215Mechanics of Time-Dependent Materials 8: 215–224, 2004.C© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

The Axial Velocity Distribution of a PolyethyleneStrand During Extrusion: Simulation andComparison with Measurements

CHRISTIAN SCHNEIDER, MARTIN SCHWETZ1, HELMUT MUNSTEDT1

and JOACHIM KASCHTA1

FCI Automotive Deutschland GmbH, Rathsbergstraße 25, D-90411 Nurnberg, Germany1Institute of Polymer Materials, University Erlangen-Nurnberg, Martensstrasse 7, D-91058Erlangen, Germany; (E-mail: polymer@ww.uni-erlangen.de)

(Received 13 May 2003; accepted in revised form 1 June 2004)

Abstract. The velocity distribution along the axis of a low-density polyethylene (LDPE) meltstrand extruded through an axisymmetric capillary and drawn by various forces is simulated usingan integral constitutive equation with a PSM damping function (Papanastasiou, Scriven, Macosko,Journal of Rheology, 27: 381–410, 1983). The simulations are performed for different drawdownforces of the strand. The numerical results are compared with experimental data obtained by velocitymeasurements using the laser-Doppler velocimetry. The strand is drawn by rotating wheels as usedin a RheotensTM testing device. At drawdown forces greater than zero the investigations show thatthe strand velocity does not increase linearly with increasing distance from the die exit. Instead, it isobserved that the acceleration of the strand increases monotonically. Except in the next vicinity ofthe die exit there is a good agreement between simulation and experiment. However, near to the diethe simulation predicts a higher strand velocity.

Abbreviations. LDV – Laser-Doppler velocimetry; PSM – Papanastasiou, Scriven, Macosko; K-BKZ –Kaye – Bernstein, Kearsley, Zapas; LDPE – Low-density polyethylene

Key words: extrusion, measurement, melt, polyethylene, simulation, velocity distribution

1. Introduction

Numerical flow simulations are widely used in industry and research as a tool to de-scribe and predict the flow behavior of polymer melts during processing. Commer-cially available simulation tools have been developed, which can be used for differ-ent purposes in injection moulding such as mould design, gate location and materialsselection. The reliability of the results for prototype moulds is good as the compari-son of computational results with experiments (e.g. filling studies) shows. However,numerical flow simulation is not that common in extrusion processes. Probably thecause for that is the fact that at extrusion the melt leaving the die flows without anyboundary conditions of fixed walls, which leads to elastic effects like the extrudateswell. Consequently, more sophisticated models are required. Furthermore, the

216 C. SCHNEIDER ET AL.

comparison of computation and experiment in general is difficult becausemonitoring the flow pattern requires sophisticated flow visualization methods.

Rauschenberger and Laun have published a recursive model for the numeri-cal calculation of the axial velocity distribution in elongated extruded strands ofa polyethylene melt under constant drawdown force (Rauschenberger and Laun,1997). This model is based on a rheological constitutive equation of the K-BKZtype (Kaye, 1962; Bernstein et al., 1963). This constitutive equation is capableof predicting the nonlinear viscoelastic behavior. Using the ideas of Fulchiron(Fulchiron et al., 1995), Rauschenberger and Laun transferred the integral constitu-tive equation into a discrete formulation assuming small cylindrical strand elementsbeing extruded during the first and further stretched during later time units. Via aniteration method the length of subsequent elements at a given drawdown forceincluding all preceding elements can be calculated. The viscoelastic behavior isseparated into a linear and a nonlinear part. The linear viscoelastic behavior reflectsthe time dependence of stress and is calculated from the relaxation time spectrum.The nonlinear deformation dependence is expressed by the damping function. Theaxial velocity distribution can be calculated from the length of each element assum-ing that the volume of each element does not change during the deformation. Thisprocedure leads to the determination of other interesting quantities such as Henckystrain and strain rate, tensile stress, strand acceleration and threadline shape.

A major advantage of the model of Rauschenberger and Laun is that the strainprehistory of the melt in the die is taken into account. Therefore, e.g. extrudateswell can be predicted by the calculation.

In order to prove the reliability of the calculations, however, suitable measuringdevices are needed. To determine the axial velocity profile along the elongatedstrand, for example, a sophisticated contactless method, namely laser-Doppler ve-locimetry (LDV), can be used which has successfully been applied to polymer melts(Kramer, 1980; Mackley and Moore, 1986; Wassner et al., 1999; Schmidt et al.,1999). It provides very accurately two-dimensional velocity measurements with ahigh resolution. This method is used to experimentally verify the calculated veloc-ity distribution along an extruded strand which is drawn down by a RheotensTM

device.

