Three-dimensional study of extrusion processes by Boundary Element Method.

In this paper we report an implementation of a Boundary Element Method (BEM) dealing with three-dimensional extrusion problems of viscous fluid. Die shapes with and without sharp corners and their influence on the final shape of extrudate have been studied: both shrinkage and swelling of the extrudates relative to the original die shapes have been discovered. The first half of this paper deals with the problem of implementing isoparametric boundary elements. It is followed by results of test problems, and finally the results of extrusion study are presented.     

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.     

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.     

Prediction of three-dimensional general shape extrudates by an implicit iterative scheme

This paper presents a numerical technique for solving three-dimensional free surface problems in extrusion applications. The method is fully implicit in the sense that a Newton-Raphson scheme is applied on all variables, and geometrically general. In particular, the die section shape may be complex and contains multiple corners: very few restrictions apply on the mesh generation because the method does not require the nodes to be located on straight lines (spines). A clear distinction is introduced between the directions associated with the kinematic condition and the remeshing rules. As a difference with respect to earlier publications, these concepts are handled separately. Only Stokes problems are solved in this paper and we have not introduced surface tension. Therefore corners in the die section propagate discontinuities in the extrudate shape, an a method for relocating corners without losing the quadratic convergence of the scheme is presented. Data structures used for the implementation are briefly discussed. We present results on the extrusion of various profiles, including a rectangular die (a benchmark problem) and various complex sections containing multiple corners.     

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.     

Annular extrudate swell of pseudoplastic and viscoplastic fluids

Numerical simulations have been undertaken for the benchmark problem of annular extrudate swell present in pipe extrusion and parison formation in blow molding. The finite element method (FEM) is used to provide numerical results for different inner/outer diameter ratios κ under steady-state conditions. The Herschel-Bulkley model of viscoplasticity is used with the Papanastasiou regularization, which reduces with appropriate parameter choices to the Bingham–Papanastasiou, power-law and Newtonian models. The present results provide the shape of the extrudate, and in particular the thickness and diameter swells, as a function of the dimensionless power-law index (in the case of pseudoplasticity) and the dimensionless yield stress (in the case of viscoplasticity). The pressures from the simulations have been used to compute the excess pressure losses in the system (exit correction). While shear-thinning leads to reduced swelling relative to the Newtonian values for all κ-values, the opposite is true for shear-thickening fluids, which exhibit considerable swelling. Viscoplasticity leads to decreased extrudate swell as the dimensionless yield stress goes from zero (Newtonian behaviour) to an asymptotic value of fully plastic behaviour. The exit correction decreases to zero with a decrease in the power-law index to zero and an increase in the dimensionless yield stress to its asymptotic limit. However, the decrease is not monotonic: for power-law fluids it has maxima in the range of power-law indices between 0.8 and 0.6, while for viscoplastic fluids it has maxima around Bingham number values of 5.     

Stretching flow instabilities at the exits of extrusion dies

As liquid leaves an extrusion die, the surface layers are rapidly stretched. Stretching flows may become unstable in two ways: by breaking, or in a ductile manner producing an uneven “necked” sample which, in continuous extrusion and drawing, is sometimes called “draw resonance”. There is a quantitative correlation between the extrusion defect known as “sharkskin” and the cohesive failure of polymer melts. By extruding under closely defined conditions, it is possible to introduce a transitory “structure” into the surface layer of the extrudate greatly enhancing its cohesive strength and eliminating this defect. A similar quantitative correlation is established between the uneven coating thickness sometimes obtained during coextrusion of a high viscosity melt on the surface of a low viscosity melt and the tensile drawing instability known as “draw resonance”. Simple criteria are established to avoid this problem in practical flow engineering.     

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.     

The compressible Newtonian extrudate swell problem

We solve the compressible Newtonian extrudate swell problem in order to investigate the effect of compressiblity on the shape of the extrudate. We employ a first-order equation of state relating the density to the pressure and use finite elements for the numerical solution of the problem. Our results show that the shape of the extrudate and the final extrudate swell ratio are not significanlty affected even at high compressibility values.     

The steady annular extrusion of a Newtonian liquid under gravity and surface tension

The steady extrusion of a Newtonian liquid through an annular die and its development outside and away from the die are studied under the influence of gravitational and surface tension forces. The finite element method (FEM) is used for the simulations. The positions of the inner and outer free surface profiles are calculated simultaneously with the other unknown fields, i.e. using the Newton–Raphson iterative scheme. The effects of three relevant parameters, i.e. the Reynolds, the Stokes and the capillary numbers, on the shape of the annular film are studied for two values of the inner to the outer diameter ratio, corresponding to a thick and a thin annular film respectively. A one‐dimensional model for the extrudate region, valid for thin annular films, is also presented, and its predictions are compared with the two‐dimensional finite element calculations. Despite the fact that it is valid away from the die exit, the one‐dimensional model predicts satisfactorily the effects of the Stokes and capillary numbers. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Influence of wall slip on extrudate swell: a boundary element investigation

This paper reports a numerical study of the extrudate swell by a transient Boundary Element Method (BEM). Furthermore the fluid, which is modeled by a differential constitutive equation, is allowed to slip at the wall, where the slip velocity is a prescribed function of the wall shear stress. This function is a curve fit of the extensive data of Ramamurthy on linear low density polyethylene which incorporates two parameters: a critical wall shear stress above which slip occurs, and another parameter which governs the shape of the slip velocity versus shear stress curve. The results show that wall slip reduces both the amount of extrudate swell and the critical Weissenberg number above which our numerical scheme no longer converges.     

