Pergamon Chemical Engineering Science, Vol. 51, No. 19, pp. 4423 4436, 1996 Copyright © 1996 Elsevier Science Ltd

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PRECIPITATION IN SMALL SYSTEMS--II. MEAN FIELD EQUATIONS MORE EFFECTIVE THAN POPULATION

BALANCE

S. MANJUNATH, K. S. G A N D H I and R. KUMAR* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

and

DORAISWAMI RAMKRISHNA* School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.

(Received 21 October 1994; accepted 27 February 1996)

Abstract--Part I (Manjunath et al., 1994, Chem. Engny Sci. 49, 1451-1463) of this paper showed that the random particle numbers and size distributions in precipitation processes in very small drops obtained by stochastic simulation techniques deviate substantially from the predictions of conventional population balance. The foregoing problem is considered in this paper in terms of a mean field approximation obtained by applying a first-order closure to an unclosed set of mean field equations presented in Part I. The mean field approximation consists of two mutually coupled partial differential equations featuring (i) the probability distribution for residual supersaturation and (ii) the mean number density of particles for each size and supersaturation from which all average properties and fluctuations can be calculated. The mean field equations have been solved by finite difference methods for (i) crystallization and (ii) precipitation of a metal hydroxide both occurring in a single drop of specified initial supersaturation. The results for the average number of particles, average residual supersaturation, the average size distribution, and fluctu- ations about the average values have been compared with those obtained by stochastic simulation techniques and by population balance. This comparison shows that the mean field predictions are substantially superior to those of population balance as judged by the close proximity of results from the former to those from stochastic simulations. The agreement is excellent for broad initial supersaturations at short times but deteriorates progressively at larger times. For steep initial supersaturation distributions, predictions of the mean field theory are not satisfactory thus calling for higher-order approximations. The merit of the mean field approximation over stochastic simulation lies in its potential to reduce expensive computation times involved in simulation. More effective computational techniques could not only enhance this advantage of the mean field approximation but also make it possible to use higher-order approximations eliminating the constraints under which the stochastic dynamics of the process can be predicted accurately. Copyright © 1996 Elsevier Science Ltd

Keywords: Precipitation stochastic population, closure hypothesis, fluctuations.

l. INTRODUCTION Of basic interest to this paper and its predecessor (Manjunath et al., 1994) is the theoretical analysis associated with the production of ceramic mixtures in minute droplets so that compositional integrity of the mixture may be assured at least within the size scale of the drop. This issue is currently of considerable tech- nological importance in the manufacture of ceramic materials including that of high-temperature super- conductors. The system of interest is a drop contain- ing an ionic mixture with a precipitant entering the drop through the surface from the exterior. The task of the theory is to predict the system evolution given the kinetic information about the nucleation and growth. The system is viewed in terms of the size distribution of the various precipitate particles and the residual supersaturation in their environment. Although our primary interest is in precipitation

* Corresponding authors.

systems, the solution of the problem is considered first in the simpler setting of an example from crystalliza- tion to demonstrate the pattern of analysis of the system evolution. It is then followed by a relatively simple example of a metal hydroxide precipitation system. A theoretical framework for the foregoing can also yield important practical information about the process such as the probability of completion of pre- cipitation as a function of time or the extent to which precipitation would have been completed with cer- tainty at a specified time and so on.

In an earlier paper (Manjunath et al., 1994), the authors addressed the analysis of precipitation reac- tions in a small system such as a droplet in which the presence of fluctuations played a significant role. The fluctuations occur in the residual supersaturation as well as in the number and sizes of precipitate particles. The authors have demonstrated the failure of deterministic populat ion balance in predicting the mean particle size distribution when high fluctu- ations are encountered in the residual supersaturation

4423

4424

accompanied by high fluctuations in particle size. This was accomplished by comparing the popula- tion balance predictions with those obtained by a stochastic simulation technique due to Shah et al. (1977).

The deviation from population balance was ob- served for low supersaturation which does not pro- vide for a buildup of particles to population levels large enough to damp out fluctuations. This is the case with small systems (such as emulsion droplets, micelles, reverse micelles, vesicles, etc.) in which super- saturation is eliminated rapidly with a relatively small number of particles.

The presence of fluctuations in particle sizes and numbers makes it essential to use stochastic simula- tion techniques. Such simulation techniques are no- tably useful because they lead to calculation of both the averages and average fluctuations. Their disad- vantage, however, lies in that they involve time con- suming computations. An alternative approach is to seek mean field equations such as those of the authors (Manjunath et al., 1994) which also have the potential to yield both the average and the average fluctuating behavior of the stochastic system. Unfortunately, the mean field equations are unclosed because they are based on an ascending hierarchy of densities (Ram- krishna and Borwanker, 1973; Ramkrishna, 1979) with equations in the lower-order density featuring the higher-order ones. Thus the alternative of the mean field equations can be realized only in the wake of suitable 'closure' approximations which yield a complete set of equations to be solved. Whether the closure approximation yields reasonable results can be ascertained by comparing them with those from stochastic simulations. The value of the mean field equations lies in reduced computation times and the possibility of analytical insights into the behavior of the system.

