$1.$ The following Dissolved Oxygen $($DO$)$ data is reported from a $2000$L prefusion reactor at a constant oxygen sparge rate of $5,000$ ccpm$.$ Note that at $100\%$ DO$,$ the solubility concentration of oxygen in water is only $8$ mg O$2/$ L$-$water.

a$.$ Determine the mass transfer co$-$efficient parameter $($kL$-$a$)$ in units of m$3/$s based on the DO profile below taken at the start of the reactor before cell inoculation.

Time$($minutes$)$ DO $\%$

Mass Transfer Oxygen

r inoculation and at a steady state cell concentration of $1$e$10$ cells$/$mL$,$ the glucose concentration in perfusion reactor should be maintained at $1\mathrm{.}5$ g$/$L$.$ If the glucose concentration of the feed media is $3\mathrm{.}5$ g$/$L and the perfusion rate is $0\mathrm{.}8$ reactor volumes per day, what is the cellular glucose consumption rate in units of $($g$/$L$-$cell$-$day$)?$

Note: you will need to perform a mass balance on glucose alone to determine the rate of glucose consumption by the cells. At steady state, the accumulation term goes to zero. The media out of the reactor has a composition equal to the reactor steady state glucose concentration.

c$.$ Use the standard cellular metabolism relationship $($Glucose $+6$O$26$CO$2+6$H$2$O$)$ to determine the standard $($theoretical$)$ required amount of oxygen for the cells in units of $($g$/$L$-$cell$-$second$).$

d$.$ The answer to part c can be converted to the oxygen uptake rate $($OUR$)$ which can be expressed in $($mols O$2/$second$)$ by accounting for the $2000$L reactor and the steady state cell concentration. What is the theoretical OUR $($in mols O$2/$second$)?$ Use data at temperature $37$C$.$

e$.$ Mammalian cell lines cannot thrive in high oxygen content. Therefore, the reactor is required to be maintained at $30\%$ DO at steady state. What will be the concentration of oxygen $($mg$/$L$)$ in the liquid at $30\%$ DO$?$

f$.$ Using your answer to part e and the measured mass transfer coefficient from part a $($kL$-$a$),$ what is the Oxygen Transfer Rate $($OTR$)$ in $($mols O$2/$second$).$ Use the given oxygen solubility concentration in the problem to solve for the OTR.

g$.$ Evaluate the difference between OTR vs$.$ OUR. In this system, is cell growth nutrient limited$?$ Why or why not.