Solution of the droplet breakage equation for interacting liquid-liquid dispersions: a conservative discretization approach

A conservative discretization approach for the population balance equation (PBE) with only droplet breakage describing the hydrodynamics of a continuously interacting liquid-liquid dispersion is presented. The approach is conservative in the sense that it conserves any two integral properties associated with the number droplet distribution and thus it is considered internally consistent. The discrete set of equations is laid down through applying the subdomain method where it is shown that this set of discrete equations is only internally consistent with respect to one integral property. The internal consistency is enforced by introducing a set of two auxiliary functions that are uniquely determined by matching the integral properties obtainable from the discrete set against those from the continuous PBE. However, it is shown that this conservation of integral properties is not exact for all the subdomains and hence it results in what we call the intrinsic discretization error (IDE). This IDE is not only associated with this approach, but also it is found inherently existing in the fixed-pivot (FP) technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996a) 1333). The derived equations of the IDE for the present discretization approach and the FP technique generalized to continuous flow systems show that the present approach enjoys a small value of the IDE. To validate the discretization approach, two analytical solutions for the continuous PBE are presented, where good agreement is found between the predicted and the analytical solutions. To assess the reliability of the present discretization approach two experimentally validated breakage frequency functions describing droplet breakage in a turbulent continuous phase as well as two daughter droplet distributions are considered. The convergence characteristics show that the present discretization approach has an identical convergence rate as that of the FP technique, and in some cases it is superior to it. This rate of convergence is found approximately proportional to the square of the inverse of the number of subdomains.