Abstract

We focus on the utility of rainbow schlieren as a tool for measuring the temperature of axisymmetric partially premixed flames (PPFs). Methane-air PPFs are established on a coannular burner. The flames involve two spatially distinct reaction zones, one in an inner premixed region that has a curved tip and a spatially planar wing portion and another that involves an outer nonpremixed zone in which intermediate species burn in air. Schlieren images are found to visualize clearly these PPF characteristics through light deflection by steep refractive-index gradients in the two reaction zone fronts. The temperature distributions of two flames established at fuel-rich mixture equivalence ratios of ϕr = 1.5 and 2.0, with bulk-averaged velocities, V reac = 60 cm s-1 and V air = 50 cm s-1, are inferred from color schlieren images, and a measurement error analysis is performed. Errors arise from two sources. One lies in the process of inferring the temperature from the refractive-index measurement by making assumptions regarding the local composition of the flame. We have shown through simulations that the average temperature deviations due to these assumptions are 1.7% for the ϕr = 1.5 flame and 2.3% for the ϕr = 2.0 flame. Another source involves the local uncertainty in the measurement of the transverse ray displacement at the filter plane that is used to determine the refractive index and thereafter the flame temperature. We have ascertained that a maximum error of 4.3% in the temperature determination can be attributed to this local measurement uncertainty. This investigation demonstrates the capability of the schlieren technique for providing not only qualitative displays of the PPFs but also full-field-of-view temperature measurements that are accurate, spatially resolved, and nonintrusive.

© 2002 Optical Society of America

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References

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  1. Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
    [CrossRef]
  2. R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
    [CrossRef]
  3. N. Peters, “Partially premixed diffusion flamelets in non-premixed turbulent combustion,” in 20th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1984), pp. 353–360.
  4. B. Rogg, F. Behrendt, J. Warnatz, “Turbulent non-premixed combustion in partially premixed diffusion flamelets with detailed chemistry,” in 21st International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1986), pp. 1533–1541.
  5. P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).
  6. H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 1, 131–136 (1985).
    [CrossRef]
  7. X. Xiao, C. W. Choi, I. K. Puri, “Temperature measurement in steady two-dimensional partially premixed flames using laser holographic interferometry,” Combust. Flame 120, 318–332 (2000).
    [CrossRef]
  8. X. Xiao, I. K. Puri, “Systematic approach based on holographic interferometry measurements to characterize the flame structure of partially premixed flames,” Appl. Opt. 40, 731–740 (2001).
    [CrossRef]
  9. G. W. Faris, R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
    [CrossRef] [PubMed]
  10. P. S. Greenberg, R. B. Klimek, D. R. Buchele, “Quantitative rainbow schlieren deflectometry,” Appl. Opt. 34, 3810–3822 (1995).
    [CrossRef] [PubMed]
  11. K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Application of rainbow schlieren deflectometry for concentration measurements in an axisymetric helium jet,” Exp. Fluids 25, 89–95 (1998).
    [CrossRef]
  12. B. W. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame 119, 84–94 (1999).
    [CrossRef]
  13. K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Quantitative measurements of laminar hydrogen gas-jet diffusion flames in a 2.25 drop tower,” in 28th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 2000), pp. 1997–2004.
  14. W. L. Howes, “Rainbow schlieren and its application,” Appl. Opt. 23, 2449–2460 (1984).
    [CrossRef]
  15. R. Rubinstein, P. S. Greenberg, “Rapid inversion of angular deflection data for certain axisymmetric refractive index distributions,” Appl. Opt. 33, 1141–1144 (1994).
    [CrossRef] [PubMed]
  16. R. J. Goldstein, T. H. Kuehn, Fluid Mechanics Measurements (Taylor and Francis, Washington, D.C., 1996).
  17. A. K. Agrawal, B. W. Albers, D. Griffin, “Abel inversion of deflectometric measurements in dynamic flows,” Appl. Opt. 38, 3394–3398 (1999).
    [CrossRef]
  18. W. Merzkirch, Flow Visualization, 2nd ed. (Academic, Orlando, Fla., 1987).
  19. X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
    [CrossRef]
  20. B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
    [CrossRef]
  21. W. S. Winthrop, M. S. Joanne, Handbook of Real-Time Fast Fourier Transforms (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1995).
  22. C. Dasch, “One-dimentional tomography: a comparison of Abel, onion-peeling and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
    [CrossRef] [PubMed]

