1. Introduction
Lithium-ion batteries (LIB) have found a wide range of applications in many consumer products in the last 25 years. LIBs are used in cell phones, tablet and laptop computers as well as in electric (EV) or hybrid electric (HEV) driven vehicles. The size and the capacity of the batteries differs with the application. A battery used in an EV or HEV is much bigger than the one used for a cell phone. However the scale up of the batteries leads to a complexity of the physical phenomenona that do not play any significant role in single or small battery systems.
The occurrence of these physical phenomena are spread over a wide range of time and length scales, from the atomic level up to the heat transfer of the whole battery. The relation of these different phenomena to each other is of great importance, especially regarding the safety of the battery systems.
The usage of model-based investigations promises a theoretical understanding of battery physics even beyond the point that is possible using experiments. Due to their general structure, the behavior of batteries is strongly affected by the interaction of physics on varying length and time scales. In the last two decades, many approaches have been reported in literature starting with the work of Newman [
1,
2,
3] based on the theory of porous electrodes [
4]. This theory is based on a electrochemical description of diffusion dynamics and charge transfer kinetics of a battery and can give a forecast of the electric response of a cell in an intercalation electrode system. This model is adequate in small battery systems. In large formatted battery cells, however, the uniformity of the electrical potential along the current collectors in the cell composites is no longer given. Additionally due to the inhomogeneity of the distribution of the temperature field with respect to the cell geometry, thermal dynamics must also be taken into account.
As consequence an extended coupled electrochemical-thermal model must be considered. First thermal approaches for Lithium batteries are suggested by Newman and Pals [
5,
6,
7] using an inhomogeneous heat equation , where the coupling of the thermal and electrochemical model is realized with the heat generation terms. This approach can be applied to single battery cells as well as to whole batteries [
8,
9,
10,
11,
12]. In parallel the multi-scale and multi-domain character of the electrochemical model is developed [
13,
14] by using the mathematical theory of homogenization resp. the spatial averaging theorem [
15]. Newer approaches are reported in the works of [
16,
17,
18].
First, coupling between the multi-scale, multi-domain electrochemical-thermal model is introduced by [
19,
20,
21] and is under development until now. A general short overview can be found in [
22] and some current results in [
23,
24,
25]. The purpose of this modeling approach is to gain a better understanding of the behavior of large LIB systems, because the transport of electrical current and heat must be considered on the length scales of the electrode particles up to the length scale of the cell composite, which is in general an anisotropic medium. One central point in the modeling of LIBs is the aspects of cell safety which corresponds to the thermal stability of a LIB. Several exothermic reactions can occur inside a cell during operation as its inner temperature increases. If the heat creation is larger than the dissipated heat in the surrounding space, this may cause heat to accumulate inside the cell and chemical reactions will be accelerated which yields in a further temperature increase until a thermal runaway is reached. In these terms, a thermal runaway describes a rapid temperature increase in a very short time interval of the typical order of
and above [
26].
This phenomenon can not be described with the mathematical models mentioned above. A first attempt to overcome this disadvantage is given in [
27]. The heat equation is coupled with ordinary differential equations (ODEs), describing the temporal evolution of the concentration of the exothermic reactions based on an Arrhenius-type law. Spotnitz
et al. [
28,
29] give a first PDE modeling of a thermal runaway including reaction kinetics. In [
30,
31,
32], the electrochemical-thermal model is extended with reaction kinetics based on an ODE formulation. Some newer results can be found in [
33,
34,
35]. In [
33] an oven test and the influence of convection is investigated, while [
34] is focused on a Lithium-Titanate Battery with a model similar as the one described in [
33]. The PhD Thesis of Tanaka [
35] gives a detailed insight in the modeling and simulation of a thermal runaway for different chemistry.
In
Section 2 of this article a model to describe the thermal behavior of a LIB in general with all sub-models is introduced and reviewed. In
Section 2.1, the thermal model is formulated and the important heat sources are identified. In the following subsections, the corresponding mathematical models behind the several heat sources are given. In
Section 2.2, the abusive exothermic kinetic reactions leading to a thermal runaway are given in terms of mathematical combustion theory [
36,
37] followed by a simplification in
Section 2.3. In
Section 2.4, the electrochemical heat source is considered and the multi-scale, multi-domain electrochemical model of LIB’s [
22,
23], which is implemented in the
Battery and Fuel Cell Module in
COMSOL Multiphysics, is introduced. In the
Section 3 some results simulated in
COMSOL Multiphysics are shown. In
Section 3.1, the model with and without exothermic contributions is compared, followed by the computation of the time, where the thermal runaway occurs in
Section 3.2. Additionally a first coarse classification of the thermal runaway is given. In
Section 3.3 the critical overall cell temperature, where above the thermal runaway occurs, is determined. Furthermore intervals for the convective heat transfer coefficient and environmental temperature are computed in this case. In the final section, we summarize our results and an outlook of possible future work is given.
