## 3 Experimental Data Processing Method  
In this experiment, we measured the droplet size and velocity of a spray field under different surface tensions and viscosities. The supporting data software system was used to collect the original experimental data, such as real-time droplet size and three-dimensional velocity of droplets. To conduct a scientific analysis and comparison of the large volume of data collected, the Sauter Mean Diameter (SMD) method was selected to analyze the original data. The SMD calculation formula is expressed as follows [35; 36]:  
\[\text{SMD}=\frac{\sum NiD_{i}^{3}}{\sum NiD_{i}^{2}}, \tag{1}\]  
where SMD is the Sauter Mean Diameter, \(V\) is the volume frequency of droplets, \(D\) is the droplet diameter, and \(\Sigma\) represents the summation of data. This method provides an accurate and reliable way to analyze the data obtained from the experiment and helps to explore the relationship between surface tension and viscosity and the atomization efficiency of solid cone nozzles. The original velocity \(X,Y,\) and \(Z\) are analyzed and processed by the transformation matrix method to obtain \(Vx\), \(Vy\), and \(Vz\) test data, and the transformation matrix is as follows:  
\[ \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}E1x&E1y&E1z\\ E2x&E2y&E2z\\ E3x&E3y&E3z\end{bmatrix}\begin{bmatrix}Vx\\ Vy\\ Vz\end{bmatrix} . \tag{2}\]  
With the data collection method described above, we experimentally measured the droplet size and velocity at 9 measuring points in the axial section of the spray field (X = 0 cm), 9 measuring points in the radial section of the spray field (Z = 15 cm), and 56 other measuring points in the spray field space. These measurements were taken to obtain a comprehensive understanding of the atomization efficiency and the distribution of droplets in the spray field under different surface tensions and viscosities. Figure 5: Experimental data processing process. When analyzing the impact of surface tension and viscosity on the atomization process, the primary approach involves the application of fluid mechanics and behavior theories. Research findings indicate that during the atomization process, spray droplets are primarily influenced by surface tension, viscous forces, inertial forces, gravity, and gas resistance [37]. The balance equation for the comprehensive force acting on the droplet is expressed in Equation (3):  
\[\mathrm{F}_{S}+\mathrm{F}_{v}+\mathrm{F}_{i}+\mathrm{F}_{g}+\mathrm{F}_{a}=m \left(\frac{d^{2}r}{dt^{2}}\right), \tag{3}\]  
where \(\mathrm{F}_{S}\) is surface tension force, \(\mathrm{F}_{v}\) is viscous force, \(\mathrm{F}_{i}\) is inertial force, \(\mathrm{F}_{g}\) is gravity, \(\mathrm{F}_{a}\) is gas resistance, \(m\) is the mass of the droplet, and \(\left(d^{2}r\right)/\left(dt^{2}\right)\) is the acceleration of the droplet. In the case of surfactant solution, the droplets formed by atomization have small droplet size and high dispersion, leading to the negligible effects of gravity and gas resistance during the spraying process. The relationship between viscous force and inertial force follows the Reynolds dimensionless constant, which can be expressed as Equation (2):  
\[Re=\frac{\rho Vd}{\mu}, \tag{4}\]  
where \(Re\) is the Reynolds number, \(\rho\) is the density of the liquid, \(V\) is the velocity of the droplet, \(d\) is the diameter of the droplet, and \(\mu\) is the viscosity coefficient of the liquid. According to the molecular dynamics theory, the dynamic viscosity coefficient is as follows:  
\[\mu\propto\rho l\bar{V}, \tag{5}\]  
where \(\bar{V}\) is the average velocity of molecules, and \(l\) is the average free path of molecules. Substituting Equation (3) into Equation (2), Equation (4) is obtained. \[Re\propto\frac{Vd}{l\bar{V}} \tag{6}\]  
During the atomization process, the droplet size is much smaller than the characteristic length, resulting in a low Reynolds number (\(Re\)). In the process of surfactant atomization, the effect of viscous force is much greater than that of inertial force due to the low \(Re\) value.