A chemostat system is being constructed that will operate for up to $4$ weeks without shutdown.

The fermentation tank is to be $50,000L$ in size and the dilution rate is to be $0\mathrm{.}5h^{-1}.$ A continuous

sterilizer operating at $121C$ is to be used to sterilize the feed medium while minimizing the

degradation of vitamins and growth factors. The unsterilized medium contains ${10}^4$ spores$/$L$.$ The

$k_d$ value of the spores has been determined to be $1mi{n}^{-1}$ at $121C.$

a$.$ Determine the total volume $($L$)$ of feed medium that needs to be sterilized over a $4-$week

production run.

b$.$ Determine the residence time $(min)$ in the holding section of the sterilizer to ensure that

there is a $99\%$ probability of operating without contamination for at least $4$ weeks.

c$.$ The most fragile of the vitamins in the medium has a first$-$order decay rate constant $(k_{d,v})$ of

$0\mathrm{.}05mi{n}^{-1}$ at $121C.$ Determine the fraction of the initial vitamin concentration that remains

active after the feed medium has been sterilized? Assume that vitamin degradation is

negligible during the rapid heat$-$up and cool$-$down periods.