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Stability in Film Casting
Olena Zavinska
Problem Statement
Project Goal
Modeling
Solution Method
Validation
Results
Conclusions
Outline
Problem Statement1. Early Film Breakage
2. Draw Resonance
Air Gap
Width
Die
Web
Chill Roll
Off-Set
Thickness
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.
Problem StatementProject Goal
ModelingSolution Method
ValidationResults
Conclusions
Outline
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.
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
Problem Statement
Project Goal
Modeling
Solution MethodValidation
Results
Conclusions
Outline
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:
Solution Procedure
Solution Method
)()0( xy
)()(),( )1()0( xyexytxy t
Extxytxydx
dftxy
dt
dM |,,,,,Scaled:
Exyxfxydx
d|)(,ˆ)(
Stationary
),',,,( euLy xxyy
+ inhomogeneous boundary conditions
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
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
Problem Statement
Project Goal
Modeling
Solution Method
Validation (Newtonian model)Results
Conclusions
Outline
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
Problem Statement
Project Goal
Modeling
Solution Method
Validation
Results (PTT model)Conclusions
Outline
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
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
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.
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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