Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

Quantitative evaluation of extrudate swell from viscoelastic properties of polystyrene

A theory of extrudate swell for short, intermediate or long dies is presented. In our experiment, we consider that the swelling phenomenon is mainly due to the recoverable elongational strain induced by the converging flow at the die entrance, as well as by recoverable shear strain originating within the die. From these concepts, an equation has been derived for the quantitative prediction of extrudate swell from the elastic material properties such as the entrance pressure drop, the relaxation modulus and the recoverable shear strain. Excellent agreement is found between predicted and measured values of extrudate swell obtained on commercial polystyrene melt, using capillaries of length-to-diameter ratios ranging from 1 to 20 and in a wide range of shear rates.     

Extrudate swell of Boger fluids

Boger fluids are dilute polymer solutions exhibiting high elasticity at low apparent shear rates, which leads to high extrudate swell. Numerical simulations have been undertaken for the flow of three Boger fluids (including benchmark Fluid M1), obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of dilute polymer solutions. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries without or with gravity. The results provide the extrudate swell and the excess pressure losses (exit correction), as well as the shape and extent of the free surface. All these quantities increase rapidly and monotonically with increasing elasticity level measured by the stress ratio, S_R. It was found that the main reason for the high extrudate swelling is high normal stresses exhibited in shear flow (namely, the first normal-stress difference, N₁). Surprisingly, the elongational parameter of the model or a second normal-stress difference N₂ do not affect the results appreciably. Gravity serves to lower the swelling considerably, and makes the simulations easier and in overall agreement with previous experiments.     

Three-dimensional non-isothermal extrusion flows

A three-dimensional (3-D) non-isothermal study of viscous free-surface flows with exponential dependence of viscosity on temperature is presented. The effects of non-isothermal conditions and/or geometry on the extrudate shape are investigated with a fully three-dimensional finite element/Galerkin formulation. Apart from the well known thermally induced extrudate swelling phenomenon, bending and distortion of the extrudate may occur because of temperature differences and/or geometric asymmetries. A temperature difference across the die can be imposed by heating or cooling the die walls, but can also arise because of asymmetric viscous heat generation due to the die geometry. Temperature differences affect velocity profiles because of the temperature dependence of viscosity and lead to extrudate bending, an effect known as kneeing in the fiber spinning industry. It is also shown numerically and confirmed experimentally that the die geometry induces extrudate bending even in the case of isothermal Newtonian flows.     

Free (open) boundary condition: some experiences with viscous flow simulations

The free (or open) boundary condition (FBC, OBC) was proposed by Papanastasiou et al. (A new outflow boundary condition, International Journal for Numerical Methods in Fluids, 1992; 14:587–608) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been tested in several benchmark problems of viscous flow in fluid mechanics. The FEM is used to provide numerical results for both cases of planar and axisymmetric domains under laminar, isothermal or non‐isothermal, steady‐state conditions, for Newtonian fluids. The effects of inertia, gravity, compressibility, pressure dependence of the viscosity, slip at the wall, and surface tension are all considered individually in the extrudate‐swell benchmark problem for a wide range of the relevant parameters. The present results extend previous ones regarding the applicability of the FBC and show cases where the FBC is inappropriate, namely in the extrudate‐swell problem with gravity or surface‐tension effects. Particular emphasis has been given to the pressure at the outflow, which is the most sensitive quantity of the computations. In all cases where FBC is appropriate, excellent agreement has been found in comparisons with results from very long domains. The formulation for Picard‐type iterations is given in some detail, and the differences with the Newton–Raphson formulation are highlighted regarding some computational aspects. Copyright © 2011 John Wiley & Sons, Ltd.     

Compressible extrudate swell

There are few computations of polymer forming processes which include compressibility. Here we estimate the effect of compressibility in Newtonian and PTT fluids on extrudate swell and stick-flip flow. Changes of the order of a few per cent occur in swelling, which is in accord with expectations.     

Die design: An implicit formulation for the inverse problem

Let us call a direct extrusion problem (DEP) the problem of finding the shape of the extrudate coming out of a die of prescribed shape. An implicit finite element formulation of the DEP which is geometrically general and for which a Newton-Raphson technique can be implemented has recently been proposed by Legat and Marchal. However, the problem posed to the die designer is frequently the inverse extrusion problem (IEP), i.e. finding the die shape which produces an extrudate of prescribed shape. This paper presents an extension of our original method for solving the IEP which avoids the ‘trial-and-error’ iteration on the die geometry itself. The advantage of the formulation lies in its capability to handle complex geometrics and in its low cost, because the CPU time and memory required to solve the IEP are almost identical to those of the DEP. We present benchmark results for squares and rectangles and new results obtained for geometries involving multiple corners. For an octagonal shape we also consider the case of a power-law fluid. For all results presented in this paper, surface tension has not been included.     

