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Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

Multi-objective optimization design of thermal management system forlithium-ion battery pack based on Non-dominated Sorting GeneticAlgorithm II

Tao Denga,b,⁎, Yan Rana, Yanli Yina, Ping Liub

a School of Mechatronics & Vehicle Engineering, Chongqing Jiaotong University, Chongqing 400074, Chinab School of Aeronautics, Chongqing Jiaotong University, Chongqing 400074, China

H I G H L I G H T S

• A novel liquid cooling system is proposed for lithium-ion battery pack.

• Multi-objective optimization of the cooling system is performed based on NSGA-II.

• Small hydraulic diameter enhances heat transfer coefficient at large friction factor.

• The cooling system achieves the desired thermal performance at a small pressure drop.

A R T I C L E I N F O

Keywords:Lithium-ion battery packBattery thermal management system (BTMS)Multi-objective optimizationNon-dominated Sorting Genetic Algorithm II(NSGA-II)Response surface approximation (RSA)

A B S T R A C T

The thermal management of batteries was a significant issue considering the safety and efficiency. Optimaldesign of a novel liquid cooling system with symmetrical double-layer reverting bifurcation channel was per-formed by combining experimental, numerical simulation and multi-objective optimization techniques. Thethermophysical parameters and heat production rate of the battery for numerical simulation were obtained byexperiments. The convective heat transfer coefficient and the surface friction coefficient were chosen as ob-jective functions to visually reflect the heat transfer process. Furthermore, batteries were confined to work at theoptimal temperature (25–40 °C) and the optimal temperature difference between cells (less than 5 °C). Theperformance values of design points obtained by Latin hypercube sampling were calculated numerically.Response surface approximation was adopted to approximate the objective function and the constraint functionto reduce computing time. The Pareto-optimal front between −h and f was obtained using Non-dominatedSorting Genetic Algorithm II. 17.19% change in heat transfer coefficient was accomplished by 85.53% change inskin friction coefficient. The results reported that the cooling system with optimized thermal performance can beobtained at low flow loss.

1. Introduction

Under the pressure of energy shortage and environmental pollution,automobile manufacturers have to turn their attention to green energyand clean cars [1,2]. The key task of developing clean energy vehicles isto find an energy storage system that can support high mileage and fastacceleration. Various batteries have been proposed, such as lead-acid,nickel-based, sodium-based and lithium-ion batteries [3]. In all theseelectrochemical systems, lithium-ion batteries are the most promisingchoice for electric vehicles (EVs) and hybrid electric vehicle (HEVs),

because they have the advantages of high specific energy and power,long cycle life, no memory effect and fast charging and dischargingspeed [4,5]. However, batteries are very sensitive to temperature. Un-even temperature can result in partial deterioration of batteries. Ex-cessive temperature can not only reduce the service life of batteries, butalso threaten the safety of batteries, and even cause permanent damage[6–8]. The optimum operating temperature range of lithium-ion bat-teries is 25–40 °C, and the maximum temperature difference betweencells is not more than 5 °C [9,10]. Therefore, to prolong the cycle life ofbatteries and maximize the performance of batteries, it is necessary to

https://doi.org/10.1016/j.applthermaleng.2019.114394Received 16 May 2019; Received in revised form 12 August 2019; Accepted 14 September 2019

⁎ Corresponding author at: School of Mechatronics & Vehicle Engineering and School of Aeronautics, Chongqing Jiaotong University, No. 66 Xuefu Rd., Nan’anDist., Chongqing 400074, China.

E-mail address: d82t722@cqjtu.edu.cn (T. Deng).

Applied Thermal Engineering 164 (2020) 114394

Available online 19 September 20191359-4311/ © 2019 Elsevier Ltd. All rights reserved.

T

adopt a thermal management system (BTMS) to make the batterieswork efficiently.

A large number of thermal management studies have focused onreducing the maximum temperature of lithium-ion batteries, pumppower and the maximum temperature difference between cells. Theymainly started from three aspects: numerical simulation scheme, ex-perimental means and optimization scheme. According to the differentcooling media, BTMS can be divided into air cooling, liquid cooling(liquid cooling plate and heat pipe cooling) and phase change materialcooling [11].

Air cooling, including natural air cooling and forced air cooling, iswidely applied because of its low cost and simple structure. Bernardoet al. [12] employed the optimization algorithm to optimize the aircooling system of the battery pack. It was found that the maximumtemperature difference between battery modules was as high as 8.36 °C.Nelson et al. [13] found that if the temperature of the battery washigher than 66 °C, it would be difficult to cool it below 52 °C by aircooling. Therefore, air cooling is mainly applicable to the cooling ofpower batteries with less heat production.

Phase change materials (PCMs) are considered to be the best energystorage method because of their high fusion latent heat [14–16].Akeiber et al. [16] found that PCMs may melt completely in hotsummer or after several continuous charging and discharging cycles.

