TWODIMENSIONAL ANALYSIS OF THE Turkey Francisco J. CRYSTALLIZATION - - PowerPoint PPT Presentation

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TWODIMENSIONAL ANALYSIS OF THE Turkey Francisco J. CRYSTALLIZATION - - PowerPoint PPT Presentation

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ICCHMT 2011, Istanbul, TWODIMENSIONAL ANALYSIS OF THE Turkey Francisco J. CRYSTALLIZATION OF HOLLOW Blanco Rodr guez, J. COMPOUND PLASTIC FIBERS I. Ramos Introduction Formulation Francisco J. BlancoRodr guez, J. I.

slide-1
SLIDE 1

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

TWO–DIMENSIONAL ANALYSIS OF THE CRYSTALLIZATION OF HOLLOW COMPOUND PLASTIC FIBERS

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

Escuela de Ingenier´ ıas, Doctor Ortiz Ramos, s/n, Universidad de M´ alaga, 29071 M´ alaga, Spain

7th International Conference on Computational Heat and Mass Transfer 18–22 July 2011, Istanbul, Turkey

1 / 27

slide-2
SLIDE 2

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Index

1 Introduction 2 Formulation 3 Results and discussion

Numerical results of the two–dimensional model Comparisons with results of one–dimensional model

4 Conclusions

2 / 27

slide-3
SLIDE 3

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Index

1 Introduction 2 Formulation 3 Results and discussion

Numerical results of the two–dimensional model Comparisons with results of one–dimensional model

4 Conclusions

2 / 27

slide-4
SLIDE 4

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Index

1 Introduction 2 Formulation 3 Results and discussion

Numerical results of the two–dimensional model Comparisons with results of one–dimensional model

4 Conclusions

2 / 27

slide-5
SLIDE 5

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Index

1 Introduction 2 Formulation 3 Results and discussion

Numerical results of the two–dimensional model Comparisons with results of one–dimensional model

4 Conclusions

2 / 27

slide-6
SLIDE 6

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

INTRODUCTION

3 / 27

slide-7
SLIDE 7

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-8
SLIDE 8

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-9
SLIDE 9

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-10
SLIDE 10

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

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SLIDE 11

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-12
SLIDE 12

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-13
SLIDE 13

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-14
SLIDE 14

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-15
SLIDE 15

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-16
SLIDE 16

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Introduction

Bi–component hollow compound fibers may be manufactured by MELT SPINNING processes. Applications

1 Chemical industry: Filtration and separation

processes.

2 Textile industry. 3 Optics: Data transmission. 4 Biomedical industry.

Necessary: modelling of the drawing process of semi–crystalline hollow compound fibers. Previous studies are based on one–dimensional models of amorphous, slender fibers at very low Re and Bi numbers. NO INFORMATION ABOUT TEMPERATURE NON–UNIFORMITIES IN THE RADIAL DIRECTION. Use of a two–dimensional model for fiber spinning.

4 / 27

slide-17
SLIDE 17

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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.

5 / 27

slide-18
SLIDE 18

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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.

5 / 27

slide-19
SLIDE 19

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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.

5 / 27

slide-20
SLIDE 20

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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.

5 / 27

slide-21
SLIDE 21

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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.

5 / 27

slide-22
SLIDE 22

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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.