2. Material Characterization

A low-density polyethylene was investigated. The material has rheologically beenwell characterized (Munstedt et al., 1998; Kurzbeck, 1999). Its mass-average molarmass is Mw = 240 000 g/mol, the molar mass distribution is Mw/Mn = 14. Themelting temperature is Tm = 108 ◦C and the activation energy determined in thelinear range of deformation was found as 58 kJ/mol. The relaxation time spectrumat T = 150 ◦C was determined by fitting the viscosity function in shear accordingto Laun (1986). The relaxation strengths gi and relaxation times τi are displayed inTable I.

THE AXIAL VELOCITY DISTRIBUTION OF A POLYETHYLENE STRAND 217

Table I. Relaxation time spectrum of the usedLDPE at 150 ◦C.

i gi (Pa) τi (s)

1 129 000 10−4

2 94 800 10−3

3 58 600 10−2

4 26 700 10−1

5 9 800 100

6 1 890 101

7 591 102

8 46 103

gi are the relaxation strengths, τi represent therelaxation times.

3. Experimental

The polyethylene melt is extruded at a constant exit velocity vExit = v(z = 0) (cf.Figure 1). The flow channel consists of a barrel (length 100 mm, diameter 19 mm)followed by an axisymmetric die (length 30 mm, diameter 3 mm). A schematicdrawing of the experimental setup together with the co-ordinate system is shownin Figure 1. In the following, the velocity component in z direction designated asv is considered. All measurements were performed in the symmetry axis of thestrand r = 0. The extruded strand is drawn by a RheotensTM test device with adefined drawdown velocity vD (Meissner, 1971, 1972). The distance between thedie exit and the axis of the wheels is defined as drawdown length S. It was chosenas 100 mm.

The velocity measurements v(z) were performed by laser-Doppler velocimetry.A detailed description of the LDV system used in the present work is given byWassner or Schmidt et al. (Wassner, 1998; Schmidt et al., 1999).

4. Numerical Method

The method of Rauschenberger and Laun is used for the calculations(Rauschenberger and Laun, 1997). It is assumed that the normal stresses trans-verse to the flow direction of the polymer melt strands are equal. Therefore, thesecond normal stress difference is assumed to be zero and a simplified K-BKZconstitutive equation for uniaxial elongation can be used:

T =∫ t

−∞m(t − t ′) · he(ε) · C−1

t(t ′)dt ′ (1)

where T is stress tensor; t, t ′ are time; m(t − t ′) is memory function; he(ε) isdamping function (for uniaxial elongation); ε is Hencky strain; C is cauchy straintensor.

218 C. SCHNEIDER ET AL.

Figure 1. Schematic drawing of the experimental setup.

For the elongational damping function he(ε), the PSM approach (the abbrevia-tion stands for the initials of the authors (Papanastasiou et al., 1983)) is chosen.

The tensile stress σ is defined by two components of the stress tensor

σ = T11 − T22 (2)

The drawdown force F is given by the following equation

F(t) = σ (t) · π · R2 · e−ε(t) (3)

Where R is the radius of the barrel. ε is the Hencky strain.For computation, the melt strand is subdivided into discrete elements of the

lengths li . Details of the procedure are given in (Rauschenberger and Laun, 1997).This leads to the following expression for the drawdown force

F = π · R2 · eε0 · lb

ln· �t ·

n∑j=−M

w j

a(Tj )

× m

(�t ·

( n∑j=−M

1

a(Ti )−

j∑j=−M

1

a(Ti )

))·(

l2n

l2j

− l j

ln

)

×n

mink= j

[α

(α + β

(l2k

l2j

− l2j

l2k

)+ 2β

(l j

lk− lk

l j

)+ l2

j

l2k

+ 2lk

l j− 3

)−1]

+ π · R2 · eε0 · lb

ln· G

(�t ·

( n∑i=−M

1

a(Ti )

))·(

l2n

l2b

− lb

ln

)

×n

mink= j

[α

(α + β

(l2k

l2b

− l2b

l2k

)+ 2β

(lb

lk− lk

lb

)+ l2

b

l2k

+ 2lk

lb− 3

)−1](4)

THE AXIAL VELOCITY DISTRIBUTION OF A POLYETHYLENE STRAND 219

whereε0 Pre-strain factor (strain of the melt at the die exit correlated to the elon-

gational deformation in the entrance flow contraction). Describes extrudateswell.

R Die radius.lb Element lengths in the barrel (lb = l0 · eε0 = vExit · �t · eε0 )�t Time interval between two observationsli Element lengths of the strand inside and outside the die (i = − M . . . n, M ...

number of elements inside the die, n... number of elements outside of thedie)

a(Ti ) Temperature shift factor for temperature in element iwi Weight factor (Due to the mathematical procedure the length of the sum of

the elements li is bigger than the length of the die plus the drawdown lengthS. In order to compensate for this discrepancy, the first and last element(l−M , ln) are weighted by the factor wi = 0.5. For all other elements wi = 1.).

For the simulations performed in this work in Equation 4 the values α = 24 andβ = 0.022 were used as PSM-parameters according to Rauschenberger and Laun.In the calculations, effects of gravity and inertia were not taken into account.

