Abstract

Our aim is to present a method of predicting light transmittances through dense three-dimensional layered media. A hybrid method is introduced as a combination of the four-flux method with coefficients predicted from a Monte Carlo statistical model to take into account the actual three-dimensional geometry of the problem under study. We present the principles of the hybrid method, some exemplifying results of numerical simulations, and their comparison with results obtained from Bouguer–Lambert–Beer law and from Monte Carlo simulations.

© 2001 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).
  2. B. Maheu, J. N. Le Toulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
    [CrossRef]
  3. M. Czerwiński, J. Mroczka, “4 flux model in real disperse media examination: the results of computer simulation,” in Proceedings on Computer Added Metrology (Polish Academy of Sciences, Warsaw, Poland, 1997), pp. 275–282.
  4. B. Maheu, J. P. Briton, G. Gouesbet, “Four flux model and a Monte-Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
    [CrossRef] [PubMed]
  5. G. Gouesbet, B. Maheu, J. N. Letoulouzan, “Simulation of particle multiple scattering and applications to particle diagnostics,” in Heat Transfer in Radiating and Combusting Systems, M. G. Carvalho, F. Lockwood, J. Taine, eds. (Springer-Verlag, New York, 1991), pp. 173–185.
    [CrossRef]
  6. B. Maheu, G. Gouesbet, “Four flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1984).
    [CrossRef]
  7. Y. P. Wang, Z. S. Wu, K. F. Ren, “Four flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
    [CrossRef] [PubMed]
  8. G. Gouesbet, B. Maheu, G. Gréhan, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
    [CrossRef] [PubMed]
  9. K. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  10. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  11. P. Chylek, “Mie scattering into the backward hemisphere,” J. Opt. Soc. Am. 63, 1467–1471 (1974).
    [CrossRef]
  12. P. Debye, “Der Lichtdruck auf Kugeln von beliebigen Material,” Ann. Phys. (Leipzig) 4, 57–136 (1909).
    [CrossRef]
  13. B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
    [CrossRef]
  14. C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
    [CrossRef]
  15. M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).
  16. M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).
  17. M. Czerwiński, “Modélisation de la turbidité spectrale d’un milieu multidiffusif et son application au problème inverse,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1998).
  18. W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
    [CrossRef]
  19. W. E. Vargas, “Two-flux transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
    [CrossRef]
  20. W. E. Vargas, “Diffuse radiation intensity propagating through a particulate slab,” J. Opt. Soc. Am. 16, 1362–1372 (1999).
    [CrossRef]

1999 (2)

W. E. Vargas, “Diffuse radiation intensity propagating through a particulate slab,” J. Opt. Soc. Am. 16, 1362–1372 (1999).
[CrossRef]

W. E. Vargas, “Two-flux transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
[CrossRef]

1998 (1)

1994 (1)

C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
[CrossRef]

1992 (1)

B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

1989 (2)

1984 (2)

1983 (1)

1974 (1)

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigen Material,” Ann. Phys. (Leipzig) 4, 57–136 (1909).
[CrossRef]

Bohren, C. F.

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

Briton, J. P.

B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

B. Maheu, J. P. Briton, G. Gouesbet, “Four flux model and a Monte-Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Chylek, P.

Czerwinski, M.

M. Czerwiński, J. Mroczka, “4 flux model in real disperse media examination: the results of computer simulation,” in Proceedings on Computer Added Metrology (Polish Academy of Sciences, Warsaw, Poland, 1997), pp. 275–282.

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

M. Czerwiński, “Modélisation de la turbidité spectrale d’un milieu multidiffusif et son application au problème inverse,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1998).

M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigen Material,” Ann. Phys. (Leipzig) 4, 57–136 (1909).
[CrossRef]

Girasole, T.

M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

Gouesbet, G.

B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

B. Maheu, J. P. Briton, G. Gouesbet, “Four flux model and a Monte-Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[CrossRef] [PubMed]

B. Maheu, J. N. Le Toulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
[CrossRef]

B. Maheu, G. Gouesbet, “Four flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1984).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, J. N. Letoulouzan, “Simulation of particle multiple scattering and applications to particle diagnostics,” in Heat Transfer in Radiating and Combusting Systems, M. G. Carvalho, F. Lockwood, J. Taine, eds. (Springer-Verlag, New York, 1991), pp. 173–185.
[CrossRef]

M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

Gréhan, G.

C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
[CrossRef]

B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

Huffman, D. R.

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

Kerker, K.

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

Le Toulouzan, J. N.

Letoulouzan, J. N.

