coagulation–fragmentation kinetics: equilibrium weight distribution of aggregates in flowing...

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Theoretical Foundations of Chemical Engineering, Vol. 35, No. 5, 2001, pp. 440–446.Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 35, No. 5, 2001, pp. 465–471.Original Russian Text Copyright © 2001 by Dolgonosov.

Fine-particle solids, which are used in various pro-cesses (production of catalysts and photomaterials,coagulation purification of liquids, etc.), are oftenobtained by precipitation from highly supersaturatedsolutions that is accompanied by the coagulationgrowth of the particles. The resulting disperse systemsoften have the following properties: a low concentra-tion of the dispersed phase (a solid volume fraction of

10

–5

–10

–3

); a wide size distribution of coagulationaggregates (each containing from several to

10

6

or moreprimary particles); and a loose, disordered, fractal-likestructure of aggregates, which have a low strength [1–3].Under certain conditions, the dispersed phase in suchsystems may fall into two fractions [4], one consistingof large aggregates of 10

4

to 10

6

particles (the coarsefraction), and the other consisting of small aggregatesand unaggregated particles (the fine fraction). In a fastflow, the large aggregates, which are unstable, readilychange their shape and break as they collide against oneanother and against the solid walls. In a slow flow, theaggregates retain their integrity despite the continuousdetachment of fragments from their surfaces. In avelocity-gradient flow, the growth of aggregates isaccompanied by an increase in the hydrodynamic stresson their surfaces; this limits the aggregate size by somevalue

R

0

, which depends on the shear rate [5, 6]. In sucha flow, the much larger aggregates will break into com-parable parts no larger than

R

0

, and the aggregates thatare a little larger than

R

0

diminish to the limiting size byfragment detachment. Actually, because the particleattachment and detachment are statistical processes, theaggregate-size distribution function does not terminateabruptly at

R

0

but extends beyond this value. Thus, it isappropriate to regard

R

0

as the equilibrium, rather thanthe limiting, aggregate size, since there is a weight bal-ance for an aggregate of this size: the weight incrementdue to particle attachment is equal to the weight decre-ment due to fragment detachment. The existence of the

equilibrium aggregate size has two consequences. Thefirst is that colliding large aggregates can coalesce onlyif the size of the forming aggregate does not exceed

R

0

(or exceeds it insignificantly). If

R

0

is substantiallyexceeded, the strong hydrodynamic stresses prevent theaggregates from coalescing by breaking those fewbonds between the aggregates that have resulted fromcollision. The second consequence relates to the varia-tions in the particle attachment and detachment fre-quencies during the aggregate growth: obviously, as

R

increases, the fragment detachment frequency growsmore rapidly than the particle attachment frequency, isequal to it at

R

=

R

0

, and exceeds it at

R

>

R

0

. Understeady-state external conditions and at a sufficient resi-dence time of the suspension in the apparatus, an equi-librium particle-size distribution is established: the sizeof the large aggregates falls within a small vicinity of

R

0

, and the small aggregates are the parts from whichlarge aggregates are formed and which have detachedfrom large aggregates under the action of the hydrody-namic stress. It is important to study this distributionand to determine the conditions for the formation of abimodal distribution pattern.

The equilibrium particle-size distribution in a sys-tem where coagulation and fragmentation occur wasconsidered previously [7, 8]. The aggregate breakdownwas regarded as either a sequential detachment of sin-gle particles from the aggregate or an instantaneousmultiple fragmentation. The dependence of the detach-ment frequency on the sizes of the aggregate anddetached fragment was not analyzed.

In this work, we examine a low-concentration dis-perse system in a shear flow. We analyze the coagula-tion–fragmentation kinetics, assuming that the particleattachment and detachment frequencies are determinedby the aggregate size and that the detachment probabil-ity depends on the size of the detached fragment.

