A study of shape non-uniformity and poly-dispersity in hopper discharge of spherical and polyhedral particle systems using the Blaze-DEM GPU code

Abstract

The importance of shape non-uniformity and the polydispersed nature of granular media in industrial hopper discharge applications has been well established experimentally. Although numerous hopper discharge simulations have been conducted over the last thirty years, the investigations into the non-uniformity of particle shape and the polydisperse nature of particle systems remains limited. These studies are usually limited to a single hopper configuration, while the number of polyhedral particles considered are usually limited to a maximum of 5000 particles. In this study we consider the polydispersed particle systems for hoppers at various angles, particle systems with non-uniform shape for hoppers at various angles and polyhedral particle systems up to 1 million particles. This is made possible by extensively utilizing the graphical processing unit (GPU) computing platform via the BlazeDEM3D-GPU code. We first perform an experimental validation of the code for mono-sized spherical and convex polyhedral shaped particles for lab-scale hoppers at three half angles using 3D printed polylactic acid material (PLA) particles. We found good agreement between the experimental, Meyrs and Sellers empirical relation and simulation discrete element method (DEM) discharge rates. We then simulate three larger square hoppers with varying half-angles to study the effect polydispersity and non-uniformity of particle shape have on the mass discharge rate. Again, good agreement between the DEM simulated mono-dispersed spherical particle systems and the Meyrs and Sellers empirical relation is obtained to verify the simulations. Finally we simulate an industrial sized silo for which we compare mono-dispersed spheres against mono-dispersed polyhedra using over a million polyhedral shaped particles. Finally, we briefly comment on the effect that the polydisperse nature of particle systems has on the loading of the supporting structure.

Introduction

An important aspect in granular material processing is the storage of particles as it goes through the material processing chain. Particles are usually stored in bins, tower silos, bunker silos and bag silos at various stages during material processing. A significant fraction (estimated around 70%) of processed material is non-spherical [1], [2]. The importance of particle shape in the dynamic behavior of particle systems is well known and has been established experimentally. For example, the comprehensive experimental study conducted by Standish [3] investigated the discharge of non-uniform particle shapes and polydispersed particle systems in hoppers. Standish’s study highlighted the importance and significance of a particle systems’ characteristics in hopper discharge and ultimately the industrial relevance of considering these particle characteristics to predict hopper discharge. Experimental results mostly provide only bulk characteristics such as discharge rates as Vlachos and Chang [4] recently demonstrated that the mass flow rate of polydispersed mixtures is affected by the particle shape, size ratio between the largest and smallest particles, the volume fraction of the smallest particle size and the initial mass flow rate. However, more sophisticated experiments include tracer particles that provides a spatial velocity field in three dimensions [5]. Based on experimental observations the complex granular rheology in silo flow is divided into three distinct regimes namely, quasi-static, transitional and inertial [6], [7]. The effect of size polydispersity and shape non-uniformity has on the shear strength of granular assemblies are well understood on a particle level [8], [9]. These experiments verify granular theories and give further insight into the flow characteristics in hoppers at particle level. However, industrial applications require these insights to be scaled to macro-scale particle assemblies to allow the organic development of the complex flow regimes within a granular assembly over a discharge event. Hence, the majority of material handling and processing is still empirically designed e.g. hopper discharge estimates are derived from Beverloo correlations [10] and Jenike yield locus (JYL) curves [11]. It is known that these approaches are not sufficient to deal with the non-uniform complexity of the majority of particle systems. Naturally, attempts have been made to modify Beverloo’s law [12] and simulated bi-dispersed or polydispersed investigations often focussed on geometrical parameters such as volume fraction and coordination number [13]. The particle system that is discharged within a hopper may frequently change, either due to changes in the bulk material or the properties of the current bulk material. Hence, the need for design tools that can quantify the effect of particle-scale changes on the macro-scale particle assembly response is becoming increasingly more important in industrial bulk material handling applications.

