Unsteady particle tracking of micro-particle deposition in the human nasal cavity under cyclic inspiratory flow

Author’s Accepted Manuscript Unsteady Particle Tracking of Micro-particle Deposition in the Human Nasal Cavity under Cyclic Inspiratory Flow Hojat Bahmanzadeh, Omid Abouali, Goodarz Ahmadi www.elsevier.com/locate/jaerosci

PII: DOI: Reference:

S0021-8502(16)30244-0 http://dx.doi.org/10.1016/j.jaerosci.2016.07.010 AS5025

To appear in: Journal of Aerosol Science Received date: 29 June 2015 Revised date: 16 June 2016 Accepted date: 6 July 2016 Cite this article as: Hojat Bahmanzadeh, Omid Abouali and Goodarz Ahmadi, Unsteady Particle Tracking of Micro-particle Deposition in the Human Nasal Cavity under Cyclic Inspiratory Flow, Journal of Aerosol Science, http://dx.doi.org/10.1016/j.jaerosci.2016.07.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Unsteady Particle Tracking of Micro-particle Deposition in the Human Nasal Cavity under Cyclic Inspiratory Flow Hojat Bahmanzadeha, Omid Aboualia, Goodarz Ahmadib a b

School of Mechanical Engineering, Shiraz University, Shiraz, Iran

Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY, USA.

Mailing address:

Dr. Abouali School of Mechanical Engineering Shiraz University Shiraz, Iran

E-mail:

[email protected]

Tel:

+98-711-613 3034

Fax:

+98-711-647 3511

Abstract In this study, a fully unsteady computational fluid dynamics (CFD) model was used and the transient airflow properties during the entire breathing cycle, including inhalation and exhalation in a human nasal cavity, were evaluated. Unsteady particle tracking was performed to find the particle motion and deposition in the nasal airway using a Lagrangian approach. In most of earlier computer simulations, the assumption of quasi-steady or steady airflows averaged over the inhalation cycle were typically used to reduce the computational cost. In the present work particular attention was given to assessing the accuracy of these assumptions and their consequence on particle deposition in a human nasal cavity with paranasal sinuses. The simulation results for the airflow field showed significant differences between the unsteady and quasi-steady cases at low breathing rates near the beginning and end of the inspiration cycle. For

1

breathing under a rest condition with a frequency of 0.25 Hz, the quasi-steady airflow assumption in the nasal cavity was found to be reasonable when the instantaneous Strouhal number was smaller than 0.2. The instantaneous deposition of micro-particles was also studied, and the effects of unsteady accelerating and decelerating flows on the deposition fraction were examined. The simulation results showed that the deposition fractions were affected by the accelerating and decelerating airflows during the inhalation phase. The simulated micro-particle depositions under the cyclic breathing condition were compared with those for steady breathing with an equivalent mean airflow rate. It was found that while the general trend was similar, the steady breathing simulation with the mean airflow rate during inhalation cannot accurately predict the total micro-particle deposition for cyclic breathing, particularly for 1-5 µm particles. Furthermore, the predicted regional depositions by the steady breathing were markedly different from those obtained under cyclic breathing.

Keywords: Human nasal cavity, CFD, micro-particle, unsteady flow, quasi-steady, Strouhal, Womersley

1. Introduction The nasal airway with its complex passages is the main entrance of the respiratory system. During low activity and resting conditions, the entire inhaled air passes through the nasal cavities. In fact, the nasal passages humidify and warm the inhaled air and remove large suspended particles before the air enters the lower airways. There have been numerous numerical studies on airflows in human nasal airways. One of the first such numerical studies was performed by Keyhani et al. (1995), where they constructed a 3D model of one side of a human nose from CT scan images. Subramaniam et al. (1998) investigated the airflow details in a model developed from MRI images of both sides of human nasal cavities and nasopharynx. Hörschler et al. (2003, 2006) simulated the steady airflow patterns in the nasal cavity for both inspiration and expiration breathing under the rest condition. Wen et al. (2008) identified flow

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features such as high velocities in the nasal valve region, high flow close to the septum walls, and vortex developments posterior to the nasal valve and olfactory regions. Xiong et al. (2008) studied a nasal model that included the paranasal sinuses and demonstrated that there is little air exchange between the paranasal sinus and the nasal cavity. Other numerical studies on nasal airflow under steady-state condition were reported also by Zachow et al. (2006), Garcia et al. (2010), Tan et al. (2012), Lintermann et al. (2013), and Kim et al. (2013). Deposition of particles in the human nasal cavity was also the subject of many studies. Zamankhan et al. (2006) simulated the nano-particle (1-100 nm) deposition for breathing rates of 4-14 L/min in a nasal cavity generated from CT scans. They used an Eulerian-Lagrangian approach in their simulation and suggested a semi-empirical equation for predicting the deposition efficiency of nano-particles on the nasal walls. Shanley et al. (2008) continued the study using the same nasal cavity model and studied the deposition of 1-10 µm particles. The effects of particle size, breathing rate and gravity under laminar flow regime were studied. They found that the deposition efficiency of larger particles is correlated with the particle relaxation time (or impaction parameter). Using different 3D models, Zhao et al. (2004) showed that anatomical alterations in the nasal valve and olfactory area influence the airflow and transport of odorants to the olfactory region. Shi et al. (2007) investigated the transport and deposition of inertial particles in the range of 1-50 µm. They showed that the deposition in the human nose increases when the impaction parameter (IP = d2Q) increases. They also observed that most particles are deposited in the nasal valve area. Wang et al. (2009) examined deposition of particles in the range of 1 nm to 50 µm for various breathing rates and compared the deposition in the left and right nasal cavities. They found that a major fraction of micron particles are deposited close to the nasal valve area and some particles are deposited on the septum wall in the turbinate area, while the deposition distribution of nano-particles was relatively uniform. Xi and Longest (2008) included the inertial and diffusive deposition mechanisms for fine particles in the size range of 100-1000 nm and used a drift flux model with a near-wall velocity correction. Schroeter et al. (2011) compared the deposition of inertial particles in a human nasal 3

