Characteristics of particle transport in an expanding or contracting ...
Aerosol Science 34 (2003) 1193 – 1215 www.elsevier.com/locate/jaerosci
Characteristics of particle transport in an expanding or contracting alveolated tube Dong Y. Lee, Jin W. Lee∗ Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Hyoja 31, Pohang, Kyunngbuk 790-784, South Korea Received 7 January 2003; accepted 11 April 2003
Abstract Features of 1ow and particle transport in a single axisymmetric alveolated tube with 30 cells in the presence of wall motion are analyzed numerically for 1ow conditions typically found in the Aerosol Bolus experiments: the tracheal breathing 1owrate of 250 ml=s and breathing period of 8 s. Di6erent characteristics of particle transport between the alveolated and the straight tube are explained based on the details of 1ow and particle motion. For both expansion and contraction the axial motion of particles in an alveolated tube is smaller than in a straight tube due to the decelerated 1ow in the alveolated region. In an expanding alveolated tube particles get accumulated near the wall and some particles enter alveolus, which make dispersion larger than that in an expanding straight tube. In a contracting alveolated tube particles consistently move toward the tube center, and 1ow velocity pro9le is much blunter than the parabolic one, making dispersion smaller than in a contracting straight tube. In the presence of wall motion, the di6erence in particle transport between alveolated tube and straight tube becomes larger as Strouhal number is increased. Limitations and physiological implications of the results are discussed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Lung airways; Particle transport; Alveolated tube; Wall motion
1. Introduction Dispersion of particles in lung airways plays an important role in such problems as nonuniform particle deposition, drug delivery, and diagnosis of lung diseases using aerosols. Especially in the assessment of pulmonary abnormalities due to exposure to particulate pollutants, an exact model for particle dispersion is essential, since the pattern of regional deposition of particles is determined ∗
Corresponding author. Tel.: +82-54-279-2967; fax: +82-54-279-3199. E-mail address: [email protected] (J.W. Lee).
0021-8502/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0021-8502(03)00097-1
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Nomenclature da diameter of an alveolus dc diameter of the central duct dmax maximum diameter of the central duct dmin minimum diameter of the central duct dB e6ective diameter of an alveolated tube, (Va + Vc )=Vc · dc De6 e6ective di6usivity De6 ; s e6ective di6usivity for the straight tube Lcell length of a cell Lin cell length at the inlet (cell 1 for expansion and cell 30 for contraction) Qa 1owrate entering or going out of an alveolus Qc 1owrate in the central duct P pressure r tube radius ra alveolus radius Re Reynolds number, uB · dc = St Strouhal number, (dc = u)=T B h t time tin characteristic time scale at the inlet (cell 1 for expansion and cell 30 for contraction), Lcell =um Th half period of breathing uc axial velocity in the central duct ue6 hypothetical axial velocity, uin mean axial velocity at the inlet of a cell um mean axial velocity uout mean axial velocity at the outlet of a cell uB mean axial velocity in the central duct, 12 · (uin − ddtx |in + uout − ddtx |out ) Va volume of an alveolus Vc volume of the central duct of one cell Vcell volume of one cell Vmax maximum volume of the whole structure Vmin minimum volume of the whole structure Wo Womersley number, dc =2 · 2=Th · x mean axial position of particles r mean radial position of particles 2 x axial dispersion of particles Greek letters
aperture angle dynamic viscosity kinematic viscosity density
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by the transport and dispersion of particles (Anderson & Dolovich, 1994; Engel & Paiva, 1985; Sarangapani & Wexler, 1999). Taylor was the 9rst to formulate the problem of dispersion for a fully developed laminar 1ow in a straight circular tube (Taylor, 1953), where the e6ective di6usivity (De6 ) becomes proportional to (u · d)2 and inversely proportional to molecular di6usivity of particles when the concentration pro9le or di6usion is fully developed. Gill showed that the nondimensional time (Dmol ·t=r 2 ) required to reach a fully developed di6usion is approximately equal to 0.5 (Gill & Sankarasubramanian, 1970), and other researches also showed that dispersion in curved tubes develops more quickly than in straight tubes due to radial mixing caused by the secondary 1ow in curved tubes (Erdogan & Chatwin, 1967; Andersson & Berglin, 1981; Nunge, Lin, & Gill, 1972; van den Berg, 1979; Johnson & Kamm, 1986). As to the particle dispersion in lung airways, various models have been suggested for the e6ective di6usivity in the successively bifurcating tubes, but most of these models considered the dispersion in lung airways as a purely di6usional process, which means that dispersions occurring in various parts of the lung airways are totally independent (Scherer, Shendalman, Greene, & Bouhuys, 1975; Ultman & Thomas, 1979; Yu, 1975). However, some recent studies found out that particle dispersion does not become fully-developed in most regions of the lung airways, but dispersion and deposition of particles in both the conducting airways and acinus are dependent on 1ow conditions and mixing state of particles in the previous generations (Sarangapani & Wexler, 1999; Lee, Lee, & Kim, 2000; Lee & Lee, 2001). Lee and Lee (2002) found out, through numerical simulations for a model airway with four successive bifurcations, that in a cyclic breathing particle dispersion can even become reduced at the start of exhalation by as much as the increase that occurred in the last generation during inhalation. But the dispersion of particles during a single or multiple complete cycles of breathing through many bifurcations or various geometric structures and 1ow patterns is not well understood yet. The acinus covers a considerable volume of the lung, and therefore transport of particles in the acinus becomes more important as breathing becomes deeper. Though the acinus has a bifurcating structure like the conducting airways, transport of particles in the acinus may be di6erent from that in the conducting airways since the ducts in the acinus are surrounded by alveoli and also they are expanding or contracting during a breathing cycle unlike the conducting airways remaining almost rigid. Tsuda (Tsuda, Butler, & Fredberg, 1994a, b) examined the e6ect of alveolated structure on the particle deposition, and Darquenne (Darquenne, 2001, 2002) showed that the distribution of particle deposition in a two-dimensional bifurcating geometry becomes much di6erent in the presence of alveoli. Federspiel (Federspiel & Fredberg, 1988) compared De6 for a duct with and without axisymmetric alveoli under the condition of the same total volume, and concluded that De6 in the alveolated tube is smaller than that in a straight tube when molecular di6usion is dominant, but is larger when convective dispersion is dominant. It was also found that the ratio of alveolar to duct volume and the aperture size are important factors for the particle dispersion in alveolated tubes. Tsuda (Tsuda, Federspiel, Grant, & Fredberg, 1991) con9rmed Federspiel’s results through experiments using an alveolated channel. The results of Federspiel and Tsuda hold good only under the conditions of relatively small Peclet number, dispersion in a steady state, and no wall motion. In the past 10 years many studies were performed on the 1ow structure and particle motion in an alveolus (Tsuda, Henry, & Butler, 1995; Tippe & Tsuda, 1999; Butler & Tsuda, 1997;
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Haber, Bulter, Brenner, Emanuel, & Tsuda, 2000). These studies were focused on the particle mixing inside an alveolus, and showed that particle motion is irreversible and chaotic in a rhythmically expanding alveolus even when Reynolds number is very small and particle di6usion is negligible. Especially, it was observed that there exists a threshold number of cycles where the mixing characteristics undergoes a transition from one with a well-de9ned structure to one with a complete mixing. It was also shown that the ratio of the change rate of the alveolar volume to the ductal 1ow rate is the dominant parameter governing the particle mixing in an alveolus. Though there are a number of studies on the 1ow pattern and particle transport in the acinus, studies on the e6ect of alveolar structure and its wall motion on the transport velocity and axial dispersion of particles are scarce. For example, in the aerosol bolus technique particle mixing during just a single cycle is important. Despite mean velocity of particles and residence time are important parameters for the mode shift of the aerosol bolus, past studies were devoted only to the dispersion or mixing of particles. Dispersion of particles of 0.5 –1 m size will not reach a steady state within one breathing cycle, but most studies treated only the dispersion in a cyclic steady state. In some studies predicting dispersion and deposition of particles, De6 obtained for the conditions typically found in the conducting airways is applied to the acinus region or even a stronger assumption is used that particles simply follow the mean 1ow at each generation (Darquenne & Paiva, 1994; Darquenne, Brand, Heyder, & Paiva, 1997; Koblinger & Hofmann, 1990; Taulbee & Yu, 1975; Hofmann, Koblinger, & Heyder, 1994; Edwards, 1994). Recently, Sarangapani (Sarangapani & Wexler, 1999) considered the characteristic 1ow structure in the alveolated tube, and modi9ed the dispersion and axial movement of particles in the acinus, but in the absence of wall motion. This study focuses on the e6ect of wall motion of an alveolated duct on the axial dispersion and mean motion of particles. Dispersion and mean axial position of particles in an axisymmetrically alveolated tube which is expanding or contracting are compared with those for a straight tube (Section 2.2). Though gravitational sedimentation may play some role in particle transport in the acinus region in the real lung, it is not considered in this study because the main objective of this study is to clarify the fundamental e6ect of the existence of alveolus and the wall motion on particle dispersion. Our principal 9ndings are that the dispersion in an alveolated tube is larger than that in a straight tube during the expanding phase but is smaller in the contracting phase, and the transport velocity of particles in an alveolated tube is always smaller than that in a straight tube. Strouhal number determines Qa =Qc , the ratio of alveolar 1owrate to duct 1owrate, and is a dominant parameter in the axial motion of particles when structural variables are 9xed. 2. Methods and model description 2.1. Model structure The model structure is a straight circular tube surrounded by axisymmetric alveoli. The whole structure is composed of 30 identical units or cells, and each single cell has one alveolus, Fig. 1. The structure is uniquely de9ned when aperture angle, Lcell =dc and da =dc are speci9ed. Parameter values used in this study are aperture angle = 50◦ , Lcell =dc = 1:2, da =dc = 0:8, and the volumetric ratio of alveolus to central duct, Va =Vc = 2:358 (Fig. 1). These structural values are approximate averages for the acinus in the lung model of Weibel (1963).
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Fig. 1. De9nition of the model structure.
