Bagley correction - Springer Link

Rheol Acta (2003) 42: 309–320 DOI 10.1007/s00397-003-0294-y

Evan Mitsoulis Savvas G. Hatzikiriakos

Received: 15 May 2002 Accepted: 13 January 2003 Published online: 12 February 2003
Springer-Verlag 2003

E. Mitsoulis School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou 157 80, Athens, Greece S.G. Hatzikiriakos (&) Department of Chemical Engineering, The University of British Columbia, Main Mall, 2216, Vancouver, BC, V6T 1Z4, Canada E-mail: [email protected]

ORIGINAL CONTRIBUTION

Bagley correction: the effect of contraction angle and its prediction

Abstract The excess pressure losses due to end effects (mainly entrance) in the capillary flow of a branched polypropylene melt were studied both experimentally and theoretically. These losses were first determined experimentally as a function of the contraction angle ranging from 10 to 150. It was found that the excess pressure loss function decreases for the same apparent shear rate with increasing contraction angle from 10 to about 45, and consequently slightly increases from 45 up to contraction angles of 150. Numerical simulations using a multimode K-BKZ viscoelastic and a purely viscous (Cross) model were used to predict the end pressures. It was found that the numerical predictions do agree well with the experimental results for small contraction angles up to 30. However, the numerical simulations under-predict the end pressure for larger contraction angles. The

Introduction In capillary flow of molten polymers generated by means of a capillary rheometer, there is a large pressure drop associated with the flow in the entrance (major) and exit (minor) regions. If the pressure in the reservoir is the quantity measured to determine the wall shear stress in the part of the capillary where flow is fully developed, these two excess pressure losses, collectively known as end pressure, should be taken into account (Dealy and

effects of viscoelasticity, shear, and elongation on the numerical predictions are also assessed in detail. Shear is the dominant factor controlling the overall pressure drop in flows through small contraction angles. Elongation becomes important at higher contraction angles (greater than 45). It is demonstrated in abrupt contractions (angle of 180) that both the entrance pressure loss and the vortex size are strongly dependent on the extensional viscosity for this branched polymer. It is suggested that such an experiment (visualisation of entrance flow) can be useful in evaluating the validity of constitutive equations and it can also be used to fitting parameters of rheological models that control the elongational viscosity. Keywords Bagley correction Æ End effects Æ K-BKZ and Cross constitutive models Æ Contraction angle

Wissbrun 1990). The end pressure is also used to estimate the extensional viscosity of polymers, a method well practised in industry (Cogswell 1972, 1981). Therefore, it is significant to understand the origin of this excess pressure and consequently be able to predict it. Figure 1 plots the axial pressure variation in a capillary die including both its entrance and exit regions. It can be seen that the total pressure drop, DP, consists of two components and may be written as

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Fig. 1 Schematic diagram of the excess pressure losses in flow through a capillary rheometer

DP ¼ D PEnt þ D PCap þ D PExit ¼ D PEnd þ D PCap

ð1Þ

where DP is the total pressure drop from the reservoir to the capillary exit, DPCap is the pressure drop over the length of the capillary where the flow is fully developed, and DPEnd=DPEnt + DPExit is the excess pressure drop due to the entrance and exit flows. Bagley (1957) has suggested an empirical method of calculating this combined excess pressure DPEnd. The results from this procedure are customarily presented in terms of the Bagley correction or end correction, nB, defined as

3. At higher shear stresses where polymers have been reported to slip in capillary flow, a pressure–dependent slip may also cause the Bagley lines to become convex (Hatzikiriakos and Dealy 1992).

In general, at high shear stresses all the three effects are present, and the data analysis becomes even more complicated. To overcome some of the difficulties in determining DPEnd, one may use orifice dies (L/R @ 0). However, this procedure has its own limitations: (i) the total load measured is small and subject to large experimental D PEnd nB ¼ ð2Þ errors and (ii) the contribution of entrance flow to the 2 rw end correction is calculated at low levels of pressure, and where rw is the wall shear stress in the die for fully de- this might be different (higher) for longer capillary dies. veloped flow. In a capillary die of radius R and length L In spite of these limitations, Hatzikiriakos and Mitsoulis (see Fig. 1), the shear stress, rw, can be related to the (1996) have demonstrated that both methods generate Bagley correction, nB, the overall pressure drop, DP, and similar results for all practical purposes. the derivative of the pressure, dP/dz, along the die axis Many studies have examined the origin of end presaccording to Bagley (1957): sure and its prediction. Feigl and O¨ttinger (1994) have simulated the axisymmetric contraction flow of an R dP 1 DP ¼ rw ¼ ð3Þ LDPE melt using a Rivlin-Sawyers (special case of the 2 dz 2 nB þ L=R K-BKZ) constitutive model that represented the availA graph of DP vs L/R for a constant apparent shear rate able rheological data well. Their numerical results well c_ A is the well-known Bagley plot, where nB can be found under-predicted the available experimental pressure daby extrapolation of straight lines to DP=0. However, ta. Barakos and Mitsoulis (1995a, 1995b) reported the data points in a Bagley plot do not always fall on similar findings for the Bagley correction in capillary straight lines, and therefore extrapolation is not always flow of the IUPAC low-density polyethylene (LDPE). possible and straightforward. Beraudo et al. (1996) studied the Bagley correction There are mainly three causes of curvature in a problem of an LLDPE melt. They used a multimode Bagley plot of a molten polymer: Phan-Thien/Tanner (PTT) constitutive relation to predict it. They found that numerical predictions signifi1. If the viscosity of the polymer is a strong function of cantly underestimated the experimental findings by two pressure, the lines in the Bagley plot exhibit a curtimes at small shear rates to about four times at higher vature upwards. In such a case, one may fit a secondrates. Guillet et al. (1996) also studied the entrance order polynomial to the data in order to calculate pressure losses for an LLDPE and an LDPE melt the DPEnd (Laun 1983; Laun and Schuch 1989). both experimentally and numerically using a multimode 2. If viscous heating effects prevail, the lines in the K-BKZ integral constitutive equation. While they found Bagley plot become concave.

