Elsevier

Powder Technology

Volume 342, 15 January 2019, Pages 356-370
Powder Technology

Understanding the varying discharge rates of lognormal particle size distributions from a hopper using the Discrete Element Method

https://doi.org/10.1016/j.powtec.2018.09.080Get rights and content

Highlights

  • Assess mechanisms underlying effect of PSD width on varying hopper discharge rates

  • Study radial velocity, angular velocity and collision force using DEM

  • Beverloo correlation mis-predicts discharge rate trends with respect to PSD width.

  • Positive correlation between PSD width and angular velocity or collision force

  • Negative correlation between hopper discharge rate and collision force

Abstract

The focus of this study is on understanding the varying discharge rates of lognormal particle size distributions (PSDs) with the same mean particle diameter. Discrete element method (DEM) is used to simulate lognormal PSDs with the same arithmetic mean of the particle diameter of 5 mm but varying PSD widths (σ/μ = 10% to 70%) in a 3D conical hopper. Four highlights are noted: (i) the Beverloo correlation and others modified to account for various particle properties predict the discharge rates of lognormal PSDs poorly, which underscores the need for more understanding on the influence of PSD width; (ii) the velocity vectors are less uniformly downwards in the hopper for the wider PSDs, which results in the slowing down of the discharge rate; (iii) the radial particle velocity, and both the radial and vertical particle angular velocity increase with PSD width throughout the first half of the hopper discharge; and (iv) the collision force magnitudes are greater for the wider PSDs, and the cross-sectional fluctuations of the collision forces at the cone height increases with PSD width. The increase of the magnitudes of these particle characteristics (namely, radial particle velocity, angular velocity, and collision force) with PSD width underlies the decreasing hopper discharge rate.

Introduction

Hoppers are common apparatuses for the storage and discharge of particles in various industries (e.g., pharmaceutical, chemical, and food). Despite the seeming simplicity of function, flow problems including erratic or stunted flow, dead zones, segregation, and dust explosions are prevalent [[1], [2], [3]]. To resolve such issues, an in-depth understanding of the mechanisms underlying the different discharge behaviors [4,5], for example, due to different hopper configurations [[6], [7], [8]] and different particle properties [[9], [10], [11], [12], [13]], is necessary. To this end, both experimental and simulation efforts have been dedicated to investigating the discharge characteristics of hoppers [1,[14], [15], [16]]. The discrete element method (DEM) provides an effective method of tracking each particle numerically and has been widely accepted to study dense particulate behavior, which was initially proposed by Cundall and Strack [17]. Due to the dominance of collisions and significant shear in the hopper, particle-particle interactions have to be appropriately accounted for [18].

Previous DEM studies focusing on discharge behaviors in the hopper have provided some understanding on monodisperse [19,20], binary-size mixtures [[21], [22], [23]] and multi-size mixtures [24,25]. Li et al. [19] found good agreement between DEM simulation and experimental results on the particle trajectory for a high-temperature monodisperse system. Yu and Saxén [20] found that the velocity distribution is uniform except near the orifice and the static particle-wall friction is important for the flow pattern of the monodisperse flow. Rahman et al. [21] concluded that, for binary mixtures, the degree of segregation is higher in a three-dimensional system compared to a two-dimensional system and flowability is closely tied to segregation patterns. Combarros et al. [22] found that particle size has a more dominant effect than particle shape on segregation and that the critical DEM parameters in segregation simulation are static and rolling friction. Ketterhagen et al. [23] investigated the effects of the particle diameter ratio, density ratio, fines mass fraction, hopper wall angle, hopper cross-sectional shape, and the initial fill conditions of a binary-size mixture, and found that the DEM model can predict well the segregation during hopper discharge. Wu et al. [24] found that the flowability of the particles is tied to the hopper slope and the mass fraction of small particles. Yu and Saxén [25] concluded the filling method, diameter ratio of fine to coarse, wall-particle static and rolling friction, inter-particle rolling friction as well as mass fraction of fine particles are the most important factors affecting the extent of segregation during the discharging process. Anand et al. [26,27] further revealed that the coefficient of restitution and hopper width have negligible influence on the discharge rate, whereas friction, hopper angle, outlet width and particle size distribution (PSD) have more significant impact. Arteaga and Tüzün [28,29] concluded that the discharge rates were found to increase with increasing mass fraction of fines, with the extent of this increase being a strong function of the size ratio of the constituents. Our previous study [30] found the lognormal PSD width has a dominant influence on the discharge rate in the hopper, with the decrease in the discharge rate significant at approximately 36% when the PSD changes from a monodisperse to a lognormal PSD width (defined as ratio of standard deviation of the particle size to mean particle diameter) of 70%. The dependence of discharge rate and wall stress on particle interactions have been investigated using the discrete element model (DEM), and it was found that the proper accounting of inter-particle interactions is critical for more accurate predictions of discharge rates [7]. The effect of differential pressure (∆P) on discharge rate in a conical hopper was experimentally studied, which indicated that increases in ∆P increased discharge rate up to a critical ∆P after which the discharge regime destabilized, and that the critical ∆P depended on hopper orifice diameter and physical properties of the particles like size distribution and particle density [5]. Recently, the discharge rate in conical hoppers was assessed using an elastoplastic model implemented with the Eulerian finite element method (FEM) approach, which demonstrated that material dilation and internal frictional angle markedly affect the discharge rate particularly for steep cones (cone angle <45°) [31]. Moreover, the behavior of super-quadric particles during hopper discharge was found to be such that the discharge rate decreases with increasing blockiness, particle friction or aspect ratio [32]. Collectively, these studies imply that particle properties like size and size distribution have a non-negligible impact on the hopper discharge rate.

