Zeitschrift Kunststofftechnik Journal of Plastics Technology

Zeitschrift Kunststofftechnik Journal of Plastics Technology archival, peer-reviewed online Journal of the Scientific Alliance of Polymer Technology archivierte, peer-rezensierte Internetzeitschrift des Wissenschaftlichen Arbeitskreises Kunststofftechnik (WAK) www.plasticseng.com; www.kunststofftech.com eingereicht/handed in: 19.09.2011 angenommen/accepted: 04.11.2011 Dipl.-Ing. Thomas Köpplmayr, Dipl.-Ing. (FH) Michael Aigner, Univ.-Prof. Dr. Jürgen Miethlinger, MBA Institute of Polymer Extrusion and Building Physics, Johannes Kepler University Linz A comparative study of viscous flow in slit-exit cross-section dies using network analysis We present a systematic approach based on Kirchhoff s laws for the analysis of complex networks that uses tensor algebra and numerical methods (such as fixed point iteration). Coathanger, fishtail, and T-manifolds are compared in terms of mass flow, residence time, and shear rate. In addition, we report the results of a three-dimensional simulation using the commercially available software ANSYS Polyflow, which were in good agreement with our numerical study. Eine vergleichende Studie viskosen Fließens in Schlitzdüsen unter Verwendung der Netzwerktheorie Wir präsentieren einen systematischen Ansatz basierend auf den Kirchhoff schen Gesetzen zur Analyse von komplexen Netzwerken, unter Verwendung von Tensor-Algebra und numerischen Methoden (z.b. Fixpunktiteration). Kleiderbügel, Fischschwanz und T-Verteiler werden in Bezug auf Massendurchsatz, Verweilzeit und Scherrate verglichen. Darüber hinaus berichten wir über die Ergebnisse einer dreidimensionalen Strömungssimulation unter Verwendung der kommerziell verfügbaren Software ANSYS Polyflow, die in guter Übereinstimmung mit unserer numerischen Studie stehen. Carl Hanser Verlag Zeitschrift Kunststofftechnik / Journal of Plastics Technology 8 (2012) 1

A comparative study of viscous flow in slit-exit cross-section dies using network analysis T. Köpplmayr, M. Aigner, J. Miethlinger 1 INTRODUCTION Flat dies (i.e., coathanger, fishtail, tee) are widely used in the industry for the production of plastic films and sheets. The flow channel cross-section of a monolayer flat die is typically a circle at the entrance and is gradually transformed into a large aspect ratio rectangle required for film and sheet extrusion. Designing such dies is a complex process requiring a thorough understanding of the flow in the die. As product quality both geometrical and material is governed by the uniformity of flow rate and residence time distributions, optimal design of extrusion dies is of great importance. Designing dies for polymer extrusion often entails trial and error improvement of the die geometry. However, in many cases the high number of variables and interactions involved makes successful optimization impossible. The general procedures and various design schemes of one-dimensional analysis were summarized by Michaeli [1]. Most of the studies based on the onedimensional model were carried out using analytical methods under the assumption that the flows in both the manifold and the slot are fully-developed in the machine direction and do not interact. Matsubara [2] developed an approximate practical method of designing the geometry of a coathanger die with both uniform flow rate and residence time across the full width of the die. However, the more complicated flow behavior in real channels can only be described adequately when the calculation is done with CFD software. The finite element method is a numerical analysis technique for obtaining approximate solutions to a great variety of engineering problems. Although originally developed to study stresses in complex airframe structures, it has since been extended and applied to the broad field of continuum mechanics. Sun and Gupta [3] made use of a finite element analysis that takes into account strain-rate and temperature dependence of shear and elongational viscosity to optimize the die geometry. Lebaal et al. [4] concentrated on a finite element approach in combination with an optimization algorithm that includes homogenization of the velocity distribution while preventing excessive pressure loss. Sienz et al. [5] studied the coupling of melt flow and die body deflection due to melt pressure. Na et al. [6] developed a three-dimensional model of the isothermal flow of a power-law fluid in a linear tapered coathanger die and conducted a parametric Journal of Plastics Technology 8 (2012) 1 54

