doi:10.1016/S0032-5910(00)00390-9 
Copyright © 2001 Elsevier Science B.V. All rights reserved.
Continuum model of mixing and size segregation in a rotating cylinder: concentration-flow coupling and streak formation
D. V. Khakhar
, 
, a, Ashish V. Orpea and J. M. Ottinob
a Department of Chemical Engineering, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India
b Department of Chemical Engineering, McCormick School of Applied Science and Engineering, Northwestern University, Evanston, IL 60201, USA
Received 1 June 2000;
revised 6 July 2000;
accepted 6 July 2000
Available online 25 May 2001.
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Abstract
The effect of segregation and concentration-flow coupling on structure development in binary mixtures of different sized particles (S-systems) in a rotating cylinder is studied. The system is a prototype of tumbling mixers widely used in industry for mixing, coating and reaction. Experiments with S-systems have shown the formation of radial streaks of the small particles when the size ratio is large; however, an explanation of this phenomenon is not available. A continuum model is presented here for the flow in the layer using mass, momentum and species balance equations averaged across the layer. The stress is assumed to be a sum of the Bagnold stress and the Coulomb frictional stress; the temperature and total solids volume fraction are assumed to be uniform across the layer. We consider the case of a large difference in particle sizes so that segregation upon flow is instantaneous and a step concentration profile exists at all points in the flowing layer with the smaller particles forming the lower layer. The velocity profile is assumed to be piecewise linear with continuity of stress at the interface between the small and large particles. The model predicts the time varying velocity, layer thickness and concentration fields in the system. The predictions are compared to experimental flow visualization studies. Conditions for the formation of streaks are investigated.
Author Keywords: Granular mixing; Rotating cylinder; Size segregation
Fig. 1. Schematic view of the flow geometry. The coordinate system used in the analysis is shown.
Fig. 2. Schematic view of the assumed shape of the velocity profile in the layer.
Fig. 3. Figure showing how the surface angles change in the moving interface model.
Fig. 4. Experimental photographs showing the dynamics of streak formation at different rotational speeds. A 50% vol/vol mixture of 3 mm (dark) and 1 mm (light) beads are used.
Fig. 5. Pattern formed after five revolutions for 50% vol/vol mixture of large (dark) and small (light) beads for a cylinder rotational speed of 0.75 rpm. (a) Mixture of 3- and 1-mm beads at a fill fraction of 0.25. (b) Mixture of 3- and 2-mm beads at a fill fraction of 0.5.
Fig. 6. Streakline photographs for a 50 % vol/vol mixture of 3 mm (dark) and 1 mm (light) glass beads for a cylinder rotational speed of 0.75 rpm. (a) Co-flowing particles in the lower part of the layer show nearly complete segregation. (b) Free surface angle is significantly smaller when the smaller (light) particles fill the layer relative to when the larger (dark) particles fill the layer.
Fig. 7. Comparison of the scaled mean velocity (u/ωR) and layer thickness (δ/R) with scaled distance along the layer (x/R) at steady state of a single component system (f=1). The solid lines are predictions of the dynamic model and the dotted lines are the predictions of the steady state model of Khakhar et al. [24].
Fig. 8. Dynamic variation of the scaled mean velocity (u/ωR) and volume flux (q/ωR2) with rotation at the mid point of the layer (x=0) for a pure system starting from rest. The time to reach steady state is about 0.02 revolutions. Notice that the asymptotic scaled volume flux is q/ωR2=1/2 as predicted by theories of Rajchenbach [20] and Khakhar et al. [24].
Fig. 9. Computational results showing the evolution of a preformed radial streak with rotation of the cylinder. The parameters used correspond to a mixture of 3- and 1-mm beads, and the cylinder rotational speed is 0.75 rpm. Left: Static interface model. Notice that the streak shortens slightly and spreads with rotation. Right: Moving interface model. The streak is preserved.
Fig. 10. Time variation of the scaled velocities (uA/ωr, uB/ωr) and volume flux (q/ωR2) at the mid point of the layer (x=0) as a streak of the smaller particles (A) passes through the layer for the fixed interface model. The computations correspond to the initial condition and parameters of Fig. 9. The velocity of the smaller particles and the volume flux increases when the streak enters the layer. Graphs of the volume flux computed with an integration step (dt) smaller by a factor of 2, and with 
smaller by a factor of 10 exactly superimpose on the solid line.
Fig. 11. Variation of the scaled velocities (uA/ωr, uB/ωr) and volume flux (q/ωR2) with cylinder rotation when a streak of the smaller particles (A) passes through the layer for the moving interface model. The computations correspond to the initial condition and parameters of Fig. 9.
Fig. 12. Variation of the upper surface angle (βu) and the scaled volume flux (q(0)/ωR2) with cylinder rotation when the streak of the smaller particles (A) repeatedly passes through the layer computed using the moving interface model. The computations correspond to the initial condition and parameters of Fig. 9.
Fig. 13. Computed patterns showing the dynamics of streak formation at different rotational speeds.
Fig. 14. Variation of the lower surface angle (βl) and the average fraction in the lower part of the layer (fl) with cylinder rotation for the first two half rotations of the cylinder computed using the moving interface model. The computations correspond to the initial condition and parameters of Fig. 13 for the 0.75-rpm case.
Fig. 15. Computed patterns formed after five revolutions for a cylinder rotational speed of 0.75 rpm. (a) Mixture of 3- and 1-mm beads at a fill fraction of 0.25. (b) Mixture of 3- and 2-mm beads at a fill fraction of 0.5.
Table 1. Measured static (βsA, βsB) and dynamic (βA, βB) angles of repose for small (A) and large (B) particles in a cylinder of diameter 32 cm. The dynamic angles are measured at a rotational speed of 0.75 rpm
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