$1.$ Consider a $4-$acre detention basin with vertical sides. Assume that thedischarge from the basin is given by Outflow $=77\mathrm{.}3$ H$0\mathrm{.}5$ where outflow has units of cfs and H is the depth of water in the basin in ft$.($Assume that the basin is able to contain the hydrograph.$)$

Given below is an inflow hydrograph:

Time $($hrs$)01234$

Inflow $($cfs$)02004002000$

Route the inflow hydrograph through the basin using the storage indication method $($without use of a computer program or spreadsheet$).$ Use a time step of $1$ hour. Plot the inflow and outflow hydrographs on the same figure.

Hints:

$1.$ Write the equation for discharge $($Q$,$ in cfs$)$ as a function of stage $($H$,$ in ft$.)$

$2.$ Write the equation for storage $($S$,$ in cfs$-$hrs$.)$ as a function of stage

$3.$ Using the previous equations, write an equation for storage $($S$)$ as a function of outflow $($Q$).$

$4.$ Using the results of step $3,$ write an equation of $2$S$/$t $+$ Q as a function of Q$,$ where t has units of hours.

The first equation in the storage$-$indication method is

I$0+$ I$1+2$ S$0/$t $-$ O$0=2$ S$1/$t $+$ O$1$

Each term in the left$-$hand side of the equation is known. Hence the right$-$hand of the equation is also known.The equation in step $4$ can then be used to solve for Q$,$ where Q in this case is O$1$Repeat for the subsequent time periods until the first decreasing outflow.

Q $=77\mathrm{.}3$ H$0\mathrm{.}5$ Q in cfs; H in ft

H $=($Q$/77\mathrm{.}3)2$

S $=4$H S in acre ft

S $=48\mathrm{.}4$ H S in cfs$-$hrs

S $=$ Q$2/123\mathrm{.}5$

For t $=1$ hr:

$2$S$/$t $+$ Q $=$ Q$2/61\mathrm{.}7+$ Q

From here on the routing is conducted in steps starting with the first two times steps $(0$ and $1$ hr$.)$ In each step computing the outflow requires solving a quadratic equation. $($You can Google this if you dont know this$)$

$2.$ Consider three triangular inflow hydrographs, each with durations of $4$ hours. The peak discharges $($cfs$)$ of the hydrographs are $400/3,400,$ and $400$x$3.$ Using the detention basin from Question $1$ and the reservoir routing program provided, route thethree inflow hydrographs.

Note the following:

The program provided uses a $5-$minute time step and depth increments of $0\mathrm{.}1$ feet.

The inflow factor can be used to obtain the first and third hydrograph.

The outlet code determines which control structure is used. $(0$ for the orifice and $1$ for the weir$).$

The only values you need to change are the inflow factor and the outlet code.Repeat, using the same basin configuration and the following stage$-$discharge relationship:

Outflow $=7$ H$1\mathrm{.}5$

Plot the two stage$-$discharge relationships on the same figure.For each of the three inflow hydrographs, plot $($on the same figure$)$ the inflow hydrograph and the outflow hydrographs corresponding to each of the two stage$-$discharge relationships. $($There will be three figures, each with one inflow and two outflow hydrographs.$)$

Give a table with the resulting outflow peaks.

Discuss the results. In particular, explain how and why the relative performance of the outlet structures varies with the hydrograph.

Area $($acres $)=4$

Delta T $($hrs$)=0\mathrm{.}08$

S$(0)($cfs$-$hrs$)=0$

O$(0)($cfs$-$hrs$)=0$

Outlet Code$=0$

Orifice $(0)=77\mathrm{.}3$H$^1/2$

Weir $(1)=7$H$^3/2$

Inflow factor $=1\mathrm{.}00$