Hindered settling calculation (Richardson–Zaki): a suspension of glass beads in ethylene glycol settles at 1.7 mm/s; a single bead settles at 17 mm/s. With index n = 4.5, what is the solids volume fraction φ?

Introduction / Context:Hindered settling in suspensions is slower than the terminal settling of a single particle because surrounding particles impede flow. The Richardson–Zaki correlation relates the hindered settling velocity to solids volume fraction.

Given Data / Assumptions:

• Single-particle terminal velocity, ut = 17 mm/s.
• Hindered settling velocity, u = 1.7 mm/s.
• Richardson–Zaki index, n = 4.5 (typical for intermediate Re).

Concept / Approach:The correlation is u/ut = (1 − φ)^n. Solve for φ from the measured ratio of velocities.

Step-by-Step Solution:Compute velocity ratio: u/ut = 1.7/17 = 0.1.Set 0.1 = (1 − φ)^4.5.Take the 4.5th root: 1 − φ = 0.1^(1/4.5) ≈ 0.599.Therefore, φ ≈ 1 − 0.599 ≈ 0.401 ≈ 0.4.

Verification / Alternative check:Back-substitute φ = 0.4 → (1 − φ) = 0.6; (0.6)^4.5 ≈ 0.1, consistent with the given velocities.

Why Other Options Are Wrong:0.1: would give u/ut ≈ (0.9)^4.5 ≈ 0.62, much larger than 0.1.0.6: would give u/ut ≈ (0.4)^4.5 ≈ 0.016, too small.“None” is unnecessary since φ ≈ 0.4 matches an option.

Common Pitfalls:Confusing mass and volume fraction; using wrong n for the Reynolds-number range; mixing cm/s and mm/s.

Final Answer:0.4