Group Title: converging flow of dilute polymer solutions
Title: The converging flow of dilute polymer solutions
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Title: The converging flow of dilute polymer solutions
Physical Description: xii, 207 leaves : ill. ; 28cm.
Language: English
Creator: Balakrishnan, Chander, 1947-
Copyright Date: 1976
 Subjects
Subject: Polymers and polymerization   ( lcsh )
Viscous flow   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Chander Balakrishnan.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 201-206.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00099396
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000170079
oclc - 02915207
notis - AAT6492

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THE CONVERGING FLOW OF DILUTE POLYMER SOLUTIONS


By

CHANDER BALAKRISHNAN














A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA



































To

Baba and Mummy














ACKNOWLEDGMENTS


The author wishes to express his gratitude and appreciation

to:

Dr. R. J. Gordon for suggesting the research topic and for

his guidance.

Dr. H. E. Schweyer, Dr. J. C. Biery, Professor R. D. Walker

and Dr. T. E. Hogen Esch for their helpful suggestions.

C. S. Chiou and M. C. Johnson for their help and suggestions.

P. M. Mathias, J. C. Noronha, J. V. Shah and E. Toko for

keeping the author in good humor during difficult times.

Mr. Jack Kalway for building the experimental apparatus and

being very patient with the author every time he broke it.

Mr. R. L. Baxley, Mr. M. R. Jones and Mr. T. Lambert for

assistance in the laboratory.

Ms. Patricia Pfeifer for typing the first draft of the

thesis.

Mrs. Jeanne Ojeda for typing the thesis in its final form.

The Chemical Engineering Department for financial support.

The National Science Foundation (Grant GK-31590) for partial

support of this research.

Ms. Jean Slobe for helping the author in numerous ways.














TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS................... .......................... iii

LIST OF FIGURES.............................................. vii

ABSTRACT.................... ................................ xi

CHAPTERS:

I

1.1 Introduction... ............................... 1

1.2 Summary of Thesis.............................. 2

1.3 Theories of Drag Reduction..................... 3

1.4 Features of Drag Reduction..................... 10

1.4.1 Dependence of DR on polymer molecular
weight and molecular weight distribu-
tion .................................... 10

1.4.2 The effect of solvent character and
polymer conformation on DR............... 11

II


2.1 Extensional Flows: Qualitative Considerations... 17

2.2 Extensional Flows: Quantitative Considerations.. 18

2.3 Literature Review: Newtonian Fluids............ 24

2.4 Literature Review: Non-Newtonian Fluids......... 26

2.4.1 Materially steady extensional flows....... 26

2.4.2 Materially unsteady flows................. 30

2.5 Kinematics of WGS Flow......................... 40










Chapters: Page

2.6 Analysis of Stress Field in WGS Flow.......... 46

III EXPERIMENTAL....................................... 58

3.1 Test Solutions ............................... 58

3.2 Polymer Solution Preparation.................. 60

3.3 Converging Flow Apparatus..................... 62

3.3.1 Details of orifices used............... 62

3.4 Photographic Equipment........................ 68

3.5 Making of a Run............................... 68

3.6 Thrust Measurements.............................. 69

3.7 Drag Reduction Measurements ................... 72

3.8 Vortex Inhibition (VI)........................ 74

IV EXPERIMENTAL RESULTS............................... 77

4.1 Drag Reduction ............................... 77

4.2 Vortex Inhibition................. ........... 77

4.3 Extensional Viscosity ......................... 77

4.4 Thrust Measurements........................... 105

4.5 Conformational Studies ....................... 108

4.6 Polymer Degradation........................... 115

4.7 Flow Instability.............................. 120

V DISCUSSION OF RESULTS ............................. 128

5.1 Extensional Viscosity......................... 128

5.2 Comparison of Results with Predictions of
Equations of Motion........................... 133

5.3 Comparison with Previous Investigations....... 138










Page

5.4 Correlation Between DR, VI, and Extensional
Viscosity ................................... 154

5.4.1 Comparison of DR, VI, and extensional
viscosity of deionized water solu-
tions................................. 156

5.4.2 Dependence of DR, VI, and extensional
viscosity on polymer conformation...... 159

5.4.3 Effect of polymer degradation on DR,
VI, and extensional viscosity.......... 164

5.5 Observations on the Converging Flow of
Poly(ethylene oxide).......................... 169

5.6 Comparison of Experimental Results with
Predictions of Constitutive Theories........... 171

5.7 Conclusions and Recommendations............... 173

5.7.1 Conclusions ........................... 173

5.7.2 Recommendations......................... 175

APPENDIX A......... ............ ...... .................... 177

BIBLIOGRAPHY................................................. 2C1

BIOGRAPHICAL SKETCH... ............... ....................... 207














LIST OF FIGURES


Figure Page

1-1 Representation of Counterrotating Eddy Pairs......... 9

2-1 Predictions of the GSE Model for Steady Extension.... 20

2-2 Ballman's Experiment................................ 20

2-3 Fano Flow.......................................... 32

2-4 Coordinate System in Fano Flow....................... 34

2-5 Wine Glass Stem Flow ................................ 36

2-6 Predictions of the Maxwell Model ..................... 38

2-7 -Division of WGS Flow into Different Regions.......... 41

2-8 Coordinates Used in Pickup's Kinematics.............. 45

2-9 The Bagley Plot.......................................... 49

2-10 Momentum Balance Sections............................ 52

2-11 Coordinate Systems Used in Sink Flow Kinematics...... 52

3-1 Converging Flow Loop ................................. 63

3-2 Details of Flow Tank................................. 64

3-3 Positioning of Camera During a Run................... 65

3-4 Details of Orifice Construction...................... 67

3-5 The Thrust Apparatus................................. 70

3-6 Calibration of Thrust Apparatus...................... 71

3-7 Flow Loop Description ................................ 73

3-8 Vortex Formation..................................... 75

3-9 Vortex Inhibition ................... .... ............. 75










Figure Page

4-1 Summary of Drag Reduction Data ....................... 79

4-2a Dependence of 6 on Orifice Velocity .................. 81

4-2b Dependence of 0 on Orifice Radius..................... 82

4-2c Dependence of 6 on Polymer Concentration........... 83

4-3 WGS Flow with 40 wppm Separan AP 273
Orifice Diameter 0.096 cm............................ 85
4-4 Dependence of Stretch Rate on Orifice Velocity........ 87

4-5 Dependence of Stretch Rate on Orifice Velocity....... 87

4-6 Dependence of Stretch Rate on Orifice Velocity....... 88

4-7 Dependence of Stretch Rate on Orifice Velocity....... 88

4-8 Dependence of Stretch Rate on Orifice Velocity....... 89

4-9 Dependence of Stretch Rate on Orifice Velocity....... 89

4-10a WGS Flow with Separan AP 273 10 wppm
Orifice Velocity = 400 cm/sec ....................... 91

4-10b WGS Flow with Separan AP 273 10 wppm
Orifice Velocity = 900 cm/sec....................... 91

4-11 Dependence of Stretch Rate on Orifice Velocity...... 92

4-12 Dependence of Entrance Pressure Drop on Orifice
Velocity ....................................... . 94

4-13 Dependence of Entrance Pressure Drop on Orifice
Velocity............................................ 94

4-14 Dependence of Entrance Pressure Drop on Orifice
Velocity......................................... . 95

4-15 Dependence of Entrance Pressure Drop on Orifice
Velocity........................................... 95

4-16 Dependence of Entrance Pressure Drop on Orifice
Velocity........................................... 96

4-17 Dependence of Entrance Pressure Drop on Orifice
Velocity .......................... .... ............. 96










Figure Page

4-18 Entrance Pressure Drop Data for Water............... 97

4-19 Dependence of Entrance Pressure Drop on Orifice
Velocity............................................ 99

4-20 Dependence of Entrance Pressure Drop on Orifice
Velocity............................................ 100

4-21 Dependence of Entrance Pressure Drop on Orifice
Velocity........................................... 100

4-22 Dependence of Entrance Pressure Drop on Orifice
Velocity........................................... 102

4-23 Dependence of Entrance Pressure Drop on Orifice
Velocity........................................... 102

4-24 Dependence of Entrance Pressure Drop on Orifice
Velocity........................................... 103

4-25 The Effect of Shear Viscosity on Entrance
Pressure Drop ..................................... 104

4-26 The Effect of L/D Ratio on Entrance Pressure Drop.. 106

4-27 Summary of Thrust Measurements ..................... 107

4-28 Effect of pH on Solution Viscosity................. 109

4-29a Effect of Polymer Conformation on Drag Reduction... 110

4-29b Effect of pH on Drag Reduction..................... 112

4-30 Stretch Rate Data for Fresh Separan AP 273 10 wppm. 114

4-31 Effect of Polymer Conformation on Entrance
Pressure Drop .................................... 116

4-32 Effect of Shear Degradation on Drag Reduction...... 118

4-33 Effect of Shear Degradation on Vortex Inhibition... 119

4-34a Flow Visualization with Fresh Separan AP 273 Solu-
tion, 10 wppm .................. ................... 121

4-34b Flow Visualization with Solution Degraded at
NRe = 20000 .................................... 121










Figure Page

4-34c Flow Visualization with Solution Degraded at
NRe = 40000 ...................................... 122
4-35 Effect of Shear Degradation on Entrance Pressure
Drop................. ............................... 123

4-36 Instability in WGS Flow: Stage 1.................... 125

4-37 Instability in WGS Flow: Stage 2................... 127

5-1 Variation of sin 6 (1 + cos e) with ................ 130

5-2 Dependence of Normal Stress Difference on Stretch
Rate...................... ......................... 134

5-3 Dependence of Normal Stress on Stretch Rate.......... 135

5-4 Dependence of Normal Stress Difference on Stretch
Rate................................................. 135

5-5 Comparison of Predictions of Equations of Motion
with Experiment.................................. 139

5-6 Comparison of Predictions of Equations of Motion
with Experiment............... ................... 139

5-7 Comparison of Predictions of Equations of Motion
with Experiment..................................... 140

5-8 Comparison of Predictions of Equations of Motion
with Experiment................................... 140

5-9 Metzner's Thrust Apparatus............... .. ... ...... 146

5-10 Momentum Balance: Control Surfaces................. 146

5-11 Momentum Balance: Control Surfaces.................. 148

5-12 Orifice Arrangement During Die Swell Measurement..... 152

5-13 Orifice Arrangement During Normal Runs............... 152

5-14 Comparison of Normal Stress Differences Calculated
by Entrance Pressure Drop and Thrust Measurements.... 155

5-15 Effect of Shear Degradation on Extensional Viscosity. 168










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy

THE CONVERGING FLOW OF DILUTE POLYMER SOLUTIONS

By

Chander Balakrishnan

March, 1976

Chairman: Ronald J. Gordon
Major Department: Chemical Engineering

Many dilute polymer solutions exhibit what is commonly known as a

wine glass stem shaped flow in flow through sudden contractions. For this

situation, flow through the contraction occurs only within a narrow conical

region, while the liquid external to this region moves in a slowly re-

circulating pattern. The flow in the conical region is predominantly

extensional in nature. In this work, flow behavior of very dilute polymer

solutions was studied in such a geometry. Extensional viscosities were

calculated from measurements of the pressure drop across the contraction

and photographs of the angle defining the extent of the conical region.

