$98-$Civ$-$A$5-2009$

reservoir $($water level of $30m).$ The pipe has an internal diameter of $400mm$

and a Darcy$-$Weisbach friction factor of $0\mathrm{.}015.$ Friction losses in the pipeline

are accurately determined by the Darcy$-$Weisbach equation. Two pumps are

connected in series at the upstream reservoir. Each pump has a

characteristic curve described by $TDH=30-10Q^2,$ in which TDH is the total

dynamic head of the pump $($in metres$)$ and $Q$ is the pump discharge in $\frac{m^3}{s}.$

Flow is controlled by a valve downstream of the pipeline and approximated

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

$\{$:$0\mathrm{.}25\frac{m^{\frac{5}{2}}}{s}),H=$ hydraulic head upstream of valve, $H_0=$ hydraulic head

downstream of valve.

Ubstream

F

Figure $2.$ Pipeline system with pumps and downstream valve.

a$)$ Calculate the flow in the pipeline when the valve is set to $=0\mathrm{.}3.$

b$)$ Re$-$calculate the flow in the pipeline when the valve is set to $=0\mathrm{.}5.$

c$)$ Compute the percent of total headloss that occurs across the valve in

parts a$)$ and b$).$ What can you say about the relationship between $-$

values and percent of total headloss that occurs across the valve?