WALL DEPOSITION OF SMALL ELLIPSOIDS FROM TURBULENT AIR FLOWS—A BROWNIAN DYNAMICS SIMULATION

J. Aerosol Sci. Vol. 31, No. 10, pp. 1205}1229, 2000( 2000 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0021-8502/00/$ - see front matterPII: S0021-8502(00)00018-5

WALL DEPOSITION OF SMALL ELLIPSOIDS FROM TURBULENTAIR FLOWS*A BROWNIAN DYNAMICS SIMULATION

Fa-Gung Fan*s and Goodarz AhmaditA

*J.C. Wilson Center for Research and Technology, Xerox Corporation, 147-59B, Webster, NY 14580, U.S.A.tDepartment of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam,

NY 13699-5725, U.S.A.

(First received 8 April 1999; and in ,nal form 20 December 1999)

Abstract*Brownian dynamics simulations of the motions of ellipsoidal particles in the turbulencenear-wall coherent vortices are carried out. The kinematics and the dynamics of an ellipsoidalparticle moving in shear #ows are outlined. Euler's four parameters (quaternions) are used fordescribing the particle orientation. The particle equation of motion includes the hydrodynamicforces and torques, the shear-induced lift force, and the Brownian forces and torques. The near-wallcoherent structures are simulated using vortical #ow models. A turbulence mean #ow is used in thestreamwise direction, and the spanwise direction is assumed to be periodic. The wall is modeled asan absorbing (a sticky surface) boundary condition. Ensembles of particle trajectories are evaluatedand statistically analyzed. For various particle sizes, aspect ratios, and densities, the depositionvelocities of elongated particles in turbulent air streams are evaluated. An empirical equation for theturbulent deposition rate of ellipsoidal particles is also developed. The results are compared withearlier studies and the available experimental data. ( 2000 Elsevier Science Ltd. All rights reserved

INTRODUCTION

Wall deposition of small particles from turbulent gas streams occurs in numerous industrial,technological, physiological, and geological processes. Air pollution control, pneumatictransport of particles, coal combustion/gasi"cation, inhalation toxicology, clean-rooms/mini-environments, and atmospheric aerosols are some typical examples of the "eldsin which particle transport and deposition are of particular interest.

Numerous #ow visualizations and numerical simulations showed that the turbulencenear-wall #ow is a region of strong dynamic interactions dominated by streamwise coherentvortices and roughly cyclic bursting phenomena. Hinze (1975) provided a description of thegeneral features of the coherent vortices formation and bursting processes. Accordingly, thevortices which are formed in the wall region are stretched in the streamwise direction andpersist for a rather long duration of time. These are responsible for the formation of thestreamwise streaks observed in numerous #ow visualizations and direct numerical simula-tions. The mean spacing between the streaks were found to be about 100 in wall length unit.

The "rst study of turbulent deposition was reported by Friedlander and Johnstone (1957).Realizing the signi"cant role of the near-wall coherent vortices in turbulent depositions,Owen (1969) suggested that the particles were carried by the #uid vortical motions directly tothe surface. Based on this idea, Cleaver and Yates (1975) developed a sublayer model forturbulent deposition process, which included the contributions due to inertia and di!usionprocesses. The in#ow region (#ow toward the wall) of coherent vortices was modeled asa plane stagnation point #ow that was used to analyze the particle motions. Their evaluationof inertia-controlled deposition relied on details of particle trajectories in the in#ow regions ofcoherent vortices. They also obtained the di!usion-controlled deposition by solving theconvective}di!usion equation for particle concentration in the near-wall vortical #ow.

The analysis of Cleaver and Yates (1975) was later further generalized and was alsoapplied to more complicated situations. Fichman et al. (1988) incorporated the Sa!man lift

sFormerly with Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam,NY 13699, U.S.A.AAuthor to whom correspondence should be addressed.

1205

force induced by the streamwise #ow shear in the particle trajectory calculations. Theyfound that the lift force could account for an adjusted constant used by Cleaver and Yates.Wu et al. (1992) applied the sublayer model to particle rebound study. Fan and Ahmadi(1993) extended the sublayer model to include the e!ects of gravity and surface roughness.They showed that particle deposition in a vertical channel depends on the #ow direction.Furthermore, the deposition rate increased signi"cantly as the wall roughness increases.Fan and Ahmadi (1994) generalized the near-wall #ow model to a set of full vortices andstudied the deposition of charged particles to a smooth surface. They found that, under theactions of external forces, the particle motion and deposition in the out#ow regions (#owaway from the wall) of the coherent vortices were signi"cant and had to be included in theanalysis. As a consequence, the earlier de"nition of limiting trajectory was also generalized.Fan and Ahmadi (1995a) also studied the e!ects of particle rebound on turbulent depositionrates using a full-vortex model. The model predictions of Fan and Ahmadi (1995a) were infavorable agreement with the available experimental data for particle rebound.

These theoretical works were exclusively concerned with idealized spherical particlesdespite the fact that most solid aerosol particles in industrial and natural processes arenonspherical. One notable example of elongated particles is the asbestos "bers that maycause serious human respiratory health problems. At the present time, however, the existingknowledge base concerning the e!ects of particle shape on the deposition rate is quitelimited. Due to the dependence of translational motion on the orientation, transport anddeposition of nonspherical particles in turbulent #ows are far more complicated than thosefor spherical particles. Recently, Fan and Ahmadi (1995b) applied the sublayer model tostudy the deposition process of ellipsoidal particles in turbulent air streams. Their modelpredictions were compared with the available experimental data for glass "ber deposition,and favorable agreements were observed. The study of Fan and Ahmadi (1995b) consideredthe deposition due to the inertia-interception mechanism. Thus, it is applicable only tointermediate and relatively large particles. For submicron particles, di!usion is expected tostrongly a!ect the deposition process. A theoretical study on turbulent deposition ofsubmicron ellipsoids is not available as yet.

In this work, turbulent depositions of ellipsoidal particles with sizes ranging fromsubmicron to several-dozen microns are studied. Instead of partitioning the deposition intodi!usion- and inertia-controlled processes as was done by Cleaver and Yates (1975),Brownian dynamics simulation which is valid for the entire range of particle sizes is used.The study is based on the detailed analysis of particle motions in the turbulence near-wallvortices. The hydrodynamic forces and torques, the Brownian forces and torques, and theshear-induced lift are included in the equation of motion. The deposition rates for ellip-soidal particles in turbulent #ows under various conditions are simulated, and the resultsare presented in several "gures. E!ects of particle size, aspect ratio, and particle-to-#uiddensity ratio on the deposition rates of ellipsoidal particles are also studied. An empiricalequation for predicting the wall deposition rate of ellipsoidal particles from turbulent airstreams is also developed.

FLOW MODELS FOR NEAR-WALL VORTICES

Figure 1 shows schematically the near-wall #ow structure of a turbulent boundary layer.In the existing literatures on sublayer analysis of particle deposition, several #ow models forsimulating this vortical #ow structure within various distances from the wall were de-veloped. (In the following discussion, the coordinate normal to and measured from the wallis indicated by y, and y`

51is the nondimensional distance of the top boundary of the #ow

model from the wall. Here, the nondimensional length is introduced by using the kinematicviscosity of the #uid and the friction velocity of turbulence.) Cleaver and Yates (1975)modeled the in#ow regions of coherent vortices by a plane stagnation point #ow. Values ofy`51

from 10 to 30 were used, and they concluded that there was no signi"cant change in theresulting deposition rate as far as y`

51was larger than 10. Fichman et al. (1988) modeled the

in#ow region of the near-wall #ow up to 30 wall units from the wall. The normal velocity

1206 F.-G. Fan and G. Ahmadi

Fig. 1. Schematics of the near-wall coherent eddies in a turbulent boundary layer.

component (in y direction) of the model at y`"30 was matched with the experimentalroot-mean-square (RMS) normal #uctuation velocity of turbulence at that level. It was alsoassumed that the particles start their near-wall #ight at y`

51"30 with their initial normal

velocities taken as the corresponding local #uid velocities. Wu et al. (1992) modeled thein#ow pattern using a stagnation point #ow with its strength selected according toa probability distribution. However, only the #ow "eld at very short distance from the wall(y`

51"5) was modeled. Using a series of stagnation point #ows, Fan and Ahmadi (1994,

1995b) modeled the full near-wall vortices within 12 wall units from the wall. They assumedthat, at y`"12, the particle velocities equal to the local #uid velocities. Fan and Ahmadi(1995a) developed a continuous full-vortex model based on an earlier stability analysis ofPhillips and Tu (1992). The RMS normal velocity at y`"12 of the model was matchedwith the experimental and direct numerical simulation data. The particle trajectories wereanalyzed by setting the initial particle velocities at y`"12 equal to those of the local #uid.

