Abstract

The temperature formally assigned to spectral electromagnetic radiance Lν by solving Planck’s equation for T permits the application of the Second Law of Thermodynamics to passive concentrating systems. (1) Concentrators operating within the limits of geometrical optics, (2) concentrators changing the frequency of the radiation but conserving the total radiant flux, and (3) systems in which the frequency is changed and part of the absorbed power transferred to the surroundings as heat are discussed. The attainable concentration ratios are given. Particularly for systems of category (3), perspectives are encouraging for further development and application. Such systems resemble, in certain respects, heat pumps. High concentration ratios are allowed by thermodynamics. In this category concentrators that use Stokes fluorescence are discussed.

© 1982 Optical Society of America

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References

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  1. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978), Sec. 3, pp. 11–14.
  2. R. Winston and W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
    [Crossref]
  3. W. B. Joyce, “Classical-particle description of photons and phonons,” Phys. Rev. D 9, 3234–3256 (1974).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec 4.8.3.
  5. L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1969), Secs. V.63 and VI.73.
  6. The argument given here in terms of frequency may be stated just as well in terms of wavelength, but then attention must be paid to the fact, that unlike frequency, wavelength depends on the refractive index.
  7. P. Wurfel and W. Ruppel, “Upper limit of thermophotovoltaic solar energy conversion,” IEEE Trans. Electron Devices ED-27, 745–750 (1980).
    [Crossref]
  8. R. T. Ross, “Thermodynamic limitations on the conversion of radiant energy into work,” J. Chem. Phys. 45, 1–7 (1966).
    [Crossref]
  9. A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
    [Crossref]
  10. An equivalent result was obtained by E. Yablonovitch, “Thermodynamics of the fluorescent planar collector,” J. Opt. Soc. Am. 70, 1362–1363 (1980).
    [Crossref]
  11. P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949), Sec. A.2, p. 3.
  12. S. I. Vavilov, “Photoluminescence and thermodynamics,” J. Phys. (Moscow) 10, 499 (1946).

1980 (2)

P. Wurfel and W. Ruppel, “Upper limit of thermophotovoltaic solar energy conversion,” IEEE Trans. Electron Devices ED-27, 745–750 (1980).
[Crossref]

An equivalent result was obtained by E. Yablonovitch, “Thermodynamics of the fluorescent planar collector,” J. Opt. Soc. Am. 70, 1362–1363 (1980).
[Crossref]

1979 (1)

1977 (1)

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

1974 (1)

W. B. Joyce, “Classical-particle description of photons and phonons,” Phys. Rev. D 9, 3234–3256 (1974).
[Crossref]

1966 (1)

R. T. Ross, “Thermodynamic limitations on the conversion of radiant energy into work,” J. Chem. Phys. 45, 1–7 (1966).
[Crossref]

1946 (1)

S. I. Vavilov, “Photoluminescence and thermodynamics,” J. Phys. (Moscow) 10, 499 (1946).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec 4.8.3.

Goetzberger, A.

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Greubel, W.

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

Joyce, W. B.

W. B. Joyce, “Classical-particle description of photons and phonons,” Phys. Rev. D 9, 3234–3256 (1974).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1969), Secs. V.63 and VI.73.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1969), Secs. V.63 and VI.73.

Pringsheim, P.

P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949), Sec. A.2, p. 3.

Ross, R. T.

R. T. Ross, “Thermodynamic limitations on the conversion of radiant energy into work,” J. Chem. Phys. 45, 1–7 (1966).
[Crossref]

Ruppel, W.

P. Wurfel and W. Ruppel, “Upper limit of thermophotovoltaic solar energy conversion,” IEEE Trans. Electron Devices ED-27, 745–750 (1980).
[Crossref]

Vavilov, S. I.

S. I. Vavilov, “Photoluminescence and thermodynamics,” J. Phys. (Moscow) 10, 499 (1946).

Welford, W. T.

R. Winston and W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
[Crossref]

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978), Sec. 3, pp. 11–14.

Winston, R.

R. Winston and W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
[Crossref]

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978), Sec. 3, pp. 11–14.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec 4.8.3.

Wurfel, P.

P. Wurfel and W. Ruppel, “Upper limit of thermophotovoltaic solar energy conversion,” IEEE Trans. Electron Devices ED-27, 745–750 (1980).
[Crossref]

Yablonovitch, E.

Appl. Phys. (1)

A. Goetzberger and W. Greubel, “Solar energy conversion with fluorescent collectors,” Appl. Phys. 14, 123–139 (1977).
[Crossref]

IEEE Trans. Electron Devices (1)

P. Wurfel and W. Ruppel, “Upper limit of thermophotovoltaic solar energy conversion,” IEEE Trans. Electron Devices ED-27, 745–750 (1980).
[Crossref]

J. Chem. Phys. (1)

R. T. Ross, “Thermodynamic limitations on the conversion of radiant energy into work,” J. Chem. Phys. 45, 1–7 (1966).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. (Moscow) (1)

S. I. Vavilov, “Photoluminescence and thermodynamics,” J. Phys. (Moscow) 10, 499 (1946).

Phys. Rev. D (1)

W. B. Joyce, “Classical-particle description of photons and phonons,” Phys. Rev. D 9, 3234–3256 (1974).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), Sec 4.8.3.

L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1969), Secs. V.63 and VI.73.

The argument given here in terms of frequency may be stated just as well in terms of wavelength, but then attention must be paid to the fact, that unlike frequency, wavelength depends on the refractive index.

