## 2 Materials and Methods
### Analytical Evaluation of the Collection Efficiency  
Many mathematical models for the particle collection process were proposed by researchers such as [36; 37; 38; 39; 40; 41]. Among them, the Deutsch-Anderson model and its variants are widely used to design and evaluate precipitator performance [42]. However, the Deutsch-Anderson model generally underpredicts particle collection efficiencies because of a simplified theoretical approach to actual operating conditions [43; 44; 45]. In order to obtain the collection efficiency in an extensive range of particle diameters, including nano-size ones, accurately, the development of a mathematical model with a relatively easy calculations procedure is an aim of the present study. A modified version of the  
Figure 4: Comparison of collection efficiency values with the experimental data. Figure 5: Comparison of collection efficiency values with the experimental data of [PERSON] et al. [9]. Deutsch-Anderson Equation (31) is considered for this objective. The collection efficiency is expressed as follows:  
\[\eta=1-exp\left(-\frac{W_{p}L}{U_{in}S_{y}}\right) \tag{24}\]  
where \(L\) (m) is the length of collecting electrodes; \(U_{in}\) (m/s) is the mean fluid velocity of the gas; and \(S_{y}\) is the distance between the charging wire and the collecting plates. \(W_{p}\) is the electrical migration velocity. In order to calculate the collection efficiency by using Equation (24), several parameters could be obtained by utilizing the Equations (25)-(37) are given below:  
\[W_{p}=\frac{n_{p}E_{ave}e\mathbb{C}_{c}}{3\pi\mu d_{p}} \tag{25}\]  
\[n_{p}(t)=n_{d}(t)+n_{f}(t) \tag{26}\]  
\[n_{d}=\frac{k_{B}d_{p}T}{2K_{E}e^{2}}ln\left(1+\frac{\pi K_{E}d_{p}c_{i}e^{2}N _{i}t}{2k_{B}T}\right) \tag{27}\]  
\[n_{f}=\ n_{s}\left(\frac{\pi K_{E}b_{ion}eN_{i}t}{1+\pi K_{E}b_{ion}eN_{i}t}\right) \tag{28}\]  
\[n_{s}=\frac{3\varepsilon_{p}}{\varepsilon_{p}+2}\frac{Ed_{p}^{2}}{4K_{E}} \tag{29}\]  
\[K_{E}=\frac{1}{4\pi\varepsilon_{0}} \tag{30}\]  
In the above equations, \(n\) stands for the number of charges on a particle subject to operating conditions. Subscripts \(d,f,\) and \(s\) indicate diffusion charge, field charge, and saturation charge, respectively. \(c_{i}\) is the mean thermal speed of gas ions (240 m/s at 298 K and 101 kPa). \(N_{i}\) is the ion concentration number in a unit volume (#/m\({}^{3}\)), and it is assumed as \(N_{i}=\rho_{ion}/e,t\) is the residence time of particles in ESPs (s). \(J_{p}\) can be calculated using Equation (31) derived by [PERSON] (1981). \[J_{p}=\frac{\varepsilon_{0}b_{i}}{16S_{y}^{3}}\left(\gamma+\sqrt{\gamma^{2}+1 92(\phi_{0}-\phi_{c})\left(S_{y}E_{1}\right)}\right) \tag{31}\]  
Figure 5:0  
Figure 5:1  
Figure 5:2  
In the above equations, \(\phi_{c}\) is the corona onset voltage (V), \(f\) is the roughness factor (assumed as one in this study), and \(E_{c}\) is the corona onset electric field defined in Equation (7). After calculating \(J_{p}\), the total current \(I\) (A) per corona charging wire can be obtained as follows [46]:  
Figure 5:3  
In order to calculate the migration velocity of the particle, the average electric strength (\(E_{ave}\)) must be estimated. It is proposed that \(E_{ave}\) can be calculated as follow:  
Figure 5:4where \(\beta\) is a constant with a value of 3.42; after \(J_{p}\) and \(E_{ave}\) are known, the average ion concentration (\(N_{i}\), #/m\({}^{3}\)) can then be calculated as:  
Figure 5:5  
Table 1 shows the comparison of the present model with [PERSON] [31] and [PERSON] [32] models for particle charging by diffusion, field, and combined charging at \(N_{i}\)\(t=10^{13}\) (s/m\({}^{3}\)), \(\varepsilon_{p}=5.1\), \(E_{ave}=500\) (kV/m). It can be seen that the agreement with the present calculations and the other models is quite good. It should be noted that [PERSON] model is incorporated in the numerical model Equations (18)-(22) of the present study.