Transcript Pressure-drop (epd) - Isaac Newton Institute for

Computational Rheology
INNFM
Isaac Newton Institute
Dynamics of Complex Fluids -10 Years on
Mike Webster
Schlumberger,
UNAM-(Mexico), INNFM
Juan P. Aguayo
Hamid Tamaddon
Mike Webster
Institute of non-Newtonian Fluid Mechanics
EPSRC Portfolio Partnership
INNFM
Computational Rheology –
Some Outstanding Challenges
 To achieve highly elastic, high strain-rate/deformation rate
solutions (polymer melts & polymer solutions)
 To quantitatively predict pressure-drop, as well as flow field
structures (vortices, stress distributions)
• To accurately represent transient flow evolution in complex
flows
• To quantitatively predict multiple-scale response (multimode)
• To achieve compressible viscoelastic representations
TRANSIENT & STEADY
INNFM
EPTT
Planar
Contraction Flows
Oldroyd
Axisymmetric
Planar
Axisymmetric
Pressure-drop vs flow-rate in
contractions
axisymmetric
planar
Fluid viscosity = 1.75Pa.s –
8:1 contraction, exit length
7.4mm
Fluid viscosity = 1.75Pa.s –
20:1 contraction, exit length
40mm
20:1 Planar contraction - Fluid v iscosity 1.7Pa.s
8:1 Axisymmetric contraction - Fluid v iscosity 1.7Pa.s
12
60
50
Newtonian syrup
Boger Fluid
10
8
Q (g/s)
Q (g/s)
40
30
6
20
4
10
2
0
0
0 10
Newtonian Syrup
Boger Fluid
4
2 10
4
4 10
4
6 10
4
8 10
5
1 10
5
1.2 10
5
1.4 10
0
0
0 10
4
5 10
5
1 10
DP (Pa)
DP (Pa)
 Newtonian syrup
 Boger fluid
5
1.5 10
5
2 10
Pressure drop (epd) vs. We,
4:1:4 axisymmetric
Szabo et al.
with FENE-CR
J. Non-Newt. Fluid Mech. 72:73-86, 1997
epd
We
Schematic diagram for a) 4:1:4
contraction/expansion, b) 4:1 contraction
a)
Symmetry line
Front-face
Mid-plane
2nd
Quadrants
Back-face
Ru

4
3rd
1st
Szabo et al.
J. Non-Newt. Fluid Mech. 72:73-86, 1997
4th
Rothstein and McKinley
Ru
Ru

4
J. Non-Newt. Fluid Mech. 86:61-88, 1999
J. Non-Newt. Fluid Mech. 98:33-63, 2001
Wapperom and Keunings
J. Non-Newt. Fluid Mech. 97:267-281, 2001
Ru

4
P
b)
Front-face
Symmetry line
End of
rounded-corner
2nd
Quadrants
1st
3
 Rc
4
Ru
Ru
Rc = 
4
Rc
Excess pressure drop (epd - P )
 P : Total pressure drop
Pfd  Pupstream  Pdownstream
(P  Pfd )Boger
P  (P  P
)
fd Newt

(Pen )Boger
(Pen )Newt
Pressure-drop (epd) vs. We, Oldroyd-B,
a, c) axisymmetric, b, d) planar
A
x
i
s
y
m
m
e
t
ri
c
4:1:4
a)a)
b)b)
P
l
a
n
a
r
4:1
c)
d)
Pressure profile around constriction zone,
4:1:4 axisymmetric and planar case
[1] Axi, We = 1
[2] Axi, We = 2
[3] Axi, We = 3
[4] Planar, We = 1
[5] Planar, We = 2
[6] Planar, We = 3
15
+
10
P
+
+
[1]
[2]
[3]
+
+
+
+
5
+
[5]
+
[6]
+
+
[4]
0
-0.6
-0.4
-0.2
0.0

0.2
Oldroyd-B, =0.9
+
0.4
+
+
0.6
+
N1p 3D view – 4:1:4
contraction/expansion
A
x
i
s
y
m
m
e
t
ri
c
P
l
a
n
a
r
Oldroyd-B, =0.9
(P - PNewt) and stress profiles along wall,
4:1 and 4:1:4 axisymmetric case
4:1
4:1:4
7
Newt
[1]
-2
5
[2]
[3]
-3
-4
[1] We = 1
[2] We = 2
[3] We = 3
Newtonian
-5
-6
P - PNewt
P - PNewt
-1
6
Front-face
Front-face
0
4:1:4 Contraction/expansion,
pressure at boundary wall
4
[1] We = 1
[2] We = 2
[3] We = 3
Newtonian
Back-face
1
Mid-plane
4:1 Contraction, pressure at boundary wall
3
2
1
0
End of rounded-corner
-1
0
1

2
3
-1
-2
4
30
2
3
4
[1] We = 1
[2] We = 2
[3] We = 3
Back-face
Mid-plane
Front-face
40
11
11
[1] We = 1
[2] We = 2
[3] We = 3
20
20
10
10
0
0
End of rounded-corner
-10
-2
1

