Condensational Growth of Polydisperse Aerosol for the Entire Particle Size Range

Aerosol Science and Technology 33:222–227 (2000)c° 2000 American Association for Aerosol ResearchPublished by Taylor and Francis0278-6826/ 00/ $12.00 + .00

Condensational Growth of PolydisperseAerosol for the Entire Particle Size Range

S. H. Park and K. W. Lee ¤

KWANGJU INSTITUTE OF SCIENCE AND TECHNOLOGY,DEPARTMENT OF ENVIRONMENTAL SCIENCE AND ENGINEERING,

1 ORYONG-DONG, PUK-GU, KWANGJU 500-712, KOREA

ABSTRACT. The condensation problem of polydisperse aerosols is investigatedtheoretically. The single particle growth rate suggested by Kulmala (1993) is ap-proximated and simpli� ed by neglecting the Kelvin effect and by adopting the har-monic mean method for the transition correction, which is a good approximation ofthe representative � ux-matching theories. An analytical solution to the particle sizedistribution change by condensation is derived in an explicit form using the modi� edgrowth rate. This study represents the � rst analytical solution to the condensationproblem of polydisperse aerosols for the entire particle size range. The derivedsolution is compared with results of the previous study of Kulmala (1993) and isshown to be reasonably accurate, except for very small particles for which the Kelvineffect cannot be neglected.

INTRODUCTIONWhen an aerosol exists in a supersaturated vaporenvironment the vapor condenses on the particlepopulation, resulting in a change of the parti-cle size distribution. Condensation is an impor-tant phenomenon in many applications, such ascloud physics, generation of test aerosols in thelaboratory, aerosol measurement, particle depo-sition in the human lung, and nuclear safetyanalysis.

The condensational growth of aerosol par-ticles is accompanied by simultaneous massand energy transport. The desired expressionsfor the vapor concentration and temperature

¤ Corresponding author.

pro� le around a growing particle can be ob-tained by solving the appropriate mass and en-ergy conservation equations. Maxwell (1877)constructed the theoretical basis for the con-densational growth of isolated particles. Fuchs(1959) took into account the fact that the par-ticle growth is based on simultaneous heat andmass transport. He also pointed out that an ana-lytical solution of the coupled equations for heatand mass transport is possible only for a smalltemperature difference between the particleand the surrounding gas medium. Mason (1971)obtained an approximate analytical expressionfor the condensational growth rate of a singleparticle in the continuum regime. More accu-rate expressions have been presented by Wagner

Aerosol Science and Technology33:3 September 2000

Condensational Growth of Polydisperse Aerosol 223

(1982), Barrett and Clement (1988), Kulmalaet al. (1989), Kulmala and Vesala (1991), andHeidenreich (1994). For accurate results, nu-merical calculations of the mass and heat � uxequations are needed (Wagner 1982; Barrett andClement 1988).

Barrett and Clement (1988) derived an ap-proximate analytical expression for the particlegrowth rate assuming D/T to be constant, whereD is the binary diffusion coef� cient and T is theabsolute temperature. Kulmala (1988) deriveda correction factor that does not assume D/Tto be constant. Kulmala et al. (1989) compareddifferent expressions for condensational parti-cle growth rate and showed that the expressionspresented by Barrett and Clement (1988) andKulmala (1988)agree quite well with each other.In their studies, Kulmala and Vesala (1991) andHeidenreich (1994) presented estimates of theimportance of various assumptions used in cal-culating the mass � ux.

Kulmala (1993) derived an analytical expres-sion for the condensational growth rate of a sin-gle aerosol particle in the transition regime. Theauthor used semiempirical transition correctionsfor mass and heat transfer adopted from Fuchsand Sutugin (1970).

The above studies are based on the assump-tion that the supersaturation in the surround-ing gas is small compared to unity and thusthe supersaturation-containing terms can be ap-proximated linearly. Williams (1995), however,investigated droplet growth for large saturationratios that can arise in studies of the safety ofpressurized water nuclear reactors. The authorshowed that signi� cant errors can arise in linearmodels for high supersaturation cases.

In this study, an analytical solution for thesize distribution change of a condensationallygrowing polydisperse aerosol in the transitionregime is presented. The growth rate derived byKulmala (1993) is used under the assumptionsthat the Kelvin effect can be neglected and thatthe supersaturation is not too great. The har-monic mean method is used for the transitioncorrection to obtain the solution in an explicitform. The derived solution is valid when the

particle diameter of aqueous particles is morethan about 50 nm because the Kelvin effect canbe ignored in these cases.

