## 3 Results and Discussion
### Secondary Breakup of Single Electrified Al/N-Decane Nanofluid Fuel Droplets  
Figure 5 shows the secondary breakup photography of single Al/n-decane nanofluid fuel droplets except for 1.0 mg/mL and 3.0 mg/mL. It demonstrates that the breakup mode belonged to the cone-jetting mode. The cone-jetting mode of the single electrified nanofluid fuel droplets can be explained by comparing the charge relaxation time and the hydrodynamics relaxation time. The charge relaxation time \(\tau_{c}\) can be expressed as [21]:  
\[\tau_{c}=\frac{\varepsilon_{nf}}{4\pi\alpha_{nf}} \tag{4}\]  
in which \(\varepsilon_{nf}\) and \(\sigma_{nf}\) are the permittivity and the electrical conductivity of Al/n-decane nanofluid fuel, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Concentration**} & \multicolumn{3}{c}{\(s_{x}\) = **At\({}^{2}\) + \(Bt\) + C(um)**} & \multicolumn{3}{c}{**q/m in These Cases**} & **Average q/m** & **Error** \\ \cline{2-7}  & **A** & **B** & **C** & **R2** & **(\(\times 10^{-6}\) C/kg)** & **(\(\times 10^{-6}\) C/kg)** & **(\(\times 10^{-6}\) C/kg)** \\ \hline
1.0 & \(-\)2.97 & \(-\)84.90 & 10.51 & 0.999 & \(-\)9.55 & 8.05 & 1.13 \\
2.0 & 1.90 & \(-\)76.62 & 7.37 & 0.999 & 6.15 & 4.46 & 0.64 \\
3.0 & \(-\)1.70 & \(-\)174.14 & 4.15 & 1.000 & \(-\)5.47 & 5.49 & 2.93 \\
4.0 & 4.57 & \(-\)46.31 & \(-\)1.12 & 0.995 & 14.7 & 6.406 & 2.05 \\
5.0 & 1.22 & \(-\)99.67 & 16.80 & 1.000 & 3.92 & 2.17 & 0.80 \\
6.0 & 2.90 & \(-\)39.86 & 46.89 & 0.982 & 9.32 & 11.4 & 0.05 \\
7.0 & 4.82 & \(-\)182.33 & 13.55 & 1.000 & 15.5 & 7.48 & 2.04 \\
8.0 & 0.23 & \(-\)77.75 & 9.54 & 1.000 & 0.74 & 2.44 & 1.70 \\
9.0 & 0.19 & \(-\)63.91 & 4.61 & 0998 & 0.61 & 3.92 & 2.42 \\
10.0 & 3.56 & \(-\)13.63 & 0.78 & 0.942 & 11.4 & 11.4 & 1.67 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The Charge-to-Mass ratio at each concentration.  
Figure 4: The X-direction displacement versus time before breakup.  
And the hydrodynamics relaxation time \(\tau_{H}\) can be expressed as  
\[\tau_{H}=\frac{\mu_{nf}D_{nf}}{2\gamma_{nf}} \tag{5}\]  
in which \(D_{nf}\) is the diameter of the single electrified droplet, \(\mu_{nf}\) and \(\gamma_{nf}\) are the viscosity and the surface tension of Al/n-decane nanofluid fuel, respectively.  
According to the properties of nanofluid fuels, for different Al nanoparticle concentrations, the ratio of the charge relaxation time \(\tau_{c}\) to the hydrodynamics relaxation time\(\tau_{H}\) was much greater than 1 (\(\tau_{c}/\tau_{H}>>\) 1) by calculating the order of magnitude relaxation time. It reveals that the surface charge transfer rate was much lower than that of the capillary wave induced by the surface tension. Therefore, the motion of the capillary wave was seriously damped by charge transfer, resulting in the breakup of low conductivity fluids, which is easy to form cone-jetting mode [21].  
The average diameters of secondary breakup sub droplets are illustrated in Figure 6, ranging from 55.4 \(\upmu\)m (at 6.0 mg/mL) \(\sim\)90.4 \(\upmu\)m (at 7.0 mg/mL). It means that fine and uniform secondary breakup sub droplets were obtained by electrostatic atomization.  
Figure 5: The secondary breakup at different Al nanoparticle concentrations.  
From experimental observations, [PERSON] [24] suggested that, for an uncharged drop in a uniform electric field, at the point of disintegration, the following correlation seemed to hold:  
\[\sqrt{\frac{D_{nf}}{2\gamma_{nf}}}=C^{\prime}/E \tag{6}\]  
In this work, the \(C^{\prime}\)/E values were calculated and listed in Table 3. It demonstrates that the \(C^{\prime}\)/E fluctuated from a minimum value of 0.180 at 6.0 mg/mL to a maximum value of 0.208 at 8.0 mg/mL with different concentrations. The fluctuation magnitude reached 13.7%. But for most cases, the \(C^{\prime}\)/E values kept constant of \(\sim\)0.20. Therefore, for the Al/n-decane nanofluid fuel, the correlation also seemed to hold, testifying that the reasonability of equilibrium dominated by the electrical force and the surface tension.