at' 0 - Semantic Scholar
Chemical
Engineering
Science.
1974, Vol.
29, pp. 987-992.
Pergamon Press.
Printed in Great Britain
DROP-BREAKAGE IN AGITATED LIQUID-LIQUID DISPERSIONS D. RAMKRISHNA Department of Chemical Engineering, Indian Institute of Technology, Kanpur 16, U.P., India (First received 3 February 1973; in revisedform 18 September 1973) Abstract-Experimental data from batch vessels on cumulative volumetric drop-size distributions at various times are shown to yield useful information on probabilities of droplet-breakup as a function of drop-size. Such information is sufficient for a priori prediction of drop-sizes in agitated dispersions in batch and continuous vessels. It may also be useful in predicting heat and/or mass transfer in liquid-liquid dispersions by accounting for the simultaneity of transport processes from individual drops and droplet breakage processes:
INTRODUCTION
in agitated vessels evolve as a consequence of droplet breakage and coalescence. The dynamic character of breakage and coalescence has been recognized by some workers[l-31 who have attempted a prediction of drop-size distributions by the method of population balances. However quantitative predictions are precluded by insufficiency of adequate information about droplet phenomena such as breakage and coalescence of droplets. The possibility of obtaining such information about droplet breakage from appropriately designed experiments has been discussed by the author [4]. If consideration is restricted to lean dispersions it becomes possible to exclude the effects of coalescence and confine attention only to droplet breakage. The purpose of this paper is to demonstrate that measurements of cumulative volume distributions of drop sizes at various instants of time obtained from a lean liquid-liquid dispersion in a batch vessel can provide all of the quantitative information required to make predictions of drop-size distributions. Experimental data of this kind obtained by Madden and McCoy [5] have been employed for this demonstration. Drop-size
distributions
Breakage may be characterized by (i) the transition probability function T(v) such that I(v)dt represents the probability that a droplet of volume o breaks in the time interval (t,t + dt) and (ii) the volume fraction of daughter droplets with volume less than v formed from breakage of a droplet of volume v’ represented by the function G(v, v’) which has the properties G(v, v’) = 0 when v ~0;
In an earlier work[4], which was based on number distribution of droplets, the cumulative distribution function G(v, v’) was replaced by two other functions v(v’) and p(v, v’); v(v’) is the mean number of daughter droplets obtained by breakage of a parent droplet of volume v’, and p(v, v’)dv is the probability that a daughter droplet arising from a parent of volume v’ has its volume between v and v + dv. It is readily seen that v”v(v’)p(v”,
is lean in the phase it is easy to show that
l3F
at’
We denote the volume fraction
F(v, t) = 0 when v 5 0;
lim F(v, t) = 1 y-m
v’)dv”.
For a batch vessel when the dispersion
dispersed
ANALYSIS
of drops of volume less than v at time t by F(v, t), which is a cumulative distribution function satisfying the conditions
G(v, v’) = 1 when v 2 v’.
I=
r(v')G( v, v’) dF(v’,
t).
(I)
0
Equation (1) can be easily established either independently or from the corresponding population balance equation written for the number density of droplets of volume v [ 1,4]. The right hand side of 987
D.
988
Eq. (1) is a Stieltjes written as
I
integral
x
r(v’)G(v, v’)
”
RAMKRISHNA
which may also be
+f(v’,t)du’
Equation
where aF/dv is the density function. Equation (1) may be solved for the function F(v, t) if I(v’) and G(v, v’) are known. Our present interest however is one of determining I(v’) and G(v, v’) when F(v, t) has been determined experimentally. Swift and Friedlander [6] in their study of coagulation of hydrosols have introduced the concept of self-preserving distributions which depends upon the existence of a similarity transformation between the particle size variable and time. Kapur[7] has provided an application of this concept to problems in size reduction. Fillipov [8] in an interesting study of particle-splitting processes has shown the existence of explicit similarity transformations under suitable conditions. Thus Eq. (1) is again the focus of discussion. It is assumed that the breakage frequency function I(v) is of the form
r(v) = Ku”
(2)
where K and n are constants. Further the assumption of similar breakage[3] is made which implies the existence of a function g(x), (0 5 x 5 1) such that (3) Thus
Eq. (1) becomes $=
1; Kv’“g (;)
dF(v’, t).