3. Simulations
First simulation results using COMSOL Multiphysics Version 4.4 are given in this chapter. The model was implemented in the Battery and Fuel Cell Module of COMSOL Multiphysics. This module includes some predefined models of a LIB as described in the last section. The user defines his simulation problem including the concrete models, their coupling, the cell geometry, the simulation parameters and the spatial-temporal discretisation.
The purpose of this first simulation is to get a better insight in the thermal behavior of a LIB under normal and abuse conditions. The standard battery model in this module is not able to resolve thermal runaway. In the following the standard model is labeled as Model A. Therefore, an extension of this model with additional contributions is required. This extension is described in the last section together with the battery model and implemented in COMSOL Multiphysics in our already existing model in the most simple case first. This model is labeled as Model B in the following.
We consider a cylindrical 18650 cell with chemistry, i.e., in the COMSOL Multiphysics material database a anode, a cathode and as electrolyte with a salt is chosen. The internal spiral wounds are not resolved in this work, which means, that the geometric considerations will be restricted to the -plane. Only the additional heat sources are used to extend the temperature equation with additional heat sources (Constant fuel assumption) from the reaction-diffusion system of the exothermic reactions inside the LIB. These heat contributions are directly implemented with the electrochemical heat sources in the MSMD approach.
It is assumed, that a single LIB is placed in an
Accelerating Rate Calorimeter (ARC) and the environment of the LIB in the ARC is filled with air. Experimental measurements inside the ARC gave typical values for
h in the range of
. During the simulations the convective heat transfer coefficient
h is varied in the interval of
. This setup represents adiabatic conditions for
. The non adiabatic conditions with positive
h describes the free flow of gases and steams for
in general, stationary gases and air in the environment are described by
h in the interval
[
47]. Therefore only natural convection is considered in this work.
In a cylindrical geometry, the equations for the particle and electrode domain model remains unchanged. Only on cell domain the corresponding equations for the potentials at the current collector and the temperature field have to be reformulated in cylindrical coordinates in the
plane,
i.e.,
:
The initial and boundary conditions of the temperature field
T are given in the Equations (
2) and (3). Since the electrodes and the separator possess a porous structure, where the pores are filled with the electrolyte, the thermal parameter in the heat Equation (19) have to be considered as an effective parameter. Therefore in each of the three porous domains the total value of the thermal parameter is computed with a mixture formula containing the parameter value of the corresponding solid phase and the liquid electrolyte. Then the effective thermal conductivity (radial-, z- direction)
, density
and heat capacity
are given as follows [
30]:
where
denotes the index set to the corresponding parameter in cathode, anode, separator and current collector.
The last four equations can be regarded as special cases from corresponding formulas of the spiral geometry restricted to the
plane [
48,
49]. Therefore, this thermal model can be considered as a serial thermal network in
direction and a parallel thermal network in
direction. With the parameters from
Table A1 in the appendix, the corresponding effective parameters are given in
Table 1. In this table, the effective parameters are compared with parameters given in literature [
50,
51].
The main physical parameters of the simulation for both models can be found in
Table A1 and in
Table A2 in the appendix. The additional simulation parameters of the exothermic reaction in
Model B for the abuse contributions are taken from [
30,
31].
The simulations were performed with the two current load profiles with a constant current density of
as seen in
Figure 2. In the first profile labeled
Profile I shown in
Figure 2a there are no relaxation periods between the charging and discharging pulses with duration of
each. In the second profile (
Profile II) relaxation periods of
between charging and discharging pulses are used as shown in
Figure 2b. A complete current cycle lasts
and
. The maximal simulation time is
and
. For the time integration in
COMSOL Multiphysics, a BDF integration scheme is chosen with a minimum order of 1 and a maximum order of 5 using a variable step size with a maximum time step of
and an absolute tolerance of
. Since the spatial discretisation in
COMSOL Multiphysics is based on the FEM method, we have used an adaptive spatial discretisation in the three models of the particle domain, the electrode domain and the cell domain respectively. The model of the particle domain is solved automatically in the battery module of
COMSOL Multiphysics. Therefore only a spatial discretisation for the electrode domain and the cell domain is required. In the electrode domain, the maximum element size in the discretisation is chosen as
In total the discretisation contains 168 elements. Quadratic basis functions were chosen for the one-dimensional finite element discretisation in the electrode domain. In the cell domain the spatial discretisation is performed in the
-plane using 2266 triangular elements with the element size in the interval
and quadratic basis functions. The FEM discretisation is shown in
Figure 3.
