Abstract

The problem of applying spectroradiometric temperature measurement techniques to hot gases containing scattering particles is considered. A temperature measurement procedure is developed. This procedure is shown to yield a significant advantage over previous techniques while requiring only minimal knowledge of the exact nature of the scattering particles.

© 1989 Optical Society of America

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References

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  1. R. M. Tourin, Spectroscopic Gas Temperature Measurement (Elsevier, Amsterdam, 1966).
  2. P. H. Paul, “Spectroradiometric Temperature Measurements in Two-Phase Combustion Plasmas,” Doctoral Dissertation, Stanford U. (1984), HTGL report T-238.
  3. D. G. Goodwin, J. L. Ebert, “Rigorous Bounds on the Radiative Interaction Between Real Gases and Scattering Particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 501–508 (1987).
    [CrossRef]
  4. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  6. P. H. Paul, S. A. Self, “Method for Spectroradiometric Temperature Measurements in Two Phase Flows. 2: Experimental Verification,” Appl. Opt. 28, 2150–2155 (1989).
    [CrossRef] [PubMed]
  7. A. G. Sviridov, N. N. Sobolev, “Flame Temperature Measurements Using the Line Reversal Method,” J. Phys. Elec. Tech. 22, 93–101 (1953).
  8. P. F. Zweifel, “Neutron Transport Theory,” in Developments in Transport Theory, Nato Advanced Study Institute on Transport Theory, E. Inonu, P. F. Zweifel, Eds. (Academic, London, 1967).
  9. K. M. Case, “Elementary Solutions of the Transport Equations and Their Applications,” Ann. Phys. 9, 1–23 (1960).
    [CrossRef]
  10. P. C. Ariessohn, Optical Diagnostic Measurements of Coal Slag Parameters in Combustion MHD Systems, Doctoral Dissertation, Stanford U. (1980), HTGL report 119.
  11. P. C. Ariessohn, S. A. Self, R. H. Eustis, “Two-Wavelength Laser Transmissometer for Measurements of the Mean Size and Concentration of Coal Ash Droplets in Combustion Flows,” Appl. Opt. 19, 3775–3781 (1980).
    [CrossRef] [PubMed]
  12. J. M. Adams, “The Spectral Comparison Method for Temperature Measurement and Control in Two-Phase Flames,” in Temperature, Its Measurement and Control in Science and IndustryH. H. Plumb, Ed. (Instrument Society of America, Pittsburgh, PA, 1972), Vol. 3, p. 627.
  13. D. W. Mackowski, R. A. M. Altenkirch, R. E. Peck, T. W. Tong, “Infrared Pyrometer Measurements of Particle and Gas Temperatures in Pulverized-Coal Flames,” presented at Western States Combustion Conference (1982), paper 82-22.
  14. D. J. Carlsen, “Static Temperature Measurements in Hot Gas-particle Flows,” in Temperature, Its Measurement and Control in Science and Industry, C. M. Herzfeld, Ed. (Reinhold, New York, 1962), Vol. 3, p. 535.
  15. D. L. Thomas, “Problems in Applying the Line Reversal Method of Temperature Measurement to Flames,” Combust. Flame 12, 541–549 (1968).
    [CrossRef]
  16. L. E. Bauman, “Gas Temperature Measurements of Particle Laden MHD Flows,” presented at 1984 ASME Heat Transfer Conference (ASME, New York, 1984), paper 84-AES-8.
  17. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  18. I. W. Busbridge, “The Mathematics of Radiative Transfer,” in Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge UP., London, 1960).
  19. C. M. Chu, S. W. Churchill, “Representation of the Angular Distribution of Radiation Scattered by a Spherical Particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
    [CrossRef]
  20. E. H. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  21. I. A. Vasileva, L. P. Deputatova, A. P. Nefedov, “Investigating the Flame with the Aid of Self-Reversed Contours of Spectral Lines,” Combust. Flame 23, 305–311 (1977).
    [CrossRef]
  22. I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).
  23. S. A. Self, P. H. Paul, P. Young, “A Packaged Fiberoptic Spectroradiometer for High Temperature Gases with Automatic Readout,” in Temperature, Its Measurement and Control in Science and Industry, J. F. Schooly, Ed. (AIP, New York, 1982) Vol. 5, p. 465.

