Lithium-Ion Battery Thermal Runaway Propagation Simulation Using Joint Model of Lumped-Parameter Method for Shell and 3D Modeling for Jelly Roll

Abstract

Models of thermal runaway propagation in lithium-ion batteries are widely used for thermal safety analysis. Current methods, primarily lumped-parameter and 3D models, face challenges in balancing accuracy with computational efficiency. Three-dimensional models offer high accuracy at high computational cost, while lumped-parameter models are faster but less accurate. For instance, the battery shell is included in lumped-parameter models but often omitted in 3D models. This study focuses on a 37 Ah ternary lithium-ion battery, with Li(NiCoMn)_1/3O₂ as the cathode material and graphite as the anode material. The propagation of thermal runaway in the battery array is triggered by nail penetration. A lithium-ion battery thermal runaway propagation model is proposed, combining the lumped-parameter method with 3D modeling. The model primarily describes the heat transfer characteristics of the shell using a series connection of thermal capacitance and several thermal resistances. The shell temperature is then calculated by weighting the temperatures associated with the thermal capacitance and thermal resistances using specific weight coefficients. The joint model is detailed and applied to study thermal runaway propagation in one- and two-dimensional battery arrays. For the one-dimensional array, based on the three-dimensional simulation data and calculation time, the joint model shows only a 1.32% average deviation in propagation time compared to full 3D simulation, while maintaining good temperature agreement. It also reduces solution time by 70.22%. These findings confirm that the proposed model effectively enhances both the efficiency and accuracy of thermal runaway simulations, supporting improved safety analysis for lithium-ion battery systems.

1. Introduction

Owing to advantages such as zero tailpipe emissions and a significant reduction in carbon emissions in the transportation sector, electric vehicles have experienced rapid development in recent years [1]. Lithium-ion batteries, benefiting from high energy and power densities, no memory effect, and a long cycle life, are widely used as the power source for electric vehicles [2]. However, lithium-ion batteries in battery arrays may experience thermal runaway due to thermal abuse, electrical abuse, and mechanical abuse [3]. As the temperature of an individual battery increases, exothermic reactions are intensified, leading to a rapid and uncontrollable temperature rise. Once the critical temperature is exceeded, thermal runaway (TR) is triggered. Moreover, because of the strong thermal coupling and electrical connections among individual batteries in a battery array, once thermal runaway occurs in a single battery, the heat released will raise the temperature of neighboring batteries and trigger their thermal runaway, resulting in a chain reaction known as thermal runaway propagation (TRP), which may lead to fire or explosion accidents. Therefore, investigating thermal runaway propagation inhibition technologies is of great importance for improving the safety and reliability of electric vehicles.

As 41% of thermal runaway propagation accidents occur under the static state of electric vehicles (with the battery system shut down) [4], and the battery operating voltage rapidly drops to zero after thermal runaway is triggered [5], passive methods are predominantly adopted to inhibit thermal runaway propagation. Passive thermal runaway propagation inhibition technologies mainly involve filling thermal insulation materials between batteries, which can block approximately 65% of heat transfer [6]. Miao et al. [7] developed a calcium chloride hexahydrate-based composite phase change material, using ceramic fibers as the supporting material and carbon nanofibers as the binder and nucleating agent. This material meets the requirements of low-temperature start-up, battery cooling, and thermal runaway propagation inhibition, while exhibiting satisfactory flame retardant performance: no combustion was observed at a limiting oxygen index of 100%. Thermal runaway propagation could be completely inhibited when the distance between the battery and the composite phase change material (CPCM) was 1.8 mm. Qiu et al. [8] developed a flame-retardant flexible composite phase change material by coating a flame-retardant paint, containing 15% of a binder composed of 70% polydimethylsiloxane, onto a composite material consisting of 80% paraffin and 20% expanded graphite doped with 40% ethylene–propylene–diene monomer. When the flame-retardant coating thickness was 265 μm, and the flexible substrate thickness was 3 mm, thermal runaway propagation could be effectively inhibited, and the thermal conductivity of the composite material was 1.123 W·m⁻¹·K⁻¹. Chen et al. [9] investigated the inhibition behavior of thermal runaway propagation using a sandwich structure composed of a flame-retardant phase change material (FRPCM) and an aerogel fiber felt (AEGF). They found that thermal runaway propagation could be completely inhibited when the FRPCM thickness was 3 mm, and the AEGF thickness was 2 mm, or when the FRPCM thickness was 1 mm, and the AEGF thickness was 3 mm. Zhou et al. [10] studied the critical conditions for inhibiting thermal runaway propagation using TCM40-based CPCM, revealing that thermal runaway propagation could be inhibited when the thickness ranged from 2 to 4 mm and the thermal conductivity was in the range of 0.50~24.57 W·m⁻¹·K⁻¹. Li et al. [11] developed a multifunctional composite thermal barrier material composed of nano-ceramic fibers, a water-based phase change material, and a mica cross-framework encapsulated in an aluminum–plastic film. The thermal barrier had a thickness of 3.44 mm and an ultra-low thermal conductivity of only 0.0349 W·m⁻¹·K⁻¹, enabling complete inhibition of thermal runaway propagation in a 153 Ah prismatic lithium-ion battery.

Although passive technologies can effectively inhibit thermal runaway propagation, the increased inter-cell thermal resistance is unfavorable for battery thermal management, which has become a major challenge for this approach. Hybrid technologies that combine passive thermal insulation with active cooling can block thermal runaway propagation while reducing the inter-cell thermal resistance. Sun et al. [12] combined active liquid cooling with passive cooling using a composite material consisting of copper foam and expanded graphite–paraffin (EG–PCM), and designed a hybrid battery thermal management system. The results revealed that thermal runaway propagation could be inhibited when the working fluid flow velocity was 0.3 m/s, using EG–PCM with a melting point of 52 °C and a thermal diffusivity of 9.68 mm²/s, together with copper foam with a porosity of 0.7~0.9. Xiao et al. [13] designed a hybrid cooling system composed of composite phase change materials and liquid cooling, and found that when the coolant flow velocity was 0.029 m/s, thermal runaway propagation was delayed by 32.9 s. In addition, reducing the thermal conductivity of the CPCM could further delay thermal runaway propagation, and when the number of cooling plate channels ranged from 5 to 1, the thermal runaway inhibition time varied between 25.5 s and 27 s. Ouyang et al. [14] proposed a battery thermal management system combining composite phase change materials, aerogel, and liquid cooling. Under thermal runaway conditions, when the liquid cooling flow rate was set to 0.42 L/min, the maximum battery temperature could be controlled at 344.36 K, and only 320 s were required to reduce the maximum battery temperature to a safe range. Yu et al. [15] verified through experiments that using a composite phase-change insulating material composed of highly entangled silica fibers and active liquid cooling can effectively prevent the spread of thermal runaway in a 40 Ah large-capacity lithium iron phosphate battery module. Xie et al. [16] designed a battery thermal management system combining composite phase change materials, aerogel, and liquid cooling, which could achieve thermal runaway propagation inhibition for lithium-ion battery arrays arranged in 4S12P, 6S8P, 8S6P, and 12S4P configurations through flexible regulation of the coolant flow rate.

