the_effect_of_flow_instability.pdf - ucl discovery

THE EFFECT

OF FLOW INSTABILITY ON RESIDENCE TIME DISTRIBUTION

OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS

IN COUETTE-FLOW

SAMSON SAU SHUN YIM, B.ENG (Hons)

A thesis submitted for the Degree of Doctor o f Philosophy

in the University o f London

Ramsay Memorial Laboratory of Chemical Engineering Torrington Place London WCIE 7 JE

March 1997

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ABSTRACT

The laminar annular flow field formed in the gap of two concentric rotating

cylinders with and without a small degree of superimposed axial flow can be

used effectively to manipulate the residence time distribution of various species

in the gap in order to promote radial mixing with negligible axial dispersion,

conditions that are considered important for good heat and mass transfer and

reaction processes.

This thesis is concerned with an experimental and theoretical study of laminar

Couette flow stability in concentric devices. Numerical simulations based on

the solution of Navier-Stokes equations are presented for Newtonian and non-

Newtonian liquids, showing the effects o f operating and geometrical parameters

on the transition of laminar to Taylor vortex flow for induced rotational-axial

flow in the gap of a pair of rotating cylinders. These simulations indicate that

annular rotational flow becomes more stable in the presence o f a small degree

of axial flow and as the gap width increases. The effect o f rotational speed on

the breakdown of laminar flow is more complex and for given radius ratio and

axial flow rate depends on both the angular speed ratio and the direction of the

rotation o f the cylinders, counter-rotating flow generally producing a more

stable flow than co-rotating.

Experimental data are provided on the residence time distribution for flow of

several Newtonian water-glycerol solutions and non-Newtonian carboxymethyl

cellulose solutions and xanthan gum solutions through the gap of two

concentric rotating cylinders operating over a range of conditions. The

equipment used consisted of a pair of horizontal cylinders 1 m long with a

stationary outer perspex shell and an inner rotating shaft with a variable speed

control. The results from these experiments indicate that the residence time

distribution of the species in the gap is a complex function of the flow structure

in the gap which in turn depends on the operating and geometrical variables.

The experimental findings are successfully analysed and discussed using the

simulations studies. Recommendations are made for future work.

DEDICATION

Œb My

"Learning, oSserving, try not to stay on tHe surface o f facts.(Do not Become tfie arcBivists o f facts.

Try to penetrate to the secret o f tBeir occurrence, persistentCy search fo r the Caws which govern them. "

Ivan (Petrovich Pavlov {1849 - 1936)

LIST OF CONTENT

Page

Title Page 1

Abstract 2

List o f Content 5

List o f Figures 9

List o f Tables 14

Acknowledgments 18

Chapter One Introduction 19

1.1 Definition 19

1.2 Historical background 20

1.3 Industrial applications 21

1.4 Objectives of the present study 22

Chapter Two Literature Survey 24

2.1 Introduction 24

2.2 Theoretical studies on flow instability of Newtonian liquids 25

2.2.1 Narrow-gap problem in Couette flow 25

2.2.2 Wide-gap problem in Couette flow 30

2.2.3 Narrow-gap problem in Couette flow

with a low axial flow 34

2.2.4 Wide-gap problem in Couette flow

with a low axial flow 37

2.3 Experimental studies on flow instability of Newtonian liquids 39

2.3.1 Flow visualisation method 39

2.3.2 Power spectra method 42

2.3.3 Dispersion measurement 44

2.4 Theoretical studies on flow instability o f non-Newtonian liquids 50

2.5 Experimental studies on flow instability of non-Newtonian liquids 51

2.6 Summary 53

Chapter Three Theory 55

3.1 Introduction 55

3.2 Flow instability of Newtonian liquids in Couette flow 56

3.2.1 Basic flow 56

3.2.2 Governing equations 57

3.2.3 Method of solution 65

3.3 Flow instability of non-Newtonian liquids in Couette flow 68

3.3.1 Basic flow 68

3.3.2 Governing equations 68

3.3.3 Method of solution 76

Chapter Four Materials and Methods 79

4.1 Introduction 79

4.2 Equipment 79

4.2.1 Couette flow device 79

4.2.2 Conductivity measuring device 82

4.2.3 Viscosity measuring device 83

4.3 Materials 85

4.3.1 Newtonian liquids 85

4.3.2 Non-Newtonian liquids 87

4.3.3 Tracer solutions 100

4.4 Methods 103

4.4.1 Experimental procedure 103

4.4.2 Determination o f residence time distribution 104

4.4.3 Data acquisition system 105

Chapter Five Theoretical results and discussion 107

5.1 Introduction 107

5.2 Newtonian liquids in Couette-flow 108

5.2.1 Neutral curve 108

5.2.2 Convergence o f numerical results 112

5.2.3 The effect o f radius ratio on the

critical T ay lor number 115

5.2.4 The effect o f angular speed ratio on the

critical Taylor number 119

5.3 Newtonian liquids in Couette-flow with axial flow 121


5.3.2 The effect o f axial Reynolds number on the


5.3.3 The effect o f radius ratio on the




5.4 Non-Newtonian liquids in Couette-flow 135


5.4.2 The effect of radius ratio on the




Chapter Six Experimental results and discussion 154

6.1 Introduction 154

6.2 Residence time distribution 154

6.2.1 Reproducibility 154

6.2.2 The effect o f axial flow velocity 157

6.2.3 The effect o f rotational speed 157

6.2.4 The effect o f electrode positions 165

6.3 Axial dispersion in Couette flow 168

6.3.1 Axial dispersion model 168

6.3.2 Peclet number 173

6.3.3 The effect o f inner cylinder geometry 197

6.4 Comparison of theoretical and experimental results 201

Chapter Seven Conclusions and Recommendations 205

7.1 Conclusions 205

7.2 Recommendations for further work 208

Nomenclature 210

References 214

Appendix 1 Galerkin Method 222

Appendix 2 Mathematica program for flow instability of

Newtonian liquids in Couette flow 224

Appendix 3 Mathematica program for flow instability of

non-Newtonian liquids in Couette flow 233

Appendix 4 Computer program for conductivity measurement

and control system 239

Appendix 5 Sample calculations of Peclet number from

experimental RTD data 245

Appendix 6 Published papers relating to this project 248

LIST OF FIGURES

page

Figure 2.1 Hydrodynamics of Couette flow 26

Figure 2.2 Taylor vortices 26

Figure 3.1 Annulus configuration and co-ordinate system 56

Figure 4.1 Experimental set-up 80

Figure 4.2 Inner cylinder shaft (with blades) 81

Figure 4.3 Conductivity probe 82

Figure 4.4 Viscometer 84

Figure 4.5 Shear stress-shear rate plot for different concentration

of Newtonian glycerol-water solutions 86

Figure 4.6 Classes of non-Newtonian behaviour 88

Figure 4.7 Typical logarithmic plot o f a non-Newtonian liquid 88


of non-Newtonian carboxymethyl cellulose solutions 93

Figure 4.9 Logarithmic shear stress-shear rate plot for different

concentration o f non-Newtonian carboxymethyl

cellulose solutions 94


of non-Newtonian xanthan gum solutions 95

Figure 4.11 Logarithmic shear stress-shear rate plot for different

concentration of non-Newtonian xanthan gum solutions 96

Figure 4.12 Concentration and rheology of non-Newtonian

carboxymethyl cellulose solutions 97

Figure 4.13 Concentration and rheology of non-Newtonian

xanthan gum solutions 98

Figure 4.14 Conductivity vs concentration of KCl solution 101

Figure 4.15 The influence of the amount of tracer injected

on the dimensionless RTD 102

Figure 4.16 Data acquisition system 106

10

Figure 5.1 Typical neutral curve in the Ta - A, plane 110

Figure 5.2 Neutral curve in the Ta - A. plane at different radius ratios 111

Figure 5.3 Convergence of neutral curve 114

Figure 5.4 Comparison of critical Taylor number with previous workers

on different radius ratios, a 116

Figure 5.5 The effect o f radius ratio on dimensionless wave number 118

Figure 5.6 The effect of radius ratio on critical Taylor number for

different cases of angular speed ratio 120

Figure 5.7 The effect o f dimensionless wave number, A., on Imaginaiy

part o f Taylor number Im[Ta] for different cases of

dimensionless disturbance growth rate, a 122

Figure 5.8 The effect o f radius ratio on critical Taylor number for

different cases of axial Reynolds number 126

Figure 5.9 The effect of axial Reynolds number on critical Taylor

number for different cases of angular speed ratio 129

Figure 5.10 Angular speed distribution for a range of flow index, n,

at different radial co-ordinates 137

Figure 5.10 Stress distribution for a range of flow index, n, at

different radial co-ordinates 138

Figure 5.12 Apparent viscosity for a range of flow index, n, at

different inner cylinder rotational speeds 139

Figure 5.13 Mean viscosity for a range of flow index, n, at different

inner cylinder rotational speeds 140

Figure 5.14 Neutral curves for a range of flow index 142

Figure 5.15 The effect of radius ratio on the critical Taylor number 143

Figure 5.16 The effect of flow index on the critical Taylor number 144

Figure 5.17 The effect of angular speed ratio on the critical

Taylor number 153

11

Figure 6.1 Reproducibility o f RTD experiments for Newtonian liquid 155

Figure 6.2 Reproducibility o f RTD experiments for

non-Newtonian liquid 156

Figure 6.3 RTD experiments at different axial flow velocities

for Newtonian liquid 15 8


for non-Newtonian liquid (0.6wt% xanthan gum solution) 159

Figure 6.5 RTD experiments at different rotational speeds

for Newtonian liquid (Ta < Ta ) 161


for Newtonian liquid (Ta > Tac) 162

Figure 6.7 RTD experiments at different rotational speeds

for non-Newtonian liquid (Ta < Tac)

(0.6wt% xanthan gum solution) 163


for non-Newtonian liquid (Ta > Tac)

(0.6wt% xanthan gum solution) 164

Figure 6.9 Influence of the position o f the sensor in the annular gap 165

Figure 6.10 Influence of the location of the sensor in the annular gap 165

Figure 6.11 RTD experiment at different positions of the sensor

(Newtonian liquid) 166

Figure 6.12 RTD experiments at different locations of the sensor

(Newtonian liquid) 167

Figure 6.13 Axial dispersion model 170

Figure 6.14 Comparison of axial dispersion model and the

experimental data (Newtonian liquid) 171

Figure 6.15 Comparison of axial dispersion model and the

experimental data (non-Newtonian liquid) 172

Figure 6.16 The effect of rotational speed of inner cylinder on

dimensionless variance difference of the tracer RTD curve 174

12

Figure 6.17 The effect o f Taylor number on Peclet number for

Newtonian liquid (a = 0.84, 1 < Re < 3) 176











Figure 6.23 The effect of Taylor number on Peclet number for

non-Newtonian liquid (a = 0.84, 0.1 - 0.5 wt%) 182















13

Figure 6.31 The effect o f axial Reynolds number on critical

Taylor number at different radius ratios, a 192

Figure 6.32 The effect o f flow index on critical Taylor number

at different radius ratios, a 196

Figure 6.33 The influence of inner shaft geometry 197

Figure 6.34 The effect o f Taylor number on Peclet number

for Newtonian liquids (Shaft SI) 199

Figure 6.35 The effect o f Taylor number on Peclet number

for Newtonian liquids (Shaft S2) 200

Figure 6.36 Comparison of theoretical and experimental results

for Newtonian liquids 203

Figure 6.37 Comparison of theoretical and experimental results

for non-Newtonian liquids 204

14

LIST OF TABLES

page

Table 2.1 Summary of narrow-gap problem in Couette flow 30

Table 2.2 Summary of critical Taylor number for

different values of radius ratio 32

Table 2.3 Summaiy o f wide-gap problem in Couette flow 33

Table 2.4 Summary of narrow-gap problem in Couette flow

with axial flow 36

Table 2.5 Summary of critical Ta number for different values of

radius ratio, a and given values o f Re number 38

Table 2.6 Summary o f wide-gap problem in Couette flow

with axial flow 39

Table 2.7 Summary of major experiments on flow visualisation

method in Couette flow device 41

Table 2.8 Summary o f major experiments on power spectra

method in Couette flow device 43

Table 2.9 Summary of major experiments on dispersion

measurement in Couette flow device 49

Table 2.10 Summary of theoretical studies of non-Newtonian liquids

in Couette flow device 52

Table 2.11 Summary of experimental studies of non-Newtonian liquids

in Couette flow device 52

Table 4.1 The concentration and rheology of glycerol-water solutions 85

Table 4.2 The concentration and rheology of carboxymethyl

cellulose (CMC) solutions 99

Table 4.3 The concentration and rheology of xanthan gum solutions 99

Table 5.1 Comparison of Ta value in narrow gap geometry 112

Table 5.2 The critical Taylor number and critical dimensionless wave

number with a resting outer cylinder (p=0) as a function

of the radius ratio, a 117

15

Table 5.3 The critical Taylor number with a different angular

speed ratio, p,as a function of the radius ratio, a 119

Table 5.4 Critical Taylor number and corresponding values o f X

and G for given values of Re when a = 0.95 and (3=0 124



Table 5.6 Critical Taylor number and corresponding values of X







and G for given values of Re when a = 0.5 and P = 0 128


and G for given values of Re when a = 0.4 and p = 0 130


and G for given values of Re when a = 0.8 and P = -0.25 131






and G for given values o f Re when a = 0.8 and p = 0 132


and G for given values o f Re when a = 0.6 and P = 0 132


and G for given values of Re when a = 0.4 and p = 0 132

16


and a for given values of Re when a = 0.8 and P = 0.25 133




and a for given values of Re when a = 0.4 and p = 0.25 133






and a for given values of Re when a = 0.4 and p = 0.5 134

Table 5.23 Critical Taylor number for given values o f flow index

and radius ratio when angular speed ratio p = 1 146

Table 5.24 Critical dimensionless wave number for given values of

flow index and radius ratio when angular speed ratio P = 1 146


and radius ratio when angular speed ratio p = 0.5 147

Table 5.26 Critical dimensionless wave number for given values

of flow index and radius ratio when angular

speed ratio P = 0.5 147


and radius ratio when angular speed ratio p = 0.25 148


of flow index and radius ratio when angular

speed ratio p = 0.25 148

Table 5.29 Critical Taylor number for given values of flow index

and radius ratio when angular speed ratio p = 0 149

17


o f flow index and radius ratio when angular

speed ratio P = 0


and radius ratio when angular speed ratio p = -0.25



speed ratio p = -0.25

Table 5.33 Critical Taylor number for given values of flow index

and radius ratio when angular speed ratio P = -0.5



speed ratio p = -0.5

Table 6.1 Summary of the experimental Pe - Ta plots

Table 6.2 Experimental results o f Tac for Newtonian glycerol

solutions (a = 0.84)

Table 6.3 Experimental results of Tac for Newtonian glycerol


Table 6.4 Experimental results o f Tac for non-Newtonian


Table 6.5 Experimental results o f Tac for non-Newtonian


Table 6.6 The specifications of different inner rotating cylinder

geometry

149

150

150

151

151

175

190

191

193

194

197

18

ACKNOWLEDGMENTS

The author would first like to acknowledge, with sincere gratitude, the

invaluable assistance, guidance, and encouragement of his supervisor. Dr. P.

Ayazi Shamlou who has given generously of his time and wise counsel during

the period of this work.

Successful operation of the stability apparatus would not have been possible

without the support of the technical staff for which the author is eternally

grateful. Gratitude is owed to Martin Town, Martyn Vale, Samuel Okagbue,

Carol Welfare and Julian Perfect for contributing the equipment and

experimental materials used in his investigation. The author also likes to thank

all his colleagues, past, and present, for providing a fiiendly and humorous

working environment.

This study could not have taken off had it not been for the inspirational,

dedicated and supportive attitude of his dear wife, Mei-Yee. Besides, the

author would like to express his deep gratitude to his parents for their

encouragement and support at all times.

The author is deeply indebted to many individuals who have contributed to the

successful completion of this work.

Finally, the author gratefully acknowledge the financial support of the

Croucher Foundation Scholarships and the Overseas Research Students Award

Scheme.

Introduction 19

CHAPTER ONE

INTRODUCTION

1.1 Definition

The production and processing of Newtonian and non-Newtonian materials

frequently involve the flow of material through concentric cylinder flow

devices and often several unit operations can occur concurrently during flow.

These unit operations include heat transfer, mixing, émulsification, dispersion,

crystallisation and chemical reaction. The Couette flow device basically

consists of a pair of concentric rotating cylinders in which the outer shell may

be jacketed to facilitate heat transfer through its wall. In this way the

temperature of the process material can be controlled as it flows through the

annular gap between the two cylinders. In the food industry, in the case of

margarine and ice-cream for example, heat transfer to the process material

during flow causes crystallisation of fat and brings about significant changes in

the rheological properties o f the final product.

Most of the unit operations that occur in a Couette flow device are affected by

the behaviour of the material during flow and in the case of steady state

continuous processes a key factor is the variation in the duration of stay within

the process equipment experienced by "particles" which entered the equipment

at the same time. This variation is normally expressed in terms o f the residence

time distribution (RTD) and as a result measurement and analysis of RTD has

become an important tool in the study of continuous processes. Understanding

the relationship between the fluid dynamics and the RTD in a Couette flow

device is therefore of basic research interest to academics and industrialists.

Introduction 20

1.2 Historical Background

There are a number of studies of liquid flow between concentric rotating

cylinders since the earliest studies were conducted by Couette in 1890, who

studied the viscosity of water by measuring the moment exerted by the rotation

o f an outer cylinder on a resting inner cylinder (cf. Donnelly, 1991). After 30

years, Taylor (1923) investigated both experimentally and theoretically the

instability o f an incompressible Newtonian liquid under a purely rotational

flow with both cylinders rotating in the same direction and in opposite

direction. He discovered a cellular motion developed with the rotation of inner

cylinder in the form of a number of counter-rotating vortices regularly spaced

along the annular gap. The formation and behaviour of these vortices have

been the subject of considerable interest in fluid mechanics. For this reason,

most investigators refer to liquid flow between concentric cylinders as Taylor-

Couette flow.

The work o f Taylor (1923) on the instability o f Couette flow between

concentric rotating cylinders has inspired numerous theoretical and

experimental investigations. The following 70 years o f research yielded a vast

amount o f information on the hydrodynamics, transport properties and

applications of Taylor-Couette flow. There are also many papers documenting

the developments o f Taylor's theme, the breadth of which may be gauged from

a review paper by Stuart (1986). These include higher-order instabilities,

eccentric annuli, super-position of natural convection (radially or axially),

forced axial convection, the effect of finite length, and non-linear and finite

amplitude analyses. However, such studies lie outside the bounds of the

present study although some transport properties, such as mass and heat

transfer (Kaye and Elgar, 1958) to the cylinder walls, have been well described

in the literature. Back-mixing or dispersion in Taylor-Couette flow system has

received considerably less attention, and is the subject of this thesis.

Introduction 21

1.3 Industrial Applications

In many industrial processing situations, from polymer processing to paper-

making, from foods to pharmaceuticals and from chemical to biochemical it is

often desirable to generate plug flow of processing liquids, in order to

maximise the driving force for transfer processes, for example. The defmition

of "plug flow" is given by Levenspiel (1972) as the flow o f liquid through the

equipment is orderly with no element of liquid overtaking or mixing with any

other element ahead or behind but there may be lateral mixing of liquid in a

plug flow equipment.

Most industrially important liquids are non-Newtonian and develop a wide

residence time distribution (RTD) in pipe flow which makes the mass and heat

transfer difficult to control. For example, polymerisation is a particularly

important class of reactions in which deviation from plug flow is

disadvantageous and difficult to prevent because of the variable apparent

viscosity of the liquid at different shear rate across the reactor. In this case, the

added degree of freedom obtained by rotating the inner cylinder or a continuous

Couette flow device means that a relatively narrow RTD may be achieved even

for high viscosity liquids. Moreover, in polymerisation reactions, another

difficulty that is frequently experienced is the deposition of very high

molecular weight solid on the inner wall o f the equipment. The unique local

radial liquid flow of Taylor-Couette flow, induced by the inner rotating

cylinder, often causes a reduction in the deposition.

There is a number of publications on the hydrodynamics, transport properties

and applications of the Couette flow device. Some practical applications

presented over the years include catalytic chemical reactors (Cohen and Maron,

1983), dynamics filtration and classification on a cylindrical surface (Tobler,

1982; Rushton and Zhang, 1988), blood plasmaphoresis devices (Beaudoin and

Introduction 22

Jaffiin, 1989), characterisation of shear-dependent rate processes as in

agglomeration and breakage of particles formed during precipitation processes

(Hoare, et al., 1982), cooling of rotating electrical machinery (Kaye and Elgar,

1958), ice crystallisation (Wey and Bstrin, 1973), electrolytic applications

(Legrand and Coeuret, 1986), offshore oil exploitation (Rosant, 1994).

1.4 Objectives of the Present Study

The objective of this work is to mathematically study and experimentally

describe the dependence of the RTD in a Couette flow device on some o f the

important of operating and material variables. Both Newtonian and non-

Newtonian liquids will be used in the experiments.

It is believed that dispersion is an important factor in Couette flow devices as

the amount of mixing in different flow regimes greatly influences the

productivity of a reactive system (Cohen and Maron, 1983). Design, scale-up

and optimisation calculations for Couette flow devices require an understanding

of the transport properties o f Taylor-Couette flow. Moreover, the stability of

flow o f non-Newtonian liquids with variable viscosity has not so far been taken

into consideration. These provided the motivation for the present study.

The theoretical section of the present study has been devoted to the treatment of

the Couette flow instability of both Newtonian and non-Newtonian liquids.

The theoretical prediction of Newtonian liquids in Couette flow has been

developed for many years as discussed later in the Section 2.2. In this

investigation, a more general Couette flow problem will be proposed to include

operational and geometrical factors and the addition of axial flow velocity. In

the case of non-Newtonian liquids, effects such as flow index, consistency

index may influence the criterion of flow instability which are different form

Introduction 23

the Newtonian case. For the system investigated, the onset o f flow instability,

defined as critical Taylor numbers, are presented as functions o f rheological

properties o f the liquid medium, operational and geometrical properties o f the

Couette flow device.

Experimental considerations have also been given in the thesis to the RTD of a

range of Newtonian glycerol water solutions and two non-Newtonian liquids

(carboxymethyl cellulose solutions and xanthan gum solutions). A stimulus

response experimental technique based on an impulse input is employed. The

experimental results are interpreted in terms of a single parameter axial

dispersion model. The data include results from experiments in which flow

transition occurred from laminar to Taylor Couette flow regime. Finally, the

findings from these experiments are analysed and assessed using the

simulations studies.

Literature Survey 24

CHAPTER TWO

LITERATURE SURVEY

2.1 Introduction

In this chapter, an overview of literature survey o f flow instability study in

Couette flow will be given both theoretically and experimentally. Section 2.2

covers the theoretical development of flow instability problem in Couette flow.

Special attention will be given to Newtonian liquids. This section will be

subdivided into four parts:

1) Narrow-gap problem in Couette flow

2) Wide-gap problem in Couette flow

3) Narrow-gap problem in Couette flow with a low axial flow

4) Wide-gap problem in Couette flow with a low axial flow

In Section 2.3, the experimental studies of flow instability of a Newtonian

liquid will be discussed. Again, this section will be subdivided into three parts:

1) Flow visualisation method

2) Power spectra method

3) Dispersion measurement

The theoretical and experimental studies on the flow instability o f non-

Newtonian liquids will be discussed in Section 2.4 and 2.5 respectively.

Finally in Section 2.6, general comments on the problem of flow instability in a

Couette flow device will be given. The major studies will be summarised at the

end o f each subsection.


2.2 Theoretical Studies on Flow Instability of

Newtonian Liquids

2.2.1 Narrow-aap problem in Couette flow

Lord Rayleigh (1916) first considered the instability o f liquid flow between two

long concentric rotating cylinders for an inviscid liquid. He derived a simple

condition for instability with respect to rotational disturbances based on energy

consideration. He assumed that, in real liquids, viscosity served to maintain the

steady flow but did not affect the occurrence o f instability. Rayleigh’s criterion

led the conclusion that, for cylinders rotating in the same direction, flow was

stable if OiRi^ > QiRi^, where and O2 were the angular velocities of the

inner and outer cylinder respectively, Ri and R2 were the corresponding radii.

In 1923, Taylor (1923) made a brilliant contribution to the theory of

hydrodynamic stability by quantitatively predicting the flow instability of a

Newtonian liquid flowing between a pair of concentric rotating cylinders. He

stated that for very low rotational speeds of the inner cylinder, the liquid simply

moved azimuthally around the cylinders (see Figure 2.1). The radial pressure

gradient o f the liquid was responsible for the centripetal acceleration that kept

the liquid moving in circular paths. If, however, a ring of liquid was perturbed

outward to a larger radius, the local pressure gradient might not be sufficient to

restore the liquid to its original path. If instability occurred, the liquid

continued to be ejected outward until it met the outer cylinder; here the liquids

forced to overturn and, hence, travelled in the helical paths that constitute

toroidal vortices, now known as Taylor vortices, spaced regularly along the

cylindrical axis (see Figure 2.2).


Ceniiifiigal force (ky rptation of inner cylinder)

Centripetal force \^ y pressure gradient of the liquid)!

Outer cylinder

A ring of liquidLiner cylinder

Figure 2.1 Hydrodynamics of Couette flow

A pair of Taylor vortices/ \

Rotating inner cyiinder

Annular gap

Stainless steel

Perspex outer cylinder

Figure 2.2 Taylor vortices


Taylor (1923) began by assuming that, superimposed upon the Couette motion,

there was a small secondary velocity perturbation which was a function of the

radial and axial coordinate. He employed this assumption in the Navier-Stokes

and continuity equations, dropped those terms involving products o f secondary

quantities, and obtained a set of "disturbance equations" which were linear and

homogeneous in the component of main Couette flow. When combined with

the homogeneous boundary conditions on small secondary velocity

perturbation, these disturbance equations defined an "eigenvalue problem" for

the angular velocity o f the inner cylinder. The minimum value of angular

velocity, over all allowable eigenvectors then gave the critical conditions for

the onset o f instability. The solution of the eigenvalue problem was expressed

in term of Bessel function formula. In order to reduce the numerical difficulties

originating from the sum of the Bessel functions he made the so-called narrow-

gap approximation, which meant that he assumed the annular gap between the

cylinders was much smaller than the inner cylinder. So that, the Bessel

functions were replaced by the trigonometrical functions. After laborious

calculations he arrived at an equation for the critical condition for instability as

a function of the rotational speed of the cylinders, the radii o f the cylinders, and

the viscosity of the liquid. From his calculations, Taylor found that the

criterion for the onset o f instability could be expressed in the following form:

Onset o f instability = -------- ?------------7t [l + b /2R i] -------------------- p ] ]0.0571 [1 + 0.652 b/RiJ + 0.00056 [l + 0.652 b / Ri l

where b corresponds to the annular gap width and Ri corresponds to the radius

of inner cylinder. By using the small-gap approximation assumption, he

determined a dimensionless parameter defined in terms of geometrical and

operational factors, which was later defined as the Taylor number. Ta. The

minimum eigenvalue interpreting the onset of instability, defined as critical

Taylor number, Ta , was found approximately to be 1700. According to

Taylor's linear theory, for values of the speed of rotation less than the critical


speed all disturbances o f the flow in the annulus were damped owing to the

action o f viscosity, whereas for speeds greater than the critical speed, some of

disturbances would be amplified with increasing time and the flow would

become unstable with the formation of a steady secondary flow in the form of

pairs o f counter-rotating vortices. Taylor’s theoretical calculations were

verified convincingly by his own laboratory experiments, which showed that,

with the inner cylinder rotating and the outer cylinder at rest, the instability of

laminar Couette flow led to the Taylor-Couette flow in which a secondary

motion with cellular toroidal vortices appeared regularly in the axial direction.

Chandrasekhar (1953) reconsidered this problem by using a procedure which

was simpler than the method used by Taylor (1923). He investigated the

instability o f viscous flow of an electrically conducting liquid in the annular

gap with an inner rotating cylinder in the presence o f a magnetic field directed

along the axis o f cylinders. Considering only the narrow-gap case, he

neglected all the non-linear terms, and the basic equations were derived to one

8-th order differential equation of the following form:

-]2(d - afjf + Qa Y(r) = -Ta a' (d - a ) T(r) [2.2]

Here 'T(r) is a perturbation on the magnetic field, Q is a dimensionless

parameter based on the magnetic and electric properties o f the system, D is a

differential operator, a is wavelength of the Taylor vortex. Equation [2.2]

represents an eigenvalue problem which Chandrasekhar solved by variational

calculus for the critical values of Taylor number. For the case where the

strength of the field was equal to zero, he found that the critical Ta number was

1708 and the critical wavelength was twice the annular gap width were in

agreement quite well to those of Taylor (1923) for comparable situations. The

effect o f an axial magnetic field on such an instability problem was also studied

later by Kurzweg (1963).


Becker and Kaye (1962) considered the effect o f a radial temperature gradient

on the stability o f Couette flow between concentric rotational cylinders. The

presence of the radial temperature gradient gave rise to convective effects

through the interaction o f the associated radial density gradient with the radial

acceleration. They showed that the heating of the inner cylinder stabilised the

flow and delayed the onset o f Taylor vortices. By applying the small-gap

approximation, the 8-th order differential equation was further reduced to a 6-th

order differential equation.

û + a^(Ta-Ray)û = 0 [2.3]

where Ray is the Rayleigh number, û is a dimensionless radial velocity

component. The equation was then solved by variational calculus and they

obtained the following expression for the critical Ta number.

Tac= 1707.8 Fg [2.4]

where Fg was the complicated geometrical factor which derived from Taylor’s

results for isothermal flow.

The above researches showed that the eigenvalue problem could be

successfully solved and simplified by using the narrow-gap approximation

condition. The flow instability problem of Couette flow was also later tackled

by many researchers using different mathematical techniques. Di Prima (1955)

simplified the method of solution of an eigenvalue problem by giving an

algebraic series solution instead of Chandrasekhar’s variation calculus solution.

Meksyn (1961) attacked the problem by seeking a separate asymptotic solutions

for all cases o f speed ratio (Ü2/f^i), he was able to derive a closer form o f the

predictions for the flow instability condition. Harris and Reid (1964) used

Runge-Kutta method to solve the eigenvalue problem of an isothermal Couette


flow. Table 2.1 summarised the method used by the previous researchers and

the critical values for the onset o f instability of Couette flow, Ta . It showed

that their numerical calculations were in agreement with each other for

comparable situations.

Mathematical method Critical Taylor number

Taylor (1923) Bessel function 1698

Chandrasekhar ( 1953) Variational calculus 1708

Di Prima (1955) Algebraic series method 1695

Mekyn (1961) Asymptotic method 1712

Becker and Kaye (1962) Variational calculus 1707.8

Kurzweg (1963) Approximation method 1750

Harris and Reid (1964) Runge-Kutta method 1700

Table 2.1 Summary of narrow-gap problem in Couette flow

2.2.2. Wide-gap problem in Couette flow

Chandrasekhar (1958) first considered the wide-gap problem by using an

expansion technique similar to that for the small-gap problem (Chandrasekhar,

1953). His calculations involved the use of Bessel function series and

ultimately led to the condition that an infinite-order determinant be zero. The

computations were rather tedious and numerical results were obtained only for

the case o f radius ratio a = 1/2, but for a wide range of values of angular speed

ratio, p, from -0.5 to 0.25. The positive value and negative value of p

represented co-rotating and counter-rotating cylinders respectively.

Chandrasekhar (1958) successfully accounted for the small-gap approximation

in order to allow for the determination of critical Taylor numbers for large

annular gap widths. Later Chandrasekhar and Elbert (1962) simplified the


annular gap widths. Later Chandrasekhar and Elbert (1962) simplified the

numerical procedure by using the adjoint eigenvalue method. They stated that

the critical Ta number decreased substantially as the angular speed ratio, P,

increased fi*om counter-rotating to co-rotating.

Kirchgassner (cf. Walowit et al., 1964) constructed the Green's function for the

wide-gap problem, and solved the resultant integral equation by using an

iteration technique. Results were obtained for 1/2 < a < 1 with P = 0, -0.4 < p

< 0.25 for a = 1/2, and for -0.4 < p < 4/9 for a = 2/3. The information relating

to the onset of instability provided by the above reference is however restricted

to a relatively narrow range of the radius ratio, a.

Walowit et al. (1964) simplified the method of solution of an eigenvalue

problem by giving an algebraic series solution instead o f Chandrasekhar's

trigonometric series solution. They showed that the eigenfunctions could be

represented as a combination of simple polynomials. With the choice of

expansion function satisfying the boundary conditions, all necessary integrals

could be evaluated easily. The results o f Walowit et al. (1964) covered a wide

range of radius ratios and angular speed ratios, and were in good agreement

with the previous theoretical results.

Sparrow et al. (1964) solved the eigenvalue problem numerically by using the

Runge-Kutta method. Critical Ta numbers were determined for wide range of

radius ratios over a wide range of angular speed ratios, p. For positive value of

P (co-rotating cylinders), computations were carried out without difficulty over

the entire range fi’om p = 0 to p = a , the latter value being the limit beyond

which the flow was stable. For negative values of P (counter-rotating

cylinders), the computations were extended to p with absolute values

substantially larger than those for positive ps. The largest negative value of P


for which computations were carried out for each radius ratio was limited by

numerical difficulties attributed to loss o f significant figures.

Roberts et al. (1965) extended the theoretical studies of Chandrasekhar (1958)

on the flow instability o f Couette flow in a viscous, electrically conducting

liquid to include the factor of finite gap width. They successfully used the

Bessel function to solve the eigenvalue problem. However, their computational

work were only restricted to the case of speed ratio, p = 0. Table 2.2

summarises the values o f critical Ta number published by the previous

researchers.

Radius ratio Critical Taylor number Radius ratio Critical Taylor number

1 1696" 0.6 2572"

0.975 1724* 0.5 3100*

0.95 1755* 0.5 3100"

0.95 1755" 0.5 3099"

0.95 1756" 0.4 3998"

0.9 1823* 0.35 4717"

0.9 1824" 0.3 6524"

0.85 1902* 0.25 7442"

0.8 1995" 0.2 10356*

0.75 2102* 0.2 10364*

0.75 2103" 0.15 16317"

0.7 2230" 0.1 32606"

0.65 2384* 0.1 32500"

Table 2.2 Summary of critical Taylor number for different values o f radius ratio P denote the results from Sparrow et al. (1964); denote the results from Walowit et a i. (1964); ‘'denote the results from Roberts et al. (1965)]


Recently, Sondalgekar et al. (1994) studied the effect of an axial magnetic field

on the flow stability in Couette flow. The eigenvalue problem was solved

using asymptotic method. Their results for the critical Ta number agreed well

with those o f Sparrow et al. (1964) for the case o f strength of the magnetic field

equal to zero. They also observed that the value of the critical Ta number

increased with the strength of the magnetic field for both co-rotating or

counter-rotating cylinders. Table 2.3 summarised the method used by the

previous researchers and the range o f operating parameters they employed to

study the flow instability o f Couette flow.


Chandrasekhar (1958) Bessel function a = 0.5;

-0.5 < P < 0.25

Chandrasekhar and

Elbert (1962)

Adjoint eigenvalue

method

a = 0.5;

-0.5 < P < 0.25

Kirchgassner

(of. Walowit et al., 1964)

Green's function 0.5 < a < 1; p = 0

a = 0.5; -0.4 < p < 0.5

a = 2/3; -0.4 < P < 4/9

Walowit et al. (1964) Trigonometric

series solution

all a; P = 0 only

Sparrow et al. (1964) Runge-Kutta method all a; P = 0 only

Robert et al. (1965) Bessel function all a; P = 0 only

Sondalgekar et al.

(1994)

Asymptotic method all a; P = 0 only

Table 2.3 Summary of wide-gap problem in Couette flow


2.2.3 Narrow-aap problem in Couette flow with a low axial flow

When a low axial flow is superimposed on the Couette flow, the problem

becomes more complicated. This combined flow, so-called Couette-Poiseuille

flow, occurs in numerous industrial applications. A number of investigations

have been conducted to establish the effect o f axial flow on the occurrence of

Taylor vortices in a laminar flow. The critical Ta number was found to be a

function o f the axial Reynolds number. Re, the radius ratio, a, and the angular

speed ratio, p, o f the concentric rotating cylinders.

Goldstein (1937) first considered this problem theoretically for the case o f the

outer cylinder at rest and the small-gap width compared to the mean radius. He

used the method of expansion in Fourier series and treated the case o f flow

between rotating cylinders in the presence of an axial pressure gradient along

the axis o f the cylinders. Goldstein (1937) showed that the critical Ta number

initially increased, as the Re number (associated with the axial flow velocity),

increased from zero to a value of about 20, and then decreased rapidly as Re

increased to 25.

Chandrasekhar (1961) showed that Rayleigh's inviscid criterion for rotational

instability, as previously applied to viscous and purely rotational flow between

concentric rotating cylinders, remained valid in the presence o f an axial

velocity components. To account for viscosity effects Chandrasekhar (1961)

and simultaneously Di Prima (1960) employed linear theory to predict critical

Ta number in narrow-gap for the case of cylinders in co-rotating fully

developed laminar Couette flow for Re number below 200. Chandrasekhar

(1960) considered the tangential velocity to be uniform across the gap, an

assumption justified by extrapolation from the findings of Taylor (1923) for

zero Re number, where a small error in the predicted critical Ta number was

reported. His predictions of critical Ta number compared well with those o f Di


Prima (1960) using the same assumptions but the latter also found the

substitution of a parabolic axial velocity distribution to have an effect o f no

more than 5.5% on the solution for Re < 80, compared with uniform axial

velocity case. At higher Re number the values of critical Ta number for the

two cases diverged. For a stationary outer cylinder, critical Ta number

increased approximately with Re " and Re ^ for a parabolic and a uniform

velocity respectively; the corresponding value of critical Ta number at Re =

120 were 15126 and 11850 respectively.

Chandrasekhar (1962) considered the case when the axial velocity profile was

parabolic and developed a perturbation theory which was valid in the very low

range of values of Re number. He found that the critical Ta number increased

more rapidly than predicted by Di Prima's (1960) solutions. Chandrasekhar

(1962) used a perturbation procedure, and found that

Tac = Tac(atRe = o) + 26.5 Re a sR e -> 0 [2.5]

Later Krueger and Di Prima (1964) re-examined the same problem, and

obtained their results by using Fourier series technique. The predictions agreed

with the earlier results of Di Prima (1960) but did not show the rapid initial

increase o f the critical Ta number with Re number as reported by

Chandrasekhar (1962). Their results were an improvement on Di Prima's

(1960) previous work. They also suggested that, while the perturbation

procedure used by Chandrasekhar (1962) was suitable, in the actual

computation not enough terms in the series had been retained to give the correct

coefficients o f Re^ in Eq. [2.5].

Hughes and Reid (1968) treated the case o f large Re number (> 200) in a

narrow-gap situation using a uniform tangential distribution. They used an

asymptotic method whereby the resultant equations could be approximated by

the Orr-Sommerfield equation; their prediction for a parabolic axial distribution


appeared consistent with those of Di Prima (1960). All the above theoretical

evidence indicates that the use o f an average axial velocity leads to

underprediction of the critical Ta number by a factor which increases with Re

number for values o f Re number above 80.

Elliott (1973) used the same approach as that o f Krueger and Di Prima (1964),

but with more terms in the algebraic series formulation. His calculations for Re

number up to 200 with a parabolic axial velocity distribution and a linear

approximation to the exact fully developed tangential velocity profile, agreed

with Di Prima's prediction. Table 2.4 summarised the method used by the

previous researchers and the range o f operating parameters they employed to

study the flow instability of Couette flow with axial flow.


Goldstein (1937) Fourier series expansion 0 < Re < 50

p = oChandrasekhar (1961) Bessel function R e -» 0

p = oDi Prima (1960) Algebraic series Re < 200

p = oChandrasekhar ( 1962) Expansion of

Bessel function

R e ^ O

Krueger and Di Prima

(1964)

Fourier series expansion 0 < Re < 40

allp

Hughes and Reid

(1968)

Asymptotic method Re > 200

Elliott (1973) Algebraic series 0 < R e < 100

Table 2.4 Summary of narrow-gap problem in Couette flow with axial flow


2.2.4 Wide-gap problem in Couette flow with a low axial flow

So far we have dealt with the narrow-gap case in Couette-Poiseuille flow only.

The corresponding wide-gap stability problem has received rather less attention

because o f its complex nature and very tedious computations involved. One of

the difficulties in treating the wide-gap problem is that the differential operators

in the eigenvalue problem have variable coefficients in contrast to the constant

coefficient operators that appear in the small gap problem.

Hasoon and Martin (1977) predicted the critical Ta number and critical wave

numbers for axial symmetrical flow. Using both a time-dependent finite-

difference procedure and solution employing the Galerkin method, they

computed results for radius ratios between 0.81 and 0.95 and for Re number up

to 1000. They questioned the use of a parabolic form for the axial velocity

profile in the stability problem. However, their predictions were restricted to

the angular speed ratio equal to zero.

Chung and Astill (1977) treatment were based on linear stability theory plus a

shooting method for the fully developed axial and tangential velocity

distributions; their theory covered initially axisymmetric disturbances only for

radius ratio, a = 0.95 and initially non-axisymmetric disturbances over the

range 0.1 < a < 0.95, Re number up to 300 and cases with co-rotating and

counter-rotation of the cylinders. In this general case, the stability analysis

required two wave numbers: the axial wave number and azimuthal wave

number. The linear stability limit is then found by determining the minimum

on the family of neutral stability curves. In this regard, the minimisation

process used by Chung and Astill (1977) was too difficult to follow. Moreover,

there was no theoretical justification for this assumption so that the treatment of

the problem with a general disturbance must be regarded as incomplete. Some


additional comments on the minimisation process used by Chung and Astill

(1977) were provided by Di Prima and Pridor (1979).

Finally, Takeuchi and Jankowski (1981) and Ng and Turner (1982) studied the

wide-gap case with radius ratios o f 0.5, 0.77 and 0.95. A divergent between

theoretical predictions and experimental data (beginning at Re number as low

as 40) was observed. They suggested that it m i^ t be because of the shifting of

Taylor vortices as Re number increases. Their studies were restricted to Re

number < 100. Table 2.5 summarises the values of the critical Ta number

published by the previous researchers and Table 2.6 summarises the methods

used by the previous researchers and the range o f operating parameters they

employed to study the flow instability of Couette flow.

Re Critical Taylor number at various radius ratio, a

a = 0.95 a = 0.77 a = 0.5

0 1754.86 2056.88 -

1 - - 3101.7

5 1788.78 2096.36 -

10 1891.32 2215.69 3329.5

20 2297.97 2687.72 4039.4

30 - - 5025.1

40 4012.69 4577.4 6274.0

50 - - 7017.8

60 6805.73 6825.28 7224.5

70 - - 7031.5

80 9263.2 8069.78 -

100 11546.0 8809.15 -

Table 2.5 Summary of critical Ta number for different values of radius ratio, a and given values o f Re number. [The results for a - 0.5 are from Takeuchi and Jankowski (1981) and for a = 0.95 and 0.77 from Ng and Turner (1982)]



Hassoon and Martin

(1977)

Finite-different method and

algebraic series

a = 0.81, 0.95

0 < R e < 1000

Chung and Astill

(1977)

Runge-Kutta method 0.1 < a < 0.95

0 < Re < 300

Takeuhi and Jankowski

(1981)

Initial value method a < 0.5

0 < R e < 100

Ng and Turner

(1982)

Compound matrix method a = 0.77, 0.95

0 < R e < 100

Table 2.6 Summary of wide-gap problem in Couette flow with axial flow

2.3 Experimental Studies on Flow Instability of

Newtonian Liquids

2.3.1 Flow visualisation method

Taylor (1923) performed several experiments designed to test the validity of his

theoretical results as part of his well known work on the Couette stability of an

incompressible Newtonian flow. The apparatus employed consisted of an

opaque inner cylinder contained with an outer glass shell; the cylinders could

be rotated in the same or in opposite directions. A coloured dye was injected

into the region near the inner cylinder in order to allow for observation of the

onset o f instability which was characterised by the formation of a cellular-

vortex pattern. Taylor (1923) confined his attention to the case of water, and

found his experimental results were in very good agreement with his theory.


Lewis (1928) extended Taylor's findings to cover a wide range o f Newtonian

liquids. The liquid motion in his study was followed by means of tiny

suspended aluminum particles rather than by dye injection. Thus, the problem

of diffusion of dye into the liquid was avoided, and each experiment could be

repeated several times for the same sample.

Coles (1965) later showed the complex flow regimes occurring beyond the

Taylor-Couette flow regime. In his study of flow between both counter and co-

rotating cylinders Coles (1965) discovered several very distinctive flow

regimes, including double periodic flow, intermittent turbulent bursts and spiral

turbulence as the rotational speed further increased. He also stated that at a

given Ta number there were several distinct stable flow states, depending upon

the Ta number history, i.e., how to reach the final Ta number.

Koschmieder (1979) later found that the wavelength increased with increasing

Ta number only up to Ta/Tac - 1 0 . He also studied turbulent vortex flow

between long concentric cylinders and measured the axial wavelength o f Taylor

vortices by suspending aluminum powder in the liquid. He found that the

wavelength became substantially larger than the critical wavelength of laminar

Taylor vortices.

Andereck et al. (1983 and 1986) reported many reproducible flow states

obtained by systematic variation of inner and outer cylinder speed. They also

observed five new flows regimes occurring in the case o f co-rotating cylinders.

Fluid medium Materials added for visualisation Radius ratio Ta number studied

Taylor (1923) Water Eosin (dye) 0.942 0 <Ta/Iac < 1

Lewis (1928) Xylene,

Nitrobenzene

Suspended particle 0.829 0 <Ta/Tac < 4

Coles (1965) Silicon oil Aluminum powder 0.888 0 < Ta/Tac <100

Koschmieder (1979) Water Aluminum powder 0.896 0 < Ta/Tac <40,000

Andereck et al. (1983 and 1986) Water Polymeric flakes 0.883 0< T a /Iac< 19

Gu and Fahidy

(1985a, b)

Electrolyte

solution (acidic)

Analytical indicators 0.714 0 < Ta/Tac <23

Benjamin (1987a, b) Glycerol Pearl substance 0.615 0.3 < Ta/Tac <3.2

Table 2.7 Suiranaiy o f major experiments on flow visualisation method in Couette flow device


Gu and Fahidy (1985a, b) investigated the changes in the structure of the

vortices with increasing axial velocity by using a visualisation technique. At

small axial flows the individual Taylor cells were inclined and partial

overlapping of cells occurred. With further increase in the axial flow rate, the

cell structure degenerated progressively to a disorderly pattern. At high axial

flow rates the Taylor cells were hardly detectable and the complete

degeneration of Taylor vortices was assumed.

Finally, Benjamin (1987a, b) observed different states in Taylor vortex flow

even in an annulus so short that only 3 to 4 vortices could be accommodated.

Table 2.7 summarised the major experiments on flow visualisation in Couette

flow device.

2.3.2 Flow spectra method

Fenstermacher et al. (1979) first used the velocity power spectra method to

study the transition to turbulence for flow between concentric cylinders. The

liquid velocity was determined from measurements o f the Doppler shift of

scattered laser light. The Doppler shifts were typically ~ 10 Hz while the

characteristic frequencies o f the liquid were -0 .1 to 10 Hz, so measurements of

the Doppler shift in short time intervals yielded essentially the instantaneous

liquid velocity. They showed that different dynamic flow regimes could be

distinguished by examining high-resolution power spectra of a time-dependent

property of the flow. Transitions that were obvious in the power spectra, such

as the broadening of a spectral line or the appearance of a new characteristic

frequency in the flow, could be undetected in a direct inspection of the time

records or flow photographs. Thus power spectra method have become a major

tool for the study of the transition from laminar to turbulent flow.

Fluid medium Methods used for obtaining

power spectra

Radius ratio Ta number studied

Fenstermacher et al. (1979) Water Laser Doppler spectra 0.877 0 < Ta/Tac < 12

Walden and Donnelly (1979) Carbon

tetrachloride

Ion current spectra 0.875 0 < T a /T a c< ll

Cognet and Bouabdallah

(1980 and 1982)

Electrolytic

solution

Electrochemical current spectra 0.909 0 < Ta/Tac < 12

Kataoka (1984) Electrolytic

solution

Velocity gradient fluctuations 0.600 0 < Ta/Tac <160

Gorman and Swinney (1982) Water Photocurrent if scattered light

by platelets

0.883 0 <T a/ Tac <8 0

Table 2.8 Summary o f major experiments on the power spectra method in Couette flow device


Walden and Donnelly (1979) simultaneously studied the transitions in the flow

between concentric cylinders by an entirely different measurement technique.

