Airflow and Deposition of Nano-Particles in a Human Nasal Cavity

A 3D computational model was developed to study the flow and the transport and deposition of nano-size particle in a realistic human nasal passage. The nasal cavity was constructed from a series of MRI images of coronal sections of a nose of a live human subject. For several breathing rates associated with low or moderate activities, the steady state flows in the nasal passage were simulated numerically. The airflow simulation results were compared with the available experimental data for the nasal passage. Despite the anatomical differences of the human subjects used in the experiments and computer model, the simulation results were in qualitative agreement with the experimental data.

Deposition and transport of ultrafine particles (1 to 100 nm) in the nasal cavity for different breathing rates were also simulated using an Eulerian-Lagrangian approach. The simulation results for the nasal capture efficiency were found to be in reasonable agreement with the available experimental data for a number of human subjects given typical anatomical differences. The computational results for the nasal capture efficiency for nano-particles and various breathing rates in the laminar regime were found to correlate well with the ratio of particle diffusivity to the breathing rate especially for the particles smaller than 20 nm. Based on the simulated results, a semi-empirical equation for the capture efficiency of the nasal passage for nano-size particles was fitted in terms of Peclet number.

INTRODUCTION

During the breathing process, an important function of the nasal passage is to capture particles from the air stream. The nose, thus, provides the first line of protection against penetration of particulate contaminants into the lower respiratory tract. Therefore, from the point of view of toxicology and human health, understanding the nature of airflow, and particle transport in the human nasal passage is important.

Figure 1a shows a schematic of the nasal cavity that extends from the nostril to the entrance of nasopharynx. The nostril is the external opening of the nasal passage through which air is inspired and exhaled. The funnel region following the nostril is the vestibule that extends to the nasal valve region. The passage has its minimum cross section area at the nasal valve region; and therefore, the axial component of airflow velocity is the highest in this region. The nasal valve region is connected to the main airway region that has a complex geometry and includes turbinate regions. At the end of main airway and the beginning of nasophyranx, both nasal cavities (left and right) are joined. Figure 1b shows a coronal section of the passage in the main airway region. The inferior, middle, and superior turbinates can be seen in this figure. Passages below the turbinates are referred to with similar names as the turbinates, (i.e., inferior airway, middle airway, and superior airway). The regions between turbinates and the lateral side of the passage corresponding to the neighboring turbinates are the inferior meatus, middle meatus, and superior meatus regions. The tiny slit in the septum side of the section is called the olfactory slit in which olfactory cells are located.

FIG. 1 (a) Schematics of a nasal cavity. (b) A coronal section in the main nose airway.

Flow and particle transport through human nasal passages have been studied experimentally and numerically by several authors. Swift and Proctor (1977) were the pioneers in quantitative measurements of flow in the nasal passages. They performed their measurements using a miniature Pitot tube on a model that was made from the nasal cast of a cadaver. They reported that the flow was laminar at a breathing rate of 15 L/min (125 mL/s from each side), but turbulence was detected downstream of the nasal valve for a breathing rate of 25 L/min (208 mL/s from each side). They also performed flow visualization that showed the major part of the flow passes through middle airway in the main airway region.

Schreck et al. (1993) reported measurements using a hot wire device on 3-times magnified plastic model of a nose based on MRI images. The experimental study indicated that the main part of the flow passes through the central part of the nasal passage and smaller portions passes through the meatuses and the olfactory region. They also reported the presence of a relatively large vortex and a smaller one in the upper and the floor of the region posterior to the nasal valve.

Hahn et al. (1993) created a 20 times magnified model and measured the flow velocity at several points in the five coronal planes throughout the model with a hot film anemometer. They reported that the flow was laminar up to a breathing rate of 24 L/min. It was also found from their experiments that approximately half of the inspired airflow passed through the middle and inferior airways and a small fraction of the flow passed through olfactory slit.

Kelly et al. (2000) used Particle Image Velocimetry (PIV) on a model fabricated from CT scans. They reported that the flow was laminar for a breathing rate of 15 L/min. Their results also showed that the inspired air attains the highest velocity in the nasal valve and the inferior turbinate regions. Furthermore, a significant part of the flow passes through inferior airway while a small portion of the flow passes through olfactory slit and meatuses.

