$(30$ points$)$ Determine the parameters below for the activated sludge system described.

The effluent from the primary treatment stage is flowing at a constant flow rate of $0\mathrm{.}23\frac{m^3}{s}$ with a $BO{D}_5$ of $56m\frac{g}{L}.$ The volume of the aeration tank in the activated sludge system is $1$ million liters. To discharge the waste to the environment, the effluent needs less than $20m\frac{g}{L}BO{D}_5$ and less than $10m\frac{g}{L}$ suspended solids. Assume the $BO{D}_5$ of the suspended solids is $45\%.$ The growth constants of the sludge are:

$K_s=60m\frac{g}{L}BO{D}_5$

$x=2000MLVSSm\frac{g}{L}$

$Y=0\mathrm{.}5mgVSS\frac{?}{mgBO{D}_5}$

${}_m=3d^{-1}$

$k_d=0\mathrm{.}1d^{-1}$

Determine:

$($a$)$ the $BO{D}_5$ removal efficiency required

$\%$

$($b$)$ the F:M ratio

$($c$)$ mean cell residence time, ${}_c-$

$2.(20$ points$)$ The effluent from the previous problem is discharged into the Trinity River, which has a flow rate of $100\frac{m^3}{s}.$ If the river is at $20C,$ what is the initial oxygen deficit just after the waste mixes with the river? Assume the treated wastewater has a DO concentration of $1m\frac{g}{L}$ and that the river is saturated with oxygen just upstream of the discharge.