model with different surface smoothness. Their results indicated that even a small difference in the surface quality has a noticeable effect on the micro-particle deposition rate. Farhadi Ghalati et al. (2012) studied the micro/nanoparticle deposition in a human upper airway from the nose to the end of the trachea and reported the regional deposition rate for different areas. Variations of particle deposition in the nasal passages after septoplasty, uncinectomy/MMA, and sphenoidotomy surgeries were studied, respectively, by Moghadas et al. (2011), Abouali et al. (2012), and Bahmanzadeh et al. (2015). Recently, Ghahremani et al. (2014) investigated the effect of airflow turbulence on particle deposition in the upper airway. Dastan et al. (2014) evaluated the deposition of fibrous particles in the nasal airway accounting for their translational, as well as, rotational motions. The facial effect and the effect of the surrounding environment have been included in the recent works of Naseri et al. (2014) and Shang et al. (2015). In reality, breathing is a transient process in which the flow rate in the nasal passage varies during the respiratory cycle. There have been only a limited number of unsteady computational efforts for examining the unsteadiness effects on respiratory airflow field (Naftali et al. 2005, Elad et al. 2006, Hörschler et al. 2010 and Bates et al. 2014). In order to reduce the computational cost, the majority of the earlier studies used a quasi-steady assumption for solving the airflow field. That is, it was assumed that the nasal flow was time-independent and the acceleration of airflow and unsteadiness effects during the breathing was neglected. Theoretical justification of the steady flow assumption is typically based on dimensionless analysis. For a specific range of the non-dimensional Womersley (Wo) and Strouhal (St) numbers, the differences between solutions of the unsteady and steady flow field become small and can be neglected. There are, however, few quantitative analyses of the quasi-steady assumption criteria in the literature for the flow in the human nasal airways. Isabey and Chang (1981) observed that a quasi-steady assumption can be made for Wo < 4 and St < 1 for flows in the central airways. Doorly et al. (2008) used these criteria for the identification of quasi-steady flow in the nasal airways. Hörschler et al. (2003) suggested only St≤1 as a criterion for the use 4

of the quasi-steady assumption. In a study performed by Shi et al. (2006), Wo ≤ 4.3 and St ≤ 0.2 were specified as the appropriate criteria. They showed that the quasi-steady assumption holds for St ≤ 0.2 although the Womersley number was greater than one. Wen et al. (2008) and Wang et al. (2009) used the average hydraulic diameter (DH) of 30 cross sections throughout the nasal cavity, instead of airway length (L), as a characteristic length for evaluating the Strouhal number, which led to St ≤ 0.01. Hörschler et al. (2010) showed that the differences between the unsteady and steady flow field solutions can be neglected only for Re ≥ 1500 when St = 0.791, where Re is the Reynolds number based on the hydraulic diameter of the throat. For simulating the transport and deposition of inhaled particles, the steady flow assumption was frequently used. Two types of steady simulations can be found in the literature. The first one is the computation of particle deposition rate at a time instant of breathing under a quasi-steady flow condition. In the second type, the equivalent steady flow assumption was intended as an approximation for the entire breathing period. It should be pointed out that the time variation of airflow during breathing can affect the motion of micro-particles in the nasal cavity, which is dominated by the inertial impaction. With a steady flow assumption, the unsteadiness effects on the particle motion are ignored. Thus, there are inherent differences for particle depositions under cyclic and steady flow conditions. Zhang et al. (2002) indicated that the deposition of micro-particles in a bronchial airway model is higher under cyclic flows compared with that predicted by the equivalent mean flow assumption. This trend was also observed by Grgic et al. (2006) in a human throat model that did not include the nasal region. The presented literature review shows that there are still uncertainties on the quasi-steady assumption for the nasal airflow. In particular, the effects of unsteady breathing on microparticle deposition in the nasal cavity have not been fully understood. Only Shi et al. (2006) performed a study for transient nano-particle deposition in the nasal cavity. In the present study the accuracy of quasi-steady flow simulation in the human nasal cavity is first assessed. Then, a criterion for the validity of the steady flow assumption based on the instantaneous Strouhal number is presented. Using the unsteady particle tracking approach, the instantaneous 5

deposition of micro-particles under cyclic breathing is presented, and the effects of unsteady accelerating and decelerating flows on micro-particle deposition are investigated. In addition, comparisons of the total and regional depositions for cyclic breathing and the equivalent mean flow breathing are presented.

2. Methods In the following sections, details of the geometry construction, breathing pattern, governing equations for airflow and particle transport, boundary conditions, and numerical method are described.

2.1 Computational model of the nasal cavity Computed tomography (CT) scan images of an adult male were used to construct a 3D model of the left nasal cavity including the paranasal sinuses. The images consisted of coronal, axial, and sagittal cross sections with a spatial resolution of 512×512 pixels spaced 0.625 mm apart. The boundaries between the airway mucosa and air in the nasal passage were identified in each CT scan’s slice (under the supervision of a specialist) by using a threshold based on gray scale values using commercial software. In particular, an appropriate range of threshold values was selected to isolate all air-filled spaces in the nasal airway as well as the paranasal sinuses. By grouping together similar gray values, the image data was segmented, and finally a 3D model was generated. The Laplacian algorithm was used for smoothing the surface. Fig. 1(a) shows the view of the constructed model including the description of different parts. The maxillary, frontal and sphenoid sinuses were included to represent a detailed nasal airway model.

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(b)

(a) Vestibule Main airway- part 1 Main airway- part 2 Sphenoid sinus Maxillary sinus Frontal sinus

prismatic layers

Nasal valve area

(c)

tetrahedral cells

Fig. 1(a) Constructed model of nasal cavity with different sinuses, (b) one coronal cross section of generated hybrid mesh, and (c) location of successive coronal cross sections. The ANSYS ICEM CFD meshing software was used for the generation of the computational grid. An unstructured grid, with a hybrid mesh containing tetrahedral elements in the nasal cavity core with four prismatic boundary layer grids along the walls, was used. The thickness of the first prismatic layer and the total thickness of four layers (growing exponentially) were, respectively, 0.05 mm and 0.27 mm. The aspect ratio for almost all cell elements was above 0.2, and the worst cell element had equiangle skewness of about 0.8. The total number of the

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tetrahedral and prism elements in the final computational grid was approximately 2,400,000. Grid sensitivity was performed and several meshes were examined (see Appendix for grid study). The selected grid leads to results independent of the further refinement of the mesh. In Fig. 1(b), a cross section of the hybrid mesh is shown. ANSYS-Fluent14.0 was used to simulate the airflow in the nasal cavity, the associated particle transport, and the deposition.

