Abstract

A simple method for determining the mean size, the concentration, and the refractive index of the monodispersion and the polydispersion of particles has been presented. The method is based on the empirical inversion of measurements of forward-angle light-scattering transmittance. The effects of particle size distribution and optical constants on forward-angle-scattering transmittance have been considered by using Mie theory. The proposed method has been used successfully for single latex spheres in water and polydispersed weakly absorbing particles of Al2O3 and SiO2 in the flow of the propane–air flame combustion products.

© 1997 Optical Society of America

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References

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  1. J. D. Felske, J. C. Ku, “A technique for determining the spectral refractive indices, size and number density of soot particles from light scattering and spectral extinction measurements in flames,” Combust. and Flame 91, 1–20 (1992).
    [CrossRef]
  2. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
    [CrossRef] [PubMed]
  3. J. C. Knight, D. Ball, G. N. Robertson, “Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution,” Appl. Opt. 30, 4795–4799 (1991).
    [CrossRef] [PubMed]
  4. M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
    [CrossRef] [PubMed]
  5. G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988–993 (1979).
    [CrossRef] [PubMed]
  6. M. R. Jones, K. H. Leong, M. Q. Brewster, B. P. Curry, “Inversion of light-scattering measurements for particle size and optical constants: experimental study,” Appl. Opt. 33, 4035–4041 (1994).
    [CrossRef] [PubMed]
  7. P. C. Ariesson, S. A. Self, R. H. Eustis, “Two-wavelength laser transmissometer for measurements of the mean size and concentration of coal ash droplets in combustion flows,” Appl. Opt. 19, 3775–3781 (1980).
    [CrossRef]
  8. G. Zaccanti, P. Bruscaglioni, “Method of measuring the phase function of a turbid medium in the small scattering angle range,” Appl. Opt. 28, 2156–2164 (1989).
    [CrossRef] [PubMed]
  9. F. D. Bryant, P. Latimer, “Real-time particle sizing by a computer-controlled transmittance photometer,” Appl. Opt. 24, 4280–4282 (1985).
    [CrossRef] [PubMed]
  10. A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media; Part 1: Monodispersions,” Appl. Opt. 17, 2900–2908 (1978).
    [CrossRef] [PubMed]
  11. A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media; Part 2: Polydispersions,” Appl. Opt. 17, 3169–3176 (1978).
    [CrossRef] [PubMed]
  12. C. F. Boren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  13. R. Hitzenberger, R. Rizzi, “Retrieved and measured aerosol mass size distributions: a comparison,” Appl. Opt. 25, 546–553 (1986).
    [CrossRef] [PubMed]
  14. L. Casperson, “Light extinction in polydisperse particulate systems,” Appl. Opt. 16, 3183–3189 (1977).
    [CrossRef] [PubMed]
  15. H. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967).
  16. S. A. El-Walkil, M. T. Attia, M. A. Madkour, “Radiative transfer in spherical and cylindrical media containing aerosols,” J. Quant. Spectrosc. Radiat. Transfer 45, 235–243 (1991).
    [CrossRef]
  17. R. A. Dobbins, L. Crocco, I. Glassman, AIAA J. 37, 1882–1886 (1963).
  18. J. S. Lindner, J. C. Luthe, J. W. Andol, “On-line evaluation of particulates in the MHD bottoming cycle,” Paper 94-2246, American Institute of Aeronautics and Astronautics, New York, 1994.
  19. K. A. Kusters, J. G. Wijers, D. Thoenes, “Particle sizing by laser diffraction spectrometry in anomalous regime,” Appl. Opt. 30, 4839–4847 (1991).
    [CrossRef] [PubMed]
  20. A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. (Russia) 30, 817–825 (1992).
  21. A. M. Lipaev, A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “The determination of the mean diameter and concentration of particles in flow of plasma from the measurements of transmittivity at small scattering angles,” in Proceedings of the XIth International Conference on MHD Electrical Power Generation (International Academic Publishers, Beijing, China, 1992), Vol. 2, pp. 999–1001.
  22. O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Radiation transfer in two phase high-temperature media of different geometry in an approximation of finite multiplicity of scattering: Part 1,” High Temp. (Russia) 32, 521–529 (1994).
  23. A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).
  24. T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
    [CrossRef]
  25. M. E. Milham, R. H. Frickel, J. F. Embury, D. H. Anderson, “Determination of optical constants from extinction measurements,” J. Opt. Soc. Am. 71, 1099–1106 (1981).
    [CrossRef]

