Abstract

A new approach for representing and evaluating the flux density distribution on the absorbers of two-dimensional imaging solar concentrators is presented. The formalism accommodates any realistic solar radiance and concentrator optical error distribution. The solutions obviate the need for raytracing, and are physically transparent. Examples illustrating the method’s versatility are presented for parabolic trough mirrors with both planar and tubular absorbers, Fresnel reflectors with tubular absorbers, and V-trough mirrors with planar absorbers.

© 2013 Optical Society of America

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References

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  1. F. Alberti, M. Cozzini, and L. Crema, “Two-zone coating strategy for CSP parabolic trough applications,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 40102.
  2. K. W. Glenn and C. K. Ho, “Impact of aperture size, receiver diameter, and loop length on parabolic trough performance with consideration of heat loss, pumping parasitics, and optics for a typical meteorological year,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 25131.
  3. F. von Reeken, G. Weinrebe, and M. Balz, “Extended Rabl method to assess the optical quality of parablic trough collectors,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 41133.
  4. H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
    [CrossRef]
  5. P. D. Jose, “The flux distribution through the focal spot of a solar furnace,” Sol. Energy 1, 19–22 (1957).
    [CrossRef]
  6. A. W. Simon, “Calculation of the concentration of energy at points outside the focal spot of a parabolic condenser,” Sol. Energy 2, 22–24 (1958).
    [CrossRef]
  7. K. E. Hassan and M. F. El-Refaie, “Theoretical performance of cylindrical parabolic solar concentrators,” Sol. Energy 15, 219–244 (1973).
    [CrossRef]
  8. D. L. Evans, “On the performance of cylindrical parabolic solar concentrators with flat absorbers,” Sol. Energy 19, 379–385 (1977).
    [CrossRef]
  9. R. O. Nicolás and J. C. Durán, “Generalization of the two-dimensional optical analysis of cylindrical concentrators,” Sol. Energy 25, 21–31 (1980).
    [CrossRef]
  10. S. M. Jeter, “Calculation of the concentrated flux density distribution in parabolic trough collectors by a semifinite formulation,” Sol. Energy 37, 335–345 (1986).
    [CrossRef]
  11. A. Rabl, Active Solar Collectors and Their Applications (Oxford University, 1985).
  12. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).
  13. A. Rabl and P. Bendt, “Effect of circumsolar radiation on performance of focusing collectors,” J. Sol. Energy Eng. 104, 237–250 (1982).
    [CrossRef]
  14. N. Fraidenaich, “Analytic solutions for the optical properties of V-trough concentrators,” Appl. Opt. 31, 131–139 (1992).
    [CrossRef]

2002 (1)

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

1992 (1)

1986 (1)

S. M. Jeter, “Calculation of the concentrated flux density distribution in parabolic trough collectors by a semifinite formulation,” Sol. Energy 37, 335–345 (1986).
[CrossRef]

1982 (1)

A. Rabl and P. Bendt, “Effect of circumsolar radiation on performance of focusing collectors,” J. Sol. Energy Eng. 104, 237–250 (1982).
[CrossRef]

1980 (1)

R. O. Nicolás and J. C. Durán, “Generalization of the two-dimensional optical analysis of cylindrical concentrators,” Sol. Energy 25, 21–31 (1980).
[CrossRef]

1977 (1)

D. L. Evans, “On the performance of cylindrical parabolic solar concentrators with flat absorbers,” Sol. Energy 19, 379–385 (1977).
[CrossRef]

1973 (1)

K. E. Hassan and M. F. El-Refaie, “Theoretical performance of cylindrical parabolic solar concentrators,” Sol. Energy 15, 219–244 (1973).
[CrossRef]

1958 (1)

A. W. Simon, “Calculation of the concentration of energy at points outside the focal spot of a parabolic condenser,” Sol. Energy 2, 22–24 (1958).
[CrossRef]

1957 (1)

P. D. Jose, “The flux distribution through the focal spot of a solar furnace,” Sol. Energy 1, 19–22 (1957).
[CrossRef]

Alberti, F.

F. Alberti, M. Cozzini, and L. Crema, “Two-zone coating strategy for CSP parabolic trough applications,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 40102.

Balz, M.

F. von Reeken, G. Weinrebe, and M. Balz, “Extended Rabl method to assess the optical quality of parablic trough collectors,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 41133.

Bendt, P.

