Transcript Document

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Extensional viscosity measurements of dragreducing polymer solutions using a Capillary
Break-up Extensional Rheometer
Robert J Poole , Adam Swift and Marcel P Escudier
Department of Engineering, University of Liverpool, UK
ESR 2nd Annual European Rheology Conference, April 21-23, Grenoble-France
Outline
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• Introduction: Drag reduction and extensional viscosity
• Fluid shear and oscillatory shear rheology
• Capillary Break-up technique
• Extensional viscosity data
• Conclusions
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Introductio
n
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• (Turbulent) drag reduction by polymer additives first discovered by
Toms (1948) (or Mysels (1949)).
• Small additions (as little as a few p.p.m) of a polymer additive to a
Newtonian solvent can reduce friction factor by up to 80%.
Major reviews by
• Lumley (1969) [185 cites]
• Virk (1975) [310 cites]
• Nieuwstadt and den Toonder (2001)*
Still significant interest (>50 papers in 2004 and 15 papers already
in 2005).
*Turbulence structure and Modulation, (ed. A. Soldati and R. Monti) Springer
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Introductio
n
0.017
0.4% CMC
0.013
0.2% XG
0.009
Friction factor
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16/Re
0.09% XG /
0.09% CMC
Blasius
0.2% PAA
0.005
Virk
0.001 3
10
10 4
10 5
Re
A keyword in most attempts to explain the mechanism of drag
f Re plot for drag-reducing polymer solutions
reduction is extensional (or elongational) viscosity
*Escudier, Presti and Smith (1999) JnNFM
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Extensional viscosity
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Why is extensional viscosity thought to play a major role in turbulent
drag reduction?
Counter-rotating eddypairs
Fluid element
Direction of flow
Fluid shear
rheology
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Polymers studied (water as solvent for all):
(a) Polyacrylamide (PAA 0.2%, 0.02% and 0.01%) [Separan AP 273 E
from Floreger] ‘Very flexible’ polymer, high molecular weight (2 x 106
g/mol)
(b) 0.2% Xanthan gum (XG) [Keltrol TF from Kelco]. Semi-rigid polymer,
high molecular weight (5 x 106 g/mol)
(c) 0.4% Sodium carboxymethylcellulose (CMC) [Aldrich Grade 9004-324] molecular weight (7 x 105 g/mol)
(d) 0.09% XG / 0.09% CMC blend [same grades as unblended
polymers].
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Fluid shear
rheology
10
14
2
Viscosity (Pa.s)
0.4% CMC
10
1
10
0
0.2% XG
0.09% XG /
0.09% CMC
PAA
10
-1
10
-2
 0.2%
 0.02%
 0.01%
10 -3 -3
10
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
Shear rate (1/s)
