Fluid-particle suspension by gas release from a granular bed

We have studied experimentally particle suspension when injecting a gas at the bottom of an immersed granular layer confined in a Hele-Shaw cell. This work focuses on the dynamics of particles slightly denser than the surrounding fluid. The gas, injected at a constant flow-rate, rises through the granular bed and then forms bubbles that entrain particles in the above liquid layer. The particles settle down on the edges of the cell, avalanche on the crater formed at the granular bed free surface, and are further entrained by the continuous bubbling at the center. We report the existence of a stationary state, resulting from the competition between particle entrainment and sedimentation. The average solid fraction in the suspension is derived from a simple measurement of the granular bed apparent area. A phenomenological model based on the balance between particle lift by bubbles at the center of the cell and their settling on its sides demonstrates that most of the particles entrained by bubbles come from a global recirculation of the suspension.

Figure 1

(a) Schematic view of the experimental setup. Air is injected at a constant flow rate $Q$ at the bottom of an immersed granular layer in a Hele-Shaw cell (see text). The subsequent suspension and the remaining granular bed are observed using shadowgraphy. The different notations of the geometrical parameters are indicated. On the right, images of different batches of grains (see Table 1): (b) nonspherical polydisperse polystyrene beads (PS 130P); (c) spherical monodisperse polystyrene beads (PS 250M); (d) nonspherical polydisperse PVC beads (PVC 110P); (e) spherical monodisperse PS beads (PS 80M).

Figure 2

Temporal evolution of the bed and suspension (PS 130P, $Q=200$ mL/min, $h_g=10\pm 0.2$ cm, $h_{\ell }=20\pm 0.2$ cm, $L_{\text{c}}=13.6$ cm, $e=2$ mm). (a) Initial loosely-packed bed. (b), (c) After turning on the gas injection, air rises through the granular bed and forms bubbles that entrain particles in their wake in the above liquid layer. (d), (e) A crater grows and the suspension becomes denser. (f)–(h) The system reaches a stationary state in which the volume of the granular bed and the average solid fraction of the suspension remain constant.

Figure 3

Snapshots of the experiments in the stationary regime for different values of the flow rate $Q$ and different cell width $L_c$ (PVC 110P, $h_g=9\pm 0.5$ cm, $h_{\ell }=18\pm 0.5$ cm). The thickness of the cell is $e=2$ mm for the lower panel ($L_c=13.6$ cm) and $e=3$ mm for the upper panel ($L_c=35.6$ cm).

Figure 4

(a) Temporal evolution of the normalized bed area, $A/A_0$, for different air flow-rate $Q$ (decreasing from light gray to dark gray). Experimental parameters are those given in the caption of Fig. 3 (PVC 110P, $h_g=9\pm 0.5$ cm, $h_{\ell }=18\pm 0.5$ cm, $L_{\text{c}}=13.6$ cm, $e=2$ mm). Inset: Normalized plot, $(A-A^{*})/(A-A_0)$, as a function of $t/{\tau }_c$. (b) Normalized bed area in the stationary state, $A^{*}/A_0$, as a function of the injected air flow-rate $Q$ for different cell widths ($e=2$ mm for $L_c=13.6$ and 24.0 cm and $e=3$ mm for $L_c=35.6$ cm).

Figure 5

Snapshots of the experiments just before turning on the flow rate (a) and in the stationary regime (b) (PS 130P; $Q=100$ mL/min; $L_c=13.6$ cm; $e=2$ mm; $h_g=9.3\pm 0.5$ cm; $h_{\ell }=14.4\pm 0.5$ cm). (c) Schematic view of the mass conservation model used to compute the solid fraction of the suspension. (d) Snapshot of the experiment after turning off the flow rate. (e) Solid fraction of the suspension ${\phi }_s^{*}$, computed using the mass conservation model [Eq. (3)], as a function of ${\phi }_s^{\text{**}}$, computed using the number of particles that have sedimented after turning off the flow rate for different ratio $h_{\ell }/h_g$. The error bars correspond to the estimation of the compaction of the granular bed $[{\varphi }_b^{*}=(1.05\pm 0.05){\varphi }_b^0]$. The dashed line corresponds to the first bisector.

Figure 6

(a) Sketch of the fluid-particle suspension by gas injection (phenomenological model; see Sec. 4). Gas is injected at a constant flow rate $Q$ at the bottom center of the cell and crosses the granular bed. Bubbles rising in the above liquid layer (velocity $U_b$) entrain particles upward in the central column (width $L_b$) and form a suspension of width $L_S$ that is of the order of $L_c$ in all the data discussed in Sec. 4. Particles on both sides settle due to both sedimentation and fluid recirculation (see text). (b) Picture of bubbles rising out of the granular bed entraining particles in their wake (PVC 110P; $h_g=9$ cm; $h_{\ell }=18$ cm; $Q=0.05$ L/min; $L_{\text{c}}=13.6$ cm; $e=2$ mm).

Figure 7

(a,b) Probability density function of the bubble equivalent diameter ($L_b^i$ indicates the equivalent diameter for bubble $i$) in the stationary regime. The inset displays a snapshot of the corresponding image sequence with bubble contour (blue line) and center of mass (red cross). (a) At low flow rate ($Q=0.1$ L/min), the bubble population displays a single characteristic size $L_b$. (b) At high flow rate ($Q=0.75$ L/min), small bubbles resulting from bubble fragmentation appear, with a typical size $L_b^s$. (c) Equivalent bubble diameter as a function of the flow rate $Q$. For $Q>0.3$ L/min, we observe two bubble populations [corresponding to the two peaks in (b)]. Small bubbles are roughly constant in size (horizontal dashed line) while large bubbles follow $L_b=0.38+2.9\sqrt{Q}$ (increasing dashed line) (PVC 110P; $L_c=13.6$ cm; $e=2$ mm; $h_g=9.3\pm 0.5$ cm; $h_{\ell }=18\pm 0.5$ cm).

Figure 8

Parameter $y$ as a function of ${\phi }_s^{*}$ (see text). Following the prediction of the phenomenological model, $y={\chi }_s{\phi }_s^{*}+{\chi }_b{\phi }_b^{*}$, where ${\chi }_s$ and ${\chi }_b$ depend on the nature of the particles only. (a) PS 130P for different $h_{\ell }/h_g$ (cf. color scale) ($L_c=13.6$ cm, $e=2$ mm). The solid gray line indicates the linear fit [reported in (b) and (c) for comparison], whose equation is indicated in gray in the figure. (b) PS 250M and PS 80M ($h_g=9$ cm, $h_{\ell }=18$ cm, $L_c=13.6$ cm, $e=2$ mm). Dashed black (orange) line: linear fit for the PS 250M (PS 80M) particles. The fit equations are indicated in black (orange) in the figure. (c) PVC 110P ($h_g=9$ cm, $h_{\ell }=16$ cm). All data collapse independently of the cell width $L_c$ (blue dotted line: linear fit, whose equation is indicated in blue) $(e=2$ mm for $L_c=13.6$ and 24 cm and $e=3$ mm for $L_c=35.6$ cm).

Figure 9

Snapshots of the experiments in the stationary regime for different granular bed heights $h_g$ and liquid heights $h_{\ell }$ (PS 130P; $L_c=13.6$ cm; $e=2$ mm; $Q=200$ mL/min).

Figure 10

Snapshots of the experiments in the stationary regime for different values of the flow rate $Q$ and the two different batches of monodisperse particles $(L_c=13.6$ cm; $e=2$ mm; $h_g=9\pm 0.5$ cm; $h_{\ell }=18\pm 0.5$ cm).