Abstract

Three-dimensional (3D) images of flame emission are reported using a single direction of optical access. A Cassegrain system was designed with narrow depth of field. Images from this system are dominated by emission from the focused object plane with defocused contributions from out-of-plane structures. Translation of one mirror in the system allows for scanning the object plane through the flame. Images were taken at various depths to create a family of images. Reconstruction of the 3D flame structure was accomplished using a maximum entropy algorithm adapted for use with 3D imaging. Spatial resolution in the direction of imaging is examined using laminar flames with variable offset.

© 2012 Optical Society of America

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2011

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

B. Zhou, J. Zhang, and S. Wang, “Reconstruction of flame temperature field with optical sectioning method,” IET Image Process. 5, 382–393 (2011).
[CrossRef]

2010

D. Veynate, G. Lodato, P. Domingo, L. Vervisch, and E. Hawkes, “Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion,” Exp. Fluids 49, 267–278 (2010).
[CrossRef]

Z. Gut and P. Wolanski, “Flame imaging using 3D electrical capacitance tomography,” Combust. Sci. Technol. 182, 1580–1585 (2010).
[CrossRef]

2007

G. Gilabert and G. Lu, “Three-dimensional tomographic reconstruction of the luminosity distribution of a combustion flame,” IEEE Trans. Instrum. Meas. 56, 1300–1306 (2007).
[CrossRef]

J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. 24, 2622–2627 (2007).
[CrossRef]

2006

J. Olofsson, M. Richter, M. Aldén, and M. Augé, “Development of high temporally and spatially (three-dimensional) resolved formaldehyde measurements in combustion environments,” Rev. Sci. Instrum. 77, 013104 (2006).
[CrossRef]

2005

A. N. Karpetis and R. S. Barlow, “Measurements of flame orientation and scalar dissipation in turbulent partially premixed methane flames,” Proc. Combust. Inst. 30, 665–672 (2005).
[CrossRef]

P. M. Brisley, G. Lu, Y. Yan, and S. Cornwell, “Three-dimensional temperature measurement of combustion flames using a single monochromatic CCD camera,” IEEE Trans. Instrum. Meas. 54, 1417–1421 (2005).
[CrossRef]

H. C. Bheemul, G. Lu, and Y. Yan, “Digital imaging-based three-dimensional characterization of flame front structures in a turbulent flame,” IEEE Trans. Instrum. Meas. 54, 1073–1078 (2005).
[CrossRef]

Y. M. Wang, H. B. Wang, F. H. Li, L. S. Jia, and X. L. Chen, “Maximum entropy image deconvolution applied to structure determination for crystal Nd1.85Ce0.15CuO4-δ,” Micron 36, 393–400 (2005).
[CrossRef]

2002

H. C. Bheemul, G. Lu, and Y. Yan, “Three-dimensional visualization and quantitative characterization of gaseous flames,” Meas. Sci. Technol. 13, 1643–1650 (2002).
[CrossRef]

2000

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

1999

B. A. van der Wege, C. J. O’Brien, and S. Hochgreb, “Quantitative shearography in axisymmetric gas temperature measurements,” Opt. Lasers Eng. 31, 21–39 (1999).
[CrossRef]

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

1998

J. H. Chen and H. G. Im, “Correlation of flame speed with stretch in turbulent premixed methane/air flames,” Proc. Combust. Inst. 27, 819–826 (1998).

J. C. Broda, S. Seo, R. J. Santoro, G. Shirhattikar, and V. Yang, “An experimental study of combustion dynamics of a premixed swirl injector,” Proc. Combust. Inst. 27, 1849–1856 (1998).

1996

D. Veynante, J. Piana, J. M. Duclos, and C. Martel, “Experimental analysis of flame surface density models for premixed turbulent combustion,” Proc. Combust. Inst. 26, 413–420 (1996).

1985

T. J. Cornwell and K. F. Evans, “A simple maximum entropy deconvolution algorithm,” Astron. Astrophys. 143, 77–83 (1985).

