Conductive and Radiative Heat TransferMECH 5407 (Fall 2013)

Electric Vehicle Battery Cells

Instructor: Dr. Seaho Song

Team#6:

Jason Vallis (100298414)Ying Sun (100905136)

Mohammed Thouseeq (100911248)Rui Zhao (100877556)

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1. Introduction

Electric Vehicles (EVs) are becoming increasingly popular on Canadian roads in response to growing concerns of carbon emissions and the impacts of climate change [1]. The majority of EVs available on the market today use lithium-ion (Li-ion) batteries. Heat is generated during charging and discharging of these batteries due to the electrochemical reactions occurring within them. If this heat is not dissipated properly, it can lead to overheating of the battery and potentially thermal runaway [2]. Tesla Motors is a leader in the electric car industry, producing the award winning Model S as well as electric power-train components for other automotive manufacturers [3].

Figure 1 - Tesla Model S [4].

The common laptop Li-ion battery cells used in the Tesla battery packs are referred to as “18650 form-factor” because of their measurements: 18mm in diameter by 65mm length. According to the company website, “the small size cell size enables efficient heat transfer” ultimately extending battery pack life [5]. Around 7,000 cells make up the battery pack of the Model S, mounted in a 4” (10.16 cm) enclosure under the floor [6].

Figure 2 - Tesla Model S floor mounted battery pack [7].

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Reviewing the specifications of the Panasonic model NCR-18650A Li-ion battery gives precise dimensions as well as a range of operating temperatures for an “18650” cell (figure 3,4). Using this information and assumptions regarding the material properties, the exact and analytical solution for the steady-state temperature distribution for one cylindrical battery cell is determined, allowing for the design of a cooling system to prevent the battery pack, consisting of hundreds of cells, from overheating.

Figure 3 - Dimensions of single Panasonic NCR-18650A Li-ion battery [8]

Figure 4 - Dimensions of single Panasonic NCR-18650A Li-ion battery [8]

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2. Theory and Assumptions

The cylindrical L-ion cell is typically constructed in a spirally-wound, multilayer structure of liquid-phase electrolyte and solid-phase electrodes as shown in Figure 5 [9].

Figure 5 - Multi-layer structure of a cylindrical Li-ion battery cell [9].

Because of the interfaces between the electrode layers, the thermal conductivity of the cell in the radial direction is usually much lower than the axial direction [10]. This is further defined in a 2006 article in the Journal of The Electrochemical Society entitled Thermal Analysis of Spirally Wound Lithium Batteries, which develops and verifies thermal conductivity formulas in the radial (kr) and axial (kz) directions [11].

Finally, we have assumed a single stack of battery cells, with the bottom of the cell wrapped in electrical insulation tape causing it to be thermally insulated. The primary mode of heat transfer from the sides of the cylindrical battery cells is assumed to be forced convection of atmospheric air at 5 m/s with the to top surface of the cylinder experiencing free convection of air.

kr = 26 W/m･K (radial thermal conductivity)

kz = 1.04 W/m･K (axial thermal conductivity)

h = 90 W/(m2･K) (forced air heat transfer coefficient [12])

he = 5 W/(m2･K) (free air convection heat transfer coefficient [12])

Tf = 293.15 K (ambient air temperature) Tb = 𝛳0 = 333.15 K (battery cell temperature)a = 0.0093 m (battery cell radius)b = 0.0652 m (battery cell height)𝝆 = 2300 kg/m3 (battery cell density [11]) cp = 900 J/(kg･K) (battery cell specific heat)

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3. Analytical Solution

The NCR-18650 Li-ion battery cells can be modeled as the following two-dimensional conduction problem in the cylindrical coordinate system when at steady state:

Figure 6 - Two- dimensional conduction problem in cylindrical coordinates [13]

The 2 dimensional heat conduction equation is therefore given by,

1r∂∂rr ∂θ

∂r⎛⎝⎜

⎞⎠⎟+ ∂∂z

∂θ∂z

⎛⎝⎜

⎞⎠⎟= 0

(1)

