EFFECT OF BROWNIAN COAGULATION ANDBROWNIAN DIFFUSION ON GRAVITATIONAL

SETTLING OF POLYDISPERSE AEROSOLS

Tetsuo YOSHIDA, Yasuo KOUSAKA, Kikuo OKUYAMA*and Shigeru NISHIODepartment of Chemical Engineering, University of Osaka Prefec-ture, Sakai, 591

The behavior of aerosol particles of sub-micron diameter undergoing Brownian coagulation,Brovvniandiffusion and gravitational settling was studied by numerically solving the equation ofpopulationbalance and by experimentally observing the change of aerosol properties with time.In calculation, two dimensionless parameters, which are determined by the initial properties ofaerosols and vessel dimension in which aerosols are suspended, were introduced to evaluate themagnitudeof influence of Brownian coagulation and Brownian diffusion on gravitational settling.The results of numerical calculation were indicated by graphical representation, which isusable for quantitative estimation of the magnitude of these two effects. These computationalresults were found to be in good agreement with the experimental results obtained by an ultra-microscopic technique of particle size analysis.

Introduction

The behavior of aerosols is influenced by coagula-tion, diffusion, sedimentation, the rate of generationof particles, particle characteristics and the dynamicsof the fluid in which particles are suspended. Foraerosol particles of sub-micron diameter, coagulationby Brownian motion and deposition by Browniandiffusion as well as gravitational settling are essentialfor characterizing the behavior of aerosols4\

As basic research in the size distribution whichchanges with time undergoing Brownian coagulationand gravitational settling, a few analytical solutions3>7)and numerical solutions1-5'M) have been obtainedunder some simplified or particular conditions. As

one of the representative studies G. G. Lindauer et al.8)and C. H. Ahn et alP showed the effect of Brownian

coagulation on gravitational settling by a numericalmethod, but they included the unrealistic assumption

that particle concentrations are always uniform in the

d

irection of settling.

The purpose of this paper is to discuss the effect ofBrownian coagulation and Brownian diffusion ongravitational settling considering the variation ofparticle concentrations in the direction of sedimenta-

tion. The equation including these three effects hasbeen solved numerically on the assumption that the

initial particle size spectrum is of log-normal form.In calculation, two dimensionless parameters wereintroduced to estimate the amount of the effect of

Brownian coagulation and of Brownian diffusion ongravitational settling. The calculated results were

arranged in graphs to predict these two effects quanti-tatively, and some of them were compared with ex-perimental results obtained by an ultramicroscopictechniquen).

Theoretical Calculation

B

asic equation

Consider an aerosol located between parallel hori-zontal walls as shown in Fig. 1 on which aerosol

particles are deposited by Brownian diffusion and bygravitational settling accompanying Brownian coagu-

lation. The basic equation expressing the time-dependent variation in particle number concentrationof aerosols can be given by the population balance ofthe element Ay' (refer to Appendix)9 12). The basic

, absorbing ,wall

o Brownian diffusion o° °

production f\ dissipation^^ I ,

Brownian coagulation \_

oO gravitationalsettling

^absorbing^wall '

r/=H

Fig. 1 Behavior of aerosols between two horizontal walls

Received July 6, 1974.

Presented at the 39th Annual Meeting of The Soc. of Chem.Engrs., Japan, at Kobe, April 4 1974 (entitled "Effect ofBrownian Coagulation on Gravitational Settling of Poly-

disperse Aerosols".)

VOL 8 NO. 2 1975 137

equation written in dimensionless form becomesdn(r(, t) Cjjr'j) d*n(n,0 CJrj) rldn{n,t) dt -=DG

CJrg0) ndy* Cm(rg0) dy r</*2" +CG £ *(Vr!-p\,Pi)n(Vr?-pi t)

Pi=rimin ^(p^ij^y -CG 2 k(ri,pMn,t)n(Pi,t)

z=zmm- à"max

where n(rt, t)=n'(ri, t')/n0, t=ut(rgO)t'/H, y=y'/H,

ri=n/rg0, k(rt, pi)=K(ri p'^jK^CJr^) 0) (2)

nr_ 3kT = l f2D(rgO)no\ f ut(rgO)no\

4(pp-p)gr*0KH 2\ H* )\ H ) (3)

K(ri9 pi) is the coagulation function of particles, andin the case of Brownian coagulation K(r^, pi) is given

by10)