Further, low thermal conductivity will hinder the heat transfer of PCM.Chen et al. [11] investigated the thermal performance of indirect

liquid cooling, and direct liquid cooling. The results showed indirectliquid cooling was more efficient than direct liquid cooling for vehiclecooling applications. Indirect cooling consists of liquid cooling based oncold plate and liquid cooling based on heat pipe. Heat pipe coolingprovides a solid-state solution, but their cost, operating temperaturerange and cooling power limit its wide application in large batterypacks [17–19]. Due to the strict space-limitation of battery pack, liquidcooling system based on cold plate is preferred [20]. Therefore, theliquid-cooling technique based on cold plate was adopted to cool thelithium-ion battery pack in our work.

Chen et al. [21] and Yu et al. [22] reported that the channelstructure, namely, the route of coolant channel extension, was anotherparameter which had significant effect on the performance of cold platebesides the geometry of the channel. It can be generalized classified asparallel channel [23,24], serpentine channel [25,26] and multi-channel[27,28]. Patil et al. [24] numerically studied the inlet coolant mass flowrate, the inlet coolant temperature and the number of cooling channelsin parallel channels. The results showed that the cooling efficiency wasenhanced by the low coolant inlet mass flow, the low coolant inlettemperature and a high number of the channels. Panchal et al. [26]performed a comparative study of serpentine mini-channels for cooling

Nomenclature

Aw convection heat transfer area (m2)cp specific heat capacity (J/kg·°C)D channel width (mm)Dh hydraulic diameter of the channel (mm)d thickness of cold plate (mm)dc thickness of all channels (mm)F(x) vector representation of objective functionsf skin friction factor (m2)g(x) vector representation of constraint functionsh convection heat transfer coefficient (W/°C)I current (A)k thermal conductivity coefficient (W/m·°C)L channel length (mm)mb mass of battery (kg)n number of channelsN number of design variablesNu Nusselt numberP pump power (W)ΔP total pressure drop between inlet and outlet (Pa)p coolant pressure (N)Qgen heat production power of the battery (W)Qgen heat absorption power of the battery (W)Qconv heat dissipated by convection (W)qgen heat production per unit volume of the battery (W/m3)qm mass flow rate (kg/s)Re Reynolds numberRj Joule resistance (Ω)R2 adj multiple regression coefficientT temperature (°C)Tmax maximum temperature of battery pack (°C)ΔTmax maximum temperature difference between cells (°C)Uocv open-circuit voltage (V)U operating voltage (V)ω width ratiou flow velocity (m/s)Vb volume of battery (m3)x set of design variables∇2 Laplacian operator

Greek symbols

ρ density (kg/m3)μ dynamic viscosity (pa. s)θ bifurcation angleγ length ratioζ parameter estimated with the least square method

Superscript

' y axes° degree

Subscripts

b batteryl liquidS solidw channel wallin inletout outletmax maximum

Abbreviation

KRN kriging

Acronyms

BTMS battery thermal management systemCFD computational fluid dynamicsEVs electric vehiclesHEVs hybrid electric vehiclesLHS Latin hypercube samplingMOPSO multi-objective particle swarm optimizationNSGA non-dominated sorting genetic algorithmPCMs phase change materialsPOF pareto-optimal frontRBNN radial basis neural networkRSA response surface approximation

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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prismatic lithium-ion batteries combining experimental and numericalsimulation methods. The results reported that the temperature of thecold plate increased with the increase of discharge rate and operatingtemperature. Stephen [29] evaluated the heat dissipation performanceof cold plates with parallel channel, spiral channel and bifurcationchannel. The results reported that the spiral channel design providedthe best thermal performance at the expense of pump pressure(> 100 kPa). The bifurcation channel achieved good thermal perfor-mance and low pressure drop (< 10 kPa). There were great hot spots inparallel channel, and its comprehensive performance was worst.

It has been shown in many literatures that the bifurcation channelachieved better heat dissipation ability than parallel straight channeland serpentine channel for microelectronic devices. Furthermore, itoffered the inherent advantage of low power consumption [30,31]. Inthe previous work [27], we designed a double-layered channel in-cluding four straight channels in the collection layer and bifurcationchannels in the dispersed layer, which proved that the tree-shaped bi-furcated cold plate design had good thermal performance and lowpressure drop. Because the battery is always sandwiched by two coldplates, a novel symmetrical double-layered reverting bifurcationchannel was proposed. In addition, the thermal performance andpressure drop of the liquid-cooled plate are always contradictory, andthere are few literatures on multi-objective optimization of the liquid-cooled plate. Therefore, multi-objective optimization of the novel de-sign was performed coupled with the surrogate model.

2. Design and analysis of battery pack

2.1. Battery pack design

In practical applications, the battery pack configuration is usuallycomposed of many thin cell units. There is a cooling plate between eachtwo cells. The heat generated by each cell is conduced into the coldplate through the contact surface and then transmitted by the coolant.

