Abstract

The expected value and the variance of the regular transmittance of a dilute dispersion of slender straight circular cylinders with a diameter much larger than the wavelength of the radiation are studied theoretically in relation to the dimensions of the cylinders and the diameter of the radiation beam used. Both monodisperse and heterodisperse dispersions are considered. The study is limited to two particle categories: (i) particles that are short and (ii) particles that are long but thin compared with the diameter of the radiation beam. The expected value is a function of concentration and particle diameter. The variance is proportional to the concentration. The variance is also proportional to particle length in the case of short particles but not in the case of long particles. The results explain interesting possibilities for the characterization of dispersions of large cylindrical particles shown earlier in industrial on-line applications.

© 1998 Optical Society of America

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References

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  1. 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).
  2. 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).
  3. 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).
  4. S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med åtminstone två 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).
  5. J. Hill, T. Pettersson, S. Rydefalk, “The STFI long-fibre content meter in process control applications,” Sven. Papperstidn. 80, 579–586 (1977).
  6. 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 Polymerisation Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.
  7. S. Rydefalk, T. Pettersson, E. Jung, I. Lundqvist, “The STFI optical fibre classifier,” in International Mechanical Pulping Conference (Comité Européen de Liaison pour laCellulose et le Papier—EUCEPA, Olso, Norway, 1981), Session III, No. 4.
  8. 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, Sweden, 1984), pp. 492–496.
  9. C. E. Hubley, A. A. Robertson, S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770–787 (1950).
  10. A. A. Robertson, S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp & Pap. Mag. Can. 55, 263–269 (1954).
  11. B. G. Norman, K. Möller, R. Ek, G. G. Duffy, “Hydrodynamics of papermaking fibres in water suspensions,” in Transactions of Fibre–Water Interactions in Paper-Making (The British Paper and Board Industry Federation, Oxford, 1977), pp. 195–249.
  12. J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
    [CrossRef]
  13. J. Gregory, D. W. Nelson, “A new optical method for flocculation monitoring,” in Solid-Liquid Separation, J. Gregory, ed. (Ellis Horwood, Chichester, UK, 1984), pp. 172–182.
  14. J. Gregory, “Turbidity fluctuations in flowing suspensions,” J. Colloid Interface Sci. 105, 357–371 (1985).
    [CrossRef]
  15. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  16. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  17. P. Latimer, “Dependence of extinction efficiency of spherical scatterers on photometer geometry,” J. Opt. Soc. Am. 62, 208–211 (1972).
    [CrossRef]
  18. B. W. Lindgren, Statistical Theory (Macmillan, New York, 1976).

1987 (1)

J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
[CrossRef]

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).

1972 (1)

1954 (1)

A. A. Robertson, S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp & Pap. Mag. Can. 55, 263–269 (1954).

1950 (1)

C. E. Hubley, A. A. Robertson, S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770–787 (1950).

Bohren, C. F.

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

Duffy, G. G.

B. G. Norman, K. Möller, R. Ek, G. G. Duffy, “Hydrodynamics of papermaking fibres in water suspensions,” in Transactions of Fibre–Water Interactions in Paper-Making (The British Paper and Board Industry Federation, Oxford, 1977), pp. 195–249.

Einarsson, J.

S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med åtminstone två 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).

Eisenlauer, J.

J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
[CrossRef]

Ek, R.

B. G. Norman, K. Möller, R. Ek, G. G. Duffy, “Hydrodynamics of papermaking fibres in water suspensions,” in Transactions of Fibre–Water Interactions in Paper-Making (The British Paper and Board Industry Federation, Oxford, 1977), pp. 195–249.

Eriksson, L.

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 Polymerisation Industries—Proceedings of the 4th IFAC Conference (International Federation of Automatic Control, Ghent, Belgium, 1980), pp. 9–22.

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 Polymerisation 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]

J. Gregory, D. W. Nelson, “A new optical method for flocculation monitoring,” in Solid-Liquid Separation, J. Gregory, ed. (Ellis Horwood, Chichester, UK, 1984), pp. 172–182.

Hemel, R.

J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
[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).

Horn, D.

J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
[CrossRef]

Hubley, C. E.

C. E. Hubley, A. A. Robertson, S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770–787 (1950).

Huffman, D. R.

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

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 laCellulose et le Papier—EUCEPA, Olso, Norway, 1981), Session III, No. 4.

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).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Latimer, P.

Lindgren, B. W.

B. W. Lindgren, Statistical Theory (Macmillan, New York, 1976).

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, Sweden, 1984), pp. 492–496.

Linhart, F.

J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
[CrossRef]

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 laCellulose et le Papier—EUCEPA, Olso, Norway, 1981), Session III, 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).

Mason, S. G.

A. A. Robertson, S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp & Pap. Mag. Can. 55, 263–269 (1954).

C. E. Hubley, A. A. Robertson, S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770–787 (1950).

Möller, K.

B. G. Norman, K. Möller, R. Ek, G. G. Duffy, “Hydrodynamics of papermaking fibres in water suspensions,” in Transactions of Fibre–Water Interactions in Paper-Making (The British Paper and Board Industry Federation, Oxford, 1977), pp. 195–249.

Nelson, D. W.

J. Gregory, D. W. Nelson, “A new optical method for flocculation monitoring,” in Solid-Liquid Separation, J. Gregory, ed. (Ellis Horwood, Chichester, UK, 1984), pp. 172–182.

Norman, B. G.

B. G. Norman, K. Möller, R. Ek, G. G. Duffy, “Hydrodynamics of papermaking fibres in water suspensions,” in Transactions of Fibre–Water Interactions in Paper-Making (The British Paper and Board Industry Federation, Oxford, 1977), pp. 195–249.

