The Effect of an External Electric Field on the Scavenging of Aerosol

The Effect of an External Electric Field on the Scavenging of Aerosol Particles by Cloud Drops and Small Rain Drops P. K. WANG AND H. R. P R U P P A C H E R Department o f Atmospheric Sciences, University o f California, Los Angeles, California 90024 Received July 2, 1979; accepted S e p t e m b e r 17, 1979 A theoretical model is described which determines the efficiency E with which aerosol particles of radius r ~< 0.5/zm are collected by water drops of radius a due to the combined action of convective Brownian diffusion, thermo- and diffusiophoresis, and electric forces c a u s e d by the presence of an external electric field and electric charges on drops and aerosol particles, but in the a b s e n c e of inertial impaction effects. T h e results of this model are combined with the results of o u r earlier model which determines the collection efficiency of drops for particles o f r > 0.5/zm due to the combined action of inertial impaction, thermo- and diffusiophoresis, and electric forces due to the presence of an external electric field and electric charges on drops and aerosol particles, but in the absence of convective Brownian diffusion. Both models combined are able to quantitatively predict E vs r for 0.001 ~< r ~< 10/zm and 8.6 1.0/zm. 1. I N T R O D U C T I O N

It is well known that a considerable fraction of the aerosol particles suspended in the atmosphere is removed from the atmosphere and returned to the earth's surface by mechanisms which involve clouds and precipitation. Substantial contributions to the theoretical description of this scavenging mechanism have been made by (1-13). A critical review of these studies has recently been given by (14). This review showed that in previous theoretical studies electric effects on the mechanisms of aerosol particle scavenging by water drops in air were disregarded. This deficiency was partially eliminated in (14) which was a study of the effects of electric charges on the efficiency with which aerosol particles are captured by drops. For this study two theoretical models were developed which, in addition to the effects of electric charges,

considered the effects of phoretic forces, hydrodynamic forces and gravity, and convective Brownian diffusion. The present article considers the effects of an external electric field. For studying these effects the two theoretical models, developed by us earlier, were used. In model I, developed in (5-7, 12, 13) the efficiency was theoretically computed with which an aerosol particle of radius r ~> 0.5 /zm collided with a water drop in air due to the combined, simultaneous action of inertial impaction (due to hydrodynamic forces and gravity), thermo- and diffusophoresis, and electric forces. With this model, the collision efficiency was deduced from a determination of the trajectory of an aerosol particle falling at its terminal velocity around the water drop. The electric effects considered were those due to the presence of electric charges on drops and aerosol particles and those due to the presence of an 286

0021-9797/80/050286-12502.00/0 Copyright © 1980by AcademicPress, Inc. All rights of reproduction in any form reserved.

Journal of Colloidand InterfaceScience, Vol. 75, No. 1, May 1980

SCAVENGING

external electric field. The trajectory of the aerosol particle was determined from the relation dv m--=mg* dt

67rr~ (1 + OtNkn)

( v - u)

+ F T h + FDf + F e.

[1]

In this equation m, r, and v are the mass, radius, and velocity of the aerosol particle, respectively, t is time, g* = g (pp - pa)/

OF AEROSOL

PP, PP is the bulk density of the aerosol particle, pa and "0a are the density and viscosity of the air, respectively, 1 + a Nk. is the Stokes-Cunningham correction to the drag on a particle with size of the order of the mean free path length ha of the air molecules, Nkn = hair is the Knudsen number, a = 1.26 + 0.40 exp(--1.10Nkn -1) from (15), u is the velocity of the air around the falling drop, FTh is the force due to thermophoresis, FDf is the force due to diffusiophoresis, and Fe is the electric force. According to (15)

12~rr~a(ka + 2.5kpNkn)kaVT FTh =

--

287

5(1 + 3Nkn)(kp + 2ka + 5kpNkn)p

[2]

and FDf =

-67r~ar

0.74DvaMaVPv

(1 + aNkn)MwPa

,

[3]

where k a and ke are the thermal conductivities of air and the aerosol particles, respectively, Dva is the diffusivity of water vapor in air, Ma and Mw are the molecular weights of air and water, respectively, Pv is the density of water vapor in air, T is the absolute temperature, and p is the air pressure. For determining the forces due to thermo- and diffusiophoresis, vapor density and temperature fields were used which were derived by us following the method of (16) to numerically solve the convective diffusion equation. The velocity u of the air around the drop was determined from the flow fields derived by us and in (17) for the flow of air around a circulating water drop. The electric force due to net charges and image charges on drops and aerosol particles was computed using the results of(18). From a knowledge of the particle's trajectory around the drop, the collision efficiency E -