Because the heat production at the electrode tab is very small, theelectrode tab is omitted in this study to simplify the battery model.Fig. 1(a) illustrates a battery pack with cooling plates. In the batterypack, there are four square lithium-ion batteries, in turn named cell 1,cell 2, cell 3, cell 4, which are taken from commercial lithium-ionbatteries. Fig. 1(b) and Fig. 1(c) display the structure of the cell andcooling plate, respectively. The cold plate in this study contains asymmetrical double-layer reverting bifurcation channel, which is im-proved on the basis of our previous work [27]. In addition to thechannel thickness for the two layers, the cold plate thickness has threesolid layer variables, namely, the thicknesses of the cover layers at thetop and bottom, and the separating solid layer between the two channellayers. The coolant enters the collection layer channel from four inletson the side and converges to the center. After flowing through the se-parating solid layer between the collection layer and the dispersionlayer, it enters the dispersion layer channel, and finally leaves the coldplate through the four exits. Please note that the collection layerchannel is exactly the same as the dispersed layer channel, only the flowdirection is opposite.

2.2. Experimental setup and parameter extraction

The parameters related to the battery are listed in Table 1. In thenumerical solution, the physical parameters of the battery includedensity (ρb), thermal conductivity (kb) and specific heat capacity (cpb).For simplicity, the battery is composed of a single material, so ρb, kb andcpb are constants. According to the mass mb and volume Vb of the cell,the density ρb of the battery is calculated to be 2049 kg/m3 (ρb=mb/Vb). The thermal conductivity of the battery is generally taken as3–5W/(m °C) [23], kb=5W/(m °C) was considered in our work. Thespecific heat capacity and heat generation expression of the battery willbe obtained by the following experiment. The test device is presented inFig. 2.

The battery tester electronic load EBC-A40L can charge and

Cold plate

Cell 4Cell 3Cell 2Cell 1

x y

z

163mm 184mm

9mm

(a) Design 1 (b) Single battery cell

184 mm

163 mm163 mm

184 mm

D4,L4

D4 ,L

4 '

D1,L1

D3,L3'

D3,L3D2,L2

D1

D1,L1' D2 ,L

2 ' D5 ,L

5 '

D5,L5

(c) Double-layered cold-plate

Fig. 1. Schematic of battery and cooling plate.

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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discharge the battery according to the test procedure. Its voltage testrange is 0–5 V and current test range is 0.1–40 A. The thermostat canprovide a stable ambient temperature for battery testing. The tem-perature range of the thermostat is 10–300 °C. The K-type thermo-couples are placed on the battery surface by Kapton tape, as shown inFig. 3. Point 2 and point 3 are placed near positive and negative elec-trodes, respectively. Point 1 is placed on the center of the battery andpoint 4 is located near the bottom of the battery.

The heat generated by the battery (Qgen) is mainly composed of twoparts. One is the irreversible heat generated by the internal resistance ofthe battery. The other is the reversible heat generated by the electro-chemical reaction inside the battery. The typical Bernadi heat genera-tion model is usually used to describe the heat generation rate as fol-lows [32]:

= ⎡⎣⎢

− + ∂∂

⎤⎦⎥

Q I U U T UT

( )gen OCV bOCV

b (1)

where I (Uocv−U) is the Joule heat generated by the internal resistanceof the battery; ITb(∂Uocv/∂Tb) is the heat generated by the electro-chemical reaction inside the battery. Thus, the thermal generation ofbatteries can also be written as follows [33]:

= + ∂∂

Q I R IT UTgen j bOCV

b

2(2)

As the temperature rises, the heat absorbed by the battery is asbelow:

=Q m c dTdtabs b pb

b(3)

In an insulated environment, the heat generated by the battery isequal to the heat absorbed by the battery. After the above Eq. (2) andEq. (3) are transformed, the Eq. (4) can be obtained.

= + ∂∂I

dTdt

Rm c

Im c

T UT

1 · · 1 · ·b j

b pb b pbb

OCV

b (4)

According the heat generation theory of the battery, as described inEq. (4), the finite element model of the battery is established. To makean adiabatic environment, the battery was wrapped with the heat in-sulated cotton. Discharging with 5 different rates (8 A, 16 A, 24 A, 32 A,40 A) for 15min, respectively. Further, five different dTb/dt values areobtained.

Chacko et al. [34] and Panchal et al. [35] suggested that the Jouleresistance Rj could be regarded as a constant when the battery operatednormally at 25–45 °C and SOC values of 0.2–0.9. The term Tb(∂Uocv/∂Tb) is related to the electrochemical reaction inside the battery. Itdepends on SOC value and battery types, which can be determined as aconstant for same battery [36]. The term dTb/dt is also a constant forconstant-current discharging within 15min [37]. Therefore, the func-tion 1/I(dTb/dt) can be regarded as a linear relation about the current Iin Eq. (4). The linear relationship of 1/I(dTb//dt) against I is obtainedby experiment, as shown in Fig. 4.

