## 1 Introduction

_{2}(LCO) cell in oven experiments. The four-step kinetics have been used widely to model the thermal behaviour of LIBs in different shapes [22], different oven temperatures [23], and a stack of cells [24‐26]. In previous work [18, 27], we used the four-step kinetics to model the self-heating behaviour of large LIB ensembles. On the other hand, He et al. [28, 29] used a one-step effective model based on Frank-Kamenetskii’s theory to simulate self-heating ignition of battery ensembles with a different number of cells. Liu et al. [4, 30] also used Frank-Kamenetskii’s theory to model their experiments on piles of LIBs. The one-step effective models and multi-step models have been developed based on different simulation software, prevailing in fields. Different simulation platforms may have a different focus on particular phenomena (like heat transfer, chemical kinetics, or fires), therefore, have different simulation performances on different physics modelling. This study conducts a benchmarking analysis on two popular simulation platforms, which support numerical modelling of self-heating ignition of LIBs: COMSOL Multiphysics and Gpyro.

## 2 Theoretical Frameworks

### 2.1 Numerical Set Up for COMOSL

_{p}, k) are assumed to be constant in this temperature range. In this study, a battery cell is simplified as a bulk that all reactants are homogeneously distributed inside the cell. Previous studies have shown that a lump thermal model [20] already provides adequate accuracy for the temperature prediction for the single-cell compared to a 3D model [21] for the focused temperature range. Therefore, this lumped simplification is acceptable and at mean time, significantly reduces the computational costs. The governing equation could be simplified as:

Chemical reactions | Equations | Equations No |
---|---|---|

SEI decomposition | \(R_{{{\text{sei}}}} = - \frac{{{\text{d}}c_{{{\text{sei}}}} }}{{{\text{d}}t}} = A_{{{\text{sei}}}} {\text{exp}}\left( { - \frac{{E_{{{\text{sei}}}} }}{RT}} \right)c_{{{\text{sei}}}}^{{n_{{{\text{sei}}}} }}\) | (5) |

\(q^{\prime\prime\prime}_{sei} = \Delta H_{sei} W_{n} R_{sei}\) | (6) | |

Negative-electrolyte reaction | \(R_{n} = - \frac{{dc_{n} }}{dt} = A_{n} \exp \left( { - \frac{z}{{z_{0} }}} \right)\exp \left( { - \frac{{E_{n} }}{RT}} \right)c_{n}\) | (7) |

\(\frac{dz}{{dt}} = A_{n} \exp \left( { - \frac{z}{{z_{0} }}} \right)\exp \left( { - \frac{{E_{n} }}{RT}} \right)c_{n}\) | (8) | |

\(q^{\prime\prime\prime}_{n} = \Delta H_{n} W_{n} R_{n}\) | (9) | |

Positive-electrolyte reaction | \(R_{p} = \frac{d\alpha }{{dt}} = A_{p} \alpha \left( {1 - \alpha } \right)\exp \left( { - \frac{{E_{p} }}{RT}} \right)\) | (10) |

\(q^{\prime\prime\prime}_{p} = \Delta H_{p} W_{p} R_{p}\) | (11) | |

Electrolyte decomposition | \(R_{e} = - \frac{{dc_{e} }}{dt} = A_{e} \exp \left( { - \frac{{E_{e} }}{RT}} \right)c_{e}^{{n_{e} }}\) | (12) |

\(q^{\prime\prime\prime}_{e} = \Delta H_{e} W_{e} R_{e}\) | (13) |

Symbol | Description | Value | Unit |
---|---|---|---|

A _{sei} | SEI decomposition frequency factor | 2.08E15 [40] | s ^{−1} |

A _{n} | Negative-electrolyte frequency factor | 1.67E6 [40] | s ^{−1} |

A _{p} | Positive-electrolyte frequency factor | s ^{−1} | |

A _{e} | Electrolyte decomposition frequency factor | 5.14E25 [21] | s ^{−1} |

E _{sei} | SEI decomposition activation energy | 1.35E5 [40] | J mol ^{−1} |

E _{n} | Negative- electrolyte activation energy | 7.72E4 [40] | J mol ^{−1} |

E _{p} | Positive- electrolyte activation energy | J mol ^{−1} | |

E _{e} | Electrolyte decomposition activation energy | 2.74E5 [40] | J mol ^{−1} |

