Introduction & Context

Constant‑power‑per‑volume scale‑up is a widely used heuristic in stirred‑tank reactor design. By keeping the mechanical power dissipated per unit liquid volume (P/V) unchanged, engineers preserve local energy‑dissipation rates, which strongly influence mixing time, heat‑ and mass‑transfer coefficients, and ultimately reaction selectivity and yield. The method is routinely applied when translating lab‑scale data (1–100 L) to pilot or production scales (1–100 m³) under fully turbulent conditions, and it can also guide the power density calculation for scale‑up needed to size mixing equipment accurately.

Methodology & Formulas

Geometric similarity
Assume length scales scale with the tank diameter D and that liquid volume V∝D³. Hence the linear scale-up factor is \[ \lambda = \left(\frac{V_{2}}{V_{1}}\right)^{1/3}. \]
Power number correlation
For turbulent flow the power P drawn by an impeller is \[ P = \mathrm{Po}\,\rho N^{3}D^{5}, \] where Po is the (constant) power number, ρ the fluid density, and N the rotational speed. Power per volume therefore becomes \[ \frac{P}{V} = \mathrm{Po}\,\rho N^{3}\frac{D^{5}}{V}. \] With V∝D³ this simplifies to \[ \frac{P}{V} \propto \rho N^{3}D^{2}. \]
Constant P/V requirement
Equating the lab- and plant-scale expressions gives \[ \rho N_{1}^{3}D_{1}^{2} = \rho N_{2}^{3}D_{2}^{2} \qquad\Longrightarrow\qquad N_{2} = N_{1}\left(\frac{D_{1}}{D_{2}}\right)^{2/3}. \]
Reynolds-number check
The impeller Reynolds number \[ \mathrm{Re} = \frac{\rho N D^{2}}{\mu} \] must remain in the turbulent regime for the above power-number correlation to hold.

Flow regime	Reynolds-number range	Remarks
Laminar	\(\mathrm{Re} < 10\)	Po becomes a function of Re; constant P/V scale-up invalid.
Transition	\(10 \le \mathrm{Re} < 10^{4}\)	Gradual deviation; proceed with caution.
Fully turbulent	\(\mathrm{Re} \ge 10^{4}\)	Po constant; constant P/V scale-up valid.

It means the motor power P divided by the liquid volume V is kept the same in the pilot and full-scale vessel. Because P/V has units of W m⁻³, you match the mechanical energy dissipation rate so the average shear, turbulence, and mass-transfer intensity stay roughly comparable. In practice you calculate P = N_p ρ N³D⁵, set (P/V)_lab = (P/V)_plant, and solve for the new shaft speed N_plant.

When the reaction is kinetically limited rather than mass-transfer limited; larger tanks have longer blending times even at the same P/V.
When heat removal becomes the bottleneck; the surface-to-volume ratio drops, so jacket area may not keep up with the heat release per unit volume.
When micromixing or tip-speed-sensitive biology (shear damage, floc break-up) is involved; P/V does not uniquely define maximum shear rate.
When geometry similarity is violated—changing D/T, impeller type, or number of impellers alters the power number and flow pattern.

Use the scaling rule N₂ = N₁ (V₁/V₂)^(1/3) (D₁/D₂)^(5/3). If you keep exact geometric similarity (D ∝ T), this simplifies to N₂ = N₁ (T₁/T₂)^(2/3). A 10-fold tank diameter increase therefore drops speed to ≈ 22 % of the pilot value while maintaining the same power per unit volume.

Yes. Torque τ = P/(2πN). Because N drops more slowly than volume rises, the torque increases roughly with (V₂/V₁)^(5/9). A 1000-fold volume jump can raise torque by an order of magnitude, so verify gearbox and shaft design early in the project.

Measure or calculate blend time and compare with process requirements; add a secondary impeller or draft tube if needed.
Confirm the heat-transfer coefficient and available jacket/coil area satisfy the worst-case duty using the actual power input.
Verify tip speed πND remains within shear or erosion limits for crystals, cells, or catalyst beads.
Run a CFD or pilot tracer study to ensure the full-scale flow number (Q/ND³) keeps adequate top-to-bottom turnover.
Check motor and utility availability: full-scale power may exceed plant substation capacity if P/V is aggressively high.

Worked Example – Constant Power per Volume Scale-Up

A pilot plant is preparing a shear-sensitive cell-culture broth in a 20 L stirred vessel. Management wants to jump to a 2000 L production scale while keeping the same power input per unit volume so that oxygen mass-transfer and mixing intensity remain unchanged. Determine the impeller diameter and rotational speed for the larger tank.

Knowns

Pilot volume, V₁ = 20 L = 0.02 m³
Pilot impeller diameter, D₁ = 100 mm = 0.1 m
Pilot speed, N₁ = 300 rpm = 5 rps
Liquid density, ρ = 1050 kg m⁻³
Liquid viscosity, μ = 5 cP = 0.005 Pa s
Production volume, V₂ = 2000 L = 2 m³
Scale-up rule: constant power per volume, P/V = const

Step-by-step calculation

Check that the pilot Reynolds number is turbulent: \[ \mathrm{Re}_1 = \frac{\rho N_1 D_1^2}{\mu} = \frac{1050 \cdot 5 \cdot 0.1^2}{0.005} = 10\,500 \] (Re > 10,000 ⇒ fully turbulent, power number Po is constant.)
Impose geometric similarity for the vessel and impeller: \[ \frac{D_2}{D_1} = \left(\frac{V_2}{V_1}\right)^{1/3} = \left(\frac{2}{0.02}\right)^{1/3} = 4.642 \] \[ D_2 = 4.642 \cdot 0.1 = 0.464\ \text{m} \]
Apply constant power per volume. For turbulent flow: \[ \frac{P}{V} \propto N^3 D^2 \Rightarrow N_1^3 D_1^2 = N_2^3 D_2^2 \] \[ N_2 = N_1 \left(\frac{D_1}{D_2}\right)^{2/3} = 5 \left(\frac{0.1}{0.464}\right)^{2/3} = 1.797\ \text{rps} \] \[ N_2 = 1.797 \cdot 60 = 107.8\ \text{rpm} \]
Verify the large-scale Reynolds number: \[ \mathrm{Re}_2 = \frac{\rho N_2 D_2^2}{\mu} = \frac{1050 \cdot 1.797 \cdot 0.464^2}{0.005} = 81\,300 \] (Still turbulent, so the assumption holds.)

Final Answer

For the 2000 L vessel, use an impeller diameter of 0.464 m running at 108 rpm. This keeps the power input per unit volume identical to the 20 L pilot and maintains the same mixing intensity.

"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

Constant Power per Volume Scale-Up

Introduction & Context

🚀 Skip the Manual Math!

Methodology & Formulas

Worked Example – Constant Power per Volume Scale-Up