3D Thermal Simulation of Lithium-Ion Battery Thermal Runaway in Autoclave Calorimetry: Development and Comparison of Modeling Approaches

The mass loss during TR Δm_cell is calculated by

where m_cell is the cell weight before and is the cell weight after the test. The total heat released during TR can be separated in two parts: heat remaining in the cell Q₁ and heat being transported by venting gases and particles Q₂. ^25,26 As the autoclave calorimetry experiment is built to separate these effects, only heat conduction from the cell to the copper block is considered in this study and consequently Q₂ will be neglected. Q₁ is estimated as follows:

where t is the time since nail penetration, m is the mass of the components copper block (index: Cu), cell after TR (index: cell,TR), and thermal insulation (index: insulation), respectively, c_p is the specific heat capacity of the respective components, and ΔT is the temperature increase compared to the test's initial condition T_init of the respective components. As there is no temperature sensor information for the thermal insulation, its temperature increase is estimated by a simulation. For t = 2000 s after nail penetration, this results in a volume averaged temperature of approximately

The initial temperature T_init is defined by the initial mean value of all temperature sensors at the cell can and the copper block, that means in the center of each cell side (except the top side) and in the center of each corresponding copper block area (compare Fig. 1c).

The duration of venting t_venting is estimated as suggested by Scharner in Ref. 25. In the autoclave calorimetry experiment, the pressure curve shows a peak followed by an asymptotic approach to an equilibrium state. Scharner ²⁵ assumed, that the venting stops at the point in time when the pressure is

where p_EoV is the pressure at end of venting (index: EoV), p_eql is the pressure measured at temperature equilibrium inside the autoclave (index: eql), and is the maximum pressure. Consequently, the duration of venting is

where t_init is the point in time of nail penetration. Another parameter estimated during the experiment is the duration of measured temperature increase of the cell t_heating. It is set to the point in time when the cell reaches its maximum mean temperature (average value of all sensors on the cell can surface):

The onset temperature of continuous self-heating T_onset and the TR trigger temperature are estimated by ARC experiments. The parameters are estimated by three standard heat-wait-seek tests conducted with the same type of battery cell as used in the autoclave calorimetry experiment. The EV+ ARC manufactured by THT® (Thermal Hazard Technology) is used to perform the experiments. The following criteria are used:

• is detected as onset of self-heating and
• is detected as onset of TR.

Specific heat capacity measurements on full cell level are performed in order to estimate the specific heat capacity of the jelly roll c_p,JR. Therefore, two cells are stacked together with a heating pad in-between and the heating power needed for a temperature increase from 25 °C to 60 °C is analyzed. Unfortunately, temperatures higher than 60 °C could not be tested due to the safety limits defined by the cell manufacturer. With the resulting specific heat capacity of the full cell c_p,cell the specific heat capacity of the jelly roll is then calculated by

with m_cell being the mass of the full cell, m_x being the mass, and c_p,x being the specific heat capacity of each component x of the full cell (except jelly roll), respectively, and m_JR being the mass of the jelly roll. Table II summarizes all experimental results used as model parameters.

Table II. Experimental results used as model parameters.

Parameter	Cell 1	Cell 2	Mean	Unit
Δmcell	409.4	416.4	412.9	g
Tinit	20.9	22.4	21.7	°C
Q1	748.8	709.7	729.3	kJ
tventing	17.2	18.2	17.7	s
theating	33.1	46.7	39.9	s
Tonset	—	—	75.2 a	°C
	—	—	137.5 a	°C
cp,JR	—	—	see Fig. 3	J kg−1 K−1

^aMean value of three cells tested in ARC experiments.

As shown in previous work, a lithium-ion battery cell loses a significant amount of its mass during the autoclave calorimetry experiment. ²⁶ According to

this mass loss will influence the temperature increase ΔT of a battery cell during TR. Most of the previous studies do not account for this effect and assume the cell to maintain its full mass. ^{12–16,18–22,27–35} Coman et al. were one of the first to consider a loss of mass in their model by adapting the jelly roll density ρ_JR depending on the amount of vented electrolyte. ³⁶ The outcome was, that the consideration of venting and hence mass loss improves the simulation results compared to experiments. Other publications confirmed this observation. ^23,37 Unfortunately, all these models include a gas phase based on flow equations and therefore are not directly applicable to solid body simulations. However, Liu et al. and Coman et al. published models of solid body simulations, that adapt ρ_JR when TR occurs. ^17,24

