An investigation of critical parameters on spontaneous combustion for lithium-ion battery using bifurcation theory

Abstract

Electric vehicles (EVs) are prevalent in the modern automotive industry, providing zero emission during their operation. However, there are numerous circumstances such as extreme temperatures and collisions, which cause the spontaneous ignition of Lithium-based batteries (LIBs), resulting in severe safety problems. Current literature mainly focuses on advancing the heat conduction of such batteries, using LIB numerical simulation based on software such as COMSOL, few explore the critical parameters, for which the spontaneous ignitions of batteries occur. Due to the extensive parameters, instability and huge computational times in running numerical simulations, it is difficult to simulate the critical parameters of LIBs. Mathematically, bifurcation theory provides a qualitative way to determine the critical parameters, which does not require complex numerical calculations to evaluate such parameters of LIB thermal runaway (TR). Henceforth, we establish two simple parallel and circular models to demonstrate how we can use bifurcation theory to determine paramount factors, paving the way for more sophisticated studies on the design of LIB cooling systems, the choice of battery materials and the curvature of LIBs to enhance their safety operations. Last but not least, the present study provides starting references for computationally extensive LIB numerical simulations.

Introduction

Lithium-ion batteries (LIBs), owing to their inherent advantages such as high energy density, long cycle life, low self-discharge rate, and absence of memory effect, have become the primary power source for electric vehicles and are widely regarded as the preferred energy storage technology (Harper et al., 2019; Kennedy et al., 2000 and Duan et al., 2020)^1–3. However, the same high energy density that makes LIBs desirable also presents significant safety risks (Shahid et al., 2022; Hu et al., 2024; EV Fire-Safe, 2023)^4–6. In the event of malfunctions or accidents, LIBs may undergo thermal runaway (TR), potentially resulting in severe fire (Nozaki et al., 2023 and Mahdy et al., 2025)^7,8. Data from the global database indicates that the global fire incidence rate for electric passenger vehicles is around 0.0012%, while that for ICEVs is approximately 0.1% making traditional fuel-powered vehicles about 80 times more likely to catch fire than EVs. Despite the lower fire incidence rate in EVs, such events may occur too rapidly for safe evacuation, resulting in casualties. As a result, research into the safety mechanisms of LIBs has become a global focus and an urgent concern in both academia and industry.

In this context, understanding the fundamental structure of LIB cells and their assembly into modules and packs is essential for evaluating both performance and safety characteristics. A single LIB cell consists of three fundamental components: a cathode region, an anode region, and a separator placed between them (Wang et al., 2012)⁹. These elements illustrated in Fig. 1 constitute the basic functional unit of LIBs. In electric vehicles, the battery system typically comprises thousands of such individual cells, which are assembled to provide sufficient energy for their operations. Depending on the design specifications and physical form of battery cells, various configurations can be employed to assemble these cells into modules and battery packs. The structure and arrangement of these cells play a crucial role in determining the overall performance, efficiency, and safety of EV battery systems.

Fig. 1.

Schematic diagram of the structure of LIBs and LIB battery packs.

With the understanding of cell structure and its influence on thermal behavior, theories of battery heat transfer and self-heating have been extensively investigated. As early as 1958, Sherfey and Brenner conducted pioneering studies on the thermal effects of simultaneous electrochemical reactions (Sherfey and Brenner, 1958)¹⁰. They derived heat generation equations based on parameters such as electric current and entropy change, and further advanced this research in subsequent works (Gross, 1969; Gibbard, 1978)^11,12. Later, Newman and his collaborators developed predictive models for battery discharge processes based on porous electrode theory and the coupled dynamics of diffusion and charge transfer (Doyle et al., 1993; Fuller et al., 1994; Doyle and Newman, 1996; Newman and Tiedemann, 1975)^13–16. However, these models are primarily applicable to small-scale batteries, as they assume uniform electromotive force within the electrode materials, an assumption that does not hold for large-format batteries where internal potential gradients become significant.

