Built on Facts

Pop quiz! The picture below is a solar power facility wherein light from the sun is collected by mirrors and focused onto the top of a collecting tower. Fluid within the tower is heated by this light and the hot fluid is used to generate power. We won’t care about that in this quiz though; we’re just assuming that all the energy goes into heating the tower until its own radiant heat output is equal to that coming in from the mirrors.

i-2445b15b370226ec93aad7e22371084b-PS10_solar_power_tower (Small).jpg

Here’s the quiz setup. There’s nothing stopping you from adding as many mirrors as you want to this installation – for the purposes of this question you can put it in space and use a million square miles of mirrors, all focused onto the tower. As you dump more and more energy into the tower, it will eventually reach some high equilibrium temperature as it emits its own radiant heat just like the heating element in a toaster.

So the question is this: how hot can this equilibrium temperature be, if you add as many mirrors as you want? Is there any limit? If not, how are you not breaking the laws of thermodynamics by spontaneously transferring energy from a colder body (the sun) to a hotter one (the tower)? If so, why?

As a hint, there is a simple and unambiguous answer, but it’s not necessarily one you heard in thermodynamics class.

Comments

#1 Blaise Pascal
May 26, 2010

Stefan-Boltzmann says P=AσT⁴ for a blackbody. If we take a ellipsoidal reflector (which, for the purposes of this discussion is transparent to hard UV and shorter wavelengths) and put the sun at one focus and our heating target at the other, almost 100% of the power output of the sun is impinging on the heating target. If we assume our heating target has a radius of 7km (compared to the 70Mm radius of the sun), then when things come to an equalibrium so that P(tower) = P(sun), and dividing out varous common factors, we get 10000T_s⁴ = T_t⁴, or the temperature of the tower is 10 times the temperature of the sun.

By making the target smaller, you can make the temperature arbitrarily high, but the temperature ratio scales as the 4th root of the area difference (or the square root of the scale length).
#2 Thomas
May 26, 2010

The simple answer is that you can reach the same temperature as the surface of the sun. If you sit at the target and look at a point on one of those mirrors you will see the reflection of the sun, i.e. a temperature of 6000 K. What those mirrors accomplish is to magnify the image of the sun so it fills a larger part of the sky as seen from the target, but even in the extreme case where you have mirrors focussing light from all points around the target you will still never have incoming radiation with a temperature larger than 6000K. The idea that you can focus unlimited amounts of energy fail because you can’t focus an extended source like the sun on an arbitrarily small target.

An even more drastic example of this principle is to ask how much energy one of those huge radio telescopes pick up at their focus. It doesn’t really matter how large the disk is you still only receive the same energy, it is just coming from a smaller part of the sky.

The caveat here is that is you surround the target with a medium with a high index of refraction you actually can get a slightly higher temperature than on the sun.
#3 TheVirtuosi
May 26, 2010

As hot as the surface of the sun (minus some small hole). I’ve always thought of this question as a neat way to teach solid angle, and how important it is in E&M and optics and the like. Ah, the strange peculiarity of 1/r^2 force laws.
#4 Pierce Nichols
May 26, 2010

Thomas is correct. It will come to the surface temperature of the sun, and no hotter. If it got that hot, it would melt the focus, of course.
#5 Eric Lund
May 26, 2010

Sunlight is incoherent, and its frequencies are many orders of magnitude above any potential resonance frequencies of matter on Earth, so Thomas’s simple answer that T_max = T_sun ~ 6000 K is a starting point. However, that exceeds the melting point of all substances known, so in practice your maximum temperature will be limited by the need to preserve the structural integrity of the tower. You also have to worry about heating the air around the tower–convection works in an atmosphere.

If you were working with lasers, there would be no theoretical upper limit. One of the proposed schemes for fusion, the Laser Ignition Facility, depends on this fact.

