1704
I n d . Eng. Chem. Res. 1989,28, 1704-1711
Generalized Model for Prediction of the Steady-State Drop Size Distributions in Batch Stirred Vessels Eleni G. Chatzi,* Asterios D. Gavrielides, and Costas Kiparissides Chemical Engineering Department and Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 19517, 540 06 University City, Thessaloniki, Greece
T h e microscopic phenomena occurring in an agitated vessel are extremely complex. The exact mechanisms of coalescence and breakage in bubble and drop systems are generally not very well understood. Several attempts have been made in the literature to improve our understanding of these phenomena in multiparticle systems. In the present work, the most comprehensive models describing the coalescence and breakage processes in a styrene/water dispersion system are incorporated in a numerical algorithm to calculate the steady-state drop size distributions in a batch stirred vessel. A new breakage distribution function is introduced that considers droplet broken into two daughter and several satellite droplets. As a result, bimodal diameter density distributions are obtained. It is shown that the proposed model has the ability t o fit reasonably well a series of experimental data obtained for a low-coalescence system a t different impeller speeds, temperatures, and dispersed phase hold-up fractions.
A liquid-liquid dispersion formed in an agitated vessel is characterized by two dynamic processes: drop breakup and coalescence. The droplet size distribution depends on the conditions of agitation and the physical properties of the system. Droplet breakup occurs either in regions of high shear stress near the agitator blades or as a result of turbulent velocity and pressure variations along the surface of a single drop. On the other hand, turbulent flow can result in either an increase or a decrease in the droplet coalescence rate. These phenomena occur in a microscale-they are determined by what happens in a very small volume of fluid around the individual droplet. Thus, the influence of the large-scale flow may be comparatively small, and if the Reynolds number of the flow is high, the interaction processes can be estimated from the concept of local isotropy. Homogeneous interaction models have been extensively used in the literature for the description of dispersions in agitated vessels. On the other hand, inhomogeneous interaction models presuppose knowledge of the local variations of flow, energy conditions, and interaction frequencies. Therefore, their use in describing liquid-liquid dispersions has been limited since a detailed analysis of the local flow and energy inhomogeneities in an agitated vessel is rather hard to obtain. The homogeneous interaction models are based on the assumptions of perfect macromixing and statistical homogeneity of the vessel content. This does not necessarily imply that the turbulence and energy conditions are uniform throughout the vessel. In general, the impeller and circulation regions are clearly distinguished, but the homogeneous interaction models can successfully be applied to vessels with short circulation times or low coalescence rates (Park and Blair, 1975). In this study, the homogeneous interaction model developed by Valentas and Amundson (1966) was used to calculate the steady-state drop size distributions of a styrene/ water dispersion system in a batch agitated vessel. Experimental Details The mixing vessel was a capped round-bottomed glass cylinder of 15-cm internal diameter fitted with four vertical, equally spaced stainless steel baffles. The width of each baffle was equal to one-tenth of the tank diameter. Water from a constant-temperature bath was steadily circulated through the vessel’sjacket to maintain a desired temperature inside the tank. A stainless steel six-blade
turbine impeller with diameter equal to one-half the tank diameter was connected to a controlled variable-speed power supply. The length of the immersed portion of the impeller was equal to one-third of the total height of the liquid-liquid dispersion, which was always maintained equal to the vessel diameter. The total volume of the dispersion was 2 L. A schematic diagram of the equipment arrangement is shown in Figure 1. Photographs were taken with a Yashica FX-3 Super camera equipped with a 50-mm lens, a full extended bellow, and a set of two close-up rings, across a vertical cell consisting of two parallel glass plates held at a distance of approximately 3 mm by an elliptical rubber ring. Samples from three different locations (Figure 1) were driven by suction first through the cell to photograph and then through a metering piston pump to be discarded. This setup minimized the hydrodynamic effects of pumping on the dispersion before the photographs were taken. The cell was illuminated by a 1000-W flashlight, and the contrast was improved by a black surface placed behind the cell. Experimental Procedure. The procedure used to obtain the experimental measurements was the following: 1. Initially the system was thoroughly cleaned with distilled water. Special care was taken for cleaning the observation cell. 2. The vessel was filled with the required amount of distilled water up to a total volume of 2 L. After the addition of the suspending agent, poly(viny1 alcohol), the temperature of the mixture was raised to a specified constant value and the impeller speed was adjusted to the desired setting. 3. Subsequently, the organic phase, styrene, was added to the water-PVA mixture through an inlet port in the vessel cap. 4. At prespecified time intervals, the sampling pump was turned on to obtain a new sample in the observation cell. Then the pump was turned off, and three photographs of the sample were taken. The same procedure was repeated two more times with samples obtained from different locations in the vessel. This was accomplished by moving the tip of the suction tube vertically inside the mixing vessel. Photographs of the dispersion were taken with camera apertures varying from f/S to f l l . 9 for different dispersed-phase fractions. Exposure times of 1/250 to 11500 were found sufficient to freeze the motion of the droplets in the cell, at the lighting conditions used.
