Abstract

The limit of concentration for cylindrical concentrators whose collector rejects rays arriving at grazing angles is calculated. Cylindrical concentrators having as cross section a 2-D optimal and ideal concentrator (CPC type) with restricted exit angle are analyzed. These concentrators do not achieve, in general, the upper limit but are very close to it. It is also shown that the upper limit is achievable. Finally, two 2-D concentrators with restricted exit angle are shown for the case when the collector is bifacial and specularly reflects the rejected rays.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. This result derives from the conservation of etendue theorem. See W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978); A. Rabl, “Comparison of Solar Energy Concentrators,” Sol. Energy 18, 93 (1976); R. E. Jones, “Collection Properties of Generalized Light Concentrators,” J. Opt. Soc. Am. 67, 1594 (1977); F. Grasso, F. Musumeci, A. Triglia, “Relation Between Concentrators and Acceptance in Solar Collectors,” Sol. Energy 22, 521 (1979); R. P. Patera, “Irradiance on the Receiver of a General Optical Concentrator,” J. Opt. Soc. Am. 70, 986 (1980); E. J. Guay, “Maximally Concentrating Collectors for Solar Energy Applications,” Sol. Energy 24, 265 (1980).
    [CrossRef]
  2. J. C. Minano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” Appl. Opt. 22, 2751 (1983).
    [CrossRef] [PubMed]
  3. A more usual expression of this equation refers to the ideal concentrators with flat entry aperture that collect all the rays which form an angle lower than ϕa with the normal to the entry aperture. In this case â = πsin2ϕa (see Ref. 6).
  4. J. C. Minano, “Application of the Conservation of Etendue Theorem for 2-D Subdomains of the Phase Space in Nonimaging Concentrators,” Appl. Opt. 23, 2021 (1984).
    [CrossRef] [PubMed]
  5. J. C. Minano, A. Luque, “Limit of Concentration for Cylindrical Concentrators Under Extended Light Sources,” Appl. Opt. 22, 2437 (1983).
    [CrossRef] [PubMed]
  6. A. Rabl, R. Winston, “Ideal Concentrators for Finite Sources and Restricted Exit Angles,” Appl. Opt. 15, 2880 (1976).
    [CrossRef] [PubMed]
  7. See Ref. 2 and R. P. Patera, “Irradiance on the Receiver of a General Optical Concentrator,” J. Opt. Soc. Am. 70, 986 (1980). In this last paper it is shown that a maximum exists for each Co value of â (independently of the entry aperture shape).
    [CrossRef]
  8. A. Luque, “Theoretical Bases of Photovoltaic Concentration for Extended Light Sources,” Sol. Cells 3, 355 (1981).
    [CrossRef]
  9. A. Luque, “Bifacial Solar Cells” in Silicon Processing for PhotovoltaicsC. P. Khattak, K. V. Ravi, Eds. (North Holland, Amsterdam, 1984), in press.
  10. W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

1984

1983

1981

A. Luque, “Theoretical Bases of Photovoltaic Concentration for Extended Light Sources,” Sol. Cells 3, 355 (1981).
[CrossRef]

1980

1976

Luque, A.

J. C. Minano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” Appl. Opt. 22, 2751 (1983).
[CrossRef] [PubMed]

J. C. Minano, A. Luque, “Limit of Concentration for Cylindrical Concentrators Under Extended Light Sources,” Appl. Opt. 22, 2437 (1983).
[CrossRef] [PubMed]

A. Luque, “Theoretical Bases of Photovoltaic Concentration for Extended Light Sources,” Sol. Cells 3, 355 (1981).
[CrossRef]

A. Luque, “Bifacial Solar Cells” in Silicon Processing for PhotovoltaicsC. P. Khattak, K. V. Ravi, Eds. (North Holland, Amsterdam, 1984), in press.

Minano, J. C.

Patera, R. P.

Rabl, A.

Welford, W. T.

This result derives from the conservation of etendue theorem. See W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978); A. Rabl, “Comparison of Solar Energy Concentrators,” Sol. Energy 18, 93 (1976); R. E. Jones, “Collection Properties of Generalized Light Concentrators,” J. Opt. Soc. Am. 67, 1594 (1977); F. Grasso, F. Musumeci, A. Triglia, “Relation Between Concentrators and Acceptance in Solar Collectors,” Sol. Energy 22, 521 (1979); R. P. Patera, “Irradiance on the Receiver of a General Optical Concentrator,” J. Opt. Soc. Am. 70, 986 (1980); E. J. Guay, “Maximally Concentrating Collectors for Solar Energy Applications,” Sol. Energy 24, 265 (1980).
[CrossRef]

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

Winston, R.

