Jamming during the discharge of granular matter from a silo

In this work, we present an experimental study of the jamming that stops the free flow of grains from a silo discharging by gravity. When the outlet size is not much bigger than the beads, granular material jams the outlet of the container due to the formation of an arch. Statistical data from the number of grains fallen between consecutive jams are presented. The information that they provide can help one to understand the jamming phenomenon. As the ratio between the size of the orifice and the size of the beads is increased, the probability that an arch blocks the outlet decreases. We show here that there is a power-law divergence of the mean avalanche size for a finite critical radius. Beyond this critical radius, no jamming can occur and the flow is never stopped. The dependence of the arch formation on the shape and the material of the grains has been explored. It has been found that the material properties of the grains do not affect the arch formation probability. On the contrary, the shape of the grains deeply influences it. A simple model to interpret the results is also discussed.

Figure 1

Sketch of the experimental setup. S, silo; W, electronic balance; B, blower; E, electrovalve; M, microphone; O, oscilloscope; PC, computer. In (a) a photograph of the orifice is shown as seen from below. In (b) a section of the orifice is shown.

Figure 2

(a) A diagram of the pasta grains and (b) two photographs of the glass beads before (right) and after (left) the treatment with fluorhidric acid (series 2 and 5, respectively).

Figure 3

(a) Histogram for the number of grains $s$ that fall between two successive jams. Data correspond to $R=3$ (beads have a diameter of 2 mm, and the circular orifice is 6 mm wide). The two different regimes are separated with a vertical dashed line. In the inset is shown the position of the mode for different $R$. (b) First return map is plotted for a series of avalanches, i.e., the avalanche size $n_t$ vs the next avalanche size $n_t+1$. Note that $t$ is just a correlative index ordering the sequence of avalanches.

Figure 4

The two regimes marked in Fig. 3a are shown in two separate graphs. In (a), in logarithmic scale, a power law fits the avalanches smaller than the mode. In (b) the exponential tail is shown in a semilogarithmic scale.

Figure 5

Histogram for the number of grains of the avalanches normalized with the mean avalanche size $⟨s⟩$. Different symbols correspond to 14 histograms from $R=1.52$ to $R=3.96$ that are plotted together. It can be seen that all the histograms collapse into a single curve.

Figure 6

Mean avalanche size $⟨s⟩$ vs $R$. The symbols ◻ represent experimental points obtained from the histograms. The solid line is the fit with Eq. (1). Inset: mean avalanche size $⟨s⟩$ vs $1∕(R_c-R)$. Note the logarithmic scale.

Figure 7

(a) Mean avalanche size $⟨s⟩$ vs $R$ in a semilogarithmic scale for spherical grains of different types: Delrin (▵), glass of two and three mm in diameter (엯), lead of 2 and 3 mm (★), steel (▿), rough glass (+), and glass with high size dispersion (◻). (b) Mean avalanche size $⟨s⟩$ vs $R$ in a semilogarithmic scale for different grain shapes. The symbols ◻, 엯, and ▵ are experimental points for spheres, rice, and pasta grains, respectively. The solid line is the fit with Eq. (1) with $\gamma =6.909$, which remains constant for all the shapes, while $R_c$ and $A$ change considerably. This can be observed in the inset, where the mean avalanche size $⟨s⟩$ vs $1∕\left(R_c-R\right)$ is plotted in a logarithmic scale.

Figure 8

(a) Normalized histogram for spherical grains with different material properties. Delrin (▵), glass of two and three mm in diameter (엯), lead of 2 and 3 mm (★), steel (▿), rough glass (+), and glass with high size dispersion (◻). All the histograms were obtained for $R$ between 2 and 3. (b) Normalized histogram for rice (◻) and lentils (엯) with $R=2.72$ and $R=2.2$, respectively. Note that both figures are plotted in a semilogarithmic scale.

Figure 9

Probability that one grain passed through the outlet $\left(p\right)$ vs $R$.

Figure 10

The avalanche size distribution for spherical glass beads. ◻, ▵, ◇, and 엯 correspond to $R=1.74$, 2.23, 2.42, and 3.07, respectively. Solid lines correspond to Eq. (5) (see text for details of the fitting procedure). The corresponding values of the parameters $p$ and $\alpha$ are $p=0.615$, $\alpha =1.94(R=1.74)$; $p=0.913$, $\alpha =2.69(R=2.23)$; $p=0.938$, $\alpha =3.70(R=2.42)$; and $p=0.992$, $\alpha =18.52(R=3.07)$. Inset: the same graph in a semilogarithmic scale.