## 4 Single-Droplet Ignition Prediction Models
### Droplet Ignition Prediction Model Under Normal Gravity  
First, it is assumed that the droplet is stationary relative to the environment, and after vaporization, the fuel molecules only undergo spherically symmetric radial one-dimensional flow caused by the Stefan flow, with the heat radiation and thermal dissociation reactions of the fuel being neglected. In the short time before ignition, the droplet remains in a pure evaporation state, with the fuel vapor diffusing outward from the droplet surface, while the air diffuses inward toward the droplet surface. When the droplet ignites, the reaction is most intense in a thin layer close to the hot boundary, and the temperature gradient at the outer boundary of this layer is zero. Based on the experience of solving ignition problems [10], the order of magnitude of the parameter changes in the reaction zone thin layer at ignition is approximately \(RT_{\infty}/E\), where \(R\) is the gas constant, \(T_{\infty}\) is the ambient temperature inside the heating vessel, \(E\) is the activation energy of the fuel, parameters with an asterisk '\(*\)' denote values at the reactive zone's inner boundary, subscript '\(w\)' indicates values at the droplet surface, and the subscript '\(\infty\)' represents ambient parameters. The boundary conditions for ignition can be approximated as follows. In the reaction zone, the average concentration of fuel gas is  
\[Y_{f}\approx Y_{fw}\frac{RT_{\infty}}{E} \tag{2}\]  
The average concentration of oxygen is  
\[Y_{ox}\approx Y_{ox,\infty} \tag{3}\]  
The boundary temperature of the reaction zone is  
\[T^{*}\approx T_{\infty}-\frac{RT_{\infty}}{E} \tag{4}\]  
Due to the extremely small actual scale of the thin layer, curvature variations can be neglected, and the convective effects caused by temperature changes are negligibly weak, allowing the convective terms in the equations to be ignored. Therefore, we have  
\[\frac{d^{2}T}{dr^{2}}\approx-\frac{w_{\rm s}}{\lambda_{\infty}}Q_{s} \tag{5}\]  
In the equation, \(Q_{s}\) represents the heat of combustion. Integrating this equation results in  
\[\left(\frac{dT}{dr}\right)_{l}=\left(\frac{2Q_{s}}{\lambda_{\infty}}\int_{T^{* }}^{T_{\infty}}w_{s}dT\right)^{\frac{1}{2}} \tag{6}\]The parameter '\(w_{s}\)' in the equation is calculated using the following expression:  
\[w_{s}=k_{os}\rho_{\infty}^{2}Y_{fw}Y_{ox,\infty}e^{-\frac{E}{kT_{\infty}}} \tag{7}\]  
In the equation, \(k_{os}\) represents the reaction kinetics parameter, and \(\rho_{\infty}\) denotes the environmental density. Integrating Equation (6) yields  
\[\int_{T^{*}}^{T_{\infty}}w_{s}dT\approx\left(1-\frac{1}{e}\right)\frac{R^{2}T_ {\infty}^{3}}{E^{2}}k_{os}\rho_{\infty}^{2}Y_{fw}Y_{ox,\infty}e^{-\frac{E}{kT_{ \infty}}} \tag{8}\]  
Introducing the analytical expression \(\left(\frac{dT}{dr}\right)_{II}\) for droplet combustion, and assuming that the temperature gradient at the outer boundary of the reaction zone's thin layer is approximately equal to that under pure evaporation without combustion, we have  
\[\left(\frac{dT}{dr}\right)_{II}=\frac{G\left[c_{p}(T_{\infty}-T_{w})+q_{e} \right]}{4\pi r_{1}^{2}\lambda_{\infty}} \tag{9}\]  
In Equation (9), \(G\) represents the total mass flux, \(c_{p}\) is the specific heat capacity at constant pressure, \(q_{e}\) denotes the latent heat of evaporation, \(\lambda_{\infty}\) is the thermal diffusivity, and \(r_{1}\) is the radius of the reaction zone's thin layer after being converted to a film. The parameters \(G\) and \(r_{1}\) can be calculated using the following expressions:  
\[G=2\pi\frac{\lambda_{\infty}}{c_{p}}Nu^{*}r_{w}ln\left[1+\frac{c_{p}(T_{\infty }-T_{w})}{q_{e}}\right] \tag{10}\]  
\[r_{1}=r_{w}\frac{Nu^{*}}{Nu^{*}-2} \tag{11}\]  
Due to the temperature gradient in the atmospheric space near the droplet surface, slight natural convection occurs.