2002 (1)

X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
[CrossRef]

2001 (1)

2000 (1)

X. Xiao, C. W. Choi, I. K. Puri, “Temperature measurement in steady two-dimensional partially premixed flames using laser holographic interferometry,” Combust. Flame 120, 318–332 (2000).
[CrossRef]

1999 (4)

B. W. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame 119, 84–94 (1999).
[CrossRef]

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
[CrossRef]

A. K. Agrawal, B. W. Albers, D. Griffin, “Abel inversion of deflectometric measurements in dynamic flows,” Appl. Opt. 38, 3394–3398 (1999).
[CrossRef]

1998 (1)

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Application of rainbow schlieren deflectometry for concentration measurements in an axisymetric helium jet,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

1995 (1)

1994 (1)

1992 (1)

1988 (1)

1985 (1)

H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 1, 131–136 (1985).
[CrossRef]

1984 (1)

1982 (1)

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

Aggarwal, S. K.

X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
[CrossRef]

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
[CrossRef]

Agrawal, A. K.

B. W. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame 119, 84–94 (1999).
[CrossRef]

A. K. Agrawal, B. W. Albers, D. Griffin, “Abel inversion of deflectometric measurements in dynamic flows,” Appl. Opt. 38, 3394–3398 (1999).
[CrossRef]

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Application of rainbow schlieren deflectometry for concentration measurements in an axisymetric helium jet,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Quantitative measurements of laminar hydrogen gas-jet diffusion flames in a 2.25 drop tower,” in 28th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 2000), pp. 1997–2004.

Akinyemi, O. C.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Alammar, K. N.

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Application of rainbow schlieren deflectometry for concentration measurements in an axisymetric helium jet,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Quantitative measurements of laminar hydrogen gas-jet diffusion flames in a 2.25 drop tower,” in 28th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 2000), pp. 1997–2004.

Albers, B. W.

B. W. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame 119, 84–94 (1999).
[CrossRef]

A. K. Agrawal, B. W. Albers, D. Griffin, “Abel inversion of deflectometric measurements in dynamic flows,” Appl. Opt. 38, 3394–3398 (1999).
[CrossRef]

Azzoni, R.

R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
[CrossRef]

Behrendt, F.

B. Rogg, F. Behrendt, J. Warnatz, “Turbulent non-premixed combustion in partially premixed diffusion flamelets with detailed chemistry,” in 21st International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1986), pp. 1533–1541.

Buchele, D. R.

Byer, R. L.

Choi, C. W.

X. Xiao, C. W. Choi, I. K. Puri, “Temperature measurement in steady two-dimensional partially premixed flames using laser holographic interferometry,” Combust. Flame 120, 318–332 (2000).
[CrossRef]

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

Dasch, C.

Dec, J. E.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Durrett, R. P.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Faris, G. W.

Flynn, P. F.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Goldstein, R. J.

R. J. Goldstein, T. H. Kuehn, Fluid Mechanics Measurements (Taylor and Francis, Washington, D.C., 1996).

Gollahalli, S. R.

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Application of rainbow schlieren deflectometry for concentration measurements in an axisymetric helium jet,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Quantitative measurements of laminar hydrogen gas-jet diffusion flames in a 2.25 drop tower,” in 28th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 2000), pp. 1997–2004.

Greenberg, P. S.

Griffin, D.

Hertz, H. M.

H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 1, 131–136 (1985).
[CrossRef]

Howes, W. L.

Hughey, B. J.

