Flow analysis network (FAN)—A method for solving flow problems in polymer processing

A finite element method is proposed for solving two dimensional flow problems in complex geometrical configurations commonly encountered in polymer processing. The method is applicable to flow in relatively narrow gaps of variable thickness and any desired shape. It was developed for analyzing flow in injection molding dies and certain extrusion dies. The fluid can be any non-Newtonian fluid which is incompressible, inelastic, and time independent. The flow field is divided into an Eulerian mesh of cells. Around each node, located at the center of the cell, a local flow analysis is made. The analysis around all nodes results in a set of linear algebraic equations with the pressures at the nodes as unknowns. The simultaneous solution of these equations results in the required pressure distribution, from which the flow rate distribution is obtained. Solution for the isothermal Newtonian flow problem is obtained by a one-time solution of the equations, whereas solution of a non-Newtonian problem requires iterative solution of the equations.     

An interface‐update‐based implicit algorithm for mold filling simulation of liquid composite molding

In this article, a new implicit numerical algorithm has been presented for mold filling simulation in liquid composite molding process. The new implicit numerical algorithm is based on the update of flow interface to track flow front for each time step. Nodes of mesh are divided into three groups, i.e. filled nodes, interface nodes (or partially filled nodes), and empty nodes. Governing equations of filled nodes are solved to obtain pressure distribution; filling fractions of interface nodes are checked to determine if any node is needed to be added to filled nodes or be removed from filled nodes. The local LU factorization solver and preconditioned conjugated gradient iterative solver were employed to investigate the new implicit algorithm. Case studies demonstrated the performance of the proposed implicit algorithm. The new implicit algorithm reduced the computational complexity to below 2.0 power order of problem sizes. POLYM. COMPOS., 27:271–281, 2006. © 2006 Society of Plastics Engineers.     

A finite element method for flow in compression molding of thin and thick parts

Based on Barone and Caulk's model and a generalized variational functional, a finite element simulation was developed for the compression molding of thin and thick parts. For solving the u-v-p type equations, an element-based penalty method and a mixed formulation were implemented. Numerical results show that the new model gives better accuracy in velocity and velocity gradient than the Hele-Shaw formulation for cases where both models are appropriate. Predictions of velocity and its gradient by the model are compared with other FEM results and BEM solutions. Using a fixed base mesh that covers the mold cavity, a new technique was developed for tracking the moving flow front. Temporary elements and nodes are generated for the filled part of elements intersected by the flow front. This method allows a smooth representation of the flow front and has exact boundary conditions on the flow front. The scheme is demonstrated for compression molding of an elliptical and an L-shaped charge.     

Finite element analysis for wavelike flow marks in injection molding

Wavelike flow marks are a kind of surface defect that can arise during the filling stage of the injection molding process. In this study, we performed a numerical analysis using a finite element method to predict the conditions under which flow marks are generated. To simplify the analysis, a two dimensional flow through a channel between two parallel plates was considered. The viscosity of the polymer melt was modeled by the Cross‐WLF equation. For the finite element analysis, a velocity–pressure formulation was used to simultaneously solve the continuity and momentum equations. The calculation domain for the numerical analysis keeps changing with time due to the advancing melt front. To handle the free surface more accurately, a moving grid method was employed. A numerical mesh was generated at each time step using an automatic mesh generation scheme. An analytical model was developed to correlate the effects of process variables to the flow mark geometry. Results of the numerical analysis were compared with the available experimental data. The estimated geometry of the flow marks were in good qualitative agreement with experimental observations. Parametric studies have been performed to examine the effects of various processing conditions and the material properties on flow mark size. POLYM. ENG. SCI., 47:922–933, 2007. © 2007 Society of Plastics Engineers     

三维熔体前沿界面的Level Set追踪

给出三维Level Set方程,采用五阶加权本质无振荡格式进行空间离散,通过算例验证了该算法的正确性及追踪三维运动界面的准确性。进而将Level Set算法和同位网格有限体积法进行耦合,模拟了注塑成型充填阶段的三维流动过程,准确追踪到了不同时刻熔体前沿界面,预测并分析了流动过程中不同时刻的压力、速度等重要流动特征。数值结果表明,该方法可追踪三维熔体前沿界面,预测充填过程中的重要流动特征。     

