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

Abstract

During glass tempering, residual stresses are caused by the temperature gradient appearing between the surface and the core of the glass. It is therefore necessary to have an accurate knowledge of the heat transfer occurring during the process to obtain an accurate estimation of the residual stresses. Since glass is a semitransparent material, radiative heat transfer takes place inside the glass in addition to the conductive heat transfer. In this paper, we implemented 1‐D thermomechanical models using different methods for the computation of the radiative effects inside the glass: Rosseland approximation, P1 approximation and back ray tracing. Thermal and residual stresses results are compared to experimental results available in the literature to investigate the effects of the different methods on the heat transfer and mechanical behavior. We found that P1 approximation and back ray tracing are in good agreement with the experimental results. Finally, a numerical investigation is done by changing different parameters for which the experimental results are not available.

The radiative transfer equation with a nonhomogeneous reflectivity boundary condition coupled with the heat conduction equation

“…We give also uniform bounds on the solution in terms of bounds on the data, which are the constant ambient temperature of the surrounding dry air T a appearing in the boundary condition ( i i ) , the initial condition T 0 (·) for the absolute temperature in Ω, and the absolute temperature T S (·) versus time of the black radiative source surrounding Ω. As a consequence, we obtain also bounds on the radiative intensities (see Corollary ), which could be used in the study of the convergence of the back ray–tracing iterative method for solving our radiative transfer boundary value problems .…”

“…Many related approximate problems have been studied in the literature (see, eg, Pinnau, Hinze et al, and Agboka et al) for numerical purposes, in view to reduce the computational complexity of the exact problem , considering instead of the radiative boundary value problems for the unknown radiative intensities I k ( x , t , v ), approximate elliptic boundary value problems for the quantities of interest in , the incident radiations ( d μ denotes the area measure on V ≡ S 2 ). Indeed, these quantities, multiplied by the linear absorption coefficients κ k , are the volumic sources of heat per unit of time, appearing in the nonlinear heat conduction equation ( i ) .…”

“…Many related approximate problems have been studied in the literature (see, eg, Pinnau, 11 Hinze et al, 12 and Agboka et al 13 ) for numerical purposes, in view to reduce the computational complexity of the exact problem (1)-(2), considering instead of the radiative boundary value problems (1) for the unknown radiative intensities I k (x, t, v), approximate elliptic boundary value problems for the quantities of interest in (2), the incident radiations 6 k (., .) ∶= ∫ V I k (., ., v)d (v) (d denotes the area measure on V ≡ S 2 ).…”