5 = (1 +
Kt)v’” ; gl (;)-g[(t)l’“].
(5) may be rewritten
4(z) =
as
I’ 4(5kl (5)d5
where 4(z) = zf’(z). For times sufficiently large the similarity variable is nearly Ktu” which suggests an interesting possibility for evaluation of the breakage probability from experimental measurements of the cumulative volumetric drop-size distribution F( v, t) at various times in a batch vessel. Madden and McCoy[5] have indeed provided such data which could be used for some preliminary calculations. Their data obtained with low dispersed phase fraction of carbon tetrachloride in water are shown in Fig. 1. For a fixed value of F(v, t) (represented by horizontal lines in Fig. (1) the similarity variable z should remain constant or for t sufficiently large v”t = constant. Thus a log-log plot of v vs t should produce a straight line with a negative slope of - l/n. Further, similar plots for different fixed values of F(v, t) (and hence fixed z) should yield a set of parallel straight lines if the assumptions made in the model are tenable. Figure 2 represents a plot of drop diameter vs time for four arbitrarily selected but widely different fixed values of the cumulative volume distribution function.? The plot in Fig. 2 appears to provide a fair vindication of the theory although considerable additional experimentation would be
(4)
Fillipov[8] has shown that the solution of Eq. (4) eventually approaches an asymptotic distribution in the similarity variable z = (1 + Kt)v”. That the grouping (1 + Kt)v” is a similarity variable is easily verified by letting f(z) = F(v, t) and substituting into Eq. (4) which yields
zf’(z) =
I=lf’(Osi (;)d5
(5)
/ 1.
where
f’ is
the derivative
of f and
’
0
100200300400500600’
7cIO
Drop diameter(microns)
tBecause of the explicit nature of the similarity transformation the test of self-preservation is more direct and reliable.
(6)
Effect
of stirring
time
on drop Fig. 1.
size
distribution
Drop-breakage in agitated liquid-liquid dispersions
CUMULATIVE
VOLUME
I
I
2
L TIME
989
larity variable u “t which we shall again denote by z for economy of notation. Equation (5) then should be replaced by
FRACTION
6 (MINUTES)
I
I
IO
15
I
Multiplying Eq. (5a) by z’-’ and integrating to 30 one gets
from 0
Fig. 2. necessary to establish these results on a firmer basis. An evaluation of the slope yields a value of about 2 for the exponent n so that
By changing the order in which the area of integration is covered by the variables z and 5 it is easy to show that I-L,= KcL,+,&,
where d is the drop diameter. The large exponent on the drop diameter indicates that the probability of drop break-up decreases rapidly as the drop-size is reduced. Using this approximate value of n = 2, the asymptotic distribution function is plotted against the similarity variable t '"d in Fig. 3. In practice of course numerical computation would be more adequate than the graphical procedure used here for the evaluation of n. Of course the value of n is most likely a function of the local turbulence energy, which in turn depends on a number of parameters associated with the agitated vessel. That such batch data can provide further vital information can be seen from what follows. For sufficiently large times one may use the simi-
(8)
where p, represents the left hand side of Eq. (7) and p, is defined by &=
I
’ x’g,(x)
dx.
(9)
x’g(x) dx
(10)
”
If one defines yr =
I’ ”
then it is readily established
that
pr = nylr+l)n-l.
(11)
Measurement of the cumulative volume distribution function should in principle provide for the moments {IL,}through evaluation of the Stieltjes integrals I P, = z’df(z). . I0 From Eq. (8) it is therefore possible to calculate the quantity KP, for various values of r, although much of this hinges on the accuracy with which the values of {CL”}can be obtained. Alternatively, one may define a function
0
IC1(Z”n) = f(z)
0
01 200
n
(12)
with moments
0
x ’ 300
I
1
400
Similarity
1
500
600
l&Id
Variable = t
Fig. 3.
I
A, =
I”
z”” d$(z”“).
