Transcript 25.107 Intro. to Engineering Session 1: Introduction

Simulation of
Polymer Processing
David O. Kazmer, P.E., Ph.D.
March 26, 2005
Progress in
Polymer Process Simulation!

General Electric 1988

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Vax 8800 cluster
E&S 3D vector graphics

UML 2005

PC
Simulation of Polymer Processing:
Agenda
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Modeling Overview
Governing Equations
Constitutive Models
Numerical Solution
Capabilities
Challenges
Motivation:
Understand Process

Polymer processing is a nasty black box


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Dynamic process
Multivariate process
Spatially distributed process
Complex 3D geometry
Thermoviscoelastic materials
Multiple quality requirements
Expensive mold tooling changes
x(t)
Polymer
Processing
y(t)



Cost
Part Weight (g)

Ba
rr e
Ba l T
rr e m
Ru el T p L
nn e m o w
Ru er p H
i
nn Te
m gh
Co er T p
ol e m Lo w
Co ant p
ol Te H ig
an m
h
Tr t Te p L
a n m ow
Tr sfer p H
a n P ig
sf os h
e
In r P Lo w
j V os
In elo H ig
c
Pa j Ve ity h
ck loc Lo
Pa Pr ity w
ck ess Hig
Pr ure h
e
Pa ssu L ow
ck re
Pa Tim H ig
ck
e h
Ti
L
m ow
e
Hi
gh
Motivation:
Virtual Development
Model and understand the process
Perform virtual development
What-if analyses
System-level optimization
Observed for Lower Cavity
7.9
8
7.8
7.7 Quality Costs
Total
7.6
7.5
7.4
7.3
7.2
7.1
7
Compliance Costs
Defect Costs
Quality Level
Motivation:
Post-Mortem Analysis


Modeling of existing processes
Inspection of internal polymer states

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Pressure, temperature, flow rate, shear
stress, shear rate, …
Development of corrective strategies

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Change process conditions
Assess material changes
Recommend mold tooling changes
Simulation provides the means
for trying the impossible
at negligible cost.
Agenda
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Motivation
Governing Equations
Constitutive Models
Numerical Solution
Capabilities
Challenges
Governing Equations:
Navier Stokes Equations

For laminar (or time-averaged turbulent) flow:


Net pressure force is the gradient of the pressure
Net viscous force is the Laplacian of the velocity

div v   0

Dv
2

 p   v
Dt
N-S assumes that all macroscopic length
and time scales are considerably larger than
the largest molecular length and time scales.
Polymer Processing Simulation:
Typical Assumptions

Viscous flow
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Known boundary conditions
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Negligible inertia
Negligible viscoelasticity
No slip at mold wall
Constant inlet resin temperature
Flow travels in a plane

No out of plane flow

“2D” simplification
Governing Equations:
Mass Equation

Conservation of mass
 
 v   0
t x

What goes in must come out


Or stay in there…
Change in density with non-steady velocity
IN

OUT
Governing Equations:
Momentum Equation

Conservation of momentum
  v  P
  
z  z  x

Change in pressure in the flow direction is
due to shear stress of flowing viscous melt
P1
P2
v
 
v
 
z
P P2  P1

x
L
Governing Equations:
Heat Equation

Conservation of energy
T 
 2T
 T
C p   v   k 2   2
x 
z
 t

Change in temperature balances heat
convection, heat conduction, and shear
heating (and others)
T1
Q   2
T,v
Q  h T  TMW 
T2
T T2  T1

x
L
Agenda
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Motivation
Governing Equations
Constitutive Models
Numerical Solution
Capabilities
Challenges
Constitutive Models:
Overview

Constitutive model: describes the
behavior of the material as a function of
polymer state
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Trade-offs between:
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Viscosity, density, …
Model form and complexity
Number of model parameters
Data redundancy in model fitting
Computational efficiency & stability
“Everything should be made as simple
as possible -but no simpler!” - Einstein
Constitutive Models:
Viscosity
0 (T , P)

