Abstract

It has been shown in a previous paper [J. Opt. Soc. Am. A 15, 1689 (1998)] that the expected value and the variance of the fluctuating regular transmittance through a dispersion of slender cylinders can be used to measure properties such as the size and concentration of the dispersed particles. The theory is valid, however, only for very low concentrations or very short path lengths. In practical applications, such as on-line applications, it is often desirable to work at higher concentration×path-length products p to avoid complicated sensor constructions. An extension of the previous theory toward higher values of p is presented. It is based on the assumption of the independence of the regular transmittance of parallel layers of the dispersion. The extended theory shows that information about the dispersion found in the expected value and variance at very low values of p can also be obtained for higher values of p with a few simple expressions. Therefore applications possible in the case of low p values are also possible for higher p values. Two experimental examples have been included to facilitate the discussion of the theory presented.

© 1999 Optical Society of America

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References

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  1. S. Rydefalk, “Fluctuations in the regular transmittance of dispersions of straight circular cylinders with a diameter much larger than the wavelength of the radiation,” J. Opt. Soc. Am. A 15, 1689–1697 (1998).
    [CrossRef]
  2. T. Pettersson, G. Fladda, L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent75 135 24–4 (September29, 1977) and “Method for determination of concentration,” U.S. patent4,110,044 (August29, 1978).
  3. T. Pettersson, H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent81 058 02–6 (April21, 1988) and “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent4,529,309 (July16, 1985).
  4. I. Lundqvist, T. Pettersson, G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent77 063 20–4 (March1, 1979) and “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent4,318,180 (March2, 1982).
  5. S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med a°tminstone tva° typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent84 007 84–8 (March3, 1986) and “Device for separately measuring particles in a suspension,” U.S. patent4,689,988 (September1, 1987).
  6. J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).
  7. G. Fladda, T. Pettersson, L. Eriksson, G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.
  8. S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et du Papier—EUCEPA, Oslo, 1981), Session III, Paper No. 4.
  9. T. Lindström, S. Rydefalk, L. Wågberg, “The development of an integrated retention control system,” in SPCI 84—The World Pulp and Paper Week. New Available Techniques (The Swedish Association of Pulp and Paper Engineers, Stockholm, 1984), pp. 492–496.
  10. A. Schuster, “Radiation through foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  11. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  12. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948); errata 1067.
    [CrossRef] [PubMed]
  13. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [CrossRef]
  14. J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
    [CrossRef]
  15. S. Karlin, A First Course in Stochastic Processes (Academic, New York, 1966).
  16. J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, UK, 1969).

1998 (1)

1985 (1)

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
[CrossRef]

1977 (1)

J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).

1954 (1)

1948 (1)

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

1905 (1)

A. Schuster, “Radiation through foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Aitchison, J.

J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, UK, 1969).

Brown, J. A. C.

J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, UK, 1969).

Einarsson, J.

S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med a°tminstone tva° typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent84 007 84–8 (March3, 1986) and “Device for separately measuring particles in a suspension,” U.S. patent4,689,988 (September1, 1987).

Eriksson, L.

G. Fladda, T. Pettersson, L. Eriksson, G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.

T. Pettersson, G. Fladda, L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent75 135 24–4 (September29, 1977) and “Method for determination of concentration,” U.S. patent4,110,044 (August29, 1978).

Fladda, G.

T. Pettersson, G. Fladda, L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent75 135 24–4 (September29, 1977) and “Method for determination of concentration,” U.S. patent4,110,044 (August29, 1978).

G. Fladda, T. Pettersson, L. Eriksson, G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.

I. Lundqvist, T. Pettersson, G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent77 063 20–4 (March1, 1979) and “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent4,318,180 (March2, 1982).

Gregory, J.

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
[CrossRef]

Hill, J.

J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).

Jung, E.

S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et du Papier—EUCEPA, Oslo, 1981), Session III, Paper No. 4.

Karlin, S.

S. Karlin, A First Course in Stochastic Processes (Academic, New York, 1966).

Karlsson, H.

T. Pettersson, H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent81 058 02–6 (April21, 1988) and “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent4,529,309 (July16, 1985).

Kubelka, P.

Lindström, T.

T. Lindström, S. Rydefalk, L. Wågberg, “The development of an integrated retention control system,” in SPCI 84—The World Pulp and Paper Week. New Available Techniques (The Swedish Association of Pulp and Paper Engineers, Stockholm, 1984), pp. 492–496.

