If the stream of water mentioned in Prob. 31 were directed...

If the stream of water mentioned in Prob. 31 were directed against the dished blades of a turbine, approximately what power is developed by the turbine if the speed of the water (with respect to the ground) is $10 \mathrm{ft} / \mathrm{sec}$ in the opposite direction from that at impact when it leaves the turbine, and the turbine blades are moving at $12 \mathrm{ft} / \mathrm{sec}$ at the point of water impact?

Key Concepts

Kinetic Energy of Fluids

This concept pertains to the energy possessed by a moving fluid due to its velocity. In fluid mechanics and energy conversion systems, the kinetic energy of a fluid is quantified as one?half the mass of the fluid times the square of its velocity. This energy is the resource that turbines aim to capture and convert into mechanical work and subsequently into other forms of energy such as electricity.

Relative Velocity

Relative velocity is the speed of one object as observed from another moving object. In turbine applications, it is critical to assess the velocity difference between the incoming fluid and the rotating blades. This difference governs how effectively energy is transferred from the fluid to the turbine, impacting the resultant force and the efficiency of power extraction.

Momentum Transfer

Momentum transfer deals with the change in momentum experienced by a fluid as it interacts with a surface, such as turbine blades. When the fluid strikes and is deflected by the blades, the change in its momentum results in an impulse exerted on the turbine. This impulse is directly related to the force acting on the blades, which, combined with the blade movement, translates into mechanical power.

Power Calculation

Power in the context of fluid turbines is the rate at which energy is extracted from the moving fluid and converted to mechanical work. It is generally calculated by considering the energy flux of the fluid stream, which incorporates the mass flow rate and the velocity (squared) of the fluid, and adjusting for the relative motion between the fluid and the turbine. Understanding this concept is fundamental in predicting and optimizing turbine performance.

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