The objective of this paper is to propose a mean field approximation for the equations presented by M anjunath et al. (1994) and to investigate whether the mean field equations are able to provide the proper mean behavior and fluctuations in situations where the population balance equation does not describe the mean behavior of the system. More specifically, it is of interest to evaluate the effectiveness of the first-order (simplest) closure scheme in replacing the population balance equation as a mean field approximation.

2. THEORY

The precipitation reactions which will be of interest have already been discussed in Part I by the authors. We shall therefore be succinct in our recapitulation of the system here. We consider a single precipitation reaction involving a metal ion with valency ct. As before, the system is closed at its boundaries to the transport of the metal salt, an assumption made only for specificity. The precipitant is ammonium hydrox- ide which supplies hydroxyl ions by dissociation and

S. MANJUNATH et al.

the precipitate is the metal hydroxide. No spatial variations of any variable within the drop are enter- tained. The analysis encompassing ionic equilibria, solubility product, and charge balance eventually leads to expressions for the rates of change of super- saturation and hydroxyl ion concentration. Thus the solution phase variables are the supersaturation A and the hydroxyl ion concentration C; their rates of change, denoted A and C, respectively, depend on the prevailing supersaturation, the hydroxyl ion concen- tration, the number of particles in the system and their sizes, all of which may be stochastic variables in view of the smallness of the system. Relying on Manjunath et al. (1994) for details, we directly excerpt the fore- going rate A and d" as

A = Ao(A,C) + AL(A,C) i I] L(I j ,A , C) (1) j = l

= C0(A,C) + CL(A,C) ~ 12 L(I~,A, C) (2) j = l

where A0, AL, 12o and CL are as given by Manjunath et al. (1994). The terms A0,12o represent the influx of ammonium ions into the system across its boundaries which enhance the supersaturation as well as the hydroxyl ion concentration whether or not any pre- cipitation has occurred in the system. The second term on the right-hand side of either equation above shows the cumulative effect of the growth rates of all the individual precipitate particles on the rate of change of supersaturation or hydroxyl ion concentra- tion. L(l, A, C) represents the growth rate of a particle of size l when the supersaturation is A and the hy- droxyl ion concentration is C, and v represents the number of particles in the system. The expression for the growth rate is given by Manjunath et al. (1994) in their eq. (16). •

The nucleation rate in the drop as given by Volmer and Weber (1926) is given by

+ 1) B = A e x p (~n)k~ I V , A > 1 (3)

where A and K are constants and V is the drop volume.

We are now armed with all that is needed for the mean field equations presented by Manjunath et al. (1994). These equations are written in terms of the following density functions. The first is a probability density function f(;~, c; t) for the supersaturation and hydroxyl ion concentration at any given instant, fol- lowed by a first-order product density function fl(l;2, c;t) for the joint probability of the solution phase variables and a precipitate particle of size l; next is a second-order product density f2(l, l'; 2, c; t) which represents the joint probability for the solution phase variables in conjunction with the existence of two particles with sizes I and l' at time t. While f ( 2 , c; t) is a true probability density, the product densities f l and fz are not, although they have a probability inter- pretation (Ramkrishna and Borwanker, 1973). The

Precipitation in small systems--II

functions fa and f2 are essential for the calculation of average behavior and fluctuations of second order. For the precipitation process, the above density func- tions have been shown by the authors (Manjunath et al., 1994) to satisfy the following equations:

~f(z ,c; t ) +--~2 Ao(,~,c)f(Lc;t)

+ AdLc)fof l ( l ;2 ,c ; t )12L(l ,2 ,c)dl 1

[d0 (A, c)f(2, c; +c3c t)

+ Q(2,c) f~(l;2,c;t)12L(l,2,c)dl = 0 (4)

ct=- fa(l;z'c;t) + ~l [L(l,/,c) f~(l;z,c;t)]

+ ~X [{Ao + A,j2L(I, 2, c)} fall; A, c; t)]

' ' 1 L(I,2,c)dl' +~2 AL f2(I,l;/o,c;t) ,2" ,

b + ~ [{~o + CLlZL(l,2,c)}f~ (l;2,c;t)]

+7- CL f2(l,l';2,c;t)l'2L(l',)~,c)dt ' =0. OC

(5)