2.2 Breathing pattern In this study, the rest breathing condition is simulated using both cyclic and quasi-steady state assumption. For cyclic breathing conditions, a sinusoidal pattern as shown in Fig. 2 is assumed for generating the unsteady airflow. It should be pointed out that the real breathing profile is not quite sinusoidal, as typically the resting nasal airflow has a longer inspiration period compared to the expiration.

However, the simplified sinusoidal waveform provides a

reasonable approximation. That is,

Q=Qpeaksin(2π f t)

(1)

where Q is the time-dependent flow rate, Qpeak is the peak airflow rate which represents the maximum breathing intensity, and f is the frequency of breathing. The total duration for one cycle of breathing is about 4 seconds, in which the first 2 seconds represent the inhalation period and the last 2 seconds correspond to the exhalation phase. For some test cases the simulation continued for three breathing cycles, and it was found that the results for the second and third cycles were quite similar. Hence for the subsequent simulations, two breathing cycles were evaluated, and the results for the second cycle were used for analysis to reduce the effect of initial conditions. The mean flow rate ( Q ) during the inhalation phase is defined as t=2s

Q=

t=0

Qpeak sin(2π f t) dt

(2)

2

Peak flow rates are assumed to be 7.5 and 10 L/min, and typically a breathing frequency of 0.25 Hz is considered. These two cyclic flow patterns, respectively, produce average flow rates

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of 4.77 and 6.37 L/min during the inhalation phase of breathing (see Fig. 2). These values are the equivalent mean airflow rates that are used for steady flow simulations.

Fig. 2 Sinusoidal time variation of the breathing patterns with their equivalent mean flow rates for the inhalation phase.

2.3 Governing equations For the airflow in the nasal airways, the theoretical critical Reynolds number for the transition from laminar flow to turbulent regime is not known. However, the experimental study of Hahn et al. (1993) showed that for flow rates less than 12 L/min in a single nasal passage, the airflow is laminar. In the present work, simulations of airflow in the nasal cavity for both cyclic and steady flow conditions are performed. The maximum breathing rates of 10 L/min or smaller, for which the airflow is in laminar regime, are used in the analysis. Since the volume fraction of particles is very small, a one-way coupling assumption is used for the particles and the airflow. That is, the airflow transports the particles, but the effects of the particles on the flow are neglected. The governing equations for the airflow are continuity and balance of momentum equations. For unsteady and incompressible laminar flows these are

9

 u  0

(3)

μ u 1  (u )u   P  2u ρ ρ t

(4)

In equations (3) and (4), u is the velocity vector, t is the time, P is the fluid pressure, ρ =1.225, kg/m3 is the constant fluid density, and μ = 1.7894×10‐5, kg/m·s is the dynamic viscosity. Trajectories of particles can be tracked by integrating a force balance equation consisting of drag, inertia and gravity force acting on the particle. Accordingly, p 3μCD Rep  p du  u u   g    2 dt  4ρp d p 

(5)

In equation (5), u p is the particle velocity vector, d p is the particle diameter, ρ p is the particle p

density, g= -gkˆ is the acceleration gravity, and Rep = ρ u j -u dp /μ is the particle Reynolds j number. The drag coefficient C is given by D

CD 

24 (1  0.15 Rep0.687 ) Rep Cslip

(6)

where

Cslip  1 

dp 2λ [1.257  0.4exp(1.1 )] dp 2λ

(7)

is the Cunningham slip correction factor. In equation (7),

λ is the air mean free path.

The method for solving these equations is described in detail in section 2.5.

2.4 Boundary conditions The airway walls are assumed to be rigid, and the no-slip (zero velocity) boundary condition is used. From the physiological point of view, breathing occurs due to a pressure difference between lung and atmosphere. Hence, it is appropriate to impose a pressure condition to the nostril inlet and the nasopharynx outlet. At the nostril, the pressure inlet boundary condition was applied with zero gauge ambient pressure (Dirichlet condition); however, the pressure at the

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nasopharynx outlet is unknown. Thus, it is more convenient to use the breathing flow rate boundary condition. Therefore, a mass flow rate was prescribed at the nasopharynx outlet (Dirichlet condition) to drive the flow into the nasal cavity. The mass flow rate was constant and periodic, respectively, for steady and cyclic breathing simulation. For the cyclic inhalation, the mass flow rate was considered as a sinusoidal function of time as given by Equation (1) and also shown in Fig. 2. That is, in the first half-period of the breathing cycle, air flows from the nostril toward the nasopharynx, simulating the inhalation phase; in the second half-period, the airflow is reversed and moves from the nasopharynx toward the nostril, which simulates the exhalation phase. The particles with the same diameter are distributed uniformly at the inlet surface at the nostril. Several runs for different particle diameters are performed. The particle-to-air density ratio 816.3 (for water droplets) is used in these simulations. The initial velocities of particles are assumed to be the same as the averaged airflow velocity at the nostril. Deposition of particles on airway walls occurs when the distance between the particle center and the adjacent wall surface is less than or equal to the particle radius. The probability for particle bouncing from the airway surfaces, which is negligibly small, is neglected. Note that during the exhalation phase, the nostril is an airflow outlet, and no particle enters the nose in this period.

2.5 Numerical method The fluid governing equations were solved using the SIMPLE algorithm (Patankar, 1980). The convective and diffusive terms were discretized, respectively, by the upwind and the central difference schemes. For unsteady calculations, the sensitivity of the solution to the time step was studied. Time steps of 0.1 s, 0.05 s and 0.02 s were used for the numerical integration for several test cases. It was found that identical results for time steps less than 0.05 s were obtained; therefore, 0.05 s was selected as the flow time step in the present study. The particle transport and deposition calculations were evaluated by the Lagrangian trajectory analysis approach. The airflow field was first simulated in each flow time step, and then the

