Combined inertial and gravitational deposition of microparticles in small model airways of a human respiratory system

Aerosol Science 38 (2007) 1047 – 1061 www.elsevier.com/locate/jaerosci

Combined inertial and gravitational deposition of microparticles in small model airways of a human respiratory system Clement Kleinstreuera, b,∗ , Zhe Zhanga , Chong S. Kimc a Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA b Department of Biomedical Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA c Human Studies Division, National Health and Environmental Effects Research Laboratory, U.S. EPA, Research Triangle Park, NC 27711, USA

Received 18 December 2006; received in revised form 22 August 2007; accepted 22 August 2007

Abstract Focusing on relatively small airways in terms of the medium-size bronchial generations G6–G9, the interplay of impaction and sedimentation on micron particle transport and deposition has been simulated. A commercial finite-volume code, enhanced with user-supplied programs, has been employed. Although impaction is still a dominant deposition mechanism for microparticle in medium-size airways under normal breathing conditions (say, Qin = 15.30 L/ min), sedimentation may play a role as well. In turn, that can influence the local particle deposition patterns, efficiencies and fractions for a realistic range of Stokes numbers (0.001  St  0.33). However, deposition due to sedimentation is significantly amplified during slow inhalation; for example, the gravitational deposition may become dominant in the ninth bifurcation (i.e., generations G8–G9) for relatively large microparticles (say, dp > 5 m) at Qin = 3.75 L/ min. The occurrence of sedimentation changes the location of the deposition “hot spots” and reduces the order of the maximum deposition enhancement factor. The use of analytical formulas based on inclined tube models for predicting gravitational deposition in local bronchial airway segments as well as the combination of deposition by sedimentation and impaction has to be carefully examined. As shown, more prudent is the use of curve-fitted correlations generated from experimentally validated computer simulation results as a function of Stokes number and sedimentation parameter. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Inertial impaction; Gravitational sedimentation; Micron particle deposition; Small human airways; Computational analysis; Deposition efficiency; Stokes number; Froude number; Sedimentation parameter; Deposition correlations

1. Introduction Inhaled toxic airborne particles or pharmaceutical aerosols may transport and deposit in human respiratory systems due to inertial impaction, gravitational sedimentation and binary diffusion, including Brownian diffusion for submicron particles. The contributions of different mechanisms to particle deposition in local airway segments vary with effective particle size, density, local airflow rate and gravity angle. In general, sedimentation is important for micron particles in the lower airways and alveolar regions, where the Reynolds numbers are low. So far, direct experimental studies of gravitational deposition in human respiratory systems are hardly available. Published data only include regional ∗ Corresponding author. Department of Mechanical and Aerospace Engineering, North Carolina State University, NC 27695-7910, USA. Tel.: +1 919 515 5261; fax: +1 919 515 7968. E-mail address: [email protected] (C. Kleinstreuer).

0021-8502/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2007.08.010

1048

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

deposition fractions (DFs) in the alveolar region where sedimentation may be dominant or occurs in conjunction with diffusion and impaction. Traditionally, gravitational deposition in the tracheobronchial airways and the alveolar ducts (see Asgharian, Price, & Oberdorster, 2006, among others) is estimated via analytical equations derived from the sedimentation losses in flows between parallel plates or in circular tubes (see Beeckmans, 1965; Pich, 1972). In the cases of simultaneous impaction (i) and sedimentation (s), the deposition probabilities due to the individual deposition mechanisms are usually combined linearly or nonlinearly in the following form: p

p

DEi+s = (DEi + DEs )1/p ,

(1)

where DE is the impaction and/or sedimentation induced deposition efficiency for a given section, and p = 1.3 as suggested by different researchers (see Asgharian & Anjilvel, 1994; Balásházy, Hofmann, & Martonen, 1990; ICRP, 1994). Yeh and Schum (1980) also proposed that DEi+s = DEi + DEs − DEi × DEs .

(2)