The above equations were determined by averaging over the particle phase statistical equations for the entire particle population presented by the authors (Manjunath et al., 1994). They can also be derived by using the probabilistic interpretations o f f ./'1 and .1"2. Either procedure used for the function f2 will give rise to the third-order product density f3 thus exposing the unclosed nature of the foregoing product density equations. Thus a mean field approach clearly requires some form of closure procedure. We propose here a closure approximation and examine the resulting mean field equation as follows. We may write

f2(I,l';2,c;t) =f2(l,l,tl,t,c)f(z,c,t) (6)

where f2(l, l'; t l2, c) represents the probability of two particles with lengths I and l' conditional on a specific supersaturation 2 and hydroxyl ion concentration c. We assume as our basic closure approximation

f2(l,l';tl2,c) =fdl;tlZ, c)f~(l';tl).,c). (7)

Substitution of eq. (7) into eq. (6) yields

4425

The foregoing expression is substituted in eq. (5) to obtain

-~fa(l;2,c;t) + ~l [L(I, Lc)f~(I;),,c; t)]

+ ~

~o fl(l';),c;t)l'2L(l',)',c)dl']~]

+ ' ~ JJJ

+~ {to+CL[12L,;,;.c) ~c [fx(l;A,c;t)

~°' fl (l'; )~'c; t) l' 2L(l" ~'c)dl']) ]

= O. (9)

Equation (9) featuring only the functions f and fl must be considered together with eq. (4) as mu- tually coupled equations for a mean field approxima- tion. They must be supplemented with initial and boundary conditions. For the initial conditions, if we assume that there are no particles to begin with and that the supersaturation and the hydroxyl ion concen- tration are initially distributed as f°(2,c), we may write

f(2, c; 0) = fo ( ) , c) (10)

f1(1;2, c;0) = 0. (11)

For the boundary condition we write

f()~, c; t) B(2) fl(O;2,c;t) ~ (12)

which is obtained by recognizing that particles of zero size are formed by nucleation at a rate as determined by the instantaneous supersaturation.

The mathematical implements for our mean field approximation are now complete. While higher-order closure approximations are conceivable it is of inter- est to investigate the full potential of the closure approximation given by eq. (7) [or by eq. (8)] in determining fluctuations and average behavior of a precipitation system. The average size distribution at time t, denoted f~ (l, t) is obtained by

f~(l,t) = f~ d2 f[ dc f~(l;;,,c;t). (13)

The expected number of particles is given by

= f o f~ (l, t) dl. (14) ( v(t)>

The average residual supersaturation (A(t)) is ob- tained from

fx (1; 2, c; t)fl (l'; 2, c; t) f2(I, 1'; 2, c; t) - (8)

f(2, c; t) , ,5,

4426

Similarly the average residual hydroxyl ion concen- tration can also be obtained. The fluctuation in the total number of particles can be calculated from the second moment of the number density.

<v=(t)>=fof,(l, nd'+ffd'fodrf tl, r;t)

where f2(l, I'; t) is defined by

f2tl, l';t) = f o d2 f o dcf2(1,1';2, c;t)

for the details of which the reader is referred to Man- junath et al. (1994). The fluctuation in v is given by

( < I v - < v > ] ~ > ) ° '~ = ( < v ~ > - < v > ~ ) ° '~.

The fluctuation in residual supersaturation is sim- ilarly obtained from its second moment

(Az)=fod2fodc22f(2,c;t). (17)

It is also possible to use the population balance equation for the case of initially distributed super- saturation to calculate the average number of par- ticles, fluctuations and so on as follows. Suppose the solution of the population balance equation for the initial condition A(0)= 2 and C(0)= c is denoted fl(I;tlA(O) = 2, C(0) = c) then the average number density is obtained as

fl(I,t) -- d2 dcf°(2,c)f~(I;tlA(O) = 2, C(0) = c).

From the average number density f~ (l, t) the average total number is obtained as usual by integrating with respect to l over the semi-infinite interval.

S. MANJUNATH et al.

differential equations. Solutions were obtained for two examples considered by the authors in Part I (Manjunath et al., 1994). They are briefly introduced in the next section.

The numerical stability of the solution of the mean field eqs (4) and (9) by finite difference depended critically on the choice of the difference schemes

(16) for the derivatives. The two-dimensional domain of particle size and supersaturation was uniformly discretized into a 32x32 grid. A forward dif- ference scheme was used for the derivatives of the dependent variables with respect to 2, while the deriv- ative of the product density with respect to particle length was obtained by using a cubic spline interpola- tion between grid points. The discretized mean field equations comprised a system of ordinary differential equations which were solved using the IMSL routine IVPAG. The quality of the solutions obtained was evaluated by comparing the solutions on a finer grid (64 x 64). Such solutions did not vary by more than 5% of those on the 32 x 32 grid.

