Abstract

The determination of the particle size distribution and the volume fraction in concentrated suspensions from the multiwavelength measurement of isotropic-scattering coefficients by use of frequency-domain photon migration techniques is demonstrated for three different polydisperse polystyrene suspensions. When a Newton-type inverse algorithm is used, the successful recovery of the particle size distribution, in the form of a Weibull function, and the volume fraction of polystyrene suspensions is achieved. Our results are in excellent agreement with dynamic light-scattering size distribution measurements. On consideration of the particle mass conservation as an additional constraint penalty term in the inverse algorithm, it is shown that the quality of the particle size distribution reconstruction can be improved. Because no calibration is needed, photon migration techniques are especially suited for on-line measurement of the particle size distribution and the volume fraction in the chemical- and the pharmaceutical-based industries.

© 1997 Optical Society of America

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References

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  1. T. Allen, Particle Size Measurement, 4th ed. (Chapman and Hall, New York, 1990).
    [CrossRef]
  2. J. Wang, F. R. Hallet, “Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum,” Appl. Opt. 35, 193–200 (1996).
    [CrossRef] [PubMed]
  3. P.-H. Wang, G. S. Kent, M. P. McCormick, L. W. Thomason, G. K. Yue, “Retrieval analysis of aerosol-size distribution with simulated extinction measurements at SAGE III wavelengths,” Appl. Opt. 35, 433–440 (1996).
    [CrossRef] [PubMed]
  4. G. E. Elicabe, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetric measurements,” in Polymer Characterization, Vol. 227 of ACS Advances in Chemistry Series, C. Carver, T. Provder, eds. (American Chemical Society, Washington, D.C., 1990), pp. 83–104.
  5. J. Vavra, J. Antalik, M. Liska, “Application of regression analysis in spectroturbidity size characterization methods,” Part. Part. Syst. Charact. 12, 38–41 (1995).
    [CrossRef]
  6. K. S. Shifrin, I. G. Zolotov, “Spectral attenuation and aerosol particle size distribution,” Appl. Opt. 35, 2114–2124 (1996).
    [CrossRef] [PubMed]
  7. F. Sun, J. W. L. Lewis, “Simplex deconvolutions of particle-size distribution functions from optical measurements,” Appl. Opt. 34, 8437–8445 (1995).
    [CrossRef] [PubMed]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1983).
  9. J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
    [CrossRef] [PubMed]
  10. M. Box, B. McKellar, “Relationship between two analytic inversion formulae for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
    [CrossRef] [PubMed]
  11. A. Fymat, C. Smith, “Analytical inversion in remote sensing of particle size distributions. 4: comparison of Fymat and Box–McKellar solutions in the anomalous diffraction approximation,” Appl. Opt. 18, 3595–3598 (1979).
    [CrossRef] [PubMed]
  12. J. Heintzenberg, H. Muller, H. Quenzel, E. Thomalla, “Information content of optical data with respect to aerosol properties: numerical studies with a randomized minimization-search-technique inversion algorithm,” Appl. Opt. 20, 1308–1315 (1981).
    [CrossRef] [PubMed]
  13. O. V. Dubovik, T. V. Lapyonok, S. L. Oshchepkov, “Improved technique for data inversion: optical sizing of multicomponent aerosols,” Appl. Opt. 34, 8422–8436 (1995).
    [CrossRef] [PubMed]
  14. E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
    [CrossRef] [PubMed]
  15. B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
    [CrossRef]
  16. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1976), pp. 103–232.
  18. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).
  19. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  20. G. F. Bohren, D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  21. M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements using phase modulation spectroscopy,” in Photodynamic Therapy: Mechanisms II, J. Dougherty, ed., Proc. SPIE1203, 62–75 (1991).
  22. J. B. Fishkin, P. T. C. So, A. E. Cerussi, S. Fantini, M. A. Franceschini, E. Gratton, “Frequency-domain method for measuring spectral properties in multiple scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. 34, 1143–1155 (1995).
    [CrossRef] [PubMed]
  23. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
    [CrossRef] [PubMed]
  24. C. Derman, L. J. Gleser, I. Olkin, A Guide to Probability Theory and Application (Holt, Rinehart & Winston, New York, 1973), Chap. 8, p. 378.
  25. D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 11, 431–441 (1963).
    [CrossRef]
  26. J. R. Alcala, E. Gratton, D. M. Jameson, “A multifrequency phase fluorimeter using the harmonic content of a mode-locked laser,” Anal. Instrum. 14, 225–250 (1985).
    [CrossRef]
  27. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, Press, New York, 1983), pp. 51–108.
    [CrossRef]

1996 (4)

1995 (4)

1994 (1)

1992 (1)

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

1991 (1)

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

1985 (1)

J. R. Alcala, E. Gratton, D. M. Jameson, “A multifrequency phase fluorimeter using the harmonic content of a mode-locked laser,” Anal. Instrum. 14, 225–250 (1985).
[CrossRef]

1984 (1)

1981 (1)

1979 (2)

1963 (1)

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Alcala, J. R.

