Aerosol particle deposition in an obstructed turbulent duct flow

Pergamon J. Aerosol Sci., Vol. 25, No. 1, pp. 91-112, 1994 Copyright © 1994 Elsevier Scion~ Ltd

Printed in Great Britain. All fights r'-~crve¢l 0021-8502/94 $6.00 + 0.00

A E R O S O L P A R T I C L E D E P O S I T I O N I N A N O B S T R U C T E D T U R B U L E N T D U C T F L O W

AMY LI,* GOODARZ AHMADI,* RAYMOND G. BAYER t a n d MICHEAL A. GAYNES t

* Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, U.S.A. * IBM Corporation, Endicott, NY 13760, U.S.A.

(First received 24 March 1993; and in final form 16 August 1993)

Al~tract--A computational scheme for simulating aerosol particle dispersion and deposition in turbulent flows in passages with complex geometry is developed. A thermodynamically consistent rate-dependent algebraic stress model is used to simulate the mean turbulent flow fields. The instantaneous turbulence fluctuation is simulated as a continuous Gaussian random field. The Brownian motion is modeled as a white noise process. The particle equation of motion including the Stokes drag, Brownian and Saffman forces is used. The computational model predictions for particle deposition velocity in a turbulent channel flow are compared with the experimental data and earlier simulation results. Several digital simulations for aerosol particle transport and deposition in a duct with an obstructing block are performed. The corresponding capture efliciencies of rectangular and trapezoidal blocks for different particle Stokes number are evaluated and discussed. It is shown that the deposition rate decreases significantly as the shape of the obstruction becomes more streamlined.

INTRODUCTION

Considerable effort has been devoted to understanding the kinetics of aerosol dispersion and deposition due to its significance in numerous industrial processes. An extensive review of particle diffusion in laminar flows was provided by Levich (1972). More recent studies in connection with microcontamination processes were presented by Cooper et al. (1989) and Liu and Ahn (1987). Cooper (1986) summarized the state of understanding of the microcontamination control in electronic industries. In turbulent flows, particles are transported by the mean motion and are dispersed by turbulence fluctuation and Brownian diffusion. Fuchs (1964), Davies (1966), Friedlander and Johnstone (1957), and Cleaver and Yates (1975) provided semi-empirical expressions for particle mass flux from a turbulent stream to smooth surfaces. Particle deposition to rough walls was studied by Browne (1974), EI- Shobokshy and Ismail (1980), and Wood (1981a). Extensive reviews on the subject were provided by Wood (1981b), Hidy (1984) and Papavergos and Hedley (1984).

Using digital simulation to study diffusion of particles in turbulent pipe or channel flows was considered by a number of authors. Ahmadi and Goldschmidt (1970) used digital simulation and analytical techniques to study the turbulent dispersion of small spherical particles. Peskin (1975) considered turbulent diffusion of particles in a channel flow. McLaughlin (1989), Pedinotti et al. (1992), Ounis and Ahmadi (1989) and Ounis et al. (1993) analyzed the aerosol particle deposition in a channel using a pseudospectral computer code to simulate the instantaneous turbulent flow field. Kallio and Reeks (1989) used a Lagrangian random-walk approach to modeling particle deposition in turbulent boundary layers. Berlemont et al. (1990) used Lagrangian approach to describe particle dispersion in turbulent flows. Ounis et al. (1991) also studied Brownian diffusion of submicron particles in the viscous sublayer. Abuzeid et al. (1991) used a simple simulation technique to study the dispersion and deposition processes of suspended particles released from point sources in a turbulent channel flow. Earlier, Crowe (1979, 1982) and Durst et al. (1984) used a com- bined Eulerian-Lagrangian approach for analyzing gas-particle flows. Sommerfeld (1992) studied confined particulate two-phase flows including particle-wall collisions. Recently, Li and Ahmadi (1992, 1993) performed a series of digital simulations on deposition of aerosols of various sizes from point sources and initially uniform concentration in turbulent channel flows.

91

92 AMY Lz et al.

In spite of numerous studies for simple channel and pipe flows, little work was done on particle transport and deposition in complex geometries of industrial interest. An example of this type of concern is the situation associated with the forced air cooling of electronic packages used in the computer industry. In these applications, air is forced through complex geometry passages which is a result of the overall packaging design and presence of individual components. Dust may then accumulate on and around various components and connectors, which can result in loss of cooling and electrical connector instabilities. Consequently, it is desirable to understand the effect of obstruction shapes on particle deposition rate. Such an understanding could also be helpful in improving the design of electrostatic precipitators. Furthermore, it could provide insight into the natural phenomena of sand dune formation and snow fences.

The present study is concerned with developing a computational model for predicting dispersion and deposition of aerosol particles in complex geometries. The effects of air stream turbulence and its anisotropic nature are included in the model. The particle equation of motion includes the Stokes drag, the Brownian, the Saffman lift, and the gravitational forces. The Brownian force is simulated as a white noise process. A thermodynamically consistent, rate-dependent algebraic stress model is used to simulate the mean turbulent flow conditions. The instantaneous fluctuating turbulent velocity field is modeled by an anisotropic Gaussian random process. Several digital simulations concerning the dispersion and deposition of aerosol particles in a duct with an obstruction are performed. The results show that a large number of particles may deposit on the rectangular block due to impaction and interception. However, the particle deposition rate decreases significantly as the shape of the obstacle becomes more streamlined.

PARTICLE EQUATION OF MOTION

The equation of motion of a small aerosol particle is given by

du[' 1 2Kvl/2d,j ( 1 ) =-(u i -u~)4 (uj-u~)+ 1--~ 9i+ni(t),

dt z Sd(dlkdkl) x/4 and

(1)

where 2 is the molecular mean free path of the gas, and the deformation rate tensor d o is defined as

dij =½ (ui,j + uj.3. (4)

The first term on the right-hand side of equation (1) is the drag force due to the relative motion between particles and fluid. The second and third terms are the Saffman lift and the gravitational forces. The fourth term is the Brownian force. The drag force is always present and is generally a dominating force.

