Abstract

Measurements of bubble size and velocity in multiphase flows are important in much research and many industrial applications. It has been found that high-order refractions have great impact on microbubble sizing by use of phase-Doppler anemometry (PDA). The problem has been investigated, and a model of phase-size correlation, which also takes high-order refractions into consideration, is introduced to improve the accuracy of bubble sizing. Hence the model relaxes the assumption of a single-scattering mechanism in a conventional PDA system. The results of simulation based on this new model are compared with those based on a single-scattering-mechanism approach or a first-order approach. An optimization method for accurately sizing air bubbles in water has been suggested.

© 2003 Optical Society of America

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References

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  1. M Saffman, “The use of polarized light for optical particle sizing,” presented at the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 14–17, 1986.
  2. H.-H. Qiu, M Sommerfeld, “The impact of signal processing on the accuracy of phase-Doppler measurements,” presented at the Sixth Workshop on Two-Phase Flow Predictions, Erlangen, Germany, March 30–April 2, 1992.
  3. S. V. Sankar, A. S. Inenaga, W. D. Bachalo, “Trajectory dependent scattering in phase Doppler interferometer: minimizing and eliminating sizing errors,” presented at the Sixth International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 20–23, 1992.
  4. G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  5. Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.
  6. G Grehan, G Gouesbet, A Naqwi, F Durst, “Evaluation of phase Doppler system using generalized Lorenz– Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, September 24–27, 1991.
  7. S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and structural resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
    [CrossRef] [PubMed]
  8. H.-H. Qiu, C. T. Hsu, “A Fourier optics method for the simulation of measurement-volume-effect by the slit constraint,” presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1996.
  9. H.-H. Qiu, C. T. Hsu, “Optimization of optical parameters for particle sizing in multiphase flows by using EPDA,” Opt. Lasers Eng. 30, 3–15 (1998).
    [CrossRef]
  10. H.-H. Qiu, C. T. Hsu, “Method of phase-Doppler anemometry free from the measurement-volume effect,” Appl. Opt. 38, 2737–2742 (1999).
    [CrossRef]
  11. N. Yokoi, Y. Aizu, H. Mishina, “Estimation of particle trajectory effects and their reduction using polarization inphase Doppler particle measurements,” Opt. Eng. 38, 1869–1882 (1999).
    [CrossRef]
  12. H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
    [CrossRef]
  13. C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.
  14. W. J. Glantschnig, S.-H. Chen, “Light scattering fromwater droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef] [PubMed]
  15. C. E. Dean, P. L. Marston, “Critical angle light scattering from bubbles: an asymptotic series approximation,” Appl. Opt. 30, 4764–4776 (1991).
    [CrossRef] [PubMed]
  16. A. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
    [CrossRef]
  17. H.-H. Qiu, M. Sommerfeld, F. Durst, “High resolution data processing for phase-Doppler measurements in a complex two-phase flow,” Meas. Sci. Technol. 2, 455–463 (1991).
    [CrossRef]

2000

H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
[CrossRef]

1999

N. Yokoi, Y. Aizu, H. Mishina, “Estimation of particle trajectory effects and their reduction using polarization inphase Doppler particle measurements,” Opt. Eng. 38, 1869–1882 (1999).
[CrossRef]

H.-H. Qiu, C. T. Hsu, “Method of phase-Doppler anemometry free from the measurement-volume effect,” Appl. Opt. 38, 2737–2742 (1999).
[CrossRef]

1998

H.-H. Qiu, C. T. Hsu, “Optimization of optical parameters for particle sizing in multiphase flows by using EPDA,” Opt. Lasers Eng. 30, 3–15 (1998).
[CrossRef]

1994

1991

C. E. Dean, P. L. Marston, “Critical angle light scattering from bubbles: an asymptotic series approximation,” Appl. Opt. 30, 4764–4776 (1991).
[CrossRef] [PubMed]

A. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

H.-H. Qiu, M. Sommerfeld, F. Durst, “High resolution data processing for phase-Doppler measurements in a complex two-phase flow,” Meas. Sci. Technol. 2, 455–463 (1991).
[CrossRef]

1988

1981

Aizu, Y

Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.

Aizu, Y.

N. Yokoi, Y. Aizu, H. Mishina, “Estimation of particle trajectory effects and their reduction using polarization inphase Doppler particle measurements,” Opt. Eng. 38, 1869–1882 (1999).
[CrossRef]

Alexander, D. R.

Bachalo, W. D.

S. V. Sankar, A. S. Inenaga, W. D. Bachalo, “Trajectory dependent scattering in phase Doppler interferometer: minimizing and eliminating sizing errors,” presented at the Sixth International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 20–23, 1992.

