A Pelton wheel is supplied with water from a lake at an elevation \(H\) above the turbine. The penstock that supplies the water to the wheel is of length \(\ell\), diameter \(D\), and friction factor \(f\). Minor losses are negligible. Show that the power developed by the turbine is maximum when the velocity head at the nozzle exit is \(2 H / 3\). The result of Problem 12.56 may be of use.

Problem 12.56