Entry Flow of Polyethylene Melts in Tapered Dies

Transcription

1 REGULAR CONTRIBUTED ARTICLES M. Ansari 1, A. Alabbas 1, S. G. Hatzikiriakos 1, E. Mitsoulis 2 * 1 Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada 2 School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, Athens, Greece Entry Flow of Polyethylene Melts in Tapered Dies The excess pressure losses due to end effects (mainly entrance) in the capillary flow of several types of polyethylenes were studied both experimentally and numerically under slip and no-slip conditions. These losses were first measured as a function of the contraction angle ranging from 158 to 908. It was found that the excess pressure loss attains a local minimum at a contraction angle of about 308 for all types of polyethylenes examined. This was found to be independent of the apparent shear rate. This minimum becomes more dominant under slip conditions that were imposed by adding a significant amount of fluoropolymer into the polymer. Numerical simulations using a multimode K-BKZ viscoelastic model have shown that the entrance pressure drops can be predicted fairly well for all cases either under slip or no-slip boundary conditions. The clear experimental minimum at about 308 can only slightly be seen in numerical simulations, and at this point its origin is unknown. Further simulations with a viscous (Cross) model have shown that they severely under-predict the entrance pressure by an order of magnitude for the more elastic melts. Thus, the viscoelastic spectrum together with the extensional viscosity play a significant role in predicting the pressure drop in contraction flows, as no viscous model could. The larger the average relaxation time and the extensional viscosity are, the higher the differences in the predictions between the K-KBZ and Cross models are. 1 Introduction When a molten polymer flows through a contraction of a given angle, there is a large pressure drop associated with such flow, known as entrance pressure or Bagley correction (Bagley, 1957; Dealy and Wissbrun, 1990). This pressure is required in order to calculate the true shear stress in capillary flow and also frequently the apparent extensional rheology of molten polymers, a method well practiced in industry (Cogswell, 1972, 1981; Binding 1988, 1991). Therefore, it is important to understand the origin of this excess pressure and consequently to be able to predict it. Accurate prediction of this pressure might serve as a strong test for the predictive capabilities of a constitutive equation. Many studies have previously attempted to examine the origin of entrance pressure and its prediction. Feigl and Öttinger * Mail address: Evan Mitsoulis, School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, Athens, Greece mitsouli@metal.ntua.gr (1994) have simulated the axisymmetric contraction flow of a low-density (LDPE) melt using a Rivlin-Sawyers constitutive model. However, their numerical results well under-predicted the available experimental pressure data for entry flows. Barakos and Mitsoulis (1995a; 1995b) reported similar findings for the Bagley correction in capillary flow of the IUPAC low-density polyethylene (LDPE). Béraudo et al. (1996) used a multimode Phan-Thien/Tanner (PTT) constitutive relation and found that numerical predictions significantly under-estimated the experimental findings. Guillet et al. (1996) also studied the entrance pressure losses for a linear low density (m- LLDPE) and a low-density polyethylene (LDPE) melt both experimentally and numerically using a multimode K-BKZ integral constitutive equation. Again significant under-estimation was reported. Using a K-BKZ constitutive relation, Hatzikiriakos and Mitsoulis (1996, 2003) and Mitsoulis et al. (1998) also found that the numerical predictions significantly under-estimated the experimental data for the various geometries used to determine the entrance pressure. From the above studies, it is clear that state-of-the-art numerical simulations cannot predict quantitatively the pressure drop in a relatively simple flow, such as the entry capillary flow of a polymer melt. Considering the fact that experimental measurements from such flows are extensively used in industrial practice to calculate the shear and extensional viscosity