What is the hottest you can make a focused beam of sunlight?

[kmarinas86]

Which is the general geometric reason that the net energy flux along the ray will never be from cold to hot.

No, this is incorrect. They must be equal in number because they are the same rays, just opposite directions. It is simply not possible to draw a ray that goes from large to small that does not also go from small to large.

I obviously didn't mean 1 ray for 1 ray.

Even though I said, "Are these rays are going to be equal in power?" what I really meant was, "Are light rays, of quantities corresponding to the areas in question, traveling such paths as those two hypothetical light rays, going to match in power?"

That assumes that all of the power emitted from the larger goes to the smaller. Tracing a few rays should convince you that is not the case. Some of the rays leaving the larger trace back to the larger.

Of course, one could, if having the time, devise an alternate geometry where 50% of rays of the larger disc hit the smaller one, and where 50% of the rays of the smaller disc hit the larger one.

A simpler geometry would be a two mirror "periscope" like arrangement at 45 angle to each disc, and replacing each disc with a simple rectangular shape object. The percentages could be modified by simply changing the distance from the mirrors until the percentages match.

An oscillator occurs whenever you have a second derivative of a quantity which is proportional to the quantity with a negative constant of proportionality. In radiative heat transfer I don't know of any second derivatives of temperature.

It can be found though in the literature:
https://www.google.com/search?q="second+derivative+of+temperature+*+time"

Regarding shape issues, you should read into the concept of "shape factor" or "view factor": http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node137.html

Unfortunately, that model assumes that each body has blackbody spectrum. Anything with enough absorption lines or emission lines can deviate signficantly from such a spectrum.

It does indicate however that if two blackbody surfaces are unequal in area but have equal temperature, then the shape factor of each should be reciprocal to their respective area, which does correspond to the point just raised by DaleSpam.

 

[Dale]

Even though I said, "Are these rays are going to be equal in power?" what I really meant was, "Are light rays, of quantities corresponding to the areas in question, traveling such paths as those two hypothetical light rays, going to match in power?"

It is not just the areas that need to match, but also the angles.

Of course, one could, if having the time, devise an alternate geometry where 50% of rays of the larger disc hit the smaller one, and where 50% of the rays of the smaller disc hit the larger one.

I doubt it. Just claiming that it can be done doesn't make it so.

A simpler geometry would be a two mirror "periscope" like arrangement at 45 angle to each disc, and replacing each disc with a simple rectangular shape object. The percentages could be modified by simply changing the distance from the mirrors until the percentages match.

Increasing the distance from the mirrors will reduce both the number of rays emitted as well as the number of rays received. It will change the rate of energy transfer, but not the balance.

It can be found though in the literature:
https://www.google.com/search?q="second+derivative+of+temperature+*+time"

Interesting, thanks. Are any of those proportional to temperature with a negative constant of proportionality? If so, then you do indeed have a thermal pendulum.

 

[kmarinas86]

It is not just the areas that need to match, but also the angles.

I doubt it. Just claiming that it can be done doesn't make it so.

Increasing the distance from the mirrors will reduce both the number of rays emitted as well as the number of rays received. It will change the rate of energy transfer, but not the balance.

Here we see:
http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node137.html

19.4 Radiation Heat Transfer Between Black Surfaces of Arbitrary Geometry

In general, for any two objects in space, a given object 1 radiates to object 2, and to other places as well, as shown in Figure 19.10.

...

For body 1, we know that $ E_b$ is the emissive power of a black body, so the energy leaving body 1 is $ E_{b1} A_1$ . The energy leaving body 1 and arriving (and being absorbed) at body 2 is $ E_{b1} A_1 F_{1-2}$ . The energy leaving body 2 and being absorbed at body 1 is $ E_{b2} A_2 F_{2-1}$ .

This doesn't seem very general to me.

http://www.lprl.org/resources/Kirchhoffs_law.pdf

On the validity of Kirchhoff’s law
B. Kraabel, M. Shiffmann, P. Gravisse
Laboratoire de Physique et du Rayonnement de la Lumière
Abstract
Kirchhoff’s law of heat radiation is a well-know law that can, under certain conditions,
lead to a relationship of complimentarity between reflectivity r and emissivity e (e = 1 –
r). However, the details of Kirchhoff’s law and the situations in which it may be applied
are not clearly understood, as evidenced by the debate in the literature. Here we outline
the law and the main points of confusion regarding this law. We also give examples of
situations in which Kirchhoff’s law is not valid, such as paints with metallic particles,
layered optical materials, or semi-infinite bodies with a large thermal gradient at the
surface.

...