They measured the ion current between a collector embedded in the outer

cylinder wall and the gold-plated inner cylinder as a function of time. The

records were Fourier-transformed to obtain ion current power spectra, in which

the results were in agreement with Fenstermacher et al. (1979).

Cognet and his coworker (1980 and 1982) measured the radial gradient of the

azimuthal velocity on the outer cylinder wall by using an electrochemical

technique and analysed the velocity-gradient spectra by means of an electronic

spectrum analyser. They found that turbulence regime originated from the

vortex outflow boundaries.

Kataoka et al. (1984) applied similar diffusion-controlled electrolytic reaction

and analysed the power spectra of the velocity-gradient fluctuations on the

outer cylinder wall for explanation o f the dynamical modes of ionic mass

transfer measured on the outer cylinder wall.

Gorman and Swinney (1982) measured the power spectra of the intensity of

light scattered by the fine platelets that aligned with the flow. They discovered

many distinct wave patterns in the doubly periodic flow. Table 2.8 gives a

summary o f the major experiments on power spectra method in Couette flow

device.

2.3.3 Dispersion measurement

Kataoka et al. (1975) first experimentally studied the mixing property of the

axially moving Taylor vortex flow in connection with the application of this

flow system to chemical equipment. They made measurements of the

intermixing over cell boundary between Taylor vortices for very small axial


flow rates by measuring the residence time distribution using two probes placed

on the inner surface o f the outer cylinder. All o f their measurements were

taken in the range of 1 < Ta/Tac <12. And 0 < Re < 23. They showed that the

toroidal vortices motion o f liquid elements caused highly effective radial

mixing within cellular vortices, whereas the cell boundaries prevented liquid

elements from being exchanged over the vortex inflow boundaries. Each pair

of vortices could be regarded as a well-mixed batch vessel, which moved

axially at a constant velocity. The vortices, whose size was approximately

equal to the annular gap, marched through the annulus at a constant velocity

equal to the mean axial velocity. Therefore, it could be considered that all the

liquid elements leaving the annulus had the same residence time in the

apparatus.

Kataoka et al. (1975) showed that Taylor-Couette flow was one of the rare flow

types combining ideal plug flow with ideal stirred tank behaviour. This

publication has become the key reference in the field of Taylor-Couette flow

mixing. Their assumption of non-intermixing vortices however was based upon

qualitative interpretations o f tracer experiments. Nevertheless, it has been

adopted in all subsequent published calculations of reactor productivity in the

laminar regime (Cohen and Maron, 1983).

Kataoka et al. (1977) later attempted to quantitatively investigate the extent to

which Taylor-Couette flow offers this unusual combination of mixing

properties. They made local measurements of mass transfer on the inner wall

of outer stationary cylinder by means of an electrochemical technique. The test

section was far down stream from entry, so that Taylor vortex structure could

be established in laminar axial flow. For a small forced axial flow (0 < Re <

130), toroidal vortices marched through in single file without breaking up. As

the Re number increased gradually, the regular variation of Sherwood number

(associated with the rate of mass exchange) was not only distorted by the


forced axial motion, but also its mean value was reduced greatly. Kataoka et

al. (1977) also showed that during the Taylor-Couette flow regime, the added

axial motion tended to stabilise the circular Couette flow and to delay the initial

formation of Taylor vortices.

Kataoka et al. (1981) identified the existence of a small mass flux between

adjacent vortices for 1 < Ta/Tac < 12 and 0 < Re < 18. They measured the rate

of exchange of liquid elements between the boundaries of vortices by using the

same method as Kataoka et al. (1975). They assumed that each vortex unit

behaved as an ideal stirred tank and quantified the inter-vortex flux with a

localised inter-vortex mass transfer coefficient. They stated that the rate of

circumferential and radial mixing was much faster compared to axial mixing.

Legrand and Coeuret (1986) and Guihard et al. (1989) published the residence

time distribution measurements, for a range of Ta number, from which they

claimed the total absence of vortex-intermixing, thereby confirming the first

conclusion of Kataoka et al. (1975). They assumed that the tracer material

would be bounded by the vortices' boundaries as the radius ratio was smaller

than 0.72. This assumption was also supported by Coles (1965) and Gu and

Fahidy (1986). So that, the circumferential dispersion coefficient De, was

calculated based on the equation of tank-in-series model. They showed the

value was negligibly small that the tracer material would be entirely consumed

in a vortex cell behaved like a closed reactor.

Recently, Pudjiono et al. (1992) restricted their experiments to the laminar flow

regime (very slow or zero rotational flow where Taylor vortices were absent)

and the Taylor-Couette flow regime. All of their measurements were carried in

the range of 0 < Ta/Tac < 2 and 0.4 < Re < 5.5. The residence time distribution

(RTD) measurements were restricted to the outer layers and the dispersion

coefficient was determined from the variance of RTD curves recorded.


Pudjiono et al. (1992) observed that the flow became fairly unstable below the

critical Ta number. When Ta number reached the critical value, Ta , a peak in

the shape of the RTD function was observed due to good radial mixing induced

by the Taylor vortex flow. In their study a minimum dispersion coefficient was

used to characterize the critical Ta number. The results were in agreement with

previous researchers by implying that 'near' plug-flow could be obtained as the

critical Ta number was reached.

Pudjiono and Tavare (1993) further obtained RTD data around the critical Ta

number in the ranges 0 < Ta < 2.5 and 0 < Re < 5.5. They found that

dispersion coefficient increased fi*om a minimum value as the Ta number

increased beyond its critical value. It was suggested that the 'near' plug flow

behaviour occurred only at critical Ta number.

Croockewit et al. (1955) studied the axial dispersion coefficient o f a Newtonian

liquid in a continuous Couette flow device based on the fi*equency response

analysis o f RTD. All o f their measurements were taken at rotational speeds

much higher than the critical Ta number. The sinusoidal frequency and phase

shift o f the electrical conductivity changes of the liquid following the injection

were recorded by using two small conductivity probes placed inside the

Couette flow-device. By comparing the records from both probes under a

given set of conditions, the values o f axial dispersion coefficient were

calculated based on the equations of dispersion model.

Croockewit et al. (1955) showed that the calculated values of dispersion

coefficient were in the range 0.3 x IC to 3.0 x 10^ m s" in their experiments

and these were independent of Re number when the axial flow velocity are low

compared to the tangential velocity of the liquid.


Tam and Swinney (1987) studied experimentally the mass transport in the

turbulent Taylor-Couette flow without axial flow, using a relatively small-scale

apparatus, the annular wide-gap of which ranged from 1.285 to 0.318 cm.

They proposed an axial dispersion model for the analysis o f mass transport and

measured the effective axial diffusion coefficient, i.e. the axial dispersion

coefficient, D, for radius ratios, a ranging from 0.494 to 0.875, and at Re

number ranging from 100 to 1000 times the onset Taylor vortex flow. They

described the dispersion coefficient by power law, D oc Ree . P varied with the

experimental conditions and geometry of the experimental apparatus. Related

torque studies (Lathrop et al., 1992) implied that the exponent p was not be a

fixed value but increased with increasing Ree.

Enokida et al. (1989) also measured axial dispersion coefficients in turbulent

vortex flow for the radius ratios of 0.593 and 0.760 with Ta number ranging

from 30 to 500 times the critical value. They found a slightly different power

law expression, D ~ (Ravg co ) for dispersion in vortex flow without axial flow.

The increase in the dispersion coefficient upon the introduction of axial flow

was roughly proportional to the azimuthal velocity component Ree u.

Moore and Cooney (1995) experimentally studied and mathematically

described the axial dispersion in vortex flow for a wide range of operational

parameters (rotation rate, axial flow rate) and design parameters (reactor

geometry). They found a value of 1.05 for Ree exponent which was slightly

higher than 0.69 to 0.86 exponents found by Tam and Swinney (1987) or

Enokida et al. (1989). Table 2.9 summarises the major experiments on

dispersion measurement in Couette flow device.

Fluid medium Method for measuring dispersion Radius ratio Ta number studied

Kataoka et al. (1975) Glycerol solution.

Sucrose

Conductivity o f ionic tracer

Dissolution of particle

0.750 1 < Ta/Tac < 12

Kataoka et al. (1981) Glycerol solution Conductivity o f ionic tracer 0.600 5 < Ta/Tac < 1000

Legrand and Coeuret (1986)

Guihard (1989)

Potassium

ferricyanide.

Sodium hydroxide

Conductivity of reduced

electrolyte

0.549 ll< T a/T ac< 25

Pudijono et al. (1992, 1993) Diethylene glycol.

Deionised water

Concentration o f dye.

Protein precipitation

0.94 0 < Ta/Tac <2.5

Croockewit et al. (1955) Water Conductivity o f ionic solution 0.114-0.680 5 < Ta/Tac < 123

Tam and Swinney (1987) Distilled water Concentration o f dye 0.494-0.875 50 < Ta/Tac < 1000

Enokida et al. (1989) Deionised water Concentration o f dye 0.593, 0.760 30 < Ta/Tac <500

Moore and Cooney (1995) Water, Glycerol Concentration o f ionic tracer 0.859 4 < Ta/Tac <200

Table 2.9 Summary o f major experiments on dispersion measurement in Couette flow device


2.4 Theoretical Studies on Flow Instability of

Non-Newtonian Liquids

Graebel (1961) studied the Couette stability of non-Newtonian liquids. By

employing the small-gap approximation to simplify the analysis, Graebel found

that the viscosity can strongly influence the results obtained for the critical

Taylor number. Although Graebel's analysis has been performed for the

somewhat unrealistic conditions, it did give some general indication o f what

one might expect for non-Newtonian liquids and showed that the critical Taylor

number depended on the non-Newtonian properties o f the liquid.

Ginn and Denn (1969) and Sun and Denn (1972) considered the general case of

stability of liquids exhibiting weak elastic properties and used the narrow-gap

approximation. Both groups showed the results obtained for one set of

rheological parameter. Their results were in good agreement with the

experimental data carried out for polymer solutions exhibiting shear thinning

behaviour. They also found that the critical Ta number increased or decreased

depending on the value of two dimensionless groups associated with the normal

stress coefficients.

Larson (1990) carried out a linear stability analysis for the case of the narrow-

gap, using the Doi-Edwards model which accounts qualitatively for both

normal stress and shear thinning effects. He recognised that the combination of

elastic and pseudoplastic effects was important for concentrated polymer

solution. The results showed that the shear thinning behaviour described by the

Doi-Edwards equation has a destabilising influence on the flow. Larson (1990)

also obtained a numerical solution for the Taylor-Couette flow for an upper-

convected Maxwell liquid neglecting the inertia.


2.5 Experimental Studies on Flow Instability of

Non-Newtonian Liquids

Giesekus (1972), using polyacrylamide solution, showed that the critical value

of the Ta number at the onset o f Taylor vortices could be smaller or larger than

the Newtonian value depending on polymer concentration. At high polymer

concentration, when elastic effects become dominate, the critical Re number

was found to decrease with the flow Weissenberg number (given as the product

of the liquid's longest relaxation time and the shear rate). However, at high

polymer concentration, solution might exhibit significant shear thinning.

Giesekus's (1972) experiments were generally in agreement with his own

theoretical studies for dilute polymer solution, but showed an unexplained

decrease in the critical Ta number of up to 50% as the polymer concentration

approached 1000 ppm. He stated that both viscoelastic and shear thinning

effects must be accounted for through a suitable constitutive model.

Sinevic et al. (1985) experimentally investigated the Couette flow instability by

means o f a power number as a function of Ta numbers. All the experiments

were performed with carboxymethyl cellulose solutions for radius ratio greater

than 0.6. Sinevic et al. (1985) also defined the Ta number on the basis of the

apparent viscosity value at the inner rotating cylinder. Experimental results

showed that the critical Ta number was dependent on the gap width and on the

flow index, n. In most cases, the critical Ta number was found to be larger

than a Newtonian liquid with similar viscosity but it approached the Newtonian

value asymptotically as the value of the critical Ta number decreased with the

decrease of pseudoplasticity. Tables 2.10 and 2.11 summarise the theoretical

and experimental studies o f non-Newtonian liquids in Couette flow device.

Non-Newtonian model Mathematical method Effect o f non-Newtonian properties

Graebel (1961) Reiner-Rivlin liquids Fourier series transformation Destabilising effect

Ginn and Denn (1969)

Sun and Denn (1972)

Weak elastic liquids Fourier series transformation Destabilising effect

Larson (1990) Doi-Edwards liquids Orthogonal shooting procedure Destabilising effect

Table 2.10 Summary o f theoretical studies o f non-Newtonian liquids in Couette flow device

Fluid medium Methods used Radius ratio Effect o f non-Newtonian properties

Giesekus (1972) Viscoelastic liquids Flow visualisation method 0.89 Stabilised at low concentration

Destabilised at high concentration

Sinevic et al. (1986) CMC solutions Dispersion measurement 0.9, and 0.7 Critical Ta number decreased as

flow index, n increased

Table 2.11 Summary o f experimental studies o f non-Newtonian liquids in Couette flow device


2.6 Summary

Instabilities in flow between concentric rotating cylinders have been studied

extensively since Taylor's classic theoretical and experimental work. However,

most of the theoretical work has been concerned with the Couette problem

without superimposed axial flow because of the mathematical complication that

the presence of axial flow introduces. Although a few researchers have dealt

with the later problem, their results were restricted, e.g. flow in narrow-gap of a

concentric cylinders, flow in a fixed gap width with a small superimposed axial

flow and without the inclusion o f angular speed ratio effect. Recently, much of

the new theoretical research has shifted on the instabilities that occur as the

inner cylinder speed is further increased (Ta > Tac), e.g. the onset o f wavy

Taylor-Couette flow in simple geometry. Therefore, a more general solution of

Couette flow stability problem with axial flow is still incomplete, and requires

further investigation.

The design and scale-up of vortex flow reactors also require a complete

understanding of the transport properties o f Taylor-Couette flow. While some

transport properties, such as mass and heat transfer to the cylinder wall, have

been described in the literature, back-mixing or dispersion in vortex flow

systems has received considerably less attention. Dispersion is an important

factor in reactor design as the extent o f mixing within the vortices greatly

influences the reaction. Mixing in Taylor-Couette flow occurs in

circumferential, radial and axial directions. Kataoka and coworkers (1975)

showed that the rate o f circumferential and radial mixing was fast compared to

axial mixing. Consequently, axial dispersion was an important parameter in

designing vortex flow reactors. Pervious studies o f axial mixing in vortex flow

have focused on operation at high or low rotation speeds, typically without

axial flow and without regard to the effect|of reactor geometry.


Relatively little work has been devoted to the case of non-Newtonian liquids.

One of main reasons is the variation of the apparent viscosity o f liquids in the

cross-section o f the annular gap. The dependence of the predicted flow

behaviour on the choice o f a constitutive model is another difficulty in

attempting to interpret an already complex flow situation. Moreover, the non-

Newtonian constitutive equations often involve rheological parameters such as

flow index and consistency index. Recently, computer-assisted linear stability

analyses have been extended to non-Newtonian liquids, even involving an

analysis o f the possible pattern of the Taylor vortices established after the onset

o f the instability. However, most o f the researches to date have been restricted

to the small-gap geometry for mathematical simplicity. Except for narrow-gap

geometry, one rarely finds the complete solution because o f the complicated

link between stress and strain of the non-Newtonian liquids. It is therefore the

aim of the present investigation to show how this analysis can be extended to

arbitrary gaps, and to discuss the influence of the rheological parameters on the

basic velocity field as well as on the onset of the Taylor-Couette flow.

There are a few experimental investigations of the properties of polymer

solution (i.e. viscoelastic liquids) on the Taylor-Couette instability problem. In

general, these workers have found that viscoelasticity increases the critical Ta

number at which a cellular flow is observed. Thus, the addition of

viscoelasticity stabilises the flow against the formation of Taylor vortices.

None o f the research works cited so far investigated the effect o f Taylor vortex

flow on mass transport (residence time distribution) on non-Newtonian

pseudoplastic liquids. Investigation on the mass transfer characteristic o f non-

Newtonian liquids can lead to an understanding of the fundamental aspects of

this process.

Theory 55

CHAPTER THREE

THEORY

3.1 Introduction

In Section 3.2, the theoretical consideration is extended to a more general

Couette flow problem between concentric rotating cylinders when, in addition

to rotation, a constant axial flow velocity is present. Previous work has shown

that in Couette flow, the streamlines continue to be circular. However, when a

Poiseuille flow is superimposed over the same distribution of rotational

velocities, the streamlines are no longer circular and the flow cannot be treated

as Couette flow. Indeed, as we shall see, the superposition of axial flow over

rotational flow introduces certain essentially new elements into the problem.

Section 3.3 is concerned with the stability of a non-Newtonian liquid in Taylor-

Couette flow which has not so far been taken under consideration. It is shown

that if the extent of non-Newtonian properties is increased, the variation of

apparent viscosity across the annular gap becomes significant in determining

the flow instability of Couette flow. In order to assess the effects of shear-

thinning, commonly exhibited by non-Newtonian liquids, we employ the power

law model to estimate the influence o f rheological properties on the flow

instability in Couette flow.

Accounting for Couette flow problem requires detail consideration of

constitutive equations. In each section the governing equations are set down

before presenting the method of numerical solutions.

Theory 56

3.2 Flow Instability of Newtonian Liquids

in Couette Flow with Axial Flow

3.2.1 Basic flow

Consider an incompressible Newtonian liquid in the presence of an axial

pressure gradient flowing through two infinitely long concentric rotating

cylinders in usual cylindrical co-ordinates r, 0, z (see Figure 3.1).

Outer cylinder

inner cylinder

Figure 3.1 Annulus configuration and co-ordinate system

If we denote the radial, tangential and axial velocity components respectively

by u, V and w, and the pressure by p. The basic flow is given by

u = 0, V = V(r), w = W(r), d p i d z = constant

where

V(r) = Ar + B/r

[3.1]

[3.2]

Theory 57

[3.3]

where A and B are two constants which are related to the angular velocities Qi

and Q2 with which the inner and the outer cylinders are rotated. Thus, if Ri

and R2 (> Ri) are the radii o f the two cylinders, then we have

where

a = R1/R2 and p = Q2/

1-a^[3.4]

[3.5]

The average axial velocity, W , can be obtained by integrating of Eq. [3.3] with

respect tor. Thus:

1 + l - ( R i / R z ) : R2 J ln(R2 /R i)

[3.6]

where p is the kinematics viscosity. The Eq. [3.2] represents a circular Couette

flow while the Eq. [3.6] is the average Poiseuille flow in an annular gap.

3.2.2 Governing equations

Assuming the flow to be axisymmetric, the hydrodynamics equations governing

an incompressible Newtonian liquid in cylindrical polar co-ordinates (r, 0, z)

may be expressed in the following forms: (Bird, Stewart and Lightfoot, 1960)

at "ar az r " arlp âû 1 au aûar

a av av av uv 0 : "5T+U— + w — + — = +v

at ar ÔZ Ta^v 1 av a^v

[3.7a]

[3.7b]

Theory 58

dw dw , , ,at ar dz ~ & lp

aw afp^ ( a^w i aw a^w —-----1 H I J. \«------- 1-------- 1------rar a,2

and, the continuity equation: au u aw .

[3.7c]

[3.7d]

To study the instability o f this steady motion, we disturb the velocity field of

the Couette flow with small disturbances;

e.g. V = V + v' [3.8]

where the primed quantities denote small perturbations on the basic motion. It

is convenient to separate the flow into a mean part and a disturbance part,

where the latter has a zero mean. It is clear that the two parts of the flow are

interdependent through the action o f the Reynolds stress on the mean flow. On

the basis o f linear theory, the disturbance is assumed to be so small that the

effect o f the Reynolds stress on the mean motion can be neglected, in which

case the mean flow is the original laminar flow. Substituting these quantities

into the Navier-Stokes and continuity equations give

r:

z:

1 au' a^u' u*"'2 r ar a_2 _2ar'

a (v + V' + (V+ viu'at ar ÔZ

= V^a^(v + v') i a (v + v') a^cv + v') (v + v')"'------ 1--------r------ 1-------- --------V ar' ar az' r y

+ _ a (w + w ')at ar az a z lp

fa^(w + w') ia (w + w’) a^(w+w')^+ V - + T-------- - +

ar' ar az'

continuity equation: au' u' a (w + w') _

[3.9a]

[3.9b]

[3.9c]

[3.9d]

Theory 59

It is assumed that the disturbance velocities u', V, w’ to be small compared to

the mean velocity V and W, we can neglect their quadratic terms to linearise

those equations. We obtain the linearised Navier Stokes and continuity

equations

r:

0:

z:

o t or T d z,2.,d 1 d v ' d v' v'

— j ■ — - - ■ — ---------

2 r d r ^ 2 _2

A

d r ‘ d z^ r v

1 d w '

continuity equation: du' u’ d w '

[3.10a]

[3.10b]

[3.10c]

[3.10d]

An examination of these equations show that they allow a solution in normal

mode form

[3.11a] [3.11b] [3.11c] [3.1 Id]

where u,v,w and p are functions of r only and t is the dimensionless time. The

two parameters a and s in Eqs. [3.1 la-d] characterise the flow perturbations in

the axial direction. Their values determine the growth rate of the disturbance

and for a given flow rate and geometry they uniquely determine whether the

imposed perturbations will decay or be able to grow exponentially in time

leading to the formation of secondary flow in the form of Taylor vortices. The

axial wave number, a, characterise the form of the disturbance. For a spatially

bounded disturbance, a must be real. In addition, with the wave number

restricted to positive values the imaginary part o f the complex growth rate, s,

determines the stability (s < 0) or instability (s > 0) of the basic flow. The

condition s = 0 corresponds to the case of neutral stability.

Theory 60

Substituting Eqs. [3.1 la-d] into Bqs.[3.10a-d] give:

r:gggi(az St) 3ue‘(az st) 2V j(az-st) _ 1 5 j(az-st)

at 0z r ~ p d ! ^

1 ggi(az-st)^+ V ---------------- 1------------------ 1-------------------------------

. ar: r ar az , 2 ^

— 2V-isu + W iau vr

J(az-st)p dr

‘ I d 2 1 ^ Li(az-st)

for A + i l Adr rJ dr

u =' d f d i] r

dr \dr

a f a n 1 2 1 s lawarlar rJ J.2 “ .2 v v

I dpU H V — -----—

r p ar

D D * - a : + ^ - ^V V

lapu + V = — —

r p ar[3.12a]

where D and D* denote and respectively

0: aveit-^-^) ray i(az-st) ât lar rJ"® az

( 1 a v e ' (^ -^ ) g 2-g i(a z -st) ~î(az-st)^

2 " 7 & & ^2 ^dr-

. ~ T7T. .w f av V 1-isv + Wiav - 1 + — I u ei(az-st):3<v a^ 1 a 2 1

^ g ,2+ rar-" \ 2 î(az-st)

- is v + Wiav - D * Vu = v| d D * -a '

V = D * Yu [3.12b]

Theory 61

z: + ggi(az-st) ^ _ w 5 5 î!^ !L ^ = _lA%^i(az-st)W pe5t dz dz p dz

/ 1 ô2~gi(az-st)^2 r ar azar

a w 1 /-isw + u - ^ + iawW = — iap + v| d * D - a4w

D . D - a : + ^ - ^V V

w - ÏÏDW = ia— [3.12c]

continuity equation:

^ei(az-st) ggi(az-st) ^gi(az-st)+ + : = 0

ar dz

=> f + “ +iaw = 0 dr r

DD * u = -law [3.12d]

To render values o f the dimensionless radial co-ordinate r independent o f gap

width, we substitute a dimensionless length scale x having values of -1/2, 0, 1/2

at R2, Ro, Ri respectively, using the following transformation:

r — Ro 4- (R 2 — R i ) x

n _ (^1 + R2 ) R o - 2

D* = — + s£, dx

( Q 1 + Q 2 )

1(1 + 8X )

[3.13]

[3.14]

[3.15]

[3.16]

[3.17]

[3.18]

Theory 62

G =

a =

2 (R 2 -R i) (R2 + R i )

s( R 2 - R i )^

A, = a(R2 -R l)

—4 2o(R2 — Ri)^ATa =

Re = W(R2 - Rl)

M = Q(r) 2^0 (1 + PXl-a^)

(1 + ôx)

|(p -a 2 ) + (l-p)>p-2j

(1- 8 / 2)

f Æ =w (l-a ^ ) + (l + a^)lna

(l-a^ )lnY + a^(Y -l)lna]

u = QoRoV

2 A (R 2 - R l) '

V = RoQoV

u

w = QoRoV

2 A ( R 2 - R i )'w

QoRopv .

P = — :-----------3-p2A(Rz-Ri)^

[3.19]

[3.20]

[3.21]

[3.22]

[3.23]

[3.24]

[3.25]

[3.26]

[3.27]

[3.28]

[3.29]

[3.30]

Execution of the foregoing operations and the introduction of dimensionless

variables, r-component (Eq. [3.12a]) becomes:

a(R 2 - R l)

=> |dD * -')? + i(a - XFRe)jû - 2M(no^Ro)

=> ^DD*-X^ + i(o - XF Re)]û - MTav = Dp

Û + 2Q(QoRo)v = Da(R 2 - R l)'

a(R2 - Rl)' v = Dp

[3.31a]

Theory 63

0: v [d D *-X ^ + io -iA F R e| [v£îoRo]v = 2Aa(R 2 - R l)

DD * - } ? + i(CT - A,FRe)|v = û [3.31b]

zi | d * D — + ÎG — ÎA.F Rej a(R2 - R l ) ' w

+ a(R 2 -R l)^ DFReû = - a (R 2 -R l)^

[ v2 J L v2 JiXp

=> | d *D-A.2 +i(a-X FR e)Jw = ReDFû + a p


[3.31c]

D*a(R 2 - Rl)'

Û = -iA,a(R 2 - Rl)'

w

=> D * Û = -iAw [3.31d]

Substituting Eq. [3.3 Id] into Eq. [3.31c] to eliminate w and rearranging the

resulting expression leads to

P = D *D -X ^ +i(cr-XFRe)^D »ûl Re(DF)

iX[3.32]

Equation [3.32] is now substituted into Eq. [3.31a] to eliminate p . Thus

DD*-A^ +i(CT-AFRe) û = Ta M V

+d | | d ♦ D - + i(CT - AF Re)J D*Û^

V A V

Re(DF)iX

[3.33]

Theory 64

It is assumed that the concentric rotating cylinders between which the liquid is

contained are of rigid material and that no slip occurs at the surfaces o f contact.

In addition to the relation D ♦ û = -iA,w (Continuity equation), the three pairs of

boundary conditions are expressed as:

û = Dû = v = 0 at X = ±1/2 [3.34]

The homogeneous system o f Eqs. [3.31b] and [3.33] coupled with the boundary

conditions [3.34] define an eigenvalue problem with the eigenvalue Ta. The

flow is unstable or according as there are or are not solutions for which the

imaginary part o f dimensionless growth rate, a, is negative. In the present

study, we consider only the neutrally stable, imaginary part o f a equals to zero.

When the axial Re number equals to zero, we have the classic Taylor (1923)

problem and the eigenvalue problem will reduce to the case of pure Couette

flow which has been discussed in Section 2.2.1. Moreover, as in many

investigations of the linear stability theory, the complicated eigenvalue problem

can be simplified by considering the case when the gap between the cylinders is

small compared to their mean radius. So that, radius ratio, a tends to 1, D*

tends to D and the operator DD* reduces to D appropriate to the narrow-gap

configuration, as following equations, considered by Chandrasekhar (1961)

which has also been considered in Section 2.2.3.

+ i(CT - XFRe)jv = û [3.35]

-X ^ +i(CT-XFRe)Jû = Ta M v

+d | ( d +i(a-W Re)) [3.36]

In our present study, the eigenvalue problem will be solved without

simplification as in Eqs. [3.35] and [3.36] so that the restriction to narrow-gap

is lifted.

Theory 65

3.2.3 Method of solution

The eigenvalue problem defined by Eqs. [3.31b] and [3.33] is difficult to treat

analytically. An obvious choice for a numerical procedure is the Galerkin's

method. The principle o f the Galerkin method is given in Appendix 1.

Basically, this method consists o f expanding û and v in sets of complete

functions (trial function) that satisfy the boundary conditions. The coefficients

in the trial function series are determined by the requirement that the errors in

Eqs. [3.31b] and [3.33] be orthogonal to the expansion functions (weighting

function) for û and v , respectively. This leads to a system o f infinitely many

linear, homogeneous equations for the coefficients in the series. For a non

trivial solution it is necessary that the determinant for the system of equation

vanishes, and this gives a determinantal equation for Taylor number. In

practice only a finite number o f terms are used in the series for û and v , say N,

and this leads to a determinant of size 2N.

According to Galerkin method, there is no restrictions on the form of the trial

function unless they satisfy the boundary conditions (Eq. [3.34]). In our study,

there is a number of possible sets o f complete functions which can be used for

the expansions o f û and v . For example, the set of functions sin(2n-I)7cx and

cos2n7ix (n = I, 2,....) could be used for v , and the set of functions Cn and Sn

tabulated by Harris and Reid (1964) could be used for û . Unfortunately, the

choice of trigonometric functions leads to a prohibitive amount of labour just to

evaluate the entries that appear in the determinantal equation. Indeed it is not

clear that many o f the integrals can even be evaluated in closed form.

For our purposes it is more convenient, to use simple polynomials in x that

satisfy the boundary conditions. Thus we choose

00 00

U = ^anUn and V = ^bnVn [3.37]n=l n=I

Theory 66

where

Un =n-1 and V n = X X

n-1 [3.38]

Here and bn are unknown coefficients which are yet to be determined. One

can now introduce the concept o f an error function by substitution of Eq. [3.37]

in Eqs. [3.31b] and [3.33] respectively. The expression Ei and E? represent

the errors occurring in the domain or on the boundary due to non-satisfaction of

the above equation. The error functions are given by

| d D*-A,^ + i ( a - À F R e ) ] ^ anUn - Ta M ^ bnVnn=l n=l

—d I^ D * D — + i(a — X,FRe)j f — anUn[ V ^ n=l -

R e ( D F ) ^ 1 ^/ , anUn = EliX

[3.39a]n=l

P , N N[DD * — + i(g — A .F R e ) J b n V n ~ anUn = £ 2

n=l n=l

where n = 1, 2,...., N

[3.39b]

In this case the weighting functions are, Um and Çv , where Ç is the appropriate

wei^ting function for DD* given by

1(1 + Ex)

where

2(R2 - R l)

(R 2 + R i)

[3.40]

[3.41]

m = l, 2,..., N

Theory 67

Then, multiplying [3.39a] by Çum and [3.39b] by Çvm for m = 1, 2,...., N, and

integrating from -1/2 to +1/2. The weighted error expressions are

N

n=lan

1/2 1/2 R fi/DF'IJÛm|DD * + i ( c - XFRe)]undx - Jûm — — Un

- 1/2

1/2

- 1/2

- 1/2

— j* ÛmD |d * D — + i ( c — XFRe)j ( —— Un

-b n T a M j^^^^ÛmVndx = 0 [3.42a]

"1/2 ■

E an J^VmUndx — bnn=l --1/2

1/2

J^Vm^DD * -X^ + i(CT - XFRe)jvndx

- 1/2

= 0 [3.42b]

m = 1, 2 ,...., N

The subsequent solution procedure consists of solving the linear algebraic

homogeneous equations [3.42a-b] for given radius ratio, a, angular speed ratio,

P, Reynolds number. Re and increasing N (beginning with N = 1) by equating

the real and imaginary parts o f their determinant to zero to yield a non-trivial

solution. For N = 1, a systematic searching technique is employed which, for

given dimensionless growth rate, a, yields a real positive root for Taylor

number, together with the corresponding value of dimensionless wave number,

X. The minimum Ta number over a range of a is taken as the approximation to

critical Ta number together with the associated values of X and a.

For the large-size determinants (N = 4) the procedure differed somewhat in

that, for given a, P, Re and X, a is so chosen that the value of Ta number for

which the secular determinant vanished is real; a is then varied to determine the

minimum positive real value of critical Ta number which, in turn, is minimised

over all real positive X. All computation are programmed in Mathematica

software as given in Appendix 2.

Theory 68

3.3 Flow Instability of non-Newtonian Liquids

in Couette Flow

3.3.1 Basic flow

Consider an incompressible non-Newtonian liquid flowing through two

infinitely long concentric cylinders in cylindrical co-ordinates r, 0, z (see

Figure 3.1). If we denote the radial, tangential and axial velocity components

respectively by u, v and w, and the pressure by p. The basic flow is given by

u = 0, V = V(r), w = 0, d ^ l d z = constant [3.43]

where

V(r) = Ar + B/r [3.44]

where A and B are two constants which are related to the angular velocities Qi

and Q2 with which the inner and the outer cylinders are rotated. Thus, if Ri

and R2 (> R l) are the radii of the two cylinders, then we have

A = -& -Î^ Q i B = [3.45]

where

a = R1/R2 and P = Q2/ 0% [3.46]

3.3.2 Governing equations

Assuming the flow to be axisymmetric, the hydrodynamics equations governing

an incompressible non-Newtonian liquid in cylindrical co-ordinates may be

expressed in the following forms: (Bird, Stewart and Lightfoot, 1960)

= [3.47a]T dr r dz dr

Theory 69

0: ôw dw dw MW I d ( 2 \ , àzzQdt ' dr ' d z ^ T ) ôrV dz(r^rt)

f dw dw 1 ^ / \ dizz dpp l - â r + " - & 4 + ^ - %

and, continuity equation:

[3.47b]

[3.47c]

[3.47d]

In the particular case o f an incompressible, non-Newtonian flow convenient

relationships have been obtained between the (viscous) stress tensor, Xÿ, and the

shear rate, ÿ ÿ, for the power-law model

'Tij =Tl(Ÿ) ÿij

where

n(ÿ) = K l l r .

n-1~2~

where

Yij = r 505u dw d z ^ dr

d~^^dr (t)]

1 5u d fw

4 # - 4dw 1 dw% +7-æ

5u dw~didw 1 dw

dw

[3.48]

[3.49]

[3.50]

Substituting the non-zero stress component into Eqs.[3.47a-d]

r:

0:

5u 5u 5u V — + u — + w — - -

2 \ 5u dw

5v 5v 5v uvPI aT

1 d 3 d d ' f^ VJ.2 dr ^ dz

dr

[3.51a]

[3.51b]

f dw dw dw i Ar dr

f dû dw [3.51c]

Theory 70

Similar to Section 3.2, small disturbances are added the velocity field of the

Couette flow in order to study the stability of this steady motion

e.g. V = V + v' [3.52]

Substituting these quantities into the Navier-Stokes and continuity equations

gives

r: du', ,au' ,gu' (v+v'yd t ^ d z r

2_ar dr r ( T i + T | ' ) ~d^

2 ,_ ddr [3.53a]

0: d(V + v') ,d(V + v') ,a(V + v') (V + v'X+ U _------ + W _----- -4-d t dr d z

j _ _ a

r: a-+ d z (n + Ti')l^(v+v’) [3.53b]

z: d w ' dw ' dw ' _ ^ r du' dw'^~d^_ r dr . d z ^

[3.53c]


u ’ d ^ ' [3.53d]

It is assumed that the disturbance velocities u', v% w' to be small compared to

the mean velocity V, we can neglect their quadratic terms to linearise those

equations. We obtain the linearised Navier Stokes and continuity equations

r: d i r

d du 2 du! — ^du dr\ 2\xv[

— d du! — d d w ' du' [3.55a]

Theory 71

0:'dw' ,av ,va ar r

1 a r 3 _ a (wYl _ a av'_ " r : ar [3.55b]

aw' a —/"au' aw'^ 1 _/"au' aw'l a — aw'z: p“aT "ar

+ - r 5-^^-arJ_ dz - f [3.55c]

continuity equation: au' u' aw' [3.55d]

An examination of these equations show that they allow a solution in normal

mode form

u'=ue^

w = we p’=pe*“

[3.56a][3.56b][3.56c][3.56d]

where u,v,w and p are functions of r only. Substituting Eqs. [3.56a-d] into

Eqs. [3.55a-d] gives:

r: aue*“ 2Vat ve a aue‘“ 2%

r arag arj iaz 2r|ue‘ ar ar

_ a aiïe^ _ a awe^ ape'ar

- ^ vr = 2t]DDu-i-—Dut] + 2DuDt| - - -H t]u + ia^Dw - Dp [3.57a]r

where

D = —ar

0: 1 ar2 ar ■’" 'It'*"

av Var r + a T|v

Theory 72

"av Var r

u = nu r ar

p[D * V]u = T[|nDD * -a^ j v + n[Dr[]D v

V 2 —~ , "a r " a n "

7 - a T]v + na r r

Va r

[3.57b]

where

'a w e ^ ' a faue‘“ a w e ^ lz: P at "ar az dr ^ + d z

1+ -r

aue*“ awe +

ÔZ drape*

d z

- I_ —awr j iA u + T] — + -^2A, t]w + — d z r

_ _awT11ÀU+T1 — -iÀp

0 = iA,T|Du + iA,uDri + t|DE>w + DwDti + 2A, t|w+ —iA,T)u+ —t|E)w - i?ip [3.57c]


aïïe*“ ue*“ awe*“ ^

— H— + l a w = 0 a r r

DD*u = -iaw [3.57d]

To render values of the dimensionless radial co-ordinate r independent of gap

width, we substitute a dimensionless length scale x having values of -1/2, 0, 1/2

at R2, Ro, Rl respectively, using the following transformation:

r — R q 4- ( R 2 — R l ) x

Ro = (R, ~"R )

[3.58]

[3.59]

Theory 73

D* = — + dx

D = - — dx

Ho =(Q l + 0 2 )

4 =

E =

1(1 + Ex)

2(R2 - R l)

(R 2 + R l)

< R 2-R i )

"k = a(R2 - R l)

-4p^O o(R z — R l) ATa =

M =

Mm

Q(r) 2

(1 + PXl-a^)

(1 + 5x)

( P -a ^ ) + ( l -P )Y -2

(1 -5 /2 )

P _ _n_ _ (2 - nXl - p) j-j2/n-l Pm

n= (1 + Ôx)(1 + 5 / 2 )

n = K K2 .2/n

n—1

R2

Pm =( R 2 -R .)

Rl

K K2

n-1

1R 2 — Rl

K2 =2(0: -ni)R2^'°

n[(R2 /R i) ^ '° - l

[3.60]

[3.61]

[3.62]

[3.63]

[3.64]

[3.65]

[3.66]

[3.67]

[3.68]

[3.69]

[3.70]

[3.71]

[3.72]

[3.73]

[3.74]

[3.75]

Theory 74

u =QoRoV

2 A ( R 2 - R i)'

V = RoQoV

u

w = OoRoV2A(R2-Riy

w

Q o R o p v .P — , . 3 P

2 A ( R 2 - R , f

[3.76]

[3.77]

[3.78]

[3.79]

Execution o f the foregoing operations and the introduction of dimensionless

variables, r-component (Eq. [3.57a]) leads to:

—2f2(OoRo )p^a(R2 - R l)

V = 2PDDÛ + 204DÛP + 2DÛDP - 2Ô ^ ÛP

-l^PÛ + iXPEXv-Dp

MTav = [P(D + Ô4)DÛ - P6^^^Û + iÀPDw] + p[dD + Ô D - ]û

+2DÛDP-2À^PÛ-Dp

MTav = [p(d * D - 52^2 J- ^ iKPD^^ + p[dD * ]û + 2DÛDP - Dp

MTav = [PDD * û + iA,PDw] + p[üD * ]û + 2DÛDP - Dp

MTav = PD[D * û + iÀw] + p[dD * - X ^ ]û + 2DÛDP - Dp

p[dD *-X ^]û +2DÛDP = MTav + Dp [3.80a]

0: p2A gmpa(R2 - R l)

u = PmP(R 2 - R l) '

[nDD*-A.^l [£2„Ro]v + nDgmPD^(R2 — Ri)^

[QoRo]i

P^nDD * -X^ jv + nDpmPD^ v = û [3.80b]

Theory 75

z: 0 = iXPDû + iXûDP + PDDw + DwDP - 2X^Pw + S iXPû + 5ÇPDw - iXp

0 = iXP[D + 8^]û - X^Pw + P[D + SÇ]Dw - X PW + DPD^ + DPiXû - iXp

0 = iXP[D * û + iXw] + p Fd * D - xMw + DP[Dw + iXÛ] - iXp

p [d * D - X |w + DP[Dw + iXû] - iXp = 0 [3.80c]


D* P<mpot(R2 — Rl)"

Û = -iA, M-m

pa(R2 — Ri)w

D * û = -iAw [3.80d]

Substituting Eq. [3.80d] into Eq. [3.80c] to eliminate w and rearranging the

resulting expression leads to

p = i [ 0 * D - X ^ ] D»Û + [D D * Û + Û] [3.81]

Equation [3.81] is now substituted into Eq. [3.80a] to eliminate p . Thus

p [d D*-A^]û +2DÛDP = M Ta v + d J ^ D * D D * û - PD * Û

+ ^ D D » û + û [3.82]

If we assume that the rotating cylinders between which the liquid is contained

are of rigid material and that no slip occurs at the surfaces of contact. In

addition to the relation D*û = -iAw (continuity equation), the three pairs of

boundary conditions are expressed as:

Û = Dû = V = 0 at X = ±1/2 [3.83]

Theory 76

3.3.3 Method of solution

The homogeneous system of Eqs. [3.80b] and [3.82] coupled with the boundary

conditions (Eq. [3.83]) define an eigenvalue problem with the eigenvalue Ta

number. When flow index, n, equals to unity, we have the classic Taylor

(1923) problem which has been discussed in Sections 2.2.1.

The conditions o f the stability limit o f the Couette flow with the power law

viscosity are described by the critical Ta number, Ta , being a function of

parameters: radius ratio, a, angular speed ratio, P and the flow index, n. All

eigenvalues Ta number of the Eqs. [3.80b], [3.82] and [3.83] are real. The set

of the smallest positive values Ta number determined at the given values of

dimensionless wave number, A,, defines the neural curves o f Ta number. The

minimum of the function Ta number at a given a, p, and n determines the

critical Ta number and the dimensionless wave number,

For our purposes it is convenient to use the Galerkin method as a simple

polynomials in x that satisfy the boundary conditions. Thus we choose

00 00

Û = ^anUn and V = ^bnVn [3.84]n=l n=l

where

2 1 _ r ..2 l _.n—1U n= |x"-^ l and V n=lx^-^ |x“- [3.85]

Here a and bn are unknown coefficients which to be determined. One can now

introduce the concept of an error function by substitution o f Eqs. [3.84] in

[3.80b] and [3.82] respectively. The expression Ei and Eo represent the errors

occurring in the domain or on the boundary due to non-satisfaction of the above

equation. The error functions are given by

Theory 11

p [d D * -X ^ ]2 ] a „ U n -T a M ^ b „ v „ - d | - ^ ( d * D -A .^ ) D * ^ a .u .n=l n=I n=l

D P (D D * + 1 )^ 1.2 [3.86a]

n=l

( X 1 \ N N|p[nDD*-X2] + nD^mPD'^|^ b n V n -^ anUn=E2

n=l n=l

[3.86b]

where n = 1, 2,...., N

In this case the weighting functions are, Um and Vm, where % is the appropriate

weighting function for DD* given by

1(1 + 8 X )

where

s =2 ( R 2 - R i ) (R 2 + R i )

[3.87]

[3.88]

m = l , 2,..., N

Then, multiplying [3.86a] by Çum and [3.86b] by for m = 1, 2,...., N, and

integrating from -1/2 to +1/2. The weighted error expressions are

N

n=l

Sin

1/2 1/2 r

J^UmP|DD*-A,^Jundx- J ^UmD — JD*D-A,^J D*Un- 1/2 - 1/2

DP(DD‘ +1) , I f/2 r,Um — Un -b n T a M| ÇUmVn- j ,

- 1/2

dx = 0 [3.89a]

Theory 78

N ' 1/2

z a n J ^ V m U n d x — b on=l - - 1/2

1/2

|4vm {p[nDD * -^^] + nDn„PD*|v„dx- 1/2

= 0

[3.89b]

m = 1, 2 , N.

Equations [3.89a-b] represent a doubly infinite set of linear homogeneous

equations in the variable a and bn for given radius ratio, a, angular speed ratio,

P, flow index, n and N. By setting the determinant of the system equal to zero,

we obtain the required eigenvalue problem for Ta number. All computation are

programmed in Mathematica software as given in Appendix 3.

Materials and Methods 79

CHAPTER FOUR

MATERIALS AND METHODS

4.1 Introduction

The main objective of the present experimental study is to determine the

residence time distribution (RTD) o f a range of Newtonian and non-Newtonian

liquids flowing through the annular gap of a Couette flow device. The

experiments were carried out by using a stimulus response technique based on

an impulse input to characterise the transition from Couette flow to Taylor-

Couette flow regime. A general arrangement of the experimental set-up is

shown in Fig. 4.1. In Section 4.2, equipment of the present study will be

described. It includes concentric cylindrical apparatus, sensors associated with

the Couette flow device and the viscosity measuring device. In Section 4.3, a

rheological properties and the preparation of a range of Newtonian and non-

Newtonian liquids and their corresponding tracer solutions will be described in

details. Finally, the experimental methods associated with the determination of

the RTD and the technique for obtaining experimental data by using the data

acquisition system will be discussed in Section 4.3.

4.2 Equipment

4.2.1 Couette flow device

A Couette flow device of the present study basically consisted of a stationary

outer shell and a rotating inner cylinder with an axial length of 1000 mm. The

diameter of inner cylinder was less than that of the shell giving a horizontal


coaxial annular gap between them. The stationary outer cylinder had a

diameters o f 90 mm and was made o f transparent perspex so as to make

possible visual observations. Two interchangeable stainless steel inner

cylinders with diameter of 76 mm and 60 mm were used to provide gap widths

of 7 mm and 15 mm respectively.

Conductivily ProbeCouette Flow DeviceInjectorDistributor SpeedfTorque Motor

T ransducer \

Conductivity M eterPump

Tank

SpeedController

Figure 4.1 Experimental set-up

In addition, a 21.3 mm diameter cylinder, as shown in Fig. 4.2, assembled with

a number o f blades were used to investigate the effect of geometry on the RTD.

Twenty-four blades were employed and were arrayed into four rows at 90

degree intervals. The length and width of each blade were 54.9 mm and 27.4

mm respectively and the space between two successive blades was 76 mm.

The rotation of the inner cylinder was achieved by joining the inner cylinder to

a 1.1 kW motor (SI GEC Small Machine Ltd., UK) with a variable speed drive


unit, giving speed of rotation in the range from 0 to 1500 rpm with an accuracy

of ±2 rpm. The rotational speed was measured continuously by a shaft-

mounted speed/torque transducer (2400AB Series, EEL Ltd., UK). The device

consisted of a miniature light source and photo transistor, housed within the

speed/torque transducer, which responded to light reflection from the incident

light beam onto the white markings placed on the transducer shaft.

Stain less Steel Shaft

21.3 mm

76 mm27.4 mm

Six Blades in each row

54.9 mm

Figure 4.2 Inner rotating shaft (with blades)

The working liquid entered the annular space between the shell and the rotating

cylinder at a constant axial flow rate, by using a peristaltic pump (HR single

phase, Watson-Marlow Ltd., UK). The pump was connected to a storage tank,

through four 9 mm diameter inlet holes distributor at one end (see Figure 4.1).

The liquid left through four 9 mm diameter outlet holes drilled in the wall o f

the outer cylinder at the exit end.


4.2.2 Conductivity measuring device

The test section had one injection point and two conductivity probe points

which were used to measure the dispersion of an impulse, which was injected

using a manual injector. The two conductivity probes were constructed from a

5 mm diameter glass rod having two 0.315 mm diameter stainless steel

electrodes mounted 2.5 mm apart as shown in Fig. 4.3. The overall length of

each probe was 50 mm. To permit the insertion of conductivity probes into the

annular gap, a number of measuring stations were provided on the stationary

outer shell. Each probe was fixed in position by using a sealing ring so that the

working liquid would not spill out from the measuring stations. The electrodes

could be lowered at the middle o f the annular gap, depending on the gap width,

giving a good contact with the working liquid.

To Conductivity Meter

Stainless Steel Electrodes Clip

50 mmG lass Rod

Yr

Figure 4.3 Conductivity probe

The injection technique involved an impulse of tracer into the feed stream and

followed by measuring the concentration of the tracer in the product stream as a

fimction of time. The volume of the tracer injected was 1.5 ml which was

about 1200 and 2300 times smaller than the volume of the liquid in the annular

space corresponding to the diameter of inner rotating cylinder. Additionally,

the injection time was of the order of a few seconds (0.5 s - 1.5 s) which was

sufficiently short for the injection to be considered as an impulse input.


In order to provide an injection without leakage as the injector was removed

from its port, a small circular piece of silicon "self sealing" compound was

placed inside the injection station.