7r(a + r) 2

[41

was deduced, where Yc is the largest initial horizontal offset a particle can have and still collide with the drop (yc being measured

from the drop's axis aligned along g and sufficiently far upstream from the drop). The collision kernel was computed from K = ETr(a + r)2(V~oa - V®,r),

[5]

where V=,a and V=,r are the terminal fall velocity of the drop and aerosol particle, respectively. The present paper introduces a second model which is complementary to model I, but applies to particles of r ~ < 0.5/zm. This model (called model II) is conceptually analogous to the model II developed in (14) for studying the effects of electric charges on the collision efficiency. However, the present model II includes the effects of an external electric field in addition to the effects of electric charges, phoresis, and convective Brownian diffusion. In this paper we shall: (1) quantitatively describe the present model II, (2) give some results derived from this model, and (3) combine the results from this model with the results of model I. In our discussion, particular emphasis shall be given to the scavenging of aerosol particles in the size range within which the efficiency is minimum, a size range which we have previously termed the "Greenfield Gap" (since Greenfield, 1957, appears to have been the first to point out Journal o f Colloid and Interface Science, Vol. 75, No. 1, May 1980

288

WANG AND PRUPPACHER

the meteorological significance of this minimum in scavenging efficiency). Both models I and II predict collision efficiency and collision kernels. However, it is reasonable to assume that the efficiency with which an aerosol particle is retained by a drop after it has collided with it is unity. Under these conditions our collision efficiencies represent collection efficiencies, and our collision kernels represent collection kernels. 2. PHYSICS AND MATHEMATICS OF MODEL II

In the present model II, the collision efficiency is found from a determination of the flux of aerosol particles to a collector drop which falls at terminal velocity in air. In this model it is assumed that the aerosol particles are sufficiently small that their inertia may be disregarded. The total particle current density jp is written as j v - - - - n v v - Dp~7n, where Yp =BvFext, n is the number concentration of aerosol particles, Vp is their drift velocity, Dp is their diffusivity, Bp = (1 + ~V.Nkn)/67r~ar is t h e i r mobility, and Fext is the vector sum of the external force acting on the particle. For the case of an aerosol particle moving toward a drop as a result of Brownian diffusion, thermophoresis, diffusiophoresis, and electric and gravitational forces

[6]

where Feq is the force due to chargecharge interaction, and FeE is the force due to charge-electric field interaction. Assuming steady-state, constant diffusivity and assuming that the aerosol particles are small such that the term rag* may be neglected in comparison to FTh, FDf, and Fe, the condition of particle continuity V.jp = 0 for steady state leads to Bp(FTh + FDf -{- Feq + FeE)

× Vn - DpV2n -- 0.

BvFext" Vn - DvV2n = O,

[8]

where

jp : n B p ( m g * + FTh + FDf + Feq + FeE)

- DpVn,

In obtaining Eq. [7] we have assumed that V'(FTh + FDf + Feq + FeE) = 0. It is obvious that this is true for V'Feq since we can assume that the electric field around an electric charge on a spherical drop or aerosol particle is spherically symmetric. Therefore we may write Feq = (Ceq/R2)OR, where Ceq = QaQr, where Qa and Qr are the electric charges on the drop and aerosol particle, respectively, and where 0a is the unit vector in radial direction. For a stationary drop V.FTh -- 0 and V'FDf : 0 since then the temperature and vapor density fields are spherically symmetric around the drop. However, for a moving drop for which the temperature and vapor density fields vary with angle around the drop these conditions do not hold. We therefore follow the arguments of (14) and assume that it is reasonable to set V "(fhFxh + ?vFDf = 0, wherefh andfv are the mean ventilation coefficients for heat and water vapor transport in air (see (15)). Then fhFTh = ( C h / R 2 ) e R and fvFof = ( C v / R 2 ) ~ R . For the force due to the interaction of the charge Qr on the aerosol particle with an external electric field of strength E0 we may write FeE = EQr. Therefore V'FeE = V'(EQr) = QrV'E = 0, from Maxwell's equation. With these, we may write Eq. [7] now as

[7]

Journal of Colloidand Interface Science, Vol.75, No. 1, May 1980

Fext = (CTh + CDf -[- Ceq) ~

1

dR "[- QrE.

