Abstract

A comparison of different correlations commonly used to describe the limb darkening effect is made. A somewhat new correlation is proposed which predicts the values to within ±1.5% of the experimental values. Using a conventional ray trace technique and assigning proper weight factors to each ray, the distribution of the local concentration ratio over a flat absorber placed in the focal plane of a cylindrical parabolic trough is also determined.

© 1985 Optical Society of America

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References

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  1. S. S. Mathur, T. C. Kandpal, R. N. Singh, Solar Concentrators, A Bibliography (Innovative Information, Inc., P.O. Box 408, Greenbelt, Md., 1982).
  2. C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithson. Inst. 4, 221 (1922).
  3. O. Kamada, “Theoretical Concentration and Attainable Temperature in Solar Furnaces,” Sol. Energy 9, 39 (1965).
    [CrossRef]
  4. R. G. Giovanelli, “The Nature of Solar Energy,” in Solar Energy, H. Messel, S. T. Butler, Eds. (Pergamon, London, 1975).
  5. P. D. Jose, “The Flux Through the Focal Spot of a Solar Furnace,” Sol. Energy 1, 19 (1957).
    [CrossRef]
  6. F. W. Lipps, M. D. Walzel, “An Analytic Evaluation of the Flux Density due to Sunlight Reflected from a Flat Mirror Having a Polygonal Boundary,” Sol. Energy 21, 113 (1978).
    [CrossRef]
  7. D. L. Evans, “On the Performance of Cylindrical Parabolic Solar Concentrators with Flat Absorbers,” Sol. Energy 19, 379 (1977).
    [CrossRef]

1978 (1)

F. W. Lipps, M. D. Walzel, “An Analytic Evaluation of the Flux Density due to Sunlight Reflected from a Flat Mirror Having a Polygonal Boundary,” Sol. Energy 21, 113 (1978).
[CrossRef]

1977 (1)

D. L. Evans, “On the Performance of Cylindrical Parabolic Solar Concentrators with Flat Absorbers,” Sol. Energy 19, 379 (1977).
[CrossRef]

1965 (1)

O. Kamada, “Theoretical Concentration and Attainable Temperature in Solar Furnaces,” Sol. Energy 9, 39 (1965).
[CrossRef]

1957 (1)

P. D. Jose, “The Flux Through the Focal Spot of a Solar Furnace,” Sol. Energy 1, 19 (1957).
[CrossRef]

1922 (1)

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithson. Inst. 4, 221 (1922).

Abbot, C. G.

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithson. Inst. 4, 221 (1922).

Aldrich, L. B.

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithson. Inst. 4, 221 (1922).

Evans, D. L.

D. L. Evans, “On the Performance of Cylindrical Parabolic Solar Concentrators with Flat Absorbers,” Sol. Energy 19, 379 (1977).
[CrossRef]

Fowle, F. E.

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithson. Inst. 4, 221 (1922).

Giovanelli, R. G.

R. G. Giovanelli, “The Nature of Solar Energy,” in Solar Energy, H. Messel, S. T. Butler, Eds. (Pergamon, London, 1975).

Jose, P. D.

P. D. Jose, “The Flux Through the Focal Spot of a Solar Furnace,” Sol. Energy 1, 19 (1957).
[CrossRef]

Kamada, O.

O. Kamada, “Theoretical Concentration and Attainable Temperature in Solar Furnaces,” Sol. Energy 9, 39 (1965).
[CrossRef]

Kandpal, T. C.

S. S. Mathur, T. C. Kandpal, R. N. Singh, Solar Concentrators, A Bibliography (Innovative Information, Inc., P.O. Box 408, Greenbelt, Md., 1982).

Lipps, F. W.

F. W. Lipps, M. D. Walzel, “An Analytic Evaluation of the Flux Density due to Sunlight Reflected from a Flat Mirror Having a Polygonal Boundary,” Sol. Energy 21, 113 (1978).
[CrossRef]

Mathur, S. S.

S. S. Mathur, T. C. Kandpal, R. N. Singh, Solar Concentrators, A Bibliography (Innovative Information, Inc., P.O. Box 408, Greenbelt, Md., 1982).

Singh, R. N.

S. S. Mathur, T. C. Kandpal, R. N. Singh, Solar Concentrators, A Bibliography (Innovative Information, Inc., P.O. Box 408, Greenbelt, Md., 1982).

Walzel, M. D.

F. W. Lipps, M. D. Walzel, “An Analytic Evaluation of the Flux Density due to Sunlight Reflected from a Flat Mirror Having a Polygonal Boundary,” Sol. Energy 21, 113 (1978).
[CrossRef]

Ann. Astrophys. Obs. Smithson. Inst. (1)

C. G. Abbot, F. E. Fowle, L. B. Aldrich, Ann. Astrophys. Obs. Smithson. Inst. 4, 221 (1922).

Sol. Energy (4)

O. Kamada, “Theoretical Concentration and Attainable Temperature in Solar Furnaces,” Sol. Energy 9, 39 (1965).
[CrossRef]

P. D. Jose, “The Flux Through the Focal Spot of a Solar Furnace,” Sol. Energy 1, 19 (1957).
[CrossRef]

F. W. Lipps, M. D. Walzel, “An Analytic Evaluation of the Flux Density due to Sunlight Reflected from a Flat Mirror Having a Polygonal Boundary,” Sol. Energy 21, 113 (1978).
[CrossRef]

D. L. Evans, “On the Performance of Cylindrical Parabolic Solar Concentrators with Flat Absorbers,” Sol. Energy 19, 379 (1977).
[CrossRef]

Other (2)

R. G. Giovanelli, “The Nature of Solar Energy,” in Solar Energy, H. Messel, S. T. Butler, Eds. (Pergamon, London, 1975).

S. S. Mathur, T. C. Kandpal, R. N. Singh, Solar Concentrators, A Bibliography (Innovative Information, Inc., P.O. Box 408, Greenbelt, Md., 1982).

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of parameters commonly used in the correlations for describing solar limb darkening.

Fig. 2
Fig. 2

Correlations describing the brightness distribution over the solar disk.

Fig. 3
Fig. 3

Distribution of local concentration ratio (LCR) over a flat absorber: constant intensity disk (CID) - - -, nonuniform disk (NUD) —.

Equations (7)

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I = I 0 ( 2 + 3 cos θ ) 5 ,
I = I 0 R 0 + 1.5641 R 0 2 R 2 2.5641 R 0 ,
I = I 0 cos ( m ξ 2 ) ,
cos ( m ξ 0 2 ) = 0.55 ,
I = I 0 + I 1 sin 2 ξ + I 2 sin 4 ξ + I 3 sin 6 ξ ,
I / I 0 = 1 1.313 × 10 4 sin 2 ξ + 4.125 × 10 8 sin 4 ξ 3.246 × 10 13 sin 6 ξ .
0.785 I 0 , uD = 0.625 I 0 , NUD .

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