# A Simplified Analysis to Predict the Fire Hazard of Primary Lithium Battery
## 3 Results and Discussion  
In 1980, [PERSON] introduced a method to estimate HRR based on oxygen consumption measurements [36]. After a combustion process has been completed, the HRR can be estimated by:  
\[\dot{q}=E(\dot{m}_{O_{2}}^{0}-\dot{m}_{O_{2}}) \tag{1}\]  
where \(\dot{q}\) is the heat release rate, \(\dot{m}_{O_{2}}^{0}\) and \(\dot{m}_{O_{2}}\) are the mass flow rates of oxygen from the entrained air, and \(E\) is the energy released per mass unit of O\({}_{2}\) consumed for a given fuel. [PERSON] validated the constant net heat release per unit mass of oxygen consumed for organic solids and obtained an average value of 13.1 kJ g\({}^{-1}\)\(\pm\) 5% for the constant. Considering incomplete combustion, the CO\({}_{2}\) and CO consumption are also needed for correcting the HRR result. The final form of the equation for HRR is then given by [37]:  
\[\dot{q}=\left[E\phi-(E_{CO}-E)\frac{1-\phi}{2}\frac{X_{CO}^{A}}{X_{O_{2}}^{A}} \right]\frac{M_{O_{2}}}{M_{a}}\frac{\dot{m}_{e}}{1+\phi(\alpha-1)}\Big{(}1-X_ {H_{2}O}^{0}\Big{)}X_{O_{2}}^{A^{0}} \tag{2}\]  
with oxygen depletion factor:  
\[\phi=\frac{X_{O_{2}}^{A^{0}}\Big{(}1-X_{CO_{2}}^{A}-X_{CO}^{A}\Big{)}-X_{O_{2} }^{A}\Big{(}1-X_{CO_{2}}^{A^{0}}\Big{)}}{\Big{(}1-X_{O_{2}}^{A}-X_{CO_{2}}^{A} -X_{CO}^{A}\Big{)}X_{O_{2}}^{A^{0}}} \tag{3}\]  
where \(E\) = 13.1 kJ/g is the heat release per mass unit of oxygen consumed; \(E_{CO}\) = 17.6 kJ/g is the heat release for the conversion from CO to CO\({}^{2}\); \(\alpha\) = 1.105 is the volumetric expansion factor; \(M_{O_{2}}\) = 32 g/mol and \(M_{a}\) = 29 g/mol are the molecular weights of oxygen and air, respectively; \(X_{O_{2}}^{A^{0}}\) and \(X_{CO_{2}}^{A^{0}}\) are measured mole fraction of O\({}_{2}\) and CO\({}_{2}\), respectively; \(X_{O_{2}}^{A}\), \(X_{CO_{2}}^{A}\), and \(X_{CO}^{A}\) are the mole fraction of O\({}_{2}\), CO\({}_{2}\), and CO in the incoming air, respectively; and \(X_{H_{2}O}^{0}\) is the mole fraction of water in the incoming air. According to the oxygen consumption principle and experimental method, the HRR of a lithium battery can be obtained. Figure 3 shows the HRR data of one Panasonic CR123A battery. The maximum  
Figure 2: Diagram of in-situ calorimeter and experimental setup. HRR value is 2.51 kW. The energy is released within a very short time, which may entail the potential risk of explosion. The integration of the HRR curve is the totally released heat (THR) of the lithium batteries fire. \[\text{THR}=\int_{0}^{t}\dot{q}(\tau)\text{d}\tau \tag{4}\]  
The detailed maximum HRR and THR data are shown in Table 1. The discussed methodology to create the designed lithium battery fire and criteria for selection of input data are, at best, based on a generalization of fire experimental data and statistics. Performing an analysis to determine a plausible designed lithium battery fire scenario for predicting fire growth was our objective. The t\({}^{2}\) fire model was used to create the designed lithium battery fire, and the required data were based on the experiment results in Table 1.  
Experimental research has shown that the HRR changes with time. For a wide range of fires, the fire growth can be accurately represented with a power law relation:  
\[\dot{q}=b\cdot t^{2} \tag{5}\]  
where \(\dot{q}\) is the heat release rate of the fire, \(b\) is the fire intensity coefficient, and \(t\) is time. National Institute of Standards and Technology (NIST) developed this model that is used in Consolidate Fire and Smoke Transport (CFAST), which is simulation software for fire and smoke spreading in multi-room buildings [38]. According to the combustion behavior and the HRR curve, lithium battery fires are quite fast, involving a jetting fuel process, which is suitable for the \(t^{2}\) fire model. The complete \(t^{2}\) fire model of a specific fire can be built by presetting the fire growth phase, maximum heat release rate, and fire attenuation phase.