] minimized entropy generation number to optimize cross-flow PFHEs. However, only the shape variables were considered as the decision variables while the fin parameters were fixed in the above studies. Mishra [6] exploited a GA to minimize the total annual cost in a cross-flow PFHE where the fin configurations were optimized. Besides many of the new methods had been introduced to heat exchangers design optimization. Peng et al. optimized the total weight and total annual cost of PFHEs by using a GA combined with back propagation neural networks [7] and an improved particle swarm optimization [8]. Hadidi et al. [9] used imperialist competitive algorithm to minimize the cost of shell-and-tube heat exchangers by varying tube length, tube outer diameter, pitch size and baffle spacing. Patel et al. [10] proposed an improved teaching–learning-based optimization algorithm to optimize a Stirling heat engine by considering two and three objective functions simultaneously for the maximization of thermal efficiency, output power and minimization of total pressure drop of the engine. Besides Berrazouane et al. [11] used cuckoo search algorithm to minimize loss of power supply probability, excess energy and levelized energy cost of an optimized fuzzy logic controller. However, no single algorithm is able to outperform the others for all engineering applications, due to continuous improvements in meta-heuristic algorithms. The optimization of PFHEs often faces with more than one objective function and among them are conflicting. Multiobjective genetic algorithm (MOGA) was also successfully applied. Najafi et al. optimized a plate-and-frame heat exchanger [12] and a plate-and-fin heat exchanger [13] by considering maximum the total rate of heat transfer and minimum the total annual cost as two objective functions. Lee et al. [14] used a MOGA to maximize heat transfer rate and minimize pressure drop in PFHEs. Gholap et al. [15] studied the PFHEs by minimizing the energy consumption and material cost as two conflicting objective functions. Liu et al. [16] optimized a recuperator for the maximum heat transfer effectiveness as well as minimum exchanger weight or pressure loss. The thermodynamic irreversibility is unavoidable in PFHEs due to finite temperature difference heat transfer in the fluid streams and the pressure drops along them. Therefore, the second law based optimization by number of entropy generation unit (Ns)

s Sg t T u V

spacing of serrated fin, m entropy production, W kg1 K1 thickness of serrated fin, m temperature, K velocity, m s1 volume flow rate, m3 s1

Greek symbols d thickness of cover plate, m e effectiveness s hours of year operation, h g compressor efficiency U rate of heat transfer in one channel, W Subscripts B exchanger (core) length c cold side h hot side H exchanger (core) depth of cold side in inlet L exchanger (core) width out outlet

minimization is a very effective method to reduce the amount of lost useful power in PFHEs. Bejan [17] optimized a PFHE through a GA, in which the Ns is optimized with the specified heat duty under given space. Hang et al. [18] considered Ns due to friction, Ns due to heat transfer and total Ns as objective functions by single objective and multi-objective optimization. Wang et al. [19] applied an improved multi-objective cuckoo search algorithm, in which minimum heat transfer and fluid friction number of entropy generation unit were considered to be two objective functions. The first law (conservation of energy) and second law (entropy generation minimization) of thermodynamics are two main categories for optimization design of PFHEs. However, studies combined with the first and second laws of thermodynamics are lack. In this study, an improved algorithm combing a Kriging response surface and multi-objective genetic algorithm is presented to find optimal four fin design parameters of PFHEs with serrated fin. The total heat flow rate, total annual cost and number of entropy production units of PFHE are considered as objection functions by single objective and multi-objectives optimization with the specified mass flow rate under given space. In addition, in order to guide the design of PFHEs, the j factor, f factor and JF factor are defined as objection functions, while four fin design parameters and channel Reynolds number are considered as optimization parameters. The principal aim of this paper is to investigate the effects of fin design parameters of serrated fin on the performance of PFHEs. 2. Thermal modeling of PFHEs In this section, the equations for calculating the total heat flow rate, pressure drop, the total annual cost and the modified number of entropy production units of the system are presented. In Fig. 1, the schematic diagram of a typical serrated fin is shown in detail, in which the fin height h, fin space s, fin thickness t and interrupted length l are considered as the four optimization design parameters. In order to cover the design range of majority of serrated fins in air–air applications, the fin height h is selected from 4.7 mm to 9.5 mm, fin space s is 1.5 mm
3.5 mm, fin thickness t is 0.1 mm
0.5 mm and interrupted length l is 3 mm
9 mm.