THEORY AND RESULTSThe change in the size distribution of particlesby condensation is represented by the followingpopulation balance equation:

@n(v , t )

@t= ¡

@{I (v ) ¢ n(v , t )}

@v, (1)

where n(v , t ) is the particle size distributionfunction at time t and I (v ) is the single par-ticle growth rate by condensation of particleswith volume v . Particle growth by condensationis a mass transfer process combined with energytransfer between the gas phase and the particu-late phase. Therefore I (v ) can be obtained bysolving simultaneous mass and energy transportequations. In this study, we use the followinggrowth rate suggested by Kulmala (1993):

drp

dt

=S ¡ Sa

q rp

µRT

b m MDPs{1 + (S + Sa )Ps / (2p)}+ Sa L2 M

b t RKT 2

¶ ,

(2)

where r p is the particle radius, S is the satura-tion ratio, Sa is the saturation ratio at the par-ticle surface representing the Kelvin effect, q

is the density of the growing particle, R is thegas constant, T is the absolute temperature ofthe surrounding gas phase, M is the molecularmass of the vapor, D is the diffusion coef� cientof the condensing vapor, Ps is the saturation va-por pressure of the condensing vapor, p is thetotal gas pressure, L is the latent heat of vapor-ization, K is the thermal conductivity of the sur-rounding gas phase, and b m and b t are the tran-sition correction factors for mass � ux and heat� ux, respectively. Throughout the derivation ofthe growth rate, Kulmala (1993) neglected radi-ation effects, which can be important in some in-stances as shown by Barrett and Clement (1988).In the present study, the Kelvin effect, i.e., the

224 S. H. Park and K. W. Lee Aerosol Science and Technology33:3 September 2000

effect of particle curvature on condensation rate,is also neglected. The Kelvin effect generallycan be neglected for aqueous particles largerthan about 50 nm (Seinfeld 1986). Pratsinis et al.(1986)quantitatively evaluated the neglect of theKelvin effect in the parameter space of a constantrate aerosol reactor.

Several semiempirical methodologies can beused to obtain the transition correction factorsb m and b t in Equation (2), which are based onthe so-called � ux-matching theory. Flux match-ing assumes that the noncontinuum effects arelimited to a region rp · r · D + r p beyond theparticle surface and that the continuum theoryapplies for r ¸ D + rp . The thickness D is thenof the order of the mean free path k and withinthis inner region the simple kinetic theory ofgases is assumed to apply. Equation (3) repre-sents the general formula of the transition cor-rection factor by the � ux-matching theory:

b =1 + B1Kn

1 + B2Kn + B3Kn2 , (3)

where Kn [= k / rp] is the Knudsen number andk is the mean free path length of gas molecules.The coef� cients B1, B2, and B3 are givenfor some theories in Table 1. Kulmala (1993)used the correction factor suggested by Fuchsand Sutugin (1970). According to Monchickand Blackmore (1988), the model of Fuchs andSutugin (1970) seems to be almost universallyvalid for mass � ux in the transition regime. Thesimplest approach for the transition correction,the harmonic mean method, is like matching the� uxes at the particle surface, i.e., D = 0, as iseasily seen from the coef� cients in Table 1. Con-sequently, the harmonic mean method is not only

TABLE 1. Coef� cients for the enhancement functionf (Kn).

B1 B2 B3

Fuchs (1934) D 12k p

43

43 ¢ D 12

k p

Fuchs and Sutugin (1970) 1 43 + 0.377 4

3

Dahneke (1983) 23

43

43 ¢ 2

3

Harmonic mean 0 43 0

FIGURE 1. Comparison of the harmonic mean methodwith that of Fuchs and Sutugin (1970).

a mathematical tool but also a � ux-matchingtheory.

Among the possible approaches shown inTable 1, the harmonic mean method was foundto be the only method to provide a solution in anexplicit analytical form. Figure 1 compares theharmonic mean correction factor as a function ofthe Knudsen number with the method of Fuchsand Sutugin (1970). Figure 1 shows that the har-monic mean is in good agreement with Fuchsand Sutugin (1970) in the transition regime(maximum deviation 13.4% at Kn = 0.87).Thus, in this study, the harmonic mean methodis used for the transition correction.

For the harmonicmean method, the correctionfactors b m and b t are expressed as the followingequations:

b m =1

1 + 4 k v3rp

, (4)

b t =1

1 + 4 k g

3r p

, (5)

where k v and k g are the effective mean free pathsof the vapor molecules and the surroundinggas molecules, respectively (Fuchs and Sutugin1970).

By substituting Equations (4) and (5) intoEquation (2), setting the value of Sa to 1(neglecting the Kelvin effect), and writing theequation in terms of the particle volume, thesingle particle growth rate I (v ) is obtained as



follows:

I (v ) =dv

dt=

S ¡ 1

C1v ¡ 1/ 3 + C2v ¡ 2/ 3, (6)

where

C1 = 3 ¡ 1/ 3(4p ) ¡ 2/ 3

£µ

q RT

MDPs{1 + (S + 1)Ps / (2p)}

+q L2 M

RKT 2

¶

and

C2 =2

3

³2

9 p

´1/ 3

£µ

k v q RT

MDPs{1 + (S + 1)Ps / (2p)}

+k g q L2 M

RKT 2

¶.

