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.
All Science Journal Classification (ASJC) codes
- Chemical Engineering(all)
- Inverse problem
- Non-Linear population balance model
- Non-first order effects
- Non-linear optimization