Abstract

An analysis of the optical design characteristics of a new high concentration solar collector is presented. This type of collector consists of spherical segments that are sections of a spherical cap by planes perpendicular to its axis. These ring-shaped spherical segments are so arranged along their common axis that the planes of their circles of least confusion are superposed. The optical characteristics and simulation of this system are developed to provide information for the engineering design of this type of solar energy collector system. The calculations are checked by a laser scanning onto a breadboard mock-up.

© 1980 Optical Society of America

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References

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  1. T. Sakurai, K. Shishido, Appl. Opt. 3, 813 (1964).
    [CrossRef]
  2. F. Trombe, C. R. Acad. Sci. 235, 704 (1952).
  3. R. M. Brown, Sky and Telescope, Aug.1964, p. 64.
  4. J. J. Burke, W. Kirchhoff, Sky and Telescope, Nov.1968, p. 284.
  5. R. Pasquetti, G. Péri, Rev. Phys. Appl. 15, 123 (1980).
    [CrossRef]
  6. J. M. Davies, E. S. Cotton, Sol. Energy 1, 16 (1963).
    [CrossRef]
  7. A. Marechal, Imagerie Geométrique (Édition de la Revue d’Optique, Paris, 1952) p. 101.
  8. B. Authier et al., Appl. Opt. 18, 3081 (1979).
    [CrossRef] [PubMed]

1980

R. Pasquetti, G. Péri, Rev. Phys. Appl. 15, 123 (1980).
[CrossRef]

1979

1964

1963

J. M. Davies, E. S. Cotton, Sol. Energy 1, 16 (1963).
[CrossRef]

1952

F. Trombe, C. R. Acad. Sci. 235, 704 (1952).

Authier, B.

Brown, R. M.

R. M. Brown, Sky and Telescope, Aug.1964, p. 64.

Burke, J. J.

J. J. Burke, W. Kirchhoff, Sky and Telescope, Nov.1968, p. 284.

Cotton, E. S.

J. M. Davies, E. S. Cotton, Sol. Energy 1, 16 (1963).
[CrossRef]

Davies, J. M.

J. M. Davies, E. S. Cotton, Sol. Energy 1, 16 (1963).
[CrossRef]

Kirchhoff, W.

J. J. Burke, W. Kirchhoff, Sky and Telescope, Nov.1968, p. 284.

Marechal, A.

A. Marechal, Imagerie Geométrique (Édition de la Revue d’Optique, Paris, 1952) p. 101.

Pasquetti, R.

R. Pasquetti, G. Péri, Rev. Phys. Appl. 15, 123 (1980).
[CrossRef]

Péri, G.

R. Pasquetti, G. Péri, Rev. Phys. Appl. 15, 123 (1980).
[CrossRef]

Sakurai, T.

Shishido, K.

Trombe, F.

F. Trombe, C. R. Acad. Sci. 235, 704 (1952).

Appl. Opt.

C. R. Acad. Sci.

F. Trombe, C. R. Acad. Sci. 235, 704 (1952).

Rev. Phys. Appl.

R. Pasquetti, G. Péri, Rev. Phys. Appl. 15, 123 (1980).
[CrossRef]

Sol. Energy

J. M. Davies, E. S. Cotton, Sol. Energy 1, 16 (1963).
[CrossRef]

Other

A. Marechal, Imagerie Geométrique (Édition de la Revue d’Optique, Paris, 1952) p. 101.

R. M. Brown, Sky and Telescope, Aug.1964, p. 64.

J. J. Burke, W. Kirchhoff, Sky and Telescope, Nov.1968, p. 284.

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Figures (15)

Fig. 1
Fig. 1

Schematic configuration of a stepped spherical collector including a spherical dish n = 0 and two spherical segments n = 1 and n = 2. P is the plane of the circles of least confusion of the three spherical elements whose centers of curvature are C0, C1, and C2 respectively.

Fig. 2
Fig. 2

Optical diagram illustrating the opposite extreme reflected rays 1e and 1i coming from the exterior and interior rims of the spherical segment n = 1, respectively.

Fig. 3
Fig. 3

Spherical mirror segment whose circle of least confusion is defined through the intersection of the caustic envelope by the reflected rays Ne coming from the exterior rim of the mirror.

Fig. 4
Fig. 4

Spherical ring mirror segment whose circle of least confusion is defined by the intersection of two opposite extreme reflected rays Ne and Ni.

Fig. 5
Fig. 5

Angular half height h° of the spherical rings of an aberrationally equilibrated perfect SSC vs mid-point spherical ring angular position α°. αo is the rim angle of the vertex dish.

Fig. 6
Fig. 6

Profile of an aberrationally equilibrated perfect SSC (αo = 12.5°) and related concentration factor C vs rim angle. The dotted line is the parabolical profile of same focal length collector; the dashed line represents the circle of the spherical dish meridian section.