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reasonably good agreement between experiment and simulation for the case of the LLDPE melt, significant underestimation was reported for the case of LDPE. The origin of this underestimation was not discussed. Hatzikiriakos and Mitsoulis (1996) studied the Bagley correction of a linear low-density polyethylene (LLDPE) both experimentally and numerically using a K-BKZ constitutive relation. It was found that the numerical predictions agreed qualitatively but significantly underestimated the experimental data for the various geometries used to determine the end effects. Mitsoulis et al. (1998) have demonstrated that for LLDPE melts the entrance pressure loss is insensitive to extensional rheology, while it depends more strongly on the shear rheology. They have demonstrated that a change in extensional viscosity of ten times (keeping the shear viscosity constant) produces a change in the predicted entrance pressure drop of about only 10%. This finding raises doubts as to the usefulness of end pressure (Bagley correction) as a method of determining the extensional viscosity of polymer melts at high extensional rates (Cogswell 1972). Rajagopalan (2000) criticised this important result based on his one-mode Phan-Thien/ Tanner numerical simulations. Essentially, he could not vary independently the effects of shear and elongation, and therefore could not study independently their effects. In spite of this, he concluded that the end pressure is sensitive to elongational viscosity. From the above studies, it is clear that state-of-theart numerical simulations cannot predict quantitatively the pressure drop in a relatively simple flow, such as the entrance capillary flow of a polymer melt. Considering the fact that experimental measurements from such flows are extensively used in industrial practice to calculate the extensional viscosity of polymer melts at high shear rates (Cogswell 1972), it is essential to understand the origin of these disagreements. It should be emphasised that the extensional part of the flow is transient, and one should know the strain before being able to calculate the extensional viscosity. This was illustrated by Padmanabhan and Macosco (1997). They have used several methods, including Cogswell’s, to determine the elongational viscosity from entrance pressure drop measurements. Comparison of these analyses revealed considerable differences in their predictions. On the other hand, improved approximate methods, such as the ones advanced by Binding (1988, 1991), have been also used with varying degrees of success. For example, Gotsis and Odriozola (1998) showed that Binding’s method gave good results for branched polymers, like LDPE and a PP melt, especially at high strain rates. In our previous work on the subject (Hatzikiriakos and Mitsoulis 1996; Mitsoulis et al. 1998), we have studied both experimentally and numerically the entrance pressure losses in the capillary flow of LLDPE

melts as discussed above. In the present work, we study the end pressure in a more systematic way. First, the end pressure of a branched polypropylene is studied as a function of contraction angle ranging from 10 to 150 in an attempt to determine the relative significance of the extensional and shear contributions to the end pressure. These results consequently are simulated in order to check the capability of rheological models to predict this apparently simple flow. Two types of constitutive models are used: a viscoelastic K-BKZ and a purely viscous model, namely the Cross model. The differences in these results will also indicate the relative importance of viscoelasticity. Finally, a sensitivity analysis of the parameters controlling the shear and elongational properties will indicate the relative importance of the shear and extensional contributions to the total excess pressure losses.

Experimental Rheological testing Several rheological experiments were carried out in order to characterise the polymer under study. This is a branched polypropylene, hereafter denoted as PP, of high molecular weight. First, a Rheometrics System IV was used in the parallel-plate geometry to determine its linear viscoelastic moduli at 200–260 C. Second, a true-shear sliding-plate rheometer (INTERLAKEN) was used to measure the absolute viscosity and the damping function of the polymer in shear, in order to determine the non-linear parameters of the viscoelastic constitutive equations used for the simulations. All data and simulation results are presented at 200 C. Bagley correction An Instron capillary rheometer (constant piston speed) was used to determine the end pressure using the Bagley method (Dealy and Wissbrun 1990) and the viscosity as a function of
the wall shear stress, rW, and apparent shear rate, c_ A ¼ 4Q pR3 . Circular dies of various radii, R, and length-toradius ratios, L=R, were used to determine the entrance pressure losses (Bagley correction). All circular dies had a tapered 90 entrance angle. Orifice dies were also used to determine the Bagley correction as a function of the contraction angle, namely, 10, 15, 30, 60, 90, and 150 (see Fig. 2). Bagley correction data, obtained by extrapolation of pressure drop vs length-to-diameter ratio to zero length and directly from orifice die for the case of 90 contraction, gave identical results. Similar observations were reported previously as well (Hatzikiriakos and Mitsoulis 1996; Mitsoulis et al. 1998). Constitutive equations and rheological modelling 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 1 s¼ 1
h

