MODELING THREE-DIMENSIONAL NON-NEWTONIAN FLOWS IN SINGLE- SCREW EXTRUDERS

Transcription

1 MODELING THREE-DIMENSIONAL NON-NEWTONIAN FLOWS IN SINGLE- SCREW EXTRUDERS Christian Marschik Wolfgang Roland Bernhard Löw-Baselli Jürgen Miethlinger Johannes Kepler University Institute of Polymer Extrusion and Compounding Altenbergerstraße Linz Austria Abstract Modeling the multidimensional non-newtonian flow of shear-thinning polymer melts in single-screw extruders generally requires the use of numerical methods We present a heuristic approach to predicting the threedimensional fully developed isothermal flow of powerlaw fluids in single-screw extruders that avoids complex and time-consuming numerical simulations By applying a heuristic optimization algorithm we approximate numerical results obtained from a comprehensive parametric design study yielding an analytical relationship for the output-pressure gradient relationship depending on four independent parameters: (i) height-towidth ratio (h/w) (ii) pitch-to-diameter ratio (t/d b ) (iii) power-law index (n) and (iv) dimensionless pressure gradient in the down-channel direction ( pz ) The new approach is demonstrated to provide a close approximation to numerical solutions Introduction Numerous theoretical studies have analyzed the meltconveying zone of single-screw extruders both analytically and numerically The most widely used model commonly referred to as the traditional pumping model was originally proposed by Rowell and Finlayson [1] and later expanded by Tadmor and Klein [2] In this early work the melt-conveying zone was modeled as a straight rectangular flow channel covered by a moving flat plate The polymer melt was assumed to be Newtonian and thus the model provided practicable analytical solutions for the flow rate in the form of two independent terms: (i) a drag flow and (ii) a pressure flow The applicability of the traditional pumping model is limited It is well known that polymers exhibit a non- Newtonian flow behavior causing substantial deviations from the original modeling theory For this reason numerical techniques have been used extensively to examine the effect of the non-newtonian flow characteristics on the pumping capacity of single-screw extruders Rotem and Shinnar [3] presented numerical solutions for a one-dimensional non-newtonian flow between two parallel plates Griffith [4] obtained numerical results for a two-dimensional non-newtonian flow in an infinitely wide rectangular channel Spalding et al [5] as well as Spalding and Campbell [6] simulated a three-dimensional non-newtonian flow in a straight rectangular screw channel From a practical viewpoint numerical analyses tend to be time-consuming making the use of fast and less complex analytical models more desirable Rauwendaal [7] refined the original theory by introducing correction factors for the non-newtonian flow behavior of polymer melts These factors however omit the ratelimiting influence of the screw flights Recently Campbell and Spalding [8] have proposed an empirical relationship for adjusting the drag flow rate in the traditional pumping model based on three-dimensional non-newtonian studies Since the pressure flow rate remains unmodified the approach continues to leave some matters unaddressed We present a heuristic method for estimating the three-dimensional output-pressure gradient relationship of single-screw extruders Instead of providing correction parameters to the traditional pumping model we propose a new relationship for the net flow in the melt-conveying zone Traditional Pumping Model The traditional pumping model is based on the flatplate approximation with moving barrel in which the helical screw channel is unwound from the screw to yield a straight rectangular flow channel located on a flat plate namely the barrel surface Hence the curvature of the screw channel is ignored Further the kinematic conditions are reversed compared to the real process assuming the screw to be stationary and the barrel surface to be moving at a constant velocity v b at an angle b with respect to the down-channel direction A schematic representation of the flat-plate system with moving barrel is shown in Fig 1 Sun and Rauwendaal [9] examined the effect of kinematic reversal and showed that the flat-plate model with stationary screw and moving barrel accurately predicts the actual process based on a cylindrical coordinate system Given the barrel diameter D b and the screw pitch t the helix angle b is obtained from: SPE ANTEC Anaheim 2017 / 1125

