Drop-breakage in Agitated Liquid-liquid Dispersions

Experimental data from batch vessels on cumulative volumetric drop-size distributions at various times are shown to yield useful information on probabilities of droplet-breakup as a function of drop-size. Such information is sufficient for a priori prediction of drop-sizes in agitated dispersions in batch and continuous vessels. It may also be useful in predicting heat and/or mass transfer in liquid-liquid dispersions by accounting for the simultaneity of transport processes from individual drops and droplet breakage processes: INTRODUCTION Drop-size distributions in agitated vessels evolve as a consequence of droplet breakage and coalescence. The dynamic character of breakage and coalescence has been recognized by some workers[l-31 who have attempted a prediction of drop-size distributions by the method of population balances. However quantitative predictions are precluded by insufficiency of adequate information about droplet phenomena such as breakage and coalescence of droplets. The possibility of obtaining such information about droplet breakage from appropriately designed experiments has been discussed by the author [4]. If consideration is restricted to lean dispersions it becomes possible to exclude the effects of coalescence and confine attention only to droplet breakage. The purpose of this paper is to demonstrate that measurements of cumulative volume distributions of drop sizes at various instants of time obtained from a lean liquid-liquid dispersion in a batch vessel can provide all of the quantitative information required to make predictions of drop-size distributions. Experimental data of this kind obtained by Madden and McCoy [5] have been employed for this demonstration. ANALYSIS We denote the volume fraction of drops of volume less than v at time t by F(v, t), which is a cumulative distribution function satisfying the conditions F(v, t) = 0 when v 5 0; lim F(v, t) = 1 y-m Breakage may be characterized by (i) the transition probability function T(v) such that I(v)dt represents the probability that a droplet of volume o breaks in the time interval (t,t + dt) and (ii) the volume fraction of daughter droplets with volume less than v formed from breakage of a droplet of volume v’ represented by the function G(v, v’) which has the properties G(v, v’) = 0 when v ~0; G(v, v’) = 1 when v 2 v’. In an earlier work[4], which was based on number distribution of droplets, the cumulative distribution function G(v, v’) was replaced by two other functions v(v’) and p(v, v’); v(v’) is the mean number of daughter droplets obtained by breakage of a parent droplet of volume v’, and p(v, v’)dv is the probability that a daughter droplet arising from a parent of volume v’ has its volume between v and v + dv. It is readily seen that v”v(v’)p(v”, v’)dv”. For a batch vessel when the dispersion is lean in the dispersed phase it is easy to show that l3F at’ 0 I= r(v')G( v, v’) dF(v’, t). (I) Equation (1) can be easily established either independently or from the corresponding population balance equation written for the number density of droplets of volume v [ 1,4]. The right hand side of