## 3 Numerical  
Mathematically, a multidimensional agglomeration population balance equation (PBE) can be expressed in terms of an integro-partial differential equation as follows:  
\[\frac{\partial n(v)}{\partial t}+\
abla\cdot(\overset{\rightarrow}{\mathbf{ u}}n(v))=B(v)-D(v) \tag{17}\]  
where \(n(v)\) is the number density function, \(v\) is the volume of a particle, and \(\overset{\rightarrow}{\mathbf{u}}\) is the gas flow velocity vector. \(B(v)\) and \(D(v)\) denote the birth and death rates of particles of volume \(v\), respectively:  
\[B(v)=\frac{1}{2}\int_{0}^{v}K_{ij}\big{(}v-v^{\prime},v^{\prime}\big{)}n\big{(} v-v^{\prime}\big{)}n(v)dv^{\prime} \tag{18}\]  
\[D(v)=\int_{0}^{\infty}K_{ij}(v,v^{\prime})n\big{(}v^{\prime}\big{)}n(v)dv^{\prime} \tag{19}\]where \(v^{\prime}\) is the volume of another particle with a different size. \(K_{ij}\) refers to the total agglomeration coefficient of bipolar charged particles by an external electric field with the effective Coulomb force and with the aerodynamic effect (see Equation (16)). The gas flow velocity \(\overrightarrow{u}\) was determined by solving the continuity and Navier-Stokes equations. The standard k-\(\epsilon\) turbulence model was utilized. Details of governing equations are presented as follows:  
\[\text{Continuity}\ \frac{\partial}{\partial t}(\rho)+\
abla.\left(\rho \overrightarrow{u}\right)=\overset{\rightarrow}{S} \tag{20}\]  
\[\text{Momentum}\ \frac{\partial}{\partial t}\left(\rho\overrightarrow{u} \right)+\
abla\left(\rho\overrightarrow{u}\overrightarrow{u}\right)=-\
abla p +\mu\
abla^{2}\overrightarrow{u}+\rho\overset{\rightarrow}{g} \tag{21}\]  
\[\text{Turbulence k}\ \frac{\partial}{\partial t}(\rho k)+\frac{\partial}{ \partial x_{i}}\left(\rho k\overrightarrow{u}\right)=\frac{\partial}{ \partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\frac{ \partial k}{\partial x_{j}}\right]+P_{k}-\rho\epsilon \tag{22}\]  
\[\text{Turbulence}\ \epsilon\frac{\partial}{\partial t}(\rho\epsilon)+\frac{ \partial}{\partial x_{i}}\left(\rho\epsilon\overrightarrow{u}\right)=\frac{ \partial}{\partial x_{i}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right) \frac{\partial\epsilon}{\partial x_{i}}\right]+C_{1e}\frac{\epsilon}{k}P_{k}- C_{2e}\rho\frac{\epsilon^{2}}{k} \tag{23}\]  
where \(\overset{\rightarrow}{S}\) is the source term, \(p\) is the static pressure, and \(\overset{\rightarrow}{g}\) is the gravitational acceleration. \(P_{k}\) is the generation of turbulent kinetic energy due to the mean velocity gradients, and \(\epsilon\) is the dissipation rate of the turbulent energy. \(C_{1e}=1.44\) and \(C_{2e}=1.92\) are model constants. The boundary conditions are presented in Table 1.  
In this study, numerical calculations were performed for a domain of the lab-scale agglomerator (see Figure 1a, with a cross-sectional area of 4 \(\times\) 4 cm\({}^{2}\) and a length of 0.6 m). The CFD grids were generated via ANSYS meshing, with a total grid number of approximately one million. The inlet air speed was set at 0.1 m/s, and the outlet was set at atmospheric pressure. In accordance with the particle distribution used in the experiment, the minimum size of the inlet particles was set to 50 nm and the maximum particle size was set to 8.2 \(\upmu\)m. There were 49 different sizes between 50 nm and 8.2 \(\upmu\)m, resulting in a polydisperse distribution. The total concentration was \(3.58\times 10^{6}\) particles per cm\({}^{3}\) of air. The time step was set to 0.01 s, the number of time steps was 300, and the number of iterations per time step was 50. In addition, the discrete phase model (DPM) was employed to observe the trajectories of different particles. The equation of motion of a single particle is as follows:  
\[m\frac{\overset{\rightarrow}{u}_{p}}{dt}=m\frac{\left(\overset{\rightarrow} {u}-\overset{\rightarrow}{u}_{p}\right)}{\tau_{r}}+m\frac{\overset{ \rightarrow}{g}\left(\rho_{p}-\rho\right)}{\rho_{p}} \tag{24}\]  
where \(\tau_{r}\) is the particle relaxation time given as  
\[\tau_{r}=\frac{{d_{p}}^{2}\rho_{p}C_{C}}{18\mu} \tag{25}\]