Coagulation–Fragmentation Kinetics: Equilibrium Weight Distribution

of Aggregates in Flowing Suspensions

B. M. Dolgonosov

Institute of Water Problems, Russian Academy of Sciences, ul. Gubkina 3, GSP-1, Moscow, 117735 Russia

Received May 17, 2000

Abstract

—Kinetic equations of the aggregation and fragmentation of particles in a coagulating suspension arereported. When the mechanisms of coagulation growth of small and large particles greatly differ (the Brownianand gradient mechanisms), the dispersed phase falls into two fractions. The equilibrium weight distribution ofaggregates in this system is derived.

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING

Vol. 35

No. 5

2001

COAGULATION–FRAGMENTATION KINETICS 441

COAGULATION–FRAGMENTATION KINETICS

Basic principles.

The formation of a fine-particleamorphous phase, which is initiated by adding a precip-itant to the solution, typically proceeds as follows. Ini-tially, a rapid chemical reaction occurs to yield primaryparticles of a new phase, which are characterized bysome size distribution. Next, these particles begin tocoagulate to form aggregates. In a shear flow, there iscompetition between two coagulation mechanisms,namely, the Brownian mechanism and the gradientmechanism. Small aggregates coagulate by the Brown-ian mechanism, and the gradient mechanism is typicalfor coagulation involving large aggregates. At the initialstage of the process, when aggregates are small, theBrownian coagulation occurs, whose rate is weaklydependent on the particle size. The aggregates grow,and, at some point in time, their size exceeds the value

R

∗

above which the gradient coagulation dominates(for aggregates of the size

R

∗

, the rates of growth bythese mechanisms are equal, and

R

∗

can therefore beregarded as a conditional boundary between the smalland large aggregates). The gradient coagulation ratedepends strongly on the aggregate size (as the cube ofthe size). Therefore, after the formation of large aggre-gates, their growth rate rapidly increases, leading to anequally rapid disappearance of small particles, whichare captured by the large aggregates. The depletion ofthe fine fraction suppresses the Brownian coagulation,whose rate is proportional to the squared number ofparticles. As a result, the formation of large aggregatesceases, and their number remains virtually invariable.At the same time, the growth of large aggregates con-tinues. As an aggregate grows, the hydrodynamic stressbuilds up, raising the frequency of fragment detach-ment from the aggregate surface. When the fragmentdetachment frequency becomes equal to the particleattachment frequency, the aggregate stops growing.This means that the equilibrium aggregate size isattained. In a steady-state hydrodynamic flow, the par-ticle-size distributions in the fine and coarse fractionsstabilize over the course of time.

The figure visualizes the particle-size distributionsat various stages of the process. In this figure, the loga-rithms of the distribution functions

f

and

F

for the fineand coarse fractions, respectively, are plotted againstthe particle weight

m

(which depends on the particlesize). In the initial state (after the chemical reaction iscomplete), there are only small particles. In the figure,the appearance of large aggregates shows itself as awave in the vicinity of the transient point

m

∗

. As thelarge aggregates grow, the wave travels toward largerweights. After the large aggregates cease to form, theirnumber remains constant, but their growth continues.The representative wave (figure) grows in amplitudeand decreases in length. Once the equilibrium isattained, the wave stops in the vicinity of the equilib-rium point

m

0

(the weights

m

∗

and

m

0

correspond to the

sizes

R

∗

and

R

0

, respectively). Simultaneously, theequilibrium particle-size distribution is established inthe fine fraction. Thereby, a bimodal particle-size distri-bution pattern of the dispersed phase forms. Clearly, itis mainly due to the different growth mechanisms of thesmall and large aggregates that the dispersed phase fallsinto two fractions.

Below, we consider the final stage of the process, atwhich the dispersed phase has already fallen into twofractions. Let us summarize the main conceptionsunderlying the two-fraction model developed here.

(1) The dispersed phase consists of two fractionscharacterized by nonoverlapping peaks in the particle-size distribution function. One of these fractions (thecoarse fraction) comprises large particles, and the other(the fine fraction) consists of small particles (primaryparticles and small aggregates).

(2) The large aggregates grow by capturing particlesof the fine fraction and decrease in size because of thedetachment of fragments from their surface under theaction of the hydrodynamic stress. The detached frag-ments join the fine fraction. The coalescence of largeaggregates (whose size is of the order of

R

0

) is impos-sible because of the hydrodynamic stress, which breaksthe bonds between the contacting aggregates.

