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Roland Winston Science of Nonimaging Optics: The Thermodynamic Connection Roland Winston Schools of Natural Science and Engineering, University of California Merced Director, California Advanced Solar Technologies Institute (UC Solar) [email protected] http://ucsolar.org SinBerBEST) annual meeting for 2013. Singapore Keynote Lecture 1:30 – 2:00 January 9, 2013 Solar Energy 2 Applications 3 The Stefan Boltzmann law s T4 is on page 1 of an optics book--- Something interesting is going on ! 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. And that is why the fusion of the science of light (optics) with the science of heat (thermodynamics), is where much of the excitement is today. During a seminar I gave some ten years ago at the Raman Institute in Bangalore, the distinguished astrophysicist Venkatraman Radhakrishnan famously asked “how come geometrical optics knows the second law of thermodynamics?” This provocative question from C. V. Raman’s son serves to frame our discussion. Nonimaging Optics 5 During a seminar at the Raman Institute (Bangalore) in 2000, Prof. V. Radhakrishnan asked me: How does geometrical optics know the second law of thermodynamics? A few observations suffice to establish the connection. As is well-known, the solar spectrum fits a black body at 5670 K (almost 10,000 degrees Fahrenheit) Now a black body absorbs radiation at all wavelengths, and it follows from thermodynamics that its spectrum (the Planck spectrum1) is uniquely specified by temperature. The well-known Stefan Boltzmann law which also follows from thermodynamics relates temperature to radiated flux so that the solar surface flux is Φs ~ 58.6 W/mm2 while the measured flux at top of the earth’s atmosphere is 1.35 mW/mm2 . That the ratio, ~ 44,000, coincides with 1/sin2 Θs 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). Nonimaging Optics is the theory of thermodynamically efficient optics and as such, depends more on thermodynamics than on optics I often tell my students to learn efficient optical design, first study the theory of furnaces. Nonimaging Optics 7 Invention of Entropy (The Second Law of Thermodynamics) • Sadi Carnot had fought with Napoleon, but by 1824 was a student studying physics in Paris. In that year he wrote: • Reflections on the Motive Power of Heat and on Machines fitted to Develop that Power. • The conservation of energy (the first law of thermodynamics) had not yet been discovered, heat was considered a conserved fluid-”caloric” • So ENTROPY (the second law of thermodynamics) was discovered first. • A discovery way more significant than all of Napoleon’s conquests! Nonimaging Optics 8 Nonimaging Optics 9 Invention of the Second Law of Thermodynamics by Sadi Carnot Nonimaging Optics 10 TdS = dE + PdV is arguably the most important equation in Science If we were asked to predict what currently accepted principle would be valid 1,000 years from now, The Second Law would be a good bet (Sean Carroll) From this we can derive entropic forces F = T grad S The S-B radiation law (const. T^4) Information theory (Shannon, Gabor) Accelerated expansion of the Universe Even Gravity! And much more modestly---- The design of thermodynamically efficient optics Nonimaging Optics 11 The second law of Thermodynamics: The invention of ENTROPY at the time of Carnot (1824), folks considered heat as a well-defined function 푄(푇, 푉) for example and called it “Caloric” which was conserved but it’s not. 푑푄 is an example of a non-perfect differential, Take 푑푄 = 퐴푑푇 + 퐵푑푉 which is not a perfect differential. If it were, then 휕2푄 휕퐴 휕퐵 = = but this needs not be true. This can be extended to more than two 휕푇휕푉 휕푉 휕푇 variables. Let’s generalize 푑푄 = 푋푑푥 + 푌푑푦 + 푍푑푧 where 푋, 푌, 푍 are functions of 푥, 푦, 푧 and 푥, 푦, 푧 can be temperature, volume etc. 휕휎 휕휎 Notice, if dQ were a perfect differential, say 푑휎, then 푋 = , 푌 = , 푍 = 휕푥 휕푦 휕휎 휕푌 휕푍 휕푍 휕푋 푎푛푑 = , = need not be true(these are called integrability conditions). 휕푧 휕푧 휕푦 휕푥 휕푧 푑푄 But sometimes, there exists an integrating factor, say 휏(푥, 푦, 푧) so = 푑휎 = 휏 휕휎 휕휎 휕휎 푑푥 + 푑푦 + 푑푧. The point is, for Q, there is such a factor and it’s called 휕푥 휕푦 휕푧 temperature. Due to Carrathodory 100 years ago! Math.Ann.67, 355(1909) and 푑푄 = 푑푆,푆 is the entropy or 푇푑푆 = 푑푄. 