The linear equation is as follows:

= × + ×− −dTdt

I1I

4.271 10 1.390 10b 6 4(5)

According to Eq. (4), the slope of liner line is Rj/mbcpb. Theequivalent specific heat of the battery (cpb) is calculated to be 847 J/(kg·°C). Therefore, combing the Eq. (3) with Eq. (4), the heat generationQgen (W) of the battery is expressed as below:

= × + ×− −Q I I 2 10 6.509 10gen3 2 2 (6)

As a result, the heat generation per unit volume of the battery (qgen)can be written as follows:

= = +qQV

I I7.409 241.139gengen

b

2(7)

Table 1Parameters of lithium ion battery.

Parameters Nominal values

Capacity 30 AhVoltage 3.7 VMass 0.553 kgDimensions 163×184×9mm (length×width× thickness)Resistance 2mΩ

Charging conditions Max charging current 1 CCharging voltage 4.25 ± 0.05 VCharging cut-off current 0.05 C

Discharging conditions Max continuous discharging current 5 CDischarging cut-off voltage 2.75 V

battery tester

electronic load

testing lithium ion

battery

computer for data

collection

thermostat

thermodetector

Fig. 2. Test platform for the lithium ion battery.

163

184

point 1

38

2242

15

point 3 point 2

15

point 4

Unit: mm

15

42

K type thermocouple

Kapton tape

Fig. 3. Thermocouple locations.

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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Uncertainty in the results of battery charging and discharging ex-periment may come from two aspects. First of all, the difference in theresults may be caused by different researchers. For example, Kaptontape was not pasted properly, resulting in the K-type thermocouple notmeasuring the battery temperature in real time. The insulation cottonwas not wrapped properly, and the battery dissipated heat in thethermostat. Secondly, the results may be different due to the thermalinsulation capacity of different thermostats.

2.3. Numerical solution

In our study, the model was meshed by Hypermesh and solvednumerically by commercial hydrodynamic software STAR-CCM+. Thepolyhedral mesh has more elements and faster computing speed toachieve accurate result. The mesh model of the battery pack is shown inFig. 5.

For simplicity, a single material of aluminum was used and waterwas chosen as the coolant. The parameters of cooling water and alu-minum are tabulated in Table 2.

2.3.1. Initial and boundary conditionThe initial conditions of the battery pack cooling system are given as

follows:

• The initial temperature of battery is equal to 25 °C, and the inlettemperature of the coolant and the ambient temperature are equalto 25 °C;

• Apply a volume heat source to all batteries. Based on the experi-mental heat generation rate Eq. (7), when the battery is dischargedat 3C (C-rate refers to the ratio of current to nominal voltage), theheat generation rate is calculated as 81715W/m3.

The boundary conditions are provided in Table 3.The details are as bellow:

• Contact surface of battery pack with air: Free convection boundarycondition with heat transfer coefficient of 5W/(m °C);

• Cold plate inlet: Mass flow rate (qm).

• Cold plate outlet: The ambient pressure is used as the referencepressure of the fluid at the outlet and is equal to 0 Pa.

• Cold plate wall: If the outer wall is in contact with the batterysurface, the solid-solid coupling is performed, and the inner channelwall and the fluid domain were fluid-solid coupled. In order to en-hance the simulation of convective heat transfer, two boundarylayers at the interface of fluid-solid interaction were added, whichmade the calculation of fluid-solid interaction more accurate.

2.3.2. AssumptionBecause the actual heating generation process inside lithium-ion

batteries is complex, the following assumptions are needed for somephysical properties of batteries [5]:

① The physical properties of various media in the battery are uni-form, and the density and specific heat capacity of batteries are con-stant.

② The thermal conductivity is anisotropic.

③ The radiation inside the battery has little effect on the heat dis-sipation and can be neglected.

④ The current density of the battery distributes uniformly duringcharging and discharging, and the heat generation of each part is uni-form.

For cooling plate simulation, the following assumptions are putforward [38].

(1) The cold plate is homogeneous and isotropic.(2) The fluid is single phase, incompressible and steady.(3) The thermophysical properties of cooling water and aluminum are

independent of temperature.(4) Ignore gravity and contact thermal resistance.(5) Ignore the influence of viscosity dissipation.

2.3.3. Governing equationBased on the above assumptions, the governing equations of the

numerical model can be written as follows [39]:Mass conservation:

∇∙ → =ρ u( ) 0l (8)

Momentum conservation:

→∙∇∙→ = −∇∙ + ∇ →ρ u u p μ u( )l2 (9)

Energy conservation in the fluid domain:

→∙∇ = ∇ρ c u T k T( )l p l l l2

l (10)

Energy conservation in the solid domain:

∇ =T 0s2 (11)

The Reynolds number at the inlet is defined as:

=Reρ uD

μl h

(12)

The maximum inlet mass flow rate of coolant is 10 g/s and thecorresponding Reynolds number is 2210.33. Thus, the laminar flowmodel is adopted.

