## 3 Results and Discussion
### Results of the Conducted Regression-Correlation Analysis  
Based on the results of the experimental studies, regression-correlation analysis (1) was carried out, during which a multiple regression equation was developed to describe the change in the gas opacity indicators as a function of the engine speed, the power of the ultrasonic emitters, and the distance between the emitters.  
\[\begin{split} D=91.7879+(-0.1175)\times N+23.2307\times L+(-0.02 63)\times n+0.0001\times N^{2}+(0.0394)\times N\times L\\ -(10)^{-7}\times N\times n+(20.5010)\times L^{2}+(-0.0016)\times L \times n+(10)^{-7}\times n^{2}\end{split} \tag{1}\]  
where \(D\) is the PM density (by experiment), %; \(N\) is the power, \(W\); \(L\) is the distance between the emitters, m; \(n\) is the engine speed, rpm.  
The obtained value of the coefficient of determination (R\({}^{2}\) = 0.9715) from the regression equation indicates that the multiple regression model effectively captures the relationship between the opacity of the exhaust gases and the engine speed, power of the ultrasonic emitters, and the distance between the emitters. This model is reliable for describing the cleaning process within these limits.  
Furthermore, the obtained correlation coefficient (r = 0.98) indicates a strong linear relationship between the predicted values from the model and the actual experimental data. This indicates that the model reliably reflects the variations in opacity resulting from changes in engine speed, emitter power, and spacing.  
According to the obtained regression equation, the values of opacity were determined for each combination of emitter activation. Based on the obtained opacity values, their deviation from the experimental values was calculated. The results of the calculation are presented in Figure 4.  
According to Table 4, the approximation error shows that, on average, the regression model predictions deviated from the experimental data by 1.54%. The largest deviation was recorded for experiment 7 at 1500 rpm (4.01%), while the smallest was at 2000 rpm without ultrasonic impact (only 0.03%) (experiment 1). This proves that the regression model can be used to predict changes in opacity depending on the parameters of ultrasonic impact. This level of error is associated with the small number of experimental trials, which limits the model's accuracy.  
Therefore, to reduce the approximation error, it was necessary to increase the number of experimental tests.  
The graph in Figure 5 reflects how accurately the regression model describes the exhaust gas cleaning process using ultrasound and how the experimental data correspond to the theoretical calculations. A comparison with the regression equation demonstrates that the calculated model matches the experimental data with a small deviation. This deviation may be due to experimental conditions, such as the inhomogeneity of pollutants or the dynamics of their removal, which were not fully accounted for in the model. The graphs also emphasize the importance of uniform ultrasound impact. Indeed, a uniform ultrasound effect throughout the length of the experimental stand led to lower opacity values.  
Therefore, the proposed regression model can be used to predict cleaning results. However, to achieve the maximum cleaning effect, further optimization of parameters such as the distance between emitters and their power is required, since, in some cases, reducing the distance between them yields better results.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Experiment**} & **Engine** & **Smoke (by Experiment)** & **Power of UE** & **Distance** & **Smoke Density** & \\  & **rpm** & **(D), \%** & **(N), W** & **Between** & **(According to the** & **Deviation,** \\  & & **(D), \%** & **(N), W** & **Ultrasonic** & **Regression** & **(\%)** \\  & & & & **Emitters** & **Equation) (D), \%** & \\ \hline
1 & & 79 & \(-\) & \(-\) & 79.68 & 0.86 \\
2 & & 73 & 100 & 0.17 & 73.69 & 0.95 \\
3 &