It should be emphasized that the Saffman lift force becomes important only in the region with a sbtrong shear field and for particles which are not too small. Recent study of McLaughlin (1991) showed that the expression given by Saffman (1965, 1968) is accurate when the particle is away from the wall. At distances very close to the wall, the lift force becomes smaller than that predicted by Saffman. Here, however, the wall effects for both

dxi p S d 2 C c

-dT=ui , T= 18--7-' (2)

where ur is the velocity of the particle, xi is its position, t is the time, d is the particle diameter, S is the ratio of particle density to fluid density, g~ is the acceleration of body force, z is the particle relaxation time, n~(t) is the Brownian force per unit mass, v is the kinematic viscosity, K=2.594 is the constant coefficient of Saffman's lift force, and ui is the instantaneous fluid velocity with u~= t/~ + u'i, where ~i is the mean velocity of the fluid, and u; is the fluctuation component of fluid velocity. In equation (1), C¢ is the Stokes-Cunningham slip correction given as

C¢ = 1 + ~ (1.257 + 0.4 e-l"la/2a), (3) a

Aerosol particle deposition in turbulent flow 93

drag and lift forces are neglected. In addition, the rarefaction correction for the lift force is also ignored. This, however, does not affect the results, since the influence of the lift force for submicron particles is negligible.

The Brownian force, which is very important for submicron particles, is modeled as a Gaussian white noise random process. The simulation procedure for the Brownian excitations was described at length by Li and Ahmadi (1992, 1993) and, hence, need not be repeated here.

Equation (1) requires the knowledge of the instantaneous fluid velocity field. The mean flow field is evaluated by using a nonlinear rate-dependent turbulence model. The instantaneous turbulent fluctuating velocities are approximated as continuous anisotropic Gaussian random fields. These are described in the subsequent sections.

TURBULENT MODEL

The two-equation k-~ model is widely used in industrial applications due to its simplicity and the relative ease that it could be incorporated in the Navier-Stokes equation solver codes. However, it is also well known that the standard k-e model suffers from several serious shortcomings. Among the obvious ones are its inability to handle unequal turbulent normal stresses and its limitation in using an isotropic eddy viscosity. Rodi (1976) developed an algebraic stress model that led to anisotropic eddy viscosity and offered certain improvements. Speziale 0987) suggested a nonlinear expression for the turbulent stress tensor that satisfies the required invariance properties. Using a multiple scale direct iteration approximation, a similar expression was obtained by Yoshizawa (1984, 1985). Ahmadi and Chowdhury (1991) used an iterative solution of the stress transport model and obtained a similar rate-dependent algebraic expression for the turbulent stress tensor. A thermodynamically consistent rate-dependent model for turbulence was developed by Ahmadi (1991) and Chowdhury and Ahmadi (1992).

For an incompressible fluid, the equations of continuity and balance of momentum for the mean motion are given as

~a, =0, (5) c~xi

~fil c~fil l c3p c~2~ 8 ~ - + fi; t~x--j. = p ~xi F v 8xjtOxj ~xj R O, (6)

where/~ is the mean pressure, p is the constant mass density, and Rq is the Reynolds stress tensor (second moment of fluctuation velocity). The thermodynamically consistent anisotropic expression for the Reynolds stress tensor as obtained by Chowdhury and Ahmadi (1992) is given as

2 __2vT3, j - v r k [0t 6 - k - 3 3 - - 1 - - 1 Rij='~ k f i j ~ ~ ~ dij +T -~ dtk kS ij--fl(dikdjk--- ~ dkldklfij) ], (7)

where k is the turbulence kinetic energy, v T the turbulent eddy viscosity, e is the energy dissipation rate and ct, fl and y are certain constants. Here,

5 d ~ d ad,j _ _ -~ i j = ~ ij+flk~xkd-dikCOkjq-djk(Ok, , (8)

is the co-rotational (Jaumann) derivative of mean deformation rate tensor, and the deformation rate tensor and spin tensors are, respectively, defined as

d,j=~ -~xj+-~xl , COjk=~ \dXk 8xj ]" (9)

The eddy viscosity v T is given by V T = CUk2/t~, (10)

where C" is a constant.

94 AMY LI et al.

The equations governing the kinetic energy of turbulence and the dissipation rate are

Ok Ok ~ + ~ ~xjJ ~xj . . . . 1 - -R i i - - - -e , (11) cgt + fij ~xj t~xj

-~+U~Oxj Oxj v + ~ i OxjJ kRiJ~xj -k" (12)

The values of constants are given as

C~=0.09, ak=l ,

~t =0.93,

a t= 1.3, C ~1 =1.44, C ~2= 1.92, (13)

fl--0.54, y=~--o.o06. (14)

This thermodynamically-consistent model leads to an anisotropic effective viscosity and is capable of predicting the expected turbulence normal stress differences. It was reported by Ahmadi and Chowdhury (1991) and Chowdhury and Ahmadi (1992) that the mean flow properties, including mean-square fluctuation velocity components, as predicted by this new model are in good agreement with the experimental data for a number of turbulent flows.

For simulating particle deposition processes, the fluctuation of turbulence perpendicular to the wall plays a crucial role. Therefore, the computational model must have the capability of accurately predicting the components of turbulence intensity. Thus, a simple k-e model, which is associated with an isotropic assumption, is inherently incapable of providing the needed information.

The requirements of the anisotropic rate-dependent turbulent model were incorporated in the STARPIC-RATE computational model (a modified version of STARPIC computer code by Lilley and Rhode (1982)) which was described at length by Ahmadi and Chowdhury (1991), and Chowdhury and Ahmadi (1992). In this study, the STARPIC-RATE computational code was used and the mean turbulent flow conditions in a duct with obstructing blocks of various shapes were evaluated. In these computations, it was assumed that the mean flow is steady and two-dimensional and a staggered grid with 96 x 48 node points was used. Uniform profiles for the mean velocities, turbulence kinetic energy and dissipation rate were specified at the inlet. Typically, values of inlet velocity V= 5 m s- 1, inlet fluctuation energy k = 0.25 m 2 s - 2 , l~-0.4 mm and e = k 3/2/1, where I is the length (integral) scale, were used. For the grids near the wall, the standard wall function boundary conditions were prescribed. At the outlet, zero normal gradient conditions were specified.

