1. Why kLa Matters
Oxygen is the most common limiting substrate in aerobic fermentation. Unlike carbon sources that can be fed on demand, oxygen has extremely low solubility in aqueous media—roughly 7–8 mg/L at 37°C and atmospheric pressure. A high-density E. coli fermentation at 50 g/L DCW can demand upwards of 200 mmol O2/L/h, yet the dissolved oxygen reservoir in your vessel would be depleted in under two seconds without continuous resupply.
This is where kLa—the volumetric mass transfer coefficient—becomes the single most important parameter characterizing your bioreactor's oxygen delivery capacity. It quantifies how rapidly oxygen can move from the gas phase into the liquid, and it governs the oxygen transfer rate (OTR):
where:
OTR = oxygen transfer rate (mmol/L/h or mol/m³/s)
kLa = volumetric mass transfer coefficient (h¹ or s¹)
C* = saturated dissolved oxygen concentration (mg/L)
CL = actual dissolved oxygen concentration (mg/L)
The term (C* − CL) is the driving force: the concentration difference between what the liquid could hold at saturation and what it actually holds. kLa is the rate constant that tells you how efficiently your bioreactor converts that driving force into actual oxygen delivery.
When OTR < OUR (oxygen uptake rate), your culture becomes oxygen-limited. The downstream effects cascade rapidly: E. coli shifts to mixed-acid fermentation producing acetate, S. cerevisiae triggers the Crabtree effect, CHO cells reduce specific productivity and viability. In every case, you lose product yield, quality, or both.
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Try the Calculator →2. What is kLa? The Physics
Diagram of a stirred-tank bioreactor cross-section. The vessel has a rounded bottom with a ring sparger at the base producing gas bubbles of varying sizes. A Rushton turbine impeller is mounted at mid-height on a central shaft, with four baffles along the vessel walls. Bubbles rise from the sparger through the liquid, becoming more dispersed at higher positions. Annotations label the gas bubbles as providing interfacial area (a), the bulk liquid as having dissolved oxygen concentration CL, and arrows indicate the dissolved oxygen concentration gradient. An inset panel shows a magnified view of the gas-liquid interface where O2 molecules cross from the bubble through the liquid film (where kL acts) into the bulk liquid.
The subscript notation tells the whole story. kLa is actually two parameters lumped together because separating them experimentally is impractical in most bioreactor systems:
- kL — the liquid-side mass transfer coefficient (m/s). This describes how fast oxygen molecules traverse the stagnant liquid film surrounding each gas bubble. It depends on diffusivity, liquid viscosity, and the degree of surface renewal at the bubble interface.
- a — the specific interfacial area (m²/m³). This is the total surface area of all gas bubbles per unit volume of liquid. Smaller bubbles, higher gas holdup, and better gas dispersion all increase a.
In practice, we almost always measure and report kLa as a combined parameter because both components respond simultaneously to changes in operating conditions. Increasing impeller speed, for example, both thins the liquid film (increasing kL) and breaks bubbles into smaller sizes (increasing a).
Typical kLa Ranges by System
| System Type | Typical kLa Range (h−1) | Primary Limiting Factor |
|---|---|---|
| Shake flask (250 mL) | 10–100 | Fill volume, shaking speed |
| Wave / rocking bioreactor | 5–30 | Rocking rate, fill level |
| STR 2–10 L (lab) | 50–200 | Impeller speed, sparger |
| STR 50–500 L (pilot) | 80–300 | Power input, gas flow |
| STR 5,000–20,000 L (production) | 100–400 | Impeller design, backpressure |
| Single-use STR | 30–150 | Lower power, polymer film |
Factors Affecting kLa
- Agitation / power input: Higher P/V increases both kL and a by generating turbulence and breaking bubbles.
- Aeration rate: More gas flow increases gas holdup and thus interfacial area.
- Media properties: Viscosity suppresses turbulence (lower kL); salts and proteins inhibit bubble coalescence (higher a); antifoam agents reduce interfacial tension but can form films that block transfer (lower kL).
- Temperature: Higher temperature increases diffusivity (higher kL) but decreases oxygen solubility (lower C*). Net effect on OTR depends on the system.
- Geometry: Impeller type, sparger design, D/T ratio, number of baffles, H/T ratio all influence the flow patterns and gas dispersion.
3. Measuring kLa: Dynamic Gassing Out Method
The dynamic gassing out (or gassing-in) method is the most widely used experimental technique for measuring kLa in stirred tank bioreactors. It requires only a DO probe and a source of nitrogen gas. Here is the step-by-step protocol:
Protocol
- Establish baseline conditions. Fill the vessel to working volume with your medium (or water for initial characterization). Set temperature, agitation, and back-pressure to your target conditions.
