RECENT PROGRESS AND FUTURE CHALLENGES
COMPUTER-AIDED POLYMER PROCESSING
ANALYSIS AND DESIGN
Centre for Advanced Polymer Processing and Design (CAPPA-D)
Department of Chemical Engineering
Hamilton, Ontario, Canada L8S 4L7
Presented at the ATV-Semapp Meeting
Funen, Odense, Denmark
The first serious efforts on modeling of polymer processing operations (especially plasticating extrusion) were carried out at DuPont, Delaware, USA and subsequently published in the early fifties. Also, Maillefer in Switzerland developed, independently from the DuPont team, some very important models for extrusion at about the same time. The contributions of McKelvey, Gore and Squires of DuPont are well known. Bernhardt's book  summarized just about everything important on process modeling until about 1958. McKelvey's book  was perhaps the first ever and very successful attempt to present a unified approach in the framework of the equations of conservation of mass, momentum and energy and the change of phase mechanisms. Klein and Marshall's book  was perhaps the first monograph ever exclusively devoted to computer modeling of plastics processing but it had very little impact, because the material was really outdated for the seventies. Tadmor and Klein's book  presented the first complete model for plasticating extrusion including transport of solids from the hopper forward, as the screw rotates, melting and melt pumping.
A simulation package for plasticating extrusion called EXTRUD  became commercially available in the early seventies. This was largely based on the models described in Tadmor and Klein's book . In the seventies many investigators in universities and industry worked on various computer models for extrusion, injection molding, calendering and other processes. However, there was little impact of the computer models on process technology till 1978 when C. Austin produced the first MOLDFLOW package for injection mold filling . The art of mold design started to become an engineering discipline heavily relying on computer predictions with the release of C-MOLD  and other software packages exclusively devoted to the injection molding process , in the early eighties.
In the eighties also, many rigorous investigations on various aspects of polymer flows through channels, dies, and process equipment were carried out by various research groups in North America and Europe. General purpose finite element packages for polymer flows such as FIDAP , POLYFLOW , NEKTON  and POLYCAD®  became commercially available. In the nineties the emphasis is on viscoelasticity, complex 3-D simulations and prediction of orientation, residual stresses and phenomena occurring after solidification. Also, in the nineties there is more emphasis on process-specific application of computer methods for such processes as twin-screw extrusion, thermoforming, compression molding, film blowing, reaction injection molding, and gas-assisted injection molding.
MODELING OF INJECTION MOLDING
The greatest technological impact of computer models was in injection molding. The reason is the ability of the Hele-Shaw flow approximation  to describe reasonably well the mold filling process. When the thickness of a flow channel is small compared to other dimensions and the flow is nearly parallel, the conservation of momentum equation simplifies to a single equation with pressure, P, the unknown. In some of the commercially available computer programs the finite element method is used to solve the momentum equation for pressure and the finite difference method for the energy equation.
The role of the fountain effect at the free flow front as shown by Mavridis et al  is responsible for complex shear and stretching motions that result in fluid element deformations (see Figure 1) and molecular orientation phenomena.
The filling stage of injection molding which might be typically of the order of a few seconds duration is followed by a very short packing stage necessary to pack more polymer in the mold to offset the shrinkage after cooling. During the packing stage there is no net fluid flow but motions due to density differences that require PVT behavior analysis. The cooling stage as well as the accompanying phenomena of crystallization, orientation, shrinkage and warpage are subjects of current research . MOLDFLOW  and C-MOLD  offer computer modules for cooling, shrinkage and warpage, but the detailed description of these phenomena is likely to remain a challenge for some time to come. There are several issues that influence shrinkage and warpage such as viscoelasticity, crystallinity, molecular orientation, and various anisotropies that are difficult to describe mathematically. To some extent the physics is lacking. Also, the necessary material parameters are difficult to obtain experimentally with a high degree of accuracy. Consequently, the predictive power of computer simulations which are based on poorly described physical interactions and imprecisely measured material properties, is likely to remain limited in the near future.
SIMULATION OF MELT FLOWS THROUGH EXTRUSION DIES
Most polymer melt flows through extrusion dies are nearly always steady and non-isothermal, sometimes planar or axisymmetric and frequently fully three-dimensional. The equations of conservation of mass, momentum and energy for steady, creeping flow (very low Reynolds number) must be solved.