Barton, J. P.

Chen, S.-H.

Dean, C. E.

Durst, F

Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.

G Grehan, G Gouesbet, A Naqwi, F Durst, “Evaluation of phase Doppler system using generalized Lorenz– Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, September 24–27, 1991.

Durst, F.

H.-H. Qiu, M. Sommerfeld, F. Durst, “High resolution data processing for phase-Doppler measurements in a complex two-phase flow,” Meas. Sci. Technol. 2, 455–463 (1991).
[CrossRef]

A. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

Glantschnig, W. J.

Gouesbet, G

G Grehan, G Gouesbet, A Naqwi, F Durst, “Evaluation of phase Doppler system using generalized Lorenz– Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, September 24–27, 1991.

Gouesbet, G.

Grehan, G

G Grehan, G Gouesbet, A Naqwi, F Durst, “Evaluation of phase Doppler system using generalized Lorenz– Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, September 24–27, 1991.

Grehan, G.

Gréhan, G

Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.

C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.

Haugen, P.

C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.

Hsu, C. T.

H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
[CrossRef]

H.-H. Qiu, C. T. Hsu, “Method of phase-Doppler anemometry free from the measurement-volume effect,” Appl. Opt. 38, 2737–2742 (1999).
[CrossRef]

H.-H. Qiu, C. T. Hsu, “Optimization of optical parameters for particle sizing in multiphase flows by using EPDA,” Opt. Lasers Eng. 30, 3–15 (1998).
[CrossRef]

H.-H. Qiu, C. T. Hsu, “A Fourier optics method for the simulation of measurement-volume-effect by the slit constraint,” presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1996.

Inenaga, A. S.

S. V. Sankar, A. S. Inenaga, W. D. Bachalo, “Trajectory dependent scattering in phase Doppler interferometer: minimizing and eliminating sizing errors,” presented at the Sixth International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 20–23, 1992.

Jia, W.

H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
[CrossRef]

Maheu, B.

Marston, P. L.

Mishina, H.

N. Yokoi, Y. Aizu, H. Mishina, “Estimation of particle trajectory effects and their reduction using polarization inphase Doppler particle measurements,” Opt. Eng. 38, 1869–1882 (1999).
[CrossRef]

Naqwi, A

G Grehan, G Gouesbet, A Naqwi, F Durst, “Evaluation of phase Doppler system using generalized Lorenz– Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, September 24–27, 1991.

Naqwi, A. A.

A. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

Onofri, F

C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.

Onofri, F.

Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.

Qiu, H.-H.

H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
[CrossRef]

H.-H. Qiu, C. T. Hsu, “Method of phase-Doppler anemometry free from the measurement-volume effect,” Appl. Opt. 38, 2737–2742 (1999).
[CrossRef]

H.-H. Qiu, C. T. Hsu, “Optimization of optical parameters for particle sizing in multiphase flows by using EPDA,” Opt. Lasers Eng. 30, 3–15 (1998).
[CrossRef]

H.-H. Qiu, M. Sommerfeld, F. Durst, “High resolution data processing for phase-Doppler measurements in a complex two-phase flow,” Meas. Sci. Technol. 2, 455–463 (1991).
[CrossRef]

H.-H. Qiu, M Sommerfeld, “The impact of signal processing on the accuracy of phase-Doppler measurements,” presented at the Sixth Workshop on Two-Phase Flow Predictions, Erlangen, Germany, March 30–April 2, 1992.

H.-H. Qiu, C. T. Hsu, “A Fourier optics method for the simulation of measurement-volume-effect by the slit constraint,” presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1996.

Saffman, M

M Saffman, “The use of polarized light for optical particle sizing,” presented at the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 14–17, 1986.

Sankar, S. V.

S. V. Sankar, A. S. Inenaga, W. D. Bachalo, “Trajectory dependent scattering in phase Doppler interferometer: minimizing and eliminating sizing errors,” presented at the Sixth International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 20–23, 1992.

Schaub, S. A.

Sommerfeld, M

H.-H. Qiu, M Sommerfeld, “The impact of signal processing on the accuracy of phase-Doppler measurements,” presented at the Sixth Workshop on Two-Phase Flow Predictions, Erlangen, Germany, March 30–April 2, 1992.

Sommerfeld, M.

H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
[CrossRef]

H.-H. Qiu, M. Sommerfeld, F. Durst, “High resolution data processing for phase-Doppler measurements in a complex two-phase flow,” Meas. Sci. Technol. 2, 455–463 (1991).
[CrossRef]

Tropea, C

C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.

Xu, T.-H.

C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.

Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.

Yokoi, N.