SIMULATION OF TURBULENT FLOW FIELD

The turbulence fluctuations are random functions of space and time. The Monte-Carlo velocity simulation techniques have been used as an economical method for generating time histories that have the random characters and statistical properties of turbulence. Kraichnan (1970) suggested a simple method for generating a Gaussian random field which resembles a pseudo-isotropic turbulence. Accordingly, the nondimensional instantaneous fluctuating velocity is given as

~1"(1", t*)= ~l(k.)cos(~c. '~*+co.t*)+ ~2(t .)sin(~.-~*+to.t*) . (15) n n = l

In this equation, ~1(~.)=(. x ~., ~2(k.)=~. x ~., (16)

with ~." ~1 (~.)=~." ~2(~.) = 0, (17)

ensures the incompressibility condition. The components of vectors ~. and ~n and the frequencies ton are picked independently from a Gaussian distribution with a standard


deviation of unity. Each component of To, is a Gaussian random number with a standard deviation of 1/2. Here, N is the number of terms in the series.

In equation (15), the dimensionless quantities are defined as

x* =--x t* =--t u!. =u[ ' (18) 1o' to' - ' up'

r is where lo, to and up are the length, the time and the velocity scales of turbulence and u i the fluctuation fluid velocity. For this pseudo-turbulent velocity field the energy spectrum E(k) is given by

E(k)= 16(2/rc)X/2 k4 e - 2k2, (19)

where k is the wave number. In this study, the fluctuation velocity given by equation (15) is modified in order to make

it suitable for generating the nonisotropic instantaneous velocity field. It is assumed that

u;=u[*ei(?~) (no sum on i), (20)

where el(~), e2(~) and es(~) are the axial, vertical and transverse root-mean-square turbulent fluctuation intensities calculating from the STARPIC-RATE (rate-dependent) computational model.

Normal component of turbulence fluctuations near a wall has a profound effect on the deposition rate of particles. Therefore, the magnitude of e2(~ ) must be correctly evaluated for short distances from the wall. In this study, a quadratic variation between the first grid and the wall is assumed.

Estimates for the length and time scales of turbulence for wall bounded duct flows were provided by Davies (1972). Accordingly,

lo = 0.1 h(2 Re)- l/s, (21) and

h to=~-p, (22)

where V is the section average velocity and h is the channel half-width. Equations (15) and (20) with N = 100 together with (21) and (22) are used for simulating

the fluctuation components of turbulent velocity in the duct. It should be emphasized that with the use of instantaneous fluid velocity field in equation (1), the full interaction of particle with the local turbulence is accounted for. As expected, small particles respond quickly to the turbulent fluctuating motions, while large particles are sluggish in their response due to inertia.

RESULTS AND DISCUSSION

In this section, simulation results for dispersion and deposition of particles which are emanating from point sources at different locations and from an initially uniform concentration at the inlet of the channel are described. Simulations are performed for a channel which is 2 cm wide and 10 cm long. An obstruction in the shape of a two-dimensional rectangular or trapezoidal block is assumed to be present. This flow geometry resembles a segment of the cage passages of a computer. The special case of unobstructed channel is also studied as a bench mark for code verification.

A temperature of 288 K, # = 1.84 x 10- s N" s m - 2 and p = 1.125 kg m- 3 for air are used. As noted before, a mean air velocity of V= 5.0 m s- 1 at the channel inlet is also assumed. Thus, the flow Reynolds number based on the channel width is 6657, and the air is in a state of turbulent motion. A density ratio of particle to fluid S = 2000 and different particle diameters ranging from 0.01 to 20 #m are used in the analysis. Ensembles of 1000 samples are employed for evaluating various particle trajectory statistics and deposition rates. While in most cases it is assumed that when a particle touches a wall it is going to stick to the surface, the possibility of particle rebound from a dry surface is also studied.

96 AMY L] et al.

Channel flow

In order to verify the accuracy of the computational model, a series of simulations for an unobstructed are performed. A uniform inlet concentration of 600 particles per 1 mm height of channel is used and the ensemble of particle trajectories were evaluated and statistically analyzed. The simulations were repeated for particles of different sizes from 0.01 to 10 #m. Deposition rate of various size particles were evaluated. Figure 1 shows variation of nondimensional deposition velocity with nondimensional particle relaxation time. The corresponding particle diameters used in the simulation are also shown in this figure. The nondimensional deposition velocity is defined as

J u~ = (23)

C O / , / * '

where J is the particle flux to the wall per unit time, Co is the particle concentration and u* is the shear velocity. The nondimensional particle relaxation time is defined as

TU . 2 S d 2 u .2 T + = _ _ _ _ v - 1 8 v 2 C c. (24)

The experimental data as collected by Papavergos and Hedley (1984) and earlier simulation results of McLaughlin (1989), Ounis et al. (1993) and Li and Ahmadi (1993) are shown in Fig. 1 for comparison. In this figure the dashed line is the result calculated from the empirical equation suggested by Wood (1981b) given as

U~" = 0 . 0 5 7 S c - 2 / 3 - t -4 .5 x 1 0 - 4 Z +2 , (25)

where Sc is the Schmidt number defined as

Sc = v/D, (26)

with D being the particle mass diffusivity. The solid line in Fig. 1 is the prediction of sublayer model of Fan and Ahmadi (1993).

It is observed that the present simulation results are in reasonable agreement with the experimental data, model prediction of Fan and Ahmadi, and earlier simulations for

1 0 °

10 - t

10 -z

0

"~ lO-S @

10--4

10

I

• . ?.;

. .".'-~. ~ .;

j i t

d ( # m )

0 . 0 1 0 . 1 1

I I

a P r e s e n t Resu l t Li & A h m o d i

• McLGughl in • P a p a v e r g o s & Hed ley

Wood - - Fan & A h m a d l

"_,

". .~: D • ~:1

,N • ~'/-I

\ \ o ./. ; \ \ • / . . .

