Abstract

A new method for the in situ optical determination of the soot-cluster monomer particle radius a, the number of monomers per cluster N, and the fractal dimension D is presented. The method makes use of a comparison of the volume-equivalent sphere radius determined from scattering–extinction measurements RSE and the radius of gyration Rg, which is determined from the optical structure factor. The combination of these data with the measured turbidity permits for a novel measurement of D. The parameters a and N are obtained from a graphical network-analysis scheme that compares RSE and Rg. Corrections for cluster polydispersity are presented. The effects of uncertainty in various input parameters and assumptions are discussed. The method is illustrated by an application to data obtained from a premixed methane–oxygen flame, and reasonable values of a, N, and D are obtained.

© 1992 Optical Society of America

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  1. A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
    [CrossRef]
  2. A. D’Alessio, “Laser light scattering and fluorescence diagnostics of rich flames,” in Particulate Carbon, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), p. 207.
  3. W. Hinds, P. C. Reist, “Aerosol measurement by laser Doppler spectroscopy,” Aerosol Sci. 3, 501–514, 515–529 (1982).
    [CrossRef]
  4. G. B. King, C. M. Sorensen, T. W. Lester, J. F. Merklin, “Photon correlation spectroscopy used as a particle size diagnostic in sooting flames,” Appl. Opt. 21, 976–978 (1982).
    [CrossRef] [PubMed]
  5. S. M. Scrivner, T. W. Taylor, C. M. Sorensen, J. F. Merklin, “Soot particles size distribution measurements in a premixed flame using photon correlation spectroscopy,” Appl. Opt. 25, 291–297 (1986).
    [CrossRef] [PubMed]
  6. W. L. Flower, “Optical measurements of soot formation in flames,” Combust. Sci. Technol. 33, 17–33 (1983).
    [CrossRef]
  7. M. E. Weill, P. Flament, G. Gousebet, “Diameters and number densities of soot particles in premixed flat flames, propane/oxygen,” Appl. Opt. 22, 2407–2409 (1983).
    [CrossRef] [PubMed]
  8. M. E. Weill, N. Lhuissier, G. Gouesbet, “Mean diameters and number densities of soot particles in premixed flat flames CH4-O2 by diffusion broadening spectroscopy,” Appl. Opt. 25, 1676–1683 (1986).
    [CrossRef] [PubMed]
  9. S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
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    [CrossRef]
  14. R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
    [CrossRef]
  15. T. Freltoft, J. K. Kjems, S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
    [CrossRef]
  16. R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
    [CrossRef]
  17. M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
    [CrossRef]
  18. J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
    [CrossRef]
  19. A. J. Hurd, W. L. Flower, “In situ growth and structure of fractal silica aggregates in a flame,” J. Colloid Interface Sci. 122, 178–192 (1988).
    [CrossRef]
  20. H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
    [CrossRef]
  21. S. Gangopadhyay, I. Elminyawi, C. M. Sorensen, “Optical structure factor measurements of soot particles in a premixed flame,” Appl. Opt. 25, 4859–4864 (1991).
    [CrossRef]
  22. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  23. G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
    [CrossRef]
  24. S. J. Harris, A. M. Weiner, “Surface growth of soot particles in premixed ethylene/air flames,” Combust. Sci. Technol. 31, 155–167 (1983).
    [CrossRef]
  25. S. J. Harris, A. M. Weiner, “Determination of the rate constant for soot surface growth,” Combust. Sci. Technol. 32, 267–276 (1983).
    [CrossRef]
  26. W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” J. Heat Transfer 91, 100–104 (1969).
    [CrossRef]
  27. J. Lahaye, G. Prado, “Morphology and internal structure of carbon blacks and soot,” in Particulate Carbon, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), p. 33.
  28. P. G. van Dongen, M. H. Ernst, “Dynamic scaling in the kinetics of clustering,” Phys. Rev. Lett. 54, 1396–1394 (1985).
    [CrossRef] [PubMed]
  29. S. C. Graham, A. Robinson, “A comparison of numerical solutions to the self-preserving size distribution for aerosol coagulation in the free-molecular regime,” J. Aerosol Sci. 7, 261–273 (1976).
    [CrossRef]
  30. F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
    [CrossRef]
  31. K. W. Lee, “Change of particle size distribution during Brownian coagulation,” J. Colloid Interface Sci. 92, 315–325 (1983).
    [CrossRef]
  32. C. M. Megaridis, R. A. Dobbins, “Comparison of soot growth and oxidation in smoking and non-smoking ethylene diffusion flames,” Combust. Sci. Technol. 66, 1–16 (1989).
    [CrossRef]
  33. E. R. Ramer, J. F. Merklin, C. M. Sorensen, “The effect of benzene doping on the sootiness of a premixed methane-oxygen flame,” Combust. Sci. Technol. (to be published).