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trajectories of particles were determined by solving the particle equation of motion on that time step. In the earlier computational studies for the steady or quasi-steady inhalation conditions, the injection of particles was typically performed only once. In the present study, in order to simulate the continuous entering of particles into the nasal airway under evolving airflow conditions, an unsteady particle tracking method was used. During the inhalation period, microparticles were injected uniformly at constant rate at the nostrils. The appropriate injection time step value was determined by carrying out a sensitivity analysis. A series of simulations with different injection time steps was performed. The decrease of time step between successive injections of particles at the nostril opening was continued until the particle deposition fraction (DF) in the nasal cavity became independent of time intervals between successive particle injections at the inlet. The injection time step of 0.1 second was selected based on this analysis. The number of injected particles at each time step was also selected such that the independence of the simulation results from the number of injected particles was within an accuracy of about 0.5%. At each injection time step, 2000 particles were injected and a total of 2000 × 19 = 38000 particles were introduced during the inhalation phase of the breathing cycle. The continuous injection of particles during the inhalation phase at the nostril was necessary for a realistic analysis of the transient particle transport and the deposition behavior during the breathing cycle. Micro-particles with aerodynamic diameters in the range of 1 to 30 μm were simulated to examine a wide range of nasal deposition efficiencies. The relaxation times for these particles are in the range of 3.1 ×10-6 – 2.8 ×10-3 s, and the corresponding Stokes numbers based on the maximum velocity at the nostril and nostril diameter are in the range of 5.2 ×10-4 – 0.47. All calculations were performed on a PC with a 3.4 GHz, Core (TM) i7 CPU, and 8 GB RAM. The simulations were run in parallel on eight processors. Typical run times for cyclic and steady inhalation conditions were, respectively, about 192 and 2 hours.

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3. Results 3.1 Investigation of quasi-steady flow assumption in the nasal cavity As illustrated in Fig. 3, in the inhalation phase there are two time instants that a flow rate less than the peak occurs, one in the accelerating phase and the other in the deceleration phase. To analyze the effects of unsteady accelerating and decelerating flows, the static pressure profiles averaged over the cross-sectional area across the nasal cavity as shown in Fig. 1 (c) are evaluated for both unsteady and steady state conditions for different instantaneous flow rates and the results are plotted in Fig. 4. For an airflow rate of 1.56 L/min which occurs at 0.1 s and 1.9 s, Fig. 4 (a) shows that the pressure variations in the nasal cavity differ markedly for the acceleration and deceleration phases of the same instantaneous breathing rate. The pressure profile for a steady inhalation of 1.56 L/min is roughly the average of the profiles for accelerating and decelerating conditions. In particular, at the beginning of inspiration (t = 0.1 s), the instantaneous pressure profile for the accelerating airflow is lower than that of the steady flow pressure, while towards the end of the inspiration (t = 1.9 s) the instantaneous pressure profile of the decelerating airflow is higher than the steady prediction. That is, the accelerating flow generates a larger pressure drop compared to that of the steady and/or decelerating flows. This trend is as expected because the accelerating flow needs more energy to increase the airflow velocity in the passage in addition to overcoming the frictional losses. For a volume flow rate of 7.07 L/min, which occurs at t = 0.5 s and t = 1.5 s, Fig. 4 (b) shows the pressure profiles for unsteady accelerating and decelerating flows, as well as the steady simulations. It is seen that for this relatively high breathing rate, the effects of unsteadiness are relatively small. A close examination of Fig. 4 (b) reveals the same general trends as those seen in Fig. 4 (a); however, the differences are small. That is, at higher volume flow rates the frictional losses increase, and the needed force for accelerating/decelerating the flow decreases as the rate of

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change of the volume flow rate near to the peak of the cyclic flow decreases; hence, the difference between the unsteady and steady simulations becomes quite small.

volume flow rate (L/min)

10 8 6 4 2 0 -2 0 t=0.1 s -4 -6 -8 -10

volume flow rate (L/min)

10 8 6 4 2 0 -2 0 -4 -6 -8 -10

(a)

1.56 L/min t=1.9 s 2

4

Time (s)

(b) 7.07 L/min

t=0.5 s

t=1.5 s

2

4

Time (s)

Fig. 3 Time instants corresponding to breathing rates of (a) 1.56 L/min and (b) 7.07 L/min.

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Fig. 4 Static pressure profile across the nasal cavity for (a) Q = 1.56 L/min and (b) Q = 7.07 L/min.

The results in Fig. 4 were for two volume flow rates of 1.56 and 7.07 L/min. More details on the role of airflow rate on the pressure profiles for unsteady and steady conditions are presented in Fig. 5. The non-dimensional pressure drop versus the volume flow rate for two breathing cycles with amplitudes of 7.5 and 10 L/min and average inhalation flow rates of, respectively, 4.77 and 6.37 L/min are shown in Fig.5. In this figure, variations of the non-dimensional pressure drops versus the instantaneous flow rate for the transient inspiratory acceleration, deceleration and also the steady-state condition are compared. For low flow rates corresponding to the beginning and the end of each inhalation period, there are considerable differences in the non-dimensional 15

pressure drops. That is, the pressure drop for the acceleration phase is markedly higher compared to that of the deceleration flow. The steady pressure drop is roughly about the average of the accelerating and decelerating phases of inhalation. This observation was also reported by Jiang and Zhao (2010) for a rat nasal cavity. As the airflow rate increases and gets close to the peak inhalation rate, the differences in the pressure drop reduce, and the transient accelerating and decelerating flows as well as the steady flow coincide. Therefore, the unsteady effects on the nasal flow for high flow rates near the peak inhalation are negligible, and the flow field can be estimated from a quasi-steady analysis. Hörschler et al. (2010) also reported that the total pressure loss for unsteady conditions roughly matches that of the steady state solutions at high mass fluxes. The deviation of the pressure drop among the three cases in the low flow rates indicates that the unsteady effect is important only in the beginning and the end of the inhalation period. This is due to high acceleration at the start and near the end of the inhalation. Furthermore, as the flow velocity increases, the frictional losses increase and overwhelm the unsteady acceleration and deceleration effects.

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Fig. 5 Dimensionless pressure drop across the nasal passage versus volume flow rate; (a) Qpeak = 7.5 L/min, (b) Qpeak = 10 L/min.