(13)
A*
(14)
700
It then follows that /-L/n
=
990
D. RAMKRISHNA
and Eq. (8) may be transformed
to
are the coefficients
I
A, = Khr+nyr-1.
(17)
where LI (u) is the kth Laguerre polynomial and {at}
(18)
may be written as
(16)
Thus from a plot of $(z”“) vs 2”” similar to Fig. 3 or numerically fitting the data to standard polynomial expansions in which the coefficients are known linear combinations of the moments one may obtain the values of the moments {A,}. Equation (16) then provides for the values of {K-y,} from which one obtains in principle the function Kg(x) again by such means as a suitable polynomial expansion [9]. Since g(1) = 1 it is clear that K is obtained from evaluating Kg(x) at x = 1. However Eq. (16) in calculating the moments of g, calls for moments of d$t/dz”” of a much higher order. Thus yr , requires A, and A,,, which in view of inaccuracies associated with the calculation of higher moments of an experimental distribution produces a note of caution. Nevertheless there are two redeeming features. Firstly, the distribution function $ is obtained as collective data of size distributions at various times which provide for a large number of points required for calculating the higher moments. Secondly, the moments of the cumulative distribution function g which is non-zero only in the unit interval decrease progressively in numerical value. Thus g is likely to be determined by a small number of leading moments through a suitable polynomial expansion. Furthermore, since g is the cumulative distribution function of daughter droplet sizes, the zeroth moment determines the average daughter droplet size and the first moment will yield the standard deviation about the mean. These in themselves represent useful information regarding the phenomena of droplet breakage. Since our interest in the moments is only insofar as they determine the distribution, it may prove expedient to expand the distribution in terms of say Laguerre functions (or any other suitable set of functions [9]); the coefficients of such an expansion are known linear combinations of the moments of the distribution. Thus denoting the density function associated with $ by I/J’(U) where u = 2”’ we may write Jl’(u) = emu2, otLk(U)
I
The kth Laguerre polynomial
with (11) yields
given by
a,, = (, Jl’(u)Lk(u)du.
(1% which in combination
of expansion
(19) The rth moment A, is readily seen to be given by the finite sum A, =
2) $ y
x (F)(j + r)!.
ak
j=O
(20)
.
Thus an accurate knowledge of the rth moment A, depends much on the accurate evaluation of the r leading coefficients {(u,}. The data available as +(u) may then be fitted in the sense of least squares to obtain the coefficients {ak} from
which was obtained by integrating Eq. (17) from 0 to u. From Eq. (21) it becomes clear that 4(u) must be obtained over a sufficiently large domain especially in the rising and tapering region of the curve of the distribution function before a reliable estimate of the higher moments can be made. Unfortunately, the data of Madden and McCoy [5], when plotted on ordinary graph paper, are somewhat sparse in the tapering region of the curve so that the higher moments cannot be made with sufficient accuracy. This could have been corrected by making a careful analysis of the left-over larger drops at all times and especially larger times. The technique of sieve analysis used by Madden and McCoy is possibly not suited to this purpose because of the lack of finer subdivisions of the coarser mesh. An estimate of the moments from their data could still have been made but the quantitative significance of such isolated data which have not been obtained for the specific purpose suggested here, is somewhat limited. Hence further treatment of the data of Madden and McCoy has been desisted. It should however be clear that the analysis presented here demonstrates the importance of obtaining size distribution data in batch vessels as a function of time, which has been uncommon practice among experimentalists in this area. It will now be shown that when the functions
Drop-breakage in agitated liquid-liquid dispersions
I(V) and g(vlv’) have been determined from batch data they may also be used to predict drop-size distributions in a continuous flow vessel. Thus consider a vessel which is fed continuously by the continuous and dispersed phases the latter in the form of droplets of known size distribution, while the well-stirred dispersion is simultaneously withdrawn maintaining a constant hold-up in the vessel. The integral equation for this situation may be readily written as follows. a
e,,
F(u, t) = F,(u) - F(u, t)
I
+0
or
l-(u’)G(u, u’) dF(v’, t)
”
(22)
where F,(V) is the cumulative volumetric size distribution of feed droplets and 8 is the average holding time for the droplets and the continuous phase. At steady state we have 6’(u) = F,(u) + 0 ccT(u’)G(u, u’)dP(u’) I” If I(u’) and G(u, u’) may be represented (3) then Eq. (23) becomes @u)=F,(u)+BKj-j’“g(;)dfi(u’).