J. M. Seitzman, G. Kychakoff, and R. K. Hanson, “Instantaneous temperature field measurements using planar laser-induced fluorescence,” Opt. Lett. 10, 439–441 (1985).
[CrossRef]

1984

S. F. Gull and J. Skilling, “Maximum entropy method in image processing,” IEE Proc. F 131, 646–659 (1984).
[CrossRef]

1982

1980

J. E. Shore and R. W. Johnson, “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,” IEEE Trans. Inf. Theory 26, 26–37 (1980).
[CrossRef]

R. K. Bryan and J. Skilling, “Deconvolution by maximum entropy, as illustrated by application to the jet of M87,” Mon. Not. R. Astron. Soc. 191, 69–79 (1980).

1972

Akamatsu, F.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Aldén, M.

J. Olofsson, M. Richter, M. Aldén, and M. Augé, “Development of high temporally and spatially (three-dimensional) resolved formaldehyde measurements in combustion environments,” Rev. Sci. Instrum. 77, 013104 (2006).
[CrossRef]

Augé, M.

J. Olofsson, M. Richter, M. Aldén, and M. Augé, “Development of high temporally and spatially (three-dimensional) resolved formaldehyde measurements in combustion environments,” Rev. Sci. Instrum. 77, 013104 (2006).
[CrossRef]

Barlow, R. S.

A. N. Karpetis and R. S. Barlow, “Measurements of flame orientation and scalar dissipation in turbulent partially premixed methane flames,” Proc. Combust. Inst. 30, 665–672 (2005).
[CrossRef]

Bheemul, H. C.

H. C. Bheemul, G. Lu, and Y. Yan, “Digital imaging-based three-dimensional characterization of flame front structures in a turbulent flame,” IEEE Trans. Instrum. Meas. 54, 1073–1078 (2005).
[CrossRef]

H. C. Bheemul, G. Lu, and Y. Yan, “Three-dimensional visualization and quantitative characterization of gaseous flames,” Meas. Sci. Technol. 13, 1643–1650 (2002).
[CrossRef]

Brisley, P. M.

P. M. Brisley, G. Lu, Y. Yan, and S. Cornwell, “Three-dimensional temperature measurement of combustion flames using a single monochromatic CCD camera,” IEEE Trans. Instrum. Meas. 54, 1417–1421 (2005).
[CrossRef]

Broda, J. C.

J. C. Broda, S. Seo, R. J. Santoro, G. Shirhattikar, and V. Yang, “An experimental study of combustion dynamics of a premixed swirl injector,” Proc. Combust. Inst. 27, 1849–1856 (1998).

Bryan, R. K.

R. K. Bryan and J. Skilling, “Deconvolution by maximum entropy, as illustrated by application to the jet of M87,” Mon. Not. R. Astron. Soc. 191, 69–79 (1980).

Byer, R. L.

Chen, J. H.

J. H. Chen and H. G. Im, “Correlation of flame speed with stretch in turbulent premixed methane/air flames,” Proc. Combust. Inst. 27, 819–826 (1998).

Chen, X. L.

Y. M. Wang, H. B. Wang, F. H. Li, L. S. Jia, and X. L. Chen, “Maximum entropy image deconvolution applied to structure determination for crystal Nd1.85Ce0.15CuO4-δ,” Micron 36, 393–400 (2005).
[CrossRef]

Cornwell, S.

P. M. Brisley, G. Lu, Y. Yan, and S. Cornwell, “Three-dimensional temperature measurement of combustion flames using a single monochromatic CCD camera,” IEEE Trans. Instrum. Meas. 54, 1417–1421 (2005).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell and K. F. Evans, “A simple maximum entropy deconvolution algorithm,” Astron. Astrophys. 143, 77–83 (1985).

Crosley, D. R.

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

Domingo, P.

