The Dynamics of Drop Enlargement

Consider a polydisperse emulsion, assuming a spatially homogeneous case, with low volume concentrations of the disperse phase. Assume further that it is possible to limit ourselves to consideration of pair interactions of drops. The dynamics of enlargement (integration) of drops due to their collision and coalescence is then described by the following kinetic equation 

The first term in the right-hand side of equation (11.1) corresponds to the formation rate of drops of volume V due to the coalescence of drops with volumes V — CO and co, and the second term - to the rate of population decrease of drops of volume V at their coalescence with other drops. 

The solution of the equation (11.1) for a given initial distribution o( V) allows to trace the evolution of volume distribution of drops and to determine the following key parameters of distribution numerical concentration of drops (the number of drops per unit volume of emulsion) 

The kernel K V,co) defines the mechanism of drop interactions, therefore the study of its general properties along with its concrete forms for various processes is an independent problem of physics of disperse medium. 

The important property of the kernel of kinetic equation is symmetry with respect to sizes of colliding drops, that is K V, co) = K(co, V). Multiply both parts of the equation (11.1) by Vand integrate the result over Vfrom 0 up to oo. Taking into account the expression (11.3), one obtains 

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Flow-induced coalescence is accelerated by the same factors that favor drop breakup, e.g., higher shear rates, reduced dispersed-phase viscosity, etc. Most theories start with calculation of probabilities for the drops to collide, for the liquid separating them to be squeezed out, and for the new enlarged drop to survive the parallel process of drop breakup. As a result, at dynamic equilibrium, the relations between drop diameter and independent variables can be derived. 

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