stability in film casting olena zavinska. problem statement project goal modeling solution method...
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Stability in Film Casting
Olena Zavinska
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Problem Statement
Project Goal
Modeling
Solution Method
Validation
Results
Conclusions
Outline
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Problem Statement1. Early Film Breakage
2. Draw Resonance
Air Gap
Width
Die
Web
Chill Roll
Off-Set
Thickness
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Project Goal
Design and implement a method for analysis of stability of the film
casting process
Determine the tolerance values of system parameters to keep the process stable
Reference: Silagy, D. et.al., Study of the Stability of the Film Casting Process, Polymer Engineering and Science, 36, no.21, 1996.
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Problem StatementProject Goal
ModelingSolution Method
ValidationResults
Conclusions
Outline
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Assumptions
• Velocity (u)Velocity (u)• Length (X)Length (X)
• Polymer flow:Polymer flow:– IsothermalIsothermal– ElongationalElongational
• Inertia, gravity, and surface tension are Inertia, gravity, and surface tension are neglectedneglected
• Kinematics’ Hypothesis (Silagy)Kinematics’ Hypothesis (Silagy)– membrane approximation membrane approximation – 1D model1D model
• Coordinates (x,y,z)Coordinates (x,y,z)• Width (L)Width (L)• Thickness (e)Thickness (e)
Reference:Reference: Silagy, D. et.al., Study of the Stability of the Film Casting Process, Polymer Engineering and Science, 36, no.21, 1996.
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Governing Equations
0
x
eLu
t
eL
1. Mass Conservation:
0
xxeLxx
F
2. Forces:
' Ip
3. Constitutive Eq.:
gex
eu
t
e
fLx
Lu
t
L
5. Kinematics F.S. Condition:
0zz
4. Stress F.S. condition:
2
x
Lzzyy
);(),0();(),0();(),0( 000 tetetututLtL
10;),0(' Dekktxx
);,0(),0( tt Nyyyy
chillroll),( utXu
6. Boundary Conditions: ?),',,,( euL xxyy Solving Unknowns
Modeling
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Problem Statement
Project Goal
Modeling
Solution MethodValidation
Results
Conclusions
Outline
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Step 1: Scaling
Solution Method
;~;~;~
000 e
ee
u
uu
L
LL ;
''~;~
0
00
0
00
F
Le
F
Le iiii
iiii
1. Unknown Variables:
;~;~
0
tu
Xt
X
xx 2. Independent Variables:
;;; 0
00
chillroll
X
uDe
L
XA
u
uDr
4. Input Parameters:
.0
000
XF
LueE
3. Unknown Parameter:
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Solution Procedure
Solution Method
)()0( xy
)()(),( )1()0( xyexytxy t
Extxytxydx
dftxy
dt
dM |,,,,,Scaled:
Exyxfxydx
d|)(,ˆ)(
Stationary
),',,,( euLy xxyy
+ inhomogeneous boundary conditions
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Step 2: Stationary Solution
Solution Method
1. Shooting method is applied to find the parameter E
2. RK4 is applied to solve the system, when E is given
Exyxfxydx
d|)(,ˆ)(
+ inhomogeneous b.c.’s
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Step 3: Dynamic Solution
Solution Method
)()()()()()( )1()1()1( xyxCxyxBxydx
dxA
+ homogeneous b.c.’s
Parameter - indicates instability)(velocity
0Re
0Re
- process is stable
- process is unstable
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Problem Statement
Project Goal
Modeling
Solution Method
Validation (Newtonian model)Results
Conclusions
Outline
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Comparison with literature reference
20 22 24 26 28 30 32 34 360
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Dr
AStability Curve: Method VS Literature
STABLE
UNSTABLE
Method for N=100
Literature
NEWTON: Method vs Literature
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Problem Statement
Project Goal
Modeling
Solution Method
Validation
Results (PTT model)Conclusions
Outline
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18 20 22 24 26 28 30 32 340
0.5
1
1.5
2
2.5
3
Dr
ALLDPE: Stability Curves
De=0.0125
De=0.012
De=0.011
De=0.010
De=0.009
De=0.008
LLDPE (eps=0.1) : Stability Curves
STABLE
UNSTABLE
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20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
Dr
A
LDPE: Stability Curves
De=0.0125
De=0.012
De=0.011
De=0.010
De=0.009
De=0.008
LDPE (eps=0.01) : Stability Curves
STABLE
UNSTABLE
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Conclusions
• A numerical algorithm for the resolution of linear stability analysis was developed
• It shows excellent performance (precision, low calculation time)
• The material rheological model explains the stabilization effect of LDPE
• The algorithm can be applied to other similarly mathematical described processes.
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Acknowledgment
• Angela Sembiring (TU/e)
• Hong Xu (TU/e)
• Andriy Rychahyvskyy (TU/e)
• Jerome Claracq (Dow)
• Stef van Eijndhoven (TU/e)
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