Dave Carlson

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Mathematical Modeling of Processes in the
Fiber and Film Industries
Dave Carlson
Staff Scientist Mitsubishi Polyester Film, LLC
Chris Cox (clcox_at_clemson.edu)
Mathematical Sciences Department Center for
Advanced Engineering Fibers and Films Clemson
University
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Outline Typical fiber/film processes Modeling
Governing equations Numerical
methods Challenges Example Industrial,
government academic collaboration Challenges
Opportunities Example
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Typical Fiber and Film Processes
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• Melt-process (e.g. melt-spun fiber, cast blown
  film) stages

Extrusion
Filtration
Forming
Free Surface Development/ Solidification
Post-draw
(Many other specialty processes/uses of polymers)
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Fiber Processes Melt Spinning Wet
Spinning Dry Spinning Film Processes Blowing
Tentered Biaxially Oriented
Increasing Complexity
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Fiber Melt-Spinning
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Solution Spinning
Dry Spinning
Wet Spinning
Spinneret
z 0 v0 , d0 T0 , ?20
z
Air and solvent
j2R
j2R
Air Ta ,ya? ,v a
z L vL , dL
Take-up Roll
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Blown Film
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Cast Film
Top view
Side view
coathanger die
chill roll
draw roll
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(No Transcript)
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Modeling Fiber and Film Processes
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Modeling

• Challenges
• 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 s -pI t -pI tn tp
• Newtonian part h(?v (?v)T) (linear)
• polymeric part tp (nonlinear)

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Typical 2D Domain

• not to scale
• round fiber (axisymmetric)
• film cross-section

confined flow region
inflow
free surface
outflow
symmetry boundary
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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)

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Governing Equations

• Constitutive Equation
• Newtonian
• Generalized Newtonian
• e.g. Carreau Model
• Viscoelastic
• e.g. Giesekus Model

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Numerical Solution

• Mixed finite element approach
• v continuous piecewise quadratic
• p continuous pw linear
• tp pw linear
• continuous with SUPG
• discontinuous with jump conditions across
  element interfaces
• additional unknown tensor for stability
• D ?v (?v)T or
• G ?v

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Numerical Solution

• Handling nonlinear terms in constitutive models
• Generalized Newtonian Newtons method
• Differential constitutive models (e.g. Giesekus)
• Newtons method, or
• (pseudo) time-dependent methods
• Theta-method series of 3 steps (each linear)
• VPG solve, t solve, VPG solve
• RK method (also involves VPG and t 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)

h
y
Elliptic mapping equations
x
x
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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
• RD 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
Scientific Advisory Board
Dean
Industrial Advisory Board
Executive Committee
Director
Coordination Council
Industrial Liaison
Thrust Leaders
Deputy Director
Visiting Researchers
Administrative Director
Topic Leaders
Administrative Staff
Research Teams
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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Example Project
Sphere which determines distance traveled

• Oxygen diffusion through nanocomposite films
  Clay platelets influence barrier properties
  without harming transparency of food wrap

Resulting trajectory
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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