094_1990_Tobin_Deter.. - Purdue Engineering - Purdue University
Chemical Printed
Engineering in Great
Science,
Vol.
45, No.
12, pp. 3491-3504,
1990.
ooos2509po 93.00 + 0.00 Q 1990 Pergamon Press plc
Britain.
DETERMINATION IN LIQUID-LIQUID
OF COALESCENCE FREQUENCIES DISPERSIONS: EFFECT OF DROP SIZE DEPENDENCE
T. TOBIN, R. MURALIDHAR, H. WRIGHT and D. RAMKRISHNA’ School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (First received 25 September 1989; accepted
in revisedform
1 March
1990)
Abstract-The size dependence of the drop coalescence frequency is investigated by measurement of transient drop size distributions in purely coalescing systems (with negligible drop break-up). A twopronged approach is employed to estimate the bivariate coalescence frequency function, using such experimental data. First, coalescence frequency expressions derived from mechanistic models of the relative motion of the drops are evaluated based on their ability to predict the experimental transient sire spectra. Second, experimental drop size distributions which exhibit a property known as self-similarity are analyzed through an inverse problem in order to extract the coalescence frequency function directly from the data. Results indicate that the coalescence. frequency of small droplets (l&SO pm in diameter)is lower than that predicted from a constant coalescence efficiency model. In addition, experimentsshow that, given favorable
initialconditions, self-similardrop size distributionscan be manifest.In the cases where similaritybehavior is observed, the frequenciesobtained from the inverse problem are in qualitative agreement with the mechanistic models that describe the data best.
1. INTRODUflrON Many engineering operations, such as liquid-liquid extraction and multi-phase reaction, involve the formation of stirred dispersions of two immiscible liquids. In such systems drop coalescence (and breakup) can profoundly influence the overall performance, by altering the interfacial area available for species exchange between the phases. In order to optimize these operations fully it is therefore desirable that engineers learn more about the dispersive process, through detailed studies of drop coalescence and break-up. A rational approach to the modelling of dispersive systems begins with the framework of population balances, and is discussed by Ramkrishna (1985). This describes the temporal evolution of a drop population due to random coalescence and break-up events. For a purely coalescing dispersion in which no mass exchange is involved, the population balance equation is given by
ai-qfi,q
1
dtl=-
2s0
-
”
g(u”-
r 0
v”‘,o’)A(a -
v”‘,E)ri(C’, t’)dC
ij(iJ, v”‘)ii(i?, F)ii(a’, 7)dv”
(1.1)
where n(& 7) is the number density (concentration) of drops of volume 6 at time ?, and q(iJ,v”) is the binary coalescence frequency for drops of volumes v”and 5’. The bivariate frequency function is a key input to the coagulation equation, and must be either determined from experiments or derived from phenomenological models of drop coalescence.
‘Author
to whom correspondence
should be addressed.
Many previous studies have sought to gain information about coalescence frequencies in a turbulent flow field. Such studies may be grouped into three basic categories, according to the experimental methods employed. The first type of study infers an average coalescence frequency from measurable changes in the physical properties of a system. An example of this is the work of Madden and Damerell (1962) in which coalescence rates of water droplets in toluene were estimated by observing the rate of extraction of iodine from the toluene into the water drops. From their experiments they reported the following relationship: 4 = fi2.*+o.504
(1.2)
where R is the stirring speed, 4 is the volume fraction of the dispersed phase, and q is the average coalescence frequency. A similar work by Miller et al. (1963) yielded basically the same results as those of Madden and Damerell. In a slightly different type of experiment, Howarth (1967) used a light absorbance technique to determine the rate of change of the mean drop size in a stirred vessel following a stepwise reduction in the impeller speed. Howarth obtained an average coalescence frequency from the change in average drop size with time, and his results were also consistent with those cited above. Although these studies can reveal the gross behavior of the coalescence frequency as a function of the energy dissipation rate and the dispersed-phase fraction, they are of very limited utility for the following reasons. First, they yield no information about the effect of drop size on the coalescence frequency, since they are built on the tacit and severe assumption that coalescing dispersions are essentially mono-disperse. Real dispersions can and often do have drops of
349 1
3492
T. TOBINet al.
extremely disparate sizes, however. Because of this, information about coalescence as a function of drop size is essential if one is to realistically model and study these dispersive systems. Second, in the experiments of Madden and Damerell and Miller et uI. the coalescence frequencies were measured under conditions where mass transfer was occurring between the phases. Groothuis and Zuiderweg (1964) have clearly shown that any such mass transfer drastically alters the coalescence process. At the very least then such an effect needs to be taken into account when attempting to interpret the results. A second basic type of experimental study has been to visually observe coalescence events, and thereby determine directly the coalescence frequency as a function of drop size, dispersed-phase fraction, etc. Kuboi et al. (1972) attempted this using high-speed photography, and their technique was employed by Park and Blair (1975) in a later study. Although this approach has potential to yield excellent estimates of the coalescence frequency in lean dispersions, the investigators who employed it invariably had difficulty in observing a sufficient number of coalescence events to obtain any reliable estimates of the frequency. A third approach of investigators in the past has been to measure steady-state drop size distributions in batch stirring experiments. Under these conditions, drop break-up and coalescence are in balance and the size spectrum is therefore time-independent. Coulaloglou and Tavlarides (1977) were among the first to use measurements of drop size distributions with the objective of understanding the size dependence of drop coalescence and break-up. They derived frequency expressions for drop break-up as well as drop coalescence, and using these expressions they were able to predict the steady-state spectra well. Although the last-mentioned approach represents a significant advance over prior efforts, there is still one serious limitation. Since drop break-up and coalescence depend on the same physical parameters, it is extremely difficult to reliably pinpoint the dependencies of these two processes on the experimental conditions and on the properties of the two phases. In fact, using qualitatively different coalescence and break-up frequency models currently found in the literature, Chatzi and Lee (1987) have shown that there is no significant difference in the ability of any combination of these models to predict steady-state drop size spectra. This conclusion is not surprising, because steady-state simulations cannot accept the burden of determining the rates of two balancing processes. Some investigators have recently reported using a slight variation of this technique to study transient drop size distributions [see Laso et al. (1987) and Konno et al. (1988)]. In these studies dispersions have been perturbed (for example by subjecting them to a step change in impeller speed) and then observed as they settle back to a steady state. While this method
yields somewhat improved results, these studies still suffer significantly from the obscuring effects of drop break-up aliuded to above. It is evident from the preceding discussion that very little exists in the literature by way of reliable information regarding the dependence of the coalescence frequency on the sizes of two coalescing drops. A major cause for this has been the lack of a robust experimental and theoretical approach to tackle the coalescence problem. Our approach for investigating coalescence hinges on measurement of transient drop size distributions in purely coalescing systems. When break-up effects are negligible, one can clearly observe drop size distributions as they evolve through coalescence events. These coalescing dispersions then carry fingerprints of the coalescence frequency. A two-pronged approach is employed to understand the drop size dependence of the coalescence frequency. First, coalescence frequencies derived from mechanistic models of coalescence are evaluated, based on their ability to predict transient coalescence size spectra. Second, in cases where the dyanamic size spectra exhibit self-similar behavior, an inverse population balance equation is employed to directly extract the bivariate coalescence frequency function. The outline of this paper is as follows. In Section 2, the theoretical framework for exploiting the transient size spectra is presented. Section 3 discusses the experimental aspects. In Section 4 we investigate the ability of different coalescence frequency expressions to predict the transient size spectra, while in Section 5 we present the coalescence frequencies obtained via an inverse problem formulation. Section 6 summarizes the salient aspects of the present work and describes the main conclusions that can be drawn.
2
THEORETICAL
FRAMEWORK
TRANSIENT
SIZE
FOR
EXPLOlTING
SPECTRA
Studies of drop coalescence made in the absence of drop break-up are amenable to two different but complementary forms of analysis. The first approach involves observing salient features of the evolving droplet size spectra and subsequently investigating the ability of coalescence frequency expressions to reproduce these same features. These coalescence frequencies may be derived from mechanistic models of the coalescence process, and several such expressions are given in Table 1. (The models in Table 1 only show dependence on drop size; since dependence on other properties of the dispersions were not studied, these dependencies have been lumped into the constants of the models.) The first of these expressions, called the constant efficiency model, is derived by considering the collision frequency between drops in a turbulent flow field in a manner analogous to that used to derive the kinetic theory of gases (Delichatsios and Probstein, 1975). Writing the collision frequency w as the product of the collision cross section and the root mean
Determination of coalescence frequenciesin liquid-liquid
dispersions
3493
Table 1. Coalescence frequency models used in coalescence rate studies Model name
Coalescence frequency expression
(1) Constant efficiency
k(*l/” +
(2) Dynamic
k(vl’”
deformation+
(3) Static deformation’
k(v
l/3
1 - exp (-
k(v 113
IiF1gives
+ U’1’3)2 [i?Pl.
161 Oc E1”Y1”
where r may be taken as the distance
+
&/3)7/3
-> ;
k, k, ~--4~3) k,t4’3)
exp
+
v’
-k,E” vv’(u1’3 + IFS)
1
(2.1)
(2.2)
between
drop
centers. Combining this with eq. (2.1), and making the assumption that a constant fraction of all collisions result in drop coalescence, the constant efficiency expression of Table L is obtained. The remaining three expressions of Table 1 are obtained by considering coalescence to be a two-step process. The first step is termed collision and describes the approach of a pair of droplets to a close proximity, at which point a viscous thin film of liquid separating the droplets resists their relative motion. The second
3.700
z
exp (-
k, < 1, k, w 0
Essentially all drops observed in these experiments were larger than the Kolmogorov dissipation length. For such drops it has been shown by Shinnar (1961) and others that
N 0
1-
(1988).
fluctuation
o(u, 0’) = (67rz)1’3 (L+
y’1/3)7/3
P)
”
(4) Kinetic collision
square relative velocity
+ ~‘~r~)‘/~ exp (-k,
a> 0
0
0.93
1
2’ w
G 0.010
0.910 0.00
2001
1
2001 1
2001-1
1.230 “0
0.927
z
0.622
-> J G
I.830 n*
1.380
1
0.917
-a. 0.316
>-
0.459
G 0.010
2001 1
2001-1
(b) Fig. 1. Typical
0.000
2001-
1
(d)
coalescence frequency surfaces: (a) constant eiliciency model, (b) dynamic model, (c) static deformation model, (d) kinetic collision model.
deformation
3494
T.
