Nicotiana tabacum cells are cultured to high density for producing polysaccharide gum. A stirred$-$tank reactor was used containing initially $100$ L medium. The maximum specific growth rate of the culture is $0\mathrm{.}18$ d$1,$ and the yield of biomass from the substrate is $0\mathrm{.}5$ g$/$g$.$ The concentration of the growth$-$limiting substrate in the medium is $3\%($w$/$v$).$ The reactor is inoculated with $1\mathrm{.}5$ g$/$L cells and operated in batch until the substrate is totally exhausted. Medium flow is then started at a rate of $4$ L$/$d$.$ Fed batch operation occurs under quasi$-$steady$-$state conditions. $($a$)$ Estimate the batch culture time and final biomass concentration.$($b$)$ Fed batch operation is carried out for $40$ d$.$ What is the final mass of cells in the reactor? $($c$)$ The fermenter is available $275$ d per year with a downtime between runs of $24$ h$.$ How much biomass is produced annually?