Population Balance Modeling of Environment Dependent Breakage: Role of Granular Viscosity, Density and Compaction. Model Formulation and Similarity Analysis

Abstract

In this work, we introduce mechanistic considerations in nonlinear milling of particulate material. We show that the mobility of the powder can have a significant effect on the breakage frequency. During grinding, the change in particle size leads to changes in the mobility of the powder which in turn alters the breakage rate. The mechanistic modeling, being able to accommodate different responses of powder mobility for wet and dry grinding, is able to distinguish naturally between dry and wet grinding, an attribute not possessed by other nonlinear models. It is shown that the rich variety of phenomena observed in grinding mills emanates from the interplay between opposing factors that influence the breakage environment in a mill. Such factors include viscosity, density, and compaction of the powder. For dry grinding, the change in powder porosity and viscosity leads to a variety of nonlinear effects. For wet grinding the viscosity and density of the slurry impart a nonmonotonic variation in the breakage rate with particle size. A population balance equation (PBE) for environment dependent breakage has been formulated by considering the foregoing effects. Similarity analysis of the model equation shows the existence of self-similar behavior even for the environment dependent breakage. Experimental investigation of such self-similarity provides an effective diagnostic process for validating model framework. Furthermore, when combined with the methodology of solving inverse problems (e.g., Sathyagal, A. N.; Ramkrishna, D. Chem. Eng. Sci. 1996, 51, 1377, an effective approach becomes available for model identification.

Homotopy perturbation method and its convergence analysis for nonlinear collisional fragmentation equations

“…Linear fragmentation results from external forces or internal stresses acting on the particles, while nonlinear fragmentation is additionally driven by collisions or interactions among particles, as illustrated in figure 1. Nonlinear fragmentation mechanisms play a crucial role in various physical and engineering processes, such as the bulk distribution of raindrops [1,2], communication systems [3,4], fluidized beds [5], and various types of crushing and milling processes [6,7]. One specific form of the nonlinear process is the pure binary collisional fragmentation , where we assume that collisions between two particles are inelastic and instantaneous, resulting in the fragmentation of either one or both of the colliding particles.…”

Experimental Characterization and Mathematical Modeling of Breakage of Needle-like Crystals in a Continuous Rotor-Stator Wet Mill

“…The main advantage of this approach is the possibility of achieving a good compromise between simplicity and detailed characterization of the fundamental mechanisms. Several examples of this model, as well as of correlations for the description of the kinetic parameters in the population balance equation, are available in the literature. , Currently most of the models available are not considering the evolution of shape of crystals during breakage, also due to the limited amount of suitable enhanced imaging devices commercially available. Nevertheless, in the past few years, several efforts have been made to characterize particle morphology by means of other descriptors, such as roundness, or to improve the daughter distributions to better characterize fragments distribution. − …”

An existence‐uniqueness result for the pure binary collisional breakage equation

“…Therefore, the reaction rate of this is taken to be proportional to the numbers of particle per unit volume of the colliding particles . Nonlinear breakage has significant impact in many engineering processes such as fluidized beds, the bulk distribution of raindrops, comminution systems, and several types of milling process . Binary collisional breakage is the specific case of the nonlinear breakage, one where the breaking behavior is influenced through the sole of collisions between two particles.…”

“…1 Nonlinear breakage has significant impact in many engineering processes such as fluidized beds, [2][3][4][5] the bulk distribution of raindrops, [6][7][8][9] comminution systems, [10][11][12][13] and several types of milling process. [14][15][16][17] Binary collisional breakage is the specific case of the nonlinear breakage, one where the breaking behavior is influenced through the sole of collisions between two particles. In this work, pure binary collisional breakage model is considered to differentiate it from the general nonlinear breakage model that allows some mass (or energy) redistribution between two colliding particles and might thus produce particles that are larger than the two colliding ones.…”