Abstract

A combined system using laser Doppler velocimetry for velocity measurements and a top-hat laser beam (THLB) technique for simultaneous sizing is described. Measurements have been carried out on various sprays produced by a TSI-3050 generator. Results are compared with scattering theory, free-fall velocity computations, and holography measurements. The conclusion is that the measurement obtained from the combined system are satisfactory.

© 1986 Optical Society of America

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  1. G. Gouesbet, G. Grehan, “Laser Doppler Systems for Plasma Diagnostics: A Review and Prospective Paper,” in Proceedings, International Symposium on Plasma Chemistry, Zürich, 27 Aug.–1 Sept. (1979).
  2. G. Gouesbet, “A Review on Measurements of Particle Velocities and Diameters by Laser Techniques, with Emphasis on Thermal Plasmas,” Plasma Chem. Plasma Process. 5, 91 (1985).
    [CrossRef]
  3. W. M. Farmer, “Measurement of Particle Size, Number Density, and Velocity Using a Laser Interferometer,” Appl. Opt. 11, 2603 (1972).
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    [CrossRef] [PubMed]
  5. W. M. Farmer, “Visibility of Large Spheres Observed with a Laser Velocimeter: a Simple Model,” Appl. Opt. 19, 3660 (1980).
    [CrossRef] [PubMed]
  6. D. Holve, S. A. Self, “Optical Particle Sizing for in situ Measurements. Part 1,” Appl. Opt. 18, 1632 (1979).
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  8. A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).
  9. N. A. Chigier, A. Ungut, J. Yule, “Particle Size and Velocity Measurements in Flames by Laser Anemometer,” paper presented at Seventeenth Symposium (International) on Combustion, Leeds, England (20–25 Aug. 1978), pp. 315–323.
  10. A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.
  11. V. Ghezzi, A. Goghe, F. Miot, “Droplet Size Measurement by Laser Interferometer,” in Proceedings, Fourth International Symposium on Air Breathing Engineering, Florida (1979).
  12. G. Gouesbet, G. Grehan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. Paris 13, 97 (1982).
    [CrossRef]
  13. G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
    [CrossRef]
  14. G. Gouesbet, B. Maheu, G. Grehan, “The Order of Approximation in a Theory of the Scattering of a Gaussian Beam by a Mie Scatter Center,” J. Opt. Paris, 16, 239 (1985).
    [CrossRef]
  15. G. Grehan, B. Maheu, G. Gouesbet, “Scattering of Laser Beam by Mie Scatter Centers. Numerical Results Using a Localized Approximation,” Appl. Opt.00, 000 (198x), submitted.
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    [CrossRef]
  18. W. J. Lentz, “Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions,” Appl. Opt. 15, 668 (1976).
    [CrossRef] [PubMed]
  19. G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
    [CrossRef] [PubMed]
  20. G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
    [CrossRef] [PubMed]
  21. G. Grehan, G. Gouesbet, C. Rabasse, “The Computer Program supermidi for Lorenz-Mie Theory and the Research of One-to-One Relationships for Particle Sizing,” in Proceedings, Symposium on Long Range and Short Range Optical Velocity Measurements, Institut Franco-Allemand de Saint Louis (15–18 Sept. 1980).
  22. G. Grehan, G. Gouesbet, C. Rabasse, “Montonic Relationships Between Scattered Powers and Diameters in Lorenz-Mie Theory for Simultaneous Velocimetry and Sizing of Single Particles,” Appl. Opt. 20, 796 (1981).
    [CrossRef] [PubMed]
  23. G. Gouesbet, G. Grehan, “Advances in Quasi-Elastic Scattering of Light with Emphasis on Simultaneous Measurements of Velocities and Sizes of Particles Embedded in Flows,” presented at AIAA Sixteenth Thermophysics Conference, Palo Alto, CA, (23–25 June 1981), paper 81-1124.
  24. D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Comparative Measurements of Calibrated Droplets Using Gabor Holography and Corrected Top-Hat Laser Beam Sizing with Discussion of Simultaneous Velocimetry,” in Proceedings, International Symposium on Applications of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, Portugal (5–7 June 1982).
  25. D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
    [CrossRef]
  26. A. Ungut, G. Grehan, G. Gouesbet, “Comparisons Between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911 (1981).
    [CrossRef] [PubMed]
  27. F. Durst, H. Umhauer, “Local Measurements of Particle Velocity, Size Distribution and Concentration with a Combined Laser Doppler Particle Sizing System,” in Proceedings, L.D.A. Conference, Copenhagen (1975).
  28. B. Ruck, “Untersuchung zur optischen messung von Teilhengrösse und Teilchengeschwindigkeit mit Streulichtmethoden,” Dissertation, U. Karlsruhe (1981).
  29. R. Kleine, G. Gouesbet, B. Ruck, “Lokale, optische Messungen von Teilchengeschwindigkeiten, Teilchengrössenverteilungen und Teilchenkozentrationen,” in Proceedings, Sensor 82, Essen, 12–14 Jan. (1982).
  30. B. Ruck, B. Pavlowski, “Combined Measurement of Particle Size and Velocity Distributions in Ducts,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).
  31. K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).
  32. C. J. Bates, H. L. Yeoman, L. E. Drain, “Development and Applications of a Two-Colour Particle Sizing Instrument,” presented at IUTAM Symposium on Measurement Techniques in Two-phase Flow, Nancy, France (1983).
  33. C. F. Hess, “Nonintrusive Optical Single-Particle Counter for Measuring the Size and Velocity of Droplets in a Spray,” Appl. Opt. 23, 4375 (1984).
    [CrossRef] [PubMed]
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  35. D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 00, 159 (1980).
  36. Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
    [CrossRef]
  37. M. R. Latta, K. Jain, “Beam Intensity Uniformization by Minor Folding,” Opt. Commun. 49, 435 (1984).
    [CrossRef]
  38. Y. Belvaux, S. P. S. Virdi, “A Method for Obtaining a Uniform Non-Gaussian Laser Illumination,” Opt. Commun. 15, 193 (1975).
    [CrossRef]
  39. Ealing Optics Catalogue, 1982.
  40. W. H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 496 (1981).
    [CrossRef]
  41. W. B. Veldkamp, C. J. Kastner, “Beam Profile Shaping for Laser Radars that Use Detector Arrays,” Appl. Opt. 21, 345 (1982).
    [CrossRef] [PubMed]
  42. M. Quintanilla, A. M. de Frutos, “Holographic Filter that Transforms a Gaussian into a Uniform Beam,” Appl. Opt. 20, 879 (1981).
    [CrossRef] [PubMed]
  43. G. Gouesbet, G. Grehan, R. Kleine, “Simultaneous Optical Measurements of Velocity and Size of Individual Particles in Flows,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).
  44. R. Dony, Graphismes scientifiques sur micro-ordinateur de la 2ème à la 3ème dimension, Masson, Paris, (1985).
  45. R. C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, 1976).
  46. R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).
  47. G. B. Wallis, One-Dimensional Two-Phase Flow (McGraw Hill, New York, 1969).
  48. G. Gouesbet et al., “Laser Optical Sizing from 100 Å to 1 mm Diameter and from 0 to 1 kg/m3 Concentration,” AIAA paper 85-1083, AIAA Twentieth Thermophysics Conference, Williamsburg, VA (19, 21 June 1985).
  49. G. Grehan, R. Kleine, G. Gouesbet, “Combined Measurements of Velocity and Size of Supermicronic Particles in Flows Using a Top-Hat Laser Beam Technique,” PSA 85, 16–19 Sept. (1985), Bedford, U.K., to be published by John Wiley & Sons.

