3-D ANALYSIS OF FULLY FLIGHTED SCREWS OF CO-ROTATING TWIN SCREW EXTRUDER

Transcription

1 3-D ANALYSIS OF FULLY FLIGHTED SCREWS OF CO-ROTATING TWIN SCREW EXTRUDER A. Lawal, S. Railkar, and D. M. Kalon Highl Filled Materials Institute Stevens Institute of Technolog Hoboken, NJ 73 Abstract A new mathematical model of the flow and heat transfer occurring in the full-flighted regular flighted screw elements of the co-rotating twin screw etruder is developed. The method avoids the unwrapping of the screws and solves the conservation equations using a realistic replication of the actual channel geometr. The numerical solution of the conservation equations is accomplished using the three dimensional Finite Element Method. The technique allows a wide range of processing features including the effects of the gap thickness between the barrel and the screws to be investigated. Here tpical isothermal results are presented for two generalied Newtonian fluids i.e., Bingham and Ostwald-de Waele "Power-Law" fluids. Introduction The co-rotating twin screw etruders are onl available commerciall in the full intermeshing mode and are used widel in the polmer processing industr for compounding, blending, reactive processing etc. The versatilit of the machine to some etent arises from the modular nature of the etruder since a number of machiner producers offer co-rotating twin screw etruders with interchangeable barrel and screw sections. Thus, in the age of fleible manufacturing where manufacturers are pressured to move from one product line to another using the same capital equipment the versatile twin screw etrusion technolog plas a significant role. However, each time a new screw and barrel configuration is assembled the etruder becomes a new machine. Therefore, optimiation and scale-up continue to remain as significant issues for most companies. Mathematical modeling of the co-rotating twin screw etrusion process is an obvious asset which can be utilied to predict apriori which screw configuration needs to be used, which operating conditions need to be emploed and the processabilit characteristics of new formulations as the are being developed. There are two principal methods of mathematical modeling of the twin screw etrusion processing. In the first method Finite Element Method is used in conjunction with a mesh which duplicates accuratel the channel geometr without unwrapping. This technique which our own research group has championed [1-5] in spite of the computational difficulties involved, has been successful in describing the flow and heat transfer occurring in such comple shaped screw elements as the lenticular kneading discs [1-7]. In the second method the channels are unwrapped and the conservation equations are solved emploing either FEM [8-12] or methods like the Flow Analsis Method [13,14]. For the regular flighted screw elements of the co-rotating twin screw etrusion process the popular method has been the unwrapping of the screw channel [8-12]. However, the methods which are applicable to the solution of the conservation equations in conjunction with the constitutive equations for generalied Newtonian fluids (onl temperature and shear rate dependence of the shear viscosit allowed, no dependence on the histor of the deformation, no time effects, elasticit or etensional flows) are equall applicable for the modeling of the full-flighted screw elements, which are important components of the co-rotating etrusion technolog [1-5]. In this paper we will demonstrate the use of the Finite Element Method in the mathematical modeling of the thermo-mechanical histor of processing occurring in the regular flighted screw elements of the co-rotating twin screw etrusion process. Materials Two tpes of generalied Newtonian fluids were used for the demonstration of the method. The first material was a Bingham plastic and was utilied as the material of a stud in a scale-up stud. The densit, specific heat capacit and thermal-conductivit of the viscoplastic material were, 155. kg/m 3, 151 J/(kg-K) and.35 W/(m-K). The ield stress, was Pa and the shear rate consistenc inde was 46 Pa-s. The

