An aerobic digester is to be designed to stabilize $600$ kg$/$d of primary sludge at $5\%$ solids and $75\%$ volatiles, along with $400$ kg$/$d of waste$-$activated sludge at $1\%$ solids and $75\%$ volatiles. The volume of the combined sludge that will be fed to the digester is $104$ m $3/$d$.$ The minimum design operating temperature is $18$ C$.$ A temperature correction coefficient $()$ of $1\mathrm{.}06$ should be used to correct the detention or digestion time for temperature. The primary sludge contains $210$ kg of BOD, with an oxygen demand of $1\mathrm{.}9$ kg of oxygen per kg of BOD, whereas use an oxygen demand of $2\mathrm{.}0$ kg of oxygen per lb of VS destroyed for the WAS. Assume that $90\%$ of the BOD contained in the primary sludge is destroyed and that $50\%$ of the volatiles solids in the WAS is destroyed. Determine the following:

a$.$ Design hydraulic detention time $()$ or digestion time $($days$).$

b$.$ Aerobic digester volume $($m $3).$

c$.$ Volatile solids loading rate $($kg$/$d $$ m $3)$ to the aerobic digester.

d$.$ Quantity of oxygen required $($kg$/$d$)$ to stabilize the primary sludge.

e$.$ Quantity of oxygen required $($kg$/$d$)$ to stabilize the secondary sludge $($WAS$).$

f$.$ Total quantity of oxygen required $($kg$/$d$)$ to stabilize the combined sludge.

g$.$ Total air required $($m $3/$min$),$ if there are $1\mathrm{.}2$ kg of air per cubic meter of air and the atmosphere consists of $23\%$ oxygen by weight $(0\mathrm{.}23$ kg O $2/$kg air$)$ and the diffusers have a transfer efficiency of $4\mathrm{.}0\%.$