Particle shape factor: what is the sphericity of a right circular cylinder with diameter 1 mm and length 3 mm?

Introduction / Context:
Sphericity is a dimensionless shape factor used in fluid–particle systems (sedimentation, packed beds, filtration). It helps correct correlations for drag, settling velocity, and pressure drop when the particle is not a perfect sphere.

Given Data / Assumptions:

• Cylinder diameter d = 1 mm, length L = 3 mm.
• Sphericity, psi = (surface area of a sphere having the same volume as the particle) / (actual surface area of the particle).
• Ends of the cylinder are flat; include both end and lateral areas.

Concept / Approach:
Compute particle volume and actual area for the cylinder; then find the sphere having the same volume and compute its surface area. Finally, take the ratio Asphere / Aparticle to obtain the sphericity.

Step-by-Step Solution:
Volume, Vcyl = (pi/4) * d^2 * L = (pi/4) * 1^2 * 3 mm^3 = 0.75pi mm^3.Equivalent sphere diameter, De such that (pi/6)De^3 = Vcyl → De = (6Vcyl/pi)^(1/3) ≈ 1.651 mm.Surface area of equivalent sphere, As = pi * De^2 ≈ 8.56 mm^2.Actual cylinder area, A = lateral + ends = pidL + 2(pid^2/4) = pi13 + pi(1^2)/2 ≈ 10.996 mm^2.Sphericity, psi = As / A ≈ 8.56 / 11.00 ≈ 0.78.

Verification / Alternative check:
Typical sphericities: spheres = 1; cylinders with L/d ≈ 3 fall around 0.75–0.8, consistent with the computed value.

Why Other Options Are Wrong:
0.9: too high; closer to near-spherical shapes.0.6 or 0.5: too low for L/d = 3; would indicate highly elongated or rough particles.

Common Pitfalls:
Forgetting to include end areas; mixing diameter and radius; using volume ratio instead of area ratio in the definition.

Final Answer:
0.78