Abstract

Total radiant power emission from various diffuse wall cavity sources is calculated without approximation. In addition, a useful radiometric quantity, O-d, the fraction of blackbody power received by a distant viewer, is precisely defined and calculated. Extensive tabulation of numerical results and tutorial background are included. A new method, faster and more accurate than traditional quadrature methods, for the numerical solution of integral equations is described in an appendix. Results are applied to the practical problems of cavity design and analysis.

© 1974 Optical Society of America

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References

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  1. M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill, New York, 1951).
  2. … on a magnitude scale of about one wavelength.
  3. See Ref. 4 for a more rigorous proof of Eq. (8).
  4. M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).
  5. F. E. Nicodemus, Appl. Opt. 4, 768 (1965).
    [CrossRef]
  6. Radiation entering the cavity from outside is neglected.
  7. M. L. Fecteau, Appl. Opt. 7, 1359 (1968).
    [CrossRef]
  8. A. Gouffe, Rev. Opt. 24 (1–3) (1945).
  9. See Ref. 10 for an alternate method based on angle factor algebra.
  10. E. M. Sparrow, V. K. Jonsson, J. Opt. Soc. Am. 53, 816 (1963).
    [CrossRef]
  11. All numerical values reported in the present work were computed independently. Although an iterative procedure was used as in Ref. 10, improved accuracy was obtained by (1) progressively increasing the number of subdivisions of the interval [0, L] until acceptable convergence occurred; (2) replacing the numerical quadrature of Ref. 10 with a more accurate integration technique especially suited to the weakly singular integrands under consideration (see Appendix); (3) using the exact analytical expression of Kelly (see Ref. 15) for ∊a(ξ = 0).
  12. The numerical quadrature used in Refs. 13 and 14 leads to some inaccuracy near x = 0. See Appendix.
  13. E. M. Sparrow, L. U. Albers, E. R. G. Eckert, J. Heat Transfer Trans. ASME Ser. C 84, 73 (1962).
    [CrossRef]
  14. E. M. Sparrow, R. P. Heinisch, Appl. Opt. 9, 2569 (1970).
    [CrossRef] [PubMed]
  15. F. J. Kelly, Appl. Opt. 5, 925 (1966).
    [CrossRef] [PubMed]
  16. For the cylindrical cavity, Sparrow and Heinisch have shown that receiver size has no significant effect on ∊o-d for d > 10R. See Ref. 14.
  17. This expression can be obtained by actually carrying out the two integrations or, more elegantly, by the methods of angle factor algebra as outlined in Ref. 13.
  18. It should be noted that ∊O-O = PO/AOWB, where AO = area of the cavity opening.
  19. See Ref. 20, Chap. 6.
  20. E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Brooks/Cole, Belmont, Calif., 1966).
  21. F. O. Bartell, W. L. Wolfe, Minutes of the Meeting of the IRIS Specialty Group on I.R. Standards 28 (Aug.1972).
  22. K. E. Atkinson, SIAM J. Numerical Anal. 4, 337 (1967).
    [CrossRef]

1970

1968

1967

K. E. Atkinson, SIAM J. Numerical Anal. 4, 337 (1967).
[CrossRef]

1966

1965

F. E. Nicodemus, Appl. Opt. 4, 768 (1965).
[CrossRef]

1963

1962

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, J. Heat Transfer Trans. ASME Ser. C 84, 73 (1962).
[CrossRef]

1945

A. Gouffe, Rev. Opt. 24 (1–3) (1945).

Albers, L. U.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, J. Heat Transfer Trans. ASME Ser. C 84, 73 (1962).
[CrossRef]

Atkinson, K. E.

K. E. Atkinson, SIAM J. Numerical Anal. 4, 337 (1967).
[CrossRef]

Bartell, F. O.

F. O. Bartell, W. L. Wolfe, Minutes of the Meeting of the IRIS Specialty Group on I.R. Standards 28 (Aug.1972).

Cess, R. D.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Brooks/Cole, Belmont, Calif., 1966).

Eckert, E. R. G.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, J. Heat Transfer Trans. ASME Ser. C 84, 73 (1962).
[CrossRef]

Fecteau, M. L.

Gouffe, A.

A. Gouffe, Rev. Opt. 24 (1–3) (1945).

Heinisch, R. P.

Jonsson, V. K.

Kelly, F. J.

Nicodemus, F. E.

