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.

[1]  J. Williams,et al.  The segregation of particulate materials. A review , 1976 .

[2]  Rajchenbach.  Flow in powders: From discrete avalanches to continuous regime. , 1990, Physical review letters.

[3]  S. Gupta,et al.  Axial segregation of particles in a horizontal rotating cylinder , 1991 .

[4]  L. Fan,et al.  Recent developments in solids mixing , 1990 .

[5]  L. T. Fan,et al.  ANNUAL REVIEW Solids Mixing , 1970 .

[6]  G. A. Kohring.  Studies of diffusional mixing in rotating drums via computer simulations , 1995 .

[7]  I. Inoue,et al.  Motion of Particle and Mixing Process in a Horizontal Drum Mixer , 1970 .

[8]  Gerald H. Ristow,et al.  Radial segregation in a two-dimensional rotating drum , 1997 .

[9]  J. K. Brimacombe,et al.  Experimental study of transverse bed motion in rotary kilns , 1983 .

[10]  Douglas W. Fuerstenau,et al.  Diffusional mixing in an ideal system , 1966 .

[11]  J. Ottino.  The Kinematics of Mixing: Stretching, Chaos, and Transport , 1989 .

[12]  Alex Hansen,et al.  Disorder and granular media , 1993 .

[13]  S. Hsiau,et al.  Granular thermal diffusion in flows of binary-sized mixtures , 1996 .

[14]  Levine,et al.  Rotationally induced segregation of granular materials. , 1994, Physical review letters.

[15]  Hani Henein,et al.  An analysis of radial segregation for different sized spherical solids in rotary cylinders , 1986 .

[16]  Baumann,et al.  Surface properties and the flow of granular material in a two-dimensional rotating-drum model. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  J. Bridgwater,et al.  Fundamental powder mixing mechanisms , 1976 .

[18]  Julio M. Ottino,et al.  Transverse flow and mixing of granular materials in a rotating cylinder , 1997 .

[19]  S. Savage,et al.  Particle size segregation in inclined chute flow of dry cohesionless granular solids , 1988, Journal of Fluid Mechanics.

[20]  P. V. Danckwerts.  The Difinition and Measurement of Some Characteristics of Mixtures , 1953 .

[21]  Julio M. Ottino,et al.  Radial segregation of granular mixtures in rotating cylinders , 1997 .

[22]  J. Bertrand,et al.  Powder mixing: Some practical rules applied to agitated systems , 1991 .

[23]  J. Ottino.  Mixing, chaotic advection, and turbulence , 1990 .

[24]  Douglas W. Fuerstenau,et al.  Transverse mixing in rotating cylinders , 1972 .

[25]  S. Hesselbo,et al.  Sequence stratigraphy in British geology , 1996, Geological Society, London, Special Publications.

[26]  Fernando J. Muzzio,et al.  Sampling practices in powder blending , 1997 .

[27]  Michael H. Cooke,et al.  Interparticle Percolation: a Statistical Mechanical Interpretation , 1979 .

[28]  F J Muzzio,et al.  Chaos, Symmetry, and Self-Similarity: Exploiting Order and Disorder in Mixing Processes , 1992, Science.

[29]  Kakalios,et al.  Reversible axial segregation of rotating granular media. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  M. Nakagawa,et al.  Non-invasive measurements of granular flows by magnetic resonance imaging , 1993 .

[31]  B. Smit,et al.  Commensurate ‘freezing’ of alkanes in the channels of a zeolite , 1995, Nature.

[32]  Volkhard Buchholtz,et al.  Simulation of rotating drum experiments using non-circular particles , 1995, cond-mat/0203589.

[33]  Suresh K. Bhatia,et al.  Axial transport of granular solids in rotating cylinders. Part 2: Experiments in a non-flow system , 1991 .

[34]  G. H. Ristow,et al.  Dynamics of granular materials in a rotating drum , 1996 .