1994

M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
[CrossRef] [PubMed]

M. R. Jones, K. H. Leong, M. Q. Brewster, B. P. Curry, “Inversion of light-scattering measurements for particle size and optical constants: experimental study,” Appl. Opt. 33, 4035–4041 (1994).
[CrossRef] [PubMed]

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Radiation transfer in two phase high-temperature media of different geometry in an approximation of finite multiplicity of scattering: Part 1,” High Temp. (Russia) 32, 521–529 (1994).

A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).

1992

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. (Russia) 30, 817–825 (1992).

J. D. Felske, J. C. Ku, “A technique for determining the spectral refractive indices, size and number density of soot particles from light scattering and spectral extinction measurements in flames,” Combust. and Flame 91, 1–20 (1992).
[CrossRef]

1991

1989

1986

1985

1981

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

M. E. Milham, R. H. Frickel, J. F. Embury, D. H. Anderson, “Determination of optical constants from extinction measurements,” J. Opt. Soc. Am. 71, 1099–1106 (1981).
[CrossRef]

1980

1979

1978

1977

1963

R. A. Dobbins, L. Crocco, I. Glassman, AIAA J. 37, 1882–1886 (1963).

Agrawal, Y. C.

Anderson, D. H.

Ariesson, P. C.

Attia, M. T.

S. A. El-Walkil, M. T. Attia, M. A. Madkour, “Radiative transfer in spherical and cylindrical media containing aerosols,” J. Quant. Spectrosc. Radiat. Transfer 45, 235–243 (1991).
[CrossRef]

Ball, D.

Boren, C. F.

C. F. Boren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Box, M. A.

Brewster, M. Q.

Bruscaglioni, P.

Bryant, F. D.

Casperson, L.

Crocco, L.

R. A. Dobbins, L. Crocco, I. Glassman, AIAA J. 37, 1882–1886 (1963).

Curry, B. P.

Deepak, A.

Dobbins, R. A.

R. A. Dobbins, L. Crocco, I. Glassman, AIAA J. 37, 1882–1886 (1963).

El-Walkil, S. A.

S. A. El-Walkil, M. T. Attia, M. A. Madkour, “Radiative transfer in spherical and cylindrical media containing aerosols,” J. Quant. Spectrosc. Radiat. Transfer 45, 235–243 (1991).
[CrossRef]

Embury, J. F.

Eustis, R. H.

Felske, J. D.

J. D. Felske, J. C. Ku, “A technique for determining the spectral refractive indices, size and number density of soot particles from light scattering and spectral extinction measurements in flames,” Combust. and Flame 91, 1–20 (1992).
[CrossRef]

Frickel, R. H.

Glassman, I.

R. A. Dobbins, L. Crocco, I. Glassman, AIAA J. 37, 1882–1886 (1963).

Gupta, R. P.

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

Hitzenberger, R.

Hottel, H.

H. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967).

Huffman, D. R.

C. F. Boren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jones, M. R.

Knight, J. C.

Kondrat’ev, A. B.

A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).

Ku, J. C.

J. D. Felske, J. C. Ku, “A technique for determining the spectral refractive indices, size and number density of soot particles from light scattering and spectral extinction measurements in flames,” Combust. and Flame 91, 1–20 (1992).
[CrossRef]

Kusters, K. A.

Latimer, P.

Leong, K. H.

