Ind. Eng. Chem. process
888
ms. D ~ V1985, . 24, 868-872
Changes of the Average Drop Sizes during the Initial Period of Liquid-Liquid Dispersions in Agitated Vessels Paul 0. Hong chemlcai Enginmring &patiment, Cleveland State University, Cleveland, Ohio 44 115
James M. Lee' Depatiment of Chemical Eng/neering, Washlngton State University, Pullman, Washington 99 164
(d32) with respect to time during the initial period of iiquid-llquici dispersions in agitated vessels was developed as fdiows: (d32 - d32+)/d32,* = a(~t)-O.'~, where the steady-state droplet size (d32.) and a! were correlated with operatlng variables and physical properties of liquid-liquid systems. The minimum time required to reach steady state was also correlated.
A general correlation for the changes of the Sauter-mean droplet diameter
Introduction The dispersion of one liquid into another immiscible liquid using a mechanically agitated system is of major importance in a number of chemical processes such as suspension polymerization and liquid-liquid extraction. Several models have been proposed to describe the fluid-fluid interactions in an agitated vessel. The models can be classified into two categories, the nonhomogeneous interaction model (Rietma, 1964) and the homogeneous interaction model (Curl, 1963; Valentas et al., 1966), depending upon whether they take into consideration the local variations of flow characteristics or not. The former is more realistic than the latter; however, its applicability is limited because it is difficult, if not impossible, to evaluate various parameters at various locations in a vessel. Valentas et al. (1966) and Valentas and Amundson (1966) have proposed a model based on the assumption of statistical homogeneity of the contents of the vessel. They considered both coalescence and redispersion as occurring a t finite rates and derived a population balance equation by introducing six functions describing the breakage frequency [g(u)],the number of daughter drops formed per breakage [ ~ ( v ) ]the , breakage distribution function [@(u',u)], the collision efficient [A(u,u?], the collision frequency [h(u,u?],and the escape frequency [f(u)]. For sufficiently small dispersed-phase fraction during the initial period of a batch agitation process, the terms for the coalescence in the droplet population balance equation developed by Valentas and Amundson (1966) can be neglected to find a simplified equation:
which shows that the change of the number of droplets with volume with respect to time is equal to the net rate of formation of droplets with volume u by breakage. The solution of the above equation with initial conditions and derived functions based on the proposed mechanisms for drop breakage can predict transitional and steady-state size distributions as a function of physical properties and operating conditions. The change of Sauter-mean diameter during the initial period of mixing can be calculated with the distribution obtained from the solution of eq 1 as follows:
Therefore, the change of Sauter-mean diameter of liquid droplets during the initial period of mixing is a function of breakage frequency, number of daughter drops, and breakage distribution function, which, in turn, are functions of the type and size of impeller, tank diameter, the fraction of dispersed phase, and physical properties of liquids. However, the mechanism of droplet breakage by an agitator is not well-known due to the complexity of the phenomenon. Coulaloglou and Tavlarides (1977) proposed a phenomenological model for drop breakup; however, numerical solution of eq 1based on their model is a lengthy procedure. Furthermore, the model contains several constants that have to be determined experimentally. It is possible, however, to obtain an empirical correlation that is useful in engineering design calculations by examining each parameter that influences the droplet breakage and coalescence with the guidance of a dimensional analysis. Recently, Hong and Lee (1983) reported the preliminary results of the studies on the changes of the average drop size and the minimum transition time required to reach steady state during the initial period of liquid-liquid dispersion by using a microphotographic technique and a light transmission method. A spectrophotometer was converted into a light transmittance unit by employing a specifically designed fiber optic probe, which allowed the investigators to follow a wide range of changing drop sizes. The average drop size during the initial period of mixing was found to change exponentially from large to small while the distribution changes less drastically from wide to narrow. As the continuation of this study, the generalized correlations for the change of the Sauter-mean droplet size during the initial period of mixing and for the minimum time required to reach steady state will be reported in this paper. Experimental Details Material and Mixing Vessels. The materials chosen from continuous phase were distilled water and 15% and 30% (by weight) sucrose solution. Dow Corning 200 fluids (5 cSt), ethyl acetate, and kerosene were used for the dispersed phase. The Dow Corning 200 fluids are clear dimethylsiloxanes with low vapor pressures and relatively 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 869 0.6 I
Table I. Systems Studied system
1 2 3 4 5
dispersed phase 5-cSt Dow Corning 200 fluid ethyl acetate 5-cSt Dow Corning 200 fluid kerosene 5-cSt Dow Corning 200 fluid