The convective heat transfer coefficient and Nusselt number in thefluid domain are calculated by the following equations:

=−

h QnA T T( )

conv

w w l (13)

=N hDku

h

l (14)

The skin friction coefficient of the fluid domain is calculated by thefollowing equation:

=f Pρ uΔ

lL

D12

2h (15)

Fig. 4. Experimental fitting results.

Fig. 5. The mesh model of battery pack.

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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To overcome the flow resistance, the pump power of driving coolantthrough all channels can be calculated as follows:

=PP qρ

Δ · m

l (16)

In this work, the double-layered bifurcation channel contains manyturning sections and variable cross-section channels. Therefore, theoverall pressure drop in the channel consists of longitudinal pressuredrop and local pressure drop.

2.3.4. Mesh independence testBased on the above initial conditions and boundary conditions, a

rigorous independent test of the volume mesh number was conducted toobtain accurate simulation results, as shown in Fig. 6.

It can be concluded that the volume grid number has little effect onthe simulation results. The mesh model with number of 2,506,874elements was selected for further study.

3. LiBTMS optimization methodology

3.1. Design variables and objective functions

The double-layered bifurcation channel with variable cross-sectioncontains many variables. According to Murray's law [40], when theflow state of coolant is laminar, the width of all branches at the splittingpoint behaves the following relationship, which can reduce the pumppower consumption.

∑ ∑=D Dinflow outflow3 3

(17)

Given the width value D1 of the main channel and the width ratio ω,the width of all channels can be calculated. The length ratio (γ) wasemployed to determine the split point position of the internal channelalong the X axes and Y axes, namely, to determine the relative length ofL1 and L3, and that of L1′ and L3′. When the bifurcation angle betweenchannels (with widths of D2 and D3) is known, the channel length L2 canbe determined, and the values of length L4 and L5 are determined. Toreduce the workload, the optimal bifurcation angle of heat transfer wasobtained by numerical simulation. Therefore, the plane structure of thedouble-layered bifurcation channel can be determined by three vari-ables: the channel width D1, the width ratio (ω), the length ratio (γ).

The width ratio (ω) and the length ratio (γ) are defined as follows:

=ω DD

4

1 (18)

=+

=′

′ + ′γ L

L LL

L L1

1 3

1

1 3 (19)

For simplicity, the thickness of the cover layers at the top andbottom was designed to be constant at 0.5mm. Consequently, thethickness of the cold plate (d) is determined by two variables, namely,the channel thickness (dc) and the separated solid layer thickness(d− dc * 2–0.5 * 2). One of the variables (channel thickness) waschosen for optimize. In addition, increasing mass flow rate can enhanceheat transfer, but increase power consumption. Accordingly, the inletmass flow rate qm can be used as one of the optimization variables.

Therefore, the channel width D1, the width ratio (ω), the lengthratio (γ), the channel thickness dc and the inlet mass flow rate qm wereselected as optimization design variables. The range of variation isgiven in Table 4.

In our work, convective heat transfer coefficient h and skin frictioncoefficient f were used as objective functions. The convective heattransfer coefficient represents the heat transfer rate. The skin frictioncoefficient reflects the flow of coolant in the channel. Increasing con-vective heat transfer coefficient usually causes an increase in skinfriction coefficient. The convective heat transfer coefficient was max-imized and the skin friction coefficient was minimized to obtain acooling system with better heat transfer capacity and less flow loss.Besides, batteries were confined to work at the optimal temperature(25–40 °C) and the optimum temperature difference between cells (lessthan 5 °C).

3.2. Response surface approximation (RSA)

Multi-objective optimization based on NSGA-II requires multipleevaluation of the objective function to find the optimal solution. In theabsence of representative response functions, the evaluation becomesvery expensive and time-consuming. Queipo et al. [41] suggested var-ious surrogate models, including response surface approximation(RSA), Kriging (KRG) and radial basis neural network (RBNN). RSAsurrogate model is the most widely adopted [42]. Therefore, RSA modelwas employed to approximate these performance functions.

RSA is a method of fitting polynomial function of discrete response.In our study, the second order polynomial with intercept term, linearterm, square term and quadratic interaction term was used as the re-sponse function. As bellow:

∑ ∑ ∑= + + += = <

f x ζ ζ x ζ x ζ x x( )i

N

i ii

N

ii ii j

N

ij i j01 1

2

(20)

3.3. Multi-objective optimization

NSGA-II evolutionary algorithm reduces the complexity of non-in-ferior sorting genetic algorithm, and provides the advantages of fastrunning speed and good convergence of solution set [43].

The above multi-objective problems can be formulated as follows:Minimize F(x)= [F1(x), F2(x), F3(x), … Fm(x)]Subject to g(x)= [g1(x), g2(x), g3(x), … gm(x)]≤ 0The work-flow of multi-objective optimization is illustrated in

Fig. 7.

Table 2Properties of water and aluminum.