The model expands or contracts in such a manner that the volume changes linearly with time while the geometry remains self-similar throughout the respiration cycle (Ardila, Horie, & Hildebrandt, 1974; Gil & Weibel, 1972; Gil, Bachofen, & Weibel, 1979; Miki, Butler, Rogers, & Lehr, 1993). The reference point for the volume change is the center of the proximal (cell 1) inlet cross-section. Flow enters through cell 1 during expansion (inhalation), and cell 30 during contraction (exhalation). The volume changes as much as 33%, which corresponds to the breathing condition from the functional residual capacity of 3000 to 4000 ml. Results of particle transport for di6erent structural parameters and breathing volumes are discussed in Section 3.3. 2.2. Flow ;elds and particle trajectory 2.2.1. Flow The equations for 1ow 9eld calculation are the continuity and the Navier–Stokes equation. ∇ · u = 0;
(1)
@u (2) + (u · ∇)u = −∇P + ∇2 u: @t These equations are numerically solved on moving grids. Used for solution is a commercial package (CFX-F3D) with hybrid scheme in 9nite di6erencing for advection and backward di6erence scheme for time di6erencing. In the aerosol bolus technique subjects follow a prescribed breathing pattern where the 1ow rate is kept constant either in inhalation or in exhalation. So a parabolic velocity pro9le is speci9ed at the inlet boundary with the 1ow rate kept constant. Also the constant pressure condition and no-slip condition are speci9ed at the outlet boundary and on the wall, respectively. Flows in the expanding and the contracting phases are obtained separately over a period of one half breathing cycle each. For hydrodynamic similarity, Reynolds number and Womersley or Strouhal number are matched. The range of nondimensional parameters covered in this study
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2.5 Uchida(1977) Numerical
1.0 0.8
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r(t)/ro(t)
Fig. 2. Axial and radial velocity pro9les in an expanding or contracting straight tube are compared with the results of Uchida (1977): (a) axial velocity in expansion; (b) radial velocity in expansion; (c) axial velocity in contraction; and (d) radial velocity in contraction. The results shown are for (dr=dt) · r= = 1, where um is the mean 1ow velocity, and vw the wall velocity.
is 0:004855 ¡ Re ¡ 1:6632; 0:0202 ¡ Wo ¡ 0:128, and 0:00355 ¡ St ¡ 0:1073, which are the average 1ow conditions in the acinus at the tracheal breathing 1owrate of 250 ml=s and breathing period of 8 s, typical 1ow conditions used in the aerosol bolus inhalation experiments. All numerical setups were tested and con9rmed satisfactory by comparing the numerical results with the analytical solution of (Uchida & Aoki, 1977) for the case of expanding or contracting straight tube (Fig. 2). Total number of active control volumes used in numerical calculation was approximately 100,000, and the use of any larger number of active control volumes than this did not result in any appreciable di6erence in the results for particle dispersion. Integration time step used was 1=50 of the breathing period, and the e6ect of using a smaller time step was con9rmed negligible. 2.2.2. Particle tracking The dispersion of particles is calculated from the positions of particles obtained by integrating the following equation of motion. Since it has been well con9rmed that molecular di6usion can be safely neglected (Lee et al., 2000), the 1ow velocity 9eld u(x; t) obtained as above is used for the
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particle velocity 9eld: dx=dt = u(x; t);
x(t = 0) = xo
(3)
Total number of 1751 particles is released at the inlet in the form of a parabolic particle 1ux, and their trajectories are tracked with time. From the positions of the 1751 particles, mean axial position, axial dispersion, and e6ective di6usivity are calculated as follows (Eqs. (4) – (6)). It was con9rmed that the use of any larger number of particles did not a6ect the calculated results appreciably. N xi x = i=1 ; (4) N N (xi − x)2 2 x = i=1 ; (5) N De6 =
1 dx2 · : 2 dt
(6)
2.3. Mean /<x>s
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0
2
4
6
8
10
12
14
16
18
time/tc
(a) 2.0 1.8 1.6 1.2 1.0
2
σx /σx, s
2
1.4
0.8 0.6 0.4 0.2 0.0
0
2
4
6
8
(b)
Fig. 3. Streamlines for a stationary alveolated tube: (a) Re = 0:01 and (b) Re = 5:0.
10
12
14
16
18
20
<x>/Lcell
Fig. 4. Comparison of the time change of (a) the mean axial position and (b) the axial dispersion between the alveolated and straight tubes in the absence of wall motion (Re = 0:01). tc is the characteristic time scale in a cell, Lcell =um .
be attributed to the fact that 1ow structure changes only negligibly with Re in so far as Re remains small. 3.2. Characteristics of particle transport in the presence of wall motion As discussed in Section 3.1, the characteristics of particle transport in a stationary alveolated duct are rather simple, and the di6erences caused by the existence of alveoli can be described in a simple
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Fig. 5. Illustration of particle bolus pro9les in stationary tubes: A for a straight tube; B for an alveolated tube at the same time as A; and C for an alveolated tube at the time of same mean position as A.
2.0
2.0 <x>/<x>s
1.5
2
1.0
1.0
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<x>/<x>s
2
σ x /σ x, s
2
σx /σx, s
1.5
0.0
0
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2
3
4
5
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Re(inlet) Fig. 6. Comparison of the steady values of mean axial position and axial dispersion between the alveolated and straight tubes in the absence of wall motion for various Reynolds numbers.
manner. But 1ow structure and particle transport become complicated when the duct wall expands or contracts. General characteristics of 1ow 9eld and particle dispersion in an alveolated tube with wall motion will be presented in this section, and various parameters a6ecting the transport of particles will be discussed in Section 3.3. As a 9rst step toward understanding the characteristics of a moving alveolated tube, the e6ect of wall motion for a straight tube will be brie1y discussed 9rst. Contrary to the case of a stationary tube, mean axial 1ow velocity at any position inside the 1ow tube is decreased with time in expansion and increased in contraction. Then axial velocity of particles in an expanding tube gets steadily decreased always lower than in a stationary tube, but in a contracting tube gets steadily increased always higher than in a stationary tube. This di6erence in the axial velocity 9eld results in a di6erent axial dispersion even when a parabolic pro9le of axial velocity is reserved. In a contracting tube particles near the tube center penetrate into a deeper region axially, and the axial distance between nearby particles near the tube center becomes larger than in a stationary tube, resulting in a much larger dispersion. On the contrary, the axial dispersion in an expanding tube becomes smaller than in a stationary tube (Fig. 7). When the mean position and dispersion of particles are de9ned based on the volume not on the axial position, absolute values are changed a little bit, but the qualitative characteristics remain almost the same.
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 20
20
15
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25
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stationary expanding contracting
60
stationary expanding contracting
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10 0 0
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Fig. 7. Comparison of particle transport characteristics in a straight tube (Re = 0:05; Wo = 0:0286 at the inlet of the 9rst cell): (a) x with time; and (b) axial dispersion at the same x:tin (=Lcell =um ) and Lin are the characteristic time scale and the cell length, respectively, at the inlet (cell 1 for expansion and cell 30 for contraction).
(b)
0
5
10
15
20
<x>/L in
Fig. 8. Comparison of particle transport characteristics of an alveolated tube in the presence of wall motion (Re = 0:05; Wo = 0:0286 at the inlet of the 9rst cell): (a) x with time; (b) axial dispersion at the same x. tin (=Lcell =um ) and Lin are the characteristic time scale and the cell length, respectively, at the inlet (cell 1 for expansion and cell 30 for contraction).
The characteristics of particle transport for an alveolated tube in the presence of wall motion are di6erent from those for a straight tube. The e6ect of wall motion on the mean particle motion for an alveolated tube is similar to that of straight tube, but the e6ect on dispersion is opposite to the case of straight tube. Axial velocity of particles in an alveolated tube becomes smaller during expansion and larger during contraction than when stationary, just as in a straight tube. But dispersion in an alveolated tube becomes larger during expansion and smaller during contraction than when stationary, which is the opposite trend to the case of straight tube (Fig. 8). The reasons for these results will be discussed in detail in the following. As shown in previous studies (Tsuda et al., 1995; Tippe & Tsuda, 1999), 1ow 9eld in the presence of wall motion is much di6erent from that in the absence of wall motion. During expansion a part
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Fig. 9. Typical streamlines in the 9rst cell of an alveolated tube with wall motion, at the time of 1=50 of the half period: (a) Re=1:663; Wo=0:0639, expansion; (b) Re=0:0493; Wo=0:0286, expansion; (c) Re=1:663; Wo=0:0639, contraction; and (d) Re = 0:0493; Wo = 0:0286, contraction.
of the streamlines near the duct wall penetrate into the alveolus terminating on the alveolar wall, and during contraction some streamlines starting from the alveolar wall penetrate into the central duct (Fig. 9). The pro9le of axial velocity in the central duct is changed due to the wall motion, but noticeable only in the distal region (near cell 30) no matter whether expanding or contracting. As is observed from the typical velocity pro9les shown in Fig. 10, velocity pro9le in the proximal region (near cell 1) is only negligibly blunter than the parabolic shape, but the velocity pro9le in the distal region (near cell 30) becomes much distorted. And the amount of 1ow exchange between the central duct and alveoli increases as Reynolds number decreases for 9xed Wo. When mean axial position of particles in an expanding or contracting alveolated tube is plotted for a typical value of Re and Wo (Fig. 11a), it is observed that, for both expansion and contraction, mean position is smaller than in a straight tube because 1ow velocity in the region surrounded by alveoli is smaller. Also the ratio of mean position in the alveolated tube to that in the straight tube (x=xs ) decreases with time during expansion but increases during contraction. This result can be easily inferred if mean radial position of particles (r) at various axial positions is plotted in Fig. 12, where the radial position is nondimensionalized by the radius of the central duct at the corresponding time. Particles inside the alveoli are assumed to stay on the wall because particle motion inside the duct area is of major concern here. During expansion r increases i.e. particles
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Fig. 10. Pro9les of the axial velocity near the inlet and outlet during (a) expansion and (b) contraction (Re = 0:0493; Wo = 0:0286). The pro9les are at the mid point of the 9rst and the last straight duct.
move toward the wall, and during contraction r decreases i.e. particles move toward the duct center. Then during expansion particles become accumulated near the wall where the 1ow velocity is low, and during contraction near the duct center where the 1ow velocity is high. The migration of particles in the radial direction is caused by the interchange of 1uid 1ow between the central duct and alveoli. Typical particle trajectories near the wall are shown in Fig. 13, where it is clearly seen that particles move closer to the wall during expansion and farther away from the wall during contraction. The decrease of x for expansion is mainly due to those particles entering the alveoli that do not proceed axially any more, and the decrease of x is accelerated due to the radial motion of particles toward the wall. For contraction there is another factor a6ecting x=xs . It was shown previously that velocity pro9le in the distal region is much di6erent from the parabolic shape i.e. 1ow velocity is higher near the wall and lower near the tube center than for the parabolic pro9le. Because particles are initially distributed in a parabolic shape, a blunter pro9le of 1ow velocity makes x smaller than for the parabolic pro9le, which results in a much larger decrease of x for contraction in the early stage. But as particles proceed toward the proximal region, this e6ect becomes negligible and axial motion of particles becomes fast due to the radial motion of particles toward the tube center.
