Coalescence of Rigid Droplets in a Stirred Dispersion-ii_ Band-limited Force Fluctuations

The coalescence of nearly rigid liquid droplets in a turbulent flow field is viewed as the drainage of a thin film of liquid under the action of a stochastic force representing the effect of turbulence. The force squeezing the drop pair is modelled as a correlated random function of time. The drops are assumed to coalesce once the film thickness becomes smaller than a critical thickness while they are regarded as separated if their distance of separation is larger than a prescribed distance. A semi-analytical solution is derived to determine the coalescence efficiency. The veracity of the solution procedure is established via a Monte-Carlo solution scheme. The model predicts a reversing trend of the dependence of the coalescence efficiency on the drop radii, the film liquid viscosity and the turbulence energy dissipation per unit mass, as the relative fluctuation increases. However, the dependence on physical parameters is weak (especially at high relative fluctuation) so that for the smallest droplets (which are nearly rigid) the coalescence efficiency may be treated as an empirical constant. The predictions ofthis model are compared with those of a white-noise force model. The resuhs of this paper and those in Muralidhar and Ramkrishna (1986, Ind. Engng Chem. Fundam. 25, 554-560) suggest that dynamic drop deformation is the key factor that influences the coalescence efficiency. INTRODUCTION Stirred liquid-liquid dispersions are often encountered in several unit operations in chemical engineering. For an accurate analysis of transport and reaction rates in such processes, it is essential to know how the droplets are distributed with respect to size, concentration and/or any other relevant physical parameter required for the description of the rate processes for a single drop immersed in the continuous phase. The framework of population balances is ideally suited to predict the states of the drop population if certain key rate functions such as the coalescence and break-up frequencies of droplets become available either from theory or experiment (see Ramkrishna, 1985). In this paper, we focus on coalescence of liquid droplets. The coalescence frequency of liquid droplets in stirred liquid-liquid dispersions is usually written as the product of the collision frequency and a correction factor commonly referred to as the coalescence efficiency. The former pertains to the frequency of contacts of the droplets while the latter is the probability that any contact leads to coalescence of the two drops. The relative velocity of approach of two droplets is significantly reduced at very small separations due to viscous hydrodynamic interaction so that two droplets may be assumed to be in contact when they are separated by a thin film of continuous-phase liquid. The subsequent coalescence of the drops requires the “draining out” of this viscous film of liquid +To whom correspondence should be addressed. to a critical thickness at which film rupture occurs followed by almost instantaneous formation of a larger drop. The contiguous turbulent flow field may separate the drops even before this critical thickness is attained so that not every drop-pair contact leads to coalescence. It therefore becomes necessary to determine the coalescence efficiency, which is the fraction of collisions that lead to coalescence. While several models exist in the literature for predicting the collision frequency (see, for example, Shinichi Yuu, 1984) very few exist for predicting the coalescence efficiency. The viscous hydrodynamic interaction of two liquid droplets in a turbulent flow field involves several coupled dynamic effects such as drop deformation, viscous retardation and stochastic forces due to the turbulent flow, not all of which are easily characterizable. Models are derived in the literature based on several simplifying assumptions. Coulaloglou and Tavlarides (1977) argue that a constant force due to the turbulent flow field should squeeze the drops for a random time interval which is exponentially distributed. If this contact time is larger than the time required for the film to drain to a critical thickness, coalescence occurs. They obtained the coalescence efficiency for drops larger than the turbulence microscale as q = exp[ -?(&)‘ I (1) where k is an empirical parameter, d, and d2 are the droplet diameters and y is the interfacial tension. While the above model incorporates the effect of turbulence