Annular Extrudate Swell of a Fluoropolymer Melt

INVITED PAPERS

2012 Carl Hanser Verlag, Munich, Germany

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E. Mitsoulis1*, S. G. Hatzikiriakos2 1 2

School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, Athens, Greece Department 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 viscoelastic one (the Kaye – Bernstein, Kearsley, Zapas/Papanastasiou, 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 taken into account. In all cases, slip at the wall is the dominant contribution reducing the swelling when compared with corresponding 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 viscoelastic melts. On the other hand, the viscoelastic (K-BKZ) simulations correctly obtain increasing swelling with increasing shear (flow) rates. It was found that due to the mild viscoelasticity 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 5 000 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 characteristics, 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 devices and laboratory ware and tubing among others (Ebnesajjad, 2003; Domininghaus, 1993; Dealy and Wissbrun, 1990). Their polymer processing involves mainly tubing extrusion of fluoroethylene copolymers (FEP), in which the melt comes out of an annular crosshead die and meets a moving wire for wire insulation (Baird and Collias, 1998; Tadmor and Gogos, 2006). The rheology of FEP melts was previously studied by Rosenbaum et al. (1995, 1998, 2000) and more recently by Mitsoulis and Hatzikiriakos (2012a), who used a viscous Cross model and a viscoelastic K-BKZ constitutive equation to model 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 * Mail address: Evan Mitsoulis, School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, 157 80, Athens, Greece E-mail: [email protected]

Intern. Polymer Processing XXVII (2012) 5

dies, and that viscoelasticity was not found important under these mild conditions. However, at high shear rates (as high as 6 000 s–1), slip was found to dominate the flow behavior, especially 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 examine what happens to the melt after it exits the annular die and acquires a free surface, giving rise to the phenomenon of annular 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 parametric study for annular dies. Since then, most of the simulation efforts for extrudate swell have been concentrated on viscoelastic 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 scientific issue that has attracted attention in the literature. Mitsoulis (2007a) examined the effect of wall slip on the extrudate in annular flows of Newtonian fluids. More recently, Chatzimina et al. (2009) examined the stability of annular flows for Newtonian 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 fitting 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 predictions and trends by the different models. Although the pressure drops in the annular crosshead die have been well simulated before by both models, their predictive capabilities differ vastly outside the die, where the viscoelastic character of the fluid most eloquently manifests itself.

Carl Hanser Verlag, Munich

535

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

Tubing extrusion experiments were performed for the FEP4100 resin (a copolymer of tetrafluoroethylene and hexafluoropropylene) 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 polydispersity index of about 2 (Ferrandino, 2004a; 2004b). The melting point of this resin is around 260 8C, determined by differential 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 details of the design. In the present work we only consider for simplicity 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 extrusion 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 apparent 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-isothermal, creeping, steady flow conditions. These are written as (Mitsoulis et al., 1988; Tiu et al., 1989; Tanner, 2000): u rq þ qðr uÞ ¼ 0;

ð1Þ

0 ¼ rp þ r s;

ð2Þ

qCp u rT ¼ kr2 T þ s : ru;

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 compressible fluid, pressure and density are connected as a first approximation through a simple linear thermodynamic equation of state (Tanner, 2000):



2 Experimental

q ¼ q0 ð1 þ bc pÞ;

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): 2 s ¼ gðjc_ jÞ c_ ðr uÞI ; ð5Þ 3 where gðjc_ jÞ is the non-Newtonian viscosity, which is a function of the magnitude jc_ j of the rate-of-strain tensor c_ ¼ ru þ ruT , which is given by: rffiffiffiffiffiffiffiffiffi 1=2 1 1 IIc_ ¼ c_ : c_ ; ð6Þ jc_ j ¼ 2 2 where IIc_ is the second invariant of c_ XX IIc_ ¼ c_ : c_ ¼ ð7Þ c_ ij c_ ij : i

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

536

j

The tensor I in Eq. 5 is the unit tensor. To evaluate the role of viscoelasticity in flow through a tubing 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): g0 ; ð8Þ g¼ _ 1n 1 þ ðkcÞ Intern. Polymer Processing XXVII (2012) 5

