- Email: [email protected]
Dynamics of discrete distribution for Smoluchowski coagulation model
Dynamics of Discrete Distribution for Smoluchowski Coagulation Model MICHAEL F R E N K L A C H Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803 Received February 11, 1985; accepted March 13, 1985 A new mathematical approach to the dynamics of a coagulation model with the collision frequency coefficient independent of particle size is presented. The technique consists in developing differential equations for the moments of the discrete distribution of the particle size, which allows an exact analytical solution for the dynamics of the size distribution to be determined. Analysis of this solution provides additional proof for the self-preserving hypothesis. The main novelty of the approach is the development of the criteria for the time required to attain the asymptotic distribution for both monodisperse and polydisperse initial distributions. © 1985AcademicPress,Inc. INTRODUCTION
The dynamics of the coagulation process has received significant attention in research literature [(1-4) and references therein]. The basic framework of the theory was formulated by Smoluchowski almost 70 years ago. In his classical work, Smoluchowski (5) described the coagulation kinetics of noninteracting particles by a system of differential equations dNi i-1 at - ½ Z ~(v:, v~_j)NsNe_: j=l
- Z N v . vj)NiNj,
[1]
j=l
where i=1,2 ..... ~; t is time; Ni is the number of particles with volume vi per unit volume of gas; v~ is the volume of ith particle; fl(vi, vs) is the collision frequency coefficient.
obtained by Smoluchowski (5), this special case of constant/3 will be referred to as the Smoluchowski coagulation model. All Ni constitute a discrete distribution of the particle size and the solution of Eq. [1] defines the time evolution of this distribution. Unfortunately, a general analytical solution for system [1] is not available and the Smoluchowski's solution (5), specifying each Ni as a function of time, does not immediately reveal the properties of the distribution as whole. Further mathematical development mainly took the directions of either numerical integration of system [ 1] of a truncated size (1, 7) or approximating the discrete distribution by a continuous one [(2-4, 8-11) and references therein]. The main conclusions drawn by various researchers using different mathematical methods (1-3, 8, 9, 12-18) is the existence of an asymptotic solution for the distribution of reduced particle size ni = v~N~/4),
[2]
where:
When the collision frequency coefficient is assumed to be independent of particle size, the exact analytical solution for all N~ is easily obtained (3, 5, 6). Since the solution (for monodisperse initial size distribution) was originally
ni is the reduced size of the/th particle; N~ is the total number of particles per unit volume at time t; q~ is the particles volume fraction. 237
0021-9797/85 $3.00 Journal of Colloid and Interface Science, Vol. 108, No. i, November 1985
Copyright © 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
238
MICHAEL FRENKLACH
The asymptotic solution is referred to in the literature as a "self-preserving" particle size distribution. It was noted (1, 3, 9) that the time required to reach the self-preserving distribution depends on the initial distribution of particle size. It was also pointed out (4, 19, 20) that the self-preserving hypothesis is not valid for broad initial size distributions, that is when some moments of finite order fail to exist. In this paper a new mathematical approach, based on the moments of discrete distribution of the particle size, is presented. The technique allows exact analytical solution for the dynamics of the Smoluchowski coagulation model (with a constant collision frequency coefficient) to be determined. A significant result obtained using this technique is a rigorous criterion for the time required to reach the self-preserving distribution, the solution to a previously unresolved problem.
METHOD
The rth moment about zero of I is defined as
irNi '=E{I')-
-2N
llr
~ Fni _
i
i
~ l'li i
r= 1,2,...,
[4]
where: ~'~ is the rth moment about zero o f / ; ni = Ni/N~(O) is the dimensionless concentration of ith particles; No(O) = N ~ ( t = 0) is the initial number of particles; E{ } is the expectation of { }; ~i indicates summation from i = 1 to i=~3. Defining m,, = ~ Fni,
STATEMENT OF THE PROBLEM
[5]
i
A system of colliding particles with a given initial distribution is considered. There are no forces between particles; each collision of particle i and particle j results in a new particle with volume vi + vj. Thus vi = iv1,
i
[3]
where v~ is the volume of the smallest particle in the system. In the following analysis, the collision frequency coefficient is assumed to be independent of particle size, i.e., 13(vi, vj) =/~. The objective is to determine the size distribution of particles at a given time. Size distribution is usually expressed in terms of par- " ticle diameter, mass, or volume. Here, based on the relationship [3], the distribution of the number of "monomer units," i = vi/vl, is considered, A distribution is completely characterized by specifying the moments of the distribution function (21). Thus, the stated objective is to determine the moments of L where I is a discrete random variable of the number of monomer units in a particle and can have values of i = 1, 2 . . . . , oo. Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
expressions [4] can be rewritten as ~'~ = rn~/rno.