D. Veynate, G. Lodato, P. Domingo, L. Vervisch, and E. Hawkes, “Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion,” Exp. Fluids 49, 267–278 (2010).
[CrossRef]

Duclos, J. M.

D. Veynante, J. Piana, J. M. Duclos, and C. Martel, “Experimental analysis of flame surface density models for premixed turbulent combustion,” Proc. Combust. Inst. 26, 413–420 (1996).

Evans, K. F.

T. J. Cornwell and K. F. Evans, “A simple maximum entropy deconvolution algorithm,” Astron. Astrophys. 143, 77–83 (1985).

Fletcher, D. A.

J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. 24, 2622–2627 (2007).
[CrossRef]

Freeman, R. H.

Frieden, B. R.

Garcia, H. R.

Gilabert, G.

G. Gilabert and G. Lu, “Three-dimensional tomographic reconstruction of the luminosity distribution of a combustion flame,” IEEE Trans. Instrum. Meas. 56, 1300–1306 (2007).
[CrossRef]

Gull, S. F.

S. F. Gull and J. Skilling, “Maximum entropy method in image processing,” IEE Proc. F 131, 646–659 (1984).
[CrossRef]

Gut, Z.

Z. Gut and P. Wolanski, “Flame imaging using 3D electrical capacitance tomography,” Combust. Sci. Technol. 182, 1580–1585 (2010).
[CrossRef]

Hanson, R. K.

Hawkes, E.

D. Veynate, G. Lodato, P. Domingo, L. Vervisch, and E. Hawkes, “Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion,” Exp. Fluids 49, 267–278 (2010).
[CrossRef]

Hochgreb, S.

B. A. van der Wege, C. J. O’Brien, and S. Hochgreb, “Quantitative shearography in axisymmetric gas temperature measurements,” Opt. Lasers Eng. 31, 21–39 (1999).
[CrossRef]

Hudgins, D.

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

Ikeda, Y.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Im, H. G.

J. H. Chen and H. G. Im, “Correlation of flame speed with stretch in turbulent premixed methane/air flames,” Proc. Combust. Inst. 27, 819–826 (1998).

Jeffries, J. B.

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

Jia, L. S.

Y. M. Wang, H. B. Wang, F. H. Li, L. S. Jia, and X. L. Chen, “Maximum entropy image deconvolution applied to structure determination for crystal Nd1.85Ce0.15CuO4-δ,” Micron 36, 393–400 (2005).
[CrossRef]

Johnson, R. W.

J. E. Shore and R. W. Johnson, “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,” IEEE Trans. Inf. Theory 26, 26–37 (1980).
[CrossRef]

Karpetis, A. N.

A. N. Karpetis and R. S. Barlow, “Measurements of flame orientation and scalar dissipation in turbulent partially premixed methane flames,” Proc. Combust. Inst. 30, 665–672 (2005).
[CrossRef]

Katsuki, M.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Kawahara, N.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Kychakoff, G.

Li, F. H.

Y. M. Wang, H. B. Wang, F. H. Li, L. S. Jia, and X. L. Chen, “Maximum entropy image deconvolution applied to structure determination for crystal Nd1.85Ce0.15CuO4-δ,” Micron 36, 393–400 (2005).
[CrossRef]

Lodato, G.

D. Veynate, G. Lodato, P. Domingo, L. Vervisch, and E. Hawkes, “Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion,” Exp. Fluids 49, 267–278 (2010).
[CrossRef]

Long, M. B.

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

Lu, G.

