Abstract

Salt concentration distribution around a potassium dihydrogen phosphate (KDP) crystal growing from its aqueous solution has been experimentally determined using a laser schlieren technique. The growth process is initiated by inserting a KDP seed into its supersaturated solution, followed by slow cooling of the solution. Fluid convection leads to a distribution of concentration around the growing crystal. The pattern and strength of convection are important factors for the determination of the crystal growth rate and quality. Experiments have been conducted in a beaker with a diameter of 16.5 cm and a height of 23 cm. A monochrome schlieren technique has been employed to image the concentration field from four view angles, namely, 0°, 45°, 90°, and 135°. By interpreting the schlieren images as projection data of the solute concentration, the three-dimensional concentration field around the crystal has been determined using the convolution backprojection algorithm. The suitability of the overall approach has been validated using a simulated convective field in a circular differentially heated fluid layer, where full as well as partial data are available. Experiments have been conducted in the convection-dominated regime of crystal growth. The noncircular shape of the crystal is seen to affect axisymmetry of the concentration field close to the crystal surface. The reconstructed concentration fields reveal symmetry of the flow field away from the growing crystal. The solute concentration contours show large growth rates of the side faces of the crystal in comparison with the horizontal faces. In this respect, the concentration profiles are seen to correlate with the crystal geometry.

© 2005 Optical Society of America

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References

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  1. R. Snyder, L. Hesselink, “Measurement of mixing fluid flows with optical tomography,” Opt. Lett. 13, 87–89 (1988).
    [CrossRef] [PubMed]
  2. L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).
  3. D. W. Sweeney, C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
    [CrossRef] [PubMed]
  4. D. W. Watt, C. M. Vest, “Turbulent flow visualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
    [CrossRef]
  5. Y. C. Michael, K. T. Yang, “Three-dimensional Mach-Zehnder interferometric tomography of the Rayleigh-Benard problem,” ASME J. Heat Transfer 114, 622–629 (1992).
    [CrossRef]
  6. L. McMackin, R. J. Hugo, “High speed optical tomography system for imaging dynamic transparent media,” Opt. Express 1, 302–311 (1997).
    [CrossRef] [PubMed]
  7. A. K. Agrawal, N. K. Butuk, S. R. Gollahalli, D. Griffin, “Three-dimensional rainbow schlieren tomography of a temperature field in gas flows,” Appl. Opt. 37, 479–485 (1998).
    [CrossRef]
  8. H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Exp. Fluids 27, 542–550 (1999).
    [CrossRef]
  9. D. Mishra, K. Muralidhar, P. Munshi, “Study of Rayleigh-Benard convection at intermediate Rayleigh numbers,” Fluid Dyn. Res. 25, 231–255 (1999).
    [CrossRef]
  10. A. G. Notcovich, I. Braslavsky, S. G. Lipson, “Imaging fields around growing crystals,” J. Cryst. Growth 198–199, 10–16 (1999).
    [CrossRef]
  11. W. Dezhong, Z. Tiange, “The measurement of 3-D asymmetric temperature field by using real time laser interferometric tomography,” Opt. Lasers Eng. 36, 289–297 (2001).
    [CrossRef]
  12. D. Mishra, J. P. Longtin, R. P. Singh, V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1–11 (2004).
    [CrossRef]
  13. K. Onuma, K. Tsukamoto, I. Sunagawa, “Role of buoyancy driven convection in aqueous solution growth: a case study of Ba(NO3)2 crystal,” J. Cryst. Growth 89, 177–188 (1988).
    [CrossRef]
  14. M. Masayuki, M. Sugiyama, T. Ogawa, “Electronic measurement of concentration gradient around a crystal growing from a solution by using Mach-Zehnder interferometer,” J. Cryst. Growth 114, 71–76 (1991).
    [CrossRef]
  15. S. Kleine, W. J. P. van Enckevort, J. Derix, “A dark field-type schlieren microscope for quantitative, in situ mapping of solute concentration profiles around growing crystals,” J. Cryst. Growth 179, 240–248 (1997).
    [CrossRef]
  16. T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
    [CrossRef]
  17. A. Srivastava, K. Muralidhar, P. K. Panigrahi, “Comparison of interferometry, schlieren and shadowgraph for visualizing convection around a KDP crystal,” J. Cryst. Growth 267, 348–361 (2004).
    [CrossRef]
  18. A. Srivastava, K. Muralidhar, P. K. Panigrahi, “A schlieren study of the effect of ramp rate and rotation on convection around a crystal growing from an aqueous solution,” J. Cryst. Growth 274, 191–208 (2005).
    [CrossRef]
  19. G. T. Herman, Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transport Media (Springer-Verlag, 2001).
  20. G. T. Herman, Image Reconstruction from Projections (Academic, 1980).
  21. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).
  22. F. Mayinger, ed., Optical Measurements: Techniques and Applications (Springer-Verlag, 1994).
    [CrossRef]
  23. P. Munshi, “Error analysis of tomographic filters. I: Theory,” NDT & E Int. 25, 191–194 (1992).
    [CrossRef]
  24. P. Munshi, “Error analysis of tomographic filters. II: Results,” NDT & E Int. 26, 235–240 (1993).
    [CrossRef]
  25. R. J. Goldstein, ed., Fluid Mechanics Measurements (Hemisphere, 1983).
  26. M. G. Velarde, C. Normand, “Convection,” Sci. Am. 243, 79–94 (1980).
  27. S. S. Leong, “Numerical study of Rayleigh-Benard convection in a cylinder,” Numer. Heat Transfer Part A 41, 673–683 (2002).
    [CrossRef]
  28. B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport (Hemisphere, 1988).

