Enhancement of brownian and turbulent diffusive deposition of charged aerosol particles in the presence of an electric field

Enhancement of Brownian and Turbulent Diffusive Deposition of Charged Aerosol Particles in the Presence of an Electric Field MANABU SHIMADA,* K I K U O OKUYAMA,* YASUO KOUSAKA,* Y O S H I N A R I O K U Y A M A , * AND J O H N H. SEINFELD t * Department of Chemical Engineering, University of Osaka Prefecture, Sakai 591, Japan, and tDepartment of Chemical Engineering, California Institute of Technology, Pasadena, California 91125

Received November 30, 1987; accepted April 26, 1988 The effect of an electric field upon diffusive deposition of aerosol particles is investigated. In the experiments, charged and uncharged monodisperse particles of 0.02-0.2 um are confined in a metal stirred tank inside which an electricfieldexists. It is found from measurementsof the particleconcentration decay inside the tank that the deposition rates of charged particles are enhanced as the electric field strength increases. To evaluate the electrostatic enhancement theoretically, the diffusion and transport ofparticlesin an electricfieldare described, accountingfor Brownianand turbulent diffusionand coulombic force acting on the particles. Deposition rates predicted by numerically solvingthe convectivediffusion equation agree with those measured. © 1989AcademicPress,Inc. 1. INTRODUCTION The enhancement of the deposition of charged particles by electrical forces is important in dust collection in devices such as electrostatic precipitators and electret fiber filters, particularly in semiconductor manufacturing processes, where contamination by the rapid deposition of charged submicron particles onto devices must be prevented. The influence of electric fields upon particle diffusive deposition onto single spherical (7, 13, 14 ), cylindrical ( 17 ), and disk ( 9, 18 ) collectors has been investigated theoretically. Previous works have been concerned with the capture efficiency of submicron particles by fibrous and granular bed filters and silicon wafers. Although these theories have been used to explain experimental results of total-capture efficiencies, few data are available for a single collector which can be compared directly with theories, especially for ultrafine particles. Several measurements of aerosol deposition rates inside channels or chambers have been reported (1, 4). Fjeld and Overcamp (6) found that the wall loss rates of 0.1- and 0.5-/~m latex

particles in a rectangular chamber increased when a dc voltage was applied between a pair of electrodes installed on the chamber walls. McMurry and Rader (10) extended the theory of C r u m p and Seinfeld (3), which accounts for particle deposition by Brownian and turbulent diffusion and gravitational sedimentation in a vessel, to successfully correlate the deposition rate of charged particles in a 0.25m 3 Teflon film bag. In their study, however, the magnitude of the electric field strength near the chamber walls must be estimated in terms of the average strength integrated over the whole inner surface, the value of which was determined empirically from the measured deposition rates. A similar assumption and calculation of the average electric field strength near the surface of dielectric substances is also found in a recent study of Emi et al. (5), who measured penetrations of charged aerosol particles through pipes of various materials. The goal of this paper is to present measurements of the effect of the electric field intensity upon the Brownian and turbulent diffusive deposition of charged particles in a closed vessel. Wall loss rates of singly charged

157

0021-9797/89 $3.00 Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989

Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

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SHIMADA ET AL.

and uncharged monodisperse particles over the diameter range 0.02-0.2 #m in metal stirred tanks having an electrode on the bottom wall will be presented. The intensity of turbulence and the electric field strength in the tank can be controlled. Predictions of the deposition rates are obtained from the solution of the convective diffusion equation, accounting for coulombic forces acting on the particles in the electric field formed inside the tank.

