Extensional viscosity measurements of drag-reducing 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
141
Outline
• Introduction: Drag reduction and extensional
viscosity
• Fluid shear and oscillatory shear rheology
• Capillary Break-up technique
• Extensional viscosity data
• Conclusions
142
• (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%.
Introduction
Major reviews by• Lumley (1969) [185 cites]• Virk (1975) [310 cites]• Nieuwstadt and den Toonder (2001)*
*Turbulence structure and Modulation, (ed. A. Soldati and R. Monti) Springer
Still significant interest (>50 papers in 2004 and 15 papers already
in 2005).
143
f Re plot for drag-reducing polymer solutions
Re
Frictionfactor
103 104 1050.001
0.005
0.009
0.013
0.017
16/Re
Virk
Blasius
Introduction
0.4% CMC
0.2% XG
0.09% XG / 0.09% CMC
0.2% PAA
A keyword in most attempts to explain the mechanism of dragreduction is extensional (or elongational) viscosity
144
*Escudier, Presti and Smith (1999) JnNFM
Extensional viscosity
Why is extensional viscosity thought to play a major role in turbulent
drag reduction? Counter-rotating
eddy-pairs
Fluid element Direction of
flow
145
Fluid shear rheology
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 9004-32-4] molecular weight (7 x 105 g/mol)
(d) 0.09% XG / 0.09% CMC blend [same grades as unblended polymers].
146
Fluid shear rheology
0.4% CMC
0.2% XG
0.09% XG / 0.09% CMC
PAA
0.2%
0.02%
0.01%
Figure 1: Viscosity versus shear rate for various polymer solutions (including Carreau-Yasuda fits)
Shear rate (1/s)
Visco
sity(Pa.s)
10-3 10-2 10-1 100 101 102 103 10410-3
10-2
10-1
100
101
102
147
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
Figure 2: Storage (G' - open) and loss (G' ' - closed) moduli for various polymer solutions.(a) 0.4% CMC, (b) 0.09%CMC/0.09% XG, (c) 0.2% XG & (d) 0.2% PAA
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
(c) 0.2% XG
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
(b) 0.09% CMC / 0.09% XG
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
(d) 0.2% PAA
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
(c) 0.2% XG
Angular frequency (rads-1)
G'(
Pa)
G''
(Pa)
10-1 100 10110-3
10-2
10-1
100
101
10-3
10-2
10-1
100
101
G’ (open symbols), G’’ (closed symbols)0.4%
CMC
0.2% XG
0.09% XG / 0.09% CMC
0.2% PAA
0.02%
0.01%
= 2.1 s
= 25 s
= 5.8 s
= 30 s
148
Capillary Break-up technique
t =- 50 ms
D = 4 mm
h0 = 2 mm
149
Capillary Break-up technique
t =- 50 ms t > 0
D = 4 mm
h0 = 2 mm
hf 8 mm
= hf / h0
DMID (t)Laser micrometer
measures DMID (t)
Surface tension drives ‘pinch off’ of liquid
thread resisted byextensional stresses
149
Capillary Break-up technique
t > 0
DMID (t)
0
MID
H D(t)D
2(t) ln
dt(t)dD(t)
(t)D2 t) ,(
MID
MID
/
E
)3/ (-t exp )/(GDD(t)DEX
1/3
00MIDσ
Single relaxation time Maxwell model gives:
alternatively you may calculate a Hencky strain at the midpoint:
and estimate an apparent ‘extensional viscosity’:
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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}
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(b) 0.09% CMC / 0.09% XG
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(a) 0.4% CMC
time (s)F
ilam
entd
iam
eter
(mm
)
0 0.05 0.1 0.15 0.210-2
10-1
100
time (s)F
ilam
entd
iam
eter
(mm
)
0 1 2 3 4 5 6 7 810-2
10-1
100
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(b) 0.09% CMC / 0.09% XG
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(a) 0.4% CMC
time (s)
Fila
men
tdia
met
er(m
m)
0 1 2 3 4 5 6 7 8
10-2
10-1
100(d) 0.2% PAA
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.210-2
10-1
100
Thinning of filament diameter
0.2% XG 0.2% PAA
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time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(b) 0.09% CMC / 0.09% XG
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(a) 0.4% CMC
time (s)
Fila
men
tdia
met
er(m
m)
0 1 2 3 4 5 6 7 8
10-2
10-1
100(d) 0.2% PAA
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.210-2
10-1
100
EX = 0.065 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}
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(b) 0.09% CMC / 0.09% XG
time (s)
Fila
men
tdia
met
er(m
m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.5
1
1.5
2
2.5
3
3.5
4
(a) 0.4% CMC
time (s)F
ilam
entd
iam
eter
(mm
)
0 0.05 0.1 0.15 0.210-2
10-1
100
time (s)F
ilam
entd
iam
eter
(mm
)
0 1 2 3 4 5 6 7 810-2
10-1
100
EX = 0.840 s
Thinning of filament diameter
0.2% XG 0.2% PAA
)3/ (-t exp )/(GDD(t)DEX
1/3
00MIDσ
Effects of inertia
Finite extensionabilit
y effects?
‘intermediate times’
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Extensional viscosity
0.2% XG 0.2% PAA
dt(t)dD(t)
(t)D2 t) ,(
MID
MID
/
E
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
(b) 0.4% CMC(a) 0.2% PAA
(d) 0.09% CMC / 0.09% XG
Hencky strainE
xten
sio
nal
visc
osi
ty(P
a.s)
0 1 2 3 4 5 6 70
250
500
750
1000
1250
1500
1750
EX 1600 Pa.s
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
(b) 0.4% CMC(a) 0.2% PAA
(d) 0.09% CMC / 0.09% XG
Hencky strain
Ext
ensi
on
alvi
sco
sity
(Pa.
s)
0 1 2 3 4 5 6 70
10
20
30
EX 1.5 Pa.s
0
MID
H D(t)D
2(t) ln
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Extensional viscosity data
FluidDR (%)* (Pa.s
)
0.2% PAA
48 1600 67178000
1660
0.2% XG
46 1.5 0.086 465 89
0.4% CMC
39 6 8.2 6000 65
CMC/XG blend
36 1 0.81 264 44*DR at Re =5000
EX
0
EX
Tr
EXTr
EXTr
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Conclusions…
• 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)
1414