(23)
by (2) and
Pfr
liquid-liquid dispersions in which they have accounted for the simultaneity of mass transfer processes in individual drops and droplet breakage phenomena. The quantitative information required for applications of such analysis to real systems can be shown to consist of the values of K, n and {yr} provided mass transfer processes do not substantially atfect the processes leading to the breakage of a droplet [4]. CONCLIJSIONS
From a preliminary analysis of data on cumulative volumetric size distribution functions of carbon tetrachloride droplets dispersed in water obtain by Madden and McCoyl51, it appears reasonable to make the following conclusions. A ‘power law’ expression for the transition probability of droplet breakage as expressed by (2) and the assumption of similar breakage of droplets defined by (3) provide a satisfactory description of the behavior of agitated lean dispersions. Batch experiments can provide a considerable amount of quantitatively significant information about droplet breakage, that will suffice for the prediction of drop-size distributions in noncoalescing dispersions. Further this information may also suffice for analysis of rate processes in dispersions in a more rational manner than what has so far been in practice.
(24)
NOTATION
d F
Defining moments
=
I
I
”
u’ dFf(u)
it is easy to show from Eq. (24) that
(26) which represents an unclosed set of equation in {CL,} regarded here as unknowns. Besides one also runs the risk of having to deal with fractional moments if n is not an integer. In such situations it may necessitate a mutilated polynomial expansion of the distribution function to express the higher (or fractional moments in terms of the lower (integral) moments such as that suggested by Hulbert and Katz[lO]. Thus it appears possible to predict the steady state size distribution when {KY,} and n are known from batch experiments. Shah and Ramkrishna [3] have analyzed a population balance model for mass transfer in lean
991
drop diameter volume fraction cumulative distribution function of drop-size cumulative distribution function of drop-size Ff in feed to continuous mixer f cumulative density function in the similarity variable z f’ derivative of f with respect to its argument G cumulative distribution function of daughter droplet size arising from breakage of a larger droplet again in terms of volume fraction g function defined by Eq. (3) ET1function defined in Eq. (5) K constant in the power law expression for transition probability function T(u) as in Eq. (2) n exponent on drop-volume in T(u) given by Eq. (2) P density function of daughter droplet-volume arising from breakage of a parent droplet in terms of number r integer, 0, 1, 2, . . . t time drop volume U
D. RAMKRISHNA variable representing the ratio of daughter drop-volume to parent drop-volume similarity variable defined just preceding Eq. (5) coefficient of expansion of (u) in terms of Laguerre functions defined by Eq. (9) defined by Eq. (10) defined in Eq. (6) defined by Eq. (12) defined by Eq. (13) moments of the cumulative distribution function f defined in Eq. (15) average residence time in continuous flow vessel mean number of droplets from breakage of a parent droplet
REFERENCES
[l] Valentas K. and Amundson N. R., Ind. Engng Chem. Fundls 1966 5 533. [2] Valentas K., Bilous 0. and Amundson N. R., Ind. Engng Chem. Fundls 1966 5 271. [3] Shah B. H. and Ramkrishna D., Chem. Engng Sci. 1973 28 389. [4] Ramkrishna D., Paper presented at 7lst National Meeting of the American Institute of Chemical Engineers, February 1972 at Dallas. Texas, U.S.A. [5] Madden A. J. and McCoy B. J., Chem. Engng Sci. 1969 24 416. [6] Swift D. L. and Friedlander S. K., J. Colloid Sci. 1964 19 621. [7] Kapur P. C., Chem. Engng Sci. 1970 25 899. [8] Fillipov A. F., Theory Prob. Applic. 1961 6 275. [9] Ramkrishna D.. Chem. Engng Sci. 1973 28 1362. [IO] Hulburt H. M. and Katz S. L., Chem. Engng Sci. 1964 19 555.