Using the separation of variable method,

θ =U(r,z) (2)

Substituting into the conduction equation

1r∂∂rr ∂U(r,z)

∂r⎛⎝⎜

⎞⎠⎟+ ∂∂z

∂U(r,z)∂z

⎛⎝⎜

⎞⎠⎟= 0

(3)

1r∂U(r,z)

∂r+ ∂2U(r,z)

∂r 2+ ∂2U(r,z)

∂z2= 0

(4)

1r∂U(r,z)

∂r + ∂2U(r,z)∂r 2

= − ∂2U(r,z)∂z2

= −λ2 (5)

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1r∂U(r,z)

∂r + ∂2U(r,z)∂r 2

= −λ2 (6)

Therefore,

− ∂2U(r,z)∂z2

= −λ2 (7)

1r

∂(R(r )Z(z))∂r

⎛⎝⎜

⎞⎠⎟+ ∂2(R(r )Z(z))

∂r 2= −λ2

(8)

On dividing R(r)Z(z) from equation (7) and (8) we could get,

1r∂(R(r ))

∂r+ ∂2(R(r ))

∂r 2+R(r )λ 2 = 0

(9)

∂2(Z(z))∂z2

= −λ 2Z(z) (10)

The general solution for equation (9) is,

R(r ) =C1J0(λnr )+C2Y0(λnr ) (11)

C2 = 0 since Y0 ∞ , as r ∞ . Thus the equation becomes,

R(r ) =C1J0(λnr ) (12)

While the solution for equation (10) is,

Z(z) =C3 cosh(λnz)+C4 sinh(λnz) (13)

And from equation (12) and (13) the combined solution of the conduction equation would be,

θ(r,z) =C1J0(λnz)(C3 cosh(λnz)+C4 sinh(λnz)) (14)

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Now applying the given boundary conditions to the above solution, starting with the 1st boundary condition for r = a.

∂θ∂r

= −C1J1(λna)Z(z) = − hkC1J0(λna)Z(z)

(15)

On solving we get,

(λna) =hakJ0(λna)J1(λna) (16)

But we have the Biot number, Bi = (ha/k). Thus we could write the above equation as,

(λna) = BiJ0(λna)J1(λna) (17)

Now applying the second boundary condition at z = b,

C1J0(λnr )(C3 sinh(λnb)+C4 cosh(λnb)) =hek(−C1J0(λnr )(C3 cosh(λnb)+C4 sinh(λnb)))

(18)

C4 = −C3

hekcosh(λnb)+ λn sinh(λnb)

λn cosh(λnb)+heksinh(λnb)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

(19)

On applying the third boundary condition at z = 0, 𝛳 = 𝛳0. Hence we could write the solution as,

θ0 = AJ0(λnr )n=1

∞

∑ (20)

Substituting the value from equations (19) and (20) in the general conduction equation, we can write the new solution for all values of n as,

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θ0 = AJ0(λn)(cosh(λnz)+Bλn sinh(λnz))n=1

∞

∑ (21)

B = −

hekcosh(λnb)+ λn sinh(λnb)

λn cosh(λnb)+heksinh(λnb)

(22)

Therefore we have the final solution as,

θ(r,z) = AJ0 cosh(λnr )+Bsinh(λnr ) (21)

In solving the temperature profile with our given parameters, we used only the forced air convective heat transfer coefficient (h) and radial thermal conductivity (kr). This is due to the fact that the sides and forced air contribute the most surface cooling of the cell.

Figure 6 - Temperature profile of 18650 cell cooling from maximum operating temperature.

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4. Numerical Solution

Steady state heat conduction equation (2-D), Taylor series method is given by,

f (x +δ ) = f (x)+ δ 2

2!f ''(x)+ δ 3

3!f '''(x)...

(22)

f (x +δ ) = f (x)− δ 2

2!f ''(x)− δ 3

3!f '''(x)...

