Ind. Eng. Chem. Res. 1991,30,835-841 11. Solution in time domain. J. Theor. Biol. 1984,106, 207-238. Lenhoff, A. M.; Lightfoot, E. N. Convective dispersion and interphase mass transfer. Chem. Eng. Sci. 1986,41, 2795-2810. Paine, M. A.; Carbonell, R. G.; Whitaker, S. Dispersion in pulsed systems-I. Heterogeneous reaction and reversible adsorption in capillary tubes. Chem. Eng. Sci. 1983,38, 1781-1793. Reis, J. F. G.; Lightfoot, E. N.; Noble, P. T.; Chiang, A. S. Chromatography in a bed of spheres. Sep. Sci. Technol. 1979, 14, 367-394.
Sankarasubramanian, R.; Gill, W. N. Unsteady convective diffusion with interphase mass transfer. Proc. R. SOC.London, A 1973,333, 115-132; Corrigendum 1974, 341, 407-408. Shankar, A.; Lenhoff, A. M. Dispersion in laminar flow in short tubes. AIChE J. 1989,35, 2048-2052. Shankar, A.; Lenhoff, A. M. Dispersion in round tubes and its implications for extra-column dispersion. J. Chromatogr. 1990, in press.
835
Taylor, G. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. SOC.London, A 1953, 317, 186-203. Wang, J. C.; Stewart, W. E. New descriptions of dispersion in flow through tubes: convolution and collocation methods. AIChE J. 1983,29, 493-498.
Westhaver, J. W. Theory of open-tube distillation columns. Znd. Eng. Chem. 1942,34, 126-130. Yu, J. S. An approximate analysis of laminar dispersion in circular tubes. J. Appl. Mech. 1976,43,537-542. Yu, J. S. On laminar dispersion for flow through round tubes. J. Appl. Mech. 1979,46, 750-756. Yu, J. S. Dispersion in laminar flow through tubes by simultaneous diffusion and convection. J. Appl. Mech. 1981, 48,217-223. Received for review June 11, 1990 Revised manuscript received November 12, 1990 Accepted November 16, 1990
Experimental Observations of Bubble Breakage in Turbulent Flow? Robert P. Hesketht Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, England CB2 3RA
Arthur W. Etchells Engineering Department, E. I. du Pont de Nemours & Co., Newark, Delaware 19714
T . W . Fraser Russell* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19711
Bubble and drop breakage in turbulent liquid flow in a pipeline was observed to determine the mechanism of breakup. Bubble breakage sequences from high-speed movie film were examined to determine a breakage time, the number of bubbles formed, and bubble sizes. Bubble breakage was observed only at dimensionless radii greater than 0.5 and only two bubbles were formed. The bubble breakage is caused by a surface disturbance which eventually deforms the bubble into a dumbbell shape. The sizes of the resulting two bubbles have a higher probability of being of unequal volume than equal volume. Preliminary results from drop breakage experiments indicate that bubbles and drops have similar breakage mechanisms and that the time to break ranges from 0.01 to 0.1 s.
Introduction Dispersion systems are primarily formed to increase the interfacial area and improve the rates of heat and mass transfer between one or more phases. The interfacial area of a dispersion can be calculated from predicted bubble or drop sizes. To maximize bubble or drop interfacial areas within a two-phase contactor, the energy input required usually assures that the continuous phase is in turbulent flow. Correlations have been developed to estimate bubble or drop size in turbulent liquid flow. The parameters in these correlations are typically obtained using a bubble or drop size distribution measured after some time in a batch experiment or measured in the exit stream of a continuously flowing system. Very little research has been done to examine the breakage process of individual bubbles or drops in turbulent liquid flow. Most of the models that predict bubble or drop size in turbulent liquid flow are based on the work of Kolmogoroff (1949) and Hinze (1955). These theories are used to predict a maximum stable bubble or drop size, d,, for a given + W ededicate this paper to Professor R. L. Pigford, who felt strongly that understanding of basic phenomena is essential to sound chemical engineering research and design. We hope that this paper added, in some small way, to the important issue of predicting interface area in gas-liquid and liquid-liquid systems, a problem of lifelong interest to R. L. Pigford. f Present address: Chemical Engineering Department, University of Tulsa, Tulsa, OK 74104.
