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CHANGE IN PARTICLE SIZE DISTRIBUTIONS OFPOLYDISPERSE AEROSOLS UNDERGOINGBROWNIAN COAGULATION
Tetsuo YOSHIDA, Kikuo OKUYAMA*,Yasuo KOUSAKAAND YOSHINORI KIDA
Department of Chemical Engineering, University of OsakaPrefecture, Sakai, 591
The time-dependent change in particle size distributions of highly concentrated polydisperseaerosols undergoing Brownian coagulation was studied by numerically solving the basic equationof coagulation for various size distributions initially having log-normal form. The results wereplotted in the forms of the change with time in cumulative size distributions and the changes innominal geometric mean radius, as well as standard deviation for various initial distributions of
aerosols. These figures showed that size distributions approached certain asymptotic ones,which might correspond to SPDF(self-preserving distribution function), almost independentlyof initial distributions as coagulation proceeded. The process of the approach to asymptoticdistributions was also made clear by the graphs. Some of these results were verified by ex-
perimental results obtained by the ultramicroscopic size analysis previously developed by theauthors.
Intro ductio nIn the previous paper15\ the change in particle
number concentration of polydisperse aerosols under-going gravitational settling, Brownian diffusion and
Brownian coagulation was discussed, and Browniancoagulation was found to be important when highconcentration aerosols were concerned. Analyticalsolutions2'n), asymptotic solutions3'5 8 13) and nu-merical solutions6'12) have been reported for thechange in paricle size distribution of polydisperse
aerosols with time undergoing Brownian coagulation.These solutions, however, have been obtained undersome simplified or specialized conditions, and seemunsatisfactory for understanding the general aspectsof time-dependent change in particle size distribu-tions of polydisperse aerosols. In experimental
studies, because of the difficulty in accurate measure-ment of particle size distribution of highly concen-
trated aerosol, sufficientamounts of available datahave not been reported7>10).
In this paper, the change in particle size distribu-tion of highly concentrated polydisperse aerosols wasstudied by numerically solving the basic equation forBrownian coagulation with various initial log-normalsize distributions. The results of the calculation were
graphically presented to show the effect of polydis-persion on the change in size distribution of aerosols,and then were confirmed by experimental resultsobtained by the technique of ultramicroscopic sizeanalysis previously developed by the authors14). Thestudy presented here gives one of the basic proper-ties ofpolydisperse aerosols, and is useful, for instance,for estimation of the extent of the contribution ofparticle growth to industrial precipitators.Theoretical Calculation
The basic equation for the time-dependent sizedistribution change of aerosols undergoing Browniancoagulation has been reported by Muller9)16) and itsdimensionless form can be written as
^^-^^K^rl-plpM^rl-pl^nip^ t)
dtPi=rimin
X 7~^¥=^r- I! kir^p^nir^nip^ t)\ V ri~Pi) Pi=rimin
i=imin... Amax (1)
where,
n(rt, t)=n'(r'i, t')lno, t=Kifl0CJrll0)t', r( =r'i/rlli),}k{ri,Px)=K{r'i,p[)IK(lCm{rg0\ K0=2kT/3/, ) ( }
K(.ri>Pi) is the coagulation function of particles and inthe case of Brownian coagulation it is given by11}
Kfa p^lQtr l +pMCJr Mr l + CJpMpl} (3)The left side of Eq. (1) is the change in particle con-centration of size Yi with time. The first term on the
Received November 6, 1974.Presented at the 8th AutumnMeeting of The Soc. of Chem.Engrs., Japan at Tokyo, Oct. 8, 1974. ("Particle Growth ofPolydisperse Aerosols by Brownian Coagulation")
VOL. 8 NO. 4 1975 317
1 "*~^W^' ' ' ' 1 ' f tobaccosmoke^A>s^ key npCpartidesfcc: Tgo CjJH OgoZ-3nfl à" %"^v ® 7.69x106 0,36 1.23
V ^P? ^ c 1.05x107 0.375 1.32.