To address the limitations of earlier models, Newman, Pals, and collaborators (Newman and Tiedemann, 1995; Pals and Newman, 1995; Pals and Newman, 1995)^17–19 proposed the first thermally coupled electrochemical model for LIBs, introducing a thermal analysis approach based on the inhomogeneous heat equation. In this framework, thermal and electrochemical processes are integrated via heat generation terms. This coupled model is applicable not only to individual battery cells but also to complete battery packs (Al Hallaj et al., 1999; Chen et al., 2005; Inui et al., 2007; Zhu et al., 2013; Hu and Jing, 2014;)^20–24. Further developments of electrochemical-thermal models for LIBs can be found in (Gu and Wang, 2000; Wang and Srinivasan, 2002; Kumaresan et al., 2008; Cai and White, 2011)^25–28. In addition, Kumaresan et al. (Kumaresan et al., 2008)²⁷ introduced a discharge performance model under varying operating temperatures, highlighting the influence of thermal conditions on battery behavior. Wang and Gu adopted a multidimensional modeling approach tailored for automotive battery systems and developed an electrochemical-thermal model using the volume-averaging method to predict both average temperature and temperature distribution within vehicle batteries (Gu and Wang, 2000; Srinivasan and Wang, 2003)^25,29. For large-scale batteries, Kim and colleagues (Kim et al., 2008; Kim et al., 2009; Kim et al., 2011)^30–32 investigated the influence of electrode configuration on LIB thermal behavior. They constructed a two-dimensional model to simulate temperature distributions by incorporating potential and current distributions across the electrodes. Together, these studies have laid the theoretical foundation for understanding heat transfer and self-heating phenomena in batteries, thereby providing a basis for subsequent advances in TR prediction and safety-oriented modeling.

Building on these theoretical foundations, attention has also been directed toward the causes and propagation of TR events. During TR accidents, a chain reaction may occur between adjacent battery cells or modules, leading to catastrophic damage. Spotnitz and Franklin conducted a detailed study on the various reactions that may occur during battery TR (Spotnitz and Franklin, 2003)³³, categorizing the common conditions leading to TR into the following five types: Short Circuit (Fang et al., 2014; Zhao et al., 2015)^34,35; Overcharge (Arora et al., 1999; Perkins et al., 2012)^36,37; Nail Penetration (Yamauchi et al., 2004; Zhao et al., 2015)^35,38; Crush (Sahraei et al., 2012)³⁹and Overheating. Among them, overheating is regarded as the most direct trigger of LIB TR. Consequently, modeling and numerical simulation have been extensively employed to capture the underlying thermal abuse mechanisms and to predict the associated failure behaviors of LIBs. Spotnitz et al. (Spotnitz et al., 2003)³³ and Hatchard et al. (Hatchard et al., 2001;)⁴⁰ developed a one-dimensional oven abuse model for LIBs to simulate the oven tests of prismatic and cylindrical lithium cobalt oxide cells, and effectively predicted the thermal behavior of the batteries under oven abuse conditions. Kim et al. (Kim et al., 2007)⁴¹ established a three-dimensional thermal abuse model for lithium cobalt oxide batteries and conducted detailed simulations of the internal temperature field evolution of cylindrical cells of various sizes during oven testing at 155 °C, as well as the diffusion process of internal chemical reactions. These numerical simulations, based on energy-balance and finite-element approaches, have effectively elucidated the thermal behavior of LIBs under overheating conditions.

Under overheating conditions, LIBs are prone to TR, which poses serious safety hazards. Existing studies primarily rely on multiphysics numerical simulations, such as finite element or finite volume methods based on COMSOL, to characterize the spatiotemporal thermal behavior inside batteries. However, these approaches are computationally expensive and time-consuming, and they exhibit high sensitivity to initial conditions and numerical perturbations near critical parameters, making it difficult to efficiently and reliably identify the key thresholds for TR (e.g., critical thickness, or critical heat generation rate). At present, comprehensive and systematic studies on the identification of LIB TR critical parameters under overheating conditions, as well as on the influence of cell geometry (prismatic versus cylindrical cells) on the critical thickness of LIBs, remain scarce. In addition, detailed investigations of TR critical values under different heat dissipation scenarios (natural convection versus forced convection) have rarely been reported.