3. SOLUTION

The coupled eqs (4) and (9) must be solved for the density functions f and f l subject to the initial conditions given by eqs (10) and (11) and the boundary condition in eq. (12) to obtain a com- plete evaluation of the mean field approximation of this paper. Both eqs (4) and (9) feature on their left-hand sides first-order partial differential oper- ators. An iterative scheme using a suitable initial ap- proximation for the first-order product density func- tion f l (1;),, c; t) would then require the solution of the two first-order partial differential equations by the method of characteristics. This appeared to be an attractive computational prospect for obtaining suc- cessive iterates but the approach was scuttled by the distortion of the characteristic curves with each itera- tion. Successive solutions required inaccurate interpo- lations and consequent accumulation of errors. Thus a finite difference scheme which served as an implicit method requiring no iteration constituted the bulk of our numerical effort. This was accomplished by a finite difference technique which discretized the problem into a large number of coupled ordinary

4. EXAMPLES

We consider a computationally simple example from crystallization to demonstrate the effectiveness of the mean field approximation. It is followed by a generic example on precipitation. The parameters used in the following examples are listed in Tables 1-3.

4.1. Crystallization Consider an isolated drop in which crystallization

of a single solute occurs (batch-wise) in such a way that both the nucleation and growth rates are linear functions of supersaturation. Furthermore, the super- saturation is the only solution phase variable of con- cern here. Following the development of Part I we write

L = kgA, B = k.AV, A = -ykgA.

Table 1. Characteristic variables used in non-dimensionaliza- tion

Crystallization Length, lm ( ~ o ) '/3

Supersaturation, A,. Ao

Time, t,~ l,,

kgAo

Precipitation

Hydroxyl ion concentration, C., x/C~KbKa

Supersaturation, A,. [M~+]oC~ Ks

Time, tm lm L(A., C~)

Precipitation in

Of particular interest to the mean field approxima- tion here are the situations presented by Manjunath e t a l . (1994) where the populat ion balance predictions are significantly at variance from those of stochastic simulation.

The most important issue for discussion concerns the validity of the mean field approximation. In as- sessing this issue, it is essential that the numerical solution for the mean field approximation be accu- rate.

The mean field equations were solved for three different initial conditions for the supersaturation dis- tribution. Using dimensionless variables the initial supersaturation distributions used were (i) 1, (ii) 322, (iii) 524, all defined over the unit interval and showing a progressively increasing degree of non- uniformity.

Figure 1 shows the results of the numerical simula- tion of the mean field approximation subject to the uniform initial condition (i). The average supersatura- tion as well as the number of particles at different dimensionless times is plotted alongside the stochastic

small systems--II 4427

Table 2. Parameters used in Example 4.1

k . 5x 101ZM icc-1 s-I kg 2 M - l c m s -1 p 16.77 M Ao 0.1 M Drop diameter 20 lain

simulation results and the populat ion balance solution. Clearly, the mean field approximation shows an excellent agreement with the results from stochastic simulation while the population balance solution deviates noticeably from both. Figure 2 shows the comparison between the cumu- lative size distribution functions at t = 1 and at t = 3. Again the predictions of the mean field approxi- mation are clearly superior to those of population balance.

The results for the initial condition (ii) are shown in Figs 3 and 4. The mean field predictions again agree well at short times (t ~< 2) with the stochastic results but deviate at larger times.

2.5

A 1.5

V1

0.5

• S I M r t - r.~ i l - - ~

- - PBE . . " ' " ~ " " "

I I I I

0 1 2 3 4 5 6 t i m e

0.6

0.4

A 0.3

V 0.2

0.1

. . . . M ~

I I i I I

0 1 2 3 4 5 6 t i m e

4 2.5

i

SIM

. . . . . . M F E

'%.

% ~ % * ' 4 % r i l lS .k ; : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I I I I I I I I I i i i i i i i i i i i i i i i i i i

1.5

U 1

0.5

°¢' t ,

.,.;~" SIM

/ - . " - ....... M F E

I I I I I I I i I I

0 1 2 3 4 5 6 0 1 2 3 4 5 t i m e t i m e

Fig. 1. Example 4.1. f°0. ) = 1; average and fluctuation of number of particles and residual supersaturation.

4428 S. MANJUNATH et al.

0.8

~ , 0.6

~ 0.4

0.2

3

2.5

2

A ~ 1.5

V 1

0.5

/ t - 1 Y PBE

S I N

N F E

I I I I

0 0.2 0.4 0.6 0.8

0.8

~ " 0.6

0.4

0.2

t " ' - • •

,:'" • S I M

0 0.2 0.4 0.6 0.8

Fig. 2. Example 4.1. f°(2) = 1; cumulative particle size distribution at two different times.

• SIM . . . . . . . . . . . . . . . . . . .

- - PBE , . . . . . . . . . . . . . . . . . . . . . . , . . . . .

I L

0 1 2 I I I

3 4 5 t i m e

0.8

0.6

A 0,4

V

0.2

[ I I I I

0 1 2 3 4 5 6 t i m e

2 3

2

0.5

0 ,

SIM

. . . . . . . . M F E

o ' , . . . . . . . . . . . . . . . . . .