N^V ° 4-16x107 0.51 1.53Xn->a <d 7.38x107 0.64 1.71
n0'6 ^Sft 1 ® I 8A1xiQ710.465 ll.36u *VV ° ~~ f~" calculated
c0.4 steanc ac.d part.cles à¬JV -ff~ *Q. _^
key nogporticies/ccj rgoc\ii Ogot-i n®s 0 35 1 50 *-oh 13.55x106 I0.32 |1.28 °N^S; I1.0 flipO j--.02 POP particles ^^ ' ' '
key nocparticies/ccj rgoc)ja ago c-3 o xXv
* 1.99x1Q6 0.37 1.22 ^^^wI <> I2.14x106 |0,37 |1.22 | i i i I i i , , ©
10~2 2 4 6810~1 2 4 6810° 2 4 68101 2
t r n
Fig. 1 Experimental and calculatedtime-dependent change in particlenumberconcentration
"| y-35Hi ri/,1å IT/T'{' itfgon-j ////7//'/ ,1 ,'n95 1,30 " /VII/à"°/' - ,
u90 1-50 _ Jil'i/T' '
90 12,00 ////////// /
IS" /////'å ' '
i t_0%/y////«i
./ .ilJlJll-J */1 /i i l l0.2 0.4 0.6 1 2 4 6 10 20
r C-1
Fig. 2 Change in particle size distribution with time
right side represents the rate of formation of particlesof size rt due to coagulation of two particles smallerthan size ri9 and the second term the rate of loss ofparticles of size rt due to their coagulation with par-ticles of any other sizes including rt.In the derivation of Eq. (1), the following assump-tions are made:(1) There exist no external forces and the mediumis in stationary state with no convection.
(2) Particles are spherical and electrically neutral.(3) Particles collide with each other to form a
new spherical particle whose mass may be the sameas the combined mass of the two smaller particles.
Initial particle size spectra are assumed to be oflog-normal distributions and to be generated in-stantaneously with spatial homogeneity. Then the
discrete expression for the spectrum is written as^' 0) =72^7
expI -lnV{2 1nV,0
-)Jlnr (4)
whered kL r={ln (r 'imax/rgO)--hi (r 'imiJrgO)}/M
Mrepresents the number of divisions of radius.
Equation (1) is approximately normalized on rg0
(refer to Appendix). Since Eq. (1) cannot be solvedanalytically, the Runge-Kutta-Merson method wasemployed to solve it. To compare the effect ofvarious initial sizes and their distributions on their
changes with time, calculations were made for valuesofrg0from0.1 ^to 1.0juand(TgQfrom 1.3 to 2.0. Inmost computations r'imin was chosen for 0.01 ju, andrimax for 10 jbt. The number of divisions of radius Mwas chosen as one hundred, the validity of which waschecked by recomputing over finer subdivisions. As a numerical check the total mass of aerosols wascalculated every few time steps and the error wasfound to be within 5 % during the progress of coagula-tion until the fall in number concentration to 10% of the initial value.
Calculation Results and Discussion
Particle number concentration The curves in Fig. 1 show the calculated relationbetween the ratio of particle number concentrationat any time to the initial value and the dimensionless
time. The effect of the degree of polydispersion onparticle number concentration seems not to exceedthe order of 10%. The curve for crff0=1.3 agrees
with the estimated one from the well-known coagula-tion equation of Smoluchowski for monodisperse
aerosols11}, which is given by
dt-=-2n2(5)
In the case of rg0>0A fi the effect of rg0 on the con-centration change seems to be negligible in the figure.
