Sphericities of non-spherical objects

Abstract

Sphericity, as one of the most important shape parameter for non-spherical objects, is extensively applied in evaluating the porosity or packing density of particles. In this paper, the sphericities of common non-spherical objects are deduced and investigated. Maximum sphericities and optimum shapes of these objects are presented as well. A decreasing order of sphericity from sphere (1.0) to regular tetrahedron (0.671) for objects with constant sphericity is given. Similar trends are found in most sphericity–aspect ratio relationships, which exhibit single peak and the sphericity increases with the growth of aspect ratio before the peak point and decreases afterward. The peak loci of aspect ratio are all around 1.0 which makes the shape approaching to a sphere. The information in the paper could be useful as literature for general application.

Introduction

The term sphericity was defined by Wadell in 1935 (Wadell, 1935), as the ratio of the surface area of a sphere which has the same volume as the given particle to the surface area of that particle, i.e., the sphericity is a shape factor to tell how round a particle is, and the sphere has the highest sphericity of unity. Sphericity, as one of the most important parameter for non-spherical particle morphology, has been widely applied in the research of many fields. For example, sphericity has been used to describe the effect of particle morphology in circulating fluidized bed combustion (CFBC) (Liu, Zhang, Wang, & Hong, 2008), to give the drag coefficient and terminal velocity of falling particles (Haider & Levenspiel, 1989), and to distinguish the aspects of the shape of rock particles in the study of sediments in geology (Barrett, 1980). Moreover, one of the most significant applications of the sphericity is to evaluate the porosity (or the packing density) of particle packing. For sphere packing, prediction model has been developed to describe the relationship between porosity and particle size distribution (Westman, 1936, Yu et al., 1996). For non-spherical particle packing, empirical methods have also been developed to predict the porosity of mono-sized objects (Zou & Yu, 1996) and binary mixtures (Yu, Standish, & Mclean, 1993). These methods are based on the concept of equivalent packing diameter (Yu and Standish, 1993, Zou and Yu, 1996) which is determined by the sphericity of the particle and the equivalent volume diameter. Consequently, the sphericity is one of the most important factors for porosity evaluation in engineering applications.

However, study on sphericity is far beyond adequate, where only a few sphericities of basic non-spherical objects were listed (http://en.wikipedia.org/wiki/Sphericity; Yang, 2003), whose sphericities are mostly constant. The purpose of this paper is to deduce and analyze the sphericity of non-spherical objects in order to (a) provide a reference for the sphericities (or sphericity functions) of common non-spherical objects; (b) show how sphericity changes with shape factors and what properties (maximum value, monotonicity, etc.) the sphericity functions may have. Such efforts should avoid duplicate derivations and can be useful in the literature for general application.

We only show the derivation of sphericity for a cube in Section 2.1.1, the derivations for other referred particles are similar. Table 1, Table 2 in Section 3 give the sphericities and volumes, surface areas of common non-spherical objects, respectively. One may quickly get the sphericity–aspect ratio relationship with the data and therefore we only give the definitions of parameters used and the results in the main text.