metal tank by a thin silicon rubber spacer (about 3 mm in thickness). Thus an electric field is formed inside the tank when adc voltage is applied to the electrode. Figure 2 shows a schematic diagram of the experimental apparatus. The particles used are unipolarly charged or uncharged monodisperse NaC1 particles, 0.02-0.2 #m in diameter, which are produced by a combination of an evaporation-condensation-type generator and a differential mobility analyzer (DMA). When the stirred tank is being filled with 2. MEASUREMENT OF PARTICLE DEPOSITION IN A STIRRED VESSEL charged particles, the outlet of the DMA is connected directly to the tank. The polarity of 2.1. Experimental Apparatus and Method the particle charge is opposite that of the DMA center rod. The size classification by the DMA The structure and dimensions of the two is carried out so as to obtain predominantly cylindrical stirred tanks used in the experiments are shown in Fig. 1. Tanks A and B singly charged particles at the exit of the DMA. That is, the particles, nearly the largest in size with similar form and different sizes are preamong the particles from the generator, are pared to investigate the effect of vessel size on designed to be classified to prevent larger muldeposition. The upper and side walls of the tiply charged particles from exiting the DMA. tank, the six-bladed turbine impeller, and the Uncharged particles are obtained by introstirrer shaft are made of metal which is ducing the charged particles into a neutralizer grounded. The bottom wall of the tank is a and plate condenser. All the metal tubing used metal disk connected to a dc voltage supply. in the experiment is grounded to reduce excess This electrode is separated from the rest of the loss of charged particles prior to the tank. The batch-type experiment starts with flowing an aerosol through the tank with the 2RT voltage applied to the base wall, and stirring " " o " - , ~ n g ~ vtaCrUUrm at a selected speed. The initial particle number concentration, no, is measured after the aerosol concentration in the tank becomes sufficiently // II I steady. In this procedure, the particle number concentration of the feed aerosol needs to be kept almost constant. The concentration is controlled at about 108 particles/m 3 to avoid the Brownian and turbulent coagulation. The measurement of particles is made by a mixingspacer II electrode r L type CNC (8), of which the metallic inlet tubing and the mixing part of saturated vapor are 0 ring grounded. The aerosol concentration is redc voltage supply measured after the tank has been closed and the aerosol has been stirred for a designated tank H RT HT DT ] WT E,.LTT A 0.15 0.0?5 HI2 (2/3)RT DT/5 time, t. B 0.09 0.045 The parameters varied in the experiments in m were the stirring speed, Ns, the voltage applied FIG. l. Stirred tank used in this study. to the tank floor, q~o, the particle diameter, dp,

Jl

L

Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989

AEROSOL DEPOSITION IN AN ELECTRIC FIELD

159

dryair motor drive r~-n tach°meterl

A•---high

voltage

~

~ ( p = * l ) /

stirred tank

DMA

voltage

generator

FIG. 2. Schematicdiagramof experimentalapparatus. and the polarity of the particles, p ( = - 1, + 1, or 0). 2.2. Experimental Results

When the aerosol inside the vessel is sufficiently well stirred so that the particle concentration can be assumed to be uniform except in a boundary layer near the wall, the particle number concentration in the core, nc(t), decays due to wall deposition according to no(t) = no e x p ( - f l t ) ,

[1]

where /3 is the deposition rate constant. This relationship has been confirmed when Brownian and turbulent diffusive deposition and gravitational sedimentation are predominant (2, 10, 11, 15, 16). Figure 3 shows representative decreases in particle number concentration measured for a particular set of conditions. It is seen that the particle number concentration indeed decays exponentially, and thus Eq. [ 1] applies in the case in which electrostatic forces influence the deposition process. The deposition rate constants obtained by varying the size and polarity of the particles, the voltage on the electrode, and the stirring speed are shown in Fig. 4. Generally, the value of/3 decreases as particle size increases in the range of particle size studied. It is also found, as expected, that/3 increases as the stirring speed increases, that is, at larger intensity of turbulence in the vessel. When the voltage applied to the lower wall of the tank 4~0 = 0, namely when all the walls

of the tank are grounded, the deposition rates show virtually no difference between charged and uncharged particles. Therefore, the effect of image forces between the charged particle and the grounded wall or the effect of electric field due to the charges of the particles cannot be discerned from the available measurements. When a dc voltage, the polarity of which is opposite to that of the particles, is applied to the lower wall, enhancement of particle loss is clearly seen. The magnitude of the enhancement increases as the particle size decreases, reaching a value about an order of magnitude higher than that without electrical effects at maximum enhancement. Virtually the same deposition rates are obtained when the signs of both 4)o and p are reversed. Therefore the enhancement of the measured deposition rates is considered to be caused by the drift of charged particles in an electrostatic field due to coulombic forces. The enhancement of deposition also appears in the case that the sign i

"U E

~ ~

,

i

,

r

,

~

,

,

,

i

Ns=1000rain-1 qb0=50V

0.021pm

0.1

p = -1

6'8 t

lb'1'2

{mini

FIG. 3. Changein particlenumberconcentrationwith stirringtime. Journal of Colloid andlnterface Science, V o l .

128,

No.

1, March

1, 1989

• 160

SHIMADA ET AL. ,

'

'

'

I ....

2oov~,

'

I

(a)

'

i ....

