Modeling and Simulation of Viscoelastic Flows in Polymer Processing with Integral Constitutive Equations Evan Mitsoulis

Transcription

1 Modeling and Simulaion of Viscoelasic Flows in Polymer Processing wih Inegral onsiuive Equaions Evan Misoulis AMP - R&D ompuer-aided Maerials Processing Rheology & Design School of Mining Engineering and Meallurgy Naional Technical Universiy of Ahens Ahens, Greece

2 Molecular Srucure Processing Produc (Srucure-Properies)

3 A. GOVERNING FLOW EQUATIONS (onservaion equaions) Α1. Equaion of conservaion of mass v = 0 Α2. Equaion of conservaion of momenum = ρv v = p + τ+ ρg Α3. Equaion of conservaion of energy 2 ρ p v T = k T + τ : v ****************************************************************** Β.. RHEOLOGIAL MODEL (onsiuive equaion) = τ = f(, v v) =

4

5 20:1 Axisymmeric onracion LDPE HDPE τ w = 0.1 MPa Bagley and Birks, J. Appl. Phys., 1960

6 INTEGRAL ONSTITUTIVE EQUATIONS τ () = M(-',I,I ) (') + M(-',I,I ) (') d' where: M, M = maerial propery funcions 1 2 = presen ime ' = pas ime I, I firs invarians = auchy Green ensor 1 = = Finger srain ensor

7 Paricle Tracking

8 INTEGRAL ONSTITUTIVE EQUATIONS K-BKZ EQUATION Kaye (1962), Bernsein-Kearsley-Zapas (1963) τ () = U ( ', I, I ) I U ( ', I, I ) ( ') I ( ') d' Facorized K-BKZ proposed by Whie and Tokia (1967) V( I, I ) V( I, I ) τ () = m ( ') ( ') ( ') d' I I where: U = poenial funcion m( ') = ime dependen funcion V = srain dependen funcion I, I firs invarians = auchy Green ensor 1 = = Finger srain ensor

9 The Faces behind he Names From Rheology: An Hisorical Perspecive, R.I. Tanner & K. Walers, Elsevier, 1998.

10 INTEGRAL ONSTITUTIVE EQUATIONS RIVLIN-SAWYERS EQUATION Rivlin and Sawyers (1971) τ () = W ( ', I, I ) ( ') + W ( ', I, I ) ( ') d' Facorized R-S 1 1 τ () = m ( ') H( I, I ) ( ') + H( I, I ) ( ') d' where:w = no he poenial funcion m( ') = ime dependen funcion H = srain dependen funcion 2 I, I firs invarians = auchy Green ensor 1 = = Finger srain ensor

11 INTEGRAL ONSTITUTIVE EQUATIONS Special ases of he Facorized RIVLIN-SAWYERS EQUATION Wagner Model (1980) M a k = ' exp( ), H = exp I + I k ( ) λk λk ( ) β α 1 1 α 3 M Papanasasiou-Scriven-Macosko (PSM) Model (1983) a k = ' exp( ), H = α k λ k λ k α 3 + βi 1 + ( 1 β) I where: λ a k k = relaxaion ime = relaxaion modulus α, β = maerial consans I, I firs invarians = auchy Green ensor 1 = = Finger srain ensor

12 Relevan Lieraure Kaye, A. (1962) Non-Newonian Flow in Incompressible Fluids, Par I: A General Rheological Equaion of Sae; Par II: Some Problems in Seady Flow, Noe No. 134, ollege of Aeronauics, ranford, UK. Bernsein, B., Kearsley, E.A., and Zapas, L. (1963) A Sudy of Sress Relaxaions wih Finie Srain, Trans. Soc. Rheol., 7, Papanasasiou, A.., Scriven, L.E., and Macosko,.W. (1983) An Inegral onsiuive Equaion for Mixed Flows: Viscoelasic haracerizaion, J. Rheol. 27, Olley, P. (2000) An adapaion of he separable KBKZ equaion for comparable response in planar and axisymmeric flow, J. Non-Newonian Fluid Mech., 95,

13 Numerical Mehods (2-D)( u-v-p formulaion (primiive variables) u-v-p formulaion (SI) mixed formulaion (u-v-p-τ) (EVSS) enhanced mixed formulaion (u-v-p-τ-g) (DEVSS-G) many ohers (EEME, SI, EVSS/SUPG, EVSS/SU, AVSS/SI, AVSS/SUPG, ec.)