1985 (3)

G. Gouesbet, “A Review on Measurements of Particle Velocities and Diameters by Laser Techniques, with Emphasis on Thermal Plasmas,” Plasma Chem. Plasma Process. 5, 91 (1985).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, “The Order of Approximation in a Theory of the Scattering of a Gaussian Beam by a Mie Scatter Center,” J. Opt. Paris, 16, 239 (1985).
[CrossRef]

1984 (4)

G. Gouesbet, G. Grehan, “Corrections for Mie Theory Given in “The Scattering of Light and Other Electromagnetic Radiations”: Comments,” Appl. Opt. 23, 4462 (1984).
[CrossRef]

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
[CrossRef]

C. F. Hess, “Nonintrusive Optical Single-Particle Counter for Measuring the Size and Velocity of Droplets in a Spray,” Appl. Opt. 23, 4375 (1984).
[CrossRef] [PubMed]

M. R. Latta, K. Jain, “Beam Intensity Uniformization by Minor Folding,” Opt. Commun. 49, 435 (1984).
[CrossRef]

1983 (2)

Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

1982 (2)

G. Gouesbet, G. Grehan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. Paris 13, 97 (1982).
[CrossRef]

W. B. Veldkamp, C. J. Kastner, “Beam Profile Shaping for Laser Radars that Use Detector Arrays,” Appl. Opt. 21, 345 (1982).
[CrossRef] [PubMed]

1981 (4)

1980 (2)

W. M. Farmer, “Visibility of Large Spheres Observed with a Laser Velocimeter: a Simple Model,” Appl. Opt. 19, 3660 (1980).
[CrossRef] [PubMed]

D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 00, 159 (1980).

