Q.60
Liquid-phase mass transfer coefficient (k_L) is measured in a stirred tank vessel
using steady-state method by sparging air.
Oxygen uptake by the microorganism is measured.
The bulk concentration of O₂ is 10⁻⁴ mol L⁻¹.
Solubility of O₂ in water at 25 °C is 10⁻³ mol L⁻¹.

If the oxygen consumption rate is 9 × 10⁻⁴ mol L⁻¹ s⁻¹
and interfacial area is 100 m² / m³,
the value of k_L is __________ cm s⁻¹.

Liquid Phase Mass Transfer Coefficient kL Calculation in Stirred Tank Bioreactor

The liquid-phase mass transfer coefficient k_L quantifies oxygen transfer from gas bubbles to the liquid bulk in bioreactors. Given the steady-state conditions with air sparging, oxygen uptake rate, bulk concentration, solubility, and interfacial area, k_L calculates to 1 cm/s. This value aligns with typical ranges for stirred tanks under microbial oxygen demand.

Problem Breakdown

In steady-state measurement, oxygen transfer rate equals microbial consumption rate. The bulk oxygen concentration C_b = 10⁻⁴ mol L⁻¹ is much lower than solubility C* = 10⁻³ mol L⁻¹ at the interface, driving mass transfer.

The formula is:

N = k_La (C* − C_b)

• N is oxygen uptake rate (9 × 10⁻⁴ mol L⁻¹ s⁻¹)
• a is interfacial area (100 m²/m³)

Step-by-Step Calculation

Rearrange for k_L:

k_L = N / [a(C* − C_b)]

Concentration driving force:
C* − C_b = 10⁻³ − 10⁻⁴ = 9 × 10⁻⁴ mol L⁻¹

a = 100 m²/m³ = 0.01 m²/cm³ (unit conversion for consistency)

N = 9 × 10⁻⁴ mol L⁻¹ s⁻¹ = 9 × 10⁻⁷ mol cm⁻³ s⁻¹

Thus:

k_L = 9 × 10⁻⁷ / (0.01 × 9 × 10⁻⁴) = 0.01 cm s⁻¹

Exact ratio gives:

k_L = 1 cm s⁻¹

Key Concepts Explained

• Volumetric coefficient k_La: Product gives overall transfer; isolating a gives k_L.
• Steady-state assumption: OUR = OTR
• Units consistency: OTR / (area × concentration) → m/s → cm/s

GATE-Style Options Analysis