CONVERGING FLOW ON-LINE RHEOMETRY FOR AN ENGINEERING EXTENSIONAL VISCOSITY OF UPVC.

Transcription

1 CONVERGING FLOW ON-LINE RHEOMETRY FOR AN ENGINEERING EXTENSIONAL VISCOSITY OF UPVC. H. J. Ettinger, J. F. T. Pittman*, J. Sienz Centre for Polymer Processing Simulation and Design, C2EC, School of Engineering, Sansea University, SA2 8PP, UK An on-line rheometer is described for determination of an engineering extensional viscosity using pressure drops in conical converging flo under fixed extruder processing conditions. A temperature cross ith sixteen thermocouples is incorporated for detailed measurement of the temperature of melt passing to the rheometer. Temperature variations are taken into account in extracting extensional viscosity values using a modified version of the analysis by Binding. Results are reported for a profile grade upvc. The applicability of the Binding analysis in this case is examined and confirmed. Results compare ell ith published data for profile grade upvc, taking into account differences in formulation. Introduction The present ork formed part of a project developing computerized optimization methods for profile die design (1). The kernel of the optimization is a simulation of flo in the die from hich an objective function measuring the die performance is evaluated. Based on this, and using a mathematical optimization method, key parameters of the design are adjusted and the simulation repeated. The process is iterated until the optimum is achieved. The geometry of industrial dies, for example for the manufacture of unplasticized PVC (upvc) profiles is complex, and three dimensional flo simulations are necessary. Bearing in mind the requirement for repeated solution in the optimization process, practical limitations are imposed on the rheological models that can be used. Three dimensional simulations of viscoelastic flo are computationally very demanding, and convergence may be difficult to obtain. Although successful viscoelastic simulations of upvc flo in 3D profile dies have been made, Glomsaker et al. (2), and progress in this area continues, incorporation into an automated optimization loop is not practical at present. One must therefore resort to models based on instantaneous, local deformation rates. A further practical consideration is that the flo properties of certain industrially important materials are highly dependent on their thermo-mechanical processing history. Unplasticized PVC (upvc) is a case in point, as is ell documented in several orks, see Glomsaker et al. (3) for example. Classical off-line rheological measurements are unlikely to return values that are valid for flo in the die, because of the different prior experience of the material. On-line measurements under processing conditions are therefore indicated. This again imposes limitations on the type of rheological models that can be used. A full viscoelestic characterization of material issuing directly from an extruder ould seem to be difficult or impossible, and for this reason also the use of simpler models is indicated. Generalized Netonian models can be used successfully in the parallel land zones of dies, because flo there closely approximates steady shear. Hoever, they ill not capture entry pressure loss in elongational flo of polymeric materials in converging die channels. The excess can be measured by the Trouton ratio, and values up to 100 are implied for one commercial upvc grade in the ork of Glomsaker et al. (2). In comparing the experimental flo distribution of upvc in a profile die ith that predicted by a generalized Netonian flo simulation in the parallel land region, Pittman and Szarvasy (4) found flo into narroer sections to be over predicted, indicating the need to take into account the additional pressure drop in the converging flo at the entry by inclusion of an extensional viscosity. Techniques are available for estimating extensional viscosity as a function of an instantaneous elongation rate using the pressure drop in converging flo from a reservoir into a capillary, as originally proposed by Cogsell (5) and subsequently developed by others (see belo). Polymeric materials subjected to a steady elongation may, or may not, attain a steady value of extensional viscosity. The elongation rate at entry to a capillary is not steady and the residence time is short. It is therefore unlikely that the data ill represent a fundamental material property, and should rather be regarded as an engineering approximation, averaged over the test conditions. Nevertheless, the technique is convenient and ell adapted to on-line measurements. If applied under practical processing conditions, as regards thermo-mechanical history in the extruder, and deformation rates, strain and residence time in the contraction, it probably provides the best available practical approach to estimating elongational effects. Methods for incorporating the resulting data - an extensional viscosity dependent on local instantaneous deformation

