# Curvature to mirror and focus light?

Thundagere
I posted earlier asking if my science fair idea—to use many mirrors to reflect light onto one point—would work. After getting a reply that it would, I now have another question.
It's clear that I have to use a curved board to mount the mirrors onto, or I would just get reflection of straight light ray parallel to each other. Kinda useless. How can I find the equation for the curvature of my board? Honestly, I don't know when to start here... if it makes a difference I plan to use between 300 and 500 mirrors of 3/4 in squared to an 1inch squared area each.
All help appreciated!

JeffKoch
How small of a point do you want to make? Obviously a 1" flat mirror will only make a 1" spot, so that would be the diameter of your focal spot - to do better you need more smaller mirrors.

Radius of board = distance from the board to the focal spot.

Thundagere
Well, I don't honestly have to heat that much, I'm heating water so I only need 100 degrees celsius.
At any rate, I was thinking 1" squared of about 500 mirrors. I suppose I could use smaller mirrors but the smallest I found online are 3/4 inch squared. What I'm looking for is the cross sectional curvature.

JeffKoch
It doesn't matter if the mirrors are flat (which is what I assumed you were doing) or curved, the only radius that matters is the distance from the mirror to the focal point. If you want to focus light to a 1" spot with a bunch of flat 1" mirrors, you mount them on a board a distance R away from the spot and bend the board to a radius R. If you use a bunch of spherical mirrors with radii R instead, you still get a spot but it's a smaller spot.

Thundagere
I meant the curvature of the board, like you said, I'm using straight mirrors.
So what you're saying is that my board has to be an arc segment of a circle, essentially?

Mentor
Radius of board = distance from the board to the focal spot.

No, the focal length is half the radius of curvature.

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/mireq.html

Beware that this works for a spherical mirror only if the size is not too large compared to the radius of curvature. When the mirror gets beyond a certain size (for a fixed radius), you get spherical aberration and incoming parallel rays don't converge to a single point.

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/aber.html

(This illustrates it for a lens with spherical surfaces, but the same thing happens with mirrors.)

A parabolic shape would be better, in principle. Maybe it won't make much difference because you're using flat mirror segments so you're going to get a fairly crude focus anyway.

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Thundagere
Hmm...I think I could probably find the radius of curvature with differentiation...so essentially any parabolic shape would work? A different parabolic equation would only change the location of the focal point?
Do you think I could hit boiling temperatures with this?

JeffKoch
No, the focal length is half the radius of curvature.

Yes, there is a 2 in there, forgot about that.

JeffKoch
Hmm...I think I could probably find the radius of curvature with differentiation...so essentially any parabolic shape would work?

For your purposes you don't need to worry about a parabola vs. a sphere.

Thundagere
Actually, parabola is probably easier for me to get ahold of in my area, I'm just wondering if a parabola will hit the minimum of 100 degrees celsius.

Gold Member
Thundagere

1] You don't need to curve the panel upon which the mirrors sit. Mount them on a flat board and simply angle each mirror. This will be way easier to construct.

2] To ensure the mirrors are focusing properly throw away your pencil and paper - do it for reals. Just shine a light on the mirror and simply adjust the mirror until its spot of light is centred on your focus.

Thundagere
How would I angle it...as in, how would I get it, practically, to stick in that position?

Gold Member
How would I angle it...as in, how would I get it, practically, to stick in that position?
How permanent do you want it? A big wad of plastercine would do it.

But permanently, what you do is attach one side to the base and the other side is held in place with a nut and bolt system that can be adjusted and fine tuned. Dial the bolt up to deflect the mirror one way, dial it down to deflect it back. Or even simpler, two mirror edges held in place with screws and one or more shims placed underneath. Tighten the screw against the shim.

The exact mechanism would be left up to you. It would be dependent on the materials you have available, the amount of angle and the size of the mirror. I could draw you a diagram of a proposed idea, but it would mostly be experimenting.

You'll need some system like this anyway, since with your method, if it is fixed in place, you would have no way of fine-tuning the mirrors. And you'll need to, I assure you.

Thundagere
How permanent do you want it? A big wad of plastercine would do it.

But permanently, what you do is attach one side to the base and the other side is held in place with a nut and bolt system that can be adjusted and fine tuned. Dial the bolt up to deflect the mirror one way, dial it down to deflect it back. Or even simpler, two mirror edges held in place with screws and one or more shims placed underneath. Tighten the screw against the shim.

The exact mechanism would be left up to you. It would be dependent on the materials you have available, the amount of angle and the size of the mirror. I could draw you a diagram of a proposed idea, but it would mostly be experimenting.

You'll need some system like this anyway, since with your method, if it is fixed in place, you would have no way of fine-tuning the mirrors. And you'll need to, I assure you.

Yeah, I think I'll probably use the shim method. It'll take longer, but it solves the problem of mirrors drooping down over time.
The only thing is, I'll hae to drill through the mirror for that and place the bolt+shim. Would that overly interfere with the light focusing? I think I'll still reached desired temperatures, but since I have no frame of reference...

