How to Calculate Sphericity

Sphericity is a measure of the roundness of a shape. A sphere is the most compact solid, so the more compact an object is, the more closely it resembles a sphere. Sphericity is a ratio and therefore a dimensionless number. It has applications in geology, where it is important to classify particles according to their shape. Sphericity may be calculated for any three-dimensional object if its surface area and volume are known.

    Define sphericity mathematically as Y = As/Ap, where Y is the sphericity, Ap is the surface area of a test particle P, and As is the surface area of a sphere S with the same volume as P. Since the volume V of the two objects is equal, we can say that Vs = Vp.

    Calculate the radius of a sphere in terms of its volume. The volume of a sphere is V = 4/3 ? r^3, where V is the volume and r is the radius. V = 4/3 ? r^3 => 3V/4? = r^3 => r = (3V/4?)^(1/3).

    Express the surface area of the sphere in terms of its volume. The surface area of a sphere is A = 4? r^2. Using the solution for r obtained in Step 2, we have A = 4? (3V/4?)^(1/3)^2 = 4? (3V/4?)^(2/3) = 4?^(1/3)(3V/4)^(2/3) = ?^(1/3)(4^(3/2)3V/4)^(2/3) = ?^(1/3)(8)3V/4)^(2/3) = ?^(1/3)(6V)^(2/3). Therefore, A = ?^(1/3)(6V)^(2/3) for all spheres.

    Substitute the equality A = ?^(1/3)(6V)^(2/3) obtained in Step 3 into the equation Y = As/Ap for the sphericity given in Step 1. This gives us Y = As/Ap = ?^(1/3)(6V)^(2/3)/Ap. Thus, the sphericity of a particle P is given by Y = ?^(1/3)(6Vp)^(2/3)/Ap, where Vp is the particle's volume and Ap is its surface area.

    Interpret the sphericity ratio. Since a sphere is the most compact three-dimensional object, As <= Ap so 0 < Y <= 1. Thus, the closer the sphericity is to 1, the more round P is.


About the Author

Allan Robinson has written numerous articles for various health and fitness sites. Robinson also has 15 years of experience as a software engineer and has extensive accreditation in software engineering. He holds a bachelor's degree with majors in biology and mathematics.

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