Annular Extrudate Swell of a Fluoropolymer Meltold-2017.metal.ntua.gr/uploads/3798/1094/A143.pdf · The tensorI in Eq. 5 is the unit tensor. To evaluate the role of viscoelasticity

E. Mitsoulis1*, S. G. Hatzikiriakos2

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

Annular Extrudate Swell of a Fluoropolymer Melt

Annular extrudate swell is studied for a fluoropolymer (FEP)

melt using a tubular die. The rheological data of the melt have

been fitted using (i) a viscous model (Cross) and (ii) a visco-

elastic one (the Kaye – Bernstein, Kearsley, Zapas/Papanasta-

siou, Scriven, Macosko or K-BKZ/PSM model). Numerical

simulations have been undertaken to study the extrudate swell

of the FEP melt in an annular die. Compressibility, thermal

and pressure effects on viscosity, and slip at the wall were tak-

en into account. In all cases, slip at the wall is the dominant

contribution reducing the swelling when compared with corre-

sponding no-slip simulations. The viscous (Cross) simulations

show that the swell decreases with increasing apparent shear

rate, which is opposite to what happens in the extrusion of vis-

coelastic melts. On the other hand, the viscoelastic (K-BKZ)

simulations correctly obtain increasing swelling with increas-

ing shear (flow) rates. It was found that due to the mild visco-

elasticity of FEP and its severe slip at the wall the swelling of

this melt is relatively small, reaching values of about 20% for

a wide range of apparent shear rates, exceeding 5000 s–1. This

is corroborated by experimental observations.

1 Introduction

Fluoropolymers are of great technological interest due to their

unique combination of properties. These include excellent

chemical stability and dielectric properties, anti-stick character-

istics, mechanical strength, and low flammability (Imbalzano

and Kerbow, 1994; Ebnesajjad, 2003). Their most important

uses are in electronics and electrical applications, such as

wiring insulation, chemical processing equipment, medical de-

vices and laboratory ware and tubing among others (Ebnesajjad,

2003; Domininghaus, 1993; Dealy and Wissbrun, 1990). Their

polymer processing involves mainly tubing extrusion of fluor-

oethylene copolymers (FEP), in which the melt comes out of an

annular crosshead die and meets a moving wire for wire insula-

tion (Baird and Collias, 1998; Tadmor and Gogos, 2006).

The rheology of FEP melts was previously studied by Ro-

senbaum et al. (1995, 1998, 2000) and more recently by Mit-

soulis and Hatzikiriakos (2012a), who used a viscous Cross

model and a viscoelastic K-BKZ constitutive equation to mod-

el the available rheological data at hand. It was found that for

shear rates below about 80 s–1, both models resulted in more

or less the same predictions for the pressure drop in capillary

dies, and that viscoelasticity was not found important under

these mild conditions. However, at high shear rates (as high as

6000 s–1), slip was found to dominate the flow behavior, espe-

cially in tubing extrusion, where an annular crosshead die was

employed at high speeds (Rosenbaum, 1998; Rosenbaum et al.,

2000). The analysis of this annular flow was undertaken in a

follow-up paper by Mitsoulis and Hatzikiriakos (2012b). A

power-law slip model was assumed, and the parameters of this

model were fitted to match the experimental pressure drops by

using the viscoelastic K-BKZ model.

A natural consequence of these two recent works is to exam-

ine what happens to the melt after it exits the annular die and

acquires a free surface, giving rise to the phenomenon of annu-

lar extrudate swell. The problem of extrudate swell has been a

major benchmark problem in Fluid Mechanics since the early

work of Middleman and Gavis (1961a, 1961b). Tanner (1973)

was the first to solve the Newtonian problem in capillary and

slit dies, while Mitsoulis (1986) performed a thorough para-

metric study for annular dies. Since then, most of the simula-

tion efforts for extrudate swell have been concentrated on vis-

coelastic fluids (Tanner, 2000), and most of these efforts refer

to flows from capillary dies (round tubes).

Surprisingly, little work has been done on annular extrudate

swell and even less so for the effect of viscoelasticity on swell.

Notable exceptions are the work by Luo and Mitsoulis (1989)

on HDPE melts using a K-BKZ integral model, and more

recently by Karapetsas and Tsamopoulos (2008), who have

used a Phan-Thien/Tanner (PTT) differential model for full

parametric studies of various parameters affecting the swell

behavior. These authors have also summarized all important

works on annular extrudate swell up to 2008.

The effect of slip on annular flow is another important scien-

tific issue that has attracted attention in the literature. Mitsoulis

(2007a) examined the effect of wall slip on the extrudate in an-

nular flows of Newtonian fluids. More recently, Chatzimina

et al. (2009) examined the stability of annular flows for Newto-

nian fluids. However, a study involving slip at the wall for a

viscoelastic melt in annular flow has not been studied before.

It is, therefore, the main objective of this work to examine

the annular extrudate swell problem of the FEP melt studied

in our previous works (Mitsoulis and Hatzikiriakos, 2012a;

2012b). Having a full rheological characterization and the fit-

ting of the parameters with a viscous (Cross) and a viscoelastic

(K-BKZ) model, as well as a slip model, it would be interesting

to find out the extent to which the melt swells, and also the pre-

dictions and trends by the different models. Although the pres-

sure drops in the annular crosshead die have been well simulat-

ed before by both models, their predictive capabilities differ

vastly outside the die, where the viscoelastic character of the

fluid most eloquently manifests itself.