(23)

Using the central method,

f 'i =fi+1 − fi−12δ

+O(δ 2 ) (24)

f ''i =fi+1 − fi−1 − 2 fi

δ 2 +O(δ 2 ) (25)

The governing equation is given as:

∂2T∂2 r

+ 1r∂T∂r

+ ∂2T∂2 z

= 0 (26)

where 0 ≤ r ≤ a, and 0 ≤ z ≤ b.

Assume,

δ a =

a / 2M (27)

δb =a / 2M (28)

where a/2 is the radius, b is the length of the cylinder and M is the number of finite parts.

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Therefore,

1δ a2 (Ti−1, j − 2Ti+1, j +Ti+1, j )+

1iδ a

12δ a

(Ti−1, j −Ti+1, j )+1δb2 (Ti−1, j − 2Ti+1, j +Ti+1, j ) = 0

(29)

Expand and collect terms:

(2i −1)(δb )2Ti−1, j + (2i +1)(δb )

2Ti+1, j − 4i((δ a )2 + (δb )

2 )Ti, j−1 + 2i(δ a )2Ti, j+1

A. For the center node at r = 0:

1rdTdr

= d2Tdr2 L’Hospital’s rule

! ! ! ! ! ! ! ! !Therefore the governing equation is:

(δb )2T1, j − ((δ a )

2 + (δb )2 )T1, j +

(δ a )2

2T1, j−1 = 0

! ! ! ! (32)

B. Boundary condition at z = 0: 𝛳 = 𝛳0 at z = 0, 0 ≤ z ≤ b.Therefore the governing equation is:

(2i −1)(δb )2Ti−1,1 + (2i −1)(δb )

2Ti+1,1 − 4i((δ a )2 + (δb )

2 )Ti,1 + 2i(δ a )2Ti,2 = −2i(δ a )

2 (T0 −T∞ )

C. Boundary condition at r = a: r = a, 0 ≤ z ≤ b

∂θ∂r

= − hkθ

(33)

Therefore the governing equation is:

4i(δb )2TM−1, j − 4i((δ a )

2 + (δb )2 + 2hδ a (2i +1)

δb2

k)TM , j−1 + 2i(δ a )

2TM , j−1 + 2i(δ a )2TM , j+1

= − 2hδ a (2i +1)δb

2

kT∞

! ! ! ! ! ! ! ! (34)

D. Boundary condition at z = b: r = a, 0 ≤ z ≤ b

(31)

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∂θ∂r

= − hekθ! ! ! ! ! ! ! ! ! (35)

Therefore the governing equation is:

(2i −1)(δb )2Ti−1,N − (2i +1)δb

2Ti+1,N − 4i((δ a )2 + (δb )

2 + 4iheδbδ a2

k)Ti,N + 2i(δ a )

2Ti,N−1

+2i(δ a )2TM ,N = − 4iheδbδ a

kT∞

! ! ! ! ! ! ! (36)

In order to numerically solve the temperature gradient, COMSOL Multophysics was used. The following figures show the grid that was used and the results of various snapshots at specific values of t.

Figure 7 - 3D rendering of generic 18650 cell with mesh overlay.

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Figure 8 - Temperature distribution within 18650 cell at maximum operating temperature.

Figure 9 - Temperature distribution within 18650 cell after sufficient cooling.

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5. Conclusion: Comparison of Analytical and Numerical Solution

The 18650 Li-ion battery cell is used in applications ranging from laptops to electric cars where approximately 7000 are employed in the Tesla Model S. Thermal management of the battery packs is critical for the safety of occupants of the electric vehicles and the longevity of the packs themselves. This paper investigated a single 18650 Li-ion battery cell to determine the temperature distribution assuming maximum operating temperature of the battery cell and forced air convection cooling along the sides. Both analytical and numerical methods were used to derive a solution. The analytical method was solved by separation of variables method using boundary conditions. The numerical solution was derived by discretization of the energy differential equation and Fourier equation and setting up a system of equations on each node of the discretized domain.

