Ind. Eng. Chem. Fundam., Vol. 17,No. 2, 1978
coefficient k that develops far downstream of the entrance in the Graetz solution.) t = time, s T = reduced time, (vt- 2)/[('2C1*Z)/(r,-,c1)], dimensionless (The product N T also serves as a reduced time for some purposes.) U = Graetz solution, dimensionless U = mking-cup-mean of Graetz solution, dimensionless U' - = dUIdN, dimensionless U = Laplace transform of dimeqionless V ( r ) = Poiseuille velocity profile, 2V(ro2 - r2)/ro2,cmls V = mean channel velocity, cmls x = reduced adsorbate concentration in fluid, c / c l , dimension 1ess Z = mixing-cup-mean value of x , dimensionless y = reduced adsorbed concentration, C/C1*, dimensionless Z = axial coordinate, cm (Note that in the formulation of the differential equations, 2 is taken to be a running variable, that is, a variable position in a channel of some fixed but unspecified length. The solutions, however, are valid at any axial position, and in particular at the exit. Thus, it is permissible and sometimes convenient in interpreting the results to view Z (or the dimensionless parameter N that contains 2) as representing the total length of the channel.)
u,
Greek Letters
8,
a,
= dimensionless A = difference operator, dimensionless { = functional relationship defining the position of the saturation boundary: NT = { ( N ) ,dimensionless .I' = d{/dN, dimensionless f = Laplace transform of j-, dimensionless 7 = reduced radius, rho, dimensionless ,I = partition ratio, (2C1*)/(rocl),dimensionless
E
101
= functional relationship defining the saturation boundary:
N = { ( N T ) ,dimensionless
Subscripts 0 = outer wall 1 = inlet 1 = mesh point index for the variable N m = mesh point index for the variable 7 7 = partial derivative, 8/87 m = constant-pattern value Superscripts j = mesh point index for the variable N T
Literature Cited Abramowitz, M., J. Math. Phys., 32, 184 (1953). Acrivos. A,, Chambre, P. L., Ind. Eng. Chem., 49, 1025 (1957). Ames. W. F., "Numerical Methods for Partial Differential Equations", 2nd ed, Academic Press, New York. N.Y., 1977. Aris. R., Proc. Roy. SOC.London, Ser. A, 252, 538 (1959). Churchill, R. V., "Modern Operational Mathematics in Engineering", McGraw-Hill, New York, N.Y., 1944. Cowherd, C., Hoelscher, H. E., lnd. Eng. Chem. Fundam., 4, 150 (1965). Giedt, W. H., "Principles of Engineering Heat Transfer", Van Nostrand, New York, N.Y., 1957. Giueckauf, E.,Coates, J. I., J. Chem. Soc., 1315 (1947). Golay, M. J. E., "Gas Chromatography", D. H. Desty, Ed., Butterworths. London, 1958. Graetz, L., Ann. Phys. Chem., XXV, 22 (1885). Jakob. M., "Heat Transfer", Vol. 1, Wiley, New York, N.Y., 1949. Langmuir, I,, J. Am. Chem. SOC.,40, 1361 (1918). Leveque, J., Ann. Mines( 73,13,201 (1928). Michaels, A. S.,lnd. Eng. Chem., 44, 1922 (1952). Nusselt, W., Z.Ver. Dtsch. lng., 54, 1154 (1910). Richardson, L. F., Phil. Trans. Roy. Soc., 226, 299 (1927). Schenk, J., Dunrnore, J. M., Appl. Sci. Res., A, 4, 39 (1953). Vermeulen, T., Klein, G.. Hiester, N. K., Section 16, "Chemical Engineers' Handbook". 5th ed,R. H.Perry and C. H. Chiiton, Ed., McGraw-Hill, New York, N.Y., 1973.
