Water falls over a dam of height h with a mass flow rate of...

Water falls over a dam of height $h$ with a mass flow rate of $I_{V}$ in units of kilograms per second. (a) Show that the power available from the water is $P=I_{V} g h$ where $g$ is the free-fall acceleration. (b) Each hydroelectric unit at the Grand Coulee Dam takes in water at a rate of $8.50 \times 10^{5} \mathrm{~kg} / \mathrm{s}$ from a height of $87.0 \mathrm{~m} .$ The power developed by the falling water is converted to electric power with an efficiency of $85.0 \% .$ How much electric power does each hydroelectric unit produce?

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Key Concepts

Gravitational Potential Energy

This is the energy that an object possesses due to its position in a gravitational field. It is quantified as the product of the mass, gravitational acceleration, and height (m*g*h), signifying the conversion potential into kinetic energy or other forms when the object is allowed to fall.

Mass Flow Rate

This refers to the amount of mass passing through a given point or cross-sectional area per unit time, usually expressed in kilograms per second (kg/s). It is essential in calculating the amount of energy transferred in systems like hydroelectric power plants.

Power Calculation

Power is the rate at which work is done or energy is transferred. In fluid dynamics and energy conversion contexts, power can be determined by multiplying the rate of energy (derived from mass flow rate and gravitational potential energy) by time, resulting in a formula such as P = I_V * g * h.

Energy Conversion Efficiency

This concept describes how effectively a system converts input energy into output energy. It is expressed as a percentage or fraction, and it plays a critical role in determining the practical output power of devices like hydroelectric generators, where not all the available energy is converted into useful work.

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