INVITED PAPERS

Intern. Polymer Processing XXVII (2012) 5 � Carl Hanser Verlag, Munich 535

* Mail address: Evan Mitsoulis, School of Mining Engineering andMetallurgy, National Technical University of Athens, Zografou,157 80, Athens, GreeceE-mail: [email protected]

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt

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

(2012a). 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.

In our previous work (Mitsoulis and Hatzikiriakos, 2012b) we

have presented a schematic of the tubing extrusion die used in the

experiments and the numerical simulations together with the de-

tails of the design. In the present work we only consider for sim-

plicity the most important part of the annular crosshead die,

namely the die land (see Fig. 1), which has an outer diameter D0

of 0.3048 cm (0.12’’) and an inner diameter Di of 0.1524 cm

(0.06’’), resulting in an annular gap h0 of 0.0762 cm (0.03’’).

Thus, the diameter ratio is defined as j = Di/D0 = 0.5. The die

land length L is 0.762 cm (0.3’’). If we scale the lengths with the

die gap h0, then we obtain D0 = 4h0, Di = 2h0, L = 10h0. If we

consider the extrudate, then the equivalent lengths are Dp for

the outer extrudate swell diameter, hp for the extrudate thickness

swell, and Lp for the extrudate length (Fig. 1).

The experimental results for the extrudate swell in annular

flow were found by inspection of the extrudate after the extru-

sion as it was difficult to exactly measure the thickness of the

annulus due to small dimensions. The extrudates had to be

microtomed in order to measure the wall thickness. At all ap-

parent shear rates the swelling was negligible, and was not

measured in detail. In most cases the values were close to about

10 to 20% more than the annular die gap.

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

1þ ðk _cÞ1�n; ð8Þ

536 Intern. Polymer Processing XXVII (2012) 5

Fig. 1. Schematic representation of extrusion through an annular dieand notation for the numerical analysis

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

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

and n is the power-law index. The experimental data of FEP

were obtained at 300 8C, 325 8C, and 350 8C (Rosenbaum et al.,

1995), while the experiments with the tubing die were per-

formed at 371 8C. The data has been shifted to 371 8C by using

the Arrhenius relationship (Dealy and Wissbrun, 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), in this case 644 K (371 8C). The activation

energy was calculated from the shift factors determined by ap-

plying the time-temperature superposition principle to obtain

the curves at 371 8C plotted in Fig. 2 (Dealy and Wissbrun,

1990; Mitsoulis and Hatzikiriakos, 2012b). It was found that

E = 50,000 J/mol. The fitted viscosity of the FEP melt by

Eq. 8 is plotted in Fig. 2 (continuous line), while the parame-

ters 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 approximately 10 s–1 giv-

ing a power-law index n = 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 below).

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� �

; ð10Þ

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, 2012a; 2012b):

bp ¼ mp�np ; ð11Þ

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

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

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; ð12Þ

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; ð13Þ

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 and have been obtained by the fitting

procedure outlined by Kajiwara et al. (1995), using linear vis-

coelastic data which can be found elsewhere (Mitsoulis and

Hatzikiriakos, 2012a; 2012b).

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; Alaie and

Papanastasiou, 1993; Barakos and Mitsoulis, 1996; Peters and

Baaijens, 1997; Beaulne and Mitsoulis, 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 and are taken from various sources (Van Krevelen,

1990; Hatzikiriakos and Dealy, 1994; Ebnesajjad, 2000; Mit-

soulis and Hatzikiriakos, 2012a; 2012b). The viscoelastic stres-

ses calculated by the non-isothermal version of the above con-


Intern. Polymer Processing XXVII (2012) 5 537

Fig. 2. The shear viscosity of the FEP melt at 371 8C fitted with theCross model (Eq. 8) using the parameters listed in Table 1. For com-parison the prediction from the K-BKZ model (Eq. 12) is also shown

Parameter Value

g0 1542.3 Pa � sk 0.0049 sn 0.316

Table 1. Parameters for the FEP melt obeying the Cross model(Eq. 8) at 371 8C

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.


stitutive equation (Eq. 12) enter in the energy equation (Eq. 3)

as a contribution to the viscous dissipation term.

The various thermal and flow parameters are combined to

form 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 ¼qCpUh0

k; ð14Þ

Na ¼�gEU2

kRgT20

; ð15Þ

where �g ¼ fðU=h0Þ is a nominal viscosity given by the Cross

model (Eq. 8) at a nominal shear rate of U/h0. In the above,

U is the average velocity in the die [U = 4Q/p(D02�Di

2)],

where Q is the volumetric flow rate. The Pe number repre-

sents the ratio of heat convection to conduction, and the Na

number represents the ratio of viscous dissipation to conduc-

tion and indicates the extent of coupling between the momen-

tum and energy equations. A thorough discussion of these

effects in non-isothermal polymer melt flow is given by Win-

ter (1977).

