Cluster Kinetics of Granular Mixing Benjamin J. McCoy Dept. of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803 Giridhar Madras Dept. of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India DOI 10.1002/aic.10338 Published online in Wiley InterScience (www.interscience.wiley.com). Granular mixing of identical particles that cluster together is a challenging and important engineering problem. Mixing requires the breakup of clusters, both by individ- ual particle detachment and cluster fragmentation. We apply a population balance (distribution kinetics) approach to describe such size-distributed cluster processes. Ex- pressions for mixing effectiveness and segregation measures are derived and expressed in terms of the rate coefficients for reversible cluster distribution kinetics. Analytical and numerical moment solutions illustrate how the novel method is implemented and also provide some realistic results. The method allows straightforward derivation of experi- mentally observed long-time power law or exponential asymptotic behavior of segregation metrics for various rate coefficient expressions. © 2005 American Institute of Chemical Engineers AIChE J, 51: 406 – 414, 2005 Keywords: granular mixing, cluster distribution, population balance, fluid flow, fragmen- tation, aggregation Introduction Granular mixing (of powders, sand, seeds, grains, gravel, slurries) is often accomplished by tumbling operations, 1-4 whereby loose particles slide down the inclined surface and/or particle clusters fragment and cascade down the incline (Figure 1). In granular mixing equipment, there are many variations on this theme of splitting and recombining, all with the objective of causing particles to move relative to one another and thus to mix. When interparticle attraction causes clustering, breaking up the clusters is essential to good mixing. When clustering does not occur because cohesion among particles is negligibly small, granular flows are restricted to thin regions. 5 Complica- tions arise when the particles are not uniform because subtle differences in physical properties can cause segregation, such as whether particles differ in density and size. 1 Here we con- sider identical particles, with some marked as tracers so that mixing can be assessed. Our premise is that the interactions of individual particles and particle clusters that constitute granular mixing can be described quantitatively by addressing the ki- netics and dynamics of size-distributed clusters. The paradigm of continuum mechanics, or mean field theory, cannot easily describe the dynamics of structures and hetero- geneities that underlie and strongly influence many processes, 6 including granular mixing, phase transitions, glass dynamics, polymer reactions, turbulence, and other complex systems. Generally, the structures and heterogeneities are dynamic: par- ticles, molecules, or clusters can merge or break apart. We postulate that for many such systems the underlying structural kinetics can be represented quantitatively by population bal- ance (distribution kinetics) modeling. The fundamental pro- cesses are breakage and aggregation for clusters and for the irreducible units (particles or monomers) of which they are composed. The monomers are the repeat units in polymeriza- tion or depolymerization, the molecules or ions in crystalliza- tion, and the particles in granular processes. Population dynam- ics govern the evolution in time and space for monomer concentration and cluster size distributions. The dynamics of clustering is a fundamental characteristic of granular systems. 7 The purpose of the current investigation is to ascertain how Correspondence concerning this article should be addressed to B. J. McCoy at bjmccoy@lsu.edu.