Equivalent packing size of spheroidal particles: A microscopic test

doi:10.1016/j.powtec.2018.04.035

Powder Technology

Volume 333, 15 June 2018, Pages 286-292

https://doi.org/10.1016/j.powtec.2018.04.035 Get rights and content

Highlights

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DEM has been used to simulate the packing of binary mixtures of spheres and spheroids.
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The packing structure is quantified by coordination number, a popular microscopic parameter.
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The equivalent packing diameter of a particle can be quantified by coordination number.
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The resulting packing sizes are consistent with those determined experimentally.

Abstract

The equivalent packing size of spheroidal particles is studied microscopically. It is done based on the structural results obtained by means of discrete element method. The packing similarity between spheres and spheroids is first established in terms of coordination number, the most popular microscopic parameter associated with packing structure. It is then shown through the analysis of the mean coordination number of each component in a binary mixture of spheres and spheroids that the equivalent packing size does exist. The results are consistent with those determined in terms of macroscopic parameters such as packing density or specific volume. The findings provide evidence to support the general application of the concept of equivalent packing size.

Graphical abstract

Packing density of binary mixtures of spheroids (d_v,e = 10 mm, and η = 0.25) and spheres of different diameters at X_s = 0.5.

Introduction

Particle packing is of fundamental importance to granular materials, where the particle-size distribution (PSD) is a key control variable [1]. In the past, a few mathematical models have been developed to predict the packing density (=1-porosity) of a mixture of particles as a function of PSD [[2], [3], [4], [5], [6], [7], [8]]. However, these models are mainly for spherical particles. How to extend such a model from spherical to non-spherical particle packing is very challenging. One way to do so is based on the so called equivalent diameter so that a mathematical model for spherical particles can be applied to non-spherical particles. In the past, different equivalent spherical diameters have been proposed [9]. For example, the equivalent volume diameter of a non-spherical particle, d_v, is defined as the diameter of a sphere with the same volume or surface area of the particle. For fluid-particle systems, it is often defined based on the interaction between a single particle with a medium such as gas or liquid, where the settling of a particle in a liquid is a good example [[9], [10], [11], [12]]. All these definitions, however, are not based on the interaction between particles, which actually control the packing of particles.

To solve this problem, the so called equivalent packing diameter is proposed [13,14]. By use of this concept, Yu and his colleagues have successfully developed mathematical models to predict the porosity of a packing of non-spherical coarse particles [15,16], or fine particles [17,18]. In their work, the equivalent packing diameter of a non-spherical particle is determined based on the similarity between spherical and non-spherical particle packings. In particular, it results from the analysis of particle-particle interactions in terms of porosity or specific volume variation. According to Yu and Standish [13], the equivalent packing size of a non-spherical particle is defined as the diameter of spheres which, when mixed with the non-spherical particles, gives the minimum packing density or zero specific volume variation. Correspondingly, it can be quantified from the study of the packing of binary mixtures of a type of non-spherical particles and spherical particles. It has been confirmed that the obtained equivalent packing diameter is independent of packing method and volume fraction of a component [14]. That is, once determined, the equivalent packing diameter of a particle can be used generally. This approach or its variants are useful in the modelling of the packing of coarse and/or fine non-spherical particles for different applications [[15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]]. However, to date, it is not clear if the concept of equivalent packing diameter is valid from the microscopic, structural perspective.

To overcome this problem, this paper provides a study of the equivalent packing diameter in term of coordination number, the most popular parameter used to describe the packing structure of particles. It is focused on the packing of binary mixtures of spherical and spheroidal particles. Discrete element method (DEM) is used to simulate a packing process, with aspect ratio and volume fraction as variables. The results of coordination numbers will be analysed to first establish the similarity between spherical and non-spherical particle packings and then quantify the equivalent packing diameter of spheroidal particles. Finally, it is shown that the resulting equivalent packing diameters are consistent with those determined macroscopically, confirming the generality of this concept in the study of particle packing.

Section snippets

DEM simulation

The DEM originally proposed by Cundall & Strack [31] has been well established in the literature. The model has been modified or further developed to simulate particle packing under different conditions [[32], [33], [34], [35], [36]]. It has been extended to non-spherical particles, particularly for spheroids [[37], [38], [39]]. The DEM model used for the present work is the same as that used by Zhou et al. [38], where its details and applicability have been discussed. As such, it is not

Results and discussion

In this section, the results will be analysed in terms of macroscopic parameters such as packing density or specific volume V, followed in terms of coordination number which is known to be the most representative microscopic, structural parameter. Then, the macro- and micro-scopic outcomes will be compared.

Conclusions

The packing of binary mixtures of spheroidal and spherical particles has been studied by means of DEM, aiming to explore the applicability of the concept of equivalent packing diameter from a microscopic, structural viewpoint. It is found that the equivalent packing diameter of spheroidal particles can be determined in terms of macroscopic parameters such as packing density or specific volume, or in terms of microscopic parameter such as mean coordination number of a component in a mixture. The

Nomenclature

radius of spheroidal particles along the polar direction, mm

b/c

radius of spheroidal particles in the Equatorial plane (b = c), mm

overall mean coordination number

c_n

normal damping coefficient

c_t

tangential damping coefficient

diameter of the cylindrical container used in simulations, mm

d_p

equivalent packing diameter, mm

d_s

diameter of spheres in binary mixtures of spheroids and spheres, mm

d_v

equivalent volume diameter of a non-spherical particle, mm

d_v,e

equivalent volume diameter of spheroids in

Acknowledgements

The authors are grateful to the Australian Research Council (ARC) and BlueScope Steel Research for the financial support of this work, and the Australian National Facility (NCI) for the support in computation.

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Equivalent packing size of spheroidal particles: A microscopic test

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

DEM simulation

Results and discussion

Conclusions

Nomenclature

Acknowledgements

J. Colloid Interface Sci.

Powder Technol.

Powder Technol.

Powder Technol.

Powder Technol.

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Powder Technol.

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Chem. Eng. Sci.

Miner. Eng.

Cem. Concr. Compos.

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J. Food Eng.

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Physica A

Physica A

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Particle Packing Characteristics