Two identical conductivity meters (CM35, WPA Co. Ltd., UK) were used to

detect the conductivity of tracer in the liquid passing through. These portable

meters were designed for routine laboratory and research measurements. Their

operating range was suitable for direct readings of conductivity between O.lp

siemen and Ip siemen and have an accuracy of ±3%. This allowed a wide

range of readings to be taken with a single cell. The conductivity measurement

versus time data were plotted and stored in a data-acquisition system for

subsequent analysis.

4.2.3 Viscosity measuring device

A standard coaxial cylinder (Contraves Rheomat 115, Contraves GmbH,

Germany) viscometer was used for the rheological measurements. The

Contraves has been used for many years as a research viscometer because of an

extensive shear rate range, high accuracy. Figure 4.4 shows an exploded view

of the sample holder from the Contraves viscometer.

The measuring bob was driven by an electromotor. The braking torque exerted

on the bob was measured in the measuring head of Rheomat 115 and indicated

on the control panel. The (shearing) speed and the shear stress were used to

obtain the rheological behaviour.

Principally, the data collected during rheological measurements were placed on

a spreadsheet for conversion from torque readings (N.m) and rotational speeds

(rpm) to the more useful rheological measurement units, i.e. the shear rates

(1/s) and shear stresses (Pa). The spreadsheet used was Lotus Corporation's


Lotus 123. Details for the conversion of the (Contraves) parameters to the

characteristic rheological parameters were supplied in the Contraves Rheomat

115 operating instructions. In general, a rheological characterisation using the

Contraves took about thirty minutes.

rW ater Outlet ■

W ater Inlet g

Stand

nr inm i i i i i inT TÎ i---

Motor Gearbox

Therm om eter Connecting S leeve

Coolant Cham ber

Cap nut for fas ten ing m easuring sy s te m

Ball Head

Fill Line

M easuring Boh

M easuring Cup

Sam ple Holder Bottom Lock Nut

Figure 4.4 Viscometer


4.3 Materials

4.3.1 Newtonian liquids

For the Newtonian liquids, three concentration (45%, 64% and 85%) of

glycerol-water solutions (GPR, BDH Ltd., UK) were chosen. They were

prepared by adding a weighted amount of glycerol into a known amount of

deionised water. The volume o f sample required for the Couette device was

approximately 25 litres. The densities o f the solutions were measured by

hydrometer whereas the viscosities were measured by the viscometer. Besides,

viscosity measurements were made before and after each experiment by taking

sample taken directly from the Couette flow device.

Figure 4.5 presents the results o f the viscosity measurements obtained for

different concentration of glycerol-water solutions; the shear stress has been

plotted against the shear rate at fixed values o f temperature and composition.

The slope o f this curve is evidently constant showing a typical rheological

behaviour o f Newtonian liquids. Table 4.1 summarises their properties.

Glycerol conc. (%wt) Temperature (°C) Density (kg/m ) Viscosity (Pa.s)

45 20 1112.8 4.7

64 20 1164.8 14.4

85 20 1221.8 113

Table 4.1 The concentration and rheology of glycerol-water solutions


(0û .

IIs zCO

90000

85% glycerol-solution



80000

70000

60000Viscosity =113 Pa s

50000

40000 Viscosity =14.4 Pa s

30000

Viscosity =4.7 Pa s20000

10000

0500 1000 1500 2000 2500 3000 3500 40000

Shear rate (s' )

Figure 4.5 Shear stress-shear rate plot for different concentrationof Newtonian glycerol-water solutions


4.3.1 non-Newtonian liquids

The following is a brief summary of some of the unusual effects exhibited by

non-Newtonian liquids. It is presented in order to acquaint the reader with the

type of material being considered and also to establish some nomenclature.

For most practical purposes we can regard rheology as the study of liquids that

do not obey Newton's law o f viscosity. This law says in effect that:

• the only stress generated when a liquid flows - ignoring inertia - is the

shear stress;

• this shear stress is directly proportional to the shear rate, and thus the

viscosity (the shear stress divided by the shear rate) is constant;

• shear stress appears and disappears instantaneously when the flow

starts or stops.

The viscosity p is then given by

T = p y [4.1]

where x and ÿ are the shear stress and shear rate respectively. Figure 4.6

indicates that a plot o f shear stress against shear rate is a straight line through

the origin with slope equal to the viscosity of a Newtonian liquid, p. Most

liquids of simple structure, composed of relatively simple molecules in a single

phase, behave as Newtonian liquids, e.g. water.

However, in non-Newtonian liquids stresses other than the shear stress can

appear. These, together with the shear stress, are rarely linearly related to the

shear rate, and they neither appear nor disappear instantaneously on start-up or

cessation o f flow.


Bin^am Plastic

PseudoplasticShear stress,!

Slope = |lp

Newtonian

DilatantSlope =

Shear rate, T

Figure 4.6 Classes o f non-Newtonian behaviour

Log!

X

Power Law Region

00

Logÿ

Figure 4.7 Typical logarithmic plot of a non-Newtonian liquid


General rheological equation o f non-Newtonian liquid is expressed as

T = f (y ) [4.2]

This equation implies that the shear stress at any point, the liquid is a function

o f the shear rate at that point. The behaviour depends on the nature o f the

liquid. Three o f the most common rheological non-Newtonian liquids are

a) Binÿiam plastic liquids

b) Pseudoplastic liquids

c) Dilatant liquids

A Bingham plastic liquid is characterised by a flow curve which is a straight

line having an intercept, Xy, on the shear stress axis as shown in Fig. 4.6. This

intercept is known as the yield stress, which is the stress that must be exceeded

before flow starts. The equations o f the flow curve for stresses above Xy is

x-Xy = Ppÿ; x>Xy [4.3]

where Pp, the plastic viscosity, is the slope of the flow curve. Common

examples o f Bingham plastic liquids are slurries, drilling muds, oil paints, tooth

paste, and sewage sludge.

The pseudoplastic liquids are the subdivision into which the majority of non-

Newtonian liquids fall. Liquids in this category show no yield values and their

flow curves indicate the ratio of shear stress to the rate o f shear, apparent

viscosity, pa, decrease with shear rate, and the flow curves become linear only

at veiy high rates o f shear. The logarithmic plot o f shear stress versus shear

rate for pseudoplastic liquids is often linear, with a slope between zero and

unity.


A typical viscosity curve for non-Newtonian liquid is shown in Fig. 4.7. As

can be seen from this figure, three distinct regions are present. At very low

rates of shear the viscosity approaches Po the "zero-shear viscosity". For

extremely high shear rates, the "upper limit viscosity", pœ, is approached. The

intermediate region is characterised by a viscosity which decreases with shear

rate, a phenomenon referred to as pseudoplastic liquid. The familiar "power

law" viscosity model is frequently found to apply in this region. An empirical

functional relation is given by

t =Ky “ [4.4]

where consistency index, K, corresponds to the viscosity o f the liquid; flow

index, n, is the power law describes three flow models.

1) Pseudoplastic, n < 1

2) Newtonian, n = 1

3) Dilatant, n > 1

Examples of pseudoplastic liquids include melts (rubbers, cellulose acetate),

mayonnaise, gelatine, clay, milk, blood and liquid cement.

Dilatant liquids display a rheological behaviour opposite to that of

pseudoplastic, in that the apparent viscosity increases with increasing shear

rate, but similar to pseudoplastic liquids, they show no yield stress. Equation

[4.4] above for pseudoplastic liquids is also applicable to dilatant liquids, but

the flow behaviour index, n, is greater than unity. Concentrated solutions of

sugar in water and aqueous suspensions of rice starch come in this category.

For the present study, carboxylmethyl cellulose (CMC) solutions (GPR, BDH

Ltd., UK) and xanthan gum solutions (GPR, BDH Ltd., UK) were chosen as

they are commonly used in the food, pharmaceutical, and agricultural industries


where they are used for thickening, rheological control, emulsion stabilisation,

and water-loss control. Their physical properties are given in details in Kirk-

Othmer (1979a, b). These non-Newtonian solutions were prepared by

dissolving the correct amount of the powder in deionised water followed by

mixing. The resulting solutions had exhibited pseudoplasticity, i.e., the

viscosity decreases as the shear rate increases.

Figures 4.8 and 4.10 present the results of the viscosity measurements of

carboxymethyl cellulose solution and xanthan gum solution respectively; the

data have been plotted as shear stress against the shear rate at constant values

of temperature and concentration. It is seen from these graphs that the non-

Newtonian aqueous solutions exhibited shear-thinning properties. Figures 4.9

and 4.11 show their logarithmic plot for non-Newtonian properties

measurement which based on Eq. [4.4].

Figure 4.9 indicates that over the range o f CMC concentrations studied the flow

behaviour of the solutions remained shear thinning. The lines plotted in Fig.

4.9 are reasonably straight indicating that the value of flow index, n, (i.e. slope

of the curves) is consistent over the range of shear rate studied. As seen from

the plots the value of the flow index, n increases gradually with an increase in

CMC concentration which indicates that the apparent viscosity of the solution

increases with increasing CMC concentration.

Figure 4.11, however, shows that for xanthan gum solutions, the slope of the

curves in reasonably constant over the shear rate range investigated provided

xanthan gum concentration is below about 0.3 (wt%). Above this value the

slope changes, i.e. the flow index, n, is shear rate dependent. However, it is

still possible to approximate the non-Newtonian behaviour by using the power

law shear thinning model over a narrow range of shear rate. For the purpose of

this thesis, the average shear rate in the gap of the two rotating cylinders was in


the range o f 286 to 1509 s"\ Over the same shear rate range, the Fig. 4.11

indicates a constant value of n which increases with an increase in xanthan gum

concentration.

Figures 4.12 and 4.13 show that with different concentrations of non-

Newtonian medium powder dissolved in deionised water, a range of solutions

with different flow behaviour are obtained. A computer program was written to

determine the consistency coefficient K and the flow index, n of the power law

model fi’om the velocity - torque data obtained fi*om the Rheomat viscometer

(see Section 4.2.3).

The densities and rheological properties o f CMC solutions and xanthan gum

solutions used are shown in Tables 4.2 and 4.3 respectively.


100000

90000

0.9 wt%80000

70000

S . 60000CO(p_ 50000CO

COQ).c 400000.3 m oCO

30000

200000.1 m o

10000

0 250 500 750 1000 1250 1500 1750

Shear rate (s‘ )

Figure 4.8 Shear stress-shear rate plot for different concentration ofnon-Newtonian carboxymethyl cellulose solutions


100000 1

0.9 wt%10000

0.4 wt°/o

1000 wt°410000100 .

Shear rate (s )

Figure 4.9 Logarithmic shear stress-shear rate plot for differentconcentration of non-Newtonian carboxymethylcellulose solutions


50000

0.9 wt%45000

40000

35000

30000 0.7 wt%

250000.5 wt% 0.4 wt%200000.3 wt%

15000

0.1 wt%10000

5000

400 800 1200 Shear rate (s' )

1600 2000

Figure 4.10 Shear stress-shear rate plot for different concentrationof non-Newtonian xanthan gum solutions


100000

0.9 wt£ 10000

0.7 wt

0.6 wt 0.5 wt

0.4 wt

0.1 wt

1000 J10000100

Shear rate (s' )

Figure 4.11 Logarithmic shear stress-shear rate plot for differentconcentration of non-Newtonian xanthangum solutions


o

I

0.7

0.65 Flow index

0.6

0.55

0.5

0.45

0.4

0.35

0.3COd

CDdd

COd

G)d

CNdo

- 5

-- 4

0

Concentration (wt%)

CO

(0Û.

0)■ Dc

ICOcoo

Figure 4.12 Concentration and rheology of non-Newtonian carboxymethyl cellulose solutions


0.5

0.45 Flow index

-- 40.4I

c

■o 0.35Ç

ILi.

0.3 -

0.25

0.2 4O)d

COd

I D

dCD

dood

CNJ

d do

CO

COÛ.

0)"OÇ

ICOcoO

Concentralon (wt%)

Figure 4.13 Concentration and rheology of non-Newtonian xanthan gum solutions


Carboxymethyl cellulose concentration (%wt)

Density, p (kg/m )

Flow index, n(-)

Consistency index, K (Pa.s")

0.1 999 0.653 0.218

0.3 998 0.569 0.625

0.4 999 0.535 0.955

0.5 999 0.510 1.472

0.6 998 0.487 2.051

0.7 999 0.476 2.849

0.9 997 0.457 5.373

Table 4.2 The concentration and rheology of carboxymethyl cellulose (CMC) solutions

Xanthan gum concentration (%wt)

Density, p (kg/m )

Flow index, n(-)

Consistency index, K (Pa.s")

0.1 998 0.493 0.268

0.3 997 0.385 0.685

0.4 999 0.342 1.105

0.5 998 0.307 1.625

0.6 998 0.285 2.289

0.7 997 0.268 3.146

0.9 997 0.254 5.598

Table 4.3 The concentration and rheology of xanthan gum solutions


4.3.3 Tracer solutions

Two different type of tracers can be used to measure the residence time

distribution (RTD), either a dye or an electrolyte. Dye has the advantage of

providing a visible marker o f its presence. However, it generally produces less

accurate results because the colorimetric detectors tends to give a non-linear

response at low concentration. This may cause a problem in defining the tail of

the distribution. Electrolyte tracer technique is preferred in the present study

because the conductivity meters can be used to detect the presence of minute

amounts o f electrolyte which enables accurate determination of the

conductance o f the liquid. Wen and Fan (1975) have given an excellent

account on the choice o f electrolyte tracers used by many previous researchers.

Strictly speaking, the measured distribution refers to the tracer particles

themselves, but it is usually assumed that the measurement also reflects the

RTD of the liquid stream. The tracer should have physical properties similar to

those of the working liquid. Ideally, the background concentration of the tracer

should be zero, and the detector should give a response that is linear with

concentration and easily detectable and measurable at all concentrations as

shown in Fig. 4.14.

The tracer, electrolyte, used in the RTD experiments is KCl solution (GPR,

BDH Ltd., UK). The tracer solution (1.5M) was prepared by dissolving KCl in

the corresponding Newtonian and non-Newtonian liquids. As the tracer is

injected into the annular gap, it readily mixed with the working liquid without

altering the velocity profile along the annular gap.

The amount of tracer injected in an experiment is an important consideration.

If it is too small, the number of tracer particles is not sufficient to provide the


800

700 -

600 --

500 -

"O

400 --

3000.2

Concentration of KOI solution (mol/litre)

0.4 0.80.6

Figure 4.14 Conductivity vs concentration o f KCl solution


16 1

14 -□ A

ooco

Tracerweight

2 o 3 ml A 2.5 ml o 2 ml □ 1.5 ml

1c8

CO

$coCOc0)Eb

1 AO O

0.8 0.9 1 1.2 1.31.1

Dimensionless time, 0 (-)

Figure 4.15 The influence of the amount of tracer injected on the dimensionless RTD


desired accuracy for analysis. If it is too large, it may generate "pseudo" flow

which can seriously distort the true flow pattern. The effect o f the amount of

tracer used on the dimensionless residence time distribution is shown in Fig.

4.15, which gives results for injected impulse o f 1.5, 2, 2.5 and 3.0 ml. It is

evident that the dimensionless outlet tracer concentration was not affected

significantly by the amount of tracer injected. Based on this result, 1.5 ml o f

tracer would be used in all experiments.

4.4 Methods

4.4.1 Experimental procedures

In every experiments, steady-state flow conditions were assured prior to the

introduction of the tracer by running liquid medium through the annular gap

until the discharge flow rate remained steady for several sampling intervals.

The conductivity probes were inserted in the measuring stations and were

connected to the conductivity meters. The electrolytic tracer was rapidly

injected manually with a hypodermic syringe and the recording of tracer

concentrations at the two stations were initiated automatically and

simultaneously. In all cases the injected volume was less than one-thousandth

of the volume o f the liquid in the gap. Therefore, the effect of the tracer upon

the flow pattern was ignored. Moreover, it was assumed that the tracer

particles was injected over a sufficient small time interval (approximately one

second) that the idealisation of an instantaneous pulse (impulse) stimulus was

suitable.

With the liquid flowing through the annular gap, conductivity probes then

detected the tracer as it passed by the electrodes. The change of the

conductivity with time was recorded at each station using the data acquisition


system. The optimum sampling period and frequency were found to be

dependent on the volumetric flow so that enough samples were taken to

describe the conductivity signal.

Finally, the above procedure was repeated by setting either a different

rotational speed or an axial flow velocity. Each tracer impulse, added to the

liquid, increased the "background" conductivity. In order to compensate for

this effect and maintain high cell sensitivity, a conductivity meter was

employed which had provision for using a compensating resistance.

4.4.2 Determination of RTD

The transport characteristic o f a liquid in a continuous reactor was assessed

using the tracer stimulus-response techniques which have been employed

successfully for many years for the description of liquid flow process systems.

In this treatment, the concepts and terminology originally developed by

Danckwerts (1953) were adopted. With the stimulus-response method, it was

assumed that steady-state flow conditions prevailed and that neither the

injection of the tracer stimulus in the inlet nor the measurement of the response

at the outlet perturbed this steady state. The response to an impulse input of

tracer gave the RTD directly. The dimensionless response curve for an impulse

was called the Ce-curve which is defined as

= ^ - - t [4.5]S Ci At

It is often convenient to analyse material transport behaviour in terms of certain

parameter of the RTD. The two most useful are the mean, t , and variance,

of the RTD of the response curve.


Z t C A t

t = [4.6]Ê C At

00.2Z r c At

------------- 1" [4.7]Z C At0

4.4.3 Data acquisition system

The electrical conductivity signal produced by the conductivity meters and the

rotational speed deduced by the speed/torque transducer with respect to time

are measured in the present study. These measurements were directed into the

data acquisition system shown in Fig. 4.16. This essentially consisted o f a

measurement and control device for data storage and processing (Keithley

Model 575 measurement and control system, KDAC 500/1 data acquisition and

control software) and a computer with a BASICA program (see Appendix 4).

The signals from conductivity meters and transducer were converted into digital

signals via an analogue/digital converter which was connected to a clock

generator allowing the operator to monitor the frequency of the samples

recorded. The sampling frequency could be adjusted by a variable dial,

attached to the analogue/digital clock, from 0 - 1,000 samples per second. The

sampling period was set via the main menu of the computer data acquisition

program. The optimum sampling period and frequency were determined by

preliminary trial experiments such that enough samples are taken with

sufficient frequency to completely describe a given fluctuating conductivity

signal. The BASICA programs enabled the operator to (a) instantaneously


display the signal output on the monitor, (b) stored and/or printed the data, (c)

converted the data for various statistical purposes.

A conductivity meter, with an accuracy of 1% full scale, was used to detect the

concentration o f tracer solution in the liquid passing by the electrodes. The

transmitter's mA signal was converted to mV. The recorded data were digitised

and the concentration versus time data were stored in computer flow for

subsequent analysis (see Figure 4.16).

Speed/Torque transducer

Rotary flow-through device||^^J^^9

Motor

Outlet

Couette flow device andConductivity meters

Conductivitymeter III Measurement and

Control systemData acquisition system

Computer for Screen monitor and Printer

Figure 4.16 Data acquisition system

Theoretical Results and Discussion 107

CHAPTER FIVE

THEORETICAL RESULTS AND DISCUSSION

5.1 Introduction

The purpose o f this chapter is to present the theoretical results o f the eigenvalue

problems which govern various modes o f flow instability of the steady Couette

flow subject to perturbations o f the form of Taylor vortices. Consideration is

given to the cases o f both Newtonian and non-Newtonian liquids.

In Section 5.2, a numerical solution to the classical flow stability o f pure

Couette flow, in which the axial Reynolds number equals to zero, is briefly

discussed. The results will be compared with the previous works in order to

assess the validity o f the present numerical method derived in Chapter 3.

Engineers often encounter the axial flow in the annulus between the rotating

cylinders. When the forced axial flow is superimposed on the Couette flow,

the stability o f liquid motion will be affected. In Section 5.3, the problem o f

flow instability o f such a flow is discussed.

The instability o f the Couette flow is also critically affected by rheological

properties o f the non-Newtonian liquids. In Section 5.4 the dependence o f the

stability limit on shear thinning behaviour of non-Newtonian material is

investigated theoretically for the case of purely viscous liquid with the power

law viscosity and the wide gap geometry.

All the theoretical results shown in this chapter were obtained by using the

Galerkin method (Appendix 1) for both Newtonian and non-Newtonian liquids

respectively. Computations were carried out by using the Mathematica

software which are presented in Appendix 2 and 3 respectively.


5.2 Newtonian Liquids in Couette-Flow

In this section, the numerical results of the investigation are presented.

Consideration is given first to the classical Taylor problem, where the effect of

axial flow is neglected (Re = 0). Thus, the disturbance equations (Eqs.[3.35]

and [3.36]) describing this situation are simplified as follows:

[DD*-X^+ic]v = û [5.1a]

[DD*-X^|DD*-A.2+ia]û = Ta M v [5.1b]

Equations [5.1a-b] are solved subject to the boundary conditions

û = Dû = v = 0 at X = ±1/2 [5.2]

5.2.1 Neutral curve

The homogeneous system o f Eqs. [5.1a-b] coupled with the boundary

conditions Eq. [5.2] lead to an eigenvalue problem of the form

F(a, P, 1, a. Ta) = 0 [5.3]

where the parameter a describes the geometry, the parameter p and Ta describe

the basic flow and the parameter X is the dimensionless wave number of the

disturbance in the axial direction; A, is 7i times the reciprocal of the height o f a

toroidal cell. The parameter a is in general complex; its real part, Gr» describes

the growth rate o f the disturbance and its imaginary part, Ci, is the frequency of

oscillation. If O i - 0, there is no oscillation and the instability is called

stationary; otherwise it is unstable. If Gr > 0, the disturbance tends to grow in

time and the base Couette flow is unstable to a disturbance resulting in the form

of a secondary motion (i.e. Taylor-Couette flow). If Gr < 0, the basic flow is

stable to the disturbance. Gr = 0 constitutes a condition of neutral stability.


In the present study, only the marginal stability is treated in which Gr and G{ are

equal to zero although the method to be described can be adapted to the study

o f the unstable modes as well. Then, the Eqs. [5.1a-b] become

[dD*-A, ]v = û [5.4a]

[DD*-X, ]^û = Ta M V [5.4b]

For a given a and P, the eigenvalue Ta number obtained from this relation,

together with the corresponding values of X,, define the conditions under which

instability will first occur. Ta number can have a sequence of possible

determinate values corresponding to each value of X. This set o f points defines

a neutral curve in the Ta-X plane as shown in Fig. 5.1. The critical Ta number

for the onset o f the instability (for a given, a and P) will be given by the

minimum point on the neutral curve. Below which all disturbances are

damped, and above which the corresponding disturbances will grow. Figure

5.2 displays Ta number as a fimction of dimensionless wave number for

different value o f radius ratios, a. It shows that the determination of the neutral

curve is dependent on the operation parameters such as a and p.

In the simplest case often of interest in applications and experiments, the outer

cylinder is chosen to be at rest and the annular gap between the concentric

rotating cylinders is small (a -> 1). This flow case was employed by Taylor

(1923) in his formulation of the stability problem and for which the following

expression for the critical Ta number

where

[5.5]V

nV(Ri + R,) [5.6]2P(R2 - R i ) ' R ]


3300

3100

2900

2700 - Neutral curve, a = 0

f 2500 -

Unstablec 2300 -

(0I - 2100

Stable1900 -

Critical Taylor num ber, Tac

1700

Crrtical dim ensionless 'wave num ber, 7

1500 J

Dimensionless wave number, X (-)

Figure 5.1 Typical neutral curve in the Ta - 1 plane


2150a = 0.9

2100 a = 0.925

a = 0.952050

a = 0.9752000

a = 0.99

1950

1900 -

1850

1800

1750

1700

1650


Figure 5.2 Neutral curve in the Ta-X plane at different radius ratios


where

P = 0.057l( l - 0.652 — + 0.00056( 1 - 0.652-1

[5.7]

According to our plot in Fig. 5.1, the critical Ta number is 1697 (the minimum

point on the curve, a = 0.99) which agrees very well with the predicted value

(Taylor, 1923) o f 1698 at a = 0.99, based on Eqs [5.3-7].

In order to illustrate the accuracy of the present numerical procedure. Table

5.1, which contains a comparison of the current findings with those of several

other authors for the Newtonian case, has been prepared. It shows that their

numerical calculations were in agreement with each other for comparable

situations.

Critical Taylor numberTaylor (1923) 1698Chandrasekhar (1953) 1708Di Prima (1955) 1695Mekyn (1961) 1712Becker and Kaye (1962) 1707.8Kurzweg (1963) 1750Harris and Reid (1964) 1700Present Study 1697

Table 5.1 Comparison of Tac value in narrow gap geometry

5.2.2 Convergence of numerical results

As mentioned earlier, the Galerkin method is used to solve numerically the

eigenvalue problem. It is a powerful tool for finding approximation solution

and yields better results compared with other approximation methods such as


method o f moment, collocation method, subdomain method and least square

method. The theory of this method is given Appendix 1. Basically, this

method consists of expansion of û and v from Eq. [5.4a-b] in complete sets o f

series functions (trial function) satisfying the boundary conditions. The

coefficients in the trial function are determined by the requirement that the

residual function be orthogonal to the expansion functions for û and v

respectively. This leads to a system o f infinitely many linear, homogeneous

equations for the coefficients in the series. For a non-trivial solution it is

necessary that the determinant o f the system o f equations vanish, and this gives

a determinant equation for Taylor number. The convergence of the numerical

results can always be improved by increasing the number of (N) functions used.

In practice only a finite number of terms are used in the series for û and v , say

N, and this leads to a determinant of size 2N. The evaluation of the

determinant programmed in the Mathematica software (Appendix 2 and 3).

For the present study, computations have been performed for increasing values

o fN until two successive values o f Taylor number are found that differ by less

than 1% (usually N = 3 or 4) as shown in Fig. 5.3. The instability problem

were carried out for a range of values o f radius ratio, a. For a = 0.995, and p =

0 the results for N = 4 are within 0.2% of the "the exact results" given by

Chandrasekhar (1961). Indeed even with N = 2, the error is within 5%. If we

bear in mind that the analytical and numerical work using the Galerkin method

with N = 2 is much less than that for the expansion procedure o f Chandrasekhar

(1961). For a = 0.5 and N = 4 the maximum difference between the present

results and those of Chandrasekhar and Elbert (1963) occurs at a = 0.5 is about

2%. With decreasing a (wider-gap geometry) it becomes more difficult to

approximate the eigenfunction, so that a < 0.2 it was not possible to determine

Ta number accurately without taking more terms in the series for û and v . By

comparing the results for N = 2, 3 and 4 it is estimated that all the results given

for N = 4 are generally correct within 2%.


3300

3100

2900

3 2700

fe 2500JQIc 2300L _

o>%CDK 2100

1900

1700

1500

a=0.995(3=0

1780 1N = 1

N = 4

1 2 3 4 5Dimensionless wave number, X (-)

3.3

Figure 5.3 Convergence of neutral curve


5.2.3 The effect of radius ratio on the critical Tavlor number

Figure 5.4 shows the effect of radius ratio on the onset o f flow instability,

expressed as critical Taylor number, Ta , in Couette flow at angular speed ratio,

P = 0. The values o f Tac were obtained from the neutral curves o f Taylor

number - dimensionless wave number plots for different radius ratios. The

figure shows that the critical Ta number can be significantly affected by the

ratio o f the radii o f the two cylinders. It is found that stable state o f the laminar

Couette flow is achieved by having a large gap width (i.e. small radius ratio).

Astill (1964) studied the development o f Taylor vortices between concentric

cylinders with the inner cylinder rotating. He found the flow instability began

as a series o f oscillating waves occurring first near the wall o f the rotating inner

cylinder. In the case o f a wide gap, a high Taylor number is needed for the

vortices to spread to the outer wall. This finding was supported later by Coney

and Simmers (1979) and Pfitzer and Beer (1992).

Figure 5.4 shows that the destabilisation of flow is very sensitive to changes in

a from 0.1 to 0.2. The destabilisation effect of a slows down as a increase

until a equals about 0.5 (i.e. Ri = 0.5 R2). Results from calculations of critical

Ta number for different radius ratios are summarised in Table 5.2 together with

the critical dimensionless wave number. Figure 5.4 also shows the theoretical

results o f Walowit et al. (1964), Sparrow et al (1964) and Roberts et al. (1965)

for comparison with the present results. In general the agreement is excellent.

At the critical Ta number the vortices are characterised by the critical

dimensionless wave number, A,, or in other words, a critical size of the vortices

which is related to the wavelength of a pair o f Taylor vortices. The wavelength

is the most characteristic feature of the Taylor vortices; it can be seen with the

naked eye on the photographs of Taylor vortices by flow visualisation

technique as discussed in Section 2.3.1. Figure 5.5 shows that the value of


35000

— Present study

□ Sparrow et al. (1964)

A Walowit et al. (1964)

o Roberts et al (1965)

30000 -

25000 -

.Q 20000

15000 -

10000 -

5000 -

0.6 0.80.2 0.4

Radius ratio, a (-)

Figure 5.4 Comparison of critical Taylor number with previous workers on different radius ratios, a


critical dimensionless wave number increases as radius ratio decreases. The

value o f X changes little from ^ = 3.12 in the range o f a = 0.45 to 1, but

increases significantly for a < 0.45. That means that for wide-gap geometry

the wavelength of the Taylor vortices can be much smaller than for the narrow-

gap geometry. It is interesting to note that the vortices can be expected to have

an almost square cross section when a -> 1 which is consistent with the

experimental observation dating back to Taylor (1923).

Radius ratio, a Critical Taylor numb^, Ta« Critical dimoisioaless wave number. A?

0.999 1697 3.120.995 1701 3.120.99 1707 3.120.95 1756 3.120.9 1824 3.12

0.82 1903 3.130.8 1995 3.13

0.75 2103 3.130.7 2231 3.14

0.65 2385 3.140.6 2573 3.14

0.55 2806 3.150.5 3101 3.16

0.45 3485 3.160.4 4000 3.17

0.35 4720 3.190.3 5778 3.20

0.25 7451 3.220.2 10375 3.26

0.15 16361 3.300.1 32549 3.33

Table 5.2 The critical Taylor number and critical dimensionless wave numberwith a resting outer cylinder (p=0) as a function of the radius ratio, a


3.35 1

3.3

5Eg 3.25

I(0

^ 3.2 +c oCOc (DE^ 3.15

3.10

d.

Ù □

□ 'd 0 0 0 0 :

0.2 0.4 0.6 0.8

Radius ratio, a (-)

Figure 5.5 The effect of radius ratio on dimensionless wave number


5.2.4 The effect of angular speed ratio on the critical Tavlor

number

Figure 5.6 and Table 5.3 show the variation of the critical Ta number with

radius ratio, a, for a range of angular speed ratio, p, from -0.5 to 1. The

negative values o f P correspond to simulation of flow of counter-rotating

whereas the positive values represent the simulation of flow of co-rotating

cylinders. It is observed from Fig. 5.6 that for given radius ratio, a, the flow in

the gap of a pair o f counter-rotating cylinders is significantly more stable

compared with the case of co-rotating cylinders. For P smaller than -0.5, the

Couette flow is generally stable based on the Rayleigh's criterion.

Critical Taylor number

a p = -0.5 p = -0.25 p = -0.125 3 = 0 3 = 0.25 3 = 0.5 3 = 0.75 3= 1

0.995 1616 1682 1694 1701 1707 1709 1709 1709

0.99 1633 1702 1702 1707 1711 1711 1710 1709

0.95 1761 1770 1724 1756 1740 1727 1717 1709

0.9 1964 1887 1763 1824 1780 1750 1727 1710

0.85 2240 2029 1807 1903 1826 1775 1739 1712

0.8 2591 2200 1852 1995 1877 1803 1752 1715

0.75 3147 2423 1958 2103 1935 1834 1767 1719

0.7 3824 2697 2081 2231 2002 1870 1785 1725

0.65 4991 3080 2234 2385 2078 1911 1805 1733

0.6 6251 3584 2416 2573 2167 1957 1828 1742

0.55 9229 4359 2651 2806 2270 2009 1856 1754

0.5 16488 5497 3868 3101 2391 2069 1887 1769

0.45 20173 7569 4663 3485 2535 2140 1925 1788

0.4 32922 11105 5842 4000 2708 2221 1967 1811

0.35 59097 18611 7958 4720 2917 2317 2020 1841

0.3 97467 33185 12003 5778 3172 2430 2080 1877

0.25 223448 90931 22499 7451 3487 2563 2155 1924

0.2 616126 160914 51461 10375 3877 2724 2245 1984

Table 5.3 The critical Taylor number with a different angular speed ratio, p asa function of the radius ratio, a


5000 1

P = -0.125

4500P = -0.25

4000 - p = oP = -0.5

c 3500

Sr 3000

P = 0.25

2500 -P = 0.5

2000 - - P = 0.75

1500 J0.80.2 0.4 0.6

Radius ratio, a (-)

Figure 5.6 The effect of radius ratio on critical Taylor number for different cases of angular speed ratio


5.3 Newtonian Liquids in Couette-Flow with Axial Flow

5.3.1 Neutral curve

When an axial flow is superimposed on the Couette flow, the streamlines in the

undisturbed flow become spiral. This complicated combined flow, so-called

Couette-Poiseuille flow, can be characterised as a function o f two independent

parameters: the axial Reynolds number. Re and the Taylor number. Ta. The

system of Eq. [3.31b] and Eq. [3.33] together with boundary conditions Eq.

[3.34] determine an eigenvalue problem of the form

F(a, p, A,, a, Ta, Re) = 0 [5.8]

Analogous with the case o f pure Couette flow, the above parameters are similar

to Eq. [5.3] except for the addition of axial Reynolds number which is defined

by using the mean axial velocity. The condition of the flow stability can be

determined in the form of functional dependence of the critical value of the Ta

number on the Re number, for a given value of a and p.

However, the parameter a is more complicated because for an arbitrarily value

o f a, the Ta number will in general be complex. Thus, unlike the previous

case, the real part of a is no longer assumed to be zero. The motion will be

stable or unstable as the imaginary part o f o is respectively positive or negative.

We shall only consider the neutrally stable case, imaginary part o f a equal to

zero. Mathematically the problem is the following: for given a, p and Re, the

minimum positive real value of Ta number is determined with respect to real

positive values o f X and real values of a. The corresponding values o f X and a

determine the dimensionless wave number and the frequency o f the secondary

motion.


250Re[Ta] = Real part of Taylor number lm[Ta] = Imaginary part of Taylor number

CT = 1.166

200 CT = 1.169

.a = 1.172

E 150 Re[Ta] I im p-a]= o - 2147.35

? 100 Re[Ta]|it7i[Ta]=o - 2140.78

50 - Re[Ta]|imfTa]=o ~ 2147.51

Q.

4.52.5 3.5D)

-50

-100 J


Figure 5.7 The effect of dimensinless wave number, À, on Imaginary part of Taylor number Im[Ta] for different cases of dimensionless disturbance growth rate, a

(marginal stability : lm[Ta] = 0)


It is clear that the determination o f the critical Ta number is a rather

complicated five-parameter problem (Re, cx, P, a and A,). In the iteration

procedure to determine the critical value, calculation is started with available

values of critical Ta number and X found from previous section with a = 0 for

Re = 0. Iterations continue for increasing Re number. For Ta number,

minimum positive determinant is obtained by iteration of a pair (a, X ) as shown

in 5.7. Simulations of the critical Ta number were found to be more sensitive

to changes in the value of a compared with X for given a and p. Completion of

the process just described yields a single point on a neutral stability curve.

Therefore, in searching for the critical Ta number, the increment Aa was kept

as small as possible in order to find the minimum positive real Ta number,

which in turn was minimised over all real positive X as shown in Fig. 5.7.

5.3.2 The effect of axial Reynolds number on the

critical Tavlor number

Figure 5.8 shows the results o f the present study o f critical Ta number versus

Re number plotted with the results of other investigations for a = 0.95 and P =

0. It shows that these simulations compare well with results published by

previous researchers for the case o f narrow gaps. They generally show that the

value of the critical Ta number increases as the Re number increases. Thus, the

axial Re number has an effect in delaying criticality and damping of the initial

formation of Taylor vortices in annuli. In other words, increased stability of

the laminar flow in the annuli can be achieved by imposing an axial flow.

Coney and Simmers (1979) and Abdallah and Coney (1988) experimentally

showed that the axial flow practically confined the vortices to a restricted

region of the annular gap until the rotational speed had increased sufficiently to

enable the vortices to extend to the outer wall. The variation of the critical Ta

number with Re number for a = 0.95 and p = 0 with associated eigenvalue o f X


and a is tabulated in Table 5.4. It is necessary to limit the range of the flow

parameters for which the calculations are carried out since parallel numerical

and experimental studies were performed. The experimental apparatus was

designed on the premise of stable Couette flow up to Re = 70. The calculations

were restricted to 0 < Re < 70.

Table 5.4 shows that the parameters X and ct remain almost unchanged for low

values o f Re number. For Re > 20, the dimensionless wave number increases

with Re number indicating the size change of the Taylor vortices as the axial

flow velocity increases. In addition, the magnitude o f g also changed as the

dimensionless wave number changed indicating the close relationship between

the two parameters.

Re Tac X a0.01 1756 3.13 1.169

1 1757 3.13 1.169

2 1761 3.13 1.169

5 1788 3.13 1.169

10 1884 3.13 1.168

20 2279 3.21 1.167

30 2964 3.30 1.165

40 3962 3.36 1.163

50 5296 3.53 1.166

60 6939 3.89 1.183

70 8711 4.63 1.227

Table 5.4 Critical Taylor number and corresponding values of X and a for given values of Re when a = 0.95 and (3 = 0



Figure 5.8 shows the effect o f radius ratio, a, on the critical Ta number with

axial Re number, as a parameter. The plot indicates that the critical Ta number

increases as the radius ratio decreases, that is, for otherwise similar conditions,

the flow becomes more stable as gap width increases. According to the

simulations shown in Fig. 5.8 the extent of the increase in critical Ta number

also depends on the Re number. The variation o f critical Ta number with Re

number is given in Tables 5.5-5.10 for a range of radius ratios 0.9 < a < 0.4

with P = 0.

Re Tac X G

0.01 1824 3.13 1.169

1 1826 3.13 1.169

2 1830 3.13 1.169

5 1857 3.13 1.169

10 1958 3.13 1.168

20 2368 3.21 1.167

30 3078 3.25 1.164

40 4116 3.39 1.164

50 5501 3.57 1.168

60 7215 3.91 1.184

70 9066 4.63 1.227

Table 5.5 Critical Taylor number and corresponding values of X and a for given values of Re when a = 0.9 and p = 0


25000

20000 - Re = 70

Re = 60

■9 15000

Re = 50

10000 - - Re = 40

Re = 30

Re = 205000 -Re = 0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Radius Ratio, a (-)

Figure 5.8 The effect of radius ratio on critical Taylor number for different cases of axial Reynolds number


Re Tac X a0.01 1995 3.13 1.170

1 1997 3.13 1.170

2 2001 3.13 1.170

5 2031 3.14 1.170

10 2141 3.14 1.169

20 2582 3.21 1.168

30 3361 3.25 1.165

40 4494 3.40 1.165

50 6007 3.58 1.169

60 7893 3.86 1.182

70 9932 4.63 1.228

Table 5.6 Critical Taylor number and corresponding values o f X and a for given values of Re when a = 0.8 and p = 0

Re Tac X a0.01 2231 3.14 1.172

1 2233 3.14 1.172

2 2238 3.14 1.172

5 2271 3.14 1.172

10 2393 3.14 1.171

20 2888 3.21 1.170

30 3747 3.27 1.167

40 5005 3.41 1.167

50 6692 3.57 1.170

60 8804 3.83 1.182

70 11105 4.62 1.229

Table 5.7 Critical Taylor number and corresponding values of X and afor given values of Re when a = 0.7 and P = 0


Re Tac X a

0.01 2573 3.15 1.175

1 2575 3.15 1.175

2 2580 3.15 1.175

5 2619 3.15 1.175

10 2757 3.15 1.175

20 3320 3.23 1.173

30 4298 3.27 1.170

40 5732 3.38 1.169

50 7660 3.57 1.173

60 10088 3.79 1.183

70 12767 4.57 1.229

Table 5.8 Critical Taylor number and corresponding values of X and a for given values of Re when a = 0.6 and p = 0

Re Tac % a0.01 3101 3.16 1.180

1 3103 3.16 1.180

2 3110 3.16 1.180

5 3155 3.17 1.180

10 3318 3.17 1.179

20 3983 3.19 1.177

30 5136 3.27 1.175

40 6830 3.38 1.174

50 9117 3.55 1.177

60 12012 3.73 1.185

70 15258 4.53 1.232

Table 5.9 Critical Taylor number and corresponding values of X and afor given values of Re when a = 0.5 and P = 0


45000

: Radius ratio = 0.8 P = -0.2540000 -

- ; Radius ratio = 0.435000

30000 -

25000 -

20000 - p = o

15000 - P = 0.25 P = -0.25 P = 0.5

P = 0.25 P = 0.5

10000

5000 -

20 40 60

Axial Reynolds number, Re (-)

Figure 5.9 The effect of axial Reynolds number on critical Taylor number for different cases of angular speed ratio


Re Tac X a0.01 4000 3.18 1.188

1 4003 3.18 1.187

2 4011 3.18 1.187

5 4068 3.18 1.187

10 4271 3.18 1.187

20 5100 3.21 1.185

30 6538 3.28 1.183

40 8651 3.38 1.182

50 11507 3.49 1.183

60 15163 3.71 1.191

70 19358 4.46 1.236

Table 5.10 Critical Taylor number and corresponding values o f X and a for given values o f Re when a = 0.4 and p = 0

5.3.4 The effect of angular speed ratio on the


Figure 5.9 shows the effect o f Re number on the critical Ta number, Ta«, with

angular speed ratio, p, as a parameter. The values of a = 0.8, 0.4 are chosen as

representative values for different radius ratios. The plots indicate that as the

Re number increases the critical Ta number increases, that is, for otherwise

similar conditions, the flow in the gap of a pair of counter-rotating cylinders is

significantly more stable compared with the case of co-rotating cylinders.

According to the simulations shown in Fig. 5.9 the extent of the increase in

critical Ta number depends critically on the Re number. The results of the

present analysis show that when the cylinders are rotating in opposite direction

(negative p) the critical Ta number increases rapidly with increasing Re

number. The variation o f critical Ta number with Re number is given in Tables

5.11-5.22 for a range of angular speed ratios -0.25 < p < 0.5 with three radius

ratios a = 0.8, 0.6 and 0.4.


Re Tac X a

1 2205 3.16 1.169

10 2363 3.17 1.168

30 3695 3.31 1.162

50 6603 3.53 1.157

70 11427 4.60 1.215

Table 5.11 Critical Taylor number and corresponding values o f X and a for given values of Re when a = 0.8 and P = -0.25

Re Tac X a1 3602 3.22 1.172

10 3849 3.25 1.171

30 5939 3.38 1.165

50 10509 3.45 1.145

70 17892 3.31 1.119

Table 5.12 Critical Taylor number and corresponding values o f X and a for given values of Re when a = 0.6 and p = -0.25

Re Tac X a

1 10311 3.81 1.161

10 11922 3.82 1.158

30 16830 3.93 1.137

50 26318 3.79 1.104

70 40461 3.59 1.071

Table 5.13 Critical Taylor number and corresponding values o f X and afor given values of Re when a = 0.4 and P = -0.25


Re Tac X a1 1997 3.13 1.170

10 2141 3.14 1.169

30 3361 3.25 1.165

50 4009 3.58 1.169

70 9932 4.63 1.228

Table 5.14 Critical Taylor number and corresponding values of X and a for given values o f Re when a = 0.8 and P = 0

Re Tac X a1 2575 3.15 1.175

10 2757 3.15 1.175

30 4298 3.27 1.170

50 7660 3.57 1.173

70 12767 4.57 1.229

T able 5.15 Critical T ay lor number and corresponding values o f X and a for given values o f Re when a = 0.6 and p = 0

Re Tac X a1 4003 3.18 1.187

10 4271 3.18 1.187

30 6538 3.28 1.183

50 11509 3.49 1.183

70 19358 4.46 1.236

Table 5.16 Critical Taylor number and corresponding values o f X and afor given values of Re when a = 0.4 and P = 0


Re Tac X a

1 1879 3.12 1.170

10 2015 3.13 1.170

30 3167 3.25 0.166

50 5652 3.57 0.171

70 9220 4.66 0.232

Table 5.17 Critical Taylor number and corresponding values o f X and a for given values o f Re when a = 0.8 and P = 0.25

Re Tac X a1 2170 3.13 1.175

10 2324 3.15 1.175

30 3631 3.27 1.171

50 6443 3.57 1.177

70 10498 4.59 1.234

Table 5.18 Critical Taylor number and corresponding values o f X and a for given values of Re when a = 0.6 and P = 0.25

Re Tac X a

1 2713 3.16 1.187

10 2897 3.18 1.187

30 4450 3.25 1.184

50 7773 3.50 1.189

70 12554 4.48 1.243

Table 5.19 Critical Taylor number and corresponding values o f X and afor given values of Re when a = 0.4 and P = 0.25


Re Tac X a1 1804 3.13 1.171

10 1935 3.13 1.170

30 3042 3.26 1.167

50 5423 3.56 1.172

70 8811 4.65 1.232

Table 5.20 Critical Taylor number and corresponding values o f X and a for given values of Re when a = 0.8 and P = 0.5

Re Tac X CT

1 1959 3.13 1.175

10 2098 3.16 1.177

30 3278 3.27 1.172

50 5809 3.57 1.178

70 9405 4.62 1.236

Table 5.21 Critical Taylor number and corresponding values o f X and a for given values of Re when a = 0.6 and p = 0.5

Re Tac À a1 2225 3.15 1.187

10 2376 3.19 1.187

30 3652 3.25 1.184

50 6367 3.46 1.188

70 10201 4.48 1.243

Table 5.22 Critical Taylor number and corresponding values o f X and afor given values of Re when a = 0.4 and p = 0.5


5.4 Non-Newtonian Liquids in Couette-Flow

All analytical and experimental papers dealt with so far are based on

Newtonian liquids. However, complex flow behaviour o f the liquid in the gap

can change the results. It is reasonable to investigate the influence of

rheological properties o f the liquid in order to obtain general description o f the

flow stability in Couette flow. In this work the dependence of the flow stability

on shear thinning behaviour of non-Newtonian liquids is investigated

theoretically for different angular speed ratios and the wide gap geometry.

The Couette flow of strongly pseudoplastic liquids in a wide gap is

characterised by a variation o f the apparent viscosity across the annular gap,

which involves additional difficulties when describing the stability limit.

Figures 5.10 and 5.11 show typical angular speed distribution and stress

distribution for different power law liquids flowing across the annular gap of

the concentric rotating Couette flow device. The angular velocity o f inner

rotating cylinder Qi = 500 s" and the outer cylinder was stationary (O2 = 0 s' ).

Both of the figures show that for flow index n = 1, the stress distribution and

the angular speed distribution are almost linear starting fi*om the inner cylinder

(bottom of the figures) to the outer cylinder (top of the figures).

Figure 5.10 shows that as the flow index decreases, radial variations of angular

speed across the annular gap become more pronounced. The angular velocity

decreases sharply with the radial position away from the inner rotating

cylinder. However, the angular velocity above the mid-gap is relatively lower

especially as the flow index, n, approaches zero. Figure 5.11 shows that the

stress distributions are more affected compared with the Newtonian case (n =

1). The maximum shear at the inner rotating cylinder (bottom of the figure) is

increased with pseudoplasticity. Above the mid-gap, the stress decreases with

Theoretical Results and Discussion 13 6

decreasing flow index. These adverse effects are due to the small angular

speed gradient as shown on Fig. 5.10.

In order to account for the effect of rheological properties on the instability of

Couette flow the definition o f Taylor number was modified to include the

variable viscosities of the non-Newtonian liquids in the Couette flow device.

According to the power law model o f non-Newtonian liquids, the apparent

viscosity of liquid is variable in a cross-section of the annular gap, and

selection of proper dimensionless Taylor number becomes rather difficult.

By introducing the dimensionless radial coordinate, x to replace the radial

coordinate r. The apparent viscosity, Pa is related to the mean viscosity, as

follows:

1=

fJ g a dr [5.1]

(R 2-R i) r,

Figures 5.12 and 5.13 show the apparent viscosity, pa and the mean viscosity.

Pm, for a range of flow indices at different inner cylinder rotational speeds

respectively. Since apparent viscosity is a function of shear rate, the mid-gap

position [r = 0.5(R2- Ri)] was selected as the representative shear rate across

the annular gap.

Both of the figures show no appreciable difference over the whole range of

flow index, so that the mean viscosity term, pm, which is independent o f shear

rate, is used in the modification of Ta number definition to include the

pseudoplastic properties of non-Newtonian liquid.