\ \ o .'/." s" ANN~ . / . - " .: o~

\ \o . I

1 0 -3 1 0 - s 1 0 -1 1 0 °

10 "5 i

1 0 -~ 101 1 0 z

./.+

Fig. 1. Var ia t ion of depos i t ion velocity with particle relaxat ion t ime (z + ) for an unobs t ruc t ed channe l flow.


particles larger than 0.5 ttm. The results are also in qualitative agreement with the empirical equation given by (25) and the earlier simulation for submicron particles in trend of variations. From Fig. 1 it is observed that u~- follows the expected V-shape curve variation. The minimum deposition velocity occurs for particle diameters in range of 1.0-3.0 gm for the flow conditions used in this study. This is because that the Brownian motion is the controlling mechanism for submicron particle deposition, while the turbulent eddy impaction becomes dominant for particle deposition with d > 5 #m.

Obstructed duct (point source)

The simulation results for point sources of particles in a duct with an obstructing block are presented in this section. Effects of modification in the shape of the block are also studied.

Rectangular obstruction. It is assumed that a rectangular 1.25 cm x 0.61 cm block is attached to the upper wall of the channel. Figure 2 shows the mean velocity vector plot, and the streamwise and vertical root-mean-square fluctuation intensities el and e2 in a duct flow with a rectangular block. It is observed that a large recirculation region behind the block is formed. In addition, one vortex upstream and a small vortex downstream in the corners along side of the block are observed. The air flow also seems to separate at the front tip of the block. Figure 2 clearly shows that the turbulence is anisotropic. In the recirculation region, the root-mean-square streamwise fluctuation el component is much larger than the vertical component e2. While near the tip of the block, the vertical fluctuation component is higher than the streamwise one.

Figure 3 displays variations of particle trajectory statistics for different diameters from a point source at a distance of 0.1 cm from the upper wall (Yo = 0.019 m). The gravitational effect is neglected in these simulations. The mean particle paths are shown by the solid lines. Here a corresponds to the standard deviation of particle trajectories in the vertical direction. The absolute maximum and minimum trajectories for an ensemble of 1000 particles are also shown in this figure for reference. While these are sample dependent, they

2.0 1.9

A

[~ 1.8 >. 1.7

1.6 1.5

2.5 3.0 3.5 4.0 4.5

X (cm)

. , * ~ t / t

. i s t • • •

, i t i i , ,

tl I I I t ,

5.0

VELOCITY FIELD

2.0

,_1.5

0.0 0 2 4 6 8 10

x (cm)

Fig. 2a. Mean flow velocity vector plot for a rectangular block.

98

g'

3.57554 3.33717 3.0988 2.86043 2.62206 2.38369 2.14532 1.90695

1.66858 1.43021 1.19185 0.953476 0.715107 0.476738 0.238369

m

~

2.0

• --, 1.5 E ~'~.0 >..

0.5

0,0 0

2.0

,_,1.5

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>" 0.5

0.0

AMY L] e t al.

S T R E A M W I S E FLUCTUATION C O M P O N E N T

0 2 4 6 8 X (cm)

VERTICAL FLUCTUATION C O M P O N E N T

2 4 6 8 X (cm)

Fig. 2b. Fluctuation velocity components for a rectangular block.

10

10 V'

4.95317 4.62296 4.29275 3.96254 3.63232 3.30211 2.9719 2.64169 2.31148 1.98127 1.65106 1.32085 0.990634 0.660423 O.33O211

provide additional information on the spreading of particles. The numbers listed on various surfaces in this figure denote the number of deposited particles. For 10 #In particles, it is observed that 755 particles are deposited on the front surface of the block and 2 particles are deposited on the bottom surface of the block. For 1/~m and 0.01/an particles, respectively, 106 and 105 particles reach the front surface of the block. In addition, 24 1 #m particles and 31 0.01 #m particles deposit on the surface of the upper wall behind the block. Figures 3b and 3c also show that about 15 1/~m and 0.01 #m particles are deposited on the back surface of block. The high deposition rate of 10 #m particles shows that the inertia impaction mechanism is quite effective for these relatively large particles. The small particles are, however, deposited mainly due to diffusion. The 1 and 0.01 #m particles are also more easily captured by the recirculation zone and are consequently deposited on the channel wall downstream of the block.

Figure 4 shows the particle trajectory statistics for the source distance of 2 mm from the wall (Y0 = 0.018 m). As the source distance from the wall increases, the numbers of 1 #m and 0.01/~m particles deposited on the block are reduced significantly. However, the number of 10 #m particles that deposit on the front surface of the block increases slightly.

It is observed from Fig. 5 that the numbers of deposited particles of all sizes decrease when the point source is moved further away from the wall to Yo =0.017 m. In this case, 155 10/~m particles deposit on the block, while only one of 1 #m and 0.01 #m particles reach the block. Additional simulation results (not shown here due to space limitation) show that the deposition rate of particles generally decreases as the point source location moves away from the wall. The slight deviation from this trend for 10/~m particles and point source locations at II0 =0.019 m and 0.018 m is due to the strong reverse flow due to the upstream corner vortex as shown in Fig. 2.

Trapezoidal obstruction. Here, it is assumed that a trapezoidal block with the parallel sides being of ~ength of 3.63 cm and 1.23 cm and a height of 0.61 cm is attached to the upper wall of the channel. The front side has a slope of 25.5 °, and back side has a slope of 29 °. The corresponding mean velocity vector plot, and the streamwlse and vertical root-mean-square

Aerosol particle deposi t ion in turbulent f low 99

0.020

0.015

0.010

0 .005

0 .000 0 .00

- - - A b s o l u t e M a x i m u m - - - - - - Mean +~r - - Mean - - - - - Mean -o" . . . . Absolute Minimum Y o : O . O 19m

I . . . . , ,

, 55 r _ A

l d=tO.m (a) i I i [ i I , I , /

0.02 0 .04 O.Ofl 0 .08 0 .10

0.020

0.015

0.01o

0 .005

0 .000 0 .00

24

d=1#m (b) ] , I , I , I l I I I

0 .02 0 .04 0 .06 0 .08 0 .10

31 O.O2O ~ (

~ _ - - ~ ~ 848 0.015 ~ - P ~ _"-S'~ . . . . .

• ~ 0 .010 t,-,

0.005 c)

0.000 0.00 0.02 0.04 0.06 0.08 0.10

x (m)

Fig. 3. Particle trajectory statistics for ro =0.019 m. (a) d= 10/am, (b) d= 1 #m, (c) d=0.01/~m.

fluctuation intensities el and e2 are shown in Fig. 6. A relatively large recirculation region behind the block and a small vortex at the upstream corner of the block are observed. This figure also shows that the turbulent fluctuation intensities are much lower than those for a rectangular block as shown in Fig. 2. Figure 6 also clearly shows that the turbulence is anisotropic with axial intensity being much larger than the cross-stream component in most of the flow region.

