Tubing Extrusion of a Fluoropolymer Meltold-2017.metal.ntua.gr/uploads/3798/1059/A139.pdf · The tensorI in Eq. 5 is the unit tensor. To evaluate the role of viscoelasticity in flow

E. Mitsoulis1*, S. G. Hatzikiriakos2

1School of Mining Engineering and Metallurgy, National Technical University of Athens, Athens, Greece2Department of Chemical and Biological Engineering, The University of British Columbia, Vancouver, Canada

Tubing Extrusion of a Fluoropolymer Melt

The tubing extrusion-forming process of a fluoropolymer

(FEP) melt was studied both experimentally and numerically.

The flow behaviour of a FEP resin was determined by using

a tubular die used in industrial-scale operations and these

data were compared with simulation results using (i) a vis-

cous model (Cross) and (ii) a viscoelastic one (the Kaye–

Bernstein, Kearsley, Zapas / Papanastasiou, Scriven, Macos-

ko or K-BKZ/PSM model) in order to assess the viscoelastic

effects. In all simulations, compressibility, thermal and pres-

sure effects on viscosity were taken into account. It was

found that the viscoelastic results for the pressure, and hence

the stresses at the wall, were always higher than the viscous

results. Both were higher than the experimental results. A

quadratic slip model plus viscoelasticity was found necessary

to reproduce the experiments. The smooth flow curves result-

ing from this industrial tubular-coating die are a further

manifestation that this is an appropriate design for coating

fluoropolymers at very high apparent shear rates, exceeding

5000 s– 1.

1 Introduction

Fluoropolymers are of great commercial and scientific interest

due to their unique combination of properties. These include

excellent chemical stability and dielectric properties, anti-stick

characteristics, mechanical strength, and low flammability

(Imbalzano and Kerbow, 1994; Ebnesajjad, 2003). Their most

important uses are in electronics and electrical applications,

especially for wiring insulation, chemical processing equip-

ment, medical devices and laboratory ware and tubing (Ebne-

sajjad, 2003; Domininghaus, 1993; Dealy and Wissbrun,

1990). The main focus of this work is on the process of tubing

extrusion of FEP copolymers (Baird and Collias, 1998; Tad-

mor and Gogos, 2006).

The capillary extrusion flow of this resin and its complete

rheological characterizations was previously studied by Rosen-

baum et al. (1995, 1998, 2000) and more recently by Mitsoulis

and Hatzikiriakos (2012). These rheological data are used in

the present study to formulate the constitutive equations

needed for the flow simulations in tubing extrusion dies. In par-

ticular a viscous Cross model and a viscoelastic K-BKZ are

used to address various effects discussed below (Rosenbaum,

2000; Mitsoulis and Hatzikiriakos, 2012).

The extrusion tubing process of polytetrafluoroethylene

(PTFE) paste has been studied by Patil et al. (2006, 2008). Spe-

cifically, Patil et al. (2006) have derived an analytical expres-

sion to predict the pressure drop in annular dies for flow of

PTFE paste. The model predictions were compared with ex-

perimental results in PTFE paste extrusion to examine form-

ability of tubing through annular dies of various geometries

(Patil et al., 2008). Due to its high melting point (greater than

340 8C), PTFE can only be processed in the form of paste at

temperatures between 35 to 50 8C. However, copolymers of

PTFE, such as the FEP resins considered in the present work,

have considerably lower melting points (less than 260 8C) and

thus their melt processing is possible. Melt tubing extrusion is

the subject of the present study.

Ferrandino (2004a, 2004b) reviewed the extrusion tubing

process from the point of view of material (melt) and control

of extrusion process, and made specific recommendations to

produce tubing with consistent properties, particularly in med-

ical applications. Guo and Stehr (2001) presented a semi-ana-

lytical method for the design of crosshead annular dies by con-

sidering simplified versions of the equations of motion and

energy as approximations of the flow channel using a series of

varying annular slits, each having constant geometric parame-

ters. The analysis was performed using a Cross model fitted to

experimental data for an isotactic polypropylene (i-PP). They

reported that the optimum die designs are related, but less sen-

sitive to extrusion conditions within a certain range, i. e. pulling

speeds of the tube. More recently, Kolitawong et al. (2011)

studied the temperature distribution of a power-law fluid in a

pressure-driven axial flow between isothermal eccentric cylin-

ders. Eccentricity is used to balance the expected sagging dur-

ing the cooling process. They found that the dimensionless

temperature profile depends upon the radius ratio of the inner

to outer cylinders, the eccentricity, the angular position, and

the power-law exponent n. They have also reported that the

temperature is a strong function of the gap between the cylin-

ders. The temperature profiles are flat in the middle of the gap

and then, near the wall, suddenly drop to the wall temperature.

The main objective of the study is to model the tubing pro-

cess of melt-processible fluoropolymers. The flow model

predictions are performed using both a viscous and a viscoelas-

tic model and these are compared with experimental results

obtained from industrial-scale tubing dies for a certain fluoro-

polymer (FEP). The tubing simulation model includes all im-

portant effects in a full numerical scheme, such as pressure-

and temperature-dependence of viscosity, compressibility, vis-

REGULAR CONTRIBUTED ARTICLES

Intern. Polymer Processing XXVII (2012) 2 � Carl Hanser Verlag, Munich 259

* Mail address: Evan Mitsoulis, School of Mining Engineering andMetallurgy, National Technical University of Athens, Zografou,15780, Athens, GreeceE-mail: [email protected]

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E. Mitsoulis, S. G. Hatzikiriakos: Tubing Extrusion of a Fluoropolymer Melt

cous dissipation, slip-at-the-wall, and viscoelasticity through

an appropriate integral constitutive equation.

