Abstract

Closed-form analytic solutions that describe the optical behavior of V-trough concentrators are derived. The procedure analyzes the angular and spatial distribution of all the rays that undergo a given number of reflections (0, 1, 2), referred to as reflection modes. Then we obtain (1) the optical efficiency for beam and diffuse radiation, and (2) spatial and angular flux distribution on the absorber, with proper account being taken of reflective losses (in addition to the geometric losses of rejected rays). These are often essential data for evaluation of V-trough solar energy collectors, in particular for photovoltaic applications where homogeneous flux distributions are desirable.

© 1992 Optical Society of America

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References

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  1. E. C. Boes, “A summary of recent photovoltaic concentrator technology developments,” in Proceedings of the 20th IEEE Photovoltaic Specialists Conference, Las Vegas, Nev. (Institute of Electrical and Electronic Engineers, New York, 1988), pp. 21–30.
  2. N. Fraidenraich, H. S. Costa, A. N. S. Fraga, “Photovoltaic water pumping with compound elastic concentrators,” in Proceedings of the Second Ibero-American Solar Energy Congress (de Oliviera Fernandes, Porto, Portugal), Suppl., 15–20.
  3. K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
    [CrossRef]
  4. N. Fraidenraich, A. N. S. Fraga, “Numerical simulation of photovoltaic systems with one-axis tracking coupled to V-trough cavities,” Internal Rep. (Research Group on Alternative Sources of Energy, Departamento de Energia Nuclear, Universidade Federal de Pernambuco, Recife PE, Brazil, 1990.
  5. D. E. Williamson, “Cone channel condenser optics,” J. Opt. Soc. Am. 42, 712–715 (1952).
    [CrossRef]
  6. J. Freilich, J. M. Gordon, “Case study of a central-station grid-intertie photovoltaic system with V-trough concentration,” Sol. Energy 46, 267–273 (1991).
    [CrossRef]
  7. K. D. Mannan, R. B. Bannerot, “Optimal geometries for one- and two-faced symmetric side-wall booster mirrors,” Sol. Energy 21, 385–391 (1978).
    [CrossRef]
  8. J. R. Howell, R. B. Bannerot, “Trapezoidal grooves as moderately concentrating solar energy collectors,” Progr. Astronaut. Aeronaut. 49, 277–289 (1976).
  9. D. G. Burkhard, G. L. Strobel, D. R. Burkhard, “Flat-sided rectilinear trough as a solar concentrator: an analytical study,” Appl. Opt. 17, 1870–1883 (1978).
    [CrossRef] [PubMed]
  10. N. Fraidenraich, G. J. Almeida, “Optical properties of V-trough concentrators,” Sol. Energy 47, 147–155 (1991).
    [CrossRef]
  11. A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
    [CrossRef]
  12. A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18, 497–511 (1976).
    [CrossRef]

1991 (2)

J. Freilich, J. M. Gordon, “Case study of a central-station grid-intertie photovoltaic system with V-trough concentration,” Sol. Energy 46, 267–273 (1991).
[CrossRef]

N. Fraidenraich, G. J. Almeida, “Optical properties of V-trough concentrators,” Sol. Energy 47, 147–155 (1991).
[CrossRef]

1978 (2)

D. G. Burkhard, G. L. Strobel, D. R. Burkhard, “Flat-sided rectilinear trough as a solar concentrator: an analytical study,” Appl. Opt. 17, 1870–1883 (1978).
[CrossRef] [PubMed]

K. D. Mannan, R. B. Bannerot, “Optimal geometries for one- and two-faced symmetric side-wall booster mirrors,” Sol. Energy 21, 385–391 (1978).
[CrossRef]

1976 (3)

J. R. Howell, R. B. Bannerot, “Trapezoidal grooves as moderately concentrating solar energy collectors,” Progr. Astronaut. Aeronaut. 49, 277–289 (1976).

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
[CrossRef]

A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18, 497–511 (1976).
[CrossRef]

1971 (1)

K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
[CrossRef]

1952 (1)

Almeida, G. J.

N. Fraidenraich, G. J. Almeida, “Optical properties of V-trough concentrators,” Sol. Energy 47, 147–155 (1991).
[CrossRef]

Bannerot, R. B.

K. D. Mannan, R. B. Bannerot, “Optimal geometries for one- and two-faced symmetric side-wall booster mirrors,” Sol. Energy 21, 385–391 (1978).
[CrossRef]

J. R. Howell, R. B. Bannerot, “Trapezoidal grooves as moderately concentrating solar energy collectors,” Progr. Astronaut. Aeronaut. 49, 277–289 (1976).

Boes, E. C.

