Thickening $($dewatering$)$ of excess biomass from a wastewater treatment plant involves a

combination of a settling tank and a centrifuge. Once the system is operating at steady state, the

settling tank produces supernatant with a biomass concentration of $100$ mg$/$L and a settled

biomass at a concentration of $20,000$ mg$/$L$.$ The settled biomass stream then enters a centrifuge

that concentrates the biomass to the final thickened biomass. The centrate $($liquid from

centrifuge$)$ is then recycled back and blended with influent as indicated below. The volumetric

flow rates of the water $+$ biosolids are represented by the symbol Q $($m $3$

$/$hr$),$ and the

concentration of biomass in the flow is represented by the symbol X $($mg$/$L$).$ The arrows

indictate the directions of flow in a pipeline system.

For an influent flow rate of $100$ m$3/$hour at a biomass concentration of $5000$ mg$/$L:

$($a$)$ Calculate the concentration of thickened sludge leaving the centrifuge $($X$4)$

$($b$)$ Calculate the flow rate of centrate from the centrifuge $($Q$5)$