TheInfoList 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, $V_p$ :$A_^3 = \left\left(4 \pi r^2\right\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left\left(4^2 \pi^2 r^6\right\right) = 4 \pi \cdot 3^2 \left\left(\frac r^6\right\right) = 36 \pi \left\left(\frac r^3\right\right)^2 = 36\,\pi V_^2$ therefore :$A_ = \left\left(36\,\pi V_^2\right\right)^ = 36^ \pi^ V_^ = 6^ \pi^ V_^ = \pi^ \left\left(6V_\right\right)^$ hence we define $\Psi$ as: :$\Psi = \frac = \frac$

Sphericity of common objects

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

References