Discrete particle simulation of particulate systems: Theoretical developments

doi:10.1016/j.ces.2006.12.089

Chemical Engineering Science

Volume 62, Issue 13, July 2007, Pages 3378-3396

https://doi.org/10.1016/j.ces.2006.12.089 Get rights and content

Abstract

Particle science and technology is a rapidly developing interdisciplinary research area with its core being the understanding of the relationships between micro- and macroscopic properties of particulate/granular matter—a state of matter that is widely encountered but poorly understood. The macroscopic behaviour of particulate matter is controlled by the interactions between individual particles as well as interactions with surrounding fluids. Understanding the microscopic mechanisms in terms of these interaction forces is therefore key to leading to truly interdisciplinary research into particulate matter and producing results that can be generally used. This aim can be effectively achieved via particle scale research based on detailed microdynamic information such as the forces acting on and trajectories of individual particles in a considered system. In recent years, such research has been rapidly developed worldwide, mainly as a result of the rapid development of discrete particle simulation technique and computer technology. This paper reviews the work in this area with special reference to the discrete element method and associated theoretical developments. It covers three important aspects: models for the calculation of the particle–particle and particle–fluid interaction forces, coupling of discrete element method with computational fluid dynamics to describe particle–fluid flow, and the theories for linking discrete to continuum modelling. Needs for future development are also discussed.

Introduction

The macroscopic behaviour of particulate matter is controlled by the interactions between individual particles as well as interactions with surrounding gas or liquid and wall. Understanding the microscopic mechanism in terms of these interactions is therefore the key leading to truly interdisciplinary research into particulate matter and producing results that can be generally used. This aim can be effectively achieved via particle scale research based on detailed microdynamic information. In recent years, such research has been rapidly developed worldwide, mainly as a result of the rapid development of discrete particle simulation technique and computer technology. Several discrete modelling techniques have been developed, including Monte Carlo method, cellular automata and discrete element method (DEM). DEM simulations can provide dynamic information, such as the trajectories of and transient forces acting on individual particles, which is extremely difficult, if not impossible, to obtain by physical experimentation at this stage of development. Consequently, it has been increasingly used in the past two decades or so.

Two types of DEMs are most common: soft-particle and hard-particle approaches. The soft-sphere method originally developed by Cundall and Strack (1979) was the first granular dynamics simulation technique published in the open literature. In such an approach, particles are permitted to suffer minute deformations, and these deformations are used to calculate elastic, plastic and frictional forces between particles. The motion of particles is described by the well-established Newton's laws of motion. A characteristic feature of the soft-sphere models is that they are capable of handling multiple particle contacts which are of importance when modelling quasi-static systems. By contrast, in a hard-particle simulation, a sequence of collisions is processed, one collision at a time and being instantaneous; often the forces between particles are not explicitly considered. Therefore, typically, hard-particle method is most useful in rapid granular flows. The two DEMs, particularly the soft-sphere method, have been extensively used to study various phenomena, such as particle packing, transport properties, heaping/piling process, hopper flow, mixing and granulation. DEM has been coupled with computational fluid dynamics (CFD) to describe particle–fluid flows such as fluidization and pneumatic conveying.

A survey of the literature indicates that many papers relating to DEM have been published in the past two decades or so, as shown in Fig. 1. The rapid development and application of DEM can also be highlighted by the large number of papers based on DEM at major international conferences, for example, from 29 in 2001 to 80 in 2005 for the International Conference on Powders and Grains, and from 34 in 2002 to 92 in 2006 for the World Congress on Particle Technology. In spite of the large bulk volume, little effort has been made to comprehensively review and summarize the progress made in the past, except for a few relatively focused reviews including, for example, Tsuji (1996) on the work in Japan, Mishra, 2003a, Mishra, 2003b on tumbling milling processes, Yu (2004) on the work done in his laboratory, Richards et al. (2004) on environmental science, and Bertrand et al. (2005) on mixing of granular materials.