2 tan b t D Furthermore the velocity of the barrel surface is decomposed into a component in the cross-channel direction v bx and a component in the down-channel direction v bz : b bx b b (1) v D N sin (2) v D N cos (3) bz b b where N is the screw speed Applying the down-channel pressure gradient p/ z needed to force the melt through the die the net flow rate results in: 3 iwhvb z iwh p V F F d p 2 12 z where i is the number of parallel flights w and h are the average width and the depth of the screw channel respectively and is the shear viscosity F d and F p are shape factors for the rate-limiting effect of the screw flights A detailed derivation is outlined in [2] Since the traditional pumping model deals with Newtonian fluids the net flow results from a linear superposition of a drag flow and a pressure flow as shown in Eq 4 Figure 1 Flat-plate system: The barrel surface moves at constant velocity v b whereas the screw is stationary Power-Law Fluid Flows Taking the non-newtonian flow behavior of polymer melts into account increases the complexity of the flow analysis as the governing transport equations must be solved numerically Only a few theoretical studies have proposed approximation methods for calculating the non- Newtonian flow of polymer melts in single-screw extruders without the need for numerical techniques Generally these analyses use a power-law equation to describe the shear-thinning nature of polymer melts: K n1 (4) (5) where K is the consistency index and n is the power-law index Potente [10] presented approximate equations for estimating the conveying characteristics of power-law fluids in straight rectangular screw channels Since the equations exhibit undefined regions their practical usefulness is limited Rauwendaal [7] modified the traditional pumping model as given in Eq 4 by replacing the shape factors with correction parameters for the non- Newtonian flow behavior These parameters were derived from numerical studies analyzing the flow of power-law fluids in infinitely wide screw channels and applied only to helix angles ranging from 15 to 25 For this reason Pachner and Miethlinger [11 12] proposed a generalized melt-conveying model for power-law fluids All of these studies are based on a two-dimensional modeling framework investigating the flow of infinitely wide screw channels Thus the reducing effect of the flight flanks on the flow rate is generally omitted Modeling We consider a straight rectangular screw channel of infinite length as schematically shown in Fig 1 In order to describe the kinematic process conditions we apply the flat-plate approximation with moving barrel The leakage flow over the screw flights and thermal effects are ignored We start the analysis by simplifying the transport equations governing the three-dimensional extruder flow: v 0 t v v v p g t where Eqs 6 and 7 are the conservation equations of mass and momentum respectively (see [13] for a detailed description) The following assumptions are made: (a) the flow is steady in time (b) the flow is laminar and isothermal (c) the fluid is incompressible (d) gravitational forces are ignored and (e) there is no slip at the wall Furthermore we consider a fully developed flow in the down-channel direction and thus the velocity vector v = (v x v y v z ) T is a function of x and y only Additionally we apply a power-law model to describe the shear-thinning nature of the polymer melt as shown in Eq 5 Since the Reynold s number in extruder flows is typically very small inertia forces are omitted Applying these assumptions we obtain a coupled system of partial differential equations: v v x y 0 x y (6) (7) (8) SPE ANTEC Anaheim 2017 / 1126

3 volume flow rate v p xx yx 0 x x y p xy yy 0 y x y p xz yz 0 z x y (9) (10) (11) Since we assume a fully developed flow in the downchannel direction the shear stress is independent of the z- coordinate Eqs 8-11 depend on four unknown quantities: the fluid velocities in each spatial direction v x v y and v z and the local pressure distribution p By rewriting the flow equations in a dimensionless form it can be shown that the following influence parameters can be varied independently: (i) the height-to-width ratio (h/w) (ii) the pitch-to-diameter ratio (t/d b ) (iii) the power-law index (n) and (iv) the dimensionless pressure gradient in the down-channel direction ( pz ) given by: p z p h z 6 K v ' 1n n b z with ' p z p z (12) In contrast to previous two-dimensional analyses our three-dimensional modeling framework considers the ratelimiting influence of the screw flights via the crosssection ratio h/w Simulation We analyzed the three-dimensional creeping flow of power-law fluids in single-screw extruders by carrying out a comprehensive numerical parametric design study To this end we solved the governing flow equations shown in Eqs 8-11 using a commercial software package based on the finite-volume method Varying all characteristic input parameters affecting the extruder flow (h/w t/d b n pz ) we simulated a total number of set-ups derived from theory of similarity The scope of variation is given in Tabs 1 and 2 Table 1 Values of h/w t/d b and n used in the simulation quantity min max increment h/w t/d b n The dimensionless pressure gradient in the downchannel direction was varied based on the power-law index For each n the interval was divided into 30 equal segments Our analysis focused on pressure-generating metering sections which are usually found in smooth-bore single-screw extruders Thus only positive values of the dimensionless pressure gradient were applied in the simulations Table 2 Values of pz used in the simulation n min max increment For each design point the dimensionless volume flow rate v was evaluated numerically: 2V v i w h v b z (13) At the beginning the simulation set-up was validated for the Newtonian case (with n = 1) by comparing the numerical solutions for the volume flow rate with the analytical results obtained from Eq 4 Simulation Results Our numerical parameter study based on simulation set-ups provides solutions for the dimensionless volume flow rate v as a function of the height-to-width ratio h/w the pitch-to-diameter ratio t/d b the power-law index n and the dimensionless pressure gradient in the down-channel direction pz In the following diagrams each point represents the simulation result for one specific design point h/w = 0 ; t/d b = 12 pressure gradient pz n = 09 n = n = 07 n = n = 05 n = n = 03 n = Figure 2 Dimensionless throughput versus dimensionless pressure gradient for h/w = and t/d b = 1 SPE ANTEC Anaheim 2017 / 1127

