Reference ID: MET-5584 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Froude number Fr is a dimensionless group that compares inertial forces to gravitational forces in a stirred tank. In process engineering, it is used to decide whether surface waves, vortexing, or gas entrainment will occur when no baffles are present. A low Fr indicates gravity dominates and the surface remains nearly flat; a high Fr indicates the impeller can create a deep vortex or draw gas from the head-space. The number is therefore critical when scaling-up unbaffled or partially-baffled agitated systems in fermentation, wastewater treatment, and crystallisation.
Methodology & Formulas
Convert rotational speed from rpm to revolutions per second:
\[ N = \frac{\text{rpm}}{60} \]
Calculate the Reynolds number to check the flow regime:
\[ Re = \frac{\rho N D^{2}}{\mu} \]
\(\rho\) = fluid density (kg m-3) \(D\) = impeller diameter (m) \(\mu\) = dynamic viscosity (Pa·s)
Compute the Froude number:
\[ Fr = \frac{N^{2} D}{g} \]
\(g\) = standard gravity (9.80665 m s-2)
Fr meaningful; use to assess vortex depth or gas entrainment
The Froude number (Fr) is a dimensionless ratio of inertial to gravitational forces, defined for stirred tanks as Fr = N²D/g, where N is impeller speed (rev/s), D is impeller diameter (m), and g is 9.81 m/s². In mixing, it predicts the onset of surface vortexing and helps scale-up geometrically similar vessels so that gravity-driven phenomena (vortex depth, swirl, solid suspension uniformity) behave the same way in the plant vessel as in the lab.
For axial-flow impellers with low-viscosity Newtonian fluids, keep Fr < 0.1 to prevent a deep vortex from reaching the impeller.
Radial turbines can tolerate Fr ≈ 0.2 before significant air entrainment.
If the tank has baffles or an anti-vortex plate, Fr can safely rise to 0.3–0.4 because the baffles break the tangential velocity field.
Always confirm with pilot tests; small changes in fill level or impeller submergence shift the critical Fr.
Convert units before inserting into Fr = N²D/g:
Speed: divide rpm by 60 to get rev/s.
Diameter: multiply inches by 0.0254 to get metres.
Example: 180 rpm and 12 in impeller → N = 3 rev/s, D = 0.305 m → Fr = 3² × 0.305 / 9.81 = 0.28.
Not always. Fr matches gravitational effects but solids suspension also depends on settling velocity and impeller flow. Use Fr as a secondary check while maintaining constant tip speed (πND) or power per unit mass (P/V). A common practice is to keep Fr within ±20 % of the pilot value and verify the just-suspended speed (Njs) with the Zweitering correlation or CFD.
Worked Example – Froude Number for a 250 mm Pitched-Blade Impeller
A process engineer is verifying the hydraulic regime inside a 1 m³ mixing vessel that is agitated by a 250 mm diameter pitched-blade impeller turning at 300 rpm. The vessel contains water-like properties at 25 °C. The engineer needs to confirm whether the inertial forces are large enough relative to gravity forces to avoid surface vortexing. The Froude number is the appropriate dimensionless group for this check.
Gravitational acceleration, g = 9.807 m s⁻²
Impeller rotational speed, N = 300 rpm = 5.000 rps
Impeller diameter, D = 0.250 m
Liquid density, ρ = 1000 kg m⁻³
Liquid dynamic viscosity, μ = 0.001 Pa·s
Convert rotational speed to consistent units:
\[ N = \frac{300}{60} = 5.000 \text{ rps} \]
Compute the characteristic velocity term:
\[ V = N\,D = 5.000 \times 0.250 = 1.250 \text{ m s⁻¹} \]
Final Answer: The Froude number for the given operating conditions is Fr = 0.637 (dimensionless). Because Fr < 1, gravity effects are still noticeable; the engineer may consider installing baffles or increasing the impeller speed if strong surface vortexing is undesirable.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
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