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ournal of Colloid and Interface Science248,295305 (2002)

oi:10.1006/jcis.2002.8212, available online at http://www.idealibrary.com on

The Self-Preserving Size Distribution Theory

I. Effects of the Knudsen Number on Aerosol Agglomerate Growth

Petrus J. Dekkers,1 and Sheldon K. Friedlander

Department of Chemical Engineering, University of Technology Delft, The Netherlands; andDepartment of Engineering,

University of California Los Angeles, Los Angeles, California

Received April 4, 2001; accepted January 2, 2002

Gas-phase synthesis of fine solid particles leads to fractal-like

ructures whose transport and light scattering properties differ

om those of their spherical counterparts. Self-preserving size dis-

ibution theory provides a useful methodology for analyzing the

symptotic behavior of such systems. Apparent inconsistencies in

revious treatments of the self-preserving size distributions in the

ee molecule regime are resolved. Integro-differential equations

or fractal-like particles in the continuum and near continuum

egimes are derived and used to calculate the self-preserving and

uasi-self-preserving size distributions for agglomerates formed by

rownian coagulation. The results for the limiting case (the con-

nuum regime) were compared with the results of other authors.

or these cases the finite difference method was in good in agree-

ment with previous calculations in the continuum regime. A new

nalysis of aerosol agglomeration for the entire Knudsen number

ange was developed and compared with a monodisperse model;

Higher agglomeration rates were found for lower fractal dimen-ons, as expected from previous studies. Effects of fractal dimen-

on, pressure, volume loading and temperature on agglomerate

rowth were investigated. The agglomeration rate can be reduced

y decreasing volumetric loading or by increasing the pressure. In

aminar flow, an increase in pressure can be used to control par-

cle growth and polydispersity. For Df= 2, an increase in pres-

ure from 1 to 4 bar reduces the collision radius by about 30%.

arying the temperature has a much smaller effect on agglomerate

oagulation. C 2002 Elsevier Science (USA)

Key Words: self-preserving size distribution; model; fractal;

actal-like dimension; agglomerate; aerosol production; Knudsen

umber.

INTRODUCTION

Most industrial aerosol processes are carried out at high tem-

eratureswith high particle volumetric loadings andtend to yield

ractal-like or power law(both terms are used interchangeably in

his paper) agglomerates composed of a large number of primary

spherical) particles. The dynamic behavior of these agglomer-

1 To whom correspondence should be addressed at DSM Research, CT&A/PT, P.O. Box 18, 6160 MD Geleen, The Netherlands.

ates is quite different from their spherical counterparts; collis

rates are higher and the resulting distribution functions are d

ferent (1, 2).

In some cases, aerosol agglomerates can be described

a power law relationship with an exponent, which is ually referred to as the fractal or Hausdorff dimension Df. F

agglomerates suspended in a gas this fractal dimension is l

than the Euclidean dimension of three (3, 4). Clustercluster

glomeration studies yield fractal dimensions ranging from

to 2.1. Mountainet al.(5) simulated coagulation in both the f

molecule and the continuum size regime and found values of

ranging between 1.7 and 1.9. By solving the Langevin equati

Mulhollandet al.(6) found values ofDfranging from 1.89

2.07.

The value of the fractal dimension has significant influence

the dynamics and the size distribution of a coagulating aerosFor spherical particles, it has been shown that a similarity tra

formationof theparticlesize distribution leads to a so-calledse

preserving form, reached after sufficiently long times (7). T

attainment of these self-preserving size distributions (SPSD

in uniform systems and local SPSDs in non-uniform system

very fast and can be on the order of microseconds in the c

of the high initial number concentrations that occur soon af

the chemical reactions in aerosol reactors. Numerical simu

tions were carried out to calculate the form of the transform

particle size distribution for both the continuum regime (8) a

the free molecule regime (9). Small deviations were found

other studies in the free molecule regime (10, 11), and for continuum regime(11,12).Studies have also been made of qua

self-preserving size distributions for spherical particles, wh

change slowly in time in the near continuum transition regi

(13), and in the continuum regime including simultaneous co

densation (14).