With the above properties and a characteristic length the die

gap h0 = 0.0762 cm, the dimensionless thermal numbers are

calculated to be in the ranges: 42 < Pe < 2952, 0.007 < Na <

11, 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-unifor-

mities generated by viscous dissipation, and a strong coupling

between momentum and energy equations. More details are

given in Table 4.

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. (10) is used to modify the viscosity. For

viscoelastic models, such as the K-BKZ model (Eq. 12), the

pressure-shift factor modifies the relaxation moduli, ak, ac-

cording to:

ak p t0ð Þð Þ ¼ ak p0ð Þap p t0ð Þð Þ: ð16Þ

This is equivalent to multiplying the stresses by ap, according

to Eq. 10. 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: ð17Þ

Similarly, the compressibility coefficient bc gives rise to the

dimensionless compressibility parameter, Bc. This is defined

as:

Bc ¼bc�gU

h: ð18Þ

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 ranges of values are: 5.2 · 10–4 < Bp <

1.1 · 10–2 and 1.7 · 10–5 < Bc < 3.5 · 10–4, showing a moder-

ate dependence of viscosity on pressure and an even weaker

compressibility effect in the range of simulations. More details

are given in Table 4.


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

k kks

akPa

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. 12) at 371 8C (� = 7.174,b = 0.6, h = –0.11, �k = 0.76 s, g0 = 1613 Pa � s)

Parameter Value

bc 0.00095 MPa–1

bp 0.03 MPa–1

m 1.39 · 10–4 Panp�1

np 0.54bsl 400 cm/(MPab · s)b 2.0q 1.492 g/cm3

Cp 0.96 J/(g ·K)k 0.00255 J/(s · cm ·K)E 50,000 J/molRg 8.314 J/(mol ·K)T0 371 8C (644 K)

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

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

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 �bslsbw; ð19Þ

where usl is the slip velocity, sw 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. The slip law and the values of bsl and b were deter-

mined in detail in our recent work (Mitsoulis and Hatzikiria-

kos, 2012b) and found to be: bsl = 400 cm/(MPab · s) and b = 2.

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; ð20Þ

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

: ð21Þ

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: 0.122 < Bsl < 0.795, show-

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

More details are given in Table 4.

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, 1990a; Barakos and

Mitsoulis, 1996).

The boundary conditions (BC’s) for the problem at hand are

well known and shown in Fig. 4. 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 vz(r) or the

axial surface traction Tz is imposed, corresponding to the flow

rate at hand, and a constant temperature T0 is assumed; at the

outlet, zero surface traction and zero heat flux q are assumed;

on the two free surfaces, no penetration and zero heat flux are

imposed. It was found that when slip was present, it was more

difficult to find numerically the inlet velocity profile corre-

sponding to a known flow rate, especially for the higher range

of flow rates. It was found much easier to just impose the axial

surface traction there and then calculate the flow rate by inte-

grating the inlet velocity profile obtained from the solution.

To do this accurately, it was found necessary to have a refined

grid near the inlet.

Typical finite element meshes are shown in Fig. 5 for the an-

nular die of Fig. 1. The entry length is L0 = 10h0, long enough

to guarantee fully-developed conditions even for the viscoelas-

tic runs. The extrudate length depended on whether we used

viscous or viscoelastic simulations. For the viscous simula-

tions, there are no memory effects and a relatively short extru-

date length Lp = 12h0 suffices (see Fig. 5A, meshes M1 and

M2). For the viscoelastic simulations, memory effects are im-

portant and longer meshes are necessary. We have tried extru-

date lengths of Lp = 17h0 and 30h0, and report the results here



Apparent shear rate,_cAs–1

Peclet number, Pe Nahme number, Na Compressibility pa-rameter, Bc

Pressure-shiftparameter, Bp

Slip parameter, Bsl

80 42 0.007 1.65 · 10–5 5.20 · 10–4 0.122320 169 0.10 5.44 · 10–5 1.72 · 10–3 0.334800 420 0.49 1.09 · 10–4 3.45 · 10–3 0.538

1600 840 1.56 1.74 · 10–4 5.49 · 10–3 0.6823200 1680 4.69 2.62 · 10–4 8.27 · 10–3 0.7735600 2952 11.07 3.52 · 10–4 1.11 · 10–2 0.795

Table 4. Range of the dimensionless parameters in the flow of FEP melt at 371 8C (based on die land gap h0 = 0.0762 cm)

Fig. 4. Schematic diagram of flow domainand boundary conditions. S1 and S2 are thetwo singular points

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.


for the longer case (Fig. 5B, C meshes M3 and M4). Mesh M3

consists of 1560 elements, 4965 nodes, and 16,598 unknown

u-v-p-T degrees of freedom (d.o.f.), while mesh M4 is 4-times

denser, having been created by subdivision of each element

into 4 sub-elements for checking purposes of mesh-indepen-

dent 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 geome-

try, the only parameter left to vary was the apparent shear rate

in the die. 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 pre-

sence of a small amount of processing aids (boron nitride),

although the pressure drop remained the same with and without

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

1998; 2000).

