## 4 Establishment and Verification of Mathematical Model
### Establishment of Mathematical Model  
In the multivariate nonlinear regression method, the function form should be known first, and then the model coefficient can be fitted. To establish the mathematical model for predicting the SMD of the nozzle, several single-factor fitting formulas were firstly obtained, allowing for the best fitting formula to be determined through variance analysis and regression analysis. Next, the single-factor fitting formulas were synthesized into the multivariate nonlinear mathematical model. According to the change law of the air pressure and SMD, eight functions, including the linear function, logarithmic function, and S-shape function etc. were used in the SPSS software for fitting. The fitting results of each function are shown in Table 3. In this table, \(R^{2}\) indicates the goodness of fit of the model, and the closer of \(R^{2}\) to 1 is fit the better. \(F\) is a statistic to test the significance of the formula, which is the ratio of the mean regression sum of squares to the mean residual sum of squares. The larger the \(F\) is, the better the fit. The closer the regression significance is to 0, the better the fit [49]. Figure 4: The relationship between SMD and three factors: (**a**) \(p_{\text{air}}\); (**b**) \(p_{L}\) and (**c**) \(d\). Figure 5: The impact of water pressure on the air–liquid flow ratio and the nozzle flow. From Table 3, among the eight functions, the values of \(R^{2}\) of the cubic function, exponential function and quadratic function are all greater than 0.99, indicating the better goodness of fit. The regression significance results of the model indicated that the exponential function shows the best fitting, with the maximum \(F\) and the minimum regression significance. Meanwhile, according to the significance of regression coefficients in Table 4, the significance of the quadratic function and cubic function is far greater than that of the exponential function. Based on the above analysis, it was concluded that the variation law of the air pressure and SMD was in accordance with the exponential function. For the change relationship between the \(p_{\text{L}}\) and \(d\) and SMD, the same analysis method was used to obtain the optimal fitting formula, and the optimal fitting formula between the three factors and SMD was summarized in Table 5.  
According to the fitting formula between the single factor and SMD in Table 5, a multivariate nonlinear regression mathematical model was established [50,51]:  
\[y=b_{1}+b_{2}e^{-1.419x_{1}}+b_{3}\ln(x_{2})+b_{4}/x_{3} \tag{1}\]  
where, \(y\) represents the SMD of the droplet, \(\mu\)m; \(x_{1}\) represents the air pressure (\(p_{\text{air}}\)), MPa; \(x_{2}\) represents the water pressure (\(p_{\text{L}}\)), MPa; \(x_{3}\) represents the outlet diameter (\(d\)), mm; \(b_{\text{k}}\) represents the regression coefficient, \(k\) = 1-4. According to the measured 25 groups of experimental data, the multivariate nonlinear regression was selected in the SPSS software \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Function**} & \multirow{2}{*}{**Fitting Formula**} & \multicolumn{4}{c}{**Significance of Regression Coefficient**} \\ \cline{3-6}  & & & **Constant** & **Coefficient 1** & **Coefficient 2** & **Coefficient 3** \\ \hline Linear & \(y=92.512-83.5x\) & 0.000 & 0.001 & & \\ Logarithmic & \(y=29.037-30.48\ln(x)\) & 0.001 & 0.001 & & \\ S-shape & \(y=e^{3.593+0.161/x}\) & 0.000 & 0.013 & & \\ Exponential & \(y=102.187e^{-1.419x}\) & 0.000 & 0.000 & & \\ Inverse function & \(y=31.055+9.675/x\) & 0.007 & 0.007 & & \\ Power function & \(y=34.95x^{-0.512}\) & 0.000 & 0.001 & & \\ Quadratic & \(y=102.712-141.786x+72.857x^{2}\) & 0.003 & 0.044 & 0.197 & \\ Cubic & \(y=80.032+58.914x-467.143x^{2}+450x^{3}\) & 0.118 & 0.727 & 0.403 & 0.359 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Significance of regression coefficients t of the function.