[9]

Great difficulties are encountered if one wants to exactly solve Eq. [8] together with Eq. [9]. Therefore, we pursue the superposition method which, although less accurate, leads to a useful solution. For implementing this method we add the flux J1, due to phoretic effects, convective Brownian diffusion, and electric charge on drops and aerosol particles, to the flux J2, exclusively due to the drift of electric charges in an external electric field. (Ac-

SCAVENGING OF AEROSOL ............ +

F ........... I

fir

+

=0

--

t I I _a_

(o)

289

f~'/2noBpQrEo 30=0

4-

× [(1 + 2aar3) COS01 (b)

FIG. 1. Arrangements of electric charges and electric fieldsfor whichthe presentcalculationswerecarded out. cording to (19), the method of superimposing fluxes has been proven useful in predicting realistic values for the charging rate of water drops of ions.) A solution to Eq. [8] together with Eq. [9] without the electric field term has been obtained in (14). Under these latter conditions the flux of particles to the drop is

X r 2 sin0d0d~.

The first integral pertains to the drop hemisphere attractive to aerosol particles, while the second integral pertains to the drop hemisphere repulsive to aerosol particles. Thus, no particles are captured on this latter hemisphere. We therefore set the second integral to zero, and integrate only over the first integral, to get

J2 =

noBpQrEo3 =0

J1 = I DPfp(Vn'dS)

~r12

× c o s 0 a 2 sin0d0d~

[10]

= 37rnoBvEoQ~a 2. 47rnoBpC [' BpC ~ exp ~ - 1 IDpfpa )

[11]

( 2°3t

Er = E0 1 + r3 ] COS0"~R

[12]

E0 = --Eo (1 - 2a3) r 3 sinO'ko,

[13]

wherein we have neglected the forces due to image charges. The total flux can be computed by evaluating the integral

J = i~ noBvQr(E'dS)

=0

foT

[14]

noBpQrEo [ ( 21 a+3 )

=~r/2

× r 2 sinO dOd~

[16] [17]

Flux addition yields

J = Ji + J2 = 37moBvEoQra z

where no is the number density of aerosol particles at R = ~. The flux J2 due to the drift of electrically charged aerosol particles toward a conducting sphere in an external electric field can be found by considering that the flux density due to this drift is noBpQrE, where E has the components

= fi~

[15]

cos0 ] F3

4rmoBvC

+

[18] exp(BvC/Dvfva)- 1 Equation [17] is identical to the equation describing the transport of ions toward a conducting sphere in a uniform external electric field due to pure condition, except that the ion mobility is now replaced by the mobility of the aerosol particles (19). From Eq. [18] we find for the collection kernel J K - 37rBpEoQra 2 no

+

4~BpC exp(BpC/Dpfpa)-

1

[19]

taking into account capture of aerosol particles due to Brownian diffusion, due to phoretic forces, and due to electric forces caused by electric charges on drop and aerosol particle and by the presence of an external electric field. Since the collision efficiency is related to the collision kernel by

E = K/rc(a + r)2(V~,a - V~,r)

[20]

Journal o f Colloid and lnterJace Science, Vol. 75, No. 1, May 1980

290

WANG AND PRUPPACHER

we find E -3BvE°Qra2

+ 4BpC/[exp(BpC/Dpfva) -

1]

[21]

(a + r)2(V=.a - Vo%r)

In using Eqs. [19] and [21] consideration has to be given to the fact that an electrically charged drop will accelerate (or decelerate) in an external electric field, thus altering its terminal velocity V=., and with it, the ventilation coefficients fv, fh, and fp. The terminal velocity of a charged drop in the presence of an external electric field may be c o m p u t e d from formulae given in (20) and the relation 6"rr'Oaa V ~ ' a \

Z W

~5

~ i0 ° u_ !3_ w

~ e

7

z

_o t0-J _J _1 0 tj

iO-2

1

0

0.001

-

0.01

PARTICLE

3

~

0.1

RADIUS

1.0 (#.m)