F. E. Nicodemus, Appl. Opt. 4, 768 (1965).
[CrossRef]

Planck, M.

M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).

Sparrow, E. M.

E. M. Sparrow, R. P. Heinisch, Appl. Opt. 9, 2569 (1970).
[CrossRef] [PubMed]

E. M. Sparrow, V. K. Jonsson, J. Opt. Soc. Am. 53, 816 (1963).
[CrossRef]

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, J. Heat Transfer Trans. ASME Ser. C 84, 73 (1962).
[CrossRef]

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Brooks/Cole, Belmont, Calif., 1966).

Wolfe, W. L.

F. O. Bartell, W. L. Wolfe, Minutes of the Meeting of the IRIS Specialty Group on I.R. Standards 28 (Aug.1972).

Zemansky, M. W.

M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill, New York, 1951).

Appl. Opt.

J. Heat Transfer Trans. ASME Ser. C

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, J. Heat Transfer Trans. ASME Ser. C 84, 73 (1962).
[CrossRef]

J. Opt. Soc. Am.

Rev. Opt.

A. Gouffe, Rev. Opt. 24 (1–3) (1945).

SIAM J. Numerical Anal.

K. E. Atkinson, SIAM J. Numerical Anal. 4, 337 (1967).
[CrossRef]

Other

All numerical values reported in the present work were computed independently. Although an iterative procedure was used as in Ref. 10, improved accuracy was obtained by (1) progressively increasing the number of subdivisions of the interval [0, L] until acceptable convergence occurred; (2) replacing the numerical quadrature of Ref. 10 with a more accurate integration technique especially suited to the weakly singular integrands under consideration (see Appendix); (3) using the exact analytical expression of Kelly (see Ref. 15) for ∊a(ξ = 0).

The numerical quadrature used in Refs. 13 and 14 leads to some inaccuracy near x = 0. See Appendix.

See Ref. 10 for an alternate method based on angle factor algebra.

Radiation entering the cavity from outside is neglected.

M. W. Zemansky, Heat and Thermodynamics (McGraw-Hill, New York, 1951).

… on a magnitude scale of about one wavelength.

See Ref. 4 for a more rigorous proof of Eq. (8).

M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).

For the cylindrical cavity, Sparrow and Heinisch have shown that receiver size has no significant effect on ∊o-d for d > 10R. See Ref. 14.

This expression can be obtained by actually carrying out the two integrations or, more elegantly, by the methods of angle factor algebra as outlined in Ref. 13.

It should be noted that ∊O-O = PO/AOWB, where AO = area of the cavity opening.

See Ref. 20, Chap. 6.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Brooks/Cole, Belmont, Calif., 1966).

F. O. Bartell, W. L. Wolfe, Minutes of the Meeting of the IRIS Specialty Group on I.R. Standards 28 (Aug.1972).

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Figures (15)

Fig. 1
Fig. 1

Two flat, diffusely radiating areas as seen by a distant observer.

Fig. 2
Fig. 2

Geometry of the diffuse wall cavity.

Fig. 3
Fig. 3

Geometry of the spherical cavity.

Fig. 4
Fig. 4

a as a function of Ac/4πRs2 for the spherical cavities.

Fig. 5
Fig. 5

Geometry of the conical cavity.

Fig. 6
Fig. 6

a(x/L) for the conical cavities.

Fig. 7
Fig. 7

Geometry of the cylindrical cavity.

Fig. 8
Fig. 8

a on the walls and end disk of the cylindrical cavities.

Fig. 9
Fig. 9

Geometry of the extended cone cavity.

Fig. 10
Fig. 10

a along the walls of the extended cone cavities.

Fig. 11
Fig. 11

The diffuse wall cavity as seen by a distant observer.

Fig. 12
Fig. 12

o-d for the conical and cylindrical cavities.

Fig. 13
Fig. 13

o-o as a function of Ao/Ac for spherical, conical and cylindrical cavities.

Fig. 14
Fig. 14

o-∞ as a function of Ao/Ac for spherical, conical and cylindrical cavities.

Fig. 15
Fig. 15

Comparison of diffuse and specular reflection from the walls of a conical cavity.