Lipaev, A. M.

A. M. Lipaev, A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “The determination of the mean diameter and concentration of particles in flow of plasma from the measurements of transmittivity at small scattering angles,” in Proceedings of the XIth International Conference on MHD Electrical Power Generation (International Academic Publishers, Beijing, China, 1992), Vol. 2, pp. 999–1001.

Lowe, A.

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

Madkour, M. A.

S. A. El-Walkil, M. T. Attia, M. A. Madkour, “Radiative transfer in spherical and cylindrical media containing aerosols,” J. Quant. Spectrosc. Radiat. Transfer 45, 235–243 (1991).
[CrossRef]

Mai-Viet, T.

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

Milham, M. E.

Nefedov, A. P.

A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Radiation transfer in two phase high-temperature media of different geometry in an approximation of finite multiplicity of scattering: Part 1,” High Temp. (Russia) 32, 521–529 (1994).

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. (Russia) 30, 817–825 (1992).

A. M. Lipaev, A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “The determination of the mean diameter and concentration of particles in flow of plasma from the measurements of transmittivity at small scattering angles,” in Proceedings of the XIth International Conference on MHD Electrical Power Generation (International Academic Publishers, Beijing, China, 1992), Vol. 2, pp. 999–1001.

Petrov, O. F.

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Radiation transfer in two phase high-temperature media of different geometry in an approximation of finite multiplicity of scattering: Part 1,” High Temp. (Russia) 32, 521–529 (1994).

A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. (Russia) 30, 817–825 (1992).

A. M. Lipaev, A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “The determination of the mean diameter and concentration of particles in flow of plasma from the measurements of transmittivity at small scattering angles,” in Proceedings of the XIth International Conference on MHD Electrical Power Generation (International Academic Publishers, Beijing, China, 1992), Vol. 2, pp. 999–1001.

Riley, J. B.

Rizzi, R.

Robertson, G. N.

Samaryan, A. A.

A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).

Sarofim, A. F.

H. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967).

Self, S. A.

Shaw, G. E.

Thoenes, D.

Vaulina, O. S.

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Radiation transfer in two phase high-temperature media of different geometry in an approximation of finite multiplicity of scattering: Part 1,” High Temp. (Russia) 32, 521–529 (1994).

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. (Russia) 30, 817–825 (1992).

A. M. Lipaev, A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “The determination of the mean diameter and concentration of particles in flow of plasma from the measurements of transmittivity at small scattering angles,” in Proceedings of the XIth International Conference on MHD Electrical Power Generation (International Academic Publishers, Beijing, China, 1992), Vol. 2, pp. 999–1001.

Wall, T. F.

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

Wibberley, L. J.

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

Wijers, J. G.

Zaccanti, G.

AIAA J.

R. A. Dobbins, L. Crocco, I. Glassman, AIAA J. 37, 1882–1886 (1963).

Appl. Opt.

K. A. Kusters, J. G. Wijers, D. Thoenes, “Particle sizing by laser diffraction spectrometry in anomalous regime,” Appl. Opt. 30, 4839–4847 (1991).
[CrossRef] [PubMed]

R. Hitzenberger, R. Rizzi, “Retrieved and measured aerosol mass size distributions: a comparison,” Appl. Opt. 25, 546–553 (1986).
[CrossRef] [PubMed]

L. Casperson, “Light extinction in polydisperse particulate systems,” Appl. Opt. 16, 3183–3189 (1977).
[CrossRef] [PubMed]

J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
[CrossRef] [PubMed]

J. C. Knight, D. Ball, G. N. Robertson, “Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution,” Appl. Opt. 30, 4795–4799 (1991).
[CrossRef] [PubMed]

M. R. Jones, B. P. Curry, M. Q. Brewster, K. H. Leong, “Inversion of light-scattering measurements for particle size and optical constants: theoretical study,” Appl. Opt. 33, 4025–4034 (1994).
[CrossRef] [PubMed]