.,oov, ~

'

,oov\\

,

I

i ....

oov\\

(b)

50V ~

10-2

200V\

I

(c)

~'

,Ul

10-3 tank B Ns= 1000 rnin-1 ~.o=5.15x10-1 m2/s 3

~min-1 xx ~.o=9.06x10-1 m2/s 3 ~

0.01

~

I

~,,,I

i

0.1

dp IIJml

~0=0

i

,

0.01

dp

i ,i,,I

0.1 I iJml

0.01

0.1 dp IiJml

p;0 p;1 p;1

P~0 <0 ~. i Pqb0>0

p:-llp=*1

I"I'01 p:-1 ,=,I

Zx

•

50

-~

-,IW-

[]

•

I00

0

I

v

•

200

~

FIG. 4. Measured and predicted deposition rate constants obtained under various conditions using two tanks.

of the charged particles and the voltage applied to the base electrode are the same, i.e., p~b0 > 0. The values of/3 are, again in this case, not changed if the signs of both @o and p are changed. However, the enhanced deposition rates are 20-50% larger than those obtained when p~bo < 0. The explanation for this phenomenon will be discussed later.

tribution of the dimensionless particle concentration, c = n/no, in a quasi-steady state within the boundary layer is governed by

3. PREDICTION OF DEPOSITION RATES OF CHARGED PARTICLES IN A VESSEL

where vr and Vz are the particle migration velocities in the radial, r, and vertical, z, directions due to forces acting on the particle, and D and DE are the particle Brownian and turbulent diffusion coefficients, respectively. Since the particle size studied is sufficiently small that the gravitational settling is negligible compared with the effects of Brownian and turbulent diffusion (16), the particle migration in the present study is regarded as a result of coulombic force in an external electrostatic field. Assuming that a particle immediately attains its terminal velocity because of its very short relaxation time, vr and Vz in Eq. [2 ] are expressed by

3.1. Governing Equations To predict the particle wall loss inside the vessel it is necessary to describe the deposition process occurring by simultaneous Brownian and turbulent diffusion and electrostatic particle migration. First, we assume the existence of a particle concentration distribution in a turbulent boundary layer (thickness 6) near the walls. The location of the boundary layer in the vessel is illustrated schematically in Fig. 5 by regions ( 1 ) - ( 5 ) . If the stirred tank is approximated as a body of rotation, the disJournal of Colloid and Interface Science, V o l .

128, No.

1, M a r c h

1, 1 9 8 9

r Or L

o{

vzc-(D+DE)-~z

=0,

[21

161

AEROSOL DEPOSITION IN A N ELECTRIC FIELD

Z

H,ff H-5

From Eqs. [ 3 ], [ 5 ], and [ 6 ] it is noted that the migration velocity due to the electrostatic field is nondivergent, that is, .

.

.

.

.

1 0 (rvr) + Ovz r Or -~z = 0.

t u r b u l e n t boundary~

layer ( regions <1>-<5>)

Substitution of Eq. [ 7 ] into Eq. [ 2 ] gives the final form of the governing equation:

i O(r O+OE, Oc} O{

q

]

7Or

,<3"

-g;r +

<6>

I

( z) +

o Ocl E)

vr O Oc r Or (rc) - Vz~zz = 0.

lurbuJenl core

<1>

0

[71

1<2>

~ F

Rr-~ RT

FIG. 5. Model of calculation domain of particle deposition inside stirred tank.

eo = NpN3sD~/VT.

[31

where Er and E~ are the electric field strengths, and Be is the particle electrical mobility, [41

with e = 1.602 × 10 -19 C. The electric field strength depends upon the location in the tank and is obtained from potential gradients in the r and z directions: Er=-O4~/Or,

Ez=-&b/Oz.

[51

The potential distribution in the vessel, q~(r, z), is governed by the following Laplace equation because the electric field formed by a lowconcentration aerosol composed of singly charged particles can be neglected relative to that due to the bottom electrode: 1O(&b) O2~b r-&r + - ~ z2 =0"

rOr

[9]

where

Vr(r, Z) = BeEr(r, z),

p-e g7 Be = -3~-#dp

[8]

An expression for the turbulent diffusion coefficient, DE, for a stirred tank with four baffle plates, the width of which is one-tenth of the tank diameter, has been experimentally obtained in terms of the average energy dissipation rate of air inside the tank, e0, and the distance from the wall, y, as (1 l) DE = Key 2"7, Ke = 7.5V~%/15v,

vz(r, z) = BeEz(r, z),

j

[6]

[101

In this study, we also apply this relationship to the tank having no baffle plate using the corresponding value of the power number, Np (19). Then the distributions of DE in the regions ( 1 ) , ( 3 ) , and ( 5 ) in Fig. 5 are given by DE(r, z) = Kez 2"7 (for region ( 1 ) ) ,

[11]

DE(r, z) = Ke(RT -- r ) 2"7 (for region ( 3 ) ) ,

[121 DE(r, z) = Ke(H - z) 2"7 (for region ( 5 ) ) .

