Particle Size Correlations in Brownian ... - Semantic Scholar
Particle Size Correlations in Brownian Agglomeration. Closure Hypotheses for Product Density Equations K E N D R E E J. SAMPSON AND D O R A I S W A M I R A M K R I S H N A
School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907 Received May 17, 1985; accepted September 3, 1985 In a previous paper the authors have demonstrated inaccuracies in the population balance equation, as applied to agglomeratingparticulate systems, which arise due to an inherent assumption regarding the statistical independence of particle pairs. When this assumption is removed, an intractable hierarchy of product density equations results. In this paper a closure hypothesisfor the product density equations is proposed, evaluated against alternatives, and used in an approximate solution to the product density equations. The closure hypothesisis shownto improve significantlyupon the population balance equation which is itself the result of the simplestclosure. © 1986AcademicPress,Inc.
INTRODUCTION The evolution of a population of particles is often modeled through the use of a kinetic equation known as the population balance equation (PBE). The PBE provides a general framework in which any n u m b e r of variables relevant to a particular problem can be included in governing equations through the use of n u m b e r densities relating the particle state and frequency functions relating the rate of change of the particle state. As applied to the problem of pure particulate agglomeration where the rate of agglomeration is governed by the particles' volumes only, it is sufficient to characterize the population using the n u m b e r density f(v, t) where v is particle volume, t is time, and f(v, t)dv represents the n u m b e r o f particles in a population with volume in the range v to v + dv at time t per unit spatial volume. An agglomeration frequency q(v, v') is also needed where q(v, v')dt represents the fraction of pairs of particles per unit spatial volume, one of size v and another of size v', which will agglomerate in a time interval dr.
The derivation of the PBE follows from a consideration of the ways in which particles of size v to v + dv can be found at time t + dt given the state of the system at time t. The n u m b e r of particles of size v to v + dv formed by agglomeration of two smaller particles is given by
q(v - v', v')f(v - v', t)f(v', t)dv'dtdv. [1] The factor ½ in [ 1] compensates for an overcounting of the n u m b e r of pairs available for agglomeration when both v' and v - v' extend from 0 to v in the integration. The n u m b e r o f particles of size v to v + dv which will survive in the time interval without agglomerating with another particle is given by
f(v, t ) d v [ 1 - fo~ q(v, v')f(v', t)dv'dtl .
[2]
If the means by which a particle can change its size is restricted to agglomeration with another particle, the sum of [1] and [2] can be equated to f(v, t + dt)dv. After rearranging
410 0021-9797/86 $3.00 Copyright © 1986 by Academic Press, Inc. All,rigl~of reproduction in any forra reserved.
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
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BROWNIAN AGGLOMERATION
problem is to predict values for n(v, t) given the state of the system at an earlier time. After realizing that the sequence of agglomeration events that a particular population might fol~ t f ( v , t) = q(v - v', v') low is not a deterministic function, it becomes clear that the value of n(v, t) can only (in gen× f ( v - v', t)f(v', t)dv' eral) be stated in a probabilistic sense. Thus q(v, v')f(v, Of(v', t)dv'. [31 the proper route toward defining a continuous number density exists only through the use o f expected quantities. The PBE [31 along with an agglomeration freA proper replacement f o r f ( v , t) is called a quency provides the means for predicting the product density of order one and denoted by evolution of the size distribution and total fl(v, t) wherefl(v, t)dv is the expected number number concentration of an agglomerating of particles with sizes between v and v + dv particulate system. located anywhere in the system at time t. It is For the case of Brownian agglomeration, defined by Smoluchowski (1) has provided a functional form for the agglomeration frequency: fi(v, t)dv = E[n(v, t)]dv [5] terms, dividing by dvdt, and letting dt ~ O, the result is
q(v, v') = (2kT/#)[2 + (v/v') 1/3 +
(vt/l)) 1/3]
[4] where k is the Boltmann constant, T is the absolute temperature, and # is the viscosity of the surrounding medium. Ramkrishna and Borwanker (2) have pointed out the fact that the determinism of the population balance equation arises from the averaging of random variations in particle behavior over a large number of particles. Thus a proper analysis of particulate agglomeration should reflect the stochastic nature of real systems. In their work, Ramkrishna and Borwanker (2) point out that since the actual population is an integral-valued function, the corresponding population density cannot be, as suggested by f ( v , t), continuous. The actual population density is denoted by n(v, t), where n(v, t)dv represents the number of particles in the range v to v + dv located anywhere in the entire system at time t. It is assumed for the sake of simplicity that no two particles can have the same volume v. As applied to real systems of agglomerating particles, this assumption is certainly valid. At any given time, the actual number density will take on a particular value corresponding to the set of particles in the system. The
where the symbol E refers to the statistical expectation over all possible particle states. The actual number density is in turn related to a sum of Dirac delta functions, one for each particle in the system, N
n(v, t) = ~ 6(v - vi) i=1
[6]
where N is the number of particles in the system and {vi}iu_l is the set of particles volumes at time t. The function f~(v, t) is the proper quantity to be used in construction of a population balance; however, because it appears after taking an expectation, it does not by itself contain any information about the stochastic nature of agglomerating populations. Instead, its advantage lies in the fact that it is derived from stochastic variables and thus will illuminate any assumptions used in the PBE. The most important limitation of the PBE is evidenced after realizing that agglomeration events require the simultaneous availability of a pair of particles. Therefore, when expected quantities are used, products of the form f ( v , t)f(v', t) in the PBE should be replaced by an expectation of the number of pairs of particles. This expectation is given by a second-order product density Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
412
SAMPSON AND RAMKRISHNA
to subpopulations where the particles are sufficiently well mixed to undergo agglomeration with any other particle in the same subpopu= E [ E a(/) -- l)i) ~ a(~) -- l)j)]v÷ff [71 lation. For the case of Brownian agglomeration i=1 j=l,÷i it is further shown that the number of particles (The exclusion of v = v' is required by the in a "well-mixed" population is inversely proderivation by Ramkrishna and Borwanker (2). portional to the square root of the dispersed It is not central to this discussion and hence phase volume fraction and that the number will not be considered further.) of particles in a well-mixed volume remains It is clear from Eqs. [5] and [7] that the as- substantially constant as the agglomeration sumption process proceeds. As the spatial number denj~(v, v', t) = fffv, t ) f f f v ' , t) [81 sity decreases due to agglomeration, the spatial volume occupied by a well-mixed population which is used in the PBE is not in general ac- increases in such a way as to maintain a concurate. Examination of Eq. [7] reveals one of stant number of particles (on the average) the origins of the inaccuracy caused by the use within. of Eq. [8] in the PBE. Exclusion o f j -- i in the The need for an improved analysis is thus second summation on the rhs of Eq. [7] is nec- established by pointing out the existence of essary to prohibit consideration of any particle small populations and the inaccuracies of the being paired with itself. As the number of par- PBE when it is applied to small populations. ticles is increased, the exclusion of j = i becomes relatively less important to the value of T H E P R O D U C T DENSITY E Q U A T I O N S the summation. This suggests the fact that the PBE will be more inaccurate when applied to The obvious route toward constructing an small populations. In addition to this inac- improved analysis which circumvents the incuracy, the use of Eq. [8] also assumes that dependence assumption is to substitute J~(v, the volumes of particles in a population are v', t) into the PBE. When this is done, the PBE uncorrelated. In other words, it is assumed that is replaced by the first-order product density the probability of finding a particle of a given equation (PDE) size is statistically independent of the existence 0 v, t) of another particle of a second size. Because ~f~( of this probabilistic interpretation, Eq. [8] is referred tO as an "independence assumption" = q'(v - v', v')f~(v - v,' v,' t)dv' with deviations from it being caused by "particle size correlations." ! In a previous paper (3) the authors have exq'(v, v ' ) f z ( v , v', t)dv' + ~ ( t ) f ( v , t). [91 amined the impact of the independence assumption through the use of Monte Carlo simulation for populations of particles ag- The term ~ ( t ) f f f v , t) is included to allow for glomerating due to Brownian motion. The in- the effect of variable mixing volume. The new vestigation revealed the existence of inaccu- variable ~(t) is defined by racies, caused by the use of the independence 1 d assumption, arising when the PBE is applied ~(t) = Vmix d t Vmix [ 101 to small populations. In addition the investigation establishes the fact that whereas most where Vm~,is the mixing volume. Its value can systems of practical importance contain vast be established for the case of a constant numnumbers of particles, a proper analysis of the ber of particles in the mixing volume by reagglomeration dynamics can only be applied quiting that f2(v, v', t) = E [ n ( v , t)n(v', t)]o#v, N
N
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413
N
fit
f~(v, t)dv = 0
[1 t]
Pr{n(v, t) = ~ 6(v - vj)} j=l
and for the case of constant size mixing volume by requiring that
= J N ( 1 ) I , l)2 . . . .
