M Entry Flow of Polyethylene Melts in Tapered Dies

M. Ansari1, A. Alabbas1, S. G. Hatzikiriakos1, E. Mitsoulis2*

1 Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada2 School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, Athens, Greece

Entry Flow of Polyethylene Melts in Tapered Dies

The excess pressure losses due to end effects (mainly entrance)in the capillary flow of several types of polyethylenes were stud-ied both experimentally and numerically under slip and no-slipconditions. These losses were first measured as a function ofthe contraction angle ranging from 158 to 908. It was found thatthe excess pressure loss attains a local minimum at a contrac-tion angle of about 308 for all types of polyethylenes examined.This was found to be independent of the apparent shear rate.This minimum becomes more dominant under slip conditionsthat were imposed by adding a significant amount of fluoro-polymer into the polymer. Numerical simulations using a multi-mode K-BKZ viscoelastic model have shown that the entrancepressure drops can be predicted fairly well for all cases eitherunder slip or no-slip boundary conditions. The clear experimen-tal minimum at about 308 can only slightly be seen in numericalsimulations, and at this point its origin is unknown. Furthersimulations with a viscous (Cross) model have shown that theyseverely under-predict the entrance pressure by an order ofmagnitude for the more elastic melts. Thus, the viscoelasticspectrum together with the extensional viscosity play a signifi-cant role in predicting the pressure drop in contraction flows,as no viscous model could. The larger the average relaxationtime and the extensional viscosity are, the higher the differencesin the predictions between the K-KBZ and Cross models are.

1 Introduction

When a molten polymer flows through a contraction of a givenangle, there is a large pressure drop associated with such flow,known as entrance pressure or Bagley correction (Bagley,1957; Dealy and Wissbrun, 1990). This pressure is required inorder to calculate the true shear stress in capillary flow and alsofrequently the apparent extensional rheology of molten poly-mers, a method well practiced in industry (Cogswell, 1972,1981; Binding 1988, 1991). Therefore, it is important to under-stand the origin of this excess pressure and consequently to beable to predict it. Accurate prediction of this pressure mightserve as a strong test for the predictive capabilities of a consti-tutive equation.

Many studies have previously attempted to examine the ori-gin of entrance pressure and its prediction. Feigl and Öttinger

(1994) have simulated the axisymmetric contraction flow of alow-density (LDPE) melt using a Rivlin-Sawyers constitutivemodel. However, their numerical results well under-predictedthe available experimental pressure data for entry flows. Bara-kos and Mitsoulis (1995a; 1995b) reported similar findings forthe Bagley correction in capillary flow of the IUPAC low-den-sity polyethylene (LDPE). Béraudo et al. (1996) used a multi-mode Phan-Thien/Tanner (PTT) constitutive relation andfound that numerical predictions significantly under-estimatedthe experimental findings. Guillet et al. (1996) also studiedthe entrance pressure losses for a linear low density (m-LLDPE) and a low-density polyethylene (LDPE) melt both ex-perimentally and numerically using a multimode K-BKZ inte-gral constitutive equation. Again significant under-estimationwas reported. Using a K-BKZ constitutive relation, Hatzikiria-kos and Mitsoulis (1996, 2003) and Mitsoulis et al. (1998) alsofound that the numerical predictions significantly under-esti-mated the experimental data for the various geometries usedto determine the entrance pressure. From the above studies, itis clear that state-of-the-art numerical simulations cannot pre-dict quantitatively the pressure drop in a relatively simple flow,such as the entry capillary flow of a polymer melt. Consideringthe fact that experimental measurements from such flows areextensively used in industrial practice to calculate the shearand extensional viscosity of polymer melts at high shear rates(Dealy and Wissbrun, 1990; Cogswell, 1972; Binding 1988,1991; Padmanabhan and Macosco, 1997), it is essential to un-derstand the origin of these disagreements and furthermore beable to predict the excess entrance pressure.

One important aspect for entry contraction flows is the var-iation of entrance pressure as a function of contraction angleat a given apparent shear rate under slip or no-slip boundaryconditions. This was studied by Mitsoulis and Hatzikiriakos(2003) for a branched polypropylene (PP) melt both experi-mentally and theoretically. The entrance pressure was first de-termined experimentally as a function of the contraction angleranging from 108 to 1508. It was found that at a given apparentshear rate, the pressure loss decreases with increasing contrac-tion angle from 108 to about 458, and consequently slightly in-creases from 458 up to contraction angles of 1508. Numericalsimulations using a multimode K-BKZ viscoelastic were usedto predict the pressures. It was found that the numerical predic-tions do agree well with the experimental results for small con-traction angles up to about 308. However, the numerical simu-lations under-predicted the end pressure for larger contractionangles. The importance of the existence of a minimum in thevariation of excess pressure as a function of contraction anglewas also mentioned by Hatzikiriakos and Mitsoulis (2009).

