Journal of Modern Physics, 2013, 4, 1-7 Published Online July 2013 (
Thermal Modeling of Cylindrical LiFePO4 Batteries
Mojtaba Shadman Rad1,2, Dmitri L. Danilov1, Morteza Baghalha2, Mohammad Kazemeini2,
Peter H. L. Notten1,3*
1Department of Chemistry and Chemical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
2Chemical & Petroleum Engineering Department, Sharif University of Technology, Tehran, Iran
3Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
Email: *
Received May 4, 2013; revised June 6, 2013; accepted July 3, 2013
Copyright © 2013 Mojtaba Shadman Rad et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Thermal management of Li-ion batteries is important because of the high energy content and the risk of rapid tempera-
ture development in the high current range. Reliable and safe operation of these batteries is seriously endangered by
high temperatures. It is important to have a simple but accurate model to evaluate the thermal behavior of batteries un-
der a variety of operating conditions and be able to predict the internal temperature as well. To achieve this goal, a ra-
dial-axial model is developed to investigate the evolution of the temperature distribution in cylindrical Li-ion cells. Ex-
perimental data on LiFePO4 cylindrical Li-ion batteries are used to determine the overpotentials and to estimate the
State-of-Charge-dependent entropies from the previously developed adaptive thermal model [1]. The heat evolution is
assumed to be uniform inside the battery. Heat exchange from the battery surfaces with the ambient is non-uniform, i.e.
depends on the temperature of a particular point at the surface of the cell. Furthermore, the model was adapted for im-
plementation in battery management systems. It is shown that the model can accurately predict the temperature distribu-
tion inside the cell in a wide range of operating conditions. Good agreement with the measured temperature develop-
ment has been achieved. Decreasing the heat conductivity coefficient during cell manufacturing and increasing the heat
transfer coefficient during battery operation suppresses the temperature evolution. This modified model can be used for
the scale-up of large size batteries and battery packs.
Keywords: Lithium-Ion Batteries; Thermal Modeling; Entropy; Energy; Electrochemistry; Heat Transfer
1. Introduction
A successful design of battery packs starts with the cor-
rect accommodation of the thermal battery properties.
Using high thermally resistant materials, such as binders
in the electrodes and polymer separators is inevitable in
constructing high performance batteries, but this limits
the heat transfer inside the cells. Consequently, higher tem-
peratures are expected at the core of the cells compared
to its surface. However high temperatures are not permit-
ted for safety and reliability reasons; they can accelerate
battery degradation and even cause thermal runaway. In a
battery management system (BMS), it is desirable to ac-
curately predict the internal temperature evolution of the
battery according to the state-of-charge (SOC), cell po-
tential, current and surface temperature. Such a system re-
quires an efficient thermal model with a limited number
of measured parameters at each state. Therefore heat dis-
sipation in a cell must be properly studied in order to
avoid elevated temperatures where decomposition reac-
tions of the electrodes and electrolyte are started [2].
A number of models have been developed to study the
thermal behavior of Li-ion or Li-polymer batteries [1,3-
22]. Theoretical models, which are usually based on a
combination of electrochemistry and physics, can give
accurate predictions for various operating conditions [3-
10]. But these are complicated and need sophisticated
measurements and estimation of transport properties, elec-
trochemical reaction constants, etc. to be accurately solv-
ed. Such models are well-suited for battery design pur-
poses, though not optimal for the low computing-power
environment of micro-controllers used in BMS [20]. On
the other hand, experimental-based models employ bat-
tery measurements on the cell level to determine equilib-
rium potentials and, consequently, overpotentials and in
some studies, entropy contributions in heat generation,
but do not go into detail of the electrochemical processes
occurring inside the cells [1,11-18,22].
