Effect of hopper angle on granular clogging

We present experimental results of the effect of the hopper angle on the clogging of grains discharged from a two-dimensional silo under gravity action. We observe that the probability of clogging can be reduced by three orders of magnitude by increasing the hopper angle. In addition, we find that for very large hopper angles, the avalanche size ($\langle s\rangle$) grows with the outlet size ($D$) stepwise, in contrast to the case of a flat-bottom silo for which $\langle s\rangle$ grows smoothly with $D$. This surprising effect is originated from the static equilibrium requirement imposed by the hopper geometry to the arch that arrests the flow. The hopper angle sets the bounds of the possible angles of the vectors connecting consecutive beads in the arch. As a consequence, only a small and specific portion of the arches that jam a flat-bottom silo can survive in hoppers.

Figure 1

(a) Photograph of the experimental setup with (b) a zoom of the outlet region. Also we show four typical arches formed in (c) a flat-bottom silo ($\beta =0^{\circ }$); (d) a hopper with $\beta ={45}^{\circ }$; (e) a hopper with $\beta ={60}^{\circ }$; and (f) a hopper with $\beta ={80}^{\circ }$.

Figure 2

Mean avalanche size $\langle s\rangle$ versus the outlet size $D$ (rescaled by the particle size $d$) for different hopper angles $\beta$ as indicated in the legend. The error bars, indicating the $95\%$ confidence interval of the values obtained, are of around the symbols size. Note the logarithmic scale on the vertical axis.

Figure 3

(a) Average number of particles that form the arches depending on the outlet-to-particle diameter ratio $D/d$ for the four values of $\beta$ explored in this work as indicated in the legend. The error bars, indicating the $95\%$ confidence interval of the values obtained, are of around the symbols size. (b)–(e) Bar diagrams illustrating the percentage of clogging arches formed by $n$ beads for each experimental condition. The colors correspond to different values of $n$, as indicated in the legend.

Figure 4

Probability of clogging $p_c$ versus the outlet size for the four hopper angles explored in this work. Empty squares account for the probability that the orifice gets clogged by any kind of arch whereas the other symbols correspond to $p_{cn}$, the probability that the orifice gets clogged by an arch of $n$ particles. Note the logarithmic scale on the vertical axes.

Figure 5

Probability density functions (pdf) of the vector angle $\theta$ that every two consecutive particles make with the horizontal, as defined in Fig. 2. Results are presented for the four hopper angles investigated in this work and different outlet sizes as indicated in legends. Dashed lines indicate the limit values of $\theta$ that would be possible for frictionless grains. Note the logarithmic scale on the vertical axes.

Figure 6

Probability density functions of the arch spans $l$. Experimental results are presented for different hopper angles and outlet sizes as indicated in legends. The shadowed regions in each plot correspond to the only arch spans that would be possible for frictionless grains according to Eq. (2).