2 Experimental

Tubing extrusion experiments were performed for the FEP-

4100 resin (a copolymer of tetrafluoroethylene and hexafluoro-

propylene) using a tubing extrusion die. This resin has been

rheologically characterized previously in detail by Rosenbaum

et al. (1995, 1998, 2000), and Mitsoulis and Hatzikiriakos

(2012). It has a molecular weight of about 208,000 and a poly-

dispersity index of about 2 (Ferrandino, 2004a; 2004b). The

melting point of this resin is around 260 8C, determined by dif-

ferential scanning calorimetry (DSC) analysis. The rheological

parameters determined by fitting the experimental results are

summarized in Tables 1 to 3. These include the parameters of

the Cross and power-law models for the viscosity of the fluoro-

polymer (Table 1), the parameters for the K-BKZ viscoelastic

constitutive model (Table 2), and the physical properties of

the FEP-4100 needed for the non-isothermal flow simulations

(Table 3). The data in these Tables are also discussed in more

detail below.

A schematic of the tubing extrusion die used in the experi-

ments and the numerical simulations appears in Fig. 1 (Buck-

master et al., 1997; Dupont de Nemours, 1998), while the

details of the design are given in Table 4. It is a special annular

crosshead die attached to the rheometer to mimic the tubing

extrusion process. This crosshead die included dies and tips

(internal diameter) with equal entry cone angles of 2a = 608

and die land length L of 7.62 mm. In the present work we re-

port data obtained with a die diameter D of 3 mm and a tip

diameter d of 1.52 mm. The molten polymer enters the tapered

die via the uppermost port and is forced around the wire tip

guide towards the die orifice. The wire tip guide serves as a

mandrel for the molten polymer, giving the extrudate a tubular

shape. The die land passage forms the exterior surface of the

tubular shape, and the exterior surface of the cylindrical exten-

260 Intern. Polymer Processing XXVII (2012) 2

Parameter Value

g0,C 1542.3 Pa · sk 0.0049 snC 0.316K 11,931 Pa · sn

n 0.5

Table 1. Parameters for the FEP melt obeying the Cross model(Eq. 8) and the power-law model (Eq. 10) at 371 8C

k kk (s ) ak (Pa)

1 0.337 · 10 – 3 0.49050 · 106

2 0.178 · 10 – 2 0.22702 · 106

3 0.865 · 10 – 2 66,2244 0.562 · 10 – 1 4823.15 0.488 172.416 2.98 18.6287 16.6 3.6235

Table 2. Relaxation spectrum and material constants for the FEPmelt obeying the K-BKZ model (Eq. 13) at 371 8C (� = 7.174,b = 0.6, h = –0.11, �k= 0.76 s, g0 ¼ 1 613Pa � sÞ

Parameter Value Reference

bc 0.00095 MPa – 1 Hatzikiriakos and Dealy (1994)bp 0.03 MPa – 1 Rosenbaum et al. (1995)m 1.39 · 10 – 4 Panp�1 Mitsoulis and Hatzikiriakos (2012)np 0.54 Mitsoulis and Hatzikiriakos (2012)bsl 400 cm/(s · MPab) This workb 2.0 This workq 1.492 g/cm3 Ebnesajjad (2000)Cp 0.96 J/(g · K) Van Krevelen (1990)k 0.00255 J/

(s · cm · K)Van Krevelen (1990)

E 50,000 J/mol Rosenbaum et al. (1995)Rg 8.3143 J/(mol · K) Dealy and Wissbrun (1990)T0 371 8C (644 K) Rosenbaum et al. (1995)

Table 3. Values of the various parameters for the FEP melt at 371 8C

Parameters Symbol Value

Overall die length Zmax 60.6 mmChannel entry gap Hen 3 mmTapered length Lt 15.25 mmHalf cone angle a 308

Minimum tapered gap f 1.2 mmDie land length L 7.62 mm

Outer die diameter D 3 mmInner die (tip) diameter d 1.524 mm

Die land gap h 0.738 mmWire diameter dw 0.574 mmDie temperature T0 371 8C (644 K)

Table 4. Design parameters of the Nokia Maillefer 4/6 crosshead die(Dupont de Nemours, 1998)

Fig. 1. Nokia Maillefer 4/6 crosshead die design for tubing coatingwith dimensions in mm (not in scale) (Dupont de Nemours, 1998). Datagiven in Table 4

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sion forms the interior surface of the tubular shape. This die

can also be used for wire-coating studies (Mitsoulis, 1986; Mit-

soulis et al., 1988; Tiu et al., 1989). However, in the present

study the pressure extrusion makes no use of wire as the die is

used in conjunction with a capillary rheometer having a barrel

of diameter equal to 25.4 mm.