A. Rabl and P. Bendt, “Effect of circumsolar radiation on performance of focusing collectors,” J. Sol. Energy Eng. 104, 237–250 (1982).
[CrossRef]

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Cohen, G.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Cozzini, M.

F. Alberti, M. Cozzini, and L. Crema, “Two-zone coating strategy for CSP parabolic trough applications,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 40102.

Crema, L.

F. Alberti, M. Cozzini, and L. Crema, “Two-zone coating strategy for CSP parabolic trough applications,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 40102.

Durán, J. C.

R. O. Nicolás and J. C. Durán, “Generalization of the two-dimensional optical analysis of cylindrical concentrators,” Sol. Energy 25, 21–31 (1980).
[CrossRef]

El-Refaie, M. F.

K. E. Hassan and M. F. El-Refaie, “Theoretical performance of cylindrical parabolic solar concentrators,” Sol. Energy 15, 219–244 (1973).
[CrossRef]

Evans, D. L.

D. L. Evans, “On the performance of cylindrical parabolic solar concentrators with flat absorbers,” Sol. Energy 19, 379–385 (1977).
[CrossRef]

Fraidenaich, N.

Gee, R.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Glenn, K. W.

K. W. Glenn and C. K. Ho, “Impact of aperture size, receiver diameter, and loop length on parabolic trough performance with consideration of heat loss, pumping parasitics, and optics for a typical meteorological year,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 25131.

Hassan, K. E.

K. E. Hassan and M. F. El-Refaie, “Theoretical performance of cylindrical parabolic solar concentrators,” Sol. Energy 15, 219–244 (1973).
[CrossRef]

Ho, C. K.

K. W. Glenn and C. K. Ho, “Impact of aperture size, receiver diameter, and loop length on parabolic trough performance with consideration of heat loss, pumping parasitics, and optics for a typical meteorological year,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 25131.

Jeter, S. M.

S. M. Jeter, “Calculation of the concentrated flux density distribution in parabolic trough collectors by a semifinite formulation,” Sol. Energy 37, 335–345 (1986).
[CrossRef]

Jose, P. D.

P. D. Jose, “The flux distribution through the focal spot of a solar furnace,” Sol. Energy 1, 19–22 (1957).
[CrossRef]

Kearney, D.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Lüpfert, E.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Mahoney, R.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Nicolás, R. O.

R. O. Nicolás and J. C. Durán, “Generalization of the two-dimensional optical analysis of cylindrical concentrators,” Sol. Energy 25, 21–31 (1980).
[CrossRef]

Price, H.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Rabl, A.

A. Rabl and P. Bendt, “Effect of circumsolar radiation on performance of focusing collectors,” J. Sol. Energy Eng. 104, 237–250 (1982).
[CrossRef]

A. Rabl, Active Solar Collectors and Their Applications (Oxford University, 1985).

Simon, A. W.

A. W. Simon, “Calculation of the concentration of energy at points outside the focal spot of a parabolic condenser,” Sol. Energy 2, 22–24 (1958).
[CrossRef]

von Reeken, F.

F. von Reeken, G. Weinrebe, and M. Balz, “Extended Rabl method to assess the optical quality of parablic trough collectors,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 41133.

Weinrebe, G.

F. von Reeken, G. Weinrebe, and M. Balz, “Extended Rabl method to assess the optical quality of parablic trough collectors,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 41133.

Winston, R.

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

Zarza, E.

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Appl. Opt. (1)

J. Sol. Energy Eng. (2)

A. Rabl and P. Bendt, “Effect of circumsolar radiation on performance of focusing collectors,” J. Sol. Energy Eng. 104, 237–250 (1982).
[CrossRef]

H. Price, E. Lüpfert, D. Kearney, E. Zarza, G. Cohen, R. Gee, and R. Mahoney, “Advances in parabolic trough solar power technology,” J. Sol. Energy Eng. 124, 109–125 (2002).
[CrossRef]

Sol. Energy (6)

P. D. Jose, “The flux distribution through the focal spot of a solar furnace,” Sol. Energy 1, 19–22 (1957).
[CrossRef]

A. W. Simon, “Calculation of the concentration of energy at points outside the focal spot of a parabolic condenser,” Sol. Energy 2, 22–24 (1958).
[CrossRef]

K. E. Hassan and M. F. El-Refaie, “Theoretical performance of cylindrical parabolic solar concentrators,” Sol. Energy 15, 219–244 (1973).
[CrossRef]