1991

S. Gangopadhyay, I. Elminyawi, C. M. Sorensen, “Optical structure factor measurements of soot particles in a premixed flame,” Appl. Opt. 25, 4859–4864 (1991).
[CrossRef]

1989

J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
[CrossRef]

C. M. Megaridis, R. A. Dobbins, “Comparison of soot growth and oxidation in smoking and non-smoking ethylene diffusion flames,” Combust. Sci. Technol. 66, 1–16 (1989).
[CrossRef]

1988

A. J. Hurd, W. L. Flower, “In situ growth and structure of fractal silica aggregates in a flame,” J. Colloid Interface Sci. 122, 178–192 (1988).
[CrossRef]

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
[CrossRef]

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

1987

R. J. Samson, G. W. Mulholland, J. W. Gentry, “Structural analysis of soot agglomerates,” Langmuir 3, 273–281 (1987).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

1986

1985

P. G. van Dongen, M. H. Ernst, “Dynamic scaling in the kinetics of clustering,” Phys. Rev. Lett. 54, 1396–1394 (1985).
[CrossRef] [PubMed]

1983

K. W. Lee, “Change of particle size distribution during Brownian coagulation,” J. Colloid Interface Sci. 92, 315–325 (1983).
[CrossRef]

S. J. Harris, A. M. Weiner, “Surface growth of soot particles in premixed ethylene/air flames,” Combust. Sci. Technol. 31, 155–167 (1983).
[CrossRef]

S. J. Harris, A. M. Weiner, “Determination of the rate constant for soot surface growth,” Combust. Sci. Technol. 32, 267–276 (1983).
[CrossRef]

M. E. Weill, P. Flament, G. Gousebet, “Diameters and number densities of soot particles in premixed flat flames, propane/oxygen,” Appl. Opt. 22, 2407–2409 (1983).
[CrossRef] [PubMed]

W. L. Flower, “Optical measurements of soot formation in flames,” Combust. Sci. Technol. 33, 17–33 (1983).
[CrossRef]

1982

1979

S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
[CrossRef]

1976

S. C. Graham, A. Robinson, “A comparison of numerical solutions to the self-preserving size distribution for aerosol coagulation in the free-molecular regime,” J. Aerosol Sci. 7, 261–273 (1976).
[CrossRef]

1972

F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
[CrossRef]

1969

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” J. Heat Transfer 91, 100–104 (1969).
[CrossRef]

Beretta, F.

A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Berry, M. V.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

D’Alessio, A.

A. D’Alessio, “Laser light scattering and fluorescence diagnostics of rich flames,” in Particulate Carbon, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), p. 207.

A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Dalzell, W. H.

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” J. Heat Transfer 91, 100–104 (1969).
[CrossRef]

DiLorenzo, A.

A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Dobbins, R. A.

C. M. Megaridis, R. A. Dobbins, “Comparison of soot growth and oxidation in smoking and non-smoking ethylene diffusion flames,” Combust. Sci. Technol. 66, 1–16 (1989).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

Elminyawi, I.

S. Gangopadhyay, I. Elminyawi, C. M. Sorensen, “Optical structure factor measurements of soot particles in a premixed flame,” Appl. Opt. 25, 4859–4864 (1991).
[CrossRef]

Ernst, M. H.

G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
[CrossRef]

P. G. van Dongen, M. H. Ernst, “Dynamic scaling in the kinetics of clustering,” Phys. Rev. Lett. 54, 1396–1394 (1985).
[CrossRef] [PubMed]

Flament, P.

Flower, W. L.

