$(22$ pts$)$ Hydroelectric power generation

Hydropower is easy to understand using the Engineering Bernoulli equation. This picture

provides a schematic of a typical scenario, in which water accumulated behind a dam is directed

through a turbine $($which extracts power$),$ and discharged at some velocity V$,$ at a vertical

distance H below the surface of the reservoir.

a$.(4$ pts$)$ Using the Engineering Bernoulli equation, derive an expression for the power

produced by the dam in terms of H$,$V and the mass flow rate of water, w$.$ Neglect losses.

For what value of V is the power output theoretically maximized?

b$.(2$ pts$)$ How do viscous losses impact power generation?