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 performed 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: g E 1 1 ¼ exp aT ðTÞ ¼ : ð9Þ g0 R g T T0 In the above, g0 is a reference viscosity at T0, E is the activation 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 applying 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 parameters of the model are listed in Table 1. We observe that the FEP melt has a wide Newtonian plateau and then shows shear-thinning for shear rates above approximately 10 s–1 giving 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): g ap ¼ exp bp p ; ð10Þ gp0 where g is the viscosity at absolute pressure p, gp0 is the viscosity at ambient pressure, and bp is the pressure-shift factor. The latter was found to depend on pressure through the following equation (Mitsoulis and Hatzikiriakos, 2012a; 2012b): bp ¼ mpnp ;

ð11Þ –4

np 1

and np = 0.54, with the pressure where m = 1.39 · 10 Pa p given in Pa (Mitsoulis and Hatzikiriakos, 2012a; 2012b). Viscoelasticity is included in the present work via an appropriate 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: Zt X N 1 ak t t0 a s¼ exp 1h k kk ða 3Þ þ bIC1 þð1 bÞIC k¼1 k 1

0 0 0 C1 t ðt ÞþhCt ðt Þ dt ;

ð12Þ

where t is the current time, kk and ak are the relaxation times and relaxation modulus coefficients, N is the number of relaxation 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 h ; ¼ N1 1 h

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

Parameter

Value

g0 k n

1 542.3 Pa s 0.0049 s 0.316

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


ð13Þ

where N1 and N2 are the first and second normal stress differences, respectively. It is noted that h is not zero for polymer melts, which possess a non-zero second normal stress difference. Its usual range is between – 0.1 and – 0.2 in accordance with experimental findings (Dealy and Wissbrun, 1990; Tanner, 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 viscoelastic data which can be found elsewhere (Mitsoulis and Hatzikiriakos, 2012a; 2012b). Fig. 3 plots a number of calculated and experimental material 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 corresponding rates (shear or extensional). The non-isothermal modeling follows the one given in earlier 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; Mitsoulis and Hatzikiriakos, 2012a; 2012b). The viscoelastic stresses calculated by the non-isothermal version of the above con537





ber, Pe, and the Nahme-Griffith number, Na. These are defined as:

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

k

kk s

ak Pa

1 2 3 4 5 6 7

0.337 · 10–3 0.178 · 10–2 0.865 · 10–2 0.562 · 10–1 0.488 2.98 16.6

0.49050 · 106 0.22702 · 106 66,224 4 823.1 172.41 18.628 3.6235

Table 2. Relaxation spectrum and material constants for the FEP melt obeying the K-BKZ model (Eq. 12) at 371 8C ( = 7.174, b = 0.6, h = – 0.11, k = 0.76 s, g0 = 1 613 Pa s)

Pe ¼

qCp Uh0 ; k

ð14Þ

Na ¼

EU2 g ; kRg T20

ð15Þ

¼ fðU=h0 Þ is a nominal viscosity given by the Cross where g 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(D02Di2)], where Q is the volumetric flow rate. The Pe number represents the ratio of heat convection to conduction, and the Na number represents the ratio of viscous dissipation to conduction and indicates the extent of coupling between the momentum and energy equations. A thorough discussion of these effects in non-isothermal polymer melt flow is given by Winter (1977). With the above properties and a characteristic length the die gap 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-uniformities 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 principle can be used to account for the pressure effect on the stresses. 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, according to: ak ðpðt0 ÞÞ ¼ ak ðp0 Þap ðpðt0 ÞÞ:

Parameter bc bp m np bsl b q Cp k E Rg T0

Value –1

0.00095 MPa 0.03 MPa–1 1.39 · 10–4 Panp 1 0.54 400 cm/(MPab · s) 2.0 1.492 g/cm3 0.96 J/(g · K) 0.00255 J/(s · cm · K) 50,000 J/mol 8.314 J/(mol · K) 371 8C (644 K)

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

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; Ansari et al., 2010). The relevant ones here are the Peclet num538

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: U bp g : ð17Þ h Similarly, the compressibility coefficient bc gives rise to the dimensionless compressibility parameter, Bc. This is defined as: Bp ¼

U bc g : ð18Þ h The value Bp = 0 corresponds to the case of pressure-independence 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 moderate dependence of viscosity on pressure and an even weaker compressibility effect in the range of simulations. More details are given in Table 4. Bc ¼






