Reference ID: MET-417B | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The volumetric mass-transfer coefficient, kLa, quantifies the rate at which a sparingly soluble gas (typically oxygen in aerobic fermentations or activated-sludge basins) crosses the gas–liquid interface per unit of liquid volume. It is the product of the true liquid-side mass-transfer coefficient kL and the specific interfacial area a. Because both kL and a are difficult to measure independently, the combined group kLa is treated as a single empirical parameter. In design and scale-up of stirred-tank reactors, packed towers, bubble columns, and aerated lagoons, kLa is the key figure of merit that links power input and superficial gas velocity to the achievable oxygen transfer rate (OTR). Accurate estimation of kLa is therefore essential for sizing aerators, compressors, and agitators, for meeting dissolved-oxygen set-points, and for avoiding either oxygen limitation or excessive energy consumption.
Methodology & Formulas
Dimensional consistency
Convert the specific power from kW m−3 to W m−3:
\[ P_{\text{V,W}} = 1000 \cdot P_{\text{V,kW}} \]
Empirical correlation
For mechanically agitated, sparged vessels operating in the turbulent regime, kLa is correlated with specific power and superficial gas velocity via:
\[ k_L a = K \cdot (P_{\text{V,W}})^\alpha \cdot (v_{\text{s}})^\beta \]
where
K is a system-dependent constant (dimensionless when PV is in W m−3 and vs in m s−1)
α and β are empirical exponents
Unit conversion
Express the result in reciprocal seconds or reciprocal hours:
\[ k_L a\ [\text{h}^{-1}] = k_L a\ [\text{s}^{-1}] \times 3600 \]
Correlation validity limits
Parameter
Minimum
Maximum
Units
Specific power, PV
0.5
10
kW m−3
Superficial gas velocity, vs
0.001
0.08
m s−1
kLa is the product of the liquid-side mass-transfer coefficient (kL) and the interfacial area per unit volume (a). It quantifies how fast oxygen (or another sparingly soluble gas) can be delivered from the gas phase into the broth. A high kLa ensures that the oxygen uptake rate (OUR) of the culture never exceeds the oxygen transfer rate (OTR), preventing growth or product-formation limitations.
Specific power input (P/V): raise impeller speed or use larger impellers; kLa generally scales with (P/V)0.7.
Superficial gas velocity (Ug): increase airflow or switch to micro-spargers; kLa ∝ Ug0.3–0.8.
Broth properties: reduce viscosity (dilution, temperature), minimize surfactants or antifoam, and control cell concentration.
Reactor internals: add baffles, use dual impellers, or switch to concave-blade or Rushton turbines depending on regime.
Pressure and oxygen enrichment: higher total pressure or enriched air increases the driving-force term without changing kLa directly, but allows lower airflow for the same OTR.
Dynamic gassing-out: stop airflow, allow dissolved oxygen (DO) to drop slightly, resume airflow and fit the re-oxygenation curve; works best at low cell densities.
Dynamic balance during exponential growth: record OUR from off-gas analysis and divide by the driving force (C*–CL) to obtain kLa in real time.
100% step-change method: switch from nitrogen to air and fit the DO response; keep mixing identical to production conditions.
Ensure probe response time < 5% of the smallest time constant measured and correct for temperature and pressure changes.
Maintain constant P/V (W m−3) and constant superficial gas velocity to keep kLa roughly constant; this often requires lower impeller speed on large tanks.
Watch for constant mixing time (turnover) which grows with tank diameter; poor top-to-bottom blending can create DO gradients even if average kLa is adequate.
Antifoam and ionic strength usually increase with scale due to longer batch times and higher carbonate levels, both can depress kLa by 20–40%.
Heat removal limits may force lower agitation; evaluate if oxygen-enriched air or higher back-pressure can compensate without exceeding impeller flooding.
Worked Example – Estimating kLa in a Pilot-Scale Aerobic Fermenter
A process engineer is scaling-up a citric acid fermentation from lab to pilot scale.
To ensure the culture remains aerobic, the vessel must deliver an oxygen transfer rate that matches the peak oxygen uptake of the fungus.
The engineer therefore needs to estimate the volumetric mass-transfer coefficient, kLa, under the chosen operating conditions.
Tank geometry: liquid height H = 2 m, tank diameter T = 2 m
Operating range: specific power input PV = 3 kW m−3 (3000 W m−3)
Superficial gas velocity vs = 0.025 m s−1
Step-by-Step Calculation
Select the correlation for kLa that matches the impeller–sparger geometry used in the pilot vessel:
\[ k_L a = K \left( \frac{P_V}{1\ \text{kW m}^{-3}} \right)^{\!\alpha} \left( \frac{v_s}{1\ \text{m s}^{-1}} \right)^{\!\beta} \quad \text{[h}^{-1}\text{]} \]
Insert the given constants and operating values:
\[ k_L a = 0.024 \left( \frac{3}{1} \right)^{0.7} \left( \frac{0.025}{1} \right)^{0.6} \]
Compute the power term:
\[ 3^{0.7} = 2.157 \]
Compute the gas-velocity term:
\[ 0.025^{0.6} = 0.158 \]
Multiply the factors:
\[ k_L a = 0.024 \times 2.157 \times 0.158 = 0.0082\ \text{s}^{-1} \]
Convert to more common engineering units:
\[ 0.0082 \times 3600 = 29.5\ \text{h}^{-1} \]
Final Answer
kLa = 29.5 h−1 (0.008 s−1)
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