(0c

I

0.045 1

0.042

© 0.039

g 0 036JD(00 1

0.033

0.03

Inner cylinder radius = 0.03 m Outer cylinder radius = 0.045 m Inner cylinder angular speed = 500 s" Outer cylinder angular speed = 0 s'^

Decreasing flow Index, n

100 200 300 400

Angular speed (s' )

n = 0.1

500

Figure 5.10 Angular speed distribution for a range of flow index, n at different radial co-ordinates


0.045Inner cylinder radius = 0.03 m Outer cylinder radius = 0.045 m Inner cylinder angular speed = 500 s Outer cylinder angular speed = 0 s'^0.042

EL-

0.039(0c

0.033

= 0.20.03

800060000 2000 4000Shear stress (Pa)

Figure 5.11 Stress distribution for a range of flow index, n at different radial co-ordinates


Inner cylinder radius = 0.03 m Outer cylinder radius = 0.045 m0.9

0.8

0.7

0.6 - -

0.5

0.4

0.2200 rpm

300 rpm500 rpm 900 rpm

0.2 0.4 0.6Flow Index, n (-)

0.8

Figure 5.12 Apparent viscosity for a range of flow index, n at different inner cylinder rotational speeds(Assume viscosity = IPa s“ when n = 1)


Inner cylinder radius = 0.03 m Outer cylinder radius = 0.045 m0.9

0.8 -

0.7 -

0.6 ■ ■

0.5 -

0.4

0.3 -

0.2

200 rpm 300 rpm500 rpm 900 rpm

0.4 0.6

Flow Index, n (-)

0.80.2

Figure 5.13 Mean viscosity for a range of flow index, n at different inner cylinder rotational speeds(Assume viscosity = IPa s° when n = 1)


5.4.1 Neutral curve

The homogeneous set o f equations Eqs. [3.80b] and [3.82] with the boundary

conditions [3.83] determine an eigenvalue problem of the form

F(a, p, T, n, a) = 0 [5.9]

For n = 1, the stability problem for the case of Newtonian liquid is obtained as

discussed in previous Section 5.2. Analogous with the case of Newtonian

liquids, the parameter a is in general complex in which Cr and ai are equal to

zero constituting a condition of neutral stability.

The conditions o f the stability limit of the Couette flow of the liquid with the

power law viscosity are described by the critical Ta number, being a function

of the parameter, radius ratio, a, angular speed ratio, P, and the flow index, n.

All (eigenvalues) Ta numbers are real. The set o f the smallest positive values

of Ta determined at the given values o f dimensionless wave number, A,, define

the neutral curves. The minimum of the function Ta number at a given a, p

and n determines the critical Ta number and critical dimensionless wave

number. The eigenvalue problem has been solved with the Galerkin method.

The details of the solution are given in the Appendix 3. Computations have

been carried out for a wide range of radius ratio, angular speed ratio and the

flow index. Numerical procedures have provided an accuracy for the Ta

number estimation better than 0.25%.

In order to determine the critical value of Ta number, a course of neutral curves

was investigated in a range of dimensionless wave numbers for different values

of flow index as shown in Fig. 5.14. For all values o f the parameter X and n,

the curves show a single minimal. Results o f the present work obtained


2600 1

n = 0.92300 -n = 0.8

n = 0.72000

n = 0.6

n = 0.51700 -

n = 0.4

n = 0.31400 --

n = 0.21100

800 J


Figure 5.14 Neutral curves for a range o f flow index


6000 1n=1.0

n=0.9

5000 -n=0.8

n=0.74000 -

n=0.63000 - -

2000 - n=0.5

1000 - n=0.4

n=0,2 n=0.1n=0.3

Radius Ratio, a (-)

Figure 5.15 The effect o f the radius ratio on the critical Taylor number


12000

a=0.210000 -

8000 -

a=0.36000

>>

a=0.44000

a=0.6a=0.7a=0.82000 - -

0.2 0.4 0.6Flow Index (n)

0.8

Figure 5.16 The effect of the flow index on the critical Taylor number


for n = 1 are in very good agreement with the earlier predictions (see Section

5.2). In the whole range o f flow index considered, the critical value of the Ta

number is an increasing function of n indicating that the shear thinning

properties o f the liquid generally has a destabilising influence on the flow.


Figure 5.15 shows the effect o f radius ratio, a, on the critical Ta number with

flow index, n, as a parameter at angular speed ratio, P = 0. The plot indicates

that the critical Ta number generally decreases as the flow index decreases over

the whole range o f radius ratio. Thus, for otherwise similar conditions, the

flow becomes less stable as shear thinning properties increase. According to

the simulations shown in Fig. 5.15 the extent o f the decreases in critical Ta

number also depends on the radius ratio.

Figure 5.16 is a cross plot o f the effect o f flow index on the critical Ta number

with radius ratio as a parameter. For flow index, n > 0.45 the critical Ta

number generally increases as the radius ratio decreases which is similar to the

case of Newtonian liquids as discussed in previous Section 5.2.3. However, for

n < 0.45, a different effect appears that is the critical Ta number increases as

the radius ratio increases. The variation of critical Ta number with flow index

is given in Tables 5.23-5.34 for a range of radius ratio 0.4 < a < 0.9 with a

range o f angular speed ratio together with dimensionless wave number. They

all show that the effect of rheology on the critical values is significant in all

cases and notably when n < 0.45. It should be noted that the predictions shown

in Fig. 5.16 are confirmed by the experimental RTD data obtained from the

coaxial cylinder apparatus as discussed later.


Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 1709 1622 1532 1441 1347 1250 1148 1039 919 7760.95 1709 1622 1533 1441 1347 1249 1146 1034 904 7100.9 1710 1623 1534 1442 1347 1250 1139 1017 856 5310.8 1715 1628 1539 1445 1345 1236 1109 942 673 1570.7 1725 1639 1548 1451 1342 1215 1052 814 427 21.00.6 1742 1657 1564 1460 1337 1181 964 640 206 2.010.5 1769 1686 1591 146 1330 1130 842 443 72.2 0.190.4 1811 1732 1632 1501 1319 1057 687 257 19.3 0.020.3 1878 1804 1699 1542 1305 957 506 117 4.37 0.000.2 1983 1923 1810 1611 1287 822 314 40.8 0.87 0.000.1 2159 2135 2015 1743 1269 633 144 11.0 0.15 0.00

Table 5.23 Critical Taylor number for given values of flow index and radius ratio when angular speed ratio p = 1

Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 3.12 3.06 3.00 2.93 2.84 2.75 2.63 2.49 2.29 1.980.95 3.12 3.06 3.00 2.93 2.85 2.75 2.63 2.49 2.30 2.000.9 3.12 3.06 3.00 2.93 2.85 2.75 2.64 2.50 2.31 2.070.8 3.12 3.06 3.00 2.93 2.85 2.76 2.65 2.52 2.38 2.360.7 3.12 3.07 3.01 2.94 2.86 2.77 2.68 2.58 2.52 2.800.6 3.13 3.07 3.02 2.95 2.88 2.79 2.72 2.67 2.75 3.280.5 3.14 3.09 3.03 2.97 2.91 2.84 2.80 2.82 3.11 3.960.4 3.15 3.10 3.05 3.00 2.95 2.90 2.91 3.07 3.54 4.850.3 3.17 3.13 3.08 3.04 3.01 3.01 3.10 3.48 4.06 5.880.2 3.21 3.16 3.13 3.10 3.10 3.17 3.46 4.02 4.77 6.990.1 3.25 3.21 3.19 3.19 3.25 3.52 4.17 4.93 5.68 8.14

Table 5.24 Critical dimensionless wave number for given values offlow index and radius ratio when angular speed ratio P = 1


Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 1711 1624 1534 1443 1349 1251 1148 1039 918 7720.95 1728 1639 1548 1455 1359 1259 1153 1037 902 6930.9 1750 1659 1567 1471 1372 1268 1154 1024 823 5110.8 1803 1709 1512 1510 1401 1281 1141 958 668 1450.7 1871 1773 1669 1558 1434 1288 1102 836 421 19.00.6 1958 1855 1744 1619 1471 1285 1031 665 202 1.850.5 2071 1965 1843 1698 1514 1267 923 466 70.8 0.180.4 2223 2133 1978 1803 1564 1229 771 273 19.4 0.020.3 2431 2322 2169 1948 1624 1162 589 127 4.63 0.000.2 2734 2628 2454 2160 1698 1053 381 47.2 1.00 0.000.1 3140 3096 2908 2497 1794 866 189 14.2 0.19 0.00

Table 5.25 Critical Taylor number for given values o f flow index and radius ratio when angular speed ratio p = 0.5

Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 3.12 3.06 3.00 2.93 2.85 2.75 2.63 2.49 2.29 1.990.95 3.12 3.06 3.00 2.93 2.85 2.75 2.64 2.49 2.30 2.010.9 3.12 3.06 3.00 2.93 2.85 2.75 2.64 2.50 2.32 2.090.8 3.12 3.06 3.00 2.93 2.86 2.76 2.66 2.54 2.40 2.400.7 3.12 3.07 3.01 2.94 2.87 2.78 2.69 2.60 2.55 2.830.6 3.13 3.08 3.02 2.96 2.89 2.81 2.74 2.68 2.81 3.300.5 3.14 3.09 3.03 2.98 2.93 2.86 2.83 2.87 3.17 3.990.4 3.15 3.11 3.05 3.01 2.96 2.93 2.95 3.14 3.59 4.870.3 3.18 3.13 3.08 3.05 3.02 3.04 3.16 3.56 4.09 5.910.2 3.21 3.17 3.13 3.11 3.12 3.22 3.55 4.09 2.82 7.010.1 3.25 3.22 3.19 3.20 3.27 3.58 4.25 4.78 5.68 8.16

Table 5.26 Critical dimensionless wave number for given values offlow index and radius ratio when angular speed ratio P = 0.5


Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 1711 1623 1534 1442 1348 1250 1147 1037 916 7680.95 1740 1650 1558 1464 1366 1265 1156 1038 899 6840.9 1781 1688 1592 1494 1391 1283 1165 1029 849 4960.8 1878 1777 1673 1564 1447 1318 1167 970 663 1370.7 2003 1894 1778 1654 1514 1351 1144 853 416 17.70.6 2168 2048 1917 1770 1596 1380 1090 685 198 1.740.5 2394 2260 2107 1926 1699 1401 996 484 69.6 0.880.4 2711 2561 2377 2143 1832 1409 859 287 19.5 0.020.3 3176 3009 2782 2463 2013 1398 675 137 1.14 0.000.2 3879 3710 3425 2966 2273 1352 457 51.1 0.85 0.000.1 4934 4836 4506 3818 2673 1224 252 18.6 0.25 0.00

Table 5.27 Critical Taylor number for given values of flow index and radius ratio when angular speed ratio P = 0.25

Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 3.12 3.06 3.00 2.93 2.85 2.75 2.63 2.49 2.30 1.990.95 3.12 3.06 3.00 2.93 2.85 2.75 2.64 2.50 2.31 2.030.9 3.12 3.06 3.00 2.93 2.86 2.76 2.65 2.51 2.33 2.110.8 3.12 3.07 3.01 2.94 2.87 2.77 2.67 2.55 2.42 2.760.7 3.13 3.07 3.01 2.95 2.88 2.79 2.71 2.61 2.58 2.860.6 3.13 3.07 3.03 2.96 2.90 2.83 2.77 2.73 2.87 3.320.5 3.14 3.09 3.04 2.98 2.93 2.88 2.86 2.91 3.22 4.110.4 3.16 3.11 3.07 3.02 2.98 2.99 3.00 3.21 3.62 4.900.3 3.18 3.14 3.10 3.07 3.05 3.08 3.24 3.64 4.13 5.930.2 3.21 3.17 3.14 3.13 3.16 3.28 3.67 4.16 4.85 7.030.1 3.25 3.22 3.20 3.22 3.31 3.70 4.37 4.83 5.74 8.18

Table 5.28 Critical dimensionless wave number for given values offlow index and radius ratio when angular speed ratio P = 0.25


Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 1707 1620 1530 1439 1344 1246 1143 1033 941 7610.95 1756 1664 1571 1475 1375 1271 1160 1037 892 6660.9 1824 1727 1628 1525 1417 1303 1178 1033 841 4730.8 1995 1885 1767 1648 1518 1373 1203 985 653 1260.7 2231 2101 1963 1814 1647 1452 1209 878 4.8 16.00.6 2573 2414 2241 2047 1820 1544 1187 714 193 1.600.5 3101 2896 2664 2374 2066 1653 1126 512 67.8 0.170.4 4000 3713 3374 2960 2442 1790 1018 309 19.5 0.020.3 5778 5320 4747 4017 3091 1977 850 154 5.22 0.000.2 10375 9443 8189 6535 4477 2271 639 68.7 1.43 0.000.1 32549 29144 24042 17095 9114 2925 492 34.8 0.48 0.00

Table 5.29 Critical Taylor number for given values of flow index and radius ratio when angular speed ratio p = 0

Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 3.13 3.07 3.01 2.94 2.86 2.76 2.65 2.50 2.31 2.000.95 3.13 3.07 3.01 2.94 2.86 2.76 2.65 2.51 2.32 2.040.9 3.13 3.07 3.01 2.94 2.86 2.77 2.66 2.53 2.35 2.140.8 3.13 3.08 3.02 2.95 2.87 2.78 2.69 2.57 2.45 2.470.7 3.14 3.09 3.03 2.97 2.90 2.82 2.73 2.6 2.63 2.890.6 3.15 3.10 3.04 2.99 2.93 2.87 2.81 2.79 2.93 3.340.5 3.16 3.12 3.07 3.02 2.97 2.93 2.92 3.00 3.29 4.010.4 3.18 3.14 3.11 3.07 3.01 3.04 3.09 3.34 3.68 4.910.3 3.22 3.18 3.16 3.15 3.16 3.23 3.43 3.79 4.19 5.950.2 3.26 3.25 3.25 3.28 3.36 3.60 4.00 4.32 4.92 7.070.1 3.33 3.35 3.41 3.56 3.95 4.56 4.82 5.04 5.86 8.24

Table 5.30 Critical dimensionless wave number for given values offlow index and radius ratio when angular speed ratio P = 0


Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 1692 1605 1516 1424 1330 1232 1129 1019 895 7430.95 1772 1678 1582 1483 1381 1273 1158 1030 875 6340.9 1888 1785 1679 1568 1452 1329 1192 1033 823 4380.8 2204 2073 1936 1791 1635 1462 1259 1003 635 1100.7 2705 2525 2234 2128 1899 1636 1319 913 192 13.80.6 3600 3320 3018 2687 2314 1182 1367 759 183 1.410.5 5544 4997 4408 3764 3053 2260 1393 556 64.7 0.150.4 11305 9726 8082 6377 4632 2913 1379 346 19.6 0.190.3 33700 27290 20804 14467 8702 4128 1297 189 6.00 0.000.2 178174 124881 79534 44118 19913 6454 1131 120 2.46 0.00

Table 5.31 Critical Taylor number for given values o f flow index and radius ratio when angular speed ratio P = -0.25

Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 3.15 3.08 3.02 2.95 2.87 2.78 2.66 2.52 2.33 2.020.95 3.15 3.09 3.03 2.95 2.88 2.79 2.67 2.53 2.35 2.080.9 3.15 3.10 3.04 2.97 2.89 2.80 2.60 2.56 2.39 2.180.8 3.17 3.11 3.05 2.99 2.91 2.83 2.73 2.62 2.51 2.530.7 3.19 3.14 3.08 3.02 2.95 2.88 2.81 2.73 2.72 2.930.6 3.24 3.19 3.14 3.08 3.03 2.97 2.92 2.91 3.03 3.370.5 3.36 3.31 3.26 3.21 3.17 3.13 3.12 3.19 3.40 4.040.4 3.81 3.70 3.64 3.57 3.50 3.46 3.49 3.62 3.81 4.960.3 4.96 4.86 4.75 4.61 4.46 4.30 4.19 4.14 4.31 6.020.2 7.58 7.35 7.09 6.77 6.39 5.90 5.30 4.87 5.18 7.25

Table 5.32 Critical dimensionless wave number for given values offlow index and radius ratio when angular speed ratio p = -0.25


Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 1633 1548 1641 1372 1280 1134 1083 974 851 6960.95 1764 1668 1570 1478 1362 125 1129 993 828 5710.9 1970 1856 1738 1615 1485 1345 1191 1010 774 3770.8 2607 2429 2243 2045 1833 1600 1333 1010 589 88.40.7 3872 3542 3195 2826 2429 1992 1504 949 356 10860.6 6674 5951 5192 4390 3539 3640 1709 834 163 1.140.5 13266 11492 9636 7700 5702 3716 1916 610 58.5 0.130.4 33181 27278 21347 15527 10068 5286 2041 400 19.6 0.020.3 111034 83544 58522 36935 19888 8301 2248 282 8.32 0.000.2 616126 439327 288497 168485 82585 30979 7641 1277 167 0.00

Table 5.33 Critical Taylor number for given values o f flow index and radius ratio when angular speed ratio p = -0.5

Flow index

a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.10.99 3.20 3.15 3.08 3.01 2.93 2.84 2.72 2.58 2.38 2.080.95 3.22 3.16 310 3.04 2.95 2.85 2.74 2.60 2.42 2.150.9 3.24 3.19 3.12 3.06 2.98 2.88 2.78 2.65 2.48 2.260.8 3.33 3.27 3.21 3.14 3.07 2.98 2.88 2.76 2.64 2.620.7 3.55 3.49 3.41 3.34 3.25 5.15 3.05 2.94 2.89 2.980.6 4.06 3.96 3.86 3.75 3.63 3.49 3.35 3.25 3.24 3.410.5 4.78 4.69 4.57 4.44 4.29 4.11 3.90 3.70 3.57 4.090.4 5.85 5.74 5.63 5.49 5.31 5.08 4.75 4.28 3.99 5.030.3 8.05 7.81 7.52 7.20 6.82 6.36 5.78 5.05 4.69 6.270.2 10.1 9.77 9.45 9.09 8.65 8.13 7.73 7.80 8.26 9.06

Table 5.34 Critical dimensionless wave number for given values offlow index and radius ratio when angular speed ratio p = -0.5


5.4.3 The effect of angular speed ratio on the


Figure 5.17 shows the variation of the critical Ta number with radius ratio, a,

for a range o f angular speed ratios, P, from -0.5 to 1. Solid lines and dotted

lines are plotted representing the simulated critical Ta number for Newtonian

liquid (n = 1) and non-Newtonian liquid (n = 0.7) respectively. The negative

values o f p correspond to simulation of flow o f counter-rotation whereas the

positive values represent the simulation of flow of co-rotating cylinders. It is

observed from Fig. 5.17 that for given radius ratio, a, the flow in the gap of a

pair o f counter-rotating cylinders is significantly more stable compared with the

case of co-rotating cylinders in both cases of Newtonian and non-Newtonian

liquids. The extent o f the increase in the critical Ta number depends on the

flow index, n.

Figure 5.17 also shown that for decreasing flow index, n, the non-Newtonian

shear thinning liquid is significantly less stable compared with the case of

Newtonian liquid in the whole range of ps. This is confirmed with previous

discussion in Section 5.4.1 that the shear thinning properties o f the liquid has a

destabilising influence on the Couette flow.


9000

: n = 0.78000

P = -0.57000 P=-0.25p = o

p = o6000

P = 0.25C 5000

>»P = 0.25

L. 4000

P = 0.5

3000 -P = 0.5

2000

1000 J0.6 0.80.2 0.4

Radius Ratio, a (-)

Figure 5.17 The effect o f angular speed ratio on the critical Taylor number

Experimental Results and Discussion 154

CHAPTER SIX

EXPERIMENTAL RESULTS AND DISCUSSION

6.1 Introduction

In this section, experimental data are presented to assess the applicability o f the

axial dispersion model to the RTD of the different liquids in a Couette flow

device. The experiments show that the rheological effects could be significant,

and that this effect was not solely due to the fact the absolute value of the

apparent viscosity changes across the annular gap but also due to variation in

the power-law index. The experimental critical Taylor numbers are discussed

for a Newtonian liquid and the two non-Newtonian power-law liquids. Finally,

the experimental results will compared with the theoretical results presented in

Chapter 5.

6.2 Residence Time Distribution

6.2.1 Reproducibility

In the tracer stimulus-response experiments, there were several potential

sources o f error which could affect the RTD. Among these were the liquid feed

system, the tracer injection technique, the sampling frequency and period, and

slight fluctuations in the ambient temperature. It was not possible to estimate

the errors due to each individual source due to their mutual dependence.

Figure 6.1 and 6.2 show the Ce-curves for some of replicate experiments for

Newtonian and non-Newtonian liquids respectively. The reproducibility is

considered good. This suggested that the RTD and its parameters provided a

reliable tool for the analysis and interpretation of liquid transport in the Couette

flow devices. In the present study, each experiment was performed twice and

the values used were the mean.


12 1

^ 10

oco% 8c8i e8COCOI 4gCOc(DEb 9

64% glycerol solution Ta number = 2649

Re number =10

ÔOo8 o 1 St trial

a 2nd trial* ° A 3 rd trial

□ o 4 th trial

o□oA

A□

0 O C X X X X X X X X X X X X X 3 9 a

0 0.5

06□O„ ^^i66fi88fiQ6»nnoaooaaa

1.5 2 2.5


Figure 6.1 Reproducibility of RTD experiments for Newtonian liquid


8CO

12 1

10<x>

Oco2 8 c8c

6 -

c 4g(/)cQ)Eb 2

0.7 wt% xanthan gum solution Ta number = 1609

Re number =0.8

8oA

A□OoA

AAO

o 1 St trial a 2 n d trial A 3 rd trial o 4 th trial

A

□A

AO

A□ 8

A

âO

I►

A coa

\ \

0 oocmooooaBBÔoS.

0 0.5 1—2§BSBB88a06oooooooi

1.5 2 2.5


Figure 6.2 Reproducibility of RTD experiments for non-Newtonian liquid

Experimental Results and Discussion 15 7

6.2.2 The effect of axial flow velocity on RTD

Figure 6.3 shows the RTD curve for Newtonian liquids for four axial flow

velocity with a rotational speed o f 56 rpm corresponding to Ta = 2649. The

tracer concentrations are presented in dimensionless form, Ce ,defined as:

t [6 .1 ]^C tA t

where t is the mean residence time given by

Ct is the tracer concentration measured at each probe at time t, following the

introduction of the tracer impulse. The plots indicate that a decrease in the

axial flow velocity results in a broadening o f the RTD curves and a

consequential deviation from plug flow condition. In experiments involving the

second probe, the mean residence time was reduced from 25 to 18s

corresponding to an increase in flow velocity from 22 to 40 mm/s. These

curves suggest a flow pattern between plug flow and mixed flow. As the axial

flow velocity increases, the curves indicate flow approaching that o f plug flow.

Experiments were repeated for a non-Newtonian liquid (0.6 wt% xanthan gum

solution) and results are shown in Fig. 6.4. These also showed similar trends.

6.2.3 The effect of rotational speed

Typical RTD curves for the two probes are shown in Fig. 6.5 for four rotational

speeds of the inner cylinder in the range of 26 to 48 rpm and axial flow velocity


St Probe

nd Probe

oCO

O)CNI

L O

O O

Time, t (s)C N

C O o N) O) ro CD hOAxial flow velocity

oCO CO

CO64% glycerol solution

Ta number = 2649 (mm/s)

Figure 6.3 RTD experiments at different axial flow velocitiesfor Newtonian liquid


1 St Probe

2 nd Probe

Time, t (s) ^ csj

0.6 wt% xanthan gum solution Ta number = 1626

(DÜcg

Î888CD

0)

s.1COc0)E

Axial flow velocity (mm/s)

Figure 6.4 RTD experiments at different axial flow velocitiesfor non-Newtonian liquid (0.6 wt% xanthan gum solution)


o f 15 mm/s corresponding to an axial Reynolds number, Re = 10 and Taylor

numbers. Ta = 582, 958, 1429 and 1992 respectively. The theoretical critical

Taylor number, Ta for the present geometry (radius ratio = 0.67) was found to

be 2061 according to the eigenvalue of Eqs. [3.42a-b]. The figure shows that

the rotational speed critically affect the spread of the RTD. The Ce-curve for

rotational speed = 26 rpm in Fig. 6.5 shows the combined effect o f rotational-

axial flow on the RTD in the laminar flow regime. The other Ce-curve

corresponding to rotational speed = 48 rpm is very different. Narrowing of the

Ce-curve is observed due to the occurrence of Taylor vortex which provides

evidence o f flow change away fi*om well-mixed to plug flow.

Also shown in Fig. 6.6 for comparison are the Ce-curves obtained for the cases

o f four higher rotational speeds of the inner cylinder in the range of 50 to 112

rpm at the same axial flow velocity (Re number = 10) and Ta number (Ta =

2139, 3608, 5883 and 10712 respectively). The figure shows that the rotational

speed significantly affects the spread o f the RTD curve. Kataoka et al. (1981)

suggested that when Ta exceeded Ta , the plug flow condition could not be

maintained owing to the occurrence of longitudinal mixing over the vortices

boundary. It was attributable to the development the wavy Taylor-Couette

flow.

Similar experiments were carried out for a non-Newtonian liquid (0.6 wt%

xanthan gum solution) in Figs. 6.7 and 6.8. The plots show similar trends

observed for Newtonian fluids. The theoretical critical Ta number was found

to be 973 for a flow index n = 0.285. It is interesting to note that the mean

residence times are approximately the same for all rotational speeds examined.


Rotational speed from) Tavlor number26 58234 95841 142948 1992

1 St Probe

Probe

Time, t (s)

W O)

Œ>Ücg

1888(DCOScgCO

5E

64% glycerol solution Re number = 10

Rotational speed (rpm)

Figure 6.5 RTD experiments at different rotational speedsfor Newtonian liquid (Ta < Ta )


Rotational speed from) Tavlor number50 213965 360883 5883112 10712

1 St Probe

2 nd Probe

00 cn

Time, t (s)

Œ>0cq2

188CD

CO

si(OcCDE

64% Qlvcerol solution Re number = 10


Figure 6.6 RTD experiments at different rotational speedsfor Newtonian liquid (Ta > Ta )


Rotational speed (rpm) Tavlor number

100140200

249833

St Probe

2 nd Probe

GO

CSJ CN ^

oOO

Time, t (s)

SCNt

§§8 g | 8

^Rotational speed (rpm)

0.6 wt% xanthan gum solution Re number = 1.0

Figure 6.7 RTD experiments at different rotational speedsfor non-Newtonian liquid (Ta < Tac)(0.6 wt% xanthan gum solution)


Rotational speed (rpm) 240 275 325 380

Tavlor number 1556 2482 4403 7527

1 S t Probe

2 nd Probe

tv. go cn

Time, t (s)

oco2c888(D0)CO0)coCOc0)E

0.6 wt% xanthan cum solution Re number = 1.0


Figure 6.8 RTD experiments at different rotational speedsfor non-Newtonian liquid (Ta > Tac)(0.6 wt% xanthan gum solution)


6.2.4 The effect of electrode positions

Figure 6.11 shows the effect of the radial position o f the electrodes in the gap

on the Newtonian RTD for the case of radius ratio a = 0.84. The results were

obtained with the electrodes located at 0.2 cm and 0.5 cm from the outer

cylinder (see Fig. 6.9). The data in Fig. 6.11 show that the RTD curves do not

depend on the position o f the sensor.

Figure 6.12 confirms that RTD is independent o f electrode's location. The test

section had one injection point I and two conductivity probe point pi and p2.

The two probes were separated by 90° on the same axial plane (see Figure

6.10). Comparing the tracer-response curves of the two conductivity probes

(pi and p2) in Figure 6.12, no appreciable difference can be found in the shape

of RTD curves indicating that the toroidal motion o f the liquid causes effective

radial mixing. This conclusion is in agreement with that made by Kataoka et

al. (1975).

0.2 cm

Inner cylinderOuter cylinder

0.5 cm

/ Outer cylinder i P Inner cylinder

Figure 6.9 Influence of the position of the sensor in the annular gap

Figure 6.10 Influence of the location of the sensor in the annular gap


8

d 7Co 2 6

§I *m2 40)(/)•I 3o(O

b

64% qlvcerol solution Ta number = 2649

Re number =10«□

* 0.2 cm from the outer cylinder

□ 0.5 cm from the outer cylinder

□O0 qOOOOOCKKHDOOOOOO-0 0.5 1

I° P^^Sâ6ùooaoooaoocKH1.5 2.5


Figure 6.11 RTD experiment on different positions of the sensor(Newtonian liquid)


10

oÜC0

1c8C8I</)8coO)c0)Eb

9 +

8

7

6

5

4

3 +

2

1

64% Qlvcerol solution Ta number = 2649

Re number =10

o Probe 1, pi

□ Probe 2, p2

□

□o0 ooaDooooaaDaooa Oe j

0 0.5 1 1.5 2 2.5


Figure 6.12 RTD experiment on different locations of the sensor(Newtonian liquid)


6.3 Axial Dispersion in Couette Flow

6.3.1 Axial dispersion model

An axial dispersion model is usually used to quantify the mixing performance

and the RTD curves of tracer material flowing through a Couette flow device.

The applicability of this model is based on several assumptions. The most

important o f these are:

1. Steady-state material flow conditions prevail and are not perturbed by the

introduction of the tracer stimulus.

2. Material holdup in the drum is independent o f axial position so that the

axial flow velocity o f the tracer is constant for fixed operating conditions.

3. Tracer concentration is uniform in each cross-section of the drum, i.e. it is

a function only of axial position and time.

4. The axial dispersion coefficient, which characterises axial transport of

tracer due to dispersive effect, is constant for fixed operating conditions.

The axial dispersion model for the tracer in its dimensionless form is

(Levenspiel, 1972)

aCe aCe 1 aXe r . -1

where 0, x. Ce correspond to dimensionless time, length, and tracer

concentration respectively. The dimensionless Peclet number is given by

P e ~ [6.4]


where W corresponds to the axial flow velocity, L corresponds to the distant

between the conductivity probes and D is the dispersion coefficient. The

solution o f Eq. [6.3] is used as the basis for characterizing the RTD for the

axial dispersion model. The solution depends on the boundary conditions of

the test section, i.e. whether closed or open, (Levenspiel, 1963) and is

expressed usually as a dimensionless concentration Ce as a unique function of

Pe number. In the present study, an open-open boundary system is considered

for which the Ce expression is:

(l-e)^Pe40

[6.5]

Typical Ce curves for axial dispersion model are shown in Fig. 6.13 for a wide

range o f Peclet number from 1 to 700.

Fig. 6.14 shows a comparison of simulated curves and the experimental RTD

curves o f a Newtonian liquid for three rotational speeds of the inner cylinder in

the range o f 48 to 64 rpm and axial flow velocity o f 21 mm/s corresponding to

an axial Re number. Re = 12 and Ta numbers. Ta = 1968, 2678 and 3498

respectively. This figure shows that, for all combination of operating

conditions considered, Eq. [6.5] provides a good simulation of the experimental

dimensionless RTD data. This is shown in Fig. 6.14, in which the simulated Ce

curves based on the axial dispersion model are in excellent agreement with the

experimental points.

Experiments were repeated for a non-Newtonian (0.6 wt% xanthan gum

solution). The results are shown in Fig. 6.15 for three rotational speeds at

Reynolds number. Re = 1.0 and Ta numbers. Ta = 973, 1270, and 1626

respectively They also show very good agreement with the simulated curves.


8.00 1

Pe = 700

7.00

Pe = 500CD 6.00 ■ ■

5.00 -Pe = 300

4.00 -

P e= 150c 3.00

Pe = 75E 2.00-

Pe = 30

1.00 Pe = 10Pe = 1

0.00 J0.5


Figure 6.13 Axial dispersion model


64% Qlvcerol solution Re number = 12

Radius ratio = 0.847

<D□ Ta number = 1968 A Ta number = 2678 o Ta number = 3498

Axial dispersion model

0.0 0.5 2.0 2.5


Figure 6.14 Comparison of axial dispersion model andthe experimental data (Newtonian liquid)


0.6 wt% xanthan gum solution Re number = 1.0

Radius ratio = 0.84□ Ta number = 973 A Ta number = 1270 o Ta number = 1626 — Axial dispersion model

o

2.50.5

Dimensionless time, 0

Figure 6.15 Comparison of axial dispersion model andthe experimental data (non-Newtonian liquid)


6.3.2 Peclet number

The Peclet number is used to measure the extent of axial dispersion. It

represents the ratio of total momentum transfer to molecular transfer. Two

limiting situations are readily appreciated from Eq. [6.3] and Fig. 6.13. When

Pe approaches zero, there exists significant dispersion and hence the flow is

mixed flow. When Pe approaches infinity, the dispersion is negligible and

hence the flow is plug flow. In the present study, the value of Pe can be

calculated from the mean residence time, 0, and variance of the curve, , by

using (Levenspiel, 1953): (Also see Appendix 5 for sample calculations)

y ti^Ci Ati ^ -0

^ Ci Ati

2 Ci Ati

y ti Ci Ati[6.7]

Aae represents the dimensionless variance difference of the tracer RTD curves

at any two points along the annular gap. In the present study Aae is evaluated

for the gap between the two probes using the response curves obtain from them,

as shown in Figs. 6.5. The value of Actg is substituted into Eq. [6.6] in order to

obtain the Pe number for the prevailing flow.

Fig. 6.16 shows the value of Aoe against the rotational speed of the inner

cylinder. The data in Fig. 6.16 are obtained from replicated experiments at

different Taylor numbers. For each value of Ta number the data points are

shown by a bar covering the range of values o f Aae obtained in the

experiments. Variations of the Aae values among replicate samples are within

±7%. There is some scatter in the experimental measurements, but a trend can

be detected and the minimum value corresponds to the transition of Couette

flow to Taylor-Couette flow regime.


I8§

0.018

0.016

0.014

0.012

0.01

0.008

8 c (U "C§CO

S 0.006c oCO

§ 0.004 E b

0.002 +

0 J20

64% Qvlcerol solution Radius ratio = 0.84

Re number = 12

30 40 50

Taylor number, Ta (-)

60 70

Figure 6.16 The effect of rotational speed of inner cylinder on dimensionless variance difference of the tracer RTD curve


The effect o f rotational speed on the Pe number for a Newtonian liquids (85%

glycerol solution), is shown in Fig. 6.17 for three axial flow rates expressed in

term of the axial Reynolds number. The plots in Fig. 6.17 show that as the

rotational speed o f the inner shaft increases, the Pe number increases fi’om an

initial value of about 100 to a maximum value of approximately 360 beyond

which any further increase in rotational speed causes a decrease in the Pe

number. The minimum and maximum values of the Pe number were found to

be functions o f the flow rate through the gap. In the present study the lowest

value of Pe was found to be about 80 corresponding to an axial flow velocity o f

26.4 mm/s (Re = 2) and the h ip est value o f Pe was 360 corresponding to an

axial flow velocity o f 39.6 mm/s (Re = 3) as shown in Fig. 6.17.

The variations in Pe number with increasing Ta number depicted in Fig. 6.17

suggests that significant deviation fi'om plug flow conditions can occur during

flow. Pudjiono e t a l (1992) and Pudjiono and Tavare (1993) in their recent

publications suggested that the maximum Pe number correspond to the critical

condition at which the Taylor vortices are fully developed.

Figures 6.17 to 6.30 show the results o f similar experiments carried out for

different geometrical and/or rheological parameters, summarized in Table 6.1.

They also show similar trends in data and confirm the results shown in Fig.

6.17.

Radius ratio, Newtonian liquids non-Newtonian liquids non-Newtonian liquidsR2/R] (Glycerol solution) (CMC solution) (Xanthan gum solution)0.84 Fig. 6.17 Fig. 6.23 Fig. 6.27

Fig. 6.18 Fig. 6.24 Fig. 6.28Fig. 6.19

0.67 Fig. 6.20 Fig. 6.25 Fig. 6.29Fig. 6.21 Fig. 6.26 Fig. 6.30Fig. 6.22

Table 6.1 Summary of the experimental Pe - Ta plots


450 1- c - Re number = 1- -A - Re number = 2 - o - Re number = 3

85% glycerol solution Radius ratio = 0.84

400 -

350 -

300

250

200

150/ / /

d / /\ \ 0

100

50 -i— 1000 2000

Taylor number, Ta (-)2500 30001500

Figure 6.17 The effect of Taylor number on Peclet numberfor Newtonian liquid


500 1

450

400 -

X 350 +(D 0_fe 300

J DIc 250%Ü 0)0_ 200

150

100

50


- - Re number = 8 - Re number = 12

-■A - Re number = 18 - - 0 - Re number = 23- « - Re number = 27

f

1000 2000 3000 4000


5000

Figure 6.18 The effect o f Taylor number on Peclet numberfor Newtonian liquid


550

500

450

400

S . 350

0)E 3003C

^ 250Ü<D

OL200


- -o - Re number = 35- -o - Re number = 50- -A - Re number = 58- -o - Re number = 62

V / V' \ \

/ / \

\ \\ \\ \\ \

I / / / \150- / / /

100 -

50

\ \\ \ \

\ \\ 4\ ^

\

\ \\

\

\ \\ \\ \

h o \\

5000 10000

Taylor number, Ta (-)15000

F igure 6.19 The effect of T ay lor number on Peclet numberfor Newtonian liquid


450

400 --

350 -

(D 300 Û.

L .3c%o 200 CL

150

100

50


- a - Re number = 1- -A - Re number = 2- o - Re number = 4

A

i//

/ / %

1000 2000 3000

Taylor number, Ta (-)4000



350 1

E 200


- -a - Re number = 7Re number = 17

” -A - Re number = 22- o - Re number = 25- •* - Re number = 29

w ww ww \v\\ \ \.

M v .

"IK------O---- o

□

0 4000 8000 12000


16000



350

300 -

2500)CL

0)E 2003C0)8Û- 150

100

50


- d - Re number = 47Re number = 55

- -A “ Re number = 61 - o - Re number = 69

/

/P N // I / P/ / / /I I I /

/ ¥ /I I I I

6 / / /

I A VT/1 \\ f f V \ \

1111 \

\ \\ \

I

\ \\ \\ \\ \\ \ \

U VV

\

\

\\

\

10000 20000 30000Taylor number, Ta (-)

40000



320

270 -

0)CL 220i_r0).aE3C

% 1708

CL

120

70

Carboxvmethvl cellulose solution Radius ratio = 0.84

fSX

h ''m \

— "O —— o —— o —— ^ —

0.1 wt% 0.3 wt% 0.4 wt% 0.5 wt%

500 1000 1500 2000 2500


3000 3500

Figure 6.23 The effect of Taylor number on Peclet numberfor non-Newtonian liquids


320

270

(DÛ. 220

5E3C

% 170 o(DQ_

120


900

&•— -A — “ "O — — Cl —

0.6 wt% 0.7 wt% 0.9 wt%

11/I t//#I

/\//

il‘ " 'l i

/ /

\ \ \\ \ \\ \ \

\\ \ \\ \ \\ \ \

w \\ \ \

V\"\ \\ \

\b

1100 1300 1500 1700


1900



370

320 -

2700)Û.

<DE 220= 3C0)ÿO. 170

120 -


r~ < — : 0.1 wt%

_ o — — " A —

0.3 wt% 0.4 wt% 0.5 wt%

1000 2000 3000 4000 5000 6000 7000


Figure 6.25 The effect o f Taylor number on Peclet numberfor non-Newtonian liquids


350

300

3 2500)Û.

E 2003C0)

0. 150

100

50


0.6 wt% 0.7 wt% 0.9 wt%

i \ \ \\ \ \ \

?\\\\

\ \\ \\ \\ \\ \

\ \ \\ V.

1000 1200 1400 1600 1800 2000 2200 2400




420

370 -

320

0)Q.

- 270Q)E3C%Ü0>Q_

220 -

170

120

70

Xanthan gum solution Radius ratio = 0.84

| / t \\ \II \\

9 'h\ w \ \ \\ \ \\ \ \\ \ \\ \ \\ \ \

if\

-o -o -

0.1 wt% 0.3 wt%

■” ■o — ; 0.4 wt% — -A — : 0.5 wt%

\ -\

/// ///

I '

\

1000 2000 3000 4000


5000



320

270

0)CL 220

iE3C

JW170ëCL

120

70


k11I II I

I II II I

I II Itn

iIIfn

I

\ \ \\ \ \\ \ \\ \ \\ \ \

\\ \ \\ \ \ \ \ '

\ \

— -A — : 0.6 wt%— ! 0.7 wt%- a - : 0.9 wt%

k\ \ \

500 1000 1500 2000


2500


Experimental Results and Discussion 18 8

370Xanthan gum solution

Radius ratio = 0.67: 0.1 wt%

0.3 wt% 0.4 wt% 0.5 wt%

1000 2000 3000 4000 5000 6000 7000

Taylor number, Ta {-)



370

320 -

2700)Û.

E 2203C0)8Û. 170

120

70200


f:( ' 'n \i\I I V '" * \II i \ \II I R \

— " A —

— ^ —

— O -

0.6 wt% 0.7 wt% 0.9 wt%

I \I II I I I

I I / I f I H I

I I I J I

&■ I '

\\ 4\ \

\ \

\

\□

600 1000 1400


1800



Figures 6.17 to 6.19 show the effect o f Ta number on the Pe number for a range

of axial Re numbers for radius ratio, a = 0.84. Three concentration of glycerol-

water solutions (85%, 64% and 45%) with different viscosities were employed

in order to cover a range of Re numbers from 1 to 62. The experimental data

show that the Re number and liquid viscosities affect the extent o f dispersion.

The experimental Pe number obtained for all experiments was in the range of

50 to 400 depending on the Ta number. The plots in Figs. 6.17 to 6.19 show

similar trends o f Pe number variation with increasing Ta number. In each case

the maximum Pe number on each curve corresponds to transition from Couette

flow to Taylor-Couette flow. The Ta number corresponding to the maximum

Pe number is defined as the critical Taylor number, Ta , for the transition.

The Pe-Ta plots from Fig. 6.17 - 6.19 over the range of Re numbers also show

that the critical Ta number is an increasing function o f Re number. The

experimental critical Ta number is increased from 1913 to 7891 as the Re

number is increased from 1 to 62 (See Table 6.1). Thus, the presence of a

small axial flow through the Couette flow device has an stabilising effect and

causes a delay for the onset o f flow instability o f Couette flow.

Glycerol solutions Reynolds number Critical Taylor number

85% 1 191385% 2 192785% 3 192564% 8 199264% 12 203464% 18 232764% 23 269764% 27 302345% 35 382345% 50 567545% 58 691645% 62 7891

Table 6.2 Experimental results o f Ta for Newtonian glycerol solution (a = 0.84)


Figures 6.20 to 6.22 show the effect o f Ta number on the Pe number for Re

numbers between 1 and 69 and a radius ratio, a = 0.67. Similar observations

are made from these plots, that is the critical Ta number increases as the axial

Re number increases. The experimental data revealed that the radius ratio also

affected the extent o f dispersion. Table 6.3 shows the critical Taylor number

for a range of Re numbers. By comparing these results with the Table 6.2 for

radius ratio = 0.84, it is seen that the critical Taylor numbers obtained for

radius ratio = 0.67 are generally higher than those for radius ratio = 0.84. (see

Fig. 6.31)

Glycerol solutions Reynolds number Critical Taylor number

85% 1 231685% 2 233885% 4 241064% 7 250864% 17 283164% 22 339464% 25 353664% 29 397845% 47 631845% 55 823445% 61 928745% 69 12196

Table 6.3 Experimental results o f Tag for Newtonian glycerol solution ( a = 0.67)

Figures 6.18 to 6.19 and Figure 6.22 to 6.23 show the effect of Ta number on

the Pe number for a range of concentrations (0.1 - 0.9 wt%) of CMC and

Xanthan gum solutions respectively. A radius ratio, a = 0.84 and seven

concentrations of non-Newtonian solutions were employed in order to cover a

wide range of rheological behaviour o f the liquid from flow index n = 0.254 to

0.653 as shown on the figures.


14000 185% glycerol solution 85% glycerol solution 64% glycerol solution 64% glycerol solution 45% glycerol solution 45% glycerol solution

12000 -

10000

8000 - a = 0.6

^ 6000

4000

a = 0.84


Figure 6.31 The effect of axial Reynolds number on criticalTaylor number at different radius ratios, a


All the experiments were carried out at low Re number ranging from 0.3 to 2.4

in order to provide a negligible effect o f axial flow on the flow instability on

Couette flow. So that the experimental results could be used to compare with

the theoretical prediction for Ta values obtained in Section 5.4.

All the plots show similar trends o f Pe number variation with increasing Ta

number with the maximum Pe number in each curve corresponding to the

transition from Couette flow to Taylor-Couette flow regime. The Ta number

corresponding to the maximum Pe number is, taken as the critical Ta number.

The experimental data revealed that the flow index affected the extent of

dispersion in the range of Pe number studied (50 to 400). The Pe-Ta plots over

the range o f flow index for two non-Newtonian liquids show that the critical Ta

number increases with an increase in the flow index. In the experiments, the

value o f critical Ta number increased from 904 to 1542 as the flow index was

changed from 0.254 to 0.653 (see Table. 6.4). Thus, the non-Newtonian

(pseudoplastic) properties have a destabilising effect on Couette flow.

CMC solution Xanthan gum solution

Flow index Critical Ta number Flow index Critical Ta number

0.457 1279 0.254 904

0.476 1310 0.268 935

0.487 1323 0.285 973

0.510 1360 0.307 1028

0.535 1392 0.342 1091

0.569 1436 0.385 1173

0.653 1542 0.493 1343

Table 6.4 Experimental results o f Tac for non-Newtonian solution (a = 0.84)


Figures 6.25 to 6.26 and 6.29 to 6.30 show the effect o f Ta number on the Pe

number for non-Newtonian carboxymethyl cellulose solutions and xanthan gum

solutions respectively. A radius ratio, a = 0.67 and seven concentration of

non-Newtonian solutions (with different flow index) were employed. Similar

results were obtained, that is the critical Ta number was found to increase with

an increase in flow index. The experimental data revealed that the radius ratio

affected the extent o f dispersion in the range of Pe number studied (50 to 400).

Table 6.5 shows the critical Ta number values for a range of flow index values

for two non-Newtonian liquids.

By comparing the results o f Table 6.5 with Table 6.4 for radius ratio = 0.84, it

is noted that for otherwise similar conditions the critical Ta number obtained

for radius ratio = 0.67 is different from that of radius ratio = 0.84 and depend

on the flow index value.

CMC solution Xanthan gum solution

Flow index Critical Ta number Flow index Critical Ta number

0.457 1387 0.254 6060.476 1422 0.268 6530.487 1451 0.285 7510.510 1513 0.307 8450.535 1579 0.342 9900.569 1632 0.385 11280.653 1835 0.493 1458

Table 6.5 Experimental results of Tac for non-Newtonian solution (a = 0.67)

In Fig. 6.32 the critical Ta number obtained for radius ratio = 0.67 (triangular

symbols) and radius ratio = 0.84 (square symbols) are plotted. Moreover, the

solid symbols represent the carboxymethyl cellulose solution whereas the


hollow symbols represent the xanthan gum solutions. It is observed from Fig.

6.32 that the Couette flow is generally destabilised with decreasing flow index,

n. The extent of the reduction in the critical Ta number depends on the flow

index.

Figure 6.32 also shows that when flow index is greater than 0.4, the critical Ta

number obtained for radius ratio = 0.67 is generally higher than that o f radius

ratio = 0.84. This is in agreement with data on the Newtonian liquids (Section

5.2.3). However, it is noted the extent of increase in the critical Ta number is

gradually decreased with decreasing flow index. When flow index is lesser

than 0.4, an adverse effect appeared, that is the critical Ta number obtained for

radius ratio = 0.67 was found to be lower than that of radius ratio = 0.84. It is

suggested that the stability of the Couette flow not only depends on the radius

ratio but also the flow index.

As shown previously in Fig. 5.11 that as the flow index decreases, the radial

variation of shear stress across the annular gap becomes more pronounced and

the shear stress distribution becomes highly hyperbolic. As the gap width

increases, the effect of flow index on the critical Ta number becomes more

prominent and the rate of decrease of critical Ta number with respect to flow

index is faster in the wide gap geometry (radius ratio = 0.67) than that of the

narrow gap geometry (radius ratio = 0.84).


2000

1800

1600 -

o1400(Ü

(D 1200Ic 1000o>%^ 800

8g 600 O

400

200 +

0

a = 0.67

a = 0.84

□a = 0.84

a = 0.67

■ carboxymethyl cellulose solution A carboxymethyl cellulose solution □ xanthan gum solution A xanthan gum solution

0.1 0.2 0.3 0.4 0.5 0.6 0.7Flow Index (n)

Figure 6.32 The effect of flow index on critical Taylornumber at different radius ratios, a


6.3.3 The effect of inner cylinder geometry

Couette-Taylor flow is claimed to induce an intense local radial mixing with

only a small amount of axial dispersion. This assumption can be tested by

investigating the effect of gap width and inner cylinder geometry on the Pe

number over a range o f Ta numbers. Figure 6.33 shows the different designs of

inner rotating unit used in the present study and their specification, given in

Table 6.6. The RTD experiments were performed by using a 85% Newtonian

glycerol water solution.

