Numerical prediction of phase-change heat conduction of injection-molded high density polyethylene thick-walled parts via the enthalpy transforming model with mushy zone

Polymer Engineering and Science, Sept, 2008 by Bin Yang, Xiao-Rong Fu, Wei Yang, Li Huang, Ming-Bo Yang, Jian-Min Feng

INTRODUCTION

Transient heat conduction problems involving solidification/melting are usually referred to as the "Stefan problems" or "moving-boundary problems," which were raised by Stefan (1) in his works in relation to the ice formation early in 1891. Heat transfer issues of this kind are significant in many engineering applications such as the solidification of metal and polymer melt in the molds, the freezing of foodstuffs, thermal energy storage system and so on. However, due to the very nature of the nonlinearity which is caused by the fact that the interface between the liquid and the solid is always moving with the absorption or liberation of the latent heat at the interface, only limited exact closed-form solutions are available for some simplified or idealized systems within one-dimensional infinite or semi-infinite domains (2). With the fundamental assumption that the solidification occurs at a discrete temperature, which gains wide applications in the solidification of pure small molecule substances, various methods were developed to settle these problems, such as the variational method (3), (4), the moving heat source method (5), (6), the perturbation method (7), (8) as well as the numerical methods (9-11). The numerical methods have been widely used in the solution of multidimensional phase-change heat transfer issues.

On the whole, the thermal numerical methods can be classified into two major types, i.e., the single region (continuum) approach and the multiple region/domain approach. In the first group (as referred to as the enthalpy method), the enthalpy is used as a dependent variable along with the temperature and the interface is eliminated from the calculation procedures in this model. Besides, the problem is simplified into that of nonlinear heat conduction with the absence of the phase-change process and separate phase conservation equations, which makes it possible to extend the model to multidimensional problems. While in the second class, the energy equations should be separately applied for each phase coupled with the energy balance as well as the continuity of temperature at the phase interface. Therefore, it will be necessary to determine the location of the interface, whereas both its position and shape vary with time (12). However, due to the fact that there is no need to explicitly track the movement of the solid-liquid interface during the phase change process, the enthalpy formulation has been widely adopted in solving these issues (13-16).

Injection molding is a common plastics manufacturing technique, accounting for 1/3 of all the processed plastics. The computer-aided engineering (CAE) technology has been proved a powerful tool for production engineers to qualify their product designs, material selection, optimization of processing parameters and so on. As the quality and performance of the final injection molded parts depend largely on the processing conditions during the real molding process, great attention has been paid to investigate the mathematical models and the computer simulations during polymer injection molding process since the end of 1980s.

Generally speaking, most of the previous researches were dedicated to the modeling and simulation of the flow field analysis (17), (18), the nonisothermal crystallization kinetics and the microstructure development (19-21) (e.g. birefringence, macromolecular orientation, mass transport, etc.) as well as the product defect predictions (22-24) (e.g. residual stress, shrinkage, warpage, sink marks, etc.) during the cavity-filling or post-filling/packing stages of the injection molding process. Actually, for industrial purposes, the optimization of the injection molding process should also be concentrated on the cooling stage, which accounts for the largest part of the total injection molding cycle (especially true for the molded parts with thick walls). Moreover, the morphology and properties of the parts are significantly affected by a number of factors during the cooling stage (25). Thus, investigations on the solidification/crystallization phenomena under real cooling conditions could supply a better insight into the optimization of thermal or pressure history during injection molding as well as the studies of microstructure evolution (26).

However, the simulation of the cooling stage in injection molding from a transient heat transfer viewpoint is relatively scarcely presented. The literature concerning the cooling analysis could be categorized into two general classes: one class aimed at the evaluations and improvements on the cooling system designs as well as optimizing the mold cooling operations (27-30). The other type mainly dealt with the modeling/simulation of the cooling stage via numerical techniques, such as the finite-element method (FEM), the finite-difference method (FDM), the boundary element method (BEM), etc., usually coupled with the control-volume method (CVM) for the region discretization, among which various CAE softwares or integral packages (C-Mold, Moldflow, COOL-3D, etc.) were widely adopted for detailed calculation and description of the molding cycle (31-35). However, it is necessary to point out that in most cases, the injection molding CAE softwares including cavity filling, packing and mold cooling show perfect simulation results for the thin-walled parts, especially when the generalized Hele-Shaw flow formulation applies in the cavity filling computation. It should be mentioned that from the viewpoint of flow and heat transfer simulations, whether a plastic part is thick-walled or thin-walled is always in close relation to the shape and dimensions of its cross-section, say, when the ratio of width/height (W/H) is larger than 10 for a rectangular cavity and the edge effects along the width direction can thus be negligible, the part has the characteristics of a two-dimensional object and can be considered to be thin-walled. Most of the above mentioned approaches, except for some simple empirical formulas, include the solution of the transient heat conduction equation in one or multidimensions with or without phase changes (for crystalline or amorphous polymers, respectively) for the part and/or the transient heat conduction equation without phase changes for the mold. As for the simulations for injection moldings of articles with complex, geometrical configurations, the discretization and computation are rather complicated since too many influencing factors should be taken into account simultaneously. As far as some existing simplified or empirical/semi-analytical formulas (36), (37) were concerned, only specific types of cooling process were discussed. Thus, a more generalized model is needed to interpret the cooling process in injection molding with some indispensable parameters, especially for the theoretical cooling estimation of the thick-walled parts (i.e., the W/H [much less than] 10). Besides, the mushy zone existing during the solidification process of polymer melt has been rarely studied. Actually, as a result of the polydispersity and hierarchical structures, polymers do not solidify at a discrete temperature as most existing models have assumed but over a temperature range. In this sense, valuable insights into the issues of the mushy region (with polymers regarded as "mixture") can be gained from works that has been done in other fields such as the solidification/crystallization analysis in the alloy researches (38), (39).