50
50
30
0
4:1:4 Contraction/expansion, 11 at boundary wall
4:1 Contraction, 11 at boundary wall
40
-1
Front-face
-7
-2
-1
0
1

2
3
4
-10
-2
-1
Oldroyd-B, =0.9
0
1

2
3
4
(P - PNewt) and stress profiles along wall –
4:1:4 planar and axisymmetric case
Planar
Axisymmetric
3
2
4
3
0
0
3
4:1:4 Planar contraction/expansion,
11 at boundary wall
40
11
30
20
10
0
0
-1
0
1

2
3
1

2
3
4
4
[1] We = 1
[2] We = 2
[3] We = 3
20
10
-10
-2
0
50
[1] We = 1
[2] We = 2
[3] We = 3
Back-face
Mid-plane
Front-face
30
-1
4:1:4 Axisymmetric contraction/expansion,
11 at boundary wall
50
40
-1
-2
4
Back-face

2
Mid-plane
1
Newt
Front-face
0
[3]
[2]
[1]
2
1
-1
[1] We = 1
[2] We = 2
[3] We = 3
Newtonian
Back-face
5
1
-1
-2
11
6
Front-face
[1] We = 1
[2] We = 2
[3] We = 3
Newtonian
Back-face
4
Mid-plane
P - PNewt
5
Front-face
6
Mid-plane
4:1:4 Axisymmetric contraction/expansion,
7
pressure at boundary wall
P - PNewt
4:1:4 Planar contraction/expansion,
pressure at boundary wall
7
-10
-2
Oldroyd-B, =0.9
-1
0
1

2
3
4
Pressure-drop (epd) vs. We,
4:1:4 axisymmetric, alternative models
epd
We
epd
We
(P - PNewt) profiles along wall –
4:1:4 axisymmetric, increasing 
15
4:1:4 Contraction/expansion, pressure at boundary wall,  = 1/9
We = 0.5
We = 1.5
Newtonian
P - PNewt
10
monotonic
decrease epd
5
0
-5
15
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9

-8
-7
-6
-5
-4
-2
-1
0
1
2
4:1:4 Contraction/expansion, pressure at boundary wall,  = 0.9
We = 1
We = 2
We = 3
Newtonian
10
P - PNewt
-3
upturn epd
5
0
-5
15
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9

-8
-7
-6
-5
-4
-2
-1
0
1
2
4:1:4 Contraction/expansion, pressure at boundary wall,  = 0.99
We = 1
We = 2
We = 5
Newtonian
10
P - PNewt
-3
upturn &
enhanced epd
5
0
-5
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9

-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
Alternative differential pressure-drop measure
(Pen )Boger  (Pen ) Newt
(Pen ) Newt
 P  1 
[PBoger  PNewt ]en
(Pen ) Newt
[PBoger  PNewt ] fd  0
[PBoger  PNewt ]en
(Pen ) Newt
Since

(Pen ) Newt  0
 P  1 
& by calibration
[PBoger  PNewt ]
(Pen ) Newt
[PBoger  PNewt ]exit  0
[P  PNewt ]entry  0 
up sp
P 1
Rate of dissipation & pressure-drop, 4:1:4
P  Pinlet  Pexit
definition
P Q  D
rate of dissipation
[Pinlet  Pexit ]Boger Q  DBoger ,
[P  PNewt ]inlet
[Pinlet  Pexit ]Newt Q  DNewt
D  DNewt
 [P  PNewt ]exit 
Q
0
 PBoger  PNewt  fd  0
[P  PNewt ]entry
up sp
 D  DNewt 

 (P  1)(Pen ) Newt

Q

 constriction
zone
Seeking {P – 1} > 0
 D  D
Newt

constriction
zone
0

P  P 
Newt
entry
upsp
0
Pressure-drop (epd) vs. a) We, b) upstream
sampling distance, 4:1:4 axisymmetric
epd
We
epd
4:1:4 axisymmetric vortex cell size, Oldroyd-B,  change
=0.99
= 0.99
We = 2
We = 1
sal = 0.552
upturn &
enhanced epd
sal = 0.553
We = 5
sal = 0.524
sal = 0.556
sal = 0.562
=1/9
= 1/9
We = 0.1
sal = 0.707

mono-dec
epd
We = 1.5
We = 1
sal = 0.507

sal = 2.06

sal = 0.656
sal = 3.08
sal = 0.091


sal = 0.041

Rheological properties: Oldroyd-B, LPTT, EPTT, SXPP
a) Oldroyd-B extensional viscosity, 
b) Shear and extensional viscosity, 0.9
c) Shear and extensional viscosity, 0.99
NEW BOGER fluid modelling
& Pressure Drop
Axisymmetric contraction
Planar contraction
Centreline pressure gradient
4:1:4 axisymmetric, Oldroyd-B
=1/9
=0.9
=0.99