T a = Taylor number defined by eq 21 U = bulk velocity u = total fluid velocity u' = fluctuating velocity V = meanvelocity
Greek Letters wavenumber ratio of u to wave number wavelength absolute viscosity kinematic viscosity fluid density u = circular frequency plus amplification factor T = shear stress tensor component 4 = amplitude function w = angular velocity
a = p = X = p = v = p =
Subscripts = radial direction = axial direction = tangential direction = inner wall (radial position) = outer wall (radial position) = amplitude function for radial component of disturbance = amplitude function for tangential component of disturbance
3 = amplitude function for axial component of disturbance Literature Cited Coles, D., J. FluidMech., 21, 385 (1965). Cornish, R . J., froc. Roy. SOC.London, Ser. A, 140, 227 (1933). couette, M., Ann. Chim. fhys., 21, 433 (1890). Goldstein, S . , froc. Cambridge Phil. SOC., 33, 41 (1937). Jeffreys, H., froc. Roy. SOC.London, Ser. A, 118, 195 (1928). Joseph, D. D., "Stability of Fluid Motions. I," Chapters 2 and 6, p 27, SpringerVerlag. New York, N.Y., 1976. Lin. C. C., "The Theory of Hydrodynamic Stability," Chapter 7, p 109, University Press, Cambridge, -1955. Lord Rayleigh, "On the Dynamics of Revolving Fluids," "Scientific Papers," Volume VI, 1916. Ludwieg, H.,2.Flugwiss. 8, 135 (1960). Mallock, A., Phil. Trans. Roy. SOC.London, Ser. A, 187, 41 (1896). Prengle, R. S.,Rothfus, R. R.. lnd. Eng. Chem., 47, 379 (1955). Schlichtina. H., "Boundary-Layer . . Theory," ChaDter 2, D 27, McGraw-Hill. New York, NTY., 1968. Schultz-Grunow, Von F., Z.Angew. Math. Mech., 39, 101 (1959). Squire, H. B.. Proc. Roy. SOC.London, Ser. A, 118, 621 (1933). Taylor, G. I., Phil. Trans. Roy. SOC.London, Ser. A, 223 (1923). Tollmien, W., NACA, Technical Memorandum, 792 (1936). van Duuren. F. A,, J. Sank Eng. Div., ASCE, 671 (1968). White, F. M., "Viscous Fluid Flow," Chapter 5,p 399, McGraw-Hill, New York, N.Y., 1974.
Received f o r review October 19, 1976 Accepted M a y 11,1977 T h e authors are grateful t o t h e N a t i o n a l Science F o u n d a t i o n a n d t o t h e Office o f Water Research a n d Technology of t h e D e p a r t m e n t o f the I n t e r i o r for financial support.
EXPERIMENTA1 TECHNIQUES
Breakage and Coalescence Processes in an Agitated Dispersion. Experimental System and Data Reduction F. H. Verhoff,'' S. L. Ross, and R. L. Curl Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48 104
A method for the rapid and accurate determination of the bivariate distributions of drop volume and a tracer dye concentration in a continuous flow agitated dispersion is described. The method involves extracting a sample of the dispersion from a vessel, protecting the sampled drops with a surfactant, and pumping the sample at a constant flow rate through a capillary smaller than the smallest drop of interest; light transmission gives dye concentration, and time-of-passage of a drop gives drop volume simultaneously. The method yields information about the rate processes of breakage and coalescence in an agitated dispersion.
