Reference ID: MET-CF7C | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Settling velocity quantifies how fast a solid fruit particle falls (or rises) through a surrounding liquid. In beverage mixing tanks this single number sets the minimum impeller speed required to keep solids uniformly suspended. If the local upward fluid velocity is less than the particle’s settling velocity, solids will accumulate at the bottom and create quality-control issues. Stokes’ law provides a rapid hand-calculation route for engineering checks during process design, HACCP trials, and agitator-specification work orders.
Methodology & Formulas
Particle diameter
\[ d = \frac{d_{\text{mm}}}{1000} \]
Particle Reynolds number for validity test
\[ Re_{\text{p}} = \frac{\rho_{\text{f}}\, v_{\text{t}}\, d}{\mu} \]
Regime criterion
Stokes law applicable?
\( Re_{\text{p}} < 0.1 \)
Yes (creeping flow assumption valid)
\( Re_{\text{p}} \geq 0.1 \)
No (use generalized drag correlation; error if applied)
If validation passes, \( v_{\text{t}} \) is the magnitude of the terminal velocity. To achieve complete off-bottom suspension for settling particles (\(\rho_p > \rho_f\)), the minimum upward fluid velocity should exceed \( v_{\text{t}} \). For rising particles (\(\rho_p < \rho_f\)), appropriate downward or circulating flow is required to prevent accumulation at the top.
Maintain impeller tip speed ≥ 3.5 m s⁻¹ for typical mineral slurries; if the computed terminal velocity magnitude \( v_t \) exceeds 0.06 m s⁻¹, increase power per unit volume to > 1 kW m⁻³. Select axial-flow impellers (pitch-blade turbines or hydrofoils) set at 20–40 % of tank diameter off-bottom clearance to keep the cloud height above 80 % of liquid level. For cohesive or fine solids, operate at a Reynolds number > 10⁴ to guarantee complete off-bottom suspension.
Use the constrained drag force equation adapted for static mixers:
with \( \psi \): shape factor ≈ 0.7 for crushed quartz, 0.9 for rounded sand
and \( C_D \) from Morsi & Alexander correlation for \( Re_p \) 0.1–10000
In brines, multiply by hindrance factor \( (1 - c_v)^{4.65} \) to account for elevated solids volume fraction \( c_v \)
Validate with bench-scale cylinder tests at field temperature and ionic strength
Check scale-up pitfalls:
Tip-speed similarity alone is insufficient—maintain constant power per mass; large tanks need higher torque
Verify impeller Reynolds number \( Re_i = \frac{\rho_f N D^2}{\mu} > 10^4 \) to avoid laminar pocket formation
Ensure cloud height sensor reads > 85 % of batch height; solids recirculation loops can help
Measure actual slurry viscosity—shear thinning polymers or fines (p80 < 30 µm) raise effective μ and drop \( Re_i \)
Confirm no dead zones—add wall baffles 4×90°; increase sweep speed at boundaries
If solids density > 3200 kg m⁻³, consider draft-tube circulator or dual impellers (bottom axial, top radial)
CFD can reduce trials but not fully replace them:
Use Euler-granular or dense discrete phase models with validated drag laws (Gidaspow, Syamlal–O’Brien)
Calibrate particle stress modulus and frictional viscosity with 10 L batch data at identical solids loading
Compare cloud height and bottom solid fraction; aim for < 5 % deviation before scaling further
Include temperature-dependent brine density and viscosity; validate at min/max process conditions
CFD is most reliable for relative sensitivity studies (e.g., different impeller types); always perform a single confirmatory pilot run at predicted optimum geometry
Worked Example: Terminal Velocity Magnitude in a Mixing Tank
A process engineer in a juice production facility needs to determine the terminal velocity magnitude of small fruit particles in a mixing tank. This velocity is used to set the agitator speed high enough to keep the particles suspended, ensuring uniform mixing. The particles are approximated as spheres with given properties in a fluid with specified density and viscosity.
Gravitational acceleration, \( g = 9.807 \, \text{m/s}^2 \) (rounded from 9.80665)
Step-by-Step Calculation:
Convert diameter to meters: \( d = 0.0005 \, \text{m} \).
Convert viscosity to SI units: \( \mu = 0.0025 \, \text{kg/(m·s)} \).
Calculate the absolute density difference: \( |\Delta \rho| = |\rho_p - \rho_f| = 10.0 \, \text{kg/m}^3 \). Note: Since \(\rho_p < \rho_f\), the particle is buoyant and would rise; the magnitude is used for velocity calculation.
Apply Stokes' law for the terminal velocity magnitude: \( v_t = \frac{g \cdot d^2 \cdot |\Delta \rho|}{18 \mu} \). Using the values, \( v_t = \frac{9.807 \times (0.0005)^2 \times 10.0}{18 \times 0.0025} = 0.0005448 \, \text{m/s} \approx 0.000545 \, \text{m/s} \).
Check the particle Reynolds number for validity: \( Re_p = \frac{\rho_f \cdot v_t \cdot d}{\mu} = \frac{1050.0 \times 0.0005448 \times 0.0005}{0.0025} = 0.1144 \approx 0.114 \). Since \( Re_p < 0.1 \), Stokes' law is marginally valid; for precise design, a generalized correlation is recommended, but this approximation suffices for initial estimates.
Final Answer:
The magnitude of the terminal velocity is approximately \( 5.45 \times 10^{-4} \, \text{m/s} \) (or 0.00055 m/s). For this buoyant particle, this represents the rising velocity magnitude; agitation must provide downward flow to maintain suspension.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