J. R. Alcala, E. Gratton, D. M. Jameson, “A multifrequency phase fluorimeter using the harmonic content of a mode-locked laser,” Anal. Instrum. 14, 225–250 (1985).
[CrossRef]

Allen, T.

T. Allen, Particle Size Measurement, 4th ed. (Chapman and Hall, New York, 1990).
[CrossRef]

Antalik, J.

J. Vavra, J. Antalik, M. Liska, “Application of regression analysis in spectroturbidity size characterization methods,” Part. Part. Syst. Charact. 12, 38–41 (1995).
[CrossRef]

Bohren, G. F.

G. F. Bohren, D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Box, M.

Cerussi, A. E.

Chance, B.

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements using phase modulation spectroscopy,” in Photodynamic Therapy: Mechanisms II, J. Dougherty, ed., Proc. SPIE1203, 62–75 (1991).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).

Derman, C.

C. Derman, L. J. Gleser, I. Olkin, A Guide to Probability Theory and Application (Holt, Rinehart & Winston, New York, 1973), Chap. 8, p. 378.

Dubovik, O. V.

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Elicabe, G. E.

G. E. Elicabe, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetric measurements,” in Polymer Characterization, Vol. 227 of ACS Advances in Chemistry Series, C. Carver, T. Provder, eds. (American Chemical Society, Washington, D.C., 1990), pp. 83–104.

Fantini, S.

Feng, T.-C.

Fishkin, J. B.

Franceschini, M. A.

Fymat, A.

Garcia-Rubio, L. H.

G. E. Elicabe, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetric measurements,” in Polymer Characterization, Vol. 227 of ACS Advances in Chemistry Series, C. Carver, T. Provder, eds. (American Chemical Society, Washington, D.C., 1990), pp. 83–104.

Gleser, L. J.

C. Derman, L. J. Gleser, I. Olkin, A Guide to Probability Theory and Application (Holt, Rinehart & Winston, New York, 1973), Chap. 8, p. 378.

Gratton, E.

Hallet, F. R.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Haskell, R. C.

Heintzenberg, J.

Hibst, R.

Hoffman, D. R.

G. F. Bohren, D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1976), pp. 103–232.

Jameson, D. M.

J. R. Alcala, E. Gratton, D. M. Jameson, “A multifrequency phase fluorimeter using the harmonic content of a mode-locked laser,” Anal. Instrum. 14, 225–250 (1985).
[CrossRef]

Kent, G. S.

Kienle, A.

Klett, J. D.

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, Press, New York, 1983), pp. 51–108.
[CrossRef]

Lapyonok, T. V.

Leigh, J.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Lewis, J. W. L.

Lilge, L.

Liska, M.

J. Vavra, J. Antalik, M. Liska, “Application of regression analysis in spectroturbidity size characterization methods,” Part. Part. Syst. Charact. 12, 38–41 (1995).
[CrossRef]

Maris, M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

McAdams, M. S.

McCormick, M. P.

McKellar, B.

Moulton, J. D.

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements using phase modulation spectroscopy,” in Photodynamic Therapy: Mechanisms II, J. Dougherty, ed., Proc. SPIE1203, 62–75 (1991).

Muller, H.

Nioka, S.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Olkin, I.

C. Derman, L. J. Gleser, I. Olkin, A Guide to Probability Theory and Application (Holt, Rinehart & Winston, New York, 1973), Chap. 8, p. 378.

Oshchepkov, S. L.

Patterson, M. S.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements using phase modulation spectroscopy,” in Photodynamic Therapy: Mechanisms II, J. Dougherty, ed., Proc. SPIE1203, 62–75 (1991).

Quenzel, H.

Sevick, E. M.

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Shifrin, K. S.

Smith, C.

So, P. T. C.

Steiner, R.

Sun, F.

Svaasand, L. O.

Thomalla, E.

Thomason, L. W.

Tromberg, B. J.

Tsay, T.-T.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1983).

