Light Management

Thermodynamically efficient solar concentrators

[+] Author Affiliations
Roland Winston

University of California Merced, School of Natural Science, 5200 North Lake Road, Merced, California 95343

J. Photon. Energy. 2(1), 025501 (Apr 27, 2012). doi:10.1117/1.JPE.2.025501
History: Received September 15, 2011; Revised November 17, 2011; Accepted December 1, 2011
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Abstract.  Nonimaging Optics is the theory of thermodynamically efficient optics and as such depends more on thermodynamics than on optics. Hence, I propose a condition for the best design based on purely thermodynamic arguments, which I believe has profound consequences for design of thermal and even photovoltaic systems. This new way of looking at the problem of efficient concentration depends on probabilities, the ingredients of entropy, and information theory while optics in the conventional sense recedes into the background. The paper takes a pedagogical and retrospective approach, as to place the new approach in perspective, it helps to first appreciate what has succeeded in nonimaging optics, and turned some devices [e.g. the compound parabolic concentrator (CPC) cone] into a commodity. I will conclude with some speculative directions on where the new ideas might lead.

Figures in this Article

Nonimaging optics is the theory of thermodynamically efficient optics and as such depends more on thermodynamics than on optics. It is by now a key feature of most concentrating photovoltaic (CPV) designs. What is the best efficiency possible? When we pose this question, we are stepping outside the bounds of a particular subject. Questions of this kind are more properly the province of thermodynamics, which imposes limits on the possible (like energy conservation) and the impossible (like transferring heat from a cold body to a warm body without doing work). Therefore, the fusion of the science of light (optics) with the science of heat (thermodynamics), is the source of much excitement today.

When I first confronted the problem of maximal concentration from extended sources1 I turned to the tools of Hamiltonian mechanics because classical geometrical optics was concerned with point sources.2,3 In this paper I cite a spectacular failure of point source optics in analyzing a well-known paradox. I decided it would be useful to illustrate this explicitly, rather than relying on an earlier reference because I recently discovered that an advanced audience was stumped by this example even after having a week to think about it.

As the field of nonimaging optics developed, practicioners gradually realized that the second law of thermodynamics was the guiding hand behind the various designs. If asked to predict what currently accepted principle would be valid 1000 years from now,4 the Second Law would be a good bet. From this Law, we can derive entropic forces F=TS, the Stefan-Boltzmann radiation law (const. T4). information theory (Shannon, Gabor), accelerated expansion of the universe, even gravity!5,6

This communication intends to show how nonimaging optics can similarly be derived. As a result, “optics” recedes into the background, and we are left with abstract probabilities, the ingredients of entropy and information theory. I will conclude with some illustrations where these ideas are already enhancing the efficiency of solar cells. This paper is organized as follows: sections two through four provide a brief review of nonimaging optics, with the emphasis on its connection to thermodynamics, as well as set the stage for the discussion of the best design, which is addressed in section five. Section six concludes with some speculative directions of where the new ideas might lead.

An example of imaging optics’ failure to confront real-world situations is shown in Fig. 1. The point object A is at the center of a spherical reflecting cavity and also one focus of an elliptical reflecting cavity. The point object B is at the other focus. If we start A and B at the same temperature, the probability of radiation from B reaching A is clearly higher than A reaching B. So we conclude that A warms up while B cools off, in violation of the second law of thermodynamics. This paradox can be resolved by making A and B extended objects, no matter how small. In fact, a physical object with temperature has many degrees of freedom and cannot be point-like. Then the correct cavity is not elliptical, but a nonimaging shape that ensures efficient equal radiation transfer between A and B.7 It is worth mentioning that the correct nonimaging design does not converge to the ellipse/sphere configuration in the limit that the size of A and B tend to zero.

Graphic Jump LocationF1 :

The ellipse paradox: the ellipse images point object B (right) at point object A (left) perfectly and the sphere images A on itself perfectly.