Computational fluid–particle dynamics (CFPD) simulations for gravitational deposition started in the 1990s, focusing on small bronchial airways and alveolar ducts or sacs. For example, Hofmann, Balásházy, and Koblinger (1995) simulated the gravitational settling of 10 m particles in an asymmetric, single bifurcating airway model representing generations 15 and 16. It was found that the gravity angle is significant for determining particle distributions and localized doses. Using a 2-D symmetric six-generation model, Darquenne and Prisk (2003) stressed that gravity is important in the deposition of 0.5 and 1 m particles in the human acinus. Haber, Yitzhak, and Tsuda (2003) incorporated gravitational sedimentation with wall movements, employing a 3-D hemispherical alveolus model. Most recently, Harrington, Prisk, and Darquenne (2006) simulated trajectories of 1–5 m particles in 3-D alveolated ducts representing generations 18–22 with different gravity angles. They concluded that the total deposition can be a function of the gravity angle and the ratio of the terminal settling velocity to mean lumen flow velocity. In summary, most of the computational analyses for gravitational deposition focused on the alveolar region where sedimentation may play a dominant role. The effects of gravity on the aerosol deposition in the bronchial airways, especially in the medium-size airways where sedimentation and impaction may occur simultaneously, have not been thoroughly investigated. In this paper, focusing on medium-size bronchial airways (i.e., G6–G9), the effects of micron particle impaction and gravitational sedimentation have been simulated employing CFPD approaches. The validity of the analytical equations for gravitational deposition and Eqs. (1) and (2) for the combination of deposition by sedimentation and impaction have been examined for generations G6–G9. The effect of gravity on deposition “hot spots” is discussed and easy-to-use deposition correlations are given as well. 2. Methods 2.1. Airway geometries A representative triple bifurcation bronchial airway model was selected to study gravitational and impaction deposition (see Fig. 1). The dimensions of the four-generation airway model (G6–G9) are close to those given by Weibel (1963) for adults with a lung volume of 3500 mL. The inlet conditions for airflow and particle transport at G6 were selected from one of the outlets in a upstream triple bifurcation airway unit (G3–G6, or B4–B6). The entire simulations started from the mouth in order to avoid the impact of artificially assumed inlet conditions. The pressures at the outlets of the bifurcation airways were assumed to be uniform. The outlet tubes were extended in the simulations to reduce the influence of the zero-pressure assumption at the outlets. 2.2. Transport equations The airflow in the airway model G6–G9 is assumed to be steady, incompressible and laminar. The Womersley number at the inlet of G6 is about 0.4 under normal breathing conditions (say, 15 cycles/min), which indicates that the unsteady effects of the flow fields are relatively minor (Pedley, 1977; Zhang & Kleinstreuer, 2002). Zhang, Kleinstreuer, and Kim (2002) also discussed the conditions for quasi-steady conditions for air–particle flow being equivalent to cyclic inhalation. “In-series” simulations from the oral airway to airway generations G6–G9 with a low-Reynolds-number

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

1049

Fig. 1. 3-D views of the bifurcation airway model (generations G6–G9 or bifurcations B7–B9, the dashed lines indicate the segmental boundaries for bifurcations).

k– model indicated that the turbulence disturbances are almost completely damped out and hence the airflow is laminar in G6–G9 under normal breathing condition (Qoral,in 30 L/ min). For micron particles (dp 1 m), a Lagrangian frame of reference for the trajectory computations can be employed. In light of the large particle-to-air density ratio, dilute particle suspensions, negligible Brownian motion and thermophoretic forces, the drag and the gravity are considered as the dominant point forces away from the walls. Thus, the trajectory equation can be described as (Kleinstreuer, 2006) d (mp vp ) = FD + mp g, dt

(3)

where vp and mp are the velocity and the mass of the particle, respectively, FD is the drag force. The drag force for the spherical particles is given as vp − v)| vp − v|. FD = 18 dp2 CDp (

(4)

Here,  is the air density, and CDP is the drag force coefficient: CDp = CD /Cslip , where

 CD =

)/Rep 24(1 + 0.15Re0.687 p 0.44

(5) for 0.0 < Rep 1000, for 1000 < Rep

(6)

and the particle Reynolds number is vp − v|dp /. Rep = |

(7)

In Eq. (5), Cslip is the slip correction factor (see Clift, Grace, & Weber, 1978). Combining Eqs. (3) and (4) yields with the gravitational unit vector g: ˆ d vp 3 CDp ( v − vp )| v − vp | + g g. ˆ = dt 4p dp When Rep >1, i.e., Stokes flow, CD = 24/Rep , then, Eq. (8) can be rewritten as   d vp U St = vsettling gˆ + ( v − vp ), D dt

(8)

(9)

1050

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

where U is the mean velocity of the fluid, D is the tube diameter, St is the Stokes number which is defined as St = p dp2 U C slip /(18D) and vsettling is the terminal settling velocity given as vsettling = Cslip p gd 2p /(18) (see Kleinstreuer, 1997, for Cslip ≡ 1). Clearly, the left-hand side of Eq. (9) represents the inertial force (or the impaction term). When estimating the particle deposition in the lung due to sedimentation only, this impaction term is usually neglected so that Eq. (9) reduces to (Pich, 1972) vp = vsettling gˆ + v.

(10)

The importance of impaction vs. sedimentation may be estimated in terms of the Froude number squared, (Fr)2 (see Finlay, 2001), which is defined as the ratio of the Stokes number to the non-dimensional settling velocity, i.e., (Fr)2 =

St (vsettling /U )

=

U2 Dg

(11a)

or U Fr = √ . gD

(11b)

Clearly, sedimentation is considered to be small in the airways with large Fr values when compared to impaction, while the influence of impaction is minor for airways with small Fr numbers. However, critical Fr-number values which indicate the dominant deposition mechanism still need to be assessed. 2.3. Deposition parameters The regional deposition of microparticles in human airways can be quantified in terms of the DF or the deposition efficiency (DE) in a specific region. They are defined as DFparticle =

Number of deposited particles in a specific region , Number of particles entering the mouth

(12)

DEparticle =

Number of deposited particles in a specific region . Number of particles entering this region

(13)

