### A Unified Approach to Mathematical Treatment of Multi-Particle Interactions during Size Reduction: Extending the Nonlinear Population Balance Model

Ecevit Bilgili, PhD, Pharmaceutical Commercialization Technology, Merck & Company, Inc., WP 78-110, P.O. Box 4, West Point, PA 19486

Breakage of particles is a complex phenomenon that is intended in size reduction processes for reducing particle size and enhancing specific surface area. As a unit operation, comminution or size reduction is widely used in a variety of industries including ceramics, composites, foods, minerals, paints and inks, pharmaceuticals, etc. The operation can be carried out under wet or dry conditions in batch or continuous modes. Among various approaches to the analysis of comminution processes, the use of population balance models (PBMs) provides a quantitative understanding at the process length scale. Not only can PBMs be used as a tool for simulation, design, and optimization, but also they can elucidate the breakage mechanism(s) such as massive fracture, cleavage, and/or attrition.

Although the traditional PBM has been described in a variety of mathematical forms (size-continuous form [1,2], size-discrete form , and matrix form ), it is always linear. The traditional model is based on the basic assumption that for particles with size x, the breakage probability p in discrete breakage events, as e.g. in roller milling, and specific breakage rate S during a non-discrete breakage process, as e.g. in ball or vibratory milling, are dependent only on the mass fraction of particles with size x at a given time and spatial location. In other words, the presence or absence of other particles with generic size z has no influence on the breakage probability or specific breakage rate of particles with size x. It is also assumed that the breakage (daughter) distribution function b is also independent of possible multi-particle interactions. These assumptions allowed researchers to find a simple yet elegant solution to the model equations. On the other hand, researchers have experimentally observed some interesting and non-intuitive breakage phenomena that cannot be explained by the traditional linear model (refer to the review in ). Gradual accumulation of fines in a batch mill is known to alter the specific breakage rate of the relatively coarse particles. Purposeful addition of fines to a particle assembly has a significant impact on the specific breakage rates. Particle bed breakage tests in the literature also show that multi-particle interactions among particles of different sizes lead to nonlinear breakage probability . Bilgili  showed that the presence of multi-particle interactions during milling processes can lead to deviations from self-similar breakage behavior. All these phenomena can be labeled as nonlinear effects and arise from the multi-particle mechanical interactions during breakage events or non-discrete breakage processes .

Bilgili and Scarlett  and Bilgili et al.  have recently proposed a nonlinear population balance model for size reduction as a rate process, which decomposes the specific breakage rate into a size-dependent apparent specific rate function k and a population density dependent functional F. In other words, the birth and death terms (constitutive relations) in the population balance equation account for the multi-particle interactions phenomenologically. The population balance equations were derived for batch and continuous milling operations in view of the non-linear constitutive relations [5,7-10], and the numerical simulation results demonstrated the predictive capability of the novel nonlinear model in explaining the experimentally observed complex breakage behavior.

Although the recent efforts by Bilgili and co-workers [5,7-10] provide a sound mathematical framework for treating the multi-particle interactions or nonlinear effects during size reduction processes, the treatment of these interactions for single or multiple breakage events has been mainly semi-empirical. Baxter et al. [6,11] extended the breakage matrix approach of Broadbent and Calcott  and developed a sound experimental methodology to quantify the degree of multi-particle interactions. However useful the breakage matrix approach is for quantifying the extent of multi-particle interactions for a given feed size distribution, its predictive capability for any possible natural feed size distribution is questionable. This is because the fundamental parameters characterizing the breakage, i.e., the apparent breakage probability parameters, the daughter distribution parameters, and the multi-particle interaction parameters are lumped in the matrix elements in some unknown manner; they are not explicitly defined. Moreover, the breakage matrix approach, as used in [6,11], suggests an infinite number of matrix elements for infinitely different feed size distributions, which, to the least, creates a philosophical modeling dilemma. As described in , clearly the breakage matrix approach is no more than a matrix representation of the fundamental population balance equation usually described in some size-discrete form. This suggests that the breakage matrix can and should be derivable from the population balance equation with nonlinear constitutive relations for the birth and death terms.

In this paper, the population balance equation is formulated for breakage events starting with the matrix representation as proposed by Bilgili et al. . After solving for the product mass fraction distribution as a function of feed mass fraction distribution and number of breakage events, we describe the nonlinear breakage matrix in terms of the apparent breakage probability matrix, daughter distribution matrix, and the multi-particle interaction matrix. For the case of single particle breakage, a simple experimental methodology is suggested for determining the elements of these matrices from a series of mono-dispersed breakage tests and natural feed breakage tests. In essence, the paper attempts to give the breakage matrix formalism [6,11] some predictive capability by deriving it from a fundamental population balance equation and to give it a more explicit mathematical structure. More importantly, a unified population modeling approach is presented for treating multi-particle interactions during both breakage events and rate processes.

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