A novel simplified multivariate PBE solution method for mass transfer problems

Interphase mass transfer estimation may require not only the accurate knowledge of the interfacial area, which depends on the information about the size of each dispersed element, but also on the driving force, that can be different if the elements of the disperse phase have different chemical composition. To take into account such polydispersity, bivariate (or multivariate) population balance model (PBM) are formulated according to physical phenomena occurring in the investigated mass transfer problem. This often includes aggregation, breakage, advection, mass transfer of the chemical species and chemical reactions of the transferring components. In this work we propose a novel and simplified method to solve the bivariate/multivariate population balance equation for a mass transfer problem, based on the high-order moment-conserving method of classes (HMMC) (Alopaeus et al., 2006). The proposed method is based on the idea of deriving additional material balance equations for the concentration of droplets belonging to each size class, reducing significantly the total number of unknown variables with respect to true bivariate/multivariate method of classes. This modeling approach is compared with two other possible solution methods for a test case in which mass transfer and chemical reactions occur in a system with two immiscible liquid phases. In the first the traditional approach is used, where a single material balance is formulated for the disperse phase along with PBM, while in the second a true bivariate/multivariate solution method is used. The results of this comparison show that the proposed method is robust and accurate, capable of properly describing the multidimensional droplet size-composition distribution needed to evaluate the mass transfer rates, in a fraction of computational time compared with more accurate methods.