Although hybrid methods can block thermal runaway propagation under relatively small thermal insulation resistance, active cooling technologies require continuous power input to maintain operation and are therefore difficult to apply under static battery conditions. In our previous studies, a pseudo-passive power battery thermal runaway propagation inhibition system was proposed, in which the key mechanism achieves pseudo-passive heat removal by integrating semiconductor thermoelectric modules installed on the side of the battery array with a pump-driven cooling device [17]. This system can operate under the static state of batteries and can significantly reduce the inter-battery thermal insulation resistance required to inhibit thermal runaway propagation [18].

To further comprehensively investigate the performance of the pseudo-passive power battery thermal runaway propagation inhibition system, it is necessary to establish models that can accurately describe the thermal propagation blocking processes within both the battery shell and the jelly roll. Since the heat dissipation devices (pump-driven cooling units and thermoelectric modules) of the pseudo-passive power battery thermal runaway suppression system are closely attached to the surface of the battery array, accurately predicting the temperature of the battery casing is crucial for analyzing the heat dissipation performance. Additionally, the battery core temperature is a primary factor influencing battery heat generation and thermal runaway propagation.

Table 1. Representative lithium-ion battery thermal runaway propagation models in recent years.

Reference	Modeling Method	Precision
A lumped
Mishra et al. [19] in 2024	electrochemical–thermal model	Error < 6%
He et al. [20] in 2024	A reduced-order TR model that distributes the TR heat source proportionally	Featuring less than a 1 min error in the TR moment
Sorensen et al. [21] in 2024	Heat accumulation with radiation and convection (no conduction)	Within 1.98% relative error
Lu et al. [22] in 2024	A lumped thermal resistance model based on electrical circuit analogy	Good temperature and thermal runaway onset time agreement
Sadeghi et al. [23] in 2025	Electrochemical/thermal coupled modeling with inverse parameter estimation via Genetic Algorithm	Optimization errors range from 0.039% to 1.531%
Sun et al. [24] in 2025	A hybrid model coupling a lumped model, a P2D model, and a 3D computational fluid dynamics (CFD) model	The simulation results are within the experimental data range and exhibit a consistent evolution trend
Menz et al. [25] in 2023	A lumped-element thermal network model with integrated heat-absorbing barrier effects	Good temperature and thermal runaway onset time agreement
A 3D model
Uwitonze et al. [26] in 2024	ignoring the casing structure and integrating the NTGK-ISC-thermal runaway kinetics model	T-test p-value = 0.461 (≫0.05)
Jindal et al. [27] in 2021	A 3D-1D coupled model with a 3D conjugate heat transfer model coupling with a 1D electrochemical model	Good temperature and thermal runaway onset time agreement
Hong et al. [28] in 2024	Ignored casing structure and simplified a two-phase boiling heat transfer model based on heat flux/superheat relationship	Maximum error on the maximum temperature is 14.8%
Zhang et al. [29] in 2024	A 3D unsteady heat conduction model based on experimental data	Acceptable simulation errors: 3.2% for TR onset temperature and 2.9% for onset time
Choi et al. [30] in 2025	An NTGK electrochemical model integrated with an NREL thermal runaway model, and analyzed via Random Forest and LSTM	Time error ≈ 3.7%, temperature error ≈ 0.3%
Zhao et al. [31] in 2026	An electrochemical/thermal model coupled with a thermal runaway model	Error < 2.8%
Liu et al. [32] in 2024	An electrochemical–thermal abuse model integrated with a discrete-phase spray model and Navier–Stokes equations	Good temperature and thermal runaway onset time agreement
Chen et al. [33] in 2024	Considering the coupling of temperature gradient, chemical reactions, and gas flow	The simulation data compared with the experimental data show an R2 ≥ 99.35%
Tan et al. [34] in 2025	Ignoring casing structure and coupling multi-stage thermal abuse reactions	Good temperature and thermal runaway onset time agreement
Zhang et al. [35] in 2023	3D heterogeneous heat source modeling	Error < 6.3%

summarizes representative models of thermal runaway propagation developed in recent years, which are mainly constructed using lumped-parameter methods and 3D modeling approaches. Consequently, existing models for lithium-ion battery thermal runaway propagation still exhibit limitations in simultaneously achieving high accuracy and short solution time. Specifically, 3D models provide high accuracy but require excessively long computational times, whereas lumped-parameter methods offer short solution times but suffer from relatively low modeling accuracy.

To address the aforementioned research gaps, this study proposes a joint modeling approach that combines the lumped-parameter method with a three-dimensional model. The aim is to accurately describe the thermal characteristics of both the jelly roll and the shell during thermal runaway propagation while maintaining acceptable solution times. The remainder of this article is organized as follows: Section 2 presents the joint lithium-ion battery thermal runaway propagation model based on the lumped-parameter method and 3D modeling, along with model validation. Section 3.1 compares the thermal runaway propagation simulation results of the joint model with those of a full 3D model for a one-dimensional array, demonstrating the advantages of the joint model in improving computational efficiency. Section 3.2 presents an investigation of the thermal runaway propagation behavior in a two-dimensional battery array using the joint model. Section 4 discusses the outlook and challenges. Section 5 summarizes the main conclusions of this article. The primary innovation of this study lies in the development of a joint modeling method for lithium-ion battery thermal runaway propagation that integrates the lumped-parameter method with a 3D model, effectively balancing prediction accuracy and solution time. At the same time, although methods such as regression analysis or machine learning techniques (e.g., neural networks) may offer certain advantages in prediction tasks, these methods require a large amount of experimental data for effective training and validation. The proposed joint model, however, does not rely on extensive experimental data; instead, it is based on the coupling of physical models, providing clear interpretability. This study contributes to the advancement of lithium-ion battery thermal runaway simulation and enhances the safety of electric vehicles.

2. Numerical Simulation

The battery thermal runaway propagation model established in this article, which integrates the lumped-parameter method with three-dimensional modeling, primarily describes the heat transfer characteristics of the battery shell and the jelly roll using the lumped-parameter approach and the 3D model, respectively (as shown in Figure 1). During thermal runaway propagation, the battery jelly roll exhibits a non-uniform temperature distribution due to its relatively low thermal conductivity, and thermal runaway is triggered when the temperature at a certain location within the jelly roll reaches the separator collapse temperature [36]. Therefore, employing a three-dimensional model to describe heat transfer within the jelly roll is beneficial for improving model accuracy. For the battery shell, which is thin and typically fabricated from metal (most commonly aluminum, with a thickness not exceeding 2 mm), the use of the lumped-parameter method to characterize the corresponding heat transfer process is advantageous for reducing the overall solution time. In this study, COMSOL Multiphysics 6.1 is primarily used for model development.

