Predicting segregation of nonspherical particles

Segregation, or demixing, of flowing mixtures of size-disperse noncohesive spherical particles is well understood. However, most particle systems in industry and geophysics involve nonspherical particles. Here, the segregation of bidisperse mixtures of millimeter-sized particles having various shapes is characterized using discrete element method simulations of gravity-driven free-surface granular heap flow. As a proxy for nonspherical particles, we study mixtures of cylindrical particles that vary widely in both their length and diameter ratios, including both disks and rods, as well as mixtures of cylindrical and spherical particles. The propensity to segregate is measured with a segregation length scale that characterizes the segregation velocity of the two species for these types of mixtures, identical to the approach for mixtures of spherical particles. Surprisingly, segregation can be predicted based solely on the volume ratio of the two species, regardless of particle shape. The segregation length scale increases linearly with the log of the volume ratio for volume ratios varying from 0.1 to 10 in the same way as it does for bidisperse mixtures of spherical particles. Thus, segregation properties based on spherical particles can be directly applied to nonspherical particles.

Figure 1

Depiction of the range of cylindrical and spherical particles sizes. The leftmost particle has length-to-diameter ratio of 9 (rodlike), and the rightmost particle has a length-to-diameter ratio of 1/9 (disklike).

Figure 2

Schematic of the 2D bounded heap with periodic spanwise boundary conditions in the $T$-wide $y$ direction. The coordinate system is rotated by the repose angle $\alpha$ and rises at the heap-rise velocity $v_r$. The flowing layer shape is idealized, and its thickness is exaggerated for purposes of illustration.

Figure 3

Heap segregation in experiment (left) and simulation (right) for validation case 1 (see Table 2) with blue disklike ($d=6.3$ mm, $l=3.2$ mm) and red rodlike ($d=3.2$ mm, $l=4.7$ mm) cylinders. Example particles and a magnified section of the sidewall are shown above each heap image. Average deposited streamwise concentration profiles are calculated in the region between the two green lines.

Figure 4

Deposited concentration at the wall of disklike particles $D1$ ($d=6.3$ mm, $l=3.2$ mm) for equal volume concentration mixtures with three different rodlike particles from experiment (solid) and DEM simulation (dashed) for validation cases (a) 1, (b) 2, and (c) 3, see Table 2. The deposited concentration for the example in Fig. 3 is shown in (a).

Figure 5

Segregation velocity, $w_{p,i}/\dot{\gamma }$, vs $1-c_i$ from DEM simulation for $d=6.35$ mm and $l=3.175$ mm cylinders (blue $\circ$) and $d=3.175$ mm and $l=4.75$ mm cylinders (red $\times$) for validation case 1, but with periodic instead of frictional sidewalls. Linear least-squares fits to Eq. (2) (lines) provide a segregation length scale value of $|S_i|=0.44$ mm. Data averaged into 0.02-wide concentration bins.

Figure 6

Nondimensional segregation length scale of the longer species $A, S_A/d_{s,eq}$, vs length ratio $R_L$ and diameter ratio $R_D$ for $R_{*}=1$. Data symbol ($\circ$) diameter is proportional to $S_A/d_{s,eq}$, and color contours are interpolated from data points. The zero segregation contour ($S=0$, black curve) corresponds closely to an equal volume ratio, $R_V=1$ (dashed line). Representative particles of species $A$ (red) and species $B$ (blue) are shown at each corner and the top of the equal-volume line.

Figure 7

$R_D, R_L$, and $R_{*}$ parameter space for mixtures of cylindrical particles where each circle represents one of the 325 simulations performed. Colored constant-ratio planes correspond to data shown in Fig. 6 (red), Fig. 8 (green), and Fig. 9 (blue).

Figure 8

Dependence of $S_A/d_{s,eq}$ on $R_L$ and $R_D$ for (a) $R_{*}=1/3$ and (b) $R_{*}=3$ (green planes in Fig. 7). Symbol ($\circ$) diameter is proportional to $S_A/d_{s,eq}$, and color contours are interpolated from data points. The zero segregation contour ($S=0$, black curve) again (as in Fig. 6) corresponds closely to an equal volume ratio, $R_V=1$ (dashed line). Representative particles of species $A$ (red) and species $B$ (blue) are shown at each corner and the top of the equal-volume line.

Figure 9

Dependence of $S_A/d_{s,eq}$ on (a) $R_{*}$ and $R_D$ for $R_L=1.5$ and (b) $R_{*}$ and $R_L$ for $R_D=0.8$ (blue planes in Fig. 7). Symbol ($\circ$) diameter is proportional to $S_A/d_{s,eq}$, and color contours are interpolated from data points. The zero segregation contour ($S=0$, black curve) again (as in Figs. 6 and 8) corresponds closely to an equal volume ratio, $R_V=1$ (dashed line). Representative particles of species $A$ (red) and species $B$ (blue) are shown at each corner.

Figure 10

Scaled segregation length scale, $S_i/d_{s,eq}$, vs volume ratio $R_V$ for data from simulations with cylindrical species mixtures indicated in Fig. 7. Symbols are semitransparent, so regions with overlapping symbols are more intense in color. Error bars correspond to the rms deviation of the fit for $S_i/d_{s,eq}$ in Eq. (2). Slope of linear least-squares fit to Eq. (10) (line) is $C_C=0.080$.

Figure 11

(a) Nondimensional segregation length scale, $S_A/d_{s,eq}$, vs $R_{L^{\prime }}$ and $R_D$ for cylindrical particle ($A$) in bidisperse mixtures of cylinders and spheres. Symbol ($\circ$) diameter is proportional to $S_A/d_{s,eq}$, and color contours are interpolated from data points. The zero segregation contour ($S=0$, black curve) again (as in Figs. 6, 8, and 9) corresponds closely to an equal volume ratio, $R_V=1$ (dashed line). Representative particles of species $A$ (cylinders, red) and species $B$ (spheres, blue) are shown at each corner. (b) $S_i/d_{s,eq}$ vs $R_V$ for bidisperse mixtures of cylinders and spheres. Error bars correspond to the rms of the fit for $S_i/d_{s,eq}$ in Eq. (2). Slope of linear least-squares fit to Eq. (10) (line) is $C_{CS}=0.084$, where the subscript $CS$ corresponds to mixtures of cylinders and spheres.

Figure 12

Nondimensional segregation length scale, $S_i/d_{s,eq}$, vs volume ratio $R_V$ for bidisperse mixtures of cylinders (red $\circ$), cylinders and spheres (blue $\square$), spheres (black $\times$) [14], and cylinders with frictional sidewalls (green $\diamond$) [11]. The linear least-squares fit to the data (line) for all types of mixtures to Eq. (10) has slope $C_{*}=0.082$.

Figure 13

Deposited concentration averaged across the span of disklike particles $D1$ ($d=6.3$ mm, $l=3.2$ mm) for equal volume concentration bidisperse mixtures with three different rodlike particles from DEM simulation (black) and from Eq. (1) using the value for $S_i$ measured directly from the simulation (solid blue) and the predicted $S_i$ from Eq. (10) (dashed blue) for validation cases 1 (a), 2 (b), and 3 (c), see Table 2.

Figure 14

DEM heap flow simulations of mixtures of spheres (blue) and cuboids (red) with (a) frictional sidewalls and (b) periodic sidewalls. Average concentration of deposited spheres $c_s$ vs streamwise position $x/L$ for (c) frictional sidewalls and (d) periodic sidewalls. Magnified particle images are shown for a portion of the flow along the sidewall (a) and along the periodic plane (b).