A. J. Hurd, W. L. Flower, “In situ growth and structure of fractal silica aggregates in a flame,” J. Colloid Interface Sci. 122, 178–192 (1988).
[CrossRef]

W. L. Flower, “Optical measurements of soot formation in flames,” Combust. Sci. Technol. 33, 17–33 (1983).
[CrossRef]

Forrest, S. R.

S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
[CrossRef]

Freltoft, T.

T. Freltoft, J. K. Kjems, S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
[CrossRef]

Friedlander, S. K.

F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
[CrossRef]

Gangopadhyay, S.

S. Gangopadhyay, I. Elminyawi, C. M. Sorensen, “Optical structure factor measurements of soot particles in a premixed flame,” Appl. Opt. 25, 4859–4864 (1991).
[CrossRef]

Gentry, J. W.

R. J. Samson, G. W. Mulholland, J. W. Gentry, “Structural analysis of soot agglomerates,” Langmuir 3, 273–281 (1987).
[CrossRef]

Gouesbet, G.

Gousebet, G.

Graham, S. C.

S. C. Graham, A. Robinson, “A comparison of numerical solutions to the self-preserving size distribution for aerosol coagulation in the free-molecular regime,” J. Aerosol Sci. 7, 261–273 (1976).
[CrossRef]

Harris, S. J.

S. J. Harris, A. M. Weiner, “Surface growth of soot particles in premixed ethylene/air flames,” Combust. Sci. Technol. 31, 155–167 (1983).
[CrossRef]

S. J. Harris, A. M. Weiner, “Determination of the rate constant for soot surface growth,” Combust. Sci. Technol. 32, 267–276 (1983).
[CrossRef]

Hidy, G. M.

F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
[CrossRef]

Hinds, W.

W. Hinds, P. C. Reist, “Aerosol measurement by laser Doppler spectroscopy,” Aerosol Sci. 3, 501–514, 515–529 (1982).
[CrossRef]

Hurd, A. J.

A. J. Hurd, W. L. Flower, “In situ growth and structure of fractal silica aggregates in a flame,” J. Colloid Interface Sci. 122, 178–192 (1988).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

King, G. B.

Kjems, J. K.

T. Freltoft, J. K. Kjems, S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
[CrossRef]

Lahaye, J.

J. Lahaye, G. Prado, “Morphology and internal structure of carbon blacks and soot,” in Particulate Carbon, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), p. 33.

Lai, F. S.

F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
[CrossRef]

Lee, K. W.

K. W. Lee, “Change of particle size distribution during Brownian coagulation,” J. Colloid Interface Sci. 92, 315–325 (1983).
[CrossRef]

Lester, T. W.

Lhuissier, N.

Mandelbrot, B.

B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1983).

Masi, S.

A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Megaridis, C. M.

C. M. Megaridis, R. A. Dobbins, “Comparison of soot growth and oxidation in smoking and non-smoking ethylene diffusion flames,” Combust. Sci. Technol. 66, 1–16 (1989).
[CrossRef]

R. A. Dobbins, C. M. Megaridis, “Morphology of flame-generated soot as determined by thermophoretic sampling,” Langmuir 3, 254–259 (1987).
[CrossRef]

Merklin, J. F.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

S. M. Scrivner, T. W. Taylor, C. M. Sorensen, J. F. Merklin, “Soot particles size distribution measurements in a premixed flame using photon correlation spectroscopy,” Appl. Opt. 25, 291–297 (1986).
[CrossRef] [PubMed]

G. B. King, C. M. Sorensen, T. W. Lester, J. F. Merklin, “Photon correlation spectroscopy used as a particle size diagnostic in sooting flames,” Appl. Opt. 21, 976–978 (1982).
[CrossRef] [PubMed]

E. R. Ramer, J. F. Merklin, C. M. Sorensen, “The effect of benzene doping on the sootiness of a premixed methane-oxygen flame,” Combust. Sci. Technol. (to be published).

Mountain, R. D.

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
[CrossRef]

Mulholland, G. W.

G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
[CrossRef]

R. D. Mountain, G. W. Mulholland, “Light scattering from simulated smoke agglomerates,” Langmuir 4, 1321–1326 (1988).
[CrossRef]

R. J. Samson, G. W. Mulholland, J. W. Gentry, “Structural analysis of soot agglomerates,” Langmuir 3, 273–281 (1987).
[CrossRef]

Nelson, J.

