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Ind. Eng. Chem. Fundam. 1986, 2 5 , 554-560
Analysis of Droplet Coalescence in Turbulent Liquid-Liquid Dispersions Ramachandran Muralldhar and Doraiswaml Ramkrlshna School of Chemical Engineering, Purdue University, West Lafayetie, Indiana 47907
The coalescence of two drops in a turbulent flow field requires the drainage of an intervening viscous film of continuous-phase fluid under the action of a turbulent force, which is a random function of time. Because the hydrodynamic problem is complicated, a detailed time scale analysis is employed to identify simple models corresponding to limiting physical situations. Analysis of these models reveals that the dependence of the coalescence efficiencies on physical parameters may reverse as one goes from one limiting physical situation to another.
1. Introduction
Coulaloglou and Tavlarides (1977) recognized the probabilistic nature of the coagulation process. They argue that the turbulent force compressing the drops together must act for a sufficient length of time for the intervening continuous-phase film to drain to a critical thickness at which instantaneous film rupture and coalescence can take place. In other words, the random contact time, T , must exceed the coalescence time, t , defined as the time required for the film to drain to a critical thickness from a specified initial separation under the action of the mean force. The mean force was estimated based on the properties of isotropic turbulence. The authors assume that the contact times are exponentially distributed and define the coalescence efficiency as 17 = exp(-f/ri) (1.1)
One of us (D.R.) fondly recalls conversations with Professor Olaf Hougen while sharing an office with him a t the University of Wisconsin about 12 years ago. In one instance, sensing his neighbor’s obsession with equations, Professor Hougen pulled out a book from his collection and asked to have it examined for equations. There were none in the entire book! It was a book from which Professor Hougen had taught chemical engineering as a young assistant professor in the 1920s. The experience was humbling to this author in the realization that today’s clarity of perception which admitted mathematical treatment had owed so much to the efforts of giants like Professor Hougen. T o him we most humbly dedicate this mathematical paper. An accurate prediction of the coalescence frequency of droplets in stirred liquid-liquid dispersions involves the analysis of relative motion between drop pairs. The analysis has proven to be fairly simple as long as the drops are sufficiently far apart for any hydrodynamic interactions to be negligible. The relative diffusion coefficient, estimated from the equation of motion of the two drops and the statistical characteristics of isotropic turbulent flow, is used to compute the collision frequency. The analysis of relative motion a t small separations is complicated by the presence of viscous interaction, drop deformation, and random flow fields, all of which contribute to retarding the coalescence process. The neglect of this analysis is manifest by the appearance of an empirical multiplicative correction of the collision frequency, commonly referred to as the coalescence efficiency. In the literature, models exist for calculating the coalescence efficiency in turbulent flow situations derived after making several simplifying assumptions. It has been convenient to view the relative motion a t small separations as the problem of drainage of the continuous-phase film entrapped between the two drops. Maraschino and Treybal (1971) derived a detailed expression for film drainage between two deformable drops by dividing the separation into three distinct regimes in which film drainage expressions are different. These authors correlated the film thickness a t maximum deformation for two equal drops to an empirically established coalescence efficiency with a collision number and the Reynolds number as parameters. A deficiency of this model is that it did not recognize the turbulence characteristics of the fluidized bed in which the particles were assumed to be present. While the film drainage expressions are very detailed, the mechanism of separation or coagulation is not evident. 0196-4313/86/1025-0554$01.50/0
where t and f are averages. If a mean force is employed a t the ends of an eddy of size equal to the sum of the drop diameters, the coalescence efficiency is given by
where k is an empirical parameter. A disadvantage of this model is that it does not recognize that film drainage occurs randomly in time owing to the stochastic nature of the squeezing force. In principle, the model tacitly assumes that the drainage time scale is much smaller than the time scale of the force fluctuations. More recently, Das et al. (1986) have proposed a stochastic version of the film drainage equation between two rigid spheres as
--(
_ dh - 2hF(t) 1 dt
-
3XP
+
i)
(1.3)
wherein the force is a rapidly fluctuating white-noise process
F(t) = E - 6 T f q ( t ) (1.4) The two drops are assumed to be initially separated by a distance, ho,larger than the film rupture thickness, h,, but smaller than a distance, h,, a t which the particles are deemed to have separated. Monte Carlo simulation of eq 1.3 and 1.4 yields the coalescence probability. While the model represents the first step in modeling film drainage as a stochastic process, it is arguable whether the whitenoise force is appropriate because this requires that the film drainage rate be much slower than the rate of the force fluctuations. The existence of white noise may be ques0
1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 555
tionable if the intervening film between rigid spheres is associated with large drainage rates. Possibly a t higher stirring speeds, the rate of force fluctuations may increase and the white-noise model may be more appropriate. From the preceding discussion, it is clear that there has not been a systematic effort to understand the last stage of the coalescence process. Further, there has been little effort to model the force squeezing the drops together as a random function of time. In this paper, a time scale analysis is employed to understand the significance of the factors affecting drop coalescence. Coalescence models are then derived for limiting physical situations for which the hydrodynamic interaction is rather simple. The objective is to observe the sensitivity of the coalescence efficiency to the nature of the force fluctuations. This is important because a high sensitivity would imply that an accurate characterization of the force is necessary to estimate the coalescence efficiency reliably. The influence of physical parameters on the efficiency is also studied in view of its importance in manipulating size distributions in physical situations such as dispersion polymerization. 2. Theory 2.1. Salient Features. In this section the focus is on turbulent dispersions in which the Reynolds numbers are high enough for Kolmogorov's principle of local isotropy to be valid (Hinze, 1959). Dispersions of interest in this paper are dilute, and any alteration of the flow field due to the presence of the particles is neglected. Only neutrally buoyant drops larger than the microscale of turbulence, which occur commonly practice, are considered. The deformability of drops may be characterized by the Weber number given by vh
vh2n
This dimensionless number is the ratio of the restoring pressure due to interfacial tension ( y / R )and the viscous stresses developed in the film (pU/h). U is a characteristic fluid velocity in the film, and Rel is the film Reynolds number. If We >> 1, the drops remain essentially spherical while the drops are easily deformed if We > 1 and > 1). In this situation. the dimensionless parameter cy > 1, drop deformation can be evaluated from the mean force alone and dynamic drop deformation is not important. On the other hand, if /3 > 1 and 0