More recent studies have shown that SPSDs exist also

fractal-like particles. Distributions for the free molecule regim

derived by van Dongen and Ernst (15) using the Langevin eq

tion were fitted by self preserving variables (6). A Monte Ca

method was used to calculate SPSDs in the free molecule regi

for different values ofDf(2, 15). SPSDs for fractal-like particin both the free molecule and the continuum regimes w

295 0021-9797/02 $35

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96 DEKKERS AND FRIEDLANDER

alculated by solving the complete population balance equation

sing a sectional method (11). The sectional method showed

ome inconsistencies with a previous Monte Carlo simulation

f Wu and Friedlander (2).

These inconsistenciesin the free molecule regime are resolved

n this paper. New calculations were made for both the conti-

uum and the near continuum transition regimes. The ordinary

ntegro-differential equations for the self-preserving size distri-utions for agglomerates in the free molecule and the continuum

egime are derived, as well as the quasi-self-preserving size dis-

ibution (changes slowly in time) for the near continuum tran-

tion regime (0.01 < Kn < 1). The self-preserving and quasi-

elf-preserving distributions, for the continuum and the near

ontinuum transition regimes, respectively, were calculated by

umerical integration of the ordinary integro-differential equa-

ons, using a technique previously employed for spherical par-

cles (8). Furthermore, equations are derived for the decay in

gglomerate number in the continuum and near-continuum tran-

tion regimes.

A harmonic mean approach is used to couplethe free molecule

xpression with the near-continuum transition expression, ex-

nding the latter into the continuum regime. Calculations with

his model are compared to the single free molecular expres-

on (16) and a monodisperse model (17). Also discussed are

he effects on agglomerate growth of the process conditions in a

article production system, including temperature, particle volu-

metric loading, and pressure. Although most aerosol production

carried out under turbulent conditions (18), much research is

urrently done in laminarflow reactors. The analysis described

bove can be used to find sets of optimal conditions, especially

egarding particle size and polydispersity, for the design of lam-nar aerosol production systems with nearly uniform velocity

rofiles.

THEORY, THE FUNDAMENTAL EQUATIONS

oagulation Equations

Earlier aerosol dynamics models were based on the assump-

on that colliding particles coalesce instantaneously. The classi-

al equation in the aerosol theory for coagulation with instanta-

eous coalescence, the Smoluchowski or coagulation equation,

an be written as follows (19):

n

t=

1

2

v0

(v, v v)n(v)n(v v) dv

0

(v, v)n(v)n(v) dv. [1]

The left-hand side describes the change in number concen-

ation n between volume element v and v + dv at any time

The first term on the right-hand side (RHS) of the equationthe gain by the collision of smaller particles and the second

term accounts for the loss of particles in the size range. For

of practical interest, the resulting integro-differential equatio

cannot usually be solved analytically.

Methods of solving this equation are summarized by Willia

and Loyalka (20). These methods range from the discrete (co

putationally intensive) and sectional models (21) in wh

Eq. [1] is transformed into a number of differential equatio

to the less accurate monodisperse models. Approximate sotions can be found using the method of moments (22, 23). Ex

solutionsof Eq. [1] for asymptotic limiting cases can be obtain

with the self-preserving theory (19).

The assumption of instantaneouscoalescence usually doesn

hold in the high-temperature aerosol-production systems wh

the colliding particles either keep their original shape or coale

with afinite rate. The coagulation equation can be extended

take into account the effect offinite sintering rates (24). Sim

taneous sintering during coagulation of the aerosol complica

solving the extended equation. The equation for agglomer

growth reduces again to Eq. [1] if sintering is completely om

ted. Several researchers have used this approach to calculate

effect of agglomerate structure on agglomeration (2, 11, 16, 2

Hence it becomes possible to calculate the self-preserving s

distributions through the different methods mentioned in the

troduction. The same approach is used here for the derivatio

and calculations. It is also possible to extend the self-preservi

fractal growth dynamics for finite sintering using an additio

differential equation as has been shown for the free molecu

regime (26, 27).

