Abstract

In two previous papers [J. Opt. Soc. Am. A 15, 1698 (1998); J. Opt. Soc. Am. A 16, 2737 (1999)] the theory of fluctuations in the regular transmittance Τ through a flowing dispersion of large slender cylindrical particles was presented. The theory covers, among other things, expressions for the expected value μΤ and the variance σΤ2 of Τ in the two extreme cases when the cylinders are much shorter or much longer than the diameter of the optical beam used. Intermediate lengths were not treated. Numerical simulation is used to demonstrate the random behavior of Τ for intermediate cylinder lengths. The simulation results are consistent with the theory and provide a reliable estimate of the measurements produced by this analysis process. The result of the simulation is summarized as a fitted Bézier function model. The advantage of the simulation lies primarily in estimating measurement errors caused by the presence of intermediate length particles in measurement applications.

© 2008 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. S. Rydefalk, “Theory of fluctuations in the regular transmittance through a dispersion of large cylindrical particles: extension to higher concentrations,” J. Opt. Soc. Am. A 16, 2737-2745 (1999).
    [CrossRef]
  3. T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 7513524-4 (29 September 1977).
  4. T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).
  5. T. Pettersson and H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent 8105802-6 (21 April 1988).
  6. T. Pettersson and H. Karlsson, “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent 4,529,309 (16 July 1985).
  7. I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 7706320-4 (1 March 1979).
  8. I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).
  9. S. Rydefalk and J. Einarsson, “Anordning för att i en suspension med åtminstone två typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent 8400784-8 (3 March 1986).
  10. S. Rydefalk and J. Einarsson, “Device for separately measuring particles in a suspension,” U.S. patent 4,689,988 (1 September 1987).
  11. J. Hill, T. Pettersson, and S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Svensk Papperstidn. 80, 579-586 (1977).
  12. G. Fladda, T. Pettersson, L. Eriksson, and G. Tidstam, “A new optical method for measuring suspended solids in pulp and paper effluents,” in Proceedings of the 4th International Federation of Automatic Control Conference--Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerisation Industries (International Federation of Automatic Control, Belgium, 1980), pp. 9-22.
  13. S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et de le Papier--European Liaison Committee for Pulp and Paper, Oslo, 1981), Session III, no. 4, p. 16.
  14. T. Lindström, S. Rydefalk, and L. Wågberg, “The development of an integrated retention control system,” in Swedish Association of Pulp and Paper Engineers 84--The World Pulp and Paper Week. New Available Techniques (The Swedish Association of Pulp and Paper Engineers, 1984), pp. 492-496.

1999

1998

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]

1977

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

J. Opt. Soc. Am. A

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]

S. Rydefalk, “Theory of fluctuations in the regular transmittance through a dispersion of large cylindrical particles: extension to higher concentrations,” J. Opt. Soc. Am. A 16, 2737-2745 (1999).
[CrossRef]

Svensk Papperstidn.

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

Other

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

S. Rydefalk, T. Pettersson, E. Jung, and I. Lundqvist, “The STFI optical fibre classifier,” in Proceedings of the International Mechanical Pulping Conference (Comité Européen de Liaison pour la Cellulose et de le Papier--European Liaison Committee for Pulp and Paper, Oslo, 1981), Session III, no. 4, p. 16.

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

T. Pettersson, G. Fladda, and L. Eriksson, “Förfarande för koncentrationsbestämning,” Swedish patent 7513524-4 (29 September 1977).

T. Pettersson, G. Fladda, and L. Eriksson, “Method for determination of concentration,” U.S. patent 4,110,044 (29 August 1978).

T. Pettersson and H. Karlsson, “Förfarande för att bestämma medelpartikelradie och/eller medelpartikellängd,” Swedish patent 8105802-6 (21 April 1988).

T. Pettersson and H. Karlsson, “Method for determining the average radius and/or the average length of particles carried by a flowing medium,” U.S. patent 4,529,309 (16 July 1985).

I. Lundqvist, T. Pettersson, and G. Fladda, “Förfarande för att indikera fraktionsfördelningen hos suspenderade ämnen i ett strömmande medium,” Swedish patent 7706320-4 (1 March 1979).

I. Lundqvist, T. Pettersson, and G. Fladda, “Method and apparatus for indicating the size distribution of particles in a flowing medium,” U.S. patent 4,318,180 (2 March 1982).

