Settling Velocity of Magnetite Slurry Particles (Stokes' Law): 04-BS-7, December 2016, Q9

Question 9: Settling Velocity of Magnetite Slurry Particles (Stokes' Law) (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.

Quantity	Value
Particle diameter d	50 μm = 50×10⁻⁶ m
Particle specific gravity (magnetite)	5.0 (ρ_p=5000 kg/m³)
Carrier fluid	water, ρ_f=1000 kg/m³, μ=1.0×10⁻³ N·s/m²

Find. Stokes settling velocity, and whether the Stokes assumption is valid for this particle (via its particle Reynolds number).

Approach. Apply Stokes' law (drag-buoyancy-weight balance for a sphere in creeping flow) to get the settling velocity directly from the given data, then compute the particle Reynolds number and compare against the Stokes-law validity range (Re < ~1) implied by the empirical drag-coefficient curve.

Stokes settling velocity. $$v = \frac{2}{9}\frac{(\rho_p-\rho_f)g\,r^2}{\mu} = \frac{2}{9}\frac{(5000-1000)(9.81)(25\times10^{-6})^2}{1.0\times10^{-3}} = \boxed{5.45\ \text{mm/s}}$$
Particle Reynolds number, to check the assumption. $$Re_p = \frac{\rho_f v d}{\mu} = \frac{1000(5.45\times10^{-3})(50\times10^{-6})}{1.0\times10^{-3}} = \boxed{0.27}$$
Compare with the empirical drag curve. The Stokes-law line (C_D=24/Re) tracks the measured drag-coefficient curve closely for Re < 1, diverging by only a few percent at Re≈0.27 — well inside the low-Re creeping-flow region of the Drag Diagram for Solid Bodies (spheres). Stokes' law was therefore a good assumption for this particle.

Quantity	Result
Settling velocity	5.45 mm/s
Particle Reynolds number	0.27
Stokes' law valid?	Yes — Re≈0.27 is within the creeping-flow (C_D=24/Re) region of the empirical curve

Topic: Stokes' law settling and its Reynolds-number validity range.

Key relations: Stokes settling v=(2/9)(ρ_p−ρ_f)gr²/μ; particle Reynolds number Re_p=ρ_fvd/μ; C_D=24/Re (Stokes regime).

Why this works. Stokes' law assumes the drag is entirely viscous (creeping flow, negligible inertia); the empirical drag-coefficient curve confirms C_D=24/Re is accurate only while Re stays well below about 1, so checking Re after the fact validates (or invalidates) the initial assumption used to derive v in the first place.

Common pitfall. Using the SLURRY's bulk specific gravity (1.5) instead of the carrier fluid's (water, SG=1.0) in the Reynolds number and drag balance — the particle settles relative to the surrounding water, not relative to the bulk slurry density, even though the slurry's bulk SG is what keeps coal afloat and rock sinking.

Source: White, Fluid Mechanics, Ch. 7 (drag on immersed bodies, Stokes flow).

Question 9 of 13: Settling Velocity of Magnetite Slurry Particles (Stokes' Law)

Question 9: Settling Velocity of Magnetite Slurry Particles (Stokes' Law) (5 marks)