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J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489

j ¼ 0:6522Re0:5403 a0:1541 d0:1499 c0:0678  ½1 þ 5:269  105 Re1:34 a0:504 d0:456 c1:055 

0:1

ð2Þ

where the colburn factor j also can be defined as:

j¼

Nu

ð3Þ

Re Pr 1=3

And Reynolds number, the hydraulic diameter and Nusselt number can be calculated as follow:

Re ¼ D¼

In the analysis, assumptions are as follows: 1. Air is adopted as working fluid and its properties are assumed to be constant. 2. The working medium is assumed as a uniform distribution, thus the total heat exchange is the sum of each channel heat exchange and the total pressure drop is the sum of each layer channel pressure drop. 3. The working medium is assumed to be constant, which has nothing to do with the pressure drop. 4. Number of fin layers for the hot side is assumed to be one more than the cold side. 5. The configuration parameters of serrated fin and mass flow rate in hot and cold channels are identical for each calculates in order to eliminate the effects of different operating conditions of hot channel on the optimization results of cold channel.

ð4Þ

v

4lðh  tÞðs  tÞ 2ðlðh  tÞ þ lðs  tÞ þ tðh  tÞÞ þ tðs  tÞ

Nu ¼

Fig. 1. The schematic diagram of serrated fins.

uD

aD

ð5Þ ð6Þ

k

Therefore, the heat transfer coefficient a and overall thermal resistance 1/(KA) are obtained as:

a¼

Nuk D

ð7Þ

1 1 ¼ KA aA1 þ Acodv k þ aA1 chan

ð8Þ

chan

When a mass flow rate in hot and cold channels are identical and the heat exchanger is a counterflow, the value of NTU and the effectiveness e can be calculated as follows [20]:

NTU ¼

KA _ p Þc ðmc

NTU 1 þ NTU

ð9Þ ð10Þ

2.1. Heat transfer performance

e¼

The e-NTU [20] method is utilized in order to determine the value of the total heat flow rate for a counterflow PFHE which the flow directions of the cold and hot fluids are in the opposite direction (Fig. 2). Heat transfer areas of the hot and cold side are identical and those are defined as:

By substituting e, the outlet temperatures of hot and cold channels are obtained as follow:

T out;c ¼ T in;c þ eðT in;h  T in;c Þ

ð11Þ

T out;h ¼ T in;h  eðT in;h  T in;c Þ

ð12Þ

Achan ¼ 2NL ð2lðh þ s  2tÞ þ 2tðs  tÞ þ tðs  2tÞÞ

Therefore, the heat flow rate in single channel and the total heat flow rate are obtained as:

ð1Þ

The thermal performance of the surface of PFHEs is used by the correlations of Mangles & Bergles [21]:

_ p Þc ðT out;c  T in;c Þ U ¼ ðmc

ð13Þ

_ p Þc ðT out;c  T in;c Þ Q ¼ N B NH ðmc

ð14Þ

where NB = LB/s is the number of channels in each layer, NH = LH/(h + d) is number of fin layers for cold sides and NL = LL/l is number of periodic serrated fin along the length. It is worth noting that the exchanger (core) depth (LH) is referred to the effective height of the cold side. 2.2. Total annual cost calculations Total annual cost includes investment cost and operating cost of compressor. A similar cost calculation of PFHEs can be seen in references [10,22,23], as follows:

C Inv ¼ Af  C A  Anchan C Ope ¼ kel s

DpV

g

C Total ¼ NB NH C Inv þ NH C Ope

Fig. 2. A schematic diagram of counterflow PFHE.