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3.0
1.8
Expansion Contraction
1.6
2.5 2.0
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σ x /σ x, s 2
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Expansion Contraction
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Deff /Deff, s
1.2 1.0 0.8 0.6 0.4 0.2 0.0
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0
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Fig. 11. Comparison of particle transport characteristics between the alveolated and straight tube in the presence of wall motion for a typical 1ow condition (Re = 0:0493; Wo = 0:0286 in the 9rst cell): (a) x; (b) dispersion at the same x; and (c) De6 at the same x. (x for the alveolated tube and xs for the straight tube).
Exchange of 1uid 1ow between central duct and alveolus also changes the pattern of particle dispersion. When dispersion is compared at the same x, dispersion in the alveolated tube is larger for expansion and smaller for contraction than in a straight tube (Fig. 11b). During expansion in an alveolated tube dispersion is small in the early stage due to a little blunter velocity pro9le in the central duct and a slow 1uid 1ow near the 9rst alveolus. But particles move gradually toward the wall where radial velocity gradient is large, and some particles enter the alveolus without ever escaping, both of which make dispersion much larger than in a straight tube. During contraction dispersion is much smaller than in a straight tube in the early stage due to a much blunter velocity pro9le in the central duct, and this trend is maintained during the whole process because particles move toward the tube center where radial velocity gradient is small. The di6erent trends of dispersion between expansion and contraction can be clearly seen in Fig. 13. At the inlet the radial distance between A and B is equal to that between C and D, but the A–B radial distance increases while the C–D radial distance decreases as particles proceed. Then the axial velocity di6erence between A and B is larger than that for C and D, and A falls farther behind B than C behind D. When the distribution of axial positions is converted into the e6ective di6usivity, De6 =De6 ; s is larger than 1.0
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1207
2.0 1.8
Expansion Contraction
1.6 1.4
Deff /Deff, s
1.2 1.0 0.8 0.6 0.4 0.2 0.0
0
5
10
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20
<x>/L cell
Fig. 12.Change of the mean radial position of particles with breathing (Re = 0:0493; Wo = 0:0286 in the 9rst cell). r = ( Ni=1 ri )=N where ri is the radial position of a particle.
Fig. 13. Typical particle trajectories near the wall for expansion (upper) and contraction (lower). (Re=0:0493; Wo=0:0286 in the 9rst cell).
and increasing with breathing for expansion, but for contraction is smaller than 1.0 and decreasing with breathing (Fig. 11c). To sum up the results, for an expanding alveolated tube x=xs is smaller than 1.0 decreasing with time, and De6 =De6 ; s is larger than 1.0 increasing with time, while for a contracting tube x=xs is smaller than 1.0 increasing with time and De6 =De6 ; s is smaller than 1.0 decreasing with time. 3.3. E?ect of various parameters There are several parameters which can a6ect the transport of particles in the alveolated tube with wall motion. Aperture angle, amount of volume change during breathing, and structural ratios,
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for example da =dc , are parameters related with the geometry of acinus, and Re, Wo, and St are nondimensional parameters related with the periodic unsteady 1ow in a duct. Previous studies, including Tsuda (Tsuda et al., 1995; Tippe & Tsuda, 1999; Butler & Tsuda, 1997; Haber et al., 2000), showed that Qa =Qc determines the 1ow structure and particle motion in a rhythmically expanding alveolated tube. In order to see in a more systematic manner how various parameters a6ect Qa =Qc , Qa =Qc is expressed, using the mass conservation condition, as a simple function of various parameters stated above (Eq. (10) for expansion and Eq. (11) for contraction) for the case of linear volume change with time (Appendix B) Qa = St · Qc
B 2 d x (dmax =dmin )3 − 1 ; · · dc dc 1 + ((dmax =dmin )3 − 1) · (t=Th )
(10)
Qa = St · Qc
B 2 d x (dmax =dmin )3 − 1 · · : 3 dc dc (dmax =dmin ) − ((dmax =dmin )3 − 1) · (t=Th )
(11)
Eqs. (10) and (11) clearly show how various parameters a6ect the amount of 1ow exchange between B c or Va ==Vc , rate of volume the central duct and alveoli. Qa =Qc increases when such parameters as d=d change during breathing (dmax =dmin ), and St increase. So Qa =Qc will be large in the distal region of acinus where Va =Vc and St are large (Weibel, 1963) and also at the start of inhalation and the end of exhalation. One conclusion drawn from Eqs. (10) and (11) is that, when Qa =Qc is the dominant parameter for the particle transport, x and dispersion should depend solely on St once the structural parameters are 9xed. In order to con9rm this conclusion, transport of particles is compared between 1ow conditions of di6erent Re and Wo but identical St. As can be seen in Fig. 14, x and dispersion are almost the same if St is identical, no matter whether St is high or low, nor whether tube is expanding or contracting. This result implying that the only important 1ow parameter to be considered is St is in agreement with Tsuda (Tsuda et al., 1995). Based on the previous result, the change of x and dispersion are examined with St varied. (Fig. 15) From the 9gures it is clear that the characteristics of particle transport for various values of St remain similar. Since St is the ratio of the characteristic 1ow time to the oscillation period, the 1ow, both for expansion and contraction, becomes similar to those in a stationary alveolated tube as St approaches zero, while the unsteady e6ect becomes larger as St increases. When St is very small, x and dispersion for both expansion and contraction are very similar to those for a stationary tube, which means that particle transport in the proximal region of acinus can be modeled using a stationary geometry model. As St is increased, the amount of 1ow exchange between central duct and alveolus increases and the characteristics of 1ow structure, and particle transport become deviated from the stationary case. A higher Qa =Qc makes radial movement of particles more active, and thus during expansion more particles get accumulated near the wall of the central duct or enter the alveolus, while more particles move near the tube center during contraction. Therefore the increase of dispersion during expansion and the decrease of dispersion during contraction become more considerable as St is increased. For both expansion and contraction, the wall motion does not make x=xs di6er much from that in a stationary tube. It is to be noted that main variation in x and dispersion due to wall motion occurs in the distal region (near the outlet for expansion and near the inlet for contraction). It is because the change of 1ow structure due to a variation of St is
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 2.0 1.8 Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
1.6
2
1.2
σ x /σ x, s
1.0
2
<x>/<x>s
1.4
0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.8
1.0
2.0 1.8
2
σ x /σ x, s 2
0.8
0.8 0.6 0.4
0.2
0.2
0.8
0.0
1.0
time/Th
(c)
20
1.0
0.4
0.6
15
1.2
0.6
0.4
10
Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
1.4
1.0
0.2
5
1.6
Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
1.2
0.0 0.0
0
<x>/L cell
1.8 1.4
Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
(b)
2.0 1.6
<x>/<x>s
0.6
time/Th
(a)
2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
1209
(d)
0
5
10
15
20
<x>/L cell
Fig. 14. Comparison of particle transport characteristics for di6erent values of Re and Wo but at the same St (St = 0:01056 in the 9rst cell): (a) x for expansion; (b) dispersion for expansion; (c) x for contraction; and (d) dispersion for contraction. For any smaller value of St, the results remain the same.
Table 1 Summary of conditions simulated in the present study for the alveolated tube with wall motion. All nondimensional parameter values are those at the start of expansion St
Re
Wo
Cell1
Cell30
Cell1
Cell30
1:5658E − 3 1:5658E − 3 6:3025E − 3 1:0563E − 2 1:0563E − 2
3:5547E − 3 3:5547E − 3 2:4201E − 2 1:0728E − 1 1:0728E − 1
1:6632E − 1 1:6632E + 0 1:6528E + 0 4:9305E − 1 4:9305E − 2
7:3261E − 2 7:3261E − 1 4:3043E − 1 4:8548E − 2 4:8548E − 3
2:0225E − 2 6:3958E − 2 1:2792E − 1 9:0450E − 2 2:8602E − 2
larger in the distal region than in the proximal region; for example, St is increased about 20-folds in the 30th cell when St is increased seven-folds in the 1st cell (Table 1). Though we conducted numerical simulations for a typical 1ow conditions found in bolus experiments, breathing 1owrate of 250 ml=s and breathing period of 8 s, the characteristics of particle transport for higher 1ow rates and shorter breathing period can also be easily inferred from these
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 2.0 2.5
1.8 St=0.00157 St=0.00630 St=0.01056
1.6 1.2
2
σ x /σ x, s
1.0
1.5
2
<x>/<x>s
1.4
St=0.00157 St=0.00630 St=0.01056
2.0
0.8
1.0
0.6 0.4
0.5
0.2 0.0
0
5
15
0.0
20
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2 1.0 0.8
0.2 0.0
0
5
10
15
1.0 0.8
0
5
10
15
20
<x>/L cell
(d) 2.00 1.75
St=0.00157 St=0.00630 St=0.01056
1.6 1.4
St=0.00157 St=0.00630 St=0.01056
1.50
Deff /Deff, s
1.2
2
σ x /σ x, s
St=0.00157 St=0.00630 St=0.01056
1.2
0.0
20
1.8
2
1.0 0.8 0.6
1.25 1.00 0.75 0.50
0.4
0.25
0.2 0.0
20
0.2
2.0
(e)
15
0.4
<x>/L cell
(c)
10
0.6
St=0.00157 St=0.00630 St=0.01056
0.4
5
<x>/Lcell
2.0
0.6
0
(b)
<x>/<x>s
Deff /Deff, s
10
<x>/L cell
(a)
0
5
10
15
<x>/L cell
20
0.00
25
(f)
0
5
10
15
20
25
<x>/d c
Fig. 15. E6ect of Strouhal number on particle transport characteristics: (a) x for expansion; (b) dispersion for expansion; (c) De6 for expansion; (d) x for contraction; (e) dispersion for expansion; and (f) De6 for expansion. St in the 9gure is for the 9rst cell.
results. For example, the e6ect of wall motion on particle transport in the acinus for a breathing condition of 500 ml=s and 4 s will be similar with the one for 250 ml=s and 8 s, because the Strouhal numbers for both conditions are the same. Since the results of this study is for a single duct with 30 alveolated cells, the quantitative results for x and De6 may not hold good for the real acinus which has a bifurcating structure. Also the di6erence in particle transport between alveolated and straight tube is thought to become larger as Va =Vc and the volume change during breathing increase due to the increased Qa =Qc , which is not con9rmed yet, either.
