Abstract

A nephelometer is presented that theoretically requires no absolute calibration. This instrument is used for determining the particle-size distribution of various scattering media (aerosols, fogs, rocket exhausts, engine plumes, and the like) from angular static light-scattering measurements. An inverse procedure is used, which consists of a least-squares method and a regularization scheme based on numerical filtering. To retrieve the distribution function one matches the experimental data with theoretical patterns derived from Mie theory. The main principles of the inverse method are briefly presented, and the nephelometer is then described with the associated partial calibration procedure. Finally, the whole granulometer system (inverse method and nephelometer) is validated by comparison of measurements of scattering media with calibrated monodisperse or known size distribution functions.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.
  2. M. E. Essawy, A. G. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
    [CrossRef]
  3. A. Delfour, B. Guillame, “Granulométrie de milieux à faible concentration de particules,” in Atmospheric Propagation Effects through Natural and Man-Made Obscurants for Visible to MM-Wave Radiation, AGARD Conference Proceedings 542 (Advisory Group for Aerospace Research and Development, Paris, 1993), pp. 35.1–35.10.
  4. H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).
  5. G. Mie, “Beitrage zur optik trüber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
    [CrossRef]
  6. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  7. D. Deirmendjian, “A survey of light scattering techniques used in remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
    [CrossRef]
  8. L. Hespel, A. Delfour, “Mie light-scattering granulometer with an adaptive numerical filtering method. I. Theory,” Appl. Opt. 39, 6897–6917 (2000).
    [CrossRef]
  9. C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
    [CrossRef]
  10. T. N. E. Greville, Theory and Applications of Spline Functions (Academic, Orlando, Fla., 1969).
  11. J. G. McWhiter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
    [CrossRef]
  12. Hj. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).
  13. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley–Interscience, New York, 1983).
  15. H. C. van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, New York, 1981).
  16. A. C. Holland, G. Gagne, “The scattering of polarized light by polydisperse systems of irregular particles,” Appl. Opt. 9, 1113–1121 (1970).
    [CrossRef] [PubMed]
  17. A. Holland, “Problem in calibrating a polar nephelometer,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), 247–254.
  18. D. E. Gray, American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1982).

2000

1983

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

1980

D. Deirmendjian, “A survey of light scattering techniques used in remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
[CrossRef]

M. E. Essawy, A. G. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

1978

J. G. McWhiter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

1970

1908

G. Mie, “Beitrage zur optik trüber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

1895

Hj. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).

Bohren, C. F.

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

De Boor, C.

C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, “A survey of light scattering techniques used in remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
[CrossRef]

Delfour, A.

L. Hespel, A. Delfour, “Mie light-scattering granulometer with an adaptive numerical filtering method. I. Theory,” Appl. Opt. 39, 6897–6917 (2000).
[CrossRef]

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

A. Delfour, B. Guillame, “Granulométrie de milieux à faible concentration de particules,” in Atmospheric Propagation Effects through Natural and Man-Made Obscurants for Visible to MM-Wave Radiation, AGARD Conference Proceedings 542 (Advisory Group for Aerospace Research and Development, Paris, 1993), pp. 35.1–35.10.

Delfour, A. G.

M. E. Essawy, A. G. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

Essawy, M. E.

M. E. Essawy, A. G. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

Gagne, G.

Girgis, W.

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

Granie, M.

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

Gray, D. E.

D. E. Gray, American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1982).

Greville, T. N. E.

T. N. E. Greville, Theory and Applications of Spline Functions (Academic, Orlando, Fla., 1969).

Guillame, B.

A. Delfour, B. Guillame, “Granulométrie de milieux à faible concentration de particules,” in Atmospheric Propagation Effects through Natural and Man-Made Obscurants for Visible to MM-Wave Radiation, AGARD Conference Proceedings 542 (Advisory Group for Aerospace Research and Development, Paris, 1993), pp. 35.1–35.10.

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Hespel, L.

Holland, A.

A. Holland, “Problem in calibrating a polar nephelometer,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), 247–254.

Holland, A. C.

Huffman, D. R.

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

Kuentzmann, P.

J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Martin, H.

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

McWhiter, J. G.

J. G. McWhiter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Mellin, Hj.

Hj. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).

Mie, G.

G. Mie, “Beitrage zur optik trüber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Morucci, J. P.

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

Pike, E. R.

J. G. McWhiter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Prévost, M.

J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Tarrin, P.