ð15Þ ð16Þ ð17Þ

where CA, Af and Achan are the heat exchanger investment cost per unit surface area, annual cost coefficient and total heat transfer areas in a single channel, respectively; n is the exponent of

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J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489

nonlinear increase with area increase; kel and s the electricity unit cost and the hours of operation per year respectively; Dp, V and g are pressure drop, volume flow rate and compressor efficiency respectively. Moreover, the annual cost coefficient Af and pressure drop Dp can be calculated as:

Af ¼

r 1  ð1 þ rÞy

ð18Þ

2f qu2 LL D

ð19Þ

Dp ¼

where r and y are interest rate and depreciation time, respectively. In addition, the friction performance of the surface of PFHEs can be found by the correlations of Mangles & Bergles [22]:

f ¼ 9:6243Re0:7422 a0:1856 d0:3053 c0:2659  ½1 þ 7:669  108 Re4:429 a0:92 d3:767 c0:236 

0:1

ð20Þ

2.3. The number of entropy production units estimation The modified number of entropy production units (Ns) is defined as [24]:

Ns ¼ NB NH

Sg ðT in;h  T in;c Þ

_ p Þc ln Sg ¼ ðmc

ð21Þ

U 

T in;c T out;c



_ c ln  ðmRÞ



Pin;c Pout;c

The Kriging method includes a multidimensional interpolation and a polynomial model, which provides a global model of the design space with local deviations determined. Moreover, it can take correlation between observations into account and uses ‘‘space-filling” designs to fit the model, in which the space-filling design can be augmented to fit gradually the potentially irregular shape of the response. Therefore, the interpolation model of Kriging response surface has a very flexible shape and is capable of modeling complex surfaces. In addition, the Kriging response surface can improve by adding points at the maximum of the Kriging predictor variance that is very useful to prevent premature convergence of MOGA. However, the second-order polynomial model is widely in the standard experimental design methodology since the methodology is simpler and cheaper method. Nevertheless, a simple polynomial model may not be appropriate for modeling complex computer processes [26]. What is more, the standard experimental design methodology may not be used due to their most design points lie on the boundary and do not allow the detection of possible irregularities inside the experimental domain. 3.2. Multi-objective Genetic Algorithm A hybrid variant of the popular NSGA-II (Non-dominated Sorted Genetic Algorithm-II) based on controlled elitism concepts was

Parameters

 ð22Þ

Initial Population Updated

where Sg is the total entropy generation rate in PFHEs due to the finite temperature difference between heat conduction temperatures and the finite pressure drop of fluid friction.

Kriging Construction 3. Optimization methods The proposed algorithm in this paper is a mathematical optimization combing a Kriging response surface and the MOGA optimization algorithm that is a hybrid variant of the popular NSGA-II (Non-dominated Sorted Genetic Algorithm-II) based on controlled elitism concepts. The general optimization approach is the same as MOGA, but a Kriging response surface is used. Compared with the conventional MOGA algorithm, the use of Kriging has three main advantages: 1. Allow for a more rapid optimization process because: (1) Except when necessary, it does not evaluate all design points. (2) Part of the population is ‘‘simulated” by evaluations of the Kriging, which is constructed of all the design points submitted by the MOGA algorithm. 2. The Kriging error predictor can reduce the number of evaluations used in finding the first Pareto front solutions that reduces the number of design points necessary for the optimization. 3. An approximation of the output values can be provided when the Kriging response surface is established. Therefore, if the initial populations and the new design points that are used for the Kriging construction can generate by finite element analysis softwares, such as Fluent. This optimization method can overcome the dependence on empirical correlations, which has been demonstrated in references [25].

MOGA : Selection, Crossover, Mutation Design Point Update Kriging Error Predictor

Predicted Error Acceptable?

No

Yes Generate New Population

Algorithm Converged? No

Yes

Output Results

3.1. Kriging response surface

No A metamodel is an approximation of the Input/Output (I/O) function that is implied by the underlying simulation model. At present, design and analysis of I/O function mainly adopt two following approaches: the standard experimental design methodology and the accurate response surface method such as the Kriging response surface. The first uses well known designs, such as factorial or composite designs, to fit a polynomial regression model. The second prefers a more sophisticated statistical model.

Maximum Number of Iterations?

Yes Not Converged Fig. 3. A flowchart of optimization process.