1979 (3)

1977 (1)

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).

1976 (2)

W. M. Farmer, “Sample Space for Particle Size and Velocity Measuring Interferometers,” Appl. Opt. 15, 1984 (1976).
[CrossRef] [PubMed]

W. J. Lentz, “Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions,” Appl. Opt. 15, 668 (1976).
[CrossRef] [PubMed]

1975 (1)

Y. Belvaux, S. P. S. Virdi, “A Method for Obtaining a Uniform Non-Gaussian Laser Illumination,” Opt. Commun. 15, 193 (1975).
[CrossRef]

1972 (1)

Allano, D.

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
[CrossRef]

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Comparative Measurements of Calibrated Droplets Using Gabor Holography and Corrected Top-Hat Laser Beam Sizing with Discussion of Simultaneous Velocimetry,” in Proceedings, International Symposium on Applications of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, Portugal (5–7 June 1982).

Atakan, S.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).

Bates, C. J.

C. J. Bates, H. L. Yeoman, L. E. Drain, “Development and Applications of a Two-Colour Particle Sizing Instrument,” presented at IUTAM Symposium on Measurement Techniques in Two-phase Flow, Nancy, France (1983).

Belvaux, Y.

Y. Belvaux, S. P. S. Virdi, “A Method for Obtaining a Uniform Non-Gaussian Laser Illumination,” Opt. Commun. 15, 193 (1975).
[CrossRef]

Chigier, N. A.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).

N. A. Chigier, A. Ungut, J. Yule, “Particle Size and Velocity Measurements in Flames by Laser Anemometer,” paper presented at Seventeenth Symposium (International) on Combustion, Leeds, England (20–25 Aug. 1978), pp. 315–323.

A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.

Clift, R.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

de Frutos, A. M.

M. Quintanilla, A. M. de Frutos, “Holographic Filter that Transforms a Gaussian into a Uniform Beam,” Appl. Opt. 20, 879 (1981).
[CrossRef] [PubMed]

Dony, R.

R. Dony, Graphismes scientifiques sur micro-ordinateur de la 2ème à la 3ème dimension, Masson, Paris, (1985).

Drain, L. E.

C. J. Bates, H. L. Yeoman, L. E. Drain, “Development and Applications of a Two-Colour Particle Sizing Instrument,” presented at IUTAM Symposium on Measurement Techniques in Two-phase Flow, Nancy, France (1983).

Durst, F.

F. Durst, H. Umhauer, “Local Measurements of Particle Velocity, Size Distribution and Concentration with a Combined Laser Doppler Particle Sizing System,” in Proceedings, L.D.A. Conference, Copenhagen (1975).

Farmer, W. M.

Felton, P. G.

A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.

Ghezzi, V.

V. Ghezzi, A. Goghe, F. Miot, “Droplet Size Measurement by Laser Interferometer,” in Proceedings, Fourth International Symposium on Air Breathing Engineering, Florida (1979).

Goghe, A.

V. Ghezzi, A. Goghe, F. Miot, “Droplet Size Measurement by Laser Interferometer,” in Proceedings, Fourth International Symposium on Air Breathing Engineering, Florida (1979).

Gouesbet, G.

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, “The Order of Approximation in a Theory of the Scattering of a Gaussian Beam by a Mie Scatter Center,” J. Opt. Paris, 16, 239 (1985).
[CrossRef]

G. Gouesbet, “A Review on Measurements of Particle Velocities and Diameters by Laser Techniques, with Emphasis on Thermal Plasmas,” Plasma Chem. Plasma Process. 5, 91 (1985).
[CrossRef]

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
[CrossRef]

G. Gouesbet, G. Grehan, “Corrections for Mie Theory Given in “The Scattering of Light and Other Electromagnetic Radiations”: Comments,” Appl. Opt. 23, 4462 (1984).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. Paris 13, 97 (1982).
[CrossRef]

