Abstract

Cylindrical concentrators illuminated by an extended source with an arbitrary distribution of radiance are analyzed taking into account basic properties derived from the Fermat principle and not from the specific concentrator shape. The upper limit of concentration achievable with this type of concentrator is obtained and it is found to be lower than that of general (3-D) concentrators. Cylindrical compound parabolic concentrators are analyzed in the light of this theory, and it is shown that they achieve the highest optical concentration possible for a cylindrical concentrator.

© 1983 Optical Society of America

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References

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  1. A. Luque, Sol. Cells 3, 355 (1981).
    [CrossRef]
  2. J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
    [CrossRef]
  3. J. C. Miñano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” accepted for publication in Applied Optics.
  4. W. T. Welford, R. Winston, Optics of Nonimaging Concentrators (Academic, New York, 1978), pp. 23 and 163.
  5. R. Winston, Sol. Energy 16, 89 (1974).
    [CrossRef]

1983

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

1981

A. Luque, Sol. Cells 3, 355 (1981).
[CrossRef]

1974

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

Luque, A.

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

A. Luque, Sol. Cells 3, 355 (1981).
[CrossRef]

J. C. Miñano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” accepted for publication in Applied Optics.

Miñano, J. C.

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

J. C. Miñano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” accepted for publication in Applied Optics.

Welford, W. T.

W. T. Welford, R. Winston, Optics of Nonimaging Concentrators (Academic, New York, 1978), pp. 23 and 163.

Winston, R.

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

W. T. Welford, R. Winston, Optics of Nonimaging Concentrators (Academic, New York, 1978), pp. 23 and 163.

Sol. Cells

A. Luque, Sol. Cells 3, 355 (1981).
[CrossRef]

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

Sol. Energy

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

Other

J. C. Miñano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” accepted for publication in Applied Optics.

W. T. Welford, R. Winston, Optics of Nonimaging Concentrators (Academic, New York, 1978), pp. 23 and 163.

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Figures (7)

Fig. 1
Fig. 1

Upper bound of optical concentration for an arbitrary concentrator of flat entry aperture whose intercept factor is I when the source illuminating the concentrator is the yearly averaged sky above Madrid, represented in Fig. 2. The upper bound increases by a factor of n2 when the collector is surrounded by a medium whose index of refraction is n (the cases n = 1 and n = 1.5 are represented here).

Fig. 2
Fig. 2

Yearly averaged sky radiance of Madrid. This map represents the celestial sphere. The points u = 0, υ = sinL (where L = 40.44° is the latitude) correspond to the local azimuth. The darkened region represents the rays coming from the ground. The line separating these rays from the sky rays is the local horizon. The albedo coefficient is assumed to be zero, so the darkened region has zero radiance. Coordinates: u is the cosine of the angle formed by the ray and an X axis that is in the local W–E direction, υ is the cosine of the angle formed by the ray and the Y axis which is the earth's axis. The Z axis is the intersection of the local meridian plane with the equatorial plane. (a) corresponds to the hemisphere Z positive and (b) corresponds to the other one. The trajectory of the sun during a day is, in practice, a line of υ constant. A region can be found of high radiance (|υ| ≤ 0.398) corresponding to the region of the sky where the sun can be found during the year. This region is bounded by two lines of infinite radiance which are due to the stop in the declination movement of the sun during the solstices and to the chosen model of direct radiance (see Ref. 2 for details). The sky region of |υ| > 0.398 emits only diffuse radiance which has been considered isotropic. The transformation of the u,υ,σ variables into the p,q variables used at the entry aperture of a concentrator can be found in Ref. 2. The levels of radiance are in W⋅m−2 sr−1.

Fig. 3
Fig. 3

Coordinates used at the entry aperture of an arbitrary concentrator.

Fig. 4
Fig. 4

Limits of the function F(p). The ellipse derives from the conditions at the entry aperture and the circle from the conditions at the collector: p1 = −p2.

Fig. 5
Fig. 5

Upper bound of the degree of isotropy for cylindrical concentrators when the collector is surrounded by a medium of index of refraction 1.5. This upper bound is also shown when the condition of the monofacial collector is added.

