Abstract

The spectral emissivity of carbon dioxide from 1800–2500 cm−1 is calculated as a function of temperature, pressure, and amount of radiating gas. Careful consideration is given to the choice of a proper model to represent the emission when the overlapping of the spectral lines is taken into account. At elevated temperatures the random Elsasser model should be used since many vibrational bands overlap. The emissivities were calculated from the usual intensity and energy expressions on an electronic computer taking into account up to 890 000 spectral lines at 2400°K. The electric moment matrix element was evaluated by using the harmonic oscillator approximation to represent the vibrational states. The emission from C13O2 and from the transitions near 2100 and 1900 cm−1 was included in the calculation. The shift of the emission to lower frequencies as the temperature increases is quantitatively explained. The emission in this frequency region from any flame can readily be obtained from the results given here.

© 1959 Optical Society of America

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References

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  1. See, e.g., E. E. Bell and et al., Final Report, Contract AF 30(602)-1047 (Ohio State University, Columbus, Ohio, 1956).
  2. D. M. Dennison, Revs. Modern Phys. 3, 280 (1931).
    [Crossref]
  3. D. M. Dennison, Revs. Modern Phys. 12, 175 (1940).
    [Crossref]
  4. Benedict, Herman, and Silverman, J. Chem. Phys. 19, 1325 (1951).
    [Crossref]
  5. S. S. Penner, J. Appl. Phys. 25, 660 (1954).
    [Crossref]
  6. C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
    [Crossref]
  7. See, e.g., L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1949).
  8. W. S. Benedict and E. K. Plyler, in Energy Transfer in Hot Gases (U. S. Government Printing Office, Washington, D. C., 1954) p. 57.
  9. G. N. Plass, Publication U-238, (Aeronutronic Systems, Inc., Glendale, California, 1958). Calculations of the emission from hot CO2 have also been made recently by R.O’B. Carpenter.
  10. G. N. Plass, J. Meteorol. 9, 429 (1952).
    [Crossref]
  11. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
    [Crossref]
  12. L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
    [Crossref]
  13. Howard, Burch, and Williams, Contract AF 19(604)-516, Air Force Cambridge Research Center (Ohio State University Research Foundation, Columbus, Ohio, 1954).
  14. R. Tourin, Tech. Rept. No. 258 (Warner and Swasey Research Corporation, New York, 1954).

1958 (1)

1957 (1)

C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
[Crossref]

1956 (1)

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[Crossref]

1954 (1)

S. S. Penner, J. Appl. Phys. 25, 660 (1954).
[Crossref]

1952 (1)

G. N. Plass, J. Meteorol. 9, 429 (1952).
[Crossref]

1951 (1)

Benedict, Herman, and Silverman, J. Chem. Phys. 19, 1325 (1951).
[Crossref]

1940 (1)

D. M. Dennison, Revs. Modern Phys. 12, 175 (1940).
[Crossref]

1931 (1)

D. M. Dennison, Revs. Modern Phys. 3, 280 (1931).
[Crossref]

Bell, E. E.

See, e.g., E. E. Bell and et al., Final Report, Contract AF 30(602)-1047 (Ohio State University, Columbus, Ohio, 1956).

Benedict,

Benedict, Herman, and Silverman, J. Chem. Phys. 19, 1325 (1951).
[Crossref]

Benedict, W. S.

W. S. Benedict and E. K. Plyler, in Energy Transfer in Hot Gases (U. S. Government Printing Office, Washington, D. C., 1954) p. 57.

Burch,

Howard, Burch, and Williams, Contract AF 19(604)-516, Air Force Cambridge Research Center (Ohio State University Research Foundation, Columbus, Ohio, 1954).

Courtoy, C. P.

C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
[Crossref]

Dennison, D. M.

D. M. Dennison, Revs. Modern Phys. 12, 175 (1940).
[Crossref]

D. M. Dennison, Revs. Modern Phys. 3, 280 (1931).
[Crossref]

Eggers, D. F.