  t
t0 a akover kk exp
k ða
3Þ þ b I k C
1 þ ð1
bÞ IC k¼1

Zt X N
1

0 0 0  ½ C
1 t ðt Þ þ h Ct ðt Þdt

ð4Þ

where 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

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Fig. 2 Schematic diagram of the orifice dies used in the experiments for determination of the excess pressure losses Cauchy-Green tensor Ct and its inverse Ct–1, the Finger strain tensor. The material constant h is given by h N2 ¼ N1 1
h

ð5Þ

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). As discussed above, experiments were performed in the parallelplate and sliding-plate rheometers for the PP to rheologically characterise the polymer. Figure 3 plots the master dynamic moduli G¢ and G¢¢ of PP at the reference temperature of 200 C. The model predictions obtained by fitting the experimental data to Eq. (4) with a spectrum of relaxation times, kk, and coefficients, ak, determined by a non-linear regression package (Kajiwara et al. 1995), are also plotted. The parameters found from the fitting procedure are listed in Table 1. The relaxation spectrum is used to find the average relaxation time k and zero-shear-rate viscosity g0 according to the formulas < k >¼

N X ak k2 k

k¼1

g0 ¼

N X

ak kk

ak kk

ð6Þ

ð7Þ

k¼1

Figure 4 plots a number of calculated and experimental material functions for PP. 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 symbols for N1 correspond to values obtained from Laun’s empirical formula (Laun 1986), i.e., "  0 2 #0:7 G 0 ð8Þ N1 ¼ 2G 1 þ G The parameter b, that controls the calculated elongational viscosity of PP, was originally fitted by simulating the vortex that is present in flow through an abrupt contraction (180-contraction

Fig. 3 Experimental data (symbols) and model predictions of storage (G¢) and loss (G¢¢) moduli for the PP melt at 200 C using six relaxation times listed in Table 1 Table 1 Relaxation spectrum for PP at 200 C (a=11.283, b= 0.05, h=)0.11, =22.1 s, g0=40,222 PaÆs) using the K-BKZ constitutive equation (Eq. 4) k

kk (s)

1 2 3 4 5 6

2.04 1.09 1.15 1.12 8.76 1.23

· · · · · ·

ak (Pa) 10–4 10–2 10–1 100 100 102

4.22 6.37 3.66 1.11 1.62 7.08

· · · · · ·

105 104 104 104 103 101

angle). Figure 5 shows a comparison between the calculated (with a single b-value) and the experimental vortex at the apparent shear rate of 32.4 s–1 in a 10:1 axisymmetric contraction. Using multiple b-values could improve the vortex fit (see, e.g., Luo and Tanner 1988), but this was beyond the scope of the present simulations. The experimental method used to determine the vortex formation can be found elsewhere (Boger 1987; Kazatchkov et al. 2000). For this branched PP it was found (see below) that the vortex size is very sensitive to the elongational viscosity. Therefore, we preferred to use the method based on predicting the vortex size in determining the elongational viscosity of PP rather than Cogswell’s method (Cogswell 1972), that is based on DPend data. In this latter method, the average extensional rate, e_, and extensional viscosity, gE ðe_Þ, as a function of end pressure, shear viscosity, gS, and the local power–law exponent, n, are as follows: e_ ¼

4c_ 2A 9ðn þ 1Þ2 ðDPEnd Þ2 ; gE ðe_Þ ¼ 3ðn þ 1ÞDPEnd 32gS c_ 2A

ð9Þ

where the local power-law exponent, n, is determined locally from a shear stress vs apparent shear rate log-log plot. To evaluate the role of viscoelasticity in the prediction of Bagley correction, purely viscous models were also used in the simulations. Namely, the Cross model was used to fit the shear viscosity data of PP. The Cross model is written as (Dealy and Wissbrun 1990)

313

Fig. 4 Experimental data (solid symbols) and model predictions of shear viscosity gS, first normal stress difference N1, and elongational viscosity gE for the PP at 200 C using the K-BKZ model (Eq. 4) with the parameters listed in Table 1. Open symbols for N1 are predictions using Eq. (8)

Fig. 6 The shear viscosity of PP at 200 C fitted with the Cross model using the parameters listed in Table 2 Table 2 Parameters of the viscous Cross model for the PP polymer Parameter