(3) The possibility of detachment of a fragment rap-idly decreases with increasing fragment weight,because the number of bonds between the fragment andthe aggregate grows.

(4) The particle attachment and detachment fre-quencies increase as the large aggregate grows; thedetachment frequency increases more rapidly than the

~~

1

2

3

0

log

f

, log

F

I II

m

*

m

0

m

3

Particle distributions at various stages of the formation ofthe dispersed phase: (

I

) the region of the fine particles thatare characterized by the weight distribution

f

(

µ

) (

µ ≡

m

)

andcoalesce by the Brownian mechanism; (

II

) the region of thecoarse particles that are characterized by the weight distri-bution

F

(

m

)

and coalesce by the gradient mechanism;(

1

) the initial particle-weight distribution, (

2

) the transientparticle-weight distribution (emergence of large aggre-gates), and (

3

) the equilibrium particle-weight distribution.

442

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING

Vol. 35

No. 5

2001

DOLGONOSOV

attachment frequency. The aggregate size at which theattachment and detachment frequencies are equal is theequilibrium aggregate size.

Kinetic equations.

Let us characterize the particlesof the fine and coarse fractions by continuous particleweight distribution functions

f

(

µ

)

and

F

(

m

)

, respec-tively, where

µ

and

m

are the weights of the particles ofthe respective fractions. Both

f

and

F

are normalized tothe number of particles in the corresponding fraction.According to the above conceptions, the coagulation ofsmall particles and their capture by aggregates, as wellas the growth and breakdown of aggregates, can bedescribed by the following kinetic equations:

(1)

(2)

where

β

(

x

,

y

)

is the coagulation kernel, which is thespecific rate of coalescence of the particles

x

and

y

intoa single aggregate

x

+

y

, and

γ

(

x

,

y

)

is the breakdownkernel, which is the specific rate of breakdown of theaggregate

x

+

y

into

x

and

y

.

Let us expand the integrands in the first and thirdintegrals in Eq. (2) into a series in the powers of

µ

andthen group the terms containing derivatives of the sameorder with respect to

m

. Instead of Eq. (2), the follow-ing equations are obtained:

(3)

(4)

∂f µ( )∂t

-------------- β µ µ'– µ',( ) f µ µ'–( ) f µ'( ) µ'd

0

µ/2

∫=

– β µ µ',( ) f µ( ) f µ'( ) µ'd

0

∞

∫

– β m µ,( )F m( ) f µ( ) md

0

∞

∫ γ m µ– µ,( )F m( ) m,d

µ

∞

∫+

∂F m( )∂t

---------------- β m µ– µ,( )F m µ–( ) f µ( ) µd

0

m/2

∫=

– β m µ,( )F m( ) f µ( ) µd

0

∞

∫

+ γ m µ,( )F m µ+( ) µd

0

∞

∫ γ m µ– µ,( )F m( ) µ,d

0

m/2

∫–

t∂∂F ∂

∂m-------

n ∂

∂m-------DnF VnF–

,n 1=

∞

∑=

Vn m( ) 1n!----- β m µ,( ) f µ( ) γ m µ– µ,( )–[ ]µ n µ,d

0

∞

∫=

(5)

Let us call Eqs. (1) and (2) or Eqs. (1) and (3)–(5) thetwo-fraction model of a disperse system where coagu-lation and fragmentation occur.

Coagulation and breakdown kernels. Let us specifythe functional form of the coagulation and breakdownkernels in the integral equations (1) and (2). The parti-cles of the fine fraction coalesce by the Brownianmechanism, to which the kernel of the following formcorresponds [9]:

For particles of similar sizes, the kernel can be approx-imated by a constant:

(6)

The small particles are captured by large aggregatesby the gradient coagulation mechanism. The gradientcoagulation kernel β(m, µ) depends on the argumentsaccording to a power law [9], which is thoroughly con-sidered below. At this point, it is sufficient to know thedependence of β on the aggregate weight m. Since typ-ical values of the weights m and µ differ by severalorders of magnitude, one can write

(7)

The exponent ν and the coefficient b in this expressionare defined below.