2005

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “A schlieren study of the effect of ramp rate and rotation on convection around a crystal growing from an aqueous solution,” J. Cryst. Growth 274, 191–208 (2005).
[CrossRef]

2004

D. Mishra, J. P. Longtin, R. P. Singh, V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1–11 (2004).
[CrossRef]

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “Comparison of interferometry, schlieren and shadowgraph for visualizing convection around a KDP crystal,” J. Cryst. Growth 267, 348–361 (2004).
[CrossRef]

2002

S. S. Leong, “Numerical study of Rayleigh-Benard convection in a cylinder,” Numer. Heat Transfer Part A 41, 673–683 (2002).
[CrossRef]

2001

W. Dezhong, Z. Tiange, “The measurement of 3-D asymmetric temperature field by using real time laser interferometric tomography,” Opt. Lasers Eng. 36, 289–297 (2001).
[CrossRef]

2000

T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
[CrossRef]

1999

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Exp. Fluids 27, 542–550 (1999).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “Study of Rayleigh-Benard convection at intermediate Rayleigh numbers,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

A. G. Notcovich, I. Braslavsky, S. G. Lipson, “Imaging fields around growing crystals,” J. Cryst. Growth 198–199, 10–16 (1999).
[CrossRef]

1998

1997

L. McMackin, R. J. Hugo, “High speed optical tomography system for imaging dynamic transparent media,” Opt. Express 1, 302–311 (1997).
[CrossRef] [PubMed]

S. Kleine, W. J. P. van Enckevort, J. Derix, “A dark field-type schlieren microscope for quantitative, in situ mapping of solute concentration profiles around growing crystals,” J. Cryst. Growth 179, 240–248 (1997).
[CrossRef]

1993

P. Munshi, “Error analysis of tomographic filters. II: Results,” NDT & E Int. 26, 235–240 (1993).
[CrossRef]

1992

P. Munshi, “Error analysis of tomographic filters. I: Theory,” NDT & E Int. 25, 191–194 (1992).
[CrossRef]

Y. C. Michael, K. T. Yang, “Three-dimensional Mach-Zehnder interferometric tomography of the Rayleigh-Benard problem,” ASME J. Heat Transfer 114, 622–629 (1992).
[CrossRef]

1991

M. Masayuki, M. Sugiyama, T. Ogawa, “Electronic measurement of concentration gradient around a crystal growing from a solution by using Mach-Zehnder interferometer,” J. Cryst. Growth 114, 71–76 (1991).
[CrossRef]