It is always of value to express the results in terms of non-dimensional numbers to remove case dependence effects and to provide a criterion for the validity of the quasi-steady assumption. Typically, the Womersley number, Wo, and Strouhal number, St, defined, respectively, as Wo=

St=

d 2π f 1/2 ( ) υ 2

(8)

2π f L U

(9)

are used to evaluate the unsteadiness in internal flows, where d is the hydraulic diameter of the nostril, f is the frequency of cyclic flow, υ is the kinematic viscosity of air, L is the length of the nasal cavity, and U is the average velocity of one cycle or instantaneous velocity at the nostril. The Womersley number is the ratio of the unsteady inertial force to the viscous force, while the Strouhal number is the ratio of unsteady inertial force (transient acceleration) to the convective inertial force (convective acceleration). For the present nasal passage, the Womersley number is 1.1 and the Strouhal numbers based on average velocity for cyclic breathing with peaks of 7.5 and 10 L/min are, respectively, 0.17 and 0.13. Jiang and Zhao (2010) suggested using a Strouhal number based on the instantaneous velocity. Fig.6 presents a comparison of the non-dimensional pressure drops versus the instantaneous Strouhal number (defined with instantaneous spatially averaged velocity at the nostril) for

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unsteady inspiratory flows and the steady flow. It is observed that the non-dimensional pressure drop generally increases as the Strouhal number increases (velocity decreases). The exception is the decelerating flows for which the non-dimensional pressure drop saturates and then decreases slightly for Strouhal numbers larger than 0.5 or 0.7 for the breathing rates studied. It is also seen that for both breathing intensities, as the instantaneous Strouhal number decreases (velocity increases) the non-dimensional pressure drops for accelerating, decelerating and steady flows converge to the same values. More specifically, for St ≤ 0.2, the steady flow analysis predicts the unsteady pressure drop with an error less than 5%. Hence, the quasi-steady approximation for the analysis of the pressure drop is acceptable for St ≤ 0.2 for the rest breathing conditions. This corresponds to about 65% and 75% of the cyclic breathing period, respectively, for peak breathing of 7.5 and 10 L/min.

40

acceleration phase- Qmax=7.5 L/min

35

deceleration phase- Qmax=7.5 L/min steady state- at each instant

∆P/0.5ρU2

30

acceleration phase- Qmax=10 L/min

25

deceleration phase- Qmax=10 L/min steady state- at each instant

20 15 10 5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Strouhal number

Fig. 6 Comparison of non-dimensional pressure drops versus instantaneous Strouhal number.

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3.2 Particle deposition In this section, we first validate the present model for particle deposition by comparing the results with earlier experimental and numerical studies. Next, the deposition rate of particles is presented and the effects of unsteadiness on particle deposition are discussed. Then, comparisons of both total and regional deposition fractions for cyclic breathings and their equivalent mean flow breathing rate are presented.

3.2.1 Validation Experimental data regarding the particle deposition in a nasal cavity under cyclic flow conditions are rather limited. Therefore, for particle model validation, the deposition rate of micro-particles in a human nasal model for constant inhalation rates of 5, 7.5 and 10 L/min (in one nasal passage) are compared with the experimental data of Kelly et al. (2004) and computer simulations of Schroeter et al. (2011) in Fig.7. In this figure, Viper and SLA represent the same nasal replicas, which are produced by two different manufacturing methods that led to different surface roughness. The surface roughness was decreasing from model A to Model B and model C in the work of Schroeter et al. (2011). The x-axis in Fig. 7 is the impaction parameter, dp2Q, which is shown in logarithmic scale. Here, Q is the airflow rate entering one nostril, and d is the particle diameter. It is seen that the deposition fraction follows a monotonically increasing Sshape trend with the impaction parameter. Fig.7 also shows that present predictions for the deposition fractions for different breathing rates versus impaction parameter collapse roughly to a single curve, which is also in reasonable agreement with the simulation results of Schroeter et al. (2011) for their Model C (smooth passage). The present simulation results are also in general agreement with the trend of the experimental data of Kelly et al. (2004) and the simulations of Schroeter et al. (2011) for Models A and B; however, there are some quantitative differences. In particular, the present simulation under-predicts the data of Kelly et al. (2004) for the impaction parameter more than 10000 µm2cm3/s. Simulation results of Schroeter et al. (2011) for Models 19

A and B are closer to the experimental data of Kelly et al. (2011). These observations show that the particle deposition in the nasal passage increases with the increase in surface roughness. The increase of particle deposition with surface roughness agrees with the earlier experimental results of Montgomery and Corn (1970) and the computer simulations of Fan and Ahmadi (1993) and Li and Ahmadi (1993) for duct flows. Therefore, it is conjectured that the differences between the present simulation results and the experimental data of Kelly et al. (2004) are due to the surface roughness in the nasal cavity replica used in the experiment.

present work- 5 L/min

100

present work- 7.5 L/min present work- 10 L/min kelly et al. 2004 (viper)- 10 L/min

Deposition fraction (%)

80

kelly et al. 2004 (viper)- 15 L/min kelly et al. 2004 (viper)- 20 L/min kelly et al. 2004 (SLA)- 10 L/min

60

kelly et al. 2004 (SLA)- 15 L/min kelly et al. 2004 (SLA)- 20 L/min Schroeter et al. 2011 (Model A)

40

Schroeter et al. 2011 (Model B) Schroeter et al. 2011 (Model C)

20

0 100

1000

10000

100000

1000000

dp2Q (µm2 cm3/s)

Fig. 7 Comparison of predicted deposition fraction versus impaction parameter with the earlier experimental data and numerical results of Kelly et al. (2004) and Schroeter et al. (2011).

20

3.2.2 Transient particle deposition The deposition rate of the particles at successive time intervals during the cyclic inhalation is shown in Fig. 8. Here particle diameters of 1, 10, 20, and 30 µm under cyclic breathing at the equivalent mean flow rate of 6.37 L/min are studied. The deposition rate at each time interval is defined as the number of particles deposited in the time interval normalized by the total number of particles inhaled throughout the entire inhalation period. As expected, it is seen that the particle deposition rate increases with the increase of particle diameter, while the 1 µm particles have the lowest deposition fraction that remains small for the entire range of time varying flow rates. The deposition rate for 10 µm particles increases somewhat compared to 1 µm particles and has two peaks. The first peak of 0.33% occurs for the time intervals of 0.4-0.5 s in the first half of the inhalation phase, and the second peak of 0.42% occurs at 1.6-1.7 s for the second half of the inhalation phase. The deposition fraction of 20 µm particles markedly increases and roughly follows the flow pattern. That is, the deposition rate increases sharply as the flow rate increases in the first half of the inhalation period and then decreases in the second half when the airflow decreases. The peak deposition fraction for 20 µm particles is about 4%, which occurs at the time interval of 0.9-1 s corresponding to the peak breathing flow rate. Fig. 8 also shows that at the time intervals of 0.2-0.3 s there is a sharp increase in the deposition fraction of 20 µm particles, while there is a secondary peak near the end of the inhalation period at 1.8-1.9 s. Fig. 8 also shows that the deposition rate of 30 µm particles is highest among the studied particle sizes. Furthermore, their peak deposition of 5.7% occurs earlier at the time interval of 0.4-0.5 s before the peak of the airflow rate. Time variation of the airflow rate during the cyclic breathing may change the relative importance of the forces acting on the particles including the particle inertia and gravity. For better understanding of transient particle deposition, it is worthwhile to specify the contribution of each force on the deposition rate over the inhalation time. For this purpose, a series of unsteady particle tracking in the absence of gravity is performed, and the corresponding 21