To address these gaps, this study proposes a qualitative analysis framework based on bifurcation theory in nonlinear dynamics (Fulford and Broadbridge, 2002; Strogatz, 2015)^42,43, which enables rapid identification and calculation of the critical thickness for TR by analyzing steady-state solutions and bifurcation points. Specifically, the main contributions of this work are as follows: (1) establishing heat source terms and constructing simplified thermal models based on four typical exothermic reactions under overheating conditions, namely SEI decomposition, negative solvent reaction, positive solvent reaction, and electrolyte decomposition; (2) developing two simplified geometric models: a parallel model for prismatic cells and a circular model for cylindrical cells, and incorporating two heat dissipation scenarios into the models: natural convection cooling and forced convection cooling; (3) performing bifurcation analysis to determine the critical thickness and construct bifurcation diagrams, thereby systematically revealing the effects of temperature, heat dissipation conditions, and curvature on TR thresholds; (4) employing the obtained qualitative critical parameters as references for initialization and parameter selection in subsequent high-dimensional numerical simulations, which significantly reduces the computational cost and improves numerical convergence in accurately determining critical parameters.

The proposed method combines computational efficiency with physical interpretability, providing theoretical guidance and initial parameter estimates for advanced numerical simulations, battery structural and material design, and optimization of thermal management strategies. The methodology is demonstrated using representative parameters of a cylindrical 18650 LIB, and its applicability and limitations are further discussed.

Thermal model of libs

LIBs rely on complex reversible chemical reactions within their cell bodies to supply electric power to electrical devices. However, under high-temperature conditions, both reversible electrochemical reactions and intense irreversible chemical reactions occur inside the battery. Batteries can operate alongside an appropriate cooling system so that timely dissipation of excess heat can maintain the battery temperature within a safe range. Conversely, if the internal heat exceeds the dissipation to the surrounding environment, the battery temperature will continuously rise, ultimately leading to TR. The key factor in LIB TR is whether relevant parameters surpass critical threshold values, above which TR occurs. This study aims to develop two one-dimensional simplified mathematical models, incorporating bottom heating and top cooling boundary conditions, to represent overheating-induced exothermic reactions in LIBs. Based on representative parameters of an 18650 LIB (Melcher et al., 2016)⁴⁴, these models are used to simulate the thermal behavior of prismatic and cylindrical cells shown in Fig. 2, with the objective of employing bifurcation theory to determine the evolution of internal temperature and the associated TR critical thickness thresholds.

Fig. 2.

LIB diagram. (a1): exterior of a prismatic LIB; (a2): internal structure of a prismatic LIB; (b1): exterior of a cylindrical LIB; (b2): internal structure of a cylindrical LIB; (b3): Cross-sectional view of a cylindrical LIB; (b4): Longitudinal sectional view of a cylindrical LIB.

The energy conservation

Neglecting heat convection inside the LIB, the governing equation can be derived from Fourier’s Law as initial-boundary problem (Melcher et al., 2016; Bebernes and Eberly, 1989; Volpert, 2014 and Frank-Kamenetskii,1969; Peng et al., 2014)^44–47,50 with the parabolic differential equation:

1

Equation (1) represents the energy conservation equation for transient heat conduction, where the rate of temperature change is governed by thermal diffusion and internal heat generation. The system is supplemented with the following initial and boundary conditions. Equation (2) specifies the initial temperature distribution inside the LIB and Eq. (3) defines the heat dissipation at the cooling boundary, which combines convective and radiative heat transfer to the ambient environment.

2

3

where () denotes a special interior or surface point of LIBs. is the complete LIB region, consisting of the interior of LIB and the boundary of LIB . represents the volumetric heat generation rate in LIBs []; is the density of the LIB []; is the heat capacity []; is the thermal conductivity of LIBs []; is the time coordinate []; is the environment temperature; is the initial temperature inside LIBs ; is the outward pointing normal vector; Furthermore is the heat transfer coefficient [], is the emissivity []; is the Stefan-Boltzmann constant . Accordingly, the relevant physical parameters of the LIB are summarized in Table 1.

Table 1.

Thermophysical properties of LIBs.

Parameter	Zhang48	Drake49	Ziebert44	Simulation
	Optimized: 0.48	0.2 ± 0.01	0.898	0.50
	Optimized:1243	1720 ± 86	1389.7	1120
	—	—	2172.99	2172.99
/ 1	—	—	0.357	0.357

The heat generation term comprises numerous chemical reactions inside LIBs, including heat generated from reversible chemical reactions during normal battery operation, Joule heating due to the current flowing through conductors, as well as heat generated from other chemical processes. The expression for the heat generation term is given as follows (Wang et al., 2012)⁹: : heat generated by overheating chemical reactions in LIBs, : Joule heating within LIBs, : heat from entropy changes, : heat generated by overpotential, : heat exchange between the system and the ambient environment.