1990

D. W. Watt, C. M. Vest, “Turbulent flow visualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

1988

R. Snyder, L. Hesselink, “Measurement of mixing fluid flows with optical tomography,” Opt. Lett. 13, 87–89 (1988).
[CrossRef] [PubMed]

K. Onuma, K. Tsukamoto, I. Sunagawa, “Role of buoyancy driven convection in aqueous solution growth: a case study of Ba(NO3)2 crystal,” J. Cryst. Growth 89, 177–188 (1988).
[CrossRef]

1980

M. G. Velarde, C. Normand, “Convection,” Sci. Am. 243, 79–94 (1980).

1973

Agrawal, A. K.

Bedarida, F.

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

Boccacci, P.

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

Braslavsky, I.

A. G. Notcovich, I. Braslavsky, S. G. Lipson, “Imaging fields around growing crystals,” J. Cryst. Growth 198–199, 10–16 (1999).
[CrossRef]

Butuk, N. K.

Dall’aglio, G. A.

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

Derix, J.

S. Kleine, W. J. P. van Enckevort, J. Derix, “A dark field-type schlieren microscope for quantitative, in situ mapping of solute concentration profiles around growing crystals,” J. Cryst. Growth 179, 240–248 (1997).
[CrossRef]

Dezhong, W.

W. Dezhong, Z. Tiange, “The measurement of 3-D asymmetric temperature field by using real time laser interferometric tomography,” Opt. Lasers Eng. 36, 289–297 (2001).
[CrossRef]

Gatti, L.

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

Gebhart, B.

B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport (Hemisphere, 1988).

Gollahalli, S. R.

Griffin, D.

Herman, G. T.

G. T. Herman, Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transport Media (Springer-Verlag, 2001).

G. T. Herman, Image Reconstruction from Projections (Academic, 1980).

Hesselink, L.

Hugo, R. J.

Inoue, T.

T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
[CrossRef]

Jaluria, Y.

B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport (Hemisphere, 1988).

Kageyama, Y.

T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
[CrossRef]

Kihm, K. D.

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Exp. Fluids 27, 542–550 (1999).
[CrossRef]

Kleine, S.

S. Kleine, W. J. P. van Enckevort, J. Derix, “A dark field-type schlieren microscope for quantitative, in situ mapping of solute concentration profiles around growing crystals,” J. Cryst. Growth 179, 240–248 (1997).
[CrossRef]

Ko, H. S.

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Exp. Fluids 27, 542–550 (1999).
[CrossRef]

Leong, S. S.

S. S. Leong, “Numerical study of Rayleigh-Benard convection in a cylinder,” Numer. Heat Transfer Part A 41, 673–683 (2002).
[CrossRef]

Lipson, S. G.

A. G. Notcovich, I. Braslavsky, S. G. Lipson, “Imaging fields around growing crystals,” J. Cryst. Growth 198–199, 10–16 (1999).
[CrossRef]

Longtin, J. P.

D. Mishra, J. P. Longtin, R. P. Singh, V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1–11 (2004).
[CrossRef]

Mahajan, R. L.

B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport (Hemisphere, 1988).

Masayuki, M.

M. Masayuki, M. Sugiyama, T. Ogawa, “Electronic measurement of concentration gradient around a crystal growing from a solution by using Mach-Zehnder interferometer,” J. Cryst. Growth 114, 71–76 (1991).
[CrossRef]

McMackin, L.

Michael, Y. C.

Y. C. Michael, K. T. Yang, “Three-dimensional Mach-Zehnder interferometric tomography of the Rayleigh-Benard problem,” ASME J. Heat Transfer 114, 622–629 (1992).
[CrossRef]

Mishra, D.

D. Mishra, J. P. Longtin, R. P. Singh, V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1–11 (2004).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “Study of Rayleigh-Benard convection at intermediate Rayleigh numbers,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

Mori, H.

T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
[CrossRef]

Mori, K.

T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
[CrossRef]

Munshi, P.