In order to derive and calculate the SPSDs it is necessary

introduce the following set of equations. The rate of the to

particle number decay (dN/dt) is affected only by collision afollows after integration of Eq. [1] over all particle sizes:

dN

dt=

1

2

0

0

(vi , vj)n(vi , t)n(vj, t) dvidvj.

If the collision kernel is a homogeneous function ofvi a

vjof some degree h , Eq. [1] can be converted into an ordina

integro-differential equation by introducing the self-preservi

dimensionless variables (19),

() =n(v)v

N, =

v

v, with v =

N.

where is a dimensionless particle volume and () is a

mensionless number distribution function and v is the numb

average agglomerate volume with being the volumetric pa

cle loading. It follows from the definition of the dimensionl

variables that the 0th and the 1st moment are equal to 1, wh

thekth moment of a distribution is defined by

k=

0

k

() d.

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ON THE SELF PRESERVING SIZE DISTRIBUTION THEORY. I 2

An expression frequently used to describe an agglomerate of

adiusrcontaining Np primary particles of radiusrp is (16)

Np =v

vp= A

r

rp

Df. [5]

he dimensionless constant A depends on the definition of the

gglomerate radius and is assumed to be 1 in this paper (28). The

alue ofAcan, however, be adjusted in the equations whenever

more accurate estimate ofAis known.

The quantitative comparison of the self-preserving size dis-

ibutions is made using number-based and volume-based GSDs

geometric standard deviations). The number-based GSD is de-

ned by

gn = exp

VAR

ln

1/3

, [6]

with

VAR

ln

1/3=

0

ln

1/3 ln

1/3gn

2 () d, [7]

nd

ln

1/3gn=

0

ln

1/3

() d. [8]

Equation [6] gives the (densified spherical equivalent diame-

er) number-based GSD (the same definition was also used by

Vemury and Pratsinis (11)). To calculate the volume-based GSDne replaces the self-preserving number distribution, i.e., ()

with a self-preserving volume distribution, the product of and

().

ree Molecule Regime (Kn 1)

The relations for the free molecular regime are only sum-

marized here. Particles are regarded to be in the free molecule

egime if the Knudsen number Kn (/r) is larger than 10. The

ollowing collision kernel can be derived from the kinetic theory

f gases for rigid spheres and after incorporation of Eq. [5],

fm(vi , vj) =

6kbT

p

1/2 3

4

h1

vi+

1

vj

1/2

v1/Dfi + v

1/Dfj

2, [9]

wherevi andvjare the volumes of the colliding particles, pis

he particle density, andh (the degree of homogeneity) is

h =2

Df

1

2. [10]

he above equation holds best for Df 2 (6, 25). Rogak andlagan (25) estimated a collision kernel for the full range of the

fractal dimension and used this equation for sectional calcu

tions. Calculations in this paper were restricted to Df 2

the free molecule regime.

Substituting the collision kernel (Eq. [9]) in Eq. [2] with

transformation Eq. [3], equations have been derived for the p

ticle number decay and particle growth in the free molec

regime (16). The total particle number decay is given by

dN

dt=

1

2chNh2. [

The dimensionless collision integral is defined by

=

0

0

1

+

1

1/21/Df+ 1/Df

2 ()() d d. [

The value of this integral is a function of the self-preserv

size distribution and the fractal dimension.c is a constant (

constant temperatureTand in absence of sintering) and is givby the following equation:

c =

6kbT

p

1/2 3

4

hr26/Dfp . [

It can be shown that after combining Eqs. [1], [2], [3], and [9] a

rearranging, the following ordinary integro-differential equat

results:

2 ()+

d

d+

0

1/Df

+ ( )1/Df2

1

+

1

1/2() ( ) d 2 ()

0

1/Df+ 1/Df

21+

1

1/2 () d= 0. [

Solution of this equation for different values ofDfshould yi

the curves obtained by the sectional method and the Monte Ca

simulation, as discussed below, but was not carried out in tpaper.