4

5

However, during TR, intense irreversible exothermic reactions occur inside LIBs due to the exceptionally high temperatures, which release a large amount of heat, accounting for the major part of total heat generations. Therefore, in the TR process, our primary focus is on the irreversible chemical reactions occurring under overheating conditions, represented by four volumetric heat generation terms: , , , and (details of these reactions will be explained below) (Kim et al., 2007; Peng et al., 2014;)^41,50.

Identifying overheating chemical exothermic reaction

Several exothermic chemical reactions can occur inside a LIB as the temperature rises. External factors lead to an increase in temperature, including external heating, overcharging or over-discharging, high current charging, nail penetration, external short circuits, and other abuse conditions. To accurately describe absurd behaviors such as TR, it is essential to identify the main exothermic chemical reactions involved. Following the works in (Hatchard et al., 2001; Peng et al., 2014; Abraham et al., 2006; Peng et al., 2014; Lisbona and Snee, 2011)^40,50–53 the general mechanism leading to TR with respect to rising temperature can be summarized in four sequential steps:

1. SEI decomposition reaction (SEI): At temperatures above , the solid-electrolyte interface (SEI) decomposes in an exothermic reaction.

2. Negative electrode solvent reaction (Neg): At temperatures above , an exothermic reaction begins between the intercalated lithium ions and the electrolyte.

3. Positive electrode solvent reaction (Pos): At temperatures above , an exothermic reaction occurs between the positive electrode material and the electrolyte, accompanied by oxygen evolution inside the cell.

4. Electrolyte decomposition reaction (Ele): Finally, at temperatures above , the electrolyte undergoes an exothermic decomposition reaction.

The exact temperature thresholds depend on the specific cell chemistry. For example, in Lithium cobalt oxide chemistry, these temperatures typically lie within the ranges (Peng et al., 2014)⁵⁰: , , and . A suitable theoretical framework to describe the kinetics of these exothermic reactions is the mathematical theory of combustion (Bebernes and Eberly, 1989; Volpert, 2014)^45,46. Assuming that these exothermic reactions occur independently and follow Arrhenius kinetics, the heat generation rate from each reaction can be modeled using a simple Arrhenius law.

SEI decomposition reaction

The decomposition of the solid electrolyte interphase (SEI) occurs within the temperature range of 363.15–393.15 K at the negative electrode of LIBs. The corresponding chemical reaction is given as follows (Kim et al., 2007; Coman et al., 2016)^41,54:

6

7

In the reaction equation, ​ represents the volumetric heat generation rate associated with the SEI decomposition reaction []; represents the specific heat release per unit mass []; ​ is the carbon content []; denotes the reaction rate []; is the pre-exponential factor []; is the activation energy of the reaction []; is the universal gas constant, with a value of ; is the reaction order; and represents the fraction of unstable lithium in the SEI.

Negative solvent reaction

When the temperature exceeds 393.15 K, the lithium intercalated in the graphite anode undergoes a direct exothermic reaction with the electrolyte. The chemical reaction is expressed as follows (Coman et al., 2016; Liu et al., 2017)^54,55:

8

9

In the reaction equation: ​ represents the volumetric heat generation rate associated with the negative electrode solvent reaction []; represents the specific heat release per unit mass []; is the carbon content []; denotes the reaction rate []; is the pre-exponential factor []; is the activation energy []; is the reaction order; represents the fraction of intercalated lithium in the carbon anode that reacts with the electrolyte; is the ratio of the SEI layer thickness to the characteristic size of active material particles; and is a constant scaling factor.

Positive solvent reaction

When the temperature exceeds 443.15 K, an exothermic reaction also occurs between the positive electrode material and the electrolyte. The corresponding reaction can be expressed as follows (Coman et al., 2016; Liu et al., 2017)^54,55:

10

11

where ​ represents the volumetric heat generation rate associated with the positive electrode solvent reaction []; represents the specific heat release per unit mass []; is the concentration of active material in the positive electrode []; is the reaction rate []; is the pre-exponential factor []; denotes the activation energy []; and are the reaction orders; and is the conversion rate of the positive electrode, defined as the ratio of reacted to total active cathode material.

