Optimization investigation on configuration parameters of sine wavy fin in plate-fin heat exchanger based on fluid structure interaction analysis

International Journal of Heat and Mass Transfer 131 (2019) 385–402

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Optimization investigation on configuration parameters of sine wavy fin in plate-fin heat exchanger based on fluid structure interaction analysis Jian Wen a, Ke Li a, Chunlong Wang a, Xing Zhang a, Simin Wang b,⇑ a b

Department of Refrigeration and Cryogenics Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Department of Process Equipment and Control Engineering, 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 4 June 2018 Received in revised form 13 October 2018 Accepted 4 November 2018

Keywords: Plate-fin heat exchanger Wavy fin Stress analysis Response surface Multi-Objective Genetic Algorithm

a b s t r a c t The comprehensive performance of sine wavy fin in plate-fin heat exchangers (PFHEs) is numerically studied based on fluid structure interaction (FSI) analysis in this paper. The analysis results of stress distribution reveal that the highest stress is located in the inlet and outlet region of fin structure, and the fluctuant stress reaches to the peak in the wave crest. By way of analyzing Full 2nd-Order Polynomial response surface (RS), the effects of inlet velocity and five configuration parameters (fin height, fin space, fin thickness, fin wavelength and double amplitude) on heat transfer, flow resistance and stress of sine wavy fin structure are quantitatively assessed. The results reveal that the j factor increases with the increase of fin space and fin height, and decreases with the increase of fin thickness, wavelength and inlet velocity. The j factor firstly increases with the increase of double amplitude and then decreases. The f factor increases with double amplitude, fin space and fin height, and decreases with fin thickness, wavelength and inlet velocity. The maximum stress increases with the increase of wavelength and fin space, and decreases with the increase of fin thickness and double amplitude. The interaction effects of input parameters on the j factor and f factor are not obvious. While the interaction effects of fin thickness and wavelength, double amplitude and wavelength on the maximum stress are obvious. Based on RS, Multi-objective Genetic Algorithm (MOGA) is performed to optimize the fin structure comprehen1=3 sively, with multiple objectives of increasing the JF factor (JF ¼ j=f ) and decreasing the maximum stress to the best. The optimization results shows that, compared with the original design, the JF factor of optimal design 1, 2, and 3 increases by 11.0%, 8.4% and 15.9% respectively, and the maximum stress decreases by 32.3%, 42.4% and 20.7% respectively. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction PFHEs are now commonly used in a wide range of chemical processes and other industrial applications [1], such as radiators, evaporators, condensers, and air conditioning systems. It has high effective heat transfer performance and compact construction, and has been developed, as a result of several studies which assess the influence of complex fin geometries on thermal performance. There are several kinds of compact heat exchangers including louver fin, offset strip fin, wavy fin, and so on [2,3]. As one sort of compact fin, wavy fin is made by corrugating a plain fin in the flow direction. It has high thermal performance because of its dynamically formed internal flow as well as extended heat transfer area [4].

⇑ Corresponding author. E-mail address: [email protected] (S. Wang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.023 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

Chang et al. [5] considered enhancing heat transfer by setting the novel longitudinal wavy ribs along a wavy two-pass square channel. The detailed Nusselt number (Nu) distributions over the ribbed wavy end wall are measured using the steady-state infrared thermography method together with the Fanning friction factors (f) evaluated from the pressure drop measurements. For wavy fin channel, in depth mechanism research is interesting and rare. Michioka [6] implemented large-eddey simulation to investigate the turbulent flow and gas dispersion over wavy walls across a wide range of the wave amplitude to wavelength. Two tracer gases are emitted from point sources located at a single crest and trough of the wavy wall. He got a lot of interesting conclusion about turbulent flow and gas dispersion in wavy fin channel. A lot of researchers focused on the heat transfer and flow resistance of wavy-fin structure, but for wavy fin PFHEs the available date based on experiment is not as much as that for serrated fin. Muley et al. [7] presented experimental j and f measurements for the flow rates in the Re range from 500 to 5000 in sinusoidal wavy channels. They

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J. Wen et al. / International Journal of Heat and Mass Transfer 131 (2019) 385–402

Nomenclature Latin symbols A double amplitude, mm secondary heat transfer area in fin channel, mm2 A2 Aw total heat transfer area of fin channel, mm2 Aw,cl wall area of the clipboard in fin channel, mm2 Ac heat transfer area in fin channel, mm2 Ain inlet area of extension, mm2 b clipboard thickness, mm specific heat at constant pressure, J kg1 K1 cp Dh equivalent diameter of wavy fin structure, m F external force located on fin surface, MPa f friction factor Gk generation of turbulent kinetic energy due to the mean velocity gradients h fin height, mm hc mean heat transfer coefficient, W m2 K1 Hn normalised helicity, demensionless HV volume-average of the absolute value of normalised helicity, demensionless j colburn factor k turbulence pulsation kinetic-energy, m2 s2 JF JF factor L length of heat exchanger, mm m mass flow rate, kg s1 Nu Nusselt number Pr Prandtl number P pressure, Pa Dp differential pressure in whole fin channel, Pa Q total rate of heat transfer, W Re Reynolds number s fin space, mm t fin thickness, mm Dtm logarithmic mean temperature difference, K T temperature, K Tout outlet temperature, K Tin inlet temperature, K Tw wall temperature, K Dtm logarithmic mean temperature difference, K U heat transmittance coefficient, W m2 K1 u velocity, m/s

also conducted the computational studies in the Re range from 100 to 1000, where the j and f predictions were in a good agreement with the experimental measurements. Kays and London [8] provided j and f factor versus Re curves for three wavy-fin structures with condensing steam heating the normal temperature air in wind tunnel experiments, but it didn’t give the concrete shape of the wavy curve. Dong et al. [9,10] conducted a set of experimental tests for 16 sets of wavy fin geometry parameters for wavy fin-andflat tube aluminum heat exchanger. Furthermore, the authors put forward experimental correlations for the Nu number and friction factor. In fact, the shape variables of the wavy fin affected heat transfer performance and pressure drop directly so several studies associated with shape variables have been conducted. Dong et al. [11] found that the waviness profile (triangular, sinusoidal and triangular round corner) had little effect on the heat transfer and pressure drop of wavy fin by numerically simulation, and he conducted his own experimental facility to validate the simulation results of triangular wavy fin. Asako et al. [12] numerically investigated the heat transfer performance of round corner shape wavy channel with Re = 100–1000. Manglik [13,14] et al. investigated the steady forced convection in Periodically developed Reynolds number (10 < Re < 1000) air

um uin v V w

maximum velocity in fin channel, m/s inlet velocity, m/s velocity vector, m/s volume of wavy channel unit, m3 wavelength, mm

Greek symbols j turbulent kinetic energy per unit mass, m2 s2 g0 surface efficiency of fin channel gf;id ideal one-dimensional fin efficiency f ratio of wavy fin passage length to wavy fin length in x-direction e turbulent energy dissipation, m2 s3 k thermal conductivity, W m1 K1 q density, kg m3 r stress, MPa s shear force, MPa l dynamic viscosity of fluid, Pa s m kinematic viscosity, m2 s1 x vorticity vector, s1 Subscripts c channel e extension f fluid in inlet out outlet s solid w wall Abbreviation CCD central composite design CFD computational fluid dynamic DOE design of experiment HTC heat transfer enhancement MOGA multi-objective genetic algorithm PFHE plate-fin heat exchanger RS response surface RSM response surface methodology RMS root mean square SST standard j  e model

