Penetration Efficiency and Concentration Distribution of Nanoparticles in a Hollow Tapered Cylinder

Abstract

Knowing particle penetration efficiencies and concentration distributions in an inlet channel of a sampling device is beneficial for the robust assessment, attribution and quantification of nanoparticles produced by various activities. The aim of this research is to evaluate the effect of the presence or absence of a conical column inside a hollow tapered cylinder on the nanoparticle penetration efficiency and its outlet concentration profile for different flow rates. The particle penetration characteristics of various sizes from 3 nm to 20 nm were numerically investigated by using the flow field and convection diffusion equations within the hollow tapered cylinder. Firstly, the proposed model of the nanoparticle penetration efficiency for the hollow tapered cylinder with the conical column is validated with the experimental data in the literature. Then, the results indicate that the concentration at the outlet of the hollow tapered cylinder with the conical column exhibits annular profiles for 3 nm and 5 nm nanoparticles at a flow rate of 2.0 L/min, which is found to avoid centralizing the particles in the exit area. In addition, the penetration efficiency of nanoparticles can be improved by increasing flow rates or removing the conical column inside the hollow tapered cylinder. Finally, the ring-shaped concentration profile of the 10 nm nanoparticles at the outlet of the hollow conical cylinder with the conical column becomes more obvious as the flow rate decreases. This study interprets and quantitatively decides the nanoparticle penetration efficiency and its exit concentration profile for the hollow tapered cylinder with or without the conical column. Therefore, the results can provide some useful design references for the transport of nanoparticles in the hollow tapered cylinder.

1. Introduction

Nanoscale airborne particles from natural or manufactured sources are abundant in our environment. Although manufactured nanoparticles have been used in many fields, they may also endanger human health due to their potential toxicity [1,2,3]. On the scale of nanoparticles, it can be compared with viruses or molecules scale. For humans, one of the main pathways of exposure to such small nanoparticles is the respiratory system, as the nanoparticles can be efficiently deposited in larger airways and in the alveolar region of the lung [4,5,6].

To study the characteristics and concentrations of a large number of the nanoparticles in the surrounding environment, it is generally feasible to collect these particles for analysis or to perform real-time measurement. Because of the low inertia of nanoparticles, it is often necessary to expose these particles to thermophoretic forces to trap them, or to collect them by adding a particle charger before reaching the collection surface [7,8,9]. Thermal precipitators have been applied in some nanoparticle samplers that require uniform collection of particles on a cooler plate. Using the transmission electron microscopy or the scanning electron microscopy, the crystallography, morphology, and chemical composition of the collected nanoparticles on the plate could be further analyzed. The uniform collection of particles within a specific area of the collection plate is important for the ideal grid position of the image generation [10,11]. Therefore, the calculation of the particle concentration profile at the exit and the avoidance of high concentrations at the center location are well worth exploring. In addition, the penetration efficiency of the particles at the inlet channel of the sampling device also greatly affects the measurement error.

Transport studies of nanoparticles have been extensively utilized for various applications involving the transportation phenomenon in different geometries, the conduction properties in fluids, and the assistance in obtaining efficient energy during heat transfer [12,13,14,15]. In many tube flow studies, theoretical models and empirical equations of Brownian diffusion were used to compare and predict the penetration efficiency of nanoparticles [16,17]. In contrast to the numerous studies of the tube flow, the Brownian diffusion of nanoparticles in the hollow tapered cylinder with the conical column was rarely reported in the literature. However, there are many applications of the nanoparticle measurement for the hollow tapered cylinder, including the sampler inlets and the needle chargers [18,19,20]. Gregson et al. [21] determined how many particles were produced through coughing, deep breathing, and singing events compared to natural breathing events by using a hollow tapered funnel to collect the particles and passed through a circular conduit to a particle measuring instrument. Respiratory events such as coughing, speaking, and breathing had been shown to produce measurable aerosol concentrations [22,23]. Due to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), social activities and many economic activities have been significantly affected. Based on strict prevention of airborne transmission, proper particulate sampling is important to assist in the identification of the particle sources [20,21]. The particle penetration efficiencies and concentration distributions at the entrance in front of the particle detection instrument affect the measurement results. Therefore, the particle penetration and concentration profile of the hollow tapered cylinder are important to the particulate measurement.