注塑成型充填过程数值模拟的隐式算法

应用有限元 /控制体积方法 ,基于熔体在控制体积内的质量守恒 ,以结点压力和充填分数为未知量 ,建立了熔体在充填阶段流动的控制方程 ,并对方程进行隐式求解 ,完成了对注塑成型充填过程的数值模拟 .算例分析证明算法是正确的     

塑料注射充模过程流动分析的隐式控制体积法

为了改善充模过程流动分析隐式方法的迭代过程,提出了一种基于界面更新的隐式方法。该方法只对已注满结点进行求解,然后更新界面结点的体积函数以便决定是否要更新界面。计算表明了隐式方法性能良好,计算速度比显式方法有量级程度的改进。     

Modeling fountain flow and filling front shape in reaction injection molding

A numerical model of the reaction injection molding process was developed to test front shape and flow approximations employed in previous models. The model was two-dimensional and simulated the flow, reaction, and heat transfer in the typically long axial dimension and the typically small thickness dimension of a mold. The filling front shape and the velocity profiles in the filling fluid were determined by numerical solution of the momentum equation with the appropriate stress boundary conditions using the method of Patankar (1980). The predicted temperature and conversion results agreed with calculations assuming that the front was flat perpendicular to the flow and that a parabolic velocity profile existed behind the fountain flow region at the front. Thus, simple assumptions about front shape and velocity in the thin dimension of a reaction injection mold can be employed without significant loss of accuracy in modeling reaction injection molding.     

Decoupled Gating and Simulation for Injection Molding

Polymer process control is limited by a lack of observability and controllability of the state of the polymer melt. A new polymer processing system is described that utilizes multiple self-regulating valves to independently and dynamically control the melt pressure at multiple locations in a hot runner injection mold. Concurrently, multiple process simulations are executed in parallel to analyze the flow rates, given the pressure drops between the outlets of self-regulating valves and each mold cavity. The developed simulation utilizes a hybrid finite difference and finite element scheme to simultaneously solve the mass, momentum, and heat equations including juncture losses according to a Cogswell model. Numerical verification indicates that the flow rate predictions of the described simulations compare well with the results from a commercial mold-filling simulation. However, empirical validation indicates that the process simulation is qualitatively useful, but does not yet possess sufficient accuracy for quantitative process and quality control. The most significant sources of variance were the calibration of process data and the modeling of the polymer rheology.     

Three‐dimensional simulation of multi‐material injection molding: Application to gas‐assisted and co‐injection molding

This paper presents an overview of the results obtained at the Industrial Materials Institute (IMI) on the numerical simulation of the gas‐assisted injection molding and co‐injection molding. For this work, the IMI's three‐dimensional (3D) finite element flow analysis code was used. Non‐Newtonian, non‐isothermal flow solutions are obtained by solving the momentum, mass and energy equations. Two additional transport equations are solved to track polymer/air and skin/core materials interfaces. Solutions are shown for different thin parts and then for thick three‐dimensional geometries. Different operating conditions are considered and the influence of various processing parameters is analyzed.     

Simulation of reacting flow during filling in reaction injection molding (RIM)

This paper deals with computer simulation of the filling stage of the Reaction Injection Molding (RIM) process for cavities of rectangular, cylindrical, and disc shapes. The computer model is in two parts: the main flow and the flow by the moving front. In the main flow part, the transient equations of axial momentum, energy and species conservation and also the continuity equation are solved numerically by finite-difference methods using a moving, changing mesh. In the flow front part, which is quite novel, the transient (parabolic) vorticity, energy and species conservation equations and the elliptic streamfunction equation are again solved by finite-difference methods. An important feature of both parts is that convection along and across the flow is included in all the transient equations. Results are presented for all three cavity shapes and those for rectangular cavities are compared with the experimental results of previous investigators.     