, 1)N, t)d1)ldV2" " " d1)N,
N= 1,2,.... ~(t) = 0.
[121
A transition probability function q'(v, v') replaces the agglomeration frequency also such that q'(v, v')dt equals the probability that particles of sizes v and v' will agglomerate during a time interval dt conditional on their existence in the population under consideration at the beginning of the interval. The two terms can be shown to be related by (4)
q(v, v') = q'(v, v'). Vmix
[131
Equation [9] contains two independent variables, fl andJ~, and hence cannot be used by itself to predict the behavior of an agglomerating population. The appearance of a second-order product density on the rhs of Eq. [9] demands the construction of a second equation forj~(v, v', t). When this is done, the resulting equation contains a third-order product density and it becomes apparent that an entire hierarchy of "unclosed" equations will have to be solved before the first-order product density is known. Unfortunately, the complete set of product density equations form an (apparently) insurmountable problem when a solution is attempted for anything but a constant transition probability. For the sake of a complete perspective it is important to note that there is an alternate route available for generating a solution for the first-order product density. Implicit in the definitions of the first- and second-order product densities is the need for a probability density which relates the likelihood that the actual population density n(v, t) takes on a particular value. Once this function is known, the product densities can be calculated from the expectations in Eqs. [5] and [6] (see Ramkrishna and Borwanker (2)). Janossy (5) defines this probability density function by
[14]
Since no two particles of the same size are distinguishable, the order of the particle volume arguments in the Janossy densities {JN}N%0is arbitrary and the normalization is expressed by Z
~
d1)2"'"
d1)NJN
N=0
× (vj, v2 . . . .
, VN, t) = 1.
[15]
In addition, the total number of particles is governed by
1
Pn(t) = ~ .
dvl
d1)2 " ° "
f0
dVNJ N
(va, v2 . . . . , vN, t).
[16]
The Janossy densities can be solved for using the associated Janossy density equations which are derived in a fashion similar to that used to construct the PBE. The Janossy density equations are given by 0 Ot JN(v~, v2 . . . . .
=
vN, t)
1 U foVi i~=l q'(1)i -- 1)', Vt)JN+I(1)I, V2 . . . . .
1)i-l, V i - V', 1)i+1, . . . , N-1
-- ~
VN, 1)', t)dv'
N
~
q'(Vi, 1)j)JN(1)I, 1)2 . . . . .
ON, t).
i=1 j = i + l
[17]
Bayewitz et aL (6) and later Williams (7) have solved a discrete form ofEq. [ 17] for a constant agglomeration kernel and initial population fixed in number and monodisperse. Their solution reveals significant deviations from the PBE when there is a small number of particles in the population (E[N(t)] < 50). The combinatoric complexity of the Janossy Journal of Colloidand InterfaceScience, Vol. 110, No. 2, April 1986
414
SAMPSON AND RAMKRISHNA
density equations makes a solution to the complete set a difficult task for anything but the constant kernel or a very small number of particles. More importantly, the resulting solution will retain the same combinatoric complexity and is therefore of limited utility. Instead, some form of closure hypothesis must be developed which will render a small set of equations to be solved simultaneously. Toward this end, Ramkrishna and Borwanker (2) point out that the product density equations may be closed at considerably lower orders than the Janossy density equations. Thus, the ultimate goal of accurately predicting the firstorder product density can be approached directly by suitably closing the PDEs. It is evident that the PBE is itself the result of the most drastic (say level one) closure hypothesis which uses the independence assumption, Eq. [8]. Our task is then to postulate an alternate closure hypothesis which avoids the use of Eq. [8] and improves upon the PBE. It would be simple (and rather naive) to postulate a level two closure analogous to Eq. [8] such as f3(v, v', v", t) =J~(v, v', t)f~(v", t);
[181
however, the arbitrariness of Eq. [18] points out the fact that the proper choice for an alternate closure hypothesis is not a simple manner. Making and evaluating a choice is, in fact, the major goal of this analysis. Closure of the PDEs (applied to agglomerating populations) has been addressed by other workers. Scott (8) postulates a closure hypothesis
how this closure hypothesis can be substituted into the PDEs to determine f~(v, t) and R(v, v', t) but does not solve the resulting equations. The closure problem is also addressed by Williams (7). Instead of dealing with the PDEs directly, he considers closure of moment equations. As in the paper by Scott (8), no solution is attempted. AN ALTERNATE CLOSURE HYPOTHESIS
The general goal of this paper is to develop a closure hypothesis which is potentially capable of modeling a wide variety of correlated situations. Because of this, specific information about the nature of the correlations existing in Brownian coagulation is not used in the development. Instead, it is hoped that a closure hypothesis can be found which is not dependent on the agglomeration kernel but ~511 be able, nevertheless, to reflect kernel-specific size correlations in the solution to the resulting PDEs (which, of course, include the agglomeration kernel). If we restrict ourselves further to the consideration of closure hypotheses which propose approximations for f3 only, a general statement of the resulting problem is then given by Eq. [9] along with ~ t A ( v , v,' t) 2
q'(v - v", v")f3(v - v , v', v", t)dv"
+ -~
q'(v - v", v " l A ( v , v' - v", v", t)
f2(vl, v2, t) = fl(vl, t)fl(vz, t)
+ R(vl, v2, t),
x dr" -
[19]
A(Vl, v2, v3, t) : f ( v l , t)f(v2, t)A(v3, t)
× f3(v, v', v", t)dv" - q'(v, v')fz(v, v', t)
+ R(vl, v2, t)A(v3, t)
+ 2~(t)f~(v, t)f~(v', t) f3(v, v', v", t) = H [20]
where R(v, v', t) is the remainder term reflecting the degree of correlation. Scott (8) shows .Iournal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
[211
and
+ R(vl, 7)3, t)A(/)2, t) + R(v2, v3, t)fl(v~, t)
[q'(v, v") + q'(v', v")]
[22]
where Eq. [22] is the closure hypothesis. Note the appearance once again of ~(t). The use of the product of first-order product
BROWNIAN AGGLOMERATION densities in the term ((t)fl(v, t)fl(v', t) in Eq. [21 ] reflects the assumed uncorrelated state of particle pairs formed when new particles enter the mixing volume due to its expansion. The functional form of H is, as of yet, unspecified; however, it will depend on the lower product densities fl and J~, just as the independence assumption approximatesj~ in terms o f f . On the other hand, the components of H are not restricted to f~ and f2. The only requirement is that the closure hypothesis renders a system of equations which can be solved simultaneously. Unlike the level one closure, the level two closure suggests no obvious choice for a form of H. Simple forms such as Eq. [18] or
f3(v, v', v", t) = HI = fl(v, t)fl(v', t)fl(v", t)
Two more forms, which are used in analogous work in statistical fluid mechanics (9, 10) are given by
H4
=
A(A)f2(B, C) + A(B)f2(A, C) + f~(C)f2(A, B) - 2f~(A)f(B)f~(C) [26]
and //5 =
f2(A, B)fz(B, C)f2(A, C) A(A)A(B)A(C)
[27]
Although the five forms H r H 5 do not exhaust the possibilities for closure hypotheses, they do cover a wide range and thus represent a good starting point for further investigations. In order to evaluate the performance of the various forms, it is instructive to define two correlation functions by
[23] are unattractive because they do not suggest significant improvement over the independence assumption. Fortunately there are some guidelines, arising from a priori considerations, which aid in the selection process. The closure hypothesis should, at the very least, be symmetric in its volume arguments. It should also contain information about the extent of correlations betweenJ~ a n d f thus eliminating Eq. [23] from serious consideration). More specifically, it should be able to predict strong correlations between three particle sizes whenever two of the three particle sizes are correlated. Several possible closure hypotheses, which observe the above guidelines, are listed below. For the sake of notational clarity, the time argument is dropped from the product densities and the volume arguments are replaced by capital letters. Along the same lines as Eq. [ 18] an arithmetic average can be used to construct
415
G2(A, B) -
fz(A, B)
Z(A)f,(B)
[28]
and H G3(A, B, C; H) = fl(A)fl(B)f~(C)"
[29]
(Note that this definition differs from the standard usage in that no correlation implies G = 1 instead of G = 0.) They can be used to evaluate the closure hypotheses by substituting the alternate forms (Eqs. [23]-[27]) and solving for G3. The results of these procedures are
G3(A, B, C; H1) = 1,
[30]
G3(A, B, C; He) = I[G2(A, B) + G2(B, C) + G2(A, C)], [31]
G3(A, B, C;//3) = [G2(A, B)G2(B, C)G2(A, C)] u2,
[32]
G3(A, B, C; H4) = G2(A, B) + G2(B, C) + G2(A, C) - 2, [331
He = ~[f(A)f2(B, C) + f~(B)f2(A, C) and
+fi(C)f2(A, B)].