Introduction
In this document, we will discuss Kirchhoff’s law of heat radiation, and the
circumstances under which it is expected to hold, as well as those circumstances where
the law is expected not to hold.
Kirchhoff’s law of heat radiation states that the emissivity of radiating bodies in thermal
equilibrium is equal to the absorptivity. More precisely, Baltes¹ summarizes Kirchhoff’s
law as follows:
“For anybody in (radiative) thermal equilibrium with its environment, the ratio between the spectral
emissive power E(n,T) and the spectral absorptivity a(n,T) for a given frequency n and temperature
T is equal to the spectral emissive power EBB(n,T) of the blackbody for the same frequency and
temperature.”

...

This distinction comes into play if one intends to use Kirchhoff’s law to derive the
directional emissivity from the bidirectional reflectance distribution function (BRDF).
Nicodemus² shows that Equation 5 holds due to the Helmhotz reciprocity law and Synder
et al.³ show that the Helmhotz reciprocity holds only for materials that are invariant under
time reversal. Synder et al. give a simple example of a Faraday isolator (an optical
device consisting of a Faraday rotator sandwiched between two linear, crossed polarizers)
as a system that violates time reversal. They also give examples of materials that may
potentially violate the directional for of Kirchhoff’s law, among which they cite effects
paints, such as paints with metal flakes, layered optical materials, and other materials in
which multiple reflections and polarization effects are present.

Other cases for which Kirchhoff’s law is observed to be violated include a freely
radiating body with a large thermal gradient at the surface,^4,3 or systems whose energy
states do not follow the Boltzman distribution.

¹ H.P. Baltes, “On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium
environment”, Progress in Optics XIII, ed. E. Wolf, North-Holland, 1976
² F. Nicodemus, “Directional Reflectance and Emissivity of an Opaque Surface”, Applied Optics, vol. 4, p.
767, 1965.
³ W.C.Synder, Z. Wan, and X. Li, “Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s
law”, Applied Optics, vol. 37, p. 3463, 1998.
⁴ J.W. Salisbury, A. Wald, and D. M. D’Aria, “Thermal-infrared remote sensing and Kirchhoff’s law”, J. of
Geophysical Research, vol. 99, no. B6, pp. 11,897 – 11,911, 1994.

 

[kmarinas86]

Kirchhoff’s Law of Thermal Emission: 150 Years
http://www.ptep-online.com/index_files/2009/PP-19-01.PDF

Relative to Planck’s equation itself, the solution remains valid. It does however, become strictly limited to the problem of radiation within cavities which are known to be black (i.e. made of graphite, lined with soot, etc). Universality is lost. As for the mathematical value of Planck’s formulation for the perfectly absorbing cavity, it is preserved. In describing blackbody radiation, Planck consistently invokes the presence of a perfect absorber. In his treatise [7], he repeatedly calls for a minute particle of carbon [8]. Planck views this particle as a simple catalyst, although it can be readily demonstrated that this is not the case: the carbon particle acted as a perfect absorber [12]. As a result, I have stated that Kirchhoff’s law is not universal [8, 12, 26, 27] and is restricted to the study of cavities which are either made from, or contain, perfect absorbers. Arbitrary cavity radiation is not black [12]. There can be no universal function. Planck’s equation presents a functional form which, far from being universal, is highly restricted to the emission of bodies, best represented on Earth by materials such as graphite, soot, and carbon black [8].

In closing, though 150 years have now elapsed since Kirchhoff and Stewart dueled over the proper form of the law of thermal emission [11, 12], little progress has been made in bringing closure to this issue. Experimentalists continue to unknowingly pump black radiation into arbitrary cavities using their detectors. Theorists replicate the approach with Monte Carlo simulations. At the same time, astrophysicists apply with impunity the laws of thermal emission [1–7] to the stars and the universe. Little pause is given relative to the formulation of these laws [1–7] using condensed matter. The fact that all of electromagnetics stands in firm opposition to the universality, instilled in Kirchhoff’s law, is easily dismissed as science unrelated to thermal emission [61, 62]. Losses in electromagnetics are usually thermal in origin. Nonetheless, electromagnetics is treated almost as an unrelated discipline. This occurs despite the reality that Kirchhoff himself specifically included other processes, such as fluorescence, provided enclosures were maintained. Though the generalization of Kirchhoff’s law is widely recognized as valid [52–55], its application to the microwave cavity has been strangely omitted [52], even though it is used in treating the waveguide. This is the case, even though waveguides and cavities are often treated in the same chapters in texts on electromagnetics. All too frequently, the simple equivalence between apparent spectral absorbance and emission is viewed as a full statement of Kirchhoff’s law [57, 65], adding further confusion to the problem. Kirchhoff’s law must always be regarded as extending much beyond this equivalence. It states that the radiation within all true cavities made from arbitrary walls is black [1, 2]. The law of equivalence [57, 65] is Stewart’s [6].