REGULAR CONTRIBUTED ARTICLES

Intern. Polymer Processing XXV (2010) 4 � Carl Hanser Verlag, Munich 287

* Mail address: Evan Mitsoulis, School of Mining Engineering andMetallurgy, National Technical University of Athens, Zografou,Athens, GreeceE-mail: [email protected]

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M. Ansari et al.: Entry Flow of PE Melts in Tapered Dies

This minimum was found to be more pronounced under slipconditions both numerically and experimentally. Experimentaldata reported by Mitsoulis and Hatzikiriakos (2003) haveshown the existence of a minimum at around 408. Such pro-nounced minima have also been reported in the flow of PTFEpastes several times (Horodin, 1998; Gibson, 1998; Ariawanet al., 2002; Ochoa and Hatzikiriakos, 2005).

It is the main objective of this work to study the dependenceof entrance pressure as a function of entrance angle for varioustypes of polyethylenes, namely high-density, low-density andlinear low-density polyethylenes. It is also our goal to studythe dependence of entrance pressure on contraction angle un-der strong slip conditions in order to understand its origin. Asmall amount of a fluoropolymer is added into one of the poly-ethylenes to promote slip (Achilleos et al., 2002). It is notedthat by promoting slip, the relative contribution of the exten-sional viscosity to flow increases relative to shear, and this willhelp us understand the nature of this flow. Finally, using theK-BKZ constitutive equation in numerical simulations, thecapability of this model to predict the entrance pressure is stud-ied, particularly in view of previous studies that well under-predicted it (Feigl and Öttinger, 1994; Barakos and Mitsoulis,1995a; 1995b; Béraudo et al. 1996; Guillet et al. 1996; Hatzi-kiriakos and Mitsoulis, 1996; and Mitsoulis et al., 1998).

2 Experimental

2.1 Materials

Three different polyethylenes were used in this work in orderto address molecular-structure effects with emphasis on the ef-fects of Long Chain Branching (LCB) on entry pressure drop.It is known that branched polymers exhibit extensional strain-hardening effects, and therefore increased entrance pressurevalues are obtained compared to those obtained for their linearcounterparts (Hatzikiriakos and Mitsoulis, 1996; Mitsouliset al., 1998; Mitsoulis and Hatzikiriakos, 2003). This is ex-pected as the entry pressure has been theoretically associatedwith extensional viscosity (Cogswell, 1972; Binding, 1988,1991; Padmanabhan and Macosco, 1997). First, a metallocenelinear low-density polyethylene (labelled here as m-LLDPE)was used, which is a butane-copolymer supplied by ExxonMo-bil (Exact 3128) of MW about 80 kg/mol and PDI of about 2. Ahigh-density polyethylene (labeled here as HDPE) was alsoused, which is a metallocene resin obtained from Chevron-Phillips Chemical Company of high molecular weight of about263 kg/mole (Ansari et al., 2009). The low-density polyethyl-ene (LDPE) resin used was also of high MW with significant

amount of Long-Chain Branching (LCB), which results in alow Melt Flow Index (MFI) of about 0.5. Some of the proper-ties of these resins are summarized in Table 1.

To study the effect of slip on the dependence of entrancepressure on contraction angle, 1.0 wt.% of a fluoropolymer(Dynamar 9613 from Dyneon) was added into m-LLDPE indry mixing form in order to enhance its slip in the capillaryflow, particularly in the entrance. A comparison of the flowcurves with and without fluoropolymer could quantify the slipvelocity of the polymer, and this is presented below.

2.2 Rheological Testing

Several rheological experiments were carried out in order tocharacterise the polymers in the molten state. First, an AntonPaar MCR-501 device was used in the parallel-plate geometryto determine their linear viscoelastic moduli over a wide rangeof temperatures from 130 to 230 8C. Master curves were ob-tained using the time-temperature superposition principle(TTS), and the results are presented at the reference tempera-ture of 160 8C. Creep experiments were performed at 160 8Cto obtain the zero-shear viscosity values. A constant shearstress of 10 Pa was used to attain very low shear rates in orderto reach the Newtonian viscosity flow regime. These valuesare reported in Table 1 and are found to compare well withthose obtained from linear viscoelastic measurements.