The dependence of extensional viscosity on molecular variables such as

molecular weight, molecular weight distribution and polymer conformation

was examined. The experimentally determined extensional stresses were

compared with predictions of the Equations of Motion and constitutive

theories of dilute polymer solutions. It was shown that previously

measured values of extensional viscosities using the thrust technique

were conservative due to the underestimation of tensile stresses by an

amount equal to the entrance pressure drop.











The experimentally measured viscosities were extremely large.

A 20 parts per million (by weight) solution of a high molecular weight

polyacrylamide was found to have an extensional viscosity as high as

30 poise, while its viscosity in shear was nearly equal to that of

water. Conformational studies carried out with polyelectrolytes

showed that the extensional viscosity is maximum when the polymer

molecules are in an expanded state.

The role of extensional viscosity in understanding the mechanism

of turbulent drag reduction and vortex inhibition was closely examined.

The results of this work offer considerable support to theories that

explain these phenomena as manifestations of the large resistance of

dilute polymer solutions to extensional motions.


_1














CHAPTER I


1.1 Introduction

It is well known that the resistance of a liquid to turbulent

flow can be considerably reduced by the addition of certain high

molecular weight polymers. This phenomenon, commonly known as drag

reduction (DR), has been observed with other substances as well,

such as soaps and fibers (Patterson et al. (1969)). DR has found

several applications both commercial and military. It has been

successfully used in firefighting, sewage disposal, transportation

of fluids through long pipelines and in reducing drag in the flow

of liquids past submerged vessels.

DR is sometimes referred to as the "Toms Effect," after

Toms (1949) who first reported it in the literature. Following

Toms, there was little interest shown in DR until the early sixties

when, spurred by potential naval applications, a systematic investiga-

tion of the effect began. The extensive research of the last fifteen

years has resulted in the elucidation of several important aspects

of DR, such as its dependence on various flow and molecular variables,

although the precise mechanism by which DR occurs is still being

debated. Several hypotheses have been proposed, and of these, the

most widely accepted explanation is that DR is a manifestation of

the large viscosity of dilute polymer solutions in extensional motions

Agoston (1954) observed DR in the turbulent flow of gasoline
thickened by napalm during World War II but this was not reported
until 1954.










such as those believed to occur in the wall region in turbulent flow.

The large resistance of dilute polymer solutions to extensional

or "stretching motions" is predicted by many constitutive theories

of dilute polymer solutions (Peterlin (1966), Denn and Marucci (1971),

Everage and Gordon (1971)), and, in recent years, has been experimentally

observed at polymer concentrations as low as 20 parts per million by

weight (Balakrishnan and Gordon (1975a)). However, a detailed examina-

tion of the correlation between extensional viscosity and drag reducing

ability under different conditions of polymer molecular weight, con-

centration, conformation and molecular weight distribution has yet to

appear in literature and is the primary goal of this thesis. Extensional

viscosities are obtained from a converging flow system and interpreted

in light of the Simple Fluid Theory due to Noll (1958/1959). The con-

verging flow of dilute polymer solutions is of interest not only as

a means of measuring extensional viscosity but also due to the remark-

able similarity in behavior between dilute polymer solutions and polymer

melts and concentrated polymer solutions. Converging flows with polymer

melts are commonly encountered in practical theological operations such

as fiber spinning and blow moulding (Petrie (1975)).


1.2 Summary of Thesis

The remainder of this chapter comprises a discussion of various

theories of DR followed by a brief literature review of pertinent

features of the phenomenon. In Chapter II, a review of extensional

flows of polymeric systems is presented. The converging flow of












dilute polymer solutions is discussed in detail, and equations

that are used later in the thesis developed.

The experimental techniques, procedures and apparatus used to

measure extensional viscosity are described in Chapter III. Chapter

IV summarizes the experimental results of this work. The implica-

tions of these results are discussed in Chapter V. It is shown

that the sink flow kinematics proposed by Uebler (1966) can be used

to describe the converging flow of dilute polymer solutions only

under specific conditions. The correlation between extensional

viscosity and DR and Vortex Inhibition (VI), a viscoelastic

effect recently discovered by our group and believed to be

closely related to DR, is discussed.


1.3 Theories of Drag Reduction

Ever since DR was discovered, the phenomenon has intrigued

and baffled some of the keenest minds in science. An understanding

of DR presents extreme difficulty because turbulence itself is not

completely understood and coupled with this is the fact that a

description of the behavior of dilute polymer solutions, at con-

centrations at which they are almost identical to the solvent, is

extremely complex. However, some consistent explanations have

been offered based on the current state of knowledge of turbulence

and dilute polymer solution rheology.

The earliest explanation of DR was proposed by Oldroyd (1949),

who suggested that the tube walls may cause preferred orientation of










polymer molecules in the wall region, thereby leading to, by some

vague mechanism, a highly mobile viscous sublayer. This would

effectively amount to slip at the wall which has not been experi-

mentally verified for very dilute polymer solutions. In 1959,

Shaver and Merrill proposed the shear-thinning viscosity theory to

explain DR. In the light of the findings of Paterson (1969), that

DR can be observed at very low polymer concentrations at which

the solutions possess viscosity that is Newtonian and equal to that

of the solvent, this theory is inadequate. Merrill et al. (1966)

suggested that polymer molecules--due to the large shear rates

near the wall region--could be elongated and oriented in the direc-

tion of flow, and thereby spatially hinder radial momentum transport.

This theory is also inadequate in explaining DR because of the

tremendous disparity between the dimensions of the macromolecules

and the smallest eddies in turbulent flow in most pipes. The

polymer molecules are so much smaller than the eddies that they

would be expected to be convected by the eddies rather than spatially

hinder the processes of turbulence production and diffusion.

In 1965, Gadd pointed out that in seeking a mechanism for DR,

one should focus on how polymer molecules affect turbulence generation

rather than turbulence dissipation. Walsh (1967), using this concept,

developed a theory that, according to a recent review by Hoyt (1972),

is the most comprehensive theory of DR that has yet to appear in

literature. In Walsh's theory, the fundamental idea is that in

turbulent flow the large disturbances that produce the Reynolds

A situation where the length scales of polymer molecules and
turbulence are equal could conceivably arise in turbulent flow in
capillaries.










stresses some distance downstream were, at an earlier time, small

disturbances at the edge of the viscous sublayer some distance

upstream. The production of turbulent energy (i.e., Reynolds

stresses) can be considered as follows: if a small scale dis-

turbance extracts more energy from the flow than it dissipates, it

begins to grow. Such.disturbances at the edge of the wall region

move out from the wall and towards the core as they are convected

downstream. After the elapse of some time, these small scale

disturbances become large scale disturbances. They then become part

of the large scale structure in the outer part of the boundary layer

that produces the Reynolds stresses. Walsh hypothesized that polymer

molecules alter this energy balance by absorbing energy from the small

scale disturbances, and after being convected downstream and outwards

from the wall, give up this energy by viscous dissipation. This would

amount to an alteration of the turbulent energy balance, resulting

in decreased production of Reynolds stresses. Walsh was able to

correlate the data of several investigators quite successfully with

this theory. However, he did not expound on how polymer molecules

absorb energy from the disturbances that grow to generate turbulence.

In the late sixties and early seventies, another "wall effect"

exhibited by drag reducing polymers began to attract the attention of

scientists, viz., the tendency of polymers to adsorb on tube walls.

This gave rise to the so-called "Adsorption Theory." In this theory,

drag reduction is explained in terms of the interfacial properties

of polymer solutions. Most polymer solutions are interfacially active,

i.e., the polymers have a tendency to concentrate at interfaces. Bryson









was perhaps the first to use this observation in explaining drag

reduction. Using a 30 ppm solution of Polyox WSR 301, Bryson et al.

(1971) measured the concentration of the polymer at the wall in pipe

flow and found it to be larger than in the bulk. It was then hypo-

thesized that only the "active" sites in the polymer molecules adsorb

on the wall leaving entangled loops of polymer chains protruding from

the wall. Fluid is trapped and its motion retarded within this layer,

thereby causing a thickening of the sublayer and consequently a smaller

velocity gradient. Although Bryson's theory is capable of qualitatively

explaining several phenomena exhibited by drag reducing systems, there

is little conclusive experimental evidence in its support. Moreover,

the observation of DR by Paterson (1969) at concentrations as low as

0.1 wppm suggests that the role of an adsorbed entangled layer of

polymer chains at the wall is unlikely to be important. Several other

investigators have reported the adsorption of drag reducing polymers

on tube walls but they do not explain how this adsorbed layer acts

to reduce friction. The adsorption theory has been challenged by a

number of workers, including Little (1971,1975) and Gyr and Mueller

(1974). Little measured the thickness of adsorbed layers of several

drag reducing polymers and found that there is no correlation between

the thickness of the adsorbed polymer layer and its drag reducing

ability. Gyr and Mueller performed experiments with identical tubes

of different materials and arrived at the conclusion that adsorption

makes little or no contribution to drag reduction.

In 1970, Gordon offered an explanation of DR based on an analysis

of the turbulence structure in the wall region proposed by Bakewell and










Lumley (1967). Bakewell and Lumley performed an extensive investigation

of the viscous sublayer in turbulent flow and found that the dominant

large scale structure of the wall region consisted of randomly dis-

tributed, counterrotating pairs of elongated eddies with their axes

in the streamwise direction. This structure was theoretically predicted

by Townsend (1956) and experimentally observed by the above mentioned

authors. This large eddy structure is consistent with the experimental

results on the so-called bursting phenomenon observed by Kline's group

at Stanford (Runstadler et al. (1963)), and also reported by Corino

and Brodkey (1969), and Brodkey et al. (1974).

The bursting phenomenon can be broken down into a sequence

consisting of the following events:

(1) Local deceleration of flow over a large area near the

wall. The region has very small velocity gradients and

almost resembles plug flow except at the edges of the

region where there are large velocity gradients. This

corresponds to the initial formation of the eddy pair.

(2) An ejection of fluid outward from the wall from

this decelerated region, occurring suddenly with a

large normal velocity component. This was the most

striking feature of Corino and Brodkey's observations.

As noted by Bakewell and Lumley, this stage corresponds

to a tilting of the plane of circulation of the counter-

rotating pair of eddies more nearly normal to flow. The

region between the eddies is characterized by a strong

updraft of low momentum fluid moving away from the wall.









Figure 1-1 reproduces the streamlines of the large eddy

structure described in Bakewell and Lumley's work.

(3) The sweep phenomenon in which, following the ejection

sequence, a mass of fluid larger than the ejection scale

and with velocity greater than the mean appears, moving

nearly parallel to the wall or slightly towards it. This

fluid swept the field of elements of retarded flow and

reestablished a normal velocity profile.