[24]
Similarly, a closure hypothesis which resembles a geometric average is given by //3
=
[f2(A, B)f2(B, C)A(A, C)] '/2. [25]
G3(A, B, C;//5) = G2(A, B)G2(B, C)G2(A, C).
[34]
The most obvious result of this transformation is a confirmation of the inadequacy of i l l . The Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
416
SAMPSON AND RAMKRISHNA
value of G3 = 1 in Eq. [30] means that HI does not allow for any correlation among particle triplets. Closer examination reveals the more important results that H4 represents a "magnification" of H2 in an arithmetic sense and //5 represents a "magnification" o f / / 3 in a geometric sense. More formally 1 -- G3(H4)
=
311 - G3(H2)]
[35]
Even though these conditions are highly restrictive, none of the other closure hypotheses shows the same or similar result. Another test stems from the expectation that as the second-order correlation becomes stronger to the point where no pairs are found at all, the third-order correlation should do likewise. In other words G2(A, B) --~ 0 ~ G3(A, B, C; H ) ~ O.
and G3(Hs) = [G3(H3)] 2.
[36]
[40]
Referring back to Eqs. [30]-[34], it is easily seen that only H3 and//5 guarantee this property (assuming G2 is bounded in all cases). The preliminary evaluation in the preceding discussion can be reinforced considerably through the use of a sample calculation designed to compare the various forms of the closure hypothesis. The sample calculation uses a Monte Carlo simulation to estimatef~, J~, and f3 which in turn are used to evaluate H1-Hs. The simulation represents an approximate solution to the entire set of Janossy density equations sacrificing precision for the sake of an unbiased solution free from the use of any closure hypothesis. Sample solutions, each consisting of one possible outcome of the agHa = 3//2 - 2H1 [37] glomeration history of a population of partiand cles, are generated using random numbers 115 = H~/HI. [381 which follow probability distributions governThe level one closure cannot copy this be- ing the initial size distribution of the particles havior (in any rational manner) because it has and others governing the likelihood of particles only J] to work with and cannot make any of different sizes agglomerating. The initial size comparisons between the different product distribution was taken to be the exponential densities. Thus it is hoped that the level two distribution and the distribution governing closure will be able to considerably reduce the agglomeration was derived from the transition difference between the kinetic equations and agglomeration probability for Brownian agthe fundamentally accurate Monte Carlo sim- glomeration. The sample solutions are averulations. aged yielding a statistically accurate but imAdditional insight into the relative merits precise estimate of the evolution of the size of the different closure hypotheses can be ob- spectrum. tained by considering their limiting behaviors. It should be noted that the results are someFor example, if for any triplet of particle sizes, what limited by the fact that no statistical tests one of the possible three pairs is uncorrelated are performed to establish the degree of imand one particle of this pair is not correlated precision of the solution. The complexity of with the remaining pair, t h e n / / 5 is exact: the interdependence between values off~, f2, and f3 prohibits a meaningful statistical analf2(A, C) = f~(A)f~(C) //5 ~f3. [391 ysis. Instead, the imprecision is minimized by f3(A, B, C) = fz(A, B)f~(C) lumping the data into three size ranges and The value of this "magnification" can be explained quite readily. Since the independence assumption underestimates the correlation in f2 (it ignores size correlation completely), it is reasonable to expect a simple level two closure such as//2 or//3 to underestimate the correlation inJ) (to a somewhat lesser extent). The real power of the level two closure stems from this realization. In b o t h / / 4 and //5, two copies of closure hypotheses, one based onf~ and the other based on f2, are combined in such a way that the result predicts a higher degree of correlation for f3 than exists for f2:
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BROWNIAN AGGLOMERATION
using the lumped data to predict values offs [more precisely
fA fB fc f (v, v', v", t)dvdv'dv"] for each distinct permutation of the three size ranges and the three volume arguments off3. The size ranges are chosen in such a way as to maintain similar values forJ~ in each range. This is done to assure that the statistical errors are of the same order of magnitude for each comparison (sampling errors are inversely proportional to the square root of the sample size) thus permitting the collective comparison of all the results. The simulation was performed on initial populations of 50 particles whose sizes followed the exponential distribution. Two hundred sample solutions were averaged to produce the estimate. The inaccuracy of the various closure hypotheses was then computed as
ei =
Hi(A,tot/Hi,tot) - A X 100%, A
i= 1,2,3,4,5,
[41]
where the notationsfl, f2, andfs are meant to emphasize the fact that all the densities are estimated from the simulation results. Values of {ei}5=1 are given in Table I for time r = 4.0 where z is a dimensionless quantity defined by ~- -
2 k T No
t
3 iz Vmix
[42]
and No is the initial number of particles. For the purpose of this preliminary study, it was assumed that the value of Vmixremained constant during the agglomeration so that the initial population of 50 particles would inevitably dwindle to only 1. The size ranges in Table I are also reported in dimensionless form
n = v/F~o,
[43]
where/Yo is the average initial volume of the particles. Entries are ordered by increasing correlation (as reflected by el). The results in-
4 17
dicate dearly the superiority of Ha and//5 over the other forms. As expected, the simpler forms//2 and H3 only partially account for the correlation effect. When the differences between//2 or//3 and//1 are "magnified," the resulting approximations H4 and//5 are very good. The problem now devolves to the choice between Ha and//5. Although the overall figure shows//5 to be better, the difference is small enough to be of dubious significance, especially considering the scatter of the individual values. A more significant comparison exists for the more correlated size range combinations. It would appear that when the correlation is small, //4 a n d / / 5 are equally accurate, but when the correlation is strong, H4 becomes inferior. The proper choice for a closure hypothesis (among those considered) is now clear. Only //5 exhibits a high degree of accuracy as well as robustness. Of course, the ultimate test remains to be made through implementation in the product density equations. SOLUTION OF THE P R O D U C T DENSITY EQUATIONS
Equations [9]-[ 12] and [21 ] along with the closure hypothesis, Eq. [27], form the basic system of equations which are needed for the evaluation of the closure hypothesis //5. It turns out, however, that the closed system can be enhanced considerably by identifying a set of naturally arising integral constraints. A system of closed moment equations is found by multiplying the PDEs by their volume arguments and integrating to give
vdvfffv, t) = ~(t)
vdvfl(v, t),
{fo vZdvfffv, t) + vdv × fo ,} qj v'dv'fz(v, o', t
= ~(l
[44]
vZdvfl(v, l)
q~ 2[fo~ Vdvf(v, t)] 2}
[45]
Journal of Colloidand InterfaceScience, Vol. 110,No. 2, April 1986
418
SAMPSON AND RAMKRISHNA TABLE I Preliminary Evaluation of Closure Hypotheses H1-H5 ~
size range
~1
e2
4~3
64
e5
(~)
(~)
(~)
(~)
(~)
3.96 4.07 4.73 4.80 5.40 2.55 4.96 3.71 4.03 2.46
19.8 21.9 27.8 31.8 42.7 46.0 62.4 78.6 109.0 258.0
13.5 13.7 19.1 21.1 25.4 34.9 40.8 51.8 66.9 141.7
10.4 9.8 14.9 15.9 17.6 29.7 31.1 37.6 47.8 98.7
0.8 -2.6 1.0 -0.2 -9.0 12.7 -2.5 -1.8 -17.1 -90.6
1.8 1.7 3.3 2.0 -3.0 15.2 5.8 6.0 4.5 10.3
115.56
51.1
32.6
24.2
-4.4
2.1
~
(A, A, B) (A, A, C) (A, B, B) (A, B, C) (B, B, B) (A, A, A) (B, B, C) (A, C, C) (B, C, C) (C, C, C)
All
"A = {r/: 0 < rj < 5.1},B = {r/: 5.1 < r j < 16.0}, C = {r/: 16.0 < rj}; ~- =4.0.