Shaft SI

Inlet \ Outlet

Inner cylinderOuter cylinder

Shaft S2

Inlet Outlet

Figure 6.33 The influence of inner shaft geometry

SI S2Gap width (mm) 7 7

R] (mm) 90 90Ri (mm) 76 76

Blades size - 55 X 27 (28 pcs)

Table 6.6 The specification of different inner rotating cylinder geometry


The effect o f rotational speed, expressed as Ta number, on the Pe, is shown in

Fig. 6.34 for three axial flow rates expressed as Re number. When the speed o f

rotation o f the inner cylinder is zero, the flow regime in the annular gap is

laminar for both shafts SI and 82, a fully developed laminar liquid velocity

profile exists in the gap giving a high value of Pe. However, with laminar flow

the radial mixing is poor, resulting in poor heat and mass transfer (Harrod,

1986). As the speed of rotation is slowly increased, the liquid flow become

unstable and the Pe number decreases. For shaft 81, Taylor vortices develop

causing an increase in the Pe number again and a narrowing of the RTD curves.

The maximum points in the figure (highest Pe) shown in Fig. 6.34 correspond

to the conditions at which the Taylor vortices are fully established along the

column and flow approaches near plug flow condition. With further increase in

the speed o f rotation of the inner cylinder intermixing and exchange of material

occurs at the boundaries between the neighbouring vortices causing an increase

in the axial dispersion and a broadening of the RTD curves, and hence a

decrease in the value of Pe number is observed. Figure 6.35 indicates that the

introduction of the blades on the inner shaft has a dramatic effect on the RTD.

The Pe number falls sharply as Ta number is increased, regardless of the axial

flow conditions. Considerable back mixing is caused by the blades. Taylor

vortices are totally eliminated by the backmixing action of the blades. The Pe

number given in Fig. 6.34 are described by using an a single parameter axial

dispersion model.


400

350 6

300 ^

0) 250 Û.

I 2003C%o 150 0_

100

50

- \ 'I

I4

85% glycerol solution Radius ratio = 0.84 Shaft S1

- -A - Re number = 1- -o - Re number = 1.5 - c - Re number = 2

\

/AI t

I''V

<=-wr--a

1000 2000 3000 4000


5000 6000

Figure 6.34 The effect of Taylor number on Peclet numberfor Newtonian liquids (Shaft SI)


400

350

300

Q) 250 CL

OE 2003Co5Ô 150 Û.

100

50

0

85% glycerol solution Radius ratio = 0.84 Shaft 82

A O-

0

- c - Re number = 1

- -o - Re number = 2

- -A - Re number = 3

1750 3500 5250


7000

Figure 6.35 The effect o f Taylor number on Peclet numberfor Newtonian liquids (Shaft S2)


6.4 Comparison of Theoretical and Experimental Results

Comparison of experimental data with theoretical results obtained in Chapter 5

are shown in Figure 6.36 for Newtonian liquids over a range of Reynolds

number from 0 to 70. The different symbols shown in Fig. 6.36 represent the

experimental critical Ta numbers obtained from the Peclet number - Taylor

number plots such as those shown in Figs. 6.17-22 for different concentration

of glycerol solutions while the solid curves are the simulations. The agreement

between theory and experiments is considered satisfactory. Both theoretical

and experimental results show that the critical Ta number is an increasing

function of radius ratio, a.

Figure 6.36 indicates the adequacy of the theoretical predictions for Newtonian

liquids. However, limited experimental data were obtained for a stationary

outer cylinder, i.e. P = 0. Theoretical prediction obtained in Chapter 5 showed

that the effect o f angular speed ratio and the direction of rotation of the

cylinders also play a significant role in determining the critical Ta number.

Comparison of experimental data with theoretical results obtained are shown in

Fig. 6.37 for non-Newtonian liquids over a range o f flow index number from 0

to 70. The different symbols shown in Fig. 6.37 represent the experimental

critical Ta numbers obtained from the Peclet number - Taylor number plots

such as those shown in Figs. 6.23-30 for different flow index obtained from

different concentration of CMC solutions and xanthan gum solutions while the

continuous solid curves are the simulations for non-Newtonian liquids. The

agreement between theory and experiments is considered satisfactory. Both

theoretical and experimental results show that pseudoplasticity has a

considerable destabilising influence on the Couette flow. Thus, a decrease in

flow index value, n, is accompanied by a decrease of the critical value of the Ta

number.


The theoretical simulations and experimental data in Fig. 6.37 also show that

the effect of flow behaviour index on the critical Ta number as a function o f a

is complex and depends on the value of n. For flow behaviour index, n, less

than 0.4 the critical Ta number increases as a increases, while o f n greater than

0.4 the reverse occurs. This was explained in terms of changes in the stress

distribution within the gap as a function of radial position for different values

of flow behaviour index (see Section 6.3.2).


E3CL_o

S*u-o

14000o : 85% glycerol solution♦ : 85% glycerol solution o : 64% glycerol solution# : 64% glycerol solution □ : 45% glycerol solution ■ ; 45% glycerol solution

a = 0.6

12000

a = 0.7

10000 a = 0.

a = 0.9

8000

a = 0.67

6000

4000

2000 a = 0.84

0806020 400


Figure 6.36 Comparison of theoretical and experimental resultsfor Newtonian liquids


2500

CMC

2000

X-Gum

a = 0.9

■ a = 0.84 A a = 0.66 □ a = 0.84 A a = 0.66

a = 0.8

500 -a = 0.7

a = 0.6

0.3 0.4Flow Index, n (-)

0.5 0.6 0.70.2

Figure 6.37 Comparison of theoretical and experimental resultsfor non-Newtonian liquids

Conclusions 205

CHAPTER SEVEN

CONCLUSIONS

7.1 Conclusions

The liquid motion induced between a pair of rotating cylinders with a small

amount of superimposed axial flow has potential applications in many chemical

and biochemical operations. In the present study, the onset o f flow instability

(defined as critical Taylor number) of a range of Newtonian and non-

Newtonian liquids in Couette flow has been considered both theoretically and

experimentally.

In the mathematical analysis, the theoretical study on the flow instability o f a

Newtonian liquid in Couette flow has been extended to include cases o f wide

radius ratio, angular speed ratio and the direction of rotation of the cylinders

with and without axial flow.

The present results indicated that the Galerkin method in conjunction with

simple polynomial expansion functions yielded accurate results for a variety of

flow stability problems. The numerical simulations showed that Couette flow

was stabilised (i.e. critical Ta number increased) due to an increase in the gap

width (radius ratio decreased). For a fixed radius ratio, the critical Ta number

decreased in the case of co-rotating and increased in the case of counter-

rotating of the rotating cylinders as a function o f angular speed ratio. In the

presence of a small amount of axial flow, the critical Ta number was found to

increase with an increase in axial flowrate when radius ratio and angular speed

ratio of the rotating cylinders were held constant.

Conclusions 206

Another important theoretical study of Couette flow concerned the effect of

non-Newtonian behaviour of the liquid on the critical Ta number. The

modified governing flow instability equations including the power law flow

index, n, were solved in a satisfactory manner by using the Galerkin method.

The theoretical results showed that critical Ta number was considerably smaller

than for the corresponding case of Newtonian liquids. It was observed that the

variation in the value of critical Ta number due to the rheological was

significant in all cases and dramatic in some, notably when the flow index was

less than 0.4. Generally, it was found that pseudoplasticity had a destabilising

influence on Couette flow. Thus, a decrease in flow index, n, was

accompanied by a decrease in the critical Ta number. In the ranges of the flow

index, O.I < n < 1, the radius ratio, 0.1 < a < 0.95, and the angular speed ratio,

-0.5 < p < 1, the effect of operating variables on the stability limit was

predicted to be almost the same for non-Newtonian liquids as for Newtonian

ones.

The present work was concerned with an experimental and theoretical

investigation o f the flow instability o f a range of Newtonian glycerol solutions

and non-Newtonian liquids in Couette flow. The data on flow instability

(critical Taylor number) were obtained by measuring the residence time

distribution of flow under various conditions of rotational-axial flows. The

operational conditions covered the transition form Couette flow to Couette-

Taylor flow regime.

All the experimental results showed that axial dispersion model could be

successfully used to study the axial mixing of the tracer solution in the main

Couette flow and in turn, the mixing properties o f the liquids could be

quantitatively interpreted in terms of a dimensionless Peclet number. When Pe

approached zero, there existed significant dispersion and hence the flow was

Conclusions 207

mixed flow. When Pe approached infinity, the dispersion was negligible and

hence the flow was near plug flow.

The experimental results indicated that Pe number was dependent upon the

rotational speed o f cylinders, axial Re number and the rheological properties of

the liquid medium with little dependence on the radius ratio. The effect of

rotational speed, expressed as Ta number, on the Pe number, indicated that

when the speed o f rotation of the inner cylinder was zero, the flow regime in

the annular gap was laminar giving a high value o f Pe number. As the speed of

rotation was slowly increased, the Pe number decreased. Formation o f Taylor

vortices caused an increase in the Pe number again. With further increased in

the speed o f rotation, a decrease of Pe number. A typical complete Pe number -

Ta number plot covering the transition from Couette flow to Taylor-Couette

was shown in Fig. 6.34. The variation in Pe number as a function of rotational

speed was successfully described qualitatively in terms of changes in flow

structure within the annular region and quantitatively by comparing the

experimental results with theoretical simulations.

In the present study, the critical Ta number was defined by the peak in the plot

of Pe number versus Ta number for a given axial Re number. Results showed

that the value of critical Ta number was an increasing function of gap width

and the Re number for both Newtonian and non-Newtonian liquids. The

critical Ta number was also reported for a wide range of non-Newtonian liquids

whose rheological properties were expressed in term of a power law flow

index, n. The results showed that decrease in the value of flow index caused a

decrease in the critical Ta number. This indicated that for otherwise similar

conditions the flow of a non-Newtonian, shear thinning, liquid in the gap of a

pair of counter rotating cylinders was significantly less stable compared with a

Newtonian liquid. Theoretical simulations and experimental observations of

Conclusions 208

flow instability with and without superimposed axial flow were in good

agreement.

7.2 Recommendations for further work

In the present study, theoretical equations were developed and were

numerically solved for the onset o f flow instability in Couette flow. The

formulation was applicable to flow of both Newtonian and non-Newtonian

liquids. The numerical simulations were presented in terms of the critical Ta

number as a function of four dimensionless groups affecting it, i.e. the Re

number, the dimensionless radius ratio, and the angular speed ratio of

concentric rotating cylinder and the power law flow index o f the liquid

properties.

The experimental observations based on the RTD data and the associated

numerical simulations presented have important implications for the design and

operation of equipment in which Taylor-Couette flow was used to enhance

transport and reaction processes. These operations should be part o f any future

practical studies o f Taylor-Couette flow.

The present theoretical model is limited to power-law shear thinning liquids

only. Many industrially important liquids have viscoelastic and time-dependent

properties. The formulation of the theory described in the present study should

serve as a basis for future studies rheologically more complex liquids in

Couette system.

Conclusions 209

Experimentally, the design o f the equipment used in the present study was such

that only the inner cylinder could be rotated. Any fiiture research in this area

should modify the apparatus to allow independent rotation o f the outer

cylinder. The data obtained from experiments with this modified device could

then be compared with theoretical simulations (Fig. 5.6) to provide further

confirmation o f the modelling presented in this thesis.

The theoretical analysis presented in this study demonstrates the mathematical

complexity of the problem even for the simple case two coaxial cylinder

geometry. As far as the author is aware no theoretical solution exist for the

case of a rotating blade in the shell. It may be possible to obtain reasonable

simulations of the flow in such a case by using a computational fluid dynamics

(CFD) and it is recommended that this should form part of any further work in

this area.

Nomenclature 2 10

NOMENCLATURE

Notations

A = Constant (s-')a = Wave number (-)

an = Coefficient of trial function (-)

B = Constant (s')b = Gap width of the rotating cylinder (m)

bn = Coefficient o f trial function (-)

Ce = Concentration (kg/m^)

Ct = Concentration (kg/m^)

D = Axial dispersion coefficient (m" s ')

D = Differential operator (-)

D = Dimensionless differential operator (-)

D* = Differential operator (-)

D* = Dimensionless differential operator (-)

= Differential operator (-)

= Dimensionless differential operator (-)

F = Dimensionless axial flow velocity (-)

Fg = Geometrical factor in Eq. [2.4] (-)

K = Consistency index (Pa s')

K2 = Parameter defined in Eq. [3.75] (-)

L = Distance between two measuring points (m)

M = Dimensionless angular speed, (-)

N = Number of term used in the trial

and weighting functions (-)

Nomenclature 211

N

n

P

P

Pe

Q

r

Ri

R2

Ray

Re

Ro

s

t

Ta

Tac

u

Um

Un

V

V(r)

Vm

Vn

W

W

W(r)

X

z

= Rotational speed of cylinder

= Flow index

= Dimensionless viscosity

= Pressure

= Peclet number [= WLD" ]

= Dimensionless parameter based on the magnetic

and electric properties in Eq. [2.2]

= Radial coordinate

= Radii of inner cylinders

= Radii of outer cylinders

= Rayleigh number

= Axial Reynolds number [= W(R2 - Ri )v“ ]

= Mean radius

= Growth rate of disturbances

= Time

= Taylor number [=-4Q o(R 2 - Ri)Av"^ ]

= Critical Taylor number

= Radial velocity component

= Weighting function

= Trial fimction

= Tangential velocity component

= Tangential velocity

= Weighting fimction

= Trial function

= Axial flow velocity

= Average axial flow velocity

= Axial velocity component

= Transformed dimensionless radial coordinate

= Axial coordinate

(rps)

(-)

(-)

(Pa)

(-)

(-)(m)

(m)

(m)

(-)

(-)

(m)

(-)

(s)

(-)

(-)(m s'')

(-)

(-)(m $■')

(m s'')

(-)

(-)(m s'')

(m s'')

(m s'')

(-)(m)

Nomenclature 212

Greeks Symbols

a = Radius ratio, (-)

P = Angular speed ratio, (-)

ÿ = Shear rate (s')

Ô = Parameter defined in Eq. [3.25] (-)

8 = Weighting function (-)

r = Dimensionless constant. (-)

n = Apparent viscosity (kgm‘ s"

Tim = Mean apparent viscosity (kgm‘ s'

X = Dimensionless wave number (m-‘)

= Viscosity o f the working liquid (kgm" s"

ILL» = Upper limit viscosity of the working liquid (kgm' s'

14 = Apparent viscosity o f the working liquid (kgm‘ s'

Ho = Zero-shear viscosity of the working liquid (kgm' s"

Hp = Plastic viscosity of the working liquid (kgm'

V = Kinematics viscosity of the working liquid (kgm' s‘^

0 = Dimensionless time (-)

0 = Tangential coordinate (m)

CT = Dimensionless growth rate (-)

cr' = Variance (s') •

= Dimensionless variance (-)

Aae^ = Dimensionless variance difference (-)

T = Shear stress (Pa)

Xy = Yield stress (Pa)

Û) = Angular velocity given by Eq. (16) (rad')

Q(r) = Angular velocity (rad')

Ü1 = Angular velocities o f inner cylinders (rad')

Nomenclature 213

Q2 = Angular velocities of outer cylinders (rad‘ )

Qo = Average angular velocity (rad' )

= Dimensionless variable (-)

v|/(r) = Perturbation on the magnetic field in Eq. [2.2] (-)

Superscripts and Subscripts

= Average value

' = Perturbed quantity

~ = Eigenfunction

= Dimensionless eigenfunction

c = Critical value

Abbreviation

CMC = Carboxymethyl cellulose

Pe = Peclet number

Re = Reynolds number

RTD = Residence time distribution

Ta = Taylor number

References 214

REFERENCES

Abdallah, Y.A.G. and Coney, J.E.R. 1988. Adiabatic and diabatic flow studies

by shear stress measurements in annuli with inner cylinder rotation, J.

Fluid Engng, Trans. ASME, 110(12), 399-405.

Andereck, C.D., Dickman, R. and Swinney, H.L. 1983. New flows in a circular

Couette system with co-rotating cylinders, Phys. Fluids, 26, 1395-1401.

Andereck, D., Liu, S. S. and Swinney, H. L. 1986. Flow regimes in a circular

Couette system with independently rotating cylinders, J. Fluid Mech.,

164, 155-183.

Astill, K.N. 1964. Studies o f the developing flow between concentric cylinders

with the inner cylinder rotating, J. Heat Transfer, Trans. ASME, C86,

383-392.

Beaudoin, G. and Jaffrin, M.Y. 1989. Plasma filtration in Couette flow

membrane devices, Artif. Organs, 13(1), 43-51.

Becker and Kaye, J. 1962. The influence of a radial temperature gradient on the

instability o f fluid flow in an annulus with an inner rotating cylinder, J.

Heat Transfer, Trans. ASME, C84, 106-111.

Benjamin, T.B. 1978a. Bifurcation phenomena in steady flow of a viscous

fluid. I. Theory, Proc. R. Soc. London, A359, 1-26.

Benjamin, T.B. 1978b. Bifurcation phenomena in steady flows o f a viscous

fluid. II. Experiments, Proc. R. Soc. London, A 359, 27-43.

Bird, R.B., Stewart, W.E. and Li^tfoot, E.N. 1960. Transport phenomena.

Chapter III, John Wiley and Sons, Singapore.

Burkhalter, J. E. and Koschmieder, E. L. 1973. Steady supercritical Taylor

vortex flow, J. Fluid Mech., 58(3), 547-560.

Burkhalter, I.E. and Koschmieder, E.L. 1974. Steady supercritical Taylor

vortex after sudden starts, Phys. Fluids, 17(11), 1929-1935.

References 215

Chandrasekhar, S. 1953. The stability of viscous flow between rotating

cylinders in the presence of a magnetic field, Proc. Roy. Soc. A216,

293-309.

Chandrasekhar, S. 1954. The stability of viscous flow between rotating

cylinders, Mathematika, 1, 5-13.

Chandrasekhar, S. 1958. The stability o f viscous flow between rotating

cylinders in the presence of a magnetic field, Proc. Roy. Soc. A 246,

301-311.

Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. Chapter

VII, Oxford, Clarendon Press.

Chandrasekhar, S. 1962. The stability of spiral flow between rotating cylinders,

Proc. Roy. Soc., A 265, 188-197.

Chandrasekhar, S. and Elbert, D. 1962. The stability o f viscous flow between

rotating cylinders - II, Proc. Roy. Soc., A 268, 145-152.

Chung, K.C. and Astill, K.N. 1977. Hydrodynamic instability of viscous flow

between rotating coaxial cylinders with fully developed axial flow, J.

Fluid Mech., 81(4), 641-655.

Cohen, S. and Maron, D. M. 1983. Experimental and theoretical study of a

rotating annular flow reactor, Chem. Engng J., 27, 87-97.

Coles, D. 1965. Transition in circular Couette flow, J. Fluid Mech., 21(3), 385-

425.

Coney, J.E.R. and Simmers, D.A. 1979. A study of fully-developed, laminar,

axial flow and Taylor vortex flow by means of shear stress

measurements, J. Mech. Engng Sci., 21(1), 19-24.

Croockewit, P., Honig, C. C. and Kramers, H. 1955. Longitudinal diffusion in

liquid flow throu^ an annulus between a stationary outer cylinder and a

rotating inner cylinder, Chem. Engng Sci., 4, 111-118.

Danckwerts, P. V. 1953. Continuous flow systems - Distribution of Residence

times, Chem. Eng. Sci., 2(1), 1-13.

References 216

Di Prima, C. and Pridor, A. 1979. The stability o f viscous flow between

rotating concentric cylinders with an axial flow, Proc. Roy. Soc.

London, Ser. A 246, 312-325.

Di Prima, R.C. 1955. Application of the Galerkin method to problems in

hydrodynamic stability, Quart. Applied Mathematics, 13, 55-62.

Di Prima, R.C. 1960. The stability o f a viscous flow between rotating cylinders

with an axial flow, J. Fluid Mech., 9, 621-631.

Di Prima, R.C. and Swinney, H.L. 1981. Instabilities and transition in flow

between concentric rotating cylinders. In Hydrodynamic Instabilities

and the Transition to Turbulence. Eds. Swinney and Gollub, Springer,

Berlin, Heidelberg, New York, 139-180.

Donnelly, R. J. 1991. Taylor-Couette flow: The early days. Physics today, 11,

32-39.

Elliott, L. 1973. Stability o f a viscous fluid between rotating cylinders with

axial flow and pressure gradient round the cylinders, Phys. Fluids, 16,

577-580.

Enokida, Y., Nakata, K. and Suzuki, A. 1989. Axial turbulent diffusion in fluid

between rotating coaxial cylinders, A.I.Ch.E. J., 35, 1211-1214.

Fenstermacher, P.R., Swinney, H.L. and Gollub, J.P. 1979. Dynamical

instabilities and the transition to chaotic Taylor vortex flow, J. Fluid

Mech., 94, 103-128.

Fletcher, C.A.J., 1984. Computational Galerkin Methods. Springer series in

computational physics. Chap I, Springer-Verlag, New York.

Giesekus, J. 1972. On instabilities in Poiseuille and Couette flows of

viscoelastic fluids. Prog. Heat Mass Transfer, 5, 187-193.

Ginn, R.F. and Denn, M.M. 1969. Rotational Stability in Viscoelastic liquids:

Theory, A.I.Ch.E. J., 15, 450-454.

Goldstein, S. 1937. The stability of viscous fluid flow between rotating

cylinders, Proc. Camb. Phil. Soc., 33,41-61.

References 217

Gorman, M. and Swinney, H.L. 1982. Spatial and temporal characteristics of

modulated waves in the circular Couette system, J. Fluid Mech., 117,

123-142.

Graebel, W.P. 1961. Stability of a Stokesian fluid in Couette flow, Phys.

Fluids, 4, 362-368.

Gu, Z.H. and Fahidy, T.Z. 1986. The effect o f geometric parameters on the

structure of combined axial and Taylor-vortex flow, Can. J. Chem. Eng.,

64, 185-189.

Gu, Z.H. and Fahidy, T.Z., 1985. Visualization of flow patterns in axial flow

between horizontal coaxial rotating cylinders. Can. J. Chem. Eng., 63,

14-21.

Guihard, L., Coeuret, F., Legrand, J., Fahidy, T.Z. and Gu, Z.H. 1989.

Circumferential mixing in the Taylor-Couette reactor, I.Chem.E. Symp.

Ser., 112, 105-117.

Harris D.L. and Reid, W.H. 1964. On the stability of viscous flow between

rotating cylinders. Part 2 Numerical analysis, J. Fluid Mech., 20, 95-

101.

Harrod, M. 1986. Scraped Surface Heat Exchangers - A literature survey of

flow patterns, mixing effects, residence time distribution, heat transfer

and power requirements, J. Food Process Eng., 9, 1-62.

Hasoon, M.A. and Martin, B.W. 1977. The stability of viscous axial flow in an

annulus with a rotating inner cylinder, Proc. Roy. Soc. London, A 382,

351-380.

Hoare, M., Narendranathan, T. J., Flint, J.R., Heywood-Waddington, D., Bell,

D.J. and Dunnill, P. 1982. Disruption of protein precipitates during

shear in Couette flow and in pumps, I & EC Fundamentals, 21(4), 402-

406.

Hughes, T.H. and Reid, W.H. 1968. The stability of spiral flow between

rotating cylinders, Phil. Trans. R. Soc. Lond., A 263, 57-91.

References 218

Kataoka, K. 1986. Taylor vortices and instabilities in circular Couette flow. In

Encyclopedia o f Fluid Mechanics, Vol. 1, 236-274, N.P. Cheremisinoff,

Gulf Publ., Houston.

Kataoka, K. and Takigawa, T. 1981. Intermixing over cell boundary between

Taylor vortices, A.I.Ch.E.J., 27(3), 504-508.

Kataoka, K., Doi, H. and Komai, T. 1977. Heat/mass transfer in Taylor vortex

flows with constant axial flow rates. Int. J. Heat Transfer, 20, 57-63.

Kataoka, K., Doi, H., Hongo, T. and Futagawa, M. 1975. Ideal plug-flow

properties o f Taylor vortex flow, J. Chem. Engng Jap., 8(6), 472-476.

Kaye, J. and Elgar, E C. 1958. Modes of adiabatic and diabatic fluid flow in an

annulus with an inner rotating cylinder, Trans. ASME., 80(4), 753-765.

Kirk-Othmer, 1979. Encyclopedia o f chemical technology, (3rd ed.) 12, John

Wiley and Sons., USA, 45-66.

Kirk-Othmer. 1979. Encyclopedia of chemical technology, (3rd ed.) 5, John

Wiley and Sons., USA, 143-163.

Koschmieder, E.L. 1979. Turbulent Taylor vortex flow, J. Fluid Mech., 93,

515-527.

Koschmieder, E.L. 1993. Taylor vortex Flow, In Bénard Cells and Taylor

Vortices, Part II, Cambridge University Press, USA.

Krueger, E.K. and Di Prima, R.C. 1964. The stability of a viscous fluid

between rotating cylinders with an axial flow, J. Fluid Mech., 19, 528-

538.

Kurzweg, U.H. 1963. The stability of Couette flow in the presence of an axial

magnetic field, J. Fluid Mech., 17, 52-60.

Larson, R.G., Shaqfeh, E.S.G. and Muller, S.J. 1990. A purely elastic

instability in Taylor-Couette flow, J. Fluid Mech., 218, 573-600.

Lathrop, D.P., Fineberg, J. and Swinney, H.L. 1992. Turbulent flow between

concentric rotating cylinders at large Reynolds number, Phys. Rev. Lett.,

68(10), 1515-1525.

References 219

Legrand, L. and Coeuret, F. 1986. Circumferential mixing in one-phase and

two-phase Taylor vortex flows, Chem. Eng. Sci., 41(1), 47-53.

Levenspiel, O. 1972. Chemical reaction Engineering, 2nd ed.. Chapter IX, J.

Wiley, New York.

Lewis, J.W. 1928. An experimental study of the motion o f a viscous liquid

contained between two coaxial cylinders, Proc. Roy. Soc. London, A

117, 388-406.

Meksyn, D. 1961. New methods in laminar boundary-layer theory. Chapter

XXIII & XXIV, London: Pergamon Press, New York.

Moore, C.M.V. and Cooney, C.L. 1995. Axial dispersion in Taylor-Couette

flow device, A.I.Ch.E.J., 41, 723-727.

Ng, S. and Turner, E.R. 1982. On the linear stability o f spiral flow between

rotating cylinders, Proc. R. Soc. Lond., A 382, 83-102.

Pfitzer, H. and Beer, H. 1992. Heat transfer in an annulus between

independently rotating tubes with turbulent axial flow. Int. J. Heat Mass

Transfer, 35(3), 623-633.

Pudjiono, P.I. and Tavare, N.S. 1993. Residence time distribution analysis

from a continuous Couette flow device around critical Taylor number,

Can. J. Chem. Engng, 71, 312-318.

Pudjiono, P.I., Tavare, N.S., Garside, J. and Nigam, K.D.P. 1992. Residence

time distribution from a continuous Couette flow device, Chem. Engng

J., 48, 101-110.

Rayleigh, O.M. 1916. On the dynamics of revolving fluids, Proc. Roy. Soc

London, A 93, 148-154.

Roberts, P.H. 1965. Appendix in experiments on the stability of viscous flow

between rotating cylinders. VI. Finite-amplitude experiments, Proc.

Roy. Soc. London, A 283, 531-556.

Rosant, J., Legrand, J., Rey, C. and Benmerzouka, A. 1994. Separated two-

phase flow in a rotating annulus: application to the extraction of liquid

phase, Chem. Engng Sci., 49(11), 1769-1781.

References 220

Rushton, A. and Zhang, G.S. 1988. Rotary microporous filtration,

Desalination, 70, 379-394.

Sinevic, V., Kuboi, R. and Nienow, A.W. 1986. Power numbers, Taylor

numbers and Taylor vortices in viscous Newtonian and non-Newtonian

fluids, Chem. Eng. Sci. 41, 2915-2923.

Snyder, A. 1970. Waveforms in rotating Couette flow. Intern. J. Nonlinear

Mech., 5, 659-685.

Snyder, H.A. 1969a. Change in waveform and mean flow associated with

wavelength variations in rotating Couette flow. Part 1, J. Fluid Mech.

35, 337-352.

Snyder, H.A. 1969b. Wave number selection at finite amplitude in rotating

Couette flow, J. Fluid Mech., 35, 273-298.

Soundalgekar, V.M., Ali, M.A. and Takhar, H.S. 1994. Hydromagnetic

stability o f dissipative Couette flow: wide-gap problem. Int. J. Energy

Res., 18, 689-695.

Sparrow, E.M., Munro, W.D. and Jonsson, V.K. 1964. Instability of the flow

between rotating cylinders: the wide gap problem, J. Fluid Mech., 20(1),

35-46.

Stuart, J.T. 1986. Taylor-vortex flow: a dynamical system, SIAM Rev., 28(3),

315-342.

Sun, Z.S. and Denn, M.M. 1972. Stability of Rotational Couette flow of

polymer solutions, A.I.Ch.E.J., 18, 1010-1015.

Takeuchi, D.I and Jankowski, D.F. 1981. A numerical and experimental

investigation o f the stability of spiral Poiseuille flow, J. Fluid Mech.,

102, 101-126.

Tam, W.Y. and Swinney, H.L. 1987. Mass transport in turbulent Couette-

Taylor flow, Phys. Rev. A36, 1374-1381.

Taylor, G.I. 1923. Stability of a viscous fluid contained between two rotating

cylinders, Phil. Trans., A 223, 289-343.

References 221

Tobler, W. 1982. Dynamic filtration: principle and application o f shear

filtration in an annular gap, Filtr. Sep., 19, 329-332.

Walden, R.W. and Donnelly, R.J. 1979. Re-emergent order of chaotic circular

Couette flow, Phys. Rev. Lett., 42(5), 301-304.

Walowit, J., Tsao, S. and Di Prima, R.C. 1964. Stability of flow between

arbitrarily spaced concentric cylindrical surfaces including the effect o f

a radial temperature gradient, Trans. ASME, 12, 585-593.

Wamecke, H., Pruss, J. and Langemann, H. 1985. On a mathematical model

for loop reactors - I. Residence time distribution, moments and

eigenvalues, Chem. Engng Sci., 40(12), 2321-2326.

Wen, Y. and Fan, L.T. 1975. Models for flow systems and chemical reactors,

Marcel Dekker, New York.

Wey, J.S. and Estrin, J. 1973. Modelling the batch crystallization process: the

ice-brine system, Ind. Eng. Chem. Process Des. Dev., 12(3), 236-246.

Appendix One 222

APPENDIX ONE

GALERKIN METHOD

The Galerkin method has been used to solve problems in structural mechanics,

dynamics, fluid flow, hydrodynamic stability, heat and mass transfer, acoustics,

microwave theory, neutron transport, etc. Problems governed by ordinary

differential equations, partial differential equations, and integral equations have

been investigated via the Galerkin formulation. Steady, unsteady, and

eigenvalue problems have proved to be equally amenable to the Galerkin

treatment. Essentially, any problem for which governing equations can be

written down is a candidate for a Galerkin method (Fletcher, 1984).

Basically, it is a method for finding the approximate solution of an eigenvalue

or eigenelement problems in the form of a linear combination of the elements

of given linear independent system. In engineering practice the exact solution

can only be known in a few simple cases and it is therefore important to see

how the solution behaves when one introduces an approximation. By

approximating the function u given by Eq. [A 1.1] as:

u = a i (})i+tt2 (|>2 +... [ A l . l ]

where Oi are unknown coefficients and the (|)i is a set of linearly independent

functions which are known. Oj are generalized coefficients although in some

cases they can be associated with nodal values of the variable under

consideration. In general fluid flow engineering problems, it is preferable to

use nodal values as they provide a clear physical meaning (such as boundary

conditions) via the finite elements, finite differences or the boundary element

method. In such cases the approximation for u can be written as:

Appendix One 223

U = U1 (j)l + U2 <j)2+... [A1.2]

u = ^uj(j)j [A1.3]j=i

where <t>j is a set o f linearly independent functions which are sometimes called

interpolation functions, uj are the nodal values of the field variable and its

derivative (which are generally any variable with physical meaning directly

related to u).

Introducing the approximation for u into the governing differential equation, an

expression is obtained which is no longer identically satisfied except for the

case where Eqs. [A l.l] or [A 1.2] can represent the exact solution. This

produces an error or residual function in the domain or the boundary due to the

inadequacy of the above equations. The error function in the domain is called

R. The numerical methods are used in controlling the value of R as small as

possible by setting its residual equal to zero for various values of the weighting

functions, i|/j, such that

jRv|/jdQ = 0 in n j = l ,2, . . . ,N [A1.4]n

where Q is the domain condition. These functions have to be linearly

independent. In the case of the Galerkin method the weighting functions are

the same as the approximating functions, i.e. (|)j = if/j; hence Eq.[A1.4] becomes:

fR<tijdQ = 0 j = l , 2 , . . . , N [A1.5]

Equation [A 1.5] will produce a system of algebraic equations from which the

unknown values o f Oi or Ui coefficients of u (in Eqs. [Al. l] or [A1.2]) can be

obtained. The approximation can always be improved by increasing the

number of N functions used, where N is the number of terms in the

approximate solution equal to the number of weighting functions required.

Appendix Two 224

APPENDIX TWO

MATHEMATICA PROGRAM FOR FLOW

INSTABILITY OF NEWTONIAN LIQUIDS

IN COUETTE FLOW

" I N P U T THE S YS TEM GEOMETRY & P R O P E R T I E S " ;" n = m a t r i x s i z e r e q u i r e d" n b a = n u m b e r o f d i m e n s i o n l e s s w a v e n u m b e r i n p u t" i a = v a l u e s o f d i m e n s i o n l e s s w a v e n u m b e r i n p u t" n b c = n u m b e r o f d i m e n s i o n l e s s d i s t u r b a n c e g r o w t h

r a t e i n p u t" i c = v a l u e s o f d i m e n s i o n l e s s d i s t u r b a n c e g r o w t h

r a t e i n p u t

C l e a r [ n , n b a , i a , n b c , i c ]

n = 4 ; n b a = 1 5 ;i a = { 2 . 8 , 2 . 9 , 3 . 0 , 3 . 1 , 3 . 1 1 , 3 . 1 2 , 3 . 1 3 , 3 . 1 4 ,

3 . 1 5 , 3 . 1 6 , 3 . 1 7 , 3 . 1 8 , 3 . 1 9 , 3 . 2 , 3 . 3 } ; n b c = 3 ;i c = { l . 1 6 6 , 1 . 1 6 9 , 1 . 1 7 2 } ;

^ ^ ' k ' k ' k ' k ' i r ' i e ' i f ' k ' k ' k ' k - k - k - k ' k - k ' k - k ' k ' i r - k - k - k ' k - k - k - k ' k ' k - k - k ' k - k ' k ' k - k ' k ' k - k ' k ' k ' k - k - k - k - k - k ' k - k - k - k ' k - i e ' k ^ ^ m

" I N P U T THE S YS TEM GEOMETRY & P R O P E R T I E S " ;" a = d i m e n s i o n l e s s w a v e n u m b e r " ;" c = d i m e n s i o n l e s s d i s t u r b a n c e g r o w t h r a t e " ;" r e x = a x i a l R e y n o l d s n u m b e r " ;" r l = r a d i u s o f i n n e r c y l i n d e r " ;" r 2 = r a d i u s o f o u t e r c y l i n d e r " ;" q l = a n g u l a r s p e e d o f i n n e r c y l i n d e r " ;" q 2 = a n g u l a r s p e e d o f o u t e r c y l i n d e r " ;

C l e a r [ a , c , r e x , r l , r 2 , q l , q 2 ]

a [ w _ ] : = i a [ [w] ]c [ m _ ] : = i c [ [m] ]r e x = 1 0 ;r l = 0 . 8 ;r 2 = 1 . 0 ;q l = l ;q 2 = 0 ;

Appendix Two 225

" D E F I N E THE SYS TEM P A R A M E T E R S " ;" h = r a d i u s r a t i o o f c o n c e n t r i c r o t a t i n g c y l i n d e r s" u = a n g u l a r s p e e d r a t i o o f c o n c e n t r i c r o t a t i n g

c y l i n d e r s " s i g = f u n c t i o n o f c , a a n d r e x " ;" k k = f u n c t i o n o f a , s i g , r e x a n d f " ;" s = g e o m e t r i c a l p a r a m e t e r "" e = g e o m e t r i c a l p a r a m e t e r "" y = g e o m e t r i c a l p a r a m e t e r "" g = g e o m e t r i c a l p a r a m e t e r "" f = g e o m e t r i c a l p a r a m e t e r "

C l e a r [ h , u , s i g , k k , s , e , y , g , f ]

h = r l / r 2 ;u = q 2 / q l ;s i g [ m _ , w _ ] : = c [ m ] a [ w ] r e x ;k k [ m _ , w _ ] : = a [ w ] ^ 2 - ( s i g [ m , w ] - a [ w ] r e x f ) I ;s = 2 ( r 2 - r l ) / ( r 2 + r l ) ;e = ( 1 + s x ) ^ ( - 1 ) ;y = ( 1 + s x ) / ( l - s / 2 ) ;g = 2 / ( ( l + u ) ( l - h " 2 ) ) ( ( u - h ^ 2 ) + ( l - u ) y " ( - 2 ) ) ;f = ( ( - 2 ) / ( ( l - h ^ 2 ) + ( ( l + h " 2 ) N [ L o g [ h ] ] ) ) ) ( ( l - h ^ 2 )N [ L o g [ y ] ] +

( h " 2 ) ( y " 2 - l ) N [ L o g [ h ] ] ) ;

" D E F I N E T HE D I F F E R E N T I A L O P E R A T O R ,T R I A L F U N C T I O N & W E I G H T I N G F U N C T I O N " ;

" d = d i f f e r e n t i a l o p e r a t o r " ;" d z = d i f f e r e n t i a l o p e r a t o r " ;" u t = t r i a l f u n c t i o n " ;" v t = t r i a l f u n c t i o n " ;" u w = w e i g h t i n g f u n c t i o n " ;" v w = w e i g h t i n g f u n c t i o n " ;^ ^ ' k - k ' k ' k - k ' k - k ' k ' k ' k - k - k ' i r - i r - k - k ' i c - k ' k ' k - k ' k - k ' k ' k ' k ' k - i e - k ' ^ ' k - k - k - k - i e ' k ' k - k ' k ' k ' k - k ' k - i f ' k ' k - k ' k - k ' k ' k - k - i ^ ' k ^ ^ »

C l e a r [ d , d z , u t , v t , u w , v w ]

d [ i _ ] : = D [ i , x ]d z [ i _ ] : = D [ i , x ] + s e i

u t [ i _ ] : = ( x ^ 2 - l / 4 ) ^ 2 x ^ ( i - l ) v t [ i _ ] : = ( x ^ 2 - l / 4 ) x ^ ( i - l )

u w [ i _ ] : = e ( x ^ 2 - l / 4 ) ^ 2 x ^ ( i - l ) v w [ i ] : = e ( x ^ 2 - l / 4 ) x ^ ( i - l )

Appendix Two 226

" D E F I N E T HE GOVE RNI NG E Q U A T I O N S " ;

C l e a r [ q x a , q x b , q y a , q y b ]

" q x a [ u ] + q x b [ v ] t = = 0 &&" ;" q y a [ v ] + q y b [ u ] = = 0 " ;

q x a [ i _ , m _ , w _ ] : = d [ d z [ u t [ i ] ] ] - k k [ m, w] u t [ i ] - l / a [ w ] ^ 2 ( d [ d z [ d [ d z [ u t [ i ] ] ] ] ] - d [ k k [ m , w ] d z [ u t [ i ] ] ] ) + r e x / ( I a [ w ] ) d [ d [ f ] u t [ i ] ]

q x b [ i _ , m _ , w _ ] : = - g v t [ i ]q y a [ i _ , m _ , w _ ] : = u t [ i ]q y b [ i , m , w ] : = - ( d [ d z [ v t [ i ] ] ] - k k [ m , w ] v t [ i ] )

" A P P L Y G A L E R K I N ME T H O D " ;

C l e a r [ f n , x a , x b , y a , y b ]

f n = F u n c t i o n [ ( i , j } , N I n t e g r a t e [ i j , { x , - 1 / 2 , 1 / 2 } ] ] ;

x a [ i _ , m _ , w _ , j _ ] : = f n [ q x a [ i , m , w ] , u w [ j ] ] x b [ i _ , m _ , w _ , j _ ] : = f n [ q x b [ i , m , w ] , u w [ j ] ] t y a [ i _ , m _ , w _ , j _ ] : = f n [ q y a [ i , m , w ] , v w [ j ] ] y b [ i _ , m _ , w _ , j _ ] : = f n [ q y b [ i , m , w ] , v w [ j ] ]

" A P P L Y M A T R I X S YSTEM TO S OL VE T a y l o r n u m b e r , t " ;

C l e a r [ r x a , r x b , r y a , r y b , m a t ]

r x a [ m _ , w _ , j _ , k _ ] : = T a b l e [ x a [ i , m , w , j ] , { i , 1 , k } ] r x b [ m _ , w _ , j _ , k _ ] : = T a b l e [ x b [ i , m , w , j ] , { i , 1 , k } ] r y a [ m _ , w _ , j _ , k _ ] : = T a b l e [ y a [ i , m , w , j ] , { i , 1 , k } ] r y b [ m _ , w _ , j _ , k _ ] : = T a b l e [ y b [ i , m , w , j ] , { i , 1 , k } ]

m a t [ m _ , w _ , k _ ] : = F l a t t e n [ { T a b l e [ F l a t t e n [ { r x a [ m , w , j , k ] , r x b [ m , w , i , k ] } ] , { ] , l , k } ] .T a b l e [ F l a t t e n [ { r y a [ m , w , j , k ] , r y b [ m , w , j , k ] } ] , { j , 1 , k } ] } , 1 ]

" F ORMAT THE O U T P U T " ;

C l e a r [ s o l s , a n s , o k l , o k 2 , r e s O l ]

Appendix Two 227

s o l s [ m _ , w _ , k _ ] : = S o l v e [ D e t [ m a t [ m, w , k ] ] = = 0 , t ] a n s [ m _ , w _ , k _ ] : = S i m p l i f y [ t / . s o l s [ m , w , k ] ]

o k l [ m _ , w _ ] : = T a b l e [ { { " C - v a l u e = " , c [ m ] } , { " A l p h a = " , a [ w ] } , { " R e y . N o . = " , r e x } , { " R a d i i % = " , h } ,{ " V e l . % = " , u } } ] / / C o l u m n F o r m

o k 2 [ m _ , w _ ] : = T a b l e [ a n s [ m , w , n ] ] / / C o l u m n F o r m

r e s 0 1 = T a b l e [ T a b l e [ T a b l e [ { o k l [ m , w ] , o k 2 [ m , w ] } , { m , 1 , n b c } ] / / T a b l e F o r m , { w , 1 , n b a } ] / / T a b l e F o r m ] / / T a b l e F o r m

T a b l e F o r m [ T a b l e F o r m

C - v a l u e = , 1 . 1 6 6 } 2 1 7 8 . 0 3 - 1 2 . 7 7 0 1 IA l p h a = , 2 . 8 } 3 7 5 3 1 . 1 - 2 8 0 7 . 5 5 IR e y . N o . = , 1 0 } 3 7 5 5 6 7 . - 4 0 6 7 5 . 3 IR a d i i % = , 0 . 8 } 6V e l . % — f 0} 2 . 7 1 0 4 1 1 0 - 2 9 8 4 7 2 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 7 6 . 8 2 - 2 6 . 8 4 1 1 IA l p h a = , 2 . 8 } 3 7 5 2 7 . 8 - 2 9 0 9 . 4 7 IR e y . N o . = , 1 0 } 3 7 5 5 5 0 . - 4 1 1 4 9 . 1 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 7 1 0 3 1 0 - 3 0 0 5 2 9 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 7 5 . 5 6 - 4 0 . 9 1 1 7 IA l p h a = , 2 . 8 } 3 7 5 2 4 . 3 - 3 0 1 1 . 3 8 IR e y . N o . = , 1 0 } 3 7 5 5 3 3 . - 4 1 6 2 3 . IR a d i i % =, 0 . 8 } 6V e l . % = , 0} 2 . 7 1 0 1 9 1 0 - 3 0 2 5 8 7 . I

C - v a l u e = , 1 . 1 6 6 } 2 1 5 9 . 2 7 - 5 . 4 1 6 9 7 IA l p h a = f 2 . 9 } 3 5 8 8 5 . 1 - 2 7 3 7 . 8 3 IR e y . N o . = , 1 0 } 3 5 4 2 3 7 . - 3 9 4 9 1 . 4 IR a d i i % = , 0 . 8 } 6V e l . % = f 0} 2 . 5 4 4 0 9 1 0 - 2 8 9 5 0 6 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 5 8 . 1 2 - 1 9 . 5 4 3 2 IA l p h a = , 2 . 9 } 3 5 8 8 1 . 8 - 2 8 3 7 . 8 6 IR e y . N o . = , 1 0 } 3 5 4 2 2 0 . - 3 9 9 5 2 . 6 IR a d i i % = , 0 . 8 } 6V e l . % = , 0 } 2 . 5 4 3 9 8 1 0 - 2 9 1 5 0 2 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 5 6 . 9 3 - 3 3 . 6 6 8 9 IA l p h a = , 2 . 9 } 3 5 8 7 8 . 3 - 2 9 3 7 . 8 9 IR e y . N o . = , 1 0 } 3 5 4 2 0 2 . - 4 0 4 1 3 . 8 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 5 4 3 8 7 1 0 - 2 9 3 4 9 9 . I

Appendix Two 228

C - v a l u e = , 1 . 1 6 6 } 2 1 4 7 . 5 1 + 2 . 5 6 1 4 5 IA l p h a = , 3 . } 3 4 4 1 7 . - 2 6 7 3 . 5 9 IR e y . N o . = , 1 0 } 3 3 5 0 3 9 . - 3 8 3 9 4 . 6 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 3 9 4 2 2 1 0 - 2 8 1 1 8 8 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 4 6 . 4 3 - 1 1 . 6 4 9 8 IA l p h a = , 3 . } 3 4 4 1 3 . 6 - 2 7 7 1 . 9 4 IR e y . N o . = , 1 0 } 3 3 5 0 2 2 . - 3 8 8 4 4 . 2 IR a d i i % = , 0 . 8 } 6V e l . % — r 0} 2 . 3 9 4 1 1 1 0 - 2 8 3 1 2 9 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 4 5 . 3 - 2 5 . 8 6 0 6 IA l p h a = , 3 . } 3 4 4 1 0 . 2 - 2 8 7 0 . 2 9 IR e y . N o . = , 1 0 } 3 3 5 0 0 5 . - 3 9 2 9 3 . 7 IR a d i i % = , 0 . 8 } 6V e l . % = , 0 } 2 . 3 9 4 1 0 - 2 8 5 0 6 9 . I

C - v a l u e = , 1 . 1 6 6 } 2 1 4 2 . 2 + 1 1 . 1 8 8 8 IA l p h a = , 3 . 1 } 3 3 1 0 4 . 8 - 2 6 1 4 . 2 9 IR e y . N o . = , 1 0 } 3 1 7 7 0 6 . - 3 7 3 7 6 . 7 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 5 8 7 1 1 0 - 2 7 3 4 5 8 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 4 1 . 1 8 - 3 . 1 3 6 0 4 IA l p h a = f 3 . 1 } 3 3 1 0 1 . 5 - 2 7 1 1 . 1 4 IR e y . N o . = , 1 0 } 3 1 7 6 8 9 . - 3 7 8 1 5 . 5 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 5 8 6 1 0 - 2 7 5 3 4 6 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 4 0 . 1 2 - 1 7 . 4 6 0 3 IA l p h a = , 3 . 1 } 3 3 0 9 8 . 1 - 2 8 0 7 . 9 9 IR e y . N o . = , 1 0 } 3 1 7 6 7 1 . - 3 8 2 5 4 . 4 IR a d i i % = , 0 . 8 } 6V e l .% — f 0} 2 . 2 5 8 4 9 1 0 - 2 7 7 2 3 4 . I