[6]
The method consists in developing differential equations for moments m0, m~, - . • (22) Usually one is interested only in the first few moments, because they are sufficient to define the basic features of the distribution (3, Chap. 1; 23, Chap. 3). In order to develop the differential equations for m0, rnl . . . . . let us rewrite Eq. [1] in dimensionless form dnl dr -
nj ~ nj J
dn2 dr - ½n~ - n2 ~ nj J dn3
dr tin4
: nln2 - n3 ~ nj J 1 2
, t - = ~ n2 + n l n 3 - n4 ~ nj J
[7.1]
[7.21
[7.3] [7.41
239
DYNAMICS OF DISCRETE DISTRIBUTION
where r = No~(O)3t is the dimensionless time. Summation of Eqs. [7] results in the wellknown equation of Smoluchowski (5)
d ~ ni i
dr
i-1 = ½ Z Z n j n i - j - Z ni E nj i j~l
i
plying Eq. [7.1 ] by 1, Eq. [7.2] by 2 z, etc. and adding them together, we obtain d Z i2ni i d-r i-I
j
[8]
= - ½(E ni) z.
= ½ ~ ~ i2njni_j -- ~ i2ni ~ nj i j=l
i
j
i
Taking into account definition [5], Eq. [8] can be rewritten as dmo _ dr
1 2 ~ mo,
[9]
= ~1 ~ ni E (i +j)2nj - ( E i2ni)(~ hi) i
j
i
= ( ~ in~)2,
[10]
dm2 _ rn~
Multiplying Eqs. [7] by i, that is, multiplying Eq. [7.1] by 1, Eq. [7.2] by 2, etc. and adding them together, we obtain
with the initial condition [16]
i
Continuing with similar procedures, we obtain
din3 - 3mlm2 dr
d ~ it'li i
[ 15]
dr
m2(~- = 0) = ~ iZni('c = O) =- m2.o.
i
[14]
or, taking into account [5],
which is the required differential equation for moment too. The initial condition for this equation is mo(r = 0) = ~ ni(~" = 0) = 1.
i
[17]
with
dz
m3(r = O) = ~ i3ni(r = 0) ~ m3,0,
i-I
= ½ Z Z injni_j -- Z ini E nj i j:-I
i
i
din4 - 4mira3 + 3rn~ dr
j
= ½ Z Z (i + j ) n i n j -
( Z i n i ) ( Z ni) = 0
j
i
[18]
i
[19]
with
i
[11]
m4('r = O) = ~ i4ni(1" = 0) -------n4,0, etc. [20] i
or
dml
dr
-
0,
[121
with the initial condition ml(r=O)=
~ini(r =O)~-ml,o.
In general, the differential equation for mr takes the form
l(r)
dr - ~ ~
[13]
i
The differential equation for m~ could be obtained, of course, without performing summations in [ 11 ], simply based on the material balance E i ini = const. Multiplying Eqs. [7] by i z, that is, multi-
l mtmr-l,
r=2,3,...,
[21]
l=l
with the initial condition m a r = O) = ~ irni(r = O) ~ mr, o.