If a mixture of two immiscible liquids is agitated in a batch or continuous flow system, a dispersion of droplets may be formed whose characteristics are functions of the geometry and size of the mixer and its materials of construction, the intensity of agitation, the phase fraction, and fluid properties including viscosities, densities, surface energies, and the concentration of dissolved solutes. These factors act to proAddress correspondence t o t h i s author a t t h e D e p a r t m e n t of Chemical Engineering, W e s t V i r g i n i a University, Morgantown, W. Va. 26506.
duce the directly observable drop size distribution as determined by breakage and coalescence. These two rate processes are called dispersed phase mixing because they mix substances between droplets in the dispersed phase. These processes may be important in the design of liquid-liquid contacting equipment. Although such properties as interfacial area and mean drop size have been measured and correlated with system parameters, a better understanding of droplet breakage and coalescence in an agitated system would permit the treatment of problems involving mass transfer and chemical reaction, Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
371
and the prediction of drop size distributions. Several recent studies illustrate this. Valentas et al. (1966, 1968) applied population balances to dispersions and, by assuming mathematical dependences of breakage and coalescence rates on drop sizes, calculated resulting drop size distributions and some effects in reactors. Bayens and Laurence (1969) subsequently predicted the effect of coalescence on mass transfer rate but used a coalescence rate expression that was independent of drop size. Zeitlin and Tavlarides (1972a,b) included spatial variation of dispersion properties in modeling an agitated vessel, but also conjectured about the breakage and coalescence rates as functions of drop sizes and position in the vessel. Finally, Shah and Ramkrishna (1973) modeled the effect of drop breakage on mass transfer rate (and distribution of solute concentration among the drops of a dispersion) in a manner similar to that of Bayens and Laurence, using the breakage models of Valentas et al. Missing in all of these studies has been a confirmed quantitative description of the rate of the breakage and coalescence processes, based upon experimental evidence. The effects of agitation and system variables on the behavior of dispersions in mixing vessels has, of course, been extensively studied. I t has been found that relatively large drops tend to break down, until a size, d,,, is reached below which the probability of breakup is low. Similarly, it has been found by Shinnar and Church (1960, 1961) and implied by other workers, that relatively small drops tend to coalesce up to a size, dmin,above which coalescence is unlikely. Obviously, no such absolute limits can exist in real systems; breakage presumably decreases with decreasing drop size and becomes small compared to some other rate process as d,,, is passed. The same must also be true of coalescence as drop size increases to dmin. A correct description of dispersion processes must predict the rates of breakage and coalescence as functions of all system parameters. The dispersed phase mixing (i.e., breakage and coalescence) is defined by the following functions: g(u), the number fraction rate of breakage of drops of volume u ; p ( u l u ’ ) , the number probability density distribution of volume u of daughter drops formed on breaking a drop of volume u’; u ( u ) , the average number of drops formed per breakage of drops of volume U ; h(u, u ’ ) , the number rate at which a drop of volume u collides with drops of volume u’, per drop of volume u’; X(u, u’), the probability of coalescence of two drops of volume u and u’, given that a collision has occurred. These functions are not spatially dependent for the experiments reported herein as will be discussed later. No experimentally determined expressions for a n y of t h e above functions haue been preuiously reported. Experiments have, of course, provided information about these functions, and much has been conjectured on the basis of these experiments. It is our purpose in this series of papers to present experimentally based expressions for all of these dispersed phase “mixing” functions. Previous Experimental Methods A variety of experimental approaches have been used to discover features of dispersion behavior. The most widely studied system is the impeller-agitated, baffled, batch, liquid-liquid dispersion. The major parameter was the interfacial area (or Sauter mean diameter, d32), and Calderbank (1958) as well as McCaughlin and Rushton (1973) used absolute methods whereas Rodger et al. (1956), Scott et al. (19581, Vermeulen et al. (1955), and Weinstein and Treybal (1973) used calibrated interfacial area measurement schemes. Additional information was obtained from the distribution of drop sizes, for which photography (e.g., Brown and Pitt (1974), Chen and Middleman (1967),van Heuven (1969),and Jeffreys et al. (1970)),encapsulation (van Heuven (1969 and 1971) and 372
