Population balance modeling offers a quantitative understanding of comminution processes at the process length-scale. The fundamental assumption of traditional population balance models for comminution processes is the first-order specific breakage rate of particles, i.e. the linearity of the process. Although this assumption has been severely criticized by many in the literature and ample experimental data have been generated invalidating this assumption, the linear theory has not been modified until recently.

Non-first-order effects have been observed in dry and wet milling of a variety of materials. Bilgili and Scarlett have classified the non-linear effects that occur during the batch milling of mono-sized fractions and binary fractions of the coarse and fines [1]. Multi-particle interactions of particles with different sizes appear to be the origin of these non-linear effects. A non-linear theory [1-3], which decomposes the specific breakage rate into a size-dependent function and population density dependent functional multiplicatively, has the capability of predicting the deviations from the linear theory.

In this paper, we present numerical simulations of batch comminution processes using the aforementioned non-linear theory. A power-law specific breakage rate function and a size-normalized form of the breakage distribution function yield self-similar size spectra when non-first-order effects are absent. In other words, particle size distributions at different milling times develop into a single curve when the size axis is normalized with the mean particle size. Interestingly, when non-first-order effects such as the slowing-down action of the relatively fine particles exist, self-similarity does not exist. As the degree of non-linearity increases, the deviation from potential self-similar spectra increases. This study suggests an alternative way of evaluating the degree of non-first-order effects from the temporal evolution of the particle size distribution with normalized size axis. The presentation will also elucidate the implications of these non-first-order effects in terms of comminution practice.

[1] E. Bilgili, B. Scarlett, "Population Balance Modeling of Non-Linear Effects in Milling Processes," Powder Technol., 153, 59–71 (2005). [2] E. Bilgili, J. Yepes, B. Scarlett, "Formulation of A Non-Linear Framework for Population Balance Modeling of Batch Grinding: Beyond First-Order Kinetics," Chem.Eng. Sci., in press (2006). [3] E. Bilgili, B. Scarlett, "Numerical Simulation of Open-Circuit Continuous Mills Using a Non-Linear Population Balance Framework: Incorporation of Non-First-Order Effects," Chem. Eng. & Technol., 28, 153–159 (2005).