8:00-8:30am 30 min.

Fill ratio distribution in a co-rotating self-wiping
twin screw extruder—theoretical andexperimental study○Kentaro Taki, Kanazawa University, Kanazawa, Ishikawa, Japan,
Shin-ichiro Tanifuji, HASL, Tokyo, Japan,Takemasa Sugiyama, Kanazawa University, Kanazawa, Ishikawa, Japan
Jun-ichi Murata, Kaneka, Osaka, Japan,
Isao Tsujimura, Kaneka, Osaka, Japan(C)Kentaro Taki, Kanazawa University

1Agenda1. Introduction
2. Theory and formulation
3. Algorithm for fill ratio distribution
4. Conclusion(C)Kentaro Taki, Kanazawa University2Numerical Simulation of TSE

Resolution

3D sim etc[1]

This study[2]
FAN method[3]

1s1-10 min

1 day-1 weekCalculation time
1. Yasuya Nakayama, Toshihisa Kajiwara, Tatsunori Masaki, AIChE J, 62(7), 2563-2569 (2016).b
2. Andreas Eitzlmayr, Johannes Khinast, Chemical Engineering Science, 134, 861-879 (2015).
3. Tadmor, Z., Broyer, E. and Gutfinger, C.: Polym. Eng. Sci., 14,660(1984)
4. Broyer, E., Gutfinger, C. and Tadmor Z. : Trans. Soc. Rheology, 19, 423(1975)
5. J. L. White, Z. Y. Chen, Polymer Engineering and Science, 34(3), 229-237 (1994).
6. Santosh Bawiskar, James L. White, Polymer Engineering & Science, 38(5), 727-740 (1998).
7. James L. White, Eung Kyu Kim, Jong Min Keum, Ho Chul Jung, Dae Suk Bang,(C)Kentaro
Polymer Engineering& Science,University41(8), 1448-1455 (2001).
Taki, Kanazawa
8. J. L. White, B. J. Kim, S. Bawiskar, J. M. Keum, Polymer-Plastics Technology and Engineering, 40(4), 385-405 (2001).

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What our simulator can do.
Hele-Shaw flow model + finite element method
Pressure distribution
Fill ratio distribution

Strain rate, velocity, flow rate distribution on whole screw elements
Strain magnitude

Droplet dispersion

Fiber attrition(C)Kentaro Taki, Kanazawa University

Plasticization

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Advantage of our software
1. Calculation cost is low as the pressure is the only unknown variable tobe determined for isothermal condition.
2. It is easy to calculate flow metrics; velocity, strain rate, viscosity
distribution by the integration along the thickness (r direction) when
the pressure and temperature are determined.
3. Flow balance satisfies with high-precision.
Q4

Q3
The total flow rate at anode becomes zero forany values of pressure.

Element

44Qα = 0∑αQ ∑ φα = 1α =1=1Q2Q1q4q3●q1q2● Internal node

The summation of flow rate forinternal nodes calculated by thepressure becomes zero.
(C)Kentaro Taki, Kanazawa University

4qα = 0∑α=15Hele-Shaw flow 2.5D model in cylinder
vr = 0, vz and vθ are solo function of r.
vθ = 0, v z = 0 at r = Rsvθ = Rb Ω, v z = 0 at r = Rb

ΩRsRbzrθRs: Screw radius
Rb: Barrel radius(C)Kentaro Taki, Kanazawa University

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Geometry(C)Kentaro Taki, Kanazawa University

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Shape function
444α =1α =1α =1p = ∑ φα pα , θ = ∑ φα θα , z = ∑ φα zα , r = ( Rb − Rs ) ζ L + Rs
φ1 =11
11(1 − ξ L )(1 − η L ) , φ2 = (1 + ξ L )(1 − η L ) , φ3 = (1 + ξ L )(1 + η L ) , φ4 = (1 − ξ L )(1 + η L )4444

Element in cylindrical coordinate

Element in computational coordinate(C)Kentaro Taki, Kanazawa University8

Eq. of Continuity in cylindrical coordinate
As ρ = constant and vr = 0,∂ρ 1 ∂1 ∂∂+( ρrvr ) +( ρvθ ) + ( ρvz ) = 0∂t r ∂rr ∂θ∂z1 ∂∂( ρvθ ) + ( ρvz ) = 0r ∂θ∂z(C)Kentaro Taki, Kanazawa University

9Eq. of Motion
θ-component

As steady state, ∂vθ/∂θ = ∂vθ/∂z = 0, vr = 0,22∂v
∂vθ 2η ∂vr 1 ∂p
∂ η ∂ ηθ=( rvθ )  + 2 2 + η 2 + 2∂r  r ∂rr ∂θ r ∂θ∂z r ∂θ1 ∂  2  ∂vθ vθ   1 ∂p
−  =
 r η2r ∂r  ∂r r   r ∂θ(C)Kentaro Taki, Kanazawa University

