Abstract

The emissivity of spherical carbon particles is calculated in both the infrared and visible regions of the spectrum. The scattering and absorption cross sections for individual particles are obtained from the Mie theory of scattering. A suitable dispersion equation is derived which represents the optical properties of carbon at flame temperatures. An expression is obtained for the radiation intensity emitted by a large number of dispersed particles which includes all higher order scattering processes. From these results the emissivity of carbon particles in flames is calculated for particle radii in the range from 50 to 800 A and for 109 to 1015 particles cm−2. In addition the emissivity is obtained for several different particle size distributions which are representative of actual flames. A quantitative explanation is given for the occurrence of the intensity maximum at shorter wavelengths than corresponds to the blackbody maximum at the same temperature.

© 1960 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. G. Gaydon and H. G. Wolfhard, Flames: Their Structure, Radiation, and Temperature (Chapman and Hall, Ltd., London, 1953); A. G. Gaydon, The Spectroscopy of Flames (Chapman and Hall, Ltd., London, 1957).
  2. G. Mie, Ann. Physik 25, 377 (1908).
    [Crossref]
  3. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 563 ff.
  4. O. Halpern and H. Hall, Phys. Rev. 73, 477 (1948).
    [Crossref]
  5. H. Senftleben and E. Benedict, Ann. Physik 54, 65 (1918).
  6. H. G. Wolfhard and W. G. Parker, Proc. Phys. Soc. (London) 62B, 523 (1949).
  7. W. G. Parker and H. G. Wolfhard, J. Chem. Soc.,  2038 (1950).
  8. P. A. Tesner, “Seventh international symposium on combustion, Royal Institution and Oxford University, England,” August 28–September 3, 1958 (to be published).

1950 (1)

W. G. Parker and H. G. Wolfhard, J. Chem. Soc.,  2038 (1950).

1949 (1)

H. G. Wolfhard and W. G. Parker, Proc. Phys. Soc. (London) 62B, 523 (1949).

1948 (1)

O. Halpern and H. Hall, Phys. Rev. 73, 477 (1948).
[Crossref]

1918 (1)

H. Senftleben and E. Benedict, Ann. Physik 54, 65 (1918).

1908 (1)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Benedict, E.

H. Senftleben and E. Benedict, Ann. Physik 54, 65 (1918).

Gaydon, A. G.

A. G. Gaydon and H. G. Wolfhard, Flames: Their Structure, Radiation, and Temperature (Chapman and Hall, Ltd., London, 1953); A. G. Gaydon, The Spectroscopy of Flames (Chapman and Hall, Ltd., London, 1957).

Hall, H.

O. Halpern and H. Hall, Phys. Rev. 73, 477 (1948).
[Crossref]

Halpern, O.

O. Halpern and H. Hall, Phys. Rev. 73, 477 (1948).
[Crossref]

Mie, G.

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Parker, W. G.

W. G. Parker and H. G. Wolfhard, J. Chem. Soc.,  2038 (1950).

H. G. Wolfhard and W. G. Parker, Proc. Phys. Soc. (London) 62B, 523 (1949).

Senftleben, H.

H. Senftleben and E. Benedict, Ann. Physik 54, 65 (1918).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 563 ff.

Tesner, P. A.

P. A. Tesner, “Seventh international symposium on combustion, Royal Institution and Oxford University, England,” August 28–September 3, 1958 (to be published).

Wolfhard, H. G.

W. G. Parker and H. G. Wolfhard, J. Chem. Soc.,  2038 (1950).

H. G. Wolfhard and W. G. Parker, Proc. Phys. Soc. (London) 62B, 523 (1949).

A. G. Gaydon and H. G. Wolfhard, Flames: Their Structure, Radiation, and Temperature (Chapman and Hall, Ltd., London, 1953); A. G. Gaydon, The Spectroscopy of Flames (Chapman and Hall, Ltd., London, 1957).

Ann. Physik (2)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

H. Senftleben and E. Benedict, Ann. Physik 54, 65 (1918).

J. Chem. Soc. (1)

W. G. Parker and H. G. Wolfhard, J. Chem. Soc.,  2038 (1950).

Phys. Rev. (1)

O. Halpern and H. Hall, Phys. Rev. 73, 477 (1948).
[Crossref]

Proc. Phys. Soc. (London) (1)

H. G. Wolfhard and W. G. Parker, Proc. Phys. Soc. (London) 62B, 523 (1949).

Other (3)

P. A. Tesner, “Seventh international symposium on combustion, Royal Institution and Oxford University, England,” August 28–September 3, 1958 (to be published).