I l l l l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I l l l l i l I I I B I i l I G i l I I I I l i l I I l I I I N I I I O I I H I I I I

I I I F

0 1 2 3 4 5 6 t i m e

U

mlI g I

i I

i I

i I

i g

i •

.'~ SIM i I . . . . . . . . ~ .,

ml •m• ...- . . . . . . . . . . . . . . . . .

w • . . . . . -*""

. j . , o , , ¢ - .....

I I I I 1

0 I 2 3 4 5 6 t i m e

Fig. 3. Example 4.1. f°(2) = 322; average and fluctuation of number of particles and residual supersaturation.

Finally, Figs 5 and 6 show the results of the mean field theory for the initial condi t ion (iii) which is the mos t non-uniform. The restr ict ion of the agreement of the mean field theory predict ions

with s tochast ic s imulat ions to increasingly shor- ter t imes as the initial supersa tura t ion becomes more and more non-uni form is evident from the above.

0.8

0.6

0.4

0.2

2.5

2

A 1.5

V 1

0,5

Precipitation in small systems II

/ I I I I

0 0.2 0.4 0.6 0.8

0.8

P N 0.6

~ 0.4

0.2

; • S I M

0 0.5 1

Fig. 4. Example 4.1. f°(2) = 3}~2; cumulative particle size distribution at two different times.

4429

1.5

• S I M . . . . . . . . . . . . . . . . . . .

- - P B E ~:.;: .: . . . . . . . . . . . . . . . . . . . . . . . . . M F E . . ~ ' ' ' " '

. . ; /

I I I I I

1 2 3 4 5

t i m e

A

0 , 8

0,6

0.4

0.2

" - - ' 4

, - - P B E

M F E

L I I t I - -

0 1 2 3 4 5 6

t i m e

3

2.5

2 S IM

........ M F E

1 , ' - .. . . . . . . . . . .....

• looilO.lll" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 . 5 " . . m . , , , , ,

0 I i i i I

20

15

10

• SIM

........ M F E

i

o

"m

. I

m i m

. s m m

• i mmmjm• ...........................

I i i i I

0 1 2 3 4 5 6 0 1 2 3 4 5 6 t i m e t i m e

Fig. 5. Example 4.1. f°(2) 4 = 52 ; average and fluctuation of number of particles and residual supersaturation.

In each of the a b o v e cases the m e a n field app rox i - m a t i o n is c lear ly supe r io r to the p red i c t i on of p o p u l a - t i on ba lance . Whi l e the p o p u l a t i o n ba l ance e q u a t i o n m a k e s no p r e d i c t i o n of the f l uc tua t ion of the super -

s a t u r a t i o n a n d the n u m b e r s , the m e a n field app rox i - m a t i o n does.

It is clear tha t if the good predictions of the m e a n field theory mus t be preserved for longer times, a higher-order

4430 S. MANJUNATH et al.

0.8

~ ' 0.6

~ , l 0.4

0.2

0.6

0.5

0.4

A ~,.o...a V

O.2

0.1

t=l

I I I I

0 0.2 0.4 0.6 0.8

0.8

~ 0 . 6

~-~ 0.4

0.2

/ / •

SIM P B E ~ ,-" ---- M F E

I ] I I

0.2 0.4 0.5 0.8 1

Fig. 6. Example 4.1. f0(~.) = 524; cumulative particle size distribution at two different times.

• SIM

-- PBE

.... MFE

y

/ J

2

• . / "

I I I

4 6 8 t i m e

0,3

0.25

0.2

A ,1~ 0.15

V 0.I

0.05

~ , , • SIM

• - - P B E

0 2 4 6 8 t i m e

0.9

3

D 2

SIM

........ M F E "',.

• "'-. ...........

0.85

~0. 8

0.75

0.7

/ /

/ i

i • i l •

n

i i i m

• SIM

........ M F E

i

I I I I I I r I

0 2 4 6 8 0 2 4 8 8 t i m e t i m e

Fig. 7. Example 4.2. A = t017 ; average and fluctuation of number of particles and residual supersatura- tion.

closure at, say, the third-order product density, must be sought. It is o f course likely that numerical solutions of such higher-order closures may be increasingly difficult.

The m e a n field equa t i ons in this p a p e r are c o m - pu ta t iona l ly economica l over the s tochas t i c s imula- t ions. F o r example , the to ta l C P U t imes for ob t a in ing

Precipitation in small systems--II

the solution of the mean field equations are 15-30 min on a Cyber C992 machine which are much less than the 3-4 h of stochastic simulation times.

4.2. Precipitation The details of this example are elucidated in Man-

junath et al. (1994). Since this problem involves two solution phase variables, the supersaturation as well as the hydroxyl ion concentration, the system is three- dimensional. An assumption was therefore introduced in order to reduce the dimensionality. This is made possible by assuming that the volume fraction of pre- cipitate in the drop at any instant is small compared to that at full precipitation. This is true at the begin- ning of the precipitation process. Since at large times the mean field approximation is itself expected to be less accurate (because of higher-order correlations) the assumption made is self-consistent. In order to see how the assumption leads to a reduction in dimen- sionality, we observe that mass balance leads to the

4431

following relationship between the supersaturation in the drop 2 and the hydroxyl ion concentration c.