∆H _{sei} | SEI decomposition heat release | J kg ^{−1} | |

∆H _{n} | Negative- electrolyte heat release | J kg ^{−1} | |

∆H _{p} | Positive- electrolyte heat release | J kg ^{−1} | |

∆H _{e} | Electrolyte decomposition heat release | 1.55E5 [21] | J kg ^{−1} |

W _{n} | Specific negative active content | 363 [20] | kg m ^{−3} |

W _{p} | Specific positive active content | 726 [20] | kg m ^{−3} |

W _{e} | Specific electrolyte content | 407 [21] | kg m ^{−3} |

n _{sei} | Reaction order for SEI decomposition | – | |

n _{e} | Reaction order for electrolyte decomposition | 1 [21] | – |

c _{sei0} | Initial value of c _{sei} | – | |

c _{n0} | Initial value of c _{n} | – | |

α _{0} | Initial value of α | – | |

z _{0} | Initial value of z | – | |

c _{e0} | Initial value of c _{e} | 1 [21] | – |

T _{0} | Initial temperature | 301.15 [20] | K |

k | Average thermal conductivity | 3.4 [20] | W m ^{−1} K^{−1} |

ρ | Average density | 2580 [20] | kg m ^{−3} |

c _{p} | Average heat capacity | 830 [20] | J kg ^{−1} K^{−1} |

h | Convective heat transfer coefficient | 7.17 [21] | W m ^{−2} K^{−1} |

ε | Surface emissivity | 0.8 [21] | – |

### 2.2 Numerical Set Up for Gpyro

_{vl}[28, 38] is the volumetric heat transfer coefficient considering the heat transfer from x and y directions. Gpyro also has various interfaces to set thermal boundaries such as constant temperature, heat flux, and, convection, and thermal insulation. The reactions scheme considered in Gpyro is similar to the classical self-heating theories which assume an effective overall Arrhenius type of reaction. For LIBs, the one-step effective reaction considered is:

_{2}(LCO) prismatic cells at 100% SOC. [29]

Parameters | Description | Value | Unit |
---|---|---|---|

A | Frequency factor | 1.42E23 | s ^{−1} |

E | Activation energy | 3.25E5 | J mol ^{−1} |

∆H | Heat of reaction | 8.87E5 | J kg ^{−1} |

k | Thermal conductivity | 1.08 | W m ^{−1} K^{−1} |

ρ | Density | 2164.7 | Kg m ^{−3} |

c _{p} | Heat capacity | 990.0 | J kg ^{−1} K^{−1} |

ε | Emissivity | 0.8 | – |

h _{c} | Surface convective heat transfer coefficient | 11 | W m ^{−2} K^{−1} |

h _{v} | Volumetric heat transfer coefficient | 902.3 | W m ^{−3} K^{−1} |

### 2.3 Set Up for Benchmarking

_{2}(LCO) prismatic cells [28, 29, 39]. The detailed experimental setup is introduced in our previous work [28, 29]. The dimensions of these cells are 34 × 10 × 50 mm, with a nominal capacity and voltage of 1.88 Ah and 3.7 V, respectively.

Case No | Shape & dimensions | Boundary conditions | Internal heat source |
---|---|---|---|

1 | Spherical (R = 4.8 mm) | Adiabatic | \(q^{\prime\prime\prime}_{tot}\) |

2 | Prismatic (34 × 40 × 50 mm) | T _{a} = T_{oven}; Convection & Radiation | \(q^{\prime\prime\prime}_{c}\) = 10 kW m ^{−3} |

3 | Prismatic (34 × 10 × 50 mm) | T _{a} = T_{oven}; Convection & Radiation | \(q^{\prime\prime\prime}_{tot}\) |

4 | Prismatic (34 × 40 × 50 mm) | T _{a} = T_{oven}; Convection & Radiation | \(q^{\prime\prime\prime}_{tot}\) |

_{0}. In this condition, all the heat generated by internal chemical reactions heats up the battery cell. Therefore, the temperature information is directly related to the chemical kinetics, and can be used to benchmark chemical kinetics, as shown in Eq. (22):

_{ref}is the electrochemical capacity of the reference battery, which is 1.65 Ah. This work tries to investigate if the adapted function based on the kinetics [20] developed 20 years ago could give a reasonable prediction of the self-heating ignition behaviour of the current LCO batteries.

_{vl}[29] to consider the heat transfer from the other two dimensions. COMSOL supports simulations on a complex 3D geometry. Therefore, both 1D and 3D simulations are conducted for COMSOL to benchmark with Gpyro. The boundary conditions consider both convective and radiative heat transfer with the ambient.

## 3 Results and Discussion

### 3.1 Microscale Chemistry, Case #1

_{p}, k, which uses the values listed in Table 3. The energy ratio, β, considers two values: β = 1, which represents the same chemical kinetics for the battery in Hatchard et al. [39] with 1.65 Ah, and β = 1.14, which is adapted to the battery with 1.88 Ah [28, 29]. Two initial temperatures are considered: T

_{0}= 130 °C, and T

_{0}= 140 °C. Focusing on self-heating igniton phenomenon of LIB, we consider temperature below 200 °C, that after this temperature, the battery involves a more complex combustion stage and is out of the scope of this research.

