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

TL;DR

This study uses DEM simulations to analyze spherical granular flow through a hopper, validating the Beverloo law's applicability limits and proposing a criterion for its validity based on system dimensions.

Contribution

The paper provides a detailed DEM-based validation of the Beverloo law, identifying conditions where it holds and where it breaks down for granular hopper discharge.

Findings

Good agreement with Beverloo law for small D/d ratios and large bed heights.

Discharge rate decays exponentially for larger D/d ratios, indicating law breakdown.

Proposes a dimensionless criterion for Beverloo law validity: h/D > 2.

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 $\Pi_h = h/D > 2$, 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.