Mathematical modeling of a rotor spinning process for Twaron

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Mathematical modeling of a rotor spinning process
for Twaron
Interim thesis

• Everdien Kolk

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Introduction

• Teijin and Teijin Twaron
• Products made of Twaron
• The rotor spinning process
• Mathematical models
• stationary case, rotating s
• stationary case, rotating r
• Comparison stationary cases
• Variable k
• Solving the systems
• Further research

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Teijin and Teijin Twaron

• Teijin
• Osaka, Japan
• Human Chemistry, Human Solutions
• Teijin Twaron
• Arnhem, The Netherlands
• Aramid polymer Twaron

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Products made of Twaron
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The Rotor Spinning Process
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Mathematical Models
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The rotor spinner
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The stationary case in a rotating coordinate
system with coordinate s
Because of Pythagoras
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The forces
with
if the polymer is Newtonian.
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Momentum balance
With
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The stationary case in a rotating coordinate
system with coordinate s
We need 6 boundary conditions.
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Boundary conditions

• Not that obvious are
• Another possibility

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The stationary case in a rotating coordinate
system with coordinate r
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The stationary case in a rotating coordinate
system with coordinate r
and unknowns
We need 5 boundary conditions.
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Boundary conditions
Maybe
but
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Comparison stationary cases
Polar coordinates
Then
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Comparison stationary cases

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Comparison stationary cases
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Comparison stationary cases
Repeating
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Comparison stationary cases
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Variable k

• So
• and
• When the momentum transport k is negative near
  the rotor and positive near the coagulator there
  is a radius at which k0.

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Solving the systems

• Initial value problem
• Eulers method
• Runge-Kutta order 4
• Boundary value problem
• Finite difference
• Non-linear systems
• Use an iterative process to solve the system

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Further research

• The model
• Comparison of the several models.
• Is it possible that the spinning line curves
  backward to the rotor?
• Research to the point rk0.
• What is the meaning of this point?

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Further research

• Boundary conditions
• What is the correct leaving angle of the spinning
  line.
• What are correct conditions on the coagulator.
• What is the value of , the viscous force?

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Further research

• Solving the systems
• Numerically.
• With perturbation theory.

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Further research

• Problem extension
• Z-direction and introduce gravity.
• Is the polymer Newtonian?
• Heat equation because of rapid change of
  viscosity possible.
• Air friction.

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Questions?

• ?