As Webster 1913 mentions, sphericity refers to the roundness of objects in solid geometry. However, in this definition sphericity is not quantified.

It is quite common in physics and engineering to approximate objects as being spherical. For instance, in calculations on planetary motion, the planets are often considered to be spherical instead of ellipsoidal. In engineering calculations, assuming a spherical geometry often reduces the mathematical complexity of models.

In Chemical Engineering, sphericity is sometimes used as a characterization parameter for solid particles. For instance, in fluidized bed reactors, particulate matter is fluidized by an upward flowing gas or liquid in a vertical column; the sphericity of the particles inside the column is one of the key design parameters that determine the operating conditions of the reactor. Another, more generic name for this parameter is shape factor

Sphericity is quantified by using a unique mathematical property of the sphere; the sphere has the lowest surface-to-volume ratio of any solid geometric objects. Thus, for an object with volume V, the external surface area A is minimal if the object is spherical. The sphericity Ψ of an object or particle can be calculated by visualizing a sphere whose volume is equal to the particle's and dividing the surface area of this sphere by the actually measured surface area of the particle:

      Ψ = As / Ap

Where: As is the surface area of the equivalent sphere and Ap is the measured surface area. The sphericity can have a value ranging from 0-1, where Ψ = 1 for an ideal sphere.

The volume of a spherical particle is:

      Vp= (1/6) π dp3

Where: dp is the diameter of the particle.

The surface area of a sphere is:

      As = π dp2 = π [ (6 Vp / π)(1/3) ]2

Thus, for a particle, Ψ can be calculated by measuring its volume and surface area:

      Ψ = As / Ap = π (6 Vp / π)(2/3) / Ap

An example: A cube measuring 1 × 1 × 1 cm has a volume of 1 cm3, and a surface area of 6 × (1 × 1) = 6 cm2. Its sphericity is:

      Ψ = π × (6 × 1 / π)(2/3) / 6 = 0.806

Another example: A cylinder with a diameter of 1 cm, and height of 1 cm has a volume of:

      Vp = 0.25 × π × 12 × 1 = 0.785 cm3.

Its surface area is:

      Ap = 2 × (0.25 × π × 12) + (π × 1 × 1) = 4.712 cm2
The sphericity of this cylinder is:
      Ψ = π × (6 × 0.785 / π)(2/3) / 4.712 = 0.874

If we compare the sphericity of the cube (Ψ = 0.806) to that of the cylinder (Ψ = 0.874), we can conclude that the cylinder is more spherical (as would be expected).

Exercise for the reader: measure the surface area and volume of a cow, and calculate how well this mammal is approximated by a sphere.

Sphe*ric"i*ty (?), n. [Cf. F. sph'ericit'e.]

The quality or state of being spherial; roundness; as, the sphericity of the planets, or of a drop of water.


© Webster 1913.

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