A few observations suffice to establish the connection. As is well-known, the solar spectrum fits a black body at 5670 K. The well-known Stefan Boltzmann law,1 which also follows from thermodynamics, relates temperature to radiated flux so that the solar surface flux is Φs58.6W/mm2 while the measured flux at top of the earth’s atmosphere is 0.135mW/mm2. The fact that this ratio, 44,000, equals 1/sin2θs where θs is the angular subtense of the sun is not a coincidence, but rather illustrates a deep connection between the two subjects (the sine law of concentration).8

Strings

Hoyt Hottel, an MIT engineer working on the theory of furnaces,9 showed a convenient method for calculating radiation transfer between walls in a furnace using strings. We now recognize this was much more than a shortcut to a tedious calculation, but instead formed the basis of an elegant algorithm for thermodynamically efficient optical design. A few examples will suffice.

Figure 2(a) is a 3-wall enclosure extruded perpendicular to the plane. Similarly, Fig. 2(b) is a 4-wall enclosure. The walls are taken to be black bodies at the same temperature T. Introduce quantities Pij as the probability that radiation originating from surface i reaches surface j. Notice that the Pij are bounded by 0 and 1. In this discussion we assume Pii=0 (convex or flat surfaces). The generalization to concave surfaces (Pii0) is straightforward. In a 3-wall enclosure there are clearly six unknown Pij, the number of ways of selecting two objects from three. Three equations follow from conservation of energy (or probability) P12+P13=1, etc. The other three from the equality of radiation transfer between any pair of walls required by the second law of thermodynamics, A1P12=A2P21, etc. Here, A is the area per unit length out of the plane. The result follows from solving these trivial equations: P12=A1+A2A3/2A1, etc. They can almost be written down by inspection, since F12 is symmetric in one and two but not all three. In any case, we have found these probabilities without a tedious calculation. The radiation transfer between walls in a four-wall enclosure [Fig. 2(b)] follow by dividing the enclosure into two three-wall enclosures and noting that P14=1P12P13, etc. The result is: P14=[(A5+A6)(A2+A3)]/2A1, which again could have almost been written down by inspection. The mnemonic, well-known by radiation transfer engineers, is: connect the four corners by strings, then subtract the sum of the short string lengths from the sum of the long string lengths normalized by twice the area/length and you have your answer. The Hottel strings are justly famous! However, Hottel never imagined that his strings would become design tools for nonimaging concentrators.

Graphic Jump LocationF2 :

(a) 3-wall enclosure. (b) 4-wall enclosure with strings decomposed into two 3-wall enclosures.

We can use the strings to derive the limit to concentration.11 Suppose the radiation source (e.g. the sun) is very far away. Then the difference between long and short string lengths is A1sin(θ) Then the fraction of radiation P14 from A1 (considered as a black body) back to the source A4 is sin(θ) or sin2(θ) for rotationally symmetrical systems. The smallest area receiver A3 that can receive this radiation without exceeding the temperature of the source A1 is given by A3=A4sin(θ) This means the largest concentration is A4/A3=1/sin(θ). For rotationally symmetrical systems this becomes 1/sin2θ. This will be explored in greater detail below.

One example is the use of Hottel strings, usually used to calculate radiation exchange between walls in a furnace to instead design efficient concentrating systems. These strings can actually be used to design the concentrator, as shown in Fig. 3. A familiar image of a "CPC cone" is illustrated in Fig  4.

Graphic Jump LocationF3 :

The string method design of a compound parabolic concentrator (CPC). Notice that in a refractive medium, the string length becomes the optical path length. This provides a nice bridge between imaging optics (Fermat’s principle) and nonimaging optics.

Graphic Jump LocationF4 :

Cover of a recent Edmunds Scientific Catalog with a compound parabolic concentrator (CPC) cone.

