J. Non-Newtonian Fluid Mech. 145 (2007) 102–108
Transient capillary rheometry: Compressibility effects Evan Mitsoulis a,∗ , Omar Delgadillo-Velazquez b , Savvas G. Hatzikiriakos b a
b
School of Mining Engineering and Metallurgy, National Technical University of Athens, Athens, Greece Department of Chemical and Biological Engineering, The University of British Columbia, Vancouver, BC, Canada Received 15 November 2006; received in revised form 1 March 2007; accepted 3 May 2007
Abstract The “rise time” required to achieve a steady pressure reading in a capillary rheometer operated at constant piston speed can be very short or very long depending on the amount of the material in the barrel, its isothermal compressibility, the flow rate and the geometrical characteristics of the dies used [S.G. Hatzikiriakos, J.M. Dealy, Start-up pressure transients in a capillary rheometer, Polym. Eng. Sci. 34 (1994) 493–499]. When a short die having a large diameter is used, a maximum can also be obtained in the pressure transient, which is solely due to the compressibility of the material. This phenomenon is explained by means of a viscous mathematical model including compressibility, whose predictions are compared with experiments for a molten linear low-density polyethylene (LLDPE). © 2007 Elsevier B.V. All rights reserved. Keywords: Compressibility; Transient rheometry; Capillary flow; LLDPE
1. Introduction When a constant piston-speed capillary rheometer is used to measure the viscosity of a material, the time required to reach a steady driving pressure can be very long when (i) the length-to-diameter ratio (L/D) of the capillary is large, (ii) the diameter is small, (iii) the apparent shear rate, γ˙ A , defined as γ˙ A ≡ 32Q/πD3 , where Q is the volumetric flow rate, is small, and (iv) when the amount of the material in the reservoir is large [1]. This long rise time is solely due to the isothermal compressibility of the material, β. An approximate mathematical model has been developed by Hatzikiriakos and Dealy [1] that accurately predicts this “rise time”. Based on this model, experiments can be planned so that the time required can be minimized. Along these lines, the works by Perez-Trejo et al. [2] and Warley [3] have also made valuable contributions to our understanding of the time required for steady state. Although the isothermal compressibility of molten polymers is very small (usually of the order 10−10 Pa−1 ), it can have a dramatic effect on the time required for the pressure to level off in a capillary flow experiment [1]. In extreme cases (long dies having a small diameter) several hours are required
∗
Corresponding author. E-mail address:
[email protected] (E. Mitsoulis).
0377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2007.05.004
to reach a steady driving pressure. This is obviously inconvenient, and it is a serious problem when the material under study is one prone to degradation or other structural changes. Following the work of Hatzikiriakos and Dealy [4] on oscillating flow of polyethylenes in a capillary, several papers have appeared to model and study the stick–spurt effect, where compressibility plays a significant part in polymer melt flow [5,6]. The multipass rheometer [7] has been effectively used to further our understanding of such oscillating flows and the various timedependent phenomena associated with the flow of polymer melts [8]. Another phenomenon that might occur during capillary rheometry is a maximum in the pressure versus time transient under certain conditions. To the best of our knowledge this phenomenon has not be studied before, particularly in the case of pressure-driven flows of molten polymers by taking into account the full reservoir in a 2D flow. A recent study [9] addresses transient pressure build-ups for a polymer melt including compressibility, albeit for a limited range of apparent shear rates where no such phenomena were observed. Therefore, we first report here the results of experiments carried out in a capillary rheometer using capillary dies of various diameters and lengths for a molten polyethylene in order to determine the parameters that influence the rate of pressure build-up with emphasis on the existence of the maximum
