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 Variable k Solving the systems Further research Questions?

Teijin and Teijin Twaron Osaka, Japan Human Chemistry, Human Solutions Teijin Twaron Arnhem, The Netherlands Aramid polymer: Twaron Human Chemistry: ontwikkelen van chemische technologien, vriendelijk voor mens en omgeving. Human Solutions: optimaal gebruik maken van deze ontwikkelde technieken.

Products made of Twaron Producten: Beschermende kleding (helmen/vesten), banden, buizen, optische kabels, remblokjes.

The Rotor Spinning Process Rotor spinnen een spin methode. Rotor 15 cm, coagulator 30 cm. airgap 15 cm. Omega = 2500 rpm. Diameter gaatje: 250 mu

Mathematical Models Proces niet geheel onder controle -> wiskundig model. Binnen teijin was er een model gemaakt, Tijdens week wiskunde en industrie in 2004 is ook aan dit probleem gewerkt….. 2D model! Hier bespreken stationare geval met s en r. instationair ook af te leiden, hier niet bespreken. Stroboscoop opname.

The rotor spinner S = boog lengte langs het spinlijntje. Modelleren van Rrot tot Rcoag.

The stationary case in a rotating coordinate system with coordinate s Forces acting on . Kijken naar klein stukje spin lijn: delta s. Coriolis kracht, centrifugaal kracht en visceuse kracht. S = boog lengte langs het spinlijntje. Because of Pythagoras:

if the polymer is Newtonian. The forces Coriolis kracht, centrifugaal kracht en visceuse kracht. S = boog lengte langs het spinlijntje. Dv/ds… -> Newtons! with if the polymer is Newtonian.

Momentum balance Momentum balance: With: and Then: Momentum centrifugal coriolis viscous flux force force force Iin entering momentum flux, (inkomende) Iout leaving momentum flux. (uitgaande)

The stationary case in a rotating coordinate system with coordinate s With mass flux and unknowns: Momentum balances; Polymeer is Newtons; Pythagoras (mom eq x *dx/ds +mom eq y *dy/ds, met dxds^2+dyds^2=1). Rho= dichtheid, eta=viscosity. Mass flux CONSTANT! Belangrijk: F-phiv=0… wat dan. We need 6 boundary conditions.

Boundary conditions Not that obvious are: Another possibility: Neem aan dat spinlijntje rotor loodrecht verlaat. F0 en ve onbekend…, L onbekend!! Daarom volgende model!

The stationary case in a rotating coordinate system with coordinate r Aannemen dat spinlijntje niet terug keert richting rotor. Dan bij 1 r, verschillende gevallen..

The stationary case in a rotating coordinate system with coordinate r centrifugal coriolis force force With: d/dt uit instationare geval negeren. and unknowns: We need 5 boundary conditions.

Boundary conditions Maybe: but Assume perpendicular leaving orifice. A loodrecht tov spinning line. Ve onbekend. but

Comparison stationary cases Polar coordinates: also: Herschrijven stationair, roterend S: Then

Comparison stationary cases Then: +

Comparison stationary cases Polar coordinates: Pythagoras says: Then:

Comparison stationary cases Repeating: With: En vermenigvuldigen met -1. ALLEEN HERSCHRIJVEN!! centrifugal coriolis force force

Comparison stationary cases centrifugal coriolis force force With and follows: Nu omschrijven naar hoeken.

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.

Solving the systems Initial value problem Boundary 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 Als IV -> coagulator heeft invloed op vorm spinlijntje. BV beter.

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?

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?

Further research Solving the systems Numerically. With perturbation theory. 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. Storings theorie -> lijkt mogelijke manier, is voorheen begin mee gemaakt, wil ik verder naar kijken.

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

Questions? ?