study [7]. The results show that the manifold angle is the design parameter with the greatest influence on the flow rate distribution. Also, the power-law index must be considered when different polymers are processed using the same die. One of the major problems of three-dimensional FEM simulations is their requirements in terms of high computational power and large memory, since great precision is essential to obtain good results. Yu et al. [8] suggested a simple numerical solution based on lubrication approximation and demonstrated that compared to the results from complete FEM simulations the predictions are reasonably accurate. In this work, a modified finite volume method termed network analysis is investigated. The general procedure has been summarized by Michaeli and Pöhler [9]. This numerical method allows a flow problem in a complex geometry to be solved by subdividing the manifold into smaller, geometrically simpler parts for which analytical formulae based on constitutive equations are available. The resulting two-dimensional flow resistance network can be solved in a manner analogous to network analysis of electrical circuits. We present a systematic approach based on Kirchhoff s laws for the analysis of complex networks that uses tensor algebra and numerical methods (such as fixed point iteration) and compare coathanger, fishtail, and T-manifolds in terms of mass flow, residence time, and shear rate. 2 GENERAL DESIGN RULES The preprocessing step includes defining the geometry and generating an Eulerian mesh consisting of a network of control volumes in which the simplified Navier Stokes equations are solved. Figure 1 shows three slit die geometries (consider symmetry) that aim to form the output of an extruder into a flat rectangular sheet with uniform thickness and velocity across its full width. The melt is distributed by a manifold followed by a flow resistance region (also referred to as land). Generally, uniform flow distribution of the extrudate is required, and the average velocity in the land area is constant. This means that the volumetric flow rate in the manifold from point of entry to end of channel falls off to zero in a linear fashion. v V land manifold V0 const. w V0 x 2 W The total pressure loss along the flow path consists of the pressure loss in the distribution channel and the pressure loss across the land. To meet the re- Journal of Plastics Technology 8 (2012) 1 55 (1)

quirement of equal flow resistance, the total pressure loss must also be constant. manifold land px p x p x p x const. x 0 The second requirement is that the melt have a relatively uniform history of deformation, temperature, and residence time. Finally, it is expected that the above requirements be met across a wide range of operating conditions and for a variety of materials. Figure 1: Three slit-exit cross-section dies: coathanger, fishtail, and T- manifold. Journal of Plastics Technology 8 (2012) 1 56 (2)

The basic relationships are valid for all design work pertaining to manifolds. Furthermore, the following restrictions apply: isothermal, steady-state, laminar, and fully developed flow incompressibility of the melt (i.e., = const.) inlet and outlet effects are ignored wall adhesion The following simplifications to the design formulae were used in this study: the manifold is considered as a pipe constant slit height in the land area The flow problem in a complex geometry can be solved by subdividing the manifold into smaller, geometrically simpler lining segments for which analytical formulae based on constitutive equations are available. The resulting twodimensional flow resistance network can be solved in a manner analogous to network analysis of electrical circuits, as illustrated in Figure 2. Figure 2: Design of a flat die using network analysis. In each control volume, simple relationships for pressure loss and flow rate can be used. The equations can be written in the following general form: V C p (3) Journal of Plastics Technology 8 (2012) 1 57

Here, C denotes the conductance, which is a function of the geometry and the melt viscosity. For a circular or rectangular cross-section the following relations are obtained: C C 4 R 8L 3 W 12 L The representative viscosity method allows the application of relationships developed for Newtonian materials to shear-thinning materials by introducing representative quantities. If a representative point in the flow channel is known, the shear rate at this location and the representative viscosity can be calculated from the flow rate V and the known viscosity curve respectively. 4 V 4 V 3n e 3 3 R R 2n 1 n1 Journal of Plastics Technology 8 (2012) 1 58 n 1n 6 V 6 V 4n e 2 2 W W 3n 1 K In the case of a coathanger manifold, the representative shear rate in the distribution channel shall remain constant, and the course of the radius can be determined. n 1n V 0 x 4 4 Vmanifold x 2 W manifold e e 3 3 R x R x 1 x 3 2 V 0 e Rx R 0, where R0 W manifold A uniform distribution is achieved only with the specific design material at the design-specific point of operation. Expanding this design by adding the condition of equal shear rates results in a procedure which is independent of the operating conditions: 3 V0 manifold land 2 e W (4) (5) (6) (7)