TOBIN
step is termed film drainage and involves the draining of the film mentioned above to permit actual coalescence of the droplets. During this stage, the forces from the contiguous turbulent flow can separate the drops; consequently, not all collisions will lead to coalescence. From this view of coalescence the coalescence frequency may be written as the product of a collision frequency and a coalescence efficiency or probability. It should be emphasized that this factorization is possible only for dilute dispersions for which the characteristic time of the collision process is much larger than the timescale of film drainage. In the latter three models the collision frequency term is the same as that used in the first model, while the coalescence efficiency is obtained from an analysis of the drainage of the viscous liquid film which separates two colliding drops. Models 2 and 3 were derived previously (Muralidhar and Ramkrishna, 1986a; Muralidhar, 1988) so they will be described here only briefly. In both models, the Reynolds film drainage equation was analyzed with a fluctuating random force depicting turbulence. Coalescence was assumed to occur if the film drains to a critical thickness under the action of the dynamic turbulent pressure fluctuations. The drops were regarded as separated if the film expanded sufficiently for hydrodynamic retardation to be negligible. As a first approximation, van der Waals attractive forces and electrical double layer forces were regarded as only determining the film rupture thickness, and did not enter the dynamic analysis. When the inertia of the colliding drops is neglected and the characteristic time of turbulent force fluctuations is much smaller than the drop deformation time, the deformation of the drops may be determined by an average turbulence force squeezing the drops. This average force (given by the product of pressure fluctuations across the drop pair and the projected area of the drops) is assumed positive because of the absence of turbulence between the contacting drops. The resulting model, referred to as the static deformation model, predicts an increase in the coalescence efficiency with increasing drop size. On the other hand if the drop shapes respond to every fluctuation in the force field, their deformation is determined by the instantaneous force squeezing the drop pair. This gives rise to the model referred to as the dynamic deformation model, which predicts a decrease in the coalescence probability with increasing drop size. In violent collisions, the inertia of the drops is the governing factor and the effectiveness of the collision depends on the adequacy of the collision kinetic energy to accomplish film drainage work. By performing a dimensional analysis on this process one may arrive at the fourth coalescence expression, termed here the kinetic collision model. This model yields a drop size dependence of coalescence frequency similar to the static deformation model, though it was obtained from a very different form of analysis. The restriction to purely coalescing systems also permits one to probe certain symmetries of the transi-
et al.
ent size spectra which can be effectively employed to directly extract the bivariate coalescence frequency function [see, for example, Swift and Friedlander (1964) or Ruckenstein and Pulvermacher (1973)-J. More specifically, the size distributions at different times can be meaningfully represented by a single scaling function V(z) defined by Y(z) = i-q, t)2,
z=&
(2.3)
where o 5 4 z 3
al.
1.2
0.9 iE
0.8
1.0
d
i
0.7
I
0.8
E F
0.6
0.6
q
. 0.s 0.4
Omin 1smm
- 3Omm
8 0.4
0.3
B
0.2
2 1
0.2
0.1
0
2
4
Drop
6
6
volume
10
12
(~1
14
16
16
30
60
x 10’)
Fig. 2. Transient size spectra for experiment I (5% benzene-carbon
90
Drop
120
lb0
diameter
160
210
240
(pm)
tetrachloride in water).
270
1
Determination
of coalescence frequencies in liquid-liquid
;
1.0
,’
5. i :a
:,’ ,-
s.
‘1
.
08
---
0.6
8 3
0.4
E s
1.0
3 a eI
Omin
-.-.-
15 min
-..-
30
mhl
% k
I..
.
.
;.c
‘.
6
0Z6
,,!
‘.
omin
‘.
‘*..i
-.-.- 15
:
.,“,
0.6
---
b, .
:
L
a Y z
-..-
l
.
2
0.2
30
60
90
120
Drop
0.4
lb0
160
diameter
210
240
270
0.2
0
!
\-.______ . 0
30
60
90
120
Drop
(Frn)
___
c. I ,
E a
i
‘. :
: 0.6
0 mill _.-._ISmin
8 z 3
5
;A “•a.,
0.8
1
0.6
0.4
E i
0.4
0.2
1
0.2’
-..-
0
30
60
90
120
Drop
210
240
270
?
(pm)
.
g
0.6
z d
1.0
Ei
,
-3 3
180
(b)
1.2
1.0
150
diameter
(a)
CL
min
30 min
E
0
i
3497
1.2
1.2
i
dispersions
150
160
diameter
30
210
min
240
I
Omi”
-..-
‘i
\\
*
5 min
30min
.
0+
270
___ -.---
+-p.__._._, C)
30
60
90
120.150
Drop
(pm)
160
diameter
210
240
270
Qun)
(d)
(cl
Fig. 3. Test of coalescence frequency expressions using data of experiment 1: (a) fit of constant efficiency model to data, (b) fit of dynamic deformation model for k, = 0.0001, (c) fit of static deformation model, (d) fit of kinetic collision model.
the shift of the number density peak to the left, the slow decrease in the number density of the smallest drops, the rapid decrease in the number density in the size range 100-l 50 and the rather slow increase for the larger drops.
1.0
-
-
q
0.9.
s
3 0 P
s < 3
-0 n
0.6. 0.7.
l
0.6.
q
0.5.
l
0.4.
‘C o zc E
0.2.
G
0.1
0.3
04
s
-
.
.
.
A repeat of the previous experiment (experiment 2) was performed to investigate the reproducibility of the experiments. The data are depicted in Fig. 4. The data show the same basic features of the previous experiment although not identical_ The differences in the
a
a
1
= 7 mill . 15 twin a 25 min
=
m
7 min 15min a 25 min
l
l
.;
‘) 0
2
. 4 Drop
. 6
8
vohme
10
. 12
(~1
14
. 16
x lo3
1
18
Fig. 4. Transient size spectra for experiment 2 (5%
0
30
60
90
120
Drop
benzene-carbon
150
diameter
160
210
240
(pm)
tetrachloride in water).
270
!
IO
3498
T. TOBIN
initial conditions could be due to two reasons: first, the time of prestirring at a high impeller speed was different, and perhaps was not of sufficient length to attain a steady-state distribution. Second, there may have been trace contaminants in the experiments, which can significantly alter the break-up and coalescence frequencies and thus introduce experimental error. The predictions using different models are depicted in Fig. S(aHd). Although none of the models describe the steep decrease in number density for drops approximately 100 pm in diameter, the static deformation and the kinetic collision models perform better than either a constant efficiency model or a dynamic deformation model. A third coalescence experiment (experiment 3) was performed on a dispersion of hexanesarbon tetrachloride in water and the results are shown in Fig. 6. By far, the most interesting and significant feature of this data set is the tendency to evolve into a bimodal distribution with time. This in itself reveals a great deal about the coalescence frequency function. The intuitive interpretation is that the coalescence fiequency of the smallest droplets is very small, as a
et al.
consequence of which such droplets linger for long times, leading to the development of bimodality. The dynamic deformation model is unable to predict the bimodality, although the fit to the data is not unreasonable [Fig. 7(a)]. In contrast, the static deformation model shows bimodality, although not nearly as pronounced as in the actual data. The simulation with this model, along with the experimental curves, is shown in Fig. 7(b). The transient coalescence experiments thus consistently show that of all the models tested the static deformation and the kinetic collision models describe the transient coalescence data best. It thus appears that the coalescence frequency of the droplets is an increasing function of the drop sizes.’ Further discrimination between the kinetic collision and static deformation models can be made by studying the dependence of the frequency on other physical parameters such as continuous-phase viscosity, interfacial tension, energy dissipation, etc. ‘This trend is in conflict with that predicted by using the coalescence frequency of Coulaloglou and Tavlarides.
2.6
2.6 r-
2.4
E a. ‘1, i
-
e! e
2.0 1.6 1.2
8 &i s 1
0.6 0.4
I ,
0
2.6 ; i
2.4 E a
3 w. E E
2.0 1.6 1.2
G $
0.6
e zz
0.4
0
30
30
i-
2.4
E 5*
2.0
.s z em f 28
__._
1.6
-.-.-..-
I: 1.2
7 min
16 min 25 min
i $
0.6
i
J 1
60
SO
Drop
120
diameter
150
460
210
0.4
{ r
0
2
30
(pm)
60
SO
Drop
120
diometcr
150
.160
210
I 240
(pm)
fb)
la) 2.8 i-
2.4 E
___
60
SO
Drop
120
diometer (cl
7 min
_._._
16 min
-..-
2S
160
160
(pm)
min
210
P
i a.
2.0
2 ~
1.6
Y ::E
1.2
zz
0.6
1
0.4
__-
0
30
60
Drop
SO
120
diameter
7 min
-.-.-
15 min
-..-
25 mm
150
160
210
240
(pm)
fd)
Fig. 5. Test of coalescence frequency expressionsusing data of experiment 2: (a) fit of constant efficiency model, (b) fit of dynamic deformation model for k, = 0.0002, (c) fit of static deformation model, (d) fit of kinetic collision model.
Determination 1 .o
5
0.9
Q
0.8
*
l
x
:
dispersions
3499
.
.
.
q
of coalescence frequencies in liquid-liquid
p
0.5
5
0.4.
.