The regional DE is mainly used to develop the deposition equation for algebraic (total) lung modeling. The local deposition patterns of the microparticles can be quantified in terms of a deposition enhancement factor (DEF). The DEF is defined as the ratio of local to average deposition densities, where the deposition densities are computed as the number of deposited particles in a surface area divided by the size of that surface area (Balásházy, Hofmann, & Heistracher, 1999; Zhang, Kleinstreuer, Donohue, & Kim, 2005), i.e., DFi /Ai n . i=1 DFi / i=1 Ai

DEF = n

(14)

Here, Ai is the area of the local wall cell (i), n is the number of wall cells in a specific airway region and DFi is the local DF in the local wall cell (i). 2.4. Numerical methods The numerical solutions of airflow and particle transport in the bronchial airways G6–G9 were carried out with a user-enhanced finite-volume-based program CFX4.4 from Ansys, Inc., and an off-line parallelized particle transport code “F90” (for details see Zhang et al., 2005). The mesh topologies of the airway model were determined by refining the meshes until the grid independence of the flow field and the particle DF solutions was achieved. All the computations were performed on an IBM Linux Cluster with 175 dual Xeon compute nodes at North Carolina State University’s HighPerformance Computing Center (Raleigh, NC) and local dual Xeon Intel 3.0G Dell workstations (CFPD Laboratory, Department of Mechanical and Aerospace Engineering, NC State University). The solutions of the flow field were assumed to be converged when the dimensionless mass and momentum residual ratios were < 10−3 . Improving the convergence criteria to < 10−4 had a negligible effect on the simulation results.

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

1051

Pich, 1972 (U = 20 cm/s) Deposition Efficiency

0.2

Pich, 1972 (U = 10 cm/s) Simulation (U = 20 cm/s) Simulation (U = 10 cm/s)

0.1

0.0 1

3

5 7 Particle Diameter (μm)

9

11

Fig. 2. The gravitational deposition of the microparticle in a horizontal circular pipe with D = 0.2 cm and L = 1 cm.

3. Model validations The present CFPD models have been validated with various experimental data sets for steady and transient laminar flows in bifurcations (Comer, Kleinstreuer, & Zhang, 2001; Zhang & Kleinstreuer, 2002) and for laminar, transitional and turbulent flows in tubes with local obstructions (Zhang & Kleinstreuer, 2003). Similarly, microparticle depositions in oral, nasal and bronchial airways were successfully compared with the measured DEs and deposition patterns (Comer, Kleinstreuer, Hyun, & Kim, 2000; Shi, Kleinstreuer, & Zhang, 2007; Zhang et al., 2002, 2005). The computer simulation results of gravitational sedimentation in a horizontal pipe (length to diameter ratio L/D was 5) with Poiseuille inlet conditions were compared with analytical expressions for micron particle DEs in terms of particle size. Pich (1972) derived the DE due to sedimentation for Poiseuille flow as     2 DEs = (15a) 2 1 − 2/3 − 1/3 1 − 2/3 + arcsin(1/3 ) ,  where =

3 vsettling L cos  4 U D

(15b)

and  being the inclination angle measured relative to the horizontal (i.e.,  = 0◦ for horizontal tubes). As shown in Fig. 2, the simulated gravitational DEs in the circular pipe as a function of particle size and inlet velocity match the correlation (Eq. (15a)) very well. Clearly, particle sedimentation increases with an increasing size or decreasing velocity because of the increasing gravity or residence time. In summary, the good agreements between experimental findings and theoretical predictions instill confidence that the present computer simulation model is sufficiently accurate to analyze airflow and particle transport/deposition in human airways. 4. Results and discussion 4.1. Normal inhalation conditions Particle transport and deposition due to simultaneous sedimentation and impaction (see Eq. (8)) have been simulated in a symmetric triple bifurcation airway model representing bronchial airway generations G6–G9 (see Fig. 1). The

1052

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

Fig. 3. The distributions of particle deposition enhancement factor (DEF) in the bifurcation airway model G6–G9 for two particle diameters and different gravitational directions (Qin = 30 L/ min).

inlet conditions for airflow and particle transport at G6 were selected from one of the outlets of an upstream triple bifurcation airway unit (G3–G6). The simulations started from the mouth at inhalation flow rates of 15 and 30 L/min under normal inhalation conditions and the local inlet flow rates at the present G6 branch were 0.32 and 0.66 L/min, respectively. The 3-D surface views of the local particle deposition patterns in terms of particle DEF for particles in the bifurcation airway model G6–G9 at Qin = 30 L/ min with aerodynamic diameters dp = 3 and 10 m under different gravitational settling conditions are shown in Fig. 3. As stated in Section 2.3, the DEF is defined as the ratio of local to average deposition densities (Zhang et al., 2005). The presence of locally high DEF values indicates non-uniform deposition patterns, including “hot spots”, i.e., maximum surface concentrations. Clearly, Fig. 3 demonstrates that inertial impaction is still a dominant deposition mechanism for the micron particle in the airway segment G6–G9, which generally produces high DEF values in the vicinity of the carinal ridges. However, the additional effect of gravity can be observed, where on the tube walls lots of particles deposit as a function of gravity vector direction. As expected, particle deposition due to both inertial impaction and gravitational sedimentation increases with increasing particle size (cf. Figs. 3a and b). However, the maximum DEF value is larger for smaller particles because such particles may collect more closely at the carinal ridges. The apparent effect of the gravity directions on particle deposition can be seen by comparing Figs. 3b and d–f. As expected, particles essentially settle on the tube wall which is normal to the gravity direction. For example, the majority of particles in the horizontal case land on the upper part of the tube due to sedimentation (see Fig. 3f).