5 / 27

slide-23
SLIDE 23

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

FORMULATION

6 / 27

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SLIDE 24

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Governing equations

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

  • = −∇pi+∇·τ i+ρi·f m

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

  • = −Ki∆Ti

i = 1, 2, Constitutive equations Rheology τ = µ

  • ∇v + ∇vT

+ τ p, where τ p = 3c kB T

  • −λ

φ F(S) + 2λ

  • ∇vT : S
  • (S + I/3)
  • 7 / 27
slide-25
SLIDE 25

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Governing equations

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

  • = −∇pi+∇·τ i+ρi·f m

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

  • = −Ki∆Ti

i = 1, 2, Constitutive equations Rheology τ = µ

  • ∇v + ∇vT

+ τ p, where τ p = 3c kB T

  • −λ

φ F(S) + 2λ

  • ∇vT : S
  • (S + I/3)
  • 7 / 27
slide-26
SLIDE 26

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Governing equations

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

  • = −∇pi+∇·τ i+ρi·f m

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

  • = −Ki∆Ti

i = 1, 2, Constitutive equations Rheology τ = µ

  • ∇v + ∇vT

+ τ p, where τ p = 3c kB T

  • −λ

φ F(S) + 2λ

  • ∇vT : S
  • (S + I/3)
  • 7 / 27
slide-27
SLIDE 27

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Governing equations

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

  • = −∇pi+∇·τ i+ρi·f m

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

  • = −Ki∆Ti

i = 1, 2, Constitutive equations Rheology τ = µ

  • ∇v + ∇vT

+ τ p, where τ p = 3c kB T

  • −λ

φ F(S) + 2λ

  • ∇vT : S
  • (S + I/3)
  • 7 / 27
slide-28
SLIDE 28

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Molecular orientation and crystallinity models

Crystallization: Avrami–Kolmogorov’s theory & Ziabicki’s model ∂Yi ∂t + vi · ∇Yi = kAi(Si) (Y∞,i − Yi) i = 1, 2, where kAi(Si) = kAi(0) exp ` a2iS2

i

´ , i = 1, 2. Molecular orientation scalar order parameter S ≡ r 3 2 (S : S) S = 1 3 · diag (Srr, − (Srr + Sxx) , Sxx) , Molecular orientation tensor equation: Doi–Edwards theory S(1) = F(S) + G(∇v, S), F(S) = −φ λ {(1 − N/3) S − N (S · S) + N (S : S) (S + I/3)} G(∇v, S) = 1 3 “ ∇v + ∇vT ” − 2 “ ∇vT : S ” (S + I/3) . where subscript (1) denote UCTD operator Λ(1) = ∂Λ ∂t + v · ∇Λ − “ ∇vT · Λ + Λ · ∇v ” 8 / 27

slide-29
SLIDE 29

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Molecular orientation and crystallinity models

Crystallization: Avrami–Kolmogorov’s theory & Ziabicki’s model ∂Yi ∂t + vi · ∇Yi = kAi(Si) (Y∞,i − Yi) i = 1, 2, where kAi(Si) = kAi(0) exp ` a2iS2

i

´ , i = 1, 2. Molecular orientation scalar order parameter S ≡ r 3 2 (S : S) S = 1 3 · diag (Srr, − (Srr + Sxx) , Sxx) , Molecular orientation tensor equation: Doi–Edwards theory S(1) = F(S) + G(∇v, S), F(S) = −φ λ {(1 − N/3) S − N (S · S) + N (S : S) (S + I/3)} G(∇v, S) = 1 3 “ ∇v + ∇vT ” − 2 “ ∇vT : S ” (S + I/3) . where subscript (1) denote UCTD operator Λ(1) = ∂Λ ∂t + v · ∇Λ − “ ∇vT · Λ + Λ · ∇v ” 8 / 27

slide-30
SLIDE 30

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Molecular orientation and crystallinity models

Crystallization: Avrami–Kolmogorov’s theory & Ziabicki’s model ∂Yi ∂t + vi · ∇Yi = kAi(Si) (Y∞,i − Yi) i = 1, 2, where kAi(Si) = kAi(0) exp ` a2iS2

i

´ , i = 1, 2. Molecular orientation scalar order parameter S ≡ r 3 2 (S : S) S = 1 3 · diag (Srr, − (Srr + Sxx) , Sxx) , Molecular orientation tensor equation: Doi–Edwards theory S(1) = F(S) + G(∇v, S), F(S) = −φ λ {(1 − N/3) S − N (S · S) + N (S : S) (S + I/3)} G(∇v, S) = 1 3 “ ∇v + ∇vT ” − 2 “ ∇vT : S ” (S + I/3) . where subscript (1) denote UCTD operator Λ(1) = ∂Λ ∂t + v · ∇Λ − “ ∇vT · Λ + Λ · ∇v ” 8 / 27

slide-31
SLIDE 31

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Boundary and initial conditions