Let us represent the breakdown kernel as the prod-uct

(8)

where the first multiplier is the frequency of detach-ment of fragments from the aggregates of weight m andthe second is the probability of detachment of a frag-ment of weight µ from the aggregates of the weight m.It is shown below that the dependence of the detach-ment frequency on the aggregate weight is described bythe power law

(9)

whose exponent is defined below. The condition that thefragment detachment frequency Eq. (9) increases morerapidly than the particle attachment frequency Eq. (7)as the aggregate weight increases implies the constraintα > 0. If this inequality is valid, there exists an equilib-rium aggregate size.

The probability γ2 is independent of the weight m ofthe large aggregate but decreases rapidly as the weightµ of a detached fragment increases, since the larger thefragment the greater the number of bonds to be broken

Dn m( ) 1n 1+( )!

-------------------=

× β m µ,( ) f µ( ) γ m µ– µ,( )–[ ]µ n 1+ µ.d

0

∞

∫

β µ µ',( ) =

2kBT / 3ηW( )( ) µ1/3 µ '1/3 ) µ–1/3 µ ' 1/3–+(+( )= .

β µ µ',( ) a 4 2kBT / 3ηW( )( ).≡≈

β m µ,( ) β m( )≈ bmν.=

γ m µ– µ,( ) γ1 m( )γ2 µ m( ),=

γ1 m( ) cmν α+ ,=

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 35 No. 5 2001


between it and the aggregate. The number of bonds isproportional to the contact area r2 between the fragmentand the aggregate, and since the weight and the size ofa fragment having a d-dimensional fractal structure areinterrelated as µ rd [10, 11], the work to be done todetach the fragment is proportional to r2 µ2/d. Thiswork is done by random flow pulses; therefore, thefragment detachment probability is related to the frag-ment weight as γ2 exp(–kµ2/d). Since d is close to 2[10, 12, 13], we can take d = 2 and simplify this rela-tionship to

(10)

Equation (10) will eliminate computational difficultiesin deriving the particle-weight distribution and willenable one to perform necessary calculations in an ana-lytical form.

EQUILIBRIUM AGGREGATE-WEIGHT DISTRIBUTION

Fine fraction. For the equilibrium state, Eq. (1)with the coagulation and breakdown kernels (6)–(10) isbrought to the form

(11)

where

N1 obeys the equation

which is obtained from Eq. (8) by integration withrespect to µ.

By applying the Laplace transformation, Eq. (11)can be reduced to a quadratic equation for the transformL{f}. By finding this transform and performing theinverse Laplace transformation, we obtain [14]

(12)

where

(13)

and In are the modified Bessel functions.

If

(14)

(i.e., if the rate of the Brownian coagulation of particlesof the fine fraction is low as compared with the rate of

∞∞

∞

γ2 µ m( ) k kµ–( ).exp=

0.5a f µ µ'–( ) f µ'( ) µ'd

0

µ

∫– aN1 bMν+( ) f µ( ) ckMν α+ kµ–( )exp+ 0,=

N1 f µ( ) µ, Mηd

0

∞

∫ mηF m( ) m.d

0

∞

∫= =

0.5aN12– bMνN1– cMν α++ 0,=

f µ( ) kBe kµ– Akµ+ I0 Akµ( ) I1 Akµ( )–[ ] ,=

A acMν α+ aN1 bMν+( ) 2– ,=

B cMν α+ aN1 bMν+( ) 1– ,=

aN1 ! bMν

capture of small particles by large aggregates in the gra-dient coagulation), then

(15)

which allows one to simplify Eq. (12):

(16)

By comparing Eqs. (16) and (10), one can infer that theparticle-weight distribution of the fine fraction in thiscase is determined by the weight distribution of frag-ments detached by the flow from large aggregates.