Solving for R 0, we obtain: R 0 2 e W 3 e The total pressure loss along the flow path consists of the pressure loss in the distribution channel and the pressure loss across the land. p x p x p x manifold x W 8 manifold V0 W 12 land V0 4 dx 3 y x R x W x Differentiating and setting the result to zero results in the following differential equation for the course of the length of the land: Journal of Plastics Technology 8 (2012) 1 59 land 3 dy x manifold 2 x dx 4 x 3 Rx W 2 1 3 4 land 3 Integration yields the shape of the die contour. However, using this design approach, the maximum length of the die is fixed and cannot be changed without violating the condition of equal shear rates. 2 x 3 manifold W yx y, where y yx W W R 3 2 0 0 4 land 0 If the maximum length of the die is to be supplied by the user rather than calculated by the design software, another approach must be taken. Assuming equal pressure gradients along different flow paths, we obtain the following relation for the radius: p x land p x manifold x 8 manifold V0 W 12 dy x land V0 4 3 R x W dx 2 3 W manifold x x Rx y0 land x W 4/3 1/4 (8) (9) (10) (11) (12)

The viscosity ratio can be expressed by the radius if the definition of a representative viscosity is used. n1 2 manifold x K manifold x manifold x 4 e x n 1 3 land x K land x land x 3 e R x Inserting equation (13) into equation (12) provides an implicit formulation of the radius. To calculate the course of the radius for a given geometry, we must use an iterative scheme. Thus, we applied a so-called fixed point iteration: given a function f defined on the real numbers with real values and given a point x 0 in x x, n 0,1, 2,, which is to Journal of Plastics Technology 8 (2012) 1 60 n1 the domain of f, the fixed point iteration is converge to a point x. The fishtail manifold was analyzed in a similar manner. However, the length of the land across the entire width was assumed to be linear. y0 y x n1 n n1 (13) x (14) W The condition of equal pressure gradients along different flow paths provides the following relationship: p p 1 y0, where arctan y l sin W land manifold (15) Using representative data in analogy to the procedure described in equation (12) yields the conditional equation for the radius: R x 1 n1 4 2 3 2 4 e x x 3 3 3 e R x sin Again, a fixed point iteration was used to find the course of the radius in the manifold. In the tee manifold, the length of the land is constant (y = y 0 ) along the entire width of the die, and sin() in equation (15) becomes 1. (16)

3 NETWORK ANALYSIS The basic concept that underlies network analysis is graph theory. From a physical point of view, network analysis deals with predicting the behavior of a system of interconnected physical elements in terms of the characteristics of the elements and the manner in which these elements are interconnected. Any electrical network can be represented by a graph a model of the physical network. In mathematical terms, a directed graph is a pair G(V,A) of a set V whose elements are called vertices or nodes, and a set A of ordered pairs of vertices called arcs, directed edges, or arrows. A subgraph is called a tree if it is connected and includes all vertices but no loops. In other words, any connected graph without cycles is a tree. A tree consists of v-1 tree branches and b-v+1 links (v and b denoting the number of vertices and branches respectively). Figure 3 illustrates how an electrical network can be characterized by a tree of a directed graph. Figure 3: Characterization of an electrical network by a tree of a directed graph. Journal of Plastics Technology 8 (2012) 1 61

