Sphericity

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

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,^[1] the sphericity, $\Psi$ , 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:

\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{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, $\Psi$ , 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 particle, $V_p$

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
Platonic Solids
tetrahedron	$\frac{\sqrt{2}}{12}\,s^3$	$\sqrt{3}\,s^2$	$\left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671$
cube (hexahedron)	$\,s^3$	$6\,s^2$	$\left( \frac{\pi}{6} \right)^{\frac{1}{3}} \approx 0.806$
octahedron	$\frac{1}{3} \sqrt{2}\, s^3$	$2 \sqrt{3}\, s^2$	$\left( \frac{\pi}{3\sqrt{3}} \right)^{\frac{1}{3}} \approx 0.846$
dodecahedron	$\frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3$	$3 \sqrt{25 + 10\sqrt{5}}\, s^2$	$\left( \frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}} \right)^{\frac{1}{3}} \approx 0.910$
icosahedron	$\frac{5}{12}\left(3+\sqrt{5}\right)\, s^3$	$5\sqrt{3}\,s^2$	$\left( \frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}} \right)^{\frac{1}{3}} \approx 0.939$
Round Shapes
ideal cone $(h=2\sqrt{2}r)$	$\frac{1}{3} \pi\, r^2 h$ $= \frac{2\sqrt{2}}{3} \pi\, r^3$	$\pi\, r (r + \sqrt{r^2 + h^2})$ $= 4 \pi\, r^2$	$\left( \frac{1}{2} \right)^{\frac{1}{3}} \approx 0.794$
hemisphere (half sphere)	$\frac{2}{3} \pi\, r^3$	$3 \pi\, r^2$	$\left( \frac{16}{27} \right)^{\frac{1}{3}} \approx 0.840$
ideal cylinder $(h=2\,r)$	$\pi r^2 h = 2 \pi\,r^3$	$2 \pi r ( r + h ) = 6 \pi\,r^2$	$\left( \frac{2}{3} \right)^{\frac{1}{3}} \approx 0.874$
ideal torus $(R=r)$	$2 \pi^2 R r^2 = 2 \pi^2 \,r^3$	$4 \pi^2 R r = 4 \pi^2\,r^2$	$\left( \frac{9}{4 \pi} \right)^{\frac{1}{3}} \approx 0.894$
sphere	${\frac {4}{3}}\pi r^{3}$	$4 \pi\,r^2$	$1\,$

References

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

External links

Look up sphericity in Wiktionary, the free dictionary.

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