Seyer (1969) analyzed the streamline pattern of the flow between

the counterrotating pairs of eddies (Figure 1-1), and found that the

fluid that is swept up and between the eddies undergoes a transient

stretching motion. It was then proposed by Gordon (1970) that the

large extensional viscosity of dilute polymer solutions would decrease

the frequency and intensity of the "bursts" from the wall region into

the bulk, thereby altering the fundamental step in the turbulence

generation process. Thus a reduction in bursts would decrease

turbulence and, therefore, drag. It is well known that constitutive

theories of dilute polymer solutions predict extremely large resistances

to stretching motions (Peterlin (1966), Gordon (1970), Denn and Marucci

(1971), Everage and Gordon (1971)). Recent experimental studies have

confirmed these predictions. Metzner and Metzner (1970) calculated

the extensional viscosity of a 100 ppm solution of Separan AP 30

(a partially hydrolyzed polyacrylamide by Dow and an excellent drag

reducer) to be as much as 6000 times the shear viscosity of the

solution (1.44 cp). Similar results have been reported by the group

headed by Oliver at the University of Birmingham (1973a),and by

Balakrishnan and Gordon (1975a).













































- Tube Wall


Figure 1-1. Representation of Counterrotating
Eddy Pairs.










Recent experimental studies conducted by Donohue (1972) and

Hanratty's group at the University of Illinois offer considerable

support to the above mechanism. Donohue used a dye injection

technique to study the bursting rate in a drag reducing fluid.

This was found to be as much as 70% less than in water at equal

mass flow rates. Also, there was a considerable decrease in the

non-dimensionless streak spacing, indicating stabilization of the

sublayer. Fortuna and Hanratty (1972), using an electrochemical

technique (Mitchell and Hanratty (1966), Sirkar and Hanratty (1970a,1970b)),

found an increase in width of longitudinally oriented eddies. As

shown by Katsibas and Gordon (1974), this can be interpreted as a

reduction in the frequency of ejection from the wall.

The mechanisms proposed by Peterlin (1970), and Lumley (1973)

are similar to that proposed by Gordon in that they also attribute

DR to the large extension (and, therefore, large intrinsic viscosity)

of polymer molecules in extensional or irrotational flow fields.


1.4 Features of Drag Reduction

Some experimental aspects of DR that are pertinent to this work

are briefly reviewed below.


1.4.1 Dependence of DR on polymer molecular weight and molecular
weight distribution

The first detailed study of the effect of molecular weight on

DR was performed by Pruitt and Crawford (1965). They concluded from

their data that the reduction in friction shown by a homologous series

of polymers increased with increasing molecular weight. However,

Paterson (1969) showed that this general rule was true only if the










molecular weight distributions were similar. He found Polyox WSR 35,

a polyethylene oxide) polymer, to be a better drag reducer than

Polyox WSRN 750 even though the latter had a higher viscosity-

average molecular weight. This immediately poses the question as

to the effect of molecular weight distribution on DR. Paterson's

study revealed that WSR 35 had a broad molecular weight distribution

and that it possessed a high molecular weight "tail" which WSRN 750

did not. He also found that the intrinsic viscosity of polymer

solutions did not correlate with either DR or shear degradation.

Mechanical degradation affects the high molecular weight species

more than the low molecular weight species of a polymer sample

(Nakano and Minoura (1975)). Paterson found that large changes in

DR due to degradation were accompanied by disproportionately small

changes in intrinsic viscosity, and he concluded that DR depended

predominantly on the high molecular weight species of a polymer.

This has been further confirmed by Gordon and Balakrishnan (1972)

and by Morgan and Pike (1.972) who used fractionated samples of

polyox.


1.4.2 The effect of solvent character and polymer conformation on DR

The early investigations of DR were mostly conducted using

concentrated polymer solutions with pronounced shear-thinning behavior.

In the studies of Pruitt and Crawford (1965) and Fabula (1965) it was

shown that DR could also be observed with dilute polymer solutions

with concentrations in the parts per million range. Later Paterson

(1969) and Fabula (1966) measured DR at extremely low polymer con-

centrations--0.1 and 0.5 wppm respectively.










In such dilute polymer solutions, the polymer molecules are

separated by large amounts of fluid, and hydrodynamic interaction

is negligible. A measure of the separation between two polymer

molecules is defined as



S= Lc/2Rg


where Lc is the center to center distance between two molecules and

R is the radius of gyration. When C is equal to unity, the domains

of the two molecules would be expected to interact each other, and

smaller values of C would indicate overlapping domains. For the

polymer solution that exhibited DR at 0.1 ppm, Paterson found

= 8.5, thereby indicating a large separation between polymer

molecules. This is, of course, only an order of magnitude estimate

because is calculated using average quantities and so its value

would be expected to change depending on the type of average used.

Even so, it points out the importance of interaction between solvent

and polymer molecules in turbulent DR.

Hershey (1965), using four polymers in three different solvents,

found that the drag reducing ability of a polymer depended on the solvent

used. For instance, it was found that poly(isobutylene) (PIB) is a

better drag reducer in cyclohexane than in benzene. It is well known

that polymer solvent interactions are favored in "good" solvents, and

as a result the polymer molecules exist in an expanded state. On the

other hand, "poor" solvents do not favor interaction with polymer

molecules and the molecules therefore assume a compact structure.










In Hershey's work, benzene was at its 9 temperature thermodynamicc

interaction between PIB molecules and benzene was a minimum at this

temperature and hence the macromolecules were of smaller dimensions

in benzene than in cyclohexane). Hershey's work showed that DR in-

creases with increasing molecular dimensions and his observations

have been confirmed by several other investigators (Merrill et al.

(1966), Liaw (1971), Pruitt and Crawford (1965), Kim et al. (1973),

White and Gordon (1975)).

In 1966 Merrill et al. attempted to determine the effect of

molecular parameters such as the rigidity and expansion factors on

DR. They defined these factors as follows:


r -2
c = expansion factor = 12 r = actual mean square
r end to end distance


2-2
s flexibility factor = 1/ r = (define) mean square
r ,f end to end distance
under 6 conditions,
and

-2
r = calculated mean square
Send to end distance for
a freely rotating chain.

It follows from the preceding discussion that c is a measure of polymer

solvent interaction with large values indicating favorable interactions.

s on the other hand is a measure of steric and other such restrictions

on rotation about the bonds on the chain backbone and small values of

s indicate greater flexibility of the molecule. Comparing the values










of s for Polyox (PEO) in water and PIB in cyclohexane, and from

experiments conducted with PEO in water (a good solvent), and in

0.4M potassium sulphate at 350C (a theta solvent), Merrill et al.

arrived at the tentative conclusion that for optimum drag reduction,

c must be maximized and s minimized. However, Lee (1966), from data

obtained with a random coiling polymer, has pointed out that there

is no simple relationship between s and effectiveness of a polymer

in reducing turbulent friction. It is therefore seen that there is

no clear agreement on the effect of rigidity on drag reduction.

With the exception of the carboxymethyl cellulose and poly-

acrylamide polymers used by Pruitt and Crawford (1965), almost all

the work on solvent character and its effect on drag reduction that

has been discussed so far was done with uncharged, random coiling

polymers. In the past four years, intriguing results of several

investigations have been reported in which charged polymers have

been used.

Charged polymers are commonly known as polyelectrolytes and

their behavior in solution is quite different from that of uncharged

polymers. As the name suggests, polyelectrolytes have fixed charges

on the backbone chain and this causes them to undergo dramatic changes

in conformation with change in their ionic environment. Thus, poly

methacrylicc acid) which has the repeating unit


CH3


E CH2 C CH2 -


C = 0


OH









is known to be highly coiled in acidic environment, perhaps due to

intramolecular hydrogen bonding, and has an extended conformation

at high pH due to ionization. Similar behavior has been noticed

with poly(acrylic acid) (PAA), partially hydrolyzed polyacrylamide

(PAM), deoxyribonucleic acid (DNA), and other polyelectrolytes (Hand

and Williams (1969,1970), Katchalsky and Eisenberg (1951), Mathieson

and MacLaren (1965)). This tendency of polyelectrolytes to change

in conformation with changes in their ionic environment makes them

ideally suitable for studying the effect of polymer conformation on

DR.

In 1969, Hand and Williams reported results obtained with a

poly(acrylic acid) (PAA) polymer showing dramatic changes in DR with

conformation. They arrived at the conclusion that PAA was most

effective as a drag reducer when it was in a compact structure

(pH between 2.8 and 3) believed to be helical. They also conducted

experiments with calf thymus DNA and found that maximum drag reduction

was obtained when the polymer was in the native state, in the form of

a double helix (1970). Thus the results obtained by Hand and Williams

seem to indicate that a helical structure is most suitable for drag

reduction.

Parker and Hedley in 1972 reported results quite contrary to

those reported by Hand and Williams. Using a very high molecular

weight PAA, Parker and Hedley found that the polymer was most

effective as a drag reducer when it was in the form of stretched

rods, i.e., in neutral or alkaline medium. They also observed DR

to decrease with decreasing pH. Kim et al. (1973), Banijamali et al.






16


(1974) and Balakrishnan and Gordon (1975b)have also obtained results

similar to those of Parker and Hedley and it appears now that an

extended conformation is indeed more suitable to drag reduction than

a compact rigid structure (White and Gordon (1975), Banijamali et al.

(1974), Frommer et al. (1974)).

There are many other important aspects of drag reduction besides

those discussed here. Excellent summaries of these can be found in the

review articles by Hoyt (1972) and by Virk (1975).














CHAPTER II


2.1 Extensional Flows: Qualitative Considerations

In Chapter I, the importance of extensional viscosity in

explaining turbulent drag reduction, as well as in characterizing

the extensional flow behavior of polymeric materials was discussed

briefly. Before embarking on a discussion of the various geometries

and techniques available for the measurement of extensional viscosity,

it is important to qualitatively distinguish extensional flows from

shearing flows.

Shearing flows are rotational in nature in the sense that the

vorticity tensor is non-zero. For example, in the case of steady

shearing flow (Lodge (1964)), the magnitudes of the second invariants

of the strain rate and vorticity tensors are equal. In such flows,

the particles in the fluid rotate about an axis that is normal to

the direction of both the velocity and the velocity gradient. Polymer

molecules in the fluid do not experience large extensions because the

different segments of the coil are alternately stretched and compressed.

This gives each segment an opportunity to relax.

Extensional flows, on the other hand, are primarily irrotational

and are characterized by a strain rate tensor that is almost diagonal,

and by negligible vorticity rates. In flow fields such as this, the

polymer molecules tend to orient themselves in the direction of flow.

Since the molecules do not rotate in extensional flows as they do in









shearing flows, molecular extension is considerably larger, and this

causes large stresses to develop in the fluid.

2.2 Extensional Flows: Quantitative Considerations

Constitutive theories of viscoelastic materials such as polymer

solutions in which the stress is entirely determined by the instantaneous

rate of deformation, have frequently been found to be inadequate in

predicting the stress response of these substances to imposed deforma-

tions. For example, consider the Reiner-Rivlin theory


L = A (IIDIIID)D + A2(IIII0,II)D2 (2-1)


where T is the extra stress tensor, D is the rate of strain tensor,

IID and IIID are the second and third invariants of D and A1 and A2
are scalar functions of the invariants of 0. For a simple shearing

flow (in right cartesian coordinate system (RCCS)),


vi = (Gx2, 0, 0) (2-2)


where G is the shear rate, the Reiner-Rivlin theory predicts the

following results for the components of r:



'12 = 21 = A1 G/2 (2-3a)

N1 z ll -22 = 0 (2-3b)


N2 T22 33 = A2 G 2/4


(2-3c)










The first normal stress difference N1 is predicted to be zero, in

contrast to the experimentally observed N1 with viscoelastic fluids.

