ENERGY SCIENCE ESSAY NO. 16
A FUTURE ENERGY OPTION
Copyright © Harold Aspden, 1993
This Essay is essentially the basis of a contribution presented at the 28th Intersociety Energy Conversion Engineering Conference (IECEC) in Atlanta in 1993
Radiation Concentration Panels as a Future Energy Option
This paper reports on the design feasibility of a thermodynamic panel in which a multi-layer array of convex reflecting surfaces transports radiant heat with the object of focusing the radiation in a way which elevates its temperature. This innovative concept was one of two themes discussed in paper No. 929474 entitled "Electronic Heat Engine" included in volume 4 of the Proceedings of the 1992 27th IECEC. This paper reports on the development of that concept.
The generally accepted theory of heat radiation has certain rules which govern the calculation of heat transfer, but these tend to be dominated by the second law of thermodynamics, which precludes the upward transfer of heat energy through a temperature gradient. The important technological question at issue is whether the action of mirror focusing can overcome the dictates of this law. The clear evidence to be presented in this paper is that the accepted mathematical rules for the analysis of heat radiation phenomena do lead us into such a conflict.
On this basis, there is scope for studying the feasibility of designing a multilayer panel which can concentrate heat radiation stage-by-stage, using appropriate surface grading of absorption and reflection properties and mirror shaping. The result of the study is that, in principle, the concept is technologically feasible and one should, at least tentatively, consider the prospect of building panels which, of their own accord, can develop a temperature difference between their outer surfaces.
Although this may seem absurd to a trained engineer or physicist, the price paid by society generally for turning our backs on this potential future environmental energy source is too great to justify sparing the effort needed just to be sure of the facts. This paper is a reasoned analysis of the author's perception of what is here suggested and the showing that, in design concept and analytical detail at least, a panel of 10 cm thickness could be fabricated which should elevate heat at outside winter temperatures to room heat temperature or conversely function to extract room heat in summer to provide cooling.
The thermal radiation from a black body surface at room temperature may seem insignifant but with a multilayer microcell structure any net thermal radiation transfer from one surface layer to the next that produces a temperature increment can be compounded to enhance overall temperature gain. The focusing geometry within each microcell of the structure determines the temperature gain per cell and that geometry does not depend upon the overall dimensions of the cell.
A microcell fabrication is conducive to design which reduces any degradation of performance caused by reverse thermal conduction and can involve cells from which air has been evacuated to thereby avoid convection problems.
Provided one is assured that there can be temperature gain on a per cell basis, one can be confident that technological implementation by multilayered panel fabrication is a feasible proposition.
A general summary of evidence of prior record is contained in the statement of the 'Background of the Invention' in the author's U.S. Patent Specification No. 5,101,632. Quoting from that: "In the journal Nature, 345, p. 802; 1990, there is a report sourced in the Enrico Fermi Institute in Chicago announcing that terrestrial sunlight has been concentrated, by a two-stage system including a mirror, to an intensity which exceeds that at the surface of the sun."
Now, we regard the sun as our prime heat source. The heat radiated to Earth comes with a frequency spectrum determined by the temperature at the sun's surface. As the radiation disperses on its way to Earth its intensity diminishes to levels commensurate with the surface temperature of the Earth. However, the scientists at the Enrico Fermi Institute have found that by mirror focusing they can concentrate that radiation from the sun back to intensity levels which exceed those at the solar surface.
This means that, by the magic of mirrors, heat from a radiating surface can be focused onto an absorbing surface to heat that absorbing surface to a temperature higher than that of the source.
This defies the spirit of the second law of thermodynamics, because that mirror does no work. It is a passive agent which merely diverts the flow of heat to cause it to go uphill against a temperature gradient.
Similarly, as many will know, light from a laser can be focused to produce temperatures that exceed those occurring within the laser. There is no sensible way of bringing the second law of thermodynamics to bear to deny the possibility, therefore, of fabricating a laminar sheet which, of its own accord, can get hotter on one side than the other. The only question is: "Hotter by how much?"
The question at issue is the technological feasibility of fabricating such a sheet with a laminar microstructure, including textured mirror-surfaces, blackened radiant areas on metal foil and possibly incorporating translucent material which is not too dispersive at the heat radiation frequencies involved. This is a design question within the discipline of materials science and not a question of fundamental scientific principle. In short, it should be technically possible, but is it commercially feasible?
For those who may still wonder why Establishment scientists insist on adhering to their belief in the second law of thermodynamics it may help to point out that that the endorsement of that belief is slowly being eroded. For example, very recently (Zhang and Zhang, 1992) have shown that even explicit mechanical models breaching the second law of thermodynamics can exist if there is what they term 'a non-vanishing robust momentum flow'. This causes the author to stress the point that there is 'a robust momentum flow' carried by radiant heat energy where a curved mirror sits at the control centre directing that flow.
What is intended here is to introduce and justify the concept to show that investment in the appropriate design effort is warranted.
