Problem $1$

A four$-$stage laboratory RBC unit with a wetted area of $1\mathrm{.}507$ ft$2/$stage was used in a treatability study for the determination of the treatability factor P $($mg$/$ft$2$d$)$ and the half$-$velocity constant Ks $($mg$/$L$).$ A synthetic wastewater with a BOD$5$ of $250$ mg$/$L was used. The study was conducted at three different flow rates and generated effluent quality data for each stage given in the table below. Use these experimental data to determine the values of the two constants P $($mg$/$ft$2$d$)$ and Ks $($mg$/$L$).$

Q$,$ L$/$d

S$1,$ mg$/$L

BOD$5$

S$2,$ mg$/$L

BOD$5$

S$3,$ mg$/$L

BOD$5$

S$4,$ mg$/$L

BOD$5$

$10$

$112$

$58$

$27$

$12$

$15$

$130$

$78$

$42$

$22$

$20$

$140$

$88$

$59$

$40$