([M"+]o - p~)c ~ 2 =

K~

where ~ is the volume fraction of the precipitate for- med and [M~+]0 is the initial concentration of the metal ion in the drop. The relationship between the two solution phase variables 2 and c involves the variable amount of precipitate in the drop. However, if e is assumed to be adequately small the foregoing relationship reduces to

[M~+]0c ~ 2

Ks

which eliminates the need for considering both 2 and c as solution phase variables. The above approxima- tion implies that the hydroxyl ion concentration is not allowed to change independently of the super- saturation. This relationship is made use of in the

0.8

0.6

0.4

0.2

L=l

i,; / - - PBE

. . . . . MFE (64x64)

q F I

0 0.5 i 1.5 2

0.8

~ 0.6

~ 0.4

0.2 . . . . . MFE (64x64)

I I I I I

0 0.5 I 1.5 2 2.5

O.fl

0.6

0.4

0.2

, ' /

• / / t - 8 - - P B E ,:,I

.... MFE (12~x128)

. . . . . MFE (,~x64)

/ / // / ,,"/

/' /

t~ 4

0 0 0 0.5 i 1,5 2 2.5 3 0 2 4 6 8 I0

t i m e

Fig. 3. Example 4.2. A = 10w; cumulative particle size distribution at three different times, and third moment of the particle size distribution.

4432

2,5

A 1.5 ;> V l

0,5

0

S . M A N J U N A T H e t al.

0.3

• SIM - - PBE .... MF E

I I I

2 3 4 t i m e

/ I

0 1

0 . 2 5

0 . 2

A ~ 0.15

V 0.1

0.05

0

MFE

i iiiiiiiiiiiii I I I I

1 2 3 4 5

t i m e

2.5

2

A

V 1

0.5

\ 'b

',,p ',,m

SIM

' , , | m l ". ........... a--w--K .......................................

2.5

2

A ~ 1.5

V 1

0.5

, .....--*'" ...--

. . - - . ," /..:"

, '"

/

, / • SIM

, ' " . . . . . . . . M ~

i .......... ..."- . . . . . . . . . . . . . . . . .

I [ I I I I I I

0 1 2 3 4 5 0 1 2 3 4 5 t i m e t i m e

Fig. 9. Example 4.2. A = 1018; average and fluctuation of number of particles and residual supersatura- tion.

Table 3. Parameters used in Example 4.2

Drop diameter 20 ~m K 900 K b 1.78 x 10 -5 M k~,,k'~ 100 cms -1 K, 8 . 4 x l 0 -22 M 4 k~ 10-14M

3 p 16.77 M [M~+]o 0.001 M

Parameters used for evaluating Fr~n., Solubility of NH3 in the external phase (Cs): 0.02 M Distribution coefficient of NH3 between water and the ex- ternal phase (Kd): 1500 Diffusivity of NH3 in the external phase: 3.14 x 10- s cm 2 s 1

expressions for Ao, AL, and L. The mean field equa- t ions under this approx imat ion will, after discarding the explicit dependence of the product density on e,

reduce to eqs (18) and (19).

~tf(2;t) + ~2 [ho(2)f(2, t) + AL(2)

fo x fl(l;2;t)lgL(1,2)dl] = 0. (18)

An initial condi t ion of the form f ( 2 , 0 ) = n(1 - 2 + Am ~)"- 1 has been used so tha t in the limit of large n, f (2 ,0 ) approximates 6(2 - A2, ~). The num- ber density equa t ion to be solved is given by

~tf~ (l; 2; t) + ~ [L(t, )Oft (l; 2; t)]

+ j j j = 0 (19)

0.8

0 .6

0 .4

0.2

Precipitation in small systems II 4433

/ i m.-

, / . SIN

I I

0 0.5 1

t=l 1 ";

0.0 /

4' ¢

• /

0.6

0.4

1.5 0 0.5

t=2

• SIM -- PBE .... MF~///

I I I I

1 1.5 2 2.5 3

1.5

0.5

t=4 • SIM - - P B E

- - - M F E

,,r/ I I I I

0.5 1 1.5 2

3 t~ :d.

2

m 0

0 2.5 3 0

I I I I

1 2 3 4 t i m e

Fig. 10. Example 4.2. A = 1018; cumulative particle size distribution at three different times, and third moment of the particle size distribution.

subject to the initial condition f1(/,2,0) = 0 and the boundary condition

f ( 2 ; t)B(2) f,(O;2;t) = L(0,2)

The mathematical problem is now similar to that of the crystallization example in form except for the significantly non-linear nucleation rate.