The common finding in the earlier experimental studies is that the flow regime inside of human nasal cavity for low to moderate breathing rates is laminar. In addition, a large portion of inspired airflow passes through the middle and inferior airways while a smaller fraction passes through olfactory and meatuses regions.

Several computer-modeling studies of the flow field inside the human nasal passages have been reported in the literature. Keyhani et al. (1995) simulated the steady flow in the nose by the finite element method using FIDAP™ commercial software. Their 3D computational model for one of the nasal cavities from the nostril to the beginning of nasopharynx was constructed from the CAT scan sections of a human subject. Their simulations were performed for flow rates of 7.5 L/min and 12 L/min that were half of the breathing rate for one side of the nose. The flow regime was assumed to be laminar and both inhalation and exhalation processes were simulated. Their computational results showed very little changes in the flow patterns for these different breathing rates. Keyhani et al. (1995) also showed that a large part of the airflow passed through the middle and inferior airways. The existence of secondary flows was also reported but it was also mentioned that the secondary flow velocities were small compared with the axial velocities.

A detailed numerical investigation of the airflow inside the human nose was performed by Subramanian et al. (1998). They constructed a 3D computational model that included both nasal passages from the nostril to the pharynx using coronal cross sections obtained from MRI of a human subject. They used FIDAP™ code, which uses a finite element solver. The flow inside the passage was assumed steady and laminar for the two simulated breathing rates of 15 L/min and 26 L/min. The computed results did not show a significant difference between the flow patterns for the cases. The results of the study also showed that the major part of the inspired flow passed through the middle and the inferior airways. Finally, it was found that the flow was streamlined in the main passage, but the flow was rather complex in vestibule and nasopharynx where some vortices were generated.

Experimental study of deposition of spherical particles in the human nasal passage has been the subject of several studies. Cheng et al. (1988), (1993), Swift et al. (1992), (1996), and Strong and Swift (1987) measured the deposition of ultrafine particles in human nasal passage for different flow rates. Cheng et al. (1993) also suggested an empirical relation for capture efficiency of the human nasal system based on particle molecular diffusivity and breathing rate for particle sizes in a range between 1 to 200 nm. They assumed that airflow was turbulent for all breathing rates. Swift and Strong (1996) measured the deposition of nano-particles in nasal passages of three human volunteers. Using the previously available data and a curve fitting based on an assumption of turbulent flow, they suggested another empirical relationship for nasal deposition. In vivo measurements of the deposition of ultrafine particles in the human nasal passages were also conducted by Cheng et al. (1996). In their work, they also developed an improved empirical equation for the nasal capture efficiency in terms of Reynolds number (Re), Schmidt number (Sc), and the ratio of average cross sectional area to the surface area of the nasal passage. Cheng (2003) suggested a revised formula of nasal deposition for both inertial and diffusional regimes based on in vivo and replica measurements.

Arguing that in different parts of the nasal passage the airflow can be in laminar, turbulent or transition regimes, Martonen et al. (2003) suggested a two-term formula for nasal deposition. Recently, Kelly et al. (2004) measured the deposition of ultrafine particles in nasal airway replicas produced by stereo-lithography with different accuracy and consequently different surface quality. They reported that the surface quality and small anatomical differences do not significantly affect the nasal capture efficiency.

Computer modeling studies of transport and deposition of ultrafine particles in human nasal cavities have been reported by a number of researchers. Yu et al. (1998) simulated the flow and transport of ultrafine spherical particles in the human upper airways. They assumed a laminar flow regime and used the Eularian-Eularian approach for particle transport analysis using CFX commercial software. They reported the capture efficiency of different organs in the upper respiratory system for two different particle sizes. Scherer et al. (1994) simulated uptake patterns of inhaled pollutants into the nasal cavity in a laminar regime with the Eularian-Eularian approach. For the parts of the cavity that are covered by the mucus layer, the effects of solubility and diffusivity on the mass transfer were considered through a first order chemical reaction model. The simulations were performed for particles with Sc ≈1, which corresponds to spherical particles smaller than 1 nm in the air. The focus of their study was on the effect of solubility on the nasal capture efficiency.

