SCIENCE AND ENGINEERING FAIR
Research Plan and/or Abstract for 2018

Category Award:   1 (First Place)

Research Plan:

Abstract:
The Brachistochrone Problem ponders the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip, without friction, from one point to another in the shortest amount of time. The Brachistochrone curve has many applications in sports engineering, including downhill skiing, surfing, skateboarding, and rollercoasters. After exploring the solution of the Brachistochrone problem, a cycloid, I would like to see how I could apply the Brachistochrone curve to hydropower systems, namely pumped storage systems. Instead of the water pipes that lead to the turbines being straight lines or L-shaped curves, I think replacing the pipes with Brachistochrone curves (cycloids or parts of cycloids) would lead to more efficient resource use and higher power generation from the hydropower plants. In addition, different types of drainage systems like those in our homes and process plants can be changed from straight line and L-shaped curves to cycloids to increase efficiency. The mathematics behind the Brachistochrone Problem is typically done using beads or some type of a single point mass and frictionless surface. However, in my project I have to deal with viscosity, friction, and fluid in motion when dealing with water.

To demonstrate the power increase of using the cycloid in the hydropower systems, I used three pipes: straight line, L-shape, and half of a cycloid. After attaching each pipe to 2 feet by 3 feet boards, with one end of each pipe at the top left corner and the other end at the bottom right, I poured 1 gallon of water through a funnel at the top of each pipe and timed how long it took for the water to fill up a bottle at the end of the pipe to .8 gallons. In addition, to compare how the distance covered affects the time differences for water to travel between two points, I created different ratios of heights to lengths of the points. So, I had 2 feet by 4 feet boards and 3 feet by 4 feet boards as well. After visually observing a few of the trials, I observed that the force of the water coming out of the Brachistochrone curve was much higher than that of the straight line and L-shaped curves, especially the straight line. In order to calculate the force of the water coming out of the pipes, I measured the horizontal distance the water coming out of the pipe reached and used basic physics equations for projectile motion to determine the corresponding forces of the water.

Substituting the forces and times that I calculated and measured into the equation for power: work/time = (force*distance)/time gave me the rate at which the water coming out of the pipes would do work on the water turbine. Comparing the times taken for water to come out of each pipe configuration allowed me to understand whether changing the shape of pipes in homes or processing plants would benefit from being changed to cycloids.