Inverse populations balance problems: heuristics for aggregation/breakage of solid clusters in liquid metals

Population balance models have been applied to the simultaneous aggregation and breakage of solid clusters in agitated liquid metals. Challenges in these systems lie in determination of kinetic constants (kernels) from measurements of experimental cluster concentrations. However, due to difficulties inherent to molten metal experimentation (severely limiting available data points and replication) and lack of literature concerning the physical phenomena involved, more advanced inverse methods are needed. Taking into account physical characteristics, theoretical approaches using reaction networks proved the existence of at least one stable positive equilibrium state. This result allowed the construction of two distinct fitting algorithms, aimed at solving the corresponding inverse problem (kernel determination from experimental data). The first of these heuristic methods accurately identifies kernel parameters from perfect steady-state data, while the second, based on transitory states, reliably leads to correct results with as few data points as 2 and measurement errors as high as 5%.