 ( , T , P) 
0 1n
1 ( * )


0
WLF temperature
dependence
A1 (T  T * )
0 (T , p)  D1 exp(
)
*
A2  (T  T )
T  Tt
10 1000
8
Exp.
Fitted
6
WLF
4
Log(aT)

Most polymers are shear thinning
Cross model
Viscosity (Pa Sec)

2 100
n
0
-2
-4
-6
10
1
10
-8
100
1000
10000
Shear Rate (1/sec)
20
70
120
*
170
Temperature ( oC)
220
270
320
Constitutive Models:
Viscoelasticity
Polymers exhibit melt elasticity
σ   p(  , T )I   M  (t )   ( )h( I , I )C ( )d
Memory effect

1
t
gi 

e
i 1 aT  i
Extremely data and
CPU intensive
Need to store and
compute on current and
all past process states!
1.E+07
1.E+06
109
109
108
108
107
107
106
106
105
105
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
G'-Exp.
G"-Exp.
G'-Fitted
G"-Fitted
104
103
104
103
102
102
101
101
100
100
-1
-1
10
10-3
1.E+00
1.E-06
10
10-2
1.E-03
10-1
100
101
Freq [rad/s]
1.E+00
1.E+03
Frequency (rad/s)
102
1.E+06
103
1.E+09
G" ( )
[Pa]

1.E+08
 ( t )  ( )
i
5 orders of magnitude!

1.E+09
Loss Modulus
dG (t )   ( ) 
M  (t )   ( )   
dt
m

2
Storage Modulus

1
G' ( )
[Pa]
t
G', G" (Pa)

Constitutive Models:
Specific Volume

Polymers exhibit thermal expansion
and compressibility
Double domain
Tait Equation
1.00
0 MPa Exp.
20 MPa Exp.
40 MPa Exp.
60 MPa Exp.
80 MPa Exp.
100 MPa Exp.
120 MPa Exp.
140 MPa Exp.
160 MPa Exp.
180 MPa Exp.
200 MPa Exp.
0.98


P 

v T , P   v0 T 1  0.0894ln1 
 
BT   


v0  b1,l  b2,l T  b5 
BT   b3,l exp b4,l T  b5 
Specific Volume (10-3m3/kg)

0.96
0.94
0.92
0 MPa Fitted
20 MPa Fitted
40 MPa Fitted
60 MPa Fitted
80 MPa Fitted
100 MPa Fitted
120 MPa Fitted
140 MPa Fitted
160 MPa Fitted
180 MPa Fitted
200 MPa Fitted
0.90
0.88
0.86
0.84
0.82
0.80
0
50
100
150
200
250
o
Temperature ( C)
300
350
Constitutive Models:
Specific Heat
Specific heat Cp
C p T   c1  c2 T  C5   c3 tanhc4 T  C5 
Specific Heat (103J/kgoC)

2.30
2.20
2.10
2.00
1.90
1.80
1.70
1.60
1.50
1.40
1.30
1.20
1.10
Exp.
Fitted
50
100
150
200
250
Temperature (oC)
300
350
Constitutive Models:
Thermal Conductivity
Thermal conductivity k
k T   1  2 T  5   3 tanh4 T  5 
0.32
Thermal Conductivity (W/moC)

Exp.
Fitted
0.30
0.28
0.26
0.24
0.22
0.20
20
70
120
170
220
o
Temperature ( C)
270
320
Agenda
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Motivation
Governing Equations
Constitutive Models
Numerical Solution
Capabilities
Challenges
Numerical Methods:
Geometric Modeling


Polymer domain decomposed into elements
2D elements across flow domain


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Plastics parts are often thin so nice assumption
Each element has defined thickness
3D elements for entire domain

Need many, many elements of higher order
shape functions
Numerical Methods:
Solution

