Hi’1 LETI\

31

L1 ZTIiEM-\TI(‘,4L VOLUMF

HI:)1 O:i\

39. 1977

PRECIPITATION OF SUBMICRON CHARGED PARTICLES IN HUMAN LUNG AIRWAYS

m C.

P. Yu and K. CHANDRA Department of Engineering Science, Aerospace Engineering and Nuclear Engineering, State University of New York at Buffalo, Buffalo. NY 14214. U.S.A.

Precipitation of charged particles in a tube by their own space charge is investigated theoretically. when the number density of the particles is large enough so that the potential is a smooth function given by Poisson’s equation, and when the number density is small so that only the image force is important. These two approaches have been applied to the data given by Weibel for the human lung. to determine the deposition probabilities at different generations for suhmicron particles when the particle density is I x 10” particles/cm”. The results indicate that the electrostatic dispersion can only lead to a small effect on the lung deposition, the predominant effect is due to the image force exerted on the particles.

I. Iritl~duction. Electrification normally occurs in aerosol generating processes. It has been long suspected that the unipolar charge carried by the particle may have a significant effect on the deposition efficiency of particles in the lung during breathing. Experimental data on the deposition of particles in the lung shows a relatively small fractional deposition under the same condition when condensation aerosols are used. Since the condensation aerosols are free of electric charge. the difference in fractional deposition may be caused by the additional deposition from electrostatic forces. There have been some but limited experimental attempts in the past for the determination of the electrostatic effect on the deposition of the particles in the lung. Earlier data indicated only a relatively minor influence by electric charges (Mercer, 19731. Recent systematic measurements by Melandri e’t crl. (1975) however showed that this effect is quite significant for submicron and micron particles 471

472

C. P. YU AND

K. CHANDRA

carrying unipolar charge of a magnitude of hundred electrons. Since the results are insensitive to the particle density but depend strongly upon the particle charge, they further reasoned that the observed changes are caused primarily by the electrostatic precipitation resulting from the image force between the particle and the wall. The purpose of this paper is to examine the problem analytically by considering only small particles where the Brownian diffusion is the only other important mechanism responsible for the lung deposition. Wilson (1947) was first to study the deposition of charged particles in circular tubes, with reference to the retention of aerosols in the human lung. He considered the case when the deposition was due to electrostatic dispersion and the deposition by other mechanisms was neglected. The results showed that the deposition is independent of the diameter of the tube but depends only upon the total time that an aerosol has spent in each tube. We shall see later that the electrostatic dispersion can only lead to a small effect on the lung deposition. The predominant influence is due to the image force exerted on the particles. For a simple calculation of the lung deposition, we may 2. Formulation. assume that all parts of the airways have the geometry of cylindrical tubes. In addition, the airflow in each tube is assumed to have a flat velocity profile. The later assumption implies that the deposition from air flowing through the tube with a given passage time is equal to the deposition from still air in the same time interval. Consider a system of charged particles of uniform size suspended in air, with uniform particle density CO suddenly brought in a long cylindrical duct of radius R with grounded conducting walls. The space charge of the particles, each carries a charge q, creates nonzero potentials within the system. If the number density, C, of particles is large, their space charge forms a continuum and the potential, 4, is a smooth function given by Poisson’s equation

which because of axial symmetry and independence of axial coordinate reduces t0

1 (7

&p

rati!r -+->=-

qc Eg ~-

3

(2)

where cOis the permittivity of the air and r is the radial distance from the axis of the cylinder.

SUBMICRON

CHARGED

PARTICLES

IN

HUMAN

LUNG

The number density C(r, t) at time t in the unsteady transport equation c?C ._ _= _vej

413

AIRWAYS

state follows the (3)

dt

If the particle flux,J, is assumed to be due only to migration and diffusion, J=KE-DVC

(4)

where 1,is the particle mobility assuming Stokes law given by .

q

AZ-----

6npr, ’ I? is the electric field, D is the coefficient of Brownian diffusion, p is the viscosity of the air, and rP is the radius of the particle. Thus we get

SC

~+~~VC+XV~-DV*C=O.

