# Penetration Efficiency and Concentration Distribution of Nanoparticles in a Hollow Tapered Cylinder
## 3 Results and Discussions
### Concentration Profile at Different Positions  
Figure 5(a) shows the nanoparticle concentration profile of the cross section at different straight positions of the hollow tapered cylinder with the conical column for the particle diameters of 3 nm at the flow rate of 2 L/min. It can be seen that the particle concentration in the radial direction increases with the increase in the radial position from the center point, after reaching a high point at about \(R_{p}/R_{ext1}=0.43\)-0.52, \(R_{p}\): the radial distance, it begins to decrease until it approaches the minimum as approaching the outer wall. The same trend exists for the different cross sections at straight positions of the hollow tapered cylinder, where the axial position of the cross section is 0.022 m, 0.024 m, 0.026 m, 0.028 m, and 0.03 m for \(z_{1}\), \(z_{2}\), \(z_{3}\), \(z_{4}\), and \(z_{out}\), respectively. The calculation results show that the variation in the concentration profiles at different axial positions of the hollow tapered cylinder gradually decreases with the increase in the axial distance, and the concentration profile tends to be indistinguishable when approaching the exit position. Figure 5(b) shows the velocity profile of the cross section at different straight positions of the hollow tapered cylinder with the conical column at the flow rate of 2 L/min. It shows that the velocity near the inner cylinder wall is lower than that near the central area for the different cross sections of the hollow tapered cylinder. Figure 7 shows the concentration profile at the outlet cross section of the hollow tapered cylinder without the conical column or with the conical column for the different particle diameters at the flow rate of 2 L/min. For the hollow tapered cylinder without the conical column, the concentrations of particles with different sizes near the center point, \(R_{p}/R_{ext1}=0\), are uniform and close to the inlet concentration, as shown in Figure 7(a). When approaching the position of \(R_{p}/R_{ext1}=0.35\), the concentration of smaller particles (3 nm and 5 nm) begins to decrease till the outer position of \(R_{p}/R_{ext1}=1\).  
In comparison, the concentration of larger particles (10 nm and 20 nm) begins to drop rapidly after approaching the outer position of \(R_{p}/R_{ext1}=0.92\). Figure 7(b) shows the concentration profile at the outlet cross section of the hollow tapered cylinder with the conical column. It can be seen that the concentration of the smaller particle size particles (3 nm and 5 nm) in the area close to the center point is significantly lower than that without the conical column. The lower concentration in the central region of the outlet is caused by the diffusion effect resulting in more loss on the wall of the conical column. In addition, the highest concentration appears at the position of about \(R_{p}/R_{ext1}=0.43\)-0.52 for the smaller particles (3 nm and 5 nm), and the concentration also decreases in the vicinity of the peripheral part similar to that without the conical column. The results show that the concentration of nanoparticles at the outlet of the hollow tapered cylinder is lower for the smaller the particle diameter under the same conditions. This is because the particles with the smaller size have a more obvious diffusion effect, resulting in a greater loss of  
Figure 5: Particle concentration distribution inside the hollow tapered cylinder (**a**) without the conical column and (**b**) with the conical column for the particle diameters of 3 nm at the flow rate of 2 L/min.  
particles adjacent to the wall surface, so some particles cannot reach the outlet of the hollow tapered cylinder.  
Figure 6: (**a**) Concentration and (**b**) velocity profile of z—z cross sections at different positions of the hollow tapered cylinder with the conical column for the particle diameters of 3 nm at the flow rate of 2 L/min.