In this study, a computer model of airflow and nano-particle transport and deposition in the nasal passage was developed and applied. The computational domain was constructed using the MRI sections of the nose of a human subject. For several breathing rates, the airflows, transport, and deposition of ultrafine particles in size range of 1 to 100 nm were evaluated. An Eulerian-Lagrangian approach was used in the simulations. The results for the nasal capture efficiency were compared with the available experimental data for the nasal cavities with different anatomies and good agreement was found. The computational data showed that in the laminar flow regime, the capture efficiency of the nasal system for small size particles is a function of D/Q which D represents particle molecular diffusivity and Q is the breathing rate. An improved fitted model for the nasal capture efficiency of nano-particles based on Peclet number, Pe = Q/DL_s, (with L_s being a characteristic length) was derived.

COMPUTATIONAL MODEL OF THE NASAL CAVITY

The first step for simulation of flow and particle transport in the nasal cavity is the creation of an accurate 3D model of the airway having a complex geometry. Here, the coronal cross sections of the left nasal cavity obtained by MRI of an anonymous adult male human subject (Figure 2) were used to construct a smooth airway passage. The computational domain was generated with the use of Gambit™ code. To construct the passage, first, coronal sections were imported into Gambit™. Then a number of key points were identified on each coronal section. These points divided each coronal section perimeter into a number of edges with roughly uniform features between the neighboring sections. Smooth curves were then created to connect the points on all coronal sections. Then surfaces were formed that passed through each two consecutive smooth curves and the coronal section edges between the two curves. The surfaces were then united, and in the final step, the surfaces were stitched to create the computational volume of the nasal passage. Figure 3 shows the resulting 3D passage extending from the nostril to the beginning of nasopharynx region.

FIG. 2 2D coronal cross sections of a nasal cavity of an adult male subject.

FIG. 3 The constructed 3D nasal cavity for computational analysis.

Since the coronal sections in the main airway have complex geometries, proper selection of the key points are critical to creation of an appropriate nasal passage. Creation of the 3D passage from the coronal sections was a rather delicate and time-consuming task. The complexity of the reconstruction process has been also noted by the other researchers who used different methods (e.g., Kimbell 2001).

GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

As noted before, the experimental studies of Swift and Proctor (1977), Hahn et al. (1993), and Kelly et al. (2000) showed that the flow in the nasal cavity was laminar for low to moderate activities corresponding to a breathing rate of 15 L/min or less. In this study, only the airflow and particle transport in the nasal cavity for low to moderate activities was of interest. Therefore, a laminar flow regime was assumed. For a breathing rate of 14 L/min, the Reynolds number based on the hydraulic diameter was about 490. Thus, the airflow in the nose was indeed in laminar flow regime.

The governing equations for airflow are: Continuity

Conservation of Momentum

In Equations (1) and (2), u is the velocity vector, p is the fluid pressure, ρ is the fluid density, and ν is the kinematic viscosity.

For boundary conditions, a no slip flow velocity on the passage surfaces was assumed. At the nostril, a uniform flow perpendicular to the inlet was specified. Assuming a uniform flow perpendicular to the nostril is an approximation. Keyhani et al. (1995) specified the velocity profile at the nostril based on the experimental data. However, they showed that for a given flow rate the downstream flow field is not significantly affected by the details of the velocity profile at the nostril. At the outlet (the beginning of nasopharynx) the outflow boundary condition was used. As noted in Fluent™ User's Manual (2003), for this boundary condition, the values of velocity and pressure at the boundary are extrapolated from within the domain at each iteration. Since the airflow was simulated only in the left nasal cavity, the airflow rate was considered half of the breathing rate.

The particle transport and deposition calculations were performed by a Lagrangian approach. Since particle concentration was low, a so-called one-way coupling was used. That is, the airflow transports the particles, but the effect of particles movements on the flow was neglected. In this approach, the airflow field was first simulated, and then the trajectories of individual particles were evaluated using particle equation of motion given by

In Equation (3), u_i ^p is the particle velocity, τ (τ = Sd²C_c/18ν) is the particle relaxation time, S is the ratio of particle density to fluid density, d is the particle diameter, C_c(C_c = 1 + 2λ/d−(1.257 + 0.4e^−.11d/2λ)) is the Cunningham slip correction factor, λ is the air mean free path, g_i is the acceleration gravity, and n_i (t) is the Brownian force per unit mass of particle.

The terms on the right hand side of Equation (3) represent, respectively, the drag force, the lift force, the gravitational force and the Brownian force per unit mass. For small size particles drag, and Brownian forces are dominant for determining particles trajectories. Note that since the Saffman lift force for ultrafine particles in the size range of interest in the present study is negligible, it was not included in Equation (3).

