Milling is an important unit operation in the production of raw materials, specialty chemicals, and value-added products. Population balance models (PBMs) have been used as a quantitative tool to model the evolution of particle size distribution (PSD) during milling processes. Most PBMs used in the last 70 years assumed the linearity of the breakage rate, i.e., first-order breakage kinetics. In this presentation, I will present some fundamental experimental studies that provide strong evidence for the emergence of non-linear breakage: deviations from first-order breakage kinetics, significant impact of initial PSD on the breakage kinetics, anomalies in binary breakage tests, etc. The non-linear breakage appears to be prevalent in dense milling systems such as in dry ball mills where enduring mechanical multi-particle interactions occur among particles. Then, I will give highlights from recent theoretical developments that address this complex phenomenon of non-linear particle breakage. The traditional linear PBM, the time-variant PBM, and the non-linear functional model [1] have been critically analyzed in view of experimental data. Extensive numerical simulations of dry ball milling systems [2,3] and particle bed breakage experiments [4] suggest that the non-linear functional model can serve as a unified framework by which non-linear particle breakage in a multitude of comminution systems can be systematically studied and quantified at the process length scale. Finally, the presentation will highlight recent efforts in exploring the origin of non-linear breakage [5] via multi-scale DEM–PBM models and how to best use such models judiciously to predict the evolution of PSD in non-linear milling processes [6].


[1] E. Bilgili, J. Yepes, B. Scarlett, "Formulation of A Non-Linear Framework for Population Balance Modeling of Batch Grinding: Beyond First-Order Kinetics," Chemical Engineering Science, Vol. 61, 2006, pp. 33–44.

[2] E. Bilgili, B. Scarlett, "Population Balance Modeling of Non-Linear Effects in Milling Processes," Powder Technology, Vol. 153, 2005, pp. 59–71.

[3] M. Capece, E. Bilgili, R. Dave, “Identification of the Breakage Rate and Distribution Parameters in a Non-Linear Population Balance Model for Batch Milling,” Powder Technology, Vol. 208, 2011, pp. 195–204.

[4] E. Bilgili, M. Capece, “A Rigorous Breakage Matrix Methodology for Characterization of Multi-Particle Interactions in Dense-Phase Particle Breakage,” Chemical Engineering Research and Design, Vol. 90, 2012, pp. 1177–1188.

[5] M. Capece, R. N. Dave, E. Bilgili, “On the Origin of Non-linear Breakage Kinetics in Dry Milling,” Powder Technology, Vol. 272, 2015, pp. 189–203.

[6] M. Capece, R. N. Dave, E. Bilgili, “A Pseudo-coupled DEM–Non-linear PBM Approach for Simulating the Evolution of Particle Size during Dry Milling,” Powder Technology, Vol. 323, 2018, pp. 374–384.