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

Hunter, G. L.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Joanne, M. S.

W. S. Winthrop, M. S. Joanne, Handbook of Real-Time Fast Fourier Transforms (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1995).

Katta, V. R.

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

Klimek, R. B.

Kuehn, T. H.

R. J. Goldstein, T. H. Kuehn, Fluid Mechanics Measurements (Taylor and Francis, Washington, D.C., 1996).

Loye, A. O.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Merzkirch, W.

W. Merzkirch, Flow Visualization, 2nd ed. (Academic, Orlando, Fla., 1987).

Peters, N.

N. Peters, “Partially premixed diffusion flamelets in non-premixed turbulent combustion,” in 20th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1984), pp. 353–360.

Puri, I. K.

X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
[CrossRef]

X. Xiao, I. K. Puri, “Systematic approach based on holographic interferometry measurements to characterize the flame structure of partially premixed flames,” Appl. Opt. 40, 731–740 (2001).
[CrossRef]

X. Xiao, C. W. Choi, I. K. Puri, “Temperature measurement in steady two-dimensional partially premixed flames using laser holographic interferometry,” Combust. Flame 120, 318–332 (2000).
[CrossRef]

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
[CrossRef]

Qin, X.

X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
[CrossRef]

Ratti, S.

R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
[CrossRef]

Rogg, B.

B. Rogg, F. Behrendt, J. Warnatz, “Turbulent non-premixed combustion in partially premixed diffusion flamelets with detailed chemistry,” in 21st International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1986), pp. 1533–1541.

Rubinstein, R.

Santavicca, D. A.

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

Shu, Z.

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

Warnatz, J.

B. Rogg, F. Behrendt, J. Warnatz, “Turbulent non-premixed combustion in partially premixed diffusion flamelets with detailed chemistry,” in 21st International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1986), pp. 1533–1541.

Westbrook, C. K.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

Winthrop, W. S.

W. S. Winthrop, M. S. Joanne, Handbook of Real-Time Fast Fourier Transforms (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1995).

Xiao, X.

X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
[CrossRef]

X. Xiao, I. K. Puri, “Systematic approach based on holographic interferometry measurements to characterize the flame structure of partially premixed flames,” Appl. Opt. 40, 731–740 (2001).
[CrossRef]

X. Xiao, C. W. Choi, I. K. Puri, “Temperature measurement in steady two-dimensional partially premixed flames using laser holographic interferometry,” Combust. Flame 120, 318–332 (2000).
[CrossRef]

Appl. Opt. (7)

Combust. Flame (5)

X. Qin, X. Xiao, I. K. Puri, S. K. Aggarwal, “Effect of composition distribution on temperature reconstruction from refractive index in flames,” Combust. Flame 128, 121–321 (2002).
[CrossRef]

X. Xiao, C. W. Choi, I. K. Puri, “Temperature measurement in steady two-dimensional partially premixed flames using laser holographic interferometry,” Combust. Flame 120, 318–332 (2000).
[CrossRef]

B. W. Albers, A. K. Agrawal, “Schlieren analysis of an oscillating gas-jet diffusion flame,” Combust. Flame 119, 84–94 (1999).
[CrossRef]

Z. Shu, C. W. Choi, S. K. Aggarwal, V. R. Katta, I. K. Puri, “Gravity effects on steady two-dimensional partially premixed methane–air flames,”Combust. Flame 118, 91–107 (1999).
[CrossRef]

R. Azzoni, S. Ratti, S. K. Aggarwal, I. K. Puri, “The structure of triple flames stabilized on a slot burner,” Combust. Flame 119, 23–40 (1999).
[CrossRef]

Combust. Sci. Technol. (1)

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

Exp. Fluids (1)

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Application of rainbow schlieren deflectometry for concentration measurements in an axisymetric helium jet,” Exp. Fluids 25, 89–95 (1998).
[CrossRef]

Opt. Commun. (1)

H. M. Hertz, “Experimental determination of 2-D flame temperature fields by interferometric tomography,” Opt. Commun. 1, 131–136 (1985).
[CrossRef]

Other (7)

K. N. Alammar, A. K. Agrawal, S. R. Gollahalli, “Quantitative measurements of laminar hydrogen gas-jet diffusion flames in a 2.25 drop tower,” in 28th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 2000), pp. 1997–2004.

