J. J. McCarthy

TRANSVERSE FLOW AND MIXING OF GRANULAR MATERIALS IN A ROTATING CYLINDER

The focus of this work is analysis of mixing in a rotating cylinder—a prototype system for mixing of granular materials—with the objective of understanding and highlighting the role of flow on the dynamics of the process. The analysis is restricted to low speeds of rotation, when the free surface of the granular solids is nearly flat, and when particles are identical so that segregation is unimportant. The flow is divided into two regions: a rapid flow region of the cascading layer at the free surface, and a fixed bed of particles rotating at the angular speed of the cylinder. A continuum model, in which averages are taken across the layer, is used to analyze the flow in the layer. Good agreement is obtained between the predictions of the flow model for the layer thickness profile and experimental results obtained by digital image analysis. The dynamics of the mixing process are studied by advecting tracer particles by the flow and allowing for particle diffusion in the cascading layer. The mixing model predictions for distribution of tracer particles and mixing rates are compared qualitatively and quantitatively to experimental data. Optimal operating conditions, at which mixing rates are maximum, are determined.

Radial segregation of granular mixtures in rotating cylinders

Simultaneous mixing and segregation of granular materials is of considerable practical importance; the interplay among both processes is, however, poorly understood from a fundamental viewpoint. The focus of this work is radial segregation—core formation—due to density in a rotating cylinder. The flow regime considered is the cascading or continuous flow regime where a thin layer of solids flows along a nearly flat free surface, while the remaining particles rotate as a fixed bed along with the cylinder. The essence of the formation of a central segregated core of the more dense particles lies in the flow, mixing, and segregation in the cascading layer. The work involves experiments and analysis. A constitutive model for the segregation flux in cascading layers is proposed and validated by particle dynamics and Monte Carlo simulations for steady flow down an inclined plane. The model contains a single parameter, the dimensionless segregation velocity (β), which is treated as a fitting parameter here. Expe...

Particle dynamics simulation: a hybrid technique applied to granular mixing

Soft-sphere particle dynamics simulations have found wide use in recent years. One application for which this technique is particularly well suited is that of granular mixing. Particle properties can be varied on a particle-by-particle basis and detailed mixed structures are easily captured and visualized. While this method has proven to be quite versatile, it is computationally intensive. Current particle dynamics simulations in the literature generally handle 3000–5000 particles, whereas a typical industrial may easily contain as many as 109 particles—six orders of magnitude more. However, in certain circumstances — such as in a tumbler mixer, where the bulk of the particle motion consists of a solid body rotation — it is not necessary to explicitly calculate the motion of all of the particles. By combining particle dynamics and geometrical insight, in essence, by focusing the particle dynamics simulation only where it is needed, a new hybrid method of simulation, which is much faster than a conventional particle dynamics method, can be achieved. This technique can yield more than an order of magnitude increase in computational speed, allowing simulations of the order of 102 particles, while maintaining the versatility of a particle dynamics simulation. Two applications of this technique are presented: a tumbler operated in the avalanching regime and a continuously flowing tumbler.

Chaotic mixing of granular materials in two-dimensional tumbling mixers

We consider the mixing of similar, cohesionless granular materials in quasi-two-dimensional rotating containers by means of theory and experiment. A mathematical model is presented for the flow in containers of arbitrary shape but which are symmetric with respect to rotation by 180 degrees and half-filled with solids. The flow comprises a thin cascading layer at the flat free surface, and a fixed bed which rotates as a solid body. The layer thickness and length change slowly with mixer rotation, but the layer geometry remains similar at all orientations. Flow visualization experiments using glass beads in an elliptical mixer show good agreement with model predictions. Studies of mixing are presented for circular, elliptical, and square containers. The flow in circular containers is steady, and computations involving advection alone (no particle diffusion generated by interparticle collisions) show poor mixing. In contrast, the flow in elliptical and square mixers is time periodic and results in chaotic advection and rapid mixing. Computational evidence for chaos in noncircular mixers is presented in terms of Poincare sections and blob deformation. Poincare sections show regions of regular and chaotic motion, and blobs deform into homoclinic tendrils with an exponential growth of the perimeter length with time. In contrast, in circular mixers, the motion is regular everywhere and the perimeter length increases linearly with time. Including particle diffusion obliterates the typical chaotic structures formed on mixing; predictions of the mixing model including diffusion are in good qualitative and quantitative (in terms of the intensity of segregation variation with time) agreement with experimental results for mixing of an initially circular blob in elliptical and square mixers. Scaling analysis and computations show that mixing in noncircular mixers is faster than that in circular mixers, and the difference in mixing times increases with mixer size. (c) 1999 American Institute of Physics.

Mixing of granular materials: A test-bed dynamical system for pattern formation

Mixing of granular materials provides fascinating examples of pattern formation and self-organization. More mixing action — for example, increasing the forcing with more vigorous shaking or faster tumbling — does not guarantee a better-mixed final system. This is because granular mixtures of just barely different materials segregate according to density and size; in fact, the very same forcing used to mix may unmix. Self-organization results from two competing effects: chaotic advection or chaotic mixing, as in the case of fluids, and flow-induced segregation, a phenomenon without parallel in fluids. The rich array of behaviors is ideally suited for nonlinear-dynamics-based inspection. Moreover, the interplay with experiments is immediate. In fact, these systems may constitute the simplest example of coexistence between chaos and self-organization that can be studied in the laboratory. We present a concise summary of the necessary theoretical background and central physical ideas accompanied by illustrative experimental results to aid the reader in exploring this fascinating new area.