C A E F F

Dave Carlson

Staff ScientistMitsubishi Polyester Film, LLC

Mathematical Modeling of Processes in the Fiber and Film Industries

Chris Cox (clcox@clemson.edu)

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