Figure 1: Viscosity versus shear rate for various polymer solutions (including Carreau-Yasuda fits)
G’ (open symbols), G’’ (closed symbols)8 14
1
1
1
1011
10
1
101 1010
1
10
10
10
0.09% CMC
-2
-2
-2
10-2-2
-2 10
10-2 1010
-2
10 10
10
-1
-2
10 -2
10
-2
10
-2
10
 = 2.1 s
 = 5.8 s
1
1
10 10
1
101 10
101
10
1
(c) 0.2% XG
(c) 0.2%
(c) 0.2%
XGXG
0
-1
-1
10 10
-1
10-1 10
-2
-2
10 10
-2
10-2 10
0
-2
-2
10 10
-2
10-2 10
0
10
100
G' (Pa)
G' (Pa)
G'' (Pa)
-1
10
10
G''
(Pa)
G'
(Pa)
G'
(Pa)
G''
(Pa)
-1
10 10
-1
10-1 10
0
10
10
0
0
10
10
G'' (Pa)
G' (Pa)
G' (Pa)
G'
G'(Pa)
(Pa)
0
10
0
10
-1
10-1
10
10
-1
10
-2
10-2
10
10
-2
10
0
1
10-3
-3
10
-3
10
-3
-1
-1
0
0
1
1010
1
10
1
10
1
0
1010
0
10
100
0
0
 0.02%
-1
-1
-2
1010
-2
-2
1010
-3
-3
 0.01%
-2
-1
-1
1010
10
-1
10
10
-2
10
 = 30 s
-3
10
-1
-1
10
10
10
10
PAA
 = 25 s
10
-3
10 -3
10
-3
10
1
1
(d) 0.2%
 PAA
0.2%
0.2% XG
0
10
1
10
-3
0
10 1
10 10 0
10
1
100 10
10
-1 -1 10
Angular
frequency
)
-1
Angular
frequency
(rads
)
-1(rads
Angular
frequency
Angular
frequency
(rads )(rads )
G'' (Pa)
G'' (Pa)
-3
-3
10-3-3
-3 10
10-3-31010
-3
-1
-1
10 10
10
10 10-1
10-1 10
G'' (Pa)
1
G'' (Pa)
0
0
1
0
1
100 1010
101 1010
-110
10
-1
Angular
frequency
(rads
)
-1
-1 )
Angular
frequency
(rads
Angular
frequency
(rads
Angular
frequency
(rads
) )
1
1
10 10
1
101 10
10
0
-1
10 -1
10
-1
10
10
-2
-3
-3
10-3-31010
-1
-1
10
-1
10-1 1010
10
G'' (Pa)
-1
-1
-1
10-1-1
-1 10
10-1 1010
-1
10 10
10
0
(Pa)
G''G''(Pa)
G''
G''(Pa)
(Pa)
-1
-1
-1
10-1 1010
10
10
0
10 0
10
0
10
10
G'G'(Pa)
G''
(Pa)
(Pa)
G'' (Pa)
G'G'(Pa)
(Pa)
0
0
1000
10
0
1000 1010
0
10
10
10
G' (Pa)
G' (Pa)
0
0
1000 1010
10
G'G'(Pa)
(Pa)
1
(b) 0.09% CMC / 0.09% XG
(b) 0.09% CMC
/ 0.09%
0.09%
XGXG/
0.4% CMC
-2
-2
10-2 1010
10
1
10 1
10
1
10
10
G'' (Pa)
1
1
1
101 1010
10
1
10
10
0
1
0
1
10
10
10
10
-1 -1) 10 10
Angular
frequency
-1
Angular
frequency
(rads
)
-1(rads
Angular
frequency
)
Angular
frequency
(rads(rads
)
1
-3
10
10
-3
-3
10
10
10
10
-3
-3
-3
10
-1 10-1
10-3 10
-1
-1
10 10
0
0 10
1 10
0
1
10
10
10
10
-1
100frequency
101
-1
Angular
(rads
)
-1
Angular
frequency
(rads
Angular frequency
(rads
) ) -1
1
-3
10
10-3
Capillary Break-up technique
D = 4 mm
h0 = 2 mm
t =- 50 ms
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Capillary Break-up technique
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Surface tension drives
‘pinch off’ of liquid
thread  resisted by
extensional stresses
hf  8 mm
= hf / h0
DMID (t)
Laser micrometer
measures DMID (t)
D = 4 mm
h0 = 2 mm
t =- 50 ms
t>0
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Capillary Break-up technique
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Single relaxation time Maxwell model gives:
D (t)  D (GD / σ ) exp (-t / 3  )
1/3
MID
DMID (t)
0
0
EX
alternatively you may calculate a Hencky
strain at the midpoint:
 DMID (t) 