PP (200 C) Cross Model (Eq. 10)

go g¥ k n

40,360 PaÆs 1 PaÆs 1.57 s 0.26

Fig. 5 Fitting the parameter b of the K-BKZ model that controls the elongational viscosity for PP at 200 C. A value of b=0.05 fixes the size to Lv/D=0.36 in flow through a 10:1 axisymmetric abrupt contraction (180). Below it is demonstrated that the vortex size is very sensitive to the value of b. Left half is streamline pattern from the simulations with the K-BKZ model, right half is experimental pattern from Kazatchkov et al. (2000) g ¼ g1 þ

go
g1

ð10Þ 1 þ ðkc_ Þ1
n The fitted viscosity of PP by Eq. (10) is plotted in Fig. 6, while the parameters of the model are listed in Table 2.

Experimental results – end pressure

Fig. 7 The end pressure of PP at 200 C as a function of apparent shear rate for various contraction angles from 10 to 150

Figure 7 plots the end pressure of PP at 200 C as a function of the apparent shear rate for all six orifice dies ranging from 10 to 150. Differences between the various pressure-drop curves can be observed. The entrance

angle has a significant effect on the end pressure. The latter does not change monotonically with entrance angle. It decreases with increasing contraction angle from 10 to about 30–45 and subsequently slightly

314

increases up to contraction angles of 150. This can be seen more clearly in Fig. 8, where the end pressure is plotted as a function of entrance angle for several values of the apparent shear rate. The values at 150 do not differ substantially from the values at 90 for the same apparent shear rate (e.g., at c_ A ¼ 70:3s
1 the values are 0.974 and 1.086, respectively, while at c_ A ¼ 234s
1 the values are 1.972 and 2.123, respectively) and are not plotted here. At about 30–45, depending on the shear rate, the minimum end pressure is obtained. This observation can be very helpful in assessing the relative importance of shear and extensional components of the flow for various contraction angles, as well as in assessing the usefulness of contraction flows in determining the elongational viscosity of polymer melts, as will be discussed below. Similar trends were found for other branched polymers in experiments performed in our lab for metallocene-LLDPEs and -LDPEs.

Numerical results Viscoelastic modelling The numerical simulations have been undertaken using the integral constitutive equation (Eq. 4) as well as the viscous model (Eq. 10). Each of these constitutive relations is solved together with the usual conservation equations of mass and momentum for an incompressible fluid under isothermal conditions (Luo and Mitsoulis 1990; Barakos and Mitsoulis 1995a). Figure 9 shows the representative grids for the various capillary dies used to simulate the flow of the PP melt. Because of symmetry, only one half of the flow domain is used. The other (symmetric) half shows the finite element grids with four

Fig. 8 The end pressure of PP at 200 C as a function of contraction angle at various values of apparent shear rate

times the number of elements used in test cases to guarantee results independent of mesh size (Barakos and Mitsoulis 1995b). It is worthwhile mentioning the significant tapered die length differences between the two extreme cases of contraction angle of 10 and 150. The contraction die length LC is calculated according to the formula shown in Fig. 2. All lengths have been made dimensionless by the orifice radius, R. The contraction ratio was D/d=Rres/R=12.5 for all dies used, except for the abrupt contraction (180) for which the ratio was 10. Having fixed the K-BKZ model parameters and the problem geometry, the only parameter left to vary was the apparent shear rate (c_ A ). Simulations were performed for a range of apparent shear rate values from 1.0 s)1 to as high as possible (700 s)1) depending on convergence. Figure 10 plots the numerical results of end pressure for PP as a function of the apparent shear rate for orifice dies having contraction angles from 10 to 90. Open symbols represent the corresponding experimental results, which were also plotted in Fig. 7. First, it can be seen that the numerically calculated end pressure decreases with increasing contraction angle (which is also intuitive) and that they agree with the experimental ones for small contraction angles, i.e., 10 and 15. On the other hand, as the contraction angle increases, the numerical results do not show the reversal of end pressures found experimentally for the 30, 60, and 90, especially for the low shear rates. However, at high shear rates they converge together as in the experiments. The simulation results are also plotted as a function of the contraction angle in Fig. 11 for different apparent shear rates together with the experimental data of Fig. 8. There is a

Fig. 9 Various die geometries (different contraction angles) along with representative grids used in the simulations. The upper half contains four times the number of elements of the lower half for test cases to establish the mesh independence of the numerical solutions

315

true for large contraction angles. For small contraction angles the flow is shear-dominated, and the modelling gives good predictions. However, as extension becomes significant there are discrepancies between the model (flow simulation) and the experiments. This is interesting to note and needs further investigation. In particular, (i) why at high contraction angles the model does not capture well this important aspect of experimental rheology, (ii) what is the sensitivity of the model predictions (pressure drop) in terms of shear and extensional properties, and (iii) what is the effect of viscoelasticity on the pressure drop predictions and how its predictions are compared with those obtained from viscous flow simulations? We make an attempt to answer these important questions below. Viscous modelling Fig. 10 Comparison between experimental data (symbols) and simulation end pressure results using the K-BKZ constitutive model for the PP at 200 C as a function of the apparent shear rate for several contraction angles

monotonic decrease of end pressure with increasing contraction angle for the same apparent shear rate, but for the highest apparent shear rates, there is a levellingoff of the end pressure for angles ‡30, as was also the case in the experiments. To understand this counter-intuitive behaviour, we need to recall that contraction flow is a mixed flow, where viscoelasticity manifests itself by interplay between shear and extensional flow components. This is