The Brownian force per unit of particle mass represented by n_i(t) in Equation (3) was modeled by a Gaussian white noise random process model. The corresponding spectral intensify of the noise, S_o, is given as

where T is the absolute temperature of air, k is the Boltzmann constant. The amplitude of the excitation at every time step is given by

G_i is selected from a population of zero-mean, unit variance independent Gaussian random numbers, and Δ t is the time step used in the simulation. The entire sample is then shifted with uniform distribution over an interval (0–Δ t). Additional details of modeling of Brownian motion may be found in the earlier work of Li and Ahmadi (1992).

In the present study, for ultrafine particle transport and deposition, the drag force as well as the Brownian effects were included in all of the cases studied. The gravity force, however, was included in some cases to investigate the effect of gravitational force on nano-particle deposition in the nasal cavity. In this study, by ultrafine particles, we mean particles in the size range of 1 to 100 nm.

For evaluating the particle deposition rates, the possibility of the particle rebounding from the passage surfaces was ignored and it was assumed that if the distance between the particle center and the surface was less than or equal to the particle radius the particle will attach to the surface. For nano-particle deposition in the nose, the assumption of sticking upon contact was expected to be quite reasonable.

NUMERICAL SCHEME AND COMPUTATIONAL MESH

For simulation of the airflow and particle transport and deposition in the nose, the FLUENT™ code, which is a finite volume based code, was used. To discretize the governing equations, the SIMPLE scheme with an upwind formulation was used and the resulting discretized equations were solved by a segregated approach in which the discretized governing equations for each quantity were solved separately and then the values for those quantities were used in a subsequent equation. The particle transport and deposition analysis was performed using the FLUENT™ code through integration of Equation (3) with Euler implicit method, which is a stable scheme regardless of the value of integration time step. It should be noted that FLUENT™ uses a variable Δ t to optimize the numerical integration for particle trajectory analysis. Additional details on the procedures for setting up the problem and features of the FLUENT™ code may be found in the FLUENT user's manual (2003).

Figure 4 shows the unstructured computational mesh that was produced by Gambit™ and includes 965,000 tetrahedral elements and 250,000 computational points. A finer computational mesh including 1,791,544 tetrahedral elements and 414,476 computational points was also used for the calculation. The effects of mesh refinement on the computational results are discussed in the next sections.

FIG. 4 Computational mesh for the nasal cavity.

The flow field calculations were conducted on a PC with 4 CPU of 3.4 GHz. The CPU time for simulating of the flow field was on average about 1 hour. The particle transport analysis was conducted after the flow simulation. The CPU time for each particle size calculation was about 5 minutes.

To test the accuracy of the numerical procedure, the flow and particle transport in the entrance region of a pipe under a laminar condition was simulated and the results were compared favorably with the analytical solutions. The details are presented in Appendix A.

FLOW FIELD SIMULATION

To study the airflow patterns inside of the nasal passage, a series of simulations for different breathing rates between 4 to 14 L/min were performed. The features of the simulated flow fields are discussed in this section.

Stream-traces in the nasal cavity for breathing rates of 4 L/min and 14 L/min are shown in Figure 5. The injection locations of the stream-traces in the nostril are shown in Figure 5a. Comparison between stream-traces in Figures 5b and 5c shows the general pattern of the airflow inside of the nasal cavity did not change when the breathing rate increases from 4 L/min to 14 L/min. This observation is in agreement with those reported earlier by Keyhani et al. (1995), Subramanian et al. (1998), and Kelly et al. (2000).

FIG. 5 Stream-traces for breathing rates of 4 and 14 L/min, (a) Injection points on nostril, (b) Stream-traces for 4 L/min, (c) Stream-traces for 14 L/min.

Most of the stream-traces are smoothly varying curves and no recirculation regions were observed for both breathing rates. The exception was a small region between the nostril and the tip of the nose in vestibule region. However, swirling flow was observed near the bottom of the nasal passage as it is depicted by stream-trace number 7 (Figures 5b and 5c). Keyhani et al. (1995) did not show any vortex formation in the nasal cavity. The existence of the recirculation areas in vestibule region was noted by Subramanian et al. (1998). The experimental visualization technique used by Kelly et al. (2000) also showed the formation of some vortices in certain regions of the nasal cavity. The differences in the observed patterns of the stream-traces could be due to anatomical differences between different cases in the experiments and simulations. The common finding in all the studies is that stream-traces have smooth variations in the main airway regions of the nose.

