Abstract

The accuracy of the particle size distribution (PSD) recovered from laser diffraction measurements by the Chin-Shifrin integral transform algorithm is reduced unless the proper angular parameters, including the lower and upper angular integration limits (θmax and θmin) and the angular resolution (Δθ), are used. To determine the selection criteria for these parameters, we use two metrics: the inversion error of the non-negative PSD, ε1, and the fitting error of the scattered laser light, ε2. By studying the variation of the minimum θmax and ε2 with the particle size at different inversion errors, and by analyzing the inversion error, as θmin and Δθ are varied, the optimized selection criteria for the minimum θmax, θmin and Δθ are obtained respectively. The inversion errors of the Chin-Shifrin algorithm with different selection criteria are compared, the different PSDs are recovered, and the optimal pixel selection range of the linear charge-coupled-device (CCD) array is determined according to the optimized selection criteria. Simulation results show that the optimized selection criteria for the angular parameters make the PSDs retrieved with the Chin-Shifrin algorithm more accurate.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. H. Chin, C. M. Sliepcevich, and M. Tribus, “Particle Size Distributions from Angular Variation of Intensity of Forward-Scattered Light at Very Small Angles,” J. Phys. Chem. 59(9), 841–844 (1955).
  2. K. S. Shifrin and A. Y. Perelman, “Calculation of particle distribution by the data on spectral transparency,” Pure Appl. Geophys. 58(1), 208–220 (1964).
  3. S. D. Coston and N. George, “Particle sizing by inversion of the optical transform pattern,” Appl. Opt. 30(33), 4785–4794 (1991).
    [PubMed]
  4. J. Vargas-Ubera, J. F. Aguilar, and D. M. Gale, “Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods,” Appl. Opt. 46(1), 124–132 (2007).
    [PubMed]
  5. Z. Cao, L. Xu, and J. Ding, “Integral inversion to Fraunhofer diffraction for particle sizing,” Appl. Opt. 48(25), 4842–4850 (2009).
    [PubMed]
  6. L. C. Chow and C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15(2), 378–383 (1976).
    [PubMed]
  7. R. Santer and M. Herman, “Particle size distributions from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22(15), 2294–2301 (1983).
    [PubMed]
  8. K. S. Shifrin and I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys 3, 749–753 (1967).
  9. L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).
  10. J. H. Koo and E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31(12), 2130–2140 (1992).
    [PubMed]
  11. Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).
  12. B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
    [PubMed]
  13. J. C. Knight, D. Ball, and G. N. Robertson, “Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution,” Appl. Opt. 30(33), 4795–4799 (1991).
    [PubMed]
  14. A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).
  15. J. B. Riley and Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30(33), 4800–4817 (1991).
    [PubMed]
  16. S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).
  17. J. J. Liu, “Essential parameters in particle sizing by integral transform inversions,” Appl. Opt. 36(22), 5535–5545 (1997).
    [PubMed]
  18. F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).
  19. G. Mie, “Beitrage Zur Optik Truber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25(3), 377–445 (1908).
  20. M. A. Popovici, N. Mincu, and A. Popovici, “A comparative study of processing simulated and experimental data in elastic laser light scattering,” Math. Biosci. 157(1-2), 321–344 (1999).
    [PubMed]
  21. G. M. Jia, G. Z. Zhang, and W. H. Xiang, “Using portable laser and CCD to do small particle sizing,” Acta Photon. Sin. 35(9), 1293–1295 (2006).
  22. Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

2017 (1)

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

2016 (1)

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

2012 (1)

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

2011 (1)

B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
[PubMed]

2009 (2)

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Z. Cao, L. Xu, and J. Ding, “Integral inversion to Fraunhofer diffraction for particle sizing,” Appl. Opt. 48(25), 4842–4850 (2009).
[PubMed]

2007 (1)

2006 (1)

G. M. Jia, G. Z. Zhang, and W. H. Xiang, “Using portable laser and CCD to do small particle sizing,” Acta Photon. Sin. 35(9), 1293–1295 (2006).

1999 (1)

M. A. Popovici, N. Mincu, and A. Popovici, “A comparative study of processing simulated and experimental data in elastic laser light scattering,” Math. Biosci. 157(1-2), 321–344 (1999).
[PubMed]

1997 (1)

1992 (1)

1991 (3)

1983 (1)

1977 (1)

A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).