1989

1987

D. G. Goodwin, J. L. Ebert, “Rigorous Bounds on the Radiative Interaction Between Real Gases and Scattering Particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 501–508 (1987).
[CrossRef]

1980

1977

I. A. Vasileva, L. P. Deputatova, A. P. Nefedov, “Investigating the Flame with the Aid of Self-Reversed Contours of Spectral Lines,” Combust. Flame 23, 305–311 (1977).
[CrossRef]

1974

E. H. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1973

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

1968

D. L. Thomas, “Problems in Applying the Line Reversal Method of Temperature Measurement to Flames,” Combust. Flame 12, 541–549 (1968).
[CrossRef]

1960

K. M. Case, “Elementary Solutions of the Transport Equations and Their Applications,” Ann. Phys. 9, 1–23 (1960).
[CrossRef]

1955

1953

A. G. Sviridov, N. N. Sobolev, “Flame Temperature Measurements Using the Line Reversal Method,” J. Phys. Elec. Tech. 22, 93–101 (1953).

Adams, J. M.

J. M. Adams, “The Spectral Comparison Method for Temperature Measurement and Control in Two-Phase Flames,” in Temperature, Its Measurement and Control in Science and IndustryH. H. Plumb, Ed. (Instrument Society of America, Pittsburgh, PA, 1972), Vol. 3, p. 627.

Altenkirch, R. A. M.

D. W. Mackowski, R. A. M. Altenkirch, R. E. Peck, T. W. Tong, “Infrared Pyrometer Measurements of Particle and Gas Temperatures in Pulverized-Coal Flames,” presented at Western States Combustion Conference (1982), paper 82-22.

Ariessohn, P. C.

P. C. Ariessohn, S. A. Self, R. H. Eustis, “Two-Wavelength Laser Transmissometer for Measurements of the Mean Size and Concentration of Coal Ash Droplets in Combustion Flows,” Appl. Opt. 19, 3775–3781 (1980).
[CrossRef] [PubMed]

P. C. Ariessohn, Optical Diagnostic Measurements of Coal Slag Parameters in Combustion MHD Systems, Doctoral Dissertation, Stanford U. (1980), HTGL report 119.

Bauman, L. E.

L. E. Bauman, “Gas Temperature Measurements of Particle Laden MHD Flows,” presented at 1984 ASME Heat Transfer Conference (ASME, New York, 1984), paper 84-AES-8.

Bohren, C. F.

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

Busbridge, I. W.

I. W. Busbridge, “The Mathematics of Radiative Transfer,” in Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge UP., London, 1960).

Carlsen, D. J.

D. J. Carlsen, “Static Temperature Measurements in Hot Gas-particle Flows,” in Temperature, Its Measurement and Control in Science and Industry, C. M. Herzfeld, Ed. (Reinhold, New York, 1962), Vol. 3, p. 535.

Case, K. M.

K. M. Case, “Elementary Solutions of the Transport Equations and Their Applications,” Ann. Phys. 9, 1–23 (1960).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chu, C. M.

Churchill, S. W.

Deputatova, L. P.

I. A. Vasileva, L. P. Deputatova, A. P. Nefedov, “Investigating the Flame with the Aid of Self-Reversed Contours of Spectral Lines,” Combust. Flame 23, 305–311 (1977).
[CrossRef]

Ebert, J. L.

D. G. Goodwin, J. L. Ebert, “Rigorous Bounds on the Radiative Interaction Between Real Gases and Scattering Particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 501–508 (1987).
[CrossRef]

Eustis, R. H.

Goodwin, D. G.

D. G. Goodwin, J. L. Ebert, “Rigorous Bounds on the Radiative Interaction Between Real Gases and Scattering Particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 501–508 (1987).
[CrossRef]

Hansen, E. H.

E. H. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Huffman, D. R.

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

Kirillov, V. V.

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

Mackowski, D. W.

D. W. Mackowski, R. A. M. Altenkirch, R. E. Peck, T. W. Tong, “Infrared Pyrometer Measurements of Particle and Gas Temperatures in Pulverized-Coal Flames,” presented at Western States Combustion Conference (1982), paper 82-22.

Malyuzhonok, G. P.

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

Masimov, I. A.

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

Nefedov, A. P.

I. A. Vasileva, L. P. Deputatova, A. P. Nefedov, “Investigating the Flame with the Aid of Self-Reversed Contours of Spectral Lines,” Combust. Flame 23, 305–311 (1977).
[CrossRef]

Novosadov, V. B.

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

Paul, P. H.

P. H. Paul, S. A. Self, “Method for Spectroradiometric Temperature Measurements in Two Phase Flows. 2: Experimental Verification,” Appl. Opt. 28, 2150–2155 (1989).
[CrossRef] [PubMed]

S. A. Self, P. H. Paul, P. Young, “A Packaged Fiberoptic Spectroradiometer for High Temperature Gases with Automatic Readout,” in Temperature, Its Measurement and Control in Science and Industry, J. F. Schooly, Ed. (AIP, New York, 1982) Vol. 5, p. 465.