G. Grehan, G. Gouesbet, C. Rabasse, “Montonic Relationships Between Scattered Powers and Diameters in Lorenz-Mie Theory for Simultaneous Velocimetry and Sizing of Single Particles,” Appl. Opt. 20, 796 (1981).
[CrossRef] [PubMed]

A. Ungut, G. Grehan, G. Gouesbet, “Comparisons Between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911 (1981).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, C. Rabasse, “The Computer Program supermidi for Lorenz-Mie Theory and the Research of One-to-One Relationships for Particle Sizing,” in Proceedings, Symposium on Long Range and Short Range Optical Velocity Measurements, Institut Franco-Allemand de Saint Louis (15–18 Sept. 1980).

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Comparative Measurements of Calibrated Droplets Using Gabor Holography and Corrected Top-Hat Laser Beam Sizing with Discussion of Simultaneous Velocimetry,” in Proceedings, International Symposium on Applications of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, Portugal (5–7 June 1982).

G. Gouesbet, G. Grehan, “Advances in Quasi-Elastic Scattering of Light with Emphasis on Simultaneous Measurements of Velocities and Sizes of Particles Embedded in Flows,” presented at AIAA Sixteenth Thermophysics Conference, Palo Alto, CA, (23–25 June 1981), paper 81-1124.

R. Kleine, G. Gouesbet, B. Ruck, “Lokale, optische Messungen von Teilchengeschwindigkeiten, Teilchengrössenverteilungen und Teilchenkozentrationen,” in Proceedings, Sensor 82, Essen, 12–14 Jan. (1982).

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of Laser Beam by Mie Scatter Centers. Numerical Results Using a Localized Approximation,” Appl. Opt.00, 000 (198x), submitted.

G. Gouesbet, G. Grehan, “Laser Doppler Systems for Plasma Diagnostics: A Review and Prospective Paper,” in Proceedings, International Symposium on Plasma Chemistry, Zürich, 27 Aug.–1 Sept. (1979).

G. Gouesbet et al., “Laser Optical Sizing from 100 Å to 1 mm Diameter and from 0 to 1 kg/m3 Concentration,” AIAA paper 85-1083, AIAA Twentieth Thermophysics Conference, Williamsburg, VA (19, 21 June 1985).

G. Grehan, R. Kleine, G. Gouesbet, “Combined Measurements of Velocity and Size of Supermicronic Particles in Flows Using a Top-Hat Laser Beam Technique,” PSA 85, 16–19 Sept. (1985), Bedford, U.K., to be published by John Wiley & Sons.

G. Gouesbet, G. Grehan, R. Kleine, “Simultaneous Optical Measurements of Velocity and Size of Individual Particles in Flows,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

Grace, J. R.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

Grehan, G.

G. Gouesbet, B. Maheu, G. Grehan, “The Order of Approximation in a Theory of the Scattering of a Gaussian Beam by a Mie Scatter Center,” J. Opt. Paris, 16, 239 (1985).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
[CrossRef]

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
[CrossRef]

G. Gouesbet, G. Grehan, “Corrections for Mie Theory Given in “The Scattering of Light and Other Electromagnetic Radiations”: Comments,” Appl. Opt. 23, 4462 (1984).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. Paris 13, 97 (1982).
[CrossRef]

G. Grehan, G. Gouesbet, C. Rabasse, “Montonic Relationships Between Scattered Powers and Diameters in Lorenz-Mie Theory for Simultaneous Velocimetry and Sizing of Single Particles,” Appl. Opt. 20, 796 (1981).
[CrossRef] [PubMed]

A. Ungut, G. Grehan, G. Gouesbet, “Comparisons Between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911 (1981).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, C. Rabasse, “The Computer Program supermidi for Lorenz-Mie Theory and the Research of One-to-One Relationships for Particle Sizing,” in Proceedings, Symposium on Long Range and Short Range Optical Velocity Measurements, Institut Franco-Allemand de Saint Louis (15–18 Sept. 1980).

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Comparative Measurements of Calibrated Droplets Using Gabor Holography and Corrected Top-Hat Laser Beam Sizing with Discussion of Simultaneous Velocimetry,” in Proceedings, International Symposium on Applications of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, Portugal (5–7 June 1982).

G. Gouesbet, G. Grehan, “Advances in Quasi-Elastic Scattering of Light with Emphasis on Simultaneous Measurements of Velocities and Sizes of Particles Embedded in Flows,” presented at AIAA Sixteenth Thermophysics Conference, Palo Alto, CA, (23–25 June 1981), paper 81-1124.

