Prediction and Correction of LFL for Multicomponent Gases Relevant to Lithium-Ion Battery Thermal Runaway

Abstract

The lower flammability limit (LFL) is critical for assessing ignition and explosion risks associated with lithium-ion battery thermal runaway gases. In this study, a primary experimental LFL database containing 58 data points for H₂–CO–CH₄–C₂H₄–CO₂–N₂–O₂ mixtures was compiled from the first author’s publicly available doctoral dissertation. Le Chatelier’s rule was first evaluated as a baseline model, and its prediction residuals were then corrected using a five-coefficient data-driven model based on physically interpretable variables, including total diluent fraction, CO₂ fraction in diluent gases, oxygen fraction, and the interaction between highly reactive fuels and dilution. Results showed that Le Chatelier’s rule provided a useful first-order estimate but systematically underestimated LFL under high-dilution and CO₂-rich conditions. The proposed correction model reduced the mean absolute error from 2.40% to approximately 1.00% while retaining an explicit mathematical form. This experimental-data-driven framework provides a preliminary and interpretable tool for rapid LFL estimation of multicomponent gases relevant to battery thermal runaway risk assessment.

1. Introduction

Lithium-ion batteries have been widely deployed in electric vehicles, electrochemical energy storage systems, and portable electronic devices because of their high energy density and long cycle life [1,2]. However, under abusive conditions such as overheating, overcharging, mechanical damage, or internal short circuit, lithium-ion batteries may undergo thermal runaway, accompanied by rapid heat release, venting, fire, and even explosion [3,4,5]. The vent gases generated during thermal runaway are not only toxic but also flammable, and their accumulation in confined or semi-confined spaces may create a serious explosion hazard. Recent reviews have emphasized that battery off-gas is a critical source of fire, explosion, and toxicity risks, and that quantitative characterization of gas composition and flammability is essential for battery safety assessment [6,7].

The composition of thermal runaway vent gases is highly complex and depends on cell chemistry, state of charge, cell format, electrolyte formulation, and failure conditions [8,9,10]. Despite these variations, several major gaseous species are frequently reported, including H₂, CO, CO₂, CH₄, and C₂H₄ [11,12,13]. For example, experimental studies on LFP and NCM lithium-ion batteries under inert atmosphere showed that the main gas components produced during thermal runaway include H₂, CO, CO₂, C₂H₄, and CH₄, with LFP cells showing a particularly high proportion of H₂ [14,15,16]. Because H₂ has a very low lower flammability limit and a wide flammability range, an increased H₂ fraction can significantly enhance the flammability hazard of the vent gas [17,18]. Therefore, reliable estimation of the flammability characteristics of thermal-runaway-relevant multicomponent gases is necessary for evaluating ignition risk, designing ventilation strategies, and establishing early warning thresholds.

Among flammability parameters, the lower flammability limit (LFL) is especially important for early-stage risk assessment [19,20]. The flammable limit is commonly defined as the upper or lower concentration limit of a flammable gas or vapor in air, at a specified temperature and pressure, that can be ignited. In battery enclosures or confined spaces, vent gas concentration usually evolves from a lean nonflammable state toward the flammable range as vent gases accumulate [21]. Therefore, the LFL is the first critical boundary for identifying the onset of ignitability and is directly relevant to ventilation activation, inerting design, and early-warning thresholds.

For multicomponent fuel mixtures, Le Chatelier’s (LC) mixing rule has been widely used as a simple and practical method for estimating the LFL when the LFL values of individual combustible components are known [19,20,22]. Previous studies have described Le Chatelier’s rule as the prevailing method for estimating the LFL of mixtures containing multiple flammable components, and it remains extensively used in industrial applications because of its simplicity and low data requirement [22]. In the context of lithium-ion battery vent gas, experimental work on premixed battery vent gas also showed that Le Chatelier’s rule can provide a useful baseline estimate. For example, the experimentally measured LFL of Li-ion premixed battery vent gas in air was reported as 7.88 ± 0.41 vol%, while the corresponding calculated value based on Le Chatelier’s rule was close to the experimental value [23,24].

