Mathematical modeling of a rotor spinning process for

Introduction Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical

Teijin and Teijin Twaron Table of contents Teijin and Teijin Twaron Products made of

Products made of Twaron Table of contents Teijin and Teijin Twaron Products made of

The Rotor Spinning Process Table of contents Teijin and Teijin Twaron Products made of

Mathematical Models Table of contents Teijin and Teijin Twaron Products made of Twaron The

The rotor spinner Table of contents Teijin and Teijin Twaron Products made of Twaron

The stationary case in a rotating coordinate system with coordinate s Table of contents

The forces Table of contents Teijin and Teijin Twaron Products made of Twaron The

Momentum balance Table of contents Teijin and Teijin Twaron Products made of Twaron The

Boundary conditions Table of contents Teijin and Teijin Twaron Products made of Twaron The

The stationary case in a rotating coordinate system with coordinate r Table of contents

Comparison stationary cases Table of contents Teijin and Teijin Twaron Products made of Twaron

Variable k Table of contents Teijin and Teijin Twaron Products made of Twaron The

Solving the systems Table of contents Teijin and Teijin Twaron Products made of Twaron

Further research Table of contents Teijin and Teijin Twaron Products made of Twaron The

Questions? Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor

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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 r Comparison stationary cases Variable k Solving the systems Further research Questions?

Teijin and Teijin Twaron Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Teijin – Osaka, Japan – Human Chemistry, Human Solutions Teijin Twaron – Arnhem, The Netherlands – Aramid polymer: Twaron 3

Products made of Twaron Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 4

The Rotor Spinning Process Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 5

Mathematical Models Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 6

The rotor spinner Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 7

The stationary case in a rotating coordinate system with coordinate s Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Forces acting on Because of Pythagoras: 8 .

The forces Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? if the polymer is Newtonian. with 9

Momentum balance Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Momentum balance: With: and Then: Momentum centrifugal coriolis flux force 10 viscous force

Boundary conditions Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Not that obvious are: Another possibility: 12

The stationary case in a rotating coordinate system with coordinate r Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 13

Comparison stationary cases Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Polar coordinates: also: Then 16

Variable k Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 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. 21

Solving the systems Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 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 22

Further research Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? 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? 23

Further research Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Boundary conditions • What is the correct leaving angle of the spinning line. • What are correct conditions on the coagulator. • What is the value of 24 , the viscous force?

Further research Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Solving the systems • Numerically. • With perturbation theory. unperturbed 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. 25

Further research Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? Problem extension • Z-direction and introduce gravity. • Is the polymer Newtonian? • Heat equation because of rapid change of viscosity possible. • Air friction. 26

Questions? Table of contents Teijin and Teijin Twaron Products made of Twaron The rotor spinning process Mathematical models stationary case, rotating r Comparison stationary cases Variable k Solving the systems Further research Questions? ? 27