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

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ICTEA 2010, Marrakesh, Morocco J.I. Ramos Crystallization of Compound Plastic Optical and Francisco J. Blanco Fibers Rodr guez Introduction Mathematical J.I. Ramos and Francisco J. BlancoRodr guez model of melt spinning

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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

Crystallization of Compound Plastic Optical Fibers

J.I. Ramos and Francisco J. Blanco–Rodr´ ıguez

Escuela de Ingenier´ ıas Universidad de M´ alaga

Fifth International Conference on Thermal Engineering: Theory and Applications May 10–14, 2010, Marrakesh, Morocco

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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

Index

1 Introduction 2 Mathematical model of melt spinning 3 Numerical method 4 Simulation results of melt spinning fibers 5 Discussion

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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

Introduction

Polymer Optical Fibers (POF) are manufactured by MELT SPINNING processes. Necessary: modelling of the drawing process for both hollow and solid compound optical fibers. Previous studies are based on one–dimensional models. NO INFORMATION ABOUT RADIAL VARIATIONS. Use of a hybrid model for melt spinning phenomena. Applications

1 Telecommunications: Data transmission. 2 Chemical industry: Filtration and separation processes. 3 Biomedical industry. 4 Textile industry.

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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

Melt spinning

Involves the extrusion and drawing of a polymer cylinder. Four zones.

1 Shear Flow

Region.

2 Flow

Rearrangement Region.

3 Melt Drawing

Zone.

4 Solidification

region.

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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

Melt spinning

Involves the extrusion and drawing of a polymer cylinder. Four zones.

1 Shear Flow

Region.

2 Flow

Rearrangement Region.

3 Melt Drawing

Zone.

4 Solidification

region.

4 / 17

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SLIDE 6

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

Melt spinning

Involves the extrusion and drawing of a polymer cylinder. Four zones.

1 Shear Flow

Region.

2 Flow

Rearrangement Region.

3 Melt Drawing

Zone.

4 Solidification

region.

4 / 17

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SLIDE 7

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

Melt spinning

Involves the extrusion and drawing of a polymer cylinder. Four zones.

1 Shear Flow

Region.

2 Flow

Rearrangement Region.

3 Melt Drawing

Zone.

4 Solidification

region.

4 / 17

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SLIDE 8

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

Melt spinning

Involves the extrusion and drawing of a polymer cylinder. Four zones.

1 Shear Flow

Region.

2 Flow

Rearrangement Region.

3 Melt Drawing

Zone.

4 Solidification

region.

4 / 17

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SLIDE 9

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

Melt spinning

Involves the extrusion and drawing of a polymer cylinder. Four zones.

1 Shear Flow

Region.

2 Flow

Rearrangement Region.

3 Melt Drawing

Zone.

4 Solidification

region.

4 / 17

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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

Problem formulation (I)

Mass conservation equation ∇ · vi = 0 i = 1, 2, where v = u(r, x) ex + v(r, x) er Linear Momentum conservation equation ρi ∂vi ∂t + vi · ∇vi

  • = −∇p+∇·τ i +ρi ·f m

i = 1, 2, where f m = g ex Energy conservation equation ρi Ci ∂Ti ∂t + vi · ∇Ti

  • = −∇ · qi,

i = 1, 2, Constitutive equations Newtonian rheology τ i = 2µeff,iDi = µeff,i

  • ∇vi + ∇vT

i

  • ,

Fourier’s law qi = −ki ∇Ti, 5 / 17

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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

Problem formulation (II)

Molecular orientation: Doi–Edwards equation ∂S ∂t + v · ∇S = −φ λ F(S) + G(∇v, S), F(S) = S (1 − N/3 (1 − S) (2 S + 1)) G(∇v, S) = (1 − S) (2 S + 1) ∂u ∂x. Crystallization: Avrami–Kolmogorov kinetics ∂θi ∂t + v · ∇θi = kAi(Si) (θ∞ i − θi) , i = 1, 2, where kAi(Si) = kAi(0) exp ` a2iS2

i

´ , i = 1, 2.