Figure 1a illustrates the three-dimensional model of a one-dimensional array in the joint thermal runaway propagation model. Except for the nail-penetrated battery Cell-0, the remaining propagation batteries are not subjected to nail penetration. Figure 1b shows the yz-direction cross-sectional view of the propagation battery modeled using the lumped-parameter method, while Figure 1c presents the xz-direction cross-sectional view of the propagation battery established using the same lumped-parameter approach.

2.1. Battery Jelly Roll Model

2.1.1. Governing Equation

The temperature distribution of the jelly roll during thermal runaway propagation is calculated using the fundamental heat transfer equation, as expressed in Equation (1) [18]:

$$\rho C_p{\displaystyle \frac{dT}{d\tau }}=q_v+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}({\lambda }_x{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }x}})+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}({\lambda }_y{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }y}})+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}({\lambda }_z{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }z}})$$

(1)

The heat generation power of battery thermal runaway induced by thermal abuse is expressed as shown in Equation (2) [18]:

$$q_{v,Bat}=q_{v,JR}={\displaystyle \frac{1}{V_{JR}}}\cdot \Delta H_{TR}\cdot {\displaystyle \frac{du}{d\tau }}$$

(2)

where the parameter ΔH_TR represents the total heat generated by Joule heating due to internal short circuits after separator collapse and by exothermic chemical reactions at elevated temperatures, with its value taken from Ref. [37]. Since solid particles and gases are ejected during the thermal runaway process, according to Ref. [38], the heat loss is 165 KJ. The specific calculation is given in Equation (3) [18]:

$$H_{TR}={\displaystyle \underset{V_{JR}}{\int }{\displaystyle {\int }_{T_1}^{T_3}{\rho }_{JR}(T)C_{p,JR}(T)\cdot \frac{du}{d\tau }}}$$

(3)

Due to the difficulty in measuring the thermophysical properties of the battery jelly roll during the TR process, despite their critical importance to the calculations, the values of the thermophysical parameters before and after TR were measured and linearly interpolated. The corresponding data are provided in Ref. [38], and the calculation formula is given in Equation (4) [18]:

$$r_i(T)=\left\{\begin{array}{ll}{r}_{i0}& (T<T_2,u>0)\\ r_{i0}-{\displaystyle \frac{T-T_2}{T_3-T_2}}\cdot \Delta r_i& (T_2<T<T_3,u>0)\\ r_{i0}-\Delta r_i& (u=0)\end{array}\right.$$

(4)

where the parameters T₁, T₂, and T₃ [38] correspond to the self-generating heat temperature, the thermal runaway triggering temperature, and the maximum thermal runaway temperature, respectively, with values of 105 °C, 250 °C, and 821 °C. The ambient temperature is 25 °C. r_i denotes the i-th thermophysical parameter, r_i0 represents the initial value of r_i, and Δr_i indicates the variation of r_i.

The parameter u denotes the normalized concentration of reactants, with an initial value of 1, which reflects the degree of battery TR and controls the heat release of the battery in the thermal runaway model. When u = 0, the TR process is considered to be terminated. As reported in Ref. [38], when T > T₂, the reaction rate remains constant at Const_TR; when T₁ < T < T₂, the reaction rate follows the Arrhenius equation. Accordingly, the expression for the reaction rate is given in Equation (5) [18]:

$${\displaystyle \frac{du}{d\tau }}=\left\{\begin{array}{ll}A\cdot \exp (-{\displaystyle \frac{E_a}{R_{\mathrm{u}}T}})& (0\le u\le 1,T_1\le T\le T_2)\\ {Const}_{TR}& (0\le u\le 1,T>T_2)\\ 0& \mathrm{other}\end{array}\right.$$

(5)

2.1.2. Nail Penetration Model

The heat generation rate of battery thermal runaway induced by nail penetration is expressed as shown in Equation (6):

$$q_{v,Bat}=q_{v,nail}+q_{v,JR}$$

(6)

The electrical power released by the nail per unit volume after penetrating the battery is calculated as shown in Equation (7) [18]:

$$q_{v,nail}={\displaystyle \frac{1}{V_{nail}}}\cdot \Delta H_{nail}\cdot f(\tau )$$

(7)

where the parameter ΔH_nail represents the total thermal power released by the nail, which is calculated as shown in Equation (8) [18]:

$$\Delta H_{nail}=\alpha \cdot \Delta H_{TR}$$

(8)

where α is the ratio of the heat generated by the nail to the total heat generation, which can be obtained through optimization and parameter identification as reported in Ref. [38], and its value is 0.02.

The parameter f(τ) denotes the function describing the energy release rate during the internal short-circuit process. The release rate reaches its peak at 0.15 s and subsequently decreases after 0.25 s. The fitted expression is given in Equation (9):

$$f(\tau )=\left\{\begin{array}{ll}{\displaystyle \frac{1}{0.15}}\tau & 0\le \tau \le 0.15\\ -{\displaystyle \frac{20}{3}}\tau +2& 0.15\le \tau \le 0.25\\ -{\displaystyle \frac{1}{15.45}}\tau +0.35& 0.25\le \tau \le 5.4\end{array}\right.$$

(9)

The f(τ) term in the needle heating source is only used to describe the transient reaction and internal short-circuit heat generation during the initial stage of acupuncture, while the subsequent thermal evolution process is solely dominated by the heat transfer within the battery system.

The thermal power released by the battery per unit volume after nail penetration is calculated as shown in Equation (10):

$$q_{v.JR}={\displaystyle \frac{1}{V_{JR}}}\cdot (\Delta H_{TR}-\Delta H_{nail})\cdot {\displaystyle \frac{du}{d\tau }}$$

(10)

2.2. Battery Shell Model

Considering the contact thermal resistance between the battery jelly roll and the shell, the conductive thermal resistance and heat capacity of the shell itself, the contact thermal resistance between the shells of neighboring batteries, and the convective and radiative thermal resistances between the shell and the ambient environment, the above lumped thermal elements are incorporated into the shell model. Each lumped thermal element contains two nodes, and the elements are connected in series, where the downstream node of the preceding element is connected to the upstream node of the subsequent element.