K(r- , pd =KJrl +pd[CJri)/ri + CJp'd/pi},

K0 =2ic T/3fi (4)

In Eq. (1), the first term accounts for Brownian dif-fusion, the second for gravitational settling, and thethird represents the number of newly formed particleswith radius ^ by collision of two particles, while thelast term represents the decrease in number of par-ticles with radius rt by collision with other particles. CG and DG are dimensionless parameters whichcan be evaluated from initial aerosol properties andphysical conditions. As seen from the definition, theparameter CG means the ratio of Brownian coagula-tion rate to deposition rate by gravitational settling. Another parameter DG, which was proposed byC. N. Davies2) to evaluate the concentration changeof monodisperse aerosols undergoing diffusion andgravitational settling in the non-coagulation field,describes the deposition rate ratio of Brownian dif-fusion to gravitational settling. Though the param-eters CG and DG are based on an initial geometricmean radius, they are convenient parameters topredict the influence of Brownian coagulation orBrownian diffusion relative to gravitational settling.

In the derivation of Eq. (1), the following assump-tions were made: (1) There exist no external forces except gravity andthe medium is in stationary state with no convection.

(2) Particles are spherical and electrically neutral. (3) Particles collide with each other to form singlenew spherical particle whose mass may be the sameas the combined mass of two smaller particles.

(4) All particles colliding with two horizontal wallsare caught by them. The initial particle size spectrum is assumed to beof log-normal distribution and to be generated in-stantaneously with spatial uniformity ; then

/ /w 1 / In2Yi \.-^ 0)=72^^^{-2W^)Jln r-

forO<j<lw(ri> 0)=0, atj;=O and 1

The boundary conditions are given asnfc,0=0,atj=0and1, when t>0

(6)

If the coefficient Cm(ri)/Cm(rg0) on the right-handside of Eq. (1) is normalized about rg0, solutions de-pend on (7g0 and two dimensionless parameters CG

and DG.F

inite difference approximation and calculation methodSince Eq. (1) is a nonlinear differential equation, it

cannot be solved analytically. It is approximated byusing central difference formulae accurate to secondorder, and then the finite difference equation is givenas follows :

nfa, y, t+Jt)=n(r,, y, t)-{Cm(r;yt/Cm(rgt))}

1 Ay ) ^ CJr^rt

x f np-j,y+Jy, Q+iifa,y-ij, Q-2h (r«,y, t) }

+CGJt\ S ^(^-^,^)»(^?-^,j,0

Vri~Pi / Pprjmin

x«(^,j,Ow(i°»>J»Of» i=i"min /max (7)

Calculation was started by assigning initial values ofparticle number concentration by Eq. (5) at every in-

terior point between two walls. Being stepped byAt, newvalues of particle number concentration werecalculated by Eq. (7), and this step was repeated until

particle number came to zero.

Calculation Results and Discussion

It can be considered that the main factors whichinfluence the time dependence of particle size distri-bution are initial properties of particles such as geo-metricmean radius, number concentration and geo-metric standard deviation. Therefore solutions wereobtained for various values of them. The results areshown in Table 1, together with the values of CG,DG and computational parameters.The effect of Brownian diffusion

According to the definition the parameter DG in-creases as particle radius decreases, and consequentlythe effect of Brownian diffusion becomes large. Fig. 2shows the time dependence on dimensionless number

138 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

Table 1 Conditions for calculationrs o (T g o n o C G D G

M [- ] [p a r t ic le s / c c ] [- ] [- ]

0 . 1 1 . 2 1 0 6 3 . 1 0 x l O " 1 3 . 9 4 x 1 0 ~ 2

0 . 2 1 . 2 4 . O x l O 6 3 . 1 0 x l O - 1 4 . 9 2 x l O ~ 3

106 3 . 4 4 x l (r 2

9 . O x 1 0 6 3 . 1 0 x l O " 1

1 0 7 3 . 4 4 x l O "