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

  • f hollow compound fiber

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

9 / 27

slide-32
SLIDE 32

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Boundary and initial conditions

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

  • f hollow compound fiber

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

9 / 27

slide-33
SLIDE 33

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Boundary and initial conditions

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

  • f hollow compound fiber

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

9 / 27

slide-34
SLIDE 34

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Boundary and initial conditions

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

  • f hollow compound fiber

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

9 / 27

slide-35
SLIDE 35

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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 σ , Pr = µ0C0 K0 , Pe = Re Pr, Bi = hR0 K0

10 / 27

slide-36
SLIDE 36

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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 σ , Pr = µ0C0 K0 , Pe = Re Pr, Bi = hR0 K0

10 / 27

slide-37
SLIDE 37

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Asymptotic analysis: 1D model

Asymptotic method using the fiber slenderness, ǫ << 1, as perturbation parameter 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. Flow regime steady ( ∂

∂ˆ t = 0) jets and

Re = ǫ ¯ R, Fr = ¯ F ǫ , Ca = ¯ C ǫ , Pe = ǫ ¯ P, Bi = ǫ2 ¯ B where ¯ Υ = O(1).

11 / 27

slide-38
SLIDE 38

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Asymptotic analysis: 1D model

Asymptotic method using the fiber slenderness, ǫ << 1, as perturbation parameter 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. Flow regime steady ( ∂

∂ˆ t = 0) jets and

Re = ǫ ¯ R, Fr = ¯ F ǫ , Ca = ¯ C ǫ , Pe = ǫ ¯ P, Bi = ǫ2 ¯ B where ¯ Υ = O(1).

11 / 27

slide-39
SLIDE 39

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

One–dimensional equations of the 2D model

Asymptotic one–dimensional mass conservation equation A1 B = V1, A2 B = V2, d dˆ x „R2 2 B « = C (ˆ x) where A1 = R2

0 − R2 1

2 , A2 = R2

2 − R2

2 , and C (ˆ x) = „ 1 2 ¯ C « ` σ1

σ

´

1 R1 + 1 R0 +

` σ2

σ

´

1 R2

< ˆ µe1,0 > “

1 R2

0 −

1 R2

1

” + < ˆ µe2,0 > “

1 R2

2 −

1 R2

”,

12 / 27

slide-40
SLIDE 40

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

One–dimensional equations of the 2D model

Asymptotic one–dimensional axial momentum equation (ˆ ρ1V1 + ˆ ρ2V2) ¯ R dB dˆ x = (ˆ ρ1A1 + ˆ ρ2A2) ¯ R ¯ F + d dˆ x „ 3 (< ˆ µe1,0 > A1+ < ˆ µe2,0 > A2) dB dˆ x « +2 C (ˆ x) „ −< ˆ µe1,0 > R1 dR1 dˆ x + < ˆ µe1,0 > − < ˆ µe2,0 > R0 dR0 dˆ x + < ˆ µe2,0 > R2 dR2 dˆ x « − „ A1 dD1 dˆ x + A2 dD2 dˆ x « , where D1 = − < ˆ µe1,0 > 2 C (ˆ x) R2

1

− 1 ¯ C “σ1 σ ” 1 R1 , D2 = − < ˆ µe2,0 > 2 C (ˆ x) R2

2

+ 1 ¯ C “σ2 σ ” 1 R2 Incompressibility condition V(ˆ r, ˆ x) = C (ˆ x) ˆ r − ˆ r 2 dB dˆ x ,

