ASME-ATI- UIT 2010, Sorrento, MODELLING OF MOLECULAR Italy ORIENTATION AND CRYSTALLIZATION IN J.I. Ramos and Francisco J. Blanco– THE MANUFACTURE OF Rodr´ ıguez SEMI–CRYSTALLINE COMPOUND FIBRES Introduction Mathematical formulation J.I. Ramos and Francisco J. Blanco–Rodr´ ıguez Numerical method Numerical Escuela de Ingenier´ ıas results Universidad de M´ alaga, Spain Two– dimensional model numerical results Conference on Thermal and Environmental Issues in Comparison with one–dimensional Energy Systems model 16–19 May 2010, Sorrento, Italy Conclusions 1 / 19
Outline ASME-ATI- UIT 2010, Sorrento, Italy 1 Introduction J.I. Ramos and Francisco J. Blanco– 2 Mathematical formulation Rodr´ ıguez Introduction 3 Numerical method Mathematical formulation 4 Numerical results Numerical method Two–dimensional model numerical results Numerical results Comparison with one–dimensional model Two– dimensional model numerical results 5 Conclusions Comparison with one–dimensional model Conclusions 2 / 19
Introduction ASME-ATI- Bi–component compound fibres are manufactured by UIT 2010, Sorrento, MELT SPINNING processes. Italy Applications J.I. Ramos and Francisco 1 Telecommunications: Data transmission. J. Blanco– Rodr´ ıguez 2 Chemical industry: Filtration and separation processes. 3 Biomedical industry. Introduction 4 Textile industry. Mathematical formulation Necessary: modelling of the drawing process for both Numerical hollow and solid semi–crystalline compound fibres. method Previous studies are based on one–dimensional models of Numerical results amorphous, slender fibres at low Re . Two– dimensional NO INFORMATION ABOUT RADIAL VARIATIONS. model numerical results Comparison Use of a hybrid model for fibre spinning. with one–dimensional model Conclusions 3 / 19
Melt spinning ASME-ATI- UIT 2010, Sorrento, Involves the extrusion and drawing of a polymer cylinder. Four Italy zones. J.I. Ramos and Francisco J. Blanco– 1 Shear Flow Rodr´ ıguez Region. Introduction 2 Flow Mathematical formulation Rearrangement Numerical Region. method 3 Melt Drawing Numerical results Zone. Two– dimensional model numerical 4 Solidification results Comparison region. with one–dimensional model Conclusions 4 / 19
Melt spinning ASME-ATI- Involves the extrusion and drawing of a polymer cylinder. Four UIT 2010, Sorrento, zones. Italy J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Region. Introduction 2 Flow Mathematical formulation Rearrangement Numerical Region. method Numerical 3 Melt Drawing results Zone. Two– dimensional model numerical results 4 Solidification Comparison with region. one–dimensional model Conclusions 4 / 19
Melt spinning ASME-ATI- Involves the extrusion and drawing of a polymer cylinder. Four UIT 2010, Sorrento, zones. Italy J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Region. Introduction 2 Flow Mathematical formulation Rearrangement Numerical Region. method Numerical 3 Melt Drawing results Zone. Two– dimensional model numerical results 4 Solidification Comparison with region. one–dimensional model Conclusions 4 / 19
Melt spinning ASME-ATI- Involves the extrusion and drawing of a polymer cylinder. Four UIT 2010, Sorrento, zones. Italy J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Region. Introduction 2 Flow Mathematical formulation Rearrangement Numerical Region. method Numerical 3 Melt Drawing results Zone. Two– dimensional model numerical results 4 Solidification Comparison with region. one–dimensional model Conclusions 4 / 19
Melt spinning ASME-ATI- Involves the extrusion and drawing of a polymer cylinder. Four UIT 2010, Sorrento, zones. Italy J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Region. Introduction 2 Flow Mathematical formulation Rearrangement Numerical Region. method Numerical 3 Melt Drawing results Zone. Two– dimensional model numerical results 4 Solidification Comparison with region. one–dimensional model Conclusions 4 / 19
Melt spinning ASME-ATI- Involves the extrusion and drawing of a polymer cylinder. Four UIT 2010, Sorrento, zones. Italy J.I. Ramos and Francisco J. Blanco– Rodr´ ıguez 1 Shear Flow Region. Introduction 2 Flow Mathematical formulation Rearrangement Numerical Region. method Numerical 3 Melt Drawing results Zone. Two– dimensional model numerical results 4 Solidification Comparison with region. one–dimensional model Conclusions 4 / 19