A fundamental loop includes a number of tree branches but only one link. The number of links is equal to the number of fundamental loops. A fundamental cut-set includes a number of links but only one tree branch. A tree of a connected graph has v-1 fundamental cut-sets. The purpose of using this systematic approach is to find a set of linearly independent equations for solving the resistance network. As in conventional network analysis, there are two possibilities: mesh analysis, which makes use of fundamental loops, and nodal analysis, which makes use of fundamental cutsets. In this contribution, nodal analysis using Kirchhoff s current law for each fundamental cut-set was employed. Therefore, all the branch constitutive relations must have admittance representations analogous to those of the die conductances in equation (4). As a consequence, all branch currents incident on a node must sum to zero. Figure 4 displays the resistance network of the current flow problem and its representation as a tree of a directed graph. In this case, we can take advantage of the symmetry and perform network analysis for half of the die. For illustration purposes, the geometry is divided into five sections (n=5), which results in a flow resistance network consisting of eleven (2n+1) elements. Figure 4: Resistance network of a slit-exit cross-section die and the tree of its directed graph. Analysis of this network yields the following cut-sets: cut-set no. 1: i i 0 1 6 cut-set no. 2: -i i i 0 1 2 7 cut-set no. 3: -i i i 0 2 3 8 cut-set no. 4: -i i i 0 3 4 9 cut-set no. 5: -i i i 0 4 5 10 cut-set no. 6: -i i 0 5 11 (17) Journal of Plastics Technology 8 (2012) 1 62

This system of cut-sets can be summarized in the form of a fundamental cut-set matrix Q. In this matrix, the rows correspond to the cut-sets and the columns to the edges of the graph. Each row of Q is called a cut-set vector. Q has (n+1)(2n+1) elements when half a die is divided into n sections. Q i 0 i1 i 2 i 3 1 0 0 0 0 1 0 0 0 0 0 0 i4 1 1 0 0 0 0 1 0 0 0 0 0 i 5 0 1 1 0 0 0 0 1 0 0 0 0 i6 0 0 1 1 0 0 0 0 1 0 0 0 i 7 0 0 0 1 1 0 0 0 0 1 0 0 i8 0 0 0 0 1 0 0 0 0 0 1 0 i 9 i 10 i11 The link voltages e i (i = 0 n) are a linear combination of the tree branch voltages u i (i = 0 2n+1). The tree branch voltages provide a set of independent voltage variables. Recall that each link is a connection between nodes, and hence each link voltage can be expressed purely in terms of tree branch voltages. 3 1 u4 0 0 0 1 1 0 e 2 u 5 0 0 0 0 1 1 e 3 u6 1 0 0 0 0 0 e4 u 7 0 1 0 0 0 0 e 5 u 8 0 0 1 0 0 0 e6 u 9 0 0 0 1 0 0 (18) Journal of Plastics Technology 8 (2012) 1 63 T u Q e u1 1 1 0 0 0 0 u 2 0 1 1 0 0 0 u 0 0 1 1 0 0 u 10 0 0 0 0 1 0 u11 0 0 0 0 0 1 e (19) A typical network branch is illustrated in Figure 5. The branch currents can be calculated if the branch voltages and the conductances are known.

Figure 5: A subnetwork comprising of a general network branch. Using tree branch currents, the characteristic equation can be formulated: i G u i Gu T 0 0 Q i Q G u Q i Q Gu 0 0 0 Q GQ e Q i Q Gu 0 0 The essential equation for analyzing an electrical network is obtained by introducing a cut-set admittance matrix Y. This matrix is invertible and allows a set of linearly independent equations to be found. q 1 Y e i e p Y i Y Q GQ i Q i Q Gu q 0 0 As in equation (4), the die conductances C i can be used instead of the electric conductances G i. By replacing the currents with the volumetric flow rate and the voltages with the pressure drop, we can use the same procedure to solve flow problems in complex geometries. (20) Journal of Plastics Technology 8 (2012) 1 64 T q (21) The boundary condition in the simulation is defined by the throughput given by either the extruder or the power source at the die inlet. 0 0 u 0, i0 0 V0 2 G n We implemented a simulation routine based on the described approaches in Wolfram Mathematica 8. The network analysis itself provides the distribution of the pressure drop along the die width. The mass flow and residence time distributions must be calculated separately. (22)

