Abstract

An algorithm is offered for finding the range within which cumulative particle size distribution functions can be located in consistency with experimental turbidimetric data at a number of wavelengths. It is based on linear programming and minimization techniques. Several tests were performed. The lower right-hand branch of the corridor was found to locate near the initial distribution function.

© 2014 Optical Society of America

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References

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  1. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 1977).
  2. S. Y. Shchyogolev, “Inverse problems of spectroturbidimetry of biological disperse systems: an overview,” J. Biomed. Opt. 4, 490–503 (1999).
    [CrossRef]
  3. T. Kourti, “Turbidimetry in particle size analysis,” in Encyclopedia of Analytical Chemistry: Instrumentation and Applications, R. A. Meyers, ed. (Wiley, 2000), pp. 5549–5580.
  4. H. Tang and S.-Y. Ding, “Inversion of particle size distribution with unknown refractive index from specific turbidimetry,” in 2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering, Vol. 3 (IEEE, 2010), pp. 116–119.
  5. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  6. B. Sedláček, “Light scattering. XVI. Particle size determination by the turbidity ratio method,” Collection of Czechoslovak Chemical Communications (1929-1939, 1947-) 32, 1374–1389 (1967).
    [CrossRef]
  7. B. Sedláček and K. Zimmerman, “Integral and differential turbidity ratio method applied to the analysis of changes in the size and number of scatterers,” Polymer Bulletin 7, 531–538 (1982).
  8. T. Kourti, J. F. MacGregor, and A. E. Hamielec, “Turbidimetric techniques. Capability to provide the full particle size distribution,” in Particle Size Distribution II, T. Provder, ed. (ACS, 1991), pp. 2–19.
  9. G. B. Dantzig and M. N. Thapa, Linear Programming, 1: Introduction (Springer, 1997).
  10. O. Platz and R. G. Gordon, “Rigorous bounds for time-dependent correlation functions,” Phys. Rev. Lett. 30, 264–267 (1973).
    [CrossRef]
  11. R. McGraw, “Sparse aerosol models beyond the quadrature method of moments,” AIP Conf. Proc. 1527, 651–654 (2013).
  12. B. D. Bunday, Basic Optimization Methods (Edward Arnold, 1984).
  13. R. L. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
    [CrossRef]
  14. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 1961).
  15. W. Heller, H. L. Bhatnagar, and M. Nakagaki, “Theoretical investigations on the light scattering of spheres. XIII. The “wavelength exponent” of differential turbidity spectra,” J. Chem. Phys. 36, 1163–1170 (1962).
    [CrossRef]
  16. D. H. Melik and H. S. Fogler, “Turbidimetric determination of particle size distributions of colloidal systems,” J. Colloid Interface Sci. 92, 161–180 (1983).
    [CrossRef]
  17. W. Heller and W. J. Pangonis, “Theoretical investigations on the light scattering of colloidal spheres. I. The specific turbidity,” J. Chem. Phys. 26, 498–506 (1957).
    [CrossRef]
  18. P. Rosin and E. Rammler, “The laws governing the fineness of powdered coal,” J. Inst. Fuel 7, 29–36 (1933).
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

2013 (1)

R. McGraw, “Sparse aerosol models beyond the quadrature method of moments,” AIP Conf. Proc. 1527, 651–654 (2013).

1999 (1)

S. Y. Shchyogolev, “Inverse problems of spectroturbidimetry of biological disperse systems: an overview,” J. Biomed. Opt. 4, 490–503 (1999).
[CrossRef]

1983 (1)

D. H. Melik and H. S. Fogler, “Turbidimetric determination of particle size distributions of colloidal systems,” J. Colloid Interface Sci. 92, 161–180 (1983).
[CrossRef]

1982 (1)

B. Sedláček and K. Zimmerman, “Integral and differential turbidity ratio method applied to the analysis of changes in the size and number of scatterers,” Polymer Bulletin 7, 531–538 (1982).

