$($a$)$ For fluid flow in a pipeline, show that the frictional dissipation per unit mass term $F$ is

given by

$F=cL{Q}^2,$ where $,c=\frac{32(\text{:}f\text{:})}{{}^2{D}^5}$

where, $Q$ is the volumetric flow rate, $($:$f$:$)$ average friction factor.

$($b$)$ Derive an expression for the flow rate $Q$ from a hydrant at node $3,$ where the required

gauge delivery pressure is $P_3.$ Your answer should give $Q$ in terms of $L,H,c,g,P_3,$ and

$.$

Hint: Use energy balance in two stages, first between nodes $1$ and $2,$ and then between

nodes $2$ and $3,$ along just one of the two identical legs. Neglect minor Losses.

$($c$)$ Repeat part $($b$)$ when the diameter of the ring main section is only $\frac{D}{2}.$