0 . 3 1 . 2 1 0 8 3 . 4 4 x l O - 1 . 4 6 x l O " 3

i n s 3 . 4 4 x l O l

1 01 0 3 . 4 4 x 1 0 2

1 0 6 1 . 2 4 x l O - 2

1 . 2 1 0 7 1 . 2 4 x l O " 1

0 . 5 1 0 8 1 . 2 4 x l O - 3 . 1 5 x l O ~ 4

1. 5 1 0 9 1 . 2 4 x l O l

1 0 1 0 1 . 2 4 x l O 2

1 0 6 3 . 1 O x I O " 3

1 0 7 3 . 1 0 x l O - 2

1. 0 1 . 2 1 0 8 3 . 1 0 x l O " 1 3 . 9 4 x l O " 5

1 0 9 3 . 1 0 x l O -

1 0 1 0 3 . 1 0 x l O l

J ln r - 0 .0 7 7 , J v - 0 .0 1 , 0 .0 2 , J * = 0 .0 1 , 0 .0 0 2 , 0 .0 0 0 4

/ > p = 1 . 0 g / c m 8 , # = 0 .2 5 c m

0 . 0

0 .2

0 .4

i i i i n ¥ i i

jI

*i > v .

f �"^ �"蝣'

sD. 2 ^

^ov X

¥

¥

¥

¥

V

¥¥

¥

¥

¥

I

I

¥ '

. 0 . 3 -

E V X x >v ^ ^.¥ N>

0 . 4- A ¥ ¥ ,> %

0 .6

0 . 8

1. 0

0

vO.5 -¥ -Q6- V

0.7 ¥ ¥ ¥

pa rtic les W * d D G C G x > SX

¥s i ¥

> .1 ¥* / ¥l¥

l'¥ I ¥

ZA - ' X A A

- ,V ' * * ¥ * ' Z ¥

s e ttlin g o n ly 0. 0 0 .0

1 0 6 0 .1 3. 94 xl 0

0 .3 1 04 0 x lO 6 0 .2 4 9 2 xl O

9 . Ox l O 6 0 .3 1. A 6x lO 310 8 1. 0 3.94xl 0

7- =- 1 V. V

0.2 0. 4 0. 6 0. 8

n r - u

Fig. 2 Distribution of aerosol concentration undergoingBrownian coagulation, Brownian diffusion and gravitationalsettling

concentration of paricles n at each height under con-stant CG. When coagulation and diffusion occur,that is CG>0 and DG>0, more rapid decrease ofparticle concentrations is seen in the figure in com-parison with those of gravitational settling (CG=0and DG=0). When the values of DG are smallenough, the difference from gravitational settlingdepends mainly on Brownian coagulation, and thenparticle concentrations are determined by the valuesofCG and ag0. It may be seen from Fig. 2 that inthe case of DG<0.004 the effect of Brownian diffusion

0 .0

0 .2

O.A

(a )

> ,0.6

0 .8

1.0

0.0

0 .2

0 .4

(b )

> *0 .6

0. 8

1.0

t = 0 . 0 5

IIIIII II

. .0 .1

v^ ~0.2

s c - JO3mvv oA ¥¥ ¥ �"¥ ¥ ^

¥ ¥ ¥ ¥

0.2 0A 0. 6 0. 8

n M

r I I �����������������������������������������������������������������������������������������������������

^ 2

'4 �"^. 0 .4 > <

.¥ ¥v 蝣*_.' lｫv ¥S

¥ ¥ x v f C x

V o .5 ^ ¥

I ¥蝣d n

C G o .0 1. 2 41 2. 4 l oj r n= 1. 2|

0. 2 0 .4 0 .6 0.8n M

Fig. 3 Effect of Brownian coagulation ongravitational settling

seems to be negligible, which almost agrees withDavies' analytical solution2) obtained for monodis-perse aerosols undergoing Brownian diffusion andgravitational settling. So far as aerosol particles of

more than several tenths micron in diameter are con-cerned, the effect of Brownian diffusion may benegligible. In the following section, the effect ofBrownian coagulation on settling aerosols in the

absenceof Brownian diffusion will be discussed.

T

he effect of Brownian coagulationThe curves of concentration change with time for

various values of CG are shown in Figs. 3 (a) and (b).

