Settling Velocity in Mixing Tank

Reference ID: MET-DE58 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

In a mixing tank, suspended solids such as fruit pieces in juice tend to settle under gravity while agitation tries to keep them uniformly dispersed. The settling velocity of a single particle is the terminal speed at which these two effects balance; it is the key scaling parameter for:

Impeller power selection to maintain suspension.
Tank height-to-diameter ratios to avoid dead zones.
Residence-time calculations for heat treatment or sterilisation.
Scale-up from pilot to industrial vessels.

Methodology & Formulas

Step 1 – Stokes regime (low particle Reynolds number)

For creeping flow around a sphere, the drag coefficient is \( C_D = 24/Re_p \). Balancing drag, buoyancy, and weight gives the classical Stokes velocity:

\[ v_{\text{Stokes}} = \frac{g\,d^2\,(\rho_p - \rho_f)}{18\,\mu} \]

with the particle Reynolds number defined as:

\[ Re_p = \frac{\rho_f\,v\,d}{\mu} \]

Regime	Range	Drag law
Stokes (laminar)	\( Re_p \leq 1 \)	\( C_D = 24/Re_p \)
Transition	\( 1 \lt Re_p \leq 1000 \)	Schiller–Naumann (see below)
Newton (turbulent)	\( Re_p \gt 1000 \)	\( C_D \approx 0.44 \)

Step 2 – Schiller–Naumann correction for \( Re_p \gt 1 \)

When the Stokes estimate yields \( Re_p \gt 1 \), the drag coefficient is corrected to:

\[ C_D = \frac{24}{Re_p}\left(1 + 0.15\,Re_p^{0.687}\right) \]

and the terminal velocity is obtained from the force balance:

\[ v_t = \sqrt{\frac{4\,g\,d\,(\rho_p - \rho_f)}{3\,C_D\,\rho_f}} \]

Because \( C_D \) itself depends on \( v_t \), the two equations are solved iteratively:

Start with \( v_t^{(0)} = v_{\text{Stokes}} \).
Compute \( Re_p^{(k)} = \rho_f\,v_t^{(k)}\,d/\mu \).
Update \( C_D^{(k)} \) with the Schiller–Naumann expression.
Recompute \( v_t^{(k+1)} \) from the square-root formula.
Repeat until successive values differ by less than the desired tolerance (20 iterations are usually sufficient).

Step 3 – Application

The converged \( v_t \) is the settling velocity used to assess suspension quality, impeller speeds, and minimum agitation power in the mixing tank.

Compare the particle settling velocity (V_s) to the average liquid circulation velocity (V_c) in the tank. A widely used rule-of-thumb is:

If V_s / V_c < 0.1, solids remain suspended with modest power.
If 0.1 < V_s / V_c < 0.5, partial suspension; increase impeller speed or switch to axial-flow impeller.
If V_s / V_c > 0.5, expect a large dead zone or settled bed; consider larger impeller, draft tube, or denser slurry preparation.

V_c can be approximated from impeller pumping capacity divided by the tank cross-sectional area. V_s is obtained from Stokes’ law or empirical correlations for the largest particle size.

For irregular, non-spherical particles, apply the following sequence:

Calculate nominal diameter from sieve analysis (d₅₀ or d₉₅).
Estimate shape factor (ψ) from image analysis or literature (typical 0.7–0.9 for crushed quartz).
Use the Richardson–Zaki equation with hindered-settling correction: V_s = V_s0 (1–φ)ⁿ, where V_s0 is the single-particle terminal velocity corrected for shape, φ is solids volume fraction, and n ≈ 4.6 for turbulent regime.
If slurry is >15% v/v solids, also check slip-velocity models (e.g., Garside–Al-Dibouni) to account for inter-particle effects.

Cylinder tests measure quiescent settling; the plant sees turbulence and non-uniform flow. Key mismatches:

Scale-up ignores turbulence damping: energy dissipation in a 1 L cylinder is 10–100× higher per unit mass than in a 100 m³ tank.
Wall effects in narrow lab cylinder reduce hindered settling, giving falsely low V_s.
Impeller pumping direction matters: radial impellers create strong bottom currents but weak upward flow at the wall, letting particles settle in the annulus.
Temperature differences change liquor viscosity; a 10 °C drop can increase V_s by 20% for <100 µm particles.

Run a pilot-scale test with matched specific power (W kg⁻¹) and measure cloud height versus time to validate.

At yield-stress (τ_y) > 3 Pa, particles smaller than the critical “plastic” size d_p ≈ τ_y / (Δρ g) remain indefinitely suspended. Practical steps:

Measure τ_y with a vane rheometer at field temperature.
If d_p > calculated critical size, add dispersant or dilute to reduce τ_y below 1 Pa.
For pseudoplastic slurries, use the effective viscosity at the shear rate experienced by the particle (typically 5–20 s⁻¹) in Stokes’ law instead of liquor viscosity.
Account for shear-induced migration: higher shear near the impeller can increase local solids fraction and hinder settling, while low-shear zones permit faster settling—verify with CFD or PEPT studies.

Worked Example – Estimating Particle Settling Velocity in a Brine Mixing Tank

A process engineer is troubleshooting a stirred brine tank that contains suspended polymer beads. To verify that the impeller keeps the solids off the bottom, the engineer needs to know the terminal settling velocity of a single bead under tank conditions.

Knowns

Gravitational acceleration, \(g = 9.81\ \text{m s}^{-2}
Bead diameter, \(d = 7.4\ \text{mm} = 0.007\ \text{m}
Bead density, \(\rho_p = 1060\ \text{kg m}^{-3}
Brine density, \(\rho_f = 1025\ \text{kg m}^{-3}
Brine dynamic viscosity, \(\mu = 0.002\ \text{Pa s}

Step-by-Step Calculation

Compute the density difference: \[ \Delta\rho = \rho_p - \rho_f = 1060 - 1025 = 35\ \text{kg m}^{-3} \]
Estimate the Stokes settling velocity (laminar assumption): \[ v_{\text{Stokes}} = \frac{g\,d^2\,\Delta\rho}{18\,\mu} = \frac{9.81\,(0.007)^2\,35}{18\,(0.002)} = 0.522\ \text{m s}^{-1} \]
Check the particle Reynolds number: \[ Re_p = \frac{\rho_f\,v_{\text{Stokes}}\,d}{\mu} = \frac{1025\,(0.522)\,(0.007)}{0.002} = 255.3 \] Because \(Re_p \gg 1\), the flow is outside the Stokes regime; a drag-coefficient correlation is required.
Use the iterative drag-coefficient method (Schiller–Naumann): \[ C_D = \frac{24}{Re_p}\left(1 + 0.15\,Re_p^{0.687}\right) = 0.729 \]
Solve the force-balance equation for terminal velocity: \[ v_t = \sqrt{\frac{4\,g\,d\,\Delta\rho}{3\,\rho_f\,C_D}} = \sqrt{\frac{4\,(9.81)\,(0.007)\,(35)}{3\,(1025)\,(0.729)}} = 0.067\ \text{m s}^{-1} \]

Final Answer

The terminal settling velocity of the polymer bead in the brine is 0.067 m s^-1 (≈ 6.7 cm s^-1). Because this value is well below the typical mean circulation velocity created by the impeller, the solids are expected to remain suspended.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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