Continuum Regime (Kn 1)

Equations for the number decay and the growth of fract

like particles in the continuum regime can be derived using

approach for spherical particles (19). The following collisi

kernel is applied for agglomerates in the continuum regim

and is assumed to hold between Df= 1 and 3.0 (1):

c(vi , vj) =

2kbT

3 1

v1/Dfi

+

1

v1/Dfj

v

1/Dfi + v

1/Dfj

. [

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pplying this collision kernel in the expression for the total

umber decay expression (Eq. [2]) together with Eqs. [3] and

4] gives

dN

dt=

2kbT

3

1+ 1/Df1/Df

N2. [16]

ntegration gives

N(t)

N0=

2kb T

3

1+ 1/Df1/Df

N0t+ 1

1, [17]

hereN0is the initial particle number concentration. Since v =

/N, the number average particle volume is given by

v(t) =2kbT

3

1+ 1/Df1/Df

t+ v0. [18]

ombining Eqs. [1], [2], [3], and [15] and applying Eq. [4]

ives after some rearrangement the following ordinary integro-

ifferential equation:

1+ 1/Df1/Df

d

d

+

21/Df1/Df 1/Df1/Df 1/Df

1/Df

()

+

0

() ( )

1+

1/Dfd = 0. [19]

n order to solve this equation by means of a finite differencemethod, a number of initial values at the lower end of the curve

must be generated. An analytical form for the lower end of the

urve is obtained by assuming that the integral term in Eq. [19]

small compared to and the derivative of with respect

o and thus can be neglected. The following solution is then

btained for the lower end of the curve;

() = Cc

exp

Df1/Df1/DfDf1/Df

1/Df

1+1/Df1/Df

21/Df1/Df

1+1/Df1/Df

. [20]

his function can be used to generate initial values for the nume-

cal integration. After the complete curve is calculated one can

heck the assumption made regarding the integral. For Df= 3

oth Eq. [19] and Eq. [20] reduce to the original equations as

an be expected (see Refs. 7 and 8).

ear-Continuum Transition Regime (0.01

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espect to and thus can be neglected. Hence after integration

f Eq. [24] the following solution is obtained for the lower end

f the curve:

()

=Cnct

exp

Df(1/Df+2/Df)

1/DfDf(1/Df+)1/Df

Df2

1/Df2/Df

1+1/Df1/Df+(1/Df+1/Df2/Df) 2

1/Df1/Df

+(1/Df

+21/Df

2/Df

)

1+1/Df1/Df

+(1/Df+1/Df

2/Df)

[25]

As in the continuum regime, both Eq. [24] and Eq. [25] reduce to

he original equations forDf= 3 derived by Wang and Friedlan-

er (13). For Df= 3 (spherical particles) they emphasized that

he special requirement for the change in mean free path during

oagulation can be relaxed. The same approach is followed here

or fractal-like particles. If the mean free path stays constant in

he coagulation process it can be shown that changes as

= 01/2Df. [26]

0 is the initial value for and is a dimensionless coagu-

ation time defined as (N/N0)2. In this case the more general

ansformation instead of the similarity transformation yields

1+ 1/Df1/Df+

1/Df2/Df+1/Df

d

d+ 2

d

d

+

21/Df1/Df+

21/Df2/Df+ 1/Df

1/Df+2/Df1/Df 1/Df+1/Df1/Df

2/Df

(, )

+

0

(, )( , )

1+

1/Df

+

1

1/Df+

2

1/Dfd = 0. [27]

he dimensionless number concentration is now a function

f both the dimensionless volumeand the coagulation time.

his equation is a partial integro-differential equation for which

general solution is difficult to obtain. But if the derivativeerm is suppressed, the equation reduces to Eq. [24], and this

quation can be solved for any fixed value of . Wang and

riedlander (13) checked this assumption, for Df= 3, by com-

aring(/) with

2

= 2

d

d. [28]

hey showed that thederivative term can be neglected for the

ower and upper end of the spectrum. In this paper the assump-

on was checked in a similar way and holds for the fractal-likearticles.