In defining the best design, one criterion should certainly be: the best design maximizes the temperature of the absorber (T3), while maintaining the source at T1 (a heat reservoir, like the sun). Introducing probabilities Pij where P13 is the probability of radiation from wall 1 reaching wall 3, and so on (Fig  5). The condition is P31=1. In case P33 is not 0 (as in a cavity receiver), the generalization is P31=1P33 which still maximizes P33. These conditions are intuitively self-evident. Here we sketch a proof: If P31<1, then some energy, say P3x is exchanged with a body x which is, in general at a lower temperature that the source 1. Therefore at equilibrium, T3 will reach a temperature intermediate between T1 and Tx. Stated the other way, if T3 reaches T1 at equilibrium, then all the energy exchange must be between A1 and A3 since the other objects will in general be at a lower temperature.Additionally, for an efficient concentrating system (P12=P13) the concentration value is 1/P21. In the case of photovoltaic systems, the condition states that light starting from the solar cell would invariably reach the source. The application of this concept should lead to the development of more thermodynamically efficient thermal designs as well as higher-efficiency solar cells.

Graphic Jump LocationF5 :

A schematic of the general concentrator problem.

These new results could have profound consequences for design of thermal, and even photovoltaic, systems. In the case of photovoltaic systems,the condition derived in section 5 states that light starting from the solar cell would invariably reach the source. In a way, this is ultimate light-trapping. Any escaping light would be collimated to within the angular subtense of the sun. The application of this concept should lead to the development of more thermodynamically efficient designs. A hint of things to come recently appeared in “Limiting acceptance angle to maximize efficiency in solar cells.”11 This very interesting paper showed that limiting the acceptance angle on solar cells (using nano-CPC cones) increases efficiency. This is an entropic effect that increases the open circuit voltage.12 Clearly, limiting the angle is equivalent to maximizing P31.

Etendue Matching Screens

We need not rely on conventional concentrator optics to achieve large values of P31. One can use a sheet of back-to-back nano CPC cones that limit the transmitted light to a predetermined angular cone; these could hypothetically approach the angle subtense of the source (the sun). The Atwater group has fabricated such cones.13 It is likely that other techniques for angularly selective coatings will be found, for example using plasmonics.

* These are variously called view factors or shape factors in the engineering literature and denoted by Fij. I prefer to call them probabilities connoting the connection with entropy.

† This interesting possibility was pointed out to me by Sayatani Ghosh.

I am grateful to SPIE for inviting the newsroom article “Thermodynamics Illuminates Solar Optics”, July 6, 2011 where many of these ideas were formulated, and to the California Community Fund for supporting the Nonimaging Optics Lab at UC Merced. Support from the DARPA PoP program and MicroLink Devices Inc. is acknowledged with appreciation.

Winston  R., “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am.. 60, (2 ), 245 –247 (2010),  0030-3941 CrossRef.
Luneburg  R. K., Mathematical Theory of Optics. , pp. 82 –128,  Univ. of California Press ,  Berkeley and Los Angeles  (1964).
Born  M., Wolf  E., Principles of Optics. , 7th ed., pp. 142 –227,  Cambridge University Press ,  Cambridge, United Kingdom  (1999).
Carroll  S., From Eternity to Here. ,  Dutton (Penguin Group) ,  New York, NY  (2009).
Smoot  G. F., “Go with the flow, average holographic universe,” Int. J. Mod. Phys.. D19, , 2247 –2258 (2010).http://arxiv.org/abs/1003.5952v1
Verlinde  E., “On the origins of gravity and the laws of Newton,” JHEP. , 04, , 29  (2011),  1126-6708 CrossRef.
Welford  W. T., Winston  R., “The ellipsoid paradox in thermodynamics,” J. Stat. Phys.. 28, (3 ), 603 –606 (1982),  0022-4715 CrossRef.
Welford  W. T., Winston  R., The Optics of Nonimaging Concentrators. , 1 –3,  Academic Press ,  New York  (1978).
Hottel  H. C., “Radiant Heat Transmission,” Chapter 4, in Heat Transmission. , McAdams  W. H., Ed.,  McGraw-Hill ,  New York  (1954).
Winston  R., Minano  J. C., Benitez  P., with contributions by Narkis Shatz and John C. Bortz, Nonimaging Optics. ,  Elsevier ,  New York  (2005).
Kosten  E. D., Atwater  H. A., “Limiting acceptance angle to maximize efficiency in solar cells,”  SPIE Optics and Photonics Paper [8124-14] ,  San Diego , August 21–22 (2011).
This was emphasized by Eli Yablonovitch in a UC Solar Seminar at UC Merced,  Merced, CA  (December 9, 2011),http://www.ucsolar.org/2011-solar-symposium, see also Owen D. Miller, Eli Yablonovitch, Sarah R. Kurtz, Intense Internal and External Fluorescence as Solar Cells Approach the Shockley-Queisser Efficiency Limit, arXiv:1106.1603v3.
Atwater  J. H. et al., “Microphotonic parabolic light directors fabricated by two-photon lithography” Appl. Phys. Lett.. 99, , 151113  (2011), CrossRef. 0003-6951 

Biography and photograph of the author are not available.