1976 (2)

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

L. C. Chow and C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15(2), 378–383 (1976).
[PubMed]

1967 (1)

K. S. Shifrin and I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys 3, 749–753 (1967).

1964 (1)

K. S. Shifrin and A. Y. Perelman, “Calculation of particle distribution by the data on spectral transparency,” Pure Appl. Geophys. 58(1), 208–220 (1964).

1955 (1)

J. H. Chin, C. M. Sliepcevich, and M. Tribus, “Particle Size Distributions from Angular Variation of Intensity of Forward-Scattered Light at Very Small Angles,” J. Phys. Chem. 59(9), 841–844 (1955).

1908 (1)

G. Mie, “Beitrage Zur Optik Truber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25(3), 377–445 (1908).

Agrawal, Y. C.

Aguilar, J. F.

Assid, K.

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Ball, D.

Bao, Z. H.

B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
[PubMed]

Boitova, L. N.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

Cao, Z.

Chen, Q.

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

Chin, J. H.

J. H. Chin, C. M. Sliepcevich, and M. Tribus, “Particle Size Distributions from Angular Variation of Intensity of Forward-Scattered Light at Very Small Angles,” J. Phys. Chem. 59(9), 841–844 (1955).

Chow, L. C.

Cornet, A.

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Coston, S. D.

Dai, B.

B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
[PubMed]

Ding, J.

Dong, L.

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

Dou, Z.

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

Gale, D. M.

George, N.

Herman, M.

Hirleman, E. D.

Houmairi, S.

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).

Jia, G. M.

G. M. Jia, G. Z. Zhang, and W. H. Xiang, “Using portable laser and CCD to do small particle sizing,” Acta Photon. Sin. 35(9), 1293–1295 (2006).

Knight, J. C.

Kolmakov, I. B.

K. S. Shifrin and I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys 3, 749–753 (1967).

Koo, J. H.

Kudryavitskii, F. A.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

Liu, J. J.

Liu, W.

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

Mei, D. Q.

B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
[PubMed]

Mie, G.

G. Mie, “Beitrage Zur Optik Truber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25(3), 377–445 (1908).

Mincu, N.

M. A. Popovici, N. Mincu, and A. Popovici, “A comparative study of processing simulated and experimental data in elastic laser light scattering,” Math. Biosci. 157(1-2), 321–344 (1999).
[PubMed]

Ming, H.

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

Nassim, A.

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Perelman, A. Y.

K. S. Shifrin and A. Y. Perelman, “Calculation of particle distribution by the data on spectral transparency,” Pure Appl. Geophys. 58(1), 208–220 (1964).

Petrov, G. D.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

Popovici, A.

M. A. Popovici, N. Mincu, and A. Popovici, “A comparative study of processing simulated and experimental data in elastic laser light scattering,” Math. Biosci. 157(1-2), 321–344 (1999).
[PubMed]

Popovici, M. A.

M. A. Popovici, N. Mincu, and A. Popovici, “A comparative study of processing simulated and experimental data in elastic laser light scattering,” Math. Biosci. 157(1-2), 321–344 (1999).
[PubMed]

Rachafi, S.

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Riley, J. B.

Robertson, G. N.

Robinskii, V. L.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

Santer, R.

Shen, J.

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

Shifrin, K. S.

K. S. Shifrin and I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys 3, 749–753 (1967).

K. S. Shifrin and A. Y. Perelman, “Calculation of particle distribution by the data on spectral transparency,” Pure Appl. Geophys. 58(1), 208–220 (1964).

Sliepcevich, C. M.

J. H. Chin, C. M. Sliepcevich, and M. Tribus, “Particle Size Distributions from Angular Variation of Intensity of Forward-Scattered Light at Very Small Angles,” J. Phys. Chem. 59(9), 841–844 (1955).

Sokolov, R. N.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

Thomas, J. C.

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

Tien, C. L.

Tribus, M.

J. H. Chin, C. M. Sliepcevich, and M. Tribus, “Particle Size Distributions from Angular Variation of Intensity of Forward-Scattered Light at Very Small Angles,” J. Phys. Chem. 59(9), 841–844 (1955).

Vargas-Ubera, J.

Wang, A. T.

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

Wang, W. J.

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

Xiang, W. H.

G. M. Jia, G. Z. Zhang, and W. H. Xiang, “Using portable laser and CCD to do small particle sizing,” Acta Photon. Sin. 35(9), 1293–1295 (2006).

Xu, L.

Xu, S. L.

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

Yang, F. G.

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

Yang, L.

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

Yuan, Y. N.

