Oxygen Transfer Sizing — A microbial population has μm = 0.2 h^-1, KO = 0.2 mg O2/L, YX/O2 = 0.5 mg dry weight per mg O2, and a critical dissolved oxygen Ccrit = 0.8 mg/L. For a required cell concentration X = 1000 mg/L and saturation concentration C* = 5.8 mg/L, the minimum required kLa must be greater than:

Introduction:
Correctly sizing oxygen transfer capacity is crucial in aerobic bioprocesses. The overall volumetric mass transfer coefficient (kLa) must be high enough that the oxygen transfer rate (OTR) meets or exceeds the oxygen uptake rate (OUR) at or above the critical dissolved oxygen level, preventing oxygen limitation of growth.

Given Data / Assumptions:

• μm = 0.2 h^-1 (maximum specific growth rate).
• KO = 0.2 mg O2/L (Monod oxygen half-saturation constant).
• YX/O2 = 0.5 mg biomass per mg O2 (yield on oxygen).
• Ccrit = 0.8 mg O2/L (minimum allowable DO).
• X = 1000 mg/L (biomass concentration), C* = 5.8 mg O2/L (saturation DO).
• Maintenance oxygen demand is neglected; steady growth near Ccrit is assumed.

Concept / Approach:

At DO = Ccrit, the attainable specific growth rate is μ = μm * C / (KO + C). The specific oxygen uptake rate is qO2 = μ / YX/O2 when maintenance is neglected. Then OUR = qO2 * X. The oxygen transfer rate is OTR = kLa * (C* − C). To avoid oxygen limitation at Ccrit, require OTR ≥ OUR evaluated at C = Ccrit, yielding a lower bound for kLa.

Step-by-Step Solution:

Verification / Alternative check:

This bound is conservative. If maintenance demand exists, required kLa would be even higher. Operating above the minimum allows safety margin for disturbances.

Why Other Options Are Wrong:

B–E underestimate the necessary mass transfer capacity; at those kLa values, OTR < OUR at Ccrit, leading to oxygen limitation and lower growth.

Common Pitfalls:

Using C* instead of (C* − C) as the driving force; forgetting to evaluate μ at Ccrit; omitting the yield relationship between oxygen uptake and growth.

Final Answer:

64 h^-1