[131 for the corners of the tank, regions ( 2 ) and ( 4 ) , cannot be formulated as simply as for the other regions, the following assumption was made. First, it is necessary for values of DE for region ( 2 ) to coincide with those for region ( 1 ) or ( 3 ) at the boundary between them. Next, the intensity of turbulence is thought to be very small in the vicinity Because DE

Journal of Colloid and Interface Science, Vol. 128,No. 1, March 1, 1989

162

SHIMADA

of the corner ( r = RT, z = 0), so that DE may be almost zero in this area. Taking into account these considerations, the value of DE at any position in region (2 } is defined to depend on the distance,/2, from the point (r, z) = (RT - 6, 6) as

Dr(r, z) = Ke(6 - / 2 ) 2"7, /2 ~< 6, Dr(r,z)=O,

12>6,

[14]

where /2 = ~/(r- RT + 6)2 + (z -- /~)2. [151 Similarly, DE for region (4 } is taken as

Dr(r, z) = Ke(6 -/4) 2"7, /4 ~ t~, Dr(r,z)=O,

/4>6,

[16]

where

14=I/(r--RT+6) 2+(z-H+6)

2

[17]

is the distance from the point (r, z) = (RT -- 6, H - 8). The thickness of the turbulent boundary layer, 6, is defined as the point at which the value of DE obtained by Eq. [9] becomes equal to the turbulent diffusivity in the core, D~ (16). That is,

6 = (DrE/Ke) 1/2'7,

[18]

where

ll/27kTI/2F,t3/2 ~,s ~T

D'r = 1.54 N~/2(2RT)I/3HI/6.

[191

3.2. Numerical Solution Procedure Equations [6] and [8] are solved numerically to obtain the potential and the particle number concentration distributions. For Eq. [6], the calculation domain is the regions ( 1 } - ( 6 } in Fig. 5 considered together. After the domain is divided into a number of rectangular meshes, the equation is transformed into finite difference form by using a central difference scheme. The value of ¢ along the base wall is set to ¢0 as a boundary condition. On all the other surfaces of the inner walls, including the stirrer and the shaft, ¢ is set to Journal of Colloid and Interface Science, Vol. 128, No. I, March 1, 1989

ET AL.

zero. Simultaneous equations from finite difference formulae are solved iteratively by a successive overrelaxation method. The electrostatic field strengths, Er and Ez, are calculated from Eq. [5 ] with the computed values of ¢(r, z). In this procedure, Er and Ez are obtained by differentiating a cubic spline interpolation function of ¢. Then the electrical migration velocities, vr and vz, which are calculated from Eq. [3], are substituted into Eq. [8]. Equation [8] is also solved numerically by a finite difference method. To avoid instabilities in the solution, an upwind difference scheme (12) is used to represent the particle migration terms. The computational domain for the concentration distribution is regions ( 1 }- ( 5 } of Fig. 5. The particle number concentration is set equal to zero at a distance half the particle diameter away from the upper, side, and lower walls of the tank and from the surface of the stirrer shaft. The gradient of the concentration in the radial direction, Oc]Or, on the central axis is also set to zero to represent the symmetry of the concentration. An order of magnitude analysis indicated that the electrical migration had much less effect than the turbulent diffusion at the outer edge of the turbulent boundary layer. Hence the concentration, c, is set to a constant value of unity along the dashed lines in Fig. 5 because the mixing of the aerosol can be regarded sufficient outside the boundary layer. After numerical calculations with various grid arrangements were carried out, a mesh of 150 Y 300 grid points within the computational domain was found to be enough to obtain the values of ¢ or c accurately. Since the typical particle number concentration and the potential near the edge of the bottom wall of the tank are expected to change steeply in the vicinity of the walls, those grid points were arranged such that the mesh interval becomes small as the walls are approached. The thickness of the turbulent boundary layer was divided into 120 meshes, and the minimum mesh size was designated as 1.3 × 10-6H. A successive solution of simultaneous equations

AEROSOL

DEPOSITION

was iterated upon until the difference between values at present and previous iterations, I ( - ~bpre)/~b[ and [c - Cpre[, becomes smaller than 10 -6 at every mesh point.

