Abstract

A new type of convex Fresnel lens is discussed capable of concentrating solar radiation very near the ultimate concentration limit. The differential equations that describe the lens are solved to provide computed solutions, which are then checked by ray tracing techniques. The performance (efficiency and concentration) of the lens is investigated and compared with that of a flat Fresnel lens, showing that the new lens is preferable for concentrating solar radiation.

© 1979 Optical Society of America

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References

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  1. A. Rabl, Sol. Energy 18, 93 (1976).
    [CrossRef]
  2. A. B. Meinel, M. P. Meinel, Applied Solar Energy (Addison-Wesley, Reading, Mass., 1976), p. 1543.
  3. C. E. Backus et al., “Considerations for Using Solar Concentrators in Photovoltaic Systems,” in Proceedings of Twelfth IECEC (American Institute of Chemical Engineers, New York, 1977), pp. 1147–1153.
  4. D. L. Evans, L. W. Florschuetz, Sol. Energy 19, 255 (1977).
    [CrossRef]
  5. L. L. VantHull, A. F. Hilderbrant, Sol. Energy 18, 31 (1976).
    [CrossRef]
  6. H. Tabor, Sol. Energy 2, 4 (1958).
  7. H. Tabor, H. Zeimer, Sol. Energy 6, 55 (1962).
    [CrossRef]
  8. R. E. Athey, Sol. Energy 18, 143 (1976).
    [CrossRef]
  9. P. Singh, L. S. Cheema, Sol. Energy 18, 135 (1976).
    [CrossRef]
  10. M. Collares-Pereira, A. Rabl, R. Winston, Appl. Opt. 16, 2677 (1977).
    [CrossRef] [PubMed]
  11. D. T. Nelson, D. L. Evans, R. K. Bansal, Sol. Energy 17, 285 (1975).
    [CrossRef]
  12. L. J. Hastings, S. L. Allums, R. M. Cosby, “An Analytical and Experimental Evaluation of the Plano Cylindrical Fresnel Lens Solar Concentrator,” in Proceedings of the Solar Energy Conference on Sharing the Sun, Winnipeg, (Pergamon Press, New York, 1976), pp. 275–290.
  13. R. L. Pendleton, ASHRAE J. 11, 47 (1976).
  14. H. Tabor, H. ZeimerSol. Energy 6, 55 (1962).
    [CrossRef]
  15. J. D. Garrison, Proc. Soc. Photo-Opt. Instrum. Eng.114 (1977).
  16. R. Winston, J. Opt. Soc. Am. 60, 245 (1970).
    [CrossRef]
  17. R. Winston, Sol. Energy 16, 89 (1974).
    [CrossRef]
  18. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  19. E. L. Burgess, Opt. Eng. 16, 305 (1977).
    [CrossRef]
  20. R. E. Barber, Sol. Energy 20, 1 (1978).
    [CrossRef]
  21. O. E. Miller, J. H. McLeod, W. T. Sherwood, J. Opt. Soc. Am. 41, 807 (1951).
    [CrossRef]
  22. E. M. Kritchman, A. A. Friesem, G. Yekutieli, Sol. Energy 22, 119 (1979).
    [CrossRef]
  23. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 166–169.
  24. J. F. Woodman, Acrylics, Modern Plastics Encyclopedia (McGraw-Hill, New York, 1971), Vol. 14.
  25. L. G. Rainhart, W. P. Schimmel, Sol. Energy 17, 259 (1975).
    [CrossRef]

1979

E. M. Kritchman, A. A. Friesem, G. Yekutieli, Sol. Energy 22, 119 (1979).
[CrossRef]

1978

R. E. Barber, Sol. Energy 20, 1 (1978).
[CrossRef]

1977

E. L. Burgess, Opt. Eng. 16, 305 (1977).
[CrossRef]

J. D. Garrison, Proc. Soc. Photo-Opt. Instrum. Eng.114 (1977).

M. Collares-Pereira, A. Rabl, R. Winston, Appl. Opt. 16, 2677 (1977).
[CrossRef] [PubMed]

D. L. Evans, L. W. Florschuetz, Sol. Energy 19, 255 (1977).
[CrossRef]

1976

L. L. VantHull, A. F. Hilderbrant, Sol. Energy 18, 31 (1976).
[CrossRef]

A. Rabl, Sol. Energy 18, 93 (1976).
[CrossRef]

R. E. Athey, Sol. Energy 18, 143 (1976).
[CrossRef]

P. Singh, L. S. Cheema, Sol. Energy 18, 135 (1976).
[CrossRef]

R. L. Pendleton, ASHRAE J. 11, 47 (1976).

1975

D. T. Nelson, D. L. Evans, R. K. Bansal, Sol. Energy 17, 285 (1975).
[CrossRef]

L. G. Rainhart, W. P. Schimmel, Sol. Energy 17, 259 (1975).
[CrossRef]

1974

R. Winston, Sol. Energy 16, 89 (1974).
[CrossRef]

1970

1962

H. Tabor, H. ZeimerSol. Energy 6, 55 (1962).
[CrossRef]

H. Tabor, H. Zeimer, Sol. Energy 6, 55 (1962).
[CrossRef]

1958

H. Tabor, Sol. Energy 2, 4 (1958).

1951

Allums, S. L.

