QUESTION $3.$ Calculating oxygen transport properties in a bioreactor. $(15$ pts$)$

We have a $10-$liter $($broth volume$)$ fermenter that is well mixed and is supporting the growth of the yeast Pichia pastoris at $30C$ using methanol as the carbon and energy source. The agitator is running at $800rpm$ and we are sparging in air at a rate of $10$ SLPM $($standard liters per minute$).$ The head pressure on the fermenter is $0\mathrm{.}5$atm $(1\mathrm{.}5$atm absolute pressure$).$ The exit oxygen concentration is being monitored by a mass spectrometer, which tells us that the exit gas has an oxygen concentration of $18\%.$ The dissolved oxygen probe was calibrated to $100\%$ with oxygen saturated medium at $30C$ and with $0\mathrm{.}5$atm head pressure and now reads $30\%.$

a$)$ What is the oxygen uptake rate $($mmoles$/$L$-$min$)?$

b$)$ What is the overall oxygen transfer coefficient, $k_{L}a($mmoles$/$L$-$min atm$)?$

c$)$ If we know from previous work with this vessel and fermentation that $k_{L}a$ is proportional to $rp{m}^3,$ what will be the new dissolved oxygen reading $(\%)$ if we increase the rpm to $1000?$