B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
[PubMed]

Zhang, G. Z.

G. M. Jia, G. Z. Zhang, and W. H. Xiang, “Using portable laser and CCD to do small particle sizing,” Acta Photon. Sin. 35(9), 1293–1295 (2006).

Acta Photon. Sin. (2)

G. M. Jia, G. Z. Zhang, and W. H. Xiang, “Using portable laser and CCD to do small particle sizing,” Acta Photon. Sin. 35(9), 1293–1295 (2006).

Q. Chen, W. Liu, Z. Dou, L. Yang, and J. Shen, “Improved Chin-Shifrin algorithm in the Measurement of particle sizing used by Fraunhofer Diffraction Method,” Acta Photon. Sin. 45(11), 118–123 (2016).

Ann. Phys. (1)

G. Mie, “Beitrage Zur Optik Truber Medien, Speziell Kolloidaler Metallosungen,” Ann. Phys. 25(3), 377–445 (1908).

Appl. Opt. (9)

S. D. Coston and N. George, “Particle sizing by inversion of the optical transform pattern,” Appl. Opt. 30(33), 4785–4794 (1991).
[PubMed]

J. Vargas-Ubera, J. F. Aguilar, and D. M. Gale, “Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods,” Appl. Opt. 46(1), 124–132 (2007).
[PubMed]

Z. Cao, L. Xu, and J. Ding, “Integral inversion to Fraunhofer diffraction for particle sizing,” Appl. Opt. 48(25), 4842–4850 (2009).
[PubMed]

L. C. Chow and C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15(2), 378–383 (1976).
[PubMed]

R. Santer and M. Herman, “Particle size distributions from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22(15), 2294–2301 (1983).
[PubMed]

J. H. Koo and E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31(12), 2130–2140 (1992).
[PubMed]

J. C. Knight, D. Ball, and G. N. Robertson, “Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution,” Appl. Opt. 30(33), 4795–4799 (1991).
[PubMed]

J. B. Riley and Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30(33), 4800–4817 (1991).
[PubMed]

J. J. Liu, “Essential parameters in particle sizing by integral transform inversions,” Appl. Opt. 36(22), 5535–5545 (1997).
[PubMed]

Combust. Explos. Shock Waves (1)

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, and R. N. Sokolov, “Particle-size distribution for the flame from a mixture containing magnesium powder,” Combust. Explos. Shock Waves 12(2), 258–262 (1976).

Guangpuxue Yu Guangpu Fenxi (1)

B. Dai, Y. N. Yuan, Z. H. Bao, and D. Q. Mei, “[An improved method for inversion of particle size distribution from scattering spectrum,” Guangpuxue Yu Guangpu Fenxi 31(2), 539–542 (2011).
[PubMed]

Izv. USSR Acad. Sci. Atmos. Oceanic Phys (1)

K. S. Shifrin and I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys 3, 749–753 (1967).

J. Phys. Chem. (1)

J. H. Chin, C. M. Sliepcevich, and M. Tribus, “Particle Size Distributions from Angular Variation of Intensity of Forward-Scattered Light at Very Small Angles,” J. Phys. Chem. 59(9), 841–844 (1955).

Math. Biosci. (1)

M. A. Popovici, N. Mincu, and A. Popovici, “A comparative study of processing simulated and experimental data in elastic laser light scattering,” Math. Biosci. 157(1-2), 321–344 (1999).
[PubMed]

Optik (Stuttg.) (1)

S. Houmairi, K. Assid, A. Nassim, S. Rachafi, and A. Cornet, “Digital optical particle sizing instrument based on Chin–Shifrin inversion,” Optik (Stuttg.) 120(3), 141–145 (2009).

Powder Technol. (1)

Q. Chen, W. Liu, W. J. Wang, J. C. Thomas, and J. Shen, “Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm,” Powder Technol. 317(15), 95–103 (2017).

Proc. IEEE (1)

A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review,” Proc. IEEE 65(11), 1565–1596 (1977).

Pure Appl. Geophys. (1)

K. S. Shifrin and A. Y. Perelman, “Calculation of particle distribution by the data on spectral transparency,” Pure Appl. Geophys. 58(1), 208–220 (1964).

Waves Random Complex Media (1)

F. G. Yang, A. T. Wang, S. L. Xu, L. Dong, and H. Ming, “Numerical studies on the Chin–Shifrin inversion method for particle sizing,” Waves Random Complex Media 22(2), 121–132 (2012).