Received for reuiew M a y 31, 1977 Accepted January 26, 1978
Droplet Breakage and Coalescence Processes in an Agitated Dispersion. 2. Measurement and Interpretation of Mixing Experiments Seymour L. ROSS, Francis H. Verhoff,'' and Hane L. Curl Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48 104
Simultaneous measurements of the drop size distributions and rate of dispersed-phasemixing were obtained for an agitated liquid-liquid dispersion in a 1.36-L vessel under steady-slate, continuous flow conditions. The impeller speed was varied from 160 to 278 rpm, and the dispersed phase fraction from 0.05 to 0.20. A regime was discovered in which the rate of dispersed-phase mixing varied as the impeller speed to about the tenth power, while the mean drop size was little altered. This is interpreted as a competition between drop breakage rate, drop coalescence rate, and drop flow-departure rate, with a very strong dependence of breakage and coalescence rates on agitation and drop size over certain ranges of these variables.
Previously (Verhoff et al., 1977), a technique has been described by which a liquid-liquid dispersion may be sampled to obtain simultaneous measurements of the volume and a tracer-dye concentration for each of a large number of drops. Address correspondence t o t h i s author a t the D e p a r t m e n t of Chemical Engineering, West V i r g i n i a University, Morgantown, W. Va. 26506.
0019-7874/78/1017-0101$01.00/0
Such measurements provide information about the fundamental rate processes occurring in an agitated dispersion breakage, producing smaller drops, and coalescence, forming larger drops and also causing mixing within the dispersed phase. Herein results of the application of the method to the study of a particular dispersion over a range of impeller speed and dispersed phase function will be presented and interpreted.
0 1978 American Chemical Society
102
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
Table I. Mixing Vessel Dimensions Vessel Diameter Height Volume
11.1cm
14.0 cm 1355. cm3 Impeller
D = 5.1 cm Dl5 X Dl4
Diameter Blades (6) Table 11. Fluid Properties (22 "C)
Dispersed phase (39.1%by vol. Dowtherm-E, 61.9% by vol. Shell No. 3747 Base Oil) Continuous Phase (water with 0.001 N Na3P04) Interfacial tension: 35 dynlcm
Density, g/cm3
Viscosity, glcm s
1.0-
0.035
1.0+
0.019
Experimental System The details of the experimenta, apparatus and procedures have been presented (Verhoff et al., 1977). In summary, the experiments used a baffled, agitated, continuous flow Rushton (Rushton et al., 1950) vessel, geometrically similar to many described by earlier workers, to which two streams of dispersed (oil) phase and one stream of continuous (water) phase were fed. The two dispersed phase streams had different concentrations of a tracer dye. Representative samples of the steady-state dispersion formed in the vessel were collected and analyzed for the volume and dye concentration of each drop in the sample. The dimensions of the vessel and impeller are given in Table I and the compositions and properties of the fluids used are given in Table 11. All experiments were conducted at a thermostated "room temperature", approximately 22 "C. The design of the equipment and the liquid properties were such that the experimental operation was only satisfactory in the impeller speed range of about 150-300 rpm. At higher
i
AGITATOR SPttD
P n l s t FRACTION
speeds, the drops in the vessel were too small to be analyzed accurately and at lower speeds, the drops exhibited buoyancy effects in the mixing vessel. The scope of the experiments that yielded acceptable results was (Ross, 1971) impeller speed: 160,174,185,195,210,227,278rpm; dispersed phase fraction: 0.025,0.05,0.10,0.20. In all of these experiments the residence time in the vessel was 19.6 min. In total, 88 experiments were performed. Many of the early experiments involved a wall-mixing effect and were not reproducible. In these cases, it was found that one or two clumps of dispersed phase became attached to the baffles, grew in size, became unstable, and broke off. This had a remarkable effect on the resulting bivariate distributions. When the walls and baffles were thoroughly cleaned, wall effects were avoided and the data were reproducible. Most earlier work makes no mention of this problem.
