1. What is the Power Number (Np)?
The power number (Np, also called Newton number) is a dimensionless parameter that characterizes the power consumption of an impeller at a given speed. It is the key link between your impeller geometry and the power-per-volume (P/V) delivered to the fluid—and P/V directly determines mixing intensity, oxygen transfer (kLa), and shear environment in any stirred bioreactor.
In the turbulent regime (Re > 10,000), Np is essentially constant for a given impeller geometry. This makes it a reliable design parameter: once you know Np, you can predict power draw at any speed and scale using a single equation.
Choosing the right impeller is one of the most consequential decisions in bioreactor design. A Rushton turbine (Np = 5.0) delivers 14 times more power than a marine propeller (Np = 0.35) at the same RPM and diameter. That difference translates directly into oxygen transfer capability—but also into shear stress on your cells.
2. Power Number Reference Table
Values are for fully baffled vessels (4 baffles, width = T/10) in the turbulent regime (Re > 10,000) unless noted otherwise. Di/DT is the ratio of impeller diameter to tank diameter.
| Impeller Type | Np (Turbulent) | Np (Transitional) | Di/DT Range | Flow Pattern | Best For |
|---|---|---|---|---|---|
| 6-blade Rushton turbine | 5.0 | 8–15 | 0.30–0.40 | Radial | Gas dispersion, microbial fermentation |
| 4-blade Rushton turbine | 3.5 | 6–10 | 0.30–0.40 | Radial | Lower-power gas dispersion |
| Smith turbine (CD-6) | 3.2 | 5–9 | 0.30–0.40 | Radial | Gas dispersion with less flooding |
| Maxblend | 2.5 | 4–8 | 0.50–0.60 | Combined axial/radial | High-viscosity, cell culture |
| Pitched blade (45°, down-pumping) | 1.3 | 2–5 | 0.30–0.50 | Mixed axial/radial | Solids suspension, blending |
| Pitched blade (45°, up-pumping) | 1.3 | 2–5 | 0.30–0.50 | Mixed axial/radial | Gas dispersion at low shear |
| Lightnin A315 (hydrofoil) | 0.75 | 1.5–3 | 0.35–0.50 | Axial | Low-shear cell culture, blending |
| Marine propeller (3-blade) | 0.35 | 0.8–2 | 0.25–0.40 | Axial | Blending, low-shear mixing |
| Intermig (Ekato) | 0.35 | 0.8–2 | 0.60–0.70 | Axial | Large-scale cell culture |
| Elephant ear (down-pumping) | 0.3 | 0.6–1.5 | 0.40–0.55 | Axial | Mammalian cell culture, single-use |
| Anchor | 0.35 | 1.0–4 | 0.90–0.98 | Tangential | High-viscosity, wall scraping |
| Helical ribbon | 0.35 | 2–10 | 0.90–0.98 | Axial (close-clearance) | Very high viscosity (>10 Pa·s) |
Microbial fermentation (E. coli, yeast, Bacillus): Use Rushton or Smith turbines for maximum gas dispersion and P/V. These organisms tolerate high shear.
Mammalian cell culture (CHO, HEK293): Use pitched blade, elephant ear, or hydrofoil impellers. Np below 1.5 keeps shear stress within safe limits for animal cells.
High-viscosity broths (filamentous fungi, polysaccharides): Consider anchor or helical ribbon impellers at very high viscosity, or Maxblend for moderate viscosity with good top-to-bottom mixing.
3. Multiple Impeller Correction
Most bioreactors above 5 L use two or more impellers on a single shaft. The total power draw depends on whether the impellers interact hydrodynamically.
Rule of thumb: When impellers are spaced at least 1 impeller diameter (Di) apart, they behave as independent units and the total Np is approximately:
where n = number of impellers on the shaft
This linear scaling is a good approximation for most configurations. However, there are two important exceptions:
- Closely spaced impellers (spacing < 0.75 Di): Hydrodynamic interaction reduces the effective Np per impeller by 10–20%.
- Mixed impeller configurations (e.g., one Rushton + one pitched blade): Calculate power for each impeller separately using its own Np, then sum. The Rushton at the bottom provides gas dispersion while the pitched blade above provides bulk mixing.
A typical pilot-scale STR (50–200 L) uses two 6-blade Rushton turbines spaced 1.0–1.5 Di apart. Total Np = 2 × 5.0 = 10.0. At production scale, a combination of one Rushton (bottom, for gas dispersion) and one or two pitched blade turbines (upper, for bulk mixing) is common, giving total Np = 5.0 + 1.3 = 6.3 for a two-impeller system.
4. How Np Changes with Reynolds Number
The impeller Reynolds number determines which flow regime your system operates in:
where:
ρ = fluid density (kg/m³)
N = impeller speed (rev/s)
Di = impeller diameter (m)
μ = dynamic viscosity (Pa·s)
| Regime | Re Range | Np Behavior | Practical Notes |
|---|---|---|---|
| Laminar | < 10 | Np ∝ 1/Re (very high) | Only relevant for very viscous fluids; use anchor or helical ribbon |
| Transitional | 10–10,000 | Np decreasing toward constant | Common in high-viscosity fungal broths; Np can be 2–3× higher than turbulent |
| Turbulent | > 10,000 | Np is constant | Most aqueous fermentations and cell cultures operate here |
A filamentous Aspergillus fermentation can start with water-like viscosity (Re > 100,000) and end at 5–10 Pa·s with Re in the transitional range. As Re drops, Np increases—but the power draw also depends on N³, and you may be unable to increase RPM further without exceeding motor torque limits. This is one reason filamentous fermentations are among the most challenging to scale up.
5. Power Draw Formula
The ungassed power draw of an impeller in the turbulent regime is calculated from:
where:
P = power draw (W)
Np = power number (dimensionless)
ρ = fluid density (kg/m³)
N = impeller speed (rev/s, NOT RPM)
Di = impeller diameter (m)
Note the fifth-power dependence on impeller diameter. Doubling Di increases power draw by a factor of 32 at the same RPM. This is why small changes in Di/DT ratio have outsized effects on P/V.
Gassed power correction
When gas is sparged beneath the impeller, the apparent power draw decreases because gas cavities form behind the impeller blades. The gassed-to-ungassed power ratio (Pg/P0) depends on impeller type:
| Impeller Type | Pg/P0 at 1 vvm |
|---|---|
| 6-blade Rushton | 0.40–0.60 |
| Smith turbine (CD-6) | 0.70–0.85 |
| Pitched blade (down) | 0.70–0.90 |
| Hydrofoil (A315) | 0.80–0.95 |
The Smith turbine's curved blades resist gas flooding much better than flat Rushton blades, which is why it maintains more of its ungassed power during aeration. This makes it increasingly popular for high-aeration microbial processes.
Calculate Power & P/V Automatically
Enter your impeller type, dimensions, and RPM. Get power draw, P/V, tip speed, Reynolds number, and scale-up predictions.
Scale-Up Calculator →6. Calculate Your P/V
Power number is the starting point for almost every bioreactor scale-up calculation. With Np and the formula P = Np × ρ × N³ × Di5, you can calculate P/V at any scale—and from P/V, estimate kLa using the Van't Riet correlation.
For more on how P/V connects to oxygen transfer and scale-up strategy, see these related resources:
- How to Calculate kLa — Van't Riet correlation using P/V to estimate kLa, with worked examples.
- kLa Reference Table — Typical kLa values by vessel type and scale.
- Scale-Up Calculator — Compare constant P/V, tip speed, kLa, Re, and mixing time criteria.