Solid Suspension Speed (Njs)

Reference ID: MET-5E8B | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The just-suspended speed (symbol \( N_{js} \)) is the minimum rotational speed at which a pitched-blade or disc-turbine impeller keeps all solid particles in a liquid medium fully suspended off the vessel bottom. Operating below \( N_{js} \) leads to particle dropout, stagnant zones, and uneven solids distribution, while any additional speed only wastes power without improving solids homogeneity. It is therefore a key design and scale-up parameter in crystallisation, slurry hydrogenation, food processing (e.g., spice-in-sauce), and mineral leaching.

Zwietering’s 1958 empirical correlation—still the industrial default—links geometry, physical properties, and gravitational forces to \( N_{js} \) provided the tank is baffled (to suppress a central vortex) and the flow is turbulent (\( Re > 10\,000 \)).

Methodology & Formulas

All variables are in consistent SI units before the final conversion to rpm. The correlation is written exactly as used in the code.

Convert practical data to base units:
- \( d \; [\text{m}] = d_{\text{mm}} \; [\text{mm}] \times 0.001 \)
- \( \mu \; [\text{Pa·s}] = \mu_{\text{cP}} \; [\text{cP}] \times 0.001 \)
- \( \nu \; [\text{m}^2\text{/s}] = \mu / \rho_L \)
- \( \Delta\rho \; [\text{kg/m}^3] = \rho_p - \rho_L \)
Zwietering correlation (absolute form): \[ N_{js}\; [\text{rps}] = S \; \nu^{\,0.1} \; d^{\,0.2} \; \left(\frac{g \; \Delta\rho}{\rho_L}\right)^{0.45} X^{\,0.13} \; D^{-0.85} \]
Convert speed to field units: \[ N_{js}\; [\text{rpm}] = N_{js}\; [\text{rps}] \times 60 \]
Check turbulent regime: \[ Re = \frac{\rho_L \; N_{js} \; D^{2}}{\mu} \quad (> 10\,000 \text{ required}) \]

Parameter	Zwietering Empirical Range
Particle size \( d \)	0.05 mm – 3 mm
Density difference \( \Delta \rho \)	50 – 2000 kg m⁻³
Solids loading \( X \)	0.1 – 20 wt %
Impeller-to-tank ratio \( D/T \)	0.15 – 0.5
Reynolds number	Re > ~10 000

Values outside these limits invalidate the correlation; alternative methods (pilot measurement or modified regressions) are required.

N_js (just-suspended speed) is the minimum impeller rpm at which no particles remain on the vessel bottom for longer than 1-2 s. Operating below N_js leads to:

Settled slurry causing dead zones, off-spec product, and potential line plugging
Reduced heat/mass transfer and increased erosion of seals or bottom head

Running slightly above N_js maximizes yield while minimizing power draw and mechanical wear.

Begin with the Zwietering correlation (1958) because it has the widest geometry database. Scale-up steps:

Keep (P/V) constant and adjust D/T ratio to maintain similar flow pattern
If particle size or density differs, recalculate N_js using the correlation’s dimensionless groups (Ar, φ)
Verify with pilot trial: visually confirm zero stationary particles; if any remain, increase rpm by 5–10% and recheck

For non-standard impellers (e.g., Intermig, A310) switch to the Baldi–Conti or Sharma–Shaikh correlations; they include power number curves for modern impellers.

Target N_js + 5–10% rpm for routine operation; this covers normal slurry variation. Additional safeguards:

Install a variable frequency drive (VFD) and link it to a torque or power meter; alarm on torque spikes that indicate viscosity rise or dumping of dense heel
Use duplex impellers (lower radial, upper axial) to spread power over two zones, cutting peak torque by ~25%
Perform a wet natural frequency check after any impeller or shaft change; keep first critical speed > 1.4 × max operating speed

Yes, but apparent viscosity near the impeller must be evaluated at the shear rate ≈ 10 × N. Procedure:

Measure flow curve (τ vs γ̇) and fit to power-law model (K, n)
Calculate Metzner–Otto constant for your impeller type (e.g., 11 for a pitched-blade turbine)
Use this effective viscosity in the Zwietering correlation instead of water viscosity; recalculate N_js
Validate in lab: if Brookfield viscosity at 10 s⁻¹ is > 500 cP, expect N_js to increase 30–60% versus water

Worked Example: Calculation of Just-Suspended Impeller Speed (N_js)

In a food processing application, solid spice particles must be maintained in suspension within a low-viscosity tomato-based sauce inside a baffled, cylindrical mixing tank. The goal is to determine the minimum impeller speed required to achieve just-suspended conditions using the Zwietering correlation.