Tables (1)

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Table I Computed Values of oo and o−∞

Equations (52)

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W B = σ T 4 [ W · c m 2 ]
= W / W B [ dimensionless ]
( λ ) W λ d λ / W λ B d λ [ dimensionless ] ,
= 1 W B 0 ( λ ) W λ B d λ [ dimensionless ] .
2 p A Ω = W cos θ π = W B cos θ π [ W · c m 2 · s r 1 ] ,
W B A = α W B A [ W ] ,
α = [ dimensionless ] .
α ( λ ) = ( λ ) [ dimensionless ] ,
α ( λ ) + ρ ( λ ) = 1 [ dimensionless ] ,
ρ ( λ ) = 1 ( λ ) [ dimensionless ] .
ρ = 1 [ dimensionless ]
W 1 = W B + ρ H 1 [ W · c m 2 ]
a 1 W 1 / W B = + ρ H 1 / W B [ dimensionless ] ,
d H 1 - j = W j cos θ j - 1 d A j d Ω j - 1 π d A 1 = a j W B cos θ 1 - j d Ω 1 - j π [ W · c m 2 ] .
a 1 = + ρ cavity a j cos θ 1 - j d Ω 1 - j π = + ( 1 ) cavity a j cos θ 1 - j d Ω 1 - j π [ dimensionless ] ,
W B d A 1 α H 1 d A 1 = d A 1 ( W B H 1 ) [ W ] ,
[ W B / ( 1 ) ] ( 1 a 1 ) d A 1 [ W ] .
P 0 = W B 1 cavity ( 1 a ) d A [ W ] .
cos θ 1 - j d Ω 1 - j π = cos 2 θ 1 - j d A j π 4 R s 2 cos 2 θ 1 - j = d A j 4 π R s 2 [ dimensionless ]
a 1 = a = + ( 1 ) a cavity d A j 4 π R s 2 = + ( 1 ) a A c 4 π R s 2 [ dimensionless ] ,
a = 1 ( 1 ) ( A c / 4 π R s 2 ) [ dimensionless ] .
cavity a j cos θ 1 - j d Ω 1 - j π = 0 L a ( ξ ) · cos 2 θ / 2 2 x sin θ / 2 × { 1 | ξ x | ( ξ x ) 2 + 6 ξ x sin 2 θ / 2 [ ( ξ x ) 2 + 4 ξ x sin 2 θ / 2 ] 3 / 2 } d ξ [ dimensionless ]
a ( x ) = + ( 1 ) 0 L a ( ξ ) cos 2 θ / 2 2 x sin θ / 2 × { 1 | ξ x | ( ξ x ) 2 + 6 ξ x sin 2 θ / 2 [ ( ξ x ) 2 + 4 ξ x sin 2 θ / 2 ] 3 / 2 } d ξ [ dimensionless ]
a ( x ) = + ( 1 ) 0 L a ( ξ ) 1 2 R × { 1 | ξ x | ( ξ x ) 2 + 6 R 2 [ ( ξ x ) 2 + 4 R 2 ] 3 / 2 } d ξ + ( 1 ) × 0 R a ( r ) 2 R r x ( R 2 + x 2 r 2 ) d r [ ( R 2 + x 2 r 2 ) 2 + 4 R 2 x 2 ] 3 / 2 [ dimensionless ]
a ( r ) = + ( 1 ) 0 L × a ( ξ ) 2 R 2 ξ ( R 2 + ξ 2 r 2 ) d ξ [ ( R 2 + ξ 2 r 2 ) 2 + 4 R 2 ξ 2 ] 3 / 2 [ dimensionless ] .
a ( x 1 ) = + ( 1 ) 0 L 1 a ( ξ 1 ) cos 2 θ 1 / 2 2 x 1 sin θ 1 / 2 [ 1 | ξ 1 x 1 | ( ξ 1 x 1 ) 2 + 6 ξ 1 x 1 sin 2 θ 1 / 2 [ ( ξ 1 x 1 ) 2 + 4 ξ 1 x 1 sin 2 θ 1 / 2 ] 3 / 2 ] d ξ 1 + ( 1 ) S 0 L 2 a ( ξ 2 ) cos θ 1 / 2 cos θ 2 / 2 2 x 1 sin θ 1 / 2 × [ 1 ( R 1 2 + R 2 2 + s 2 ) [ s 4 + 2 s 2 ( R 1 2 + R 2 2 2 R 1 R 2 tan θ 1 / 2 tan θ 2 / 2 ) ] 4 s R 1 R 2 × ( R 1 tan θ 2 / 2 + R 2 tan θ 1 / 2 ) + ( R 1 2 R 2 2 ) 2 8 s R 1 2 R 2 2 ( s R 1 tan θ 1 / 2 R 2 tan θ 2 / 2 ) [ ( R 1 2 + R 2 2 + s 2 ) 2 4 R 1 2 R 2 2 ] 3 / 2 ] d ξ 2 [ dimensionless ] .
