a) The minimum mass transfer coefficient necessary to sustain this culture can be calculated using the equation:
kLa(C* - C)/H = Q/V
where kLa is the mass transfer coefficient, C* is the saturation concentration of oxygen in the liquid, C is the actual concentration of oxygen in the liquid, H is the Henry's law constant, Q is the gas flow rate, and V is the liquid volume.
Assuming a typical value of H = 0.03 mol/L·atm, and substituting the given values, we get:
kLa(0.004*0.9 - C)/0.03 = Q/V
where we have used 0.9 as an estimate of the solubility of O2 in the fermentation broth relative to water (i.e., 10% lower).
Solving for kLa, we get:
kLa = (Q/V)(0.004*0.9 - C)/0.03
The O2 requirement is given as 80 mmol/l/h, which is equivalent to 0.08 mol/L/h. Assuming a typical value of 21% O2 in air, and using the ideal gas law, we can calculate the gas flow rate required as:
Q = (0.21)(1 atm)(0.08 mol/L/h)(22.4 L/mol) = 3.53 L/h
Assuming a liquid volume of V = 1 L, and a critical dissolved O2 concentration of 0.004 mM, we can calculate the corresponding concentration of O2 in the gas phase as:
C = (0.004 mM)(0.8)/(8.05x10^-3 kg/m^3) = 0.396 mol/m^3
Substituting these values, we get:
kLa = (3.53/1)(0.004*0.9 - 0.396)/(0.03) = 1.24 h^-1
Therefore, the minimum mass transfer coefficient necessary to sustain this culture is approximately 1.24 h^-1.
b) If pure O2 is used instead of air, the solubility of O2 in the fermentation broth increases by a factor of 4.8 (i.e., solubility in water is multiplied by 4.8). Assuming all other parameters remain the same, we can calculate the new gas flow rate required as:
Q = (1)(1 atm)(0.08 mol/L/h)(22.4 L/mol)/(0.21)(4.8) = 0.98 L/h
Substituting this value, and the new concentration of O2 in the gas phase (which is now 5 times higher), we get:
kLa = (0.98/1)(0.02*0.9 - 1.98)/(0.03) = 1.96 h^-1
Therefore, the mass transfer coefficient required if pure O2 is used instead of air is approximately 1.96 h^-1.