While the analytical approach provided a relatively quick method of obtaining a solution, the assumptions used to plot the final temperature gradient were proven incorrect by the numerical method. Assuming the radial thermal conductivity (kr) for the entire cylindrical cell essentially ignored heat transferring from the top of the cell which was quickly shown using the numerical modeling software COMSOL. These plots indicated a much lower temperature gradient, or more uniform temperature within the battery cell. This is verified by the very low Biot number of 0.032 that essentially tells us that this will be the case. This leads to the final conclusion that modeling one individual cell tells us very little about the thermal management requirements in a large battery pack and that modeling of the entire pack and configuration is necessary.

Appendix 1 - MATLAB code and COMSOL settings.

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References:

[1] Plug’n Drive Ontario (2013, October 18). Electric Cars on Canadian Roads: 4,491 [Online]. Available: http://www.plugndriveontario.ca

[2] K. Somasundaram, E. Birgersson, A.S. Mujumdar, “Thermal-electrochemical model for passive thermal management of a spiral-wound lithium-ion battery,” Journal of Power Sources, vol 203 (2012), pp. 84-96, Dec, 2011.

[3] Business Summary - Tesla Motor Inc (2013, October 18). Mergent Online [Online]. Available: http://www.mergentonline.com.proxy.library.carleton.ca/companydetail.php?compnumber=129614&pagetype=synopsis

[4] J. Lingeman. (2013, May 13). Tesla Model S outsells luxury competition (Autoweek) [Online]. Available: http://www.autoweek.com/article/20130513/carnews/130519937

[5] Tesla Motors. (2013, October 18). Increasing Energy Density Means Increasing Range [Online]. Available: http://www.teslamotors.com/roadster/technology/battery

[6] D. Lavrinc. (2013, April 16). Breaking Down Tesla’s Captivating Model S (Wired) [Online]. Available: http://www.wired.com/magazine/2013/04/electriccars/

[7] K. Bulls (2013, August 7). How Tesla Is Driving Electric Car Innovation (MIT Technology Review) [Online]. Available: http://www.technologyreview.com/news/516961/how-tesla-is-driving-electric-car-innovation/

[8] NCR18650A 3.1Ah Lithium Ion Specification Sheet, Panasonic Batteries, Panasonic Corporation of America (2011) [Online]. Available: http://www.panasonic.com/industrial/includes/pdf/ACA4000CE254-NCR18650A.pdf

[9] D. Linden, Handbook of Batteries, second ed., McGraw-Hill, New York, 1994.

[10] R. Mahamud, & C. Park, “Spatial-resolution, lumped capacitance thermal model for cylindrical Li-ion batteries under high Biot number conditions,” Applied Mathematical Modelling, vol. 37, pp. 2787-2801, Jun. 2012.

[11] S.Chen, Y. Wang, & C. Wan, “Thermal Analysis of Spirally Wound Lithium Batteries,” Journal of The Electrochemical Society, vol 153 (4), pp. A637-A648 (2006).

[12] D.W. Hahn, & M.N. Özişik, Heat Conduction, third ed.,Wiley, New Jersey, 2012.

[13] S. Song (2013, September 25). Conductive and Radiative Heat Transfer Project. Carleton University.

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Appendix

Analytical Solution Code (MATLAB)

function[r_A,z_A,Tmtx_A]=analytical(r_size,z_size)%define datar_size=50;z_size=50;a=0.0093;%mz_length=0.0652;h=90;%w/m^2Kk=26;%W/mkrootguess=1:4:60;Theta_o=60;%k T_inf=293.15;%k %create a meshTheta=zeros(r_size,z_size);z=linspace(0,z_length,z_size);r=linspace(0,a,r_size); %define biot numberBi=h*a/kBi_e=Bi;%define"x-axis"x=linspace(0,50,20000);yfun=(Bi*besselj(0,x)./besselj(1,x))-x; %plot the functionplot(x,yfun);axis([0 10 -10 10]);grid on;title('finding the root');xlabel('lambda');ylabel('error'); j=1; %find all the roots including the asymptones for i=1:20000if(i<19999) if(yfun(i)*yfun(i+1)<0) r_prime(j)=(x(i)+x(i+1))/2; j=j+1; end endend %separate out all asymptotes from rootsi_real=1:2:length(r_prime);lamda=r_prime(i_real)

