Beverloo law for hopper flow derived from self-similar profiles
- PDF / 2,066,786 Bytes
- 8 Pages / 595.276 x 790.866 pts Page_size
- 66 Downloads / 138 Views
ORIGINAL PAPER
Beverloo law for hopper flow derived from self‑similar profiles Fernando Alonso‑Marroquin1 · Peter Mora2 Received: 8 February 2020 / Accepted: 7 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We use particle simulations to investigate the mass flow in two-dimensional hopper flow and to analyze the dependency of the flow rate with the bottleneck width and the particle diameter. A flow rate law is derived from self-similar velocity and density profiles at the neck. The resulting relation is an enhancement of the Beverloo relation that incorporates the dependency of the density with the neck width. The parameters of the Beverloo relation are interpreted by coupling the hourglass theory with the free-fall arch theory using non-zero arch velocity as accounted by the hourglass theory. Keywords Granular flow · Beverloo relation · Hopper flow · Hourglass theory
1 Introduction The description of the two-dimensional flow of particles through bottlenecks has been instrumental in the understanding of granular flow in conveyor belts [1], pedestrian flow [10], sheep flow [8], and flow of other complex biological entities [23]. Observations of the flow rate when the diameter of the bottleneck is a few times the particle diameter have raised some questions about the applicability of the widely used Beverloo relation near the regime of intermittent flow [11, 12, 14, 16]. For gravity-driven flow, the hourglass theory (HGT) has been used the explain the Beverloo relation [15]. The theory predicts that the mass flow rate scales with the diameter D of the exit as D5∕2 for three-dimensional hoppers. It is also concluded that the flow rate depends on the hopper angle 𝜃 as sin1∕2 𝜃 which is consistent with experiments. On the other hand, the prediction of the dependency of flow rate on the neck diameter fails when the diameter is less than six times the diameter of the particles. To solve the governing equations of the hourglass flow, the HGT borrows from the Free Fall Arch Concept (FFAC): * Fernando Alonso‑Marroquin [email protected] 1
School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia
College of Petroleum Engineering and Geosciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2
there is a spherical free-fall arch where the radial stress vanishes; below the free-fall arch, the particles lose contact and accelerate freely by gravity [15]. Based on this concept, the Free-Fall Arch Theory (FFAT) has been developed [11, 16]. Recent micromechanical observations of the forces and displacement of the grains near the exit provide some corrections to this FFAC: particle image velocimetry analysis on the flow of bunkers (flat-bottom silos) shows self-similarity in both density and velocity profiles [11]. The analysis of the velocity profile with the FFAC suggests that the arch is not perfectly spherical but rather parabolic [11]. In more recent work, a detailed micromechanical analysis of the stress field concludes
Data Loading...