Most troubling is the realization that the physical cause of blackbody radiation remains as elusive today as in the days of Kirchhoff. Physicists speak of mathematics, of Planck’s equation, but nowhere is the physical mechanism mentioned. Planck’s frustration remains: “Therefore to attempt to draw conclusions concerning the special properties of the particles emitting rays from the elementary vibrations in the rays of the normal spectrum would be a hopeless undertaking” [7; x111]. In 1911, Einstein echoes Planck’s inability to link thermal radiation to a physical cause: “Anyway, the h-disease looks ever more hopeless” [66; p. 228]. Though he would be able to bring a ready derivation of Planck’s theorem using his coefficients [67], Einstein would never be able to extract a proper physical link [68]. In reality, we are no closer to understanding the complexities of blackbody radiation than scientists were 150 years ago.

 

[kmarinas86]

Recycling thermal sources
http://images3.freshpatents.com/pdf/US20110203767A1.pdf

Recycling systems have been demonstrated for a variety of optical systems. Localized areas of higher photon flux are generated in these systems. Optical systems enhance brightness and power density using recycling optical cavities. In this case, non-blackbody radiators such as LEDs, phosphors, and fluorescent lamps are used within highly reflective cavities. If these sources exhibit sufficient reflectivity to the photons they emit, it is possible to generate enhanced radiance within the cavity and/or at the output aperture of the cavity relative to the source radiance these sources emit outside the recycling cavity. Enhancements of over 15× have been demonstrated in highly reflective systems such as phosphor based sources. These sources operate outside the basic assumptions and boundary conditions of equilibrium and blackbody radiators used to form conservation of optical extent theory and Kirchhoff\'s Law.

Numerous articles and papers have been written over the last 150 years pointing out experimental and theoretical sources which do not obey Kirchhoff\'s Law, especially sources which are non-blackbody radiators (Kirchhoff\'s Law of Thermal Emission: 150 Years, Pierre-Marie Robitaille, Progress in Physics October 2009, volume 4). Kraabel (On the validity of Kirchhoff\'s law, B. Kraabel, M. Shiffmann, P. Gravisse, Laboratoire de Physique et du Rayonnement de la Lumière) as well as others have demonstrated numerous situations, in which Kirchhoff\'s law cannot be used effectively, such as paints with metallic particles, layered optical materials, or semi-infinite bodies with a large thermal gradient at the surface. In general, the basic concept that cavities are always black regardless of the properties of the materials from which they are constructed has been proven invalid and the formation of blackbody cavities which even approach blackbody radiators requires specialized materials and form factors. A wide range of recycling products enhance brightness, radiance, and energy/power density which clearly operate outside present day understanding of Kirchhoff\'s Law and the conservation of optical extent theory. While, alternate interpretations can be used to try and overcome these deficiencies, the reality is that a great deal of confusion and misuse of these theories has resulted. It is reasonable to state that both these theories are only strictly valid for blackbody radiators at thermal equilibrium. It is also reasonable to state that the improper use of these theories has been used to set limits which can be overcome in the case of sources and optical systems which deviate significantly from blackbody behavior. As such an alternate theory based on Heisenberg\'s uncertainty principle has been developed.

This new theory requires only that there be a change in the uncertainty of at least one property of a photon or assemblage of photons (momentum, polarization, wavelength, position, etc.) within a given system to allow for localization of energy density within the system. This theory accurately predicts the effects measured in recycling optical cavities presently being created by Goldeneye, Inc. The use of Heisenberg uncertainty principles are already used in commercial ray tracing algorithms to accurately predict wave based effects such as edge diffraction from companies such as Lambda Research (Edge Diffraction in Monte Carlo ray tracing, Feniere, Gregory, Hasler, Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999.). In this case Heisenberg\'s Uncertainty principle is used to modify the direction of each ray based on its position as it passes in proximity to an edge. Heisenberg states that if there is a decrease in the uncertainty in position there will be a corresponding increase in momentum. In the Lambda Research\'s ray tracing software, the distance of each ray from the edge is used to modify the momentum of the ray by bending the ray towards the edge. The algorithm accurately predicts the diffraction of light at an edge, which is clearly a wave based mechanism. This application proposes that Heisenberg\'s Uncertainty principles can be used to overcome the deficiencies found in Kirchhoff\'s Law and the theory of optical extent. Because uncertainty relationships exist between all the properties of actinic radiation, this alternate theory has broad applicability. In addition Heisenberg\'s Uncertainty Principles represent the ultimate limits for actinic radiation so their use as performance boundaries for optical systems is appropriate.