[221
i
Solving a system of the developed differential equations with the given initial conditions and using relationship [6], the required Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
240
MICHAEL FRENKLACH
moments are determined as a function of time, thus establishing the dynamics of the size distribution. For example, a general solution of the first five differential equations [9], [12], [ 15], [ 171, and [ 19] with corresponding initial conditions [101, [13], [15], [181, and [201 is
mZo(m2 mJ 2(i ) = __ ~2(n) = m~ rn~ ~oo mom2 -
mZl] ~oo1
#~(I) 1
m2
-
-
-
[/.{t(])] 2
1
[34]
E{. E(n)} [aa(n)] 3/2 -
I
mo -
1+-
[231
T
2
ml = ml,o
[24]
m2 = m2,o + m2,or
[251
m3 = rn3,0 + 3m2,or + 3r2
[261
m4 = m4,o +
v l ( n ) ---
=
~- E { n
'~2(n)
(4rn3,o + 3m~,o)r
skewness
~ = ~'~
[29]
fz~ - 3#'~/~ + 2(/£03 ~3
=
4
kmll
DISCUSSION For a monodisperse initial size distribution, that is ml,o = m2,o . . . . .
Similar properties for other size measures are easily determined via [28]-[31 ]. For example, let us consider the reduced particle size defined in [2]. Relationship [2] can be rewritten, noting that vi = iv1, Noo =- m o , and 4~ = v i r a l , as rn~
[37]
[31 ]
if4
mo --
ml,o = 1,
the first four moments for the distribution of I at a time r are given as
#~, - 4#'1/~ + 60z'1)2#~ - 3(#i) 4
/7i =
[36]
T2(/)-
[30]
kurtosis "Y2 =
=
-- E('~')}4
(mm~°l)4E{/- E(/)} 4
[28]
~2 = ~ _ (#~)2 Yl =
[351
[27]
Substitution of [23]-[27] into [6] determines the first four moments for the distribution of L which in turn, specifies the time evolution of the four basic features of the distribution function (23, Chap. 3):
variance
= yrI)
[~r2(r/)]2
+ 9rn2,or 2 + 3r 3.
mean
3
(mm~°l) [02(1)13/2
. I.
~=(1
+r)(1 +2)
[39]
/~=(l+7"r+9r2+3r3)(l+2).
[41]
[32]
Then we obtain u(n) = E { n } = mo E ( n
m~ _ mo u(I) ml
-
mo m l
_
1
[331
ml mo
Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
Upon substitution of [381-[41 ] into [281-[31 ] and then into [34]-[36], the time evolution of the distribution of n is determined. It is of interest to analyze the behavior of the solution at the limit r ~ oe. Thus,
241
DYNAMICS OF DISCRETE DISTRIBUTION
lim
O'2(n)= lim
.
.
.
.
.
2
- 1
.
=2-!=1.
[421
Similarly, it can be shown that lim Yl01) = 2
[43]
lim Y2(~) = 9.
[44]
T~oo
r~ao
These results indicate (and prove) the existence of the asymptotic self-preserving size distribution. Furthermore, the asymptotic values of the first four moments [42]-[44] are those of the Exponential Distribution function (23, p. 137) ~(n) = e-". [45] It is interesting to compare this result with that of Friendlander and Wang (8) ~P(n) = 0.915e -°'9%,
[461
which is an approximate expression for the upper end of the asymptotic distribution spectrum in the case of coagulation by Brownian motion. The technique developed allows one to answer the as yet unresolved problem regarding the time required to attain the self-preserving size distribution. The analysis of expression [42] leads to the following statement for the time when the variance of o approaches its asymptotic limit ~> 2. [471 Similarly, for
"{l(n) r >> 1.
[481
r ~> 2.
[49]
and for "ra(n) This time requirement is in agreement with the result of a numerical study of Hidy (1) that for a monodisperse initial size distribution the asymptotic ~b(n) develops after r _-__24. In the case of initially polydisperse size distribution, the solution is provided by Eqs.
[23]-[27], where the initial conditions, m~,0, m2,0, • • • are actually the moments of the initial distribution. Subjecting the solution of this case to r ~ o0, reveals that the asymptotic values of#, ~2, y~, and 3'2 for n are exactly the same as in the case of monodisperse initial size distribution. The time required to attain these values, taking into account that m~ <~ m o m 2 (2), is specified by the following inequalities: r >> max 2, mz°~ m2,oj
for ~201)
[50]
r~>max 2,3C5--2
foryffn)
[511
for 3'201).
