Boerseun's Perpetual Motion Machine...

[modest]

I got back after having had a look at it, to say that it hasn't convinced me yet.

 

That's fine. I've given my conclusion and the reason for it and a source saying the same,

Not coincidentally, such a tight focus would also violate the second law of thermodynamics. If you could focus the sun’s rays more tightly than permitted by Liouville’s theorem, it would be possible to create a focal spot hotter than the surface of the sun. This is perfectly consistent with the first law of thermodynamics (conservation of energy), but would immediately violate the second law of thermodynamics, since you could in principle run a heat engine using the focal spot as the "hot" side of the engine, and the surface of the sun as the "cold" (!) side, thereby producing work using only one heat bath (the sun) rather than the conventional two...

 

Years later I learned about Liouville’s theorem, and realized that no such system is possible. Each of the deep-sky objects I was interested in is intrinsically so dim that even if you travelled there and stood next to the object, you couldn’t see it with the naked eye. (The beautiful pictures you see come from long time exposures, which the eye cannot do.)

No system of lenses (or any other passive system) can produce an image with more surface brightness than the original object

.

Phase Space of a Thin Lens

This seems very intuitive to me and seems exactly the reason the OP is not possible. But, perhaps it is wrong. I would certainly welcome being shown otherwise. I'd, in fact, be eager to be shown otherwise as I am always eager to learn anything, but I find unacceptable the 'you are wrong, but too unlearned for me to explain why' attitude.

 

Your claim,

With a parabolic mirror sufficiently extended, you can get arbitrarily close to 50% (semispherical surface) onto a much smaller conjugate figure near the paraboloid's focal point. With a flat mirror enclosing the paraboloid, you get the equivalent of a source being two semispheres. A surface near the conjugate figure can be quite small and intercept almost all of the EM radiation form that source, except for the longest wavelength portion of the spectrum.

cannot be disproved by example. I can't prove a negative. I've gone as far as drawing a diagram and providing a source explaining and supporting why I think it is in error. That's all I can do.

 

 

As far as the Stefan–Boltzmann law. There is, perhaps, a very relevant loophole that we have failed to consider. The law is given by the integration of Plank's law in which [math]c[/math] and [math]\lambda[/math] depend on the medium. Assuming the medium is a vacuum this gives:

[math]j^{\star} = \sigma T^4[/math]

 

Stefan–Boltzmann law - Wikipedia, the free encyclopedia

for a perfect black body. But, where the medium has a refractive index >1 (or where the refractive index is considered) it gives:

[math]j^{\star} = n^2 \sigma T^4[/math]

 

Perry's chemical engineers' handbook - Google Books

A black body of a given temperature will therefore have a higher total emissive power if it is immersed in a medium with refractive index > 1 (as, for example, in water).

 

Now then, I wouldn't want to generalize this to the case of a parabolic mirror or lens, but it certainly is a worthwhile addendum to what I said previously,

If you propose that all of the sun's radiated energy is caught by a smaller object (a premise I think breaks the second law) then the smaller object must emit a similar amount of power. The temperature of the two bodies would then be:

[math]T_{sun} = \left( \frac{P}{\sigma A_{sun}} \right)^{1/4}[/math]

[math]T_2 = \left( \frac{P}{\sigma A_2} \right)^{1/4}[/math]

Power being constant for both, the temperature will be larger for the body with less surface area.

This would more-properly be,

[math]T_{sun} = \left( \frac{P_{sun}}{n_{sun}^2 \sigma A_{sun}} \right)^{1/4}[/math]

[math]T_2 = \left( \frac{P_{2}}{n_{2}^2 \sigma A_2} \right)^{1/4}[/math]

If P_sun = P₂ and A_sun > A₂ this would no longer necessitate T₂ > T_sun. This would also allow the abstract I quoted to remain valid,

...a corollary to the second law is proposed for focusing systems. The states that 'no system may focus from one object, through an aperture onto a second identical object, more radiation than may be emitted by the second object through the same aperture'...

