Exercise 4.V Head, flow rate, and output of a hydroelectric power plant

Exercise 4. V Head, flow rate, and output of a hydroelectric power plant 

A pipe draws water from a mountain lake and channels it to the tuibine of a hydraulic power plant located 200 m below. The (cast iron) pipe has a length of 500 m and a diameter D = 0.5 m. The turbine discharges water into the open air through an aperture of the same diameter. 

Express the head variation, A//turbme, between the inlet and the outlet of the turbine, taking into account the head losses in this hydraulic circuit. 

Express the hydraulic power of the turbine as a function of the area-averaged streamwise velocity in the pipe. 

Determine the streamwise velocity and the flow rate through the pipe under the condition where the hydraulic power of the turbine is at its maximum. It will be assumed that the flow in the pipe is of rough type, an assumption whose validity will be checked a posteriori. 

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