FIG. 2. Effect of particle size, electric charge, and external electric field on the efficiency with which aerosol particles of various radii collide with a water drop of radius a = 8.6 /xm in air of 900 m b a r and 10°C. Solid lines represent results derived fi'om model II. Dashed lines represent results derived from model I. R H is relative humidity of the air; qa = Qa/a2, qr = Q~/rZ; curves (1) to (4) are for [qa[ = ]qrJ = 0, E0 = 0, and R H = 50, 75, 95, and 100%, respectively; curves (5) to (7) are for Iqal = ]q,I = 2 esu c m -2, E0 = 0, and for 50, 75, and 95%, respectively; curves

(8)to(It)are for Iqol = [qr] = 2esacm-2, E0 = 3000 V/cm, and for RH = 50, 75, 95, and 100%,respectively.

SCAVENGING OF AEROSOL

10z

i0 i I

0=18.6~m

~ io° _ ,7

? mo _o2 _J 0

LO -2

i0 "3-

i i l lllllI

0.001

I I I IIIJii

001 PARTICLE

I l~k.lltllJ

0.I 1.0 RADIUS (M.m)

291

ever, in this size range phoretic and electric forces play an important role. (2) Curves 1 to 4 in Figs. 2 to 10 demonstrate that, in absence of electric forces, phoretic forces may raise E by more than one order of magnitude as the relative humidity R H decreases from 100 to 50%. Also, with decreasing R H the minimum in E shifts to increasingly larger values of r. (3) With decreasing size of the collector drop the Greenfield Gap becomes progressively deeper and narrower, and shifts progressively toward larger values of r. This result is due to the fact that phoretic forces affect the Greenfield Gap more strongly on the small particle side than on the large particle side. We also note that the collision efficiencies in the Greenfield Gap increase rapidly with decreasing radius of the collector drop, reaching a value of E ~ 1 for drops of a = 10 /xm, if R H 1 /~m. For particles ofO.O1 ~< r ~< 1 /zm, n e i t h e r o f these two processes is very effective. How-

I0°

w

~

iO-2

5 o

I0 -3

i 0 "4]

4

I

0.001

~ fllll~l

I

0.01 PARTICLE

I Ililld

I

I JaJIJII

0.I 1.0 RADIUS (it.m)

FIG. 4. Same as Fig. 2 except for a drop radius of 30 /zm. Journal of Colloid and Interface Science, Vol. 75, No. 1, May 1980

292

WANG AND PRUPPACHER

i0°

I0 ° o=lO6/zm

10" z

iiio

r ,~\

I0-z

u. laJ

\\ \\\,

~, 10-3

3~, \x

j--

10"2

J I0-4

10.5 i ] i IIIHI 0.001 0.01 RADIUS

I I I llllll 0.I

OF AEROSOL

{ I

illllll

l!l

i Illlll

1.0

I

I0.0

I IIIIIII

0,001 RADIUS

P A R T I C L E (H.m)

FIG. 5. Same as Fig. 2 except for a drop radius of 42 /xm.

gap region is not only the result of the competitive action of Brownian diffusion and inertial impaction. Figures 5 to 10 show that inertial impaction creates its own gap

I

I lllllll

0.01 OF AEROSOL

I

I llllllt

I

I lllllll

0.I 1.0 P A R T I C L E (/J.m)

I0.0

FiG. 7. Same as Fig. 2 except for a drop radius of 106 /zm.

(see dashed curves) d u e t o the fact that

particles of sufficiently small sizes are caught in the rear of the drop, as trajectory analysis reveals (5, 6, 13).

i0 °

i0 °

Q=72F.m

0 =173/J,m

.,

i

/

I°l

/

i0 -I >-

f 8 9

z bJ

j

,?

~, iO-z w

~

I i

$

I

\\ ix

rl

\ \\

G

io"2

.

8

,:,,o

EbJ

fl

II

10-3

io-~

(/I

7

du

8

4 10- 4

~0-5|

I0"

I

0.001 RADIUS

I I IIIIII

I

I I Illlll

0.01 OF A E R O S O L

I

I I l llIIl

0.I PARTICLE

I

I I lJllld

1,0

I0.0

(/J.m)

FIG. 6. Same as Fig. 2 except for a drop radios of 72 ~ m . Journal of Colloid and Interface Science, Vol. 75, N o . 1, M a y 1980

[0 "S

I

I

0.001 RADIUS

I llllll

l

i I111111

0.01 OF AEROSOL

I

I I lllllI

0.I PARTICLE

1.0 (/J.m)

I

I I IIIIII

I0.0

FIG. 8. Same as Fig. 2 except for a drop radius of 173 /xm.