The polymers were also rheologically characterized in sim-ple extension using the SER-2 Universal Testing Platform fromXpansion Instruments (Sentmanat, 2003, 2004). Details ofsample preparation and operating principles of this rheometerto obtain reliable results are given in Delgadillo et al. (2008a,2008b). Uniaxial extension tests at the temperature of 160 8Cwere performed for all polymers.

2.3 Entrance Pressure

An Instron capillary rheometer (constant piston speed) was usedto determine the entrance pressure (known also as Bagley meth-od) (Dealy and Wissbrun, 1990) and the viscosity as a functionof the wall shear stress, rW, and apparent shear rate,_cA ¼ 32Q=pD3 all at 160 8C, where Q is the volumetric flowrate and D is the capillary diameter. A circular die of diameterequal to 0.02’’ (0.0508 cm), length-to-diameter ratio, L/D = 20,and a tapered entrance angle of 908 was used to determine theflow curves of the resins. A series of orifice dies (L/D = 0) wereused to determine directly the entrance pressure (Bagley correc-tion) as a function of the apparent shear rate and contraction an-

288 Intern. Polymer Processing XXV (2010) 4

Sample ID Resin type Melt index (190 8C)g/10 min

Density (25 8C)g/cm3

Zero-shear-rate viscosity(160 8C) Pa · s

m-LLDPE Exact 3128 1.3 0.900 10,594HDPE CPChem – 0.935 254,460LDPE DOW 662I 0.47 0.919 185,620

Table 1. Properties of polyethylene resins used in this study

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gle with angles of 108, 308, 458, 608, and 908. Hatzikiriakos andMitsoulis (1996) and Kim and Dealy (2001) clearly demon-strated that by using the standard Instron orifice dies, the en-trance pressure drop can be artificially high when the melt fillsthe exit region, which often happens with elastic melts that pos-sess high extrudate (die) swell. Therefore, precautions were tak-en during the experiments so that the exit region remains un-filled without the melt touching its walls. Kim and Dealy(2001) proposed a new design for orifice dies to easily avoid fill-ing of the exit region of the die from happening.

3 Constitutive Equation and Rheological Modeling

The constitutive equation used in the present work to solve theusual conservation equations of mass and momentum for an in-compressible fluid under isothermal conditions is a K-BKZequation proposed by Papanastasiou et al. (1983) and modifiedby Luo and Tanner (1988). This is written as:

s ¼ 11� h

Z t

�1

XN

k¼1

ak

kkexp � t� t0

kk

� �a

ða � 3Þ þ bIC�1þð1� bÞIC

� C�1t ðt0ÞþhCtðt0Þ

� �dt0; ð1Þ

where kk and ak are the relaxation times and relaxation modu-lus coefficients, N is the number of relaxation modes, a and bare material constants, and IC, IC

– 1 are the first invariants ofthe Cauchy-Green tensor Ct and its inverse Ct

– 1, the Fingerstrain tensor. The material constant h is given by

N2

N1¼ h

1� h; ð2Þ

where N1 and N2 are the first and second normal stress differ-ences, respectively. It is noted that h is not zero for polymermelts, which possess a non-zero second normal stress differ-ence. Its usual range is between –0.1 and – 0.2 in accordancewith experimental findings (Dealy and Wissbrun, 1990).

For the capillary flow simulations the effect of pressure onviscosity should be taken into account as this becomes moreevident below. This effect is quite significant for LDPE com-pared to m-LLDPE and less significant for HDPE (Zetlaceket al., 2004; Carreras et al., 2006). This effect can be taken intoaccount by multiplying the constitutive relation with a shiftfactor, ap, given by the following equation (Carreras et al.,2006)

ap ¼ exp bPpð Þ; ð3Þwhere bP is the pressure coefficient and p is the absolute pres-sure. The values of bP used in this work are 18.33 GPa – 1,11.72 GPa – 1 and 10.36 GPa – 1 for LDPE, m-LLDPE andHDPE, respectively (Zetlacek et al., 2004; Carreras et al., 2006).