There are other ways of getting temperatures higher than T_sun. These methods, which include ion cyclotron resonance heating, depend on using plasma waves to heat the ambient plasma. The example most familiar to most people is the solar corona with its temperature of 2 MK, but there is an even hotter plasma closer at hand: the Earth’s plasma sheet routinely has temperatures an order of magnitude higher than the corona. Again, there is a fusion application: ion cyclotron resonance heating is often used to heat plasmas to temperatures suitable for fusion.
#6 rob
May 26, 2010

i was gonna go with you can only heat it up until it melts.
#7 Taylor
May 26, 2010

I’m in agreement with the majority of the people in the comments. Assuming you could build a large enough array of reflectors in such a way that you could capture and redirect every photon the sun could emit, the total energy transfered to the water tank would be equal to the total solar output.
#8 eddie
May 26, 2010

I’m assuming the point of the tower is to collect heat energy to turn into electricity, so the liquid, presumably water, would be boiled and the steam would drive a turbine. Done properly, the temp would be a steady 100 degrees C. The practical difficulty would be taking the heat (latent heat of vapourisation) away as quickly as it arrives, and condensing it afterwards.
#9 eddie
May 26, 2010

Not done properly, the temp would be that of the next lowest melting point material in the tower.
#10 Matt Springer
May 26, 2010

The tower is just to serve to illustrate the idea. For the purpose of the quiz, you can assume the target of the focused light is a magic blackbody that remains solid at any temperature.

Lots of interesting discussion thus far, but I’m going to hold off answering until tomorrow.
#11 Ryan
May 27, 2010

I agree with #1

The thermodynamic argument (heat flowing from a cold thing to a hot thing) would be valid if the method that the tower got hot was to be physically immersed in the sun.

I think of it as an energy balance thing, if the blackbody tower is in equilibrium then the power per unit area it gets from the sun has to be the power per unit area which it radiates. The temperature of the blackbody adjusts accordingly, if you increase the absorbed (and hence emitted) power per unit area, you increase the temperature.
#12 miller
May 27, 2010

If the sun were a point source of energy, you could theoretically focus all its energy to a single point. One way to do this is by having elliptical mirror walls with the sun at one focus and the collecting tower at the other focus. Since the energy is focused on a single point, the collecting tower can be point-sized, with zero surface area. The energy lost by black-body radiation is proportional to surface area, so no energy is lost. Therefore the temperature increases without bounds.

However, this doesn’t work because the sun is not a point source. This limits how much you can focus the sun’s energy. The only way to collect the total solar output is by having a tower with as much surface area as the sun. And since the energy lost to blackbody radiation is proportional to the tower’s surface area, it all works out so that the tower never gets hotter than the sun.
#13 LordAddy
May 27, 2010

By confining the photon gas with all of those mirrors, would we be increasing the entropy of the photon gas?

Would this allow us to raise the temperature of the solar tower above the sun’s temperature while not violating the 2nd law?

This question is hard.
#14 Anonymous
May 27, 2010

how hot can this equilibrium temperature be, if you add as many mirrors as you want? as hot as you want.

Is there any limit? the only limit is due to ratio of the surface area of the sun and the collector, and you can make the collector very very small*.

If not, how are you not breaking the laws of thermodynamics by spontaneously transferring energy from a colder body (the sun) to a hotter one (the tower)? the collector is not in equilibrium.

*how small can you make a black body before it isn’t a black body anymore?
#15 Uncle Al
May 27, 2010

White diamond’s enormous Debye temperature, greater than 1600 C (even higher for pure C-12 diamond), offers a body that does not efficiently radiate through about 1000 C. One can naively play some naughty First Law games (but first, propose a mechanism for absorption). Otherwise, you cannot heat a target by blackbody radiation hotter than the emitter. If you did, the excess radiation would follow the same path in reverse.

Feel free to play with one-way dielectric reflective coatings (e.g., hot and cold mirrors in the IR).

Solar surface temp is 5500 C. The most refractory solid is substoichiometric TaC(0.89) near 4000°C versus TaC 3880°C. melting points, 5500°C boiling point. The sun example thus suffers from containment issues.
#16 Tom
May 27, 2010

Thomas is right — the temperature of the sun. This is the brightness theorem.
#17 Glen
June 7, 2010

The ultimate collector would be completely surrounded with mirrors, which would be like having the collector inside a hollow sphere at the temperature of the Sun.

Although the interior surface would emit enough power to raise the collector temp to a much higher temp if it was all absorbed, most radiation would just be absorbed by another part of the hot sphere (the collector has a smaller surface area and most radiation will just miss it.)

It will reduce to just a Zeroth Law equilibrium – same temperatures.
#18 Nomen Nescio
June 15, 2010

assuming these are two-way mirrors, no hotter than the sun — or else it’d just heat up the sun to match itself, of course.
#19 Mike Williams
August 31, 2010

I have always been interested in doing a DIY solar installation. It is a great topic that everyone should get involved in. Keep the articles coming!!