In order to accurately describe the behavior of viscoelastic

fluids, the stress must be allowed to depend not only on the instanta-

neous value of the deformation rate but also the deformation history

the fluid has experienced. For a fluid with a quickly fading memory,

the stress at any particular instant is determined by the history of

deformation only in the immediate past. When the fluid has been in

the same deformation field for a sufficiently long time, such that it

no longer remembers any other deformation it might have experienced

at some earlier past time, the flow is said to be one of constant

stretch history. To illustrate this point, consider the response

of a viscoelastic fluid in a steady extensional flow. The constitutive

equation chosen for this purpose is that due to Gordon, Schowalter and

Everage. The GSE model was obtained from a continuum modification of

the dumbbell theory of dilute polymer solutions and it predicts both

a non-Newtonian viscosity and non-zero primary and secondary normal

stress differences.


T + A 1t =- 2 ckTA(1-E)Dj (2-4)


STi = T i + vki iki + EDmj + T Dm} (2-5)
DTh ta t ,k sk ,kte r is de d


The total stress tensor rij is defined by


Ti pgij + 2sDi + ij


(2-6)









20










4-

w
E

CL
x














I'D 0



E-



U3 CD
CDM
Ln









COL C

X
0 >


ci
CD

CD
0 X




S E o


E I- cii-
-c






CD
oo
CMJ

[I







CO Ch



(SI _LISO1 E
Ual









In equations (2-4) to (2-6), c is a phenomenological constant subject

to the condition 0 s e < 1. X, the relaxation time, is related to
molecular parameters by the equation

us kM1+a
Ui:kT -c (2-7)


k and a are constants in the Mark-Houwink-Sakurada equation for zero

shear intrinsic viscosity.


[n]G=O = kMa (2-8)


M is the molecular weight of the polymer.
For the extensional flow field v = [(t)x, (t)x2/2, F(t)x3/2],
the predictions of the GSE model for the limiting case of constant stretch
rate are (Everage and Gordon (1971)):


T11 =T11 T 22 3
T T + 3pSF


3NckTX(1-E)F 2NckTX(1-c)F e-[l-21(l-c)F]t/
M[I{-2X(1-c)r 7-1+T-c)F}]T M[-2A(1-e)rJ

NckTA(1-r)r e-[l+X(1-C)F t/1
M[1+X(1-c)F] + 3usF (2-9)


The extensional viscosity nex is defined by the equation,


nex (T11 3us )/F (2-10)

In Figure 2-1 nex is plotted versus time for different molecular weights










of a 200 wppm solution of poly(ethylene oxide), an excellent drag

reducer with' = 0.4 and stretch rate G = 2000 sec-1. It is seen

that for low values of molecular weight (and, therefore, relaxation

time, from equation (2-7)), the extensional viscosity reaches a

constant value at large values of time. The flow is one of constant

stretch history, and the extensional viscosity in the asymptotic

region is a unique quantity for that material at the stretch rate

2000 sec-1. In the transient region where the extensional viscosity

is a function of time, the flow is not of constant stretch history

because the stress in the fluid has not reached a steady value

corresponding to the stretch rate. The extensional viscosity in

this region is not uniquely determined by the stretch rate 1.

In contrast to the predictions of a memory fluid, a fluid

that has no memory at all (such as a Newtonian fluid), attains a

steady value of stress as soon as the deformation is imposed, and

the viscosity is constant at all times. For such fluids, the stress

response is independent of the stretch history. However, in defining

an extensional viscosity for a viscoelastic fluid, the history of

deformation is extremely important, and nex is a unique quantity only

when the stress in the fluid has reached a steady value corresponding

to the deformation rate the fluid is experiencing.

In a flow field in which the deformation rate is spatially

dependent (such as in converging flows (Section 2.4.2)), the material

does not experience the same stretch rate as it flows from one point

to another. For a viscoelastic fluid in such a flow field, the

stresses vary with position, and the stress at any point in the










fluid may not attain the steady value associated with the local velocity

field. This is illustrated by considering the predictions of the GSE

model for spherical sink flow kinematics (Section 2.4.2) given by


(2-11)


vr = Q'/r2


v = = 0


Q' is the sink strength. The rate of strain

is given by


2Q'/r3

Di 0

0


0

- Q/r3
0


The primary normal stress difference for the

equations (2-11) and (2-12) is


tensor D is diagonal and


0

0

- Q'/r3


(2-12)


flow field defined by


F 5.8c 8F
NE(F) 2N -e) ( 4( 1 (4e3
0


4e r 4 4E
+ (2) 3 2(1-:) (3)3 3 exp d
0


7)/3 exp 3-C d


(2-13)


where r is the stretch rate at any point and is equal to 2Q'/r3

The integration in equation (2-13) over all past stretch rates

implies that the tensile stress (NE(F)) depends not only on the stretch










rate at that point but also on all past values of I that the fluid has

experienced. Clearly, this is not a flow of constant stretch history

and the extensional viscosity measured in such a flow will not generally

be a unique quantity in the sense that it was in the case of steady

extension. Thus the extensional viscosity measured at a particular

stretch rate in a materially steady flow cannot, in general, be expected

to be the same as that measured in an unsteady flow at the same value

of stretch rate.

Ballman (1965) has pointed out that results of extensional flow

experiments should be reported in terms of extensional viscosity only

when the stress in the fluid has reached a steady value, a situation

that can occur only in materially steady flows. In this work, however,

results are reported in terms of extensional viscosities even though

the flow field in which they are measured (converging flow) is

materially unsteady. This, we believe, is acceptable so long as it

is borne in mind that these values of extensional viscosities may not

be equal to those measured in steady extension, and that they are used

to give the reader only an indication of the large values of stresses

that are developed in the fluid in such a flow.


2.3 Literature Review: Newtonian Fluids

The first measurement of extensional viscosity was by Trouton

who, in 1906, measured the "coefficient of viscous traction" (i.e.,

extensional viscosity) of pitch and related it to its viscosity

measured in shear. From his experiments he arrived at the relationship


(2-14)


nex = 3sh










where nex and sh are the extensional and shear viscosities respectively.

Equation (2-14) is known as Trouton's rule, and the ratio of extensional

viscosity to shear viscosity, the Trouton ratio. Trouton's rule is

known to be obeyed by Newtonian fluids, and by viscoelastic fluids at

extremely low values of stretch rate. To illustrate this point, consider

the predictions of the GSE model for the steady extension of an initially

unstressed fluid (equation (2-9)). The extensional viscosity defined

by equation (2-10) can be written as


nex 1 2 exp [1-2X(1-c)F]t/A
3(P1osh- O [1-2(l-c)rj[[1(1-c)rF]- 3[1-2X(1-)2Fj


exp [-{l (- ) ]t/ (2-15)
3[1+A(1-c)F


where posh = + kTX(l-c) is the zero shear viscosity and ps is the

solvent viscosity.

For large values of time, the GSE model predicts



Snex = 1 (2-16)
3(osh s


Since Posh ps is the contribution to zero shear viscosity due to the

presence of polymer molecules, equation (2-16) is equivalent to Trouton's

rule.

Experiments performed at extremely low values of stretch rate have

shown that viscoelastic materials approach Newtonian behavior in extension

when the deformation rates are small enough (Spearot and Metzner (1972),

Everage and Ballman (1975), Cogswell (1972)). At larger stretch rates,










the Trouton ratio for many polymeric materials such as melts and solutions

is several orders of magnitude larger than that predicted by equation

(2-14).


2.4 Literature Review: Non-Newtonian Fluids

As first discussed, extensional flows of polymeric materials can

be conveniently divided into two categories--those which are materially

steady and those which are materially unsteady--and the extensional

viscosity measured in the two flows would not in general be expected

to be equal.


2.4.1 Materially steady extensional flows

Materially steady flows are characterized by a constant stretch

history. Mathematically, a materially steady extensional flow can be

defined (Coleman and Noll (1962)) as a flow where there is a fixed

cartesian coordinate system in which the following equations are

obeyed:


v = Fx (2-17)



v are the components of velocity of a material point at x'

and ri are constants subject to the condition


3
ri = 0 (2-18)
i=l


There are several other definitions of materially steady extensional

flows. A brief summary of these definitions can be found in Kanel

(1972).










The steady extensional flows of polymeric materials has been

experimentally investigated and theoretical analyses have also been

made in order to compare theory with experiment. The typical experiment

is that due to Ballman (1965) in which the material is clamped at one

end and is extended in some programmed manner, usually to yield a

constant stretch rate. This is schematically described in Figure 2-2.

Most of the earlier extensional flow experiments at constant stretch

rates were performed at low stretch rates, and in these experiments

a constant value of extensional viscosity, independent of stretch rate

was obtained. Using molten polystyrene, Ballman found extensional

viscosity to vary only slightly from 9.975x103 poise at a stretch

rate of 0.001 sec-1 to 9.5x103 poise at a stretch rate of 0.02 sec-1

This value was nearly 160 times greater than the shear viscosity, and

was found to be independent of the stretch rate.

The results of these experiments are consistent with the predictions

of constitutive theories for low values of stretch rate. However, most

of these theories, such as the convected Maxwell model and other rate

models of this type, and also integral models such as the network

theory of Lodge (1964), predict the growth of stress to very large

values when the stretch rate-exceeds a critical value. Experiments

performed over a wide range of stretch rates have shown these predic-

tions to be true. Specific examples of such observations are those

of Meissner (1972), Spearot and Metzner (1972), Stevenson (1972), and

Everage and Ballman (1975). Table 2.1 summarizes the results of some

of the extensional viscosity measurements that have been discussed in

this section.