3 Governing Equations and Rheological Modelling

We consider the conservation equations of mass, momentum

and energy for weakly compressible fluids under non-isother-

mal, creeping, steady flow conditions. These are written as

(Mitsoulis et al., 1988; Tiu et al., 1989; Tanner, 2000):

�u � rqþ q r � �uð Þ ¼ 0; ð1Þ

0 ¼ �rpþr � ��s; ð2Þ

qCp�u � rT ¼ kr2Tþ ��s : r�u; ð3Þ

where q is the density, �u is the velocity vector, p is the pressure,��s is the extra stress tensor, T is the temperature, Cp is the heat

capacity, and k is the thermal conductivity. For a weakly com-

pressible fluid, pressure and density are connected as a first ap-

proximation through a simple linear thermodynamic equation

of state (Tanner, 2000):

q ¼ q0 1þ bcpð Þ; ð4Þ

where bc is the isothermal compressibility with the density to

be q0 at a reference pressure p0 (= 0).

The viscous stresses are given for inelastic non-Newtonian

compressible fluids by the relation (Tanner, 2000):

��s ¼ g _cj jð Þ ��_c�2

3r � �uð Þ��I

� �

; ð5Þ

where g _cj jð Þ is the non-Newtonian viscosity, which is a func-

tion of the magnitude _cj j of the rate-of-strain tensor��_c ¼ r�uþr�uT, which is given by:

_cj j ¼

ffiffiffiffiffiffiffiffiffi

1

2II _c

r

¼1

2_c : _c

� �

� �1=2

; ð6Þ

where II _c is the second invariant of ��_c

II _c ¼ _c : _c� �

¼X

i

X

j

_cij _cij: ð7Þ

The tensor ��I in Eq. 5 is the unit tensor.

To evaluate the role of viscoelasticity in flow through a tub-

ing die, it is instructive to consider first purely viscous models

in the simulations. Namely, the Cross model was used to fit

the shear viscosity data of the FEP melt. The Cross model is

written as (Dealy and Wissbrun, 1990):

g ¼g0;C

1þ ðk _cÞ1�nC; ð8Þ

where g0,C is the Cross zero-shear-rate viscosity, k is a time

constant, and nC is the Cross power-law index. The experimen-

tal data of FEP were obtained at 300 8C, 325 8C, and 350 8C

(Rosenbaum et al., 1995), while the experiments with the tub-

ing die were performed at 371 8C. The data has been shifted to

371 8C by using the Arrhenius relationship (Dealy and Wiss-

brun, 1990) for the temperature-shift factor aT:

aTðTÞ ¼g

g0

¼ expE

Rg

1

T�

1

T0

� ��

: ð9Þ

In the above, g0 is a reference viscosity at T0, E is the activa-

tion energy, Rg is the ideal gas constant, and T0 is a reference

temperature (in K). The activation energy was calculated from

the shift factors determined by applying the time-temperature

superposition principle to obtain the curves at 371 8C plotted

in Fig. 2 (Dealy and Wissbrun, 1990; Mitsoulis and Hatzikiria-

kos, 2012). It was found that E = 50,000 J/mol. The newly

fitted viscosity of the FEP melt by Eq. 8 is also plotted in

Fig. 2 (line), while the parameters of the model are listed in

Table 1. We observe that the FEP melt has a wide Newtonian

plateau and then shows shear-thinning for shear rates above

10 s – 1 giving a power-law index nC = 0.32. The Cross model

fits the data well over the range of experimental results. The

same is also true for the viscoelastic K-BKZ model (see be-

low).

For easy checks and simple analytical formulas, the high

shear-rate range of the viscosity data can be fitted to the

power-law model for the viscosity (Dealy and Wissbrun,

1990):

g ¼ Kj _cjn�1; ð10Þ

where K is the consistency index and n is the power-law index.

These values are also given in Table 1. We note here that the

Cross model will turn into a power-law model for high shear

rates so the power-law exponent in the Cross model should be

the same as in the power-law model. We have tried to do this,

but for narrow molecular weight distribution FEP melts with a

Newtonian plateau over a wide range of shear rates and a small

k (0.0049 s), the power law gives a good fit only at extremely

high shear rates beyond the range of available experimental

data (above 5000 s – 1). Then the fitting misses all the points


Intern. Polymer Processing XXVII (2012) 2 261

Fig. 2. Shear viscosity of the FEP melt at 371 8C fitted with the Cross(Eq. 8) and the power-law (Eq. 10) models using the parameters listedin Table 1. For comparison the predictions from the K-BKZ model(Eq. 13) is also shown

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for which runs were made. This is not the case with polydis-

perse polyethylenes (Ansari et al., 2010), where the onset of

shear-thinning occurs early, k is above 1 s, and the power-law

fitting does a good job over a wide range of shear rates. We

have thus fitted the power-law constants as shown in Table 1

and Fig. 2.

The viscosity of this resin also depends on the pressure for

which the Barus equation can be used (Dealy and Wissbrun,

1990; Baird and Collias, 1998; Tadmor and Gogos, 2006;

Sedlacek et al., 2004; Carreras et al., 2006):

ap �g

gp0

¼ exp bpp� �

; ð11Þ

where g is the viscosity at absolute pressure p, gp0 is the visc-

osity at ambient pressure, and bp is the pressure-shift factor.