As discussed above, experiments were performed in the par-allel-plate and extensional rheometers for all polyethylenes torheologically characterise them. Figs. 1A to C plot the masterdynamic moduli G’ and G’’ of all three polyethylenes at the ref-erence temperature of 160 8C. The model predictions obtainedby fitting the experimental data to Eq. 1 with a spectrum ofrelaxation times, kk, and coefficients, ak, determined by anon-linear regression package (Kajiwara et al., 1995), are also


Intern. Polymer Processing XXV (2010) 4 289

A)

B)

C)

Fig. 1. Experimental data (symbols) and model predictions of storage(G’) and loss (G’’) moduli for the polyethylenes at 160 8C using the re-laxation times listed in Table 2, (A) m-LLDPE, (B) HDPE and (C) LDPE

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plotted. The parameters found from the fitting procedure arelisted in Table 2. The relaxation spectrum is used to find theaverage relaxation time, �k, and zero-shear-rate viscosity, g0,according to the formulas:

�k ¼XN

k¼1

akk2k

akkk; ð4Þ

g0 ¼XN

k¼1

akkk: ð5Þ

Figs. 2A to C plot a number of calculated and experimental ma-terial functions for the three melts at the reference temperatureof 160 8C. Namely, data for the shear viscosity, gS, the elonga-tional viscosity, gE, and the first normal stress difference, N1,


m-LLDPE HDPE LDPE

a = 1.319, b = 0.02, h = 0 a = 10.15, b = 0.6, h = 0 a = 1.51, h = 0

�k = 0.46 s, g0 = 11,048 Pa · s �k = 18.9 s, g0 = 235,393 Pa · s �k = 298 s, g0 = 201,804 Pa · s

k kk (s ) ak (Pa) kk (s ) ak (Pa) kk (s ) ak (Pa) bk

1 1.00 · 10 – 3 5.00 · 105 0.902 · 10 – 8 0.236 · 109 1.00 · 10 – 5 4.00 · 106 12 7.39 · 10 – 3 3.83 · 105 0.131 · 10 – 4 0.576 · 107 1.47 · 10 – 3 1.08 · 105 13 4.52 · 10 – 2 87,778 0.442 · 10 – 2 99,508 1.09 · 10 – 2 53,017 0.184 0.1829 11,602 0.17797 92,960 6.96 · 10 – 2 28,623 0.455 0.7735 1,241 0.0282 0.104 · 106 0.4723 14,623 0.49 · 10 – 2

6 5.683 117.6 1.1834 46,958 4.465 5,246 0.0267 7.0756 12,169 45 1,180 0.24 · 10 – 2

8 51.243 1,438.6 500 231 0.014

Table 2. Relaxation spectra and material constants for polyethylene resins obeying the K-BKZ model (Eq. 1) at 160 8C

A)

B)

C)

Fig. 2. Experimental data (solid symbols) and model predictions of shearviscosity, gS, first normal stress difference, N1, and elongational viscosity,gE, for the polyethylenes at 160 8C using the K-BKZ model (Eq. 1) with theparameters listed in Table 2, (A) m-LLDPE, (B) HDPE, and (C) LDPE

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are plotted as functions of corresponding rates (shear or exten-sional). The parameter b that controls the calculated elonga-tional viscosity was fitted by using the extensional behaviourof the three polymers. For the m-LLDPE and HDPE, the tensilestress growth coefficients follow the linear viscoelastic envel-ope defined by 3gþ (not plotted here), consistent with their lin-ear macrostructure (Hatzikiriakos, 2000; Dealy and Larson,2006). Fig. 3 shows the extensional behaviour of the LDPE atseveral Hencky strain rates at 160 8C and the model predictionsof Eq. 1 using multiple b-values listed in Table 2. It can be seenthat the overall rheological representation of all resins is excel-lent.

To evaluate the role of viscoelasticity in the prediction ofBagley correction, it is instructive to also consider purely vis-cous models in the simulations. Namely, the Cross model wasused to fit the shear viscosity data of the three melts. The Crossmodel is written as (Dealy and Wissbrun, 1990)

g ¼g0;C

1þ ðk _cÞ1�n ; ð6Þ

where g0,C is the zero-shear-rate viscosity of the Cross model,k is a time constant, and n is the power-law index. The fittedviscosity of the three melts by Eq. 6 is plotted in Fig. 4, whilethe parameters of the model are listed in Table 3. We observethat of the three melts, HDPE is the most viscous, followed byLDPE and then by m-LLDPE at low range of shear rates. Them-LLDPE melt is more viscous than the LDPE at high shearrates, while the LDPE melt is the least viscous for shear ratesabove 1 s – 1 due to its significant shear thinning that is due to

the presence of long chain branching (Dealy and Larson,2006). The Cross model fits the data well over the range of ex-periment results.

