Introduction & Context

Power-density scale-up keeps the volumetric power input P/V constant when a mixing process is transferred from a pilot vessel to a full-scale production tank. Maintaining P/V preserves the average shear, turbulence intensity, and blend time that were validated at bench scale, so it is the default strategy for fermentations, precipitations, and fast competitive reactions in the chemical, biotech, and water-treatment industries.

Methodology & Formulas

Pilot measurements
Tank volume: V₁ (m³)
Impeller speed: N₁ (s⁻¹)
Impeller diameter: D₁ (m)
Power drawn: P₁ = Po ρ N₁³ D₁⁵ (W)
Volumetric power: P/V│₁ = P₁/V₁ (W m⁻³)
Reynolds number: Re₁ = (ρ N₁ D₁²)/μ
Geometric scale-up
Length scale factor: λ = (V₂/V₁)^1/3
Production tank: T₂ = λ T₁, D₂ = λ D₁
Speed selection (constant P/V)
P/V│₂ = P/V│₁ ⇒ N₂ = N₁ λ^−2/3
Production power and motor sizing
Production power: P₂ = (P/V)│₁ V₂
Motor rating: P_motor = sf P₂ (with safety factor sf)

Flow regime limits (Rushton turbine, Po≈5)
Regime	Reynolds criterion	Implication
Laminar	Re < 10	Power number rises; scale-up must keep Po Re = const
Transitional	10 ≤ Re < 10⁴	Use full Po(Re) curve or CFD
Fully turbulent	Re ≥ 10⁴	Po constant; constant-P/V rule valid

Power density (P/V) is simply the mechanical power of the impeller divided by the batch volume.

Measure shaft torque (τ) and rotational speed (N) at the pilot scale: P = 2πNτ.
Convert P to kW and V to m³; P/V has units kW m⁻³.
Keep P/V constant when you scale-up; adjust impeller diameter, speed, or blade type to hit the same value in the larger tank.

Yes, but you must also check that the impeller tip speed and flow number remain in an acceptable range.

Constant P/V keeps shear similar, yet a taller aspect ratio may reduce top-to-bottom turnover; compensate by adding a second impeller or increasing pumping capacity.
Verify blend time with a tracer test or CFD so the larger tank still meets process specifications.

No. Broth viscosity, oxygen transfer rate (k_La), and CO₂ removal often become limiting.

Scale-up on constant P/V first, then check k_La using the correlation k_La ∝ (P/V)^0.7 (u_s)^0.6.
If k_La is too low, raise aeration rate or switch to a micro-sparger while watching foam control.
For shear-sensitive cultures, cap tip speed below ~3.5 m s⁻¹ even if P/V is constant.

Use the power number (N_p) correlation: P = N_p ρ N³ D⁵.

Look up N_p for the new impeller geometry under turbulent flow (Re > 10,000).
Adjust N or D to deliver the same P/V; a high-efficiency hydrofoil needs lower N_p, so you can run at lower N for the same power, reducing shear.
Always validate with a bench-scale torque measurement before finalizing the full-scale design.

Worked Example – Scaling Up a Stirred Reactor

A process team must duplicate the mixing performance of a 200 L pilot reactor in a 4 m³ production vessel. The pilot unit uses a 7.5 kW motor and achieves a power density of 0.0375 kW L⁻¹. The larger vessel must match this value while including a 15 % design margin.

Knowns

Target power density: 0.0375 kW L⁻¹
Production volume: 4000 L
Design margin: 15 %

Step-by-Step Calculation

Calculate the required shaft power without margin:
P_base = 0.0375 kW L⁻¹ × 4000 L = 150 kW
Apply the 15 % margin:
P_motor = 150 kW × (1 + 0.15) = 172.5 kW
Select the next standard motor size: 185 kW.

Final Answer

Motor ≥ 185 kW (with 15 % margin included).

"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