Particle size distributionFigure 2 indicates the calculated results for the
change of particle size distribution with time forthree different initial size distributions. Size distri-butions of aerosols tend not to remain log-normal butasymptotically approach equilibrium states as timeproceeds. The shapes of asymptotic distributions
318 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
Q Q. å
**| calculated I/fjrj Irgocpo ltfgoc-3 J/l >
£qc 0.35 1.5 /// -
^à"90 Q,1 I]3 \^l!I ~^ j^^
*70- PZ/b0' -
I S: #_^
I 5- /fa I"//// I-"SPPF I
J /^ I I II
0.1 0.2 0.4 0.60.8 1
Fig. 3 Comparison of self-preserving size distributions
2-5| tobaccosmoke 1' H } r~7 7©key n0Cparticles/CCD Tgo c^u Cfgo C-j - / J ~
» 7.69xIP6 O36~1,23 ' ' /O 1,05x1Q7 Q.375 1.32 / IO 4.16x107 0.51 1.53 ' / /
<D 7.38x 107 (164 1.71 / ' /QI 8.A1x1Q7 I~~g465 I 1.36 ~~~ / I
2Q stearic acid particles j_ / /keyn0Cparticles/CCJ rg0C^l3 (JgoC-3 ' / /
à"O-l 3.55x106I "0T2 I1.28 / / /7 «*å DOP particles / / /1-1 keyn0Cparticles/cc: fgoCjaD cTgoC-D / / /'
o <)1 2.1Ax1Q6I'o_37- [1.22 \/ //!r calculated T~~ ' > ,'Q/Q& rgpCfi^icTgoC-il y / /
1.5 0.1 1.3 ^ o'-©
0.35 1.5 /O J'/(D1.0 I 2.o I-----p '' ' m>
10*~å ^^-^^^===r6"-1"""T"cl-> 6 Q>. i l l 1 I 1 1 1
'10"2 2 A 6 81CT1 2 A 6 810° 2 A 6 8101 2
Fig. 4 Experimental and calculated variation of nominalgeometric mean radius with time
after long periods of time resemble each other ir-respective of initial geometric standard deviations.These asymptotic distributions at equilibrium statecorrespond to self-preserving distribution function(SPDF) which was derived by Friedlander13) using thesimilarity theory. To compare asymptotic size dis-tributions numerically obtained in this study withSPDF given by Friedlander the following common
variables used amongmost investigators are intro-duced1'1^
Vr =r 'ln'l<f)^ 4>(Vr)=n'(r'i, t'WnM'-rn' i/3 (6)where
rimax
n'= S n'(r'{,t%A rimax
The correlation of cp(j]r) and r]r after sufficiently longperiods of coagulation is shown in Fig. 3 together
with SPDF given by Wang and Friedlander13) forBrownian coagulation without the Stokes-Cunning-ham correction. Though the agreement amongthem is fairly good, <p(j]r) numerically obtained inthis study seems to depend slightly on initial rg0 andag0. This dependence on initial rg0 and ag0 seems tobe caused by Cunningham's correction CJr^jCm(rg0), which increases with decreasing rg0 and in-
creasing ag0.The change of size distribution with time will benext discussed by the aid of two parameters describedbelow. Representative particle size and width ofdistribution will be characterized by two well knownparameters, geometric mean radius and geometricstandard deviation, when the particle size distributionfollows log-normal form. However, when size dis-
tribution does not follow any laws of size distribution,it cannot be characterized by any simple parameters,but only by the curve of frequency against particle
size or cumulative percentage against particle size.It is complicated, however, to discuss this by the
curves themselves, and here two parameters, a nominalgeometric mean radius rg and a nominal geometricstandard deviation ag9 are introduced as follows,
even if some deviation from log-normal form in sizedistribution exists
T(rimax ) "I1
lnag=\ \ E n'irU nhx'irifrMn' l(7)
In Fig. 4 the ratio of the nominal geometric meanradius to the initial one is presented as a function ofdimensionless time. It may be understood that rg ofthe larger standard deviation grows faster than thatof the smaller one. Figure 5 shows the variation ofthe nominal geometric standard deviation with time.It is found that nominal geometric standard devia-tions decrease with time when ag0 equals 1.5 and 2.0,while they increase when ag0 equals 1.3. They seemto converge to certain equilibrium values which willbe expected to lie between 1.3 and 1.5.