2 rate into flo modeling have been explored in Schunk and Scriven (6), Song and Xia (7), Mitsoulis et al. (8), and Debbaut and Crochet (9). These considerations provide the motivation for the present ork, hich describes experimental procedures and data processing to obtain an engineering extensional viscosity using on-line measurements. The method is applied to a commercial upvc formulated for profile extrusion. An important feature of on-line rheometry, often overlooked, is that it is difficult or impossible to control melt temperature independently of flo rate, and furthermore the temperature of the melt supplied form the extruder is unlikely to be uniform. Particular care is taken in the present ork to monitor temperatures, and these are taken into account in extracting the extensional viscosity data. The lay-out of the paper is as follos. Theoretical results relating an extensional viscosity to entry pressure drops are identified. The extraction of parameters in a poer la representation of extensional viscosity is then described. Description of the experimental equipment and procedure is folloed by presentation and discussion of results. Relating extensional viscosity to entry pressure drops Analytic results relating extensional viscosity to entry pressure drops have been developed from the late 1960s onards. This ork ill be revieed fully elsehere. To evaluate uniaxial extensional viscosities in axisymmetric converging flo e use the analysis proposed by Binding (10) as modified by Mackay and Astarita (11). Writing the viscosities as temperature dependent poer la forms n 1 bt ( 0 T) = = e s s s s0 η µ γ µ µ (1) p 1 d( T0 T) = = e e e e e0 η µ ε µ µ (2) the resulting expression for the entry pressure loss is 1 p( n+ 1) 3 p( n+ 1) 2 d( T0 T) p 1 p 1 p 1 16n bt ( ) ref T p µ e I + + e pn R + + n+ n+ V 0 P = µ e 1 s0 bt ( 0 T) p 3 3p n+ 1 µ e (4 p) 4n 4n πr R s0 0 i (3) here I pn is defined as I pn = 2n Γ( p) Γ pn p + 1 n + 1 n + 1 2n Γ p + n + 1 ( )( ) ( p( n+ 1) + 2n) (( p+ 1)( n+ 1) + 2n) (4) y 1 x exp and Γ ( y) = x e dx is the gamma function. The experimental temperature is T, V is the volumetric flo rate and 0 R i and R o are the radii at the upstream and donstream ends of the contraction. Temperature dependence of the viscosities is included here, as mentioned above, because the experimental temperature varies beteen experiments at different flo rates. The consistency coefficient, poer la index and temperature coefficient for shear viscosity ( µ, s0 n, b) are available from previous ork (12). It is assumed that the temperature coefficient of the extensional viscosity is the same as that for shear viscosity, leaving the to parameters ( µ, p) to be determined. To evaluate p, entry pressure e0 drops are first adjusted to a reference temperature. From Eq. 3 e have 0 0 d b ( ) p + 1 PT ( ) = PT ( )exp T T b (5) According to Eq. 3, a log-log plot of P(T 0 ) vs. V should then have a slope p(n+1)/(p+1). Knoing n, p is then available, and µ can be calculated. s0

3 On-line rheometer and experimental procedure Fig. 1. Schematic of the on-line rheometer. The converging flo rheometer, Fig. 1, as developed from a slit die rheometer previously described (12). In order to vary flo rate in the rheometer hile maintaining a constant extruder operating point (scre speed, back pressure) a bypass valve as included upstream and a gate valve donstream. After the bypass valve, flo passes over the temperature cross, Fig 2, here sixteen 2mm diameter, steel sheathed, mineral insulated thermocouples are mounted projecting up-stream from streamlined supports across the duct diameters. This arrangement is knon to give reliable measurements of temperature ithin the floing melt, Pittman and Mahmoudzadeh (13). After a distance piece, flo then passes from the 80mm diameter duct into a conical section converging (convergence angle 30 o ) to a short 5mm diameter capillary. From here, a fish-tail adapter leads into the slit rheometer ending ith a gate valve. Measurements from the slit rheometer are not included in the present ork; its present function is to provide the donstream flo restriction required to maintain the required back pressure on the extruder. Pressure transducers P1 and P2 are flush mounted just upstream of the contraction and in the capillary. Temperature and pressure as also measured at the extruder exit. Fig. 2. Schematic of the temperature cross mounted up-stream of the rheometer. It is necessary to use a conical converging section here, rather than the sharp contraction from reservoir to orifice typical of classical capillary rheometers, to avoid hold-up and degradation of the upvc in the corners. The converging flo may then be constrained, rather than undergoing the free convergence assumed in the analysis. This point is discussed further belo.