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J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489

used. This genetic algorithm is characterized by fast nondominated sorting approach, fast crowded distance estimation procedure and simple crowded comparison operator [27,28]. The optimization process is as follows: (1) Optimization starts with a set of initial population that is used for the Kriging construction. (2) Value of objective function of initial population is calculated. Thus Kriging response surfaces is created for each output, based on the first population and can improved during simulation with the addition of new design points. (3) MOGA is run and generates a new population via selection, crossover and mutation, in which the Kriging is used as an evaluator to check the error for each design point. If the error for a given point is acceptable, the approximated value is included in the next population to be run through the MOGA algorithm, if the error is not acceptable, the point is promoted as new design points to improve the Kriging. (4) MOGA converges when the maximum allowable Pareto percentage has been reached. Thus, the optimized values are output. However, if the optimization is not converged, the process is returns to step 3 until the maximum number of iterations. A flowchart of optimization process was shown in Fig. 3. Where the initial population size was set up to be 150 and the number of individuals in each generation was 100. In addition, the maximum allowable Pareto percentage was put up to be 90%, the crossover was selected to be 0.98 and the possibility of the mutation was chosen to be 0.01.

4. Results and discussions 4.1. Algorithm effectiveness The effectiveness of the proposed algorithm is assessed by analyzing an example of PFHEs which was earlier analyzed using the NSGA-II by Sanaye and Hajabdollahi [24] and using a modified TLBO by Rao and Patel [10]. The PFHE having serrated fins was designed and optimized for maximum effectiveness and minimum annual cost. More detailed operating conditions could be seen in reference [24]. Fig. 4 represents the Pareto optimal curve obtained by using the proposed algorithm for multi-objective optimization and its comparison with the results obtained by Sanaye [24] and Rao [10]. As seen from the results, under the same effectiveness, the Pareto optimal design points of annual cost, using the proposed algorithm, are 10.5% and 10.8% lower compared to the NSGA-II and modified TLBO approach, respectively. In addition, the results show that the proposed algorithm performs great in terms of

1400

Krining + MOGA NSGA2[24] Modified TLBO[10]

Annual Cost /$ year-1

1200

1000

800

E

600

D C

400 0.55

A 0.60

0.65

0.70

B 0.75

0.80

0.85

0.90

Effectiveness Fig. 4. The distribution of Pareto-optimal points solutions for PFHE using proposed algorithm, NSGA-II and modified TLBO.

convergence which is converged after 498 evaluations. The average run time of the proposed algorithm is 6.39 s. However, it is 8.98–24.47 s consumed with different strategies by Rao [10]. Therefore, the optimization results of the proposed algorithm are considered a higher accuracy and a faster convergent rate compared to the earlier studies. Besides, to guide the design of PFHE, Table 1 presented five optimal values of the design parameters (A
E). 4.2. Thermal hydraulic performance optimization In this section, the fin height h, fin space s, fin thickness t and interrupted length l of serrated fin and channel inlet Reynolds number were considered as optimization parameters, while the j factor, f factor and JF factor were optimized using MOGA by single objective and multi-objective optimization, respectively. The range of four geometric variables of serrated fin were selected as described in Section 2 and the channel inlet Reynolds number was selected from 200 to 3000 that covers the majority of the operating conditions of PFHEs. 4.2.1. Single objective optimization Table 2 presents the optimal values of the design parameters and the predicted error by the correlations of Manglik & Bergles [22]. As depicted, it shows that a larger fin height, smaller fin space, thicker fin thickness and shorter interrupted length are beneficial to improve heat transfer. On the contrary, a shorter fin height, smaller fin space, thinner fin thickness and longer interrupted length will decrease the pressure drop. In addition, it can be understand that j factor, f factor and JF factor increase with the decrease of inlet Reynolds. Therefore, maximum j factor and JF factor have a smaller inlet Reynolds, while minimum f factor has a bigger inlet Reynolds. Furthermore, comparing with the values of the correlations, the results illustrated a good agreement within ±5% error. Therefore, the optimization method is verified to be successful. Under the operating conditions, the optimized results are obtained by fin height h = 9.2 mm, fin space s = 1.5 mm, fin thickness t = 0.4 mm, interrupted length l = 5.6 mm and inlet Reynolds Re = 217. In order to identify the key parameters of the design and how they influence the performance of PFHEs, a sensitivity of the design parameters was studied. On each of these results, only one input parameter is varying while the other input parameters are constant. These results show in Fig. 5. Clearly, the inlet Reynolds number has a major influence on the performance of PFHEs and it is negative growth. The effect of the fin height on j factor, f factor and JF factor shows positive growth. The effect of the fin space on j factor and JF factor shows negative growth and that on f factor shows positive growth. The effect of fin thickness on j factor and f factor shows positive growth and that on JF factor shows negative. The effect of interrupted length on j factor and f factor shows negative growth and that on JF factor shows positive. Besides, regardless of the effect of Reynolds number, it is concluded that the interrupted length has the biggest impact on j factor, the interrupted length and fin thickness have more important influence on f factor and the fin thickness has the greatest impact on JF factor. The influence degrees of design parameters on thermal hydraulic performance show detailed in Fig. 5. Therefore, the sensitivity of design parameters on the thermal hydraulic performance of PFHEs has been studied, which will have an important guiding role in the selection of design parameters by the designer. 4.2.2. Multi-objective optimization Since the j factor and f factor are mutually conflicting, there is no single design parameter that yielded the best value for the