Pettersson, T.

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

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 Polymerisation 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).

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 laCellulose et le Papier—EUCEPA, Olso, Norway, 1981), Session III, No. 4.

Robertson, A. A.

A. A. Robertson, S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp & Pap. Mag. Can. 55, 263–269 (1954).

C. E. Hubley, A. A. Robertson, S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770–787 (1950).

Rydefalk, S.

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

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, Sweden, 1984), pp. 492–496.

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

S. Rydefalk, J. Einarsson, “Anordning för att i en suspension med åtminstone två 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).

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 Polymerisation 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, Sweden, 1984), pp. 492–496.

Can. J. Res. (1)

C. E. Hubley, A. A. Robertson, S. G. Mason, “Flocculation in suspensions of large particles,” Can. J. Res. 28, 770–787 (1950).

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. (1)

Nord. Pulp Paper Res. J. (1)

J. Eisenlauer, D. Horn, F. Linhart, R. Hemel, “Fiber-optic flocculation sensor for on-line control of retention and drainage aids efficiency,” Nord. Pulp Paper Res. J. 2, 132–138 (1987).
[CrossRef]

Pulp & Pap. Mag. Can. (1)

A. A. Robertson, S. G. Mason, “Flocculation in flowing pulp suspensions,” Pulp & Pap. Mag. Can. 55, 263–269 (1954).

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).

Other (12)

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 Polymerisation 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 laCellulose et le Papier—EUCEPA, Olso, Norway, 1981), Session III, 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, Sweden, 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 åtminstone två 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).

B. G. Norman, K. Möller, R. Ek, G. G. Duffy, “Hydrodynamics of papermaking fibres in water suspensions,” in Transactions of Fibre–Water Interactions in Paper-Making (The British Paper and Board Industry Federation, Oxford, 1977), pp. 195–249.

J. Gregory, D. W. Nelson, “A new optical method for flocculation monitoring,” in Solid-Liquid Separation, J. Gregory, ed. (Ellis Horwood, Chichester, UK, 1984), pp. 172–182.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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

B. W. Lindgren, Statistical Theory (Macmillan, New York, 1976).

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Figures (6)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Sensor model. The sample particles are placed at fixed but randomly chosen positions and orientations in a cell of infinite extension. The position of the radiation beam is random with a uniform probability distribution.

Fig. 3
Fig. 3

Region in which interaction is possible: the region surrounding the cylinder within which the optical beam must fall if the fiber is to contribute to the radiation attenuation—(a) exact region, (b) simplified region.

Fig. 4
Fig. 4

Cross section of the radiation beam at different positions close to the projection of the cylinder. When the optical axis passes inside the rectangle formed by the dashed lines, the geometrical cross section of the particle is set equal to the length of the chord across the beam multiplied by the particle width (positions 1–3). When the optical axis passes outside the region, then the geometrical cross section of the particle is set to zero (position 4).

Fig. 5
Fig. 5

Relation between vΤ and cylinder length. The different types of linear behavior for the short-particle and long-particle categories have been extrapolated toward each other. The lines intersect at lb, which is closely related to the beam diameter Db.

Fig. 6
Fig. 6

Relation between cylinder length and three linear combinations of vΤ from three different photometers with different beam diameters. The coefficients of the linear combinations have been chosen such that the combination gives a measure of the concentration at particle lengths corresponding to the maximum response.

Equations (38)

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τ=1-1Cb(c1+c2++cn),
ci=Qigi.
Τ=1-1Cb(C1+C2++CN),
EΤ[Τ|N=n]=1-1CbμCn,
varΤ[Τ|N=n]=nvCCb2.
EΤ[Τ]=EN{ET[Τ|N]},
varΤ[Τ]=EN{varΤ[Τ|N]}+varN{EΤ[Τ|N]};
μΤ=1-μNμCCb,
vΤ=vCμNCb2+μC2vNCb2,
μNvNaCbS/(ρpVp),
μΤ=1-aSμCρpVp,
vΤ=aSρpVpCb(vC+μC2)=aSρpVpCbEC[C2].
G=dlΛ+d2π(1-Λ)/4,
μΤ=1-4aSQπρpardπ4+ar-π4μΛ,
vΤ=16aQ2Sdπ2ρparDb2π216+π2ar-π4μΛ+ar-π42EΛ[Λ2].
μΤ=1-4aQSμΛπρpd,
vΤ=16aQ2Slπ2ρpDb2EΛ[Λ2],
μN=vN=aSNμΑ=aSμΑ/(ρpVp).
μΤ=1-aSμΑμCρpCpVp,
vΤ=aSμΑρpCb2VpEC[C2].
μC=(π/4)QdDb,
EC[C2]=(2/3)Q2d2Db2.
μΤ=1-4aQSμΛπρpd,
vΤ=128aQ2SμΛ3π3ρpDb.
μΤ=1-kμSμ1/D
vΤ=kV μLlbS1+lbkμ2Sv1/DkVμL
vΤ=kVS1+kμ2Sv1/DkV
kμ=4aQμΛ/(πρp),
kV=128aQ2μΛ/(3π3ρpDb),
lb=8μΛDb/[3π(vΛ+μΛ2)].
μΤ=1-kμS/d
vΤ=kVSl/lblDbkVSlDb,
μΤ=1-kμSμ1/D,
vΤ=kVSμL/lbLDbkVSLDb,
μΤ=1-3aQS2ρpd,
vΤ=3πaSQ2d8ρpCb.
μΤ=1-3aS/(2ρpd),
vΤ=3aS/(2ρpd).

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