A Discrete Population Balance Model for Particle Breakage In Dense-Phase Particulate Systems

Population balance models (PBMs) provide a quantitative understanding at the process length scale for comminution processes. Not only can PBMs serve as a quantitative tool for simulation, design, and optimization [1,2], but they can also elucidate the breakage mechanism(s) such as massive fracture, cleavage, and/or attrition [3]. Time-discrete or space-discrete PBMs are commonly used for comminution processes in which the average residence (retention) time of the particles is relatively short and the process can be described by considering a series of elementary breakage events each with a small duration of Dt or with a small length DL, respectively [1]. 

Traditional discrete linear PBMs (DL-PBMs) assume the independence of particle breakage from the surrounding particle population. However, at the particle ensemble scale, particles of all sizes mechanically interact with each other. Particles, which have a dynamically varying size distribution and inter-particle contacts, deform and transmit forces among themselves. The distribution of contact forces (stresses) evolves with the continuously changing population density and thus leads to a population-dependent breakage probability. These multi-particle interactions become dominant especially in dense-phase comminution processes. Toward accounting for these multi-particle interactions explicitly, a non-linear kinetics framework for rate-based comminution processes has been proposed [4], which decomposes the specific breakage rate into an apparent first-order breakage rate and a population-dependent functional. The functional describes different types of non-first order breakage kinetics.

In this study, using the aforementioned framework, we formulate a fully discrete, non-linear population balance model (DNL-PBM) in an attempt to analyze and model the multi-particle interactions for dense-phase comminution processes. Numerical simulations considering the breakage of binary- and poly-dispersed particle mixtures were performed to assess the predictive capability of the DNL-PBM over the traditional DL-PBM. They provided significant insight into the retardation effect of fines on the breakage probability of the coarser particles experimentally observed in particle bed compression tests. With a flexible mathematical structure able to account for multi-particle interactions explicitly, the DNL-PBM can serve as a framework to model dense-phase comminution processes. The practical applicability, accuracy, and stability aspects of the model are also discussed.

References

[1] L.G. Austin, A review: introduction to the mathematical description of grinding as a rate process, Powder Technol. 5 (1971) 1–17.

[2] A.D. Randolph, M.A. Larson, Theory of Particulate Processes, Academic Press, San Diego, 1988.

[3] E. Bilgili, R. Hamey, B. Scarlett, Nano-milling of pigment agglomerates using a wet stirred media mill: elucidation of the kinetics and breakage mechanisms, Chem. Eng. Sci. 61 (2006) 149–157.

[4] 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. 61 (2006) 33–44.