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?

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?

Key Concepts

Power

Power is the rate at which energy is converted or transferred, typically measured in watts (W). In hydroelectric systems, the power available from falling water is determined by the rate at which gravitational potential energy is converted into kinetic energy, expressed as the product of the mass flow rate, gravitational acceleration, and height (I_v * g * h).

Gravitational Potential Energy

This is the energy stored by an object due to its position in a gravitational field. For an object of mass m at a height h above a reference point, the gravitational potential energy is given by the product mgh, where g is the acceleration due to gravity. This principle underlies the calculation of the energy available from water falling over a dam.

Mass Flow Rate

Mass flow rate is the measure of mass passing through a given point per unit time, typically expressed in kilograms per second (kg/s). In the context of hydroelectric power, it quantifies the amount of water that flows over the dam every second, which in turn determines the total gravitational potential energy converted to kinetic energy per unit time.

Energy Conversion Efficiency

Efficiency in energy conversion quantifies the fraction of input energy that is successfully converted into the desired form of output energy, with the remainder typically lost as waste (e.g., heat). In hydroelectric power generation, the efficiency determines how much of the gravitational potential energy is actually transformed into electrical energy, affecting the final electric output of the system.

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