 

Focused radiation, the second law of thermodynamics and temperature measurements

 

If the two objects exist in areas of differing refractive indices then the above should remain true. But, again, I do not know if a system of mirrors and lenses would necessarily constitute an area of increased refractive index.

 

~modest

[Qfwfq]

That's fine. I've given my conclusion and the reason for it and a source saying the same,

I've given mine and I've made some points about what you quoted and:

I find unacceptable the 'you are wrong, but too unlearned for me to explain why' attitude.

That is not what I said.

 

If you don't make a bona fide effort in your manner of debate, if you reply to my point with "Prove it." and follow with an argument in which I see flaws, I can't complete your understanding of the optics in here, neither do I have the time. I'm sorry but there's no use getting upset, crack a good textbook of optics until you can follow my argument and avoid flaws in yours.

 

Your claim,

....

cannot be disproved by example. I can't prove a negative. I've gone as far as drawing a diagram and providing a source explaining and supporting why I think it is in error. That's all I can do.

If that's all you can do, make no more effort. Of course my argument cannot be disproved by example but we're not talking about black or white swans here, it's a very analytic matter and the thermodynamic statements are cast in analytic terms. You can't prove a negative by example but you can in analytic matters. I sure would be interested in seeing exactly how these folks infer the negative about focussing the flux from Liouville and compare it with my own arguments. It seems to me your quotes are considering the flat lense approximation with its limitations and then generalizing; it also strikes me their argument falls short of the optical reversibility principle.

 

As far as the Stefan–Boltzmann law. There is, perhaps, a very relevant loophole that we have failed to consider.

The point about refractive index is relevant indeed, now that you bring it in! As far as I can see offhand, on one account it would lend some good support to my opinion, but there's also a matter I will need to make sure of in order to clear up an apparent paradox.

 

This would also allow the abstract I quoted to remain valid,

...a corollary to the second law is proposed for focusing systems. The states that 'no system may focus from one object, through an aperture onto a second identical object, more radiation than may be emitted by the second object through the same aperture'...

 

If the two objects exist in areas of differing refractive indices then the above should remain true. But, again, I do not know if a system of mirrors and lenses would necessarily constitute an area of increased refractive index.

It proposes a corollary, I proposed something different and I became even more convinced as I thought it over through the evening. Over the weekend I found a bit of time to look into these matters and I'm hoping to go further into them sometime presently, in the meantime I can't say much more.

[modest]

If you don't make a bona fide effort in your manner of debate, if you reply to my point with "Prove it." and follow with an argument in which I see flaws, I can't complete your understanding of the optics in here, neither do I have the time...

 

If that's all you can do, make no more effort. Of course my argument cannot be disproved by example but we're not talking about black or white swans here, it's a very analytic matter and the thermodynamic statements are cast in analytic terms. You can't prove a negative by example but you can in analytic matters.

 

I do apologize if asking you to prove or support your assertion came across as offensive or inconvenient. I was honestly interested in any demonstration, source, link, or reference you could give to prove what you had said especially after I had failed to accomplish it myself after some effort and found a source explicitly disagreeing. I really was not meaning to be confrontational, but asked out of curiosity.

 

I sure would be interested in seeing exactly how these folks infer the negative about focussing the flux from Liouville and compare it with my own arguments. It seems to me your quotes are considering the flat lense approximation with its limitations and then generalizing; it also strikes me their argument falls short of the optical reversibility principle.

 

Nonimaging Optics by Roland Winston gives on page 5 the theoretical maximum thermodynamic concentration, [math]C_{max}[/math], for a 3D passive optic flux concentrator where input and output media both have a refractive index of unity, the source is circular and Lambertian and subtends a semiangle [math]\theta_1[/math] as,

[math]C_{max} = 1/sin^2\theta_1[/math]

Where [math]C[/math] is the ratio of illumination on the entrance and exit apertures of the concentrator.

[math]C = \frac{L_2 sin^2 \theta_2}{L_1 sin^2 \theta_1}[/math]

where L is flux per unit solid angle per unit projected area.

