A small hydraulic impulse turbine is supplied with water through a penstock with diameter \(D\) and length \(L\); the jet diameter is \(d\). The elevation difference between the reservoir surface and nozzle centerline is \(Z\). The nozzle head loss coefficient is \(K_{\text {nozzle }}\) and the loss coefficient from the reservoir to the penstock entrance is \(K_{\text {entrance }}\). Determine the water jet speed, the volume flow rate, and the hydraulic power of the jet, for the case where \(Z=300 \mathrm{ft}, L=1000 \mathrm{ft}\) \(D=6\) in., \(K_{\text {entrance }}=0.5, K_{\text {nozzle }}=0.04\) and \(d=2\) in., if the pipe is made from commercial steel. Plot the jet power as a function of jet diameter to determine the optimum jet diameter and the resulting hydraulic power of the jet. Comment on the effects of varying the loss coefficients and pipe roughness.