Figures 5b and 5c show that most of the stream traces passes through the middle airway, the inferior airway and the region between the middle and the inferior airways in the septum side of the section. To further study the distribution of airflow in the nose, the section was divided into several subsections as shown in Figure 6. The flow rate across each subsection was evaluated, and results are tabulated in Table 1. This table shows that more than 70% of the flow passes through the inferior airway, the middle airway, and the region in between on the septum side of the section. About 7% of the flow passes through the olfactory slit, 13% across the three meatuses areas, and around 10% across the superior airway. The simulations were repeated with the finer mesh. No major difference in the flow pattern was observed. The values for the ratio of the flow passing from different subsections of a coronal section in the mean airway region were nearly the same with the ones tabulated in Table 1 for the coarser mesh. The differences were in the range of 0.3% to 2% depending on the subsections.

FIG. 6 Division of the coronal section into different parts for sectional flow rate analysis.

TABLE 1 Flow rate from different parts of a coronal section in the main-airway region

Region	Area (cm^2)	Flow rate (L/min) Breathing rate = 4 L/min	Flow rate (L/min) Breathing rate = 14 L/min
A	9.19 (8.5%)	0.08 (4%)	0.27 (3.9%)
B	20.52 (19%)	0.32 (16.0%)	1.18 (16.9%)
C	18.59 (17.2%)	0.48 (24%)	1.59 (22.7%)
D	11.54 (10.7%)	0.27 (13.5%)	1.02 (14.6%)
E	5.05 (4.7%)	0.17 (8.5%)	0.53 (7.6%)
F	6.05 (5.6%)	0.12 (6%)	0.41 (5.9%)
G	5.73 (5.3%)	0.19 (9.5%)	0.61 (8.7%)
H	10.22 (9.5%)	0.07 (3.5%)	0.25 (3.6%)
I	9.13 (8.5%)	0.18 (9.0%)	0.62 (8.9%)
J	11.90 (11.0%)	0.14 (7.0%)	0.50 (7.1%)

These quantitative findings are in general agreement with the observations of the earlier works that most of the airflow passes through the inferior and the middle airways with a smaller fraction crossing the olfactory and meatuses. The numerical results of Keyhani et al. (1995) and Subramanian et al. (1998), and the experimental observation of Kelly et al. (2000) lead to slightly higher share of the flow rate in the inferior airway than the present model prediction. The reason for the small differences could be due to the anatomical differences of the nasal models used. A close comparison suggests that the inferior airway of the present model as shown in Figure 2 is noticeably narrower than the ones in the models used by Keyhani et al. (1995), Subramanian et al. (1998), and Kelly et al. (2000). With narrower passage, a higher resistance is expected and consequently the flow rate becomes smaller.

Figure 7 shows the contours of z-component of velocity (V_z) in three coronal sections located in vestibule, nasal valve and the main airway regions for the different breathing rates. It is seen that the contours patterns are nearly the same for the two breathing rates studied. The axial velocity has its minimum value in the vestibule where the flow is bending sharply into the nasal cavity and it reaches its maximum value in the nasal valve region that has the minimum cross sectional area. In the section located in the main airway region, the distribution of axial velocities is non-uniform. The peak of axial velocity occurs in the middle airway while the value of axial velocity becomes quite small in meatuses. These observations are in agreement with the previous works (e.g., Keyhani et al. 1995).

FIG. 7 Axial velocity contours in different coronal sections of the nasal cavity for different breathing rates, (a) Vestibule, (b) Nasal Valve, (c) Main Airway.

Figure 8 shows contours of the coronal (in plane) component of the velocity, () in three sections across the nasal passage. The contour patterns for both breathing rates are nearly the same. In the vestibule region where air enters upwardly into the nasal cavity, the coronal velocity has its maximum value. In the nasal valve and main airway regions, however, the value of coronal velocity decreases sharply indicating that the flow is mainly axial. Comparison of Figures 7 and 8 shows that in the nasal valve and the main airway regions, the value of axial velocity is much greater than the coronal velocity. In the vestibule, the coronal velocity has a greater value than the axial velocity. The magnitudes of the velocity components evaluated with the finer mesh were with in 3% of those calculated with the coarser mesh.