Drop-Size Distributions The marginal distributions of drop diameters are shown in Figures 1 to 4 for selected agitation rates (160,174,227, and 278 rpm) and all dispersed-phase fractions used. The average input drop diameters were in the range of 2.1 to 2.4 mm, depending on the phase fraction of the flows. The steady-state distributions were found to be independent of the input drop size for the given experimental conditions, meaning that the large feed drops were broken too rapidly to be found in the samples taken. The evolution of the drop-size distributions with changes in impeller speed and dispersed-phase fraction are as expected: increased impeller speed making smaller drops and increased dispersed-phase fraction forming larger drops. The values of the Sauter mean diameter (d32), as well as the number and volume average drop volumes (u10 and u p l ) are indicated in the figures. The dependence of the Sauter mean diameter, d32, upon impeller speed is shown in Figure 5, and upon dispersed-phase fraction in Figure 6. These figures show all the experimental results obtained. Concentration Distributions The (marginal) volume distributions of tracer concentration, normalized from 0 to 1,are shown in Figures 7 to 10 for A G I T A T O R SPttO
160 rpm 0.05
2.1
3.0
AGITATOR SPEED
*
Pnist FMCTIDI 2.5
-
.
: - 2.0 . w
I
.
1.5
. .
150 rpm
30
0.15 1.5
ORDPS 3 1 5 5 d12 Vi0
1.090 ,531
E .
1.0 I
)
ORDPS 3 0 4 6
.
. .
:1.1 . I
I
m
160 rpm
AGITATOR S P t i D
Pnrsr FRACTION
160 rpm
0 20
DROPS 1 6 5 6 d12
1.367
Y10
,962
Y21
1.012
.
: .. 8.5 . 2
1.0
A , . . . .
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
t
ICITITOR SPttO P N I S t FRICTlON
25.
?-
f 20.
-
w
z
-
i
.
vlo
1a2
Y21
317
-
2.0
J
.
05.
. -
H
I
OROPS I S 9 3
:: 1.5 . I
2
.
.
1.0
: . 0.5
r,.
. 0.8
0.6
0.4
DROP
OS
_1
-2 I. . . .
0.2
0.4
0.1
.
,
.
,
0.8 1.0 1.2 1.4 D R O P D I l Y i T t R . mm
836
di2
.
0.2
0.5
174 rpm 0 10
I
i
“L
-
IGITPTOR SPttO PWISt iRPCllON
*
I
5 IO. >
75a
I
15.
I
. 2.5 . 3.0
DROPS 1 2 6 5
1’1
.
=
I
I 1 4 rpm
0 OS
103
1.8
1.1
2.0
.
rl
0.1
.?7.
0.4
0.6
1.0 1.2 1.4 O l I M t T t R mm
1.6
2.0
1.1
L L-
.
,
,
,
.
,
,
0.1 1.0 1.2 1.4 D R O P D l l M t l t R , mm
,
,
,
1.6
1.1
,
2.0
Figure 2. Marginal distribution of volume over drop diameter a t N = 174 rpm.
u
l
.
6 0 1.5
. 5 1.0 . - .
Y
.
I
i
0
0.5
0.2
0.4
0.1
0.1 OROP
1.0 1.2 1.4 O I l Y t T t l , mm
1.6
1.8
2.0
*
r
[ . .I . 0.2
0.4
. 0.6
. .
.
. . . .%-.
0.8 1.0 1.2 1.4 D R O P O l A Y t T t R mm
. 1.6
r.
1.8
.
2.0
Figure 3. Marginal distribution of volume over drop diameter a t N = 227 rpm. the experiments corresponding to the drop size distributions in Figures 1to 4.The normalized average concentration, F, was obtained from the sample drops on a volume average basis. For all the experiments (except one), F = 0.528 f 0.017, at 95% confidence. This should be the same as the fraction of the “dark” (higher tracer concentration) oil constituting the feed, which was measured to be 0.525 to 0.530. This agreement is satisfactory. At the lower impeller speeds, most of the drops in the vessel have the feed tracer concentrations, which means that little mixing by coalescence is occurring. At higher impeller speeds, or at higher dispersed-phase fractions, the degree of dispersed phase mixing is considerably increased. The concentration variance in the feed, on a volume basis,
is given by 002
= E(1
- F)
(1)
while the concentration variance in the vessel (or in the effluent) can be calculated from the sums over the normalized concentrations and volumes of all drops in a sample as
2 vi
i= 1
The ratio of these two variances, C, = u2/uo2, is a measure of the degree of mixing of the dispersed phase. For no coales-
104
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
0 OS
PHPSI FRDCIION
? .