of polymer melts at high shear rates (Dealy and Wissbrun, 1990; Cogswell, 1972; Binding 1988, 1991; Padmanabhan and Macosco, 1997), it is essential to understand the origin of these disagreements and furthermore be able to predict the excess entrance pressure. One important aspect for entry contraction flows is the variation of entrance pressure as a function of contraction angle at a given apparent shear rate under slip or no-slip boundary conditions. This was studied by Mitsoulis and Hatzikiriakos (2003) for a branched polypropylene (PP) melt both experimentally and theoretically. The entrance pressure was first determined experimentally as a function of the contraction angle ranging from 108 to It was found that at a given apparent shear rate, the pressure loss decreases with increasing contraction angle from 108 to about 458, and consequently slightly increases from 458 up to contraction angles of Numerical simulations using a multimode K-BKZ viscoelastic were used to predict the pressures. It was found that the numerical predictions do agree well with the experimental results for small contraction angles up to about 308. However, the numerical simulations under-predicted the end pressure for larger contraction angles. The importance of the existence of a minimum in the variation of excess pressure as a function of contraction angle was also mentioned by Hatzikiriakos and Mitsoulis (2009). Intern. Polymer Processing XXV (2010) 4 Ó Carl Hanser Verlag, Munich 287

2 This minimum was found to be more pronounced under slip conditions both numerically and experimentally. Experimental data reported by Mitsoulis and Hatzikiriakos (2003) have shown the existence of a minimum at around 408. Such pronounced minima have also been reported in the flow of PTFE pastes several times (Horodin, 1998; Gibson, 1998; Ariawan et al., 2002; Ochoa and Hatzikiriakos, 2005). It is the main objective of this work to study the dependence of entrance pressure as a function of entrance angle for various types of polyethylenes, namely high-density, low-density and linear low-density polyethylenes. It is also our goal to study the dependence of entrance pressure on contraction angle under strong slip conditions in order to understand its origin. A small amount of a fluoropolymer is added into one of the polyethylenes to promote slip (Achilleos et al., 2002). It is noted that by promoting slip, the relative contribution of the extensional viscosity to flow increases relative to shear, and this will help us understand the nature of this flow. Finally, using the K-BKZ constitutive equation in numerical simulations, the capability of this model to predict the entrance pressure is studied, particularly in view of previous studies that well underpredicted it (Feigl and Öttinger, 1994; Barakos and Mitsoulis, 1995a; 1995b; Béraudo et al. 1996; Guillet et al. 1996; Hatzikiriakos and Mitsoulis, 1996; and Mitsoulis et al., 1998). 2 Experimental 2.1 Materials Three different polyethylenes were used in this work in order to address molecular-structure effects with emphasis on the effects of Long Chain Branching (LCB) on entry pressure drop. It is known that branched polymers exhibit extensional strainhardening effects, and therefore increased entrance pressure values are obtained compared to those obtained for their linear counterparts (Hatzikiriakos and Mitsoulis, 1996; Mitsoulis et al., 1998; Mitsoulis and Hatzikiriakos, 2003). This is expected as the entry pressure has been theoretically associated with extensional viscosity (Cogswell, 1972; Binding, 1988, 1991; Padmanabhan and Macosco, 1997). First, a metallocene linear low-density polyethylene (labelled here as m-lldpe) was used, which is a butane-copolymer supplied by ExxonMobil (Exact 3128) of MW about 80 kg/mol and PDI of about 2. A high-density polyethylene (labeled here as HDPE) was also used, which is a metallocene resin obtained from Chevron- Phillips Chemical Company of high molecular weight of about 263 kg/mole (Ansari et al., 2009). The low-density polyethylene (LDPE) resin used was also of high MW with significant Sample ID Resin type Melt index (1908C) g/10 min amount of Long-Chain Branching (LCB), which results in a low Melt Flow Index (MFI) of about 0.5. Some of the properties of these resins are summarized in Table 1. To study the effect of slip on the dependence of entrance pressure on contraction angle, 1.0 wt.