o - d P o - d / P B - d [ dimensionless ] ,
P o - d = W B cavity receiver a 1 cos θ 1 - r d Ω 1 - r d A 1 π [ W ] .
a 1 = 1
o - d = cavity receiver a 1 cos θ 1 - r d Ω 1 - r d A 1 π cavity receiver cos θ 1 - r d Ω 1 - r d A 1 π [ dimensionless ] .
a 1
P o - d = π 2 W B 0 L a ( x ) × [ Z Z Z 4 x L 2 sin 4 θ / 2 ( Z 2 4 x 2 L 2 sin 4 θ / 2 ) 1 / 2 ] d x [ W ] ,
a ( x ) [ 4 x L 2 sin 4 θ / 2 Z 2 Z x 2 L 2 sin 4 θ / 2 Z 2 + 8 x 3 L 4 sin 8 θ / 2 Z 3 + ] [ cm ]
o - = 2 L 2 0 L a ( x ) x d x [ dimensionless ] ,
o - = 1 ( 1 ) sin θ / 2 o - o [ dimensionless ] .
P o - d = π 2 W B 0 L a ( x ) [ Y Y Y ( Y 2 4 R 4 ) 1 / 2 ] d x + π 2 W B 0 R a ( r ) [ V V V 4 r R 2 ( V 2 4 R 2 r 2 ) 1 / 2 ] d r [ W ] ,
Y ( d + L x ) 2 + 2 R 2 , Y Y / x , V ( d + L ) 2 + R 2 + r 2 , V V / r ,
o - = 2 R 2 0 R a ( r ) r d r [ dimensionless ] .
o - = 2 sin 2 θ 1 S 0 2 sin 2 θ 2 0 s 0 sin θ 2 sin θ 1 a ( x ) x d x [ dimensionless ]
a = o - d = / { 1 ( 1 ) [ 1 ( A o / A c ) ] } [ dimensionless ]
W λ 1 d λ = ( λ ) W λ B d λ + ρ ( λ ) ( H λ ) d λ [ W · c m ] .
a j ( λ ) = ( λ ) + [ 1 ( λ ) ] cavity × a j ( λ ) cos θ 1 - j d Ω 1 - j π [ dimensonless ]
P o - d = cavity receiver cos θ 1 - r 0 × a 1 ( λ ) W λ B d λ d Ω 1 - r d A 1 [ W ] .
0 L a ( ξ ) F ( ξ , x ) d ξ ,
( ξ j ξ ) a ( ξ j 1 ) + ( ξ ξ j 1 ) a ( ξ j ) ξ j ξ j 1 .
j = 1 N [ α j a ( ξ j 1 ) + β j a ( ξ j ) ] ,
α j = 1 ξ j ξ j 1 ξ j 1 ξ j ( ξ j ξ ) F ( ξ , x ) d ξ , β j = 1 ξ j ξ j 1 ξ j 1 ξ j ( ξ j ξ j 1 ) F ( ξ , x ) d ξ .
a ( 0 ) = + lim x 0 ( 1 ) 0 L a ( ξ ) cos 2 θ / 2 2 x sin θ / 2 [ 1 | ξ x | ( ξ x ) 2 + 6 ξ x sin 2 θ / 2 [ ( ξ x ) 2 + 4 ξ x sin 2 θ / 2 ] 3 / 2 ] d ξ .
R δ ( a x ) f ( x ) d x = f ( a ) ,
a ( 0 ) = + ( 1 ) 0 L a ( ξ ) δ ( ξ ) ( 1 sin 3 θ / 2 ) d ξ = + ( 1 ) a ( 0 ) ( 1 sin 3 θ / 2 ) .
a ( 0 ) = / [ 1 ( 1 ) ( 1 sin 3 θ / 2 ) ] .
a ( 0 ) = + ( 1 ) 0 L a ( ξ ) 1 2 R × [ 1 ξ ( ξ 2 + 6 R 2 ) ( ξ 2 + 4 R 2 ) 3 / 2 ] d ξ + a ( R ) / 2 .

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