The present paper represents combination of both ap-
*Corresponding author.
opyright © 2013 SciRes. JMP
proaches. An experimental-based methodology for accu-
rate two-dimensional thermal modeling of cylindrical cells
is proposed. Commercially available high power 2.3 Ah
LiFePO4 batteries, which are an interesting option for
hybrid electrical vehicles (HEVs) and full electrical ve-
hicles (EVs), are used in this study. Previously developed
lumped thermal model [1] is implemented for this cell
chemistry to provide part of the input data of the two-
dimensional model based on the experimental data and
then for extension of the model. In the previous work, it
was shown how one can explicitly determine the equi-
librium potentials by polynomial extrapolation towards
zero current, and obtain the overpotentials at various cur-
rents and SoC. Then a back-estimation method was in-
troduced to find the values of the entropy changes from
the adapted thermal model.
The main challenge and contribution of this work is to
implement a proper two-dimensional model of heat dis-
sipation by using overpotentials and temperature deriva-
tives of the equilibrium potentials, which gives a proper
rate of the heat generation. The developed model is ex-
ported into Matlab, enabling a user-friendly implemen-
tation into a BMS. COMSOL multiphysics is used for fi-
nite element modeling, and then the model is exported to
MATLAB for estimation of the various parameters.
2. Theoretical Considerations
2.1. Model Development
Based on the differential energy balance in cylindrical
coordinates, the following heat transport partial differ-
rential equation (PDE) is obtained
pTk T
 , (1)
is the density [g·m3],
C the specific heat
capacity [J·g1·K1], T the temperature [K], t time [s], k
the thermal conductivity [W·m1·K1], r radial coordinate
and z is the axial coordinate. Note that derivatives with
respect to
, the tangential coordinate, are absent since
the considered problem possesses radial symmetry. The
heat source term
 
[W·m3] includes all bat-
tery processes that contributes to the heat evolution, ac-
cording to
 
[W] represents the overpotential heat,
trz C
 
[W] the entropic heat and V [m3] is the
volume of the battery.
, (3)
where the overpotential
t [V] is defined as the
difference between equilibrium cell potential and
the operating cell potential
 
 , (4)
t is always defined positive during charging and
negative upon discharging. I(t) [A] specifies the current
flowing through the individual electrodes which is by de-
finition positive during charging and negative upon dis-
charging. So
is always positive.
the entropic heat [W] which can be represented by
,, ,,
H trzItTtrznF
)( tS
, (5)
where n is the number of electrons that are transferred in
the electrochemical reaction during the battery operation
and F is the Faraday constant [96,486 C·mol1].
[J·mol1·K1] expresses the entropy change of the indi-
vidual electrochemical charge transfer reaction during
battery operation and has been derived at our previous
work [1], according to
St nFT
, (6)
represents, in general terms, the reaction pro-
gress, in our specific case the SoC of the cell. The initial
condition for the partial differential equation (Equation
(1)) for the whole cell is
0, ,a
TrzT, (7)
where a [K] is the ambient temperature. Denoting the
radius of the cell as R and length of the cell as L energy
conservation requires that on the surface of the cell there
is a balance between heat conduction and sum of con-
vection and radiation heat transfer, according to
 n, (8)
where h is the average convective heat transfer coeffi-
cient [W·m2·K1] between the cell and the environment,
= 5.67 × 10-8 [J·s1·m2·K4] is the Stefan-Boltzmann
constant and
is the emissivity factor of the surface
material. T
represents temperature gradient and
is the outer normal vector of unit length.
2.2. Experimental
2.3 Ah cylindrical A123 (ANR26650m1) LiFePO4 cells
were investigated in this study. Temperature develop-
ment as a function of applied current, potential and oper-
ating regime was recorded. The experimental data were
used to determine the overpotential and SoC-dependent
entropy changes and to validate the thermal model. Cy-
cling was performed with an 8-channel MACCOR 4000
battery tester. The temperature development was meas-
ured on the middle of the surface of the cell by the poten-
tial drop across PT100 type thermo-resistors connected to
Copyright © 2013 SciRes. JMP
battery tester via the auxiliary potential input. A water
bath was used to control the ambient temperature in a cli-
mate control box (Figure 1). To evaluate the thermal be-
havior of the cells, three different temperatures were in-
vestigated: 0˚C, 20˚C and 40˚C.