Figure 7 displays variations of particle trajectory statistics for different diameters from a point source at a distance of 0.1 cm from the upper wall (Yo = 0.019 m) for the trapezoidal obstruction. For 10/am particles, it is observed that 489 particles are deposited on the front surface of the block and 509 particles leave the duct. For 1 #m and 0.01/am particles, respectively, 42 and 52 particles reach the front surface of the block. In addition, 55 1/am particles and 49 0.01/am particles deposit on the surface of the upper wall behind the block. Figures 7b and 7c also show that 34 1/am and 39 0.01/am particles are deposited on the back surface of block.

Comparing Figs 3 and 7, it is observed that the number of 1/am and 0.01/am particles deposited on the front side of the trapezoidal block are roughly about one-half of those for the rectangular block. This implies that the presence of the slope in the front side of block reduces the effectiveness of the inertia impaction mechanism for 10/am particles. The numbers of particles deposited on the back side of the trapezoidal block and on the upper wall downstream of the block are, however, almost doubled. These results show that when

100 AMY Li et al.

A

0.020

0.015

0.010

0.005

0.000 0.00

- - - A b s o l u t e M a x i m u m

- - - - - - M e a n +o"

- - M e a n

M e a n - u

. . . . A b s o l u t e M i n i m u m Y o = O . O 1 8 m

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

0 . 0 2 0

0 . 0 1 5

.~ 0.010

0 . 0 0 5

0.000 0.00

I 0

d = lUlqo_ ( b )

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . I 0

13 0.020 _ _ . . - . , - _ ~ . . . . . . , . . . . • . . . . .

.~ 0.010 .... L-= L---- .....

0.005 d=O.OI/xm (e) t 0 . 0 0 0 , ~ , ~ , L , ~ , ~

0.00 o.oe 0.04 0.05 0.08 0.10

x (m)

Fig. 4. Pa r t i c le t r a j e c t o r y s ta t is t ics for Yo=O.O18 m. (a) d = l O # m , (b) d = I #m, (c) d = O . O l #m.

the shape of the obstruction becomes more streamlined the capture efficiency of its front side reduces significantly. In this case, however, more particles may be entrained in the recirculation region because particle trajectories are not deflected to a major extent by the block.

Rebound. In the performed simulation studies, so far, the particle.rebound effect was neglected. For large particles, when the impact velocity is larger than a certain critical value, the particles may rebound from the surface. The critical approach velocity was given by Friedlander (1977)

V~= - - \ r2 //_] , (27)

where m is the particle mass, r is the coefficient of restitution, E is the surface potential energy given as

Ad E = - - (28)

12yo"

Here, A is the Hamaker constant, yo=4h, is the equilibrium separation distance. For a silicon particle and a gold surface, the Hamaker constant is 31.6 x 10-2°J. For 10 #m particles, the critical approach velocity is about 0.9 cm s- 1 and 6 cm s- 1 for the coefficients of restitution of 0.96 and 0.5, respectively.

A e r o s o l pa r t i c l e d e p o s i t i o n in t u r b u l e n t f l ow 101

0 . 0 2 0

0.015

,~ 0.010 t-

0.005

0.000 0 . 0 0

- - - A b s o l u t e M a x i m u m .

- - - - M e a n + ~

- - M e a n

-- - - - M e a n - o .... Absolute Minimum Yo=O.OlTm

- - - - - _ 3 _ - _ 84s

-$2--_ d=10Nm (a)

I I I I

0 . 0 2 0 . 0 4 0 . 0 0 0 . 0 8 0 . 1 0

0 . 0 2 0

0 . 0 1 5

~,~ 0 . 0 1 0

0 . 0 0 5

0 . 0 0 0 0 . 0 0

6

0 . 0 2 0 . 0 4 0 . 0 8 0 . 0 8 0 . 1 0

6 0 . 0 2 0 . , . , , . . . . , - - . , . , , . . . .

O.OLO =

0 . 0 0 5 d 0.01?m . " , . , , , .(c) t 0 . 0 0 0 ?

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 8 0 . 0 8 0 . 1 0

X ( m )

Fig. 5. P a r t i c l e t r a j e c t o r y s ta t i s t ics for Yo = 0 . 0 1 7 m . (a) d = I 0 / a n , (b) d = 1 pro, (c) d = 0 . O l pro.

The computational model was upgraded to account for the particle rebound effects, and several simulations were performed. Different shape blocks and various coefficients of restitution were used. Sample results for deposition of 10/~m particles are shown in Figs 8 and 9. It is observed that rebound effect significantly reduces the particle deposition rate. Figure 8 shows that the number of deposited particles on the rectangular block decreases to 57 and 56 as r increases to 0.5 and 0.96, respectively. For the trapezoidal obstruction as shown in Fig. 9, only 4 and 2 particles deposit. Additional simulation results (not shown here) show that particle rebound effects are insignificant for particles less than 1/tm.

It should be emphasized that the simplistic model used in this section shows the general features of the effect of particle bounce during impact in reducing the deposition rate. There are, however, more advance models available which could account for plastic deformation, velocity dependent energy losses and other parameters (Rogers and Reed, 1984). In addition, particle resuspension effects were not included in this study. Particle re-entrainment is important for many industrial applications. In particular, when surfaces are covered with particles, resuspension could become the main source for particle emission into the fluid stream.

Obstructed duct (uniform concentration)

To provide an understanding of the deposition rates of particles with a uniform inlet concentration a series of simulations are performed. With 2440 particles released randomly

AS 25:1-H

102 AMY L] et al.

2.0

,_1 .5

i i ' ,-., 1.9 ", ,

~ 1 . 8 ~ - U > . 1 . 7 - - - - - - - ~ ~ .

