An Effect of Hold-Up on Drop Sizes in Liquid-Liquid Dispersions
With the help of the Kolmogoroff theory of universal equilibrium an expression is derived to account for the increase in mean drop size with volume fraction of dispersed phases, in absence of coalescence. The relative increase in the mean drop size between dilute and concentrated dispersions agrees well with the data from literature when the dispersion viscosity is expressed by the Einstein equation.
Drop sizes in agitated liquid-liquid dispersions a t low hold-up values (4 < 0.01) can be described by the expression derived by Shinnar and Church (1960). In agitated tanks where the average energy dissipation per unit mass can be expressed by E =
KN3L2
(1)
the equation for the mean drop size in dilute dispersions takes the form
For the same dispersion a t t l let the viscosity be changed from ul to u2 so that the micro-scale '72 corresponding to t1 and v z equals that a t t2 and u1. Thus from eq 5 it follows that
q2 =
( U ~ ~ / E ~ )= ' ~ '
(7)
=
(U,/UI)3
(8)
lk2/lk1 = ( u ~ / u , ~ ~
(9)
or E1/€2
which is considered to be valid up to a hold-up value 4 0.01 (Chen and Middleman, 1967). At high hold-up values of the dispersed phase the mean drop size formed under a break-up mechanism can be correlated empirically by the equation
(3) C Z is a dimensionless coefficient for which the following values have been reported by several authors: 2.5 (Vermeulen et al., 1955), 5.4 (Mlynek and Resnick, 1972), 3.14 (Brown and Pitt, 1970), and 9.0 (Calderbank, 1958). In the absence of coalescence, the drop sizes a t high hold-up values, like those a t low hold-up values, should be independent of hold-up values. The functional dependence of the mean drop size on hold-up in eq 3, even in absence of coalescence, has led some authors (Van Heuvan and Beek, 1971; Brown and Pitt, 1972) to believe that the observed hold-up function is due to a reduction of turbulence intensity in the presence of the dispersed phase. The object of this communication is to show, using Kolmogoroff's theory of universal equilibrium (Kolmogoroff, 1941) that, because the dispersion viscosity depends upon hold-up and because turbulent scales are affected, drop sizes in concentrated dispersions depend on dispersion viscosity. When this effect is analyzed eq 3 can be deduced. Assume that the turbulence in agitated tanks is homogeneous and described by the Kolmogoroff energy-spectrum function, E ( k ) ,which has the form E(k) =
Equations 6 and 8 yield
Equation 9 shows that the change in the viscosity of a turbulent medium causes a corresponding change in the average length scales of energetic eddies even at a constant level of external power input into the system. As viscosities of dilute and concentrated dispersions differ due to differences in dispersed-phase volumes, the turbulence structures originally present in concentrated dispersions a t 6 1 rearrange in a way such that drop sizes formed there correspond to an energy level c2 if viscosity is changed from u l to U Z . Drop sizes obtained in dispersions have a statistical size distribution. The maximum size in a distribution can be represented by the expression developed by Shinnar and Church (1960). For dilute dispersions a t an energy input el
domax= C3(y/p,)3/5E*45
(10)
For the reasons already stated, the maximum drop size in concentrated dispersions of the same dispersed and continuous phases as in dilute dispersions, formed a t the energy input rate t l and in absence of coalescence can be represented by
From eq 10 and 11 it follows that
(12)
~ y ~ ~ n k ~ ~(4)
and microscale
Substitution for t l / t z in eq 12 from eq 8 leads to
Since the drop size is greater than the micro-scale where viscous action is negligible, the effect of the dispersion viscosity on drop sizes is not easily discernible. However, the viscosity effect on the average lengths of the energetic eddies responsible for drop break-up can be found. From this the effect on drop sizes can be obtained. Consider a dispersion formed in the universal equilibrium range of turbulence with viscosity V I and a t two levels of energy input rates, 6 1 and € 2 . For equal fluctuation intensities the average length scales of energetic eddies can be related by
Since d32 = ad,,, (Brown and Pitt, 1970), and d 0 3 2 = Pldomax(Chen and Middleman, 1967) and also $ = p1 (in absence of coalescence), it follows from eq 13 that
The viscosity of concentrated dispersions of oil and water up to 4 = 0.2 can be represented by the Einstein equation (Ward and Knudsen, 1967).