Iterative solution method
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Flow field
Temperature field
Read Input
Solve Flow
Update BC’s
Solve Heat
Advance Time
Done?
Write Output
Numerical Methods:
Finite Element Solution of Flow
K P  Q
k35 
2 W 
0
2 W 
k35 
h
 P h ~
z 
    d~z dz
 x z  
x
n _ layers
 rho[ j ]z[ j ]  dz[ j ] / visc[ j ]dz[ j ]
j 0
k35
length
k12
 Q1   k12  k13  k14
0 
k k25kk!426
 0  k12 k23k12
k12
 k
24 13
  
Q  
0 
k13
k 23k12
 1  
 
k14
0 
Q2   k24k13
Q5  
0
k 25
  
  
Q
 3   k26k14
0
Q6  
 100
 P 
  P1  k26
k 23k12
k 24 k13
k 25k14
 2 



  P3 
kk1212 
k 23k23k35
0 k 23
k35k 24
0
 k 24
 0 
 
0
 k14  k 24  k 46
0
k 46
  P4 
 0 
k 23
 k13  k 23
0
 0 
k35
0
 k 25  k35  k56    k56
k
0
 kk14  k 24   0k  k  k   0 
0 24
k 46
56
26
46
56  

k13
k14
0
0
Numerical Methods:
Finite Difference Solution of Heat
Change
in Temp
Heat
Convection
Viscous
Heating
Heat
Conduction
Adiabatic
Compression
T 
T
 T
2

v

k



  PdV

2
t1
x 
z
 t
C p 
2
v
t2
 FoTi t 1t  1  2 FoTi t  t  FoTi t 1t
0
1  2 Fo  2 Fo
  Fo 1  2 Fo  Fo

 0
 Fo 1  2 Fo

0
0
 0
Q
t
 T 
2 t

 Ti t  t  v




i  PV
 x i C p
t
 
t
T
t


0 T1t  t  T1t  t  v

 2 1  PV 
x C p
 t  t  


0 T2 











  t  t  

1 Tn 



 
Agenda
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Motivation
Governing Equations
Constitutive Models
Numerical Methods
Capabilities
Challenges
Capabilities:
Optical Media Molding


Optical media: CDs & DVDs
Injection-compression molding (coining)
Numerical Algorithm:
Coining Process

Coining Process




Partly open mold
Inject polymer
Profile clamp force
Simulation
Calculate
Temperature
Adjust Thickness
Change Element Properties
Restore Old Profiles
Calculate
Pressure
Calculate Cavity Force
Force
N
Cavity
Force=Clamp
Force?
Y
Move on to Next Time Step
1.45
1.45
1.40
1.40
Mold Displacement (mm)
Mold Displacement (mm)
Coining Process Validation:
Displacement Profiles
1.35
300oC
310oC
1.30
1.25
320oC
1.20
1.15
300oC
310oC
320oC
1.35
1.30
1.25
1.20
1.15
1.10
1.10
0
1
2
Time (s)
3
4
0
1
2
Time (s)
3
Effect of melt temperature: experiment vs. simulation
4
Birefringence Models

Constitutive model for flow induced stress
(Wagner, M. H. et al)
σ   P(  , T )I   M  (t )   ( )h( I 1 , I 2 )Ct1 ( )d
t

dG (t )   ( )  m g i 
M  (t )   ( )   

e
dt
i 1 aT  i
 ( t )  ( )
i
h(I1 , I 2 )  m* exp(n1 I  3)  (1  m* ) exp(n2 I  3)
1   2 (t   )2  (t   ) 0