and
The term ~'(t) in Eq. [47] is specified at each time t by requiring Eqs. [44]-[46] to be satd v3dvfl(v, t) + 3 v2dv isfied. dt The appearance of ~'(t) in Eq. [47] is, of course, not dependent on the specific choice x v'dvN(v, v', t) + vdv of a closure hypothesis. It could have been used to enhance the accuracy of the various forms On the other hand, results from the X l~ v'dv' l~ v"dv'~(v, v', v", t)} H1-Hs. Monte Carlo simulation used to generate Table I can be used to show that even if the overall error in the closure hypotheses is eliminated =~(t){l~v3dvfl(v,t)+6l~v2dvfl(v,t) by normalization, the advantages o f / / 5 are preserved. The size-specific correlations can only be modeled using a size-dependent cloX l°~ vdvf~(v, t) +3 l~ vdvf~(v, t) sure hypothesis which is more sophisticated than H~, or the independence assumption. X i °° vdv i °~ v'dv'f2(v, v', t)}. [46] It should also be noted that a normalization factor could be used in the level one closure. It would improve the results of the PBE subAn additional constraint on the closure hystantially but only for the overall agglomerapothesis is indicated in Eqs. [44]-[46] because f3 appears in this closed set of equations (in tion rate (or the predicted total number of contrast to the unclosed PDEs). The constraint particles). No size specific improvements can be satisfied by adding an additional term would be found. T h e augmented problem, Eqs. [9]-[12], into the closure hypothesis: [21], [44]-[47], and the Brownian transition f3(v, v', v", t) agglomeration probability is now ready to be solved. Due to the complexity of the solution = ~(t)f2(v, v', t)fz(v, v", t)f2(v', v", t). [47] strategy and the desired focus of this paper, f(v, t)f(v', Of(v", t) extensive details of the approach will not be
{f0
i
£
f/
Journal of Colloidand InterfaceScience, Vol. 110,No. 2, April1986
BROWNIAN
AGGLOMERATION
presented here. Instead a brief summary is presented. The reader interested in more detail should consult the work of Sampson (4). The relevant equations are first nondimensionalized leaving a set of equations in which the initial number of particles in the population appears explicitly. Thus the equations are not truly nondimensional. This circumstance is unavoidable due to the fact that different order densities appear in the second- (and higher-) order PDEs. Separate solutions have to be generated for different values of No.
The solution strategy uses orthogonal collocation to provide an approximate solution to the resulting set of equations. However, in order to avoid the appearance of quotients composed of series expansions in both the numerator and denominator when the closure hypothesis is substituted into the PDEs, an additional transformation involving the product densities is made before specifying the expansion formula. The variables representing the transformed product densities are then approximated by truncated formulas. At this
~(
PBE
a
N O = i0 "~ Monte Carlo NO 50 J Simulation
,
N O = iO ,
--
I
NO
50
J
419
PDEs
Confidence Limits (95%l
5,
u
4,
o~ 3. =
o~
O,
O.
I
[
I
.5
l,O
1,5
I
2,0 Dimensionless time T
I
I
2.5
3.0
3,5
FIG. 1. The average particle v o l u m e as a function o f time. Results f r o m the constant m i x i n g v o l u m e case with No = 10, 50.
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
420
SAMPSON AND RAMKRISHNA
point the solution strategy becomes very similar to one reported by the authors for the PBE (11). The accuracy of the resulting solution suffers somewhat in comparison with the solution to the PBE due to the fact that fewer terms in the expansion were included in the solution to the PDEs. This was necessitated by the additional combinatoric complexities in the level two closure and the limitation in available precision on the Purdue University CDC 6500/6600 computer (single precision carries 15 significant figures). However, the
governing equations are still satisfied to within 2.0% for all values of reduced volume V/f above 0.03. An approximate solution to the closed PDEs was generated with No = 10 and No = 50 for the constant mixing volume case and Nmi~ = 5, 10, and 20 for the constant population size case (where Nmix is the number of particles in the constant-sized population). The initial size distribution was taken to be exponential. In Figs. 1 and 2 a comparison is made between the solutions to the PBE, the level two closure,
.9999 -
-- --
_
---
: .999 - -
PBE
-
a A
I
NO = i0 ~
Monte Carlo 50 ~ Simulation
NO
////////
I~
]
- ........... NO - i0 PDEs _
- .......
,'
NO = 50;
T
,," j
T
Confidence
~
Limits (95%)
I ~ /
.99 - -
~
~T
/~ ///
,'
,"
/¢
,"
.9
.0
O.
i.
2.
3,
4.
5.
6.
7,
Reduced volume v/v
~G. 2:Thepa~cle ~ e ~ e ~ m .
Res~ts~om ~ e c o n ~ a n t m i ~ volumec~e ~th ~ = 10,50;
r=l.O.
Journal of Colloid and InterfaceScience, Vol. 110, No. 2, April 1986
BROWNIAN AGGLOMERATION and results from a Monte Carlo simulation for the constant mixing volume [3, 4]. The Monte Carlo simulation differs from the one used in the preliminary evaluation of the closure hypothesis in that no attempt was made to estimate the second- and higher-order product densities. It is similar in that it offers an unbiased, but imprecise, estimate of the evolution of a population of agglomerating Brownian particles. The imprecision is conveyed by confidence limits in the figures. Fig-
421
ure 1 plots the dimensionless average particle volume as a function of dimensionless time and shows that the level two closure tends to overcorrect for the error in the PBE. In addition, both the error in the PBE and the error in the level two closure increase as the initial population size decreases. The match between the size spectrum predicted by the level two closure and the simulation results, depicted in Fig. 2, is better than that for the average particle volume; however, the same trend of in-
45,
/ /
PBE
×
/
Nmi x = 5 "i Monte 40.-
35.
A
Nmi x = i 0 ~ 5 Simulation Carlo
0
Nmlx = 20
*
Nmlx 5]1
/ //
///
/!
T
Confidence
i
Limits (95%)
..
,'
//f I/ ,""
~/I// ,'
///, ,./
-~
Z4v 7
d ,
~
~' "
/,4/ ,," ////,"
----,---- Nmi x = i0~, PDEs
i~'- 30.
,,
'ii)"''///'"
20,
///- /
~_
#i/
~_
7//
/?,/ :
/
3.0.
z,/
5.
__
0.
Q.
I 5.
1~
I
iO, 15. Dimensionless time 7"
I 20.
25.
FIG. 3. The average particle v o l u m e as a function o f time. Results f r o m the constant-sized population case with N ~ = 5, 10, 20.
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
422
SAMPSON A N D RAMKRISHNA
creasing error with decreasing initial population size is repeated. Figures 3 and 4 plot the same variables as in Figs. 1 and 2, respectively, except that the solution is for the case of a constant-sized population. In both figures, the agreement between the level two closure hypothesis and the simulation results is very good. The seeming disparity between the ability of the closure hypothesis to accurately solve the constant volume and constant number cases may be explained by differences in the extent of devia-
tion from the PBE rather than a more subtle inherent effect. CONCLUSIONS
The development and solution of an alternate closure to the product density equations represents an elegant and appropriate solution to the dilemma raised when the PBE is known to be inaccurate. Using a level two closure which is general wrt the agglomeration mechanism instead of a general level one closure,
,9999
- -
PBE o
............
I
,999
Monte Carlo Simulation PDEs Confidence Limits (95%)
--
,~ ,"' > I.>
~o .99 _,,,'"
,,,"
~
.9
__ __ I
.0
O.
z
i.
~ 2.
L 3.
I 4. Reduced volume v l q
5.
6.
2.
FIG. 4. The particle size spectrum. Results from the constant-sized population case with N~x = 5, 10.0.
=
Journal of Colloid and Interface Science, Vol. 110, N o . 2, April 1986
BROWNIAN AGGLOMERATION the PBE, the errors, i n c l u d i n g size-specific errors, in the s o l u t i o n o f the closed p r o d u c t density e q u a t i o n s can be r e d u c e d considerably. T h e general n a t u r e o f the s o l u t i o n strategy suggests that it can be used for o t h e r agglome r a t i o n m e c h a n i s m s as well. REFERENCES 1. Smoluchowski, M. V., Physik Z. 17, 557 (1916). 2. Ramkrishna, D., and Borwanker, J. D., Chem. Eng. Sci. 28, 1423 (1973). 3. Sampson, K. J., and Ramkrishna, D., J. Colloid Interface Sci. 104, 269 (1985).
423
4, Sampson, K. J., Ph.D. dissertation, Purdue Univ., West Lafayette, Indiana, 1981. 5. Janossy, L., Proc. Roy Irish. Acad. Ser. A 53, 181 (1950). 6. Bayewitz, M. H., Yerushalmi, J., Katz, S., and Shinnar, R., J. Atmos. Sci. 31, 1604 (1974). 7. Williams, M. M. R., J. Phys. A. 12, 983 (1979). 8. Scott, W. T., J. Atmos. Sci. 24, 221 (1967). 9. Lundgren, T. S., in "'Statistical Models and Turbulence" (M. Rosenblatt and C. Van Atta, Eds.), Springer-Verlag, Berlin, 1972. 10. Rice, S. A., and Gray, P., "The Statistical Mechanics of Simple Liquids," lnterscience, New York, 1965. 11. Sampson, K. J., and Ramkrishna, D., J. Colloid Interface Sci. 103, 245 (1985).
