Abstract

We present an efficient approach to determine the axial temperature profile in tube furnaces by analyzing the diameter profile after stretching from a silica glass rod. Given the temporal load for stretching, the effective axial temperature profile can be deduced. This approach neglects diameter change due to surface tension by using rods. It also considers homogeneous cross sectional temperatures as effective axial temperatures. Since the effective temperature is derived from the rod’s viscous behavior, it yields optimized temperature profiles for heat treatment with glass rods, such as all-solid fiber drawing. This provides better temperature profiles than using thermal couples, because they are measured by the glass rods from the working material. Silica glass was used in this approach for its well-studied viscosity-temperature relationship. The derived method was carried out in an inductive graphite furnace under argon flow at 1400 °C and 1800 °C, respectively. The results will be validated by the mean of saphire fiber Bragg grating sensor measurements. Finally, the problems of the stretched axis and spatial resolution will be discussed.

© 2015 Optical Society of America

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References

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  1. Laboratory Tube furnaces, (Carbolite, 2015), http://www.carbolite.com/products/furnaces/laboratory-tube-furnaces/ .
  2. J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
    [Crossref]
  3. J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
    [Crossref]
  4. K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
  5. Y. Nakayama and R. F. Boucher, Introduction to Fluid Mechanics (Elsevier, 2002), Chap. 4.
  6. R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
    [Crossref]
  7. M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
    [Crossref]
  8. J. R. Davis, Tensile Testing (ASM international, 2004).
  9. Y. Ling, “Uniaxial true stress-strain after necking,” in AMP Journal of Technology 5. (AMP incorporated, 1996), pp. 37–48.
  10. P. W. Bridgman, Studies in Large Plastic Flow and Fracture (McGraw-Hill, 1952).
  11. T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
    [Crossref]
  12. G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
    [Crossref]
  13. D. H. Johnson, “Signal-to-noise ratio,” http://www.scholarpedia.org/article/Signal-to-noise_ratio .
    [Crossref]
  14. E. L. Bourhis, Glass Mechanics and Technology (Wiley-VCH, 2007).
  15. Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
    [Crossref]

2014 (2)

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

2013 (1)

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

2010 (1)

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

2007 (1)

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

2002 (1)

R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
[Crossref]

1986 (1)

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

1982 (1)

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Aichele, C.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Bartelt, H.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Bierlich, J.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Bottinga, Y.

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Buchak, P.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Couderc, V.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Crowdy, D. G.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Doremus, R. H.

R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
[Crossref]

Ebendorff-Heidepriem, H.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Elsmann, T.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Funke, A.

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

Graf, A.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Grimm, S.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Habisreuther, T.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Hand, R. J.

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Jamier, R.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Kirchhof, J.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

Kobelke, J.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Labruyere, A.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Leproux, P.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Lindner, F.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Litzkendorf, D.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Ojovan, M. I.

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Pan, Z.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Pißler, H.-J.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Richet, P.

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Rothhardt, M.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Schmidt, M. A.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Schuster, K.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Schwuchow, A.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Stokes, Y. M.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Tombelaine, V.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Travis, K. P.

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Unger, S.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Urbain, G.

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Willsch, R.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Wondraczek, K.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Adv. Opt. Technol. (1)

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Cryst. Res. Technol. (1)

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

Geochim. Cosmochim. Acta (1)

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

J. Appl. Phys. (1)

R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
[Crossref]

J. Fluid Mech. (1)

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

J. Phys. Condens. Matter (1)

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Opt. Mater. (1)

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Proc. SPIE (1)

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Other (7)

D. H. Johnson, “Signal-to-noise ratio,” http://www.scholarpedia.org/article/Signal-to-noise_ratio .
[Crossref]

E. L. Bourhis, Glass Mechanics and Technology (Wiley-VCH, 2007).

Laboratory Tube furnaces, (Carbolite, 2015), http://www.carbolite.com/products/furnaces/laboratory-tube-furnaces/ .

Y. Nakayama and R. F. Boucher, Introduction to Fluid Mechanics (Elsevier, 2002), Chap. 4.

J. R. Davis, Tensile Testing (ASM international, 2004).

Y. Ling, “Uniaxial true stress-strain after necking,” in AMP Journal of Technology 5. (AMP incorporated, 1996), pp. 37–48.

P. W. Bridgman, Studies in Large Plastic Flow and Fracture (McGraw-Hill, 1952).

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Figures (12)

Fig. 1
Fig. 1 Schema of viscous stretching in temperature profiles with different length of gage section.
Fig. 2
Fig. 2 Stress-strain relationship in uniaxial tensile stretching for viscoelastic material.
Fig. 3
Fig. 3 Schema of glass rod stretched from one side at normal temperature distribution T(x).
Fig. 4
Fig. 4 Schema of a cylindrical rod after stretching with localized necking has occurred. The axial position at x is shifted to x’ after stretching. The volume determined by position x before stretching (light gray) is the same as that determined by x’ after stretching (dark gray) for incompressible flow.
Fig. 5
Fig. 5 Schema of the experimental setup for demonstration of the rod stretching method for temperature profile measurement.
Fig. 6
Fig. 6 Diameter profile of a 1 mm silicate glass rod before and after stretching at 1400°C in a gage section of about 80 mm.
Fig. 7
Fig. 7 Load-time curve and its integration over the plastic strain area (shaded area) at 1400 °C at 1 mm/min. The bright area under the curve can be attributed to the elasitc contribution at the initial stage of stretching.
Fig. 8
Fig. 8 Stress-strain data from the necking method with silica glass rod specimen of 9 mm in diameter at different temperatures.
Fig. 9
Fig. 9 Comparison of temperature profile measured via the viscous stretching method (circle) using a 1 mm silica glass rod and scanning with an FBG sensor (triangle), respectively.
Fig. 10
Fig. 10 Spatial resolution of viscous stretching measurement at 1400°C with fixed furnace setup and moving furnace setup (half the stretching rate).
Fig. 11
Fig. 11 Temperature profile measured via the viscous stretching method using a 9 mm silica glass rod at 1800°C for the inductive tube furnace.
Fig. 12
Fig. 12 Spatial resolution of viscous stretching at 1800°C with large total strain.

Tables (1)

Tables Icon

Table 1 Fitting parameters for viscosity-temperature relationship for silica glass in Arrhenius form.

Equations (17)

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

σ = 3 η d ε d t
ε ˙ = σ ( x , t ) 3 η ( T ( x , t ) )
σ ( x , t ) = F A ( x , t )
ε ˙ ( x , t ) = 1 L d L d t = 1 A ( x , t ) d A ( x , t ) d t
σ ( x , t ) = F A 0
ε ˙ ( x , t ) = Δ L L 0
d A ( x , t ) d t = F ( t ) 3 η ( T ( x , t ) )
A 0 A ( x , t ) = 0 t F ( t ) 3 η ( T ( x , t ) )
T ( x , t ) T ( x )
A 0 A ( x , t ) = M 3 η ( T ( x ) )
η ( T ( x ) ) = C exp ( B / R T ( x ) )
η ( T ( x ) ) = D T ( x ) ( 1 + E exp ( L / R T ( x ) ) ) ( 1 + F exp ( H / R T ( x ) ) )
f ( x ) = A 0 A ( x , t ) = M 3 C exp ( B / R T ( x ) )
T ( x ) = B R ( ln ( M / 3 C ) ln f ( x ) )
ε ( t ) = Δ L ( t ) L 0 = v x t L 0
V p = x A 0 = 0 x ´ A ( x , t ) d x
ε ( x , t ) = ln D 0 D 90 D 0 ( x , t ) D 90 ( x , t )

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