% of a fluoropolymer (Dynamar 9613 from Dyneon) was added into m-lldpe in dry mixing form in order to enhance its slip in the capillary flow, particularly in the entrance. A comparison of the flow curves with and without fluoropolymer could quantify the slip velocity of the polymer, and this is presented below. 2.2 Rheological Testing Several rheological experiments were carried out in order to characterise the polymers in the molten state. First, an Anton Paar MCR-501 device was used in the parallel-plate geometry to determine their linear viscoelastic moduli over a wide range of temperatures from 130 to 230 8C. Master curves were obtained using the time-temperature superposition principle (TTS), and the results are presented at the reference temperature of 160 8C. Creep experiments were performed at 1608C to obtain the zero-shear viscosity values. A constant shear stress of 10 Pa was used to attain very low shear rates in order to reach the Newtonian viscosity flow regime. These values are reported in Table 1 and are found to compare well with those obtained from linear viscoelastic measurements. The polymers were also rheologically characterized in simple extension using the SER-2 Universal Testing Platform from Xpansion Instruments (Sentmanat, 2003, 2004). Details of sample preparation and operating principles of this rheometer to obtain reliable results are given in Delgadillo et al. (2008a, 2008b). Uniaxial extension tests at the temperature of 160 8C were performed for all polymers. 2.3 Entrance Pressure An Instron capillary rheometer (constant piston speed) was used to determine the entrance pressure (known also as Bagley method) (Dealy and Wissbrun, 1990) and the viscosity as a function of the wall shear stress, r W, and apparent shear rate, _c A ¼ 32Q=pD 3 all at 1608C, where Q is the volumetric flow rate and D is the capillary diameter. A circular die of diameter equal to 0.02 ( cm), length-to-diameter ratio, L/D = 20, and a tapered entrance angle of 908 was used to determine the flow curves of the resins. A series of orifice dies (L/D = 0) were used to determine directly the entrance pressure (Bagley correction) as a function of the apparent shear rate and contraction an- Density (25 8C) g/cm 3 Zero-shear-rate viscosity (160 8C) Pa s m-lldpe Exact ,594 HDPE CPChem ,460 LDPE DOW 662I ,620 Table 1. Properties of polyethylene resins used in this study 288 Intern. 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3 gle with angles of 108, 308, 458,608, and 908. Hatzikiriakos and Mitsoulis (1996) and Kim and Dealy (2001) clearly demonstrated that by using the standard Instron orifice dies, the entrance pressure drop can be artificially high when the melt fills the exit region, which often happens with elastic melts that possess high extrudate (die) swell. Therefore, precautions were taken during the experiments so that the exit region remains unfilled without the melt touching its walls. Kim and Dealy (2001) proposed a new design for orifice dies to easily avoid filling of the exit region of the die from happening. 3 Constitutive Equation and Rheological Modeling The constitutive equation used in the present work to solve the usual conservation equations of mass and momentum for an incompressible fluid under isothermal conditions is a K-BKZ equation proposed by Papanastasiou et al. (1983) and modified by Luo and Tanner (1988). This is written as: s ¼ 1 Z t X N a k t t0 a exp 1 h k k¼1 k k k ða 3ÞþbI C 1þð1 bþi C 1 C 1 t ðt 0 ÞþhC t ðt 0 Þ dt 0 ; ð1þ where k k and a k are the relaxation times and relaxation modulus coefficients, N is the number of relaxation modes, a and b are material constants, and I C,I 1 C are the first invariants of the Cauchy-Green tensor C t and its inverse C 1 t, the Finger strain tensor. The material constant h is given by N 2 ¼ h N 1 1 h ; ð2þ where N 1 and N 2 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). For the capillary flow simulations the effect of pressure on viscosity should be taken into account as this becomes more evident below. This effect is quite significant for LDPE compared to m-lldpe and less significant for HDPE (Zetlacek et al., 2004; Carreras