E. C. Boes, “A summary of recent photovoltaic concentrator technology developments,” in Proceedings of the 20th IEEE Photovoltaic Specialists Conference, Las Vegas, Nev. (Institute of Electrical and Electronic Engineers, New York, 1988), pp. 21–30.

Burkhard, D. G.

Burkhard, D. R.

Costa, H. S.

N. Fraidenraich, H. S. Costa, A. N. S. Fraga, “Photovoltaic water pumping with compound elastic concentrators,” in Proceedings of the Second Ibero-American Solar Energy Congress (de Oliviera Fernandes, Porto, Portugal), Suppl., 15–20.

Fraga, A. N. S.

N. Fraidenraich, H. S. Costa, A. N. S. Fraga, “Photovoltaic water pumping with compound elastic concentrators,” in Proceedings of the Second Ibero-American Solar Energy Congress (de Oliviera Fernandes, Porto, Portugal), Suppl., 15–20.

N. Fraidenraich, A. N. S. Fraga, “Numerical simulation of photovoltaic systems with one-axis tracking coupled to V-trough cavities,” Internal Rep. (Research Group on Alternative Sources of Energy, Departamento de Energia Nuclear, Universidade Federal de Pernambuco, Recife PE, Brazil, 1990.

Fraidenraich, N.

N. Fraidenraich, G. J. Almeida, “Optical properties of V-trough concentrators,” Sol. Energy 47, 147–155 (1991).
[CrossRef]

N. Fraidenraich, A. N. S. Fraga, “Numerical simulation of photovoltaic systems with one-axis tracking coupled to V-trough cavities,” Internal Rep. (Research Group on Alternative Sources of Energy, Departamento de Energia Nuclear, Universidade Federal de Pernambuco, Recife PE, Brazil, 1990.

N. Fraidenraich, H. S. Costa, A. N. S. Fraga, “Photovoltaic water pumping with compound elastic concentrators,” in Proceedings of the Second Ibero-American Solar Energy Congress (de Oliviera Fernandes, Porto, Portugal), Suppl., 15–20.

Freilich, J.

J. Freilich, J. M. Gordon, “Case study of a central-station grid-intertie photovoltaic system with V-trough concentration,” Sol. Energy 46, 267–273 (1991).
[CrossRef]

Gordon, J. M.

J. Freilich, J. M. Gordon, “Case study of a central-station grid-intertie photovoltaic system with V-trough concentration,” Sol. Energy 46, 267–273 (1991).
[CrossRef]

Hollands, K. G. T.

K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
[CrossRef]

Howell, J. R.

J. R. Howell, R. B. Bannerot, “Trapezoidal grooves as moderately concentrating solar energy collectors,” Progr. Astronaut. Aeronaut. 49, 277–289 (1976).

Mannan, K. D.

K. D. Mannan, R. B. Bannerot, “Optimal geometries for one- and two-faced symmetric side-wall booster mirrors,” Sol. Energy 21, 385–391 (1978).
[CrossRef]

Rabl, A.

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
[CrossRef]

A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18, 497–511 (1976).
[CrossRef]

Strobel, G. L.

Williamson, D. E.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Progr. Astronaut. Aeronaut. (1)

J. R. Howell, R. B. Bannerot, “Trapezoidal grooves as moderately concentrating solar energy collectors,” Progr. Astronaut. Aeronaut. 49, 277–289 (1976).

Sol. Energy (6)

N. Fraidenraich, G. J. Almeida, “Optical properties of V-trough concentrators,” Sol. Energy 47, 147–155 (1991).
[CrossRef]

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
[CrossRef]

A. Rabl, “Optical and thermal properties of compound parabolic concentrators,” Sol. Energy 18, 497–511 (1976).
[CrossRef]

K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
[CrossRef]

J. Freilich, J. M. Gordon, “Case study of a central-station grid-intertie photovoltaic system with V-trough concentration,” Sol. Energy 46, 267–273 (1991).
[CrossRef]

K. D. Mannan, R. B. Bannerot, “Optimal geometries for one- and two-faced symmetric side-wall booster mirrors,” Sol. Energy 21, 385–391 (1978).
[CrossRef]

Other (3)

N. Fraidenraich, A. N. S. Fraga, “Numerical simulation of photovoltaic systems with one-axis tracking coupled to V-trough cavities,” Internal Rep. (Research Group on Alternative Sources of Energy, Departamento de Energia Nuclear, Universidade Federal de Pernambuco, Recife PE, Brazil, 1990.

E. C. Boes, “A summary of recent photovoltaic concentrator technology developments,” in Proceedings of the 20th IEEE Photovoltaic Specialists Conference, Las Vegas, Nev. (Institute of Electrical and Electronic Engineers, New York, 1988), pp. 21–30.