10Eq. of Motion (Cont’)
z-component

As steady state, ∂vz/∂θ = ∂vz/∂z = 0,∂vz ∂p
1 ∂  ∂vz  η ∂ vzrη
+ 2+η 2 =
2r ∂r ∂r  r ∂θ∂z∂z
21 ∂  ∂vz  ∂p
 rη
=r ∂r  ∂r  ∂z(C)Kentaro Taki, Kanazawa University11

Strain rate
θr-component∂  vθ  1  ∂p  1  Ω  1  Ω γ&θ r ≡ r   =   1 − 2    + 2  ∂r  r  2η  ∂θ   r  γ c   η r  γ c zr-component∂vz
1  ∂p   1  α c  =γ&zr ≡   r −   ∂z 2η  ∂z  r  βc  Integration constant
3
Rb 1
Rb r
1
αc = ∫dr , β c = ∫dr , γ c = ∫dr , δ c = ∫drRs ηRs η rRs η r 3
Rs ηRbr

Rb(C)Kentaro Taki, Kanazawa University12

Velocity
θ-component  Ωr  r 1
 βc  r 1r  ∂p   r 1vθ =    ∫dr −   ∫dr  + dr33∫RR2  ∂θ   Rs η r
 γ c  s ηr
  γ c  s ηrz-component

αc r 1 1  ∂p   r r
vz =    ∫dr − ∫dr β c Rs η r 2  ∂z   Rs η

Integration constant

3Rb 1
Rb r
1
αc = ∫
dr , β c = ∫dr , γ c = ∫dr , δ c = ∫drRs ηRs η r
Rs η r 3Rs ηRbr

Rb(C)Kentaro Taki, Kanazawa University13Flow rateθ-component

2βc  Ω  2 βc 1  ∂p qθ ≡ ∫ vθ dr = − 
α c −  +  Rb − Rs4  ∂θ  γc  2 γc z-component

α c2 1 ∂p qz ≡ ∫ vz rdr = −δc − 
Rs4 ∂z βc RbRbIntegration constant
3
Rb 1
Rb r
1
αc = ∫dr , β c = ∫dr , γ c = ∫dr , δ c = ∫dr3
Rs ηRs η r
Rs η rRs ηRbr

Rb(C)Kentaro Taki, Kanazawa University

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Flow balance of each element(eθ)

Qα = − S αβ + S αβ p β + Dαe

Flow through anode of element

ez

Pressure-gradientdriven flow

Qαe = ∫ φα ( nθ qθ + n z q z ) dΓ,eDrag-force
driven flow
β c2 α c2 1
1
z



,, S = δ c −S = α c −γ c 44β c θΓe∂φα ∂φβ∂φ ∂φ βez
= Sz ∫ αdS , SαβdS ,∂θ ∂θ∂z ∂zSeSeeθ= Sθ ∫Sαβe

Dθ =Ω  2 βc  Rb − 2 γc ∂φαdS∂θSeDαe = Dθ ∫3Rb 1Rb r
1dr , γ c = ∫dr , δ c = ∫drα c = ∫ dr , β c = ∫Rs ηRs η r
Rs η r 3
Rs ηRb

φ1 =r

Rb

1
1
11
(1 − ξ L )(1 − η L ) , φ2 = (1 + ξ L )(1 − η L ) , φ3 = (1 + ξ L )(1 + η L ) , φ4 = (1 − ξ L )(1 + η L )44
44(C)Kentaro Taki, Kanazawa University15

Filled and unfilled calculation
Completely filled screws
Press. at feed
0MPa（given）

Head press.
10MPa（given）
Q = 20kg/h (Calc. result)Press. at feed0MPa（given）

Head press.
5MPa（given）
Q = 40kg/h (Calc. result)Partially filled screw
Press. at feed
0MPa（given）？
Q = 20kg/h(given)(C)Kentaro Taki, Kanazawa University

One degree of freedom is added.

Head press.
5MPa（given）
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Pressure downstream update scheme
Screw inlet position

Screw head position

Z：Element to be renew：Downstream elements

1st step pressure distribution

ZPressure downstream update schemep ( z − ∆z , θ − ∆θ ) = p ( z , θ ) −Corrected pressure distribution

f=1.0
f=0Z

Filling ratio distribution

FAN method (1D)∂p∂p∆z −∆θ∂z∂θAs the pressure is need to be determined according to thetwo coordinates (z, θ), the pressure downstream update
scheme is developed.