As shown in Table II, the mean mass loss of the autoclave calorimetry experiments is Δm_cell = 412.9 g. To show the influence of this phenomenon, it is assumed that this mass loss originates from the jelly roll only and consequently, the mass loss is modeled by adapting the jelly roll density ρ_JR. In this study, three different (constant) jelly roll masses will be modeled (Table V, No. 1–3):

• "Full mass" assumes no mass loss and therefore, the standard jelly roll density of ρ_JR = 2620.6 kg m⁻³ is applied.
• "End mass" assumes that the mass loss already occurred before the heat release of TR and hence the jelly roll density is set to match the jelly roll mass after TR.
• "Half mass" is the mean value of the "Full mass" and "End mass" case.

As shown in Eq. 15, the specific heat capacity c_p is supposed to have the same influence on the temperature increase ΔT as the mass m. However, the mass (loss) during TR is time dependent, whereas a material's specific heat capacity is usually depending on temperature. Previous studies do not account for a temperature dependent specific heat capacity of the jelly roll. ^{13,14,17–21,23,24,27–32,34–38} Although there are some measurements of temperature dependent values as summarized in Ref. 39, a major issue with these is the limited temperature range (usually T < 60 °C). Due to the onset of TR at higher temperatures, there is a lack of experimental investigations and consequently an analysis of the specific heat capacity's influence on TR in simulation is necessary.

In order to analyze the influence of an increasing specific heat capacity of the jelly roll c_p,JR with rising temperature, three different dependencies according to Fig. 3 are implemented in this study. The black crosses show the results of specific heat capacity measurements on full cell level that are converted to values for the jelly roll by Eq. 8. The temperature dependency of c_p,JR is chosen to match these measurements. Outside of the measurement's temperature range, an asymptotic value of could be suggested. However, it is expected that a maximum value of would not lead to a significant change of the simulation results in comparison to the standard case of . Therefore, c_p,JR is set constant to three different (purely empirical) maximum values of , 1400 J kg⁻¹ K⁻¹, and 1600 J kg⁻¹ K⁻¹ (Table V, No. 4–6). It should be pointed out that for the sensitivity analysis of the mass loss during TR the specific heat capacity is set to .

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The third varied material property is the through-plane thermal conductivity of the jelly roll λ_JR. For prismatic cells with NMC cathode, values between 0.82 W m⁻¹ K⁻¹ < λ_JR < 1.1 W m⁻¹ K⁻¹ are reported. ^40,41 However, these values are estimated under normal operating conditions with temperatures of T < 60 °C and therefore not necessarily applicable to TR simulations. Comparing the experimentally measured values to assumptions made in previous publications for TR modeling shows a greater range. For example, Parhizi et al. and Mishra et al. used a value of λ_JR = 0.2 W m⁻¹ K⁻¹, ^27–30 whereas others use values up to λ_JR = 3.4 W m⁻¹ K⁻¹. ^{17,31–33,36,38}

In this study, the median value of λ_JR = 1.034 W m⁻¹ K⁻¹ as reported by Steinhardt et al. will be used as a baseline. ³⁹ This is also in accordance with other publications. ^{13,14,21,24,34,35} In order to evaluate the influence of lower and higher values, this baseline value will be halved and doubled (Table V, No. 7–8).

The first approach is based on an empirical time dependent heat release function that is derived from experimental data of ARC tests. A comparable approach was also used by Coman et al. in Ref. 17. Figure 4 shows the heat release over time t since TR initiation for three ARC experiments (colored dots) and the assumed heat release function for the simulation model (black curve). In the ARC experiments, the heat release is reaching a maximum directly after TR initiation for several seconds followed by a decrease to lower values. Comparing the venting duration of the autoclave calorimetry experiments of t_venting = 17.7 s to the ARC data shows that there is still a heat release detected for t > t_venting.

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This observed behavior is approximated by a superposition of a Gaussian distribution () and a linear function ():

with the normalization factor β:

The result of this superposition is the black curve shown in Fig. 4: as soon as the TR is triggered, heat is released in form of a Gaussian distribution (Eq. 18) with a subsequent linear decrease (Eq. 20) for the duration of (measured) temperature increase t_heating (see Table II):

and

Due to the normalization, it is possible to implement on several parts of a model and assign certain fractions of the total heat released. In this study, the heat release of method 1 is therefore modeled for each jelly roll part x as follows:

with α_x being the volume fraction of the jelly roll part x related to the total volume of the jelly roll (that means both middle parts and all 4 sides - compare Fig. 1b), Q₁ being the heat remaining in the cell as measured in the autoclave calorimetry experiments, Q_nail being the heat released in the nail for triggering the TR and being the maximum temperature in the jelly roll part x.