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1211
3.4. Limitations and physiological implication of the results Axial movement and dispersion of particles in an alveolated tube with wall motion are very di6erent from the case of a straight tube, and the di6erences come from consistent motion of particles in the radial direction due to 1ow exchange between the central duct and alveoli. The only mechanism causing the radial motion in a single tube is the exchange 1ow induced by the wall motion, so the radial motion is uni-directional. In a bifurcating structure of the real acinus, however, the radial position of particles can change abruptly at the bifurcations, and the geometric parameters of the alveolated duct such as the diameter and the diameter-to-length ratio varies with bifurcation, all of which factors may cause changes in the characteristics of particle transport. Despite these limitations, the features of particle transport in an alveolated tube observed in this study present meaningful clues to the solution of several problems still in question to date. One of them is the prediction of particle deposition when the particle size is in the range where particles follow 1ow streamlines very well. Some of theoretical models (Darquenne et al., 1997; Sarangapani & Wexler, 2000), underestimate particle deposition, especially in the range between 0.1 and 1:0 m. Sarangapani (Sarangapani & Wexler, 2000) suggested that this problem might be overcome if particles remaining in the lung after exhalation are assumed to deposit but this correction overestimates deposition. Based on the results of this study it may be suggested that the consistent motion of particles toward the wall during expansion (inhalation) is a possible mechanism for the enhanced deposition of particles, especially in the aerosol bolus experiments. Once entering an alveolus by this mechanism, particles cannot escape the alveolus completely due to the irreversible motion of particles in an alveolus (Tsuda et al., 1995), and may eventually get deposited. The other problem is the mode shift in the aerosol bolus experiments. Darquenne (Darquenne & Paiva, 1994) tried to predict bolus parameters using various models of e6ective di6usivity, but all the existing models underestimated the mode shift. As discussed in Section 3.2, axial motion of particles is decelerated during expansion or inhalation and is accelerated during contraction or exhalation, which can result in a minus mode shift. But a direct comparison or check is not possible in the present state because the model structure used in this study is much di6erent from the real acinus. 4. Conclusions Fluid 1ow and particle motion in an alveolated tube in the presence of wall motion are analyzed numerically, and mean axial position and dispersion of particles are compared between the alveolated and the straight tube. In the absence of wall motion, the existence of alveoli makes the e6ective 1ow cross-section in the alveolated region larger. Then the particle migration velocity becomes lower, by a factor of 0.85 for the conditions of this study, and dispersion becomes slightly larger, and these characteristics are independent of Re. In the presence of wall motion, the characteristics of 1uid 1ow and particle transport in an alveolated tube are much di6erent from those in a straight tube due to interchange of 1uid 1ow between central duct and alveolus. First of all, 1ow velocity pro9le becomes blunter than the parabolic pro9le in the distal region, no matter whether expanding or contracting. For both expansion and contraction the axial motion of particles in an alveolated tube is smaller than that in a straight tube, due to the decelerated 1ow in the alveolated region. In an expanding alveolated tube
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
particles get accumulated near the wall and some particles enter alveolus, which make dispersion larger than in an expanding straight tube. In a contracting alveolated tube particles consistently move toward the tube center, making dispersion smaller than in a contracting straight tube. For the conditions of this study Strouhal number is the dominant parameter determining particle dispersion. In the presence of wall motion the di6erence in particle transport between alveolated tube and straight tube becomes larger as Strouhal number is increased. Acknowledgements This work was supported by the Brain Korea 21 project in 2002. Appendix A. When the total volume changes linearly with time during the expansion phase while the geometry remains self similar, the volume and each length of the model structure will change with time as Vmax − Vmin · t; (A.1) V (t) = Vmin + Th d(t) =
V (t) Vmin
= 1+
1=3
Vmax t 1=3 · dmin = 1 + −1 · · dmin Vmin Th
dmax dmin
3
−1
t · Th
1=3 · dmin :
(A.2)
B and cell length (x) can be written as Then the time-change of e6ective diameter (d) B = (1 + · t)1=3 · dB min ; d(t)
(A.3)
x(t) = (1 + · t)1=3 · xmin ;
(A.4)
where dB and are de9ned as follows: 3 V + V d c a max · dc ; = −1 Th : dB = Vc dmin
(A.5)
Now the volume change can be written in terms of the 1owrates at the inlet and outlet of a cell using the continuity condition, as follows. dVcell d(dB 2 · x) dx · d2c : (A.6) = · = · uin − uout − dt 4 dt 4 dt B d x d x d d 2 · x + dB · = uin − uout − · d2c : (A.7) 2dB · dt dt dt
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1213
Since (A.3) and (A.4) can be rewritten as d dB 1 B = · · d(t); dt 3 1+·t
dx 1 = · · x(t) dt 3 1+·t
(A.8)
by inserting Eq. (A.8) into (A.7), the ratio of mean velocities at the inlet and the outlet is obtained as B 2 uout (t) 1 d 1 =1+ − · : (A.9) ·x· uin (t) 3 dc 1 + · t uin (t) A similar equation can be easily written for the contraction phase as 2 uout (t) dB 1 1 =1+ ·x· · ; − uin (t) dc 3 − · t uin (t) where
=
dmax dmin
3
:
(A.10)
(A.11)
Appendix B. The ratio of the duct-alveolous 1ow exchange to the 1owrate in the central duct can be obtained using the results of Appendix A. Flowrate in the central duct can be written as Eq. (B.1), and the 1owrate entering or escaping the alveolus can be obtained as Eq. (B.3) using the di6erence of 1owrates between inlet and outlet (Eq. (B.2)).
1 d x
d x
Qc = · uin − · · d2c = uB · · d2c ; + uout − (B.1) 2 dt in dt out 4 4
d x
d x
PQ = uin − · · d2c ; − uout +
dt in dt out 4
(B.2)
2 Va dc Qa = PQ · : = PQ · 1 − Va + V c dB
(B.3)
When nondimensional 1ow parameters are de9ned as in Eq. (B.4), using Eq. (A.9), the ratio of the duct-alveolous 1ow exchange to the 1owrate in the central duct for expansion is obtained as uB · dc 1 4 · Wo2 2 dc Re = · ; St = · ; (B.4) ; Wo =
2 Th · 2 Re Qa = St · Qc
B 2 d x (dmax =dmin )3 − 1 · · : dc dc 1 + ((dmax =dmin )3 − 1) · (t=Th )
(B.5)
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
The ratio for contraction is obtained through a similar procedure as B 2 Qa d x (dmax =dmin )3 − 1 : = St · · · Qc dc dc (dmax =dmin )3 − ((dmax =dmin )3 − 1) · (t=Th )
(B.6)
References Anderson, P. J., & Dolovich, M. B. (1994). Aerosols as diagnostic tools. Journal of Aerosol Medicine, 7, 77–88. Andersson, B., & Berglin, T. (1981). Dispersion in laminar 1ow through a circular tube. Proceedings of the Royal Society of London, 377A, 251–268. Ardila, R. T., Horie, T., & Hildebrandt, J. (1974). Macroscopic isotropy of lung expansion. Respiration Physiology, 20, 105–115. Butler, J. P., & Tsuda, A. (1997). E6ect of convective stretching and folding on aerosol mixing deep in the lung, assessed by approximate entropy. Journal of Applied Physiology, 83, 800–809. Darquenne, C. (2001). A realistic two-dimensional model of aerosol transport and deposition in the alveolar zone of the human lung. Journal of Aerosol Science, 32, 1961–1974. Darquenne, C. (2002). Heterogeneity of aerosol deposition in a two-dimensional model of human alveolated duct. Journal of Aerosol Science, 33, 1261–1278. Darquenne, C., Brand, P., Heyder, J., & Paiva, M. (1997). Aerosol dispersion in human lung: Comparison between numerical simulation and experiments for bolus tests. Journal of Applied Physiology, 83, 966–974. Darquenne, C., & Paiva, M. (1994). One-dimensional simulation of aerosol transport and deposition in the human lung. Journal of Applied Physiology, 77, 2889–2898. Edwards, D. A. (1994). A general theory of the macrotransport of nondepositing particles in the lung by convective dispersion. Journal of Aerosol Science, 25, 543–565. Engel, L. A., & Paiva, M. (1985). Gas mixing and distribution in the lung. New York: Marcel Dekker. Erdogan, M. E., & Chatwin, P. C. (1967). The e6ects of curvature and buoyancy on the laminar dispersion of solute in a horizontal tube. Journal of Fluid Mechanics, 29, 465–484. Federspiel, W. J., & Fredberg, J. J. (1988). Axial dispersion in respiratory bronchioles and alveolar ducts. Journal of Applied Physiology, 64, 2614–2621. Gil, J., Bachofen, P. G., & Weibel, E. R. (1979). Alveolar volume-surface area relation in air and saline-9lled lungs 9xed by vascular perfusion. Journal of Applied Physiology, 45, 990–1001. Gil, J., & Weibel, E. R. (1972). Morphological study of pressure volume hysteresis in rat lungs 9xed by vascular perfusion. Respiration Physiology, 15, 190–213. Gill, W. N., & Sankarasubramanian, R. (1970). Exact analysis of unsteady convective di6usion. Proceedings of the Royal Society of London, 316, 341–350. Haber, S., Bulter, J. P., Brenner, H., Emanuel, I., & Tsuda, A. (2000). Shear 1ow over a self-similar expanding pulmonary alveolus during rhythmical breathing. Journal of Fluid Mechanics, 405, 243–268. Hofmann, W., Koblinger, L., & Heyder, J. (1994). The e6ect of convective mixing on particle transport in the human respiratory tract. The Annals of Occupational Hygiene, 38, 167–174. Johnson, M., & Kamm, R. D. (1986). Numerical studies of steady 1ow dispersion at low Dean number in a gently curving tube. Journal of Fluid Mechanics, 172, 329–345. Koblinger, L., & Hofmann, W. (1990). Monte carlo modeling of aerosol deposition in human lungs. Part 1: Simulation of particle transport in a stochastic lung structure. Journal of Aerosol Science, 21, 661–674. Lee, D. Y., & Lee, J. W. (2001). Dispersion during exhalation of an aerosol bolus in a double bifurcation. Journal of Aerosol Science, 32, 805–815. Lee, D. Y., & Lee, J. W. (2002). Dispersion of aerosol bolus during one respiration cycle in a model of lung airways. Journal of Aerosol Science, 33, 1219–1234. Lee, J. W., Lee, D. Y., & Kim, W. S. (2000). Dispersion of an aerosol bolus in a double bifurcation. Journal of Aerosol Science, 31, 491–505. Miki, H., Butler, J. P., Rogers, R. A., & Lehr, J. (1993). Geometric hysteresis in pulmonary surface to volume ratio during tidal breathing. Journal of Applied Physiology, 75, 1630–1636.