=:
k
0.7
i 3 g > al .Z! g
0.6
3 h
0.5 0.4 0.3
i
0.2
3
0.1 0
.= E!
1
a
a
1 min
. 30 a 60
m.
8 5
min min
0.2
2
7 E
i
0.5
-
I 10
5
15
01
20
(~1 x
‘.
‘8 ,
,’ I’
o.4
.
40
80
1 mill
m-e
120
Drop
I-.
,’ a’
0
10’)
I
a_
-3
240
260
*
0.1
Fig. 6. Transient size spectra for experiment 3 (5%
,
min min
E
Drop volume
0.6
30 a 60
l
0.3
-.-.-
30
mm
-..-
60
mln
hexane-carbon
160
200
diometer
?
I
(pm)
tetrachloride in water).
forces. We are currently investigating the effect of salt concentration on coalescence rates as salt can diminish the influence of any electrical double layer forces. The mechanistic models will also be subject to refinement to include such colloidal forces if they turn out to be significant. 5. SELF-SIMILAR SlZE SPECTRA AND THE INVERSE PROBLEM
0
40
80
120
Drop
160
diameter
200
280
240
320
(pm)
(0)
-!-! a
0.4
,’ 1
___
:
:
-.-..
:
1 min 30
min
I
iamb
The inverse coalescence problem is directed at deriving the coalescence frequency function from transient drop size spectra of purely coalescing systems. The procedure involves deriving an integral equation for the coalescence frequency function. While a detailed description of the inverse problem is given in Wright et al. (1990), we present the salient aspects in this section. The direct inversion of the population balance equation [eq. (Ll)] for the unknown bivariate frequency function using the transient data is difficult. The inverse problem procedure is considerably simplified in situations where the transient size spectra are self-preserving. We make the population balance equation [eq. (l.l)] dimensionless by defining the following dimensionless quantities: n(o, t) = fi(~,7)v’,/rj,,
q(v, If) = 4”(6, v”)/E, ,20
Drop
160
diameter
200
240
280
320
(*ml
(b) Fig. 7. Test of coalescence frequency expressions using data of experiment 3: (a) fit of dynamic deformation model, (b) fit of static deformation model.
An interesting feature of the experiments is the slow rates of coalescence compared to that observed by Delichatsios and Probstein (1975) for particles much smaller than the microscale of turbulence. This indicates a very small coalescence efficiency for droplets in a turbulent flow field. This could be due to hydrodynamic interaction as well as repulsive electrostatic
v(u’) = E(O’)/v-,,
t = (EN~)f.
(5.1)
In the above, @,, is the initial total number density, and uOis the initial average volume. These two quantities are given by
No=
,=--n(r, r,)d6
s0
m fi,i&
=
s0
iFi@, t’,) dB. (5.2)
The constant E is a multiplicative factor having dimensions of volume per time and may be set equal to unity without loss of generality. It is useful to define the cumulative volume fraction distribution function when self-similar size distributions are observed: F(v, t) =
u’n(u’, t) dv’.
(5.3)
3500
T. TOBIN et al.
In terms of dimensionless variables the coagulation equation [eq. (1. l)] written in terms of the cumulative volume
fraction
aFti t)
law
becomes
s(t) oc t
(5.4)
the average size s(t). On substituting eq. (5.9) into eq. (5.8) and using eq. (5.10), we obtain
We introduce the transformation F(v,
c) -f(z),
z = w(f)
(5.3
where s(t) is the inverse of a mean droplet size, , at time t. Frequently employed choices for s(t) are
$$$, (b) s(t) = 1
M,(t) M,(t)”
(c) s(t) = s 3
(5.6) although in certain cases not all the above equations yield similarity behavior [see von Dongen and Ernst (1988)]. For all the data presented in this work, eq. (56b) was employed in the similarity analysis. In the above, M,(t) is the pth moment of the distribution at time t and is given by m M,(t)
=
(5.11)
The parameter m can be determined by a fit of eq. -S:dF(v’,t)~~“,dF~~‘r)q(u’,u’~). (5.10) using an experimentally measured reciprocal of
dt=
(a) s(t) =
C&l.
I0
urn(u, t) du
p = 0, 1,
...
M,(t)
=
1. (5.7)
The similarity distribution in terms of the number density ‘Y(z), referred to in Section 2, is given by f’(z) On substituting the scaling transformation -. z given by eq. (5.5) into eq. (5.4), we obtain
where the primes on s and f denote the derivatives of these functions with respect to their arguments. Under conditions where similarity behavior are observed, it has been shown by Wright et 01. (1989) that the coalescence frequency must be of the form 4 [&,
G]
= &s(t)-“lb(Y, 0.
(5.9)
If the function b(z’, z”) is the same as q(z’, z”), the coalescence frequency function is homogeneous of degree m. The coalescence frequency is hence completely specified by the exponent m and the function b(z’, z”). During self-similar behavior, the transient spectra can be succinctly described in terms of the scaling spectrum f(z) and the size scaling variable s(t). The rate of evolution of the population is described by the decay law for the reciprocal of the average size s(t) which is shown to be (Wright et al., 1989) s’s(t)m-Z = -a,.
(5.10)
Physically, this equation implies that, for large times, the average drop size grows according to the scaling
a,zf’(z) =
I ss0
m
b(z’, z”)f’(z’)f’(z”)/z”dz”dz’
z-z’
(5.12)
which is an integral equation for the unknown function b(z’, z”) in terms of the similarity spectrum. The solution procedure for b(z’, z”) from eq. (5.12) has been outlined in Wright et al. (1989). The coalescence frequency was determined via the inverse problem for two different transient coalescence experiments that exhibited similarity behavior. In case 1, the transients were obtained from a 5% dispersion of hexane-carbon tetrachloride in water. At t = 0 the impeller speed was reduced from 800 to 200 rpm. Case 2 refers to experiment 1, outlined in Section 4. It is pertinent to note that self-similar behavior is not always observed in experiments of this type. Thus experiment 3 of Section 4 did not lead to a self-similar distribution during the course of the experiment. This is possibly because of an initial preponderance of very small droplets; the slow coalescence of these droplets would cause the initial distribution to be retained for a long period, perhaps longer than the length of the actual experiment. Figure 8(a) shows the transient drop size distributions for case 1. The transients did not possess any distinct characteristics that would permit discrimination between the different model frequencies by the strategy described in the previous section. The size spectra are self-similar with respect to the scaling size in eq. (5.6b), however, as is evident from Fig. 8(b); thus the data can be subjected to the inverse problem methodology. The exponent m was estimated from the asymptotic property of the scaling spectra derived by von Dongen and Ernst (1988):
f ‘cd
~
z
x
zme(-orr) for large z
(5.13)
as well as the decay law given by eq. (5.10). A value of 0.8 was obtained. The value is close to the degree of homogeneity of the constant efficiency model (7/9). The integral equation for b(z’, z”) was solved [see Wright et al. (1989) for details] and the solution is shown in Fig. 8(c). Although not identical to the static deformation model (or kinetic collision model) frequency surface, it is also an increasing function of the size. The coalescence frequency obtained from the inverse problem was assessed by forward simulation using eq. (1.1) using the data at size distribution at the earliest time of similarity behavior as the initial condition. It is evident from Fig. 8(d) that the frequency correctly predicts the transients.
Determination l.O--
of coalescence frequencies in liquid-liquid
-
.. . . . . .
m % .ti
e
M
0.8
0.9 L
.
.
L .
-1
s
-.
E
0.7.
E
OS
‘, >
0.5’
g
0.4’
i x 5 aa >
m 15 min
....1
“_yiy-.
3501
q.0,
1
.-
0.9
dispersions
a 30 min L 60 min
0.8 0.7 I
0.6
15 mill
* 30 min - 60 min
0.5 0.4
4J
4
0.3
i s
0.2 0.1
0
0.050
0.150
0.100 Drop
vohme
0.200
0
0.250
to-3
10-z
(~1)
lo-’
Similority
variable
100
(v/v,,
1
(b)
(a 1
0.9
.E
0.8
3.46149
t; e
0.7
2.33492
2
0.6
E :
3 g >
= Iu _
-N
I. 20034
g u 4 i 3
-
Predicted II lbmin l 30min - 60min
0.5 0.4 0.3 0.2 0.t 0
10-6
10-J
10-4 Drop
lo-'
volume
10-z
lo-'
100
1~1)
(d)
(cl
Fig. 8. Inverse problem for experiment4 (5% hexsne- carbon tetrachloridein water).
The drop size distribution of case 2 also exhibit similarity behavior as is evident from Fig. 9(a) and (b), although the spectra have not yet converged for small z. As there were only two times at which s(t) was available, it was difficult to estimate the parameter m from a least-square fit of eq. (5.10). This parameter was estimated to be 0.8 from eq. (5.13). The function b(z’, z”) obtained from the solution of eq. (5.12) is depicted in Fig. 9(c). The coalescence frequency is an increasing function for all but the largest drops. The simulation results are shown in Fig. 9(d). Important conclusions can be drawn from the results of this section. The frequency obtained from the inverse problem for cases 1 and 2 shows that coalescence of small drops with small drops is inhibited and that coalescence of small drops with large drops is promoted, thus corroborating the size dependence of the frequency established in Section 4. It should be acknowledged that the inverse problem for case 2 is less conclusive, which underscores the fact that many transients are required to accurately determine the similarity spectrum.
It is important to note that we have tested the predictive capabilities of these frequencies only with respect to the cumulative volume fraction distribution F(v, t). This is because the similarity distribution,f(z), converges slowly for small z (corresponding to the smallest drops) as a consequence of which the coalescence frequencies of the smallest droplets are not known accurately. This is evident from Fig. 9(b). We have also confirmed this by numerical experiments with several coagulation frequencies including the Brownian coagulation frequency and the static deformation model coalescence frequency. The cumulative volume fraction gives more importance to the larger drops, whose coalescence frequencies are accurately known from the inverse problem and hence serves as a means of evaluating the frequencies.