0888-5885/91/2630-0835$02.50/0
turbulent flow field and fluid physical properties. Davies (1987) presents an equation that predicts the largest drop size that can survive within a turbulent flow field for energy dissipation rates varying over 5 orders of magnitude. The work by Hesketh et al. (1987) extends the range of this work by using a single equation for both bubbles and drops in turbulent pipeline flow
where the critical Weber number, Wecrit,is approximately equal to 1 (Hesketh, 1987). The maximum stable bubble or drop size, d,, predicted by the above equation can only be achieved if the bubble or drop is within a turbulent flow field for a sufficient period of time (usually on the order of seconds). In commercial-scale two-phase contactors there are active zones in which bubbles will break up and inactive or neutral zones in which the turbulence level is too low for breakup. In designing two-phase contactors, one must be able to calculate the bubble residence times in the active and neutral zones of the process equipment. Some bubbles may be in the active zone for too short a time for complete break up, and they will be larger than the maximum stable bubble size. Recently investigators have examined bubble and drop breakage using population balances that describe the change in drop size due to either coalescence or breakage. 0 1991 American Chemical Society
836 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 These population balances contain breakage rate expressions that can only be verified by studying rate of droplet or bubble breakage. This paper contains quantitative data on individual bubble and drop breakup in turbulent liquid flow which should help formulate proposed drop or bubble breakage rate expressions. Sleicher (1962),Collins and Knudson (1970),and Walter and Blanch (1986) have photographed drops and bubbles breaking up in turbulent pipeline flow. Sleicher photographed both the horizontal and vertical planes of the pipeline using a mirror. He concluded that liquid drop breakage occurred very close to the wall and observed two types of breakage. A breakage type similar to the cigarshaped deformation postulated by Hinze (1950) was observed to produce two approximately equal volume drops. Sleicher observed that the drop stretched to considerable lengths before breakage, and in some cases he observed drops stretching to lengths of 4 times their original diameter without breaking. In some cases drops larger than his predicted d, passed near the wall without departing much from a spherical shape and did not break up. The second type of breakage observed by Sleicher in which a very small drop formed from a much larger drop also occurred near the wall, but at a much lower frequency than the cigar shaped deformation breakage. Collins and Knudson (1970) photographed drops using a single lens reflex camera at dimensionless radii, r / R , of 0.96,0.9, and 0.6. These still photographs showed an assortment of highly deformed drops, most of which had a dumbbell shape. In many cases the two spherical ends of the dumbbell were connected with a long liquid cylinder that extended from 1to 4 diameters of one of the spherical ends. These still photographs support the breakage descriptions of Sleicher (1962). Walter and Blanch (1986) observed bubbles breaking up in a 0.0508-m-diameter pipeline. They identified the bubble breakage mechanism as having three stages: oscillation, dumbbell stretching, and pinching off. They observed the bubble surface oscillating and rippling and then the bubble stretching into the shape of a dumbbell which pinches off into two equal-sized bubbles. Bubble size measurements were not reported. The time required to stretch the bubble into a dumbbell shape was deters. They stated that mined to be approximately 25 X most of the breakage events occurred near the center of the pipe, but since they only used a single-plane image of the pipeline the exact radial location of the breaking bubble could not be determined. In order to quantify the bubble breakage process in turbulent liquid flow, high-speed motion photographs were taken of the bubble breakage process. Individual breakage sequences were located and quantified using frame by frame analysis of the cine film. The quantitative data included measurements of a characteristic breakage time, sizes of bubbles before and after breakage, and position of breakage with respect to the pipeline walls.