The main purpose of this section is to compute the overall mean cell temperature, below which the LIB shows normal behavior and above which the cell shows a thermal runaway with respect to some system parameters like the environmental temperature or the convective heat transfer coefficient. Furthermore, critical values for such parameters will be computed from the simulations and a first coarse classification of the thermal runaway will be given.
3.1. Model Comparison
In a first step
Model A and
Model B are compared under normal and abusive conditions with
Profile (I). The following three cases are considered:
- (1)
: In this case the LIB is unable to dissipate any generated heat to the environment. Therefore, a thermal runaway will always occur after a sufficiently large time.
- (2)
h positive and a thermal runaway occurs after a sufficiently large time: In this case, more heat is generated inside the LIB, as can be dissipated to the environment.
- (3)
h positive and no thermal runaway occurs after a sufficiently large time: In this case the generated heat can be dissipated completely to the environment.
The first case coincides with adiabatic boundary conditions while the other two cases coincide with isoperibolic boundary conditions. Additionally for this comparison an environmental temperature of
is chosen as a representative temperature in the interval of
. This temperature coincides with the minimum temperature where stable environmental conditions can be adjusted in the ARC. As corresponding convective heat transfer coefficients the values
are chosen. The spatial averaged mean cell temperature
of all interior points is considered W.l.o.g. the overall mean cell temperature
can be replaced by the mean temperature
of the cell surface or the temperature
at an arbitrary interior point of the cell. ,
i.e.,
Since the simulation results are stored in a non-equidistant time grid, the mean temperature
is interpolated to an equidistant time grid with
using splines with Matlab
® 2015b. In
Figure 4a–c the
Time- Temperature plots for
Model A (blue) and
Model B (red) are shown with respect to
. In
Figure 4d–f the corresponding
Temperature-Heating Rate plots are given on a semi-logarithmic scale. From
Figure 4 one can see that in (a) and (b) both models shows similar behavior until
resp.
. The corresponding cell temperature is approximately
(Lower horizontal line in (a), (b), first vertical line in (d), (e)). This temperature is near to the start temperature of SEI decomposition, which is given in [
26] to
. After this time
Model B exhibits a greater increase in temperature as
Model A. Finally, the simulation of
Model B shows a
blow-up in solution which corresponds to a thermal runaway in a LIB. For
, the reaction for the electrolyte decomposition starts [
26]. This is marked with the upper horizontal line in (a), (b) and the second vertical line in (d), (e) in
Figure 4. The corresponding heating rate is
which correspond very well with the start of the thermal runaway given in [
26] with
. In (c), both models have no significant difference in their time evolution of the temperature and the both solutions remain bounded. Therefore, in the graphs (a), (b) and (d), (e) of
Figure 4 one can split the time evolution of the temperature in the three zones (I),(II) and (III).
First consider the plots (a) and (b) of
Figure 4. In Zone (I) the cell shows a normal behavior. In this zone, the heat generation is governed by the electrochemical, Joule and Ohmic heat contributions. Exothermic contributions are negligible. In Zone (II), the cell temperature is high enough, so that the exothermic heat contributions are no longer negligible. This zone can be considered as an intermediate zone or the onset of the thermal runaway. In Zone (III), the exothermic heat contributions are dominant and thermal runaway takes place. Electrochemical heat contributions can be neglected in Zone (III).
Next consider in
Figure 4 in the plots (d), (e), (f) the corresponding
Temperature-Heating Rate trajectory. Since the current profiles
Profile I and
Profile II are of a rectangular shape, the first derivative
will be a piecewise continuous function with discontinuities at the time, where the current profiles jumps. Therefore, we consider
in the sense of distributions. At the time of the discontinuities one has to take the limit from right and left to get a derivation from left and right.
The graphs in (d) and (e) can be separated into three zones again (Vertical black lines). In the first zone (I)
Model A and
Model B show similar behavior with an increase in the mean cell temperature
while the heating rate
remains almost constant or has a small decrease with respect to some outlier points. These points coincide with the time, where the current profile jumps from the charging state to the discharging state and vice versa. At the end of zone (I) in
Model B, the heating rate begins to rise. This is justified due to heat contribution of the exothermic reaction, which is no longer negligible. The general shape of the curve from
Model B remains unchanged. This is the transition to zone (II). In the zones (II) and (III) one observes an increase in temperature
T and an increase in the heating rate
, which is justified in the accelerating of the exothermic reactions. The transition from zone (II) to (III) is reached, when the heating rate exceeds the critical value of
[
26]. The corresponding temperatures are computed from the plots (d) and (e) in
Figure 4 as the intersection the horizontal black line with the graph of
Model B (red). This result is compared with the upper black line in the plots (a) and (b) with the graph of
Model B (red). The results are given in the last column in
Table 2. Next, we compare our simulation results for
Model B with the experimental results from Abraham
et al. [
26]. In their work, the thermal abuse of a 18650 cell with a
chemistry is considered. The time evolution of the temperature is also grouped into three zones. A comparison between the experimental results from [
26], the results given in [
32] and our simulation results is given in
Table 2. From a physical point of view one can interpret the zones (I)–(III) as follows: If the
curve remains in zone (I) and stays below certain temperature values the cell works in safe conditions. If the
-curve changes to zone (II) thermal runaway starts, and both the heating rate and temperature rise. In zone (III) the heating rate exceeds a critical heating rate value. This is the thermal runaway, in which the heating rate tends to infinity. In
Figure 4c,f both models stay in zone (I), no thermal runaway occurs and no significant difference is to be observed in both models.