Nevertheless, the direct application of Le Chatelier’s rule to thermal-runaway-related gas mixtures still requires caution. The rule was originally developed for mixtures of combustible gases and is based on simplifying assumptions related to combustion heat release, flame temperature, and the independent contribution of each flammable component [25,26,27]. Chen et al. noted that the validity of Le Chatelier’s rule is closely related to the assumption that the adiabatic flame temperature rise at the LFL is approximately the same for the flammable components; deviations may occur when this assumption is not satisfied [22]. Moreover, lithium-ion battery vent gases often contain not only multiple combustible components such as H₂, CO, CH₄, and C₂H₄, but also substantial amounts of diluent gases such as CO₂ and N₂ [28,29,30]. These inert or semi-inert components may alter flame propagation, heat capacity, and radical chemistry, thereby causing systematic deviations between calculated and experimentally measured LFL values [31]. Therefore, although Le Chatelier’s rule is valuable as a baseline model, its prediction bias under high-dilution and battery-relevant multicomponent conditions remains to be further quantified.

Existing studies on battery vent gas flammability have mainly focused on gas composition measurement, explosion characteristics, numerical calculation of flammability limits, or specific battery chemistries and states of charge [32,33,34,35]. However, experimental LFL data for H₂–CO–CH₄–C₂H₄–CO₂–N₂–O₂ mixtures remain scattered across different research fields, including battery safety, syngas combustion, industrial gas explosion, and fundamental flammability studies. In addition, in some battery-safety studies, flammability limits are estimated using Le Chatelier’s rule rather than measured directly, which limits the availability of directly measured LFL data for model validation. This creates a methodological gap: there is still a need to construct a unified experimental database, evaluate the baseline error of Le Chatelier’s rule, and develop a correction model that remains explicit, interpretable, and suitable for limited experimental data.

To address this gap, the present study develops an experimental-data-driven framework for predicting and correcting the LFL of multicomponent gases relevant to lithium-ion battery thermal runaway. A literature-derived experimental LFL database was first constructed using gas mixtures containing CO₂, CO, H₂, CH₄, C₂H₄, N₂, and O₂. Only experimentally measured LFL data were included in the main database, whereas values derived solely from Le Chatelier’s rule or numerical simulations were excluded from model training. Le Chatelier’s rule was then used as the baseline method, and its prediction residuals were analyzed with respect to fuel composition, total diluent concentration, and CO₂/N₂ dilution characteristics. Finally, a data-driven correction model was established using physically interpretable variables, including total diluent fraction, CO₂ fraction in the diluent gases, oxygen fraction, and the interaction between highly reactive fuel components and total dilution. The corrected model was compared with conventional regression models, random forest, and a shallow neural network to evaluate whether an explicit correction equation can provide improved accuracy while retaining interpretability.

The main contribution of this work is threefold. First, a curated literature-derived experimental LFL database was compiled for multicomponent gas mixtures relevant to lithium-ion battery thermal runaway. Second, the applicability and systematic bias of Le Chatelier’s rule were quantitatively evaluated under dilution-dominated mixture conditions. Third, a parsimonious residual correction model was proposed to improve LFL estimation while retaining an explicit and physically interpretable form. This framework provides a preliminary tool for rapid flammability-risk screening of battery-thermal-runaway-relevant gas mixtures.

2. Materials and Methods

2.1. Experimental Database and Data Preprocessing

The primary experimental LFL database used for model development consisted of 58 data points compiled from the first author’s publicly available doctoral dissertation [36], which is traceable through its DOI and institutional repository record. These data were not newly measured in the present study and are not presented here as an independently journal-published database. The novelty of the present work lies in the standardized organization of this primary database, the quantitative diagnosis of the systematic bias of Le Chatelier’s rule under dilution-dominated conditions, and the development of a physically interpretable and potentially software-implementable residual-correction equation for rapid LFL estimation. The source dissertation and related experimental descriptions provided, where available, information on ignition source, vessel geometry, test temperature, pressure, and flame-propagation criteria. Because the present study focused on standardized gas-composition and measured LFL data, differences in experimental apparatus, ignition conditions, and flame-propagation criteria were treated as sources of database heterogeneity and are further discussed as limitations.

Each data point contained the volume fractions of CO₂, CO, H₂, CH₄, C₂H₄, N₂, and O₂, together with the corresponding experimentally measured LFL. These 58 data points were used for all main analyses in this study, including Le Chatelier baseline evaluation, residual correction, coefficient fitting, cross-validation, uncertainty analysis, and comparison with machine-learning models. Only experimentally measured LFL values were included in the primary database; values derived solely from Le Chatelier’s mixing rule, numerical simulations, chemical equilibrium calculations, or other theoretical estimations were excluded from model training. All gas compositions were expressed as volume percentages, and components not present in a reported mixture were assigned a value of 0. The composition range and LFL distribution of the primary database are summarized in

Table 1. Summary of the primary experimental LFL database used for model development.