P. H. Paul, “Spectroradiometric Temperature Measurements in Two-Phase Combustion Plasmas,” Doctoral Dissertation, Stanford U. (1984), HTGL report T-238.

Peck, R. E.

D. W. Mackowski, R. A. M. Altenkirch, R. E. Peck, T. W. Tong, “Infrared Pyrometer Measurements of Particle and Gas Temperatures in Pulverized-Coal Flames,” presented at Western States Combustion Conference (1982), paper 82-22.

Self, S. A.

Sobolev, N. N.

A. G. Sviridov, N. N. Sobolev, “Flame Temperature Measurements Using the Line Reversal Method,” J. Phys. Elec. Tech. 22, 93–101 (1953).

Sviridov, A. G.

A. G. Sviridov, N. N. Sobolev, “Flame Temperature Measurements Using the Line Reversal Method,” J. Phys. Elec. Tech. 22, 93–101 (1953).

Thomas, D. L.

D. L. Thomas, “Problems in Applying the Line Reversal Method of Temperature Measurement to Flames,” Combust. Flame 12, 541–549 (1968).
[CrossRef]

Tong, T. W.

D. W. Mackowski, R. A. M. Altenkirch, R. E. Peck, T. W. Tong, “Infrared Pyrometer Measurements of Particle and Gas Temperatures in Pulverized-Coal Flames,” presented at Western States Combustion Conference (1982), paper 82-22.

Tourin, R. M.

R. M. Tourin, Spectroscopic Gas Temperature Measurement (Elsevier, Amsterdam, 1966).

Travis, L. D.

E. H. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vasileva, I. A.

I. A. Vasileva, L. P. Deputatova, A. P. Nefedov, “Investigating the Flame with the Aid of Self-Reversed Contours of Spectral Lines,” Combust. Flame 23, 305–311 (1977).
[CrossRef]

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

Young, P.

S. A. Self, P. H. Paul, P. Young, “A Packaged Fiberoptic Spectroradiometer for High Temperature Gases with Automatic Readout,” in Temperature, Its Measurement and Control in Science and Industry, J. F. Schooly, Ed. (AIP, New York, 1982) Vol. 5, p. 465.

Zweifel, P. F.

P. F. Zweifel, “Neutron Transport Theory,” in Developments in Transport Theory, Nato Advanced Study Institute on Transport Theory, E. Inonu, P. F. Zweifel, Eds. (Academic, London, 1967).

Ann. Phys.

K. M. Case, “Elementary Solutions of the Transport Equations and Their Applications,” Ann. Phys. 9, 1–23 (1960).
[CrossRef]

Appl. Opt.

Combust. Flame

D. L. Thomas, “Problems in Applying the Line Reversal Method of Temperature Measurement to Flames,” Combust. Flame 12, 541–549 (1968).
[CrossRef]

I. A. Vasileva, L. P. Deputatova, A. P. Nefedov, “Investigating the Flame with the Aid of Self-Reversed Contours of Spectral Lines,” Combust. Flame 23, 305–311 (1977).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Elec. Tech.

A. G. Sviridov, N. N. Sobolev, “Flame Temperature Measurements Using the Line Reversal Method,” J. Phys. Elec. Tech. 22, 93–101 (1953).

J. Quant. Spectrosc. Radiat. Transfer

D. G. Goodwin, J. L. Ebert, “Rigorous Bounds on the Radiative Interaction Between Real Gases and Scattering Particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 501–508 (1987).
[CrossRef]

Space Sci. Rev.

E. H. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Teplofiz. Vys. Temp.

I. A. Vasileva, V. V. Kirillov, G. P. Malyuzhonok, V. B. Novosadov, I. A. Masimov, “Measurements of Plasma Temperature by Spectroscopic Method with a Continuous Automatic Reading,” Teplofiz. Vys. Temp. 4, 838–843 (1973).

Other

S. A. Self, P. H. Paul, P. Young, “A Packaged Fiberoptic Spectroradiometer for High Temperature Gases with Automatic Readout,” in Temperature, Its Measurement and Control in Science and Industry, J. F. Schooly, Ed. (AIP, New York, 1982) Vol. 5, p. 465.

P. F. Zweifel, “Neutron Transport Theory,” in Developments in Transport Theory, Nato Advanced Study Institute on Transport Theory, E. Inonu, P. F. Zweifel, Eds. (Academic, London, 1967).

