Acknowledgment
SUBSCRIPTS
The authors wish to express gratitude to the Department of Chemical Engineering and the School of Engineering of Illinois Institute of Technology for the general encouragement given for conducting the research.
i
Nomenclature
SUPERSCRIPTS
total interfacial area, m 2 interfacial area per unit volume, m-l molar concentration, mol-l.-l particle diameter, m dS2 = Sauter mean diameter, m d,, = maximum particle diameter, m D = impeller diameter, m D a = coefficient of molecular diffusion, m2/sec g = acceleration of gravity, m/sec2 H = continuous phase fill height, m kl = continuous phase mass transfer coefficient, m/sec k , = reaction rate constant, sec-l or m3mol-1/sec K = combined mass transfer coefficient, m/sec KCR = coalescence efficiency in circulation regions K I R = coalescence efficiency in impeller region K O = overall mass transfer coefficient, m/sec A; = impeller speed, rps ~ Y s , = impeller Keber number t = time, see T = tank diameter, m C’ = root mean square of fluctuating velocity component, m/sec rav= average velocity, m/sec CF = fluid velocity, m/sec C v p = particle velocity, m/sec r R = random velocity, m/sec = terminal particle velocity, m/sec v1 = continuous phase volume, m a
+
a
= a* = c = d =
-
GREEKLETTERS PB,CR = breakage efficiency function in circulation regions OB,IR = breakage efficiency function in impeller region p = continuous phase density, kgm/m3 u = interfacial tension, dyn/m + = continuous phase viscosity, cP
= value of concentration or time, a t a chosen value for mass transfer calculations o = value of concentration a t the instant the mass transfer starts
*
= =
equilibrium value in continuous phase a t interface
literature Cited
Bird, It. B., Stevart, W. E., Lightfoot, E. N., “Trantport Phenomena,” Wiley, Sew York, N.Y., 1960. Flynn, .4.W., Trej-bal, It. E., A f C h E J . , 3, 324 (1955). Karr, il., Scheibel, E. G., Chem Erzg. Progr. Symp. Ser., 50 ( l o ) , 73 (19*54). - - -,\ -
Keev, R. B., Glen, J. B., A f C h E J., 15, 912 (1969). kIok, Y. I., Treybal, R . E., ibicl., 17,916 (1971). Xagata, S., Yamaguchi, I., Mem. Fac. Erig. Kyoto l’niv., 22, 249 (1960).
Overcashier, R . H., Kingslej-, €I. A,, Olney, It. B., A f C h E J., 4 , 529 il93Al. -, Rushton, J. H., Xagata, S., Roonev, T. B., ibid., 10, 298 (1964). Schindler, H. D., Treybal, R . E.,’ibid., 14, 790 (1968). Vanderveen, J . H., &IS thesis, Univ. of Calif., Berkeley, 1960. Zeit,lin.31. -4.. PhD thesis, IIT, Chicago, 1971. Zeitlin; 11. A., Tavlarides, L. L., “Ilispersed Phaye Reactor Model for Predicting Mass Transfer and Mixing Effects,’’ presented at 64th Annual JIeetiIig, American Institute of Chemical Engineers, San Francisco, Calif., Xovember 1971, in press, AIChE J . (1972). Zeitlin, 11. A,, Tavlarides, L. L., ,‘On Fluid-Fluid Int,eract,ions and Hydrodynamics in Dispersed Phase CSTR’s: Prediction of Local Concentrations, Transfer Rateq, and Reaction Conversions,” Proc. Fifth European (Second Int,ernational) Symposium on Chemical Reaction Engineering, Amsterdam, 1Iay 1972, 1972a. Zeitlin, M. A., Tavlarides, L. L., “Flitid-Fluid Interact iolls and Hydrodynamics i r i ilgitated Dispersions: -2 Simulation Model,” Can. J . Chem. Eiig., 50 (21, 207 (197213). \ - -
~
_
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RECEIVED for review Janiiary 26, 1972 ACCEPTED June 16, 1972 In part work was supported by the Satioiial Science Foundation through Research Initiation Grant GK-27903 and by a predoctoral traineeship granted to M.4Z.
