Sphericity φ for a non-spherical particle: choose the correct expression in terms of particle volume V, particle surface area S, and the diameter of a sphere with the same volume.

Introduction / Context:
Sphericity is a shape factor widely used in fluid-particle systems to correct drag, settling velocity, and packed-bed correlations. It normalizes any particle against a sphere of the same volume.

Concept / Approach:
By definition, φ = (surface area of the volume-equivalent sphere) / (actual particle surface area). For a sphere of diameter D_eq having the same volume V as the particle, D_eq = (6V/π)^(1/3). The sphere’s surface area is A_sph = π D_eq^2. Therefore φ = A_sph / S = π D_eq^2 / S. Substituting for D_eq yields φ = π * [(6V/π)^(2/3)] / S = (π^(1/3) * (6V)^(2/3)) / S.

Step-by-Step Solution:
Compute D_eq from V: D_eq = (6V/π)^(1/3).Compute sphere area: A_sph = π D_eq^2.Form ratio: φ = A_sph / S.

Why Other Options Are Wrong:
S / (π D_eq^2): this is the reciprocal (shape factor >= 1).V / S: wrong dimensions; not bounded by 1.(6V/π) / S: incorrect exponent on V; ignores square power for area.

Common Pitfalls:
Confusing φ with its reciprocal (often termed shape factor). Remember φ ≤ 1 and equals 1 only for a sphere.

Final Answer:
φ = (surface area of equal-volume sphere) / S = π · D_eq^2 / S, where D_eq = (6V/π)^(1/3)