et al., 2006). This effect can be taken into account by multiplying the constitutive relation with a shift factor, a p, given by the following equation (Carreras et al., 2006) a p ¼ expðb P pþ; ð3þ where b P is the pressure coefficient and p is the absolute pressure. The values of b P used in this work are GPa 1, GPa 1 and GPa 1 for LDPE, m-lldpe and HDPE, respectively (Zetlacek et al., 2004; Carreras et al., 2006). As discussed above, experiments were performed in the parallel-plate and extensional rheometers for all polyethylenes to rheologically characterise them. Figs. 1A to C plot the master dynamic moduli G and G of all three polyethylenes at the reference temperature of 160 8C. The model predictions obtained by fitting the experimental data to Eq. 1 with a spectrum of relaxation times, k k, and coefficients, a k, determined by a non-linear regression package (Kajiwara et al., 1995), are also Intern. Polymer Processing XXV (2010) A) B) C) Fig. 1. Experimental data (symbols) and model predictions of storage (G ) and loss (G ) moduli for the polyethylenes at 160 8C using the relaxation times listed in Table 2, (A) m-lldpe, (B) HDPE and (C) LDPE

4 m-lldpe HDPE LDPE a = 1.319, b = 0.02, h =0 a = 10.15, b = 0.6, h =0 a = 1.51, h =0 k = 0.46 s, g 0 = 11,048 Pa s k = 18.9 s, g0 = 235,393 Pa s k = 298 s, g0 = 201,804 Pa s k k k (s ) a k (Pa) k k (s ) a k (Pa) k k (s ) a k (Pa) b k , , , , , , , , , , , , , Table 2. Relaxation spectra and material constants for polyethylene resins obeying the K-BKZ model (Eq. 1) at 160 8C A) B) plotted. The parameters found from the fitting procedure are listed in Table 2. The relaxation spectrum is used to find the average relaxation time, k, and zero-shear-rate viscosity, g 0, according to the formulas: k ¼ XN k¼1 g 0 ¼ XN k¼1 a k k 2 k a k k k ; a k k k : Figs. 2A to C plot a number of calculated and experimental material functions for the three melts at the reference temperature of 160 8C. Namely, data for the shear viscosity, g S, the elongational viscosity, g E, and the first normal stress difference, N 1, 290 Intern. Polymer Processing XXV (2010) 4 C) Fig. 2. Experimental data (solid symbols) and model predictions of shear viscosity, g S, first normal stress difference, N 1, and elongational viscosity, g E, for the polyethylenes at 1608C using the K-BKZ model (Eq. 1) with the parameters listed in Table 2, (A) m-lldpe, (B) HDPE, and (C) LDPE ð4þ ð5þ

5 are plotted as functions of corresponding rates (shear or extensional). The parameter b that controls the calculated elongational viscosity was fitted by using the extensional behaviour of the three polymers. For the m-lldpe and HDPE, the tensile stress growth coefficients follow the linear viscoelastic envelope defined by 3g þ (not plotted here), consistent with their linear macrostructure (Hatzikiriakos, 2000; Dealy and Larson, 2006). Fig. 3 shows the extensional behaviour of the LDPE at several Hencky strain rates at 160 8C and the model predictions of Eq. 1 using multiple b-values listed in Table 2. It can be seen that the overall rheological representation of all resins is excellent. To evaluate the role of viscoelasticity in the prediction of Bagley correction, it is instructive to also consider purely viscous models in the simulations. Namely, the Cross model was used to fit the shear viscosity data of the three melts. The Cross model is written as (Dealy and Wissbrun, 1990) g 0;C g ¼ ; ð6þ 1 n 1 þðk_cþ where g 0,C is the zero-shear-rate viscosity of the Cross model, k is a time constant, and n is the power-law index. The fitted viscosity of the three melts by Eq. 6 is plotted in Fig. 4, while the parameters of the model are listed in Table 3. We observe that of the three melts, HDPE is the most viscous, followed by LDPE and then by m-lldpe at low range of shear rates. The m-lldpe melt is more viscous than the LDPE at high shear rates, while the LDPE melt is the least viscous for shear rates above 1 s 1 due to its significant shear thinning that is due to Fig. 3. Extensional data for LDPE 662I at 160 8C and their best fit using the K-BKZ model (Eq. 1) with the parameters listed in Table 2 Parameter m-lldpe HDPE LDPE g 0,C 11,034 Pa s 245,153 Pa s 201,000 Pa s k s s 100 s n Table 3. Parameters for the three polymer resins obeying