Gold Member
Yeah, I think I'll probably use the shim method. It'll take longer, but it solves the problem of mirrors drooping down over time.
The only thing is, I'll hae to drill through the mirror for that and place the bolt+shim. Would that overly interfere with the light focusing? I think I'll still reached desired temperatures, but since I have no frame of reference...
No need to put any holes in the mirrors. Hold them by their edges.

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Thundagere
Ah, I'm not quite sure what you mean...could you elaborate?
Sorry, I'm in middle school. Studied theoretical physics, but... I don't have much practical experience.

JeffKoch
I hope you've considered the time and effort required to fine-align 500 mirrors.

Thundagere
Don't worry. I have at least 6 months. Maybe a bit more ;).

Gold Member
See diagram attached.

One edge of the mirror rests on a fulcrum so it can tilt.
Two other edges have washers glued to the back of the mirror.
Screws or bolts run through washers. Use double nuts to clamp washer.
Adjust mirror by dialing nuts up or down.

The reason you need so many nuts is because
1] every nut must be a pair, so they lock in place
2] You need 2 nut pairs for each screw, top and bottom
3] You need two screws for two axes.
A total of 8 nuts per mirror.

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Gold Member
I hope you've considered the time and effort required to fine-align 500 mirrors.
Tru dat!

Thundagere
And the fact that with 500+ mirrors, I would need about 4000 nuts :-o. That's a lotta nuts.
Cheapness is a major factor in my project...I'm wondering if there's a way to make it slightly less. 500 fulcrums is a lot to cut, too. Maybe decrease the number of mirrors? I'm only boiling water...not sure what the bear minimum I could get by with would be though.
Do you guys think that maybe just 100 small mirrors would do it. That's only 10 inches by 10 inches, but it might be able to heat water.
Otherwise, I might just use the plasticine method, and then cover the edges (space between mirrors and board) with some sort of plastic material to stop any water or stuff from getting in.

Gold Member
And the fact that with 500+ mirrors, I would need about 4000 nuts :-o. That's a lotta nuts.
Cheapness is a major factor in my project...I'm wondering if there's a way to make it slightly less. 500 fulcrums is a lot to cut, too. Maybe decrease the number of mirrors? I'm only boiling water...not sure what the bear minimum I could get by with would be though.
Do you guys think that maybe just 100 small mirrors would do it. That's only 10 inches by 10 inches, but it might be able to heat water.
Otherwise, I might just use the plasticine method, and then cover the edges (space between mirrors and board) with some sort of plastic material to stop any water or stuff from getting in.
Ah. If it's only a one-time use thing then go with plastercine or somesuch. But plastercine will sag if kept warm, such as near a stove.

If you adjusted them while attaching them, you could use a glue gun instead.

As for scaling it down to 10x10, you can calculate how big it would need to be to do the job. My instinct is that 10x10 would be much too small.

How much water do you need to boil? How big is your pot?

Thundagere
Well, now that's the thing. My project is using solar power to cheaply, safely, and efficiently distill polluted water. With that in mind, probably a gallon or two gallons would be my goal. I've considered fresnel lenses, solar parabolic collectors, but I liked this idea best (it's like an offspring of a parabolic collector).
It actually has to be a multiple use thing, when you can use it several times. Maybe cotton balls, stuffed around the plasticine, to insulate it?

Gold Member
Well, now that's the thing. My project is using solar power to cheaply, safely, and efficiently distill polluted water. With that in mind, probably a gallon or two gallons would be my goal. I've considered fresnel lenses, solar parabolic collectors, but I liked this idea best (it's like an offspring of a parabolic collector).
It actually has to be a multiple use thing, when you can use it several times. Maybe cotton balls, stuffed around the plasticine, to insulate it?
I don't understand what you're insulating...

One of the things you will have to do before any design or construction is to calculate how much solar influx you will need to boil that amount of water. Look up the wattage of a square metre of solar flux and compare that to how much energy it would require to boil a gallon of water. That will determine how big your array will have to be. Too small, and all that work will be for naught before you even begin.

Thundagere
I actually researched that when I was looking at the Stefan-Boltzmann law, I'm pretty sure it's about 1000 W / m^2
I mean you insultate the plasticine so that it doesn't sag...or maybe I got something wrong there.

Gold Member
I actually researched that when I was looking at the Stefan-Boltzmann law, I'm pretty sure it's about 1000 W / m^2
Ah. Good.
I mean you insultate the plasticine so that it doesn't sag...or maybe I got something wrong there.
Oh. I don't think cotton balls will insulate from heat.

Thundagere
Ah. Good.

Oh. I don't think cotton balls will insulate from heat.

I dunno, I just threw that out there as an idea.
But the way I see it, if I want it to be adjustable, I have two options:

The nut-bolt schematic you mentioned
and putting plasticine underneath, then insulating it so that it doesn't sag as easily?
Would that work?

JeffKoch
A gallon of hot tap water is maybe 50C and is about 4 liters in volume, which is 4000 mL, which is 4000 cm3. The heat capacity of water is about 4.2 J/cm3/K, so you need to dump 840 kJ of energy into the water to get it to 100C and start to boil. This neglects losses from conduction of heat into the bucket, the ground, the air surrounding the bucket, etc.

