Combined size and density segregation and mixing in noncircular tumblers

Flowing granular materials segregate due to differences in particle size (driven by percolation) and density (driven by buoyancy). For noncircular tumblers the additional interaction between the segregation mechanisms and chaotic advection complicates the physics. Experiments are performed using a bidisperse mixture of equal volumes of different sizes of steel and glass beads in a quasi-two-dimensional square tumbler. Mixing is observed instead of segregation when the denser beads are larger than the lighter beads so that the ratio of particle sizes is greater than the ratio of particle densities. This can be expressed in terms of the particle diameters and masses as $d_{\text{heavy}}∕d_{\text{light}}>{(m_{\text{heavy}}∕m_{\text{light}})}^{1∕4}$. Segregation patterns vary from a semicircular core when the fill level is below 50% to more complicated patterns including lobes and streaks for fill levels above 50%. Temporal evolution of segregated patterns is quantified in terms of a “segregation index” (based on the area of the segregated pattern) to capture both the rate and extent of segregation at different particle properties. The circular and noncircular tumblers have no significant difference in the segregation index, even though the segregation patterns differ significantly.

Figure 1

Schematic of the square rotating tumbler geometry with the coordinate system and other parameters. Streamlines are sketched as dotted curves.

Figure 2

Granular segregation in half-filled circular and square tumblers for 3 mm glass and steel beads ($D$ system) resulting in “classical” core formation for the circular tumbler and lobes for the square tumbler. Experiments are performed at 2 rpm and images are taken after 20 rotations.

Figure 3

Segregation patterns when buoyancy and percolation reinforce each other $(S+D)$ in a half-filled square tumbler. (a)–(c) for 1 mm steel beads and 2, 3, and 4 mm glass beads, respectively ($d_s∕d_g=0.5$, 0.33, 0.25); (d), (e) for 2 mm steel beads and 3 and 4 mm glass beads, respectively ($d_s∕d_g=0.66$, 0.5); and (f) 3 mm steel beads and 4 mm glass beads $(d_s∕d_g=0.75)$. Experiment is performed at 2 rpm and images are taken after 20 rotations.

Figure 4

Segregation patterns when buoyancy and percolation oppose each other $(S-D)$ in a half-filled square tumbler. 4 mm steel beads tumbled with (a) 0.2 mm glass beads $(d_s∕d_g=20)$—no segregation; (b) 1 mm glass beads $(d_s∕d_g=4)$—band of high concentration steel beads at the tumbler periphery; (c) 2 mm glass beads $(d_s∕d_g=2)$—small lobes of glass beads, a band of steel beads, and glass beads at the periphery; and (d) 3 mm glass beads $(d_s∕d_g=1.33)$—core of steel beads with a slight tendency toward lobes. Experiments are performed at 2 rpm and images are taken after 20 rotations.

Figure 5

Effect of fill level on granular segregation. Segregation patterns in the left column are for 1 mm steel and 2 mm glass beads ($S+D$ system, $d_s∕d_g=0.5$) while segregation patterns in the right column are for 3 mm steel and 2 mm glass beads ($S-D$ system, $d_s∕d_g=1.5$). Experiments are performed at 2 rpm and images are taken after 20 rotations.

Figure 6

Effect of angular velocity on granular segregation. Segregation patterns in the left column are for 1 mm steel and 2 mm glass beads ($S+D$ system, $d_s∕d_g=0.5$) while segregation patterns in the right column are for 3 mm steel and 2 mm glass beads ($S-D$ system, $d_s∕d_g=1.5$). Experiments are performed for half-filled tumbler and images are taken after 20 rotations. The angle of the tumbler with respect to horizontal varies somewhat to best display the segregation pattern.

Figure 7

Segregation index $\left(\sigma \right)$ as a function of the number of rotations $\left(N\right)$. (a) 1 mm steel beads tumbled with 1, 2, 3, and 4 mm glass beads ($d_s∕d_g=1.0$, 0.5, 0.33, 0.25); (b) 4 mm steel beads tumbled with 1, 2, 3, and 4 mm glass beads ($d_s∕d_g=4$, 2, 1.33, 1.0). $d_g=●1$, ▴ 2, ∎ 3, ◆ 4 mm.

Figure 8

Steady state segregation index as a function of the ratio of steel beads to glass beads $(d_s∕d_g)$ for both circular and square tumblers. $d_s=◆1$, ∎ 2, ● 3, ▴ 4 mm. Open symbols, circular tumbler; black symbols, square tumbler.

Figure 9

Summary of segregation regimes for different tumbler shape and different size ratios $(d_s∕d_g)$. Patterns are for half-filled tumblers and dark areas represent the regions where the concentration of steel beads is significantly higher than concentration of glass beads. Symbol below each segregation pattern represents the dominant segregation mechanism, $S$ (percolation or size); $D$ (buoyancy or density); $S+D$ or $S-D$ (percolation and buoyancy); and the final pattern shape (in italics), no segregation (Mixed), a core or lobe of glass or steel beads (Core/Lobe), streaks of steel beads (Streaks), and a core or lobe of glass beads with a band of glass beads at the periphery $\mathit{(}\mathit{\text{Core}}∕\mathit{\text{Lobe}}+\mathit{\text{Band}}\mathit{)}$.