$(25$ points$)$ A circular pipe empties into a long river which opens into a reservoir with two outlets.

The reservoir is kept in constant motion.

a$.$ Draw the control volume for each component of the system showing the inlet$($s$)$ and outlet$($s$).$

$(2$ points$)$

b$.$ A chemical, $A$ is introduced at the inlet of the pipe which undergoes no reaction in the pipe

but once released into the river, first order decay occurs. The reactant and the product, $B$

produced as a result enter into the reservoir where both react with another reactant $C$ to

produce two different products, $P_1$ and $P_2.$ Beginning with In $-$ Out $=$ Accumulation $-r_{\text{p}\text{r}\text{o}\text{d}},$

derive the unsteady$-$state material balance for each reactor. You should systematically

demonstrate the steps in the derivation. $(8$ points$)$

c$.$ Draw a plot showing the change of the dimensionless concentration $($s$)$ over time at each

location of the system. $(2$ points$)$

d$.$ A tracer is released at the end of the pipe as the water enters the river. Draw the output

functions in the river and in the reservoir if the tracer release represents a Dirac delta

function. $(3$ points$)$

e$.$ Write out the Navier $-$ Stokes equation which represents the flow in the pipe and the river.

Explain each term of the equation relative to the respective flow. $(10$ points$)$