G. Gilabert and G. Lu, “Three-dimensional tomographic reconstruction of the luminosity distribution of a combustion flame,” IEEE Trans. Instrum. Meas. 56, 1300–1306 (2007).
[CrossRef]

P. M. Brisley, G. Lu, Y. Yan, and S. Cornwell, “Three-dimensional temperature measurement of combustion flames using a single monochromatic CCD camera,” IEEE Trans. Instrum. Meas. 54, 1417–1421 (2005).
[CrossRef]

H. C. Bheemul, G. Lu, and Y. Yan, “Digital imaging-based three-dimensional characterization of flame front structures in a turbulent flame,” IEEE Trans. Instrum. Meas. 54, 1073–1078 (2005).
[CrossRef]

H. C. Bheemul, G. Lu, and Y. Yan, “Three-dimensional visualization and quantitative characterization of gaseous flames,” Meas. Sci. Technol. 13, 1643–1650 (2002).
[CrossRef]

Luque, J.

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

Martel, C.

D. Veynante, J. Piana, J. M. Duclos, and C. Martel, “Experimental analysis of flame surface density models for premixed turbulent combustion,” Proc. Combust. Inst. 26, 413–420 (1996).

Mizutani, Y.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Nakajima, T.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

O’Brien, C. J.

B. A. van der Wege, C. J. O’Brien, and S. Hochgreb, “Quantitative shearography in axisymmetric gas temperature measurements,” Opt. Lasers Eng. 31, 21–39 (1999).
[CrossRef]

Olofsson, J.

J. Olofsson, M. Richter, M. Aldén, and M. Augé, “Development of high temporally and spatially (three-dimensional) resolved formaldehyde measurements in combustion environments,” Rev. Sci. Instrum. 77, 013104 (2006).
[CrossRef]

Piana, J.

D. Veynante, J. Piana, J. M. Duclos, and C. Martel, “Experimental analysis of flame surface density models for premixed turbulent combustion,” Proc. Combust. Inst. 26, 413–420 (1996).

Richter, M.

J. Olofsson, M. Richter, M. Aldén, and M. Augé, “Development of high temporally and spatially (three-dimensional) resolved formaldehyde measurements in combustion environments,” Rev. Sci. Instrum. 77, 013104 (2006).
[CrossRef]

Santoro, R. J.

J. C. Broda, S. Seo, R. J. Santoro, G. Shirhattikar, and V. Yang, “An experimental study of combustion dynamics of a premixed swirl injector,” Proc. Combust. Inst. 27, 1849–1856 (1998).

Seitzman, J. M.

Seo, S.

J. C. Broda, S. Seo, R. J. Santoro, G. Shirhattikar, and V. Yang, “An experimental study of combustion dynamics of a premixed swirl injector,” Proc. Combust. Inst. 27, 1849–1856 (1998).

Shaevitz, J. W.

J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. 24, 2622–2627 (2007).
[CrossRef]

Shirhattikar, G.

J. C. Broda, S. Seo, R. J. Santoro, G. Shirhattikar, and V. Yang, “An experimental study of combustion dynamics of a premixed swirl injector,” Proc. Combust. Inst. 27, 1849–1856 (1998).

Shore, J. E.

J. E. Shore and R. W. Johnson, “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,” IEEE Trans. Inf. Theory 26, 26–37 (1980).
[CrossRef]

Skilling, J.

S. F. Gull and J. Skilling, “Maximum entropy method in image processing,” IEE Proc. F 131, 646–659 (1984).
[CrossRef]

R. K. Bryan and J. Skilling, “Deconvolution by maximum entropy, as illustrated by application to the jet of M87,” Mon. Not. R. Astron. Soc. 191, 69–79 (1980).

Smith, G. P.

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

Smooke, M. D.

J. Luque, J. B. Jeffries, G. P. Smith, D. R. Crosley, K. T. Walsh, M. B. Long, and M. D. Smooke, “CH(A-X) and OH(A-X) optical emission in an axisymmetric laminar diffusion flame,” Combust. Flame 122, 172–175 (2000).
[CrossRef]

Tsushima, S.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Upton, T.

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

van der Wege, B. A.

B. A. van der Wege, C. J. O’Brien, and S. Hochgreb, “Quantitative shearography in axisymmetric gas temperature measurements,” Opt. Lasers Eng. 31, 21–39 (1999).
[CrossRef]

Verhoeven, D.