Recently, Chen (1995) simulated spherical particle deposition using the direct numericalsimulation method. He also examined the particle and the #uid velocities at the particlecentroids. His results showed that the majority of particles that are deposited by inertialmechanism start their "nal #ight toward the wall at y` of about 30. However, between 15and 30 wall units from the wall, the particle normal velocities are generally less than those ofthe local #uid. The particles accelerate and gain momenta from the #uid, and reach to theirmaximum normal velocities at y` of about 14 to 15. At this level, the RMS normal velocityof particles is equal to that of the local #uid. The particles then continue their #ights towardthe wall and eventually deposit on the surface. The observation of Chen (1995) concerningthe equality of particle and #uid RMS #uctuation velocity at y`K14 substantiates theassumptions used by Fan and Ahmadi (1994, 1995a,b) in their sublayer models. Therefore,these two models are used in the present study.

Flow Model I

According to Fan and Ahmadi (1994, 1995b), the normal and the spanwise componentsof the near-wall #ow may be simulated using a series of stagnation point #ows, i.e.

u`y"G

A'(s) for !

"`

2)z`(!

"`

4,

!A'(s) for !

"`

4)z`)

"`

4,

A'(s) for"`

4(z`)

"`

2,

(1)

Wall deposition of small ellipsoids 1207

Fig. 2. Velocity vector plot generated by: (a) Flow Model I, (b) Flow Model II.

u`z"G

!A2Az`#

"`

2 B'@ for !

"`

2)z`(!

"`

4,

A2z`'@ for !

"`

4)z`)

"`

4,

A2A"`

2!z`B'@ for

"`

4(z`)

"`

2,

(2)

where s"Ay`, A"0.27, a prime stands for a derivative with respect to s, a superscriptplus denotes a quantity stated in wall units, and "`"100 is the mean spacing between theturbulence streaks. The numerical values of the function '(s) for plane stagnation point#ow were tabulated by Schlichting (1979). Figure 2a shows the velocity vector plotgenerated by equations (1) and (2).

Flow Model II

Based on the stability analysis of Phillips and Tu (1992), Fan and Ahmadi (1995a)developed a model for the near-wall vortices. Accordingly, the cross-stream velocitycomponents may be described by

u`y"!BA

2n"`BA

2y`

"` B11@16

sinA2ny`

"` BcosA2nz`

"` B , (3)

u`z"

11

16A2B

"`BA2y`

"` B~5@16

sinA2ny`

"` BsinA2nz`

"` B#BA

2n"`BA

2y`

"` B11@16

cosA2ny`

"` BsinA2nz`

"` B . (4)


Fig. 3. Schematics of an ellipsoid of revolution and the corresponding coordinate systems.

Equation (3) with B"34.7 results in a velocity variation

u`y"v`

ccosA

2nz`

"` B , (5)

at y`"12, where v`c"!0.56. The RMS normal velocity of this model at 12 wall units

from the wall is then equal to 0.4 which is about the measured value. Figure 2b shows thevelocity vector plot as generated by equations (3) and (4).

A mean streamwise pro"le of turbulent #ow given as

u`x"

1

qtanh(q y`), q"0.075, (6)

is superposed on the cross-stream vortical #ow given by equations (1)}(4).

EQUATIONS OF MOTION

Formulation for the motion of an ellipsoidal particle suspended in a #ow "eld wasdescribed at length by Fan and Ahmadi (1995b,c). Here, only a brief outline is provided.Figure 3 shows an ellipsoid of revolution and the corresponding coordinate systems. Thecoordinates x"[x, y, z] is the inertial frame, x("[x( , y( , z( ] is the particle coordinates, andx(("[x(( , y(( , z(( ] is a coordinate system (co-moving frame) with its origin coinciding with that ofthe particle frame and its axes being parallel to the corresponding axes of the inertial frame.

The coordinates transformation between the co-moving and the particle frames isgiven by

x("Ax(( , (7)

where A is the transformation matrix. It is common to describe the rotational motionof a rigid body using Euler angles and to write the transformation matrix as


(Goldstein, 1980)

A"Cct c/!ch s/ st ct s/#ch c/ st st sh

!st c/!ch s/ ct !st s/#ch c/ ct ct sh

sh s/ !sh c/ ch D , (8)

where /, h, and t are Euler angles and the shorthand notations s/"sin /, c/"cos/, etc.,are used. (Here, Euler angles are de"ned following the x-convention of Goldstein.) However,due to an inevitable singularity in evaluating their time rate of change, Euler angles are notsuited for dynamic simulations of the motion of a nonspherical aerosol particle whichgenerally undergoes full rotations (Fan and Ahmadi, 1995c). This singularity di$cultyof using Euler angles is avoided by using a four parameter set known as Euler parameters(or quaternions). The transformation matrix in terms of Euler parameters is given by(Hughes, 1986),

A"C1!2(e2

2#e2

3) 2(e

1e2#e

3g) 2(e

1e3!e

2g)

2(e2e1!e

3g) 1!2(e2

3#e2

1) 2(e

2e3#e

1g)

2(e3e1#e

2g) 2(e

3e2!e

1g) 1!2(e2

1#e2

2)D , (9)

where e1, e

2, e

3, and g are Euler parameters. These parameters are related to Euler angles by

(Goldstein, 1980)

e1"cos

/!t2

sinh2, e

2"sin

/!t2

sinh2

, (10)

e3"sin

/#t2

cosh2, g"cos

/#t2

cosh2

. (11)

Note that equations (10) and (11) express a set of four parameters to one of threeparameters. Therefore, Euler parameters are subject to an identity given by

e21#e2

2#e2

3#g2"1. (12)

In the present work, Euler angles which are mutually independent are used for assigningthe initial particle orientations. The corresponding initial Euler parameters are obtainedfrom equations (10) and (11). In the subsequent simulations, the time histories of Eulerparameters are calculated, and equation (9) is used for evaluating A. It should be empha-sized that, although the use of four Euler parameters involves one more variable thanthat with Euler angles, the computational e!ort with the former may actually be less thanthat with the later. This is because the elements of A matrix expressed in Euler angles,equation (8), involves trigonometrical functions which are usually computationally moreintensive to evaluate than the simple algebraic manipulations required by equation (9).

For a nonspherical particle moving in an arbitrary #ow "eld, the dynamics for thetranslational and the rotational motions are governed by,

m1dv

dt"f )#f L#f B(t), (13)

Ix(

dux(

dt!u

y(u

z((I

y(!I

z()"¹)

x(#¹B

x((t), (14)

Iy(

duy(

dt!u

z(u

x((I

z(!I

x()"¹)

y(#¹B

y((t), (15)

Iz(

duz(

dt!u

x(u

y((I

x(!I

y()"¹)

z(#¹B

z((t). (16)


In these equations, m1 is the mass of the particle, t the time; v the translational velocityvector of the particle mass center, [v

x, v

y, v

z]; f ) the hydrodynamic drag acting on the

particle, [ f )x, f )

y, f )

z]; f L the shear-induced lift acting on the particle, [ f L

x, f L

y, f L

z]; f B the

Brownian #uctuating force, [ f Bx, f B

y, f B

z]; I

x(, I

y(, I

z(the particle moments of inertia about the

principal axes; ux(,u

y(, u

z(the particle angular velocities about the principal axes; ¹)

x(,¹)

y(,¹)

z(the hydrodynamic torques acting on the particle; and ¹B

x(,¹B

y(,¹B

z(the Brownian #uctuating

torques.Note that equation (13) for the translational motion is expressed in the inertial frame,

whereas equations (14)}(16) for the rotational motion are stated in the particle frame.The hydrodynamic drag and torque acting on an ellipsoidal particle for the continuum

limit were obtained by Oberbeck (1876) and Je!ery (1922), respectively. (These expressionswere used earlier by Fan and Ahmadi, 1995a,b). In this work, for simulating the motion ofsubmicron ellipsoids, the slip-correction factors for hydrodynamic drag and torque areincluded. For an ellipsoid of revolution with its major axis in the z( direction, the hy-drodynamic drag is given by

f )"knaK) ) (u!v), (17)

where k is the dynamic viscosity of the air, a is the semi-minor axis of the particle, and u isthe undisturbed #uid velocity vector at the particle centroid. Here, the translation tensor isgiven by

K))"A~1K) A, (18)

where the particle-frame translation tensor for an ellipsoid of revolution is a diagonalmatrix with the diagonal elements given by

kx( x("k

y( y("

16(b2!1)

C5x( C

2b2!3

Jb2!1ln(b#Jb2!1)#bD

, (19)

kz( z("

8(b2!1)

C5z( C

2b2!1

Jb2!1ln(b#Jb2!1)!bD

. (20)

Here, b"b/a is the particle aspect ratio (ratio of the semi-major to the semi-minor axes),and C5

i)'s are the translational slip-correction factors where the superscript t denotes

translation. The exact expressions for the slip-correction factors of an ellipsoidal particle arenot available. In Appendix A, an approximate method is used for evaluating these correc-tion factors.