Journal of Colloid and Interface Science, Vol, 1 i0, No. 2, April 1986
School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907 Received May 17, 1985; accepted September 3, 1985 In a previous paper the authors have demonstrated inaccuracies in the population balance equation, as applied to agglomeratingparticulate systems, which arise due to an inherent assumption regarding the statistical independence of particle pairs. When this assumption is removed, an intractable hierarchy of product density equations results. In this paper a closure hypothesisfor the product density equations is proposed, evaluated against alternatives, and used in an approximate solution to the product density equations. The closure hypothesisis shownto improve significantlyupon the population balance equation which is itself the result of the simplestclosure. © 1986AcademicPress,Inc.
INTRODUCTION The evolution of a population of particles is often modeled through the use of a kinetic equation known as the population balance equation (PBE). The PBE provides a general framework in which any n u m b e r of variables relevant to a particular problem can be included in governing equations through the use of n u m b e r densities relating the particle state and frequency functions relating the rate of change of the particle state. As applied to the problem of pure particulate agglomeration where the rate of agglomeration is governed by the particles' volumes only, it is sufficient to characterize the population using the n u m b e r density f(v, t) where v is particle volume, t is time, and f(v, t)dv represents the n u m b e r o f particles in a population with volume in the range v to v + dv at time t per unit spatial volume. An agglomeration frequency q(v, v') is also needed where q(v, v')dt represents the fraction of pairs of particles per unit spatial volume, one of size v and another of size v', which will agglomerate in a time interval dr.
The derivation of the PBE follows from a consideration of the ways in which particles of size v to v + dv can be found at time t + dt given the state of the system at time t. The n u m b e r of particles of size v to v + dv formed by agglomeration of two smaller particles is given by
q(v - v', v')f(v - v', t)f(v', t)dv'dtdv. [1] The factor ½ in [ 1] compensates for an overcounting of the n u m b e r of pairs available for agglomeration when both v' and v - v' extend from 0 to v in the integration. The n u m b e r o f particles of size v to v + dv which will survive in the time interval without agglomerating with another particle is given by
f(v, t ) d v [ 1 - fo~ q(v, v')f(v', t)dv'dtl .
[2]
If the means by which a particle can change its size is restricted to agglomeration with another particle, the sum of [1] and [2] can be equated to f(v, t + dt)dv. After rearranging
410 0021-9797/86 $3.00 Copyright © 1986 by Academic Press, Inc. All,rigl~of reproduction in any forra reserved.
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
411
BROWNIAN AGGLOMERATION
problem is to predict values for n(v, t) given the state of the system at an earlier time. After realizing that the sequence of agglomeration events that a particular population might fol~ t f ( v , t) = q(v - v', v') low is not a deterministic function, it becomes clear that the value of n(v, t) can only (in gen× f ( v - v', t)f(v', t)dv' eral) be stated in a probabilistic sense. Thus q(v, v')f(v, Of(v', t)dv'. [31 the proper route toward defining a continuous number density exists only through the use o f expected quantities. The PBE [31 along with an agglomeration freA proper replacement f o r f ( v , t) is called a quency provides the means for predicting the product density of order one and denoted by evolution of the size distribution and total fl(v, t) wherefl(v, t)dv is the expected number number concentration of an agglomerating of particles with sizes between v and v + dv particulate system. located anywhere in the system at time t. It is For the case of Brownian agglomeration, defined by Smoluchowski (1) has provided a functional form for the agglomeration frequency: fi(v, t)dv = E[n(v, t)]dv [5] terms, dividing by dvdt, and letting dt ~ O, the result is
q(v, v') = (2kT/#)[2 + (v/v') 1/3 +
(vt/l)) 1/3]
[4] where k is the Boltmann constant, T is the absolute temperature, and # is the viscosity of the surrounding medium. Ramkrishna and Borwanker (2) have pointed out the fact that the determinism of the population balance equation arises from the averaging of random variations in particle behavior over a large number of particles. Thus a proper analysis of particulate agglomeration should reflect the stochastic nature of real systems. In their work, Ramkrishna and Borwanker (2) point out that since the actual population is an integral-valued function, the corresponding population density cannot be, as suggested by f ( v , t), continuous. The actual population density is denoted by n(v, t), where n(v, t)dv represents the number of particles in the range v to v + dv located anywhere in the entire system at time t. It is assumed for the sake of simplicity that no two particles can have the same volume v. As applied to real systems of agglomerating particles, this assumption is certainly valid. At any given time, the actual number density will take on a particular value corresponding to the set of particles in the system. The
where the symbol E refers to the statistical expectation over all possible particle states. The actual number density is in turn related to a sum of Dirac delta functions, one for each particle in the system, N
n(v, t) = ~ 6(v - vi) i=1
[6]
where N is the number of particles in the system and {vi}iu_l is the set of particles volumes at time t. The function f~(v, t) is the proper quantity to be used in construction of a population balance; however, because it appears after taking an expectation, it does not by itself contain any information about the stochastic nature of agglomerating populations. Instead, its advantage lies in the fact that it is derived from stochastic variables and thus will illuminate any assumptions used in the PBE. The most important limitation of the PBE is evidenced after realizing that agglomeration events require the simultaneous availability of a pair of particles. Therefore, when expected quantities are used, products of the form f ( v , t)f(v', t) in the PBE should be replaced by an expectation of the number of pairs of particles. This expectation is given by a second-order product density Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
412
SAMPSON AND RAMKRISHNA
to subpopulations where the particles are sufficiently well mixed to undergo agglomeration with any other particle in the same subpopu= E [ E a(/) -- l)i) ~ a(~) -- l)j)]v÷ff [71 lation. For the case of Brownian agglomeration i=1 j=l,÷i it is further shown that the number of particles (The exclusion of v = v' is required by the in a "well-mixed" population is inversely proderivation by Ramkrishna and Borwanker (2). portional to the square root of the dispersed It is not central to this discussion and hence phase volume fraction and that the number will not be considered further.) of particles in a well-mixed volume remains It is clear from Eqs. [5] and [7] that the as- substantially constant as the agglomeration sumption process proceeds. As the spatial number denj~(v, v', t) = fffv, t ) f f f v ' , t) [81 sity decreases due to agglomeration, the spatial volume occupied by a well-mixed population which is used in the PBE is not in general ac- increases in such a way as to maintain a concurate. Examination of Eq. [7] reveals one of stant number of particles (on the average) the origins of the inaccuracy caused by the use within. of Eq. [8] in the PBE. Exclusion o f j -- i in the The need for an improved analysis is thus second summation on the rhs of Eq. [7] is nec- established by pointing out the existence of essary to prohibit consideration of any particle small populations and the inaccuracies of the being paired with itself. As the number of par- PBE when it is applied to small populations. ticles is increased, the exclusion of j = i becomes relatively less important to the value of T H E P R O D U C T DENSITY E Q U A T I O N S the summation. This suggests the fact that the PBE will be more inaccurate when applied to The obvious route toward constructing an small populations. In addition to this inac- improved analysis which circumvents the incuracy, the use of Eq. [8] also assumes that dependence assumption is to substitute J~(v, the volumes of particles in a population are v', t) into the PBE. When this is done, the PBE uncorrelated. In other words, it is assumed that is replaced by the first-order product density the probability of finding a particle of a given equation (PDE) size is statistically independent of the existence 0 v, t) of another particle of a second size. Because ~f~( of this probabilistic interpretation, Eq. [8] is referred tO as an "independence assumption" = q'(v - v', v')f~(v - v,' v,' t)dv' with deviations from it being caused by "particle size correlations." ! In a previous paper (3) the authors have exq'(v, v ' ) f z ( v , v', t)dv' + ~ ( t ) f ( v , t). [91 amined the impact of the independence assumption through the use of Monte Carlo simulation for populations of particles ag- The term ~ ( t ) f f f v , t) is included to allow for glomerating due to Brownian motion. The in- the effect of variable mixing volume. The new vestigation revealed the existence of inaccu- variable ~(t) is defined by racies, caused by the use of the independence 1 d assumption, arising when the PBE is applied ~(t) = Vmix d t Vmix [ 101 to small populations. In addition the investigation establishes the fact that whereas most where Vm~,is the mixing volume. Its value can systems of practical importance contain vast be established for the case of a constant numnumbers of particles, a proper analysis of the ber of particles in the mixing volume by reagglomeration dynamics can only be applied quiting that f2(v, v', t) = E [ n ( v , t)n(v', t)]o#v, N
N
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
BROWNIAN AGGLOMERATION
413
N
fit
f~(v, t)dv = 0
[1 t]
Pr{n(v, t) = ~ 6(v - vj)} j=l
and for the case of constant size mixing volume by requiring that
= J N ( 1 ) I , l)2 . . . .
