Oxygen Transfer Rate (OTR) Calculation

Reference ID: MET-8D7C | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Oxygen Transfer Rate (OTR) is a critical parameter in aerobic fermentation, representing the mass transfer flux of oxygen from the gas phase (air bubbles) into the liquid phase (culture medium). In process engineering, maintaining an adequate OTR is essential to prevent oxygen limitation, which can lead to metabolic shifts, reduced biomass yield, or cell death in high-density microbial cultures.

This calculation is typically employed during the design and scale-up of Stirred Tank Reactors (STRs). It allows engineers to balance the oxygen demand of the microorganisms (Oxygen Uptake Rate, or OUR) with the physical oxygen supply capacity of the bioreactor system, ensuring that the dissolved oxygen concentration (C_L) remains above the critical threshold for the specific organism.

Methodology & Formulas

The calculation follows a steady‑state mass‑transfer model, where the volumetric oxygen transfer coefficient (k_La) is estimated using the van’t Riet empirical correlation that relates power input and superficial gas velocity to mass‑transfer performance; for a detailed method on determining the required power input, see the gassed power number calculation.

1. Equilibrium Saturation Concentration (C*)

The saturation concentration is adjusted for the presence of solutes (salts) using a correction factor:

\[ C^{*} = C^{*}_{\text{water}} \cdot f_{\text{salt}} \]

2. Driving Force

The concentration gradient driving the mass transfer is defined as:

\[ \Delta C = C^{*} - C_{L} \]

Where C_L is the target dissolved oxygen concentration in the bulk liquid.

3. Superficial Gas Velocity (v_s)

The superficial gas velocity is calculated based on the volumetric airflow rate and the cross‑sectional area of the reactor; see our detailed superficial gas velocity calculation for the full methodology.

\[ v_{s} = \frac{\dot{V}_{\text{air}}}{A_{\text{tank}}} \]

4. Volumetric Mass Transfer Coefficient (k_La)

Using the van't Riet correlation for coalescing Newtonian fluids:

\[ k_{L}a = 0.026 \cdot \left( \frac{P}{V} \right)^{0.4} \cdot (v_{s})^{0.5} \]

5. Oxygen Transfer Rate (OTR)

The final OTR is the product of the volumetric mass transfer coefficient (kLa) and the driving force, reflecting how efficiently oxygen moves from the gas to the liquid phase.

\[ OTR = k_{L}a \cdot (C^{*} - C_{L}) \]

Parameter	Condition / Regime	Constraint / Limit
Power Input (P/V)	Empirical Validity	0.5 kW/m³ ≤ P/V ≤ 5.0 kW/m³
Superficial Velocity (v_s)	Empirical Validity	0.002 m/s ≤ v_s ≤ 0.05 m/s
Fluid Regime	Newtonian	Viscosity < 100 cP

To calculate the OTR, you must evaluate the mass transfer coefficient and the concentration gradient. Follow these steps:

Determine the volumetric mass transfer coefficient (k_La) using the dynamic gassing-out method or the sulfite oxidation method.
Measure the dissolved oxygen concentration in the liquid phase (C_L).
Identify the saturation concentration of oxygen in the medium (C*), which is dependent on temperature, pressure, and medium composition.
Apply the formula: OTR = k_La * (C* - C_L).

The k_La value is highly sensitive to the physical and operational parameters of the vessel. Key factors include:

Agitation speed and impeller geometry, which dictate the power input per unit volume.
Gas flow rate and bubble size distribution, which influence the total interfacial surface area.
Rheological properties of the fermentation broth, such as viscosity, which can hinder oxygen diffusion.
Presence of antifoaming agents, which often reduce the mass transfer coefficient.

The OUR represents the metabolic demand of the microorganisms, while the OTR represents the supply capacity of the system. For a stable process, the following conditions must be met:

The OTR must be greater than or equal to the OUR to prevent oxygen limitation.
If OUR exceeds OTR, the dissolved oxygen concentration will drop to zero, potentially leading to anaerobic metabolism or cell death.
Process engineers should maintain a safety margin where OTR is consistently higher than the peak OUR of the culture.

Worked Example: Oxygen Transfer Rate (OTR) for Aerobic Fermentation

Scenario: A 10 m³ stirred tank reactor (STR) is used for aerobic fermentation of a water-based medium containing 0.5 M NaCl and 50 g/L glucose at 30°C and 1 bar total pressure. Air is sparged at 1 vvm (volume air per volume liquid per minute). The impeller (Rushton turbine) delivers a specific power input of 2 kW/m³. The target dissolved oxygen (DO) is maintained at 20% of air saturation. We calculate the volumetric oxygen transfer rate (OTR) to verify it meets typical microbial demands.

