Question
Which hydroelectric dam would provide the most electrical power if they all have the same volume of water falling over a given period but differing in the height from which the water falls?
Answers
GPT-4o mini
The amount of electrical power that a hydroelectric dam can generate is largely determined by the height from which the water falls, known as the "head." The potential energy of the water is given by the equation:
\[
P = \eta \cdot \rho \cdot g \cdot h \cdot Q
\]
where:
- \( P \) is the power (in watts),
- \( \eta \) is the efficiency of the turbine and generator (expressed as a decimal),
- \( \rho \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the height (in meters) of the water fall (head),
- \( Q \) is the volume flow rate of water (in cubic meters per second).
Given that all dams have the same volume of water falling over a given period, the key variable in the equation is the height \( h \). Thus, among the different hydroelectric dams, the one with the greatest height (head) will provide the most electrical power.
In summary, the hydroelectric dam that provides the most electrical power will be the one with the highest elevation drop for the water.
\[
P = \eta \cdot \rho \cdot g \cdot h \cdot Q
\]
where:
- \( P \) is the power (in watts),
- \( \eta \) is the efficiency of the turbine and generator (expressed as a decimal),
- \( \rho \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the height (in meters) of the water fall (head),
- \( Q \) is the volume flow rate of water (in cubic meters per second).
Given that all dams have the same volume of water falling over a given period, the key variable in the equation is the height \( h \). Thus, among the different hydroelectric dams, the one with the greatest height (head) will provide the most electrical power.
In summary, the hydroelectric dam that provides the most electrical power will be the one with the highest elevation drop for the water.