G. Gouesbet, B. Maheu, J. N. Letoulouzan, “Simulation of particle multiple scattering and applications to particle diagnostics,” in Heat Transfer in Radiating and Combusting Systems, M. G. Carvalho, F. Lockwood, J. Taine, eds. (Springer-Verlag, New York, 1991), pp. 173–185.
[CrossRef]

Maheu, B.

C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
[CrossRef]

B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

B. Maheu, J. P. Briton, G. Gouesbet, “Four flux model and a Monte-Carlo code: comparisons between two simple and complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[CrossRef] [PubMed]

B. Maheu, G. Gouesbet, “Four flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1984).
[CrossRef]

B. Maheu, J. N. Le Toulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, J. N. Letoulouzan, “Simulation of particle multiple scattering and applications to particle diagnostics,” in Heat Transfer in Radiating and Combusting Systems, M. G. Carvalho, F. Lockwood, J. Taine, eds. (Springer-Verlag, New York, 1991), pp. 173–185.
[CrossRef]

Menard, J.

C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
[CrossRef]

Mroczka, J.

M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).

M. Czerwiński, J. Mroczka, “4 flux model in real disperse media examination: the results of computer simulation,” in Proceedings on Computer Added Metrology (Polish Academy of Sciences, Warsaw, Poland, 1997), pp. 275–282.

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

Ren, K. F.

Y. P. Wang, Z. S. Wu, K. F. Ren, “Four flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
[CrossRef] [PubMed]

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

Rozé, C.

C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
[CrossRef]

Vargas, W. E.

Wang, Y. P.

Wu, Z. S.

Ann. Phys. (Leipzig) (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigen Material,” Ann. Phys. (Leipzig) 4, 57–136 (1909).
[CrossRef]

Appl. Opt. (7)

Atmos. Environ. (1)

C. Rozé, B. Maheu, G. Gréhan, J. Menard, “Evaluations of the distance of visibility in a foggy atmosphere by Monte-Carlo simulation,” Atmos. Environ. 25, 769–775 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

P. Chylek, “Mie scattering into the backward hemisphere,” J. Opt. Soc. Am. 63, 1467–1471 (1974).
[CrossRef]

W. E. Vargas, “Diffuse radiation intensity propagating through a particulate slab,” J. Opt. Soc. Am. 16, 1362–1372 (1999).
[CrossRef]

Part. Part. Syst. Charact. (1)

B. Maheu, J. P. Briton, G. Gréhan, G. Gouesbet, “Monte-Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Other (8)

M. Czerwiński, J. Mroczka, K. F. Ren, T. Girasole, G. Gréhan, G. Gouesbet, “Scattered light predictions under multiple scattering conditions with application to inversion scheme,” in Proceedings of the Seventh European Symposium on Particle Characterisation (NurnbergMesse GmbH, Nürnberg, Germany, 1998).

M. Czerwiński, J. Mroczka, T. Girasole, G. Gréhan, G. Gouesbet, “Hybrid method to predict scattered light transmittances under multiple scattering conditions,” in Proceedings of the Fifth International Congress on Optical Particle Sizing, (University of Minnesota, Minneapolis, Minn., 1998).

M. Czerwiński, “Modélisation de la turbidité spectrale d’un milieu multidiffusif et son application au problème inverse,” Ph.D. dissertation (Université de Rouen, Rouen, France, 1998).

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

M. Czerwiński, J. Mroczka, “4 flux model in real disperse media examination: the results of computer simulation,” in Proceedings on Computer Added Metrology (Polish Academy of Sciences, Warsaw, Poland, 1997), pp. 275–282.

G. Gouesbet, B. Maheu, J. N. Letoulouzan, “Simulation of particle multiple scattering and applications to particle diagnostics,” in Heat Transfer in Radiating and Combusting Systems, M. G. Carvalho, F. Lockwood, J. Taine, eds. (Springer-Verlag, New York, 1991), pp. 173–185.
[CrossRef]

K. 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).

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Figures (23)

Fig. 1
Fig. 1

Geometry of the four-flux model.

Fig. 2
Fig. 2

Experimental geometry for the Monte Carlo simulations.

Fig. 3
Fig. 3

Relationship between the coefficient K and the transmittance τ CD (λ = 0.4 × 10-6 m, d = 0.4 × 10-6 m).

Fig. 4
Fig. 4

Relationship between the coefficient K and the transmittance τ CD (λ = 0.6 × 10-6 m, d = 0.4 × 10-6 m).

Fig. 5
Fig. 5

Relationship between the coefficient K and the transmittance τ CD (λ = 0.4 × 10-6 m, d = 0.6 × 10-6 m).

Fig. 6
Fig. 6

Relationship between the coefficient K and the transmittance τ CD (λ = 0.6 × 10-6 m, d = 0.6 × 10-6 m).