D. Mishra, K. Muralidhar, P. Munshi, “Study of Rayleigh-Benard convection at intermediate Rayleigh numbers,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

P. Munshi, “Error analysis of tomographic filters. II: Results,” NDT & E Int. 26, 235–240 (1993).
[CrossRef]

P. Munshi, “Error analysis of tomographic filters. I: Theory,” NDT & E Int. 25, 191–194 (1992).
[CrossRef]

Muralidhar, K.

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “A schlieren study of the effect of ramp rate and rotation on convection around a crystal growing from an aqueous solution,” J. Cryst. Growth 274, 191–208 (2005).
[CrossRef]

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “Comparison of interferometry, schlieren and shadowgraph for visualizing convection around a KDP crystal,” J. Cryst. Growth 267, 348–361 (2004).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “Study of Rayleigh-Benard convection at intermediate Rayleigh numbers,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Normand, C.

M. G. Velarde, C. Normand, “Convection,” Sci. Am. 243, 79–94 (1980).

Notcovich, A. G.

A. G. Notcovich, I. Braslavsky, S. G. Lipson, “Imaging fields around growing crystals,” J. Cryst. Growth 198–199, 10–16 (1999).
[CrossRef]

Ogawa, T.

M. Masayuki, M. Sugiyama, T. Ogawa, “Electronic measurement of concentration gradient around a crystal growing from a solution by using Mach-Zehnder interferometer,” J. Cryst. Growth 114, 71–76 (1991).
[CrossRef]

Onuma, K.

K. Onuma, K. Tsukamoto, I. Sunagawa, “Role of buoyancy driven convection in aqueous solution growth: a case study of Ba(NO3)2 crystal,” J. Cryst. Growth 89, 177–188 (1988).
[CrossRef]

Panigrahi, P. K.

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “A schlieren study of the effect of ramp rate and rotation on convection around a crystal growing from an aqueous solution,” J. Cryst. Growth 274, 191–208 (2005).
[CrossRef]

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “Comparison of interferometry, schlieren and shadowgraph for visualizing convection around a KDP crystal,” J. Cryst. Growth 267, 348–361 (2004).
[CrossRef]

Prasad, V.

D. Mishra, J. P. Longtin, R. P. Singh, V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1–11 (2004).
[CrossRef]

Sammakia, B.

B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport (Hemisphere, 1988).

Singh, R. P.

D. Mishra, J. P. Longtin, R. P. Singh, V. Prasad, “Performance evaluation of iterative tomography algorithms for incomplete projection data,” Appl. Opt. 43, 1–11 (2004).
[CrossRef]

Snyder, R.

Solitro, F.

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

Srivastava, A.

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “A schlieren study of the effect of ramp rate and rotation on convection around a crystal growing from an aqueous solution,” J. Cryst. Growth 274, 191–208 (2005).
[CrossRef]

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “Comparison of interferometry, schlieren and shadowgraph for visualizing convection around a KDP crystal,” J. Cryst. Growth 267, 348–361 (2004).
[CrossRef]

Sugiyama, M.

M. Masayuki, M. Sugiyama, T. Ogawa, “Electronic measurement of concentration gradient around a crystal growing from a solution by using Mach-Zehnder interferometer,” J. Cryst. Growth 114, 71–76 (1991).
[CrossRef]

Sunagawa, I.

K. Onuma, K. Tsukamoto, I. Sunagawa, “Role of buoyancy driven convection in aqueous solution growth: a case study of Ba(NO3)2 crystal,” J. Cryst. Growth 89, 177–188 (1988).
[CrossRef]

Sweeney, D. W.

Tiange, Z.

W. Dezhong, Z. Tiange, “The measurement of 3-D asymmetric temperature field by using real time laser interferometric tomography,” Opt. Lasers Eng. 36, 289–297 (2001).
[CrossRef]

Tsukamoto, K.

K. Onuma, K. Tsukamoto, I. Sunagawa, “Role of buoyancy driven convection in aqueous solution growth: a case study of Ba(NO3)2 crystal,” J. Cryst. Growth 89, 177–188 (1988).
[CrossRef]

van Enckevort, W. J. P.

