Particle shape — sphericity using equivalent diameter, volume, and surface area If dp is the volume-equivalent diameter of a non-spherical particle, Vp its actual volume, and Sp its actual surface area, what is the correct expression for sphericity Φs?

Introduction / Context:
Sphericity Φs quantifies how closely a particle's shape approaches a sphere. It is essential in calculating drag, settling velocity, and permeability in packed beds, where deviations from sphericity strongly affect hydrodynamics.

Given Data / Assumptions:

• dp is the diameter of a sphere that has the same volume as the particle (volume-equivalent).
• Vp is the particle volume; Sp is the particle surface area.
• By definition, Φs = (surface area of volume-equivalent sphere) / (actual particle surface area).

Concept / Approach:
Volume-equivalent sphere volume = (π/6) dp^3; its surface area = π dp^2. Therefore Φs = (π dp^2)/Sp. Rearranging using Vp and dp yields an equivalent expression not requiring π explicitly: 6 Vp/dp = π dp^2 for a volume-equivalent sphere, hence Φs = (6 Vp/dp)/Sp.

Step-by-Step Solution:
Start with Φs = (surface area of sphere with volume Vp) / Sp.For volume-equivalent dp: Vp = (π/6) dp^3 ⇒ 6 Vp/dp = π dp^2.Thus Φs = (6 Vp/dp)/Sp.

Verification / Alternative check:
Dimensional check: 6Vp/dp has dimensions of area; dividing by Sp yields a dimensionless Φs (0 < Φs ≤ 1).

Why Other Options Are Wrong:

• Options (b), (c), (d) are not dimensionally consistent or invert the relation, giving Φs > 1 for many shapes, which is impossible.

Common Pitfalls:

• Confusing dp as surface-area equivalent diameter instead of volume-equivalent; that changes formulas.

Final Answer:
Φs = 6 Vp/(dp Sp)