Vavra, J.

J. Vavra, J. Antalik, M. Liska, “Application of regression analysis in spectroturbidity size characterization methods,” Part. Part. Syst. Charact. 12, 38–41 (1995).
[CrossRef]

Wang, J.

Wang, P.-H.

Wilson, B. C.

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements using phase modulation spectroscopy,” in Photodynamic Therapy: Mechanisms II, J. Dougherty, ed., Proc. SPIE1203, 62–75 (1991).

Yue, G. K.

Zolotov, I. G.

Anal. Biochem. (1)

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, M. Maris, “Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxygenation,” Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Anal. Instrum. (1)

J. R. Alcala, E. Gratton, D. M. Jameson, “A multifrequency phase fluorimeter using the harmonic content of a mode-locked laser,” Anal. Instrum. 14, 225–250 (1985).
[CrossRef]

Appl. Opt. (11)

A. Fymat, C. Smith, “Analytical inversion in remote sensing of particle size distributions. 4: comparison of Fymat and Box–McKellar solutions in the anomalous diffraction approximation,” Appl. Opt. 18, 3595–3598 (1979).
[CrossRef] [PubMed]

M. Box, B. McKellar, “Relationship between two analytic inversion formulae for multispectral extinction data,” Appl. Opt. 18, 3599–3601 (1979).
[CrossRef] [PubMed]

J. Heintzenberg, H. Muller, H. Quenzel, E. Thomalla, “Information content of optical data with respect to aerosol properties: numerical studies with a randomized minimization-search-technique inversion algorithm,” Appl. Opt. 20, 1308–1315 (1981).
[CrossRef] [PubMed]

J. D. Klett, “Anomalous diffraction model for inversion of multispectral extinction data including absorption effects,” Appl. Opt. 23, 4499–4508 (1984).
[CrossRef] [PubMed]

O. V. Dubovik, T. V. Lapyonok, S. L. Oshchepkov, “Improved technique for data inversion: optical sizing of multicomponent aerosols,” Appl. Opt. 34, 8422–8436 (1995).
[CrossRef] [PubMed]

F. Sun, J. W. L. Lewis, “Simplex deconvolutions of particle-size distribution functions from optical measurements,” Appl. Opt. 34, 8437–8445 (1995).
[CrossRef] [PubMed]

J. B. Fishkin, P. T. C. So, A. E. Cerussi, S. Fantini, M. A. Franceschini, E. Gratton, “Frequency-domain method for measuring spectral properties in multiple scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. 34, 1143–1155 (1995).
[CrossRef] [PubMed]

P.-H. Wang, G. S. Kent, M. P. McCormick, L. W. Thomason, G. K. Yue, “Retrieval analysis of aerosol-size distribution with simulated extinction measurements at SAGE III wavelengths,” Appl. Opt. 35, 433–440 (1996).
[CrossRef] [PubMed]

K. S. Shifrin, I. G. Zolotov, “Spectral attenuation and aerosol particle size distribution,” Appl. Opt. 35, 2114–2124 (1996).
[CrossRef] [PubMed]

J. Wang, F. R. Hallet, “Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum,” Appl. Opt. 35, 193–200 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Part. Part. Syst. Charact. (1)

J. Vavra, J. Antalik, M. Liska, “Application of regression analysis in spectroturbidity size characterization methods,” Part. Part. Syst. Charact. 12, 38–41 (1995).
[CrossRef]

Proc. IEEE (1)

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Other (10)

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, Press, New York, 1983), pp. 51–108.
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1983).

T. Allen, Particle Size Measurement, 4th ed. (Chapman and Hall, New York, 1990).
[CrossRef]

G. E. Elicabe, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetric measurements,” in Polymer Characterization, Vol. 227 of ACS Advances in Chemistry Series, C. Carver, T. Provder, eds. (American Chemical Society, Washington, D.C., 1990), pp. 83–104.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1976), pp. 103–232.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).

G. F. Bohren, D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements using phase modulation spectroscopy,” in Photodynamic Therapy: Mechanisms II, J. Dougherty, ed., Proc. SPIE1203, 62–75 (1991).

C. Derman, L. J. Gleser, I. Olkin, A Guide to Probability Theory and Application (Holt, Rinehart & Winston, New York, 1973), Chap. 8, p. 378.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the frequency-domain instrumentation used to make measurements of phase shift and amplitude demodulation.