1980 (1)

R. L. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

1973 (1)

O. Platz and R. G. Gordon, “Rigorous bounds for time-dependent correlation functions,” Phys. Rev. Lett. 30, 264–267 (1973).
[CrossRef]

1967 (1)

B. Sedláček, “Light scattering. XVI. Particle size determination by the turbidity ratio method,” Collection of Czechoslovak Chemical Communications (1929-1939, 1947-) 32, 1374–1389 (1967).
[CrossRef]

1962 (1)

W. Heller, H. L. Bhatnagar, and M. Nakagaki, “Theoretical investigations on the light scattering of spheres. XIII. The “wavelength exponent” of differential turbidity spectra,” J. Chem. Phys. 36, 1163–1170 (1962).
[CrossRef]

1957 (1)

W. Heller and W. J. Pangonis, “Theoretical investigations on the light scattering of colloidal spheres. I. The specific turbidity,” J. Chem. Phys. 26, 498–506 (1957).
[CrossRef]

1933 (1)

P. Rosin and E. Rammler, “The laws governing the fineness of powdered coal,” J. Inst. Fuel 7, 29–36 (1933).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Bhatnagar, H. L.

W. Heller, H. L. Bhatnagar, and M. Nakagaki, “Theoretical investigations on the light scattering of spheres. XIII. The “wavelength exponent” of differential turbidity spectra,” J. Chem. Phys. 36, 1163–1170 (1962).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Bunday, B. D.

B. D. Bunday, Basic Optimization Methods (Edward Arnold, 1984).

Dantzig, G. B.

G. B. Dantzig and M. N. Thapa, Linear Programming, 1: Introduction (Springer, 1997).

Ding, S.-Y.

H. Tang and S.-Y. Ding, “Inversion of particle size distribution with unknown refractive index from specific turbidimetry,” in 2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering, Vol. 3 (IEEE, 2010), pp. 116–119.

Fogler, H. S.

D. H. Melik and H. S. Fogler, “Turbidimetric determination of particle size distributions of colloidal systems,” J. Colloid Interface Sci. 92, 161–180 (1983).
[CrossRef]

Gordon, R. G.

O. Platz and R. G. Gordon, “Rigorous bounds for time-dependent correlation functions,” Phys. Rev. Lett. 30, 264–267 (1973).
[CrossRef]

Hamielec, A. E.

T. Kourti, J. F. MacGregor, and A. E. Hamielec, “Turbidimetric techniques. Capability to provide the full particle size distribution,” in Particle Size Distribution II, T. Provder, ed. (ACS, 1991), pp. 2–19.

Heller, W.

W. Heller, H. L. Bhatnagar, and M. Nakagaki, “Theoretical investigations on the light scattering of spheres. XIII. The “wavelength exponent” of differential turbidity spectra,” J. Chem. Phys. 36, 1163–1170 (1962).
[CrossRef]

W. Heller and W. J. Pangonis, “Theoretical investigations on the light scattering of colloidal spheres. I. The specific turbidity,” J. Chem. Phys. 26, 498–506 (1957).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 1961).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 1961).

Kourti, T.

T. Kourti, “Turbidimetry in particle size analysis,” in Encyclopedia of Analytical Chemistry: Instrumentation and Applications, R. A. Meyers, ed. (Wiley, 2000), pp. 5549–5580.

T. Kourti, J. F. MacGregor, and A. E. Hamielec, “Turbidimetric techniques. Capability to provide the full particle size distribution,” in Particle Size Distribution II, T. Provder, ed. (ACS, 1991), pp. 2–19.

MacGregor, J. F.

T. Kourti, J. F. MacGregor, and A. E. Hamielec, “Turbidimetric techniques. Capability to provide the full particle size distribution,” in Particle Size Distribution II, T. Provder, ed. (ACS, 1991), pp. 2–19.

McGraw, R.

R. McGraw, “Sparse aerosol models beyond the quadrature method of moments,” AIP Conf. Proc. 1527, 651–654 (2013).

Melik, D. H.

