Harman Patil (Editor)

Sphericity

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Sphericity

Sphericity is the measure of how closely the shape of an object approaches that of a mathematically 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, the sphericity, Ψ , 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:

Contents

Ψ = π 1 3 ( 6 V p ) 2 3 A p

where V p is volume of the particle and A p 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, Ψ , of an oblate spheroid (similar to the shape of the planet Earth) is:

Ψ = π 1 3 ( 6 V p ) 2 3 A p = 2 a b 2 3 a + b 2 a 2 − b 2 ln ⁡ ( a + a 2 − b 2 b ) ,

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, A s in terms of the volume of the particle, V p

A s 3 = ( 4 π r 2 ) 3 = 4 3 π 3 r 6 = 4 π ( 4 2 π 2 r 6 ) = 4 π ⋅ 3 2 ( 4 2 π 2 3 2 r 6 ) = 36 π ( 4 π 3 r 3 ) 2 = 36 π V p 2

therefore

A s = ( 36 π V p 2 ) 1 3 = 36 1 3 π 1 3 V p 2 3 = 6 2 3 π 1 3 V p 2 3 = π 1 3 ( 6 V p ) 2 3

hence we define Ψ as:

Ψ = A s A p = π 1 3 ( 6 V p ) 2 3 A p

References

Sphericity Wikipedia
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