Abstract

The absorption of radiation by molecular bands is calculated when the source emits its energy in discrete spectral lines. The results are compared with those for a blackbody source. The absorption of flame radiation by the molecular bands of an absorbing medium is derived for a number of different conditions when the spectral lines do not overlap. These results are then extended to the case when the overlapping of the spectral lines in either the source or absorber can be described either by the Elsasser model or by the statistical model. When the same spectral lines are involved in the emission and absorption processes, the absorptance of flame radiation is always greater than the absorptance of blackbody radiation by the same medium. All of the above results are derived both for the case when the absorbing medium is homogeneous and when the pressure, temperature, and amount of absorbing gas vary along the path.

© 1965 Optical Society of America

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References

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  1. J. A. Jamieson, R. M. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (McGraw-Hill, New York, 1963), pp. 43–101.
  2. G. N. Plass, Appl. Opt. 4, 69 (1965).; referred to in text as I.
    [CrossRef]
  3. G. N. Plass, J. Opt. Soc. Am. 50, 868 (1960).
    [CrossRef]
  4. P. J. Wyatt, V. R. Stull, G. N. Plass, J. Opt. Soc. Am. 52, 1209 (1962).
    [CrossRef]
  5. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
    [CrossRef]
  6. G. N. Plass, Appl. Opt. 2, 515 (1963).
    [CrossRef]

1965

1963

1962

1960

1958

Grube, R. H.

J. A. Jamieson, R. M. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (McGraw-Hill, New York, 1963), pp. 43–101.

Jamieson, J. A.

J. A. Jamieson, R. M. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (McGraw-Hill, New York, 1963), pp. 43–101.

McFee, R. M.

J. A. Jamieson, R. M. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (McGraw-Hill, New York, 1963), pp. 43–101.

Plass, G. N.

Richards, R. G.

J. A. Jamieson, R. M. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (McGraw-Hill, New York, 1963), pp. 43–101.

Stull, V. R.

Wyatt, P. J.

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Figures (5)

Fig. 1
Fig. 1

The absorptance of radiation from a line source by nonoverlapping spectral lines as a function of xa = Saua/2παa, where the subscript a refers to the absorbing medium. These curves are valid when the line source by itself can be represented by the weak line approximation (xl < 0.2). Curves for various values of q = βa/αl are shown. The uppermost limiting curve is the weak line limit.

Fig. 2
Fig. 2

The absorptance of radiation from a line source by nonoverlapping spectral lines as a function of q2xa. These curves are valid when the line source by itself can be represented by the weak line approximation (xl < 0.2). The uppermost limiting curve is the strong line limit.

Fig. 3
Fig. 3

The absorptance of radiation from a line source by nonoverlapping spectral lines as a function of q2xa. Since it is assumed that xl = 100, the line source by itself can be represented by the strong line approximation. The uppermost limiting curve is the strong line limit.

Fig. 4
Fig. 4

The ratio of the absorptance for a line source to the absorptance for a blackbody source as a function of xa. It is assumed that q = 1 and βl = βa = 0.0314. For one curve, xl < 0.2, and for the other curve, xl = 10. It is assumed that the spectral lines do not overlap.

Fig. 5
Fig. 5

The absorptance of radiation from a line source by a band of spectral lines which obeys the Elsasser model as a function of χ a = ( 1 2 β a 2 x a ) 1 / 2. For a given value of χa the absorptance is always greater than that from a blackbody source. It is assumed in this figure that both media can be represented by the strong line approximation. Thus, the following conditions must both be satisfied: xl > 2; xa > 2.