푇 Nonimaging Optics 12 The proof is based on the impossibility of certain conditions, for example heat flows from a cold body to a warm body( free cooling) 푑푄 is 0 but I cool while you warm up. Other impossible conditions are converting heat to work with 100% efficiency and so on. (푑푄 = 0 is called adiabatic). The proof goes like this: we know 푑푄 is not a perfect differential. The adiabatic equation: 푑푄 = 0 = 푋푑푥 + 푌푑푦 + 푍푑푧, the points near (푥, 푦, 푧) cannot fill a volume, if they did, all neighboring points are accessible, they don’t just lie on a curve because 푋푑푥 + 푌푑푦 + 푍푑푧 = 0 is a tangent plane, so connecting the tangent plane that is a surface 휎 푥, 푦, 푧 푠푢푐ℎ 푡ℎ푒 휎 = 푐표푛푠푡, passes through point (푥, 푦, 푧). Then since 푑푄 = 0 confines points to the surface 푑휎 푥, 푦, 푧 = 0 푑푄 = 휏 푥, 푦, 푧 푑휎 푥, 푦, 푧 And 휏 is called 푛푡푒푔푟푎푡푛푔 푓푎푐푡표푟. For thermodynamics 휏 is the temperature 푇 and 휎 is the entropy 푆 푑푄 = 푇(푥, 푦, 푧)푑푆 make them 푑휎 푋 푌 푍 푇 = 휏 = = = 푑푥 휕푆 휕푆 휕푆 휕푥 휕푦 휕푧 The differential of the heat, 푑푄 for an infinitesimal quasi-static change when divided by the absolute 푇 is a perfect differential 푑푆 of the entropy function. Nonimaging Optics 13 Example, connect them to information by Shannon’s theorem 푆 = −푘퐵 푃푛 log 푃푛 푛 Example, consider a string of bit 0, 1 such a string can carry information, say 1,0,0,0,1,1 etc. can represent all the letters of any alphabet hence any book content etc. Suppose we know what each bit is, the 푃푛 = 1, log 푃푛 = 0 푎푛푑 푆 = 0. Perfect knowledge Suppose we are clueless- Like tossing a coin, 푃푛 = 1/2 all 푛 and 1 1 log 2 푆 = −푘 log = 푘 푁 = 푘 0.346푁 퐵 2 2 퐵 2 퐵 푛 ′ −23 푁 = 푛푢푚푏푒푟 표푓 푏푡푠, 푘퐵 = 퐵표푙푡푧푚푎푛푛 푠 푐표푛푠푡푎푛푡 ≈ 1.38 × 10 (푀퐾푆 푢푛푡푠) Nonimaging Optics 14 푑2푠 = 푑2푥 + 푑2푦 + 푑2푧 푑2푠 = 푑2푥 + 푑2푦 + 1 푑2푧 Define 푥 = 푑푥/푑푧,푦 = 푑푦/푑푧 According to Fermat Principle 훿∫ 푛 푑푠 = 0 푃2 훿 푛 푑2푥 + 푑2푦 + 1 푑푧 = 0 푃1 Define Lagrangian 퐿 = 푛 푥, 푦, 푧 푑2푥 + 푑2푦 + 1 휕퐿 푛푥 Then = = 푝푥 휕푥 푑2푥 +푑2푦 +1 휕퐿 푛푦 Then = = 푝푦 휕푦 푑2푥 +푑2푦 +1 For a system that conserves its Hamiltonian, similar to Lagrangian equation: 푑 휕퐿 휕퐿 − = 0, 푑푧 휕푥 휕푥 휕퐿 푝 = 푥 휕푥 Therefore 휕퐿 휕퐿 휕퐿 휕퐿 = 푝 , = 푝 , 푝 = , 푝 = 휕푥 푥 휕푦 푦 푥 휕푥 푦 휕푦 Nonimaging Optics 15 The Hamiltonian of the system is: 퐻 = 푝푥푥 + 푝푦푦 − 퐿 푥, 푦, 푥 , 푦 휕퐿 휕퐿 휕퐿 휕퐿 푑퐻 = 푝 푑푥 + 푑푝 푥 + 푝 푑푦 + 푑푝 푦 − 푑푥 + 푑푥 + 푑푦 + 푑푦 푥 푥 푦 푦 휕푥 휕푥 휕푦 휕푦 = −푝 푥푑푥 + 푥 푑푝푥 − 푝 푦푑푦 + 푦 푑푝푦 휕퐻 휕퐻 휕퐻 휕퐻 = −푝 푥, = −푝 푦, = 푥 , = 푦 휕푥 휕푦 휕푝푥 휕푝푦 휕푝 휕퐻 휕퐻 휕퐻 휕퐻 휕푥 푥 = − = = 휕푝푥 휕푝푥 휕푥 휕푥 휕푝푥 휕푥 Define a four vector field 푊 = 푥 , 푝 푥, 푦 , 푝 푦 then the 4 dimensional divergence of 푊 is 0. If we use W on a closed surface 휎(푥, 푃푥, 푦, 푃푦) of a four space 푣(푥, 푃푥, 푦, 푃푦), then the incremental volume along 푑푧 covered by such a surface is 푑푉 = ∫휎 푊푑푧푑휎 According to Gauss’s theorem,∫휎 푊푑휎 = ∫푣 퐷푣 푊 푑푣 = 0 , therefore the enclosed volume 푑푉 = 0.∫휎 푑푥푑푦푑푝푥푑푝푦 = 푐표푛푠푡푎푛푡 Nonimaging Optics 16 Nonimaging Optics 17 Limits to Concentration • from l max sun ~ 0.5 m we measure Tsun ~ 6000° (5670°) Without actually going to the Sun! • Then from s T4 - solar surface flux~ 58.6 W/mm2 – The solar constant ~ 1.35 mW/mm2 – The second law of thermodynamics – C max ~ 44,000 – Coincidentally, C max = 1/sin2q – This is evidence of a deep connection to optics Imaging Devices and Their Limitations • If one were to ask the proverbial “man on the street” for a suggestion of how one might attain the highest possible level of concentration of, say solar flux a plausible response would be to use a good astronomical telescope, perhaps the 200 inch telescope on Mt. Palomar, or whatever one’s favorite telescope might be. * Of course such an experiment had better remain in the realm of imagination only, since beginning astronomers are admonished never to point their telescope at the sun or risk catastrophic consequences to the instrument. • But … Nonimaging Optics 19 Concentration Limit of a Telescope • Concentration limit sin22f /4sin2q – f: rim angle of the telescope • Best concentration achieved 1/4sin2q when f = 450 • Falls short of the fundamental limit by a factor 4! – Now factors of 4 are significant in technology (and many other forms of human endeavor) Nonimaging Optics 20 Concentration Limit in 2-D Cases • Entirely similar considerations can be applied to 2-D or trough concentrators.
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