## 2 Theory  
The rate of change of number concentration in particles suspended in stagnant air caused by particle agglomeration can be expressed as follows:  
\[\frac{\mathrm{d}N_{i}}{\mathrm{d}t}=-K_{i}{N_{i}}^{2} \tag{1}\]  
where \(N_{i}\) denotes the number concentration of the particle (of size \(d_{i}\)) and \(K_{i}\) is the agglomeration coefficient defined as the effective particle collision volume per second. As the actual particles are mostly polydisperse particles, the agglomeration coefficient can be obtained by the following equation:  
\[K_{i}=\sum_{j=1}^{n}K_{ij}f_{i}f_{j} \tag{2}\]The fraction of concentration, \(f_{i}\), is defined as follows:  
\[f_{i}=\frac{N_{i}}{N_{total}} \tag{3}\]  
where \(N_{total}\) denotes the total concentration of all particles and \(K_{ij}\) is the agglomeration coefficient between a particle of diameter \(d_{i}\) and a particle of diameter \(d_{j}\). [PERSON] et al. [19] determined the agglomeration coefficient for bipolar charged particles of various sizes. Their model, based on the equation of motion of a charged single particle, calculated particle trajectories using DC/AC electric fields as key variables. The results indicated that the AC electric field had a more significant coagulation effect compared to the DC electric field. Additionally, [PERSON] et al. [20] developed an approximate formula for the agglomeration coefficient of bipolar charged particles under an AC electric field. According to [PERSON] et al. [20], \(K_{CF-ij}\) is the agglomeration coefficient of bipolar charged particles by the Coulomb force without the effect of an external electric field between particles of diameters \(d_{i}\) and \(d_{j}\):  
\[K_{ij}=K_{CF~{}ij}=\frac{q_{i}q_{j}\big{(}d_{i}+d_{j}\big{)}}{3\pi\epsilon_{0 }ln{d_{i}d_{j}}} \tag{4}\]  
where \(\mu\) is the air viscosity and \(\epsilon_{0}\) is the permittivity of free space. The charging quantity of i-th particle, \(q_{i}\), is  
\[q_{i}=n_{i}e \tag{5}\]  
where \(e\) is the elementary unit of charge (1.6 \(\times\) 10\({}^{-19}\) C). The charge number of i-particle, \(n_{i}\), can be calculated using the following equation:  
\[n_{i}=\frac{d_{i}kT}{2K_{\text{E}}e^{2}}\text{ln}\bigg{[}1+\frac{\pi K_{\text{ E}}d_{i}\bar{\epsilon}_{i}e^{2}N_{ion}t_{c}}{2 kT}\bigg{]}+\bigg{(}\frac{3e_{R}}{e_{R} +2}\bigg{)}\bigg{(}\frac{E{d_{i}}^{2}}{4K_{\text{E}}e}\bigg{)}\bigg{(}\frac{ \pi K_{\text{E}}eZ_{ion}N_{ion}t_{c}}{1+\pi K_{\text{E}}eZ_{ion}N_{ion}t_{c}} \bigg{)} \tag{6}\]  
where k denotes the Boltzmann constant (\(\sim\)1.38 \(\times\) 10\({}^{-23}\) J/K), \(T\) is the temperature, K\({}_{\text{E}}\) is the electrostatic constant of proportionality (9 \(\times\) 10\({}^{9}\) Nm\({}^{2}\)/C\({}^{2}\)), \(\bar{\epsilon}_{ion}\) is the mean thermal speed of the ions (240 m/s) at standard condition, \(Z_{ion}\) is the electrical mobility of an ion (1.5 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/V\(\cdot\)s ), \(N_{ion}\) is the ion number concentration, \(t_{c}\) is the time passing through the charger, and \(e_{R}\) is the relative permittivity of the particle. According to [PERSON] et al. [20], the agglomeration coefficient of bipolar charged particles by the alternating current electric field without the effect of the Coulomb force is as follows:  
\[K_{ij}=K_{EF~{}ij}=\frac{\big{(}d_{i}+d_{j}+2\gamma_{AC}\big{)}^{2}}{6\pi\mu}E _{AC}\Bigg{(}\frac{q_{i}}{d_{i}}C_{Ci}+\frac{q_{j}}{d_{j}}C_{Ci}\bigg{)} \tag{7}\]  
where \(E_{AC}\) is the alternating electric field. \(Y_{AC}\), which is the distance between the particles, can be approximated using the following equation:  
\[Y_{AC}=\sqrt[3]{C_{5}\frac{q_{i}q_{j}}{4\pi^{2}\mu\epsilon_{0}\big{(}X_{i}+X _{j}\big{)}}\Bigg{(}\frac{C_{Ci}}{d_{i}}+\frac{C_{Ci}}{d_{j}}\Bigg{)}} \tag{8}\]  
where \(C_{Ci}\) is the Cunningham slip correction coefficient of i-particle, \(C_{5}\)\(\approx\) 0.5 \(\times\) 10\({}^{-8}\) (dimensionless). [PERSON] et al. [20] proposed an approximate expression for the agglomeration coefficient of bipolar charged particles with the effect of an AC electric field.