Introduction & Context

In stirred-tank reactors, blending vessels and crystallisers the mechanical power introduced by the impeller controls the turbulence intensity, mixing time, heat- and mass-transfer coefficients, and the size of gas bubbles or solid agglomerates. The specific power ε (energy dissipated per unit volume) is therefore the key scaling parameter used by process engineers to guarantee identical hydrodynamic environments when moving from pilot to industrial scale. The present sheet provides a rapid, first-principle route to translate a target ε into the required shaft speed, Reynolds number, and motor rating for Newtonian liquids in the turbulent regime.

Methodology & Formulas

Tank power from specific power
The total mechanical power P that must reach the liquid is
\[ P = \varepsilon \, V \]
where V is the liquid volume.
Impeller power draw
For a given impeller of diameter D rotating at speed N (rev s⁻¹) the power is expressed by the dimensionless power number P_o:
\[ P = P_{\!o} \, \rho \, N^{3} D^{5} \]
Re-arranging gives the speed required to deliver the target power:
\[ N = \left( \frac{P}{P_{\!o} \, \rho \, D^{5}} \right)^{\!1/3} \]

Flow regime check
Calculate the impeller Reynolds number

\[ Re = \frac{\rho N D^{2}}{\mu} \]

and compare with the turbulent threshold.

Regime	Re range	Power-number behaviour
Laminar	Re ≲ 10	P_o ∝ Re⁻¹
Transitional	10 < Re < 10,000	P_o decreases roughly linearly
Fully turbulent	Re ≥ 10,000	P_o = constant

When Re is below 10,000 the code applies a linear correction factor to P_o and iterates until N converges.

Shaft power and motor sizing
Account for gearbox efficiency η_g and a service margin f_m:
\[ P_{\text{motor}} = \frac{P}{\eta_{\text{g}}} \; f_{\text{m}} \]
Typical values are η_g = 0.93 for a helical gearbox and f_m = 1.15 (15% margin).

The calculation returns the required speed (in rps or rpm), Reynolds number, effective power number, and the recommended motor power in kW. All relations are algebraic; substitute the symbols with your own dimensional data to obtain plant-specific numbers.

The quickest way is to use the power number (N_p) correlation: P = N_p · ρ · N³ · D⁵.

Measure or look up N_p for your impeller type and Reynolds range.
Insert density (ρ), rotational speed (N in rps), and impeller diameter (D in m).
Result is power in watts; divide by 1000 for kW and add at least 20% margin for process upsets.

Yes, but only if the fluid is Newtonian and you have an effective viscosity.

Calculate the mean shear rate: γ ≈ 11 · N (for a pitched-blade turbine).
Read viscosity at that shear from the flow curve.
Use the same N_p correlation, keeping Reynolds number above 10,000 for fully turbulent N_p values.

Viscosity dampens turbulence, so N_p becomes Reynolds-dependent.

Below Re ≈ 10,000, N_p rises sharply; use the curve for your impeller.
Yield stress fluids need torque to initiate motion—include a “start-up” power term.
Always verify with bench-scale tests; scale-up with constant torque per unit volume rather than constant P/V.

Keep power per unit volume (P/V) constant if the controlling regime is turbulent mixing.
Keep tip speed (πND) constant when shear-sensitive or when mass-transfer at the interface limits the rate.
Keep Reynolds number constant only when N_p is still changing rapidly with Re (viscous systems).
Always check torque at the shaft; motor size is usually torque-limited, not power-limited.

Worked Example – Power Requirement for a Fermenter Agitator

A small-scale antibiotic fermenter needs to keep 1000 L of broth in turbulent suspension. The broth behaves like water at 30 °C (ρ = 1050 kg m⁻³, μ = 0.05 Pa·s). A Rushton turbine with D = 0.4 m is available. Determine the motor size that will deliver an average specific energy dissipation rate ε = 1000 W m⁻³ while allowing a 15% motor margin. Gear efficiency is 93%.

Volume, V = 1.0 m³
Fluid density, ρ = 1050 kg m⁻³
Fluid viscosity, μ = 0.05 Pa·s
Impeller diameter, D = 0.4 m
Impeller type: Rushton turbine (Power number, Po = 5.5)
Required specific energy dissipation, ε = 1000 W m⁻³
Gear efficiency, η = 0.93
Motor margin factor, f = 1.15

Calculate the required impeller power from ε and V:
\[ P = \varepsilon \cdot V = 1000 \cdot 1.0 = 1000 \text{ W} \]
Relate power to impeller speed using the power-number correlation:
\[ P = Po \cdot \rho \cdot N^3 \cdot D^5 \]
Re-arrange for N:
\[ N = \left( \frac{P}{Po \cdot \rho \cdot D^5} \right)^{1/3} \]
Insert values:
\[ N = \left( \frac{1000}{5.5 \cdot 1050 \cdot 0.4^5} \right)^{1/3} = 3.09 \text{ rps} \approx 185 \text{ rpm} \]
Check Reynolds number to confirm turbulent regime:
\[ Re = \frac{\rho N D^2}{\mu} = \frac{1050 \cdot 3.09 \cdot 0.4^2}{0.05} = 10400 \gt 10000 \]
Flow is fully turbulent, so the assumed power number is valid.
Account for gear losses and motor margin:
\[ P_{\text{motor}} = \frac{P}{\eta} \cdot f = \frac{1000}{0.93} \cdot 1.15 = 1237 \text{ W} \]

Final Answer: Select a 1.5 kW (1500 W) motor; the calculated requirement is 1.24 kW.

"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