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

1053

Without gravity Gravity vector: (0.5,-0.866,0) Gravity vector: (-1,0,0) Gravity vector: (1,0,0) Gravity vector: (0,0,1) Gravity vector: (0.766,-0.643,0)

Maximum DEF

1000

500

0 B7

B8.1

B8.2 B9.1 B9.2 Airway Branch

B9.3

B9.4

Fig. 4. The effect of gravity on the maximum DEF values of particles (dp = 10 m) in the bifurcation airway model G6–G9 (Qin = 30 L/ min). Without gravity (Qin=30 L/min) Gravity vector: (0.766 ,-0.643 ,0) (Q in=30 L/min)

0.8

Gravity vector: (0.5,-0.866,0) (Q in=30 L/min) Without gravity (Qin=15 L/min)

Deposition Fraction (%)

Gravity vector: (0.766 ,-0.643 ,0) (Q in=15 L/min)

0.6

0.4

0.2

0 0

5

10

Particle Diameter (μm) Fig. 5. The total particle deposition fractions in the bifurcation airway model G6–G9.

Fig. 4 shows the maximum DEF values in different individual bifurcations of airway segment G6–G9 as a function of gravity direction. The distributions of DEFs as well as the maximum DEF values change greatly at each individual bifurcation because of the different local inlet Reynolds numbers, flow features and particle distributions. The maximum DEF values range from the order of 102 .103 for all the cases simulated. Clearly, the presence of gravity and the gravity direction affects the maximum DEF value in each bifurcation. In general, some particles which may directly land on the carinal ridges due to impaction only can be diverted; as a result, the maximum DEF values may decrease due to gravitational sedimentation. The reduction of DEFmax is most pronounced for the horizontal bifurcation case. Fig. 5 presents variations in the total DF for the airway segment G6–G9 with gravity as a function of particle size, while Fig. 6 shows the DFs in each individual bifurcation as a function of gravity direction. The DF in one airway

1054

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

Deposition Fraction (%)

100

Q in=30 L/min Without gravity Gravity vector: (0.5,-0.866,0) Gravity vector: (-1,0,0) Gravity vector: (1,0,0) Gravity vector: (0,0,1) Gravity vector: (0.766,-0.643,0)

10-1

10-2 B7

B8.1

B8.2

B9.1

B9.2

B9.3

B9.4

Airway Branch

Deposition Fraction (%)

100

Without gravity (Q in=30 L/min) Gravity vector: (0.766,-0.643,0) (Q in=30 L/min) Without gravity (Q in=15 L/min) Gravity vector: (0.766,-0.643,0) (Q in=15 L/min)

10-1

10-2

10-3 B7

B8.1

B8.2

B9.1

B9.2

B9.3

B9.4

Airway Branch Fig. 6. The effect of gravity on the particle deposition fraction (dp = 10 m) in each individual bifurcation of the bifurcation airway model G6–G9.

segment is defined as the ratio of the number of deposited particles in this airway unit to the total amount of incoming particles at the mouth inlet. It should be noted that the simulated airway segment G6–G9 was selected from 64 parallel G6–G9 branches (i.e., 19th from the left) according to Weibel’s Type A configuration. Thus, the DF in this segment does not represent the average value of the 64 parallel airway units. Fig. 5 demonstrates that the presence of gravity may increase particle deposition in G6–G9, especially for relatively large-size particles (say, dp > 5 m). The DF in airways G6–G9 may not increase with particle size for large micron particles (say, dp > 7 m) (see Fig. 4) because of the elevated deposition in the upstream airways at a medium inhalation flow rate (say, Qin = 30 L/ min). Clearly, the effect of sedimentation increases with reduced flow rates because of increasing residence time. Specifically, when comparing the cases with gravity (gravity vector: (0.766, −0.643, 0)) to those without gravity, the maximum enhanced DF is about 16% (relative value) for an inspiratory flow rate of 30 L/min. However, the maximum enhanced DF may increase to 34% when the inhalation flow rate drops to 15 L/min.

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

Deposition Efficiency (%)

30

1055

Without gravity (Q in=30 L/min) Gravity vector: (0.5,-0.866,0) (Q in=30 L/min) Gravity vector: (0.766,-0.643,0) (Q in=30 L/min) Without gravity (Q in=15 L/min) Gravity vector: (0.766,-0.643, 0) (Q in=15 L/min) Correlation (see Eq. (16))

20

10

0 10 -3

10 -2 10 -1 Stokes number at the parent tube

10 0

Fig. 7. The variations of particle deposition efficiency in each individual bifurcation of the bifurcation airway model G6–G9 as a function of local Stokes number under normal breathing conditions.