, 1)N, t)d1)ldV2" " " d1)N,
N= 1,2,.... ~(t) = 0.
[121
A transition probability function q'(v, v') replaces the agglomeration frequency also such that q'(v, v')dt equals the probability that particles of sizes v and v' will agglomerate during a time interval dt conditional on their existence in the population under consideration at the beginning of the interval. The two terms can be shown to be related by (4)
q(v, v') = q'(v, v'). Vmix
[131
Equation [9] contains two independent variables, fl andJ~, and hence cannot be used by itself to predict the behavior of an agglomerating population. The appearance of a second-order product density on the rhs of Eq. [9] demands the construction of a second equation forj~(v, v', t). When this is done, the resulting equation contains a third-order product density and it becomes apparent that an entire hierarchy of "unclosed" equations will have to be solved before the first-order product density is known. Unfortunately, the complete set of product density equations form an (apparently) insurmountable problem when a solution is attempted for anything but a constant transition probability. For the sake of a complete perspective it is important to note that there is an alternate route available for generating a solution for the first-order product density. Implicit in the definitions of the first- and second-order product densities is the need for a probability density which relates the likelihood that the actual population density n(v, t) takes on a particular value. Once this function is known, the product densities can be calculated from the expectations in Eqs. [5] and [6] (see Ramkrishna and Borwanker (2)). Janossy (5) defines this probability density function by
[14]
Since no two particles of the same size are distinguishable, the order of the particle volume arguments in the Janossy densities {JN}N%0is arbitrary and the normalization is expressed by Z
~
d1)2"'"
d1)NJN
N=0
× (vj, v2 . . . .
, VN, t) = 1.
[15]
In addition, the total number of particles is governed by
1
Pn(t) = ~ .
dvl
d1)2 " ° "
f0
dVNJ N
(va, v2 . . . . , vN, t).
[16]
The Janossy densities can be solved for using the associated Janossy density equations which are derived in a fashion similar to that used to construct the PBE. The Janossy density equations are given by 0 Ot JN(v~, v2 . . . . .
=
vN, t)
1 U foVi i~=l q'(1)i -- 1)', Vt)JN+I(1)I, V2 . . . . .
1)i-l, V i - V', 1)i+1, . . . , N-1
-- ~
VN, 1)', t)dv'
N
~
q'(Vi, 1)j)JN(1)I, 1)2 . . . . .
ON, t).
i=1 j = i + l
[17]
Bayewitz et aL (6) and later Williams (7) have solved a discrete form ofEq. [ 17] for a constant agglomeration kernel and initial population fixed in number and monodisperse. Their solution reveals significant deviations from the PBE when there is a small number of particles in the population (E[N(t)] < 50). The combinatoric complexity of the Janossy Journal of Colloidand InterfaceScience, Vol. 110, No. 2, April 1986
414
SAMPSON AND RAMKRISHNA
density equations makes a solution to the complete set a difficult task for anything but the constant kernel or a very small number of particles. More importantly, the resulting solution will retain the same combinatoric complexity and is therefore of limited utility. Instead, some form of closure hypothesis must be developed which will render a small set of equations to be solved simultaneously. Toward this end, Ramkrishna and Borwanker (2) point out that the product density equations may be closed at considerably lower orders than the Janossy density equations. Thus, the ultimate goal of accurately predicting the firstorder product density can be approached directly by suitably closing the PDEs. It is evident that the PBE is itself the result of the most drastic (say level one) closure hypothesis which uses the independence assumption, Eq. [8]. Our task is then to postulate an alternate closure hypothesis which avoids the use of Eq. [8] and improves upon the PBE. It would be simple (and rather naive) to postulate a level two closure analogous to Eq. [8] such as f3(v, v', v", t) =J~(v, v', t)f~(v", t);
[181
however, the arbitrariness of Eq. [18] points out the fact that the proper choice for an alternate closure hypothesis is not a simple manner. Making and evaluating a choice is, in fact, the major goal of this analysis. Closure of the PDEs (applied to agglomerating populations) has been addressed by other workers. Scott (8) postulates a closure hypothesis
how this closure hypothesis can be substituted into the PDEs to determine f~(v, t) and R(v, v', t) but does not solve the resulting equations. The closure problem is also addressed by Williams (7). Instead of dealing with the PDEs directly, he considers closure of moment equations. As in the paper by Scott (8), no solution is attempted. AN ALTERNATE CLOSURE HYPOTHESIS
The general goal of this paper is to develop a closure hypothesis which is potentially capable of modeling a wide variety of correlated situations. Because of this, specific information about the nature of the correlations existing in Brownian coagulation is not used in the development. Instead, it is hoped that a closure hypothesis can be found which is not dependent on the agglomeration kernel but ~511 be able, nevertheless, to reflect kernel-specific size correlations in the solution to the resulting PDEs (which, of course, include the agglomeration kernel). If we restrict ourselves further to the consideration of closure hypotheses which propose approximations for f3 only, a general statement of the resulting problem is then given by Eq. [9] along with ~ t A ( v , v,' t) 2
q'(v - v", v")f3(v - v , v', v", t)dv"
+ -~
q'(v - v", v " l A ( v , v' - v", v", t)
f2(vl, v2, t) = fl(vl, t)fl(vz, t)
+ R(vl, v2, t),
x dr" -
[19]
A(Vl, v2, v3, t) : f ( v l , t)f(v2, t)A(v3, t)
× f3(v, v', v", t)dv" - q'(v, v')fz(v, v', t)
+ R(vl, v2, t)A(v3, t)
+ 2~(t)f~(v, t)f~(v', t) f3(v, v', v", t) = H [20]
where R(v, v', t) is the remainder term reflecting the degree of correlation. Scott (8) shows .Iournal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
[211
and
+ R(vl, 7)3, t)A(/)2, t) + R(v2, v3, t)fl(v~, t)
[q'(v, v") + q'(v', v")]
[22]
where Eq. [22] is the closure hypothesis. Note the appearance once again of ~(t). The use of the product of first-order product
BROWNIAN AGGLOMERATION densities in the term ((t)fl(v, t)fl(v', t) in Eq. [21 ] reflects the assumed uncorrelated state of particle pairs formed when new particles enter the mixing volume due to its expansion. The functional form of H is, as of yet, unspecified; however, it will depend on the lower product densities fl and J~, just as the independence assumption approximatesj~ in terms o f f . On the other hand, the components of H are not restricted to f~ and f2. The only requirement is that the closure hypothesis renders a system of equations which can be solved simultaneously. Unlike the level one closure, the level two closure suggests no obvious choice for a form of H. Simple forms such as Eq. [18] or
f3(v, v', v", t) = HI = fl(v, t)fl(v', t)fl(v", t)
Two more forms, which are used in analogous work in statistical fluid mechanics (9, 10) are given by
H4
=
A(A)f2(B, C) + A(B)f2(A, C) + f~(C)f2(A, B) - 2f~(A)f(B)f~(C) [26]
and //5 =
f2(A, B)fz(B, C)f2(A, C) A(A)A(B)A(C)
[27]
Although the five forms H r H 5 do not exhaust the possibilities for closure hypotheses, they do cover a wide range and thus represent a good starting point for further investigations. In order to evaluate the performance of the various forms, it is instructive to define two correlation functions by
[23] are unattractive because they do not suggest significant improvement over the independence assumption. Fortunately there are some guidelines, arising from a priori considerations, which aid in the selection process. The closure hypothesis should, at the very least, be symmetric in its volume arguments. It should also contain information about the extent of correlations betweenJ~ a n d f thus eliminating Eq. [23] from serious consideration). More specifically, it should be able to predict strong correlations between three particle sizes whenever two of the three particle sizes are correlated. Several possible closure hypotheses, which observe the above guidelines, are listed below. For the sake of notational clarity, the time argument is dropped from the product densities and the volume arguments are replaced by capital letters. Along the same lines as Eq. [ 18] an arithmetic average can be used to construct
415
G2(A, B) -
fz(A, B)
Z(A)f,(B)
[28]
and H G3(A, B, C; H) = fl(A)fl(B)f~(C)"
[29]
(Note that this definition differs from the standard usage in that no correlation implies G = 1 instead of G = 0.) They can be used to evaluate the closure hypotheses by substituting the alternate forms (Eqs. [23]-[27]) and solving for G3. The results of these procedures are
G3(A, B, C; H1) = 1,
[30]
G3(A, B, C; He) = I[G2(A, B) + G2(B, C) + G2(A, C)], [31]
G3(A, B, C;//3) = [G2(A, B)G2(B, C)G2(A, C)] u2,
[32]
G3(A, B, C; H4) = G2(A, B) + G2(B, C) + G2(A, C) - 2, [331
He = ~[f(A)f2(B, C) + f~(B)f2(A, C) and
+fi(C)f2(A, B)].