[521
f/21,0 /T/2,0
r ~> 4 ~
- 2
ml,0
Inspection of expressions [50]-[52] indicates that in the polydisperse case the time for the attainment of the self-preserving size distribution is determined by the variance of the initial distribution: the larger the variance the longer it takes to reach the asymptotic distribution. Inequality [50] also indicates that there is no asymptotic solution for the initial size distributions with an infinitely large variance. It should be pointed out, however, that it is quite unrealistic to assume such broad initial distributions in practical applications. The broad distributions have attracted attention (4, 20) due to experimental observations that the size distribution of atmospheric aerosols have a v -2 size dependence. However, first, it should be noted that this dependence is valid only over a narrow range of particle sizes (24) and, second, there must always be an upper bound for an initial particle size, which necessarily leads to infinite moments of the initial distribution function. It would be more realistic to consider coagulation occurring simultaneously with nucleation, sedimentation, and other processes (25, 26). Inclusion of additional rate terms in kinetic equations [9] and [211, with initial conditions mr(r = 0) = 0, r = 2, 3 . . . . , would generally Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
242
MICHAEL FRENKLACH
lead to a time-dependent solution for the distribution of 7, for which an asymptotic solution may not exist. CONCLUSIONS
The method presented in this paper provides an additional proof for the self-preserving hypothesis. The main novelty of the approach is the development of the criteria for the time required to attain the asymptotic distribution for both monodisperse and polydisperse initial size distributions. ACKNOWLEDGMENT The work was supported by NASA-Lewis Research Center, Grant NAG 3-477. REFERENCES 1. Hidy, G. M., J. ColloidSci. 20, 123 (1965). 2. Pich, J., in "Assessment of Airborne Particles: Fundamentals, Applications, and Implications to Inhalation Toxicity" (T. T. Mercer, P. E. Morrow, and W. Stober, Eds.), p. 5. Thomas, Springfield, Illinois, 1972. 3. Friedlander, S. K., "Smoke, Dust, and Haze: Fundamentals of Aerosol Behavior," Chap. 7. Wiley, New York, 1977. 4. Mulholland, G. W., Lee, T. G., and Baum, H. R., J. Colloid Interface Sci. 62, 406 (1977). 5. Smoluchowski, M. V., Z. Phys. Chem. 92, 129 (1917). 6. Overbeek, J. T. G., in "Colloid Science" (H. R. Kruyt, Ed.), p. 278. Elsevier, New York, 1952.
Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
7. Jensen, D. E., Proc. R. Soc. London Ser. A 388, 375 (1974). 8. Friedlander, S. K., and Wang, C. S., J. Colloid Interface Sci. 22, 126 (1966). 9. Lai, F. S., Friedlander, S. K., Pich, J., and Hidy, G. M., J. Colloid lnterface Sci. 39, 395 (1972). 10. Brock, J. R., Symp. Faraday Soc. 7, 198 (1973). 11. Peterson, T. W., Gelbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 63, 426 (1978). 12. Schumann, T. E. W., Q. J. R. Meteorol. Soc. 66, 195 (1940). 13. Melzak, Z. A., Q. Appl. Math. 11, 231 (1953). 14. Martynov, G. A., and Bakanov, S. P., in "Research in Surface Forces" (B. Derjaguin, Ed.). Trans. by The Consultants Bureau, New York, 1963. 15. Scott, W. L., J. Atmos. Sci. 25, 54 (1968). 16. Wang, C. S., J. Inst. Chem. Eng. 2, 101 (1971). 17. Lushnikov, A. A., J. Colloid Interface Sci. 45, 549 (1973). 18. Lushnikov, A. A., J. Colloid Interface Sci. 48, 400 (1974). 19. Junge, C. E., in "Advances in Geophysics" (H. E. Landsberg and J. van Mieghem, Eds.), Vol. 4, p. 9. Academic Press, New York, 1958. 20. Mulholland, G. W., and Baum, H. R., Rev. Phys. Lett. 45, 761 (1980). 21. Hudson, D. J., "Lectures on Elementary Statistics and Probability." CERN, Geneva, 1963. 22. Frenklach, M., Chem. Eng. Sci., in press. 23. Johnson, N. J., and Leone, F. C., "Statistics and Experimental Design in Engineering and the Physical Sciences," Wiley, New York, 1977. 24. Witby, K. T., Husar, R. B., and Liu, B. Y. H., J. Colloid Interface Sci. 39, 177 (1972). 25. White, W. H., J. ColloMInterface Sei. 87, 204 (1982). 26. Crump, J. G., and Seinfeld, J. H., J. Colloid Interface Sci. 90, 469 (1982).