28

>-

0 0 -- r> -- o. C o .
:3 3 C00 0 L
.- 3 3 >> C *,-- 0
-0 0- 4- 0 0 O u C O
> II '- > -,-




00S- 0
X 300 4 -PU4-
C 4-' C 4-' 4-'0 000 3C
S C O> 0 0 0 CC o o C -i.C
C 0.S --. C 00 0 0 u< o **-
00 3- 0 S.- >- V S.- C
4 a) 4 0- r 04a) 0 = 4J a)3>4
C 3 ~C Cn c 4- 0 '- -a r C r-
S 0 a>1 /u >a 0 >o0 0 3: O4 -
C z > 4 Z 4 -- a.
0 CVO r-- C4l CO 4 0 U') 5- ") -
4-' c 00 4-' 00 0U O u 4-l fO M U
0 0 a0 C 0 CO CO 0
5- 4 4- OJ 4-' O O O CO C 0 0
o o Uoo 4- S- P L n U 5- 4-S- i- 0 3 3 > n >
E C c CE C 3 0 4-' 0.--- s- 0










in C0 CM 2 -

S0 0 ,- 0 0







ax 5) LO l, lU-) "0
mo o o C c ] on CM .-
4 0 C) C) C) I I I C C I
S 0 0 0 0


5- 4) "
4- 0 0 0 4
U'
0C
C 0

0 L in io
C O O O
C O 0 0 O 0- 0- -
C -- -- -- JX X X0 X< I
O O I X )X LO O LO *- I
LU *r- -C I LO o .



OCU CC C
- 4-2 -
LU

o C C C c cu Co c-
0 0 0 0 0' 0 f0 0u


5-. U U) U 0 4' C a) a U)
> o C O C C C C C 1-- C .O
E E 00 0 0 04-' 0-p O4-' 0 =>r-


C



< a ) .a

0 >
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0 *r i- >'. <0 r- i- 0. a
L S '> U>, >N L Q
o U > CC CC C > 0
O 4-l O L 4- 0- 4- 44
L- 0 -C U' DO 0 D O 0



0 0 >- >N >t >N >- >4 >4 >i
4-' U' 4-' r- 3,-- l- L 3 r-- r-





o .- 0 0 0 O >0 0

Sin -- 0- -IE O- */ -J 0 O O -
+-p




.- 0) C) C'C) 0C) i-







-< I^ Cfl -I 5-) in3; LUCQ Din































































4-


0
--
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SI L

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*r- > -
E0 E
'Ul r>,-








0
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0 0
L 'U C0




4-' a or
1- d


C0
CO






E
C C
)U C 0



L ,- '- C m

L u wn 4-'
u>, CO

r4 c O

0. *r- S -"--


"u


'U
5-



U
'U



5:.J



'U
r)



O_4J
S--





fLC
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a0


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..
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~CD
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rn
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*1










ro


u

Cn 0
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L C
1 5 -
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o o
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C C

3 C



o o
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0 0
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f0
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Wao
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(0

L 00
m Cc




>N N N
CD 4.1 4-1
* I SI
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0 -


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r0



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OrC





'0



'-5-
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m c









A common feature of all these experiments is the low deforma-

tion rates that can be attained and their limitation to high viscosity

systems. For low viscosity systems, the techniques that are available

for measuring extensional viscosity are different and they come under

the category of materially unsteady flows.

2.4.2 Materially unsteady flows

In materially unsteady flows, the history of deformation is

not constant (Section 2.2 ) and therefore the stress and other

dependent variables depend on the instant of observation. Materially

unsteady flows commonly occur in accelerative flows--flows that are

steady from an Eulerian sense but not steady from a Lagrangian sense.

As pointed out by Kanel (1972), there are several frame in-

different criteria that can be used to characterize the nature of

a flow--whether it is predominantly shearing or extensional--when

it is materially steady. For materially unsteady flows, however,

it is difficult to find a frame indifferent quantity that can serve

as a measure of the character of the flow. In view of the fact that

extensional flows are characterized by a rate of strain tensor that

is almost diagonal, and by negligible vorticity, it is convenient to

use the magnitude of the ratio of the second invariants of the vorticity

tensor W and the strain rate tensor D to determine the nature of an

unsteady flow.


IIW
E rr= (2-19)
IID









For a shearing flow defined by

extensional flow defined by vi

spherical sink flow kinematics

(Section 2.5), and is described



2Q'/r3

0= 0

0



0

W= 0

0


equation (2-2), 5 = 1, and for the
2 3
(r, Tx2 For
= - T-), { = 0. For

which is of interest to this work

by equation (2-11),


0

- Q1/r3

0


0

0

- Q'/r3


(2-12)







(2-20)


(2-21)


The flow is therefore purely extensional, by the above criterion.

Detailed investigations of two materially unsteady flows--FANO

FLOW AND CONVERGING FLOW--have appeared in the literature.

FANO FLOW: The Fano flow (also known as the tubeless siphon or self-

siphon) experiments utilize the ability of some viscoelastic fluids

to maintain a siphon even when the liquid surface is below the siphoning

tube. This is pictorially described in Figure 2-3. In a flow such as

this, the material is subjected to extensional modes of deformation

(Astarita (1970)). From photographs of the liquid column (to measure

stretch rates in the liquid) and measurement of the force required to

support the siphoning tube (to determine the tensile stress in the

















































Figure 2-3. Fano Flow









liquid at the tube entrance), one can calculate an extensional viscosity.

Pickup (1970), using aqueous solutions of Separan AP 30, a par-

tially hydrolyzed polyacrylamide made by Dow, measured Trouton ratios

as large as 160. Similar results were reported by Astarita (1970)

who also used aqueous solutions of the same polymer.

In a very detailed investigation of Fano flow, Kanel (1972) found

that the kinematics of the flow are described by the equation



m cx + c2 (2-22)


D is the diameter of the liquid column at any height x from the free

surface of the liquid (Figure 2-4). c1, c2 and m are constants that

depend on the liquid. For a flow with kinematics described by equation

(2-22), it can be easily shown that the rate of change of stretch rate

experienced by the fluid depends only on the instantaneous value of

r and the value of m.



D= vI dv = [(2-m)/2]2 p2 (2-23)
dx

Thus two fluids having the same value of m have identical stretch

histories when the stretch rates overlap. Kanel's observations on

Fano flow are important in that the kinematics of the flow as described

by equation (2-22) have been found to describe the thread profile in

many fiber spinning operations (Kanel (1972), Baid (1973)). Equation

(2-23) implies that the deformation histories of fluids in km flow





















































Figure 2-4. Coordinate System in Fano Flow










(i.e., flow described by equation 2-22) are identical when they have

the same values of m and r, regardless of the value of cl and c2.

When m, the history parameter, is equal to 2, the stretch rate does

not change with time in a Lagrangian sense, and the flow is materially

steady.

CONVERGING FLOW: The term converging flow is commonly used to describe

the flow of a fluid through a sudden contraction. The dramatic dif-

ference between the converging flow of Newtonian and viscoelastic

fluids was noticed as early as 1956 by Tordella. Clegg (1957) and

Bagley and Birks (1960) have described this difference in detail.

In the case of a Newtonian fluid, the flow entering the contraction

does so from all locations upstream of the contraction, and the

secondary eddy flow at the 90' corner of the contraction occupies a

minor portion of the flow. In the case of some viscoelastic fluids,

however, the flow through the contraction is restricted to a narrow

conical region upstream, and the liquid surrounding this region slowly

recirculates, resembling a large vortex (Figure 2-5). This phenomenon

is commonly referred to as Wine Glass Stem Shaped Flow (or simply WGS

flow) and has been observed with some polymer melts. It is to be noted

that not all polymer melts show this behavior. Thus, low density

polyethylene has been known to exhibit WGS flow whereas high density

polyethylene has not. Though it is believed that WGS flow occurs due

to the elasticity of the polymer under consideration, the precise

reason for its occurrence is not well understood.

In recent years, WGS flow has also been observed with relatively

dilute polymer solutions (Giesekus (1968), Uebler (1966), Pickup (1970)).



















































Figure 2-5. Wine Glass Stem Flow









This observation has now been extended to extremely dilute polymer

solutions of concentrations in the ppm range (Oliver and Bragg

(1973a,1973b), Balakrishnan and Gordon (1975a).

Uebler found that WGS flow can be modeled as a spherical

sink flow towards a point sink on the axis of the contraction,

with velocity field


vr = Q'/r2 (2-11)


v = v = 0


vr is the velocity measured at any distance r from the sink, and Q',

the sink strength, is proportional to the flow rate Q. For a flow

such as this the rate of strain tensor is diagonal (equation (2-12)),

and the.material is therefore subjected to extensional deformation.

Thus nex, the extensional viscosity, can be measured for polymer melts

and solutions that exhibit WGS flow if the stretch rate and the normal

stress difference arising from such deformations can be measured. For

the case of extremely dilute polymer solutions, this is in fact the

most widely used method to determine nex (Metzner and Metzner (1969),

Oliver and Bragg (1973a), Balakrishnan and Gordon (1975a)).

In a very detailed analysis of WGS flow, Murch (1970), using

0.5% solutions of Separan AP 30, determined the extensional normal

stress difference by using measurements of Tee in conjunction with

the r component equation of motion in spherical coordinates.











































CL
a

w" a.,


3 Io 30

r, sec"1


Figure 2-6. Predictions of the Maxwell Model












NE(r) = pv2(r) + -( dr) d (2-24)
r

From measurements of T as a function of r, Murch was able to calculate

the integral in equation (2-24), and thereby obtain NE. The extensional

viscosities obtained from this procedure were constant, indicating

Newtonian behavior, and were roughly three orders of magnitude greater

than the shear viscosity! Comparison of these values with those predicted

by the White-Metzner model for sink flow kinematics showed vast disagree-

ment (Figure 2-6). It is premature to attribute this to any inadequacy

on the part of the model because of experimental difficulties associated

with Murch's techniques as well as invalid assumptions made in calculating

stretch rates. A detailed discussion of sources of error in Murch's work

is presented in Section 5.3.

Metzner and Metzner used the jet thrust technique to measure

normal stress differences in WGS flow. In this method, the thrust

exerted by a jet of dilute polymer solution issuing from a reservoir

through a sharp-edged orifice was measured and was related to the

tensile stress NE in the liquid by the equation



NE = pv2 T/A (2-25)


where vo is the orifice velocity and T/Ao is the thrust per unit area

of the orifice. Metzner and Metzner used a Separan AP 30 solution that

was identical to that used by Murch and they obtained very large values

of extensional viscosity. Metzner and Metzner made the same assumptions


I










in calculating stretch rates as Murch--that the stretch rate at the

orifice was a linear function of orifice velocity.



r, = kvo (2-26)



The results of the present work show that equation (2-26) is

not generally true (Section 5.1). Furthermore, the thrust technique

used by Metzner and Metzner actually yields die swell and not tensile

stress at the orifice (Section 5.3).

Despite all the uncertainties in these methods, the results

obtained by Metzner and Metzner and by Murch clearly demonstrate the

enormous resistance exhibited by dilute polymer solutions to extensional

motions. For a 100 wppm solution of Separan AP 30 the former group

of authors estimated a conservative value of nex that was 500 times

larger than the shear viscosity (1.44 cp).


2.5 Kinematics of WGS Flow

Extensional viscosities were measured in the present work in

WGS flow. A detailed description of the kinematics of WGS flow is

presented in this section.

A WGS flow can be conveniently divided into four sections

(Figure 2-7).

(1) The section upstream of the actual WGS flow. In this

region, velocities are extremely small as compared

to the region close to the contraction. Pickup (1970)

observed simple laminar shearing flow in this region.


















































Figure 2-7. Division of WGS Flow into
Different Regions.










(2) As one travels downstream with the flow in region (1),

one observes a transition from shearing flow to WGS

flow. The liquid begins to accelerate in the direction

of flow, and converges towards the centerline. The

distance from the contraction where this transition

occurs would be the distance to which the large toroidal

vortices surrounding the central flow penetrate into

the experimental apparatus.

(3) Following region (2), the flow is more or less well

defined and resembles flow through a "cone" with a

liquid wall. As described by Pickup (1970), and

Uebler (1966), the velocity vector at any point in

this region is directed towards the contraction, and

the fluid accelerates rapidly as it approaches the

apex of the cone. There is little or no shear in the

major portion of this region. At the interface between

region (3) and the liquid surrounding it (region (4)),

there is intense shearing. The deformation levels in

region (3) are extremely large compared to the rest of

the liquid.