Experimental Analysis A horizontal pipeline with a diameter of 0.038 m and a total length of 7.62 m was chosen to examine the breakage process, since the turbulence is relatively homogeneous in the axial and circumferential directions. Compressed air was introduced into the pipeline by use of a 0.00635-mdiameter nozzle concentric with the pipeline for 1.62 m. The remaining 6 m of the horizontal pipeline was used for visual observation. Tap water was recirculated through the system. The bubbles were photographed with a high-speed 16mm motion camera designed by the Mechanical Devel-
opment Laboratory at Du Pont. The lighting for the 16mm camera was provided by an EG&G Type 501 highspeed stroboscope. The film speed for the 16-mm camera was chosen to be 1200 frames/s. All photographic information is stored in the Multi-Phase Fluid Mechanics Film Archives at the Department of Chemical Engineering, University of Delaware. A mirror wm used to obtain a two-plane image of bubble size and position within the pipeline. Distortion created by the pipeline curvature was minimized by surrounding the view section of the pipeline with a water-filled plastic box. Photographs were taken at nine axial positions along the pipeline starting from the nozzle. The photographs were analyzed by use of two separate digitizers. The 16-mm film was analyzed by use of a variable-speed projector and digitizer system. The bubble sizes were corrected for perspective reduction of the image size. A more detailed description of the apparatus and procedures is given by Hesketh (1987). Quantitative results from the high-speed movie film were obtained by digitizing bubble sizes, bubble locations, and identifying deformation and breakage events on selected frames of the film. Approximately 840 bubbles were observed in over 1000 frames of high-speed film to obtain 56 breakage events. The velocity of the continuous phase for these experiments was 3.98 m/s and the gas flow rate was 1.7 X m3/s. The minimum information required for a breakage event to be digitized was the position and size of the original and resulting bubbles. In addition each breakage event was examined to determine if the initiation of the deformation that led to breakage could be observed. This enabled the breakage time to be measured. All bubble deformation and breakage events were digitized for distances less than 1.33 m from the nozzle. A t positions greater than 1.33 m the number of bubble breakages is small, and in many cases the bubbles were partially obscured by other bubbles, making quantitative observation difficult. Bubble diameters reported are based on a prolate or oblate spheroid model. In each plane, length and angle measurements of the major and minor axes were measured. These measurements were converted to the three major axes of the spheroid model with the equivalent spherical diameter given by d = (412Z3)1/3 (2) A limited number of drop breakage events were recorded by use of similar techniques to those for bubbles. A dispersed phase of silicone oil was chosen since it has a density of 949 kg/m3, which is within 5% of the value of water at lo3 kg/m3. The low density difference results in a rise velocity of the drops which is much smaller than that of comparable-sized bubbles. In this manner the position of drop breakage could be more fully examined since the drops would be well-dispersed throughout the pipeline cross section. A continuous-phase velocity of 2.22 m/s and dispersed-phase flow rate of 600 cm3/min were chosen to study the breakage process of the silicone oil drops. Motion pictures were taken at eight axial positions starting at the nozzle and ending 5.85 m from the nozzle. Bubble and Drop Deformations. The surface of either a bubble or drop is observed to be in constant irregular motion caused by the turbulent eddies interacting with its surface. Both large- and small-scale deformations with respect to the bubble or drop diameter are observed. The small-scale deformations were observed primarily in the large-diameter bubbles. They have a lower internal pressure than small bubbles and are more easily deformed. The small deformations were restored very rapidly and did
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 837 -.
-
---
c
1
1
L
G
a
b
e
d
C
Figure 1. Stable bubble deformation. do = 6.1 mm; r / R = 0.7; 8 = 140’; f,, = 30 s-l.
.
a
b
d
C
e
Figure 2. Stable drop deformation. do = 3.4 mm; r / R = 0.5; 8 = 50°; f, = 21 s-l.
not increase in amplitude and cause breakage. The elapsed time between the appearance of the surface deformation and its restoration is an order of magnitude smaller than the equivalent elapsed time of the large deformations. Both the small time of existence and small dimensions of these deformationsindicate that they are caused by eddies much smaller than the bubble or drop diameter. This observation is in agreement with Hinze’s 1950 breakage theory which assumed that eddies smaller than the bubble diameter do not cause breakage. Large-scale deformations are present in bubbles with diameters ranging from the maximum size produced in the introduction region of the pipeline to sizes much smaller than the predicted maximum stable bubble size given by eq 1. The process of large-scale bubble and drop deformation is shown in the series of photographs in Figures 1and 2, respectively. Each figure consists of a sequence of frames from the high-speed cine film. Only the portion
of the toval frame ,hat contains the bubble or drop of interest is shown. Each of the photographs contains a two-plane image; the top image is the side view and the lower image is the mirror image of the bottom of the horizontal pipeline. The fluid motion is from left to right. The large bubble in Figure l a has a prolate shape and appears to be forming a dumbbell shape (Figure IC).In this example the bubble does not break up; instead the deformation travels toward the upstream end of the bubble and the bubble returns to a prolate shape. Examining both the mirror image and the pipe image reveals that instead of a dumbbell shape, in which the diameter of the center cylinder is small compared to either end, the bubble has been flattened. This bubble deformation sequence suggests that gas flow within the bubble may be significant in determining whether or not a bubble breaks up. The drop deformation shown in Figure 2 has a large amplitude of deformation but again returns to a stable
838 Ind. Eng. Chem. Res., Vol. 30,No. 5,1991
~~~
a
b
C
d
e
Figure 3. Unequal-volume bubble breakage. do = 5.7 mm; r / R = 0.8;8 = looo; f, = 28 s-l; dl = 3.4 mm; d2 = 5.0 mm. c - -
.