The occurrence of periodic distortions in the extrusion of polymeric melts

In the extrusion of polymeric melts at high flow rates, often flow instabilities observed as surface distortions of the extrudate occur. The flow instability ‘spurt’ in piston-driven flows is accompanied by persistent oscillations in the pressure. In this paper spurt is explained in terms of constitutive instabilities (mechanical failure of the polymeric fluid itself), while the no-slip boundary condition at the wall of the die is maintained. The influence of compression on the onset of spurt is investigated. The polymeric melt is modelled as a JSO-fluid, but two other constitutive models are also considered. Numerical computations disclose that persistent oscillations in the pressure as well as in the volumetric flow rate occur for a bounded range of prescribed plunger speeds. The occurrence of the persistent oscillations is also explained by a linearized stability analysis. The frequency of the persistent oscillations is determined both from the linearized stability theory as well as by means of a Fourier spectral analysis. In conclusion, the theory is validated by a qualitative comparison with experimental results. Received March 3, 1998     

The effect of temperature gradients on the sharkskin surface instability in polymer extrusion through a slit die

The sharkskin surface instability is commonly observed in the extrusion of polymer melts. We present a series of experiments in which a specifically designed rectangular slit die with insulated and independently heated sides and is used to induce precise temperature gradients across a flowing polyethylene melt. Our previous experiments demonstrated that the character of the surface distortions produced by the sharkskin instability was a function of the die wall temperature and therefore the extrudate had viscoelastic properties at the surface. In this paper, we explore the role of temperature and viscoelastic property gradients near the capillary wall. The amplitude of the sharkskin instability is quantified and plotted against apparent shear and extension rates. Analysis of the data demonstrates that the amplitude and frequency of the instability is independent of bulk temperature and temperature gradient and is dependent only on wall temperature. The data are normalized using a dimensionless Weissenberg number based on the extension rate to collapse the data collected over all temperatures and gradients onto a single master curve. We conclude with an example of a rectangular extrudate exhibiting varying surface roughness due to differential die heating and discuss the implications of our observations on the sharkskin surface instability mechanism and on commercial applications.     

The shape of the strongest column

The problem of determining that shape of column which has the largest critical buckling load is solved, assuming that the length and volume are given and that each cross section is convex. The strongest column has an equilateral triangle as cross section, and it is tapered along its length, being thickest in the middle and thinnest at its ends. Its buckling load is 61.2% larger than that of a circular cylinder. For columns all of whose cross sections are similar and of prescribed shape-not necessarily convex—the best tapering is found to increase the buckling load by one third over that of a uniform column. This result, which was independently obtained by H. F. Weinberger, is originally due to Clausen (1851). For a uniform column, triangularizing is shown to increase the buckling load by 20.9% over that of a circular cylinder. The results lead to isoperimetric inequalities for the buckling loads of arbitrary columns. The research reported in this paper has been sponsored by the Office of Naval Research under Contract No. (285) 46.     

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．     

Numerical simulation of delayed die swell

In a recent paper, Joseph et al. showed that, for a number of viscoelastic fluids, one can observe the phenomenon of delayed die swell beyond a critical extrusion velocity, or beyond a critical value of the viscoelastic Mach number. Giesekus had also observed that delayed die swell is a critical phenomenon.In the present paper, we find a set of material and flow parameters under which it is possible to simulate delayed die swell. For the viscoelastic flow calculation, we use the finite element algorithm with sub-elements for the stresses and streamline upwinding in the discretized constitutive equations. For the free surface, we use an implicit technique which allows us to implement Newton's method for solving the non-linear system of equations. The fluid is Oldroyd-B which, in the present problem, is a singular perturbation of the Maxwell fluid. The results show very little sensitivity to the size of the retardation time. We also show delayed die swell for a Giesekus fluid.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Effect of secondary zone dimensions on melt fracture manifestations of high density polyethylene

An experimental study was performed on melt fracture phenomena in extrusion of high density polyethylene. The purpose of the work was to study the sensitivity of melt fracture driven roughness to the size of recirculation zones, viz. secondary zones. The experimental apparatus consists of a right angle die and a hypodermic needle used as a capillary. The position of the needle relative to the die was adjusted using a special fixture. The roughness of the extrudate was studied as a function of penetration depth. A developed procedure provides a comparison between profile lengths of extruded strands. The computed mean, median, and mode values for roughness were presented as a function of capillary position. A qualitative analysis was conducted for the force oscillations during extrusion with a separate set of dies, equipped with the fixed capillaries of identical lengths and different depths of penetration. It was observed that the oscillatory pattern is sensitive to the sizes of the secondary zones. This qualitative observation supports the conclusions from the quantitative analysis that the roughness of the extrudate can be controlled through an adjustment of the secondary zone sizes.Partly presented at the 58th Annual Meeting of the Society of Rheology, Tulsa, Oklahoma, October 20–23, 1986     

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.