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%plot bessel functionfor i=1:z_size for j=1:r_size %initiate theta Theta(j,i)=0; %sum up the i'th element for n=1:10gamma=((lamda(n)*z_length/a)*sinh((lamda(n)*z_length/a))+Bi_e*cosh((lamda(n)*z_length/a)))/((lamda(n)*z_length/a)*cosh((lamda(n)*z_length/a))+Bi_e*sinh((lamda(n)*z_length/a))); term_n=(2*lamda(n)*Theta_o*besselj(1,lamda(n))*besselj(0,lamda(n)*r(j)/a)/((lamda(n)^2+Bi^2)*(besselj(0,lamda(n)))^2))*(cosh(lamda(n)*z(i)/a)-gamma*sinh(lamda(n)*z(i)/a)); Theta(j,i)=Theta(j,i)+term_n; end endend %solve for temperatureTmtx_A=Theta+T_inf;surf(r,z,Tmtx_A')xlabel('radius(m)')ylabel('length(m)')zlabel('temperature(K)')title('Temperature vs Position of a 18650 Li-ion battery cell')

Numerical Analysis Settings with COMSOL:

1. Model 1 (mod1)1.1. DEFINITIONS1.1.1. Probes1.1.1.1. Maximum T

Probe type Domain probe

SELECTION

Geometric entity level Domain

Selection Domain 1

SETTINGS

Name Value

Type maximum

Expression T

16

Table and plot unit K

Description Temperature

Output table Probe Table 1

Plot window Probe Plot 1

1.1.1.2. Minimum T

Probe type Domain probe

SELECTION

Geometric entity level Domain

Selection Domain 1

SETTINGS

Name Value

Type minimum

Expression T

Table and plot unit K

Description Temperature

Output table Probe Table 1

Plot window Probe Plot 1

1.1.2. Coordinate Systems1.1.2.1. Boundary System 1

Coordinate system type Boundary system

Identifier sys1

SETTINGS

Name Value

Coordinate names {t1, t2, n}

Create first tangent direction from

Global Cartesian

1.2. GEOMETRY 1

17

Geometry

UNITS

Length unit m

Angular unit deg

GEOMETRY STATISTICS

Property Value

Space dimension 3

Number of domains 1

Number of boundaries 6

Number of edges 12

Number of vertices 8

1.2.1. Cylinder 1 (cyl1)SETTINGS

Name Value

Position {0, 0, 0}

Axis {0, 0, 1}

Axis {0, 0, 1}

Radius 0.018

Height 0.065

1.3. MATERIALS

18

1.3.1. Material 1

Material 1

SELECTION

Geometric entity level Domain

Selection Domain 1

MATERIAL PARAMETERS

Name Value Unit

Density 2300 kg/m^3

Heat capacity at constant pressure

900 J/(kg*K)

BASIC SETTINGS

Description Value

Density 2300

Heat capacity at constant pressure

900

1.4. HEAT TRANSFER IN SOLIDS (HT)

Heat Transfer in Solids

SELECTION

Geometric entity level

Domain

Selection Domain 1

19

EQUATIONS

SETTINGS

Description Value

Show equation assuming

std1/time

USED PRODUCTS

COMSOL Multiphysics

Heat Transfer Module1.4.1. Heat Transfer in Solids 1

Heat Transfer in Solids 1

SELECTION

Geometric entity level Domain

Selection Domain 1

1.4.1.1. Equations1.4.1.2. Settings

SETTINGS

Description Value

20

Thermal conductivity User defined

Thermal conductivity {{1.04, 0, 0}, {0, 1.04, 0}, {0, 0, 26.1}}

1.4.1.3. Used Products

COMSOL Multiphysics

PROPERTIES FROM MATERIAL

Property Material Property group

Density Material 1 Basic

Heat capacity at constant pressure

Material 1 Basic

1.4.1.4. Variables1.4.1.5. Shape Functions

Name Shape function Unit Description Shape frame Selection

T Lagrange K Temperature Material Domain 1

1.4.1.6. Weak Expressions

Weak expression Integration frame

Selection

-(ht.k_effxx*Tx+ht.k_effxy*Ty+ht.k_effxz*Tz)*test(Tx)-(ht.k_effyx*Tx+ht.k_effyy*Ty+ht.k_effyz*Tz)*test(Ty)-(ht.k_effzx*Tx+ht.k_effzy*Ty+ht.k_effzz*Tz)*test(Tz)