(L.c--~
0'
a
f
--____IC
1
i
-
b
C
d
e
Figure 4. Equal-volume bubble breakage. do = 6.9 mm; r / R = 0.8; 8 = No; f,, = 19 s-l; dl = 5.4 mm; d2 = 5.5 mm.
shape. At least two examples of drop deformation can be seen in Figure 2. The 3.4-mm drop located above the 26-mm mark (in the upper half of the pipeline) in Figure 2a begins to elongate in Figure 2b, reaching a maximum length in Figure 2c (located above the 77-mm mark). The drop returns to an oblate shape in Figure 2e (locatedabove the 140-mm mark) without breaking. Other drops can be seen deforming such as the large drop above the 44-mm mark in Figure 2a. Breakage. Two types of bubble breakage are observed in turbulent pipeline flow. The most prevalent type of breakage is a result of a bubble or drop undergoing a large-scale deformation. The second type of breakage appears to be caused by some type of tearing mechanism and results in a very small volume of gas being torn from the original bubble. Examples of breakage caused by large-scale deformations are shown in Figures 3-5 for both bubbles and drops. This type of breakage produces a wide range of bubble and drop sizes as illustrated in the examples shown in these photographs. Unequal volume
breakages are shown in Figures 3 and 5, and an equal volume breakage is shown in Figure 4. In all cases before breakage, the bubble or drop has either a prolate or an oblate shape which is subsequently deformed into an elongated prolate shape. In most cases of breakage, both ends of the elongated prolate evolve into nearly spherical ends and the bubble shape is similar to that of a dumbbell. The dumbbell shape continues to be deformed until the bubble or drop breaks up. A tearing breakage is caused by a localized deformation of one end of a deformed bubble or drop and produces one bubble or drop of essentially the same volume and a second bubble or drop size less than 0.5% of the original bubble or drop volume. An example of this type of breakage is shown in the series of photographs in Figure 4. In this series of photographs, a second deformation is superimposed on a larger deformation, causing the left end of the bubble to be torn away from the remaining part of the bubble. In Figure 4d, a small bubble is being torn off of the upstream end of the large bubble, and in Figure 4e
Ind. Eng'.(>hem. Res., Vol. 30, No.5, 1991 839
1
f.
ia b
a
d
C
Figure 5. Unequal-volume drop breakage. do = 3.9 mm;r / R = 0.7;8 = 205';
f,, = 10 s-'; dl = 1.9 mm; d2 = 3.4 mm.
Table I. Drop Breakage Positions and Sizesa
/
/
/-*.
\
\
*\
1 2 2 5 6 7 11 12
3.95 4.05 3.63 3.88 3.48 2.47 3.73 2.68
0.68 0.90 0.81 0.70 0.60 0.58 0.75 0.73
0.95 1.16 1.26 1.90 2.78 1.60 1.04 1.08
3.60 3.88 3.28 3.39 2.22 3.22 3.80 2.83
0.02 0.03 0.05 0.15 0.33 0.11 0.02 0.05
aDispersed phase, silicone oil (Dow Corning 200); continuous phase, water; pd = 949 kg/m3; pe = 109; u = 30.3 X lo4 N/m; u, = 2.22 m/s; Qd = IOa m3/s; pd = 0.019 Pa s.
Figure 6. Radial bubble breakage positions in pipeline c ~ 8 section. 8 The position in which the bubble broke up is marked with a star. The solid line is the inside pipeline wall, and the dashed line is located at r / R = 0.5.