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
McCoy and Madden (196911, screening (e.g., McCoy and Madden (1969), freezing of drops (e.g., Church and Shinnar (1961) and Shinnar (1961)), and the Coulter Counter (e.g., Doulah et al. (1975), Sprow (1967a,b), have been used. The briefest summary of average drop size information may be expressed by the Kolmogoroff relationship (after Mlynek and Resnick (1972)) d32 = 0.058We-0.6(1
+ 5.44)D
(1)
which also includes an empirical dependence upon the dispersed phase fraction 4. The consequences of coalescence have also been studied primarily in the impeller-agitated, baffled, batch, liquidliquid system. Of course, coalescence is, in part, responsible for the interfacial areas and drop-size distributions measured in the experimental programs mentioned above, but generally the objectives of coalescence studies in dispersions have been to determine the coalescence rate. All previous studies of coalescence involved integral rates, such as the volume fraction of the dispersed phase entering into coalescence per unit time. Experimental methods include mass transfer to a reactive solute spreading among the drops by coalescences (Madden and Damerell(1962)),the spread of a tracer dye (van Heuven (1971), Komasawa (1969), Komasawa (1971), and Miller et al. (1963)), drop density changes due to coalescences of drops of different densities (Groothuis and Zuiderweg (1964)), the conversion for a chemical reaction occurring in the drop phase (Komasawa et al. (1969)), and transients in interfacial area as measured by light transmission (Howarth (1967) and Mlynek and Resnick (1972)). A few of the above studies have also investigated continuous flow agitated vessels (Groothuis and Zuiderweg (1964),Jeffreys et al. (1970), and Komasawa et al. (1971))but the results have been similar to those found in batch systems. The rate relation found in nearly all these studies is of the form (after Miller et al. (1963)) = N(2.5-3.2)4(0.5-1.1)
(2)
The variability, found also by others, is indicated for the exponents. This type of relationship says nothing, of course, about the dependence of coalescence rate on coalescing drop pair sizes. The less often performed type of measurement, in coalescing systems, tells us something about d,in, the drop size above which coalescence becomes negligible. Experiments to determine this have used dye-spreading and subsequent freezing of the drops, e.g. (Shinnar (1961)),the measurement of drop sizes away from the impeller region in strongly coalescing dispersions, e.g., Jeffreys et al. (1970). The usual relationship found is It is not the purpose here to expand upon the conjectures and models that have been derived from measurements of drop sizes and coalescence rate, but rather to summarize the methods that have been used and to conclude that they do not permit an unequivocal interpretation of the rate processes of breakage and coalescence in terms of the five functions previously listed. A new experimental approach appeared to be required. Experimental Motivation The five “mixing” functions occur in the general population balance equation (e.g., Bayens and Laurence (1969), Jeffreys et al. (1970), and Valentas et al. (1966)) describing the evolution of drop sizes in a dispersion and, if a solute is present, the distribution of solute concentration among the drops. This is an integral equation from which, in principle, the joint number distribution of drop volume and solute concentration
Air
Back- Pressure
Figure 1., Schematic drawing of flow system for producing liquid-liquid dispersions with known feed drop sizes.
p(u,c) can be obtained ifg(u), p ( u l u ’ ) , v(u), h(u,u’) and X(u,o’) are known. Conversely, if p(u,c) were to be measured, it should in principle be possible to deduce the rate functions by solving the integral equation. I t had already been shown (Verhoff (1969)) that a measurement of p ( u ) alone cannot uniquely determine these functions. Therefore it was necessary to introduce into the experiment a solute whose concentration distribution would reflect the coalescence process. The initial objective was to establish a dispersion in which detectable dispersed phase mixing was occurring, and then to measure the joint (bivariate) distribution of drop volume and tracer solute concentration. It was hoped that this would yield information that would permit the mixing functions to be either deduced of induced, although this was by no means assured. If two streams of drops of the dispersed phase, each with a different concentration of a tracer, are introduced into a continuous flow agitated vessel, drop breakage always leads to smaller drops without change in tracer concentration, and coalescence leads to larger drops of an intermediate concentration. Thus, breakage and coalescence leads to dispersed phase mixing. If the volume and tracer concentration of each drop in the vessel were measured, we would certainly have a great amount of information reflecting the rate processes occurring in the system. This measurement was therefore the chosen objective.