Equation (6) is theoretically valid for theentire particle size range covering the free-molecule regime, the transition regime, and thecontinuum regime. Neglect of the Kelvin effect,however, may prohibit the application of Equa-tion (6) to very small particles.

By integrating Equation (6), one can obtainthe volume change of a single particle by con-densation as follows:

v =µ½ ³

C2

C1+ v1/ 3

o

´2

+2(S ¡ 1)t

3C1

¾ 1/ 2

¡C2

C1

¶3

,

(7)

where vo is the initial particle volume. Usingtheories other than the harmonic mean methodsuch as Fuchs (1934), Fuchs and Sutugin (1970),or Dahneke (1983), it was found tobe impossibleto obtain an integrated equation in explicit formsuch as Equation (7).

In Figure 2, Equation (7) is compared withthe analytical solution of Kulmala (1993), whichused the transition correction factor of Fuchsand Sutugin (1970) and included the Kelvin ef-fect. It is shown that a signi� cant discrepancy(more than 10%) exists between the results ofthis study and that of Kulmala (1993) for the ini-

FIGURE 2. Comparison of thevolumechange of a singleparticle used in this study with the solution of Kulmala(1993).

tial particle radius rpo = 0.05 l m. However, thesolution of Kulmala (1993) for this particle sizeunderestimated the growth time by about 10%compared to the exact numerical results; thusthe difference between the result of this studyand the exact numerical results is believed to bemuch smaller. For the initial particle radius rpo =0.5 l m, the discrepancy is shown to be verysmall (< 3%).

As a next step, let us consider condensationof a polydisperse aerosol with an initial size dis-tribution, no(vo), which becomes n(v ) as time telapses. In this case, particles with initial vol-umes vo and vo + dvo will grow to have volumesv and v + dv , respectively. In addition, if thereis no source or sink, the number concentrationof the particles whose volume lies between vo

and (vo + dvo), n(vo)dvo will not change duringcondensation. Thus one can write

no(vo)dvo = n(v )dv . (8)

By integrating Equation (6) with arbitrarygrowth rate I (v ), one obtains the followingequation:

t =Z v

vo

dv

I (v )= F(v ) ¡ F (vo), (9)

where F(v ) is an inde� nite integral of 1/I (v ).Equation (9) can be rearranged with respect tov as follows:

v = F ¡ 1[F(vo) + t], (10)

v + dv = F ¡ 1[F(vo + dvo) + t]. (11)

226 S. H. Park and K. W. Lee Aerosol Science and Technology33:3 September 2000

After eliminating v , we have

dv =@

@voF ¡ 1[F(vo) + t] ¢ dvo

=@v(vo, t )

@vo¢ dvo. (12)

Differentiating Equation (9) with respect to vo,we have

@v

@vo=

F 0 (vo)

F 0 (v )=

I (v )

I (v )o. (13)

After substituting Equation (13) into Equation(12) and using Equations (6), (7), and (8), onecan � nally obtain the following equation for thesize distribution change:

n(v ) = no(vo) ¢C1v ¡ 1/ 3 + C2v ¡ 2/ 3

C1v ¡ 1/ 3o + C2v

¡ 2/ 3o

, (14)

where vo is calculated from Equation (7) as thefollowing:

vo =µ½ ³

C2

C1+ v1/ 3

´2

¡2(S ¡ 1)t

3C1

¾ 1/ 2

¡C2

C1

¶3

.

(15)

The evolution of the size distribution ofan aerosol due to condensation is plotted inFigure 3. The initial size distribution of the aero-sol was chosen to be log-normal with the geo-metric mean radius rgo = 0.1 l m and the geo-metric standard deviation r o = 1.5.

FIGURE 3. Size distribution change of an aerosol dueto condensation.

CONCLUSIONSAn analytical study for the condensation prob-lem was performed. The condensational growthrate suggested by Kulmala (1993) was modi� edfor the purpose of this study. To obtain an ana-lytical solution for the particle size distributionchange, the Kelvin effect was neglected and theharmonic mean method was used for the transi-tion correction. The harmonic mean correctionfactor was shown to be in good agreement withthat of Fuchs and Sutugin (1970) with a maxi-mum error of 13.4%. The modi� ed growth rateshowed good agreement with that of Kulmala(1993) or exact numerical results.