Electrolyte decomposition reaction

When the temperature exceeds 473.15 K, the electrolyte decomposition reaction may occur in the positive and negative electrodes as well as in the separator region. The reaction is expressed as follows (Coman et al., 2016; Liu et al., 2017)^54,55:

12

13

In the reaction, ​ represents the volumetric heat generation rate associated with the electrolyte decomposition reaction []; represents the specific heat release per unit mass []; denotes the electrolyte content []; is the reaction rate []; is the pre-exponential factor []; refers to the activation energy []; is the reaction order; indicates the fraction of electrolyte remaining undecomposed. Table 2 lists the parameters of four reactions relevant to a 18650 LIB, including reaction heat , frequency factor , activation energy ​, and volume content .

Table 2.

LIB-Exothermic simulation parameters (Ziebert et al., 2016)⁴⁴.

Overheating Reaction	Reaction heat [J/kg]	Frequency factor [1/s]	Activation energy [J/mol]	Volume content ]
SEI reaction
Neg. solvent reaction
Pos. solvent reaction
Electrolyte decomp.

Simplification of thermal runaway model

Under high-temperature conditions, the consumption of reactant concentrations within LIBs remains relatively small from the onset of reactions until the occurrence of TR (Ziebert et al., 2016)⁴⁴. Therefore, the solid cell model is simplified into a constant fuel model (Frank-Kamenetskii, 1969)⁴⁷ by neglecting all time and spatial variations of the concentrations , where . In this framework,  ​. The model thus focuses on the LIB steady-state analysis, while neglecting the temporal variation of concentrations (Ziebert et al., 2016; Frank-Kamenetskii, 1969; Semenoff, 1928)^44,47,56. This assumption enables a qualitative analysis framework that directly targets steady-state equilibria. Similar spatial simplifications have been adopted in previous studies, where the concentration distribution is assumed to be spatially uniform (Spotnitz and Franklin, 2003; Hatchard et al., 2001; Kim et al., 2007)^35,40,41. Regarding heat generation, since chemical reactions under overheating dominate the total heat release, only the overheating chemical reactions term is retained, with its major contributions from SEI decomposition reaction (​), negative solvent reaction (​), and positive solvent reaction (​). Furthermore, the Frank-Kamenetskii approximation (Fulford and Broadbridge, 2002)⁴² is applied to express the temperature dependence of these heat generation terms:

14

where , is a constant overheating temperature and the relevant parameters are shown in Table 3.

Table 3.

Simplification of relevant parameters (Peng et al., 2014)⁵⁰.

0.15	0.75	0.04	1	0.033	1	1	1	1	1

Determination of critical parameters using bifurcation theory

The concept of a critical thickness arises from the competition between internal heat generation and external heat dissipation in LIBs under overheating conditions. When the bottom of the cell is subjected to heating while the top surface is cooled, the internal temperature eventually reaches a steady equilibrium as long as heat generation and dissipation remain balanced. However, as the thickness of the cell increases, the efficiency of heat dissipation decreases, making it increasingly difficult for the system to release accumulated heat. Once the cell thickness exceeds a critical threshold, the dissipation capacity can no longer offset the internal heat generation, causing the internal temperature to rise uncontrollably and triggering TR, which may ultimately lead to catastrophic events such as fire. The parameter therefore represents the limiting cell thickness beyond which a stable thermal equilibrium no longer exists, and it serves as a theoretical safety boundary for assessing LIB stability under overheating scenarios.

In this section, ​ is determined using a qualitative bifurcation analysis framework (Fulford and Broadbridge, 2002; Strogatz, 2015)^42,43 that focuses on steady-state solutions of the governing energy conservation equation Eq. (1). Specifically, for the parallel and circular models, the problem reduces to solving Eqs. (15) and (18) subject to the boundary conditions in Eqs. (16)-(17) and (19)-(20), under the assumption that the system has reached thermal equilibrium (). The maximum internal temperature of the LIB at steady state is taken as the representative solution. Numerical calculations reveal that, as the cell thickness approaches ​, two distinct steady-state solutions exceeding the overheating temperature ​ (denoted as ​ and ​) coexist in a bifurcation region. With further increase in thickness, these two equilibria gradually approach each other and eventually coalesce. Beyond this threshold, no steady-state solution exists, signifying that the LIB undergoes TR. The corresponding thickness at which this merging occurs is defined as the critical thickness . It is important to note that the bifurcation-based approach adopted here is a qualitative approximation that intentionally neglects transient processes and instead focuses on identifying equilibrium states. This greatly reduces computational cost while still providing rapid and physically interpretable estimates of TR thresholds.