As stated in Eqs. 21–24, the heat release in each jelly roll part is initiated by exceeding the TR trigger temperature . In the autoclave calorimetry experiments, this is caused by the electrical short circuit due to the nail, that penetrates the jelly roll. To model the heat release of this short circuit, the approach of Feng et al. ¹⁸ is adopted:

with f(t) being the short circuit release rate in the nail as defined by Feng et al. ¹⁸ and α_nail = 0.01 being the fraction of heat that is released in the nail.

The second approach is based on a model proposed by Yeow et al. ¹⁵ in 2013. The heat source is modeled as a function depending on the volume averaged jelly roll temperature T_JR,avg. Yeow et al. ¹⁵ used a Gaussian distribution to approximate the self-heating rate of the cell vs. the cell temperature measured in ARC experiments. This approach is applied to the experimental data of this study: Fig. 5 shows the heat release over (measured) cell temperature T_cell of the three ARC tests (colored dots) and the assumed heat release function (black curve). The measured heat release is approximated by the mean value of the three tests and is directly implemented in the simulation model as table data set. The simulation software then linearly interpolates between these table values. After reaching in the ARC experiments, the number of measured data points is low. Therefore, the heat release rate is set to which is equivalent to a temperature rate of as conventionally measured during ARC experiments.

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As the heat release depends on the volume averaged temperature of the jelly roll, the electrical short circuit in the nail is not releasing enough heat to trigger the TR. Therefore, the initial temperature of the middle part of the nailed jelly roll is set to T_JR1mid,init = 150 °C. The heat release of method 2 is then modeled as a separate heat source for the middle and both side parts of each jelly roll:

where is the heat release interpolated from the curve shown in Fig. 5 depending on the volume averaged temperature T_x,avg of the jelly roll part x and Q_init is the amount of cummulated heat that is assumed to be already released due to the initial condition. It should be pointed out that there is no more heat release in the nail itself, as the initial condition triggers the TR in the middle part of the nailed jelly roll. The heat release in each part x stops as soon as the equivalent amount of Q₁ is released. This is achieved by the integration of over time t and monitoring the amount of released heat.

The third approach is based on a model proposed by Feng et al. ¹⁸ in 2016. The main difference to method 2 is that the heat release is depending on the local temperature in each volume element of the jelly roll mesh and therefore, the heat release is spatially resolved. Feng et al. ¹⁸ splitted the volumetric heat source into three parts:

with being caused by the electric short circuit in the nail, being caused by exothermal chemical reactions at elevated temperatures, and being caused by the electrical energy release after separator collapse followed by an internal short circuit in the cell. While is released in the nail only, and are released in the whole jelly roll material (compare Fig. 1b). In contrast to the original publication, the heat released during TR Q₁ is known from the autoclave calorimetry experiments. Therefore, and are simplified and put together into a single source term for the JR . Adopting from the original publication leads to: ¹⁸

with V_nail and V_JR being the volume of the nail and the jelly roll(s), respectively, α_nail = 0.01 being the fraction of heat released in the nail, f(t) being the short circuit release rate in the nail as defined by Feng et al. in Ref. 18, and c_e being the normalized concentration of energy in each volume element of the mesh. The reaction rate dc_e/dt is then defined as follows:

with m_JR being the mass of the jelly roll(s), A = 0.0185 K s⁻¹ being a pre-exponential factor, B = 24.94 being the reaction index, being used to normalize the temperature T, being the reaction rate during TR, and T_onset = 75.2 °C being the onset temperature of continuous self-heating in the ARC experiments. It should be pointed out that A, B, C, T_onset, and are adapted to fit the experimental results of this study and consequently differ from the values used by Feng et al. in Ref. 18.

Figure 6 shows the mean cell can temperature T_cell,mean (Fig. 6a) as well as the mean copper temperature T_Cu,mean (Fig. 6b) over time t since nail penetration for both experiments (black and black dashed) and the three simulations with different densities ρ_JR (colored). A variation of ρ_JR is equivalent to a variation of the jelly roll mass and consequently shows the influence of the mass loss during TR Δm_cell on the simulation results (orange for "Full mass", green for "Half mass" and blue for "End mass").