The method of solution adopted for the crystalliza- tion problem, however, runs into some difficulties because of the non-linearities in the growth rate and the nucleation rate. The solution for the size distribu- tion of the particles obtained as

CX3

So fl(l,2,t)d2 So So fl(I,).,t)d2dl

displayed spurious oscillations arising from numerical instability. Hence, in order to eliminate such numer-

ical instability, the derivative with respect to I is evalu- ated as a simple backward difference. However, good numerical solutions to the problem necessitated finer discretization of the (l, 2) domain than used before. Thus solutions have been obtained for grid sizes of 64 x 64 and 128 x 128. The computations are per- formed for the exponent n = 3 appearing in the initial condition. The preexponential parameter A appearing in the nucleation rate was varied. The average number of particles (and fluctuation in number), average supersaturation (and fluctuation in supersaturation) are plotted as functions of time in Figs 7, 9 and 11 for three different values of the parameter A whose effect is to change the total number of particles in the drop. The size distributions are displayed as cumulative particle fractions at specific times in Figs 8, 10 and 12. The choice of the cumulative fraction was inspired by the variability (particularly for small populations) of the size distribution density with the discrete size interval. Also included in the above figures are the

4434

30

25

20

A 15

V 1o

5

0

S. MANJUNATH et al.

0.3

.$ 4

- - PBE , "

I I I I 1

0 0.05 0.1 0.15 0.2 0.25 t i m e

0.25

0.2

A ~ 0 . 1 5

V 0.1

0.05

0

0.3

m m m a m l m m ~ m _ ~ ~

" ' . m •

• S I M

- - PBE .... MFE

I i I I I

0.05 0.I 0.15 0.2 0.25 0.3 t i m e

1.5

1

l:)

0.5

0.8

0.75

0.7

~ 0.65 0.6

0.55

0.5

¥'&-~ ' ; - i - ; ' ; ' -&--~-~- ; ........

u ' , ,

m',,

• S I N .

........ M ~

I I I I I I I I I I

0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 t i m e t i m e

Fig. 11. Example 4.2. A = 102°; average and fluctuation of number of particles and residual supersatura- tion.

third moment of the size distribution for reasons to be elucidated presently.

Figure 7 shows the results of calculations for the average values for A = 1017 which corresponds to the smallest nucleation rate used. Thus the average num- ber of particles formed for this case is less than unity with rather large fluctuations. It is evident that the period of agreement of the mean field theory with simulation significantly outlasts that of population balance. At large enough times, however, the mean field theory tends to deviate, as expected, from the results of simulations. In this connection, the fluctua- tions as predicted by the mean field theory tend to deviate somewhat earlier than the average values. A higher-order closure would be required to extend the period of agreement with the simulations. Figure 8 shows the (average) cumulative size distribution of the precipitate particles at different times. In interpret- ing these results we first observe that the dimension- less particle size has been defined with respect to the

maximum size to which a single particle could have grown by absorbing all the available supersaturation. Thus particles cannot grow to dimensionless sizes exceeding unity. At t = 1, the average number of par- ticles is about 0.1 from Fig. 7 which implies that in only 10 realizations a particle would have appeared by nucleation. From Fig. 8, at t = 1, the simula- tions show that the earliest born particle has had enough time to grow to its maximum size although with high likelihood there could have been slightly smaller particles. At larger times (such as t = 2 and t = 8) the likelihood of a particle smaller than unity falls considerably because prompt nucleation would have given the particle enough time to grow to its full potential of unity. Thus only particles which must have been born much later (the likelihood of which is small) can have sizes much smaller than one at the time in question. These considerations imply that the size distribution must rise steeply from small values to unity near the size of unity. The evaluation of the

Precipitation in

/ " / +

t=O.05 0.8 0.8 /

f - ~ 0.6 ~ 0.6 aT0.,I , 0.2 0.2

0 0

0 0.5

SIN 0.8 PBE

NFE

--~ o.13

~ 0.4

0.2

0

1 1.5 0

t=0.2

0.6

small systems--II

t=0.1

0.08

0.06

~ . 0 4

0.02

0 0.5 1 1.5 0

• SIM - - PBE .... MFE

0.5 1 1.5

- - PBE ,,/~,,,,,,,,,,,,i { { { [

0.05 0.1 0.15 0.2 O.Z t i m e

Fig. 12. Example 4.2. A = 102°; cumulative particle size distribution at three different times, and third moment of the particle size distribution.

4435

mean field theory must be regarded in the light of this background. The size distribution at t = 1 shows that the prediction using population balance does not de- viate significantly from the simulations. The mean field equation, on the other hand, when solved with a (64 x 64) grid is close to the population balance result. On refining the grid, however, to a (128 × 128) grid the prediction of the size distribution improves to agreeing very closely with the simulation. At t = 2 the population balance deviates from the simulation sig- nificantly while the present mean field equation pre- dictions for both grids are close to the simulation at sizes smaller than unity. Although the requirement of a steep rise near unity is met somewhat better with a finer grid (yielding more accurate results), it is clear that the mean field equation is unable to meet strictly the constraint of particle size being less than unity. The defense of the mean field theory lies in the fact that the size distribution is described rather well with respect to the simulation for sizes in the range below

unity following the precipitous jump by the cumulat- ive fraction. This inference is also evident from the predictions at t = 8, where the population balance prediction becomes totally irrelevant. This strong support for the mean field prediction also comes from comparing the third moment of the size distribution which must not exceed unity (from conservation of mass). Comparison of the third moment prediction by the mean field equation with that by population bal- ance shows the extent to which the latter violates the foregoing constraint.