Engineering
Science.
1974, Vol.
29, pp. 987-992.
Pergamon Press.
Printed in Great Britain
DROP-BREAKAGE IN AGITATED LIQUID-LIQUID DISPERSIONS D. RAMKRISHNA Department of Chemical Engineering, Indian Institute of Technology, Kanpur 16, U.P., India (First received 3 February 1973; in revisedform 18 September 1973) Abstract-Experimental data from batch vessels on cumulative volumetric drop-size distributions at various times are shown to yield useful information on probabilities of droplet-breakup as a function of drop-size. Such information is sufficient for a priori prediction of drop-sizes in agitated dispersions in batch and continuous vessels. It may also be useful in predicting heat and/or mass transfer in liquid-liquid dispersions by accounting for the simultaneity of transport processes from individual drops and droplet breakage processes:
INTRODUCTION
in agitated vessels evolve as a consequence of droplet breakage and coalescence. The dynamic character of breakage and coalescence has been recognized by some workers[l-31 who have attempted a prediction of drop-size distributions by the method of population balances. However quantitative predictions are precluded by insufficiency of adequate information about droplet phenomena such as breakage and coalescence of droplets. The possibility of obtaining such information about droplet breakage from appropriately designed experiments has been discussed by the author [4]. If consideration is restricted to lean dispersions it becomes possible to exclude the effects of coalescence and confine attention only to droplet breakage. The purpose of this paper is to demonstrate that measurements of cumulative volume distributions of drop sizes at various instants of time obtained from a lean liquid-liquid dispersion in a batch vessel can provide all of the quantitative information required to make predictions of drop-size distributions. Experimental data of this kind obtained by Madden and McCoy [5] have been employed for this demonstration. Drop-size
distributions
Breakage may be characterized by (i) the transition probability function T(v) such that I(v)dt represents the probability that a droplet of volume o breaks in the time interval (t,t + dt) and (ii) the volume fraction of daughter droplets with volume less than v formed from breakage of a droplet of volume v’ represented by the function G(v, v’) which has the properties G(v, v’) = 0 when v ~0;
In an earlier work[4], which was based on number distribution of droplets, the cumulative distribution function G(v, v’) was replaced by two other functions v(v’) and p(v, v’); v(v’) is the mean number of daughter droplets obtained by breakage of a parent droplet of volume v’, and p(v, v’)dv is the probability that a daughter droplet arising from a parent of volume v’ has its volume between v and v + dv. It is readily seen that v”v(v’)p(v”,
is lean in the phase it is easy to show that
l3F
at’
We denote the volume fraction
F(v, t) = 0 when v 5 0;
lim F(v, t) = 1 y-m
v’)dv”.
For a batch vessel when the dispersion
dispersed
ANALYSIS
of drops of volume less than v at time t by F(v, t), which is a cumulative distribution function satisfying the conditions
G(v, v’) = 1 when v 2 v’.
I=
r(v')G( v, v’) dF(v’,
t).
(I)
0
Equation (1) can be easily established either independently or from the corresponding population balance equation written for the number density of droplets of volume v [ 1,4]. The right hand side of 987
D.
988
Eq. (1) is a Stieltjes written as
I
integral
x
r(v’)G(v, v’)
”
RAMKRISHNA
which may also be
+f(v’,t)du’
Equation
where aF/dv is the density function. Equation (1) may be solved for the function F(v, t) if I(v’) and G(v, v’) are known. Our present interest however is one of determining I(v’) and G(v, v’) when F(v, t) has been determined experimentally. Swift and Friedlander [6] in their study of coagulation of hydrosols have introduced the concept of self-preserving distributions which depends upon the existence of a similarity transformation between the particle size variable and time. Kapur[7] has provided an application of this concept to problems in size reduction. Fillipov [8] in an interesting study of particle-splitting processes has shown the existence of explicit similarity transformations under suitable conditions. Thus Eq. (1) is again the focus of discussion. It is assumed that the breakage frequency function I(v) is of the form
r(v) = Ku”
(2)
where K and n are constants. Further the assumption of similar breakage[3] is made which implies the existence of a function g(x), (0 5 x 5 1) such that (3) Thus
Eq. (1) becomes $=
1; Kv’“g (;)
dF(v’, t).
