Abstract

A nonintrusive laser technique, based on the detection of a rainbow, is presented that permits one to determine simultaneously the temperature and size of droplets. Therefore the Airy theory for a rainbow and a calibration rainbow pattern at isothermal conditions are applied. Rainbow patterns coming from droplets in the millimeter range have been recorded on a linear CCD array. It has been found that the sphericity of the droplets plays an important role for this rainbow-based technique.

© 1995 Optical Society of America

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Errata

J. P. A. J. van Beeck and M. L. Riethmuller, "Nonintrusive measurements of temperature and size of single falling raindrops: erratum," Appl. Opt. 34, 7337-7337 (1995)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-34-31-7337

References

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  1. W. D. Bachalo, M. J. Houser, “Development of the phase/Doppler spray analyzer for liquid drop size and velocity characterizations,” in Proceedings of the 20th Joint Propulsion Conference (AIAA-84-1199, AIAA/SAE/ASMECincinnati, Ohio, 1984).
  2. E. Alessandri, “Hydrodynamic interactions and heat transfer phenomena in liquid sprays,” Tech. Rep. PR 1991-25 (Von Karman Institute, Rhode-Saint-Genese, Belgium, 1991).
  3. N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
    [CrossRef] [PubMed]
  4. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  5. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  6. J. P. A. J. van Beeck, “Nonintrusive measurement technique for temperature and size of droplets,” Tech. Rep. PR 1993-22 (Von Karman Institute, Rhode-Saint-Genese, Belgium, 1993).
  7. I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
    [CrossRef]
  8. M. St-Georges, “Etude Hydrodynamique des Pulvérisations Liquide pour Application aux Rideaux d’Eau,” Ph.D. dissertation (Université Claude Bernard—UniversitéLyon 1, France, 1993).
  9. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987).
  10. J. P. A. J. van Beeck, M. L. Riethmuller, “Determination non intrusive de la dimension et de la temperature des gouttes dans une pluverisation,” in Proceedings of the Congrés Francophone de Vélocimétrie Laser (CNRS Poitiers, France, 1994).
  11. R. Clift, J. R. Grace, M. E. Weber, Bubbles, Drops and Particles (Academic, New York, 1978).
  12. W. Möbius, “Zur theorie des regenbogens und ihrer experimentellen prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

1991

1985

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

1976

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

1910

W. Möbius, “Zur theorie des regenbogens und ihrer experimentellen prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

Alessandri, E.

E. Alessandri, “Hydrodynamic interactions and heat transfer phenomena in liquid sprays,” Tech. Rep. PR 1991-25 (Von Karman Institute, Rhode-Saint-Genese, Belgium, 1991).

Anders, K.

Bachalo, W. D.

W. D. Bachalo, M. J. Houser, “Development of the phase/Doppler spray analyzer for liquid drop size and velocity characterizations,” in Proceedings of the 20th Joint Propulsion Conference (AIAA-84-1199, AIAA/SAE/ASMECincinnati, Ohio, 1984).

Clift, R.

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

Frohn, A.

Grace, J. R.

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

Grigull, U.

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987).

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Development of the phase/Doppler spray analyzer for liquid drop size and velocity characterizations,” in Proceedings of the 20th Joint Propulsion Conference (AIAA-84-1199, AIAA/SAE/ASMECincinnati, Ohio, 1984).

Möbius, W.

W. Möbius, “Zur theorie des regenbogens und ihrer experimentellen prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

Riethmuller, M. L.

J. P. A. J. van Beeck, M. L. Riethmuller, “Determination non intrusive de la dimension et de la temperature des gouttes dans une pluverisation,” in Proceedings of the Congrés Francophone de Vélocimétrie Laser (CNRS Poitiers, France, 1994).

Roth, N.

St-Georges, M.

M. St-Georges, “Etude Hydrodynamique des Pulvérisations Liquide pour Application aux Rideaux d’Eau,” Ph.D. dissertation (Université Claude Bernard—UniversitéLyon 1, France, 1993).

Straub, J.

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Thormählen, I.

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

van Beeck, J. P. A. J.

J. P. A. J. van Beeck, “Nonintrusive measurement technique for temperature and size of droplets,” Tech. Rep. PR 1993-22 (Von Karman Institute, Rhode-Saint-Genese, Belgium, 1993).

J. P. A. J. van Beeck, M. L. Riethmuller, “Determination non intrusive de la dimension et de la temperature des gouttes dans une pluverisation,” in Proceedings of the Congrés Francophone de Vélocimétrie Laser (CNRS Poitiers, France, 1994).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Walker, J. D.

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

Weber, M. E.

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

Am. J. Phys.

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

Ann. Phys. (Leipzig)

W. Möbius, “Zur theorie des regenbogens und ihrer experimentellen prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

Appl. Opt.