(Pr = 0.7) flows in three-dimensional wavy-plate-fin cores and reported the effects of corrugation angle, pitch, and fin density on thermal performance. Some researchers put emphasize on how to enhance heat transfer. Kim et al. [4] numerically studied the heat transfer enhancement by cross-cut induced flow control in a wavy fin heat exchanger. Tian at al. [15] proposed a new fin pattern by punching delta winglets on the wavy fin surface. The heat transfer and fluid flow characteristicas of the wavy fin-andtube heat exchanger with delta winglets are numerically studied and the comparisons between staggered and in-line arrangements are performed. Khoshvaght-Aliabadi [16] et al. numerically studied laminar convection of water and 1% vol. Al2O3-water nanofluid through the straight mini-channel (SMC) and wavy mini-channel (WMC) with various cross-section geometries. Futhermore, Khoshvaght-Aliabadi et al. [17,18] investigated the influence of using three passive heat transfer enhancement (HTE) techniques, namely perforations, winglets, and nanofluids on heat transfer and flow specifications of WPFs. The waviness aspect ratios are 0.33, 0.42 and 0.51, the diameter of the perforation is 5 mm, the width/height of the winglets is 5 mm, and Reynolds number changes from 3900 to 11400. Sheik et al. [19] numerically analyzed three offset strip fin and 16 wavy fin geometries used in the

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compact PEHEs in turbulent flow region. They provided suitable baffle plates for improvement in flow distribution to enhance heat transfer. Tao et al. [20] considered extending the Field synergy principle for enhancing convective heat transfer to a twodimensional wavy channel by reasonable design which reducing the intersection angle between velocity and temperature. When outdoor heat exchangers operate with low air temperature and high humidity, frost will appears on the fin surfaces, which increases the thermal resistance, therefore Ma et al. [21] experimentally and numerically investigated the frost formation on wavy plates. The simulated frost profile, frost weight and frost thickness agree well with the experimental results, which means the frosting model can be used to predict frost formation on wavy plates. Above researchers focused on the passive HTC technique, but sometimes the so-called HTC techniques will obviously increase pump power consumption. In factual operating condition, the HTC techniques should take the flow resistance into account. In relevant investigation, Zhang et al. [14], Vasil’ev et al. [22], Kim et al. [23], Bhowmik et al. [24], Li et al. [25] and Wang et al. [26] implemented the assessment crierion the area goodness factor j/f or some other factors such as Nu/f, j/f1/3 to evaluate the the enhancement of heat transfer performance for all kinds of effective channel such as wavy plate fin cores, rectangular interrupted ducts, louvered fin, offset strip fin, multi-region louvered fin. The assesssment criterion offers the theoretical basement for further optimization. Wen et al. [27] adopted genetic algorithm to optimize the serrated fin structure in specific range of geometric parameters, while the j factor, f factor and JF factor are considered as three single objective functions for a specified Reynolds. In another reference Wen et al. [28] set the total heat flow rate, total annual cost and number of entropy production units of serrated fin structure as optimization objectives. As for the heat transfer and flow resistance of PFHE, the relative investigations have been developed for a lot, but there is few researches referring to loading capacity of PFHE. However, with the development of large-scale air separation system, there is higher demand for the loading capacity of PFHE, so the solution to optimize geometric structure of PFHE for a higher loading capacity is urgently needed. Jiang et al. [29] analyzed the brazed residual stress and its high-temperature redistribution for stainless steel plain plate-fin structure applied to recuperative heat exchanger

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in gas-turbine power generation, based on the finite element ABAQUS code. Zhang et al. [30] established an open loop tunnel to research full-size serrated PFHE in oil-side, and carried out the CFD simulation, fluid-structure-interaction (FSI) and finite element method (FEM) to investigate the stress distribution in PFHE structure. As a matter of fact, the past literatures referring to RS and MOGA were only based on the objectives of maximizing heat transfer and minimizing pressure drop. The past literatures involved in stress distribution calculation were few, and the relevant Ref. [29] were not from the perspective of FSI, or not combined with heat transfer and flow resistance. Ref. [30] investigated the heat transfer, pressure drop and stress distribution, but it didn’t provide the optimal structure based on the three objectives and MOGA. In a word, the relative references referring to wavy fin structure is less, and there are almost no references discussing the loading capacity of wavy fin structure. Considering that the wavy structure has a more enhanced heat transfer than plain fin structure and a lower pressure drop than serrated fin structure, we deem that it is necessary to make a comprehensive investigation for wavy fin structure. In this paper, the fin height, fin space, fin thickness, fin wavelength and double amplitude are selected as five input variable parameters to drive the construction of wavy fin structure. (Central Composite Design) CCD is implemented to allocate experimental design points in the variable parameters space containing six input parameters, where the inlet velocity is operating condition while the other five inputs are configuration parameters. Based on the numerical simulation results of CCD experiment points and other refinement points, the RS is constructed, to investigate the effects and interaction effects of six input parameters. RS offers the date for the following optimization, with increasing JF factor and decreasing maximum stress to the best set as objectives. 2. Geometric structure and numerical model 2.1. Geometric model and boundary condition Wavy fin structure is constructed in the form of sine curve in UG. The schematic diagram of a typical sine wavy-fin PFHE is shown in Fig. 1. Dong et al. [11] and Tian et al. [15] proposed the inlet extension to make the velocity profile uniform at the entrance

Fig. 1. Computational domain.

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Table 1 Configuration parameters and inlet velocity. Input parameters

Variation range

Fin height h (mm) Fin space s (mm) Fin thickness t (mm) Wavelength w (mm) Double amplitude A (mm) Inlet velocity uin (m/s)

4.7–9.5 1.4–3.0 0.1–0.4 6–12 1–3 1–3

and the outlet extension to avoid the recirculation. Therefore, computation domain is expanded upstream 20 mm and it is also extended downstream 40 mm. In this paper, the structure of different sine wavy fin is obtained by the variation of configuration input parameters (fin height h, fin space s, fin thickness t, wavelength w and double amplitude A) through central composite design (CCD). The variation ranges of configuration parameters and inlet velocity are listed in Table 1. In the thermal analysis, the velocity inlet and pressure outlet condition (absolute pressure 0.1 MPa) are set respectively. The fluid inlet temperature is 80 K. On the surface of upper and lower clipboard constant temperature (81 K) is set. The working fluid is cryogenic gas nitrogen and its properties are considered to be constant because of its small change of temperature. Its density, specific heat at constant pressure, thermal conductivity, viscosity and Prandtl number are 4.3489 kg m3, 1.1102 kJ kg1K, 0.0074918 W m1 K1, 5.6567e6 Pa s, and 0.83825 respectively. The symmetry boundary condition is applied to both sides of the computational domain. No slip wall and coupled thermal boundary condition are adopted at the interface between fluid and solid. The material of solid domain is set as aluminum6061, and thermal radiation and nature convection are neglected. (Fig. 1). The computational domain is meshed with hexahedral grids using ANSYS mesh as is shown in Fig. 2(a). In order to obtain faster computation speed, only the locations with significant flow changes such as velocity boundaries are meshed by concentrated grid density. After the numerical simulation of flow field, temperature distribution in solid domain and pressure distribution on inner channel surface can be obtained, which is applied to stress distribution calculation in wavy-fin structure. A new set of hexahedral mesh is generated in solid by use of mesh module in ANSY mechanical, and it is uniformly distributed in solid domain