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

Vervisch, L.

D. Veynate, G. Lodato, P. Domingo, L. Vervisch, and E. Hawkes, “Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion,” Exp. Fluids 49, 267–278 (2010).
[CrossRef]

Veynante, D.

D. Veynante, J. Piana, J. M. Duclos, and C. Martel, “Experimental analysis of flame surface density models for premixed turbulent combustion,” Proc. Combust. Inst. 26, 413–420 (1996).

Veynate, D.

D. Veynate, G. Lodato, P. Domingo, L. Vervisch, and E. Hawkes, “Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion,” Exp. Fluids 49, 267–278 (2010).
[CrossRef]

Wakabayashi, T.

F. Akamatsu, T. Wakabayashi, S. Tsushima, M. Katsuki, Y. Mizutani, Y. Ikeda, N. Kawahara, and T. Nakajima, “The development of a light-collecting probe with high spatial resolution applicable to randomly fluctuating combustion fields,” Meas. Sci. Technol. 10, 1240–1246 (1999).
[CrossRef]

Walsh, K. T.

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Figures (10)

Fig. 1.
Fig. 1.

Cassegrain optical system used in this work, including optical path, pertinent dimensions, and components: (a) burner, (b) convex mirror and mount, (c) camera and mount, (d) concave mirror and mount, (e) translation stage and micrometer.

Fig. 2.
Fig. 2.

Ray-tracing results for a point source at two locations: (a) in focus, and (b) 1 mm out of focus. Note that the two plots have different scales.

Fig. 3.
Fig. 3.

Ray-tracing results for a point source at three configurations: (a) in focus, object and concave mirror at original locations (Table 1), (b) concave mirror moved 1 mm out of focus, and (c) mirror in same location as (b) but object moved 4 mm to refocus. Note that the plots have different scales.

Fig. 4.
Fig. 4.

(a) Dual flames with 9 mm separation, as viewed from top, and (b) perspective of imaging system. Approximate region of interrogation is defined by the box.

Fig. 5.
Fig. 5.

Image created from a backlit target placed at three locations: (a) in focus, (b) 1 mm out of focus, (c) 2 mm out of focus.

Fig. 6.
Fig. 6.

Pixel counts, averaged radially from the center, for the focused image from Fig. 5(a). The measured pixel counts are overlaid with a corresponding Gaussian fit (R2=0.974).

Fig. 7.
Fig. 7.

Images of dual-flame structure at 10 mm separation, taken with Cassegrain system. Images are ordered from the near side to the far side of the flame as detailed in Fig. 8. Intensities have been inverted for easier viewing (dark=higher pixel count).

Fig. 8.
Fig. 8.

Approximate image locations with respect to flames with 10 mm separation.

Fig. 9.
Fig. 9.

Images of dual-flame structure at 10 mm separation, after MEM processing. Images are ordered from the near side to the far side of the flame as detailed in Fig. 8. Colors have been inverted for easier viewing (dark=higher pixel count).

Fig. 10.
Fig. 10.

Location of flame fronts determined from MEM processed images. Flame separation from left to right: 7, 8, 9, 10, 11, and 12 mm. +’s represent the location of the flame front. The gray lines represent the nominal flame front locations based on the known flame geometry and diameter.

Tables (1)

Tables Icon

Table 1. Dimensions of Cassegrain System

Equations (14)

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

I=O*P+N,
B=(Bijk|i=1,X;j=1,Y;k=1,Z),
H(B|I)=ijkBijkln(BijkIijk),
H(B|I)=ijkBijkln(Bijk).
I=B*P.
χ2=ijk(IijkIijk)2.
ijkBijk=ijkIijk=F.
J=Hα·χ2β·F,
ΔB=(J)1·J,
J=Hαχ2β1,
(J)iik111Bijk+2α,
(J)ijk10,ij.
In=m(Pmn·Om)+N.
ijkΔBijk=0.

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