Abstract

A problem on the steady flow of heat through glass is treated mathematically. It is shown that though infrared energy is absorbed within a short distance by almost any glass, the reradiation of energy by the glass itself is sufficient at glass-melting temperatures to cause a considerable flow of heat. At 1200°C the radiant heat flow within a common window glass is of the order of fifty times the ordinary conductive flow. The effect is much less important at lower temperatures because the total radiant energy is smaller and because most glasses are very opaque to the longer infrared wavelengths involved.

© 1952 Optical Society of America

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Corrections

B. S. Kellett, "Errata: The Steady Flow of Heat through Hot Glass.," J. Opt. Soc. Am. 43, 1231-1231 (1953)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-43-12-1231

References

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  1. H. C. Hamaker, Philips Research Repts. 2, 103 (1947).
  2. H. O. McMahon, J. Opt. Soc. Am. 40, 376 (1950).
    [Crossref]
  3. W. J. R. Merren, J. Soc. Glass Technol. (to be published).
  4. G. V. McCauley, J. Am. Ceram. Soc. 8, 493 (1925).
    [Crossref]
  5. A. F. Van Zee and C. L. Babcock, J. Am. Ceram. Soc. 34, 244 (1951).
    [Crossref]

1951 (1)

A. F. Van Zee and C. L. Babcock, J. Am. Ceram. Soc. 34, 244 (1951).
[Crossref]

1950 (1)

1947 (1)

H. C. Hamaker, Philips Research Repts. 2, 103 (1947).

1925 (1)

G. V. McCauley, J. Am. Ceram. Soc. 8, 493 (1925).
[Crossref]

Babcock, C. L.

A. F. Van Zee and C. L. Babcock, J. Am. Ceram. Soc. 34, 244 (1951).
[Crossref]

Hamaker, H. C.

H. C. Hamaker, Philips Research Repts. 2, 103 (1947).

McCauley, G. V.

G. V. McCauley, J. Am. Ceram. Soc. 8, 493 (1925).
[Crossref]

McMahon, H. O.

Merren, W. J. R.

W. J. R. Merren, J. Soc. Glass Technol. (to be published).

Van Zee, A. F.

A. F. Van Zee and C. L. Babcock, J. Am. Ceram. Soc. 34, 244 (1951).
[Crossref]

J. Am. Ceram. Soc. (2)

G. V. McCauley, J. Am. Ceram. Soc. 8, 493 (1925).
[Crossref]

A. F. Van Zee and C. L. Babcock, J. Am. Ceram. Soc. 34, 244 (1951).
[Crossref]

J. Opt. Soc. Am. (1)

Philips Research Repts. (1)

H. C. Hamaker, Philips Research Repts. 2, 103 (1947).

Other (1)

W. J. R. Merren, J. Soc. Glass Technol. (to be published).

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Figures (5)

Fig. 1
Fig. 1

Radiant energy arriving per second on an element of glass of unit cross section and thickness dx.

Fig. 2
Fig. 2

Calculated temperature distribution through common window glass 100 cm thick separating blackbody zones at 1400°C and 1000°C. The heat flow is expressed as a percentage of the heat that would be transferred by radiation if the glass were removed.

Fig. 3
Fig. 3

Heat flow through glass 100 cm thick vs absorption coefficient of the hot glass for wavelengths near to 2μ. The curves for the two sets of boundary conditions are indistinguishable.

Fig. 4
Fig. 4

Calculated temperature distribution through common window glass 1 cm thick separating blackbody zones at 1400°C and 1000°C. The heat flow is expressed as a percentage of the heat that would be transferred by radiation if the glass were removed.

Fig. 5
Fig. 5

Heat flow through glass 1 cm thick vs absorption coefficient of the hot glass for wavelengths near to 2μ.

Equations (36)