_{0}= 130 °C, a small increase of temperature (less than 2.5 °C) after 200 min is predicted for both one-step models in COMSOL and Gpyro. The internal reaction rates are limited, with low heat generation at this initial temperature. For T

_{0}= 140 °C, both models predict the exponential increase of temperature after 75 min, while the temperature predicted by COMSOL one-step model is a little faster (3 min) ahead than Gpyro. Those small differences may result from the different numerical methods (FEM for COMSOL and FDM for Gpyro) and different criteria for numerical convergence by the two tools. Overall, COMSOL and Gpyro agree well with the one-step modelling of chemical dynamics.

_{0}= 130 °C. For β = 1, which represents the same kinetic parameters as Table 1, the temperature is predicted to grow exponentially at t = 37 min, which is even earlier than the result from the one-step models at T

_{0}= 140 °C. The temperature for β = 1.14, which is modified by considering the variation of energy density, are predicted to grow faster, around 10 min ahead of that for β = 1. These fast temperature increases by the four-step model are caused by the SEI decomposition reaction, which is predicted to take place at around 100 °C and generates heat. The heat generated by the SEI decomposition reaction will raise the temperature quickly in the adiabatic condition and initiate other reactions, resulting in a rapid temperature increase in a short period. Since the four-step reactions model already predicts exponential temperature growth at T

_{0}= 130 °C, the scenario for T

_{0}= 140 °C is no longer needed to be compared. In regards to the simulation time required, it took COMSOL around 8 min to run a one-step simulation and 10 min for a four-step simulation, while Gpyro took around 20 min for a one-step simulation. COMSOL has higher computational efficiency.

### 3.2 Mesoscale Heat Transfer, Case #2

^{−3}) is assumed for the solid bulk to eliminate chemical dynamics. Convective and radiative heat transfer is considered for the solid bulk and the ambient at T

_{a}= 140 °C. The thermophysical properties and heat transfer coefficients are the same as Table 3.

_{vl}as a source term to consider the heat transfer from other dimensions. This treatment assumes the same temperature distribution for each cross-section perpendicular to the dimension analysed. While the COMSOL 3-D model directly solves equations in three dimensions, which can consider the temperature gradience in the cross-sections. Overall, the final equilibrium temperatures by all three models are in agreement, which shows the same capability of modelling mesoscale heat transfer for COMSOL and Gpyro. For the computational efficiency, COMSOL has a better performance and took around 10 min (5 min faster than Gpyro).

### 3.3 Single-Cell Comparison, Case #3

_{sub}= 147 °C (subcritical condition) and undertook thermal runaway at T

_{sup}= 149 °C (supercritical condition).

_{sub}= 146 °C for the subcritical condition and T

_{sup}= 147 °C for the supercritical condition. The temperature profiles for the subcritical condition by the two models agree well with the experiment, with an error below 3 °C. While for the supercritical condition, both models underestimate the time needed for the cell to develop into thermal runaway. Using temperature equals 200 °C as a criterion, it took around 150 min for the experiment to reach that temperature, while Gpyro predicts around 50 min and the COMSOL one-step model predicts around 100 min at T

_{a}= 147 °C. These results show that the reactivity of the one-step global reaction is a little overestimated and results in an earlier prediction of thermal runway. However, both models have a good prediction of the transition from the subcritical condition to the supercritical condition, which is the key parameter to assess self-heating ignition. Therefore, both COMSOL and Gpyro are capable of predicting self-heating ignition of LIBs with the one-step global reaction assumption.

_{sub}= 153 °C and thermal runaway at T

_{sup}= 158 °C. While for β = 1.14, which adjusts the heat generation by considering the energy difference between the cylindrical cell in Hatchard et al. [39] and prismatic cell in this work, the predicted subcritical and supercritical conditions are T

_{sub}= 150 °C and T

_{sup}= 155 °C, respectively. The 3D heat transfer model predicts a faster temperature increase for the first 30 min, and the early-stage reactions such as SEI decomposition further accelerate this temperature increase rate. These results indicate that the prismatic cell analysed in this study may have a large difference in early-stage reactions compared with the cylindrical cell analysed in Hatchard et al. [39]. This may also result from the improvement of the safety of LIB techniques. For the time to reach 200 °C, it took around 40 min for the scenario of β = 1 and 60 min for the scenario of β = 1.14, which are all lower than the prediction by one-step models. This is because the four-step models all predict a higher supercritical condition (T

_{sup}= 158 °C and 155 °C for β = 1 and 1.14), which is 5–10 °C higher than the real experiment, and therefore a much faster reaction rate. For the computational time, it took COMSOL around 15 min for the 1-D simulation and 30 min for the 3D simulation. While Gpyro took around 30 min for the 1D simulation.