The latter was found to depend on pressure through the follow-

ing equation (Mitsoulis and Hatzikiriakos, 2012):

bp ¼ mp�np ; ð12Þ

where m = 1.39 · 10 – 4 Panp�1 and np = 0.54, with the pressure

p given in Pa (Mitsoulis and Hatzikiriakos, 2012).

Viscoelasticity is included in the present work via an appro-

priate rheological model for the stresses. This is a K-BKZ

equation proposed by Papanastasiou et al. (1983) and modified

by Luo and Tanner (1988). This is written as:

s ¼1

1� h

Z

t

�1

X

N

k¼1

ak

kkexp �

t� t0

kk

� �

a

ða� 3Þ þ bIC�1þð1� bÞIC

� C�1t ðt0ÞþhCtðt

0Þ�

dt0; ð13Þ

where t is the current time, kk and ak are the relaxation times

and relaxation modulus coefficients, N is the number of relaxa-

tion modes, a and b are material constants, and IC, IC–1 are the

first invariants of the Cauchy-Green tensor Ct and its inverse

Ct– 1, the Finger strain tensor. The material constant h is given

by

N2

N1

¼h

1� h; ð14Þ

where N1 and N2 are the first and second normal stress differ-

ences, respectively. It is noted that h is not zero for polymer

melts, which possess a non-zero second normal stress differ-

ence. Its usual range is between –0.1 and –0.2 in accordance

with experimental findings (Dealy and Wissbrun, 1990; Tan-

ner, 2000). The various parameters needed for this model are

summarized in Table 2 (Kajiwara et al., 1995), where the re-

laxation times have been shifted to the new temperature of

371 8C by diving the values at the reference temperature of

300 8C by aT = 3.4.

Fig. 3 plots a number of calculated and experimental materi-

al functions for the FEP melt at 371 8C. Namely, data for the

shear viscosity, gS, the elongational viscosity, gE, and the first

normal stress difference, N1, are plotted as functions of corre-

sponding rates (shear or extensional).

The non-isothermal modeling follows the one given in ear-

lier publications (see, e. g., Luo and Tanner, 1987; Luo and

Mitsoulis, 1990; Alaie and Papanastasiou, 1993; Barakos and

Mitsoulis, 1996; Peters and Baaijens, 1997; Beaulne and Mit-

soulis, 2007) and will not be repeated here. Suffice it to say that

it employs the Arrhenius temperature-shifting function, aT,

given by Eq. 9. The various thermal parameters needed for the

simulations are gathered in Table 3 together with their source

(Van Krevelen, 1990; Hatzikiriakos and Dealy, 1994; Ebnesaj-

jad, 2000; Mitsoulis and Hatzikiriakos, 2012). The viscoelastic

stresses calculated by the non-isothermal version of the above

constitutive equation (Eq. 13) enter in the energy equation

(Eq. 3) as a contribution to the viscous dissipation term.

The various thermal and flow parameters are combined to

give appropriate dimensionless numbers (Winter, 1977; An-

sari et al., 2010). The relevant ones here are the Peclet num-

ber, Pe, and the Nahme-Griffith number, Na. These are de-

fined as:

Pe ¼qCpUh

k; ð15Þ

Na ¼�gEU2

kRgT20

; ð16Þ

where �g ¼ fðU=hÞ is a nominal viscosity given by the Cross

model (Eq. 8) at a nominal shear rate of U/h. In the above, U

is the average velocity in the die [U = 4Q/p(D2�d2)], where Q

is the volumetric flow rate, and h is the die gap [h = (D–d)/2].

The Pe number represents the ratio of heat convection to con-

duction, and the Na number represents the ratio of viscous

dissipation to conduction and indicates the extent of coupling

between the momentum and energy equations. A thorough dis-

cussion of these effects in non-isothermal polymer melt flow is

given by Winter (1977).

With the above properties and a characteristic length the die

gap h = 0.0738 cm, the dimensionless thermal numbers are in

the ranges: 41 < Pe < 2860, 0.007 < Na < 10.9, showing a

strong convection (Pe >> 1), and a moderate to strong coupling

between momentum and energy equations (Na > 1). A value of

Na > 1 indicates temperature non-uniformities generated by

viscous dissipation, and a strong coupling between momentum

and energy equations. More details are given in Table 5.


Fig. 3. Experimental data (solid symbols) and model predictions ofshear viscosity, gS, first normal stress difference, N1, and elongationalviscosity, gE, for the FEP melt at 371 8C using the K-BKZ model(Eq. 13) with the parameters listed in Table 2

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As a check, we report here the fully-developed values for

temperature in a cylindrical die occurring for L/D =1 and

giving a maximum temperature rise DTmax at the centreline of

the tube according to the formula for a power-law fluid (Bird

et al., 1987):

DTmax ¼K

k

nR

3nþ 1

� �2nþ 1

n

� �

Vmax

R

� �nþ1

: ð17Þ

According to Eq. 17 and for R = h = 0.0738 cm, a

DTmax = 921 8C is obtained for the highest apparent shear rate

of 5600 s – 1. This unrealistically high temperature rise is pri-

marily due to the high K-value (&12,000 Pa · sn) of the

power-law model needed to capture well the shear-thinning be-

haviour at high shear rates, when it is known that the zero-shear

rate viscosity is about 1500 Pa · s (see Fig. 2). In any case, the

fully-developed values are expected for very long dies, but for

shorter dies as the ones used here, we expect a smaller although

substantial increase in temperature.