487

J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489 Table 1 The selected optimal design parameters of PFHE. Points

Height (mm)

Space (mm)

Fin length (mm)

Hot length (m)

Cold length (m)

No-flow length (m)

Effectiveness

Annual cost $ year1

A B C D E

4.53 5.06 2.10 1.52 1.63

1.85 1.77 1.40 1.60 2.42

2.11 3.17 2.32 2.65 2.84

0.21 0.23 0.23 0.31 0.34

1.05 0.99 0.99 0.95 1.16

0.27 0.31 0.25 0.28 0.34

0.6905 0.7230 0.7647 0.8021 0.8326

474.82 511.04 622.15 733.81 803.56

Table 2 The optimum results of thermal hydraulic performance and the predicted error. Objectives

Re

h (mm)

s (mm)

t (mm)

l (mm)

j

f

JF

Max j factor Validate Predicted error

214

8.90

1.61

0.47

3.06

0.0393 0.0394 0.2%

0.2058 0.2045 0.6%

0.0665 0.0668 0.5%

Min f factor Validate Predicted error

2971

4.87

1.62

0.11

8.78

0.0070 0.0073 4.2%

0.0233 0.0224 4.1%

0.0264 0.0260 1.4%

Max JF factor Validate Predicted error

217

9.18

1.51

0.43

5.60

0.0358 0.0357 0.3%

0.1573 0.1543 1.9%

0.0668 0.0665 0.5%

17.1

20

4.4

Local Sensitivity /%

0

4.33.2

3.2

-0.5

-3 -9.5

-9.4

-20

4.1

3.5

4.3. Thermal-economic and irreversibility optimization

-24.9 -40

Height Space Thickness Interrupted Re

-60 -80

-50.5

-79.9

-82.5

j

the j factor and f factor. Therefore, a set of optimal solutions, each of which is a trade-off the two conflicting objective functions in an appropriate level, are obtained. These results can be selected by the designer regarding the project’s limits and the available investment.

f

JF

Fig. 5. Sensitivity of design parameters on thermal hydraulic performance.

two objectives. MOGA was also conducted by taking maximum the j factor and minimum f factor as the multi-objective functions. Ten of the selected optimal design parameters are given in Table 3. Results show that when the channel flow is in the laminar flow (Re < 1000), it is beneficial to trade off the j factor and f factor. In addition, it can obtain that a thinner fin thickness, a higher fin height and a longer interrupted length are also advisable to balance

Table 3 The selected optimal design parameters of thermal hydraulic performance by multiobjective optimization. h (mm)

s (mm)

t (mm)

l (mm)