S. Rydefalk and J. Einarsson, “Anordning för att i en suspension med åtminstone två typer av suspenderade ämnen var för sig mäta halten av varje ämnestyp,” Swedish patent 8400784-8 (3 March 1986).

S. Rydefalk and J. Einarsson, “Device for separately measuring particles in a suspension,” U.S. patent 4,689,988 (1 September 1987).

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Figures (11)

Fig. 1
Fig. 1

Experimental setup for the measurement of regular transmittance of optical radiation through a dispersion of particles.

Fig. 2
Fig. 2

Simplified response curve for φ based on Eqs. (11, 13), where l b is the intersection point between the models for short and long particles, and k B L is the k B value for long particles. For small values of k B L S a , this response curve is valid also for V 2 .

Fig. 3
Fig. 3

Transmittance sensor model. The sample particles were generated at fixed but randomly chosen positions and orientations in a cell with a larger cross section than the beam. The cell area A 0 must provide the possibility to generate particles just reaching the rim of the beam from outside. The area A 0 is centered on the axis of the radiation beam. Cartesian coordinates x, y, and z were defined such that z coincides with the optical axis.

Fig. 4
Fig. 4

Definition of the coordinates used. The projection plane was parallel with the x y plane and perpendicular to the optical axis (the z axis). The azimuthal and the polar angles are defined in relation to the x and z axes, respectively.

Fig. 5
Fig. 5

Mean regular transmittance as a function of concentration of a D b = 0.1 mm beam through a dispersion of cylinders having two orientation states, 1 mm length, and four different diameters.

Fig. 6
Fig. 6

(a) Mean regular transmittance as a function of d, and (b) as a function of l, for different beam diameters, D b , orientation states, and relations between d and l. Simulated data in (b) are marked + for particles having a diameter depending on the length and a flat orientation, and × is used for the same particles but for isotropic orientation. For the particles having a constant diameter, the symbol ∇ is used for flat orientation and · for isotropic orientation.

Fig. 7
Fig. 7

Coefficient of variation of the regular transmittance of a D b = 0.1 mm beam through a dispersion of cylinders having two orientation states, 1 mm length, and different diameters.

Fig. 8
Fig. 8

Response functions for φ as a function of particle length of the transmittance of beams having different diameters through a dispersion of cylindrical particles having the diameter 10 μm and different orientation states. In (a) the orientation state is flat, and in (b) it is isotropic. The simulated values are compared with the simple theoretical model built on extrapolation, cf. Fig. 2.

Fig. 9
Fig. 9

Variance of the regular transmittance of a D b = 0.1 mm beam through a dispersion of cylinders having two orientation states, 1 mm length and different diameters.

Fig. 10
Fig. 10

Bézier curve is controlled by four control points. Two of these are the positions of the ends of the curve. The other two define the direction of the curve at the ends. The distance between the end point and the directional point indicates the degree of dominance of the initial direction.

Fig. 11
Fig. 11

Response function for φ as a function of particle length for different beam diameters and for different dispersions of cylindrical particles. In (a) and (b) the particle diameter is constant. In (c) and (d) the particle diameter is proportional to the particle length. In (a) and (c) the orientation state is flat, and in (b) and (d) it is isotropic. The curvefitting method is described in the text.

Equations (17)

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τ = 1 - 1 C b ( c 1 + c 2 + + c n ) ,
c i = Q i g i .
C i = Q i G i .
Τ = 1 - 1 C b ( C 1 + C 2 + + C N ) ,
μ Τ = 1 - k A S a ,
σ Τ 2 = k B S a .
μ Τ = exp ( - k A S a ) ,
σ Τ 2 = exp ( - 2 k A S a ) [ exp ( k B S a ) - 1 ] ,
V 2 = exp ( k B S a ) - 1.
φ = ln ( V 2 + 1 ) ,
φ = k B S a .
k A = 4 Q μ Λ / ( π ρ p d ) ,
k B = { 16 Q 2 l E [ Λ 2 ] / ( π 2 ρ p D b 2 ) l l b 128 Q 2 μ Λ / ( 3 π 3 ρ p D b ) l l b ,
S N = 4 S π ρ p d 2 l .
f A = G p C b , p .
τ = 1 - Q f A ,
( l / l b φ / k B L ) = ( 1 - t ) 3 ( 0 0 ) + 3 t ( 1 - t ) 2 ( 0.69 0.98 ) + 3 t 2 ( 1 - t ) ( 2.30 0.92 ) + t 3 ( 7.2 1 ) ,

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