Thermal iimits of solar power.

[Radar]

I presume you are aware that with a paraboloid those averages would be considerably and consistently varied, unlike a spheroid, which would suffer spherical abberation. I'm also not at all sure that you'd want the focus that close to the surface.

I simplified the math to make it tractable and get the key point across concerning target size and the resulting temperature.

I'm still not understanding why we can't use mirrors and lenses to make this incoming light more "ordered". We could theoretically boil water and make an electric light from that which could be as hot as we want, so I'm not seeing at all that that is significantly different from making temperature directly from the light. It's all energy, which can be measured in Watts, and more light is more Watts, so I don't see a thermodynamic problem.

You could for example produce electricity and use that to heat up your target arbitrarily high (within technological constrains), but this would incur inevitable losses that are rooted in thermodynamics: when you produce electricity, some of the source energy goes off as heat - this is inevitable as electrical current has very low entropy and sunlight (or any kind of blackbody radiation) has a comparatively higher entropy. So it has to go somewhere and take part of the energy with it. This way you do extract low entropy energy which you can use later to heat something how high our technology allows us regardless of the original source of energy. That being said, the lower the temperature of the original incoming radiation, the lower the efficiency of producing electricity as the entropy of light is related to its frequency distribution.

That being said, the initial problem was, if it is possible to use mirrors and lenses to heat something to a temperature higher than the source. Those are passive tools (as in they do not require any energy to operate) and the thermodynamic limits are applied directly. But even without involving thermodynamics, geometrical optics shows why it is not possible to do it as I tried to explain with my examples.

[halfeye]

Thermodynamically reversible doesn't just mean 'you can't make more than one beam worth of energy going backwards', it also means 'you can't make less than one beam worth of energy going backwards'. Idealized lenses and mirrors are reversible in that sense - its not just that they don't produce additional energy, they don't convert coherent energy to heat either.

Are any real world mirrors or lenses ideal in that sense? There is always some absorbtion as heat in either, and no lenses don't also reflect.

[Radar]

Are any real world mirrors or lenses ideal in that sense? There is always some absorbtion as heat in either, and no lenses don't also reflect.

That's not the point: they do not require energy to operate nor do they have to push out part of the incoming energy aside as a leftover the way heat engines, solar panels or other active devices do. There are always some losses beyond the ideal situation but it only means that in the the real world, you would get even worse results.

[halfeye]

That's not the point: they do not require energy to operate nor do they have to push out part of the incoming energy aside as a leftover the way heat engines, solar panels or other active devices do. There are always some losses beyond the ideal situation but it only means that in the the real world, you would get even worse results.

How is that not the point? They do absorb energy while operating, as heat perhaps, but how is absorbtion as heat different from other devices?

[Radar]

How is that not the point? They do absorb energy while operating, as heat perhaps, but how is absorbtion as heat different from other devices?

Because it is not inherent to their function - it is just a loss due to some imperfections. Moreover, those imperfection will not allow you to do something that an ideal mirror or lens could not do - they will only worsen the outcome and make the maximum temperature limit even lower.

On the other hand, the other devices even at their theoretical ideal will have inevitable losses that cannot be mitigated. Take Carnot cycle as one of the simplest examples: it has to dump heat into the cold reservoir in order to keep working and the ratio between energy used for work and that lost is determined by the temperatures of the reservoirs. Whatever you do, you cannot do better and actual engines obviously have even worse efficiency as nothing is ideal - especially since the isothermal compression and expansion would in theory take infinite time.

So the word "reversible" refers to processes that in ideal conditions are indeed reversible. In practice, you always have losses due to various imperfections, noise, etc. And if something is not possible in pure theory, it will not work in more realistic conditions either.

[halfeye]

Because it is not inherent to their function - it is just a loss due to some imperfections. Moreover, those imperfection will not allow you to do something that an ideal mirror or lens could not do - they will only worsen the outcome and make the maximum temperature limit even lower.

However, if the reason the theoretical items could not perform was because they were theoretical, the real items might be able to do what the theoretical ones couldn't?