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of Laser Beam by Mie Scatter Centers. Numerical Results Using a Localized Approximation,” Appl. Opt.00, 000 (198x), submitted.

G. Gouesbet, G. Grehan, “Laser Doppler Systems for Plasma Diagnostics: A Review and Prospective Paper,” in Proceedings, International Symposium on Plasma Chemistry, Zürich, 27 Aug.–1 Sept. (1979).

G. Grehan, R. Kleine, G. Gouesbet, “Combined Measurements of Velocity and Size of Supermicronic Particles in Flows Using a Top-Hat Laser Beam Technique,” PSA 85, 16–19 Sept. (1985), Bedford, U.K., to be published by John Wiley & Sons.

G. Gouesbet, G. Grehan, R. Kleine, “Simultaneous Optical Measurements of Velocity and Size of Individual Particles in Flows,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

Hess, C. F.

Hironaga, K.

K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).

Hisida, K.

K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).

Holve, D.

Imaru, J.

K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).

Itagaki, Y.

Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
[CrossRef]

Jain, K.

M. R. Latta, K. Jain, “Beam Intensity Uniformization by Minor Folding,” Opt. Commun. 49, 435 (1984).
[CrossRef]

Kano, H.

K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).

Kastner, C. J.

W. B. Veldkamp, C. J. Kastner, “Beam Profile Shaping for Laser Radars that Use Detector Arrays,” Appl. Opt. 21, 345 (1982).
[CrossRef] [PubMed]

Kawamura, Y.

Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
[CrossRef]

Kerker, M.

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

Kleine, R.

R. Kleine, G. Gouesbet, B. Ruck, “Lokale, optische Messungen von Teilchengeschwindigkeiten, Teilchengrössenverteilungen und Teilchenkozentrationen,” in Proceedings, Sensor 82, Essen, 12–14 Jan. (1982).

G. Gouesbet, G. Grehan, R. Kleine, “Simultaneous Optical Measurements of Velocity and Size of Individual Particles in Flows,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

G. Grehan, R. Kleine, G. Gouesbet, “Combined Measurements of Velocity and Size of Supermicronic Particles in Flows Using a Top-Hat Laser Beam Technique,” PSA 85, 16–19 Sept. (1985), Bedford, U.K., to be published by John Wiley & Sons.

Kreuzer, J. L.

J. L. Kreuzer, “Coherent Light Optical System Yielding on Output Beam of Derived Intensity Distribution and a Derived Equiphase Surface,” U.S. Patent3,476,463 (1969).

Latta, M. R.

M. R. Latta, K. Jain, “Beam Intensity Uniformization by Minor Folding,” Opt. Commun. 49, 435 (1984).
[CrossRef]

Lee, W. H.

W. H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 496 (1981).
[CrossRef]

Lentz, W. J.

Lisiecki, D.

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
[CrossRef]

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Comparative Measurements of Calibrated Droplets Using Gabor Holography and Corrected Top-Hat Laser Beam Sizing with Discussion of Simultaneous Velocimetry,” in Proceedings, International Symposium on Applications of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, Portugal (5–7 June 1982).

Maeda, M.

K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).

Maheu, B.

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, “The Order of Approximation in a Theory of the Scattering of a Gaussian Beam by a Mie Scatter Center,” J. Opt. Paris, 16, 239 (1985).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of Laser Beam by Mie Scatter Centers. Numerical Results Using a Localized Approximation,” Appl. Opt.00, 000 (198x), submitted.

Miot, F.

V. Ghezzi, A. Goghe, F. Miot, “Droplet Size Measurement by Laser Interferometer,” in Proceedings, Fourth International Symposium on Air Breathing Engineering, Florida (1979).

Pavlowski, B.

B. Ruck, B. Pavlowski, “Combined Measurement of Particle Size and Velocity Distributions in Ducts,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

Quintanilla, M.

M. Quintanilla, A. M. de Frutos, “Holographic Filter that Transforms a Gaussian into a Uniform Beam,” Appl. Opt. 20, 879 (1981).
[CrossRef] [PubMed]

Rabasse, C.