A. G. Gaydon and H. G. Wolfhard, Flames: Their Structure, Radiation, and Temperature (Chapman and Hall, Ltd., London, 1953); A. G. Gaydon, The Spectroscopy of Flames (Chapman and Hall, Ltd., London, 1957).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 563 ff.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Geometry for calculation of radiation through surface dS emitted by volume dV.

Fig. 2
Fig. 2

Geometry for calculation of radiation emitted in volume dV′ scattered within volume dV.

Fig. 3
Fig. 3

Total cross section, σt, as a function of wavelength for carbon particles of various radii.

Fig. 4
Fig. 4

Scattering cross section, σs, as a function of wavelength for carbon particles of various radii.

Fig. 5
Fig. 5

Two particle size distributions used for the calculation of the emissivity of carbon particles.

Fig. 6
Fig. 6

Emissivity as a function of wavelength for carbon particles of 50 A radius. L is the number of particles per cm2.

Fig. 7
Fig. 7

Emissivity as a function of wavelength for carbon particles of 200 A radius. The results are shown when the scattering terms are included in the calculations and when they are omitted.

Fig. 8
Fig. 8

Emissivity as a function of wavelength for carbon particles of 500 A radius.

Fig. 9
Fig. 9

Emissivity as a function of wavelength for carbon particles of 800 A radius.

Fig. 10
Fig. 10

Emissivity as a function of wavelength for particle size distribution II.

Fig. 11
Fig. 11

Emissivity as a function of wavelength for particle size distribution III.

Fig. 12
Fig. 12

Emitted intensity at 2250°K as a function of wavelength for carbon particles of 200 A radius.

Fig. 13
Fig. 13

Emitted intensity at 2250°K as a function of wavelength for particle size distribution II.

Fig. 14
Fig. 14

Emitted intensity at 2250°K as a function of wavelength for particle size distribution III.

Fig. 15
Fig. 15

Four regions used for calculation of scattering contribution.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