N. Yokoi, Y. Aizu, H. Mishina, “Estimation of particle trajectory effects and their reduction using polarization inphase Doppler particle measurements,” Opt. Eng. 38, 1869–1882 (1999).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

H.-H. Qiu, M. Sommerfeld, F. Durst, “High resolution data processing for phase-Doppler measurements in a complex two-phase flow,” Meas. Sci. Technol. 2, 455–463 (1991).
[CrossRef]

H.-H. Qiu, W. Jia, C. T. Hsu, M. Sommerfeld, “High accuracy optical particle sizing in phase-Doppler anemometry,” Meas. Sci. Technol. 11, 142–151 (2000).
[CrossRef]

Opt. Eng.

N. Yokoi, Y. Aizu, H. Mishina, “Estimation of particle trajectory effects and their reduction using polarization inphase Doppler particle measurements,” Opt. Eng. 38, 1869–1882 (1999).
[CrossRef]

Opt. Lasers Eng.

H.-H. Qiu, C. T. Hsu, “Optimization of optical parameters for particle sizing in multiphase flows by using EPDA,” Opt. Lasers Eng. 30, 3–15 (1998).
[CrossRef]

Part. Part. Syst. Charact.

A. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

Other

C Tropea, T.-H. Xu, F Onofri, G Gréhan, P. Haugen, “Dual-mode phase Doppler anemometry,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1994.

M Saffman, “The use of polarized light for optical particle sizing,” presented at the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 14–17, 1986.

H.-H. Qiu, M Sommerfeld, “The impact of signal processing on the accuracy of phase-Doppler measurements,” presented at the Sixth Workshop on Two-Phase Flow Predictions, Erlangen, Germany, March 30–April 2, 1992.

S. V. Sankar, A. S. Inenaga, W. D. Bachalo, “Trajectory dependent scattering in phase Doppler interferometer: minimizing and eliminating sizing errors,” presented at the Sixth International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, Portugal, July 20–23, 1992.

Y Aizu, F Durst, G Gréhan, F. Onofri, T.-H. Xu, “PDA systems without Gaussian beam defects,” presented at the Third International Conference on Optical Particle Sizing, Yokohama, Japan, August 23–26, 1993.

G Grehan, G Gouesbet, A Naqwi, F Durst, “Evaluation of phase Doppler system using generalized Lorenz– Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, September 24–27, 1991.

H.-H. Qiu, C. T. Hsu, “A Fourier optics method for the simulation of measurement-volume-effect by the slit constraint,” presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 8–11, 1996.

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Figures (12)

Fig. 1
Fig. 1

Optical layout of four-detector PDA. APD, avalanche photodiode.

Fig. 2
Fig. 2

Scattering rays from a microbubble: (a) surface reflection, (b) first-order refraction, (c) second-order refraction, (d) second-order refraction from the opposite side.

Fig. 3
Fig. 3

Description of different scattering mechanisms at a 34.5° scattering angle ( m = 1 / 1.33 ) .

Fig. 4
Fig. 4

Description of different scattering mechanisms at the optimized orientation angle ( m = 1 / 1.33 ) .

Fig. 5
Fig. 5

Phase-size-conversion factors for different scattering mechanisms.

Fig. 6
Fig. 6

Simulation of phase-measurement results among the single-scattering mechanism, the first-order approach, and the new approach.

Fig. 7
Fig. 7

Simulation of phase-measurement results among the single-scattering mechanism, the first-order approach, and the new approach.

Fig. 8
Fig. 8

Simulation of phase-measurement results among the single-scattering mechanism, the first-order approach, and the new approach.

Fig. 9
Fig. 9

Simulation of phase-measurement results among the single-scattering mechanism, the first-order approach, and the new approach.

Fig. 10
Fig. 10

Micro air bubble in glass for a validation experiment.

Fig. 11
Fig. 11

Experimental measurements of phase differences of an air bubble in Plexiglas moving through the measurement volume.

Fig. 12
Fig. 12

Comparison of measurement results between the single-scattering mechanism, the first-order, and the new approaches.