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Figure 6a illustrates that the mass of the jelly roll has an influence on both and the heat dissipation behavior of the cell after TR, which is indicated by the course of the temperature curve after reaching . The "Full mass" simulation shows the lowest maximum temperature with °C, followed by the "Half mass" case with °C. Assuming the jelly roll lost the vented mass before TR ("End mass") results in °C. Comparing these values to the mean of both experiments shows that is lower compared to the experiments for the "Full mass" case and higher for the "Half mass" as well as the "End mass" case. This observed behavior can be explained by Eq. 15: with a given Q and c_p , less mass m leads to a higher ΔT.

Another possible cause for the deviations is the assumption of constant values for ρ_JR. There are previous studies which considered the mass loss during TR by implementing a time dependency. ^{17,23,24,36,37} However, in the experiments, the investigated cells lose the vented mass over a time span of t_venting = 17.7 s (mean value). If it is considered that t_venting ≪ t_total = 2000 s, the "End mass" case is replicating the reality for approximately the entire amount of t_total. Therefore, it is hypothesized that the influence of the time dependent character of Δm_cell is negligible. Nevertheless, the heat dissipation behavior for this case is not matching the experimental data. A possible reason is the assumption of the specific heat capacity of the jelly roll c_p,JR being constant. Considering, in particular, that c_p,JR increases with rising temperatures, ³⁹ both the "Half mass" and "End mass" case show potential to be in better agreement with the experiments. A modeling of the full jelly roll mass is not reasonable as ΔT would be even lower.

Figure 6b shows the temperature increase over time t for the copper block. For the "Full mass" and the "Half mass" case, this increase in temperature is slower compared to the experiments, whereas the temperature increase of the "End mass" case is faster. Quantifying the deviation at t = 500 s results in

Consequently, the "Half mass" case shows the smallest deviation from the experiments. This is confirmed by analyzing the MAPE and RMSPE (see Table VI). However, if it is taken into account that c_p,JR is assumed constant instead of temperature dependent, it is not reasonable to model the "Full mass" or the "Half mass" case. This would result in the copper temperatures being lower and consequently in a underestimation of the heat transfer to adjacent components. This is a problem, in particular, for TR propagation simulations as the predicted propagation time would be higher compared to reality and hence would give a distorted impression of safety. Therefore, the "End mass" case is used as a baseline for further sensitivity analysis.

Table VI. Mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE) between mean cell can temperature and mean copper temperature of the simulations and the mean experimental data.

		MAPE	MAPE	RMSPE	RMSPE
No.	Name	Tcell,mean	TCu,mean	Tcell,mean	TCu,mean
		[%]	[%]	[%]	[%]
1	Full mass	5.9	6.6	7.0	8.1
2	Half mass	4.2	2.1	4.9	3.1
3	End mass	14.2	2.9	16.2	3.9
4		9.3	1.2	10.6	1.7
5		4.8	1.1	5.6	2.1
6		1.4	2.4	2.5	3.8
7	λJR = 0.517	2.7	3.3	4.0	4.9
8	λJR = 2.068	9.7	1.4	11.0	1.8

In conclusion, the mass loss during TR has to be considered in the simulation model to reproduce the experimental results. The "End mass" case shows the highest potential, although both the cell and copper block temperatures are higher than the experimental results. However, when accounting for a specific heat capacity of the jelly roll that increases with rising temperatures, the temperatures are lowered.

Figure 7 shows the mean cell can temperature T_cell,mean (Fig. 7a) as well as the mean copper temperature T_Cu,mean (Fig. 7b) over time t since nail penetration for both experiments (black and black dashed) and the four simulations with different specific heat capacities of the jelly roll c_p,JR. It should be pointed out that all shown simulations are calculated with ρ_JR = 1360.1 kg m⁻³ and therefore assume the cell to have the mass after a TR ("End mass" case).

