## 3 Dynamic Model Based on Composite Electric Field Separation Method  
In order to investigate the effect of composite electric field method on the separation of fine particles, the force on the particle is considered in the plane. It is assumed that the particles are spherical, and the electric field is evenly distributed. When the particles accelerate in the acceleration room, we need to change the type of drag on the particles according to a different Reynolds number. Since the particles enter the acceleration room at a very slow speed, the Reynolds number is calculated according to particle size and velocity. When the particle velocity is slow, the particle is first in the Stokes region and then changes according to the state. The force patterns at different stages in the acceleration zone are shown in Figure 3. In the Stokes region, that is Ret \(<2\), the force on particles is as follows:  
\[M_{a}=\rho_{p}\frac{\pi}{6}d_{p}^{3}, \tag{8}\]  
where \(\rho_{p}\) is particle density. Introducing Equations (1), (3)-(9) into Equation (2), we can get the dynamic equations as follows, and the subsequent power \(q_{0}\) is calculated from Equation (1):  
\[ \begin{array}{ll}\frac{\pi}{6}d_{p}^{3}\rho_{p}\ddot{y}_{p}=&-3\pi\mu d_{p} \Big{(}\dot{v}_{p}-v_{g}\Big{)}-\frac{\pi}{12}d_{p}^{3}\rho_{p}\frac{d}{dt} \Big{(}\dot{y}_{p}-v_{g}\Big{)}\\ &\frac{3}{2}d_{p}^{2}\sqrt{\pi\rho_{g}\pi}\frac{t}{0}\int\limits_{0}^{\frac{t} {dt}\big{(}\dot{y}_{p}-v_{g}\big{)}}d\tau\\ &+\frac{4\pi}{3}d_{p}^{3}\rho_{p}g+q_{0}E\end{array}  \tag{9}\]  
In the acceleration room, the air is in a static state, and \(v_{g}=0\); When calculating the Basset force, the particle acceleration is assumed to be proportional to the relative velocity. It can be shown that:  
\[\frac{dv}{dt}=Cv. \tag{10}\]  
Here, \(v\) can be get by the following Equation (11):  
\[v=v_{0}e^{Ct}. \tag{11}\]  
Here, we mainly focus on the calculation of \(F_{B}\) and substituting Equation (12) into \(F_{B}\) and performed substitution calculation, that is \(\tau=t-\mu^{2}\), to get the expression of \(F_{B}\) as:  
\[F_{B}=\frac{3\pi}{2}d_{p}^{2}\sqrt{\rho_{g}\mu}v_{0}Ce^{Ct}erf\Big{(}\sqrt{Ct} \Big{)}, \tag{12}\]  
where \(v_{0}\) is initial velocity into the acceleration room; \(C\) is a constant; \(erf(x)\) is the error function of \(x\), that can be expressed as follows:  
\[erf(x)=\frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-u^{2}}du. \tag{13}\]  
Figure 3: Particles force in accelerating electric field. In the Stokes region, the kinetic equations of the particles are expressed as:  
\[ \begin{array}{c}\frac{\pi}{6}d_{p}^{3}\rho_{p}\ddot{y}_{p}=-\frac{1}{8}\pi d_{p}^ {2}\rho_{g}C_{D}\big{|}u_{p}-u_{g}\big{|}\Big{(}u_{p}-u_{g}\big{)}+q_{0}E-\rho_{ g}V_{p}g\\ -\frac{\pi}{12}d_{p}^{3}\rho_{g}\frac{d}{dt}\big{(}\dot{y}_{p}-v_{g}\big{)}- \frac{3}{2}d_{p}^{2}\sqrt{\pi\rho_{g}}t\int\limits_{0}^{t}\frac{\frac{d}{dt} \big{(}\dot{y}_{p}-v_{g}\big{)}}{\sqrt{t-\tau}}d\tau\end{array}  \tag{14}\]  
As the particles accelerate, they move out of the Stokes region and enter into the Allen region. When the velocity of particles is faster, some particles will enter the turbulent zone and drag force will change. The variation form can be expressed as respectively:  
\[ \begin{array}{c}\Big{(}\frac{\pi}{6}d_{p}^{3}\rho_{p}+\frac{\pi}{12}C_{I}d_{p }^{3}\rho_{g}\Big{)}\ddot{y}_{p}-\frac{37\pi}{16}\rho_{g}^{0.4}u^{0.6}d_{p}^{1.4}\Big{|}\dot{y}_{p}-v_{g}\Big{|}^{0.4}\big{(}\dot{y}_{p}-v_{g}\big{)}\\ =q_{0}E+\frac{3\pi}{2}C_{B}d_{p}^{2}\sqrt{\rho_{g}}t\upsilon_{0}Ce^{C}erf \Big{(}\sqrt{Ct}\Big{)}\end{array}  \tag{15}\]  
\[ \begin{array}{c}\Big{(}\frac{\pi}{6}d_{p}^{3}\rho_{p}+\frac{\pi}{12}C_{I}d_{p }^{3}\rho_{g}\Big{)}\ddot{y}_{p}-\frac{11\pi}{200}\rho_{g}d_{p}^{2}\Big{|}\dot{ y}_{p}-v_{g}\Big{|}\Big{(}\dot{y}_{p}-v_{g}\Big{)}\\ =\frac{3\pi}{2}C_{B}d_{p}^{2}\sqrt{\rho_{g}}t\upsilon_{0}Ce^{C}erf\Big{(}\sqrt {Ct}\Big{)}+q_{0}E\end{array}  \tag{16}\]  
In the separation room, due to the action of compound electric field, the particle will be deflected when moving.