Similarly with the time-temperature superposition principle

where the stresses are calculated at a different temperature

using the shift factor aT, the time-pressure superposition princi-

ple can be used to account for the pressure effect on the stres-

ses. In both cases of viscous or viscoelastic models, the new

stresses are calculated using the pressure-shift factor ap. For

viscous models, Eq. 11 is used to modify the viscosity. For vis-

coelastic models, such as the K-BKZ model (Eq. 13), the pres-

sure-shift factor modifies the relaxation moduli, ak, according

to:

ak p t0ð Þð Þ ¼ ak p0ð Þap p t0ð Þð Þ: ð18Þ

This is equivalent to multiplying the stresses by ap, according

to Eq. (13). It should be noted that ap is an exponential function

of bp.The pressure-dependence of the viscosity gives rise to the

dimensionless pressure-shift parameter, Bp. This is defined as:

Bp ¼bp�gU

h: ð19Þ

Similarly, the compressibility coefficient bc gives rise to the di-mensionless compressibility parameter, Bc. This is defined as:

Bc ¼bc�gU

h: ð20Þ

The value Bp = 0 corresponds to the case of pressure-indepen-

dence of the viscosity, and Bc = 0 to incompressible flow. For

the present data, the range of values is: 5.4 · 10 – 4 < Bp

< 1.1 · 10 – 2 and 1.7 · 10 – 5 < Bc < 3.6 · 10 – 4, showing a

moderate dependence of viscosity on pressure and an even

weaker compressibility effect in the range of simulations. More

details are given in Table 5.

In the case of slip effects at the wall, the usual no-slip veloc-

ity at the solid boundaries is replaced by a slip law of the fol-

lowing form (Dealy and Wissbrun, 1990):

aTusl ¼ �bslrbw; ð21Þ

where usl is the slip velocity, rw is the shear stress at the die

wall, bsl is the slip coefficient, and b is the slip exponent. It

should be noted that the shift-factor aT is used here to take into

account the temperature effects of the slip law at different tem-

peratures.

In 2-D simulations, the above law means that the tangential

velocity on the boundary is given by the slip law, while the nor-

mal velocity is set to zero, i. e.,

bsl �t�n : ��sð Þb¼ aT �t � �uð Þ; �n � �u ¼ 0; ð22Þ

where �n is the unit outward normal vector to a surface, �t is the

tangential unit vector in the direction of flow, and the rest of

symbols are defined above. Implementation of slip in similar

flow geometries for a polypropylene (PP) melt has been pre-

viously carried out by Mitsoulis et al. (2005) and more recently

in annular dies by Chatzimina et al. (2009).

The corresponding dimensionless slip coefficient, Bsl, is a

measure of fluid slip at the wall:

Bsl ¼bsl�g

b

U

U

h

� �b

: ð23Þ

The value Bsl = 0 corresponds to the no-slip boundary condi-

tions and Bsl & 1, to a macroscopically obvious slip. For the

present data, the range of values is: 0.129 < Bsl < 0.821, show-

ing a moderate to strong slip effect in the range of simulations.

More details are given in Table 5. We also note in passing that

the creeping flow approximation (for which the Reynolds num-

ber, Re ¼ qUh=�g & 0) is justified due to the high viscosity of

the melt, which gives even for the highest apparent shear rate

of experiments, Re < 10 – 3.

4 Method of Solution

The solution of the above conservation and constitutive equa-

tions is carried out with two codes, one for viscous flows (u-v-

p-T-h formulation) (Hannachi and Mitsoulis, 1993) and one

for viscoelastic flows (Luo and Mitsoulis, 1990; Barakos and



Apparent shear rate,_cA (s – 1)

Peclet number, Pe Nahme number, Na Compressibilityparameter, Bc

Pressure-shiftparameter, Bp

Slip parameter, Bsl

80 41 0.007 1.69 · 10 – 5 5.35 · 10 – 4 0.129320 163 0.10 5.59 · 10 – 5 1.76 · 10 – 3 0.351800 407 0.49 1.12 · 10 – 4 3.53 · 10 – 3 0.563

1600 814 1.54 1.77 · 10 – 4 5.60 · 10 – 3 0.7103200 1628 4.62 2.66 · 10 – 4 8.41 · 10 – 3 0.8015600 2859 10.89 3.57 · 10 – 4 1.13 · 10 – 2 0.821

Table 5. Range of the dimensionless parameters in the flow of FEP melt at 371 8C (die land gap h = 0.0738 cm)

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Mitsoulis, 1996). The boundary conditions (BC) for the prob-

lem at hand are well known. Briefly, we assume no-slip (or slip

when included) and a constant temperature T0 at the solid

walls; at entry, a fully-developed velocity profile is imposed,

corresponding to the flow rate at hand and a constant tempera-

ture T0 is assumed; at the outlet, zero surface traction and zero

heat flux are assumed.

The viscous simulations are extremely fast and are used as a

first step to study the whole range of parameter values and die

designs. The viscoelastic simulations admittedly are harder to

do and they need good initial flow fields to get solutions at ele-

vated apparent shear rates. In our recent work (Ansari et al.,

2010), we explained how it was possible for the first time to

do viscoelastic computations up to very high apparent shear

rates (1000 s – 1) with good results. Briefly, the solution strat-

egy starts at a given apparent shear rate from the viscous non-

isothermal solution without pressure dependence, and then

using this as an initial solution it continues for the non-isother-

mal viscoelastic solution with all effects present. When slip is

present, the simulations are much faster as they require fewer

iterations due to the less drastic conditions encountered in the

flow field.