G. Grehan, G. Gouesbet, C. Rabasse, “Montonic Relationships Between Scattered Powers and Diameters in Lorenz-Mie Theory for Simultaneous Velocimetry and Sizing of Single Particles,” Appl. Opt. 20, 796 (1981).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, C. Rabasse, “The Computer Program supermidi for Lorenz-Mie Theory and the Research of One-to-One Relationships for Particle Sizing,” in Proceedings, Symposium on Long Range and Short Range Optical Velocity Measurements, Institut Franco-Allemand de Saint Louis (15–18 Sept. 1980).

Ruck, B.

R. Kleine, G. Gouesbet, B. Ruck, “Lokale, optische Messungen von Teilchengeschwindigkeiten, Teilchengrössenverteilungen und Teilchenkozentrationen,” in Proceedings, Sensor 82, Essen, 12–14 Jan. (1982).

B. Ruck, B. Pavlowski, “Combined Measurement of Particle Size and Velocity Distributions in Ducts,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

B. Ruck, “Untersuchung zur optischen messung von Teilhengrösse und Teilchengeschwindigkeit mit Streulichtmethoden,” Dissertation, U. Karlsruhe (1981).

Self, S. A.

Seng, C. A.

A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.

Shafer, D.

D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 00, 159 (1980).

Toyoda, K.

Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
[CrossRef]

Umhauer, H.

F. Durst, H. Umhauer, “Local Measurements of Particle Velocity, Size Distribution and Concentration with a Combined Laser Doppler Particle Sizing System,” in Proceedings, L.D.A. Conference, Copenhagen (1975).

Ungut, A.

A. Ungut, G. Grehan, G. Gouesbet, “Comparisons Between Geometrical Optics and Lorenz-Mie Theory,” Appl. Opt. 20, 2911 (1981).
[CrossRef] [PubMed]

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).

N. A. Chigier, A. Ungut, J. Yule, “Particle Size and Velocity Measurements in Flames by Laser Anemometer,” paper presented at Seventeenth Symposium (International) on Combustion, Leeds, England (20–25 Aug. 1978), pp. 315–323.

A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.

Veldkamp, W. B.

W. B. Veldkamp, C. J. Kastner, “Beam Profile Shaping for Laser Radars that Use Detector Arrays,” Appl. Opt. 21, 345 (1982).
[CrossRef] [PubMed]

Virdi, S. P. S.

Y. Belvaux, S. P. S. Virdi, “A Method for Obtaining a Uniform Non-Gaussian Laser Illumination,” Opt. Commun. 15, 193 (1975).
[CrossRef]

Wallis, G. B.

G. B. Wallis, One-Dimensional Two-Phase Flow (McGraw Hill, New York, 1969).

Wamba, S.

Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
[CrossRef]

Weber, M. E.

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

Yeoman, H. L.

C. J. Bates, H. L. Yeoman, L. E. Drain, “Development and Applications of a Two-Colour Particle Sizing Instrument,” presented at IUTAM Symposium on Measurement Techniques in Two-phase Flow, Nancy, France (1983).

Yule, A. J.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).

A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.

Yule, J.

N. A. Chigier, A. Ungut, J. Yule, “Particle Size and Velocity Measurements in Flames by Laser Anemometer,” paper presented at Seventeenth Symposium (International) on Combustion, Leeds, England (20–25 Aug. 1978), pp. 315–323.

AIAA J. Energy (1)

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle Size and Velocity Measurement by Laser Anemometry,” AIAA J. Energy 1, 220 (1977).

Appl. Opt. (6)

W. M. Farmer, “Sample Space for Particle Size and Velocity Measuring Interferometers,” Appl. Opt. 15, 1984 (1976).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, “Corrections for Mie Theory Given in “The Scattering of Light and Other Electromagnetic Radiations”: Comments,” Appl. Opt. 23, 4462 (1984).
[CrossRef]

G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

W. B. Veldkamp, C. J. Kastner, “Beam Profile Shaping for Laser Radars that Use Detector Arrays,” Appl. Opt. 21, 345 (1982).
[CrossRef] [PubMed]

M. Quintanilla, A. M. de Frutos, “Holographic Filter that Transforms a Gaussian into a Uniform Beam,” Appl. Opt. 20, 879 (1981).
[CrossRef] [PubMed]

Appl. Opt. (8)

J. Opt. Paris (1)

G. Gouesbet, G. Grehan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. Paris 13, 97 (1982).
[CrossRef]

J. Opt. Paris (2)

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, “The Order of Approximation in a Theory of the Scattering of a Gaussian Beam by a Mie Scatter Center,” J. Opt. Paris, 16, 239 (1985).
[CrossRef]