Variable	Meaning	Non-Zero Samples, n	Minimum, %	Median, %	Maximum, %
CO2	Carbon dioxide/diluent gas	43	0.000	8.200	78.581
CO	Carbon monoxide/combustible gas	44	0.000	24.240	50.030
H2	Hydrogen/combustible gas	56	0.000	37.045	100.000
CH4	Methane/combustible gas	53	0.000	2.194	100.000
C2H4	Ethylene/combustible gas	21	0.000	0.000	100.000
N2	Nitrogen/diluent gas	41	0.000	21.710	80.000
O2	Oxygen/oxidizer	24	0.000	0.000	0.950
LFL	Experimental lower flammability limit	58	0.524	8.000	54.035

Note: LFL = lower flammability limit. For LFL, the number of non-zero samples equals the total number of experimental data points.

.

In addition to the primary database, several external experimental flammability-limit datasets were identified from the literature during data screening. These external data are summarized in the Supplementary Materials to improve transparency and to provide candidate data for future external validation, but they were not used for model fitting, coefficient estimation, cross-validation, or performance metrics because of differences in mixture systems, reporting formats, experimental conditions, and data completeness. Therefore, all main analyses in the present study, including Le Chatelier baseline evaluation, residual correction, coefficient fitting, cross-validation, uncertainty analysis, and model comparison, were conducted using only the 58 data points in the primary database. The complete primary database, processed variables, Le Chatelier predictions, residuals, corrected predictions, pure-component LFL values, and data-screening criteria are provided in Supplementary Tables S1–S4, while the additional external and excluded candidate datasets are provided in Supplementary Tables S8–S10.

The total gas fraction was close to 100% for nearly all samples, indicating that the collected compositions were internally consistent. Among the 58 data points, three were pure combustible-gas reference data, whereas the remaining samples were multicomponent mixtures containing two or more combustible components and/or diluent gases. This distribution provided a basis for evaluating the applicability of conventional LFL prediction methods under multicomponent and dilution conditions.

The original gas composition variables were organized as Equation (1):

X = [CO₂, CO, H₂, CH₄, C₂H₄, N₂, O₂]

(1)

For mixtures in which a given component was not reported or not present, its volume fraction was set to 0. For example, in an H₂–air system, CO₂, CO, CH₄, and C₂H₄ were set to 0. However, if a study reported additional combustible components outside the selected variable set, such as C₂H₆, C₃H₆, C₃H₈, or C₄H₁₀, these components were recorded in the notes and were not treated as true zero values during interpretation.

Before model development, duplicate and near-duplicate gas compositions were checked. When repeated values were identified from the same source, only one representative value was retained. When similar compositions were reported in different studies, they were retained because differences in apparatus, ignition energy, flame-propagation criterion, and experimental conditions may lead to different measured LFL values.

A general workflow of data screening, baseline calculation, residual correction, and model validation is shown in Figure 1.

Le Chatelier’s mixing rule was used as the baseline method for estimating the LFL of multicomponent combustible gas mixtures. For a mixture containing several combustible components, the baseline LFL was calculated as Equation (2) [16,37]:

$$LF{L}_{\mathrm{LC}}={\left({\displaystyle \sum _i\frac{y_i}{LF{L}_i}}\right)}^{-1}$$

(2)

where LFL_LC is the LFL predicted by Le Chatelier’s rule, y_i is the normalized volume fraction of the i-th combustible component in the combustible-gas fraction, and LFL_i is the pure-component LFL of the corresponding combustible gas.

In this study, H₂, CO, CH₄, and C₂H₄ were treated as combustible components in the baseline calculation. CO₂ and N₂ were treated as diluent gases, whereas O₂ was not included as a combustible component. The pure-component LFL values used for H₂, CO, CH₄, and C₂H₄ are listed in Table S3.

The prediction residual of the baseline model was defined as Equation (3):

$$\Delta LFL=LF{L}_{\exp }-LF{L}_{\mathrm{LC}}$$

(3)

where LFL_exp is the experimentally measured LFL. A positive residual indicates that Le Chatelier’s rule underestimates the experimental LFL, whereas a negative residual indicates overestimation.

Unless otherwise stated, the term “primary database” refers to the 58 experimental LFL data points used for model development, whereas “external literature data” refers to additional data collected for reference and potential future validation but not included in the main analysis.

