Inverse Problems in Population Balances: Growth and Nucleation from Dynamic Data

Particulate process modeling is critical for system design and control used widely in the chemical industly. Previous methods have focused on the assumption of appropriate models that can capture system behavior. A new technique presented is based on viewing the population balance from an inverse problem perspective that allows to determine appropriate models directly from experimental data. Under suitable assumptions (deterministic growth rate, no aggregation), the population balance equation may be solved by the method of characteristics, which associates the number density for any size at any time with a single point from the initial or boundary condition. The key to using this is the recognition that these characteristics correspond to the size history of individual particles and can be associated with constant cumulative number densities (quantiles) of the population. These quantiles are easily identifiable from experimental data. The variation of size and number density along these characteristics provides decoupled equations used to determine the growth rate. Validity of the determined growth law is checked by the collapse of the experimental data onto initial and boundary conditions.