fill ratio distribution in a co-rotating self-wiping twin ... · performing simulation...

Fill ratio distribution in a co-rotating self-wiping

twin screw extruder—theoretical and

experimental studyKentaro 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

8:00-8:30am 30 min.

(C)Kentaro Taki, Kanazawa University1

Agenda

1. Introduction

2. Theory and formulation

3. Algorithm for fill ratio distribution

4. Conclusion


Numerical Simulation of TSE

Calculation time

Re

solu

tio

n

3D sim etc

FAN method

This study

[1]

[3]

[2]

1 s 1-10 min 1 day-1 week

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, Polymer Engineering & Science, 41(8), 1448-1455 (2001).

8. J. L. White, B. J. Kim, S. Bawiskar, J. M. Keum, Polymer-Plastics Technology and Engineering, 40(4), 385-405 (2001).(C)Kentaro Taki, Kanazawa University

3

What our simulator can do.

Hele-Shaw flow model + finite element method

Fill ratio distribution

Pressure distribution

Strain magnitude Droplet dispersion Fiber attrition Plasticization

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


Advantage of our software

5

Q1Q2

Q3Q4

1. Calculation cost is low as the pressure is the only unknown variable to

be 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.

The total flow rate at a

node becomes zero for

any values of pressure.

The summation of flow rate for

internal nodes calculated by the

pressure becomes zero.

Internal node

q1 q2

q3q4

1 04

1

4

1

== ∑∑== α

αα

α φQQ

04

1

=∑=α

αq

Element

(C)Kentaro Taki, Kanazawa University

Hele-Shaw flow 2.5D model in cylinder

rRs

Rb

Ω

Rs: Screw radius

Rb: Barrel radius

θ

sz Rratvv === 0 ,0θ

bzb RratvRv ==Ω= 0 ,θ

z

vr = 0, vz and vθ are solo function of r.


Geometry


Shape function

( )4 4 4

1 1 1

, , , b s L sp p z z r R R Rα α α α α αα α α

φ θ φ θ φ ζ= = =

= = = = − +∑ ∑ ∑

( )( ) ( )( ) ( )( ) ( )( )1 2 3 4

1 1 1 11 1 , 1 1 , 1 1 , 1 1

4 4 4 4L L L L L L L Lφ ξ η φ ξ η φ ξ η φ ξ η= − − = + − = + + = − +

Element in cylindrical coordinate Element in computational coordinate


Eq. of Continuity in cylindrical coordinate

( ) ( ) ( )1 10r zrv v v

t r r r zθ∂ρ ∂ ∂ ∂+ ρ + ρ + ρ =∂ ∂ ∂θ ∂

As ρ = constant and vr = 0,

( ) ( )10zv v

r zθ∂ ∂ρ + ρ =∂θ ∂


Eq. of Motion

22

1 1v v pr

r r r r rθ θ ∂ ∂ ∂ η − = ∂ ∂ ∂θ

θ-component

( )2 2

2 2 2 2

2 1rv v v prv

r r r rr z rθ θ

θη η ηη

θ θθ∂ ∂ ∂∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂∂ ∂

As steady state, ∂vθ/∂θ = ∂v

θ/∂z = 0, vr = 0,


Eq. of Motion (Cont’)

1 zv pr

r r r z

∂∂ ∂ η = ∂ ∂ ∂

z-component

2

2 2 2

1 z z zv v v pr

r r r zr z

ηη ηθ

∂ ∂ ∂∂ ∂ + + = ∂ ∂ ∂∂ ∂

As steady state, ∂vz/∂θ = ∂vz/∂z = 0,


Strain rate

2 2

1 1 11

2rc c

v pr

r r r r

∂ ∂ Ω Ω ≡ = − + ∂ ∂ &

θθγ

η θ γ γη

1 1

2cz

zrc

v pr

z z r

αγη β

∂ ∂ ≡ = − ∂ ∂ &

θr-component

zr-component

3

3

1 1, , ,

b b b b

s s s s

R R R R

c c c cR R R R

r rdr dr dr dr

r rα β γ δ

η η ηη= = = =∫ ∫ ∫ ∫

Integration constant


Velocity

3 3

1 1 1

2 s s s

r r rc

R R Rc c

r p rv dr dr dr

r r rθβ

θ η γ γη η ∂ Ω = − + ∂ ∫ ∫ ∫

1 1

2 s s

r rc

z R Rc

p rv dr dr

z r

αη β η

∂ = − ∂ ∫ ∫

θ-component

z-component

3

3

1 1, , ,

b b b b

s s s s

R R R R

c c c cR R R R

r rdr dr dr dr

r rα β γ δ

η η ηη= = = =∫ ∫ ∫ ∫



Flow rate

221

4 2b

s

Rc c

c bRc c

pq v dr Rθ θ

β βαθ γ γ

∂ Ω ≡ = − − + − ∂ ∫

21

4b

s

Rc

z z cRc

pq v rdr

z

αδβ

∂≡ = − − ∂ ∫

θ-component

z-component

3

3

1 1, , ,

b b b b

s s s s

R R R R

c c c cR R R R

r rdr dr dr dr

r rα β γ δ

η η ηη= = = =∫ ∫ ∫ ∫



Flow balance of each element

15

( ) eeezee DpSSQ αβαβθ

αβα ++−=

dSDD

dSzz

SSdSSS

dqnqnQ

e

ee

e

S

e

S

zez

S

e

zze

∫

∫∫

∫

∂∂=

∂∂

∂∂=

∂∂

∂∂=

Γ+=Γ

θφ

φφθφ

θφ

φ

αθα

βααβ

βαθθαβ

θθαα

, ,

,)(

drr

drr

drr

drr b

s

b

s

b

s

b

s

R

Rc

R

Rc

R

Rc

R

Rc ∫∫∫∫ ====η

δη

γη

βη

α3

3 ,

1 ,

1 ,

−Ω=

−=

−=

c

cb

c

cc

z

c

cc

RD

SS

γβ

βαδ

γβα

θ

θ

2

22

2

,4

1 ,

4

1

Flow through a

node of element

Pressure-gradient

driven flow

Drag-force

driven flow

( )( ) ( )( ) ( )( ) ( )( )1 2 3 4

1 1 1 11 1 , 1 1 , 1 1 , 1 1

4 4 4 4L L L L L L L Lφ ξ η φ ξ η φ ξ η φ ξ η= − − = + − = + + = − +


Filled and unfilled calculation

16

Head press.