P. Pringsheim, Fluorescence and Phosphorescence (Interscience, New York, 1949), Sec. A.2, p. 3.

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978), Sec. 3, pp. 11–14.

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Figures (2)

Fig. 1
Fig. 1

Two blackbodies P1 and P2 of equal temperature in radiation contact across concentrator C. A system of mirrors and diaphragms M restricts the radiation of P2 so that the etendues of the emitted and received beams are equal. Reradiation through a is possible.

Fig. 2
Fig. 2

Temperature of diffuse solar radiation of irradiance E = 50, 200, and 1000 W/m2. The spectral distribution of solar energy approximated by blackbody radiation of Ts = 5762 K is given in arbitrary units.

Tables (2)

Tables Icon

Table 1 Maximum Optical Concentration Ratio for Conservation with Conserved Radiant Fluxa

Tables Icon

Table 2 Maximum Optical Concentration Ratio for Fluorescent Concentrator in Units of 103a

Equations (42)

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L ν = ( 2 h ν 3 / c 2 ) [ exp ( h ν / k T ) - 1 ] - 1 .
L = d 2 ϕ / d ν d U .
d U = d a n 2 cos ϑ d Ω = d a d R .
L ν = ( 2 h ν 3 / c 2 ) exp ( - h ν / k T ) .
E = L ν Δ R Δ ν .
L ν e / L ν a = ( E e / E a ) ( Δ R a / Δ R e ) ( Δ ν a / Δ ν e ) .
L ν e / L ν a = optical concentration K 0 , E e / E a = area concentration K a , Δ R a / Δ R e = directional concentration K R , Δ ν a / Δ ν e = frequency concentration K ν ,
K 0 = K a K R K ν .
K a 1 / K R .
K a = n 2 π / Δ R .
K a = n 2 2 / n 1 2 .
T ( ν , L ν ) = ( h ν / k ) [ log ( 2 h ν 3 / c 2 L ν + 1 ) ] - 1 .
T ( ν , p , L ν , p ) = ( h ν / k ) [ log ( h ν 3 / c 2 L ν , p + 1 ) ] - 1 .
T ( ν , L ν ) = ( h ν / k ) [ log ( 2 h ν 3 / c 2 L ν ) ] - 1 .
L ν = f ( 2 h ν 3 / c 2 ) [ exp ( h ν / k T s ) - 1 ] - 1 .
f = E / ( σ T s 4 ) .
f = ( R s / D s ) 2 = 2 × 10 - 15 ,
T = ( Q / S ) v .
α ( ν ) = ( ν ) = { 0 for T ( ν , L ν ) < T a 1 for T ( ν , L ν ) > T a .
d S / d t = S ˙ a + S ˙ e + S ˙ i ,
- Q ˙ e / T e Q ˙ a / T r .
T e T r .
K 0 = ( ν e / ν a ) 3 [ exp ( h ν α / k T r ) - 1 ] / [ exp ( h ν e / k T r ) - 1 ] ,
K 0 = ( ν e / ν a ) 3 exp [ h ( ν α - ν e ) / k T r ] .
d S / d t = S ˙ a + S ˙ e + S ˙ s + S ˙ i .
- Q ˙ e / T e Q ˙ a / T r + Q ˙ s / T s .
T e = - Q ˙ e ( Q ˙ a / T r + Q ˙ s / T s ) - 1 ,
K 0 = ( ν e / ν a ) 3 [ exp ( h ν a / k T r ) - 1 ] / { exp [ ( - h ν e / k Q ˙ e ) ( Q ˙ a / T r + Q ˙ s / T s ) ] - 1 } .
Q ˙ s = Q ˙ a ( ν e - ν a ) / ν a .
T e = ν e [ ν a / T r - ( ν a - ν e ) / T s ] - 1 ,
K 0 = ( ν e / ν a ) 3 [ exp ( h ν a / k T r ) - 1 ] / { exp [ ( h / k ) ( ν a / T r - ( ν a - ν e ) / T s ] - 1 } .
K 0 = ( ν e / ν a ) 3 { exp [ - h ( ν a - ν e ) / k T s ] - exp ( - h ν a / k T r ) } - 1 .
ν a ν e + phonon
ν e + phonon ν a .
S ˙ i = ( 1 / T a - 1 / T r ) Q ˙ a ,
T e T a < T r ,
T e - Q ˙ e ( Q ˙ a / T a + Q ˙ s / T s ) ,
T e ν e [ ν a / T a - ( ν a - ν e ) / T s ] - 1 .
η a = Q ˙ a / ϕ r = 1 - ϕ a / ϕ r = 1 - L ν ( ν a , T a ) / L ν ( ν a , T r ) .
K 0 = ( 1 - η a ) ( ν e / ν a ) 3 [ exp ( h ν a / k T a ) - 1 ] / × [ exp ( h ν e / k T a ) - 1 ] ,
K 0 = ( 1 - η a ) ( ν e / ν a ) 3 [ exp ( h ν a / k T a ) - 1 ] { exp [ - ( h ν e / k Q ˙ e ) ( Q ˙ a / T a + Q ˙ s / T s ) ] - 1 } ,
K 0 = ( 1 - η a ) ( ν e / ν a ) 3 [ exp ( h ν a / k T a ) - 1 ] / ( exp { ( h / k ) [ ν a / T a - ( ν a - ν e ) / T s ] } - 1 ) .