Experimental Technique Flow System. The experimental system, shown in Figure 1,consisted of storage tanks for oil and water, drop generators, the mixing vessel, and the exit storage tank. All were constructed from stainless steel or brass except for the glass portions of the mixing vessel and drop generators. In steadystate operation the back-pressure valves in the pump recycle loops provided a constant 25 psi supply pressure for each fluid. The rotameters were calibrated for the fluids used. Each oil (dispersed phase) stream and half of the water stream flowed through a drop generator which consisted of a hypodermic needle inserted into a glass venturi. These produced drops of nearly equal size and counted the frequency a t which drops were produced by changes in conductivity in
the tube immediately after the generators. The oil flow rate divided by the drop frequency gave the feed drop volumes. The mixing vessel was a cylindrical section composed of two glass cylinders separated by a stainless-steel ring and closed a t the top and bottom. The vessel was 11.1cm i.d. and 14 cm high inside, with a volume of 1355 cm3. Four baffles were equally spaced around the vessel, each with a width of onetenth the vessel diameter. A six-bladed flat-blade turbine impeller, with a diameter of D = 5.08 cm, was located a t the center of the vessel. The impeller blades were D/5 high and D/4 wide. Similar dimensions have been used in many previous studies (permits comparison between experiments) and were originally recommended by Rushton et al. (1950). However, the mixing results obtained in small vessels are not necessarily extrapolatable to larger vessels and proper care should be taken in generalizing the results contained herein. Four of the six ports in the center ring were used for feed, effluent, and sampling flows. The ports in the top and bottom plate facilitated filling and emptying the vessel. Fluids and Tracer. All experiments were performed with a mixture of Dowthern E (primarily orthochlorobenzene), 39.1% by volume, and Shell No. 3747 Base Oil, for the oil phase. This mixture has a density slightly less than that of water. Since this oil mixture is similar to kerosene, which previous workers have found to have a low relative coalescence rate, a drop would be expected to circulate many times in the vessel before coalescing (or breaking), permitting the assumption of statistical homogeneity throughout the vessel (see Sprow (1971) for an example of rapidly coalescing drops and a resulting spatial dependence of drop size distribution). The continuous phase was water with 0.001 N Na3P04. The latter was found to prevent oil drops from attaching to the vessel, baffles, and impeller, and it also provided increased electrical conductivity for the drop counters (generators). The tracer dye in the dispersed phase was that used by Miller et al. (1963), recrystallized A-g,g’-bifluorene. This has an extinction coefficient of 2.5 X L/cm mol at 455 mp. The dye concentration was 1.0 mg/mL in the more concentrated (“dark”) oil stream, and 0.2 mg/mL in the more dilute (“light”) oil stream. Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
373
To Surfactant
Surfactant Inject ion F i g u r e 2.
Cross section schematic of sampling probe. Important dimensions are shown.
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F i g u r e 3. Schematic drawing of sampling and measuring system. Three-way valves are shown positioned for taking a dispersion sample. Arrows show flow directions during dispersion sampling.