J. Nelson, “Test of a mean field theory for the optics of fractal clusters,” J. Mod. Opt. 36, 1031–1057 (1989).
[CrossRef]

Olivier, B. J.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Percival, I. C.

M. V. Berry, I. C. Percival, “Optics of fractal clusters such as smoke,” Opt. Acta 33, 577–591 (1986).
[CrossRef]

Pich, J.

F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
[CrossRef]

Prado, G.

J. Lahaye, G. Prado, “Morphology and internal structure of carbon blacks and soot,” in Particulate Carbon, D. C. Siegla, G. W. Smith, eds. (Plenum, New York, 1981), p. 33.

Ramer, E. R.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

E. R. Ramer, J. F. Merklin, C. M. Sorensen, “The effect of benzene doping on the sootiness of a premixed methane-oxygen flame,” Combust. Sci. Technol. (to be published).

Reist, P. C.

W. Hinds, P. C. Reist, “Aerosol measurement by laser Doppler spectroscopy,” Aerosol Sci. 3, 501–514, 515–529 (1982).
[CrossRef]

Robinson, A.

S. C. Graham, A. Robinson, “A comparison of numerical solutions to the self-preserving size distribution for aerosol coagulation in the free-molecular regime,” J. Aerosol Sci. 7, 261–273 (1976).
[CrossRef]

Samson, R. J.

G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
[CrossRef]

R. J. Samson, G. W. Mulholland, J. W. Gentry, “Structural analysis of soot agglomerates,” Langmuir 3, 273–281 (1987).
[CrossRef]

Sarofim, A. F.

W. H. Dalzell, A. F. Sarofim, “Optical constants of soot and their application to heat-flux calculations,” J. Heat Transfer 91, 100–104 (1969).
[CrossRef]

A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Scrivner, S. M.

Sinha, S. K.

T. Freltoft, J. K. Kjems, S. K. Sinha, “Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering,” Phys. Rev. B 33, 269–275 (1986).
[CrossRef]

Sorensen, C. M.

S. Gangopadhyay, I. Elminyawi, C. M. Sorensen, “Optical structure factor measurements of soot particles in a premixed flame,” Appl. Opt. 25, 4859–4864 (1991).
[CrossRef]

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

S. M. Scrivner, T. W. Taylor, C. M. Sorensen, J. F. Merklin, “Soot particles size distribution measurements in a premixed flame using photon correlation spectroscopy,” Appl. Opt. 25, 291–297 (1986).
[CrossRef] [PubMed]

G. B. King, C. M. Sorensen, T. W. Lester, J. F. Merklin, “Photon correlation spectroscopy used as a particle size diagnostic in sooting flames,” Appl. Opt. 21, 976–978 (1982).
[CrossRef] [PubMed]

E. R. Ramer, J. F. Merklin, C. M. Sorensen, “The effect of benzene doping on the sootiness of a premixed methane-oxygen flame,” Combust. Sci. Technol. (to be published).

Taylor, T. W.

van Dongen, P. G.

P. G. van Dongen, M. H. Ernst, “Dynamic scaling in the kinetics of clustering,” Phys. Rev. Lett. 54, 1396–1394 (1985).
[CrossRef] [PubMed]

Venitozzi, C.

A. D’Alessio, A. DiLorenzo, A. F. Sarofim, F. Beretta, S. Masi, C. Venitozzi, “Soot formation in methane-oxygen flames,” in Fifteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1975), p. 1427.
[CrossRef]

Weill, M. E.

Weiner, A. M.

S. J. Harris, A. M. Weiner, “Determination of the rate constant for soot surface growth,” Combust. Sci. Technol. 32, 267–276 (1983).
[CrossRef]

S. J. Harris, A. M. Weiner, “Surface growth of soot particles in premixed ethylene/air flames,” Combust. Sci. Technol. 31, 155–167 (1983).
[CrossRef]

Witten, T. A.

S. R. Forrest, T. A. Witten, “Long-range correlations in smoke-particle aggregates,” J. Phys. A 12, L109–L117 (1979).
[CrossRef]

Zhang, H. X.

