Silo outflow of soft frictionless spheres

Outflow of granular materials from silos is a remarkably complex physical phenomenon that has been extensively studied with simple objects like monodisperse hard disks in two dimensions (2D) and hard spheres in 2D and 3D. For those materials, empirical equations were found that describe the discharge characteristics. Softness adds qualitatively new features to the dynamics and to the character of the flow. We report a study of the outflow of soft, practically frictionless hydrogel spheres from a quasi-2D bin. Prominent features are intermittent clogs, peculiar flow fields in the container, and a pronounced dependence of the flow rate and clogging statistics on the container fill height. The latter is a consequence of the ineffectiveness of Janssen's law: the pressure at the bottom of a bin containing hydrogel spheres grows linearly with the fill height.

Figure 1

Experimental setup: The small panels at the right (a),(b) show details of the hexagonal lattice formed by the soft spheres far from the container side walls in the upper half of the bin (a), and directly above the outlet (b); the positions are indicated by dashed squares. Panel (c) shows a detail of (a), where we have marked the contact areas of spheres with the front glass wall, as a measure of the force exerted on the beads by the front wall.

Figure 2

(a) Outflow through different orifice sizes. When $W>2.2D$, no clogging is observed. With decreasing orifice size, the discharge rate decreases, the fluctuations of the outflow velocity increase, and intermittent clogs occur. At a certain fill height, the system clogs permanently [horizontal lines in the $m\left(t\right)$ discharge characteristics]. (b) Remaining fill level $h$ in the permanently clogged state for different outlet sizes $W/D$. A satisfactory linear fit yields $h_c=(2.2-W/D)\times 95$ cm, or, a critical pressure $p_c=(2.2-W/D)\times 6.6$ kPa. The bar at $W=1.83D$ indicates the standard deviation of the measurements. For the pressure scale at the right-hand side, see Fig. 4.

Figure 3

Plot of the probability $p_h$ that the system permanently clogs at a remaining fill height larger than $h$ in a silo with an orifice size of $W=1.83D$ [see Fig. 2]. The graph was constructed from 26 independent runs of the experiment. The mean height where the system clogged is 29 cm, the standard deviation of clogging heights is 5 cm. The solid hyperbolic tangent curve guides the eye.

Figure 4

Static pressure $p_0$ at the container bottom as a function of the fill height in the center of the bottom (solid symbols) and near the side of the silo bottom (open symbols). Different colors symbolize independent runs of the experiment. The scattering of the data reflects natural variations of the individual experiments. The experimental error is smaller than the size of the symbols. The pressure at the bottom of the container filled with hydrogel spheres increases with fill height as $p\left(h\right)\approx h\times 7$ kN/${\mathrm{m}}^3$. Pressure measurements for peas and airsoft balls (details see text) are shown for comparison.

Figure 5

Flow profiles (mm/s) at an orifice size $W=60$ mm ($W/D=6.5$) in the completely filled silo ($h=80$ cm): (a) horizontal flow, the yellow (bright) region indicates flow to the right, the dark (blue) region to the left, green indicates zero horizontal flow; (b) vertical flow. The images show a 23 cm $\times$ 34 cm region of the container above the orifice.

Figure 6

Flow profiles at orifice sizes $W=60$ mm and 20 mm in the completely filled silo, $h=80$ cm. In levels $y$ higher than $\approx 2.5W$ above the orifice, the vertical flow profile becomes smooth. In the upper part, we find simple plug flow. (Only 20 cm of the full bin width is shown.)

Figure 7

Comparison of the flow rates vs $W/D$ for soft hydrogel spheres and hard airsoft balls in the completely filled silo, $h=80$ cm. It is evident that both can be fitted with Eq. (1), but the soft particles show a substantially higher flow rate at given relative orifice sizes, and the coefficient $k=0.3$ for the soft spheres is extremely small.