Mathematical modelling of glass flow during a pressing operation
The aim of this project was to develop a mathematical model of the behaviour of molten glass during a pressing (hot forming) operation. An outline of glass manufacturing is given in chapter 1, together with a discussion of the factors influencing the behaviour of the molten glass and the advantages of a mathematical model over direct experimentation. Chapter 2 introduces the mathematical description of the glass behaviour. The molten glass was modelled by an imcompressible Newtonian liquid undergoing slow flow, it was also assumed that the finished article would be axisymmetric. The governing equations and boundary conditions were cast into the appropriate non-dimensional form. Ways of solving the equations are considered in chapter 3. Initially the analytical solutions to simplified forms of the equations were considered, but these proved inadequate. Therefore numerical methods were used. An outline of the finite element method is given before details of its application to this problem. The results of using the finite element method to solve the isothermal flow equations are presented in chapter 4. The model was able to cope with a range of parameters, though numerical instablities manifested themselves at low Reynolds numbers. Viscosity is strongly temperature dependent, hence the flow of heat in and around the glass is important. Temperature variations were introduced into the model in chapter 5. In molten glass a thin cooled 'skin' is formed, this physical phenomenon was exploited and an alternative boundary condition which encapsulated this effect was developed. Results from the combined model are given in chapter 6. The predicted behaviour of the liquid is qualitatively correct. Physical parameters for 'real life' glass forming operations are collated in chapter 7. Consideration is also given to the actual computing power needed to fully model a pressing operation. Chapter 8 gives an overview of the work and includes suggestions for further study.