H. X. Zhang, C. M. Sorensen, E. R. Ramer, B. J. Olivier, J. F. Merklin, “In-situ optical structure factor measurements of an aggregating soot aerosol,” Langmuir 4, 867–871 (1988).
[CrossRef]

Aerosol Sci.

W. Hinds, P. C. Reist, “Aerosol measurement by laser Doppler spectroscopy,” Aerosol Sci. 3, 501–514, 515–529 (1982).
[CrossRef]

Appl. Opt.

Combust. Sci. Technol.

S. J. Harris, A. M. Weiner, “Surface growth of soot particles in premixed ethylene/air flames,” Combust. Sci. Technol. 31, 155–167 (1983).
[CrossRef]

S. J. Harris, A. M. Weiner, “Determination of the rate constant for soot surface growth,” Combust. Sci. Technol. 32, 267–276 (1983).
[CrossRef]

W. L. Flower, “Optical measurements of soot formation in flames,” Combust. Sci. Technol. 33, 17–33 (1983).
[CrossRef]

C. M. Megaridis, R. A. Dobbins, “Comparison of soot growth and oxidation in smoking and non-smoking ethylene diffusion flames,” Combust. Sci. Technol. 66, 1–16 (1989).
[CrossRef]

J. Aerosol Sci.

S. C. Graham, A. Robinson, “A comparison of numerical solutions to the self-preserving size distribution for aerosol coagulation in the free-molecular regime,” J. Aerosol Sci. 7, 261–273 (1976).
[CrossRef]

J. Colloid Interface Sci.

F. S. Lai, S. K. Friedlander, J. Pich, G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39, 395–405 (1972).
[CrossRef]

K. W. Lee, “Change of particle size distribution during Brownian coagulation,” J. Colloid Interface Sci. 92, 315–325 (1983).
[CrossRef]

A. J. Hurd, W. L. Flower, “In situ growth and structure of fractal silica aggregates in a flame,” J. Colloid Interface Sci. 122, 178–192 (1988).
[CrossRef]

J. Energy Fuels

G. W. Mulholland, R. J. Samson, R. D. Mountain, M. H. Ernst, “Cluster size distribution for free molecular aggregation,” J. Energy Fuels 2, 481–486 (1988).
[CrossRef]

J. Heat Transfer

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Figures (8)

Fig. 1
Fig. 1

Radius of gyration Rg or scattering–extinction radius RSE as a function of the height above burner for two different flames. Two runs are shown to demonstrate the reproducibility of the data.

Fig. 2
Fig. 2

Plot of the data in accordance with Eq. (22) for the C/O = 0.69 flame. Two runs are shown to demonstrate reproducibility. The slope of this graph is D/3 to imply D = 1.75 ± 0.10. The open diamond represents the correction of the closed diamond that joins it on the dashed line, which accounts for the variation in flame temperature with the height above burner. The dashed line of slope = 1 is where the correction would occur for any temperature variation.

Fig. 3
Fig. 3

Plot of data in accordance with Eq. (22) for the C/O = 0.75 flame. Two runs are shown to demonstrate reproducibility. The slope of this plot is D/3 to yield a best fit of D = 1.70 ± 0.10. Lines corresponding to D = 1.60 and D = 1.80 are drawn to illustrate the approximate error in the D measurement.

Fig. 4
Fig. 4

RSE versus Rg to use the network analysis of Eqs. (19) and (20) for the C/O = 0.69 flame. Lines of constant a and N are drawn. The filled circles represent the precision, determined from reproducibility, of our measurements. The error bars for RSE represent accuracy estimates.

Fig. 5
Fig. 5

RSE versus Rg to use the network analysis of Eqs. (19) and (20) for the C/O = 0.75 flame. Lines of constant a and N are drawn. The filled circles represent the precision, determined from reproducibility, of our measurements. The error bars for RSE represent accuracy estimates.

Fig. 6
Fig. 6

Monomer radius a under the assumption of a monodisperse size distribution as a function of height above burner h for each flame. A typical error, representing an accuracy estimate propagated from the accuracy of RSE, is given. Correction factors for polydispersity are given in Fig. 8.

Fig. 7
Fig. 7

Number of monomers per cluster N under the assumption of a monodisperse size distribution as a function of height above burner h for each flame. A typical error, representing an accuracy estimate propagated from the accuracy of RSE, is given. Correction factors for polydispersity are given in Fig. 8.