The shear-induced lift force for an arbitrary-shaped particle in the continuum limit wasobtained by Harper and Chang (1968). The lift force generated by the streamwise #ow shearon an ellipsoid of revolution is given by

f L"n2ka2

l1@2 SLu

xLy

(K)) )L )K)) ) ) (uL!v), (21)

where

L"C0.0501 0.0329 0.00

0.0182 0.0173 0.00

0.00 0.00 0.0373D (22)

is the lift tensor, K) ) is given by equation (18), and uL"[ux, 0, 0] is the reference #ow velocity.

The expression of Harper and Chang (1968) with K) ) corrected for the slip e!ect is used in thepresent study. The e!ect of lift force on particle deposition is signi"cant only for relatively


large size particles. For these particles, the slip-correction factors reduce to about unities. Itshould also be emphasized that, for the #ow models considered, the streamwise shear rateLu

x/Ly in equation (21) is always positive.

For an ellipsoid of revolution with its major axis along the z( direction, the expressions forhydrodynamic torques are given as

¹)x("

16nka3b3(b

0#b2c

0)C3

x([(1!b2)d

z( y(#(1#b2)(w

z( y(!u

x()], (23)

¹)y("

16nka3b3(b2c

0#a

0)C3

y([(b2!1)d

x( z(#(b2#1)(w

x( z(!u

y()], (24)

¹)z("

32nka3b3(a

0#b

0)C3

z((w

y( x(!u

z(), (25)

where

diK jK"

1

2 ALu

iK

LxjK#

LujK

LxiKB, w

iK jK"

1

2 ALu

iK

LxjK!

LujK

LxiKB (26)

are elements of the deformation rate and the spin tensors. Here, the subscript indices iK andjK run from 1 to 3 (corresponding to x( , y( , and z( ) and x

iKis the component of x("[x( , y( , z( ]. In

equations (23)}(25), C3iK's are the rotational slip-correction factors where the superscript

r denotes rotation. An approximate expression for the rotational slip-correction factor isderived in Appendix A, and its e!ect on particle deposition rate is studied. It is shown thatthe e!ect of rotational slip-correction factor on the deposition velocity is insigni"cant.Therefore, in the following simulations, C3

x("C3

y("C3

z("1 is assumed.

The dimensionless parameters in equations (23)}(25) were given by Gallily and Cohen(1979) as

a0"b

0"

b2

b2!1#

b2(b2!1)3@2

lnCb!Jb2!1

b#Jb2!1D , (27)

c0"!

2

b2!1!

b(b2!1)3@2

lnCb!Jb2!1

b#Jb2!1D . (28)

The velocity gradient in the particle frame needed in equation (26) may be obtained by usingthe transformation

CLu

iK

LxjKD"AC

Lui

LxjDA~1. (29)

The Brownian #uctuating force, f B(t), in equation (13) is obtained by using

f B"fKK B"A~1 fK B, (30)

where fK B"[ f Bx(, f B

y(, f B

z(] is the Brownian force with respect to the particle coordinate system.

The components of Brownian force and torques in the particle frame may be modeled asindependent zero-mean Gaussian white-noise processes as

Sf) B(t)T"0, Sf) B(t)?f) B(t#t@)T"nS5d(t@), (31)

ST) B(t)T"0, ST) B(t)?T) B(t#t@)T"nS3d(t@), (32)

where t@ is the time lag, d( ) is Dirac delta function, TK B"[¹Bx(,¹B

y(,¹B

z(] is the torque vector,

a pair of angular brackets S T stands for the expected value (ensemble average), andcorrelations are obtained by averaging the time-shifted dyadic products. In these equations,S5 and S3 are the spectral intensity tensors for the translational and the rotational Brownianmotions, respectively.


For an ellipsoidal particle with the particle coordinate axes being the principal axes, thespectral intensity tensor for Brownian force is a diagonal matrix with the diagonal elementsgiven by

S5iK iK"2i0R5

iK iK/n, (33)

as required by the principle of equal partition of energy. Here, i is the Boltzmann constant,and 0 is the absolute temperature. The resistance coe$cients for an ellipsoid translatingalong its principal axes in a quiescent #uid are given by

R5x( x("nkak

x( x(, R5

y( y("nkak

y( y(, R5

z( z("nkak

z( z(. (34)

The equations for rotational motion given by equations (14)}(16) are nonlinear. However,Goren (1979) showed that the spectral intensity tensor for Brownian torques may beapproximated as

S3iK iK"2i0R3

iK iK/n, (35)

where R3iK iK's are the resistance coe$cients for an ellipsoid rotating about its principal axes in

a quiescent #uid, i.e.

R3x( x("

16nka2

3k3x(, R3

y( y("

16nka2

3k3y(, R3

z( z("

16nka2

3k3z(, (36)

where

k3x("

ab(1#b2)

(b0#b2c

0)C3

x(, k3

y("

ab(b2#1)

(a0#b2c

0) C3

y(, k3

z("

2ab(a

0#b

0)C3

z(. (37)

Using equation (27), it follows that R3x( x("R3

y( y(which is as expected.

NONDIMENSIONALIZED EQUATIONS

Since the #ow models given by equations (1)}(4) and (6) are in wall units, it is convenientto nondimensionalize the equations of motion accordingly. This is achieved by using the#uid kinematic viscosity l and the turbulence friction velocity uH. The details were describedby Fan and Ahmadi (1995b). Therefore, only a summary of the nondimensional equationsincluding the Brownian terms is provided here. These are

dx`

dt`"v`, (38)

Cde

1/dt`

de2/dt`

de3/dt`

dg/dt` D"1

2Cgu`

x(!e

3u`

y(#e

2u`

z(

e3u`

x(#gu`

y(!e

1u`

z(

!e2u`

x(#e

1u`

y(#gu`

z(

!e1u`

x(!e

2u`

y(!e

3u`

z(D , (39)

dv`

dt`"A

S!1

S Bg`#

3

4bSa`2K)) ) (u`!v`)

#

3n4bSa`S

Lu`x

Ly`(K)) )L )K)) ) ) (uL`!v`)#A~1fK B`(t`), (40)

du`x(

dt`"u`

y(u`

z( A1!2

1#b2B#

20[(1!b2)d`z( y(#(1#b2)(w`

z( y(!u`

x()]

(b0#b2c

0)(1#b2)Sa`2C3

x(#¹B`

x((t`), (41)


du`y(

dt`"u`

z(u`

x( A2

1#b2!1B

#

20[(b2!1)d`x( z(#(b2#1)(w`

x( z(!u`

y()]

(b2c0#a

0)(1#b2)S a`2C3

y(#¹B`

y((t`), (42)

du`z(

dt`"

20(w`y( x(!u`

z()

(a0#b

0)Sa`2C3

z(#¹B`

z((t`). (43)

Here, equation (38) describes the particle displacement, equation (39) relates the time ratesof Euler parameters to the particle angular velocities, and equations (40)}(43) are thenondimensionalized forms of the dynamic equations given by equations (13)}(16). In theseequations, S is the particle-to-#uid density ratio (o1/o&), fK B`(t`) is the nondimensionalizedBrownian #uctuating force (per unit mass) along the ellipsoid principal axes, and[¹B`

x(,¹B`

y(,¹B`

z(] is the nondimensionalized Brownian #uctuating torque (per unit mo-

ment-of-inertia) about the ellipsoid principal axes.In the derivation of equations (40)}(43), the following de"nitions are used:

fK B`(t`)"3

4nbSa`3

1

klfK B(t`), (44)

¹B`x(

(t`)"15

4n(1#b2)bSa`5

uHkl2

¹Bx((t`), (45)

¹B`y(

(t`)"15

4n(1#b2)bSa`5

uHkl2

¹By((t`), (46)

¹B`z(

(t`)"15

8nbSa`5

uHkl2

¹Bz((t`). (47)

Note that the Brownian force and torque on the right-hand sides of equations (44)}(47) arefunctions of t`, and the spectral intensities increase by a factor of uH2/l as the time scalechanges from the dimensional to the nondimensional forms.

