A population balance model for mass transfer in lean liquid-liquid dispersions

A population balance model is considered for the description of mass transfer in liquidliquid dispersions in a mechanically agitated vessel operated continuously. The dispersions considered are of low dispersed phase fraction in which coalescence processes may be assumed to be negligible. The model incorporates the dynamic nature of breakage processes among the droplet population together with mass transfer processes at the droplet level into the analysis which is accomplished by the use of a trivariate number density using size, concentration of diffusing solute and age of the droplets. Mass transfer in single drops is assumed to be by diffusion. Droplet breakage is assumed to be binary and even. The results of the model are compared with predictions made from an alternative approach (reflective of methods used in the past) which ignores the fact that droplet size distributions are attained in time. Several interesting deviations are predicted in the concentration distributions for drops of definite sizes and in the overall mass transfer rates. It is concluded that the dynamics of droplet phenomena may play an important role in the analysis of mass transfer in dispersions. INTRODUCTION MASS transfer in liquid-liquid dispersions is a complex process involving droplet phenomena such as breakage and coalescence, and mass transfer from individual drops, both of which depend on a large number of a diverse variety of factors. These factors have been reviewed by Olney and Miller [ll. Various attempts in the past towards a quantitative description of mass transfer in liquid-liquid dispersions have been reviewed by Ramkrishna [2]. These attempts range from the coarse method of the overall volumetric mass-transfer coefficients to those which seek to incorporate mass transfer processes at the droplet level in the prediction of the overall mass transfer from the drop population. It is with this latter approach, in its application to liquid-liquid dispersions produced in a mechanically agitated vessel, that this paper will be concerned. Moreover, attention is focussed on lean liquid-liquid dispersions in which droplet coalescence is likely to be negligible so that droplets undergo only breakage. The necessity to incorporate the aspect of simultaneity of droplet processes (breakage and coalescence) and mass transfer processes (from single drops) has been pointed out [2,31. To accomplish this one must consider multivariate number densities in which must appear as variables the size of the droplet, concentration of the solute in the droplet, and all others pertaining to the drop, whose specification should be vital to the prediction of the instantaneous mass transfer rate from a given drop. The population balance equations which the multivariate number density must satisfy have been discussed in detail by Ramkrishna[21. In this paper we consider a specific model which employs a multivariate density including as variables, the size (radius), solute concentration and “age” of the droplet. This model is analyzed to demonstrate the interesting deviations of its predictions from those that arise from neglecting the dynamic nature of droplet phenomena. THE MODEL The system under consideration consists of a mechanically agitated vessel which continually