10MPa（given）

Press. at feed

0MPa（given）

Q = 20kg/h (Calc. result)

Q = 40kg/h (Calc. result)

Head press.

5MPa（given）Press. at feed

0MPa（given）

Completely filled screws

Partially filled screw

Head press.

5MPa（given）Press. at feed

0MPa（given）

Q = 20kg/h(given)

？

One degree of freedom is added.


Pressure downstream update scheme

17

Screw head position

Z

Screw inlet position

Z

1st step pressure distribution

Corrected pressure distribution

Z

Filling ratio distribution

f=0

f=1.0

FAN method (1D) 2.5D FEM

：Element to be renew

：Downstream elements

Pressure downstream update scheme

θθ

θθθ ∆∂∂−∆

∂∂−=∆−∆− p

zz

pzpzzp ),(),(

As the pressure is need to be determined according to the

two coordinates (z, θ), the pressure downstream update

scheme is developed.


Criteria of filled and unfilled

18

dp/dx<0

Pressure gradient Flow balance Fill or UnfillPressure

Q>Qd p>0 Filled1) fe=1

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

dp/dx>0 Q<Qd p<0 p=0 Unfilled3) fe=0

1) 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).


−=dx

dpWHHWVQ b

ηθ

12)2sin(

4

3

Pressure gradient driven flow, QpDrag-force driven flow Qd

Z

Fill ratio control line

∆∆=

∆=

∑ φφ

φφ

α

α cossin

sincos

Z

XD

XHVQ

D

aved

zd

∑=

zd

extz

Q

Qf

∑∑=

e

eeav

V

fVf

Correction of pressure distribution

z-axis fill ratio

determined by

screw geometry

↑Qext

Average fill ratio

determined by

pressure distribution

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

19

Performing simulation

Viscosity-strain rate model e.g., Cross model

Operation variables (Feed rate, Q, Head pressure P, Rot speed N)

Automatic meshing

Calculation of pressure distribution by FEM.


Simple wizard for constructing screw geometry

1E-3 0.01 0.1 1 10 100 1000100

1000

10000

180°C

190°C

200°CComplex viscosity [Pa·s]

Strain rate [s-1

]


Effect of Screw Speed (rpm)

50 rpm

100 rpm

30 rpm

Decrease of screw speed expands filled region backwards.

Fill ratio

Un-filled Filled

Feed rate 0.75kg/h

head


Effect of feed rate (kg/h)

Speed 30 rpm

Fill ratio

Un-filled Filled

0.25 kg/h

0.75 kg/h

0.50 kg/h

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

22

Experiments

( ) 01

0*

, ,

1c

T Pηη γη γτ

−= +

&

&

0 exp273.15

bTa

Tη = +

1E-3 0.01 0.1 1 10 100 1000100

1000

10000

180°C

190°C

200°CCo

mp

lex

vis

cosi

ty [

Pa

·s]

Strain rate [s-1

]

Material

Homo polypropylene (F-704NP, Prime Polymer, Japan)

Viscosity-strain rate curve was obtained by the Cross model

Cross model


Screw geometry and barrel temp.

Self-wiping co-rotating parallel twin screw extruder

L/D = 90, φ = 15 mm

(Technovel, Japan)



30 rpm

0.5 kg/h

Fill ratio

Low (0) High (1)

Pulled-out screw

Simulation


Screw speed (rpm) vs. Fill ratio

The calculation results

agree with the theoretical

definition, Q/Qd as the fill

ratio became half when the

screw speed becomes

double.

x2

x0.5


40 60 80 100 1200.0

0.1

0.2

0.3

Calc.

Fil

l ra

tio

[-]

Screw rotational speed [rpm]

Feed rate vs. fill ratio

Calculation agrees with

the theoretical definition

of fill ratio, Q/Qd as the

plots exist on the straight

line from origin.


0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.1

0.2

0.3

0.4

Calc.

Fil

l ra

tio

[-]

Feed rate [kg/h]

Throughput vs. fill ratio

Calculation agrees with

the theoretical definition

of fill ratio, Q/Qd as the

plots exist on the straight

line from origin.


0.0000 0.0025 0.0050 0.0075 0.01000.0

0.1

0.2

0.3

0.4

Calc.Fil

l ra

tio

[-]

Feed rate and Rotational spped ratio, Q/N [kg/(h rpm)]

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 also

fiber attrition, plasticization etc are demanded.

• Devolatilization model will be added.


Contact information

• This software is proprietary and commercially available by the

HASL Co. ltd., Japan.

• Please contact

– Dr. Shin-ichiro Tanifuji, CEO of HASL.

– URL: http://www.hasl.co.jp

– E-mail: [email protected]

Thank you for your kind attention.

CEO Dr. Shin-ichiro Tanifuji


fill ratio distribution in a co-rotating self-wiping twin ... · performing simulation...

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