Fig. 11 Comparison between experiment data (symbols) and simulation results using the K-BKZ constitutive model for the end pressure of PP at 200 C as a function of contraction angle at various values of apparent shear rate

To examine whether or not viscoelasticity plays a role in predicting the end pressure (Bagley correction), it is necessary to examine also the predictions by purely viscous models having the same shear viscosity. This was done in the present work using the purely viscous Cross model to perform simulations. Figure 12 plots results obtained for contraction angles of 10, 15 and 30 for the PP. It can be seen that the results between the two models (K-BKZ and Cross) agree only for the lowest shear rates and contraction angle of 10, while the deviation between the two models increases substantially with contraction angle, particularly at the higher shear rates as expected. These results suggest that the flow for

Fig. 12 Comparison between predictions of K-BKZ and Cross models for the end pressure of PP at 200 C as a function of the apparent shear rate for several contraction angles

316

this branched PP melt even at small contraction angles possesses a strong viscoelastic character and that only at the lowest angle of 10 the flow is predominantly a shear flow, since in this case the differences in predictions between viscoelastic and viscous models are small. The differences in the results between the two models at higher apparent shear rates and at larger contraction angles are attributed to a combination of elongational and first normal stress difference effects. The viscous model does not possess N1 in shear and has an elongational viscosity pffiffiffi that follows its shear counterpart (gE ðe_Þ ¼ 3gS ð 3e_Þ for all rates) (Barnes et al. 1989). The N1 force helps the material flow through, hence it reduces the pressure as calculated with the K-BKZ model (Mitsoulis et al. 1985; Binding 1991). On the other hand, the elongational viscosity is higher for the K-BKZ than the Cross model and it increases the pressure drop, substantially. These two extra viscoelastic forces act in opposite directions. Their combined effect produces the predictions plotted in Fig. 12. From these results it is also expected that elongational effects are significant for this branched polymer in contrast to linear polymers, such as LLDPEs. Hatzikiriakos and Mitsoulis (1996) had performed similar simulations for an LLDPE (Dowlex) melt to address the effect of a viscoelastic model and compare this to the effect of a viscous model on the flow simulations. They found no differences, which was due to insignificant effects of extensional rheology on the pressure drop. In fact, a sensitivity analysis of the pressure drop on the extensional rheology of the Dowlex LLDPE melt has shown that the pressure drop was insensitive to changes in the extensional viscosity (Hatzikiriakos and Mitsoulis 1996). In the present case, however, we expect that the extensional viscosity will become very significant.

Fig. 13 The effect of the elongational parameter b in the K-BKZ constitutive equation on the rheological properties of PP at 200 C

b=0.05, while the new value of b=0.5 decreases the uniaxial extensional viscosity significantly (line with crosses). Figure 14 shows simulation results for the two different values of b and three different contraction angles. It can be seen that the value of b does not play any significant role in the simulations for the smallest contraction angle. The flow field for a contraction angle of

The effect of extensional viscosity on the end pressure Steady behaviour Simulations were performed to determine the sensitivity of the end pressure (Bagley correction) on the extensional viscosity of PP. Figure 13 indicates the effect of the elongational parameter b of the K-BKZ equation on the rheological properties of PP at 200 C. As explained above, only single b-values have been used to make the results more amenable to parametric studies. It can be seen that by changing b by one order of magnitude (0.05 to 0.5), the shear viscosity gS and the first normal stress difference N1 remain exactly the same, while the uniaxial extensional viscosity gE changes by one order of magnitude at high rates (e.g., at e_ ¼ 10s
1 , gE=270,000 PaÆs for b=0.05 and gE=50,000 PaÆs for b=0.5). It is noted that the original fitting (continuous line) corresponds to

Fig. 14 Simulation results showing the effect of the elongational parameter b in the K-BKZ constitutive equation on the end pressure for flow of PP at 200 C through dies with different contraction angles