Coarse fraction. In approximation (14)–(16), thekinetic coefficients (4) and (5) are written as

(17)

and

(18)

respectively, regardless of the specific form of the func-tions β(m) and γ1(m). Substitution of Eqs. (17) and (18)into Eq. (3) for the equilibrium (i.e., time-independent)state gives

Hence,

and the constant ë is determined from the normaliza-tion condition.

Using the functional form of kernels (7) and (9) andexpression (15) for N1, one can obtain

(19)

where

(20)

The constant C is found from the condition of normal-ization of Φ(x) to 1.

The number N2 of particles of the coarse fraction isdetermined from the conservation condition: the sum ofthe weights of the fine and coarse fractions should beequal to the weight of the dispersed phase in the initialstate. This condition is concretized below.

AaN1

bMν---------- ! 1, B N1

cMν α+

bMν----------------,≈ ≈≈

f µ( ) N1ke kµ– 1 O A2( )+[ ] .=

Vn k n– N1β m( ) γ1 m( )–[ ] k–n 1+ V1,= =

Dn k–n 1– N1β m( ) γ1 m( )+[ ] k–n 1+ D1,= =

k–n 1+ ddm-------

n d

dm-------D1F V1F–

n 1=

∞

∑ 0.=

ddm-------D1F V1F– 0,=

FCD1------

V1

D1------ md

0

m

∫

,exp=

N1 m0αcb 1– , V1 k 1– cm0

ν α+ xν 1 xα–( ),= =

D1 k 2– cm0ν α+ xν 1 xα+( ),=

F m( )dm N2Φ x( )dx,=

Φ x( ) C

xν 1 xα+( )------------------------- λ 1 xα–

1 xα+-------------- xd

0

x

∫

,exp=

x m/m0, λ km0, m0 Mν α+ /Mν( )1/α .== =

444


DOLGONOSOV

The parameter λ is equal to the ratio of the charac-teristic weights m0 and k–1 of the coarse and fine frac-tions, respectively; therefore, λ @ 1.

The maximum of the function Φ(x) is attained at thepoint

At large λ, the function Φ(x) is localized within asmall vicinity of unity and can be approximated by theGaussian distribution

(21)

The function F(m) is then represented as

(22)

(23)

DISTRIBUTION PARAMETERS

The particle-weight distribution functions of the fineand coarse fractions contain the parameters k, m0, b, c,ν, α, and N2, which have to be expressed in terms of theproperties of the dispersed phase, dispersion medium,and flow.

Parameters of the fine fraction. In the absence ofthe coarse fraction (F = 0), the solution of the coagula-tion equation (1) at a constant coagulation kernel (6)and the exponential initial particle-weight distribution

is well known [15]:

(24)

This distribution is established at the initial stage ofcoagulation. After the large aggregates are formed, theyswitch from the Brownian to the gradient mechanism.The mechanism changes once the point m∗ is passed,where the Brownian and gradient coagulation kernelsbecome equal; i.e., a = b . Since that moment, thenumber of small particles rapidly decreases (because oftheir capture by large aggregates) and the particle-weight distribution of the fine fraction stabilizes: µaτbecomes equal to m∗ and subsequently remains virtu-ally unchanged. Comparing Eq. (16) with the relationobtained from Eq. (24) by replacing µaτ by m∗ , one canfind the parameter k of the equilibrium particle-weightdistribution of the fine fraction:

xc 1 2ν α+( )λ 1– .–≈

Φ x( ) 1

2πσx2

----------------x xc–( )2

2σx2

--------------------– , σx2 2α 1– λ 1– .≈exp≈

F m( )N2

2πσm2

-----------------m mc–( )2

2σm2

-----------------------– ,exp≈

mc m0 2ν α+( )k 1– , σm2– 2α 1– k 1– m0.= =

f 0 µ( ) N0µa1– µ/µa–( )exp=

f µ t,( )N0

µaτ2

---------- µµaτ--------–

, τexp 1 0.5aN0t.+= =

m*ν

k m*1– b/a( )1/ν.= =

Unlike the shape of the distribution function of the finefraction, the number of its particles does not stabilizeand keeps on decreasing, since the growing large aggre-gates continue to capture small particles until theyreach their equilibrium size. In this process, the numberof particles of the fine fraction approaches equilibriumvalue N1 Eq. (19). In the (m) plot (figure), the sta-bilization of the particle-weight distribution of the finefraction means the conservation of the slope of thestraight line (m) and the decreasing number of par-ticles in the fine fraction shows itself as the parallel dis-placement of the straight line (m) toward smallervalues (as the shift from position 2 to position 3).

