Mathematical modeling of transport phenomena in lithium ion batteries

MATHEMATICAL MODELING OF TRANSPORT
PHENOMENA IN LITHIUM-ION BATTERIES
TONG WEI
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2014

Acknowledgements
This work would not have been possible to be achieved without the help of many
people. I would like to express my deep est gratitude to my supervisors, Professor Arun S.
Mujumdar, Assistant Professor Erik Birgersson and Associate Professor Christopher Yap
for their excellent and tireless guidance for the work done in this thesis. It has been a
great honor and pleasure to work with them during the past four years. Prof. Mujumdar
paved the way for me as a PhD student to pursue my research in National University of
Singapore (NUS). He is a nice and thoughtful mentor and always provides rich knowledge
and experience. Prof. Erik Birgersson opened the door for me to the world of an exciting
research area - mathematical modeling of electrochemical energy storage system. Prof.
Erik Birgersson always helped me to build my con…dence. His dedication and enthusiasm
for research and work impress me very much and set a positive example for me during my
four-year study. Prof. Yap, although only having a short time working with him, helped
me a lot. The discussion with him, especially in how to strategically organize research
papers and the thesis, has been crucial to the completion of my research work.
In addition, I would like to thank all my colleagues and friends for their assistance which
has facilitated the completion of this work. I would thank our group members: Dr Agus
Pulung Sasmito, Dr Jundika Candra Kurnia for the discussion with them. Specially, I
would like to thank Dr Karthik Somasundaram for his kind help in developing the math-
ematical modeling.
Furthermore, I extend my gratitude to the China Scholarship Council and National Uni-
versity of Singapore for their …nancial support.

Last but not least, I dedicate this thesis to my parents for their endless love, support and
i
encouragement during the past 29 years of my life. Without them, I de…nitely could not
have reached where I am today.
ii
Contents
Acknowledgements i
Summary ix
Preface xii
List of Tables xiv
List of Figures xv
List of Symbols xx
1 Introduction 1
1.1 Overview of lithium-ion batteries . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Structure and operation principles of a lithium-ion battery cell . . . 4
1.1.2 Types and applications . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Heat generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Capacity fade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Mathematical modeling of lithium-ion batteries . . . . . . . . . . . . . . . 10
v
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Literature review 15
2.1 Theories of mathematical modeling . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Porous electrode theory . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Concentrated solution theory . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 Electrode kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4 Solid phase di¤usion . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Review of mathematical models . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Coupled electrochemical-thermal models . . . . . . . . . . . . . . . 19
2.2.2 Capacity fade models . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Models for thermal management systems . . . . . . . . . . . . . . . 24
3 Mathematical formulation 27
3.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Numerical methodology 41
4.1 COMSOL Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 A Monte Carlo simulation of a lithium-ion battery model - Correlating
the uncertainties of system parameters with safety issues 46
vi
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Mathematical formulation of lithium-ion battery model . . . . . . . . . . . 49
5.3 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5.1 Sample selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5.3 Final cell temperature distribution . . . . . . . . . . . . . . . . . . 65
6 Analysis of capacity fade distribution of a cylindrical lithium-ion battery 69
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4.2 Global behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4.3 Distribution of capacity fade . . . . . . . . . . . . . . . . . . . . . . 81
7 Numerical investigation of air cooling for a lithium-ion battery module 89

