.

1. Conductive and Radiative Heat Transfer
MECH 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)
1

2. 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].
2

3. 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]
3

4. 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)
4

5. 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,
1
r
∂
∂r
r
∂θ
∂r
⎛
⎝⎜
⎞
⎠⎟ +
∂
∂z
∂θ
∂z
⎛
⎝⎜
⎞
⎠⎟ = 0
(1)
Using the separation of variable method,
θ = U(r,z) (2)
Substituting into the conduction equation
1
r
∂
∂r
r
∂U(r,z)
∂r
⎛
⎝⎜
⎞
⎠⎟ +
∂
∂z
∂U(r,z)
∂z
⎛
⎝⎜
⎞
⎠⎟ = 0
(3)
1
r
∂U(r,z)
∂r
+
∂2
U(r,z)
∂r 2
+
∂2
U(r,z)
∂z2
= 0
(4)
1
r
∂U(r,z)
∂r
+
∂2
U(r,z)
∂r2
= −
∂2
U(r,z)
∂z2
= −λ2
(5)
5

6. 1
r
∂U(r,z)
∂r
+
∂2
U(r,z)
∂r2
= −λ2
(6)
Therefore,
−
∂2
U(r,z)
∂z2
= −λ2
(7)
1
r
∂(R(r)Z(z))
∂r
⎛
⎝⎜
⎞
⎠⎟ +
∂2
(R(r)Z(z))
∂r2
= −λ2
(8)
On dividing R(r)Z(z) from equation (7) and (8) we could get,
1
r
∂(R(r))
∂r
+
∂2
(R(r))
∂r 2
+ R(r)λ2
= 0
(9)
∂2
(Z(z))
∂z2
= −λ2
Z(z)
(10)
The general solution for equation (9) is,
R(r) = C1
J0
(λn
r) +C2
Y0
(λn
r)
(11)
C2 = 0 since Y0 ∞, as r ∞ . Thus the equation becomes,
R(r) = C1
J0
(λn
r)
(12)
While the solution for equation (10) is,
Z(z) = C3
cosh(λn
z) +C4
sinh(λn
z)
(13)
And from equation (12) and (13) the combined solution of the conduction equation
would be,
θ(r,z) = C1
J0
(λn
z)(C3
cosh(λn
z) +C4
sinh(λn
z))
(14)
6

7. Now applying the given boundary conditions to the above solution, starting with the 1st
boundary condition for r = a.
∂θ
∂r
= −C1
J1
(λn
a)Z(z) = −
h
k
C1
J0
(λn
a)Z(z)
(15)
On solving we get,
(λn
a) =
ha
k
J0
(λn
a)
J1
(λn
a)
(16)
But we have the Biot number, Bi = (ha/k). Thus we could write the above equation as,
(λn
a) = Bi
J0
(λn
a)
J1
(λn
a)
(17)
Now applying the second boundary condition at z = b,
C1
J0
(λn
r)(C3
sinh(λn
b) +C4
cosh(λn
b)) =
he
k
(−C1
J0
(λn
r)(C3
cosh(λn
b) +C4
sinh(λn
b)))
(18)
C4
= −C3
he
k
cosh(λn
b) + λn
sinh(λn
b)
λn
cosh(λn
b) +
he
k
sinh(λn
b)
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
(19)
On applying the third boundary condition at z = 0, 𝛳 = 𝛳0. Hence we could write the
solution as,
θ0
= AJ0
(λn
r)
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,
7

8. θ0
= AJ0
(λn
)(cosh(λn
z) + Bλn
sinh(λn
z))
n=1
∞
∑
(21)
B = −
he
k
cosh(λn
b) + λn
sinh(λn
b)
λn
cosh(λn
b) +
he
k
sinh(λn
b)
(22)
Therefore we have the final solution as,
θ(r,z) = AJ0
cosh(λn
r) + Bsinh(λn
r)
(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.
8

9. 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−1
2δ
+O(δ 2
)
(24)
f ''i =
fi+1 − fi−1 − 2 fi
δ 2
+O(δ 2
)
(25)
The governing equation is given as:
∂2
T
∂2
r
+
1
r
∂T
∂r
+
∂2
T
∂2
z
= 0
(26)
where 0 ≤ r ≤ a, and 0 ≤ z ≤ b.
Assume,
δa =
a / 2
M (27)
δb =
a / 2
M (28)
where a/2 is the radius, b is the length of the cylinder and M is the number of finite
parts.
9