The viscous simulations are extremely fast and are used as a

first step to study the whole range of parameter values. The vis-

coelastic simulations admittedly are harder to do and they need

good initial flow fields to get solutions at elevated apparent

shear rates. In our recent work (Ansari et al., 2010), we ex-

plained 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. Here, an extra complication arises from the

presence of free surface (actually two of them), for which se-

vere under-relaxation (factor xf = 0.1) must be used to avoid

particle tracking occurring outside the domain.

Briefly, the solution strategy starts from the Newtonian solu-

tion at the lowest apparent shear rate (10 s–1) for the base case

(bc = bp = aT = bsl = 0). Then at the given apparent shear rate,

the viscoelastic model is turned on and the solution is pursued

in the given domain until the norm of the error is below 10–4.

Then the free-surface update is turned on, and the u-v-p-T solu-

tion is alternated with the h-solution (free surface location) un-

til the maximum free surface change is less than 10–5. Meeting

this criterion gives a very good solution for the problem at

hand. Using this solution as an initial guess, the apparent shear

rate is then raised slowly to get a new solution at an elevated

value. This way it was possible to achieve solutions for as high

as 5600 s–1. It must be noted that FEP is not strongly viscoelas-

tic, unlike HDPE (Luo and Mitsoulis, 1989) and LDPE (Luo

and Mitsoulis, 1990b; Barakos and Mitsoulis, 1996), for which

it was not possible to reach apparent shear rates greater than

10 s–1 for the extrudate swell problem.

When all effects are present, we follow the same procedure.

Now the biggest contribution comes from slip, since tempera-

ture-dependence and pressure-dependence of the viscosity

have opposite effects. With slip present, the simulations are

much faster as they require fewer iterations due to the less dras-

tic flow conditions encountered in the flow field (actual shear


A)

B)

C)

Fig. 5. Finite element meshes used in the computations: (A) for vis-cous simulations; (B) and (C) for viscoelastic simulations; (C) blow-up meshes near the die exit. In (a), the upper half shows mesh M2 con-taining 4560 quadrilateral elements, while the lower half shows meshM1 containing 1140 elements. In (b) and (C), the upper half showsmesh M4 containing 6240 quadrilateral elements, while the lower halfshows mesh M3 containing 1560 elements

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

rates at the die walls are an order of magnitude less with slip

present). Also the swell is reduced compared with the base

case, which makes it easier to solve the nonlinear problem.

All velocities have been made dimensionless with the aver-

age velocity U. The lengths can be made dimensionless either

with the die gap h0 or the outer radius R0. In the former case

the pressures and stresses are made dimensionless by g0U/h0,where g0 is the zero-shear rate viscosity.

5 Numerical Results

We perform our simulations in the die land of the tubing (annu-

lar) die having a Di = 1.524 mm tip and D0 = 3.048 mm die at

T = 371 8C. Instead of the flow rate Q, the apparent shear rate

was used and calculated by using the formula which applies to

slit dies (Dealy and Wissbrun, 1990):

_cA ¼6Q

0:25ðD0 � DiÞ20:5pðD0 þ DiÞ

: ð22Þ

The results are given in terms of the three dimensionless swell

ratios, B, only two of which are independent. Referring to

Fig. 1, these are defined as follows (Orbey and Dealy, 1984;

Mitsoulis, 1986):

(i) the (outer) diameter swell, B1, defined by

B1 ¼Dp

D0

; ð23Þ

(ii) the thickness swell, B2, defined by

B2 ¼hp

h0; ð24Þ

(iii) the inner diameter swell, B3, defined by

B3 ¼Dp � 2hp

D0 � 2h0: ð25Þ

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 viscous

model and with a viscoelastic one so that the differences be-

come evident. The numerical simulations have been undertaken

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

coelastic K-BKZ model (Eq. 12). 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-depen-

dence of the viscosity. The viscous results have been checked

against each other either with a viscous code (Hannachi and

Mitsoulis, 1993) or with the viscous option of the viscoelastic

code (Luo and Mitsoulis, 1990b) giving the same results.

First, runs were carried out for the base case of no effects at

all (bc = bp = bsl = aT = 0), referred in the graphs as no slip, be-

cause slip is the predominant effect in the current problem.

Then, all effects were turned on, referred in the graphs as slip,

so that the differences become evident.

Due to the high range of simulations, all effects had a contri-

bution, although the effects of compressibility and tempera-

ture- and pressure-dependence of the viscosity are small (see

also the range of the corresponding dimensionless numbers).

First, we note in Fig. 6 that the pressures in the system are high-

est for the viscoelastic (K-BKZ) base case (no slip) followed by

the viscous (Cross) base case (no slip). Slip brings down the

pressures appreciably, especially at the higher apparent shear

rate range, although the viscoelastic slip simulations are still

higher than the viscous ones. Symbols are used in Fig. 6 and

the subsequent figures to show the simulation continuation

steps. In Fig. 6 and subsequent figures, we show the results in

the range above 10 s–1, because the extrusion of FEP is carried

out at elevated shear rates (usually above 1000 s–1). In the

range below 10 s–1, FEP behaves as a Newtonian fluid (see also

rheological data in Figs. 2 and 3), and all the results tend to-

wards their Newtonian limits, known from previous works

(Mitsoulis, 1986; 2007a).