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2.2 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . 95
7.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
vii
7.4.1 Discharge curves and current density . . . . . . . . . . . . . . . . . 99
7.4.2 E¤ect of air inlet velocity . . . . . . . . . . . . . . . . . . . . . . . 101
7.4.3 E¤ect of cell arrangement . . . . . . . . . . . . . . . . . . . . . . . 104
7.4.4 E¤ect of cell distance . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4.5 Reversal ‡ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Numerical investigation of water cooling for a lithium-ion bipolar bat-
tery pack 111
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.4.1 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.4.2 Thermal management with water cooling . . . . . . . . . . . . . . . 126
9 Concluding remarks 136
9.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.2 Contributions of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.3 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . 141
viii
Summary
Worldwide energy shortage and environment problems have necessitated more e¢ -
cient, reliable and sustainable techniques for energy transfer and storage. Electricity,
which could produce electrical energy, must be reliably and continuously available for
many applications. Therefore, electricity storage devices are critical for the e¤ective uti-

lization of these energy sources.
The lithium-ion battery is an electrochemical energy storage system that has attracted
increasing attention in recent years because of many advantages over competing technolo-
gies. These include high operating voltage, high energy density, no memory e¤ect and
low self-discharge rate. However, the performance of lithium-ion batteries is closely as-
sociated with thermal and degradation e¤ects. The identi…cation and quanti…cation of
the relationship of battery system properties and operation conditions with the thermal
issues as well as how does the degradation develop inside the battery cell are therefore
critical for the e¤ective and safe utilization of lithium-ion batteries. Furthermore, better
understanding of the parameters and mechanisms involved will enable the improvement
in design of battery thermal management systems.
In tandem with experimental investigations of lithium-ion battery systems, compu-
tational study has become an e¤ective tool for identifying the salient features that can
be found in lithium-ion battery systems. Mathematical modelling not only captures the
transport phenomena occurring inside the battery cells which are generally di¢ cult to
quantify experimentally, but also saves time and cost in experimental setup. Generally,
the transport phenomena occurring inside lithium-ion batteries are modelled. Transient
conservation of species, charge and energy in both solid and liquid phases is based on the
porous electrode theory. In this thesis, the following work has been undertaken using the
lithium-ion battery model.
Firstly, safety issues arising from a lithium-ion battery during operation can be at-
tributed to the variation of its temperature which is, in turn, associated with the uncer-
tainties in the parameters such as system properties and operating conditions. Hence, a
Monte Carlo simulation (MCS) of a lithium-ion battery model is conducted to capture the
probabilistic nature of uncertainties in the parameters and their relative importance to the
temperature of a lithium-ion battery cell. Sensitivity analysis is statistically performed
and the varied parameters are ranked according to their contributions to the variation of
the battery temperature.
Besides studying thermal e¤ects, a simulation is conducted that aims to determine if
non-uniform distributions of capacity fade will develop during the cycling of a cylindrical

lithium-ion battery. It is observed that locally non-uniform distributions of capacity fade
will develop across the surface of a single electrode during cycling while the average ca-
pacity fade among electrodes of di¤erent wounds is uniform.
As part of the applied research, the lithium-ion battery model is used to evaluate
thermal management systems for lithium-ion batteries at a module or pack level. Di¤er-
ent active thermal management systems-forced air or liquid cooling are evaluated for two
designs (cylindrical batteries and cells with bipolar con…gurations) of lithium-ion batter-
x
ies. Parametric studies have been conducted to study the e¤ect of various factors to the
performance of the thermal management system.
xi
Preface
This thesis presents the study on the modeling of transport phenomena in lithium-ion
battery systems. The following publications are based on research carried out for this
doctoral thesis
Journals:
1. Richard Hong Peng Liang, Tangsheng Zou, Karthik Somasundaram, Wei Tong and
Erik Birgersson, Mathematical Modeling and Reliability Analysis of a 3D Li-ion
Battery, Journal of Electrochemical Science and Engineering, 4(1), 1-17, 2014.
2. Wei Tong, Erik Birgersson, Arun S. Mujumdar and Christopher Yap. Correlating
uncertainties of a lithium-ion battery - A Monte Carlo simulation. (Accepted by
International Journal of Energy Research).
3. Wei Tong, Wei Qiang Koh, Erik Birgersson, Arun S. Mujumdar and Christopher
Yap. Numerical Investigation of Water Cooling for a Lithium-ion Bipolar Battery
Pack. (Resubmitted to International Journal of Thermal Sciences).
4. Wei Tong, Erik Birgersson, Arun S. Mujumdar and Christopher Yap. Analysis
of capacity fade distribution of a cylindrical lithium-ion battery. (Manuscript in
preparation).
5. Wei Tong, Erik Birgersson, Arun S. Mujumdar and Christopher Yap. Compu-
tational Study of Air Cooling for a Lithium-ion Battery Module. (Manuscript in