J. Phys. Chem. Ref. Data

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Other

M. St-Georges, “Etude Hydrodynamique des Pulvérisations Liquide pour Application aux Rideaux d’Eau,” Ph.D. dissertation (Université Claude Bernard—UniversitéLyon 1, France, 1993).

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987).

J. P. A. J. van Beeck, M. L. Riethmuller, “Determination non intrusive de la dimension et de la temperature des gouttes dans une pluverisation,” in Proceedings of the Congrés Francophone de Vélocimétrie Laser (CNRS Poitiers, France, 1994).

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

W. D. Bachalo, M. J. Houser, “Development of the phase/Doppler spray analyzer for liquid drop size and velocity characterizations,” in Proceedings of the 20th Joint Propulsion Conference (AIAA-84-1199, AIAA/SAE/ASMECincinnati, Ohio, 1984).

E. Alessandri, “Hydrodynamic interactions and heat transfer phenomena in liquid sprays,” Tech. Rep. PR 1991-25 (Von Karman Institute, Rhode-Saint-Genese, Belgium, 1991).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

J. P. A. J. van Beeck, “Nonintrusive measurement technique for temperature and size of droplets,” Tech. Rep. PR 1993-22 (Von Karman Institute, Rhode-Saint-Genese, Belgium, 1993).

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Figures (14)

Fig. 1
Fig. 1

First-order rainbow in the sky formed by Sun rays that have experienced one internal reflection inside the raindrops.

Fig. 2
Fig. 2

Geometrical optical rays that form a first-order rainbow.

Fig. 3
Fig. 3

Fringes of the monochromatic rainbow in the laboratory. Roth et al.3 took a good picture of the image on the screen.

Fig. 4
Fig. 4

Theoretical scattered-light intensity distribution as a function of the normalized angular deviation from the geometric rainbow angle θ rg .

Fig. 5
Fig. 5

(a) Angular positions of the first five extrema in the rainbow pattern as a function of droplet temperature; the droplet diameter is 1 mm. (b) Angular positions of the first five extrema in the rainbow pattern as a function of droplet diameter; the droplet temperature equals 20 °C.

Fig. 6
Fig. 6

Setup for measuring the rainbow pattern.

Fig. 7
Fig. 7

Y refers to the position at the linear CCD array. Theoretical droplet temperature at 0.5 m from the needle exit as a function of the droplet diameter for different humidities and for a thermocouple temperature equal to the ambient temperature.

Fig. 8
Fig. 8

Theoretical droplet temperature at the position where the droplet crosses the laser beam as a function of the thermocouple temperature for humidity of 60%, an ambient temperature T ambient = 20 °C, and two different droplet diameters.

Fig. 9
Fig. 9

Calibration rainbow pattern in isothermal conditions. We aim to relate each pixel number of the CCD array to a certain scattering angle.

Fig. 10
Fig. 10

(a) Typical rainbow pattern coming from a heated droplet. (Note the agreement between the rainbow temperature and the corrected thermocouple temperature.) (b) Comparison between the smoothed rainbow pattern of (a) and the Airy rainbow pattern.

Fig. 11
Fig. 11

(a) Typical rainbow pattern coming from a heated droplet. (Note the large difference between the rainbow temperature and the corrected thermocouple temperature.) (b) Comparison between the smoothed rainbow pattern of (a) and the Airy rainbow pattern.

Fig. 12
Fig. 12

Absolute difference between the corrected thermocouple temperature and the rainbow temperature as a function of droplet diameter.

Fig. 13
Fig. 13

Definition of axis diameters a and b of an ellipsoidal droplet. Ψ is the angle between the incident beam and the larger axis of the ellipsoid. For his computations Möbius12 assumed rotational symmetry around the smaller axis; as such a hamburger-shaped droplet appears.

Fig. 14
Fig. 14

Angular difference Δθ rg between the geometric rainbow position of an ellipsoidal droplet and a spherical droplet according to Möbius.12 Angle Ψ and axes a and b are defined in Fig. 13.

Tables (1)

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Table 1 Maxima in the Square of the Rainbow Integral (α)

Equations (6)

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θ r g = 2 τ r g + 4 arccos ( 1 m cos τ r g ) ,
sin τ r g = ( m 2 1 3 ) 1 / 2 .
α = ( θ θ r g ) ( 16 D 2 λ 2 cos τ r g ) 1 / 3 sin τ r g ,
F ( α ) = 0 cos 1 2 π ( α t t 3 ) d t .
D = λ 4 [ ( cos τ r g sin 3 τ r g ) 1 / 2 ( α i α j θ i θ j ) ] 3 / 2 .
y y 0 = f ( θ θ 0 ) .

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