Experimental Apparatus and MethodA schematic diagram of experiment is shown in
Fig. 6. Aerosols used in this study were tobaccosmoke, stearic acid and DOP. Aerosols of both
stearic acid and DOPwere generated by a La Mer-Sinclair type generator and tobacco smoke wasgenerated by a simple apparatus14} by which numberconcentration of particles was controlled from 106 to108 particles/cc. Aerosols thus generated were cooledto room temperature by a heat exchanger to preventthermal convection, and continuously led into anacrylic pipe with diameter of 4.2 cm to make verticalupward flow. Most experiments were carried out
VOL. 8 NO. 4 1975 319
:;*_=å =;-^J___ ' ' tobacco smoke~"-- """~-* --^ keylno:particies/ccj| rgo. cpL] 1 agoc-:~
"""^ "*">~""*^ à"769x.106 0.36 1.23"- ^"^"- C 1-05x 107 0.375 1.32
S-^ v."^ O 4.16x10? 0.51 L53_^.^JgoMPu <D 7.38 x 1Q7 P.64 1.71
1-8 I calculated ~ ^-. _V35>j^-^JJa^^ | 0.465 1.36-r---- ^vjU.Jbp^ N stearic acid particles
j1" =P1fix- "^ "Nx key|no"cparticies/cc3| rgo Z]i3 Ig-^1:-]
å '=TZZ=r~t50 ^V.^-^^Pf 1|°6| ' a3,2 I,1-2p |-
I I 2.PQ xv
1.6 _o.---,o ^.^:: I- -n. o O O Oo^"-Z"e»C^
*-> t rgo=i.oM "^-c -- ^
-O" -^1^3 ~Z^m # -| POP particles
I key lr^rpcrticies/cc]! rgo z\xi Icfgo c-3à"<^ <J> <{) (J) <J) I012JA;t1Q6I p.37 |1.22
, , ^ ^r .-1 1 L-L_L 1 J I lJ i \ i ll 1 I 1 ll
102 2 A 68101 2 A 6810° 2 A 681Q1 2 A 68102t C-3
Fig. 5 Experimental and calculatedvariation of nominal geometric standard
deviation with time
Fig. 6 Schematic diagram of experimental apparatus
with flow rates ranging from 0.25 to 0.5 //min, and atthese flow rates the flow in the pipe was observed tobe plug flow rather than laminar flow. The loss effectby Brownian diffusion was evaluated to be negligi-ble^. Aerosols sampled at every given residencetime were introduced into the observation cell installedon the stage of an ultramicroscope to measure theirparticle size distributions and particle number con-centration. The measurement method using anultramicroscope was developed previously by theauthors14), and the procedure is as follows: After thefocus of the ultramicroscope is preliminarily set at agiven depth of the observation cell, the flow of intro-duced aerosol is instantaneously stopped by closing
valves. From that moment particles existing in thefield of vision are recorded until all particles disap-pear from sight, and the numberof particles at anygiven time are counted by reproducing the recorder.From these data, size distributions are obtained by amethod almost same as the Andreasen Pipette method,and initial number concentration, n0, is given by the
initial particle number of images N(0) and the ob-
servation volumev 'no =N(O)/vn(8)
Experimental Results and DiscussionFigure 1 shows a comparison of the relative con-
centration change with time between experimentaldata and theoretical curves. Agreement betweenthem for rg0>03 /u, and (7gQ<l.l is fairly good. Theeffect of rg0 and agQ on the concentration change isfound to be negligible, as discussed in theory. Thisfact shows, in a sence, that probability of contactmay be regarded as unity, or that no repulsion mayoccur when they collide. A detailed comparison ofthe theoretical calculations with experiments shownin Fig. 1 will be made by turning the point to thechange in particle size distribution with time.
