## 3 Results and Discussion
### Results of the Mathematical Analysis of the Regression Equation  
To establish the optimal values of the parameters, such as the distance between the emitters and their power, a mathematical analysis of the equation was conducted based on the regression model. The analysis allowed us to determine how changes in the engine speed (\(n\)), power (\(N\)), and the distance between the emitters (\(L\)) affected the PM density (\(D\)) values. According to this analysis, partial derivatives of the equation were determined, which made it possible to establish the rate of change in PM density (\(D\)) when varying \(n\), \(N\), and \(L\):  
Regarding power (\(N\)):  
\[\frac{dD}{dN}=0.0394\times L+0.0002\times N-0.1175. \tag{2}\]  
Regarding distance (\(L\)):  
\[\frac{dD}{dL}=-41.002\times L+0.0394\times N-0.0016\times\mathrm{n}+23.2307. \tag{3}\]  
Regarding the number of revolutions (\(n\)):  
\[\frac{dD}{dn}=-0.0016\times L-0.0263. \tag{4}\]  
Figure 5: Comparison of the smoke indicators determined during the experimental tests and those calculated using the regression equation. Next, the critical points of the equation were determined, under the condition that:  
\[\frac{dD}{dN}=0, \tag{5}\]  
and  
\[\frac{dD}{dL}=0, \tag{6}\]  
and  
\[\frac{dD}{dn}=0. \tag{7}\]  
Based on the obtained derivatives, it follows that an increase in power N reduced the smoke emissions (due to the negative linear term \(-0.1175\)). However, this influence decreased with larger values of \(L\), as the interaction term (\(0.0394\times L\)) weakened the effect. The distance \(L\) had a strong negative influence on the smoke emissions (\(-41.002\times L\)). Increasing the number of revolutions reduced the smoke emissions, but the effect was weak, as the coefficients were small (\(-0.0016\) and \(-0.0263\)). The interaction with distance \(L\) also (\(-0.0016\times L\)) added a negligible influence on the smoke emissions. Next, the critical points of the equations were determined under the conditions that the values of Equations (5) and (7) are equal to zero. The solution to the system of equations was  
\[ \begin{array}{c}0.0394\times L+0.0002\times\mbox{N}-0.1175=0\\ -41.002\times L+0.0394\times N-0.0016\times n+23.2307=0\\ -0.0016\times L-0.0263=0\end{array}  \tag{8}\]  
This allowed us to obtain the critical values of the parameters: \(N=3825.69\), \(L=16.4375\), and \(n=529.958.23\). However, the values of the critical points lie outside the real ranges, making this point physically insignificant. In this case, the critical point has no practical application. This is because the regression model describes parameters within a limited range, and the mathematical analysis yields extreme values. The second derivative of the equation describes the nature of the critical point:  
For power:  
\[\frac{d^{2}D}{dN^{2}}=0.0002. \tag{9}\]  
Regarding distance:  
\[\frac{d^{2}D}{dL^{2}}=-41.002. \tag{10}\]  
Regarding the number of revolutions:  
\[\frac{d^{2}D}{dn}=0, \tag{11}\]  
\[\frac{d^{2}D}{dNdL}=0.0394, \tag{12}\]  
\[\frac{d^{2}D}{dNdn}=0, \tag{13}\]  
\[\frac{d^{2}D}{dLdn}=-0.0016. \tag{14}\]  
According to the calculated Hessian determinant \(H=-0.00000305\), it is negative (\(H<0\)), indicating a saddle point. This means that the function behaves differently in different directions. The saddle point confirms that the parameters (\(N\), \(L\), \(n\)) have opposite effects on the smoke density in different directions, and it is necessary to find a balance. Therefore, for maximum cleaning efficiency, the distance \(L\) between the emitters must be optimized, as both too small and too large values can lead to an increase in exhaust smoke. Additionally, it is important to consider the effect of power \(N\), but its increase should be balanced with the optimal placement of the emitters. This balance between the distance \(L\) and power \(N\) was the key factor in minimizing the smoke density, while the number of revolutions \(n\) had a less significant impact on the process efficiency.