Sensitivity analysis on resin transfer molding processes with edge effect and curing reaction characteristics

The mold filling time and resin flow front shape are of fundamental importance during resin transfer molding (RTM) processes, because the former influences productivity and the latter affects composites quality. In this article, considering both edge effect and curing reaction characteristics of the resin flow process, the sensitivity analysis method is introduced to investigate the sensitive degree of mold filling time and resin flow front shape to the key material and processing parameters. The function employed to describe the resin flow front shape is defined, and the mathematical relationships of the key physical parameters, such as fluid pressure sensitivity, flow velocity sensitivity, mold filling time sensitivity, and resin flow front shape sensitivity, are established simultaneously. In addition, then the resin infiltration process is simulated by means of a semi‐implicit iterative calculation method and the finite volume method. The simulated results are in agreement with the analytical ones. The results show that under constant injection velocity conditions, both the change in the resin temperature and the alteration of the inlet velocity hardly affect the resin flow front shape, whereas the influence of edge permeability on the resin flow front shape is the greatest. This study is helpful for designing and optimizing RTM processes. POLYM. COMPOS., 36:2008–2016, 2015. © 2014 Society of Plastics Engineer     

A numerical simulation of the cavity filling process with PVC in injection molding

Filling cold mold cavities with hot polymer melts at high pressures is of great practical interest. The transport approach to this process of solving the general equations of change with suitable equations of state to describe the flowing material has been largely ignored. No analytic solution is possible, and the non-steady state flow adds a dimension which makes digital computation discouraging because of the core storage and execution time requirements. The mold filled in this simulation is a disk which hot polymer melt enters through a tubular entrance located at the center of the top plate. The tube is 2.54 cm. long and has a radius of 0.24 cm. The plate separation and outer radius of the disk cavity may be varied. A constant pressure applied at the entrance of the tube causes the flow. The cavity walls are kept at various low temperatures. The reported results are for rigid polyvinyl chloride (PVC). The general transport equations, i.e. continuity, momentum, and energy, for a constant density power law fluid are used to solve the flow problem. Convergence to the differential solutions is guaranteed but since a lower limit was imposed on the time increment by the core storage limit of the computer facilities (27K) and long execution times, all results are semiquantitative for the problem as stated. Using the results obtained it is possible to predict “fill times”. The formation of a frozen polymer skin as the cavity fills may be followed via the velocity profiles. The temperature profiles which reflect cooling and the amount of viscous heat generated provide the basis for studying resin thermal degradation effects. Finally, because so much of the total pressure drop is disispated in the entrance tube, and so much viscous heat is generated there, this study indicates that the design of the gate and runner system is perhaps the most important facet of success in mold filling.     

Numerical simulation and experimental verification of the filling stage in injection molding

The linear low‐density polyethylene melt is described by the modified Cross model, the dependence of melt viscosity on temperature incorporated with the Arrhenius equation, and the Moldflow second‐order model in this investigation. The mass, momentum conservation, and constitutive equations are discretized and solved by using the iterative stabilized fractional step algorithm along with the Crank–Nicolson implicit difference scheme. The energy conservation equation is discretized with the characteristic Galerkin approach. The free surface of molten polymer flow front is tracked by the arbitrary Lagrangian–Eulerian (ALE) method. It is demonstrated that good agreement of the numerical predictions given by the proposed ALE method with the results obtained by the injection short‐shot experiments is achieved in the locations and shape of the melt front. Furthermore, when the melt front completely reaches the wall of the mold cavity, the horizontal velocity distribution of counterflow at the section near the finally filling wall is exhibited in the present simulation. POLYM. ENG. SCI., 2012. © 2011 Society of Plastics Engineers     

Reactive mold filling in resin transfer molding processes with edge effects

Reactive mold filling is one of the important stages in resin transfer molding processes, in which resin curing and edge effects are important characteristics. On the basis of previous work, volume‐averaging momentum equations involving viscous and inertia terms were adopted to describe the resin flow in fiber preform, and modified governing equations derived from the Navier–Stokes equations are introduced to describe the resin flow in the edge channel. A dual‐Arrhenius viscosity model is newly introduced to describe the chemorheological behavior of a modified bismaleimide resin. The influence of the curing reaction and processing parameters on the resin flow patterns was investigated. The results indicate that, under constant‐flow velocity conditions, the curing reaction caused an obvious increase in the injection pressure and its influencing degree was greater with increasing resin temperature or preform permeability. Both a small change in the resin viscosity and the alteration of the injection flow velocity hardly affected the resin flow front. However, the variation of the preform permeability caused an obvious shape change in the resin flow front. The simulated results were in agreement with the experimental results. This study was helpful for optimizing the reactive mold‐filling conditions. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Steady‐State Modeling and Simulation of an Axial‐Radial Ammonia Synthesis Reactor

