# Penetration Efficiency and Concentration Distribution of Nanoparticles in a Hollow Tapered Cylinder
## 3 Results and Discussions
### Penetration Efficiency  
The study then evaluates the effect of the conical column within the hollow tapered cylinder on the nanoparticle penetration efficiency. Figure 8 shows the numerical nanoparticle penetration efficiencies of the hollow tapered cylinder with the conical column compared with the experimental data in the literature [24]. The results demonstrate that the present  
Figure 7: Concentration profile at outlet of the hollow tapered cylinder (**a**) without the conical column and (**b**) with the conical column for different particle diameters at the flow rate of 2 L/min.  
numerical model has a good agreement with the experimental data for the flow rate of 2.0 L/min. It should be noted that the experimental data in the literature is the nanoparticle loss of the needle charger without applied voltage under the same conditions as in this study. The particle penetration efficiency was obtained by subtracting the nanoparticle loss from 100% and comparing it to the numerical results. Figure 9 represents the nanoparticle penetration efficiency as a function of the particle size for the hollow tapered cylinder without or with the conical column at different flow rates. It can be seen that the particle penetration efficiency increases with the increasing particle diameter inside the hollow tapered cylinder without the conical column. In addition, the penetration efficiency can be improved by increasing the flow rate. This is because the residence time is not sufficient for the particles to reach the walls of the hollow tapered cylinder for the higher flow rates. This finding can be applied to the measurement of nanoparticles in environments using hollow tapered cylinders. For many particle measuring instruments, the hollow funnel is required in front of it to collect nanoparticles [21; 22]. Therefore, knowing the nanoparticle penetration efficiency and concentration distribution of the hollow tapered cylinder is helpful for the correctness of nanoparticle measurements and the determination of nanoparticle sources. Although the size of the tapered funnel used to collect the particles is larger than that of the hollow tapered cylinder without the conical column calculated in this study. However, from a dimensionless parameter point of view, they have the similar geometries and the penetration parameter \(\delta\) (\(\delta=sD/q\), \(D\): the nanoparticle diffusion coefficient, \(s\): the length, \(q\): the flow rate). For example, the penetration parameter of 3 nm particles for the tapered funnel with the inlet length of 40 cm at the flow rate of 5.5 L/min and the hollow tapered cylinder without the conical column at the flow rate of 0.5 L/min is 0.0025 and 0.0021, respectively. Therefore, the present numerical models are considered suitable for the application to the geometric similarity inlets.  
Figure 8: Comparison of numerical penetration efficiency of the hollow tapered cylinder with experimental data [24].  
Additionally, a similar trend can be found from the penetration efficiency of the hollow tapered cylinder with the conical column. It should be noted that the penetration efficiency of the hollow tapered cylinder with the conical column is lower than that of the hollow tapered cylinder without the conical column. For particles with a diameter of 3 nm, the penetration efficiency of the hollow tapered cylinder with the conical column is about 66.3% at a flow rate of 0.5 L/min, which can be improved to about 73.4% by removing the conical column. This is due to the large loss of the diffusional nanoparticles on the column wall inside the hollow tapered cylinder, resulting in the lower penetration efficiency.