As the values of CG increase the decrease of particleconcentration proceeds more rapidly because of theloss of particles by coagulation and because of en-

hanced settling velocity due to the growth of particlesby coagulation. Moreover, for a constant value ofCG, the discrepancy in concentration change fromthat of gravitational settling (CG=0) increases with

time and settling depth. This is also due to the lossand the growth of particles by coagulation which

VOL. 8 NO. 2 1975 139

0.8

0.6

C OA

0.2

0.01

'= o .2 A ¥

& X A ¥

N ^ ¥ ¥ ¥ �"ｻ

=0.4

y = 0 .8

¥ ¥ l

I I I :1.2 ¥

CG = 0.0(se ttling o nly]

CG =0 .2 5

C G = 2.5

6 8 10 4 6 8 10t . c- n

Fig. 4 Dependence of aerosol concentration on settlingdepth

1.0

08

0.6

ccu

0.2

0.0

蝣

v¥ W ?X eN

% w z oｻo

Obo =1.2

¥ - I

V ^A 6 810~2 2 U 6 8 10"1

t/y H

(a) ogo= 1.2

1.0

0.8

r-i 0.6ill

COA

0.2

0.0

s >

I

Ifi S B S S i

o ¥ o¥ <?¥o

QO- '�"

!a s^ 9H H ffiK a ra N@

ck ^

xv > ,

¥

2 4 6 8 10~2 2 4 6 8 10 ' 2 4 6 8 10- 2 4 6 8

t / y C- :

(b ) (7, o= 1 .5

Fig. 5 Decrease in aerosol concentration with time

g l a s s w a l l o b j e c t i v e o f m i c r o s c o p e ¥ v a l v e _ _ _ J

in le t a c ry l ic o u tle t

m m

U L LJgroove w it h s e c t io nof 2 . 5 x lO

6 5

in

Fig. 6 Observation cell

proceed as aging time grows longer. This tendencymay be obvious in Fig. 4, which shows the dependenceof particle concentration on height. It is impossibleto predict the local effect of Brownian coagulation at

any height by using the parameter CG. However,CG shows the overall effect of coagulation between

o b je c t iv e o fm .c ro s c o p e a e ro s o | jn

g la s sb e a m / U ' /

S IS Z"a

A . & A F i g . 7 I l l u s t r a t i o n o f o b s e r v a t i o n c e l l

/ a e r o s o l o u t v m = rc D e4 n/ A

two walls, and at values of CG below 0.02 the effectof coagulation seems to be negligible. To estimate

the variation of particle concentration with time atany height, CG and t are transformed into the follow-ing forms.

re v ( 9f H K <>n- ¥ (j^_¥ y ' oC m(rgo)L ij -y - { 2(pp- p)grL ) ¥ h ) ut(rｫ )

t/y = / 2(p 9- P)gr*tc j r10y¥ 9 uH H .Ut(rg{

9uH J¥H}(8)

Thus the concentration change of aerosols is arrangedas shown in Figs. 5(a) and (b). It may be seen fromthese graphs that at values of CG-y below 0.02 theeffect of coagulation may be ignored and the change

in concentration of particles depends mainly ongravitational settling. With the values of CGà"y, the

curves have more gentle slopes than that of gravita-tional settling only. At values of CGà"y above 20.0

the concentration change is dominated only by co-agulation, almost regardless of the initial standarddeviation. Figs. 5 (a) and (b) for two initial geo-

metric standard deviations enabled one to predict theconcentration change of aerosols having various

initial size distributions.

E

xperimental Apparatus and Method

In this study it is necessary to measure accuratelythe variation in number concentration of aerosolsover short periods of time. For this purpose, anultramicroscopic techniquen) was used. The ob-

servation cell installed on the stage of an ultramicro-scope, as shown in Fig. 6, has a small sectional areato prevent the effect of thermal convection. Aerosolsused in this study were stearic acid particles and tobac-co smoke. Stearic acid aerosols were generated by aLa Mer-Sinclair type generator. Tobacco smoke wasgenerated by a simple apparatus11} by which number

concentration was controlled from 106 to 108 par-

ticles/cc. The experimental procedure is as follows.After the focus of the ultramicroscope is preliminarilyset at the depth h shown in Fig. 7, aerosol is intro-duced into the observation cell and the flow of aerosol

is instantaneously stopped by closing the valves.From that moment aerosol particles existing in thevolume vm, which are recognized because of their

140 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

Table 2 Experimental conditionsKinds o f a e r o s o ls : s t e a r ic a c i d p a r ti c l e s , to b a c c o s m o k e

Initial g e o m e t r ic m e a n r a d iu s : r g o = 0 3 u ~ 0 . 5 u

Initial g e o m e t ri c s t a n d a r d d e v ia t io n : a g o = -1 .2 ~ l A

Density o f p a r t ic l e s : ^ = 0 . 8 5 g / c m 3 ( s t e a r ic a c id p a r t ic le s)