2.2. Construction of Combined Variables and Residual Correction Model

Because the original gas composition variables were partially collinear and most mixtures had total fractions close to 100%, a small set of physically interpretable combined variables was constructed. The aim was to correct the systematic residual of the Le Chatelier baseline while avoiding a high-dimensional empirical equation.

The total diluent fraction was defined as Equation (4):

$$I={\mathrm{CO}}_2+{\mathrm{N}}_2$$

(4)

where I represents the combined dilution effect of CO₂ and N₂. To distinguish the different dilution characteristics of CO₂ and N₂, the CO₂ fraction in the total diluent gases was defined as Equation (5):

$$R_{\mathrm{CO}2}={\mathrm{CO}}_2/\left({\mathrm{CO}}_2+{\mathrm{N}}_2\right)$$

(5)

When CO₂ + N₂ = 0, R_CO2 was set to 0 to avoid division by zero. The highly reactive combustible fraction was defined as Equation (6):

$$F_{\mathrm{r}}={\mathrm{H}}_2+{\mathrm{C}}_2{\mathrm{H}}_4$$

(6)

This variable was introduced because H₂ and C₂H₄ generally have lower LFL values and higher flame reactivity than CO and CH₄. To represent the interaction between highly reactive fuels and dilution intensity, the following interaction term was used in Equation (7):

$$F_{\mathrm{r}}I=\left({\mathrm{H}}_2+{\mathrm{C}}_2{\mathrm{H}}_4\right)\left({\mathrm{CO}}_2+{\mathrm{N}}_2\right)/100$$

(7)

The oxygen fraction O₂ was retained as an additional variable because variations in oxidizer concentration may affect flame propagation near the lower flammability boundary. The definitions and physical meanings of the combined variables are summarized in

Table 2. Combined variables used in the final correction model.

Symbol	Definition	Physical Meaning
I	CO2 + N2	Total diluent fraction.
RCO2	CO2/(CO2 + N2)	CO2 fraction in the diluent gases.
O2	O2	Oxidizer-related component.
Fr	H2 + C2H4	Highly reactive combustible components.
FrI/100	(H2 + C2H4)(CO2 + N2)/100	Interaction between highly reactive fuels and total dilution.

Note: All gas fractions are expressed in volume percent, except R_CO2, which is dimensionless.

.

Instead of directly replacing Le Chatelier’s rule, a residual correction strategy was adopted. Le Chatelier’s rule provided the baseline LFL prediction, and the data-driven model was used only to correct its systematic residual as Equation (8):

$$LF{L}_{\mathrm{corr}}=LF{L}_{\mathrm{LC}}+\Delta LF{L}_{\mathrm{pred}}$$

(8)

where ΔLFL_pred is the predicted residual obtained from the correction model.

Considering the limited sample size and the need for engineering interpretability, a five-coefficient correction model was selected as Equation (9):

$$\Delta LF{L}_{\mathrm{pred}}={\beta }_0+{\beta }_{1}I+{\beta }_2{R}_{\mathrm{CO}2}+{\beta }_3{\mathrm{O}}_2+{\beta }_4{\displaystyle \frac{F_{r}I}{100}}$$

(9)

Thus, the corrected LFL was calculated as Equation (10):

$$LF{L}_{\mathrm{corr}}=LF{L}_{\mathrm{LC}}+{\beta }_0+{\beta }_{1}I+{\beta }_2{R}_{\mathrm{CO}2}+{\beta }_3{\mathrm{O}}_2+{\beta }_4{\displaystyle \frac{F_{\mathrm{r}}I}{100}}$$

(10)

where β₀ is the intercept, and β₁ to β₄ are regression coefficients determined from the experimental database. The final five-coefficient correction equation was fitted using ordinary least-squares regression to obtain an explicit and directly reproducible formula. Ridge regression was additionally evaluated during model selection as a regularized comparative model, considering the limited sample size and potential collinearity among variables.

The conceptual framework of the Le Chatelier baseline prediction and residual correction strategy is shown in Figure 2.

2.3. Model Selection and Comparative Models

A hierarchical model simplification strategy was adopted to avoid overfitting. Candidate correction models with different levels of complexity were first evaluated, including models based only on dilution descriptors, models including oxygen concentration, and models including fuel–diluent interaction terms. A term was retained only when it improved or maintained cross-validation performance while preserving physical interpretability.

The final correction model was selected because it provided a favorable balance between prediction accuracy and model simplicity. More complex models containing additional terms, such as CO + CH₄, (CO + CH₄)I/100, F_rR_CO2, or F_sR_CO2, were not selected as the main model because their improvement in validation performance was limited relative to the increase in parameter number.

