A fermentation is carried out in a single stage CSTR$,$ which contains $1\mathrm{.}50m^3$ of broth. The effluent stream is passed to a centrifuge, where $90\%$ cells are separated and recycled back to the CSTR$.$ The cell growth rate of the micro$-$organism in the vessel is

$\frac{d{C}_C}{dt}=r_C=C_C$

where $C_C=$ cell concentration, $kg{m}^{-3}$;

$r_{}=$ cell growth rate

The sterile fresh feed, with substrate concentration $15kg{m}^{-3},$ enters the reactor at flow rate $28\mathrm{.}0{10}^{-6}{m}^3{s}^{-1}.$ The recycle flow rate is $2\mathrm{.}8{10}^{-6}{m}^3{s}^{-1}$ and has substrate concentration $4\mathrm{.}5kg{m}^{-3}.0\mathrm{.}55kg$ of cells are formed per $kg$ of substrate consumed, calculate:

$($a$)$ the concentration of micro$-$organism, $kg{m}^{-3},$ in the flow stream leaving the CSTR;

$($b$)$ Total cell production rate in the CSTR

$($c$)$ Total substrate consumption rate in the CSTR