Sphericity Explained

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,

\Psi , of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:\Psi = (\pi ^{(1)/(3)} ( 6 V ^{(2)/(3)} _{p)})/(A _{p})

where

V _{p} is volume of the object and A _{p} 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

See also: Flattening.

The sphericity,

\Psi , of an oblate spheroid (similar to the shape of the planet Earth) is:\Psi = (\pi ^{(1)/(3)} ( 6 V ^{(2)/(3)} _{p)})/(= A _{p}) (2 \sqrt[ 3 ]{ab ^{2})/() },

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 object being measured, V _{p} A ^{3} _{s} = \left( 4 \pi r ^{2\right)} ^{3} = 4 ^{3} \pi ^{3} r ^{6} = 4 \pi \left( 4 ^{2} \pi ^{2} r ^{6\right)} = 4 \pi ⋅ 3 ^{2} \left( (4 ^{2} \pi ^{2})/(3 ^{2}) r ^{6\right)} = 36 \pi \left( (4 \pi)/(3) r ^{3\right)} ^{2} = 36 \pi V ^{2} _{p}

therefore

A _{s} = \left( 36 \pi V ^{2 \right)} _{p} ^{(1)/(3)} = 36 ^{(1)/(3)} \pi ^{(1)/(3)} V ^{(2)/(3)} _{p} = 6 ^{(2)/(3)} \pi ^{(1)/(3)} V ^{(2)/(3)} _{p} = \pi ^{(1)/(3)} \left( 6 V _{p} \right) ^{(2)/(3)}

hence we define

\Psi as:\Psi = (A _{s})/(A _{p}) = (\pi ^{(1)/(3)} \left( 6 V _{p} \right) ^{(2)/(3)})/(A _{p})

Sphericity of common objects

Name	Picture	Volume	Surface area	Sphericity
Sphere		(4)/(3) \pi r ^{3}	4 \pir ^{2}	data-sort-value=1	1
Disdyakis triacontahedron		(180)/(11) \left( 5+4 \sqrt{ 5 }\right)s ^{3}	(180)/(11) \sqrt{ 179-24 \sqrt{ 5 }}s ^{2}	data-sort-value=0.9857	(\left(\left( 5+4 \sqrt{ 5)/(\right)) ^{2} (11 \pi)/(5) \right) ^{(1)/(3)} }\approx0.9857
Rhombic triacontahedron		4 \sqrt{ 5+2 \sqrt{ 5 }}s ^{3}	12 \sqrt{ 5 }s ^{2}	data-sort-value=0.9609	(\pi ^{(1)/(3)} \left( 24 \sqrt{ 5+2 \sqrt{ 5)/() }\right)^}\approx0.9609
Icosahedron		(5)/(12) \left( 3+ \sqrt{ 5 }\right)s ^{3}	5 \sqrt{ 3 }s ^{2}	data-sort-value=0.939	\left( (\left( 3+ \sqrt{ 5)/(\right)) ^{2\pi}{60\sqrt{3}}\right)} ^{(1)/(3)} ≈ 0 . 939
Dodecahedron		(1)/(4) \left( 15 + 7 \sqrt{ 5 }\right) s ^{3}	3 \sqrt{ 25 + 10 \sqrt{ 5 }} s ^{2}	data-sort-value=0.910	\left( (\left( 15+7 \sqrt{ 5)/(\right)) ^{2} \pi}{ 12 \left( 25+10 \sqrt{ 5 }\right) ^{(3)/(2)} }\right)^\approx0.910
Ideal torus (R = r)		2 \pi ^{2Rr} ^{2=2\pi} ^{2r} ^{3}	4 \pi ^{2Rr=4\pi} ^{2r} ^{2}	data-sort-value=0.894	\left( (9)/(4 \pi) \right) ^{(1)/(3)} ≈ 0 . 894
Ideal cylinder (h =2 r)		\pir ^{2h=2\pir} ^{3}	2 \pir(r + h) =6 \pir ^{2}	data-sort-value=0.874	\left( (2)/(3) \right) ^{(1)/(3)} ≈ 0 . 874
Octahedron		(1)/(3) \sqrt{ 2 }s ^{3}	2 \sqrt{ 3 }s ^{2}	data-sort-value=0.846	\left( (\pi)/(3 \sqrt{ 3) }\right)^\approx0.846
Hemisphere (half sphere)		(2)/(3) \pir ^{3}	3 \pir ^{2}	data-sort-value=0.840	\left( (16)/(27) \right) ^{(1)/(3)} ≈ 0 . 840
Cube (hexahedron)		s ^{3}	6 s ^{2}	data-sort-value=0.806	\left( (\pi)/(6) \right) ^{(1)/(3)} ≈ 0 . 806
Ideal cone (h =2 \sqrt{ 2 }r)		(1)/(3) \pir ^{2 h = (2 \sqrt{ 2)/()} }\pi\,r^3	\pir(r + \sqrt{r ^{2+h} ^{2})=4\pir} ^{2}	data-sort-value=0.794	\left( (1)/(2) \right) ^{(1)/(3)} ≈ 0 . 794
Tetrahedron		(\sqrt{ 2)/() }\,s^3	\sqrt{ 3 }s ^{2}	data-sort-value=0.671	\left( (\pi)/(6 \sqrt{ 3) }\right)^\approx0.671

See also

• Equivalent spherical diameter
• Flattening
• Isoperimetric ratio
• Rounding (sediment)
• Roundness
• Willmore energy

External links

• Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

Notes and References

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