deposition rates are compared in Fig. 9 with the case that gravity is present for different particle sizes. In this figure, the case with no gravity shows the deposition rate caused only by the inertia force through the impaction process. In addition, the differences between the curves with and without gravity are due to the deposition rates caused by gravitational sedimentation. Fig. 9a shows that the deposition rate of 1µm particles does not change noticeably by the inclusion of the gravity. That is, these small particles deposit mainly by the inertia impaction mechanism. Fig. 9b, however, shows marked differences for 10 µm particles indicating that gravity plays an important role in the deposition of these larger particles. In fact, it appears that most of the 10 µm particles deposit by the gravitational sedimentation mechanism. In the absence of gravity, the peak of deposition occurs between 0.9-1 s when the flow rate is close to it maximum point. When gravity is present, high deposition occurs at the beginning and near the end of the inhalation period. According to Fig. 9c, the deposition rate of 20 µm particles is mainly due to inertia impaction; nevertheless, the gravity enhances deposition during the entire inhalation period. In fact, as will be shown later, 33.7% of the 20 µm particles are captured in the absence of gravity and 47.0% are deposited when the gravity is present. In the absence of gravity the deposition rate appears to be smoother; therefore, it may be concluded that the sharp increase and small peaks in the deposition fraction near the beginning and end of the inhalation phase is due to the gravitational force when the inertia effect is small. Fig. 9d shows that 70.4% of the 30 µm particles are deposited in the absence of gravity and 80.7% are deposited when gravity is present. That is, almost all the 30 µm particles are deposited even in the absence of gravity. Fig. 9d suggests that the influence of the gravity on the deposition of 30 µm particles is small, but this is misleading because the majority of particles are deposited and only a few are left to be affected by gravity. It is also seen that the peak deposition rate does not match the peak flow rate even for the case without gravity. That is because the 30 µm particles have sufficient inertia to deposit even before reaching the peak flow rate, so fewer particles remain suspended. 22

Fig. 8 Deposition rates of the particles at different time intervals for 6.37 L/min breathing rate.

Fig. 9 Comparison of deposition rates with and without gravity for a) 1 µm particles, b) 10 µm particles, c) 20 µm particles, and d) 30 µm particles during the inhalation phase of a cyclic breathing with equivalent mean flow rate of 6.37 L/min.

23

To better investigate the effect of unsteady accelerating and decelerating flows, Fig. 10 shows the percentage of the particles deposited during the acceleration phase (0-1 s) and the deceleration phase (1-2 s) under breathing rates of 4.77 L/min and 6.37 L/min. To study the effects of acceleration and deceleration on the distribution of deposited particles caused by the inertia, the effect of gravity is ignored in Figs. 10a and 10b. As shown in Fig. 10a for the flow rate of 4.77 L/min, 1µm particles deposit mostly in the deceleration phase. This is because these small particles with their low inertia follow the flow and the majority of them cannot deposit in the acceleration phase. During the deceleration phase their deposition fraction increases, and their concentration in the nasal cavity increases. For 10 µm particles with higher inertia, the amount of deposition for the accelerating and decelerating phase is roughly the same. For larger particles such as 20 µm and 30 µm with higher inertia, the percent of the particles deposited during the acceleration phase is larger. A reason for this is that the large particles inhaled during the acceleration phase gain sufficient speed and inertia such that they deviate from the flow streamlines and deposit in large numbers. During the deceleration phase there is a smaller amount of particles left to be captured. For an increased breathing rate of 6.37 L/min, Fig. 10b shows that a larger proportion of deposition occurs through the acceleration phase for all particles in the range of 1-30 µm. Therefore, it can be concluded that in the absence of the gravity effect, by an increase of the impaction parameter (IP), the percentage of the particles deposited in the acceleration phase increases. As shown before, the gravity increases the deposition of particles larger than 1µm. Comparing Fig. 10c, 10d and Fig. 10a, 10b indicates that the enhancement of deposition by the gravity alters the deposition proportion in the acceleration and deceleration phases. In actual fact, the gravitational sedimentation increases the deposition rate during the deceleration phase and in some cases leads to a higher percentage for deposition during the deceleration phase. This is perhaps due to the availability of more particles to sediment during the deceleration phase.

24

Thus, in summary, the present study shows that the deposition rates during the acceleration and deceleration phases are not the same.

Fig. 10 Comparison of the particles deposited during acceleration and deceleration phases with and without the gravity.

The deposition process continues during the entire inhalation period and a number of particles deposit at the low flow rates and others deposit at high flow rates. Fig. 11 compares the fraction of particles that are deposited in low and high flow rates. The low flow rates correspond to the first and last 0.5 seconds, and the high flow rate occurs in the time of 0.5-1.5 seconds of the inhalation period. Here the results are presented with and without gravity in order to show the effect of gravity in low and high flow rates. In the absence of gravity, Figs.11a and 11b show that more than two-thirds of depositions occur by the inertia impaction in the time interval 0.525

1.5 s under high flow rate conditions. Considering the effect of gravity, Figs.11c and 11d show that the percentage of deposition of particles in the time intervals of 0-0.5 and 1.5- 2 s under low flow rates increases. This result highlights the significance of gravitational sedimentation at low flow rates.

Fig. 11 Comparison of the particles deposited in the time intervals for low (0-0.5 and 1.5-2 s) and high (0.5-1.5 s) flow rates with and without the gravity.

