The "Pi in the Sky" math challenge gives students a chance to take part in recent discoveries and upcoming celestial events, all while using math and pi just like NASA scientists and engineers. In this problem from the ninth set, students use the mathematical constant pi to calculate the water output from a dam to assess its potential environmental impact.
The Surface Water and Ocean Topography, or SWOT mission will conduct NASA's first global survey of Earth's surface water. SWOT’s state-of-the-art radar will measure the elevation of water in major lakes, rivers, wetlands, and reservoirs while revealing unprecedented detail on the ocean surface. This data will help scientists track how these bodies of water are changing over time and improve weather and climate models. In Dam Deduction, students learn how data from SWOT can be used to assess the environmental impact of dams. Students then use pi to do their own analysis, finding the powered output of a dam based on the water height of its reservoir and inferring potential impacts of this quick-flowing water.
Water exiting a hydropower dam is called non-powered or powered outflow. Non-powered outflow exits via a spillway, on top of the dam. Powered outflow, which is used to generate electricity, travels through penstocks, pipes at the bottom of a dam. Powered outflow is usually colder and travels at a higher velocity, so it can disturb sediments, temperatures, and water quality of downstream rivers, especially when it’s a high percentage of the total outflow.
The SWOT mission, a satellite designed to survey all of Earth’s surface water, including lakes, rivers, oceans, and reservoirs, can help scientists better analyze these impacts.
A dam has 3 penstocks with diameters of 6.2 meters and a measured total outflow of 1,350 m3/s. If SWOT measured the reservoir’s water depth (H) at 100 m above the penstocks, compute the velocity (m/s) of the powered outflow using V=√(2gH).
What is the powered outflow if 1 penstock is open?
Is this a high or low percentage of the total outflow?
What can this tell you about the potential environmental impacts?
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