Consider a narrow, straight, earth$-$lined $($Manning$\text{'}$s $n=0\mathrm{.}015,$ slope $S=\frac{1}{500})$ river

having a rectangular cross$-$section of width $20m.$ During a flood, where the flow rate is

estimated to be $Q=100\frac{m^3}{s},$ a chemical factory accidently spills $600kg$ of a pollutant

into the river.

a$)$ Determine the water depth for the flood condition

$[8$ marks$]$

b$)$ Compute the cross$-$sectional average velocity for $Q=100\frac{m^3}{s}$

$[2$ marks$]$

c$)$ Determine a sampling interval $(t_{\frac{1}{2}})$ at a measuring station $($i$.$e$.0\mathrm{.}6km$

downstream of the spill$)$ for which the tracer concentration is vanishingly small

$($i$.$e$.$ for when the exponent term of the solution is about $0\mathrm{.}0005).$

$[4$ marks$]$

d$)$ The turbulent diffusion coefficient $D$ is $10\frac{m^2}{s}.$ Determine and plot the

variation of tracer concentration $C($in $m\frac{g}{l})$ with time $t$ at the measuring station.

Calculations should be done every $60$ s between $t_1=240s$ and $t_2=540s.$

$[8$ marks$]$

e$)$ From the graph, estimate the peak concentration and time of arrival.

$[4$ marks$]$

f$)$ Estimate the time of arrival of the peak if turbulent diffusion was negligibly

small.

$[2$ marks$]$