Abstract

The analysis of radiation transfer in specular reflector configurations, with multiple or distorted images of an extended radiation source, appears to be so complicated that the need for detailed ray tracing is usually considered inescapable. I show that the number of calculations can be greatly reduced when the source is isotropic. Starting from the fact that the radiation received from an isotropic source depends only on the emissive power of the source and on the angular contour subtended by the source, I show that only edge rays need to be traced, no matter what the curvature of the reflector. Edge rays are rays that pass through the edge of the source or of the reflector. This approach makes it possible to obtain the exact solution for configurations for which the number of reflections is not impractically high. For the solution of general configurations I propose a fast numerical procedure. Based on interpolation between the impact points of a small number of edge rays, it offers high accuracy and unlimited resolution; the convergence is rapid as the number of rays is increased. A test that uses the example of a generalized compound parabolic concentrator demonstrates that with this method one can achieve accuracies that are far better than with conventional ray-trace methods while tracing a number of rays that are orders of magnitude smaller.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. M. Sparrow, R. D. Cess, Radiation Heat Transfer, augmented ed. (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.
  2. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981), Chap. 9, pp. 281–318.
  3. J. P. Holman, Heat Transfer (McGraw-Hill, New York, 1990), Chap. 9, pp. 385–483.
  4. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 3, pp. 77–101; Chap. 10, pp. 274–288.
  5. N. Fraidenreich, I. H. Salcedo, “Multimode analysis of compound parabolic concentrators with flat absorber,” Appl. Opt. 32, 2891–2900 (1993).
    [CrossRef]
  6. A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, New York, 1985), Chap. 6, pp. 147–169.
  7. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, Orlando, Fla., 1989).
  8. R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
    [CrossRef]
  9. A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation, in particular infrared, comprising a plane source of rays and a reflector,” U.S. patent4,922,107 (1May1990); A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation,” U.S. patent4,990,788 (5February1991).
  10. W. B. Elmer, The Optical Design of Reflectors, 3rd ed. (TLA Lighting Consultants, Inc., Salem, Mass., 1989), Sec. 6.2.5, pp. 92–95;W. B. Elmer, The Optical Design of Reflectors, 2nd ed., Wiley, New York, 1980).
  11. D. K. Edwards, “The plating algorithm for radiation script-F transfer factor,” J. Heat Transfer 108, 237–238 (1986).
    [CrossRef]
  12. W. Cai, “Développement et applications de modèles d’échanges radiatifs par suivi de rayons,” Ph.D. dissertation (Centre d’Energétique, Ecole des Mines, Paris, 1992).

1993 (1)

N. Fraidenreich, I. H. Salcedo, “Multimode analysis of compound parabolic concentrators with flat absorber,” Appl. Opt. 32, 2891–2900 (1993).
[CrossRef]

1991 (1)

R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
[CrossRef]

1986 (1)

D. K. Edwards, “The plating algorithm for radiation script-F transfer factor,” J. Heat Transfer 108, 237–238 (1986).
[CrossRef]

Cai, W.

W. Cai, “Développement et applications de modèles d’échanges radiatifs par suivi de rayons,” Ph.D. dissertation (Centre d’Energétique, Ecole des Mines, Paris, 1992).

Cess, R. D.

F. M. Sparrow, R. D. Cess, Radiation Heat Transfer, augmented ed. (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.

Clodic, D.

A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation, in particular infrared, comprising a plane source of rays and a reflector,” U.S. patent4,922,107 (1May1990); A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation,” U.S. patent4,990,788 (5February1991).

Dehausse, R.

A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation, in particular infrared, comprising a plane source of rays and a reflector,” U.S. patent4,922,107 (1May1990); A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation,” U.S. patent4,990,788 (5February1991).

Edwards, D. K.

D. K. Edwards, “The plating algorithm for radiation script-F transfer factor,” J. Heat Transfer 108, 237–238 (1986).
[CrossRef]

Elmer, W. B.