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value in the pressure transients. Secondly, a full 2D axisymmetric flow simulation is performed that takes compressibility and the amount of the material in the reservoir into account. The flow simulation is found to capture the main features of the observed phenomena, such as the rise time as well as the shape of the P(t) curve as a function of the shear rate and geometric characteristics of the die. In addition it predicts the maximum in the pressure transients that occurs under certain conditions. 2. Experimental 2.1. Material and experimental methods The experiments were carried out for a linear low-density polyethylene (LL3001, hexane copolymer from ExxonMobil of melt index 1 and density 0.917 g/cm3 at 25 ◦ C) on an Instron, piston-driven, constant-speed, capillary rheometer. Circular dies of various diameters and L/D ratios were used in order to study the effects of these geometric characteristics of the die on the pressure transients with emphasis on the existence of the maximum. The rheology of the material was determined by using the parallel-plate geometry of 25 mm in diameter (Bohlin, CVOR). Experiments were carried out at four different temperatures, namely 130, 150, 170, and 190 ◦ C. The data were superposed by using the time–temperature superposition at the reference temperature of 190 ◦ C. The isothermal compressibility, β, of the material was also measured by means of capillary rheometry at 190 ◦ C. A known amount of the material, m, was introduced into the reservoir of the rheometer. The material was compressed inside the barrel by using a blank die in order to determine how pressure changes with volume decrease. Plotting the inverse of the volume versus absolute pressure, a straight line results (after the initial transients die out). The slope can be used to determine the isothermal compressibility from its definition (see [1] for more details): β=−
1 dυ˜ υ˜ dP
(1)
where υ˜ is the specific volume, and P is the pressure. It was found that the compressibility of the polyethylene was 1.05 × 10−9 Pa−1 , a value similar to those reported by Hatzikiriakos and Dealy [1] for several other polyethylene melts. Valette et al. [9] give a value of 1.26 × 10−9 Pa−1 for a similar LLDPE, and this value has been used here as a more elaborate technique was used in [9]. For a given resin and temperature, the density as a function of pressure can be represented as (to a first order approximation): ρ(P) = ρo (1 + βP)
(2)
where ρ is the density at pressure P, and ρo is the density at atmospheric pressure. Note that in the work of Valette et al. [9] an exponential state law has been used of the form: ρ(P) = ρo exp(βP)
(3)
Fig. 1. The dynamic viscosity η* of the LLDPE as a function of frequency ω at 190 ◦ C and its fit with three different viscosity models. The Carreau–Yasuda model represents the viscosity best compared to Carreau and Cross models.
However, in the range of pressures applied in the experiments (0.001 < P < 30 MPa), the two laws give virtually the same results. 2.2. Rheological characterization Fig. 1 plots the dynamic viscosity of the LLDPE at the reference temperature of 190 ◦ C over a wide range of frequencies. Using the Cox–Merz rule, we can substitute the dynamic viscosity η* for the steady shear viscosity η and the frequency ω ˙ Three different models were used to reprefor the shear rate γ. sent the data, namely a Cross, a Carreau and a Carreau–Yasuda model. It can be seen that both the Cross and Carreau–Yasuda models represent the data very well, and therefore either could be used in the simulations below. For the bulk of the simulations we have used the Carreau–Yasuda model, which can be written as η=
η0
(4)
˙ α ](1−n)/α [1 + (λγ)
where η0 is the zero-shear-rate viscosity, λ a characteristic material time, n the power-law exponent and α is a fitting parameter. The values of the parameters for this model along with those for the Cross and Carreau models are listed in Table 1 and they all refer to 190 ◦ C.
Table 1 Parameters for various viscosity models of LLDPE at 190 ◦ C Parameter
Cross
Carreau
Carreau–Yasuda
η0 (Pa s) λ (s) n (–) α (–)
8642 0.1377 0.4226 –
7568 0.884 0.6652 –
9021 0.046 0.2512 0.48
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Table 2 Die design characteristics used in the experiments (contraction angle 2φ = 180◦ , Dres = 9.525 mm, Lres = 300 mm) Die #
D (mm)
L (mm)
L/D
R (mm)
L/R
Lres /R
Rres /R
8 9 14
2.6162 2.6162 1.0668
5.0546 39.9542 4.1656
2 15 4
1.3081 1.3081 0.5334
3.86 30.54 7.81
229.34 229.34 562.43
3.64 3.64 8.93
Fig. 2. Die designs used in this study. The die characteristics are given in Table 2.
2.3. Pressure transients Three dies were used for studying the effect of compressibility during start-up flow in a capillary. The die characteristics are given in Table 2. Fig. 2 shows a schematic representation in scale of the dies used. The main features are an abrupt contraction in all cases (2φ = 180◦ ), a long reservoir filled up with polymer for a distance of 30 cm, and different contraction ratios (CR = Dres /D) and die lengths (L/D). In all cases, the polymer was packed slowly and carefully in the barrel, thus avoiding any starting voids. Fig. 3 presents the pressure transients obtained for the LLDPE using a capillary die having a length-to-diameter ratio, L/D = 2, a diameter D = 2.6 mm and a contraction angle 2φ = 180◦ at 190 ◦ C. It can be seen that the use of a short die having a large diameter causes the appearance of maximum in the pressure transients at high shear rates.