Using equations (31) and (44), components of the nondimensional Brownian force arezero-mean Gaussian white-noise processes with the spectral intensity tensors given by

S5`x( x(

"S5`y( y(

"

9

8nAkx( x(

bSa`2B2

Ai0

lR5x( x(B , (48)

S5`z( z(

"

9

8nAkz( z(

bSa`2B2

Ai0

lR5z( z(B , (49)

where k's and R5's are given by equations (36) and (37). The o!-diagonal elements of thespectral tensor are zeros. The group of parameters in the last parentheses of equations (48)and (49) are the inverses of the Schmidt numbers for the translational motions along variousprinciple axes.

Using equations (32) and (45)}(47), it can be shown that the nondimensional Browniantorques in equations (41)}(43) are zero-mean Gaussian white-noise processes with thespectral intensity tensors given by

S3`x( x(

"S3`y( y(

"

800

n Ck3x(

b(1#b2)Sa`3D2

Ai0lR3

x(B , (50)

S3`z( z(

"

800

n Ak3z(

2bSa`3B2

Ai0lR3

z(B . (51)

The last parentheses in equations (50) and (51) are the inverses of the Schmidt numbers forthe rotational motion about the principal axes.


At the limit of spherical particles, b"1 and kx( x("k

y( y("k

z( z("6/C

#(where C

#is the

Cunningham correction factor), and equations (48) and (49) reduce to

S5ìj

"

2

nq`2Scdij, (52)

where dij

is the Kronecker delta function. The nondimensional particle relaxation time andthe Schmidt number for a spherical particle are de"ned as

q`4"

Sd`24

C#

18, Sc"

lD"

3nlkd4

i0C#

. (53)

Here, d4is the sphere diameter and D is the Brownian di!usivity. Equation (52) is identical

to that used by Ounis and Ahmadi (1993) for Brownian dynamics simulation of sphericalparticles in a channel #ow.

SIMULATION PROCEDURE

In this study, homogeneity in the x direction, periodic boundary condition in thez direction, and absorbing boundary condition at y`"0 are assumed. The computationdomain for the simulations is !"`/2)z`)"`/2, 0)y`)12, and x`*0. Particlesthat cross z`"!"`/2 and leave are considered as entering from z`""`/2. Likewise,particles that leave at z`""`/2 are considered as crossing z`"!"`/2 boundary andentering the domain. Reentrainment and rebound of particles are not included in theanalyses. Particles that touch the wall at y`"0 are assumed to stick to it (due to the vander Waals forces).

Initially, 8000 particles are uniformly distributed on the x`"0 plane of computationdomain. The initial distribution of particle orientation is assumed to be isotropic. That is,the probability densities of /

0and t

0are uniformly distributed from 0 to 2n; while that of

h0

is a sine distribution from 0 to n. Such distribution of particle orientation is generated by

/0"2nX

1, t

0"2nX

3, (54)

h0"cos~1(1!X

2) or n!cos~1(1!X

2), (55)

where X1, X

2, and X

3are random numbers with uniform distributions between 0 and 1.

The choice of h0

is alternated between the two possibilities indicated in equation (55).The numerical scheme as described by Fan and Ahmadi (1995b) is used for solving

equations (38)}(43). Accordingly, the fourth-order Runge}Kutta method is used for equa-tions (38) and (39). A mixed di!erencing procedure is used for equations (40)}(43). Duringthe computation, the square sum of Euler parameters is calculated and compared withequation (12). This may serve as an indictor for the accuracy of numerical integration.A time step *t` of between 0.01 and 0.025, depending on the particle size considered, is usedfor the numerical integration of the equations.

The method for generating Gaussian white-noise processes as used by Ounis et al. (1993)is used here for simulating the Brownian force and torque acting on an ellipsoidal particle.Accordingly, an appropriate time increment dt` (larger than *t`) is "rst selected. At everyincrement, six independent zero-mean Gaussian random numbers (X5

x(, X5

y(, X5

z(and X3

x(, X3

y(,

X3z() are numerically generated. Amplitudes of the components of Brownian force and

torque are then given by

f BìK

(t`)"X5iKJn(S5

iK iK)`/dt`, (56)

¹BìK

(t`)"X3iKJn(S3

iK iK)`/dt`, (57)

where iK runs through 1}3 (corresponding to x( , y( and z( ). Each sample excitation is thenshifted in time by an amount X4*t`, when X4 is a random number uniformly distributedbetween 0 and 1.


As a particle is transported by the #uid #ow, it may either be intercepted by the wall orbeing carried back to y`"12 level by the vortical #uid motion without encountering thewall. The condition for deposition of an ellipsoid of revolution on a #at smooth wall wasdeveloped by Fan and Ahmadi (1995b). That is, when

y`"

mH(1/a`2!1/b`2)sin a cos a#J(cos a/a`)2#(sina/b`)2!(mH/a`b`)2

(cos a/a`)2#(sin a/b`)2, (58)

where y` is the distance of the particle centroid from the wall, and mH is given by

mH"S(b`2!a`22[(cos a/a`)2#(sin a/b`)2]sin2a cos2 a

[1#a`2b`2(1/a`2!1/b`2)2sin2 a cos2 a]. (59)

In the numerical simulations, equation (58) is checked at every time step for particles withy`(b`.

At the end of each time step, the loss of particles from the computational domain (thetotal number of those particles that are deposited and those left the #ow region through thetop boundary) is evaluated. The same number of particles that are lost are assumed to enterthe vortical #ow, so that the total number of particles in the computational domain remains"xed. These new particles are added to the in#ow region of the top boundary such that tosimulate a uniform inlet concentration. More speci"cally, the following procedure is used.The initial z coordinates of the new particles are given by

z`0"

"`

4(J1!X1!1) or !

"`

4(J1!X1!1), (60)

for Flow Model I, and by

z`0"!

"`

4 C1!2 cos~1(1!X1)

n D or"`

4 C1!2 cos~1(1!X1)

n D , (61)

for Flow Model II. Here, X1 is a random number with a uniform distribution from 0 to 1.Equation (60) represents a bi-linear probability density with a peak at z`"0 and zeros atz`"!"`/4 and "`/4, while equation (61) is a sine probability density from z`"!"`/4to "`/4. The orientations of the added particles are assumed to follow an isotropicdistribution as given by equations (54) and (55). Fan and Ahmadi (1995b) had shown earlierthat isotropic orientation distribution for particles at y`"12 is a good approximation forturbulent deposition analysis.

The initial translational velocities of the added particles are given by

v`0x

"u`0x

, v`0y"<`

# A1!4

"`Dz`0DB, v`

0z"u`

0z, (62)

for Flow Model I, and by

v`0x

"u`0x

, v`0y"u`

0y, v`

0z"u`

0z, (63)

for Flow Model II. Here, <`#"!0.7 is the characteristic downward velocity of the #ow

model, and u`0x

and u`0z

are the x and z component of #uid velocity at the initial position ofthe particle as given by equations (60) or (61). The y component of the initial particlevelocity as given by equation (62) varies linearly in both sides of z`"0 axis and becomeszero at z`"$"`/4. Note that equations (60) and (62), and similarity equations (61) and(63), together constitute a uniform particle concentration on the in#ow region of the topboundary.

For small particles, the particle inertia is insigni"cant. Therefore, a uniform particledistribution on the top boundary for Flow Model I and a uniform particle downwardvelocity instead of equations (60) and (62) were used in the simulations.


