Determination of Coalescence Frequencies in Liquid-liquid Dispersions: Effect of Drop Size Dependence

The size dependence of the drop coalescence frequency is investigated by measurement of transient drop size distributions in purely coalescing systems (with negligible drop break-up). A twopronged approach is employed to estimate the bivariate coalescence frequency function, using such experimental data. First, coalescence frequency expressions derived from mechanistic models of the relative motion of the drops are evaluated based on their ability to predict the experimental transient sire spectra. Second, experimental drop size distributions which exhibit a property known as self-similarity are analyzed through an inverse problem in order to extract the coalescence frequency function directly from the data. Results indicate that the coalescence. frequency of small droplets (l&SO pm in diameter) is lower than that predicted from a constant coalescence efficiency model. In addition, experiments show that, given favorable initial conditions, self-similar drop size distributions can be manifest. In the cases where similarity behavior is observed, the frequencies obtained from the inverse problem are in qualitative agreement with the mechanistic models that describe the data best. 1. INTRODUflrON Many engineering operations, such as liquid-liquid extraction and multi-phase reaction, involve the formation of stirred dispersions of two immiscible liquids. In such systems drop coalescence (and breakup) can profoundly influence the overall performance, by altering the interfacial area available for species exchange between the phases. In order to optimize these operations fully it is therefore desirable that engineers learn more about the dispersive process, through detailed studies of drop coalescence and break-up. A rational approach to the modelling of dispersive systems begins with the framework of population balances, and is discussed by Ramkrishna (1985). This describes the temporal evolution of a drop population due to random coalescence and break-up events. For a purely coalescing dispersion in which no mass exchange is involved, the population balance equation is given by ai-qfi, q 1 ” dtl=s 2 0 g(u” v”‘, o’)A(a v”‘, E)ri(C’, t’)dC r ij(iJ, v”‘)ii(i?, F)ii(a’, 7)dv” (1.1) 0 where n(& 7) is the number density (concentration) of drops of volume 6 at time ?, and q(iJ, v”) is the binary coalescence frequency for drops of volumes v” and 5’. The bivariate frequency function is a key input to the coagulation equation, and must be either determined from experiments or derived from phenomenological models of drop coalescence. ‘Author to whom correspondence should be addressed. Many previous studies have sought to gain information about coalescence frequencies in a turbulent flow field. Such studies may be grouped into three basic categories, according to the experimental methods employed. The first type of study infers an average coalescence frequency from measurable changes in the physical properties of a system. An example of this is the work of Madden and Damerell (1962) in which coalescence rates of water droplets in toluene were estimated by observing the rate of extraction of iodine from the toluene into the water drops. From their experiments they reported the following relationship: 4 = fi2.*+o.504 (1.2) where R is the stirring speed, 4 is the volume fraction of the dispersed phase, and q is the average coalescence frequency. A similar work by Miller et al. (1963) yielded basically the same results as those of Madden and Damerell. In a slightly different type of experiment, Howarth (1967) used a light absorbance technique to determine the rate of change of the mean drop size in a stirred vessel following a stepwise reduction in the impeller speed. Howarth obtained an average coalescence frequency from the change in average drop size with time, and his results were also consistent with those cited above. Although these studies can reveal the gross behavior of the coalescence frequency as a function of the energy dissipation rate and the dispersed-phase fraction, they are of very limited utility for the following reasons. First, they yield no information about the effect of drop size on the coalescence frequency, since they are built on the tacit and severe assumption that coalescing dispersions are essentially mono-disperse. Real dispersions can and often do have drops of