W. B. Elmer, The Optical Design of Reflectors, 3rd ed. (TLA Lighting Consultants, Inc., Salem, Mass., 1989), Sec. 6.2.5, pp. 92–95;W. B. Elmer, The Optical Design of Reflectors, 2nd ed., Wiley, New York, 1980).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 3, pp. 77–101; Chap. 10, pp. 274–288.

Fraidenreich, N.

N. Fraidenreich, I. H. Salcedo, “Multimode analysis of compound parabolic concentrators with flat absorber,” Appl. Opt. 32, 2891–2900 (1993).
[CrossRef]

Holman, J. P.

J. P. Holman, Heat Transfer (McGraw-Hill, New York, 1990), Chap. 9, pp. 385–483.

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981), Chap. 9, pp. 281–318.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 3, pp. 77–101; Chap. 10, pp. 274–288.

Rabl, A.

A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, New York, 1985), Chap. 6, pp. 147–169.

A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation, in particular infrared, comprising a plane source of rays and a reflector,” U.S. patent4,922,107 (1May1990); A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation,” U.S. patent4,990,788 (5February1991).

Salcedo, I. H.

N. Fraidenreich, I. H. Salcedo, “Multimode analysis of compound parabolic concentrators with flat absorber,” Appl. Opt. 32, 2891–2900 (1993).
[CrossRef]

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981), Chap. 9, pp. 281–318.

Sparrow, F. M.

F. M. Sparrow, R. D. Cess, Radiation Heat Transfer, augmented ed. (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 3, pp. 77–101; Chap. 10, pp. 274–288.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 3, pp. 77–101; Chap. 10, pp. 274–288.

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, Orlando, Fla., 1989).

Winston, R.

R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
[CrossRef]

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, Orlando, Fla., 1989).

Appl. Opt. (1)

N. Fraidenreich, I. H. Salcedo, “Multimode analysis of compound parabolic concentrators with flat absorber,” Appl. Opt. 32, 2891–2900 (1993).
[CrossRef]

J. Heat Transfer (1)

D. K. Edwards, “The plating algorithm for radiation script-F transfer factor,” J. Heat Transfer 108, 237–238 (1986).
[CrossRef]

Sci. Am. (1)

R. Winston, “Nonimaging optics,” Sci. Am. 264, 76–81 (1991).
[CrossRef]

Other (9)

A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation, in particular infrared, comprising a plane source of rays and a reflector,” U.S. patent4,922,107 (1May1990); A. Rabl, D. Clodic, R. Dehausse, “Apparatus emitting an electromagnetic radiation,” U.S. patent4,990,788 (5February1991).

W. B. Elmer, The Optical Design of Reflectors, 3rd ed. (TLA Lighting Consultants, Inc., Salem, Mass., 1989), Sec. 6.2.5, pp. 92–95;W. B. Elmer, The Optical Design of Reflectors, 2nd ed., Wiley, New York, 1980).

A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, New York, 1985), Chap. 6, pp. 147–169.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, Orlando, Fla., 1989).

F. M. Sparrow, R. D. Cess, Radiation Heat Transfer, augmented ed. (Hemisphere, Washington, D.C., 1978), Chap. 5, pp. 137–156.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1981), Chap. 9, pp. 281–318.

J. P. Holman, Heat Transfer (McGraw-Hill, New York, 1990), Chap. 9, pp. 385–483.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1989), Chap. 3, pp. 77–101; Chap. 10, pp. 274–288.

W. Cai, “Développement et applications de modèles d’échanges radiatifs par suivi de rayons,” Ph.D. dissertation (Centre d’Energétique, Ecole des Mines, Paris, 1992).

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 (11)

Fig. 1
Fig. 1

Configuration of source S, reflector R, and target T · S′ is the image of S in reflector element dR.

Fig. 2
Fig. 2

Edge rays from T to S and to the one-reflection active zone (“image”) R 1. Points R and R+ mark the boundary of R 1 on the reflector R.