xii
preparation).
Conferences
1. 1) Tong Wei, Erik Birgersson, Arun S. Mujumdar and Christopher Yap. Numerical
Study of Passive Thermal Management of a Cylindrical Lithium-ion battery, The
3rd International Conference on Informatics, Environment, Energy and Applications
(IEEE 2014), Shanghai, China, March 27-28, 2014, DOI: 10.7763/IPCBEE. 2014.
V66. 11.
xiii
List of Tables
5.1 List of varied parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Physical and geometry parameters. . . . . . . . . . . . . . . . . . . . . . . 53
5.3 p
KS
-value of K-S test on samples distribution. . . . . . . . . . . . . . . . . 59
6.1 Physical and geometry parameters. . . . . . . . . . . . . . . . . . . . . . . 78
6.2 Values of parameters used for the capcity fade model. . . . . . . . . . . . . 79
7.1 Geometry and air properties. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Standard deviations of temperature of cells at di¤erent positions at the end
of discharge (base-case conditions). . . . . . . . . . . . . . . . . . . . . . . 103
7.3 Standard deviations of temperature of cells at di¤erent positions at the end
of discharge: staggered arrangement with 9 mm cell distance. . . . . . . . . 105
8.1 Coolant and insulator properties. . . . . . . . . . . . . . . . . . . . . . . . 118
xiv
List of Figures
1.1 Schematic of (a) a simpli…ed Ragone plot showing energy density vs. power
density for various energy storage devices; (b) a comparision of energy
storage capability of common rechargeable battery systems [2, 3]. . . . . . 3
1.2 Schematic of a lithium-ion battery operation principle. . . . . . . . . . . . 5
1.3 Lithium-ion batteries of various shapes and components: a. cylindrical; b.

coin; c. prismatic; d. pouch; …gure cited from Ref. [2]. . . . . . . . . . . . 7
3.1 Schematic of (a) a single lithium-ion battery cell with various functional
layers; (b) agglomerate structure of lithium in active material in the elec-
trodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 Applied discharge current with a smoothed Heaviside function (time scale
equals to 0.01s) - N: the time current starts to ramp up from zero, H: the
time current reaches its speci…ed value. . . . . . . . . . . . . . . . . . . . . 44
5.1 Schematic of the computational domain of the lithium-ion battery. . . . . . 51
5.2 Flow chart of Monte Carlo simulation. . . . . . . . . . . . . . . . . . . . . 54
5.3 Rankings of varied parameters at (a) 1 C-rate; (b) 5 C-rate using the
coe¢ cient of variance method. . . . . . . . . . . . . . . . . . . . . . . . . . 61
xv
5.4 Rankings of varied parameters at (a) 1 C-rate; (b) 5 C-rate using the sigma-
normalized derivative method. . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Scatter plots of battery temperaturevs. top three ranked parameters in the
sensitivity analysis at 5 C-rate: (a, c, e) varied each parameter individually;
(b, d, f) varied all the parameters simultaneously. . . . . . . . . . . . . . . 64
5.6 Sample distribution of battery temperature for the scenario of all the pa-
rameters varied simultaneously at (a) 1 C-rate; (b) 5 C-rate. Solid curve:
normal density function …tted from sample mean and standard deviation. . 67
6.1 Schematic of (a) a cylindrical 18650 lithium-ion battery; (b) an axisymmetric-
section of the spiral-wound battery with the structure of various functional
layers; (c) a basic unit of the jelly roll comprising a single cell; (d) ag-
glomerate structure of lithium in active material in the electrodes. Roman
numerals: indicating the interfaces and boundaries of these layers. . . . . . 72
6.2 Validation of capacity fade model with experimental data [52]. . . . . . . . 80
6.3 Cell potential (solid line) / current density (dotted line) vs. time during
the 50th cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 History of side reaction current density for the negative electrode of the
outermost wound during the 50th cycle: current collector side (solid line);