6. SUMMARY
AND CONCLUSIONS
The drop size dependence
of coal esazllce frequencies
dispersions has been determined from transient coalescence experiments. Two in turbulent
liquid-liquid
3502
T.
TOBIN
et al.
0.9
.
0.8
0.8
.
0.7
0.7
.
E
0.6
0.6
.
2 >
0.5
$
0.4
.g ” ::
.k
z s0
0.3
E 3
0.2
.
0
.
.
.
.
0.012
0.006
Drop
.
min
.
0.018
volume
min
15mln
A 30
oe
- 30
Omln
q
0
0 rnlrl e 15 min q
. 0.0 24
oc
I 0.030
10-q
-
.
q
10-3
(~1)
10-Z
Similarity
(al
10-l
vorioble
100
10’
(v/v.,,)
(b) 1.0 0.9 g z::
1.449830
t a
0.6
1.012140
E3 2
0.5
E 9 N = -, 9
0.6 0.7
0.574446
g z
0.136744
z
0.3
5
0.2
-
Prsdictod
0.4
0.1 0 10-q
10-S
Drop
10-z
volume
(~~11
(d)
(c)
Fig. 9. Inverse problem for experiment 1 (5% benzenexarbon tetrachloridein water).
0.7 0.6
2 2:
0.5 0.4 0.3 0.2 0.1 0
0
5
10
15
20
25
30
35
40
45
v/v.. Fig. 10. 2% dispersion of o-xylen-rbon tetrachloride in 0.1 M salt water. Impeller speed reduction from 7.0 to 2.3 rps. Transient distributions taken from Konno et al.
(1988) and replotted vs the similarity variable U/Y.,.
distinct approaches have been employed to ascertain the bivariate coalescence frequency function. The indirect approach involves testing coalescence frequency functions derived from detailed models of drop-pair relative motion with the transient data. Basically, the three frequency functions listed in Table 1 were evaluated based on their ability to reproduce qualitative features in the transient data which contain information on the salient features of the coalescence frequency function. The necessity to minimize break-up effects stems from the fact that these can obscure the characteristic features of size spectra arising due to coalescence. The direct approach involves inverting the population balance equation [eq. (l.l)] for the coalescence frequency function in situations where the drop size distributions are self-similar. Such self-similar spectra cannot be obtained if drop break-up occurs in conjunction with coalescence. Coalescing transients in neutrally buoyant dispersions were obtained by prestirring at high impeller speeds (6OOrpm or greater) followed by a sudden
Determination
of coalescence frequencies in liquid-liquid dispersions
reduction in the same to 200 rpm, under which conditions the experiments were performed. Drop size distributions were periodically withdrawn, immobilized immediately with surfactant, and photographed. Detailed histograms were constructed by computer image analysis of a large number of droplets (1ooo-2ooo). The frequencies derived from the kinetic collision model and the static deformation model, which show very similar drop size dependence, predict the evolving transients best. These frequencies are in particular able to track the drift in the peak of the drop size distribution [see Fig. 3(c) and (d)] and also at least qualitatively reproduce the bimodal feature of the spectra in experiment 3 [see Fig. 7(b)]. The coalescence frequency expression derived from the dynamic deformation model predicts opposite trends in the displacement of the peak and fails to predict the bimodality in experiment 3. The quantitative agreement is also poor. Interestingly, the coalescence frequency function obtained by assuming the coalescence efficiency to be a constant (collision frequency) predicts the transients satisfactorily although not as well as the frequencies from the kinetic collision and the static deformation models. This is explained from the fact that the latter models predict an efficiency of nearly unity for the larger drops. In other words, these models reduce to the constant efficiency model for the larger drops. These results show that the coalescence efficiency as well as the coalescence frequency are increasing functions of the drop sizes for the situations considered. The transient size spectra were tested for self-similar behavior as outlined in Sections 2 and 5. The size spectra showed similarity behavior in two experiments. This is the first instance where such behavior has been observed in stirred liquid-liquid dispersions. Such self-similar behavior is observed typically only after a considerable amount of coalescence has occurred unless the initial size spectrum at the onset of the experiment is favorable for the attainment of similarity. This explains why such scaling spectra are not observed in all the experiments. From our experiments it appears that the presence of very small droplets delays the attainment of similarity behavior. The similarity spectra have been used to extract the coalescence frequency function by solving the inverse problem discussed in Section 5. The coalescence frequency functions obtained from the inverse problem show an increasing trend with drop size and thus corroborate the results of the indirect approach. As noted earlier, we are not the first to report transient drop size distributions. We have, however, applied a criteria for transient experiments under which break-up is negligible and thus can be neglected. Because of our work with purely coalescing dispersions, we are the first to notice similarity behavior in coalescing liquid-liquid dispersions subjected to turbulent agitation. Interestingly, in three of the nine
sets of transient
distributions
obtained
by
3503
Konno et al. (1988), break-up could have been neglected for the first few distribution measurement times, based on the criteria of eq. (3.2). Upon replotting the transient distributions contained in the above work vs the similarity variable (v/v,,), we found that the data showed similarity behavior for those three cases. This strengthens our belief that similarity behavior is fairly common in transient drop size distributions where break-up is negligible. The transient coalescence experiments indicate that the coalescence frequency is an increasing function of the drop pair sizes. This is in contrast to earlier approaches of fitting batch steady-state size spectra (due to balance of break-up and coalescence) with models for break-up and coalescence, which yield no definitive conclusions on the size dependencies of coalescence and break-up [see Chatzi and Lee (1987)]. Visual observations of coalescence by Park and Blair (1975) led them to believe that the efficiencies increase with drop size although the data were too scanty to make definite conclusions. No surface active materials were deliberately added and hence the results are valid only for such conditions. It is known that the presence of surfactants can alter the coalescence dynamics considerably. In the present work, we have been mainly concerned with the drop size dependence of coalescence rates. The results obtained establish the usefulness of the framework employed to determine coalescence frequencies. Moreover, these same techniques may be applied to experiments with variation of relevant physical properties of the system (such as dispersedand continuous-phase viscosities, interfacial tension, surface viscosity, etc.). Such experiments are essential for a more complete characterization of drop coalescence frequencies. When the properties of the disperse systems have been correctly accounted for, one may expect the correlating parameters to be “universal”. It is unclear whether this has been an issue in correlations attempted on droplet processes in the past. Acknowledgement-The authors gratefully acknowledge the support of the National Science Foundation through grants CBT-8414204 and CBT-8611858. NOTATION a,
Hz, 2’) d % F(r, 0 Mi ii(4
2)
N(t) I+, v”‘) s(t)
similarity parameter similarity coalescence frequency function drop diameter Sauter mean drop diameter cumulative drop volume fraction as a function of the similarity variable cumulative drop volume fraction as a function of drop volume and time ith moment of a drop distribution drop number density as a function of drop volume and time total drop number density as a function of time binary coalescence frequency function inverse mean droplet size [1/(0(t) >]
T. Tom
3504
drop volume similarity variable (scaled drop volume,
i7,u”’
2
4) Greek E
letters
u 9 ‘y(z) 4% R
4
turbulent energy dissipation per unit mass relative velocity fluctuation in a turbulent flow dispersed phase volume fraction scaling function for a self-similar distribution binary collision frequency function impeller stirring speed
REFERENCES
Chatzi, E. and Lee, J. M., 1987, Analysis of interactions for liquid-liquid dispersions in agitated vessels. Znd. Engng Chem. Res. 26, 2263-2267. Coulalonlou. C. A. and Tavlarides, L. L., 1977, Description of inter&on processes in agitated liquid-liquid d&e=sions. Chem. Enana Sci. 32. 1289-1297. Delichatsios and ‘Pzc>bstein,. R. F., 1975, Coagulation in turbulent flow: theory and experiment. J. Colloid InterfLee sci. 51, 394-405. Groothuis, H. and Zuiderweg, F. J., 1964, Coalescence rates in a continuous-flow dispersed phase system. Chem. Engng Sci. 19, 63-66. Hinze, J. O., 1959, Tz&ulence. McGraw-Hill, New York. Howarth, W. J., 1967, Measurement of coalescence frequency in an agitated tank. A.Z.Ch.E. J. 13, 1007-1013. Konno, M., Muto, T. and Saito, S., 19X8, Coalescence of dispersed drops in an agitated tank. J. Chem. Engng Japan 21, 335-338. Ku&, R.. Komasawa, I. and Otake, T., 1972, Collision and coalescence of dispersed drops in turbulent liquid flow. J. Chem. Engng Japan 5,423424.
:N
et d. Laso, M., Steiner, L. and Hartland, S., 1987, Dynamic simulation of agitated liquid-liquid dispersions--II. Experimental determination of breakage and coalescence rates in a stirred tank. Chem. Engng SC: 42, 2437-2445. Madden, A. J. and Damerell, G. L., 1962, Coalescence frequencies in agitated liquid-liquid systems. A.I.Ch.E. J. 8, 233-239. Miller, R. S., Ralph, J. L., Curl, R. L. and Towell, G. b., 1963, Dispersed phase mixing: measurements in organic dispersed systems. A.1.Ch.E. J. 9, 196-202. Muralidhar, R., 1988, Ph.D. thesis, Purdue University. Muralidhar, R. and Ramkrishna, D., 1986a, Analysis of droplet coalescence in turbulent liquid-liquid dispersions. Znd. Engng Chem. Fundam. 25, 554-560. Muralidhar, R. and Ramkrishna, D., 1986b, An inverse problem of agglomeration kinetics. J. Coiloid Interface Sci. 112, 348-361. Muralidhar. R. and Ramkrishna, D., 1988, Coalescence nhenomena in stirred liquid-liquid dispersions. 6th European Conference on Mixing, 24-26 May 1988, Pavia. Muralidhar. R. and Ramkrishna, D., 1989, Inverse problems of agglomeration kinetics-II: binary clustering coeffi-cients from self-preserving spectra. J. Colloid Interface Sci.