All velocities have been made dimensionless with the aver-

age velocity U. The lengths could have been made dimension-

less either with die gap h or with the wire radius Rw, which is

0.4 h. We have opted for the latter, since in wire coating it is

customary to use the wire radius for dimensionalization. Then

the pressures and stresses are made dimensionless by g0U/Rw,

where g0 is the zero-shear rate viscosity.

5 Experimental Results

Fig. 4 shows the apparent flow curves for the resin (FEP 4100)

obtained with the tubular die using a d = 1.524 mm

tip and D = 3 mm die at T = 371 8C. The apparent shear rate

was calculated by using the formula which applies to slit dies

(Dealy and Wissbrun, 1990):

_cA ¼6Q

0:25ðD� dÞ20:5pðDþ dÞ; ð24Þ

where Q is the volumetric flow rate, d and D is the tip and die

diameter, respectively. The apparent wall shear stress was esti-

mated as the average of the stress at the inner and outer walls

by using the following formula for a power-law fluid (Hanks

and Larsen, 1979):

srz ¼DpD

4L

2r

D� c2

D

2r

� �

; ð25Þ

where srz is the shear stress at radius r, Dp is the pressure drop

(measured experimentally by means of a load cell that mea-

sures the force needed to displace the polymer melt through

the reservoir and die), L is the length of the die land, and c is a

parameter which depends on the geometry and the power-law

index. For a power-law index of n = 0.5 and a diameter ratio

j = d/D = 0.508, we find from tables c = 0.7283 (Hanks and

Larsen, 1979).

A smooth flow curve is observed in Fig. 4 which increases

monotonically, an indication of a good design. Together with

the experimental data are shown simulation results by the

purely viscous Cross model with no effects present (base case,

bc = bp = bsl = aT = 0, see below). Obviously, the various ef-

fects at play must be taken into account to predict correctly

the flow curve. This is done below.

6 Numerical Results

6.1 No Slip at the Wall

The first simulations were performed with no slip at the wall,

although it is noticed that at such elevated apparent shear rates

slip is almost always present (Rosenbaum et al., 1995; 1998;

2000). It is instructive to show results both with a purely vis-

cous model and with a viscoelastic one so that the differences

become apparent. The numerical simulations have been under-

taken using either the purely viscous Cross model (Eq. 8) or the

viscoelastic K-BKZ model (Eq. 13). Each constitutive relation

is solved together with the conservation equations of mass and

momentum either for an incompressible or compressible fluid

under isothermal or non-isothermal conditions (conservation

of energy equation) without or with the effect of pressure-de-

pendence of the viscosity. The viscous results have been

checked against each other either with a viscous code (Hanna-

chi and Mitsoulis, 1993) or with the viscous option of the vis-

coelastic code (Luo and Mitsoulis, 1990) giving the same re-

sults.

A typical finite element grid is shown in Fig. 5 for the Nokia

Maillefer 4/6 crosshead die of Fig. 1. The vertical entry of the

crosshead has been converted into a parallel entry for ease of

setting there a fully developed velocity profile at the desig-

nated apparent shear rate _cA. Enough entry length has been giv-en to guarantee fully-developed conditions even for the visco-

elastic runs. Then the crosshead tapers with an entrance angle

2a = 608 at the inner wall and an outer angle of 2a = 648 at the

outer wall. The grid consists of 1568 elements, 6501 nodes,

and 26,462 unknown u-v-p-T degrees of freedom (d.o.f.), while

a 4-times denser grid was also used, having been created by


Fig. 4. The apparent wall shear stress of FEP 4100 melt as a functionof the apparent shear rate using the Nokia Maillefer 4/6 crosshead ofFig. 1 at 371 8C. The solid line corresponds to predictions by the vis-cous Cross model (base case, no effects present), open symbols corre-spond to smooth extrudates

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subdivision of each element into 4 sub-elements for checking

purposes of grid-independent results. This checking consists

of reporting the overall pressures in the system (pressure at

the entry to the domain) from the two meshes and making sure

that the differences are less than 1% between the two results.

Having fixed the model parameters and the problem geometry,

the only parameter left to vary was the apparent shear rate in

the die (Eq. 24). Simulations were performed for the whole

range of experimental apparent shear rates, namely from

80 s – 1 to 5600 s – 1, where smooth extrudates were obtained

with the presence of a small amount of processing aids,

although the pressure drop remained the same with and without

processing aid (Rosenbaum et al., 1995; 1998; 2000).

First, runs were carried out to study each effect separately,

namely: (a) the effect of compressibility alone, (b) the effect

of a pressure-dependent viscosity alone, (c) the effect of a tem-

perature-dependent viscosity alone, and these were compared

with the base case of no effects at all (bc = bp = bsl = aT = 0).

Due to the high range of simulations, all effects were evident.

However, compressibility played a minor role, as the results

were little affected and they are not shown here. Admittedly,

this is expected because the value of bc = 0.00095 MPa – 1 for

FEP is small, giving rise to small values for the dimensionless

compressibility parameter Bc in Table 5 in the range of simula-

tions. The same was true for the LDPE and HDPE melts in our

previous work (Ansari et al., 2010).

