Ind. Eng. Chem. Fundam. 1980, 79, 275-281
Denson, C. D., Trans. Rheol., 16(4), 697 (1972). Deryagin, B. V., Levi, S. M., "Film Coating Theory", Focal Press, New York, 1964. FltzJohn, J. L., UnpubilshedRepat, Department of Chemical Engineering, Drexel University, Philadelphia, Pa., 1974. Gutfinger, C., Tallmadge, J. A., AIChE J., 11, 403 (1965). Huppler. J. D., Ashare, E., Holms, L. A,, Trans. Soc. Rheol., 11(2), 159 (1967). Jeffreys, H. R o c . Camb. Phil. Soc., 26, 204 (1930). Khilar, K. C., M.S. Thesis, Department of Chemical Engineering, Drexel University, Philadelphia, Pa., 1977. Lang, K. C., Unpublished Report, Drexel University, Philadelphia, Pa., 1969. Lang, K. C., Tallmadge, J. A., Ind. Eng. Chem. Fundam., 10, 648 (1971). Lange, N. A., Ed., "Handbook of Chemistry", 9th ed, p 1669, Handbook Publishing CO.,Inc., Sandusky, Ohio, 1956; taken from Sheeley, M. L., Ind. Eng.
275
Chem., 24, 1060 (1932). Middleman, S.,Polym. Eng. Sci., 18, 355 (1978). Pendergrass, J., AIChE J . , 21, 487 (1975). Rajanl, M.S. Thesis, Department of Chemlcal Engineering, Drexel University, Philadelphia, Pa., 1968. Satterly, J., Givens, G., Trans. Roy. Soc. Canada III, 27, 145 (1933). Tallmadge, J. A., Gutfinger, C., Ind. Eng. Chern., 59(11), 19 (1967). Weinberger, C. B., Goddard, J. D., Int. J . Muniphase Flow, 1, 456, (1974). White, D. A., Ph.D. Dissertation, Yale Unlversity, New Haven, Conn., 1965. White, D. A., Talimadge, J. A., Chem. Eng. Sci., 20, 33 (1965).
Received for review July 16, 1979 Accepted May 23, 1980
Droplet Breakup and Distribution in Stirred Immiscible Two-Liquid Systems Menso Molag, Geert
E. H. Joosten, and Adelbert A. H. Drlnkenburg"
Laboratory for Chemical Engineering, Rijksuniversiteit Groningen, The Netherlsnds
In a stirred vessel the equilibrium droplet size distribution for a liquid/liquid system was investigated: benzenehrbon tetrachloride was the dispersed phase and water was the continuous phase. Experiments were performed with different types of agitators in baffled and unbaffled vessels. The droplet break-up mechanisms was followed in a special experimental setup, in which uniform small droplets were passed through the agitator. A model is set up that describes the droplet size distribution in an agitated system in the absence of coalescence.
Introduction Droplet size distribution in a two-phase liquid system is important in chemically reacting systems and in extraction processes, in the former because of the distribution and redistribution of chemical reagents, and in the latter since the efficiency of the process depends on the ratio of droplet surface to volume and on the rate of elutration, which also depends on the droplet size. Also the compounding of products is often connected to the droplet size, e.g., in foodstuffs. Many investigators have measured the average droplet size as a function of agitator speed, vessel dimensions, and physical parameters of the system like surface tension, density, viscosity, and fraction dispersed phase. Vermeulen et al. (1955) experimentally found a dependency on the Weber number defined as pn2D3 We = Q
Results showed that
d
- = e(@).We4.6
D
in which d = mean droplet diameter and e(@)is a function of the fraction dispersed phase only, accounting for coalescense effects. Rodger et al. (1956) found, at equal volume fractLons continuous and dispersed phase, a dependency of d / D t o Weber number to the exponent -0.3 and dependency upon the ratio of the diameter of the stirrer to the tank diameter to an exponent which was also a function of the Weber number. The same dependency on the Weber number as given by Vermeulen was found by Calderbank (19581, and Ro-