(in press).
Ramkrishna, D., 1985, The status of population balances. Rev. them. Engng 3, 49-95. Ruckenstein, E. and Pulvermacher, 1973, J. Catal. 29, 224-245. Shinnar, R., 1961, On the behavior of liquid dispersions in mixing vessels. J. Fluid Mech. 10, 259-275. Sprow, F. B., 1967, Drop size distributions in strongly coalescing agitated liquid-liquid systems. A.Z.Ch.E. J. 13, 995-998. Swift, D. L. and Friedlander, S. K., 1964, The coagulation of hydrosols by Brownian motion and laminar shear flow. J. Colloid Sci. 19, 621-647. Tobin, T. G.. 1989, M.S. thesis, Purdue University. von Dongen, P. G. J. and Ernst, M. H., 1988, Scaling solutions of Smoluchowski’s coagulation equation. J. statist. Phys. 50, 295-329. Wright, H. A., Muralidhar, R., Tobin, T. G. and Ramkrishna, D., 1989, Inverse problems of aggregation processes. J. statist. Phys. (submitted for publication).
Engineering in Great
Science,
Vol.
45, No.
12, pp. 3491-3504,
1990.
ooos2509po 93.00 + 0.00 Q 1990 Pergamon Press plc
Britain.
DETERMINATION IN LIQUID-LIQUID
OF COALESCENCE FREQUENCIES DISPERSIONS: EFFECT OF DROP SIZE DEPENDENCE
T. TOBIN, R. MURALIDHAR, H. WRIGHT and D. RAMKRISHNA’ School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (First received 25 September 1989; accepted
in revisedform
1 March
1990)
Abstract-The size dependence of the drop coalescence frequency is investigated by measurement of transient drop size distributions in purely coalescing systems (with negligible drop break-up). A twopronged approach is employed to estimate the bivariate coalescence frequency function, using such experimental data. First, coalescence frequency expressions derived from mechanistic models of the relative motion of the drops are evaluated based on their ability to predict the experimental transient sire spectra. Second, experimental drop size distributions which exhibit a property known as self-similarity are analyzed through an inverse problem in order to extract the coalescence frequency function directly from the data. Results indicate that the coalescence. frequency of small droplets (l&SO pm in diameter)is lower than that predicted from a constant coalescence efficiency model. In addition, experimentsshow that, given favorable
initialconditions, self-similardrop size distributionscan be manifest.In the cases where similaritybehavior is observed, the frequenciesobtained from the inverse problem are in qualitative agreement with the mechanistic models that describe the data best.
1. INTRODUflrON Many engineering operations, such as liquid-liquid extraction and multi-phase reaction, involve the formation of stirred dispersions of two immiscible liquids. In such systems drop coalescence (and breakup) can profoundly influence the overall performance, by altering the interfacial area available for species exchange between the phases. In order to optimize these operations fully it is therefore desirable that engineers learn more about the dispersive process, through detailed studies of drop coalescence and break-up. A rational approach to the modelling of dispersive systems begins with the framework of population balances, and is discussed by Ramkrishna (1985). This describes the temporal evolution of a drop population due to random coalescence and break-up events. For a purely coalescing dispersion in which no mass exchange is involved, the population balance equation is given by
ai-qfi,q
1
dtl=-
2s0
-
”
g(u”-
r 0
v”‘,o’)A(a -
v”‘,E)ri(C’, t’)dC
ij(iJ, v”‘)ii(i?, F)ii(a’, 7)dv”
(1.1)
where n(& 7) is the number density (concentration) of drops of volume 6 at time ?, and q(iJ,v”) is the binary coalescence frequency for drops of volumes v”and 5’. The bivariate frequency function is a key input to the coagulation equation, and must be either determined from experiments or derived from phenomenological models of drop coalescence.
‘Author
to whom correspondence
should be addressed.
Many previous studies have sought to gain information about coalescence frequencies in a turbulent flow field. Such studies may be grouped into three basic categories, according to the experimental methods employed. The first type of study infers an average coalescence frequency from measurable changes in the physical properties of a system. An example of this is the work of Madden and Damerell (1962) in which coalescence rates of water droplets in toluene were estimated by observing the rate of extraction of iodine from the toluene into the water drops. From their experiments they reported the following relationship: 4 = fi2.*+o.504
(1.2)
where R is the stirring speed, 4 is the volume fraction of the dispersed phase, and q is the average coalescence frequency. A similar work by Miller et al. (1963) yielded basically the same results as those of Madden and Damerell. In a slightly different type of experiment, Howarth (1967) used a light absorbance technique to determine the rate of change of the mean drop size in a stirred vessel following a stepwise reduction in the impeller speed. Howarth obtained an average coalescence frequency from the change in average drop size with time, and his results were also consistent with those cited above. Although these studies can reveal the gross behavior of the coalescence frequency as a function of the energy dissipation rate and the dispersed-phase fraction, they are of very limited utility for the following reasons. First, they yield no information about the effect of drop size on the coalescence frequency, since they are built on the tacit and severe assumption that coalescing dispersions are essentially mono-disperse. Real dispersions can and often do have drops of
349 1
3492
T. TOBINet al.
extremely disparate sizes, however. Because of this, information about coalescence as a function of drop size is essential if one is to realistically model and study these dispersive systems. Second, in the experiments of Madden and Damerell and Miller et uI. the coalescence frequencies were measured under conditions where mass transfer was occurring between the phases. Groothuis and Zuiderweg (1964) have clearly shown that any such mass transfer drastically alters the coalescence process. At the very least then such an effect needs to be taken into account when attempting to interpret the results. A second basic type of experimental study has been to visually observe coalescence events, and thereby determine directly the coalescence frequency as a function of drop size, dispersed-phase fraction, etc. Kuboi et al. (1972) attempted this using high-speed photography, and their technique was employed by Park and Blair (1975) in a later study. Although this approach has potential to yield excellent estimates of the coalescence frequency in lean dispersions, the investigators who employed it invariably had difficulty in observing a sufficient number of coalescence events to obtain any reliable estimates of the frequency. A third approach of investigators in the past has been to measure steady-state drop size distributions in batch stirring experiments. Under these conditions, drop break-up and coalescence are in balance and the size spectrum is therefore time-independent. Coulaloglou and Tavlarides (1977) were among the first to use measurements of drop size distributions with the objective of understanding the size dependence of drop coalescence and break-up. They derived frequency expressions for drop break-up as well as drop coalescence, and using these expressions they were able to predict the steady-state spectra well. Although the last-mentioned approach represents a significant advance over prior efforts, there is still one serious limitation. Since drop break-up and coalescence depend on the same physical parameters, it is extremely difficult to reliably pinpoint the dependencies of these two processes on the experimental conditions and on the properties of the two phases. In fact, using qualitatively different coalescence and break-up frequency models currently found in the literature, Chatzi and Lee (1987) have shown that there is no significant difference in the ability of any combination of these models to predict steady-state drop size spectra. This conclusion is not surprising, because steady-state simulations cannot accept the burden of determining the rates of two balancing processes. Some investigators have recently reported using a slight variation of this technique to study transient drop size distributions [see Laso et al. (1987) and Konno et al. (1988)]. In these studies dispersions have been perturbed (for example by subjecting them to a step change in impeller speed) and then observed as they settle back to a steady state. While this method
yields somewhat improved results, these studies still suffer significantly from the obscuring effects of drop break-up aliuded to above. It is evident from the preceding discussion that very little exists in the literature by way of reliable information regarding the dependence of the coalescence frequency on the sizes of two coalescing drops. A major cause for this has been the lack of a robust experimental and theoretical approach to tackle the coalescence problem. Our approach for investigating coalescence hinges on measurement of transient drop size distributions in purely coalescing systems. When break-up effects are negligible, one can clearly observe drop size distributions as they evolve through coalescence events. These coalescing dispersions then carry fingerprints of the coalescence frequency. A two-pronged approach is employed to understand the drop size dependence of the coalescence frequency. First, coalescence frequencies derived from mechanistic models of coalescence are evaluated, based on their ability to predict transient coalescence size spectra. Second, in cases where the dyanamic size spectra exhibit self-similar behavior, an inverse population balance equation is employed to directly extract the bivariate coalescence frequency function. The outline of this paper is as follows. In Section 2, the theoretical framework for exploiting the transient size spectra is presented. Section 3 discusses the experimental aspects. In Section 4 we investigate the ability of different coalescence frequency expressions to predict the transient size spectra, while in Section 5 we present the coalescence frequencies obtained via an inverse problem formulation. Section 6 summarizes the salient aspects of the present work and describes the main conclusions that can be drawn.
2
THEORETICAL
FRAMEWORK
TRANSIENT
SIZE
FOR
EXPLOlTING
SPECTRA
Studies of drop coalescence made in the absence of drop break-up are amenable to two different but complementary forms of analysis. The first approach involves observing salient features of the evolving droplet size spectra and subsequently investigating the ability of coalescence frequency expressions to reproduce these same features. These coalescence frequencies may be derived from mechanistic models of the coalescence process, and several such expressions are given in Table 1. (The models in Table 1 only show dependence on drop size; since dependence on other properties of the dispersions were not studied, these dependencies have been lumped into the constants of the models.) The first of these expressions, called the constant efficiency model, is derived by considering the collision frequency between drops in a turbulent flow field in a manner analogous to that used to derive the kinetic theory of gases (Delichatsios and Probstein, 1975). Writing the collision frequency w as the product of the collision cross section and the root mean
Determination of coalescence frequenciesin liquid-liquid
dispersions
3493
Table 1. Coalescence frequency models used in coalescence rate studies Model name
Coalescence frequency expression
(1) Constant efficiency
k(*l/” +
(2) Dynamic
k(vl’”
deformation+
(3) Static deformation’
k(v
l/3
1 - exp (-
k(v 113
IiF1gives
+ U’1’3)2 [i?Pl.