3.3. Calculated Deposition Rates Figure 6 shows the calculated distributions of the potential and the electrostatic field strength in the stirred tank. The results shown in this figure are obtained when one of the six blades of the turbine impeller exists in the calculation domain. The electric field was also computed for the domain including no blades since the stirrer was actually rotating. In these calculations, however, only the electric field near the impeller was found to differ and almost the same result was given inside the turbulent boundary layer in which the particle deposition was taken to be influenced by the electric field. Therefore, the electric field obtained in either calculation is considered to be sufficient to predict the particle concentration distribution inside the boundary layer.

IN AN ELECTRIC

163

FIELD

It is seen in the left half of Fig. 6 that the distances between equipotential curves near the base electrode decrease as the side wall is approached. The length of the arrows in the right half of Fig. 6 indicates the magnitude of the dimensionless electric field, E + = V-~--r~r+E2z(2RT/qbo), while the direction denotes that of the force acting on a particle, the polarity of which is opposite to that of the electrode. It is found from this figure that a particle near the tank floor experiences a force toward the floor. The force opposite to the vertical wall increases as the side wall is approached. It is also noted that the forces in the upper part of the tank are several orders of magnitude smaller than those near the bottom electrode because of the grounded stirrer installed between the upper and the lower wall. The particle deposition velocity is defined as the particle flux per unit concentration in the core. It can be obtained for each of the lower, the side, and the upper walls by using the dimensionless concentration gradient with respect to the direction normal to each wall as

1)d =

D Oc -cgZIz=dp/2

(for lower wall), [20]

v~

- D Oc

(for side wall),

Or Ir=Rr-dp/2 Vd

- D Oc

[21]

(for upper wall). [22]

Og Iz=H-dp/2

°

° °

0.7

E* =

~ErZ÷EzZ(2Rr/qb0)

I-I

=~

E ÷ =10

--~ --"

E*= 1 E*=0.1

FIG. 6. Calculated results of equipotential curves (left half) and distribution of normalized electric field (right half).

Figure 7 shows the dependence of predicted deposition velocities upon the location of the wall surface. In this case, the deposition velocity for the lower wall is much larger than that for the upper or side wall because of the force toward the electrode. The thickness of the turbulent boundary layer in the case of Fig. 7 (tank A, Ns -- 1000 min - l ) is calculated from Eqs. [18] and [19] as 6 = 5.5 mm. The largest deposition velocity onto the lower wall occurs at a position near the side wall within the boundary layer. From Fig. 6 we see that this maximum value of Vd is a result of the Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989

164

SHIMADA ET AL.

bulent intensity for a tank having four baffle plates, Eq. [9], can be concluded to be applicable for tanks with different sizes which have no baffle plates. Each predicted value of/3 for a certain value ~ 0 [ . . . . . . . .2. . . . . . 4 6 RT of [~0[ is shown by a single curve in Fig. 4 r [ x10-2 rnl because the computed values of/~ for p ~o > 0 and p ~0 < 0 are identical. It is seen in Fig. 4 that the calculated deposition rates also agree well with the electrically enhanced deposition rates measured when P~0 < 0. From the com°[ ' s ~o ' ' ' . parison of Figs. 4a and 4b, the differences in Z (xlO-2ml the deposition rates due to the turbulent inFIG. 7. Dependence of calculated deposition velocities tensity are seen to become less significant as for tank A upon location of wall surface(dp = 0.043 #m, the particle size decreases and the electric field p = -1, N~ = 1000 min -l, 4~o= 50 V): (a) va for lower strength increases. This indicates that the dewall; (b) vd for side wall. position due to electrostatic effect becomes predominant for smaller particles because of the larger electrical mobility. Slightly less large field strength near the joint of the side agreement between the measured and preand the lower wall. The decrease in vd in the dicted deposition rates shown in Fig. 4c for vicinity o f the side wall despite the increasing the smaller tank B is thought to be due to a attractive force is thought to be caused by a little uncertainty included in the measurement low concentration of particles due to repulsion of relatively large deposition rates for tank B: from the side wall. This might be caused by the experimental procedure in which the amount of aerosol 4. COMPARISON BETWEEN MEASURED AND partially sampled from the smaller tank was PREDICTED DEPOSITION RATES needed to be that smaller. However, the calSince the distribution of particles deposited culated deposition rates still lie close to those on the vessel surface could not be measured measured, and thus the effect of the vessel size in the experiments, comparison between the is sufficiently well described by the present measured and calculated deposition rates is calculation. carded out in terms of the deposition rate The deposition rates measured for P~0 > 0 constant, 8. The predicted value of/3 is given are seen to exceed the prediction. The depoby sition in the case of P~0 > 0 is expected to occur primarily on the lower part of the side (3 = -~T Vd d S , [231 wall in accordance with the direction of the electric field. Since the silicon rubber spacer where the integration is over the internal sur- between the side and lower wall of the tank face of the tank (11, 15 ). The particles depos- has been constructed so as to cover a small ited onto the stirrer and the shaft will be ne- portion of the lower part of the side wall as glected because these comprise only about 3% shown in Fig. 1, additional experiments were of the total surface area. The predicted values carried out in which the configuration of the of/~ are shown by the curves in Fig. 4. It is spacer was changed so that it covered a slight seen that the predicted deposition rate con- portion of the edge of the bottom wall. Howstants for ~o = 0 coincide with those measured ever, measurement was not changed to show when the effect of the coulombic forces is not that the difference in the deposition was not included. Therefore the description of the tur- attributable to the effect of the spacer position. '