In Fig. 7, changes of size distribution of tobacco
smoke are shown together with corresponding theo-retical curves for comparison. The agreement isfairly good. A comparison of the change of nominalgeometric mean radius with time between experi-mental data and calculation curves is shown in Fig. 4.Figure 5 shows a comparison of the change of experi-mental nominal geometric standard deviations withtheoretical ones. Although some scatter is found inexperimental results, the effect of polydispersion onthe change of particle size distribution with time isapproximately confirmed for <jg0 less than 1.6. Theo-retical curves for larger ag09 however, cannot beconfirmed by experiment. This is due to the diffi-culty of constantly generating aerosols having largergeometric standard deviation. Keith and Derrick7)performed experiments to obtain data on the changeof size distribution. In their experiments tobaccosmoke produced by a burning cigarette was used,
320 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
u'l' Li TTTI P77 ' I ' 99- tobacco /$// /// Ismoke,I JSjk II
95- / g/S%L calculated
90- $1 IS- rg0=a35fl " ^5 /©/ p / // t7g0-i.oH.80- L//// U\ i i i - 70 - e /fi° HI Jno[:pcirticies/cc3h o5x1O7S60- hg./0? // - rg° C^J °-375 U)50 |v5p»/-/t " <3go C-J ll.32140- JlfyhiI ~ keyI tC-3
I10- mill oI 1.38
i 5~ $//h1 " rgoL^3 0A65 1 /1/ C3go C-3 I1.36
,J//// . h:Ia
0.40.60.81 2 4 6 810 20 r D-J
Fig. 7 Comparison of experimental particle size distri-bution changes with calculated ones
991 calculated 1' JTP ' ^key |rgOcKj|(jgoC-] à"//
n95 -| 0.r>60 |-jffi^90 -Kelth et ai:sdata 8^/u tobacco smokeA*/
270- .à"///no=1.8.xio7 -å S60- //e7 Particles/cc_
§50 MiJ- rV> = á"*
£'- /JJ: %"m :
s'°-yoV/ - à" o5-7*'S* --^- °-28
/^y0/^ e Q.57J/1//1.0iI ~~« I 1.13 ~
0.4 0.60.81 2 A 6 810r C-3
Fig. 8 Comparison of calculated particle size distributionchanges with Keith et al. 's experimental data
q I_\i^\i r~n [ i.i ii i i nrj]PO \ NX O6- V\ -6
olO2 \ ^V^a- 1°3 ^
^2- *%cX\ -2'1-101- rgo-0.35M \ XX^ =102^
2- \^ \21Cpl 1 LJ_J 1 L_L_J 1 \llli(j106 2 4 68107 2 A68108 2 468io9
n0 [particles/cc)Fig. 9 Dependence of values of r»=<i/a)»0 and Ua=%TQX
on initial particle number concentration
and size distribution as well as concentration wasmeasured by a conifuge. Figure 8 shows a comparisonof Keith and Derrick's data with calculation resultsin this study. Good agreement is found again. Itis interesting that particle size distribution is foundto grow wider with lapse of time in Fig. 7, while theygrow narrower in Fig. 8. These facts suggest that anequilibrium size distribution may exist.
In these experiments no difference between solidparticles such as stearic acid particles and liquidparticles such as DOPparticles and tobacco smoke(semi-liquid) is found.
Figure 9 shows the half-life of particle numberconcentration and also shows the time required togrow to a geometric mean radius twice the initial
one for various initial numberconcentrations and forvarious initial geometric standard deviations. It canbe understood that at n0 above 107 particles/cc par-ticles grow in a short time.
Conclusion
The change in size distribution of highly concen-trated polydisperse aerosols undergoing Browniancoagulation was studied. The results of numericalcalculation for aerosols having various initial sizedistributions which followed log-normal form werepresented graphically. So far as the change in totalnumber concentration of polydisperse aerosols withtime was concerned, the results of calculation almostcoincided with that of monodisperse aerosols, whichwas simply predicted from Smoluchowski's theory.This was also confirmed experimentally. The man-ner of change in size distribution with time, on theother hand, was found to be different among initial
size distributions, which can not be evaluated fromSmoluchowski's theory. Geometric standard devi-
ation decreased with lapse of time when initial devi-ations were larger than about 1.5, and increasedwhen initial deviations were smaller than about 1.3.Some of these results were also ascertained by ex-
periment within experimental error, and the existenceof SPDF was suggested.
The graphs presented in this paper are useful forindustrial purpose where the extent or the rate ofparticle growth of highly concentrated polydisperseaerosols is important.
