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), except the anchor and helical ribbon, which run unbaffled in laminar flow. Di/DT is the ratio of impeller diameter to tank diameter. Where a range is given, Np moves with blade and disc thickness, off-bottom clearance and D/T, so treat the single number as the textbook value and the range as what real impellers measure.
| Impeller Type | Np (Turbulent) | Di/DT Range | Flow Pattern | Best For |
|---|---|---|---|---|
| 6-blade Rushton turbine | 5.0 (about 4.5–6) | 0.30–0.40 | Radial | Gas dispersion, microbial fermentation |
| 4-blade Rushton turbine | 3.5 | 0.30–0.40 | Radial | Lower-power gas dispersion |
| Concave-blade turbine (Smith, Chemineer CD-6) | 2.8–3.2 | 0.30–0.40 | Radial | Gas dispersion with less flooding |
| Maxblend (Sumitomo) | Geometry-specific: use vendor data | 0.50–0.60 | Combined axial/radial | High-viscosity, cell culture |
| Elephant ear (3-blade, 45°) | 1.7–2.2 | 0.40–0.55 | Axial/mixed | Mammalian cell culture, single-use |
| Pitched blade (4-blade 45°, down-pumping) | 1.3 (1.2–1.7) | 0.30–0.50 | Mixed axial/radial | Solids suspension, blending |
| Pitched blade (4-blade 45°, up-pumping) | 1.3 (1.2–1.7) | 0.30–0.50 | Mixed axial/radial | Gas dispersion at low shear |
| Lightnin A315 (wide-blade hydrofoil) | 0.75 | 0.35–0.50 | Axial | Low-shear cell culture, aerated blending |
| Intermig (Ekato) | 0.35–0.6 | 0.60–0.70 | Axial | Large-scale fermentation, blending |
| Marine propeller (3-blade, pitch = D) | 0.35 | 0.25–0.40 | Axial | Blending, low-shear mixing |
| Narrow-blade hydrofoil (Lightnin A310, Chemineer HE-3) | 0.3 | 0.30–0.50 | Axial | Low-power blending, upper impeller positions |
| Anchor (unbaffled) | Laminar use only* | 0.90–0.98 | Tangential | High-viscosity, wall scraping |
| Helical ribbon (unbaffled) | Laminar use only* | 0.90–0.98 | Axial (close-clearance) | Very high viscosity (>10 Pa·s) |
*Anchors and helical ribbons are close-clearance impellers for viscous fluids. They are characterised in the laminar regime by the constant Kp = Np × Re rather than by a turbulent Np, and Kp depends strongly on wall clearance. Sources for the ranges: Rushton thickness and scale effects from Bujalski, Nienow and Chatwin (1987); pitched-blade geometry effects from Chapple et al. (2002); the elephant-ear impeller from Delbridge et al. (2023), who measured Np = 1.81 (up-pumping) and 2.09 (down-pumping) in a standard cylindrical tank; Intermig from Szalai et al. (2004); A310, HE-3 and CD-6 from vendor data.
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. Axial impellers spread a given P/V over a larger pumped flow, so the peak energy dissipation near the blades is lower than with a Rushton. Np is a power-draw constant, not a shear rating, so check tip speed and P/V as well.
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 below about 1 Di): the upper impeller runs in the discharge stream of the lower one and draws less than its standalone Np, so the simple sum overestimates the total. The shortfall grows as the spacing shrinks, which is why dual-Rushton studies such as Hudcova, Machon and Nienow (1989) report power for the specific spacing tested.
- 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 = Kp/Re, so Np × Re is constant (Kp ≈ 70 for a Rushton) | Only relevant for very viscous fluids; use anchor or helical ribbon |
| Transitional | 10–10,000 | Impeller-dependent, approaching the turbulent constant | Common in high-viscosity fungal broths. A baffled Rushton changes little through this range, while axial impellers and hydrofoils draw noticeably more than their turbulent Np as Re falls |
| 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 power draw falls because gas-filled cavities form behind the blades. The gassed-to-ungassed power ratio (Pg/P0) depends on impeller type and, above all, on the gas flow number FlG = Qg/(N Di3), not on vvm alone. The same 1 vvm gives a very different FlG at bench and production scale. Typical ratios under heavy aeration:
| Impeller Type | Pg/P0 under heavy aeration |
|---|---|
| 6-blade Rushton | 0.40–0.60 |
| Concave-blade turbine (Smith, CD-6) | 0.70–0.85 |
| Pitched blade (down) | 0.70–0.90 |
| Hydrofoil (A315) | 0.80–0.95 |
| Elephant ear, up-pumping | about 1.0 |
| Elephant ear, down-pumping | down to about 0.7 at high gas flow |
The concave-blade turbine resists gas flooding much better than flat Rushton blades, which is why it keeps more of its ungassed power during aeration and is increasingly popular for high-aeration microbial processes. The elephant-ear rows come from Zhu et al. (2009), who found the up-pumping impeller's power number unaffected by aeration while the down-pumping one lost up to 30%.
When you need a number rather than a range, two classic correlations are still used for Rushton-type disc turbines. Michel and Miller (1962) give Pg ∝ (P02 N D3 / Qg0.56)0.45. It is dimensional, so its constant changes with the units, and it breaks down at very low gas rates, where it predicts a Pg larger than P0. Hughmark (1980) is dimensionless, Pg/P0 = 0.10 (Qg/(N V))−0.25 (N2 D4 / (g W V2/3))−0.2, with W the blade height and V the liquid volume. Both were fitted to flat-blade disc turbines in low-viscosity, mostly coalescing liquids. Do not apply them to hydrofoils, elephant-ear or concave-blade impellers, or to viscous broths.