Tables (1)

Tables Icon

Table 1 Optical Parameters

Equations (37)

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n m sin   β p = n b sin   β p ; m = n b / n m ,
sin   β p = y p r ,
ϕ p = π ( 1 - p ) - 2 β p + 2 p   arcsin 1 m sin   β p
for y > 0 ,
ϕ p = π ( p - 1 ) + 2 β p - 2 p   arcsin 1 m sin   β p
for y < 0 ,
C p ± = 720   tan ( ψ ) sin ( β p ± ) sin ( θ / 2 ) λ   sin ( ϕ ) ,
β 0 + = π 2 - ϕ 2 ,
β 1 + = π 2 - arccos m 2 sin 2 ( ϕ / 2 ) 1 + m 2 - 2 m   cos ( ϕ / 2 ) 1 / 2 ,
cos   τ 2 = m   cos 2 τ 2 + ϕ 2 4                           for y > 0 ,
cos   τ 2 = m   cos 2 τ 2 - ϕ 2 4                           for y < 0 ,
cos   τ 2 = m   cos 2 τ 2 ± ϕ 2 4 .
2   cos 2 τ 2 2 - 1 - m   cos ϕ 4 cos τ 2 2
= ± m   sin ϕ 4 1 - cos 2 τ 2 2 1 / 2 .
x 4 - m   cos ϕ 4   x 3 + m 2 - 4 4   x 2 + m   cos ( ϕ / 4 ) 2   x
+ 1 - m 2 sin 2 ( ϕ / 4 ) 4 = 0 .
β 2 + = π 2 - 2   arccos ( x 2 + ) ,
β 2 - = π 2 - 2   arccos ( x 2 - ) ,
x 2 + = 1 2 - ( a - 8 y 1 + a 2 - 4 b ) 2 + ( a - 8 y 1 + a 2 - 4 b ) 2 4 - 4 y 1 - ( ay 1 - c ) 8 y 1 + a 2 - 4 b 1 / 2 ,
x 2 - = 1 2 - ( a - 8 y 1 + a 2 - 4 b ) 2 - ( a - 8 y 1 + a 2 - 4 b ) 2 4 - 4 y 1 - ( ay 1 - c ) 8 y 1 + a 2 - 4 b 1 / 2 ,
a = - m   cos ϕ 4 , b = m 2 - 4 4 ,
c = 1 2   m   cos ϕ 4 ,
d = 1 - m 2 sin 2 ( ϕ / 4 ) 4 ,
p = 3 ( ac - 4 d ) - b 2 12 ,
q = - 2 b 3 + 9 b ( ac - 4 d ) + 27 [ d ( 4 b - a 2 ) - c 2 ] 216 ,
Δ = q 2 4 + p 3 27 ,
y 1 = - q 2 + Δ 1 / 3 + - q 2 - Δ 1 / 3 + b 6 .
ϕ p = arccos ( 2 m 2 - 1 ) = π - 2   arcsin ( m ) .
ϕ p = π - 2   arcsin ( m ) = π ( 1 - p ) - 2 β p + 2 p   arcsin 1 m sin   β p .
1 m sin   β p = cos 1 p   [ arcsin ( m ) - β p ] .
β p = arcsin ( m ) .
φ 14 = 2   arctan I 1 2 - sin C 1 2 - D 2 - I 1 0 sin C 1 0 D 2 I 1 2 - cos C 1 2 - D 2 + I 1 0 cos C 1 0 D 2 ,
φ 23 = 2   arctan I 2 2 - sin C 2 2 - D 2 - I 2 0 sin C 2 0 D 2 I 2 2 - cos C 2 2 - D 2 + I 2 0 cos C 2 0 D 2 ,
φ 14 = 2   arctan I 2 - sin C 1 2 - D 2 - I 0 sin C 1 0 D 2 I 2 - cos C 1 2 - D 2 + I 0 cos C 1 0 D 2 = 2   arctan I 2 - I 2 - + I 0 sin C 1 2 - D 2 - 1 - I 2 - I 2 - + I 0 sin C 1 0 D 2 I 2 - I 2 - + I 0 cos C 1 2 - D 2 + 1 - I 2 - I 2 - + I 0 cos C 1 0 D 2 ,
φ 23 = 2   arctan I 2 - sin C 2 2 - D 2 - I 0 sin C 2 0 D 2 I 2 - cos C 2 2 - D 2 + I 0 cos C 2 0 D 2 = 2   arctan I 2 - I 2 - + I 0 sin C 2 2 - D 2 - 1 - I 2 - I 2 - + I 0 sin C 2 0 D 2 I 2 - I 2 - + I 0 cos C 2 2 - D 2 + 1 - I 2 - I 2 - + I 0 cos C 2 0 D 2 .
- cos C 1 2 - D 2 tan φ 14 2 - sin C 1 2 - D 2 tan φ 14 2   cos C 1 0 D 2 - cos C 1 2 - D 2 - sin C 1 0 D 2 - sin C 1 2 - D 2
- - cos C 2 2 - D 2 tan φ 23 2 - sin C 2 2 - D 2 tan φ 23 2   cos C 2 0 D 2 - cos C 2 2 - D 2 - sin C 2 0 D 2 - sin C 2 2 - D 2 = 0 .

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