W. Merzkirch, Flow Visualization, 2nd ed. (Academic, Orlando, Fla., 1987).

R. J. Goldstein, T. H. Kuehn, Fluid Mechanics Measurements (Taylor and Francis, Washington, D.C., 1996).

N. Peters, “Partially premixed diffusion flamelets in non-premixed turbulent combustion,” in 20th International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1984), pp. 353–360.

B. Rogg, F. Behrendt, J. Warnatz, “Turbulent non-premixed combustion in partially premixed diffusion flamelets with detailed chemistry,” in 21st International Symposium on Combustion (The Combustion Institute, Pittsburgh, Pa., 1986), pp. 1533–1541.

P. F. Flynn, R. P. Durrett, G. L. Hunter, A. O. Loye, O. C. Akinyemi, J. E. Dec, C. K. Westbrook, SAE Paper 1999-01-0509 (Society of Automotive Engineers, Warrendale, Pa., 1999).

W. S. Winthrop, M. S. Joanne, Handbook of Real-Time Fast Fourier Transforms (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1995).

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Figures (11)

Fig. 1
Fig. 1

Methane-air PPF and the coannular burner: (a) image of a PPF established on the burner; (b) schematic diagram of the burner.

Fig. 2
Fig. 2

Rainbow schlieren deflectometry experimental setup.

Fig. 3
Fig. 3

Color filter with a resolution of 180 pixels/mm.

Fig. 4
Fig. 4

Filter calibration curve.

Fig. 5
Fig. 5

Geometry of light-ray deflection in an axisymmetric refractive-index field n(r); ε(y 1) denotes the exit angle of a ray originating at the coordinate y = y 1.

Fig. 6
Fig. 6

Schlieren images for two double-flame PPFs established at overall equivalence ratios of ϕ r = 1.5 and 2.0. Here the inner and the outer bulk-averaged velocities are V reac = 60 cm s-1 and V air = 50 cm s-1, respectively.

Fig. 7
Fig. 7

Angular deflection distribution at axial plane z = 10 mm for the ϕ r = 1.5 flame.

Fig. 8
Fig. 8

Angular deflection distribution at (a) z = 2 mm and (b) z = 10 mm for the ϕ r = 1.5 flame.

Fig. 9
Fig. 9

Temperature distribution at (a) z = 2 mm and (b) 10 mm for the ϕ r = 1.5 flame.

Fig. 10
Fig. 10

Angular deflection contours of flames corresponding to images in Fig. 6.

Fig. 11
Fig. 11

Temperature contours of flames corresponding to images in Fig. 6.

Equations (15)

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

I=R+G+B/3,
S=1-minR, G, B/I,
H=cos-1(2R-G-B/2R-G2+R-BG-B1/2).
εy=2y ydδdrdrr2-y21/2.
dy=εyf.
δr=-1πr εydyy2-r21/2,
δi=δri=-1πj=iN01εj+εj+1-εjldlj+l2-i21/2,
δri=j=1N Dijεj,
Dij=0 for j<i, Dij=1π Ai,j for j=i;Dij=1πAi,j-Bi,j-1 for j>i,
Ai,j=j+12-i21/2-j2-i21/2-j+1lnj+1+j+12-i21/2j+j2-i21/2, Bi,j=j+12-i21/2-j2-i21/2-j lnj+1+j+12-i21/2j+j2-i21/2.
nr-1=ρrKmix=ρrΣyirki,
Tr=nr-1-1βr,
Tr=T0n0-1/nr-1.
Δδi=j=iNDijΔεj21/2,
Δεj=Δdjf

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