In summary one can see the difference between
Model A and
Model B due to the exothermic heat sources in the temperature development over the time and corresponding heating rate. Furthermore
Model B gives a good agreement with experimental results from literature [
26,
32] and a coarse classification of the time evolution of a thermal runaway event has been made. This topic will be considered in more detail in a subsequent publication.
3.2. Computing the Thermal Runaway Time
In this and the following section only Model B is considered. The environmental temperature is and steps of were employed. In the simulations for each , the heat transfer coefficient h is in the interval and the emissivity is equal to . The time required for thermal runaway to occur with respect to the environmental temperature and convective heat transfer coefficient is computed in the simulation. For this purpose, the term Thermal runaway time has to be precisely defined.
Definition 1. The
Thermal Runaway Time of a Lithium ion cell is defined as the time, at which the heating rate exceeds a predefined threshold
i.e.,
such that:
for sufficiently high cell temperatures.
For an equidistant time grid of
, the heating rate
is approximated by the Equation (26).
The threshold is chosen here as:
The thermal runaway time is chosen as the time at which the condition is achieved.
In
Table 3 and
Figure 5 the time
for the current profiles (I) and (II) is given with respect to
and
h. The following two cases are considered:
- (1)
For
with
the environmental temperature is fixed, then for
the thermal runaway temperature
is determined (
Figure 5a,c).
- (2)
For
the convective heat transfer coefficient is fixed, the environmental temperature
with
is varied and the thermal runaway time
is computed (
Figure 5b,d).
In case (1) for a fixed environmental temperature, the thermal runaway time increases with increasing convective heat transfer coefficient h. In case (2) for a fixed convective heat transfer coefficient, the thermal runaway time decreases with increasing environmental temperature.
3.3. Critical parameter intervals
In the last section the thermal runaway time
was computed. In this section the time evolution of the mean cell temperature
was computed, again in the cases (1) and (2) from
Section 3.2. The aim is to calculate the critical mean cell temperature
with respect to the parameter
and
h in the sense, that below the critical temperature the LIB is under safe conditions and above the LIB will show a thermal runaway. Therefore, multiple simulations of the current profiles (I) and (II) are performed. In all simulations the environmental temperature is in the interval
and the convective heat transfer coefficient
h is in the interval
. The purpose of the simulations was to compute numerically approximately intervals for the critical environmental temperature
and the critical convective heat transfer
h, analog to the cases (1) and (2) from
Section 3.2 for fixed environmental temperature and variable heat transfer coefficient
h (case(1)) and vice versa (case (2)). The critical parameter intervals are determined with a bisection procedure, where successive the intervals of the parameter under investigations is shrinking with respect to the fact, if a thermal runaway in the simulation occurs or not.
The results are plotted in the
Figure 6 and
Figure 7. In the subplots (a) and (c) for
, the convective heat transfer coefficient is varied in the interval
for the current profiles (I) and (II). In the subplots (b) and (d), the environmental temperature is varied over the interval
for
and
for the current profiles (I) and (II). In
Table 4 some approximate intervals for the critical parameter are given.
Moreover from the temperature curves of the mean cell temperature
in the
Figure 6 and
Figure 7 the critical mean cell temperature is
. Since the development of a complete mathematical theory is far beyond the scope of this work an extension of the theories of Semenov [
52] and Frank-Kamenetskii [
38] and the corresponding critical parameter will be considered in future work. In summary the following general statement can be given from the observation of the simulation results:
Statement 1. Let
be a the asymptotic temperature profile if the limit
exists. Then the following holds:
(a) If the environmental temperature
is constant and fixed and the convective heat transfer coefficient
h is allowed to vary, then
with:
- -
- -
(b) If the convective heat transfer coefficient
h is constant and fixed and the environmental temperature
is allowed to vary, then
with:
- -
- -
If or is bounded then it is (almost) constant for the current profile (I) and periodic for the current profile (II).