Abstract

The shape of droplets in liquid–liquid systems influences their mass and momentum transfer processes. The deviation from sphericity of rising droplets in liquid–liquid systems was investigated for different droplet sizes. Rainbow refractometry permits one to test, in this case, whether the use of laser-optical particle sizing will be correct or faulty. Since the assumption of spherical particle geometry is a general basis of laser-optical particle-sizing techniques such as rainbow refractometry or phase Doppler anemometry, deviation from the spherical shape results in a measuring error. A sphericity check based on rainbow refractometry is introduced.

© 1999 Optical Society of America

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References

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  1. H. Lohner, E. H. Schombacher, K. Bauckhage, “Characterization of droplets in liquid–liquid extraction by laser-optical measurement techniques,” Chem. Eng. Technol. 21, 337–341 (1998).
    [CrossRef]
  2. E. H. Schombacher, H. Lohner, K. Bauckhage, “Ein laseroptisches Messverfahren zur Untersuchung von Tropfen in der Flüssig/Flüssig-Extraktion,” Chem. Ing. Tech. 70, 713–717 (1998).
    [CrossRef]
  3. H. Schombacher, “Laseroptische Messtechniken zur Bestimmung prozessrelevanter Grössen in der Flüssig/Flüssig-Extraktion,” VDI-Fortschrittsber. 8, 79–118 (1997).
  4. P. Lehmann, E. H. Schombacher, H. Lohner, K. Bauckhage, “Characterization of nonspherical and oscillating droplets by phase-Doppler-anemometry and rainbow refractometry,” in PARTEC 98, Seventh European Symposium on Particle Characterization, R. Weichert, ed. (Nürnberg Messe GmbH Nürnberg/Germany, 1998), pp. 109–119.
  5. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
    [CrossRef]
  6. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  7. D. S. Langley, P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
    [CrossRef]
  8. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–530 (1984).
    [CrossRef]
  9. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
    [CrossRef] [PubMed]
  10. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [CrossRef] [PubMed]
  11. J. van Beeck, M. L. Riethmuller, “Rainbow interferometry with wire diffraction for simultaneous measurement of Droplet temperature, size, and velocity,” Part. Part. Charact. 14, 186–192 (1997).
  12. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
    [CrossRef]
  13. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, England, 1992).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
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    [CrossRef]
  16. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  19. J. van Beeck, M. L. Riethmuller, “Rainbow thermometry with a pulsed laser,” in Proceedings of Ninth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Vol. 1, D. F. G. Durao, ed. (Instituto Superior Técnico, Lisbon, 1998), pp. 18.6.1–7.
  20. R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops and Particles (Academic, New York, 1978).

1998 (3)

H. Lohner, E. H. Schombacher, K. Bauckhage, “Characterization of droplets in liquid–liquid extraction by laser-optical measurement techniques,” Chem. Eng. Technol. 21, 337–341 (1998).
[CrossRef]

E. H. Schombacher, H. Lohner, K. Bauckhage, “Ein laseroptisches Messverfahren zur Untersuchung von Tropfen in der Flüssig/Flüssig-Extraktion,” Chem. Ing. Tech. 70, 713–717 (1998).
[CrossRef]

D. S. Langley, P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
[CrossRef]

1997 (2)

H. Schombacher, “Laseroptische Messtechniken zur Bestimmung prozessrelevanter Grössen in der Flüssig/Flüssig-Extraktion,” VDI-Fortschrittsber. 8, 79–118 (1997).

J. van Beeck, M. L. Riethmuller, “Rainbow interferometry with wire diffraction for simultaneous measurement of Droplet temperature, size, and velocity,” Part. Part. Charact. 14, 186–192 (1997).

1996 (1)

1995 (1)

1991 (1)

1984 (1)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–530 (1984).
[CrossRef]

1980 (1)

1979 (1)

1977 (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

1976 (1)

J. D. Walker, “Multiple rainbows from single drops and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1910 (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

Bauckhage, K.