There are few remaining challenges in the solution of creeping flow problems involving shear thinning (purely viscous) viscosity models. The commercially available packages (FIDAP, POLYFLOW, NEKTON and POLYCAD®) can handle the majority of problems for 1-D, 2-D and 3-D flows. Some simulations using Polycad are shown in Figure 2. The determination of free surfaces or interfaces is the subject of current research for 3-D flows. Karagiannis et al [16, 17] have addressed some questions relating to 3-D flow simulation. In coextrusion there is a tendency of the less viscous fluid to encapsulate the more viscous one. What happens at the contact line (intersection of the interface with a die wall) has been studied by Torres [18, 19]. It is not known whether the fluids stick or slip at the contact line and whether interfacial tension has any influence.
Generally speaking it is not known how polymeric liquids flow in the immediate vicinity of solid surfaces. The phenomenon of sharkskin occurs at wall shear stress of about 0.14 MPa in flows through capillary dies and it is believed to originate near the die exit. With currently available numerical methods we are unable to predict the onset of this important phenomenon. The physics of how polymeric liquids flow over or lose contact with a solid surface is not well understood. Another problem that is difficult to tackle with the currently available simulation methodologies is the phenomenon of die lip build-up, which is also called die drool . The continuum approach is perhaps inadequate for these kinds of problems. Alternative approaches would have to be developed in the future that take into account the molecular nature of the polymers. Öttinger's CONNFFESSIT method  seems to be promising, but it is in its infancy at the present time.
Viscoelasticity is a major challenge in modeling of molten polymers. Many of the viscoelastic constitutive equations that have been used in the past give poor predictions of certain viscoelastic properties. Tanner  gives a synoptic evaluation of several well known models. The so-called high Weissenberg Number problem (i.e. non-convergence at high Ws) has been solved using Streamline Upwinding Petrov-Galerkin (SUPG) formulations, Tucker . Many challenges lie ahead relating to viscoelastic simulations that agree with experimental data for polymer solutions and melts.
Simulation of viscoelastic flows, thus far, has had a disappointingly little impact on polymer processing technology because of unreliable predictions. Perhaps some verbatim quotations will convey the frustration felt by many researchers in this area. Hinch  studied the entry flow problem and stated "For the standard constitutive equation of an Oldroyd-B fluid, the numerical results have recirculating eddies which are too small compared with experiments, and further there is less pressure drop than a Newtonian fluid (not observed)". Goublomme et al  in a numerical study of extrudate swell from capillaries using the K-BKZ constitutive equation concluded that "inclusion of a converging upstream conical section, as in the experimental set-up, grossly exaggerates the calculated swelling ratio". In a subsequent study, Goublomme and Crochet  determined that with a normal stress ratio N2/N1 = -0.3 they could obtain relatively good agreement between simulations and experiments, but they stated that "it would be premature to claim that the decrease of the swelling ratio is due unambiguously to the increase of the ratio -N2/N1 and more work is required to confirm these provocative findings". Other researchers [27, 28] have obtained relatively good agreement between viscoelastic simulations and experiments under certain conditions. However, it is difficult to decide whether the successes would be applicable outside the range of problems examined.
From the practical point of view, the axisymmetric extrudate swell simulation is of little importance, because it is much easier to make an extrudate swell measurement at whatever shear rates required than to determine the viscoelastic parameters needed to fit the K-BKZ constitutive equation. Where simulation would be needed is in prediction of swell from profile dies, fully three-dimensional. Since the K-BKZ equation, like all the others, is not reliable for planar end and axisymmetric problems, it would be even less reliable for 3-D simulations. In addition, the computational cost required for 3-D implementations would be prohibitive for most practical applications. Alternative approaches would have to be explored and perhaps the experience in turbulence modeling  might have a parallel in viscoelastic modeling. Rigorous statistical theories of turbulence have been shown to have little predictive power for shear flows. The current trend is to use models based on experimental evidence and correlations of data together with theoretical formulations, such as the k-e turbulence model. For viscoelastic creeping flows Mitsoulis and Vlachopoulos  and Matsunaga et al  have proposed heuristic models having separate contributions for shear rate and extensional viscosity and normal stresses. These material properties can be measured experimentally, fitted by power-law expressions and introduced into the conservation equations. Several questions remain pertaining to frame invariance with this approach.