FIG. 8 Coronal velocity contours in different coronal sections of the nasal cavity for different breathing rates, (a) Vestibule, (b) Nasal Valve, (c) Main Airway.

Figure 9 shows variation of friction coefficient defined as

with flow Reynolds number,

where Δp is the average pressure drop between the nostril and nasopharynx, L is the passage length, d_h is the average hydrodynamic diameter of the coronal sections and u _m is the average flow velocity at the nostril. This figure shows that the friction coefficient decreases rapidly with Reynolds number. Inspection shows that a fitted expression given by

provides an excellent fit to the simulation data for the friction coefficient in the nose. It should be noted here that for curve fitting, the airflow fields in the nose for the breathing rates between 0.8 to 14 L/min were simulated and the corresponding pressure drop were used as the data set.

FIG. 9 Variation of nasal friction coefficient with airflow Reynolds number.

For small Re (Re < 0.5), Equation (8) leads to f ≅47.78/Re that implies that the pressure drop varies linearly with the breathing rate. As Reynolds number increases, the flow inertia becomes important and the pressure drop deviates from linear dependence on the breathing rate. It should be emphasized that the airflow is still laminar and the nonlinear variation is due to tortuosity of the flow path in the nasal passage. For the low breathing rate of 4 L/min, Re ≈ 49, and the nonlinear correction is about 85%. Thus, the nonlinear effect of flow inertia due to the tortuous passage on pressure drop in the human nasal airway is important for all realistic breathing rates.

The simulation results for the friction coefficient evaluated with the finer mesh were also shown in Figure 9. It is seen that the values for the friction factor calculated from the finer mesh are nearly the same as the ones for the coarser mesh.

PARTICLE TRANSPORT AND DEPOSITION

In this section, transport and deposition of nano-particles in the nasal airway are studied. Trajectories of ultrafine particles in the size range of 1 to 100 nm for different breathing rates are simulated using the Lagrangian discrete phase analysis of the FLUENT™ code.

To study the effect of particle size and breathing rate on the nasal capture efficiency, particles of a given size were injected uniformly at the nostril for each case for both computational grids described in the previous sections. Particle trajectories were simulated and the corresponding capture efficiencies were evaluated. Regional deposition of different size particles for different breathing rate was also studied, and the effects of particle density and the gravitational force on the capture efficiency were analyzed.

To simulate a uniform inlet concentration, particles were injected from the center of each computational cell at the nostril. The numbers of the injected particles for the coarser and the finer meshes in each run were 762 and 1162, respectively. For each size and breathing rate, the capture efficiency was calculated based on an average of five to ten separate simulations. The maximum differences between the five and ten simulations were less than 0.2% and 1%, respectively, for large and small particles.

Figure 10 shows the capture efficiencies of the nasal passage for 1 to 30 nm particles for different breathing rates for the coarser mesh. This figure shows that the capture efficiency decreases with increasing particle size. The capture efficiency also decreases as the breathing rate increases. These observations suggest that for particles smaller than 30 nm, diffusion was the main deposition mechanism.

FIG. 10 Simulated nasal passage capture efficiencies for different particle sizes and different breathing rates.

For breathing rates of 4 and 10 L/min, Figures 11a and 11b, respectively, compare the simulation results with the available experimental data reported by a number of authors for nasal passages with different anatomical features. Here the solid lines correspond to the model prediction with the finer mesh and the dashed lines are for the coarser mesh. While there are some scatters in the experimental data, these figures show that the predicted capture efficiencies are in good agreement with the experimental data, particularly, for particles smaller than 20 nm. For larger particles, the experimental data and the computer simulation lead to slightly decreasing or roughly constant capture efficiencies. The model prediction for particles larger than 20 nm is, however, somewhat higher than the average of the experimental data.

FIG. 11 Comparison between simulation and experimental data for nasal capture efficiency for different breathing rates, (a) 4 L/min, (b) 10 L/min.