(Fig. 2(b)). The pressure load is the main source to generate stress in solid domain, for the reason that the differential temperature between inlet and wall temperature is 1 K. Furthermore, considering the pressure variation in fin channel, we find that its order of magnitude is only several hundreds of Pascal, which is very low, comparing with absolute pressure in fin channel. Therefore, the constant pressure load (0.1 MPa) is directly loaded onto the inner channel surface. On right and left side of the model the symmetry boundary condition are set, and on upper surface and lower surface the pressure load 0.6 MPa and displacement condition are set. The displacement condition, which can specify whether a portion of the model displaces relative to its original location or not, is applied to constrain the movement normal to bottom surface, while the movement in other two directions is allowable. 2.2. Mathematical model Dong et al. [11] compared the results of standard j  e turbulence model with that of RNG j  e model, finding that the former more agreed well with experimental data. For the improved predictions of flow in wavy-fin channel, the standard j  e model is adopted in this paper. Governing equations mainly include fluid dynamic equation, solid mechanics equations. Fluid dynamic equations include continuity equation, momentum equation and energy equation [31]. Continuity equation:

@ ðqui Þ ¼ 0 @xi

ð1Þ

Momentum equation:

 @
@p @ qui uj ¼  þ @xj @xj @xi

 

l

@ui @uj 2 @uk þ  dij @xj @xi 3 @xk

 þ

@  qu0i u0j @xi ð2Þ

where dij is the Kronecker delta, and:

qu0i u0j ¼ lt

    @ui @uj 2 @u  þ qk þ lt k dij 3 @xj @xi @xk

Energy equation:

 @
@ quj T ¼ @xi @xj

Fig. 2. Mesh grid.







lt @T 1 þ ðU þ qeÞ Cp Pr rT @xj l

þ

ð3Þ

ð4Þ

J. Wen et al. / International Journal of Heat and Mass Transfer 131 (2019) 385–402

Turbulent kinetic energy k equation:

@ @ ðqkui Þ ¼ @xi @xj



lþ



Turbulent enrgy dissipation

@ @ ðqeui Þ ¼ @xi @xj



lþ

Turbulent viscosity

lt ¼ q C l

k

The convective heat transfer coefficient in fin channel is calculated by:



lt @k þ Gk  qe rk @xj

ð5Þ

U¼

lt can be expressed as:

2

e

ð7Þ

@ syx @y

ð8Þ

The j factor, f factor and maximum stress are quantitative indexes of heat transfer, flow resistance and loading capacity respectively. The three output parameters are regarded as objective functions. The j factor is defined as:

RePr1=3

ð9Þ

Reynolds number, Prandtl number and Nusselt number are respectively defined as:

um Dh

Re ¼ Pr ¼

m

lc p

Nu ¼

kf hc D h kf

ð10Þ ð11Þ ð12Þ

The hydraulic diameters of wavy fin are given as follows [34]:

Dh ¼

2ðs  tÞðh  tÞ ðs  t Þ þ fðh  t Þ

ð16Þ

The logarithmic mean differential temperature Dt m is calculated by:

Dt m ¼

T out  T in 

ð17Þ

T wall T in T wall T out

ln

g0 ¼ 1 

ð13Þ

A2  1  gf;id Aw

ð18Þ

The ideal fin efficiency (gf;id ) in fin channel is calculated as follows:

gf;id ¼ tanh

    1 1 me h = me h 2 2

sffiffiffiffiffiffiffiffi 2hc me ¼ ks t

ð19Þ

ð20Þ

The j factor is iteratively solved by Eqs. (9)–(20). Eqs. (21) and (22) are applied to calculate the f factor and JF factor:

f ¼

DpDh 2qu2m L

JF ¼ j=f

2.3. Date reduction

Nu

ð15Þ

The surface efficiency of fin channel (g0 ) is calculated by:

Standard j  e turbulence model is high Reynolds number model, which is not appropriate for viscous sublayer near wall, so near-wall region should be managed by wall function. Ismail et al. [32] adopted enhanced wall function. However, when verifying the reliability of CFD software, it adopted a rectangular channel. In this paper, the standard wall function is adopted. The governing equations are iteratively solved by the finite-volumemethod with SIMPLE pressure-velocity coupling algorithm and discretized by the second-order upwind scheme. The convergence criterion is that the normalized residuals are less than 106 for the flow equation and 1  108 for the energy equation. Under equilibrium state, the stress of fin balances to external force located on fin surface, and the equilibrium equations are as follows [33]:

j¼

Q A w Dt w

Q ¼ mcp ðT out  T in Þ

re ¼ 1:3; rT ¼ 0:85

9 þ @@zzx þ F x ¼ 0 > > = @ sxy @ ry @ zy þ þ þ F ¼ 0 y @x @y @z > > ; @ syz @ xz @ rz þ þ þ F ¼ 0 z @x @y @z

ð14Þ

Aw 2Aw;cl

The heat transfer amount is calculated by:

C 1e ¼ 1:44; C 2e ¼ 1:92; C l ¼ 0:09; g0 ¼ 4:38; rk ¼ 1:0;

þ

1

g0 U1  kbs

The heat transmittance coefficient U is calculated by:

ð6Þ

Gk reflects the generation of turbulent fluctuation kinetic energy due to gradient of time averaged velocity, and its experssion is similar to that of dissipation functionU. The empirical constants for the standard j  e are recommended as following values:

@ rx @x

1

hc ¼

e equation:

ly  @ e  e e2 þ C 1e G k  C 2e q re @xj k k

389

ð21Þ

1=3

ð22Þ

The maximum stress is assumed to be an objective function. The stress here referring to equivalent (von-mises) stress is calculated according to fourth strength theory as is shown in Eq. (23), where r1 ; r2 ; r3 refers to principal stress in three directions.

rr4

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1h ¼ ðr1  r2 Þ2 þ ðr2  r2 Þ2 þ ðr3  r1 Þ2 2

ð23Þ

Levy et al. [35] define a ‘‘normalised helicity” to detect and visulise vrotex cores as:

Hn ¼

vx j v j j xj

ð24Þ

This quantity has limiting values of ±1 when the angle between the velocity and vorticity vectors is zero, its sign depending on the direction of rotation about the flow direction. Rosaguti et al. [36] adopted this definition to study fully developed laminar flow and heat transfer behaviour in periodic supentine channels with a semi-circular cross-section. It adopted the volume-average of the absolute value of helicity, given by:

HV ¼

1 V

Z jHn jdV ¼ V

1 V

Z V

jv  xj dV jv jjxj

ð25Þ

The absolute value of helicity is used to exclude rotation direction information, and the increase of volume-averaged helicity crresponds to the continual increase in heat transfer enhancement.

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J. Wen et al. / International Journal of Heat and Mass Transfer 131 (2019) 385–402

Fig. 3. Grid independence.

2.4. Model validation Grid independence is implemented before the numerical simulation to improve calculation accuracy. The geometric parameters and inlet velocity of the verified wavy-fin structure are as follows: h = 7.1 mm, s = 2.2 mm, t = 0.3 mm, b = 8 mm, A = 2 mm, uin = 2 m/ s. In Fig. 3(a) it can be find that when the grid number reaches to 2,387,700, the j,f factor almost no longer changes (changing rate is smaller than 1%). Section 2.1 mentions that a new set of hexahedral mesh is generated in solid by use of mesh module in ANSY mechanical, so it is necessary to verify grid independence in solid domain. Fig. 3(b) reveals that, when the grid number reaches to 2,590,100, the maximum stress almost no longer changes (changing rate is smaller than 1%). Therefore, the subsequent calculation uses the corresponding grid size. Also, the numerical results are compared with the experimental date of wavy fin structure (11.5–3/8W) in Ref. [8], to demonstrate accuracy of the present simulation model. Before experimental validation, a question about how to define laminar region and turbulent region should be noted. Dong et al. [11] and Zhang et al. [14] adopted Re smaller than 1000 as laminar state, which is identical to that of serrated fin structure. Yet it is not a fixed solution. Ismail et al.[32] and Khoshvaght-Aliabadi et al. [37] adopted 800 and 1900 as a boundary respectively. To verify the transition range of Re and appropriate regime, both the laminar and the turbulent model are computed for Re range of 200–2500. The results in Fig. 4 show that the numerical results of the laminar regime have a good coincidence with the experimental data when Re is smaller

Fig. 4. Experimental verification.

than 600. When Re is larger than 600, the standard j  e turbulent model with standard wall function is more appropriate. The RMS values of the percent difference are 10.39% for j factor and 8.03% for f factor. The simulation results based on RNG turbulent model with standard wall function is attached here. Although the RMS value of percent difference for j factor is only 3.59%, for f factor, it is high to 18.48%.Trade off made, the mathematical model based on standard j  e model is more appropriate to predict the j and f factor of wavy fin structure.

