Practical goal:
I'd like to concentrate the low-grade heat from the back of solar panels, which I've measured is in the range of 20-50F greater than the ambient temperature throughout the day, and store it in a thermal battery, like a sand battery, as high-grade heat under the melting point of the thermal storage medium (sand, graphite, etc.), to use for cooking and heating.
As a side effect, I'd like to cool the solar panels to increase their PV efficiency.
My particular solar panels convert only 11% of available solar power into electricity; the rest is reflected or converted to heat. I would like to use some of that heat.
Theoretical (and some practical) constraints:
Heat is a low grade form of energy because it dissipates, and takes work to concentrate; this is the second law of thermodynamics. Many concentration methods involve converting it to a higher-grade form of energy which dissipates less and so can be controlled more, like electricity, pressure, or mechanical motion.
Imaging optics concentrate heat indirectly by changing the angle of rays of light from (theoretically) parallel to focused on a smaller focal (theoretical) point. They operate very efficiently; little energy is lost in this conversion. Non-imaging optics lose some practical efficiency because they do not focus light on the sink, but they gain some efficiency because they can collect light from multiple directions, from a wider angle.
The principle of the conservation of etendue is that imaging optics cannot raise a heat sink to a higher temperature than the surface of the heat source, so, for example, solar radiation cannot be focused to a temperature higher than the sun's surface through imaging optics. It seems possible that non-imaging optics does not have this constraint, so in regard to overall system performance and efficiency in some practical applications, could be more efficient than imaging optics as a heat concentrator.
I don't desire a temperature higher than the sun's surface, but do want to convert and store the available heat, and concentrate it to a high temperature, efficiently. Due to the practical consideration of the solar panels' dimensions, and my desire to collect heat from the back of the panels, I want a solar/thermal concentrator that does not require as much 3-dimensional space or weight as a system using lenses or mirrors, so I'm considering whether nonimaging optics can concentrate the heat efficiently.
Theoretical question
This is my main question. I don't know if answers on other points are allowed here, but so far as I'm concerned, I welcome them.
Can nonimaging optics exceed etendue's imaging optics thermal limit, and so concentrate heat to a higher temperature than the source, given the same light source? Why, or why not?
I know this question is imprecise because an imaging optic has a narrower acceptance angle, so a smaller source surface in that sense, than a nonimaging optic in the same environment.
Nevertheless, I think the answer is "Yes, because imaging optics only concentrate 'parallel' rays, so concentrate less light, but nonimaging optics concentrate rays from many directions, so concentrate more light."
If the answer is "Yes," then why didn't my experiment work? 😂 Maybe part of the reason is the 2nd law of thermodynamics implies "passive" concentrators fail to provide the work needed to decrease entropy. Is my quest for a solid state heat concentrator that is not actually passive, yet is fairly directly powered by the heat flux (some of which it concentrates), contrary to the laws of physics? Thermoelectric generators, Brayton cycle heat engines, and the like, make me think the answer here is "no," and so high-grade heat may yet someday be concentrated from PV solar panels.
Practical experiment
I did an experiment to answer this question.
The experiment does not answer the question perfectly, but it attempts to get in the ballpark. I'm most interested in achieving the practical result of concentrating heat, but I think something like the theoretical question above needs to be answered for me to accomplish that goal, so it's my question here, and this experiment can definitely help me better understand the answer to my theoretical question. Feel free to tell me if I'm confusing categories; that's part of why I'm asking a question.
Theory:
This is the theory I tested: a hollow black body (black on the outside) with a mirrored cavity inside will concentrate heat into any black body inside the cavity.
Assumptions:
- A black surface converts multiple wavelength ranges of light into heat.
- A heated mirror emits thermal radiation in the infrared range.
- A small black body inside the cavity will absorb infrared radiation faster, so will be hotter, than the mirrored surface.
Methods:
I coated the outside of a clear glass salsa jar with silver mirror paint, then added a coat of flat black paint on top of that. So the jar is a mirror on the inside, and a black body radiation collector on the outside; reflective on the inside and absorptive (non-reflective) on the outside. The jar's reflective interior reflects, but does not focus, any infrared radiation reflected or emitted from any (imperfect) black body inside the jar; as such, the jar's reflective interior, and any black body inside the jar, are the non-imaging optical components of this particular system.