Regarding the extrudate swell, the results are given in

Figs. 7 to 9. We observe that the viscous Cross model predicts

a decrease in all swell ratios with increasing apparent shear

rate, something which is well known in the literature (see, Mit-

soulis et al., 1984; Mitsoulis, 2007b). However, this finding is

not physically observed for polymeric liquids. The swell ratios

have a tendency to go down to 0 (no swell), and this is en-

hanced by slip at the wall, which reduces the swell even more

(Mitsoulis, 2007a).

Viscoelasticity on the other hand, as applied with the K-

BKZ model, after an initial decrease for the thickness swell,

increases the swelling, and this is more evident at the higher

range of apparent shear rates. It is to be noted that the visco-

elasticity of FEP is not strong and not as high as for other

polymer melts, notably polyethylenes, and this is shown in

Table 2 by the average relaxation time, �k < 1 s. Therefore,

swelling is not enhanced. Again, slip brings down the swel-



Fig. 6. Pressure drop DP as a function of the apparent shear rate forFEP melt at 371 8C. No slip refers to the base case (bc =bp = bsl = aT = 0), while slip refers to all effects accounted for. Sym-bols are put to show the simulation continuation steps

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.


ling as it should. The same trend was found in slit planar

flows by Phan-Thien (1988). An interesting phenomenon oc-

curs in thickness swell in Fig. 7, where the results are not

monotonic but go through a maximum both for slip and with-

out slip. The same trend was first found by Luo and Mitsoulis

(1989) for an HDPE melt and was confirmed recently by Kar-

apetsas and Tsamopoulos (2008) with the Phan-Thien/Tanner

(PTT) model. B2 is a manifestation of the elasticity of the

melt, which is similar to swell through a capillary die, and

thus increases monotonically and in some cases significantly

with shear rate. As B2 increases significantly, B1 should in-

crease mildly or even decrease, and this is dictated by conser-

vation of mass.

Details of the shape of the free surface at different apparent

shear rates are given in Fig. 10 for the base case

(bc = bp = bsl = aT = 0), which gives higher swellings. The case

when all effects are accounted for (slip) is shown in Fig. 11.

Now the swelling is less and takes a shorter distance to fully

develop. Higher apparent shear rates give higher swellings in

both cases.

In all cases the viscoelastic slip results confirm the experi-

mental findings of moderate swelling around 10 to 20%. But

of course, the simulations give a much more detailed picture

of this phenomenon, which is of interest in polymer processing

applications.

This detailed picture is given in Figs. 12 and 13 as contours

of various field variables near the exit of the annular die for a

typical apparent shear rate ( _cA = 3000 s–1) when all effects

are accounted for (slip). Fig. 12 shows the streamlines and the

contours of the primitive variables of the finite element formu-

lation, namely the velocities-pressures-temperatures (u-p-T).

The contour values are given in equally spaced 11 intervals be-

tween the maximum and minimum values. The streamlines

(STR) are smooth (Fig. 12A) and are obtained from the inte-

gration of the velocity field. Its maximum value corresponds

to the flow rate Q. The axial velocity (U) contours (Fig. 12B)

show the change from a confined flow inside the die into a

shear-free flow in the extrudate. The pressure (P) contours (iso-

bars, Fig. 12C) show that inside the die there are curved con-


Fig. 7. Thickness swell B2 as a function of the apparent shear rate forFEP melt at 371 8C. No slip refers to the base case (bc =bp = bsl = aT = 0), while slip refers to all effects accounted for. Sym-bols are put to show the simulation continuation steps

Fig. 8. Diameter swell B1 as a function of the apparent shear rate forFEP melt at 371 8C. No slip refers to the base case (bc =bp = bsl = aT = 0), while slip refers to all effects accounted for. Sym-bols are put to show the simulation continuation steps

Fig. 9. Inner diameter swell B3 as a function of the apparent shearrate for FEP melt at 371 8C. No slip refers to the base case(bc = bp = bsl = aT = 0), while slip refers to all effects accounted for.Symbols are put to show the simulation continuation steps

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

tours due to a non-zero second normal stress difference N2. To

get the actual values of pressure in MPa, the contour values

have to be multiplied by g0 = 0.001613 MPa · s. The tempera-

ture (T) contours (isotherms, Fig. 12D) show that the flow is

almost isothermal, as the differences from the wall reference

temperature of 371 8C are minimal (less that 1 8C), with the

higher differences being near the walls due to viscous dissipa-

tion.

Fig. 13 shows contours of the four stresses obtained from

the K-BKZ integral model (Eq. 12) for the same conditions

as in Fig. 12. Points to notice are the high values of the axial

stresses szz (Fig. 13A), while the radial stresses srr (Fig. 13B)are much lower. The shear stresses szr (Fig. 13C) are both

negative and positive from the shear flow inside the die, while

the azimuthal stresses shh (Fig. 13D) are the lowest and obtaintheir highest values just after the die exit in the free extrudate.

Due to severe slip at the wall, the actual stress values

(obtained in MPa by multiplying the contour values by

g0 = 0.001613 MPa · s) are small, and this also helps in obtain-

ing converged solutions fast.