N. Fraidenraich, H. S. Costa, A. N. S. Fraga, “Photovoltaic water pumping with compound elastic concentrators,” in Proceedings of the Second Ibero-American Solar Energy Congress (de Oliviera Fernandes, Porto, Portugal), Suppl., 15–20.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the V-trough cavity.

Fig. 2
Fig. 2

Schematic of the V-trough angular acceptance function.

Fig. 3
Fig. 3

Illustration of characteristic angles α2, β0, and γ3.

Fig. 4
Fig. 4

Illustration of the three states (ascending, full, and descending) for the first-order reflection mode (C = 2, ψ = 7.5°).

Fig. 5
Fig. 5

Diagram showing the composition of V-trough reflection modes in terms of the three mode states and the characteristic angles (α, β, γ) for each mode (C = 2, ψ = 7.5°).

Fig. 6
Fig. 6

Schematic representation of angles α, β, and γ in the neighborhood of the Nth reflection mode.

Fig. 7
Fig. 7

Illustration of the three states of the reflection modes and their relative geometric weights (C = 2, ψ = 7.5°, and θi = 20°).

Fig. 8
Fig. 8

(a) Nonuniform illumination by an even-order (k = 2) reflection mode for the ascending state with θi = 10°. (b) Same as (a) but for an odd-order (k = 1) mode for the ascending state with θi = 3°. In both cases, C = 2 and ψ = 7.5°.

Tables (1)

Tables Icon

Table I Angular Interval δ, Angular Acceptance Function Fl), and Optical Efficiency η(θl) in the Region of Uniform Absorber Illuminationa

Equations (62)

