What Is the Reynolds Number?
The Reynolds number is a dimensionless ratio of inertial forces to viscous forces in a fluid, and it determines whether flow around a bioreactor impeller is smooth and orderly (laminar) or chaotic and well-mixed (turbulent). In stirred-tank bioreactors, achieving turbulent flow is essential for adequate mass transfer, heat transfer, and suspension of cells or particles.
Named after Osborne Reynolds, who published his pipe-flow experiments in 1883, the Reynolds number concept was extended to stirred vessels by Rushton, Costich, and Everett in 1950. Their work established that power consumption in agitated vessels could be correlated through a dimensionless power number plotted against the impeller Reynolds number — a framework that remains the basis of bioreactor design today.
Understanding the Reynolds number in a bioreactor context matters for three practical reasons:
- Mixing adequacy — turbulent conditions (Re > 10,000) ensure rapid blending of nutrients, pH-adjusting agents, and dissolved oxygen throughout the vessel.
- Power prediction — once you know the flow regime, you can read the power number from standard correlations and calculate the power draw P = NpρN³D&sup5;.
- Scale-up confidence — verifying that Re stays well above 10,000 at production scale confirms that power-number correlations developed at bench scale still apply.
The Impeller Reynolds Number Formula
The impeller Reynolds number for a stirred tank is defined as Re = ρND²/μ, where each variable has a specific physical meaning and unit requirement.
| Symbol | Variable | SI Unit | Typical Range |
|---|---|---|---|
| ρ | Fluid density | kg/m³ | 1,000–1,050 (aqueous media) |
| N | Impeller rotational speed | rev/s (= RPM ÷ 60) | 1–10 s&supmin;¹ |
| D | Impeller diameter | m | 0.03–3.0 m |
| μ | Dynamic viscosity | Pa·s (= kg/m·s) | 0.001–0.5 Pa·s |
A common error is using RPM directly without converting to rev/s: an impeller at 200 RPM is N = 200/60 = 3.33 rev/s. Failing to convert inflates Re by a factor of 60, which can mask a transitional-flow problem.
For non-Newtonian fluids (shear-thinning broths), the apparent viscosity depends on shear rate. The Metzner–Otto method replaces μ with μapp evaluated at an effective shear rate γeff = ksN, where ks is a geometry-dependent constant (typically 10–13 for Rushton turbines). This yields a modified Reynolds number ReMO = ρND²/μapp(γeff).
Flow Regimes: Laminar, Transitional, and Turbulent
Three distinct flow regimes exist in stirred-tank bioreactors, each defined by the impeller Reynolds number. Turbulent flow (Re > 10,000) is the target for virtually all cell culture and microbial fermentation processes because it provides the chaotic mixing needed for uniform nutrient distribution and gas dispersion.
In the laminar regime, fluid moves in smooth layers around the impeller and mixing is extremely slow — regions far from the impeller may remain completely stagnant. The transitional regime produces partial turbulence near the impeller blades but poor bulk mixing, making it unreliable for bioprocesses that require homogeneous conditions. Only in the turbulent regime does the impeller generate sufficient energy dissipation to produce uniform mixing throughout the vessel.
For standard baffled stirred-tank bioreactors with four baffles (width = T/10, where T is tank diameter), the critical thresholds are:
- Re < 10 — Fully laminar. Power number inversely proportional to Re. Poor mixing, dead zones.
- 10 < Re < 10,000 — Transitional. Power number decreasing with increasing Re. Inconsistent mixing.
- Re > 10,000 — Fully turbulent. Power number constant. Standard correlations for mixing time, kLa, and heat transfer apply.
Power Number and Its Dependence on Re
The power number (Np) is the dimensionless ratio of power drawn by the impeller to the inertial force of the fluid: Np = P / (ρN³D&sup5;). In the turbulent regime, Np reaches a constant value that depends only on impeller geometry and the vessel configuration (baffles, D/T ratio), not on fluid properties or agitation rate.
This constancy is the practical payoff of ensuring turbulent flow: once you know Np for your impeller type, calculating power is straightforward. Below Re = 10,000, Np varies with Re and predictions become less reliable.
| Impeller Type | Flow Pattern | Np (turbulent) | Typical Use |
|---|---|---|---|
| Rushton turbine (6-blade) | Radial | 5.0 | Gas dispersion, microbial |
| Smith turbine (6-blade concave) | Radial | 3.2 | Gas dispersion, high-aeration |
| Pitched-blade turbine (4-blade, 45°) | Mixed axial-radial | 1.3–1.7 | Blending, suspension |
| Elephant ear (down-pumping) | Axial | 1.5–1.7 | Mammalian cell culture |
| Hydrofoil (Lightnin A315) | Axial | 0.75–0.85 | Low-shear cell culture |
| Hydrofoil (Lightnin A320) | Axial | 0.6–0.7 | Blending, suspension |
| Marine propeller (3-blade) | Axial | 0.3–0.4 | Low-viscosity blending |
| Anchor | Tangential | 0.35 | High-viscosity broths |
The chart below shows how Np varies with Reynolds number for four impeller types. In the laminar regime, all impellers follow the relationship Np ∝ 1/Re (a straight line on a log-log plot). As Re increases through the transition, each impeller settles to its characteristic constant turbulent Np.