317

10 is essentially a shear field, and the extensional viscosity has a small effect (but not negligible when compared with purely viscous simulations). Recalling Fig. 12, it becomes evident that any differences between viscous and viscoelastic simulations are caused by both N1 and elongational effects, and the viscoelastic model has a much higher elongational viscosity than a purely viscous model. For contraction angles of 90 and 180 the differences increase substantially by using different bvalues. It is then obvious that the extensional component of the flow field increases with the contraction angle, and this is reflected upon the end pressure. The relative differences are larger for the larger contraction angle. The flow patterns predicted by the simulations at the upstream region of the entrance in an abrupt contraction (180) for the two values of b are also different. This is shown in Fig. 15. It is noted that the value of b was fitted by matching the size of the vortex (discussed in a previous section). In this case by decreasing the extensional viscosity by several times (almost one order of magnitude), the impact on the flow field is noticeable, i.e., there is a dramatic vortex size reduction. At the same time, this difference is reflected upon the end pressure losses, which are changed by 43%. Previously, for an LLDPE melt, Mitsoulis et al. (1998) have shown the opposite, that is, the end pressure was insensitive to the elongational viscosity of the polymer. By varying the extensional viscosity by several times, the end pressure remained essentially the same.

that the fluid elements experience on the centreline. These range from e_max ¼ 0:19s
1 and t0=413.8 s at c_ A ¼ 4s
1 for the 10-angle, to e_max ¼ 123:2s
1 and t0=0.33 s at c_ A ¼ 640s
1 for the 90-angle. For the case shown in Fig. 15, we have e_max ¼ 4:91s
1 and t0=6.3 s at c_ A ¼ 32:4s
1 for the 180-angle. Also, the total Hencky strain experienced by a fluid element moving along the centreline of the 12.5:1 axisymmetric contraction region from far upstream to the die exit is given by (Rothstein and McKinley 1999) "  # vzZ ðz¼LÞ Zt0 dvz D 2 e  e_dt ¼ ¼ 5:05 ð11Þ ¼ ln d vz 0

vz ð
1Þ

where t0 is the time spent in the centreline and D/d is the contraction ratio. For the case of a 10:1 contraction ratio,
=4.605. Numerical integration of the centreline strain rates or velocities according to Eq. (11) gave results very close to this theoretical value. The transient elongational viscosity of PP at 200 C is shown in Fig. 16 for different strain rates covering the range encountered in the experiments, namely 0:01s
1  e_  100s
1 . We note that for small strain rates, e_ ¼ 0:01s
1 and 0:1
1 , the curves are very close together and the effect of b is negligible. It takes more than 10 s to reach steady-state conditions. However, at e_  1s
1 , the differences for the two b-values are dra-

Transient behaviour At this point it is instructive to look at the transient elongational viscosity of the polymers. A detailed look at the simulation results reveals that in the geometries studied there is a wide range of strain rates and times

Fig. 15a, b The effect of changing the elongational parameter b in the K-BKZ constitutive equation on the vortex size and the end pressure losses for flow of PP at 200 C in a 10:1 abrupt axisymmetric contraction

Fig. 16 Transient elongational viscosity for different elongational rates predicted by the K-BKZ constitutive equation for PP at 200 C. The effect of the elongational parameter b is also shown (solid vs dashed lines). Open symbols are experimental data provided independently (Gotsis 2002) by using an RME apparatus

318

matic, while a steady state is achieved much earlier. In the same figure, we have collected elongational experimental data measured independently at the University of Delft (Gotsis 2002) with an RME apparatus at several elongational rates. Except for small times where there are severe discrepancies from the model predictions, the bulk of the experimental data follow more or less the model predictions. For e_ ¼ 0:01s
1 (circles) and 0.1 s–1 (squares), the experimental data show strain-hardening, while at e_ ¼ 1:0s
1 (triangles), there is no evidence of strain hardening. Thus the degree of strain hardening seems to decrease with increasing strain rates. The experimental data are closer to the continuous curves corresponding to b=0.05, except for e_ ¼ 1:0s
1 where they seem to follow the curve for b=0.5. This data also seems to suggest that multiple b-values are needed to best capture the behaviour of the PP melt in elongation. By contrast, the equivalent curves for LLDPE (Dowlex 2049) used in our previous publication (Hatzikiriakos and Mitsoulis 1996; Mitsoulis et al. 1998) show a much reduced difference in behaviour for the two b-values used, as evidenced in Fig. 17. As the contraction flow is a transient elongational flow along the centreline and the strains involved are relatively small, a steady state is never reached. Therefore, if the transients are essentially the same for the two values of b, this would result in about the same end pressure (case of LLDPE reported by Mitsoulis et al. 1998). On the other

