(209f) Granular Attrition as a Diffusion Phenomenon
Attrition of solid particles is a commonly encountered but usually undesirable occurrence in processes involving granular material. For example, attrition of particles in fluidization and pneumatic transport systems is an issue of considerable industrial concern and research efforts have been devoted to the understanding and quantification of this phenomenon. However, granular attrition is also a complex and poorly understood process for which no general theory currently exists. The most popular approach taken by a number of research workers in deriving theoretical models for granular attrition processes has been through the application of a chemical kinetics analogy. The success of this approach has been limited so far. With the advent of powerful computers in recent years, numerical simulations employing the Discrete Element Method (DEM) have also become a popular alternative to theoretical analyses and experimental investigations.
Paramanathan and Bridgwater (1983b) reported that a simple first order kinetics model where the rate of disappearance of particles in a given size interval due to breakage is proportional to the weight of particles present gives limited agreement with experimental results. This was largely due to anomalously high initial attrition rates which cannot be satisfactorily accounted for by the model. They proposed a modified first order kinetics model with three parameters where the rate of attrition depends on the deviation from attrition at infinite strain. However, it was found that this three-parameter model was not suitable for describing attrition of some of the materials tested in their annular shear cell experiments and no significant benefits in terms of modeling accuracy could be obtained even at the expense of having an additional parameter in the model. They suggested that this might be due to inherent inaccuracies in the theory used. Ayazi Shamlou et al. (1990) carried out experiments on attrition of soda glass beads in a gas-fluidized bed and proposed a model which states that the rate of attrition is first order with respect to the total initial mass of intact particles and about 0.8th order with respect to time. Cook et al. (1996) performed attrition tests in a circulating fluidized bed and selected a second-order kinetics model as one which best fit their experimental data. The model may be interpreted to describe attrition as a process whose driving force is the deviation of the weight of solids remaining in a bed raised to the second power from the corresponding squared minimum weight required for attrition to become negligible. The most successful and versatile model for describing granular attrition so far has been the empirical formulation due to Gwyn (1969) which states that the weight fraction of particles attrited is proportional to the shear strain (or equivalently, time under constant strain rate conditions) raised to a certain power. Bridgwater (1987) found the formulation to be more satisfactory for describing attrition of high density polyethylene in an annular shear cell than a first-order kinetics model. Neil and Bridgwater (1994, 1999) also found the same formulation to describe well their experimental data for attrition of molecular sieve beads, heavy soda ash and tetra-acetyl-ethylene-diamine particles in various systems such as the annular shear cell, fluidized bed and screw pugmill. However, though mostly successful, the Gwyn formulation is not without limitations. Bridgwater (1987) commented that Gwyn's formulation is incomplete in the sense that it implies an infinite initial attrition rate at zero shear strain and also allows the amount of attrited material to increase without bound at large strains. Ghadiri et al. (2000) carried out annular shear cell experiments with porous silica catalyst beads and observed deviations of their experimental results from the Gwyn formulation under some operating conditions such as at high normal stress loads.
It seems that theoretical models based on chemical kinetics analogies or their variants are limited in their capabilities to describe granular attrition processes quantitatively. It was also verified during the initial phase of this study that models derived based on more complex kinetics than those mentioned were inadequate as well. A general and coherent theory of granular attrition capable of unifying all experimental data collected to date using various types of material, systems and operating conditions is still lacking in the literature. In this study, a novel approach is proposed for the theoretical study of granular attrition by treating such processes as analogs to mass diffusion. The diffusing quantity is weight fraction of particles while the ?medium' over which this diffusion takes place is defined by the size of particles such that over time, the proportion of large particles decreases while that of small particles increases. The governing equation for granular attrition based on this analogy is necessarily the diffusion equation. The similarity method was applied to transform the governing equation into an ordinary differential equation. The transformed equation and corresponding boundary conditions were then solved analytically to give a relation between the weight fraction of particles which has undergone attrition and time. Despite its apparent simplicity and the fundamental difference between granular attrition and mass diffusion, it was found that the model derived was remarkably adequate for correlating with experimental data reported in the literature. The attrition data obtained by Paramanathan and Bridgwater (1983a, b) using granular NaCl and molecular sieve beads as the granular material in an annular shear cell and those reported by Ghadiri et al. (2000) for porous silica catalyst particles in the same type of equipment were used. The model was successful in describing quantitatively the attrition processes in terms of the weight fraction of particles attrited at various times for these different types of granular material in an annular shear cell. Similarly, attrition data obtained using gas-fluidized beds (Ayazi Shamlou et al., 1990; Cook et al., 1996; Stein et al., 1998; Kage et al., 2000) were compared with predictions made using the model and good quantitative agreement was also observed. This seems to suggest possible correspondences in statistical characteristics between granular attrition and diffusion of material which have gone unnoticed so far.
In order to further substantiate the validity of the theoretical model developed in this study, computer simulations of granular attrition during pneumatic conveying around a sharp bend using the Discrete Element Method (DEM) were carried out. The computational model used combines DEM with Computational Fluid Dynamics (CFD) for modeling the fluid phase and Ghadiri's attrition model for simulating particle fragmentation or chipping based on inter-particle or particle-wall impact velocities. It was observed that the time scale of the attrition process was different from those seen previously. In most physical experiments, attrition may be allowed to occur over long time intervals and to large extents while in the present DEM simulations, attrition due to the flow of granular material around a sharp bend occurred over a very short time interval and resulted in smaller extents of breakage. However in all cases, good quantitative agreements were obtained between theoretical predictions and experimental or simulation results, confirming the validity of the previous analysis and theoretical model derived.
Finally, it may also be shown that the present model exhibits correct asymptotic behaviors in contrast to the Gwyn formulation and may be a generalization of this empirical correlation. The results obtained in this study strongly suggest that a diffusion analogy for granular attrition is very much adequate for the theoretical study of such processes. This is a remarkable observation given the fundamental difference between the two types of phenomena. It also reveals possible similarities in statistical characteristics or fundamental properties between generally disordered systems which may well be the basis of a more general unifying theory for such systems.
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