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As shown in Fig. 7a, c_p,JR has a similar influence on the investigated temperatures as ρ_JR, although c_p,JR is depending on temperature instead of being constant as ρ_JR. The maximum mean cell can temperature is reduced by adding a temperature dependency for c_p,JR:

Comparing these values to the experimental data results in the temperature of the case to be in-between both experiments. For the other cases, is higher than both experiments. Interestingly, comparing the constant c_p,JR (orange) to the temperature dependent c_p,JR (green, blue and red) shows no significant difference in the course of the mean temperature curve of the cell can. This can be explained as follows: the temperature dependent curves of c_p,JR differ in a temperature range of approximately 40 °C < T < 120 °C (compare Fig. 3). The experimental data shows that the mean temperature is outside of this range at t ≈ 4 s. Consequently, the c_p,JR-curves are in the constant regime for a large amount of the simulation time and therefore, the temperature dependency has only a minor effect. Nevertheless, can be lowered by increasing . At the same time, the heat dissipation and therefore, the temperature decrease of the cell slows down. This behavior was also observed in a previous study. ²⁷

Figure 7b shows the effect of c_p,JR on T_Cu,mean. An increasing leads to a slower increase of T_Cu,mean as the cell temperatures are lower and therefore, the driving force of the heat transfer, which is the temperature difference between cell surface and copper block, is smaller. Quantifying the deviation at t = 500 s results in

Consequently, the case with shows the smallest deviation. Taking into account the MAPE and RMSPE values confirms this result for T_Cu,mean, whereas for T_cell,mean, the case with shows the smallest deviation from the experimental data (compare Table VI).

In conclusion, the specific heat capacity of the jelly roll is found to have a strong impact on the temperatures during TR. The curve with is assumed to be the best fit to the test results. Although the mean cell temperature is not matching as good as for the case, the mean copper block temperature is in better agreement. Therefore, is set as baseline for the further parameter study.

Figure 8 shows the mean cell can temperature T_cell,mean (Fig. 8a) as well as the mean copper temperature T_Cu,mean (Fig. 8b) over time t since nail penetration for both experiments (black and black dashed) and the three simulations with different through-plane thermal conductivity of the jelly roll λ_JR (colored). It should be pointed out that all shown simulations are calculated with ρ_JR = 1360.1 kg m⁻³ and c_p,JR(T) with .

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As shown in Fig. 8a, a variation of λ_JR (through-plane) has the same effect on the simulation results as a variation of ρ_JR or c_p,JR. The baseline case with λ_JR = 1.034 W m⁻¹ K⁻¹ results in . Reducing the thermal conductivity to λ_JR =0.517 W m⁻¹ K⁻¹ shows a reduction of the maximum mean cell can temperature to , whereas an increase to λ_JR = 2.068 W m⁻¹ K⁻¹ leads to . Consequently, increasing λ_JR leads to higher temperatures at the cell can. At the same time, this results in lower core temperatures as shown in previous studies. ^27,42 This behavior can be explained by a higher heat conduction from the center of the jelly roll to the cell can. In contrast, reducing λ_JR slows down the internal jelly roll heat conduction and therefore, the heat is dissipated slower. In comparison to the experiments, the low thermal conductivity is in best agreement for T_cell,mean (compare Table VI).

Figure 8b shows the effect of λ_JR on T_Cu,mean. A reduction of λ_JR leads to a slower increase of T_Cu,mean, as the heat released in the jelly roll is transferred slower to the cell can and eventually to the copper block. High values of λ_JR consequently lead to a faster increase of T_Cu,mean. Quantifying the deviation t = 500 s results in

The case with λ_JR = 1.034 W m⁻¹ K⁻¹ shows the smallest deviation between simulation and test results. Taking into account the MAPE and RMSPE values does confirm this observation for T_Cu,mean, whereas for T_cell,mean the case with λ_JR = 0.517 W m⁻¹ K⁻¹ shows the smallest deviation from the experimental data (compare Table VI).

In conclusion, the through-plane thermal conductivity is found to have a significant influence on the temperatures during TR. For the investigated case, a lower λ_JR leads to better results for T_cell,mean, but the temperature increase of the copper block is too slow. Therefore, a thermal conductivity of λ_JR = 1.034 W m⁻¹ K⁻¹ is used for the subsequent comparison of modeling approaches for the heat release during TR.

In summary, all of the three investigated parameters show a similar influence on the simulation results. Consequently, there are various possibilities to fit the simulation results to experimental data. However, the authors recommend to choose the investigated parameters within a physically reasonable range and want to stress that further experimental studies on temperature dependent material parameters of the jelly roll in the temperature range of a TR are crucial for the improvement of TR simulation accuracy.