5 = (1 +
Kt)v’” ; gl (;)-g[(t)l’“].
(5) may be rewritten
4(z) =
as
I’ 4(5kl (5)d5
where 4(z) = zf’(z). For times sufficiently large the similarity variable is nearly Ktu” which suggests an interesting possibility for evaluation of the breakage probability from experimental measurements of the cumulative volumetric drop-size distribution F( v, t) at various times in a batch vessel. Madden and McCoy[5] have indeed provided such data which could be used for some preliminary calculations. Their data obtained with low dispersed phase fraction of carbon tetrachloride in water are shown in Fig. 1. For a fixed value of F(v, t) (represented by horizontal lines in Fig. (1) the similarity variable z should remain constant or for t sufficiently large v”t = constant. Thus a log-log plot of v vs t should produce a straight line with a negative slope of - l/n. Further, similar plots for different fixed values of F(v, t) (and hence fixed z) should yield a set of parallel straight lines if the assumptions made in the model are tenable. Figure 2 represents a plot of drop diameter vs time for four arbitrarily selected but widely different fixed values of the cumulative volume distribution function.? The plot in Fig. 2 appears to provide a fair vindication of the theory although considerable additional experimentation would be
(4)
Fillipov[8] has shown that the solution of Eq. (4) eventually approaches an asymptotic distribution in the similarity variable z = (1 + Kt)v”. That the grouping (1 + Kt)v” is a similarity variable is easily verified by letting f(z) = F(v, t) and substituting into Eq. (4) which yields
zf’(z) =
I=lf’(Osi (;)d5
(5)
/ 1.
where
f’ is
the derivative
of f and
’
0
100200300400500600’
7cIO
Drop diameter(microns)
tBecause of the explicit nature of the similarity transformation the test of self-preservation is more direct and reliable.
(6)
Effect
of stirring
time
on drop Fig. 1.
size
distribution
Drop-breakage in agitated liquid-liquid dispersions
CUMULATIVE
VOLUME
I
I
2
L TIME
989
larity variable u “t which we shall again denote by z for economy of notation. Equation (5) then should be replaced by
FRACTION
6 (MINUTES)
I
I
IO
15
I
Multiplying Eq. (5a) by z’-’ and integrating to 30 one gets
from 0
Fig. 2. necessary to establish these results on a firmer basis. An evaluation of the slope yields a value of about 2 for the exponent n so that
By changing the order in which the area of integration is covered by the variables z and 5 it is easy to show that I-L,= KcL,+,&,
where d is the drop diameter. The large exponent on the drop diameter indicates that the probability of drop break-up decreases rapidly as the drop-size is reduced. Using this approximate value of n = 2, the asymptotic distribution function is plotted against the similarity variable t '"d in Fig. 3. In practice of course numerical computation would be more adequate than the graphical procedure used here for the evaluation of n. Of course the value of n is most likely a function of the local turbulence energy, which in turn depends on a number of parameters associated with the agitated vessel. That such batch data can provide further vital information can be seen from what follows. For sufficiently large times one may use the simi-
(8)
where p, represents the left hand side of Eq. (7) and p, is defined by &=
I
’ x’g,(x)
dx.
(9)
x’g(x) dx
(10)
”
If one defines yr =
I’ ”
then it is readily established
that
pr = nylr+l)n-l.
(11)
Measurement of the cumulative volume distribution function should in principle provide for the moments {IL,}through evaluation of the Stieltjes integrals I P, = z’df(z). . I0 From Eq. (8) it is therefore possible to calculate the quantity KP, for various values of r, although much of this hinges on the accuracy with which the values of {CL”}can be obtained. Alternatively, one may define a function
0
IC1(Z”n) = f(z)
0
01 200
n
(12)
with moments
0
x ’ 300
I
1
400
Similarity
1
500
600
l&Id
Variable = t
Fig. 3.
I
A, =
I”
z”” d$(z”“).
