Axial band scaling for bidisperse mixtures in granular tumblers

Axial banding in rotating tumblers has been experimentally observed, but the dependence of band formation on the relative concentration of the bidisperse particles has not been thoroughly examined. We consider axial band formation and coarsening for dry and liquid granular systems of bidisperse mixtures of glass beads where the small particle volume fraction ranges from 10% to 90% in half-filled tumblers for several rotation rates. Single bands form for small particle volume fractions as low as 10% and as high as 90%, usually near the end walls. Band formation along the entire length of the tumbler is less likely at very low or very high volume fractions. After many rotations the segregation pattern coarsens, and for small particle volume fractions of 50% and greater, the coarsening is logarithmic. For very low or very high small particle volume fractions, the rate of coarsening is either not logarithmic or coarsening does not occur within the duration of the experiment (600 rotations). When bands form, the width of the band for either the small or large particles scales with the tumbler diameter.

Figure 1

The matrix of space-time portraits for both of the liquid granular systems (LGS) show band formation as a function of the fraction of small particle volume and angular velocity for 600 tumbler rotations. In each cell, time progresses from top to bottom. (a) Mixture is $0.6\mathrm{mm}$ (light) and $2.0\mathrm{mm}$ (dark) glass beads $(d_L∕d_S=3.33)$. (b) Mixture is $1.0\mathrm{mm}$ (light) and $2.0\mathrm{mm}$ (dark) glass beads $(d_L∕d_S=2)$.

Figure 2

The matrix of space-time portraits for the dry granular system (DGS) shows band formation as a function of the fraction of small particle volume and angular velocity for 600 tumbler rotations. Mixture is $0.6\mathrm{mm}$ (light) and $2.0\mathrm{mm}$ (dark) glass beads $(d_L∕d_S=3.33)$.

Figure 3

Evolution of the bands in terms of (a) the fraction of surface area of small particles and (b) the total number of bands for the LGS mixture of 0.6 and $2\mathrm{mm}$ at 10 RPM. (c) Fraction of surface area of small particles and (d) total number of bands for the LGS mixture of 1 and $2\mathrm{mm}$ at 10 RPM. Dark curves are for small particle fractions that undergo logarithmic coarsening. Volume fractions are indicated on the plots.

Figure 4

Evolution of the bandwidth and average wavelength for a LGS of 0.6 and $2.0\mathrm{mm}$ particles at a 50% small particle volume fraction is shown for 10 RPM. (a) After an initial transient, the average small and large bandwidths increase with time. (b) The average wavelength increases as a function of time after the initial transient. (c) A scatter plot of the large and small average bandwidths shows that the bandwidths are of the order of the tumbler diameter. All values are normalized by the tumbler diameter.

Figure 5

Space-time plots, the average bandwidth of small and large particles, and the wavelength depend on the relative volume fractions of small and large particles. The average small and large bandwidth are plotted against one another in the lowest row. All cases are LGS.

Figure 6

(a) Scatter plot of the average band width of small and large particles $W_{S,L}$ divided by the particle diameters, $d_S,d_L$ for all systems studied does not collapse the data. (b) The average band widths of small and large particles $W_{S,L}$ divided by the tumbler diameter, $D$, are quite similar suggesting that band widths scale with the tumbler diameter. Symbols: $d_L∕d_s=3.33$, ● LGS, ▲ DGS; $d_L∕d_s=2$: ○ LGS.