Solution approaches to the dynamic population balance modeling of a closed grinding network

Abstract

The solution of the dynamic population balance model (PBM) of a closed grinding network consisting of a mill, sump, and cyclone is problematic. The fact that the dynamic PBM model equation solution to the Inverse Problem for grinding systems is degenerate or underspecified is demonstrated. Two numerical solution approaches to the Inverse Problem are used. These are: (1) providing additional constraints on breakage selection functions or (2) performing the Arbiter-Bhrany (or other) normalization of the selection functions. Actual experimental anthracite batch grinding data is used to demonstrate the non-unique functionality of the batch dynamic mill selection and breakage functions for a real physical system. The Levenberg-Marquardt algorithm for systems of constrained non-linear equations is used to solve the batch dynamic PBM grinding equations to obtain the grinding selection and breakage rate functions. Different solutions to the same PBM transport equations are provided. The mill, sump and hydrocyclone were modeled as a CSTR operating at various retention times. Batch dynamic PBM data was used to provide the mill kinetic and breakage selection function data. Two different solutions were obtained depending on the numerical solution approach. The severity of the non-uniqueness problem for dynamic grinding is demonstrated. Each solution approach to a dynamic PBM with transport, while giving the same prediction for a single batch grinding time, gives different solutions or predictions for mill composition for other grinding times. This fact makes dynamic nodal analysis and control problematic. The fact that the constraint solution approach gives a solution may suggest that normalization for closed networks is not necessary. Differences in solutions to the PBM cannot be excused away by inaccuracies in the data used to model the grinding phenomenon.