Re

j

f

JF

6.03 6.38 9.17 8.32 6.78 8.40 5.30 8.44 7.46 7.42

2.51 3.04 2.70 1.50 1.51 1.55 1.52 1.60 1.51 1.67

0.11 0.28 0.39 0.14 0.14 0.13 0.44 0.13 0.12 0.14

7.68 7.87 8.73 6.69 6.56 6.51 8.93 7.97 5.89 8.07

241.6 288.6 394.8 752.8 752.8 747.7 628.3 600.6 758.6 888.5

0.0253 0.0239 0.0214 0.0162 0.0158 0.0163 0.0175 0.0167 0.0161 0.0141

0.1187 0.1100 0.0924 0.0539 0.0520 0.0546 0.0639 0.0600 0.0529 0.0453

0.0508 0.0517 0.0517 0.0419 0.0410 0.0418 0.0453 0.0450 0.0411 0.0399

The fin height h, fin space s, fin thickness t and interrupted length l of serrated fin, describing in Section 2, were considered as optimization parameters, while the total heat flow rate, the total annual cost and the number of entropy production units of PFHE were optimized by MOGA in order to achieve design parameters which lead to the highest possible total heat flow rate, the least total annual cost and the minimum number of entropy production units. The assumptions were described in Section 2 and the operating conditions and the parameters needed for cost evaluation were given in Tables 4 and 5 respectively. It is worth noting that the volume flow rate is referred to the flow rate of the cold side and that is similar with the exchanger (core) depth. The effects of fin design parameters on total heat flow rate, total annual cost and number of entropy production units of PFHE are given in Fig. 6. It is obvious that the effects of fin design parameters on objective functions have a same tendency. The effects of the fin height, fin space and fin thickness on objective functions show negative growth and that of the interrupted length shows positive growth. Besides the fin space has the biggest impact on total heat flow rate and total annual cost, while the fin thickness has an important influence on number of entropy production units. Unfortunately, the purpose of the total heat flow rate is to achieve

Table 4 Operating conditions. Parameters

Values

Volume flow rate of cold side (m3/h) Exchanger(core) length (m) Exchanger(core) width (m) Exchanger(core) depth of cold side (m) Inlet temperature of cold side (K) Inlet temperature of hot side (K)

600 1.2 0.16 0.2 293.15 373.15

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J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489

200

Table 5 Cost coefficients of heat exchanger. Values 2

Cost per unit area, CA ($/m ) Depreciation time, y (year) Inflation rate, r Hours of operation, s (hour/year) Electricity price, ket ($/kW h) Compressor efficiency, g Exponent of non-linear increase factor, n

90 10 0.1 6000 0.15 60% 0.6

160

140

120

60

Height Space Thickness Interrupted

40

Local Sensitivity /%

Pareto-optimal points

180

CTotal/$ year-1

Economic parameters

20

16

100 14000

35

14500

15000

15500

16000

Q/W

12

Fig. 7. Pareto-optimal points of heat flow rate and annual cost.

0 -20

-8 -16

-8

-11

-16 -31

-40 -60

-60

-26

-61

Q

Ctotal

Ns

Fig. 6. Sensitivity of fin design parameters on three objectives.

maximum, while that of total annual cost and number of entropy production units are to achieve minimum. Therefore, it is able to understand that the heat flow rate are conflict with total annual cost and the number of entropy production units, while the total annual cost is consistent with the number of entropy production units. These results in Table 6 clearly demonstrated the conflict between objectives, in which the total heat flow rate, total annual cost and number of entropy production units were optimized using GA by single objective optimization. Fig. 7 shows the results for Pareto-optimal points of heat flow rate and annual cost. As depicted, the total heat flow rate increases with the increase of the total annual cost. Therefore, any one of higher total heat flow rate in place of another will always sacrifice quality of total annual cost. Similarly, ten of the selected optimal design parameters are given in Table 7. The results show that a higher fin height is advisable to balance the total heat transfer rate and the total annual cost. Besides the values of other fin parameters in the middle are beneficial to achieve the two conflicting objective functions in an appropriate level. As seen from the Fig. 7, it can find that the effects of fin design parameters on total heat transfer rate and total annual cost are very similar, while their goal is the opposite. Consequently, the fin space, fin thickness and interrupted length are in the middle value. Fig. 8 shows the results for Pareto-optimal points of heat flow rate, annual cost and number of entropy production units. As shown, the heat flow rate increases along with the total annual

Table 7 The selected optimal design parameters of heat flow rate and annual cost. h (mm)

s (mm)

t (mm)

l (mm)

Q (W)

Ctotal ($ year1)

9.46 6.82 8.04 8.91 9.24 8.45 8.35 8.17 9.37 9.37

1.97 1.79 1.76 1.81 1.95 2.15 2.09 2.09 2.13 2.13

0.44 0.43 0.37 0.32 0.31 0.41 0.37 0.37 0.28 0.28

7.56 8.31 7.11 5.79 6.39 5.72 7.88 7.88 4.52 4.52

15431.3 15536.4 15553.1 15522.6 15381.9 15350.5 15282.2 15283.4 15318.1 15318.1

151.9 158.8 160.1 158.7 150.9 149.3 145.4 145.6 147.7 147.7

Fig. 8. Pareto-optimal points of three objectives.

