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)