# A Simplified Analysis to Predict the Fire Hazard of Primary Lithium Battery
## 3 Results and Discussion
Its mathematical description is:  
\[\dot{Q}=b_{1}t^{2}\qquad\qquad 0\leq t\leq t_{1} \tag{6}\]  
\[\dot{Q}=\dot{Q}_{max}\qquad\qquad t=t_{1} \tag{7}\]  
\[\dot{q}=\left[E\phi-(E_{CO}-E)\frac{1-\phi}{2}\frac{X_{CO}^{A}}{X_{O_{2}}^{A}} \right]\frac{M_{O_{2}}}{M_{a}}\frac{\dot{m}_{e}}{1+\phi(\alpha-1)}\Big{(}1-X_ {H_{2}O}^{0}\Big{)}X_{O_{2}}^{A^{0}} \tag{2}\]0 \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Time (s)**} & \multicolumn{2}{c}{**Maximum HRR (kW)**} & \multicolumn{1}{c}{**Heat (kJ)**} \\ \cline{2-5}  & \(t_{1}\) & \(t_{2}\) & & \\ \hline Experiment & 17 & 44 & 2.51 & 32.4 \\ Simulation & 17 & 44 & 2.51 & 33.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental and simulation values for single battery. HRR denotes heat release rate. Figure 3: Experimental and simulated heat release rate (HRR) result for single battery. where \(b_{1}\) is the fire growth rate coefficient, \(\dot{Q}_{max}\) is the maximum HRR, \(b_{2}\) is the fire attenuation rate coefficient, \(t_{1}\) is the maximum HRR time, and \(t_{2}\) is the fire end time. The fire of single lithium battery was designed according to the specific \(t_{1}\), \(t_{2}\), and maximum HRR values listed in Table 1.  
The designed fire for one lithium battery is shown in Figure 3. The designed lithium battery fire data were compared with the experimental result with a correlation coefficient of 0.977, which supports a good statistical relationships between the two data values. The designed lithium battery fire was assumed the same \(t_{1}\), maximum HRR, and \(t_{2}\). The difference was the calculated released heat, which was 32.4 kJ for the experiment and 33.6 kJ for the designed simulation, and the margin of error was about 3.7%. The simulation can be used as a fundamental element for fire prediction of multiple batteries and bundled packs. The energy released in the combustion process should increase in proportion to the amount of fuel, which has a certain combustion efficiency. Unlike normal combustion, lithium battery fire is a fast and jetting fuel process. The released energy is correlated with the complete degree of the combustion. Previous discussion indicated that the energy released is not proportionally related to the battery numbers (\(n\)) of a simple cell [10]. It can be assumed that an exponential relationship between thermal energy released and the number of batteries is:  
\[Q_{n}=Q_{1}\cdot n^{a} \tag{9}\]  
where \(n\) is the battery number, \(Q_{n}\) is the energy released from \(n\) batteries, and \(Q_{1}\) is the energy released from one battery. The fitting result by [PERSON] et al. was \(Q_{1}\) = 31.77 kJ, \(a\) = 1.26, and fitting coefficient of 0. 997. The equation should be added into the simulation of a multiple lithium battery fire. The most difficult problem is that the ignition time of each battery in the bundle is random. Even in the same external environment and with the same heating power, batteries do not always ignite at the same time. We observed that the time of ignition was quite random or varied significantly under the same external heating condition. The non-simultaneous ignition of batteries resulted in differences in heat release rate data. Figure 4 shows the two different experiment results of two batteries under the same heating condition. The two batteries ignited at the same time in Case A and separately in Case B. The time gap in Case B was about 19 s. Given the sporadic ignition of individual batteries in a multi-cell bundle, the maximum heat release rate of the bundle is not equal to the product of the single battery maximum heat release rate multiplied by the number of batteries in the bundle. The time gap in ignitions should be considered in the mathematical calculation process. A time random variable could be added into the HRR simulation for bundled batteries.