In order to calculate the residence time using equation (3), we must determine the mass flow both at the outlet of the die and inside the manifold. The total time in each cell can be determined by elementwise division of the volume by the volumetric flow rate. To calculate the residence time, the time steps in manifold and outlet must be summed up in the correct manner. 4 FINITE ELEMENT SIMULATION To verify the results obtained by the analysis described above, we used the commercially available software ANSYS 13 Polyflow to carry out a threedimensional simulation. The material used in the simulation was Borealis HC600TF, which is a polypropylene homopolymer intended for packaging applications. The viscosity was measured by High Pressure Capillary Rheometry using a Göttfert Rheograph 2002. The end pressure drop was determined by means of Bagley s end correction method [10]. In order to fit the experimental data, the Ostwald-de Waele power law was applied. To generate the complex geometry, a series of points was created using MATLAB (computational accuracy 10-8 ) and exported to a journal file processed further in GAMBIT 2.4.6. The meshed geometry is shown in Figure 6. It consists of approximately 245 000 cells, and the maximum equisize skew was 0.7, which is acceptable in finite element simulations. The manifold and the land area were meshed using the pave algorithm and a mapping scheme respectively. Figure 6: Meshed geometry of coathanger die. For the manifold boundary layers were used as a refinement near the wall. In addition, the cooper scheme was applied to generate a volume mesh based on source faces. The pave algorithm only allows little scope for change by the user and produces significant cell size changes in the transition area of the circular section manifold to the rectangular slit. Journal of Plastics Technology 8 (2012) 1 65

The total number of cells can only be varied in a relatively narrow range: A refinement of the mesh leads to highly skewed cells at the edge of the flat die. However, coarsening of the mesh reduces the number of cells in the rectangular section slot gap. To ensure convergence, the under-relaxation factors were changed to 0.2 for pressure and 0.5 for momentum. A first-order upwind scheme was used to calculate an initial condition for the second-order simulation, which was performed for half of the die (taking symmetry into account). The boundary conditions were: mass flow is 250 kg/h at the inlet, the static pressure is zero at the outlet, and a symmetry condition is applied at the symmetry plane. All parameter settings are summarized in Table 1. Table 1: Geometry: half width of flat die slit height total length of land area maximum radius of pipe die lip Process data: mass throughput List of parameters used in the simulation. W = 500 mm = 4 mm y 0 = 70 mm R 0 = 20 mm y lip =15 mm and lip =1.5 mm ṁ = 250 kg/h density = 730 kg/m³ @ T = 210 C consistency factor K = 26 500 Pas n @ T = 210 C flow index n = 0.22 Both the velocity distribution and the pressure loss were thus obtained. Stagnation areas, which are caused by the abrupt change in geometry between the circular section manifold and the rectangular section slot gap, may exist in the coathanger die. Compensatory measures such as switching from circular section to tear-drop section manifolds are possible. Journal of Plastics Technology 8 (2012) 1 66

5 COMPARISON OF RESULTS The network analysis used the same parameters as the finite element simulation in order to make the results comparable. A grid independency test was also performed. The geometry was divided into different numbers of sections. In order to evaluate the influence of the mesh on the obtained results, pressure drops and residence times were compared. It was found out that a number of 150 sections per width is adequate which results in a mesh with 299 cells. As mentioned before, symmetry was taken into account. The developed simulation routine provides four types of results: the mass flow rate at a specific position at the outlet, the residence time of a tracking particle at the same position, the shear rate in the distribution channel, and the total pressure drop of the die. The configuration of the T-type manifold requires the melt to travel a much greater distance, which increases the residence time. The performance of the tee manifold is worse than that of the coathanger or that of the fishtail manifold in terms of uniformity of both mass flow and residence time, as illustrated in Figure 7 and Figure 8. The coathanger die surpasses both the T- and the fishtail manifold in terms of shear rate (Figure 9). Determination of the contour of the distribution channel requires the representative shear rate in the coathanger manifold to be constant. Unfortunately, it is near impossible to achieve uniform wall shear rate and uniform residence time simultaneously. Figure 7: Comparison of slit-exit cross-section dies by mass flow rate. Journal of Plastics Technology 8 (2012) 1 67