How Viscosity Shifts the Flow Regime
Viscosity is the single most important variable that can push a bioreactor out of the turbulent regime. Because μ appears in the denominator of Re = ρND²/μ, even a moderate increase in broth viscosity can reduce Re by orders of magnitude.
Water-like cell culture media (CHO, HEK293, E. coli in early exponential phase) have a viscosity of approximately 0.001 Pa·s, similar to water at 25°C. Under standard agitation conditions, these broths achieve Re > 30,000 even in bench-scale vessels. However, several common bioprocess scenarios produce much higher viscosities:
| Broth Type | Apparent Viscosity (Pa·s) | Re at 200 RPM, D = 0.05 m | Flow Regime |
|---|---|---|---|
| Water / dilute media | 0.001 | 8,300 | Transitional–turbulent |
| CHO fed-batch (day 12) | 0.002–0.003 | 2,800–4,200 | Transitional |
| E. coli (OD > 100) | 0.003–0.005 | 1,700–2,800 | Transitional |
| Aspergillus mycelial (mid-batch) | 0.05–0.2 | 42–170 | Transitional (low end) |
| Xanthan gum production | 0.5–5.0 | 1.7–17 | Laminar |
For shear-thinning organisms like Aspergillus, viscosity varies with position in the vessel. Near the impeller tip where shear rates exceed 100 s&supmin;¹, the apparent viscosity may be 0.05 Pa·s, but in the bulk region between baffles, where shear rates are 1–10 s&supmin;¹, apparent viscosity can exceed 0.5 Pa·s. This creates local turbulent zones around the impeller surrounded by poorly mixed, near-laminar bulk fluid — one of the most challenging mixing problems in bioprocessing.
Strategies to maintain turbulent flow in viscous broths include:
- Increasing impeller diameter (D appears as D² in Re — the strongest geometric lever)
- Adding a second impeller to extend turbulent zones
- Using close-clearance impellers (anchor, helical ribbon) designed for low-Re operation
- Reducing broth viscosity by diluting with water or adjusting feed strategy
Worked Examples
These two examples demonstrate Reynolds number calculations for a bench-scale CHO culture and a production-scale E. coli fermentation, showing how vessel geometry and broth properties affect the flow regime.
Example 1: Bench-Scale CHO Fed-Batch (2 L Bioreactor)
Given:
- Impeller type: 3-blade elephant ear (down-pumping)
- Impeller diameter D = 0.045 m (Di/Dt = 0.33 for a 2 L vessel with Dt = 0.136 m)
- Agitation speed N = 150 RPM = 150/60 = 2.50 rev/s
- Fluid density ρ = 1,010 kg/m³ (cell culture media at 37°C)
- Dynamic viscosity μ = 0.0007 Pa·s (water at 37°C)
Calculation:
Re = ρND² / μ
Re = 1,010 × 2.50 × (0.045)² / 0.0007
Re = 1,010 × 2.50 × 0.002025 / 0.0007
Re = 5.113 / 0.0007
Re = 7,300
Interpretation: Re = 7,300 falls in the upper transitional regime — close to turbulent but not fully there. At this Re, the power number has not yet reached its constant turbulent value, and mixing time correlations may be unreliable. Increasing agitation to 200 RPM would push Re to 9,800, and 250 RPM would yield Re = 12,200, entering the fully turbulent regime.
Example 2: Production-Scale E. coli High-Cell-Density (1,000 L Bioreactor)
Given:
- Impeller type: Rushton turbine (6-blade)
- Impeller diameter D = 0.30 m (Di/Dt = 0.33 for a vessel with Dt = 0.91 m)
- Agitation speed N = 300 RPM = 300/60 = 5.0 rev/s
- Fluid density ρ = 1,020 kg/m³
- Dynamic viscosity μ = 0.0008 Pa·s (fermentation broth at 37°C)
Calculation:
Re = 1,020 × 5.0 × (0.30)² / 0.0008
Re = 1,020 × 5.0 × 0.09 / 0.0008
Re = 459 / 0.0008
Re = 574,000
Interpretation: Re = 574,000 is deeply turbulent. The Rushton turbine operates at its constant Np = 5.0, so power draw is:
P = Np × ρ × N³ × D&sup5;
P = 5.0 × 1,020 × 125 × 2.43 × 10&supmin;³
P = 5.0 × 1,020 × 0.304
P = 1,550 W (1.55 kW)
For a 700 L working volume, this corresponds to P/V = 1,550 / 0.7 = 2.2 W/L — typical for high-cell-density E. coli fermentation.
Reynolds Number in Scale-Up
Maintaining a constant Reynolds number during scale-up is rarely a practical strategy because Re naturally increases with vessel size, and holding it constant would require reducing agitation speed so severely that mixing and mass transfer become inadequate.