The dynamics of the coagulation process has received significant attention in research literature [(1-4) and references therein]. The basic framework of the theory was formulated by Smoluchowski almost 70 years ago. In his classical work, Smoluchowski (5) described the coagulation kinetics of noninteracting particles by a system of differential equations dNi i-1 at - ½ Z ~(v:, v~_j)NsNe_: j=l
- Z N v . vj)NiNj,
[1]
j=l
where i=1,2 ..... ~; t is time; Ni is the number of particles with volume vi per unit volume of gas; v~ is the volume of ith particle; fl(vi, vs) is the collision frequency coefficient.
obtained by Smoluchowski (5), this special case of constant/3 will be referred to as the Smoluchowski coagulation model. All Ni constitute a discrete distribution of the particle size and the solution of Eq. [1] defines the time evolution of this distribution. Unfortunately, a general analytical solution for system [1] is not available and the Smoluchowski's solution (5), specifying each Ni as a function of time, does not immediately reveal the properties of the distribution as whole. Further mathematical development mainly took the directions of either numerical integration of system [ 1] of a truncated size (1, 7) or approximating the discrete distribution by a continuous one [(2-4, 8-11) and references therein]. The main conclusions drawn by various researchers using different mathematical methods (1-3, 8, 9, 12-18) is the existence of an asymptotic solution for the distribution of reduced particle size ni = v~N~/4),
[2]
where:
When the collision frequency coefficient is assumed to be independent of particle size, the exact analytical solution for all N~ is easily obtained (3, 5, 6). Since the solution (for monodisperse initial size distribution) was originally
ni is the reduced size of the/th particle; N~ is the total number of particles per unit volume at time t; q~ is the particles volume fraction. 237
0021-9797/85 $3.00 Journal of Colloid and Interface Science, Vol. 108, No. i, November 1985
Copyright © 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
238
MICHAEL FRENKLACH
The asymptotic solution is referred to in the literature as a "self-preserving" particle size distribution. It was noted (1, 3, 9) that the time required to reach the self-preserving distribution depends on the initial distribution of particle size. It was also pointed out (4, 19, 20) that the self-preserving hypothesis is not valid for broad initial size distributions, that is when some moments of finite order fail to exist. In this paper a new mathematical approach, based on the moments of discrete distribution of the particle size, is presented. The technique allows exact analytical solution for the dynamics of the Smoluchowski coagulation model (with a constant collision frequency coefficient) to be determined. A significant result obtained using this technique is a rigorous criterion for the time required to reach the self-preserving distribution, the solution to a previously unresolved problem.
METHOD
The rth moment about zero of I is defined as
irNi '=E{I')-
-2N
llr
~ Fni _
i
i
~ l'li i
r= 1,2,...,
[4]
where: ~'~ is the rth moment about zero o f / ; ni = Ni/N~(O) is the dimensionless concentration of ith particles; No(O) = N ~ ( t = 0) is the initial number of particles; E{ } is the expectation of { }; ~i indicates summation from i = 1 to i=~3. Defining m,, = ~ Fni,
STATEMENT OF THE PROBLEM
[5]
i
A system of colliding particles with a given initial distribution is considered. There are no forces between particles; each collision of particle i and particle j results in a new particle with volume vi + vj. Thus vi = iv1,
i
[3]
where v~ is the volume of the smallest particle in the system. In the following analysis, the collision frequency coefficient is assumed to be independent of particle size, i.e., 13(vi, vj) =/~. The objective is to determine the size distribution of particles at a given time. Size distribution is usually expressed in terms of par- " ticle diameter, mass, or volume. Here, based on the relationship [3], the distribution of the number of "monomer units," i = vi/vl, is considered, A distribution is completely characterized by specifying the moments of the distribution function (21). Thus, the stated objective is to determine the moments of L where I is a discrete random variable of the number of monomer units in a particle and can have values of i = 1, 2 . . . . , oo. Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
expressions [4] can be rewritten as ~'~ = rn~/rno.