At the maximum of the size distribution function the te

containing the and thederivatives in Eq. [24] are negligi

compared to the two other terms. Thus Eq. [27] can be reduced

Eq. [24] and therefore the solutions to Eq. [24] can be regard

as quasi-self-preserving size distributions.

RESULTS OF THE CALCULATIONS

Results for the Free Molecule Regime

The self-preserving size distributions in the free molec

regime for several fractal dimensions are shown in Fig. 1. T

first SPSD in the free molecule regime forDf= 3 (solid squa

in Fig. 1) was calculated by solving Eq. [14] for the lower end

the distribution while the upper end of the distribution was fou

with a discrete set of differential equations (9). Several ot

calculations for the SPSD for Df= 3 were made. Mulholla

et al. (6) fitted data from several Monte Carlo simulations

equations given by van Dongen and Ernst (15). Their solut

coincides with the original solution of Lai et al. (9) and witdiscrete solution of Graham and Robinson (10).

Grahamand Robinson (10) also normalized the distribution

Laiet al.(9), using two requirements:0 = 1 and1 = 1. T

distribution coincides with the curves of Wu and Friedlander

and Vemury and Pratsinis (11).

The higher curves are probably more exact (of which two

shown in Fig. 1), for the following reasons: Earlier calculatio

(9) were made with a large step size to decrease calculation tim

The 0th moment was also off by about 2% from the de finit

(0 = 1). After normalization by Graham and Robinson (1

the curve from (9) matches with the Monte Carlo simulation and the sectional method (11). The discrete method of Grah

FIG. 1. SPSDs in the free molecule regime for different fractal dimensi

The solid squares represent the earliest results of Lai et al.(9). The lines sh

the Monte Carlo simulations of Wu and Friedlander (2), the open symbols

the results from the sectional method of Vemury and Pratsinis (11). The das

line represents the results of the Monte Carlo simulation of Mulholland e

forDf= 1.9 (6). The Monte Carlo simulations in (2) and the sectional met

(11) are in good agreement. The curve of (6) is relatively low. The resultGraham and Robinson (10) and of Mulholland et al.(6) are omitted for clar

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FIG. 4. Quasi-SPSDs for Df= 2.2 and different values of in the near

ontinuum transition regime obtained with Eq. [24] using the finite difference

ethod. The distribution becomes more narrow for higher values of.

FIG. 5. Quasi-SPSDs for Df= 1.8 and different values of in the near

ontinuum transition regime obtained with Eq. [24] using the finite difference

ethod. The distribution becomes more narrow for higher values of.

FIG. 6. GSDfor different fractal dimensions as a functionof theKn number.

he lines with open symbols represent the number-based GSDs and the lines

ith solid symbols represent the volume-based GSDs. The most narrow size

stributions are found in the near continuum transition regime whereas the freeolecule regime shows the broadest distributions.