### 3.4 Four-Cell Comparison, Case #4

_{sub}= 143 °C) and supercritical (T

_{sup}= 138 °C) conditions as the results obtained by the oven experiment. Again, the temperature profiles predicted by both COMSOL one-step model and Gpyro agree well for both subcritical and supercritical conditions, except the temperature predicted by COMSOL grows a little bit faster than Gpyro for the supercritical condition. The temperature measured in the experiment grows faster than the prediction by the two models in the initial 100 min for both subcritical and supercritical conditions, which may result from the variation of geometry and heat transfer. Comparing times taken to reach 200 °C, it took around 230 min for the experiment to reach that temperature, while Gpyro predicts around 200 min and the COMSOL one-step model predicts around 160 min. Again, the predicted time by both models is lower than those measured in the experiments. Nevertheless, the accurate predictions on the transition from the subcritical to the supercritical condition demonstrate that both COMSOL and Gpyro are capable to predict self-heating ignition of LIBs with a one-step global reaction.

_{sub}= 145 °C for β = 1, and T

_{sub}= 142 °C for β = 1.14) and supercritical (T

_{sup}= 150 °C for β = 1, and T

_{sup}= 147 °C for β = 1.14) conditions, as shown in Fig. 5. The temperature at the initial stage is still overestimated (maximum 25 °C) compared to the experiments. For the time to reach 200 °C, it took around 80 min for the scenario of β = 1 and 100 min for the scenario of β = 1.14. These large differences show that the LIB contains more complex reactions, which significantly affect the internal temperature change, especially for the supercritical conditions. The simplified one-step and four-step models are insufficient to predict the time to thermal runaway. For the computational time, it took COMSOL around 18 min for the 1-D simulation and 35 min for the 3D simulation. While Gpyro took around 40 min for the 1D simulation.

_{a,cr}, predicted by the COMSOL four-step model is around 7 °C (β = 1) to 4 °C (β = 1.14) higher than the experimental results for both single cell and the four-cell ensemble. This difference is acceptable considering the large differences in geometry, size, and improved techniques between the old generation of cylindrical cells from Hatchard et al. [39] and the prismatic cell in this study. Moreover, the decline of T

_{a,cr}for the single cell to the four-cell ensemble is accurately predicted for all models, that the experiments showed a 7.5 °C drop and numerical models predict a 6 to 8 °C decline. These results demonstrate that both COMSOL and Gpyro with effective one-step reaction schemes and four-step reaction schemes are capable of predicting the self-heating ignition of LIBs.

Type of model | Case 3- single cell comparison | Case 4- four-cell comparison | ||||
---|---|---|---|---|---|---|

T _{sub} (°C) | T _{sup} (°C) | T _{a,cr} (°C) | T _{sub} (°C) | T _{sup} (°C) | T _{a,cr} (°C) | |

Oven experiments | 147 | 149 | 148 | 138 | 143 | 140.5 |

Gpyro | 147 | 148 | 148 | 138 | 143 | 140.5 |

COMSOL one-step | 146 | 147 | 146.5 | 138 | 143 | 140.5 |

COMSOL four-step β = 1 | 153 | 158 | 155.5 | 145 | 150 | 147.5 |

COMSOL four-step β = 1.14 | 150 | 155 | 152.5 | 142 | 147 | 144.5 |

## 4 Conclusions

_{2}battery cells, one-step (Gpryo) and four-step (COMSOL) reaction kinetics for chemistry, 3D modelling (COMSOL) and Pseudo 3D modelling (Gpyro) for heat transfer, and FEM (COMSOL) and FDM (Gpyro) for simulation performance. For physical modelling, Gpyro assumes a 1-D simplification for geometry while COMSOL supports direct 3D simulation on complex geometry. For mathematical modelling, Gpyro considers more complex physics which solve heat and mass conservation equations for both gas and solid phases but consider an overall global one-step reaction, while COMSOL adopts simplifications to solve conservation equations for solid-phase but support simulations with multi-step reactions. For numerical modelling, Gpyro and COMSOL use different numerical discretization methods, numerical calculations, algorithms, and computational efficiency. However, this study proves that both tools can accurately predict the critical ambient temperature to trigger thermal runaway, which is essential for predicting self-heating ignition of LIBs. This paper validates their use to study the safety of LIBs. Although this study focuses on self-heating, the validation also indicates the tools could be used for other LIB ignition events.