1.6 r--'-- - - - - - - - - "--- ~ . ~ . ~ . ~ . ~ . ~ , ~ . \

1 . 5 . . . . . . . . . . . . . . . . • ~ ' '

2 3 4 5 6

X (cm)

VELOCITY FIELD

0 2 4 6 8 10

x (c=n)

Fig. 6a. Mean flow velocity vector plot for a trapezoidal block.

u'

1.75139 1.63463

1.51787 2.0 1,40111 1.28435 ....1.5

1.16759 ~5 1.05084 1.0 0.934076 >"

0 . 5 0.817316

0.700557 0.0 0.583797 0.467038 0.350278 0.233519 0.116759

S T R E A M W I S E FLUCTUATION C O M P O N E N T

0 2 4 6 8

X (cm)

VERTICAL FLUCTUATION C O M P O N E N T

2 .0

,,., 1 .5

~1.0 0.5

0.0 0 2 4 6 8 10

X (cm)

Fig. 6b. Fluctuation velocity components for a trapezoidal block.

iiii il i

10 v'

3.00934 2.80872 2.6081 2.40747 2.20685 2.00623 1.8056 1,60498 1,40436 1,20374 1.00311 0.802491 0.60186~ 0.40124~ 0.20062~

within 6.1 mm region (height of the block) from the upper wall at the inlet of the duct, the inlet concentration is uniform at 400 particles per 1 mm height of the channel. The initial velocities of the particles are assumed to be identical to those of the flow field at their locations. Ensembles of particle trajectories are generated and their transport and deposition are analyzed.

Aeroso l pa r t i c le d e p o s i t i o n in t u r b u l e n t f low 103

0 . 0 2 0

0 . 0 1 5

0 . 0 1 0

0 . 0 0 5

0 . 0 0 0 0 . 0 0

- - - A b s o l u t e M a x i m u m - - - - - - M e a n + o r

- - M e a n

- - - - - M e a n - a

. . . . A b s o l u t e M i n i m u m Yo=0.0 1 9 m 2

d=10/zm (a) , I ~ I , I , I ~ /

0 . 0 2 0 . 0 4 0 . 0 6 0 .06 0 . 1 0

0 . 0 2 0

0 . 0 1 5

0 . 0 1 0

0 . 0 0 5

0.000 0.00

5 5

= m - " " " 2 " ~ " ~ ' - - -- - - -] 854

, , , ( b )

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

49 0 . 0 2 0

• ~ 0 . 0 1 0

0.005 d=0.01/zm (c)

0.000 , L I 0 . 0 0 0 .02 0 ,04 0 . 0 8 0 . 0 8 O. 10

x (m)

Fig. 7. Pa r t i c l e t r a j e c t o r y s ta t is t ics for Y0 = 0 . 0 1 9 m ( t r apezo ida l block). (a) d = lO #m, (b) d = 1 #m, (c) d=O.O1 #m.

Rectangular block. The results for the rectangular obstructing block are described in this section. Figure 10 shows variation of particle capture efficiency by the front surface of the rectangular block with particle Stokes number. The corresponding particle diameters for the present flow conditions are also shown in this figure for reference. The Stokes number is defined as the ratio of the stopping distance of a particle to the block height H, i.e.

Stk =~V. (29) H

The capture efficiency is defined as Nd

~/= NH' (30)

where Nd is the number of deposited particles and NH is the number of particles swept by the block (2440 in these simulations).

The effect of particle rebound on particle capture efficiency is also studied. It is assumed that the particles are silicon and the surface is gold (gold covered). In Fig. 10, the corresponding variation of capture efficiency of the front side of the block, r/f, with Stokes number for several coefficients of restitution are shown. It is observed that for r =0 (no rebound case), as the particle Stokes number increases from 10- 5 to 10-1 (particle diameter from 0.01 to 1 #m), r/f increases only slightly. For larger particles the effect of inertia impaction becomes significant, and the capture efficiency increases rapidly as the particle

104 AMY LI et al.

0 . 0 2 0

0 , 0 1 5

.~ 0.010 >,

0 . 0 0 5

0,000 0.00

- - - Absolute Maximum

.... Mean +e

- - M e a n .

-- - -- Mean -~

.... Absolute Minimum Yo=O.O19m 1

1

r=O , I L - I

0 . 0 2 0 . 0 4 0 . 0 6

(a)

242

0 . 0 8 0 . 1 0

0,020

0.015

• ~ 0 , 0 1 0 >.,

0.005

0 . 0 0 0

2

r=0.5 (b) £ . - t J

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

941

2 1

0 . 0 1 5 x , - _ - - " " " - -

-~ 0.010 . . . ~ _ 939

o,oo r:o.oo i,;;7 0.000 B ~ L

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 0 0 . 0 8 0 . 1 0

x (m)

Fig. 8. P a r t i c l e t r a j e c t o r y s ta t i s t ics i n c l u d i n g r e b o u n d effects ( r e c t a n g u l a r b lock) . (a) r = 0 , (b) r = 0 . 5 , (c) r = 0 . 9 6 .

Stokes number increases beyond 0.2. The solid line in Fig. 10 is the theoretical prediction by Langmuir and Blodgett (as reported by Fuchs (1964)) for a potential flow about an infinitely long ribbon with no flow breakaway at the edge. The dashed line and the solid circles are, respectively, the theory and experimental data of Vincent and Humphries (1978) for airborne dust deposition on a circular disk in cross turbulent air flows. The present simulation results appear to be in good agreement with the theory of Langmuir and Blodgett, but overestimate the experimental data of Vincent and Humphries. The import- ance of blow-off of deposited particles was noted by Vincent and Humphries. However, particle re-entrainment has not been accounted for in the present computational model which could be the reason for the discrepancy.

From Fig. 10 it is observed that the particle rebound affects the capture efficiency for Stk>0.02 (d>2/tm). For the coefficient of restitution larger than 0.5, r/f decreases as particle Stokes number increases for Stk larger than 0.02. The simulation results also show that no rebound occurs for particles smaller than 0.3 pm. For d < 1/tm, the effect of rebound is, generally, very small.