Fig. 3. Pressure transients obtained for the LLDPE using a capillary die (die #8) having a length-to-diameter ratio, L/D = 2, a diameter D = 2.6 mm and a contraction angle 2φ = 180◦ at 190 ◦ C. The use of a short die having a large diameter causes the appearance of the maximum in the pressure transients at high shear rates.
Fig. 4. Pressure transients obtained for the LLDPE using a capillary die (die #9) having a length-to-diameter ratio, L/D = 15, a diameter D = 2.6 mm and a contraction angle 2φ = 180◦ at 190 ◦ C. The use of a long die having a large diameter can only suppress the appearance of the maximum in the pressure transients to a certain degree. Maxima are seen at higher shear rates only.
Fig. 4 presents the pressure transients obtained for the LLDPE using a capillary die having a length-to-diameter ratio, L/D = 15, a diameter D = 2.6 mm and a contraction angle 2φ = 180◦ at 190 ◦ C. It can be seen that the use of a long die having a large diameter can only suppress the appearance of the maximum in the pressure transients to a certain degree. Maxima are seen at higher shear rates only. Fig. 5 presents the pressure transients obtained for the LLDPE using a capillary die having a length-to-diameter ratio, L/D = 4, a diameter D = 1 mm and a contraction angle 2φ = 180◦ at 190 ◦ C. The use of a short die having a relatively small diameter can suppress completely the appearance of the maximum in the pressure transients.
Fig. 5. Pressure transients obtained for the LLDPE using a capillary die (die #14) having a length-to-diameter ratio, L/D = 4, a diameter D = 1 mm and a contraction angle 2φ = 180◦ at 190 ◦ C. The use of a short die having a relatively small diameter can suppress completely the appearance of the maximum in the pressure transients.
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In what follows, a mathematical model is developed and numerical simulations are undertaken to study these effects in order to gain a better understanding of the origin of the existence of maximum in the pressure transients. 3. Numerical simulation 3.1. Governing equations and mathematical modelling We consider the time-dependent Navier–Stokes equations for weakly compressible materials under isothermal creeping flow conditions. These are written as [10–12]: (mass conservation)
∂ρ + u¯ · ∇ρ + ρ(∇ · u) ¯ =0 ∂t
(5)
∂u¯ = −∇P + ∇ · τ¯ ∂t
(6)
(momentum conservation) ρ
where u¯ is the velocity vector and τ¯ is the extra stress tensor. For a compressible fluid, we also need a state-law that relates the density to the pressure. For isothermal flows, pressure and density are connected as a first approximation through a simple thermodynamic equation of state equilibrium. This is done with a simple linear equation that has the form of Eq. (2). The viscous stresses are given for inelastic non-Newtonian compressible fluids by the relation [10–12]: 2 ¯ ¯ ¯τ = η(|γ|) ˙ γ˙ − (∇ · u) ¯ I (7) 3 ˙ is the apparent non-Newtonian viscosity, which where η(|γ|) ˙ of the rate-of-strain tensor is a function of the magnitude |γ| γ¯˙ = ∇ u¯ + ∇ u¯ T , which is given by 1 1 ¯ ¯ 1/2 ˙ = ˙ |γ| (8) IIγ˙ = (γ˙ : γ) 2 2 ¯˙ where IIγ˙ is the second invariant of γ: ¯˙ = IIγ˙ = (γ¯˙ : γ)
i
γ˙ ij γ˙ ij
(9)
j
The tensor I¯ in Eq. (7) is the unit tensor. The relationship between the stresses and the rates-of-strain is given by an appropriate rheological constitutive equation. In the present work we use the Carreau–Yasuda model of pseudo˙ given by Eq. (4). Sample runs plasticity for the viscosity η(γ) with the Cross model did not affect the results in any visible way. The above rheological model is introduced into the conservation of momentum (Eq. (6)) and closes the system of equations. Boundary conditions are necessary for the solution of the above system of equations. Fig. 6 shows the solution domain and boundary conditions for the abrupt contraction geometry. Because of symmetry only one half of the flow domain is considered, as was done in our previous work [13]. The boundary conditions are therefore (see Fig. 6):
Fig. 6. Flow domain and boundary conditions. The shaded part is the solution domain due to symmetry.