13 / 27

slide-41
SLIDE 41

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

One–dimensional equations of the 2D model

Asymptotic one–dimensional axial momentum equation (ˆ ρ1V1 + ˆ ρ2V2) ¯ R dB dˆ x = (ˆ ρ1A1 + ˆ ρ2A2) ¯ R ¯ F + d dˆ x „ 3 (< ˆ µe1,0 > A1+ < ˆ µe2,0 > A2) dB dˆ x « +2 C (ˆ x) „ −< ˆ µe1,0 > R1 dR1 dˆ x + < ˆ µe1,0 > − < ˆ µe2,0 > R0 dR0 dˆ x + < ˆ µe2,0 > R2 dR2 dˆ x « − „ A1 dD1 dˆ x + A2 dD2 dˆ x « , where D1 = − < ˆ µe1,0 > 2 C (ˆ x) R2

1

− 1 ¯ C “σ1 σ ” 1 R1 , D2 = − < ˆ µe2,0 > 2 C (ˆ x) R2

2

+ 1 ¯ C “σ2 σ ” 1 R2 Incompressibility condition V(ˆ r, ˆ x) = C (ˆ x) ˆ r − ˆ r 2 dB dˆ x ,

13 / 27

slide-42
SLIDE 42

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional energy equation B ∂ ˆ T1 ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂ ˆ T1 ∂ˆ r = 1 ¯ P1 1 ˆ r ∂ ∂ˆ r ˆ r ∂ ˆ T1 ∂ˆ r ! R1 ≤ ˆ r ≤ R0, ˆ T1(ˆ r, 0) = ˜ T1(ˆ r), − ˆ K1 ∂ ˆ T1 ∂ˆ r (R1, ˆ x) = 0, ˆ T1 (R0, ˆ x) = ˆ T2 (R0, ˆ x) B ∂ ˆ T2 ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂ ˆ T2 ∂ˆ r = 1 ¯ P2 1 ˆ r ∂ ∂ˆ r ˆ r ∂ ˆ T2 ∂ˆ r ! R0 ≤ ˆ r ≤ R2, ˆ T2(ˆ r, 0) = ˜ T2(ˆ r), − ˆ K2 ∂ ˆ T2 ∂ˆ r (R2, ˆ x) = ¯ B ˆ h2 “ ˆ T2(R2, ˆ x) − ˆ T2,∞ ” , − ˆ K1 ∂ ˆ T1 ∂ˆ r (R0, ˆ x) = − ˆ K2 ∂ ˆ T2 ∂ˆ r (R0, ˆ x)

14 / 27

slide-43
SLIDE 43

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional degree of crystallinity equation B ∂Yi ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂Yi ∂ˆ r = kAi(0) exp ` a2iS2

i

´ (Y∞,i − Yi) , i = 1, 2, Yi(ˆ r, 0) = ˜ Yi(ˆ r), i = 1, 2, Effective dynamic viscosity ˆ µei,0 (ˆ r, ˆ x) = ˆ Gi exp „ ˆ Ei “ 1 − ˆ Ti ” + βi „ Yi Y∞,i «ni« +2 3 αi ˆ λiS2

i

i = 1, 2. Cross-sectionally averaged effective dynamic viscosity < ˆ µe1,0 > (ˆ x) = Z R0

R1

ˆ µe1,0(ˆ r, ˆ x) ˆ r dˆ r/A1, < ˆ µe2,0 > (ˆ x) = Z R2

R0

ˆ µe2,0(ˆ r, ˆ x) ˆ r dˆ r/A2.

