Question $6-$ Turbulent Flows

Water flows from a large high reservoir to a large low reservoir by gravity through a

pipeline $10cm$ in diameter and $150m$ long. The water surface elevation of the upper

reservoir is $2m$ while that of the lower reservoir is $-4m.$ A meter fitted to the pipeline $20$

$m$ from the high reservoir measures a steady flow of $0\mathrm{.}002m\frac{3}{s}.$ A valve is fitted on the

pipeline a distance of $50m$ from the low reservoir. The head losses are given by:

$h_i=k\frac{v^2}{2g}$

Where,

and the friction loss in the pipe is given by Darcy's equation as

$h_f=\frac{l}{d}\frac{v^2}{2g}$

i$)$ Deduce the value of the Darcy friction factor $.$

ii$)$ Tabulate and plot the energy gradient and the hydraulic gradient along the pipeline.