Figure 9 shows the calculations for a larger nuclea- tion rate (A = 10 TM) which yields a larger number of particles with smaller fluctuations. Although the fluc- tuations in particle number are predicted well by the mean field theory, it does not do as well for the fluctuation in residual supersaturation at larger times. The observations made of Figs 7 and 8 are valid for Figs 9 and 10, respectively. The predictions of the mean field equation for the size distribution are

4436

significantly better than those of population balance f3 particularly at larger times. F1

Figures II and 12 represent the results for the FNu3 largest chosen nucleation rate with A = 1020 which show substantially larger number of particle popula- kg tions with relatively small fluctuations about the km, k'~ mean. Consequently, both the mean field theory and population balance predictions are close to the simu- k, lations. The constraint on the third moment is ob- K served even by the population balance equation in this case. Kb

S. MANJUNATH et al.

Kd 5. CONCLUSIONS

We have presented here a mean field theory for the stochastic population balance of crystallization and Ks precipitation systems represented by eqs (4) and Kw (9). The theory is a significant improvement over l population balance in predicting the mean behavior lm of the system. In addition, it also predicts the average fluctuations about mean values which are, of course, L, G outside the purview of deterministic population bal- [M '+]° ance. The chief merit of the theory lies in its computa- tional advantage over time consuming stochastic t simulations. Its disadvantage lies in the need for high- tm er-order closures for improved predictions parti- cularly at larger times and when steeply varying initial V supersaturations are encountered. Higher-order clo- sures may require considerably more computation times.

The theoretical development here and in Part I (Manjunath et al., 1994) is significant to the design of chemical reactors for the production of fine particles in which control of chemical composition and particle size are important.

Acknowledgements--One of us (D. R.) gratefully acknowl- edges the National Science Foundation for travel grants INT-9312243, INT-9401120, and the Jawaharlal Nehru Centre for Advanced Scientific Research for a Visiting Pro- fessorship. The authors also acknowledge the support of the Department of Science and Technology, India, for the sup- port of this Research.

NOTATION preexponential parameter in Volmer nu- cleation rate expression nucleation rate hydroxyl ion concentration rate of change of hydroxyl ion concentra- tion solubility of ammonia in the organic phase probability density for state of the solu- tion phase initial distribution of the solution phase supersaturation first-order product density function second-order product density function

A

B

c, C

C~

f

TO

.A A

third-order product density function cumulative particle size distribution rate of supply of ammonia per unit vol- ume of the system growth rate parameter in Example 4.1 mass transfer coefficient of ions to crystal surface nucleation rate parameter in Example 4.1 parameter in Volmer nucleation rate ex- pression in Example 4.2 dissociation constant of ammonia in water distribution coefficient of ammonia be- tween water and the organic phase solubility product of the precipitate ionic product of water length variable (particle radius) characteristic length used in non-dimen- sionalization crystal growth rate initial concentration of cation in the drop phase time characteristic time used in non-dimen- sionalization Volume of the drop phase

Greek letters stoichiometric coefficient

y 4gp /V 6 Dirac delta function

volume fraction of the precipitate 2, A supersaturation A rate of change of supersaturation Am characteristic supersaturation used in non-

dimensionalization v total number of particles p molar density of precipitate a standard deviation a, relative fluctuation in number, a/ (v ) tr~ relative fluctuation in supersaturation,

~/(;)

REFERENCES

Manjunath, S., Gandhi, K. S., Kumar, R. and Ramkrishna, D., 1994, Precipitation in small systems--I. Stochastic analysis. Chem. Enong Sci. 49, 1451 1463.

Ramkrishna, D., 1979, Statistical models of cell populations. Adv. Biochem. Engng 11, 1 47.

Ramkrishna, D. and Borwanker, J. D., 1973, A puristic analysis of population balance--I. Chem. Engng Sci. 28, 1423-1435.

Ramkrishna, D. and Borwanker, J. D., 1974, A puristic analysis of population balance--II. Chem. Engng Sci. 29, 1711-1721.

Shah, B. H., Ramkrishna, D. and Borwanker, J. D., 1977, Simulation of particulate systems using the concept of the interval of quiescence. A.I.Ch.E.J. 23 (6), 897-904.

Volmer, M. and Weber, A., 1926, Keimbildung in ubersattig- ten Gebilden. Z. Phys. Chem. (Leipzig) 119, 277-301.