15 / 27

slide-44
SLIDE 44

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional degree of crystallinity equation B ∂Yi ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂Yi ∂ˆ r = kAi(0) exp ` a2iS2

i

´ (Y∞,i − Yi) , i = 1, 2, Yi(ˆ r, 0) = ˜ Yi(ˆ r), i = 1, 2, Effective dynamic viscosity ˆ µei,0 (ˆ r, ˆ x) = ˆ Gi exp „ ˆ Ei “ 1 − ˆ Ti ” + βi „ Yi Y∞,i «ni« +2 3 αi ˆ λiS2

i

i = 1, 2. Cross-sectionally averaged effective dynamic viscosity < ˆ µe1,0 > (ˆ x) = Z R0

R1

ˆ µe1,0(ˆ r, ˆ x) ˆ r dˆ r/A1, < ˆ µe2,0 > (ˆ x) = Z R2

R0

ˆ µe2,0(ˆ r, ˆ x) ˆ r dˆ r/A2.

15 / 27

slide-45
SLIDE 45

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional degree of crystallinity equation B ∂Yi ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂Yi ∂ˆ r = kAi(0) exp ` a2iS2

i

´ (Y∞,i − Yi) , i = 1, 2, Yi(ˆ r, 0) = ˜ Yi(ˆ r), i = 1, 2, Effective dynamic viscosity ˆ µei,0 (ˆ r, ˆ x) = ˆ Gi exp „ ˆ Ei “ 1 − ˆ Ti ” + βi „ Yi Y∞,i «ni« +2 3 αi ˆ λiS2

i

i = 1, 2. Cross-sectionally averaged effective dynamic viscosity < ˆ µe1,0 > (ˆ x) = Z R0

R1

ˆ µe1,0(ˆ r, ˆ x) ˆ r dˆ r/A1, < ˆ µe2,0 > (ˆ x) = Z R2

R0

ˆ µe2,0(ˆ r, ˆ x) ˆ r dˆ r/A2.

15 / 27

slide-46
SLIDE 46

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional equations for the molecular orientation tensor components B ∂Si rr ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂Si rr ∂ˆ r = (1 − Si rr) (1 + Si xx) dB dˆ x −φi ˆ λi  Si rr + Ni 3 [(Si rr − 1) (Si rr − Oi s)] ff , i = 1, 2, B ∂Si rr ∂ˆ x + „C (ˆ x) ˆ r − ˆ r 2 dB dˆ x « ∂Si rr ∂ˆ r = (1 + Si xx) (2 − Si xx) dB dˆ x −φi ˆ λi  Si xx − Ni 3 [(Si xx + 1) (Si xx − Oi s)] ff , i = 1, 2, Oi s(ˆ r, ˆ x) = 2 3 ` S2

i rr + S2 i xx − Si rr Si xx

´ . Si rr(ˆ r, 0) = − ˜ Si(ˆ r), Si xx(ˆ r, 0) = 2 ˜ Si(ˆ r).

16 / 27

slide-47
SLIDE 47

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Mapping: 2D model

(ˆ r, ˆ x) → (ξ, η) maps Ωˆ

rˆ x = {[R1(ˆ

x), R2(ˆ x)] × [0, 1]} into a rectangular domain Ωξη = {[0, 1] × [0, 1]}

17 / 27

slide-48
SLIDE 48

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Mapping: 2D model

(ˆ r, ˆ x) → (ξ, η) maps Ωˆ

rˆ x = {[R1(ˆ

x), R2(ˆ x)] × [0, 1]} into a rectangular domain Ωξη = {[0, 1] × [0, 1]}

17 / 27

slide-49
SLIDE 49

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional energy equation ∂ ˆ T1 ∂η = 1 2 (V1 + V2) 4 ¯ P1 ∂ ∂ξ „ ξ + R2

1

R2

2 − R2 1

« ∂ ˆ T1 ∂ξ ! 0 ≤ ξ ≤ V1 V1 + V2 , ˆ T1(ξ, 0) = ˜ T1(ξ), − ˆ K1 ∂ ˆ T1 ∂ξ (0, η) = 0, ˆ T1 „ V1 V1 + V2 , η « = ˆ T2 „ V1 V1 + V2 , η « ∂ ˆ T2 ∂η = 1 2 (V1 + V2) 4 ¯ P2 ∂ ∂ξ „ ξ + R2