To evaluate the effectiveness of the proposed correction model, several comparative models were constructed, as summarized in

Table 3. Models compared in this study.

Model	Input Variables	Purpose
Le Chatelier rule	Pure-component LFL values and combustible fractions.	Baseline model
Linear Ridge regression	Original gas components	Regularized regression benchmark
Five-coefficient correction model	I, RCO2, O2, FrI/100	Main model
Random forest regression	Combined variables	Nonlinear machine-learning benchmark
Shallow neural network	Combined variables	Small-sample black-box comparison

. The sample-wise prediction results for the baseline and corrected models are provided in Table S5, and the comparison of candidate correction models with different complexity levels is provided in Table S6.

The random forest and shallow neural network models were used only as comparative models. They were not intended to replace the proposed explicit correction equation. The random forest model used 300 trees with a fixed random seed of 42. The shallow neural network was deliberately kept small, with one hidden layer containing four neurons, a tanh activation function, an L2 regularization parameter of 0.001, a maximum of 5000 iterations, and a fixed random seed of 42. This design was used to reduce overfitting and to test whether a simple nonlinear black-box model could outperform the explicit correction model under small-sample conditions. The detailed parameter settings for the random forest, shallow neural network, and cross-validation procedures are provided in Table S7.

Model performance was evaluated using the coefficient of determination (R²) as Equation (11), root mean squared error (RMSE) as Equation (12), and mean absolute error (MAE) as Equation (13) [38,39].

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$$

(11)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}$$

(12)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|}$$

(13)

where $y_i$ is the experimental LFL, ${\widehat{y}}_i$ is the predicted LFL, $\overline{y}$ is the mean experimental LFL, and n is the number of samples.

Two validation strategies were used. Random five-fold cross-validation was first performed to evaluate interpolation performance within the available composition space. Grouped five-fold cross-validation was then used as a more conservative validation strategy by preventing identical or near-identical gas compositions from appearing simultaneously in the training and testing sets. The grouping was defined according to rounded gas-composition vectors of CO₂, CO, H₂, CH₄, C₂H₄, N₂, and O₂.

For the proposed correction model, performance was evaluated in two ways. First, the corrected LFL, LFL_corr, was compared directly with the experimental LFL. Second, the reduction in prediction error relative to the Le Chatelier baseline was quantified. The correction model was considered effective only if it reduced RMSE and MAE while maintaining stable performance under both validation strategies.

3. Results and Discussion

3.1. Characteristics of the Experimental LFL Database

The primary database used for model development contained 58 literature-derived LFL data points covering H₂–CO–CH₄–C₂H₄ mixtures with varying amounts of CO₂, N₂, and O₂. As summarized in

Table 4. Summary of the primary database used for model fitting and validation.

Metric	Value
Sample size	58
LFL range/%	0.524–54.035
LFL median/%	8.000
Mixtures with I > 0, n	55
Mixtures with O2 > 0, n	24
Pure-gas samples, n	3

Note: I = CO₂ + N₂. Pure-gas samples refer to data points with only one non-zero combustible component among CO, H₂, CH₄, and C₂H₄.

, the measured LFL values ranged from 0.524% to 54.035%, with a median of 8.000%. The database included both pure-gas reference points and diluted multicomponent mixtures; specifically, 55 samples contained CO₂ and/or N₂, and 24 samples contained explicitly reported O₂. These characteristics indicate that the database is suitable for evaluating whether a conventional mixing-rule calculation remains reliable when the mixture contains substantial dilution or oxidizer variation.

Figure 3 further illustrates the distribution of the collected data. The LFL values were concentrated mainly in the low-concentration region, whereas a smaller number of highly diluted mixtures exhibited much higher LFL values. The broad distributions of CO₂, H₂, CO, and N₂ further confirm that the dataset covers both fuel-rich and dilution-dominated compositions. This composition diversity provides the basis for diagnosing the residual error of the Le Chatelier baseline model.

3.2. Baseline Performance of Le Chatelier’s Rule

Le Chatelier’s rule was first evaluated as the baseline method. As shown in

Table 5. Error diagnostics of the Le Chatelier baseline in different composition subsets.

Subset	n	Baseline MAE
Low I	29	0.751
High I	29	4.054
Low RCO2	30	1.423
High RCO2	28	3.452

Note: Subsets were defined by median splitting of I and R_CO2. Baseline MAE refers to the mean absolute error between experimental LFL and LFL predicted by Le Chatelier’s rule.