driquez et al. (1961), although the function c($) is quite different for some of the results, sometimes caused by different stirrer and vessel geometries but also by differences in analyzing methods. All authors mentioned used some type of light-absorption technique to measure the average specific surface and from there the mean droplet diameter. Mlynek and Resnick (1972a,b) and Van Heuven et al. (1977) analyzed the droplet distribution by using a polymerization reaction in order to coat the droplets instantaneously with a nylon jacket, after which the dispersed phase could be separated and photographed. These authors also measured a dependency of the average droplet size on the Weber number to the exponent -0.6. Droplet size distributions were measured by Chen and Middleman (1958);the standard deviation (SD)found was 0.23; by Sprow (19671, SD = 0.5; Olney (1964), SD = 0.45; Huttig and Sadler (1957) SD = 0.35. Collins and Knudsen (1970) measured the droplet size in a turbulent pipe flow and Middleman (1974) in a static mixer (Kenics type). Hinze (1955) applied the Kolmogoroff theory to droplet breakup. He showed that droplets oscillate around an equilibrium: kinetic energy which is brought into the droplet by viscous shear or by turbulent inertial effects is counterbalanced by the extra surface energy when the droplet is removed from its equilibrium sphere shape. This counterbalance is represented by pd3U2
We'= d2u
the ratio between kinetic energy due to oscillations and the surface energy. Above a critical Weber number the droplet becomes unstable and splits. For larger droplets, that is for diam-
0196-4313/80/1019-0275$01.00/0@ 1980 American Chemical Society
276
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 - .P " - p k'C'
I
1
!i5
\--
I
1
-0
-- -- -
0 90
j L 5
)I L
',, ~
6~ 55
J
_
_
Figure 1. Stirred vessel and agitators; measures in mm: (1) glass wall; (2) stainless steel top and bottom plate; ( 3 ) connections to thermostat; (4) Perspex outer wall; (5) four baffles; (a) turbine; (b) inclined blade; (c) flat disk; (d) cone, 30'; (e) cone, 10'.
eters above the inertial subrange in Kolmogoroff s theory, the maximum stable droplet diameter indeed is shown to be a linear function of Weber to the exponent -0.6. Narsimhan et al. (1979) use the same theoretical concept to predict an equilibrium drop size distribution. With the aid of a few arbitrary choices for some of the parameters they arrive at a close fit between theory and experiments. In view of the incomplete knowledge on liquidlliquid dispersions, it was decided to investigate the resulting distribution of droplet sizes when different types of agitators are used for dispersing. Equipment and Experimental Procedure (a) Experiments i n a Well-Stirred vessel. Experiments were done in a baffled and unbaffled tank with four types of agitators, viz: a flat disk stirrer (high shear rate), a cone stirrer (high shear rate), a turbine stirrer (high radial pumping rate), and an inclined blade stirrer (high axial pumping rate). Figure 1provides the dimensions of vessel and stirrers. The vessel could be thermostated. The dispersed phase was injected near the stirrer by means of a metering pump, so that the injected liquid was immediately broken up. The vessel was mounted on three flat springs, connecting the outside wall with a frame. Strain gauges were pasted on the springs. Through these the moment of forces acting on the vessel wall was measured and thus the energy dissipation, P = Mw, could be measured, in which M is the moment of forces and w is the angular velocity of the stirrer. In Figure 2 a plot is given of the power number Po = P/pn3D5 as a function of the Reynolds number Re = pnD2/v. All these measurements were done with only the continuous phase being present in the vessel. The results compare favorably with the literature. Figure 2 also shows the theoretical function of Po vs. Re for the flat disk, according to Von Karman. The deviations for high Re are caused by the transition from laminar to turbulent flow. Droplet Size Distribution in a Well-Stirred Vessel. The droplet size distribution measurements were per-
_
Figure 2. Power number vs. Reynolds number: (1) turbine baffled; (2) inclined blade, baffled; (3) turbine, not baffled; (4) inclined blade, not baffled; (5) cone, top angle 10'; (6) cone, top angle 30'; (7) flat disk, baffled; (8) flat disk, not baffled; (9) experiments Rushton turbine, not baffled; (10) data of van der Vusse (19531, cone, top angle 10'; (11) theory of Von Karmann, Po = 12/(Re)'/*.