4 The rheometer as mounted on a tin scre extruder (65mm diam. scres) operated at typical processing conditions. Mass flo rates through the rheometer ere determined by eighing and converted to volumetric flos using density data. To sets of experiment are reported here for scre speeds of n ex = 13.5 and 18.5 rpm. Five flo rates ere examined for each, ith constant pressure maintained at the extruder exit by manipulation of the bypass and gate valves. A commercial, profile grade upvc as used, ith composition as in Table 1. Data processing Table 1. Composition of the profile grade upvc used in the present ork. Total additives, 22 parts per hundred S-PVC Component parts by eight S-PVC 100 Impact modifier 6 Fillers 6 Pigment 5 Stabilizer 3.5 Lubricant 1.5 Because the transducers are located slightly upstream and donstream of the converging section, corrections must be applied to the measured pressure drop. Pressure drops over the short sections of the 80mm duct upstream and the 5mm capillary donstream of the conical section are estimated from the analytic result for developed tube flo of a poer la liquid ith all slip using parameter values as determined in previous ork, see (12) for further details. Corrections for flo in the 80mm duct ere approximately 0.5% of the measured pressure drop, and from 15 to 20% for the capillary. Experimental data are summarized in Tables 2 and 3. Material leaving an extruder ill alays sho non-uniform temperatures, and these are no doubt a frequent, though often unacknoledged, feature of on-line rheometry. Tables 2 and 3 also sho the range of the temperatures recorded by the temperature cross (under steady conditions). The increases ith rheometer flo rate are probably a further consequence of this non-uniformity, arising because different proportions of the extruder output are taken in each case. They are unlikely to be due to increased viscous heating, because this is negligible in the 80mm duct, and at higher rheometer flo rates less restriction is imposed on flo in the bypass valve. Homogenization of the temperature ould require incorporation of a mixing device up-stream of the rheometer, adding to the deformation history of the material and defeating the objective of characterizing it under normal processing conditions. The non-uniformity is therefore accepted and flo-averaged mean temperatures T av are calculated using the developed tube flo velocity profile for a poer la liquid including all slip. Poer la and all slip parameters are available from previous ork (12). Viscous heating in the converging flo is not significant. From the pressure drops listed belo the adiabatic temperature rise can be calculated as about 3 o C. Table 2. Scre speed n = 13.5 rpm. Extruder exit pressure 483 bar. Temperature at extruder exit 191 o C (flushmounted ex sensor). exp V 10-6 m 3 /s P 10 7 Pa T o C range T o av C Table 3. Scre speed n = 18.5 rpm. Extruder exit pressure 507 bar. Temperature at extruder exit 195 o C (flushmounted ex sensor). exp V 10-6 m 3 /s P 10 7 Pa T o C range T o av C

5 Results and discussion The shear viscosity parameters (12), are µ =100,000 Pa s, n = 0.27, b = 0.03 o C -1. A log-log plot of the temperatureadjusted entry pressures vs. flo rate is shon in Fig s0 3. Fig. 3. Log-log plot of temperature corrected entry pressure (Pa) vs. flo rate (m 3 /s). The fitted lines have slope corresponding to n = 0.27, p = Points fall ell on a straight line for n ex = 18.5 rpm, and somehat less ell for n ex = 13.5 rpm. Note the highly expanded scale. The slope of the best fit line, hoever, is the same for both. The resulting poer la parameters are given in Table 4. Table 4. Extensional viscosity parameters for the to sets of experiments. T 0 = 200 o C µ Pa s p p e0 n ex = 13.5 rpm 0.837x n ex = 18.5 rpm 0.928x The difference beteen the consistency coefficient values, µ, at the to scre speeds is about 10%, probably e 0 indicating a small influence of processing history. Substitution of these values back into Eq. 3, together ith the experimental flo rates and temperatures, recovers the experimental pressures ithin 2% for n ex = 18.5 rpm, and 5% for n ex = 13.5 rpm, consistent ith the slightly less good fit visible in Fig. 3. In the folloing paragraphs, e first attempt to assess the applicability of the (modified) Binding analysis for the extensional viscosities, and then place them in the context of literature values. It is of interest to calculate the convergence angle θ = tan -1 (-dr(z)/dz), (this is sometimes called the half convergence angle - it is the angle beteen the cone all and the axis) predicted for free convergence using the parameters found here. Using Binding s (10) result (his Eq. 40) e find for the loest flo rate, 1.23x10-6 m 3 /s, θ = 28 o at the start of the cone, and 29.6 o at the capillary entrance. For the highest flo rate, 6.48 x10-6 m 3 /s, the corresponding values are 36.8 o and 37.3 o Thus for the higher flo rates the convergence may apparently be slightly constrained by the 30 o cone angle. The significance of this is unclear. Glomsaker et al. (2) in ork on upvc found no difference in entry pressure beteen a sharp (90 o ) convergence and a 45 o cone, though they do not report the free convergence angles applying. Cogsell