The effect of temperature-dependent viscosity (hence vis-

cous dissipation) was strong in the range of simulations and

gave high values for the temperature at the exit, reaching as

high as 420 8C at 5600 s – 1 (viscoelastic run). This is not sur-

prising given the high Na numbers of Table 5 and the predic-

tions of Eq. 17 above. The results for the maximum tempera-

ture at the exit for different apparent shear rates are shown in

Fig. 6 for both the viscous and viscoelas-

tic runs, where it becomes evident that

viscoelasticity produces higher tempera-

ture rises due to the enhanced stresses.

The high temperature increase due to

strong viscous dissipation effects at high

extrusion rates in wire-coating applica-

tions is the reason for melt fracture sup-

pression resulting in a smooth and stable

process (Tadmor and Gogos, 2006).

The effect of pressure-dependent

viscosity, with a variable value of bpobeying Eq. 12, also played a consider-

able role, which is counteracting the ef-

fect of temperature. The pressure (hence

the shear stress) is higher than the base

case for all apparent shear rates, due to

the dominant effect of pressure-depen-

dence of the viscosity, which is however

counterbalanced by the effect of tem-

perature.

Viscoelasticity enhances these trends

as the viscoelastic stresses are higher

than the viscous ones, and this becomes

more evident at higher apparent shear

rates. These trends are shown in Fig. 7,

where the axial pressure distribution

along the inner walls is plotted for both viscous and viscoelas-

tic models. The pressure falls smoothly to zero at the exit, thus

giving a further evidence of a good crosshead design. The

small kinks around z/Rw = –25 are due to the sudden change

in slope for the last part of the crosshead where it meets the

die land. This is an indication that the die should be designed

by rounding this entry to avoid the pressure kinks.

Collecting all these results together gives the apparent flow

curves shown in Fig. 8. The experimental results are shown as

symbols while the numerical results are shown as lines. These

simulation results have been obtained with all parameters

switched on, i. e., compressibility, pressure- and temperature-

dependence of viscosity, but with no slip at the wall. We note

that the curve from the K-BKZmodel is higher than the one ob-



A)

B) C)

Fig. 5. A typical finite element grid for the simulations in the crosshead die (A). Details of thegrid: (B) at the entry and (C) in the die land and exit

Fig. 6. Viscous vs. viscoelastic results for the maximum temperature atexit with various effects present for the FEP melt at 371 8C at variousvalues of apparent shear rate (no slip)

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tained from the Cross model, although the shear behaviour is

similar for the two models (see Figs. 2 and 3). Thus, one would

expect the wall shear stresses predictions for the two different

models to be nearly the same. But the apparent shear stresses

are computed from the pressure drops, which are different for

the two models, as shown in Fig. 7. The differences in the pres-

sure drop are mainly dominated by the elongational viscosity in

the K-BKZ model. The decreasing diameter in the crosshead

will cause such normal stresses that become important at the

high throughput values. The values for the pressure drops be-

come of the same order as the differences in the pressure drops

for the two different models (Fig. 7).

The results overestimate the experimental data appreciably.

It is then at this point that we turn our attention to the slip re-

sults.

6.2 Slip at the Wall

Due to lack of different dies it was not possible to measure the

slip effects in the crosshead die. It becomes then a task for the

simulations to find the correct slip law that reproduces the ex-

perimental data. It is known that slip effects of FEP are present

in flows through conventional capillary dies used in rheometry

(Rosenbaum et al., 1995; 1998; 2000). In fact, Rosenbaum

et al. (1995) have used capillary dies of various diameters to

determine slip. The slip behaviour was found to be a sigmoidal

double-valued function, which also explained the origin of the

oscillating (stick-slip) melt fracture for this linear resin. In ca-

pillary flow, the flow curve consists of two distinct branches,

namely a low flow rate and a high flow rate that are separated

by a range of shear rates where stable flow is not possible

(stick-slip). However, with the crosshead die the stick-slip melt

fracture is absent, and the flow curve is a monotonically in-

creasing function of shear rate. Thus, the slip velocity deter-

mined from capillary flow does not apply in annular. It should

be stressed here that the flow curves, slip velocities and flow

instabilities, such as melt fracture, obtained from various geo-

metries, such as capillary, slit and annular dies, differ signifi-

cantly; the origin of these differences is an open question in

the literature (Delgadillo-Velazquez et al., 2008; Ansari et al.,

2009; 2011).

Therefore in the absence of a slip velocity model, simula-

tions are used to determine one by matching experimental re-

sults with simulations. We started off with a linear slip law

(b = 1 in Eq. 21), but the curvature of the experimental data

was not obtained. On the contrary, a quadratic law (b = 2) gave

the correct curvature. Then the bsl–slip coefficient was

allowed to vary to match the experimental data from the vis-

coelastic simulations. Its value was found to be 400 cm/

(s · MPab) and the matching was excellent as will be shown be-

low. Obviously, the same could be done for the viscous simula-

tions (Cross model), but we have chosen here to match the vis-

coelastic simulations because we believe that they more

accurately reflect the viscoelastic character of the FEP melt.

