## 3 Results
### Evaluation of Droplet Size for Twin Nozzles Condition  
The REGRESS, LS-SVM, ELM, and RBFNN models were built using spray heights, nozzle spacing, and horizontal positions for quantitatively evaluating the VMD distribution of twin nozzles. The numbers of calibration sets and prediction sets were 72 and 24, respectively with a scale of 3:1. Table 4 shows the results of quantitative models based on four different methods. The scatter plot of measured values vs. predicted values of twin nozzles are shown in Figure 6. The result of REGRESS model was almost the same as the LS-SVM model and showed the best performance, with \(Rc^{2}\) and \(Rp^{2}\) both above 0.9. The F value of 36.8578 was greater than its corresponding critical value, indicating the effectiveness of REGRESS. ELM performed with a \(Rc^{2}\) value of 0.8727, and RMSEP of 3.5839 \(\upmu\)m, RBFNN with an \(Rc^{2}\) value of 0.8667 and RMSEP of 3.3721 \(\upmu\)m, both worse than models based on REGRESS and LS-SVM, indicating these two methods were not suitable for VMD prediction. As for the REGRESS model, the study also found that increasing the order of polynomials can improve the \(Rc^{2}\) but significantly reduced the accuracy of prediction set; a better result could be obtained by Quartic polynomial regressing with fewer terms and coefficients. LS-SVM had better performance with \(Rc^{2}\) and relatively small RMSEP, while quartic polynomial based on the REGRESS model was the best to express the relationship between the independent variable and dependent variable in a simpler way with formula and cured surface graphs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Parameter [a]**} & \multicolumn{3}{c}{**Accuracy of Calibration Set**} & \multicolumn{2}{c}{**Accuracy of Prediction Set**} \\ \cline{3-6}  & & \(\mathbf{Rc^{2}}\) & **RMSEC (\(\upmu\)m)** & \(\mathbf{R_{p}}^{2}\) & **RMSEP(\(\upmu\)m)** \\ \hline REGRESS & 4 & 0.9206 & 1.9949 & 0.9204 & 2.0602 \\ LS-SVM & (7.159, 1.490 \(\times\) 10\({}^{6}\)) & 0.9382 & 1.7634 & 0.9154 & 2.2450 \\ ELM & 36 & 0.8727 & 2.5271 & 0.7602 & 3.5839 \\ RBFNN & 130 & 0.8667 & 2.5843 & 0.7850 & 3.3721 \\ \hline \hline \end{tabular} Parameters of different models: the highest degree of the polynomial for REGRESS, the bandwidth of kernel function (sig2), and the trade-off between minimum model complexity and minimum training error(gam) for LS-SVM, the number of hidden nodes for ELM, the spreading coefficient for RBFNN.  
\end{table} 
Table 4: Results of VMD modeling for the twin nozzle condition. Figure 6: Scatter plot of the measured VMD values vs. the predicted VMD values of twin nozzles. (**a**) REGRESS; (**b**) LS-SVM; (**c**) ELM; (**d**) RBFNN. We used the quantitative evaluation model based on the regress function to express the formula and visualize the VMD distribution of the twin nozzle, the formula was as follows:  
\[ \begin{split}\text{y}&=127.4970{-5.6276x_{1}}^{4}+154. 6896{x_{2}}^{4}-1.0552\times 10^{-5}{x_{3}}^{4}-0.1914x_{1}x_{2}x_{3}\\ &+4.1692x_{2}x_{3}-0.8158x_{1}x_{3}+57.0344x_{1}x_{2}-450.9652x_{1 }x_{2}^{2}\\ &-0.0019x_{1}{x_{3}}^{2}-0.0438x_{2}{x_{3}}^{2}+163.9057x_{2}{x_{ 1}}^{2}+0.2832x_{3}{x_{1}}^{2}\\ &-3.5307x_{3}{x_{2}}^{2}-22.5764{x_{1}}^{2}+191.8288x_{2}^{2}+0. 0505{x_{3}}^{2}-0.4886x_{3}\end{split}  \tag{5}\]  
\(x_{1}\) stands for spray height (m), \(x_{2}\) stands for nozzle spacing (m), \(x_{3}\) stands for horizontal position (cm) and y stands for VMD (\(\mu\)m). (four decimal places are reserved for each coefficient). As shown in Figure 7, the quantitatively modeling result of VMD for the twin nozzles condition using REGRESS was presented visually. In Figure 7a-c, referring to the relationships between horizontal position, nozzle spacing, and VMD, VMD gradually increased when the measuring point is approaching the bottom of the two nozzles (the edge of two spray fans), presenting symmetrical distribution at the same height.