(i) symmetry along AB (ur = 0, τ rz = 0), (ii) no slip along the walls CSDE (uz = ur = 0), (iii) along the inflow boundary EA, fully developed velocity profile (uz (r) = f(r), ur = 0), (iv) along the outflow boundary BC, a zero radial velocity and a zero axial surface traction are imposed (ur = 0, Fz = 0). The reference pressure is also set to zero at point B. Here we ignore any possible (but small) exit pressure losses due to extrudate swell after the die exit. The initial conditions are: (i) u(r, ¯ z, 0) = 0, (ii) uz (r, zen , 0) = f (r), (iii) ur (r, zen , 0) = 0, where zen is the axial entry position (set at −30 cm upstream of the die). All lengths are scaled with the die radius R, all velocities with the average velocity V at the die exit, and all pressures and stresses with η0 (V/R)n . Then the characteristic parameter, which controls the flow phenomena is the apparent shear rate γ˙ A = 4V/R. Also due to compressibility, we can define the compressibility parameter B: V γ˙ A B = βη0 = βη0 (10) R 4 The compressibility parameter ranges from 0 (incompressible fluids) to 0.001 (weakly compressible fluids). 3.2. Method of solution The numerical solution is obtained with the Finite Element Method (FEM), employing as primary variables the two velocities and pressure (u-v-p formulation) as explained in our previous work [13]. We use Lagrangian quadrilateral elements with biquadratic interpolation for the velocities and bilinear interpolation for the pressures. The densities, appearing in the compressible conservation equations, are substituted by the equation of state (Eq. (2)) through the pressures and their linear interpolation per element. The compressible formulation
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has been tested against analytical solutions [14] with excellent results. For the solution in the time domain, we use the standard fully implicit Euler method with backward finite differences in time, which is unconditionally stable [15]. After many tests and checks, and knowing the time ranges of the problem from the experiments, the time step was set constant and equal to 0.005 s. For the solution in the space domain, we perform Picard iterations (direct substitutions) to the non-transient set of equations. The tolerance for convergence in the space domain has been set equal to 10−4 for the norm-of-the-error vector and the norm of the residuals. The tolerance for reaching steady state in the time domain has again been set equal to 10−4 for the norm-of-the-error vector between successive time solutions. A partial view of the finite element meshes used in the simulations is shown in Fig. 7. The entry has been set at −30 cm, which corresponds to 229R for dies #8 and #9, and to 562R for die #14, as reported in Table 2. The die exit varies for each die design. Mesh M1 has 900 elements, 3741 nodes and 8053 unknown degrees of freedom (DoF) with 41 points in the radial direction at entry and 21 points at exit. A more dense mesh M2 with 3600 elements (resulting by subdividing M1 into four subelements) was also used for test cases to ensure that the results are mesh-independent. Another feature of the solution is the imposition of the flow rate at entry and based on that flow rate, the inlet velocity profile is found by one-dimensional FEM taking into account the model under study, i.e. [12]: Rres m ˙ = 2π ρuz (r)r dr (11) 0
where m ˙ is the mass flow rate and Rres is the reservoir radius. For incompressible fluids, m ˙ may be substituted by Q, the volumetric flow rate, which is equal at entry and exit. For compressible fluids, it is m ˙ which is constant at entry and exit (assuming no instabilities are present), but not Q, because of density changes according to the equation of state. However, due to weak compressibility here, the values m ˙ and Q are not very different (at most 0.1% difference), and an inlet velocity profile based on Q gives approximately the same results. The imposition of a numerically found 1D velocity profile at entry results in a smooth development of the flow field, since the same order of interpolation is used for the velocities both for the 1D and the 2D problems.
Fig. 7. Partial view of the finite element meshes used in the simulations: M1 (900 elements, upper half), M2 (3600 elements, lower half).