L. J. Hastings, S. L. Allums, R. M. Cosby, “An Analytical and Experimental Evaluation of the Plano Cylindrical Fresnel Lens Solar Concentrator,” in Proceedings of the Solar Energy Conference on Sharing the Sun, Winnipeg, (Pergamon Press, New York, 1976), pp. 275–290.

Athey, R. E.

R. E. Athey, Sol. Energy 18, 143 (1976).
[CrossRef]

Backus, C. E.

C. E. Backus et al., “Considerations for Using Solar Concentrators in Photovoltaic Systems,” in Proceedings of Twelfth IECEC (American Institute of Chemical Engineers, New York, 1977), pp. 1147–1153.

Bansal, R. K.

D. T. Nelson, D. L. Evans, R. K. Bansal, Sol. Energy 17, 285 (1975).
[CrossRef]

Barber, R. E.

R. E. Barber, Sol. Energy 20, 1 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 166–169.

Burgess, E. L.

E. L. Burgess, Opt. Eng. 16, 305 (1977).
[CrossRef]

Cheema, L. S.

P. Singh, L. S. Cheema, Sol. Energy 18, 135 (1976).
[CrossRef]

Collares-Pereira, M.

Cosby, R. M.

L. J. Hastings, S. L. Allums, R. M. Cosby, “An Analytical and Experimental Evaluation of the Plano Cylindrical Fresnel Lens Solar Concentrator,” in Proceedings of the Solar Energy Conference on Sharing the Sun, Winnipeg, (Pergamon Press, New York, 1976), pp. 275–290.

Evans, D. L.

D. L. Evans, L. W. Florschuetz, Sol. Energy 19, 255 (1977).
[CrossRef]

D. T. Nelson, D. L. Evans, R. K. Bansal, Sol. Energy 17, 285 (1975).
[CrossRef]

Florschuetz, L. W.

D. L. Evans, L. W. Florschuetz, Sol. Energy 19, 255 (1977).
[CrossRef]

Friesem, A. A.

E. M. Kritchman, A. A. Friesem, G. Yekutieli, Sol. Energy 22, 119 (1979).
[CrossRef]

Garrison, J. D.

J. D. Garrison, Proc. Soc. Photo-Opt. Instrum. Eng.114 (1977).

Hastings, L. J.

L. J. Hastings, S. L. Allums, R. M. Cosby, “An Analytical and Experimental Evaluation of the Plano Cylindrical Fresnel Lens Solar Concentrator,” in Proceedings of the Solar Energy Conference on Sharing the Sun, Winnipeg, (Pergamon Press, New York, 1976), pp. 275–290.

Hilderbrant, A. F.

L. L. VantHull, A. F. Hilderbrant, Sol. Energy 18, 31 (1976).
[CrossRef]

Kritchman, E. M.

E. M. Kritchman, A. A. Friesem, G. Yekutieli, Sol. Energy 22, 119 (1979).
[CrossRef]

McLeod, J. H.

Meinel, A. B.

A. B. Meinel, M. P. Meinel, Applied Solar Energy (Addison-Wesley, Reading, Mass., 1976), p. 1543.

Meinel, M. P.

A. B. Meinel, M. P. Meinel, Applied Solar Energy (Addison-Wesley, Reading, Mass., 1976), p. 1543.

Miller, O. E.

Nelson, D. T.

D. T. Nelson, D. L. Evans, R. K. Bansal, Sol. Energy 17, 285 (1975).
[CrossRef]

Pendleton, R. L.

R. L. Pendleton, ASHRAE J. 11, 47 (1976).

Rabl, A.

Rainhart, L. G.

L. G. Rainhart, W. P. Schimmel, Sol. Energy 17, 259 (1975).
[CrossRef]

Schimmel, W. P.

L. G. Rainhart, W. P. Schimmel, Sol. Energy 17, 259 (1975).
[CrossRef]

Sherwood, W. T.

Singh, P.

P. Singh, L. S. Cheema, Sol. Energy 18, 135 (1976).
[CrossRef]

Tabor, H.

H. Tabor, H. Zeimer, Sol. Energy 6, 55 (1962).
[CrossRef]

H. Tabor, H. ZeimerSol. Energy 6, 55 (1962).
[CrossRef]

H. Tabor, Sol. Energy 2, 4 (1958).

VantHull, L. L.

