A continuous culture system is being constructed. The fermentation tank is to be $50,0001$ in size and the residence time is to be $2h.$ A continuous sterilizer is to be used. The unsterilized medium contains ${10}^4$ spores$/$I$.$ The value $k_d$ has been determined to be $1mi{n}^{-1}$ at $121C$ and $61mi{n}^{-1}$ at $140C.$ For each temperature and $($:$140C\},$ determine the required residence time in the $$ o as to ensure that $99\%$ of the time four weeks of continuous operation can be obtained without contamination $($due to contaminants in the liquid medium$).$ Make use of the direct calculation method.

$[11]$