$8\mathrm{.}17$ Dissolved oxygen is very important to support fish populations in rivers, lakes,

and streams. In particular, a fish population can become at risk if the dissolved

oxygen content falls below $4m\frac{g}{1}$ in the water. The oxygen deficit $D$ is defined

as

$D=C^{\text{s}\text{u}\text{t}}-C$

where $C$ is the concentration of oxygen in the water and $C^{is}$ is the concentration

of oxygen in the water at saturation. $C$ is about $8m\frac{g}{L}$ at ambient conditions,

so we would like to maintain $D$ at less than $8-4=4m\frac{g}{L}.$

Oxygen depletion can occur in highly polluted rivers when microorganisms

in the water use up the available oxygen in breaking down organic matter. The

biological oxygen demand $L$ is a measure of how much oxygen is required to break

down the organic material and is therefore an indirect measure of the degree of

contamination of the river.

The rate at which the organic material will use up the dissolved oxygen in

the water is given by $k_{il}L.$ The rate at which oxygen can be transferred from the

atmosphere to water is given by $K_{R}D.$ Reasonable values of $K_R$ and $k_d$ are $\frac{0\mathrm{.}2}{h}$

and $\frac{0\mathrm{.}04}{h},$ respectively.

A river saturated with oxygen, that is$,D(0)=0m\frac{g}{L},$ is suddenly exposed to

organic waste such that $L(0)=35m\frac{g}{L}.$

a$.$ Develop the model and obtain the analytical solution that describes the oxy$-$

gen concentration in the water. $($Hint: Consider a volume of liquid, e$.$g$.,1L,$

and describe what happens to the oxygen in that volume. Another hint: As

the microorganisms break down the organic material, the biological oxygen

demand, $L,$ will reduce; thus, you may need to write a model of how $L$ changes

with time.$)$

b$.$ If the stream flows at $1\frac{m}{s},$ for what range of distances $($in $km)$ downstream

will the oxygen level in the river be unacceptable?