What Parabolic Shape Most Efficiently Reflects Solar Rays to a Focal Point?

Teachers: Mrs. Cengia and Sister Joseph Barden

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Abstract:
Parabolic shapes are used to collect or project light and heat in many products within the home such as lighting fixtures, electric space heaters, and solar water heaters. All algebra students study the parabola, but little is mentioned of the best parabolic shape to use in these items. The purpose of this research project was to determine which parabolic shape most efficiently reflects solar rays to a focal point

The quadratic equation, y = ax2 + bx + c, was initially used and the values of “b”and “c” were made equal to zero to simplify the equation to y = ax2. Five parabolic curves were then selected by solving for “a” at 4-inch height increments at the maximum width of x=8 inches. The focal point for each parabolic curve was then determined using the equation, p=1/(4a). Five foamboard, parabolic troughs, 16 inches wide by 12 inches long, were constructed using graphs of the five curves. Reflective silver tape covered the surface of each trough.

On a clear, sunny day, each parabolic trough was pointed at the sun, and tap water at a constant flow was run through a 12" pipe centered at the focal point. Water temperature readings were taken before and after each trial. The change was calculated. Five trials were conducted for each parabolic trough. Data collection was repeated on another day under similar weather conditions.

The middle-sized parabolic curve, y=3/16x2, had the greatest average temperature changes, and is considered the most efficient curve for the 3/8 inch heating pipe.

Hypothesis:
The researcher hypothesized that narrower, deeper parabolic shapes will reflect solar rays to a focal point more efficiently than wider, shallower parabolic shapes as measured by water temperature changes before and after passing through the parabolic trough.

Conclusion:
The data did not support the hypothesis. The most efficient parabolic trough was y =3/16x2, or the middle trough. The least efficient trough was the wider, shallower trough (y= 1/16 x 2). The next least efficient was the narrower, deeper trough (y= 5/16x 2) which is what the researcher picked in the hypothesis.

Results were discussed with a researcher* from the Florida Solar Energy Center. He said that because the sun is a large disc and not a single point, all solar rays do not strike the parabola evenly and that some of the solar rays enter the solar trough at very small angles. These small changes in angles help evenly heat the water pipe centered at the focal point as the parabolic trough changes shape from shallow to deep. However, as the parabolic trough continues to change to a deeper shape, the solar rays that enter the parabolic trough at the small angles overshoot and miss the pipe resulting in a loss of heat. Therefore, the diameter of the pipe affects results.

The 3/8”pipe was the perfect match for the y=3/16 x2 parabolic trough causing it to be the most efficient curve.

(Note: The researcher was Dr. Ross McCluney from the Florida Solar Energy Center)