I was thinking about how to reach high temperatures inexpensively and safely during this quarantine. Not, like, really high temperatures, but hotter than the oven.
Carbon foam made by carbonizing bread is probably the easiest and most accessible insulating refractory material for this kind of thing; it doesn’t tolerate oxidizing conditions (it slowly burns above 700°), but in reducing conditions it gradually converts to graphite, which sublimes at 3642°.
“One sun”, the solar constant, is standardly approximated as 1000 W/m², which is the Stefan–Boltzmann emissivity of a black body at 91.3°. So a perfectly insulated object in full sunlight will eventually heat up to 91.3°. Because at that temperature all the thermal radiation it emits is in the infrared, you can get it to heat up to higher temperatures by painting it with paint that is highly reflective in the infrared, or by putting infrared-reflecting glass in front of it, but for simplicity I’m going to be considering the blackbody case for now.
The 1368 W/m² on orbit corresponds to 121°. “Two suns”, 2000 W/m², only corresponds to 160°, which is enough to cook, barely; you can reach this level of illuminance with a single flat mirror. Reaching 260° like this gas oven requires 4600 W/m², 4.6 suns, which is enough for soldering electronics. 600°, enough to fire some red clays and almost cast aluminum, emits 33 kW/m², 33 suns. 1000° is 150 suns, 1100° is 202 suns, 1600° (to melt quartz or pure iron) is 698 suns, and 2072° (to melt sapphire) is 1715 suns. Subliming graphite (3642°) would probably be impractical at 13300 suns. Quartic growth is a bitch. 5500° (63000 suns) is the absolute limit.
A small hole leading into a large cavity, sometimes called a cavity absorber, behaves as a very good approximation of a blackbody, one you can’t paint. At low temperatures, convection of air is a significant way to lose heat, but at the higher temperatures I’m interested in, almost all the heat loss is through radiation.
Probably the smallest hole it’s practical to make in carbon foam and concentrate sunlight through is about 10 microns in diameter. Reaching 256 suns (1184°) then requires concentrating the sunlight from a 256-times greater area on this hole: a circle of 0.16 mm in diameter, for example, gathering about 20 microwatts.
The material inside the cavity mostly “sees” other material inside the cavity; nothing short of a cat’s eye will send a significant fraction of the light coming in the hole the hole directly back out the hole. Almost all light that gets in needs to bounce around many times, losing energy each time, before it can get back out. So even the hole in the top of an opened empty beer can looks black, even though the beer can is 95%-reflective aluminum on the inside.
Unless the cavity is meter-scale or larger, parts of the cavity that aren’t the hole need to be well insulated to prevent the loss of more heat through conduction through the walls than from radiation through the hole.
So if you can concentrate 256 suns on a 10-micron hole into a sufficiently-well-insulated cavity, you should in theory be able to heat it up to 1184° with those 20 microwatts. This suggests that solar furnaces can perhaps be made fairly small, though see below about insulation thickness scaling.
It isn’t sufficient to focus the sunlight from an 0.16-mm-diameter lens of any focal length whatsoever, though. If the focal length of the lens is too long, then the focused image of the sun will be too large and therefore diffuse. From the point of view of an ant passing through the projected image, the whole lens is as bright as the sun, but the lens is only a few times bigger than the sun from her point of view, so the power density is not that high. The f-stop of the lens needs to be wide enough to get to 256 suns — specifically the lens needs to look 16 times as wide as the sun, which is 0.53° (about 32’), so the lens needs to subtend 8.53°, which means any lens with 256 suns needs to have an aperture of f/6.72. So if its focal length is 10 mm, the lens needs to be at least 1.49 mm in diameter, at which point it (like any other lens with a 10-mm focal length) will project an image of the sun some 93 microns in diameter. You can only get 256 suns with an 0.16 mm diameter lens if its focal length is about 1.1 mm.
If you use a lens that’s bigger and further away — for example, the 10-mm-focal-length, 1.5-mm lens suggested above — then most of the energy gathered by the lens will not enter the cavity. A 93-micron-diameter sun image with a 10-micron hole in the middle of it will gather about 100× as much energy as is actually put into the cavity. You might think that, in exchange, you don’t have to constantly track the sun. No such luck! The Earth turns 360° per 24 hours, which is 0.25° per clock minute, so your sun image gets displaced by a sun diameter every 2.1 minutes, whether that’s 10 microns or 90 microns. (It’s slightly less when the sun is further from the equator, but what’s important here is that it’s 2 minutes, not 20 minutes or 2 hours.)
