## 4 Single-Droplet Ignition Prediction Models
### Droplet Ignition Prediction Model Under Microgravity  
In the microgravity environment, thermal motion is the primary cause of the formation of diffusion mass flow. Therefore, the Lewis number is [10] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Fuel**} & \multicolumn{2}{c}{**Standard Error of**} & \multicolumn{1}{c}{**Standard Error of**} & \multirow{2}{*}{**R\({}^{2}\)**} \\  & **Parameter \(A\)** & **Parameter \(n\)** & \\ \hline Ethanol & 2.56661 \(\times\) 10\({}^{-13}\) & 0.05071 & 0.98227 \\ N-heptane & 8.43382 \(\times\) 10\({}^{-17}\) & 0.06301 & 0.97683 \\ N-dodecane & 1.93909 \(\times\) 10\({}^{-17}\) & 0.04884 & 0.99209 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Error range of fitting results. Figure 14: Studentized residuals vs. fitted values plot for ethanol. Figure 15: Studentized residuals vs. fitted values plot for n-heptane. \[Le=\frac{D_{\rho}}{\lambda_{\infty}/c_{p}}\approx 1 \tag{19}\]  
The generalized Reynolds analogy between the temperature field and the fuel vapor concentration field is expressed as  
\[\frac{c_{p}(T_{\infty}-T_{w})}{q_{e}}=\frac{Y_{f}}{1-Y_{fw}} \tag{20}\]  
Through the Reynolds analogy, combined with the distribution trends of fuel vapor concentration, by referring to Figure 9, approximate parameter profiles in the gas-phase region surrounding the droplet in the microgravity environment can be plotted, as in Figure 17. It can be seen that the variation trends of all parameters are basically the same as those in the normal-gravity environment, but their scales are compressed due to reduced diffusion speed. In the microgravity environment, the convective effects are extremely weak. Figure 16: Studentized residuals vs. fitted values plot for n-dodecane. Figure 17: Schematic representation of parameter distribution curves for different gravity environments. Therefore, the empirical expression for the Nusselt number calculated using Equation (14) is no longer applicable. In the droplet evaporation process dominated by low Reynolds numbers and Stefan flow, the Nusselt number can be modified using the Spalding number correction. The modified empirical formula is [40]  
\[Nu_{\mu g}^{*}=\frac{ln(1+B)}{B}\left[2+\frac{Nu^{*}-2}{F}\right] \tag{21}\]  
\[F=(1+B)^{0.7}\frac{ln(1+B)}{B} \tag{22}\]  
Figure 14:0  
By substituting the modified \(Nu_{\mu g}^{*}\) into Equation (17), the ignition condition for stationary air droplets in the microgravity environment can be derived as  
Figure 14:1  
In microgravity, the scale of the reaction zone has changed; however, the concentration of fuel vapor within the reaction zone remains largely unaffected. Because the parameters \(q_{e}\), \(c_{p}\), and \(\lambda_{\infty}\) in Equation (16) are determined by the physical properties of the fuel, the parameters \(Q_{s}\), \(k_{os}\), and \(E\) are determined by the chemical reaction pathway of the fuel and the enthalpy changes in the reactants and products. The influence of variations in gravity on these parameters is relatively minor and can be considered negligible. Therefore, the empirical coefficients \(A\) and \(n\) remain applicable in the microgravity environment. By combining the empirical coefficients derived from experiments conducted in normal gravity with the modified \(Nu_{\mu g}^{*}\), a predictive model for droplet ignition in microgravity can be established. Comparisons between the ignition models in different gravitational environments and experimental data for the three fuels are presented in Figures 18-20. By comparing the results, it was observed that under identical droplet diameter conditions, all three fuels exhibited lower ignition threshold temperatures in the microgravity environment. This indicates that droplets are more readily ignited in this environment. This conclusion aligns with phenomena observed in numerous prior microgravity combustion experiments [15; 16; 18; 19; 40; 41]. Therefore, the proposed model effectively predicts the ignition boundaries of droplets in microgravity.