Further Problems In The PBM Comminution Model: Analysis Of The Dynamic PBM

As the flaws in the population balance model (PBM) comminution model are examined more deeply, further errors and problems were discovered. The Levenberg-Marquardt algorithm for systems of constrained non-linear equations was used to solve the dynamic PBM grinding equations to obtain the grinding selection and breakage rate functions. The fact that the PBM model equation Inverse Problem is degenerate or underspecified is demonstrated. Multiple solutions to the same PBM equations are provided. It is shown that there is no unique solution to the Inverse Problem unless additional constraints are provided or assumptions are made such as the Arbiter-Bhrany normalization assumption. The severity of the non-uniqueness problem for dynamic grinding is demonstrated. Each solution to a dynamic PBM, while giving the same prediction for a given grinding time interval, gives different solutions or predictions for mill composition for other grinding times. Actual experimental grinding data was assessed to determine the functionality of mill selection and breakage functions.