0888-5885/89/2628-1704$01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1705 AG). The amount of PVA was sufficient for saturation of the total droplet interfacial area generated under different experimental conditions. The interfacial tension was measured at 25 and 50 "C with a KRUSS surface tensiometer Model K10 by using the ring method. For PVA concentrations less than 10 g/L, the physical properties of the continuous phase do not significantly vary from those of pure water (Mark et al., 1964). In Table I, we report the viscosity and density values of the continuous and dispersed phases. The average density and viscosity of the dispersion were calculated as weighted averages of the corresponding values of the dispersed and continuous phases. The impeller speed was always above the minimum impeller speed for complete liquid-liquid dispersion. The maximum value of the minimum impeller speed was 74 rpm at 25 "C and 4 = 0.03. Our experiments were conducted for impeller speeds of 150, 200, 250, and 300 rpm. The speed of 300 rpm was just below the level at which air entrainment occurred. The various experimental conditions are shown in Table 11.
Figure 1. Schematic diagram of the apparatus and sampling positions 1, 2, and 3. Table I. Numerical Values of the System's Physical Prouerties interfacial density, viscosity, tension, CP mN/m fluid dcm3 water, 25 O C 0.9971 0.9147 11.5
styrene, 25 "C water, 50 O C
0.9014 0.9881
0.7303 0.5502
styrene, 50 "C
0.8792
0.4591
7.4
5. The film negatives were projected onto a screen, and 300-500 drops were counted at diameter increments of 20 pm. A computer program was used for data manipulation. Physical Properties and Operational Conditions. The experiments were carried out at atmospheric pressure and temperatures of 25 and 50 "C. The volume fraction of the dispersed-phasestyrene (Fluka AG) varied from 0.01 to 0.03. The continuous phase consisted of distilled water and 0.5 g/L of poly(viny1alcohol) (PVA) used as a suspending agent. The PVA had a degree of polymerization of 500 and a hydrolysis ratio of 97.5-99.5 mol ?% (Fluka Table 11. Experimental Conditions and Model Parameters exp 6 N * , rpm T," C (Nw.)T (N& 1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16 17 18 19
0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.03
150 150 200 200 250 300 300 150 150 200 200 300 300 150 150 200 200 300 300
25 50 25 50 25 25 50 25 50 25 50 25 50 25 50 25 50 25 50
228 350 405 622 657 912 1401 228 350 405 622 912 1401 228 350 405 622 912 1401
15 060 24 758 20 080 33 047 25 100 30 120 50 022 14 790 24 368 19 761 32 491 29 060 48 736 15111 23 923 19 373 31 897 29 060 47 846
Experimental Results Vessel Homogeneity. In general, the flow field in an agitated vessel is far from homogeneous, even for a fully developed turbulent flow field. In terms of the dispersed-phase interactions, the flow field in an agitated vessel can be divided into two regions; the impeller region and the circulation region. Drop breakup is observed near the impeller, and drop coalescence usually occurs far from the impeller. Practically, the inhomogeneous interaction models are of limited use. As a result, the assumptions of perfect macromixing and statistical homogeneity must be invoked. Perfect macromixing implies that any effluent is representative of the vessel content, which is typical of a turbulent mixing process with no molecular diffusion. Statistical homogeneity means that the time-average position distributions of drops in any two small volumes in the vessel are approximately the same. So, homogeneous interaction models neglect the local