Material Domain 1

-ht.C_eff*Tt*test(T) Material Domain 1

1.4.2. Thermal Insulation 1

21

Thermal Insulation 1

SELECTION

Geometric entity level Boundary

Selection Boundary 3

1.4.2.1. Equations1.4.2.2. Used Products

COMSOL Multiphysics

1.4.3. Initial Values 1

Initial Values 1

SELECTION

Geometric entity level Domain

Selection Domain 1

1.4.3.1. SettingsSETTINGS

Description Value

Temperature 333.15[K]

1.4.3.2. Used Products

COMSOL Multiphysics

1.4.4. Convective Cooling 1

22

Convective Cooling 1

SELECTION

Geometric entity level

Boundary

Selection Boundary 41.4.4.1. Equations1.4.4.2. Settings

SETTINGS

Description Value

Heat transfer coefficient

5

1.4.4.3. Variables

Name Expression Unit Description Selection

ht.ccflux -ht.h_cc1*(ht.Text_cc1-T)

W/m^2 Convective heat flux

Boundary 4

ht.Text_cc1 293.15[K] K External temperature

Boundary 4

ht.h_cc1 5 W/(m^2*K)

Heat transfer coefficient

Boundary 4

1.4.4.4. Weak Expressions

Weak expression Integration frame Selection

ht.h_cc1*(ht.Text_cc1-T)*test(T)

Material Boundary 4

1.4.5. Convective Cooling 2

23

Convective Cooling 2

SELECTION

Geometric entity level Boundary

Selection Boundaries 1-2, 5-6

1.4.5.1. Equations1.4.5.2. Settings

SETTINGS

Description Value

Heat transfer coefficient 20

1.4.5.3. Variables

Name Expression Unit Description Selection

ht.ccflux -ht.h_cc2*(ht.Text_cc2-T)

W/m^2 Convective heat flux

Boundaries 1-2, 5-6

ht.Text_cc2

293.15[K] K External temperature

Boundaries 1-2, 5-6

ht.h_cc2 20 W/(m^2*K)

Heat transfer coefficient

Boundaries 1-2, 5-6

1.4.5.4. Weak Expressions

Weak expression Integration frame Selection

24

ht.h_cc2*(ht.Text_cc2-T)*test(T)

Material Boundaries 1-2, 5-6

1.5. MESH 1MESH STATISTICS

Property Value

Minimum element quality

0.2259

Average element quality 0.7722

Tetrahedral elements 22938

Triangular elements 1804

Edge elements 136

Vertex elements 8

Mesh 11.5.1. Size (size)

SETTINGS

Name Value

Maximum element size 0.00358

Minimum element size 2.60E-04

25

Resolution of curvature 0.4

Resolution of narrow regions

0.7

Maximum element growth rate

1.4

Predefined size Finer

2. Study 12.1. TIME DEPENDENTTimes: range(0,10,3600)

MESH SELECTION

Geometry Mesh

Geometry 1 (geom1) mesh1

PHYSICS SELECTION

Physics interface Discretization

Heat Transfer in Solids (ht)

physics

2.2. SOLVER CONFIGURATIONS2.2.1. Solver 12.2.1.1. Compile Equations: Time Dependent (st1)

SETTINGS

Name Value

Use study Study 1

Use study step Time Dependent

2.2.1.2. Dependent Variables 1 (v1)SETTINGS

Name Value

Defined by study step Time Dependent

Solution Zero

Solution Zero

2.2.1.2.1. Mod1.T (mod1_T)SETTINGS

Name Value

Field components mod1.T

2.2.1.3. Time-Dependent Solver 1 (t1)

26