## 3 Theoretical Analysis
### Acoustic Wave Field  
In the acoustic field, sound waves propagate through the medium vibration. The particles in the dusty gas vibrate back and forth because of acoustic entrainment. The force that drives the particle movement is called acoustic driving force, also known as Stokes viscous force. It is expressed as follows [23]:  
\[F_{st}=\frac{3\pi\eta_{g}d(u_{0}\sin\omega t-v_{a})}{C_{c}}, \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**T (\({}^{\circ}\)C)** & 160 & 180 & 200 & 220 & 240 & 260 & 280 \\ \hline
**Particle Size (\(\mu\)m)** & 0.5 & 1.3 & 2.0 & 2.7 & 4.0 & 6.3 & 8.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relationship between temperature and particle size in the PALAS generator. Figure 2: Original particle size distribution (PSD) and particle number concentration (PNC) of monodisperse particles. where \(\eta_{g}\) is the air dynamic viscosity, \(u_{0}\) is the air velocity, \(\omega\) is the angular frequency \((\omega=2\pi\eta),f\) is the acoustic wave frequency, \(d\) is particle diameter, \(v_{a}\) is the velocity of particles moving in the sound transmission direction, and \(C_{c}\) is the Cunningham correction coefficient given by as follows [24]:  
\[C_{c}=1+\frac{\lambda}{d}\bigg{[}2.514+0.8\exp\left(-\frac{0.55d}{\lambda} \right)\bigg{]}, \tag{2}\]  
where \(\lambda\) refers to average free path of air molecules. In acoustic wave field, fine particles collide and agglomerate into large ones because of the acoustic wave driving force. Furthermore, the acoustic entrainment coefficient is the ratio of the vibration amplitude of particles to that of air. It is given by the following formula:  
\[\mu_{p}=\frac{1}{\sqrt{1+\left(\omega\tau\right)^{2}}}, \tag{3}\]  
where \(\tau\) is the relaxation time calculated by the following expression:  
\[\tau=\frac{\rho d^{2}C_{c}}{18\eta_{g}}, \tag{4}\]  
where \(\rho\) is the density of particles. The effects of acoustic frequencies and particle size on the entrainment coefficient is shown in Figure 3. A higher entrainment coefficient indicates that the particles vibrate together with air, whereas a lower entrainment coefficient indicates that particles are practically motionless. In Equation (3), it can be observed that \(\upup\) is affected by the acoustic frequency and it is independent of the SPLs. As shown in Figure 3, \(\upup\) ranges from 0 to 1. When the particles are less than 1 \(\upup\), approaches 1; however, as the particle size increases, \(\upup\) drops sharply. The particle size of monodisperse particles in this study was less than 5 \(\upup\), and the entrainment effect was evident within the 1000-2200-Hz acoustic frequency range in which the experiment was conducted. Figure 3: Relationship between entrainment coefficient, particle size, and frequency of particles. The particle source used in this experiment was the DEH particles. All particles were practically in the same diameter range, and the relative speed was practically zero. Most studies considered orthokinetic interaction as the theoretical support of acoustic agglomeration [25]. In fact, hydrodynamics, especially the acoustic wake effect, also has an important function in the movement of particles in the acoustic field, as shown in Figure 4.  
In the first half-cycle, two particles follow the medium to the left. Particle 1 generates a low-pressure wake area in the section it passes through, and, because of the pressure difference, particle 2 approaches particle 1 in the movement. In the latter half-cycle, two particles move to the right with the air medium. Similarly, a low-pressure wake area is generated at the tail of particle 2, and particle 1 approaches particle 2 because of the differential pressure. After several cycles, the two particles eventually collide and aggregate. At the same time, the monodisperse aerosol particles produced by the PALAS are not completely of the same size. The speeds and effects of orthokinetic interaction differ among different particles. In view of the foregoing, particle agglomeration is intensified.