25
PHPSt F R P C I I O N OROPS 2903
DROPS 4 0 8 9
,542
d32 Vi0
0610
Vzl
I52
Y
Et
t.
- .
0.2
. f i , . .
0.6 1.0 1.2 1.4 O R O P DIIYtItR. mm
.
3.0
J
-
.
U
.
z
.
1.5 Y
.
3
1
+
.
0.5 .
2.0
111 rpm 0 1s
I -
Y,o
,0851
Y ~ I
.I9k
H
i .
,
P
,
r ‘ L - 4
.
1.0
,
,
1.6
1.8
2.0
216 r p m 0 20
OROPS 4 2 2 9
.
: .
&L-,
1 1
0.1 1.0 1.2 1.4 O R O P O l l Y t l I R mm
0.6
PClIPIOR S P t i O PHPSt f R P C I I O N
:: 1.1 .
L
? L C L ._ . L
%--.e. . . . , . . .
0.4
. E - 2.0 . Y
I
1
I 0.2
1.1
,589
I
I
3.0.
DROPS 3 5 2 8 d32
8
.
5 1.0 0 = . .
1.1
‘7
I
2.0
. . .
,
1.1
ACITPTOR S P t i O P H D S I FRDCIION
r-
2.5.
.
-
. \ . , 0.6
0.4
ii
1.1
’
d32
646
VI0
,111
v2,
252
k
..
.
,
,
,
,
k,-;, ,
,
,
1.3
/
Lz
1.2
1.1
1.1
1.0
1.o
u
a 0.9 x
€0.8
4
.p
0.8
0.7
0.7
Ob
0.6
0.5
0.5
a4 150
L
170
190
210
230
250
270
N , rpm Figure 5. Dependence of Sauter mean diameter, d32, upon impeller speed a t various phase fractions.
cence, it would have a value of 1.0, and for complete dispersed phase mixing, it would become zero. Since there is no direct way to obtain an absolute measure of the “rate of coalescence”, such as the volume fraction of the dispersed phase engaging in coalescences per unit time, various relative measures have been defined. The one used herein is based upon the variance ratio, C, and an “equal drop size” dispersed phase mixing model. Komasawa et al. (1971) and Verhoff (1969) have derived the effect of dispersed phase mixing on the concentration variance in a steady continuous flow system using the Curl (1963) model that assumed equal drop size, and coalescence rate limiting mixing. The result for the ratio of outlet to inlet concentration variance is
0.4 I
0.05
0.10
0.15
0.20
E Figure 6. Dependence of Sauter mean diameter, fraction a t various impeller speeds.
1 c, = -
d32,
upon phase
(3)
1 + k W 7
2 where w is the fractional rate of coalescence (min-l) and T is the residence time (min). For unequal-size drops, the net variance reduction should still be a good integral measure of the degree (or relative rate) of coalescence, compared to the residence time, especially if the feed concentration distribution is symmetric and the feed drop size is large and approximately constant. Rearranging eq 3 we obtain w =
2 1 ; - 1)
(4)
Ind. Eng. Chern. Fundam., Vol. 17, No. 2, 1978 a.
N
=
0
=
C
1 6 0 rpm 0.05
=
0
= = 0 534
c cr=
= 0 509
I
b. 1 6 0 rpm 0.10
N
1' d
0 820
N
9
c
=
a. 2 2 7 rpm 0.05
=
0.501
=
I N
cr=
1
N
0
w
= =
0
-1
c
.
Cr
.25'
=
0.538
=
0.500
a 5
'ION
0.537
.'I 1 L~
N
a C
N
rpm
0
= =
c= cr=
= = =
0528
t
a5
b. 1 7 4 rpm 0.10
Figure 9. Marginal distribution of volume over tracer concentration a t N = 227 rpm.
I N
c= cr=
0 511 0.664
N
=
c
= =
cr =
0.540
278 rpm
=
0 537
0 199
cr=
0 I63
1 1I
n
i t
0
1 d. 1 7 4 rpm 0.20
=
0 499
.1
0
c
d. 227rpm 0.20
1
I
a. = 174 = 0.05 = 0 380
1 N @
t
t
j.1 t
I
.