Fig. 3
Fig. 3

Simple example of a reflector with two portions that produce disjoint “images” R 1,1 and R 1,2 separated by a portion that does not reflect anything to T.

Fig. 4
Fig. 4

“Image” R 1(R) of the source, as seen from a point R on the reflector.

Fig. 5
Fig. 5

Sketch of the boundaries of the active zones for 1, 2, etc. reflections, as seen from T.

Fig. 6
Fig. 6

Example of a graph of the boundary functions r = r n +(t) and r = r n (t) for two-dimensional configurations. Dotted lines at t 1, t 2, t 3, and t 4 show examples of target points and the determination of the n-reflection active zones on the reflector. Active zones for these target points are indicated by solid vertical lines: ○, edge rays that have been traced; ×, critical points.

Fig. 7
Fig. 7

Simple example of the boundary functions of Fig. 6. (a) The geometry: flat horizontal source S, flat vertical reflector extending along the r axis from r = 1 to r = 3, flat horizontal target along the t axis. (b) The one-reflection boundary functions r 1−(t) and r 1+(t). For example, seen from t = 1 the active zone R 1 extends from r 1−(1) = 2.4 to r 1+(1) = 1.33.

Fig. 8
Fig. 8

Schematic sketch of two situations that can occur: (a) there is no caustic between impact points (first-impact point labeled j); (b) there is a caustic between impact points (first-impact point labeled k).

Fig. 9
Fig. 9

Schematic sketch of boundary coordinate r as a function of target point coordinate t. For the situation in Fig. 8(a), r decreases with t. For the situation in Fig. 8(b), r increases with t. For the transition between, t has a minimum at the critical point.

Fig. 10
Fig. 10

Geometry of generalized CPC for the flat one-sided source of width s with acceptance half-angle θ a and with gap g = S + B between the source and reflector. Origin of the coordinate system at focus S of the right parabola. The symmetry axis is between S + and S .

Fig. 11
Fig. 11

Results for the CPC reflector of Fig. 10 with θ a = 45° and g/s = 0.5, plotted versus the tan(θ) = t = coordinate of target point (θ = θ+ of Fig. 10): (a) The boundaries of the one-reflection active zone, as seen from the target point; (b) irradiance (normalized by maximum).

Tables (1)

Tables Icon

Table 1 Comparison of Accuracy Obtained by Different Methods for the Configuration in Fig. 10 a

Equations (22)

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

A 1 F 1 2 = A 2 F 2 1 ,
d A d T G 0 = A S E S d F S d T ,
A S d F S d T = d A d T F d T S ,
G 0 = E S F d T S .
d G 1 = ρ E S d F d T d R
G 1 = ρ E S R f d F d T d R ,
f = { 1 if T is inside the field of view of d R , 0 if T is outside the field of view of d R .
G 1 = ρ E S R 1 d F d T d R
G 1 = ρ E S F d T R 1 .
G = E S ( F d T S + ρ F d T R 1 ) .
G = E S n = 0 N ρ n F d T R n ,
F d T S S = n = 0 N ρ n F d T R n .
u = 2 f 1 + cos ( ϕ ) ,
f = ( s + g ) 1 + sin ( θ a ) 2 .
θ - = θ a ,
α = θ - - θ + .
sin ( α ) s = sin [ π - α - ( ϕ + θ a - π / 2 ) ] u .
θ + = θ a + arctan [ cos ( ϕ + θ a ) u / s + sin ( ϕ + θ a ) ] , u / s = ( 1 + g / s ) 1 + sin ( θ a ) 1 + cos ( ϕ ) .
tan ( β ) = y - - y + x - - x + ,
s 1 = [ ( y - - y + ) 2 + ( x - - x + ) 2 ] 1 / 2 .
F d T R 1 = K s 1 cos 2 ( θ ) cos ( β - θ ) .
F d T S = K min [ s , x A - y A tan ( θ ) ] cos 3 ( θ ) .

Metrics