161 Oc E1”Y1”
where r may be taken as the distance
+
&/3)7/3
-> ;
k, k, ~--4~3) k,t4’3)
exp
+
v’
-k,E” vv’(u1’3 + IFS)
1
(2.1)
(2.2)
between
drop
centers. Combining this with eq. (2.1), and making the assumption that a constant fraction of all collisions result in drop coalescence, the constant efficiency expression of Table L is obtained. The remaining three expressions of Table 1 are obtained by considering coalescence to be a two-step process. The first step is termed collision and describes the approach of a pair of droplets to a close proximity, at which point a viscous thin film of liquid separating the droplets resists their relative motion. The second
3.700
z
exp (-
k, < 1, k, w 0
Essentially all drops observed in these experiments were larger than the Kolmogorov dissipation length. For such drops it has been shown by Shinnar (1961) and others that
N 0
1-
(1988).
fluctuation
o(u, 0’) = (67rz)1’3 (L+
y’1/3)7/3
P)
”
(4) Kinetic collision
square relative velocity
+ ~‘~r~)‘/~ exp (-k,
a> 0
0
0.93
1
2’ w
G 0.010
0.910 0.00
2001
1
2001 1
2001-1
1.230 “0
0.927
z
0.622
-> J G
I.830 n*
1.380
1
0.917
-a. 0.316
>-
0.459
G 0.010
2001 1
2001-1
(b) Fig. 1. Typical
0.000
2001-
1
(d)
coalescence frequency surfaces: (a) constant eiliciency model, (b) dynamic model, (c) static deformation model, (d) kinetic collision model.
deformation
3494
T.
TOBIN
step is termed film drainage and involves the draining of the film mentioned above to permit actual coalescence of the droplets. During this stage, the forces from the contiguous turbulent flow can separate the drops; consequently, not all collisions will lead to coalescence. From this view of coalescence the coalescence frequency may be written as the product of a collision frequency and a coalescence efficiency or probability. It should be emphasized that this factorization is possible only for dilute dispersions for which the characteristic time of the collision process is much larger than the timescale of film drainage. In the latter three models the collision frequency term is the same as that used in the first model, while the coalescence efficiency is obtained from an analysis of the drainage of the viscous liquid film which separates two colliding drops. Models 2 and 3 were derived previously (Muralidhar and Ramkrishna, 1986a; Muralidhar, 1988) so they will be described here only briefly. In both models, the Reynolds film drainage equation was analyzed with a fluctuating random force depicting turbulence. Coalescence was assumed to occur if the film drains to a critical thickness under the action of the dynamic turbulent pressure fluctuations. The drops were regarded as separated if the film expanded sufficiently for hydrodynamic retardation to be negligible. As a first approximation, van der Waals attractive forces and electrical double layer forces were regarded as only determining the film rupture thickness, and did not enter the dynamic analysis. When the inertia of the colliding drops is neglected and the characteristic time of turbulent force fluctuations is much smaller than the drop deformation time, the deformation of the drops may be determined by an average turbulence force squeezing the drops. This average force (given by the product of pressure fluctuations across the drop pair and the projected area of the drops) is assumed positive because of the absence of turbulence between the contacting drops. The resulting model, referred to as the static deformation model, predicts an increase in the coalescence efficiency with increasing drop size. On the other hand if the drop shapes respond to every fluctuation in the force field, their deformation is determined by the instantaneous force squeezing the drop pair. This gives rise to the model referred to as the dynamic deformation model, which predicts a decrease in the coalescence probability with increasing drop size. In violent collisions, the inertia of the drops is the governing factor and the effectiveness of the collision depends on the adequacy of the collision kinetic energy to accomplish film drainage work. By performing a dimensional analysis on this process one may arrive at the fourth coalescence expression, termed here the kinetic collision model. This model yields a drop size dependence of coalescence frequency similar to the static deformation model, though it was obtained from a very different form of analysis. The restriction to purely coalescing systems also permits one to probe certain symmetries of the transi-
et al.
ent size spectra which can be effectively employed to directly extract the bivariate coalescence frequency function [see, for example, Swift and Friedlander (1964) or Ruckenstein and Pulvermacher (1973)-J. More specifically, the size distributions at different times can be meaningfully represented by a single scaling function V(z) defined by Y(z) = i-q, t)2,
z=&
(2.3)
where o 5 4 z 3
al.
1.2
0.9 iE
0.8
1.0
d
i
0.7
I
0.8
E F
0.6
0.6
q
. 0.s 0.4
Omin 1smm
- 3Omm
8 0.4
0.3
B
0.2
2 1
0.2
0.1
0
2
4
Drop
6
6
volume
10
12
(~1
14
16
16
30
60
x 10’)
Fig. 2. Transient size spectra for experiment I (5% benzene-carbon
90
Drop
120
lb0
diameter
160
210
240
(pm)
tetrachloride in water).
270
1
Determination
of coalescence frequencies in liquid-liquid
;
1.0
,’
5. i :a
:,’ ,-
s.
‘1
.
08
---
0.6
8 3
0.4
E s
1.0
3 a eI
Omin
-.-.-
15 min
-..-
30
mhl
% k
I..
.
.
;.c
‘.
6
0Z6
,,!
‘.
omin
‘.
‘*..i
-.-.- 15
:
.,“,
0.6
---
b, .
:
L
a Y z
-..-
l
.
2
0.2
30
60
90
120
Drop
0.4
lb0
160
diameter
210
240
270
0.2
0
!
\-.______ . 0
30
60
90
120
Drop
(Frn)
___
c. I ,
E a
i
‘. :
: 0.6
0 mill _.-._ISmin
8 z 3
5
;A “•a.,
0.8
1
0.6
0.4
E i
0.4
0.2
1
0.2’
-..-
0
30
60
90
120
Drop
210
240
270
?
(pm)
.
g
0.6
z d
1.0
Ei
,
-3 3
180
(b)
1.2
1.0
150
diameter
(a)
CL
min
30 min
E
0
i
3497
1.2
1.2
i
dispersions
150
160
diameter
30
210
min
240
I
Omi”
-..-
‘i
\\
*
5 min
30min
.
0+
270
___ -.---
+-p.__._._, C)
30
60
90
120.150
Drop
(pm)
160
diameter
210
240
270
Qun)
(d)
(cl
Fig. 3. Test of coalescence frequency expressions using data of experiment 1: (a) fit of constant efficiency model to data, (b) fit of dynamic deformation model for k, = 0.0001, (c) fit of static deformation model, (d) fit of kinetic collision model.
the shift of the number density peak to the left, the slow decrease in the number density of the smallest drops, the rapid decrease in the number density in the size range 100-l 50 and the rather slow increase for the larger drops.
1.0
-
-
q
0.9.
s
3 0 P
s < 3
-0 n
0.6. 0.7.
l
0.6.
q
0.5.
l
0.4.
‘C o zc E
0.2.
G
0.1
0.3
04
s
-
.
.
.
A repeat of the previous experiment (experiment 2) was performed to investigate the reproducibility of the experiments. The data are depicted in Fig. 4. The data show the same basic features of the previous experiment although not identical_ The differences in the
a
a
1
= 7 mill . 15 twin a 25 min
=
m
7 min 15min a 25 min
l
l
.;
‘) 0
2
. 4 Drop
. 6
8
vohme
10
. 12
(~1
14
. 16
x lo3
1
18
Fig. 4. Transient size spectra for experiment 2 (5%
0
30
60
90
120
Drop
benzene-carbon
150
diameter
160
210
240
(pm)
tetrachloride in water).
270
!
IO
3498
T. TOBIN
initial conditions could be due to two reasons: first, the time of prestirring at a high impeller speed was different, and perhaps was not of sufficient length to attain a steady-state distribution. Second, there may have been trace contaminants in the experiments, which can significantly alter the break-up and coalescence frequencies and thus introduce experimental error. The predictions using different models are depicted in Fig. S(aHd). Although none of the models describe the steep decrease in number density for drops approximately 100 pm in diameter, the static deformation and the kinetic collision models perform better than either a constant efficiency model or a dynamic deformation model. A third coalescence experiment (experiment 3) was performed on a dispersion of hexanesarbon tetrachloride in water and the results are shown in Fig. 6. By far, the most interesting and significant feature of this data set is the tendency to evolve into a bimodal distribution with time. This in itself reveals a great deal about the coalescence frequency function. The intuitive interpretation is that the coalescence fiequency of the smallest droplets is very small, as a
et al.
consequence of which such droplets linger for long times, leading to the development of bimodality. The dynamic deformation model is unable to predict the bimodality, although the fit to the data is not unreasonable [Fig. 7(a)]. In contrast, the static deformation model shows bimodality, although not nearly as pronounced as in the actual data. The simulation with this model, along with the experimental curves, is shown in Fig. 7(b). The transient coalescence experiments thus consistently show that of all the models tested the static deformation and the kinetic collision models describe the transient coalescence data best. It thus appears that the coalescence frequency of the droplets is an increasing function of the drop sizes.’ Further discrimination between the kinetic collision and static deformation models can be made by studying the dependence of the frequency on other physical parameters such as continuous-phase viscosity, interfacial tension, energy dissipation, etc. ‘This trend is in conflict with that predicted by using the coalescence frequency of Coulaloglou and Tavlarides.