293

S C A V E N G I N G OF A E R O S O L

F a=310/zm

I0°

/

IO'Z r

a :42/.tm

// //

Id~ k

//

10-3 i m

mE

_~ iO-Z _

u v

uJ

g io-3

.J bJ Z

5,1

bJ v

03

g I0 "'~

.

10-4

~f

10- s

.J J Q

10"o IO"

I

I IIIIIII

0.0(

I

I Illllll

0.01

RADIUS

I

I I IIIII]

0. I

I

I IIIIIII

IOD

1.0

OF AEROSOL PARTICLE (Fcm)

El6. 9. Same as Fig. 2 except for a drop radius of 310 /zm.

1

0 O.OOI RADIUS

OF

O.(31

7

~

AEROSOL

O.I

1.0

PARTICLE

(/zm)

FIG. i 1. Effect of particle size, electric charge, and external electric field on the collision kernel of a water drop of radius a = 42 /zm colliding with aerosol particles in air of 900 mbar, 10°C, and R H = 100%, from model II. (1) = Iq, I = 2 esu cm -2, E0 = 3 0 0 0 V/cm; (2) Iqol = Iq, I = 2 esu cm -z, E0

i0 -I _-- 0:458/~m

Iqol

>-

= 500 y/cm; (3) Iqol = Iq, I = 0.2 esa cm-2, eo

10- 2

=3000 z LIJ

g

II

~. iG 3

5

Z

10-4

O3 .J 0 0

10-5 I

10-6/

~ ~ ,,~ltJl

O.OOI

O.OI

i ~ ~J~.l

O.I

esu cm -=, E0

e s u c m -2, E0 = 0,

where q. = Q./a 2, and qr = Q~/r 2.

LI. ~J

2

V/cm; (4) ]q.I = lqr[ = 0 . 2

= 500 V/cm; (5) [qo I = [q,I = 0

t i ill~.l

I.O

PARTICLE RADIUS (/~m) FIG. 10. Same as Fig. 2 except for a drop radius of 438 /xm.

(4) If electric charges reside on drops and aerosol particles and an external electric field is present of a magnitude which is typical in thunderstorm situations, E is raised, and this raise is greater the larger the relative humidity and smaller the collector drop. In this manner the Greenfield Gap is substantially reduced and transformed into a shallow minimum whose position is located at a considerably smaller value of r than that of uncharged drops and particles. Also, this shift is the more pronounced the smaller the size of the collector drop. We further notice that the collision efficiency enhancing effect of electric charges and fields is confined to Journal o f Colloid and Interface Science, Vo|. 75,

N o . 1, M a y 1980

294

WANG AND PRUPPACHER "3 I0.~,

0=42~m

i0-I -

(0)

(b)

a =310/zm

v

r =O.O01u.m

r :O.O01~rn z w

w

=o.,:°°'°5~

~ I0"

=0.005

~,o-3

_J

_1 0 u I0"

iO-Z

=

=O. Ip.m

,

10.4 0. I

I

J

I0

OJ

I

i

t

, I IJl,I

i

L ~

~ , ~=I

50

L

I00

i

L

I 1 , , , I

500 (voltslcm)

ELECTRIC

I

I000

i

I

,

I i ,11

i

I i i iii

50

3000

j

I0

I (esu)

(esu)

i

I

I00

I , ,,ll

500

I

IO00

I

3000

(voltslcrn]

ELECTRIC

FIELD

FIELD

FIG. 12. Variation with electric field strength of the collision kernel of a water drop of radius: (a) 4 2 / z m and (b) 310 tzm, colliding with aerosol particles in air of 900 mbar, 10°C, a n d R H = 100%, for lq, [ = Iqr ] = 2.0 esu cm-Z; f r o m m o d e l II.

with charged aerosol particles in the presence of an external electric field is actually reduced b e l o w the efficiency in the absence of an external electric field. This result is due to a substantial increase of the moving speed of the drops and aerosol particles, causing a reduction of the interaction time b e t w e e n the drops. H o w e v e r , this reduction

aerosol particles of 0.01