Forced silo discharge: Simulation and theory

TL;DR

This study uses discrete element method simulations to analyze forced silo discharge through a circular orifice under overweight loading. It identifies two regimes: an initial flow rate $Q_{ m ini}$ that follows Beverloo-like free-discharge scaling and a final flow rate $Q_{ m end}$ that scales with $\sqrt{\rho_b P}$ and exhibits unusual $D_o$ and $D_s$ dependencies, consistent with viscous-like behavior under strong forcing. A work–energy framework is developed to derive a differential equation for the time evolution of the mass flow rate, incorporating gravity, overweight input, outflow, dissipation via $\mu(I)$-rheology, and pressure from Walters’ model; this approach accurately captures the initial Beverloo scaling and the qualitative pressure- and density-dependence of the final flow but struggles to reproduce the precise end-flow geometry and the $D_o$ and $D_s$ scalings, highlighting the need for nonlocal rheology or refined end-flow modeling in strongly forced granular flows. Overall, the work highlights new challenges in understanding forced granular discharges, where dissipation and confinement strongly alter flow behavior beyond classical Beverloo gas- or liquid-like pictures.

Abstract

We study, through discrete element simulations, the discharge of granular materials through a circular orifice on the base of a cylindrical silo forced by a surcharge. At the beginning of the discharge, for a high granular column, the flow rate $Q_{\rm ini}$ scales as in the Beverloo equation for free discharge. However, we find that the flow rate $Q_{\rm end}$ attained at the end of the forced discharge scales as $\sqrt{ρ_b P}D_o^3/D_s$, with $ρ_b$ the bulk density, $P$ the pressure applied by the overweight, $D_o$ the orifice diameter and $D_s$ the silo diameter. We use the work$-$energy theorem to formulate an equation for the flow rate $Q_{\rm end}$ that predicts the scalings only in part. We discuss the new challenges offered by the phenomenology of strongly forced granular flows.

Figures (12)

• Figure 1: Sketch of the axial cross section of a cylindrical silo with granular material and overweight.
• Figure 2: Mass flow rate $Q$ as a function of time (a) and as a function of the instantaneous mass inside the silo (b) for various overweight masses $M_{\rm p}$. In part (b) $Q$ is scaled by the mass flow rate $Q_{\rm free}$ of a free discharge through the same orifice. Simulations correspond to $D_{\rm s}=30d$, $D_{\rm o}=6d$ and $\rho=2500$ kg/m$^3$. The solid lines correspond to fitting the function $Q(M) = (Q_{\rm end}-Q_{\rm ini}) \exp[-M/M_{\rm decay}]+ Q_{\rm ini}$.
• Figure 3: Mass flow rate for a tall column $Q_{\rm ini}$ and at the end of the discharge $Q_{\rm end}$ scaled by that of a free discharge $Q_{\rm free}$ as a function of the pressure $P$ applied by the overweight. Simulations correspond to $D_{\rm o}=6d$, $D_{\rm s}=30d$ and $\rho_{\rm b}=2500$ kg/m$^3$. The yellow dotted line corresponds to Peng et al. model Peng2021. Thick lines correspond to different power laws.
• Figure 4: $Q_{\rm end}$ and $Q_{\rm ini}$ scaled by $Q_{\rm free}$ of glass as a function of the material density of the particles $\rho$ scaled by the density of glass ($\rho_{\rm glass}=2500$ kg/m$^3$). Simulations correspond to $D_{\rm o}=6d$, $D_{\rm s}=30d$ and $P=85.0$ kPa. Lines correspond to different power laws.
• Figure 5: $Q_{\rm end}$ and $Q_{\rm ini}$ scaled by $Q_{\rm free}$ at $D_{\rm o}=6d$ as a function of the empty annulus corrected orifice diameter $D_{\rm o}-kd$ scaled by $d$. Simulations correspond to $\rho=2500$ kg/m$^3$, $D_{\rm s}=30d$ and $P=85.0$ kPa. Lines correspond to different power laws.