(4) Region (4) surrounds region (3). Here the liquid simply

recirculates slowly, and the deformation levels in this

region are extremely small compared to those in region

(3) (Uebler (1966)).










Several velocity profiles have been proposed to approximate

the kinematics of WGS flow. Notable among these are those of

Uebler (1966), Pickup (1970), and Kanel (1972).

Uebler conducted experiments with concentrated solutions of

Separan AP 30 in water. Using air bubbles in the viscous solution

as tracer particles, he studied the kinematics of WGS flow. The

results obtained by Uebler suggested that the flow in the region

upstream of the contraction was radially directed toward the origin

of a spherical coordinate system located at the apex of the liquid

cone. This yielded the velocity profile described by the equations

(2-11):


vr = Q'/r2 (2-11a)

vo = v0 = 0 (2-11b)


Q' = Q'(6) (2-11c)


where 8 is the half angle of the liquid cone. Uebler also found that

Q'(e) was approximately constant and equal to r2 where
2nr (1-cos 8)
Q is the volumetric flow rate, r is the position at which the velocity

is measured. Thus the velocity profile becomes


vr = Q (2-27a)
2Tr2 (1-cos 8)


v = v0 = 0 (2-27b)









The above equations are good for about 70% of the flow that enters

the contraction (Uebler (1966)).

Uebler's measurements were carried out at values of generalized

Reynolds number ranging from moderate to high, and the radial flow

kinematics described by equations (2-27) are therefore a good approxima-

tion to the actual kinematics of the flow under these conditions.

The kinematic measurements of Pickup (1970) yielded results

very similar to those of Uebler. Pickup found the velocity in the

central region to be approximated by the following equation:



vx = v eBx (2-28)


where vx is the velocity at any point x, vo is the velocity in the

region upstream of the WGS flow and x is the distance measured from

the origin of a cartesian coordinate system located at the point

where the flow enters WGS flow (Figure 2-8).

Yet another velocity field that has been used to describe the

kinematics of WGS flow is that due to Kanel (1972). In many fiber

spinning operations and Fano flows, the diameter of the column of

the liquid is given by equation (2-22). The shape of the central

region in WGS flow resembles the shape of liquid column in Fano flow.

However, attempts made in this work to fit an equation of the form

of (2-22) showed that (1) an equation with single values for cl, c2

and m does not describe the entire flow and (2) that the flow is best

described by using equation (2-22) for a minor part of the flow, at

the beginning of WGS flow (i.e., to describe regions (2) and part of















































Figure 2-8. Coordinates Used in
Pickup's Kinematics










(3)), and sink flow kinematics to describe the major part of the flow

in region (3).

In this work, the sink flow kinematics of Uebler have been used

to arrive at relationships for stretch rates and normal stress dif-

ferences at the orifice. This is only an approximation to the actual

flow because equations (2-27) are not valid in region (2) (Figure 2-7),

where the flow field resembles the mouth of a trumpet. However, they

are applicable for the most part of region (3) where the fluid rapidly

accelerates and deformation levels are their largest. The extensional

behavior of a material in WGS flow would be expected to depend largely

on the kinematics of this region.


2.6 Analysis of Stress Field in WGS Flow

In order to analyze the stress field in a flow described by

equation (2-27), one can adopt the following techniques:

(i) Point measurement of stresses by the use of probes.

This technique is now known to be inaccurate in the case

of viscoelastic fluids. As pointed out by Astarita and

Metzner (1967), the fluid approaching a probe introduced

to measure the stress at a point in the flow experiences

large changes in the deformation rate with position from

an Eulerian sense and with time from a Lagrangian sense.

Such deformational processes are characterized by large

values of the Deborah Number



NDe = _'-/ II (2-29)










where A = relaxation time of the fluid and IID = second invariant

of the rate of strain tensor. Large values of NDe imply a solidlike

response of the liquid irnediately adjacent to the surface of the

probe (Astarita and Metzner (1967)), leading to inaccurate and sluggish

response of the probe. Also, the use of a probe with a hole in it

fails in the case of viscoelastic fluids due to hole error (Kaye et al.

(1968)).

Murch used solid probes and probes with holes in them to measure

T in the central region (region 3) of WGS flow, and from T" vs. r

profiles was able to calculate NE(r). It is obvious from the factors

stated above that Murch's results could be considerably in error.

(ii) Stress birefringence technique: Adams et al. (1965)

have used the birefringent properties of viscoelastic fluids

to analyze stresses in a converging flow. By using stress-

optical relations derived for slow flows, they obtained

estimates of shear stresses and normal stress differences

in converging and diverging flows. The weakness in this

approach is the extension of equations derived for slow

flows to accelerative flows such as those encountered in

converging flow, with no justification for this assumption.

(iii) Momentum balances. In the absence of techniques to

explicitly measure stresses at different points in WGS flow,

one can resort to the use of natural laws as continuity and

momentum balance equations to yield expressions relating

pressure drop to flow rate, both of which are easily

measurable quantities.









Bagley in 1957 noticed that the converging flow of certain

polymer melts gave rise to large entrance pressure losses, in excess

of those predicted for a purely viscous material. This excess pressure

drop is commonly attributed to the elasticity of the melt and is deter-

mined from the Bagley plot. In the typical experiment that is performed

in the capillary viscometry of polymer melts, the melt is placed in a

reservoir and forced through an attached capillary tube. The pressure

drop in the reservoir is commonly neglected in comparison with that

in the capillary. Thus the pressure drop across the length of the

capillary is simply the pressure in the reservoir minus the loss in

pressure at the entrance of the capillary due to rearrangement of the

velocity profile and viscous and elastic effects. This loss in pressure

at the entrance to the capillary AP is determined by measuring the

pressure in the reservoir P at a fixed value of the apparent shear

rate D- for different values of L/D, the length to diameter ratio of

the capillary. AP is then obtained by plotting P vs. L/D and extra-

polating the plot to zero L/D (Figure 2-9). Attempts have been made

to separate the entrance pressure drop into viscous and elastic parts

(Bagley (1961)), and it is generally believed that the entrance pressure

drop is closely related to the elasticity of the material. In particular,

large entrance pressure drops are believed to be due to extensional

viscosity effects (Metzner et al. (1969), Balakrishnan and Gordon (1975a)).

The converging flows of polymer melts and concentrated solutions

have received a fair amount of attention due to their common occurrence

in polymer processing applications. In recent years, certain anomalies

have been reported in association with these flow fields for dilute







49




























p




A P




Le

LID -t


Figure 2-9. The Bagley Plot.










polymer solutions. Pruitt and Crawford (1965) observed in their

experiments on turbulent drag reduction that dilute polymer solutions

exhibit pressure drops across contractions in flow much greater than

water alone. Giles (1969) found orifice discharge coefficients to

be much lower for a 300 wppm solution of Polyox WSR 301 than for water,

indicating an increase in pressure drop across the orifice. In a

novel experiment in which a liquid was made to flow out of an orifice

located on the wall of a tube (this was done in order to reproduce a

feature that is characteristic of turbulence--motion normal to the

wall), Morgan (1971) found the addition of small amounts of polymer

to water to reduce flow rates through the orifice. This again in-

dicates larger pressure drops across the orifice for the polymer

solution than for water alone. Bilgen (1973), using an orifice of

diameter 0.1 mm, found significant increase in pressure drop across

the orifice due to the addition of as little as 10 wppm of Polyox

FRA (a higher molecular weight homolog of WSR 301). This pressure

loss was found to increase with increasing molecular weight of the

polymer. Similar results have been reported by several other

investigators (Sylvester and Rosen (1970), Oliver et al. (1970),

Balakrishnan and Gordon (1975a)). Since the viscosities of the

solutions used in most of these studies were nearly equal to the

viscosity of water, the increased pressure loss cannot be attributed

to viscous effects, and is presumably due to elastic effects at the

entrance region to the orifice. It will be shown presently from a

momentum balance that in the case of WGS flow the entrance pressure

drop is explicitly related to the primary normal stress difference









at the orifice. The control surfaces used in making the momentum

balance are shown in Figure 2-9.

The apparatus is a large reservoir fitted with a sharp-edged

orifice. Section 1 is taken far upstream of the orifice so that it

is located in region 1 described in Section 2.5. Section 0 is located

in the plane of contraction. Section 2 is taken at the free surface

of the jet emerging from the orifice. In writing momentum balances,

the following assumptions are utilized (these are mostly identical

to the observations made in describing WGS flow):

(1) At section 1, the velocities and stresses arising from

deformations are very small. This would be a fair

approximation for a large reservoir fitted with a small

orifice.

(2) The velocities and deformation levels in the recirculating

region (region 4 of Figure 2-7) are also very small (Murch

(1970)). Therefore, the stresses in this region would
not be expected to vary much with position.

(3) Viscous drag in the reservoir is negligible. In view of

the magnitude of velocities near the walls, this is a

valid assumption.

Let the area of the reservoir perpendicular to the direction of flow

be A1, and those of the orifice and the free jet Ao and A2 respectively.

The subscripts o, 1 and 2 identify locations at which various quantities

are measured. A momentum balance between sections 1 and 0 yields

Ro Ro
I 2Trpv: rdr pvA1 = 2T rdr PR(Al- Ao) + plA1 (2-30)
0 0










52










El












sJ wI

(-J





CMj
air












:3















0~














0
C)L

Ci


U








:3


0,




LL












:3
0r
_______ U










T11 is the total stress in the 1 direction at the orifice (this would

coincide with the z direction of a cylindrical coordinate system with

its origin located at the orifice (Figure 2-10)). pi and pR are the

isotropic pressures at section 1 and in the recirculating region

respectively. For small values of 0, the half angle of convergence,

the velocity is fairly uniform across the diameter of the orifice

and vz is nearly equal to vr (in the spherical coordinate system with

origin located at the sink). Also v1, the velocity in the reservoir

upstream of the WGS flow, is very small compared to velocities near

the orifice. Equation (2-30) then becomes


PVoA = TAo PR(A- Ao) + 1A1 (2-31)


Resolving Tz into isotropic and deviatoric components,


Tz p + zz (2-32)



Now Trr (in cylindrical coordinates) is nearly equal to Tee (in spherical

coordinates) which, following Metzner et al. (1969),can be assumed equal

to PR' the isotropic pressure in the recirculating region.


Trr = T R + rr2-33)


Eliminating the unknown isotropic pressure term from equations (2-32)

and (2-33), we obtain the following relationship:

A A
o 2 o
NE 1 -pv0 PR p1 (2-34)









where NE = (zz- Trr). In the case where inertial terms are negligible
as would occur with polymer melts, equation (2-34) predicts an adverse

pressure gradient. Metzner et al. have suggested that the upstream

flow near the walls of the reservoir might be due to this pressure

gradient.