Known Input Parameters:

Tank working volume, \(V = 10.0\) m³
Temperature, \(T = 30.0\) °C
Total pressure, \(P_{\text{total}} = 1.0\) bar
Salt molarity, \(M_{\text{salt}} = 0.5\) M
Glucose concentration, \(C_{\text{glucose}} = 50.0\) g/L
Airflow, \(Q_{\text{air}} = 1.0\) vvm
Specific power input, \(P/V = 2.0\) kW/m³
Tank diameter, \(D = 2.2\) m
Target DO fraction, \(x_{\text{DO}} = 0.20\)
Pure water oxygen saturation at 30°C, \(C^{*}_{\text{water}} = 7.6\) mg/L
Salt correction factor, \(f_{\text{salt}} = 0.78\)
Molar mass of O₂, \(M_{\text{O₂}} = 32.0\) mg/mmol
Hours-to-seconds conversion, \(t_{\text{conv}} = 3600.0\) s/h
\(\pi = 3.14159\)

Step-by-Step Calculation:

Determine \(C^{*}\) (Oxygen Saturation Concentration in Liquid)
Compute \(C^{*}_{\text{mg/L}} = C^{*}_{\text{water}} \cdot f_{\text{salt}} = 7.6 \cdot 0.78 = 5.928\) mg/L.
Convert to mmol/L using \(M_{\text{O₂}}\): \(C^{*}_{\text{mmol/L}} = \frac{C^{*}_{\text{mg/L}}}{M_{\text{O₂}}} = \frac{5.928}{32.0} = 0.1853\) mmol/L (rounded from 0.18525).
Compute Actual DO Concentration \(C_{L}\)
\(C_{L} = x_{\text{DO}} \cdot C^{*}_{\text{mmol/L}} = 0.20 \cdot 0.1853 = 0.0371\) mmol/L.
Driving Force for Mass Transfer
\(\Delta C = C^{*}_{\text{mmol/L}} - C_{L} = 0.1853 - 0.0371 = 0.1482\) mmol/L.
Calculate Superficial Gas Velocity \(v_{s}\)
Airflow rate in m³/h: \(Q_{\text{air,m³/h}} = Q_{\text{air}} \cdot V \cdot 60 = 1.0 \cdot 10.0 \cdot 60 = 600.0\) m³/h.
Tank cross-sectional area: \(A = \frac{\pi}{4} D^{2} = \frac{3.14159}{4} \cdot (2.2)^{2} = 3.801\) m² (exact: \(3.80132\) m²).
Superficial gas velocity: \(v_{s} = \frac{Q_{\text{air,m³/h}} / t_{\text{conv}}}{A} = \frac{600.0 / 3600.0}{3.80132} = 0.0438\) m/s (rounded to 0.044 m/s).
Estimate \(k_{L}a\) using van't Riet Correlation
Formula: \(k_{L}a \, (\text{s}^{-1}) = 0.026 \cdot (P/V)^{0.4} \cdot (v_{s})^{0.5}\).
Using exact values: \(0.026 \cdot (2.0)^{0.4} \cdot (0.0438)^{0.5} = 0.007184\) s⁻¹.
Convert to h⁻¹: \(k_{L}a \, (\text{h}^{-1}) = 0.007184 \cdot 3600.0 = 25.86\) h⁻¹.
Calculate OTR
OTR in mmol/L/h: \(\text{OTR}_{\text{mmol/L/h}} = k_{L}a_{\text{h}^{-1}} \cdot \Delta C = 25.86 \cdot 0.1482 = 3.833\) mmol/L/h.
Convert to mass units: \(\text{OTR}_{\text{kg/m³/h}} = \frac{\text{OTR}_{\text{mmol/L/h}} \cdot M_{\text{O₂}}}{1000} = \frac{3.833 \cdot 32.0}{1000} = 0.123\) kg O₂/m³/h.

Final Answer:

The volumetric oxygen transfer rate is \(\text{OTR} = 3.833\) mmol O₂/L/h, which is equivalent to \(\text{OTR} = 0.123\) kg O₂/m³/h. This rate is sufficient for moderate-density bacterial fermentations (typical OUR 0.05–0.15 kg O₂/m³/h).

Empirical Range Check: The specific power input \(P/V = 2.0\) kW/m³ lies within the van't Riet correlation bounds (0.5–5 kW/m³). The superficial gas velocity \(v_{s} = 0.0438\) m/s is within the valid range (0.002–0.05 m/s).

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