Acetobacter aceti is grown under effective anaerobic fermentation to obtain a starter

culture for the production of vinegar. The cells synthesis acetic acid as a by$-$product of

energy metabolism. The system has the properties below:

$Yxs=0\mathrm{.}23kgk{g}^{-1}$

$Ks=0\mathrm{.}15kg{m}^{-3}$

${}_{\text{m}\text{a}\text{x}}=0\mathrm{.}35h^{-1}$

$m_s=0\mathrm{.}135kgk{g}^{-1}{h}^{-1}$

Fed$-$batch fermentation is operated in stirred tank fermenter under quasi steady state

conditions with a feeding flow rate of $4m^3{h}^{-1}$ and feeding substrate concentration of

$80kg{m}^{-3}.$ After $6h,$ the liquid medium is $40m^3.$

a$.$ Calculate the initial culture volume.

b$.$ Calculate the substrate concentration at quasi steady state.

c$.$ Calculate the cell concentration at quasi steady state.

d$.$ Calculate the cell mass produced after $6h$ of feeding operation.