## 2 Materials and Methods  
The air in the ESP is assumed as incompressible, and the airflow is considered turbulent. A corona discharge develops under the effect of an electrostatic field. Created ions cling to suspended particles. As a result, the electrostatic force is exerted on charged particles, which impacts air and charged particle movement. Under the Eulerian framework, the governing equations explaining airflow can be stated as follows:  
\[\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial}{\partial x_{i}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p}{ \partial x_{i}}+\frac{\partial}{\partial x_{i}}\bigg{[}(\mu+\mu_{t})\frac{ \partial u_{j}}{\partial x_{i}}\bigg{]}+\rho_{ion}E_{j}+S_{c} \tag{2}\]  
where \(x_{i}\) is the coordinate component, \(u_{i}\) is the fluid velocity component (m/s), \(\rho\) is the fluid density (kg/m\({}^{3}\)), \(\mu\) is the dynamic viscosity (kg/ms), \(\mu_{t}\) is the turbulent viscosity (kg/ms), \(\rho_{ion}\) is the ionic charge density (C/m\({}^{3}\)), \(p\) is the pressure (Pa), and \(E_{j}\) is the electric field strength (V/m). The electrostatic force is described by the third term on the right-hand side of Equation (2). The electric field strength and ion charge density are required for its description. \(S_{c}\) represents the other source terms. The electric field strength is defined as follows:  
\[E_{i}=-\frac{\partial\phi}{\partial x_{i}} \tag{3}\]  
where \(\phi\) is the electric potential (V); the Poisson equation can be used to describe the electric potential in the ESP. \[\frac{\partial^{2}\phi}{\partial x_{i}^{2}}=-\frac{\big{(}\rho_{ion}+\rho_{p} \big{)}}{\varepsilon_{0}} \tag{4}\]  
where \(\varepsilon_{0}\) is the permittivity of air and \(\rho_{p}\) is the particle space charge density (C/m\({}^{3}\)). On the other hand, the movement of particle charge is affected by the airflow and the electric field. The current continuity equation and current density (\(J_{i}\)) can be described as follows:  
\[\frac{\partial}{\partial x_{i}}\bigg{(}\rho_{ion}(b_{ion}E_{i}+u_{i})-D_{i} \frac{\partial\rho_{ion}}{\partial x_{i}}\bigg{)}=0 \tag{5}\]  
\[J_{i}=\rho_{ion}(b_{ion}E_{i}+u_{i})-D_{i}\frac{\partial\rho_{ion}}{\partial x _{i}} \tag{6}\]  
where \(D_{i}\) is the ion diffusion coefficient (m\({}^{2}\)/s) and \(b_{ion}\) is the ion mobility (1.5 \(\times\) 10\({}^{-4}\) (m\({}^{2}\)/Vs) of positive charging). Steady-state Reynolds Averaged Navier-Stokes and energy equations are solved. Turbulence is modelled with the two-equation RNG k-\(\varepsilon\) approach, which was used and validated in the previous studies such as [27; 28]. At the input, the fluid velocity, turbulence intensity, and turbulent viscosity ratio are provided. At the outlet, the pressure is indicated. At all walls, a no-slip condition is applied. At the wire electrode, the electric potential is specified as the operating potential. At the particle collecting plate, the electric potential is set to zero. At all other boundaries, the gradient of the electric potential is zero. At the inlet, outflow, and particle collection plate, the gradient of ion charge density is zero. The ion charge density is calculated as assuming that the charge density is uniform in the corona region. Hence, the following [PERSON]'s equation was satisfied on the corona wire surface. \[E_{c}=3\times 10^{6}f\bigg{(}\frac{T_{0}P}{TP_{0}}+0.03\sqrt{\frac{T_{0}P}{ TP_{0}r_{w}}}\bigg{)} \tag{7}\]  
where \(f\) is the roughness factor (assumed as one in this study), \(T\) is fluid temperature (K), \(P\) is operating pressure (kPa), \(T_{0}\) and \(P_{0}\) are temperature and pressure at the reference state of 101 kPa and 293 K, and \(r_{w}\) is the radius of the corona wire. The particle transport equation can be derived by applying the conservation of momentum for the particles.