Gas-assisted discharge flow of granular media from silos

We studied experimentally the discharge of a vertical silo filled by spherical glass beads and assisted by injection of air from the top at a constant flow rate, a situation which has practical interest for nuclear safety or air-assisted discharge of hoppers. The measured parameters are the mass flow rate and the pressure along the silo, while the controlled parameters are the size of particles and the flow rate of air. Increasing the air flow rate induces an increase in the granular media flow rate. Using a two-phase continuum model with a frictional rheology to describe particle-particle interactions, we reveal the role played by the air-pressure gradient at the orifice. Based on this observation, we propose a simple analytical model which predicts the mass flow rate of a granular media discharged from a silo with injection of gas. This model takes into account the coupling with the gas flow as well as the silo geometry, position, and size of the orifice.

Figure 1

Sketch of setup with (a) cylindrical silo with orifice at the bottom, (b) cylindrical silo with lateral orifice, (c) rectangular silo with orifice at the bottom, and (d) rectangular silo with lateral orifice. Distances are given in mm. Gravity is aligned with but opposite to the $z$ axis.

Figure 2

Cylindrical silo with a bottom orifice with $L=60$ mm and $D=10$ mm. (a) Temporal evolution of instantaneous mass flow rate for $d=538\mu \mathrm{m}$ and different volumetric flow rates of air $Q_{\mathrm{air}}$. (b) Mass flow rate $Q$ vs the particle diameter $d$ for different volumetric flow rates of air $Q_{\mathrm{air}}$. The full line represents Eq. (1) for an opened-top silo. The dashed lines represent the analytical model [Eq. (23)]. (c) Mass flow rate $Q$ vs volumetric flow rate of air $Q_{\mathrm{air}}$ for different sizes of particles. The solid line represents the mass flow rate of opened-top silo $Q_o$ with $d=1129\mu \mathrm{m}$ [Eq. (1)], whereas the dashed line represents Eq. (3).

Figure 3

Cylindrical silo with a bottom orifice with $L=60$ mm and $D=10$ mm and $Q_{\mathrm{air}}=12$ L/min [corresponding to the purple curve in Fig. 2]. (a) Temporal evolution of the instantaneous pressure along the cylindrical silo. (b) Mean pressure in the granular bed along the vertical axis. The black dashed (red dash-dotted) line corresponds to the estimated pressure gradient close to (far from) the orifice. The corresponding equation is $y=58010x$ ($y=3856x+541.6$).

Figure 4

Continuum simulation of the discharge of a silo with orifice at the bottom for $L=90d, D=0.25L$, and $W=0.25L$. (a) Temporal evolution of the dimensionless instantaneous flow rate $Q_i/\left(\rho \sqrt{g{L}^3}\right)$ as a function of the dimensionless time $t/\sqrt{L/g}$. (b) $Q/\rho \sqrt{g{L}^3}$ as a function of $Q_{\mathrm{air}}/\sqrt{g{L}^3}$. The lines represent the simple analytical model [Eq. (23)] with $A=1$ (black dashed line) and $A=0.68$ (red line).

Figure 5

Continuum simulation with orifice at the bottom for $L=90d, D=0.25L, W=0.25L$, and $Q_{\mathrm{air}}/\sqrt{g{L}^3}=0.5$ [corresponding to black line in Fig. 4]. (a) Pressure $p^f/\rho gL$ along the center of the silo vs time $t/\sqrt{L/g}$ for different positions. (b) Mean pressure in the granular bed along the vertical axis. The black dashed line corresponds to a linear adjustment of the pressure close to the orifice with $y=\frac{\eta {\beta }_l}{S_o}(Q_{\mathrm{air}}-Q/{\rho }_p)x=1.23x$. The red dash-dotted line corresponds to a linear adjustment of the pressure far from the orifice with $y=\frac{\eta {\beta }_l}{S_b}(Q_{\mathrm{air}}-Q/{\rho }_p)x+0.098=0.308x+0.098$, where the numerical parameter is obtained using the least-squares method.

Figure 6

Continuum simulation of discharge of a silo with an orifice at the bottom for $L=90d, D=0.25L, W=0.25L$, and $Q_{\mathrm{air}}/\sqrt{g{L}^3}=0.5$ [corresponding to the black line in Fig. 4]. (a) Streamlines for the particle phase (red dashed lines) and for the fluid phase (black full lines). (b) Air mean pressure gradient $\langle \partial p^f/\partial z\rangle$ averaged along the orifice ($\square$) and in the accelerated zone near the orifice ($\times$) vs $\frac{{\eta }_f}{{\kappa }_v{S}_o}(Q_{\mathrm{air}}-Q/{\rho }_p)$. The dashed (full) line corresponds to the best linear fit.

Figure 7

Silo with orifice at the bottom. (a) Mass flow rate of particles vs volume flow rate of air $Q_{\mathrm{air}}$ for the cylindrical silo with $L=60\mathrm{mm}, D=10\mathrm{mm}$ for different sizes of particles. The dashed lines represent Eq. (23) with the adjusted parameter ${\varphi }_o$ by using the least-squares method. (b) ${\varphi }_o/{\varphi }_b$ vs $D/d$, with different sizes of $L$ and $D$, where the solid line represents Eq. (2) with the fitting parameters given in Table 1. (c) Mass flow rate vs different sizes of particles $d$ for different volume flow rates of air $Q_{\mathrm{air}}$ for the rectangular silo with $L=60\mathrm{mm}, D=10\mathrm{mm}$, and $W=3.5\mathrm{mm}$. The full line represents Eq. (1) for an opened-top silo. The dashed lines represent Eq. (23) with the fitting parameters given in Table 1.

Figure 8

Continuum numerical simulation for a silo with a lateral orifice with $D=0.40625L$ and $W=0.25L$. (a) Streamlines for the granular flow (red dashed line) and for the air flow (black lines). (b) $Q/\rho \sqrt{g{L}^3}$ as a function of $Q_{\mathrm{air}}/\sqrt{g{L}^3}$. The lines represent Eq. (23) with $A=1$ and $\lambda =1$ ($--$), $A=0.67$ and $\lambda =1$ ($--$), and $A=0.67$ and $\lambda =0.36$ ($-$).

Figure 9

Mass flow rate vs size of particles $d$ for different volume flow rates of air $Q_{\mathrm{air}}$, for lateral orifices. (a) Cylindrical silo with $L=60$ mm, $D=20$ mm, and $W=10$ mm. (b) Rectangular silo with $L=60$ mm, $D=10$ mm, and $W=3.5$ mm. The full lines represent Eq. (24) and the dashed lines represent the generalized model (23), with the adjusted parameters given in Table 1.