# Analysis of Plasma Ion Distribution and Dust Collection Efficiency of Carbon-Brush Air Purifiers
## 2 Mathematical Basis for Simulation  
The simulation was performed as shown in Figure 1. To calculate the dust collection efficiency of an air purifier that applies a high voltage, it is necessary to calculate the charging number for dust. Several variables are required to calculate the charging number. Table 1 shows the independent variables determined under indoor conditions. Dependentvariables to be calculated from simulation included the ion concentration (\(N_{i}\)) and the electric field (E). Plasma analysis was conducted to determine the ion concentration distribution (\(N_{i}\)), and AC/DC analysis was conducted to obtain the electric field (E). The charging number for particles was calculated using the dependent variables (\(N_{i}\) and \(E\)). Finally, particle tracing analysis was conducted using the flow field and the streamline obtained for dust collection efficiency evaluation. The number of dust particles filtered by the flow field and electric field was identified, and the dust collection efficiency was calculated. The dust collection efficiency of the air purifier was compared with the experimental values under different conditions. For dust particles to be sufficiently charged, a high ion concentration is required in the electric field. When a voltage higher than the firing voltage is applied between the brush and the ground, electrons are separated from gas molecules in the air by the strong electric field and dust particles are charged as the gas becomes cations and anions are discharged. When the energy of the electrons is lower than the ionization potential, it can excite atoms or gas molecules. For atoms or molecules that were already in the excited state before the collision, ionization can even be caused by a collision with electrons with energy levels lower than the ionization energy through the mechanism shown in Equation (1) or (2). \(A^{*}\) represents a neutral gas molecule and \(K_{e}\) is the kinetic energy of an electron [25; 26]. \[A^{*}+e^{-}+K_{e}\to A^{+}+2e^{-} \tag{1}\]  
\[A^{*}+A^{*}\to A^{+}+A+e^{-} \tag{2}\]  
Determining the number of electric charges on dust requires the diffusion charging relation given by Equation (3) and the field charging relation given by Equation (4). Equation (3) shows the number of electric charges (\(n(t)\)) acquired by a particle with a diameter of \(d_{p}\) by diffusion charging during time \(t\)[20; 27; 28]:  
\[n_{1}(t)=\frac{d_{p}kT}{2K_{E}e^{2}}ln[1+\frac{\pi K_{E}d_{p}\overline{c_{i}} e^{2}N_{i}t}{2 kT}] \tag{3}\]  
Figure 1: Simulation flow chart. where \(\overline{c_{i}}\) is the average thermal velocity of the ions (\(\overline{c_{i}}=240m/s\) at standard conditions), \(N_{i}\) is the ion concentration, \(k\) is the Boltzmann constant, \(T\) is the atmospheric temperature, \(K_{E}\) is the electrical proportional constant, and \(e\) is the electron charge. Equation (4) is an equation for calculating field charging. When diffusion charging is negligible, it represents the number of electric charges (\(n(t)\)) acquired by a particle during time \(t\) in the electric field \(E\) with an ion concentration of \(N_{i}\). \(Z_{i}\) is the mobility of ions and \(\varepsilon\) is the relative permittivity [27; 28]. \[n_{2}(t)=\left[\frac{3e}{\varepsilon+2}\right]\left[\frac{Ed^{2}}{4K_{E} \varepsilon}\right]\left[\frac{\pi K_{E}eZ_{i}N_{i}t}{1+\pi K_{E}eZ_{i}N_{i}t}\right] \tag{4}\]  
Obtaining the total number of electric charges on one dust particle (\(n_{1}(t)+n_{2}(t)\)) requires the independent variables shown in Table 1. The time \(t\) required to charge the particle can be obtained as the ratio of the height \(h\) of the spatial field to the inlet flow velocity \(v\). An attempt was made to determine the variables that can be calculated (\(N_{i}\) and \(E\)) through simulation. For the plasma analysis used to obtain the ion concentration (\(N_{i}\)), it is difficult to interpret ions as particles because of the limited analysis capacity and speed of the computer because the density of charged particles is \(10^{8}\) cm\({}^{-3}\).