Finally, the time variations of the cumulative deposition fraction (CDF) of 1 µm, 10 µm, 20 µm, and 30 µm particles for breathing rates of 4.77 and 6.37 L/min are presented in Fig. 12. The CDF is defined as a ratio of the number of deposited particles from the start of the inhalation phase up to a certain time to the total number of particles entering into the nasal cavity during the entire inhalation period. For a given breathing rate, the total deposition fraction of particles

26

of different sizes at any time during the inhalation can be identified clearly in this figure. It is seen that the total deposition fraction of particles increases sharply as the particle size increases. For example for a breathing rate of Q = 6.37 L/min when gravity is present, the deposition fraction of 1µm particles at the end of the inhalation time is about 1.4%, while 80.7% of 30 µm particles deposit. The deposition fraction of 20 µm particles is about 47% at the end of the inhalation period.

Fig. 12 The time variation of cumulative deposition fractions for different particle diameters for flow rates of 4.77 and 6.37 L/min.

3.2.3 Comparison of micro-particle deposition for cyclic breathing and equivalent steady inhalation conditions Total particle deposition Fig. 13 compares the predicted deposition fraction of different size particles for two cyclic breathing rates (unsteady particle tracking method) with those obtained for equivalent mean

27

airflow rates (steady tracking approach). For unsteady tracking, all particles are tracked simultaneously in a certain inhalation period, and the amount of deposited particles is evaluated. For steady tracking, particles are tracked one by one until all inhaled particles either deposit or exit the nasal passage. Fig. 13 shows that the absolute difference between the results for cyclic flow and equivalent steady flow increases with the increase of particle diameter. It is observed that the deposition fractions of particles smaller than 10 µm are overestimated by the steady flow analysis compared to those of the cyclic flow simulations. On the contrary, the deposition fraction of particles in the diameter range of 15 to 20 µm for cyclic flows are higher (up to 7 %) compared to those for the steady flow analysis for this breathing intensity. In addition, the deposition fraction of particles larger than 25 µm under cyclic flow also is less than those for the mean flow condition. For example, the difference between the deposition behavior for cyclic and mean flows is about 14% for 30 µm particles for Qeq = 6.37 L/min. A similar trend is also observed for particle deposition at a lower breathing intensity of Qeq = 4.77 L/min, but the differences are smaller. It should be pointed out that generally the absolute differences between the unsteady and steady simulations are not significant and the general trends of the results are the same. However, the situation is different when the relative differences are considered. Relative differences of DFs for equivalent mean flow approximation compared to the cyclic flow are listed in Table 1 and also plotted in Fig.14. It is seen that for both breathing rates, the relative error of the deposition fractions for finer particles is quite high. That is, using steady flow assumption leads to an error of about 33-64% for particles smaller than 5 µm. For larger particles (20 µm and over), the relative error is in the range of 8-18%.

28

94.6 69.6 76.6

60

Equivalent mean flow

47.0 39.6

Cyclic flow

3

5

7

10

16.6 14.8

5.6 6.1

1

3.5 4.5

2.6 3.4

20

1.7 2.6

40 1.4 2.2

Deposition fraction (%)

80

80.7

(a) 6.37 L/min

100

0 15

20

25

30

particle diameter (µm)

61.0 55.5

Cyclic flow Equivalent mean flow 35.2 28.8

60

2.6 3.8

1

3

5

7

10

14.7 14.6

6.1 7.6

1.7 2.8

20

3.7 5.2

40

1.3 2.1

Deposition fraction (%)

80

75.9 82.2

(b) 4.77 L/min

100

0 15

particle diameter (µm)

20

25

30

Fig. 13 Total particle deposition fractions under cyclic breathing and the equivalent mean flow breathing. (a) 6.37 L/min, (b) 4.77 L/min.

Table 1 Relative difference of deposition fractions predicted by the equivalent mean flow approximation compared to the cyclic breathing. 4.77 L/min

6.37 L/min

Deposition fraction (%) Particle size (μm) 1 3 5 7 10 15 20 25 30

Deposition fraction (%)

Cyclic flow

Equivalent mean flow

Relative difference (%)

Cyclic flow

Equivalent mean flow

Relative difference (%)

1.3 1.7 2.6 3.7 6.1 14.7 35.2 61.0 75.9

2.1 2.8 3.8 5.2 7.6 14.6 28.8 55.5 82.2

64.0 61.3 45.4 39.9 25.2 0.4 18.2 9.1 8.3

1.4 1.7 2.6 3.5 5.6 16.6 47.0 69.6 80.7

2.2 2.6 3.4 4.5 6.1 14.8 39.6 76.6 94.6

54.4 50.6 33.1 29.6 8.6 10.6 15.6 10.1 17.3

29

Relative difference (%)

70 60

4.77 L/min

50

6.37 L/min

40 30 20 10 0 0

5

10

15

20

25

30

Particle diameter (μm)

Fig. 14 Comparison of relative differences of deposition fraction as predicted by the equivalent mean flow approximation compared with the cyclic breathing.

Regional particle deposition As shown in the previous section, the total deposition fraction (DF) for cyclic breathing is different from that obtained by the steady flow approximation. In this section, the regional particle deposition in different parts of the nasal airway for cyclic breathing rates is additionally evaluated and the results are compared with those obtained by using the equivalent mean flow assumption. The predicted DF versus particle size in the range of 1-30 μm for different parts of the nasal passage are shown in Fig. 15. Consistent with the earlier definition, the regional DF is defined as the ratio of the number of particles deposited in a region to the total number of inhaled particles. It is seen that the trend of DF curves for cyclic and steady models are similar; however, the predicted values of the regional DFs for large particles by unsteady and steady analyses in all regions including vestibule and different parts of the main airway are quite different. These differences are also more pronounced compared to those of the total DF. In addition, since the regional DF is a function of the local passage configuration, particle size, and local flow field, the predictions for cyclic flow can be higher or lower than those predicted by the steady flow model. While the differences are typically larger for larger particles which have more inertia and higher deposition rate, similar to the total deposition trends, the relative errors

30

for regional DF of fine particles are quite large and in the range of 50-108% for particles smaller than 5 µm. It should also be emphasized that an extremely small fraction of particles if any enters the sinuses (maxillary, frontal, and sphenoid) under both cyclic and mean flow conditions. This is because the paranasal sinuses are poorly ventilated organs.

