Abstract

The radiant interchange within a diffuse conical cavity has been formulated without approximation for both cases of prescribed wall temperature and prescribed wall heat flux. Highly accurate numerical solutions have been obtained for a wide range of cone opening angles and surface emissivities. Results are presented for the radiant efflux from the cavity as a whole and also for the distributions along the cavity surface of such quantities as the apparent radiant emittance, local heat flux (for prescribed surface temperature), and temperature (for prescribed heat flux). A comparison with the approximate analysis of Gouffe disclosed large errors in this prior work.

© 1963 Optical Society of America

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References

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  1. C. S. Williams, J. Opt. Soc. Am. 51, 566 (1961).
  2. E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
    [Crossref]
  3. K. S. Krishnan, Proc. Roy. Soc. (London) A257, 302 (1960).
    [Crossref]
  4. During the review by the papers committee, unpublished analytical work by Page (Ref. 5) on the cone problem was brought to the attention of the authors, who wish to acknowledge it here.
  5. C. H. Page, National Bureau of Standards (private communication, 1963).
  6. A. Gouffe, Rev. Opt. Nos.1–3 (1945).
  7. Energy entering the cavity from outside is generally not included in studies of the emission characteristics of cavities.
  8. The geometrical factor represents the fraction of the radiant energy leaving one surface element which arrives at another surface element.
  9. M. Jakob, Heat Transfer (John Wiley & Sons, Inc., New York, 1957), Vol. 2.
  10. This method of analysis has been employed in Refs. 11 and 12.
  11. A. C. Bartlett, Phil. Mag. Suppl. 6,  40, 111 (1920).
    [Crossref]
  12. H. Buckley, Phil. Mag. Suppl. 7,  4, 753 (1927).

1962 (1)

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

1961 (1)

C. S. Williams, J. Opt. Soc. Am. 51, 566 (1961).

1960 (1)

K. S. Krishnan, Proc. Roy. Soc. (London) A257, 302 (1960).
[Crossref]

1945 (1)

A. Gouffe, Rev. Opt. Nos.1–3 (1945).

1927 (1)

H. Buckley, Phil. Mag. Suppl. 7,  4, 753 (1927).

1920 (1)

A. C. Bartlett, Phil. Mag. Suppl. 6,  40, 111 (1920).
[Crossref]

Albers, L. U.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

Bartlett, A. C.

A. C. Bartlett, Phil. Mag. Suppl. 6,  40, 111 (1920).
[Crossref]

Buckley, H.

H. Buckley, Phil. Mag. Suppl. 7,  4, 753 (1927).

Eckert, E. R. G.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

Gouffe, A.

A. Gouffe, Rev. Opt. Nos.1–3 (1945).

Gregg, J. L.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

Jakob, M.

M. Jakob, Heat Transfer (John Wiley & Sons, Inc., New York, 1957), Vol. 2.

Jonsson, V. K.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

Krishnan, K. S.

K. S. Krishnan, Proc. Roy. Soc. (London) A257, 302 (1960).
[Crossref]

Page, C. H.

C. H. Page, National Bureau of Standards (private communication, 1963).

Sparrow, E. M.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

Williams, C. S.

C. S. Williams, J. Opt. Soc. Am. 51, 566 (1961).

J. Heat Transfer (1)

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, V. K. Jonsson, and J. L. Gregg, J. Heat Transfer C84, 73, 188, 270 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

C. S. Williams, J. Opt. Soc. Am. 51, 566 (1961).

Phil. Mag. Suppl. 6 (1)

A. C. Bartlett, Phil. Mag. Suppl. 6,  40, 111 (1920).
[Crossref]

Phil. Mag. Suppl. 7 (1)

H. Buckley, Phil. Mag. Suppl. 7,  4, 753 (1927).

Proc. Roy. Soc. (London) (1)

K. S. Krishnan, Proc. Roy. Soc. (London) A257, 302 (1960).
[Crossref]

Rev. Opt. Nos. (1)

A. Gouffe, Rev. Opt. Nos.1–3 (1945).

Other (6)

Energy entering the cavity from outside is generally not included in studies of the emission characteristics of cavities.