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Nunge, R. J., Lin, T. S., & Gill, W. N. (1972). Laminar dispersion in curved tubes and channels. Journal of Fluid Mechanics, 51, 363–383. Sarangapani, R., & Wexler, A. S. (1999). Modeling aerosol bolus dispersion in human airways. Journal of Aerosol Science, 30, 1345–1362. Sarangapani, R., & Wexler, A. S. (2000). The role of dispersion in particle deposition in human airways. Toxicological Sciences, 54, 229–236. Scherer, P. W., Shendalman, L. M., Greene, N. M., & Bouhuys, A. (1975). Measurement of axial di6usivities in a model of the bronchial airways. Journal of Applied Physiology, 38, 719–723. Taulbee, D. B., & Yu, C. P. (1975). A theory of aerosol deposition in the human respiratory tract. Journal of Applied Physiology, 38, 77–85. Taylor, G. I. (1953). Dispersion of solute matter in solvent 1owing slowly through a tube. Proceedings of the Royal Society of London, 219A, 186–203. Tippe, A., & Tsuda, A. (1999). Recirculating 1ow in an expanding alveolar model: Experimental evidence of 1ow-induced mixing of aerosols in the pulmonary acinus. Journal of Aerosol Science, 31, 979–986. Tsuda, A., Butler, J. P., & Fredberg, J. J. (1994a). E6ects of alveolated duct structure on aerosol kinetics 1: Di6usional deposition in the absence of gravity. Journal of Applied Physiology, 76, 2497–2509. Tsuda, A., Butler, J. P., & Fredberg, J. J. (1994b). E6ects of alveolated duct structure on aerosol kinetics 2: Gravitational sedimentation and inertial impaction. Journal of Applied Physiology, 76, 2510–2516. Tsuda, A., Federspiel, W. J., Grant, P. A. Jr., & Fredberg, J. J. (1991). Axial dispersion of inert species in alveolated channels. Chemical Engineering Science, 46, 1419–1426. Tsuda, A., Henry, F. S., & Butler, P. J. (1995). Chaotic mixing of alveolated duct 1ow in rhythmically expanding pulmonary acinus. Journal of Applied Physiology, 79, 1055–1063. Uchida, S., & Aoki, H. (1977). Unsteady 1ows in a semi-in9nite contracting or expanding pipe. Journal of Fluid Mechanics, 82, 371–387. Ultman, J. S., & Thomas, M. W. (1979). Longitudinal mixing in pulmonary airways: Comparison of inspiration and expiration. Journal of Applied Physiology, 46, 799–805. van den Berg, J. H. M. (1979). Measurement of axial dispersion in laminar 1ow through coiled capillary tubes. Chemical Engineering Science, 34, 1345–1347. Weibel, E. R. (1963). Morphometry of the human lung. New York: Academic Press. Yu, C. P. (1975). On equation of gas transport in the lung. Respiration Physiology, 23, 257–266.
Characteristics of particle transport in an expanding or contracting alveolated tube Dong Y. Lee, Jin W. Lee∗ Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Hyoja 31, Pohang, Kyunngbuk 790-784, South Korea Received 7 January 2003; accepted 11 April 2003
Abstract Features of 1ow and particle transport in a single axisymmetric alveolated tube with 30 cells in the presence of wall motion are analyzed numerically for 1ow conditions typically found in the Aerosol Bolus experiments: the tracheal breathing 1owrate of 250 ml=s and breathing period of 8 s. Di6erent characteristics of particle transport between the alveolated and the straight tube are explained based on the details of 1ow and particle motion. For both expansion and contraction the axial motion of particles in an alveolated tube is smaller than in a straight tube due to the decelerated 1ow in the alveolated region. In an expanding alveolated tube particles get accumulated near the wall and some particles enter alveolus, which make dispersion larger than that in an expanding straight tube. In a contracting alveolated tube particles consistently move toward the tube center, and 1ow velocity pro9le is much blunter than the parabolic one, making dispersion smaller than in a contracting straight tube. In the presence of wall motion, the di6erence in particle transport between alveolated tube and straight tube becomes larger as Strouhal number is increased. Limitations and physiological implications of the results are discussed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Lung airways; Particle transport; Alveolated tube; Wall motion
1. Introduction Dispersion of particles in lung airways plays an important role in such problems as nonuniform particle deposition, drug delivery, and diagnosis of lung diseases using aerosols. Especially in the assessment of pulmonary abnormalities due to exposure to particulate pollutants, an exact model for particle dispersion is essential, since the pattern of regional deposition of particles is determined ∗
Corresponding author. Tel.: +82-54-279-2967; fax: +82-54-279-3199. E-mail address: [email protected] (J.W. Lee).
0021-8502/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0021-8502(03)00097-1
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
Nomenclature da diameter of an alveolus dc diameter of the central duct dmax maximum diameter of the central duct dmin minimum diameter of the central duct dB e6ective diameter of an alveolated tube, (Va + Vc )=Vc · dc De6 e6ective di6usivity De6 ; s e6ective di6usivity for the straight tube Lcell length of a cell Lin cell length at the inlet (cell 1 for expansion and cell 30 for contraction) Qa 1owrate entering or going out of an alveolus Qc 1owrate in the central duct P pressure r tube radius ra alveolus radius Re Reynolds number, uB · dc = St Strouhal number, (dc = u)=T B h t time tin characteristic time scale at the inlet (cell 1 for expansion and cell 30 for contraction), Lcell =um Th half period of breathing uc axial velocity in the central duct ue6 hypothetical axial velocity, uin mean axial velocity at the inlet of a cell um mean axial velocity uout mean axial velocity at the outlet of a cell uB mean axial velocity in the central duct, 12 · (uin − ddtx |in + uout − ddtx |out ) Va volume of an alveolus Vc volume of the central duct of one cell Vcell volume of one cell Vmax maximum volume of the whole structure Vmin minimum volume of the whole structure Wo Womersley number, dc =2 · 2=Th · x mean axial position of particles r mean radial position of particles 2 x axial dispersion of particles Greek letters
aperture angle dynamic viscosity kinematic viscosity density
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1195
by the transport and dispersion of particles (Anderson & Dolovich, 1994; Engel & Paiva, 1985; Sarangapani & Wexler, 1999). Taylor was the 9rst to formulate the problem of dispersion for a fully developed laminar 1ow in a straight circular tube (Taylor, 1953), where the e6ective di6usivity (De6 ) becomes proportional to (u · d)2 and inversely proportional to molecular di6usivity of particles when the concentration pro9le or di6usion is fully developed. Gill showed that the nondimensional time (Dmol ·t=r 2 ) required to reach a fully developed di6usion is approximately equal to 0.5 (Gill & Sankarasubramanian, 1970), and other researches also showed that dispersion in curved tubes develops more quickly than in straight tubes due to radial mixing caused by the secondary 1ow in curved tubes (Erdogan & Chatwin, 1967; Andersson & Berglin, 1981; Nunge, Lin, & Gill, 1972; van den Berg, 1979; Johnson & Kamm, 1986). As to the particle dispersion in lung airways, various models have been suggested for the e6ective di6usivity in the successively bifurcating tubes, but most of these models considered the dispersion in lung airways as a purely di6usional process, which means that dispersions occurring in various parts of the lung airways are totally independent (Scherer, Shendalman, Greene, & Bouhuys, 1975; Ultman & Thomas, 1979; Yu, 1975). However, some recent studies found out that particle dispersion does not become fully-developed in most regions of the lung airways, but dispersion and deposition of particles in both the conducting airways and acinus are dependent on 1ow conditions and mixing state of particles in the previous generations (Sarangapani & Wexler, 1999; Lee, Lee, & Kim, 2000; Lee & Lee, 2001). Lee and Lee (2002) found out, through numerical simulations for a model airway with four successive bifurcations, that in a cyclic breathing particle dispersion can even become reduced at the start of exhalation by as much as the increase that occurred in the last generation during inhalation. But the dispersion of particles during a single or multiple complete cycles of breathing through many bifurcations or various geometric structures and 1ow patterns is not well understood yet. The acinus covers a considerable volume of the lung, and therefore transport of particles in the acinus becomes more important as breathing becomes deeper. Though the acinus has a bifurcating structure like the conducting airways, transport of particles in the acinus may be di6erent from that in the conducting airways since the ducts in the acinus are surrounded by alveoli and also they are expanding or contracting during a breathing cycle unlike the conducting airways remaining almost rigid. Tsuda (Tsuda, Butler, & Fredberg, 1994a, b) examined the e6ect of alveolated structure on the particle deposition, and Darquenne (Darquenne, 2001, 2002) showed that the distribution of particle deposition in a two-dimensional bifurcating geometry becomes much di6erent in the presence of alveoli. Federspiel (Federspiel & Fredberg, 1988) compared De6 for a duct with and without axisymmetric alveoli under the condition of the same total volume, and concluded that De6 in the alveolated tube is smaller than that in a straight tube when molecular di6usion is dominant, but is larger when convective dispersion is dominant. It was also found that the ratio of alveolar to duct volume and the aperture size are important factors for the particle dispersion in alveolated tubes. Tsuda (Tsuda, Federspiel, Grant, & Fredberg, 1991) con9rmed Federspiel’s results through experiments using an alveolated channel. The results of Federspiel and Tsuda hold good only under the conditions of relatively small Peclet number, dispersion in a steady state, and no wall motion. In the past 10 years many studies were performed on the 1ow structure and particle motion in an alveolus (Tsuda, Henry, & Butler, 1995; Tippe & Tsuda, 1999; Butler & Tsuda, 1997;
1196
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
Haber, Bulter, Brenner, Emanuel, & Tsuda, 2000). These studies were focused on the particle mixing inside an alveolus, and showed that particle motion is irreversible and chaotic in a rhythmically expanding alveolus even when Reynolds number is very small and particle di6usion is negligible. Especially, it was observed that there exists a threshold number of cycles where the mixing characteristics undergoes a transition from one with a well-de9ned structure to one with a complete mixing. It was also shown that the ratio of the change rate of the alveolar volume to the ductal 1ow rate is the dominant parameter governing the particle mixing in an alveolus. Though there are a number of studies on the 1ow pattern and particle transport in the acinus, studies on the e6ect of alveolar structure and its wall motion on the transport velocity and axial dispersion of particles are scarce. For example, in the aerosol bolus technique particle mixing during just a single cycle is important. Despite mean velocity of particles and residence time are important parameters for the mode shift of the aerosol bolus, past studies were devoted only to the dispersion or mixing of particles. Dispersion of particles of 0.5 –1 m size will not reach a steady state within one breathing cycle, but most studies treated only the dispersion in a cyclic steady state. In some studies predicting dispersion and deposition of particles, De6 obtained for the conditions typically found in the conducting airways is applied to the acinus region or even a stronger assumption is used that particles simply follow the mean 1ow at each generation (Darquenne & Paiva, 1994; Darquenne, Brand, Heyder, & Paiva, 1997; Koblinger & Hofmann, 1990; Taulbee & Yu, 1975; Hofmann, Koblinger, & Heyder, 1994; Edwards, 1994). Recently, Sarangapani (Sarangapani & Wexler, 1999) considered the characteristic 1ow structure in the alveolated tube, and modi9ed the dispersion and axial movement of particles in the acinus, but in the absence of wall motion. This study focuses on the e6ect of wall motion of an alveolated duct on the axial dispersion and mean motion of particles. Dispersion and mean axial position of particles in an axisymmetrically alveolated tube which is expanding or contracting are compared with those for a straight tube (Section 2.2). Though gravitational sedimentation may play some role in particle transport in the acinus region in the real lung, it is not considered in this study because the main objective of this study is to clarify the fundamental e6ect of the existence of alveolus and the wall motion on particle dispersion. Our principal 9ndings are that the dispersion in an alveolated tube is larger than that in a straight tube during the expanding phase but is smaller in the contracting phase, and the transport velocity of particles in an alveolated tube is always smaller than that in a straight tube. Strouhal number determines Qa =Qc , the ratio of alveolar 1owrate to duct 1owrate, and is a dominant parameter in the axial motion of particles when structural variables are 9xed. 2. Methods and model description 2.1. Model structure The model structure is a straight circular tube surrounded by axisymmetric alveoli. The whole structure is composed of 30 identical units or cells, and each single cell has one alveolus, Fig. 1. The structure is uniquely de9ned when aperture angle, Lcell =dc and da =dc are speci9ed. Parameter values used in this study are aperture angle = 50◦ , Lcell =dc = 1:2, da =dc = 0:8, and the volumetric ratio of alveolus to central duct, Va =Vc = 2:358 (Fig. 1). These structural values are approximate averages for the acinus in the lung model of Weibel (1963).