Constant current-constant voltage (CCCV) charging
was applied in this study. Charging was carried out at 1.6
A (= 0.7 C-rate) till the maximum cell potential of 3.6 V
was attained. Charging was continued under constant vol-
tage conditions at 3.6 V until the charging current drop-
ped below 0.3 A (End-of-Charge). Subsequently, CCCV-
charging was followed by a relaxation period of 180 min.
Constant current (CC) was then applied to discharge the
battery at various C-rates. Discharging was terminated at
2.0 V. The following rated currents were employed for
discharging: 0.05, 0.10, 0.15, 0.20, 0.30, 0.50, 0.75, 1.00,
1.50 and 2.00 C-rates.
2.3. Model Parameters and Assumptions
The battery specifications are summarized in Table 1.
The battery consists of several components: organic elec-
trolyte, dissolved lithium salts, polymer separator, cur-
rent collectors, graphite and composite electrodes, sup-
porting materials, metal casing, etc. So it is difficult to
estimate the overall physical properties of the cell based
on the physical properties of its constituents. Table 2
provides some proposed and estimated values of requir-
ed physical properties of the main cell components.
Surface emissivity factor
of the batteries is as-
sumed to be 0.65. The small observed variation in eq
upon battery aging is neglected. Considering heat capaci-
ties of the components, the average heat capacity of the
cells is estimated to be 1.36 [J·g1·K1].
Considering natural convection of air around a hori-
zontally positioned cylinder, the following expression
has been adopted for the heat transfer coefficient [23]
1 0.559
Nu k
, (9)
Figure 1. Experiment set-up.
Table 1. Battery specifications.
Parameter Value or description
Diameter 25.85 mm
Height 65.15 mm without terminals
Mass Approx. 70 g
Positive Electrode LiFePO4
Negative Electrode Porous Graphite
Nominal Capacity 2.3 Ah at 0.2 C
Table 2. Physical properties of the cell components.
Cell (Avg.) Separator (PP) Cathode Anode
CP [J·g1·K1]1.36 1.93-2.00 1.25 1.20
k [W·m1·K1]0.4 0.12-0.22 0.4 1.4
[g·m3]2.047·1069.02 - 9.06·105 2.208·106 1.410·106
Ra Gr
is the average dimensionless Nusselt num-
ber, h is the average natural (free) convection heat trans-
fer coefficient [W·m2·K1], d the diameter of the bat-
tery [m] and kamb is thermal conductivity of air [W·m1·K1].
Ra is the dimensionless Rayleigh number, defined as
in which
Gr is the dimensionless Gra-
shof number,
Gr and Pr p
the Prandtl number, g [m·s2] is the gravity constant,
[K1] is the volumetric thermal expansion coefficient,
which is defined as 1Ta for ideal gases. Parameters
represent the viscosity and kinematic viscosity of
air, respectively.
3. Results and Discussion
The thermal behavior of the batteries has been evaluated
up to 2C-discharge rates at ambient temperature 20˚C.
The results at this temperature are presented here as a
common ambient temperature for the battery operations
in electrical vehicles. However the battery measurements
and simulation have also been performed at 0˚C and
40˚C to determine equilibrium potentials and derive en-
tropy change values to further validation of the model.
3.1. EMF, Overpotential and Entropy
The measured cell potentials during cycling with various
discharge C-rates at 20˚C are shown in Figure 2(a). The
battery is charged to 3.6 V according to the standard
charging regime in every cycle. During the subsequent 3
hours of relaxation under open-circuit conditions, the cell
potential drops to about 3.4 V, after that discharging
Copyright © 2013 SciRes. JMP
Figure 2. (a) Cell potential development during charging (1),
relaxation (2) and discharging (3) at various C-rates and (b)
the corresponding temperature development. Ambient tem-
perature is 20˚C.
starts. The discharge cut-off potential is 2 V. Figure 2(b)
represents the temperature development during cycling.