Fig. 8
Fig. 8

Correction factors Ca as shown in Eq. (41), and CN, as shown in Eq. (42), as a function of the fractal dimension D.

Tables (1)

Tables Icon

Table 1 Comparison of the Scaling Function Moments, mi = ∫xiϕ(x)dx, of the Self-Preserving Size Distribution

Equations (43)

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N ~ R g D ,
S ( q ) ~ q - D ,
σ abs mon = 4 π k a 3 E ( m ) ,
d σ s mon d Ω = k 4 a 6 F ( m ) ,
E ( m ) = - Imag ( m 2 - 1 m 2 + 2 ) ,
F ( m ) = | m 2 - 1 m 2 + 2 | 2 .
σ abs c = N σ abs mon = 4 π N k a 3 E ,
d σ s c ( q ) d Ω = N 2 S ( x ) d σ s mom d Ω = k 4 a 6 F N 2 S ( x ) .
q = 2 k sin θ / 2 = 4 π λ - 1 sin θ / 2 ,
P s = I 0 c k 4 a 6 F N 2 n .
I T = I 0 exp ( - τ l ) ,
τ = n σ abs c + n σ scat c ,
τ = n σ abs c = n 4 π N k a 3 E
R S E 3 = a 3 N = 4 π E F k 3 P s / I 0 c τ
n = k 2 F E 2 τ 2 P s / I 0 c .
I ( q ) = I ( 0 ) S ( q R g ) .
I ( q ) = I ( 0 ) ( 1 - q 2 R g 2 )
N = k 0 ( R g / a ) D ,
R S E = k 0 1 / 3 a 1 - D / 3 R g D / 3 .
R S E = k 0 1 / D N 1 / 3 - 1 / D R g .
k 0 = ( 5 / 3 ) D / 2 .
R S E R S E , 0 ( τ 0 τ ) 1 / 3 = [ R g R g , 0 ( τ 0 τ ) 1 / 3 ] D / 3 .
M i = 0 N i n ( N ) d N .
P s = I 0 c k 4 a 6 F M 2 ,
τ = 4 π k a 3 E M 1 ,
R S E 3 = a 3 M 2 / M 1 .
I ( q ) = I ( N ) n ( N ) d N .
I ( N ) = N 2 S ( q R g )
N 2 ( 1 - q 2 R g 2 )
I ( q ) I ( 0 ) = N 2 ( 1 - q 2 R g 2 ) n ( N ) d N N 2 n ( N ) d N .
I ( q ) I ( 0 ) = 1 - 1 3 q 2 a 2 k 0 - 2 / D M 2 + 2 / D M 2 .
R g 2 = a 2 k 0 - 2 / D M 2 + 2 / D M 2 .
n ( N ) = M 1 s 1 - 2 ϕ ( N / s 1 ) = M 1 s 1 - 2 ϕ ( x ) ,
ϕ ( x ) = exp ( - x )
s 1 = M 1 / M 0 .
M i = M 1 s 1 i - 1 x i ϕ ( x ) d x = M 1 s 1 i - 1 Γ ( i + 1 ) ,
R S E 3 = 2 a 3 s 1 ,
R g 2 = a 2 k 0 - 2 / D s 1 2 / D Γ ( 3 + 2 / D ) Γ ( 3 ) ,
R S E 3 = a 3 - D k 0 [ 2 1 / D / 2 / Γ ( 3 + 2 / D ) ] D / 2 R g D .
R S E 3 = k 0 3 / D s 1 1 - 3 / D 2 5 / 2 Γ - 3 / 2 ( 3 + 2 D ) R g 3 .
a = [ 2 2 + D Γ - D ( 3 + 2 / D ) ] 1 / 2 ( D - 3 ) a mono = C a a mono ,
s 1 = [ Γ 3 ( 3 + 2 / D ) / 2 5 ] D / 2 ( D - 3 ) N mono = C N N mono .
( R S E R S E , 0 ) ( τ 0 τ ) 1 / 3 ( M 1 M 1 , 0 ) 1 / 3 = [ ( R g R g , 0 ) ( τ 0 τ ) 1 / 3 ( M 1 M 1 , 0 ) 1 / 3 ] D / 3 .

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