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
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
Vp= (1/6) π dp3
Where: dp is the diameter of the particle.
The surface area of a
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