Abstract

A theoretical model, based on the geometrical optics approach, has been developed to simulate various aspects of the phase Doppler particle analyzer (PDPA). The model has taken into consideration the nonuniform (Gaussian) illumination of the particles as they pass through the measurement probe volume. Instrument response curves have been generated for various scattering angles by performing spatial and temporal integration of the scattered intensity distribution over the receiver surface. Experimental and theoretical investigations have established the applicability of this instrument to both forward scattered and backscattered angles.

© 1991 Optical Society of America

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References

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  1. W. M. Farmer, “The Interferometric Observation of Dynamic Particle Size, Velocity and Number Density,” Ph.D. Thesis, U. Tennessee (1973).
  2. W. D. Bachalo, “Method for Measuring the Size and Velocity of Spheres by Dual-Beam Light-Scatter Interferometry,” Appl. Opt. 19, 363–370 (1980).
    [CrossRef] [PubMed]
  3. J. D. Pendleton, “Mie and Refraction Theory Comparison for Particle Sizing with the Laser Velocimeter,” Appl. Opt. 21, 684–688 (1982).
    [CrossRef] [PubMed]
  4. W. D. Bachalo, M. J. Houser, “Phase Doppler Spray Analyzer for Simultaneous Measurements of Drop Size and Velocity Distributions,” Opt. Eng. 23, 583–590 (1984).
    [CrossRef]
  5. K. Bauckhage, H. H. Flogel, “Simultaneous Measurements of Droplet Size and Velocity in Nozzle Sprays,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).
  6. M. Saffman, P. Buchhave, H. Tanger, “Simultaneous Measurement of the Size, Concentration and Velocity of Spherical Particles by a Laser Doppler Method,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).
  7. S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of Calibration Curves for the Phase Doppler Technique: Comparison Between Mie Theory and Geometrical Optics,” in Proceedings, International Symposium on Optical Particle Sizing: Theory and Practice, Mont Saint-Aignan, France (1987), pp. 10.1–10.14.
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  9. W. J. Glantschnig, S.-H. Chen, “Light Scattering from Water Droplets in the Geometrical Optics Approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef] [PubMed]
  10. W. D. Bachalo, M. J. Houser, “Analysis and Testing of a New Method for Drop Size Measurement Using Laser Light Scatter Interferometry,” NASA Contract. Rep. 174636, NASA Lewis Research Center (1984).
  11. M. Saffman, “The Use of Polarized Light for Optical Particle Sizing,” in Proceedings, Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1986).

1984 (1)

W. D. Bachalo, M. J. Houser, “Phase Doppler Spray Analyzer for Simultaneous Measurements of Drop Size and Velocity Distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

Al-Chalabi, S. A. M.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of Calibration Curves for the Phase Doppler Technique: Comparison Between Mie Theory and Geometrical Optics,” in Proceedings, International Symposium on Optical Particle Sizing: Theory and Practice, Mont Saint-Aignan, France (1987), pp. 10.1–10.14.

Bachalo, W. D.

W. D. Bachalo, M. J. Houser, “Phase Doppler Spray Analyzer for Simultaneous Measurements of Drop Size and Velocity Distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

W. D. Bachalo, “Method for Measuring the Size and Velocity of Spheres by Dual-Beam Light-Scatter Interferometry,” Appl. Opt. 19, 363–370 (1980).
[CrossRef] [PubMed]

W. D. Bachalo, M. J. Houser, “Analysis and Testing of a New Method for Drop Size Measurement Using Laser Light Scatter Interferometry,” NASA Contract. Rep. 174636, NASA Lewis Research Center (1984).

Bauckhage, K.

K. Bauckhage, H. H. Flogel, “Simultaneous Measurements of Droplet Size and Velocity in Nozzle Sprays,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

Buchhave, P.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous Measurement of the Size, Concentration and Velocity of Spherical Particles by a Laser Doppler Method,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

Chen, S.-H.

Farmer, W. M.

W. M. Farmer, “The Interferometric Observation of Dynamic Particle Size, Velocity and Number Density,” Ph.D. Thesis, U. Tennessee (1973).

Flogel, H. H.

K. Bauckhage, H. H. Flogel, “Simultaneous Measurements of Droplet Size and Velocity in Nozzle Sprays,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

Glantschnig, W. J.

Hardalupas, Y.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of Calibration Curves for the Phase Doppler Technique: Comparison Between Mie Theory and Geometrical Optics,” in Proceedings, International Symposium on Optical Particle Sizing: Theory and Practice, Mont Saint-Aignan, France (1987), pp. 10.1–10.14.

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase Doppler Spray Analyzer for Simultaneous Measurements of Drop Size and Velocity Distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

W. D. Bachalo, M. J. Houser, “Analysis and Testing of a New Method for Drop Size Measurement Using Laser Light Scatter Interferometry,” NASA Contract. Rep. 174636, NASA Lewis Research Center (1984).

Jones, A. R.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of Calibration Curves for the Phase Doppler Technique: Comparison Between Mie Theory and Geometrical Optics,” in Proceedings, International Symposium on Optical Particle Sizing: Theory and Practice, Mont Saint-Aignan, France (1987), pp. 10.1–10.14.

Pendleton, J. D.

Saffman, M.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous Measurement of the Size, Concentration and Velocity of Spherical Particles by a Laser Doppler Method,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

M. Saffman, “The Use of Polarized Light for Optical Particle Sizing,” in Proceedings, Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1986).

Tanger, H.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous Measurement of the Size, Concentration and Velocity of Spherical Particles by a Laser Doppler Method,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

Taylor, A. M. K. P.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of Calibration Curves for the Phase Doppler Technique: Comparison Between Mie Theory and Geometrical Optics,” in Proceedings, International Symposium on Optical Particle Sizing: Theory and Practice, Mont Saint-Aignan, France (1987), pp. 10.1–10.14.

van de Hulst, H. C.

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

Appl. Opt. (3)

Opt. Eng. (1)

W. D. Bachalo, M. J. Houser, “Phase Doppler Spray Analyzer for Simultaneous Measurements of Drop Size and Velocity Distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Other (7)

K. Bauckhage, H. H. Flogel, “Simultaneous Measurements of Droplet Size and Velocity in Nozzle Sprays,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous Measurement of the Size, Concentration and Velocity of Spherical Particles by a Laser Doppler Method,” in Proceedings, Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1984).

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of Calibration Curves for the Phase Doppler Technique: Comparison Between Mie Theory and Geometrical Optics,” in Proceedings, International Symposium on Optical Particle Sizing: Theory and Practice, Mont Saint-Aignan, France (1987), pp. 10.1–10.14.

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

W. M. Farmer, “The Interferometric Observation of Dynamic Particle Size, Velocity and Number Density,” Ph.D. Thesis, U. Tennessee (1973).

W. D. Bachalo, M. J. Houser, “Analysis and Testing of a New Method for Drop Size Measurement Using Laser Light Scatter Interferometry,” NASA Contract. Rep. 174636, NASA Lewis Research Center (1984).

M. Saffman, “The Use of Polarized Light for Optical Particle Sizing,” in Proceedings, Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal (1986).

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Figures (16)

Fig. 1
Fig. 1

Ray trace for a water droplet.

Fig. 2
Fig. 2

Division of scattering angles into scattering regions.

Fig. 3
Fig. 3

Coordinate system.

Fig. 4
Fig. 4

Schematic of the receiving aperture showing the three areas from which useful information is obtained.

Fig. 5
Fig. 5

Calibration curves for a 30° mean scattering angle.

Fig. 6
Fig. 6

Comparison between experimental observations and theoretical predictions for a 30° mean scattering angle.

Fig. 7
Fig. 7

Computer-generated spatial fringe pattern for a 40-μm water droplet. Ideal case of pure refraction (p = 1 only) at a mean scattering angle of 30°.

Fig. 8
Fig. 8

Computer-generated spatial fringe pattern for a 40-μm water droplet. Simultaneous presence of external reflection, refraction, and second internal reflection at a mean scattering angle of 30°.

Fig. 9
Fig. 9

Experimental comparison of backscattered and forward scattered results.

Fig. 10
Fig. 10

Computed calibration curves for a 150° mean scattering angle, with no integration over the lens surface.

Fig. 11
Fig. 11

Computed calibration curves for a 150° mean scattering angle showing the effect of performing spatial and temporal integration of the spatial intensity pattern.

Fig. 12
Fig. 12

Computer generated spatial fringe pattern for a 40-μm water droplet traveling through the center of the probe volume. Interference of external reflection and two components of first internal reflection at a mean scattering angle of 150°.

Fig. 13
Fig. 13

Trajectory dependent scattering.

Fig. 14
Fig. 14

Trajectory dependence of the calibration curve for a mean scattering angle of 30°.

Fig. 15
Fig. 15

Trajectory dependence of the calibration curve for a mean scattering angle of 150°.

Fig. 16
Fig. 16

Computer generated spatial fringe pattern for a 40-μm water droplet traveling through an edge trajectory (η = 1.0). Interference of external reflection and two components of internal reflection at a mean scattering angle of 150°.

Equations (17)

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| S i p n ( a R , m , I inc , n , θ i ) | = a R I inc ( α i p ) | n ( m , α i p ) | G ( θ i , α i p ) ,
p = 0 , 1 , 3 , and n = 1 , 2 ,
n ( m , α i p ) = r n ( m , α i p ) for p = 0 = [ 1 r n ( m , α i p ) 2 ] [ r n ( m , α i p ) ] p 1 for p = 1 , 2 , 3 ,
r n = 2 ( m , α i p ) = m cos α i p cos [ sin 1 ( sin α i p m ) ] m cos α i p + cos [ sin 1 ( sin α i p m ) ] ,
G ( θ i , α i p ) = sin α i p cos α i p sin θ i | d θ i d α i p | ,
θ i = π + 2 α i p
θ i = ( p 1 ) π 2 p sin 1 ( sin α i p m ) + 2 α i p
θ i = 2 p sin 1 ( sin α i p m ) + 2 α i p
δ i p = 2 x { cos α i p m p cos [ sin 1 ( sin α i p m ) ] } ,
x = π d λ .
i = 1 2 p = 0 3 | S i p n | cos ( ω i t + σ i p n ) = i = 1 2 | E i n | cos ( ω i t + η i n )
I scat , n ( x , z ) = | E 1 n | 2 + | E 2 n | 2 + 2 | E 1 n | | E 2 n | cos ( ω D t + η 1 n η 2 n )
x z I scat , j ( x , z ) = A j + B j cos ( ω D t + ϕ j ) for j = 1 , 2 , 3 ,
ϕ 12 = ϕ 1 ϕ 2 ,
ϕ 13 = ϕ 1 ϕ 3 ,
I ( x , z ) = I 0 exp [ 2 ( x 2 + z 2 b 0 2 ) ] .
η = z b 0 .

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