The initial angular velocities of all the added particles are given by

u`0x(

"u`0y("u`

0z("0. (64)

DEPOSITION RATE

Deposition velocity is de"ned as the ratio between the particle #ux to the wall and theparticle number concentration at some distance away from the wall. For the presentsimulations, this may be stated as

u`$"

n$

"C0

, (65)

where u`$

is the nondimensional deposition velocity, n$

is the number of particles depositingin the computational domain per unit time, and C

0is the particle number concentration at

y`"12.The number of particles supplied to the computational domain (entering from the top

boundary) per unit time may be obtained by multiplying the normal velocity component ofequation (62) or (63) with C

0and integrating from z`"!"`/4 to "`/4. The results are

n4"1

4D<`

#D"`C

0for Flow Model I, (66)

and

n4"

1

nDv`#D"`C

0for Flow Model II, (67)

where <`#"!0.7 and v`

#"!0.56.

Using equations (66) and (67) in equation (65), the nondimensional deposition velocitymay be restated as

u`$"

D<`#D

4

n$

n4

for Flow Model I (68)

and

u`$"

Dv`#

Dn

n$

n4

for Flow Model II. (69)

The deposition velocity for every one wall unit of time period is obtained from equations(68) and (69). The mean deposition velocity is then evaluated by a time averaging after thevariation of u`

$reaches a stochastically stationary state. However, for certain particle sizes,

the number of deposition in one wall unit of time period is very small. An alternativeprocedure which is more convenient may then be used. After the initial transient state, thetotal numbers of deposited (N

$) and supplied (N

4) particles for a large time period

are evaluated. Equations (68) or (69), with n$

and n4

being replaced by N$

and N4,

respectively, are then used to evaluate the deposition velocity. This later procedure intro-duces some errors in the average process. However, when the time series n

4(t) has a large

mean and small #uctuations, the error is insigni"cant.

SIMULATION RESULTS

In this section, simulation results for deposition velocities as functions of equivalentrelaxation time are presented and compared with the earlier model predictions. Theequivalent particle relaxation time as de"ned by

q`%2"

4bSa`2

9(k~1

x( x(#k~1

y( y(#k~1

z( z() (70)

is used. Equation (70) is obtained by averaging the mobility tensor (inverse of the transla-tion tensor) with an isotropic orientation distribution. Kinematic viscosity ofl"1.502]10~5 m2 s~1 and dynamic viscosity of k"1.84]10~5 kg m s~1 for air at


Fig. 4. Variations of nondimensional deposition velocity with particle relaxation time for variousaspect ratios for density ratios of (a) S"1000 and (b) S"2000 as obtained from Flow Model I.

153C are used in this study. A friction velocity uH"0.3 m s~1 which corresponds toa Reynolds number of 9588 for a channel with a hydraulic diameter of 3 cm is considered.

Figure 4 shows the simulated nondimensional deposition velocities obtained by usingFlow Model I. The results for S"1000 are presented in Fig. 4a whereas those for S"2000are presented in Fig. 4b. The deposition velocities for various particle aspect ratios aredisplayed by the di!erent symbols. The sublayer model predictions of Fan and Ahmadi(1995b) are shown here by the solid lines for comparison. In this "gure, the small dots arethe experimental data for spherical particles obtained by various researchers as collected byPapavergos and Hedley (1984). It is observed that, for q`

%2'1, the simulated deposition

velocities are in close agreement with the model predictions of Fan and Ahmadi (1995b).The deposition velocities increase rapidly with the particle relaxation time up to q`

%2K10.

Beyond this range, the deposition rate is saturated and increases only slightly with q`%2

. Fora "xed value of q`

%2, the deposition velocity increases with particle aspect ratio, b. This e!ect

is most signi"cant for particles between 1(q`%2(10. In this range, particle deposition

is mainly due to the inertia-interception mechanism. The interception of elongated part-icles by the wall is far more e$cient than that for spherical particles. A comparison of theresults for S"1000 and 2000 shows that, in the range of q`

%2'1, the deposition velocity

decreases with increasing in density ratio for a "xed value of q`%2

.Figure 4 also shows that, for q`

%2(0.1, the Brownian motion is the main mechanism for

deposition and the simulated deposition velocities increase with decreasing particle relax-ation time. For this size range, the particle inertia is insigni"cant. As noted before, a uniformparticle distribution on the top boundary for Flow Model I and a uniform particledownward velocity instead of equations (60) and (62) were used in the simulations.Although the simulation data shows some scatters in this range of particle sizes, for a "xedvalue of q`

%2, the deposition velocity tends to decrease with an increase in aspect ratio. The

reason for this behavior is described in the following. As elongated particles being trans-ported in the vicinity of the wall, the #ow shear tends to orient the particles to move parallelto the wall. The Brownian #uctuation motions perpendicular to the wall, which are themain deposition mechanism for q`

%2(0.1, are smaller for particles with larger aspect ratios.

Thus, the deposition rate decreases with increasing b.Figure 5 shows the simulated deposition velocities for S"1000 as obtained by using

Flow Model II. For this #ow model, equations (61) and (63) are used for the entirerange of particle sizes. It is observed that the general trends are similar to thoseshown in Fig. 4a. However, the deposition rates for particles with q`

%2(1 are higher than

those obtained using Flow Model I and empirical model prediction of Fan and Ahmadi(1993). This observation suggests that the Flow Model I provides a somewhat betterrepresentation of the velocity "eld of turbulence near wall eddies at short distances from thewall.


Fig. 5. Variations of nondimensional deposition velocity with particle relaxation time for densityratio of S"1000 and various aspect ratios as obtained from Flow Model II.

Fig. 6. Particle distributions as functions of y` at t`"1000 for S"1000 and various relaxationtimes. (a) Flow Model I, and b"1, (b) Flow Model I, b"5, (c) Flow Model II, b"5.

Although the main objective of the present study is to simulate the particle depositionrates, it is also of interest to know the particle distribution in the vortical #ow models.Figure 6 shows the particle concentrations at t`"1000 as functions of the distance fromthe wall for b"1 and 5 and various q`

%2. A particle-to-#uid density ratio of 1000 is used, and

the results for spherical particles (b"1) in Flow Models I is presented in Fig. 6a. It isobserved that, particles tend to accumulate in the neighborhood of the wall at y`K0.5.


This e!ect is most signi"cant for q`%2"2.3 for which the particle number concentration near

the wall is about six times that at y`"12. For q`%2"0.0032 and 8, the accumulation e!ects

are less signi"cant, and particle concentrations near the wall are about twice those aty`"12.

Chen and McLaughlin (1995) used a direct numerical simulation method to study theconcentration and deposition of spherical particles (with nondimensional particle relax-ation time in the range of 0.1}50) in a turbulent channel #ow (with uH"0.6m s~1). Theyshowed that inertial particles tend to pile up in the region of y`"0.4 to 0.5, and theaccumulation e!ect increases with particle relaxation time for 0.1(q`

%2(10. The reason

for accumulation of particles with q`%2"10 is due to an e!ect similar to turbophoresis.

That is, particles are projected toward the wall by the inertia they gain in the bu!er/outerregion, and lose their momenta in the viscous wall region. They are then trapped becauseof the relatively low level of turbulence #uctuation in the viscous wall region. Chen andMcLaughlin found that the accumulation e!ect was most signi"cant for q`

%2"10. Particles

with smaller relaxation times may not have enough inertia to penetrate into the deep regionof the viscous sublayer. These particles lose their momenta in the bu!er region and areswept away from the wall by the vortical #uid motion.

Even though the present vortical #ow model is stationary, Fig. 6 shows that particles tendto pile up at y`K0.5 though the extent of accumulation is much less than that observed byChen and McLaughlin (1995). This suggests that the vortical #ow structure contributessomewhat to the particle accumulation e!ect. The present simulation, however, shows thatthe accumulation e!ect is most signi"cant for intermediate size particles (q`

%2K2). It

appears that, for the stationary #ow model used, the inertia of particles of q`%2K8 are too

large for the particles to accumulate in the vicinity of the wall without being deposited.These particle literally penetrate through the viscous sublayer and deposit on the wall.