, the baseline model achieved an overall R² of 0.848, with an RMSE of 4.42% and an MAE of 2.40%. However, the average residual (LFL_exp − LFL_LC) was 2.07 percentage points, indicating an overall tendency to underestimate the experimental LFL. The subset analysis in Table 5 shows that this error was not uniform across the composition space: the MAE increased from 0.75% in the low-I subset to 4.05% in the high-I subset, and from 1.42% in the low-R_CO2 subset to 3.45% in the high-R_CO2 subset. These results suggest that both total dilution intensity and CO₂-dominated dilution are major contributors to the systematic deviation of the baseline model.

The residual trends are visualized in Figure 4. The residual increased with the total diluent fraction, implying that Le Chatelier’s rule increasingly underpredicted the LFL as the amount of CO₂ and/or N₂ increased. The residual also tended to increase when the diluent fraction was more CO₂-dominated. This observation supports the need for an explicit correction term that accounts for both the amount and type of dilution, rather than relying only on the combustible-gas fractions.

3.3. Performance of the Five-Coefficient Correction Model

Based on the residual patterns identified above, a parsimonious five-coefficient correction model was established. The fitted coefficients are listed in

Table 6. Coefficients and uncertainty analysis of the five-coefficient correction model.

Term	Definition	Coefficient	SE	95% CI	p Value
Intercept	constant	−0.234	0.600	−1.438 to 0.970	0.698
I	CO2 + N2	0.144	0.010	0.123 to 0.164	<0.001
RCO2	CO2/(CO2 + N2)	1.362	0.551	0.256 to 2.469	0.017
O2	O2	−2.112	0.698	−3.512 to −0.711	0.004
FrI/100	(H2 + C2H4)(CO2 + N2)/100	−0.264	0.032	−0.327 to −0.200	<0.001

Note: F_rI = (H₂ + C₂H₄)(CO₂ + N₂)/100. All composition variables are expressed in volume percent except R_CO2, which is dimensionless.

, and the resulting correction equation was expressed as Equation (14):

$$\begin{array}{cc}\hfill LF{L}_{\mathrm{corr}}& \hfill =LF{L}_{\mathrm{LC}}-0.234+0.144I+1.362R_{\mathrm{CO}2}\\ & \hfill - 2.112{\mathrm{O}}_2-0.264{\displaystyle \frac{F_{r}I}{100}}\end{array}$$

(14)

All gas fractions in Equation (14) are expressed in volume percent, except R_CO2, which is dimensionless. As shown in Table 6, the correction term increased with the total diluent fraction and the CO₂ fraction in the diluent gases, but decreased with O₂ concentration and the interaction between highly reactive fuels and total dilution. These coefficient signs are physically consistent with the interpretation that CO₂/N₂ dilution raises the LFL beyond the Le Chatelier estimate, whereas highly reactive fuels such as H₂ and C₂H₄ partly offset the dilution effect. To further evaluate coefficient robustness, standard errors and 95% confidence intervals were calculated. The coefficients of I, RCO2, O₂, and FrI/100 were statistically significant, whereas the intercept was not significant but was retained to avoid forcing the residual correction through zero.

The prediction improvement is shown in Figure 5. Compared with the baseline prediction, the corrected predictions were much closer to the y = x reference line across both low- and high-LFL regions. Quantitatively, the direct-fit performance improved from R² = 0.848, RMSE = 4.42%, and MAE = 2.40% for the baseline model to R² = 0.986, RMSE = 1.33%, and MAE = 1.00% after correction. The corrected model also remained stable under random five-fold cross-validation and grouped five-fold cross-validation, supporting its robustness within the available composition space.

3.4. Comparison with Black-Box Models and Discussion of Physical Meaning

To determine whether a more complex black-box method provided additional benefit, the five-coefficient correction model was compared with random forest regression and a shallow neural network. The complete performance comparison is provided in

Table 7. Performance comparison of the baseline, correction, and machine-learning models.

Model	Evaluation	R2	RMSE	MAE
Le Chatelier (direct)	Direct	0.848	4.421	2.403
Five-coefficient correction	Direct	0.986	1.332	1.001
Five-coefficient correction	Random 5-fold	0.984	1.424	1.075
Five-coefficient correction	Grouped 5-fold	0.984	1.444	1.073
Random forest	Grouped 5-fold	0.954	2.437	1.282
Shallow neural network	Grouped 5-fold	0.977	1.725	0.951

Note: The correction model predicts the residual of the Le Chatelier baseline.