The solar constant is indeed around 1000 W/m2, which is 0.1 J/s/cm2. A 1" mirror is 5 cm2, 500 of them is 2500 cm2, and assuming you want to achieve boiling in some reasonable period of time before everyone gets bored and walks away (say, 10 minutes = 600 seconds), the most you'll get out of the mirror array is 150 kJ.

Which says that, absolute best case, it would take an hour of waiting for the water to boil before it starts going. But in reality it won't ever happen, because you will lose most of every joule of energy you collect into heating the bucket, ground and air.

You might go through these numbers yourself, consider a much smaller volume of water, and think about how you could keep the heat in the water as it warms up, rather than lose it to the surroundings. This seems to be your biggest challenge.

johnbbahm
How about this, There seems to a lot of Satellite Dish Antenna out there that are already parabolic. Buy a used one and glue all of your 1" square mirrors on. There will be some error, but well within the 1" accuracy you have. example dish below.
I expect you real issue, will be it getting too hot. The next trick is to keep it aimed at the sun.
http://compare.ebay.com/like/360402232138?var=lv&var=sbar [Broken]

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Thundagere
A gallon of hot tap water is maybe 50C and is about 4 liters in volume, which is 4000 mL, which is 4000 cm3. The heat capacity of water is about 4.2 J/cm3/K, so you need to dump 840 kJ of energy into the water to get it to 100C and start to boil. This neglects losses from conduction of heat into the bucket, the ground, the air surrounding the bucket, etc.

The solar constant is indeed around 1000 W/m2, which is 0.1 J/s/cm2. A 1" mirror is 5 cm2, 500 of them is 2500 cm2, and assuming you want to achieve boiling in some reasonable period of time before everyone gets bored and walks away (say, 10 minutes = 600 seconds), the most you'll get out of the mirror array is 150 kJ.

Which says that, absolute best case, it would take an hour of waiting for the water to boil before it starts going. But in reality it won't ever happen, because you will lose most of every joule of energy you collect into heating the bucket, ground and air.

You might go through these numbers yourself, consider a much smaller volume of water, and think about how you could keep the heat in the water as it warms up, rather than lose it to the surroundings. This seems to be your biggest challenge.

How about this, There seems to a lot of Satellite Dish Antenna out there that are already parabolic. Buy a used one and glue all of your 1" square mirrors on. There will be some error, but well within the 1" accuracy you have. example dish below.
I expect you real issue, will be it getting too hot. The next trick is to keep it aimed at the sun.
http://compare.ebay.com/like/360402232138?var=lv&var=sbar [Broken]

You guys are awesome :).
I've been thinking about it, so basically i have to find a way to retain the heat inside the container?
My thinking was to use a metal tub of maybe 1000 g (about a quart) and surround it with insulating material? Then put a sheet of polycarbonate on the top? It's not that expensive and it would allow light in. The heat transfer outside would not occur easily because of insulation but would tend to remain within the pot and boil the water. I'm going to run some math through that and see how it works out.

EDIT: Incidentally, I also did research into parabolic troughs and such. I discarded the idea of using these, as I believe the focal point is INSIDE the thing, and additionally, on a small scale you can't boil water, you can only get to 200 degrees F.

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Thundagere
Allright, an update. I ran through the equations during some free time today, this is what I came up with.
Q = mc∆T
m= 1000 grams (slightly more than a quart)
c = 3.98 (in the polluted water that I am handling, since I'm distilling it).
∆T = 75 degrees C, because from 100 degrees celsius to 25 degrees (room temp), is 75 degrees.
Keep in mind, I'm not bothering about heat of vaporization here, I'm just raising it to that temperature and then adding a couple of arbitrary minutes.

Q = 298500
Insolation of sun =1000 W / m^2
1 inch mirror tile = 6.45 cm ^2 = 6.45 * 10^4 m^2
Solar energy in a one inch mirror tile = 0.645 W (found by proportions).
Thus, 289000 = 0.645 J/s (time) (number of mirror tiles)
462790.697 = (time) ( number of mirror tiles)
Now, I put in 500 mirror tiles. I'll probably linear program it later (Never mind, I did linear program it, got about 500, give or take. I can't rely on my mental linear programming :) ) That leaves me with 925.5 seconds, or about 15 minutes. That isn't bad, all things considered,
Now, I know I haven't considered energy leaving the system. However, I'm working on a container that will minimize this. Only looking at an assumption that energy is constant within the system, is my math correct?

Thundagere
Any advice on the previous calculations? I'm wondering, with these calculations in mind, should I put it via plasticine on a rectangular board, or on a parabolic satellite dish?

Gold Member
Can't help you with the numbers but if you can get your hands on a parabolic dish, that would certainly go a ways towards getting your mirrors aligned.

daqddyo1
Re: satellite dishes: there are two types: one where the focal point (pickup location) is on the axis if symmetry (out where you think it should be) and the second type has a non-symmetric shape. These very common types have the pickup out from the antenna but closer to the bottom of it than the centre.

Thundagere
I'll try to get the second type of satellite dish and glue my 500 mirrors on. Thanks!