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C A E F F
Dave Carlson
Staff ScientistMitsubishi Polyester Film, LLC
Mathematical Modeling of Processes in the Fiber and Film Industries
Chris Cox ([email protected] )
Mathematical Sciences Department & Center for Advanced Engineering Fibers and FilmsClemson University
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Outline
Typical fiber/film processes
ModelingGoverning equationsNumerical methods
Challenges Example
Industrial, government & academic collaborationChallengesOpportunitiesExample
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Typical Fiber and Film Processes
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Fiber Processes
Melt Spinning
Wet Spinning
Dry Spinning
Film Processes
Blowing
Tentered Biaxially Oriented
Increasin
g C
omp
lexity
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filter
air quench
spinneret
metering pump
finish applicator
bobbin
convergence guide drawing
Fiber Melt-Spinning
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Dry SpinningWet Spinning
gel
solution
coagulantbath
spinneret
AirTa ,ya ,v a
Spinneret
Take-up Roll
z = 0v0 , d0
T0 , 20
z = LvL , dL
Air and solvent
z
j2|Rj2|R
Solution Spinning
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Blown Film
air
cooling ring
extruder
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Cast Film
chill roll
coathanger die
draw roll
Top view
Side view
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Modeling Fiber and Film Processes
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ModelingChallenges
– Nonlinearities– Domain-related complexity, e.g.
• vortices• singularities• interfaces
– polymer-polymer– air-polymer
– Stability issues– Multi-scale
• spatial – e.g. crystalline regions • transient – relaxation times
– Solvers (many unknowns)
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Modeling
Dependent Variables - (standard) continuum level
• velocity v
• pressure p
• temperature T
• stress
– total stress ppnp
– Newtonian part v +
(v)T(linear)
– polymeric part p (nonlinear)
n
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Typical 2D Domain
confined flow region
free surface
symmetry boundary
• not to scale
• round fiber (axisymmetric)
• film cross-section
outflow
inflow
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Governing Equations
Conservation of Momentum
Conservation of Mass
Conservation of Energy
Cp: heat capacity k: heat conductivity
Typically assume
– incompressible
– creeping flow (drop inertial terms)
fdtd v
0 v
TkdtdTC 2
p
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Governing Equations
Constitutive Equation
– Newtonian
– Generalized Newtonian
e.g. Carreau Model
– Viscoelastic
e.g. Giesekus Model
2
1-n
})({1 2
)
} {
λ ppp
p
1p(1)1p
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Mixed finite element approach
• v : continuous piecewise quadratic
• p : continuous pw linear
• p: pw linear
• continuous with SUPG
• discontinuous with jump conditions across element interfaces
• additional unknown tensor for stability
• D ≈ v + (v)T or• G ≈ v
Numerical Solution
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Numerical Solution
Handling nonlinear terms in constitutive models
• Generalized Newtonian: Newton’s method
•
Differential constitutive models (e.g. Giesekus)
– Newton’s method, or
– (pseudo) time-dependent methods
– Theta-method – series of 3 steps (each linear)
VPG solve, solve, VPG solve
– RK method (also involves VPG and solves)
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Numerical Solution
Other nonlinearities
• Inflow boundary (Giesekus) - no closed-form expression
• Free surface
• physical domain mapped into rectangular computational domain
• Computing Jacobian
• analytically (exact) • using finite differences (approximate)
Elliptic mapping equations
x
y
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Industry, Government & Academic Collaboration
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Industry – Academic Collaboration
Challenges
Cultural differences
Industry Academia
short term deliverables long term efforts
team effort individual effort
dedicated projects multitasking (teaching, committees, . . .)
trade secrets free exchange of ideas/publication
Other differences
- evaluation criteria
- financial resources
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Industry – Academic Collaboration Opportunities
– R&D facilities in (certain) industries are scaling back or closing
– Faculty being encouraged to
• show relevance
• broaden horizons (esp. interdisciplinary)
• raise funding
– Interesting problems for faculty & students
– Potential hires for industry
– Industry has sharpened skills in
• teamwork
• leadership
• time management
– Academia offers fresh approach/problem-solving skills
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Center for Advanced Engineering Fibers & Films• An NSF Engineering Research Center Since 1998 (Award #EEC-9731680)
• Partner Institution - MIT
• Subawards – Lehigh, Ga. Tech, UIUC, SUNY Stonybrook, McGill
• Departments
Chem. Eng., Mech. Eng., Materials Sci. & Eng., Physics,Chemistry, Comp. Sci., Math Sci., Elec. & Comp. Eng., Dig. Prod. Arts
• 17 Industrial Members
• Organized into 2 Research Thrusts (formerly 3)
• 90 students (undergraduate and graduate)
• 30 faculty
• Adm. Offices: Rhodes Hall, Clemson Univ.
• http://www.clemson.edu/caeff
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Vision
The Center for Advanced Engineering Fibers and Films (CAEFF) provides
an integrated research and education environment for the systems-oriented
study of fibers and films. CAEFF promotes the transformation from trial-
and-error development to computer-based design of fibers and films. This
new paradigm for materials design -- using predictive numerical and visual
models that comprise both molecular and continuum detail -- will
revolutionize fiber and film development.
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Center Organization
Dean
Director
Deputy DirectorIndustrial LiaisonThrust Leaders
Topic Leaders
Research Teams
Scientific Advisory Board
Industrial Advisory Board
Coordination Council
Executive Committee
Administrative Director Visiting Researchers
Administrative Staff
Center Oversight• Fall Research Review (SAB & IAB)• Annual Report• Spring Site Visit (NSF)
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Research Thrusts
Thrust 1
Computer-Based Design of Materials
Thrust 2
Precursors and Processes
2.1 Liquid Crystals
2.2 Polymer Architecture
2.3 Surface Modification
2.4 Supercritical Processing
2.5 In Situ Processing
1.1 Model Development
1.2 Experimental Verification
1.3 Computer Architecture
1.4 Software/Visualization
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Recent Industry Membership
A Division of Eastman Chemical Co
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Industry Interaction
• Directed projects
• REU projects
• Plant trips
• Sabbatical visits
• Research Review & Site Visit
• Adjunct faculty/dissertation committee member
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Sphere which determines distance traveledExample Project
Resulting trajectory
• Oxygen diffusion through nanocomposite films
Clay platelets influence barrier properties without harming transparency of food wrap
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Recommended references
• Agassant, Avenas, Sergent and Carreau: Polymer Processing- Principles and Modeling, Hanser Publishers, Oxford University Press
• F. P. T. Baaijens, Mixed finite element methods for viscoelastic flow analysis: a review, J. Non-Newt. Fluid Mech. 79, (1998), 361-385.
• D.G. Baird and D.I. Collias, Polymer Processing Principles and Design, Butterworth- Heinemann, 1995.
• R. Bird, R. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume One, Wiley, second edition, 1987.
• M. Crochet, A. Davies, K. Walters, Numerical Simulation of Non-Newtonian Flows, Elsevier, 1984.
• M. Renardy, Mathematical Analysis of Viscoelastic Flows, SIAM, 2000.
• Journal of Non-Newtonian Fluid Mechanics
• Journal of Rheology