In the case of slip effects, the usual no-slip velocity at thesolid boundaries is replaced by a slip law of the following form

usl ¼ �kslrbw; ð7Þ

where usl is the slip velocity, rw is the shear stress at the diewall, ksl is a slip coefficient, and b is the slip exponent. In 2-Dsimulations, this means that the tangential velocity on theboundary is given by the slip law, while the normal velocity isset to zero, i. e.,

kslðtn : sÞb ¼ t � v; n � v ¼ 0; ð8Þwhere n is the unit outward normal vector to a surface, t is thetangential unit vector in the direction of flow, s is the extrastress tensor and v is the velocity vector. Implementation ofslip in similar flow geometries for a polypropylene (PP) melthas been also carried out in one of our previous works (Mitsou-lis et al., 2005).

4 Experimental Results

4.1 Entrance (End) Pressure

Figs. 5A to C plot the entrance pressure (or end pressure due toL/D = 0) of all polyethylenes at 160 8C as a function of the en-trance angle for an extended range of values of the apparentshear rate from 75 s – 1 to 1000 s – 1. The entrance pressure de-creases with increasing contraction angle from 108 to about308–458 and subsequently increases up to contraction anglesof 608 with a small drop at 908 (more significant in the case ofm-LLDPE). This behavior is consistent for all polyethylenesand also in agreement with other reported observations in theliterature, i. e., for a PP melt (Mitsoulis and Hatzikiriakos,



Fig. 3. Extensional data for LDPE 662I at 160 8C and their best fitusing the K-BKZ model (Eq. 1) with the parameters listed in Table 2

Fig. 4. The shear viscosity of the three polymer melts at 160 8C fittedwith the Cross model using the parameters listed in Table 3

Parameter m-LLDPE HDPE LDPE

g0,C 11,034 Pa · s 245,153 Pa · s 201,000 Pa · sk 0.067 s 4.577 s 100 sn 0.26 0.30 0.35

Table 3. Parameters for the three polymer resins obeying the Crossmodel (Eq. 6) at 160 8C

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2003). The increase of the entrance pressure at high contractionangles is associated with the increased significance of exten-sional contributions relative to shear ones at these high en-trance angles (Cogswell, 1972; Binding, 1988, 1991).

Fig. 6 plots the entrance pressure of m-LLDPE with the addi-tion of 1.0 wt.% fluoropolymer (Dynamar 9613) at 160 8C as afunction of the entrance angle for several values of the apparentshear rate in the same range as before (75 s – 1 to 1000 s – 1). Thisgraph is to be compared with Fig. 5A. The decrease of the en-trance pressure at 908 is not as dominant as it was in the absenceof fluoropolymer or absence of wall slip. It seems that exten-sional rheology plays a significant role here and the entrancepressure under slip is predominantly due to extensional defor-mation (Collier, 1994; Collier et al., 1998; Shaw, 2003). Thedrop is observed under weak extensional elements of the flowas it is expected for the case of an m-LLDPE melt. However, un-der slip conditions, extensional deformation becomes dominantover shear in the contraction region, and this is the reason thatat 908 the drop in pressure is not as significant. The same is ob-served for the cases of LDPE (strong extensional strain-harden-ing deformation due to LCB) and HDPE (stronger extensionaleffects compared to m-LLDPE due to a much higher molecularweight) (Hatzikiriakos, 2000; Dealy and Larson, 2006).

4.2 Wall Slip

Fig. 7 plots the flow curves for m-LLDPE with and withoutfluoropolymer to assess slip. For the flow curve of m-LLDPEwith no fluoroelastomer, the Bagley and Rabinowitch correc-tions as well as the pressure coefficient correction (Eq. 3) havebeen applied (Dealy and Wissbrun, 1990) in order to compareit with the flow curve obtained from linear viscoelastic measure-ments. The linear viscoelastic data are plotted in the form offlow curve as shear stress (|g*(x)|x) vs. shear rate (rotationalfrequency, x, in rad/s). It can be seen that this flow curve agreeswith that obtained from capillary flow with no fluoropolymer upto high shear rates close to the region where sharkskin melt frac-


A)

B)

C)

Fig. 5. The end pressure of all polyethylenes at 160 8C as a function ofcontraction angle at various values of apparent shear rate, (A) m-LLDPE,(B) HDPE, and (C) LDPE