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1197
Fig. 1. De9nition of the model structure.
The model expands or contracts in such a manner that the volume changes linearly with time while the geometry remains self-similar throughout the respiration cycle (Ardila, Horie, & Hildebrandt, 1974; Gil & Weibel, 1972; Gil, Bachofen, & Weibel, 1979; Miki, Butler, Rogers, & Lehr, 1993). The reference point for the volume change is the center of the proximal (cell 1) inlet cross-section. Flow enters through cell 1 during expansion (inhalation), and cell 30 during contraction (exhalation). The volume changes as much as 33%, which corresponds to the breathing condition from the functional residual capacity of 3000 to 4000 ml. Results of particle transport for di6erent structural parameters and breathing volumes are discussed in Section 3.3. 2.2. Flow ;elds and particle trajectory 2.2.1. Flow The equations for 1ow 9eld calculation are the continuity and the Navier–Stokes equation. ∇ · u = 0;
(1)
@u (2) + (u · ∇)u = −∇P + ∇2 u: @t These equations are numerically solved on moving grids. Used for solution is a commercial package (CFX-F3D) with hybrid scheme in 9nite di6erencing for advection and backward di6erence scheme for time di6erencing. In the aerosol bolus technique subjects follow a prescribed breathing pattern where the 1ow rate is kept constant either in inhalation or in exhalation. So a parabolic velocity pro9le is speci9ed at the inlet boundary with the 1ow rate kept constant. Also the constant pressure condition and no-slip condition are speci9ed at the outlet boundary and on the wall, respectively. Flows in the expanding and the contracting phases are obtained separately over a period of one half breathing cycle each. For hydrodynamic similarity, Reynolds number and Womersley or Strouhal number are matched. The range of nondimensional parameters covered in this study
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 1.2
2.5 Uchida(1977) Numerical
1.0 0.8
1.5
v/vw
u/um
2.0
1.0
Uchida(1977) Numerical
0.4
0.5 0.0 0.0
0.6
0.2
0.2
0.4
0.6
0.8
r(t)/ro(t)
(a)
0.0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
r(t)/ro(t)
(b)
2.0
1.2 1.0
1.5
v/vw
u/um
0.8 1.0 Uchida(1977) Numerical
0.6
Uchida(1977) Numerical
0.4
0.5
0.2 0.0 0.0
(c)
0.2
0.4
0.6
r(t)/ro(t)
0.8
0.0 0.0
1.0
(d)
0.2
0.4
0.6
0.8
1.0
r(t)/ro(t)
Fig. 2. Axial and radial velocity pro9les in an expanding or contracting straight tube are compared with the results of Uchida (1977): (a) axial velocity in expansion; (b) radial velocity in expansion; (c) axial velocity in contraction; and (d) radial velocity in contraction. The results shown are for (dr=dt) · r= = 1, where um is the mean 1ow velocity, and vw the wall velocity.
is 0:004855 ¡ Re ¡ 1:6632; 0:0202 ¡ Wo ¡ 0:128, and 0:00355 ¡ St ¡ 0:1073, which are the average 1ow conditions in the acinus at the tracheal breathing 1owrate of 250 ml=s and breathing period of 8 s, typical 1ow conditions used in the aerosol bolus inhalation experiments. All numerical setups were tested and con9rmed satisfactory by comparing the numerical results with the analytical solution of (Uchida & Aoki, 1977) for the case of expanding or contracting straight tube (Fig. 2). Total number of active control volumes used in numerical calculation was approximately 100,000, and the use of any larger number of active control volumes than this did not result in any appreciable di6erence in the results for particle dispersion. Integration time step used was 1=50 of the breathing period, and the e6ect of using a smaller time step was con9rmed negligible. 2.2.2. Particle tracking The dispersion of particles is calculated from the positions of particles obtained by integrating the following equation of motion. Since it has been well con9rmed that molecular di6usion can be safely neglected (Lee et al., 2000), the 1ow velocity 9eld u(x; t) obtained as above is used for the
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1199
particle velocity 9eld: dx=dt = u(x; t);
x(t = 0) = xo
(3)
Total number of 1751 particles is released at the inlet in the form of a parabolic particle 1ux, and their trajectories are tracked with time. From the positions of the 1751 particles, mean axial position, axial dispersion, and e6ective di6usivity are calculated as follows (Eqs. (4) – (6)). It was con9rmed that the use of any larger number of particles did not a6ect the calculated results appreciably. N xi x = i=1 ; (4) N N (xi − x)2 2 x = i=1 ; (5) N De6 =
1 dx2 · : 2 dt
(6)
2.3. Mean /<x>s
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0
2
4
6
8
10
12
14
16
18
time/tc
(a) 2.0 1.8 1.6 1.2 1.0
2
σx /σx, s
2
1.4
0.8 0.6 0.4 0.2 0.0
0
2
4
6
8
(b)
Fig. 3. Streamlines for a stationary alveolated tube: (a) Re = 0:01 and (b) Re = 5:0.
10
12
14
16
18
20
<x>/Lcell
Fig. 4. Comparison of the time change of (a) the mean axial position and (b) the axial dispersion between the alveolated and straight tubes in the absence of wall motion (Re = 0:01). tc is the characteristic time scale in a cell, Lcell =um .
be attributed to the fact that 1ow structure changes only negligibly with Re in so far as Re remains small. 3.2. Characteristics of particle transport in the presence of wall motion As discussed in Section 3.1, the characteristics of particle transport in a stationary alveolated duct are rather simple, and the di6erences caused by the existence of alveoli can be described in a simple
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
Fig. 5. Illustration of particle bolus pro9les in stationary tubes: A for a straight tube; B for an alveolated tube at the same time as A; and C for an alveolated tube at the time of same mean position as A.
2.0
2.0 <x>/<x>s
1.5
2
1.0
1.0
0.5
0.5
2
<x>/<x>s
2
σ x /σ x, s
2
σx /σx, s
1.5
0.0
0
1
2
3
4
5
0.0
Re(inlet) Fig. 6. Comparison of the steady values of mean axial position and axial dispersion between the alveolated and straight tubes in the absence of wall motion for various Reynolds numbers.
manner. But 1ow structure and particle transport become complicated when the duct wall expands or contracts. General characteristics of 1ow 9eld and particle dispersion in an alveolated tube with wall motion will be presented in this section, and various parameters a6ecting the transport of particles will be discussed in Section 3.3. As a 9rst step toward understanding the characteristics of a moving alveolated tube, the e6ect of wall motion for a straight tube will be brie1y discussed 9rst. Contrary to the case of a stationary tube, mean axial 1ow velocity at any position inside the 1ow tube is decreased with time in expansion and increased in contraction. Then axial velocity of particles in an expanding tube gets steadily decreased always lower than in a stationary tube, but in a contracting tube gets steadily increased always higher than in a stationary tube. This di6erence in the axial velocity 9eld results in a di6erent axial dispersion even when a parabolic pro9le of axial velocity is reserved. In a contracting tube particles near the tube center penetrate into a deeper region axially, and the axial distance between nearby particles near the tube center becomes larger than in a stationary tube, resulting in a much larger dispersion. On the contrary, the axial dispersion in an expanding tube becomes smaller than in a stationary tube (Fig. 7). When the mean position and dispersion of particles are de9ned based on the volume not on the axial position, absolute values are changed a little bit, but the qualitative characteristics remain almost the same.
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 20
20
15
<x>/L in
25
<x>/L in
15 10
stationary expanding contracting
5
10
0 5
10
15
20
25
30
35
0
5
10
15
20
25
time/tin
(a)
time/t in
(a)
stationary expanding contracting
5
0 0
1203
80
120
70 100
2
50
2
40
σ x /L in
60
2
σ x /L in
2
80
stationary expanding contracting
60
stationary expanding contracting
30
40
20 20 0
(b)
10 0 0
5
10
15
20
25
<x>/L in
Fig. 7. Comparison of particle transport characteristics in a straight tube (Re = 0:05; Wo = 0:0286 at the inlet of the 9rst cell): (a) x with time; and (b) axial dispersion at the same x:tin (=Lcell =um ) and Lin are the characteristic time scale and the cell length, respectively, at the inlet (cell 1 for expansion and cell 30 for contraction).
(b)
0
5
10
15
20
<x>/L in
Fig. 8. Comparison of particle transport characteristics of an alveolated tube in the presence of wall motion (Re = 0:05; Wo = 0:0286 at the inlet of the 9rst cell): (a) x with time; (b) axial dispersion at the same x. tin (=Lcell =um ) and Lin are the characteristic time scale and the cell length, respectively, at the inlet (cell 1 for expansion and cell 30 for contraction).