Figure 8: Figure 9: Comparison of slit-exit cross-section dies by residence time. Comparison of slit-exit cross-section dies by shear rate in the distribution channel. Journal of Plastics Technology 8 (2012) 1 68

The results obtained by network analysis were compared with those of the Polyflow simulation to verify the validity of the developed code in practice. With respect to velocity distribution, the results are in good agreement. Polyflow gives a maximum velocity of 0.1 m/s at the center of the die and a minimum velocity of 0.02 m/s at the edge. The velocities obtained by network analysis are between 0.07 m/s and 0.05 m/s which fits with the range above, even though the melt is distributed more uniformly (Figure 10). Figure 11 illustrates the contours of static pressure in the coathanger die. The total pressure drop is 66 bar. The pressure drop depends on both the quality of the mesh and the applicability of the rheological model used in the simulation. Figure 10: Velocity distribution at the outlet of the coathanger die. Journal of Plastics Technology 8 (2012) 1 69

Figure 11: Total pressure drop in the coathanger die. The pressure drop predicted by network analysis is 51 bar, which is lower than that obtained by finite element analysis. This difference may be caused by the method of a representative viscosity which was applied in the present simulation. In order to find a second reference value, the flat die was considered as a slit, and the total pressure drop was calculated according to the following analytical equation: 12 V0 12 V0 p p y p y y 3 3 2 W 2 W total land 0 lip 0 lip lip Thus, the pressure loss in the coathanger manifold can be estimated using equation (9) and setting y(x) to y 0. This gives an analytical estimate of 47 bar, which is very close to the result of the network analysis. The analytical analysis of coathanger dies is two-dimensional, which implies that the influences of side walls and of a sudden decrease in channel height are ignored. Since the network analysis approach is based on the same assumptions, obtaining comparable results is reasonable. However, network analysis is the only means using analytical formulae to compute data distributions across the width. (23) Journal of Plastics Technology 8 (2012) 1 70

6 CONCLUSION The isothermal flow of material in three types of flat dies was simulated numerically using an in-house code based on network analysis. For comparison, we carried out a three-dimensional simulation using the commercially available CFD software ANSYS 13 Polyflow, which is based on the finite element method. The results of these simulations are in good agreement with the finite element analysis, although some improvements may be possible. The differences are related to the quality of the mesh created in GAMBIT 2.4.6 and to the assumptions made in the network analysis code. Coathanger manifolds provide the advantage of a constant shear rate in the distribution channel. In fishtail and tee manifolds, compromises must be made regarding mass flow, residence time, shear rate, and total length of the die. Network analysis provides an efficient alternative to complex FEM software in terms of computing power and memory, since high precision is required to obtain good results. Although the focus of this contribution is on die design, this technique can also be applied to the design of other components, such as barrier screws and static mixers. Furthermore, some improvements concerning non-isothermal flow might be possible. Molten polymers are highly viscous but have low thermal conductivity, and heat generated by viscous forces represents an efficient mode of heating. In order to take viscous heating into account, a coupled momentum and energy equation must be included. Journal of Plastics Technology 8 (2012) 1 71

NOMENCLATURE e rep. distance for slit K consistency factor e rep. distance for pipe Q fundamental cut-set matrix e i link voltage R 0 maximum radius of manifold i i branch current V 0 half volumetric flow rate m mass throughput W half width of the die n flow index Y cut-set admittance matrix t res residence time representative shear rate u i branch voltage slit height of die land y 0 total length of the die lip slit height of die lip y lip length of die lip representative viscosity C die conductance density G electric conductance p pressure loss ACKNOWLEDGEMENTS Financial support by the Austrian Center of Competence in Mechatronics GmbH (ACCM) is gratefully acknowledged. Part of this work was performed within K- Project "Advanced Polymeric Materials and Process Technologies" (APMT). ACCM and APMT are funded by the Austrian Government and the State Government of Upper Austria within the COMET program. Journal of Plastics Technology 8 (2012) 1 72