 H (t)  2 ln 
 D0 
and estimate an apparent ‘extensional
viscosity’:
 ( , t) 
E
t>0
2 / DMID (t)


dDMID (t)
(t)
dt
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Thinning of filament diameter
0.2% XG
4
4
(a) 0.4% CMC
1
0.5 -1
10
0
0
0
0.05
0.1
0.15
0.2
time (s)
10-1
0.25
0.3
0.35
0
10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
time (s)
-2
-2
0
0.05
0.1
3
2.5
0
10 2
2
1.5
1
0.5
10
3
Filament diameter (mm)
Filament diameter (mm)
Filament diameter (mm)
0
(d) 0.2% PAA
2.5
Filament diameter (mm)
Filament diameter (mm)
Filament diameter (mm)
10
1.5
0
3.5
Filament diameter (mm)
10
3
3
2.5
(b) 0.09% CMC / 0.09% XG
3.5
3.5
2
4
(a) 0.4% CMC
(b) 0.09% CMC / 0.09% XG
3.5
0
time (s)
1
2
3
0.15
4
time (s)
Figure 4: Filament diameter versus time for various polymer solutions.
(a) 0.4% CMC, (b) 0.09%CMC/0.09% XG, (c) 0.2% XG & (d) 0.2% PAA
{solid line represents polynomial fit}
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70.2 8
Filament diameter (mm)
4
0.2% PAA
100
1.5
2.5
2
1.5
1
1
0.5
0.5
0
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.05
0.1
0.15
time (s)
10-1 10
-1
10-2 10
0
-2
0
0.05
1
0.1
time (s)
0.2
0.25
0.3
0.35
time (s)
2
0.15
3
0.2
4
5
6
time (s)
Figure 4: Filament diameter versus time for various polymer solutions.
(a) 0.4% CMC, (b) 0.09%CMC/0.09% XG, (c) 0.2% XG & (d) 0.2% PAA
{solid line represents polynomial fit}
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11
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Thinning of filament diameter
0.2% XG
4
4
(a) 0.4% CMC
1
EX = 0.065 s
3
0.5 -1
10
0
0
0
0.05
0.1
0.15
0.2
0.25
0.3
time (s)
10-1
0.35
0
10
10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
time (s)
-2
-2
0
0.05
0.1
3
2.5
0
10 2
2
1.5
1
0.5
Filament diameter (mm)
Filament diameter (mm)
Filament diameter (mm)
0
(d) 0.2% PAA
2.5
Filament diameter (mm)
Filament diameter (mm)
Filament diameter (mm)
10
1.5
0
3.5
Filament diameter (mm)
10
3
3
2.5
(b) 0.09% CMC / 0.09% XG
3.5
3.5
2
4
(a) 0.4% CMC
(b) 0.09% CMC / 0.09% XG
3.5
0
1
time (s)
2
3
0.15
4
5
6
70.2 8
Filament diameter (mm)
4
0.2% PAA
100
1.5
1
0.5
0
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
10-1 10
-1
10-2 10
0
-2
0
0.05
1
0.1
0.2
0.25
0.3
0.35
time (s)
2
0.15
3
0.2
4
5
6
7
8
time (s)
Figure 4: Filament diameter versus time for various polymer solutions.
(a) 0.4% CMC, (b) 0.09%CMC/0.09% XG, (c) 0.2% XG & (d) 0.2% PAA
{solid line represents polynomial fit}
‘intermediate times’
D (t)  D (GD / σ ) exp (-t / 3  )
1/3
MID
0.35
time (s)
time (s)
Figure 4: Filament diameter versus time for various polymer solutions.
(a) 0.4% CMC, (b) 0.09%CMC/0.09% XG, (c) 0.2% XG & (d) 0.2% PAA
{solid line represents polynomial fit}
Effects of inertia
2
1.5
EX = 0.840 s
1
0.5
time (s)
2.5
0
0
EX
Finite extensionability
effects?
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Extensional viscosity
0.2% XG
0.2% PAA
(a) 0.2% PAA
1750
(b) 0.4% CMC
30
(b) 0.4% CMC
1500
Extensional viscosity (Pa.s)
Extensional viscosity (Pa.s)
(a) 0.2% PAA
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EX  1600 Pa.s
1250
20
1000
(d) 0.09% CMC / 0.09% XG
10
EX  1.5 Pa.s
0
0
1
2
3
4
Hencky strain
5
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Figure 6: Extensional viscosity versus strain rate for various polymer solutions.
(a) 0.2% PAA, (b) 0.4% CMC, (c) 0.2% XG (d) 0.09% CMC / 0.09% XG
 Global polynomial fit
Symbols local cubic fit
 DMID (t) 

 H (t)  2 ln 
 D0 
7
750
(d) 0.09% CMC / 0.09% XG
500
250
0
0
1
2
3
4
Hencky strain
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Figure 6: Extensional viscosity versus strain rate for various polymer solutions.
(a) 0.2% PAA, (b) 0.4% CMC, (c) 0.2% XG (d) 0.09% CMC / 0.09% XG
 Global polynomial fit
Symbols local cubic fit
2 / DMID (t)

 E ( , t) 

dDMID (t)
(t)
dt
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Extensional viscosity data
Fluid
DR (%)* EX
(Pa.s)
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EX
EX
EX
Tr 
Tr 
Tr 
0


0.2%
PAA
48
1600
67
178000
1660
0.2% XG
46
1.5
0.086
465
89
39
6
8.2
6000
65
36
1
0.81
264
44
0.4%
CMC
CMC/XG
blend
*DR at Re =5000
Conclusions…
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• Capillary-thinning behaviour of PAA significantly different
to XG, CMC and a XG/CMC blend
• Extensional viscosity of PAA two orders of magnitude
greater than XG (despite very similar levels of DR)
• Biaxial not uniaxial extensional flows which are created by
streamwise vortical structures?
(Shaqfeh et al (2004) ICR)