 

If such a concentrator were parabolic and near the sun similar to the thought experiment you propose [math]\theta_1[/math] would be [math]\pi / 2[/math] and flux could be concentrated,

[math]C_{max} = 1/sin^2(\pi / 2) = 1[/math]

the luminosity at the entrance aperture of the concentrator would therefore be equal to the luminosity at the exit aperture. If the concentrator were at 1 au the maximum concentration would be,

[math]C_{max} = 1/sin^2(0.00465) \approx 46,000[/math]

Which is the amount that flux is lessened by the inverse square distance law from the sun to the earth (the surface area of 1 au is about 46,000 times greater than the surface area of the sun). It should be evident that the maximum concentration does not allow for a target to receive a flux beyond what the surface of the source gives. (r/d)² is the factor by which flux is smaller by distance where r is the radius of the radiating body and d is the distance to the point. Via trig [math]r/d = sin^2 \theta[/math] making the max concentration allowable without increasing flux beyond the source [math]1 / sin^2 \theta[/math].

 

The point about refractive index is relevant indeed, now that you bring it in!

 

The above discussion assumes that the refractive index is unity at both entrance and exit aperture for the light concentrator. As is discussed in Thermodynamic limits of light concentrators, it is possible:

If the concentrator is made of a medium of refractive index, n, and the exit plane is immersed in this medium as well, then it is necessary to modify the concentrator equation. The edge ray [math]\theta_1[/math] will be refracted to [math]\theta^\prime_1[/math] in the concentrator, where [math]sin \theta_1 = n sin \theta^\prime_1[/math] from Snell’s law. For a concentrator with the exit aperture immersed in the medium, with [math]\theta_2[/math] unchanged or unrefracted, the concentration is characterized by

[math]C \geq \frac{sin^2 \theta_2}{sin^2 \theta^\prime_1} = \frac{n^2 sin^2 \theta_2}{sin^2 \theta_1}[/math]

For convenience, the concentration is defined by the maximum incident external angel [math]\theta_1[/math], and the final exit angle [math]\theta_2[/math]. Comparing this to eq. (2), one concludes that [math]n^2L_1 \geq L_2[/math] for a passive system. This means that upon crossing into a medium of higher index of refraction, the radiation is confined to a smaller solid angle, and thus will have a higher radiance.

In this case I agree with you that the target can receive a flux per area beyond that of the source (by a factor of n²) and for reasons I gave in my last post the temperature of the second body as given by S&B's law including the refractive index ([math]j^{\star} = n^2 \sigma T^4[/math]) would not pass that of the first .

 

~modest

[Qfwfq]

OK sorry if I found it annoying it must have been a combination of things. :)

 

I've been reckoning on how the Liouville theorem is relevant and, even without actually doing the math but by nexus with other things, I see what the argument would be; it ain't trivial. My considerations about a [imath]\lambda[/imath] value such that, for a given temperature, a given fraction of the energy would all be at shorter waves, would seem to be a kind of loophole but I've got to think it over better, there's a somewhat counter-intuitive matter to understand and I'll try to sort it out. I also want to see some details of the black body topics in an old textbook I have, before saying more about the bearing of refractive index at the source.

 

If such a concentrator were parabolic and near the sun similar to the thought experiment you propose

Actually in my example the source is far from the parabolid's focal point and the further the better. The further it is along the axis, the nearer the conjugate figure is to the focal point and also smaller, with the limitations imposed by wavelength. For a given [imath]\lambda[/imath], however, the ratio can be made what you like if you can go as far as you like. Extending the paraboloid past the source and closing with a mirror allows well over have the surface to be used. I need to think better about whether my considerations are actually a loophole, there's a freaky aspect, meantime I've gotta quit for today.

[modest]

Actually in my example the source is far from the parabolid's focal point and the further the better.

I was referring to having the open end of the concentrator (the entrance aperture) close to the sun.

The further it is along the axis, the nearer the conjugate figure is to the focal point and also smaller, with the limitations imposed by wavelength. For a given [imath]\lambda[/imath], however, the ratio can be made what you like if you can go as far as you like.

