Identification of the breakage rate and distribution parameters in a non-linear population balance model for batch milling

Non-linear population balance models (PBMs), which have been recently introduced due to the limitations of the classical linear time-invariant (LTINV) model, account for multi-particle interactions and thus are capable of predicting many types of complex non-first order breakage kinetics during size reduction operations. No attempt has been made in the literature to estimate the non-linear model parameters by fitting the model to experimental data and to discriminate various models based on statistical analysis. In this study, a fully numerical back-calculation method was developed in the Matlab environment to determine the model parameters of the non-linear PBM. Not only does the back-calculation method identify the parameters of complicated non-linear PBMs, but also it gives the goodness of fit and certainty of the parameters. The performance of the back-calculation method was first assessed on computer-generated batch milling data with and without random error. The back-calculation method was then applied to experimental batch milling data exhibiting non-first order effects using both the LTINV model and two separate non-linear models. The back-calculation method was able to correctly determine the model parameters of relatively small sets of batch milling data with random errors. Applied to experimental batch milling data, the back-calculation method with a two-parameter non-linear model yielded parameters with reasonable certainty and accurately predicted the slowing-down phenomenon during dry batch milling. This study encourages experimenters to use advanced non-linear population balance models along with the back-calculation method toward estimating the breakage rate and distribution parameters from dense batch milling data sets.

Graphical AbstractBy using a fully-numerical back-calculation method, this study identifies non-linear population balance model parameters for batch milling while elucidating the slowing-down phenomenon due to multi-particle interactions and discriminates various models based on statistical analysis. A two-parameter non-linear model was discriminated as the best model among the models studied.Figure optionsDownload full-size imageDownload as PowerPoint slideResearch Highlights► Breakage kinetics is inherently non-linear due to multi-particle interactions. ► Bilgili–Scarlett kinetics accounts for the non-linear effects during size reduction. ► An optimization-based back-calculation method was developed and applied to data. ► Various models were discriminated based on statistical analysis of the fitting. ► A two-parameter non-linear model was shown to be superior to the linear model.