another small bubble has been torn off of the large bubble. Location of Bubble and Drop Breakage. Bubble breakage was only observed at dimensionless radii, r / R , greater than 0.5 (Figure 6), which agrees with similar observations made by Sleicher (1962). At distances from the nozzle less than 0.28 m, most of the bubbles are located at values of r / R less than 0.5 and no breakage was observed in this region. The bubbles in this region initially have a prolate shape caused by the gas-jet breakage, but instead of forming a dumbbell shape and breaking up, they return to a nearly spherical shape as they rise upward. From these observations a neutral zone for bubble breakage can be defined between the end of the jet (approximately6 cm from the nozzle tip) and a distance of approximately 0.3 m. The values of r / R for drop breakage are listed in Table I. Due to a small change in the size distributions over the 5.85-m viewing section (d- = 5.6 mm initially and d, = 4.2 mm at 5.85 m), the number of breakage and deformation events was much lower than bubble breakage ev-
ents, and only 12 deformation and breakage events were identified and digitized. In all drop breakage observations, only drops larger than the value from eq 1of d, = 1 mm were observed to break up. These drops were observed to break up at dimensionlessradii, r / R , greater than 0.58. Figure 7 illustrates the degree of dispersion of the drops within the pipeline at two axial positions of 0.2 and 5.85 m from the nozzle. Each point represents the approximate center of mass of a drop of size greater than approximately 1.3 mm. As can be seen by the location of the drops in Figure 7, drops initially at the center of the pipeline are dispersed throughout the pipeline cross section as they travel down the pipeline. The mean values of r / R shown in Figure 7 are 0.29 and 0.59, respectively. Since the maximum drop size of the size distribution of drops measured at the end of the pipeline ( z = 5.85m) is larger than the maximum stable drop size for this level of turbulence (d, = 1 mm), then drops that should break up exist in the pipe. If the pipeline were longer than 5.85 m, these drops would eventually travel from the neutral region to the active region near the wall and break up. The observations of both the bubble and drop breakage position demonstrates the existence of active and neutral zones for bubble and drop breakage. In the pipeline the active zone is an annular region with r / R > 0.5 and the neutral zone is the central core with r / R < 0.5. The active zone for breakage contains the highest fluctuating velocities within the pipeline cross section. Buoyancy forces cause the bubbles to rise to the active zone of the pipeline and remain there until they break up. Thus, the time in the active zone for bubbles in a horizontal pipeline is about equal to the residence time of bubbles within the pipeline.
840 Ind. Eng. Chem. Res., Vol. 30,No. 5,1991
(b)
Figure 7. Location of drops with d > 1.3 mm in pipeline cross section. The position of the drop is marked with a star. The solid line is the inside pipeline wall, and the dashed line is located at r / R = 0.5. (a) A t 0.2 m from the nozzle; (b) at 5.85 m from the nozzle.
In the liquid-liquid dispersion experiments drops are distributed throughout the pipeline cross section and the mean residence time of the drops is not equal to the time in the active zone. To predict bubble or drop size, the residence times in the active zones of process equipment must be estimated. Bubble and Drop Breakage Number and Sizes. All of the observed bubble breakages are binary, in which no more than two bubbles are formed within the length of time between two frames of film (=0.0008s). A dimensionless bubble breakage volume ratio, VR, can be defined as VR = d,S/(d: + dj9) (3) where the subscripts i and j refer to the two bubbles formed from a breakage. Using the above definition of V R , the two volume ratios will be normalized and are symmetric about VR = 0.5. All bubbles break into a range of bubble sizes. For example, large bubbles of size 6 mm have been observed to break into bubbles with VR values of 0.1 and 0.9, and 0.5 and 0.5. The same values of V R were also obtained for a small bubble of size 2.5 mm. Hesketh (1987)
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 841
t
i
t
u > w T)
s /L
010
00 1 BUBBLE DIAMETER, (m)
Figure 9. Experimental bubble oscillation frequency compared with u = 53.8 X N/m.
theory.
Table 11. Drop Deformation and Breakage Times' drop series 103d0,m elapsed time, s period freq, Hz 1 3.95 13 0.039 112 2 3.63 26 0.019 112 3 4.55 0.036 I 28 5 3.88 10 0.050 112 6 3.48 9.4 0.053 112 8 3.40 0.049 1 21 9 3.74 0.059 1 17 11 3.73 0.040 112 12 a Dispersed phase, silicone oil (Dow Corning 200); continuous phase, water; Pd = 949 kg/m3; p e = lo3; u = 30.3 X N/m;v, = 2.22 m/s; Qd = m3/s.
breakage and can be located on the cine film within &one frame. The location of the onset of an oscillation and the end of a stable oscillation requires a more subjective judgement. In this case, the onset of an oscillation was defined as the beginning of a sustained change in length or width of a bubble that initiated a large deformation (i.e., small-scale deformations were ignored). Experimentally observed times for stable and unstable oscillations are plotted as frequencies versus their size in Figure 9. The deformation time of an unstable oscillation was considered to be one-half the period of an oscillation and the frequency is the reciprocal of twice the deformation time. The experimental frequencies are within a factor of 0.5-1.8 of the theoretical frequency, and the majority of the experimental frequencies are lower than the theoretical frequencies. Both deformation and breakage times of drops were measured from the high-speed movie films. The elapsed time and frequencies are presented in Table 11. The values for the frequency vary by a factor of 3 and range from 9 to 28 Hz for a narrow range of drop sizes of 3.5-4.5 mm. These measured frequencies are within a factor of 0.3-1.5 of the theoretical frequencies using eq 4 with n = 2 and Q = 30.3 X N/m.