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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

Problem formulation (III)

Kinematic, dynamic and thermal boundary conditions are required: Symmetry conditions (r = 0) Die exit conditions (x = 0) Take–up point conditions (x = L) Conditions on free surfaces

  • f compound fiber

(r = R1(x) and r = R2(x))

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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

Non–dimensionalize

Non–dimensional variables ˆ t = t L/u0 ˆ r = r R0 ˆ x = x L ⇒ ǫ = R0 L ˆ u = u u0 ˆ v = v (u0 ǫ) ˆ p = p (µ0u0/L) ˆ T = T T0 ˆ ρ = ρ ρ0 ˆ C = C C0 ˆ µ = µ µ0 ˆ k = k k0 Non–dimensional numbers Re = ρ0u0R0 µ0 , Fr = u2 gR0 , Ca = µ0u0 σ2 , Pe = ρ0C0 k0 u0R0, Bi = hR0 k0

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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

Asymptotic analysis: 1D model

Perturbation method using the fiber slenderness (ǫ << 1) Ψi = Ψi,0 + ǫ2Ψi,2 + O

  • ǫ4

, for the variables ˆ Ri, ˆ ui, ˆ vi, ˆ pi and ˆ Ti where i = 1, 2. Steady–state flow regime considered Re = ǫ ¯ R, Fr = ¯ F ǫ , Ca = ¯ C ǫ , Pe = ǫ ¯ P, Bi = ǫ2 ¯ B where ¯ Υ = O(1).

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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

One–dimensional equations of the 1 + 1/2D model

Asymptotic one–dimensional mass conservation equation d dˆ x “ Ai ˆ U ” = 0, i = 1, 2, A1 = ˆ R2

1

2 , A2 = ˆ R2

2 − ˆ

R2

1

2 , Asymptotic one–dimensional linear momentum equation ¯ R(ˆ ρ1A1 + ˆ ρ2A2) ˆ U d ˆ U dˆ x = d dˆ x 3 (< ˆ µeff,1 > A1+ < ˆ µeff,2 > A2) d ˆ U dˆ x ! + 1 2 ¯ C d ˆ R2 dˆ x + σ1 σ2 d ˆ R1 dˆ x ! + (ˆ ρ1A1 + ˆ ρ2A2) ¯ R ¯ F Effective dynamic viscosity ˆ µeff,i = ˆ Gi exp „ ˆ Ei “ 1 − ˆ Ti ” + βi „ θi θ∞,i «ni« +2 3 αi λi S2

i ,

i = 1, 2.

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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

Mapping: 2D model

(ˆ r, ˆ x) → (ξ, η) maps Ωˆ

rˆ x =

n [0, ˆ R2(ˆ x)] × [0, 1]

  • into a rectangular

domain Ωξη = {[0, 1] × [0, 1]}

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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

Two–dimensional equations of the 1 + 1/2D model

Two–dimensional energy equation ∂ ˆ Ti ∂η = 1 2 Q 1 ¯ Pi 1 ξ ∂ ∂ξ

  • ξ ∂ ˆ

Ti ∂ξ

  • i = 1, 2,

Two–dimensional molecular orientation parameter equation ˆ U ∂Si ∂η = −φi λi Si (1 − Ni/3 (1 − Si) (2 Si + 1)) + (1 − Si) (2 Si + 1) d ˆ U dη , i = 1, 2. Two-dimensional degree of crystallinity equation ˆ U ∂θi ∂η = kAi(0) exp

  • a2iS2

i

  • (θ∞,i − θi) ,

i = 1, 2,

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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

Influence of Biot number on cooling process

¯ B = 0,5 ¯ B = 5,0

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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

Influence of Biot number on the radially averaged results

(− · −) ¯ B = 0,5 and (–) ¯ B = 5,0

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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

Degree of crystallinity near the die exit

ˆ x = 0,04 ˆ x = 0,10

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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

Discussion

Contributions of the present work:

1 Development of a 1 + 1/2D model for both amorphous and

semicrystalline fibers with Newtonian rheology.

2 Validation of applicability range of the 1D model with the

1 + 1/2D one.

3 Determination of the two–dimensional fields of temperature,

molecular orientation parameter and degree of crystallinity for solid compound fibers.

4 Find substantial temperature non–uniformities (affect the degree

  • f crystallization and have great effects on the properties of

compound fibers) in the radial direction exist even at small Biot numbers.

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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

About the authors...

Francisco J. Blanco–Rodr´ ıguez e-mail: fjblanco@lcc.uma.es website: http://www.lcc.uma.es/~fjblanco

  • J. I. Ramos

e-mail: jirs@lcc.uma.es Document created by L

AT

EX(Beamer class).

THANK YOU FOR YOUR ATTENTION

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