continuous phase water water 15% sucrose solution water 30% sucrose solution
0.5
1
I
0.4
w
Table 11. Physical Prowrties at 23 'C
system 1 2 3 4 5
N/m 0.0425 0.006 0.032 0.0419 0.023
cont lo00 1000 1087 lo00 1131
disp 920 894 920 787 920
cont 0.0010 0.0010 0.0020 0.0010 0.0029
disp 0.00460 0.00046 0.00460 0.00169 0.00460
flat viscosity-temperature curves. The physical properties of the systems are listed in Table I and 11. The range of the fraction of dispersed phase studied was 0.05 and 0.20. The mixing vessels were two sizes (0.292 and 0.387 m in diameter) of flat-bottom glass cylinders fitted with four equally spaced, radial, vertical wall baffles. The height of the total liquid volume was always equal to the vessel diameters. A Model ELB experimental agitator kit manufactured by Bench Scale Equipment Co. was used for mixing the liquids. The impellers chosen for this study were six-bladed flat turbines of two different sizes (0.076 and 0.102 m in diameter), since this type showed the best dispersion performance in preliminary runs. D r o p Size Measurement. A microphotographic technique described by Hong and Lee (1983) was employed to take clear pictures of liquid-liquid dispersion during agitation with a magnification ratio of 3.87. It consisted of a camera (Nikon F-2A), an extension tube, an objective lens (plane-chromatic) with 4X magnification, and a protective cup to eliminate the interference of dispersed drops between the objective lens and the objective plane. An electronic flash (Sunpak Auto 322) with a fiber optic light guide was used to light the object plane effectively and at the same time to catch fast moving drops. Kodak Tri-X Pan film (ASA 400) was used with a flash duration of 1/13300 or 1/18500 s. The negatives were developed and made into slides to be projected on a piece of graph paper to estimate the drop size spectrum by counting about 200-300 drops. A light transmittance technique described by Hong and Lee (1983) was used for the average drop sizes and the minimum transition time required to reach steady state. It was calibrated on the basis of the result of the photographic measurement. Experimental Procedures. The continuous and dispersed phases of each system were prepared separately by presaturating each liquid with the other. The continuous phase was placed in the tank, and the agitator was started. The impeller speed was always above the minimum impeller speed for 98% uniformity as defined and correlated by Skelland and Lee (1978). The light transmittance probe was placed in a desired location before the measured amount of dispersed phase was injected near the impeller shaft. In some experiments, the dispersed phase was placed in the vessel with the continuous phase before the agitator was started. The light transmittance was measured and recorded from the moment the agitator was started until no further decrease could be observed. About 20 photographs were taken a t intervals of 1to 60 s during each run. All runs were performed at a temperature of 23 f 1 "C.
0.1
1
/
TOTAL PTS = 134 I
0.0
0.1
0.2
0.3
J
1
04
05
0.6
dz2( m m ) , EXPERIMENTAL Figure 1. Experimental vs. predicted d3** from eq 4.
Minimum Transition Time. The minimum transition time (tmh)is the minimum time required to reach steady state. It was arbitrarily defined by Hong and Lee (1983) as the time when light transmittance reached a steady value and no further change in reaching could be observed within a specified time period (1-3 min). The location of the light transmittance probe was halfway between the impeller tip and the vertical vessel wall. Results a n d Discussion Generalized Correlation for Steady-State Droplet Size. Various investigators have found that the Sautermean diameter varies as N4I5by use of the theory of local isotropic turbulence and assumption that drop breakup is based on a critical Weber number for drop diameter greater than the microscale of turbulence (Vermeulen et al., 1955; Calderbank, 1958; Chen and Middleman, 1967; Sprow, 1967; Brown and Pitt, 1970; Mlynek and Resnick, 1972; Coulaloglou and Tavlarides, 1976). Their correlations can be expressed as the following general form:
d32*/dI = a(1 + bqb)
(3)
The effect of the dispersed-phase fraction on the Sautermean diameter was negligible for q5 < 0.015 (Sprow, 1967) and linear for qb > 0.15 (Sprow, 1967; Brown and Pitt, 1970; Mlynek and Resnick, 1972; Coulaloglou and Tavlarides, 1976). The experimental data from this study were compared to the various correlations published that had the general form of eq 3. As much larger discrepancies were found with other investigators' correlations, Coulaloglou and Tavlarides's (a = 0.081, b = 4.47) provide a satisfactory fit of the data for this study with the average absolute relative deviation (lobserved - predicted(/observed) of 25.1%. A further investigation with other variables to improve the steady-state correlation resulted in a selection of two more variables, the d I / T ratio and the Froude number (p,d12W/ApHg)as follows:
d 3 2 * / d I= 0.05(1 + 2.316~)(dI/T)-0.75Fr-0.13
(4)
Equation 4 was based on the experiments with five liquid-liquid systems (Table I). The SAS (Statistical Analysis System) was used to get the best least-squares correlation. The steady-state Sauter-mean diameters predicted from eq 4 are compared to the experimental ones in Figure 1. The average absolute relative deviation is 18.5 % .
