Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-19T06:22:39.912Z Has data issue: false hasContentIssue false

Edge behaviour in the glass sheet redraw process

Published online by Cambridge University Press:  17 November 2015

D. O’Kiely*
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
C. J. W. Breward
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
I. M. Griffiths
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
P. D. Howell
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
U. Lange
Affiliation:
Schott AG, Hattenbergstrasse 10, 55122 Mainz, Germany
*
Email address for correspondence: okiely@maths.ox.ac.uk

Abstract

Thin glass sheets may be manufactured using a two-part process in which a sheet is first cast and then subsequently reheated and drawn to a required thickness. The latter redrawing process typically results in a sheet with non-uniform thickness and with smaller width than the cast glass block. Experiments suggest that the loss of width can be minimized and the non-uniformities can be essentially confined to thickening at the sheet edges if the heater zone through which the glass is drawn is made very short. We present a three-dimensional mathematical model for the redraw process and consider the limits in which (i) the heater zone is short compared with the sheet width, and (ii) the sheet thickness is small compared with both of these length scales. We show that, in the majority of the sheet, the properties vary only in the direction of drawing and the sheet motion is one-dimensional, with two-dimensional behaviour and the corresponding thick edges confined to boundary layers at the sheet extremities. We present numerical solutions to this boundary-layer problem and demonstrate good agreement with experiment, as well as with numerical solutions to the full three-dimensional problem. We show that the final thickness at the sheet edge scales with the inverse square root of the draw ratio, and explore the effect of tapering of the ends to identify a shape for the initial preform that results in a uniform rectangular final product.

Type
Papers
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ansys Inc. Polyflow, 2013 Release 15.0.Google Scholar
Beaulne, M. & Mitsoulis, E. 1999 Numerical simulation of the film casting process. Int. Polym. Proc. 14 (3), 261275.Google Scholar
Buellesfeld, F., Lange, U., Biertuempfel, R., Pudlo, L. & Jung, H.2014 Method for production of glass components. US Patent 20,140,342,120.Google Scholar
Debbaut, B., Marchal, J. M. & Crochet, M. J. 1995 Viscoelastic effects in film casting. In Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, pp. 679698. Springer.Google Scholar
Dewynne, J., Ockendon, J. R. & Wilmott, P. 1989 On a mathematical model for fiber tapering. SIAM J. Appl. Maths 49 (4), 983990.Google Scholar
Dobroth, T. & Erwin, L. 1986 Causes of edge beads in cast films. Polym. Engng Sci. 26 (7), 462467.Google Scholar
Filippov, A. & Zheng, Z. 2010 Dynamics and shape instability of thin viscous sheets. Phys. Fluids 22 (2), 023601.CrossRefGoogle Scholar
Fortin, M. 1981 Old and new finite elements for incompressible flows. Intl J. Numer. Meth. Fluids 1 (4), 347364.CrossRefGoogle Scholar
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181206.CrossRefGoogle Scholar
d’Halewyu, S., Agassant, J. F. & Demay, Y. 1990 Numerical simulation of the cast film process. Polym. Engng Sci. 30 (6), 335340.Google Scholar
Howell, P. D.1994 Extensional thin layer flows. PhD thesis, University of Oxford.Google Scholar
Howell, P. D. 1996 Models for thin viscous sheets. Eur. J. Appl. Maths 7 (4), 321343.Google Scholar
Logg, A. & Wells, G. N. 2010 DOLFIN: automated finite element computing. ACM Trans. Math. Softw. 37 (2), 20:1–20:28.Google Scholar
Logg, A., Wells, G. N. & Hake, J. 2012 DOLFIN: A C++/Python finite element library. In Automated Solution of Differential Equations by the Finite Element Method, pp. 173225. Springer.CrossRefGoogle Scholar
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline. Steady-state isothermal viscous flows. Ind. Engng Chem. Fundam. 8 (3), 512520.CrossRefGoogle Scholar
Modest, M. F. 2013 Radiative Heat Transfer. Academic.Google Scholar
Pearson, J. R. A. & Matovich, M. A. 1969 Spinning a molten threadline. Stability. Ind. Engng Chem. Fundam. 8 (4), 605609.Google Scholar
Scheid, B., Quiligotti, S., Tran, B., Gy, R. & Stone, H. A. 2009 On the (de)stabilization of draw resonance due to cooling. J. Fluid Mech. 636, 155176.CrossRefGoogle Scholar
Shah, Y. T. & Pearson, J. R. A. 1972 On the stability of nonisothermal fiber spinning. Ind. Engng Chem. Fundam. 11 (2), 145149.CrossRefGoogle Scholar
Silagy, D., Demay, Y. & Agassant, J. F. 1999 Numerical simulation of the film casting process. Intl J. Numer. Meth. Fluids 30 (1), 118.Google Scholar
Smith, S. & Stolle, D. 2000 Nonisothermal two-dimensional film casting of a viscous polymer. Polym. Engng Sci. 40 (8), 18701877.CrossRefGoogle Scholar
Taroni, M., Breward, C. J. W., Cummings, L. J. & Griffiths, I. M. 2012 Asymptotic solutions of glass temperature profiles during steady optical fibre drawing. J. Engng Maths 80, 120.CrossRefGoogle Scholar
Trouton, F. T. 1906 On the coefficient of viscous traction and its relation to that of viscosity. Proc. R. Soc. Lond. A 77 (519), 426440.Google Scholar