## 2 Method
### Analytical Temperature Profile of Battery Cell  
In battery cell heating tests, the temperature profile can be divided into three main stages: the time period during which the battery temperature increases due to the external heating before the thermal decomposition reaction begins (stage 1), the time during which the thermal decomposition reaction occurs (stage 2), and the time after the reaction is completed (stage 3).  
The energy balance at stage 1 is  
\[\rho C_{p}\frac{\partial T}{\partial t}=q_{\ heating}-ah(T-T_{env}) \tag{11}\]  
Therefore, the battery cell temperature at this stage can be written as  
\[T=(T_{0}-M)exp\bigg{(}-\frac{ah}{\rho C_{p}}t\bigg{)}+M \tag{12}\]  
\[M=\frac{q_{heating}+ahT_{env}}{h} \tag{13}\]  
Since the temperature at which the thermal decomposition reaction begins is \(T_{1}=-k_{0}/k_{1}\), the time t1 to reach this value is  
\[t_{1}=-\frac{\rho C_{p}}{ah}ln\left[\frac{q_{heating}-ah(T_{1}-T_{env})}{q_{ heating}-ah(T_{0}-T_{env})}\right] \tag{14}\]In stage 2, using Equation (10) for the temperature dependence of the linearized heating value, the following is obtained:  
\[\rho C_{p}\frac{\partial T}{\partial t}=q_{heating}+\ K_{1}T+K_{0}-ah(T-T_{env}) \tag{15}\]  
Therefore, the battery cell temperature \(T\) is  
\[T=(T_{0}-Mt)exp\bigg{(}\frac{K_{1}-\mathrm{a}h}{\rho C_{p}}t\bigg{)}-Mt \tag{16}\]  
\[M^{\prime}=\frac{q_{heating}+ahT_{env}+K_{0}}{K_{1}-\mathrm{a}h} \tag{17}\]  
Here, the end time of stage 2 is denoted as t\({}_{2}\). Since the integrated heat generation and the total reaction latent heat \(Q_{tot}\) in t\({}_{1}\) to t\({}_{2}\) coincide,  
\[\int_{t_{1}}^{t_{2}}q_{m}dt=\int_{t_{1}}^{t_{2}}(\ K_{1}T+K_{0})dt=Q_{tot} \tag{18}\]  
Substituting Equation (16) into Equation (18) and performing the integral, we obtain  
\[K_{0}(t_{2}-t_{1})+\frac{\rho C_{p}}{K_{1}-ah}(T_{1}+Mt)\bigg{[}exp\bigg{\{} \frac{K_{1}-ah}{\rho C_{p}}(t_{2}-t_{1})\bigg{\}}-1\bigg{]}=Q_{tot} \tag{19}\]  
Equation (19) is a nonlinear equation for t\({}_{2}\) and it is difficult to find an analytical solution. In the case that the heat generation duration t\({}_{2}\)-t\({}_{1}\) is sufficiently small, the first term on the left-hand side, \(K_{0}(t_{2}-t_{1})\), can be neglected and the following equation is obtained.  
\[\frac{\rho C_{p}}{K_{1}-ah}(T_{0}+Mt)\bigg{[}exp\bigg{\{}\frac{K_{1}-ah}{\rho C _{p}}(t_{2}-t_{1})\bigg{\}}-1\bigg{]}=Q_{tot} \tag{20}\]  
However, it should be noted that this assumption is not valid when the heating rate is low or the thermal decomposition reaction rate \(k_{j}\) is small. Solving this for t\({}_{2}\) gives  
\[t_{2}=t_{1}+\frac{\rho C_{p}}{K_{1}-ah}ln\bigg{[}\frac{Q_{tot}(\ K_{1}-ah)}{K _{1}\rho C_{p}(T_{0}+Mt)}+1\bigg{]} \tag{21}\]  
Assuming that in stage 3 the heater is damaged and only heat dissipation occurs,  
\[\rho C_{p}\frac{\partial T}{\partial t}=-ah(T-T_{env}) \tag{22}\]  
Therefore, the battery cell temperature at this stage can be written as  
\[T=(T_{2}-T_{env})exp\bigg{[}-\frac{ah}{\rho C_{p}}(t-t_{2})\bigg{]}+T_{env} \tag{23}\]  
where T\({}_{2}\) is the final temperature of stage 2.  
Figure 2b describes an example of the temperature profile of the battery cell in the external heating test using Equations (12), (16), and (23). In stage 1, the battery temperature increases gradually due to the heating, rises rapidly by the thermal decomposition in stage 2, and decreases gradually in stage 3. This temperature profile is typically seen in the external heating tests [19; 28].