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C 1 + 2 cos ψ .
C = aperture width/absorber width = A / a
θ a θ c ψ,
F ( θ i ) = 1 , 0 < | θ i | < θ a , F ( θ i ) = 0 , | θ i | > θ a ,
η ( θ i ) = k = 0 M P k ( θ i ) ρ k ,
F ( θ i ) = k = 0 M P k ( θ i ) .
η ( θ i ) F ( θ i ) ρ n ( θ i ) / F ( θ i ) .
n ( θ i ) = k = 0 M k P k ( θ i ) .
sin ( α k + ψ ) cos ( α k + k ψ ) = 2 ( C 1 ) sin [ ( k 1 ) ψ ] ,
β k = α k + 2 + 2 ψ,
γ k = π 2 2 k ψ .
tan α k = 2 ( C 1 ) sin [ ( k 1 ) ψ ] cos ( k ψ ) sin ψ 2 ( C 1 ) sin [ ( k 1 ) ψ ] sin ( k ψ ) + cos ψ .
tan α 0 = ( C + 1 ) ( C 1 ) tan ψ,
α 1 = ψ,
tan α 2 = tan ψ 4 sin 2 ψ + C 3 4 sin 2 ψ + C 1 .
γ N > α N , γ N + 1 < α N + 1 .
γ N < β N 1 , α N + 1 < γ N .
P k ( θ i ) = A k / A .
P k ( θ i ) = P k R ( θ i ) + P k L ( θ i ) , n 1 ,
P k R ( θ i ) = P k L ( θ i ) , n 1 .
P k ( θ i ) = P k R ( θ i ) + P k R ( θ i ) , n 1 .
P 0 ( θ i ) = P 0 R ( θ i ) , P 0 L ( θ i ) = 0 .
P k R ( α k ) = 0 , 0 k N 1 , P k R ( β k ) = 0 , 0 k N 1 ,
P N R ( α N ) = 0 , P N R ( γ N ) = 0 .
a 1 = A 1 a = cos ( θ i + 2 ψ ) cos θ i ,
a 2 = A 2 a = cos ( θ i + 4 ψ ) cos θ i .
P 1 ( θ i ) = cos ( θ i + 2 ψ ) C cos θ i , α 2 θ i β 0 ,
P 2 ( θ i ) = cos ( θ i + 4 ψ ) C cos θ i , α 3 θ i β 1 .
P k ( θ i ) = cos ( θ i + 2 k ψ ) C cos θ i , α k + 1 θ i β k 1 .
P 0 ( θ i ) = 1 C , α 1 θ i β 1 .
P 3 ( θ i ) = A 3 A = cos ( θ i + 6 ψ ) C cos θ i + ( 1 + w ) C j = 0 3 cos ( θ i + 2 j ψ ) C cos θ i , α 3 θ i α 4 ,
w = W a = ( C 1 ) sin ( θ i + ψ ) 2 cos θ i sin ψ .
P 3 ( θ i ) = 1 C + ( C 1 ) 2 C sin ( θ i + ψ ) cos θ i sin ψ + cos ( θ i + 6 ψ ) C cos θ i j = 0 3 cos ( θ i + 2 j ψ ) C cos θ i , α 3 θ i α 4 .
P k ( θ i ) = 1 C + ( C 1 ) 2 C sin ( θ i + ψ ) cos θ i sin ψ + cos ( θ i + 2 k ψ ) C cos θ i j = 0 k cos ( θ i + 2 j ψ ) C cos θ i , α k θ i α k + k + 1 .
P k ( θ i ) = ( C 1 ) 2 C sin ( α k + ψ ) cos θ i sin ψ × [ sin ( ψ + θ i ) sin ( ψ + α k ) cos ( k ψ + θ i ) cos ( k ψ + α k ) ] , α k θ i α k + 1 .
P k ( α k + 1 ) = cos ( α k + 1 + 2 k ψ ) C cos α k + 1 .
P k ( θ i ) = ( C 1 ) 2 C sin ( β k ψ ) cos θ i sin ψ × [ sin ( θ i ψ ) sin ( β k ψ ) + cos ( k ψ + θ i ) cos ( k ψ + β k ) ] , β k 1 θ i β k .
P 0 ( θ i ) = ( C 1 ) 2 C sin ( β 0 ψ ) cos θ i sin ψ × [ sin ( θ i ψ ) sin ( β 0 ψ ) + cos θ i cos β 0 ] , β 1 θ i β 0 ,
P k ( θ i ) = cos ( 2 k Ψ θ i ) C cos θ i , α k + 1 < θ i < β k 1 ,
P k ( θ i ) = ( C 1 ) 2 C sin ( α k + Ψ ) sin Ψ cos θ i [ sin ( Ψ θ i ) sin ( Ψ + α k ) cos ( k Ψ θ i ) cos ( k Ψ + α k ] ,
I a I = j = l + 1 k 1 cos ( θ i + 2 j Ψ ) cos θ i .
x k R = P k R ( θ i ) P k f R ( θ i ) 1 2 ,
P k R ( θ i ) ( k ) [ 1 S ( x x k R ) ] + P k R ( θ i ) [ 1 ( k ) ] S ( x + x k R ) ,
P l R ( θ i ) ( l ) S ( x + x l R ) + P l R ( θ i ) [ 1 ( l ) ] [ 1 S ( x x l R ) ] .
P m L ( θ i ) ( m ) S ( x + x m L ) + P m L ( θ i ) [ 1 ( m ) ] [ 1 S ( x x m L ) ] .
j = 1 m 1 cos ( 2 j Ψ θ i ) C cos θ i .
I a ( C I ) = j = 1 m 1 cos ( 2 j Ψ θ i ) C cos θ i + j = l + 1 k 1 cos ( 2 j Ψ + θ i C o s θ i + P k R ( θ i ) ( k ) [ 1 S ( x x k R ) ] + P k R ( θ i ) [ 1 ( k ) ] S ( x + x k R ) + P l R ( θ i ) ( l ) S ( x + x l R ) + P l R ( θ i ) [ 1 ( l ) ] [ 1 S ( x x l R ) ] + P m L ( θ i ) ( m ) S ( x + x m L ) + P m L ( θ i ) [ 1 ( m ) ] × [ 1 S ( x x m L ) ] ,
C 1 + 2 sin ( N Ψ ) cos [ ( N + 1 ) Ψ ] sin Ψ .
C 1 + 2 cos ( 2 Ψ ) .
tan δ = 2 ( C 1 ) sin Ψ cos 2 Ψ sin Ψ 2 ( C 1 ) sin Ψ sin 2 Ψ + cos Ψ .
F ( θ i ) = ( 1 + 2 cos 2 Ψ ) C , | θ i | δ .
γ N > α N ,
γ N + 1 < α N + 1 ,
α N + 1 < γ N < β N 1 ,
2 ( C 1 ) sin [ ( N 1 ) Ψ ] sin ( N Ψ ) cos [ ( 2 N 1 ) Ψ ] .
{ 2 ( C 1 ) sin ( N ψ ) cos [ ( N + 1 ) ψ ] sin ψ } sin ( 2 N ψ ) < { 2 ( C 1 ) sin ( N ψ ) sin [ ( N + 1 ) ψ ] cos ψ } cos ( 2 N ψ ) ,
α N + 1 < γ N .
γ N + 1 = γ N 2 ψ,
α N + 1 = β N 1 2 ψ .
γ N < β N 1 .
α N + 1 < γ N < β N 1 .
A 3 cos θ i cos ( θ i + 6 ψ ) = P 3 R ( θ i ) P 3 f R ( θ i ) .

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