Granular flow through an aperture: Influence of the packing fraction

For the last 50 years, the flow of a granular material through an aperture has been intensely studied in gravity-driven vertical systems (e.g., silos and hoppers). Nevertheless, in many industrial applications, grains are horizontally transported at constant velocity, lying on conveyor belts or floating on the surface of flowing liquids. Unlike fluid flows, that are controlled by the pressure, granular flow is not sensitive to the local pressure but rather to the local velocity of the grains at the outlet. We can also expect the flow rate to depend on the local density of the grains. Indeed, vertical systems are packed in dense configurations by gravity, but, in contrast, in horizontal systems the density can take a large range of values, potentially very small, which may significantly alter the flow rate. In the present article, we study, for different initial packing fractions, the discharge through an orifice of monodisperse grains driven at constant velocity by a horizontal conveyor belt. We report how, during the discharge, the packing fraction is modified by the presence of the outlet, and we analyze how changes in the packing fraction induce variations in the flow rate. We observe that variations of packing fraction do not affect the velocity of the grains at the outlet, and, therefore, we establish that flow-rate variations are directly related to changes in the packing fraction.

Figure 1

Sketch of the experimental setup.

Figure 2

Number of grains $N\left(t\right)$ vs time $t$ for $V=9.6$ mm/s and different initial packing fractions $C_i$. The number $N\left(t\right)$ is linear in $t$ for initially dense systems ($\langle C_i\rangle \sim 0.81$) indicating a constant flow rate. For initially loose systems, $N\left(t\right)$ exhibits a nonlinear dependence on time $t$, which is explained by the increase of the packing fraction in the outlet region. Solid lines correspond to fitting curves obtained with Eq. (7). $\langle C_i\rangle \sim 0.66$ is fitted with $C_i=0.65\pm 0.01$, $\alpha =(0.90\pm 0.06)$ s${}^{-1}$ leading to $\lambda =(1.1\pm 0.1)$ cm and $\beta /V=(31\pm 1)$ cm${}^{-1}$. $\langle C_i\rangle \sim 0.46$ is fitted with $C_i=0.49\pm 0.01$, $\alpha =(0.34\pm 0.1)$ s${}^{-1}$ leading to $\lambda =(2.8\pm 0.2)$ cm and $\beta /V=(30\pm 1)$ cm${}^{-1}$. $\langle C_i\rangle \sim 0.38$ is fitted with $C_i=0.40\pm 0.01$, $\alpha =(0.31\pm 0.04)$ s${}^{-1}$ leading to $\lambda =(3.5\pm 0.5)$ cm and $\beta /V=(31\pm 1)$ cm${}^{-1}$.

Figure 3

Snapshots of the system during the discharge process for a system with $\langle C_i\rangle =0.46\pm 0.03$ driven at $V=(8.7\pm 0.3)$ mm/s. The arrows indicate the flow direction. Panels (a) and (b) correspond to $t=0$ s and $t=2$ s, the first stage of the process (transient stage): the grains are piling progressively, and the flow rate depends on time. Panels (c) and (d) correspond to $t=9.3$ s and $t=12.8$ s, the second stage of the process: the system has reached a steady packing fraction, and the flow rate $Q$ remains constant. The solid box in each image encloses the region of surface area $2DA$ upstream of the outlet (of size $A$) in which the packing fraction is measured. Note that (d) corresponds to the last image considered for the analysis. The dashed box in (b) indicates the upstream region in which we define $C_{\mathrm{vic}}.$ in the model.

Figure 4

Packing fraction $C$ in the outlet region vs time $t$. We observe that the temporal evolution of $C$ depends strongly on its initial value $C_i$ ($V=9.6$ mm/s).

Figure 5

Packing fraction $C$ in the outlet region as a function of the distance traveled by the belt $x=Vt$. A nice collapse of the experimental results is observed. The dotted line corresponds to the logistic model [Eq. (6)] with $C_{\infty }=0.8$, $C_i=0.45$, and $\lambda =(2.5\pm 0.5)$ cm.

Figure 6

Velocity $V_{\mathrm{g}}$ of the grains upstream of the outlet vs. $C_i$. The velocity is averaged over the duration of the discharge and normalized with the belt velocity. Even if the packing fraction increases during the discharge, i.e., for systems with $C_i<0.8$, grain velocities oscillate within $7\%$ of the velocity of the conveyor belt.

Figure 7

$\langle \lambda \rangle$ as a function of $\langle C_i\rangle$ is presented. ($\circ$) corresponds to values of $\lambda =\frac{V}{\alpha }$ obtained with $\alpha$ values from fitting experimental data with Eq. (7) and ($\blacksquare$) corresponds to $\lambda$ values obtained from Eq. (8). Values of $\langle C_i\rangle$ are mean values obtained from fitting experimental data with Eq. (7). Inset: Experimental values $\lambda =\frac{V}{\alpha }$ are fitted with Eq. (8), slope is found to be $1.7\pm 0.1$ in accordance with the expected value $\frac{W}{W-A}=1.8\pm 0.1$ (solid line), it can be observed that for $C_i=C_{\infty }$ effectively $\lambda =0$.