2.6
2.6 r-
2.4
E a. ‘1, i
-
e! e
2.0 1.6 1.2
8 &i s 1
0.6 0.4
I ,
0
2.6 ; i
2.4 E a
3 w. E E
2.0 1.6 1.2
G $
0.6
e zz
0.4
0
30
30
i-
2.4
E 5*
2.0
.s z em f 28
__._
1.6
-.-.-..-
I: 1.2
7 min
16 min 25 min
i $
0.6
i
J 1
60
SO
Drop
120
diameter
150
460
210
0.4
{ r
0
2
30
(pm)
60
SO
Drop
120
diometcr
150
.160
210
I 240
(pm)
fb)
la) 2.8 i-
2.4 E
___
60
SO
Drop
120
diometer (cl
7 min
_._._
16 min
-..-
2S
160
160
(pm)
min
210
P
i a.
2.0
2 ~
1.6
Y ::E
1.2
zz
0.6
1
0.4
__-
0
30
60
Drop
SO
120
diameter
7 min
-.-.-
15 min
-..-
25 mm
150
160
210
240
(pm)
fd)
Fig. 5. Test of coalescence frequency expressionsusing data of experiment 2: (a) fit of constant efficiency model, (b) fit of dynamic deformation model for k, = 0.0002, (c) fit of static deformation model, (d) fit of kinetic collision model.
Determination 1 .o
5
0.9
Q
0.8
*
l
x
:
dispersions
3499
.
.
.
q
of coalescence frequencies in liquid-liquid
p
0.5
5
0.4.
.
=:
k
0.7
i 3 g > al .Z! g
0.6
3 h
0.5 0.4 0.3
i
0.2
3
0.1 0
.= E!
1
a
a
1 min
. 30 a 60
m.
8 5
min min
0.2
2
7 E
i
0.5
-
I 10
5
15
01
20
(~1 x
‘.
‘8 ,
,’ I’
o.4
.
40
80
1 mill
m-e
120
Drop
I-.
,’ a’
0
10’)
I
a_
-3
240
260
*
0.1
Fig. 6. Transient size spectra for experiment 3 (5%
,
min min
E
Drop volume
0.6
30 a 60
l
0.3
-.-.-
30
mm
-..-
60
mln
hexane-carbon
160
200
diometer
?
I
(pm)
tetrachloride in water).
forces. We are currently investigating the effect of salt concentration on coalescence rates as salt can diminish the influence of any electrical double layer forces. The mechanistic models will also be subject to refinement to include such colloidal forces if they turn out to be significant. 5. SELF-SIMILAR SlZE SPECTRA AND THE INVERSE PROBLEM
0
40
80
120
Drop
160
diameter
200
280
240
320
(pm)
(0)
-!-! a
0.4
,’ 1
___
:
:
-.-..
:
1 min 30
min
I
iamb
The inverse coalescence problem is directed at deriving the coalescence frequency function from transient drop size spectra of purely coalescing systems. The procedure involves deriving an integral equation for the coalescence frequency function. While a detailed description of the inverse problem is given in Wright et al. (1990), we present the salient aspects in this section. The direct inversion of the population balance equation [eq. (Ll)] for the unknown bivariate frequency function using the transient data is difficult. The inverse problem procedure is considerably simplified in situations where the transient size spectra are self-preserving. We make the population balance equation [eq. (l.l)] dimensionless by defining the following dimensionless quantities: n(o, t) = fi(~,7)v’,/rj,,
q(v, If) = 4”(6, v”)/E, ,20
Drop
160
diameter
200
240
280
320
(*ml
(b) Fig. 7. Test of coalescence frequency expressions using data of experiment 3: (a) fit of dynamic deformation model, (b) fit of static deformation model.
An interesting feature of the experiments is the slow rates of coalescence compared to that observed by Delichatsios and Probstein (1975) for particles much smaller than the microscale of turbulence. This indicates a very small coalescence efficiency for droplets in a turbulent flow field. This could be due to hydrodynamic interaction as well as repulsive electrostatic
v(u’) = E(O’)/v-,,
t = (EN~)f.
(5.1)
In the above, @,, is the initial total number density, and uOis the initial average volume. These two quantities are given by
No=
,=--n(r, r,)d6
s0
m fi,i&
=
s0
iFi@, t’,) dB. (5.2)
The constant E is a multiplicative factor having dimensions of volume per time and may be set equal to unity without loss of generality. It is useful to define the cumulative volume fraction distribution function when self-similar size distributions are observed: F(v, t) =
u’n(u’, t) dv’.
(5.3)
3500
T. TOBIN et al.
In terms of dimensionless variables the coagulation equation [eq. (1. l)] written in terms of the cumulative volume
fraction
aFti t)
law
becomes
s(t) oc t
(5.4)
the average size s(t). On substituting eq. (5.9) into eq. (5.8) and using eq. (5.10), we obtain
We introduce the transformation F(v,
c) -f(z),
z = w(f)
(5.3
where s(t) is the inverse of a mean droplet size, , at time t. Frequently employed choices for s(t) are
$$$, (b) s(t) = 1
M,(t) M,(t)”
(c) s(t) = s 3
(5.6) although in certain cases not all the above equations yield similarity behavior [see von Dongen and Ernst (1988)]. For all the data presented in this work, eq. (56b) was employed in the similarity analysis. In the above, M,(t) is the pth moment of the distribution at time t and is given by m M,(t)
=
(5.11)
The parameter m can be determined by a fit of eq. -S:dF(v’,t)~~“,dF~~‘r)q(u’,u’~). (5.10) using an experimentally measured reciprocal of
dt=
(a) s(t) =
C&l.
I0
urn(u, t) du
p = 0, 1,
...
M,(t)
=
1. (5.7)
The similarity distribution in terms of the number density ‘Y(z), referred to in Section 2, is given by f’(z) On substituting the scaling transformation -. z given by eq. (5.5) into eq. (5.4), we obtain
where the primes on s and f denote the derivatives of these functions with respect to their arguments. Under conditions where similarity behavior are observed, it has been shown by Wright et 01. (1989) that the coalescence frequency must be of the form 4 [&,
G]
= &s(t)-“lb(Y, 0.
(5.9)
If the function b(z’, z”) is the same as q(z’, z”), the coalescence frequency function is homogeneous of degree m. The coalescence frequency is hence completely specified by the exponent m and the function b(z’, z”). During self-similar behavior, the transient spectra can be succinctly described in terms of the scaling spectrum f(z) and the size scaling variable s(t). The rate of evolution of the population is described by the decay law for the reciprocal of the average size s(t) which is shown to be (Wright et al., 1989) s’s(t)m-Z = -a,.
(5.10)
Physically, this equation implies that, for large times, the average drop size grows according to the scaling
a,zf’(z) =
I ss0
m
b(z’, z”)f’(z’)f’(z”)/z”dz”dz’
z-z’
(5.12)
which is an integral equation for the unknown function b(z’, z”) in terms of the similarity spectrum. The solution procedure for b(z’, z”) from eq. (5.12) has been outlined in Wright et al. (1989). The coalescence frequency was determined via the inverse problem for two different transient coalescence experiments that exhibited similarity behavior. In case 1, the transients were obtained from a 5% dispersion of hexane-carbon tetrachloride in water. At t = 0 the impeller speed was reduced from 800 to 200 rpm. Case 2 refers to experiment 1, outlined in Section 4. It is pertinent to note that self-similar behavior is not always observed in experiments of this type. Thus experiment 3 of Section 4 did not lead to a self-similar distribution during the course of the experiment. This is possibly because of an initial preponderance of very small droplets; the slow coalescence of these droplets would cause the initial distribution to be retained for a long period, perhaps longer than the length of the actual experiment. Figure 8(a) shows the transient drop size distributions for case 1. The transients did not possess any distinct characteristics that would permit discrimination between the different model frequencies by the strategy described in the previous section. The size spectra are self-similar with respect to the scaling size in eq. (5.6b), however, as is evident from Fig. 8(b); thus the data can be subjected to the inverse problem methodology. The exponent m was estimated from the asymptotic property of the scaling spectra derived by von Dongen and Ernst (1988):
f ‘cd
~
z
x
zme(-orr) for large z
(5.13)
as well as the decay law given by eq. (5.10). A value of 0.8 was obtained. The value is close to the degree of homogeneity of the constant efficiency model (7/9). The integral equation for b(z’, z”) was solved [see Wright et al. (1989) for details] and the solution is shown in Fig. 8(c). Although not identical to the static deformation model (or kinetic collision model) frequency surface, it is also an increasing function of the size. The coalescence frequency obtained from the inverse problem was assessed by forward simulation using eq. (1.1) using the data at size distribution at the earliest time of similarity behavior as the initial condition. It is evident from Fig. 8(d) that the frequency correctly predicts the transients.
Determination l.O--
of coalescence frequencies in liquid-liquid
-
.. . . . . .
m % .ti
e
M
0.8
0.9 L
.
.
L .
-1
s
-.
E
0.7.
E
OS
‘, >
0.5’
g
0.4’
i x 5 aa >
m 15 min
....1
“_yiy-.
3501
q.0,
1
.-
0.9
dispersions
a 30 min L 60 min
0.8 0.7 I
0.6
15 mill
* 30 min - 60 min
0.5 0.4
4J
4
0.3
i s
0.2 0.1
0
0.050
0.150
0.100 Drop
vohme
0.200
0
0.250
to-3
10-z
(~1)
lo-’
Similority
variable
100
(v/v,,
1
(b)
(a 1
0.9
.E
0.8
3.46149
t; e
0.7
2.33492
2
0.6
E :
3 g >
= Iu _
-N
I. 20034
g u 4 i 3
-
Predicted II lbmin l 30min - 60min
0.5 0.4 0.3 0.2 0.t 0
10-6
10-J
10-4 Drop
lo-'
volume
10-z
lo-'
100
1~1)
(d)
(cl
Fig. 8. Inverse problem for experiment4 (5% hexsne- carbon tetrachloridein water).