i

,

i

,

~

,

i

,

i

.

i

.

i

_f............. 1

Journal of Colloid and Interface Science,

Vol. 128,No. 1, March 1, 1989

165

AEROSOL DEPOSITION IN AN ELECTRIC FIELD

Thus the underprediction in the present calculation may be due to disregarding of the particle deposition onto the stirrer. Since the attractive force toward the impeller will act on the particles if the polarity of the particle charge and the electrode is the same, the amount deposited onto the turbine impeller, which at present cannot be evaluated, should be taken into account although the surface area of the impeller is only a few percent of the total area.

rection, E+z (=E~(2Rr/cko)), near the base wall of the stirred tank upon the distance from the wall surface. It is seen that the field strength along the central axis (r = 0) is almost constant within the turbulent boundary layer (e.g., 6/ 2RT = 0.037 for tank A, N~ = 1000 min-1). Recalling E, = 0 at r = 0 and neglecting partide diffusion in the radial direction, the basic equation governing particle diffusion and transport near the wall, Eq. [2], and its boundary conditions are reduced to

5. DISCUSSION

0~ ( D + D E ) ~

In previous works for electrically enhanced deposition of particles (5, 10), a value of an average electric field normal to the wall was estimated experimentally from measured deposition rates and utilized to represent the unknown field near the wall surface. In this section, an analysis similar to these is described. This simple method is tested to determine whether it is adequate from a viewpoint of particle deposition to represent the electric field in the vessel of the present study by using an average value of the field strength. Figure 8a shows the dependence of the dimensionless electric field strength in the z di80

c=latz=~,

-~(VzC)=0,

[241

c=0atz=0.

[251

Setting v~ = BeEz to a constant value, the deposition velocity, VdE, obtained by analytically solving Eq. [ 241 together with Eq. [ 25 ] takes the form of that derived by Crump and Seinfeld (3), in which the gravitational settling velocity is replaced by the constant electrical migration velocity, Vz: --1) z l)dE =

exp [2.7

- ~rv~. sin(Tr/2.7) 2 . 7 ~ j

]_

1 [261

I

--7

I

... 60 o

O I'M

(--- 2 RT

-~ 4C hl

0 ~ 9 7 2 1 %

i

II +N

II ÷N

IM

w 2O

0.01

0.02

zl2RT

(a)

I-I

0 . 0 3 0.04

0.01

0.02

z/2RT

0.03

0.04

I-I

(b)

FIG. 8. Dependence of electric field strength normal to surface, Ez, upon distance from wall: (a) for base electrode of stirred tank; (b) for conductive disk. Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989

166

SHIMADA ET AL.

The value OfVdEby Eq. [26] can be compared differs from the case of the tank because of with those predicted by the numerical solution the absence of grounded walls. of Eqs. [8] and [20]. Several calculations of Applying the analysis utilizing the average deposition velocity, carded out by substituting electric field to the whole surface of the stirred the value of Ez along r = 0 into Eq. [26 ], in- tank, we can calculate the average electric field dicated that the value of vo thus obtained was strengths near the lower, side, and upper walls equal to that computed by the numerical cal- of the tank, El, Es, and Eu, respectively, inculation for the central point of the base wall, stead of determining them experimentally. the value of which is, for example, shown at The computations of the average field are carr = 0 in Fig. 7a. Therefore, the electrostatic ded out within a region bounded by the outer enhancement of deposition can be evaluated edge of the turbulent boundary layer and a by Eq. [26 ] when the electric field strength is position half the particle diameter away from uniform above the wall onto which the de- the wall: position occurs and the particle transport in the lateral direction can be neglected. If we El -Ezrdrdz /2 t define the enhancement factor due to the electrostatic effect fE as the ratio of enhanced deposition velocity Valeto deposition velocity in fall2 Jofm-aP/Zrdrdz' [29] the absence of an electrical effect v~0,j~ is expressed as fH-dp/2 ~R-r-dp/2 Errdrdz /