[6]
The method consists in developing differential equations for moments m0, m~, - . • (22) Usually one is interested only in the first few moments, because they are sufficient to define the basic features of the distribution (3, Chap. 1; 23, Chap. 3). In order to develop the differential equations for m0, rnl . . . . . let us rewrite Eq. [1] in dimensionless form dnl dr -
nj ~ nj J
dn2 dr - ½n~ - n2 ~ nj J dn3
dr tin4
: nln2 - n3 ~ nj J 1 2
, t - = ~ n2 + n l n 3 - n4 ~ nj J
[7.1]
[7.21
[7.3] [7.41
239
DYNAMICS OF DISCRETE DISTRIBUTION
where r = No~(O)3t is the dimensionless time. Summation of Eqs. [7] results in the wellknown equation of Smoluchowski (5)
d ~ ni i
dr
i-1 = ½ Z Z n j n i - j - Z ni E nj i j~l
i
plying Eq. [7.1 ] by 1, Eq. [7.2] by 2 z, etc. and adding them together, we obtain d Z i2ni i d-r i-I
j
[8]
= - ½(E ni) z.
= ½ ~ ~ i2njni_j -- ~ i2ni ~ nj i j=l
i
j
i
Taking into account definition [5], Eq. [8] can be rewritten as dmo _ dr
1 2 ~ mo,
[9]
= ~1 ~ ni E (i +j)2nj - ( E i2ni)(~ hi) i
j
i
= ( ~ in~)2,
[10]
dm2 _ rn~
Multiplying Eqs. [7] by i, that is, multiplying Eq. [7.1] by 1, Eq. [7.2] by 2, etc. and adding them together, we obtain
with the initial condition [16]
i
Continuing with similar procedures, we obtain
din3 - 3mlm2 dr
d ~ it'li i
[ 15]
dr
m2(~- = 0) = ~ iZni('c = O) =- m2.o.
i
[14]
or, taking into account [5],
which is the required differential equation for moment too. The initial condition for this equation is mo(r = 0) = ~ ni(~" = 0) = 1.
i
[17]
with
dz
m3(r = O) = ~ i3ni(r = 0) ~ m3,0,
i-I
= ½ Z Z injni_j -- Z ini E nj i j:-I
i
i
din4 - 4mira3 + 3rn~ dr
j
= ½ Z Z (i + j ) n i n j -
( Z i n i ) ( Z ni) = 0
j
i
[18]
i
[19]
with
i
[11]
m4('r = O) = ~ i4ni(1" = 0) -------n4,0, etc. [20] i
or
dml
dr
-
0,
[121
with the initial condition ml(r=O)=
~ini(r =O)~-ml,o.
In general, the differential equation for mr takes the form
l(r)
dr - ~ ~
[13]
i
The differential equation for m~ could be obtained, of course, without performing summations in [ 11 ], simply based on the material balance E i ini = const. Multiplying Eqs. [7] by i z, that is, multi-
l mtmr-l,
r=2,3,...,
[21]
l=l
with the initial condition m a r = O) = ~ irni(r = O) ~ mr, o.
[221
i
Solving a system of the developed differential equations with the given initial conditions and using relationship [6], the required Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
240
MICHAEL FRENKLACH
moments are determined as a function of time, thus establishing the dynamics of the size distribution. For example, a general solution of the first five differential equations [9], [12], [ 15], [ 171, and [ 19] with corresponding initial conditions [101, [13], [15], [181, and [201 is
mZo(m2 mJ 2(i ) = __ ~2(n) = m~ rn~ ~oo mom2 -
mZl] ~oo1
#~(I) 1
m2
-
-
-
[/.{t(])] 2
1
[34]
E{. E(n)} [aa(n)] 3/2 -
I
mo -
1+-
[231
T
2
ml = ml,o
[24]
m2 = m2,o + m2,or
[251
m3 = rn3,0 + 3m2,or + 3r2
[261
m4 = m4,o +
v l ( n ) ---
=
~- E { n
'~2(n)
(4rn3,o + 3m~,o)r
skewness
~ = ~'~
[29]
fz~ - 3#'~/~ + 2(/£03 ~3
=
4
kmll
DISCUSSION For a monodisperse initial size distribution, that is ml,o = m2,o . . . . .