Fig. 10. Shapes of the calculated free surfaces of the annular extru-date obtained at different apparent shear rates with the K-BKZ modelfor the base case (bc = bp = bsl = aT = 0)

Fig. 11. Shapes of the calculated free surfaces of the annular extru-date obtained at different apparent shear rates with the K-BKZ modelwith all effects accounted for

A) B)

C) D)

Fig. 12. Contours of various field variablesobtained at _cA = 3000 s–1 with the K-BKZmodel with all effects accounted for (slip):(A)~streamlines (STR), (b) axial velocity (U),(C) pressure (P), (D) temperature (T). To getthe pressure values in MPa, multiply contourvalues by g0 = 0.001613 MPa · s

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.


Fig. 14 shows axial distributions along the walls and the free

surfaces of the four stresses (given in MPa) obtained from the K-

BKZ integral model (Eq. 12) for the same conditions as in

Fig. 13. Points to notice are the high peak values of all the stresses

near the die exit singularities (at z/R0 = 5) and their subsequent

decrease to zero along the two free surfaces. The axial stresses

szz (Fig. 14A) are highest near the singularity and then exponen-

tially decay to zero, while the radial stresses srr (Fig. 14B) arenegative and much lower. The shear stresses szr (Fig. 14C) arenegative on the outer wall and positive on the inner due to shear


A) B)

C) D)

Fig. 13. Contours of various field variablesobtained at _cA= 3000 s–1 with the K-BKZmodel with all effects accounted for (slip): (A)axial stress szz (TXX), (B) radial stress srr(TYY), (C) shear stress szr (TXY), (D) azi-muthal stress shh (T33). To get the stress val-ues in MPa, multiply contour values by g0 =0.001613 MPa · s

A) B)

C) D)

Fig. 14. Stress distributions (in MPa) alongthe walls and the free surfaces (f. s.) obtainedat _cA = 3000 s–1 with the K-BKZ model withall effects accounted for (slip): (A) axial stressszz, (B) radial stress srr, (c) shear stress szr,(D) azimuthal stress shh

�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

flow inside the annular die, and they approach zero in the extru-

date much faster than the normal stresses. The azimuthal stresses

shh (Fig. 14D) are much lower than the other stresses and obtain

their highest values just after the die exit in the free extrudate,

again showing an exponential decay to zero much further down.

6 Conclusions

A FEP melt has been studied in an annular 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.

Extrudate swell simulations with the viscous model were

found to give reduced swelling with increasing apparent shear

rates, opposite to what is expected and experimentally found

for viscoelastic polymer melts, such as FEP. Viscoelastic simu-

lations gave the correct trends, as the swelling increased with in-

creasing shear rates. However, the viscoelasticity of FEP is not

very strong (average relaxation time �k < 1 s), and therefore the

swelling is not dramatic. Furthermore, due to slip at the wall,

which is the most significant effect for this melt, the swelling

was reduced appreciably when compared with the no-slip vis-

coelastic simulations. The viscoelastic simulations with slip

gave swellings in the order of 10 to 20% in the range of apparent

shear rates from 10 to 5600 s–1. These results are in qualitative

agreement with experimental observations that found no appre-

ciable swelling, reaching about 10–20%. This behavior is in

contrast with other viscoelastic polymer melts, notably poly-

ethylenes, such as HDPE (Luo and Mitsoulis, 1989; Kiriakidis

and Mitsoulis, 1993) and LDPE (Barakos and Mitsoulis, 1995,

1996) that have shown high swell ratios even at apparent shear

rates as low as 10 s–1, due to their enhanced viscoelasticity.

References

Alaie, S. M., Papanastasiou, T.C., \Modeling of Non-isothermal FilmBlowing with Integral Constitutive Equations", Int. Polym. Proc.,8, 51–65 (1993)

Ansari, M., et al., \Entry Flow of Polyethylene Melts in Tapered Dies",Int. Polym. Proc., 25, 287–296 (2010),DOI:http://dx.doi.org/10.3139/217.2360

Baird, D. G., Collias, D. I.: Polymer Processing: Principles and Design,John Wiley, New York (1998)

Barakos, G., Mitsoulis, E., \Numerical Simulation of Extrusionthrough Orifice Dies and Prediction of Bagley Correction for anIUPAC-LDPE Melt", J. Rheol., 39, 193–209 (1995),DOI:http://dx.doi.org/10.1122/1.550700

Barakos, G., Mitsoulis, E., \Non-Isothermal Viscoelastic Simulationsof Extrusion through Dies and Prediction of the Bending Phenomen-on", J. Non-Newtonian Fluid Mech., 62, 55–79 (1996),DOI:http://dx.doi.org/10.1016/0377-0257(95)01385-7

Beaulne, M., Mitsoulis, E., \Effect of Viscoelasticity in the Film-Blowing Process", J. Appl. Polym. Sci., 105, 2098–2112 (2007),DOI:http://dx.doi.org/10.1002/app.26325

Carreras E. S., et al., \Pressure Effects on Viscosity and Flow Stabilityof Polyethylene Melts during Extrusion", Rheol. Acta, 45, 209–222(2006), DOI:http://dx.doi.org/10.1007/s00397-005-0010-1

