Parabola and focal point assistance please!

[Bad at Math Mike]

Hello!

I am designing a linear solar concentrator for my (future) greenhouse. These devices use parabolic mirrors to reflect sunlight on to a heat absorption tube mounted longitudinally at the focal point. The tube is filled with a fluid that absorbs heat; the heated fluid is then pumped to thermal storage tanks for future use (steam turbines, or in my case, radiators). See pics if you are not familiar with these.

My reflector will be made of a flexible rectangular mirror (acrylic mirror sheet or tiles) with a dimension of either 24", 30" or 36" height (width TBD, not important right now, probably 48"), depending on my limited budget. Unlike the commercial versions as seen in the pics, for protection against the elements, my concentrator will be housed in a glazed (side facing the sun will be glass) box of a size and weight that can be easily handled, mounted, and adjusted to match the sun's angle over the year. This also means, if possible, a minimum distance to the focal point to minimize the box size.

(I realize the glazing will reduce the sunlight available, up to 10%, but I think mounting the reflector and absorption tube in an insulated glazed box will reduce heat loss and somewhat offset the loss of light)

The math help I need: I need parabola profiles given the mirror dimension of 24", 30" or 36" height and if possible, that minimize the distance to the focal point. There are limits to the flexibility of the mirror material, but I think based on the images I've seen of these collectors, any working parabola will not be severely curved enough to present a problem for the mirrors.

Is there a graphing program for math dummies like me so I can create different parabolas? Ultimately, I need a scalable image of a suitable parabola and focal point that I can print actual size for a template when building the mirror support and placing the heat absorption tube.

Thanks in advance for any assistance!

Mike

 

[Bad at Math Mike]

Follow up: I went on youtube and found a great video that taught (reminded me) of the equation for a parabola, how to find the focal point, and how to graph it. However, I think my dilemma still rests with the limitations of the mirror dimension - and how to "back calculate" a parabola given the limitations of the mirror width - 24, 30 or 36" and the desire for a short focal length.

I guess my goal now is (in order to minimize the box size) to calculate parabolas with those mirror width limitations that have a focal point just below a straight line drawn between the ends of the parabola. That would keep the heat absorption tube close enough to minimize the box size but keep the parabola from being too severely curved for the mirrors to be bent. Does that make sense?

TIA,

Mike

 

[Deleted member 4993]

that minimize the distance to the focal point.

From where do you want to measure this distance?

 

[blamocur]

In the first approximation the problem is pretty straightforward: parabola with equation [imath]y=ax^2[/imath] has focal distance [imath]f=\frac{1}{4a}[/imath]. And vice versa, given the focal distance [imath]f[/imath] one gets equation [imath]y= \frac{x^2}{4f}[/imath]. As you can see, it does not depend on the size of your reflectors.

Another thing to consider is the direction of the sun rays. An ideal parabola focuses rays which are parallel to its axis (Y axis in the equation above). But the Sun moves, and the farther away it is from the axis the less focused the reflect rays become. If your heat absorption tube is too small then most of the reflected rays will miss it when the Sun is not align with the mirror axis.

You might find it interesting to play with a simulator, like https://ricktu288.github.io/ray-optics/simulator/

 

[Dr.Peterson]

Follow up: I went on youtube and found a great video that taught (reminded me) of the equation for a parabola, how to find the focal point, and how to graph it. However, I think my dilemma still rests with the limitations of the mirror dimension - and how to "back calculate" a parabola given the limitations of the mirror width - 24, 30 or 36" and the desire for a short focal length.

I guess my goal now is (in order to minimize the box size) to calculate parabolas with those mirror width limitations that have a focal point just below a straight line drawn between the ends of the parabola. That would keep the heat absorption tube close enough to minimize the box size but keep the parabola from being too severely curved for the mirrors to be bent. Does that make sense?

You seem to have conflicting goals. To make the focal length as short as possible, you want the most bend!

​

Here, the box is a square, with ratio AA':AB = 1:1, with a tiny focal length of 0.25.

To put the focus exactly on the line between ends, that line AA' will be the latus rectum, making a box with ratio 4:1:

​

To move the focus in a little, you'd make it a little smaller, like this 3.75:1:

​

@blamocur's suggestions are also useful.

 

[Bad at Math Mike]

That's great information, thank you! I am learning more about these solar heaters and getting closer to a final design, this greatly helps!

 

[Bad at Math Mike]

In the first approximation the problem is pretty straightforward: parabola with equation [imath]y=ax^2[/imath] has focal distance [imath]f=\frac{1}{4a}[/imath]. And vice versa, given the focal distance [imath]f[/imath] one gets equation [imath]y= \frac{x^2}{4f}[/imath]. As you can see, it does not depend on the size of your reflectors.

Another thing to consider is the direction of the sun rays. An ideal parabola focuses rays which are parallel to its axis (Y axis in the equation above). But the Sun moves, and the farther away it is from the axis the less focused the reflect rays become. If your heat absorption tube is too small then most of the reflected rays will miss it when the Sun is not align with the mirror axis.

You might find it interesting to play with a simulator, like https://ricktu288.github.io/ray-optics/simulator/

Thanks for the input!! Greatly appreciated, learning more and more everyday! Will try out that simulator