It is interesting to note that the simulated particle deposition rates for spherical particles(b"1) as shown in Figs 4 and 5 are consistent with the direct simulation results of Chenand McLaughlin (1995) and the experimental data, while there are di!erences in the particleconcentration pro"les shown in Fig. 6a. This contraversy may be explained as follows. Chen(1995) analyzed the trajectories of particles deposited on the wall from a turbulent channel#ow. He found that, for particles with nondimensional relaxation time about 10, themajority of particles that deposit on the wall begin their "nal unidirectional #ights to thewall from outside the viscous sublayer. Chen and McLaughlin (1995) also concluded thatinertial deposition is the dominant mechanism for deposition of q`

%2K10 particles. The

contribution to wall deposition by the di!usion of those particles within the viscoussublayer is always less signi"cant than that of inertial impaction of particles originatingoutside the viscous sublayer. That is, although the direct numerical simulation showsa signi"cant number concentration at y`K0.5 for particles with q`

%2K10, this near-wall

particle population is not a major contributor to the wall deposition process.Figures 6b and c show the concentration pro"les for particles with b"5 in Flow Model

I and II, respectively. Variations of concentration pro"les for ellipsoidal particles are quitesimilar to those of the spherical ones shown in Fig. 6a. Particles with q`

%2"2.3 have strong

tendency of accumulating in the neighborhood of the wall. For q`%2"0.0032 and 8, the

number concentration remains roughly uniform except for the region very close to the wallwhere the particle concentration reduces sharply and forms a thin boundary layer.

Figure 7 shows the number concentrations in 1(y`(4 region at t`"1000 as func-tions of z` for particles with S"1000. The simulated results for spherical particles (b"1)and Flow Model II is presented in Fig. 7a, whereas those for elongated particles (b"5) andFlow Model I are shown in Fig. 7b. As mentioned before, a periodic boundary condition inthe spanwise direction is used in the present simulations. Thus, the particle concentrationvariation across the span of the #ow is obtained by repeating the concentration pattern fora period of 100 wall length units. It is observed that, for q`

%2"1, particles are uniformly

distributed across the span of the #ow region. For q`%2"4, however, the particles tend to

accumulate in the out#ow regions of the vortices. These out#ow regions correspond to thelow-speed streaks observed by #ow visualizations and direct numerical simulations.


Fig. 7. Particle number concentrations as functions of z` in the region 1(y`(4 at t`"1000 forS"1000 and various relaxation times. (a) Flow Model I, and b"1, (b) Flow Model II, b"5.

Pedinotti et al. (1992), Rouson and Eaton (1994), and Chen (1995) studied the preferentialparticle concentration using direct numerical simulation method, and reported similarbehaviors. The #ow structure a!ects particle concentration by the following process. Theparticles are carried toward the wall by the in#ow, and transported to the out#ow regionsby (the z component of ) the vortical motion. Because of their y component inertia, theparticles deviate from the #ow streamlines, penetrate closer to the wall, and enterthe regions near the stagnation points at z`"!"`/2 and "`/2. As a result, particles inthe out#ow regions have longer residence times than those in the in#ow regions. Therefore,particle concentrations in the out#ow regions are higher than those in the in#ow regions.Particles with small inertia (q`"1), however, simply follow the #uid motion and do notshow signi"cant concentrating e!ect. Particles with intermediate sizes (q`"4) show strongtendency of accumulating in the out#ow region. For large sizes (q`'10), the depositionrate is signi"cant, and many particles reach the wall and deposit. Hence, they leave the #owdomain and are not brought to the out#ow regions.

EMPIRICAL EQUATION

For practical engineering applications, a convenient empirical equation for particledeposition rate is of great value. In this section, an empirical equation for turbulentdeposition of elongated (as well as spherical) particles in vertical ducts is developed. Earlier,based on a sublayer model and a perturbation analysis, Fan and Ahmadi (1993) developeda semi-empirical equation for evaluating turbulent deposition rate on smooth and roughsurfaces in vertical ducts. Here, this empirical equation is further generalized for applicationto deposition of elongated particles.

For submicron ellipsoids, Brownian di!usion is the dominant deposition mechanism andthe deposition velocity is determined by the Schmidt number

Sc"lD

, (71)

where D is the Brownian di!usivity normal to the wall. It is assumed that the Browniandi!usivity of ellipsoidal particles normal to the wall in the vicinity of a surface is given by

D"0.2Dz(#0.8D

x(. (72)

Here, the directional di!usivities are de"ned as

Dz("

i0nkak

z( z(, D

x("

i0nkak

x( x(, (73)

where the elements of translation tensor are given by equations (19) and (20).


For the intermediate-sized and relatively large particles, the particle inertia and the liftforce are important. The deposition of these particles are determined by the particlerelaxation time and the interception condition. Here, the relevant parameter is the equiva-lent relaxation time de"ned by equation (70). Based on the expression for q`

%2, an equivalent

particle diameter may be de"ned, i.e.

d`%2"S

18q`%2

S. (74)

The lift coe$cient then is given by

¸`"

3.08

Sd`%2

. (75)

The equation of Fan and Ahmadi (1993) is now modi"ed by using y`"a`b(0.6`0.04q`) asthe interception condition for deposition of elongated particles. (Here, for the convenienceof presentation, the subscript èqa for the equivalent relaxation time is omitted, i.e.q`,q`

%2.) This yields

u`$"G

ASc~23#

1

2 C(0.64k`#a`b(0.6`0.04q`))2#

q`2g`¸`

0.01085(1#q`2¸`)

3.42#q`2g`¸`

0.01085(1#q`2¸`) D1

(1`q`2L`)

][1#8 exp(!(q`!10)2/32)]

]0.037

1!q`2¸`(1#(g`/0.037))

if u`$(0.148,

0.148 otherwise,

(76)

where Sc is given by equation (71), q` is given by equation (70), ¸` is given by equation (75),and k` is the mean roughness of the surface. In this equation, g` is the nondimensionalacceleration of gravity, i.e. g`"lg/uH3 where g"9.81 and !9.81 m s~2 for down- andup-#owing duct #ows, respectively.

In the current literature, it is generally agreed that the turbulent deposition rate ofsubmicron spherical particles varies as Sc~2@3. The coe$cient A is, however, much lessde"nite. Wood (1981) proposed a correlation u`

$"0.057Sc~2@3 for deposition of spherical

particles due to Brownian di!usion. Based on a sublayer model and certain approximation,Cleaver and Yates (1975) suggested that di!usion-controlled deposition of spherical par-ticles follows u`

$"0.084Sc~2@3.

Figure 8a shows the predictions of equation (76) with the results obtained from theBrownian dynamics simulation using Flow Model I. The deposition velocities predicted bythe empirical equation are displayed by the di!erent lines. Here, A"0.07 in equation (76)is used. The solid lines are for the condition that the gravity is absent (g"0). Accompanyingeach solid line are two dashed (or dot-dashed) lines. The one above the corresponding solidline is for the case that the gravity acting in the direction of the air stream (i.e. down-#owingvertical duct). The one below the corresponding solid line is for the system that the gravityopposing the air #ow (i.e. a up-#owing duct). For comparison, the simulation depositionvelocities are also shown in Fig. 8a by the di!erent symbols. The sublayer model predictionsof Fan and Ahmadi (1995b) as shown in Fig. 4 are identical to the simulation results forq`'0.2 and, therefore, are not shown here. It is observed that equation (76) is in favorableagreement with the simulation results over the entire range of particle size considered.

The comparison of equation (76) with the simulation data obtained using Flow Model IIis shown in Fig. 8b. As mentioned above, for q`

%2(0.1, the simulated deposition velocities


Fig. 8. Comparison of empirical model predictions with the simulation results for S"1000 anddi!erent aspect ratios. (a) Flow Model I, (b) Flow Model II.

Fig. 9. Deposition velocities predicted by the empirical model for the parameters listed in Table 1.

Table 1. Parameters used for the empirical equation predictions shown in Fig. 9

CurveWall roughnessk (mm)

Mean #ow velocity;M (m s~1)

Reynolds numberRe

Friction velocityuH (m s~1)

k`

A 0 4.80 9588 0.3 0B 0.01 4.80 9588 0.302 0.2C 0.04 10.66 21289 0.62D 0.1 2.99 5974 0.21

obtained from Flow Model II are higher than those obtained from Flow Model I. It is alsonoticed that the simulation for q`

%2(0.1 using Flow Model II seems to overpredict the

experimental available data. It is found that equation (76) with A"0.14 agrees with thesimulated results.