the Cross model (Eq. 6) at 160 8C Fig. 4. The shear viscosity of the three polymer melts at 160 8C fitted with the Cross model using the parameters listed in Table 3 the presence of long chain branching (Dealy and Larson, 2006). The Cross model fits the data well over the range of experiment results. In the case of slip effects, the usual no-slip velocity at the solid boundaries is replaced by a slip law of the following form u sl ¼ k sl r b w ; ð7þ where u sl is the slip velocity, r w is the shear stress at the die wall, k sl is a slip coefficient, and b is the slip exponent. In 2-D simulations, this means that the tangential velocity on the boundary is given by the slip law, while the normal velocity is set to zero, i. e., k sl ðtn : sþ b ¼ t v; n v ¼ 0; ð8þ where n is the unit outward normal vector to a surface, t is the tangential unit vector in the direction of flow, s is the extra stress tensor and v is the velocity vector. Implementation of slip in similar flow geometries for a polypropylene (PP) melt has been also carried out in one of our previous works (Mitsoulis et al., 2005). 4 Experimental Results 4.1 Entrance (End) Pressure Figs. 5A to C plot the entrance pressure (or end pressure due to L/D = 0) of all polyethylenes at 160 8C as a function of the entrance angle for an extended range of values of the apparent shear rate from 75 s 1 to 1000 s 1. The entrance pressure decreases with increasing contraction angle from 108 to about and subsequently increases up to contraction angles of 608 with a small drop at 908 (more significant in the case of m-lldpe). This behavior is consistent for all polyethylenes and also in agreement with other reported observations in the literature, i. e., for a PP melt (Mitsoulis and Hatzikiriakos, Intern. Polymer Processing XXV (2010) 4 291

6 A) B) C) Fig. 5. The end pressure of all polyethylenes at 1608C as a function of contraction angle at various values of apparent shear rate, (A) m-lldpe, (B) HDPE, and (C) LDPE Fig. 6. End pressure of m-lldpe with the addition of 1 wt.% fluoropolymer at 1608C as a function of the entrance angle for several values of the apparent shear rate 2003). The increase of the entrance pressure at high contraction angles is associated with the increased significance of extensional contributions relative to shear ones at these high entrance angles (Cogswell, 1972; Binding, 1988, 1991). Fig. 6 plots the entrance pressure of m-lldpe with the addition of 1.0 wt.% fluoropolymer (Dynamar 9613) at 160 8C as a function of the entrance angle for several values of the apparent shear rate in the same range as before (75 s 1 to 1000 s 1 ). This graph is to be compared with Fig. 5A. The decrease of the entrance pressure at 908 is not as dominant as it was in the absence of fluoropolymer or absence of wall slip. It seems that extensional rheology plays a significant role here and the entrance pressure under slip is predominantly due to extensional deformation (Collier, 1994; Collier et al., 1998; Shaw, 2003). The drop is observed under weak extensional elements of the flow as it is expected for the case of an m-lldpe melt. However, under slip conditions, extensional deformation becomes dominant over shear in the contraction region, and this is the reason that at 908 the drop in pressure is not as significant. The same is observed for the cases of LDPE (strong extensional strain-hardening deformation due to LCB) and HDPE (stronger extensional effects compared to m-lldpe due to a much higher molecular weight) (Hatzikiriakos, 2000; Dealy and Larson, 2006). 4.2 Wall Slip Fig. 7 plots the flow curves for m-lldpe with and without fluoropolymer to assess slip. For the flow curve of m-lldpe with no fluoroelastomer, the Bagley and Rabinowitch corrections as well as the pressure coefficient correction (Eq. 3) have been applied (Dealy and Wissbrun, 1990) in order to compare it with the flow curve obtained from linear viscoelastic measurements. The linear viscoelastic data are plotted in the form of flow curve as shear stress ( g*(x) x) vs. shear rate (rotational frequency, x, in rad/s). It can be seen that this flow curve agrees with that obtained from capillary flow with no fluoropolymer up to high shear rates close to the region where sharkskin melt frac- 292 Intern. Polymer Processing XXV (2010) 4

7 5 Numerical Results 5.1 Viscous Modeling Fig. 7. Flow curves for m-lldpe with and without fluoropolymer to assess slip ture is observed. In this regime, a weak slip is usually observed (Hatzikiriakos and Dealy, 1992). We refer to the flow curve g*(x) x vs. x (in rad/s) as the linear viscoelasticity LVE no slip data. The shear rate that corresponds to the uncorrected flow curve (Rabinowitch and pressure effects) of pure m-lldpe (not plotted for the sake of clarity as it is close to the corrected one) is referred to as _c A;s (apparent shear rate corrected for the effect of slip). Then using the flow curve that corresponds to the fluoropolymer additive, the slip velocity, u s, can be calculated as a function of wall shear stress by u s ¼ D _c A _c A;s : ð9þ 8 Fig. 8 plots the slip velocity as a function of shear stress. A linear relationship is obtained and the slip model then becomes u s ¼ k s ðr W 0:03Þ for r W 0:03 MPa; ð10þ where the value of 0.03 MPa represents the critical shear stress for the onset of slip in the presence of fluoropolymer at the interface, and k s is the proportionality slip constant, which turns out to be equal to m/mpa s, with u s in m/s and r W in MPa. Fig. 8. Slip velocity of the m-lldpe in the presence of fluoropolymer It is instructive to perform first calculations with a purely viscous model, so that the effect of viscoelasticity will become evident later. The numerical simulations have been undertaken using the viscous Cross model (Eq. 6). This constitutive relation is solved together with the usual conservation equations of mass and momentum for an incompressible fluid under isothermal conditions. The finite element grids are the same as the ones used in our previous publication (see Fig. 9 of Mitsoulis and Hatzikiriakos, 2003). Having fixed the Cross model parameters and the problem geometry, the only parameter left to vary was the apparent shear rate (_c A ). Intern. Polymer Processing XXV (2010) A) B) Fig. 9. Simulation results with the K-BKZ model (Eq. 1) for LDPE at 1608Cina908-tapered die at apparent shear rate of 1000 s 1. Dimensionless axial distributions along the wall and the centreline for (A) pressure, (B) shear stress

8 Simulations were performed for the whole range of experimental apparent shear rates, namely from 75 s 1 to as high as 1000 s 1. The results from the simulations are shown in Figs. 5A to C for the three melts (m-lldpe, HDPE, LDPE, respectively). For clarity we only present numerical results for the lowest and highest apparent shear rates of 75 s 1 and 1000 s 1, respectively. For the other rates, the results scale accordingly, as in the experiments. We observe that the viscous results always under-predict the experimental results. The numerical results show a monotonic decrease with increasing angle for all apparent shear rates. All results do not pass through a minimum, in sharp contrast with the experiments. The least under-estimation occurs for the most viscous HDPE melt, while the worst occurs for the most elastic LDPE melt (based on their average relaxation time of Table 2), with the m-lldpe lying in between. The best results are obtained for the lowest angles as expected since the flow is shear-dominated, and at small contraction angles the lubrication approximation is nearly valid. As the angles increase, the extensional components are having a stronger effect, thus increasing the discrepancies between predictions from a purely shear model and the measured experimental data. Including slip in the case of m-lldpe with 1.0 wt.% Dynamar 9613 does cause a drop in the pressure drop as shown in Fig. 6. Again the results for the viscous model under-estimate the experimental ones, and this discrepancy increases with apparent shear rate and contraction angle. 5.2 Viscoelastic Modeling The viscoelastic numerical simulations have been undertaken using the integral constitutive equation (Eq. 1) as described above. In the course of the current work, we tested several ways to try and find out why for some melts, such as LLDPE (Hatzikiriakos and Mitsoulis, 1996) and PP (Mitsoulis and Hatzikiriakos, 2003), the pressure losses were adequately predicted, while for the most elastic LDPE this had never been the case (Barakos and Mitsoulis, 1995a); on the contrary the calculated pressure losses were about half the experimental values (discrepancies of 100 %). First of all, we verified again that the best way to proceed in the viscoelastic simulations was to increase slowly the apparent shear rates from 0.01 s 1 to 1000 s 1, essentially using a continuation scheme. We found out that 25 intermediate steps were sufficient and necessary to get results according to the logarithmic scale in the range of simulations. However, these results were achievable under some conditions, which were not the same for all three melts. Namely, to obtain solutions for L/D = 0, the following three strategies were tested: (a) use L/D = 0 for the die and add an extrudate length L/D = 5 or 10 to accommodate the free surface and thus determine the extrudate swell of the polymers (this worked only for the m-lldpe; for the HDPE and the LDPE the results diverged at low apparent shear rates < 10 s 1 ); (b) use L/D = 0 for the die and add an extrudate length L/D = 5 or 10 to accommodate the free surface using stick-slip conditions (no extrudate swell allowed, based on the assumption than no significant pressure drop occurs in the extrudate); this worked for all three polymers for all shear rates, although it gave disappointedly low pressure drops, the lower pressures the more elastic the polymer (LLDPE < HDPE < LDPE in elasticity as seen via k); (c) use L/D = 50 for the capillary die, and based on the fully developed shear stress wall values s w = 4(dP/dz)(L/D), subtract the pressure drop in the straight die DP 0 from the overall pressure drop in the system DP to obtain the entry pressure corresponding to L/D = 0; this worked for all three polymers and all shear rates and produced acceptable results even for the most elastic LDPE. Apparently the long die length was necessary to get a full relaxation of the stresses and determine the extra pressure drop due to elasticity. Sample tests with L/D > 50 showed that the entry pressure was not affected appreciably (in any discernible way in the graphs), which is an indication that the extra L/D length amounts to a fully-developed flow with no memory effects adding to the stresses and hence the pressure. Also, the long lengths made the simulations easier to converge, as more distance was given for the stresses to relax naturally. To better understand the above point (c), we present in Fig. 9 the axial pressure and shear stress distributions for the LDPE melt for the 908-tapered die and the highest apparent shear rate of 1000 s 1. The distributions are made dimensionless by dividing the pressure and the stress by the nominal stress s* = P* = g 0 U/R, where U is the average exit velocity. We observe that in the die length of 100 R, the pressure drop becomes linear after a rearrangement length of about 10 R, and the shear stress at the wall is constant up to the exit, which is tantamount to a fully-developed flow in the die. It is this constant value of s w, which is used to calculate the DP 0, to obtain the pressure drop in the die itself and subtract this value from the overall pressure in the system. Also to be noticed is the very good behaviour of the viscoelastic solution at such an elevated apparent shear rate for a very elastic melt (LDPE), where apart from the usual oscillations around the singularity to the entry of the die, the solution is smooth and well behaved both for the pressure and the stresses. The simulations have also taken into account the pressure dependency of viscosity as given by Eq. 3 (not taken into account in Fig. 9). In that case, it was found that the easiest way to proceed was to just multiply the pressure results under no pressure dependence by the factor a p of Eq. 3. It should be noted that for high apparent shear rates and low contraction angles, this correction can be important (between 10% and 30%). The results from the simulations are shown in Figs. 5A to C for the three melts (m-lldpe, HDPE, LDPE, respectively) and in Fig. 6 for m-lldpe with slip due to the presence of 1.0 wt.% Dynamar An overall good agreement can be observed between experiments and simulations, although some differences still exist. The results for the m-lldpe melt do not pass through a minimum, in contrast with the experiments. However, viscoelasticity serves to increase the pressure drop substantially, and this is enhanced for the higher angles and apparent shear rates. This is expected, as increasing the contraction angle leads to an increase of the elongational contributions to entry pressure. Also the nonlinear effect of this increase is helped by increasing the flow rate, namely for higher apparent shear and elongational rates, where the rheology of the melts shows a markedly nonlinear behaviour. 294 Intern. Polymer Processing XXV (2010) 4