3.3. Flow simulations Flow simulations have been performed using the Carreau–Yasuda model with the parameters listed in Table 1, for all three capillary dies with the geometrical characteristics listed in Table 2. Both incompressible (β = 0) and compressible conditions (β = 0.00126 MPa−1 ) have been assumed in order to assess the compressibility effects separately. The simulations have been carried out for the same range of apparent shear rates as in the experiments, namely γ˙ A = 10, 20, 50, 100, 150, 200, 250, 300 and 350 s−1 . The compressible results for the transient wall pressure at entry are shown for die #8 in Fig. 8 together with the respective experimental data. We observe an overall agreement between experiments and simulations. For example, there is a monotonic increase of the pressure towards its steady-state value for the lower rates of 10 and 20 s−1 . For higher rates, there appear maxima in the pressure transients, which are much sharper computationally than experimentally. In fact, computationally there are pressure oscillations in the first 5 s, before the pressure assumes its steady-state value. The steady-state values scale with the shear rates, as they should. The pressure values for this die design with the short die land (L/D = 2) range from 0.5 to 8 MPa. Consequently, the simulations are carried out and presented for die #9, which has a longer die land (L/D = 15) for the same contraction ratio as die #8. The numerical results and their comparison with experimental data are depicted in Fig. 9. The pressure values now are higher in the range of 3–25 MPa. The pressure transients show a monotonic increase with time for the low shear rates of 10–100 s−1 , and it also takes a much longer time for the low shear rates to reach steady state, in the order of 10 s. From shear rates ≥200 s−1 , there appear maxima in the pressure transients, which are again much sharper compared to the corresponding experimental ones. However, the agreement between simulations and experiments is now better than before. This is not surprising, since due to the longer die land and thus the higher measured pressures, the experimental errors are much smaller than before.
Fig. 8. Transient pressure simulation results against experimental data for die #8 (L/D = 2, contraction ratio 3.64:1).
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Fig. 9. Transient pressure simulation results against experimental data for die #9 (L/D = 15, contraction ratio 3.64:1).
Finally, the simulations are carried out for die #14, which has a somewhat short die land (L/D = 4) and a much larger contraction ratio compared to dies #8 and #9. The corresponding results are shown in Fig. 10. The pressure values now are again low in the range of 0.5–7 MPa, due to the short die land. Now all pressure transients show a monotonic increase with time for all shear rates in the range of 10–350 s−1 . It also takes a much longer time to reach steady state, in the order of 30–50 s mainly due to the smaller die diameter [1]. For the highest shear rates of 300 and 350 s−1 , the simulations show again a sharper increase in the pressure transients than the experiments. Overall, the agreement between simulations and experiments is considered satisfactory. To understand some of the discrepancies found above, it is instructive to look at the effect of reservoir length on the pressure transients. This is done in Fig. 11 for die #9 (long die land) and for the highest apparent shear rate of 350 s−1 . Increasing the reservoir length from 25 to 45 cm increases the time at which the maximum occurs by about 2 s and also decreases the degree of oscillations. The same effect occurs when increasing the com-
Fig. 10. Transient pressure simulation results against experimental data for die #14 (L/D = 4, contraction ratio 9:1).
Fig. 11. The effect of reservoir length in transient pressure simulation results for die #9 (L/D = 15, contraction ratio 3.64:1).
pressibility (from β = 0.00095 to 0.00126 MPa−1 for the same reservoir length Lres = 45 cm). Thus, more compressible material in the reservoir causes the maximum to occur at longer times, while the pressure transient becomes smoother. However, to reach the experimentally observed maxima a much longer reservoir would be needed if the length were the only parameter affecting the pressure maxima. It should be mentioned that the experimental error might also be a factor as the time scales involved in these capillary experiments are relatively short.
Fig. 12. The effect of compressibility in transient pressure simulation results for die #9 (L/D = 15, contraction ratio 3.64:1).
