Crystallization of Compound Plastic Optical and Francisco J. Blanco - PowerPoint PPT Presentation

ICTEA 2010, Marrakesh, Morocco J.I. Ramos Crystallization of Compound Plastic Optical and Francisco J. Blanco– Fibers Rodr´ ıguez Introduction Mathematical J.I. Ramos and Francisco J. Blanco–Rodr´ ıguez model of melt spinning Numerical Escuela de Ingenier´ ıas method Universidad de M´ alaga Simulation results of melt spinning fibers Fifth International Conference on Thermal Engineering: Discussion Theory and Applications May 10–14, 2010, Marrakesh, Morocco 1 / 17

Index ICTEA 2010, Marrakesh, Morocco J.I. Ramos 1 Introduction and Francisco J. Blanco– Rodr´ ıguez 2 Mathematical model of melt spinning Introduction Mathematical model of melt 3 Numerical method spinning Numerical method 4 Simulation results of melt spinning fibers Simulation results of melt spinning fibers Discussion 5 Discussion 2 / 17

Introduction ICTEA 2010, Polymer Optical Fibers (POF) are manufactured by MELT Marrakesh, Morocco SPINNING processes. J.I. Ramos Necessary: modelling of the drawing process for both and Francisco J. Blanco– hollow and solid compound optical fibers. Rodr´ ıguez Previous studies are based on one–dimensional models. Introduction NO INFORMATION ABOUT RADIAL VARIATIONS. Mathematical model of melt Use of a hybrid model for melt spinning phenomena. spinning Applications Numerical method 1 Telecommunications: Data transmission. 2 Chemical industry: Filtration and separation processes. Simulation results of melt 3 Biomedical industry. spinning fibers 4 Textile industry. Discussion 3 / 17

Melt spinning ICTEA 2010, Marrakesh, Morocco Involves the extrusion and drawing of a polymer cylinder. Four J.I. Ramos zones. and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Region. Introduction Mathematical 2 Flow model of melt spinning Rearrangement Numerical Region. method 3 Melt Drawing Simulation results of melt Zone. spinning fibers Discussion 4 Solidification region . 4 / 17

Melt spinning ICTEA 2010, Involves the extrusion and drawing of a polymer cylinder. Four Marrakesh, Morocco zones. J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Introduction Region. Mathematical 2 Flow model of melt spinning Rearrangement Numerical Region. method Simulation 3 Melt Drawing results of melt spinning fibers Zone. Discussion 4 Solidification region . 4 / 17

Melt spinning ICTEA 2010, Involves the extrusion and drawing of a polymer cylinder. Four Marrakesh, Morocco zones. J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Introduction Region. Mathematical 2 Flow model of melt spinning Rearrangement Numerical Region. method Simulation 3 Melt Drawing results of melt spinning fibers Zone. Discussion 4 Solidification region . 4 / 17

Melt spinning ICTEA 2010, Involves the extrusion and drawing of a polymer cylinder. Four Marrakesh, Morocco zones. J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Introduction Region. Mathematical 2 Flow model of melt spinning Rearrangement Numerical Region. method Simulation 3 Melt Drawing results of melt spinning fibers Zone. Discussion 4 Solidification region . 4 / 17

Melt spinning ICTEA 2010, Involves the extrusion and drawing of a polymer cylinder. Four Marrakesh, Morocco zones. J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Introduction Region. Mathematical 2 Flow model of melt spinning Rearrangement Numerical Region. method Simulation 3 Melt Drawing results of melt spinning fibers Zone. Discussion 4 Solidification region . 4 / 17

Melt spinning ICTEA 2010, Involves the extrusion and drawing of a polymer cylinder. Four Marrakesh, Morocco zones. J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Introduction Region. Mathematical 2 Flow model of melt spinning Rearrangement Numerical Region. method Simulation 3 Melt Drawing results of melt spinning fibers Zone. Discussion 4 Solidification region . 4 / 17

Problem formulation (I) ICTEA 2010, Mass conservation equation Marrakesh, Morocco ∇ · v i = 0 i = 1 , 2 , J.I. Ramos and Francisco where v = u ( r, x ) e x + v ( r, x ) e r J. Blanco– Rodr´ ıguez Linear Momentum conservation equation Introduction � ∂ v i � = −∇ p + ∇· τ i + ρ i · f m ρ i ∂t + v i · ∇ v i i = 1 , 2 , Mathematical model of melt where f m = g e x spinning Energy conservation equation Numerical method � ∂T i � ρ i C i ∂t + v i · ∇ T i = −∇ · q i , i = 1 , 2 , Simulation results of melt spinning fibers Constitutive equations Discussion Newtonian rheology � ∇ v i + ∇ v T � τ i = 2 µ eff,i D i = µ eff,i , i Fourier’s law q i = − k i ∇ T i , 5 / 17

Problem formulation (II) ICTEA 2010, Molecular orientation: Doi–Edwards equation Marrakesh, Morocco ∂S ∂t + v · ∇ S = − φ J.I. Ramos λ F ( S ) + G ( ∇ v , S ) , and Francisco J. Blanco– F ( S ) = S (1 − N/ 3 (1 − S ) (2 S + 1)) Rodr´ ıguez G ( ∇ v , S ) = (1 − S ) (2 S + 1) ∂u ∂x. Introduction Mathematical model of melt Crystallization: Avrami–Kolmogorov kinetics spinning ∂θ i Numerical ∂t + v · ∇ θ i = k Ai ( S i ) ( θ ∞ i − θ i ) , i = 1 , 2 , method Simulation results of melt where spinning fibers ` a 2 i S 2 ´ k Ai ( S i ) = k Ai (0) exp , i = 1 , 2 . Discussion i 6 / 17