Y"

fE = l)d''~E= Vd0

--~)z/l)dO

exp(--Vz/Vdo)- 1'

[27]

Ja~/2 ~_~

where

VdO= (2.7/¢r)sin(zc/2.7)K~/2"7D1"7/2"7

[28] --

The value of J~ is unity at v~ = 0, and approaches Vz/Vdoas Vzincreases. Consequently, the deposition velocity caused by simultaneous Brownian and turbulent diffusion and electrical migration approaches the electrical migration velocity itself when the migration velocity v~ becomes sufficiently larger than Vdo. However, the electric field normal to the bottom electrode in general depends upon the distance from the surface as the radial coordinate increases. Since v~depends on z in this case, the deposition velocity is no longer expressed as simply as Eq. [26]. Such a nonuniform profile of an etectric field as that above the bottom of the tank is not a special case and can be, e.g., also seen in that near a conductive disk (20) around which no grounded wall is present, as shown in Fig. 8b. The electric field strength E~ near the disk edge varies with the distance from the wall. The profile of it shows a tendency similar to that for the stirred tank, although the magnitude of the values Journalof ColloidandInterfaceScience,Vol. 128, No.

/

1, March 1, 1989

r2?" r

Jo

Ezrdrdz

rdrdz,

[301

/

f~_:a'/2~r-a~/2rdrdz.

[31]

The electrical migration velocity of the parti= cles perpendicular to each wall is calculated from each corresponding average field. By replacing vz in Eq. [ 26] by those velocities, the average deposition velocities VdE for each of the lower, side, and upper walls are obtained. Figure 9 shows measured and calculated dependences of the enhancement of ~ upon the voltage applied to the electrode. The dashed line indicates the enhancement computed by substituting the average deposition velocities into Eq. [23], while the solid line is that obtained by solving Eq. [ 8 ]. It is seen that the analysis utilizing the average value of electric field normal to the wall overestimates the measured deposition rates. The explanation of this discrepancy is considered as foUows. The magnitude of the overall deposition rate

AEROSOL DEPOSITION IN AN ELECTRIC FIELD

167

6. CONCLUSION

o

t 01",

. . . . . .

0

:.

I

. . . . . . . . .

100

~0

I/

200

IVl

FIG. 9. Comparison between measured and predicted factors of deposition enhancement due to voltage applied to electrode. ~(4~0) denotes value of fl obtained when voltage on electrode is 00. Symbols are the same as those in Fig. 4.

is contributed primarily by the deposition near the lower comer of the tank, as shown in Fig. 7. As mentioned previously, it is predicted by the numerical calculation that the largest deposition occurs on a portion a little away from the position of the maximum attractive force because the motion of the particles is also affected by the force parallel to the wall. However, the analysis based on the average values of electric field does not account for the particle transport in the lateral direction. In the analysis, the simple arithmetic average of field strength leads to inclusion of the contribution of the large attractive force which does not enhance the deposition effectively. Thus the prediction of fl in this simple method is considered to exceed the deposition rates obtained by the solution of Eq. [ 8 ]. It is, therefore, found insufficient only to account for particle transport toward walls in order to analyze particle deposition in a nonuniform electric field. This type of electric field will be seen in the existence of an electrode and might not have appeared inside film bags or pipe channels investigated in the previous works. As demonstrated, it is not always possible to predict particle deposition rates in the presence of an electric field by simply using a single representative value of electric field strength, such value as, for example, measured with an electrostatic fieldmeter. Depending on circumstances, structures of electric field near the walls need to be taken into consideration in the analysis.

The influence of an electric field upon the Brownian and turbulent diffusive deposition of 0.02- to 0.2-~tm particles is investigated experimentaUy and theoretically, The deposition rates of charged and uncharged particles are found to exhibit no differences when the walls of the chamber are grounded, that is, the effect of electrical forces on the deposition is negligible in the absence of an electric field. The enhancement of de, position rates in the presence of an electric field is systematically measured. This enhancement is found to increase as the electric field strength increases, or as the particle size decreases. The deposition rates can be predicted by calculating the electric field in the vessel and solving the convective diffusion equation of the particles numerically, accounting for Brownian and turbulent diffusion and coulombic forces acting on the particles. The resuits show good agreement with the measured deposition rates. APPENDIX: NOMENCLATURE Be

particle electrical mobility, m 2 S-I V - 1

c

Cprc

G D

dimensionless particle number concentration ( = n / n c) previous value of c at iterative calculation Cunningham correction factor Brownian diffusion coefficient, m 2 s-I