REFERENCES [1] Michaeli, W. Extrusion dies for plastics and rubber: design and engineering computations Hanser Publishers, Munich, 2003 [2] Matsubara, Y. Geometry design of a coathanger die with uniform flow rate and residence time across the die width [3] Sun, Y.; Gupta, M. [4] Lebaal, N.; Schmidt, F.; Puissant, S. [5] Sienz, J.; Bates, S.J.; Pittman, J.F.T. [6] Na, S.Y.; Kim, D.H. [7] Na, S.Y.; Lee, T-Y. [8] Yu, Y.-W.; Liu, T.-J. [9] Michaeli, W.; Pöhler, F. Polymer Engineering & Science 19 (1979) 3, pp. 169 172 Optimization of a flat die geometry In: Proceedings at ANTEC 04, Chicago, Illinois (2004) Design and optimization of three-dimensional extrusion dies, using constraint optimization algorithm Finite Elements in Analysis and Design 45 (2009), pp. 333 340 Flow restrictor design for extrusion slit dies for a range of materials: Simulation and comparison of optimization techniques Finite Elements in Analysis and Design 42 (2006), pp. 430 453 Three-dimensional modeling of non-newtonian fluid flow in a coathanger die Korean Journal of Chemical Engineering 12 (1995) 2, pp. 236 243 Parametric study in design of coathanger die Korean Journal of Rheology 10 (1998) 1, pp. 38 42 A simple numerical approach for the optimal design of an extrusion die Journal of Polymer Research 5 (1998) 1, pp. 1 7 Die Verwendung der Netzwerktheorie bei der Werkzeugauslegung Plaste und Kautschuk 40 (1993) 2, pp. 61 65 [10] E.B. Bagley End corrections in the capillary flow of polyethylene Journal of Applied Physics 28 (1957), pp. 624 627 Journal of Plastics Technology 8 (2012) 1 73

Keywords: network analysis, coathanger manifold, fishtail manifold, tee manifold, flat die, finite element method, computational fluid dynamics Stichworte: Netzwerktheorie, Kleiderbügelverteiler, Fischschwanzverteiler, T-Verteiler, Breitschlitzdüse, Finite Elemente Methode, numerische Strömungssimulation Autor/author: Dipl.-Ing. Thomas Köpplmayr Dipl.-Ing. (FH) Michael Aigner Univ.-Prof. Dipl.-Ing. Dr. Jürgen Miethlinger, MBA Institute of Polymer Extrusion & Building Physics Johannes Kepler University Altenbergerstr. 69 4040 Linz, Austria Herausgeber/Editor: Europa/Europe Prof. Dr.-Ing. Dr. h.c. Gottfried W. Ehrenstein, verantwortlich Lehrstuhl für Kunststofftechnik Universität Erlangen-Nürnberg Am Weichselgarten 9 91058 Erlangen Deutschland Phone: +49/(0)9131/85-29703 Fax.: +49/(0)9131/85-29709 E-Mail-Adresse: ehrenstein@lkt.uni-erlangen.de Verlag/Publisher: Carl-Hanser-Verlag Jürgen Harth Ltg. Online-Services & E-Commerce, Fachbuchanzeigen und Elektronische Lizenzen Kolbergerstrasse 22 81679 Muenchen Tel.: 089/99 830-300 Fax: 089/99 830-156 E-mail-Adresse: harth@hanser.de thomas.koepplmayr@jku.at michael.aigner@jku.at office.ipeb@jku.at Internet: www.jku.at/ipeb Phone: +43 (0) 732 2468 6572 Amerika/The Americas Prof. Prof. h.c Dr. Tim A. Osswald, responsible Polymer Engineering Center, Director University of Wisconsin-Madison 1513 University Avenue Madison, WI 53706 USA Phone: +1/608 263 9538 Fax.: +1/608 265 2316 E-Mail-Adresse: osswald@engr.wisc.edu Beirat/Editorial Board: Professoren des Wissenschaftlichen Arbeitskreises Kunststofftechnik/ Professors of the Scientific Alliance of Polymer Technology Journal of Plastics Technology 8 (2012) 1 74