Figure 11 displays variation of particle capture efficiency of the back surface of the rectangular block, r/b, with particle Stokes number. The effect of rebound is included in the analysis. The experimental data of Vincent and Humphries (1978) for the disk are also shown in this figure for comparison. It is observed that the present simulation results are in reasonable agreement with the experimental data of Vincent and Humphries. This figure

A e r o s o l p a r t i c l e d e p o s i t i o n in t u r b u l e n t f l o w 105

0 . 0 2 0

0 . 0 1 5

0 . 0 1 0

0 . 0 0 5

0 . 0 0 0

0 . 0 0

- - - A b s o l u t e M a x i m u m - - - - - - M e a n +(r

- - M e a n

- - - - - M e a n - ~

. . . . A b s o l u t e M i n i m u m Yo=O.O19m 2

~ . ~ . . ~ _ ~_ _ • , . , . , •

, ° °

r=O (a) 1 , t , i , i , I , I

0 . 0 2 0 . 0 4 0 . 0 8 0 . 0 8 0 . 1 0

0 . 0 2 0

0 . 0 1 5

0,010

0 . 0 0 5

0 . 0 0 0 0 . 0 0

r=O.5 , , , , , (b) ]

0 . 0 2 0 . 0 4 0 . 0 8 0 . 0 8 0 . 1 0

1 0 . 0 2 0 - . - .._7_ , , , , , . ,

0.010 " - . . . .

0.005 r=0.96 (c) J

0 . 0 0 0 ' ~ ' ' , ~ , ' ,

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

X ( m )

F i g . 9. P a r t i c l e t r a j e c t o r y s t a t i s t i c s i n c l u d i n g r e b o u n d ef fec ts ( t r a p e z o i d a l b lock) . (a) r = 0 , (b) r = 0 . 5 , (c) r = 0 . 9 6 .

also shows that the capture efficiency of the back surface is quite low, and decreases as the Stokes number increases. Figure 11 shows that the rebound has no significant effect on the capture efficiency of the back side of the block. For small coefficients of restitution, a large number of particles are captured by the front side of the block. Therefore, fewer particles are available for being captured by the back side. On the other hand, for a coefficient of restitution close to one, a large fraction of particles remain suspended in the air stream, but their chance for being captured decreases. The opposing effects of rebound on particle deposition on the back side of the block appears to roughly cancel out, so that the variation of r does not have a discernable trend.

Figure 12 displays variation of particle capture efficiency of the upstream wall (between the inlet and the block), r/u, with particle Stokes number. As expected, the capture efficiency is high for submicron particles due to their Brownian motion. As the Stokes number increases from 10- s to 10- 3, t/u decreases, and the minimum capture efficiency is reached at the Stokes number of about 10-3 to 10-1 (particle diameter of about 0.5-3 #m). For larger Stk, the capture efficiency increases due to turbulent eddy impaction. The V-shape variation of r/, is similar to the deposition velocity shown in Fig. 1 and was also observed by Li and Ahmadi (1993) for turbulent flows in an unobstructed channel. The capture efficiency is reduced significantly by the rebound effect for the Stokes number larger than 0.5 (particle diameter larger than 10/xm). This behavior was also observed in the earlier simulation results of Li and Ahmadi (1992, 1993).

106 AMY L! et al.

d (/~m)

0.8

%-.

0 . 6

o 0.4

~2

~ o.e

0 . 0

10 -6"° 10 a°

0 . 0 1 0 .1 1 10

I I i u Longmuir-Blodgett

o r=0 Vineent-Humphrie 4 \

A r = 0 . 5 \ / [ 0 r=0.96 ~ , / / • Vincent-Humphries ~ /o

(experiments) / /

0 °°I°

10-4.0 10 -z ' ° 10 °'°

Stokes Number

Fig. !0. Variation of capture efficiency of the front side of the rectangular block, r/t, with particle Stokes number.

i 0 -l

d(/zm)

0.01 0.1 1 10 I I F [

0

o

o

i 0 -z

10 -s

o r=O r = 0 . 5

¢ r=0.96 0 0 " " • Vincent-

A n ° • 0 O ~ ~ ~ . Humphries

D

e* • o

D

1 0 - 4 ~ I I

1 0 - e 1 0 -4 10 -2 10 ° 10 z

Stokes Number

Fig. 11. Variation of capture efficiency of the back side of the rectangular block, r/b, with particle Stokes number.

Figure 13 shows the variation of the total capture efficiency of the rectangular block and the upper surface of the channel, r/,. Because most of the particles are captured by the block itself, the variation of r/t with the Stokes number is very similar to that of 'h for the block surfaces. This implies that impaction on the obstruction is the dominating particle deposition mechanism.


d(/zrn)

0.08 0 . 0 1 0 . 1 1 10

I ] I I

v

¢J

r j

0.06

0.04

0.02

o r=O

A r = 0 . 5

o r=0.96

[] ,x

A

0

0.00 - . v . 0 U i

10 -6 i0 -~ 10-2 I0 ° I0 z

Stokes Number

Fig. 12. Variation of capture efficiency of the upstream watt, ~/u, with particle Stokes number.

d(~zm)

0.8 0 . 0 1 0 . 1 1 1 0

b i I I

0.6 O

"~ 0.4

~ 0.2 L )

[]

o r=0 A r = 0 . 5 []

¢ r=0.96

/ x

[] [] ¢

0.0 , , t 10 -6 10-4 10-z I0 o I0 z

Stokes Number

O0

Fig. 13. Variation of total capture efficiency of the rectangular block and the entire upper wall, T/t, with particle Stokes number.

F o r r>~0.5, Figs 10 and 13 show tha t the to ta l cap tu re efficiency of the b lock and the upper surface increases with Stk, while tha t of the front surface of the b lock decreases. This indica tes tha t par t ic le r ebound has a more significant effect on the front surface cap tu re efficiency.

