プラスチック成形加工学会第30回年次大会

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溶融樹脂の定常非等温非ニュートン
純粘性多層流動に関する2.5D FEM 定式化
2.5D FEM formulation forsteady non-isothermal non-Newtonianviscous multi-layer flow of polymer melt

2019/6/13
株式会社HASL○谷藤 眞一郎
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Purpose

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Practical 2.5D FEM flow simulation for multilayer extrusion dies
3D FEM: Large computational cost,difficult operation
2D FEM: Low resolution

Multilayer spiral mandrel die

Feed block type multilayer coat hanger die
Multi-manifold coat hanger dieFig. 1 Multi layer extrusion dies
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Merits of 2.5D FEM formulation

3/20◇ Easy modelling
Layer 1Layer 2Standard FEM

2.5D FEM

Fig. 2 Analysis model of multi layer flow

◇ Small computational storage◇ Flexibility of shape representation
by modification of thickness information◇ High resolution solution in the thickness direction

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Break through of 2.5D FEM formulation

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Table 1 Boundary conditions on the interface of multi layer fluid I and II

Thermal flow
field

Previous formulation

Present formulation

Fully developed state

Includingunder-developing state

Velocity
continuity

V I  V IIShear stresscontinuity I  I   II  II

Normal stresscontinuity

p pIIIII
1 Dh IIIII 1 Dh
 p  2 I  p  2 IIh Dt
h Dt
II(I, II : Layer number, V : tangential velocity vector,
・η : viscosity, γ：shear
strain rate, p:pressure, h:layer thickness)
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Mathematical modeling of multilayer flow

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Momentum equation  l vil  p ll
for0hh,l  1 ~ nh  h  xi

2.5D FEM elementh

Shear stress
Layer-n,pn,hn,vni
・・・
Layer-l,pl,hl,vli
・・・
Layer-1,p1,h1,v1i

nn-1
ll-1
10xi
Fig.3 Multilayer flow configuration(layer number :n)
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reservedvilp lh  Ailhxi
lvelocitylh hh 1pllvi dh  Ai  l dh Bill0 
xi 0 pl:pressure
hl:thickness
vli:velocity
ηl:viscosity

Ali,Bli :integral constanti=1,2,3

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Continuity conditions on the interface
Shear stress continuityll 1l vi
l 1 vih h  hl
h h  0p l lh  Ail  Ail 1
xi

Velocity continuity
li h  hlv

l 1ih 0v

hl 1
p l hl hlll 1dh

Adh
B
Bi 0iixi 0  ll

Layer-l+1,pl+1,hl+1,vl+1ilth interface;l=1~n-1

Layer-l,pl,hl,vli
Fig.4 Interface of multilayer flow
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved

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Table 2 Simultaneous equations
derived by the boundary conditions
Condition

Equation
numberLower wall
non-slip3

Velocity continuity
on interface
Shear stress continuity
on interface
Upper wall
non-slip

Simultaneousconditional equations

Bi1  0

A   B  Bli3(n-1)

ll 1ili

p l l=hxil 1iA  Ali3(n-1)

p l lxi

p n nA   B 
xini

3nni

Total 6nhl

10 lCopyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reservedhl

h0ldh,   lldh

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Normal stress continuity
 p l  2 l

1 lll 1l 1 1V
h

p

2
V l 1  hl 1 for l  1 ~ n  1ll 1hh

Constraintnl
Element thickness : H e   hl 1

Energy equation

 l C lp Vl  T l   l T l   l  l  for l  1 ~ n2

Tl:temperatureρl:density
Clp:heat capacity
κl:thermal conductivity

SUPG(Streamline Upwind Petrov-Galerkin) FEM discretization

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Galerkin FEM discretization for continuity equationlSpl  Ql  Fl  0 for l  1 ~ nPressure
gradientlLayerinteraction Jd  d ,1 1 x xii
1S    l

Flux

1Ql    qil nil d ,e lF   f i Jd  d1 1 xil11Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reservedfi l  Ail  l  hl Bil 1 for l  1 ~ n  1,fi n  Ain  n ,hl( h) 20l

 ldhα，β：node number(ξ，η）：local coordinate
J：Jacobian ：interpolation function□ Theoretical verification of 2.5D FEM formulation 10/20

1Q   , H1
1H  H1  H 2

Q 2  , H 2
2

Fig.5 Steady viscous flow between parallel plate

H2h  1
HQ2
q  1Q2  1
：thickness ratio：flux ratio：viscous ratio

 h4  4  h3  3(    q  )  h2  4 q   h   q 2

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved

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χq=1

Fig.7 Viscosity ratio dependency for thickness ratio

Fig.6 Theoretical verification modelχη=0.5
Fig.8 Flow rate ratio dependency for thickness ratio

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12/20χq=32, χη=0.5

Fig.9 Velocity distribution
near the multi layer flow junction

Fig.10 Velocity distribution at outlet
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Fig.11 Pressure distribution along the machine direction□ Application of 2.5D FEM formulation

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Modeling of multi layer die① Mesh generation for mono layer flow region using template② Append import of mesh information
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved③ Geometrical representation for multi layer flow region④ Mesh generation for multi layer flow region

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Test analysis(3-material 3-layer flow analysis in the spiral mandrel die)2.5D FEM model

3D visualization modelFig. 12 Multilayer spiral mandrel die

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15/20Fig. 13 Pressure distribution
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16/20Fig. 14 Pressure distribution in machine direction
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17/20Fig. 15 Velocity distribution in thickness direction

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18/20Fig. 16 Thickness distribution in machine direction
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19/20Fig. 17 Thickness distribution
in circumferential direction

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□Conclusions

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成果：
・Hele-Shaw 薄流れの定式化を一般化することで多層押出ダイ内の未発達状態を含む流動状況及び界面形成状態を効率的に評価可能な解析法を構築した。

・理論検証解析を通じて解析結果の妥当性を検討した。

今後の課題：
・検証解析の継続。
・2.5D FEM 粘弾性多層流動解析への定式化拡張。

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved