Settling Velocity of Magnetite in a Separation Slurry: 04-BS-7, December 2013, Q9

Question 9: Settling Velocity of Magnetite in a Separation Slurry (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Particle diameter d = 50 µm; magnetite specific gravity 5.0 (ρ_p = 5000 kg/m³), settling through water (ρ = 1000 kg/m³, μ = 1.0×10⁻³ Ns/m²).

Find. Terminal settling velocity by Stokes' Law, and whether the empirical (sphere) drag-coefficient curve confirms Stokes' Law as a good assumption.

Approach. Compute the Stokes-Law terminal velocity directly, then check the resulting particle Reynolds number against the empirical drag-coefficient curve (the same curve the exam's attachment plots) to see how closely the true $C_D$ at that Re agrees with the Stokes value $24/Re$.

Stokes' Law terminal velocity. $$V_t = \frac{(\rho_p-\rho)g\,d^{2}}{18\mu} = \frac{(5000-1000)(9.81)(50\times10^{-6})^{2}}{18\times10^{-3}} = \boxed{5.45\times10^{-3}\ \text{m/s} = 5.45\ \text{mm/s}}$$
Particle Reynolds number at this settling speed. $$Re = \frac{\rho V_t d}{\mu} = \frac{1000\times5.45\times10^{-3}\times50\times10^{-6}}{1.0\times10^{-3}} = \boxed{0.272}$$
Check against the empirical drag curve. Stokes' Law implies $C_D=24/Re=24/0.272=88.1$. Reading the empirical sphere-drag curve at $Re=0.272$ (in the attachment's low-Re "Stokes law" branch) gives $C_D\approx88.4$ — within about 0.4% of the Stokes value.

Quantity	Value
Stokes' Law settling velocity	5.45 mm/s
Particle Reynolds number	0.272
Stokes vs. empirical C_D deviation	≈0.4%

Conclusion: at Re = 0.272, well inside the Stokes-Law range (conventionally Re < 1), the empirical drag curve lies essentially on top of the $C_D=24/Re$ line, so Stokes' Law was an excellent assumption for these magnetite particles.

Topic: Stokes' Law settling and its range of validity.

Key relations: Stokes' Law $V_t=(\rho_p-\rho)gd^2/18\mu$; particle Reynolds number $Re=\rho V_t d/\mu$; drag coefficient $C_D=24/Re$ in the Stokes regime.

Why this works: Stokes' Law is the closed-form solution of the creeping-flow (very low Re) equations of motion around a sphere; the empirical drag curve merges with $C_D=24/Re$ only while Re stays well below about 1, so checking the resulting Re after the fact is the standard way to validate the assumption used to get the answer in the first place.

Common pitfall: assuming Stokes' Law is valid without checking Re afterwards — for larger or denser particles the same formula can silently be used far outside its valid range, where the empirical $C_D$ departs sharply from $24/Re$ (see Practice Q9 for a case where this happens).

Source: Crowe, Elger & Roberson, Engineering Fluid Mechanics, Ch. 9–10 (drag on immersed bodies, Stokes' Law).

Question 9 of 13: Settling Velocity of Magnetite in a Separation Slurry

Question 9: Settling Velocity of Magnetite in a Separation Slurry (5 marks)