C - v a l u e = , 1 . 1 6 6 } 2 1 4 2 . 0 1 + 1 2 . 0 8 8 1 IA l p h a = , 3 . 1 1 } 3 2 9 8 1 . 5 - 2 6 0 8 . 6 2 IR e y . N o . = , 1 0 } 3 1 6 0 6 6 . - 3 7 2 7 9 . IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 4 5 8 8 1 0 - 2 7 2 7 1 5 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 4 0 . 9 9 - 2 . 2 4 9 6 I;A l p h a = , 3 . 1 1 } 3 2 9 7 8 . 1 - 2 7 0 5 . 3 3 IR e y . N o . = , 1 0 } 3 1 6 0 4 9 . - 3 7 7 1 6 . 8 IR a d i i % =, 0 . 8 } 6V e l . % = , 0} 2 . 2 4 5 7 7 1 0 - 2 7 4 5 9 8 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 3 9 . 9 4 - 1 6 . 5 8 6 8 IA l p h a —f 3 . 1 1 } 3 2 9 7 4 . 7 - 2 8 0 2 . 0 3 IR e y . N o . = , 1 0 } 3 1 6 0 3 1 . - 3 8 1 5 4 . 6 IR a d i i % =, 0 . 8 } 6V e l . % = , 0} 2 . 2 4 5 6 6 1 0 - 2 7 6 4 8 1 . I

Appendix Two 229

C - v a l u e = , 1 . 1 6 6 } 2 1 4 1 . 8 7 + 1 2 . 9 9 4 2 IA l p h a = , 3 . 1 2 } 3 2 8 5 9 . 5 - 2 6 0 2 . 9 8 IR e y . N o . = , 1 0 } 3 1 4 4 4 2 . - 3 7 1 8 2 . IR a d i i % = , 0 . 8 } 6V e l . % = f 0} 2 . 2 3 3 1 7 1 0 - 2 7 1 9 7 7 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 4 0 . 8 6 - 1 . 3 5 6 6 9 IA l p h a = , 3 . 1 2 } 3 2 8 5 6 . 1 - 2 6 9 9 . 5 5 IR e y . N o . = , 1 0 } 3 1 4 4 2 5 . - 3 7 6 1 8 . 7 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 3 3 0 6 1 0 - 2 7 3 8 5 5 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 3 9 . 8 2 - 1 5 . 7 0 7 IA l p h a =, 3 . 1 2 } 3 2 8 5 2 . 7 - 2 7 9 6 . 1 2 IR e y . N o . = , 1 0 } 3 1 4 4 0 8 . - 3 8 0 5 5 . 5 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 3 2 9 5 1 0 - 2 7 5 7 3 3 . I

C - v a l u e = , 1 . 1 6 6 } 2 1 4 1 . 7 9 + 1 3 . 9 0 7 IA l p h a = , 3 . 1 3 } 3 2 7 3 8 . 8 - 2 5 9 7 . 4 IR e y . N o . = , 1 0 } 3 1 2 8 3 4 . - 3 7 0 8 5 . 6 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 2 0 5 9 1 0 - 2 7 1 2 4 5 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 4 0 . 7 9 - 0 . 4 5 7 2 9 2 IA l p h a = , 3 . 1 3 } 3 2 7 3 5 . 5 - 2 6 9 3 . 8 3 IR e y . N o . = , 1 0 } 3 1 2 8 1 7 . - 3 7 5 2 1 . 4 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 2 0 4 8 1 0 - 2 7 3 1 1 8 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 3 9 . 7 5 - 1 4 . 8 2 1 IA l p h a = , 3 . 1 3 } 3 2 7 3 2 . - 2 7 9 0 . 2 6 IR e y . N o . = , 1 0 } 3 1 2 8 0 0 . - 3 7 9 5 7 . 1 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 2 0 3 7 1 0 - 2 7 4 9 9 1 . I

C - v a l u e = , 1 . 1 6 6 } 2 1 4 1 . 7 8 + 1 4 . 8 2 6 6 IA l p h a = , 3 . 1 4 } 3 2 6 1 9 . 4 - 2 5 9 1 . 8 5 IR e y . N o . = , 1 0 } 3 1 1 2 4 2 . - 3 6 9 9 0 . IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 0 8 1 2 1 0 - 2 7 0 5 1 7 . I

C - v a l u e = , 1 . 1 6 9 } 2 1 4 0 . 7 8 + 0 . 4 4 8 6 2 1 IA l p h a —f 3 . 1 4 } 3 2 6 1 6 . 1 - 2 6 8 8 . 1 5 IR e y . N o . = , 1 0 } 3 1 1 2 2 5 . - 3 7 4 2 4 . 7 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 0 8 0 1 1 0 - 2 7 2 3 8 6 . I

C - v a l u e = , 1 . 1 7 2 } 2 1 3 9 . 7 5 - 1 3 . 9 2 8 8 IA l p h a —f 3 . 1 4 } 3 2 6 1 2 . 7 - 2 7 8 4 . 4 4 IR e y . N o . = , 1 0 } 3 1 1 2 0 8 . - 3 7 8 5 9 . 5 IR a d i i % = , 0 . 8 } 6V e l . % = , 0} 2 . 2 0 7 9 1 0 - 2 7 4 2 5 4 . I

Appendix Two 230

{ C - v a l u e = , 1 . 1 6 6 } 2 1 4 1 . 8 2 + 1 5 . 7 5 3 I{ A l p h a = , 3 . 1 5 } 3 2 5 0 1 . 4 - 2 5 8 6 . 3 5 I( R e y . N o . = , 1 0 } 3 0 9 6 6 6 . - 3 6 8 9 5 . I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0 } 2 . 1 9 5 7 8 1 0 - 2 6 9 7 9 5 . I

{ C - v a l u e = , 1 . 1 6 9 } 2 1 4 0 . 8 3 + 1 . 3 6 1 0 7 I{ A l p h a = , 3 . 1 5 } 3 2 4 9 8 . 1 - 2 6 8 2 . 5 1 I{ R e y . N o . = , 1 0 } 3 0 9 6 4 8 . - 3 7 3 2 8 . 8 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 9 5 6 7 1 0 - 2 7 1 6 5 8 . I

{ C - v a l u e = , 1 . 1 7 2 } 2 1 3 9 . 8 -- 1 3 . 0 3 0 3 I{ A l p h a = , 3 . 1 5 } 3 2 4 9 4 . 6 - 2 7 7 8 . 6 7 I{ R e y . N o . = , 1 0 } 3 0 9 6 3 1 . - 3 7 7 6 2 . 5 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0 } 2 . 1 9 5 5 6 1 0 - 2 7 3 5 2 2 . I

{ C - v a l u e = , 1 . 1 6 6 } 2 1 4 1 . 9 1 + 1 6 . 6 8 6 2 I{ A l p h a = , 3 . 1 6 } 3 2 3 8 4 . 6 - 2 5 8 0 . 8 9 I{ R e y . N o . = , 1 0 } 3 0 8 1 0 4 . - 3 6 8 0 0 . 8 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 8 3 5 5 1 0 - 2 6 9 0 7 8 . I

{ C - v a l u e = , 1 . 1 6 9 } 2 1 4 0 . 9 3 + 2 . 2 8 0 0 8 I{ A l p h a = , 3 . 1 6 } 3 2 3 8 1 . 3 - 2 6 7 6 . 9 2 I{ R e y . N o . = , 1 0 } 3 0 8 0 8 7 . - 3 7 2 3 3 . 5 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 8 3 4 4 1 0 - 2 7 0 9 3 7 . I

{ C - v a l u e = , 1 . 1 7 2 } 2 1 3 9 . 9 1 - 1 2 . 1 2 5 5 I{ A l p h a = , 3 . 1 6 } 3 2 3 7 7 . 9 - 2 7 7 2 . 9 5 I{ R e y . N o . = , 1 0 } 3 0 8 0 7 0 . - 3 7 6 6 6 . 3 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 8 3 3 3 1 0 - 2 7 2 7 9 5 . I

{ C - v a l u e = , 1 . 1 6 6 } 2 1 4 2 . 0 7 + 1 7 . 6 2 6 3 I{ A l p h a = f 3 . 1 7 } 3 2 2 6 9 . 2 - 2 5 7 5 . 4 7 I{ R e y . N o . = , 1 0 } 3 0 6 5 5 8 . - 3 6 7 0 7 . 2 I{ R a d i i % = , 0 . 8 } 6{ V e l . % 0 } 2 . 1 7 1 4 5 1 0 - 2 6 8 3 6 6 . I

{ C - v a l u e = , 1 . 1 6 9 } 2 1 4 1 . 1 4 3 . 2 0 5 6 8 I{ A l p h a = , 3 . 1 7 } 3 2 2 6 5 . 8 - 2 6 7 1 . 3 7 I{ R e y . N o . = , 1 0 } 3 0 6 5 4 1 . - 3 7 1 3 8 . 9 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0 } 2 . 1 7 1 3 4 1 0 - 2 7 0 2 2 0 . I

{ C - v a l u e = , 1 . 1 7 2 } 2 1 4 0 . 0 8 - 1 1 . 2 1 4 4 I{ A l p h a =, 3 . 1 7 } 3 2 2 6 2 . 4 - 2 7 6 7 . 2 7 I{ R e y . N o . = , 1 0 } 3 0 6 5 2 3 . - 3 7 5 7 0 . 7 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 7 1 2 2 1 0 - 2 7 2 0 7 3 . I

Appendix Two 231

{ C - v a l u e = , 1 . 1 6 6 } 2 1 4 2 . 2 8 + 1 8 . 5 7 3 3 I{ A l p h a = , 3 . 1 8 } 3 2 1 5 4 . 9 - 2 5 7 0 . 0 9 I{ R e y . N o . = f 1 0 } 3 0 5 0 2 7 . - 3 6 6 1 4 . 2 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 5 9 4 5 1 0 - 2 6 7 6 5 9 . I

{ C - v a l u e = , 1 . 1 6 9 } 2 1 4 1 . 3 2 + 4 . 1 3 7 8 8 I{ A l p h a =, 3 . 1 8 } 3 2 1 5 1 . 6 - 2 6 6 5 . 8 7 I{ R e y . N o . = , 1 0 } 3 0 5 0 1 0 . - 3 7 0 4 5 . I{R a d i i % — f 0 . 8 } 6{ V e l . % = , 0} 2 . 1 5 9 3 4 1 0 - 2 6 9 5 0 8 . I

{ C - v a l u e = , 1 . 1 7 2 } 2 1 4 0 . 3 1 - 1 0 . 2 9 6 9 I{ A l p h a = , 3 . 1 8 } 3 2 1 4 8 . 2 - 2 7 6 1 . 6 4 I{ R e y . N o . = , 1 0 } 3 0 4 9 9 2 . - 3 7 4 7 5 . 8 I{R a d i i % = , 0 . 8 } 6{ V e l . % = , 0 } 2 . 1 5 9 2 3 1 0 - 2 7 1 3 5 7 . I

{ C - v a l u e = , 1 . 1 6 6 } 2 1 4 2 . 5 5 + 1 9 . 5 2 7 1 I{ A l p h a = , 3 . 1 9 } 3 2 0 4 2 . -- 2 5 6 4 . 7 6 I{ R e y . N o . = , 1 0 } 3 0 3 5 1 0 . - 3 6 5 2 2 . I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 4 7 5 7 1 0 - 2 6 6 9 5 7 . I

{ C - v a l u e = , 1 . 1 6 9 } 2 1 4 1 . 5 9 + 5 . 0 7 6 7 1 I{ A l p h a = f 3 . 1 9 } 3 2 0 3 8 . 7 - 2 6 6 0 . 4 I{ R e y . N o . = , 1 0 } 3 0 3 4 9 3 . - 3 6 9 5 1 . 8 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 4 7 4 6 1 0 - 2 6 8 8 0 1 . I

{ C - v a l u e = , 1 . 1 7 2 } 2 1 4 0 . 5 9 - 9 . 3 7 3 1 3 I{ A l p h a = f 3 . 1 9 } 3 2 0 3 5 . 2 - 2 7 5 6 . 0 4 I{ R e y . N o . = , 1 0 } 3 0 3 4 7 6 . - 3 7 3 8 1 . 6 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 4 7 3 5 1 0 - 2 7 0 6 4 5 . I

{ C - v a l u e = , 1 . 1 6 6 } 2 1 4 2 . 8 7 + 2 0 . 4 8 7 9 I{ A l p h a = , 3 . 2 } 3 1 9 3 0 . 2 - 2 5 5 9 . 4 7 I{ R e y . N o . = , 1 0 } 3 0 2 0 0 9 . - 3 6 4 3 0 . 4 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 3 5 8 1 0 - 2 6 6 2 5 9 . I

{ C - v a l u e = , 1 . 1 6 9 } 2 1 4 1 . 9 2 + 6 . 0 2 2 1 9 I{ A l p h a = , 3 . 2 } 3 1 9 2 6 . 9 - 2 6 5 4 . 9 8 I{ R e y . N o . = , 1 0 } 3 0 1 9 9 1 . - 3 6 8 5 9 . 2 I{ R a d i i % 0 . 8 } 6{ V e l . % = , 0} 2 . 1 3 5 6 9 1 0 - 2 6 8 0 9 9 . I

{ C - v a l u e = , 1 . 1 7 2 } 2 1 4 0 . 9 3 - 8 . 4 4 2 9 3 I{ A l p h a = , 3 . 2 } 3 1 9 2 3 . 5 - 2 7 5 0 . 5 I{ R e y . N o . = , 1 0 } 3 0 1 9 7 4 . - 3 7 2 8 8 . 1 I{ R a d i i % = , 0 . 8 } 6{ V e l . % = , 0} 2 . 1 3 5 5 8 1 0 - 2 6 9 9 3 8 . I

Appendix Two 232

C - v a l u e = , 1 . 1 6 6 } 2 1 4 9 . 1 6 + 3 0 . 4 8 0 6 IA l p h a = , 3 . 3 } 3 0 8 7 7 . 5 - 2 5 0 8 . 6 9 IR e y . N o . = , 1 0 } 2 8 7 7 5 3 . - 3 5 5 4 9 . 1 IR a d i i % = , 0 . 8 } 6V0 I ,% = f 0} 2 . 0 2 4 1 0 - 2 5 9 5 4 6 .

C - v a l u e = , 1 . 1 6 9 } 2 1 4 8 . 2 8 + 1 5 . 8 4 7 7 IA l p h a = , 3 . 3 } 3 0 8 7 4 . 1 - 2 6 0 3 . 0 2 IR e y . N o . = , 1 0 } 2 8 7 7 3 6 . - 3 5 9 6 8 . 8 IR a d i i % = , 0 . 8 } 6Vg I .% —f 0} 2 . 0 2 3 8 9 1 0 - 2 6 1 3 4 0

C - v a l u e = , 1 . 1 7 2 } 2 1 4 7 . 3 5 + 1 . 2 1 5 4 8 IA l p h a = , 3 . 3 } 3 0 8 7 0 . 7 - 2 6 9 7 . 3 6 IR e y . N o . = , 1 0 } 2 8 7 7 1 8 . - 3 6 3 8 8 . 4 IR a d i i % —f 0 . 8 } 6V e l . % = f 0 } 2 . 0 2 3 7 8 1 0 - 2 6 3 1 3 4

Appendix Three 233

APPENDIX THREE

MATHEMATICA PROGRAM FOR FLOW

INSTABILITY OF NON-NEWTONIAN LIQUIDS

IN COUETTE FLOW

f" I N P U T THE SYS TEM GEOMETRY & P R O P E R T I E S " ;" n = m a t r i x s i z e r e q u i r e d " ;" n b a = n u m b e r o f d i m e n s i o n l e s s w a v e n u m b e r i n p u t " ;" t r i a l = v a l u e s o f d i m e n s i o n l e s s w a v e n u m b e r i n p u t " ;***********************************************************

C l e a r [ n , n b a , t r i a l ]

n = 4 ; n b a = 1 5 ;t r i a l = { 2 . 7 1 , 2 . 7 2 , 2 . 7 3 , 2 . 7 4 , 2 . 7 5 , 2 . 7 6 , 2 . 7 7 , 2 . 7 8 ,

2 . 7 9 , 2 . 8 0 , 2 . 8 1 , 2 . 8 2 , 2 . 8 3 , 2 . 8 4 , 2 . 8 5 } ;

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * .

" I N P U T THE SYS TEM GEOMETRY & P R O P E R T I E S " ;" a = d i m e n s i o n l e s s w a v e n u m b e r " ;" r l = r a d i u s o f i n n e r c y l i n d e r " ;" r 2 = r a d i u s o f o u t e r c y l i n d e r " ;" q l = a n g u l a r s p e e d o f i n n e r c y l i n d e r " ;" q 2 = a n g u l a r s p e e d o f o u t e r c y l i n d e r " ;" i n = p s e u d o p l a s t i c f l o w i n d e x " ;

C l e a r [ r l , r 2 , q l , q 2 , a , i n ]

a [ w _ ] : = t r i a l [ [ w ] ]r l = 0 . 9 ;r 2 = 1 . 0 ;q l = l ;q 2 = 0 ;i n = 0 . 5 ;

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

" D E F I N E THE SYSTEM P A R A M E T E R S " ;" h = r a d i u s r a t i o o f c o n c e n t r i c r o t a t i n g c y l i n d e r s " ;" u = a n g u l a r s p e e d r a t i o o f c o n c e n t r i c r o t a t i n g

c y l i n d e r s " ;" s = g e o m e t r i c a l p a r a m e t e r "" e = g e o m e t r i c a l p a r a m e t e r "" y = g e o m e t r i c a l p a r a m e t e r "

Appendix Three 234

" g = g e o m e t r i c a l p a r a m e t e r "" z = g e o m e t r i c a l p a r a m e t e r "" f = g e o m e t r i c a l p a r a m e t e r "

C l e a r [ h , u , s , e , y , g , z , f ]

h = r l / r 2 ; u = q 2 / q l ;s = 2 ( r 2 - r l ) / ( r 2 + r l ) ; e = ( 1 + s x ) ^ ( - 1 ) ; y = ( 1 + s x ) / ( l - s / 2 ) ;g = 2 / ( ( l + u ) ( l - h ^ 2 ) ) ( ( u - h " 2 ) + ( l - u ) y " ( - 2 ) ) ;z = ( 1 + s x ) / ( l + s / 2 ) ;f = ( 2 - i n ) / i n ( 1 - h ) / ( 1 - h ^ ( 2 / i n - l ) ) z T ( 2 / i n - 2 ) ;

**********************************************************" D E F I N E THE D I F F E R E N T I A L O P E R A T O R ,

T R I A L F U N C T I O N & W E I G H T I N G F U N C T I O N " ;" d = d i f f e r e n t i a l o p e r a t o r "" d y = d i f f e r e n t i a l o p e r a t o r "" d z = d i f f e r e n t i a l o p e r a t o r "" u t = t r i a l f u n c t i o n " ;" v t = t r i a l f u n c t i o n " ;" u w = w e i g h t i n g f u n c t i o n" v w = w e i g h t i n g f u n c t i o n " ; * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C l e a r [ d , d z , d y , u t , v t , u w , v w ]

d [ i ] : = D [ i , x ]d z [ i _ d y [ i *

u t [ i _ v t [ i _

u w [ i _v w [ i

: = D [ i , x ] + s e i : = D [ i , x ] - s e i

: = ( x " 2 - l / 4 ) " 2 x " ( i - l ) : = ( x ^ 2 - l / 4 ) x ^ ( i - l )

: = e ( x ^ 2 - l / 4 ) ^ 2 x ^ ( i - l ) : = e ( x ^ 2 - l / 4 ) x ^ ( i - l )

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

" D E F I N E THE GOVE R NI NG E Q U A T I O N S " ; * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C l e a r [ q x a , q x b , q y a , q y b ]

" q x a [ u ] + q x b [ v ] t = = 0 &&" ; " q y a [ v ] + q y b [ u ] = = 0 " ;

q x a [ i _ , w _ ] : = f d [ d z [ u t [ i ] ] ] - f a [ w ] ^ 2 u t [ i ] +2 d [ f ] d [ u t [ i ] ] - l / a [ w ] " 2 d [ f

d z [ d [ d z [ u t [ i ] ] ] ] ] +

Appendix Three 235

d [ f d z [ u t [ i ] ] ] - l / a [ w ] " 2 d [ d [ f ]d [ d z [ u t [ i ] ] ] ] -

l / a [ w ] ^ 2 d [ d [ f ] u t [ i ] ] q x b [ i _ , w _ ] : = - g v t [ i ] q y a [ i _ , w _ ] : = u t [ i ]q y b [ i _ , w _ ] : = - ( f i n d [ d z [ v t [ i ] ] ] - f a [ w ] ^ 2 v t [ i ] ) - ( i n d [ f ] d y [ v t [ i ] ] )

^ ^ • k ' k - k " k - ) r ' k - k - i r - k ' k " k ' k ' k ' k ' k ' k ' i e - k - k ' k ' k ' k ' k - k - k ' k ' k - k ' k - k ^ - k ' k - k ' k ' k - k - k ' k - k - k ' k - k ' k - k - i c - k - i ( ' k ' k - k - k ' k - k ^ ^ j

" A P P L Y GALERKEN M E T H O D " ;II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

C l e a r [ f n , x a , x b , y a , y b ]

f n = F u n c t i o n [ { i , j } , N I n t e g r a t e [ i j , { x , - 1 / 2 , 1 / 2 } ] ] ;

x a [ i _ , w _ , j _ ] : = f n [ q x a [ 1 , w ] , u w [ j ] ] x b [ i _ , w _ , j _ ] : = f n [ q x b [ i , w ] , u w [ j ] ] t y a [ i _ , w _ , j _ ] : = f n [ q y a [ i , w ] , v w [ j ] ] y b [ i _ , w _ , j _ ] : = f n [ q y b [ i , w ] , v w [ j ] ]

II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I I .

" A P P L Y M A T R I X SYS TEM TO S OL VE T a y l o r n u m b e r , t " ;II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I I .

C l e a r [ r x a , r x b , r y a , r y b , m a t , u p ]

r x a [ w _ , j _ , k _ ] : = T a b l e [ x a [ i , w , j ] , { i , 1 , k } ] r x b [ w _ , j _ , k _ ] : = T a b l e [ x b [ i , w , j ] , { i , 1 , k} ] r y a [ w _ , j _ , k _ ] : = T a b l e [ y a [ i , w , j ] , { 1 , 1 , k } ] r y b [ w _ , j _ , k _ ] : = T a b l e [ y b [ i , w , j ] , { i , 1 , k} ]

u p [ w _ , k _ ] : = T a b l e [ { r x a [ w , j , k ] , r x b [ w , j , k ] } , { j , 1 , k} ]

m a t [ w _ , k _ ] : = F l a t t e n [ { T a b l e [ F l a t t e n [ { r x a [ w , j , k ] , r x b [ w , j , k ]} ] f { ] , l , k } ] .T a b l e [ F l a t t e n [ { r y a [ w , j , k ] , r y b [ w , j , k ] }] , { j , 1 , k} ] } , 1 ]

II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I I .

" FORMAT THE O U T P U T " ;II * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I I .

C l e a r [ s o l s , a n s , o k l , o k 2 , o v e r a l l ]

s o l s [ w _ , k _ ] : = S o l v e [ D e t [ m a t [ w , k ] ] = = 0 , t ] a n s [ w _ , k _ ] : = S i m p l i f y [ t / . s o l s [ w , k ] ]o k l [ w _ ] : = T a b l e [ { { " T H E VALUE O F AL P HA I N P U T = " , a [ w ] } ,

{ " T H E VALUE O F FLOW I N D E X = " , i n } ,{ " T H E C Y L I N D E R R A D I I R A T I O = " , h } ,{ " T H E R E L . V E L O C I T Y R A T I O = " , u } } ] / / C o l u m n F o r m

o k 2 [ w _ ] : = T a b l e [ { k , a n s [ w , k ] } , { k , n , 1 , - 1 } ] / / C o l u m n F o r m

o v e r a l l = T a b l e [ { o k l [ w ] , o k 2 [ w ] } , { w , l , n b a } ] / / T a b l e F o r m

Appendix Three 236

{ THE VALUE O F A L P H A I N P U T = , 2 . 7 1 }( T H E VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 6 2 , 2 0 2 4 7 . 4 , 1 8 3 9 0 3 . , 1 . 5 0 0 3 3 1 0 }} { 3 , { 1 3 0 6 . 8 1 , 2 2 3 3 7 . 4 , 2 2 6 2 9 7 . } }{ 2 , { 1 3 3 3 . 0 6 , 2 4 2 3 0 . 2 } }{ 1 , { 1 3 6 4 . 9 9 } }

{THE VALUE O F A L P H A I N P U T = , 2 . 7 2 }{THE VALUE O F FLOW I N D E X = , 0 . 5 }{THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 3 9 , 2 0 1 6 4 . , 1 8 2 8 4 1 . , 1 . 4 9 0 7 6 1 0 }} { 3 , { 1 3 0 6 . 5 9 , 2 2 2 4 2 . 1 , 2 2 5 0 0 8 . } }{ 2 , { 1 3 3 2 . 8 2 , 2 4 1 2 8 . 7 } }{ 1 , { 1 3 6 4 . 8 2 } }

{THE VALUE O F A L P H A I N P U T = , 2 . 7 3 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 2 , 2 0 0 8 1 . 7 , 1 8 1 7 9 1 . , 1 . 4 8 1 2 9 1 0 }}{ 3 , { 1 3 0 6 . 4 1 , 2 2 1 4 8 . , 2 2 3 7 3 5 . } }{ 2 , { 1 3 3 2 . 6 3 , 2 4 0 2 8 . 4 } }{ 1 , { 1 3 6 4 . 7 1 } }


6{ 4 , { 1 3 0 3 . 0 5 , 2 0 0 0 0 . 3 , 1 8 0 7 5 3 . , 1 . 4 7 1 9 2 1 0 }} { 3 , { 1 3 0 6 . 2 7 , 2 2 0 5 5 . 1 , 2 2 2 4 7 5 . } }{ 2 , { 1 3 3 2 . 4 8 , 2 3 9 2 9 . 5 } }{ 1 , { 1 3 6 4 . 6 4 } }


6{ 4 , { 1 3 0 2 . 9 5 , 1 9 9 2 0 . , 1 7 9 7 2 6 . , 1 . 4 6 2 6 6 1 0 }} { 3 , { 1 3 0 6 . 1 8 , 2 1 9 6 3 . 3 , 2 2 1 2 3 0 . } }{ 2 , { 1 3 3 2 . 3 8 , 2 3 8 3 1 . 7 } }{ 1 , { 1 3 6 4 . 6 2 } }

Appendix Three 237

{ THE VALUE O F ALP HA I N P U T = , 2 . 7 6 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 2 . 8 9 , 1 9 8 4 0 . 7 , 1 7 8 7 1 1 . , 1 . 4 5 3 5 1 0 }} { 3 , { 1 3 0 6 . 1 3 , 2 1 8 7 2 . 6 , 2 1 9 9 9 8 . } }{ 2 , { 1 3 3 2 . 3 2 , 2 3 7 3 5 . 1 } }{ 1 , { 1 3 6 4 . 6 4 } }

{ THE VALUE O F AL P HA I N P U T = , 2 . 7 7 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 2 . 8 8 , 1 9 7 6 2 . 4 , 1 7 7 7 0 7 . , 1 . 4 4 4 4 4 1 0 }}{ 3 , { 1 3 0 6 . 1 2 , 2 1 7 8 3 . , 2 1 8 7 8 0 . } }{ 2 , { 1 3 3 2 . 3 , 2 3 6 3 9 . 8 } }{ 1 , { 1 3 6 4 . 7 1 } }

{ THE VALUE O F AL P HA I N P U T = , 2 . 7 8 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 2 . 9 , 1 9 6 8 5 . , 1 7 6 7 1 4 . , 1 . 4 3 5 4 8 1 0 }} { 3 , { 1 3 0 6 . 1 5 , 2 1 6 9 4 . 5 , 2 1 7 5 7 6 . } }{ 2 , { 1 3 3 2 . 3 3 , 2 3 5 4 5 . 6 } }{ 1 , { 1 3 6 4 . 8 2 } }

{ THE VALUE O F A L P H A I N P U T = , 2 . 7 9 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{THE C Y L I N D E R R A D I I R A T 1 0 = , 0 . 9 }{THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 2 . 9 7 , 1 9 6 0 8 . 6 , 1 7 5 7 3 2 . , 1 . 4 2 6 6 1 1 0 }} { 3 , { 1 3 0 6 . 2 3 , 2 1 6 0 7 . 1 , 2 1 6 3 8 5 . } }{ 2 , { 1 3 3 2 . 4 , 2 3 4 5 2 . 6 } }{ 1 , { 1 3 6 4 . 9 7 } }

{ THE VALUE O F AL P HA I N P U T = , 2 . 8 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0 }

6{ 4 , { 1 3 0 3 . 0 8 , 1 9 5 3 3 . 1 , 1 7 4 7 6 1 . , 1 . 4 1 7 8 5 1 0 }} { 3 , { 1 3 0 6 . 3 4 , 2 1 5 2 0 . 8 , 2 1 5 2 0 7 . } }{ 2 , { 1 3 3 2 . 5 2 , 2 3 3 6 0 . 7 } }{ 1 , { 1 3 6 5 . 1 8 } }

Appendix Three 238

{ THE VALUE O F A L P H A I N P U T = , 2 . 8 1 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 2 3 , 1 9 4 5 8 . 6 , 1 7 3 8 0 0 . , 1 . 4 0 9 1 7 1 0 }} { 3 , { 1 3 0 6 . 5 , 2 1 4 3 5 . 5 , 2 1 4 0 4 2 . } }{ 2 , { 1 3 3 2 . 6 8 , 2 3 2 7 0 . } }{ 1 , { 1 3 6 5 . 4 2 } }

{ THE VALUE O F A L P H A I N P U T = , 2 . 8 2 }{ THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 4 2 , 1 9 3 8 5 . , 1 7 2 8 5 0 . , 1 . 4 0 0 5 9 1 0 }} { 3 , { 1 3 0 6 . 7 , 2 1 3 5 1 . 3 , 2 1 2 8 9 0 . } }{ 2 , { 1 3 3 2 . 8 8 , 2 3 1 8 0 . 4 } }{ 1 , { 1 3 6 5 . 7 1 } }

{ THE VALUE O F A L P H A I N P U T = , 2 . 8 3 }{THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 6 5 , 1 9 3 1 2 . 3 , 1 7 1 9 1 0 . , 1 . 3 9 2 1 1 0 }} { 3 , { 1 3 0 6 . 9 4 , 2 1 2 6 8 . 1 , 2 1 1 7 5 0 . } }{ 2 , { 1 3 3 3 . 1 2 , 2 3 0 9 1 . 8 } }{ 1 , { 1 3 6 6 . 0 4 } }

{ THE VALUE O F A L P H A I N P U T = , 2 . 8 4 }{THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 3 . 9 3 , 1 9 2 4 0 . 5 , 1 7 0 9 8 0 . , 1 . 3 8 3 7 1 0 }} { 3 , { 1 3 0 7 . 2 3 , 2 1 1 8 5 . 8 , 2 1 0 6 2 3 . } }{ 2 , { 1 3 3 3 . 4 , 2 3 0 0 4 . 4 } }{ 1 , { 1 3 6 6 . 4 2 } }

{ THE VALUE O F AL P H A I N P U T = , 2 . 8 5 }{THE VALUE O F FLOW I N D E X = , 0 . 5 }{ THE C Y L I N D E R R A D I I R A T I O = , 0 . 9 }{ THE R E L . V E L O C I T Y R A T I O = , 0}

6{ 4 , { 1 3 0 4 . 2 4 , 1 9 1 6 9 . 5 , 1 7 0 0 6 1 . , 1 . 3 7 5 3 9 1 0 }} { 3 , { 1 3 0 7 . 5 5 , 2 1 1 0 4 . 6 , 2 0 9 5 0 8 . } }{ 2 , { 1 3 3 3 . 7 3 , 2 2 9 1 8 . } }{ 1 , { 1 3 6 6 . 8 4 } }

Appendix Four 239

APPENDIX FOUR COMPUTER PROGRAM FOR CONDUCTIVITY

MEASUREMENT AND CONTROL SYSTEM

C o n d u c t i v i t y p r o b e d a t a - l o g g i n g p r o g r a m

S e t - u p s y s t e m p a r a m e t e r s

10 3 0 5 0 7 0 9 0 110 1 3 01 5 0 V O L T S ( 0 ) = 0 1 7 0 B I N ( 0 ) = 01 9 0 B I T S l ( 0 ) = 0 : B I T S 2 ( 0 ) = 0 : B I T S 3 ( 0 ) = 0 2 1 0 C R L F $ = C H R $ ( 1 3 ) + C H R $ ( 1 0 )2 3 0 S A M P L E S = 3 5 0 2 5 0 C ONVR AT E %= 1 0 2 7 0 KEY 1 , " ! "2 9 0 KEY 2 , " œ "3 1 0 KEY 3 , " $ "3 3 0 KEY 4 , " % "3 5 0 KEY 1 0 , " ) "3 7 0 KEY 5 ,3 9 0 ON ERROR GOTO 5 0 7 0 4 1 0 CALL K D I N I T 4 3 0 '4 5 0 ' S e t - u p g r a p h p a r a m e t e r s 4 7 0 '4 9 0 F U L L S C A L E = 2 0 0 0 !5 1 0 WI ND0 WS%=15 3 0 DI M D A T A S L 0 T % ( 1 6 )5 5 0 D A T A S L O T % ( 0 ) = 1 : D A T A S L O T % ( 1 ) = 2 :

D A T A S L O T % ( 2 ) = 3 : D A T A S L O T % ( 3 ) = - l 5 7 0 DI M MI N Y ! ( 1 6 ) : DI M MAXYI ( 1 6 ) : N P T S ! = - 1 !5 9 0 Y M I N % = 0 : YMAX%=FULLSCALE6 1 0 FOR I N D X = 0 TO 1 6 : M I N Y I ( I N D X ) = Y M I N % : N E X T I NDX 6 3 0 FOR I N D X = 0 TO 1 6 : M A X Y I ( I N D X ) = Y M A X % : N E X T I NDX 6 5 0 '6 7 0 ' A l t e r c h a n n e l 0 t o l O O O m v F u l l s c a l e 6 9 0 MAXY!(0)=1000 7 1 0 '7 3 0 DI M C 0 L 0 U R % ( 1 6 )7 5 0 '7 7 0 ' M a i n p r o g r a m 7 9 0 '8 1 0 '

Appendix Four 240

8 3 08 5 08 7 08 9 09 1 09 3 09 5 09 7 0

9 9 01010

1 0 3 01 0 5 0

1 0 7 01 0 9 011101 1 3 01 1 5 01 1 7 01 1 9 012101 2 3 01 2 5 01 2 7 01 2 9 01 3 1 01 3 3 01 3 5 01 3 7 0

1 3 9 01 4 1 01 4 3 01 4 5 01 4 7 01 4 9 0

1 5 1 01 5 3 01 5 5 01 5 7 01 5 9 01 6 1 01 6 3 01 6 5 01 6 7 01 6 9 01 7 1 0

1 7 3 01 7 5 01 7 7 0

GOSUB 1 2 5 0L O C A T E 5 , 4 7 : P R I N T T I M E $I

' s a m p l e d o t p r o b e s 1 - 3I

C ALL F G R E A D ' ( " A N I N O " , " N O N E " , V O L T S ( ) , " C . M I L V L T " , " N T " )L OC AT E 1 8 , 1 2 : P R I N T U S I N G" # # # . # # " ; ( V O L T S ( 0 ) / l O O ) ; : P R I N T S P C ( 3 ) ;C AL L F G R E A D ' ( " A N I N l " , " N O N E " , V O L T S ( ) , " C . M I L V L T " , " N T " )L OC AT E 1 8 , 3 5 : P R I N T U S I N G" # # . # # # " ; ( V O L T S ( 0 ) / l O O O ) ; : P R I N T S P C ( 3 ) ;C AL L F G R E A D ' ( " A N I N 2 " , " N O N E " , V O L T S ( ) , " C . M I L V L T " , " N T " ) LOC AT E 1 8 , 5 8 : P R I N T U S I N G " # # # # " ; V O L T S ( 0 ) ; :P R I N T S P C ( 3 ) ;I

O P T K $ = I N K E Y $I F O P T K $ = " ) " THEN C L S : S Y S T E M : E N D I F O P T K $ = " o e " THEN GOSUB 1 7 7 0 I F O P T K $ = " % " THEN GOSUB 2 0 7 0 I F O P T K $ = " $ " THEN GOSUB 2 3 5 0 I F O P T K $ = " ! " THEN GOSUB 2 6 3 0 I F O P T K $ = " _ " THEN GOSUB 4 3 1 0 GOTO 8 7 0I

' M a i n m e n u s c r e e n

" T I M E $ ;

N u m b e r o f S a m p l e s " ;

CL SCOLOR 0 , 7P R I N T " C o n d u c t i v i t y D a t a A q u i s i t i o n P r o g r a m ( C) MRV 9 4

COLOR 7 , 0P R I N T : P R I N T : P R I N T : P R I N T P R I N T " D a t e " ; D A T E $ ; " T i m e P R I N T : P R I N T : P R I N T : P R I N T P R I N T " C o n v e r s i o n R a t eL OC AT E 9 , 5 8 : P R I N T S A M P L E S ; : LOCATE 9 , 2 2 : P R I N T CONVRATE%;L OC AT E 1 5 , 1 P R I N T "P R I N T "P R I N T "P R I N T "P R I N T "P R I N T "P R I N T "P R I N T : P R I N T COLOR 0 , 7 P R I N T " f l R u n F 5 R e v i e w f 1 0COLOR 7 , 0 RETURN

V I . 0 6 c

E I I I I I I I I I »°AAAAAAAAA°o oo o°AAAAAAAAA° ° P r o b e 1 ° E I I I I I I I I I ^

E I I I I I I I I I »° â a a a a M a a °o oo o° a M a a a a a a ° ° P r o b e 2 °

E I I I I I I I I I » " ° AAAAAAAAAA° "o o IIo ou° AAAAAAAAAA° " ° S p e e d R P M ° " È I I I I Ï I I I I I ^ "

f 2 F i l e s E x i t " ;

f 3 R a t e f 4 S a m p l e s

Appendix Four 241

1 7 9 0 ' F i l e s o p t i o n 1 8 1 0 '1 8 3 0 '1 8 5 0 C L S1 8 7 0 P R I N T : P R I N T : P R I N T " C u r r e n t D a t a F i l e s "1 8 9 0 P R I N T " ____________________________________________________1 9 1 0 P R I N T1 9 3 0 S H E L L " c d \ k d a c \ d a t a 1 9 5 0 F I L E S1 9 7 0 H O L D $ = I N K E Y $ : I F H O L D $ = " " THEN 1 9 7 0 1 9 9 0 O P T K $ = " 9 9 "2 0 1 0 S H E L L " C D \ K D A C \ R U N 2 0 3 0 GOSUB 1 2 5 0 2 0 5 0 RETURN 2 0 7 0 '2 0 9 0 ' S e t s a m p l e s o p t i o n 2110 '

2 1 3 0 '2 1 5 0 L OCATE 9 , 5 8 : P R I N T S P C ( 6 )2 1 7 0 COLOR 0 , 72 1 9 0 L OCATE 9 , 5 8 : P R I N T S P C ( 4 )2 2 1 0 LOCAT E 9 , 5 8 : I N P U T " " , C S A M P L E S $2 2 3 0 COLOR 7 , 0 2 2 5 0 C S A M P L E S = V A L ( C S A M P L E S $ )2 2 7 0 I F ( C S A MP L E S > 9 9 9 9 ) OR ( C S A M P L E S = 0 ) THEN

S A M P L E S = S A M P L E S E L S E S A M P L E S = C S A M P L E S 2 2 9 0 L OCATE 9 , 5 8 : P R I N T S P C ( 1 5 )2 3 1 0 L OCATE 9 , 5 8 : P R I N T S AMP LES 2 3 3 0 RETURN 2 3 5 0 '2 3 7 0 ' S e t s a m p l e r a t e o p t i o n 2 3 9 0 '2 4 1 0 '2 4 3 0 LOCATE 9 , 2 2 : P R I N T S P C ( 6 )2 4 5 0 COLOR 0 , 72 4 7 0 LOCATE 9 , 2 2 : P R I N T S P C ( 4 )2 4 9 0 LOCATE 9 , 2 2 : I N P U T " " , CCONVRATE$2 5 1 0 COLOR 7 , 02 5 3 0 C C O N V R A T E = V A L ( C C O N V R A T E $ )2 5 5 0 I F ( CCONVRATE > 9 9 9 9 ) OR ( C C O N V R A T E = 0 ) THEN

CONVRATE%=CONVRATE% E L S E CONVRATE%=CCONVRATE 2 5 7 0 LOCATE 9 , 2 2 : P R I N T S P C ( 1 5 )2 5 9 0 LOCATE 9 , 2 2 . «PRI NT CONVRATE%2 6 1 0 RETURN2 6 3 0 ' R u n t i m e r o u t i n e 2 6 5 0 2 6 7 0 2 6 9 0 2 7 1 02 7 3 0 CALL BG R E A D '

( " A N D A T A " , s a m p l e s , " A N I N O , A N I N l , A N I N 2 " , 1 , " N O N E " , 1 , " N T " , " T A S K l " )

S e t u p b a c k g r o u n d r e a d f o r n s a m p l e s & n o t r i g

Appendix Four 242

2 7 5 0 '2 7 7 0 ' S e t u p i n t e r u p t i n t e r v a l f o r n m i l i s e c s

t o g i v e r e q u i r e d s a m p l e 2 7 9 0 ' p e r i o d 2 8 1 0 '2 8 3 0 RATE%=CONVRATE%2 8 5 0 C AL L I N T O N ' ( r a t e % , " M I L " )2 8 7 0 '2 8 9 0 ' G r a p h d a t a 2 9 1 0 '2 9 3 0 S C R E E N 22 9 5 0 C ALL G R L A B E L ' ( " C o n d u c t i v i t y / R P M " , 1 , 1 , " L E F T " , " C T R " ) 2 9 7 0 C A L L G R L A B E L ' ( " R e a l t i m e p l o t o f c o n d u c t i v i t y

p r o b e s " , 1 , 1 , " T O P " , " C T R " )2 9 9 0 C AL L H G R A P H R T ' ( " A N D A T A " , D A T A S L O T % ( ) , " F A S T " ,

M I N Y ! 0 , M A X Y ! ( ) , " C . M I L V L T " , N P T S ! , W I N D O W S % , " G R I D " ) 3 0 1 0 '3 0 3 0 ' T u r n o f f i n t e r u p t s , s a v e & d e l e t e a r r a y 3 0 5 0 '3 0 7 0 C AL L I N T O F F3 0 9 0 S C R E E N 03 1 1 0 GOSUB 3 2 3 03 1 3 0 C AL L A R D E L ' ( " a n d a t a " )3 1 5 0 '3 1 7 0 C ALL K D I N I T 3 1 9 0 GOSUB 1 2 5 0 3 2 1 0 RETURN 3 2 3 0 '3 2 5 0 ' F i l e s a v e3 2 7 0 ' '3 2 9 0 '3 3 1 0 C L S 3 3 3 0 P R I N T3 3 5 0 P R I N T " F i l e S a v e P r e s s e n t e r t o q u i t "3 3 7 0 P R I N T " "3 3 9 0 P R I N T3 4 1 0 I N P U T " E n t e r F i l e n a m e " ; F I L E $3 4 3 0 F I L E $ = L E F T $ ( F I L E $ , 8)3 4 5 0 I F F I L E $ = " " THEN 4 2 1 0 3 4 7 0 S F I L E $ = F I L E $ + " . S 5 0 "3 4 9 0 A F I L E $ = F I L E $ + " . T X T "3 5 1 0 '3 5 3 0 ' U s e r e n t e r e d v a r i a b l e s 3 5 5 0 DATA C o m m e n t , C o n d u c t i v i t y R a n g e # 1 ,

C o n d u c t i v i t y R a n g e # 2 3 5 7 0 '3 5 9 0 ' C r e a t e h e a d e r o n d a t a f i l e i n CS V f o r m a t 3 5 9 5 P R I N T3 6 1 0 P R I N T " S a v i n g a s c i i d a t a t o " ; T A B ( 3 0 ) A F I L E $3 6 3 0 S H E L L " C D \ K D A C \ A S C I I3 6 5 0 OP EN A F I L E $ FOR OUTP UT AS # 13 6 7 0 P R I N T # 1 , C F I L E $ : P R I N T3 6 9 0 FOR LOOP = 1 TO 33 7 1 0 READ I N F O $ : P R I N T I N F O $ ; : P R I N T T A B ( 3 0 ) ;

Appendix Four 243

3 7 3 0 I N P U T " > " , P A R A M $3 7 5 0 P R I N T # 1 , P A R A M $3 7 7 0 NEXT L OOP 3 7 9 0 '3 8 1 0 ' A d d d a t e & T i m e3 8 3 0 P R I N T # ! , " D a t e " ; D A T E $ ; " , " ; " T i m e " T I M E $3 8 5 0 '3 8 7 0 ' C r e a t e a s c i i d a t a f i l e i n CSV f o r m a t3 8 7 5 P R I N T3 8 7 6 P R I N T " S a v i n g D a t a > " ;3 8 9 0 M I L V l ( 0 ) = 0 : M I L V 2 ( 0 ) = 0 : M I L V 3 ( 0 ) = 0 : D O N E % = 03 9 1 0 P R I N T # ! , " C o n d u c t i v i t y # 1 , C o n d u c t i v i t y # 2 , M o t o r , T i m e "3 9 3 0 P R I N T # ! , " u S , u S , R P M , m s "3 9 5 0 FOR L 0 0 P = 1 TO S AMP LES3 9 7 0 CALL A R G E T ' ( " A N D A T A " , L O O P , L O O P , " A N I N O " ,

1 , M I L V l 0 , " C . M I L V L T " )3 9 9 0 C ALL A R G E T ' ( " A N D A T A " , L O O P , L O O P , " A N I N l " ,

1 , M I L V 2 0 , " C . M I L V L T " )4 0 1 0 C ALL A R G E T ' ( " A N D A T A " , L O O P , L O O P , " A N I N 2 " ,

1 , M I L V 3 0 , " C . M I L V L T " )4 0 3 0 P R I N T # 1 , M I L V 1 ( 0 ) / 1 0 0 ; " , " ; M I L V 2 ( 0 ) / 1 0 0 0 ; " , " ;

M I L V 3 ( 0 ) ( L O O P - 1 ) * R A T E %4 0 3 5 D0NE%=D0NE% + 1 : I F DONE %= 1 0 THEN P R I N T " . " ; : DONE%=04 0 5 0 NEXT LOOP4 0 5 5 R E S T O R E4 0 7 0 C L O S E # 14 0 9 0 '4 1 1 0 ' C r e a t e r a w d a t a f i l e 4 1 3 0 S H E L L " C D \ K D A C \ D A T A 4 1 3 5 P R I N T : P R I N T4 1 5 0 P R I N T " S a v i n g r a w d a t a t o " ; T A B ( 3 0 ) S F I L E $4 1 7 0 C ALL A R S A V E ' ( " A N D A T A " , S F I L E $ , " C . M I L V L T " ,

" F T . K D A C " , RAT E %, " H M I C " )4 1 9 0 '4 2 1 0 S H E L L " C D \ K D A C \ R U N "4 2 3 0 RETURN 4 2 5 0 '4 2 7 0 ' R e v i e w d a t a4 2 9 0 ' '4 3 1 0 '4 3 3 0 C L S4 3 5 0 P R I N T : P R I N T4 3 7 0 P R I N T " F i l e R e v i e w P r e s s e n t e r t o q u i t "4 3 9 0 P R I N T " "4 4 1 0 P R I N T4 4 3 0 I N P U T " E n t e r F i l e n a m e " ; F I L E $ 4 4 5 0 F I L E $ = L E F T $ ( F I L E $ , 8 )4 4 7 0 I F F I L E $ = " " THEN 4 9 1 0 4 4 9 0 F I L E $ = F I L E $ + " . S 5 0 "4 5 1 0 A R R A Y $ = S P A C E $ ( 2 5 5 )4 5 3 0 S H E L L " C D \ K D A C \ D A T A 4 5 5 0 D A T A D E P T H ! = 0 ! : DAT AWI DT H%=0

: L A S T P O I N T ! = 0 ! : L A B L $ = S P A C E $ ( 2 5 5 ) 4 5 7 0 '

Appendix Four 244

4 5 9 0 C O L O U R % ( 0 ) = 3 : C O L O U R ! ( 1 ) = - 1 4 6 1 0 '4 6 3 0 ' R e a d a r r a y f r o m d i s k 4 6 5 0 '4 6 7 0 CALL A R L O A D ' ( A R R A Y $ , F I L E $ )4 6 9 0 '4 7 1 0 ' G r a p h d a t a 4 7 3 0 '4 7 5 0 CALL A R S T A T U S ' (A R R A Y $ , D A T A D E P T H ! ,

D A T A W I D T H ! , L A S T P O I N T ! , L A B L $ )4 7 7 0 P R I N T : P R I N T : P R I N T " -------- D a t a f i l e l o a d e d ------ " : P R I N T4 7 9 0 P R I N T : P R I N T : P R I N T " P r e s s a n y k e y w h e n r e a d y "4 8 1 0 H O L D $ = I N K E Y $ : I F H O L D $ = " " THEN 4 8 1 0 4 8 3 0 S C R E E N 2 4 8 5 0 C ALL G R A P H '

( A R R A Y $ , D A T A S L O T ! ( ) , C O L O U R ! ( ) , " S C R O L L " , 0 . , F U L L S C A L E , " N O R M A L " , 1 , 1 . , D A T A D E P T H ! , " C . M I L V L T " )

4 8 7 0 CALL A R D E L ' ( A R R A Y $ )4 8 9 0 '4 9 1 0 S C R E E N 0 4 9 3 0 S H E L L " C D \ K D A C \ R U N 4 9 5 0 GOSUB 1 2 5 0 4 9 7 0 RETURN 4 9 9 0 '5 0 1 0 ' E R R O R T r a p 5 0 3 0 5 0 5 0

I

5 0 7 0 P R I N T : P R I N T " s y s t e m e r r o r @ l i n e " ; E R L ;" i n v a l i d f i l e n a m e s y s t e m w i l l r e t r y "

5 0 9 0 FOR P A U S E = 0 TO 9 0 0 0 : NEXT P AUS E 5 1 1 0 I F E R L = 4 6 7 0 THEN RESUME 4 3 1 0 5 1 3 0 I F E R L = 3 6 5 0 THEN RESUME 3 2 9 0 5 1 5 0 P R I N T : P R I N T " u n a b l e t o i d e n t i f y e r r o r

s y t e m w i l l r e - s t a r t "5 1 7 0 FOR P A U S E = 0 TO 9 0 0 0 : NEXT P AUS E 5 1 9 0 S C R E E N 0 : R U N 5 2 0 0 END

Appendix Five 245

APPENDIX FIVE

SAMPLE CALCULATIONS OF PECLET NUMBER

FROM EXPERIMENTAL RTD DATA

The concentration readings in the following table represent a continuous

response to an impulse input into a Couette flow device.