Lundqvist, I.

S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et du Papier—EUCEPA, Oslo, 1981), Session III, Paper No. 4.

I. Lundqvist, T. Pettersson, G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent77 063 20–4 (March1, 1979) and “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent4,318,180 (March2, 1982).

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Pettersson, T.

J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).

I. Lundqvist, T. Pettersson, G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent77 063 20–4 (March1, 1979) and “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent4,318,180 (March2, 1982).

S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et du Papier—EUCEPA, Oslo, 1981), Session III, Paper No. 4.

T. Pettersson, G. Fladda, L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent75 135 24–4 (September29, 1977) and “Method for determination of concentration,” U.S. patent4,110,044 (August29, 1978).

T. Pettersson, H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent81 058 02–6 (April21, 1988) and “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent4,529,309 (July16, 1985).

G. Fladda, T. Pettersson, L. Eriksson, G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.

Rydefalk, S.

S. Rydefalk, “Fluctuations in the regular transmittance of dispersions of straight circular cylinders with a diameter much larger than the wavelength of the radiation,” J. Opt. Soc. Am. A 15, 1689–1697 (1998).
[CrossRef]

J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).

S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et du Papier—EUCEPA, Oslo, 1981), Session III, Paper No. 4.

S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med a°tminstone tva° typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent84 007 84–8 (March3, 1986) and “Device for separately measuring particles in a suspension,” U.S. patent4,689,988 (September1, 1987).

T. Lindström, S. Rydefalk, L. Wågberg, “The development of an integrated retention control system,” in SPCI 84—The World Pulp and Paper Week. New Available Techniques (The Swedish Association of Pulp and Paper Engineers, Stockholm, 1984), pp. 492–496.

Schuster, A.

A. Schuster, “Radiation through foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Tidstam, G.

G. Fladda, T. Pettersson, L. Eriksson, G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.

Wågberg, L.

T. Lindström, S. Rydefalk, L. Wågberg, “The development of an integrated retention control system,” in SPCI 84—The World Pulp and Paper Week. New Available Techniques (The Swedish Association of Pulp and Paper Engineers, Stockholm, 1984), pp. 492–496.

Astrophys. J. (1)

A. Schuster, “Radiation through foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

J. Colloid Interface Sci. (1)

J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Sven. Papperstidn. (1)

J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (9)

S. Karlin, A First Course in Stochastic Processes (Academic, New York, 1966).

J. Aitchison, J. A. C. Brown, The Lognormal Distribution (Cambridge U. Press, Cambridge, UK, 1969).

G. Fladda, T. Pettersson, L. Eriksson, G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.

S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et du Papier—EUCEPA, Oslo, 1981), Session III, Paper No. 4.

T. Lindström, S. Rydefalk, L. Wågberg, “The development of an integrated retention control system,” in SPCI 84—The World Pulp and Paper Week. New Available Techniques (The Swedish Association of Pulp and Paper Engineers, Stockholm, 1984), pp. 492–496.

T. Pettersson, G. Fladda, L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent75 135 24–4 (September29, 1977) and “Method for determination of concentration,” U.S. patent4,110,044 (August29, 1978).

T. Pettersson, H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent81 058 02–6 (April21, 1988) and “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent4,529,309 (July16, 1985).

I. Lundqvist, T. Pettersson, G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent77 063 20–4 (March1, 1979) and “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent4,318,180 (March2, 1982).

S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med a°tminstone tva° typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent84 007 84–8 (March3, 1986) and “Device for separately measuring particles in a suspension,” U.S. patent4,689,988 (September1, 1987).

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Figures (8)

Fig. 1
Fig. 1

Regular radiation transmission through a dispersion. Ideally, the incident radiation beam is parallel and the detector is insensitive to scattered radiation.

Fig. 2
Fig. 2

Experimental statistical parameters of the transmittance. The upper plot shows the sample mean, and the lower plot shows the sample variance plotted against the product of concentration and path length p. The sample is a dispersion of thermomechanical pulp fibers in water. The dots indicate experimental values. The equations derived in this paper are used for the curves fitted to the data. This means that two parameters have been determined to make the two curves fit. In this experiment the path was 1 mm and the beam diameter was ∼0.1 mm.

