Abstract

An equation is derived that relates the emissivity of a spherical cavity to the temperature distribution along the axis normal to the opening. The mathematical formulation of the problem rests on De Vos’s method, and is carried out exactly, by means of a suitable serial development. The solution enables one to estimate the influence of the temperature gradient on the radiant emission of the reference spherical blackbody.

© 1966 Optical Society of America

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References

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  1. H. Buckley, Phil. Mag. 4, 753 (1927).
  2. H. Buckley, Phil. Mag. 6, 447 (1928).
  3. H. Buckley, Phil. Mag. 17, 576 (1934).
  4. A. Gouffé, Rev. Opt. 24, 1 (1945).
  5. J. C. De Vos, Physica 20, 669 (1954).
    [CrossRef]
  6. E. M. Sparrow, Appl. Opt. 4, 41 (1965).
    [CrossRef]
  7. P. Campanaro, T. Ricolfi, Appl. Opt. 5, 929 (1966).
    [CrossRef] [PubMed]

1966 (1)

1965 (1)

1954 (1)

J. C. De Vos, Physica 20, 669 (1954).
[CrossRef]

1945 (1)

A. Gouffé, Rev. Opt. 24, 1 (1945).

1934 (1)

H. Buckley, Phil. Mag. 17, 576 (1934).

1928 (1)

H. Buckley, Phil. Mag. 6, 447 (1928).

1927 (1)

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

Buckley, H.

H. Buckley, Phil. Mag. 17, 576 (1934).

H. Buckley, Phil. Mag. 6, 447 (1928).

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

Campanaro, P.

De Vos, J. C.

J. C. De Vos, Physica 20, 669 (1954).
[CrossRef]

Gouffé, A.

A. Gouffé, Rev. Opt. 24, 1 (1945).

Ricolfi, T.

Sparrow, E. M.

Appl. Opt. (2)

Phil. Mag. (3)

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

H. Buckley, Phil. Mag. 6, 447 (1928).

H. Buckley, Phil. Mag. 17, 576 (1934).

Physica (1)

J. C. De Vos, Physica 20, 669 (1954).
[CrossRef]

Rev. Opt. (1)

A. Gouffé, Rev. Opt. 24, 1 (1945).

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Figures (3)

Fig. 1
Fig. 1

Geometrical factors used in the paper.

Fig. 2
Fig. 2

Emissivity of cavity when = 0.7: —— left ordinate; – – – – right ordinate.

Fig. 3
Fig. 3

Nomograph for computing the error entailed by the temperature gradients in determining the emissivity of materials. Solid curves: α = 10. Dashed curves: α = 20.

Equations (20)

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c = J A A J B A = Φ A A d A × cos ϑ A A d Ω A A σ T A 4
Φ A A = J B A ( T A / T A ) 4 d A cos ϑ A A d Ω A A + d A d Ω A A 0 π / 2 - ϑ A A J A 1 A ( T A 1 / T A ) 4 × cos ϑ A A 1 r A A 1 A d Ω A A 1 ,
J A 1 A = J B A ( 1 - r A 1 A A d Ω A 1 A - r A 1 A A 2 r A 2 A 1 A d Ω A 1 A 2 d Ω A 2 A 1 - ) .
Φ A A = J B A ( T A / T A ) 4 d A cos ϑ A A d Ω A A + J B A d A cos ϑ A A d Ω A A 0 π / 2 - ϑ A A ( T A 1 T A ) 4 r A A A 1 d Ω A A 1 - J B A d A cos ϑ A A d Ω A A 0 π / 2 - ϑ A A ( T A 1 T A ) 4 × r A A A 1 r A 1 A A d Ω A A 1 d Ω A 1 A - J B A d A cos ϑ A A d Ω A A 0 π / 2 - ϑ A A ( T A 1 / T A ) 4 × r A A A 1 r A 1 A A 2 r A 2 A 1 A d Ω A A 1 d Ω A 1 A 2 d Ω A 2 A -
ρ = 1 - , r A A A 1 = ( ρ / π ) cos ϑ A A 1 r A 1 A A = ( ρ / π ) sin ϑ A A 1 r A 1 A A 2 = ( ρ / π ) cos ϑ A 1 A 2 r A 2 A 1 A = ( ρ / π ) cos ( ϑ A A 1 - ϕ A 1 A 2 ) ,
d Ω A A 1 = 2 π sin ϑ A A 1 d ϑ A A 1 , d Ω A 1 A 2 = 2 π sin φ A 1 A 2 d φ A 1 A 2 ,
d Ω A 2 A = π r 2 D 2 cos ( ϑ A A 1 - φ A 1 A 2 ) ,
c = ( T A / T A ) 4 + 2 ρ F T - ( 2 ρ 2 / α 2 ) F T - ( 4 ρ 3 / α 2 ) F T E α - - ( 2 n ρ n + 1 / α 2 ) F T E α n ,
α = D / r = ( 2 / sin 2 ϑ A A ) E α = 0 π / 2 - ϑ A A sin ϑ cos ϑ d ϑ = ( 1 + cos 2 ϑ A A / 4 ) F T = 0 π / 2 - ϑ A A ( T A 1 / T A ) 4 sin ϑ A A 1 cos ϑ A A 1 d ϑ A A 1 .
c = ( T A / T A ) 4 + 2 ρ F T - 2 ρ 2 F T α 2 ( 1 - 2 ρ E α ) .
c = + 2 ρ F T * - [ 2 ρ 2 F T * / α 2 ( 1 - 2 ρ E α ) ] ,
F T * = 0 π / 2 - ϑ A A ( T A 1 / T A ) 4 sin ϑ A A 1 cos ϑ A A 1 d ϑ A A 1 .
T A 1 = T A ( 1 + ω sin 2 ϑ A A 1 ) T A / T A = 1 + ω F T = [ ( 1 + 2 ω E α ) 5 - 1 ] / 10 ω E α + 4 ω E α 2 ,
( c ) ω = ( c ) ω = 0 + 4 ω + 8 ( 1 - ) ω E α 2 ,
c = + 2 ρ E α - [ 2 ρ 2 E α / α 2 ( 1 - 2 ρ E α ) ]
= c H c ,
T A 1 = T A ( 1 + ω cos 2 ϑ A A 1 ) ,
F T * E α + 4 ω ( E α - E α 2 ) ,
d c = [ 8 ρ ( E α - E α 2 ) - 8 ρ 2 ( E α - E α 2 ) / α 2 ( 1 - 2 ρ E α ) ] d ω
d c = 8 ρ ( E α - E α 2 ) d ω = 2 ρ d ω .

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