(261h) Mathematical Modeling of Continuous Milling Processes Using a Cell-Based PBM Approach: A Parametric Study

Particles with a high specific surface area usually exhibit desirable functional responses, such as high reactivity, fast dissolution, and good content uniformity. For this reason, various industries usually increase the specific surface area of particles, like pharmaceuticals, pigments, and minerals, using milling processes that decrease the size of particles for downstream processing or end use. However, current milling technologies are both energy-intensive and energy-inefficient [1]. Therefore, designing a milling process that can operate at or near an optimal condition would benefit the overall performance, but finding optimal milling operational conditions by trial and error is costly. For this reason, computer simulations of the process could provide significant operational insights while enhancing understanding of the milling process, thereby aiding in rational process design and optimization at a minimum cost.

Several modeling approaches, such as the discrete element method and the population balance model (PBM), have attempted to develop a fundamental understanding of the milling processes. Among these approaches, only the PBM provides a quantitative analysis of the spatio-temporal evolution of the particle size distribution (PSD) during milling. In this theoretical study, we developed a cell-based PBM that takes into account the complex nonlinear breakage kinetics and the degree of mixing in an open-circuit continuous dry mill with a discharge screen, unlike previous studies [2–6]. In the context of dry milling, the proposed mathematical model is the first in literature that accounts for (i) the elusive nonlinear breakage kinetics that originate from the multi-particle interactions, (ii) a non-ideal degree of mixedness, and (iii) particle separation via an ideal discharge screen. Various back-mixing ratios (R) and number of cells (n) were used to emulate different extents of particle mixing in the mill including the idealized cases of perfect mixing and no back-mixing (plug flow). Other simulations were performed to examine the impact of cushioning action of fine particles and the screen size. The simulation results were reported in terms of cumulative PSD at steady-state and during the transient period. Apart from the PSD of the product at the mill outlet, spatial variations of the PSD inside the mill were also examined.

For continuous mills without a screen, the cell-based PBM with n = 1 exactly emulated the PBM solution of an idealized perfect mixing scenario, whereas the cell model with n = 60 nearly represented the ideal plug-flow (no back-mixing) scenario, with an increase in n converging to the PBM solution for the plug-flow obtained previously in the literature. These findings demonstrated the consistency of the cell-based PBM. The simulation results also suggest that an increase in the number of cells and/or a decrease in the back-mixing ratio, emulating lower extent of back-mixing, led to finer product. While the cushioning action of finer particles led to coarser product PSD in the absence of a screen, this nonlinear impact becomes less important when a screen is present and especially when the screen opening size is small. This study has demonstrated for the first time in literature that the use of a discharge screen with a fine opening not only allows for passage of the desired fine particles and retention of the relatively coarse particles, but also it reduces the retardation impact of the fines on breakage kinetics, thus ultimately resulting in a finer product PSD.

References

[1] B.A. Wills, An introduction to the Practical Aspects of Ore Treatment and Mineral, Butterworth-Heinemann, Oxford, 2016.

[2] L.G. Austin, P.T. Luckie, D. Wightman, Steady-state simulation of a cement-milling circuit, Int. J. Miner. Process. 2 (1975) 127–150.

[3] H. Benzer, L. Ergun, M. Oner, A.J. Lynch, Simulation of open circuit clinker grinding, Miner. Eng. 14 (2001) 701–710.

[4] 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 (2005) 153–159.

[5] O. Genc, S.L. Ergun, A.H. Benzer, The dependence of specific discharge and breakage rate functions on feed size distributions, operational and design parameters of industrial scale multi-compartment cement ball mills, Powder Technol. 239 (2013) 137–146.

[6] C. Mihalyko, T. Blickle, B.G. Lakatos, A simulation model for analysis and design of continuous grinding mills, Powder Technol. 97 (1998) 51–58.