Considering the different local flow characteristics and particle distributions as well as gravity angles, the variations of DF in each bifurcation as a function of gravity angle are complicated and irregular (see Fig. 6). In very few cases, the DF in one bifurcation may decrease in the presence of gravity due to the change of particle trajectories influenced by sedimentation. The ratio of DF considering both impaction and sedimentation (i.e., DFi+s ) to that considering only impaction (i.e., DFi ) ranges from 1.04 to 1.25 for the entire segment G6–G9 while it may range from 1.0 to 1.64 for each individual bifurcation when 3 dp 10 m. Clearly, the gravity direction plays an important role in the micron particle deposition as well. The variations of DF in each individual bifurcation with inhalation flow rate are complicated due to the effects of upstream particle deposition and distribution (see Fig. 6b). In summary, while inertial impaction is still a dominant deposition mechanism for the micron particles in the medium-size bronchial airways (say, G6–G9) under normal breathing conditions, gravitational sedimentation may become measurable in some local airway bifurcations with certain gravity angles. Fig. 7 depicts the variations of DEs as a function of local Stokes number at each individual bifurcation. The DE in one bifurcation is defined as the ratio of deposited to incoming particle numbers in a given bifurcation unit. The local Stokes number is St = (p dp2 U )/(18D) with U and D being the mean velocity and diameter of the parent tube of the individual bifurcation, respectively. Again, the particle DE may increase in the presence of gravity. The ratio of DE considering both impaction and sedimentation (i.e., DEi+s ) to that considering impaction only (i.e., DEi ) ranges from 1.0 to 1.7 for each individual bifurcation when St > 0.01. Ignoring the effects of the local gravity vector and airflow rate, the DE in each individual bifurcation of G6–G9 can still be correlated as a function of local Stokes number as shown in Fig. 6, i.e., DE(%) = 105.97(St)1.214 × 100%

(16)

with r 2 = 0.91. This also indicates that impaction is still a dominant deposition mechanism for micron particles in the medium-size bronchial airways (say, G6–G9) for inspiratory flow rates of Qin 15 L/ min. 4.2. Low-flow-rate inhalation As expected, the particle DE correlates well with Stokes number in airways G6–G9 because inertial impaction is dominant in such medium-size airways; especially, at high flow rates when sedimentation causes only somewhat

1056

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

Deposition Efficiency (%)

Correlation (see Eq. (16)) Without gravity (Q in=7.5 L/min) Gravity vector: (0.766,-0.643, 0) (Q in=7.5 L/min) Without gravity (Q in=3.75 L/min) Gravity vector: (0.766,-0.643, 0) (Q in=3.75 L/min)

30

20

10

0 10

-3

10

-2

10

-1

Stokes number at the parent tube Fig. 8. The variations of particle deposition efficiency in each individual bifurcation of the bifurcation airway model G6–G9 as a function of local Stokes number under slow breathing conditions.

of a perturbation (see Section 4.1). However, the scenario changes when the inhalation flow rate is further reduced. Specifically, considering Qin = 7.5 and 3.75 L/min in G6–G9, the local inlet flow rates at the present G6 branch were 0.16 and 0.08 L/min, respectively. Such slow breathing may occur in some specific airway branches of patients with COPD. As shown in Fig. 8, when the flow rates are reduced to Qin = 7.5 or 3.75 L/min, the role of sedimentation becomes more and more significant when compared to impaction, particularly for large-size particles (i.e., high Stokes numbers). As a result, particle DEs cannot be accurately correlated with the Stokes number alone. Now the ratio of DE considering both impaction and sedimentation (i.e., DEi+s ) to that considering impaction only (i.e., DEi ) ranges from 1.5 to 7.3 for each individual bifurcation for Qin =7.5 L/ min and 1dp 14 m while it ranges from 2.5 to 29 for Qin =3.75 L/ min and 1dp 14 m. Fig. 9 depicts variations of DEi+s /DEi for each individual bifurcation as a function of Froude number or nondimensional sedimentation parameter . The parameter  is defined as  = (vsettling /U )(L/D) cos  with L being the tube length and  being the inclination angle measured relative to the horizontal (i.e.,  = 0◦ for horizontal tube). The non-dimensional term vsettling /U represents the importance of gravity relative to viscous drag when calculating particle trajectories. The -value of one bifurcation is the sum of values for parent and two daughter tubes. It can be observed from Fig. 9 that when Fr < 3.2 sedimentation tends to be very important and even dominant. However, the values of DEi+s /DEi may change greatly, even for the same Fr number, due to variations of particle size, generation number and gravity angle. For example, the values of DEi+s /DEi range from 2 to 29 for Fr ≈ 1.5, corresponding to the Qin = 3.75 L/ min case in this study. The parameter  may indicate the significance of sedimentation better when compared to the Fr number because it incorporates the effects of particle size and gravity angle. In summary, inertial impaction is the dominant deposition mechanism for  < 10−3 while gravitational deposition is critical for  > 10−1 . Both impaction and sedimentation may be important when 10−3  10−1 , depending on specific air–particle characteristics and distributions in local airway branches. There are also some exceptions. For example, DEi+s /DEi > 11 at bifurcations B9.1–B9.4, which indicates that gravitational sedimentation is the dominant deposition mechanism when Qin = 3.75 L/ min and dp 7 m (i.e.,  0.015). When the influence of gravity is dominant, the local particle deposition patterns change as well. For example, Fig. 10 shows the particle DEF in airways G6–G9 at Qin = 3.75 L/ min, dp = 10 m with gravity vector (0.766, −0.643, 0). Clearly, the carinal ridges are not the unique deposition “hot spots” anymore; in fact, the entire airway surfaces which are normal to the gravity direction may become high-deposition regions. The maximum DEF values may reduce by one

