Annular Extrudate Swell of a Fluoropolymer Meltold-2017.metal.ntua.gr/uploads/3798/1094/A143.pdf ·...
Transcript of Annular Extrudate Swell of a Fluoropolymer Meltold-2017.metal.ntua.gr/uploads/3798/1094/A143.pdf ·...
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]
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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
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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-
E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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
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E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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.
538 Intern. Polymer Processing XXVII (2012) 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
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
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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
E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
Intern. Polymer Processing XXVII (2012) 5 539
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
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E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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
540 Intern. Polymer Processing XXVII (2012) 5
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
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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-
E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
Intern. Polymer Processing XXVII (2012) 5 541
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
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E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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-
542 Intern. Polymer Processing XXVII (2012) 5
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
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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.
E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
Intern. Polymer Processing XXVII (2012) 5 543
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
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E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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
544 Intern. Polymer Processing XXVII (2012) 5
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
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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
E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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.
E. Mitsoulis, S. G. Hatzikiriakos: Annular Extrudate Swell of a Fluoropolymer Melt
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
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internet
orintran
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