That does not help you increase the flux/area of the image. I tried to explain before, as the ratio of focal distance to source distance (F/D) goes to zero, the angle of incidence of P₁ ([math]\theta[/math]) goes to zero.

 

 

The tighter you make the parabola and thus the shorter the focal distance, the smaller you make the solid angle subtended by the target from the sun. You would do better with a compound parabolic concentrator which my last post remarks on.

 

~modest

[Qfwfq]

Friday I found time to go to my old physics dep't library and I've had a good look at that book about non-imaging optics. The use of Liouville's theorem in optics is described quite well in appendix A and is pretty much as I imagined, though to the purpose of optics it considers the phase space corresponding to a 2D cross sectional surface. I was supposing the spectral distribution was being considered but instead it is applied for a given frequency. This leaves a few things for me to figure out yet because there might be a tacit assumption in this use of the theorem.

 

As I expected, those authors are for the most considering a flat entrance surface and the source quite far from it with a quite simply defined ètendue. This does not suit the example I was considering, which requires a more complicated analysis, but there is stuff in chapter 6 which might be more relevant. I have had a few ideas for refinement but it would take a while to work things out and I'm needing to shake a few more cobwebs out of these topics, so I won't say much else for now.

 

In this case I agree with you that the target can receive a flux per area beyond that of the source (by a factor of n²) and for reasons I gave in my last post the temperature of the second body as given by S&B's law including the refractive index ([math]j^{\star} = n^2 \sigma T^4[/math]) would not pass that of the first .

As I said, I needed to think more about the bearing of the refractive index, to clear up an apparent paradox, offhand I couldn't see any fundamental reason why the flux outside the medium would be limited to that emitted by a BB in the vacuum. It did however strike me odd that nobody would have thought of it in more than a century! When I got around to it, thinking of total reflection in combination with flux isotropy, it isn't difficult to work out that the flux outside is what's appropriate for the external medium. Total reflection shaves off exactly the excess for a perfect BB and reduces in the same fraction for a grey body.

 

The reason you are recalling attention to seems to make an assumption about the refractive index for which there are no grounds, it doesn't solve the problem if [imath]n[/imath] is the same for both. If hypothetically it were possible to have a flux greater than that for a BB with [imath]\epsilon=1[/imath] at the same temperature and in the same medium, the conjecture I proposed would become necessary to match things up with the 2nd principle; there would be some sticky matters, but perhaps only definitional. I still haven't had time to review the necessary topics and unfortunately I never did much of stat mech.

 

Apart from these very very hypothetical and ludic amenities, I think you should understand that the Stefan-Boltzmann formula mustn't be taken to mean that the flux determines temperature. It is enough to see the distinction in the more realistic case of decreasing the flux from a BB (or from one having [imath]\epsilon<1[/imath]), it doesn't mean the temperature of the radiation is lower, it just means less thermal power flux density.

 

I tried to explain before, as the ratio of focal distance to source distance (F/D) goes to zero, the angle of incidence of P₁ ([math]\theta[/math]) goes to zero.

 

The tighter you make the parabola and thus the shorter the focal distance, the smaller you make the solid angle subtended by the target from the sun. You would do better with a compound parabolic concentrator which my last post remarks on.

I get the impression you still don't quite follow my point, I think it's better if I just try to work it out with analytical geometry and with the further ideas I've been thinking of, if I can find enough time, It would be interesting to get it worked out fully, even though I doubt any practical use being possible.

[Pyrotex]

.... Actually in my example the source is far from the parabolid's focal point and the further the better. The further it is along the axis, the nearer the conjugate figure is to the focal point and also smaller, with the limitations imposed by wavelength. ...

If we take a lens and put a bright object on one side (LED or candle flame), and then vary the focal length of the lens, we find that the image created by the lens gets larger and larger with increasing focal length.

 

Switching to a paraboloid mirror creates much the same thing. For a fixed distance between object and mirror, the focused image will be small if the focal length is small and large if the focal length is large.

 

Does this help?

[Qfwfq]

Does this help?

Already considered. :phones: