Transcript 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
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 s
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 s
stationary case, rotating r
Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
4
The Rotor Spinning Process
Table of contents
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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?
5
Mathematical Models
Table of contents
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?
6
The rotor spinner
Table of contents
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?
Rrot  0.15m,
Rcoag  0.30m,
ω  2500rpm,   1200Pa s,
7
d  250μm,
  1700kg / m3 .
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 s
stationary case, rotating r
Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
Forces acting on
2
s .
2
Because of Pythagoras:  dx    dy   1.
 ds   ds 
8
The forces
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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?







dy 

ds
Fcor  2mω v  sAv
,
 dx 
ds 
 x
Fcentr  mω (ωr)  sA 2 ,
 y
 



dx
 d 

F



ds
ds

F
 s 
.

visc
 d  F dy  

 

ds
ds



dv v F

with
ds  
if the polymer is Newtonian.
9
Momentum balance
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Teijin and Teijin Twaron
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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?
Momentum balance:
With:
I in  I out  ve S  ve
e  e x
Then:
Iin  I out  Fcentr  Fcor  Fvisc  0
dx
dy
 ey
ds
ds
and
S  S
 s
d
ve ,
ds
  Av  constant .

d  dx 
dy d  dx 
2
 v   A x  2 Av   F   0
ds  ds 
ds ds  ds 

d  dy 
dx d  dy 
2
v


A

y

2

A

v
  F   0.


ds  ds 
ds ds  ds 
Momentum centrifugal coriolis
flux
force
force
10
viscous
force
The stationary case in a rotating coordinate
system with coordinate s
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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?
d 2x
 2
dy dx d
F  v  2  
F  v ,
x  2 
ds
v
ds ds ds
d2y
 2
dx dy d
F  v  2  
F  v ,
y  2 
ds
v
ds ds ds
dv v F

,
ds  
dF
dv  2  dx
dy 
 
 x  y .
ds
ds
v  ds
ds 
With mass flux   Av and unknowns: F, x, y, v.
We need 6 boundary conditions.
11
Boundary conditions
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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?
x(0)  Rrot ,
v(0)  1,
y (0)  0,
F (0)  F0 .
Not that obvious are:
dx
 1,
ds s 0
dy
 0.
ds s 0
Another possibility:
2
x( L) 2  y ( L) 2  Rcoag
,
12
v( L)  ve
The stationary case in a rotating coordinate
system with coordinate r
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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?
~


cos

p
~

pr
,
 sin ~ 
p 

~


cos

~s  
s
 sin ~ ,
s 

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~



sin

~
s

m
~ ,
cos

s 

~


cos

~
s
~

v V 
~ .
sin

s 

The stationary case in a rotating coordinate
system with coordinate r
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~
~ ~
d p  tan( p  s )
Teijin and Teijin Twaron

,
dr
r
Products made of Twaron
~ ~
~ 


The rotor spinning process

A
cos(

d  ~

 ~
p  s ) dV ~ 
~ .

V

s

p

2
m

~
~ ~ ~
Mathematical models
dr  
dr   cos( p  s )  V

V



stationary case, rotating s
centrifugal coriolis
stationary case, rotating r
 F  v 
force
force
Comparison stationary cases
Variable k
With:
Solving the systems
~
~
~






cos

cos


sin

Further research
~
p
~  r
~s  
~ 
s
s
p
,
,
m
,



A
V
,
 sin ~ 
 cos~ 
 sin ~ 
Questions?
p
s 
s 




and unknowns:
~
~
~
s ,  p , V .
We need 5 boundary conditions.
14
Boundary conditions
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Products made of Twaron
The rotor spinning process
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stationary case, rotating s
stationary case, rotating r
Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
~
V ( Rrot )  1,
~
s ( Rrot )  0,
~
 p ( Rrot )  0,
~
V ( Rcoag )  ve ,
??
Maybe:
 
 tan 
 p
Rcoag    2  ,
r
Rcoag
~
15
but
 
tan   
2
Comparison stationary cases
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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?
Polar coordinates:
~
x  r cos p ,
~
y  r sin  p ,
~
d p
dx dr
~
~
 cos p  r
sin  p ,
ds ds
ds
also:
dy dr
~
~
 sin  p  r
cos p .
ds ds
ds
dx
~
 coss ,
ds
dy
~
 sin s .
ds
~
d p
dr
~
~
~
Then
cos p  r
sin  p  coss ,
ds
ds
~
d p
dr
~
~
~
sin  p  r
cos p  sin s .
ds
ds
16
~
d p
Comparison stationary cases
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stationary case, rotating s
stationary case, rotating r
Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
~
d p
dr
~
~
~
~
cos p  r
sin  p  coss
* sin s
ds
ds
~
d p
dr
~
~
~
~
sin  p  r
cos p  sin s
* - coss
ds
ds
~
d p dr
~ ~
r
 tan( p  s )
ds ds
Then:
~
d p
ds
17

~ ~
tan( p  s )
r
+
Comparison stationary cases
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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?
Polar coordinates:
~
d p
dx
~
~
 cos p  r
sin  p ,
dr
dr
~
d

dy
~
~
p
 sin  p  r
cos p .
dr
dr
Pythagoras says:
2
2
 ds   dx   dy 
     
 dr   dr   dr 
ds
1

~ ~
dr cos( p  s )
Then:
18
2
Comparison stationary cases
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stationary case, rotating s
stationary case, rotating r
Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
Repeating:
d 2x
 2
dy dx d
F  v  2  
F  v ,
x  2 
ds
v
ds ds ds
d2y
 2
dx dy d
F  v  2  
F  v .
y  2 
ds
v
ds ds ds
With:
F
 dv
:
v ds


 dx  
 dy  





x    ds  

d  
 dv  ds 


       2
 v 
ds   v ds  dy  
 v  y   dx  
 




 ds  
 ds  


 F  v 
19
centrifugal
force
coriolis
force
Comparison stationary cases
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stationary case, rotating s
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Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?


 dx  
 dy  



x    ds  

d  
 dv  ds  


       2
 v 
dy
dx
ds  
v ds   
 v  y 

 




ds
ds







 F  v 
With
centrifugal
force
coriolis
force
ds
1

~ ~ ,
dr cos( p  s )
dx
dy
~
~
 coss and
 sin s follows:
ds
ds
~ ~
~
d    cos( p  s ) dv  coss  

 v
~   




dr 
v
dr  sin s 


~
~
   cos p    sin   

s

~   2
~ ~  
~  
cos( p  s )  v  sin  p   coss  
20
Variable k
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stationary case, rotating s
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Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
~

~

d
V
~
~
~
s
k~s  V  ~ cos( p  s )
dr 
V

So
~
~ 
~ ~ dV 
,
k  V  ~ cos( p  s )
dr 
V

and
k  F  v .
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
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stationary case, rotating s
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 s
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 s
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 F0, the viscous force?
24
Further research
Table of contents
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?
Solving the systems
•
Numerically.
•
With perturbation theory.
unperturbed problem:
d 2x
 x  0,
2
dt
perturbed problem:
d 2x
dx

2

 x  0,
2
dt
dt
0    1.
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
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Teijin and Teijin Twaron
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Mathematical models
stationary case, rotating s
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 s
stationary case, rotating r
Comparison stationary cases
Variable k
Solving the systems
Further research
Questions?
?
27