Mathematical formulation (I) ASME-ATI- Mass conservation equation UIT 2010, Sorrento, Italy ∇ · v i = 0 i = 1 , 2 , J.I. Ramos where v = u ( r, x ) e x + v ( r, x ) e r and Francisco Linear Momentum conservation equation J. Blanco– Rodr´ ıguez � ∂ v i � = −∇ p + ∇· τ i + ρ i · f m ρ i ∂t + v i · ∇ v i i = 1 , 2 , Introduction where f m = g e x Mathematical Energy conservation equation formulation Numerical � ∂T i � method ρ i C i ∂t + v i · ∇ T i = − k i ∆ T i i = 1 , 2 , Numerical results Constitutive equations Two– Rheology dimensional model numerical ∇ v + ∇ v T � results τ = µ eff � + τ p , Comparison with where one–dimensional model � − λ � Conclusions ∇ v T : S � � τ p = 3 c k B T φ F ( S ) + 2 λ ( S + I / 3) 5 / 19
Mathematical formulation (II) ASME-ATI- Molecular orientation tensor equation: Doi–Edwards theory UIT 2010, Sorrento, Italy S (1) = F ( S ) + G ( ∇ v , S ) , F ( S ) = − φ J.I. Ramos λ { (1 − N/ 3) S − N ( S · S ) + N ( S : S ) ( S + I / 3) } and Francisco G ( ∇ v , S ) = 1 ∇ v T : S “ ∇ v + ∇ v T ” “ ” J. Blanco– − 2 ( S + I / 3) . 3 Rodr´ ıguez where subscript (1) denote UCTD operator Introduction Λ (1) = ∂ Λ ∇ v T · Λ + Λ · ∇ v “ ” Mathematical ∂t + v · ∇ Λ − formulation Molecular orientation scalar order parameter Numerical method r 3 Numerical S ≡ 2 ( S : S ) S = diag ( S rr , S θθ , S xx ) , results Two– Crystallization: Avrami–Kolmogorov’s theory & Ziabicki’s model dimensional model numerical results ∂ X i Comparison ∂t + v · ∇X i = k Ai ( S i ) ( X ∞ ,i − X i ) i = 1 , 2 , with one–dimensional model where Conclusions a 2 i S 2 ` ´ k Ai ( S i ) = k Ai (0) exp , i = 1 , 2 . i 6 / 19
Mathematical formulation (III) ASME-ATI- Kinematic, dynamic and UIT 2010, thermal boundary conditions Sorrento, Italy are required: J.I. Ramos and Francisco Initial conditions ( t = 0 ) J. Blanco– Rodr´ ıguez Symmetry conditions ( r = 0 ) Introduction Die exit conditions ( x = 0 ) Mathematical formulation Take–up point conditions Numerical ( x = L ) method Conditions on free surfaces Numerical results of compound fibre Two– dimensional model numerical ( r = R 1 ( x ) and results Comparison r = R 2 ( x ) ) with one–dimensional model Conclusions 7 / 19
Non–dimensionalize ASME-ATI- Non–dimensional variables UIT 2010, Sorrento, t r = r x = x ǫ = R 0 Italy ˆ t = ˆ ˆ ⇒ ( L/u 0 ) R 0 L L J.I. Ramos and Francisco u = u v p T = T ˆ J. Blanco– ˆ v = ˆ p = ˆ Rodr´ ıguez u 0 ( u 0 ǫ ) ( µ 0 u 0 /L ) T 0 ρ = ρ C = C µ = µ k = k ˆ ˆ Introduction ˆ ˆ ρ 0 C 0 µ 0 k 0 Mathematical formulation 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 , Numerical µ 0 results Pr = µ 0 C 0 Bi = hR 0 Two– , Pe = Re Pr, dimensional k 0 k 0 model numerical results Comparison with one–dimensional model Conclusions 8 / 19
Asymptotic analysis: 1 D model ASME-ATI- UIT 2010, Sorrento, Asymptotic method using the fibre slenderness, ǫ << 1 , as Italy perturbation parameter 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 Flow regime considered for steady ( ∂ formulation t = 0 ) jets ∂ ˆ Numerical method ¯ ¯ F C Re = ǫ ¯ R, Fr = ǫ , Ca = ǫ , Numerical results Two– dimensional Bi = ǫ 2 ¯ Pe = ǫ ¯ P, B model numerical results Comparison where ¯ with Υ = O (1) . one–dimensional model Conclusions 9 / 19
One–dimensional equations of the 1 + 1 / 2 D model ASME-ATI- Asymptotic one–dimensional mass conservation equation UIT 2010, Sorrento, A 1 U = Q 1 , A 2 U = Q 2 Italy J.I. Ramos where and Francisco A 1 = R 2 A 2 = R 2 2 − R 2 J. Blanco– 1 1 2 , , Rodr´ ıguez 2 Asymptotic one–dimensional linear momentum equation Introduction Mathematical „ « formulation ρ 2 A 2 ) U d U d µ eff, 2 > A 2 ) d U ¯ 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 1 „ d R 2 x + σ 1 d R 1 « + 2 ¯ C d ˆ σ 2 d ˆ x Numerical results ¯ R + (ˆ ρ 1 A 1 + ˆ ρ 2 A 2 ) Two– ¯ dimensional F model numerical results Radial velocity field Comparison with one–dimensional model x ) = − ˆ r d U V (ˆ r, ˆ x , Conclusions 2 d ˆ 10 / 19
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