TABLE 1

GSDs for Different Calculations of the Self-Preserving Size

Distributions in the Near Continuum Regime ( = 1.257 Kn

Df gn gv Method Refere

3 0 1.44 1.29 OIDE 8

3 0.050 1.43 1.29 OIDE 13

3 0.275 1.39 1.28 OIDE 13

3 0.495 1.37 1.27 OIDE 13

3 0 1.44 1.29 SM 32

3 0.050 1.43 1.29 SM 32

3 0.275 1.39 1.28 SM 32

3 0.495 1.36 1.27 SM 32

3 1.257 1.34 1.27 SM 32

3 0 1.445 1.303 OIDE This w

3 0.050 1.429 1.300 OIDE This w

3 0.275 1.391 1.290 OIDE This w

3 0.495 1.373 1.285 OIDE This w

3 1.257 1.346 1.272 OIDE This w

2.2 0 1.415 1.300 OIDE This w

2.2 0.050 1.398 1.299 OIDE This w2.2 0.275 1.359 1.287 OIDE This w

2.2 0.495 1.340 1.278 OIDE This w

2.2 1.257 1.315 1.265 OIDE This w

2.0 0 1.405 1.300 OIDE This w

2.0 0.050 1.388 1.298 OIDE This w

2.0 0.275 1.348 1.281 OIDE This w

2.0 0.495 1.330 1.274 OIDE This w

2.0 1.257 1.306 1.263 OIDE This w

1.8 0 1.395 1.299 OIDE This w

1.8 0.050 1.375 1.291 OIDE This w

1.8 0.275 1.337 1.280 OIDE This w

1.8 0.495 1.320 1.272 OIDE This w

1.8 1.257 1.296 1.260 OIDE This w

Note.OIDEs, ordinary integro-differential equation; SM, sectional meth

of Kn between 1 and 10 still cannot be described with the se

preserving theory. Currently only interpolation equations for

collision kernel exist for this range and these do not allow

simple transformations of Eq. [1]. (See for instance the Fuc

interpolation equation (33) or the interpolation equation of D

neke (34).)Figure 6 shows that narrow size distributions arem

likely in the near-continuum transition regime, whereas the f

molecule regime shows the broadest distribution.The behavior in the transition region may also explain the h

value ofDf= 2.5 of Matsoukas and Friedlander (16), found

fitting experimental data on self-preserving size distributions

agglomerates. The self-preserving size distributions for low

fractal dimensions (from 1.7 to 2.1 which are more accepted

the literature) in the free molecule regime were too broad fo

goodfit of their data. Although the Knudsen numbers for th

experiments were between 1 and 10 (corresponding to the n

free molecule transition region between the free molecule a

near continuum transition regime) the curves in Fig. 6 sugg

that the GSD decreases in the transition region. This can be complished by more narrow distribution functions in this regi

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TABLE 2

Collision Integral as a Function of the Fractal Dimension,

Taken from Wu and Friedlander (2), Necessary in Eq. [29]

actal dimension, Df 3.0 2.8 2.5 2.2 2.0

ollision integral 6.552 6.560 6.607 6.748 7.037

-Eq. [9]

hus it seems likely that the actual value ofDffor the agglom-

rates is less than 2.5.

nterpolation Equation for the Particle Collision Rate

over All Size Ranges

The expression derived for the particle number decay in the

ear continuum transition regime (which extends into the con-

nuum regime) differs significantly from the expression for the

ee molecule regime. To express the results in one equation, the

armonic mean is taken of Eqs. [11] and [23]:

dN

dt=

dN

dt

1fm

+

dN

dt

1nct

1. [29]

The resulting interpolation formula yields the original expres-

ons in the limits. Furthermore, if approaches zero Eq. [23]

educes to the particle number decay equation in the continuum

egime (Eq. [16]). Equation [29] was solved using a fourth-

rder RungeKutta integration method. The values of the inte-

ral (Eq. [12]) in the free molecule expression (Eq. [11]) were

dapted from (2) and can be found in Table 2. The values of

1/Df, 1/Df, and2/Df, in Eq. [23], which are functions of

n or are found in Table 3 and were approximated with auadratic interpolation equation for Kn = 0.01 to 1. The mo-

ments were held constant for Kn smaller than 0.01 and larger

han 1.