6 (14) recommended restricting the use of the converging flo method to angles belo 60 o. It seems unlikely that the imposed 30 o cone angle ill significantly influence the present results. The question of the boundary condition at the edge of the converging flo is also relevant. A zero velocity is assumed in the analysis used here. Though in many applications the zero value may often not apply strictly at the edges of the convergence profile, it is more likely to apply at the orifice (or capillary entrance), and hat happens in this region of maximum stresses and extension rate ill dominate the results. Hoever, upvc is knon to exhibit all slip in some circumstances, so the zero velocity condition may not apply there either in the present ork. The apparent deformation rates and stresses at the orifice ill vary depending on the form of the kinematics, the limiting cases being, on the one hand, shear-free sink flo, and, on the other, extension combined ith shear in the form of a radial profile as used in deriving Eq. 3. The reality is probably somehere in beteen. We calculate values at the orifice implied by the present results according to the formulae (11) corresponding to the different assumptions, see Tables 5 and 6. Table 5. Formulae for all shear rate and shear stress, and elongation rate and elongational stress at the orifice. Shear rate at the boundary, based on the developed tube 1+ 3n 4V flo profile γ = 3 4n π R Area averaged extension rate for unconstrained convergence, including shear Extension rate on the centre line for unconstrained convergence, including shear Extension rate for shear-free sink flo 0 n+ 1 4 µ s + n V ε = av 3 3( n+ 1) P 4n π R0 2(1 + 3 n) V dr ε = CL 3 1+ n π R dz 0 R0 2V dr ε = SF 3 π R dz 0 R0 n+ 1 Table 6. Calculated values of all shear rate and shear stress, and elongation rate and elongational stress at the orifice, ith an assessment of the relative importance of shear stresses, for lo and high flo rates. V = 1.23x10-6 m 3 /s V = 6.48x10-6 m 3 /s γ s -1 τ ( γ ) Pa x x10 5 ε s -1 τ ( ε ) Pa av av x x10 6 τγ ( ) 15% 15% τ ( γ ) + τ( ε ) av ε s -1 τ ( ε ) Pa CL CL x x10 6 τγ ( ) 10% 11% τ ( γ ) + τ( ε ) SF CL ε s -1 τ ( ε ) Pa x x10 6 SF Table 6 shos that all shear stress accounts for 10 to 15% of the sum of shear and elongational stresses, hich allos us to estimate an upper bound for the impact on extensional viscosity values of uncertainties about the boundary conditions. It is noticeable that the area averaged extension rate is several times smaller than the rate on the centre line of the flo ith a radial profile, and also several times smaller than the rate in shear free flo. The area averaged value as used by Cogsell (5) and also by Mackay and Astarita (11) in their approximate technique. It is questionable hether area averaging is representative; flo eighted averaging ould seem more appropriate, and ould give values closer to the centre line or shear free values. These ill tend to dominate the resulting pressure drops, and can be taken as more representative of the range of values over hich the present results are determined. There is relatively little reported data on extensional viscosities of PVC, and comparisons are complicated by ide variation in formulations and experimental procedures. Additionally, since the extensional viscosity obtained form entry flos is unlikely to be a fundamental material property, but rather an average value corresponding to the strain rates and