To see why, consider a geometric scale-up where all linear dimensions scale by a factor S (Dlarge = S × Dsmall). At constant Re:
Re = ρND²/μ = constant
⇒ N ∝ 1/D² ∝ 1/S²
⇒ P/V ∝ N³D² ∝ S&supmin;&sup4;
For a 10× linear scale-up: P/V drops by 10,000×
A 10,000-fold reduction in P/V would leave the large-scale vessel essentially unmixed. This is why constant P/V, constant tip speed (πND), or constant kLa are the preferred criteria for bioreactor scale-up — each keeps mixing intensity within a useful range across scales.
The following table compares how different scale-up criteria affect Re, P/V, and tip speed as a vessel scales geometrically from 2 L to 2,000 L (S = 10):
| Criterion Held Constant | N Scaling | Re Scaling | P/V Scaling | Tip Speed Scaling |
|---|---|---|---|---|
| Constant Re | ∝ S&supmin;² | 1× | ∝ S&supmin;&sup4; (0.0001×) | ∝ S&supmin;¹ (0.1×) |
| Constant tip speed (πND) | ∝ S&supmin;¹ | ∝ S (10×) | ∝ S&supmin;² (0.01×) | 1× |
| Constant P/V | ∝ S&supmin;⅔ | ∝ S⅔ (4.6×) | 1× | ∝ S⅓ (2.15×) |
| Constant kLa | Case-dependent | Increases | ≈ 1× | 1–2× |
The practical guidance is straightforward: choose your primary scale-up criterion (usually constant P/V or constant kLa) and then verify that the resulting Re remains above 10,000. If it does — and it almost always will for aqueous media — your turbulent power-number correlations still apply at the larger scale.
Scale-Up Calculator
Calculate P/V, tip speed, Re, and kLa across scales with our free bioreactor scale-up tool.
Heat Transfer Calculator
Estimate jacket and coil heat transfer in bioreactors — turbulent Re enables standard Nu correlations.
OTR/kLa Estimator
Estimate oxygen transfer rates from P/V and superficial gas velocity using Van't Riet correlations.
Frequently Asked Questions
What Reynolds number is needed for turbulent flow in a bioreactor?
An impeller Reynolds number above 10,000 indicates fully turbulent flow in a baffled stirred-tank bioreactor. Below Re = 10, flow is laminar; between 10 and 10,000 is the transitional regime. Most production-scale bioreactors operate at Re = 50,000 to 500,000, well into the turbulent regime.
How do you calculate Reynolds number for a bioreactor impeller?
The impeller Reynolds number is Re = ρND²/μ, where ρ is fluid density (kg/m³), N is rotational speed (rev/s), D is impeller diameter (m), and μ is dynamic viscosity (Pa·s). Always convert RPM to rev/s by dividing by 60 before substituting into the equation.
Why does power number become constant at high Reynolds numbers?
In fully turbulent flow (Re > 10,000), form drag on the impeller blades dominates over viscous drag. Because form drag scales with ρN²D² — the same dependence as the inertial term in the power number definition — the ratio (power number) becomes constant. Power then scales as P = NpρN³D&sup5; regardless of viscosity.
How does viscosity affect the Reynolds number in a bioreactor?
Viscosity appears in the denominator of Re = ρND²/μ, so higher viscosity decreases Re. Mycelial fermentations (apparent viscosity 0.05–0.5 Pa·s) can reduce Re by 50–500× compared to water-like media, potentially shifting flow from turbulent into the transitional regime and requiring higher impeller speeds or larger impellers to maintain adequate mixing.
Can you use Reynolds number as a scale-up criterion for bioreactors?
Constant Reynolds number is rarely used as a primary scale-up criterion because it leads to decreasing P/V at larger scales (P/V ∝ S&supmin;&sup4;), resulting in inadequate mixing. More common criteria are constant P/V, constant tip speed, or constant kLa. However, checking that Re stays above 10,000 at all scales is essential for ensuring turbulent conditions.
Related Tools
- Scale-Up Calculator — Calculate P/V, tip speed, Re, and kLa across bioreactor scales with constant-criterion matching.
- OTR/kLa Estimator — Estimate oxygen transfer rate from P/V and superficial gas velocity using Van't Riet correlations.
- Heat Transfer Calculator — Size jackets and coils using Nusselt number correlations that depend on the impeller Reynolds number.
References
- Rushton J.H., Costich E.W. & Everett H.J. (1950). Power characteristics of mixing impellers, Part I. Chemical Engineering Progress, 46(8), 395–404.
- Nienow A.W. (1998). Hydrodynamics of stirred bioreactors. Applied Mechanics Reviews, 51(1), 3–32. DOI: 10.1115/1.3098990
- Kaiser S.C., Werner S., Jossen V., Kraume M. & Eibl D. (2017). Development of a method for reliable power input measurements in conventional and single-use stirred bioreactors at laboratory scale. Engineering in Life Sciences, 17(5), 500–511. DOI: 10.1002/elsc.201600096
- Nienow A.W. (2014). Stirring and stirred-tank reactors. Chemie Ingenieur Technik, 86(12), 2063–2074. DOI: 10.1002/cite.201400087