TABLE 3

The Moments of the Size Distributions in the Continuum and

Near Continuum Transition Regime, Necessary in Eq. [29]

0 0.050 0.275 0.495 1.257

f= 3

1/Df 0.905 0.908 0.918 0.922 0.9291/Df 1.266 1.250 1.214 1.199 1.176

2/Df 1.905 1.832 1.677 1.614 1.527

f= 2.2

1/Df 0.902 0.906 0.916 0.922 0.930

1/Df 1.388 1.360 1.299 1.272 1.236

2/Df 2.544 2.377 2.058 1.934 1.778

f= 2.0

1/Df 0.905 0.907 0.919 0.924 0.932

1/Df 1.434 1.404 1.328 1.299 1.259

2/Df 2.828 2.619 2.207 2.064 1.885

f= 1.8

1/Df 0.902 0.913 0.923 0.928 0.935

1/Df

1.388 1.451 1.368 1.333 1.287

2/Df 2.544 2.911 2.414 2.234 2.018

FIG. 7. Particle number density decay as a function of time. Three mo

are compared, Eq. [29], a monodisperse model (17) (not shown in thisfigure

reasons of clarity), and the single free molecule expression Eq. [11]. The so

lines are the results forDf= 3,the dashed lines are the results for Df= 2. T

SPHM lines are calculated with the self-preserving harmonic mean model,

SPfm lines are calculated with the self-preserving free molecule model. N

the crossing of theharmonic mean curves forDf= 2 andDf= 3. Results are

rp = 20 nm, T= 1500 K, = 6.4 107, p = 2000 kg/m3, = 350

= 6 105 kg/m/s, and p = 1 bar.

The results of this calculation were compared with the s

gle free molecule expression (Eq. [11]) and with the monod

perse calculation of Kruiset al.(17), in the absence of sinteri

Figures 7 and 8 show that the free molecule expression overp

dicts the growth rate after a very short time (roughly after t

0.001 s). ForDf= 3, the monodisperse model is in good agr

ment with the harmonic mean expression (see Fig. 8). Howev

for Df= 2 the harmonic mean expression predicts a collisi

radius which is roughly 25% higher than the one calculated

FIG. 8. Agglomerate collision radius as a function of time. Three mo

are compared, calculation from Eq. [29], a monodisperse model of (17)

absence of sintering), and calculation from the single free molecule express

Eq.[11]. The solid linesare theresultsforDf= 3,the dashedlines arethe res

for Df= 2. TheSPHM lines are calculated with the self-preserving harmo

meanmodel,the SPfm lines are calculated withthe self-preserving free molec

model and the MD lines are calculated with the monodisperse model. The f

molecule expression is overpredicting the growth rate after a very short ti

Results are forr0 = 20 nm, T= 1500 K, = 6.4 107,p = 2000 kg/ = 350 nm, = 6 105 kg/m/s, and p = 1 bar.

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ressure increase results from an increase of the partial pressure

f an inert gas, with the particle volume loading held constant.

The effect of increasing pressure forDf= 3 is less severe and

noticed only after a longer time; for example, a 20% decrease

found after 100 s.

A change in volume loading does not affect viscosity (under

ilute conditions) or mean free path. The collision rate depends

n particle concentration. If the volume loading is increased byfactor of 10, the initial particle concentration is also higher by

factor of 10. Therefore, the change in volume loading affects

he growth rate significantly. If the particle volume loading is

ecreased by a factor of 10, the collision radius is decreased

y about 60% after 1 s (Fig. 10a, Df= 2). The dependence of

he collision radius on the volume loading for Df= 3 is again

elayed and less substantial, as also was found with the pressure

ependence.

FIG. 10. Agglomerate growth for different volumetric loadings. (a) Particle

ollision radius for Df= 2; (b) particle collision radius for Df= 3. In each

gure, the dashed line (right axis) shows the particle collision radius under

andard conditions, and the solid lines (left axis) represent the particle collision

dii of nonstandard conditions divided by the particle collision radius under

andard conditions. Volumetric loading has a large influence on collision rate.

esults for rp = 5nm,p = 1 bar, T= 1500 K, p = 2000 kg/m3, s = 350nm,nds = 6 10

5 kg/m/s.