(13)
A*
(14)
700
It then follows that /-L/n
=
990
D. RAMKRISHNA
and Eq. (8) may be transformed
to
are the coefficients
I
A, = Khr+nyr-1.
(17)
where LI (u) is the kth Laguerre polynomial and {at}
(18)
may be written as
(16)
Thus from a plot of $(z”“) vs 2”” similar to Fig. 3 or numerically fitting the data to standard polynomial expansions in which the coefficients are known linear combinations of the moments one may obtain the values of the moments {A,}. Equation (16) then provides for the values of {K-y,} from which one obtains in principle the function Kg(x) again by such means as a suitable polynomial expansion [9]. Since g(1) = 1 it is clear that K is obtained from evaluating Kg(x) at x = 1. However Eq. (16) in calculating the moments of g, calls for moments of d$t/dz”” of a much higher order. Thus yr , requires A, and A,,, which in view of inaccuracies associated with the calculation of higher moments of an experimental distribution produces a note of caution. Nevertheless there are two redeeming features. Firstly, the distribution function $ is obtained as collective data of size distributions at various times which provide for a large number of points required for calculating the higher moments. Secondly, the moments of the cumulative distribution function g which is non-zero only in the unit interval decrease progressively in numerical value. Thus g is likely to be determined by a small number of leading moments through a suitable polynomial expansion. Furthermore, since g is the cumulative distribution function of daughter droplet sizes, the zeroth moment determines the average daughter droplet size and the first moment will yield the standard deviation about the mean. These in themselves represent useful information regarding the phenomena of droplet breakage. Since our interest in the moments is only insofar as they determine the distribution, it may prove expedient to expand the distribution in terms of say Laguerre functions (or any other suitable set of functions [9]); the coefficients of such an expansion are known linear combinations of the moments of the distribution. Thus denoting the density function associated with $ by I/J’(U) where u = 2”’ we may write Jl’(u) = emu2, otLk(U)
I
The kth Laguerre polynomial
with (11) yields
given by
a,, = (, Jl’(u)Lk(u)du.
(1% which in combination
of expansion
(19) The rth moment A, is readily seen to be given by the finite sum A, =
2) $ y
x (F)(j + r)!.
ak
j=O
(20)
.
Thus an accurate knowledge of the rth moment A, depends much on the accurate evaluation of the r leading coefficients {(u,}. The data available as +(u) may then be fitted in the sense of least squares to obtain the coefficients {ak} from
which was obtained by integrating Eq. (17) from 0 to u. From Eq. (21) it becomes clear that 4(u) must be obtained over a sufficiently large domain especially in the rising and tapering region of the curve of the distribution function before a reliable estimate of the higher moments can be made. Unfortunately, the data of Madden and McCoy [5], when plotted on ordinary graph paper, are somewhat sparse in the tapering region of the curve so that the higher moments cannot be made with sufficient accuracy. This could have been corrected by making a careful analysis of the left-over larger drops at all times and especially larger times. The technique of sieve analysis used by Madden and McCoy is possibly not suited to this purpose because of the lack of finer subdivisions of the coarser mesh. An estimate of the moments from their data could still have been made but the quantitative significance of such isolated data which have not been obtained for the specific purpose suggested here, is somewhat limited. Hence further treatment of the data of Madden and McCoy has been desisted. It should however be clear that the analysis presented here demonstrates the importance of obtaining size distribution data in batch vessels as a function of time, which has been uncommon practice among experimentalists in this area. It will now be shown that when the functions
Drop-breakage in agitated liquid-liquid dispersions
I(V) and g(vlv’) have been determined from batch data they may also be used to predict drop-size distributions in a continuous flow vessel. Thus consider a vessel which is fed continuously by the continuous and dispersed phases the latter in the form of droplets of known size distribution, while the well-stirred dispersion is simultaneously withdrawn maintaining a constant hold-up in the vessel. The integral equation for this situation may be readily written as follows. a
e,,
F(u, t) = F,(u) - F(u, t)
I
+0
or
l-(u’)G(u, u’) dF(v’, t)
”
(22)
where F,(V) is the cumulative volumetric size distribution of feed droplets and 8 is the average holding time for the droplets and the continuous phase. At steady state we have 6’(u) = F,(u) + 0 ccT(u’)G(u, u’)dP(u’) I” If I(u’) and G(u, u’) may be represented (3) then Eq. (23) becomes @u)=F,(u)+BKj-j’“g(;)dfi(u’).