Trapezoidal block. Figure 14 shows var ia t ion of par t ic le cap ture efficiency by the front surface of the t r apezo ida l b lock with par t ic le Stokes n u m b e r for a uni form inlet concen t ra -

108

0.8

AMY LI e t al.

d(/~m)

0.01 0.1 1 10

I I I I

0 .6

0

0 ~ 0 .4

r ~

0.0

10 -6 10 2

o r = 0 ¢

r = 0 . 5

o r=0.96 • • I::]

10 -4 10-~ 10 °

S t o k e s N u m b e r

Fig. 14. Variation of capture efficiency of the front side of the trapezoidal block, t/t, with particle Stokes number.

tion. The corresponding particle diameters for the present flow condition are also shown in this figure. Different values of coefficients of restitution are used and the effect of particle rebound on particle capture efficiency is studied. The theoretical prediction by Langmuir and Blodgett (solid line) and the corresponding simulation result for the rectangular block with r = 0 (solid square) are also shown in this figure for comparison. It is observed that for Stk <0.05 (d< 3 #m), r h changes very slightly. However, as Stk increases beyond 0.2 the capture efficiency increases rapidly. The present simulation result appears to be in reasonable agreement with the theory of Langmuir and Blodgett for potential flow around a ribbon. This figure also shows that r/r for the trapezoidal block is roughly about one-half of that for the rectangular block.

Figure 14 shows that the particle rebound affects the capture efficiency for particle Stokes numbers larger than 0.02. Furthermore, for r~>0.5 and Stk >0.02, t/t decreases as particle Stokes number increases. For submicron particles, the effect of rebound is very small. These trends of variation are very similar to those observed for the rectangular block.

Variations of the capture efficiency of the back side of the trapezoidal block, r/b, with particle Stokes number are shown in Fig. 15. The experimental data of Vincent and Humphries (1978) for the capture efficiency of the back surface of a disk are also shown in this figure for comparison. It is observed that the capture efficiency is about 0.01 for Stk <0.001 and it decreases as the Stokes number increases for Stk > 0.01. Comparison of Figs 11 and 15 shows that r/b of the trapezoidal block is larger than that of the rectangular block and the data of Vincent and Humphries. This is because for the trapezoidal block, fewer particles are captured by the front surface. Therefore, more particles are left in the stream which could deposit on the back surface of the block.

The simulation results for the particle capture efficiency of the upstream wall (between the inlet and the block), t/u, are shown in Fig. 16. The V-shape variation ofr/M is quite similar to those shown in Fig. 12. The capture efficiency also reduces significantly by the rebound effect for particle Stokes numbers larger than 0.5 (particle diameter larger than 10 #m).

Figure 17 shows the variation of the total capture efficiency of the trapezoidal block and the upper surface of the channel, r/t. The corresponding results for t/t of rectangular block (with r--0) is also shdwn in this figure for comparison. It is observed that the total capture

1 0 -1

Aerosol particle deposition in turbulent flow

d ( , m )

0.01 0.1 1 10 I I I t

109

¢o

o3

m r ~

1 0 - ~

i0 -s

O 0 0 o ~

r=O a r=0.5

g o r=0.96 • $ 0 • Vincent-

• Humphries

• • o

o •

1 0 -4 J , L

1 0 -e 1 0 - 4 1 0 -~ 1 0 ° 1 0 z

Stokes Number

Fig. 15. Variation of capture efficiency of the back side of trapezoidal block, t/b, with particle Stokes n u m b e r .

dOzm)

0.08 0.01 0.1 1 10

I I I I

o.o6

~ 0.o4

~2 r.r.l

0.0z

o r=O A r = 0 . 5

o r=0.96

o.oo * ~ 8 . . . . . - . . " " 1 0 -s 1 0 - 4 1 0 - ~ 1 0 z

Stokes Number

o

oo o i

10 °

Fig. 16. Variation of capture efficiency of the upstream wall, ~/M, with particle Stokes number.

efficiency of the trapezoidal block is very close to that of the rectangular block for Stk < 10 -z (d < 1 #m). However, t/t of the trapezoidal block is significantly lower than that for rectangular block for particles larger than 2/am. This is because diffusion is the dominan t mechanism for deposi t ion of very small particles, while impact ion and intercep- t ion mechanisms are most significant for deposi t ion of large particles. The present results also show as the obstruct ion becomes more streamlined, the effectiveness of the inertial

110

0.8

A M Y L I et al.

d(/~m)

0.01 0.1 1 10

[ I I I

0 . 6 0

0

• ~ 0.4

~ 0 .2 r~

[] r=O

= r=0.5

0 r = 0 . 9 6 • o

D

0 . 0 t ,

10 "6 10 -4 10 -s 10 ° 10"

Stokes Number

Fig. 17. Variation of total capture efficiency of the trapezoidal block and the entire upper wall, qt, with particle Stokes number.

impaction mechanism reduces. This is particularly noticeable for particles larger than 10/~m and when the rebound effect is included.

C O N C L U S I O N S

In this work, a digital simulation method for analyzing the deposition rate of aerosol particles in turbulent flows in passages with complex geometry is described. A thermodynamically consistent rate-dependent algebraic stress model is used to simulate the mean turbulent field. The instantaneous turbulence fluctuation is simulated as a continuous anisotropic Gaussian random field. The Brownian motion is modeled as a white noise process. The particle equation of motion used includes the Brownian force, the Saffman lift force, and the turbulent dispersion effects. Based on the presented results, the following conclusions may be drawn.

• Impaction is the dominant mechanism for particles captured by obstacles in the flow. • The presence of the obstruction significantly increases the particle deposition rate. • The deposition rate on the obstruction decreases as the shape of the obstruction becomes

streamlined. • Turbulent dispersion is the dominating mechanism for particle spreading. • Rebound effect reduces the capture efficiency for particles larger than 3/~m. • The minimum deposition rate on the upstream wall occurs for 0.5-3 #m particles. • The digital simulation procedure provides a viable method for studying particle transport

and deposition in turbulent flows. • Anisotropy of turbulent fluctuating velocity is important and has to be included in the

simulation.

While present computational model includes a detailed account of turbulence fluctuating velocity field, and allows for particle rebound from the surface, it has a number of limitations. Most importantly, it does not consider the particle resuspension process. That is, the deposited particles which are re-entrained into the flow at a later time are not accounted for. The bounce model used is also a simplistic one and could be further


improved. However, these improvements and consideration of electrostatic effects are left for future studies.

Acknowledgements--Thanks are given to Professor J. H. Vincent for his many helpful suggestions. The financial support for this work was provided by the International Business Machines Corporation (IBM-Endicott). The later stages of the work was also supported by the New York State Science and Technology through the Center for Advance Material Processing (CAMP) of Clarkson University.

R E F E R E N C E S

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Aerosol particle deposition in an obstructed turbulent duct flow

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