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Figures (10)

Fig. 1
Fig. 1 The influence of different angular parameters on the PSDs recovered via the C-S algorithm: (a) The influence of the upper limit of angular integration, θmax, on inverted PSDs. (b)The influence of the lower limit of angular integration, θmin, on inverted PSDs. (c) The influence of the angular resolution, Δθ, on inverted PSDs.
Fig. 2
Fig. 2 Simulated diffracted intensity patterns for the model PSDs: (a) Normal particle size distribution with μ = 80 μm and σ = 5 μm. (b) Rosin-Rammler particle size distribution with D0 = 60 μm and n = 13.
Fig. 3
Fig. 3 (a). The fitting error, ε2, of the scattered light for different inversion errors, ε1. (b). The minimum upper limit of angular integration, θmax, for different inversion errors, ε1.
Fig. 4
Fig. 4 (a). Variation of the inversion error, ε1, with θmin. (b). The relationship between the optimized θmin and size.
Fig. 5
Fig. 5 The variation of Δθ with particle diameter: (a) Ideal angular resolution. (b) Optimized angular resolution.
Fig. 6
Fig. 6 NNPSDs recovered by the C-S algorithm for particles obeying the Rosin-Rammler distribution: (a) D0 = 60 μm, n = 10. (b) D0 = 100 μm, n = 13.
Fig. 7
Fig. 7 NNPSDs recovered via the C-S algorithm for different PSDs from intensity patterns by Mie theory with three types of asymmetrical PSDs.
Fig. 8
Fig. 8 PSD results for the C-S algorithm for the multiple peak particle size distribution for different angular parameters: (a) PSD recovered with angular parameters determined based on D0 = 60 μm. (b) PSD recovered with angular parameters determined based on D0 = 130 μm.
Fig. 9
Fig. 9 The optimal pixel range determined by the optimized selection criteria: (a) The end pixel determined by Criterion 1. (b) The start pixel determined by Criterion 2. (c) The pixel spacing determined by Criterion 3.
Fig. 10
Fig. 10 The recovered NNPSD obtained by the C-S algorithm based on a linear CCD detector.

Tables (4)

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Table 1 The variation of inversion error ε1 with particle diameter for the different minimum θmax.

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Table 2 Main peak particle sizes of the recovered NNPSDs for the different minimum θmax.

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Table 3 The variation of inversion error ε1 with particle diameter for the different maximum θmax.

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Table 4 The inversion error ε1 of recovered NNPSDs for different Δθ.

Equations (15)

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I( θ )= 0 I( θ,x,m )f( x )dx .
I( θ,x,m )=I( θ,x )= I 0 k 2 F 2 [ x J 1 ( xsinθ ) sinθ ] 2 ,
I( θ,x )= I 0 k 2 F 2 0 [ x J 1 ( xθ ) θ ] 2 f( x )dx .
f( x )= 2π k 2 F 2 I 0 x 2 0 J 1 ( xθ ) Y 1 ( xθ )xθ d dθ [ I( θ,x ) θ 3 ]dθ .
f( d )= 4π d 3 3 × 2π k 2 F 2 I 0 ( πd/λ ) 2 0 J 1 ( πdθ/λ ) Y 1 ( πdθ/λ )( πdθ/λ ) d dθ [ I( θ,πd/λ ) θ 3 ]dθ =Cd 0 H cs ( πdθ/λ ) E cs ( πdθ/λ )dθ
f( d )=Cd θ= θ min θ= θ max H cs ( πdθ/λ ) E cs ( πdθ/λ )Δθ .
f NOR ( d )= 1 2π σ 1 exp[ ( d μ 1 ) 2 2 σ 1 2 ],
f RR (d)= n d n1 D 0 n exp[ ( d D 0 ) n ],
f G ( d )= d s1 Γ( s ) t s exp[ d t ],
f L ( d )= 1 2π d σ 2 exp[ ( lnd μ 2 ) 2 2 σ 2 2 ],
ε 1 = j=1 j=N [ f nncs ( d j ) f in ( d j ) ] 2 / j=1 j=N f in 2 ( d j ) ,
ε 2 = θ= θ min θ= θ max ( I fit,θ I in,θ ) 2 / θ= θ min θ= θ max I in,θ 2 ,
θ max 0.0365 0.001 1+exp(5× 10 4 d) .
θ min =2.502× 10 3 × d 0.1 5.23× 10 3 .
Δθ= 1.341× 10 8 /d 1.577× 10 5 .

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