Fig. 6. End pressure of m-LLDPE with the addition of 1 wt.% fluoro-polymer at 160 8C as a function of the entrance angle for several valuesof the apparent shear rate

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https://www.researchgate.net/publication/230384451_Flow_of_polymer_melts_through_a_welllubricated_conical_die?el=1_x_8&enrichId=rgreq-ca19b30b-8ea7-4c80-9b6e-291c6dc57355&enrichSource=Y292ZXJQYWdlOzI2MjA3MDMxMTtBUzoxMDI5Mzc1Nzc2NTYzMzJAMTQwMTU1MzYzOTU3Nw==

https://www.researchgate.net/publication/238994531_Measuring_the_Extensional_Rheology_of_Polymer_Melts?el=1_x_8&enrichId=rgreq-ca19b30b-8ea7-4c80-9b6e-291c6dc57355&enrichSource=Y292ZXJQYWdlOzI2MjA3MDMxMTtBUzoxMDI5Mzc1Nzc2NTYzMzJAMTQwMTU1MzYzOTU3Nw==

ture is observed. In this regime, a weak slip is usually observed(Hatzikiriakos and Dealy, 1992). We refer to the flow curve|g*(x)|x vs. x (in rad/s) as the linear viscoelasticity LVE – noslip data. The shear rate that corresponds to the uncorrected flowcurve (Rabinowitch and pressure effects) of pure m-LLDPE (notplotted for the sake of clarity as it is close to the corrected one) isreferred to as _cA;s (apparent shear rate corrected for the effect ofslip). Then using the flow curve that corresponds to the fluoro-polymer additive, the slip velocity, us, can be calculated as afunction of wall shear stress by

us ¼ D_cA � _cA;s

8: ð9Þ

Fig. 8 plots the slip velocity as a function of shear stress. A lin-ear relationship is obtained and the slip model then becomes

us ¼ ksðrW � 0:03Þ for rW � 0:03 MPa; ð10Þwhere the value of 0.03 MPa represents the critical shear stressfor the onset of slip in the presence of fluoropolymer at the inter-face, and ks is the proportionality slip constant, which turns outto be equal to 0.075 m/MPa · s, with us in m/s and rW in MPa.

5 Numerical Results

5.1 Viscous Modeling

It is instructive to perform first calculations with a purely vis-cous model, so that the effect of viscoelasticity will becomeevident later. The numerical simulations have been undertakenusing the viscous Cross model (Eq. 6). This constitutive rela-tion is solved together with the usual conservation equationsof mass and momentum for an incompressible fluid under iso-thermal conditions. The finite element grids are the same asthe ones used in our previous publication (see Fig. 9 of Mitsou-lis and Hatzikiriakos, 2003). Having fixed the Cross model pa-rameters and the problem geometry, the only parameter left tovary was the apparent shear rate ( _cA).



Fig. 7. Flow curves for m-LLDPE with and without fluoropolymer toassess slip

Fig. 8. Slip velocity of the m-LLDPE in the presence of fluoropolymer

A)

B)

Fig. 9. Simulation results with the K-BKZ model (Eq. 1) for LDPE at160 8C in a 908-tapered die at apparent shear rate of 1000 s – 1. Dimension-less axial distributions along the wall and the centreline for (A) pressure,(B) shear stress

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Simulations were performed for the whole range of experi-mental apparent shear rates, namely from 75 s – 1 to as high as1000 s – 1.

The results from the simulations are shown in Figs. 5A to Cfor the three melts (m-LLDPE, HDPE, LDPE, respectively).For clarity we only present numerical results for the lowest andhighest apparent shear rates of 75 s – 1 and 1000 s – 1, respec-tively. For the other rates, the results scale accordingly, as inthe experiments. We observe that the viscous results always un-der-predict the experimental results. The numerical results showa monotonic decrease with increasing angle for all apparentshear rates. All results do not pass through a minimum, in sharpcontrast with the experiments. The least under-estimation occursfor the most viscous HDPE melt, while the worst occurs for themost elastic LDPE melt (based on their average relaxation timeof Table 2), with the m-LLDPE lying in between. The best re-sults are obtained for the lowest angles as expected since theflow is shear-dominated, and at small contraction angles the lu-brication approximation is nearly valid. As the angles increase,the extensional components are having a stronger effect, thus in-creasing the discrepancies between predictions from a purelyshear model and the measured experimental data.

Including slip in the case of m-LLDPE with 1.0 wt.% Dyna-mar 9613 does cause a drop in the pressure drop as shown inFig. 6. Again the results for the viscous model under-estimatethe experimental ones, and this discrepancy increases with ap-parent shear rate and contraction angle.