Aluminum Water

Specific heat capacity cp (J·kg−1 K−1) 903.0 4181.72Thermal conductivity k (W·m−1·K−1) 237.0 0.62Dynamic viscosity μ (Pa·s) 8.8871E-4Density ρ (kg·m−3) 2702.0 997.561

Table 3Boundary Conditions for Battery Packets.

Boundary conditions

Ambient temperature (°C) 25Volume heat source (W/m3) 81,715Initial inlet mass flow rate (g/s) 10Outlet pressure (Pa) 0Boundary condition Free convection

Fig. 6. Independent test of volume mesh number.

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

6

3.4. K-means Clustering

K-means clustering is based on the minimum distance method toclassify samples. Cluster centers and objects allocated to them representclusters. Therefore, K-means clustering was adopted to cluster Paretooptimal solution, and representative solutions are obtained to analyzethe relationship between objective function and design variables.

4. Result and discussion

4.1. Effect of bifurcation angle on battery pack cooling system

As mentioned above, the double-layered reverting bifurcationchannel is determined by five design variables and the bifurcation anglebetween the channels with widths of D2 and D3. In order to reduce theworkload, the influence of bifurcation angle on cooling system wasstudied by numerical simulation before multi-objective optimization.The maximum bifurcation angle (θ) is calculated as 57°. Therefore,other variables remained unchanged and bifurcation angles of 0°, 25°,35°, 45° and 55° were used for numerical simulation to obtain a θ valuewith better performance, respectively.

The variations of the maximum temperature of battery pack (Tmax),the maximum temperature difference between cells (ΔTmax), and thetotal pressure drop of cooling system (ΔP) with bifurcation angle (θ) aredepicted in Fig. 8. It can be clearly noticed that when θ=25°, Tmax isthe smallest, and ΔTmax and ΔP are also lower. Thus, the bifurcationangle with better performance is equal to 25°.

4.2. Multi-objective optimization design

There is neither functional relationship between structural variablesand objective functions, nor that between structural variables andconstraint functions. Therefore, RSA surrogate model was employed toapproximate the objective function and the constraint function. Beforethat, enough design points were selected in the design space using LatinHypercube Sample (LHS), and their performance values were solvednumerically based on STAR-CCM+, as shown in Table 5.

According to the discrete points and response values in Table 5,approximate fitting of h, f, Tmax and ΔTmax was obtained. As bellow:

= − +

− ++ + − +

−+ + + ++ + − −

+ − −

h D ω γ d q x x

x xx x x x

xx x x x x x x

x x x x x x x xx x x x x x

( , , , , ) 22.4913387 10.2257151 13.80329777

21.83124491 48.0103517.41384 0.02776014 27.40754702 8.053342345

2.4616369921.921379319 3.981526936 2.637045 0.9154182672.151664 4.134870985 27.87379937 0.27080943

3.897244377 2.940499588 15.14795376

c m1 1 2

3 4

5 12

22

32

42

52

1 2 1 3 1 4

1 5 2 3 2 4 2 5

3 4 3 5 4 5

(21)

= + +

+ −+ − − +

+

+ + − −

+− + −

−+ −

f D ω γ d q x x

x xx x x x

x

x x x x x x x

x xx x x x x x

x xx x x x

( , , , , ) 0.30995608 0.239938735 1.064487442

0.890910891 3.711680.181992 0.0142359 1.765911196 0.176740708

2.470607888

0.007706868 0.081577525 0.09053 0.09729628

0.0050440.504475025 0.680317108 0.051902656

0.089186380.015309988 0.237350428

c m1 1 2

3 4

5 12

22

32

42

52

1 2 1 3 1 4

1 5

2 3 2 4 2 5

3 4

3 5 4 5

(22)

= − −

− −− + + +

+ + − −+ + + −

− + +

+

T D ω γ d q x x

x xx x x x

x x x x x xx x x x x x x x

x x x x x x

x x

( , , , , ) 141.636654 5.93505637 1.30539756

56.56715109 36.54916.6221 0.38867563 27.64085406 18.20891082

6.649952656 1.058502118 3.133999579 0.339983.429127712 0.047321 59.91604969 21.2755918

2.178375373 6.630932012 0.289351024

0.743196591

c mmax 1 1 2

3 4

5 12

22

32

42

52

1 2 1 3

1 4 1 5 2 3 2 4

2 5 3 4 3 5

4 5

(23)

Table 4Design variables and design spaces.

Limit Variables

D1 (mm) ω γ dc (mm) qm (g/s)

Upper 4 0.15 0.05 0.2 1Lower 10 0.79 0.95 1.4 10

Fig. 7. Work-flow of the multi-objective NSGA-II algorithm.

P

Fig. 8. Effect of bifurcation angle on the performance of cooling system.

