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. Defined by Wadell in 1935, 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:

:$\Psi = \frac$

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:

:$\Psi = \frac = \frac,$

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, $V_p$

:$A_^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac r^6\right) = 36 \pi \left(\frac r^3\right)^2 = 36\,\pi V_^2$

therefore

:$A_ = \left(36\,\pi V_^2\right)^ = 36^ \pi^ V_^ = 6^ \pi^ V_^ = \pi^ \left(6V_\right)^$

hence we define $\Psi$ as:

:$\Psi = \frac = \frac$

Sphericity of common objects

See also

* Equivalent spherical diameter
*Flattening
* Index of sphericity
* Isoperimetric ratio
*Rounding (sediment)
*Roundness
*Willmore energy

References

External links

{{Wiktionary, sphericity
Grain Morphology: Roundness, Surface Features, and Sphericity of Grains

Geometric measurement
Spheres
Metrology