formed as follows. The vessel was filled with continuous phase (0.1 N NaOH) and the stirrer was switched on and its speed adjusted; a certain amount of dispersed phase was added to the vessel (fraction: 4.2 vol 5%). The dispersed phase was 71.5 wt 5% benzene, 25.8 w t 5% carbon tetrachloride, and 2.7 wt 5% sebacoyl dichloride (SDC) and had a specific gravity equal to the continuous phase. At the end of a run (16 min of stirring proved to be enough to reach equilibrium) 40 mL of hexamethylenediamine (HMD) was added to the continuous phase, providing a nylon 6,lO coating at the surface of the droplet by polymerization with the SDC. A sample of the stabilized droplets was then taken and photographed and the droplet size distribution was measured. Alternatively, samples could be drawn from the vessel, after which Hh4D was added. On probability graph paper the normalized droplet size volume distribution sa! plotted as a function of the relative droplet diameter, d/d32, d32being the Sauter mean diameter, defined as k
-
ENid?
i=l
d32 = k E Nidi2 i=l
Figure 3 gives an example of the curves obtained. The distribution of the volume is well described by the normal (Gaussian) distribution function; other distribution functions, like the y distribution, will fit the curves equally well, as was shown by a test. (b) Droplet Size Distribution in a One-Pass Vessel. Since it was not clear how many passes of the liquid through the stirrer area were needed to obtain a fairly distributed droplet size, it was decided in a second run of experimentsto let the dispersed phase flow only once along and through the agitator. Therefore the stirrer was put
277
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980
Table I P
0
0
w D O
X ,
agitator speed, rpm
d,,,
1300 700 700 400 400 225 225 800 800 800 700 500 500 500 400 400 225 1300 1300 700 400 800 600 600 500 500 400 400 1300 1000 700 400 1300 1000 700 400 1300 1000 700 700 700 700 1300 700 700 500
50 117 115 281 266 586 644 231 196 202 260 255 282 262 414 401 911 118 103 306 701 171 261 281 373 339 729 639 55 71 164 382 75 108 240 509 215 356 444 372 431 389 226 693 632 1734
m
-
std dev
J/m3s
0.29 0.38 0.40 0.30 0.28 0.28 0.29 0.24 0.25 0.23 0.22 0.24 0.20 0.22 0.21 0.22 0.29 0.37 0.34 0.30 0.31 0.26 0.25 0.23 0.24 0.23 0.30 0.28 0.34 0.38 0.36 0.36 0.37 0.35 0.34 0.38 0.34 0.33 0.28 0.28 0.27 0.32 0.32 0.33 0.29 0.30
2120 315 315 57 57 10 10 164 164 164 116 43 43 43 22 22 4.5 1087 1087 167 38 238 112 112 70 70 38 38 724 332 113 21 394 189 67 15 271 125 49 49 49 49 230 38 38 16
€ 9
0
turbine, baffled
turbine, not baffled
flat disk, baffled
flat disk, not baffled
Figure 3. Cumulative volume fraction of the normalized droplet distribution function for the turbine agitator, stirred vessel.
into a channel through which the continuous phase was pumped at a certain flow rate (1.0 X lo4 m3/s), well in the range of the regular pumping capacity of a stirrer. The dispersed phase was injected just above the stirrer; its flow rate was 4.25 X lo* m3/s. It can be shown that the mean residence time of the emulsion in the near vicinity of the stirrer is then of the order of the time needed for two revolutions of the stirrer. In some experiments these flow rates were lowered by a factor of 2 or 4 in order to notice the effect of the residence time near the stirrer. Just below the agitator HMD solution was added and the nylon jacket was formed around the droplets; samples were analyzed in the same way as in the stirred vessel experiments. (c) Droplet Breakup in a One-Pass Vessel, Small Droplets of Uniform Diameter Being Added above the Agitator. In a third series of experiments a number of modifications were made. Benzene was substituted by toluene in the dispersed phase, SDC and HMD were left out, and consequently no nylon jackets were made. The droplet size was measured with a Coulter counter attached to a digital computer. Small droplets of dispersed phase of uniform diameter were made by means of an oscillating pressure on the liquid passing through a needle of 150 pm diameter which served as the injector of the dispersed phase. To increase the electric conductivity to the level required for operation of the Coulter counter 2% of sodium chloride was added to the continuous phase. It was then possible to study the break-up process, since only a fraction of the droplets would split during the contact time with the stirrer. Measurements were only made with the cone stirrer (30° top angle).