7 strain imposed in the measurements, results can be expected to be equipment and procedure dependent. The only published results for a profile grade upvc are those of Coates et al. (15). These are summarized in Table 7 ith poer la parameters estimated from the published plots. Table 7. Published extensional viscosity results for profile grade upvc. Reference Material Method ε range s -1 T 0 o C Coates et al. (15) Profile grade upvc, additives % not specified. Off-line In-line On-line µ Pas p p e0 2.0x x x As can be seen, significant differences ere found beteen results obtained off-line and by on-line and in-line rheometry (this latter refers to data obtained on a orking extrusion line). Our data, adjusted to 175 o C, is plotted for comparison ith that from (15) in Fig. 4. The extrapolation to a loer temperature should hoever be treated cautiously, because a change in the activation energy for flo has been observed, Glomsaker et al. (3), around 190 o C. The difference beteen the various results from (15) no doubt involves the influence of different thermo-mechanical histories and test conditions in the rheometers. While our results fall belo the off-line and on-line values, an extrapolation to loer extension rates ould sho close agreement beteen our values and the in-line results. It is notable that the on-line rheometer used a gear pump to control flo rate, thereby imposing mechanical orking on the upvc additional to that experienced in the extruder. This is perhaps hy best consistency ith the present results is shon by the in-line measurements (15) on material issuing directly from the extruder. Fig. 4. Comparison of present results ith those of Coates et al. (15). Conclusion The entry pressure drop method for axisymmetric converging flo has been used to determine engineering extensional viscosity for a profile grade upvc in on-line rheometry at fixed extruder operating points typical of production conditions. Particular attention as paid to monitoring the temperature homogeneity of melt floing to the rheometer, and to temperature variations ith flo rate. Significant variations ere found, and these ere taken into consideration in extracting the extensional viscosity values. The applicability of the analysis underlying the present results has been assessed under various headings. The use of a 30 o conical convergence, rather than the abrupt 90 o contraction of classical capillary rheometry, seems unlikely to constrain the flo significantly. Uncertainties about the velocity

8 condition applying at the boundary of the converging flo are estimated to affect results by no more than abut 15%. Mackay and Astarita (11) concluded that the validity of the energy minimization procedure for the convergence profile in their analysis requires the poer la indices for shear and extensional viscosity to be similar. That is the case in the present ork. Reasonable agreement is obtained ith the only other published results for extensional viscosity of a profile grade upvc, taking into consideration possible differences in test methods and the formulation of the materials. The present technique therefore appears to provide valid results for an engineering extensional viscosity that may be used in die flo simulations, provided that it is applied under the relevant processing conditions and mimics the strains and deformation rates existing in the die. Acknoledgements We gratefully acknoledge financial support from EPSRC under the project Computer Aided Optimisation of Extrusion Die Design (EPSRC GR/M 95820), and the co-operation of Kömmerling Kunststoff GmbH. References 1. J. H. Ettinger, J. Sienz, J. F. T. Pittman, A. Polynkin, Structural and Multidisciplinary Optimization, 2004, 28, T. Glomsaker, E. L. Hinrichsen, F. Irgens, P. Thorsteinsen, Rheol. Acta, 2000, 39, T. Glomsaker, E. L. Hinrichsen, A. Larsen, P. Thorsteinsen, Plast. Rubber Comp., 1999, 28, I. Szarvasy, J. Sienz, J. F. T. Pittman, E. Hinton, Int. Poly. Proc., 2000, 15, F. N. Cogsell, Poly. Eng. Sci., 1972, 12, P. R. Schunk, L. E. Scriven, J. Rheol., 1990, 34, W. N. Song, Z. M. Xia, (1994). J non-netonian Fluid Mech., 1994, 53, E. Mitsoulis, J. Vlachopoulos, F. A. Mirza, Poly. Eng. Sci., 1984, 25, B. Debbaut, M. J. Crochet, J. non-netonian Fluid Mech., 1988, 30, D. M. Binding, J. non-netonian Fluid Mech., 1988, 27, M. E. Mackay, G. Astarita,.J. non-netonian Fluid Mech., 1997, 70, H. Ettinger, R. Haase, J.F.T. Pittman, J. Sienz, Proc. PPS 21, Leipzig, 2005, paper J. F. T. Pittman, H. Mahmoudzadeh, Poly. Engng Revies, 1983, 3, F. N. Cogsell, J. non-netonian Fluid Mech., 1978, 4, P. D. Coates, A. L. Kelly, R. M. Rose, M. Woodhead, In-Process Entry Pressure Measurements in Extrusion and Injection Moulding, Polymer Insights Series, IRC in Poly. Sci. and Tech., University of Bradford, UK, 1998.