Introduction

In a well aerated and agitated microbial culture, oxygen transfer from the gas phase
to the liquid phase is essential for microbial growth. The parameter
K_La (volumetric mass transfer coefficient) measures the efficiency of oxygen
transfer in a bioreactor.

At steady state, the oxygen supply equals the oxygen demand of the growing culture.
This balance helps derive the correct expression for K_La.

Given Parameters

• K_La – Volumetric mass transfer coefficient
• C^* – Dissolved oxygen concentration in equilibrium with gaseous oxygen
• C – Instantaneous dissolved oxygen concentration
• r – Specific oxygen uptake rate per unit weight of cells
• X – Dry weight of cells per unit volume

Oxygen Supply Rate

The rate of oxygen transfer from gas to liquid is given by:

K_La (C^* − C)

Oxygen Demand Rate

The oxygen uptake rate by microorganisms is:

rX

Steady-State Oxygen Balance

In a well aerated microbial culture, oxygen supply equals oxygen demand:

K_La (C^* − C) = rX

Expression for K_La

Rearranging the above equation:

K_La = rX / (C^* − C)

Correct Answer

Option (A):
rX / (C^* − C)

Option-wise Explanation

Option (A): rX / (C^* − C) — Correct

This expression is directly derived from the oxygen balance equation.
It correctly represents oxygen demand divided by the driving force
for mass transfer.

Option (B): r / X(C^* − C) — Incorrect

Biomass concentration is incorrectly placed in the denominator.
Oxygen demand is proportional to r × X, not r / X.

Option (C): (C^* − C) / rX — Incorrect

This is the inverse of the correct expression and does not represent
the volumetric mass transfer coefficient.

Option (D): X / r(C^* − C) — Incorrect

This option misrepresents oxygen uptake kinetics and is dimensionally incorrect.

Final Answer

K_La = rX / (C^* − C)

Key Takeaways

• K_La quantifies oxygen transfer efficiency in bioreactors
• At steady state, oxygen supply equals oxygen uptake
• Always start from the oxygen balance equation
• Frequently asked in bioprocess and GATE examinations