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[Crossref]

Herman,

Benedict, Herman, and Silverman, J. Chem. Phys. 19, 1325 (1951).
[Crossref]

Howard,

Howard, Burch, and Williams, Contract AF 19(604)-516, Air Force Cambridge Research Center (Ohio State University Research Foundation, Columbus, Ohio, 1954).

Kaplan, L. D.

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[Crossref]

Penner, S. S.

S. S. Penner, J. Appl. Phys. 25, 660 (1954).
[Crossref]

Plass, G. N.

G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).
[Crossref]

G. N. Plass, J. Meteorol. 9, 429 (1952).
[Crossref]

G. N. Plass, Publication U-238, (Aeronutronic Systems, Inc., Glendale, California, 1958). Calculations of the emission from hot CO2 have also been made recently by R.O’B. Carpenter.

Plyler, E. K.

W. S. Benedict and E. K. Plyler, in Energy Transfer in Hot Gases (U. S. Government Printing Office, Washington, D. C., 1954) p. 57.

Schiff, L. I.

See, e.g., L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1949).

Silverman,

Benedict, Herman, and Silverman, J. Chem. Phys. 19, 1325 (1951).
[Crossref]

Tourin, R.

R. Tourin, Tech. Rept. No. 258 (Warner and Swasey Research Corporation, New York, 1954).

Williams,

Howard, Burch, and Williams, Contract AF 19(604)-516, Air Force Cambridge Research Center (Ohio State University Research Foundation, Columbus, Ohio, 1954).

Can. J. Phys. (1)

C. P. Courtoy, Can. J. Phys. 35, 608 (1957).
[Crossref]

J. Appl. Phys. (1)

S. S. Penner, J. Appl. Phys. 25, 660 (1954).
[Crossref]

J. Chem. Phys. (2)

L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).
[Crossref]

Benedict, Herman, and Silverman, J. Chem. Phys. 19, 1325 (1951).
[Crossref]

J. Meteorol. (1)

G. N. Plass, J. Meteorol. 9, 429 (1952).
[Crossref]

J. Opt. Soc. Am. (1)

Revs. Modern Phys. (2)

D. M. Dennison, Revs. Modern Phys. 3, 280 (1931).
[Crossref]

D. M. Dennison, Revs. Modern Phys. 12, 175 (1940).
[Crossref]

Other (6)

See, e.g., E. E. Bell and et al., Final Report, Contract AF 30(602)-1047 (Ohio State University, Columbus, Ohio, 1956).

See, e.g., L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1949).

W. S. Benedict and E. K. Plyler, in Energy Transfer in Hot Gases (U. S. Government Printing Office, Washington, D. C., 1954) p. 57.

G. N. Plass, Publication U-238, (Aeronutronic Systems, Inc., Glendale, California, 1958). Calculations of the emission from hot CO2 have also been made recently by R.O’B. Carpenter.

Howard, Burch, and Williams, Contract AF 19(604)-516, Air Force Cambridge Research Center (Ohio State University Research Foundation, Columbus, Ohio, 1954).

R. Tourin, Tech. Rept. No. 258 (Warner and Swasey Research Corporation, New York, 1954).

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Figures (9)

F. 1
F. 1

The quantity β2x/pu=2πα0Sd−2 as a function of wave number for CO2 at 300°, 600°, 1200°, 1800°, 2400°K. The pressure is measured in atmospheres and u is the product of the pressure times the geometrical path length in the radiating gas.

F. 2
F. 2

The quantity S/d as a function of wave number for CO2 at 300°, 600°, 1200°, 1800°, 2400°K.

F. 3
F. 3

Emissivity of CO2 at 300°K, when x=Su/2πα>1.63. The units of p2X are atmos2 cm, where p is the pressure and X is the geometrical path length in the radiating gas.

F. 4
F. 4

Emissivity of CO2 at 600°K, when x=Su/2πα>1.63.

F. 5
F. 5

Emissivity of CO2 at 1200°K, when x=Su/2πα>1.63.