Fig. 7
Fig. 7

Relationship between the coefficient K and the transmittance τ CD (λ = 0.4 × 10-6 m, d = 0.8 × 10-6 m).

Fig. 8
Fig. 8

Relationship between the coefficient K and the transmittance τ CD (λ = 0.6 × 10-6 m, d = 0.8 × 10-6 m).

Fig. 9
Fig. 9

Transmittance as a function of particle concentration, d = 0.4 × 10-6 m, λ = 0.4 × 10-6 m.

Fig. 10
Fig. 10

Transmittance as a function of particle concentration, d = 0.4 × 10-6 m, λ = 0.6 × 10-6 m.

Fig. 11
Fig. 11

Transmittance as a function of particle concentration, d = 0.6 × 10-6 m, λ = 0.4 × 10-6 m.

Fig. 12
Fig. 12

Transmittance as a function of particle concentration, d = 0.6 × 10-6 m.

Fig. 13
Fig. 13

Transmittance as a function of particle concentration, d = 0.8 × 10-6 m, λ = 0.4 × 10-6 m.

Fig. 14
Fig. 14

Transmittance as a function of particle concentration, d = 0.8 × 10-6 m, λ = 0.6 × 10-6 m.

Fig. 15
Fig. 15

Relationship between the coefficient K and the transmittance τMC(λ = 0.4 × 10-6 m, d = 0.4 × 10-6 m).

Fig. 16
Fig. 16

Relationship between the coefficient K and the transmittance τMC(λ = 0.6 × 10-6 m, d = 0.4 × 10-6 m).

Fig. 17
Fig. 17

Relationship between the coefficient K and the transmittance τMC(λ = 0.4 × 10-6 m, d = 0.6 × 10-6 m).

Fig. 18
Fig. 18

Relationship between the coefficient K and the transmittance τMC(λ = 0.6 × 10-6 m, d = 0.6 × 10-6 m).

Fig. 19
Fig. 19

Relationship between the coefficient K and the transmittance τMC(λ = 0.4 × 10-6 m, d = 0.8 × 10-6 m).

Fig. 20
Fig. 20

Relationship between the coefficient K and the transmittance τMC(λ = 0.6 × 10-6 m, d = 0.8 × 10-6 m).

Fig. 21
Fig. 21

Comparison between the transmittance calculated from hybrid method 1 and that from the Monte Carlo method for λ = 0.4 × 10-6 m.

Fig. 22
Fig. 22

Comparison between the transmittance calculated from hybrid method 1 and that from the Monte Carlo method for λ = 0.75 × 10-6 m.

Fig. 23
Fig. 23

Comparison between the transmittance calculated from hybrid method 1 and that from the Monte Carlo method for λ = 0.9 × 10-6 m.

Tables (4)

Tables Icon

Table 1 Some Exemplifying Values of A and B (Hybrid Method 1)

Tables Icon

Table 2 Some Exemplifying Values of C and D (Hybrid Method 2)

Tables Icon

Table 3 Examined Particle Size Distribution

Tables Icon

Table 4 Constant Coefficients E and F

Equations (23)

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

ε=c1gc2exp1g-1c3+c4zo+c51-g3 exp-zo+12+1,
k=NCabs=NCext-Csca=NCext1-a,
s=NCsca=aNCext,
a=CscaCext=QscaQext=sk+s,
Csca=λ22πn=12n+1|an|2+|bn|2,
Cext=λ22πn=12n+1Rean+bn,
Cscaf=12 Csca-λ22πn=1 o m=12n+12m+1mm+1-nn+1-1n+m-1/2n!!m-1!!n-1!!m!!Reanam*+bnbm*+n=1 o m=1 o 2n+1nn+12m+1mm+1-1n+m-1/2n!!m!!n-1!!m-1!!Reanbm*,
0!!=1!!=1, n!!=n-2!!n.
g=4α2Qscan=1nn+2n+1Rean*an+1+bn*bn+1+2n+1nn+1Rean*bn,
an=ψnαψnβ-mψnβψnαξnαψnβ-mψnβξnα,
bn=mψnαψnβ-ψnβψnαmξnαψnβ-ψnβξnα,
dIcdz=k+sIc,
dJcdz=-k+sJc,
dIddz=εkId+ε1-ςsId-ε1-ςsJd-1-ςsJc-ςsIc,
dJddz=-εkJd-ε1-ςsJd+ε1-ςsId+1-ςsIc+ςsJc,
τtot=τCC+τCD.
l=-log rkext.
φ=2πr,  r=0θ PθdΩ0π PθdΩ,
τMC=τMC0+τMCm.
τMC=τCC+KτCD.
K=B expτCDA,
K=CτMCD,
τtot=1i=1N τhybi1-EFE

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