Abstract

We consider the limit of geometric concentration for a focusing concave mirror, e.g., a parabolic trough or dish, designed to collect all radiation within a finite acceptance angle and direct it to a receiver with a flat or circular cross-section. While a concentrator with a parabolic cross-section indeed achieves this limit, it is not the only geometry capable of doing so. We demonstrate that there are infinitely many solutions. The significance of this finding is that geometries which can be more easily constructed than the parabola can be utilized without loss of concentration, thus presenting new avenues for reducing the cost of solar collectors. In particular, we investigate a low-cost trough mirror profile which can be constructed by inflating a stack of thin polymer membranes and show how it can always be designed to match the geometric concentration of a parabola of similar form.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Bader, A. Pedretti, and A. Steinfeld, J. Sol. Energy Eng. 133, 031016 (2011).
    [CrossRef]
  2. R. Winston, J. Opt. Soc. Am. 60, 245 (1970).
    [CrossRef]
  3. D. A. Harper, R. H. Hildebrand, R. Stiening, and R. Winston, Appl. Opt. 15, 53 (1976).
    [CrossRef]
  4. R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).
  5. A. Rabl, Sol. Energy 18, 81 (1976).
    [CrossRef]
  6. D. E. Spencer, L. L. Montgomery, and J. F. Fitzgerald, J. Opt. Soc. Am. 55, 5 (1965).
    [CrossRef]
  7. J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
  8. L. Wang, “Advances in the simultaneous multiple surface optical design method for imaging and nonimaging applications,” Ph.D. dissertation (Universidad Politécnica de Madrid, 2012).

2011

R. Bader, A. Pedretti, and A. Steinfeld, J. Sol. Energy Eng. 133, 031016 (2011).
[CrossRef]

1976

1970

1965

Bader, R.

R. Bader, A. Pedretti, and A. Steinfeld, J. Sol. Energy Eng. 133, 031016 (2011).
[CrossRef]

Benitez, P. G.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Bortz, J. C.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Chaves, J.

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).

Fitzgerald, J. F.

Harper, D. A.

Hildebrand, R. H.

Minano, J. C.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Montgomery, L. L.

Pedretti, A.

R. Bader, A. Pedretti, and A. Steinfeld, J. Sol. Energy Eng. 133, 031016 (2011).
[CrossRef]

Rabl, A.

A. Rabl, Sol. Energy 18, 81 (1976).
[CrossRef]

Shatz, N.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Spencer, D. E.

Steinfeld, A.

R. Bader, A. Pedretti, and A. Steinfeld, J. Sol. Energy Eng. 133, 031016 (2011).
[CrossRef]

Stiening, R.

Wang, L.

L. Wang, “Advances in the simultaneous multiple surface optical design method for imaging and nonimaging applications,” Ph.D. dissertation (Universidad Politécnica de Madrid, 2012).

Winston, R.

D. A. Harper, R. H. Hildebrand, R. Stiening, and R. Winston, Appl. Opt. 15, 53 (1976).
[CrossRef]

R. Winston, J. Opt. Soc. Am. 60, 245 (1970).
[CrossRef]

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Appl. Opt.

J. Opt. Soc. Am.

J. Sol. Energy Eng.

R. Bader, A. Pedretti, and A. Steinfeld, J. Sol. Energy Eng. 133, 031016 (2011).
[CrossRef]

Sol. Energy

A. Rabl, Sol. Energy 18, 81 (1976).
[CrossRef]

Other

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).

L. Wang, “Advances in the simultaneous multiple surface optical design method for imaging and nonimaging applications,” Ph.D. dissertation (Universidad Politécnica de Madrid, 2012).

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Cross-sectional geometry of a generic concave focusing mirror for (a) a flat receiver and (b) a circular receiver. The designs are symmetric about the optical axis (z-axis).

Fig. 2.
Fig. 2.

Canonical solutions zC (dots), zR (solid line) and zL (dashed line) of Eq. (7) for a flat receiver, shown here for Φ=60° and θi=5°. (a) Mirror profile, z(x); (b) mirror slope, z(x); (c) focal function, u(x;+θi)(◂) and u(x;θi) (◂◂) from Eq. (5). Also shown as a thin solid line is the later-discussed test curve Eq. (12), which fails Eq. (8) but passes Eq. (7), as seen in the inlays of (b) and (c).

Fig. 3.
Fig. 3.

Canonical solutions zC (dots), zR (solid line) and zL (dashed line) of Eq. (7) for a circular receiver shown here for Φ=90° and θi=5°. (a) Mirror profile, z(x); (b) mirror slope, z(x); (c) focal function, u(x;+θi) (◂) and u(x;θi) (◂◂) from Eq. (6).

Fig. 4.
Fig. 4.

Arcspline designed to reach the concave limit (Φ=60°, θi=0.5°, flat receiver Cg,max,2D,concave,flat=48.6) with a minimum number of arcs (n=3). (a) Mirror profile, z(x). (b) Focal function from Eq. (14). Edge rays at the points, where u(x;±θi)=±1 are shown. Solid lines are θi edge rays; dashes are +θi edge rays.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Cg,ideal,2D=ai/ao=sinθo/sinθi,
Cg,max,2D,concave,flat=ai/ao=sinΦcosΦ/(sinθicosθi)1,
Cg,max,2D,concave,circular=ai/(πro)=sinΦ/(πsinθi),
r⃗(φ,t;θ)=P(φ)+v^(φ)t,
uflat(φ;θ)=x(φ)+tan(2φθ)z(φ),
uflat(x;θ)=xzz(zsinθ+2cosθ)sinθz(zcosθ2sinθ)cosθ.
ucirc(φ;θ)=x(φ)cos(2φθ)+z(φ)sin(2φθ),
ucirc(x;θ)={[xz(xz2z)]cosθ[zz(zz+2x)]sinθ}/(1+z2).
u(x;θ=+θi)1u(x;θ=θi)1}x[xT,x0]subject to z(x0)=tan(Φ/2),
zRzzLx[xT,x0].
zLzzRx[xT,x0].
zR=(x1)sinθi+zRcosθi+[(x1)2+zR2]1/2(x1)cosθizRsinθiz.
xzz(zsinθi+2cosθi)sinθiz(zcosθi2sinθi)cosθi10,
z(x)=zC(x)+0.0029exp[0.75(xx0)]sin[1.8π(xx0)].
Qj(φ)=Cj+rj(sinφ,cosφ),
uflat,ASC(φ;θ)=Cx,j+rjsinφ+tan(φ2θ)(Cz,jrjcosφ)=Nx,j1+rj(sinφsinφout,j)+tan(φ2θ)[Nz,j1rj(cosφcosφout,j)].

Metrics