Parameters of the coarse-fraction. The coarsefraction comprises aggregates having a fractal struc-ture. The equilibrium radius of such an aggregate inshear flow is approximated by [5, 6]

(25)

Here, Ga = fa/(η ) characterizes the strength of thebond between a particle and the aggregate, and fa is theforce of interaction between two particles in the aggre-gate. For two particles of radius ra such that the widthof the gap between the surfaces of these particles is h,one can write [16]

The weight and the radius of a fractal aggregate areinterrelated by [10, 11]

(26)

Substituting the equilibrium values m0 and R0 for m andR, respectively, gives, in view of Eq. (25)

Let us consider the gradient coagulation kernel,which characterizes the growth of large aggregates bycapturing particles of the fine fraction. The dependenceof this kernel on the radii of the coalescing particles isexpressed as [9]:

(27)

Replacing R by m in Eq. (27) with the use of Eq. (26)yields

By comparing this equation with Eq. (7), we canexpress the parameters that are involved in Eq. (7) butnot defined there:

(28)

flog

flog

flog

R0 ra Ga/G( )p.≈

ra2

f a AHra/ 12h2( ).=

m µa R/ra( )d.≈

m0 µa Ga/G( )pd.≈

β m µ,( ) 43---W 1– G R r+( )3 4

3---W 1– GR3 β m( ),≡≈=

R @ r.

β m( )µaG

πρaW-------------- m

µa

----- 3/d

.=

ν 3/d , b G/ πρaµa3/d 1– W( ).= =



Let us now turn to the breakdown kernel. Thedetachment of fragments from the surface of a largeaggregate is due to the shear flow (whose rate is G)around this aggregate. The detachment frequency isproportional to the hydrodynamic stress on the aggre-gate surface and the area of this surface. In turn, thestress is proportional to GR, and the area is proportionalto RD, where D is the fractal dimension of the surface,2 ≤ D ≤ 3 [17–19]. Thus, the detachment frequency is

Only large aggregates whose weight is on the order ofm0 break down. Therefore, the following relation isvalid up to a constant of the order of 1:

(29)

A comparison between Eqs. (29) and (9) shows that

(30)

From Eqs. (28) and (30), one can find

Since D ≥ 2, the inequality α ≥ 0 is obeyed. At D = 2and α = 0, the surface is smooth; consequently, theaggregate is compact (nonfractal), and its dimension isequal to the dimension of the space: d = 3. In this case,only newly attached particles are detached, but oneshould keep in mind that coagulation always yieldsaggregates that have a fractal structure and, hence, arough surface, for which D > 2 and α > 0. Conse-quently, in all physically significant cases, the detach-ment frequency grows more rapidly than the attach-ment frequency as the aggregate weight increases. Thisis the reason why the equilibrium aggregate size exists.

The number of large aggregates can be found fromthe weight balance of the dispersed phase. According toEq. (16), the weight of the fine fraction is k–1N1. As onecan see from Eqs. (22) and (23), the weight of thecoarse fraction can be estimated at m0N2. The totalweight of the dispersed phase is µaN0. From the weightbalance equation k–1N1 + m0N2 = µaN0, one can find

NUMERICAL ESTIMATES

The fractal dimension of the aggregates is d ≤ 3; itsactual value depends on the aggregation mode and var-ies within a range of 1.7–2.5 [10, 12, 13]. The fractaldimension of the surface is 2 ≤ D ≤ 3 [17, 18]; there isevidence [19] that D = 2.7. The exponent p in Eq. (25)ranges between 0.3 and 0.7, as follows from theobserved data [6], and between 0.4 and 0.5, as deter-mined by calculations [5]. In barrierless coagulation(e.g., in fixation of particles in a remote potential well),the retardation coefficient is W = 1. In calculations, weuse the following typical values of the initial parame-