D. H. Melik and H. S. Fogler, “Turbidimetric determination of particle size distributions of colloidal systems,” J. Colloid Interface Sci. 92, 161–180 (1983).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Nakagaki, M.

W. Heller, H. L. Bhatnagar, and M. Nakagaki, “Theoretical investigations on the light scattering of spheres. XIII. The “wavelength exponent” of differential turbidity spectra,” J. Chem. Phys. 36, 1163–1170 (1962).
[CrossRef]

Pangonis, W. J.

W. Heller and W. J. Pangonis, “Theoretical investigations on the light scattering of colloidal spheres. I. The specific turbidity,” J. Chem. Phys. 26, 498–506 (1957).
[CrossRef]

Platz, O.

O. Platz and R. G. Gordon, “Rigorous bounds for time-dependent correlation functions,” Phys. Rev. Lett. 30, 264–267 (1973).
[CrossRef]

Rammler, E.

P. Rosin and E. Rammler, “The laws governing the fineness of powdered coal,” J. Inst. Fuel 7, 29–36 (1933).

Rosin, P.

P. Rosin and E. Rammler, “The laws governing the fineness of powdered coal,” J. Inst. Fuel 7, 29–36 (1933).

Sedlácek, B.

B. Sedláček and K. Zimmerman, “Integral and differential turbidity ratio method applied to the analysis of changes in the size and number of scatterers,” Polymer Bulletin 7, 531–538 (1982).

B. Sedláček, “Light scattering. XVI. Particle size determination by the turbidity ratio method,” Collection of Czechoslovak Chemical Communications (1929-1939, 1947-) 32, 1374–1389 (1967).
[CrossRef]

Shchyogolev, S. Y.

S. Y. Shchyogolev, “Inverse problems of spectroturbidimetry of biological disperse systems: an overview,” J. Biomed. Opt. 4, 490–503 (1999).
[CrossRef]

Tang, H.

H. Tang and S.-Y. Ding, “Inversion of particle size distribution with unknown refractive index from specific turbidimetry,” in 2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering, Vol. 3 (IEEE, 2010), pp. 116–119.

Thapa, M. N.

G. B. Dantzig and M. N. Thapa, Linear Programming, 1: Introduction (Springer, 1997).

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 1977).

Zimmerman, K.

B. Sedláček and K. Zimmerman, “Integral and differential turbidity ratio method applied to the analysis of changes in the size and number of scatterers,” Polymer Bulletin 7, 531–538 (1982).

Zollars, R. L.

R. L. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

AIP Conf. Proc. (1)

R. McGraw, “Sparse aerosol models beyond the quadrature method of moments,” AIP Conf. Proc. 1527, 651–654 (2013).

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Collection of Czechoslovak Chemical Communications (1929-1939, 1947-) (1)

B. Sedláček, “Light scattering. XVI. Particle size determination by the turbidity ratio method,” Collection of Czechoslovak Chemical Communications (1929-1939, 1947-) 32, 1374–1389 (1967).
[CrossRef]

J. Biomed. Opt. (1)

S. Y. Shchyogolev, “Inverse problems of spectroturbidimetry of biological disperse systems: an overview,” J. Biomed. Opt. 4, 490–503 (1999).
[CrossRef]

J. Chem. Phys. (2)

W. Heller, H. L. Bhatnagar, and M. Nakagaki, “Theoretical investigations on the light scattering of spheres. XIII. The “wavelength exponent” of differential turbidity spectra,” J. Chem. Phys. 36, 1163–1170 (1962).
[CrossRef]

W. Heller and W. J. Pangonis, “Theoretical investigations on the light scattering of colloidal spheres. I. The specific turbidity,” J. Chem. Phys. 26, 498–506 (1957).
[CrossRef]

J. Colloid Interface Sci. (2)

D. H. Melik and H. S. Fogler, “Turbidimetric determination of particle size distributions of colloidal systems,” J. Colloid Interface Sci. 92, 161–180 (1983).
[CrossRef]