Parallel model

The simplified structure of the prismatic LIB cell is illustrated in Fig. 3. This study focuses on the intense chemical reactions, occurring in different regions of the LIB during bottom overheating. In this model, the lower boundary of the LIB cell is represented by a plane with a constant temperature , resembling the high-temperature bottom environment under extreme conditions. Above the LIB, a cooler regime with a natural convection cooling environment is modeled to absorb the heat generated by chemical reactions. In addition, thermal radiation is considered, such that the upper boundary condition is governed by both convective and radiative heat transfer, as expressed in Eq. (3). In the following chapters, we also consider cases in which a forced convection cooling system is employed. Different reaction types were considered by selecting several representative temperatures to simulate the overheating environment. These include the negative electrode reaction at 395 K and 445 K (Neg-395 K & Neg-445 K), the SEI decomposition reaction at 380 K and 390 K (SEI-380 K & SEI-390 K), and the positive electrode reaction at 445 K (Pos-445 K). Under these conditions, a critical thickness exists for the positive and negative electrolyte regions. When the thickness exceeds this critical value, the total heat generated in the electrode region surpasses the heat dissipated, causing a thermal imbalance inside the LIB. Different cooler regime are simulated by varying the cooler regime coefficient and maintaining a constant cooler regime temperature .

Fig. 3.

Parallel LIB thermal behavior model.

Based on the described LIB cell structure, this paper develops a simple one-dimensional mathematical model and employs bifurcation analysis to determine the critical thickness. Here, we focus on the equilibrium state of the LIB, where the internal temperature no longer changes with time. By solving the steady-state equation, the existence of a numerical solution to the governing equation indicates that TR has not occurred. Conversely, the absence of an equilibrium solution implies that the LIB has entered TR. For equilibrium conditions of the parallel model, /, so the temperature depends only on the spatial coordinate . Therefore, the equilibrium temperature satisfies:

15

Equation (15) represents the steady-state heat conduction equation for the parallel model, where the temperature distribution is assumed to vary only along the thickness direction . The corresponding boundary conditions are given as follows. Specifically, Eq. (16) fixes the bottom surface () temperature at the overheating value ​, while Eq. (17) describes heat dissipation at the top surface () through a combination of convection and radiation to the environment.

16

17

This subsection presents simulation results using MATLAB Version R2023a (Math Works, 2023)⁵⁷. The model was implemented using the shooting method with the ode45 solver. This module incorporates several predefined LIB models described in the previous section. The simulation equations, including the specific models, their coupling, and parameter settings, were determined based on actual reaction mechanisms (see Tables 2 and 3).

Temperature distribution and critical parameter

The simulation aims to better understand the thermal behavior of LIBs under overheating conditions and to identify the critical thickness for TR, providing a valuable reference for future studies. The following section presents simulation results from the parallel model. First, the negative solvent reaction is simulated as shown below. The overheating boundary condition is set at , the cooler regime temperature , and the heat transfer coefficient . Figure 4 illustrates the variation of the maximum temperature and its location under different negative solvent region thicknesses, under conditions where TR has not yet occurred.

Fig. 4.

The parallel LIB heat distribution diagram.

Although a highly exothermic negative solvent reaction has occurred in the LIB, the governing equations still admit the steady-state solution when the LIB thickness is relatively small. This indicates that TR does not occur when the system remains below the critical thickness. It is evident that as the thickness of the negative electrode region increases, the maximum steady-state temperature under overheating conditions gradually decreases. Moreover, the location of this peak temperature shifts from the vicinity of the cooler regime toward the overheating heat source.

Figure 4 illustrates the one-dimensional temperature distribution in the negative solvent region of the LIB for various thicknesses andm. At , the peak temperatures are 430.7 K and 416.8 K, respectively, demonstrating a clear decreasing trend in as the thickness increases. Figure 5 presents a bifurcation diagram, showing the relationship between the steady-state maximum temperature and the thickness . The diagram indicates that as the reaction region approaches a critical thickness, two distinct equilibrium solutions for maximum temperature may exist. Whenis sufficiently large, no steady-state solution is available, implying that the temperature will continue to rise, ultimately resulting in TR. The small segment on the lower left of Fig. 5 is magnified on the right to clearly display the onset of bifurcation behavior.