- Strip dissolved oxygen. Switch gas supply to 100% N2 and sparge until the DO reading drops below 5% of air saturation (ideally to 0%).
- Switch to air. At time t = 0, switch the gas supply from N2 to air (or your target gas blend) at the desired flow rate. Keep agitation constant.
- Record DO vs. time. Log DO readings every 1–5 seconds until the DO stabilizes near saturation (C*).
- Analyze the data. The oxygen balance in the liquid phase (without cells consuming O2) is:
Integrating this first-order ODE from the initial condition CL = CL0 at t = 0:
Plot ln[(C* − CL0) / (C* − CL)] versus time. If the system is well-mixed, this plot will be linear and the slope is kLa.
Worked Example: 10 L Bioreactor
Consider a 10 L bioreactor with 7 L working volume at 37°C. After nitrogen stripping, air is switched on at t = 0. C* at 37°C = 6.7 mg/L. The initial CL0 = 0 mg/L.
We record the following DO data:
| Time (s) | DO (% sat) | CL (mg/L) | C* − CL | ln[C* / (C* − CL)] |
|---|---|---|---|---|
| 0 | 0% | 0.00 | 6.70 | 0.000 |
| 15 | 16% | 1.07 | 5.63 | 0.174 |
| 30 | 30% | 2.01 | 4.69 | 0.357 |
| 45 | 42% | 2.81 | 3.89 | 0.543 |
| 60 | 52% | 3.48 | 3.22 | 0.733 |
| 90 | 68% | 4.56 | 2.14 | 1.143 |
| 120 | 80% | 5.36 | 1.34 | 1.609 |
| 150 | 87% | 5.83 | 0.87 | 2.041 |
Probe Response Time Correction
DO probes are not instantaneous. Polarographic probes typically have response times (τprobe) of 8–30 seconds; optical probes are faster at 3–10 seconds. If τprobe is comparable to 1/kLa, the measured kLa will be systematically underestimated.
Rule of thumb: If τprobe < 1/(5 × kLa), no correction is needed. For our example above, 1/(5 × 0.0139) = 14.4 s. So a probe with τ = 10 s is borderline—an optical probe is recommended.
When correction is needed, the true kLa can be estimated from the apparent kLa using:
Pitfalls
- Foam interference: Foam layers can trap gas and give misleading readings. Ensure the probe is submerged well below the liquid surface.
- Probe lag: As described above—always verify that your probe is fast enough for the kLa you are measuring.
- Non-ideal mixing: In large vessels, DO gradients can exist. Measure at multiple positions or use multiple probes.
- Dead volume around the sparger: N2 trapped in the headspace or sparger can re-dissolve during the air-on phase, distorting early time points. Allow 2–3 headspace volumes to flush before starting your measurement.
Line chart with time in seconds on the x-axis (0 to 180) and dissolved oxygen percent air saturation on the y-axis (0 to 100). Two curves start at 0 percent DO at time zero when nitrogen sparging stops and air is switched on. The solid teal curve (kLa = 72 per hour or 0.02 per second) rises steeply, reaching approximately 63 percent by 50 seconds and 95 percent by 150 seconds. The dashed blue curve (kLa = 36 per hour or 0.01 per second) rises more slowly, reaching approximately 63 percent by 100 seconds and 83 percent by 180 seconds. The steeper curve indicates better oxygen transfer. These data points are used to calculate kLa from the slope of the natural log plot of (C star minus CL) over (C star minus CL0) versus time.
4. Estimating kLa: Correlations
When you cannot measure kLa experimentally—during process design, scale-up calculations, or equipment selection—empirical correlations let you estimate it from operating parameters. The two most widely used are the Van't Riet correlation for stirred tanks and the Büchs correlation for shake flasks.
4a. Van't Riet Correlation (Stirred Tanks)
The Van't Riet correlation (1979) relates kLa to the volumetric power input (P/V) and the superficial gas velocity (vs):
Coalescing media (water-like):
C1 = 0.026, α = 0.4, β = 0.5
Non-coalescing media (salts, proteins):
C1 = 0.002, α = 0.7, β = 0.2
Units: P/V in W/m³, vs in m/s, kLa in s¹
The non-coalescing correlation yields higher kLa at the same P/V because salts and proteins prevent bubble coalescence, maintaining a larger interfacial area. Most cell culture and fermentation media are non-coalescing—unless you have added antifoam, which can shift behavior toward coalescing.
