Abstract

Static and quasi-static concentrators present interesting characteristics for obtaining photovoltaic solar energy. In this work we study the characteristics of the crossed compound parabolic concentrator, formed by the intersection of two cylindrical compound parabolic concentrators (CPC). Bifacial cells are used in this concentrator as a requirement for obtaining higher concentrations. Static and quasi-static concentrators see the sun as an extended source, so a simplified source model of radiance for the sky of Madrid is used. The figures of merit of a lossless concentrator are studied and the most important parameters influencing its optical behavior are discussed. We conclude that these concentrators obtain results that lead to a decrease in the cost of photovoltaic energy.

© 1984 Optical Society of America

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References

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  1. A. Luque, in Bifacial Solar Cells, C. Khattak, K. V. Ravi, Eds. (North-Holland, Amsterdam, 1984), in press.
  2. A. Luque, “Static Concentrators: A Venture to Achieve the Low Cost Target in Photovoltaics,” Sol. Cells 12, 141 (1984).
    [CrossRef]
  3. J. C. Mĩnano, A. Luque, “Static Concentrator Theory for Nonhomogeneous Extended Sources,” Sol. Cells 8, 297 (1983).
    [CrossRef]
  4. A. Luque, “Theoretical Bases of Photovoltaic Concentrators for Extended Light Sources,” Sol. Cells 3, 355 (1981).
    [CrossRef]
  5. J. C. Minano, A. Luque, “Limit of Concentration Under Extended Nonhomogeneous Light Sources,” Appl. Opt. 22, 2751 (1983).
    [CrossRef] [PubMed]
  6. J. C. Mi˜ano, A. Luque, “Limit of Concentration for Cylindrical Concentrators Under Extended Light Sources,” Appl. Opt. 22, 2437 (1983).
    [CrossRef]
  7. W. T. Welford, R. Winston, Optics of Nonimaging Concentrators (Academic, New York, 1978).

1984

A. Luque, “Static Concentrators: A Venture to Achieve the Low Cost Target in Photovoltaics,” Sol. Cells 12, 141 (1984).
[CrossRef]

1983

1981

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

Luque, A.

A. Luque, “Static Concentrators: A Venture to Achieve the Low Cost Target in Photovoltaics,” Sol. Cells 12, 141 (1984).
[CrossRef]

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

J. C. Mĩnano, A. Luque, “Static Concentrator Theory for Nonhomogeneous Extended Sources,” Sol. Cells 8, 297 (1983).
[CrossRef]

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

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

A. Luque, in Bifacial Solar Cells, C. Khattak, K. V. Ravi, Eds. (North-Holland, Amsterdam, 1984), in press.

Mi˜ano, J. C.

Minano, J. C.

J. C. Mĩnano, A. Luque, “Static Concentrator Theory for Nonhomogeneous Extended Sources,” Sol. Cells 8, 297 (1983).
[CrossRef]

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

Welford, W. T.

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

Winston, R.

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

Appl. Opt.

Sol. Cells

A. Luque, “Static Concentrators: A Venture to Achieve the Low Cost Target in Photovoltaics,” Sol. Cells 12, 141 (1984).
[CrossRef]

J. C. Mĩnano, A. Luque, “Static Concentrator Theory for Nonhomogeneous Extended Sources,” Sol. Cells 8, 297 (1983).
[CrossRef]

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

Other

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

A. Luque, in Bifacial Solar Cells, C. Khattak, K. V. Ravi, Eds. (North-Holland, Amsterdam, 1984), in press.

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Figures (14)

Fig. 1
Fig. 1

Yearly averaged sky radiance of Madrid, Spain; the hatched region represents the rays coming from the ground: (a) hemisphere containing the local sun positions, (b) opposite hemisphere (levels in Wm−2 sr−1).

Fig. 2
Fig. 2

Crossed compound parabolic concentrator with a half-cylinder as second stage where the cell is placed vertically (crosshatched region).

Fig. 3
Fig. 3

Equatorial half-acceptance angle (ϕp) vs meridian one (ϕq) for different values of aspect relation (L/W) of the cell. The isogeometrical gain lines are presented from 6 to 30 for n = 1.425.

Fig. 4
Fig. 4

Geometrical gain vs height of the concentrator for different meridian half-acceptance angles. Also presented are the meridian and equatorial truncation factors, fq and fp, respectively. For an equatorial half-acceptance angle ϕp = 62.778, refractive index n = 1.425, aspect relation of the cell L/W = 6.

Fig. 5
Fig. 5

Concentrator’s geometrical gain vs height for different meridian half-acceptance angles. Also presented are the meridian and equatorial truncation factors fq and fp, respectively. For an equatorial half-acceptance angle ϕp = 62.778, refractive index n = 1.425, aspect relation of the cell L/W = 8.

Fig. 6
Fig. 6

Concentrator’s geometrical gain vs height for different meridian half-acceptance angles. Also presented are the meridian and equatorial truncation factors fq and fp, respectively. For an equatorial half-acceptance angle ϕp = 62.778, refractive index n = 1.425, aspect relation of the cell L/W = 10. Note: L/w = 10 should read L/W = 10 in figure.

Fig. 7
Fig. 7

Three-level source for static concentrators in Madrid, Spain, with the yearly averaged radiance for each level, the diffuse radiation (the white region in the circle with 44.97 W · m−2 sr−1), the direct beam source plus the diffuse radiation (the crosshatched region with 213.69 W · m−2 · sr−1), and the albedo radiation (with 0 W · m−2 · sr−1). This hemisphere contains the local noon sun positions (σ = +1).