Fig. 5.

Bifurcation diagram of maximum temperature in parallel LIB model.

Two views of the bifurcation diagram. On the left of Fig. 5, we have searched for temperature values 430.7 K to 395 K, which show only the upper branch of the bifurcation diagram. On the right of Fig. 5, a more refined search is made which yielded both branches. The critical value can now be read off as ≈ 0.07167 m. From the right diagram of Fig. 5, the value ofwhere the curve begins to turn back towards the vertical axis.

Influence of heat transfer coefficient on

In the previous subsection, the critical thickness for TR was determined. Here, we investigate the effect of different cooling efficiencies by varying the heat transfer coefficient , aiming to evaluate the relationship between and under a fixed ambient temperature . In this context, a LIB operates under safe conditions with thickness below , while exceeding this threshold leads to TR. Simulations were performed for the following reaction scenarios: (1) SEI decomposition reactions with initial temperatures and ; (2) negative solvent reactions with and ; (3) positive solvent reactions at . In all simulations, the ambient temperature was fixed at , and the heat transfer coefficient was varied within the range . The corresponding results are presented in Fig. 6. The simulation results indicate that, within the present numerical framework, the influence of the heat transfer coefficient on TR behavior exhibits trends consistent with those reported by Ziebert et al. (Ziebert et al., 2016)⁴⁴, thereby lending further support to the reliability of the proposed model.

Fig. 6.

Thermal runaway thickness as function of the heat transfer coefficient .

The small region at the lower end of profile (B), marked by the blue strip in Fig. 6, is magnified on the left to better illustrate the trend. When , there is no heat exchange between the LIB and the cooler regime, and under this condition, the critical thicknesses for all reaction types remain below 0.02 m. As begins to increase, the critical thickness rises significantly within the range . This indicates that improved heat dissipation delays the onset of TR, allowing the LIB to tolerate greater thickness before reaching thermal instability. However, when , the rate of increase in gradually slows down despite further enhancement in heat transfer.

Circular model

This section presents a theoretical analysis of the critical thicknesses of various reactive regions within a cylindrical LIB under overheating conditions, taking into account different structural bending in the LIB unit. Figure 7 illustrates a one-dimensional heat conduction model for the cylindrical LIBs. For the cylindrical cell, due to the central symmetry of its cross-section, the problem can be simplified into a one-dimensional circular model based on the axisymmetric principle. The corresponding steady-state heat conduction equation can be expressed as Eq. (18), where denotes the radial distance from the center. The lower part of the model represents a curved surface with a radius , where high temperatures are applied to simulate extreme overheating conditions. The temperature in this region is fixed at , which varies depending on the type of reactions (SEI decomposition, negative solvent, or positive solvent reaction). The central layer represents the electrolyte region, where the thermal reactions occur. The upper part of the model with a radius, features a cooler regime maintained at a constant temperature of and a heat transfer coefficient .

Fig. 7.

Circular LIB thermal behavior model.

Keeping all other conditions identical to that of the parallel model, this study focuses on how varying the degree of curvatures in the LIB unit affects the critical thickness in the one-dimensional cylindrical LIB model. In this circular model, each reactive region forms a concentric annular layer with a distinct radius. At thermal equilibrium, the system exhibits radial symmetry, allowing the temperature distribution to be expressed as a function of the radial coordinate , defined over the interval , which satisfies the corresponding governing Eq. 

18

Equation (18) represents the steady-state heat conduction equation for the circular model, where the temperature distribution is assumed to vary only along the radial direction due to axisymmetry. The associated boundary conditions are given as follows. Equation (19) fixes the temperature at the inner curved surface (​) to the overheating value ​, simulating localized heating at the bottom of the cylindrical cell. Equation (20) describes the boundary condition at the outer curved surface (​), where heat is dissipated to the environment through a combination of convection and radiation.

19

20

For the parallel model, the temperature distribution under overheating conditions and the relationship between the critical thickness and the heat transfer coefficient are analyzed for LIBs. In this subsection, the ambient temperature of the cooler regime and the heat transfer coefficient are kept constant. While maintaining all other parameters intact with the parallel model, we vary only the radius in the circular model to simulate different circular curvatures of LIBs. The curvature of the overheating surface is defined as , providing a more intuitive way to examine how the critical thickness changes with LIB curvatures, where the numerical results are illustrated in Fig. 8.