To prove, as an academic exercise, that there is temperature gain in a cell, a mirror-in-cell configuration will be chosen, not because it is optimum from a design viewpoint, but because it is easy to summarize here the simple but rigorous analysis which avoids the computer calculation of a developed design.
Analysis of Radiation Concentration
It will be shown that, given no absorption by any intermediary substance filling the cell or depletion of heat generated, a heat radiating configuration, as governed by a concave mirror, can, in theory, set up a 15% temperature differential (temperature in Kelvin) when in thermal equilibrium.
The diagram in Fig. 1 shows the cross section of two long cavities separated by a partition having a slit at A. The upper cavity is empty and enclosed by a surface at uniform temperature, with the result, as is well known, that the radiation emerging from the cavity through the slit is blackbody radiation. The lower cavity is also empty but is bounded by a concave mirror on the side facing the slit so that in this cavity heat radiation is absorbed and emitted by the blackbody surface of the partition but reflected by the mirror. The area of the partition is that of two coplanar sections, each of length B, whereas the slit has an aperture width A.
To simplify our analysis of the radiant heat transfer within this system, the mirror is assumed to have a parabolic form with the slit at the focus of the parabola. The dimensions of the upper cavity are not relevant to the problem because the radiation through the slit from this cavity is that expected from a blackbody surface having an area equal to that of the aperture formed by the slit and a temperature equal to that within the cavity. It suffices, therefore, to consider a section of unit length and to now assume that A and B are radiating surfaces, and though we begin by deeming that both sides of the partition are at the same temperature, it is presumed the partition contains heat insulation which permits the two sides to adjust to different temperatures.
Fig 1. Microcell cavity arrangement
Virtually all heat radiation from the surface A goes to surfaces B because A is much smaller than B. Consider heat radiation from surfaces B, confining attention to that bounded by two nearby planes of the cross section and radiated from an elemental strip of section dx, where x is the distance shown. The proportion ë/ã of the total radiation from section dx of each B surface is radiated from B to A if radiation is uniformly spread over the angular field, ë being measured in radians. However, we know from the observed temperature uniformity of the solar disc that the angular distribution of heat radiation from a radiating surface has to be a cosine function, being of greatest intensity normal to that surface. Accordingly, since the analysis of radiation from B to A involves a path normal to the radiating surface, we so need to qualify ë/ã by the form factor ã/2 to obtain ë/2 as the proportion to be evaluated.
As shown below, the parabola is characterized by p+q being constant, in this case equal to B. The value of ë/2 is therefore (A/2B)cosþ. Therefore the total heat radiation to A from the two equal B areas (assuming A and B are at the same temperature) is, in relation to radiation A from A to B:
and there is imbalance in the rates of heat transfer between A and B if the above integral differs from A when integrated for all elemental strips dx.
Since we can rely on the validity of the assumption that net radiation is, by symmetry, confined to bounds set by notional non-absorbing and non-reflecting cross-sectional planes, the question concerning the validity of the second law of thermodynamics then reduces to whether the above expression equals A.
Our onward analysis concerns the geometry of a parabola having its focus at A and we need to formulate a value for þ.
The shape of the parabolic mirror is specified as being such that B/2-x2/2B is the distance p from the radiating surface to the mirror at x. The gradient or slope of the mirror at x is the differential of this distance with respect to x and so is -x/B as seen from the radiating surface. The angle þ is the angle through which the heat ray from B is reflected at x to focus onto A. Accordingly þ is double the angle between the normal to the mirror surface at x and the normal to the radiating surface. From this it follows that:
þ = 2tan-1(x/B)
The angle þ is also the angle subtended by the side of length x of a right-angled triangle formed by corresponding ray paths p and q by reflection of the mirror at x so that:
x2 = q2-p2 = (p+q)2-2p(p+q)
However, from the formula for the mirror contour:
2p = B-x2/B
we can then match x2 of these two equations to deduce that:
B2-2pB = (p+q)2-2p(p+q)
and this clearly shows that q+p is equal to B, as relied upon earlier.
Since tan(þ/2) is x/B:
dx = (B/2)sec2(þ/2)dþ
and the criterion we are examining then reduces to whether:
which, by the expression:
Upon evaluation this becomes, simply, a requirement that ã-2 is equal to 2, and, since this is not the case, by a ratio factor of 4:7, there is, in theory, a breach of the second law of thermo-dynamics, if that law is asserted where there is mirror focusing.
In principle, however, from this analysis, A should cool down relative to B until there is a 15% temperature differential in Kelvin. This applies the fourth-power Stefan-Boltzmann radiation law.
The second law of thermodynamics can therefore be disproved by theory alone. Furthermore, that temperature differential can, in principle, be harnessed in a regenerative process using input heat at room temperature.
Practical Design Considerations
In order to exploit this situation, the primary task is to devise an improved optical geometry which enhances the temperature differential of an equilibrium state under circumstances where there is less reflected heat transfer from B back to B and so better overall throughput of heat. Here one must keep in mind that we have assumed the area A to be much smaller than the area of B and this is a practical consideration in a power application.