On the other hand, the nanoparticle penetration efficiency of the hollow tapered cylinder with the conical column affects the intrinsic charging efficiency and the performance of the needle charger. The intrinsic efficiency of needle chargers can be increased by increasing the number and charging rate of nanoparticles within the needle charger [19,24,25]. Intra et al. [19] evaluated a corona-based cylindrical ionizer with the conical column for the charging efficiency, and results showed that smaller particles have higher diffusion loss than larger particles, resulting in a lower nanoparticle penetration. While the methods for increasing particle penetration by increasing the flow rate are well known, a quantitative analysis of the improvement of the nanoparticle penetration has not been fully reported in the literature. Furthermore, the effect of adding the conical column on the particle penetration and the outlet concentration profile is also lacking. In the previous study [26], the nanoparticle deposition was investigated by simulating the flow field and the particle trajectory within a needle charger. The results showed that most of the particles inside the needle charger without applied voltage were attached to the outer wall instead of the electrode wall. One of the achievements in the previous study is to solve the movement of nanoparticles while considering the corrected Stokes drag force and Brownian force, and to provide a comparison of the two forces for nanoparticles, so as to understand the underlying physics more easily. However, it is important to see if the conical column has an effect on the concentration distribution when the nanoparticles move toward the outlet of the hollow tapered cylinder. Therefore, this study solves the nanoparticle concentrations by modeling the nanoparticle diffusion inside the hollow tapered cylinder with or without the conical column, which differs remarkably from previous research. The aim of this study is to evaluate the effect of the presence or absence of the conical column within the hollow tapered cylinder on the nanoparticle penetration efficiency and the concentration distribution.

In this paper, we investigated the nanoparticle concentration distribution of the hollow tapered cylinder with or without the conical column by using the flow field and convective diffusion methods, which constitutes a novelty of the present work. The penetration efficiencies of nanoparticles in the hollow tapered cylinder were then evaluated numerically under different flow rates and the results are discussed comprehensively. The outlet concentration profiles of the cross sections for a range of particle sizes were also calculated by using the axisymmetric computational model. In the presence of the conical column, the model was validated for the nanoparticle penetration efficiency of the hollow tapered cylinder. The Brownian diffusion and convection of nanoparticles in the hollow tapered cylinder with or without the conical column has not been described in the works earlier to our knowledge. It very well may be trusted that the investigation of the nanoparticle concentration distributions and penetration efficiencies of the hollow tapered cylinder is useful for the robust assessment, attribution, and quantification of nanoparticles produced by various activities.

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 × 420 in the z-direction) non-uniform rectangular grids.

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 °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:

$$\left(\overrightarrow{V}·\nabla \right)\overrightarrow{V}=-\frac{1}{\rho }\nabla P+\gamma {\nabla }^2\overrightarrow{V}$$

(1)

$$\nabla ·\overrightarrow{V}=0$$

(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:

$$\frac{\partial \left(U_{i}C\right)}{\partial r}+\frac{\partial \left(U_{j}C\right)}{\partial z}=D\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)+\frac{{\partial }^{2}C}{\partial z^2}\right]$$

(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:

$$D=\frac{kT}{\phi }$$

(4)

where T is the absolute temperature, k is the Boltzmann constant, and $\phi$ is the coefficient of friction, which depends on the particle size and the physical properties of the surrounding gas.

The transport behavior of small particles and the interaction between the surrounding gas molecules can be characterized by considering the Knudsen number, Kn; it is the relationship between the mean free path of the gas molecules, $l_g$, and the particle diameter, d_p.

$$Kn=\frac{2l_g}{d_p}$$

(5)

For very small nanoparticles (Kn >> 1), the expression for the friction coefficient of the kinetic behavior between the particle and the surrounding gas can be derived using kinetic theory of gases as follows:

$$\phi =\frac{2}{3}\rho d_p^2\left(1+\frac{\alpha \pi }{8}\right){\left(\frac{2\pi kT}{m}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(6)

where m is the mass of the gas molecule and α is the accommodation coefficient, which is usually near 0.9 [28]. The relationship can be generalized using the following expression to cover a wider range of Kn numbers [29].

$$\phi =\frac{3\pi \mu d_p}{1+\frac{Kn}{2}\left(2.34+1.05exp\left(-0.78/Kn\right)\right)}$$

(7)

Obtaining the number concentration of particles at the outlet of the hollow tapered cylinder, the particle penetration efficiency, P_eff, of the hollow tapered cylinder with or without the conical column can be calculated using the following equation:

$$P_{eff}=\frac{{{\displaystyle \int }}_{R_{ex0}}^{R_{ex1}}{C}_{ex}\left(r\right)·U_{ex}\left(r\right)·2\pi rdr}{{{\displaystyle \int }}_{R_{in0}}^{R_{in2}}{C}_{in}\left(r\right)·U_{in}\left(r\right)·2\pi rdr}$$

(8)

where $R_{in2}$ represents the radius of the hollow tapered cylinder at the inlet, $R_{ex1}$ is the radius of the hollow tapered cylinder at the outlet, $C_{in}$ is the inlet concentration, $C_{ex}$ is the concentration at different cross sections, $U_{in}$ is the inlet velocity, and $U_{ex}$ is the axial velocity at different cross sections.

Several meshes were tested in this study to obtain an appropriate number of grids to ensure optimal grid independent solutions. Figure 3 shows the variations in the penetration efficiency of the hollow tapered cylinder without conical column for different grid sizes of 21,000, 84,000, and 336,000 at the flow rate of 2 L/min. When the number of grids was increased from 21,000 to 84,000, the numerical penetration efficiency of the hollow tapered cylinder without the conical column for 5 nm particles changed from 83.3% to 84.9%. While the number of grids was further increased from 84,000 to 336,000, the numerical penetration efficiency only changed from 84.9% to 85.1%. Therefore, 84,000 grids were used in the simulation.