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
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Figure 4. Dimensionlessdrop size vs. dimensionless time (T= 0.292 m,dI = 0.102 m, 6 = 0.05, N = 4.67 rps).
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LOg,o( Nt 1 Figure 3. Dimensionless drop size vs. dimensionless time (system = 1, T = 0.292 m, dI = 0.102 m, 6 = 0.05).
Change of Drop Size during Transition Period. Figure 2 is a typical plot of the change of the drop size with respect to time. The Sauter-mean diameter decreases exponentially during the initial period of mixing as large drops break up into small ones, and it reaches a steadystate drop size. Figure 2 can be linearized by employing dimensionless time, N t , and dimensionless drop size, (d32 - d*32)/d*32 as shown in Figure 3, where d*32 is the predicted steady-state Sauter-mean diameter by eq 4. This dimensionless drop size and time can be correlated as where d*32,a, and j3 may be functions of physical properties and operating conditions. Figures 4 and 5 show the dimensionless drop size vs. time curves for systems 3 and 4 in comparison with system 1. The solid lines were drawn
-2.0
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2
3
4
5
LOg,o( Nt 1 Figure 5. Dimensionlessdrop size w.dimensionless time (2' = 0.292 m,d~ = 0.102 m, 4 = 0.10,N = 5.83 rps).
with the slope of 0.70. On the basis of the three systems studied, the value of 0 remains constant, while that of a was affected by the different systems. The effect of the fraction of dispersed phase on a and j3 was not definitive (Figure 6) beyond the effect already included in d32* correlation (eq 4). Figure 7 shows the effect of the impeller to tank diameter ratio on a and j3. Again j3 remained constant while a was affected by the ratio. General correlation selected on the basis of three liquid-liquid systems (systems 1, 3, and 4 in Table I) is as follows:
The absolute relative deviation (ld32, observed - d32, pre-
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 871 Table 111. Experimental t h vs. Predicted Minimum Time To Reach within 12%, lo%, and 5% of the Steady-State Drop Size (Syetem = 1, Q = 0.05)
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826 784 748 497 472 450 1052 997 979
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Figure 6. Dimensionless drop size vs. dimensionless time (system = 1, T = 0.292 m, dl = 0.102 m,N = 3.6 rps).
dictedlld,?, observed) was 11.1%. Liquid-Liquid Interfacial Vortex. An interesting phenomenon has been qbserved during the experimental runs. Although the system was considered to be fully baffled in conventional terms and the liquid surface in fact showed no appreciable vortex observed from the top, a clearly visible vortex was forming a t the interface of the dispersed and the continuous phases, especially when the impeller speed was low. This was true regardless of the method used to add a dispersed phase into a continuous phase. When the dispersed phase was poured in, the addition time was kept minimum but was just long enough to avoid any splashing out of the tank. The added phase quickly formed a separate layer on top of the continuous-phase liquid as shown in Figure 8. At the completion of dispersed-phase addition by pouring, the layer looked almost as if the experiment had started with an initial layer on the top, taking into account the addition time (normally about 3-10 s depending on the amount added). This phenomenon is similar to the findings of Flynn and Treybal (1955). They observed the vortex formation in the absence of +liquid interface with a single-phase liquid and reported that more baffling was required in vessels completely filled with liquid than was needed where an air-liquid interface existed to ensure a fully baffled condition. This work was extended by Laity and Treybal (1957), who confirmed that a 16.7% baffle width was required to give the same power as a width of 10% baffle to tank diameter having a free air-liquid surface. The standard 10% baffle (baffle width/tank
A) t=O
BAFFLE
f DISPERSED PHASE
CONTINUOUS PHASE
CONTINUOUS PHASE DISPERSED PHASE
Figure 8. Interfacial vortex in fully baffled tank.