When using slip at the wall, the velocity vectors at the wall

are tangential to it and are found as part of the solution. This

is shown in Fig. 9, where the velocity vectors are given from

the viscoelastic K-BKZ simulations for the lowest and highest

apparent shear rates of 80 s – 1 and 5600 s – 1. Slip changes dras-

tically the velocity profiles across the die. Thus, we observe

that at 80 s – 1 slip at the die wall is small and the general annu-

lar profile is maintained, while at 5600 s – 1 slip is strong and

makes the profiles much flatter (more plug-like) due to the se-

vere flow conditions taking place in the die. Again, due to the

decreasing diameter in the crosshead the normal stresses be-

come important at the high throughput values. Moreover, when

slip dominates (Fig. 9B) these stresses will overrule shear


Fig. 7. Viscous vs. viscoelastic results for the axial pressure distribu-tion along the inner walls with various effects present for the FEP meltat 371 8C at various values of apparent shear rate (no slip)

Fig. 8. The apparent flow curve and its predictions for the FEP 4100melt in the Nokia Maillefer 4/6 crosshead at 371 8C. Symbols are ex-perimental data. Lines are simulation results by the viscous Crossmodel (base case, no effects present), the Cross and the K-BKZ modelswith various effects present (no slip)

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stress contributions and add to the pressure drop. We also ob-

serve that due to the abrupt change of slope at the entrance to

the die, the slip velocities there are much higher than the rest.

A curved design at that point is desirable in order to eliminate

such abnormally high velocities.

It is again instructive to compare temperature and pressure

distributions between the viscous and viscoelastic models with

all effects present including slip. This is done in Fig. 10, where

we show the maximum temperature at the exit from the Cross

and the K-BKZ models. Now, the maximum temperatures have

been reduced substantially, reaching at most 376 8C for

5600 s – 1 for the viscoelastic case, a rise of only 5 8C. In rea-

lity, this value is expected to be somewhat higher as the bound-

ary condition of constant temperature at the wall is not exactly

right, and a thermal balance would be more appropriate (Win-

ter, 1977; Mitsoulis et al., 1988).

Fig. 11 shows the axial pressure distribution along the inner

walls for different apparent shear rates. As with the tempera-

tures, the pressures are much lower now, reaching at the high-

est rate 20 MPa as opposed to 60 MPa when no slip was as-

sumed, a very substantial change. The viscoelastic pressures

are higher than the viscous ones, and this becomes more appar-

ent at elevated shear rates (maximum difference of 25% at

5600 s – 1). Again, the pressure falls smoothly to zero at exit,

thus giving evidence of a good crosshead design. The kinks

around z/Rw = –25 are now a bit more pronounced due to slip

because of the sudden change in slope for the last part of the

crosshead where it meets the die land.

Collecting again all these results together gives the apparent

flow curves shown in Fig. 12. As expected, the agreement is

excellent from the viscoelastic simulations, since the slip law

was based on matching these simulations with the experiments.

On the other hand, the same slip law with the viscous model

underestimates appreciably the experimental data.



A)

B)

Fig. 9. Velocity vectors near the entrance tothe die showing the acceleration of the veloc-ity and the non-zero vectors at the walls dueto slip for the FEP melt at 371 8C with the K-BKZ model: (A) _cA= 80 s – 1, (B) _cA=5600 s – 1

Fig. 10. Viscous vs. viscoelastic results for the maximum temperatureat exit with all effects present for the FEP melt at 371 8C at various val-ues of apparent shear rate

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7 Conclusions

A FEP melt has been studied in a crosshead die used in tubing

extrusion flows. Full rheological characterization was carried

out both with a viscous (Cross) and a viscoelastic (K-BKZ)

model. All necessary material properties data were collected

for the simulations.

Both viscous and viscoelastic models were found to over-

predict significantly the experimental apparent flow curve

(shear stress vs. shear rate) in a wide range of extrusion rates

reaching up to 5600 s – 1, where the experiments produced

smooth extrudates. The temperature rises were substantially

high for such high apparent shear rates. Viscoelasticity gave

higher results than the purely viscous Cross model.

The smooth experimental curve and the good condition of

the extrudates made the addition of slip at the wall a necessity,

given the fact that this melt appears to slip in capillary dies.

Due to lack of experimental data, a quadratic slip law was as-

sumed, which gave the correct predictions for the experiments

by using the viscoelastic K-BKZ model. Then the same slip

law used with the Cross model gave under-predictions as ex-

pected. Obviously, the opposite would have happened if we

had chosen to match the viscous simulations (Cross model),

but we believe that the K-BKZ model more accurately reflects

the viscoelastic character of the FEP melt. The numerical simu-

lations gave a good understanding of the various effects at play

(weak compressibility, temperature- and pressure-dependence

of viscosity, slip effects) and provided hints for die design opti-

mization. Furthermore, it has been verified that this is a good

design producing smooth results at high shear rates with small

to moderate temperature rises due to significant slip effects.

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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: June 23, 2011

Date accepted: September 1, 2011

BibliographyDOI 10.3139/217.2534Intern. Polymer ProcessingXXVII (2012) 2; page 259–269ª Carl Hanser Verlag GmbH & Co. KGISSN 0930-777X

You will find the article and additional material by enter-ing the document number IIPP2534 on our website atwww.polymer-process.com


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