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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= + +

+ −− + − −

+ + + −− − − ++ − + +

T D ω γ d q x x

x xx x x x

x x x x x xx x x x x x x x

x x x x x x

x x

Δ ( , , , , ) 0.89258697 0.223926378 0.835638613

1.004206556 0.137880.12543 0.00046346 0.63274644 0.381194153

0.336081887 0.006566682 0.008235639 0.046540.109254953 0.01511 0.649101521 0.0096388970.00125952 0.060993526 0.014109447 0.051017004

c mmax 1 1 2

3 4

5 12

22

32

42

52

1 2 1 3

1 4 1 5 2 3 2 4

2 5 3 4 3 5

4 5

(24)

The multivariate regression coefficient (R2 adj) was adopted tomeasure the uncertainty of polynomial coefficients. When 0.9 < R2adj < 1, it suggests that the RSA surrogate model meets the accuracyrequirement [44,45]. The regression coefficients of h, f, Tmax and ΔTmax

were 0.9995, 0.9991, 0.9812 and 0.9733, respectively, which guaran-teed the accuracy of the surrogate model.

The Pareto-optimal front based on NSGA-II evolutionary algorithm

is plotted in Fig. 9. Because the convective heat transfer coefficient h ismaximized and the surface friction coefficient f is minimized, a concavefront can be seen in the Pareto front of ((−h)−f). From this figure, it isevident that the improvement of one objective always comes at theexpense of the other, which can be attributed to the natural contra-diction between convective heat transfer coefficient and skin frictioncoefficient. In addition, because each solution is Pareto's global optimalsolution, it can be found that none of the solutions in the Pareto front iscompletely superior to the others.

To validate the prediction accuracy of the surrogate model and to

Table 5Numerical calculation of objective function values for LHS design points.

No. Design variables Objective functions Constraint functions

D1 ω γ dc (mm) qm h (W/°C) f (m2) Tmax (°C) ΔTmax (°C)

1 4.281 0.339 0.138 0.817 8.840 291.825 0.448 35.220 1.2112 8.331 0.315 0.768 0.851 2.101 48.934 0.079 58.799 2.1263 10.000 0.790 0.050 1.000 4.000 106.666 0.108 42.848 1.7824 4.797 0.528 0.226 1.195 2.202 50.294 0.313 53.116 1.8155 9.178 0.766 0.653 1.289 8.052 251.512 0.142 35.099 1.2356 6.000 0.150 0.350 1.000 10.000 345.277 0.313 34.785 1.1787 6.707 0.298 0.601 0.889 4.094 106.528 0.151 42.047 1.6018 4.000 0.150 0.050 0.200 1.000 22.788 0.675 105.010 1.5209 7.415 0.374 0.930 1.337 6.607 181.820 0.059 36.726 1.32010 8.599 0.608 0.902 1.208 4.403 116.260 0.086 40.863 1.56711 6.829 0.771 0.484 0.422 6.194 208.929 0.985 37.677 1.36012 6.000 0.363 0.050 1.400 7.000 182.672 0.118 37.447 1.37813 9.982 0.263 0.712 0.602 3.930 110.466 0.283 45.046 1.92114 4.000 0.576 0.650 1.000 7.000 184.504 0.252 36.222 1.21215 6.133 0.586 0.064 0.682 5.329 155.060 0.333 38.879 1.47816 5.374 0.416 0.629 0.567 3.468 86.994 0.320 45.524 1.69617 8.000 0.150 0.650 1.400 4.000 99.576 0.059 42.421 1.63918 5.612 0.552 0.841 1.085 5.990 157.947 0.163 37.412 1.32719 5.966 0.184 0.416 0.371 3.161 78.851 0.646 50.725 1.89920 10.000 0.150 0.950 0.600 7.000 267.472 0.507 36.298 1.39621 7.346 0.697 0.819 0.965 1.047 22.967 0.031 89.103 1.81522 4.000 0.363 0.350 0.600 4.000 101.702 0.356 42.368 1.55423 6.546 0.512 0.352 0.655 2.728 64.910 0.179 51.289 1.90924 4.584 0.640 0.314 0.769 7.440 227.470 0.412 35.327 1.21125 10.000 0.576 0.350 1.400 1.000 21.879 0.014 93.774 2.26126 9.661 0.428 0.580 1.241 1.591 35.666 0.028 65.883 2.16527 6.000 0.790 0.650 0.600 1.000 21.700 0.083 95.883 1.95628 8.884 0.214 0.260 0.237 4.767 154.944 1.924 45.679 1.92229 6.000 0.576 0.950 0.200 4.000 111.692 1.804 47.997 1.85130 8.724 0.462 0.686 1.131 9.830 362.966 0.222 77.335 1.12031 7.659 0.363 0.504 0.269 4.992 158.666 1.774 43.382 1.79032 7.002 0.396 0.241 1.376 6.982 195.855 0.116 36.695 1.29533 4.000 0.790 0.950 1.400 10.000 235.780 0.225 36.558 1.59534 4.892 0.707 0.090 1.076 1.809 40.323 0.051 59.707 1.94235 8.000 0.363 0.050 1.000 1.000 21.976 0.027 94.311 2.008

Fig. 9. Pareto-optimal front using the NSGA-II and clustering points obtainedthrough K-means clustering.