L. E. Bauman, “Gas Temperature Measurements of Particle Laden MHD Flows,” presented at 1984 ASME Heat Transfer Conference (ASME, New York, 1984), paper 84-AES-8.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

I. W. Busbridge, “The Mathematics of Radiative Transfer,” in Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge UP., London, 1960).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

R. M. Tourin, Spectroscopic Gas Temperature Measurement (Elsevier, Amsterdam, 1966).

P. H. Paul, “Spectroradiometric Temperature Measurements in Two-Phase Combustion Plasmas,” Doctoral Dissertation, Stanford U. (1984), HTGL report T-238.

P. C. Ariessohn, Optical Diagnostic Measurements of Coal Slag Parameters in Combustion MHD Systems, Doctoral Dissertation, Stanford U. (1980), HTGL report 119.

J. M. Adams, “The Spectral Comparison Method for Temperature Measurement and Control in Two-Phase Flames,” in Temperature, Its Measurement and Control in Science and IndustryH. H. Plumb, Ed. (Instrument Society of America, Pittsburgh, PA, 1972), Vol. 3, p. 627.

D. W. Mackowski, R. A. M. Altenkirch, R. E. Peck, T. W. Tong, “Infrared Pyrometer Measurements of Particle and Gas Temperatures in Pulverized-Coal Flames,” presented at Western States Combustion Conference (1982), paper 82-22.

D. J. Carlsen, “Static Temperature Measurements in Hot Gas-particle Flows,” in Temperature, Its Measurement and Control in Science and Industry, C. M. Herzfeld, Ed. (Reinhold, New York, 1962), Vol. 3, p. 535.

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Figures (4)

Fig. 1
Fig. 1

Gain and loss of radiation due to scattering.

Fig. 2
Fig. 2

Relative error in measured temperature in a scattering medium with no correction (λ = 766.5 nm, TG = 2800 K).

Fig. 3
Fig. 3

Relative error in measured temperature in a scattering medium using the method of Thomas (λ = 766.5 nm, TG = 2800 K).

Fig. 4
Fig. 4

Relative error in measured temperature in a scattering medium using a first-order isotropic correction (λ = 766.5 nm, TG = 2800 K).

Equations (34)

