Abstract
The population balances offer a continuum framework to model fragmentation of discrete size drops, bubbles, particles, crystals, polymer chains, etc. We have examined the available approaches to solve the resulting population balances and the older discrete kinetic equations. The analytical solutions have enhanced our understanding of breakup processes. We offer in this work new analytical solutions using a fresh and counter-intuitive approach. The approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We looked for scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. The approach provides analytical solutions for a family of breakup functions, including evolving populations of infinite number of particles. The success of the new approach is contingent on the discretization step having internal consistency concerning particle number and mass. The approach holds promise for empirical breakage functions.
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