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’:

1410

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

1411

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’

1411

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

1412

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

1413

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