hand, if the transients were significantly different for the two values of b, this would result in significant differences in the end pressure (PP reported and studied in the present paper). This finding raises an important question in experimental rheology–that of the usefulness of the end pressure (Bagley correction) in estimating the extensional viscosity of polymer melts through Cogswell’s analysis. Laun and Schuch (1989) have shown that the extensional viscosity of certain polyethylenes obtained using true extensional rheometers were in good agreement with those obtained through Cogswell’s analysis over narrow ranges of strain rate. However, in general, there was no quantitative correlation between the two functions for many other resins. It is worthwhile to mention, and in view of the results discussed above, Cogswell’s analysis might only be used to classify polymers in terms of transient viscosities at best. This classification will depend heavily on the transient elongational viscosity of the melts, whether they grow fast or not. It is apparent from our experience that in the presence of long-chain branching (LCB), the elongational viscosity is a strong function of strain and b, and this results in fast transients. In these cases, the end pressure might be used to classify resins (Hatzikiriakos 2000). In the absence of LCB, we believe that the use of end pressure for evaluating the elongational viscosity might not be safe to draw conclusions. Indeed, Hatzikiriakos (2000) has shown the sensitivity of the end pressure on the presence and degree of LCB. From the above results, it is clear that the issue of predicting the pressure drop in orifice dies is far from resolved. We believe that its correct qualitative and quantitative description would lead to the establishment of better rheological models. Such flows (contraction flows) may serve as strong tests for evaluating rheological models as well as fitting model parameters that depend on the extensional viscosity, i.e., the sensitivity of vortex size on the extensional parameter b. Furthermore, since the vortex size is sensitive to the presence of LCB, it is another support that the end pressure here can be used for qualitative evaluation of the elongational behaviour of polymers. For linear polymers vortices are not formed, and thus the use of the end pressure is not safe for such comparisons. The effect of shear viscosity on the end pressure

Fig. 17 Transient elongational viscosity for different elongational rates predicted by the K-BKZ constitutive equation for LLDPE (Dowlex 2049) at 200 C (Hatzikiriakos and Mitsoulis 1996). The effect of the elongational parameter b is also shown (solid vs dashed lines)

Simulations were performed to determine the sensitivity of the end pressure (Bagley correction) on the shear viscosity of PP. Figure 18 indicates the effect of the parameter a of the K-BKZ model on the rheological properties of PP at 200 C. It can be seen that by changing a by one order of magnitude (11.283 to 1.1283), all material functions change. The shear vis-

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Fig. 18 The effect of changing the parameters a and b in the KBKZ constitutive equation on the rheological properties of PP at 200 C. Solid symbols are experimental data. Solid lines are predictions from the original fitting with a=11.283 and b=0.05. Lines with crosses are predictions with a=1.1283 and b=0.003. Changing the shear parameter a changes all functions, so a new value of b is necessary to have approximately the same response in uniaxial elongation (gE)

cosity changes by a factor of 1.4–1.5 in the range of experimental measurements. It is noted that the original fitting (continuous lines) corresponds to a=11.283 and b=0.05, while a new value of b=0.003 was necessary to keep the extensional viscosity about the same (lines with crosses). Figure 19 shows the simulation results for the two different values of a for the contraction angle of 90. It can be seen that the effect of shear viscosity is significant even at these large contraction angles. A decrease of shear viscosity by a factor of 1.4–1.5 results in a pressure-drop reduction by a factor of 2. Comparing these effects to the corresponding effects of extensional viscosity at 90, the former are stronger. Figures 13 and 16 show that a decrease of extensional viscosity by a factor of 10, results in a decrease of pressure drop by a factor of only 1.3–1.4. These differences can be interpreted in terms of differences in extensional viscosity of only 2 (1:42  2) in view of Eq. (9), where gE / DPEnd . In view of the fact that the extensional viscosity has been changed by one order of magnitude, one would have expected to pobtain a much larger pressure drop differffiffiffiffiffi ence, i.e., 10  3:16 compared with only 1.3–1.4. Therefore, our discussion for the use of entrance pressure drop for evaluation of resins in terms of extensional viscosity are in order. Such comparisons should be used with caution and only when LCB is a factor and when the resins exhibit strain-hardening effects. In these cases

Fig. 19 The effect of the parameters a and b in the K-BKZ constitutive equation on the end pressure for flow of PP at 200 C through a 90-die

the resins can be classified qualitatively in terms of strain-hardening and extensional flow behaviour.

Conclusions Experiments and numerical simulations have been performed for the axisymmetric flow of a PP melt through capillary dies of different contraction angles ranging from 10 to 150. The emphasis has been on calculating the pressure drops and establishing the end pressure losses present in the flow of polymer melts in capillary rheometry. Of particular interest in this study was to predict the sensitivity of the end pressure to contraction angle both qualitatively and quantitatively. Comparison of numerical simulation results using a K-BKZ integral constitutive relation with experimental data resulted in some quantitative disagreement, at least for large contraction angles. Further simulations and comparisons with experimental data have demonstrated that the extensional viscosity plays a role for large contraction angles, whereas the effects of shear are more dominant even at large contraction angles, as large as 90. This points out the dominance of shear viscosity in converging flows. The extensional viscosity has a relatively smaller effect on the pressure drop, as has been demonstrated by simulations. In particular, a decrease of the extensional viscosity by about one order of magnitude decreases the end pressure by about 40–50%. However, the effect of decreasing the extensional vis-

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cosity is more significant in the flow pattern development. A decrease of the extensional viscosity decreases the vortex size significantly. This latter observation may be used to fit parameters of rheological constitutive models that depend on the extensional properties. In addition, it has been argued that Cogswell’s analysis can be used to classify resins in terms of extensional behaviour only in the presence of LCB, where strainhardening effects might be dominant. Therefore, one can

get qualitative results by using Cogswell’s analysis at best. 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. The authors are indebted to Prof. Alexander Gotsis of the University of Delft for providing rheological experimental data, especially on the elongational viscosity.