2 one dimensional form of Herschel-Bulkle constitutive equation for simple shear flow is given as: 12 m Ý 12 n for 12 (1a) Ý 12 for 12 (1b) where m and n are the consistenc inde and the shear rate sensitivit (power law inde) parameters. For Bingham fluid n=1. A modified form of the Bingham fluid behavior was used in the simulations as follows [15]: m Ý n 1 (1 ep( n b )) Ý ep( c' (T Ý To )) (2) here Ý is the second invariant of the rate of deformation tensor, nb is an eponent which provides non-singular variation of the shear stress with shear rate and c' is the temperature coefficient of viscosit. To is the entrance temperature and is also taken as the reference temperature. Furthermore, the power law (Ostwald-de Waele) model was used to probe the effects of changes in the shear rate sensitivit inde (power law inde) n on the characteristic operating curve of the regular flighted screw section of the etruder. Analsis The co-rotating screw elements, used for the purpose of modeling, were double flighted and with relativel small flight width as compared with the channel width as shown in Figure 1. A finite element mesh of the geometr is also provided. As consistent with our earlier work a single mesh is used and we differentiate between the fluid and the screw as a function of time as the screws rotate and as the boundar conditions change [1-5]. Flow Equations The inertia effects can be ignored, because of the highl viscous nature of the materials generall processed in continuous processors and the creeping nature of the resulting flow. Here, we consider the isothermal creeping flow of an incompressible fluid in a twin screw etruder with two-tipped co-rotating screw elements as shown in Figure 1. The equations of conservation of momentum and mass are: p 2 (3a) (3b) where p is the pressure, is the velocit. At the barrel walls and the screw surface, the no-slip condition is used. barrel (4a) screw r (4b) whereis angular velocit and r is position vector of the boundar of the screw with respect to its center. As boundar conditions at the entrance and eit planes of the flow domain the traction conditions can be used [16]. The penalt/galerkin Finite Element Method is emploed in the discretiation wherein the continuit equation is used as a constraint for the equations of conservation of momentum, and the pressure p is approimated b using a large positive number p (penalt parameter) such that [1-5]: p p v v v (5) When Eq. 5 is substituted into Eq. 3, the equations of momentum in component form, become: p v v v 2 v v v v v p v v v v v 2 v v v p v v v v v v v 2 v (6a) (6b) (6c)

3 In dimensionless forms of these equations the screw radius, Rs, screw rotational speed,, and the volumetric flow rate, Q, are used. For FEM solution of the 3-D momentum equations, the domain is discretied b the trilinear brick elements. The discretied equations take the matri form: A f (7) where A is the stiffness matri, f is a force vector resulting from source terms, and the nodal unknowns are recovered from the vector. Results and Discussion A three dimensional finite element mesh of the full-intermeshing full flighted screw elements of the corotating twin-screw etruder was prepared. The finite element mesh has eight noded isoparametric brick elements and is shown in Figure 1. The full-flighted screw elements are considered to be a stack of infinitesimall small thickness kneading blocks staggered at almost ero stagger. There are man advantages of modeling the screw elements in three-dimensions over the unwrapped model. One of the most obvious ones is that in the threedimensional analsis the clearance between the barrel and the screws can be incorporated accuratel and can be treated as a separate parameter. This is especiall important in the evaluation of the screw and barrel wear effects and the resulting deterioration of the processing capacit. Here, for the purpose of studing the effect of clearance on the flow characteristics clearances from.2 % of the screw radius to as high as 2% of the screw radius were used. The mesh is graded in the transverse direction depending on the barrel and twin-screw clearance as shown in Figure 1. This mesh encompasses the flow domain from the screw-root to the barrel surface. The mesh covers one lead of the screw aiall. Here a 3.46" lead screw element with a heli angle of o was used for the simulation. The complete threedimensional mesh of the co-rotating self-wiping twin screws has nodes and 192 elements. There are simultaneous equations in the model and the global stiffness matri contains 22 million non-ero entries. At a given position of the screw elements, the global stiffness matri must be inverted to recover the unknown vector. On three parallel processors this model takes about 73 min per iteration. Due to the shear rate dependent viscosit the solution of the nonlinear problem takes several iterations to converge. To verif that the numerical solution is not dependent on the mesh densit and tpe of grading used meshes with differing numbers of nodes were utilied i.e., 12432, and over one lead of aial distance of 87.9 mm. The computations for the Newtonian fluid generated volumetric flow rates which were within 1-8% of each other at constant pressuriation rate indicating that the mesh-dependence was negligible. Once the number of cross-sections in the mesh are fied, the program uses a single mesh for the purpose of simulation as the twin screws rotate. The positions of the screws at an time are known and it is possible to identif the elements, which belong to the solid and the fluid for a given clearance between the barrel and the screw [1-5]. Tpical simulation results for the Bingham plastic fluid using a clearance between the screws and the barrel surface of.8 % of the screw radius are shown in Fig. 2. Here the dimensionless -velocit distribution indicates both positive and negative values. Integrating this velocit distribution over each cross-sectional area generates the total mass flow rate in the etruder. Net, the clearance is varied and the pressuriation versus mass flow rate histor is obtained. as shown in Figure 3. As the clearance is increased the pressuriation abilit of the etruder drops for a constant volume flow rate. Conversel, for a given pressuriation rate, as the clearance is increased, total mass flow rate is reduced. Tpical results are also provided in Figure 4 for the power law tpe generalied Newtonian fluid as a function of the power law inde, n. The results are given in the dimensionless form to allow comparisons with analsis results where the channel was unwrapped [8-11]. The total flow rate, Q* is given as a function of the aial distance (not helical). Q * Q 2R s Cos b WH and dp * d * H n H dp (9) 2R s Cos b m d where, b is the heli angle of the screw at the tip of the flights, W and H are the channel width and channel depth, and dp/d is given along the aial and not helical channel direction and Q is the total flow rate. It should be noted that in our case there are three channels in parallel and the heli angle at the tip of the screw flights is degrees. Comparison of these results to earlier published two dimensional FEM (8)