It is clearly visible that the temperature evolution becomes
more pronounced at higher C-rates. The temperature nor-
mally increases during (dis)charging. However at some
specific stages the temperature decreases. Since the over-
potential heat is always positive, this can be explained
only by a negative entropic heat contribution, which will
be discussed later.
The methodology developed in previous work [1] is
used to determine the overpotentials and SoC-dependent
entropy changes from the experimental data. Modeling
starts with estimation of the equilibrium potential. Extra-
polation of the measured cell potential towards 0 C-rate
discharge current which has introduced in the previous
works is used to determine the equilibrium potentials
[1,24,25]. Figure 3 shows the measured potential curves,
the determined dependence (black upper potential
curve) and the overpotential curves (insert).
According to Equation (4) the overpotential can be ob-
tained from the EMF and the cell potential discharge
curves (see inset of Figure 3). The as-determined over-
potentials serve as important input parameter to calculate
the heat evolution Equation (3). Another heat contribu-
tion (see Equation (4)) is related to the entropy. In order
to reduce the complexity of the calculations 11 Qout-re-
gions are considered. Values of in 11 subsequent
nodes of the discrete grid, i.e. SoC from 0 to 1 with step
0.1 are treated as unknown optimizing parameters. These
values are determined from an optimization proce-
dure by a lumped thermal model based on battery meas-
urements [1], considering the best agreement between
experimentally measured and theoretical evolution of the
surface cell temperature as criteria. Optimization has been
performed across 10 cycle of (dis)charge as the cell mea-
surements have been performed. values as a func-
tion of SoC are displayed in Figure 4. The obtained val-
ues for entropy changes are in consistence with the re-
ported values for this battery by C. Forgez et al. [20].
3.2. 2D-Model
It is assumed, that rate of heat generation is uniform
across the cell and determined by overpotential and state-
of-charge (SoC)-dependent entropic heat sources. In con-
trast, the heat exchange at the battery surface with the
ambient environment depends on temperature, and is
therefore not uniform. Contributions from convective and
radiation cooling are considered. The model is construc-
ted in COMSOL 3.5a in radial-axial coordinates using
the physical cell properties listed in Tables 1 and 2. The
overpotential tS
values are used to calculate
heat generation (Equations (2)-(5)). These values are us-
ed as input parameter to radial-axial model for the heat
generation. Natural heat convection of air around the cell
and radiation are taken into consideration knowing that
radiation is comparable to natural heat convection amounts.
By using Equation (9) at an estimated average tempera-
ture of 24˚C (which is estimated according to the experi-
mental measurements of temperature) on the boarder of
the cell, h is calculated to be 9 [W·m2·K1]. The physi-
cal properties of the cell and air are assumed to be tem-
perature-independent over a narrow temperature range
upon each test which is less than 10˚C and an average ex-
tent for each property is chosen. The grid used in the cal-
culations is shown in Figure 5.
Equation (1) with initial condition Equation (7) and
boundary conditions Equation (8) can then be solved by a
time-dependent solver.
3.3. Temperature Profiles
The result of a COMSOL simulation is illustrated in Fig-
re 6. As expected, the core temperature of the cell is u
Copyright © 2013 SciRes. JMP
Copyright © 2013 SciRes. JMP
Figure 3. Measured cell potential discharge curves re-plotted as a function of Qout. The extrapolated EMF curve (upper black
curve) yields the overpotential curves by subtracting the measured potential curves from the EMF curve (insert).
Figure 4. Entropy change values as a function of SoC at 20˚C.