A two‐dimensional model was developed for an axial‐radial ammonia synthesis reactor of the Shiraz petrochemical plant. In this model, momentum and continuity equations as well as mass and energy balance equations are solved simultaneously by orthogonal collocation on the finite element method to obtain pressure, velocity, concentration and temperature profiles in both axial and radial directions. For the catalyst particle, the effectiveness factor is calculated by solving a two‐point boundary value differential equation. The boundary conditions for the Navier‐Stokes and continuity equations are obtained by using equations representing the phenomena of gases splitting or joining in different streams and going through holes in a thin wall. The results of the mathematical model have been compared with the plant data and a good agreement is obtained.     

3D finite element method for the simulation of the filling stage in injection molding

During the molding of industrial parts using injection molding, the molten polymer flow through converging and diverging sections as well as in areas presenting thickness and flow direction changes. A good understanding of the flow behavior and thermal history is important in order to optimize the part design and molding conditions. This is particularly true in the case of automotive and electronic applications where the coupled phenomena of fluid flow and heat transfer determine to a large extent the final properties of the part. This paper presents a 3D finite element model capable of predicting the velocity, pressure, and temperature fields, as well as the position of the flow fronts. The velocity and pressure fields are governed by the generalized Stokes equations. The fluid behavior is predicted through the Carreau Law and Arrhenius constitutive models. These equations are solved using a Galerkin formulation. A mixed formulation is used to satisfy the continuity equation. The tracking of the flow front is modeled by using a pseudo-concentration method and the model equations are solved using a Petrov-Galerkin formulation. The validity of the method has been tested through the analysis of the flow in simple geometries. Its practical relevance has been proven through the analysis of an industrial part.     

Viscous dissipation in capillary flow of rigid PVC and PVC degradation

The steady state, non-isothermal behavior of rigid polyvinyl chloride melt, flowing in capillaries of circular cross-section, was investigated by solving, with the aid of a digital computer, the momentum and energy balance equations. It was assumed that the polymer melt can be described by the “Power Law” constitutive equation. The shear rate, temperature and pressure dependent properties of the fluid were obtained experimentally. The effects of the thermal degradation of PVC on its viscosity, were also introduced in the equations of momentum and energy. The velocity, temperature and pressure profiles, obtained for both adiabatic flow and flow through a tube of constant wall temperature, indicate that considerable heating of the melt, due to viscous dissipation, can be achieved at moderate flow rates. Thermal degradation occurs in the capillary under certain conditions of temperature history and residence time of the fluid. The results of this work are in fair agreement with experimental results in this area.     

Tube flow of a particulate-filled thermosetting polymer

The conservation equations of momentum, energy, and mass are numerically solved for the flow of filled thermosets reacting In a tube. The flow is assumed to be laminar and adiabatic with a constant volumetric flow rate. The critical radii are parameters that define the processability limits. The lower one is the value of the radius where an undesirable advance in the reaction extent takes place at the wall or where viscous heating leads to degradation. The upper critical radius is the radius where wall velocity is low and gelation takes place. The effects of filler volumetric fraction, wall slip velocity, and different inlet conditions are taken into account. Increasing wall slip velocity or filler fraction and decreasing inlet temperature or tube length amplify the processability zone.     

Computer simulation of injection mold filling for viscoelastic melts with fountain flow

The ultimate properties of a molded article arc directly related In the microstructure of the article and are consequently influenced by the thermomechanical history experienced by the melt during processing. The mold filling behavior of thermopalastic polymer melts has been analyzed quantitatively by means of a computer simulation. The mathematical model is based on the equations of continuity, motion, and energy, along with appropriate constitutive relations and relevant initial and boundary conditions. The governing system of equations is solved numerically by means of a Marker-and-Cell computational scheme. One to the significant implications for microstructure development, the fountain effect at the advancing free surface is explicitly taken into consideration in the simulation. The model yields data on filling time and melt front position as well as velocity, temperature, pressure, and shear stress distributions within the mold cavity. The rearrangement of the velocity and temperature profiles in the vicinity of the melt front are considered in detail. Experimental studies have also been undertaken in order to verify the predictions of the computer Simulation.