In most computer simulations reported in the literature the finite element method is used for the solution of the appropriate differential equations. Recently, however, the control volume method seems to offer a number of advantages for fully 3-D simulations .
MODELING OF SINGLE AND TWIN SCREW EXTRUSION
With the development of Tadmor's melting model  based on Maddock's experiments at Union Carbide, it became possible to simulate the process of single screw extrusion. The extruder is subdivided into three sections for solids conveying, melting and melt pumping. The solid bed is usually modeled by applying a force and torque balance. In the melting zone, melting occurs in a thin film between the solid bed and the heated barrel. A melt pool forms in front of the rear flight. The melt channel is unwound and the flow is assumed to be two-dimensional. These are the basic ideas which led to the development of commercially available software packages EXTRUD , SSD , REX , CHEMEXTRUD , and EXTRUCAD . The most recent version of EXTRUCAD has capabilities for simulation of extruders having standard single flighted screws, multistage screws with venting, barrier screws, and screws with mixing elements . Some typical results are shown in Figure 3. The weakest link in EXTRUCAD as well as all other commercially available packages and extrusion models is perhaps the solids conveying zone. Transport of a packed bed of solid pellets, powders or flakes is difficult to describe mathematically. The friction coefficients between solid bed and barrel and between solid bed and screw are difficult to measure experimentally. Pressure, temperature, speed of rotation, material of construction and surface machining have considerable influence on these coefficients. The whole approach of balancing forces and torques which necessitates the introduction of the friction coefficients is just a convenient empiricism. There is need of better understanding of the contact problem between a solid bed of packed particles and screw and barrel surface.
Twin screw extruders come in a great variety of configurations. There are differences in screw placement, shape of the flights, and direction of rotation that makes extruders as different from each other as they are, as a group, different from single-screw extruders . There are co-rotating, counter-rotating, intermeshing, non-intermeshing, conjugated, and non-conjugated machines. Strictly speaking, twin screw extrusion is a 3-D, unsteady state process. As such, the problem is extremely difficult. Various simplifications have been introduced by various groups, i.e. 3-D steady flow (by Manas-Zloczower at Case Western Reserve University, Gogos and co-workers at Stevens Institute of Technology, Kiani at Werner-Pfleiderer, Hrymak at McMaster University, and others). Simpler models have been developed by White at University of Akron and Potente at the University of Paderborn in Germany. For a comprehensive review of the various modeling approaches up to about 1990 the reader is referred to White's book . AKRO-CO-TWIN SCREW  is a commercially available package for simulation of co-rotating screw extruders. Bang and White  have recently proposed a model for modular tangential counter-rotating twin screw extruders, based on the lubrication approximation.
The complexity of twin screw extruder geometries, screw configurations and the variety of the operational characteristics will pose a computational challenge for many years to come. There is no unique and recommended methodology to follow. There is certainly a need for new ideas.
MODELING OF OTHER PROCESSES
Numerous publications have appeared in various journals dealing with simulation of other processes such as film blowing , thermoforming and injection blow molding , reaction injection molding , compression molding , calendering , and rotational molding . Commercially available software packages are available for thermoforming and injection blow molding, C-PITA from A.C. Tech  and T-FORMCAD from POLYDYNAMICS . C-PITA has been originally developed at GE in the U.S.A. by DeLorenzi and Nied, but the package is now sold through A.C. Tech. T-FORMCAD in many respects follows the same methodology as C-PITA (i.e. finite element analysis of membrane deformation) but FE gridding techniques and constitutive models have differences. Sheet thinning predictions by T-FORMCAD are shown in Figure 4. There are several issues that require further investigation with both these packages. Simulation of plug-assisted thermoforming seems to give instability problems, probably due to poor understanding of the polymer-plug contact. Also, the viscoelastic nature of polymers must be described very well in order for the simulation to predict actual thermoforming or injection blow molding. In many simulations, the Ogden model of rubber elasticity is used which does not include the viscous creep behavior that sometimes dominates the deformation processes. In summary, modeling of plug-assisted and viscoelastic behavior are the present challenges in simulation of thermoforming and injection blow molding.