Ct1 ( )    (t   )
1
0

0
0
1

Birefringence Models (Cont.)
• Shear stress:
t
t

1
t
 rz   M (t   , T )h(3  { dt' } )[ dt' ]d
2

1  
1
0
• First normal stress difference:
N1   rr   zz  
t

t
if   0
if   0
t
M (t   , T )h(3  { dt' } )[ dt' ]2 d
2
1
1
• Integral stress-optical rule
(birefringence constitutive model):
n   C  (t )   ( ) 
t


d

• Path difference (retardation):
  N12  4 rz2 for vertic al birefirnen ce nrz
  N2 for in - planebirefringence nr
z  
d /2
d / 2
(nr  n ) z dz
Numerical Algorithm
• Incremental formulation for the integral equations:
m
2
 13,n1   

i 1 j 1
m
2
tn 1
N1,n1   
tn 1

i 1 j 1
Gi (n 1  ( )) / i
n (  ( ))
e
m j e j n 1
( n1   ( ))d
aTi
m1  m m2  1  m
Gi (n 1  ( )) / i
n (  ( ))
e
m j e j n 1
( n1   ( ))2 d
aTi
• Solved by FDM in time domain:
 13ij ,n1  Gi e
n 1 / i
m je
N1ij,n1  Gi en1 /i m j e
 n j n 1
 n1   13ij ,n1  e 
 n j n 1
 n21  N1ij,n1  e 
n 1 / i
n 1 / i
e
e
 n j  n 1
 n j  n 1

N
ij
13,n
ij
1,n
 Gi en /i m j e
 Gi en /i m j e
Gi (n 1  ( )) /i
n (  ( ))
e
m j e j n 1
( n1   ( ))d
tn a 
T i
tn 1 G
n (  ( ))
i
N1ij,n1  
e(n 1  ( )) /i m j e j n 1
( n1   ( ))2 d
tn a 
T i
tn G
n (   ( ))
i
 n1   n1   n
n  
e(n  ( )) / i m j e j n
d
0 a 
T i
 13ij ,n1  
 n j n
 n j n
 n  n

 n2  2 n1 13ij ,n   n21n
tn 1
n1  n1  n

In-plane Birefringence Validation
z
t
80

80
70
70
Exp.
Sim.--Total
Sim.--Flow
Sim.--Cooling
50
40
30
20
10
50
40
30
20
10
0
0
-10
-10
-20
-20
23
28
33
38
43
Radius (mm)
48
53
Exp.
Sim.--Total
Sim.--Flow
Sim.--Cooling
60
Path Difference (nm)
60
Path Difference (nm)
r
58
23
28
33
38
43
Radius (mm)
Validation: experiment vs. simulation
48
53
58
Vertical Birefringence Prediction
z
t

25
25
Total
Flow Induced
Thermally Induced
15
10
20
15
-4
nrz (×10 )
20
-4
nrz (×10 )
r
5
0
0
0
z/d
0.1 0.2 0.3 0.4 0.5
Flow Induced
Thermally Induced
10
5
-5
-0.5 -0.4 -0.3 -0.2 -0.1
Total
-5
-0.5 -0.4 -0.3 -0.2 -0.1
0
0.1 0.2
0.3 0.4 0.5
z/d
Effect of mold temperature (low-high): simulation
Simulation of Internal Stress
and Post-Molding Deformation

Thermal stress/warpage
 In-mold: FDM (Baaijens, F. P. T. et al)
σ   ph I  σ d
t 
1
1

p h   tr(σ)    T  tr(ε ) d
0 
3



m
t
i 1
0
σ   2 g i e  ( t ) ( )  /i ε d d
d
– Out-of-mold: FEA (plate bending)
T
w( r )  w ( r )
D  u w  
u(r, z )  u (r )  z (r )
 ( r )  dw ( r ) / dr
Finite Element Discretization

