Segregation of combined size and density varying binary granular mixtures in a slowly rotating tumbler

Abstract

Granular mixtures composed of particles which differ in size or density readily segregate when there is relative motion between particles, such as in a slowly rotating tumbler. In the past, several studies have considered segregation due to different sized particles or different density particles. The physical mechanisms that lead to segregation are different in these two cases. Few studies have considered the important (practical) situation where particles differ both in size and density, which is what we consider here. Depending on particle sizes and densities the two mechanisms either oppose or enhance each other. We consider both cases here and find when the two mechanisms oppose each other size segregation is generally stronger than density segregation. We also determine stability regions for the various patterns that form. When the mechanisms enhance each other we find, under some specific conditions, streak patterns form. We argue that one requires a bed microstructure which fluctuates in strength which subsequently leads to avalanching. We find stable streaks only occur when the smaller, denser particles are blocky in shape while the larger, less dense particles are spherical.

Appendix: Segregation in S \({-}\) D systems for \(r_s=1.5, 2.0, 3.0\)

In this Appendix we describe the simulation results for S ${-}$ D systems for various $r_s$ values (1.5, 2.0 and 3.0).

1.1 Segregation for $r_s=1.5$

Figure 10 shows the particles distributions for a size ratio of 1.5, with a similar shading scheme to Fig. 2. Particle distributions for density ratios of 2, 3 and 4.38 are given in Fig. 10a, b and c respectively. They generally show a higher concentration core of smaller particles (light grey) and a higher concentration outer corona of large particles (dark). The radial distribution function for $r_d$ = 2 and 3 (Fig. 11) show a maximum volume fraction for smaller particles of about 0.82 and 0.77 at a radius of 0.3. For $r_d=4.38$, this maximum has decreased to around 0.75 and there is an upturn in the small particle volume fraction curve closer to the cylinder walls. One could argue for $r_d=4.38$ we are close to a mixed pattern, however based on the centroid curves (Fig. 12c) which are quite similar to $r_d=3$ (Fig. 12b) we classify both these density ratios as S-Core patterns.

Fig. 10

Particle distributions, taken at $z=0$, for a size ratio of 1.5 (dark spheres correspond to larger particles and lighter spheres correspond to smaller particles): a $r_d=2.0$ (top-left) b $r_d=3.0$ (top-right) c $r_d=4.38$ (bottom-left) d $r_d=6.0$. (bottom-right)

Fig. 11

Radial distributions for a size ratio of 1.5 (dark curves correspond to larger particles and lighter spheres correspond to smaller particles) from top to bottom: a $r_d=2.0$ b $r_d=3.0$ c $r_d=4.38$ d $r_d=6.0$

Fig. 12

Centroid curves for a size ratio of 1.5 (dark curves correspond to larger particles and lighter spheres correspond to smaller particles) from top to bottom: a $r_d=2.0$ b $r_d=3.0$ c $r_d=4.38$ d $r_d=6.0$

The particle distribution for a density ratio of 6 (Fig. 10d) shows a higher concentration of smaller particles (light grey) near the cylinder walls, while in the core region there appears higher concentrations of the smaller particles (light grey). A band of dark particles separate the core and corona. The radial distribution plot (Fig. 11d) shows that the region near the cylinder wall is almost completely small particles (with a volume fraction of 1). In the core region there is a maximum volume fraction for the small particles of about 0.62, which is not large. The centroid curves for this density ratio (Fig. 12d) tend to be quite close (especially near the end of the simulation). Given these measures, it appears most appropriate to classify this pattern as Core-and-band.

In summary for this case we find (on increasing density) the change in segregation pattern is: S-Core, S-Core, S-Core, Core-and-band.

1.2 Segregation for $r_s=2.0$

Figure 13 shows the particle distribution for a size ratio of 2, with a similar shading scheme to Fig. 2. For this size ratio, the percolation effect is relatively strong and appears to overcome the buoyancy mechanism for the smaller two density ratios of 2 and 3. In both these cases (Fig. 13a, b) there is a higher concentration of smaller particles (light grey) in the core region and a higher concentration of larger particles (dark) close to the cylinder container. This observation correlates with the radial distributions (Fig. 14a, b) where we see the maximum volume fraction of the smaller particles is above 0.8 in both cases at a scaled radius of approximately 0.2, while the maximum volume fraction for the large particles is also above 0.8 adjacent to the cylinder walls. Thus the radial distributions indicate for $r_d = 2$ and 3 that the percolation mechanism is in control of segregation leading to a small particle core (S-Core). The centroid curves (Fig. 15a, b) corresponding to smaller particles (light grey) are mostly negative, which indicates their centroid is close in to the center of the cylinder for most of the simulation. The separation between the large and small particle centroid curves in both cases is large (between 0.1 and 0.15) indicating good segregation.