It is conjectured that the small discrepancy is due to the finite size of the computational grid and the error introduced by the interpolation scheme for regions very close to the wall. That is, the flow velocity at the location of particle within a computational cell is evaluated by linear interpolation using the values at the nodal points of the cell. While the velocities are zero for nodal points attached to the wall, presence of small but finite coronal velocities toward the wall for nodes away from the wall leads to slight overestimation of the normal velocity toward the wall because of the errors associated with the linear interpolation. (Because of the continuity equation, the normal velocity has to approach zero proportional to the square of the distance from the wall.) As a result, the computer model leads to additional superfluous deposition. The discrepancy, however, can be reduced by using a finer mesh. Figures 11a and 11b show that the model predictions with the finer mesh improved the agreement with the experimental data. In addition to numerical errors due to the linear interpolation, the anatomical differences in the nasal passages used in different experiments could be another source of discrepancy.

Since the airflow in the nose is in laminar regime, one may speculate that the capture efficiency of the nasal cavity is a function of the same parameters and the same dimensionless groups as for a laminar duct flow. Such a relationship would be expected in the diffusional regime. For a developing laminar airflow in a pipe, Ingham (1984) and Martonen et al. (1996), respectively, suggested that the capture efficiency of small particles is proportional to D^0.66/Q^0.55 and D^0.66/Q^0.5. For fully developed flow in a pipe, however, the capture efficiency is proportional to (D/Q)^0.66. The simulation results for the nasal capture efficiencies for different particle sizes and various breathing rates are examined respectively, in terms of parameters (D/Q)^0.66, D ^0.66/Q^0.5, and D^0.66/Q^0.55. Although based on all three parameters, the data collapsed to nearly single curves, the data plotted versus (D/Q)^0.66 had shown the least amount of scatter as presented in Figure 12. This result suggests that, for laminar airflow condition with moderate to low breathing rates, the capture efficiency in the nasal passage correlate well with (D/Q), which is proportional to inverse of the Peclet number. The Peclet number is typically defined as Pe = U_o L _s /D, where U _o is the characteristic velocity and L_s is the characteristic length scale. For nose, an effective Peclet number is defined as,

Peclet number given by Equation (9) is expected to provide a more appropriate representation of the condition in the nose that can also be conveniently evaluated.

FIG. 12 Variations of nasal capture efficiency for different particle sizes and different breathing rates versus (D/Q)^0.66.

Figure 13 shows the simulated data for the finer mesh for the capture efficiency for different cases versus Peclet number as defined by Equation (9), where for L_s the length of the nasal passage was used. It is seen that the simulation data for a range of parameters collapse to a single curve. A fitted curve to data is given as,

Figure 13 shows that the fitted equation given by Equation (10) is in good agreement with the simulation results. As the Peclet number increases, which is associated with larger particle size, some scatters appear in the simulation data. This result suggests that for larger particles, diffusion is not the only mechanism for deposition. Therefore, in addition to the Peclet number, the capture efficiency could also depend on Reynolds number and possibly Stokes number.

FIG. 13 A fitted equation for capture efficiency (η = 100(1 − 0.88e ^{− 218Pe− 0.75} ) based Pe = Q/DL_s.

Experimental data shown in Figures 11a and 11b also indicates that for particles in the range of 50–100 nm, the capture efficiency remains constant or increases slightly. This implies that diffusion is not the only mechanism for deposition in this size range. That is, there must be some contributions from deposition by other mechanisms such as inertia impaction and interception for the capture efficiency not to decrease with size. It should be emphasized that Equation (10) was found for one human subject. For application of this fitted expression for estimating the capture efficiency of other human subjects, additional comparisons with computer simulations will be required.

As noted before, the length of the nasal passage was used as the characteristic length in Equation (9) that can be easily measured. Other length scales such as average hydraulic diameter of the coronal sections, or the ratio of the nasal surface area to the average hydraulic diameter are also possible. It is conjectured that these various length scales are proportional; therefore, the form of the empirical expression given by Equation (10) remains the same irrespective of the choice for the characteristic length, and only the numerical value of the coefficient of Pe in the exponent will change.