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
137
As vc
= ul and ue
= v2 it follows from eq 14 and 15 that
After series expansion, neglecting the terms containing powers greater than one and for the condition of p c = p e , eq 16 leads to
The hold-up coefficient, C2, derived in eq 17 is 3.0, nearly equal to the value obtained by Brown and Pitt (1970) for a noncoalescing system of kerosene and water. This value may represent the maximum possible contribution of the dispersion viscosity.
Greek Letters CY = constant, defined by eq 4 @,PI = proportionality constants y = interfacial tension, dyn/cm c = average energy input rate per unit mass, cm2 sec-3 c = energy dissipation rate per unit mass, cm2 sec-3 7 = Kolmogoroff microscale, cm pc = viscosity of continuous phases, P pe = viscosity of dispersion, P u = kinematic viscosity, cm2 sec-I uc = kinematic viscosity of continuous phase, cm2 sec-I p c = density of continuous phase, g cm-3 p e = density of dispersion, g cm --3 4 = dispersed phase volume fraction Literature Cited
Nomenclature C1,c13 = constants Cz = coefficient used in eq 3 d03? = Sauter mean diameter of drops in a dilute dispersions, cm d32 = Sauter mean diameter of drops in a concentrated dispersion, cm do!,,,, = maximum drop size in dilute dispersion, cm d,,,, = maximum drop size in concentrated dispersion, cm E ( k ) = energy spectrum function, cm3 sec k = wave number, cm K = empirical constant, defined by eq 1 1, = turbulent length scale, cm L = impeller diameter, cm N = impeller speed, sec-l N , , = Weber number, L3WpL/y
Brown, D. E . , Pitt, K., Chem. Eng S c i . . 27, 577 (1972). Brown, D. E . , Pitt, K., paper presented at CHEMCA. Australia, 1970. Calderbank, P. H., Trans inst C h e m . Eng . 36, 443 (1958). Chen, Hs. Ts., Middleman. S.,A I.Ch E . J 13, 989 (1967). Kolmogoroff, A. N., C R A c a d . Sci U R S S . . 30, 301 (1941). Mlynek, Y . . Resnick, W . , A i Ch.E. 3 , 18, 122 (1972). Shinnar, R . , Church, J. M., Ind Eng Chem , 52, 253 (1960). Van Heuvan. J. W.. Beek, W. J.. f r o c ISEC. Amsterdam, April (1971). Vermeulen, T., Williams, G. M., Langlois, G. E , Chem. Eng. f r o g , 2, 85F (1955). Ward. J. A.. Knudsen, J. G . , A I.Ch E J . 13, 356 (1967)
Department of Chemical Mohammed S. Doulah Engineering Glamorgan Polytechnic Treforest, Pontypridd, Glam., S o u t h Wales Receiued for reuieur March 22, 1974 Accepted J a n u a r y 31, 1975
Vapor-Liquid Equilibrium in the Critical Region. Concentration Limits of Binary Systems The vapor-liquid equilibrium of binary system concentration limits in the critical region has been studied. I t has been shown that the following equations are generally valid: limx+l N = Iimx-, ( d y / d x ) = 1; [ T = Tell and m io,l, N = lim,+o ( d y l d x ) = 1; [ T = T C 2 ] , where cy i s the relative volatility, y and x denote the mole fractions of the more volatile component in vapor and liquid phase, and r,,, Tc2 are critical temperatures of pure components 1 and 2. These equations are important from the practical point of view and have to be taken into account in the design of high-pressure rectification columns.
An experimental investigation of the behavior of vaporliquid equilibrium in the neighborhood of the critical region of binary hydrocarbon systems made by Wichterle and Kobayashi (1971a, 1972) showed that the “y-x” curve measured exactly at the critical temperature of the more volatile component approaches 100% of the more volatile component with a slope equal to 1.
where y, ( x ) is the mole fraction of component 1 in the vapor, (liquid) phase and the symbol [Tcl] denotes that the temperature is constant and equal to the critical temperature of the pure component 1. It is shown in this contribution that eq 1 is generally valid and can be derived from the coexistence equation for a two-phase binary system. Let us consider the binary coexistence equation a t constant temperature (see HBla et al. (1967) for a derivation of eq 2) 138
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
where V is the molar volume of the phase, P is the total pressure of the system, G is the molar Gibbs function, the double prime denotes the vapor, the single prime the liquid phase and the symbol [qshows that the temperature is to be held constant. The second term of the left-hand side of eq 2 can be expressed as follows
where p ’ and p’ are chemical potentials of components 1 and 2 in the liquid phase; on differentiating eq 3 a t constant temperature and pressure, while considering the Gibbs-Duhem equation, we obtain
