# Penetration Efficiency and Concentration Distribution of Nanoparticles in a Hollow Tapered Cylinder
## 2 Methods  
In this study, a numerical investigation of the nanoparticle penetration characteristics within the hollow tapered cylinder was carried out. Figure 1 outlines a schematic diagram of the hollow tapered cylinder. The selecting conditions of the numerical simulation for the hollow tapered cylinder with the conical column were based on the studies of nanoparticle diffusion deposition measurements in the needle chargers. This study adopted a geometric design similar to that of the needle charger without applied voltage [24]. The numerical penetration efficiency of the hollow tapered cylinder with the conical column was therefore compared with the experimental data. For the boundary conditions of the computational domain, zero particle concentration is applied to all the solid walls of the hollow tapered cylinder with or without the conical column. The number concentration adopts the uniform distribution at the inlet of the hollow tapered cylinder, and the zero concentration gradient in the flow direction at the outlet. When calculating the nanoparticle transport in the hollow tapered cylinder with or without the conical column, it is assumed that interactions between particles can be ignored. When particles are introduced into the hollow tapered cylinder, the particles are assumed to be spherical and their presence does not influence the flow field. Furthermore, the flow field within the hollow tapered cylinder with or without the conical column is assumed to be axisymmetric with no flow in the azimuthal direction. The main numerical domain used to calculate the hollow tapered cylinder is shown in Figure 2, where the wall of the hollow tapered cylinder was set as the solid boundary conditions. This computational domain used a total of 84,000 (200 in the r-direction \(\times\) 420 in the z-direction) non-uniform rectangular grids. Figure 1: Sketch of the hollow tapered cylinder, unit: mm. The flow field inside the hollow tapered cylinder with the conical column or without the conical column was calculated using two-dimensional Navier-Stokes and continuity equations under the assumption of incompressible, steady, axisymmetric, viscous, and laminar fluid flow, while air was assumed to be 20 \({}^{\circ}\)C and 1 atm. This study then solved the governing equations that have been discretized using the finite volume method by the SIMPLE algorithm [27]. With the above assumptions, the Navier-Stokes and continuity equations used in this study can be expressed in the following form:  
\[(\overrightarrow{V}\cdot\
abla)\overrightarrow{V}=-\frac{1}{\rho}\
abla P+ \gamma\
abla^{2}\overrightarrow{V} \tag{1}\]  
\[\
abla\cdot\overrightarrow{V}=0 \tag{2}\]  
where \(\overrightarrow{V}\) is the air velocity vector, \(\rho\) is the air density, \(P\) is the pressure, and \(\gamma\) is the air kinematic viscosity. The approach used to simulate the particle transport behavior within the hollow tapered cylinder with or without the conical column treated the nanoparticles as a single mixed fluid of chemicals. The convection diffusion equation applied to the number concentration, \(C\), is given as:  
\[\frac{\partial(U_{i}C)}{\partial r}+\frac{\partial(U_{j}C)}{\partial z}=D \bigg{[}\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial C}{ \partial r}\bigg{)}+\frac{\partial^{2}C}{\partial z^{2}}\bigg{]} \tag{3}\]  
where \(U_{i}\) and \(U_{j}\) are the air velocities in the radial direction and axial direction, respectively. Using the Stokes-Einstein relation, the Brownian motion of small particles can be linked to gas molecules, and the diffusion coefficient expression is:  
\[D=\frac{kT}{\varphi} \tag{4}\]  
where \(T\) is the absolute temperature, \(k\) is the Boltzmann constant, and \(\varphi\) is the coefficient of friction, which depends on the particle size and the physical properties of the surrounding gas.