[24]
Similarly, a closure hypothesis which resembles a geometric average is given by //3
=
[f2(A, B)f2(B, C)A(A, C)] '/2. [25]
G3(A, B, C;//5) = G2(A, B)G2(B, C)G2(A, C).
[34]
The most obvious result of this transformation is a confirmation of the inadequacy of i l l . The Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
416
SAMPSON AND RAMKRISHNA
value of G3 = 1 in Eq. [30] means that HI does not allow for any correlation among particle triplets. Closer examination reveals the more important results that H4 represents a "magnification" of H2 in an arithmetic sense and //5 represents a "magnification" o f / / 3 in a geometric sense. More formally 1 -- G3(H4)
=
311 - G3(H2)]
[35]
Even though these conditions are highly restrictive, none of the other closure hypotheses shows the same or similar result. Another test stems from the expectation that as the second-order correlation becomes stronger to the point where no pairs are found at all, the third-order correlation should do likewise. In other words G2(A, B) --~ 0 ~ G3(A, B, C; H ) ~ O.
and G3(Hs) = [G3(H3)] 2.
[36]
[40]
Referring back to Eqs. [30]-[34], it is easily seen that only H3 and//5 guarantee this property (assuming G2 is bounded in all cases). The preliminary evaluation in the preceding discussion can be reinforced considerably through the use of a sample calculation designed to compare the various forms of the closure hypothesis. The sample calculation uses a Monte Carlo simulation to estimatef~, J~, and f3 which in turn are used to evaluate H1-Hs. The simulation represents an approximate solution to the entire set of Janossy density equations sacrificing precision for the sake of an unbiased solution free from the use of any closure hypothesis. Sample solutions, each consisting of one possible outcome of the agHa = 3//2 - 2H1 [37] glomeration history of a population of partiand cles, are generated using random numbers 115 = H~/HI. [381 which follow probability distributions governThe level one closure cannot copy this be- ing the initial size distribution of the particles havior (in any rational manner) because it has and others governing the likelihood of particles only J] to work with and cannot make any of different sizes agglomerating. The initial size comparisons between the different product distribution was taken to be the exponential densities. Thus it is hoped that the level two distribution and the distribution governing closure will be able to considerably reduce the agglomeration was derived from the transition difference between the kinetic equations and agglomeration probability for Brownian agthe fundamentally accurate Monte Carlo sim- glomeration. The sample solutions are averulations. aged yielding a statistically accurate but imAdditional insight into the relative merits precise estimate of the evolution of the size of the different closure hypotheses can be ob- spectrum. tained by considering their limiting behaviors. It should be noted that the results are someFor example, if for any triplet of particle sizes, what limited by the fact that no statistical tests one of the possible three pairs is uncorrelated are performed to establish the degree of imand one particle of this pair is not correlated precision of the solution. The complexity of with the remaining pair, t h e n / / 5 is exact: the interdependence between values off~, f2, and f3 prohibits a meaningful statistical analf2(A, C) = f~(A)f~(C) //5 ~f3. [391 ysis. Instead, the imprecision is minimized by f3(A, B, C) = fz(A, B)f~(C) lumping the data into three size ranges and The value of this "magnification" can be explained quite readily. Since the independence assumption underestimates the correlation in f2 (it ignores size correlation completely), it is reasonable to expect a simple level two closure such as//2 or//3 to underestimate the correlation inJ) (to a somewhat lesser extent). The real power of the level two closure stems from this realization. In b o t h / / 4 and //5, two copies of closure hypotheses, one based onf~ and the other based on f2, are combined in such a way that the result predicts a higher degree of correlation for f3 than exists for f2:
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
BROWNIAN AGGLOMERATION
using the lumped data to predict values offs [more precisely
fA fB fc f (v, v', v", t)dvdv'dv"] for each distinct permutation of the three size ranges and the three volume arguments off3. The size ranges are chosen in such a way as to maintain similar values forJ~ in each range. This is done to assure that the statistical errors are of the same order of magnitude for each comparison (sampling errors are inversely proportional to the square root of the sample size) thus permitting the collective comparison of all the results. The simulation was performed on initial populations of 50 particles whose sizes followed the exponential distribution. Two hundred sample solutions were averaged to produce the estimate. The inaccuracy of the various closure hypotheses was then computed as
ei =
Hi(A,tot/Hi,tot) - A X 100%, A
i= 1,2,3,4,5,
[41]
where the notationsfl, f2, andfs are meant to emphasize the fact that all the densities are estimated from the simulation results. Values of {ei}5=1 are given in Table I for time r = 4.0 where z is a dimensionless quantity defined by ~- -
2 k T No
t
3 iz Vmix
[42]
and No is the initial number of particles. For the purpose of this preliminary study, it was assumed that the value of Vmixremained constant during the agglomeration so that the initial population of 50 particles would inevitably dwindle to only 1. The size ranges in Table I are also reported in dimensionless form
n = v/F~o,
[43]
where/Yo is the average initial volume of the particles. Entries are ordered by increasing correlation (as reflected by el). The results in-
4 17
dicate dearly the superiority of Ha and//5 over the other forms. As expected, the simpler forms//2 and H3 only partially account for the correlation effect. When the differences between//2 or//3 and//1 are "magnified," the resulting approximations H4 and//5 are very good. The problem now devolves to the choice between Ha and//5. Although the overall figure shows//5 to be better, the difference is small enough to be of dubious significance, especially considering the scatter of the individual values. A more significant comparison exists for the more correlated size range combinations. It would appear that when the correlation is small, //4 a n d / / 5 are equally accurate, but when the correlation is strong, H4 becomes inferior. The proper choice for a closure hypothesis (among those considered) is now clear. Only //5 exhibits a high degree of accuracy as well as robustness. Of course, the ultimate test remains to be made through implementation in the product density equations. SOLUTION OF THE P R O D U C T DENSITY EQUATIONS
Equations [9]-[ 12] and [21 ] along with the closure hypothesis, Eq. [27], form the basic system of equations which are needed for the evaluation of the closure hypothesis //5. It turns out, however, that the closed system can be enhanced considerably by identifying a set of naturally arising integral constraints. A system of closed moment equations is found by multiplying the PDEs by their volume arguments and integrating to give
vdvfffv, t) = ~(t)
vdvfl(v, t),
{fo vZdvfffv, t) + vdv × fo ,} qj v'dv'fz(v, o', t
= ~(l
[44]
vZdvfl(v, l)
q~ 2[fo~ Vdvf(v, t)] 2}
[45]
Journal of Colloidand InterfaceScience, Vol. 110,No. 2, April 1986
418
SAMPSON AND RAMKRISHNA TABLE I Preliminary Evaluation of Closure Hypotheses H1-H5 ~
size range
~1
e2
4~3
64
e5
(~)
(~)
(~)
(~)
(~)
3.96 4.07 4.73 4.80 5.40 2.55 4.96 3.71 4.03 2.46
19.8 21.9 27.8 31.8 42.7 46.0 62.4 78.6 109.0 258.0
13.5 13.7 19.1 21.1 25.4 34.9 40.8 51.8 66.9 141.7
10.4 9.8 14.9 15.9 17.6 29.7 31.1 37.6 47.8 98.7
0.8 -2.6 1.0 -0.2 -9.0 12.7 -2.5 -1.8 -17.1 -90.6
1.8 1.7 3.3 2.0 -3.0 15.2 5.8 6.0 4.5 10.3
115.56
51.1
32.6
24.2
-4.4
2.1
~
(A, A, B) (A, A, C) (A, B, B) (A, B, C) (B, B, B) (A, A, A) (B, B, C) (A, C, C) (B, C, C) (C, C, C)
All
"A = {r/: 0 < rj < 5.1},B = {r/: 5.1 < r j < 16.0}, C = {r/: 16.0 < rj}; ~- =4.0.
