D. V. Khakhar

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...

A case study of chaotic mixing in deterministic flows: The partitioned-pipe mixer

Abstract We exploit the connection between the kinematics of mixing and the theory of dynamical systems. The presentation takes the form of a case study of a novel continuous flow mixer—the partitioned-pipe mixer—to exemplify the application of theoretical concepts relating fluid mixing in deterministic chaotic systems. Two general points are stressed: firstly, the complexities that are invariably encountered during the course of analysis limit the detail to which it may be carried out, and secondly, naive analysis based on direct use of the theory may result in misleading conclusions. Starting with an approximate Stokes flow velocity field in the partitioned-pipe mixer, we study the mixing in terms of the flow patterns in the cross-section (Poincare sections) and their relation to the conventionally used continuous mixing diagnostic, the residence time distribution, as well as to the local specific rate of stretching of material lines and the mixing efficiency. Some of the limitations of each of these methods of characterizing the mixing are exposed; however, together they provide a broad description of the mixing in the partitioned-pipe mixer, and indicate the utility of the theory. The applications of the ideas, within and outside chemical engineering, are many. The most obvious are the mixing of viscous liquids (such as molten polymers), the design of mixing devices for shear sensitive molecules and cells under non-turbulent conditions, prototype models of porous media, enhanced mass transfer devices, etc. Other applications can be expected in geophysics, environmental fluid mechanics, and condensed matter and plasma physics.

Analysis of chaotic mixing in two model systems

We study the chaotic mixing in two periodic model flows, the ‘tendril–whorl’ flow and the ‘Aref-blinking-vortex’ flow, with the objective of supplying evidence for the primary mechanisms responsible for mixing in two-dimensional deterministic flows. The analysis is based on tools of dynamical systems theory but it is clear that the mixing problem generates several questions of its own: low periodic points and horseshoes dominate the picture, since we want to achieve mixing quickly; Poincare sections, popular in dynamical systems analyses, might give misleading information with regard to dispersion at short times. Our analysis shows that both flows are able to stretch and fold material lines well below the lengthscale of the flows themselves. The inner workings of the two systems are revealed by studying the local and global bifurcations. Computations for the blinking-vortex system indicate the existence of an optimum period at which the average efficiency is maximized, whereas the intensity of segregation – a classical parameter in mixing studies – decays rapidly to an asymptotic value in the globally chaotic region. Even though our flows are not turbulent the results might have some implications for pointing to the limits of similar studies in actual turbulent flows (e.g. line stretching).

Chaotic mixing in a bounded three-dimensional flow

Even though the first theoretical example of chaotic advection was a three-dimensional flow (Henon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincare section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.

Mixing and Dispersion of Viscous Liquids and Powdered Solids

Publisher Summary Mixing and dispersion of viscous fluids—blending in the polymer processing literature—is the result of complex interaction between flow and events occurring at drop length-scales: breakup, coalescence, and hydrodynamic interactions. Similarly, mixing and dispersion of powdered solids in viscous liquids is the result of complex interaction between flow and events—erosion, fragmentation and aggregation—occurring at agglomerate length scales. Important applications of these processes include the compounding of molten polymers and the dispersion of fine particles in polymer melts. The modeling of mixing processes has undergone exciting progress in the past few years. Computations have reached maturity and exploitation of concepts and results in the context of realistic devices is now a reality. However, the advances have been restricted to single-phase fluids. Further research is needed to gain advancements in the dispersion of solids and liquids in viscous flows.

Computational studies of granular mixing

Particulate systems have proven difficult to probe experimentally in many instances. Simulations of granular flows, and mixing flows in particular, provide a useful means of studying particulate behavior. Mixing flows generate large scale patterns and structures which can be easily visualized. Thus, mixing studies provide a means of indirectly examining granular flows. In this paper we review recent computational studies of tumbler mixing, focusing on two very different, yet complementary, techniques: Particle Dynamics and Lagrangian Simulation. We discuss mixing in different tumbler geometries, as well as segregation and cohesive effects.

Breakup of liquid threads in linear flows

Abstract We study, theoretically, the surface-tension-driven breakup of a long filament of fluid in a general linear flow, v = L·x. By analyzing the problem in a moving frame and assuming a circular cross section we find that the flow around the filament is an axisymmetric extensional flow with a time-dependent strength, which can be calculated from the rate of rotation of the filament and a contribution to the axial velocity which varies with the azimuthal angle. The analysis of the axisymmetric time-dependent case does not appear to be overly restrictive: the asymmetric variation may be small even in the case of a simple shear flow, in which the asymmetry is the greatest among all possible linear flows, depending on the initial orientation of the filament. We present calculations for two special cases: hyperbolic extensional flow and simple shear flow. The results indicate that under similar conditions, the drop fragments produced on breakup in simple shear flow are larger than those in hyperbolic extensional flow. The predictions of the theory also compare reasonably well with some previous experimental data in hyperbolic extensional flow and simple shear flow.

AXIAL SEGREGATION OF PARTICLES IN A HORIZONTAL ROTATING CYLINDER

We report on an experimental study of the axial segregation process for particle mixture of different sizes. The effect of particle volume fraction and rotational speed on band formation is considered. A mechanism for the segregation process is presented based on which criteria for axial segregation are discussed

Mixing and segregation of granular materials in chute flows

Mixing of granular solids is invariably accompanied by segregation, however, the fundamentals of the process are not well understood. We analyze density and size segregation in a chute flow of cohesionless spherical particles by means of computations and theory based on the transport equations for a mixture of nearly elastic particles. Computations for elastic particles (Monte Carlo simulations), nearly elastic particles, and inelastic, frictional particles (particle dynamics simulations) are carried out. General expressions for the segregation fluxes due to pressure gradients and temperature gradients are derived. Simplified equations are obtained for the limiting cases of low volume fractions (ideal gas limit) and equal sized particles. Theoretical predictions of equilibrium number density profiles are in good agreement with computations for mixtures of equal sized particles with different density for all solids volume fractions, and for mixtures of different sized particles at low volume fractions (nu<0.2), when the particles are elastic or nearly elastic. In the case of inelastic, frictional particles the theory gives reasonable predictions if an appropriate effective granular temperature is assumed. The relative importance of pressure diffusion and temperature diffusion for the cases considered is discussed. (c) 1999 American Institute of Physics.