Abstract

The luminous efficiency of a system consisting of a spherical, thermal radiator at the center of a selectively reflecting spherical shell is investigated from a theoretical point of view. Various spectral reflectances for the spherical shell have been assumed, and computations are made both for a blackbody and a pure tungsten radiator. The effect of variation in radiator temperature and visible bandwidth transmitted by the shell, as well as the question of stability, is considered. Expressions for the luminous efficiency are derived and used for computation of specific examples. These indicate that to obtain a high luminous efficiency for a radiator of emissivity substantially smaller than 1, a perfect, or nearly perfect reflectance of the shell in a fairly small band around the region of high-energy output from the radiator, is more important than a uniform and fairly high reflectance over a larger spectral region. It is also shown that the introduction of a transparent spherical shell of reflectance 0.1, between the radiator and the reflecting shell, leads to a relatively small decrease in efficiency.

© 1962 Optical Society of America

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References

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  1. H. Rubens, Ann. Physik 20, 593 (1906).
    [Crossref]
  2. J. C. De Vos, thesis, University of Amsterdam (1953).
  3. F. J. Studer and D. A. Cusano, J. Opt. Soc. Am. 43, 522 (1953).
    [Crossref]

1953 (1)

1906 (1)

H. Rubens, Ann. Physik 20, 593 (1906).
[Crossref]

Cusano, D. A.

De Vos, J. C.

J. C. De Vos, thesis, University of Amsterdam (1953).

Rubens, H.

H. Rubens, Ann. Physik 20, 593 (1906).
[Crossref]

Studer, F. J.

Ann. Physik (1)

H. Rubens, Ann. Physik 20, 593 (1906).
[Crossref]

J. Opt. Soc. Am. (1)

Other (1)

J. C. De Vos, thesis, University of Amsterdam (1953).

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Figures (6)

Fig. 1
Fig. 1

Luminous efficiency as a function of temperature for a freely radiating blackbody (solid curve), and for a blackbody in a spherical shell of perfect reflectance outside, and perfect transmittance inside the spectral regions indicated.

Fig. 2
Fig. 2

Spectral energy distribution of freely radiating blackbody at 1760°K (broken curve), and of this blackbody at the center of a spherical shell of perfect reflectance outside, and perfect transmittance inside the visible region (solid curve). The power supplied to the blackbody is the same in both cases.

Fig. 3
Fig. 3

Luminous efficiency of a blackbody radiator (BB) and a tungsten radiator in a spherical shell as a function of cutoff wavelength. The shell has perfect reflectance above, and perfect transmittance below the cutoff wavelength.

Fig. 4
Fig. 4

H=infrared fraction of the total energy radiated by a blackbody in free space, as a function of cutoff wavelength.

Fig. 5
Fig. 5

Luminous efficiency as a function of cutoff wavelength for blackbody and tungsten radiators in a reflecting shell of reflectance 0.1 below and 0.9 above the cutoff wavelength. The dotted curve is computed according to Eq. (5.12).

Fig. 6
Fig. 6

Transmittance of heat-reflecting film on glass (solid curve), and of glass coated on both sides with a TiO2 film (broken curve).

Tables (1)

Tables Icon

Table I Luminous efficiency (in lu/w) computed by means of Eq. (5.11) at a radiator temperature of 3000°K.

Equations (34)

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W B = 2 π c 1 S 0 d λ λ 5 [ exp ( c 2 / λ T ) - 1 ] ,
ϕ B = η m 2 π c 1 S λ 1 λ 2 V λ d λ λ 5 [ exp ( c 2 / λ T ) - 1 ] ,
η B = ϕ B / W B .
W i = 2 π c 1 S λ c d λ λ 5 [ exp ( c 2 / λ T ) - 1 ] .
H = W i / W B .
W B , 0 , 1 = W B ( 1 - H ) .
η B , 0 , 1 = ϕ B / W B ( 1 - H ) ,
Δ W B = σ ( T 2 4 - T 1 4 ) .
W B , 0 , 1 = σ T 1 4 ( 1 - H ) .
Δ W B , 0 , 1 = σ ( T 3 4 - T 1 4 ) ( 1 - H ) .
T 2 4 - T 1 4 = ( T 3 4 - T 1 4 ) ( 1 - H ) .
4 T 1 3 ( T 2 - T 1 ) 4 T 1 3 ( T 3 - T 1 ) ( 1 - H ) , T 3 - T 1 ( T 2 - T 1 ) / ( 1 - H ) ,
Δ W B W B = B , 0 , 1 Δ W W B ( 1 - H ) T 2 4 - T 1 4 σ T 1 4 = ( T 3 4 - T 1 4 ) ( 1 - H ) σ T 1 4 ( 1 - H )
W ab = ( 1 - r ) W i R × [ ( 1 - r ) + r R ( 1 - r ) + + r n R n ( 1 - r ) + ] = ( 1 - r ) R 1 - r R W i ( 1 - r ) = k ( 1 - r ) W i = k H W B ( 1 - r ) ,
k = ( 1 - r ) R / ( 1 - r R ) ;             0 < k < R
W G , 0 , R = W B ( 1 - r ) - W ab = W B ( 1 - r ) ( 1 - k H ) .
η G , 0 , R = ϕ B / W B ( 1 - k H ) .
η G , 0 , R = [ ( 1 - H ) / ( 1 - k H ) ] η B , 0 , 1 .
d W B ( 1 - r ) ( 1 - R 1 ) ( 1 - R 2 ) ( 1 - γ 1 ) ( 1 - γ 2 ) .
d W B ( 1 - r ) A ,
A = ( 1 - R 1 ) ( 1 - R 2 ) 1 - R 1 R 2 ( 1 - γ 1 ) ( 1 - γ 2 ) .
d W B ( 1 - r ) B ,
B = g ( 1 - r ) [ R 1 + ( 1 - R 1 ) 2 ( 1 - γ 1 ) 2 R 2 1 - R 1 R 2 ] .
d W B ( 1 - r ) C ,
C = g r [ R 1 + ( 1 - R 1 ) 2 ( 1 - γ 1 ) 2 R 2 1 - R 1 R 2 ] = r 1 - r B .
d W ab = d W B ( 1 - r ) [ B + B · C + B · C 2 + ] = d W B ( 1 - r ) B 1 - C ,
d W = d W B ( 1 - r ) [ 1 - B 1 - C ] .
d ϕ = d ϕ B ( 1 - r ) [ A / ( 1 - C ) ] .
η = 0 λ c d ϕ ( λ ) 0 d W ( λ ) = 0 λ c d ϕ B ( 1 - r ) A 1 - C 0 d W B ( 1 - r ) [ 1 - B 1 - C ] .
A = 1 - R 1 ,             B = ( 1 - r ) R 1 ,             C = r R 1 .
0 λ c d ϕ ( λ ) = 1 - R v 1 - r R v ( 1 - H ) η B , 0 , 1 W B .
0 d W ( λ ) = 0 λ c d W ( λ ) + λ c d W ( λ ) = 1 - R v 1 - r R v ( 1 - H ) W B + 1 - R i 1 - r R i H W B .
η G , R v , R i = ( 1 - k v ) ( 1 - H ) ( 1 - k v ) ( 1 - H ) + ( 1 - k i ) H ,
k v = ( 1 - r ) R v 1 - r R v k i = ( 1 - r ) R i 1 - r R i .