(6)

Also noting that the Gauss law states that v.&

_v’(&E. Eo

(7)

From (6) and (7) we get

CC -

~3)dC

_j~__-___

ar ar

?t

iqC*

(8)

Eo

(9) Using the transformations

to nondimensionalize

(8) and (9), and dropping the asterisks, we get

?‘cj

TJ+---=

1214

-c

(11)

474

C. P. YU AND

K. CHANDRA

ac

a+ac

a’c

~+QI,,,,+QJ-~=~+;~

tsc

(12)

where (13) Boundary conditions are taken to be:

C=O=#.J C=l

at

at

r=l t=O

4=$(1-r’)

at

for all for all t=O.

t>O, r, and (14)

(1 l)-(14) represent the case when the number density of charged particles is so large that the continuum approach is valid for charge distribution. If the number density of the particles is small, deposition due to electrostatic dispersion will be very small in comparison to that due to the image force exerted on the particles. In this case, the particle flux is due only to convection and diffusion. Thus, _i= Cv, - DVC

(15)

where

ii,=lF=

lqr2 167rs,(R - r)‘R2 ’

(16)

and the force F on the particle is the image force acting in the radial direction indicated by the unit vector f,. The final equation in nondimensional form reduces to:

where

(18) and the boundary conditions for C are given by ( 14).

SUBMICRON

CHARGED

PARTICLES IN HUMAN

LU%G AIRWAYS

475

3. ~~~rn~~i~~~ ~~rnpu~&~t~~)l~. Using the Crank-Nicolson method, finitedifference equations corresponding to (111, (12) and (17) were obtained and solved numerically in an iterative fashion. After convergence was achieved in the major iterations, the mean number density C of particles defined by

c c,=

C2rdr

(19)

was calculated, to give deposition probability P in the form

For the case when the particle number density is large, we obtained from (1 1), (12) and (19) the solution of 23versus r* for different values of Q1 as shown in Figure 1. In the other case, Figure 2 gives the value of the mean particle density versus time for different values of Q2. 4. App~jcut~~~ro the ~urnaFz Lung. The theory developed in the previous sections is applied to the deposition prediction in the human lung. In order to compare with the experimental data reported by Melandri et a/. (1975), their breathing condition and particle properties have been used in our calculation. The lung geometry adopted is that given by Wiebel’s symmetric model A (1963).

Figure

1.

Mean pnrticle density versus time for different values of QI

476

C. p. YU AND K. CHANDRA too

3

I lO-s

Id'

IO-2

I

t*

Figure 2.

Mean particle density versus time for different values of Q2

For the experimental condition of 12 resp/min and 1000cm3 tidal volume without pause, values of Q,, Qz and the nondimensional passage time t* are calculated for each generation. The particle considered has a diameter of 0.33 ,um with unit density, each carrying a charge of 29 electrons. The particle concentration at mouth is 1 x 10’ particles/~m3. Table I gives the results of Qr, Q2, t* as well as the deposition probabilities for different generations. In Table I, column 5 is the deposition probability of the electrostatic dispersion model, PED, obtained by the numerical solution of (11) and (12) for given values of Qi and t* in the generation. Column 6 gives the deposition probability of the image force model, Plo, by solving (17) with listed values of Q2 and t*. The probability due to pure diffusion for small t* was obtained by Ingham (1975) in the following form

Its value for different generations is listed in column 4 for comparison. following conclusions can be reached from this study:

The

6) When the particle density is low as that was used in the experiment of Melandri et al. (1975), Table I shows that the electrostatic dispersion model gives negligible effect on deposition and the increase of deposition is due to the image force between particles and the airway surface. (ii) In the image-force model, the deposition probabilities are independent of the particle density. (iii) Deposition probabilities increase with the increase of the amount of charge carried by each particle.

SLIBMK’RON

/,I/

Precipitation of submicron charged particles in human lung airways.

Hi’1 LETI\ 31 L1 ZTIiEM-\TI(‘,4L VOLUMF HI:)1 O:i\ 39. 1977 PRECIPITATION OF SUBMICRON CHARGED PARTICLES IN HUMAN LUNG AIRWAYS m C. P. Yu and K...
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