J. Phys. D (1)

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Droplet Sizing Using a “Top-Hat” Laser Beam Technique,” J. Phys. D 17, 43 (1984).
[CrossRef]

Opt. Commun. (2)

Y. Belvaux, S. P. S. Virdi, “A Method for Obtaining a Uniform Non-Gaussian Laser Illumination,” Opt. Commun. 15, 193 (1975).
[CrossRef]

W. H. Lee, “Method for Converting a Gaussian Laser Beam into a Uniform Beam,” Opt. Commun. 36, 496 (1981).
[CrossRef]

Opt. Commun. (2)

Y. Kawamura, Y. Itagaki, K. Toyoda, S. Wamba, “A Simple Optical Device for Generating Square Flat-Top Intensity Radiation from a Gaussian Laser Beam,” Opt. Commun. 48, 44 (1983).
[CrossRef]

M. R. Latta, K. Jain, “Beam Intensity Uniformization by Minor Folding,” Opt. Commun. 49, 435 (1984).
[CrossRef]

Opt. Laser Technol. (1)

D. Shafer, “Gaussian to Flat-Top Intensity Distributing Lens,” Opt. Laser Technol. 00, 159 (1980).

Plasma Chem. Plasma Process. (1)

G. Gouesbet, “A Review on Measurements of Particle Velocities and Diameters by Laser Techniques, with Emphasis on Thermal Plasmas,” Plasma Chem. Plasma Process. 5, 91 (1985).
[CrossRef]

Other (24)

G. Gouesbet, G. Grehan, “Laser Doppler Systems for Plasma Diagnostics: A Review and Prospective Paper,” in Proceedings, International Symposium on Plasma Chemistry, Zürich, 27 Aug.–1 Sept. (1979).

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of Laser Beam by Mie Scatter Centers. Numerical Results Using a Localized Approximation,” Appl. Opt.00, 000 (198x), submitted.

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

N. A. Chigier, A. Ungut, J. Yule, “Particle Size and Velocity Measurements in Flames by Laser Anemometer,” paper presented at Seventeenth Symposium (International) on Combustion, Leeds, England (20–25 Aug. 1978), pp. 315–323.

A. J. Yule, C. A. Seng, P. G. Felton, A. Ungut, N. A. Chigier, “A Laser Tomographic Investigation of Liquid Fuel Sprays,” in Proceedings, Eighteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1980, pp. 1501–1510.

V. Ghezzi, A. Goghe, F. Miot, “Droplet Size Measurement by Laser Interferometer,” in Proceedings, Fourth International Symposium on Air Breathing Engineering, Florida (1979).

Ealing Optics Catalogue, 1982.

J. L. Kreuzer, “Coherent Light Optical System Yielding on Output Beam of Derived Intensity Distribution and a Derived Equiphase Surface,” U.S. Patent3,476,463 (1969).

F. Durst, H. Umhauer, “Local Measurements of Particle Velocity, Size Distribution and Concentration with a Combined Laser Doppler Particle Sizing System,” in Proceedings, L.D.A. Conference, Copenhagen (1975).

B. Ruck, “Untersuchung zur optischen messung von Teilhengrösse und Teilchengeschwindigkeit mit Streulichtmethoden,” Dissertation, U. Karlsruhe (1981).

R. Kleine, G. Gouesbet, B. Ruck, “Lokale, optische Messungen von Teilchengeschwindigkeiten, Teilchengrössenverteilungen und Teilchenkozentrationen,” in Proceedings, Sensor 82, Essen, 12–14 Jan. (1982).

B. Ruck, B. Pavlowski, “Combined Measurement of Particle Size and Velocity Distributions in Ducts,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

K. Hisida, M. Maeda, J. Imaru, K. Hironaga, H. Kano, “Measurements of Size and Velocity of Particle in Two-Phase Flow by a Three Beam LDA System,” in Proceedings, International Symposium on Application of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon (5–7 July 1982).

C. J. Bates, H. L. Yeoman, L. E. Drain, “Development and Applications of a Two-Colour Particle Sizing Instrument,” presented at IUTAM Symposium on Measurement Techniques in Two-phase Flow, Nancy, France (1983).

G. Gouesbet, G. Grehan, “Advances in Quasi-Elastic Scattering of Light with Emphasis on Simultaneous Measurements of Velocities and Sizes of Particles Embedded in Flows,” presented at AIAA Sixteenth Thermophysics Conference, Palo Alto, CA, (23–25 June 1981), paper 81-1124.