Fig. 7
Fig. 7

Basic configuration illustrating the ray-trace simulation method.

Fig. 8
Fig. 8

Diagram illustrating the solar disk simulation. Point source q on ring p is characterized by polar coordinates.

Fig. 9
Fig. 9

Diagram illustrating a spherical ring simulation.

Fig. 10
Fig. 10

Impinging energy variation onto the receiver aperture vs the aperture disk area illustrating determination of the thermal optimum aperture area of the receiver. Solar incident flux = 800 W/m2 and heat loss 25 kW/m2 and 80 kW/m2 are typical values for aperture temperatures of 900°C and 1,600°C, respectively.

Fig. 11
Fig. 11

Theoretical concentration ratio vs focal ratio for perfect spherical stepped collectors including 4, 5, and 6 spherical segments. In the plane of the circles of least confusion the output aperture disk is impinged by all the reflected rays. Practical concentrations of Tables I and II for f/ϕ = 0.7 and 0.95 can easily be compared to the corresponding theoretical concentrations.

Fig. 12
Fig. 12

Half meridian part of a stepped spherical collector. This demonstration model consists of spherical mirrors parallel to a 3-m long reflecting panel of the same radius of curvature (5.70 m). Except the rectangular surface on the left (representing the vertex dish of the collector), the panel is made white, υ is the vertex of the collector.

Fig. 13
Fig. 13

Interception factors vs the radius length of the output aperture disk of the experimental SSC mounting including n = 4–5 and 6 spherical rings (f/ϕ = 0.81–0.65 and 0.55, respectively). The dashed lines are the corresponding computed interception factors.

Fig. 14
Fig. 14

Slope error variation of the experimental mounting mirrors along the meridian line. d is the distance in centimeters from the mirror’s vertex. The exaggerated difference between a typical mirror profile and the related circle segment is drawn below.

Fig. 15
Fig. 15

Same radius of curvature mirrors mounting of the Rowland circle type in a meridian section including the vertex of the mirrors. C0, C1 and C2 are the centers of curvature of mirrors 0, 1, and 2, respectively.

Tables (3)

Tables Icon

Table I Computed Concentrations for a Stepped Spherical Collector According to the Number of Spherical Segmentsa

Tables Icon

Table II Computed Concentrations for a Stepped Spherical Collector According to the Number of Spherical Segmentsa

Tables Icon

Table III Computed Concentrations for Spherical Elementary Mirror Collectors Including One Dish and Four Circular Rows of Mirrors (ϕ = 8 m, f = 5.85 m)

Equations (23)

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r n , e = r n + 1 , i = r n .
y = ( x + 1 2 1 2 cos ω ) tan ( 2 ω ) ,
y = 2 ω ( x + ω 2 4 ) .
d y d ω + 2 ω d x d ω + 2 x + 3 ω 2 2 = 0.
( x = 3 4 ω 2 y = ω 3 ,
x = 3 4 | y | 2 / 3 ,
y = ω n , e ( x + ω n , e 2 4 ) .
2 ω 1 3 3 ω n , e ω 1 2 + ω n , e 3 = 0 ,
ω 1 = ω n , e 2 .
x = 3 16 ω n , e 2 .
L n = 1 2 + 3 16 r n , e 2
x = 1 2 ( 1 cos ω n , e 1 ) tan ( 2 ω n , e ) ( 1 cos ω n , i 1 ) tan ( 2 ω n , i ) tan ( 2 ω n , i ) tan ( 2 ω n , e ) .
L n = 1 2 ( r n , i + r n , e ) ( 1 2 r n , i r n , e ) r n , i ( 1 2 r n , e 2 ) 1 r n , i 2 + r n , e ( 1 2 r n , i 2 ) 1 r n , e 2
t n = L 0 1 2 ( ( r n 1 + r n ) ( 1 2 r n 1 · r n ) r n 1 ( 1 2 r n 2 ) 1 r n 1 2 + r n ( 1 2 r n 1 2 ) 1 r n 2 ) R ,
L 0 = 1 2 + 3 16 r 0 2 .
x = 1 2 1 y 2 1 y 2
ϕ p , q = 180 ( 2 q 1 ) / n p ( in ° ) ρ p = θ p 2 + θ p + 1 2 2 , ,
ρ p , + i = ρ p + 2 α i ρ p , i = ρ p 2 α i .
ρ p , i , δ c = ρ p , i + sgn ( α i ) · | δ c | , sgn ( α i ) = ( 1 if 0 if + 1 if i > 0 i = 0 i < 0.
δ n , k = d n , k 1 2 + d n , k 2 2 ,
K ( p , q , i , n , k ) = P i σ n , k = K .
E l = m = 1 l j = 1 N m K m , j ,
C l = F l · A a l ,

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