10. Therefore,
1
δa
2
(Ti−1, j − 2Ti+1, j +Ti+1, j )+
1
iδa
1
2δa
(Ti−1, j −Ti+1, j )+
1
δb
2
(Ti−1, j − 2Ti+1, j +Ti+1, j ) = 0
(29)
Expand and collect terms:
(2i −1)(δb )2
Ti−1, j + (2i +1)(δb )2
Ti+1, j − 4i((δa )2
+ (δb )2
)Ti, j−1 + 2i(δa )2
Ti, j+1
A. For the center node at r = 0:
1
r
dT
dr
=
d2
T
dr2
L’Hospital’s rule
! ! ! ! ! ! ! ! !
Therefore the governing equation is:
(δb )2
T1, j − ((δa )2
+ (δb )2
)T1, j +
(δa )2
2
T1, 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 )2
Ti−1,1 + (2i −1)(δb )2
Ti+1,1 − 4i((δa )2
+ (δb )2
)Ti,1 + 2i(δa )2
Ti,2 = −2i(δa )2
(T0 − T∞ )
C. Boundary condition at r = a: r = a, 0 ≤ z ≤ b
∂θ
∂r
= −
h
k
θ
(33)
Therefore the governing equation is:
4i(δb )2
TM −1, j − 4i((δa )2
+ (δb )2
+ 2hδa (2i +1)
δb
2
k
)TM , j−1 + 2i(δa )2
TM , j−1 + 2i(δa )2
TM , j+1
= −
2hδa (2i +1)δb
2
k
T∞
! ! ! ! ! ! ! ! (34)
D. Boundary condition at z = b: r = a, 0 ≤ z ≤ b
(31)
10

11. ∂θ
∂r
= −
he
k
θ
! ! ! ! ! ! ! ! ! (35)
Therefore the governing equation is:
(2i −1)(δb )2
Ti−1,N − (2i +1)δb
2
Ti+1,N − 4i((δa )2
+ (δb )2
+ 4iheδb
δa
2
k
)Ti,N + 2i(δa )2
Ti,N−1
+2i(δa )2
TM ,N = −
4iheδbδa
k
T∞
! ! ! ! ! ! ! (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.
11

12. Figure 8 - Temperature distribution within 18650 cell at maximum operating
temperature.
Figure 9 - Temperature distribution within 18650 cell after sufficient cooling.
12

13. 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.
13

14. 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.
14

15. Appendix
Analytical Solution Code (MATLAB)
function[r_A,z_A,Tmtx_A]=analytical(r_size,z_size)
%define data
r_size=50;
z_size=50;
a=0.0093;%m
z_length=0.0652;
h=90;%w/m^2K
k=26;%W/mk
rootguess=1:4:60;
Theta_o=60;%k
T_inf=293.15;%k
%create a mesh
Theta=zeros(r_size,z_size);
z=linspace(0,z_length,z_size);
r=linspace(0,a,r_size);
%define biot number
Bi=h*a/k
Bi_e=Bi;
%define"x-axis"
x=linspace(0,50,20000);
yfun=(Bi*besselj(0,x)./besselj(1,x))-x;
%plot the function
plot(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=1:20000
if(i<19999)
if(yfun(i)*yfun(i+1)<0)
r_prime(j)=(x(i)+x(i+1))/2;
j=j+1;
end
end
end
%separate out all asymptotes from roots
i_real=1:2:length(r_prime);
lamda=r_prime(i_real)
15

16. %plot bessel function
for i=1:z_size
for j=1:r_size
%initiate theta
Theta(j,i)=0;
%sum up the i'th element
for n=1:10
gamma=((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
end
end
%solve for temperature
Tmtx_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. DEFINITIONS
1.1.1. Probes
1.1.1.1. Maximum T
Probe type Domain probe
SELECTION
Geometric entity level Domain
Selection Domain 1
SETTINGS
Name Value
Type maximum
Expression T
16

17. 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 Systems
1.1.2.1. 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

18. 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

19. 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

20. EQUATIONS
SETTINGS
Description Value
Show equation
assuming
std1/time
USED PRODUCTS
COMSOL Multiphysics
Heat Transfer Module
1.4.1. HeatTransfer in Solids 1
Heat Transfer in Solids 1
SELECTION
Geometric entity level Domain
Selection Domain 1
1.4.1.1. Equations
1.4.1.2. Settings
SETTINGS
Description Value
20

21. 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. Variables
1.4.1.5. 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

22. Thermal Insulation 1
SELECTION
Geometric entity level Boundary
Selection Boundary 3
1.4.2.1. Equations
1.4.2.2. Used Products
COMSOL Multiphysics
1.4.3. InitialValues 1
Initial Values 1
SELECTION
Geometric entity level Domain
Selection Domain 1
1.4.3.1. Settings
SETTINGS
Description Value
Temperature 333.15[K]
1.4.3.2. Used Products
COMSOL Multiphysics
1.4.4. Convective Cooling 1
22

23. Convective Cooling 1
SELECTION
Geometric entity
level
Boundary
Selection Boundary 4
1.4.4.1. Equations
1.4.4.2. 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_c
c1-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

24. Convective Cooling 2
SELECTION
Geometric entity level Boundary
Selection Boundaries 1-2, 5-6
1.4.5.1. Equations
1.4.5.2. 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_cc
2
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

25. ht.h_cc2*(ht.Text_cc2-
T)*test(T)
Material Boundaries 1-2, 5-6
1.5. MESH 1
MESH 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 1
1.5.1. Size (size)
SETTINGS
Name Value
Maximum element size 0.00358
Minimum element size 2.60E-04
25

26. Resolution of curvature 0.4
Resolution of narrow
regions
0.7
Maximum element
growth rate
1.4
Predefined size Finer
2. Study 1
2.1. TIME DEPENDENT
Times: 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 CONFIGURATIONS
2.2.1. Solver 1
2.2.1.1. 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)
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