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ASME-ATI- UIT 2010, Sorrento, MODELLING OF MOLECULAR Italy ORIENTATION AND CRYSTALLIZATION IN J.I. Ramos and Francisco J. Blanco THE MANUFACTURE OF Rodr guez SEMICRYSTALLINE COMPOUND FIBRES Introduction Mathematical

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SLIDE 1

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

MODELLING OF MOLECULAR ORIENTATION AND CRYSTALLIZATION IN THE MANUFACTURE OF SEMI–CRYSTALLINE COMPOUND FIBRES

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

Escuela de Ingenier´ ıas Universidad de M´ alaga, Spain

Conference on Thermal and Environmental Issues in Energy Systems 16–19 May 2010, Sorrento, Italy

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SLIDE 2

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Outline

1 Introduction 2 Mathematical formulation 3 Numerical method 4 Numerical results

Two–dimensional model numerical results Comparison with one–dimensional model

5 Conclusions

2 / 19

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Introduction

Bi–component compound fibres are manufactured by MELT SPINNING processes. Applications

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

Necessary: modelling of the drawing process for both hollow and solid semi–crystalline compound fibres. Previous studies are based on one–dimensional models of amorphous, slender fibres at low Re. NO INFORMATION ABOUT RADIAL VARIATIONS. Use of a hybrid model for fibre spinning.

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SLIDE 4

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 / 19

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SLIDE 5

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 / 19

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SLIDE 6

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 / 19

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SLIDE 7

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 / 19

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SLIDE 8

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 / 19

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SLIDE 9

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 / 19

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SLIDE 10

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Mathematical 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

  • = −ki∆Ti

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

  • ∇v + ∇vT

+ τ p, where τ p = 3c kB T

  • −λ

φ F(S) + 2λ

  • ∇vT : S
  • (S + I/3)
  • 5 / 19
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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Mathematical formulation (II)

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 ” Molecular orientation scalar order parameter S ≡ r 3 2 (S : S) S = diag (Srr, Sθθ, Sxx) , Crystallization: Avrami–Kolmogorov’s theory & Ziabicki’s model ∂Xi ∂t + v · ∇Xi = kAi(Si) (X∞,i − Xi) i = 1, 2, where kAi(Si) = kAi(0) exp ` a2iS2

i

´ , i = 1, 2. 6 / 19

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Mathematical formulation (III)

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

  • f compound fibre

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

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 , Pr = µ0C0 k0 , Pe = Re Pr, Bi = hR0 k0

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SLIDE 14

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Asymptotic analysis: 1D model

Asymptotic method using the fibre 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 considered for steady ( ∂

∂ˆ t = 0) jets

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

9 / 19

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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

Asymptotic one–dimensional mass conservation equation A1 U = Q1, A2 U = Q2 where A1 = R2

1

2 , A2 = R2

2 − R2 1

2 , Asymptotic one–dimensional linear momentum equation ¯ R(ˆ ρ1A1 + ˆ ρ2A2)U dU dˆ x = d dˆ x „ 3 (< ˆ µeff,1 > A1+ < ˆ µeff,2 > A2) dU dˆ x « + 1 2 ¯ C „dR2 dˆ x + σ1 σ2 dR1 dˆ x « + (ˆ ρ1A1 + ˆ ρ2A2) ¯ R ¯ F Radial velocity field V(ˆ r, ˆ x) = − ˆ r 2 dU dˆ x ,

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Mapping: 2D model

(ˆ r, ˆ x) → (ξ, η) maps Ωˆ

rˆ x = {[0, R2(ˆ

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

11 / 19

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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 degree of crystallinity equation ˆ U ∂Xi ∂η = kAi(0) exp

  • a2iS2

i

  • (X∞,i − Xi) ,

i = 1, 2, Effective dynamic viscosity ˆ µeff,i = ˆ Gi exp

  • ˆ

Ei

  • 1 − ˆ

Ti

  • + βi

Xi X∞,i ni i = 1, 2.

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Molecular orientation tensor components equations

U ∂Si rr ∂η =

  • Si rr + 1

3

  • (Si rr + Si θθ + Si xx − 1) dU

dη −φ ˆ λ

  • Si rr − N
  • Si rr + 1

3

  • (Si rr − Πi s)
  • ,

i = 1, 2, U ∂Si θθ ∂η =

  • Si θθ + 1

3

  • (Si rr + Si θθ + Si xx − 1) dU

dη −φ ˆ λ

  • Si θθ − N
  • Si θθ + 1

3

  • (Si θθ − Πi s)
  • ,

i = 1, 2, U ∂Si xx ∂η =

  • Si xx + 1

3

  • (Si rr + Si θθ + Si xx + 2) dU

dη −φ ˆ λ

  • Si xx − N
  • Si xx + 1

3

  • (Si xx − Πi s)
  • ,

i = 1, 2, Πi s = S2

i rr + S2 i θθ + S2 i xx. 13 / 19

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Two–dimensional velocity field ( ¯ B = 5)

U(ˆ r, ˆ z) V(ˆ r, ˆ z)

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SLIDE 20

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Influence of Biot number on cooling process

¯ B = 1 ¯ B = 5

15 / 19

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SLIDE 21

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Analysis of the 2D model

ˆ r = 0 (–), ˆ r = R−

1 (– –) and ˆ

r = R2 (− · −) η = 0 (–), η = 0,1 (– –) and η = 1 (− · −) 16 / 19

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

1D model vs 2D model

1D (–) and 2D (− · −)

17 / 19

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SLIDE 23

ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

Conclusions

Contributions of the present work:

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

semicrystalline fibres with modified Newtonian rheology.

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

1 + 1/2D one for slender fibres.

3 Determination of the two–dimensional fields of temperature,

molecular orientation tensor and degree of crystallinity for solid compound fibres.

4 Find substantial temperature non–uniformities (affect the degree

  • f crystallization and have great effects on the properties of

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

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ASME-ATI- UIT 2010, Sorrento, Italy J.I. Ramos and Francisco

  • J. Blanco–

Rodr´ ıguez Introduction Mathematical formulation Numerical method Numerical results

Two– dimensional model numerical results Comparison with

  • ne–dimensional

model

Conclusions

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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