Fig. 3
Fig. 3

The probability that the transmittance T of the continuum model exceeds unity as a function of p, the product of path length and concentration. In this example, kA has been set to 250 kg/m2, kB to 25 kg/m2, and the path length to 1 mm. The range of p in the example in Fig. 2 is indicated. The parameters kA and kB determine the shape of the log-normal distribution of T through Eqs. (33) and (34).

Fig. 4
Fig. 4

Experimental values of V2 and φ plotted against p, the product of path length and concentration. V2 is the square of the coefficient of variation of the transmittance T, and φ is a function of V2 and is defined in Eq. (38). The sample is a dispersion of thermomechanical pulp fibers in water. The dots are the experimental values, and the curves are fitted to the data on the basis of the average mkB of the experimental values of kB. The parameter kB was determined with Eq. (39).

Fig. 5
Fig. 5

Linear light attenuation A plotted against p, the product of path length and concentration. The sample is a dispersion of thermomechanical pulp fibers in water. The dots are the experimental values, and the curve is fitted to the data on the basis of the average of the experimental values of kA, which are determined with Eq. (37).

Fig. 6
Fig. 6

Experimental values μ^Τ and σ^Τ2 of the expected value μΤ and the variance σ^Τ2 of the transmittance T plotted against p, the product of path length and concentration. The sample is a dispersion of bleached kraft pulp fibers in water. The large dots are the experimental values, and the solid curves are fitted to the data on the basis of the averages of the experimental values of kA and kB, which are determined with Eqs. (37) and (39). For comparison, the corresponding curves from example 1 are plotted as dotted curves.

Fig. 7
Fig. 7

Experimental values of V2 and φ plotted against p, the product of path length and concentration. V2 is the square of the coefficient of variation of the transmittance T, and φ is a function of V2 and is defined in Eq. (38). The sample is a dispersion of bleached kraft pulp fibers in water. The dots are the experimental values, and the curves are fitted to the data on the basis of the average mkB of the experimental values of kB. The parameter kB was determined with Eq. (39).

Fig. 8
Fig. 8

Comparison between φ defined in Eq. (38) and φν defined in Eq. (41). The sample is a dispersion of bleached kraft pulp fibers in water. The solid dots are the experimental φ values, and the open dots are the experimental φν values. The curve fitted to the φ data is the same as that in Fig. 7, and the curve fitted to the φν data is a regression line forced to pass through the origin.

Equations (41)

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Τ(x)=Φ(x)/ϕ0,
C=QG.
Τ=1-1Cb (C1+C2++CN),
τ=1-1Cb (c1+c2++cn).
τ=1-SNxc;
dτ=-SNcdx.
τ(x)=exp(-SNcx),
μΤτ(μN),
σΤ12 [τ(μN+σN)-τ(μN-σN)],
μΤexp(-SNCx),
σΤexp(-SNCx)sinh[(SNCbx)1/2C/Cb].
V2=σΤ2/μΤ2sinh2[(SNCbx)1/2C/Cb].
μΤ=1-kAxS,
σΤ2=kBxS.
kA=3Q/(2ρpd),
kB=3πQ2d/(8ρpCb),
kA=4QμΛ/(πρpd),
kB=16Q2lE[Λ2]/(π2ρpDb2),
kA=4QμΛ/(πρpd),
kB=128Q2μΛ/(3π3ρpDb).
Τ(p0)=i=1mΤi(Δp),
E[Τi(Δp)]=1-kAΔp,
var[Τi(Δp)]=kBΔp.
W(p0)=i=1mWi(Δp).
E[Τi(Δp)]=exp{E[Wi(Δp)]+12 var[Wi(Δp)]},
var[Τi(Δp)]=exp{2E[Wi(Δp)]+var[Wi(Δp)]}(exp{var[Wi(Δp)]}-1).
E[Τi(Δp)]=1+E[Wi(Δp)]+12 var[Wi(Δp)],
var[Τi(Δp)]=var[W(Δp)].
E[Wi(Δp)]=-(kA+12kB)Δp,
var[Wi(Δp)]=kBΔp
E[W(p0)]=-(kA+12kB)p0,
var[W(p0)]=kBp0.
μΤ=exp(-kAp)
σΤ2=exp(-2kAp)[exp(kBp)-1].
V2=exp(kBp)-1
A=-ln(μΤ),
A=kAp.
φ=ln(V2+1),
φ=kBp.
ΔV2-3πd2Q4128ρp2Cb2 p2;
φν=ν ln(V2/ν+1),

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