G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988–993 (1979).
[CrossRef] [PubMed]

M. R. Jones, K. H. Leong, M. Q. Brewster, B. P. Curry, “Inversion of light-scattering measurements for particle size and optical constants: experimental study,” Appl. Opt. 33, 4035–4041 (1994).
[CrossRef] [PubMed]

P. C. Ariesson, S. A. Self, R. H. Eustis, “Two-wavelength laser transmissometer for measurements of the mean size and concentration of coal ash droplets in combustion flows,” Appl. Opt. 19, 3775–3781 (1980).
[CrossRef]

G. Zaccanti, P. Bruscaglioni, “Method of measuring the phase function of a turbid medium in the small scattering angle range,” Appl. Opt. 28, 2156–2164 (1989).
[CrossRef] [PubMed]

F. D. Bryant, P. Latimer, “Real-time particle sizing by a computer-controlled transmittance photometer,” Appl. Opt. 24, 4280–4282 (1985).
[CrossRef] [PubMed]

A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media; Part 1: Monodispersions,” Appl. Opt. 17, 2900–2908 (1978).
[CrossRef] [PubMed]

A. Deepak, M. A. Box, “Forwardscattering corrections for optical extinction measurements in aerosol media; Part 2: Polydispersions,” Appl. Opt. 17, 3169–3176 (1978).
[CrossRef] [PubMed]

Combust. and Flame

J. D. Felske, J. C. Ku, “A technique for determining the spectral refractive indices, size and number density of soot particles from light scattering and spectral extinction measurements in flames,” Combust. and Flame 91, 1–20 (1992).
[CrossRef]

Combust. Sci. and Technol.

T. F. Wall, A. Lowe, L. J. Wibberley, T. Mai-Viet, R. P. Gupta, “Fly ash characteristics and radiative heat transfer in pulverized-coal-fired furnaces,” Combust. Sci. and Technol. 26, 107–121 (1981).
[CrossRef]

High Temp. (Russia)

O. S. Vaulina, A. P. Nefedov, O. F. Petrov, “Radiation transfer in two phase high-temperature media of different geometry in an approximation of finite multiplicity of scattering: Part 1,” High Temp. (Russia) 32, 521–529 (1994).

A. B. Kondrat’ev, A. P. Nefedov, O. F. Petrov, A. A. Samaryan, “Optical diagnostics of the conversion of coal particles in a flow of plasma of combustion products,” High Temp. (Russia) 32, 425–431 (1994).

A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “Calculation of the scattering and absorbing characteristics of spherical particles in high-temperature flows,” High Temp. (Russia) 30, 817–825 (1992).

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

S. A. El-Walkil, M. T. Attia, M. A. Madkour, “Radiative transfer in spherical and cylindrical media containing aerosols,” J. Quant. Spectrosc. Radiat. Transfer 45, 235–243 (1991).
[CrossRef]

Other

H. Hottel, A. F. Sarofim, Radiative Transfer (McGraw-Hill, New York, 1967).

A. M. Lipaev, A. P. Nefedov, O. F. Petrov, O. S. Vaulina, “The determination of the mean diameter and concentration of particles in flow of plasma from the measurements of transmittivity at small scattering angles,” in Proceedings of the XIth International Conference on MHD Electrical Power Generation (International Academic Publishers, Beijing, China, 1992), Vol. 2, pp. 999–1001.

J. S. Lindner, J. C. Luthe, J. W. Andol, “On-line evaluation of particulates in the MHD bottoming cycle,” Paper 94-2246, American Institute of Aeronautics and Astronautics, New York, 1994.

C. F. Boren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (9)

Fig. 1
Fig. 1

Diagram of transmission measurement scheme: 1, laser source; 2, particles; 3, aperture diaphragm; 4, aperture lens; 5, photodetector.

Fig. 2
Fig. 2

Angular distribution of the relative cross sections q = σext *(θd)/σ ext for polystyrene-latex particles in water: 1, d = 2 µm; 2, d = 4 µm; 3, d = 3 µm (λ = 0.633 µm).

