Sphericity
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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.
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.
Definition
Defined by Wadell in 1935,[1] the sphericity, , of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:
where is volume of the object and is the surface area. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any shape which is not a sphere will have sphericity less than 1.
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 object being measured,
therefore
hence we define as:
Sphericity of common objects
Name | Picture | Volume | Surface area | Sphericity |
---|---|---|---|---|
Sphere | ![]() | 1 | ||
Disdyakis triacontahedron | ![]() | |||
Rhombic triacontahedron | ![]() | |||
Icosahedron | ![]() | |||
Dodecahedron | ![]() | |||
Ideal torus | ![]() | |||
Ideal cylinder | ![]() | |||
Octahedron | ![]() | |||
Hemisphere (half sphere) | ![]() | |||
Cube (hexahedron) | ![]() | |||
Ideal cone | ![]() | |||
Tetrahedron | ![]() |
See also
- Equivalent spherical diameter
- Flattening
- Isoperimetric ratio
- Rounding (sediment)
- Roundness
- Willmore energy
References
- ^ 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.
External links
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png)
- Grain Morphology: Roundness, Surface Features, and Sphericity of Grains