Kirchhoff thin-plate elements
Finite Element Formulation
• Strain-displacement relationship
 1
  rr    s
   1  
   
 r
6z
(1  2 )
s2
6z
(   2 )
rs
z
( 4  6 )
s
1
s
z

( 1  4  3 2 )
r
r
6z
( 1  2 )
s2
6z
(    2 )
rs
 u1 
w 
z
 1 
( 2  6 )   
1
s



z
u
2
2


( 2  3 )
r
  w2 
 
 2 
2
ε   Bi ui
i 1
• Stress-strain relationship
 rr  a 
  
   b
b    rr   rr 
 

a        
σ  Hε  h
• Element stiffness matrix and element right-hand-side vector
k e   BT HBdV 2 
V
d /2

r2
d / 2 r1
KD  R
R e   ( NT f  BT h)dV 2 
V
BT HBrdrdz
d /2

r2
d / 2 r1
( NT f  BT h)rdrdz
Relaxation Modeling:
Truncated WLF Equation

WLF Fit by data at 150-280oC
Truncated at at 140, 135, 130, 125oC
1E+8
1E+6
Exp.
Fitted
1E+4
1E+2
aT

1E+0
Tg
1E-2
1E-4
1E-6
1E-8
80
100 120 140 160 180 200 220 240 260 280 300
o
Temperature ( C)
Effect of the Truncation
Warpage at different truncation temperatures

Could fudge any desired result!
150
100
50
Warpage (micro meter)

z
T
0
-50
t
-100
Ttrunc=140
Ttrunc=135
Ttrunc=130
Ttrunc=125
Exp. Data
-150
-200
-250
Radial Direction
-300
23
28
33
38
43
Radius (mm)
48
53
58
r

Proposed Function for
Relaxation Model, aT

For T<Tref
log(aT ) 

ad(1  e
d e
 ( b( Tref T ))c
( b( Tref T ))c
For T>Tref
log(aT ) 
ad(1  e
d e
( b ( T Tref )) c
( b ( T Tref )) c
)
)
Results for Implemented
Relaxation Function, aT
Model fit & performance in simulation
1E+6
Exp.
Fitted
1E+4
1E+2
1E+0
1E-2
1E-4
1E-6
1E-8
Vertical Displacement (micro meter)
1E+8
aT

5
Exp.
Sim.
-15
-35
-55
-75
-95
-115
70
100 130 160 190 220 250 280 310
Temperature (oC)
23
28
33
38
43
Radius (mm)
48
53
58
Optical Molding Simulation:
Results Summary
Optical media simulation used for



Process development and optimization
Development of new polymeric materials
Higher data density & lower costs
120
Exp.
Sim.
100
Vertical Displacement (micro meter)
Vertical Displacement (micro meter)
120
80
60
40
20
0
295
120
Vertical Displacement (micro meter)

Exp.
Sim.
100
80
60
40
20
80
60
40
20
0
0
300
305
310
315
Melt Temperature (oC)
320
325
Exp.
Sim.
100
95
100
105
110
Mold Temperature ( oC)
115
120
0
10
20
30
Packing Pressure (kgf/cm2)
40
50
Agenda






Motivation
Governing Equations
Constitutive Models
Numerical Solution
Capabilities
Challenges
Challenges:
Process Controllability

What are the boundary conditions for
analysis?



Is melt temperature constant?
What is the mold wall heat transfer?
Is a no-slip condition at mold wall valid?
Wall
Center
Wall
0
1
Time
2
3
Challenges:
Constitutive Models

N-S assumes a continuum

Is a continuum approach valid on the
nano-level? If not:



What are the governing equations?
What are the constitutive models?
How to apply thermodynamics & statistics?
Challenges:
Numerical Methods

Modeling on the atomic scale?



Sandia Labs Atomic weapons
Crystal-level modeling of metals
Protein folding
Final Thoughts:
Modeling Principles

Pritsker’s Modeling Principles, from Handbook of
Simulation, edited by Jerry Banks for Wiley
Interscience, 1998


Model development requires system knowledge,
engineering judgment, and model-building tools.
The modeling process is evolutionary because the act of
modeling reveals important information piecemeal.


A model should be evaluated according to its usefulness.


The secret to being a good modeler is the ability to remodel.
From an absolute perspective, a model is neither good or bad,
nor is it neutral.
All truths are easy to understand once they are
discovered; the point is to discover them.

Galileo