The geometrical factor represents the fraction of the radiant energy leaving one surface element which arrives at another surface element.

M. Jakob, Heat Transfer (John Wiley & Sons, Inc., New York, 1957), Vol. 2.

This method of analysis has been employed in Refs. 11 and 12.

During the review by the papers committee, unpublished analytical work by Page (Ref. 5) on the cone problem was brought to the attention of the authors, who wish to acknowledge it here.

C. H. Page, National Bureau of Standards (private communication, 1963).

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Figures (6)

Fig. 1
Fig. 1

Schematic of a conical cavity.

Fig. 2
Fig. 2

Efflux of radiation through cavity opening, uniform wall temperature.

Fig. 3
Fig. 3

Surface distribution of heat flux and apparent radiant emittance, uniform wall temperature, = 0.3 and 0.5.

Fig. 4
Fig. 4

Surface distribution of heat flux and apparent radiant emittance, uniform wall temperature, = 0.7.

Fig. 5
Fig. 5

Surface distribution of heat flux and apparent radiant emittance, uniform wall temperature, = 0.9.

Fig. 6
Fig. 6

Surface distribution of temperature and brightness, uniform wall heat flux.

Equations (23)

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B ( x ) = σ T 4 ( x ) + ρ I ( x ) .
d A ξ d F ξ x = d A x d F x ξ ,
I ( x ) = ξ = 0 L B ( ξ ) d F x ξ .
B ( x ) = σ T 4 ( x ) + ρ ξ = 0 L B ( ξ ) d F x ξ .
a ( x ) = + ( 1 ) ξ = 0 L a ( ξ ) d F x ξ ,
a ( x ) = B ( x ) / σ T w 4 .
q ( x ) = σ T 4 ( x ) α I ( x ) .
q ( x ) = [ / ( 1 ) ] [ σ T 4 ( x ) B ( x ) ] ;
q ( x ) σ T w 4 = 1 [ 1 B ( x ) σ T w 4 ] = 1 [ 1 a ( x ) ] .
Q = 0 L q ( x ) 2 π x sin ( θ / 2 ) d x ,
Q σ T w 4 A 0 = 2 sin ( θ / 2 ) 0 1 q ( x ) σ T w 4 x L d ( x L ) ,
σ T 4 ( x ) = [ ( 1 ) / ] q w + B ( x ) .
β ( x ) = 1 + ξ = 0 L β ( ξ ) d F x ξ ; β = B q w .
F d d = { h 2 + r 1 2 + r 2 2 [ ( h 2 + r 1 2 + r 2 2 ) 2 4 r 1 2 r 2 2 ] 1 2 } / 2 r 1 2 .
r 1 = ξ sin ( θ / 2 ) , r 2 = x sin ( θ / 2 ) , h = | ξ x | cos ( θ / 2 ) .
F d d ( ξ , x ) = x 2 + ξ 2 2 x ξ cos 2 ( θ / 2 ) | ξ x | [ ( x + ξ ) 2 4 x ξ cos 2 ( θ / 2 ) ] 1 2 2 ξ 2 sin 2 ( θ / 2 ) .
F d d ( ξ , x + d x ) = F d d ( ξ , x ) + d F d r ( ξ , x ) ,
d F d r ( ξ , x ) = ( F d d / x ) d x .
F r d ( x , ξ ) = [ ξ 2 sin ( θ / 2 ) / 2 x ] ( F d d / x ) .
F r d ( x , ξ ) = F r d ( x , ξ + d ξ ) + d F r r ( x , ξ ) ,
d F x ξ = F r d ξ d ξ = sin ( θ / 2 ) 2 x ξ [ ξ 2 F d d x ] d ξ .
d F x 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 .
Q = q w π L 2 sin ( θ / 2 ) .