variations of flow, energy conditions, and interaction frequencies. The validity of the spatial homogeneity assumption with regard to the size distribution for a dispersion depends on the experimental conditions. It can be applied to vessels with short circulation times and for low coalescing systems, that is, systems with a low dispersed-phase volume fraction, especially in the presence of protective colloids (Park and Blair, 1975). A measure of the homogeneity of a dispersion is the ratio of coalescence frequency, w, to the circulation of frequency, f,. At the lower limit, w / f c 0,
-
x
Ndn
50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
N, 25 10 10 10 3 1 1 25 13 12 4 1 1 25 15 6 3 1 1
10-10C703 5 5 5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5
C W ~ 0.40 0.40 0.38 0.40 0.38 0.40 0.47 0.40 0.46 0.41 0.44 0.40 0.47 0.43 0.46 0.42 0.45 0.43 0.50
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1706 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
the dispersion behaves as an homogeneous medium. However, as the ratio w / f , increases, the dispersion becomes more and more inhomogeneous. The circulation frequency is given by (Coulaloglou and Tavlarides, 1976; Holmes et al., 1964) f, = l / t c = N/0.85(DI/DT)' (1)
Alternatively, assuming that the drop size was determined by a coalescence process, they obtained the following relation:
d32/D1 = ~ ( c J D I ) ~ .We~)'T4.375 ~(N
(5)
These derivations are valid under the assumption of local isotropy. At high Reynolds numbers of the main flow A model for the collision frequency proposed by Coulalo((Nb)T= N*Df/vm > lOOOO), droplet diameters are much glou and Tavlarides (1977) assumes that the mechanism smaller than the scale of the main flow or macroscale of of collision in a locally isotropic flow field is analogous to turbulence (d > 7). The macroscale of gases, that is, turbulence, L, is given approximately by the width of the fluid ejected by the agitator (Coulaloglou and Tavlarides, h(u,u') = and the microscale of turbulence, 7, is given by CIII(U~ +/ (~~ ' ) ~ / ~ ) ( u+2 /(9U ' ) ~ / ~ ) ~ / ~ D I ~4)/ (2) ~ N * / (1976), ~
+
where Cm is a parameter of the order 10*-104. Assuming that all collisions are effective for coalescence to occur, the coalescence frequency was obtained for the experimental conditions of Table 11. The calculated value for w was s-l. From eq 1, the circulation approximately equal to frequency was found to vary in the range 1.43-0.71 s-l under the same experimental conditions. On the basis of these results, the dispersion can be assumed statistically homogeneous, which is consistent with our low-dispersed-phase fraction and the presence of protective colloids. The degree of inhomogeneity of the vessel contents was tested by examining samples obtained from three different vessel positions, as shown in Figure 1. The deviation of the average drop size, defined as [(d32)mm- (d32)mhl/ (d3Jmh,ranges from 1.2% to 7.6% and increases with the dispersed-phase volume fraction. For most of our experiments, the Sauter mean diameter changes in a random way with the distance from the tip of the impeller, probably indicating that the degree of inhomogeneityobserved cannot be clearly differentiated from the experimental error inherent in the measurement technique. For comparison, in the system MIBK-water (Park and Blair,1975), the average drop size between sampling positions 2 and 3 changed less than 5% for C$= 0.005 and more than 16% for 4 = 0.10. An isooctane-carbon tetrachloride/water system (Mlynek and Reshnick, 1972) for the same sampling positions gave changes up to 5% for 4