A momentum balance between sections 0 and 2 yields

2Ro Ro
pv2A2 27 pv rdr 2 J Tzz rdr (2-35)
0 0

Once again for small values of 0, the half angle of convergence,

equation (2-35) becomes



pv2A2 pv2Ao = T A (2-36)


In the absence of vena contract or die swell effects, the diameters

of the jet at section 2 would be the same as the orifice diameter, and

equation (2-36) becomes


T = 0 (2-37)


Once again, using equations (2-32) and (2-33), the above equation can

be rewritten as



NE (zz- rr)o PR (2-38)




A situation such as this would be expected for dilute polymer
solutions with small values of e.










Thus the normal stress difference at the orifice is simply equal to

the pressure in the recirculating region, which is actually the

entrance pressure drop as would be measured by a transducer located

on the wall of the reservoir in the recirculating region. Designating

this as AP ,



NE = APe (2-39)
o


For the case where the half angle of convergence is not small, and

the velocity profile is therefore not uniform across the diameter

of the orifice, the integrals in equation (2-35) have to be explicitly

evaluated. This can be done as follows:

Let vz be the velocity in the direction of flow at any point

across the diameter of the orifice. Referring to Figure 2-11 it is

clear that vz is related to vr (in spherical coordinates) by the

relationship


v = vr cos 4 (2-40)



where P is the angle from the orifice axis at which vz is measured.

It can be easily shown that the momentum influx into the

orifice is given by

R
o 9 v2 2 R2
2Tp rv2 dr = 2Tp 0 (1 + cos e) o cos6 tan sec2 4 d
0 0 4 cos4 tan2
0 0 (2-41)










Carrying out the integration gives the following result

o 2
p rv2 Adr v o (1 + cos e)2 (1 + cos2 6)
j cos2 e
0


Following a similar procedure for

hand side of equation (2-35), and

contract occurrences, we obtain

2
2 P 2o2
PVoo 2 0 (1 + cos2 e)(1
8 cos2


integrating the term on the right

neglecting die swell or vena




2 ATrr cos 6
+ cos 0) = 2A (1 + cos 0)

(2-43)


Resolving Trr into isotropic and deviatoric components


Trr (- + Trr)


and eliminating the isotropic pressure term

(2-38) we obtain


using equations (2-33) and


p2 (os + cos )2( + os2 8)
NE = P + (+ cos e) 1 -
E e 2 cos 8 cos2
o 8 cos a
(2-45)
Equations (2-45) and (2-39) give a simple means of measuring

NE from measurements of entrance pressure drops and angles of con-

vergence.

Pickup (1970) has performed a similar analysis to determine the

normal stress difference at the orifice in the two-dimensional converging

flow of polymer solutions. The final result obtained by Pickup was


(2-42)


(2-44)







57



NE = APe psh (2-46)
0


where Ap sh is pressure drop due to shear in flow through the slit die.













CHAPTER III
EXPERIMENTAL

3.1 Test Solutions

The polymers used in this study are listed in Table 3.1.

Separans AP 273 and AP 30 are very high molecular weight synthetic

water soluble polymers made from the polymerization of acrylamide



E CH2 -- CH -- 3n

C=O
C = 0


NH2


For both AP 273 and AP 30, a small portion (about 20%) of the amide

units are hydrolyzed to COOH (Burkholder (1971)). The Separans may

be classified as anionic in neutral and alkaline solutions. Under

acidic conditions, the ionization of the carboxyl groups is suppressed

and the molecules assume a non-ionic character (Ref. Technical Service

Report (1969)). Due to the anionic character of these two polymers,

their solution viscosity is extremely sensitive to changes in ionic

environment. The ionic charge of the carboxylate group disappears

at pH = 3 and gradually increases to a maximum at pH = 8. Viscosity

is likewise maximum at pH = 8 and a minimum at pH = 3. Of the two

Separans the AP 273 grade is of a higher molecular weight grade than

AP 30 but both polymers are believed to have similarly shaped molecular

weight distributions (Penzenstadler (1975)).










59















-Co

CD ko k
C) -- D CD
x x - (


0 L


















w D X -a 0


cici (O~i Cb
4- -0
S-J














44-
(0 ms M C C
Z, Qj a-














410


00 i<

C
LI (J- '













a) ~ ~ U) cia) w
oa ao m 0









aci
E-) n
0-










(0 (0 0 >,(0 >
0- 0 ClS Cl c-
0 C j co LO U-) c
























-- ci c a Ci 0
CL) 4) CI 0- LO 0
,;




C3 a
a) c ai
Ln L) a x









Percol 155 and Polyhall 295 are also non-ionic polyacrylamides

that dissolve readily in water, and are very similar to the Separans

in their properties.

Versicol S25 is a very high molecular weight partially neutralized

poly(acrylic acid) (PAA). It dissolves readily in water yielding a white

translucent solution of pH 5-6. Due to the presence of carboxyl groups

along its chain, Versicol shows dramatic changes in its solution prop-

erties with change in pH, much the same as the polyacrylamides do. The

viscosity of Versicol solutions is fairly stable in the range of pH 5.5-

12.5 (Ref. Versicol Catalog). Addition of electrolytes to Versicol

solutions suppresses the viscosity of the solutions.

The only non-ionic polymer used in this work was Polyox WSR 301

(PEO), a poly(ethylene oxide) polymer. Due to the ether groups present

along its chain, the polymer has the potential of forming hydrogen bonds

with water. Its viscosity is very stable over a wide range of pH (Gordon

et al. (1973)). PEO is extremely susceptible to shear degradation, much

more so than the other polymers used in this study.

The range of polymer concentrations used was 10 to 200 wppm, and

all solutions were made in deionized water. Conformational studies were

carried out with Separan AP 273 and the pH of these solutions was varied

by using HC1 and dilute NaOH solutions.


3.2 Polymer Solution Preparation

All the polymers used in this study have a tendency to form gel-like

agglomerates when they come in contact with water. To avoid the formation

of these agglomerates and the consequent increase in dissolution time,

the following procedure was adopted in making polymer solutions:










The required amount of polymer was weighed and placed in a

dry 4 1/2 gallon Nalgene jar. To this, about 50 ml of isopropanol

was added. The isopropanol served as a non-solvent medium in which

the polymer formed a dispersion, and in the case of Polyox FRA, it

also acted to prevent chemical degradation. Demineralized water

was then added to the dispersion in the jar and the resulting

solution allowed to sit for about 8 hours before it was used. In

order to expedite the dissolution process, the solution was mixed

periodically by hand with a glass rod.

The procedure described above yielded polymer solutions that

were clear and free of undissolved polymer or agglomerates and was

ideally suited for dissolving small amounts of polymer (less than

10 gms.). For the case where larger amounts of polymer were required,

the dissolution process was slightly different. Once again the

required amount of polymer was weighed and was placed in a beaker

and about 50 ml of isopropanol added to it. The Nalgene jar was

filled with some water and a vortex created in it by the use of a

low speed mixer. The polymer-isopropanol mixture was then poured

into the vortex, resulting in immediate dispersion of the polymer

in water, and the mixer was turned off. Periodical mixing by hand

(with a glass rod) and keeping the solution for about 8 hours yielded

a clear solution that was ready for use. In all the cases the con-

centration of the solution made in the Nalgene jar was about 1000 wppm.

This solution was then diluted in the storage tank of the converging

flow apparatus (Section 3.3), or the DR apparatus (Section 3.6) to

yield solutions of the desired concentration level.

The use of a mechanical mixer was kept to a minimum in order
to avoid shear degradation of the polymer solutions.










3.3 Converging Flow Apparatus

The apparatus used to study the converging flow of dilute

polymer solutions is shown in Figure 3-1. It consisted of four

major components--the storage tank, the pressure tank, the flow

(visualizing) tank, and the dye reservoir. The dimensions of the

first three are given in Figure 3-1. The storage and flow tanks

were made of Plexiglas while the pressure tank was made of stainless

steel. The flow tank had the capability of having orifices of dif-

ferent sizes fitted to it. It also had two baffles in it to prevent

channeling of the test liquid (i.e., to ensure uniform distribution

of the liquid before it entered the converging flow field). Dye from

a reservoir could be injected into the flow tank by means of small

holes drilled into a 3 mm stainless steel tube that spanned the width

of the tank. The dye injection tube could be positioned at different

distances from the orifice. Details of the dimensions and arrangement

of the various fittings on the flow tank are given in Figure 3-2.

During a run, the flow tank was set up in such a way that the jet

emerging from the orifice was in a horizontal plane (in other words,

the flow tank was laid on its back with its face up, facing the

camera that was used to take photographs of the flow pattern in

the tank). The positioning of the flow tank and camera during a

run is shown in Figure 3-3.


3.3.1 Details of orifices used

All the orifices used in this study were made of Plexiglas.

Their dimensions are given in Table 3.2, and the details of their

construction in Figure 3-4. There were three major components



























O3 I- 0
F- scS-

s ot
tot) E


V) [I. LL- !:.


1- C- en Zz-


---- I99
ha99




SWs-

























Dye Injection
Grid


To Manometer


Distributor
Plate


Orifice Plate


Figure 3-2. Details of Flow Tank























































CL






L




c







F-







L
E






r

I-

0
0
F-




o Io


m


cn
c:
0,
.13

iL






















TABLE 3-2


Dimensions of Orifices


Orifice diameter
cms


0.040
0.051
0.109
0.132
0.178
0.244


Orifice Length
cms


0.150
0.150
0.347
0.284
0.347
0.400


L/D ratio


3.75
2.94
3.18
2.15
1.96
1.64









67













a



U,
S- 10



00
a





4-' -
m 0











'7
0 a

l -U







U


, i ^





C^-




U-
i ^ \
-^ L'\ '--










constituting the orifice plates--(i) the base plate made of 1.27 cm

thick Plexiglas, (ii) the orifice bearing plate which was a 1.27 cm

thick, 2.54 cm dia Plexiglas plate glued to the base plate, and (iii)

the orifice disk that was a 0.16 cm thick, 2.54 cm dia Plexiglas plate

that was mounted on the orifice bearing plate. The orifice hole was

drilled into the orifice disk. This hole then flared into a 0.62 cm

hole that went through part of the disk and through the orifice bearing

and base plates. All drilling processes were carried out extremely

slowly in order to avoid distortion of the hole surfaces due to heat

generation.


3.4 Photographic Equipment

As mentioned earlier in Chapter II, the angle defined by the

streamlines converging into the orifice is required for the calculation

of stretch rates at the orifice. These angles were measured by taking

photographs of the streamlines and projecting the developed film on

tracing paper through an enlarger. The camera used was a Bell and Howell

FD 35 single lens reflex camera equipped with a 50 mm lens and close-up

lenses. The film used was Kodak Tri-X rated at 400 ASA. The camera.

was mounted on a tripod and positioned over the flow tank of the

converging flow apparatus such that the field of view covered all of

the flow from the dye injection grid to the orifice.


3.5 Making of a Run

The concentrated polymer solution made in the Nalgene jar was

first poured into the storage tank of the converging flow apparatus

and water added to it to bring the polymer concentration to the










desired level. Following slow mixing of the solution with a paddle

(to ensure uniformity of the solution), the pressure tank and the

flow tanks were filled with the polymer solution. Some polymer

solution was also taken in the dye reservoir and a few drops of dye

(food color) added to it. The pressure tank was then pressurized

with compressed air in order to force the polymer solution through

the orifice. Dye was injected into the flow through the dye injection

grid and photographs of the streamlines were taken. Flow rates were

measured by collecting the liquid issuing out of the orifice and

weighing it. Entrance pressure drops were measured by a mercury

manometer connected to the flow tank (Figure 3-2). Several photo-

graphs were taken at each flow setting to make sure that the flow

was steady, and fully developed.