Fig. 15 Comparison of the regional deposition fraction for cyclic breathing patterns and equivalent mean flow conditions for different parts of the nasal airway; (a) Vestibule, (b) Main airway- Part 1, (c) Main airway- Part 2.

31

To further highlight the unsteady flow effects, the spatial deposition patterns for 20 μm particles under both cyclic and mean flows for a breathing rate of 6.37 L/min are presented in Fig.16. The red and blue small spheres shown on the surfaces represent particles that are deposited, respectively, under cyclic and mean flow conditions. It is seen that the particles deposit roughly in the same regions of the nasal airways for cyclic and steady flows. The major difference is due to the formation of five deposition hot spots for cyclic breathing. These hot spots are marked by circles in Fig. 16. The first hot spot is located at the junction of the vestibule and nasal valve region where about 1.18% of the 20 μm particles deposit under cyclic breathing. The second region is located at the medial wall (nasal septum) near the middle turbinate where about 8.11% of particles are deposited. The third region is in the frontal area of the turbinates with a DF of 4.21%. Finally, the fourth and fifth sites are again on the nasal septum downstream of the main airway- part 2, respectively, with DF of 7.50% and 3.71%. In cyclic flows, as shown before, the effects of particle inertia and gravity evolve as particles travel through the passage during the inhalation. This variation seems to increase the scattering of the deposition sites of the particles as is seen in Fig. 16. It should be emphasized that the particles that were not deposited in the nasal cavity fall into two groups. The first group of particles exit through the nasopharynx and here it is assumed that they will be deposited in the lower parts of the respiratory tract. That is, they do not come back into the nasal region during the exhalation phase of breathing. The second group of particles remains suspended in the nasal cavity at the end of the breathing cycle, and these particles are tracked as the breathing cycle continues. In reality it is possible that a small fraction of fine particles returns back into the nasal cavity during exhalation. However, for particles larger than a few microns, the amount is expected to be negligibly small. Nevertheless, neglecting the return of particles from the outlet is an approximation that is used in the present analysis.

32

1

3

2 4 5

Fig. 16 Deposition pattern of 20 μm particles under both cyclic breathing and equivalent mean flow of 6.37 L/min. The red and blue small spheres correspond to the particles deposited, respectively, in cyclic and steady flow analyses.

33

4. Conclusions Airflow and deposition of micro-particles in the human nasal cavity for cyclic inspiratory breathing under the rest condition were studied. To assess the airflow inertial effects under unsteady conditions, the pressure drops across the nasal cavity for inspiratory transient accelerating, decelerating, and steady flow conditions were evaluated and compared. It was shown that there are marked differences in the pressure drops at low airflow rates corresponding to the beginning and end of the inhalation cycle. In the time intervals of the cyclic breathing in which the airflow rate was high, the differences between unsteady and steady state pressure dropped sharply and became negligibly small. An evaluation of the unsteady effects based on the instantaneous Strouhal number, based on the instantaneous spatially averaged velocity at the nostril, was performed. It was shown that the quasi-steady assumption for evaluation of the airflow pressure drop in the nasal cavity provided a reasonable approximation when the Strouhal number was smaller or equal to 0.2. In addition, using the unsteady particle tracking, the instantaneous deposition of micro-particles under cyclic inhalation was evaluated and the effects of accelerating and decelerating airflow on micro-particle deposition were investigated. In addition, the contributions of inertia and gravity on the particle deposition were discussed. The results show that in the absence of gravity, the deposition rate followed the breathing rate and increased/decreased with the variation in the flow rate. Gravity significantly increased the deposition rate of large particles, and the effect of gravity appeared mainly at the time intervals in which the flow rate was low. Examining the impact of the flow acceleration/deceleration, it was shown that the deposition rates in the accelerating and decelerating flow phases were not same and their proportion varied depending on particle size and inhalation flow rate. In addition, the total and regional deposition fractions under cyclic breathing and the equivalent mean flow breathing were evaluated and compared. The comparison of the results showed that the total deposition fractions obtained for cyclic breathing were different from those obtained from a mean flow analysis. Particularly, for particles smaller than 10 µm, the steady flow analysis consistently over predicted the cyclic 34

flow analysis by relative errors of about 10 to 60%. For larger particles (20 µm and over), relative errors in the range of 8 to 18 % were observed. Similar errors were seen for the regional deposition; that is, the absolute differences between the predictions of the steady and unsteady analyses for particle deposition were not high, but the relative error was large especially for small particles in the studied range. Based on the presented results, it is concluded that the steady simulation with the equivalent mean flow rate of the inhalation cannot accurately predict the total micro-particle deposition under cyclic breathing.

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Appendix A A.1. Grid study For investigating the grid size sensitivity on the airflow solution, different grids with approximately 648000, 1230000, 1560000, and 1800000 tetrahedral cells were used. In Fig. A1, the grid sensitivity study results for estimating the normalized velocities at four line segment in different region of the airflow field are shown. It is seen that the increase of grid resolution from 1560000 cells to 1800000 cells did not noticeably change the results. Therefore, the grid with approximately 1560000 cells was selected for simulating the airflow field. In addition, fourprismatic boundary layer grids were added along the walls to accurately capture the near-wall particle trajectories. As a result of the near-wall refinements, the mesh size was increased to approximately 2400000 cells, which was used in the present simulations.

38

Cˊ C Bˊ

B

Aˊ

D

Dˊ

A

Fig. A1 Gird sensitivity study results for prediction of the flow field.

39



Under rest condition, the quasi-steady airflow assumption in the nasal cavity is reasonable when the instantaneous Strouhal number is smaller than 0.2



Deposition fraction is different in the acceleration and deceleration phases of the inhalation.



The quantitative predictions of the total micro-particles deposition between steady airflow simulation with mean of the inhalation and unsteady simulation are different although the general trend is similar.



The relative difference is higher for the particles with diameter smaller than 10 µm but the absolute difference is higher for larger particles.



The gravity plays an important role in deposition of larger particles in the rest condition.



The difference between unsteady simulation and steady case with equivalent flow rate is more noticeable for the regional depositions.

40

Graphical abstract

Deposition rate of the particles in a single nasal passage at different time intervals for the inhalation phase.

41