Sampling System. The heart of the experimental method was the sampling system for obtaining measurements of the joint distribution of drop volume and tracer dye concentration. The sampling probe is shown in Figure 2. This was inserted through one of the ports in the central ring and sampled the stream coming from the impeller. The sampling probe was part of the sampling system shown in Figure 3. It worked as follows. Normally, a small flow of dispersion left the vessel through the center tube of the sampling probe (the remainder flowing from the vessel through the back-pressure valve shown in Figure 1).When it was desired to take a sample of dispersion, the sampling piston was driven outward by a hydraulic cylinder system at a rate such that the sample was taken at approximately the velocity of the impinging flow (this was not found to be important, however, in obtaining a representative sample of dispersion). The space behind the sampling piston was filled with a dyed (methylene blue) solution of 2.9% by volume of Tween 40, a surfactant for stabilizing the sample of dispersion, and this was therefore “pumped” out through the surfactant ports, near the sampling probe tip, to mix with the incoming dispersion. The dimensions of the sampling probe were such that the injected surfactant solution flow was about 35% of the dispersion sampled. The surfactant did not 374
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
ba-k-mix into the vessel. The tube (hypodermic tubing) t h ough which the sample left the probe was a slide-fit into 1.‘;5 mm i.d. glass capillary tubing, which carried the dispersion sample to the reservoir. The dispersion sample, indicated by the methylene blue dye, was trapped in the reservoir (Figure 3) by the manipulation of the stopcocks. The volume of the reservoir was about 2 mL. Since the dispersed phase was intentionally slightly lighter than the continuous phase, the drops settled slowly to the top of the reservoir, nearest to the measuring capillary. The surfactant prevented coalescence during and after settling. Drop volume and dye concentration were measured simultaneously as the concentrated dispersion was forced through the capillary a t a constant flow rate by means of the infusion pump, using a 0.05 N water solution of NaSP04. The capillary was imbedded in epoxy resin, between two microscope cover slips, to reduce refraction caused by its curved sides. A logarithmic photometer focused on the center of the capillary, through a microscope, measured the light transmittance as the drops passed. The diameter of the capillary, 0.022 cm, was smaller than the smallest drops of interest in this study, and therefore each drop was extended into a cylinder while passing through the capillary. Neither breakage nor coalescence of drops occurred during this process. The output of the photometer, using a 455-mp interference filter to match the absorption peak of the tracer dye, was a series of “random” rectangular pulses; the height of each pulse was proportional to the dye concentration, and the width of each pulse to the drop volume. The “light” oil phase contained dye to distinguish drops from the continuous phase between drops. High-speed photographs of drops as they flowed through the capillary indicated no appreciable distortion of the leading or trailing minisci. The pulses from the photometer were recorded on magnetic tape in the FM mode (Ampex SP 300 tape recorder). These data were subsequently digitized and recorded on digital tape for processing on a computer. The rate at which drops could be sampled for their volume and tracer concentration depended upon their sizes, the phase fraction used, and the frequency response limitations of the entire system. The flow velocities through the capillary when measuring drops ranged from about 2 to 10 cm/s in different experiments. At 10 cm/s, a 1.0-mm drop requires about 0.14 s for both its volume and tracer concentration to be measured. In general, it took 5 to 10 min to measure the sizes and tracer concentrations of a sample of 500 drops. Procedure. T o begin an experiment, the vessel was thoroughly cleaned, first with pure acetone and then with a satu-
0
Concentration
1
Figure 4. Fractions of dispersed phase volume in drop volume-concentration intervals. The concentration interval is normalized (0,l)and drop diameter intervals are equal between 0.372 and 1.541 mm. N = 174 rpm. 4 = 0.1. rated solution of Na3P04. If the inside of the mixing vessel was not absolutely clean, oil drops tended to stick to the glass and stainless steel internal components. These drops, even just one or two of them, had a marked effect on the measured bivariate distribution and on the reproducibility of the experiments. After cleaning, the mixing vessel was filled with the water phase, and all air pockets were purged. The impeller speed was set using a tachometer (and checked periodically thereafter). The oil and water flows were then started and the backpressure valve adjusted to maintain the vessel under a slight positive pressure. Because the relative flows of “dark” and “light” oil were not important to the experiment, they were held a t identical rotometer readings, but the actual “dark” oil flow was usually slightly greater than the “light” (52.5% vs. 47.5%). Fluctuations in flow had little effect on the average phase ratio in the vessel but did cause a slight variation in input drop dize. While the system was approaching steady state, surfactant stabilized samples were taken and used to calibrate the sampling system. The pulses produced by the drops were viewed on an oscilloscope and the infusion-pump rate, or electronic signal filtering, was adjusted to give sharp rectangular pulses. The infusion rate selected was the largest possible to measure satisfactorily the “smallest” drop size in the sample. This maximized