Scatter and line chart on logarithmic scales. The x-axis shows power input per unit volume (P/V) in watts per cubic metre from 10 to 10000. The y-axis shows kLa in per hour from 10 to 1000. A solid teal line with data points represents coalescing media using the Van't Riet correlation kLa equals 0.026 times P/V to the 0.4 times vs to the 0.5 times 3600 with superficial gas velocity 0.005 metres per second. Data points are plotted at P/V values of 50, 100, 200, 500, 1000, 2000, and 5000. A dashed blue line represents non-coalescing media using kLa equals 0.002 times P/V to the 0.7 times vs to the 0.2 times 3600. The non-coalescing line has a steeper slope, crossing above the coalescing line at higher P/V values. A shaded region between P/V 500 and 3000 indicates the typical stirred-tank reactor operating range.
Worked Example: 50 L Bioreactor
Given: 50 L vessel, 35 L working volume. Two 6-blade Rushton turbines, Di = 0.08 m. Np = 5.0 (Rushton). N = 300 RPM. Air at 1 vvm. Vessel cross-section diameter DT = 0.30 m. Culture medium (non-coalescing).
A kLa of ~360 h−1 is within the expected range for a well-agitated pilot-scale STR. At a DO setpoint of 30% (CL = 0.3 × 6.7 = 2.0 mg/L), this gives an OTR of ~0.047 mg/L/s or about 5.3 mmol/L/h.
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Scale-Up Calculator →4b. Büchs Correlation (Shake Flasks)
Shake flasks lack mechanical agitation, so oxygen transfer depends entirely on orbital shaking parameters. Büchs et al. (2001) developed a widely-used correlation:
where:
n = shaking speed (RPM)
VL = liquid fill volume (mL)
VF = nominal flask volume (mL)
d0 = shaking / orbital diameter (mm)
Typical empirical values:
C ≈ 0.6–1.3, a ≈ 1.6–1.9, b ≈ −0.6 to −1.0, c ≈ 0.5–0.7
The key insight: kLa in shake flasks is strongly dependent on the fill ratio (VL/VF) and shaking speed. Overfilling a flask drastically reduces kLa because the liquid cannot form the thin film needed for efficient gas transfer.
Worked Example: 250 mL Shake Flask
Given: 250 mL Erlenmeyer flask with baffles, 50 mL fill volume, 200 RPM shaking speed, 25 mm orbital diameter.
For non-baffled flasks, kLa drops by roughly 40–60%. A non-baffled 250 mL flask at the same conditions would give kLa of 25–45 h−1. If you are screening cultures that will move to stirred tanks, baffled flasks give much more representative oxygen transfer.
4c. Other Correlations
Several alternative correlations exist for specific applications:
- Linek correlation: Extends Van't Riet for viscous fluids and non-Newtonian broths. Important for filamentous fungal fermentations where broth viscosity can increase 100-fold during cultivation.
- Gezork correlation: Accounts for different impeller types (e.g., pitched blade, hydrofoil) and multi-impeller configurations more accurately than Van't Riet.
- Garcia-Ochoa & Gomez (2009): Comprehensive review and updated correlations incorporating modern impeller designs and scale-up data.
- Wave bioreactors: No universal correlation exists. Empirical data from Cytiva (formerly GE Healthcare) publications suggests kLa = 3–30 h−1 depending on rocking rate (6–42 rpm), rocking angle (2–12°), and fill level (40–80%). At maximum rocking parameters, 20 L WAVE Cellbag systems can reach ~25 h−1.
5. Scale-Up Using kLa
Constant kLa is one of the five major criteria for bioreactor scale-up, alongside constant P/V, constant tip speed (N·Di), constant Reynolds number, and constant mixing time. The choice of criterion depends on what aspect of the process is most critical to preserve.
When to Use Constant kLa
Scale-up on constant kLa is appropriate when:
- Oxygen transfer is the identified bottleneck or the process operates near oxygen limitation
- The organism has a high specific oxygen uptake rate (qO2)
- You are scaling a high-cell-density E. coli or yeast fermentation
- DO control has been problematic during development
How to Apply It
If you know the kLa at small scale (from measurement or correlation), you can back-calculate the required P/V or agitation rate at the larger scale to match it:
(P/V)large = [ kLatarget / (C1 × vs,largeβ) ]1/α
When NOT to Use Constant kLa
Constant kLa scale-up is not ideal for:
- Shear-sensitive cells: Maintaining kLa at large scale can require very high impeller tip speeds that damage mammalian cells, insect cells, or fragile filamentous organisms. For CHO processes, constant P/V or constant tip speed is usually preferred.
- Mixing-limited processes: If substrate gradients (e.g., pH excursions from fed-batch feeding) are more detrimental than oxygen limitation, constant mixing time is a better criterion.
