## 2 Materials and Methods
### Methods
Then, a diagram of the change in the ratio of the gas smoke indicators from the values of the angular velocity of the engine crankshaft at an ultrasonic frequency of 25 kHz, 28 kHz, 40 kHz was recorded. The values of the angular speed of the engine crankshaft were converted from the values of the engine crankshaft speed (Table 7). From the diagram in Figure 10, the following picture emerges: the lowest values of the ratio of gas smokiness indicators were recorded at 78.5 rad/s. Then, at 99.5 rad/s, there was a significant increase in the values of the ratio of the gas smoke indicators, which reached the maximum points in the diagram at an ultrasound frequency of 25 and 40 kHz. However, a gradual decrease in the values of the ratio of the gas smoke indicators was already observed at 130.9 rad/s and an ultrasound frequency of 25 and 40 kHz. Meanwhile, at an ultrasound frequency of 28 kHz, the ratio of the smoke content of the gas increased further. These results can be explained by the fact that the coagulation process in the gas stream flowing through the muffler does not start immediately, and it takes some time for the particles to stick together and settle, such that particles of different sizes come into contact with each other and form large agglomerates at the same time. Depending on particle size and ultrasound frequency, the duration of the coagulation process may vary. For example, if, at a frequency of 40 kHz, small particles vibrate more intensely and combine more quickly to form large particles, the process of forming large particles is slower at a frequency of 25 kHz. At a frequency of 25 kHz, however, large particles settle more quickly at the bottom of the unit, so that the smoke values of the gas tend to a minimum more quickly than at a frequency of 40 kHz. And, at a frequency of 28 kHz and the angular velocities indicated, the coagulation process is not yet complete and the settling phase of the particles has not yet taken place. Therefore, the coagulation process takes longer at this frequency. To analyze the complex effect of the ultrasonic frequency parameters and angular velocity of the engine crankshaft on the indicators of the gas-smoke ratio, the following three-dimensional graph was created (Figure 11). \begin{table}
\begin{tabular}{l l l l} \hline Engine crankshaft speed, rpm & 750 & 950 & 1250 \\ \hline Angular speed of the engine crankshaft \(\omega\), rad/s & 78.5 & 99.5 & 130.9 \\ \hline \end{tabular}
\end{table} 
Table 7: Values for the angular velocity of the engine crankshaft. Figure 10: The change in the relationship between the smoke content of the gas and the values of the angular velocity at the ultrasound frequency of 25 kHz, 28 kHz, and 40 kHz. The obtained diagram shows that under the complex influence of the considered parameters, the values of the gas smoke ratio gradually decrease at the ultrasound frequency of 40 kHz. In addition, at this frequency, we recorded relatively minimal values of the gas smoke ratio for various indicators of angular velocity of gas. According to the graph, these values are in the range of 0.87-0.81 at angular velocity 78.5 rad/s, 9.5, and 130.9 rad/s. The data obtained roughly coincide with the results of theoretical calculations. Furthermore, in order to establish the dependencies between the ratios of the gas smokiness indicators (\(D_{2}/D_{1}\)) and the ratios of the ultrasound frequency to the angular rotation speed of the engine crankshaft, \(f/\omega\) calculations were carried out to determine the values of \(f/\omega\). The results of calculations to determine the ratio of \(f/\omega\) indicators are presented in Table 8.  
Using the calculated indicators, an \(f/\omega\) diagram was created showing the dependence of the ratio of the indicators for the smokiness of the gas (\(D_{2}/D_{1}\)) on their values (Figure 12). The diagram in Figure 12 shows that with an increase in the value of \(f/\omega\), the ratio of the flue gas indicators decreases. This modification meets the requirements for optimal functioning of the ultrasonic muffler.