D. Allano, G. Gouesbet, G. Grehan, D. Lisiecki, “Comparative Measurements of Calibrated Droplets Using Gabor Holography and Corrected Top-Hat Laser Beam Sizing with Discussion of Simultaneous Velocimetry,” in Proceedings, International Symposium on Applications of Laser-Doppler Anemometry to Fluid Mechanics, Lisbon, Portugal (5–7 June 1982).

G. Grehan, G. Gouesbet, C. Rabasse, “The Computer Program supermidi for Lorenz-Mie Theory and the Research of One-to-One Relationships for Particle Sizing,” in Proceedings, Symposium on Long Range and Short Range Optical Velocity Measurements, Institut Franco-Allemand de Saint Louis (15–18 Sept. 1980).

G. Gouesbet, G. Grehan, R. Kleine, “Simultaneous Optical Measurements of Velocity and Size of Individual Particles in Flows,” in Proceedings, Second International Symposium on Application of Laser Anemometry to Fluid Mechanics, Lisbon (2–4 July 1984).

R. Dony, Graphismes scientifiques sur micro-ordinateur de la 2ème à la 3ème dimension, Masson, Paris, (1985).

R. C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Cleveland, 1976).

R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops, and Particles (Academic, New York, 1978).

G. B. Wallis, One-Dimensional Two-Phase Flow (McGraw Hill, New York, 1969).

G. Gouesbet et al., “Laser Optical Sizing from 100 Å to 1 mm Diameter and from 0 to 1 kg/m3 Concentration,” AIAA paper 85-1083, AIAA Twentieth Thermophysics Conference, Williamsburg, VA (19, 21 June 1985).

G. Grehan, R. Kleine, G. Gouesbet, “Combined Measurements of Velocity and Size of Supermicronic Particles in Flows Using a Top-Hat Laser Beam Technique,” PSA 85, 16–19 Sept. (1985), Bedford, U.K., to be published by John Wiley & Sons.

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Figures (23)

Fig. 1
Fig. 1

Scattering geometry for the intimi or genmie program

Fig. 2
Fig. 2

Percentage of error on the scattered intensities for a particle lighted by a plane or a Gaussian wave.

Fig. 3
Fig. 3

Overall setup.

Fig. 4
Fig. 4

Module LD-OK-302.

Fig. 5
Fig. 5

Module LD-OG-10.

Fig. 6
Fig. 6

Absorption law of the Gaussian filter.

Fig. 7
Fig. 7

Module LD-OK-501.

Fig. 8
Fig. 8

Module LD-0-701.

Fig. 9
Fig. 9

General presentation of the electronics.

Fig. 10
Fig. 10

Dual probe.

Fig. 11
Fig. 11

Dual signal.

Fig. 12
Fig. 12

Digitized size signal.

Fig. 13
Fig. 13

Medium under study: geometry.

Fig. 14
Fig. 14

Three-dimensional representation: water experiment 1.

Fig. 15
Fig. 15

Signal amplitude vs the measured velocity: water experiment 1.

Fig. 16
Fig. 16

Three-dimensional representation: methanol experiment 1.

Fig. 17
Fig. 17

Linear regression for five experiments with water and five others with methanol.

Fig. 18
Fig. 18

Calibration curve.

Fig. 19
Fig. 19

Histogram for 500 water droplets.

Fig. 20
Fig. 20

Free-fall velocity vs measured velocity: orifice 100 μm.

Fig. 21
Fig. 21

Free-fall velocity vs measured velocity: orifice 50 μm.

Fig. 22
Fig. 22

Free-fall velocity vs measured velocity: orifice 35 μm.

Fig. 23
Fig. 23

Comparison between histogram obtained with the dual size/velocity apparatus or the size alone device.

Tables (3)

Tables Icon

Table I Typical Water Droplet Measurementsa

Tables Icon

Table II Summary of Results of Various Experiments for Water Droplets Where N Is the Number of Particles, S is the Slope of the Distribution, and Z is the Ordinate at Zero Velocity

Tables Icon

Table III Summary of Results of Various Experiments for Methanol Droplets Where N is the Number of Particles, S is the Slope of the Distribution, and Z is the Ordinate at Zero Velocity

Equations (3)

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

v f f = g ρ p d 2 18 μ f ,
Re d = v m d ρ f μ f ,
F = 3 π μ f d v ( 1 + 0.15 Re d 0 . 687 ) ,

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