Figure 9 shows the mean cell temperature T_cell,mean (Fig. 9a) as well as the mean copper temperature T_Cu,mean (Fig. 9b) over time t since nail penetration for both experiments (black and black dashed) and the three simulations with different modeling approaches for heat release during TR (colored). It should be pointed out that all shown simulations are calculated with ρ_JR = 1360.1 kg m⁻³, c_p,JR(T) with (see Fig. 3), and λ_JR = 1.034 W m⁻¹ K⁻¹.

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As shown in Fig. 9a, the modeling approach of heat release during TR has a minor influence on the temperature curves for t ≫ 0. This is explained by the short duration of heat release compared to the heat dissipation process. However, there is an influence on the maximum mean cell temperatures: method 1 shows the lowest temperatures with °C, followed by method 2 with °C, and method 3 with °C.

Figure 9b shows the temperature increase over time t for the copper block. As also observed for T_cell,mean, the modeling approach of heat release during TR has a minor influence on T_Cu,mean for t ≫ 0. Comparing the MAPE and RMSPE values confirms this observation (compare Table VII), as the difference between the three methods is ΔMAPE_cell,mean = 0.9 % for the cell and ΔMAPE_Cu,mean = 0.5 % for the copper block. The difference for the RMSPE is slightly higher with ΔRMSPE_cell,mean = 2.0 % for the cell and ΔMAPE_Cu,mean = 0.8 % for the copper block. Therefore, it is concluded that for processes with focus on the heat dissipation after a TR (t ≫ 0) it is more critical to ensure a correct implementation of boundary parameters such as mass loss during TR or material parameters of the jelly roll than to precisely model the time dependent character of heat release. However, if the focus is on the single cell and its TR process, the modeling approach of heat release can be of significant importance.

Table VII. Mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE) between mean cell can temperature and mean copper temperature of the simulations and the mean experimental data.

		MAPE	MAPE	RMSPE	RMSPE
No.	Name	Tcell,mean	TCu,mean	Tcell,mean	TCu,mean
		[%]	[%]	[%]	[%]
Method 1		4.8	1.1	5.6	2.1
Method 2		4.4	1.4	6.2	1.7
Method 3		5.3	0.9	7.6	1.3

Figure 10 shows the cell can temperatures T_cell,k at the positions k (Figs. 10a–10e) as well as the copper temperatures T_Cu,k at the equivalent positions k (Figs. 10f–10j) over time t since nail penetration for both experiments (black and black dashed) and the three simulations with different modeling approaches for the heat release during TR (colored). It should be pointed out that all shown simulations are calculated with ρ_JR = 1360.1 kg m⁻³, c_p,JR(T) with (see Fig. 3), and λ_JR = 1.034 W m⁻¹ K⁻¹. In addition, Table VIII summarizes the three characteristics for sensor curve comparison as described above.

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Table VIII. Comparison of temperature curve characteristics of each sensor on the cell can for both experiments and the three different modeling approaches for heat release during TR.

Cell can temperature on nail side Tcell,nail
No.	Name	Time tbegin at T > 50 °C		Mean ΔT/Δt
Exp. 1	—	t = 1.3 s	faulty	faulty
Exp. 2	—	t = 1.7 s	696 °C	135 K s−1
Method 1		t = 0.7 s	688 °C	37 K s−1
Method 2		t = 0.0 s	809 °C	71 K s−1
Method 3		t = 0.7 s	771 °C	122 K s−1
Cell can temperature on bottom side Tcell,bot
Exp. 1	—	t = 3.4 s	439 °C	8 K s−1
Exp. 2	—	t = 5.3 s	460 °C	10 K s−1
Method 1		t = 6.9 s	538 °C	14 K s−1
Method 2		t = 4.3 s	548 °C	15 K s−1
Method 3		t = 4.8 s	543 °C	18 K s−1
Cell can temperature on side of negative terminal Tcell,neg
Exp. 1	—	t = 6.3 s	582 °C	94 K s−1
Exp. 2	—	t = 5.9 s	585 °C	37 K s−1
Method 1		t = 3.0 s	631 °C	35 K s−1
Method 2		t = 6.0 s	740 °C	148 K s−1
Method 3		t = 9.9 s	779 °C	141 K s−1
Cell can temperature on side of positive terminal Tcell,pos
Exp. 1	—	t = 6.2 s	531 °C	20 K s−1
Exp. 2	—	t = 6.4 s	489 °C	12 K s−1
Method 1		t = 3.0 s	631 °C	34 K s−1
Method 2		t = 6.0 s	740 °C	147 K s−1
Method 3		t = 9.9 s	779 °C	141 K s−1
Cell can temperature on opposite side of nailing Tcell,opp
Exp. 1	—	t = 10.1 s	652 °C	32 K s−1
Exp. 2	—	t = 10.5 s	573 °C	15 K s−1
Method 1		t = 3.4 s	662 °C	30 K s−1
Method 2		t = 11.2 s	761 °C	63 K s−1
Method 3		t = 8.3 s	804 °C	214 K s−1

