Current theories of drop breakup mechanisms do not even explain the observed DSD trends qualitatively for high-shear mixers. The population balance equations (PBE) framework quantifies the changes in DSDs. PBEs are integral-partial differential equations that describe the behaviour of a population (drops, microbes etc.) which is vital in many chemical engineering, biological and particulate processes. Due to the complex nature of PBE, very few analytical solutions are available for these equations, and mostly are solved numerically by discretization. A new alternative approach to constructing analytical solutions for pure breakup has been developed by studying the patterns in the eigenvectors and eigenvalues of the matrix generated when these coupled ODEs are written in matrix-vector notation The method led to the construction of analytical solutions for an entire class of breakup problems. A series solution is developed for a widely used breakup function where conventional mathematical technique fails. Second part of the problem deals with solving inverse problem. The population balance inverse problem involves extracting these breakup functions in PBE using transient experimental DSDs. This is challenging because inverse problems are ill-posed, and the experimental data available is very limited and noisy. We propose to develop a new approach in this work, where a minimum set of basic breakup modes is chosen such that in different combinations, it can produce all possible PSDs. Genetic algorithm is being used to estimate the parameters of basic modes as it has proved accurate for simulated cases.
