and closed $($tau $=0)$ at time $4$ hours, determine the design height of the

reservoir to prevent overflow. The initial water depth in the reservoir is $0\mathrm{.}0m.$

Figure $5.$ Rainwater harvesting system for single$-$family house.A rainwater harvesting and re$-$use system in a single$-$family house is being

designed and is indicated in Figure $5.$ The system consists of roof gutters that

collect rainwater from the roof and convey it $(Q_{in})$ to a small reservoir at the side

of the house. The reservoir is cylindrical with a $4\mathrm{.}5m$ diameter and has a free

water surface. The reservoir is connected to Type $\text{'}L\text{'}$ copper pipe with a $3$ inch

$(76mm)$ internal diameter that conveys the rainwater by gravity to an outside

tap $(Q_{\text{o}\text{u}\text{t}})$ for outdoor lawn and garden irrigation. The total length of the copper

pipe is $30\mathrm{.}5m$ and has a $\text{'}C\text{'}$ factor of $110.$ The flow through the copper pipe is

controlled by a tap. The flow through the tap is estimated with the valve

equation

$Q_v=E_s\sqrt[\phantom{2}]{H-H_0}$

where $Q_v=$ valve discharge $($in $\{$:$\frac{m^3}{s}),E_s=$ valve discharge constant

$\{$:$\frac{m^{\frac{5}{2}}}{s}),H=$ hydraulic head upstream of valve, $H_0=$ hydraulic head downstream

of valve $($tap discharges to the atmosphere$).$ Both the bottom of the reservoir

and the centerline of the tap are at the same elevation.

a$)$ Write the governing equations that describe the quasi$-$steady state conditions

in the reservoir$-$pipe$-$valve system.

b$)$ The inflow hydrograph to the reservoir for a $6-$hour period is indicated in

Figure $5.$ Assuming that the tap is fully opened $($tau $=1\mathrm{.}0$ at time $2$ hours