On the other hand, the other devices even at their theoretical ideal will have inevitable losses that cannot be mitigated. Take Carnot cycle as one of the simplest examples: it has to dump heat into the cold reservoir in order to keep working and the ratio between energy used for work and that lost is determined by the temperatures of the reservoirs. Whatever you do, you cannot do better and actual engines obviously have even worse efficiency as nothing is ideal - especially since the isothermal compression and expansion would in theory take infinite time.

I don't understand that. I'm reminded of the real case of the wind powered vehicle that was faster than the wind driving it:

https://en.wikipedia.org/wiki/Blackb...wered_vehicle)

So the word "reversible" refers to processes that in ideal conditions are indeed reversible. In practice, you always have losses due to various imperfections, noise, etc. And if something is not possible in pure theory, it will not work in more realistic conditions either.

In theory there is no difference between theory and practice. In practice there is.

[NichG]

However, if the reason the theoretical items could not perform was because they were theoretical, the real items might be able to do what the theoretical ones couldn't?

Well, if the reason you think it should be possible to heat something up above the source's temperature using optics doesn't actually make use of those specific dissipative interactions, it doesn't help resolve the state of your understanding I think... E.g. if the argument is 'I can just move closer', that argument doesn't actually depend on the degree of dissipation of your optics.

On the other hand, if you'd said something like 'I'm going to use a lens doped with a phosphorescent substance, so it retains and re-emits some photons at a different frequency' or 'I'm going to use a multi-photon process crystal to perform frequency doubling of the incoming sunlight' or things like that, those are mechanics which would explicitly make use of the dissipation to make a difference in the argument of how you could attain that higher temperature.

[Radar]

However, if the reason the theoretical items could not perform was because they were theoretical, the real items might be able to do what the theoretical ones couldn't?

No, at least not in this case. Let's get back to geometrical optics and start from the simplest possible case, which is a flat mirror (how it is relevant will be written later in this post):

1. Reflection means that the angle of reflected ray has the same angle with respect to the surface as the incident ray. That's an ideal mirror. A realistic mirror would scatter the incident light somewhat, so you get light going off in more or less a cone around that ideal reflection direction.

2. Let's take an ideal mirror and a light source that is not a single point (being infinitely far is the same thing as only the angular size matters), so if we look at a single point on the mirror surface, there is a whole cone of rays coming in from different spots on the source. When each ray is reflected ideally, the reflected light will form a cone with the same angular size as the incident one.

Now, what will happen with a realistic mirror? Each ray from the incoming cone of light will be reflected into many different direction roughly gathered in a cone as I wrote before. When you sum it all up, the incoming cone will be reflected into a wider cone (and without sharp edge, but that's not important) and the image of whatever is reflected becomes blurry.


So in my previous post I was working with ideal mirrors and the temperature at the focus was limited by the angular size of the light source, right? Once you account for the inaccuracies of your mirrors, the focal spot simply be even bigger as the angular size of the reflected beam will be inevitable larger than that of the incident one.

[georgie_leech]

No, at least not in this case. Let's get back to geometrical optics and start from the simplest possible case, which is a flat mirror (how it is relevant will be written later in this post):

1. Reflection means that the angle of reflected ray has the same angle with respect to the surface as the incident ray. That's an ideal mirror. A realistic mirror would scatter the incident light somewhat, so you get light going off in more or less a cone around that ideal reflection direction.

2. Let's take an ideal mirror and a light source that is not a single point (being infinitely far is the same thing as only the angular size matters), so if we look at a single point on the mirror surface, there is a whole cone of rays coming in from different spots on the source. When each ray is reflected ideally, the reflected light will form a cone with the same angular size as the incident one.

Now, what will happen with a realistic mirror? Each ray from the incoming cone of light will be reflected into many different direction roughly gathered in a cone as I wrote before. When you sum it all up, the incoming cone will be reflected into a wider cone (and without sharp edge, but that's not important) and the image of whatever is reflected becomes blurry.


So in my previous post I was working with ideal mirrors and the temperature at the focus was limited by the angular size of the light source, right? Once you account for the inaccuracies of your mirrors, the focal spot simply be even bigger as the angular size of the reflected beam will be inevitable larger than that of the incident one.

In other words, if a perfect mirror can only get you to a given temperature, an imperfect mirror that is inherently less efficient won't be able to do any better? And a similar argument applies for lenses?

[Radar]

In other words, if a perfect mirror can only get you to a given temperature, an imperfect mirror that is inherently less efficient won't be able to do any better? And a similar argument applies for lenses?

