Abstract

The optical thermodynamic efficiency is a comprehensive metric that takes into account all loss mechanisms associated with transferring flux from the source to the target phase space, which may include losses due to inadequate design, non-ideal materials, fabrication errors, and less than maximal concentration. We discuss consequences of Fermat’s principle of geometrical optics and review étendue dilution and optical loss mechanisms associated with nonimaging concentrators. We develop an expression for the optical thermodynamic efficiency which combines the first and second laws of thermodynamics. As such, this metric is a gold standard for evaluating the performance of nonimaging concentrators. We provide examples illustrating the use of this new metric for concentrating photovoltaic systems for solar power applications, and in particular show how skewness mismatch limits the attainable optical thermodynamic efficiency.

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References

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  1. W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
    [CrossRef]
  2. P. Benitez, and J. C. Miñano, “Concentrator Optics for the next generation photovoltaics,” Chap. 13 of A. Marti and A. Luque, Next Generation Photovoltaics: High Efficiency through Full Spectrum Utilization, Taylor & Francis, CRC Press, London (2004).
  3. V. I. Arnold, Mathematical Methods of Classical Mechanics, 88–91 & 161–270, Springer Verlag (1989).
  4. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics, Elsevier Academic Press, New York (2005).
  5. H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14(10), 2855–2862 (1997).
    [CrossRef]
  6. J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001).
    [CrossRef]
  7. L. D. Landau, and E. M. Lifshitz, Statistical Physics, Pergamon, London (1958).
  8. E. Yablonovitch, “Thermodynamics of the fluorescent planar concentrator,” J. Opt. Soc. Am. 70(11), 1362–1363 (1980).
    [CrossRef]

2001

J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001).
[CrossRef]

1998

W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
[CrossRef]

1997

1980

Bortz, J.

J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001).
[CrossRef]

H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14(10), 2855–2862 (1997).
[CrossRef]

Muschaweck, J.

W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
[CrossRef]

Ries, H.

W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
[CrossRef]

H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14(10), 2855–2862 (1997).
[CrossRef]

Shatz, N.

J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001).
[CrossRef]

H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14(10), 2855–2862 (1997).
[CrossRef]

Spirkl, W.

W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
[CrossRef]

H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14(10), 2855–2862 (1997).
[CrossRef]

Winston, R.

J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001).
[CrossRef]

W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
[CrossRef]

Yablonovitch, E.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001).
[CrossRef]

Sol. Energy

W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998).
[CrossRef]

Other

P. Benitez, and J. C. Miñano, “Concentrator Optics for the next generation photovoltaics,” Chap. 13 of A. Marti and A. Luque, Next Generation Photovoltaics: High Efficiency through Full Spectrum Utilization, Taylor & Francis, CRC Press, London (2004).

V. I. Arnold, Mathematical Methods of Classical Mechanics, 88–91 & 161–270, Springer Verlag (1989).

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics, Elsevier Academic Press, New York (2005).

L. D. Landau, and E. M. Lifshitz, Statistical Physics, Pergamon, London (1958).

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Figures (6)

Fig. 1
Fig. 1

Refractive solar concentrator with 40-mm-diameter entrance aperture and 6-mm-square solar cell.

Fig. 2
Fig. 2

Flux-transfer efficiency as a function of the source’s angular half width for refractive concentrator (solid line) and hypothetical ideal concentrator (dashed line).

Fig. 3
Fig. 3

Boundaries of direction-cosine regions corresponding to source (red line) and target (blue line) for second example. The unit circle is depicted as a dashed black line. The daily operation time is T = 6 hr and the maximum allowed incidence angle on the target is θ trg = 60°.

Fig. 4
Fig. 4

Source and target translational skewness distributions for the second example. The source has unit area and the target étendue equals that of the source. The daily operation time is T = 6 hr and the maximum allowed incidence angle on the target is θ trg = 60°.

Fig. 5
Fig. 5

Source and target translational skewness distributions for the second example. The source has unit area and the target area has been adjusted to the smallest value that allows the target’s skewness distribution to completely enclose the source’s skewness distribution. The daily operation time is T = 6 hr and the maximum allowed incidence angle on the target is θ trg = 60°.

Fig. 6
Fig. 6

Upper limit on optical thermodynamic efficiency as a function of daily operation time for the second example. The maximum allowed incidence angle on the target is θ trg = 60°.

Equations (28)

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C A P C g sin ( α ) ,
C A P i d e a l = n t r g n s r c sin ( θ t r g m a x ) ,
S = k log E + Thermal term,
η s y s t e m * = η o p t i c s * η c e l l s * ,
L s r c a v e = Φ s r c ε s r c .
Φ t r g m a x = L s r c a v e ε t r g .
Φ t r g m a x = Φ s r c ε t r g ε s r c .
η o p t i c s * Φ t r g Φ s r c ,  for  ε t r g ε s r c .
η o p t i c s * Φ t r g Φ t r g m a x ,  for  ε s r c < ε t r g .
η o p t i c s * = Φ t r g Φ s r c ε s r c ε t r g ,  for  ε s r c < ε t r g .
η o p t i c s * = η o p t i c s min ( 1 , ε s r c ε t r g ) ,
η o p t i c s Φ t r g Φ s r c
ε s r c = π A s r c sin 2 ( θ s r c ) = 29.99 mm 2 sr
ε t r g = π A t r g = 113.10 mm 2 sr,
η o p t i c s = 60.48 % .
η o p t i c s * = 16.04 % .
| k y | sin ( θ y )
| k z | sin [ θ z ( T ) ] 1 k y 2 ,
θ y = θ e a r t h + θ s u n
θ z ( T ) = T 12 hr 90 ° + θ s u n ,
k y 2 + k z 2 sin 2 ( θ t r g ) ,
ε s r c ( T ) = 2 A s r c sin [ θ z ( T ) ] [ sin ( θ y ) cos ( θ y ) + θ y ] ,
ε t r g = π A t r g sin 2 ( θ t r g ) ,
d ε s r c ( S z ) d S z = [ 2 A s r c sin ( θ y ) ,    for  | S z | sin [ θ z ( T ) ] cos ( θ y ) 2 A s r c 1 S z 2 sin 2 [ θ z ( T ) ] ,      for  sin [ θ z ( T ) ] cos ( θ y ) < | S z | sin [ θ z ( T ) ] 0 ,      for  sin [ θ z ( T ) ] < | S z | ,
d ε t r g ( S z ) d S z = [ 2 A t r g sin 2 ( θ t r g ) S z 2 ,    for  | S z | sin ( θ t r g ) 0 ,    for  sin ( θ t r g ) < | S z | .
d ε s r c d S z ( sin [ θ z ( T ) ] cos ( θ y ) ) = d ε t r g d S z ( sin [ θ z ( T ) ] cos ( θ y ) ) ,
A t r g , r e q = A s r c sin ( θ y ) sin 2 ( θ t r g ) sin 2 [ θ z ( T ) ] cos 2 ( θ y ) ,
T θ t r g θ s u n 90 ° 12 hr .

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