## 2 Materials and Methods
### Validation of Numerical Model  
Validation of the numerical model is performed with the available experimental data in the literature. The schematic of the ESP system used in the present study is illustrated in Figure 1. Mesh structure and the geometric parameters are given in Figure 2. Considering the variations of the electrical variables near the wire electrodes, high mesh intensity surrounding the wires were generated. Consequently, the generated mesh was a much denser vicinity of the wire electrodes to increase the accuracy of the calculated gradients in the high current density zones. Since the mesh structure significantly affects the accuracy of the results and the solution time, a grid dependence test was also performed. It is found that about two million elements in total are enough for the grid-independent analysis. The solution domain was discretized with tetrahedral and hexahedral cells. These structures are adjusted with the available experimental data given in the literature. The single-channel ESP system consisted of a simple channel with two plate grounded electrodes, and five discharge supply wire electrodes were used for the validation of the present simulation model. The discharge supply electrodes were placed in the midplane of the channel along the positive y-direction, with the flow of the particle-injected gas in the positive x-direction (Figure 1). The total length of the rectangular ESP channel was 900 mm length with a width of either 230 mm or 180 mm. The discharge supply electrodes that were modelled had a diameter of 0.3, 1, and 2 mm with a length of 300 mm. The present numerical model is validated against the experimental measurement data of [PERSON] and [PERSON] (2016). The operating voltages are 25.5 and 43.5 kV, the corona wires diameter is 0.305 mm, the distance between the wire and the plate is 115 mm, and the distance between the wires (\(S_{x}\)) is 152 mm. The comparative results of the model  
Figure 1: Schematic of the ESP model. Figure 2: Mesh structure and geometric parameters. verification study for the potential variations between the wire and plates are shown in Figure 3. Figure 3a shows the potential variation versus the distance from the wire electrode to the collection plate. Figure 3b shows the potential variation versus the distance from the midplane (at the point of a quarter distance between the wires) to the collection plate. Figure 3c shows the potential variation versus the distance from the midplane (at the middle point between the wires) to the collection plate. In general, the current simulation agrees well with the experimental results. It demonstrates that the existing model is very good at simulating the electric field distribution in the ESP. The present model is also validated against the experimental data of [PERSON] [35]. The experimental set-up had eight corona discharge wires. The operating voltages vary between 6 and 13 kV. Each corona wires' diameter (\(d_{w}\)) is 0.1 mm, the distance between the wire and the plate (\(S_{y}\)) is 25 mm, the distance between the wires (\(S_{x}\)) is 50 mm, and the length (\(L\)) and height (\(H\)) of plates are 400 mm and 50 mm, respectively. Particle density (\(\rho_{p}\)) and relative dielectric constant (\(\varepsilon_{p}\)) are 893.5 kg/m\({}^{3}\) and 2.5. The injected particles' diameter is 4 \(\upmu\)m. Gas velocity (\(U_{int}\)) is 2 m/s. Particles are injected into the computational domain from the inlet surface, and the motions and charging process of particles were calculated once through the domain. Collection efficiency as a function of the particle diameter was obtained as follow:  
\[\eta_{p}(d_{p})=\left(1-\ \frac{N_{c}}{N_{tot}}\right) \tag{23}\]  
where \(N_{c}\) represents the trapped particles on the collection plates, \(N_{tot}\) defines the total number of particles injected into the channel. Figure 4 shows the comparison of the present model computation of the collection efficiency values with the experimental data of [PERSON] [35].