Stability analysis of a polymer film casting problem

The polymer cast film process consists of stretching a molten polymer film between a flat die and a drawing roll. Drawing instabilities are often encountered and represent a drastic limitation to the process. Newtonian fluid film stretching stability is investigated using two numerical strategies. The first one is a ‘tracking’ method, which consists of solving Stokes equations in the whole fluid area (extrusion die and stretching path) by finite elements. The interface is determined to satisfy a kinematic equation. A domain decomposition meshing technique is used in order to account for a flow singularity resulting from the change in the boundary conditions between the die flow region and the stretching path region. A linear stability method is then applied to this transient kinematic equation in order to investigate the stability of the stationary solution. The second method is a direct finite element simulation in an extended area including the fluid and the surrounding air. The time‐dependent interface is captured by solving an appropriate level‐set function. The agreement between the two methods is fair. The influence of the stretching parameters (Draw ratio and drawing length) is investigated. For a long stretching distance, a critical Draw ratio around 20 delimitating stable and unstable drawing conditions is obtained, and this agrees well with the standard membrane models, which have been developed 40 years ago. When decreasing the stretching distance, the membrane model is no longer valid. The 2D models presented here point out a significant increase of the critical Draw ratio, and this is consistent with experimental results. Copyright © 2015 John Wiley & Sons, Ltd.     

液晶高分子-各向异性流体挤出拉伸分岔研究

应用共转导数型本构方程研究了液晶高分子纺丝挤出过程的拉伸黏度,应用计算机符号运算软件 Maple得出解析表达式,拉伸黏度与拉伸率之间关系(随剪切速率变化)表明存在分岔现象,得出拉伸黏度显著高于相应的剪切黏度,解释了液晶高分子熔体挤出时不发生挤出胀大的物理机制．     

Extrudate swell of linear and branched polyethylenes: ALE simulations and comparison with experiments

Extrudate swell is a common phenomenon observed in the polymer extrusion industry. Accurate prediction of the dimensions of an extrudate is important for appropriate design of dies for profile extrusion applications. Prediction of extrudate swell has been challenging due to (i) difficulties associated with accurate representation of the constitutive behavior of polymer melts, and (ii) difficulties associated with the simulation of free surfaces, which requires special techniques in the traditionally used Eulerian framework. In a previous work we had argued that an Arbitrary Lagrangian Eulerian (ALE) based finite element formulation may have advantages in simulating free surface deformations such as in extrudate swell. In the present work we reinforce this argument by comparing our ALE simulations with experimental data on the extrudate swell of commercial grades of linear polyethylene (LLDPE) and branched polyethylene (LDPE). Rheological behavior of the polymers was characterized in shear and uniaxial extensional deformations, and the data was modeled using either the Phan–Thien Tanner (PTT) model or the eXtended Pom–Pom (XPP) model. Additionally, flow birefringence and pressure drop measurements were done using a 10:1 contraction–expansion (CE) slit geometry in a MultiPass Rheometer. Simulated pressure drop and contours of the principal stress difference were compared with experimental data and were found to match well. This provided an independent test for the accuracy of the ALE code and the constitutive equations for simulating a processing-like flow. The polymers were extruded from long (L/D = 30) and short (L/D = 10) capillaries dies at 190 °C. ALE simulations were performed for the same extrusion conditions and the simulated extrudate swell showed good agreement with the experimental data.     

Numerical study of the combined effects of inertia,slip, and compressibility in extrusion of yield stress fluids

The axisymmetric extrudate swell flow of a compressible Herschel–Bulkley fluid with wall slip is solved numerically. The Papanastasiou-regularized version of the constitutive equation is employed, together with a linear equation of state relating the density of the fluid to the pressure. Wall slip is assumed to obey Navier’s slip law. The combined effects of yield stress, inertia, slip, and compressibility on the extrudate shape and the extrudate swell ratio are analyzed for representative values of the power-law exponent. When the Reynolds number is zero or low, swelling is reduced with the yield stress and eventually the extrudate contracts so that the extrudate swell ratio reaches a minimum beyond which it starts increasing asymptotically to unity. Slip suppresses both swelling and contraction in this regime. For moderate Reynolds numbers, the extrudate may exhibit necking and the extrudate swell ratio initially increases with yield stress reaching a maximum; then, it decreases till a minimum corresponding to contraction, and finally, it converges asymptotically to unity. In this regime, slip tends to eliminate necking and may initially cause further swelling of the extrudate, which is suppressed if slip becomes stronger. Compressibility was found to slightly increase swelling, this effect being more pronounced for moderate yield stress values and wall slip.     

Numerical analysis of three-dimensional Newtonian extrudate swell

The present paper considers the problem of predicting extrudate shapes from asymmetrical dies for Newtonian fluids. The flow is fully three-dimensional and an exploration of finite elements is made with a view to finding accurate, stable and economical schemes. A number of elements are compared and we conclude that some of the Fortin elements are most useful on the grounds of computational overhead and solution accuracy. These are used to investigate some symmetrical (square dies) and asymmetrical (unequal lip) planar and general L-shaped die flows. Finally, we show that in an unconstrained extrudate the final shape must be such that particles describe a helix in space; special cases include circular flow and rectilinear flow.     