Fig. 3
Fig. 3

Dependencies of qd = 11.5°, m, D) on the real part n of the refractive index for Gaussian distributions of transparent particles with various mean Sauter diameters, D32 (1, 2, and 4 µm; λ = 0.633), and standard deviation, Δ1/2 [1, Δ1/2 = 0 (single particle); 2, Δ1/2 = 15%; 3, Δ1/2 = 25%].

Fig. 4
Fig. 4

Model PSD functions f2 (r) and f3 (r) with the Sauter diameter D32 = 3.191 µm and with the various parameters given in Table 1.

Fig. 5
Fig. 5

The angular distribution qd, m, D) for model PSD functions f2 (r) and f3 (r) with the Sauter diameter of D32 = 3.191 µm, with the various parameters given in Table 1, and with m = 1.8–0.0 i and λ = 0.633 µm. The dashed line marks the qd, m, D) for the single particle of the diameter d = 3.191 µm.

Fig. 6
Fig. 6

Dependencies of qd, m, D) on the Sauter diameter D32 for single particles (Δ = 0) and Gaussian distributions of particles with various refractive indexes, m(λ = 0.633 µm): 1, m = 1.8 (Δ = 0); 2, m = 1.8 (Δ = 30%); 3, m = 1.6–0.6 i(Δ = 30%).

Fig. 7
Fig. 7

Dependence of the extinction efficiency ext on the Sauter diameter D32 (λ = 0.633 µm) for f2 (r): 1, b is varied, g = 0.5; 2, b = 0.6,g is varied; for f3 (r): 3, a is varied, σ1 = σ2 = 0.3a/(2 ln 2) 1/2; and for particles with differing refractive indices: (a) m = 1.58–0 i, (b) m = 1.8–0.6i.

Fig. 8
Fig. 8

Experimental setup.

Fig. 9
Fig. 9

Relative uncertainty, ε = Δd/d, calculated for different measurement errors, δmax(λ = 0.633 µm): 1, δmax = 5%; 2, δmax = 2%; *, the relative uncertainty of the retrieved size obtained from the FAST measurements (Table 2) for: (a) for latex spheres and ε′ = Δn/n, (b) for transparent particles.

Tables (3)

Tables Icon

Table 1 Parameters of Distributions f2(r) and f3(r) with a Sauter diameter, D32 = 3.191 µm

Tables Icon

Table 2 Results of Measurements of Monodispersions of Latex Particles in Water

Tables Icon

Table 3 Results of Measurements of Polydispersions of Weak-Absorbing Particles of Al2O3 and SiO2 in the Flow of the Combustion Products of the Propane–Air Flame

Equations (18)

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

T=I/I0=exp-npσ¯extl,
σ¯ext=0σextfrdr,
σext*θd=0σext-1/2σsca0θdpθsin θ dθfrdr,
Smin=i=1Nqicalc-qimeasNδiqimeas2,
f2r=c2rb exp-gr, c2=gb+1/b!,
f3r=22πσ1+σ2expr-a2/2σ1raexpr-a2/2σ2ra,
D32=20r3frdr/0r2frdr.
Cm=2τD32ρ0S¯/3lσ¯ext,
S¯=π0r2frdr.
δm=qθd, m, d-qm,
qm=1NKj=1Ki=1Nqθd, ni, kj, d.
θdθr=0.478/ρ.
θdθ*1.396/ρ.
θd2θr=0.956/ρ.
ρn-1<10.
qθdi, m, D=σext*θdiσext=lnIbi0/Iri0/Ibi/IrilnIai0/Iri0/Iai/Iri,
σext*θdi=lnIbi0/Iri0/Ibi/Irinpil, σ¯ext=lnIai0/Iri0/Iai/Irinpil.
qicalc1+δmaxqimeasqicalc1-δmax,

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