The drop size distribution of case 2 also exhibit similarity behavior as is evident from Fig. 9(a) and (b), although the spectra have not yet converged for small z. As there were only two times at which s(t) was available, it was difficult to estimate the parameter m from a least-square fit of eq. (5.10). This parameter was estimated to be 0.8 from eq. (5.13). The function b(z’, z”) obtained from the solution of eq. (5.12) is depicted in Fig. 9(c). The coalescence frequency is an increasing function for all but the largest drops. The simulation results are shown in Fig. 9(d). Important conclusions can be drawn from the results of this section. The frequency obtained from the inverse problem for cases 1 and 2 shows that coalescence of small drops with small drops is inhibited and that coalescence of small drops with large drops is promoted, thus corroborating the size dependence of the frequency established in Section 4. It should be acknowledged that the inverse problem for case 2 is less conclusive, which underscores the fact that many transients are required to accurately determine the similarity spectrum.
It is important to note that we have tested the predictive capabilities of these frequencies only with respect to the cumulative volume fraction distribution F(v, t). This is because the similarity distribution,f(z), converges slowly for small z (corresponding to the smallest drops) as a consequence of which the coalescence frequencies of the smallest droplets are not known accurately. This is evident from Fig. 9(b). We have also confirmed this by numerical experiments with several coagulation frequencies including the Brownian coagulation frequency and the static deformation model coalescence frequency. The cumulative volume fraction gives more importance to the larger drops, whose coalescence frequencies are accurately known from the inverse problem and hence serves as a means of evaluating the frequencies.
6. SUMMARY
AND CONCLUSIONS
The drop size dependence
of coal esazllce frequencies
dispersions has been determined from transient coalescence experiments. Two in turbulent
liquid-liquid
3502
T.
TOBIN
et al.
0.9
.
0.8
0.8
.
0.7
0.7
.
E
0.6
0.6
.
2 >
0.5
$
0.4
.g ” ::
.k
z s0
0.3
E 3
0.2
.
0
.
.
.
.
0.012
0.006
Drop
.
min
.
0.018
volume
min
15mln
A 30
oe
- 30
Omln
q
0
0 rnlrl e 15 min q
. 0.0 24
oc
I 0.030
10-q
-
.
q
10-3
(~1)
10-Z
Similarity
(al
10-l
vorioble
100
10’
(v/v.,,)
(b) 1.0 0.9 g z::
1.449830
t a
0.6
1.012140
E3 2
0.5
E 9 N = -, 9
0.6 0.7
0.574446
g z
0.136744
z
0.3
5
0.2
-
Prsdictod
0.4
0.1 0 10-q
10-S
Drop
10-z
volume
(~~11
(d)
(c)
Fig. 9. Inverse problem for experiment 1 (5% benzenexarbon tetrachloridein water).
0.7 0.6
2 2:
0.5 0.4 0.3 0.2 0.1 0
0
5
10
15
20
25
30
35
40
45
v/v.. Fig. 10. 2% dispersion of o-xylen-rbon tetrachloride in 0.1 M salt water. Impeller speed reduction from 7.0 to 2.3 rps. Transient distributions taken from Konno et al.
(1988) and replotted vs the similarity variable U/Y.,.
distinct approaches have been employed to ascertain the bivariate coalescence frequency function. The indirect approach involves testing coalescence frequency functions derived from detailed models of drop-pair relative motion with the transient data. Basically, the three frequency functions listed in Table 1 were evaluated based on their ability to reproduce qualitative features in the transient data which contain information on the salient features of the coalescence frequency function. The necessity to minimize break-up effects stems from the fact that these can obscure the characteristic features of size spectra arising due to coalescence. The direct approach involves inverting the population balance equation [eq. (l.l)] for the coalescence frequency function in situations where the drop size distributions are self-similar. Such self-similar spectra cannot be obtained if drop break-up occurs in conjunction with coalescence. Coalescing transients in neutrally buoyant dispersions were obtained by prestirring at high impeller speeds (6OOrpm or greater) followed by a sudden
Determination
of coalescence frequencies in liquid-liquid dispersions
reduction in the same to 200 rpm, under which conditions the experiments were performed. Drop size distributions were periodically withdrawn, immobilized immediately with surfactant, and photographed. Detailed histograms were constructed by computer image analysis of a large number of droplets (1ooo-2ooo). The frequencies derived from the kinetic collision model and the static deformation model, which show very similar drop size dependence, predict the evolving transients best. These frequencies are in particular able to track the drift in the peak of the drop size distribution [see Fig. 3(c) and (d)] and also at least qualitatively reproduce the bimodal feature of the spectra in experiment 3 [see Fig. 7(b)]. The coalescence frequency expression derived from the dynamic deformation model predicts opposite trends in the displacement of the peak and fails to predict the bimodality in experiment 3. The quantitative agreement is also poor. Interestingly, the coalescence frequency function obtained by assuming the coalescence efficiency to be a constant (collision frequency) predicts the transients satisfactorily although not as well as the frequencies from the kinetic collision and the static deformation models. This is explained from the fact that the latter models predict an efficiency of nearly unity for the larger drops. In other words, these models reduce to the constant efficiency model for the larger drops. These results show that the coalescence efficiency as well as the coalescence frequency are increasing functions of the drop sizes for the situations considered. The transient size spectra were tested for self-similar behavior as outlined in Sections 2 and 5. The size spectra showed similarity behavior in two experiments. This is the first instance where such behavior has been observed in stirred liquid-liquid dispersions. Such self-similar behavior is observed typically only after a considerable amount of coalescence has occurred unless the initial size spectrum at the onset of the experiment is favorable for the attainment of similarity. This explains why such scaling spectra are not observed in all the experiments. From our experiments it appears that the presence of very small droplets delays the attainment of similarity behavior. The similarity spectra have been used to extract the coalescence frequency function by solving the inverse problem discussed in Section 5. The coalescence frequency functions obtained from the inverse problem show an increasing trend with drop size and thus corroborate the results of the indirect approach. As noted earlier, we are not the first to report transient drop size distributions. We have, however, applied a criteria for transient experiments under which break-up is negligible and thus can be neglected. Because of our work with purely coalescing dispersions, we are the first to notice similarity behavior in coalescing liquid-liquid dispersions subjected to turbulent agitation. Interestingly, in three of the nine
sets of transient
distributions
obtained
by
3503
Konno et al. (1988), break-up could have been neglected for the first few distribution measurement times, based on the criteria of eq. (3.2). Upon replotting the transient distributions contained in the above work vs the similarity variable (v/v,,), we found that the data showed similarity behavior for those three cases. This strengthens our belief that similarity behavior is fairly common in transient drop size distributions where break-up is negligible. The transient coalescence experiments indicate that the coalescence frequency is an increasing function of the drop pair sizes. This is in contrast to earlier approaches of fitting batch steady-state size spectra (due to balance of break-up and coalescence) with models for break-up and coalescence, which yield no definitive conclusions on the size dependencies of coalescence and break-up [see Chatzi and Lee (1987)]. Visual observations of coalescence by Park and Blair (1975) led them to believe that the efficiencies increase with drop size although the data were too scanty to make definite conclusions. No surface active materials were deliberately added and hence the results are valid only for such conditions. It is known that the presence of surfactants can alter the coalescence dynamics considerably. In the present work, we have been mainly concerned with the drop size dependence of coalescence rates. The results obtained establish the usefulness of the framework employed to determine coalescence frequencies. Moreover, these same techniques may be applied to experiments with variation of relevant physical properties of the system (such as dispersedand continuous-phase viscosities, interfacial tension, surface viscosity, etc.). Such experiments are essential for a more complete characterization of drop coalescence frequencies. When the properties of the disperse systems have been correctly accounted for, one may expect the correlating parameters to be “universal”. It is unclear whether this has been an issue in correlations attempted on droplet processes in the past. Acknowledgement-The authors gratefully acknowledge the support of the National Science Foundation through grants CBT-8414204 and CBT-8611858. NOTATION a,
Hz, 2’) d % F(r, 0 Mi ii(4
2)
N(t) I+, v”‘) s(t)
similarity parameter similarity coalescence frequency function drop diameter Sauter mean drop diameter cumulative drop volume fraction as a function of the similarity variable cumulative drop volume fraction as a function of drop volume and time ith moment of a drop distribution drop number density as a function of drop volume and time total drop number density as a function of time binary coalescence frequency function inverse mean droplet size [1/(0(t) >]
T. Tom
3504
drop volume similarity variable (scaled drop volume,
i7,u”’
2
4) Greek E
letters
u 9 ‘y(z) 4% R
4
turbulent energy dissipation per unit mass relative velocity fluctuation in a turbulent flow dispersed phase volume fraction scaling function for a self-similar distribution binary collision frequency function impeller stirring speed
REFERENCES
Chatzi, E. and Lee, J. M., 1987, Analysis of interactions for liquid-liquid dispersions in agitated vessels. Znd. Engng Chem. Res. 26, 2263-2267. Coulalonlou. C. A. and Tavlarides, L. L., 1977, Description of inter&on processes in agitated liquid-liquid d&e=sions. Chem. Enana Sci. 32. 1289-1297. Delichatsios and ‘Pzc>bstein,. R. F., 1975, Coagulation in turbulent flow: theory and experiment. J. Colloid InterfLee sci. 51, 394-405. Groothuis, H. and Zuiderweg, F. J., 1964, Coalescence rates in a continuous-flow dispersed phase system. Chem. Engng Sci. 19, 63-66. Hinze, J. O., 1959, Tz&ulence. McGraw-Hill, New York. Howarth, W. J., 1967, Measurement of coalescence frequency in an agitated tank. A.Z.Ch.E. J. 13, 1007-1013. Konno, M., Muto, T. and Saito, S., 19X8, Coalescence of dispersed drops in an agitated tank. J. Chem. Engng Japan 21, 335-338. Ku&, R.. Komasawa, I. and Otake, T., 1972, Collision and coalescence of dispersed drops in turbulent liquid flow. J. Chem. Engng Japan 5,423424.
:N
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