Time (min) t

Tracer concentration (g m' ) C

0 03 206 509 10112 18115 28218 30021 20124 14627 6030 0

The following quantities are first determined

^ Ci =1341 g m"

^ Ci ti =22416 g s :

CiAt = 4023 g s m

- 3

y c* t-Mean residence time = t= ^ ^ =16.716 min

Z Ci

The following table is constructed to evaluate the dimensionless scale residence

time distribution curve (Ce-curve);

Appendix Five 246

Dimensionless Time 0 = t / t

Ce-curve Ce= ( C i / ^ C i A t ) X t

0 00.18 0.080.36 0.710.54 0.420.72 0.750.90 1.171.08 1.251.26 0.841.44 0.601.62 0.251.79 0

The axial dispersion model is used as the basis for characterising the RTD

curve. The Ce-curve is a unique function of Peclet number and by matching

experimental curves with theoretical curves, the values of the Pe number, and

hence, dispersion coefficient, is obtained. The comparison can also be

achieved, or aided, by using appropriate expressions derived from the

theoretical response curves, relating number to mean residence time and

variance. The variance is calculated by using the following derived values:

^ Ci = 1341 g

y Ci ti = 22416 g s

^ ti^Ci = 413352 g

Therefore the variance:

CT = 4133521341

224161341

= 28.82

The dimensionless variance is:

28.82

(16.716)'= 0.10314

Appendix Five 247

For Peclet number:

Pe =ae

= 1934

This low value of the Pe number corresponds to high dispersion condition. In

order to ascertain the validity o f Pe number the shape of the theoretical curve is

expected to show a good fit with the experimental Ce-curve as indicated in Fig.

A5.1. The predicted Ce-curve obtained by using the expression:

Ce _U n e exp

( 1 - e f P e4 6

[A5.1]

1.4

1.2 - -

0.8

0.6 -

0.4 -

0.2 -

0.5 1 1.5Dimensionless time, e (-)

Figure A5.1 Ce-curve for the dispersion model

Appendix Six 248

APPENDIX SIX

PUBLISHED PAPER RELATING TO

THIS PROJECT

Summary of Research Outputs

Papers Published

Samson S. S. Yim and P. Ayazi Shamlou, 1996. Residence TimeDistribution In A Rotary Flow-Throush Device. Fluid Mixing 5: IChemE Symp. Ser. 140, 191-201.

Samson S. S. Yim and P. Ayazi Shamlou, 1996. The Effect of Geometry On Mixing Performance In Continuous Couette-flow Devices. Proceedings: 5th World Congress Of Chemical Engineering, 1, 647-652.

Samson S. S. Yim and P. Ayazi Shamlou, 1996. Residence TimeDistribution In A Rotary Flow-through Device. Proceedings: IChemE Research Events/Second European Conference for Young Researchers, 2, 955-957.

Samson S. S. Yim and P. Ayazi Shamlou, 1996. Residence TimeDistribution (RTD) In A Rotary Flow-through Device with non-Newtonian Fluids. AIChE Annual Meeting (Oral presentation and hill paper).

Papers Accepted

Samson S. S. Yim , N. Titchener-Hooker and P. Ayazi Shamlou, 1997. Prediction o f Critical Taylor Number For Non-Newtonian Fluids. The First European Congress on Chemical Engineering. (Oral presentation and full paper).

Samson S. S. Yim and P. Ayazi Shamlou, 1997. Prediction of Critical Tavlor Number in Rotating Flow-through Devices. Proceedings: 1997 Jubilee Research Events.

Appendix Six 249

Papers Submitted

Samson S. S. Yim and P. Ayazi Shamlou, 1996, Factors affecting flow instability of Newtonian liquids in Couette-Poiseuille flow. Full paper submitted for publication to Chemical Engineering Science.

Research Awards

University College London-Graduate School Exhibition -1993 M A P S , E n g i n e e r i n g a n d T h e B u i l t E n v i r o n m e n t P r i z e A w a r d

University College London-Graduate School Exhibition - 1995 M A P S , E n g i n e e r i n g a n d T h e B u i l t E n v i r o n m e n t P r i z e A w a r d

Appendix Six 250

Papers PublishedSamson S. S. Yim and P. Ayazi Shamlou, 1996. Residence Time Distribution In A Rotary Flow-Through Device. Fluid Mixing 5: IChemE Symp. Ser. 140, 191-201.

ICHEME SYMPOSIUM SERIES NO. 140

R e s id e n c e T im e D is t r ib u t io n In A R o t a r y F l o w -T h r o u g h D e v ic e

Samson S. S. Yim and Ayazi Shamlou P.Chemical and Biochemical Engineering, University College London. London WCIE 7JE

Residence time distribution (RTD) experiments were earned out to investigate the critical conditions for the establishment of near plug flow of a Newtonian liquid flowing through the gap of a horizontal coaxial cylinder device. The operating conditions in the annulus could be varied between the laminar flow and Taylor-Couette-Poiseuille flow. The RTD curves were described using a single parameter axial dispersion model. The results suggested significant variations from plug flow conditions depending on the axial Reynolds number and Taylor number.

Keywords: Residence Time Distribution, Coaxial Cylinder,Taylor-Couette-Poiseuille flow, Taylor number.Axial dispersion model.

IN T R O D U C T IO N

Many operations in the chemical, biochemical and food process industries involve the continuous transport of materials through rotating equipmenL Examples include émulsification, polymerization, solvent extraction, crystallization, precipitation, mixing and heat transfer operations (1,23,4,5). A coaxial cylinder device consisting o f a stationary outer shell and a rotating iiuier cylinder with or without blades and wipers is Aequently used in these operations because of its ability to promote good radial mixing and negligible axial dispersion (6,7). These flow characteristics are prerequisite when trying to optimize any operations occurring in the rotating flow through device. In solvent extraction applications, for example, deviations from plug flow can result in a significant reduction in the mass transfer driving forces and thus the separation efficiency of the equipment can be determintally affected (8).

The rational design of a rotating coaxial cylinder device with superimposed axial flow requires knowledge of the fluid flow panems and mixing in the equipment The complex flow panems produced in the gap between the two cylinders strongly depend on the rotational speed, axial flow rate, gap width and the theological properties o f the material (9,10,11,12). In the absence of any superimposed axial flow, for relatively low rotational speeds of the inner cylinder the liquid flow in the aimular gap is laminar (Couette) flow and the prevailing

191

Appendix Six 251


radial velocity profile of the liquid causes significant dispersion. As the rotational speed increases the flow structure becomes unstable characterized by the appearance of a series of (Taylor) vortex cells formed at regular intervals along the axis of the cylinder (12). The condition at which the vortices first appear is usually expressed in terms of a critical Taylor number, Ta . In the prevailing laminar vortex (Taylor-Couene) flow regime, there is negligible interaction between the neighboring cells but within each cell the toroidal motion of the fluid causes good radial mixing (1,7). When a small axial flow is added to the Taylor- Couette flow, the cells maintain their separate identity but move along the gap at a constant velocity determined by the flow rate. This type of flow is generally referred to as Taylor- Couette-Poiseuille flow and provides hydrodynamic conditions in the equipment which closely approach those of plug flow(7,l 1). With further increase in the rotational speed of the inner cylinder or the axial flow rate the vortices develop circumferential waves causing exchange of material at their boundaries (1). As a result axial dispersion increases and flow deviates from the ideal plug flow conditions. The flow regime in the armulus gradually changes from wavy vortex flow to turbulent vortex flow and fully turbulent flow as rotational speed of the inner cylinder increases (11).

The transitions between the various flow regimes are strong functions of the dimensionless

axial Reynolds number. Re dimensionless rotational Taylor number. Ta

= , and the dimensionless gap width, n [= • Of particular interest

to this study is the effects of Re and Ta numbers on the extent of mixing in the Taylor- Couette-Poiseuille (TCP) flow. Surprisingly few publications have addressed this aspect (1,2). The aim of the present study is to provide axial dispersion data obtained from residence time distribution (RTD) experiments carried out in a horizontal coaxial cylinder device operating in the TCP flow regime.

EXPERIMENTAL

The residence time distribution experiments were performed by using an impulse tracer injection technique. The experiments were directed towards determining the influence of the rotational speed of the inner cylinder and axial flow rate on the RTD of a 60 % glycerol-water solution. The rotational speed was varied between 30 and 350 rpm and the volumetric flow rate was changed between 1x10 and 4x10^ nun Vs.

All the experiments reported in this study were carried out in a 1 m long horizontal coaxial cylinder device which is shown in Fig. 1. The iiuier rotating cylinder was constructed from a

192

Appendix Six 252

ICHEM E S Y M PO SIUM S E R I E S NO. 140

stainless steel tube having an outside diameter of 76 mm and driven by a variable speed electric motor with rotational speed in the range of 0 • 1000 rpm. The speed of rotation was measured by using a shaft-mounted speed transducer (EEL Ltd., Westland Aerospace, East Cowes, Isle of Wight, UK). The outer shell was fabricated from Perspex to allow visual observations of the flow structure and had an inner diameter of 90 mm. The annular gap formed by the two cylinders was 7 mm wide giving a ratio of the outer to inner tube diameter o f 1.18.

The outer shell was equipped with two electrical conductivity probes and a single injection point. The conductivity probes were constructed from 35 rtun diameter glass rod, having two0.315 mm diameter platinum wire electrodes mounted 2.5 mm apart. The two probes were separated 600 mm apart in a horizontal line with the first probe positioned 50 mm from the injection point. When in position, the tip of the electrodes protruded 3 mm from the inside wall of the outer shell. The microinjector was a standard hypodermic syringe having a stainless tube of 0.7 mm outside diameter. The injection point was 200 mm h’om the entrance in the same horizontal line as the conductivity probes and the tip of the injection tube was positioned at the centre of annular gap. The tracer solution was prepared from concentrated KCl (BDH Supplies) and the working liquid. Injection was carried out manually within a period of approximately 1.0 s and the tracer concentrations were determined simultaneously and continuously at the two probe positions as a function of time. The data from both conductivity probes were stored using a data acquisition system (Keithley Model 575 measurement control system attached to a KDAC 500/1 data collection unit and a IBM PC2 computer) for analysis and evaluation of the various parameters including the first and second moments of the distributions.

Prior to injection of the tracer materiaL steady state flow condition was established in the device by continually pumping (Watson-Marlow Ltd., UK) the working liquid through the aruiular gap at a fixed volumetric flow rate until the efflux remained constant with time. The efflux was obtained manually by measuring the time required to collect a known volume of the discharge liquid.

Each experiment was replicated three times and the mean value used for parameter estimation.

RESULTS AND DISCUSSION

The experimental RTD C-curves obtained from the second probe are shown in Fig. 2 for a range of rotational speed of the inner cylinder. Initially the same experiment was repeated three times to gain experience in conducting the measurements and also to check the

193

Appendix Six 253


reproducibility of the data. It was found that the time taken for the injection of the tracer material had a significant effect on the reproducibility of the results and the subsequent fit of data. By trial and error, the best injection time was found to be less than 2 seconds for which the RTD measurements were reproducible to ±5%. All data reported in the present study fall in this range. The data in Fig. 2 refer to a flow rate of 1.5x10^ mm^/s. The tracer concentrations are presented in dimensionless form defined as C(t)/Co, where C(t) is the tracer concentration measured by each probe at time t following the introduction of the tracer impulse. C, is the total concentration of the tracer material obtained from the area under the tracer response curve. Figure 3 shows the effect of superimposed axial flow rate on the RTD curves for a rotational speed of 100 rpm. The plots in Figs 2 and 3 indicate that either an increase in the rotational speed or a decrease in the axial flow rate results in a broadening of the RTD curves and a consequential deviation from plug flow conditions.

According to previous studies (2,13,14.15) the transition from Couette flow to Taylor vortex flow occurs in the range of Taylor numbers between 40 and 80 depending on the axial Reynolds number and the gap width. An examination of the data shown in Figs 2 and 3 indicate the existence of Taylor vortex flow under most of the experimental conditions.

Typical RTD curves for the two probes are shown in Fig. 4 for a rotational speed of 110 rpm and axial flow velocity of 20 mm/s corresponding to an axial Reynolds number. Re = 25 and Taylor number. Ta = 121. The two response curves in Fig. 4 show that a vortex cell takes approximately 25 s (peak to peak time) to travel between the two probes and noting that the distance between the probes was fixed at 600 mm gives a (vortex) shift velocity of about 24 mm/s which is reasonably close to the. average axial liquid velocity in the gap. This is in good agreement with previous results (1,7).

The RTD curve can be mathematically modeled by using its various moments. In many cases however, the first and second moments provide an adequate description of the distribution and will therefore be used in the present study. The first moment of the RTD curve gives the mean residence time and is defined as

(Tcdt

jpd.

The mean residence time, i , can also be obtained directly from the axial flow rate and the working volume of the equipment. Thus

194

Appendix Six 254


V (2)

The dimensionless time, 9, is defined as

T L(3)

where v is the axial liquid velocity and L the distance between the two probe positions.The second moment of the RTD curve gives the variance of the distribution and is defined as

a" (t) =fC d t

(4)

and the dimensionless variance, o(6), defined as

o " (0 )= c - ( t )(5)

The flow of the tracer material in the annular gap of a coaxial cylinder device has been described adequately by an axial dispersion model ( 17,18). For the experiments carried out in the present investigation an impulse was injected into the gap at an arbitrary time, t = 0. Prior to the injection, the tracer concentration in the gap was zero and following the input of the tracer, its concentration was measured at two positions in the gap. These are the boundary conditions that describe an open-open system ( 17) and the partial differential equation of the dispersion concerned with an impulse measurement technique in this type of system has been solved analytically ( 17). The dimensionless tracer concentration, C(6), at the measuring point for the prevailing conditions is given by

(i-ey40

Pe (6)

assuming no stagnant region and negligible short circuiting of the liquid in the gap. In Eq. 6, the dimensionless Peclet number. Pe = vL/D, and the dimensionless tracer concentration, C(0), at dimensionless time 0, C(0) = CftyCp.

195

Appendix Six 255

ICHEME SY M P O S IU M S E R I E S NO. 140

The axial dispersion coefficient. D, can be obtained from the definition of the Pe number.

(vL / D ), which in turn is related to the variance of the distributions by (17)

Ac^(0) = — (7)

Aa^(6) represents the variance difference of the tracer RTD curves at any two points along the annular gap. In the present study Aa^(0) was determined between the two probe positions using the experimental response curves obtained from them, as shown in Fig. 4. The value of Ao^(0) was substituted into Eq. 7 in order to obtain the Pe for the prevailing flow and hence the axial dispersion coefficient, D. Finally, the Pe number was used together with Eq. (6) to fits the observed time variation of the tracer concentration C(t) at the two probe positions. The agreement of the experimental tracer RTD curves for the liquid in the gap with Eq. 6 was satisfactory, as shown in Fig. 5 for all the conditions examined in this work.

The effect of the rotational speed on the Pe. obtained from the impulse response curves at the two conductivity probe positions is shown in Fig. 6 for two axial flow rates expressed as Reynolds number. The plots on Fig. 6 shows that as the rotational speed of the inner shaft increases, the Pe number increases from an initial value of about 30 to a maximum value of approximately 250 beyond which any further increase in rotational speed causes a decrease in the Pe number. The minimum and maximum values of the Pe number were found to be a function of the flow rate through the device. In the experiments carried out in the present study the lowest value of Pe was found to be about 30 corresponding to a axial flow rate of 7.2 mm/s (Re =11) and the highest value of Pe was 250 as shown in Fig. 6.

The variations in Pe number with increasing Ta number depicted in Fig. 6 suggest that significant deviations from plug flow conditions can occur in the vessel during flow. Coney and Simmers (18), Ogawa ei al.(]9) and Astill (20) carried out detailed studies of the flow instability in the Taylor vortex flow as function of both rotational speed and axial flow rate. These investigations revealed that the vortices first appear in the annulus next to the wall of the rotating shaft. As the speed of the inner shaft increases the vortices grow in size until they eventually reach the walls of the outer stationary cylinder (18,19,20). Pudjiono ei al.(2) in their recent publication argued that the maximum Pe number corresponded to the conditions at which the Taylor vortices were fully developed. These observations regarding the birth and growth of vortices suggest that the transition from Couette flow to fully developed Taylor vortex flow occurs over a period of time and range of speed. In other words there is not a single speed at which flow regime suddenly changes as that speed is reached. The data presented in Fig. 6 are insufficient to allow firm conclusions to be made about the causes for the observed dependency of Pe on Re and Ta, but the general trend in

196

Appendix Six 256

ICHEME SYM POSIUM S E R I E S NO. 1 4 0

data shown in Fig. 6 appear to support the work Pudjiono ei a/.(2) and the general observations o f others (18,19^0).

CONCLUSIONS

Residence time distribution data were obtained for the dispersion of a impulse injection of tracer in the gap of a coaxial cylinder device operating in the Taylor-Couette-Poiseuille regime. The RTD data were described by using an axial dispersion model and the results suggested that significant deviations from plug flow can occur depending on the prevailing Taylor number and axial Reynolds number.

NOMENCLATURE

C(t) = tracer concentration at time t (mol/litre)Co = total tracer concentration (mol/litre)C(9) = dimensionless tracer concentration (Eq. 6) (-)d = annular gap (Ro-R;) (m)D = axial dispersion coefficient (m^ s ')L = distance between two measuring points (m)

n = dimensionless gap width (-)

N = rotational speed of inner cylinder (rps)

fPe = Peclet number. I (-)

Q = volumetric flow rate of liquid (m^ s ')Ro = radius of the outer shell (m)Rj = radius of the inner cylinder (m)

Re = axial Reynolds number. (.)

Ta = Taylor number.p (2 n N )R o à I d (-)P \ Ro

Ta, = critical Taylor number (-)t = time (s)V = working volume of the equipment (m^)V = axial flow velocity through the gap (m s ')T = mean residence time (s)

197

Appendix Six 257

ICHEME SYM POSIUM S E R I E S NO. 14 0

6 = dimensionless time (Eqs 1 and 2) (-)a(t) = variance of the distribution at time t (-)<y(0) = dimensionless variance (Eq. 6) (-)p * density of the working liquid (kg m'^)H - viscosity of the working liquid (kg m ' s ')

REFERENCES

1. Kataoka, K. and Takigawa, T., 1981. AlChE J.. 27; 504-508.2. Pudjiono. P.I., Tavare, N.S.. Garside. J. and Nigam. K.D.P.. 1992. Chem. Engng J..

101 - 110 .

3. Abichandani, H. and Sarma, S.C.. 1988. Chem. Engns Sci.. 43: 871-881.4. Hoare. M., Narendranathan. TJ.. Flint, J.R.. Heywood-Waddington. D., Bell. DJ. and

Dunnill. P., 1982, l&EC Fundam.. 21: 402-406.5. Dongaonkar, K.R., Pratt, H.R.C. and Stevens. G.W., 1991. AlChE J.. 37: 694-704.6. Legrand. J. and Coeuret, F., 1986, Chem. Engng Sci.. 4 1 : 47-53.7. Kataoka, K., Doi, H.. Hongo, T. and Futagawa, M., 1975, J. Chem. Engng Japan. 8:

472-476.8. von Stockar, U. and Lu, X.. 1991. Ind. Engng. Chem. Res.. 30: 1248-1257.9. Coles, D., 1965, J. Fluid Mech.. 21: 385-425.10. Croockewit, P., Honig, C.C. and Kramers. H.. 1955. Chem. Engng Sci.. 4: 111-118.11. Harrod. M.. 1986, J. Food Proc. Engng.. 9: 1 -62.12. Taylor. G.I., 1923. Phil. Trans.. A233: 289-343.13. Ng. S. and Turner. E. R., 1982, Proc. Rov. Soc. Lond.. A 382: 83-102.14. Takeuci, D. I. and Jankowski, D. F., 1981. J. Fluid Mech.. 102: 101-126.15. Hasoon, M. A., and Martin, B. W., 1977, Proc. Rov. Soc. Lond.. A 352: 351-380.16. Danckwerts. P.V.. 1953. Chem. Engng Sci.. 2: 1-13.17. Levenspiel. O., 1979, Chemical Reaction Engineering, Wiley, New York. Chapter IX.18. Coney, J.E.R. and Simmers. D.A., 1979. J. Mech. Engng Sci.. 21: 19-24.19. Ogawa, A., Fujita, V. and Nagabayashi, N.. 1985, Chem. Engng Commun.. 37; 209-

222.

20. Astill. K.N., 1964, J. Heat Transfer. 8: 383-392.

198

Appendix Six 258


I ConducbvKy Sp.eb/ro™» transducer

ConductMty meter

Rotary Sowinrouch device

Outlet

_ Rotary Hovnnrough device Æji and

motor ConducDvty meters

Measurement Data acquisition system and

Control system

Computer torScreen monitor

Pnnter

Fig. 1 Experimental Equipment

0.2Re = 25

0.15

ü° 0.1 Ü

0.05

: Ta = 201

0 10 20 30 40 50 60

t (s)

Fig.2 Dimensionless concentration vs time for a range of Taylor number

1 9 9

Appendix Six 259

OO

IC HEM E SY M PO SIUM S E R I E S NO. 1 4 0

0.3Ta =1250.25

0.2Re = 14 Re = 170.15

.1

0.05

0

10 20 30 40

t (s)

50 60

Fig.3 Dimensionless concentration vs time for a range of Reynolds number

0.25Ta = 121 : Re = 250.2 . .

0.15 -- 1st ProbeI2nd Probe

0.05 --

0 10 20 30 40 50 60

t(s)

Fig.4 RTD curves for the two probes in the TCP flow regime

200

Appendix Six 260

ICHEME SY M PO SIU M S E R I E S NO. 14 0

o ; Experimental data

— : Axial dispersion model70rpm

I 265rpm2 --

C D

o 330rpm

30.5 21 t 2.5

Fig. 5 Comparison of axial dispersion model and experimental data

250■ : Re = 26•200

100 - -

200150100Taylor number ( -')

Fig. 6 Peclet num ber vs Taylor num ber

201

Appendix Six 261

Papers PublishedSamson S. S. Yim and P. Ayazi Shamlou, 1996. The Effect of Geometry On Mixing Performance In Continuous Couene-flow Devices. Proceedings: 5th World Congress Of Chemical Engineering. 1, 647-652.

The Effect of Geometry on Mixing Performance in Continuous Couette-Flow DevicesS. s. Yim Samson and P. Ayazi ShamlouChemical and Biochemical Engineering, University College London, Loncon WCIE 7]E

Abstract

Experimental results are reported on the residence time distribution (RTD) for Newtonian liquids in a continuous Couette-flow device. The effects o f rotational speed, axial fiowrate and the design o f inner rotating shafts are examined using stimulus-response technique. An axial dispersion model is used to describe the RTD dfM The results suggest significant variations fi*om plug fiow conditions depending upon the geometry of inner rotating design and the operating parameters.

The production and processing of viscous materials fi-equently involve the fiow of material through rotating devices. Examples include émulsification, polymerisation, solvent extraction, crystallisation, mixing and heat transfer operations (1,2,3,4). A continuous Couette-flow device consisting o f a stationary outer shell and a rotating inner cylinder with and without blades and wipers is firquently used in these operations because of its ability to promote good radial mixing and negligible axial dispersion (1,5). However, the mixing performance in these devices is afiected by the behaviour of the material during fiow and in the case o f steady state continuous processes a key foctor is the variation in the duration of stay within the process equipment experienced by "particles' which entered the equipment at the same time. In solvent extraction applications, for example, deviation fit)m plug fiow can result in a significant reduction in the mass transfer driving forces and thus the separation efficiency of the equipment can be detrimentally afiected (6). This variation is normally expressed in terms o f the residence time distribution (RTD) and as a result measurement and analysis o f RTD has become an important tool in the study o f continuous processes. Understanding the interaction between the RTD and process parameters in a continuous Couette-flow device is therefore o f basic research interest to academics and industrialists.

Experimental Detail and Methods

All the experiments reported in this study were carried out in a 1 m long horizontal coaxial cylinder device which is shown in Figure 1. The outer shell was fobricated from Perspex to allow visual observations of the fiow structure and had an inner diameter of 90 mm. Figure 2 shows the two different designs of the inner rotating shafts (SI and S2) used in this study and their specifications respectively. Both of the inner shafts were constructed from a stainless steel tube having an outside diameter of 76 mm and driven by a variable speed electric motor. The outer shell was equipped with two conductivity probes and a single injection point. Two probes, with platinum wire electrodes, were separated 400 mm apart in a horizontal line with the first probe positioned 200 mm from the injection point. The tracer solution was prepared fi’om concentrated KCl and the working liquid. The data from both conductivity probes were stored in a data acquisition system for analysis.

647

Appendix Six 262

Yim S. S. Sam son an d P. Ayazi Sham lou

The residence time distribution experiments were performed by using an impulse tracer injection technique. The experiments were conducted to determine the influence of rotational speed o f the inner cylinder and axial velocity on the residence time distribution o f a highly viscous glycerol-water solution (~ 85% w/w). In every experiment, steady-state flow conditions were assured prior to introduction o f the tracer by continuing to feed bulk material until the discharge flow rates were identical for several sampling intervals. It was assumed that the tracer was injected over a sufBciently small time interval (approximately one second) that the idealisation of an instantaneous inçulse stimulus was suitable.

Taylor (7), investigated the instability o f flow in the annulus between two concentric cylinders and found that when the rotational speed exceeded a critical value, there appeared pairs of counter- rotating vortices spaced regularly along the cylinder axis. This type o f flow is generally referred to as T^for-Couette-Poiseuille (TCP) flow. The aim of the present study was to provide axial dispersion data obtained from RTD experiments carried out in a continuous Couette-flow device operating before and after TCP flow regime.

Results and Discussion

Figure 3 shows the experimental RTD curves obtained from the second probe for a range of rotational speed o f the inner shaft at a steady axial flow velocity o f 33 mm/s corresponding to an axial Reynolds number. Re = 2.1. The tracer concentrations are presented in dimensionless form as (8)

Cl Si(tiCi)At (ZiCi)'

c b = J [1]

Figure 4 shows the effect of a superimposed axial flow rate on the RTD curves at rotational speed o f 60 rpm and corresponding to Taylor number. Ta = 10, respectively. The plots in Figures 3 and 4 indicate a similar trends for different geometry o f inner shaft design (SI and S2). They show that either an increase in the rotational speed or a decrease in the axial fiow rate results in a broadening o f the RTD curves and a consequential deviation from plug fiow conditions. Although Figure 3 shows that the effea of axial fiowrate on RTD is relatively small it may be because the range of axial fiowrate used in the present study was not sufficiently wide to identify the effect of axial fiowrate.

Typical RTD curves for the two probes are shown in Figure 5. This figure shows the experimental results performed at a steady axial flow velocity of Re = 1.8 and Ta = 42, respectively. The results shown in Figure 5 are typical o f experiments in which Taylor vortices were observed. These vortices were seen to move through the annulus in single file with no overtaking or intermixing between the neighbouring vortices. Comparing the tracers o f the two conductivity cells.

648

Appendix Six 263

T he Effect o f G eom etry on Mixing Perform ance in C ontinuous C ouette-F low D evices

Couette-now device

Conductivity I meter

Inlet0>W\Ci.\0V»

Data acquisition system

Figure 1 Experimental Setup

Shaft 81

t

Shaft S2

tfii Td IRo

J p j i n ,\ nmnirir~irnn I

"HT

S t S 2d I m m ) 7 7

R o i m m ) 9 0 9 0R i ( m m ) 7 6 7 6

B l a d e s i z e - 55x27 (280CS)

Figure 2 Different geometry of inner shafts and their specifications

a : Ta = 14 (81) o; Ta = 32 (31) D : Ta = 19 (82) o ; Ta = 40 (82)

0(- )

Figure 3 Ce vs 0 for a range of Taylor number

3.5Re = 1.5 (81) Re = 3.5 (81) Re = 1.4 (82) Re = 2.9 (82)

2.5

d

0.5

0 21 3

6(-)

Figure 4 0» vs e for a range of Reynolds number

649

Appendix Six 264

Yim S. S. Sam son an d P. Ayazi Sham lou

no appreciable difference could be seen in the shape of response curves except for a phase shift caused fy the distance-velocity lag.

For the experiments carried out in the present investigation a pulse was injected into the gap at an arbitrary time, t = 0. Following the injection, the tracer its concentration was measured at two positions in the gq>. These are the boundary conditions that describe an open-open system and the partial differential equation of the dispersion concerned with a pulse measurement technique in this type o f system has been solved analytically (8). The dimensionless tracer concentration. Ce, at the measuring point for the prevailing conditions is given by (8)

_ [Pe Pe ( i - e r4 6 [3]

where the dimensionless Peclet number, Pe is defined as

P . - ^ [4]

When Pe tends to infinity, the dispersion rate is negligible con^ared to the convection rate. The flow is defined as plug flow. However, wdien Pe approaches zero the comection rate is much slower that the dispersion rate and the flow is completely mixed.

Figure 6 shows that the comparison between experimental results and the axial dispersion model The agreement of the experimental tracer RTD curves for the liquid in the gap with Equation [3] is satisfactory. Equation [4] was used to obtain the individual longitudinal dispersion coefficietn, Q, ty fitting the observed time variation of the tracer concentration at the two probe positions.

The Peclet number was also evaluated by using the variance difference of the tracer response curves (1,2). The effect of rotational speed, expressed as Taylor number, on the Pe, is shown in Figure 7 for three axial flow rates expressed as Reynolds nunmer. When the speed of rotation o f the inner cylinder is zero, the flow regime in the aimular gap is laminar for both shafts SI and S2, a fiilfy developed laminar liquid velocity profile exists in the gap giving a high value of Pe. As the speed of rotation is slowly increased, the fluid flow become unstable and the Pe number decreases. For shaft SI, Taylor vortices develop causing an increase in the Peclet number again and a narrowdng o f the RTD curves. The maximum points in the figure (highest Pe) shown in Figure 7 correspond to the conditions at which the Taylor vortices are fully established along the cohunn and flow approaches near plug flow condition. With further increase in the speed of rotation of the inner cylinder intermixing and exchange o f material occurs at the boundaries between the neighbouring vortices causing an increase in the axial dispersion and a broadening of the RTD curves, and hence a decrease in the value o f Pe is observed as flow becomes increasingly turbulent with shaft S2. Figure 8 suggest that the introduction of blades on the inner shaft promotes the occurrence of turbulent flow and TCP flow could not be detected.

650

Appendix Six 265

T h e Effect o f G eom etry on Mixing Perform ance in C ontinuous C ouette-F iow D evices

Ta *42 : Re* 1.4.5

1st Probe I 2nd Probe3.5

U 2.5

0.5

10 20 30 40 50 600

□ : Expenmental data — : Axial dtspeision model

4.5

3.6iTa * 0

u 2.5Ta* 13

1.5ITa * 26

0.5

32.51.5 20.5 10t(s) 0 ( - )

Figures RTD curves for the two probes Figure 6 Comparison between experimental dataand axial dispersion model

400

Re = 1.4

Re = 1.8300

Re* 1.9

c 200

I100

100806040200

400

Re = 2.0

Re = 2.3300

-o -R e = 2.9

I 200

100

3020100Taylor number ( - ) Taylor number ( - )

Figure 7 Peclet number vs Taylor numberfor shaft SI

Figure 8 Peclet number vs Taylor number for shaft 32

651

Appendix Six 266

Yim S. S. Samson and P. Ayazi Shamlou

The data in Figure 7 were supported by visual observations o f the flow structure in the annulus Wiich indicated the presence of small vortices. These vortices first appeared in the region close to the walls of the inner rotating cylinder and gradually grew radially outwards until they reached the inner walls o f the outer shell (9). Unfortunately, a tten ^ to photograph these vortices have so 6 r seen unsuccessful

Conclusions

The RTD results were described by using an a single parameter axial dispersion model The results showed that the RTD was strongly dependent upon the geometry o f inner rotating device the operating parameters including rotational speed. The results also indicated that the width o f RTD narrowed only for shaft SI when the transition of flow from laminar to Taylor-Couette-PoiseuiHe regime over a range o f rotational speed. For shaft S2, it never show the occurrence o f TCP flow regime.

Nomenclature

C = Concentration (molL*')Ce = Dimensionless concentration (-)d = Annular gap (L)D » Axial dispersion coeflScient ( L ^ ‘)L = Distance between two probes (L)N = Rotation speed (T ‘)Pe = Peclet number ( - )R , Ro = Radius o f the inner and outer cylinder (L'')Re = Reynolds number = 2dv/v ( - )t =Time(T)Ta = Taylor number = (2:iN)Ro*^d^ /v ( - )V = Aj flow velocity (LT*)6 = Dimensionless time ( - )V = Kinematic viscosity ( L ^ ‘)

References

1. Kataoka, K., DoL H., Hongo, T. and Futagawa, M.,J. Chem. Engng Japan, 8:472-476 (1975).2. Pudjiono, PJ., Tavare, N.S., Garside, J. and Nigam, K.D.P.,CAem. Engng J., 48:101-110(1992).3. Abichandani H. and Sarma, S.C., Chem. Engng Sci, 43:871-881(1988).4. Daongaonkar, K.R., Pratt, H.R.C. and Steven, G. W., AIChE J., 37:694-704 (1991).5. Legrand, J. and Coeuret, F., Chem. Engng ScL, 41:47-53 (1986).6. von stockar, U. and Lu, X., Ind. Engng Chem. Res., 30:1248-1257 (1991).7. Taylor, G. I., Phil. Trans., A233:289-343 (1923).8. Levenspiel 0 ., Chemical Reaction Engineering, Wiley, New York, Chapter IX (1965).9. Ogawa, A., Fujita, V. and Nagabayashi N., Chem. Engng Commun., 37:209-222 (1985).

652

Appendix Six 267

Papers PublishedSamson S. S. Yim and P. Ayazi Shamlou, 1996. Residence Time Distribution In A Rotary Flow-through Device. Proceedings: IChemE Research Events/Second European Conference for Young Researchers, 2, 955-957.

THE 1 9 9 6 ICHEME RESEARCH E V EN T/SEC O N D E U R O PE A N CO NFERENCE FOR YOUNG R E SEA R C H E R S

R E S ID E N C E TIM E D ISTR IB U TIO N

IN A R o t a r y F l o w - T h r o u g h D e v i c e

S A M S O N s . s . YIM A N D A Y A ZI S H A M L O U P .Chemcal and Biochemcol Engineering. University College London. London WCIE 7JE

A B S T R A C TReadcnoc tune dmnbotioD (RTD) experunents woe caried oat to investigate the critical oondiaons for the estabiisfament <tf tiear ping fknv of a Newtonian ixghd flowing through the gap of a bonzontal coaxial cylinder device. The npwanng condmons in the annulus could be varied between the laminaT flow 10 the TayiorCouette"fQiseaille flow The RTD carves were described «mg a s tn ^ axial dispersion model The resultsQiggffOfri ngmflcant variations from plug flow conditions depending on the axial Revnolds number and Tavlor number.

I n t r o d u c t i o nThe continuous processing of Newtonian and non-Newtonian materials frequently involves the flow of material through rotating devices. An «jnipi*? of an indnsmaliy unporum rotary flow-through device is a scraped surface heat prrhawgw (i) which is used routinely in the production of many food stuffr e.g., margarine, icecream and many dairy prnfocts. Most unit operations that occur m a rotary flow-through device are affected by the behavMur of the material during fiow and tn the case of steady gate comumoui processes a key factor is the variation m the duration of stay within the process wpnpw«<tit experienced by ‘parades’ which entered the mynpwwn, g the ame time. This Variation is normally expressed in terms of the residence ttme distnbtmon (RTD) and as a result measuremem and analysis of RTD has became an unponam tool in the study of coniinnoas processes. Understanding the interaction between the RTD and process parametos m a flow through rotary device IS therefore of basic research interest to academics and industnaiists.

E x p e r im e n ta l D e ta i l sThe residence nmg distribution

were performed by usmg an imppiq» tncet tiymrm techtuqoe. The

jiinfnn were directed towards determining the iwWu nr of the rotatimal speed of the umer cylinder and axial velocity on the RTD of a 60% glycerol-water solution. All the « p eiiments reported tn this study were earned out m a 1 m long honzomal coaxial cylinder device which u shown m Fig. 1. The mner rotating cybnder was constructed from a ged tubehaving an outside of 76 anddriven by a variable speed electnc motor.

Fig 1 Expenmental setup

955

Appendix Six 268

THE 1 9 9 6 ICHEME RESEARCH EVENT/SECOND EUROPEAN CONFERENCE FOR YOUNG RESEARCHERS

The Older f h e l l was f a h n r a tn r i from Ptispcji ID allow visual observauoss of the How strocmre and had an m n w

ihwm^i-T of 90 mm. The outer shell was equipped with two conductivity probes and a single mjecoon poinl Two probes, with ptatiimm wire dectiodes. were tepaiated 400 mm apart in a honzomal line with the first probe poeitiooed 200 mm fiom the mjecoon poinL The tracer sohmon was prepared from concwmated KQ and the working liquid. The data fiom both oooductivity probes were stored m a dau acqmsmon system fiir analysis.

Results and DiscussionTypical RTD curves for the two probes are shown in Fig. 2 This figure shows the expenmental results performed at a steady axial fiow velocity of 20 mm/s and at a rotational speed of 110 rpm oonespondtng to an asal Reynolds number. Re ■ 24.7 and Taylor number. Ta - 121. respecnveiy.

For the experiments earned out m the present investi ganon a pulse was mjeaed into the gap at an arbitrary tfine, t « 0. Following the tracer, its oonceatiauon was mei stired at two positions m the gq>. These are the boundary condmons that describe an opennipen system and the partial difEdenaal rqnannn of the dispersioo concerned with a pulse measurement «fhnwpp in this type of system has been solved analytically (2). The dùnensiooless tracer onrwiiTration, C*. at the measanng pomt for the prevailing conditions is given by (2)

Cf = (Pe / 4 ie^ ' : cxp(-Pe(l - / 4 ^ (IJ

where the thmensuailess Pedet number. Pe is defined as

Pe-vU D (2J

When Pt tends to infinity, the dispersion rate is negligible compared to the convection rate. The flow is «tiHÿiwuri as plug flow. However, when Pe approaches zero the comecoon rate is nmch slower that the (hspersioo rate and the flow is completely mixed.

Fignre 3 shows that the comparison between expenmental results and the axial dispersion model The agreemem of the «TyrmKnwai nacer RTD curves for the liquid m the gap with Eq [I] is saosfsctory. Equation [2] was used to obtam the individnal longrtudinal dispersion cntrffirimt D. bv fttnng the observed time variation of the tracer conoemration at the two probe posmons.

The Pedet muntaer was also evaluated by usmg the varunce difference of the tracer response curves (3.4). The effed of rotaoooal speed, expressed as Taylor number, on the Pe. is shown m Fig 4 for two axial flow rates eoqaessed as Reynolds nmnber. When the speed ctf rotauon of the mner cybnder is low. the flow regime in the «itmiar gap g lamnar (Couette) flow. a fully developed laminar liquid vdocity profile exists m the gap giving a low value of Pe. As the speed of rotanon is slowly increased. Taylor vortices develop ransmg a gradual m the «tmI dispersion and a narrowing of the RTD curves The mammmw points m the data(highest Pe) shown in Fig 4 correspond to the conditions at which the Taylor vortices are fully established along the column and flow near plug flow condmoa With further increase m the speed of rotationof the inner cylinder intermixing and exchange of material occurs at the boundaries between the neighbouring vortices «n«ng an increase m the axial ch^rrsion and a broadetung of the RTD curves, and hence a decrease m the value of Pe IS observed.

The data in Fig 4 were supported tw visual observations of the flow structure m the annulas which inAtcateA the presence of flow mstabüity in the form of small vortioes These vortices first appeared m the region close to the walls cS the mner rotating cylinder and gradually grew radially outwards until they reached the inner walls of the outer shell The buth and growth of the Taylor vomces occurred over a penod of time and a range of speed dependmg on the axrai flow rate In the presem studv . the cnucai speed at which the Pe had Its maximum value was considered arbitranly to represent the condmons at which the Taylor vortices were fully formed

The critical Taylor number. Ta., necessary for the estahlishmeni of Taylor voraces plotted against the nperaiing axial Reynolds is shown m Fig S. The change m the cnucai Tavior number. Ta., with axial Reynolds number follow the trend reported bv previous groups which are also shown. These plots suggest that the cnucai Taylor «nmivr |$ a function of the dimensionless gap as well as the axial Revnolds number.

956

Appendix Six 269

THE 1 9 9 6 ICHEME RESEARCH E V E N T /SE C O N D EU RO PEA N CONFERENCE FOR YOUNG R E SE A R C H E R S

C o n c l u s i o n sThe RTD resulu were descnbed by usmg an a smgie paiameter axial dispenmn model and the results

that tfac Width of RTD nanowed dunng the transmon of flow flom laminar to Taylor-Couene- Potsenille ttgune over a range of rotational speed. The critical Taylor nmnber was artanrary defined as the point when the vomces were fully dei'eloped and was found to be deprnrtrnt on the axial Reynolds number

N o m e n c la tu r eC - ConoemxaQon (molL'^ R. R«D ” Axial dispersion coeffiaent ( L ^ ') ReL " Distance between two probes (L) TaN -Rotanon speed (T ‘) vPe - Peclet number ( - ) v

- Radius of the mwer and outer cylinder (L'') » Axial Revnolds number - 2(R.-Rjv/v ( - )- Tayiw number - (2*N)Re*'’(Ro-RJ^/v ( - ) « Axial flow vekxaty (LT*)- Kinematic viscosity ( L ^ )

R e f e r e n c e s1. Harrod. \L ,J . Food Proc. Engng., 9. 1-62 (1986).2. LevenspieL O.. Chemical Reaction Engineering, Wiley, New York. Chapter DC (1965).3. PD(^ono.P.L, Tavare. N.S.. Garside. J. and Nigam. K.DJ>.,CAeffl. Engng y., 101-110(1992). 4 Kataoka. K. Dot. R . Hongo, T. and Futagawa. h/L,J. Chem. Engng Japan, S. 472-476 (1975).5. Ng, S. and Turner, E R.. 1982. Proc. Roy. Soc. Lond, A 382. 83-102 (1982).6. Takeuci. D. L and Jankowski. D. P., J. Fluid Mech., 102.101-126 (1981).7. Hasoon. M A . and Mamn, B. W., Proc. Roy. Soc. Lond, A 352. 351-380 (1977).8. AstilL K.N.. J. Heat Transfer, 8, 383-392 (1964).

C/Co(->0 .2 5

Ta = 121 ; R* = 24.70.2

1st Prêt»2nd

0.1

0.05

0 10 20 30 40 50 60t(S)

Fip2 RTD cunwa tor tha IWD prabaa

250.

200.>

/150. /

1100. 150. I

AA / " \

o na-atar A: R«>10J7

'A

50 100 150 200Ta ( - )

np. 4 Padat marmar va Ta>«or nunear

C .(-)

3.5

25

1.5

0.5

0 0.5 1.5 2 25 31

80706050403020100

Ta. ( - )

M| ■» T««v naa w ■ ar?) tmh- JMM» naai) «I ■ 0 s) OM» M MM narr) «I ■ aw

■>■11»—«MMoarnu-aw

10 20 30Re ( - )

Fi05 Companaen et neawM d >mtuaa

and ma praaam study

957

Appendix Six 270

Papers PublishedSamson S. S. Yim and P. Ayazi Shamlou, 1996. Residence Time Distribution (RTD) In A Rotan' Flow-through Device with non-Newtonian Fluids. AlChE Annual Meeting (Oral presentation and full paper).

R e s i d e n c e T im e D i s t r i b u t i o n ( R T D ) In A R o t a r y F l o w - t h r o u g h D e v i c e W it h

N o n - N e w t o n i a n F l u id s

SAMSON S. S. YIM AND AYAZI SHAMLOU P.Chemical and Biochemical Engineering, University College London, London WCIE 7JE

ABSTRACT

The flow in the gap of a pair of rotating cylinders with and without superimposed axial flow has many potential applications in situations where good radial mixing and negligible axial dispersion are required. In the present study, RTD experimental data are reported for Non-Newtonian liquids flowing in the gap between two horizontal coaxial cylinders in the presence of a small superimposed axial flow. Liquid motion was achieved by the rotation of the inner cylinder and the action of a pump which allowed independent control of the axial flow in the gap. The RTD curves were described using an axial dispersion model and the experimental data were interpreted in terms of a Peclet number. The data were used to identify the critical Taylor number at or close to the point when flow changed from Couette to Couette-Taylor flow regime. The analysis showed that significant variation from plug flow occurred as the flow approached the point of criticality.

MATERIALS AND METHODS

The RTD data were obtained by using an impulse tracer injection technique. The experiments were carried out in a 1 m long horizontal coaxial cylinder device (Figure 1). The inner cylinder had a diameter o f 76 mm and driven by a variable speed electric motor. The outer shell, with an diameter of 90 mm, was equipped with two conductivity probes. 400 mm apart, and a single injection point. The tracer solution was prepared from concentrated KCl and the working liquid. The fluids employed in this study were various concentrations of CMC and Xanthan gum (X-gum) solutions (see Table 1).


Typical RTD curves for the two probes at a low axial flow rate are shown in Figure 2. The data were analysed in terms of a single parameter axial dispersion model used previously by the authors to describe the flow of Newtonian liquids in the same device [1]. Thus, the dimensionless tracer concentration. C©, at the measuring point is given by [2]

C e = ( P c / 4 7 1 0 ) '^ 2 e x p [ - P e ( I - 0 ) ’ /40] ( 1 )

where the dimensionless Peclet number, Pe = vLD, was evaluated by using the variance difference of the two tracer response curves. The agreement between the predictions by the dispersion model and the experimental data is satisfactory as shown in Figure 3.

Appendix Six 271

Figure 4 shows the effect of rotational Taylor number on the Peclet number. The low value of the Peclet number at low rotational speeds and low axial flow rates is caused by the laminar velocity profile in the gap. As the speed of rotation is slowly increased, Taylor vortices develop causing a gradual decrease in the axial dispersion and a narrowing of the RTD curves. The maximum points in the data (highest Pe) shown in Figure 4 correspond to the conditions at which the Taylor vortices are fully established along the column and flow approaches near plug flow condition. The critical rotational speed was expressed in terms of the critical Taylor number, Ta [1]. With further increase in the speed of rotation of the inner cylinder intermixing and exchange of material occurs at the boundaries between the neighbouring vortices causing an increase in the axial dispersion and a broadening of the RTD curves. This was supported by visual observations of the flow structure in the annulus which indicated that the birth and growth of the Taylor vortices occurred over a period of time and a range of speed.

Table 2 gives the critical Taylor numbers, Ta^ for the different non-Newtonian liquids used and for comparison data are also included for Newtonian fluids reported previously by the authors [1]. The calculations of the Taylor numbers were based on the apparent fluid viscosity evaluated at the average shearing conditions in the apparatus. Experimental values showed that the critical Taylor number increased with a decrease in the flow index n showing the stabilizing influence of non- Newtonian (shear thinning) properties of liquids on the rotational flow.