. The proposed correction model achieved R² = 0.984, RMSE = 1.44%, and MAE = 1.07% under grouped five-fold validation. In comparison, the random forest model yielded R² = 0.954, RMSE = 2.44%, and MAE = 1.28%, whereas the shallow neural network yielded R² = 0.977, RMSE = 1.72%, and MAE = 0.95%.

It should be noted that the performance metrics in Table 7 were calculated exclusively using the 58 data points in the primary database listed in Table S1. The sample-wise predictions are provided in Table S5. The external literature data listed in Tables S8 and S9, as well as the excluded external candidates listed in Table S10, were not used in the main model fitting, coefficient estimation, cross-validation, or primary performance metrics.

To further assess external applicability, the final correction equation was applied without refitting to selected external literature data with complete gas-composition information and experimentally measured lean-limit values. This analysis was treated as an exploratory external applicability check rather than a formal external validation, because the external data differed in mixture system, propagation direction, apparatus, ignition criterion, and reporting format. As summarized in Table S11, the corrected model reduced the MAE from 0.945% to 0.624% and the RMSE from 1.310% to 0.691% for the exploratory subset including Burgess upward lean-boundary multicomponent data and one Li-ion premixed battery vent-gas case. These results suggest potential external applicability of the correction equation, but further validation using newly measured real battery vent-gas data is still required.

Figure 6 presents the same comparison graphically. Figure 6a shows that the corrected model retained a high R² under grouped validation, while Figure 6b,c show that it also produced low RMSE and MAE. Although the shallow neural network achieved a slightly lower MAE than the explicit correction model, it showed a lower R² and a higher RMSE under grouped validation. This suggests that the neural network reduced the average absolute error for some samples but produced larger deviations for other underrepresented or high-LFL mixtures. Therefore, the proposed correction model should not be interpreted as universally superior to black-box models. Its main advantage lies in its explicit mathematical form, low computational cost, and physical interpretability. For safety-oriented preliminary LFL estimation under small-sample conditions, transparency and reproducibility are as important as average error reduction. Recent battery-machine-learning studies have emphasized the importance of cross-condition prediction, standardized modeling platforms, and chemistry-aware modeling for improving generalizability and reproducibility in battery data science [40,41,42]. Compared with these large-scale battery-ML frameworks, the present study focuses on a small experimental LFL database and therefore prioritizes an explicit, interpretable, and potentially software-implementable correction equation rather than a highly flexible black-box model. This positioning is consistent with the objective of developing a preliminary screening-level tool for rapid LFL estimation under limited experimental data.

The signs of the fitted coefficients also support the physical consistency of the proposed correction. The positive coefficient of I indicates that stronger dilution increases the correction term, which is consistent with the observed underestimation of the baseline formula in highly diluted mixtures. The positive coefficient of R_CO2 suggests that CO₂-rich dilution requires a larger upward correction than N₂-rich dilution, reflecting the stronger flame-suppressing effect of CO₂. Compared with N₂, CO₂ can more effectively reduce the adiabatic flame temperature because of its higher heat capacity and may also influence the active radical pool near the flammability boundary. In contrast, the negative coefficient of O₂ indicates that a higher oxidizer fraction reduces the upward correction required for the Le Chatelier estimate, which is consistent with the tendency of oxygen enrichment to facilitate flame propagation near the lower flammability boundary. The negative coefficient of F_rI/100 suggests that the LFL-raising effect of dilution is partly offset when highly reactive fuels, especially H₂ and C₂H₄, are present at higher fractions. This interaction term should be interpreted as an empirical composition-dependent correction within the current dataset rather than as a universal kinetic parameter.

Another limitation is related to data provenance. The main model was developed using 58 experimental LFL data points compiled in the first author’s doctoral dissertation [36]. Although additional experimental flammability-limit data were identified from external literature and are listed in the Supplementary Materials, these data were not used for model fitting in the present study because their mixture systems, experimental apparatus, ignition criteria, and reporting formats were not fully consistent with those of the primary database. Future studies should use these external data, after further harmonization, as an independent validation set or as part of an expanded database.