Two approaches are readily available for simulation of particulate systems. Namely, continuum models [6] and discrete element models [14]. The inability of continuum models to accurately predict the macroscopic response for variations at the particle-scale has only seen limited continuum solutions in isolation [15]. A number of strategies combine the continuum approaches with discrete approaches to supplement the shortcoming of the continuum models [15], [16] and alleviate the cost of the discrete approaches. However, they are not generally applicable and can be time consuming to setup properly. Continuum approaches fail to reliably predict the transitional flow regime in which both the collisional and frictional interactions drive flow behavior [17], in addition to being unable to predict bulk discharge rates in hopper flow when particle-scale features such as particle size and shape are varied. The discrete element model is the only simulation approach that is capable of capturing microscopic and particle-scale effects to accurately predict the macro-scale of bulk behavior but requires significant computational resources [15], [16]. The discrete element method however is able to reliably predict all three flow regimes and in turn bulk discharge rates. As a result the discrete element method (DEM) has become the de facto standard to simulate particulate materials for which the discrete nature of particles and importance of quantifying particle-scale effects cannot be neglected. Consequently, the need for large scale discrete element simulations that can quantify the effect of particle-scale changes on the macro-scale response is becoming increasingly more important in industrial bulk material handling applications, where computational efficient continuum models fail to give reliable predictions [15]. This extends to particle-scale effects on the most basic and fundamental macro-scale responses such as for example bulk discharge rates [6]. The need to consider the polydispersed nature and shape non-uniformity of particulate systems in simulation has been well-established for a number of years by Clearly and Sawley [14], and has recently been reiterated by Höhner et al. [18] and Zhong et al. [2]. However, numerical studies to date that consider shape non-uniformity and polydispersity of polyhedral particle systems have been limited in scope due the large computational cost and lack of suitable codes to efficiently handle the associated geometrical complexity [19].

To compute industrially relevant discrete element simulations within a realistic time frame remains a challenging computational endeavor. As a result the particle shape is often simplified as either spheres or by the multi-sphere approach that approximates a non-spherical particle as multiple spheres that are glued together [20], as depicted in Fig. 6.1(b). Although the multi-sphere particles is an improved description for complex particle shapes and allows for particle fragmentation to be taken into account [21], it is limited in the number of spheres used to represent complex shapes and is unable to accurately capture the angularity of particles. Both the spherical and multi-spherical simplifications significantly reduce the complexity of inter-particle contact detection that results in a significant computational saving. In addition the constitutive models of interacting spheres has been well established and as a result has been used extensively in hopper simulations [22], [23]. Since the influence of particle shape and angularity on the mechanical behavior particle systems is well known [14], [24], the validity of approximate particle representations remains refutable [1], [25], [26]. An additional complexity is shape non-uniformity and the polydispersed nature of particle systems, in that particle size distributions are often simplified and modeled as mono-dispersed systems and shape non-uniformity simplified to a uniform shape for a particle system. Although recent modeling efforts have started to consider improved particle representations [2], [18], [27], the consideration of non-uniform polydispersed particle systems for large scale simulations remain elusive and understudied. This is due to the additional computational burden that different particle sizes and shapes place on the already computational demanding single particle shape problem.

Höhner conducted a purely numerical study that investigated the bulk discharge rates for non-cohesive, mono-dispersed spherical and in addition uniform shape polyhedral particle systems for hoppers at various inclinations (0° , 30° and 60° from the horizontal) [18]. This study clearly highlighted the importance of particle shape. Höhner conducted two follow-up studies that supplemented the numerical observations by experiment [28] and highlighted the importance of angularity and quantified the effects of aspect ratio [29], when only flat-bottomed (0° from the horizontal) hoppers are considered. Similarly, a detailed review of non-spherical particulate studies by Zhong et al. [2], highlighted the prevalence of mono-dispersed and uniform shape studies and the lack of more realistic and complex systems. The review by Zhong et al. [2] further highlighted the debilitating associated computational cost by highlighting the fact that the number of polyhedral particles considered in particulate assemblies have been mainly limited to under 10 000 particles to date. Specifically, the polyhedral particles in previous studies varied between 300 by Mack et al. [30] and 5000 particles by Höhner et al. [18]. Simulations using a limited number of particles is highlighted as one of the critical limitations and important challenges for future discrete element studies [2], and one of three aspects this study aims to investigate. In particular this study investigates for the first time hopper discharge for

• polydispersed particle systems for hoppers at various angles, namely 0° , 30° and 60° from the horizontal,

• particle systems with non-uniform shape for hoppers at various angles, namely 0° , 30° and 60° from the horizontal,

• polyhedral particle systems that exceeds 10 000 particles as we consider up to 1000000 particles.

This is achieved by the parallelization benefits of the graphical processing unit (GPU) [5], [31], [32], [33], [34], which is becoming increasingly more important as an alternative computational platform for discrete element simulations [19]. In this study we specifically consider BlazeDEM3D-GPU that has been demonstrated to simulate (i) tens of millions of mono-dispersed spherical particles and (ii) millions of shape uniform and mono-dispersed polyhedral particles within a realistic time frame on a desktop computer using a single K6000 GPU [27], [31], [35]. BlazeDEM3D-GPU [27] has been extensively validated in the simulation of tumbling mills [35].