# Analysis of Plasma Ion Distribution and Dust Collection Efficiency of Carbon-Brush Air Purifiers
## 2 Mathematical Basis for Simulation
Therefore, the behavior of electrons and ions was assumed to be a continuum, i.e., a flow. By using this continuum assumption, the governing equation for gas discharge can be expressed considering the diffusion of electrons, as shown in the following equation [31; 32]:  
\[\frac{\partial n_{e}}{\partial t}=-\frac{\partial n_{e}v_{e}}{\partial x}+ \frac{\partial^{2}(D_{e}n_{e})}{\partial x^{2}}+S \tag{5}\]  
where \(n_{e}\) is the electron density, \(v_{e}\) is the electron movement speed, \(D_{e}\) is the electron diffusion coefficient, and \(S\) is the density of the electrons newly generated per hour by photoionization at position \(x\). The electrons or ions calculated by this process act as space charges in the space between the electrode and the ground. The average velocity of the charged particles is derived from the Boltzmann equation. If the inertia of ions and electrons is neglected, it is expressed as [33; 34]  
\[\frac{\partial}{\partial t}(n_{e})+\
abla\cdot\left[-n_{e}(\mu_{e}\cdot E)-D_{ e}\cdot\
abla n_{e}\right]=R_{e} \tag{6}\]  
\[R_{e}=\sum_{j=1}^{M}x_{j}k_{j}N_{i}n_{e} \tag{7}\]  
where \(x_{j}\) is the mole fraction of the species and \(k_{j}\) is the scale factor. From this basic theory, the ion concentration (\(N_{i}\)) can be calculated by performing simulation using the Plasma module of COMSOL. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Expression** & **Unit** & **Description** & **Reference** \\ \hline \(k\) & \(1.38\times 10^{-23}\) & J/K & Boltzmann constant & [28; 29] \\ \(K_{E}\) & \(9\times 10^{-9}\). & N \(*\) m\({}^{2}\)/C\({}^{2}\) & Electrical proportional constant & [28; 29] \\ \(\overline{c}\) & 240 & m/s & Thermal speed of ion & [28] \\ \(E\) & \(1.6\times 10^{-19}\). & C & Electron charge & [28; 29] \\ \(\varepsilon\) & 4.81 & - & Relative Permittivity of KCl & [28; 30] \\ \(Z_{i}\) & 0.00015 & m\({}^{2}\)/V \(*\) s & Mobility of Ions & [28] \\ \(u\) & 0.9 & m/s & Velocity & measurement \\ \(h\) & 380 & mm & Height of the charged space & measurement \\ \(t\) & 0.42 & s & Charging time & (h/u) \\ \(d_{p}\) & 0.3 & um & Particle diameter & measurement \\ \(T\) & 300 & K & Temperature & measurement \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Charging number equation independent variables. In the field charging equation, the electric field (\(E\)) value is the basic equation of Coulomb's law, and the electric force (\(F_{E}\)) can be obtained from Equation (8). Here, \(q\) is the quantity of electric charge, \(q^{\prime}\) is the quantity of electric charge of the charged dust particles, \(r\) is the charged radius, and \(\epsilon_{0}\) is the permittivity [20; 28]. \[F_{E}=-\frac{qq^{\prime}}{4\pi\epsilon_{0}r^{2}} \tag{8}\]  
When the intensity of the electric field acting on dust particles is \(E\), the electric force \(F_{E}\) can be expressed as Equation (9). For simple geometry, the intensity of the electric field (\(E\)) can be expressed as Equation (10) according to [PERSON]'s law. Using these equations, the AC/DC module was used to calculate the \(E\) value [19]. \[F_{E}=qE \tag{9}\]  
\[E=\frac{F_{E}}{q^{\prime}}=\frac{q}{r^{2}} \tag{10}\]  
The charging number for dust was calculated using \(N_{i}\) and \(E\), which were obtained through simulation, and the particle trajectory was traced using the Particle Tracing module. In CFD analysis, the Navier-Stokes equation was used as a basic governing equation and a streamline was used in the result. Particles flow along the streamline. Charged particles are collected on the ground because of the attractive or repulsive force caused by the potential of the collector while flowing along the streamline. The motion of a charged particle can be explained as Lagrangian for an electric charge in an electromagnetic field. \[L=-d_{mp}c^{2}\sqrt{1-v\cdot vc^{2}}+ZeA\cdot v-ZeV \tag{11}\]  
where \(d_{mp}\) is the mass of the particle, \(c\) is the velocity of light, \(v\) is the particle velocity, \(Z\) is the particle charge, \(A\) is the magnetic vector potential, and \(V\) is the electric scalar potential.