Question $3$

$($a$)$ A company plans to install a single Pelton Wheel turbine on a river with a mean flow rate of $\backslash (15\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash )$ and an available head of $380$ m $.$ Local legislation states the maximum allowable flow diversion for energy generation is $10\backslash (\backslash$mathrm$\{$m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash ).$ Assuming this maximum flow is diverted, and assuming the turbine is $85\backslash \%$ efficient, determine:

$($i$)$ the power output from the turbine

$($ii$)$ the wheel diameter

$($iii$)$ the number of jets needed

$($iv$)$ the jet diameter and flow

Ignore energy losses through the penstock and assume the following:

$-$ the ratio of jet diameter to wheel diameter is $1$:$12,$

$-$ the ratio of cup speed to jet speed is $0\mathrm{.}45,$

$-\backslash (\backslash$quad N$\_\{$s$\}=35\backslash )$ for the turbine.