Similar properties for other size measures are easily determined via [28]-[31 ]. For example, let us consider the reduced particle size defined in [2]. Relationship [2] can be rewritten, noting that vi = iv1, Noo =- m o , and 4~ = v i r a l , as rn~
[37]
[31 ]
if4
mo --
ml,o = 1,
the first four moments for the distribution of I at a time r are given as
#~, - 4#'1/~ + 60z'1)2#~ - 3(#i) 4
/7i =
[36]
T2(/)-
[30]
kurtosis "Y2 =
=
-- E('~')}4
(mm~°l)4E{/- E(/)} 4
[28]
~2 = ~ _ (#~)2 Yl =
[351
[27]
Substitution of [23]-[27] into [6] determines the first four moments for the distribution of L which in turn, specifies the time evolution of the four basic features of the distribution function (23, Chap. 3):
variance
= yrI)
[~r2(r/)]2
+ 9rn2,or 2 + 3r 3.
mean
3
(mm~°l) [02(1)13/2
. I.
~=(1
+r)(1 +2)
[39]
/~=(l+7"r+9r2+3r3)(l+2).
[41]
[32]
Then we obtain u(n) = E { n } = mo E ( n
m~ _ mo u(I) ml
-
mo m l
_
1
[331
ml mo
Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
Upon substitution of [381-[41 ] into [281-[31 ] and then into [34]-[36], the time evolution of the distribution of n is determined. It is of interest to analyze the behavior of the solution at the limit r ~ oe. Thus,
241
DYNAMICS OF DISCRETE DISTRIBUTION
lim
O'2(n)= lim
.
.
.
.
.
2
- 1
.
=2-!=1.
[421
Similarly, it can be shown that lim Yl01) = 2
[43]
lim Y2(~) = 9.
[44]
T~oo
r~ao
These results indicate (and prove) the existence of the asymptotic self-preserving size distribution. Furthermore, the asymptotic values of the first four moments [42]-[44] are those of the Exponential Distribution function (23, p. 137) ~(n) = e-". [45] It is interesting to compare this result with that of Friendlander and Wang (8) ~P(n) = 0.915e -°'9%,
[461
which is an approximate expression for the upper end of the asymptotic distribution spectrum in the case of coagulation by Brownian motion. The technique developed allows one to answer the as yet unresolved problem regarding the time required to attain the self-preserving size distribution. The analysis of expression [42] leads to the following statement for the time when the variance of o approaches its asymptotic limit ~> 2. [471 Similarly, for
"{l(n) r >> 1.
[481
r ~> 2.
[49]
and for "ra(n) This time requirement is in agreement with the result of a numerical study of Hidy (1) that for a monodisperse initial size distribution the asymptotic ~b(n) develops after r _-__24. In the case of initially polydisperse size distribution, the solution is provided by Eqs.
[23]-[27], where the initial conditions, m~,0, m2,0, • • • are actually the moments of the initial distribution. Subjecting the solution of this case to r ~ o0, reveals that the asymptotic values of#, ~2, y~, and 3'2 for n are exactly the same as in the case of monodisperse initial size distribution. The time required to attain these values, taking into account that m~ <~ m o m 2 (2), is specified by the following inequalities: r >> max 2, mz°~ m2,oj
for ~201)
[50]
r~>max 2,3C5--2
foryffn)
[511
for 3'201).