One type of recycling optical cavity based on this theory is constructed using highly reflective LEDs in which the area of emission is greater than the exit aperture of the cavity. Based on the reflectivity of the LEDs and cavity and the area relationship of the emitter area and the output aperture area, it is easy to calculate the brightness/radiance gain of the cavity relative to a LED external to the cavity. It is also very easy to model this optical system using standard ray tracing techniques. If the optical path length of the rays exiting through the aperture of the cavity is tabulated and a histogram of optical path length is created, it can be shown that the brightness/radiance enhancement of the recycling optical cavity (gain) at the output aperture exactly corresponds to the average increase in optical path length. Optical path length can then be correlated to the temporal distribution of the optical rays passing through the aperture of the cavity. The corresponding Heisenberg relationship is ΔtΔE≧. In other words, as the uncertainty of when a particular ray exits the aperture of this type of system is increased (e.g. rays spend time bounces around in the cavity), then an equivalent decrease in the uncertainty that energy is present at the aperture is allowed (e.g. more photons per unit area at the exit aperture of the cavity). This increase of energy density within the cavity and at the output aperture translates into higher watts per unit area at the aperture of the cavity than is being emitted by the emitting LEDs if they were just emitting external to cavity. This is a clear violation of the optical extent theory unless an additional term is added which takes into account the temporal effects discussed earlier. Interestingly, a temporal term already exists within the optical extent theory based on the effects of refractive index. The proposed new theory simply expands refractive index term to include other temporal effects created by recycling. In the extreme, if photons are being continuously emitted from a source in a cavity which do not absorb any of the photons emitted, eventually all those photons must exit the cavity or the conservation of energy law is violated. If the cavity output aperture is small relative to the emitting area within the cavity, the density of photons per unit area at the aperture must increase to a level determined solely by the area ratio of the emitting source and aperture.

In the case of recycling optical cavities based on LEDs, the emitting sources are highly reflective to the light they emit, while the aperture represents a perfect absorber. In addition, a wide range of materials including air can be used within the cavity which does not absorb the radiation emitted by the LEDs. The limited wavelength range of operation also further enables the effectiveness of this type of recycle optical cavity.

However, based on the proposed uncertainty theory, even low level thermal sources can also be enhanced. The requirements are the same (e.g. low absorption in the emission range and a recycling means), but the wavelength range is greatly expanded which limits the materials which can be used effectively and imposes the need for a low absorption means within the cavity (e.g. vacuum or equivalent) to reduce absorption losses from the air itself. It is proposed that these losses are the limiting factor to enhancing low level thermal sources and the reason there appears to be a fundamental restriction of creating high quality thermal sources from low quality thermal sources. Based on this new theory, a large area source exhibiting non-blackbody properties can be coupled to a smaller area with much different radiative properties and the smaller area can have a higher temperature than the large area source.

From a practical standpoint, several hurdles exist. The wavelength range of low level thermal radiators extends from microns down into the microwave region. No one material exists which exhibits low absorption over this wide wavelength range. KBr and other binary inorganics are transparent from the visible region down to 10 s of microns, while organic polymers like CTFE exhibit low absorption losses from microwave up to 100 s of microns. Not only does no single material exhibits low absorption throughout the entire wavelength range of thermal radiation, there also exists a gap of low absorption materials centered within the emission spectra of most thermal sources. In addition, water vapor and even the air can strongly absorb throughout this range of wavelengths. As such there is little wonder that the perception is that thermal sources cannot be enhanced.

Recently however, Sandia Labs has demonstrated that photonic bandgap structures can be constructed which restrict the spectral range of blackbody radiators. In their work, tungsten filaments were constructed to contain photonic bandgaps which could only radiate a specific range of wavelengths. Using these structures, researchers were able to create incandescent light sources which emitted more visible light because longer wavelengths were forbidden to emit by the photonic bandgap structure itself. As stated earlier, the criteria for localization of energy within an optical system based on the new theory is simply that there be two surfaces which exhibit significantly different radiative properties and that they be connects via a low loss optical system. This invention generally discloses methods by which two surfaces which differ substantially in their radiative characteristics can be coupled via a low loss optical means to enhance the energy density of surface relative to the other. More specifically, this invention relates to the use of photonic bandgap radiators in vacuum recycling thermal cavities. In this case, a large photonic bandgap surface would be coupled to a smaller absorptive surface. The radiative nature of the photonic bandgap would be significantly different than the smaller absorptive surface. The ability of the smaller absorptive surface to radiate energy back to photonic bandgap surface will be significantly hindered by the photonic bandgap itself. To reduce absorption losses within the system, a vacuum enclosure is a preferred embodiment of this invention. This eliminates gas and water vapor absorption losses. This disclosure covers apparatus and uses of recycling systems that localize the energy density within thermal systems down to and including ambient environment and below.

