$(20$ pts $)$ Showing all your work and reasoning, find the volume of aeration tank in a water treatment process. The water treatment plant with a completely mixed activated sludge system needs to meet an effluent standard of $20m\frac{g}{L}$ of BOD $5$ and $15m\frac{g}{L}$ of suspended solids. Before the activated sludge tank, there is a primary tank which has a flow rate of $0\mathrm{.}9\frac{m^3}{s}$ and the effluent has BODs of $85m\frac{g}{L}.$ Using the assumptions provided, estimate the required volume of the aeration tank.

BOD$5$ of the effluent suspended solids is $80$ percent of suspended solids concentration.

Growth constants are estimated to be; $Ks=100m\frac{g}{L}$ BOD$5$; $Kd=0\mathrm{.}070d-1$;$m=2\mathrm{.}5d-1$;$Y=0\mathrm{.}5mg$ VSS$/$mg BOD$5$ removed.

The design MLVSS is $2,000m\frac{g}{L}$

$S=$BOD$5$ allowed $-$ BOD$5$ in suspended solids