The idea is to make a Parabolic Trough with a pipe through the length of
its focus, to have sunlight heat liquid in the pipe.
In this particular development I wanted to be able to fully utilize the entire 48 inch width of a roll of reflective mylar to surface the trough.
I calculated the arc length of a 2-D slice of the parabola using Calculus.
I used my nested parabolas solution http://www.stonetabernacle.com/NESTED_PARABOLAS3.html to account for 1/4 inch plywood to create the parabolic trough surface. I chose the focus of the parabola at (0., 5 inches).
The resulting equation was complex, so I drew a graph of it and
qualitatively deduced where the curve intersected the abscissa, which was
the solution to the width at the top of the parabola, which resulted in 34 3/4 inches, well within the use of a 4' x 8' sheet of 3/4 inch thick
plywood to make the frame to the trough.
While there are infinite combinations that will work to construct a Solar Parabolic Trough, I chose this one for depth, range and the availability
of materials. A flatter parabola doesn't have to be adjusted as much to
face the sun but suffers from lesser heat. A deeper parabola has to
follow the sun more closely but produces more concentrated heat.
There may be infinite combinations, but the equation remains the same for
all of them.
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