Apparent shear rate, c_ A s–1

Peclet number, Pe

Nahme number, Na

Compressibility parameter, Bc

Pressure-shift parameter, Bp

Slip parameter, Bsl

80 320 800 1 600 3 200 5 600

42 169 420 840 1 680 2 952

0.007 0.10 0.49 1.56 4.69 11.07

1.65 · 10–5 5.44 · 10–5 1.09 · 10–4 1.74 · 10–4 2.62 · 10–4 3.52 · 10–4

5.20 · 10–4 1.72 · 10–3 3.45 · 10–3 5.49 · 10–3 8.27 · 10–3 1.11 · 10–2

0.122 0.334 0.538 0.682 0.773 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)

In the case of slip effects at the wall, the usual no-slip velocity at the solid boundaries is replaced by a slip law of the following form (Dealy and Wissbrun, 1990): aT usl ¼ bsl sbw ;

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 temperatures. The slip law and the values of bsl and b were determined in detail in our recent work (Mitsoulis and Hatzikiriakos, 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 normal velocity is set to zero, i. e., bsl ðt n : sÞb ¼ aT ðt uÞ;

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 previously 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: b U b b g : ð21Þ Bsl ¼ sl h U The value Bsl = 0 corresponds to the no-slip boundary conditions and Bsl & 1, to a macroscopically obvious slip. For the present data, the range of values is: 0.122 < Bsl < 0.795, showing 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 equations is carried out with two codes, one for viscous flows (u-vp-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 corresponding 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 integrating 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 annular die of Fig. 1. The entry length is L0 = 10h0, long enough to guarantee fully-developed conditions even for the viscoelastic runs. The extrudate length depended on whether we used viscous or viscoelastic simulations. For the viscous simulations, there are no memory effects and a relatively short extrudate length Lp = 12h0 suffices (see Fig. 5A, meshes M1 and M2). For the viscoelastic simulations, memory effects are important and longer meshes are necessary. We have tried extrudate lengths of Lp = 17h0 and 30h0, and report the results here

Fig. 4. Schematic diagram of flow domain and boundary conditions. S1 and S2 are the two singular points


539





A)

B)

C)

Fig. 5. Finite element meshes used in the computations: (A) for viscous simulations; (B) and (C) for viscoelastic simulations; (C) blowup meshes near the die exit. In (a), the upper half shows mesh M2 containing 4 560 quadrilateral elements, while the lower half shows mesh M1 containing 1 140 elements. In (b) and (C), the upper half shows mesh M4 containing 6 240 quadrilateral elements, while the lower half shows mesh M3 containing 1 560 elements

for the longer case (Fig. 5B, C meshes M3 and M4). Mesh M3 consists of 1 560 elements, 4 965 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-independent results. This checking consists of reporting the overall pressures in the system (pressure at the entry to the domain) from the two meshes and making sure that the differences are less than 1 % between the two results. Having fixed the model parameters and the problem geometry, the only parameter left to vary was the apparent shear rate in the die. Simulations were performed for the whole range of experimental apparent shear rates, namely from 80 s–1 to 5 600 s–1, where smooth extrudates were obtained with the presence 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 viscoelastic 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 explained how it was possible for the first time to do viscoelastic computations up to very high apparent shear rates (1 000 s–1) with good results. Here, an extra complication arises from the

presence of free surface (actually two of them), for which severe 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 solution 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 solution is alternated with the h-solution (free surface location) until 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 5 600 s–1. It must be noted that FEP is not strongly viscoelastic, 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 temperature-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 drastic flow conditions encountered in the flow field (actual shear

540






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 average 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 (annular) 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): c_ A ¼

6Q 0:25ðD0 Di Þ2 0:5pðD0 þ Di Þ

:

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 ¼

Dp ; D0

ð23Þ

(ii) the thickness swell, B2, defined by B2 ¼

hp ; h0

ð24Þ

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 highest 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 1 000 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 towards 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, Mitsoulis 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 enhanced by slip at the wall, which reduces the swell even more (Mitsoulis, 2007a). Viscoelasticity on the other hand, as applied with the KBKZ 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 viscoelasticity of FEP is not strong and not as high as for other polymer melts, notably polyethylenes, and this is shown in < 1 s. Therefore, Table 2 by the average relaxation time, k swelling is not enhanced. Again, slip brings down the swel-