the number of drops analyzed per length of analogue or digital tape. About 30 min to 1h after the flows were started (compared to 19.6 min residence time in all experiments to be reported), a time sufficient for steady state to be reached, and after the sampling system was calibrated, a stabilized sample was withdrawn and collected in the sample reservoir. The sample size was usually about 200-1000 drops, depending on the phase fraction of dispersed phase in the tank and the agitator speed. These drops were allowed to settle upward, and then studied with a flashlight to check for the “sparkling” that accompanies coalescences. If the sample was deemed stable, a signal was placed on the analog tape to indicate the start of a sample, and the sample was forced through the capillary. The recording time was logged with a stopwatch. The taping was stopped when all but a few small drops had passed through the capillary. This procedure was repeated until sufficient samples had been collected to make up about 1000-3000 drops. A signal was then put on the tape to indicate the end of the last sample. A t the end of an experiment the vessel contents were col-
le ed, allowed to settle, and the phase fraction measured. 7 : i u was always within 5%of the estimated input value. IFata Processing. After digitizing the analog record, the da. I were processed on a digital computer to yield the volume and dye concentration of each drop sampled. The programs for this are listed in Verhoff (1969) and Ross (1971). The essential features are the following. (1)The “base line” is established by averaging the digitized values with only water in the capillary. (2) So long as no drop is detected, base-line averaging continues, compensating for drift in the analog recording. (3) If a drop is detected by the signal exceeding a specified threshold, the computer counts the number of digital samples, and accumulates the sum of the values, until the signals caused by the drop fall below the threshold. (4) The “concentration” of the drop is calculated by dividing the accumulated sum by the total number of digital samples corresponding to that drop. The “volume is the total number of such digital samples. The values are stored if the “voltime” is larger than a prescribed minimum, equivalent to a drop diameter of 0.3 to 0.5 mm, depending on the experimental conditions. Fast electrical noise pulses are thereby rejected. “Concentrations” are later normalized, but “volume” is calibrated by knowing the infusion rate during measuring, the analog and digital tape speeds, and the digital sampling rate. The “volume” and “concentration” of each drop in a sample of a dispersion is then used to calculate the bivariate distributions of volume-concentration, log volume-concentration, and diameter concentration (stored as matrices) and various properties of the sample such as Sauter mean diameter (dsz), volume average drop volume ( u z l ) , number average drop volume ( u ~ o )maximum , and minimum drop sizes, second and cross moments of drop size and concentration, the volume average concentration variance, etc. Because of noise in the recorded signal there is a random error associated with the measurement of the dye concentration in each drop. This tends to decrease the apparent concentration variance if the measured concentration is normalized between the maximum and minimum concentrations found (while true mixing, of course, also decreases it). Although this is negligible with substantial dispersed phase mixing, it is important with a small degree of mixing. With no coalescing, it was found that the two measured apparent concentrations were still each distributed approximately normally with a range of about 5%of the nominal (“dark” or Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
375
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“light”) feed concentrations. These apparent distributions were “condensed” into impulses a t the mean concentration extremes that represented uncoalesced drops (see Ross (1971) for details). Verhoff (1969) and Ross (1971) also considered other sources of error and found them to be negligible. Bivariate Data. Figure 4 shows the bivariate display of the fractions of dispersed phase volume occurring in joint diameter-concentration intervals. The concentration has been normalized (0,l); the diameter range (0.372 mm, 1.541 mm) has been divided into ten equal intervals. One thing is certain: the contemplation of data in this form is particularly unenlightening. Marginal distributions of drop size or concentration are more comprehensible. A marginal distribution is the distribution of either the drop volume or the concentration by 376
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
themselves and does not involve the interrelationships of drop volume and concentration. For example, the number distribution of concentration in the drops would be the marginal concentration distribution of the bivariate number distribution of concentration and drop volume. The term “marginal” only indicates that there are other random variables which are not included in the distribution. Sub-Samples. As has already been explained, several individual (“component”) samples were usually combined to obtain net samples of 2000-4000 drops. In Figure 5 are shown two such measured component drop size distributions consisting of 690 and 1352 drops, respectively. C, is the ratio of volume average concentration variance in the vessel to that in the feed. The value shown represents a considerable degree
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0 CONCENTRATION 1 Figure 7. E x a m p l e o c m a r g i n a l volume d i s t r i b u t i o n of dispersed phase concentration. C is mean feed concentration (equivalent t o volume f r a c t i o n o f o i l phase w i t h higher tracer dye concentration).