The empirical equation given by equation (76) is very versatile. As noted above, itincludes the e!ects of nonsphericity of particles, particle-to-#uid density ratio, #ow direc-tion, #ow Reynolds number, and the surface roughness. In Fig. 9, the predictions ofequation (76) for di!erent combinations of these parameters are presented. The e!ect of #owdirection which was shown in Fig. 8 is omitted in here (i.e., g`"0 is used). As stated before,wall deposition of small ellipsoids in vertical duct #ows is considered in this study. Fora duct #ow, the friction velocity is related to the friction factor f by

uH"Jf;M 2/8, (77)


Fig. 10. Comparison of empirical model predictions with the glass "ber deposition data of Shapiroand Goldenberg (1993).

where ;M is the mean bulk #ow velocity. The friction factor may be estimated from anempirical equation given as (White, 1986),

1

f1@2K!1.8 log

10C6.9

Re#A

k/D)

3.7 B1.11

D , (78)

where

Re";M D

)l

, (79)

is the #ow Reynolds number. In these equations, D)

is the hydraulic diameter of the duct(D

)"3 cm). Table 1 lists the parameters used for the curves presented in Fig. 9. This "gure

shows that, for q%2(10, the deposition rate of small ellipsoids is strongly dependent on the

wall roughness.In the remaining part of this section, the present empirical equation is compared with the

deposition data of glass "bers. Current available experimental data for nonsphericalparticle deposition in turbulent #ows is very limited. Shapiro and Goldenberg (1993)studied depositions of glass "bers (o1"2230 kg m~3) with averaged diameters of 0.93 and1.86 km and various length ranging from 2 to 50 km in a horizontal pipe. Depositionvelocities for the #oor, the ceiling, and the side wall were measured. In Fig. 10, thepredictions of equation (76) for g`"0 are compared with the side wall deposition data ofShapiro and Goldenberg (1993). The #ow Reynolds number used in these depositionexperiments was Re"3.0]104 which corresponded to a friction velocity ofuH"0.6 m s~1. In the empirical equation, a particle-to-#uid density ratio S"1820 andsemi-minor axes of ellipsoids a"0.465 and 0.93 km, respectively, for simulating the twodi!erent "ber diameters are used. It is observed that the present empirical equation predictsthe general trend of the experimental data. For the 1.86 km diameter "bers, however, theempirical model underestimates the experimental deposition rate.

CONCLUSIONS

In this work, the motion and deposition of ellipsoidal particles ranging from sub toseveral dozen micrometers in turbulent air streams are considered. Brownian dynamicssimulation method is used to study the particle trajectories and the deposition statistics. Thedeposition velocities for ellipsoidal particles in turbulent air streams are evaluated. It isshown that the deposition velocity follows a V-shape variation with particle relaxation time.The minimum deposition rate occurs at q`

%2K0.1. For intermediate size particles

(0.1(q`%2(10), the deposition rate increases with the aspect ratio. For particles with

q`%2

larger than 10, the e!ect of aspect ratio is insigni"cant. For small particles (q`%2(0.1), the


deposition rate decreases with an increase in b. It is also shown that the present simulationresults closely agree with the earlier sublayer model predictions for q`

%2'1.

Variations of particle concentration in the vortical #ow models are also studied. Theresults for particle concentration variation normal to the wall show that particles ofintermediate size (q`

%2"2.3) tend to accumulate in the vicinity of the wall. This e!ect,

however, is not observed for large (q`%2"8) and small (q`

%2"0.0032) size particles. The

results for particle concentration variation parallel to the wall show that particles withintermediate size (q`

%2"4) tend to accumulate in the low-speed streaks of a turbulent

boundary layer #ow.Based on the present Brownian dynamics simulation results and an earlier sublayer

model analysis, an empirical equation for evaluating the turbulent deposition rate ofellipsoidal (as well as spherical) particles on smooth and rough wall in vertical duct #ow isalso developed. This equation includes the e!ects of particle nonsphericy, particle-to-#uiddensity ratio, direction of air #ow, #ow Reynolds number, and wall roughness. Comparisonof the empirical equation with the available experimental data shows that the model iscapable of predicting the deposition trend of glass "bers, but certain discrepancies exist.

In this study, transport and deposition of ellipsoids with aspect ratios up to 5 wereanalyzed. Most natural and man made "bers, however, have higher aspect ratios. Theapproach described in this paper can be easily extended to elongated "bers. E!ort in thisdirection is under way and the results will be reported in a future communication.

Acknowledgements*The "nancial supports of the US Department of Energy (University Coal Research Program,PETC) through grant DE-FG26-99FT-40584, and the New York State Science and Technology Foundation(through the Center for Advanced Materials Processing, CAMP, of Clarkson University) are gratefully acknow-ledged.

REFERENCES

Chen, M. (1995). Aerosol deposition, dispersion, and interaction in a turbulent channel #ow. Ph.D. Thesis,Clarkson University.

Chen, M. and McLaughlin, J. B. (1995) A new correlation for the aerosol deposition rate in vertical ducts. J. ColloidInterface Sci. 169, 437}455.

Cleaver, J. W. and Yates, B. (1975) A sub layer model for the deposition of particles from a turbulent #ow. Chem.Engng. Sci. 30, 983}992.

Dahneke, B. E. (1973) Slip correction factors for nonspherical bodies*III. The form of the general law. Aerosol Sci.4, 163}170.

Edwards, D. (1928) Ann. Phys. 86, 628.Eisner, A. D. and Gallily, I. (1981) On the stochastic nature of the motion of nonspherical aerosol particles III. The

rotational di!usion diadic and applications. J. Colloid Interface Sci. 81, 214}233.Fan, F.-G. and Ahmadi, G. (1993) A sublayer model for turbulent deposition of particles in vertical ducts with

smooth and rough surfaces. J. Aerosol Sci. 24, 45}64.Fan, F.-G. and Ahmadi, G. (1994) On the sublayer model for turbulent deposition of aerosol particles in the

presence of gravity and electric "elds. Aerosol Sci. Technol. 21, 49}71.Fan, F.-G. and Ahmadi, G. (1995a) Analysis of particle motion in the near-wall shear layer vortices*application to

the turbulent deposition process. J. Colloid Interface Sci. 172, 263}277.Fan, F.-G. and Ahmadi, G. (1995b) A sublayer model for wall deposition of ellipsoidal particles in turbulent

streams. J. Aerosol Sci. 26, 831}840.Fan, F.-G. and Ahmadi, G. (1995c) Dispersion of ellipsoidal particles in an isotropic pseudo-turbulent #ow "eld.

ASME J. Fluids Engng. 117, 154}161.Fichman, M., Gut"nger, C. and Pnueli, D. (1988) A model for turbulent deposition of aerosols. J. Aerosol Sci. 19,

123}136.Friedlander, S. K. and Johnstone, H. T. (1957) Deposition of suspended particles from turbulent gas streams. Ind.

Engng Chem. 49, 1151}1156.Goldstein, H. (1980) Classical Mechanics, 2nd Edition. Addison-Wesley, Reading, MA.Goren, S. L. (1979) E!ective di!usivity of nonspherical sedimenting particles. J. Colloid Interface Sci. 71, 209}215.Hinze, J. O. (1975) Turbulence, 2nd Edition. McGraw-Hill, New York, NY.Hughes, P. C. (1986) Spacecraft Attitude Dynamics. Wiley, New York, NY.Je!ery, G. B. (1922) The motion of ellipsoidal particles immersed in a viscous #uid. Proc. Roy. Soc. A 102, 161}179.Oberbeck, H. A. (1876) Crelles J. 81, 79.Owen, P. R. (1969) Pneumatic transport. J. Fluid Mech. 39, 407}432.Papavergos, P. G. and Hedley, A. B. (1984) Particle deposition behaviour from turbulent #ows. Chem. Engng. Res.

Des. 62, 275}295.Pedinotti, S., Mariotti, G. and Banerjee, S. (1992) Direct numerical simulation of particle behaviour in the wall

region of turbulent #ows in horizontal channels. Int. J. Multiphase Flow 6, 927}941.Phillips, W. R. C. and Tu, H. Y. (1992) Why Riblets work. Bull. Am. Phys. Soc. 37, 1794.


Rouson, D. W. and Eaton, J. K. (1994) Direct numerical simulation of turbulent channel #ow with immersedparticles. In Numerical Methods in Multiphase Flows, ASME FED-Vol. 185. ASME, New York, pp. 47}57.

