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.

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 ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{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

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

\Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}},

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_{s}^{3}=\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 \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{p}^{2}

therefore

A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}

hence we define $\Psi$ as:

\Psi ={\frac {A_{s}}{A_{p}}}={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}

Sphericity of common objects

Name	Volume	Surface area	Sphericity
Sphere	${\frac {4}{3}}\pi r^{3}$	$4\pi \,r^{2}$	1
Disdyakis triacontahedron	${\frac {180}{11}}\left(5+4{\sqrt {5}}\right)\,s^{3}$	${\frac {180}{11}}{\sqrt {179-24{\sqrt {5}}}}\,s^{2}$	${\frac {\left(\left(5+4{\sqrt {5}}\right)^{2}{\frac {11\pi }{5}}\right)^{\frac {1}{3}}}{\sqrt {179-24{\sqrt {5}}}}}\approx 0.9857$
Rhombic triacontahedron	$4{\sqrt {5+2{\sqrt {5}}}}\,s^{3}$	$12{\sqrt {5}}\,s^{2}$	${\frac {\pi ^{\frac {1}{3}}\left(24{\sqrt {5+2{\sqrt {5}}}}\right)^{\frac {2}{3}}}{12{\sqrt {5}}}}\approx 0.9609$
Icosahedron	${\frac {5}{12}}\left(3+{\sqrt {5}}\right)\,s^{3}$	$5{\sqrt {3}}\,s^{2}$	$\left({\frac {\left(3+{\sqrt {5}}\right)^{2}\pi }{60{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.939$
Dodecahedron	${\frac {1}{4}}\left(15+7{\sqrt {5}}\right)\,s^{3}$	$3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}$	$\left({\frac {\left(15+7{\sqrt {5}}\right)^{2}\pi }{12\left(25+10{\sqrt {5}}\right)^{\frac {3}{2}}}}\right)^{\frac {1}{3}}\approx 0.910$
Ideal torus $(R=r)$	$2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}$	$4\pi ^{2}Rr=4\pi ^{2}\,r^{2}$	$\left({\frac {9}{4\pi }}\right)^{\frac {1}{3}}\approx 0.894$
Ideal cylinder $(h=2\,r)$	$\pi \,r^{2}h=2\pi \,r^{3}$	$2\pi \,r(r+h)=6\pi \,r^{2}$	$\left({\frac {2}{3}}\right)^{\frac {1}{3}}\approx 0.874$
Octahedron	${\frac {1}{3}}{\sqrt {2}}\,s^{3}$	$2{\sqrt {3}}\,s^{2}$	$\left({\frac {\pi }{3{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.846$
Hemisphere (half sphere)	${\frac {2}{3}}\pi \,r^{3}$	$3\pi \,r^{2}$	$\left({\frac {16}{27}}\right)^{\frac {1}{3}}\approx 0.840$
Cube (hexahedron)	$\,s^{3}$	$6\,s^{2}$	$\left({\frac {\pi }{6}}\right)^{\frac {1}{3}}\approx 0.806$
Ideal cone $(h=2{\sqrt {2}}r)$	${\frac {1}{3}}\pi \,r^{2}h={\frac {2{\sqrt {2}}}{3}}\pi \,r^{3}$	$\pi \,r(r+{\sqrt {r^{2}+h^{2}}})=4\pi \,r^{2}$	$\left({\frac {1}{2}}\right)^{\frac {1}{3}}\approx 0.794$
Tetrahedron	${\frac {\sqrt {2}}{12}}\,s^{3}$	${\sqrt {3}}\,s^{2}$	$\left({\frac {\pi }{6{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.671$

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

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

[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.

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