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ω λ = σ λ α λ G + α λ P + σ λ .
d I λ d τ λ + I λ = I λ ( B ) .
τ λ ( z ) 0 z α λ G d z ,
I λ ( d ) = I λ ( B ) ( T L ) exp ( Γ λ ) + I λ ( B ) ( T L ) [ 1 exp ( Γ λ ) ] .
T G = T L 1 λ T L C 2 ln ( S G S G + S L S G + L ) .
I λ ( B ) = 2 h c 2 λ 5 exp ( C 2 λ T ) .
Γ λ ( α λ G + α λ P + σ λ ) d .
Δ T G T G = [ 1 + C 2 λ T G / ln Φ ( ω λ , Γ λ ) ( 1 exp ( Γ λ ) ) ] 1 .
μ I λ z + ( α λ G + α λ P + σ λ ) I λ = α λ G I λ ( B ) ( T G ) + α λ P I λ ( B ) ( T P ) + σ λ 2 1 1 I λ ( z , μ ) p ( μ , μ ) d μ .
χ λ ω λ U λ U λ α λ P σ λ exp [ C 2 λ ( 1 T P 1 T G ) ] ,
μ I λ τ λ + I λ = ( 1 + χ λ ω λ ) I λ ( B ) ( T G ) + ω λ 2 1 1 I λ ( τ λ , μ ) p ( μ , μ ) d μ .
I = i = 0 ( ω ) i I ± ( i ) [ I + ( i 1 ) , I ( i 1 ) ] .
I + ( τ , μ ) I λ ( τ λ , μ ) I ( τ , μ ) I λ ( τ λ , μ ) μ 0 .
I + ( τ , μ ) = exp ( τ / μ ) I λ ( B ) ( T W ) + ( 1 + χ ω ) × [ 1 exp ( τ / μ ) ] I λ ( B ) ( T G ) + ω 2 0 τ exp [ ( τ τ ) / μ ] μ [ 0 1 p ( μ , μ ) I ( τ , μ ) d μ + 0 1 p ( μ , μ ) I + ( τ , μ ) d μ ] d τ .
I + 0 = ( 1 + χ ω ) [ 1 exp ( τ / μ ) ] I λ ( B ) ( T G ) , I 0 = ( 1 + χ ω ) [ 1 exp ( ( Γ τ ) / μ ) ] I λ ( B ) ( T G ) ,
I + 1 = ( 1 + χ ω ) I λ ( B ) ( T G ) ( 1 exp ( τ / μ ) l = 0 b l ( μ ) μ × { exp ( τ / μ ) F l + 2 ( τ , μ ) + ( 1 ) l exp [ ( Γ τ ) / μ ] × F l + 2 ( Γ , μ ) F l + 2 ( Γ τ , μ ] } ) .
F l ( τ , μ ) 0 τ exp ( t / μ ) E l ( t ) d t 1 μ 1 .
p ( μ , μ ) = n = 0 a n P n ( μ ) P n ( μ ) = n = 0 b n ( a n , μ ) ( μ ) n .
ω λ 1 = ω λ 2 Γ λ 2 Γ λ 1 .
I λ ( d ) = [ I + 0 ( W ) ( Γ , 1 ) + ω I + 1 ( W ) ( Γ , 1 ) + ] I λ ( B ) ( T W ) + ( 1 + χ ω ) [ I + 0 ( G ) ( Γ , 1 ) + ω I + 1 ( G ) ( Γ , 1 ) + ] I λ ( B ) ( T G ) ,
I λ ( d ) = exp ( Γ ) I λ ( B ) ( T L ) + I λ ( B ) ( T G ) ( 1 ω λ ) × ( [ 1 exp ( Γ ) ] ω λ l = 0 L b l ( μ = 1 ) { [ 1 exp ( Γ ) ] [ exp ( Γ ) F l + 2 ( Γ , 1 ) + F l + 2 ( Γ , 1 ) ] / 2 } ) .
Λ ( Γ λ ) 1 exp ( Γ ) , Λ l ( Γ λ ) b l ( μ = 1 ) [ exp ( Γ ) F l + 2 ( Γ , 1 ) + ( 1 ) l F l + 2 ( Γ , 1 ) ] .
F n ( Γ , ± 1 ) = ± [ F n 1 ( Γ , ± 1 ) + exp ( ± Γ ) E n ( Γ ) 1 / ( 1 n ) ] , n 1 .
F 1 ( Γ , 1 ) = γ + ln ( 2 ) + exp ( Γ ) E 1 ( Γ ) , F 1 ( Γ , 1 ) = ln ( 2 ) + E 1 ( 2 Γ ) exp ( Γ ) E 1 ( Γ ) ,
I λ ( d ) = exp ( Γ ) I λ ( B ) ( T L ) + I λ ( B ) ( T G ) ( 1 ω ) [ ( 1 + ω ) Λ ( Γ ) Λ 0 ( Γ ) / 2 ] .
Γ 1 = ln S G + L S G S L ,
Γ i = ln S G + L S G s L | λ i , i = 1 , 2 ,
Φ i exp [ C 2 λ i ( 1 T G 1 T L ) ] = S G S L | λ i , i = 1 , 2 .
Φ 1 = ( 1 ω 1 ) [ ( 1 + ω 1 ) Λ ( Γ 1 ) ω 1 Λ 0 ( Γ 1 ) ] , Φ 2 = ( 1 ω 1 Γ 1 / Γ 2 ) [ ( 1 + ω 1 Γ 1 / Γ 2 ) Λ ( Γ 2 ) ω 1 Γ 1 / Γ 2 Λ 0 ( Γ 2 ) ] .
Φ 12 Φ 1 / Φ 2 = S G / S L | λ 1 S G / S L | λ 2 .
[ Λ ( Γ 1 ) Φ 12 ( Γ 1 / Γ 2 ) Λ ( Γ 2 ) ] [ Λ ( Γ 1 ) Φ 12 ( Γ 1 / Γ 2 ) Λ ( Γ 2 ) ] ω 1 + { Λ ( Γ 1 ) Λ ( Γ 1 ) Φ 12 ( Γ 1 / Γ 2 ) 2 [ Λ ( Γ 2 ) Λ ( Γ 2 ) ] } ω 1 2 = 0 .
T G = T L 1 λ 1 T L C 2 ln ( S G 1 S L 1 1 Φ 1 ( Γ 1 , ω 1 ) ) ,
Φ 1 = Λ ( Γ 1 ) = 1 exp ( Γ 1 ) = S G + S L S G + L S L | λ i .
Φ i = [ 1 ω i ( 1 U ) ] [ ( 1 + ω i ) Λ ( Γ i ) ω i Λ ( Γ i ) ] , i = λ 1 , λ 2 , λ 3 ,

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