## 6 Discussion  
When considering the agglomerator with a vortex generator but no electric field (see Figure 5a) and the electric agglomerator without the vortex generator (see Figure 7a), the overall efficiency of a system consisting of these two agglomerators arranged in series is given by the following equation:  
\[\eta_{overall}=1-(1-\eta_{VG})(1-\eta_{E}) \tag{29}\]  
where \(\eta_{E}\) is the particle reduction efficiency in passing through the electric agglomerator (without the vortex generator). The number concentration at a position downstream of the system is represented as follows:  
\[N_{Down}=(1-\eta_{VG})(1-\eta_{E})N_{Up}\ \approx\big{\{}1-(\eta_{VG}+\eta_{E}) \big{\}}N_{Up} \tag{30}\]  
To determine whether the particles were significantly reduced purely by the electric field (i.e., to determine \(\eta_{E}\)), additional experiments were performed when the vortex generator was removed from the agglomerator where the electric field was applied (Figure 6a). The results are shown in Figure 7b. When the vortex generator was absent (Figure 7a), the number concentration at a position downstream decreased to 2.71 \(\times\) 10\({}^{6}\) particles per cm\({}^{3}\) of air volume from the upstream concentration of 3.69 \(\times\) 10\({}^{6}\) particles per cm\({}^{3}\) of air volume (26% decrease, \(\eta_{E}\) = 0.26). The results of Figure 7b can be simplified using the relation between upstream and downstream number concentrations (Figure 7c):  
\[N_{Down}=(1-\eta_{E})N_{Up} \tag{31}\]  
\(\eta_{VG+E}\) (=0.5) is greater than \(\eta_{VG}+\eta_{E}\) (=0.065 + 0.26 = 0.325). Additional CFD simulation was performed to investigate the cause of the difference between \(\eta_{VG+E}\) and \(\eta_{VG}+\eta_{E}\). In the simulation, the DPM was employed to calculate trajectories of different particles. Figure 8a illustrates positions of a representative single particle at different times inside the agglomerator without a vortex generator installed. It takes 2.33 s for the particle to pass through the agglomerator from the inlet to the outlet. Meanwhile, Figure 8b illustrates positions of a particle at different times inside the agglomerator with a vortex generator installed. The time required for the particle to pass through the agglomerator is 6.13 s. This increase in residence time by 2.33 s ensures more chances of exposure to the electric field. Therefore, the application of an electric field to a space where a vortex generator is located results in an efficiency of \(\eta_{VG+E}\), which is higher than the summation of two individual efficiencies, \(\eta_{VG}+\eta_{E}\). [PERSON] et al. [25] also reported an extension of residence time for fine particles and an increase in dust collection efficiency by installing a vortex generator. The increase in residence time when the vortex generator was added to the space in which an electric field was applied led particles to be exposed to the electric field for a longer duration, thus generating a synergistic effect.  
Figure 7: (a) Overview: with an electric field applied to the agglomerator (without a vortex generator), (b) particle size distributions upstream and downstream of the agglomerator, (c) AC electric field as an agglomeration factor.