Three-dimensional streamlined finite elements: Design of extrusion dies

A method to determine three-dimensional die shapes from extrudate swell and vice versa is presented using a three-dimensional Galerkin finite element method based on a streamlined formulation with the fluid velocities and pressures represented by triquadratic and trilinear basis functions respectively. The three-dimensional streamlined method, an extension of the two-dimensional formulation, uses successive streamsurfaces to form a boundary-conforming co-ordinate system. This produces a fixd, computational domain leaving the spatial location of the elements as unknowns to be determined with the standard primary variables (u, v, w, p). The extrudate produced by a die of a given shape is considered for moderate Reynolds numbers. Finally, the method is extended to address the problem of die design, where a die profile is sought to produce a target extrudate shape.     

Similarity solution of free convective boundary-layer behavior at a stretching surface

In the present study, free convection and heat transfer behavior of electrically conducting fluid in the boundary layer over a vertical continuously stretching surface is investigated. The effects of free convection, magnetic field, suction/blowing at the surface and the stretching speed of the surface on the flow and heat transfer characteristics are considered. By applying one-parametric group theory to analysis of the problem, a similarity solution is found. The governing equations of continuity, momentum and energy are solved numerically by a fourth-order Runge-Kutta scheme. The numerical results, which are obtained for the flow and heat transfer characteristics, reveal the influences of the parameters. Received on 9 September 1998     

Flow of a viscoelastic fluid over a stretching sheet

This paper presents a study of the flow of an incompressible second-order fluid past a stretching sheet. The problem has a bearing on some polymer processing application such as the continuous extrusion of a polymer sheet from a die.     

Flow birefringence study of sharkskin and stress relaxation in polybutadiene melts

In this paper we report rheo-optical and rheological observations made through a transparent slit die attached to a capillary rheometer. We find that the flow birefringence signal oscillates periodically near the die exit when sharkskin-like extrudate distortion is present. In contrast, steady behavior is observed in the die inland region. Specifically, the flow birefringence varies at the die exit with a period identical to that measured directly from the sharkskin extrudate. We also show that the exit flow instability leading to sharkskin can be observed directly through cross-polarizers in terms of the temporal change of the retardation order. We demonstrate that the same kind of interfacial flow instability can occur at a boundary discontinuity within the die land where the upper portion of a clean die wall meets the lower portion of a polysiloxane-coated die wall. Finally, stress relaxation upon the cessation of the slit die flow of two polybutadiene melts is studied through time-dependent flow birefringence measurements. The stress relaxation is then correlated with sharkskin time scales to describe the role of relaxation in sharkskin ridge formation. Received: 8 February 1999 Accepted: 28 July 1999     

Extrudate swell through an orifice die

The extrudate swell of a viscoelastic fluid through an orifice die is investigated by using a mixed finite element and a streamline integration method (FESIM), using a version of the K-BKZ model. The free surface calculation is based on a local mass conservation scheme and an approximate numerical treatment for the contact point movement of the free surface. The numerical results show a vortex growth and an increasing swelling ratio with the Weissenberg number. Convergence with mesh refinement is demonstrated, even at a high Weissenberg number of O(587), where the swelling ratio reaches a value of about 360%. In addition, it is found that the effective flow channel at the entrance region next to the orifice die is reduced due to the enhanced vortex growth, which may be a source of flow instability.     

On the behavior of viscoelastic free jets with elliptical cross section

In [1,2] we develop a comprehensive theory of one space dimensional closure models (closed systems of 1-D equations for the unknown, or modal, variables) for free viscoelastic jets. These closure models are derived via asymptotics from the full 3-D boundary value problem under the conditions of a Von Kármán-like flow geometry, a Maxwell-Jeffreys constitutive model, elliptical free surface cross section, and a slender jet scaling. The focus of the present paper is to determine the consequences and predictions of the lowest order system of equations in this asymptotic analysis. For the special cases of elliptical inviscid and Newtonian free jets, subject to the effects of surface tension and gravity, our model predicts oscillation of the major axis of the free surface elliptical cross section between perpendicular directions with distance down the jet, and draw-down of the cross section, in agreement with observed behavior. In the absence of surface tension the transformation from a cross section with major axis in one direction to a cross section with major axis in the perpendicular direction occurs only once, in agreement with the observation of Taylor [4]. In viscoelastic regimes, our model predicts swell of the elliptical extrudate and distortion of the elliptical extrudate cross section from the dimensions of the die aperture.     

A finite difference analysis of the extrudate swell problem

The extrudate swell phenomenon of a purely viscous fluid is analysed by solving simultaneously the Cauchy momentum equations along with the continuity equation by means of a finite difference method. The circular and planar jet flows of Newtonian and power-law fluids are simulated using a control volume finite difference method suggested by Patankar called SIMPLER (semi-implicit method for pressure-linked equations). This method uses the velocity components and pressure as the primitive variables and employs a staggered grid and control volume for each separate variable. The numerical results show good agreement with the analytical solution of the axisymmetric stick-slip problem and exhibit a Newtonian swelling ratio of 13.2% or 19.2% for a capillary or slit die respectively in accordance with previously reported experimental and numerical results. Shear thinning results in a decrease in swelling ratio, as does the introduction of gravity and surface tension.