AppendixFrom Eqs. (2) and (3), it follows that
^'^+^^+<§sy <a-j>If in Eq. (A-l) values of Cm(ri)ICm(rgo) do not depend on rg0,
VOL. 8 NO. 4 1975 321
|\\ ' \' """H ' I IfrQO\ \ \ 0.1
2-° \T V -"035^ , \\ \ crgo=2.0 _0j_
u 1.0-*-*- :this range is"^^,^chosen such that ^s^-'^^.-rrr-90% aerosolsexist ^"^.^^
0.5'-L 1 1 1-i-I 1 i i l l101 2 4 6810° 2 4 6 8101
n=rAo MFig. A Dependence of values of Cm{Yi )lCm{rg^ on initialparticle size distribution
Eq. (1) can be normalized on rg0. Fig. A shows the depend-ence of Cm(ri)ICm(rgo) for various rg0 against dimensionlessparticle radius. So far as small ago or large rgo is concerned,values of Cm(ri)/Cm(rg0) are approximately the same, Eq. (1)
thus can be considered to be almost normalized on rg0.Acknowledgment
S. Nishio was very helpful in the experimental work.Nomenclature
Cmiri) = Cunningham's correction factor of radius ri [-]Ko = coefficient in Eq. (2) (=2kT/3ju) [cm3/sec]k(r*, Pi) = dimensionless coagulation function
( =K(ri 9 PMKoCm(rgO)) [-]Kir'i, Pi) = coagulation function for two particles of
sizes ri and pi [cm3/sec]Jlnr = dimensionless size width between ri and ri+1
[-]M == number of divisions in radius [-]n'ir'i, t'\ n(n, t) = number and dimensionless number
concentration of aerosol particles(=n'(ri , f)lno) [particles/cc] [-]
n\ n à"= total and dimensionless total numberconcentration (=E«'W, fO/«o)
[particles/cc] [-]wo = total particle numberconcentration at
time zero [particles/cc]N(0) = number of aerosol particles in vm at
time zero [particles]r\ r - particle radius and dimensionless particle
radius ( =rf\rg?) [cm] [/u] [-]Ar' = particle size width between r[ and r'i+1 [cm]rg = geometric mean radius [ju]
T = absolute temperature [°K]t', t = time and dimensionless time
( = noKoCm,(rgo)t /) [sec] [-]vm = observation volume of ultramicroscope [cm3]
yjr = size scaling function for self-preservingfuncti on [-]
k = Boltzman's constant (=1.38xlO~16) [erg/°K]ju = viscosity of fluid [g/cm- sec]pi = another dimensionless particle radius in
Eq. (1) [-]
og = geometric standard deviation [-]^ = total volume of particles per unit volume
[cm3 /cc](p(y]r) = self-preserving number density for aerosol
size distribution [-]<Subscripts>i = refers to the number of particle sizemin = minimummax = maximum0 = at time zero
Literature Cited1) Clark, W. and K. Whitby: /. Atmos. ScL, 24, 677 (1967).2) Cohen, E. C. and E. V. Vaughan: /. Colloid Int. ScL, 35,
612 (1971).
3) Friedlander, S. K. and C. S. Wang: /. Colloid ScL, 22,126 (1966).
4) Gormley, P. G. and M. Kennedy: Proc. Roy. Irish. Acad.,52-A, 163 (1949).
5) Hidy, G. M.: /. ColloidScL, 20, 123 (1965).6) Huang, C. M., M. Kerker and E. Matijevic: /. Colloid
Int. ScL, 33, 529 (1970).7) Keith, C. H. and J. C. Derrick: /. Colloid ScL, 15, 340
(1960).
8) Lai, F. S., S. K. Friedlander, J. Pich and G. M. Hidy: /.Colloid Int. ScL, 39, 395 (1972).
9) Miiller, H. : Kolloidbeihefte, 27, 233 (1928).10) Nicolaon, G., M. Kerker, D. D. Cooke and E. Matijevic:/. Colloid Int. ScL, 38, 460 (1972).
ll) Smoluchowski, M von Z: Phys. Chem., 92, 129 (1917).12) Takahashi, K. and M. Kasahara: Atmos. Environ., 2, 441
(1968).
13) Wang, C. S. and S. K. Friedlander.: «/. Colloid Int. ScL,24, 170 (1967).
14) Yoshida, T., Y. Kousaka and K. Okuyama: Ind. Eng.
Chem. Fundam., 14, 47 (1975).15) Yoshida, T., Y. Kousaka, K. Okuyama and S. Nishio: /.
Chem. Eng. Japan, 8, 137 (1975).
16) Zebel, G.: Kolloid-Z, 156, 102 (1958).
322 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN
