Abstract

A model based on geometric optics for predicting the response of interferometric (phase Doppler) instruments for size measurements of particles with radially symmetric but inhomogeneous internal refractive index profiles is developed. The model and results are important for applications in which heat or mass transfer from the particles or droplets is significant, for example, in liquid-fuel combustion. To quantify the magnitude of potential bias errors introduced by the classical assumption of uniform internal properties on phase Doppler measurements, we compute calibration curves for a sequence of times during the evaporation of a decane droplet immersed in an environment of T = 2000 K and p = 10 bars. The results reveal considerable effects on the relation between phase difference and droplet diameter caused by the refractive index gradients present. The model provides an important tool to assess sizing uncertainties that can be expected when applying conventional (based on uniform properties) phase Doppler calibration curves in spray combustion and similar processes.

© 1994 Optical Society of America

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References

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  1. S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
    [CrossRef] [PubMed]
  2. S. K. Aggarwal, “Modeling of a dilute vaporizing multicomponent fuel spray,” Intl. J. Heat Mass Transfer 30, 1949–1961 (1987).
    [CrossRef]
  3. R. Kneer, M. Schneider, B. Noll, S. Wittig, “Effects of variable liquid properties on multicomponent droplet vaporization,” ASME paper 92-GT-131 (American Society of Mechanical Engineers, New York, 1992).
  4. R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 3, pp. 121–124.
  6. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 4, pp. 28–29.
  7. S. V. Sankar, B. J. Weber, W. D. Bachalo, “Sizing fine particles with the phase-Doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1981).
    [CrossRef]
  8. A. A. Naqwi, F. Durst, “Analysis of the laser light scattering interferometric devices for the inline diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
    [PubMed]
  9. Fig. 41 on p. 229 of Ref. 6 has the magnitude of K consistent with our work but differs in the sign; K will need to be negative for θ′ to be negative.
  10. E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991).
    [CrossRef] [PubMed]
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, pp. 57–63.
  12. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4, p. 192.
  13. M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

1993 (2)

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

A. A. Naqwi, F. Durst, “Analysis of the laser light scattering interferometric devices for the inline diagnostics of moving particles,” Appl. Opt. 32, 4003–4018 (1993).
[PubMed]

1991 (2)

1987 (1)

S. K. Aggarwal, “Modeling of a dilute vaporizing multicomponent fuel spray,” Intl. J. Heat Mass Transfer 30, 1949–1961 (1987).
[CrossRef]

1981 (1)

Aggarwal, S. K.

S. K. Aggarwal, “Modeling of a dilute vaporizing multicomponent fuel spray,” Intl. J. Heat Mass Transfer 30, 1949–1961 (1987).
[CrossRef]

Bachalo, W. D.

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4, p. 192.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, pp. 57–63.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 3, pp. 121–124.

Chowdhury, D. Q.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

Durst, F.

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4, p. 192.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

Hirleman, E. D.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, pp. 57–63.

Kneer, R.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Effects of variable liquid properties on multicomponent droplet vaporization,” ASME paper 92-GT-131 (American Society of Mechanical Engineers, New York, 1992).

Naqwi, A. A.

Noll, B.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Effects of variable liquid properties on multicomponent droplet vaporization,” ASME paper 92-GT-131 (American Society of Mechanical Engineers, New York, 1992).

Saleheen, H. I.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

Sankar, S. V.

Schneider, M.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Effects of variable liquid properties on multicomponent droplet vaporization,” ASME paper 92-GT-131 (American Society of Mechanical Engineers, New York, 1992).

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 4, pp. 28–29.

Weber, B. J.

Wittig, S.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Effects of variable liquid properties on multicomponent droplet vaporization,” ASME paper 92-GT-131 (American Society of Mechanical Engineers, New York, 1992).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 3, pp. 121–124.

Appl. Opt. (4)

Intl. J. Heat Mass Transfer (2)

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Intl. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

S. K. Aggarwal, “Modeling of a dilute vaporizing multicomponent fuel spray,” Intl. J. Heat Mass Transfer 30, 1949–1961 (1987).
[CrossRef]

Other (7)

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Effects of variable liquid properties on multicomponent droplet vaporization,” ASME paper 92-GT-131 (American Society of Mechanical Engineers, New York, 1992).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 3, pp. 121–124.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 4, pp. 28–29.

Fig. 41 on p. 229 of Ref. 6 has the magnitude of K consistent with our work but differs in the sign; K will need to be negative for θ′ to be negative.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, pp. 57–63.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4, p. 192.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 269–286 (1993).

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Figures (10)

Fig. 1
Fig. 1

Time variation of radial refractive index gradionts. Data are based on a conduction-limit model.3,4

Fig. 2
Fig. 2

Ray in a medium with spherical symmetry,6 where P is an arbitrary point on the ray path.

Fig. 3
Fig. 3

Ray tracing for a droplet without (top) and with (bottom) refractive index gradients. The relative refractive index profiles are shown on the right-hand side.

Fig. 4
Fig. 4

Relevant angles for a droplet without (top) and with (bottom) refractive index gradients. The closest approach of the ray to the droplet center occurs at the point (r h , β h ) in polar coordinates, and the path is symmetric around the line hh.

Fig. 5
Fig. 5

Path length for a droplet without (top) and with (bottom) refractive index gradients. The contribution to the phase difference caused by path-length differences is calculated relative to the reference ray.

Fig. 6
Fig. 6

Geometrical setup and receiving aperture of the PDA used for calculations in the present study (after Ref. 1); DET, detector.

Fig. 7
Fig. 7

Calibration curves for a θ = 30° mean scattering angle: phase differences between (a) detectors 1 and 2, (b) detectors 1 and 3.

Fig. 8
Fig. 8

Calibration curves for a θ = 160° mean scattering angle: phase differences between (a) detectors 1 and 2, (b) detectors 1 and 3.

Fig. 9
Fig. 9

Normalized slopes based on linear regression for θ = 30° calibration curves.

Fig. 10
Fig. 10

Normalized slopes based on linear regression for θ = 160° calibration curves.

Equations (31)

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d d s ( n d r d s ) = grad n .
n r sin ϕ = const . = c .
β = c 0 r [ 1 / r ( n 2 r 2 c 2 ) 1 / 2 ] d r .
p j = { r j for p = 0 ( 1 r j 2 ) ( r j ) p 1 for p > 0 ,
I p j = ( a / R ) 2 I inc p j 2 D p ,
D p = sin τ cos τ sin θ | d θ / d τ | .
| S p j | = ( I p j ) 1 / 2 .
θ = 2 τ 2 p τ , n = const ,
θ = 2 τ 2 p Δ β, n = n ( r ) .
Δ β = β o β h = c r h a [ 1 / r ( n 2 r 2 c 2 ) 1 / 2 ] d r .
c = n ( a ) a sin ( τ + π / 2 ) = a cos τ .
c = n ( r h ) r h .
θ = 2 π K + q θ .
| d θ d τ | = | d θ d τ | .
d θ d τ = 2 2 p tan τ tan τ , n = const ,
d θ d τ = 2 2 p d ( Δ β ) d τ , n = n ( r ) ,
phase contribution of p j = { 0 if p j > 0 π if p j < 0 .
δ = OPL R OPL ,
OPL = k ( X 1 + p n L + X 2 ) , n = const ,
OPL = k [ X 1 + p n ( s ) d s + X 2 ] , n = n ( r ) .
δ p = 2 k a k [ ( a a sin τ ) + p n ( 2 a sin τ ) + ( a a sin τ ) ] = 2 x ( sin τ p n sin τ ) , n = const ,
δ p = 2 k a k [ ( a a sin τ ) + p n ( s ) d s + ( a a sin τ ) ] = 2 x ( sin τ ( p / 2 a ) n ( s ) d s ) , n = n ( r ) ,
n ( s ) d s = 2 | { β h β 0 n [ r ( β ) ] | r ( β ) | d β } | ,
σ p j = π / 2 + ( phase of p j ) + ( π / 2 ) [ p 2 K + ( s / 2 ) ( q / 2 ) ] + δ p ,
σ p j * = π / 2 σ p j .
| E j ( θ ) | cos [ ω t + η j ( θ ) ] = all p | S p j ( θ ) | cos [ ω t + σ p j ( θ ) ] ,
I scat = | E 1 | 2 + | E 2 | 2 + 2 | E 1 | | E 2 | cos ( ω D t + η 1 η 2 ) ,
η = η 1 η 2 .
x y I scat , m = A m + B m cos ( ω D t + ϕ m ) ,
ϕ 12 = ϕ 1 ϕ 2 ,
ϕ 13 = ϕ 1 ϕ 3 .

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