[521
f/21,0 /T/2,0
r ~> 4 ~
- 2
ml,0
Inspection of expressions [50]-[52] indicates that in the polydisperse case the time for the attainment of the self-preserving size distribution is determined by the variance of the initial distribution: the larger the variance the longer it takes to reach the asymptotic distribution. Inequality [50] also indicates that there is no asymptotic solution for the initial size distributions with an infinitely large variance. It should be pointed out, however, that it is quite unrealistic to assume such broad initial distributions in practical applications. The broad distributions have attracted attention (4, 20) due to experimental observations that the size distribution of atmospheric aerosols have a v -2 size dependence. However, first, it should be noted that this dependence is valid only over a narrow range of particle sizes (24) and, second, there must always be an upper bound for an initial particle size, which necessarily leads to infinite moments of the initial distribution function. It would be more realistic to consider coagulation occurring simultaneously with nucleation, sedimentation, and other processes (25, 26). Inclusion of additional rate terms in kinetic equations [9] and [211, with initial conditions mr(r = 0) = 0, r = 2, 3 . . . . , would generally Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
242
MICHAEL FRENKLACH
lead to a time-dependent solution for the distribution of 7, for which an asymptotic solution may not exist. CONCLUSIONS
The method presented in this paper provides an additional proof for the self-preserving hypothesis. The main novelty of the approach is the development of the criteria for the time required to attain the asymptotic distribution for both monodisperse and polydisperse initial size distributions. ACKNOWLEDGMENT The work was supported by NASA-Lewis Research Center, Grant NAG 3-477. REFERENCES 1. Hidy, G. M., J. ColloidSci. 20, 123 (1965). 2. Pich, J., in "Assessment of Airborne Particles: Fundamentals, Applications, and Implications to Inhalation Toxicity" (T. T. Mercer, P. E. Morrow, and W. Stober, Eds.), p. 5. Thomas, Springfield, Illinois, 1972. 3. Friedlander, S. K., "Smoke, Dust, and Haze: Fundamentals of Aerosol Behavior," Chap. 7. Wiley, New York, 1977. 4. Mulholland, G. W., Lee, T. G., and Baum, H. R., J. Colloid Interface Sci. 62, 406 (1977). 5. Smoluchowski, M. V., Z. Phys. Chem. 92, 129 (1917). 6. Overbeek, J. T. G., in "Colloid Science" (H. R. Kruyt, Ed.), p. 278. Elsevier, New York, 1952.
Journal of Colloid and Interface Science, Vol. 108, No. 1, November 1985
7. Jensen, D. E., Proc. R. Soc. London Ser. A 388, 375 (1974). 8. Friedlander, S. K., and Wang, C. S., J. Colloid Interface Sci. 22, 126 (1966). 9. Lai, F. S., Friedlander, S. K., Pich, J., and Hidy, G. M., J. Colloid lnterface Sci. 39, 395 (1972). 10. Brock, J. R., Symp. Faraday Soc. 7, 198 (1973). 11. Peterson, T. W., Gelbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 63, 426 (1978). 12. Schumann, T. E. W., Q. J. R. Meteorol. Soc. 66, 195 (1940). 13. Melzak, Z. A., Q. Appl. Math. 11, 231 (1953). 14. Martynov, G. A., and Bakanov, S. P., in "Research in Surface Forces" (B. Derjaguin, Ed.). Trans. by The Consultants Bureau, New York, 1963. 15. Scott, W. L., J. Atmos. Sci. 25, 54 (1968). 16. Wang, C. S., J. Inst. Chem. Eng. 2, 101 (1971). 17. Lushnikov, A. A., J. Colloid Interface Sci. 45, 549 (1973). 18. Lushnikov, A. A., J. Colloid Interface Sci. 48, 400 (1974). 19. Junge, C. E., in "Advances in Geophysics" (H. E. Landsberg and J. van Mieghem, Eds.), Vol. 4, p. 9. Academic Press, New York, 1958. 20. Mulholland, G. W., and Baum, H. R., Rev. Phys. Lett. 45, 761 (1980). 21. Hudson, D. J., "Lectures on Elementary Statistics and Probability." CERN, Geneva, 1963. 22. Frenklach, M., Chem. Eng. Sci., in press. 23. Johnson, N. J., and Leone, F. C., "Statistics and Experimental Design in Engineering and the Physical Sciences," Wiley, New York, 1977. 24. Witby, K. T., Husar, R. B., and Liu, B. Y. H., J. Colloid Interface Sci. 39, 177 (1972). 25. White, W. H., J. ColloMInterface Sei. 87, 204 (1982). 26. Crump, J. G., and Seinfeld, J. H., J. Colloid Interface Sci. 90, 469 (1982).