1

R2

2 − R2 1

« ∂ ˆ T2 ∂ξ ! V1 V1 + V2 ≤ ξ ≤ 1, ˆ T2(ξ, 0) = ˜ T2(ξ), − ˆ K2 ∂ ˆ T2 ∂ξ (1, η) = R2

2 − R2 1

2 R2 ¯ B ˆ h2 “ ˆ T2(1, η) − ˆ T2,∞ ” , − ˆ K1 ∂ ˆ T1 ∂ξ „ V1 V1 + V2 , η « = − ˆ K2 ∂ ˆ T2 ∂ξ „ V1 V1 + V2 , η « 18 / 27

slide-50
SLIDE 50

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional degree of crystallinity equation B ∂Yi ∂η = kAi(0) exp ` a2iS2

i

´ (Y∞,i − Yi) , i = 1, 2, Yi(ξ, 0) = ˜ Yi(ξ), i = 1, 2, Effective dynamic viscosity ˆ µei,0 (ξ, η) = ˆ Gi exp „ ˆ Ei “ 1 − ˆ Ti ” + βi „ Yi Y∞,i «ni« +2 3 αi ˆ λiS2

i

i = 1, 2. Cross-sectionally averaged effective dynamic viscosity < ˆ µe1,0 > (η) = V1 + V2 V1 Z

V1 V1+V2

ˆ µe1,0(ξ, η) dξ, < ˆ µe2,0 > (η) = V1 + V2 V2 Z 1

V1 V1+V2

ˆ µe2,0(ξ, η) dξ.

19 / 27

slide-51
SLIDE 51

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Two–dimensional equations of the 2D model

Two–dimensional equations for the molecular orientation tensor components B ∂Si rr ∂η = (1 − Si rr) (1 + Si xx) dB dη −φi ˆ λi  Si rr + Ni 3 [(Si rr − 1) (Si rr − Oi s)] ff , i = 1, 2, B ∂Si xx ∂η = (1 + Si xx) (2 − Si xx) dB dη −φi ˆ λi  Si xx − Ni 3 [(Si xx + 1) (Si xx − Oi s)] ff , i = 1, 2, Oi s(ξ, η) = 2 3 ` S2

i rr + S2 i xx − Si rr Si xx

´ . Si rr(ξ, 0) = − ˜ Si(ξ), Si xx(ξ, 0) = 2 ˜ Si(ξ).

20 / 27

slide-52
SLIDE 52

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

RESULTS AND DISCUSSION

21 / 27

slide-53
SLIDE 53

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Thermal boundary layer

E2 = 10

22 / 27

slide-54
SLIDE 54

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Effect of the cladding’s viscosity parameters

G2 = 1 (–), G2 = 0,01 (– –) and G2 = 100 (− · −) E2 = 100 (–), E2 = 50 (– –) and E2 = 10 (− · −) 23 / 27

slide-55
SLIDE 55

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Effect of the cladding’s viscosity parameters

G2 = 1 (–), G2 = 0,01 (– –) and G2 = 100 (− · −) E2 = 100 (–), E2 = 50 (– –) and E2 = 10 (− · −) 23 / 27

slide-56
SLIDE 56

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

1D model vs 2D model

1D (–) and 2D (− · −) 24 / 27

slide-57
SLIDE 57

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

CONCLUSIONS

25 / 27

slide-58
SLIDE 58

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Conclusions

Contributions of the present work:

1 Development of a quasi–two–dimensional model for

semi–crystalline hollow compound fibers with modified Newtonian rheology.

2 1D model (asymptotic analysis for ǫ << 1) for Ri and B and

2D equations for ˆ Ti, Si and Yi.

3 Determination of the two–dimensional fields of temperature,

  • rder parameter for molecular orientation and degree of

crystallinity for hollow compound fibers.

4 Integro–differential model highly dependent on E2 and G2. 5 Find substantial temperature non–uniformities (may affect the

degree of crystallization and have great effects on the mechanical, optical,... properties of hollow compound fibers) in the radial direction.