 

[Dale]

Do any of these references claim to have experimental evidence that you can transfer heat from a colder body to a hotter body via radiation without doing work, or otherwise claim that the second law of thermodynamics can be violated for radiation?

 

[kmarinas86]

Do any of these references claim that you can transfer heat from a colder body to a hotter body via radiation without doing work, or otherwise claim that the second law of thermodynamics can be violated for radiation?

No, they don't make that claim, but they claim that Kirchhoff's law of radiation is violated. In other words, an object may absorb more thermal energy than it emits, and vice versa.

http://en.wikipedia.org/wiki/Talk:Kirchhoff's_law_of_thermal_radiation#Perfect_black_body

The Einstein theory of A and B coefficients shows how absorptivity must depend on the actual present state of the body, not merely on its elementary characteristics. The absorptivity depends on the occupation numbers of the possible quantum levels of the atomic or molecular species. This was not understood even in the early days of the Planck law, but was pointed out by Einstein in papers of 1916, with a near-re-publication in 1917. A translation of the 1917 paper is to be found in The Old Quantum Theory by D. ter Haar (1967). If the state of the body is such that nearly all the atoms are in upper level quantum states, there is 'negative absorption'. 'Negative absorption' means that stimulated emission exceeds primary absorption. Thermodynamic analysis, to which Kirchhoff's law refers, must consider stimulated emission as a component of absorption because it is directly proportional to the radiation density. See for example The Quantum Theory of Light, third edition, Loudon, R. (2000), Oxford University Press, Oxford UK, ISBN 0–19–850177–3.

Kirchhoff's law strictly stated applies to thermodynamic equilibrium. There are conditions away from strict thermodynamic equilibrium in which Kirchhoff's law is an extremely good approximation, and local thermodynamic equilibrium is one such condition. See for example Foundations of Radiation Hydrodynamics, Mihalas, D. and Weibel-Mihalas, B., (1984), Oxford University Press, Oxford UK, ISBN 0–19–503437–6. There are, however, conditions far from thermodynamic equilibrium in which Kirchhoff's law does not apply, for example in a laser. Such conditions were hardly considered before 1916, and even at the time of the invention of the laser, there was scepticism. It is fundamental to Kirchhoff's law that in its strict statement it refers to thermodynamic equilibrium. Discussion of departure from thermodynamic equilibrium is a development of the law, but is not primary to it.Chjoaygame (talk) 23:32, 26 December 2011 (UTC)

http://en.wikipedia.org/wiki/Thermodynamic_equilibrium

In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance. In a state of thermodynamic equilibrium, there are no net flows of matter or of energy, no phase changes, and no unbalanced potentials (or driving forces), within the system. A system that is in thermodynamic equilibrium experiences no changes when it is isolated from its surroundings.

http://en.wikipedia.org/wiki/Thermal_equilibrium

Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space.[1][2][3] When the temperatures of interest are just those in the different parts of one body, the concept also requires that any flow of heat by thermal conduction or by thermal radiation into or out of one part of the body be balanced by a flow of heat in the opposite sense into or out of another part of the body. When the temperatures of interest belong to several bodies, the concept also requires that flows of heat between each pair of bodies balance to a zero net flow, but it allows the several bodies to gain or lose heat to several external reservoirs provided that their total rate of inflow from all reservoirs is equal to their total rate of outflow to all reservoirs and that each flow is unchanging in time. For some situations, the definition of transfer of heat can be problematic.[4]

Some writers use the term thermal equilibrium in a different sense. They mean by it that the spatial temperature distribution of the body is not necessarily uniform, and indeed is likely to be non-uniform, but is maintained unvarying in time, by flows of energy; for example they mean that there is spatially distributed radiative cooling of the body and equal and opposite spatially distributed energy addition by condensation of water vapour, just so as on average to keep the spatial distribution of temperature time-invariant.

Thermal equilibrium does not mean the same as thermodynamic equilibrium, because the latter requires that there be equilibrium of all kinds, not only thermal, and that there be no flow of any kind, in the system of interest.

For entropy to be irreversible, a move toward thermal equilibrium must not lead to an increase of mechanical, radiative, and/or chemical disequilibrium of sufficient amount to undo the thermal equilibrium.

 

[Dale]

an object may absorb more thermal energy than it emits, and vice versa.

Yes, that is clear.