R. L. Zollars, “Turbidimetric method for on-line determination of latex particle number and particle size distribution,” J. Colloid Interface Sci. 74, 163–172 (1980).
[CrossRef]

J. Inst. Fuel (1)

P. Rosin and E. Rammler, “The laws governing the fineness of powdered coal,” J. Inst. Fuel 7, 29–36 (1933).

Phys. Rev. Lett. (1)

O. Platz and R. G. Gordon, “Rigorous bounds for time-dependent correlation functions,” Phys. Rev. Lett. 30, 264–267 (1973).
[CrossRef]

Polymer Bulletin (1)

B. Sedláček and K. Zimmerman, “Integral and differential turbidity ratio method applied to the analysis of changes in the size and number of scatterers,” Polymer Bulletin 7, 531–538 (1982).

Other (8)

T. Kourti, J. F. MacGregor, and A. E. Hamielec, “Turbidimetric techniques. Capability to provide the full particle size distribution,” in Particle Size Distribution II, T. Provder, ed. (ACS, 1991), pp. 2–19.

G. B. Dantzig and M. N. Thapa, Linear Programming, 1: Introduction (Springer, 1997).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 1977).

T. Kourti, “Turbidimetry in particle size analysis,” in Encyclopedia of Analytical Chemistry: Instrumentation and Applications, R. A. Meyers, ed. (Wiley, 2000), pp. 5549–5580.

H. Tang and S.-Y. Ding, “Inversion of particle size distribution with unknown refractive index from specific turbidimetry,” in 2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering, Vol. 3 (IEEE, 2010), pp. 116–119.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 1961).

B. D. Bunday, Basic Optimization Methods (Edward Arnold, 1984).

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Figures (2)

Fig. 1.
Fig. 1.

Cumulative PSD curves: source (thin) and calculated right-hand lower limiting ones (thick) for the set of wavelengths 400, 440, 490, 550, and 585 nm. Simulation: Rosin–Rammler’s functions, k=10, R=450nm.

Fig. 2.
Fig. 2.

Cumulative PSD curves: source (thin) and calculated right-hand lower limiting ones (thick) for the set of wavelengths 400, 440, 490, 550, and 585 nm. Simulation: a monodisperse system, R=450nm.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

lnI0(λ)Il(λ)=τ(λ)l,
τ=N2πr2K(α,m)andτ=N2πr2K(α,m)f(r)dr,
T1=τ(λ1)τ(λ0),T2=τ(λ2)τ(λ0)TL1=τ(λL1)τ(λ0).
κ1=λ0λ1=α1α0,
T1=K(κ1α0)K(α0).
T1=r2K(2πμ1r/λ1)f(r)drr2K(2πμ1r/λ0)f(r)dr
[T1K(2πμ1r/λ0)K(2πμ1r/λ1)]r2f(r)dr=0.
[T1K(α0)K(κ1α0)]α02f(α0)dα0=0.
α¯0=α0f(α0)dα0,
{wi=1,[T1K(αi)K(κ1αi)]αi2wi=0,[T2K(αi)K(κ2αi)]αi2wi=0,[TL1K(αi)K(κL1αi)]αi2wi=0,αiwi=α¯0max.
±U(α0αi)wi=F(α0)max,
n=lnτlnλ=lnK(α,m)lnα,
n¯=n(α)α2K(α)f(r)drα2K(α)f(r)dr,
[n¯n(α)]α2K(α)f(α)dα=0.
[n¯n(αi)]αi2K(αi)wi=0.
Φ=N243πr3f(r)dr,
g(α)=τΦ=3πμ12λ·α2K(α)f(r)drα3f(r)dr=3πμ12λ·α2K(α)f(α)dαα3f(α)dα.
{αi3wi=1,[n¯n(αi)]αi2K(αi)wi=0,±αi2K(αi)wimax.
F(r)=1exp{(rR)k},f(r)=kR(rR)k1exp{(rR)k}.
T=ri2K(2πμ1ri/λ1)wiri2K(2πμ1ri/λ0)wi

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