Taken together, the results in Table 5, Table 6 and Table 7 and Figure 4, Figure 5 and Figure 6 indicate that the conventional Le Chatelier calculation can serve as a useful first-order screening tool, but its direct use may introduce non-negligible underestimation when the gas mixture is strongly diluted, especially under CO₂-rich dilution. For lithium-ion battery thermal runaway scenarios, such underestimation may affect the predicted concentration threshold at which accumulated vent gases first enter the flammable range. The proposed correction equation retains the simplicity of the baseline method while explicitly accounting for dilution intensity, CO₂/N₂ dilution characteristics, oxygen fraction, and the interaction between highly reactive fuels and dilution. This makes the model suitable for preliminary engineering calculations related to ventilation design, inerting assessment, and early warning threshold estimation.

Several limitations should be noted. First, although the primary database was compiled from experimentally measured LFL data, the original references may have used different experimental apparatuses, ignition sources or energies, vessel geometries, flame-propagation criteria, temperatures, pressures, and gas-preparation procedures. Detailed experimental protocols were reported in the original references; however, not all parameters could be fully standardized across different sources. Such differences may introduce unavoidable experimental heterogeneity into the measured LFL values and may partly affect the fitted residual-correction relationship.

Second, the primary database contained only 58 experimental data points, which is relatively limited for regression fitting and machine-learning comparison. Although grouped five-fold validation was used to reduce the risk of overly optimistic evaluation, it cannot replace independent external validation using newly measured battery-vent-gas data. Under small-sample conditions, the fitted coefficients may also be affected by data distribution, composition imbalance, and experimental uncertainty, and such uncertainty may propagate into the corrected LFL values, especially for mixtures located near the boundary of the current composition space.

Third, the present model was developed for mixtures represented by H₂, CO, CH₄, C₂H₄, CO₂, N₂, and O₂. Therefore, its applicability to mixtures containing substantial fractions of additional hydrocarbons, electrolyte vapors, fluorinated gases, or other volatile organic species remains limited. In addition, the correction model should not be regarded as a substitute for standard flammability testing, particularly under elevated temperature, elevated pressure, oxygen-deficient, or confined-space conditions outside the range of the current database.

Despite these limitations, the experimental-data-driven residual correction strategy provides a useful compromise between empirical accuracy and physical interpretability. Rather than replacing Le Chatelier’s rule, the model adapts it to dilution-dominated gas mixtures relevant to lithium-ion battery thermal runaway. Future work should expand the directly measured LFL database, standardize experimental-condition reporting, perform independent external validation, and quantify uncertainty propagation under controlled and application-relevant battery vent-gas conditions.

4. Conclusions

This study developed an experimental-data-driven correction framework for estimating the LFL of multicomponent gases relevant to lithium-ion battery thermal runaway. A literature-derived experimental database was compiled for H₂–CO–CH₄–C₂H₄–CO₂–N₂–O₂ mixtures, and Le Chatelier’s mixing rule was evaluated as the baseline prediction method. The results showed that Le Chatelier’s rule provided a useful first-order estimate, but it tended to underestimate the experimentally measured LFL, particularly for mixtures with high total diluent fractions and CO₂-dominated dilution.

To reduce this systematic deviation, a parsimonious five-coefficient residual correction model was proposed. The final model used total diluent fraction, CO₂ fraction in the diluent gases, oxygen fraction, and the interaction between highly reactive fuel components and total dilution as physically interpretable predictors. Compared with the Le Chatelier baseline, the corrected model substantially improved prediction accuracy, reducing the MAE from 2.40% to approximately 1.00%. Stable performance was also observed under random and grouped five-fold cross-validation, indicating that the selected variables captured the dominant residual patterns within the available experimental composition space.

Compared with random forest regression and a shallow neural network, the proposed correction model retained an explicit mathematical form and clearer physical interpretability while achieving competitive prediction performance. These results suggest that classical Le Chatelier estimation can be adapted to thermal-runaway-relevant multicomponent gases through a simple residual correction strategy based on experimental data.

It should be emphasized that the proposed model is not intended to replace standard flammability-limit experiments or serve as a universal LFL formula. Owing to the limited size and composition range of the current database, as well as the heterogeneity of experimental conditions in the literature, the model should be regarded as a preliminary screening tool for gas mixtures compositionally relevant to lithium-ion battery thermal runaway. From an engineering perspective, reducing the MAE from 2.40% to approximately 1.00% may decrease the uncertainty in estimated flammability thresholds for preliminary ventilation design, gas-warning settings, and explosion-prevention strategies. Nevertheless, appropriate safety factors should still be retained, and further validation using real battery vent gases under controlled temperature, pressure, oxygen-concentration, and confinement conditions is needed before broader application.