14 Numerical Mehods (3-D)( 3-D x-y-z-p- formulaion (primiive variables) Lagrangian Inegral Mehod (LIM) H.K. Rasmussen, Time-dependen finie-elemen mehod for he simulaion of hree-dimensional viscoelasic flow wih inegral models, JNNFM, 84, (1999).

15 Relaive Difficuly of Pom-Pom vs. K-BKZ K Models (or differenial vs. inegral models) 2-D dof / elemen K-BKZ Pom-pom u (serendipiy) 9 u (Lagrangian) 8 v 9 v 4 p 4 p 8 T 9 T 20 (28) 22 (31) planar 4 (τ xx, τ yy, τ xy ) = 12/mode x 8 = 96 4 (g xx, g yy, g xy )= 12/mode x 8 = (55) 214 (223) axisymmeric 4 (τ rr, τ zz, τ rz, τ θθ ) = 16/mode x 8 = (g rr, g zz, g rz, g θθ )= 16/mode x 8 = (63) 278 (287) dof/elem/8 modes

16 Parameers for Fiing Daa for he IUPA-LDPE Mel Pom-Pom K-BKZ (PSM) k λ k (s ) a k (Pa) q k (τ b/ τ s ) k x x x x x x x x k λ k (s ) a k (Pa) α k β k x x x x x x x x Inkson, McLeish, Harlen, Groves, J. Rheol. (1999) Barakos and Misoulis, J. Rheol. (1995)

17 Elongaional Viscosiy Pom-Pom K-BKZ (PSM) N 1

18 Simulaions LDPE Mels γ α = 84 s -1, τ w = 0.1 MPa Experimens

19 HDPE Mel ( a 150 o ) K-BKZ Model Original Fi A non-linear regression analysis is performed on he K-BKZ inegral consiuive equaion o deermine he parameers in he model.

20 HDPE Mel ( a 150 o ) K-BKZ Model Parameers Original Fi k λ k (s ) a k (Pa) α k β k x x x x x θ = 0 P. Wood-Adams, PhD Thesis, McGill Universiy, 1999.

21 Simulaions HDPE Mels γ α = 248 s -1, τ w = 0.1 MPa Experimens

22 NON-ISOTHERMAL ONSTITUTIVE EQUATION TIME-TEMPERATURE TEMPERATURE SHIFTING ONEPT τ ξ ( ) 1 ( ) = 1 θ MH ( ξ) ('( ξ')) + θ( ξ) ('( ξ')) dξ' M ak ξ = ξ' α exp( ), H = k λ k λk α 3 + βi + ( 1 β) I where: λ k k = relaxaion ime a = relaxaion modulus α, β, θ = maerial Noe: consans N N 2 θ = θ 1 1 I, I firs invarians = auchy Green ensor 1 = = Finger srain ensor

23 Non-Isohermal Analysis Pecle Number: Nahme Number: where or Pe Na VR p = ρ k a V o = η 2 k η E o 1 1 at ( ) = = exp ηo Rg T To Eo at ( )= o 2 RT g o

24 Pseudo-Time Inegral Mehod dξ = d' ( (')) at Δ i d ' = dl dv l i x i ( ) at η E = = exp η 0 R0 T T0 O : observer s ime ξ: paricle s ime

25 UPWIND FEM ξ +1 r' r = a h 2 η y a = coh Pe 2 x Pe = u h k ξ +1 h=lengh of sreamline -1 r r' +1 η r r' k=diffusion coefficien -1 y x Suppor of DOF Suppor of Inegraion Node

26 Exrudae Swell and Bending Phenomenon

27 Polymer Processes Polymer processes where he K-BKZ model has been used: oexrusion Fiber Spinning Film Blowing Film asing Blow Molding

28 New Developmens 3-D D Simulaions Pom-Pom vs K-BKZ From Sirakov (2006)

29 ONLUSIONS ompuaional rheology has made good progress in he las 15 years due o rapid growh in compuing power. From he modelling poin of view, progress has been achieved mainly by using inegral consiuive equaions of he K-BKZ ype, which are quie capable of fiing rheological daa for polymer soluions and mels in simple flows. In he processing of non-newonian, complex maerials, many unsolved problems sill remain; however here are now available appropriae means of ackling hem, so ha he analysis and design of several processes becomes feasible.

30 FURTHER WORK The subjec of viscoelasiciy requires furher sudy for more appropriae damping funcions for fas deformaions and beer (faser) compuaional mehods. Time-dependen flows Muliple-layer layer flows Three-Dimensional flows