In COMSOL, a conductive thermal resistance element is selected to represent the contact thermal resistance, and the specific formulation is given in Equation (11):

$$q=-{\displaystyle \frac{\Delta T}{R}}$$

(11)

In COMSOL, a thermal capacitance element is selected, and the corresponding formulation is given in Equation (12):

$$q=-C{\displaystyle \frac{\mathsf{\partial }\Delta T}{\mathsf{\partial }\tau }}$$

(12)

The lumped thermal elements between neighboring jelly rolls are configured as follows: the thermal resistance between the upstream jelly roll and the shell, the shell thermal capacitance, the shell conductive thermal resistance, the thermal resistance between adjacent shells, the shell conductive thermal resistance, the shell thermal capacitance, and the thermal resistance between the downstream jelly roll and the shell. The thermal components on the side of the battery in contact with the environment are configured as follows: the thermal resistance between the jelly roll and the shell, the shell thermal capacitance, the shell thermal resistance, convective heat transfer, and radiative heat transfer. Because multiple lumped thermal elements are present in the shell model, it is not feasible to select the temperature at a single node as the shell surface temperature. To obtain an equivalent temperature that represents the overall thermal state of the shell, it is necessary to account for the differences among various heat dissipation paths and the thermal characteristics of different shell surfaces. In this study, the node temperatures on each side, taken prior to the action of thermal capacitance, convection, and radiation, are separately selected. Different weighting factors are assigned according to their corresponding characteristics to characterize the contribution of each side to the equivalent shell temperature, on the basis of which the shell temperature is calculated, as shown in

Table 2. Thermal elements selected for temperature extraction at each side surface and their corresponding weighting factors.

T	Thermal Elements and Weighting Factors
Tb	$-0.006934T_R-0.00559T_C-0.39163T_h+1.25T_{rad}+0.000793T_h\times T_{rad}+0.001917time$
Tt	$-0.050726T_R+0.013999T_C-5.985457T_h+6.74169T_{rad}+0.005976T_h\times T_{rad}-0300544time$
Tl	$-0.062956T_R-0.014944T_C-3.253918T_h+4.081257T_{rad}+0.003428T_h\times T_{rad}+0.006231time$
Tsc	$\begin{array}{l}0.2418R_{0\_1}+0.2422C_0-0.0309R_x-0.0085R_{0\_2}+0.0094{R_{0\_1}}^2+0.0145R_{0\_1}\times C_0\\ -0.022R_{0\_1}\times R_x+0.091R_{0\_1}\times R_{0\_2}+0.019{C_0}^2-0.011C_0\times R_x+0.08C_0\times R_{0\_2}-0.109{R_x}^2-0.106{R_{0\_2}}^2\end{array}$

.

2.3. Boundary Conditions

The geometric model of thermal runaway propagation consists of the jelly roll, the battery shell, the electrodes, the nail, and the air layer. In this model, the thermal resistance involved in heat conduction is treated as an equivalent thermal resistance characterized solely by its resistance value.

The thermal resistances in the model are categorized into two types: one is the equivalent thermal resistance between the jelly roll and the shell, and the other is the equivalent thermal resistance between neighboring battery shells. The specific heat transfer equation is given in Equation (13):

$$-{\lambda }_n{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }\overrightarrow{n}}}={\displaystyle \frac{\lambda }{\delta }}(T_{front}-T_{back})$$

(13)

Figure 2 illustrates the locations of various boundary conditions, and

Table 3. Numerical values of the various types of thermal resistances in the model; data is from Ref. [18].

Type	λ/W·m−1·K−1	δ/mm	h/W·m−2·K−1	εrad
	0.22	0.4	/	/
	0.04	0.2	/	/
	/	/	10	/
	/	/	10	/
	/	/	8	/
	/	/	1	/
	/	/	/	0.03

presents the specific settings for each boundary condition.

2.4. Model Solving

The thermal runaway propagation model is implemented in COMSOL Multiphysics 6.1. After constructing the 3D models of components such as the jelly roll and the nail, and appropriately defining the heat sources and boundary conditions, a lumped-parameter model of the shell is established. Through lumped connectors, the temperature and heat flux at corresponding nodes between the jelly roll model and the shell model are bidirectionally transferred, thereby achieving the integration of the 3D model and the lumped-parameter model. A schematic diagram of the lumped connector is shown in Figure 3, and the main parameters of the battery array are summarized in

Table 4. Main parameters required for the thermal runaway propagation model of the battery array; data is from Ref. [18].

Parameter	Value
Material of cathode/anode	Li(NiCoMn)1/3O2/graphite
Capacity of battery	37 Ah
SOC of battery	100%
Cut-off voltages for charging/discharging	4.2 V/2.8 V
Weight of battery/mass loss after TR	830 g/210 g
Dimensions of battery	146 × 91 × 26.5 mm
Thickness of shell	0.7 mm
Height of jelly roll/gas layer	80.3 mm/9.3 mm
Density of jelly roll	Before TR: 2680 kg·m−3 After TR: 1990.7 kg·m−3
Heat capacity of jelly roll	Before TR: 1100 J·kg−1·K−1 After TR: 788.47 J·kg−1·K−1
Thermal conductivity of jelly roll in the x direction/y direction/z direction	Before TR: 0.84/15.30/15.30 W·m−1·K−1 After TR: 0.35/8.18/8.18 W·m−1·K−1
Density of shell/nail/cathode/anode	2700/7850/8522/2700 kg·m−3
Heat capacity of shell/nail/cathode/anode	900/475/385/900 J·kg−1·K−1
Thermal conductivity of shell/nail/cathode/anode	160/44.5/146/160 W·m−1·K−1
Heat capacity of equivalent thermal resistance between shell and shell	66 J·kg−1·K−1
Temperature of ambiance/self-generating heat/TR triggering/TR maximum	25/102/250/821 °C
Pre-exponential factor	1.75 × 1012 s−1
Universal gas constant	8.3145 J/(mol·K)
Activation energy of the chemical reaction	1.20227 × 105 J/mol
Diameter of nail	8 mm

.

The temperatures of various feature surfaces in the three-dimensional model (including the top surface, bottom surface, left side, and contact with the large side) and the temperatures of each thermal element node in the joint model (including T_b, T_t, T_l, and T_sc) are extracted. Additionally, a time term (time) is introduced, and nonlinear fitting is performed using the least squares method. These variables are used to fit the battery shell temperature, aligning it with the temperature from the three-dimensional model. To accurately describe the three stages of temperature change (slow heating, rapid heating, and cooling), weights are assigned to each stage: the first stage is given a weight of 1, the second stage a weight of 12, and the third stage a weight of 3.

Next, the coefficients of the basic terms (T_R, T_C, T_h, T_rad), interaction terms, squared terms, and time are calculated, obtaining the fitting coefficients. The weight coefficients proposed in Table 2 are only applicable to the specific type of battery examined in this study (a 37 Ah ternary lithium-ion battery with Li(NiCoMn)_1/3O₂ as the cathode material and graphite as the anode material, 146 × 91 × 26.5 mm). If the battery model changes, each new type of battery requires a re-fitting. Finally, the thermal runaway propagation simulation is extended from the one-dimensional battery array to the two-dimensional battery array, validating the generalizability of the proposed modeling method. The specific flowchart is shown in Figure 4.