Fig. 6
Fig. 6

Upper bound of optical concentration vs intercept factor for cylindrical concentrators (of flat entry aperture) illuminated by the yearly averaged sky above Madrid COMc(I) when n = 1.5. This can be compared with the general upper bound COM(I) when n = 1.5 and when n = 1. The two straight lines (which cross the origin) are lines of constant geometrical concentration. The Co and I of several cylindrical CPCs of n = 1.5 are also shown. The number associated with each dot is the semiacceptance angle in degrees.

Fig. 7
Fig. 7

Adaptation factor of several cylindrical CPCs filled with a medium of n = 1.5 when they are illuminated by the yearly averaged sky above Madrid.

Equations (31)

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C O M ( I ) = C O M ( I ) n 2 n 2 ,
C o = g γ C O M ( I ) .
dxdydpdq = d x d y d p d q .
( x , y , p , q ) ( x , y , p , q ) = 1 = | x x x y x p x q y x y y y p y q 0 0 1 0 q x q y q p q q | = ( x , y , q ) ( x , y , q ) .
dxdydq = d x d y d q .
F ( p ) = 1 4 A c e dxdy ( 1 p 2 ) 1 / 2 ( 1 p 2 ) 1 / 2 t ( x , y , p , q ) d q ,
F ( p ) = 1 4 A c σ c d x d y ( n 2 p 2 ) 1 / 2 ( n 2 p 2 ) 1 / 2 t 1 ( x , y , p , q , σ ) d q ,
F ( p ) C g 4 ( 1 p 2 ) 1 / 2 ( 1 p 2 ) 1 / 2 d q = C g 2 ( 1 p 2 ) 1 / 2 ,
F ( p ) 2 1 4 ( n 2 p 2 ) 1 / 2 ( n 2 p 2 ) 1 / 2 d q = n 2 p 2 .
g = R sph e H t ( x , y , p , q ) dxdydpdq n 2 R sph A c 2 π ,
g = 4 A c 2 π n 2 A c 1 1 F ( p ) d p .
F 2 ( C g / 2 ) 2 + p 2 = 1 ,
F 2 + p 2 = n 2 ,
g g m ( C g , n ) = 2 π n 2 ( C g 2 sin 1 [ n 2 1 ( C g 2 ) 2 1 ] 1 / 2 + n 2 sin 1 { 1 n [ ( C g 2 ) 2 n 2 ( C g 2 ) 2 1 ] 1 / 2 } ) .
g 1 2 g m ( 2 C g , n ) .
g m ( C g , n ) < lim C g g m ( C g , n ) = 2 π n 2 [ ( n 2 1 ) 1 / 2 + n 2 sin 1 ( 1 / n ) ] .
C OMc ( I ) = C O M ( I ) g m ( C g , n ) .
C g = C o I
C OMc ( I ) = C O M ( I ) g m ( C OMc ( I ) I , n ) .
C i + 1 = [ C O M ( I ) n 2 ] [ n 2 g m ( C i I , n ) ] .
C 1 = C O M n = 1 ( I ) = C O M ( I ) n 2 ,
n 2 g m ( C 1 I , n ) > 1
C 2 > C 1 .
C i < [ C O M ( I ) n 2 ] n 2 [ lim C g g m ( C g , n ) ] ,
C 1 ( I 1 ) = C O M ( I 1 ) n 2 ,
C 1 ( I 2 ) = C O M ( I 2 ) n 2 ,
C OMc ( I 1 ) > C OMc ( I 2 ) ,
C 1 = C OMc n = n 1 ( I ) .
C OMc n = n 1 ( I ) = [ C O M n = n 1 ( I ) n 1 2 ] n 1 2 g m ( C OMc n = n 1 ( I ) I , n 1 ) .
C 2 = [ C O M n = n 2 ( I ) n 2 2 ] n 2 2 g m ( C OMc n = n 1 ( I ) I , n 2 ) .
C g = 2 n sin ϕ .

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