H. Lohner, E. H. Schombacher, K. Bauckhage, “Characterization of droplets in liquid–liquid extraction by laser-optical measurement techniques,” Chem. Eng. Technol. 21, 337–341 (1998).
[CrossRef]

E. H. Schombacher, H. Lohner, K. Bauckhage, “Ein laseroptisches Messverfahren zur Untersuchung von Tropfen in der Flüssig/Flüssig-Extraktion,” Chem. Ing. Tech. 70, 713–717 (1998).
[CrossRef]

P. Lehmann, E. H. Schombacher, H. Lohner, K. Bauckhage, “Characterization of nonspherical and oscillating droplets by phase-Doppler-anemometry and rainbow refractometry,” in PARTEC 98, Seventh European Symposium on Particle Characterization, R. Weichert, ed. (Nürnberg Messe GmbH Nürnberg/Germany, 1998), pp. 109–119.

Clift, R.

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

de Boer, J. H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grace, J. R.

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

Können, G. P.

Langley, D. S.

Lehmann, P.

P. Lehmann, E. H. Schombacher, H. Lohner, K. Bauckhage, “Characterization of nonspherical and oscillating droplets by phase-Doppler-anemometry and rainbow refractometry,” in PARTEC 98, Seventh European Symposium on Particle Characterization, R. Weichert, ed. (Nürnberg Messe GmbH Nürnberg/Germany, 1998), pp. 109–119.

Lohner, H.

E. H. Schombacher, H. Lohner, K. Bauckhage, “Ein laseroptisches Messverfahren zur Untersuchung von Tropfen in der Flüssig/Flüssig-Extraktion,” Chem. Ing. Tech. 70, 713–717 (1998).
[CrossRef]

H. Lohner, E. H. Schombacher, K. Bauckhage, “Characterization of droplets in liquid–liquid extraction by laser-optical measurement techniques,” Chem. Eng. Technol. 21, 337–341 (1998).
[CrossRef]

P. Lehmann, E. H. Schombacher, H. Lohner, K. Bauckhage, “Characterization of nonspherical and oscillating droplets by phase-Doppler-anemometry and rainbow refractometry,” in PARTEC 98, Seventh European Symposium on Particle Characterization, R. Weichert, ed. (Nürnberg Messe GmbH Nürnberg/Germany, 1998), pp. 109–119.

Marston, P. L.

Möbius, W.

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, England, 1992).
[CrossRef]

Riethmuller, M. L.

J. van Beeck, M. L. Riethmuller, “Rainbow interferometry with wire diffraction for simultaneous measurement of Droplet temperature, size, and velocity,” Part. Part. Charact. 14, 186–192 (1997).

J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

J. van Beeck, M. L. Riethmuller, “Rainbow thermometry with a pulsed laser,” in Proceedings of Ninth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Vol. 1, D. F. G. Durao, ed. (Instituto Superior Técnico, Lisbon, 1998), pp. 18.6.1–7.

Schombacher, E. H.

E. H. Schombacher, H. Lohner, K. Bauckhage, “Ein laseroptisches Messverfahren zur Untersuchung von Tropfen in der Flüssig/Flüssig-Extraktion,” Chem. Ing. Tech. 70, 713–717 (1998).
[CrossRef]

H. Lohner, E. H. Schombacher, K. Bauckhage, “Characterization of droplets in liquid–liquid extraction by laser-optical measurement techniques,” Chem. Eng. Technol. 21, 337–341 (1998).
[CrossRef]

P. Lehmann, E. H. Schombacher, H. Lohner, K. Bauckhage, “Characterization of nonspherical and oscillating droplets by phase-Doppler-anemometry and rainbow refractometry,” in PARTEC 98, Seventh European Symposium on Particle Characterization, R. Weichert, ed. (Nürnberg Messe GmbH Nürnberg/Germany, 1998), pp. 109–119.

Schombacher, H.

H. Schombacher, “Laseroptische Messtechniken zur Bestimmung prozessrelevanter Grössen in der Flüssig/Flüssig-Extraktion,” VDI-Fortschrittsber. 8, 79–118 (1997).

Trinh, E. H.

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–530 (1984).
[CrossRef]

van Beeck, J.

J. van Beeck, M. L. Riethmuller, “Rainbow interferometry with wire diffraction for simultaneous measurement of Droplet temperature, size, and velocity,” Part. Part. Charact. 14, 186–192 (1997).

J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

J. van Beeck, M. L. Riethmuller, “Rainbow thermometry with a pulsed laser,” in Proceedings of Ninth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Vol. 1, D. F. G. Durao, ed. (Instituto Superior Técnico, Lisbon, 1998), pp. 18.6.1–7.

van de Hulst, H. C.