FIG. 11. Agglomerate growth for different temperatures. (a) Particle co

sion radius for Df= 2; (b) particle collision radius for Df= 3. In eachfig

the dashed line (right axis) shows the particle collision radius under stand

conditions, and the solid lines (left axis) represent the particle collision radi

nonstandard conditions divided by the particle collision radius under stand

conditions. Temperature has a relatively small influence on collision rate.

sults for rp = 5 nm, p = 1 bar, = 1 108, p = 2000 kg/m

3, s = 350

ands = 6 105 kg/m/s (s and sfor use in Eq. [30]).

The influence of temperature on coagulation was a

checked. Several changes take place when the temperature

changed, some have opposite effects. The kinetic energies

the particles, the mean free path, and the viscosity of the gchange. The effects of a change in the temperature are small

temperature increase of 500 K changes the collision radius

about 15% (see Fig. 11a, Df= 2). Again the effect on parti

growth is less for Df= 3 (see Fig. 11b).

The temperature significantly affects the sintering rate a

particle morphology, which was not taken into account in the

calculations. Particle coalescence is enhanced for higher te

peratures so increasing temperature could in fact decrease t

collision radius. The rate of coalescence strongly depends on

sintering mechanism and its temperature dependence. Theref

it can be concluded that changes in particle growth at other teperatures are accounted for by sintering and not by coagulatio

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SUMMARY

Monte Carlo simulations used to calculate the self-preserving

ize distributions in the free molecule (2) and the continuum

egime (29) show excellent agreement with a sectional method

f (11). A numerical integration procedure gave the same dis-

ibution for the continuum regime. The curves for Df= 3 in

oth regimes were slightly higher than the results of earlier cal-

ulations (8, 9). The quasi-self-preserving curves for the near-

ontinuum transition regime withDf= 3 were about 5% higher

han earlier results (13).

In the free molecule regime the self-preserving distributions

roaden at the lower end as the fractal dimension decreases.

n the continuum regime the opposite effect was found; i.e.,

he distribution becomes more narrow at the lower end of the

istribution with decreasing Df. The qualitative effect of the

Knudsen number on the quasi-self-preserving size distributions

n the near-continuum transition regime is similar for all fractal

imensions; that is, the quasi-self-preserving size distributions

roaden as the Knudsen number decreases. For a fixed Df theistributions are therefore the narrowest in the near continuum

ansition regime, which is of interest for the design of aerosol

eactors.

Equations for particle number decay were derived and com-

ined by harmonic averaging into a single equation for the whole

article size distribution. Calculations with the harmonic mean

ormula covering the complete size distribution of an aerosol

howed a faster number decay and growth rate than the monodis-

erse aerosol model, especially for low Df. This effect is in

ualitative agreement with more accurate models which predict

aster growth rates for polydisperse systems (36). The SPSDree molecule expression (Eqs. [11]) overestimates the growth

gnificantly after a very short time if high initial numberconcen-

ations are used and therefore cannot be used solely for longer

mes and high particle concentrations.

Studies were made of the effects of temperature, volume load-

ng, and pressure over ranges likely to be of interest for aerosol

roduction. The effect of changing the parameters was more sig-

ificant forDf= 2 thanDf= 3. The temperature is not effective

or controlling growth, because the temperature is usually de-

ermined by the chemical properties of the system, i.e., vapor

ressure of the reactants and chemical reaction rate for instance.

While the most common method of controlling particle growth

through the volumetric loading, the pressure has a significant

ffect on both particle growth and polydispersity.

ACKNOWLEDGMENTS

We acknowledge support from the U.S. National Science Foundation from

rant #CTS9527999. S.K. Friedlander is Parsons Professor of Chemical Engi-

eering.

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