Mathematical modeling of a rotor spinning process for Twaron

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? • ?