=0.78 g /c m 3 ( to b a c c o s m o k e )

Concentration o f a e r o s o ls : n o = 1 0 6 ~ 1 0 8 p a r t ic l e s /c c

h : 2 0 0 /^ - 1 5 0 0 ^

v m : 1 .4 x l O " 5 c m 3 ( e y e p ie c e x o b j e c t iv e , 1 0 x 2 0 )

1 .2 x l O - 6 c m 3 ( // , 1 0 x 4 0 )

CG : 0 . 0 3 - 1 . 7

D G : 4 .7 x l O " 5 - 1 . 7 x l O - 3

shining, are recorded by a video recorder, until allparticles disappear in the sight. The number of

particles at any given time are counted by reproduc-ing the recorder.

Initial number concentration of particles, n0, isgiven as follows by the initial particle number of

i

mages N(0) and the observation volume vm:no=N(O)/vm (9)

The particle size spectrum was also determined bythe ultramicroscopic technique developed previously

by the authorsn), whose principle is almost the sameas that of the Andreasen-Pipette method. Experi-

mental conditions and physical properties of aerosolsare shown in Table 2.

Experimental Results and Discussion

Fig. 8 shows a comparison of the relative con-centration change with time between experimentaldata and theoretical curves. In this comparison the

decreases of the experimental values seem to be slightlyslower than those of the predicted ones. However,

overall agreement is good within experimental error.Asseen from the values of CG and DG, the effect ofBrownian coagulation and diffusion on particle con-centration can be almost ignored. This agreement

indicatesthat convection, photophoresis and thermo-phoresis do not occur in the observation cell.

Fig. 9 shows the experimental data of the relativeconcentration change on tobacco smoke and stearic

acid particles, together with corresponding theoreticalcurves. Experimental data are in good agreementwith the tendency of the calculated curves, and the

effect of Brownian coagulation increases as the valuesof CG-y increase. This kind of experimental data

has not been reported because of the difficulty in ac-curate measurement of changing number concentra-tion of particles with time. In this experiment the

values of CG y were rather small, but the effect ofBrownian coagulation on gravitational settling wasclearly found to be characterized. These experi-

mental results might indicate, in a certain sense, thatthe conception used in deriving the basic equation,above all the assumption that particles stick together

upon impact and will not repel, are basically correct.

0

5 0 0

10 0 0nｱLJ

SI 15 0 0

2 0 0 0

25 0 00

crflo=1.22 stearic acidcG=7.72xlOo o3=9.15xl O " '

�"

S : t=15 s ec*

"> v 1

*^ 50 ^ ¥ N

������������������������������������������������

6 0

7 08 0

C Gi ¥ n i i ¥

D G

O . Q 0 . 0

- o - 1 0 0 7.72xlO 2 9.15xlO "

0. 2 0 .4 0 .6 0 . 8n M

Fig. 8 Comparison of experimental aerosolconcentration with calculated ones

1. 0

0. 8

. 0 .6J l

? ^ ^ u ¥

o^ ^ %oo

^ Ioomc 0. 4M W

o¥

�"¥<?l0 .2

0 .0

C G -y o¥o 0 .3 3 6

�" 0 .0 3 6 2

Q 0 .O U 9

8 10 4 6 8 10

t / y [ - I

F

ig. 9 Comparison of experimental aerosolconcentration with calculated ones

One more sensitive test of the coagulation theorycould be made by the change of concentration of eachspecies of particle size distribution.

Conclusion

The behavior of aerosols undergoing Browniancoagulation, diffusion and gravitational settling be-tween two horizontal walls was studied theoretically

and experimentally.The basic equation considering the above three

effects and the variation of particle concentration inthe direction of sedimentation was solved numeri-cally for various initial particle size spectrums. The

overall effect of diffusion or coagulation on particleconcentration of settling aerosol particles was esti-mated using the values of two dimensionless param-eters DG and CG respectively. At values of CGless than 0.02, the effect of coagulation was then

found to be negligible, and in the regions of DG<0.004 the effect of diffusion could be ignored. Inthe absence of Brownian diffusion the change of

particle concentration with time at every height could

VOL 8 NO. 2 1975 141

be estimated by introducing a new dimensionlessparameter CG-y. Some of these theoretical resultswere examined by a technique of ultramicroscopicsize analysis and were found to be in good agreementwith experimental ones.