1 CONCENl Figure 7. Marginal distribution of volume over tracer concentration a t N = 160 rpm. 0
0 I94
0
--.1JI
.15.
227 rpm 0.10
n
I
1
g .35.
= =
e
.l
d. 1 6 0 rpm a20
105
g.35.
N
=
-1
O
=
0
r
.
0 345
.25
c=
cr =
218 rpm 0.15 0 545
*
.
1 N
=
Q = '
0 155
t
c=
c, =
d. 278 rpm 0.20
-
0532
0 I22
-l5M *15M ,.1;111 1
8 50
1
ti
il
I
8 50
1 CONCENT RTlON Figure 8. Marginal distribution of volume over tracer concentration a t N = 174 rpm.
1 0 1 CONCENTRATION Figure 10. Marginal distribution of volume over tracer concentration a t N = 278 rpm.
The values of C, and w obtained from our experiments are shown in Figure 11and Table 111. The minima occurring in Figure 11is thought due to experimental error. The curves are shown only to join related points.
tory, investigations of Verhoff (1969). I t was found immediately that, at a phase fraction of 0.05, when the impeller speed was lowered by lo%, the mixing frequency decreased by more than a factor of 2. This was pursued further and the impeller speed was lowered in steps of about lo%, to 160 rpm, below which the larger drops displayed noticeable bouyancy. In order to find the region where the mixing rate has the more moderate dependence upon agitation noted by other researchers, experiments were conducted up to 278 rpm.
Summary of Results A Dramatic Mixing Effect. The experimental work was begun by using an impeller speed of 195 rpm, primarily to compare the results presented here with the earlier, explora-
106
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978 I.”
&
0.9
0- 01
2-
E
:: 3
1.5-
0.5
g
LO
0.4
yL
u
5
-
9-
0.3
8-
-.:
1-
0.2
E
E
0.1
‘150
I10
190
210
230
270
250
4-
5 .5c y
AGITATOR SPEEO, N , rpm
Figure 11. Concentrationvariance ratio C, as a function of impeller speed at various phase fractions.
2=
,4-
Table 111. Mixing Frequency and Concentration Variance Ratio as a Function of Imoeller h e e d and Phase Fraction Impeller speed, rPm
Phase fraction, 4
C,
min-l
160
0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 0.05 1.10 0.15 0.20 0.05 0.10 0.15 0.20
0.905 0.820 0.665 0.500 0.728 0.664 0.440 0.345 0.548 0.415 0.267 0.191 0.351 0.307 0.225 0.132 0.323 0.286 0.259 0.180 0.255 0.194 0.210 0.191 0.199 0.163 0.155 0.122
0.0102 0.0225 0.0475 0.102 0.0386 0.0524 0.132 0.196 0.0874 0.146 0.284 0.440 0.191 0.234 0.358 0.682 0.216 0.260 0.296 0.472 0.298 0.423 0.385 0.433 0.410 0.525 0.555 0.735
174
185
195
210
227
278
a
w,
From eq 4.
Between 160 and 195 rpm there is an enormous effect of impeller speed. This result was quite reproducible and occurs, though to a less marked degree, at phase fractions above 0.05. Expressed as a power law, we would have to write something like w a N10. At higher impeller speeds, the effect is more like N 2 - 3 ) , a result often found in the past in dispersed-phase mixing experiments. Another aspect of this phenomenon should be noted. It occurs at absolute coalescence rates that are close to or smaller than the residence frequence (UT). For example, while u equalled 0.0386 min-l at N = 174 rpm and 4 = 0.05, and was very sensitive to the agitation rate, the residence frequency was 1/r = 0.051 min-1. The significance of this will be developed later. The mixing frequency varied less dramatically with phase fraction. At the lower impeller speeds, as the phase fraction increased fourfold from 0.05 to 0.20, the mixing frequency increased fivefold to tenfold. At the higher impeller speeds
D
Figure 12. Sauter mean diameter and various diameter percentiles (volume fractions) as a function of impeller speed at @ = 0.05.