For lower concentrations, you can use a one-dimensional concentrator like a solar trough (or a glass rod), running parallel to the sun’s path in the sky, but reaching hundreds of suns that way is not practical, though in theory it’s possible.
Non-imaging optics such as a compound parabolic concentrator are said to improve the situation dramatically, permitting much wider input angles. You can use two developable compound parabolic concentrators made of aluminum foil (reflectivity 95%) on cardboard, at right angles to each other, to funnel light into the hole over a wider range of sun angles; the disadvantage over using a CPC that is a solid of revolution is that most of the light will be reflected from the aluminum twice instead of once before going in the hole, thus reducing efficiency.
The overall principle limiting the performance of NIO is conservation of étendue: the intensity of illuminance times the angle it’s coming from. The thermodynamic limit is that you can’t use the sunlight to heat things hotter than the sun’s surface (5500°); you would reach that limit by arranging optics so that the poor ant sees solar surface in every direction, 4π steradians of nuclear flaming death, 63000 suns†. Conservation of étendue says that the reflection the ant sees is only as bright as the sun, and you can only do that if all those optics would direct any light the ant emits into some part of the sun’s disc, which means that such optics necessarily have a very narrow angle of acceptance: 2.1 minutes later, the ant’s remains will see only cool blue sky.
So it seems like you ought to be able to shape the optics such that you get 256 suns for 1/256 of the day before you have to reorient them; any light emitted from the hole would be redirected onto the sun’s daytime path. Unfortunately, 1/256 of the day is only 5.625 minutes. So this doesn’t help as much as you’d hope for these ceramic-firing applications; you need to use feedback control.
5 suns, 271°, enough for soldering or baking, can be achieved by optically coupling the hole to 4.8 hours of the sun’s path. A one-dimensional trough CPC focused on a slit might be adequate; four flat mirrors spaced at angles around a hole might also work.
I’m not sure if I’m thinking this through correctly. Sunlight on the ground gives varying amounts of illuminance depending on the sun’s angle; it’s only a whole sun at noon (and only twice a year at that, and only if you’re in the tropics). Sunlight reflected in a mirror surely does look just as bright as the regular sun when you’re looking at the mirror (from an angle where you can see the sun in the mirror, anyway), but the mirror can be angled to spread it across a lot of ground.
† this 63000 ought to be 4π steradians divided by however many steradians the sun subtends, but I haven’t calculated that.
Above I said that you can get your cavity to 1000° with 20 microwatts of sunlight focused through a 10-micron hole if the cavity is well enough insulated and you have good feedback control orienting the reflector. But it turns out to be impractical to insulate the cavity well enough.
If your insulation material conducts heat at 0.3 W/m/K (typical for refractory bricks), your cavity’s surface area is 6 cm², and you have a 1000 K temperature difference, then at 1 mm of insulation thickness you would lose 180 watts. Not microwatts or milliwatts, but entire watts. So you’re seven orders of magnitude away from being able to reach 1000° with a millimeter of insulation. Exotic vacuum panels might be able to gain you some of those orders of magnitude back, but charred bread won’t.
You can get maybe two or three of them back by making the cavity smaller and the insulation thicker, but I think that at some point it’s sort of a lost cause because once the heat diffuses a few millimeters through the insulation it’s diffusing through a much larger surface area again. So microwatt-scale kilns would need building-sized insulation.
A more practical approach is to scale up to, say, 100 watts, which is about 320 mm × 320 mm of sunlight. If you concentrate that down to 20 mm × 20 mm (or a 23-mm-diameter hole), you have your 256 suns. The hole can be at the end of a bit of a bottleneck leading into a chamber of, say, 50 mm diameter, which is 65 mℓ and has a surface area of 7900 mm², 0.0079 m². This would require 24 mm of insulation thickness to lose the 100 W through conduction, so 100 mm or so should be adequate to get it most of the way up to that temperature. This ends up being 250 mm in total diameter, which is probably about as big as I can bake a loaf of bread.
So a solar furnace of subcentimeter total size probably isn’t practical without vacuum multilayer insulation, but submeter is totally feasible.
Insulation stops being a difficult problem with large cavities. Consider scaling up by a factor of 200: a 10-meter-diameter cavity. Let’s scale the hole up only by a factor of 50: it’s a 1.15-meter-diameter circle, swallowing 256 kilowatts fed to it by hundreds of square meters of mirrors. It holds 524'000 liters, and its surface area is 314.15927 m². To keep its conduction losses down to 256 kilowatts, it only needs 400 mm of insulation! Now the chamber dwarfs the insulation; if you can dig it into the ground, you don’t need any further insulation, although you might need to line it with a sturdy refractory in case it turns the ground into lava.