プラスチック成形加工学会第33回年次大会

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押出成形 CAE における新規 2.5D FEM 解析技術の活用と今後の展望
Applications of new 2.5D FEM Analysis Technologyin Extrusion CAE and Future Prospections
2022/6/15((株)HASL）○谷藤 眞一郎，依藤 大輔(芝浦機械(株)）尾原 正俊((株)プラスチック工学研究所 ）鬼防 崇，辰巳 昌典(金沢大学）瀧 健太郎
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Purpose

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Practical 2.5D FEM simulation for polymer extrusion process

Single screw
Multi layer spiral mandrel die

Twin screw

Octa screw

Screw extruder

Feed block type multi die

Multi manifold dieExtrusion dieFig. 1 Various equipments used in the extrusion moldingCopyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Recently published papers concerning 2.5D FEM 3/30
1)

Ohara, M., Tanifuji, S., Sasai, Y., Sugiyama, T., Umemoto, S.,
Murata, J., Tsujimura, I., Kihara, S. and Taki, K.
「Resin Distribution along Axial and Circumferential Directions ofSelf-Wiping Co-Rotating Parallel Twin-Screw Extruder」
AIChE J.,66,11,e17018(2020)

2)

Ohara, M., Sasai, Y., Umemoto, S., Tanifuji, S., Kihara, S. and Taki, K.
「Experimental and Numerical Simulation Study of Devolatilization
in a Self-Wiping Corotating Parallel Twin-Screw Extruder」
Polymers, 12,11,2728(2020)

3)

4)
5)

Tanifuji, S.，Yorifuji, D.，Kibou, T. and Tatsumi, M.
「A Generalized 2.5D FEM Formulation for Steady Non-Newtonian Viscous Multi Flow
Part 1: Formulation and Theoretical Verification」
Seikei-Kakou，33,2,60(2021)

Twin Screw

Twin Screw

Multi Layer Die

Tanifuji, S.，Yorifuji, D.，Kibou, T. and Tatsumi, M.
「A Generalized 2.5D FEM Formulation for Steady Non-Newtonian Viscous Multi Flow Multi Layer DiePart 2: Implementation of Pseudo Encapsulation Model and Experimental Verification」
Seikei-Kakou，33,12,447(2021)
Liu, C., Mikoshiba, S., Kobayashi, Y., Ishigami, A., Yorifuji, D.,Tanifuji, S. and Ito, H.
「Experimental Investigation and Numerical Simulationof a Self-Wiping Corotating Parallel Octa-Screw Extruder」
Polymers, 14,6,1201(2022)

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Octa Screw

□ Back ground of 2.5D FEM formulation
3D FEM

2.5D FEM(u, v, w) : Flow velocity vector
p:Flow pressure
Fig. 2 Comparison of 3D and 2.5D FEM discretization procedures
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□ Generalization of 2.5D FEM formulation

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Pressure equation
Original Form since 1970’~ (Cornell university)
Flow analysis in injection molding
Flow analysis in mono layer extrusion die

Q = S p
Flux

Pressure gradient flowGeneralizationin semi analytical integration

Generalized Form
Flow analysis in screw extruder

Flow analysis in multi layer extrusion dieQ = S p + DlQl = Spl + Fl = 0 for l = 1 ~ nDrag flow

Interaction flow with adjacent layers

α，β：Nodal number
S αβ ：Flow conductance matrix D , Fl ：Load vector

Fig. 3 Generalized form of the pressure equation
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Free (Layer) surface tracking method
Multi purpose method

Eulerian method :VOF
Level-Set
CIP

Problem oriented method in 2.5D FEM
Un-fill state in screw extruder :Rearrangement of the pressure distributionp( , z ) = p( +  , z + z ) −Arbitrary Lagrangian Eulerian method :ALEpp − zzMulti layer surface :Particle method :DEM

Balance equation of the normal stresson multi layer surface
II1 Dh III
II 1 Dh− p + 2 I
= − p + 2 II
h Dt
h Dt

SPHMPSI

Free surface updated by using
flow velocity vector information

IFree surface updated by using
flow pressure information

Fig. 4 Various free surface tracking methods
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Application for the twin screw extruder

7/30「Resin Distribution along Axial and Circumferential Directions ofSelf-Wiping Co-Rotating Parallel Twin-Screw Extruder」1)

Experimental information

Numerical prediction

Homo polypropylene (F-704NP)
supplied by Prime Polymer (Japan)

Transport direction

Fig.5 Qualitative comparison of the resin fill distribution measured during
the experiment and predicted by the simulation for 1 kg/h and 88 rpm
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Fig.6 Open-view port of TSE and line laser
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9/30Theoreticallyf =Q 1Q=Qd g NExp.
Sim.

Fig.7 Comparison of simulation and experimental results
for the effect of throughput on the degree of fill
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10/30「Experimental and Numerical Simulation Study of Devolatilization
in a Self-Wiping Corotating Parallel Twin-Screw Extruder」2)

Q＝3 kg/h

Fig.8 Screw configuration and barrel temperature settings

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S: mC ( zs ) − C *1/2== exp ( SDb Dm N ) ( ze − zs ) *C ( ze ) − C Qw
Fig.9 Schematic illustration of Latinen’s model*)for devolatilization in a single-screw extruder*) Latinen, G.A. Devolatilization of viscous polymer systems. In
Polymerization and Polycondensation Processes; American Chemical
Society: Washington, DC, USA, 1962; Volume 34, pp. 235–246.