Chatzimina, M., et al., \Stability of the Annular Poiseuille Flow of aNewtonian Fluid with Slip along the Walls", J. Non-NewtonianFluid Mech., 159, 1–9 (2009),DOI:http://dx.doi.org/10.1016/j.jnnfm.2008.10.008

Dealy, J. M., Wissbrun, K. F.: Melt Rheology and its Role in PlasticsProcessing – Theory and Applications, Van Nostrand Reinhold,New York (1990)

Domininghaus, H.: Plastics for Engineers: Materials, Properties, Ap-plications, Hanser, Munich (1993)

Ebnesajjad, S.: Fluoroplastics: Vol. 2, Melt Processing Fluoroplastics,William Andrew, New York (2000)

Ebnesajjad, S.: Melt Processible Fluoropolymers, Elsevier, New York(2003)

Ferrandino, M., \Tubing Extrusion Made Easier, Part I", Medical De-vice Technology, 15.8, 12–15 (2004a)

Ferrandino, M., \Tubing Extrusion Made Easier, Part II", Medical De-vice Technology, 15.9, 20–23 (2004b)

Hannachi, A., Mitsoulis, E., \Sheet Coextrusion of Polymer Solutionsand Melts: Comparison between Simulation and Experiments",Adv. Polym. Technol., 12, 217–231 (1993),DOI:http://dx.doi.org/10.1002/adv.1993.060120301

Hatzikiriakos, S. G., Dealy, J. M., \Start-up Pressure Transients in aCapillary Rheometer", Polym. Eng. Sci., 34, 493–499 (1994),DOI:http://dx.doi.org/10.1002/pen.760340606

Imbalzano, A. E., Kerbow, D. L., \Advances in Fluoroplastics",Trends Polym. Sci., 2, 26–30 (1994)

Kajiwara, T., et al., \Rheological Characterization of Polymer Solu-tions and Melts with an Integral Constitutive Equation", Int. J.Polym. Anal. Charact., 1, 201–215 (1995),DOI:http://dx.doi.org/10.1080/10236669508233875

Karapetsas, G., Tsamopoulos, J., \Steady Extrusion of ViscoelasticMaterials from an Annular Die", J. Non-Newtonian Fluid Mech.,154, 136–152 (2008),DOI:http://dx.doi.org/10.1016/j.jnnfm.2008.04.007

Kiriakidis, D. G., Mitsoulis, E., \Viscoelastic Simulations of ExtrudateSwell for an HDPE Melt through Slit and Capillary Dies", Adv.Polym. Technol., 12, 107–117 (1993),DOI:http://dx.doi.org/10.1002/adv.1993.060120201

Luo, X.-L., Mitsoulis, E., \Memory Phenomena in Extrudate SwellSimulations for Annular Dies", J. Non-Newtonian Fluid Mech., 33,1307–1327 (1989)

Luo, X.-L., Mitsoulis, E., \An Efficient Algorithm for Strain HistoryTracking in Finite Element Computations of Non-Newtonian Fluidswith Integral Constitutive Equations", Int. J. Num. Meth. Fluids, 11,1015–1031 (1990a),DOI:http://dx.doi.org/10.1002/fld.1650110708

Luo, X.-L., Mitsoulis, E., \A Numerical Study of the Effect of Elonga-tional Viscosity on Vortex Growth in Contraction Flows of Poly-ethylene Melts", J. Rheol., 34, 309–342 (1990b),DOI:http://dx.doi.org/10.1122/1.550131

Luo, X.-L., Tanner, R. I., \A Pseudo-time Integral Method for Non-Isothermal Viscoelastic Flows and its Application to Extrusion Sim-ulation", Rheol. Acta, 26, 499–507 (1987),DOI:http://dx.doi.org/10.1007/BF01333733

Luo, X.-L., Tanner, R. I., \Finite Element Simulation of Long andShort Circular Die Extrusion Experiments Using Integral Models",Int. J. Num. Meth. Eng., 25, 9–22 (1988),DOI:http://dx.doi.org/10.1002/nme.1620250104

Middleman, S., Gavis, J., \Expansion and Contraction of Capillary Jetsof Newtonian Liquids", Phys. Fluids, 4, 355–359 (1961a),DOI:http://dx.doi.org/10.1063/1.1706332

Middleman, S., Gavis, J., \Expansion and Contraction of Capillary Jetsof Viscoelastic Liquids", Phys. Fluids, 4, 963–969 (1961b),DOI:http://dx.doi.org/10.1063/1.1706446

Mitsoulis, E., \Extrudate Swell of Newtonian Fluids from AnnularDies", AIChE J., 32, 497–500 (1986),DOI:http://dx.doi.org/10.1002/aic.690320317

Mitsoulis, E., \Annular Extrudate Swell of Newtonian Fluids: Effectsof Compressibility and Slip at the Wall", J. Fluids Eng., 129,1384–1393 (2007a), DOI:http://dx.doi.org/10.1115/1.2786491

Mitsoulis, E., \Annular Extrudate Swell of Pseudoplastic and Visco-plastic Fluids", J. Non-Newtonian Fluid Mech., 141, 138–147(2007b), DOI:http://dx.doi.org/10.1016/j.jnnfm.2006.10.004

Mitsoulis, E., et al., \Numerical Simulation of Entry and Exit Flows inSlit Dies", Polym. Eng. Sci., 24, 707–715 (1984),DOI:http://dx.doi.org/10.1002/pen.760240913


Intern. Polymer Processing XXVII (2012) 5 545�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.