A. Rabl, R. Winston, “Ideal Concentrators for Finite Sources and Restricted Exit Angles,” Appl. Opt. 15, 2880 (1976).
[CrossRef] [PubMed]

This result derives from the conservation of etendue theorem. See W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978); A. Rabl, “Comparison of Solar Energy Concentrators,” Sol. Energy 18, 93 (1976); R. E. Jones, “Collection Properties of Generalized Light Concentrators,” J. Opt. Soc. Am. 67, 1594 (1977); F. Grasso, F. Musumeci, A. Triglia, “Relation Between Concentrators and Acceptance in Solar Collectors,” Sol. Energy 22, 521 (1979); R. P. Patera, “Irradiance on the Receiver of a General Optical Concentrator,” J. Opt. Soc. Am. 70, 986 (1980); E. J. Guay, “Maximally Concentrating Collectors for Solar Energy Applications,” Sol. Energy 24, 265 (1980).
[CrossRef]

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

Appl. Opt.

J. Opt. Soc. Am.

Sol. Cells

A. Luque, “Theoretical Bases of Photovoltaic Concentration for Extended Light Sources,” Sol. Cells 3, 355 (1981).
[CrossRef]

Other

A. Luque, “Bifacial Solar Cells” in Silicon Processing for PhotovoltaicsC. P. Khattak, K. V. Ravi, Eds. (North Holland, Amsterdam, 1984), in press.

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

A more usual expression of this equation refers to the ideal concentrators with flat entry aperture that collect all the rays which form an angle lower than ϕa with the normal to the entry aperture. In this case â = πsin2ϕa (see Ref. 6).

This result derives from the conservation of etendue theorem. See W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978); A. Rabl, “Comparison of Solar Energy Concentrators,” Sol. Energy 18, 93 (1976); R. E. Jones, “Collection Properties of Generalized Light Concentrators,” J. Opt. Soc. Am. 67, 1594 (1977); F. Grasso, F. Musumeci, A. Triglia, “Relation Between Concentrators and Acceptance in Solar Collectors,” Sol. Energy 22, 521 (1979); R. P. Patera, “Irradiance on the Receiver of a General Optical Concentrator,” J. Opt. Soc. Am. 70, 986 (1980); E. J. Guay, “Maximally Concentrating Collectors for Solar Energy Applications,” Sol. Energy 24, 265 (1980).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Upper limit of degree of isotropy for cylindrical concentrators whose collector is surrounded by a medium of refractive index n = 1.5 vs the geometrical concentration when the collector accepts all the arriving rays (dashed line) and when the collector rejects the rays forming an angle with the normal to the collector surface ϕm > 70° (solid line).

Fig. 2
Fig. 2

Upper limit of geometrical concentration for concentrators whose collectors are surrounded by a medium of refractive index n = 1.5 vs the average acceptance area for a, general case, b, the collector rejects rays with ϕm > 70°; c, the concentrator is cylindrical; d, the concentrator is cylindrical and the collector rejects rays with ϕm > 70°.

Fig. 3
Fig. 3

Two-dimensional optimal and ideal concentrator (CPC type) with restricted exit angle to ϕm. The slope from A to B is a straight line.

Fig. 4
Fig. 4

Regions representing several bundles of rays defined by their optical direction cosines at the entry aperture p,q (left) and at the collector p′,q′ (right).

Fig. 5
Fig. 5

Cross section of a cylindrical concentrator which achieves the upper limit of g and Cg obtained in Sec. II.

Fig. 6
Fig. 6

Two-dimensional optimal and ideal concentrator with restricted exit angle to ϕm for a bifacial flat collector which specularly reflects the rejected rays; acceptance angle ϕa.

Fig. 7
Fig. 7

Concentrator for the same conditions as Fig. 6 when the collector is placed horizontally.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

C g π n 2 a ^ = C g m ( a ^ ) .
g π A c n 2 sin 2 ϕ m π A c n 2 = sin 2 ϕ m ,
C g sin 2 ϕ m C g m ( a ^ ) = sin 2 ϕ m n 2 π a ^ .
g m ( C g , n ) = 2 π n 2 { C g sin - 1 ( n 2 - 1 C g 2 - 1 ) 1 / 2 + n 2 × sin - 1 [ 1 n ( C g 2 - n 2 C g 2 - 1 ) 1 / 2 ] } .
g = 1 A c π n 2 x = 0 x = 1 d x p = - 1 p = 1 E 2 - D ( p ) d p ,
E 2 - D ( p ) = B ( p ) d y d q 2 A e ( 1 - p 2 ) 1 / 2 ,
E 2 - D ( p ) = B ( p ) d y d q 2 A c ( n 2 sin 2 ϕ m - p 2 ) 1 / 2 ,
g m r ( C g , n , ϕ m ) = sin 2 ϕ m g m ( C g , n sin ϕ m ) ,
n 2 g m ( C g , n ) n > 0
n 2 sin 2 ϕ m g m ( C g , n sin ϕ m ) ( n sin ϕ m ) > 0 ,
g m r ( C g , n , ϕ m ) ϕ m > 0.
C g g m r ( C g , n , ϕ m ) C g m ( a ^ ) .
C o g m r ( C o I , n , ϕ m ) C O M ( I ) .
C g = A e 2 A C ¯ = n sin ϕ a .
A e O C ¯ = 2 n sin ϕ m sin ϕ a ,

Metrics