The characteristics of particle transport for an alveolated tube in the presence of wall motion are di6erent from those for a straight tube. The e6ect of wall motion on the mean particle motion for an alveolated tube is similar to that of straight tube, but the e6ect on dispersion is opposite to the case of straight tube. Axial velocity of particles in an alveolated tube becomes smaller during expansion and larger during contraction than when stationary, just as in a straight tube. But dispersion in an alveolated tube becomes larger during expansion and smaller during contraction than when stationary, which is the opposite trend to the case of straight tube (Fig. 8). The reasons for these results will be discussed in detail in the following. As shown in previous studies (Tsuda et al., 1995; Tippe & Tsuda, 1999), 1ow 9eld in the presence of wall motion is much di6erent from that in the absence of wall motion. During expansion a part
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
Fig. 9. Typical streamlines in the 9rst cell of an alveolated tube with wall motion, at the time of 1=50 of the half period: (a) Re=1:663; Wo=0:0639, expansion; (b) Re=0:0493; Wo=0:0286, expansion; (c) Re=1:663; Wo=0:0639, contraction; and (d) Re = 0:0493; Wo = 0:0286, contraction.
of the streamlines near the duct wall penetrate into the alveolus terminating on the alveolar wall, and during contraction some streamlines starting from the alveolar wall penetrate into the central duct (Fig. 9). The pro9le of axial velocity in the central duct is changed due to the wall motion, but noticeable only in the distal region (near cell 30) no matter whether expanding or contracting. As is observed from the typical velocity pro9les shown in Fig. 10, velocity pro9le in the proximal region (near cell 1) is only negligibly blunter than the parabolic shape, but the velocity pro9le in the distal region (near cell 30) becomes much distorted. And the amount of 1ow exchange between the central duct and alveoli increases as Reynolds number decreases for 9xed Wo. When mean axial position of particles in an expanding or contracting alveolated tube is plotted for a typical value of Re and Wo (Fig. 11a), it is observed that, for both expansion and contraction, mean position is smaller than in a straight tube because 1ow velocity in the region surrounded by alveoli is smaller. Also the ratio of mean position in the alveolated tube to that in the straight tube (x=xs ) decreases with time during expansion but increases during contraction. This result can be easily inferred if mean radial position of particles (r) at various axial positions is plotted in Fig. 12, where the radial position is nondimensionalized by the radius of the central duct at the corresponding time. Particles inside the alveoli are assumed to stay on the wall because particle motion inside the duct area is of major concern here. During expansion r increases i.e. particles
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1205
2.0
u/um
1.5
1.0 1st cell 30th cell Parabolic
0.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
r/ro
(a) 2.0
u/um
1.5
1.0 1st cell 30th cell Parabolic
0.5
0.0 0.0
(b)
0.2
0.4
r/ro
Fig. 10. Pro9les of the axial velocity near the inlet and outlet during (a) expansion and (b) contraction (Re = 0:0493; Wo = 0:0286). The pro9les are at the mid point of the 9rst and the last straight duct.
move toward the wall, and during contraction r decreases i.e. particles move toward the duct center. Then during expansion particles become accumulated near the wall where the 1ow velocity is low, and during contraction near the duct center where the 1ow velocity is high. The migration of particles in the radial direction is caused by the interchange of 1uid 1ow between the central duct and alveoli. Typical particle trajectories near the wall are shown in Fig. 13, where it is clearly seen that particles move closer to the wall during expansion and farther away from the wall during contraction. The decrease of x for expansion is mainly due to those particles entering the alveoli that do not proceed axially any more, and the decrease of x is accelerated due to the radial motion of particles toward the wall. For contraction there is another factor a6ecting x=xs . It was shown previously that velocity pro9le in the distal region is much di6erent from the parabolic shape i.e. 1ow velocity is higher near the wall and lower near the tube center than for the parabolic pro9le. Because particles are initially distributed in a parabolic shape, a blunter pro9le of 1ow velocity makes x smaller than for the parabolic pro9le, which results in a much larger decrease of x for contraction in the early stage. But as particles proceed toward the proximal region, this e6ect becomes negligible and axial motion of particles becomes fast due to the radial motion of particles toward the tube center.
1206
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 2.0
3.0
1.8
Expansion Contraction
1.6
2.5 2.0
1.2
σ x /σ x, s 2
1.0 0.8
1.5
2
<x>/<x>s
1.4
Expansion Contraction
0.6 0.4
1.0 0.5
0.2 0.0 0.0
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0
5
10
(b)
time/Th
15
20
<x>/L cell
2.0 1.8
Expansion Contraction
1.6 1.4
Deff /Deff, s
1.2 1.0 0.8 0.6 0.4 0.2 0.0
(c)
0
5
10
15
20
<x>/L cell
Fig. 11. Comparison of particle transport characteristics between the alveolated and straight tube in the presence of wall motion for a typical 1ow condition (Re = 0:0493; Wo = 0:0286 in the 9rst cell): (a) x; (b) dispersion at the same x; and (c) De6 at the same x. (x for the alveolated tube and xs for the straight tube).
Exchange of 1uid 1ow between central duct and alveolus also changes the pattern of particle dispersion. When dispersion is compared at the same x, dispersion in the alveolated tube is larger for expansion and smaller for contraction than in a straight tube (Fig. 11b). During expansion in an alveolated tube dispersion is small in the early stage due to a little blunter velocity pro9le in the central duct and a slow 1uid 1ow near the 9rst alveolus. But particles move gradually toward the wall where radial velocity gradient is large, and some particles enter the alveolus without ever escaping, both of which make dispersion much larger than in a straight tube. During contraction dispersion is much smaller than in a straight tube in the early stage due to a much blunter velocity pro9le in the central duct, and this trend is maintained during the whole process because particles move toward the tube center where radial velocity gradient is small. The di6erent trends of dispersion between expansion and contraction can be clearly seen in Fig. 13. At the inlet the radial distance between A and B is equal to that between C and D, but the A–B radial distance increases while the C–D radial distance decreases as particles proceed. Then the axial velocity di6erence between A and B is larger than that for C and D, and A falls farther behind B than C behind D. When the distribution of axial positions is converted into the e6ective di6usivity, De6 =De6 ; s is larger than 1.0
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1207
2.0 1.8
Expansion Contraction
1.6 1.4
Deff /Deff, s
1.2 1.0 0.8 0.6 0.4 0.2 0.0
0
5
10
15
20
<x>/L cell
Fig. 12.Change of the mean radial position of particles with breathing (Re = 0:0493; Wo = 0:0286 in the 9rst cell). r = ( Ni=1 ri )=N where ri is the radial position of a particle.
Fig. 13. Typical particle trajectories near the wall for expansion (upper) and contraction (lower). (Re=0:0493; Wo=0:0286 in the 9rst cell).
and increasing with breathing for expansion, but for contraction is smaller than 1.0 and decreasing with breathing (Fig. 11c). To sum up the results, for an expanding alveolated tube x=xs is smaller than 1.0 decreasing with time, and De6 =De6 ; s is larger than 1.0 increasing with time, while for a contracting tube x=xs is smaller than 1.0 increasing with time and De6 =De6 ; s is smaller than 1.0 decreasing with time. 3.3. E?ect of various parameters There are several parameters which can a6ect the transport of particles in the alveolated tube with wall motion. Aperture angle, amount of volume change during breathing, and structural ratios,
1208
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
for example da =dc , are parameters related with the geometry of acinus, and Re, Wo, and St are nondimensional parameters related with the periodic unsteady 1ow in a duct. Previous studies, including Tsuda (Tsuda et al., 1995; Tippe & Tsuda, 1999; Butler & Tsuda, 1997; Haber et al., 2000), showed that Qa =Qc determines the 1ow structure and particle motion in a rhythmically expanding alveolated tube. In order to see in a more systematic manner how various parameters a6ect Qa =Qc , Qa =Qc is expressed, using the mass conservation condition, as a simple function of various parameters stated above (Eq. (10) for expansion and Eq. (11) for contraction) for the case of linear volume change with time (Appendix B) Qa = St · Qc
B 2 d x (dmax =dmin )3 − 1 ; · · dc dc 1 + ((dmax =dmin )3 − 1) · (t=Th )
(10)
Qa = St · Qc
B 2 d x (dmax =dmin )3 − 1 · · : 3 dc dc (dmax =dmin ) − ((dmax =dmin )3 − 1) · (t=Th )
(11)
Eqs. (10) and (11) clearly show how various parameters a6ect the amount of 1ow exchange between B c or Va ==Vc , rate of volume the central duct and alveoli. Qa =Qc increases when such parameters as d=d change during breathing (dmax =dmin ), and St increase. So Qa =Qc will be large in the distal region of acinus where Va =Vc and St are large (Weibel, 1963) and also at the start of inhalation and the end of exhalation. One conclusion drawn from Eqs. (10) and (11) is that, when Qa =Qc is the dominant parameter for the particle transport, x and dispersion should depend solely on St once the structural parameters are 9xed. In order to con9rm this conclusion, transport of particles is compared between 1ow conditions of di6erent Re and Wo but identical St. As can be seen in Fig. 14, x and dispersion are almost the same if St is identical, no matter whether St is high or low, nor whether tube is expanding or contracting. This result implying that the only important 1ow parameter to be considered is St is in agreement with Tsuda (Tsuda et al., 1995). Based on the previous result, the change of x and dispersion are examined with St varied. (Fig. 15) From the 9gures it is clear that the characteristics of particle transport for various values of St remain similar. Since St is the ratio of the characteristic 1ow time to the oscillation period, the 1ow, both for expansion and contraction, becomes similar to those in a stationary alveolated tube as St approaches zero, while the unsteady e6ect becomes larger as St increases. When St is very small, x and dispersion for both expansion and contraction are very similar to those for a stationary tube, which means that particle transport in the proximal region of acinus can be modeled using a stationary geometry model. As St is increased, the amount of 1ow exchange between central duct and alveolus increases and the characteristics of 1ow structure, and particle transport become deviated from the stationary case. A higher Qa =Qc makes radial movement of particles more active, and thus during expansion more particles get accumulated near the wall of the central duct or enter the alveolus, while more particles move near the tube center during contraction. Therefore the increase of dispersion during expansion and the decrease of dispersion during contraction become more considerable as St is increased. For both expansion and contraction, the wall motion does not make x=xs di6er much from that in a stationary tube. It is to be noted that main variation in x and dispersion due to wall motion occurs in the distal region (near the outlet for expansion and near the inlet for contraction). It is because the change of 1ow structure due to a variation of St is
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 2.0 1.8 Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
1.6
2
1.2
σ x /σ x, s
1.0
2
<x>/<x>s
1.4
0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.8
1.0
2.0 1.8
2
σ x /σ x, s 2
0.8
0.8 0.6 0.4
0.2
0.2
0.8
0.0
1.0
time/Th
(c)
20
1.0
0.4
0.6
15
1.2
0.6
0.4
10
Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
1.4
1.0
0.2
5
1.6
Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
1.2
0.0 0.0
0
<x>/L cell
1.8 1.4
Re=0.0494, Wo=0.0286 Re=0.494, Wo=0.0905
(b)
2.0 1.6
<x>/<x>s
0.6
time/Th
(a)
2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
1209
(d)
0
5
10
15
20
<x>/L cell
Fig. 14. Comparison of particle transport characteristics for di6erent values of Re and Wo but at the same St (St = 0:01056 in the 9rst cell): (a) x for expansion; (b) dispersion for expansion; (c) x for contraction; and (d) dispersion for contraction. For any smaller value of St, the results remain the same.