L. L. VantHull, A. F. Hilderbrant, Sol. Energy 18, 31 (1976).
[CrossRef]

Welford, W. T.

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

Winston, R.

M. Collares-Pereira, A. Rabl, R. Winston, Appl. Opt. 16, 2677 (1977).
[CrossRef] [PubMed]

R. Winston, Sol. Energy 16, 89 (1974).
[CrossRef]

R. Winston, J. Opt. Soc. Am. 60, 245 (1970).
[CrossRef]

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 166–169.

Woodman, J. F.

J. F. Woodman, Acrylics, Modern Plastics Encyclopedia (McGraw-Hill, New York, 1971), Vol. 14.

Yekutieli, G.

E. M. Kritchman, A. A. Friesem, G. Yekutieli, Sol. Energy 22, 119 (1979).
[CrossRef]

Zeimer, H.

H. Tabor, H. Zeimer, Sol. Energy 6, 55 (1962).
[CrossRef]

H. Tabor, H. ZeimerSol. Energy 6, 55 (1962).
[CrossRef]

Appl. Opt.

ASHRAE J.

R. L. Pendleton, ASHRAE J. 11, 47 (1976).

J. Opt. Soc. Am.

Opt. Eng.

E. L. Burgess, Opt. Eng. 16, 305 (1977).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

J. D. Garrison, Proc. Soc. Photo-Opt. Instrum. Eng.114 (1977).

Sol. Energy

R. Winston, Sol. Energy 16, 89 (1974).
[CrossRef]

L. G. Rainhart, W. P. Schimmel, Sol. Energy 17, 259 (1975).
[CrossRef]

R. E. Barber, Sol. Energy 20, 1 (1978).
[CrossRef]

E. M. Kritchman, A. A. Friesem, G. Yekutieli, Sol. Energy 22, 119 (1979).
[CrossRef]

H. Tabor, H. ZeimerSol. Energy 6, 55 (1962).
[CrossRef]

A. Rabl, Sol. Energy 18, 93 (1976).
[CrossRef]

D. L. Evans, L. W. Florschuetz, Sol. Energy 19, 255 (1977).
[CrossRef]

L. L. VantHull, A. F. Hilderbrant, Sol. Energy 18, 31 (1976).
[CrossRef]

H. Tabor, Sol. Energy 2, 4 (1958).

H. Tabor, H. Zeimer, Sol. Energy 6, 55 (1962).
[CrossRef]

R. E. Athey, Sol. Energy 18, 143 (1976).
[CrossRef]

P. Singh, L. S. Cheema, Sol. Energy 18, 135 (1976).
[CrossRef]

D. T. Nelson, D. L. Evans, R. K. Bansal, Sol. Energy 17, 285 (1975).
[CrossRef]

Other

L. J. Hastings, S. L. Allums, R. M. Cosby, “An Analytical and Experimental Evaluation of the Plano Cylindrical Fresnel Lens Solar Concentrator,” in Proceedings of the Solar Energy Conference on Sharing the Sun, Winnipeg, (Pergamon Press, New York, 1976), pp. 275–290.

A. B. Meinel, M. P. Meinel, Applied Solar Energy (Addison-Wesley, Reading, Mass., 1976), p. 1543.

C. E. Backus et al., “Considerations for Using Solar Concentrators in Photovoltaic Systems,” in Proceedings of Twelfth IECEC (American Institute of Chemical Engineers, New York, 1977), pp. 1147–1153.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 166–169.

J. F. Woodman, Acrylics, Modern Plastics Encyclopedia (McGraw-Hill, New York, 1971), Vol. 14.

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

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Figures (12)

Fig. 1
Fig. 1

Concentrating optics with entrance aperture A and exit (absorbing) aperture a.

Fig. 2
Fig. 2

Artist’s view of a linear concentrating system based on a concave Fresnel lens.

Fig. 3
Fig. 3

Thin convex Fresnel lens. One of the grooves is magnified.

Fig. 4
Fig. 4

Shape y(x), primary angle ϕ(x), and secondary angle ψ(x) of thin Fresnel lenses for various θ0.

Fig. 5
Fig. 5

y(x), ϕ(x), and ψ(x) of thin Fresnel lenses with different indices of refraction n.

Fig. 6
Fig. 6

Efficiency of (a) whole and (b) half lenses of various θ0. On the bottom appear the losses for a lens of θ0 = 30°.

Fig. 7
Fig. 7

Efficiency and loss of the aplanatic and flat lenses as a function of f/No.

Fig. 8
Fig. 8

Efficiency and loss of the aplanatic and flat lenses as a function of the incident beam direction θ.