- Low oxygen demand: Slow-growing mammalian cell cultures often operate well below the oxygen transfer capacity of even modest bioreactors.
Compare All 5 Scale-Up Criteria
See how constant kLa, P/V, tip speed, Re, and mixing time predict different operating conditions at your target scale.
Scale-Up Calculator →6. Common Mistakes
Antifoam agents (silicone, polypropylene glycol) can reduce kLa by 30–50%. Complex media components, high protein concentrations, and cell debris all alter interfacial properties. Always measure kLa in your actual medium—or at minimum, apply a correction factor of 0.5–0.7 to water-measured values when antifoam is present.
A polarographic DO probe with a 20-second response time will underestimate kLa by 20–40% in a well-agitated bioreactor. Use optical (fluorescence-based) probes for kLa measurements, or apply the correction described in Section 3.
Most correlations and literature values are reported at 20°C. If your process runs at 37°C, you need to correct:
At 37°C: correction factor = 1.02217 = 1.45
A kLa of 100 h¹ at 20°C becomes ~145 h¹ at 37°C
As cell density increases, broth viscosity changes (especially with filamentous organisms or high-cell-density cultures). Viscosity increases of 5–10× can reduce kLa by 50–80%. Monitor DO throughout your fermentation and be prepared to increase agitation or oxygen enrichment as the culture progresses.
If you measure kLa in a vessel containing live cells, the OUR of those cells will offset your OTR measurement. The equation becomes dCL/dt = kLa × (C* − CL) − OUR. Either measure in cell-free conditions, or independently determine OUR (e.g., by a brief gas-off measurement) and include it in the analysis.
7. Quick Reference Table
Use this table for rapid estimation during process design. Values assume aqueous media without antifoam, at 20–37°C, and standard atmospheric pressure.
| System | Typical kLa (h−1) | Notes |
|---|---|---|
| Shake flask 250 mL | 20–80 | Fill volume dependent; baffled flasks 1.5–2× higher |
| Wave bioreactor (2–50 L) | 5–30 | Rocking rate and angle dependent |
| STR 2 L (lab) | 50–200 | Typical lab bioreactor with Rushton turbine |
| STR 50 L (pilot) | 80–300 | Higher P/V achievable at this scale |
| STR 10,000 L (production) | 100–400 | Back-pressure and O2 enrichment extend range |
| Single-use STR (50–2000 L) | 30–150 | Lower than SS equivalent; limited by bag/impeller design |
| Airlift bioreactor | 20–100 | No mechanical agitation; gas flow dependent |
| Microtiter plate (96-well) | 100–400 | Very small volumes; shaking speed critical |
8. Try It Yourself
This guide gives you the theoretical foundation to calculate and interpret kLa for any bioreactor system. But doing these calculations by hand every time you adjust conditions is tedious and error-prone. That is why we built free calculators to handle it for you.
OTR & kLa Estimator
Enter your vessel type, dimensions, and operating conditions. Get kLa, OTR, and see whether your setup can meet your culture's oxygen demand.
Calculate kLa Now →You might also find these tools useful for related calculations:
- Fed-Batch Calculator — Optimize feeding strategies to manage oxygen demand and avoid overflow metabolism.
- Scale-Up Calculator — Compare all five scale-up criteria and see how your P/V, tip speed, and kLa translate across scales.
- Heat Transfer Calculator — High kLa conditions generate more metabolic heat. Make sure your cooling system can handle it.
References
- Van't Riet, K. (1979). "Review of measuring methods and results in nonviscous gas-liquid mass transfer in stirred vessels." Industrial & Engineering Chemistry Process Design and Development, 18(3), 357–364. doi:10.1021/i260071a001
- Büchs, J., Maier, U., Milbradt, C., & Zoels, B. (2000). "Power consumption in shaking flasks on rotary shaking machines: I. Power consumption measurement in unbaffled flasks at low liquid viscosity." Biotechnology and Bioengineering, 68(6), 589–593. doi:10.1002/bit.1
- Garcia-Ochoa, F. & Gomez, E. (2009). "Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview." Biotechnology Advances, 27(2), 153–176. doi:10.1016/j.biotechadv.2008.10.006
- Linek, V., Vacek, V., & Benes, P. (1987). "A critical review and experimental verification of the correct use of the dynamic method for the determination of oxygen transfer in aerated agitated vessels to water, electrolyte solutions, and viscous liquids." Chemical Engineering Journal, 34(1), 11–34.
- Gezork, K.M., Bujalski, W., Cooke, M., & Nienow, A.W. (2001). "Mass transfer and hold-up in an agitated vessel with standard and novel impellers." 10th European Conference on Mixing, Elsevier, 137–144.