5.2 Viscoelastic Modeling

The viscoelastic numerical simulations have been undertakenusing the integral constitutive equation (Eq. 1) as describedabove. In the course of the current work, we tested severalways to try and find out why for some melts, such as LLDPE(Hatzikiriakos and Mitsoulis, 1996) and PP (Mitsoulis and Hat-zikiriakos, 2003), the pressure losses were adequately pre-dicted, while for the most elastic LDPE this had never beenthe case (Barakos and Mitsoulis, 1995a); on the contrary thecalculated pressure losses were about half the experimentalvalues (discrepancies of 100%).

First of all, we verified again that the best way to proceed inthe viscoelastic simulations was to increase slowly the appar-ent shear rates from 0.01 s – 1 to 1000 s – 1, essentially using acontinuation scheme. We found out that 25 intermediate stepswere sufficient and necessary to get results according to thelogarithmic scale in the range of simulations. However, theseresults were achievable under some conditions, which werenot the same for all three melts. Namely, to obtain solutionsfor L/D = 0, the following three strategies were tested:(a) use L/D = 0 for the die and add an extrudate length L/D = 5

or 10 to accommodate the free surface and thus determinethe extrudate swell of the polymers (this worked only forthe m-LLDPE; for the HDPE and the LDPE the results di-verged at low apparent shear rates < 10 s – 1);

(b) use L/D = 0 for the die and add an extrudate length L/D = 5or 10 to accommodate the free surface using stick-slip condi-tions (no extrudate swell allowed, based on the assumptionthan no significant pressure drop occurs in the extrudate);this worked for all three polymers for all shear rates, although

it gave disappointedly low pressure drops, the lower pres-sures the more elastic the polymer (LLDPE < HDPE < LDPEin elasticity as seen via �k);

(c) use L/D = 50 for the capillary die, and based on the fullydeveloped shear stress wall values sw = 4(dP/dz)(L/D),subtract the pressure drop in the straight die DP0 from theoverall pressure drop in the system DP to obtain the entrypressure corresponding to L/D = 0; this worked for allthree polymers and all shear rates and produced acceptableresults even for the most elastic LDPE. Apparently thelong die length was necessary to get a full relaxation ofthe stresses and determine the extra pressure drop due toelasticity. Sample tests with L/D > 50 showed that the en-try pressure was not affected appreciably (in any discern-ible way in the graphs), which is an indication that the ex-tra L/D length amounts to a fully-developed flow with nomemory effects adding to the stresses and hence the pres-sure. Also, the long lengths made the simulations easierto converge, as more distance was given for the stressesto relax naturally.

To better understand the above point (c), we present in Fig. 9the axial pressure and shear stress distributions for the LDPEmelt for the 908-tapered die and the highest apparent shear rateof 1000 s – 1. The distributions are made dimensionless by di-viding the pressure and the stress by the nominal stresss* = P* = g0U/R, where U is the average exit velocity. We ob-serve that in the die length of 100 R, the pressure drop becomeslinear after a rearrangement length of about 10 R, and the shearstress at the wall is constant up to the exit, which is tantamountto a fully-developed flow in the die. It is this constant value ofsw, which is used to calculate the DP0, to obtain the pressuredrop in the die itself and subtract this value from the overallpressure in the system. Also to be noticed is the very good be-haviour of the viscoelastic solution at such an elevated appar-ent shear rate for a very elastic melt (LDPE), where apart fromthe usual oscillations around the singularity to the entry of thedie, the solution is smooth and well behaved both for the pres-sure and the stresses.

The simulations have also taken into account the pressuredependency of viscosity as given by Eq. 3 (not taken into ac-count in Fig. 9). In that case, it was found that the easiest wayto proceed was to just multiply the pressure results under nopressure dependence by the factor ap of Eq. 3. It should benoted that for high apparent shear rates and low contraction an-gles, this correction can be important (between 10% and 30%).

The results from the simulations are shown in Figs. 5A to Cfor the three melts (m-LLDPE, HDPE, LDPE, respectively)and in Fig. 6 for m-LLDPE with slip due to the presence of1.0 wt.% Dynamar 9613. An overall good agreement can beobserved between experiments and simulations, although somedifferences still exist. The results for the m-LLDPE melt do notpass through a minimum, in contrast with the experiments.However, viscoelasticity serves to increase the pressure dropsubstantially, and this is enhanced for the higher angles and ap-parent shear rates. This is expected, as increasing the contrac-tion angle leads to an increase of the elongational contributionsto entry pressure. Also the nonlinear effect of this increase ishelped by increasing the flow rate, namely for higher apparentshear and elongational rates, where the rheology of the meltsshows a markedly nonlinear behaviour.

