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Mathematical modeling of a rotor spinning process for Twaron

Mathematical modeling of a rotor spinning process for Twaron. Interim thesis. Everdien Kolk. 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

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Mathematical modeling of a rotor spinning process for Twaron

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  1. Mathematical modeling of a rotor spinning process for Twaron Interim thesis Everdien Kolk

  2. 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 Questions?

  3. Teijin Osaka, Japan Human Chemistry, Human Solutions Teijin Twaron Arnhem, The Netherlands Aramid polymer: Twaron Teijin and Teijin Twaron

  4. Products made of Twaron

  5. The Rotor Spinning Process

  6. Mathematical Models

  7. The rotor spinner

  8. The stationary case in a rotating coordinate system with coordinate s Forces acting on . Because of Pythagoras:

  9. The forces if the polymer is Newtonian. with

  10. Momentum balance Momentum balance: and Then: Momentum centrifugal coriolis viscous flux force force force With:

  11. The stationary case in a rotating coordinate system with coordinate s • With mass flux and unknowns: We need 6 boundary conditions.

  12. Boundary conditions • Not that obvious are: • Another possibility:

  13. The stationary case in a rotating coordinate system with coordinate r

  14. The stationary case in a rotating coordinate system with coordinate r centrifugal coriolis force force With: and unknowns: We need 5 boundary conditions.

  15. Boundary conditions Maybe: but

  16. Comparison stationary cases also: Polar coordinates: Then

  17. Comparison stationary cases Then: +

  18. Comparison stationary cases Polar coordinates: Pythagoras says: Then:

  19. Comparison stationary cases With: centrifugal coriolis force force Repeating:

  20. Comparison stationary cases With and follows: centrifugal coriolis force force

  21. 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 k=0.

  22. Solving the systems • Initial value problem • Euler’s method • Runge-Kutta order 4 • Boundary value problem • Finite difference • Non-linear systems • Use an iterative process to solve the system

  23. 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 rk=0. • What is the meaning of this point?

  24. 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?

  25. Further research unperturbed problem: perturbed problem: The introduction of small perturbations triggers off qualitatively and quantitatively behaviour of the solutions which diverges very much from the behaviour of the solutions of the unperturbed problem. • Solving the systems • Numerically. • With perturbation theory.

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

  27. Questions? • ?

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