Yes, with trying to explain why is that.

[Lord Torath]

I'm still not understanding why we can't use mirrors and lenses to make this incoming light more "ordered". We could theoretically boil water and make an electric light from that which could be as hot as we want, so I'm not seeing at all that that is significantly different from making temperature directly from the light. It's all energy, which can be measured in Watts, and more light is more Watts, so I don't see a thermodynamic problem.

Perhaps a visual indication?
You want a lens that does this:


And that’s fine, it will take the rays coming from the indicated directions, and converge them into a beam. But the light from each point on the sun’s surface goes in all directions. The green lines show some other rays from the closest point of the star, and the lens will tend to scatter those rays, rather than aligning them with the others:


There is no lens that can take two rays that hit the same point from different directions and direct them both in the same direction out the other side.

[halfeye]

In other words, if a perfect mirror can only get you to a given temperature, an imperfect mirror that is inherently less efficient won't be able to do any better? And a similar argument applies for lenses?

Except that an imperfect lens or mirror does not contradict the second law of thermodynamics which a perfect one might.

this is a diagram of the solar furnace I have been discussing, I cannot understand why this model would not work at half an AU, and if it would not as it stands, why it could not be modified (with for instance helistats altered to get more sunlight to the parabolic mirror)?

[NichG]

Except that an imperfect lens or mirror does not contradict the second law of thermodynamics which a perfect one might.

this is a diagram of the solar furnace I have been discussing, I cannot understand why this model would not work at half an AU, and if it would not as it stands, why it could not be modified (with for instance helistats altered to get more sunlight to the parabolic mirror)?

Well, as I said before, you're using things like a parallel rays approximation to understand how the device works. The reduction in intensity as you move away from a light source is a geometric consequence of the non-parallel nature of the rays from that source. So as you get closer to the sun, the parallel rays which all go perfectly into that parabolic focus will start to become more and more non-parallel, meaning that the size of the 'image' of those rays at the focus will grow and grow as you get closer. That's not to say you wouldn't get any improvement going from 1 AU to 0.5 AU, but as you get closer and closer, there are going to be diminishing returns. You won't just see a clean 1/r^2 scaling up of the power at the focus of the parabolic mirror. Instead, you'll see something that looks like 1/r^2 from far away, that eventually goes to a constant at r=0 rather than going to infinity.

Essentially, you're misinterpreting the 1/r^2 scaling as being something like 'the light itself is getting dimmer (intrinsic property)' when really it just comes from 'the same light is being spread out over a larger area (extrinsic property)'. Ultimately, the best you can do passively is take 'the entire output of the surface of the sun' and re-concentrate it down onto an area that is exactly the same as the surface area of the sun.

Here, this might be useful to get an intuition for these things. It's a web-based optics bench simulator: https://ricktu288.github.io/ray-optics/simulator/

[Radar]

Except that an imperfect lens or mirror does not contradict the second law of thermodynamics which a perfect one might.

None of them can contradict the second law of thermodynamics. The point is, an imperfect optical device obviously cannot focus things better than a perfect one - that's kind of there in the adjective and I think I described it in detail before.

this is a diagram of the solar furnace I have been discussing, I cannot understand why this model would not work at half an AU, and if it would not as it stands, why it could not be modified (with for instance helistats altered to get more sunlight to the parabolic mirror)?

You will get higher power, but not temperature - the focus spot will simply get bigger. Why is that? That helistat setup makes the angular size of the incident beam substantially bigger, which inevitably cuts down your ability to focus it into a small area.

[DavidSh]

this is a diagram of the solar furnace I have been discussing, I cannot understand why this model would not work at half an AU, and if it would not as it stands, why it could not be modified (with for instance helistats altered to get more sunlight to the parabolic mirror)?

Are the heliostats supposed to be flat mirrors? Then the angle of incidence equals the angle of reflection, and the width of the incoming beamlets should equal the width of the outgoing beamlets. Or is this just a drawing issue that shows them otherwise?

[Quizatzhaderac]

Are the heliostats supposed to be flat mirrors? Then the angle of incidence equals the angle of reflection, and the width of the incoming beamlets should equal the width of the outgoing beamlets. Or is this just a drawing issue that shows them otherwise?

