## 4 Establishment and Verification of Mathematical Model
### Verification of Mathematical Model  
The established model was used to calculate the SMD under 25 experimental conditions to verify the accuracy of the prediction model. Then, the calculation results were compared with the orthogonal experimental data, as shown in Figure 6. From the figure, it can be seen that the calculated value of SMD based on the prediction model was consistent with the orthogonal experimental results, with an average relative error of only 4.39%.  
In order to further verify the validity of the established prediction model, a nozzle was selected for experiment. In the experiment, the differences between the predicted SMD values of the nozzle and the experimental values were compared under different air pressure and water pressure. Figure 7 shows the change curve of the predicted SMD value and the experimental value of the nozzle with \(d=2.0\) mm. It can be seen from the figure that the predicted value has a consistent change trend with the experimental value, indicating that the model can qualitatively reflect the change rule of droplet particle size with the operating parameters. At the same time, the average relative error between the predicted value and the experimental value is only 4.89%. Considering the influence of the  
Figure 6: Comparison between the predicted values and orthogonal experimental values results of SMD.  
experimental environment on the spray field as well as the complex environment of actual engineering site, there are many environmental factors affecting the particle size of droplets, thus, the model error is within the acceptable range. Therefore, it can be concluded that the prediction model established in this study has high accuracy and can be used for theoretical prediction and calculation of the SMD of nozzles.  
The droplet size is an important parameter affecting the spray dust reduction. In practical application, the influencing factors of droplet size include air pressure, water pressure and outlet diameter. When the values of the three factors are known, the SMD of the nozzle can be predicted according to the prediction model. At the same time, the three factors can be changed according to the prediction model to obtain the appropriate droplet size. Due to the complexity and variability of the environment in practical application, arbitrary changes of the three factors are limited. Therefore, in order to obtain the appropriate droplet size, other factors can be adjusted through the prediction model under the condition that one factor is limited, as shown in Figure 8. It can be seen from the figure that the prediction model can be used to determine the nozzle droplet size and provide guidance for the actual application of nozzles. Overall, the variation law of the three factors and SMD plays an important role in the engineering application of nozzles, and the relationship between the three factors and SMD can be established through the prediction model.  
Figure 8: The relationship between the SMD of a single-factor constant nozzle and various factors: (**a**) \(p_{\text{air}}\) = 0.4 MPa; (**b**) \(p_{\text{L}}\) = 0.3 MPa; and (**c**) \(d\) = 1.2 mm.  
Figure 7: The predicted and experimental values of SMD for different air-water pressure nozzles: (**a**) \(p_{\text{air}}\) = 0.2 MPa; (**b**) \(p_{\text{air}}\) = 0.3 MPa; (**c**) \(p_{\text{L}}\) = 0.5 MPa and (**d**) \(p_{\text{L}}\) = 0.6 MPa.