Fig. 13

Particle distributions, taken at $z$ = 0, for a size ratio of 2.0 (dark spheres correspond to larger particles and lighter spheres correspond to smaller particles): a $r_d=2.0$ (left-top) b $r_d=3.0$ (right-top) c $r_d=4.38$ (left-bottom) d $r_d=6.0$ (right-bottom)

Fig. 14

Radial distributions for a size ratio of 2.0 (dark curves correspond to larger particles and grey curves correspond to smaller particles) from top to bottom: a $r_d=2.0$ b $r_d=3.0$ c $r_d=4.38$ d $r_d=6.0$

Fig. 15

Centroid curves for a size ratio of 2.0 (dark curves correspond to larger particles and grey curves correspond to smaller particles) from top to botttom: a $r_d=2.0$ b $r_d=3.0$ c $r_d=4.38$ d $r_d=6.0$

For the density ratio of 6 (Fig. 13d), we see a larger concentration of small particles close to the cylinder walls and also closer to the center (core region). The larger particles are more homogeneously distributed throughout. The radial distribution plot for this case (Fig. 14d) gives a relatively small maxima (for the volume fraction of smaller particles) at a scaled radius of 0.2 of about 0.7 and a larger volume fraction of 0.8 (for the smaller particles) close to the cylinder walls. The centroid curve for this case (Fig. 15d) shows overlapping of the two curves throughout the simulation, indicating the particles are not segregated. It is difficult to definitively classify this case, based on all the measures, but it is either a Core-and-band pattern or a mixed pattern. Based on the radial distribution, we classify it as a Core-and-band pattern.

The density ratio of 4.38 (Fig. 13c) appears to be predominantly a core of small (light grey) particles, although there are larger particles distributed in the inner core region. The radial distribution function (Fig. 14c) takes the shape of small particle core with a maximum volume fraction (for the smaller particles) of about 0.75. The centroid curve for this case (Fig. 15c) shows large oscillations in the smaller particle curve and the separation between the curves tends to vary quite a lot (from zero up to 0.12). Thus at this density ratio the percolation mechanism and buoyancy mechanism seem to be competing quite closely, although the percolation mechanism is slightly stronger leading to an S-Core pattern.

In summary for this case we find (on increasing density) the change in segregation pattern is: S-Core, S-Core, S-Core, Core-and-band.

1.3 Segregation for $r_s=3.0$

The particle distributions for a size ratio of 3 are given in Fig. 16, with a similar shading scheme to all previous cases. For density ratios of 2 and 3, both particle distributions (Fig. 16a, b) show a high concentration inner core of smaller particles (light grey) and a concentrated outer corona of larger particles (dark). (Note, there appears to be a slight asymmetry in the distribution of dark particles in these figures. This is because, as the size ratio gets larger the number of large particles decreases and hence at a particular instant the distribution of these large particles may not appear as radially symmetric as for the smaller size ratios.) In both cases the radial distribution plots (Fig. 17a, b), have typical size-segregation curves. That is a maximum volume fraction (for smaller particles) at a small radius (around 0.3 or so) above 0.8. At large radii, the large particle curve has a maximum of 0.8–0.9. The centroid curves for these two density ratios (Fig. 18a, b) also show that the smaller particles are found closer in to the center (hence they are negative) while the large particles are found further out. The separation between these curves is also quite large at 0.1–0.15. Hence these two density ratios can be classified as S-Core patterns.

Fig. 16

Particle distributions, taken at $z=0$, for a size ratio of 3.0 (dark spheres correspond to larger particles and lighter spheres correspond to smaller particles): a $r_d=2.0$ (left-top) b $r_d=3.0$ (right-top) c $r_d=4.38$ (left-bottom) d $r_d=6.0$ (right-bottom)

Fig. 17

Radial distributions for a size ratio of 3.0 (dark curves correspond to larger particles and grey curves correspond to smaller particles) from top to bottom: a $r_d=2.0$ b $r_d=3.0$ c $r_d=4.38$ d $r_d=6.0$

Fig. 18

Centroid curves for a size ratio of 3.0 (dark curves correspond to larger particles and grey curves correspond to smaller particles) from top to bottom: a $r_d=2.0$ b $r_d=3.0$ c $r_d=4.38$ d $r_d=6.0$

The next density ratio of 4.38 is more difficult to classify. It appears from the particle distribution (Fig. 16c) that the larger particles (dark) are more or less distributed throughout the particle bed, although there appears to be slightly more large particles closer to the cylinder wall on the left side. The radial distribution plot (Fig. 17c) yields a maximum volume fraction (for smaller particles) of 0.75, while the maximum for large particles is 0.7 (adjacent to the cylinder walls). The centroids (Fig. 18c) curves tend to overlap (with a large oscillation amplitude in the small particle centroid curve) for significant durations throughout the simulation. So while the radial distribution indicates a weakly segregated S-Core, the centroids indicate a mixed pattern (since the centroids overlap). In this case, we shall classify this pattern as transitional, since it is not definitively one or the other.

The particle distribution for the final density ratio of 6 shows large particles (Fig. 16d) distributed throughout the particles bed. The radial distribution function has a very small volume fraction maximum (for the smaller particles) at a scaled radius 0.15 of 0.62. Further out (close to the cylinder walls) the small particle volume fraction increases up to 0.75. Based on this, one may argue a Core-and-band pattern, even though the maxima are small. On the other hand, the centroid curves for this density ratio (Fig. 18d) overlap throughout the simulation, with no indication of any deviation from overlapping. This might indicate a mixed particle bed. Since the two measures are not definitive we classify this case as transitional. For the last two density ratios ($r_d=4.38$ and 6), the average position of each centroid curve does not deviate significantly throughout the simulation indicating an insignificant amount of segregation. Conversely the radial distribution plots indicate (marginally) S-Core or Core-and-band patterns. Because of this conflict between measures, we have classified both as transitional patterns.

In summary, for this case, we find (on increasing density) the change in segregation pattern is: S-Core, S-Core, transitional and finally transitional.