To overcome this gap, we have recently reviewed the major work in this area with special reference to the soft-particle model. It is based on the publications from the Web of Science available until the mid of 2006, and covers the development of simulation techniques and their application to the study of particle packing, particle flow and particle–fluid flow. This article, as part of the review effort, presents a summary of the major theoretical developments in DEM, which are for convenience grouped into three important aspects: models for the calculation of the particle–particle and particle–fluid interaction forces, coupling of DEM with CFD to describe particle–fluid flow, and technique for linking discrete to continuum modelling. Needs for future development are also discussed.

Section snippets

Governing equations and force models

A particle in a granular flow can have two types of motion: translational and rotational. During its movement, the particle may interact with its neighbouring particles or walls and interact with its surrounding fluid, through which the momentum and energy are exchanged. Strictly speaking, this movement is affected not only by the forces and torques originated from its immediate neighbouring particles and vicinal fluid but also the particles and fluids far away through the propagation of

Particle–fluid flow

Particle flow is often coupled with fluid (gas and/or liquid) flow. In fact, coupled particle–fluid flow can be observed in almost all types of particulate processes. Understanding the fundamentals governing the flow and formulating suitable governing equations and constitutive relationships are of paramount importance to the formulation of strategies for process development and control. This necessitates a multiscale approach to understand the phenomena at different length and time scales

Transition from discrete to continuum

By use of a proper averaging procedure, a discrete particle system can be transferred into a corresponding continuum system. Extensive research has been carried out to develop such averaging methods. Various methods have been proposed to derive the balance equations of the continuum system. Earlier work often ignored the effect of the rotational motion of particles. Thus, the resultant balance equations are only for mass and linear momentum. These equations are the same as those in the

Concluding remarks

The multiscale phenomena associated with particulate matter poses a need for multiscale modelling and analysis. Fig. 7 schematically shows the approaches at different time and length scales. Since the bulk behaviour of particulate matter depends on the collective interactions among individual particles, it is the particle scale modelling and analysis that plays a crucial role in elucidating the underlying fundamentals and linking fundamental to applied research. Indeed, this has been the major

Notations

$A$	Hamaker constant, J
$c_{0}$	normalized constant of weighting function, dimensionless
$C_{d}$	fluid drag coefficient on an isolated particle, dimensionless
$C_{n}$	damping coefficient, dimensionless
$C_{r}$	viscosity coefficient, dimensionless
$C_{t}$	tangential damping coefficient, dimensionless
$C_{Vm}$	virtual mass force coefficient, dimensionless
$d_{p}$	particle diameter, m
$d_{ij}$	branch vector connecting the mass centres of particles i and j, m
$d_{i}^{b}$	ray from the mass centre of particle $i$ to $\partial Ω$ , m
$E^{*}$	reduced Young's modulus, dimensionless
$f^{c}$	contact

Acknowledgement

The authors are grateful to the Australian Research Council for the financial support of their work.

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Discrete particle simulation of particulate systems: Theoretical developments

Abstract

Introduction

Section snippets

Governing equations and force models

Particle–fluid flow

Transition from discrete to continuum

Concluding remarks

Notations

Acknowledgement

Powder Technology

International Journal of Engineering Science

Chemical Engineering Science

Advanced Powder Technology

Powder Technology

Applied Mathematical Modelling

Journal of Colloid and Interface Science

Journal of Colloid Interface Science

International Journal of Multiphase Flow

Chemical Engineering Science

Chemical Engineering Science

Chemical Engineering Science

Powder Technology

Chemical Engineering Science

Chemical Engineering Science

Computer Methods in Applied Mechanics and Engineering

Physica (Amsterdam)

Powder Technology

Powder Technology

Chemical Engineering Science

Powder Technology

Powder Technology

Advances in Colloid and Interface Science

International Journal of Multiphase Flow

Powder Technology

Chemical Engineering Science

Chemical Engineering Science

Powder Technology

Chemical Engineering Science

Powder Technology

Powder Technology

Computer Methods in Applied Mechanics and Engineering

Powder Technology

Chemical Engineering Science

Chemical Engineering Science

Powder Technology

Chemical Engineering Science

Powder Technology

Chemical Engineering Science

Powder Technology

Chemical Engineering Science

Chemical Engineering Science

Chemical Engineering Science

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