References Bagley EB (1957) End corrections in the capillary flow of polyethylene. J Appl Phys 28:193–209 Barakos G, Mitsoulis E (1995a) Numerical simulation of extrusion through orifice dies and prediction of Bagley correction for an IUPAC-LDPE melt. J Rheol 39:193–209 Barakos G, Mitsoulis E (1995b) A convergence study for the numerical simulation of the IUPAC-LDPE extrusion experiments. J Non-Newtonian Fluid Mech 58:315–329 Barnes HA, Hutton JF, Walters K (1989) An introduction to rheology. Elsevier, Amsterdam Beraudo C, Coupez T, Fortin A, Demay Y, Vergnes B, Agassant JF (1996) Viscoelastic computations in 2-D flow geometries: comparison with experiments on molten polymers. In: Ait-Kadi A, Dealy JM, James DF, Williams MC (eds) Proceedings of the XIIth International Congress on Rheology, Quebec City, pp. 417–418 Binding DM (1988) An approximate analysis for contraction and converging flows. J Non-Newtonian Fluid Mech 27:173–189 Binding DM (1991) Further considerations of axisymmetric contraction flows. J Non-Newtonian Fluid Mech 41:27–42 Boger DV (1987) Viscoelastic flows through contractions. Annu Rev Fluid Mech 19:157–182 Cogswell FN (1972) Measuring the extensional rheology of polymer melts. Trans Soc Rheol 16:383–403 Cogswell FN (1981) Polymer melt rheology–a guide to industrial practice. Wiley Dealy JM, Wissbrun KF (1990) Melt rheology and its role in plastics processing– theory and applications. Van Nostrand Reinhold, New York

Feigl K, O¨ttinger HC (1994) The flow of a LDPE melt through an axisymmetric contraction: a numerical study and comparison to experimental results. J Rheol 38:847–874 Gotsis AD (2002) Private communication to the authors, Univ. of Delft, the Netherlands Gotsis AD, Odriozola A (1998) The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts. Rheol Acta 37:430– 437 Guillet J, Revenue P, Bereaux Y, Clermont J-R (1996) Experimental and numerical study of entry flow of low-density polyethylene melts. Rheol Acta 35:494– 507 Hatzikiriakos SG (2000) Long chain branching and polydispersity effects on the rheological properties of polyethylenes. Polym Eng Sci 40:2279–2287 Hatzikiriakos SG, Dealy JM (1992) Wall slip of molten high density polyethylene. II. Capillary rheometer studies. J Rheol 36:703–741 Hatzikiriakos SG, Mitsoulis E (1996) Excess pressure losses in the capillary flow of molten polymers. Rheol Acta 35:545– 555 Kajiwara T, Barakos G, Mitsoulis E (1995) Rheological characterization of polymer solutions and melts with an integral constitutive equation. Int J Polym Anal Charact 1:201–215 Kazatchkov IB, Yip F, Hatzikiriakos SG (2000) The effect of boron nitride on the rheology and processing of polyolefins. Rheol Acta 39:583–594 Laun HM (1983) Polymer melt rheology with a slit die. Rheol Acta 22:171–185 Laun HM (1986) Prediction of elastic strains of polymer melts in shear and elongation. J Rheol 30:459–501

Laun HM, Schuch H (1989) Transient elongational viscosities and drawability of polymer melts. J Rheol 33:119–175 Luo X-L, Mitsoulis E (1990) An efficient algorithm for strain history tracking in finite element computations of nonNewtonian fluids with integral constitutive equations. Int J Num Meth Fluids 11:1015–1031 Luo X-L, Tanner RI (1988) Finite element simulation of long and short circular die extrusion experiments using integral models. Int J Num Meth Eng 25:9–22 Mitsoulis E, Vlachopoulos J, Mirza FA (1985) A numerical study of the effect of normal stresses and elongational viscosity on entry vortex growth and extrudate swell. Polym Eng Sci 25: 677–689 Mitsoulis E, Hatzikiriakos SG, Christodoulou K, Vlassopoulos D (1998) Sensitivity analysis of the Bagley correction to shear and extensional rheology. Rheol Acta 37:438–448 Padmanabhan M, Macosco CW (1997) Extensional viscosity from entrance pressure drop measurements. Rheol Acta 36:144–151 Papanastasiou AC, Scriven LE, Macosco CW (1983) An integral constitutive equation for mixed flows: viscoelastic characterization. J Rheol 27:387–410 Rajagopalan D (2000) Computational analysis of techniques to determine extensional viscosity from entrance flows. Rheol Acta 39:138–151 Rothstein JP, McKinley GH (1999) Extensional flow of a polystyrene Boger fluid through a 4:1:4 axisymmetric contraction/expansion. J Non-Newtonian Fluid Mech 86:61–88