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α d x ( R + S ) - 2 α d x σ θ 4 + k d x · d 2 θ / d x 2 = 0 ,
k d 2 θ / d x 2 - 2 α σ θ 4 + α ( R + S ) = 0.
d R = - α d x R + α d x σ θ 4                         d R / d x = α ( σ θ 4 - R ) .
d S / d x = α ( S - σ θ 4 ) ,
k d 2 θ / d x 2 = α ( 2 σ θ 4 - R - S ) = d R / d x - d S / d x .
d R / d x + d S / d x = α ( S - R ) = - α k d θ / d x - α H .
R = 1 2 ( k d θ / d x + H - α k θ - α H x + A )
S = 1 2 ( - k d θ / d x - H - α k θ - α H x + A ) .
k d 2 θ / d x 2 - 2 α σ θ 4 - α 2 k θ - α 2 H x + α A = 0.
θ 4 = θ ¯ 4 [ 1 + ( θ - θ ¯ ) / θ ¯ ] 4 = 4 θ θ ¯ 3 - 3 θ ¯ 4 ,
k d 2 θ / d x 2 - θ ( α 2 k + 8 α σ θ ¯ 3 ) = α 2 H x - α A - 6 α σ θ ¯ 4 ,
θ = B e - m x + C e m x - ( α 2 H x - α A - 6 α σ θ ¯ 4 ) / m 2 k ,
m 2 = α 2 + 8 α σ θ ¯ 3 / k .
R = ( 1 - ρ ) σ θ 1 4 + ρ S .
S = ( 1 - ρ ) σ θ 2 4 + ρ R             at             x = d .
- m B + m C - α 2 H / m 2 k = 0
- m B e - m d + m C e m d - α 2 H / m 2 k = 0 ,
α H [ e - m x - e - m ( d - x ) ] / m ( 1 + e - m d ) + 2 σ θ 4 + α k θ + α H x = A .
α H ( 1 - e - m d ) / m ( 1 + e - m d ) + 2 σ θ 0 4 + α k θ 0 = A
- α H ( 1 - e - m d ) / m ( 1 + e - m d ) + 2 σ θ d 4 + α k θ d + α H d = A ,
2 σ θ 1 4 = H ( 1 + ρ ) / ( 1 - ρ ) - α k θ 0 + A
2 σ θ 2 4 = - H ( 1 + ρ ) / ( 1 - ρ ) - α k θ d - α H d + A .
H = σ ( θ 1 4 - θ 2 4 ) [ ( 1 + ρ ) / ( 1 - ρ ) + 1 2 α d ] + α k ( θ 0 - θ d ) 2 + α d
H [ 1 + ρ 1 - ρ + α ( 1 - e - m d ) m ( 1 + e - m d ) ] = 2 σ ( θ 1 4 - θ 0 4 ) = 2 σ ( θ d 4 - θ 2 4 ) .
θ = B [ e - m x - e - m ( d - x ) ] - ( α 2 H x - α A - 6 α σ θ ¯ 4 ) / m 2 k .
B = ( θ 1 - θ 2 - α 2 H d / m 2 k ) / 2 ( 1 - e - m d ) .
d θ / d x = - m B [ e - m x + e - m ( d - x ) ] - α 2 H / m 2 k
d 2 θ / d x 2 = m 2 B [ e - m x - e - m ( d - x ) ] .
m 2 k B [ e - m x - e - m ( d - x ) ] / α - 2 σ θ 4 - α k θ - α H x + A = 0 ,
m 2 k ( θ 1 - θ 2 - α 2 H d / m 2 k ) / 2 α - 2 σ θ 1 4 - α k θ 1 + A = 0
- m 2 k ( θ 1 - θ 2 - α 2 H d / m 2 k ) / 2 α - 2 σ θ 2 4 - α k θ 2 - α H d + A = 0.
m 2 = 2 α σ ( θ 1 4 - θ 2 4 ) / k ( θ 1 - θ 2 ) + α 2 .
2 σ θ 1 4 = k 1 + ρ 1 - ρ [ - m B ( 1 + e - m d ) - α 2 H m 2 k ] + H 1 + ρ 1 - ρ α k θ 1 + A
2 σ θ 2 4 = k 1 + ρ 1 - ρ [ m B ( 1 + e - m d ) + α 2 H m 2 k ] - H 1 + ρ 1 - ρ α k θ 2 - α H d + A ,
H = 2 σ ( θ 1 4 - θ 2 4 ) + α k ( θ 1 - θ 2 ) + m k ( θ 1 - θ 2 ) ( 1 + ρ ) ( 1 + e - m d ) / ( 1 - ρ ) ( 1 - e - m d ) 2 ( 1 + ρ ) / ( 1 - ρ ) + α d - 2 α 2 ( 1 + ρ ) / m 2 ( 1 - ρ ) + α 2 d ( 1 + ρ ) ( 1 + e - m d ) / m ( 1 - ρ ) ( 1 - e - m d ) .
d = 100 cm ;             θ 1 = 1673 ° K ; θ 2 = 1273 ° K ; α = 0.3 cm - 1 ;             ρ = 0.04 ;             k = 0.0022 cal / cm ° K sec ; σ = 1.37 × 10 - 12 cal / cm 2 ° K 4 sec .