## 4 Single-Droplet Ignition Prediction Models
### Droplet Ignition Prediction Model Under Normal Gravity
The Nusselt number under natural convection conditions can be calculated using an empirical formula, and its expression is  
\[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]0  
In this expression, \(Ra\) denotes the Rayleigh number, representing the influence of buoyancy induced by the temperature gradient. It can be calculated using the following expressions:  
\[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]1  
In this equation, \(\beta\) represents the chemical equivalence ratio. Given the relatively low concentration magnitude of fuel vapor within the thin-film reaction zone, and considering that thermal condition \(T_{\infty}\gg T_{w}\), for ease of application in engineering contexts, certain thermophysical parameters such as \(c_{p}\), \(\rho\), and \(\lambda_{\infty}\) are approximated by the corresponding properties of air at the same temperature, as shown in \[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]2 The parameters within temperature ranges are fitted using power functions, and interpolation is performed accordingly. The ignition conditions for droplets given by Zhou are [10]  
\[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]3By equating Equations (6) and (9), and substituting Equation (12) into them, the semi-empirical ignition condition for a single fuel droplet in the stationary air environment in normal-gravity conditions can be derived as follows:  
\[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]4  
Due to the challenges in directly measuring parameters such as fuel vapor and oxygen concentrations within the reaction layer surrounding the droplet under practical conditions, and considering that determining the reaction kinetics parameters and latent heat of vaporization in Equation (15) involves complex calculations and measurement processes, alternative approaches are often employed. These include utilizing simplified models or indirect measurement techniques to estimate the necessary parameters [10; 35]. To solve engineering problems, the complex components of Equation (15) are simplified in this study by employing a power model based on ambient temperature and summarizing it in the form of a semi-empirical formula. \[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]5  
\[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]6  
The empirical constants \(A\) and \(n\) can be determined through experimental data. Regarding the selection of activation energy values for chemical reactions, some existing studies indicate that fuels exhibit negative temperature coefficient (NTC) behavior within certain temperature ranges [36]. This phenomenon arises from the complex mechanisms of fuel oxidation reactions and may be accompanied by cool-flame combustion [37]. However, the focus of this study is on the autoignition phenomena of fuel droplets. The flames observed in the experiments are not characteristic of cool flames. Therefore, based on the experimental data obtained in the high-temperature reaction regime [37; 38; 39], three activation energies for chemical reactions were determined, as shown in \[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]7 \begin{table}
\begin{tabular}{c c c c c} \hline
**T/K** & **\(\rho\)/kg\(\cdot\)m\({}^{-1}\)** & **Cp/kJ**-(kg\(\cdot\)K)\({}^{-1}\)** & **\(\lambda\)/W**-(m\(\cdot\)K)\({}^{-1}\)** & **Pr** \\ \hline
673 & 0.524 & 1.068 & 0.0520 & 0.678 \\
773 & 0.456 & 1.093 & 0.0574 & 0.687 \\
873 & 0.404 & 1.114 & 0.0622 & 0.699 \\
973 & 0.362 & 1.135 & 0.0671 & 0.706 \\
1073 & 0.329 & 1.156 & 0.0718 & 0.713 \\
1173 & 0.301 & 1.172 & 0.0763 & 0.717 \\ \hline \end{tabular}
\end{table} 
Table 4: Thermophysical properties of dry air under atmospheric pressure (\(p=1.013\) kPa) [34]. \begin{table}
\begin{tabular}{c c c} \hline
**Ethanol** & **N-Heptane** & **N-Dodecane** \\ \hline
130 kJ/mol & 120 kJ/mol & 155 kJ/mol \\ \hline \end{tabular}
\end{table} 
Table 5: Activation energies for three fuels.