Shape factor in particle technology: For a non-spherical particle, how is “sphericity” (ψ) correctly defined?

Introduction / Context:
Sphericity quantifies how closely a particle’s shape approximates a sphere. It is widely used in correlations for drag, heat transfer, and packed-bed pressure drop (e.g., Ergun equation). A correct definition is essential for consistent calculations.

Given Data / Assumptions:

• Compare an arbitrary particle with a sphere that has the same volume.
• Use surface areas to construct a dimensionless ratio.
• Particles are rigid and nonporous for the definition.

Concept / Approach:
Sphericity ψ is defined as ψ = A_sphere,eq / A_particle, where A_sphere,eq is the surface area of a sphere having the same volume as the particle. Because the sphere has the minimum surface area for a given volume, ψ ≤ 1, with ψ = 1 for a perfect sphere.

Step-by-Step Solution:
Let V_p be the particle volume.Define sphere of equal volume: diameter d_v so that (pi/6) d_v^3 = V_p.Compute A_sphere,eq = pi * d_v^2, compare to actual surface area A_p.Therefore ψ = A_sphere,eq / A_p (dimensionless).

Verification / Alternative check:
Textbook definitions and Perry’s handbook align with this expression and note that ψ approaches 1 as shapes near spherical.

Why Other Options Are Wrong:

• Dimension of length: ψ is dimensionless.
• “Always less than 1”: incorrect because spheres have ψ = 1 exactly.
• Volume-based inversion (Option D) is not the standard definition.
• Option E is not a recognized sphericity form.

Common Pitfalls:
Mixing sphericity with aspect ratio, or confusing equal-volume vs equal-surface reference spheres.

Final Answer:
ψ = surface area of a sphere having the same volume as the particle divided by the actual particle surface area