Fig. 2
Fig. 2

Experimental measurements (symbols) of relative phase shift θrel (in degrees) versus modulation frequency (in megahertz) for 1.055% diluted PP722 samples interrogated at 800 nm by use of source–detector separations of 2, 1.5, and 0.5 cm. Relative phase-shift measurements of 0.5 and 1.0 are fit to Eq. (3) (solid curve).

Fig. 3
Fig. 3

Isotropic-scattering coefficient μ s′ (in inverse centimeters) versus wavelength λ (in nanometers) derived from experimental measurements of the phase shift (symbols) and that computed from Mie theory and DLS measurements.

Fig. 4
Fig. 4

PSD f(x) as a function of diameter x (in micrometers) for the PP722 sample as inverted from photon migration (PM) measurements without the mass balance constraint (solid curve) and as measured from DLS (dashed curve).

Fig. 5
Fig. 5

PSD f(x) as a function of diameter x (in micrometers) for the PP755 sample as inverted from photon migration (PM) measurements without the mass balance constraint (solid curve) and as measured from DLS (dashed curve).

Fig. 6
Fig. 6

PSD f(x) as a function of diameter x (in micrometers) for the PP788 sample as inverted from photon migration (PM) measurements without the mass balance constraint (solid curve) and as measured from DLS (dashed curve).

Fig. 7
Fig. 7

PSD f(x) as a function of diameter x (in micrometers) for the PP722 sample as inverted from photon migration (PM) measurements with the mass balance constraint (solid curve) and as measured from DLS (dashed curve).

Fig. 8
Fig. 8

PSD f(x) as a function of diameter x (in micrometers) for the PP755 sample as inverted from photon migration (PM) measurements with the mass balance constraint (solid curve) and as measured from DLS (dashed curve).

Fig. 9
Fig. 9

PSD f(x) as a function of diameter x (in micrometers) for the PP788 sample as inverted from photon migration (PM) measurements with the mass balance constraint (solid curve) and as measured from DLS (dashed curve).

Tables (2)

Tables Icon

Table 1 Solids Volume Fractions for Suspension PP722, PP755, and PP788 Obtained without Mass Balance Constraint

Tables Icon

Table 2 Solids Volume Fractions for Suspensions PP722, PP755, and PP788 Obtained with Mass Balance Constraint

Equations (24)

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

-1L lnIλI0λ=μsλ=03Qscatx, n, λ2xϕfxdx,
-Dλ2Φr, ω+iωcnλΦr, ω+μaλΦr, ω=δr-rsm,
Dλ=13μaλ+1-gμsλ.
1-gμsλ=μsλ=03Qscatx, n, λ1-gx, n, λ2x×ϕfxdx,
fxjdxj=VxjNxji=1VxiNxi,
θrs-rd, ω, λ=tan-1Im Φrs-rd, ω, λRe Φrs-rd, ω, λ,
Mrs-rd, ω, λ=Im Φrs-rd, ω, λ2+Re Φrs-rd, ω, λ21/2.
θrd, ω, λ-rs-rd×3μsλμaλcn2+ω212cn12×sintan-1ωμaλcn.
θrelλ=θrd1-θrd2=rd1-rd2cn2μa2λ+ω2cn2D2λ1/4×sin12 tan-1ωcnμaλ.
χ2=j=λ1λMμsjo-μsjc2,
fx=cbx-abc-a exp-x-abc,
JTJΔζ=JTμso-μsc,
J=μsλ1caμsλ1cbμsλ1ccμsλ1cdμsλ2caμsλ2cbμsλ2ccμsλ2cdμsλMcaμsλMcbμsλMccμsλMcd,
μso=μsλ1o, μsλ2o, , μsλMo,  μsc=μsλ1c, μsλ2c, , μsλMc.
Δζ=δa, δb, δc, δdT.
JTJ+αIΔζ=JTμso-μsc,
χ2=j=λ1λMμsjo-μsjc2+wMt-0ρ112πx3ϕfxdx2,
JTJ++αIΔζ=JTμso-μsc+V,
=V1aV1bV1cV1dV2aV2bV2cV2dV3aV3bV3cV3dV4aV4bV4cV4d,
V=V1, V2, V3, V4T,
V1=wMt-0ρ112πx3ϕfxdx×0ρ112πx3ϕfxadx,
V2=wMt-0ρ112πx3ϕfxdx×0ρ112πx3ϕfxbdx,
V3=wMt-0ρ112πx3ϕfxdx×0ρ112πx3ϕfxcdx,
V4=wMt-0ρ112πx3ϕfxdx×0ρ112πx3fxdx,

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