(23)
by (2) and
Pfr
liquid-liquid dispersions in which they have accounted for the simultaneity of mass transfer processes in individual drops and droplet breakage phenomena. The quantitative information required for applications of such analysis to real systems can be shown to consist of the values of K, n and {yr} provided mass transfer processes do not substantially atfect the processes leading to the breakage of a droplet [4]. CONCLIJSIONS
From a preliminary analysis of data on cumulative volumetric size distribution functions of carbon tetrachloride droplets dispersed in water obtain by Madden and McCoyl51, it appears reasonable to make the following conclusions. A ‘power law’ expression for the transition probability of droplet breakage as expressed by (2) and the assumption of similar breakage of droplets defined by (3) provide a satisfactory description of the behavior of agitated lean dispersions. Batch experiments can provide a considerable amount of quantitatively significant information about droplet breakage, that will suffice for the prediction of drop-size distributions in noncoalescing dispersions. Further this information may also suffice for analysis of rate processes in dispersions in a more rational manner than what has so far been in practice.
(24)
NOTATION
d F
Defining moments
=
I
I
”
u’ dFf(u)
it is easy to show from Eq. (24) that
(26) which represents an unclosed set of equation in {CL,} regarded here as unknowns. Besides one also runs the risk of having to deal with fractional moments if n is not an integer. In such situations it may necessitate a mutilated polynomial expansion of the distribution function to express the higher (or fractional moments in terms of the lower (integral) moments such as that suggested by Hulbert and Katz[lO]. Thus it appears possible to predict the steady state size distribution when {KY,} and n are known from batch experiments. Shah and Ramkrishna [3] have analyzed a population balance model for mass transfer in lean
991
drop diameter volume fraction cumulative distribution function of drop-size cumulative distribution function of drop-size Ff in feed to continuous mixer f cumulative density function in the similarity variable z f’ derivative of f with respect to its argument G cumulative distribution function of daughter droplet size arising from breakage of a larger droplet again in terms of volume fraction g function defined by Eq. (3) ET1function defined in Eq. (5) K constant in the power law expression for transition probability function T(u) as in Eq. (2) n exponent on drop-volume in T(u) given by Eq. (2) P density function of daughter droplet-volume arising from breakage of a parent droplet in terms of number r integer, 0, 1, 2, . . . t time drop volume U
D. RAMKRISHNA variable representing the ratio of daughter drop-volume to parent drop-volume similarity variable defined just preceding Eq. (5) coefficient of expansion of (u) in terms of Laguerre functions defined by Eq. (9) defined by Eq. (10) defined in Eq. (6) defined by Eq. (12) defined by Eq. (13) moments of the cumulative distribution function f defined in Eq. (15) average residence time in continuous flow vessel mean number of droplets from breakage of a parent droplet
REFERENCES
[l] Valentas K. and Amundson N. R., Ind. Engng Chem. Fundls 1966 5 533. [2] Valentas K., Bilous 0. and Amundson N. R., Ind. Engng Chem. Fundls 1966 5 271. [3] Shah B. H. and Ramkrishna D., Chem. Engng Sci. 1973 28 389. [4] Ramkrishna D., Paper presented at 7lst National Meeting of the American Institute of Chemical Engineers, February 1972 at Dallas. Texas, U.S.A. [5] Madden A. J. and McCoy B. J., Chem. Engng Sci. 1969 24 416. [6] Swift D. L. and Friedlander S. K., J. Colloid Sci. 1964 19 621. [7] Kapur P. C., Chem. Engng Sci. 1970 25 899. [8] Fillipov A. F., Theory Prob. Applic. 1961 6 275. [9] Ramkrishna D.. Chem. Engng Sci. 1973 28 1362. [IO] Hulburt H. M. and Katz S. L., Chem. Engng Sci. 1964 19 555.
