Sphericity


Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape. Defined by Wadell in 1935,[1] the sphericity, , of a particle is the ratio of the surface area of a sphere with the same volume as the given particle to the surface area of the particle:

Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

where is volume of the particle and is the surface area of the particle. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any particle which is not a sphere will have sphericity less than 1.

Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft, is called roundness.

Ellipsoidal objects


The sphericity, , of an oblate spheroid (similar to the shape of the planet Earth) is:

where a and b are the semi-major and semi-minor axes respectively.

Derivation


Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.

First we need to write surface area of the sphere, in terms of the volume of the particle,

therefore

hence we define as:

Sphericity of common objects


Name Picture Volume Surface Area Sphericity
Platonic Solids
tetrahedron
cube (hexahedron)

octahedron

dodecahedron

icosahedron
Round Shapes
ideal cone


hemisphere
(half sphere)

ideal cylinder

ideal torus

sphere

Other Shapes
rhombic triacontahedron
disdyakis triacontahedron

See also


References


  1. Wadell, Hakon (1935). "Volume, Shape, and Roundness of Quartz Particles". The Journal of Geology. 43 (3): 250–280. Bibcode:1935JG.....43..250W. doi:10.1086/624298.