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved:Devolatile efficiency
ze − z s : Partial fill interval m : Melt density
Qw :Mass flux
Db :Barrel diameter
Dm : Diffusion coefficient
Rotation speed
N:12/30

Qw : 3kg / h

Fig.10 Simulation results of toluene concentration along
the machine direction of twin screw extruder

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13/30(a) Volatile concentration(b) Resin fill region

Fig.11 Comparison of the experiment and the predicted
information ((a) volatile concentration, (b) resin fill region)

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□ Application for the multi extrusion die

14/30「A Generalized 2.5D FEM Formulationfor Steady Non-Newtonian Viscous Multi Flow : Part 1 & 2」3),4)
Surface layer : LDPE (LC720),4 kg/h,215℃
Mid layer : LDPE (LF405M),16 kg/h,215℃

Fig. 12 Feed block type coat hanger die used in the experimental verification
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15/30Fig. 13 Snapshots of the extruded multilayer film((a) flat and (b) variated)
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16/30Viscoelastic phenomena neglected in the traditional 2.5D FEM

Transport direction

Main effecting factor :first normal stress difference 1  0, 2 = 0

Main effecting factor :second normal stress difference 1 = 0, 2  0

Transportdirection

Secondary flowat contraction of channel
Secondary flow in the rectangular cross section channel
Die swell

Variation of multi layer flow interface in the circular cross section channel(blue region : high viscosity flow, gray region : low viscosity flow )Fig. 14 Various viscoelastic phenomena
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17/30Derivation of Pseudo Encapsulation Model*) Criminale, Jr. W.O., Ericksen, J.L. and Filbey, Jr. G. L. :Arch. Rat. Mech. Anal., 1,410(1985)

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TD: Transverse Direction

p  xz  yz  zz =++= ( 2 xy xz ) + 2 xz2zxyz
yz()

 , 2 , H : constant
Averaging in thickness direction
Using analytical solutionof mono layer flow

Approximate representation of pressurep( z ) = p(0) + 2 W2 g f ( z ),

W =W p,2 x (2n − 1) z  sinh 2  
32  H 1H 
g f ( z) = 4   4  W  n =1 (2n − 1)  (2n − 1) W   cosh 2H

2Fig. 15 Dependency of aspect ratio
and TD position for the shape factor : gf
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Representation of the encapsulation using PEM    h2  1 + 2  u 1 = p1 − p2 +  2,2  W2 2 − 2,1  W2 1 g f ( z ) h1 h2  x(h1pp→
02
 1x p  p → h1  02
 1
x
Assumption:)22 1 p2  1 p1 h1
 2,2 
0   2,1 
 →
x
x
x
 2

 1


22
 1 p2  1 p1 
h1


→
0

2,1
 2,2x
x
x
21



p1  p2 , 
 = 1  2,1  1
2  2,2?

Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved2   →h1
0x

Low-viscosity fluid encapsulates highviscosity fluid under limited condition.

!!!

20/302.5D FEM model

Developing
flow region & PEM

Fully developed
flow region

Scale unit : [mm]Fig. 16 Multi layer flow analysis model
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21/30Material properties
Surface layer

Table 1 Second normal difference
coefficients of PEM

Mid layerLDPE (LC720,LF405M)
supplied by Japan Polyethylene Corp.

Fig.17 Flow curve fitted using the Cross model
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22/30 2  Pa • s 2 
Layer1 Layer 2 Layer3
Case4： -18 -60 -180
Case0： 0
Case1：

Fig. 18 Predicted surface layer thickness at the outlet position (B-B’)of the feed block channel under the flat condition
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23/30Fig. 19 Predicted layer thickness ((a) surface layer, (b) mid layer : flatand (c) surface layer, (d) mid layer : variated)
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24/30Mysterious nature of multi layer flow
Multi layer flow (  surafce  mid )Flow velocity

Flow velocity

Mono layer flow

Hsurface：Hmid=1:1
Hsurface：Hmid=1:2
Hsurface：Hmid=1:5

HeightΔp < Δp < Δp2 w
p =H
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights ReservedHeightΔp =Δp = Δp !!!
h4 = −4 h3 − 3( − q  )h2 + 4q  h + q 2for two layer flow under fully developed state25/30  pl  Sl = Flxi  xi Elliptic typeDiffusionpl = p for l = 1 ~ nvmi

 qlmxi=0

hyperbolic type
Convection

Fig. 20 Predicted flux ratio χ12(= q1/q2 = χ32) : contour plot of (a) flat and (b) variated ,graph plot of (c) inlet (B-B’) and (d) outlet (C-C’)
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26/30Fig. 21 Comparison of the predicted pressure((a) flat and (b) variated)
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27/30Fig. 22 Comparison of the measured and the predicted flux rate
at the die outlet position (C-C’), ((a) flat and (b) variated)
Copyright© 2010~ Hyper Advanced Simulation Laboratory Co., Ltd. All Rights Reserved□Conclusions

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・Various phenomena in the polymer extrusion process can be quantified
in a short compulational time with an easy operation using 2.5D FEM.
・The unfilled state in the twin screw extruder can be represented
with moderate accuracy by the pressure rearrangement algorithm.

・The implementation of PEM reduces the discrepancy between the predicted
layer thickness and the measured value in a feed block type coat hanger die.

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

29/30「 Experimental Investigation and Numerical Simulationof a Self-Wiping Corotating Parallel Octa-Screw Extruder 」5)

Transport direction

Fig. 23 Measured and calculated temperature distribution in an octa screw extruder
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30/30Experimental verifications and enhancement of applications
For screw extruder

Side feed, Reactive extrusion, etc.

For multilayer extrusion die

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