For a solid particle that is a perfect cube, what is its sphericity ψ (defined as area of a sphere with equal volume divided by actual particle area)? Provide the canonical closed form.

Introduction / Context:Sphericity ψ is a fundamental shape factor in particle technology, influencing drag, heat and mass transfer, and packing. It is defined as the surface area of a sphere having the same volume as the particle divided by the actual surface area of the particle. Spheres have ψ = 1 and all other shapes have ψ < 1.

Given Data / Assumptions:

• Particle shape: perfect cube with side length a.
• Definition: ψ = A_sphere(V_cube) / A_cube.

Concept / Approach:Compute the area of the equivalent-volume sphere and divide by the area of the cube. For a cube, volume V = a^3 and surface area A_cube = 6 * a^2. The sphere with equal volume has radius r such that (4/3) * π * r^3 = a^3. Its surface area is A_sphere = 4 * π * r^2.

Step-by-Step Solution:

Verification / Alternative check:Numerically, (π/6)^(1/3) ≈ 0.806, which is less than 1 as expected for a cube and matches handbook tables.

Why Other Options Are Wrong:

Common Pitfalls:Using the wrong definition (e.g., volume shape factor) or forgetting to compute the equivalent-volume sphere correctly.

Final Answer:(π/6)^(1/3)