Problem $5$: Real World Problem$-25\backslash \%$

Since first order differential equations are ubiquitous in engineering, knowing how to solve first order circuits can give profound insight into all kinds of engineering disciplines. In this problem, we consider the completely mixed flow reactor $($CMFR$)\backslash (\{\}^\{1\}\backslash ),$ a common system in chemical, civil, and environmental engineering, in its most important application: processing human waste.

Sewage contains high quantities of organic molecules that encourage bacteria growth that can cause illness in humans. In addition, if left untreated the bacteria can damage the environment by depleting water oxygen levels to the point that kills aquatic wildlife. The amount of organic matter in water is quantified using the biochemical oxygen demand $($BOD$).$ The BOD in raw sewage is around $\backslash (200\backslash$mathrm$\{$mg$\}$ O$^\{2\}/\backslash$mathrm$\{$L$\}\backslash )$ while the BOD in river water is around $\backslash (2\backslash$mathrm$\{$mg$\}\backslash$mathrm$\{$O$\}\_\{2\}/\backslash$mathrm$\{$L$\}\backslash ).$ Thus, the BOD of sewage must be reduced by two orders of magnitude for it to be safe to dump into a river. To achieve this, civil and environmental engineers use CMFRs at water treatment plants.

Figure $5$ shows a diagram of a CMFR at a local water treatment plant. A tank with a $\backslash (1,000,000\backslash$mathrm$\{$~L$\}\backslash )$ volume is filled with water containing helpful bacteria that consume organic molecules. The tank is constantly stirred to ensure uniform BOD concentration. The bacteria clean the water in the tank at a rate $\backslash ($ k$=0\mathrm{.}7\backslash )$ day $\backslash (^\{-1\}\backslash ).$ An inlet pipe pours sewage into the tank at a rate of $\backslash ($ Q$=50,000\backslash$mathrm$\{$~L$\}/\backslash )$ day and an outlet pipe transports water to the next treatment stage of the plant at the same rate. The input sewage has a BOD concentration of $\backslash ($ C$\_\{$i n$\}=200\backslash$mathrm$\{$mg$\}$ O$\_\{2\}/\backslash$mathrm$\{$L$\}\backslash )$ and the output treated water has an unknown BOD concentration.

$\backslash (\{\}^\{1\}\backslash )$ Also known as the continuous stirred$-$tank reactor.