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 $\mathrm{kg} / \mathrm{s} .$ (a) Show that the power available from the water is $$\mathscr{P}=R g h$$ where $g$ is the frec-fall acceleration. (b) Each hydroclectric unit at the Grand Coulce Dam takes in watcr 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 clectric power with an efficiency of $85.0 \% .$ How much electric power is produced by each hydroelectric unit?

Water falls over a dam of height $h$ with a mass flow rate of $R,$ in units of $\mathrm{kg} / \mathrm{s} .$ (a) Show that the power available from the water is
$$\mathscr{P}=R g h$$
where $g$ is the frec-fall acceleration. (b) Each hydroclectric unit at the Grand Coulce Dam takes in watcr 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 clectric power with an efficiency of $85.0 \% .$ How much electric power is produced by each hydroelectric unit?

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. In contexts such as a dam, water falling from a certain height possesses an amount of potential energy determined by its mass, the gravitational acceleration, and the height from which it falls. This energy is the primary source that can be converted into mechanical or electrical energy in hydroelectric power plants.

Power Conversion

Power is defined as the rate at which energy is converted from one form to another or transferred over time. When water falls from a height, the gravitational potential energy is transformed into kinetic energy, and through turbines into electrical energy. The power available from the falling water is determined by multiplying the rate of energy conversion by the mass flow rate, gravitational acceleration, and height, leading to an expression such as Rgh.

Hydroelectric Efficiency

Efficiency in a hydroelectric system refers to the ratio of the useful electrical power output to the total power available from the falling water. It accounts for energy losses in the mechanical conversion process, friction, and other system inefficiencies. This concept is critical in real-world applications since the actual electrical power generated is less than the theoretical energy available due to such losses.

Mass Flow Rate

Mass flow rate is a measure of the quantity of mass passing through a given cross-sectional area per unit time. In the context of hydroelectric power generation, it quantifies how much water is being used to generate energy, thereby directly affecting the total gravitational potential energy available for conversion into electrical power.

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