DEM Simulations of Spheres Flowing Through a Hopper: Validation of Beverloo Law

TL;DR

The paper investigates whether Beverloo scaling accurately predicts mass flow rate through a hopper for spherical granular media. It employs Discrete Element Method (DEM) simulations of $Al_{95}Fe_2Cr_2Ti_1$ alloy spheres across varying particle size $d$ and bed height $h$, with validation against gas-atomized polydisperse powders. A key finding is that Beverloo scaling, $Q \propto (D - k d)^{5/2}$, holds only when the column height is sufficiently large, quantified by the dimensionless criterion $\Pi_h = h/D > 2$ (or $N = h/d > 20$); otherwise the discharge becomes height-dependent and transient effects dominate. The study demonstrates the DEM model's predictive capability for confined granular discharge and offers a practical guideline for the onset of Beverloo-type behavior in engineering hoppers.

Abstract

This work presents a detailed investigation of the discharge behavior of spherical granular materials through a conical--cylindrical hopper using \emph{Discrete Element Method (DEM)} simulations. The aim is to assess the applicability limits of the empirical \emph{Beverloo law}. The system was modeled with a monodisperse particles whose mechanical properties correspond to the $Al_{95}Fe_2Cr_2Ti_1$ alloy, and interparticle contacts were described using the Hertz--Mindlin (no slip) model. The simulations systematically explored the influence of particle diameter ($d$) and bed height ($h$) on the resulting mass flow rate ($Q$). The results reveal the coexistence of transient and steady-state discharge regimes. Good agreement with the Beverloo scaling was observed for relatively small diameter ratios ($D/d = 10$) and sufficiently large bed heights, where the flow stabilizes rapidly. For larger $D/d$ ratios, the discharge rate decays exponentially, indicating a breakdown of the constant-hydrostatic-pressure assumption underlying the Beverloo model. A dimensionless criterion for the validity of the Beverloo law is proposed as $Π_h = h/D > 2$, or equivalently $N = h/d > 20$. The quantitative agreement between DEM simulations and experimental measurements for polydisperse particle size distributions further validates the computational model and demonstrates its predictive capability for granular discharge in confined geometries.

Figures (6)

• Figure 1: Representative snapshot of the granular-flow simulations through a funnel-shaped hopper. The variable $R_{max}$ denotes the maximum radius of the upper container, while $R_{min}$ corresponds to the minimum radius at the outlet. The parameter $D$ represents the diameter of the opening through which the particles exit. The coordinate axes $(x, \, y, \,z)$ indicate the orientation used in the simulations.
• Figure 2: Particle size distributions used in the experiments:a narrow distribution (blue) and a broader one (red).
• Figure 3: Flow rate as a function of time for two different ratios between the funnel diameter $D$ and particle diameter $d$.
• Figure 4: Flow rate as a function of particle diameter $d$ obtained from DEM simulations (black points) and compared with the Beverloo law (red line).
• Figure 5: Initial bed height $h$ as a function of the particle diameter ratio $D/d$. The red dashed line indicates the approximate threshold where the Beverloo law becomes valid.