F. 6
F. 6

Emissivity of CO2 at 1800°K, when x=Su/2πα>1.63.

F. 7
F. 7

Emissivity of CO2 at 2400°K, when x=Su/2πα>1.63.

F. 8
F. 8

Emissivity of CO2 at 1200°K, when x=Su/2πα<0.2.

F. 9
F. 9

Emissivity of CO2 as a function of temperature at selected wave numbers, when p2X=100 atmos2 cm and x=Su/2πα>1.63.

Tables (2)

Tables Icon

Table I Number of terms included in intensity calculation.

Tables Icon

Table II High-temperature emissivities for 0.5 cm-atmos.

Equations (19)

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E r / h c = F ( J , l , υ 1 , υ 2 , υ 3 ) = ( J 2 + J l 2 ) × [ 0.3925 0.00058 ( υ 1 + 1 2 ) + 0.00045 ( υ 2 + 1 ) 0.00307 ( υ 3 + 1 2 ) ] + 1.7 ( l 2 + 1 ) ,
E υ / h c = G ( υ 1 , υ 2 , υ 3 ) = 1351.2 ( υ 1 + 1 2 ) + 672.2 ( υ 2 + 1 ) + 2396.4 ( υ 3 + 1 2 ) 0.3 ( υ 1 + 1 2 ) 2 1.3 ( υ 2 + 1 ) 2 12.5 ( υ 3 + 1 2 ) 2 + 5.7 ( υ 1 + 1 2 ) ( υ 2 + 1 ) 21.9 ( υ 1 + 1 2 ) ( υ 3 + 1 2 ) 11.0 ( υ 2 + 1 ) ( υ 3 + 1 2 ) ,
S = 8 π 3 ν N T 3 h c Q υ Q r exp [ E υ ( υ 1 , υ 2 , υ 3 ) + E r ( J , l ) k T ] × g J l ( R J l J l ) 2 β 2 ( 1 e h ν / k T ) ,
α ( υ 1 , υ 2 l , υ 3 υ 1 , υ 2 l , υ 3 ) = J , J S ( υ 1 , υ 2 l , υ 3 ; J υ 1 , υ 2 l , υ 3 ; J ) .
G ( υ 1 , υ 2 , υ 3 ) G ( υ 1 , υ 2 , υ 3 1 ) = 2374.4 21.9 υ 1 11.0 υ 2 25.0 υ 3 .
F ( υ 1 , υ 2 0 , υ 3 ; J ) F ( υ 1 , υ 2 0 , υ 3 1 ; J 1 ) = ( 0.7792 0.00307 J ) J ;
F ( υ 1 , υ 2 0 , υ 3 ; J ) F ( υ 1 , υ 2 0 , υ 3 1 ; J + 1 ) = ( 0.7822 + 0.00307 J ) J .
d I ν = I b ν k ν d u ν I ν k ν d u ν ,
I ν ( u ) = I ν ( 0 ) τ ν ( 0 , u ) + 0 u k ν ( υ ) I b ν ( υ ) τ ν ( υ , u ) d υ ,
τ ν ( υ , u ) = exp ( υ u k ν d u ) .
I ν = I b ν [ 1 exp ( k ν u ) ] ,
I = Δ ν I ν d ν = I b Δ ν [ 1 exp ( k ν u ) ] d ν .
x = S u / 2 π α
β = 2 π α / d ,
I = I b S u / d ,
I = I b ϕ [ ( 1 2 β 2 x ) 1 2 ] ;
I = I b { 1 exp [ ( 2 π 1 β 2 x ) 1 2 ] } ;
I = I b { 1 i = 1 N ( 1 ϕ [ ( 1 2 β i 2 x i ) 1 2 ] ) } ,
I ν ( u ) = I b 0 , ν [ 1 exp ( k 0 , ν u 0 ) ] exp [ k 1 , ν ( u u 0 ) ] + I b 1 , ν [ 1 exp ( k 1 , ν ( u u 0 ) ] ,