The impeller Reynolds number is defined as Re = ρND²/μ, where ρ is the fluid density (kg/m³), N is the rotational speed in revolutions per second (RPM ÷ 60), D is the impeller diameter (m), and μ is the dynamic viscosity (Pa·s). For example, a Rushton turbine with D = 0.1 m spinning at 200 RPM in water (ρ = 1000 kg/m³, μ = 0.001 Pa·s) gives Re = 1000 × 3.33 × 0.01 / 0.001 = 33,333.

For stirred-tank impellers: Re < 10 is laminar flow where the power number (Np) varies inversely with Re (Np × Re = constant). Between Re 10 and 10,000 is the transitional regime where flow patterns are developing. Above Re 10,000 is fully turbulent flow where Np is constant and independent of Re. Most bioreactors operate in the turbulent regime. High-viscosity fermentations (mycelial, polysaccharide-producing) may operate in the transitional regime.

The power number Np = P/(ρN³D&sup5;) relates impeller power draw to fluid properties and agitation speed. In turbulent flow, Np is constant for a given impeller geometry: Rushton turbine ≈ 5.0, pitched-blade 45° ≈ 1.27, marine propeller ≈ 0.35, hydrofoil ≈ 0.75. Knowing Np lets you calculate power consumption P = Np × ρ × N³ × D&sup5;, which determines motor sizing, heat generation, and the P/V parameter used for scale-up.

Re is inversely proportional to viscosity. As broth viscosity increases during fermentation (due to cell growth, exopolysaccharides, or mycelial morphology), Re decreases proportionally. A culture starting at Re = 100,000 in water-like media can drop to Re = 2,000 if viscosity increases 50-fold during a filamentous fermentation. This transition from turbulent to transitional flow reduces mixing efficiency and oxygen transfer. Use the viscosity slider to simulate this effect and identify when to increase RPM.

During scale-up, Re naturally increases because impeller diameter increases (Re ∝ D²). This means turbulent flow is generally maintained at larger scales. However, maintaining constant Re as a scale-up criterion leads to impractically high power inputs. Instead, engineers verify that Re remains above 10,000 at the target scale, then use constant P/V (0.5–2 W/L for cell culture, 2–5 W/L for microbial) or constant tip speed (1–2 m/s for mammalian cells) as the primary criterion.

For shear-sensitive mammalian cells (CHO, HEK293), tip speed should be kept below 1.5–2.0 m/s. For microcarrier cultures, the limit is lower (0.5–1.0 m/s) to avoid cell detachment. Microbial fermentations tolerate much higher tip speeds (3–7 m/s). Tip speed = π × N × D, where N is in rev/s and D in metres. This calculator shows tip speed for all impeller types at your current conditions.

Re always rises at larger scale under constant P/V, because impeller speed N scales as T^−2/3 while diameter D scales as T, so Re = ρND²/μ scales as T^4/3. Worked example for a Rushton turbine (Np = 5.0, D/T = 0.33, water) held at 1 W/L:

• 2 L vessel (T = 0.137 m, D = 0.045 m): 680 RPM → Re ≈ 23,000, tip ≈ 1.6 m/s.
• 100 L vessel (T = 0.5 m, D = 0.165 m): 290 RPM → Re ≈ 130,000, tip ≈ 2.5 m/s.
• 10,000 L vessel (T = 2.34 m, D = 0.77 m): 105 RPM → Re ≈ 1,030,000, tip ≈ 4.2 m/s.

Turbulent flow is preserved across the whole scale-up in aqueous media; tip speed only rises about 2.6× for a 5,000× volume increase.

For pseudoplastic (shear-thinning) broths — common in mycelial and polysaccharide fermentations — apparent viscosity depends on local shear rate, so the standard Re formula must use an effective viscosity. The Metzner-Otto (1957) approach gives μ_app = K × (k_sN)ⁿ⁻¹, where K is the consistency index (Pa·sⁿ), n is the flow behaviour index (n < 1 for shear-thinning), and k_s ≈ 11 for a Rushton turbine (about 10 for pitched-blade, 12–14 for hydrofoils).

Example: K = 0.1 Pa·sⁿ, n = 0.5, 200 RPM (N = 3.33 rev/s) gives μ_app = 0.1 × 36.7^−0.5 ≈ 0.0165 Pa·s — roughly 16× water. Enter this apparent viscosity above; you will see Re can drop into the transitional regime even at modest viscometer readings.