J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Traineau, J. C.

J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, New York, 1981).

Acta Soc. Sci. Fenn.

Hj. Mellin, “Über die fundamentale Wichtigkeit des Stazes von Cauchy für die Theorien der Gamma und hypergeometrischen Functionem,” Acta Soc. Sci. Fenn. 20, 1–115 (1895).

AIAA J.

M. E. Essawy, A. G. Delfour, “Determining size distribution of liquid nitrogen particles flowing in an airstream by scattered light detection,” AIAA J. 18, 665–668 (1980).
[CrossRef]

Ann. Phys. (Leipzig)

G. Mie, “Beitrage zur optik trüber medien speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Appl. Opt.

Innovations Tech. Biol. Med.

H. Martin, W. Girgis, M. Granie, J. P. Morucci, A. Delfour, “Laser nephelometer and applications,” Innovations Tech. Biol. Med. 4, 385–391 (1983).

J. Phys. A

J. G. McWhiter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A 11, 1729–1745 (1978).
[CrossRef]

Rev. Geophys. Space Phys.

D. Deirmendjian, “A survey of light scattering techniques used in remote monitoring of atmospheric aerosols,” Rev. Geophys. Space Phys. 18, 341–360 (1980).
[CrossRef]

Other

A. Holland, “Problem in calibrating a polar nephelometer,” in Light Scattering by Irregularly Shaped Particles, D. W. Schuerman, ed. (Plenum, New York, 1980), 247–254.

D. E. Gray, American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1982).

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Vol. 1 of the Series on Automatic Computation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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

H. C. van de Hulst, Light Scattering by Small Particles, 2nd ed. (Dover, New York, 1981).

J. C. Traineau, P. Kuentzmann, M. Prévost, P. Tarrin, A. Delfour, “Particle size distribution measurements in a subscale motor for the ARIANE 5 solid rocket booster,” presented at the AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference and Exhibit, Nashville, Tenn., 6–8 July 1992.

A. Delfour, B. Guillame, “Granulométrie de milieux à faible concentration de particules,” in Atmospheric Propagation Effects through Natural and Man-Made Obscurants for Visible to MM-Wave Radiation, AGARD Conference Proceedings 542 (Advisory Group for Aerospace Research and Development, Paris, 1993), pp. 35.1–35.10.

C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
[CrossRef]

T. N. E. Greville, Theory and Applications of Spline Functions (Academic, Orlando, Fla., 1969).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

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 (20)

Fig. 1
Fig. 1

Block diagram of the nephelometer.

Fig. 2
Fig. 2

Measurement of (a) transmitted and (b) scattered light.

Fig. 3
Fig. 3

Light-scattering collection system.

Fig. 4
Fig. 4

Notation for the ellipse displayed in Fig. 1: A, major axis; B, minor axis; C, interfoci distance; θ i , scattering angle; α i , encoder angular position.

Fig. 5
Fig. 5

Reference medium composed of two Lambertian diffusers (LD1, LD2) and a field stop aperture.

Fig. 6
Fig. 6

PMT signal (and its uncertainties) measured in the calibration medium described above (solid curve) and the best cosine function fitting the measured data (dashed curve) as a function of scattering angle.

Fig. 7
Fig. 7

Representation of the nephelometer calibration function K(θ) (and its uncertainties) as a function of scattering angle.

Fig. 8
Fig. 8

Relative variations as a time function of a Gaussian PSD subject to sedimentation effects described by Eq. (18).

Fig. 9
Fig. 9

Quartz cell geometry.

Fig. 10
Fig. 10

Representation of the relative variation ΔT/ T = [Tin) - T(0)]/T(0) of transmission function Tin) as a function of the internal scattering angle.

Fig. 11
Fig. 11

Representation of the corrective function Cext) as a function of the external scattering angle.

Fig. 12
Fig. 12

Normalized scatter diagrams for the monodisperse PSD ϕ = 1.02 µm ± 0.02 µm: Measurement and its uncertainties (open circles with error bars) and calculations (solid curve) as a function of the internal scattering angle.

Fig. 13
Fig. 13

Normalized scatter diagrams for the monodisperse latex PSD ϕ = 4.998 µm ± 0.035 µm: Measurement and its uncertainties (open circles with error bars) and calculations (solid curve) as a function of the internal scattering angle.

