# A Review of Experimental and Numerical Studies of Lithium Ion Battery Fires
## 6 Modeling of LIBs  
Attempts at numerical modeling LIB fire is increasing due to difficulty in conducting experiments. Furthermore, numerical modeling provides a clearer understanding of the chemical-physics associated with the LIBs. Generally, numerical modeling techniques such as CFD and finite element analysis are used to model various stages of fire and or the effect of fire (production of heat and smoke, and their transfer/transport) on surrounding materials. Modeling programs such as FDS, FireFOAM, and ANSYS Fluent allow researchers and manufacturers alike to simulate fire scenarios and estimate parameters such as HRR, heat flux, temperature rise, and smoke production. Many modeling studies on the TR of LIBs can be found in the literature [10; 141; 142; 143]. Owen exposure testing of LIBs was replicated using one-dimensional [142] and three-dimensional [143] predictive models. [PERSON] et al. [144] presented an experimental validation method which was able to predict the critical temperature (\(T_{critical}\)) to TR based on the thermal balance between temperature-dependent heat generation, thermal conduction within the cell, and heat dissipation to the cell surface. A thermal test cell with the geometry of a cylindrical 26650 Li-ion cell was used to determine the value of \(T_{critical}\) in different conditions to compare against the theoretical model. In this study, \(T_{critical}\) was first evaluated theoretically by solving a nonlinear transcendental equation as shown in Equation (4):  
\[\frac{Q_{0}R^{2}E_{a}}{\text{k}\;\mu^{2}R_{u}T_{critical}^{2}}\;e^{(-\frac{E_{a }}{R_{u}T_{critical}})}-1=0 \tag{4}\]  
where \(Q_{0}\) is the pre-exponential coefficient, \(E_{a}\) are activation energies, \(R_{u}\) is the universal gas constant, \(R\) is the cell radius, \(k\) is the cell radial thermal conductivity, and \(\mu\) is the first root of the eigenfunction corresponding to the convective boundary condition at the outer surface of the cell. Using different sets of heat generation and heat dissipation conditions, the theoretical results yielded \(T_{critical}\) temperatures of 35.7 and 44.8 \({}^{\circ}\)C. The theoretical results were considered lower than the usually recorded temperatures leading to the onset of TR. The lower \(T_{critical}\) temperatures were selected and utilized in order to safely carry out the experimental works within the thermal test cell. For both \(T_{critical}\) results, the corresponding Arrhenius parameters of combustion or pyrolysis were used as software inputs to ensure that the heat generation followed the corresponding Arrhenius kinetics. The cell temperature was measured as a function of time for different oven temperatures and from this, the oven temperature that results in TR was determined. [PERSON] et al. [144] concluded that when the oven temperature is lower than the theoretically predicted \(T_{critical}\), the cell temperature remains in a steady state. However, when the oven temperature is in excess of theoretically predicted \(T_{critical}\), the cell temperatures increase in an uncontrolled fashion, which indicates the onset of TR. Although the temperatures in this study were somewhat below the temperatures associated with and proven to induce TR, the results still provided validation of the theoretical model at low values of \(T_{critical}\). It is expected that by following the Arrhenius nature of the heat generation, accurate values of larger \(T_{critical}\)can be anticipated. [PERSON] et al. [145] completed a study attempting to assess the risk of TR/fire propagating between LIBs. Finite element modeling using the COMSOL multiphysics software that can model conjugate heat transfer to calculate the propagation of heat between adjacent cells was used. The cells were exposed to a 15 kW heat source and the results of the two models (conjugate heat transfer and thermal model) utilized in this study were correlated to experimental data from fire tests where the measured HRR was appropriately used as the input (source term) for the simulations.