## 2 Method
### Linearizing Heat Generation of Battery Cell  
This study focuses on the external heating tests of the battery. When the representative temperature of the cell is \(T\), the energy balance can be written as follows. \[\rho C_{p}\frac{\partial T}{\partial t}=q_{heating}+q_{reac}-ah(T-T_{env}) \tag{1}\]  
where \(\rho\) is density, \(C_{p}\) is specific heat, \(q_{heating}\) is the amount of heating by the heater, h is the heat transfer coefficient assuming heat dissipation to the environment, and \(T_{env}\) is the environmental temperature. \(q_{reac}\) is the amount of heat generated by the thermal decomposition reaction, expressed as a linear sum of the heating values of the positive electrode, negative electrode, and electrolyte. Namely,  
\[q_{reac}=\sum_{j}q_{reac,j} \tag{2}\]  
Figure 1: Examples of the temperature changes in a single cell (**left**) and multiple cells (**right**). Each heat quantity has the cell temperature and the degree of nonreaction dependence and is usually assumed to be the Arrhenius or modified-Arrhenius quantity with the degree-of-nonreaction term introduced in the pre-factor. \[q_{reac,j}=Q_{tot,j}k_{j}exp\left(-\frac{E_{j}}{RT}\right)\alpha_{j} \tag{3}\]  
\[\frac{\partial\alpha_{j}}{\partial t}=-k_{j}exp\left(-\frac{E_{j}}{RT}\right) \alpha_{j} \tag{4}\]  
where \(Q_{tot,j}\) is the latent heat of reaction, \(E_{j}\) is the activation energy, and \(R\) is the gas constant. \(kj\) is the reaction rate for the modified-Arrhenius-type equation. \[k_{j}=k_{0,j}\alpha_{j}^{m_{j}}\left(C_{j}-\alpha_{j}\right)^{n_{j}} \tag{5}\]  
where \(m\), \(n\), and \(C_{1}\) are constants, and \(k_{0,j}\) is the frequency factor. However, it is generally difficult to obtain an analytical solution for Equation (1) because the heat generation term (Equation (3)) is nonlinear. In the present study, an attempt is made to linearize the reaction term with respect to temperature. According to Equation (4), the degree of nonreaction \(\alpha_{j}\) changes with time t and temperature T. In order to simplify the equations, \(\alpha_{j}\) is assumed to be a representative constant value (herein \(\alpha_{j}\) is assumed to be 1). Note that this assumption makes \(k_{j}\) a constant value (Equation (5)), and thus the thermal behavior at the maximum value of \(k_{j}\) (at \(\alpha=0.5\) if m = n) and the minimum values (near \(\alpha=0\) and \(\alpha=C_{1}\)) is not considered. In particular, the maximum value of \(k_{j}\) can cause a rapid increase in cell temperature, which a constant \(k_{j}\) may not be able to accurately capture. Furthermore, the influence of the representative value of \(\alpha_{j}\) adopted should also be considered. These points will be explored in future work. Figure 2a shows the schematic image of the relationship between the heat generation and the battery temperature based on Equation (3). Since the maximum slope of \(q_{max}\) can be derived from the condition that the second derivative of Equation (3) is zero,  
\[\frac{\partial^{2}q_{reac,j}}{\partial T^{2}}=\frac{E_{j}}{R}Q_{tot,j}k_{j} \alpha_{j}\left[\frac{1}{T^{3}}exp\left(-\frac{E_{j}}{RT}\right)\left(-2+ \frac{E_{j}}{RT}\right)\right]=0 \tag{6}\]  
Figure 2: (a) Arrhenius-type heat generation functions (blue line) and linearized function (dotted red line). Total latent heat of the former (area surrounded by blue line and x-axis) coincide with those of the latter (orange colored area). (b) Typical temperature profile of battery cell in external heating test using Equations (12), (16), and (23).