Determination of Droplet Size Distribution in Liquid-Liquid Dispersions Gershon Grossman Fluid Physics Department, Avco Systems Dicision, TT-ilniington, Mass. 01587
T h e separation of a mixture of two immiscible liquids into its constituents has been studied extensively, and several separation processes have been developed for a wide variety of applications (Treybal, 1963; Hanson, 1971). Perhaps the simplest and most commonly used are gravity decanters, Ivhere separation takes place because of the difference in density between t h e dispersed and continuous phase. Decanters are only efficient for mixtures of relatirely large drop size. For small droplets and foamlike dispersions, other devices
are required. The mixture can be passed through a coalescer prior to the decanter to increase its drop size. Also, c,eiitrifuges of various kinds may he used t,o increase t h e droplet velocity. The present study has been concerned with the separation of dispersions of liquid Freon in water for applic,ation i n a freeze crystallizat’ion process for n-ater purification (Emmermaim et a]., 1972; Fraser and Gibson, 19i1). Several basic properties of the dispersion must be knon-11in order to design Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
537
We define a distribution function of droplet size N ( r ) such that
is the number of droplets per unit volume having radii between T I and r2. T h e total number of droplets per unit volume is therefore given by
and t h e total volume fraction of the dispersed phase in t h e mixture is LY
f C(t)
C
r
Figure 1. Model for settling of liquid-liquid dispersions
a n effective decanter for it. I n particular i t is important to know the droplet size distribution in the mixture. The size and the properties of the two liquids determine the settling velocity of the individual droplets, which in turn determines the total settling rate and residence time required in the decanter. Various methods have been developed and used by different investigators t o determine droplet size distribution i n their mixtures. A good survey of such methods was given by Browning (1958). They include centrifugal separation, light diffraction techniques, measurement of electrical charge, high-speed photographs, and more. Most of these methods are fairly complex, and many of them inappropriate for t h e type of dispersions considered here. T h e purpose of t h e present study has been to obtain some basic information on t h e settling properties of Freon dispersions in water. A simple method has been developed t o determine droplet size distribution in a liquid-liquid dispersion, based on measurements of its settling rate in a stationary vertical column. T h e method was tested in mixtures of Freon 113in water and applied t o samples of t h e Freon-water dispersions taken from the freeze-crystallization system. T h e information obtained on droplet size distribution, combined with data on density and viscosity of t h e two liquids, mas used to calculate droplet settling velocities and total separation rates.
=
lm 4/3
dN(r)dr
(3)
The analysis is based on t h e following two assumptions : The settling velocity of a droplet is a function of its size and the properties of both liquids and is fixed with time. The initial acceleration of t h e droplet is neglected. I n most cases the terminal velocity is achieved after a very short distance of fall; and N o collisions or interactions between the droplets. This assumption is good when t h e volume fraction of t h e dispersed phase is small and the variations in size between the droplets are not too large. Observations to this effect were made in t h e experimental part of this study and will be described in detail later. Corrections can be introduced in the model t o account for moderate deviations from t h e conditions of this assumption. At time t = 0 t h e droplet size distribution N ( r ) is uniform throughout the column. After time t, a certain portion of the dispersed phase has settled and is collected a t the bottom of the column a t height I ( t ) , as shown in Figure 1. The interface between the dispersion and t h e already settled portion moves up a t a velocity dl dt
u(t) = -
(4)
A droplet of radius r, which hits the interface a t time t, was a t t = 0 a t height h(r,t)
=
1
+ v(r)t
(5)
above the bottom of the column. All the droplets of the same size, r, which were below this droplet a t t = 0, would have already settled a t time t. Therefore the total volume of droplets of size r decanted after time t i s
A[2
+ v(r)t]
"3
ar311'(r)dr
(6)
The largest droplet size, R, still in the dispersion a t time t i s one for which
h(R,t) = L
(7)
Therefore the total volume of dispersed phase decanted in time tis
Theoretical Model
T h e purpose of t h e following analysis is to calculate the settling rate in a stationary liquid-liquid dispersion of given droplet size distribution. Consider a vertical column of length L and constant cross-section area, A , containing a n initially uniform mixture as shown in Figure 1. The mixture contains a large number of droplets of t h e dispersed phase in various sizes. 538 Ind.
Eng. Chern. Process Des. Develop., Vol. 1 1 , No. 4, 1972
where R is the droplet radius for which
I(t)
+ v(R)f= L
(9)
A study has been conducted on fhe separation of mixtures of water and liquid Freon refrigerant for application in a freeze-crystallization process. A theoretical model was developed which describes the settling of a stationary, initially uniform mixture in a vertical column. The analysis provides a simple experimental method to determine droplet size distribution in a liquid-liquid dispersion based on measurements of its settling rate. Experimentally, mixtures of water and Freon 1 13 were prepared and studied. The droplet size distribution was determined and defined in terms of three characteristic parameters: a characteristic drop size (corresponding to the mode of the distribution) and the variance of the distribution and its deviation from symmetry. The method has been applied to samples of Freon-water dispersions taken from the freeze-crystallization process. The information obtained on droplet size distribution, combined with data on density and viscosity of the two liquids, i s used to calculate droplet settling velocities and total separation rates.