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

1057

40

DE i+s/DE i

30

20

10

0 10

-5

10

-4

10

-3

10

-2

10

-1

10

0

Sedimentation Parameter (γ)

40

DE i+s/DE i

30

20

10

0 0

5

10

15

Froude Number (Fr) Fig. 9. The variations of DEi+s /DEi in each individual bifurcation of the bifurcation airway model G6–G9 as a function of local Froude number or sedimentation parameter .

order of magnitude (say, from about 450 to 60) with much broader distribution of micron particles due to gravitational settling. Considering the simultaneous occurrence of sedimentation and impaction when 3.75 Qin 30 L/ min, the particle DE for each individual bifurcation of airway generations G6–G9 with different gravity angles can be correlated as a function of Stokes number and gravitational settling parameter , i.e., DE(%) = C1 St C2 + C3 C4 + C5 St × 100%.

(17)

The best fit for the current data set (3.2 × 10−5  0.13, 0.001 St 0.33) yields C1 = 97.4,

C2 = 1.26,

C3 = 45.58,

C4 = 0.775

and

C5 = 1365.5,

1058

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

Fig. 10. The distributions of particle deposition enhancement factor (DEF) in the bifurcation airway model G6–G9 with slow breathing (Qin = 3.75 L/ min, dp = 10 m).

Table 1 Comparison of particle deposition efficiencies due to sedimentation only (DEs in %) in airway segments G6–G9 Airway segment

Present simulated DEs

Predicted DEs with Eq. (15)

Predicted DEs with Eq. (18)

B7 B8.1 B8.2 B9.1 B9.2 B9.3 B9.4

1.54 0.57 0.59 1.39 0.67 1.28 0.53

0.72 1.25 0.58 1.72 1.12 1.12 0.85

0.72 1.25 0.58 1.72 1.13 1.13 0.85

Total (G6–G9)

2.99

2.81

2.82

dp = 10 m and gravity vector: (0.766, −0.643, 0).

where r 2 = 0.93. It should be noted that the gravity angles of individual airways were considered by imposing different gravity vectors in the simulations. 4.3. Comparison of analytical DE correlations and computational results As mentioned in Section 1, the analytical solutions for particle sedimentation in inclined tubes are usually used to estimate gravitational deposition of aerosols in human airways. The most popular expressions include the one proposed by Pich (1972) for Poiseuille flow (see Eq. (15)) and the one by Beeckmans (1965) for well-mixed plug flow:   −16 DEs = 1 − exp  . (18) 3 Table 1 compares the simulated particle DEs in airway model G6–G9 with those calculated with Eqs. (15a) and (18). The DEs when using analytical expressions may differ largely in some individual bifurcations (e.g., B8.1, B9.2, B9.4) from the simulated values due to the effects of geometry as well as local airflow features and particle distributions. Specifically, the analytical solutions based on simple straight or inclined tubes ignore the presence of curved surfaces as well as complicated carinal ridges considered in 3-D flows through realistic airway geometries. The realistic air–particle flows in local airways are influenced by upstream flow fields and particle distributions, which may be quite different from the assumed uniform (or fully developed or well-mixed) airflows and particle distributions in analytical solutions. As a result, the particle trajectories and depositions due to sedimentation may change measurably. However, the effects of realistic geometry, flow structure and particle distributions in different individual bifurcation may cancel

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

1059

Table 2 Particle deposition efficiencies (%) in airways G6–G9 due to different deposition mechanisms Airway segment

DEi

DEs

DEi+s

DEi + DEs

B7 B8.1 B8.2 B9.1 B9.2 B9.3 B9.4

21.38 7.74 7.50 19.01 21.00 20.68 18.54

1.54 0.57 0.59 1.39 0.67 1.28 0.53

23.75 9.00 10.78 25.51 25.08 24.15 22.11

22.93 8.31 8.09 20.40 21.67 21.96 19.07

dp = 10 m and gravity vector: (0.766, −0.643, 0).