of dispersed phase mixing (although this is not displayed, of course, by the drop size distribution). In general, it was concluded by statistical analysis (likelihood ratio test) that the component samples do not come from the same population. This reflects the fact that in an agitated dispersion local conditions, even in a statistical sense, follow a stochastic process. In our systems the rate of breakage and coalescence were so low compared to the circulation (vessel mixing) time that it was appropriate to consider the average statistical properties in developing models. Reproducibility. The reproducibility of the procedure is very important in terms of reaching conclusions from a series of experiments performed over a period of time. Changes might occur in the fluids, the vessel surface properties, and in the various unknown variables that make much liquidliquid dispersion work “frustrating”. In Figure 6 are shown the results of two duplicate experiments at nominally the same conditions, performed about 2 months apart. Most integral measures (d32, etc.) differ by less than 1%between the two samples. Concentration Distributions. A typical marginal volume distribution of concentration (normalized between 0 and 1) is shown in Figure 7. The intervals at the extremes of concentration include also the impulses at 0 and 1,representing uncoalesced drops. The feed-vessel variance reduction ratio (C,) was computed from the bivariate data, however, and does not reflect the distortion caused by the histogram plotting convention. Acknowledgments The experimental apparatus was built with the aid of a National Science Foundation Institutional Grant provided by the Rackham School of Graduate Studies. The logarithmic photometer was built with the aid of an Eastman Kodak Company departmental grant. The Dow Chemical Company donated the Dowtherm-E used as a component fluid. Analog-Digital data conversion and advice was provided by the Cooley Laboratory of Electronics at the University of Michigan. Additional Fellowship support was provided by The National Science Foundation. Nomenclature C = mean feed tracer concentration, normalized (0,l)
-
= ratio of effluent (vessel) volume average tracer concentration variance to that for the feed D = impeller diameter, mm d32 = Sauter mean diameter, mm or cm d, = maximum stable drop size in a stirred dispersion, mm dmin = drop size above which coalescence is unlikely, mm g ( u ) = breakage rate of a drop of volume u , s-l h(u,u’) = number rate at which a drop of volume u collides with drops of volume u’, per drop of volume u’ per unit volume of dispersion N = impeller speed, rpm or rps p ( u ) = marginal number distribution of drop volume = So- d u , c ) d c p(u,c) = joint number distribution of drop volume and tracer concentration ul0 = number average drop volume, mm3 uZ1 = volume average drop volume, mm3 We = pN2D3/(r,Weber number fl(u Iu’) = number probability density distribution of volume u of drops produced by breakage, cm-3 4 = dispersed phase fraction h(u,u’) = probability of coalescence of two drops of volume v and u’, given that a collision has occurred ~ ( u )= average number of drops formed per breakage of drops of volume u p = continuous phase fluid density, g/cm3 (r = interfacial tension, g/cm s w = coalescence rate, s-1
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Received f o r review January 21, 1974 Resubmitted July 6,1976 Accepted M a r c h 16,1977
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
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