## 2 Design and Simulation
### Impactor Design  
Inertial separation, gravity settling, thermal precipitation, and centrifugal settling were the main techniques of particle sampling technology. The most commonly used aerosol sampling is inertial separation, which includes conventional impactors, virtual impactors, and cyclone samplers [23]. Figure 8 shows the schematic diagram of a conventional impactor with a circular nozzle. Particles smaller than the cut-size of the impactor will follow the streamlines, while larger particles will slip across the streamlines and impact upon the impaction plate to be collected. Theoretical studies of conventional impactors have been relatively well developed and have achieved theoretically accurate predictions of the impactor's cut-off diameter, such as [PERSON] [24] and [PERSON] [25], their theories have been compared to the experimental results with good agreement.  
Figure 7: The normalized distribution of particles at different impact distances.  
The conventional impactor's cut-off diameter depends on the flow rate \(Q\), Stokes number \(Stk\), Reynolds number \(Re\), nozzle diameter \(D_{0}\), and jet-to-plate distance \(S\). Among these parameters, the dimensionless \(Stk\) is a critical parameter, its relationship with cut-off point (\(D_{50}\)) is:  
\[D_{50}=\sqrt{\frac{9\eta D_{0}}{\rho_{p}C_{c}U}}\sqrt{Stk_{50}} \tag{1}\]  
where \(U\) is the average air velocity in the nozzle outlet, \(\eta\) is the air viscosity coefficient, \(\rho_{p}\) is the particle density, and \(C_{c}\) is the Cunningham slip correction factor. For 10, 2.5, and 1 \(\upmu\)m particles, their Cunningham slip correction factors were calculated to be 1.0164, 1.0658, and 1.1644, respectively. Substituting the average flow velocity of the circular nozzle into the Equation (1), a design formula for the diameter of the circular nozzle can be derived:  
\[D_{0}=\sqrt{\frac{\rho_{p}Re}{9\rho_{a}Stk_{50}}}\sqrt{C_{c}}D_{50} \tag{2}\]  
According to the above theory and considering the size of the optimal response area of the resonator for particle attachment, we designed a three-stage cascade impactor with a total flow of 2 LPM, combined by the LCP stage, PM\({}_{\text{2.5-10}}\) stage, and PM\({}_{\text{1-2.5}}\) stage. The design parameters of the impactor were shown in the Table 4 below. We calculated the Reynolds numbers of each stage as 563.8, 1398.4, and 2210, respectively, which meet the requirement of \(500\leq Re\leq 3000\) for a steep collection efficiency curve.  
We used the computational fluid dynamics module of the commercial software COMSOL MULTIPHYSICS to simulate each stage of the impactor. Figure 9 shows the 3D simulation of flow velocity distribution inside the impactor. The average flow velocity at the acceleration nozzle was increased in successive stages, and the values were 1.68, 10.8, 34.2 m/s, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stage** & \(D_{0}\) **(mm)** & \(T\) **(mm)** & \(S\) **(mm)** \\ \hline LCP & 4.94 & 9 & 5 \\ PM\({}_{\text{2.5-10}}\) & 1.99 & 5 & 3 \\ PM\({}_{\text{1-2.5}}\) & 1.12 & 5 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Design parameters of cascade impactor.  
Figure 8: Schematic of the conventional impactor.  
In order to evaluate the impactor's selecting effect on particles, we simulated the particle collection efficiency curve for each stage using the particle tracking module of COMSOL MULTIPHYSICS, as shown in Figure 10. When the particle collection efficiency was 50%, the fitted cut-off diameter for each stage was 1.06, 2.48, and 9.94 \(\upmu\)m, with a small deviation from the designed value, indicating that the cascade impactor has a good particle collection efficiency, meeting the sampling requirements.