Table 1 Summary of conditions simulated in the present study for the alveolated tube with wall motion. All nondimensional parameter values are those at the start of expansion St
Re
Wo
Cell1
Cell30
Cell1
Cell30
1:5658E − 3 1:5658E − 3 6:3025E − 3 1:0563E − 2 1:0563E − 2
3:5547E − 3 3:5547E − 3 2:4201E − 2 1:0728E − 1 1:0728E − 1
1:6632E − 1 1:6632E + 0 1:6528E + 0 4:9305E − 1 4:9305E − 2
7:3261E − 2 7:3261E − 1 4:3043E − 1 4:8548E − 2 4:8548E − 3
2:0225E − 2 6:3958E − 2 1:2792E − 1 9:0450E − 2 2:8602E − 2
larger in the distal region than in the proximal region; for example, St is increased about 20-folds in the 30th cell when St is increased seven-folds in the 1st cell (Table 1). Though we conducted numerical simulations for a typical 1ow conditions found in bolus experiments, breathing 1owrate of 250 ml=s and breathing period of 8 s, the characteristics of particle transport for higher 1ow rates and shorter breathing period can also be easily inferred from these
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215 2.0 2.5
1.8 St=0.00157 St=0.00630 St=0.01056
1.6 1.2
2
σ x /σ x, s
1.0
1.5
2
<x>/<x>s
1.4
St=0.00157 St=0.00630 St=0.01056
2.0
0.8
1.0
0.6 0.4
0.5
0.2 0.0
0
5
15
0.0
20
2.0
1.8
1.8
1.6
1.6
1.4
1.4
1.2 1.0 0.8
0.2 0.0
0
5
10
15
1.0 0.8
0
5
10
15
20
<x>/L cell
(d) 2.00 1.75
St=0.00157 St=0.00630 St=0.01056
1.6 1.4
St=0.00157 St=0.00630 St=0.01056
1.50
Deff /Deff, s
1.2
2
σ x /σ x, s
St=0.00157 St=0.00630 St=0.01056
1.2
0.0
20
1.8
2
1.0 0.8 0.6
1.25 1.00 0.75 0.50
0.4
0.25
0.2 0.0
20
0.2
2.0
(e)
15
0.4
<x>/L cell
(c)
10
0.6
St=0.00157 St=0.00630 St=0.01056
0.4
5
<x>/Lcell
2.0
0.6
0
(b)
<x>/<x>s
Deff /Deff, s
10
<x>/L cell
(a)
0
5
10
15
<x>/L cell
20
0.00
25
(f)
0
5
10
15
20
25
<x>/d c
Fig. 15. E6ect of Strouhal number on particle transport characteristics: (a) x for expansion; (b) dispersion for expansion; (c) De6 for expansion; (d) x for contraction; (e) dispersion for expansion; and (f) De6 for expansion. St in the 9gure is for the 9rst cell.
results. For example, the e6ect of wall motion on particle transport in the acinus for a breathing condition of 500 ml=s and 4 s will be similar with the one for 250 ml=s and 8 s, because the Strouhal numbers for both conditions are the same. Since the results of this study is for a single duct with 30 alveolated cells, the quantitative results for x and De6 may not hold good for the real acinus which has a bifurcating structure. Also the di6erence in particle transport between alveolated and straight tube is thought to become larger as Va =Vc and the volume change during breathing increase due to the increased Qa =Qc , which is not con9rmed yet, either.
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1211
3.4. Limitations and physiological implication of the results Axial movement and dispersion of particles in an alveolated tube with wall motion are very di6erent from the case of a straight tube, and the di6erences come from consistent motion of particles in the radial direction due to 1ow exchange between the central duct and alveoli. The only mechanism causing the radial motion in a single tube is the exchange 1ow induced by the wall motion, so the radial motion is uni-directional. In a bifurcating structure of the real acinus, however, the radial position of particles can change abruptly at the bifurcations, and the geometric parameters of the alveolated duct such as the diameter and the diameter-to-length ratio varies with bifurcation, all of which factors may cause changes in the characteristics of particle transport. Despite these limitations, the features of particle transport in an alveolated tube observed in this study present meaningful clues to the solution of several problems still in question to date. One of them is the prediction of particle deposition when the particle size is in the range where particles follow 1ow streamlines very well. Some of theoretical models (Darquenne et al., 1997; Sarangapani & Wexler, 2000), underestimate particle deposition, especially in the range between 0.1 and 1:0 m. Sarangapani (Sarangapani & Wexler, 2000) suggested that this problem might be overcome if particles remaining in the lung after exhalation are assumed to deposit but this correction overestimates deposition. Based on the results of this study it may be suggested that the consistent motion of particles toward the wall during expansion (inhalation) is a possible mechanism for the enhanced deposition of particles, especially in the aerosol bolus experiments. Once entering an alveolus by this mechanism, particles cannot escape the alveolus completely due to the irreversible motion of particles in an alveolus (Tsuda et al., 1995), and may eventually get deposited. The other problem is the mode shift in the aerosol bolus experiments. Darquenne (Darquenne & Paiva, 1994) tried to predict bolus parameters using various models of e6ective di6usivity, but all the existing models underestimated the mode shift. As discussed in Section 3.2, axial motion of particles is decelerated during expansion or inhalation and is accelerated during contraction or exhalation, which can result in a minus mode shift. But a direct comparison or check is not possible in the present state because the model structure used in this study is much di6erent from the real acinus. 4. Conclusions Fluid 1ow and particle motion in an alveolated tube in the presence of wall motion are analyzed numerically, and mean axial position and dispersion of particles are compared between the alveolated and the straight tube. In the absence of wall motion, the existence of alveoli makes the e6ective 1ow cross-section in the alveolated region larger. Then the particle migration velocity becomes lower, by a factor of 0.85 for the conditions of this study, and dispersion becomes slightly larger, and these characteristics are independent of Re. In the presence of wall motion, the characteristics of 1uid 1ow and particle transport in an alveolated tube are much di6erent from those in a straight tube due to interchange of 1uid 1ow between central duct and alveolus. First of all, 1ow velocity pro9le becomes blunter than the parabolic pro9le in the distal region, no matter whether expanding or contracting. For both expansion and contraction the axial motion of particles in an alveolated tube is smaller than that in a straight tube, due to the decelerated 1ow in the alveolated region. In an expanding alveolated tube
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
particles get accumulated near the wall and some particles enter alveolus, which make dispersion larger than in an expanding straight tube. In a contracting alveolated tube particles consistently move toward the tube center, making dispersion smaller than in a contracting straight tube. For the conditions of this study Strouhal number is the dominant parameter determining particle dispersion. In the presence of wall motion the di6erence in particle transport between alveolated tube and straight tube becomes larger as Strouhal number is increased. Acknowledgements This work was supported by the Brain Korea 21 project in 2002. Appendix A. When the total volume changes linearly with time during the expansion phase while the geometry remains self similar, the volume and each length of the model structure will change with time as Vmax − Vmin · t; (A.1) V (t) = Vmin + Th d(t) =
V (t) Vmin
= 1+
1=3
Vmax t 1=3 · dmin = 1 + −1 · · dmin Vmin Th
dmax dmin
3
−1
t · Th
1=3 · dmin :
(A.2)
B and cell length (x) can be written as Then the time-change of e6ective diameter (d) B = (1 + · t)1=3 · dB min ; d(t)
(A.3)
x(t) = (1 + · t)1=3 · xmin ;
(A.4)
where dB and are de9ned as follows: 3 V + V d c a max · dc ; = −1 Th : dB = Vc dmin
(A.5)
Now the volume change can be written in terms of the 1owrates at the inlet and outlet of a cell using the continuity condition, as follows. dVcell d(dB 2 · x) dx · d2c : (A.6) = · = · uin − uout − dt 4 dt 4 dt B d x d x d d 2 · x + dB · = uin − uout − · d2c : (A.7) 2dB · dt dt dt
D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
1213
Since (A.3) and (A.4) can be rewritten as d dB 1 B = · · d(t); dt 3 1+·t
dx 1 = · · x(t) dt 3 1+·t
(A.8)
by inserting Eq. (A.8) into (A.7), the ratio of mean velocities at the inlet and the outlet is obtained as B 2 uout (t) 1 d 1 =1+ − · : (A.9) ·x· uin (t) 3 dc 1 + · t uin (t) A similar equation can be easily written for the contraction phase as 2 uout (t) dB 1 1 =1+ ·x· · ; − uin (t) dc 3 − · t uin (t) where
=
dmax dmin
3
:
(A.10)
(A.11)
Appendix B. The ratio of the duct-alveolous 1ow exchange to the 1owrate in the central duct can be obtained using the results of Appendix A. Flowrate in the central duct can be written as Eq. (B.1), and the 1owrate entering or escaping the alveolus can be obtained as Eq. (B.3) using the di6erence of 1owrates between inlet and outlet (Eq. (B.2)).
1 d x
d x
Qc = · uin − · · d2c = uB · · d2c ; + uout − (B.1) 2 dt in dt out 4 4
d x
d x
PQ = uin − · · d2c ; − uout +
dt in dt out 4
(B.2)
2 Va dc Qa = PQ · : = PQ · 1 − Va + V c dB
(B.3)
When nondimensional 1ow parameters are de9ned as in Eq. (B.4), using Eq. (A.9), the ratio of the duct-alveolous 1ow exchange to the 1owrate in the central duct for expansion is obtained as uB · dc 1 4 · Wo2 2 dc Re = · ; St = · ; (B.4) ; Wo =
2 Th · 2 Re Qa = St · Qc
B 2 d x (dmax =dmin )3 − 1 · · : dc dc 1 + ((dmax =dmin )3 − 1) · (t=Th )
(B.5)
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D.Y. Lee, J.W. Lee / Aerosol Science 34 (2003) 1193 – 1215
The ratio for contraction is obtained through a similar procedure as B 2 Qa d x (dmax =dmin )3 − 1 : = St · · · Qc dc dc (dmax =dmin )3 − ((dmax =dmin )3 − 1) · (t=Th )
(B.6)
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