The optimum design structure will be one for which heat energy is transferred forward from surface layer to surface layer to convey a much greater proportion of heat energy, though accepting a smaller temperature increment in each stage or cell layer of a multi-layered fabricated sheet assembly.
Computer modelling of such a design arrangement shows that the theoretical net transfer of energy can be quite substantial but it is critically dependent upon the design parameters. Very extensive analysis of this kind is needed before determining the optimum design and the data now to be presented can be taken only as an indication of the potential we can expect.
In order that the analysis should have a certain and secure foundation it was decided to run the calculations for a worst-case scenario, so far as the radiation theory is concerned.
It is known that thermal radiation from a distant surface is emitted normal to the surface, even though many think that energy is transferred by photons each following their own trajectories. Energy quanta, when emitted from a radiation surface, do not take their bearings from a prior survey of that radiating surface. Therefore, if energy really is radiated by photons those photons must set off in directions which have a whole spectrum of possibilities governed by the way the emitting atoms sit as part of an emitting surface. If, on the other hand, the emission is regulated by standing waves and wave overlap then there is a case for understanding how the radiation is generally normal to the emitting surface. From the viewpoint of our proposed mirror focusing action, the randomly directed emission is 'worst-case' and, though solar radiation as such does concentrate in the way assumed in the above analysis, being sensed by absorbers well removed in terms of wavelength, it is a prudent design precaution to assume that the 'worst-case' scenario applies in the close-range microcell technology here investigated.
A microcell-mirror configuration as shown in Fig. 2 is the one chosen as offering practical possibilities. It is reproduced four times to show different modes of heat radiation energy transfer, the angle subtended by each element of radiating heat sink surface being a measure of the heat rate emerging from that source. The integration of these various angles over the range x applicable in a particular design configuration gives a measure of the total heat radiated in that mode. The graphical indication of these angles by the side diagrams serves to show, by the area below the curve, the heat energy transfer rate involved, assuming temperature equilibrium.
Note that a concentration of heat in the absorbing cavity will occour if the integration over an appropriate range of x (depending upon the spacing D) of à plus þ is greater than þ plus î plus ë, assuming that the underside of the barrier is a blackbody radiator. Otherwise, if this latter surface is a reflecting surface, í will replace ë. It can be seen by inspection that the design having the dimensions shown, with the reflecting underside of the apertured barrier, gives a better performance and that there is certainly a significant balance indicating transfer of heat from the upper cavity to the lower cavity.
Rigorous analysis shows that a much greater heat transfer rate can be obtained if the apertured barrier is placed nearer to the upper radiating surface. By using dimensions which have the ratios: W = 2.5, r = 1.75, A = 0.5, B = 3.5 and d = 0.25 the summation of the energy radiation rate balance (angles in degrees with x at 0.05 intervals) is 2162.89 from x = 0 to x = 3.0. This uses the reflecting lower surface of the apertured barrier. With the blackbody undersurface of this barrier the summation is 1850.31.
To calculate the total heat radiation rate from the first heat sink surface over the same range of x from 0 to 3.0 one simply needs to multiply 180 by 3.0 by 20 to obtain a base reference of 10800. It follows that some 20% of the heat radiated is captured in the lower absorbing cavity, meaning that, if allowed to, the temperature of that smaller absorbing and re-radiating surface of the lower heat sink will rise, by as much as 4% on the absolute Kelvin scale, until there is equilibrium.
A design aspect that needs to be addressed in the onward development of what is proposed is that the Stefan-Boltzmann radiation constant sets a limit on blackbody radiation of 460 watts per square metre at 17o C. One may need, therefore, to devise ways of enhancing this to take the fullest advantage of this new technology.
It can, however, be concluded from such analysis that it is feasible to contemplate a multilayered panel assembled by microfabrication techniques so that the temperature could be elevated by a few degrees in each of several stages. A one cm thick panel sandwiched between supporting metal plates with five layers should suffice for normal room heating and cooling purposes. The panel assembly would comprise alternate layers of metallic film having blackened stripes on both sides interspersed between layers of translucent material having reflecting areas on both sides apart from the aperture portions. The metallic film would have contoured reflecting ribs providing the convex mirror surfaces or the mirrors could be formed by silver wire set in juxtaposition with the apertures. The emissivity or areas of the blackened surfaces would need to be graded from layer to layer to assure the progressive temperature profile with suitable spacer means locating the layers in a composite structure.
To enhance the heat transfer capacity one could use a 10 cm thick assembly with the radiating areas enhanced internally by staggered heat sink couplings and transverse multilayered components.
However, one may conclude that a practical end product in the form of a thermodynamically active panel does seem possible.
Zhang, Kechen; Zhang, Kezhao, Physical Review A, 1992, Vol. 46, pp. 4598-4605.