and
The term ~'(t) in Eq. [47] is specified at each time t by requiring Eqs. [44]-[46] to be satd v3dvfl(v, t) + 3 v2dv isfied. dt The appearance of ~'(t) in Eq. [47] is, of course, not dependent on the specific choice x v'dvN(v, v', t) + vdv of a closure hypothesis. It could have been used to enhance the accuracy of the various forms On the other hand, results from the X l~ v'dv' l~ v"dv'~(v, v', v", t)} H1-Hs. Monte Carlo simulation used to generate Table I can be used to show that even if the overall error in the closure hypotheses is eliminated =~(t){l~v3dvfl(v,t)+6l~v2dvfl(v,t) by normalization, the advantages o f / / 5 are preserved. The size-specific correlations can only be modeled using a size-dependent cloX l°~ vdvf~(v, t) +3 l~ vdvf~(v, t) sure hypothesis which is more sophisticated than H~, or the independence assumption. X i °° vdv i °~ v'dv'f2(v, v', t)}. [46] It should also be noted that a normalization factor could be used in the level one closure. It would improve the results of the PBE subAn additional constraint on the closure hystantially but only for the overall agglomerapothesis is indicated in Eqs. [44]-[46] because f3 appears in this closed set of equations (in tion rate (or the predicted total number of contrast to the unclosed PDEs). The constraint particles). No size specific improvements can be satisfied by adding an additional term would be found. T h e augmented problem, Eqs. [9]-[12], into the closure hypothesis: [21], [44]-[47], and the Brownian transition f3(v, v', v", t) agglomeration probability is now ready to be solved. Due to the complexity of the solution = ~(t)f2(v, v', t)fz(v, v", t)f2(v', v", t). [47] strategy and the desired focus of this paper, f(v, t)f(v', Of(v", t) extensive details of the approach will not be
{f0
i
£
f/
Journal of Colloidand InterfaceScience, Vol. 110,No. 2, April1986
BROWNIAN
AGGLOMERATION
presented here. Instead a brief summary is presented. The reader interested in more detail should consult the work of Sampson (4). The relevant equations are first nondimensionalized leaving a set of equations in which the initial number of particles in the population appears explicitly. Thus the equations are not truly nondimensional. This circumstance is unavoidable due to the fact that different order densities appear in the second- (and higher-) order PDEs. Separate solutions have to be generated for different values of No.
The solution strategy uses orthogonal collocation to provide an approximate solution to the resulting set of equations. However, in order to avoid the appearance of quotients composed of series expansions in both the numerator and denominator when the closure hypothesis is substituted into the PDEs, an additional transformation involving the product densities is made before specifying the expansion formula. The variables representing the transformed product densities are then approximated by truncated formulas. At this
~(
PBE
a
N O = i0 "~ Monte Carlo NO 50 J Simulation
,
N O = iO ,
--
I
NO
50
J
419
PDEs
Confidence Limits (95%l
5,
u
4,
o~ 3. =
o~
O,
O.
I
[
I
.5
l,O
1,5
I
2,0 Dimensionless time T
I
I
2.5
3.0
3,5
FIG. 1. The average particle v o l u m e as a function o f time. Results f r o m the constant m i x i n g v o l u m e case with No = 10, 50.
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
420
SAMPSON AND RAMKRISHNA
point the solution strategy becomes very similar to one reported by the authors for the PBE (11). The accuracy of the resulting solution suffers somewhat in comparison with the solution to the PBE due to the fact that fewer terms in the expansion were included in the solution to the PDEs. This was necessitated by the additional combinatoric complexities in the level two closure and the limitation in available precision on the Purdue University CDC 6500/6600 computer (single precision carries 15 significant figures). However, the
governing equations are still satisfied to within 2.0% for all values of reduced volume V/f above 0.03. An approximate solution to the closed PDEs was generated with No = 10 and No = 50 for the constant mixing volume case and Nmi~ = 5, 10, and 20 for the constant population size case (where Nmix is the number of particles in the constant-sized population). The initial size distribution was taken to be exponential. In Figs. 1 and 2 a comparison is made between the solutions to the PBE, the level two closure,
.9999 -
-- --
_
---
: .999 - -
PBE
-
a A
I
NO = i0 ~
Monte Carlo 50 ~ Simulation
NO
////////
I~
]
- ........... NO - i0 PDEs _
- .......
,'
NO = 50;
T
,," j
T
Confidence
~
Limits (95%)
I ~ /
.99 - -
~
~T
/~ ///
,'
,"
/¢
,"
.9
.0
O.
i.
2.
3,
4.
5.
6.
7,
Reduced volume v/v
~G. 2:Thepa~cle ~ e ~ e ~ m .
Res~ts~om ~ e c o n ~ a n t m i ~ volumec~e ~th ~ = 10,50;
r=l.O.
Journal of Colloid and InterfaceScience, Vol. 110, No. 2, April 1986
BROWNIAN AGGLOMERATION and results from a Monte Carlo simulation for the constant mixing volume [3, 4]. The Monte Carlo simulation differs from the one used in the preliminary evaluation of the closure hypothesis in that no attempt was made to estimate the second- and higher-order product densities. It is similar in that it offers an unbiased, but imprecise, estimate of the evolution of a population of agglomerating Brownian particles. The imprecision is conveyed by confidence limits in the figures. Fig-
421
ure 1 plots the dimensionless average particle volume as a function of dimensionless time and shows that the level two closure tends to overcorrect for the error in the PBE. In addition, both the error in the PBE and the error in the level two closure increase as the initial population size decreases. The match between the size spectrum predicted by the level two closure and the simulation results, depicted in Fig. 2, is better than that for the average particle volume; however, the same trend of in-
45,
/ /
PBE
×
/
Nmi x = 5 "i Monte 40.-
35.
A
Nmi x = i 0 ~ 5 Simulation Carlo
0
Nmlx = 20
*
Nmlx 5]1
/ //
///
/!
T
Confidence
i
Limits (95%)
..
,'
//f I/ ,""
~/I// ,'
///, ,./
-~
Z4v 7
d ,
~
~' "
/,4/ ,," ////,"
----,---- Nmi x = i0~, PDEs
i~'- 30.
,,
'ii)"''///'"
20,
///- /
~_
#i/
~_
7//
/?,/ :
/
3.0.
z,/
5.
__
0.
Q.
I 5.
1~
I
iO, 15. Dimensionless time 7"
I 20.
25.
FIG. 3. The average particle v o l u m e as a function o f time. Results f r o m the constant-sized population case with N ~ = 5, 10, 20.
Journal of Colloid and Interface Science, Vol. 110, No. 2, April 1986
422
SAMPSON A N D RAMKRISHNA
creasing error with decreasing initial population size is repeated. Figures 3 and 4 plot the same variables as in Figs. 1 and 2, respectively, except that the solution is for the case of a constant-sized population. In both figures, the agreement between the level two closure hypothesis and the simulation results is very good. The seeming disparity between the ability of the closure hypothesis to accurately solve the constant volume and constant number cases may be explained by differences in the extent of devia-
tion from the PBE rather than a more subtle inherent effect. CONCLUSIONS
The development and solution of an alternate closure to the product density equations represents an elegant and appropriate solution to the dilemma raised when the PBE is known to be inaccurate. Using a level two closure which is general wrt the agglomeration mechanism instead of a general level one closure,
,9999
- -
PBE o
............
I
,999
Monte Carlo Simulation PDEs Confidence Limits (95%)
--
,~ ,"' > I.>
~o .99 _,,,'"
,,,"
~
.9
__ __ I
.0
O.
z
i.
~ 2.
L 3.
I 4. Reduced volume v l q
5.
6.
2.
FIG. 4. The particle size spectrum. Results from the constant-sized population case with N~x = 5, 10.0.
=
Journal of Colloid and Interface Science, Vol. 110, N o . 2, April 1986
BROWNIAN AGGLOMERATION the PBE, the errors, i n c l u d i n g size-specific errors, in the s o l u t i o n o f the closed p r o d u c t density e q u a t i o n s can be r e d u c e d considerably. T h e general n a t u r e o f the s o l u t i o n strategy suggests that it can be used for o t h e r agglome r a t i o n m e c h a n i s m s as well. REFERENCES 1. Smoluchowski, M. V., Physik Z. 17, 557 (1916). 2. Ramkrishna, D., and Borwanker, J. D., Chem. Eng. Sci. 28, 1423 (1973). 3. Sampson, K. J., and Ramkrishna, D., J. Colloid Interface Sci. 104, 269 (1985).
423
4, Sampson, K. J., Ph.D. dissertation, Purdue Univ., West Lafayette, Indiana, 1981. 5. Janossy, L., Proc. Roy Irish. Acad. Ser. A 53, 181 (1950). 6. Bayewitz, M. H., Yerushalmi, J., Katz, S., and Shinnar, R., J. Atmos. Sci. 31, 1604 (1974). 7. Williams, M. M. R., J. Phys. A. 12, 983 (1979). 8. Scott, W. T., J. Atmos. Sci. 24, 221 (1967). 9. Lundgren, T. S., in "'Statistical Models and Turbulence" (M. Rosenblatt and C. Van Atta, Eds.), Springer-Verlag, Berlin, 1972. 10. Rice, S. A., and Gray, P., "The Statistical Mechanics of Simple Liquids," lnterscience, New York, 1965. 11. Sampson, K. J., and Ramkrishna, D., J. Colloid Interface Sci. 103, 245 (1985).
Journal of Colloid and Interface Science, Vol, 1 i0, No. 2, April 1986