separator side (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.5 Surface distributions of side reaction overpotential and side reaction current
density for the negative electrode of the outermost wound during the 50th
cycle: (a, b) 5000 s (halfway of constant current charge); (c, d) 7000 s
(halfway of constant voltage charge). . . . . . . . . . . . . . . . . . . . . . 84
xvi
6.6 Comparison of side reaction current density during the 50th cycle: outer-
most electrode (solid line); innermost electrode (dotted line). . . . . . . . . 86
6.7 Comparison of the normalized capacity loss during the 50th cycle: outer-
most electrode (solid line); innermost electrode (dotted line). . . . . . . . . 86
6.8 Comparison of (a) cumulative resistance increase and (b) cumulative nor-
malized capacity loss with cycles of the outermost (solid line) and innermost
(dotted line) electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.1 Schematic of a lithium-ion battery module. . . . . . . . . . . . . . . . . . . 91
7.2 Schematic of the computational domain of two-dimensional battery module
of di¤erent con…gurations: (a) in-line; (b) staggered. . . . . . . . . . . . . . 93
7.3 Coupling of the one-dimensional and the two-dimensional models. . . . . . 95
7.4 Cell voltage curves/current densities vs. time for the cells at P1 (solid
lines) and P11 (dotted lines). . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.5 Maximum temperature variation vs. air inlet velo city of the battery module
operated under base-case conditions. . . . . . . . . . . . . . . . . . . . . . 102
7.6 Maximum temperature variation of the two arrangements of the battery
module operated under the base-case conditions. . . . . . . . . . . . . . . . 104
7.7 E¤ects of cell distance on the maximum temperature variation for the bat-
tery with staggered con…guration. . . . . . . . . . . . . . . . . . . . . . . . 106
7.8 Average temperature history of cells at di¤erent positions under di¤erent
frequencies of reversal ‡ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.9 Instantaneous average temperature of cells at di¤erent positions. . . . . . . 109
xvii
8.1 Schematic of the lithium-ion battery (a) a pack; (b) cross-section of a repre-

sentative bipolar pack (named repeating-module) with the roman numerals
indicating the interfaces and boundaries of stacks and coolant plates; (c)
cross-section of a stack with the roman numerals indicating the interfaces
and boundaries of cells and various functional layers; (d) cross-section of a
single cell; (e) agglomerate structure of active material in the electrodes. . 114
8.2 Voltage vs. capacity of the battery pack during discharge at various C-
rates under two limiting conditions: cooling (solid lines); without cooling
(dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.3 Concentration pro…le of Li
+
in the electrolyte at the cut-o¤ voltage at a
5 C-rate under two limiting conditions (y = 3  10
2
m): cooling (solid
line) and without cooling (dashed line). . . . . . . . . . . . . . . . . . . . . 124
8.4 Temperature window of the battery pack: upper limit (dashed line); lower
limit (solid line); operating line of temperature (dotted line). . . . . . . . . 125
8.5 Average temperature di¤erence vs. number of stacks at the cut-o¤ voltage
at various C-rates (1 C: H; 2 C: ; 5 C: N) with u
c
= 2.5  10
3
m s
1
and w
c
= 5  10
2
cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.6 Local temperature distribution at the cut-o¤ voltage at 5 C-rate for the

battery pack of (a) m
s
= 5, u
c
= 2.5  10
3
m s
1
, w
c
= 5  10
2
cm;
(b) m
s
= 15, u
c
= 2.5  10
3
m s
1
, w
c
= 5  10
2
cm; (c) m
s
= 15, u
c
= 2.5  10