2.5. Mesh Independence Verification

Figure 5 illustrates the variation in the deviation of the first thermal runaway time interval between simulation and experiment under different mesh numbers, namely, 64,821, 399,609, and 7,656,513. When the number of grids increases from 399,609 to 7,656,513, the deviation of the first thermal runaway time interval is reduced by only 0.25%, and the deviation of the first battery maximum temperature is reduced by only 0.84% (with reference data sourced from Ref. [18]), while the model solution time increases from 9 h and 21 min to 25 h and 18 min. The computation is performed using a high-performance server equipped with dual Intel Xeon Platinum 8370C processors, each with a base frequency of 2.80 GHz, providing a total of 64 physical cores and 128 logical threads. The computer is configured with 256 GB of DDR4 memory, ensuring that large-scale datasets can be fully resident in memory during computation. All simulations are performed under the Windows Server operating system. The solver’s tolerance factor is set to 0.1, the initial time step is 0.001, and the maximum time step is automatically constrained. Therefore, to balance computational time and efficiency, a mesh number of 399,609 is adopted.

2.6. Model Validation

The model validation is conducted based on the nail penetration-induced thermal runaway propagation experiment detailed in Ref. [18]. Figure 6 shows the comparison of the temperature change curves between the 3D model established in this study and the heat runaway propagation model induced by a nail in Ref. [18]. The results indicate that the data are very close. Furthermore,

Table 5. Comparison of thermal runaway propagation times and maximum temperature of each battery, data is from Ref. [18].

Battery	This Study Time Interval [s]	Ref. [18] Time Interval [s]	This Study Maximum Temperature [°C]	Ref. [18] Maximum Temperature [°C]
Cell-0	26.06	29.8	959.6	824.4
Cell-1	149	154	831.84	805.1
Cell-2	100	107.5	898.7	865.9
Cell-3	108.7	109.3	895.4	858.3
Cell-4	102	111.1	855.3	853.8
Cell-5	100.7	109.9	840.2	859.9
Cell-6	116.3	111.6	841.8	855.1
Cell-7	-	-	843.4	856.8
Average error	6.30%		3.71%

summarizes the relative deviation of the thermal runaway triggering time interval for each cell in the battery array, as well as the relative deviation of the maximum temperature of each battery. The average deviations are 6.30% and 3.71%, respectively, indicating a high level of consistency between the experimental results and the simulation results. This confirms that the proposed modeling approach can accurately capture the process of thermal runaway propagation.

Figure 7 presents the temperature–time curves at the monitoring locations obtained from the 3D model and the joint model, where the solid lines represent the data from the 3D model and the dashed lines represent the data from the joint model. The results indicate that the data are very close.

Table 6. Comparison of thermal runaway triggering time, maximum temperature, and time to reach maximum temperature for each battery between the 3D model and the joint model.

Battery	Trigger Time [s]		Maximum Temperature [°C]		Time to Reach Maximum Temperature [s]
	Joint Model	3D Model	Joint Model	3D Model	Joint Model	3D Model
Cell-0	0.64	0.64	913.0	959.6	3.2	5.1
Cell-1	26.1	26.7	865.9	831.84	66.3	78.1
Cell-2	168.9	175.7	905.4	898.7	195.9	209.3
Cell-3	278.6	275.7	899.3	895.4	312.4	299.8
Cell-4	388.2	384.4	908.6	855.3	420.9	414.7
Cell-5	493.6	486.4	907.0	840.2	511.3	527.4
Cell-6	592.5	587.1	922.1	841.8	624.2	628.8
Cell-7	712.7	703.4	920.8	843.4	735.1	738.2
Average error	1.32%		4.78%		7.63%

summarizes the comparison of the thermal runaway triggering time, the maximum temperature, and the time to reach maximum temperature for each battery between the 3D model and the joint model. The deviations are 1.32%, 4.78%, and 7.63%, respectively, confirming that the proposed joint model can accurately capture the process of thermal runaway propagation.

3. Results and Discussion

3.1. Thermal Runaway Propagation in a One-Dimensional Battery Array

Figure 8, Figure 9, Figure 10 and Figure 11 present the variations in the shell temperatures at the bottom surface, top surface, left-side surface, and large-side contact surface of the five cells in the one-dimensional battery array, respectively (as shown in Figure 1a; the selection of thermal element nodes on the battery’s top, bottom, y-direction (left surface), and x-direction surfaces (contact surface), along with their corresponding weighting factors, are listed in Table 2). These results are compared with the shell temperatures obtained from the 3D model (the heat transfer of battery shell and jelly roll is described using 3D simulation). For the 3D model, the shell temperature is taken as the temperature at the center point of the corresponding surface. In the figures, the dashed lines represent the results of the joint model, while the solid lines denote those of the 3D model.

As shown in Figure 8, Figure 9 and Figure 10, after performing nonlinear fitting using the least squares method, the temperatures on the various sides of the joint model match very well with the corresponding temperatures of the 3D model. Moreover, it can be observed that after nail penetration induces thermal runaway, the temperature of the penetrated battery rises rapidly to the maximum temperature. During the self-heating stage of the propagation batteries, their temperature evolutions are essentially identical, with the temperature increasing slowly until the jelly roll temperature reaches the thermal runaway triggering temperature, followed by a rapid rise to the maximum temperature and then a gradual decrease. For example, in Figure 8, the temperature of Cell-2 in the y-direction surface, denoted as T_b2, shows a sharp increase at 220.00 s. From the temperature curve T_t2 of Cell-2 in Figure 9, it can be seen that the joint model predicts a sharp temperature rise at 221.66 s. In Figure 10, the left-side surface temperature of Cell-2, denoted as T_l2, shows a sharp increase at 224.57 s in the joint model.

For the x-direction surface of batteries, there is a slow variation stage after the initial fast growth, since the heat inflow to the shell is approximately equal to the heat outflow. During this period, the jelly roll continues to heat up until it reaches the thermal runaway triggering temperature. Subsequently, the heat generated by thermal runaway of the jelly roll further heats the battery shell, causing the shell temperature to increase until the maximum temperature is reached. For example, the third x-direction surface center temperature in Figure 11, denoted as T_sc3, shows that the joint model and the 3D model begin to increase almost simultaneously at 226.03 s, and enter the stage of slow temperature variation at 231.15 s and 235.68 s, respectively. Thereafter, the temperatures predicted by the joint model and the 3D model rise again at 314.33 s and 320.58 s, respectively.