The calculation results presented in this paper arean elementary step in providing a theoretical approachto predict the behavior of aerosols in a closed vesseland also will be useful to estimate the influence ofBrownian coagulation or diffusion on particle sizeanalysis of aerosols by sedimentation methods.

AppendixThe basic equation expressing the time dependenceof the size

distribution of aerosols undergoing Brownian coagulation,Browniandiffusion and gravitational settling is given4'9-12) by:

dn'(r\t') ( d*n'{r\t') dn\r\tf)

+\r'^2K( Vr'*-p'\p')n\ Vr'*-p'*91')

-\ K(r',pr)n\r\f)n{p\t')dprJ|0'=O

where D(r') is the diffusion coefficient and ut(r') is the terminalsettling velocity.

Ack nowl edgment

T. Miyazaki and Y. Kida were very helpful in the experimentalwork.

Nomencl atur e

C

G =dimensionless parameter defined in Eq. (3)( = 9vHKonol2(pp - p)grlo) [-]

Cm(n') =Cunningham's correction factor of r%' [-]

D

e =effective diameter of microscopic sightshown in Fig. 7 [cm]

D

G =dimensionless parameter defined in Eq. (3)( = 3,cTI4(pP - p)grlo7iH) [-]

D

{n') =diffusion coefficient( = Cm(n /)/cT/6Ti:iuri /) [cm2/sec]

=acceleration of gravity [cm/sec2]=height between two horizontal walls [cm]=values shown in Fig. 7 M[cm]=coefficient in Eq. (4) [cm3/sec]=coagulation function for two particles of

sizes nf and pi [cm3/sec]=dimensionless coagulation function

( =K(n '9 Pi ')IK0Cm(rg0)) [-]

=dimensionless size width between n and n+i [-]

n'(ri\ t'), n(ri, t) = number and dimensionless numberof aerosol particles (=«'(>*', f)/no)

[p articles/cc] [-]n = dimensionless total number of particles

(= Zn\n\ nino) [-]no = total number of particles at time

zero [p articles/cc]N(0) = numberof aerosol particles in vm at

time zero [particles]r\ r = particle radius and dimensionless

particle radius [/*][-]rgo =geometric meanradius at time zero [/u]

T = absolute temperature [°K]t', t = time and dimensionless time [sec][-]At = dimensionless time step [-]

u

t(ri') =terminal settling velocity( = 2Cm(n ')(Pp - P)n '2gl9[i) [cm/sec]

vm =volumeshown in Fig. 7 [cm3]

y

\ y = vertical and dimensionless verticaldistance from top of the cell [cm][-]

A

y\ Ay - meshwidth and dimensionless mesh widthalong the vertical direction [cm][-]

Ogo = geometric standard deviation at time zero [-]

r

=Boltzman's constant (=1.38 x 10~16) [erg/°K]li = viscosity of fluid [g/cm. sec]

pt pp = fluid and particle density [g/cm3]pi =dimensionless particle radius [-]

<Subscripts>

/

=refers to the number particle sizemin = minimummax =maximum

Literature Cited1) Ahn, C. H. and J. W. Gentry: Ind. Eng. Chem., Fundam.,

ll, 483 (1972).

2) Davies, C. N.: Proc. Roy. Soc, A200, 100 (1949).3) Friedlander, S. K. and C. S. Wang.: /. Coll. Int. Set, 22,

126 (1966).4) Fuchs, N. A.: "The Mechanics of Aerosols," Pergamon

Press, Oxford, (1964).5) Greenfield, M. A., R. L. Koontz and D. F. Hausknecht:

/

.Coll. Int. ScL, 35, 102 (1971).

6) Hauck, H. : Staub-Reinhalt. Luft., 33, 1233 (1973).7) Langstroth, G. and T. Gillespie: Cand. J. Research, 25B,

455 (1947).8) Lindauer, G. C. and A. W. Castleman, Jr. : Aerosol Science,

2, 85 (1971).9) Mori, Y. and A. Yoshizawa: Kagaku Kogaku, 32, 1233

(1968).10) Smoluchwoski, M.: Phys. Chem. (Leipzig), 92, 129 (1917).

ll) Yoshida, T., Y. Kousaka and K. Okuyama: Ind. Eng.

Chem.Fundam., 14, 47 (1975).12) Zebel, G.: Kolloid-Z, 156, 102 (1958).

142 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN