Introduction & Context

The just-suspended speed, denoted N_js, is the minimum impeller rotational speed at which no solid particle remains stationary on the vessel base for more than 1–2 s. Knowing N_js is essential for crystallisers, precipitators, hydrogenation loops, ore-leaching tanks and any other solid–liquid operation where:

mass-transfer between solid and liquid must be maximised,
particle attrition or energy consumption has to be minimised,
scale-up from lab to plant is performed on a geometrical and hydrodynamic basis.

Zwietering’s correlation is the industrial standard for mechanically agitated vessels equipped with radial-flow impellers (Rushton turbines, flat-blade or pitched-blade turbines). The correlation links N_js to the physical properties of the two phases, the particle size, the solid concentration and the impeller geometry.

Methodology & Formulas

Dimensional input conversion
All quantities are converted to SI units before substitution.

Just-suspended speed (Zwietering, 1958)
The dimensional correlation reads: \[ N_{\text{js}} = S \; \nu^{0.1} \; d_{\text{p}}^{0.2} \left[ \frac{g (\rho_{\text{p}} - \rho_{\text{f}})}{\rho_{\text{f}}} \right]^{0.45} X^{0.13} D^{-0.85} \] where:

Symbol	Meaning	Unit
\(N_{\text{js}}\)	just-suspended speed	rps
\(S\)	Zwietering constant (geometry factor)	—
\(\nu\)	kinematic viscosity of liquid	m² s⁻¹
\(d_{\text{p}}\)	particle sieve diameter	m
\(g\)	gravitational acceleration	m s⁻²
\(\rho_{\text{p}}\)	particle density	kg m⁻³
\(\rho_{\text{f}}\)	fluid density	kg m⁻³
\(X\)	mass ratio solid/liquid × 100 %	%
\(D\)	impeller diameter	m

Impeller Reynolds number at N_js
\[ Re_{\text{imp}} = \frac{N_{\text{js}} D^{2}}{\nu} \] is used to confirm that the correlation is applied in the turbulent regime.

Validity envelope

Parameter	Lower limit	Upper limit	Remarks
\(Re_{\text{imp}}\)	10 000	—	Correlation derived in turbulent regime
\(X\)	0.1 %	20 %	Mass ratio solid/liquid
\(d_{\text{p}}\)	0.15 mm	5 mm	Particle size tested by Zwietering
\(D/T\)	0.25	0.60	Impeller-to-tank diameter ratio

Outside these ranges the exponent constants may deviate and experimental validation is recommended.

N_js is the minimum impeller speed at which no solid particles remain stationary on the vessel base for more than 1–2 seconds. Operating below N_js risks incomplete reaction, fouling, or blocked outlets; running far above it wastes power and may cause unwanted attrition or gas entrainment.

Zwietering’s equation is still the industry standard: N_js = S (g Δρ/ρ_L)^0.45 X^0.13 ν^0.10 D^–0.85
Use the published S values for your impeller type (PBT, A310, Rushton, etc.) and D/T ratio; interpolate rather than extrapolate.
For very high solids (>40 wt %) or non-Newtonian slurries, apply the Hicks-Morton correction or CFD validation.

Geometric similarity with constant D/T and C/T keeps S nearly constant, so N_js scales roughly with T^–0.85. A 6-fold increase in tank diameter therefore drops N_js from 300 rpm to about 70 rpm, but tip speed and power per unit volume must be checked to ensure solids are still lifted and mass-transfer requirements are met.

Switch to a narrow particle-size distribution or reduce median size; finer solids lift more easily.
Increase slurry density toward neutral buoyancy if chemistry allows.
Install a concave bottom or angled baffles to eliminate dead zones.
Use an axial-flow impeller with swept-back blades (A310, HE-3) instead of a radial disc turbine.

Worked Example – Estimating the Just-Suspended Speed for a Pharmaceutical Slurry

A process engineer needs to select an agitator for a 200 mm-diameter pilot reactor that will keep 0.4 mm API crystals fully suspended in a 2.5 cSt broth. The target is to suspend 95 % of the solids (X = 5 % allowed on the tank bottom). The vessel uses a standard pitched-blade impeller with a geometric factor S = 6.5. Determine the minimum impeller speed N_js.

Knowns

Particle diameter, d_p = 0.4 mm
Particle density, ρ_p = 1150 kg m⁻³
Fluid density, ρ_f = 1050 kg m⁻³
Fluid kinematic viscosity, ν = 2.5 cSt = 2.5 × 10⁻⁶ m² s⁻¹
Impeller diameter, D = 0.200 m
Geometric factor, S = 6.5
Gravitational acceleration, g = 9.81 m s⁻²

Step-by-step calculation

Convert particle size to metres: d_p = 0.4 mm × 0.001 = 0.0004 m.
Calculate the dimensionless settling-term exponent: \[ \text{term}_1 = S = 6.5 \]
Compute the buoyancy ratio: \[ \text{term}_2 = \left(\frac{\rho_p - \rho_f}{\rho_f}\right)^{0.33} = \left(\frac{1150 - 1050}{1050}\right)^{0.33} = 0.275 \]
Evaluate the viscosity correction: \[ \text{term}_3 = \left(\frac{\nu}{g^{0.33} d_p^{1.5}}\right)^{0.1} = \left(\frac{2.5\times10^{-6}}{(9.81)^{0.33}(0.0004)^{1.5}}\right)^{0.1} = 0.209 \]
Compute the gravitational Froude-type term: \[ \text{term}_4 = \left(\frac{g}{d_p}\right)^{0.25} = \left(\frac{9.81}{0.0004}\right)^{0.25} = 0.970 \]
Combine the above to obtain the just-suspended speed in rps: \[ N_{js} = \text{term}_1 \times \text{term}_2 \times \text{term}_3 \times \text{term}_4 = 6.5 \times 0.275 \times 0.209 \times 0.970 = 1.757\ \text{rps} \]
Convert to rpm for impeller specification: \[ N_{js} = 1.757 \times 60 = 105.4\ \text{rpm} \]

Final Answer

The minimum impeller speed required to just suspend 95 % of the 0.4 mm crystals is 105 rpm. Select a variable-speed drive capable of at least this value to ensure reliable solid suspension during the batch.

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

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