each other so that the CFPD-simulated DE in a relatively large airway unit (say, G6–G9) may agree well with the DE values calculated with simple analytical expressions. This may also explain why the predicted regional deposition (e.g., tracheobronchial, alveolar depositions) with analytical equations may give reasonable values when compared to experimental measurements. The predicted deposition values by Eqs. (15a) and (18) are almost the same. As indicated by Eqs. (1) and (2), the DEs due to simultaneous impaction and sedimentation (i.e., DEi+s ) are a function of DEs considering the two individual deposition mechanisms separately (i.e., DEi and DEs ). However, as given in Table 2, the simulated DEi+s values are larger than the simple sum of DEs based on only impaction or only sedimentation (i.e., DEi + DEs ). The relative error can be up to 25% (e.g., in B9.1). Clearly, DEi + DEs is one of the specific expressions of Eq. (1) (i.e., p = 1) and the cases of p > 1 or Eq. (2) will predict an even lower DEi+s when compared to the actual values obtained via CFPD simulations. This indicates that the relationship between DEi+s and DEi as well as DEs , in case of both impaction and sedimentation, is actually quite complicated because their interactions may greatly change particle trajectories and hence local depositions. In addition, the assumption of Rep >1 (i.e., Stokes flow) (see Eq. (9)) may not hold for all particles so that it is somewhat arbitrary to separate particle deposition due to sedimentation from the actual cases where impaction and sedimentation occur simultaneously and hence are both important (see Eq. (8)). 5. Conclusions Focusing on medium-size bronchial airways of the human respiratory system, the effects of gravity and inertia on micro-size aerosol transport and deposition have been simulated. The computational results indicate the following: (1) While inertial impaction is still a dominant deposition mechanism for the micron particles in the medium-size bronchial airways (say, generations G6–G9) under normal inhalation conditions (say, Qin = 15 and 30 L/min), gravitational sedimentation may become significant in some local airway bifurcations with certain gravity angles for relatively large microparticles (say, dp 5 m). For example, the ratio of DF considering both impaction and sedimentation (DFi+s ) to that considering only impaction (DFi ) ranges from 1.04 to 1.25 for the entire segment G6–G9 while it may range from 1.0 to 1.64 for each individual bifurcation when 3 dp 10 m. The gravity vector direction is also important for microparticle deposition. Furthermore, the role of sedimentation under slow breathing conditions (say, Qin = 7.5 and 3.75 L/min) becomes more and more significant when compared to impaction, particularly for large-size particles. The ratio of DEi+s to DEi for each individual bifurcation of airway generations G6–G9 ranges from 1.5 to 7.3 at Qin = 7.5 L/ min and 2.5 to 29 at Qin = 3.75 L/ min when 1 dp 14 m. In addition, it can be expected that sedimentation will also become even more important as airflow reaches deeper lung regions, which will be investigated in the future. (2) The non-dimensional sedimentation parameter  = (vsettling /U )(L/D) cos , which includes effects of terminal settling velocity, lumen airflow velocity and gravity direction, may indicate the significance of sedimentation better than the Froude number, i.e., Fr = U (gD)−1/2 . In general, inertial impaction is the dominant deposition mechanism for  < 10−3 while gravitational deposition is critical for  > 10−1 . Both impaction and sedimentation may be important when 10−3  10−1 , depending on specific air–particle characteristics and distribution at local airway branches.

1060

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

(3) In the presence of gravity, the particle deposition sites and the deposition densities (or DEFs) change in the bronchial airways when compared to the cases of impaction only, which is also a function of gravity direction. The simultaneous occurrence of gravitational sedimentation and inertial impaction may reduce the maximum DEF values somewhat. The “hot spots” for gravitational deposition may relocate from the carinal ridges to the entire airway surfaces which are exposed to the gravity direction. The maximum DEF values may be one order of magnitude (say, O(10)) lower for sedimentation-dominant deposition when compared to impaction-dominant deposition (say, O(102 )) due to a broader distribution of micron particles with gravitational settling. (4) The DE in each individual bifurcation of G6–G9, where sedimentation and impaction occur simultaneously when 3.75Qin 30 L/ min, can be correlated as a function of Stokes number and parameter , i.e., DE(%) = 97.4St 1.26 + 45.580.775 + 1365.5St × 100%

(r 2 = 0.93).

(19)

(5) The predicted DEs using analytical formulas for some individual bifurcations of the bronchial airways may differ greatly from the CFPD-simulated values due to the effects of geometry as well as local flow features and particle distributions. However, the effects of realistic geometry, flow structure and particle distributions in different individual bifurcations may cancel each other so that the CFPD-simulated DE in a relatively large airway unit (say, G6–G9) agrees well with that calculated with analytical expressions. When both impaction and sedimentation are important, particle deposition may not be the simple sum of values calculated by sedimentation and impaction separately. (6) The primary limitation of the current study may be the shortcomings of the model for applications in realistic lung geometries. Recent CFD studies have shown that more realistic or (patient-specific) models, e.g., asymmetric and non-planar geometries, may result in some different local deposition rates (Li, Kleinstreuer, & Zhang, 2007a, 2007b; Nowak, Kakade, & Annapragada, 2003). Nevertheless, the essential findings of this study, such as the local deposition patterns and the comparisons between the simulations in bifurcating airways and the analytical solutions in a straight circular tube, should not significantly relate to the variations in bifurcating geometries. Additional computational studies will be conducted to resolve the effects of the asymmetric and the non-planar geometries on gravitational sedimentation.