Figure 10a presents the cell can temperature on the nailed side with a focus on the time span of the TR and the early heat dissipation. There is a high deviation between both experiments for this certain position. The data of experiment 1 shows temperatures that are approximately 200 °C higher compared to experiment 2. A possible reason for this behavior is that gas vented through the nail penetration hole and influenced the temperature measurement. Therefore, only experiment 2 is considered to show plausible results for this particular position.

In contrast to the long term behavior, the focus on the first seconds after nail penetration reveals clear differences between the single modeling approaches. The temperature increase on the nail side starts earlier compared to the experiment for all three modeling approaches (compare Fig. 10a and Table VIII). Method 2 stands out in particular: as the initial temperature of one jelly roll has to be set high enough to trigger the heat source, the temperature at t = 0 s is already at an elevated level. With respect to the maximum temperature , method 1 only shows a small deviation, whereas method 2 and 3 result in higher values. Analyzing the mean temperature gradient ΔT/Δt leads to method 3 being in best agreement with the experiment, while the temperature increase of method 2 and, in particular, method 1 is too low for the sensor on the nailed side.

This observed behavior can be explained by the different modeling approaches. The more heat is released in less time, the higher the maximum temperature becomes due to less time for heat dissipation. For method 1, the total "heating power" of the jelly roll is directly set by the modeling approach with a maximum of at (see Fig. 4). For method 2, the maximum power is defined as , but it can occur in each of the four jelly roll parts independently. For the specific case presented in this paper, only two jelly roll parts release heat at the same time, which leads to . This maximum value is already reached at t ≈ 6 s and therefore, the gradient is higher compared to method 1. In addition, the duration of heat release for method 2 is Δt_M2,heating ≈ 21 s, while the total time of heat release for method 1 is set to Δt_M1,heating = 39.9 s. The duration of heat release for method 3 is Δt_M3,heating ≈ 12 s, which leads to a maximum "heating power" of and hence the highest temperature gradient as well as maximum temperature.

Focusing on the other positions on the cell can and comparing the begin of temperature increase reveal a disadvantage of method 1: Due to the time dependent heat release in the whole jelly roll material, the temperature increase starts to early, especially on the short sides and the opposite side of nailing (compare Figs. 10a–10e and Table VIII). For method 2, the begin of temperature increase fits to the experimental data. Although both method 1 and 2 release the heat homogeneously, method 2 triggers the second jelly roll not as fast as method 1. This is due to the different trigger mechanisms. For method 1, the maximum temperature is decisive for the begin of heat release, while method 2 uses the volume averaged temperature of the jelly roll as trigger criterion. Therefore, the second jelly roll is triggered at a later point in time and consequently the temperature on the opposite side of nailing increases later. The heat release of method 3 is spatially resolved and therefore, the TR reaction is propagating through the jelly roll in form of a reaction front. The velocity of this front is mainly depending on the reaction rate C. ²⁰ On the one hand, decreasing the reaction rate C would help to match the simulation results with the sensor on the opposite side of nailing, but on the other hand it would also shift the short sides to a later point in time. That is why it is hypothesized, that the reaction rate is actually direction dependent, that means that the through-plane direction is too fast in this case, but the in-plane direction is too slow. Another possible reason is a wrong assumption for the in-plane thermal conductivity of the jelly roll, as the latter also has an influence on the velocity of the reaction front. ²⁰

Comparing the maximum temperatures of simulations and experiments shows that method 2 and method 3 overestimate for all sensor positions on the cell can. However, this overestimation is occurring as a temperature peak and therefore, the deviation exists only over a short amount of time. Method 1 better estimates the maximum temperatures.