diameter = 0.1) was used as fully baffled condition in this work. The formation of a vortex results from the influence of gravitational forces. The influence of these forces is measured by means of the Froude number. Therefore, the condition for vortex formation is that the centrifugal acceleration due to tangential acceleration produced by the
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985
rotating impeller predominates over the gravitational acceleration. Furthermore, the general form of a free vortex can be determined through the use of the momentum balance and the Bernoulli theorem. This general form is shown to be hyperbolic (see Hong, 1983),as expected from Figure 8. The role of the vortex during an initial period of dispersion may be viewed as a force that feeds the dispersed phase into the impeller breakage region as if someone is manually injecting a controlled amount of dispersed phase into the high shear field. The initial period is as much influenced by the time-dependent main flow as the drop breakage mechanism itself. Therefore, the interfacial vortex is responsible for the inclusion of the Froude number for the correlation of either steady-state drop size or minimum transition time as shown in eq 4 and 7 . Minimum Transition Time. The important variables that influence the minimum transition time to reach a steady-state liquid-liquid dispersion were identified by Hong and Lee (1983). General correlation selected on the basis of 181 runs with five liquid-liquid systems by using SAS computer analysis is as follows: Ntmin= 1995.3(dI/T)-2.371;0.97(~d/~c)F~~’66 (7)
The average absolute relative deviation is 38%. The high deviation is due to the difficulty of locating the minimum time. The minimum transition time can also be estimated from the transition drop size model, eq 5, which can be rearranged to give Nt
(1/CY)[(d32 - d 3 2 * ) / d 3 2 * ] - l i P
(8)
Hence a new definition of the minimum transition time is
t = tm,cfor (d32- d*32)/d*32 = c
(9)
where c can be arbitrarily determined. For example, if c = 0.05, then tm,0.05 is the time when the average transition drop size has reached within 5% of the steady-state average drop size. Table I11 shows the results of calculation for tm,0.12, t,,,,0.10, and tm,0.05 at three different impeller speeds approximately corresponding to constant power input per volume using two sizes of tank and impeller. The average observed values (tmin)obtained from light transmittance recorder outputs were generally found to fall near the calculated values of tm,O,l.In light of the definition of tmh,the transition drop sizes reached within about 10% of the steady-state sizes when no noticeable changes in light transmittance could be observed within 1min. From that time on, the change in the light transmission curve is so slow that pinpointing tm,cfor c < 0.05 would be difficult without a large error. Acknowledgment This material is based upon work supported by the National Science Foundation under Grant CPE-800666.
Nomenclature A(u,t) = probability density of droplet size, u, in vessel C = impeller distance off tank bottom, m c = defined in eq 9 dI = impeller diameter, m dS2= Sauter-mean droplet diameter, m d32* = steady-state Sauter-mean droplet diameter, m f ( u ) = escape frequency, i.e., fraction of drops with volume between u and u + du flowing out of the vessel per unit time, S-1
F = Taylor number = W e / R e Fr = impeller Froude number, defined by p,d?W/(ApHg) g(u) = breakage frequency of drops having volume u, s-l h(u,u’) collision frequency of drops of volume u and u’, s-l H = height of liquid in the vessel, m N = impeller speed, rps N ( t ) = total number of dropleb at time t Re = impeller Reynolds number, defined by p,hTdf/pc T = vessel diameter, m t- = minimum transition time required to reach steady-state drop size, s = minimum transition time required to reach (d32 d32*)/&* = c,s u = volume of droplet, m3 W e = impeller Weber number, defined by M d T p c / o Greek Letters a,/3 = defined in eq 5
p(u’,u) = number fraction of droplets with volume u to u + du formed by breakage of a drop of volume u’ X(u,u? = coalescence efficiency of drops of volume u with drops of volume u’ ~1 = viscosity, Ns/m3 v(u) = number of drops formed per breakage of drop of volume U p =
density, kg/m3
= interfacial tension, N/m
4 = volume fraction of dispersed phase Subscripts c = continuous phase d = dispersed phase o = inlet condition
Literature Cited Brown, D. E.: Pltt, K. Proceedlngs, Chemeca ‘70, Melbourne, 1970, p 83. Calderbank, P. H. Trans. Inst. Chem. Eng. 1958, 36, 443. Chen, H. T.; Middleman, S. AIChE J . W67, 13, 989. Coulaloglou, C. A.; Tavlarldes, L. L. Chem. Eng. Sci. 1977, 32, 1289. Curl, R. L. AIChE J . 1963, 9 , 175. Flynn. A. W.; Treybal, R. E. AIChE J . 1955, 1 , 324. Hong, P. 0.;Lee, J. M. h d .Eng. Chem. Process Des. D e v . 1983.22, 130. Hong. P. 0. Ph.D. Dissertation, Cleveland State University, Cleveland, OH, 1983. Law, D. S.; Treybal. R. E. AIChE J . 1957, 3 , 2. Mlynek, Y.; Resnick, W. AIChE J . 1972, 78, 122. Rletma, K . A&. Chem. Eng. 1964, 5 , 273. Skelland, A. H. P.: Lee. J. M. I n d . Eng. Chem. Process Des. Dev. 1978, 17, 473. Sprow, F. B. AJChE J . 1967, 13, 995. Valentas, K. J.; Bllous, 0.; Amundson, N. R. I n d . Eng. Chem. Fundam. 1866, 5 , 271. Valentas, K. J.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1966, 5 , 533. Vermeulen, T.; Williams, G. M.: Langlols, G. E. Chem. Eng. Prog. 1955, 51, 85-F.
Received for review May 7 , 1984 Revised manuscript receiued October 29, 1984 Accepted November 8, 1984