The Beverloo correlation [33] represents one of the most popular for predicting hopper discharge rates, which was subsequently modified to account for varying particle properties. Because it was experimentally found that the correlation only worked well for coarser particles but not the finer ones (d ≤500 μm), the correlation was modified based on the different momentum and drag of the particles [34]. Another group investigated finer particles (50 μm≤ d ≤500 μm) and found that, although the developed correlation for discharge rates has the right relationship with particle size, it gives quantitative values about twice larger than the experimentally measured values [13]. The shape of the particles was also experimentally found to have a significant effect on the flow rate, and therefore the Beverloo correlation was further incorporated with the shape parameters [35]. A recent work [36] investigated the influence of orifice shape on the flow rate in flat-bottom hoppers numerically and experimentally, and found the data can be well-fitted with a modified Beverloo's formula. Also, with regards to a wider range of particle sizes, the Beverloo correlation was extended to binary-size mixtures [12] based on earlier experimental data [9] and also polydisperse mixtures by including the differential pressure gradients [14,37].

The reports to date provide limited understanding on the granular motion of continuous particle size distributions (PSDs), which are ubiquitous both naturally and industrially. Our recent DEM study found that the discharge rate decreases as the PSD width increases, which is tied to increasing extents of segregation but surprisingly not tied to the vertical particle velocity [30]. To obtain a more in-depth understanding, the Discrete Element Method (DEM) is used here to further understand the mechanisms underlying the different discharge rates of the lognormal PSDs with the same mean diameter but different PSD widths. The specific objectives of the current study are: (1) assess if the available correlations can predict the discharge rate of lognormal PSDs well; (2) understand the impact of PSD width on the particle characteristics during hopper discharge, namely, radial particle velocity (vr), radial and vertical particle angular velocity (ωr and ωz), and collision force (F); (3) armed with the understanding that the vertical particle velocity (vz) has no direct relationship with the discharge rate [30], attempt to relate the discharge rate to these other particle behaviors to provide an explanation for the varying discharge rates resulting from the different PSD widths.

Section snippets

Discrete element method (DEM)

In the current work, the discrete element method (DEM) is adopted to track the motion of each particle in the Lagrangian framework. In DEM simulations, the motion of each particle is governed by Newton's second law of motion [17], with the translational and rotational motion of a particle generally formulated based on the force-displacement relationship. The equations governing the motion of each particle i are expressed as follows:midVidt=j=1KFc,ijn+Fc,ijt+Fd,ijn+Fd,ijt+migIidωidt=j=1KMt,ij+M

Beverloo correlation

Fig. 4 presents the simulation results, which show that the mass discharge rate (W) decreases as the PSD width (σ/μ) increases. This trend agrees qualitatively with two correlations [12,37] despite some quantitative discrepancies, but disagrees with the well-known Beverloo equation [33] that instead exhibits an opposite trend. The disparity in the predictions presumably stems from the lack of consideration of polydispersity effects and warrants a closer inspection.

The most widely accepted

Conclusion

In view of the ubiquity of continuous PSDs in both natural and industrial granular flow, this study aims at getting a better mechanistic understanding of the negative correlation between PSD width and discharge rate, based off a previous study that indicated that, although size-segregation increases with PSD with, no direct relationship exists between PSD width and vertical particle velocity [30]. By means of DEM, the particle velocities and collision forces of lognormal PSDs with the same

Nomenclature

    d

    particle diameter (m)

    d50

    50th percentile of the cumulative mass-based particle size distribution (m)

    di

    diameter of the large or small particles (m)

    (dP/dz)0

    pressure gradient across the particle bed (Pa/m)

    dVM

    de Brouckere mean diameter (m)

    e

    restitution coefficient (dimensionless)

    G

    shear modulus (Pa)

    H

    particle bed height in the cylindrical section (0.422 m)

    h

    height in the cylindrical section assessed (m)

    Fnc, ij

    normal contact force (N)

    Ftc, ij

    tangential contact force (N)

    Fnd, ij

    normal damping force (N)

    Ftd,

Acknowledgement

The authors thank the financial support from the National Research Foundation (NRF), Prime Minister's Office, Singapore, under its Campus for Research Excellence and Technological Enterprise (CREATE) program, and also the 2nd Intra-CREATE Seed Collaboration Grant (NRF2017-ITS002-013).

References (42)

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