γ1 GRD 1+ Gm D 1+( )/d.∞ ∞

γ1 m( ) G m/m0( ) D 1+( )/d.≈

ν α+ D 1+( )/d , c Gm0D 1+( )/d– .= =

α D 2–( )/d .=

N2 µaN0 k 1– N1–( )/m0.=

ters: p = 0.5, d = 2, D = 2.7, ρa = 2 g/cm3, ra = 10–5 cm,h = 10–7 cm, AH = 10–14 erg, η = 0.01 P, kBT = 4 ×10−14 erg, G = 10 s–1, and N0 = 1010 cm–3. The calculatedresults are the following: Ga = 8 × 105 s–1, R0 = 3 ×102ra, m0 = 8 × 104µa, σm = 7 × 103µa, k–1 = 90µa, N1 =3 × 107 cm–3, and N2 = 9 × 104 cm–3.

The weight fractions of the dispersed phase in thefine and coarse fractions are 0.3 and 0.7, respectively.The parameter A in Eqs. (15) is 10–2. Hence, the condi-tion A ! 1 is met. The characteristic weights m0 and k–1

of the particles of the coarse and fine fractions, respec-tively, substantially differ, and their ratio is λ = km0 ≈103. Since λ @ 1, the Gaussian approximation (21),(22) is applicable. The ratio of m0 to σm suggests thatthe points representing the coarse fraction are locatedwithin a narrow region of m0 ± σm = (1 ± 0.08)m0. Acomparison between k–1 and m0 demonstrates that thepoints representing the fine fraction are located near theorigin of the weight axis and that the peaks of the fineand coarse fractions do not overlap. Thus, the basicprinciples of the two-fraction coagulation–fragmenta-tion model are corroborated by numerical estimates.

NOTATION

A, B—see Eqs. (13);AH—Hamaker constant;a—parameter of the Brownian coagulation kernel;b—parameter of the gradient coagulation kernel;c—parameter of the frequency of detachment of

fragments from an aggregate;D—fractal dimension of the aggregate surface;Dn—see Eq. (5);d—fractal dimension of an aggregate;G—shear rate of the flow;Ga—quantity that characterizes the strength of a

bond between a particle and an aggregate;h—width of the gap between the surfaces of the con-

tacting particles in an aggregate;k—parameter of the probability of detachment of a

fragment from an aggregate;kB—Boltzmann constant;m—aggregate weight;m0—equilibrium aggregate weight;mc—center of the weight distribution of large aggre-

gates;m∗ —transient aggregate weight;

Mν—moment of the distribution of the large aggre-gates of the order ν;

N0—number of particles in the dispersed phase inthe initial state;

N1—number of particles in the fine fraction;

446


DOLGONOSOV

N2—number of large aggregates;

R—aggregate radius;

R0—equilibrium aggregate radius;

r—radius of a particle of the fine fraction;

ra—average radius of the particles of the fine frac-tion;

T—temperature;

t—time;

Vn—see Eq. (4);

W—coagulation retardation coefficient;

β—coagulation kernel;

γ—breakdown kernel;

γ1—frequency of detachment of fragments from anaggregate;

γ2—frequency of detachment of a fragment of a cer-tain weight;

η—viscosity of the medium;

λ—see Eqs. (20);

µ—weight of a particle of the fine fraction;

µa—average particle weight in the initial weight dis-tribution;

ρa—particle density;

—variance of the weight distribution of largeaggregates.

REFERENCES

1. Vygornitskii, N.V., Lebovka, N.I., and Mank, V.V.,Local Structure of Three-Dimensional Clusters Result-ing from Diffusion-Controlled Aggregation, Kolloidn.Zh., 1995, vol. 57, no. 6, p. 788.

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σm2

coagulation–fragmentation kinetics: equilibrium weight distribution of aggregates in flowing...

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