2
m s
1
, w
c
= 5  10
2
cm; (d) m
s
= 15, u
c
= 2.5  10
3
m
s
1
, w
c
= 2  10
1
cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
xviii
8.7 Average temperature di¤erence vs. coolant velocity at the cut-o¤ voltage
at various C-rates (1 C: H; 2 C: ; 5 C: N) with m
s
= 15 and w
c
= 5 
10
2

cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.8 Average temperature di¤erence vs. thickness of coolant plate at the cut-o¤
voltage at various C-rates (1 C: H; 2 C: ; 5 C: N) with m
s
= 15 and u
c
=
2.5  10
3
m s
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xix
List of Symbols
A
s
speci…c interfacial area per unit volume, m
1
c
l
electrolyte concentration, mol m
3
c
0
l
initial electrolyte concentration, mol m
3
c
avg
s

average concentration of lithium in the active material, mol m
3
c
max
s
maximum concentration of lithium in the active material, mol m
3
c
surf
s
surface concentration of lithium in the active material, mol m
3
c
0
s
initial concentration of lithium in the active material, mol m
3
c
v;i
coe¢ cient of variation
C
p
speci…c heat capacity, J kg
1
K
1
D
l
di¤usion coe¢ cient of electrolyte, m
2

s
1
D
s
di¤usion coe¢ cient of lithium in the active material, m
2
s
1
E
a
activation energy for a variable, J mol
1
Er
ab
absolute error
Er
re
relative error
F Faraday’s constant, 96,487 C mol
1
i
app
applied current density, A m
2
i
fara
faradaic transfer current density, A m
2
i
0;fara

exchange current density, A m
2
xx
i
l
liquid phase current density, A m
2
i
s
solid phase current density, A m
2
j sequence number of simulation runs
J local charge transfer current per unit volume, A m
3
k e¤ective thermal conductivity, W m
1
K
1
k
0
reaction rate constant, mol
2:5
m
0:5
s
1
l
s
di¤usion length, m
n total number of varied parameters

N
critical
critical sample size
N
l
species ‡ux, mol m
2
s
1
p probability
Q volumetric heat generation, W m
3
q conductive heat ‡ux, W m
2
R gas constant, J mol
1
K
1
R
s
radius of active material in the electrodes, m
s
x
i
standard deviation of sample (input)
s
y
standard deviation of sample (output)
S


X
i
sigma-normalized derivative
t time, s
t
0
+
transference number of cation
T temp erature, K
T
ref
reference temperature, K
U
ref; i
open circuit potential of the electrode i, V
w
i
thickness of the functional layer i, m
x
i
mean of sample (input)
X
i
population (input)
xxi
y mean of sample (output)
Y population (output)
xxii
Greek letters
 signi…cance level


a
anodic transfer coe¢ cient

c
cathodic transfer coe¢ cient
 Bruggemann constant
"
f
volume fraction of conductive …ller additive
"
l
volume fraction of electrolyte
"
p
volume fraction of polymer phase
 overpotential, V

ne
state of charge of negative electrode

pe
state of charge of positive electrode
 e¤ective density, kg m
3

l
ionic conductivity of electrolyte, S m
1


s
electronic conductivity of solid matrix, S m
1

X
i
standard deviation of population (input)

Y
standard deviation of population (output)

l
liquid phase potential, V

s
solid phase potential, V

X
i
mean of population (input)

Y
mean of population (output)
xxiii
Subscripts
ab absolute value
critical critical value
fara faradaic process related value
l liquid phase
ncc negative current collector

ne negative electrode
pcc positive current collector
pe positive electrode
re relative value
ref reference value
s solid phase
sp separator
xxiv
Superscripts
avg average value
eff e¤ective value
f …nal state
max maximum value
surf surface
0 initial value
xxv