4 volume flow rate v volume flow rate v volume flow rate v Fig 2 illustrates the effect of the shear-thinning behavior of the polymer melt on the conveying characteristics of a screw with h/w = and t/d b = With decreasing power-law index the output-pressure gradient relationship becomes increasingly non-linear Especially at low power-law indices substantial deviations from the linear output-pressure gradient behavior known from Newtonian fluids are evident Furthermore given the same pressure gradient the higher the power-law index of the polymer melt becomes the lower the dimensionless volume flow rate n = 03 ; h/w = pressure gradient pz t/d b = t/d b = t/d b = t/d b = 12 t/d b = 14 t/d b = 16 t/d b = 18 t/d b = 20 Figure 3 Dimensionless throughput versus dimensionless pressure gradient for n = 03 and h/w = Fig 3 shows the influence of the pitch-to-diameter ratio on the pumping capacity of a screw with h/w = assuming a power-law fluid with n = 03 It can be seen that the dimensionless volume flow rate increases with decreasing t/d b -ratio (and thus helix angle) over the entire range of pressure gradients This result is strongly related to the down-channel velocity of the moving barrel surface which is higher at smaller helix angles as shown in Eq 3 h/w = 5 h/w = 010 h/w = 015 h/w = 0 h/w = 5 h/w = 030 h/w = 035 h/w = 0 h/w = 5 h/w = 050 The reducing effect of the screw flights on the dimensionless volume flow rate for a power-law fluid with n = 03 is shown in Fig 4 It is obvious that the ratelimiting influence of the screw flights is more pronounced at greater h/w ratios For small h/w ratios the results can be compared to the two-dimensional flow analysis presented in [11 12] in which the screw channels were assumed to be infinitely wide Heuristic Model To derive a relationship that predicts the numerical results of the parametric design study we applied a heuristic optimization algorithm that uses symbolic regression based on genetic programming Minimizing the mean squared deviation between the simulation results and the estimated solutions we sought an analytical approximation for the dimensionless flow rate as a function of the characteristic influence parameters h/w t/d b n and pz Analyzing the simulation results of set-ups we identified an empirical relationship for the three-dimensional output-pressure gradient behavior in the form of a third-order polynomial function: 2 3 a a a a v 0 1 p z 2 p z 3 p z (14) where a 0 to a 3 are the coefficients of the polynomial function summarized in the appendix Additionally the approximate solution depends on the 38 constants listed in Tab 3 Considering all design points the coefficient of determination (R 2 ) was calculated to be which demonstrates that our new approach predicts the numerical results accurately h/w = 015 ; t/d b = n = 09 (sim) n = 09 (appr) n = (sim) n = (appr) n = 07 (sim) n = 07 (appr) n = (sim) n = (appr) n = 05 (sim) n = 05 (appr) n = (sim) n = (appr) n = 03 (sim) n = 03 (appr) n = (sim) n = (appr) n = 03 ; t/d b = pressure gradient pz Figure 4 Dimensionless throughput versus dimensionless pressure gradient for n = 03 and t/d b = 1 12 pressure gradient p Figure 5 Comparison of numerical solutions and approximated results for h/w = 015 and t/d b = Comparisons between simulation results and approximated solutions for three different parameter setups are plotted in Figs 5 to 7 It can be seen that our new output-pressure gradient model approximates the SPE ANTEC Anaheim 2017 / 1128