Stretch blow molding is widely used for enhancement of the physical properties of PET bottles. Simulation of this process would require a good understanding and mathematical description of stress-induced crystallinity. There have not been any well acceptable approaches for stretch blow molding simulation thus far.
In extrusion blow molding a semi-molten tube, called the parison, is formed by forcing the polymer melt to flow through an annular die. Depending on whether the annular channel is straight , converging, or diverging and on the viscoelastic polymer properties, different parison and thickness swellings are obtained. Such processes are difficult to describe with the current knowledge of viscoelastic flow and there remain several challenges ahead for simulation of extrusion blow molding .
For film blowing, a simulation package called B-FILMCAD has been recently made available from POLYDYNAMICS , that predicts bubble formation. Some results are shown in Figure 5. The model thus far is purely viscous and requires accurate knowledge of dependence of viscosity on temperature. It predicts all kinematic variables very well, but the predictions are not very accurate for stresses or bubble pressure . Again, an appropriate viscoelastic constitutive equation would be required. Current plans include the introduction of stress relaxation parameters to improve the agreement between simulation and experiments.
THE ART AND SCIENCE OF DESIGN
The 1978 Nobel Prize winner in Economics, Herbert Simon, called design the science of the artificial . Simon explains that design is much more complex than optimization. In fact, the optimization problem is a standard mathematical problem - to maximize a function subject to constraints. "Traditional engineering design methods make much more use of inequalities - specifications of satisfactory performance - than of maxima and minima". He introduced the term "satisficing" to refer to such procedures. "The design process itself involves management of the resources of the designer, so that his efforts will not be dissipated unnecessarily in following lines of inquiry that prove fruitless".
It is tempting to think that all one needs for design of polymer process equipment such as dies, molds, extruders, etc., is a reliable, user-friendly fully 3-D FEM package. While it is indeed possible to carry out a detailed flow simulation with a 3-D package under the restrictions mentioned above, it is extremely difficult to design with it. Searching for a satisfactory design with a fully 3-D FEM package will require powerful supercomputers and practically prohibitive computer times. A 3-D package does not provide the designer with any hints on how to choose the appropriate channel shapes and flow geometries to perform the simulation. Of course, the simulation must be repeated many times until a satisfactory design is found. This, of course, will be a poor management of resources, in disagreement with Simon's ideas  and common engineering sense.
Some attempts have been made to formulate the mathematical inverse problem . For example, for the case of profile extrusion, the output profile is specified and a computer program is used to determine a channel that would produce such a profile. Such philosophy has been applied by POLYFLOW . For simple profiles this might be possible, but how could one design a window profile with dozens of sides of hollow channels? Inverse problems are mathematically "ill-posed" and solutions are not unique. Certainly, there is room for progress in this area and new mathematical methodologies are called for.
The use of the Hele-Shaw flow approximation for simulation of cavity filling has been very successful for mold design purposes. This approximation in one form or another is used by the well known software packages MOLDFLOW, C-FLOW, as well as smaller competitors. These packages include not only mold filling analysis, but also modules for runner balancing, packing, cooling, shrinkage and warpage based on simplified mathematical models. There are numerous companies around the world using these packages. Although many users demand more predictive power and accuracy than is currently available, it is reasonable to conclude that computer aided analysis of injection molding of plastics has been a very successful endeavor.
For extrusion die design several computer programs have been developed and marketed by POLYDYNAMICS , under the general name of the POLYCAD® family such as PROFILECAD for profile dies, FLATCAD for flat dies (sheet or film), SPIRALCAD for spiral dies (blown film or tubing), and LAYERCAD for multimanifold co-extrusion. These programs are easy to use and the simulations require very short computer times. Iterations can be easily performed around a chosen geometry until a satisfactory design is obtained. Refinements can easily be made and if necessary, subsequent fully 3-D FEM simulations can be carried out to determine if there are any problematic regions. Among the various types of extrusion dies, the design of profile dies presents the greatest challenges. There are many possibilities in designing these dies. The primary objective of flow balancing between narrow and wide gaps can be achieved by using completely different channels, separating the flow by webs or die profiling in such a way as to minimize cross-flows [50, 51]. The use of flow simulation software can reduce significantly the number of times that steel needs to be cut for the manufacture of a die. The replacement of trial-and-error on the factory floor by trial-and-error on the computer screen results in significant reduction of the extrusion die design cost and time. However, several challenges lie ahead, one of them relating to die swell determination from complex profile dies.