In order to determine the pre-strain factor ε0 in Equation 4 the drawdown forcewas measured as a function of the drawdown velocity vD by a RheotensTM tester(cf. Figure 1). The thin lines in Figure 2 show this function for the LDPE atT = 180 ◦C and the exit velocity vExit = 10 mm/s. vExit was measured by LDV.The parameter ε0 follows from fitting the function F(vD) (cf. Equation 4 withvD = ln/�t) to the measured RheotensTM curve. The fitted function is shown asbold line in Figure 2. It describes the RheotensTM curve rather well except fordrawdown velocities between about 5 and 40 mm/sec.

As a next step simulations of the strand velocity v(z) were performed for four dif-ferent drawdown velocities (5,50,100 and 150 mm/sec) as shown in the next section.The selected drawdown velocities correspond to the drawdown forces representedas black dots in Figure 2.

5. Results of the Numerical Simulations

Using the parameters according to the last section the velocity of the strand v(z) asa function of the distance from the die exit z can be calculated for a given drawdownforce F . In Figure 3 the results are shown for the four different drawdown forceschosen from Figure 2.

The velocity of the undrawn strand, i.e. F = 0, decreases within the drawdownlength S from 10 mm/sec to about 5 mm/sec. This effect can be related to theextrudate swell. At F > 0 the strand is accelerated with increasing z. The curvesare nonlinear with a slope increasing with growing distance z from the die exit.

220 C. SCHNEIDER ET AL.

Figure 2. Measured and modeled RheotensTM curves.

Figure 3. Modeled axial strand velocity.

6. Comparison of Simulations with LDV Measurements

In order to investigate the validity of the simulations, the velocity functions v(z)were measured by LDV for different drawdown velocity vD. The velocity distri-butions v(z) for the four different drawdown forces are shown in Figure 4. Thesymbols in Figure 4 represent the velocities measured by LDV, the lines representthe simulated values. For the applied drawdown forces there is a good agreement

THE AXIAL VELOCITY DISTRIBUTION OF A POLYETHYLENE STRAND 221

Figure 4. Comparison of the simulated and measured strand velocities as a function of distancefrom the die exit.

between modeling and experiment except in the region very close to the die exit. AtF = 0 the velocity increases over proportional with the distance z. This finding is inagreement with the results of Wagner et al. (2002) who’s similarity model predictsa nonlinear velocity increase, too.

As mentioned above, the simulation for the drawdown force F = 0 shows adecreasing strand velocity v with increasing distance z from the die exit. Thisbehavior could be observed by the LDV measurements, too, as is shown in Figure 4.However, the agreement becomes worse approaching the die exit as the higherresolution of the velocity profiles in Figure 5 demonstrates. The measured strandvelocity v is represented by the symbols, the simulation by the full line. The symbolsrepresent the average of three measurements, the error bars give an indication oftheir accuracy. Measurement and simulation show qualitatively the same behavior,a steep velocity decrease near the die exit, followed by a more or less constantvelocity with increasing z. This behavior can be attributed to the extrudate swellwhich is very pronounced at the die exit. The agreement between simulation andexperiment is poor in the vicinity of the die exit. The simulated velocities are toohigh.

At the drawdown force of F = 21 cN the LDV measurements show a velocitydecrease very near to the die exit followed by an increase at larger distances z,as shown in Figure 6. Qualitatively the same behavior was observed for the otherdrawdown forces.

As for F = 0, the velocity minimum observed can be attributed to the extrudateswell. Again, the modeled velocities near the die exit at drawdown forces F > 0are too high and they do not show a minimum (cf. Figure 6).

222 C. SCHNEIDER ET AL.

Figure 5. Measured and calculated strand velocities without drawdown force.

Figure 6. Measured and calculated strand velocity at drawdown force 21 cN.

7. Conclusion

As a general result of the investigations performed in this work, there is goodaccordance of axial strand velocity profiles measured by LDV and the ones modeledby means of the Rauschenberger and Laun method. The velocities of strands pulledby a RheotensTM device do not increase linearly with distance from the die exit.The velocity growth is over proportional. Near to the die exit the velocity decreasesor exhibits a minimum, respectively, depending on the drawdown force. This effectcan be attributed to the extrudate swell. The velocity decrease in case of drawdownforce F = 0 is predicted fairly well. However, for nonvanishing drawdown forces

THE AXIAL VELOCITY DISTRIBUTION OF A POLYETHYLENE STRAND 223

the model does not describe the velocity minimum near to the die exit. Further workis needed for modeling these findings.Generally, it is shown that laser-Doppler velocimetry is a very powerful tool tocheck modeling predictions concerning the flow of polymer melts. Moreover, theresults demonstrate how difficult it is to describe the flow behavior of polymerseven in the relatively simple deformation process investigated in this paper.

Acknowledgements

One of the authors, Christian Schneider, would like to acknowledge the financialsupport of Siemens AG, Erlangen, Germany.

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