Problem formulation (III) ICTEA 2010, Marrakesh, Kinematic, dynamic and Morocco thermal boundary conditions J.I. Ramos and Francisco are required: J. Blanco– Rodr´ ıguez Symmetry conditions ( r = 0 ) Introduction Die exit conditions ( x = 0 ) Mathematical model of melt Take–up point conditions spinning ( x = L ) Numerical method Conditions on free surfaces Simulation results of melt of compound fiber spinning fibers ( r = R 1 ( x ) and Discussion r = R 2 ( x ) ) 7 / 17

Non–dimensionalize ICTEA 2010, Non–dimensional variables Marrakesh, Morocco t r = r x = x ǫ = R 0 ˆ t = ˆ ˆ ⇒ J.I. Ramos L/u 0 R 0 L L and Francisco J. Blanco– u = u v p T = T Rodr´ ıguez ˆ ˆ ˆ v = p = ˆ u 0 ( u 0 ǫ ) ( µ 0 u 0 /L ) T 0 Introduction ρ = ρ C = C µ = µ k = k ˆ ˆ ˆ ˆ Mathematical ρ 0 C 0 µ 0 k 0 model of melt spinning Non–dimensional numbers Numerical method Fr = u 2 Re = ρ 0 u 0 R 0 Ca = µ 0 u 0 0 , gR 0 , σ 2 , Simulation µ 0 results of melt spinning fibers Pe = ρ 0 C 0 Bi = hR 0 u 0 R 0 , Discussion k 0 k 0 8 / 17

Asymptotic analysis: 1 D model ICTEA 2010, Marrakesh, Morocco Perturbation method using the fiber slenderness ( ǫ << 1) J.I. Ramos and Francisco J. Blanco– Ψ i = Ψ i, 0 + ǫ 2 Ψ i, 2 + O � ǫ 4 � , Rodr´ ıguez Introduction for the variables ˆ p i and ˆ R i , ˆ u i , ˆ v i , ˆ T i where i = 1 , 2 . Mathematical model of melt Steady–state flow regime considered spinning Numerical ¯ ¯ F C method Re = ǫ ¯ R, Fr = ǫ , Ca = ǫ , Simulation results of melt Bi = ǫ 2 ¯ spinning fibers Pe = ǫ ¯ P, B Discussion where ¯ Υ = O (1) . 9 / 17

One–dimensional equations of the 1 + 1 / 2 D model ICTEA 2010, Asymptotic one–dimensional mass conservation equation Marrakesh, Morocco d “ ” A i ˆ U = 0 , i = 1 , 2 , d ˆ x J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez ˆ ˆ 2 − ˆ R 2 R 2 R 2 1 1 A 1 = 2 , A 2 = , 2 Introduction Asymptotic one–dimensional linear momentum equation Mathematical model of melt ! U d ˆ µ eff, 2 > A 2 ) d ˆ spinning U d U ¯ ρ 2 A 2 ) ˆ R (ˆ ρ 1 A 1 + ˆ = 3 ( < ˆ µ eff, 1 > A 1 + < ˆ d ˆ x d ˆ x d ˆ x Numerical method d ˆ d ˆ ! 1 R 2 x + σ 1 R 1 + Simulation 2 ¯ d ˆ σ 2 d ˆ x C results of melt ¯ spinning fibers R + (ˆ ρ 1 A 1 + ˆ ρ 2 A 2 ) ¯ F Discussion Effective dynamic viscosity „ θ i « n i « „ +2 “ ” µ eff,i = ˆ ˆ 1 − ˆ 3 α i λ i S 2 ˆ G i exp E i T i + β i i , i = 1 , 2 . θ ∞ ,i 10 / 17

Mapping: 2 D model n o [0 , ˆ (ˆ r, ˆ x ) �→ ( ξ, η ) maps Ω ˆ x = R 2 (ˆ x )] × [0 , 1] into a rectangular ICTEA 2010, r ˆ Marrakesh, domain Ω ξη = { [0 , 1] × [0 , 1] } Morocco J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez Introduction Mathematical model of melt spinning Numerical method Simulation results of melt spinning fibers Discussion 11 / 17

Two–dimensional equations of the 1 + 1 / 2 D model ICTEA 2010, Two–dimensional energy equation Marrakesh, Morocco J.I. Ramos � � ∂ ˆ ξ ∂ ˆ and Francisco T i 1 1 1 ∂ T i ∂η = i = 1 , 2 , J. Blanco– ¯ 2 Q P i ξ ∂ξ ∂ξ Rodr´ ıguez Introduction Two–dimensional molecular orientation parameter equation Mathematical model of melt spinning U ∂S i − φ i ˆ Numerical = S i (1 − N i / 3 (1 − S i ) (2 S i + 1)) ∂η λ i method (1 − S i ) (2 S i + 1) d ˆ Simulation U + dη , i = 1 , 2 . results of melt spinning fibers Two-dimensional degree of crystallinity equation Discussion U ∂θ i ˆ a 2 i S 2 � � ∂η = k Ai (0) exp ( θ ∞ ,i − θ i ) , i = 1 , 2 , i 12 / 17

Influence of Biot number on cooling process ICTEA 2010, Marrakesh, Morocco J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez Introduction Mathematical model of melt spinning Numerical method Simulation results of melt spinning fibers Discussion ¯ ¯ B = 0 , 5 B = 5 , 0 13 / 17