Title: Mathematical modeling of a rotor spinning process for Twaron
1Mathematical modeling of a rotor spinning process
for Twaron
Interim thesis
2Introduction
- 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
3Teijin and Teijin Twaron
- Teijin
- Osaka, Japan
- Human Chemistry, Human Solutions
- Teijin Twaron
- Arnhem, The Netherlands
- Aramid polymer Twaron
4Products made of Twaron
5The Rotor Spinning Process
6Mathematical Models
7The rotor spinner
8The stationary case in a rotating coordinate
system with coordinate s
Because of Pythagoras
9The forces
with
if the polymer is Newtonian.
10Momentum balance
With
11The stationary case in a rotating coordinate
system with coordinate s
We need 6 boundary conditions.
12Boundary conditions
- Not that obvious are
- Another possibility
13The stationary case in a rotating coordinate
system with coordinate r
14The stationary case in a rotating coordinate
system with coordinate r
and unknowns
We need 5 boundary conditions.
15Boundary conditions
Maybe
but
16Comparison stationary cases
Polar coordinates
Then
17Comparison stationary cases
18Comparison stationary cases
19Comparison stationary cases
Repeating
20Comparison stationary cases
21Variable 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.
22Solving 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
23Further 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?
24Further 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?
25Further research
- Solving the systems
- Numerically.
- With perturbation theory.
26Further research
- Problem extension
- Z-direction and introduce gravity.
- Is the polymer Newtonian?
- Heat equation because of rapid change of
viscosity possible. - Air friction.
27Questions?