2.5. Optimization theory and methods Parametric modeling, mesh grid, flow field calculation, stress analysis are conducted by means of ANSYS DesignModeler, ANSYS mesh, ANSYS fluent and ANSYS mechanical respectively. When sufficient data points are obtained, the optimization process is run with the help of ANSYS DesignXplorer. A flowchart of optimization process is explained as follows: (1) The fin height, fin space, fin thickness, wavelength and double amplitude are set as input parameters to drive generation of the parameterized model of wavy fin structure, and the j factor, f factor, JF factor and maximum stress are obtained by numerical simulation. (2) Set the range of input parameters and employ CFD (computational dynamic fluid) and CCD (central composite design) to generate many input/output combinations, which is a set of initial population that is used for response surface construction. (3) Based on step (2), the Full 2nd-Order Polynomial response surface is created, and can be improved during simulation with the addition of new design points. (4) The prediction error of response surface is assessed by the Goodness of fit information of verification points. If the error is unacceptable, the verification points of large prediction error are inserted to refine the response surface. If the error is acceptable, the process is going to continue. (5) MOGA is run and a set of pareto-optimal points are generated via selection, crossover and mutation. The CFD method is implemented to guarantee the correctness of optimization results. If the error is acceptable, the optimized values are output. If not, the points are promoted as new design points to improve response surface. Again, based on new response surface, MOGA is run. CCD is used to generate the experimental design for it is characterized by fewer tests, a high precision and good predictability [27,38]. Central composite designs are five-level factorial designs that are suitable for calibrating the quadratic response model. A CCD consists of 1 center point, 2 N(N represents the number of input parameters) axis points located at the –1 and +1 positions on each axis of the selected input parameter and 2 N-f factorial

J. Wen et al. / International Journal of Heat and Mass Transfer 131 (2019) 385–402

Fig. 5. A CCD of two input system.

points located at the 1 and +1 positions along the diagonals of the input parameter space. Fig. 5 shows the full factorial CCD of a twoinput system. Factorial (f) is used to get rid of some diagonal points, to restrict the number of design points to a reasonable number. What should be pointed out is that the experimental points generated by full factorial CCD can’t sufficiently express the performance of wavy fin structure in the whole input parameter space, so additional experiment points should be added, not only in preliminary experimental design, but also in the subsequent construction of RS. Response surfaces are functions of different nature where the output parameters are described in terms of the input parameters [39]. It provides the approximated values of the output parameters, every one of which is in the analyzed design space, without need of performing a complete solution. RS is constructed based on the simulation results of above experimental design points. Full 2ndOrder polynomial method is applied to predict unkown parameter points based on known parameter points, which expresses space change by way of variance change and guarantees the smaller error of predicted value from spatial distribution [40]. It is the default RS type based on a modified quadratic formulation like ‘‘Output = f (inputs)”, where f is a second order polynomial. In general, this regression model is an approximation of the true input-to-output relationship and only in special cases does it yield a true and exact relationship. ANSYS DesignXplorer provides coefficient of determination (R) for mean heat transfer coefficient (hc), maximum

391

velocity in fin channel (um), differential pressure (Dp), ideal onedimensional fin efficiency (gf;id ) which are significant transition parameters to calculate j factor and f factor. They are 0.9031, 0.9989, 0.9985, 0.9692. R for maximum stress is 0.9012. Based on RS, MOGA is run to optimize the wavy fin structure. It is a hybrid variant of the popular NSGA-Ⅱ (Non-dominated sorted Genetic Algorithm-Ⅱ) based on controlled elitism concepts, which is characterized by fast non-dominated sorting approach, fast crowded distance estimation precedure and simple crowded comparison operator [41,42]. MOGA goes through several iterations retaining the ‘‘elite” percentage of the samples and through each iteration it makes the samples ‘‘genetically” evolve the best pareto front. It is ideally suited for calculating global maxima/minima (designed to avoid local optima traps). The basic genetic operators are selection, crossover and mutation. The selection process copies parent chromosomes into new population. The number of copies reproduced for the next generation by an individual is expected to be directly proportional to its fitness value. The crossover combines two chromosomes (parents) to produce a new chromosome (offspring), in which parent solutions are recombined to generate the offspring solutions. For continuous parameters the crossover operator linearly combines two parent chromosome vectors to produce new offspring according to the following equations:offspring = a  parent1 + (1-a)  parent2. A smaller value indicates a less change with more stable population and a faster (but less accurate) solution. The mutation operator alters one or more gene values in a chromosome from its initial state which results in entirely new gene values being added to the gene pool. Mutation reintroduces genetic diversity back into the population and assists the search to escape from local optima [43]. 3. Reuslts and discussion 3.1. Stress distribution The geometric structure parameters of wavy fin structure used for illustrating the stress distribution are as follows: fin height h is 7.1 mm, fin space s is 2.2 mm, fin thickness t is 0.3 mm, wavelength w is 8 mm, and double amplitude A is 2 mm. In Fig. 6, the upper

Fig. 6. Path A, B, C, D in wavy-fin structure.

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Fig. 7. Stress distribution in four paths.

clipboard is hided to more clearly see the stress distribution in the region of right angle of fin structure, where paths A, B, C and D are highlighted, and the stress distribution in these four paths are investigated as is shown in Fig. 7. Fig. 7 indicates that the largest stress is located at point MAX1, where the largest stress is 14.87 MPa. Due to antisymmetry, another point of the largest stress can be observed in point MAX2, as is revealed in Fig. 6. Fig. 7 reveals that the stress increases in wave crest and decreases in wave trough, and the fluctuant stress reaches to the largest in the wave crest peak (C1) and reaches to the lowest in the wave trough nadir (C2). The peak stress in wave crest of path A and path B are identical, but the valley stress in path B is much smaller than that in path A. Additionally, there is no obviously abrupt stress change in inlet region and outlet region of path B, while in inlet region of path A, the stress sharply increases. The phenomenon that the stress distribution in path A and path B are different is due to the fact that path A is in the joint region of fin structure and clipboard, where the clipboard influences the stress distribution in path A. Fig. 7 reveals that the stress distribution in path C is chaotic, which is thoroughly different from that in path A and B. The chaotic in path C is for the reason that path C is in the middle of joint region of clipboard and fin, and the stress distribution on

this region is easily influenced by the corrugated surface. In path D, the stress changes like that in path A and path B, but the variation rate is far less. To conclude, the concentrated stress in nearinlet region, near-outlet region and wave crest of wavy-fin structure ought to attract sufficient attention. 3.2. The effects of configuration parameters and inlet velocity RS expresses the variation of one output parameter with the change of two input parameters. If we fix one input in RS, the effects of single one input parameter can be investigated. However, the variation trend of one output with the change of one input sometimes can’t accurately express the effect degree, due to the two-order interaction effects, or in another way of saying, the changing of one input will change the capacity of the other input affecting the output. The two-order interaction effects of six input parameters can be observed by 3D response surface or 2D slice as what follows in the subsequent illustration depict. 3.2.1. The effects of configuration structure parameters on j factor Common characteristics in Figs. 8 and 9 can be observed that the j factor is not sensitive to inlet velocity when the inlet velocity

Fig. 8. The effects of fin thickness (a), double amplitude (b) and wavelength (c) on j factor.