(iii) the inner diameter swell, B3, defined by B3 ¼

Dp 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 become evident. The numerical simulations have been undertaken using either the purely viscous Cross model (Eq. 8) or the viscoelastic 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-dependence 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, because 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 contribution, although the effects of compressibility and temperaIntern. Polymer Processing XXVII (2012) 5

Fig. 6. Pressure drop DP as a function of the apparent shear rate for FEP melt at 371 8C. No slip refers to the base case (b c = b p = b sl = aT = 0), while slip refers to all effects accounted for. Symbols are put to show the simulation continuation steps

541

Fig. 7. Thickness swell B2 as a function of the apparent shear rate for FEP melt at 371 8C. No slip refers to the base case (b c = b p = b sl = aT = 0), while slip refers to all effects accounted for. Symbols are put to show the simulation continuation steps





Fig. 8. Diameter swell B1 as a function of the apparent shear rate 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

ling as it should. The same trend was found in slit planar flows by Phan-Thien (1988). An interesting phenomenon occurs in thickness swell in Fig. 7, where the results are not monotonic but go through a maximum both for slip and without slip. The same trend was first found by Luo and Mitsoulis (1989) for an HDPE melt and was confirmed recently by Kar542

Fig. 9. Inner diameter swell B3 as a function of the apparent shear rate 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

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 increase mildly or even decrease, and this is dictated by conservation 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 experimental 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 (c_ A = 3 000 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 formulation, namely the velocities-pressures-temperatures (u-p-T). The contour values are given in equally spaced 11 intervals between the maximum and minimum values. The streamlines (STR) are smooth (Fig. 12A) and are obtained from the integration 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 (isobars, Fig. 12C) show that inside the die there are curved conIntern. Polymer Processing XXVII (2012) 5





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

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

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 temperature (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 dissipation. 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 obtain their 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 obtaining converged solutions fast.

A)

C)

B)

D)


Fig. 12. Contours of various field variables obtained at c_ A = 3 000 s–1 with the K-BKZ model with all effects accounted for (slip): (A)~streamlines (STR), (b) axial velocity (U), (C) pressure (P), (D) temperature (T). To get the pressure values in MPa, multiply contour values by g0 = 0.001613 MPa · s

543





A)

B)

C)

D)

A)

B)

C)

D)

Fig. 14 shows axial distributions along the walls and the free surfaces of the four stresses (given in MPa) obtained from the KBKZ 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 544

Fig. 13. Contours of various field variables obtained at c_ A = 3 000 s–1 with the K-BKZ model with all effects accounted for (slip): (A) axial stress szz (TXX), (B) radial stress srr (TYY), (C) shear stress szr (TXY), (D) azimuthal stress shh (T33). To get the stress values in MPa, multiply contour values by g0 = 0.001613 MPa · s

Fig. 14. Stress distributions (in MPa) along the walls and the free surfaces (f. s.) obtained at c_ A = 3 000 s–1 with the K-BKZ model with all effects accounted for (slip): (A) axial stress szz, (B) radial stress srr, (c) shear stress szr, (D) azimuthal stress shh

decrease to zero along the two free surfaces. The axial stresses szz (Fig. 14A) are highest near the singularity and then exponentially decay to zero, while the radial stresses srr (Fig. 14B) are negative and much lower. The shear stresses szr (Fig. 14C) are negative on the outer wall and positive on the inner due to shear Intern. Polymer Processing XXVII (2012) 5





flow inside the annular die, and they approach zero in the extrudate 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 simulations gave the correct trends, as the swelling increased with increasing shear rates. However, the viscoelasticity of FEP is not < 1 s), and therefore the very strong (average relaxation time k 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 viscoelastic 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 5 600 s–1. These results are in qualitative agreement with experimental observations that found no appreciable swelling, reaching about 10 – 20 %. This behavior is in contrast with other viscoelastic polymer melts, notably polyethylenes, 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.

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Acknowledgements Financial assistance from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the programme \PEBE 2009 – 2011" for basic research from NTUA are gratefully acknowledged. Date received: January 16, 2012 Date accepted: April 11, 2012

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