5 volume flow rate v volume flow rate v numerical results well It is therefore possible to quickly predict the three-dimensional conveying characteristics of power-law fluids in smooth-bore single-screw extruders without the need for numerical simulations h/w = 0 ; t/d b = 12 pressure gradient pz n = 09 (sim) n = 09 (appr) n = (sim) n = (appr) n = 07 (sim) n = 07 (appr) n = (sim) n = (appr) n = 05 (sim) n = 05 (appr) n = (sim) n = (appr) n = 03 (sim) n = 03 (appr) n = (sim) n = (appr) Figure 6 Comparison of numerical solutions and approximated results for h/w = and t/d b = 1 h/w = 035 ; t/d b = 15 n = 09 (sim) n = 09 (appr) n = (appr) n = (sim) n = 07 (sim) n = 07 (appr) n = (sim) n = (appr) n = 05 (sim) n = 05 (appr) n = (sim) n = (appr) n = 03 (sim) n = 03 (appr) n = (sim) n = (appr) predicts the simulation results accurately yielding a coefficient of determination (R 2 ) of In contrast to previous two-dimensional analyses our approach considers the rate-limiting influence of the screw flights Furthermore it enables fast and stable computational modeling without the need for cost-intensive and timeconsuming simulations Appendix Table 3 Coefficients of the polynomial function coefficient value coefficient value c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c pressure gradient pz Figure 7 Comparison of numerical solutions with approximated results for h/w = 035 and t/d b = 15 Conclusion We have developed an analytical model for estimating the three-dimensional output-pressure gradient behavior of power-law fluids in single-screw extruders with pressure-generating metering zones The analysis is based on an extensive numerical design study in which the characteristic input parameters affecting the extruder flow namely h/w (height-to-width ratio) t/d b (pitch-todiameter ratio) n (power-law index) and pz (dimensionless pressure gradient in down-channel direction) were varied A total number of set-ups derived from theory of similarity were simulated By applying a heuristic optimization algorithm we approximated the numerically evaluated output-pressure gradient relationship by means of a third-order polynomial function An error analysis showed that our new approach d h t and d (15) w D 1 2 a 0 v0 (16) f c c c p0 37 p0 37 p0 (17) a f 7c 4c 3c p p0 v c c 37 p0 37 p0 v1 c c 37 p0 37 p0 v2 b (18) SPE ANTEC Anaheim 2017 / 1129

6 c a f c c c p0 37 p0 v c 2c 37 p0 37 p0 v1 3 3 c 37 p0 37 p0 v a f 3c c c p p0 v0 2c 4 c 2 2 c 37 p0 37 p0 v1 2 2 c 37 p0 37 p0 v2 cn 1 cd v0 0 1 c d c c d c d c c nd c d d c n c d c v c d c d / n c n c d n c n c d c n c n c d c d c c n c d c n c nd c c v c d c n c p c d c nc nd c Acknowledgements c 10 (19) (20) (21) (22) (23) (24) 4 RM Griffith Ind Eng Chem Fundam (1962) 5 MA Spalding GA Campbell F Carlson and K Nazrisdoust SPE ANTEC Tech Papers (2006) 6 MA Spalding and GA Campbell SPE ANTEC Tech Papers (2008) 7 C Rauwendaal Polymer Extrusion 5 th Ed Hanser Publishers Munich MA Spalding and GA Campbell SPE ANTEC Tech Papers (2011) 9 J Sun and C Rauwendaal SPE ANTEC Tech Papers (2002) 10 H Potente Rheol Acta (1983) 11 S Pachner and J Miethlinger A Generalized 2- Dimensional Output Model of the Metering Section of Single-Screw Extruders presented at: PPS Graz S Pachner B Loew-Baselli M Affenzeller and J Miethlinger A Generalized 2-Dimensional Output Model of the Polymer Melt Flow in Single-Screw Extrusion submission accepted: International Polymer Processing (2016) 13 RB Bird WE Stewart and EN Lightfood Transport Phenomena 2 nd Ed Wiley & Sons Inc New York 2002 This work was supported by the Austrian COMET K2 program of the Linz Center of Mechatronics (LCM) and was funded by the Austrian federal government and the federal state of Upper Austria References 1 HS Rowell and D Finlayson Engineering (1922) 2 Z Tadmor and I Klein Engineering Principles of Plasticating Extrusion Van Nostrand Reinhold Co New York Z Rotem and R Shinnar Chem Eng Sci (1961) SPE ANTEC Anaheim 2017 / 1130