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Regarding the HDPE melt, we observe that solution could beobtained even for the highest apparent shear rate of 1000 s – 1

(cases (b) and (c) above), despite the very elastic nature of thispolymer melt (note that its �k = 18.9 s compared with �k = 0.46 sfor m-LLDPE). The results between the envelopes of the low-est apparent shear rate ( _cA = 75 s – 1) and the highest one( _cA = 1000 s – 1) capture, in general, the experimental data, butthey show a monotonic decrease with increasing contractionangle.

Regarding the most elastic LDPE melt, we observe that solu-tion could be again obtained even for the highest apparent shearrate of 1000 s – 1 (cases (b) and (c) above), despite its strong elas-tic nature in the melt state (note that its �k = 298 s compared with�k = 18.9 s for HDPE and �k = 0.46 s for m-LLDPE). It is interest-ing to note that a small minimum is obtained at around 2a = 308.These minima were present for all shear rates and are bettershown in a Cartesian plot in Fig. 10. For the first time for LDPEwe get entrance pressure results in agreement with experimentaldata for all shear rates and angles. Previous results have shownunder-estimates for the entrance pressure by almost 100 % com-pared with experimental data (see simulations for the IUPAC-LDPE melt by Barakos and Mitsoulis, 1995b, and the other re-ferences in the introduction). It is to be noted that a differentelongational parameter b-value (b = 0.002 for all 8 modes),which corresponds to an order-of-magnitude higher response inelongation, did increase somewhat the results but not substan-tially (5 to 10 %). The present simulations with a die length ofL/D = 50 showed that such a length is necessary for the melt torelax its stresses and unravel its full viscoelastic character. Thesimulation results were well behaved for all shear rates with nooscillations or other defects. This is also the first time that simu-lations for LDPE have been carried out up to 1000 s – 1. It shouldbe emphasized that all previous studies on LDPE, combining ex-periments and simulations did not exceed _cA = 125 s – 1 (Mitsou-lis, 2001).

The results for m-LLDPE with slip in Fig. 6 also show asmall minimum for all cases of apparent shear rates around2a = 308. Slip is known to enhance the minimum in the entrypressure vs. contraction angle as found out both theoreticallyand computationally by the Hatzikiriakos and Mitsoulis(2009).

6 Conclusions

Three commercial polyethylene melts (m-LLDPE, HDPE,LDPE) have been studied in entry flows through tapered diesof various angles. The experiments have shown a minimumfor tapered dies with a taper of about 308 for all three melts.After that the pressures increase. Numerical simulations em-ploying the K-BKZ model showed a minimum for the LDPEmelt and for the m-LLDPE melt with slip. Then the results lev-el off, which is not always the case with the experiments. It isinteresting to note that the simulations for the LDPE followthe ups and downs in the experimental data (albeit in a smallerscale) and for the first time are in agreement with the experi-mental data. It should be noted that previous simulations forthe IUPAC-LDPE melt (Barakos and Mitsoulis, 1995a) gavevalues about half the experimental ones. The problem was re-solved by using a long die length (L/D = 50) and then subtract-ing the pressure drop in the die to get the end pressure. Also thecorrection due to the pressure dependency of viscosity was im-portant in matching the values up to the experimental levels.The present viscoelastic results are the first ones to reach levelsof apparent shear rates as high as 1000 s – 1, and showed the im-portance of including long lengths to fully relax the viscoelas-tic stresses for highly viscoelastic polymer melts, such as theHDPE and LDPE melts.

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Acknowledgements

Financial assistance from the Natural Sciences and Engineer-ing Research Council (NSERC) of Canada and the programme“PEBE 2009– 2011” for basic research from NTUA are grate-fully acknowledged.

Date received: March 13, 2010Date accepted: May 23, 2010

BibliographyDOI 10.3139/217.2360Intern. Polymer ProcessingXXV (2010) 4 page 287–296ª Carl Hanser Verlag GmbH & Co. KGISSN 0930-777XDOI-Nr.

You will find the article and additional material by enter-ing the document number IIPP2360 on our website atwww.polymer-process.com

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M Entry Flow of Polyethylene Melts in Tapered Dies