please solve and explain all parts $($a$-$D$).$ C$.$ acetobutylicum is used for biobutanol production in a batch anaerobic fermenter,

which is operated with a microbubble distillation sidearm that recycles the liquid

media completely. The sidearm distiller is designed to achieve a two$-$minute

contact time. In two minutes of contact, half of whatever concentration of butanol

is in the sidearm is stripped away. The distiller sidearm has a flowrate $S=20\frac{L}{\text{m}\text{i}\text{n}}.$

The fermentation in a volume $V=1m^3$ starts with an initial concentration of butanol

of $c_0=24\mathrm{.}3g{L}^{-1}.$ The distiller is sparged with bubbles of nitrogen. The system

initialisation is when the distiller sparge is turned on$.$ Pure water flows into the

sidearm recycle stream after the distiller to maintain liquid flowrate returning to the

fermenter at the same rate as the sidearm flow $S=20\frac{L}{(min)}.$

$($a$)$ Draw a schematic of the system labelling all streams.

$($b$)$ Suppose there is no $C.$ acetobutylicum metabolism to produce butanol. List your

assumptions, then write a butanol material balance for the no butanol

producing system, and solve it for an arbitrary time after the diluting feed stream

starts. What will the concentration $c$ of butanol be after $30$ minutes?

$($c$)$ Butanol attacks the cell membrane of $C.$ acetobutylicum, inhibiting its metabolism and

therefore its production. This inhibition effect is modelled by a rate law for the

production of butanol $r.$

$r=k(c_0-c)$

If $k=0\mathrm{.}2mi{n}^{-1},$ what is the rate of butanol production at the initial condition?

$($d$)$ Write a material balance on the system that takes into account the inhibition on the

fermenter. Does it have a steady state solution? If so$,$ calculate the value of the

steady concentration $c_{}.$