I'm pretty sure it's a simplification to make drawing easier.

this is a diagram of the solar furnace I have been discussing...

THis diagram has some simpliciations and is analogous to Toarath's first diagram.

A more complete diagram would have two exta sets of lines.
1) Lines hitting the helostats but missing the parabolic mirror.
2) Lines hitting the helostats and parabolic mirrors, but missing the focus.

As you move the whole thing closer to the sun you get more of (1) and (2). Once you're close enough to the sun, moving the furnace closer just results in more light going to (1) and (2).
If you add more helostats and make the parabolic mirror bigger, you get a higher proportion of (2). With enough mirrors you eventually reach a point where additional mirrors are only increasing the amount of light going to (2).

[Mastikator]

Except that an imperfect lens or mirror does not contradict the second law of thermodynamics which a perfect one might.

this is a diagram of the solar furnace I have been discussing, I cannot understand why this model would not work at half an AU, and if it would not as it stands, why it could not be modified (with for instance helistats altered to get more sunlight to the parabolic mirror)?

Sun rays are not parallel though. Not even to us on the surface of the Earth 149597871 kilometers away from the sun are they parallel. The sun occupies ~1/4 square degrees of the sky. If you want it to be more parallel it would have to be further away, which means you get less energy.

The way the math works out is that no matter how far away you are you can at theoretically best focus all of the sun's light into an area equally big as the surface of the sun. And when you do that you focus the sunlight into a temperature equal to the surface of the sun.

For sunlight to be parallel it would have to be observed at an infinite distance. And with an infinitely large lens at infinite distance you could focus the light to a mathematical point, however even then you couldn't get arbitrarily high temperature because at infinite distance it would take infinite time for the light to reach, meaning you'd end up with no energy actually. So no, not even then.

Edit-

Consider this. In this diagram you're showing the focus is only using the portion of the sun's light that happens to be parallel, which also happens to be only a fraction of the sun's emitted light. Light is emitted at 180 degrees squared from the surface and you're only getting the light that is traveling in parallel. Those foci never reach the surface temperature of the sun, at best they're a thousand degrees Celsius off.

[NichG]

(Bold formatting added by me)

Sun rays are not parallel though. Not even to us on the surface of the Earth 149597871 kilometers away from the sun are they parallel. The sun occupies ~1 square degrees of the sky. If you want it to be more parallel it would have to be further away, which means you get less energy.

The way the math works out is that no matter how far away you are you can at theoretically best focus all of the sun's light into an area equally big as the surface of the sun. And when you do that you focus the sunlight into a temperature equal to the surface of the sun.

For sunlight to be parallel it would have to be observed at an infinite distance. And with an infinitely large lens at infinite distance you could focus the light to a mathematical point, however even then you couldn't get arbitrarily high temperature because at infinite distance it would take infinite time for the light to reach, meaning you'd end up with no energy actually. So no, not even then.

Edit-

Consider this. In this diagram you're showing the focus is only using the portion of the sun's light that happens to be parallel, which also happens to be only a fraction of the sun's emitted light. Light is emitted at 180 degrees squared from the surface and you're only getting the light that is traveling in parallel. Those foci never reach the surface temperature of the sun, at best they're a thousand degrees Celsius off.

This is correct, except for the argument in the bold bit, which (whether it's true or not) is not the reason you can't get arbitrarily high temperature in that case.

When you have multiple things going to infinity, you have to be careful about how you take the limit. Here you've got a lens whose size grows to infinity, whose focal length grows to infinity, and whose distance from the source grows to infinity. However, you're taking the limit of 'the rays become parallel' first, then taking the infinite lens limit.

What you should instead do is look at the difference in angle between the upper-most ray impacting the lens and the lower-most ray impacting the lens. As you take both the distance and scale of the lens to infinity, that angular deviation can either go to zero or to 90 degrees depending on whether the lens gets larger faster than it gets further away, or slower. In the limit that the lens gets large slower than it becomes distant, the focus improves but the total amount of energy captured over the surface of the lens decreases with distance. In the limit that the lens gets large faster than it becomes distant, the quality of the focus decreases as you take the limit, so even though you eventually capture all of the energy coming out of one side of the sun, you end up concentrating it less and less.