3. Results and Discussions

3.1. Concentration Distribution of Nanoparticles

This study first explored the distribution of nanoparticle concentrations of the hollow tapered cylinder with or without the conical column by calculating the flow field and convection diffusion equations. Figure 4 shows the streamlines in the hollow tapered cylinder without the conical column for the flow rate of 2 L/min. When the airflow enters the inlet of the hollow conical cylinder, the flow velocity gradually increases in the upstream due to the reduction in the inner tapered diameter, and it attains a higher value in the downstream of the hollow conical cylinder which remains constant to the exit. Figure 5 shows the concentration distribution within the hollow tapered cylinder in the axial direction for the particles with diameter of 3 nm at the flow rate of 2 L/min. For the hollow tapered cylinder without the conical column, one can find out that the concentration distribution is not homogenous throughout the hollow tapered cylinder, as shown in Figure 5a. The concentration in the vicinity of the tapered wall is less, while it is higher near the symmetry axis of the cylinder. Figure 5b shows the nanoparticle concentration distribution for the hollow tapered cylinder with the conical column. It can be observed that the concentration is lower near both the tapered wall and the column wall, while it is higher in their middle position. Compared to the case without the conical column, the nanoparticle concentration near the centerline of the hollow tapered cylinder with the conical column is lower in the straight positions.

3.2. Concentration Profile at Different Positions

Figure 6a 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 $R_p/R_{ex1}$ = 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 6b 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, $R_p/R_{ex1}=0$, are uniform and close to the inlet concentration, as shown in Figure 7a. When approaching the position of $R_p/R_{ex1}$ = 0.35, the concentration of smaller particles (3 nm and 5 nm) begins to decrease till the outer position of $R_p/R_{ex1}$ = 1.

In comparison, the concentration of larger particles (10 nm and 20 nm) begins to drop rapidly after approaching the outer position of $R_p/R_{ex1}=0.92$. Figure 7b 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 $R_p/R_{ex1}$ = 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 particles adjacent to the wall surface, so some particles cannot reach the outlet of the hollow tapered cylinder.

3.3. 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 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.

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.

3.4. Influence of Conical Column

Figure 10 represents the effect of the conical column on the concentration profile at the outlet of the hollow tapered cylinder for the particle sizes of 3 nm and 10 nm at different flow rates. For particles with a particle diameter of 3 nm, the conical column results in a lower concentration in the central region than in the middle region of the hollow tapered cylinder at the flow rates of 2 L/min and 5 L/min, as shown in Figure 10a. A phenomenon can be observed, which avoids the centralization of particles in the central region and presents an annular profile of the radial concentration at the outlet of the hollow tapered cylinder. This phenomenon is not obvious when the flow rate is reduced to 0.5 L/min, since the penetration efficiency of the hollow tapered cylinder is reduced. In contrast, Figure 10b shows that the influence of the conical column on the concentration profile in the central region of the hollow tapered cylinder is insignificant for 10 nm particles at flow rates of 2 L/min and 5 L/min. Furthermore, the annular profile of the radial concentration at the outlet of the hollow tapered cylinder becomes apparent when the flow rate is reduced to 0.5 L/min. Using the characteristics of particle concentration distribution, the difficulty of excessive or insufficient nanoparticle concentration at the outlet of the hollow conical cylinder can be solved. For example, when the nanoparticle concentration at the outlet is excessively concentrated in certain locations, it causes a bias in the downstream detector. At this point, the concentration distribution can be adjusted by the central column and the flow rate to homogenize the particle distribution at the outlet. Conversely, as the concentration of particles at the outlet is insufficient at certain locations, resulting in insufficient collection for quantitative analysis, it can be improved by collecting particles at specific locations in this way.

4. Conclusions

This work has identified and assessed the effect of the conical columns within the hollow tapered cylinder on the nanoparticle penetration efficiency and its outlet concentration profile for different particle sizes and flow rates. This has been carried out by calculating both the flow field and convection diffusion equations for the distribution of nanoparticle concentrations of the hollow tapered cylinder with or without the conical column. A good agreement is found between the current numerical results and experimental data in the literature. It is observed that the conical column avoids the particle centralization in the central region and the radial concentration at the outlet presents an annular profile within the hollow tapered cylinder for 3 nm and 5 nm nanoparticles with a flow rate of 2.0 L/min. This provides an important performance feature of the hollow tapered cylinder for the robust assessment, attribution, and quantification of nanoparticles produced by various activities. This work also shows that the particle penetration efficiency increases with the particle size and the flow rate of the hollow conical cylinder. These results, albeit limited in applicability by the small geometry calculated in this study, have important practical implications for the collecting the nanoparticles of the conical funnels. Furthermore, the annular profile of the radial concentration of 10 nm nanoparticles at the outlet of the hollow conical cylinder with conical columns becomes apparent when the flow rate was reduced to 0.5 L/min. Finally, this paper has provided useful conditions for the transport of nanoparticles in a hollow tapered cylinder as operational references.