Figure 8, Figure 9, Figure 10 and Figure 11 indicate that the joint model achieves satisfactory accuracy in predicting battery thermal runaway and its propagation, with the temperature variation curves closely overlapping. Moreover, in the above simulations of thermal runaway propagation in the one-dimensional battery array, the computational time of the joint model is only 6 h 23 min, whereas that of the full 3D model is 21 h 26 min, resulting in a time saving of 70.22%.

3.2. Thermal Runaway Propagation Characteristics of a Two-Dimensional Array

Based on the validation of the joint model in the preceding section, the same modeling framework is employed to simulate the thermal runaway propagation processes in 3 × 3 and 5 × 5 battery arrays composed of 9 and 25 cells, respectively. Nail penetration is adopted to trigger thermal runaway in the initial battery, while the remaining batteries undergo thermal runaway as a result of thermal abuse induced by the runaway battery.

3.2.1. A 3 × 3 Two-Dimensional Array

A 3 × 3 two-dimensional battery array was established, and the triggering location of nail penetration-induced thermal runaway was selected at the central Cell-0, as illustrated in Figure 12.

The temperature variation curves at the core center of the 3 × 3 two-dimensional battery array obtained from the 3D model and the joint model are compared, as shown in Figure 13. It can be observed that, in the two-dimensional array, the results of the joint model are in good agreement with those of the 3D model. Meanwhile,

Table 7. Comparison of the thermal runaway triggering times and maximum temperatures of each battery in a 3 × 3 two-dimensional battery array between the joint model and the 3D model.

Battery	Joint Model Trigger Time [s]	3D Model Trigger Time [s]	Joint Model Maximum Temperature [°C]	3D Model Maximum Temperature [°C]
Cell-0	0.64	0.64	1078.93	1035.45
Cell-1	131.37	121.45	911.31	983.18
Cell-2	120.81	115.65	1000.12	976.84
Cell-3	131.16	124.43	985.95	983.81
Cell-4	24.07	8.31	965.64	934.42
Cell-5	23.9	21.3	991.45	965.23
Cell-6	135.91	122.14	999.13	977.26
Cell-7	126.99	116.35	1005.96	978.2
Cell-8	138.33	124.79	977.32	983.8
Average error	27.91%		2.88%

presents a comparison of thermal runaway propagation time and maximum temperature between the 3D model and the joint model. Taking the values of the 3D model as the reference for error calculation, it can be seen that the average error of the thermal runaway triggering time is 27.91%, and the average error of the maximum temperature is 2.88%, demonstrating that the joint model can accurately predict the thermal runaway propagation characteristics of the two-dimensional battery array. From Table 7, we can observe that the triggering time of Cell-4 varies significantly between the joint model and the 3D model. This is because the joint model simplifies the battery shell from a three-dimensional distributed thermal model to a zero-dimensional centralized thermal model. In the proposed joint model, the shell temperature is represented by the spatially averaged centralized temperature rather than the temperature field with spatial resolution. In the original puncture penetration configuration, the exposed surface area of the needle tip is very small, accounting for approximately 3.7% of the large-side shell surface area. Therefore, in the complete three-dimensional model, the heat transferred by the high-temperature needle to the entire shell surface is relatively slow, and the temperature distribution of the shell during the early stage of thermal runaway is very uneven. The temperature rise of the large-side shell is mainly driven by the thermal runaway of the battery cell rather than the direct heating of the needle. After simplifying the shell to a zero-dimensional centralized model, the local high-temperature areas near the needle penetration point are spatially averaged within the entire shell range. Therefore, compared to the full three-dimensional model, the predicted center temperature of the shell near the needle is lower, which makes it more difficult to trigger thermal runaway in Cell-4, thereby delaying the predicted propagation process. In the 3D model, since the temperatures of corresponding grids are directly transferred, the temperature in the area where the Cell-4 shell is in close contact with the needle can rise rapidly, and the thermal runaway can be triggered more quickly. To eliminate the individual error issues caused by special working conditions, after eliminating the Cell-4 error, the calculation error of the thermal runaway triggering time was only 6.83%, indicating that this combined model can well meet the calculation accuracy requirements. To solve this problem and improve the physical authenticity of the model configuration, we reconsidered the position of the needle electrode puncture, adjusting the puncture position to the center of the top of the battery (in practical applications, it is difficult to insert the needle into the center of the large-side surface of the front of Cell-0 without affecting other batteries). Appendix A compares the differences in the thermal runaway propagation results of the 3D model and the combined model in a 3 × 3 two-dimensional battery array when the needle electrode punctures from the top. The results show that under this condition, the results of the joint model are very close to those of the 3D model. Specifically, the time when Cell-4 experienced thermal runaway in the joint model was 40.29 s, while in the 3D model it was 41.22 s, with an error of only −2.26%.

Figure 13 presents the temperature evolution curves at the centers of the jelly rolls of each battery. It can be observed that Cell-5 and Cell-4 are the first to undergo thermal runaway, followed by Cell-2 and Cell-7. These four cells are those directly adjacent to the four side surfaces of the penetrated battery. Cell-5 and Cell-4 initiate thermal runaway earlier because they are neighboring the large-side surfaces (x direction) of the penetrated battery; compared with the small-side surfaces (y direction), under the same temperature difference, the large- side surfaces receive more heat, thereby triggering thermal runaway at an earlier stage. Subsequently, Cell-3, Cell-1, Cell-6, and Cell-8 undergo thermal runaway in sequence.

In addition, the 3D model requires 38 h 40 min of computational time, whereas the joint model requires only 9 h 44 min, resulting in a 74.84% reduction in computational cost.

The thermal runaway propagation process of the 3 × 3 two-dimensional array is illustrated in Figure 14. It can be observed that nail penetration induces Cell-0 to trigger thermal runaway first, and the entire cell undergoes thermal runaway at 15 s. Subsequently, heat is transferred to the surrounding cells through the side surfaces of the jelly roll. At 20 s, the side surfaces of Cell-5 and Cell-4 that are in direct contact with Cell-0 trigger thermal runaway and continuously propagate heat to the remaining regions; by 35 s, both cells have completely entered thermal runaway. Through heat transfer across the small-side surfaces of Cell-0, Cell-2 and Cell-7 trigger thermal runaway at 120 s and 123 s, respectively, and further propagate heat to the remaining parts. At 126 s, Cell-3 and Cell-1 trigger thermal runaway at the small-side surfaces adjacent to Cell-5 and Cell-4, followed by the successive initiation of thermal runaway in Cell-6 and Cell-8. By 135 s, thermal runaway has propagated throughout the entire battery array. Thereafter, thermal runaway continues to develop: at 140 s, Cell-5 and Cell-4 are completely in thermal runaway; at 145 s, Cell-3 and Cell-1 fully undergo thermal runaway; and at 150 s, Cell-6 and Cell-8 completely enter thermal runaway. At this point, the entire battery array is fully in a thermal runaway state. To analyze the thermal runaway propagation path, the heat fluxes at various contact surfaces are extracted. Taking the heat fluxes of the typical battery cells Cell-0, Cell-1, Cell-2, and Cell-4 as examples, the analysis is conducted. The center of the battery is taken as the origin, with the side in the positive x direction named as +x, the side in the negative x direction named as −x, the side in the positive y direction named as +y, and the side in the negative y direction named as −y. In Figure 15, a positive Q value indicates heat flux flowing out of the corresponding side, while a negative Q value indicates heat flux flowing into this side.