The integral Equation 8 can now be solved for the droplet size distribution S ( T in ) terms of the settling rate Z(t). From Equation 3 we obtain
tween the droplet size and velocity have been found (Treybal, 1963; Soo, 1967). The drop Reynolds numbers in the present study were in the low range where Equation 15 applies. Substituting Equation 15 in Equation 9, we obtain the relationship between R and t:
Equation 8 can then be rewritten as
L - Z(t) R = ( X - )
1'2
which can be substituted in Equation 14 to give by differentiating Equation 11 with respect to t we obtain:
u = [2
+
+ v(R)t - L ] 43 nR3N(R)dR + dt -
-
s,"
+
[u
A'(R) = - --3 (1 - ff) (kZt4/L3)(dU/dt) 2 n (I - 2/L) [l (ut - I)/L]8
4 U(T) -
3
Tr3'17(i-)dr
(12)
The first term in Equation 12 is zero, from Equation 9. Substituting the result of Equation 12 into Equation 11, we find:
(18)
An experiment can now be run on a given dispersion in a decanting column, where 1 is recorded as a function of t . From Equations 4 and 17, u(t) and R(t) are respectively calculated and substituted in Equation 18 to give N ( R ) . The analysis then provides a method to obtain the initial droplet size distribution in a mixture from its settling rate. Experimental
and by differentiating Equation 13 with respect to t , we finally obtain
The experimental apparatus used in the study is shown in Figure 2. h mixture of Freon 113 (CClzF-CClFZ) and water was prepared in a cylindrical mixing chamber (4 in. in diamST\RRER
where R is given in terms of t through Equation 9. Equation 14 provides the desired relationship between the droplet size distribution, S ( r ) , and the settling rate I(t). Xote that the solution is independent of the type of relationship between the droplet size and its settling velocity. This relationship varies, depending on the drop Reynolds number (based on the droplet diameter and velocity and the viscosity of the continuous phase). For low Reynolds numbers (Re < l), the settling velocity can be computed analytically from the Savier-Stokes equations, and is proportional t o the drop radius
v
=
kr2
MIXING CHAMBER DRAIN VALVE
(15)
where h. is given by Hadamard's Equation (Hadamard, 1911) :
in which p D and p are the densities of the dispersed and continuous phases, respectively, and f i D and p are their viscosities. When p D >> p , Equation 16 reduces to Stokes law for rigid spheres. For large Reynolds numbers, where t h e above relationships do not hold, empirical correlations be-
/DECANTING COLUMN
Figure 2. Experimental apparatus Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4 , 1972
539
35
et g 20 I5
0
20
40
60
80
100
120
140
160
180
200
220
240
TIME.=
Figure 3. Typical experimental curve time
of interface height vs.
eter by 5 in. high). T h e chamber was equipped with a stirrer, 11/2in. in diameter with four blades a t a 30" pitch mounted on a l/d-in. shaft. T h e colorless Freon was dyed with a trace of iodine to make i t distinct from t h e water, and the two liquids were stirred together long enough to obtain a uniform dispersion. T h e mixture was then drained into a decanting column, and its settling rate measured as a function of time. T h e draining time was of t h e order of 5 sec, very short compared to t h e characteristic settling times. The rectangular in.) immediately elimcross section of t h e column (2 x inated any residual spin introduced into t h e mixture while in the mixing chamber. T h e position of the interface was observed by eye relative to a fixed scale, and signals for dif-
2
I I
Figure 4. Interaction between two settling droplets
5 4 0 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 ,
No. 4, 1972
Figure 5. Droplet size distribution function and distribution parameters
ferent heights were recorded correspondingly on a time recorder. T h e relative proportions of t h e tlvo liquids and the mixing conditions (propeller speed and position) were varied in order to obtain different mixtures. Figure 3 shows a typical plot of settling rate vs. time. I n this particular experiment t h e mixture was prepared from 25 cc of Freon and 575 cc of water. The stirrer was "4 in. above the bottom of t h e chamber, and 1 in. from its center and rotated a t 1000 rpm. As expected, the curve shows a monotonic increase in the height of the interface, asymptotically approaching the maximum value of 1 = La. The settling rate decays with time as the large droplets are gradually depleted from t h e dispersion. Note t h a t for the limited accuracy of simple observations by eye, the scatter of the data around the curve is relativelysmall. Experimental curves similar to the one of Figure 3 were obtained in the same way for various other mixtures. During some of t h e experiments, t h e motion of the individual droplets in t h e dispersion was observed and photographs were taken. T h e velocity of the droplets was found to be constant with time, in agreement with the first assumption of the model. Drop Reynolds numbers based on t h e measured velocity were of order unity or less. The assumption about no collisions between droplets was checked too. Figure 4 describes qualitatively what happened. When two droplets falling at different velocities approach each other, the water between them has to be squeezed out through a narrow gap, thereby applying a lateral hydrodynamic force which pushes t h e droplets away from each other. It was concluded on t h e basis of many such observations that if the difference in size between the droplets is not too large, collisions hardly ever occur. Several phenomena were observed which could have caused deviation from t h e ideal behavior described in t h e model. A relatively small amount of tiny gas bubbles (mostly air and perhaps some Freon gas) were always present in t h e dispersion. Some of t h e liquid Freon originally introduced into the mixing chamber evaporated in t h e course of the experiment (this seems to have occurred primarily in the mixing process). Occasionally, a large liquid drop which had been trapped at the interface on top of the dispersion was seen falling through t h e column and causing some local stirring. These effects are probablythe reason for part of the scatter in the data. The droplet size distribution in each of the above experiments can now be found by applying Equaticn 18 to t h e data t ( t ) of the type sholvn in Figure 3. A difficulty in doing this lies in the fact that the use of Equation 18 requires taking
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