Fig. 8.

Critical thickness as function of the circular overheating radiusand curvature .

In profiles (I) and (II), as increases, all curves (except for the Neg-445 K case) show a decreasing trend in the critical thickness across different reaction regions (Neg, SEI, and Pos) under various temperatures. Notably, decreases rapidly within the range , indicating that greater curvature (smaller) significantly raises the threshold of thicknesses for TR. As continues to increase, the curvature effect becomes negligible, leading to converge toward the value predicted by the parallel model.

The relationship between the critical TR thicknessand the curvature of the overheating arc surface is presented in profiles (III) and (IV) in Fig. 8. The right-hand figure shows the range of , with a detailed view of the blue strip provided in the left profile (III). Asapproaches infinity, the curvatureapproaches zero. Since the overheating surface in the parallel model has zero curvature (), the critical thicknessesfor various reaction regions and temperatures converge toward the corresponding values predicted by the parallel model asapproaches zero.

Forced convective model

The previous parallel and circular models do not take into account the effects of forced fluidic convection in the cooling system. With the circulation of an internal coolant, actively cooled systems can significantly enhance heat dissipation. Taking the parallel model as an example, the cooler regime is replaced with forced convective cooling. At the bottom, the overheating temperature , negative solvent reaction (Neg-395 K) occurs, with and the heat transfer coefficient . Where the boundary condition for this system, incorporating convective cooling is modified as follows:

21

where is the convection coefficient of the active cooling system [], which primarily depends on the uniform flow velocity of the cooling medium. To better illustrate the effect of active convective cooling, the following approximation is adopted:

22

where represents the effective depth generated by the cooling system. We then combine with the heat transfer coefficient , and treat the sum as an effective heat transfer coefficient, denoted by ​, then Eq. (21) becomes:

23

where . The effective heat transfer coefficient is considered to represent the combined effects of symmetric heat conduction and fluidic convection from the active cooling system. Below presents the variation in the critical thickness of the LIB cell under the influence of convective cooling during a negative solvent reaction at an overheating condition of 395 K.

In Fig. 9, curve represents the critical thickness of the LIB without considering the fluidic convection from the active cooling system. Two distinctive, i.e. andcurves correspond to the bifurcation curves of fluidic convection. It is evident that active convective cooling significantly increases the critical thicknesses of the LIB under overheating conditions.

Fig. 9.

Bifurcation diagram of in parallel LIB model influence by convection, i.e..

Conclusion

This paper reviews the existing studies on the thermal behavior and TR of LIBs. Considering that current numerical simulation approaches often suffer from high computational complexity, long processing times, and strong sensitivity to initial conditions and numerical errors near critical thresholds, this work proposes a qualitative bifurcation-based framework to rapidly estimate the critical thickness for TR in LIBs. Two one-dimensional thermal models, the parallel model and the circular model, were developed to analyze the critical thickness of three representative exothermic reactions under overheating conditions, namely negative electrolyte decomposition, positive electrolyte decomposition, and SEI decomposition. The results indicate that applying forced convection at the LIB cooling boundaries and increasing the structural curvature can significantly raise the critical thickness threshold for TR. This work shows that qualitative bifurcation-based analysis can capture the onset of TR through steady-state solutions and bifurcation points, providing a computationally efficient and physically interpretable approach. While approximate in nature, the method reveals key qualitative trends, offers initial parameter estimates for high-dimensional numerical simulations, and contributes theoretical guidance for improving LIB safety design and thermal management strategies. However, due to the assumption of a constant fuel source, the model is unable to capture the decay phase that follows complete fuel consumption during TR, and it may overestimate the period of continuous heat release. Additionally, it assumes that the exothermic reactions at different temperatures occur independently, neglecting the potential coupling of multiple reactions within the LIB under three-dimensional overheating conditions. Therefore, future work will focus on developing higher-dimensional coupled models of LIB TR that incorporate evolving fuel concentrations. This advancement will enable more realistic and accurate predictions of critical parameters for LIB safety.

Author contributions

Y.C.: Supervision, methodology, writing-review and editing, modification for the final layout. H.Y.W.: Conceptualization of this study, methodology, software, data curation, writing-original draft. All authors have read and agreed to the published version of the manuscript and agree to be accountable for their contributions.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.