this same increase in phase fraction produces less than a twofold increase in w. Drop Size Effects. In contrast to the dramatic mixing effects, the drop sizes in the vessel were observed to be only moderately dependent on the impeller speed and phase fraction. This is shown generally in Figures 5 and 6. A plot of diameter distribution vs. N 2 in Figure 12 (this plotting scale helps to separate the individual data groups) shows the Sauter mean diameter volume percentiles d x % .It should be noted that this result was obtained in a continuous-flow agitated dispersion and is therefore not immediately comparable to the “conventional” N-1.2result found in agitated batch dispersions. Also, using the relation obtained by Mlynek and Resnick (1972) and the fluid properties and system dimensions in our experiments, one would predict d32 = 0.523 mm at 160 rpm and C#J = 0.05. Our experimental result, d32 = 0.843 mm (four times the volume), suggests that the drops do not stay in the vessel long enough to break down to the “batch” steadystate sizes. The changes in drop size distributions with agitation rate and disperse phase fraction are shown somewhat more clearly in Figure 13, where the results shown in Figures 2 and 4 have been smoothed and combined. It is seen that, as one might expect, the drops become larger and spread in the distribution increases as the phase fraction is increased. This effect is more pronounced a t the lower impeller speed. Spatial Homogeneity of Experiments. The mixing frequencies in our experiments ranged from about 0.01 min-1 to 0.7 min-1. This implies that an average time between coalescences is of the order of magnitude from 3 to 200 min. Since an average drop makes many journeys around the vessel before coalescing, the dispersion can be considered approximately statistically homogeneous. Thus the location of the sampling point is not critical. This is not the case in other systems such as Sprow’s strongly coalescing experiments (Sprow, 1967) where drop sizes were significantly different near and away from the impeller. Rushton et al., (1950) have shown that for Reynolds numbers greater than about 5 X lo3, the power input for a flat, six-blade impeller in a four-baffle mixing vessel is (in consistent units)
P = 6pN3D5
(5)
Ind. Eng. Chern. Fundarn., Vol. 17, No. 2, 1978
AGITATOR SPEED PHASE FRACTION
= =
278rpm 0.05-0.20
\
0.2
-
-
E = v)
Z
3.0
-
2.5
-
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.1
=
174 rpm 0.05- 0.20
=
by breakage or coalescence and their intersections (after Shinnar, 1961).
-
0 >
0.5
-
0.8 1.0 1.2 1.4 1.6 1.8 2.0 DROP DIAMETER, mm Figure 13. Smoothed distributions of volume over drop diameter.
0.2
0.4
0.6
The Reynolds number for our system and conditions is Re=--ND 2 p - 42.ON P
(6)
for N in rpm. Thus, for N = 160 rpm, Re = 6900, and eq 5 applies for our range of agitator speeds. The power input per unit mass of vessel content is then
7 = 7.6 x 10-3
N 3 ($)
N
Figure 14. Schematic representation of the various regions dominated
1.5 -
3 1.0
do
2.0
2.0
Y
Feed--r
log
AGITATOR SPEED PHASE FRACTION
107
(7)
with N again in rpm, and ? in hp/1000 gal of fluid of specific gravity 1.0. The range of power inputs from N = 160 to 278 rpm were 0.15 to 0.76 hp/1000 gal. The power inputs used in this study, and the fluid properties, gave a Kolmogoroff turbulence microscale b 3 / p 3 7)1’4 ranging from 0.08 mm ( N = 278 rpm) to 0.12 mm ( N = 160 rpm). Since only a very small volume fraction of the dispersion occurred as drops smaller than 0.2 mm, the drop breakage processes, as well as the vessel hydrodynamics, were in the inertial regime.