DE DT

G

Er, E~

E1, Es, Eu

turbulent diffusion coefficient, m 2 s-I diameter of stirrer, m particle diameter, m electric field strength, V/m, in radial and axial directions dimensionless electric field strength in z direction (=Ez X (2RT/~b0)) average electric field strength for lower, side, and upper walls of tank, V / m

Journal of Colloid and lnterface Science, Vol. 128, No. 1, March 1, 1989

168 E+

H

H~ K~ LT

No N~ n

SHIMADA ET AL.

dimensionless electric field strength ( = ~ + Ez2 × (2RT/~bo)) height of stirred tank, m fitting up height of stirrer, m coefficient of turbulent diffusivity, m -°'7 S -1 width of turbine impeller, m power number stirring speed, s -1 particle number concentration, m-3

/'/c no r

RT

ST

particle number concentration in turbulent core, m-3 initial particle number concentration, m -3 radial coordinate, m radius of cylindrical stirred tank, m total inner surface area of stirred tank, m 2

time, s l)d, /)dE, l)dO deposition velocity, m svolume of stirred tank, m 3 VT electrical migration velocity, m Dr, Vz t

S-I

WT Y z

~0 # P

~pre

~0

length of turbine impeller, m distance from wall, m axial coordinate, m deposition rate constant, s-1 thickness of turbulent boundary layer, m energy dissipation rate, m 2 S - 3 viscosity of gas, Pa. s kinematic viscosity of gas, m 2 S-1 potential, V previous value of ~b at iterative calculation, V potential on the electrode, V

Journalof Colloidand InterfaceScience,Vol.128~No. 1,March1, 1989

ACKNOWLEDGMENTS This work was supported in part by the Japan Grantin-Aid for Scientific Research under Grant 62602031 and the National Science Foundation under Grant ATM8503103. REFERENCES 1. Bergin, M. H., Microcontamination 5, 22 (1987). 2. Crump, J. G., Flagan, R. C., and Seinfeld, J. H.,AerosolSci. Technol. 2, 303 (1983). 3. Crump, J. G., and Seinfeld, J. H., J. Aerosol Sci. 12, 405 (1981). 4. Ehlich, R. M., and Melcher, J. R., Ind. Eng. Chem. Res. 26, 456 (1987). 5. Emi, H., Kanaoka, C., Otani, Y., and Fujiya, S., J. AerosolRes. Japan 2, 304 (1987). 6. Fjeld, R. A., and Overcamp, T. J., Nucl. Technol. 65, 402 (1984). 7. Gupta, D., and Peters, M. H., J. Colloid Interface Sei. 104, 375 (1985). 8. Kousaka, Y., Okuyama, K., Niida, T., Hosokawa, T., and Mimura, T., Part. Charact. 2, 119 (1985). 9. Liu, B. Y. H., Fardi, B., and Ahn, K. H., in "Proceedings, 33rd Annual Technical Meetings, Inst. Environ. Sci., San Jose, CA, 1987," p. 461. 10. McMurry, P. H., and Rader, D. J., AerosolSci. Technol. 4, 249 (1985). 11. Okuyama, K., Kousaka, Y., Yamamoto, S., and Hosokawa, T., J. Colloid Interface Sci. 110, 214 (1986). 12. Patankar, S. V., "Numerical Heat Transfer and Fluid Flow." McGraw-Hill, New York, 1980. 13. Peters, M. H., Jalan, R. K., and Gupta, D., Chem. Eng. Sci. 40, 723 (1985). 14. Shapiro, M., and Laufer, G., J. Colloidlnterface Sci. 99, 256 (1984). 15. Shimada, M., Okuyama, K., Kousaka, Y., and Ohshima, K., J. Chem. Eng. Japan. 20, 57 (1987). 16. Shimada, M., Okuyama, K., Kousaka, Y., and Seinfeld, J. H., J. ColloidlnterfaceSci. 125, 198 (1988). 17. Wang, P. K., J. AerosolSci. 17, 201 (1986). 18. Yost, M., and Steinman, A., Microcontamination 4, 18 (1986). 19. "Kagaku Kogaku Binran," 4th ed., p. 1316. Maruzen, Tokyo, 1978. 20. "Seidenki Hando Bukku," p. 172. Ohm, Tokyo, 1981.