26 / 27

slide-59
SLIDE 59

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Conclusions

Contributions of the present work:

1 Development of a quasi–two–dimensional model for

semi–crystalline hollow compound fibers with modified Newtonian rheology.

2 1D model (asymptotic analysis for ǫ << 1) for Ri and B and

2D equations for ˆ Ti, Si and Yi.

3 Determination of the two–dimensional fields of temperature,

  • rder parameter for molecular orientation and degree of

crystallinity for hollow compound fibers.

4 Integro–differential model highly dependent on E2 and G2. 5 Find substantial temperature non–uniformities (may affect the

degree of crystallization and have great effects on the mechanical, optical,... properties of hollow compound fibers) in the radial direction.

26 / 27

slide-60
SLIDE 60

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Conclusions

Contributions of the present work:

1 Development of a quasi–two–dimensional model for

semi–crystalline hollow compound fibers with modified Newtonian rheology.

2 1D model (asymptotic analysis for ǫ << 1) for Ri and B and

2D equations for ˆ Ti, Si and Yi.

3 Determination of the two–dimensional fields of temperature,

  • rder parameter for molecular orientation and degree of

crystallinity for hollow compound fibers.

4 Integro–differential model highly dependent on E2 and G2. 5 Find substantial temperature non–uniformities (may affect the

degree of crystallization and have great effects on the mechanical, optical,... properties of hollow compound fibers) in the radial direction.

26 / 27

slide-61
SLIDE 61

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Conclusions

Contributions of the present work:

1 Development of a quasi–two–dimensional model for

semi–crystalline hollow compound fibers with modified Newtonian rheology.

2 1D model (asymptotic analysis for ǫ << 1) for Ri and B and

2D equations for ˆ Ti, Si and Yi.

3 Determination of the two–dimensional fields of temperature,

  • rder parameter for molecular orientation and degree of

crystallinity for hollow compound fibers.

4 Integro–differential model highly dependent on E2 and G2. 5 Find substantial temperature non–uniformities (may affect the

degree of crystallization and have great effects on the mechanical, optical,... properties of hollow compound fibers) in the radial direction.

26 / 27

slide-62
SLIDE 62

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Conclusions

Contributions of the present work:

1 Development of a quasi–two–dimensional model for

semi–crystalline hollow compound fibers with modified Newtonian rheology.

2 1D model (asymptotic analysis for ǫ << 1) for Ri and B and

2D equations for ˆ Ti, Si and Yi.

3 Determination of the two–dimensional fields of temperature,

  • rder parameter for molecular orientation and degree of

crystallinity for hollow compound fibers.

4 Integro–differential model highly dependent on E2 and G2. 5 Find substantial temperature non–uniformities (may affect the

degree of crystallization and have great effects on the mechanical, optical,... properties of hollow compound fibers) in the radial direction.

26 / 27

slide-63
SLIDE 63

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

Conclusions

Contributions of the present work:

1 Development of a quasi–two–dimensional model for

semi–crystalline hollow compound fibers with modified Newtonian rheology.

2 1D model (asymptotic analysis for ǫ << 1) for Ri and B and

2D equations for ˆ Ti, Si and Yi.

3 Determination of the two–dimensional fields of temperature,

  • rder parameter for molecular orientation and degree of

crystallinity for hollow compound fibers.

4 Integro–differential model highly dependent on E2 and G2. 5 Find substantial temperature non–uniformities (may affect the

degree of crystallization and have great effects on the mechanical, optical,... properties of hollow compound fibers) in the radial direction.

26 / 27

slide-64
SLIDE 64

ICCHMT 2011, Istanbul, Turkey Francisco J. Blanco– Rodr´ ıguez, J.

  • I. Ramos

Introduction Formulation Results and discussion

Numerical results of the two–dimensional model Comparisons with results of

  • ne–dimensional

model

Conclusions About the authors...

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

27 / 27