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. Worked Example: Impeller Power for a 50 L Bioreactor
For a single Rushton turbine (Np ≈ 5.5) with an impeller diameter Di = 0.15 m running at N = 100 rpm (1.667 rev/s) in a broth of density ρ = 1000 kg/m³, the ungassed power draw is about 1.93 W, giving a power-per-volume of roughly 39 W/m³ across a 50 L (0.05 m³) working volume. Here is the full calculation, step by step.
Start from the turbulent power equation and substitute each value:
Step 1 — convert speed to rev/s:
N = 100 rpm ÷ 60 = 1.667 rev/s
Step 2 — cube the speed:
N³ = 1.667³ = 4.63 (rev/s)³
Step 3 — raise diameter to the 5th power:
Di5 = 0.15⁵ = 7.59 × 10⁻⁵ m⁵
Step 4 — multiply through:
P = 5.5 × 1000 × 4.63 × 7.59 × 10⁻⁵
P ≈ 1.93 W
Step 5 — divide by working volume for P/V:
P/V = 1.93 W ÷ 0.05 m³ ≈ 39 W/m³
A P/V near 40 W/m³ is a modest, cell-culture-friendly intensity. Microbial fermentations typically push P/V much higher (often several hundred to a few thousand W/m³) to meet oxygen demand, which is why they favour high-Np Rushton or Smith turbines run at higher RPM.
The 1.93 W above is the ungassed power. Under heavy sparging, a Rushton turbine loses roughly 40–60% of its power draw as gas cavities form behind the blades, so the gassed power for this example would be closer to 0.8–1.2 W. At 1 vvm this example is heavily aerated: Qg = 0.05 m³/min = 8.3 × 10⁻⁴ m³/s, so FlG = Qg/(N Di3) = 8.3 × 10⁻⁴ / (1.667 × 0.15³) ≈ 0.15, well into the range where a Rushton running at only 100 rpm is likely to flood. Always apply the Pg/P0 ratio from the table above when sizing motors or estimating kLa under aeration. Bear in mind that the reported kLa is a volume average: local kLa near the impeller discharge zone is typically 3–8× the tank mean.
7. 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.
Frequently Asked Questions
What is the power number of an ambr 250 impeller?
The ambr 250 has no single catalogued power number, because the impeller depends on the vessel type: cell-culture vessels use small pitched, elephant-ear-style impellers (single or dual, up- or down-pumping) and microbial vessels use Rushton-type turbines. A 2026 study of six elephant-ear configurations (Miranda et al.) found that down-pumping raised the cumulative Np by 14–22% over up-pumping, and that a single impeller had a 33% higher cumulative Np than a dual set. For a cell-culture vessel expect a per-impeller value in the pitched-blade to elephant-ear range of roughly 1.3–2.2, and use the measured value for your exact vessel and impeller set when the answer matters.
How sensitive is the impeller power number Np to D/T ratio?
Np is defined at fully turbulent Reynolds numbers (Re > 104) and is nominally a geometry-only constant, but it is not one number for a named impeller. For a Rushton the larger effect is disc and blade thickness: Bujalski, Nienow and Chatwin (1987) showed Np falling as the disc gets thicker relative to the impeller, which is why measured values span roughly 4.5 to 6. For a 45° pitched-blade turbine, Chapple et al. (2002) found Np moves with blade thickness, D/T and off-bottom clearance, giving roughly 1.2 to 1.7. Over the usual D/T window of 0.33–0.5, D/T alone is a secondary effect next to those. Use the tabulated value for first estimates, and a torque-measured value for your own impeller when the power figure drives a decision such as motor sizing.
Rushton or pitched-blade impeller for mammalian cell culture?
Pitched-blade axial-flow impellers (Np ≈ 1.3) and lower-Np hydrofoils such as the Lightnin A315 (≈ 0.75) or Chemineer HE-3 (≈ 0.3) are the standard choice for shear-sensitive mammalian cell culture (CHO, HEK293, Vero) because at an equivalent power-per-volume of 30–50 W/m³ they produce lower peak strain rates than a Rushton (Np ≈ 5.5). To hit the same P/V a lower-Np impeller must be spun faster, but the resulting flow is smoother and more axial, so the maximum energy dissipation rate near the blade tip is smaller. Rushton radial-flow turbines remain the default for microbial fermentations (E. coli, yeast, Bacillus) at 1–5 kW/m³ where oxygen mass transfer dominates and the cells tolerate the higher shear.
How do I use the power number to estimate agitator power draw?
In the fully turbulent regime the ungassed shaft power is P = Np × ρ × N³ × D5, where P is in watts, ρ is fluid density in kg/m³, N is impeller speed in rev/s (not rpm) and D is impeller diameter in metres. Worked example: a 6-blade Rushton with D = 0.3 m at N = 200 rpm (3.333 rev/s) in water (ρ = 1000 kg/m³) draws P = 5.5 × 1000 × 3.333³ × 0.35 ≈ 495 W (about 0.5 kW). Two things bite people here: N must be converted to rev/s before it is cubed, and the fifth-power dependence on D means small changes in impeller diameter dominate the answer. Under heavy aeration a Rushton keeps only about 0.4–0.6 of that (Pg/P0, set by the gas flow number rather than vvm alone). Size the motor on the ungassed figure, because the impeller must be able to start and run before the gas is on.