Results (a) Experiments in the Stirred Tank. Equilibrium Distribution. Table I presents the results of these experiments. The conclusion may be drawn that the standard deviation of the normalized distribution function (see, e.g.,
inclined blade, baffled inclinded blade, not baffled cone, 30°, slit width
4mm 4mm 4mm 4mm 2mm 2mm
cone, IO", slit width 4mm
Figure 3) does not depend very much upon either the stirrer type or its speed; its mean value is about 0.30. Somewhat broader distributions are found for the inclined blade stirrer and in the case of a baffled vessel. From the distributions itself, it becomes clear that no droplets are found with a diameter larger than twice d32. This is a reason to suggest that d = 2d32 is near the maximum stable droplet diameter, belonging to the critical Weber number, We,,. Hinze's theory suggests that We, must be constant for each agitator. If, however, We,, is plotted vs. Z, the average power dissipation per unit volume, no such constancy is found, especially not at low stirrer speeds. A possible explanation could be coalescence in the parts of the vessel with a lower energy dissipation rate, away from stirrer and baffles, but this seems unlikely since the shape of the droplet size distribution curve does not change markedly at lower speeds. A better explanation probably is that the pattern of liquid flow, and thus the relative local power dissipation, changes at lower speeds. The maximum stable droplet diameter is determined by the maximum local energy dissipation rate, which occurs near the stirrer. Changes in distribution of the power dissipation will certainly in-
278
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980
a,? -
Table I1
5 -s
agitator speed, rpm
type of stirrer turbine
500 500 600 600 600 750 750
1000 inclined blade ?&,
pv E .
Figure 4. Plots of d S 2 / Dvs. Weber number: *, cone, top angle loo; @ cone, top angle 3 0'; 0,flat disk, not baffled; X, flat disk, baffled. +, turbine, not baffled; 0,turbine, baffled; 0,inclined blade paddle, not baffled; A, inclined blade paddle, baffled. c IJ \r
flat disk
h'!
0LYI,
cone, 30" slit width t
1 mm 1 mm lmm 1 mm 4mm
1250 1250 750 750 1000 1250 1250 750' 1250' 1000 1250 1500 1750 750 1250 1250' 1280b 1250
' Throughout continuous phase is halved. put continuous phase is quartered.
?.
I
TLRBINE
ONE P A S S
+
750
RDM
x
1000
Y
l25C
RPM FlPM
.I
I
:'
54
f
1.
I
-
* 02
C6
13
18
IL
--
"/al2 Figure 5. Cumulative volume fraction vs. d / & for turbine stirrer, one-paw vessel.
fluence this maximum value. Cutter (1966) calculates on the basis of experimental data with a turbine stirrer, a ratio of 270 between the energy dissipation rate in the region of the stirrer and the energy dissipation rate in the more quiet parts of the vessel. In Figure 4 the value of d 3 2 / Dis plotted vs. the Weber number. For the turbine and inclined blade agitators the dependency is indeed equivalent to an exponential function with an exponent -0.6. For the high shear rate agitators, cone and flat disk, the exponent is about equal to -1, but in these cases Hinze's underlying assumption of isotropic energy dissipation can be doubted. (b) Experiments in t h e One-Pass Vessel. Table I1 presents the results for the Sauter mean diameter and the standard deviation. Figure 5 presents an example of some of the experimental runs. It shows that the distribution in this case cannot be represented very well by a normal distribution curve; only 80% of the droplets are randomly spread.
d,,,
std m dev
707 787 568 630 735 435 375 184 133 168 290 279 193 143 126 267 146 300 308 216 137 870 502 710 494 691
0.23 0.25 0.25 0.25 0.26 0.26 0.26 0.25 0.26 0.26 0.25 0.26 0.27 0.27 0.25 0.27 0.25 0.26 0.26 0.27 0.28 0.23 0.27 0.26 0.22 0.26
Through-
What can be seen by direct inspection of Tables I and I1 is that in the one-pass vessel the breaking-up process if far from equilibrium. For example, for the turbine agitator the average droplet diameter is still 2-3 times larger than the equilibrium diameter. However, the standard deviation in the one-pass vessel is only slightly lower than in the stirred tank. (c) Droplet Breakup in the One-Pass Vessel, Small Droplets of Uniform Diameter Being Added above the Agitator. The experiments in sections a and b start with such large droplets that every droplet must be broken up approximately 20 times before reaching the equilibrium size. We therefore started in new experiments with small droplets of uniform diameter of 400 pm, which is near the critical size. The droplets were injected upstream the stirrer. After passing the agitator the droplets are sampled through a Coulter counter measuring tube. The resistance of the capillary opening in the tube is increased when a droplet moves through; the resulting voltage jump is, in first approximation, a linear function of the droplet volume. By mounting a glass ring of appreciable size (ca. 2 x 2 mm) around the outside of the capillary opening, disturbances caused by passing but not entering droplets can be prevented. The voltage over the capillary was continuously monitored electronically and by sampleand-hold-circuitsthe size of each passing droplet was found and passed to the memory of a digital computer. Measurements were made with the cone agitator only, at slit widths of 1and 2 mm. Flow rates of the continuous phase ranged from 2.65 to 11.7 lo4 m3/s, the speed of the agitator was varied between 175 and 1400 rpm. Figure 6 shows the results of a series of experiments. Below 485 rpm no appreciable fraction broken-up droplets can be observed, but this fraction increases strongly above 700 rpm. According to Bird et al. (1963) the transition from laminar to turbulent flow for this configuration occurs at approximately 900 rpm. Hence it seems unlikely
