## 2 Materials and Methods  
### Modeling of Purification of Waste Gases Using the Ultrasonic Method  
According to the proposed hypothesis, exhaust gas cleaning should take place through acoustic coagulation of the particles directly in the muffler. The schematic diagram is shown in Figure 1.  
The existing theoretical solution to describe the proposed process was analyzed. The kinetics of coagulation of small spherical particles that fuse upon collision was developed by [PERSON] [45]. He suggested that in a large volume, compared to the volume of the particles, the particles diffuse (adhere) to each other. In this case, each particle is surrounded by a sphere with a diffusion radius S to which the other particle adheres, and the diffusion coefficient is equal to \(D\). If two particles act as centers of absorption of particles, i.e., coagulation takes place, then the rate of disappearance of particles per unit volume is determined using the formula  
\[-\frac{dn}{dt}=2\pi DSn^{2} \tag{1}\]  
where \(n\) is the number of particles per unit volume. The diffusion process was described by [PERSON] [46], and it showed that the value of \(D\) is equal to  
\[D=\frac{RT}{6\pi\mu rN} \tag{2}\]  
where \(R\) is the gas constant; \(T\) is the absolute temperature; \(N\) is the Avogadro number; \(\mu\) is the viscosity of the medium; and \(r\) is the radius of the particles. Assuming that diffusion is the same for all particles, the radii of the particles are the same, and the ratio \(S\) between the radius of the sphere and the radius of the particle is the same [39]:  
\[\frac{1}{n}-\frac{1}{n_{0}}=\frac{2}{3}\cdot\frac{RTs}{\mu N}\cdot t \tag{3}\]  
where \(n_{0}\) is the initial number of particles  
or  
\[\sigma-\sigma_{0}=\frac{2}{3}\cdot\frac{RTs}{\mu N}\cdot t \tag{4}\]  
where \(\sigma\) and \(\sigma_{0}\) are, accordingly, the volumes of particles. It follows from Equation (4) that there is a linear relationship between the volume produced per particle fraction and the coagulation time. The results obtained, despite their great importance, are not suitable to describe the process we propose. First, these results are obtained for Brownian motion; in our case, the ultrasonic pressure acts from the engine wall. Secondly, [PERSON]'s theory applies to a much larger volume of coagulation space compared to the total volume of particles [45]. The linear velocity of the particles is not affected by the pressure of the active force. In our case, the so-called acoustic coagulation takes place under the influence of a sound field [46]. Gas particles of different sizes have a certain frequency of oscillation. In the beginning, the particle follows the motion of the gas as it sticks together and increases in size. Afterward, the particle  
Figure 1: Schematic representation of the gas flow in the muffler: 1—the output manifold of the engine; 2—the muffler body; 3—the vector of exhaust gas movement at a speed of \(\theta\); 4—a single particle of gas mass \(m\); 5—ultrasonic wave; 6—exhaust tube. increases in size due to chaotic oscillations. This process takes place at low oscillation frequencies. As the oscillation frequency increases, there is an optimal frequency range where particles of different sizes have different amplitudes, collide with each other, and coagulate. This type of coagulation is called orthokinetic. As the frequency increases, the coagulation becomes hydrodynamic and occurs through friction. This process is described using the Bjerkness equation [47]. The equation describing acoustic coagulation was established by [PERSON] [47]. A gas with viscosity \(\mu\), oscillating with amplitude \(X_{g}\) and frequency \(f\), and in which there is a particle with radius \(r\) and density \(\rho\), was considered. According to [PERSON] law, the frictional force acts on a particle:  
\[F_{s}=6\pi\mu r\theta \tag{5}\]  
where \(\theta\) is the difference between the velocities of a particle and a gas.