Table 6Comparison between numerical and RSA predicted values for representativePareto-optimal solutions.

Design A B C D

D1 (mm) 9.771 9.894 9.967 10.000ω 0.789 0.789 0.774 0.790γ 0.514 0.087 0.050 0.050dc (mm) 0.917 0.723 0.551 0.376qm (g/s) 9.866 9.845 9.999 9.917

RSA predicted resultsh (W/°C) 411.703 440.568 461.711 482.460f (m2) 0.324 0.904 1.495 2.238

Simulation resultsh (W/°C) 416.536 429.225 447.538 494.926f (m2) 0.329 0.842 1.426 2.177

Error (%)h 1.16 2.64 3.17 2.52f 1.70 7.28 4.82 2.80

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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(a) POF-A

(b) POF-B

(c) POF-C

(d) POF-D Fig. 10. Velocity profile and temperature profile of representative Pareto-optimal solutions.

T. Deng, et al. Applied Thermal Engineering 164 (2020) 114394

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analyze the relationship between the design variables and the objectivefunction, K-means clustering algorithm was used to select re-presentative solutions (with red-marked) from Pareto-optimal frontier(POF), as shown in Fig. 9. The representative solutions and corre-sponding design variables are tabulated in Table 6.

The predicted objective function values of representative solutionswere compared with those calculated by CFD. The maximum relativeerrors in the predictions of convective heat transfer coefficient (h) andskin friction coefficient (f) are 3.17% and 7.28%, respectively, whichdemonstrates that the predicted values for the representative solutionshave reached an agreement with the CFD results. It can be noticed fromTable 6 that when the maximum temperature of the battery pack rangesfrom 25 °C to 40 °C and the maximum temperature difference betweencells is less than 5 °C, the values of D1, ω and qm tend to be near theupper bound, the γ value is near the middle and its lower bounds, butthe dc value fluctuates widely. In other words, the design variables D1, ωand qm are less sensitive than that γ and dc. Besides, it can be observedthat the convective heat transfer coefficient increases as the length ratioγ and channel thickness dc decrease. This is because the increase in γvalue makes the length of L2 channel larger. The surface integral valuesof convective heat transfer coefficient and skin friction coefficient en-large with the expansion of heat transfer area. Decreasing the channelthickness can reduce the hydraulic diameter of the channel and enlargethe flow velocity of the coolant, thus increasing the convective heattransfer coefficient and the skin friction coefficient. 17.19% change inheat transfer coefficient is accomplished by 85.53% change in skinfriction coefficient.

Fig. 10 illustrates the temperature profile and velocity profile ofrepresentative solutions. It can be seen that the flow velocity of coolingsystem corresponding to representing solutions A-D is increasing. Inresponse the heat transfer is improved and the maximum temperatureof the battery pack is decreasing. In addition, it can be observed that theflow velocity of coolant in inner channel is higher than that in per-ipheral channel, which results in heat concentration areas appear on theperiphery of battery.

Behavior of pressure drop (ΔP) versus skin friction factor (f) ispresented in Table 7. It can be concluded that the trend of pressure dropis consistent with that of skin friction coefficient, which is in ac-cordance with Eq. (15). The flow loss of the cooling system was eval-uated based on the transient indicator f and the result indicator ΔP. Asexpected, the pressure drop of the cooling system is much smaller thanthat of the cold system with serpentine channel [25], which reflects theinherent advantage of low power consumption in bifurcation channel.

5. Conclusion

Based on the evolutionary algorithm, the multi-objective optimiza-tion design of the novel battery pack liquid-cooling system was per-formed to maximize the convective heat transfer coefficient and mini-mize the skin friction coefficient. Furthermore, the maximumtemperature of the battery pack and the maximum temperature dif-ference between cells were considered as constraints. The design vari-ables included the geometric variables of the channel cross section (themain channel width D1, the width ratio ω, the length ratio γ, and thechannel thickness dc) and the inlet flow rate qm. Latin hypercube sam-pling was used for the selection of enough design points. Further, re-sponse surface approximation models of the performance functionswere constructed. The tradeoff Pareto front between the two objective

functions indicated that the thermal performance can be improved athigher flow loss. The representative solutions illustrated that thechannel thickness and length ratio had a great effect on the perfor-mance of the cooling system. As expected, the pressure drop of thestudied channel was less than 1/2 of that of the traditional serpentinechannel, which highlighted the inherent advantages of the bifurcationchannel structure.

Declaration of Competing Interest

The authors declare that there is no conflict of interest.

Acknowledgements

This work was supported by National Natural Science Foundation ofChina (51305473), Science and Technology Project Affiliated to theEducation Department of Chongqing Municipality (KJ1600538),Natural Science and Frontier Technology Research Program of theChongqing Municipal Science and Technology Commission(cstc2016jcyjA0582), Chongqing Postgraduate Research andInnovation Project (CYS18225), and Scientific and TechnologyResearch Program of Chongqing Municipal Education Commission(KJQN201800718).

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