Using the modi� ed growth rate, an analyticalsolution for the particle size distribution changeby condensation for the entire particle size rangewas derived. This study represents the � rst ana-lytical solution for the size distribution changeof a condensationally growing aerosol for theentire particle size range. Although the solu-tion may derive signi� cant errors for very smallparticles or for very high supersaturation con-ditions, the solution is believed to be useful invarious applications as it is presented in an ex-plicit analytical form.

ReferencesBarrett, J. C., and Clement, C. F. (1988).Growth Rates

for Liquid Drops, J. Aerosol Sci. 19:223–242.Dahneke, B. (1983). Simple Kinetic Theory of

Brownian Diffusion in Vapors and Aerosols. InTheory of Dispersed Multiphase Flow, edited byR. E. Meyer. Academic Press, New York, pp. 97–

133.Fuchs, N. A. (1934). Zur Theorie der Koagulation,

Z. Phys. Chemie 171:199–208.Fuchs, N. A. (1959). Evaporation and Droplet

Growth in Gaseous Media, Pergamon Press, Lon-don.

Fuchs, N. A., and Sutugin, A. G. (1970). Highly Dis-persed Aerosols. Ann Arbor Science Pub., Ann Ar-bor, MI.

Heidenreich, S. (1994). Condensational DropletGrowth in the Continuum Regime: A Critical Re-view for the System Air-Water, J. Aerosol Sci.25:49–59.

Kulmala, M. (1988). Condensation in the Contin-uum Regime: Integration of the Mass Flux, Report



Series in Aerosol Science 8, Finnish Associationfor Aerosol Research, Helsinki, pp. 2–5.

Kulmala, M. (1993). Condensational Growth andEvaporation in the Transition Regime: An Ana-lytic Expression, Aerosol Sci. Tech. 19:381–388.

Kulmala, M., Majerowicz, A., and Wagner, P. E.(1989). Condensational Growth at Large VaporConcentration: Limits of Applicability of theMason Equation, J. Aerosol Sci. 20:1023–1026.

Kulmala, M., and Vesala, T. (1991). Condensation inthe Continuum Regime, J. Aerosol Sci. 22:337–

346.Mason, B. J. (1971). The Physics of Clouds, 2nd Edi-

tion, Clarendon Press, Oxford.Maxwell, J. C. (1877). Diffusion. In Encyclopedia

Britannica, Vol. 2, p. 82; reprinted in The Scienti� cPapers of James Clerk Maxwell, Vol. 2, edited byW. D. Niven. Cambridge University Press, Cam-bridge, 1890, p. 625.

Monchick, L., and Blackmore, R. (1988). A VariationCalculation of the Rate of Evaporation of SmallDroplets, J. Aerosol Sci. 19:273–286.

Pratsinis, S. E., Kodas, T. T., Dudukovic , M. P., andFriedlander, S. K. (1986).Aerosol Reactor Design:Effect of Reactor Type and Process Parameters onProduct Aerosol Characteristics, Ind. Eng. Chem.Process Des. Dev. 25:634–642.

Seinfeld, J. H. (1986). Atmospheric Chemistry andPhysics of Air Pollution, Wiley, New York,pp. 416–422.

Wagner, P. E. (1982). Aerosol Growth by Conden-sation, Aerosol Microphysics II, edited by W. H.Marlow. Springer, Berlin, pp. 129–178.

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NOMENCLATUREB1, B2, B3 constants appearing in Equation (3)

[–];C1 constant appearing in Equation (6)

[s/m2];C2 constant appearing in Equation (6)

[s/m];D diffusion coef� cient of condensing

vapor [m2/s];F inde� nite integral of 1/ I appearing

in Equation (9) [s];

I single particle growth rate by con-densation [m3/s];

K thermal conductivity of the sur-rounding gas phase [W/m/K];

Kn Knudsen number [–];L latent heat of vaporization [J/kg];

M molecular mass of condensing va-por [kg/mole];

N total number concentration of par-ticles [particles/m3];

n particle size distribution densityfunction [particles/m3/m3];

p total gas pressure [N/m2];Ps saturation vapor pressure of con-

densing vapor [N/m2];R gas constant [J/mole/K];rp particle radius [m];rg geometric mean particle radius

[m];S saturation ratio [–];

Sa saturation ratio at the particle sur-face representing the Kelvin effect[–];

t time [s];T absolute temperature [K];v particle volume [m3];

vg geometric mean particle volume[m3];

b transition correction factor [–];D thickness of the shell within which

the simple kinetic theory of gasesapplies [m];

k mean free path of gas molecules[m];

q particle density [kg/m3];r geometric standard deviation based

on particle radius [–].

Subscript

o refers to initial condition.

Received March 19, 1999; accepted July 15, 1999.

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Condensational Growth of Polydisperse Aerosol for the Entire Particle Size Range