CONCLUSIONS

Experimental data were reported on the RTD of non-Newtonian fluids flowing between the gap of two concentric rotating cylinders. It was found that the rotational speed of the (inner) cylinder and the shear thinning properties of the non-Newtonian fluids significantly affected the flow structure in the gap and the critical condition at which flow changed from Couette to Couette-Taylor flow regime. Further work is planned to model the point of criticality as function of non-Newtonian properties.

NOMENCLATURE

A = -a^û)/2(l-a‘) (rad"') Ce = Dimensionless concentration ( - )D = Axial dispersion coefficient (L 'T ') K = Consistency index (Pa s")L = Distance between two probes (L) n = Flow index ( - )Pe = Peclet number ( - ) Ri, Ro = Radius of the inner and outer cylinder (L '')Ta = Taylor number = -4A(o6'*/|ia ( - ) v = Axial flow velocity (L T ')a = Radius ratio ( - ) Ô = Gap width = Rg -Rj (L)

= Apparent viscosity (L^T ') e = Dimensionless time ( - )(0 = Angular velocity (rad ')

REFERENCES

1. S. S. Yim and P. Ayazi Shamlou. Chem. Engng Sci.. (submitted).2. Levenspiel, 0 ., Chemical Reaction Engineering, Wiley, New York, Chapter IX (1965).3. Pudjiono. P. I., Tavare. N. S., Garside, J. and Nigam, K. D. P., Chem. Engng J., 101-110 (1992).

Appendix Six 272

Fluid Medium n ( - ) K(Pa s")

60wt% Glycerol (N ew tonian) 1 -

0.4 w t% CMC 0.543 0.456

0.5 wt% CMC 0.511 0.772

0.6 wt% CMC 0.495 1.012

0.7 wt% CMC 0.456 1.759

0.8 wt% CMC 0.414 3.201

0.4 w t% X-gum 0.365 0.653

0.5 w t% X-gum 0.331 1.079

0.6 wt% X-gum 0.26 1.885

0.7 wt% X-gum 0.258 2.341

0.8 w t% X-gum 0.243 2 .903

fluids u se d In this work

—O —Taylof number * 2675

Taykx number * 3876

- O —Taylor number * 5870

2nd Probe1st Probe

20 40Tim e (s)

Figure 2 - RTD cu rv es for the two p robes (0.6 wt% CMC solution)

— 0.8 wt'/oCMC

1— 0 6 wt%CMC

' — 0 4wt%CMC

TTO

m otor

Couette-flow device

OutletInlet

C3

Figure 1 - Expenm ental eq ipm en t

7— : Axial dispersion model . ; Expenmental data6

Ta = 10857 |5

4

Ta = 149443

2Ta = 18923 |

10

0.5 1 1.5 2

D im ensionless tim e ( - )

2.5

Figure 3 - C om parison of axial d ispersion m odel and experim ental d a ta

(0.6 wt% X-gum solution)

2000 4000 6000 8000 10000

Taylor number ( - )

Fluid M edium Ta,

85 wt% Glycerol 1913

0.4 wt% CMC 1993

0.5 wt% CMC 2582

0.6 wt% CMC 2675

0.7 wt% CMC 4164

0.8 wt% CMC 4925

0.4 wt% X-gum 7065

0.5 wt% X-gum 8465

0.6 wt% X-gum 10857

0.7 wt% X-gum 11247

0.8 wt% X-gum 13827

Figure 4 - P ec le t num b er vs Taylor num ber Table 2 - Ta^ for non-N ew tonian solutions

Appendix Six 273

Papers AcceptedSamson S. S. Yim , N. Titchener-Hooker and P. Ayazi Shamlou, 1997. Prediction of Critical Tavlor Number For Non-Newtonian Fluids. The First European Congress on Chemical Engineering.

P r e d i c t i o n O f C r i t i c a l T a y l o r N u m b e r In C o u e t t e F l o w

F o r N o n - N e w t o n i a n F l u id s

S A M S O N s . s . y im , N . T IT C H E N E R -H O O K E R A N D P . AYAZI S H A M L O U

Chemical and Biochemical Engineering, University College London, London WCIE 7JE

ABSTRACT

A numerical method is presented for solving the eigenvalue problem which governs the stability o f non-Newtonian liquids in Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with various modes o f instability. Consideration is given to the cases of radius ratio from 0.1 to 0.99 and the non-Newtonian flow index from 0.2 to 1. The Taylor number, being a criterion of the stability, has been defined using the mean q^parcnt viscosity. Results showed that the critical value o f Taylor number Ta« was an increased as either the flow index or radius ratio increased.

INTRODUCTION

Couette flow describes the steady circular flow o f a liquid contained between two concentric rotating cylinders. The parameters usually used to specify the flow are the radius ratio, a (=Ri/R2), angular speed ratio, |3 (=Q2/Qi) and the Taylor number. Ta, which is a non- dimensional measure o f the gradient of centrifugal force. G. I. Taylor (1923) provided the first experimental and theoretical evidence on the breakdown o f a Newtonian liquid in Couette flow with increasing speed o f rotation of the cylinders, in which the laminar fiow would be replaced by a cellular pattern around the cylinders in layers o f vortices - now known as Taylor vortices - (see Figure 1). The instability of Couette flow has since been studied by a number o f workers yielding a vast amount o f information on the hydrodynamics (Chandrasekhar, 1961) and applications o f Taylor-Couette flow (Cohen and Marom, 1983). In the present study, a more general Couette fiow instability problem will be investigated to include the effect of non-Newtonian properties on onset o f fiow instability.

THEORY

Consider an inconçressible non-Newtonian liquid fiow between two infinitely long concentric cylinders in the usual cylindrical co-ordinates r, 6, z (See Figure 1). If we denote the radial, tangential and axial velocity components respectively by u,, ue and Uz, and the pressure by p, the Navier-Stokes equations give an exact time-independent solution of the form

Appendix Six 274

A p a ir o f T a y lo r v o rL c e a O u te r c y l in d e r

A n n u la r g q )

I W a v e le n g t h I

where

U r = 0

p-a^

Figure 1 Annulus conâguration and co-ordinate system

U e = rQ (r) = Ar + B/r ^ = constant

A =

ÔZ

and a = R1/R2 and P =

(1)

(2)1-a ' 1-aA power law model given by Equation (3) is used to described the rheological behaviour o f the material to be simulated. Thus

0 -1

Xii =K

where tÿ and ÿij corresponds to the shear stress and shear rate, respectively

Yü (3)

To study the stability of the steady fhiid motiotL we superimpose on the basic flow an infinitesimal perturbation that is periodic in z and 6 . Thus we inqx)se small perturbations on the mean fiow which are assumed to be axisymmetrical and to have the following form,

[ u r , U 0 , U z ] = [O,V,O] + [u,v,w]e'“ ; p = 0 + p e “ (4)

The equations o f the disturbed motion are derived by substituting Equation (4) into the Navier-Stokes equations and neglecting terms that are quadratic in the disturbance. The Navier Stokes equations are made dimensionless based on the following dimensionless variables and dimensionless operators. D and D*. Thus

r = R + (R ,- R ,)x D = -^ and D* = -^ + e4 (5)dx dx

The pressure and axial velocity components are eliminated fi’om the governing equations by substitution to give:

p[dD*-X^1u +2DuDP = M Ta v + d| 4 -D * D D * u -P D * u + ^ D D * u + ^ iI J X

P nDD * -X^ jv+nDpmPD* v = u

2 ( R 2 - R i )where =(1 + ex)

X = a(R2 - R ,)

-4p ^n o (R 2 - R i ) V

e =

Qo =

(R2 +R i) (O1 +O2 )

Ta = P =Pm Pm

( 2 - n K l - n ) « 2/D-1

(6)

(7)

(8)

(9)

(10)

Appendix Six 275

M Y =

Pm = K*ü 1R2 - R 1

n=

(1-1-ex) ( l - e / 2 )

(1 + ex) (I + e / 2 )

p = K K2Lr 2/n

n—IK2 =

10-1

2(n:-0,)R2 2/n

( 1 1 )

( 12)

(13)

(14)are solved numerically by applying the Galerkin's technique (Walowit, et aL, 1964). This method consists o f expanding u and v in sets o f complete function, preferably orthogonal, that satisfy the boundary conditions and then require the error in the equations for u and v to be orthogonal to the expansion function for u and v. Details o f the calculation procedure and the algorithms are given elsewhere (Yim et aL, 1996). It should be noted that wbm flow index, n, equals to unity, we have the classical Couette flow instability problem as given by Taylor ( 1923).

n |(R j/R i)^ '- l]

Equations (6) and (7) together with the associated boundary conditions given by u = Du = v = o at X = ±1/2


Equations (6), (7) and (14) define an eigenvalue problem of the formF(a, p. Ta, n, X) = 0

The parameter a describes the geometry, the parameter p and Ta describe the basic flow, the parameter n is power law flow index, and the parameter X is the dimensionless wave number o f the disturbance in the axial direction. For a given a, P and n, Taylor number can have a sequence of possible determinate value corresponding to each value of X. This set of points defines a neutral curve in the T-X plane. The critical T^lor number. Ta*, for the onset o f the instability (for a given, a , P and n) will be given by the minimum point on the neutral curve. Below which all disturbance are damped, and above which the corresponding disturbance will grow. The critical value o f Taylor number also determines the critical wave number a * . Figure 2 displays Ta as a function o f dimensionless wave number for different value of flow index, n. It shows that as the flow index decreases; the neutral curve is shifted toward a lower values o f Ta.

The case o f flow index equals to unity. Le. the Newtonian case, represents the conditions used by T^lor (1923) in his formulation o f the stability problem. According to our plot in Fig. 2, flow becomes unstable at a Taylor number of approximately 1840 (the minimum point on the curve, n = 1) which agrees well with the Taylor's predicted value o f 1950 under a conçarable situation.

Figure 3 shows the effect o f radius ratio, a, on the critical Taylor number, Ta*, with the flow index, n, as a parameter. The plots indicate that as the radius ratio decreases the critical Taylor number increases, that is, for otherwise similar conditions, the flow becomes more stable as gap width increases. According to the simulations shown in Fig. 3, the extent o f the increases in Tac depends critically on the flow index, n. It is observed fiom these plots that for decreasing flow index, n, the non-Newtonian shear thinning liquid flows in the gap of a pair o f counter- rotating cylinders is significantly more stable conpared with the case o f Newtonian liquid flow.

Appendix Six 276

CONCLUSIONS

Theoretical equations are developed and are numerically solved for the onset o f flow instability o f non-Newtonian in Couette flow. The numerical simulations are presented in terms o f the critical Taylor number as a function of two parameters affecting it. Le. the flow index and the radius ratio, a. The theoretical prediction show that an increase of non-Newtonian properties o f liquids (decreasing value o f n) and/or an increase in radius ratio result in a decrease in the critical value of the Taylor number as well as the dimensionless wave number. The predictions for the special case o f n =1 and a close to unity (small g ^ ) are also in good agreement with the results reported by previous researchers.

NOMENCLATURE

A s Constant (s ') a = Wave number (-)B * Constant (s ') D, D* = Dimensionless differential operatw (-)K = Consistency index (Pas) n = Flow index (•)r s Radial coKxdinate (m) Ri, Rj = Inner and outer radius (m ')R, - Average radius (m) s = Growth rate of disturbances (-)Ta = Ta)dQr number (-) V = Tangential flow velocity (ms')X » Dimensionless radial co-cxdinate (-) z = Axial co-ordinate (m)a = Radius ratio. (-) P = Angular qieed ratio. (-)f * Shear rate (s ') X = Dimensionless wave number (-)

“ Viscosity (kgm ''s')n„ = Mean viscosity (kgm's*')P “ Density (kgm 'l T = Shear stress (Pa)Qi, O] » Inner and outer angular velocities (rad ) a, = Average angular velocity (rad')

REFERENCES

1. Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability^ Oxford: Clarendon Press, London.

2. Cohen, S. andMarom, D.M 1983. The Chem. EngngJ., 27, 87-97.3. Taylor, G.1.1923. Phil Trans. Roy. Soc. (London), A233,289-343.4. Walowh, J. Tsao, S. and Di Prima, R.C. 1964. Trans. ASME, 12, 585-593.5. Yim, S.S. and Ayazi Shamlou. P. 1996. Chem. EngngSci. (submitted).

2600

T 2300 mT 2000

J 1400

H 1100

900

/ n - 1.0/ n - 0 9

Æ n -0 .8n -0 .7n -0 .6

D - 0 4. - 0 3

%1 2 3 4 5 6Dimensioniess wavenunter. A. (-)

^ 35000

£ 30000

25000

20000

^ 15000

H 10000

5000

Radius Ratio, a (-)

Figure 2 Typical simulated neutral curves Figure 3 Simulated Tac vs a (at different n)

Appendix Six 277

Papers AcceptedSamson S. S. Yim and P. Ayazi Shamlou, 1997. Prediction of Critical Tavlor Number in Rotating Flow-through Devices, Proceedings; 1997 Jubilee Research Events.

Prediction Of Critical Taylor Number In Rotating Flow-Through Devices

SAMSON s . s . YIM AND AYAZI SHAMLOU P.Chemical and Biochemical Engineering, University College London, London WCIE 7JE

ABSTRACTTheoredcal and experimental results are presented showing the effects of operating and geometrical parameters tm the transition of laminar to Taylw vortex flow for induced rotational-axial flow in the gap of a pair of rotating cylinders. The simulations indicated that annular rotational flow became more stable in the presence of a small d^ree of axial flow and as gap width increases. Limited experimental data also were provided on the residence time distribution for flow of Newtonian liquids through the gap of two rotating cylinders. The results were successfully analysed and assessed using the simulations studies.Keyword: Rotating flow-through devices, Navier-Stokes equations,

Taylor number, Tayira- vortex flow,Residence time distribution, Peclet number.

IntroductionThe rotational flow generated ly a pair of coaxial cylinders, especially with a small superimposed axial flow, can be used to manipulate the residence time distributicm (RTD) of various species in the gap in order to promote radial mixing with negligible axial dispersim (Kataoka et al., 1981), conditions that are considered prerequisite for good mass and beat transfa and (bio)chemical reactions in many process engineering operations. In these unit operations, the dependence of the RTD of the different species on the fluid flow field occurring in the annulus is of great oq>erimental and theoretical interests.

TheoryFor an incompressible Newtonian liquid flowing under laminar flow cmdition between two infinitely long concentric cylinders (qdindrical co-ordinates r, 8, z). The radiaL tangential and axial velocity components respectively hyu^ue and Uz, and the pressure, p, can be obtained uniquely from the Navier-Stokes equations.

Ur * 0

u. = = —

ue * r Q (r) * Ar + B/r

apdz

= constant

, P — ccwhere A = - —l-o/ '

1 +1-(R,/Rj)^hXRj/R,)

(1)

(2)

(3)

(4)

where a = Ri/Ri and P = Qj/Qi. (5)

To study the condition leading to the instability of the laminar modem, we superimposed on the basic flow an infinitesimal perturbation that was periodic in z and 6. The small perturbations imposed on the mean flow were assumed to be axisymmterical and to have the following form.

Appendix Six 278

[DD*+nju = Ta M v + D-^[D*D + n]

= ; p = 0 ^ p (6)

The equations of the disturbed motion were derived by substituting (6) and a similar expression for p into the Navier-Stokes equations, neglecting terms that were quadratic. The resulting equations were made dimensioaiess by using the following:

r = R + ( R , - R , ) x D = — and D* = — rc^ (7)° ' dx dx

The pressure, axial velocity component were eliminated by substituuon. reducing the number of cquatims as follows

D * u l Re(DF) i

— J-— ®[DD*+n]v = u (9)

wheren = i ( G - A f R e ) - % \ (10)

A , - R . fc X = a ( R , - R , ) (II)

v‘ v“

F = --------:----- :------- f(l-OL^)lnT + a ^ ( Y - l ) l n a | (14)( 1 - 0 ) + ( l + o ) ln a i J

Equaticms (8) and (9) together with the associated boundary conditions given by:

u = Du = v = 0 at X = ±1/2 (15)

were solved numerically by applying the Galerkin's technique (WalowiL et al., 1964). Details of the calculation procedure and the algorithms are given elsewhere (Yim et al., 1996) and arc only briefly outlined here. For a fixed flow rate (axial Reynolds number) and condition given by a , and fi, Eqs. (8) and (9) were solved to yield values of Taylor numbers for a range of values of the dimensionless wave number k and growth rate a.

Results and DiscussionThe residence time distribuiicm experiments were dircacd towards examming the eflfccts of axial flow rate, gap width and the speed of rotation on the RTD of a range of glyccrol-water solutions of different viscosities (Table 1). A general arrangement of the experimental sa-up is shown schematically in Figure 1. The expcnmems were performed under operating conditions that included the pomt of cnucality. Full details of the flow device and experimental procedure have been reported elsewhere (Yim, 1997).

Figure 2 shows typical simulated neutral curves for the case of zoo axial flow, a range of narrow gap (a =0.9 - 0.99) and speed ratio, P = 0. The minimum point on the neutral curve defines a critical value of the Taylor number, below which all disturbance arc damped, and above which the disturbance will grow to become vortices. Figure 3 shows the variation of the critical Taylor number as a function of the axial Reynolds number. From these plots it can be seen that for fixed a as the flow rate through the gap increases the critical Taylor number increases indicatmg that by introducing a small degree of axial flow, the induced rotaucmal flow becomes more stable. The predictions shown in Fig. 3 arc confirmed by the experimental RTD data obtained from the rotating flow-through device as discussed below.

Appendix Six 279

Hie eflfect of Reynolds number on the Peclet number, obtained from the impulse response is shown in Fig. 4 for four axial flow rates. For the present data the Peclet numbers were obtained for each experiment by using the definition of the dimensionless variance of the residence time distributions given by Levenspiel (1977).

Pe = 2/Aoe^ (16)

Hie plots in Fig. 4 show that as the rotatitmal speed of the inner .shaft increases, the Pe number increases from an initial value of about 100 to a maximum value of approximately 380 beyond which any further increase in rotational speed causes a decrease in the Pe number. Hie minimiim and mariniimi values of the Pe number were found to be fimctions of the flow rate through the gap. Hie variations in Pe number with increasing Ta number depicted in Fig. 5 suggests that significant deviation from plug flow conditions can occur during flow and the maximum Pe number correspond to the conditiœs at which the Taylor vortices are fully develc ied (Pudjiono et ai., 1992)

In Fig. 5, data on the critical Taylor numbers are plotted as a function of the axial Reynolds number for the experimental conditions used in the present investigation, Hie different symbols shown in Fig. 5 represent the experimental critical Taylor numbers obtained from the Peclet number • Taylw number plots sudi as those shown in Fig. 4 while the solid curves are the simulations. The agreement between theory and experiments is considered satisfactory.

ConclusionsTheoretical equations are developed and are numerically solved for the onset of flow instability in combined Couette and Poiseuille flow. The formulation is applicable to flow of Newtonian liquids through wide gaps formed between two concentric rotating cylinders. The numerical simulations are presented in terms of the critical Taylor number as a functifxi of three dimensionless groups affecting it, i.e. the axial Reynolds number, the dimensionless radius ratio and the relative speed of rotation of the two cylinders. The results indicate that for otherwise similar conditions, rotational flow in the gap becomes more stable as the axial Reynolds increases and as the dimensionless radius ratio decreases. The analysis of experimental RTD data showed a systematic variation in the Peclet number. This variation, was explained adequately in terms of the changing flow structure in ihe gap as flow approached the point of cridcality.

NomenclatureA «Constant Eq.(4) (s ') a = Wave number (-)B «Constant Eq.(4) (s ') D,D*' = Dimensionless differential operator (-)F = Dimensionless constant Eq. (14) (-) M « Dimensionless constant Eq. (13) (-)Pe » Peclet number. (-) r K Radial coordinate (m)Ri, Rj « Inner and outer radius (m'> Re = Axial Reynolds number Eq. (12) (-)R, = Average radius (m) s - Growth rate of disturbances (-)t « Time (s) Ta = Tayicx numbc Eq. (12) (-)Ue « Tangential velocity component (m s ') Ur « Radial velocity cmnponent (m s ')Uz « Axial velocity component (m s ') V « Tangential flow velocity (m s ')w „ » Axial flow velocity (m s ') X « Dimensionless radial coordinate (•)z * Axial coordinate (m) a s Radius ratio. (-)P , « Angular qieed ratio. (-) X « Dimensionless wave number Eq. (11)(-)Ao# « Dimensionless variance difierence(-) a = Dimensionless disturbance growth rate(-)

« Inner and outer angular velocities (rad ') a , » Average angular velocity (rad')

References1. Harris, J. 1977, Rheology and non-Newtonian flow, Longman. London.2. Kataoka, K. And Takegawa, T. 1981, A.I.Ch.E. J, 27(3), 504-508.3. Levenspiel 0. 1965, Chemical Reaction Engineering, Wiley, New Ywk, Chapter DC4. Pudjiono, P.I., Tavare, N.S., Garsidc, J. and Nigam, K.D.P. 1992. Chem. Engng J„ 101-110.5. Walowit, J. Tsao, A. And Di Prima, R.C. 1964, J. Applied Mechanics Trans. ASME, 31,585-593.6. Yim, S. and Ayazi Shamlou, P. 1996. Proc. IChemE Research Events, 2,955-957.7. Yim, S. 1997, PhD Thesis, University of London. (In preparation).

OWcerol Tcnptnniic Oomcy ViKMtyrPt.s)

43 20 1112* 47

64 20 1164* 14 4

15 20 1221 * 113

Tabic 1; The physical propcrocs of glycerol solunœ s

Appendix Six 280

Wet

Figure I : Experimental Equipment

21502100205020001950190018501800175017001650

nw w enunter, >.(-)

20000

15000

F 10000

■5 5000

02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rkéu» Rat». a H

Figure 2; Simulated neutral curves Figure 3 : The of axial flow on tbeTa,

400 — RB * 8— Ô — Ra *12— 6 — Rb *18— — Rb * 23

350

300

250

200

150

100

2000 4000 6000 8000Taylor number. Ta (-)

14000

10000

5 6000

Ï 4000

5 2000

0

] o ASk&wmoi p a»0£^ tm otfanto :64«Ghen4 / Ê (%m0.7• S4«Qnnm / a«0.8□ 49ftO»t«anM a*09m 4MOMmo<

1 a « O j 6 7 ___ ,

i

0 20 40 60 80 100Reynold* nirber. Re (-)

Figure 4: Experimental Pe vs Ta Figure 5; Comparison of experimental and theoreitcalTae

Appendix Six 281

Papers SubmittedSamson S. S. Yim, M. Y. A. Lo, N. Titchener-Hooker and P. Ayazi Shamlou, 1996. Factors affecting flow instability of Newtonian liquids in Couette-Poiseuille flow. Full paper submitted for publication to Chemical Engineering Science

Factors affecting flow instability of Newtonian liquids in Couette-Poiseuille flow

S. S. S. Yim, M. V. A. Lo, N. Titchener-Hooker and P. A\3zi Shamlou^

Department of Chemical and Biochemical Engineering, University College London,

Torrington Place, London WCIE 7JE U. K.

ABSTRACT

Numerical simulations are presented showing the effects o f operating and geometrical

parameters on the transition of laminar to Taylor vortex flow for induced rotational-axial

flow in the gap of a pair of rotating cylinders. These simulations indicate that annular

rotational flow becomes more stable in the presence of a small degree of axial flow and

as gtq> width increases. The effect of rotational speed on the breakdown of laminar flow

is more conplex and for given radius ratio and axial flow rate depends on both the speed

ratio and the direction of the rotation of the cylinders, counter-rotating flow generally

producing a more stable flow than co-rotating.

Limited experimental data are provided on the residence time distribution for flow of

Newtonian liquids through the gap of two rotating cylinders. The data include results

from experiments in which flow transition occured from latninar to Taylor vortex flow.

The findings from these experiments are successfully analyzed and assessed using the

simulations studies.

* The author to whom correspondence should be addressed

Appendix Six 282

INTRODUCTION

The laminar flow field formed in the gap of two rotating coaxial cylinders has been

exploited fully by rheologists for a long time in the study of flow behaviour of Newtonian

and non-Newtonian liquids (Fredrickson, 1964; Astarita and MamiccL, 1974 and Harris,

1977). What is less known is that the induced rotational flow generated by a pair of

coaxial cylinders, especially with a small superimposed axial flow, can be used to

manipulate the residence time distribution (RTD) of various species in the g ^ in order to

promote radial mixing with negligible axial dispersion (Kataoka et al., 1975), conditions

that are considered prerequisite for good mass and heat transfer and (bio)chemical

reactions in many process engineering operations. In these unit operations, the

dependence of the RTD of the different species on the fluid flow field occurring in the

annulus is of great experimental and theoretical interests and has been the subject of

numerous studies in the past (Becker and Kaye. 1962; Coney and Simmer, 1979;

Kataoka and Takigawa, 1981; Moore and Cooney, 1995).

G. I. Taylor (1923) provided the first experimental and theoretical evidence on the

breakdown of Couette flow with increasing speed of rotation of the cylinders. With

superimposed axial flow (Couette-Poiseuille flow), the breakdown of laminar flow is a

strong function of the dimensionless axial Reynolds number. Re [ = j , the

rotational Tavlor number. Ta I = - ] . the relative speed and theI v* ;

direction of rotation of the cylinders as well as the dimensionless radius ratio, a

Appendix Six 283

The interdependence of these parameters and their combined effect on the onset of flow

instability is worthy of further investigation and is the subject of the present study. From

a process engineering point of view it is the changes in the flow structure resulting from

the breakdown of laminar flow that afreet the residence time distribution of species in the

gap and are thus thought to be responsible for the observed in^rovements in transport

and reaction processes occurring in annular rotational flow fields (Astill, 1964; Kaye and

Elgar, 1975; Cohen and Marom, 1983: Meyashita and Senna, 1993).

THEORY

The Navier-Stokes and the continuity equations governing the stability of rotational flow

o f an incompressible Newtonian liquid in the gtqs of two coaxial cylinders have so fru

been solved only for a limited number of cases, e.g. flow in narrow and wide g^ s with

zero axial flow (Walowit et a i, 1964) and flow in narrow gap with a small superimposed

axial flow (Takeuchi and Jankowski, 1981 ). In the present study, the stability problem is

formulated for the case of viscous incompressible flow of a Newtonian liquid flowing at a

low velocity through the gap of two co-rotating and counter-rotating coaxial cylinders.

The classical approach is adopted in im posing small velocity perturbations of an

exponential form on the mean flow described by the general equations, products and

squares of the perturbations are neglected and the linearised equations are made

dimensionless and are solved numerically using the Galerkin's method (DiPrima, 1955;

DiPrima and Swiimey, 1981) in order to determine the critical Taylor number at which

lam inar flow breaks down as a function of gap width, axial flow rate and the relative

speed and direction of rotation of the two cylinders.

Appendix Six 284

Thus, for these assunptions the equations for the mean flow through the gap of two

rotating cylinders are as follows.

U (r) - 0 V(r) = r CÎ (r) = A t + B/r

where

- i f

B = — Q,Rr ]-or

( 1)

(2)

and

a - Ri/Ri and 3 = Oi/Q,.

The average axial velocity, W , can be obtained by integration of Eq. (2). Thus:

, r R , y i - ( R i / r 2)=iRzV ln(R2 /R i)

(3)

(4)

(5)

The small perturbations imposed on the mean flow are assumed to be axisymmterical and

to have the following form, in accordance with most previous works (Chandrasekhar,

1961; Chung and Astill, 1977; DiPrima and Swinney, 1981)

u'(r,z,t) = ïïe’ "*’> (6a)

v'(r.z,t) = (6b)

w'(r,z,t) = we’ ' "** (6c)

p'(r.z.t) = pe‘<^'“> (6d)

where u .v .w and p are functions of r only. The two parameters s and X in Eqs. (6a) -

(6d) characterize the flow perturbations in the axial direction. Their values determine the

growth rate of the disturbance and for a given flow rate and geometry they uniquely

Appendix Six 285

determine whether the imposed perturbations will decay or be able to grow exponentially

in time leading to the formation of secondary flow in the form of Taylor vortices.

Using the cylindrical coordinate system, introducing Eqs. (6a) - (6d) into the Navier-

Stokes and continuity equations, linearising the resulting expressions and making them

dimensionless gives

^DD * -a* + i(a - aT Re)ju = Ta M v + Dp (7)

^DD • -a^ + i(a - aT Re)jv = û (8)

♦ D - a* + i(a - aT Re)jw = iap + Re(DT)û (9)

D*û + iaw = 0 (10)

where

D = -^and D* = -^ + e (11a)dx dx

r = R o + (R :-R i)x (11b)

= , 1 ,0

0 = ——— —— a = X(R2 - R i ) (lie)

M . f ! r Æ

Ù = ------— - Û V = R o Q o V ( l l g )2 A ( R 2 - R i)‘

OoRo V . __ OoRopv . . , .p = — -------------- " 7 p ( l l h )

2 A ( R 2 - R j ) 2 A ( R 2 - R i )

Appendix Six 286

Substituting Eq. (12) into Eq. (9) to eliminate w and rearranging the resulting expression

leads to

p = (d ♦ D - a + i(a - aT - - 4 ^ - ^ û (12)

Equation ( 12) is now substituted into Eq. (7) to eliminate p . Thus

(DD * -a^ + i(a - aT Re))u = Ta M v

+ D |( D * D - a = + i ( o - a T R e ) ) ( ^ ] - ^ ^ ® û | (13)

Equations (8) and (13) together with the associated boundary conditions given by;

û = Dû = v = 0 at X = ±1/2 (14)

are solved numerically by qaplying the Galerkin's technique (DiPrima, 1955) and the

calculations are performed by using the Mathematica software (version 2.2) on a Elonex

PC 560. Details of the calculation procedure and the algorithms are given elsewhere

(Yim. 1996) and are only briefly outlined here. For a fixed flow rate (axial Reynolds

R,1 . „ r «2number) and condition given by a, , and P, | = , Eqs. (8) and (13) are

solved to yield values of Taylor numbers for a range of values of the dimensionless wave

number, a (2.5 - 4.25) and a (1.169-1.236). Typical results which will be discussed later

are shown in Fig. 2 in the form of “neutral” curves for a specific radius ratio and speed

ratio at zero axial flow. The minimum point on each neutral curve defines the critical

Taylor number. Tac for the specified conditions (Walowit et al., 1964).

Appendix Six 287

EXPERIMENTAL

The residence time distribution experiments were directed towards examining the effects

of axial flow rate, gap width and the speed of rotation on the RTD of a range of glycerol-

water solutions o f different viscosities. The experiments were performed under operating

conditions that included the point of criticality. Because of limitations of the apparatus

the outer cylinder remained stationary in the experiments and flow in the gap was

induced by varying the speed of rotation of the inner cylinder which was varied between

0.0 and 400 rpm. The volumetric flow rate was varied between 2x10 and 9x10 mmVs.

The experiments were carried out in a 1 m long horizontal coaxial cylinder device in

order to minimise end effects (Fig. 1 ). The outer shell was Abricated from Perspex to

allow visual observations of the flow structure and had an inner diamete r of 90 mm,

Photographs were not taken during the experiments, but visual observations of the flow

structure was carried out to conflrm the condition of criticality. The anmilar gap width

was varied by using two inner cylinders of diameters 0.076 and 0.06 m. The inner

cylinders were constructed from stainless steel tubes and were driven by a variable speed

electric motor with rotational speed in the range of 0 - 1500 rpm. The speed of rotation

was measured by using a shaft-mounted speed transducer (2400AB Series, EEL Ltd.,

UK).

The outer shell was equipped with two electrical conductivity probes and a single

injection point. The conductivity probes were constructed from 35 mm diameter glass

rod, having two 0.315 mm diameter platinum wire electrodes mounted 2.5 mm apart.

Appendix Six 288

The two probes were separated 400 mm apart in a horizontal line with the first probe

positioned 50 mm fi*om the injection point. When in position, the tip of the electrodes

protruded 3 mm fi-om the inside wall of the outer shell. The microinjeaor was a standard

hypodermic syringe having a stainless tube of 0.7 mm outside diameter. The injection

point was 200 mm fi*om the entrance in the same horizontal line as the conductivity

probes and the tip of the injection tube was positioned at the center of the annular gap.

The tracer solution was prepared fi'om concentrated KCl (AnalaR, BDH Limhed-Poole,

England) and the woridng liquid. Injection was carried out manually within a period of

approximately 1.0 s and the tracer concentrations were recorded simultaneously and

continuously at the two probe positions as a function of time. The data fiom both

conductivity probes were stored using a data acquisition system (Keithley Model 575

measurement control system) attached to a KDAC 500/1 data collection unit and an IBM

PC2 computer for analysis and evaluation of the various parameters including the first

and second moments o f the distributions.

Prior to injection of the tracer material, steady state flow condition was established in the

apparatus by continually pumping (H.R. Flow-Inducer, Watson-Marlow Ltd., UK) the

working liquid through the annular g ^ at a fixed volumetric flow rate until the efiQux

remained constant with time. The efifiux was obtained manually by measuring the time

required to collect a known volume of the discharge liquid. Each experiment was

replicated three times and the mean value used for parameter estimation.

Appendix Six 289


Figure 2 shows typical simulated neutral curves for the case of zero axial flow, a range o f

narrow gzy (a = 0.9 - 0.99) and speed ratio, p = 0. These are the conditions used by

Taylor (1923) in his formulation of the stability problem and for which the following

expression for the critical Taylor number has been recommended (Becker and Kaye,

1962). Thus

.p m/Rrn(R2 -Ri)^i c - ^ (15 )

where

(1 6 )2P (R :-R ,)'R ,

where

P = 0.0571 1 -0.0571 + 0.00056^1 -0.652 (17)

According to our plot in Fig. 2, flow becomes unstable at a Taylor number of

approximately 1700 (the minimum point on the curve, a = 0.99) which agrees well with

the predicted value (Taylor, 1923) of 1697 at a = 0.99, based on Eq. (15) to (17).

In Fig. 3, our simulations con iare well with results published by previous researchers for

the case of narrow gaps (a = 0.95). Limited experimental data reported by Snyder

( 1962) for the case of annular flow in a small gap with superimposed axial flow are also

shown in Fig. 3 and these also support the numerical simulations obtained from our

analysis of the instability problem

Appendix Six 290

Figure 4 shows the eflfea of radius ratio, a, on the critical Taylor number, Tac, with

velocity ratio, P, as a parameter. The plots are for the case of zero axial flow and

indicate that as the radius ratio decreases the critical Taylor number increases, that is, for

otherwise similar conditions, the flow becomes more stable as gap width increases.

According to the simulations shown in Fig. 4 the extent o f the increases in Tac depends

critically on the velocity ratio, p. The values o f P used in the calculations procedure are

given in Fig. 4; the negative values correspond to simulation o f flow of counter-rotating

and the positive values represent the simulation o f flow of co-rotating cylinders. It is

observed from these plots that for given radius ratio, a. the flow in the gap of a pair of

counter-rotating cylinders is significantly more stable con^ared with the case of co-

rotating cylinders.

Figure 5 shows the variation of the critical Taylor number as a function of the axial

Reynolds number. From these plots it can be seen that for fixed a and p as the flow rate

through the gap increases the critical Taylor number increases indicating that by

introducing a small degree of axial flow, the induced rotational flow becomes more

stable.

The predictions shown in Fig. 5 are confirmed by the experimental RTD data obtained

from the coaxial cylinder apparatus as discussed below. It should be noted however that

all the RTD data reported in this work are limited by the condition of zero speed ratio

since - 0.

Appendix Six 291

Typical RTD curves for the two probes are shown in Fig. 6 for four rotational speeds of

the inner cylinder in the range o f 60 to 260 rpm and axial flow velocity of 26.5 mm/s

corresponding to an axial Reynolds number. Re = 15 and Taylor numbers, Ta = 3074,

16738, 41332 and 57728 respectively. The RTD data for flow in the gap of a coaxial

cylinder apparatus has been described adequately in the past by an axial dispersion model

characterized by the dimensionless axial Peclet number, Pe^= . For the present

data the axial Peclet numbers were obtained for each experiment ly using the definition

of the variance of the distributions given by (Levenspiel 1972)

A oe^=-^ (18)

Ace represents the variance difference of the tracer RTD curves at any two points along

the annular gap. In the present study Aae' was evaluated for the g ^ between the two

probes using the response curves obtained fix)m them, as shown in Fig. 6. The value of

Aoe was substituted into Eq. (18) in order to obtain the Pe number fbr the prevailing

flow.

The effect of rotational speed on the Pe number, obtained fi-om the induise response

curves at the two conductivity probe positions is shown in Fig. 7 for four axial flow rates

expressed in term of the axial Reynolds number. The plots in Fig. 7 show that as the

rotational speed of the inner shaft increases, the Pe number increases fi-om an initial value

of about 100 to a maximum value of ^proximately 380 beyond which any further

increase in rotational speed causes a decrease in the Pe number. The minimum and

m aximum values of the Pe number were found to be functions of the flow rate through

the gap. In the present study the lowest value of Pe was found to be about 100

Appendix Six 292

corresponding to an axial flow velocity of 14 mm/s (Re = 8) and the highest value of Pe

was 380 corresponding to an axial flow velocity o f 47 mm/s (Re = 27) as shown in Fig.

7.

The variations in Pe number with increasing Ta number depicted in Fig. 7 suggests that

significant deviation fi-om plug flow conditions can occur during flow. Pudjiono et al.

(1992) and Pudjiono and Tavare (1993) in their recent publications suggested that the

maximum Pe number correspond to the conditions at which the Taylor vortices are fully

developed. In Fig. 8 we have plotted the critical Taylor number as a function of the axial

Reynolds number for the e?q)enmental conditions used in the present investigation. The

difierent symbols shown in Fig. 8 represent the experimental critical Taylor numbers

obtained fi-om the Peclet number - Taylor number plots such as those shown in Fig. 7

while the solid curves are the simulations. The agreement between theory and

experiments is considered satisâctory.

CONCLUSIONS

Theoretical equations are developed and are numerically solved for the onset of flow

instability in combined Couette and Poiseuille flow. The formulation is applicable to flow

of Newtonian liquids through wide gaps formed between two co-rotating and counter-

rotating cylinders. The numerical simulations are presented in terms of the critical Taylor

number as a function of three dimensionless groups afiecting it, i.e. the axial Reynolds

number, the dimensionless radius ratio and the relative speed of rotation of the two

cylinders. The results indicate that for otherwise similar conditions, rotational flow in the

Appendix Six 293

gap becomes more stable as the axial Reynolds increases and as the dimensionless radius

ratio decreases. The efifect of speed of rotation is more complex and depends on the

direction as well as the absolute values of the speed of the two cylinders.

Limited residence time distribution data were obtained fbr the axial dispersion of a tracer

fluid in the gap of a coaxial cylinder apparatus operating close to the Taylor-Couette-

Poiseuille regime. The analysis of RTD data showed a systematic variation in the

prevailing axial Peclet number. This variation was explained adequately in terms of the

changing flow structure in the gap as flow approached the point o f criticality. The

experimental observations based on the RTD data and the associated numerical

simulations presented have inqx)rtant implications for the design and operation of

equipment in which Taylor-Couette-Poiseuille flow is used to enhance transport and

reaction processes.

Appendix Six 294

NOMENCLATURE

A_ B -a _= Constant, ------;-Gi

l i - o / ;(s*‘)

a = Dimensionless wave number (-)

B = Constant, |~ - y f i iR i^ j (s')

D,D* = Dimensionless differential operator (-)

D = Axial dispersion coefBcient (m's*')

L = Distance between two measuring points (m)

M = Dimensionless angular velocity, (-)

N = Rotational speed of cylinder (rps)

P = Pressure (Pa)

? = Parameter defined by Eq. (17) (-)

Pe = Peclet number, (-)

Re = Axial Reynolds number, I —— I (-)

Rj, Ri = Radii of outer and inner cylinders (m)

R= = Mean radius, (m)

r = Radial coordinate (m)

s = Growth rate of disturbances (-)

Te = Critical Taylor number given by Eq. ( 15) (-)

Ta = Taylor number, (-)

Tac = Critical Taylor number (-)

t = Time (s)

U(r) = Radial velocity component (m s')

V(r) = Tangential velocity component (ms"')

w = Axial flow velocity (m s ')

W(r) = Axial velocity conyonent (m s’’)

X = Transformed dimensionless radial coordinate (-)

z = Axial coordinate (m)

Greeks Symbols

Appendix Six 295

= Radius ratio'& )

XnV

aAoe

= Angular speed ratio,

= Weighting function

= Dimensionless constant,

= Wave number

= Viscosity of the working liquid

= Kinematics viscosity of the working liquid

= Dimensionless growth rate

= Dimensionless variance difference

Q2, = Angular velocities of outer and inner cylinders

Qo = Average angular velocity

O (r) = Angular velocity

Û) = Angular velocity given by Eq. ( 16)

4 = Dimensionless variable

(-)

(-)

(')

(-)

(m*‘)

(kg m*' s"’)

(kg m’’ s ')

(■) —

(-)

(rad-')

(rad-')

(rad*')

(rad-')

(-)

Superscripts and Subscripts

= Average value

= Perturbed quantity

= Eigenfunction given by Eqs. (4) - (7)

= Dimensionless eigenfunction

= Critical value

Appendix Six 296

REFERENCES

1. Astrarita, G. and Mamicci, G., 1974, Principles o f non-Newtonian fluid

mechanisms, McGraw-Hill, London.

2. Astill, K. N., 1964, Studies of the developing flow between concentric cylinders

with the inner cylinder rotating, J. Heat Transfer Trans. ASME., C86, 383-392.

3. Becker, K. M. and Kaye, J., 1962, The influence of a radial tenqxrature gradient

on the instability of fluid flow in an annulus with an inner rotating cylinder. J.

Heat Tranter Trans. ASME, C84,383-392.

4. Chandrasekhar, S., 1960, The hydrohynamic stability o f viscid flow between

coaxial cylinders, Proc. Nat. Acad. Sci., 46, 141.

5. Chandrasekhar, S., 1961, Hydrodnamic and Hydromagnetic Stability, Oxford:

Clarendon Press, London.

6. Chung, K. C. and Astill, K. N., 1977, Hydrodynamic instability o f viscous flow

between rotating coaxial cylinders with fully developed axial flow, J. Fluid

Meek, 81(4), 641-655.

7. Cohen, S. and Marom, D. M., 1983, Experimental and theoretical study of a

rotating annular flow reactor, Chem. Engng J., 27, 87-97.

8. Coney, J. E. R. and Simmers, D. A., 1979, A study of fully-developed, laminar,

axial flow and Taylor vortex flow by means of shear stress measurements, J. Meek Engng Sci., 19-24.

9. DiPrima, R. C., 1955, Application of the Galerkin method to problems in

hydrodynamic stability. Quart. Applied Mathematics, 13, 55-62.

10. DiPrima, R. C., and Swinney, H. L., 1981. Stabilities and Transition in Flow

between Concentric Rotating Cylinders, In Hydroefynamic Instabilties and the

Transition to Turbulence (ed. H. L. Swinney and J. P. Gollub), Chap. VI,

Springer-Verlag, Berlm.

11. Elliott, L., 1973, Stability of a viscous fluid between rotating cylinders with axial

flow and pressure gradient round the cylinders, Phys. Fluids, 16(5), 577-580.

12. Fredrickson, A. G., 1964, Principles and applications o f rheology, Prentic-Hall,

London.

13. Harris, J., 1977, Rheology and non-Newtonian flow, L ongm an, London.

Appendix Six 297

14. Kataoka, K, and Takigawa. T.. 1981. Intermixing over cell boundary between

Taylor vortices, ,/4/CA£ 7., 27, 504-508.

15. Kataoka, K., Doi, H., Hongo, T. and Futagawa, M , 1975, Ideal plug-flow

properties o f Taylor vortex flow, J. Chem. Engng Japan, 8,472-476.

16. Kaye, J. and Elgar, E. C., 1958, Modes of adiabatic and diabaiic fluid flow in an

annulus with an inner rotating cylinder, Trans. ASME, 80, 753-765.

17. Levenspiel, O., 1972, Chemical Reaction Engineering, 2nd ed.. Chuter IX,

Wiley, New York.

18. Miyashita, T. and Senna, M., 1993, Development of Taylor vortices in a

concentrated suspension conqmsing monodispersed microspheres, J. Colloid

Interface Sci., 155,290-296.

19. Moore, C. M. V. and Cooney C. L., 1995, Axial dispersion in Taylor-Couette

flow, AIChEJ., 42(3), 723-727.

20. Pudjiono, P.I., Tavare, N.S., Garside, J. and Nigam, K.D.P., 1992, Residence

time distribution from a continuous Couette flow device, Chem. Engng J., 48,

101- 110.

21. Pudjiono, P.I. and Tavare, N.S., 1993, Residence time distribution analysis from a

continuous Couette flow device around ciritical Taylor number. Can. J. Chem.

Engng, 71(4), 312-318.

22. Snyder, H. A., 1962, Stability of spiral flow at low Reynolds number, Proc. Roy.

Soc., A265, 198-213.

23. Takeuchi D. I. and Jankowski, D. P., 1981, A numerical and experimental

investigation of the stability of spiral Poiseuille flow, J. Fluid Mech., 102. 101-

126.

24. Taylor, GJ., 1923, Stability of viscous liquid contained between two rotating

cylinders. Phil. Trans. Roy. Soc. (London), A233,289-343.

25. Walowh, J., Tsao, A. and DiPrima, R. C., 1964, Stability of flow between

arbitrarily spaced concentric cylinderical surfrces including effect of a radial

tenqjerature gradient, J. Applied Mechanics Tran. ASME, 31, 585-593.

26. Yim, S. S. S., 1996, Residence time distribution (RTD) studies in a rotary flow

through device, PhD Thesis, Unverisity College London, (in preparation).

Appendix Six 298

CAPTIONS

Figure 1 Typical simulated neutral curves obtained from

the solution of Eq. (8) and Eq. (13)

Figure 2 Experimental equipment

Figure 3 Conqaarison of numerical simulations and experimental data

for rotational-axial flow in the small gap of a pair of rotating cylinders

Figure 4 Simulated critical Taylor number vs Radius Ratio

(at different speed ratio)

Figure 5 The effect of axial flow on the critical Taylor number

as a function of radius ratio

Figure 6 The effect of rotational speed on residence time distribution

Figure 7 Experimental Peclet number vs Taylor number at different

axial Reynolds numbers Radius Ratio = 0.84

Figure 8 Comparison of experimental and theoretical critical Taylor numbers

as function of the axial Reynolds number

Appendix Six 299

Speed/T orque transducer

ÎInlet

* uonoucuvny

Rotary flowHhrough device

Conductivitymeters

Outlet

Rotary AowMhrough device ^ and

motor ConductMy meters

Data acquisition system

n = = ^ É .

MeasurementandControl system

Computer for Screen monitor and Printer

Appendix Six 300

2150 1a = 0.92100a = 0.925

2050a = 0.95

2000 a = 0.975 a = 0.99

^ 1950 0)E 1900

i. 1850 - *JO

1800

p = 01750

Re = 01700

1650

Wavenumber, a (-)

Appendix Six 301

14000

12000

^ 10000eo

o>• | 8000

o! • 6000

sg 4000

2000

o Chandrasekhar(1960)

A Elliott (1973)

Q Snyder (1962)

X Chung and Astlll (1977)

o Present analysis

a i ■ a = 0.95

15 30 45 60

Reynolds number. Re (-)

75 90

Appendix Six 302

5000

Re = 04500

4000 p = o

.-3500 - ■

c 3000 -

= 0.25

2500 -

2000 ■ - 0.75

1500 -I0.80.4 0.6

Radius ratio, a (-)0.2

Appendix Six 303

25000P = 0

20000 - - Re s 70

R e » 6 0

E 15000 -R e * 5 0

P 10000 - R3.40

R R e * 3 0

O 5000 ■ * 20R e * 0

Radius Ratio, a (-)

Appendix Six 304

14

2nd Probe1st Probe

Rotational Speed = 60 rpm

"Rotational Speed = 140 rpm



10 20 30 40Time (s)

50 60

Appendix Six 305

400— Re * 8

350

250

E 200

150

(L 100 / / / ^

50Radius ratio « 0.84

60002000Taylor number, Ta (-)

4000

Appendix Six 306

14000o : 85% Glycerol

♦ : 85% Glycerol

o : 64% Glycerol

• : 64% Glycerol

□ : 45% Glycerol

■ : 45% Glycerol

a = 0 . 6

12000a = 0 . 7

10000 a = 0.8■ /

a = 0 . 9

8000

a = 0.67

^ 6000

4000 ■

2000 a = 0.84

Reynolds number, Re (-)

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