S. Kleine, W. J. P. van Enckevort, J. Derix, “A dark field-type schlieren microscope for quantitative, in situ mapping of solute concentration profiles around growing crystals,” J. Cryst. Growth 179, 240–248 (1997).
[CrossRef]

Velarde, M. G.

M. G. Velarde, C. Normand, “Convection,” Sci. Am. 243, 79–94 (1980).

Vest, C. M.

D. W. Watt, C. M. Vest, “Turbulent flow visualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

D. W. Sweeney, C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
[CrossRef] [PubMed]

Watt, D. W.

D. W. Watt, C. M. Vest, “Turbulent flow visualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

Yang, K. T.

Y. C. Michael, K. T. Yang, “Three-dimensional Mach-Zehnder interferometric tomography of the Rayleigh-Benard problem,” ASME J. Heat Transfer 114, 622–629 (1992).
[CrossRef]

Zefiro, L.

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

Appl. Opt.

ASME J. Heat Transfer

Y. C. Michael, K. T. Yang, “Three-dimensional Mach-Zehnder interferometric tomography of the Rayleigh-Benard problem,” ASME J. Heat Transfer 114, 622–629 (1992).
[CrossRef]

Cryst. Res. Technol.

T. Inoue, K. Mori, Y. Kageyama, H. Mori, “Various growth shapes of Na2S2O3.5H2O crystals,” Cryst. Res. Technol. 35, 587–593 (2000).
[CrossRef]

Exp. Fluids

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Exp. Fluids 27, 542–550 (1999).
[CrossRef]

D. W. Watt, C. M. Vest, “Turbulent flow visualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

Fluid Dyn. Res.

D. Mishra, K. Muralidhar, P. Munshi, “Study of Rayleigh-Benard convection at intermediate Rayleigh numbers,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

J. Cryst. Growth

A. G. Notcovich, I. Braslavsky, S. G. Lipson, “Imaging fields around growing crystals,” J. Cryst. Growth 198–199, 10–16 (1999).
[CrossRef]

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “Comparison of interferometry, schlieren and shadowgraph for visualizing convection around a KDP crystal,” J. Cryst. Growth 267, 348–361 (2004).
[CrossRef]

A. Srivastava, K. Muralidhar, P. K. Panigrahi, “A schlieren study of the effect of ramp rate and rotation on convection around a crystal growing from an aqueous solution,” J. Cryst. Growth 274, 191–208 (2005).
[CrossRef]

K. Onuma, K. Tsukamoto, I. Sunagawa, “Role of buoyancy driven convection in aqueous solution growth: a case study of Ba(NO3)2 crystal,” J. Cryst. Growth 89, 177–188 (1988).
[CrossRef]

M. Masayuki, M. Sugiyama, T. Ogawa, “Electronic measurement of concentration gradient around a crystal growing from a solution by using Mach-Zehnder interferometer,” J. Cryst. Growth 114, 71–76 (1991).
[CrossRef]

S. Kleine, W. J. P. van Enckevort, J. Derix, “A dark field-type schlieren microscope for quantitative, in situ mapping of solute concentration profiles around growing crystals,” J. Cryst. Growth 179, 240–248 (1997).
[CrossRef]

NDT & E Int.

P. Munshi, “Error analysis of tomographic filters. I: Theory,” NDT & E Int. 25, 191–194 (1992).
[CrossRef]

P. Munshi, “Error analysis of tomographic filters. II: Results,” NDT & E Int. 26, 235–240 (1993).
[CrossRef]

Numer. Heat Transfer Part A

S. S. Leong, “Numerical study of Rayleigh-Benard convection in a cylinder,” Numer. Heat Transfer Part A 41, 673–683 (2002).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

W. Dezhong, Z. Tiange, “The measurement of 3-D asymmetric temperature field by using real time laser interferometric tomography,” Opt. Lasers Eng. 36, 289–297 (2001).
[CrossRef]

Opt. Lett.

Sci. Am.

