Storage and discharge of a granular fluid

Experiments and computational simulations are carried out to study the behavior of a granular column in a silo whose walls are able to vibrate horizontally. The column is brought to a steady fluidized state and it behaves similar to a hydrostatic system. We study the dynamics of the granular discharge through openings at the bottom of the silo in order to search for a Torricelli-like behavior. We show that the flow rate scales with the wall induced shear rate, and at high rates, the granular bed indeed discharges similar to a viscous fluid.

Figure 1

(Color online) Photograph of the real silo. See the text.

Figure 2

Plot of the mass measured at the bottom of the silo as a function of the mass poured into it. The triangular points were taken using static walls. The circular points were obtained experimentally with vibrating walls. With static walls, the measured mass follows Janssen’s law, which is indicated by the dashed curved black line. With vibrating walls, a hydrostatic behavior is obtained, which is indicated by the dashed straight black line. The dark gray line is obtained in a simulation with vibrating walls. The light gray line is obtained using static walls. Both the total mass and the mass on the bottom obtained in the simulation are multiplied by five for comparison with the experimental results.

Figure 3

A silo discharge experiment. The mass leaving the silo is plotted as a function of time, for different scaled outlet diameters ($f=20\mathrm{Hz}$, $a=1.5\mathrm{mm}$). The linear behavior means that the flow rate is constant.

Figure 4

Flow rates plotted as a function of the outlet diameter. The black markers indicate the results obtained from the experiments and the gray markers indicate the results obtained from the simulation, both for static and vibrating walls ($f=20\mathrm{Hz}$, $a=1.5\mathrm{mm}$). The solid lines are curves fitted with Beverloo’s model. The parameters used are shown in the body of the figure, where $C=\zeta {\rho }_B{g}^{1∕2}$.

Figure 5

Mass carried by the bottom (light gray line) and walls (dark gray line) during the discharge. Mass leaving the silo (black line). The vibration conditions are frequency $20\mathrm{Hz}$ and amplitude $1.5\mathrm{mm}$. The dashed lines resembles Beverloo’s and Torricelli’s discharge.

Figure 6

Mass carried by the bottom (light gray line) and walls (dark gray line) during the discharge. Mass leaving the silo (black line). The vibration conditions are: frequency $50\mathrm{Hz}$ and amplitude $3\mathrm{mm}$. The dashed lines resembles Beverloo’s and Torricelli’s discharge.

Figure 7

Silo with frictionless walls. Mass carried by the bottom (light gray line) and walls (dark gray line) during the discharge. Mass leaving the silo (black line). The dashed line resembles Beverloo’s and Torricelli’s discharge.

Figure 8

(Color online) (a) Cross section of the granular column with walls vibrating at a frequency of $20\mathrm{Hz}$ and amplitude of $3\mathrm{mm}$. The color indicates the force on the particles. Force chains can be identified. (b) and (c) Cross section of the granular column with walls vibrating at a frequency of $50\mathrm{Hz}$ and amplitude of $3\mathrm{mm}$ at two instances during a vibration cycle. In (b), the column is detached from the wall. The closure of the gap causes a shockwave through the column, as can be seen in (c). However, this shockwave does not influence the outflow rate since this rate scales with the shear rate. A movie of these simulations can be seen in Ref. 21.

Figure 9

Flow rates plotted as a function of the pressure on the bottom. All the results are obtained in simulations using a box with a depth and width of $0.1\mathrm{m}$, except the dark gray triangles, for which a box with a depth and width of $0.15\mathrm{m}$ was used. Note that, with the same vibration parameters, the flow rate in the large box is lower than the flow rate in the standard box. The restitution and friction coefficients are, respectively, 0.9 and 0.3.

Figure 10

Scaled flow rates plotted as a function of the pressure on the bottom. The flow rates from Fig. 9 are scaled by subtracting the constant flow rate with static walls divided by the walls induced shear rate. The data collapses reasonably well for low and high shear rates, indicating that the flow rate scales with the wall induced shear rate.

Figure 11

Scaled flow rates plotted as a function of the pressure on the bottom. The restitution and friction coefficients are, respectively, 0.7 and 0.3.

Figure 12

Scaled flow rates plotted as a function of the pressure on the bottom. The restitution and friction coefficients are, respectively, 0.9 and 0.5.