Two-dimensional finite element modeling of glass forming and tempering processes, including radiative effects

Abstract

The entire glass sheet forming process consists of heating and forming a glass sheet and cooling and tempering it afterwards. For the first step, the glass sheet is heated using a local radiative source and deforms by sagging. In the thermo-mechanical calculations, temperature dependent glass viscosity, heat exchange with the ambient air and radiative source effects should be considered. A two-dimensional finite element model with plane deformation assumptions is developed. Using the PI-Approximation, the formulation and numerical resolution of the Radiative Transfer Equation (RTE) are performed on the glass domain as it changes over time to estimate the flux of the radiative body at each position in the glass. In the next step, the sheet is cooled. Narayanaswamy's model is used to describe the temperature dependent stress relaxation and the structural relaxation. The RTE is again solved using the P1-Approximation to consider the internal radiative effects during the cooling. There is a discussion using the P1-Approximation and comparing the results to other existing methods for the temperature changes of the glass throughout the forming process, for the deformed shape at the end of the forming step and for the residual stresses after tempering

Influence of radiative heat transfer model on the computation of residual stresses in glass tempering process

“…Combining Equations 20 and 21 the radiative transfer equation for the P1 approximation method in axisymmetric coordinate system is given as 7,22 :…”

“…The first one is the convection with the surrounding air at temperature T ∞ and α is the convection coefficient. The second term is the radiative heat flux on the glass surface in the opaque wavelength region …”

“…Combining Equations and the radiative transfer equation for the P1 approximation method in axisymmetric coordinate system is given as: …”

“…The second term is the radiative heat flux on the glass surface in the opaque wavelength region. 17,22,23,25 Equations 3 are the symmetry conditions imposing a null flux on the revolution axis at r = 0 and in the middle plane of the disk at z = 0. To have a 1-D problem, a null flux is imposed on the cylindrical surface at r = R (Equation 4).…”

“…In this paper, we investigated the effects of thermal radiation models on the temperature evolution and the residual stresses. Thermomechanical models with radiative and conductive heat transfer were developed using the following methods to solve the RTE: Rosseland approximation, P1 approximation, and back ray tracing to solve radiative transfer equation. In the first section, an axisymmetric thermomechanical model will be exposed for these different methods with boundary conditions leading to a 1‐D problem.…”