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Fig. 9. The effects of fin space (a) and fin height (b) on j factor.

is in the medium range of 1.5–2.5 m/s, and when the inlet velocity is smaller than 1.5 m/s or larger than 2.5 m/s, the j factor decreases rapidly with the increase of inlet velocity. For a plain fin structure, its j factor decreases with the increase of Re (or fluid velocity) [8]. However, the corrugated effect will enhance the heat transfer, especially in the range of 1.5–2.5 m/s, which offsets the decreasing trend of j factor. Also, it can be concluded that, the j factor is the most sensitive to wavelength. In Fig. 8(a), when inlet velocity is 1 m/s, the j factor is the most reduced, by about 16.0%. Larger fin thickness means that the fin

channel is narrower, and the fluid can be heated by wall adequately. However, the differential temperature between inlet and wall is only 1 K, and the fluid far from wall may be adequately heated already, so the so-called adequate heating can’t promote heat transfer performance. This phenomenon can be verified by Fig. 10 which graphs the temperature distribution for flows in wavy fin channel with t = 0.1, 0.2, 0.3, 0.4 mm in wavy-fin structure with h = 7.1 mm, s = 2.2 mm, A = 2 mm, w = 9 mm. It shows that, with the increase of fin thickness, more and more fluid in middle region is heated. Besides that, the narrow fin channel will suppress

Fig. 10. The temperature distribution under condition of different fin thickness (uin = 2 m/s).

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Fig. 11. The pathline under condition of different fin thickness (uin = 2 m/s).

the further generation of helical vortex. The development and growth of steady lateral vortices in trough regions of wavy-fin channels with changing fin thickness are depicted in Fig. 11, where streamlines for flows with t = 0.1, 0.2, 0.3, 0.4 mm in wavy-fin structure with h = 7.1 mm, s = 2.2 mm, A = 2 mm, w = 9 mm are graphed. With the increase of fin thickness, the vortex in second trough region almost fades away, and the vortex in first trough region is no longer obvious. Therefore, the j factor decreases. Although visual contour offers the different vortex in different channels, it reflect the vortex locally, from the qualitative point of view. Therefore, volume-average of the absolute value of helicity in the whole fin channel is used to quantitatively detect the vortex, as is shown in Fig. 11. It reveals that, the influence of fin thickness on vortex in the whole fin channel is not very obvious, so the effect of the other factors except the helical vortex on heat transfer performance should be taken into consideration. In Fig. 8(b), the j factor firstly increases by 11.8% and then decreases by 5.7%, when double amplitude increases from 1 mm to 3 mm under condition of inlet velocity of 1 m/s. It reveals that the influence of the increase of double amplitude on heat transfer enhancement is becoming weaker, with the increase of inlet velocity. Although the intuitive thought tell us that the increase of double amplitude will much more obviously leads to the growth of vortices in the trough region, as a result leading to greater mixing within the fluid layer near the hot wall and also fluid layers away from wall, the enhancing effect only can be observed under condition of low velocity and small double amplitude (smaller than 2 mm). It seems that, under condition of high velocity, the increase of double amplitude can’t provide satisfied heat transfer enhancement. In Fig. 8(c), the j factor decreases by 18.7%, 17.9%, 17.1%, 16.1% and 16.5% when wavelength increases from 6 mm to 12 mm under condition of inlet velocity of 1 m/s, 1.5 m/s, 2m/s, 2.5 m/s and 3.0 m/s respectively. It is deemed that the interaction effects of inlet velocity and wavelength are weak, for the five lines representing five different inlet velocities are almost paralleled. The increase of wavelength make the transition between wave crest and wave trough in wavy fin structure

more smoothly, which weakens the flow separation effect and finally impairs the turbulent intensity resulting from helical vortex. Therefore, the heat transfer is weakened, and the j factor decreases. To intuitively depict the flow field variation, the streamlines for flows with w = 6, 8, 10, 12 mm in wavy-fin structure with h = 7.1 mm, s = 2.2 mm, A = 2 mm, t = 0.3 mm are graphed in Fig. 12. It can be observed that, when wavelength increases from 8 mm to 10 mm, the helical vortex fades away, leading to the sharper decrease of heat transfer in the range of 8–10 mm compared with that in the range of 6–8 mm and 10–12 mm, which is reflected in Fig. 8(c). Volume-averaged helicity as a quantitative index to evaluate vertical structures under condition of different wavelength is calculated under condition of different wavelength, and it can be observed that the volume-averaged helicity decreases monotonically across the range of wavelength studied, corresponding to the continual decrease in heat transfer enhancement as is shown in Fig. 8(c). Also, we list the temperature distribution under condition of different wavelength, as is shown in Fig. 13. It reveals that there is more and more blue1 region, representing the insufficient heating of fluid, obviously reflecting the decline of heat transfer ability. Fig. 9(a) indicates that there are almost no interaction effects of inlet velocity and fin space on j factor. Under condition of different inlet velocity, the increment of j factor is almost identical which is for about 11.6% when the fin space increases from 1.4 mm to 3.0 mm. As a matter of fact, the larger fin space means that more fluid is far from wall, and this part of fluid can take away more heat from wall, which further enhanced heat transfer, so the j factor increases with fin space. Fig. 9(b) shows that the j factor changes sharply under condition of inlet velocity of 3 m/s, and it increases by 48.5% when fin height increases from 4.7 mm to 9.5 mm. This phenomenon can also be ascribed to the further heating of fluid in intermediate region in fin structure, when the fin height increases. Especially under con1 For interpretation of color in Fig. 13, the reader is referred to the web version of this article.

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Fig. 12. The pathline under condition of different wavelength (uin = 2 m/s).

Fig. 13. The temperature distribution under condition of different wavelength (uin = 2 m/s).

dition of high inlet velocity, the fluid in far-wall region can take away more heat from wall surface and fin surface. In fact, the interaction effects of two input parameters on j factor are observed to be very weak, by observing 3D response surface results. Fig. 14(b) exhibits the relatively obvious interaction effects. Under circumstance of different wavelength, the variation trend of the j factor with double amplitude changes for a little. However, in general, the interaction effects on j factor are like that in Fig. 14(a).

According to the weak interaction effects of inputs on j factor, the conclusion that the performance of wavy fin structure can be approximately expressed by superimposed effects of plain fin structure performance and corrugated impact can be drawn. 3.2.2. The effects of configuration parameters on f factor A common phenomenon can be observed in Figs. 15 and 16 that the f factor is negatively correlated with inlet velocity, and that the

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Fig. 14. The relatively obvious interaction effects of geometric parameter (uin = 2 m/s) (a) double amplitude and fin space and (b) double amplitude and wavelength.

