Fraction of Clogging Configurations Sampled by Granular Hopper Flow

We measure the fraction $F$ of flowing grain configurations that precede a clog, based on the average mass discharged between clogging events for various aperture geometries. By tilting the hopper, we demonstrate that $F$ is a function of the hole area projected in the direction of the exiting grain velocity. By varying the length of slits, we demonstrate that grains clog in the same manner as if they were flowing out of a set of smaller independent circular openings. The collapsed data for $F$ can be fit to a decay that is exponential in hole width raised to the power of the system dimensionality. This is consistent with a simple model in which individual grains near the hole have a large but constant probability to precede a clog. Such a picture implies that there is no sharp clogging transition, and that all hoppers have a nonzero probability to clog.

Figure 1

Average mass $⟨m⟩$ discharged before a clog forms, illustrated as a function of (a) hole diameter $D$ and (b) $D^3$ for $d=2\mathrm{mm}$ glass beads in hoppers with different tilt angles, $\theta$. These data are from Ref. [23], and are well fit by a variety of functions as labeled. Vertical error bars are based on measured $m$ values for 10 or more discharge events; horizontal error bars reflect uncertainty in $D$.

Figure 2

(a) $\rho A_{\text{eff}}d/⟨m⟩$ versus $(A_{\text{eff}}-A_g{)}^{3/2}$ for tilted hoppers, where $\rho$ is the bulk density of the grains, $d=2\mathrm{mm}$ is the grain diameter, $⟨m⟩$ is the average mass discharged before clogging, $A_{\text{eff}}$ is the effective hole area for a tilted hopper, and $A_g$ is the cross-sectional grain area. According to Eq. (1), the quantity plotted is equal to $Fd/\ell$, where $\ell$ is the sampling length. (b) Sampling length $\ell /d$ versus tilt angle $\theta$, as found from fits as displayed in part (a). The average values of $⟨\ell ⟩$ for linear and exponential fits are displayed; combining these, we henceforth take the sampling length as $\ell =(0.75\pm 0.20)d$.

Figure 3

Fraction of configurations that precede a clog, deduced from measurements of $⟨m⟩$, versus the normalized effective hole area. In (a), data points for circular holes are shaded by $\widehat{\mathit{v}}·\widehat{\mathit{n}}$. In (b), data points for hole-equivalent subsections of rectangular slits are shaded by slit aspect ratio $L/D$. In both the grain sizes are indicated in the legend. The clogging transition locations found from critical power law fits in Ref. [23] for the 2 mm glass spheres are indicated by $A_{\text{eff},\mathrm{c}}/A_g$ at bottom right. The best fits to Eq. (4) (upper) and Eq. (4) (lower) are overlaid as dashed and solid curves, respectively.

Figure 4

Ratio of the stable to total single-grain microstates versus normalized effective hole area, obtained for all the data shown in Fig. 3 by computing $F^{1/N}$ where $N=(\widehat{\mathit{v}}·\widehat{\mathit{n}})(D/d{)}^3$. For moderately large holes, $V_{pc}/V_1$ falls to a constant value $0.87\pm 0.03$, as illustrated by the exponential fit shown in blue.