4 solutions using unwrapping of the screw elements [8-11], reveals good general agreement between the 2-D and 3-D results eplaining the reason as to wh the 2-D approach has been so popular at least for screw geometries where the flight width is small in comparison to the channel width of the screw element. However, the 2-D approach is epected to break down for cases where the ratio of the flight width over the channel width becomes appreciable and the intermesh and the relative velocit between the two screws can no longer be ignored. Conclusions A three-dimensional finite element method based source code is presented to model the full-flighted regular-flighted" screw elements of the fullintermeshing and co-rotating twin screw etrusion process. The mathematical model is especiall appropriate for cases where the unwrapping of the channel is no longer acceptable including relativel high ratios of flight width over the channel gap. Furthermore, the effects of various geometrical parameters including the clearance between the screws and the barrel can be investigated with ease. Together with our earlier presented methods for mathematical modeling of the flow and heat transfer occurring in kneading disc section of the etruder [1-5] the presented method would allow for the first time the simulation of the entire continuous processing operation using 3-D FEM. 7. H. Yang and I. Manas-Zlocower, Polm. Eng. Sci., 32, (1992). 8. C. D. Denson and B. K. Hwang, Polm. Eng. Sci., 2, 965 (198). 9. D. M. Kalon, A. D. Gotsis, U. Yilmaer and C. Gogos, Sangani, Aral, and Tsenoglou, Adv. Polm., Techn., 8, (1988). 1. D. M. Kalon, A. D. Gotsis, C. G. Gogos and C. Tsenoglou, SPE ANTEC Technical Papers, 34, 64 (1988). 11. R. Lai-Fook, A. Senouci, A. C. Smith and D. P. Isherwood, Polm. Eng. Sci., 29, 433 (1989). 12. A. Lawal and D. M. Kalon, Numerical Heat Transfer, 26, 13 (1994). 13. W. Sdlowski and J. L. White, Adv. Polm. Tech., 7, 177 (1987). 14. Y. Wang and J. L. White, J. Non-Newtonian Fluid Mech., 32, 19 (1989). 15. T. C. Papanastasiou, J. Rheol., 31, 385 (1987). 16. A. Lawal, S. Railkar, D. M. Kalon, SPE ANTEC Technical Papers, 42, (1996) KEY WORDS: Etrusion, Twin-screw, Modeling, Simulation ACKNOWLEDGEMENT We are grateful for the partial support of ONR under contract # N REFERENCES 1. A. D. Gotsis, and D. M. Kalon, SPE ANTEC Technical Papers, 35, (1989). 2. A. D. Gotsis, Z. Ji and D. M. Kalon, SPE ANTEC Technical Papers, 36, 139 (199). 3. A. Lawal and D. M. Kalon, SPE ANTEC Technical Papers, 39, (1993). 4. A. Lawal and D. M. Kalon, J. Appl. Polm. Sci, 58, (1995) 5. A. Lawal and D. M. Kalon, Polm. Eng. & Sci, 35, (1995) 6. A. Kiani and H. J. Samann, SPE ANTEC Technical Papers, 39, 2758 (1993)

5