M. G. Velarde, C. Normand, “Convection,” Sci. Am. 243, 79–94 (1980).

Other

B. Gebhart, Y. Jaluria, R. L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport (Hemisphere, 1988).

R. J. Goldstein, ed., Fluid Mechanics Measurements (Hemisphere, 1983).

L. Gatti, F. Solitro, F. Bedarida, P. Boccacci, G. A. Dall’aglio, L. Zefiro, “Three-dimensional measurements of concentration fields in crystal growth by multidirectional holographic interferometry,” in Laser Interferometry: Quantitative Analysis of Interferograms, Proc. SPIE1162, 126–131 (1989).

G. T. Herman, Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transport Media (Springer-Verlag, 2001).

G. T. Herman, Image Reconstruction from Projections (Academic, 1980).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

F. Mayinger, ed., Optical Measurements: Techniques and Applications (Springer-Verlag, 1994).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the four-view crystal growth chamber placed in the path of the laser beam in a Z-type schlieren setup. (1) Growing crystal on a platform, (2) growth chamber, (3) outer chamber, (4) heating element, (5) thermocouple, (6) optical window, (7) seed holder (platform configuration), (8) covering lid, (9) temperature controller unit.

Fig. 2
Fig. 2

Definition of partial projection data.

Fig. 3
Fig. 3

Buoyancy-driven convection in a differentially heated circular cavity. (a) Complete projection data in the form of isotherms for the differentially heated circular fluid layer; (b) reconstructed temperature contours at y/H = 0.65 for full (i) 100% and partial [(ii) 60%, (iii) 30%] projection data; (c) comparison of original and reconstructed nondimensional temperature distribution along the radial direction for the three different combinations of rays and views.

Fig. 4
Fig. 4

Schlieren images of the convective field above the growing crystal as recorded by turning the crystal at increments of 10°/step while keeping the growth chamber fixed. The projection data for the view angles from 70° to 120° could not be recorded because of the interference of the seed holder glass rod with the laser beam.

Fig. 5
Fig. 5

Reconstructed concentration fields at various horizontal planes above the growing crystal. The projection data in the form of two-dimensional schlieren images were recorded by turning the growing crystal at increments of 10°/step while keeping the growth chamber fixed.

Fig. 6
Fig. 6

Comparison of the reconstructed concentration field over two planes (y/H = 0.05 and 0.90) above the growing crystal with two sets of projection data. Combinations used are (a) 0°, 10°, 20°, 30°, 40°, 140°, 150°, 180° and (b) 0°, 10°, 20°, 30°, 50°, 130°, 150°, 180°. (c) Concentration profiles along the central sector in the horizontal plane for the two combinations of projection data at y/H = 0.05 and y/H = 0.90.

Fig. 7
Fig. 7

Schlieren images of the convective field around the crystal growing from its aqueous solution as recorded from four different view angles at two different time instants (2 and 30 h).

Fig. 8
Fig. 8

Concentration contours around a growing crystal in the presence of a well-defined convective plume for the four view angles at two different time instants.

Fig. 9
Fig. 9

Reconstructed concentration profiles over five horizontal planes (y/H = 0.05–0.90) above the crystal growing in the presence of a stable convection plume. Time instants of 2 and 30 h of experimental run time are presented.

Fig. 10
Fig. 10

Photograph of the grown KDP crystals in diffusion-dominated (after 70 h) and stable convection regimes (after 45 h).

Tables (1)

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Table 1 Comparison of the Original and Reconstructed Temperature Fields in Terms of Errors E1 and E2 for Buoyancy-Driven Convection in a Circular Cavity

Equations (8)

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δ = 1 n a 0 L n ( ln n ) y d z .
Δ I I k = f a k n a 0 L n y d z .
Δ I I k = f a k n y L .
N y = 9 n 2 α ( n 2 + 2 ) 2 n y .
f ( r ,     ϕ ) = 0 π - D / 2 D / 2 p [ ( s - s ) ; θ ] q ( s ) d s d θ ,
q ( s ) = R c R c R W ( R ) exp ( i 2 π R s ) d R .
W ( R ) = 0.54 + ( 1 - 0.54 ) cos ( π R R c ) .
E 1 = max ( T orig - T recon ) ( absolute maximum temperature difference ) , E 2 = 1 N ( T orig - T recon ) 2 ( rms error ) .

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