### Two-dimensional heat transfer and thermal stress analysis in float glass processing

Ines Busuladzic, Dr. Clemons, Dr. Golovaty, and Dr. Young

Two-dimensional heat transfer in a float glass process is considered. Specifically, we consider the processing region where the glass is floating on a tin layer and the temperature of the cooling air above the glass sheet is controllable. Two float glass systems, a one-layer and a multi-layer system, are considered. The one layer system consists of a pure glass layer. The multi-layer system consists of three layers. Here, a contaminated layer of glass is placed between two pure layers to simulate possible accumulation of contaminants in the glass during processing. For both systems we solve the governing heat equations with boundary conditions. For each system an asymptotic analysis is performed. The small parameter is the ratio of the glass thickness to length. Further, the heat transfer across the upper and lower surfaces of the flat sheet of glass is taken to be small. The asymptotic analysis results in a simpler heat transfer model that is solved numerically.

Once the temperature distribution in the glass sheet is determined, a two-dimensional analysis of the thermal stresses through the thickness and length of the sheet is considered. The thermal stresses are expressed as the particular solution of the thermoelastic displacement potential. This potential is set up analytically, but solved for numerically. Once the thermal stresses are determined, we select the temperature of the cooling air to minimize the thermal stresses in order to possibly reduce crack propagation through the glass. For this optimization problem we apply constraints to the temperature of the air so that the temperature of the glass ranges between 1100 and 600 K, from the beginning to the end of the furnace, respectively.