Knowns (Input Parameters):

Particle diameter, \( d = 0.2 \, \text{mm} \)
Particle density, \( \rho_p = 1150 \, \text{kg} \cdot \text{m}^{-3} \)
Liquid density, \( \rho_L = 1050 \, \text{kg} \cdot \text{m}^{-3} \)
Liquid dynamic viscosity, \( \mu = 1 \, \text{cP} \)
Solids loading, \( X = 2 \, \text{wt%} \)
Tank diameter, \( T = 0.6 \, \text{m} \)
Impeller diameter, \( D = 0.09 \, \text{m} \)
Zwietering geometric constant for a 6-blade disc turbine, \( S = 6.0 \)
Gravitational acceleration, \( g = 9.81 \, \text{m} \cdot \text{s}^{-2} \)

Step-by-Step Calculation:

Convert input parameters to SI units where necessary:
- Particle diameter: \( d = 0.2 \, \text{mm} = 2.000 \times 10^{-4} \, \text{m} \)
- Dynamic viscosity: \( \mu = 1 \, \text{cP} = 1.000 \times 10^{-3} \, \text{Pa} \cdot \text{s} \)
- Kinematic viscosity: \( \nu = \mu / \rho_L = 1.000 \times 10^{-3} / 1050 = 9.524 \times 10^{-7} \, \text{m}^{2} \cdot \text{s}^{-1} \)
- Density difference: \( \Delta\rho = \rho_p - \rho_L = 1150 - 1050 = 100.000 \, \text{kg} \cdot \text{m}^{-3} \)
Compute each term of the Zwietering correlation \( N_{js} = S \cdot \nu^{0.1} \cdot d^{0.2} \cdot (g \Delta\rho / \rho_L)^{0.45} \cdot X^{0.13} \cdot D^{-0.85} \):
- \( \nu^{0.1} = (9.524 \times 10^{-7})^{0.1} = 0.250 \)
- \( d^{0.2} = (2.000 \times 10^{-4})^{0.2} = 0.182 \)
- \( (g \Delta\rho / \rho_L)^{0.45} = \left( \frac{9.81 \times 100.000}{1050} \right)^{0.45} = 0.970 \)
- \( X^{0.13} = 2^{0.13} = 1.094 \)
- \( D^{-0.85} = (0.090)^{-0.85} = 7.743 \)
Calculate \( N_{js} \) in revolutions per second (rps): \( N_{js,\text{rps}} = 6.000 \times 0.250 \times 0.182 \times 0.970 \times 1.094 \times 7.743 = 2.243 \, \text{rps} \)
Convert \( N_{js} \) to practical units (revolutions per minute, rpm): \( N_{js,\text{rpm}} = 2.243 \, \text{rps} \times 60 = 134.580 \, \text{rpm} \)
Verify the turbulent flow regime by calculating the impeller Reynolds number: \( Re = \frac{\rho_L \cdot N_{js,\text{rps}} \cdot D^{2}}{\mu} = \frac{1050 \times 2.243 \times (0.090)^{2}}{1.000 \times 10^{-3}} = 19077.000 \) Since \( Re = 19077.000 > 10000 \), the flow is turbulent, validating the use of the correlation.

Final Answer: The minimum impeller speed to achieve just-suspended conditions is \( N_{js} = 134.580 \, \text{rpm} \).

"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

Solid Suspension Speed (Njs)

Introduction & Context

Methodology & Formulas

Worked Example: Calculation of Just-Suspended Impeller Speed (Njs)

Worked Example: Calculation of Just-Suspended Impeller Speed (N_js)