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 $R,$ in units of kilograms per second. (a) Show that the power available from the water is $$ \mathscr{P}=R g h $$ 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 $R,$ in units of kilograms per second. (a) Show that the power available from the water is
$$
\mathscr{P}=R g h
$$
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

Mass Flow Rate

Mass flow rate is the measure of the mass of water passing through a given section of the system per unit time. It is crucial in power generation because it determines the continuity of the energy conversion process; a higher flow rate means more energy is available per unit time.

Conversion Efficiency

Conversion efficiency is the fraction of total available energy that is successfully converted into usable electrical energy. It accounts for energy losses due to friction, heat, and other inefficiencies in the system, and is essential for calculating the actual power output from the theoretical energy available.

Power

Power is the rate at which energy is transferred or converted from one form to another. In the context of hydroelectric power, it is determined by the product of the mass flow rate, gravitational acceleration, and the height through which the water falls, representing the rate of energy conversion from gravitational potential energy to mechanical energy.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. When water falls from a certain height, its gravitational potential energy (calculated as mass × g × height) is converted into kinetic energy, which can then be harnessed to do work, such as driving turbines in a hydroelectric dam.

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