Fig. P$3\mathrm{.}13$ Pumping installation.

Prove that the pressure drop $($psi$)$ in the pipeline between points $2$ and $3$ is given closely in terms

of the flow rate $Q(f\frac{t^3}{s})$ by:

$p_2-p_3=10\mathrm{.}83+10,265f_F{Q}^2,$

in which $f_F$ is the Fanning friction factor.

The performance curve for the pump has been determined, and relates the pressure increase $p$

$($psi$)$ across the pump to the flow rate $Q(f\frac{t^3}{s})$ through it:

$p=19\mathrm{.}2-133\mathrm{.}4Q^{4\mathrm{.}5}.$

If the viscosity of the water is $1cP,$ determine:

$($a$)$ The flow rate $Q(f\frac{t^3}{s}).$

$($b$)$ The pressure increase across the pump, $p($psi$).$

$($c$)$ The Reynolds number Re in the pipeline.

$($d$)$ The Fanning friction factor, $f_F.$