As shown in Figure 15a, thermal runaway is first triggered in Cell-0 by nail penetration, and the battery begins to heat rapidly. The heat (Q) is transferred to neighboring cells. Due to the smaller scale of the battery in the x direction compared to the y direction, the time for Q to reach its peak value is relatively shorter. The maximum heat transferred from Cell-0 to Cell-4 and Cell-5 in the x direction is 1144.03 W, while in the y direction, the maximum heat transferred to Cell-2 and Cell-7 is 338.64 W. The heat ratio is approximately 3.38, as the surface area in the x direction is larger than that in the y direction. After the Q in the x direction reaches its peak, it rapidly declines to the minimum value of −934.7 W. This is because Cell-4 and Cell-5 receive a large amount of heat and are triggered into thermal runaway, leading to rapid heating. After Q in the y direction reaches its peak, it decreases slowly. Due to the smaller amount of heat transferred, the temperature increase in Cell-2 and Cell-7 is slower, and they remain in the self-heating phase. Subsequently, at 118 s and 120 s, Cell-2 and Cell-7 trigger thermal runaway sequentially. Afterward, the exothermic process of thermal runaway in the battery ends, and thermal equilibrium is gradually reached with the surrounding cells.

Since Cell-4 and Cell-5 are almost simultaneously triggered into thermal runaway by Cell-0, with Cell-4 triggering slightly earlier, Cell-4 is selected for further analysis. The difference in the thermal runaway triggering times of Cell-4 and Cell-5 may be attributed to the differences in the thermal properties of the anode and cathode materials. As shown in Figure 15b, Cell-4 first receives heat from the −x direction and enters the self-heating phase at 13 s. It then heats rapidly, reaching the highest temperature and simultaneously heating Cell-0, as the temperature of Cell-4 exceeds that of Cell-0. The maximum heat transferred from Cell-4 to Cell-0 is 890.28 W, which is smaller than the maximum heat transferred from Cell-0 to Cell-4. This is because when Cell-0 transfers heat to Cell-4, Cell-4 is initially at an ambient temperature, creating a large temperature difference between the two. When Cell-4 transfers heat to Cell-0, although the exothermic process of thermal runaway in Cell-0 has ended, its temperature is still high, resulting in a smaller temperature difference. The heat transfer from Cell-4 to Cell-1 and Cell-6 in the y direction is similar to that of Cell-0. Cell-4 also has a side in contact with the environment, namely the +x side. This side dissipates heat through convection and radiation, but the heat dissipation is small, only 55.28 W.

As shown in Figure 15c, Cell-2 is heated by the heat flux coming from the +y direction, which is the heat transferred from Cell-0. Due to the smaller surface area of the smaller sides, less heat is transferred, resulting in a delayed thermal runaway trigger. Cell-2 enters the self-heating phase at 14 s, and the heat received from Cell-0 starts to decrease. At 113 s, thermal runaway is triggered, and Cell-2 starts heating Cell-1 and Cell-3 while also heating Cell-0 in reverse. The maximum heat transferred from Cell-2 to Cell-1 and Cell-3 is 511.1 W and 458.32 W, respectively, both smaller than the maximum heat transferred from Cell-4 to Cell-0. This is because Cell-1 and Cell-3 trigger thermal runaway around 124 s, leading to a smaller temperature difference between the cells. Cell-2 also has a side in contact with the environment, namely the −y side, which dissipates heat to the environment. The maximum heat dissipation is 10.94 W, approximately one-fifth of the heat dissipation from the larger sides, due to the smaller surface area.

As shown in Figure 15d, the temperature rise of Cell-1 is caused by the heat transferred from the +y direction, which is induced by Cell-4. Cell-1 enters the self-heating phase at 30 s, and then, through the combined effect of Cell-4 and Cell-2, thermal runaway is triggered at 123 s.

3.2.2. A 5 × 5 Two-Dimensional Array

A 5 × 5 two-dimensional battery array was established, with the nail penetration-induced thermal runaway triggering location selected at the central Cell-0 of the array, as shown in Figure 16. Figure 17 presents the temperature evolution curves at the centers of the jelly rolls of individual batteries, with the legend ordered according to the sequence of thermal runaway initiation. Due to the large number of batteries triggering thermal runaway during the same stage, only 11 representative locations are shown (Cell-0, Cell-13, Cell-12, Cell-14, Cell-11, Cell-17, Cell-15, Cell-4, Cell-23, Cell-14, and Cell-20); the complete temperature curve is shown in Figure A4. These correspond to the initially triggered battery Cell-0; four batteries along the x-axis direction (Cell-11, Cell-12, Cell-13, and Cell-14); three batteries along the y-axis direction (Cell-17, Cell-4, and Cell-23); and three batteries located at the edge of the battery pack (Cell-1, Cell-15, and Cell-20).

After nail penetration induces thermal runaway in Cell-0, the batteries arranged along the x-axis, namely Cell-13 and Cell-12, trigger thermal runaway earlier due to contact with the large-side surfaces. However, thermal runaway propagation in the negative x-axis direction is faster, resulting in Cell-13 triggering thermal runaway earlier than Cell-12. Subsequently, along the x-axis direction from Cell-0, Cell-14 and Cell-11 successively trigger thermal runaway. Thereafter, the two batteries in contact with the small-side surfaces of Cell-0, Cell-17, and Cell-8 trigger thermal runaway in sequence; because their triggering times are close, only the temperature curve of Cell-17 is presented. Subsequently, along the x-axis direction from Cell-17, the batteries arranged in the positive direction trigger thermal runaway earlier than those arranged in the negative direction. Thermal runaway then propagates successively to Cell-4 and Cell-23, and finally spreads gradually to all batteries along the x-axis direction.