Fig. 15. The effects of fin thickness (a), double amplitude (b) and wavelength (c) on f factor.

five curves representing five different inlet velocity conditions are almost paralleled which indicates that the interaction effects of inlet velocity and configuration parameters on f factor are very weak. Also, it can be concluded that the f factor is the most sensitive to wavelength, only the geometric parameters taken into consideration. In Fig. 15(a), the decrement of f factor is 11.9%, 14.8%, 15.9%,16.3%,16.3% respectively when fin thickness increases from 0.1 mm to 0.4 mm under circumstance of inlet velocity of 1 m/s, 1.5 m/s, 2.0 m/s, 2.5 m/s and 3.0 m/s. For wavy fin structure, the

increase of fin thickness promotes the flow stability, for it will suppress the extent of lateral swirl in some local region (as is shown in Fig. 11), leading to the decrease of f factor consequently. It should be paid attention to that the so-called decrement is not obvious, which can be explained by the inconspicuous change of Hv discussed in above analyses. In Fig. 15(b), the f factor increases by 45.0%, when double amplitude increases from 1 mm to 3 mm, under condition of inlet velocity of 1 m/s. The variation trend of f factor is similar when on

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Fig. 16. The effects of fin space (a) and fin height (b) on f factor.

condition of other inlet velocity. This phenomenon can be attributed to the promotion of the augment of the double amplitude to the transverse vortices, which will obviously weak the flow stability and lead to the augment of flow resistance. Fig. 15(c) reveals that the f factor is negatively related to wavelength, the f factor decreases for 56.0% under condition of inlet velocity of 1 m/s when the wavelength changes from 6 mm to 12 mm, and there are almost no interaction effects of the wavelength and inlet velocity on f factor. The negative effect of wavelength on f factor is for the reason that the increase of wavelength makes the flow transition smoother, which impairs the flow resistance. As a result, the f factor declines. This variation trend can also be explained by Fig. 12. Fig. 16(a) reveals that the f factor increases almost linearly with the increase of fin space, for about 39.1%, when fin space increases from 1.4 mm to 3.0 mm under condition of inlet velocity of 1 m/s. The increase of fin space means more spacious channel, which will promote the generation of helical vortex. As a result, flow resistance increases. Fig. 16(b) shows that the effects of fin height and inlet velocity on f factor are very weak. The most evident increment, when the fin height changes from 4.7 mm to 9.5 mm, is 17.7% under condition of inlet velocity of 3 m/s. The rangeability of the f factor in the whole range of fin height is very narrow, compared with that, in the whole range of other geometric parameters. If the fin height increases, the fluid in intermediate region is under less influence of wall shear stress from up and down wall, and flow stability of this part of fluid descends. Therefore, the f factor increases. The response surface results of relatively obvious two-order interaction effects are listed in Fig. 17. Fig. 17(a) reveals that the f factor increases more rapidly with fin space when the wavelength is smaller. Under condition of wavelength of 6 mm, the f factor increases by 110.4% when the fin space increases from 1.4 mm to 3.0 mm. In comparison, when wavelength is 12 mm, the f factor increases by only 25.1% in the whole range of the fin space. In fact, the very large wavelength means that viscous force dominates and

fully developed duct flow type behavior prevails, when the fin channel is much more like a plain fin channel. Therefore the increment of the f factor is not obvious. When wavelength is sufficiently small, the boundary-layer separation downstream of corrugation peaks gives rise to a vortex flow structure in the valley region [14]. At this time, the increase of the fin space will drastically promote the generation of recirculation region, which obviously lead to the increase of the f factor. Fig. 17(b) shows that the f factor increases sharply by 82.9% when fin space increases from 1.4 mm to 3.0 mm and double amplitude is 3 mm. When double amplitude is 1 mm, the f factor almost do not change with fin space, because at this time, the wavy fin structure is much more similar to plain fin structure. The increase of fin space provides more space for the generation of recirculation of the region, which leads to the increase of the f factor. To sum up, the interaction effects of wavelength and fin space on f factor are much more distinct than that of double amplitude and fin space. 3.2.3. The effects of configuration parameters on JF factor Firstly, the intrinsic physical meaning of JF factor, which is regarded as an evaluation criterion, should be discussed. Eq. (22) is proposed for the reason that the objective of heat transfer enhancement simultaneously requires to decrease flow resistance. It is a comprehensive performance index, which evaluates the heat transfer capacity under the unit pump power. According to Figs. 18 and 19, the JF factor has a gentle change in inlet velocity range of 1.5–2.5 m/s, which is for the reason that the j factor is not sensitive to inlet velocity in this range as is revealed in above analysis. The JF factor increases with the decrease of inlet velocity, which indicates that PFHE has a better performance of heat transfer under the unit pump power when the inlet velocity decreases. Fig. 18(a) reveals different variation trend of JF factor under condition of different inlet velocity. Even in inlet velocity of 1 m/ s, the peak point for JF factor can be obtained. This phenomenon

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Fig. 17. The relatively obvious interaction effects of geometric parameters on f factor (uin = 2 m/s) (a) fin space and wavelength and (b) fin space and double amplitude.

Fig. 18. The effects of fin thickness (a), double amplitude (b), wavelength (c) on JF factor.

may be due to different intense degree of helical vortex in fin channel. In Fig. 18(b), the peak point can also be observed, but it presents a declining trend on the whole. With the increase of double amplitude, the pump power will obviously increases, and it seems that heat transfer capacity increases more slowly. Fig. 18(c) shows the effects of wavelength on JF factor. Similarly, the peak point for JF factor can be observed under condition of high inlet velocity, and

it can be concluded that, with the increase of wavelength, the declining speed of heat transfer capacity sometimes is slower than pump power consumption, and sometimes is more quickly than pump power. Fig. 19(a) reveals that the JF factor doesn’t change with fin space. Above analysis reveals that the j factor and f factor both increase with the increase of fin space, and the rate of descent

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Fig. 19. The effects of fin space (a), fin height (b) on JF factor.

of two factor satisfies the specific condition, make JF factor change a little. Fig. 19(b) manifests that the JF factor increases with the increase of fin height. The changing trend of the j factor and f factor with the change of fin height can explain this phenomenon. 3.2.4. The effects of configuration parameters on maximum stress The pressure distribution on fin channel surface is analyzed, to conclude that, the pressure distributions on fin channel are similar

under circumstance of different inlet velocity. This phenomenon is for the reason that the gas nitrogen of low density is adopted as working fluid, and that the inner fin channel is a smooth transition. Therefore, the maximum stress is not sensitive to inlet velocity, and in this section, only the effects of configuration parameters are taken into consideration. It can be concluded that the maximum stress is the most sensitive to wavelength. In Fig. 20(a), the decreasing trend of maximum stress with fin thickness is much smoother when wavelength is larger. It

Fig. 20. The effects of geometric parameters on maximum stress (a) wavelength and fin thickness, (b) double amplitude and wavelength, (c) fin height and fin space.

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making per fin undertake much more stress, leading to the increase of the maximum stress. 3.3. Optimization results As is analyzed above, to increase the j factor, the wavy-fin structure of larger fin space, larger fin height should be chosen. However, increasing fin space and fin height will augment the f factor and maximum stress. The three objective functions are mutually conflicting, and there is no single design parameter of wavy fin that yields the best value for the three objectives. Multi-objective optimization using genetic algorithm is conducted by taking increasing j factor, decreasing f factor and decreasing maximum stress to the best as the three objectives. Here, the j factor and f factor are combined into a JF factor 1=3

Fig. 21. The pareto-optimal results.