The mean temperature gradient ΔT/Δt of method 1 shows in general the smallest deviations, but there is still room for improvement, such as the nail side sensor (compare Figs. 10b–10e and Table VIII). Method 2 results in higher gradients compared to the experiments except for the sensor on the nailed side, which is directly depending on . Consequently, this value is assumed to be too high. Method 3 shows the highest mean ΔT/Δt except for the short sides. Therefore, the reaction rate during TR C = 1 s⁻¹ is also assumed to be too high.

Figures 10f–10j shows the copper block temperatures at the given positions with a focus on the time span of TR and the early temperature increase. The observations on the cell side also match to the behavior on the copper block side. A fast temperature increase on the cell side leads to a fast temperature increase on the copper block side. Except for the nailed side, all three simulation methods are in good agreement with the experimental results. As stated before, on the nailed side, there may be some deviations due to the venting gases flowing out of the nail penetration hole.

There are further observations that have an influence on the results of all three methods:

• Melting of the aluminum can is not considered in the model. Especially on the nailed side the temperature sensors show temperatures higher than the melting point of aluminum and therefore, the phase change of the can absorbs additional heat.
• The assumptions for the material parameters of the bottom insulator may be inaccurate, for example there is no temperature dependency implemented in the simulation model. A rising specific heat capacity of the bottom insulator with increasing temperature would result in lower temperatures.
• Internal venting gas flow is not considered. This could either cause an additional convective heat transfer to the cell can or transport energy out of the cell through the vent.
• A possible reason for the difference of positive and negative side in the experiments is the different material of the current collectors. It is safe to conclude, that for the negative side (copper current collector) the heat transfer into the side of the can is higher, as the thermal conductivity is higher compared to aluminum. In addition, the specific heat capacity of copper is lower compared to aluminum. A third possible reason is the melting of the aluminum current collector which results in a heat dissipation due to the phase change. As the simulation model is symmetrical (except from the material of the simplified terminals) there is no significant difference between the simulation results on the short cell sides and hence, this behavior cannot be replicated.
• The gas and particle flow out of the vent is not considered in the simulation model. Both could result in an additional heat transfer to the copper block, for example (hot) particles rebounding from the retainer and sticking to the copper block or convective heat transfer from gas to the retainer and subsequent heat conduction into the copper block.

In conclusion, all modeling approaches are able to replicate the TR behavior in the autoclave calorimetry experiment. However, all methods show potential for further improvement, which can be achieved by adapting the heat source parameters. Another opportunity for improvement of the simulation results are temperature dependent material parameters for all components or the consideration of melting.

Table IX compares the modeling approaches for heat generation during TR regarding the accuracy _Q according to Eq. 36, the needed time step Δt_step to reach this accuracy, and the total computational time needed for the simulation. In addition, the input as well as the fitting parameters and further advantages or disadvantages are summarized.

Table IX. Comparison of the modeling approaches for heat release during TR.

Criteria	Method 1	Method 2	Method 3
Accuracy Q	0.60%	1.91%	1.06%
Time step Δtstep	0.05 s	0.02 s	0.005 s
Computational time	9954 s	36091 s	149555 s
Needed input parameters	• Q1	• Q1	• Q1
	•	•	•
			•
			• Tonset
(Optional) fitting parameters	• tventing	• Tonset	• C
	• theating	•	• αnail
	• αnail	•
Miscellaneous	Lowest computational effort	Easy to implement and parameterize, but high initial temperature necessary	Spatial resolution of TR reaction, but small time step necessary

Method 1 shows the best accuracy _Q with the biggest time step Δt_step and therefore, the lowest computational time. To maintain _Q < 2% for method 2 and 3, the time step has to be reduced and subsequently the computational time is longer. The number of needed input parameters is higher for method 3 in comparison to method 1 and 2. However, all input parameters can be estimated by a calorimetric measurement of Q₁ and an ARC test with the heat-wait-seek method. The main advantage of method 1 is the low computational effort and therefore, it is suitable for simulations of large models such as battery packs. On the other hand, unlike the other methods, method 1 is not able to reproduce the self-heating of the battery call that occurs for . Whether this is crucial for TR propagation simulation needs to be investigated in further work. Method 2 is easy to implement and parameterization requires a minor effort, as all fitting parameters can be estimated by ARC tests. However, the necessity of a high initial temperature can cause deviations. Method 3 is the only method that is able to model a spatial resolution of the TR reaction, but requires a higher computational time.