Discussion Impeller Speed Effect. Consider the experiments carried out a t 4 = 0.05 and a t the various impeller speeds. At the lowest speed of 160 rpm, it is evident from Figure 7 and Table 111that very little dispersed phase mixing is occurring. The average drop size is dS2 = 0.843 mm. When the speed is increased to 174 ppm, the average size decreases to 0.758 mm, while the mixing rate increases more than 300%. I t can be assumed that the drops in the vessel are smaller at the higher N because the overall breakage rate is higher. If it is further assumed that all the drops in the vessel be-
have similarly with respect to coalescence, it is impossible to explain the 300% increase in coalescence rate. The only aspects that are different between the case at N = 160 rpm and that a t N = 174 rpm are (1)the relative velocity between drops is 10%larger and (2) the drops are 10% smaller. In addition, previous experiments (Church and Shinnar, 1961; Mylnek and Resnick, 1972; Shinnar, 1961; Sprow, 1967) have suggested that the coalescence rate decreases with an increase in agitation rate, for a given drop size. We are left with the conclusion that the dramatic increase in coalescence rate is due to the creation of small, rapidly coalescing drops by a dramatic increase in the breakage rate. These small drops probably coalesce rapidly with all drop sizes present. That is, there must be very strong effects of both agitation rate and drop size on the rates of the breakage and coalescence processes. It is well known that if a dispersion is fed into a region of higher energy dissipation the most important effect is an increased breakage rate, whether or not coalescence of the initial drops is also increased or decreased. The same should be true in the present instance-increasing the impeller speed must primarily increase the rate of breakage of the drops initially present (to a dramatic degree) and it follows that coalescence must be most affected by the smaller drops being generated. Consider the possible behavior of drops in the agitated vessel operating a t a given power input. As shown by the drop-size distribution curves, there is generally a wide range of drop sizes in the vessel. Let us assume, for argument’s sake, that the small drops in the vessel have a much greater probability of coalescing with all sizes of drops than do the large drops, and that the large drops have a much greater breakage frequency than small drops. In addition, in a continuous flow system, each drop in the agitated vessel not only has a finite probability of breaking or coalescing, as in batch systems, but also has an opportunity of leaving the vessel. That is, there are three rate processes (breakage, coalescence, departure) in action, not just two. Given suitable conditions, large drops entering a vessel (as in our system) can break down to smaller drops but the probability of leaving the vessel could remain much greater than the probability of coalescing. In such a system, coalescence or mixing would be negligible. Thus it should be possible, in a continuous flow system, to form essentially noncoalescing dispersions which are “stabilized” by the flow component. This simplified argument is similar to that used by Church and Shinnar (Church and Shinnar, 1961; Shinnar and Church, 1960; Shinnar, 1961) to explain “turbulence-stabilized” dis-
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persions. Their argument can be restated with the aid of Figure 14. According to Shinnar, there is a maximum drop size, d,,, below which the breakage rate is zero (or very a N4l5 (from the Kolsmall), which depends on N as d,, mogoroff inertial turbulence drop stability theory (Shinnar and Church, 1960). Similarly, there is a minimum drop size, d,i,, above which the coalescence rate is zero (or very small). This diameter was found by Shinnar et al. (Church and Shinnar, 1961;Shinnar, 1961) to depend on N as dmina N-3/4. Since these two functions must cross, as shown in Figure 14, four special regions are formed. Above both curves, most drops break and few coalesce. Below both curves, most drops codrops alesce and few break. In the region where dmin> d,, readily break and coalesce, and therefore a finite rate of dispersed phase mixing, w, is establkhed. In the region where d, > d,i,, little breakage or coa’escence can occur, yielding what Shinnar called a “turbulen :e stabilized dispersion”. Shinnar observed this phenomenon in an agitated batch dispersion, and others have not, because he used a system loaded with the surfactant polyvinyl alcohol. While this may down slightly, it should even more dramatically shift d,, lower d,,,; that is, coalescence is suppressed until drops are much smaller than in a “clean” system. He also operated at relatively low energy input rates. The combination of these enlarged the “turbulence stabilized” regime so that it extended to “normal” impeller speeds. The phenomenon should be observable in clean dispersions except that such low agitation rates are required that bouyancy effects dominate. Our experiments were conducted in a clean system at “normal” impeller speed, and therefore our results should lie in the region of high drop interaction rates, if they had been conducted at steady state but without flow. The effect of a flow system may be seen by imagining the drops being introduced into the system at point X in Figure 14, at a diameter of do. Because they are unstable, they will break down but, as they work their way toward the sizes at which coalescence can occur, they have a high probability of leaving the system. Therefore a negligible number of drops reach a suitable size for coalescence, and negligible dispersed phase mixing occurs. In order, therefore, for a dramatic dependence of coalescence rate on agitation rate to be observed, it is necessary that there be first and foremost a dramatic effect of agitation rate on the breakage rate of drops larger than the nominal d,,. That is, we must conclude that the observed dramatic mixing rate effect is actually caused by an inherent dramatic dependence of the breakage rate on agitation, such as NIO. Phase Fraction Effect. The results suggest that as phase fraction is increased, at a given impeller speed, the small and the large drops within the system tend to coalesce and form larger drops. For example, at N = 160 rpm, the drops at q!J = 0.20 are four or five times larger in volume than those found at 0 = 0.05, and the mixing rate is ten times larger as well. This observation is in accordance with the concept that the increased number of drops would yield an increased probability of drop-drop collisions. The increased number of drops would also have an effect on the breakage rates.