decreases by 72.6% in the whole range of fin thickness when the wavelength is 6 mm, and it almost doesn’t change when the wavelength is 12 mm. That the maximum stress is negatively related to fin thickness is on account of directly proportional relationship between the stress in fin region and fin section area. The large wavelength means small curvature change, leading to the phenomenon that the maximum stress is not sensitive to fin thickness. It can be concluded that to promote the loading capacity of wavyfin structure of sufficiently large wavelength cannot be realized by increasing its fin thickness. Fig. 20(b) reveals that the maximum stress augments more rapidly with wavelength when double amplitude is smaller. When double amplitude is 1 mm, the maximum stress increases by 219.1% in the whole range of the wavelength, and when double amplitude is 3 mm, the maximum stress increases by 31.5%. It is an interesting phenomenon that the maximum stress in wavy-fin structure of larger wavelength and smaller double amplitude, which is more similar to plain fin structure, is higher. In fact, the wavy fin structure is corrugated by plain fin structure, and it is similar to the fold sandwich structure, or the corrugated paper, which has excellent mechanical properties. Refs. [44,45] explained the corrugated effect in detail. Therefore, the wavy fin structure which is more similar to plain fin structure, obtains the worse loading capacity. Fig. 20(c) shows that the maximum stress almost doesn’t change with fin height, and it linearly correlates with fin space. On both sides of fin the pressure load is almost equivalent besides the pressure on root segment of fin, which generates very small moment of force increasing with the increase of the fin height. As a result, the maximum stress almost doesn’t change. The lager fin space means more fin number along the direction of width,

(JF ¼ j=f ), so the three objectives are transformed into two objectives of increasing JF factor and decreasing maximum stress. In process of optimization, we fix the inlet velocity to 1 m/s, only considering the effects of geometric parameters. Fig. 21 shows the optimization results, which are a series of pareto-optimal results, and every point is a trade-off between two objectives, and these points can be selected by the designer regarding the project’s limits and the available investment. Table 2 lists three optimal designs based on two objective functions, and these output values are obtained from RS, so the results are validated by CFD simulation. To demonstrate the effectiveness of optimal results, three pareto-optimal points are extracted to be compared with a original design (as is revealed in Fig. 21 and Table 2). Compared with original design, the JF factor of optimal design 1, 2, and 3 increases by 11.0%, 8.4% and 15.9% respectively, and the maximum stress decreases by 32.3%, 42.4% and 20.7% respectively. To intuitively reflect the geometric parameters of optimal structure, we plot the distribution of geometric parameters in feasible domain, as is shown in Fig. 22, where two sets of input parameters are selected as a group and the horizontal and vertical axis ranges are respectively optimization intervals of these two input parameters. Fig. 22 reveals that the parameters of optimal structures locate in a relative wide range, especially for fin thickness (from 0.1 mm to 0.4 mm). When inlet velocity is 1 m/s, JF factor increases with the increase of fin thickness, and maximum stress has a reversible trend. Therefore, it can be concluded that the wide range of fin thickness of optimal structure results from the trade off of JF factor and maximum stress. In comparison, the distribution interval of fin height, fin space, wavelength and double amplitude has a smaller range. The results encourage designers to choose wavy fin structure in fin height range of 8–9.5 mm, fin space range of 1.4–1.6 mm, wavelength range of 6–8.2 mm, and double amplitude of 1.6–2.1 mm.

Table 2 The original design and three optimization results based on objective functions. Items

Height/mm

Space/mm

Wavelength/mm

Thickness/mm

Double amplitude/mm

Inlet velocity/m/s

JF

Maximum stress/MPa

Original design Optimal design 1 CFD Validation error Optimal design 2 CFD Validation error Optimal design 3 CFD Validation error

7.1 8.89

3 1.45

9 6.78

0.25 0.15

2

1

2.03

1

1.44

1

1.80

1

0.0347 0.0392 0.0385 1.8% 0.0384 0.0395 2.8% 0.0397 0.0409 2.9%

21.7 15.3 14.7 4.1% 13.3 12.5 6.4% 16.5 17.2 4.1%

8.86

8.86

1.48

1.48

6.78

7.28

0.14

0.19

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Fig. 22. Parameters distribution of optimized structures.

4. Conclusion In this paper, based on FSI the comprehensive performance of wavy-fin structure is investigated from perspective of heat transfer, flow resistance and stress. The analysis results of stress distribution reveal that the highest stress is located in the inlet and outlet of fin structure, and that the fluctuant stress reaches to the peak in the wave crest peak. Subsequently, based on RSM, the effects of the configuration parameters (fin space, fin thickness, fin height, wavelength, and double amplitude) and inlet velocity heat transfer, flow resistance and stress of sine wavy fin structure are quantitatively analyzed. The results reveal that the j factor is positively correlated with fin space and fin height, and the relationship between the j factor and fin thickness, wavelength and inlet velocity is negative. The variation trend of the j factor for wavy fin structure is different from that for serrated fin structure, in which the j factor is positively correlated with the j factor. It is for the reason that the helix vortex in trough region is less obvious with the increase of fin thickness. The j factor firstly increases with the increase of double amplitude and then decreases. For the f factor, the increase of double amplitude, fin space and fin height are positive to it while the increases of fin thickness, wavelength and inlet velocity are negative to it. The increasing trend of the f factor also can be ascribed to the fading helix vortex with the increase of fin thickness. The maximum stress increases with the increase of wavelength and fin space, and decreases with the increase of fin thickness and double amplitude. The interaction effects of input parameters on the j factor and f factor are not obvious. The maximum stress considered as objective function, the interaction effects of fin thickness and wavelength, double amplitude and

wavelength are the most obvious. When considering the effect degree of single geometric parameter, it can be concluded that the j, f factor and the maximum stress are the most sensitive to wavelength. By taking increasing the JF factor and decreasing the maximum stress to the best as two objectives, MOGA is performed to obtain optimization results, which are expressed by a set of pareto-optimal points. The results shows that, compared with the original design, the JF factor of optimal structure 1, 2, and 3 increases by 11.0%, 8.4% and 15.9% respectively, and the maximum stress decreases by 32.3%, 42.4% and 20.7% respectively. The distribution of geometric parameters of optimal structures are investigated, the results show that the parameters of optimal structures locate in a relative wide range, especially for fin thickness (from 0.1 mm to 0.4 mm). Fin height is in the range of 8–9.5 mm; fin space is in the range of 1.4–1.6 mm; wavelength is in the range of 6–8.2 mm; double amplitude is in the range of 1.6–2.1 mm. Conflict of interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 51676146), for which the authors are thankful.