Table 6 The optimum results of total heat flow rate, total annual cost and number of entropy production units. Objectives

h (mm)

s (mm)

t (mm)

l (mm)

Q (W)

Ctotal ($ year1)

Nstotal

Maximum Q Minimum Ctotal Minimum Nstotal

4.81 9.30 8.72

1.50 3.50 2.77

0.42 0.16 0.10

3.05 8.95 8.94

15908.9 14072.4 14494.7

203.18 111.55 123.23

0.2872 0.2481 0.2467

J. Wen et al. / Energy Conversion and Management 117 (2016) 482–489 Table 8 The selected optimal design parameters of three objectives.

489

Acknowledgments

h (mm)

s (mm)

t (mm)

l (mm)

Q (W)

Ctotal ($ year1)

Nstotal

9.40 9.26 9.15 8.95 8.96 9.37 9.29 8.84 7.71 8.81

2.20 1.92 1.87 2.16 1.90 1.82 2.11 2.13 2.02 1.94

0.11 0.14 0.18 0.15 0.26 0.23 0.20 0.14 0.13 0.16

4.41 5.80 6.95 5.57 8.87 8.10 6.72 3.84 4.70 8.68

15087.2 15263.4 15301.6 15110.6 15298.9 15355.5 15147.7 15216.2 15243.1 15179.3

139.13 148.23 150.02 140.72 148.72 151.80 141.91 144.43 147.19 146.19

0.2483 0.2488 0.2493 0.2487 0.2501 0.2501 0.2493 0.2502 0.2495 0.2483

cost and the number of entropy production units. Ten of the selected optimal design parameters are given in Table 8. Compared with candidate points of heat flow rate and annual cost, the optimized design parameters of three objectives are very similar except that the interrupted length is smaller slightly. One reason is that the effects of fin design parameters on the total annual cost are consistent with the number of entropy production units, thus the optimized values of heat flow rate and annual cost are similar to that of three objectives. Another reason is that the effect of the fin thickness on the number of entropy production units has more important influence on that of annual cost, which results in a thinner fin thickness. 5. Conclusions In this paper, an improved algorithm combing a Kriging response surface and the MOGA optimization algorithm is successfully introduced for the multi-objective optimization of a counterflow PFHE. Based on this algorithm, the fin height h, fin space s, fin thickness t and interrupted length l of serrated fin and channel inlet Reynolds number are considered as optimization parameters. While the j factor, f factor and JF factor are optimized by unconstrained multi-objective optimization. Furthermore, the total heat flow rate, the total annual cost and the number of entropy production units of PFHEs are optimized with the specified mass flow rate under given space. A sensitivity analysis is carried out in order to investigate the effect of design parameters on each objective function. Besides to demonstrate the proposed algorithm’s accuracy and performance, a case study is carried out to compare with the earlier studies. The main conclusions of this study are as follows: (1) Compared with the previous studies, the proposed algorithm shows a higher accuracy and a faster convergent rate with an average run time of 6.39 s. (2) Ten Pareto-optimal points to balance between j factor and f factor are achieved as selectable to design the PFHEs since each of them is a trade-off on improves heat transfer and reduces friction. Besides when the channel flow is in the laminar flow (Re < 1000), it is beneficial to trade off the j factor and f factor. (3) The fin design parameters are opposite to achieve a higher total heat flow rate or to less total annual cost and number of entropy production units, while it is consistent for total annual cost and number of entropy production units. (4) Compared with candidate points of heat flow rate and annual cost, the optimized design parameters of heat flow rate, annual cost and number of entropy production units of PFHEs are very similar except that the interrupted length is smaller slightly. These obtained Pareto-optimal points can be selected by the designer regarding the project’s limits and the available investment.

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