Fig. 14
Fig. 14

Comparison of normalized PSD (by volume) for the ϕ = 5 µm polydisperse latex solution: Log-normal distribution nvl(D) (solid curve), Gaussian distribution nvg(D) (curve with triangles), optimized bounding distributions fminv(D) (open circles) and fmaxv(D) (dashed curve).

Fig. 15
Fig. 15

Comparison of normalized PSD (by area) for the ϕ = 5 µm polydisperse latex solution: Log-normal distribution nal(D) (solid curve), Gaussian distribution nag(D) (curve with triangles), optimized bounding distributions fmina(D) (open circles) and fmaxa(D) (dashed curve).

Fig. 16
Fig. 16

Comparison of normalized PSD (by volume) for the ϕ = 10 µm polydisperse latex solution: Log-normal distribution nvl(D) (solid curve), Gaussian distribution nvg(D) (curves with triangles), optimized bounding distributions fminv(D) (open circles) and fmaxv(D) (dashed curve).

Fig. 17
Fig. 17

Comparison of normalized PSD (by area) for the ϕ = 10 µm polydisperse latex solution: Log-normal distribution nal(D) (solid curve), Gaussian distribution nag(D) (curve with triangles), optimized bounding distributions fmina(D) (open circles) and fmaxa(D) (dashed curve).

Fig. 18
Fig. 18

Comparison of normalized log-normal PSD’s for the bimodal polydisperse latex solution.

Fig. 19
Fig. 19

Comparison of normalized PSD’s (by area) for the bimodal polydisperse latex solution: log-normal distribution nal(D) (solid curve), Gaussian distribution nag(D) (curve with triangles), optimized bounding distributions fmina(D) (open circles) and fmaxa(D) (dashed curve).

Fig. 20
Fig. 20

Comparison of normalized PSD’s (by volume) for the bimodal polydisperse latex solution: log-normal distribution nvl(D) (solid curve), Gaussian distribution nvg(D) (curve with triangles), optimized bounding distributions fminv(D) (open circles) and fmaxv(D) (dashed curve).

Tables (2)

Tables Icon

Table 1 Theoretical Mean Diameters and SD’s (µm) of Polydisperse Distributions by Area and Volumea

Tables Icon

Table 2 Optimized Mean Diameters and SD’s (µm) of Polydisperse Distributions by Area and Volumea

Equations (35)

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

βmeasθi, ν=0 fDσDν, m*ν, θidD,
Kextν=0 fDσextDν, m*νdD,
βθi, ν=DminDmax hDKθiνDdD,  i1, N,
KθiνD=4σDν, θiexp-γνDπD2,
hD=πD2 expγνDfD4.
hpD=j=1Mp cjpSjpD.
tkp=lnDmin+k+1Δtp,
Δtp=1/ωmaxp-1.
tMp+2p=lnDmax.
βθi, ν=j=1Mp uijpcjp,
uijp=Dj-2pDj+2p KθiνDSjDdD,  j1, Mp,  i1, N.
hD=j=1M cjSjD.
Wθi=1Pj=1PWjθi,
ΔWθi=1P-1j=1PWjθi-Wθi21/2.
Wθi  0 fD|S2D, θi|2dD.
Kext=-1LlnWtWref,
ΔKext=1LΔWtWt2+ΔWrefWref21/2,
Kext=λ2π0 fDReS20, DdD.
Kθ=WmeasrefθWtheorrefθ.
Wmeascorrθ=WmeasθKθ.
ξt, ϕ=12500 ϕ2t.
Tθin=1-Rwqθin×1-Rqaθq×1+Rwq2θinexp-2τcos θin×1+RqwθqRqaθq,
Wmeasθin=WmeasθextTθin.
Cθext=WmeaswaterθextWmeaswater0.
Wmeasθin=Wmeasθext-CθextWmeas0TθinKθext.
WmeasθinN=WmeasθinWmeasθinN,
βθinN=βθinβθinN.
ngD=12πσgexp-D-Dg22σg2,
nlD=12πσl1Dexp-ln D-ln Dl22σl2,
nal,gD=πD24 nl,gD,
nvl,gD=πD36 nl,gD.
Dji= njiDDdD njiDdD,  j=a, v, i=g, l.
σji= njiDD-Dji2dD njiDdD1/2.
fminv,aD=fminv,aD×maxDnv,alDmaxDfminv,aD,
fmaxv,aD=fmaxv,aD×maxDnv,alDmaxDfmaxv,aD.

Metrics