3.6 Thrust Measurements

A jet thrust apparatus similar to that used by Oliver (1966)

was built to measure normal stress differences at the orifice. A

photograph of the apparatus is given in Figure 3-5. The jet issuing

from the orifice was made to strike a small Plexiglas plate attached

to one end of a brass rod that was vertically suspended by bearings.

The other end of the brass rod was kept in contact with a Statham

force transducer (Model UC3 Gold Cell) that was connected to a

recorder through a Sanborn (Model 311A) transducer-indicator-

amplifier. The force exerted by the jet on the Plexiglas plate

was transmitted to the transducer by the brass rod, and by cali-

brating the system by suspending weights from the Plexiglas plate

(Figure 3-6), these forces could be measured.









































Figure 3-5. The Thrust Apparatus


777
~J~C

~___













3.7 Drag Reduction Measurements

Drag reduction measurements were made in two kinds of apparatus--

a flow loop and a gravity flow setup. The flow loop (Figure 3-7) con-

sisted of (i) a Moyno pump (Model 2L6) with a Carter variable speed

drive, and (ii) four stainless steel tubes of inside diameters 0.457 cm,

1.092 cm, 1.656 cm, and 2.146 cm. Details of entrance length, exit

length and location of pressure drop taps are given in Figure 3-7.

Flow rates were measured by collecting the liquid and weighing it and

pressure drops by using manometers containing mercury, Meriam Red Fluid

(sp. gr. 2.95) and Meriam Blue Fluid (sp. gr. 1.75).

The gravity flow apparatus was identical to that described in

the author's master's thesis. It consisted of a 100 liter 44 cm dia

68.5 cm tall Plexiglas tank to the bottom of which a 0.45 cm i.d.,

185 cm long stainless steel tube was attached. A schematic representa-

tion of the apparatus can be found in Figure 3-7. The tank was filled

with water and the water allowed to drain. The times for the water

level to drop from 922 to 872 and 91k to 862 were measured and

averaged. Similar measurements were then made for the polymer

solutions and percent drag reduction was defined as


% DR = 100(tw tp)/tw (3-1)


where tw = average efflux time for water, and

t = average efflux time for polymer solution.

By measuring t at different polymer concentrations DR was determined

as a function of concentration. Successive lowering of concentration



































F- H-



4-' 4-' 4-'
(A L/1 (/A

U, U) Uf)
cccr





4-' 4) 41-






E E=
flml



LO 0)O





(Ni3
C:) C\
< C L.












was achieved by diluting the polymer solution in situ.


3.8 Vortex Inhibition (VI)

Vortex Inhibition, a viscoelastic phenomenon believed to be

related to DR was discovered during the course of some experiments

on DR, and is discussed in detail in Balakrishnan (1972). The

effect and its measurement are briefly described here:

When a square Plexiglas tank was filled with water and the

plug at the bottom of the tank was removed after stirring the water,

a vortex was found to form immediately. The vortex was quite stable

to perturbations and it remained intact until all the water in the

tank had drained. In the presence of drag reducing polymers above

a critical concentration (that depended on the polymer in question),

this was not the case--the vortex was not complete in that the air

core did not reach the drain hole at the bottom of the tank. The

appearances of the vortex with water and with polymer solution are

shown in Figures 3-8 and 3-9. At polymer concentrations slightly

below the critical, the vortex reached the bottom of the tank inter-

mittently and with further reduction in concentration the liquid

became indistinguishable from water. When the polymer concentration

was increased above the critical, the vortex became progressively

smaller until at very high concentrations only a very small dip formed

at the surface of the liquid. At polymer concentrations equal to or

very near the critical the vortex was extremely unstable in that its


















































0 -- --wo 9,t9


C
0























41
CL























X

0











C
':1-










length fluctuated (the tip of the vortex kept moving up and down).

Experimental investigation of VI showed that it correlated extremely

well with the drag reducing ability of polymers and that it was a

sensitive measure of polymer degradation at concentrations less than

10 wppm. Table 4.1 summarizes the vortex inhibiting ability of the

polymers used in this study. In the table CVI is the critical polymer

concentration below which vortex formation occurs, and it was determined

as follows:

The square Plexiglas tank illustrated in Figures 3-9 and 3-10

was filled with the polymer solution to a level of 65 liters, the

solution stirred vigorously and allowed to drain. The liquid level

was allowed to drop to 58 liters before the vortex was inspected for

completion. This drop in level from 65 to 58 liters was arbitrarily

chosen to remove the effects of initial stirring. If the vortex

completed when the liquid level was between 58 and 53 liters CVI was
*
taken as the next highest concentration in the dilution sequence.

Even though the procedure was arbitrary, the results were extremely

reproducible.













The dilution sequence used was (in wppm): 3000, 2000, 1500,
1000, 800, 600, 400, 300, 200, 150, 100, 75, 50, 40, 30, 20, 15, 10,
7.5, 5, 4, 3, 2, 1. For most polymers the starting concentration was
well below 500 wppm.














CHAPTER IV
EXPERIMENTAL RESULTS

All experimental data are summarized in this chapter.


4.1 Drag Reduction

Drag reducing ability was determined by using the flow loop

described in Chapter III. The polymer concentration chosen for DR

runs was 5 wppm. Data obtained in the 1.09 cm tube are given in

Figure 4-1 in which the friction factor f is plotted versus Reynolds

Number, NRe, based on solvent viscosity.


4.2 Vortex Inhibition

The vortex inhibition of the polymers used in this study is

listed in Table 4.1.


4.3 Extensional Viscosity

As discussed in Chapter II, values of entrance pressure drop

and the angle of convergence are required for an extensional viscosity

to be calculated.


Angle of Convergence Measurements

Stretch rates at the orifice are calculated from measurements

of 6, the half angle of convergence in WGS flow, using equations (2-12)

and (2-27). In this section, results are presented for the polyacry-

lamide solutions (PAMS), followed by those for PAA (Versicol S25), and






















Table 4.1

Summary of VI Data

Polymer CVI wppm

Separan AP 273 2

Separan AP 30 10

Percol 155 2

Polyhall 295 15

Versicol S25 3

Polyox WSR 301 3






















































10i
Reynolds Number, NRe

Figure 4-1. Summary of Drag Reduction Data










PEO (Polyox WSR 301). Values of 0 for all the runs are tabulated

in Appendix A.

PAM: As mentioned in Chapter III, the polyacrylamides used in this

study were Separan AP 273, Separan AP 30, Percol 155 and Polyhall 295.

All of these exhibit pronounced WGS flow that extends far upstream

from the orifice. In the case of Separan AP 273, WGS flow was observed

at concentrations as low as 10 wppm. At polymer concentrations below

10 wppin, traces of WGS flow were still visible but were not clear

enough for the angle of convergence to be accurately defined.

The angle of convergence a was found to depend on orifice

velocity vo, orifice radius R and polymer concentration. With most

polymer solutions, 0 was found to decrease slightly with increasing

v at low flow rates, increase slightly with increasing vo at moderate

to high flow rates, and, in some cases, decrease slightly with increasing

v at very high flow rates. This trend was observed with polymer solu-

tions that were of moderate or high concentrations (e.g., 20 wppm or

above for Separan AP 273) and is illustrated in Figure 4-2a in which

6 is plotted versus v0 for a 20 wppm solution of Separan AP 273. The

limits on the data points in the figure correspond to + 2% error in

measurement of 9 from photographs. The orifice diameter was 0.132 cm.

At concentrations lower than 20 wppm, 0 did not decrease with increasing

v at very large flow rates.

Considering next the influence of orifice radius, it was observed

that e tended to increase slightly with increasing R This trend is

illustrated in Figure 4-2b in which 0 is plotted against Ro for a

40 wppm solution of Separan AP 273. 0 also showed some dependence








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Separan AP 273

Orifice Diameter, v cm/sec
cms

50- 0.109 415

S0.132 485
-s 0.178 450
C 40-

c-

I 30-
0


20.




10-
0


*

0 1 I,,!
0 10 20 30 40
Concentration, wppm


Figure 4-2c. Dependence of 6 on Polymer Concentration










on polymer concentration. For a fixed orifice radius and velocity,

it decreased with increasing polymer concentration. Figure 4-2c

illustrates this trend.

Most of the data presented here were taken in the flow region

where e was a slightly increasing function of v In making measure-

ments of stretch rates at the orifice, problems were encountered with

the determination of 0. The most important of these are listed below.

(a) Nature of the flow field and flow visualization technique:

Accurate measurement of 0 was difficult due to the nature

of the flow field as well as the flow visualization technique used.

As described in Section 2.5, WGS flow is characterized by a central

converging region shaped like a cone, surrounded by slowly recirculating

fluid. The fluid in the space that borders these two regions experiences

shear rates nearly equal to the stretch rates in the central flow and

is dragged down towards the orifice with the fluid in the central region.

Close to the orifice, the liquid in this thin shearing region bounces

away from the orifice while the fluid in the central converging region

goes through it. It is extremely difficult to delineate the boundary

between these two regions. Moreover, dye in the thin shearing region

had a tendency to diffuse into a broad band (due to shear) instead of

forming well-defined streamlines and this obscured the border between

the shearing region and the central converging flow. Figure 4-3 il-

lustrates these effects for a 40 wppm solution of Separan AP 273.

I' the stretch rate at the orifice, was calculated from the

equation






















































Figure 4-3. WGS Flow with 40 wppm Separan AP 273
Orifice Diameter 0.096 cm









V
o = sin e (1 + cos 0) (4-1)
0

Equation (4-1) follows from equations (2-12) and (2-27) and the value

of r at the orifice.



f(r) = dvr (2-12)
dr


vr= -Q (2-27)
2r2(l-cos 0)

R
orifice sin 0(4-2


Plots of 1' vs. v0 for various polyacrylamide solutions are

given in Figures 4-4 to 4-9. Figures 4-4 to 4-6 are for Separan AP 273

in deionized water at concentrations of 10, 20, and 40 wppm and Figures

4-7 to 4-9 are for solutions of Separan AP 30 (20 wppm), Percol 155

(40 wppm) and Polyhall 295 (100 wppm), respectively in deionized water.

It is important to note that the flow field observed with the

lowest polymer concentration studied (Separan AP 273 at 10 wppm) in the

largest orifice (0.178 cm) was quite different from the WGS flow described

earlier. At large velocities, the streamlines entering the orifice

resembled the bottom of a wine glass rather than a wine glass stem.

The appearance of the flow field at low and high velocities is illus-

trated by photographs in Figures4-10a and b. At large velocities, the

flow angle became quite large and with the exception of a small re-

circulating region, the flow field bore little resemblance to WGS flow.

In the case of Percol 155, large gel-like particles were noticed

in the solution which did not disappear even when the solution was kept







87





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