D. L. Evans, “On the performance of cylindrical parabolic solar concentrators with flat absorbers,” Sol. Energy 19, 379–385 (1977).
[CrossRef]

R. O. Nicolás and J. C. Durán, “Generalization of the two-dimensional optical analysis of cylindrical concentrators,” Sol. Energy 25, 21–31 (1980).
[CrossRef]

S. M. Jeter, “Calculation of the concentrated flux density distribution in parabolic trough collectors by a semifinite formulation,” Sol. Energy 37, 335–345 (1986).
[CrossRef]

Other (5)

A. Rabl, Active Solar Collectors and Their Applications (Oxford University, 1985).

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Elsevier, 2005).

F. Alberti, M. Cozzini, and L. Crema, “Two-zone coating strategy for CSP parabolic trough applications,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 40102.

K. W. Glenn and C. K. Ho, “Impact of aperture size, receiver diameter, and loop length on parabolic trough performance with consideration of heat loss, pumping parasitics, and optics for a typical meteorological year,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 25131.

F. von Reeken, G. Weinrebe, and M. Balz, “Extended Rabl method to assess the optical quality of parablic trough collectors,” in Proceedings of 18th Solar PACES2012 Conference, Marrakech, Morocco (2012), p. 41133.

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Figures (4)

Fig. 1.
Fig. 1.

Parabolic trough of unit focal length, with a flat absorber in the focal plane. (a) Geometry: angular and spatial coordinates (described in the text). (b) Characteristic iso-line diagram relating the angular coordinates θ and ϕ. Each curve is for a different spatial coordinate y. θm=45° is chosen as the value typically maximizing averaged flux concentration at maximum collection efficiency for the flat absorber [11]. Then with ϕs=0.267°=4.66mrad, ym=0.0076. (c) Local flux concentration along the absorber (relative to the incident irradiance), for three different effective input solar radiance distributions explained in the text. The rms error for the Gaussian distribution is 0.0060.

Fig. 2.
Fig. 2.

Parabolic trough of unit focal length with a tubular absorber. (a) Geometry: angular and spatial coordinates (described in the text). (b) Iso-line diagram relating the angular coordinates θ and ϕ. Each curve is for a different absorber position angle α, where, relative to the downward optic axis, α is positive in the counterclockwise direction and negative in the clockwise direction. θm=90° is chosen as the value typically maximizing averaged flux concentration at maximum collection efficiency [11]. rabs=0.01. (c) Local flux concentration along the absorber (relative to the incident irradiance), for different effective input solar radiance distributions.

Fig. 3.
Fig. 3.

Fresnel reflector with a tubular absorber, with the unit length defined as the height of the absorber’s center above the plane of the mirrors. (a) Geometry: angular and spatial coordinates (described in the text). (b) Iso-line diagram relating the angular coordinates θ and ϕ. Each curve is for a different absorber position angle α, where, relative to the downward optic axis, α is positive in the counterclockwise direction and negative in the clockwise direction. θm=60°. rabs=0.0025. (c) Local flux concentration along the absorber (relative to the incident irradiance), for different effective input solar radiance distributions. (The “constant” and “Rabl” curves are essentially indistinguishable.)

Fig. 4.
Fig. 4.

Single-reflection V-trough with a flat absorber of unit width. (a) Geometry: angular and spatial variables (described in the text). (b) Iso-line diagram relating x and ϕ. Each curve is for a different absorber spatial coordinate y. ψ=30°.

Equations (7)

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y=Sec(θϕ)Sec2(θ/2)Sin(ϕ),
I(y)=θmθmB(φ(θ,y))cos(θ+φ(θ,y))dθ.
tan(ϕ)=rabsSin(θα)2Sec2(θ/2)rabsCos(θα),
I(α)=θmθmB(ϕ(θ,α))cos(θ+ϕ(θ,α))dθ.
tan(ϕ)=rabsSin(θα)Cos(θ)1rabsCos(θ)Cos(θα),
Tan(ϕ)=1Tan(2ψ)+k(x,y,ψ)Sin(2ψ)Cos(2ψ)wherek(x,y,ψ)=[(2x1)Cos2(ψ)Cos2(ψ)xyCos(2ψ)],
I(y)=21/2x0B[ϕ(x,y)]cos[2ψ+ϕ(x,y)]dx

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