Schlichting, H. (1979) Boundary Layer Theory, 7th Edition. McGraw-Hill, New York.Wood, N. B. (1981) The mass transfer of particles and acid vapour to cooled surfaces. J. Inst. Energy 76, 76}93.Wu, Y.-L., Davidson, C. I. and Russel, A. G. (1992) A stochastic model for particle deposition and bounceo!.

Aerosol Sci. Technol. 17, 231}244.

APPENDIX A

This appendix provides the derivations of the translational and rotational slip-correction factors for an ellipsoidof revolution which are used in equations (19), (20) and (23)}(25). The derivations are based on the proceduresuggested by Dahneke (1973).

¹ranslational slip-correction factors

In this section, the adjusted-sphere-approximation (ASA) method as developed by Dahneke (1973) is used forderiving the translational slip-correction factors. The elements of translational tensor for an ellipsoid of revolutionin the continuum regime are obtained from Stokes #ow calculations with no-slip boundary condition applied onthe rigid particle surface. These are given by

kx( x("k

y( y("

16(b2!1)

2b2!3

Jb2!1ln(b#Jb2!1)#b

, (A1)

kz( z("

8(b2!1)

2b2!1

Jb2!1ln(b#Jb2!1)!b

, (A2)

and for the continuum regime, of course, C5x("C5

y("C5

z("1. In the free molecular regime, using statistical

mechanics, Dahneke (1973) obtained the translational tensor components for ellipsoidal particles as

kx( x("k

y( y("

bKnGE1C4#A

n2!1BaD#

G1

F21A2#

4F21#n!6

4aBH , (A3)

kz( z("

bKnG2E

1a#

G1

F21AF2

1(4!2a)!4#A3!

n2b2BaDH , (A4)

where

E1"

sin~1F1

F1

, F1"S1!

1

b2, G

1"

1

b!E

1,

(A5)

Kn"j/a is the Knudson number and j denotes the mean free path of air.Using equations (A1) and (A3), the x( direction slip-correction factor in the free molecule regime C5

x(D&.

may beobtained as

C5x(D&.

"

16(b2!1)Kn

bG2b2!3

Jb2!1ln(b#Jb2!1)#bHGE1C4#A

n2!1BaD#

G1

F21A2#

4F21#n!6

4aBH

. (A6)

Similarly, the z( direction slip-correction factor under the free molecule condition is obtained form equations (A2)and (A4) as

C5z(D&.

"

8(b2!1)Kn

bG2b2!1

Jb2!1ln(b#Jb2!1)!bHG2E

1a#

G1

F21CF2

1(4!2a)!4#A3!

n2b2BaDH

. (A7)

It is observed that the slip-correction factors in the free molecule regime are linear functions in Kn.According to Dahneke (1973), the slip-correction factors for an ellipsoid in the three principal axes may each be

expressed in the form of Cunningham correction for a sphere with an adjusted sphere radius, i.e.

C"1#[email protected]#0.4 expA!1.1

Kn@ BD , Kn@"aKn/r@ (A8)

where a is the semi-minor axis of the ellipsoid, and r@ is the radius of the adjusted sphere. At the limit of freemolecule regime (KnPR), equation (A8) reduces to

C"1.657aKn/r@, (A9)

which linear in Kn.


Fig. A1. Translational slip-correction factors for di!erent aspect ratios.

The radius of adjusted sphere for the ellipsoid translating in the x( -direction may be obtained from equations (A6)and (A9). This leads to

r5x("

1.657ab16(b2!1) C

2b2!3

Jb2!1ln(b#Jb2!1)#bD

]GE1C4#An2!1BaD#

G1

F21A2#

4F21#n!6

4aBH . (A10)

Similarly, the radius of adjusted sphere for the ellipsoid translating in the z( direction is obtained by using equations(A7) and (A9) as

r5z("

1.657ab8(b2!1)C

2b2!1

Jb2!1ln(b#Jb2!1)#bD

]G2E1a#

G1

F21CF2

1(4!2a)!4#A3!

n2b2BaDH. (A11)

Using the radii of adjusted spheres as given by equations (A10) and (A11) to replace r@ in equation (A8), thetranslation slip-correction factors are obtained. This leads to

C5x("C5

y("1#

a

r5x(KnC1.257#0.4 expA

!1.1r5x(

aKn BD (A12)

and

C5z("1#

a

r5z(KnC1.257#0.4 expA

!1.1r5z(

aKn BD, (A13)

where Kn"j/a. The resulting variations of translational slip-correction factor with Kn for b"5 are shown inFig. A1.

Rotational slip-correction factors

In the current literature, studies on the slip-correction factor for the rotational motion of a rigid particlesuspended in #uid are rather scarce, those on for the translational motion. A complete and accurate expression forthe rotational slip-correction factors of an ellipsoid of revolution is not available at present due to the lack oftheoretical studies and experimental evidences. In this section, the ASA method which was originally developed forevaluating the translational slip-correction factors is extended to the rotational motion. As noted before, in orderto apply the ASA method to the rotational motion, knowledges about the moments acting on the particle in boththe continuum and the free molecule regimes and an empirical expression for the variation of rotationalslip-correction factor of a sphere are necessary. That is, the rotational counterparts of equations (A1)}(A4)and (A8).

While the moments acting on an ellipsoid of revolution in the continuum regime was obtained by Je!ery (1922),expressions of these quantities in the free molecular regime is not available. In this section, the rotationalslip-correction factors for a cylindrical particle is discussed. For b<1, these rotational slip-correction factors mayserve as approximations to those for the elongated ellipsoids of revolution. For a cylindrical particle rotating ina quiescent #uid, the hydrodynamic moments acting on the particle are given by

Mx("f 3

x(u

x(, M

y("f 3

y(u

y(, M

z("f 3

z(u

z(, (A14)


Fig. A2. Rotational slip-correction factors for di!erent aspect ratios.

Fig. A3. E!ect of the slip-correction factor on the simulation results.

where f 3iK's are the components of rotational resistant tensor with respect to the iK particle axes. It is also noted that

f 3x("f 3

y(due to the geometrical symmetry of the particle.

In the continuum regime, using the slender body theory, the value of f 3x(

was obtained by Edwards (1928) as

f 3x("

(6n/3)ka3b3

ln(2b)#0.1931. (A15)

At the free molecular limit, the value of f 3x(

was given by Eisner and Gallily (1981) as

f 3x("

8nka3b3

Kn CA1

4#

1

4b#

1

4b2#

1

8b3B#aAn!6

48!

1

8b!

1

8b2#

n!4

64b3 BD . (A16)

The rotational slip-correction factor in the continuum regime is C3x("1, and that for the free molecular regime may

be obtained using equations (A15) and (A16) as

C3x(D&.

"

8Kn

3[ln(2b)#0.1931]CA1

4#

1

4b#

1

4b2#

1

8b3B#aAn!6

48!

1

8b!

1

8b2#

n!4

64b3 BD. (A17)

Assuming that the empirical expression for variation of slip-correction factors as given by equation (A8) is alsovalid for the rotational motion, the adjusted radius for rotational motion, r3

x(, may be obtained from equations (A9)

and (A17). This leads to

r3x("

4.971a

8[ln(2b)#0.1931]CA

1

4#

1

4b#

1

4b2#

1

8b3B#aA

n!6

48!

1

8b!

1

8b2#

n!4

64b3 BD . (A18)

Using the radius of adjusted spheres as given by equation (A18) to replace r@ in equation (A8), the rotationalslip-correction factor C3

x(is obtained. Due to the geometrical symmetry of the particle, C3

x("C3

y(and the rotational


slip-correction factor with respect to the z( -axis is not needed. Figure A2 shows that variations of C3x(

("C3y() with

Kn for b"5.The e!ects of rotational slip-correction factors on deposition rate are investigated by comparing the resulting

deposition velocities as obtained with rotational slip correction to those obtained without using rotational slipcorrection. Figure A3 shows the comparison. The open symbols display the deposition velocities obtained withoutthe rotational slip correction, whereas the "lled symbols are those obtained with rotational slip correction. It isobserved that, over a broad range of particle sizes, the e!ects of rotational slip-correction factors on the resultingdeposition velocities are within the simulation scatters. Therefore, it is assumed that the slip e!ect of rotationalmotion does not a!ects the overall deposition rate signi"cantly and is not included in the simulations.


Documents

WALL DEPOSITION OF SMALL ELLIPSOIDS FROM TURBULENT AIR FLOWS—A BROWNIAN DYNAMICS SIMULATION