Fig. 8
Fig. 8

Three-level source for quasi-static concentrators in Madrid, Spain, with two adjustments a year. The yearly averaged radiances for each level are the diffuse radiation (white region inside the circle, with 44.97 W · m−2 · sr−1), the direct beam source plus the diffuse radiation (crosshatched region with 382.41 W · m−2 · sr−1), and the albedo radiation (darkened region with 0 W · m−2 · sr−1). This hemisphere contains the local noon sun positions (σ = +1) (approximation).

Fig. 9
Fig. 9

Map of the directional intercept factor α(p,q); p and q are the directional cosines of the radiation. The isodirectional intercept factor lines are plotted.

Fig. 10
Fig. 10

Power flux (W · m−2) collected by optimally oriented concentrators vs the averaged acceptance area (â) for the three-level static source in Madrid, Spain.

Fig. 11
Fig. 11

Power flux (W · m−2) collected by optimally oriented concentrators vs the average acceptance area (â) for the three-level source in Madrid, Spain.

Fig. 12
Fig. 12

Optical concentration Co vs the averaged intercept factor I for the three-level source of static concentration. The results obtained in Table III are plotted, as well as the lines of maximum optical concentration for n = 1.425 and n = 1.

Fig. 13
Fig. 13

Optical concentration Co vs the averaged intercept factor I for the three-level source of quasi-static concentration with two adjustments a year. The results of Tables IV and V are plotted, as well as the lines of maximum optical concentration for n = 1.425 and n = 1.

Fig. 14
Fig. 14

Truncated section and its significant parameters: ϕ q half-acceptance angle; ϕ a q half-aperture angle of the truncated concentrator; L′,W′ and L″,W″, aperture of nontruncated and truncated, respectively; hq, p, and h are the nontrucated and truncated heights of the section.

Tables (5)

Tables Icon

Table I Geometrical Characteristics for the Static Crossed Concentrators with ϕq = 30° and ϕp = 62.778°

Tables Icon

Table II Geometrical Characteristics for Quasi-Static Concentrators with Two Seasonal Adjustments for ϕq = 15° and ϕp = 60°

Tables Icon

Table III Figures of Merit for the Static Crossed Concentrator (ϕq = 30°, ϕp = 62.778°, and n = 1.425) a

Tables Icon

Table IV Figures of Merit of the Quasi-Static Crossed Concentrator, with Two Adjustments a Year (ϕq = 15° and ϕp = 60°), for the Heights Considered for Static Crossed Concentrators

Tables Icon

Table V Figures of Merit of the Quasi-Static Crossed Concentrator, with Two Adjustments a Year (ϕq = 20° and ϕp = 60°), for the Same Heights Considered in the Previous Cases

Equations (21)

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h q / W = n + sin ϕ q sin 2 ϕ q ( n 2 - sin 2 ϕ q ) 1 / 2 ,
h p / W = n + sin ϕ p 2 sin 2 ϕ p ( n 2 - sin 2 ϕ p ) 1 / 2 * ( L / W ) ,
h q / W = h p / W .
C g = 2 n 2 sin ϕ p sin ϕ q .
L W = 2 1 - f q 1 - f p sin 2 ϕ p sin 2 ϕ q n + s i n ϕ q n + sin ϕ p ( n 2 - sin 2 ϕ q ) 1 / 2 ( n 2 - sin 2 ϕ p ) 1 / 2 .
C g = 2 ( h W · tan ϕ aq - 1 ) ( 2 · h L · tan ϕ a p - 1 ) ,
I = A e s α ( p , q ) d p d q ,
I = I b f b + I d f d ,
C o = I C g .
C o = g γ C o m ( I ) ,
C o m ( I ) = 2 π n 2 I ψ c - 1 [ I ψ e ( π ) ] ,
r = f / [ sin 2 ( ψ / 2 ) ] ,
f = W ( 1 + sin ϕ q ) .
h = r cos ϕ a q = W 1 + sin ϕ q sin 2 ( ϕ q + ϕ a q 2 ) cos ϕ a q .
h q = W ( 1 + sin ϕ q ) sin 2 ϕ q cos ϕ q .
h = L 2 1 + sin ϕ p sin ( ϕ p + ϕ a p 2 ) cos ϕ a p .
ϕ a q = 2 tan - 1 ( - sin ϕ q + { 4 W h ( n + sin ϕ q ) [ 2 W h ( n + sin ϕ q ) + 2 ( n 2 - sin 2 ϕ q ) 1 / 2 ] } 1 / 2 4 [ 2 W n ( n + sin ϕ q ) + n + ( n 2 - sin 2 ϕ q ) 1 / 2 ] ) .
ϕ a p = 2 tan - 1 ( - sin ϕ p + { 2 L h ( n + sin ϕ p ) [ W h ( n + sin ϕ p ) + 2 ( n 2 - sin 2 ϕ p ) 1 / 2 ] } 1 / 2 4 [ W h ( n + sin ϕ p ) + n + ( n 2 - sin 2 ϕ p ) 1 / 2 ] )
W = 2 ( h tan Φ a q - W ) ,
L = 2 ( h tan Φ a p - L / 2 ) ,
C g = 4 ( h W tan Φ a q - 1 ) ( h L tan Φ a p - 1 / 2 ) .

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