9 Regarding the HDPE melt, we observe that solution could be obtained even for the highest apparent shear rate of 1000 s 1 (cases (b) and (c) above), despite the very elastic nature of this polymer melt (note that its k = 18.9 s compared with k = 0.46 s for m-lldpe). The results between the envelopes of the lowest apparent shear rate (_c A =75s 1 ) and the highest one (_c A = 1000 s 1 ) capture, in general, the experimental data, but they show a monotonic decrease with increasing contraction angle. Regarding the most elastic LDPE melt, we observe that solution could be again obtained even for the highest apparent shear rate of 1000 s 1 (cases (b) and (c) above), despite its strong elastic nature in the melt state (note that its k = 298 s compared with k = 18.9 s for HDPE and k = 0.46 s for m-lldpe). It is interesting to note that a small minimum is obtained at around 2a =308. These minima were present for all shear rates and are better shown in a Cartesian plot in Fig. 10. For the first time for LDPE we get entrance pressure results in agreement with experimental data for all shear rates and angles. Previous results have shown under-estimates for the entrance pressure by almost 100 % compared with experimental data (see simulations for the IUPAC- LDPE melt by Barakos and Mitsoulis, 1995b, and the other references in the introduction). It is to be noted that a different elongational parameter b-value (b = for all 8 modes), which corresponds to an order-of-magnitude higher response in elongation, did increase somewhat the results but not substantially (5 to 10 %). The present simulations with a die length of L/D = 50 showed that such a length is necessary for the melt to relax its stresses and unravel its full viscoelastic character. The simulation results were well behaved for all shear rates with no oscillations or other defects. This is also the first time that simulations for LDPE have been carried out up to 1000 s 1. It should be emphasized that all previous studies on LDPE, combining experiments and simulations did not exceed _c A = 125 s 1 (Mitsoulis, 2001). Fig. 10. Simulation results with the K-BKZ model (Eq. 1) for the end pressure of LDPE at 160 8C as a function of contraction angle at various values of apparent shear rate plotted in Cartesian coordinates. Symbols are put for the different die designs The results for m-lldpe with slip in Fig. 6 also show a small minimum for all cases of apparent shear rates around 2a =308. Slip is known to enhance the minimum in the entry pressure vs. contraction angle as found out both theoretically and computationally by the Hatzikiriakos and Mitsoulis (2009). 6 Conclusions Three commercial polyethylene melts (m-lldpe, HDPE, LDPE) have been studied in entry flows through tapered dies of various angles. The experiments have shown a minimum for tapered dies with a taper of about 308 for all three melts. After that the pressures increase. Numerical simulations employing the K-BKZ model showed a minimum for the LDPE melt and for the m-lldpe melt with slip. Then the results level off, which is not always the case with the experiments. It is interesting to note that the simulations for the LDPE follow the ups and downs in the experimental data (albeit in a smaller scale) and for the first time are in agreement with the experimental data. It should be noted that previous simulations for the IUPAC-LDPE melt (Barakos and Mitsoulis, 1995a) gave values about half the experimental ones. The problem was resolved by using a long die length (L/D = 50) and then subtracting the pressure drop in the die to get the end pressure. Also the correction due to the pressure dependency of viscosity was important in matching the values up to the experimental levels. 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