A differential equation for the flow rate during silo discharge: Beyond the Beverloo rule

TL;DR

This paper introduces a differential equation modeling silo discharge flow rate based on energy balance, offering a theoretical foundation that aligns with the Beverloo rule and provides new insights into pressure behavior during discharge.

Contribution

It develops a novel differential equation for granular flow during silo discharge grounded in energy principles, extending beyond heuristic models.

Findings

Equation aligns with Beverloo rule and estimates its universal prefactor.

Provides an analytic expression for pressure during discharge.

Offers a theoretical framework for granular flow in silos.

Abstract

We present a differential equation for the flow rate of granular materials during the discharge of a silo. This is based in the energy balance of the variable mass system in contrast with the traditional derivations based on heuristic postulates such as the free fall arch. We show that this new equation is consistent with the well known Beverloo rule, providing an independent estimate for the universal Beverloo prefactor. We also find an analytic expression for the pressure under discharging conditions.