σ s ( ρ , ν ) = 2 π a 2 ρ - 2 l = 1 ( 2 l + 1 ) ( a l 2 + b l 2 ) ,
σ t ( ρ , ν ) = 2 π a 2 ρ - 2 Re l = 1 ( 2 l + 1 ) ( a l + b l ) ,
a l = - { 1 + i N Y l ( ρ ) j l + 1 ( N ρ ) - j l ( N ρ ) Y l + 1 ( ρ ) N j l ( ρ ) j l + 1 ( N ρ ) - j l ( N ρ ) j l + 1 ( ρ ) } - 1
b l = - { 1 + i ( l + 1 ) ( N 2 - 1 ) Y l ( ρ ) j l ( N ρ ) + N ρ Y l ( ρ ) j l + 1 ( N ρ ) - N 2 ρ j l ( N ρ ) Y l + 1 ( ρ ) ( l + 1 ) ( N 2 - 1 ) j l ( ρ ) j l ( N ρ ) + N ρ j l ( ρ ) j l + 1 ( N ρ ) - N 2 ρ j l ( N ρ ) j l + 1 ( ρ ) } - 1 .
k 2 = μ ω 2 + i σ μ ω
m d 2 r d t 2 + m g d r d t + ω 0 2 m r = e E 0 e - i ω t ,
r = e m E 0 ω 0 2 - ω 2 - i ω g e - i ω t .
σ = j E = e n c E ( d r d t ) c = i e 2 m n c ω + i g c ,
P = e j n j r j = ( e 2 E / m ) j n j / ( ω 0 j 2 - ω 2 - i ω g j ) ,
n = j n j + n c ,
= D / E = 0 + ( P / E ) = 0 + ( e 2 / m ) j n j / ( ω 0 j 2 - ω 2 - i ω g j ) 1 - 1 3 ( e 2 / 0 m ) j n j / ( ω 0 j 2 - ω 2 - i ω j ) .
= 0 + ( e 2 / m ) j n j / ( ω 0 j 2 - ω 2 - i ω g j ) .
R = ( N 1 - 1 ) 2 + N 2 2 ( N 1 + 1 ) 2 + N 2 2 .
N 1 2 - N 2 2 = 1 + e 2 m 0 j n j ( ω 0 j 2 - ω 2 ) ( ω 0 j 2 - ω 2 ) 2 + ω 2 g j 2 - e 2 m 0 n c ω 2 + g c 2
2 N 1 N 2 = e 2 m 0 j n j ω g j ( ω 0 j 2 - ω 2 ) 2 + ω 2 g j 2 + e 2 m 0 n c g c ω ( g c 2 + ω 2 ) .
N 1 2 - N 2 2 = 1 + 6.448 × 10 32 4.062 × 10 35 - ω 2 + 3.224 × 10 32 9.549 × 10 33 - ω 2 + 3.224 × 10 32 5.217 × 10 33 - ω 2 + 6.348 × 10 32 ( 1.966 × 10 32 - ω 2 ) ( 1.956 × 10 32 - ω 2 ) 2 + 1.369 × 10 33 ω 2 - 3.05 × 10 31 2.323 × 10 31 + ω 2
2 N 1 N 2 = 6.347 × 10 32 × 3.70 × 10 16 ω ( 1.956 × 10 32 - ω 2 ) 2 + 1.369 × 10 33 ω 2 + 3.05 × 10 31 × 4.82 × 10 15 ( 2.323 × 10 31 + ω 2 ) ω .
d E = I b ( T , ν ) σ a ( ν ) d ν ,
σ t = σ a + σ s .
d E ( ν ; θ 0 , ϕ 0 ) = I b ( T , ν ) σ a ( ν ) ρ - 2 d S × cos θ 0 N d V exp ( - σ t N ρ ) ,
I 0 ( ν ; θ 0 , ϕ 0 ) = d E / d Ω d S = I b ( T , ν ) N σ a ( ν ) 0 z max exp ( - N σ t z sec θ 0 ) d z ,
I b ( T , ν ) σ a ( ν ) N d V d ω exp ( - N σ t ρ ) ,
I b ( T , ν ) ( N σ a / 4 π ) V ρ - 2 exp ( - N σ t ρ ) d V σ s ρ - 2 d S × cos θ 0 N d V exp ( - N σ t ρ ) .
I 0 ( ν ; θ 0 , ϕ 0 ) + I 1 ( ν ; θ 0 , ϕ 0 ) = I b ( T , ν ) N σ a 0 z max exp ( - N σ t z sec θ 0 ) × { 1 + ( N σ s / 4 π ) V ρ - 2 exp ( - N σ t ρ ) d V } d z .
I ( ν ; θ 0 , ϕ 0 ) = I b ( T , ν ) N σ a 0 z max exp ( - N σ t z sec θ 0 ) × { 1 + ( N σ s / 4 π ) V ρ - 2 exp ( - N σ t ρ ) × { 1 + ( N σ s / 4 π ) V ρ - 2 exp ( - N σ t ρ ) × { 1 + } d V } d V } d z .
σ ( ν ) Av N = 0 σ ( ν , r ) N ( r ) d r .
I 0 ( ν ; θ 0 ) = I b ( T , ν ) ( σ a / σ t ) × cos θ 0 { 1 - exp [ - σ t L ( θ 0 , ϕ 0 ) ] } ,
L ( θ 0 , ϕ 0 ) = N z max sec θ 0 .
N ( r ) d r = ( N / 159.52 ) exp { - [ ( r - 200 ) / 90 ] 2 } d r ,
N ( r ) d r = 4.75 × 10 5 N r - 3 e - 640 / r d r
I ( ν ; θ 0 ) = I b ( T , ν ) ( σ a / σ t ) cos θ 0 { 1 - exp [ - σ t L ] } + 1 2 I b ( T , ν ) ( σ a σ s / σ t 2 ) cos 2 θ 0 × { σ t L [ 1 - exp ( - σ t L ) ] E 1 ( σ t L 0 ) - [ 1 + exp ( - σ t L ) ] E 1 ( σ t L 0 ) + E 0 [ σ t ( L 0 + L ) ] - exp ( - σ t L ) E * [ σ t ( L - L 0 ) ] + sec θ 0 [ 1 - exp ( - σ t L 0 ) ] [ 1 - exp ( - σ t L ) ] + ln ( 1 + sec θ 0 ) + exp ( - σ t L ) ln ( sec θ 0 - 1 ) } ,
E 1 ( x ) = x e - t t - 1 d t , E * ( x ) = - - x e - t t - 1 d t ,
I ( ν ; θ 0 = ) = I b ( T , ν ) ( σ a / σ t ) [ 1 - exp ( - σ t L 0 ) ] + 1 2 I b ( T , ν ) ( σ a σ s / σ t 2 ) { [ 1 - exp ( - σ t L 0 ) ] 2 + ln 2 - γ exp ( - σ t L 0 ) - exp ( - σ t L 0 ) ln ( σ t L 0 ) + σ t L 0 [ 1 - exp ( - σ t L 0 ) ] E 1 ( σ t L 0 ) - [ 1 + exp ( - σ t L 0 ) ] E 1 ( σ t L 0 ) + E 1 ( 2 σ t L 0 ) } ,