Mitsoulis, E., et al., \Numerical Simulation of Wire-Coating Low-Density Polyethylene: Theory and Experiments", Polym. Eng. Sci.,28, 291–310 (1988), DOI:http://dx.doi.org/10.1002/pen.760280505

Mitsoulis, E., et al., \The Effect of Slip in the Flow of a Branched PPPolymer: Experiments and Simulations", Rheol. Acta, 44, 418–426 (2005), DOI:http://dx.doi.org/10.1007/s00397-004-0423-2

Mitsoulis, E., Hatzikiriakos, S. G., \Capillary Flow of a FluoropolymerMelt", Int. J. Mater. Form., (2012a),DOI:10.1007/s12289-011-1062-7

Mitsoulis, E., Hatzikiriakos, S. G., \Tubing Extrusion of a Fluoropoly-mer Melt", Int. Polym. Proc., 27, 259–269 (2012b),DOI:http://dx.doi.org/10.3139/217.2534

Orbey, N., Dealy, J. M., \Isothermal Swell of Extrudate from AnnularDies; Effects of Die Geometry, Flow Rate, and Resin Characteris-tics", Polym. Eng. Sci., 24, 511–518 (1984),DOI:http://dx.doi.org/10.1002/pen.760240710

Papanastasiou, A. C., et al., \An Integral Constitutive Equation forMixed Flows: Viscoelastic Characterization", J. Rheol., 27, 387–410 (1983), DOI:http://dx.doi.org/10.1122/1.549712

Peters, G. W. M., Baaijens, F. P. T., \Modelling of Non-IsothermalViscoelastic Flows", J. Non-Newtonian Fluid Mech., 68, 205–224(1997), DOI:http://dx.doi.org/10.1016/S0377-0257(96)01511-X

Phan-Thien, N., \Influence of Wall Slip on Extrudate Swell: A Bound-ary Element Investigation", J. Non-Newtonian Fluid Mech., 26,327–340 (1988),DOI:http://dx.doi.org/10.1016/0377-0257(88)80024-7

Rosenbaum, E., \Rheology and Processability of FEP Teflon Resinsfor Wire Coating", PhD thesis, University of British Columbia,Vancouver, Canada (1998)

Rosenbaum, E., et al., \Flow Implications in the Processing of TeflonResins", Int. Polym. Proc., 10, 204–212 (1995)

Rosenbaum, E., et al., \Rheological Characterization of Well DefinedTFE/HFP Copolymers", Rheol. Acta, 37, 279–288 (1998),DOI:http://dx.doi.org/10.1007/s003970050115

Rosenbaum, E., et al., \Boron Nitride as a Processing Aid for the Ex-trusion of Polyolefins and Fluoropolymers", Polym. Eng. Sci., 40,179–190 (2000), DOI:http://dx.doi.org/10.1002/pen.11151

Sedlacek, T., et al., \On the Effect of Pressure on the Shear and Elon-gational Viscosities of Polymer Melts", Polym. Eng. Sci., 44,1328–1337 (2004), DOI:http://dx.doi.org/10.1002/pen.20128

Tadmor, Z., Gogos, C. G.: Principles of Polymer Processing, 2nd Ed.,John Wiley, New York (2006)

Tanner, R. I., \Die-swell Reconsidered: Some Numerical SolutionsUsing a Finite Element Program", Appl. Polym. Symp., 20, 201–208 (1973)

Tanner, R. I.: Engineering Rheology, 2nd Ed., Oxford University Press,Oxford (2000)

Tiu, C., et al., \Chapter ?? Process and Simulation of Wire Coating", inHandbook of Polymer Science and Technology, Applications andProcessing Operations, Cheremisinoff, N. P. (Ed.), Vol. 3, MarcelDekker, New York, p. 609–647 (1989)

Van Krevelen, D. W.: Properties of Polymers, 3rd Ed., Elsevier, NewYork (1990)

Winter, H. H., \Viscous Dissipation in Shear Flows of Molten Poly-mers", Adv. Heat Transfer, 13, 205–267 (1977),DOI:http://dx.doi.org/10.1016/S0065-2717(08)70224-7

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: January 16, 2012

Date accepted: April 11, 2012

BibliographyDOI 10.3139/217.2601Intern. Polymer ProcessingXXVII (2012) 5; page 535–546ª Carl Hanser Verlag GmbH & Co. KGISSN 0930-777X

You will find the article and additional material by enter-ing the document number IIPP2601 on our website atwww.polymer-process.com

546 Intern. Polymer Processing XXVII (2012) 5�20

12CarlH

anserVerlag,

Mun

ich,

German

ywww.polym

er-process.com

Not

forusein

internet

orintran

etsites.

Not

forelectron

icdistrib

ution.

Documents

Annular Extrudate Swell of a Fluoropolymer Meltold-2017.metal.ntua.gr/uploads/3798/1094/A143.pdf · The tensorI in Eq. 5 is the unit tensor. To evaluate the role of viscoelasticity