Q.61 A batch cultivation of E. coli follows zeroth order Monod’s growth kinetics.
The cell growth is terminated when the residual dissolved oxygen concentration
attains 10% of its saturation value and oxygen mass transfer coefficient
(kLa) reaches its maximum value (80 h−1).
The saturation value of dissolved oxygen concentration is 0.007 kg m−3.
If the maximum specific growth rate and yield coefficient
(YX/O2) are 0.2 h−1 and
1.5 (kg cells)(kg O2)−1, respectively, then the final cell
concentration in kg m−3 (rounded off to two decimal places) at the end
of the batch cultivation is _________.
Final Cell Concentration Calculation in E. coli Batch Cultivation
Zeroth-order Monod kinetics in E. coli batch cultivation simplifies growth rate to a constant maximum under oxygen limitation. Cultivation ends at 10% saturation dissolved oxygen (DO) with maximum kLa of 80 h-1, yielding a final cell concentration of 8.40 kg m-3.
Key Problem Parameters
Saturation DO (C*) equals 0.007 kg m-3, so critical DO (CL) is 0.10 × 0.007 = 0.0007 kg m-3. Maximum specific growth rate (μmax) is 0.2 h-1, and yield coefficient (YX/O2) is 1.5 kg cells per kg O2. At termination, oxygen transfer rate (OTR) balances oxygen uptake rate (OUR) exactly at the critical DO point.
Zeroth-Order Kinetics Explanation
Zeroth-order Monod implies oxygen is fully saturating until limitation, making growth rate μ = μmax = 0.2 h-1 constant. OUR for biomass growth is OUR = (μmax × Xfinal) / YX/O2, where Xfinal is final cell concentration. At steady-state limitation, OUR = kLa × (C* – CL).
Final Concentration Derivation
Substitute values: (0.2 × Xfinal) / 1.5 = 80 × (0.007 – 0.0007). Simplify: 0.1333 × Xfinal = 80 × 0.0063 = 0.504. Thus, Xfinal = 0.504 / 0.1333 ≈ 8.40 kg m-3 (rounded to two decimals).
Verification Steps
- Calculate DO difference: 0.0063 kg m-3.
- OTRmax: 80 h-1 × 0.0063 = 0.504 kg O2 m-3 h-1.
- Required OUR: μmax / YX/O2 = 0.2 / 1.5 = 0.1333 h-1.
- Xfinal = OTRmax / 0.1333 = 8.40 kg m-3. This matches GATE BT 2021 solution confirming oxygen-limited termination.
Common Distractors
No options provided, but common distractors include:
- Ignoring critical DO (using full C*): yields ~9.00 kg m-3
- Misapplying first-order kinetics: incorrect OUR calculation
- Forgetting yield coefficient: overestimates biomass