One finer point is that I painted the jar's metal lid with the silver mirror paint on the inside, and the flat black paint on the outside, so the lid might operate slightly differently from the glass jar.
Refinement:
Can the inside get hotter than the outside? This is a bit difficult to test, because we want the inside to be a mirror emitter, not a collector. An internal thermometer is a collector; measuring changes the thing measured. So I refined my guiding question and theory to test: if a small black body thermal collector is placed inside the jar, can it collect most of the infrared radiation inside the jar (before it gets re-absorbed by the mirror), and so become a higher temperature than the jar's inside mirrored surface, and so higher than the jar's outside black surface (which is the goal)?
So I put a small dot of flat black paint in the center of the inside of the jar's metal lid. So the inside surface of the lid was mostly a silver mirror, with a small black dot (about 1/2 cm in diameter) in the center. I assumed if the black dot got hot enough, its heat would conduct through the metal lid and become visible as a localized hot spot in the thermal camera's image.
I tested the theory by putting the jar in the sun for an hour, with the lid in the shade, then viewing the jar through a thermal camera.
Results:
- Jar's outside surface. The jar's sunlit outside surface was about 30-40F hotter than the ambient air temperature (and temperature of nearby shaded surfaces). This is in the same range I have measured the solar panels heat up in the sun, so in this regard the experiment was similar to the intended application. The jar's shaded surface was cooler and likely emitted heat back to the surroundings, so decreased the concentrator's efficiency and somewhat confounded the experiment's results.
- Jar's inside surface & collection point. The jar lid's outside surface was all the same temperature and similar in temperature to other shaded parts of the jar. The lid's outside surface where the black dot was located on the inside of the jar was the same temperature as the rest of the lid's surface.
Reflection
Perhaps the black dot did heat up faster than other components (and so is still useful to promote more heat flux in that location and so concentrate the heat), but after an hour had reached equilibrium with the mirror's emissions to the black dot, the heat flux from the black dot, through the metal lid, to the outside air, and the heat conducted through all the air, glass & metal, so no "hot spot" was visible. To test this theory, a heat sink could be attached temporarily to various points on the exterior of the jar, and the heat flux through the black dot could be measured and compared with heat flux through other parts of the lid or jar to verify whether the black dot absorbs infrared faster than the jar's mirrored interior.
Conclusion:
The experiment did not confirm my theory. I'm not surprised. I think something like etendue is still making my theory impossible, but I don't know what law or principle explains why my theory, and experiment, did not work. Could someone explain why this is?
For further research:
I'd still like to collect and concentrate the heat from my solar panels! Especially in an efficient and affordable way. Any recommendations? Hope springs eternal in the human breast, and I think nonimaging optics can exceed etendue's limit; it seems some heat concentration methods can; heat pumps exist.
Some practical options:
- DIY compression (air conditioner) heat pumps have a reasonably good COP (efficiency) and affordability, but a lower thermal lift than I am hoping for, and are mechanically complex. Multi-stage (so multi-refrigerant) systems can achieve the desired thermal lift.
- Some Stirling engines can make the cold head make liquid air or glow red-hot, so their thermal lift is in my desired range, but their COP/efficiency seems low, and they are mechanically complex. See https://youtu.be/Jml027c1uWc?t=846
- Thermoacoustic heat pumps are mostly solid-state. A relatively solid-state concentrator would be attractive because it is mechanically simple and low-maintenance. My experiment was passive and low-maintenance, but it didn't work!
- Thermoelectric generators have a very low efficiency, and are mechanically simple.
- A variable pressure cold steam cryophorus with a Stirling engine or Tesla turbine between hot and cold thermal stores could convert heat to electricity (which could be stored as high-grade heat) at any common day or nighttime temperature, and could convert electricity to heat (operating as a heat pump). This is interesting because it could be integrated into a combined heat & power (CHP) system for efficiency gains, but seems to have a lower COP than refrigerant compression heat pumps, and is complex!
