Modelling of the blowing stage of the glass container forming process Christina G. Giannopapa

Overview Introduction Blowing model requirements Level-set methods Results Conclusions

From Craft to Mathematics

Glass Manufacturing Process Melting Forming Conditioning TreatmentAutomatic inspection Packaging

Glass container forming process Pressing step Blowing step

Motivation Minimise unwanted variations in wall thickness of containers Maintain stregth Reduce weight Optimise cooling conditions Increase production speed

Blowing Model Requirements Features –Free surface flow –Flow with a high temperature-dependent viscosity Input (e.g. from TNO Pressing Model or CASA Press Model) –Glass preform shape –Temperature –Air pressure –Cooling of moulds Output –Container product shape –Product thickness distribution –Temperature distribution in Product Mould –Stresses and thermal deformations

Blowing Model Incompressible Stokes flow Temperature-dependent viscosity for glass Pressure prescribed at inflow boundaries No slip for glass at mould walls Free stress for air at mould walls Energy equation for temperature Convection-diffusion equation Prescribed temperature at inflow boundaries Heat transfer coefficients at mould walls Two level set functions Glass-air interface

Level Set Method Equations for level set functions advancement Convection equations Integrated for time step Re-initialisation of level set functions Re-initialisation of L1 and L2 Each time, or only after several time steps, to establish as much as possible, without changing positions of interfaces But difficult to accomplish!!

Reinitialisation FMM Algorithm Starts with computed levels set function for time step Distinguishes between three type of nodes Red ones: Have an updated value Purple ones: Have already a trial value Yellow ones: Have not been considered

Reinitialisation FMM Algorithm First all nodes next to the interface turn red Get the computed values of L, slightly adapted such that Interface position does not change Length of gradients as close to 1 as possible within elements

Reinitialisation FMM Algorithm Second all nodes connected to red ones get trial Turn to purple Here one uses as much as possible that Length of gradients as close to 1 as possible within elements

Reinitialisation FMM Algorithm Iterative: Purple node with lowest absolute value Turns to red Connected purple and yellow nodes Turn to purple when yellow Receive an (updated) trial value Again: Length of gradients as close to 1 as possible within elements

TNO Blowing Model Example1: Test example for fine-tuning

TNO Blowing Model Mass losses < 5% in general depend on time step for fixed mesh size

TNO Blowing Model Temperature dependent viscosity

TNO Blowing Model Example2: Axi-symmetric blowing of bottle Prescribed inflow pressure on top Air is allowed to “flow out” No flow of glass when in contact with boundary Small layer of air on top of domain

TNO Blowing Model Example2: Axi-symmetric blowing of bottle Results for different meshes

Final remarks and discussion TNO Blow Model applies level set techniques for surface tracking Algorithms currently applied in 2D axi symmetric problems So far reasonable results –Although some fine-tuning left Next months focus on: –Speed-up –Efficiency –Enhanced (re-) initialisation –Extension to 3D