Rational screw design for single screw extruders is possible through the use of EXTRUCAD . Of course, accurate viscosity and friction coefficient data are necessary. Since friction coefficients are seldom measured it is possible to tune EXTRUCAD by comparing throughput and pressure profile measurements to computer predictions. Then, by adjusting the friction coefficients, prediction of performance of new screw designs can be carried out.
BEYOND DIFFERENTIAL EQUATIONS
In the field of polymer processing, the term computer-aided engineering (CAE) usually refers to the numerical solution of partial differential equations describing the transport phenomena, the physical or chemical interactions and subsequent analysis of the results. CAE, of course, includes the interfacing with mesh generators and solid modeling packages like Pro/ENGINEER  and HYPERMESH . In a manufacturing environment, the objective is to produce a product of the highest quality at the lowest possible cost. CAE, as it is applied today, is certainly very helpful towards meeting this objective, but computers can also be used to apply artificial intelligence techniques (AI) for problem solving and enhanced productivity. AI has made significant impact in the petroleum refining industry and in the chemical process industry . Unfortunately, there have been very few and limited applications in polymer processing and virtually no related publications on this subject . There are three powerful AI techniques that are likely to have great impact on polymer processing in the near future: expert systems, neural networks, and fuzzy logic. Expert systems are knowledge-based systems which through an inference engine locate existing knowledge and infer new knowledge through a search strategy. Neural networks are computer programs capable of recognizing patterns through a training or learning process by being exposed to large amounts of input and output data. Fuzzy logic is a method of representing continuous processes by the use of rules, or heuristics, rather than via precise mathematical models.
AC Tech  recently released the Dr. C-MOLD software package for helping the design or process engineers with material selection, optimization of part thickness and determination of the molding process window. This package can be classified as an expert system.
POLYDYNAMICS , in cooperation with POLYEXPERT S.A. , has recently developed the XTRU-XPERT system for fault diagnosis and troubleshooting in extrusion. This expert system utilizes two levels of knowledge: practical knowledge from the factory floor and quick calculations via algebraic equations and approximate formulas. XTRU-XPERT can be used together with in-depth knowledge based on solution of differential equations (i.e. from simulations with POLYCAD, EXTRUCAD, FLATCAD, SPIRALCAD, PROFILECAD, etc.). Here is an example: the problem of extruder surging cannot be diagnosed just by solving the differential equations for single screw extrusion. It might be due to moisture in the feed stream, sticky or slippery additives, solid bed break-up, poor screw design, etc. XTRU-XPERT will be able to infer the root cause of the problem by detailed questioning, calculations, or EXTRUCAD simulations. Analogous methodologies can be followed for diagnosis and troubleshooting of gels in film processing, barrel and screw wear, sharkskin, die lip build-up, interfacial instabilities in coextrusion, flow lines in blown film extrusion and thickness non-uniformities. XTRU-XPERT like other expert systems can also be used for training purposes of extrusion design or process engineers.
Computer-aided engineering methods developed over the past thirty years have had considerable impact in the analysis and design of polymer processes and equipment. The greatest impact has been in injection molding. Considerable progress has been made recently in understanding the flow of melts through extrusion dies and single screw extruders. Current efforts by several groups are aimed towards improving the simulation of thermoforming, blow molding, reaction injection molding, gas-assisted injection molding, twin screw extrusion, and other processes. For the creation of meshes for geometrically complex flow domains the interfacing with various solid modeling packages is rapidly expanding and becoming easier. Among the challenging unsolved problems are modeling of viscoelastic flow behavior, contact between polymer melt and metal wall and instability phenomena such as sharkskin, melt fracture and draw resonance . To improve productivity in the future, more emphasis should be given towards developing new methodologies for equipment design and exploring artificial intelligence tools, such as expert systems, neural networks and fuzzy logic.
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