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The effect of compressibility is more evident in Fig. 12, where a comparison is made for an incompressible simulation (β = 0) and a compressible one (β = 0), all other parameters being the same. In the case of incompressible flow there is a very fast increase towards the steady-state values, which have also been independently calculated by a steady-state solution of the same problem. The effect of compressibility is more dramatic. Its effect is not only to delay the reaching of this steady-state value, but most importantly to produce an overshoot and then an undershoot before reaching steady state. Thus, it takes a split second to reach 99% of the steady-state value without compressibility (β = 0) and about 5 s with compressibility (β = 0.00126 MPa−1 ). Compressibility is important then in the start-up experiments, and the higher it is the longer the time it takes to achieve steady state. However as noted above and also shown by Georgiou [10], high compressibility values reduce the maxima and suppress oscillations in the start-up of transient flows. One may argue that part of the discrepancies is due to the viscoelastic nature of the polymer. It is, however, worth mentioning that LLDPE is not highly viscoelastic (as much as a branched LDPE melt would be) and that in start-up of steady shear the maximum in the stress response at high shear rates occurs in times considerably less than 1 s. More importantly, the compressibility effects are enhanced if there is a large reservoir. The volume of the material in the reservoir slows down the pressure increase tremendously. Here it has been argued that the time at which the pressure maximum appears scales with the amount of the material in the reservoir. Therefore it is clear that the maximum in the pressure transient has nothing to do with viscoelasticity. 4. Conclusions Transient capillary rheometry of an LLDPE melt was studied both experimentally and theoretically. Different die designs were employed that have demonstrated the effect of compressibility in the transient pressure distributions for a wide range of shear rates. Short dies produce a rapid increase to a maximum, followed by a steady-state plateau. Long dies or dies with small radii do not produce such pronounced maxima, but need significantly longer times to reach their steady-state values. The simulations based on the compressible conservation equations have shown that the main effect is a combination of die geometry and compressibility. Parametric studies have shown that without compressibility the pressure jumps to its steady-state value,
while compressibility serves to delay this accordingly. The effect of compressibility is thus the most important in understanding these pressure build-ups in start-up capillary experiments of polymer melts. Other issues worth exploring are the effects of viscoelasticity and viscous heating for the higher die length and the higher shear rates on top of compressibility. Acknowledgements Financial assistance from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the General Secretariat for Research and Technology (GGET) of Greece are gratefully acknowledged. References [1] S.G. Hatzikiriakos, J.M. Dealy, Start-up pressure transients in a capillary rheometer, Polym. Eng. Sci. 34 (1994) 493–499. [2] L. Perez-Trejo, J. Perez-Gonzalez, L. de Vargas, About the determination of the steady state flow for polymer melts in capillary rheometers, Polym. Test. 20 (2001) 523–531. [3] R. Warley, Modeling the pressure decay curve of a capillary rheometer, Appl. Rheol. 13 (2003) 8–13. [4] S.G. Hatzikiriakos, J.M. Dealy, Role of slip and fracture in the oscillating flow of HDPE in a capillary, J. Rheol. 36 (1992) 845–884. [5] J. Molenaar, R.J. Koopmans, Modeling polymer melt-flow instabilities, J. Rheol. 38 (1994) 99–109. [6] V. Durand, B. Vergnes, J.F. Agassant, E. Benoit, R.J. Koopmans, Experimental study and modeling of oscillating flow of high density polyethylenes, J. Rheol. 40 (1996) 383–394. [7] M.R. Mackley, R.T.J. Marshall, J.B.A.F. Smeulders, The multipass rheometer, J. Rheol. 39 (1995) 1293–1309. [8] M. Ranganathan, M.R. Mackley, P.H.J. Spitteler, The application of the multipass rheometer to the time-dependent capillary flow measurements of a polyethylene melt, J. Rheol. 43 (1999) 443–451. [9] R. Valette, M.R. Mackley, G. Hernandez Fernandez del Castillo, Matching time dependent pressure driven flows with a Rolie Poly numerical simulation, J. Non-Newtonian Fluid Mech. 136 (2006) 118–125. [10] G.C. Georgiou, M.J. Crochet, Compressible viscous flow in slits with slip at the wall, J. Rheol. 38 (1994) 639–654. [11] G.C. Georgiou, M.J. Crochet, Time-dependent extrudate swell problem with slip at the wall, J. Rheol. 38 (1994) 1745–1755. [12] G.C. Georgiou, The time-dependent, compressible Poiseuille and extrudate-swell flows of a Carreau fluid with slip at the wall, J. NonNewtonian Fluid Mech. 109 (2003) 93–114. [13] E. Mitsoulis, S.G. Hatzikiriakos, Bagley correction: the effect of contraction angle and its prediction, Rheol. Acta 42 (2003) 309–320. [14] C.R. Beverly, R.I. Tanner, Compressible extrudate swell, Rheol. Acta 32 (1993) 526–531. [15] K.M. Huebner, E.A. Thornton, The Finite Element Method for Engineers, Wiley, New York, 1982.