Summary We have presented here the results of an experimental program to study the effect of agitation rate and dispersed phase fraction on drop coalescence and breakage rates. The simultaneous measurements of drop size distribution and volume concentration distribution permitted the separation of the breakage from the coalescence phenomena. Through
a combination of fortunate circumstances, a low impeller rpm regime was discovered where the apparent dispersed phase mixing i a t e depended on the agitation rate in a dramatic fashion ( m N 1 0 ) with little simultaneous affect on the mean drop size. It has been argued that such a result calls for a dramatic dependence of the breakage rate on agitation. A t higher rpm the mixing results corresponded to those of other investigators. The drop sizes observed in this investigation were larger than that calculated from correlations batch experiments. This deviation was attributed to the outflow of drops in the continuous stirred tank flow vessel employed.
Nomenclature c = concentration of tracer in an individual drop (normalized) F = normalized (0,l) volume average dye concentration C, = ratio of vessel volume average tracer concentration variance to that for the feed d = drop diameter,mm d32 = Sauter-mean drop diameter, mm d,, = maximum stable drop size, mm dmin = drop size above which coalescence is unlikely, mm D = impeller diameter, mm dxv; = x-percentile (by volume) drop diameter, mm n = number of drops in a sample of dispersion N = impeller speed, rpm or rps P = total power input to vessel, hp/1000 gal, or consistent units Re = ND2p/k = impeller Reynolds number u = volume of an individual drop, mm3 u10 = number average drop volume, mm3 u 2 1 = volume average drop volume, mm3 Greek Letters 5 = power input per unit mass of vessel contents, hp/1000 gal ( p = 1.0) q!J = dispersed phase fraction = continuous phase viscosity, g/cm s p = (r2 T
continuous phase density, g/cm3
= volume average concentration variance = residence time, min
w = dispersed phase mixing rate, min-’ (eq 4)
Subscripts i = ithdrop 0 = feed condition Literature Cited Church, J . M.. Shinnar, R., Ind. Eng. Chem., 53, 479 (1961). Curl, R. L., AlChEJ., 9, 175 (1963). Komasawa, I., Morioka, S., Kuboi, R., Otake, T., J. Chem. SOC.Jpn., 4, 319 (1971). Mlynek, Y., Resnick, R., AfChEJ., 18, 122 (1972). Ross, S. L., Ph.D. Dissertation, University of Michigan, Ann Arbor, Mich., 1971. Rushton, J. H.. Costich. E. W., Everett, M. J., Chem. Eng. Prog., 46, 395-376 ( 1950). Shinnar, R., Church, J. M., fnd. Eng. Chem., 52, 253 (1960). Shinnar, R., J. FfuidMech., IO, 259 (1961). Sprow, F. E., AlChEJ., 13, 995 (1967). Verhoff, F. H., Ph.D. Dissertation, University of Michigan, Ann Arbor, Mich., 1969. Verhoff, F. H., Ross, S. L., Curl, R. L., lnd. Eng. Chem. Fundam., 16, 371 (1977). Received for review February 4,1974 Resubmitted August 25, 1977 Accepted January 26,1978 Acknowledgment is made t o the donors o f t h e Petroleum Research Fund, administered by t h e American Chemical Society, for a Graduate Fellowship G r a n t awarded t o R. L. C u r l for t h e support o f L.
s.
Ross.