© 2012 Society of Photo-Optical Instrumentation Engineers

Citation

Roland Winston
"Thermodynamically efficient solar concentrators", J. Photon. Energy. 2(1), 025501 (Apr 27, 2012). ; http://dx.doi.org/10.1117/1.JPE.2.025501


Figures

Graphic Jump LocationF5 :

A schematic of the general concentrator problem.

Graphic Jump LocationF4 :

Cover of a recent Edmunds Scientific Catalog with a compound parabolic concentrator (CPC) cone.

Graphic Jump LocationF3 :

The string method design of a compound parabolic concentrator (CPC). Notice that in a refractive medium, the string length becomes the optical path length. This provides a nice bridge between imaging optics (Fermat’s principle) and nonimaging optics.

Graphic Jump LocationF2 :

(a) 3-wall enclosure. (b) 4-wall enclosure with strings decomposed into two 3-wall enclosures.

Graphic Jump LocationF1 :

The ellipse paradox: the ellipse images point object B (right) at point object A (left) perfectly and the sphere images A on itself perfectly.

Tables

References

Winston  R., “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am.. 60, (2 ), 245 –247 (2010),  0030-3941 CrossRef.
Luneburg  R. K., Mathematical Theory of Optics. , pp. 82 –128,  Univ. of California Press ,  Berkeley and Los Angeles  (1964).
Born  M., Wolf  E., Principles of Optics. , 7th ed., pp. 142 –227,  Cambridge University Press ,  Cambridge, United Kingdom  (1999).
Carroll  S., From Eternity to Here. ,  Dutton (Penguin Group) ,  New York, NY  (2009).
Smoot  G. F., “Go with the flow, average holographic universe,” Int. J. Mod. Phys.. D19, , 2247 –2258 (2010).http://arxiv.org/abs/1003.5952v1
Verlinde  E., “On the origins of gravity and the laws of Newton,” JHEP. , 04, , 29  (2011),  1126-6708 CrossRef.
Welford  W. T., Winston  R., “The ellipsoid paradox in thermodynamics,” J. Stat. Phys.. 28, (3 ), 603 –606 (1982),  0022-4715 CrossRef.
Welford  W. T., Winston  R., The Optics of Nonimaging Concentrators. , 1 –3,  Academic Press ,  New York  (1978).
Hottel  H. C., “Radiant Heat Transmission,” Chapter 4, in Heat Transmission. , McAdams  W. H., Ed.,  McGraw-Hill ,  New York  (1954).
Winston  R., Minano  J. C., Benitez  P., with contributions by Narkis Shatz and John C. Bortz, Nonimaging Optics. ,  Elsevier ,  New York  (2005).
Kosten  E. D., Atwater  H. A., “Limiting acceptance angle to maximize efficiency in solar cells,”  SPIE Optics and Photonics Paper [8124-14] ,  San Diego , August 21–22 (2011).
This was emphasized by Eli Yablonovitch in a UC Solar Seminar at UC Merced,  Merced, CA  (December 9, 2011),http://www.ucsolar.org/2011-solar-symposium, see also Owen D. Miller, Eli Yablonovitch, Sarah R. Kurtz, Intense Internal and External Fluorescence as Solar Cells Approach the Shockley-Queisser Efficiency Limit, arXiv:1106.1603v3.
Atwater  J. H. et al., “Microphotonic parabolic light directors fabricated by two-photon lithography” Appl. Phys. Lett.. 99, , 151113  (2011), CrossRef. 0003-6951 

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