(C)Kentaro Taki, Kanazawa University

2.5D FEM

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Criteria of filled and unfilled
Pressure gradient Flow balance

Pressure Fill or Unfill

dp/dx<0Q>Qdp>0

Filled1) fe=1

dp/dx>0Q<Qdp>0Filled2) fe=1

dp/dx>0Q<Qdp<0p=0Unfilled3) fe=01) Q>Qd, unfilled state is impossible.
2) Q<Qd , p>0 : Filled sate and back flow (pressure gradient driven flow) is larger than the net flow.
3) Q<Qd , p<0 : Unfilled state and correction of p=0,W=fW are applied for FAM method (f=Q/Qd).

WVb H
WH 3  dp 
Q=sin( 2θ ) − 412η  dx Drag-force driven flow Qd

Pressure gradient driven flow, Qp(C)Kentaro Taki, Kanazawa University18

Correction of pressure distribution
Fill ratio control line

↑QextZ

Average fill ratio
determined bypressure distributionz-axis fill ratio
determined byscrew geometry
Qdz = Vdave cos φ sin φ∆XHfav=e eV∑ feV∑ ∆X= ∑ Dα sin φ cos φ  ∆ZDαQextf =zQ∑ dz

Adjusting pressure distribution to agree f av with f z.
(C)Kentaro Taki, Kanazawa University19

Performing simulation
Simple wizard for constructing screw geometry

Automatic meshing

Viscosity-strain rate model e.g., Cross model
Operation variables (Feed rate, Q, Head pressure P, Rot speed N)

Complexv
iscosity[Pa·s]

10000

1000

180°C
190°C
200°C

100
1E-3

0.01

0.1

1101001000

Strain rate [s-1]Calculation of pressure distribution by FEM.

Fill ratio distribution(C)Kentaro Taki, Kanazawa University

20Effect of Screw Speed (rpm)Fill ratio
Un-filled

FilledFeed rate 0.75kg/h
head

100 rpm
50 rpm30 rpmDecrease of screw speed expands filled region backwards.
(C)Kentaro Taki, Kanazawa University

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Effect of feed rate (kg/h)
Fill ratio
Filled

Un-filled

Speed 30 rpm
0.25 kg/h0.50 kg/h

0.75 kg/h

Increase of feed rate expands filled region backwards.
(C)Kentaro Taki, Kanazawa University22

Experiments
Material
Homo polypropylene (F-704NP, Prime Polymer, Japan)
Viscosity-strain rate curve was obtained by the Cross model

Complex viscosity [Pa·s]

10000

Cross modelη ( γ&, T , P ) =

1000

0.01

0.1

1101− c η0γ& 1+  * τ Tb
η0 = a exp T+
273.15


180°C
190°C
200°C

100
1E-3

η0100

1000

Strain rate [s-1](C)Kentaro Taki, Kanazawa University23Screw geometry and barrel temp.
Self-wiping co-rotating parallel twin screw extruder
L/D = 90, φ = 15 mm(Technovel, Japan)(C)Kentaro Taki, Kanazawa University24Fill ratio distribution
30 rpm
0.5 kg/hFill ratio
Low (0)High (1)Pulled-out screw
Simulation(C)Kentaro Taki, Kanazawa University

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Screw speed (rpm) vs. Fill ratio
0.3

Calc.
Fill ratio [-]

0.2

0.1

The calculation results
agree with the theoreticaldefinition, Q/Qd as the fillratio became half when thescrew speed becomesdouble.

x0.5x20.0
406080100120Screw rotational speed [rpm](C)Kentaro Taki, Kanazawa University

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Feed rate vs. fill ratio
0.4

Calculation agrees withthe theoretical definitionof fill ratio, Q/Qd as theplots exist on the straightline from origin.

Fill ratio [-]

0.3

0.2

0.1

Calc.

0.0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

Feed rate [kg/h](C)Kentaro Taki, Kanazawa University27

Throughput vs. fill ratio
0.4

Calculation agrees withthe theoretical definitionof fill ratio, Q/Qd as theplots exist on the straightline from origin.

Fill ratio [-]

0.3

0.2

Calc.
0.1

0.0
0.00000.00250.00500.00750.0100Feed rate and Rotational spped ratio, Q/N [kg/(h rpm)](C)Kentaro Taki, Kanazawa University28

Conclusion• We achieved a development of 2.5D Hele-Shaw model for
calculation of pressure distribution in twin screw extruder.
• This model has advantage for short-calculation time, possible
to whole screw elements.
• The fill ratio was calculated based on the FAN method.
• Further experimental validations not only the fill ratio but alsofiber attrition, plasticization etc are demanded.
• Devolatilization model will be added.
(C)Kentaro Taki, Kanazawa University

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Contact information• This software is proprietary and commercially available by theHASL Co. ltd., Japan.
• Please contact– Dr. Shin-ichiro Tanifuji, CEO of HASL.
– URL: http://www.hasl.co.jp– E-mail: tanifuji@hasl.co.jp
CEO Dr. Shin-ichiro Tanifuji

Thank you for your kind attention.
(C)Kentaro Taki, Kanazawa University

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