After creating a 2D-COMSOL model it is saved as a
MATLAB compatible file (M-file). Then resulting
MATLAB function is run in MATLAB, calculating tem-
perature distributions for given set of parameters. These
data are used to find the maximum and minimum tempe-
ratures upon cycling and plot them simultaneously with
the measured temperatures from the experiments to vali-
date the model. The average temperature of the cell is
also defined and plotted. To have a better view over the
battery operation, the r-z surface temperature combined
with a colored contour and 1-dimensional time-series tem-
Figure 5. Cell structure and grid.
higher than that at the surface. A maximum temperature
of about 28˚C inside the cell is observed at the end of
discharge (2 C-rate). Obviously, the edges are coolest
with a temperature of approximately 25˚C.
perature plots were calculated, then time-series movie
was created. The simulated temperature profiles of the
cell at 2C-discharge and upon 3 hours of relaxation in the
following period are shown in Figure 7. The temperature
is measured on the surface of the cell during the experi-
ments, while the heat generation occurs inside the cell.
So, it is expected that measurements remain between the
Figure 6. Temperature distribution inside the cell at the end
of 2C-rate discharging at an ambient temperature of 20˚C.
Figure 7. Minimum, maximum and average temperature
profiles versus measured temperatures on the surface of the
cell at 2C-rate discharge from MATLAB model.
minimum and average temperature of the cell. A maxi-
mum temperature difference around 3˚C is observed in
the cell at the end of discharge. It is seen that even with
moderately high discharge rate, which is not extreme
case, the core temperature increases by 9˚C.
The benefit of running the model in MATLAB is the
ability to use it for optimization purposes. Moreover the
physical properties of the cell or adjustable parameters
for BMS can easily be changed and the results be evalu-
ated individually in-line with other part of the model. To
cease or moderate the temperature rise in the cell upon
high discharge rates, it is possible to increase the heat
transfer coefficient h. For instance it is possible to use a
fan to apply force convection of air around the cell. On
the other hand, heat conduction of the cell seems another
limiting factor for effective temperature control over the
cell. Therefore it is better to increase the k value during
manufacturing the cell. A combination of different k and
h values and the consequences on the minimum and ma-
ximum temperatures and maximum temperature differ-
ence across the cell are presented in Table 3. It is con-
cluded that by increasing the thermal conductivity from
0.4 to 2 W·m1·K1, the gap between the minimum and
maximum temperature in the cell decreases from 3˚C to
less than 1˚C by running the MATLAB model with the
new k value. However, it seems that increasing h can
control the temperature rise. By increasing h by a factor
of 4, the maximum internal temperature of the cell de-
creases to 25.2˚C. However, the maximum temperature
difference in the cell increases to 3.9˚C. More effective
cooling can be done if the heat transfer coefficient in-
creases, but the heat conduction of the cell decreases.
A 3-dimensional view from the r-z surface temperature
profile and contours and comparison with the maximum
temperature at each moment has been plotted in Figure 8.
The animated movie from the r-z surface temperature, con-
tours and maximum temperature plot during the whole
cycle is available as electronic file.
4. Conclusion
A simple radial-axial model has been developed to eva-
luate the thermal behavior of LiFePO4 batteries. The over-
potential and SoC-dependent entropy change heating are
two important sources of heat generation in the cell. The
parameter values for these sources are obtained by im-
plementing the previously developed lumped thermal
model. The as-determined values for entropy changes are
consistent with recent literature [20]. The model provides
a good agreement with the experimentally measured tem-
peratures upon cycling. The model is constructed and run
in COMSOL 3.5a. It is combined with MATLAB to use
optimization capabilities for validating the model with
measured data. It is shown that the core temperature of
the cell is much higher than that at the surface of the
Copyright © 2013 SciRes. JMP
Copyright © 2013 SciRes. JMP
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Case 1 2 3 4
k [W·m1·K1] 0.4 2 0.4 2
h [W·m2·K1] 9 9 45 45
Cp [J·g1·K1] 1.36 1.36 1.36 1.36
Min. T [˚C] 25.1 25.5 21.3 22.2
Max. T [˚C] 28.0 26.4 25.2 23.4
Max. T [˚C] 2.9 0.9 3.9 1.1
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