Equations (39)

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I = I b ( ν ) [ 1 - exp ( - i k l ( i ) u l ) ] ,
A = Δ ν [ 1 - exp ( - i k l ( i ) u l ) ] [ 1 - exp ( - i k a ( i ) u a ) ] d ν Δ ν [ 1 - exp ( - i k l ( i ) u l ) ] d ν ,
A l Δ ν = Δ ν [ 1 - exp ( - i k l ( i ) u l ) ] d ν .
A a Δ ν = Δ ν [ 1 - exp ( - i k a ( i ) u a ) ] d ν .
A l a Δ ν = W l a = Δ ν [ 1 - exp ( - i k l ( i ) u l - i k a ( i ) u a ) ] ,
A = ( A l + A a - A l a ) / A l .
τ = ( τ a - τ l a ) / ( 1 - τ l ) ,
A = S a u a π ( α l + α a ) = x a 2 1 + ( β l / β a ) ,
x = S u / 2 π α ,             β = 2 π α / d ,
A = 1 + β a x a 1 / 2 β l x l 1 / 2 - ( 1 + β a 2 x a β l 2 x l ) 1 / 2 .
A = 1 - e η [ 1 - ϕ ( η 1 / 2 ) ] ,
ϕ ( x ) = 2 π - 1 / 2 0 x e - y 2 d y , η = 2 β l - 2 β a 2 x a = S a α a u a / π α l 2 .
A = 2 π - 1 / 2 η 1 / 2 ,             η 1 ,
A = 1 - π - 1 / 2 η - 1 / 2 ,             η 1.
A = π 1 / 2 ξ - 1 / 2 x a { 1 - e ξ [ 1 - ϕ ( ξ 1 / 2 ) ] } ,
A = 2 x a ,             ξ 1 ,
A = π 1 / 2 ξ - 1 / 2 x a ( 1 - π - 1 / 2 ξ - 1 / 2 ) ,             ξ 1.
A a = ϕ [ ( ½ β a 2 x a ) 1 / 2 ] ,
A = { ϕ ( χ l ) + ϕ ( χ a ) - ϕ [ ( χ l 2 + χ a 2 ) 1 / 2 ] } / ϕ ( χ l ) ,
χ = ( ½ β 2 x ) 1 / 2 .
τ = Δ ν Δ ν d ν ( 1 ) d ν ( N ) 0 0 P ( S ( 1 ) ) d S ( 1 ) P ( S ( N ) ) × d S ( N ) [ 1 - exp ( - i = 1 N k l ( i ) u l ( i ) ) ] exp [ - i = 1 N k a ( i ) u a ( i ) ] , Δ ν Δ ν d ν ( 1 ) d ν ( N ) 0 0 P ( S ( 1 ) ) d S ( 1 ) P ( S ( N ) ) d S ( N ) [ 1 - exp ( - i = 1 N k ( i ) u ( i ) ) ]
τ = ( τ ¯ a N - τ ¯ l a N ) / ( 1 - τ ¯ l N ) ,
τ ¯ = 0 P ( S ) τ d S = 1 Δ ν Δ ν d ν 0 P ( S ) d S e - k u .
τ = exp ( - W ¯ a / d ) - exp ( - W ¯ l a / d ) 1 - exp ( - W ¯ l / d ) ,
A = 1 - τ = W / d = β f ( x ) ,
f ( x ) = x e - x [ I 0 ( x ) + I 1 ( x ) ] ,
W a / d = 1 - τ a = β a x a - ½ β a x a 2
A = Δ ν [ 1 - exp ( - i k l ( i ) u l ) ] [ 1 - exp ( - i 0 u a k a ( i ) d u ) ] d ν Δ ν [ 1 - exp ( - i k l ( i ) u l ) ] d ν ,
A a Δ ν = Δ ν [ 1 - exp ( - i 0 u a k a ( i ) d u ) ] d ν
A l a Δ ν = Δ ν [ 1 - exp ( - i k l ( i ) u l - i 0 u a k a ( i ) d u ) ] d ν ,
A = ( A l + A a - A l a ) / A l ,
A l a = A l + A a - S l u l π Δ ν 0 u a S a ( u ) α + α a ( u ) d u ,
A = 1 π 0 u a S a ( u ) α l + α a ( u ) d u .
A l a = ( A l 2 + A a 2 ) ½ ,
A l a Δ ν = 2 ( S l α l u l + 0 u a S a α a d u ) ½ ,
A l a = A a + A l e η [ 1 - ϕ ( η ) ½ ] ,
η = π - 1 α l - 2 0 u a S a u a d u ,
A = 1 - e η [ 1 - ϕ ( η ½ ) ] .
A l a = ϕ [ ( π / d ) ½ ( S l α l u l + 0 u a S a α a d u ) ½ ] .

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