Figures 14a, 14b, 14c, 14d show the ratio of the regional capture efficiency to the total capture efficiency in the vestibule, the nasal valve and the main airway for two different breathing rates and four different particle sizes, calculated from the coarser mesh. For 1 nm particles and a breathing rate of 4 L/min, more than 50% of the captured particles are deposited in the vestibule and the nasal valve which constitute about 17% of the total interior surface area of the nasal passage boundary for this case. As the flow rate increases to 14 L/min, the fraction of the deposited 1 nm particles in vestibule and the nasal valve decreases. However, for 5, 10, and 100 nm particles, the capture efficiency in the vestibule and the nasal valve increases as the breathing rate increases from 4 L/min to 14 L/min. Since on average about 40% of the captured particles are deposited in the vestibule and the nasal valve that constitute only a small fraction of the total capturing area, it is concluded that the vestibule and the nasal valve regions are highly efficient areas in the nasal cavity for capturing the smallest sized particles. No major differences in the trend of regional deposition were observed when the finer mesh was used for the calculation. The maximum differences observed were less than 4%.

FIG. 14 Regional capture efficiencies for different particles sizes, (a) 1 nm, (b) 5 nm, (c) 10 nm, (d) 100 nm.

To examine the effect of particle density and gravitational sedimentation on the capture efficiency, a series of simulations were performed for different particle densities in a range of 775 to 3100 kg/m³ with and without the presence of gravitational force. The simulation results showed that the particle density and gravitational force did not have a noticeable effect on the nose capture efficiency for particles smaller than 100 nm. The graphical results, however, are not presented due to space limitation.

CONCLUSIONS

Using a series of MRI images of coronal sections of a nose of a human subject a smooth 3D computational model for analysis of the nasal passage was constructed and the flow and particle transport in the model were simulated. Considering the anatomical differences among individuals, the simulated results for the flow and the particle transport were in reasonable agreement with the available experimental data.

Based on the presented results, the following conclusions may be drawn:

• Changing the breathing rate (in a range from 4–14 L/min) does not change the main features of the flow in the nasal passage.

• Most of the inspired flow passes through the middle, the inferior airways, and the region in between; only a small portion of the flow passes through olfactory region and meatuses.

• For smallest range of particles (1–30 nm), diffusion is the dominant deposition mechanism. For larger particles, both the experimental and numerical results show a deviation from diffusion as the only mechanism for deposition.

• Mesh refinement has important effect in evaluating of accurate capture efficiency of the nasal passage for larger particles.

• The nasal capture efficiency for nano-particles (less than 20 nm) in a laminar flow is proportional to D/Q.

• Per unit area, the vestibule and the nasal valve regions are more efficient in capturing suspended particles.

• For 1–100 nm particles, the variation of density and gravitational force do not affect the nasal capture efficiency.

The results were for laminar flow regime in the nose. For higher breathing rates, the flow in the nasal passage may become turbulent. In this case, the effect of turbulence dispersion on particle dispersion needs to be included in the analysis. This additional effort is left for a future study.

APPENDIX A: PARTICLE DEPOSITION IN PIPES—COMPARISON BETWEEN SIMULATION RESULTS AND ANALYTICAL SOLUTIONS

To test the accuracy of the numerical procedure, the capture efficiency of the entrance region of a pipe with a diameter of 1 cm and a length of 1 cm under laminar regime was analyzed and the results are presented in this Appendix. For the flow simulation, 480,000 elements and 497,000 nodes were used and 7912 particles were injected uniformly from the inlet of the pipe.

Earlier, analytical expressions for the capture efficiency of the pipe entrance region were developed by Ingham (1991) and Martonen et al. (1996). These, respectively, are given as

and

where

Here, L_pipe is the pipe length, U_inlet is the inlet velocity and R is the pipe radius.

Cohen and Asgharian (1990) developed an empirical expression for the capture efficiency given as

that is applicable to particles larger than 10 nm.

Figure A1 compares the computed capture efficiency with the analytical formulas for a case in which the flow velocity at the pipe inlet was 1 m/s. While there are some discrepancies, it is seen that the computed results are in general agreement with the analytical solutions. The presence of some discrepancies between the capture efficiency computed by the FLUENT and CFX codes using an Eulerian-Lagrangian method and the analytical expressions was also reported by Snyder and Robinson (2005). Figure A1 shows that the absolute value of the difference between the simulation capture efficiencies and the analytical models is quite small. Therefore, if the pipe length is not very long, the absolute error would remain quite small. Since the human nasal is relatively short (between 10–20 cm), then using the Lagrangian approach of the FLUENT code for calculating the capture efficiency is reasonable.

FIG. A1 Comparison of the computed capture efficiency with the analytical and empirical equations.

The support of the National Institute for Occupational Safety and Health (NIOSH) through Grant R01 OH003900 is gratefully acknowledged.