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References [1] Z. Wang, Y. Li, Layer pattern thermal design and optimization for multistream plate-fin heat exchangers—a review, Renew. Sustain. Energy Rev. 53 (53C) (2016) 500–514. [2] S. Kakaç, A.E. Bergles, F. Mayinger, H. Yüncü, Heat transfer enhancement of heat exchangers, Nato Asi (1999). [3] H. Auracher, Principles of enhanced heat transfer, Int. J. Refrig. 18 (8) (1995) 565. [4] G.W. Kim, H.M. Lim, G.H. Rhee, Numerical studies of heat transfer enhancement by cross-cut flow control in wavy fin heat exchangers, Int. J. Heat Mass Transf. 96 (2016) 110–117. [5] S.W. Chang, T.M. Liou, K.-C. Yu, S.-S. Huang, Thermal-hydraulic performance of longitudinal wavy rib along wavy two-pass channel, Appl. Therm. Eng. 133 (2018) 224–236. [6] T. Michioka, Large-eddy simulation for turbulent flow and gas dispersion over wavy walls, Int. J. Heat Mass Transf. 125 (2018) 569–579. [7] A. Muley, J.B. Borghese, S.L. White, R.M. Manglik, Enhanced Thermal-Hydraulic Performance of a Wavy-Plate-Fin Compact Heat Exchanger: Effect of Corrugation Severity, in: ASME 2006 International Mechanical Engineering Congress and Exposition, 2006, pp. 243–249. [8] W.M. Kays, A.L. London, Compact heat exchangers, Mech. Eng., ASME 86 (1964) 31–34. [9] J. Dong, L. Su, Q. Chen, W. Xu, Experimental study on thermal–hydraulic performance of a wavy fin-and-flat tube aluminum heat exchanger, Appl. Therm. Eng. 51 (1–2) (2013) 32–39. [10] J. Dong, Y. Zhang, G. Li, W. Xu, Experimental study of wavy fin aluminum plate fin heat exchanger, Exp. Heat Transf. 26 (4) (2013) 384–396. [11] J. Dong, J. Chen, W. Zhang, J. Hu, Experimental and numerical investigation of thermal-hydraulic performance in wavy fin-and-flat tube heat exchangers, Appl. Therm. Eng. 30 (11–12) (2010) 1377–1386. [12] A. Yutaka, N. Hiroshi, M. Faghri, Heat transfer and pressure drop characteristics in a corrugated duct with rounded corners, Int. J. Heat Mass Transf. 31 (6) (1988) 1237–1245. [13] R.M. Manglik, J. Zhang, A. Muley, Low Reynolds number forced convection in three-dimensional wavy-plate-fin compact channels: fin density effects, Int. J. Heat Mass Transf. 48 (8) (2005) 1439–1449. [14] J. Zhang, J. Kundu, R.M. Manglik, Effect of fin waviness and spacing on the lateral vortex structure and laminar heat transfer in wavy-plate-fin cores, Int. J. Heat Mass Transf. 47 (8) (2004) 1719–1730. [15] L. Tian, Y. He, Y. Tao, W. Tao, A comparative study on the air-side performance of wavy fin-and-tube heat exchanger with punched delta winglets in staggered and in-line arrangements, Int. J. Therm. Sci. 48 (9) (2009) 1765– 1776. [16] M. Khoshvaght-Aliabadi, S.E. Hosseini Rad, F. Hormozi, Al2O3–water nanofluid inside wavy mini-channel with different cross-sections, J. Taiwan Inst. Chem. Eng. 58 (2016) 8–18. [17] M. Khoshvaght-Aliabadi, A. Jafari, O. Sartipzadeh, M. Salami, Thermal– hydraulic performance of wavy plate-fin heat exchanger using passive techniques: perforations, winglets, and nanofluids, Int. Commun. Heat Mass Transf. 78 (2016) 231–240. [18] M. Khoshvaght-Aliabadi, M. Tatari, M. Salami, Analysis on Al2O3/water nanofluid flow in a channel by inserting corrugated/perforated fins for solar heating heat exchangers, Renew. Energy 115 (2018) 1099–1108. [19] L.S. Ismail, C. Ranganayakulu, R.K. Shah, Numerical study of flow patterns of compact plate-fin heat exchangers and generation of design data for offset and wavy fins, Int. J. Heat Mass Transf. 52 (17) (2009) 3972–3983. [20] W.Q. Tao, Z.Y. Guo, B.X. Wang, Field synergy principle for enhancing convective heat transfer––its extension and numerical verifications, Int. J. Heat Mass Transf. 45 (18) (2002) 3849–3856. [21] Q. Ma, X. Wu, F. Chu, B. Zhu, Experimental and numerical investigations of frost formation on wavy plates, Appl. Therm. Eng. 138 (2018) 627–632.

[22] V.Y. Vasil’ev, An experimental investigation into rational enhancement of convective heat transfer in rectangular interrupted ducts of plate-fin heattransfer surfaces, Therm. Eng. 53 (12) (2006) 1006–1016. [23] J.H. Kim, J.H. Yun, S.L. Chang, Heat-transfer and friction characteristics for the louver-fin heat exchanger, J. Thermophys Heat Transf. 18 (1) (2012) 58–64. [24] H. Bhowmik, K.S. Lee, Analysis of heat transfer and pressure drop characteristics in an offset strip fin heat exchanger, Int. Commun. Heat Mass Transf. 36 (3) (2009) 259–263. [25] J. Li, S. Wang, W. Cai, W. Zhang, Numerical study on air-side performance of an integrated fin and micro-channel heat exchanger, Appl. Therm. Eng. 30 (17) (2010) 2738–2745. [26] J. Dong, J. Chen, Z. Chen, W. Zhang, Y. Zhou, Heat transfer and pressure drop correlations for the multi-louvered fin compact heat exchangers, Energy Convers. Manage. 48 (5) (2007) 1506–1515. [27] J. Wen, H. Yang, X. Tong, K. Li, S. Wang, Y. Li, Optimization investigation on configuration parameters of serrated fin in plate-fin heat exchanger using genetic algorithm, Int. J. Therm. Sci. 101 (2016) 116–125. [28] J. Wen, H. Yang, X. Tong, K. Li, S. Wang, Y. Li, Configuration parameters design and optimization for plate-fin heat exchangers with serrated fin by multiobjective genetic algorithm, Energy Convers. Manage. 117 (2016) 482–489. [29] J. Guo, L. Cheng, M. Xu, Entransy dissipation number and its application to heat exchanger performance evaluation, Chin. Sci. Bull. 54 (15) (2009) 2708–2713. [30] L. Zhang, Z. Qian, J. Deng, Y. Yin, Fluid–structure interaction numerical simulation of thermal performance and mechanical property on plate-fins heat exchanger, Heat Mass Transf. 51 (9) (2015) 1337–1353. [31] W.Q. Tao, Numercial Heat Transfer, second ed., 2001 (in Chinese). [32] L.S. Ismail, R. Velraj, Studies on fanning friction (f) and Colburn (j) factors of offset and wavy fins compact plate fin heat exchanger–a CFD approach, Numer. Heat Transf., Part A: Appl. 56 (12) (2009) 987–1005. [33] E. Suhir, Strength Theories, Springer, Netherlands, 1991. [34] Y. Zhu, Y. Li, Three-dimensional numerical simulation on the laminar flow and heat transfer in four basic fins of plate-fin heat exchangers, J. Heat Transf.Trans. ASME 130 (11) (2008) 1617–1620. [35] Y. Levy, Graphical visualization of vortical flows by mean of helicity, AIAA J. 28 (1990). [36] N.R. Rosaguti, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic serpentine channel with semi-circular cross-section, Int. J. Heat Mass Transf. 50 (17) (2007) 3471–3480. [37] M. Khoshvaght Aliabadi, F. Hormozi, E. Hosseini Rad, New correlations for wavy plate-fin heat exchangers: different working fluids, Int. J. Numer. Meth. Heat Fluid Flow 24 (5) (2014) 1086–1108. [38] T.J. Santner, B.J. Williams, W.I. Notz, The design and analysis of computer experiments, J. Am. Stat. Assoc. 99 (468) (2003) 1203–1204. [39] N.M. Hariharan, P. Sivashanmugam, S. Kasthurirengan, Optimization of thermoacoustic primemover using response surface methodology, Hvac & R Res. 18 (5) (2012) 890–903. [40] J. Sacks, W.J. Welch, T.J. Mitchell, H.P. Wynn, Design and analysis of computer experiments, Asta Adv. Stat. Anal. 4 (4) (1989) 409–423. [41] K. Deb, A fast elitist multi-objective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2) (2000) 182–197. [42] S.H. Yang, U. Natarajan, Multi-objective optimization of cutting parameters in turning process using differential evolution and non-dominated sorting genetic algorithm-II approaches, Int. J. Adv. Manuf. Technol. 49 (5–8) (2010) 773–784. [43] H.Z. Huang, J. Qu, M.J. Zuo, A new method of system reliability multi-objective optimization using genetic algorithms, Rams 06 Reliability & Maintainability Symposium, IEEE Computer Society, 2006, pp. 278–283. [44] A. Lebée, K. Sab, Transverse shear stiffness of a chevron folded core used in sandwich construction, Int. J. Solids Struct. 47 (18–19) (2010) 2620–2629. [45] A. Lebée, K. Sab, A bending-gradient model for thick plates. Part I: theory, Int. J. Solids Struct. 48 (20) (2011) 2878–2888.