Thermal Development in Injection Mould Filling

The heat transfer in the filling stage of injection moulding is researched, determined by Gunnar Aronsson’s distance model for flow expansion ([Aronsson], 1996). The option of a thermoplastic materials model is inspired by basic physical properties, admitting temperature and pressure dependence. Two-phase, per-phase-incompressible, power-law fluids are considered. The shear rate expression considers pseudo-radial flow from a point inlet. Rather than using a finite element (FEM) solver for the momentum equations a general analytical viscosity expression is applied, adjusted to current axial temperature profiles and yielding expressions for axial velocity profile, pressure distribution, frozen layer expansion and special front convection. The nonlinear energy partial differential equation is changed into its conservative form, depicted by the internal energy, and is solved in a different way in the regions of streaming and stagnant flow, respectively. A finite difference (FD) scheme is selected using control volume discretization to keep truncation errors small in the presence of non-uniform axial node spacing. Time and pseudo-radial marching is used…

Contents: Computation of Thermal Development in Injection Mould Filling, based on the Distance Model

1 Introduction
1.1 Purpose and limitations
1.2 Method principles
1.3 Structure of the thesis
2 Injection moulding and temperature modelling
2.1 Modes of heat transfer
2.2 Temperature dependent material properties
2.2.1 Heat capacity and latent heat
2.2.2 Density and thermal conductivity
2.2.3 Viscosity
2.2.4 Dimensionless groups and asymptotic temperature profiles
2.2.5 Assumptions
2.3 The governing equations
2.3.1 General notation
2.3.2 Mass and momentum balance
2.3.3 Energy balance
2.4 Boundary conditions
2.4.1 Symmetry, points of injection and mould walls
2.4.2 Flow front
3 Model and method
3.1 Analytical sub-models
3.1.1 Vertical velocity profile
3.1.2 Pressure distribution
3.1.3 Freezing layer
3.1.4 Fountain flow
3.2 PDEs and solution method
3.2.1 General and regional melt PDEs
3.2.2 Time marching and pseudo-radial marching
3.2.3 Outer iteration: Surface of frozen layer
3.2.4 Inner iteration: Vertical temperature profile
3.2.5 Cooling PDE and its series solution
Flowchart 3.1 Data processing
Flowchart 3.2 Solution routine for active flow. Radial symmetry
Flowchart 3.3 Solution routine for passive flow
3.3 FD scheme
3.3.1 Control volume approach and truncation error
3.3.2 Convergence of inner iterations
4 Application: Circular plate
4.1 Special modelling: Radial flow
4.2 Materials data
4.3 Comparison runs
4.3.1 Pressure distribution
4.3.2 Temperature distribution
4.4 Variation of physical model
4.4.1 Latent heat of crystallization
4.4.2 Heat conductivity
4.4.3 Viscosity dependence of pressure
4.5 Method performance
4.5.1 Relations to the number of vertical levels (control volumes)
4.5.2 Wall series solution
4.5.3 Control volume at the frozen layer
5 Application: Triangular plate
5.1 Special modelling: Geometry
5.2 Materials data
5.3 Comparison runs
5.3.1 Average temperature
5.3.2 Temperature profiles
5.4 Method performance
5.4.1 Square-root parameter
5.4.2 Iteration statistics
5.4.3 Velocity profiles and residence time
6 Conclusions…

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