Fig. 9
Fig. 9

Tracing rays near the collector of the half-aplanatic lens. Input collimated monochromatic beams are of three colors.

Fig. 10
Fig. 10

Tracing rays near the collector of an aspherical flat lens of f/No. equal to that of the half-aplanatic lens.

Fig. 11
Fig. 11

Aplanatic design for various values of the index of refraction.

Fig. 12
Fig. 12

Ray trace in the aplanatic lens with infinite index of refraction.

Tables (2)

Tables Icon

Table I f/No. and Concentration C of Lenses with Different Indices of Refraction n

Tables Icon

Table II f/No. and Concentration C of Lenses Designed for Different Acceptance Angles ±θ0; n = 1.49

Equations (39)

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

C ( θ 0 ) 1 / sin θ 0 ,
C ( θ 0 ) A / a .
[ ϕ ( x ) is included in η θ 0 ( x ) ]
- ( F - w ) · tan θ 0 - w 2 · tan β θ 0 = ( x - Δ x ) - ( y - Δ y ) · tan η θ 0 ( x ) ,
β θ 0 = arcsin ( sin θ 0 n ) .
Δ x = w · sin [ α ( x ) - β ( x ) ] cos β ( x ) ,
Δ y = w · cos [ α ( x ) - β ( x ) ] cos β ( x ) ,
α ( x ) = - arctan d y ( x ) d x
β ( x ) = { arcsin { sin [ α ( x ) - θ 0 ] n }             α - θ 0 π / 2 arcsin ( 1 n )             α - θ 0 > π / 2.
η θ 0 ( x )
η θ 0 ( x ) = arcsin { n sin [ ϕ ( x - Δ x ) - β ( x ) ] } + α ( x ) - ϕ ( x - Δ x ) .
y ( 0 ) = F .
ψ ( x - Δ x 0 ) = ½ [ β 0 ( x ) + α ( x ) - η 0 ( x - Δ x 0 ) + π ] ,
- F · tan θ 0 = x - y ( x ) · tan η θ 0 ( x ) ,
η θ 0 ( x ) = arcsin { n sin [ ϕ ( x ) - β ( x ) ] } + α ( x ) - ϕ ( x ) .
- F · tan θ 0 = - x + y ( x ) · tan η - θ 0 ( x ) .
2 F · tan θ 0 = y ( x ) · [ tan η θ 0 ( x ) - tan - θ 0 ( x ) ] 2 x = y ( x ) · tan η θ 0 ( x ) + tan - θ 0 ( x ) ] } .
F = y ( x ) tan η θ 0 ( x ) θ 0 | θ 0 = 0 .
tan η θ 0 ( x ) θ 0 | θ 0 = 0 = cos [ ϕ ( x ) - β 0 ( x ) ] · cos α ( x ) cos 2 η 0 ( x ) · { 1 - n 2 · sin 2 [ ϕ ( x ) - β 0 ( x ) ] } 1 / 2 [ 1 - [ sin α ( x ) / n ] 2 } 1 / 2 .
η - θ 0 ( x ) θ 0 0 η θ 0 ( x ) ,
x = y ( x ) tan η 0 ( x ) .
f / No . max = cot η max 2 = 0.45.
F x = 1 sin [ η 0 ( x ) ] ,
y 2 ( x ) = F 2 - x 2 .
η 0 ( x ) = arcsin [ n · ϕ ( x ) - sin α ( x ) ] + α ( x ) ,
η 0 ( x ) = α ( x ) .
n · ϕ ( x ) = sin α ( x ) = x / F .
tan η θ 0 ( x ) θ 0 | θ 0 = 0 = cos α ( x ) cos 2 η 0 ( x ) · { 1 - [ n · ϕ ( x ) - sin α ( x ) ] 2 } 1 / 2 .
F = y ( x ) / cos α ( x ) ,
1 ( X ) = 1 1 ( X ) + 1 2 ( X ) + 1 3 ( X ) ,
1 1 ( X ) = F - y ( X ) .
w ( X ) = x = F X ϕ ( x ) d s = F X ϕ ( x ) d s ( x ) d x d x ,
s ( x ) = F · [ π 2 - α ( x ) ] ,
d s ( x ) d x = - F · d α ( x ) d x .
d α ( x ) d x = 1 F · cos α ( x ) = F F ( F 2 - x 2 ) 1 / 2 = 1 ( F 2 - x 2 ) 1 / 2 .
w ( X ) = F X x n F · ( - F ) 1 ( F 2 - x 2 ) 1 / 2 d x = ( F 2 - X 2 ) 1 / 2 n .
1 2 ( X ) = ( F 2 - X 2 ) 1 / 2 = y ( X ) .
1 3 ( X ) = F .
1 ( X ) = F - y ( X ) + y ( X ) + F = 2 F ,

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