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 175 r p m
279
m
2b
f z -b
2 Figure 7. Notation for modeling breakup.
I.
,
-,..’:
I 16;
.;, 8
0
I 1 0 0 rpni
‘
,
~~
*‘.---.\ .
,
,
6
8
~~
~
,
the Static Ellipsoidal Random Cut model (SERC). When both axes of the ellipsoidal drop are equal, the model degenerates in that for a randomly cut-through sphere. The second model describes the breaking-up process as a dynamic competition between surface energy and kinetic energy, resulting in an oscillatorial behavior which ends, if enough energy is added to the kinetic movement, in the breaking up of the mother droplet into two equal daughter droplets. This model is called Oscillatory Produced Equal Daughters (OPED). The chance that a droplet breaks up is taken to be proportional to the droplets equilibrium (= spherical) diameter. SERC Model. The equivalent diameter of the ellipsoidal droplet is d ; then the volume of the droplet (see Figure 7 ) is
*. c 2
L
1
__+
0 1 2 CHANNEL NUMBER
Figure 6. Series of experiments, uniform droplets being added to the one-pass-vessel; input droplet diameter 400 wm; flow rate continuous phase: 5.3 X lo4 m3/s: slit width 1mm; fraction dispersed phase 38 X
that breaking up of the droplets occurs in the turbulent regime only. In some of the experiments, notably at the lower speeds, coalescense of the initial droplets occurs in the slit, due to the increased probability of contact between the droplets in the slit because of the velocity gradient. Table I11 gives the experimental results. The peaks in the distribution curves are analyzed in a way analogous to gas chromatographic peaks. The standard deviation in the size of injected droplets is about 0.05-0.1 in volume, corresponding to 0.02--0.04 in diameter. The coalesced droplets have somewhat smaller standard deviation, as should be the case: considering the law of large numbers, the standard deviation of the coalesced droplets has to be smaller by a factor 2lI2. The broken up droplets have a much larger standard deviation in volume, although lower than found in the sections a and b. It can also be seen that the droplets tend to break up into two nearly equal daughter droplets, especially at higher agitator speeds. When the fraction of broken-up droplets in Table I11 is extrapolated to zero, the minimum rotation speed is found at which droplets of 400 pm can be broken up. This stirrer speed is, for all experiments, 600 rpm and not dependent upon the liquid flow rate and the slit width. It corresponds to a critical Weber number of about 60.
Theoretical Considerations For the droplet break-up process two theoretical models have been developed. In the first model the droplet is considered to have a steady-state elliptical shape. The droplet moves through a turbulent zone and is broken up perpendicular to the main axis as though the droplet was cut into two parts by a knife. The knife cuts randomly along the axis. The chance that a droplet is cut is proportional to the length of the main axis. We call this model
The chance that the droplet with the largest main axis amax and volume corresponding to do breaks up in a certain time interval is taken to be m. The chance E for any other droplet to be broken up in this time interval is then a
E=-m %ax
For a constant value off this gives E = mdj/do. The droplet splits into two new ones. Suppose that the droplet is split on position z on the main axis; then the volume of one of the new droplets is
By subtraction the volume of the second daughter droplet is found. If we take x i = di+Jdi, then
Introducing h = [ ( z + a)/2a].di gives z / a = (2h/di)- 1and consequently
($)-:($)
2 + s 1x ? = O
OIh