Conclusions Bubbles and drops were observed by high-speed photographs to undergo both large- and small-scale deformations in a turbulent flow field. A large-scale deformation can cause unstable oscillations leading to break up. A second type of tearing mechanism which produced a very small bubble was also observed with bubbles, but such events were much less frequent than the unstable oscillation mode of breakup. The breakage events observed produced two bubbles which had a much higher probability of unequal size than equal size. In the pipeline, as in all mass contactors, there is an active zone where the turbulence level is great enough to cause breakage and a neutral zone where no breakage
occurs. The active zone in the pipeline is the annular region between r / R = 0.5 and the pipe wall. The neutral zone is the central core defined by r / R < 0.5. Bubbles rise into the active zone of the pipeline within rather short distances, and the maximum size measured at the end of pipe is the same as that predicted by eq 1. The almost neutrally buoyant drops distributed themselves in both the active and neutral zones by the end of the experimental pipeline, and the maximum size measured was found to be greater than that predicted. The experimentally observed time for a bubble or drop of a given diameter to break can also be estimated within a factor of 2 from a calculation of frequency of oscillation of a sphere. For bubble diameters ranging from about 0.002 to 0.007 m, breakage times ranged from approximately 0.01 to 0.1 s.
Nomenclature d = equivalent spherical bubble or drop diameter (m) di,dj = individual bubble diameter (m) do = bubble or drop size before breakage (m) d, = maximum bubble or drop diameter of a distribution (m) d, = maximum stable bubble or drop diameter in a given fluid
flow field (m) f,, = frequency of oscillation (s-l) 1 = axis of an ellipse-spheriod (m) n = mode of oscillation Q d = flow rate of dispersed phase (m3/s) r = radial position (m) R = radius of pipeline (m) u, = continuous-phase average velocity (m/s) V , = breakage volume ratio defined by eq 3 z = axial distance from nozzle (m) Greek Letters
density (kg/m3) surface tension (N/m) E = local energy dissipation rate per unit mass (W/kg) 0 = angular location of bubble position
p = u =
Subscripts c = continuous phase d = dispersed phase
Literature Cited Collins, S. B.; Knudson, J. B. Drop-size Distributions Produced by Turbulent Pipe Flow of Immiscible Liquids. AZChE J. 1970,16, 1072. Davies, J. T. A Physical Interpretation of Drop Sizes in Homogenizers and Agitated Tanks, Including the Dispersion of Viscous Oils. Chem. Eng. Sci. 1987, 42. Hesketh, R. P. Bubble Breakage in Turbulent Liquid Flow. Ph.D. Dissertation, University of Delaware, 1987. Hesketh, R. P.; Etchells, A. W.; Russell, T. W. F. Bubble Size in Horizontal Pipelines. AIChE J. 1987, 33, 663. Hinze, J. 0. Fundamentals of the Hydrodynamic Mechanism of Splitting Dispersion Processes. AZChE J. 1955, 1 , 289. Kolmogoroff, A. N. About the Breaking of Drops in Turbulent Flow. Dokl. Akad. Nauk SSSR 1949,66, 825. Lamb, H. Hydrodynamics, 6th ed.; Cambridge University Press: Cambridge, England, 1932. Sleicher, C. A. Maximum Stable Drop Size in Turbulent Flow. AIChE J. 1962,8, 471. Tavlarides, L. L.; Stamatoudis, M. The Analysis of Interphase Reactions and Mass Transfer in Liquid-Liquid Dispersions. Adv. Chem. Eng. 1981,11, 199. Walter, J. G.; Blanch, H. W. Bubble Break-Up in Gas-Liquid Bioreactors: Break-Up in Turbulent Flows. Chem. Eng. J. 1986,32, B7. Received for review June 14, 1990 Revised manuscript received October 11, 1990 Accepted October 22, 1990