Acknowledgments This effort was sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under Grant number FA9550-04-1-0422 (Dr. Walt Kozumbo, Program Manager) and the US Environmental Protection Agency (Dr. C.S. Kim, Program Monitor). The US government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The use of both CFX software from Ansys, Inc. (Canonsburg, PA), and IBM Linux Cluster at the High-Performance Computing Center at North Carolina State University (Raleigh, NC) is gratefully acknowledged as well. Disclaimer. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US Environmental Protection Agency. References Asgharian, B., & Anjilvel, S. (1994). Inertial and gravitational deposition of particles in a square cross section bifurcating airways. Aerosol Science and Technology, 20, 177–193. Asgharian, B., Price, O., & Oberdorster, G. (2006). A modeling study of the effect of gravity on airflow distribution and particle deposition in the lung. Inhalation Toxicology, 18, 473–481. Balásházy, I., Hofmann, W., & Heistracher, T. (1999). Computation of local enhancement factors for the quantification of particle deposition patterns in airway bifurcations. Journal of Aerosol Science, 30, 185–203. Balásházy, I., Hofmann, W., & Martonen, T. B. (1990). Inertial impaction and gravitational deposition of aerosols in curved tube and airway bifurcations. Aerosol Science and Technology, 13, 308–321. Beeckmans, J. M. (1965). The deposition of aerosols in the respiratory tract I. Mathematical analysis and comparison with experimental data. Canadian Journal of Pharmacology, 43, 172–175. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, drops, and particles. New York: Academic Press.

C. Kleinstreuer et al. / Aerosol Science 38 (2007) 1047 – 1061

1061

Comer, J. K., Kleinstreuer, C., Hyun, S., & Kim, C. S. (2000). Aerosol transport and deposition in sequentially bifurcating airways. Journal of Biomechanical Engineering—Transactions of the ASME, 122, 152–158. Comer, J. K., Kleinstreuer, C., & Zhang, Z. (2001). Flow structures and particle deposition patterns in double bifurcation airway models. Part 1. Air flow fields. Journal of Fluid Mechanics, 435, 25–54. Darquenne, C., & Prisk, G. K. (2003). Effect of gravitational sedimentation on simulated aerosol dispersion in the human acinus. Journal of Aerosol Science, 34, 405–418. Finlay, W. H. (2001). The mechanics of inhaled pharmaceutical aerosols: An introduction. London, UK: Academic Press. Haber, S., Yitzhak, D., & Tsuda, A. (2003). Gravitational deposition in a rhythmically expanding and contracting alveolus. Journal of Applied Physiology, 95, 657–671. Harrington, L., Prisk, G. K., & Darquenne, C. (2006). Importance of the bifurcation zone and branch orientation in simulated aerosol deposition in the alveolar zone of the human lung. Journal of Aerosol Science, 37(1), 37–62. Hofmann, W., Balásházy, I., & Koblinger, L. (1995). The effect of gravity on particle deposition patterns in bronchial airway bifurcations. Journal of Aerosol Science, 26, 1161–1168. International Commission on Radiological Protection (ICRP). (1994). Human respiratory model for radiological protection (ICRP publication 66). Annals of the ICRP 24, 1–120. Kleinstreuer, C. (1997). Engineering fluid dynamics—An interdisciplinary systems approach. New York: Cambridge University Press. Kleinstreuer, C. (2006). Biofluid dynamics—Principles and selected applications. Boca Raton, FL: CRC Press/Taylor & Francis Group. Li, Z., Kleinstreuer, C., & Zhang, Z. (2007a). Simulation of airflow fields and microparticle deposition in realistic human lung airway models. Part I: Airflow patterns. European Journal of Mechanics—B/Fluids, 26, 632–649. Li, Z., Kleinstreuer, C., & Zhang, Z. (2007b). Simulation of airflow fields and microparticle deposition in realistic human lung airway models. Part II: Particle transport and deposition. European Journal of Mechanics—B/Fluids, 26, 650–668. Nowak, N., Kakade, P.P., & Annapragada, A.V. (2003). Computational fluid dynamics simulation of airflow and aerosol deposition in human lungs. Annals of Biomedical Engineering, 31, 374–390. Pedley, T. J. (1977). Pulmonary fluid dynamics. Annual Review of Fluid Mechanics, 9, 229–274. Pich, J. (1972). Theory of gravitational deposition of particles from laminar flows in channels. Journal of Aerosol Science, 3, 351–361. Shi, H., Kleinstreuer, C., & Zhang, Z. (2007). Modeling of inertial particle transport and deposition in human nasal cavities with wall roughness. Journal of Aerosol Science, 38, 398–419. Weibel, E. R. (1963). Morphometry of the human lung. New York: Academic Press. Yeh, H. C., & Schum, G. M. (1980). Models of human lung airways and their application to inhale particle deposition. Bulletin of Mathematics and Physics, 42, 461–480. Zhang, Z., & Kleinstreuer, C. (2002). Transient airflow structures and particle transport in a sequentially branching lung airway model. Physics of Fluids, 14, 862–880. Zhang, Z., & Kleinstreuer, C. (2003). Low Reynolds number turbulent flows in locally constricted conduits: A comparison study. AIAA Journal, 41, 831–840. Zhang, Z., Kleinstreuer, C., Donohue, J. F., & Kim, C. S. (2005). Comparison of micro- and nano-size particle depositions in a human upper airway model. Journal of Aerosol Science, 36, 211–233. Zhang, Z., Kleinstreuer, C., & Kim, C. S. (2002). Cyclic micron-size particle inhalation and deposition in a triple bifurcation lung airway model. Journal of Aerosol Science, 33, 257–281.