Computation of Thermal Development in Injection Mould Filling, based on the Distance Model
Abstract: The heat transfer in the filling phase of injection moulding is studied, based on Gunnar Aronsson’s distance model for flow expansion ([Aronsson], 1996).The choice of a thermoplastic materials model is motivated by general physical properties, admitting temperature and pressure dependence. Two-phase, per-phase-incompressible, power-law fluids are considered. The shear rate expression takes into account pseudo-radial flow from a point inlet.Instead of using a finite element (FEM) solver for the momentum equations a general analytical viscosity expression is used, 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 transformed into its conservative form, expressed by the internal energy, and is solved differently in the regions of streaming and stagnant flow, respectively. A finite difference (FD) scheme is chosen 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. A local system of nonlinear FD equations is solved. In an outer iterative procedure the position of the boundary between the “solid” and “liquid” fluid cavity parts is determined. The uniqueness of the solution is claimed. In an inner iterative procedure the axial node temperatures are found. For all physically realistic material properties the convergence is proved. In particular the assumptions needed for the Newton-Mysovskii theorem are secured. The metal mould PDE is locally solved by a series expansion. For particular material properties the same technique can be applied to the “solid” fluid.In the circular plate application, comparisons with the commercial FEM-FD program Moldflow (Mfl) are made, on two Mfl-database materials, for which model parameters are estimated/adjusted. The resulting time evolutions of pressures and temperatures are analysed, as well as the radial and axial profiles of temperature and frozen layer. The greatest differences occur at the flow front, where Mfl neglects axial heat convection. The effects of using more and more complex material models are also investigated. Our method performance is reported.In the polygonal star-shaped plate application a geometric cavity model is developed. Comparison runs with the commercial FEM-FD program Cadmould (Cmd) are performed, on two Cmd-database materials, in an equilateral triangular mould cavity, and materials model parameters are estimated/adjusted. The resulting average temperatures at the end of filling are compared, on rays of different angular deviation from the closest corner ray and on different concentric circles, using angular and axial (cavity-halves) symmetry. The greatest differences occur in narrow flow sectors, fatal for our 2D model for a material with non-realistic viscosity model. We present some colour plots, e.g. for the residence time.The classical square-root increase by time of the frozen layer is used for extrapolation. It may also be part of the front model in the initial collision with the cold metal mould. An extension of the model is found which describes the radial profile of the frozen layer in the circular plate application accurately also close to the inlet.The well-posedness of the corresponding linearized problem is studied, as well as the stability of the linearized FD-scheme.
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