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

Weber, M. E.

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

Am. J. Phys. (1)

J. D. Walker, “Multiple rainbows from single drops and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Ann. Phys. (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. 33, 1493–1558 (1910).
[CrossRef]

Appl. Opt. (6)

Chem. Eng. Technol. (1)

H. Lohner, E. H. Schombacher, K. Bauckhage, “Characterization of droplets in liquid–liquid extraction by laser-optical measurement techniques,” Chem. Eng. Technol. 21, 337–341 (1998).
[CrossRef]

Chem. Ing. Tech. (1)

E. H. Schombacher, H. Lohner, K. Bauckhage, “Ein laseroptisches Messverfahren zur Untersuchung von Tropfen in der Flüssig/Flüssig-Extraktion,” Chem. Ing. Tech. 70, 713–717 (1998).
[CrossRef]

Nature (London) (1)

P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–530 (1984).
[CrossRef]

Part. Part. Charact. (1)

J. van Beeck, M. L. Riethmuller, “Rainbow interferometry with wire diffraction for simultaneous measurement of Droplet temperature, size, and velocity,” Part. Part. Charact. 14, 186–192 (1997).

Sci. Am. (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

VDI-Fortschrittsber (1)

H. Schombacher, “Laseroptische Messtechniken zur Bestimmung prozessrelevanter Grössen in der Flüssig/Flüssig-Extraktion,” VDI-Fortschrittsber. 8, 79–118 (1997).

Other (6)

P. Lehmann, E. H. Schombacher, H. Lohner, K. Bauckhage, “Characterization of nonspherical and oscillating droplets by phase-Doppler-anemometry and rainbow refractometry,” in PARTEC 98, Seventh European Symposium on Particle Characterization, R. Weichert, ed. (Nürnberg Messe GmbH Nürnberg/Germany, 1998), pp. 109–119.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, England, 1992).
[CrossRef]

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. van Beeck, M. L. Riethmuller, “Rainbow thermometry with a pulsed laser,” in Proceedings of Ninth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Vol. 1, D. F. G. Durao, ed. (Instituto Superior Técnico, Lisbon, 1998), pp. 18.6.1–7.

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

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Figures (10)

Fig. 1
Fig. 1

Ray-tracing computations for oblate, spherical, and prolate droplets.

Fig. 2
Fig. 2

Deviation of the angular position of the rainbow, depending on the angle of incidence and the axis ratio d h /d v of a spheroidal water droplet according to Möbius.5

Fig. 3
Fig. 3

Rainbow pattern of a 2-mm toluene droplet fitted by computation of the Airy integral (dashed curve).

Fig. 4
Fig. 4

Comparison between Airy intensity patterns and Mie computations for a toluene droplet in a continuous water phase, assuming that d = 2 mm for parallel and perpendicular polarization (with respect to the scattering plane).

Fig. 5
Fig. 5

Experimental setup.

Fig. 6
Fig. 6

Diameter of fixed toluene droplets determined from Eq. (3) compared with the diameter determined by evaluation of CCD images.

Fig. 7
Fig. 7

Experimental results for rising toluene droplets in a continuous water phase. The difference between the droplet diameters was obtained by two receivers at different locations.

Fig. 8
Fig. 8

Rising toluene droplets for different droplet diameters taken by a high-speed video technique: (a) 2 mm, (b) 3 mm, (c) 4.5 mm.

Fig. 9
Fig. 9

Experimental results for rising CHA droplets in a continuous water phase. The difference between the droplet diameters is obtained by two receivers at different locations.

Fig. 10
Fig. 10

Rising CHA droplets for different droplet diameters taken by a high-speed video technique: (a) 2 mm, (b) 3 mm, (c) 4.5 mm.

Equations (9)

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

φrg=2τrg-4 arccoscos τrgmr
sin τrg=mr2-131/2,
ΔφAiry=1.0845sin τrgλd2cos τrg161/3.
FAiry=sin τrg2.379416d2λ2 cos τrg1/3.
L  Δv2λ,
Δv=d2 tan ΔΘ4h1/2,
h=944-m2m2-131/2.
Frip=d2λcos τrg+cosφrg2.
Re=wddρcηc,

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