The modeling of transverse solids motion in rotary kilns
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I.
INTRODUCTION
THE representation of transverse solids motion in a rotary kiln by a Bed Behavior Diagram (plot o f bed depth v s rotational speed or percent fill v s Froude number) has been described in a previous paper) Experimentally it was shown that the boundary between slumping and rolling beds in this diagram was a function o f particle size, particle shape, static angle of repose, kiln diameter, rotational speed, and local bed depth in a rotary kiln. This study also revealed that the slumping/rolling boundary o f given granular solids could be determined only from the boundary o f another material by comparing their static angles of repose, their slumping frequencies, or by applying the scale-up criteria: [ D ] 1/2
f O q l/2
[Fr]M[ZJM
= [Fr]e
and (pct fill)M = (pct fill)p The purpose of this paper is to provide a fundamental basis for the slumping/rolling behavior of solids beds as well as for the other modes of bed motion--slipping, cascading, cataracting, and centrifuging--in a rotary cylinder. The analysis to be described makes it possible to predict bed behavior quantitatively from measured particle characteristics and the bed/wall friction coefficients.
II.
PREVIOUS WORK
Modeling of transverse solids flow in rotary cylinders previously has focused on the characterization of the thickness o f the active layer on the bed surface, the rate and H. HENEIN, formerly Graduate Student, Department of Metallurgical Engineering, University of British Columbia, is now with the Department of Metallurgical Engineering and Materials Science, CarnegieMellon University, Pittsburgh, PA 15213 as Assistant Professor. J.K. BRIMACOMBE, Stelco Professor of Process Metallurgy, Department of Metallurgical Engineering, and A.P. WATKINSON, Professor, Department of Chemical Engineering, are both with The University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada. Manuscript submitted October 8 , 1982. METALLURGICAL
extent of transverse mixing and, using rigid-body mechanics, the conditions for slipping and centrifuging at the kiln wall. The active-layer thickness has been described mathematically for a rolling or a cascading bed,2'3 but owing to the complex nature o f particle kinematics in real systems, the models are o f limited use for predictive purposes. The primary aim o f the mixing studies 3-~2 was to develop, using statistical techniques or diffusion analogies, a coefficient of mixedness by which the mixed or unmixed state o fthe solids could be evaluated. Many of these coefficients have been proposed, but none has found wide application. CarleyMacauly and Donald5'6 and Hogg3 have studied transverse mixing of cascading solids and shown that transverse mixing is at least two orders o f magnitude faster than axial mixing. Their results as well as those o f Lehmberg e t a l4 indicate that the mixing kinetics are first order. However, using these modeling approaches little insight is gained on the mechanisms involved or on the influence o f many of the material, operating, and design variables o ft
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