The thermal runaway propagation process of the 5 × 5 two-dimensional array is illustrated in Figure 18. It can be observed that nail penetration induces Cell-0 to trigger thermal runaway first, and the entire cell undergoes thermal runaway at 15 s. Subsequently, heat is transferred to the surrounding cells through the side surfaces of the jelly roll. At 20 s, the side surfaces of Cell-13 and Cell-12 that are in close contact with Cell-0 successively trigger thermal runaway and continuously propagate heat to the remaining regions; by 35 s, both cells have completely entered thermal runaway. At 39 s and 43 s, Cell-14 and Cell-11 trigger thermal runaway, respectively, and by 60 s, both cells have fully undergone thermal runaway. At this point, all batteries arranged along the x-axis direction of Cell-0 have completely entered thermal runaway. At 136 s and 143 s, Cell-17 and Cell-8, which are tightly adjacent to the small-side surfaces of Cell-0, successively trigger thermal runaway. At 145 s, Cell-16 and Cell-18 also trigger thermal runaway. By 150 s, thermal runaway has propagated to Cell-9 and Cell-7, and at 156 s and 159 s, Cell-19, Cell-15, Cell-10, and Cell-6 successively trigger thermal runaway. At 162 s, Cell-17 and Cell-8 completely enter thermal runaway; at 167 s, Cell-16, Cell-18, Cell-9, and Cell-7 fully undergo thermal runaway; and at 173 s and 176 s, Cell-19, Cell-15, Cell-10, and Cell-6 are all completely in a thermal runaway state. At 264 s, Cell-4 and Cell-3 successively trigger thermal runaway, and at 270 s, thermal runaway propagates to Cell-23 and Cell-22. By 278 s, Cell-5, Cell-2, Cell-24, and Cell-20 have all triggered thermal runaway. At 287 s, thermal runaway has propagated to all batteries in the array, and by 300 s, all batteries in the array are completely in a thermal runaway state. The thermal analysis of the 5 × 5 battery array is similar to that of the 3 × 3 array and will not be discussed further here. The joint model’s calculation for a 5 × 5 two-dimensional battery array takes 38 h and 27 min, while the 3D model requires 111 h and 33 min (4 days, 15 h, and 33 min) for calculation. Therefore, the joint model saves 65.5% of the calculation time.

4. Outlook and Challenges

This study proposes a joint modeling approach that combines the lumped-parameter method with a three-dimensional model, aiming to accurately describe the temperature characteristics of both the jelly roll and the shell during thermal runaway propagation in battery arrays while maintaining reasonable computational efficiency. However, the joint model presented in this article still has some limitations:

(1)

The current model has only completed the accuracy verification for the single battery model, fixed electrode material, specific cell size, and module arrangement form used in this experiment. It has not yet fully expanded the verification and parameter adaptation under various cell specifications, different electrode materials, and different module structure forms. In future research, more common models and different arrangement methods of batteries will be studied for their thermal runaway propagation.

(2)

The thermal properties of the battery core vary nonlinearly with temperature. To simplify the model calculation, linear interpolation is used in this model to describe these properties. Future work will involve experimental measurements of the changes in the thermal properties during thermal runaway propagation to improve the nonlinear model of the battery core’s thermal properties. For example, we will use the methodological approach for inverse thermophysical parameter identification [39] to improve the physical consistency and predictive robustness of the 3D model and joint model during thermal runaway.

(3)

During the process of heat runaway propagation, the battery shell may deform or undergo phase changes, and the physical properties such as conductivity and density of each component will also change, thereby causing a change in its heat capacity. In this study, for the sake of simplicity, these changes were ignored, and it was assumed that the heat capacity remained constant. Future work will incorporate mechanical deformation simulations into the combined model to achieve more accurate predictions.

(4)

In this study, it is assumed that the environmental conditions during thermal runaway propagation in the battery array remain constant, using natural cooling boundary conditions. Future modeling will take into account the effects of airflow in open environments or the influence of closed environments.

(5)

This model takes into account the radiation heat exchange between the battery and the environment, but it does not consider internal radiation. In subsequent research, we will further improve the calculation of radiation heat exchange to enhance the model’s completeness and prediction accuracy.

(6)

This study on the thermal runaway propagation path in two-dimensional arrays is limited to small-scale arrays and does not consider scenarios beyond the 5 × 5 array. Future research will include larger-scale two-dimensional arrays to better represent the number of cells in real-world battery storage systems.

(7)

This study focuses solely on the thermal runaway propagation characteristics of the array and does not apply these characteristics to guide the battery pack design process. In future work, some key components will be added, including busbars and connection pieces, to provide guidance for the design and practical application of the battery pack.

(8)

This study adopts a hierarchical indirect verification approach: First, the full three-dimensional model is verified based on the measured data; then, using the verified and reliable three-dimensional simulation results as the benchmark, the accuracy of the combined model under one-dimensional array is checked; on this basis, all subsequent studies on two-dimensional arrays will be conducted based on the fully verified three-dimensional model as the reference benchmark. The deficiency of this study in model verification lies in not directly comparing and verifying with the experimental data of two-dimensional arrays. In subsequent research, we will conduct a thermal runaway propagation simulation experiment for two-dimensional arrays to lay a foundation for model verification.

(9)

This study clearly defines the temperature at which the battery enters the self-generated heat stage and the temperature at which the actual heat runaway is determined. However, the process of battery heat runaway is complex. Future research will strictly distinguish the temperature ranges corresponding to the four different stages of the battery heat runaway, namely, separator failure, self-generated heat initiation, internal short circuit occurrence, and heat runaway explosion. It will also clearly state the precise numerical temperature standard used in this article to determine the triggering moment of heat runaway, thereby improving the basis for judgment.

5. Conclusions

This study proposes a joint modeling approach that combines the lumped-parameter method with a three-dimensional model, aiming to accurately describe the temperature characteristics of both the jelly roll and the shell during thermal runaway propagation in battery arrays while maintaining reasonable computational efficiency. One-dimensional and two-dimensional battery arrays are taken as representative cases for investigation, and the main conclusions are summarized as follows:

(1)

The joint modeling approach primarily simplifies the shell model by representing it as a thermal capacitance connected in series with multiple thermal resistances, and the shell temperature is calculated through a weighted summation based on the temperatures of the thermal capacitance and the thermal resistances. For a one-dimensional battery array, compared with the three-dimensional model, the average deviation of the thermal runaway propagation time is 1.32%, the average deviation of the maximum temperature is 4.78%, and the average error of the time reaching the highest temperature is 7.63%. The solution time of the joint model is 6 h and 23 min, while that of the 3D model is 21 h and 26 min. Thus, the solution time has been shortened by 70.22%.

(2)

For the two-dimensional battery array, when the central cell of the array is the first to undergo thermal runaway, the propagation initially occurs along the direction of the large-side surfaces, extending to all cells aligned in this direction. Subsequently, thermal runaway spreads outward along the direction perpendicular to the large-side surfaces toward both sides. This process is repeated until all cells in the array experience thermal runaway. Additionally, for a 3 × 3 battery array, the 3D model calculation took 38 h and 40 min, while the combined model only required 9 h and 44 min. For a 5 × 5 battery array, the 3D model calculation took 111 h and 33 min, and the combined model only needed 38 h and 27 min, saving 74.84% and 65.5% of the calculation time, respectively.