the effect of the particle size distribution on non-newtonian turbulent ...
the effect of the particle size distribution on non-newtonian turbulent ... the effect of the particle size distribution on non-newtonian turbulent ...
THE EFFECT OF THE PARTICLE SIZE DISTRIBUTION ON NON-NEWTONIAN TURBULENT SLURRY FLOW. IN PIPES by GARY SVEN THORVALDSEN Thesis presented for
- Page 2 and 3: ABSTRACT THE EFFECT OF THE PARTICLE
- Page 4 and 5: Preamble DEDICATION to my parents P
- Page 6 and 7: Preamble PUBLICATIONS The following
- Page 9 and 10: Preamble Page ix 2.14 Conclusions 2
- Page 11 and 12: Preamble Page xi 5.4 Temperature <s
- Page 13 and 14: Preamble Page xiii 2.18 Comparison
- Page 16 and 17: Preamble LIST OF TABLES Page xvi Pa
- Page 18 and 19: Preamble n' apparent flow behaviour
- Page 20: CHAPTER 1
- Page 24 and 25: Chapter 1 Introduction Page lA flow
- Page 26 and 27: Chapter 1 Introduction Page 1.6 1.4
- Page 28: 2.1 INTRODUCTION CHAPTER 2 LITERATU
- Page 35 and 36: Chapter 2 Literature Review Page 2.
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- Page 49: Chapter 2 Literature Review Page 2.
THE EFFECT OF THE PARTICLE SIZE<br />
DISTRIBUTION ON NON-NEWTONIAN<br />
TURBULENT SLURRY FLOW. IN PIPES<br />
by<br />
GARY SVEN THORVALDSEN<br />
Thesis presented for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
M Tech Degree in Chemical Engineering,<br />
School <str<strong>on</strong>g>of</str<strong>on</strong>g> Mechanical and Process Engineering,<br />
Cape Technik<strong>on</strong> .<br />
October 1996
ABSTRACT<br />
THE EFFECT OF THE PARTICLE SIZE<br />
DISTRIBUTION ON NON-NEWTONIAN<br />
TURBULENT SLURRY FLOW IN PIPES<br />
The handling <str<strong>on</strong>g>of</str<strong>on</strong>g> solid-liquid suspensi<strong>on</strong>s is an important c<strong>on</strong>cern within <str<strong>on</strong>g>the</str<strong>on</strong>g> chemical and<br />
processing industries and many <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models have been proposed to try and explain and<br />
predict <strong>turbulent</strong> flow behaviour. However, <str<strong>on</strong>g>the</str<strong>on</strong>g> predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>turbulent</strong> flow from <strong>on</strong>ly <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
viscous properties <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian suspensi<strong>on</strong>s has over <str<strong>on</strong>g>the</str<strong>on</strong>g> years been questi<strong>on</strong>ed by<br />
researchers. This <str<strong>on</strong>g>the</str<strong>on</strong>g>sis c<strong>on</strong>siders <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models well established in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature and<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model, which uses both <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspensi<strong>on</strong> and <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g><br />
<str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> solids. These models are used to analyze <str<strong>on</strong>g>the</str<strong>on</strong>g> experimental data and <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>effect</str<strong>on</strong>g> that <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> and <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> has <strong>on</strong> <strong>turbulent</strong> flow behaviour.<br />
The literature c<strong>on</strong>cerning <str<strong>on</strong>g>the</str<strong>on</strong>g> rheological fundamentals relevant to fluid flow in pipes has<br />
been examined. The Newt<strong>on</strong>ian <strong>turbulent</strong> flow model as well as <str<strong>on</strong>g>the</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian models<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g>Dodge & Metzner, Torrance, Kemblowski & Kolodziejski, Wils<strong>on</strong> & Thomas and Slatter<br />
have been reviewed.<br />
Test work was c<strong>on</strong>ducted at <str<strong>on</strong>g>the</str<strong>on</strong>g> University <str<strong>on</strong>g>of</str<strong>on</strong>g> Cape Town's Hydrotransport Research<br />
Laboratory using a pumped recirculating pipe test rig. The test apparatus has been fully<br />
described and calibrati<strong>on</strong> and test procedures to enable collecting <str<strong>on</strong>g>of</str<strong>on</strong>g> accurate pipeline data<br />
have been presented. Three slurries were used in test work namely kaolin clay, mixture I<br />
(kaolin clay and rock flour) and mixture 2 (kaolin clay, rock flour and sand) with ad,s<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> ranging from 24/Lm to 170/Lm.<br />
The yield pseudoplastic model has been used to model and predict <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar flow <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
suspensi<strong>on</strong>s that were tested and <str<strong>on</strong>g>the</str<strong>on</strong>g> meth9J adopted by Neill (1988) has been used to<br />
determine <str<strong>on</strong>g>the</str<strong>on</strong>g> rheological c<strong>on</strong>stants. The pipeline test results have been presented as pseudoshear<br />
diagrams toge<str<strong>on</strong>g>the</str<strong>on</strong>g>r with <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical model lines providing a visual appraisal <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
performance <str<strong>on</strong>g>of</str<strong>on</strong>g> each model. The Slatter model predicts <str<strong>on</strong>g>the</str<strong>on</strong>g> test data best with <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models that were c<strong>on</strong>sidered tending to under predict <str<strong>on</strong>g>the</str<strong>on</strong>g> head loss. The reas<strong>on</strong><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model performs better than <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models is because this model can<br />
account for <str<strong>on</strong>g>the</str<strong>on</strong>g> wall roughness and <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>. Evidence to support this<br />
statement has been presented.<br />
This <str<strong>on</strong>g>the</str<strong>on</strong>g>sis highlights <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> is a vitally important property<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspensi<strong>on</strong> and that it does influence <strong>turbulent</strong> flow behaviour. It shows that<br />
turbulence modelling using <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> (eg Slatter, 1994) is valid and can<br />
be adopted for n<strong>on</strong>-Newt<strong>on</strong>ian slurries. It is c<strong>on</strong>cluded that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> must<br />
be used to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> and this <str<strong>on</strong>g>effect</str<strong>on</strong>g> must be incorporated in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<strong>turbulent</strong> flow analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian slurries.
Preamble<br />
DECLARATION<br />
Page iii<br />
I, Gary Sven Thorvaldsen, hereby declare that to <str<strong>on</strong>g>the</str<strong>on</strong>g> best <str<strong>on</strong>g>of</str<strong>on</strong>g> my knowledge and belief, this<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>sis is essentially my own work and has not been submitted for <str<strong>on</strong>g>the</str<strong>on</strong>g> award <str<strong>on</strong>g>of</str<strong>on</strong>g> any o<str<strong>on</strong>g>the</str<strong>on</strong>g>r<br />
diploma or degree at any university or technik<strong>on</strong>.<br />
G S Thorvaldsen<br />
October 1996
Preamble<br />
DEDICATION<br />
to my parents<br />
Page iv<br />
and to those who have shown and taught me to celebrate life to <str<strong>on</strong>g>the</str<strong>on</strong>g> fuIIest, to seize <str<strong>on</strong>g>the</str<strong>on</strong>g> day,<br />
to be educated - not simply to fiII <str<strong>on</strong>g>the</str<strong>on</strong>g> mind with knowledge but to open <str<strong>on</strong>g>the</str<strong>on</strong>g> mind to all<br />
possibilities, for learning produces change, change produces surprise and surprise, joy.<br />
"I am a survivor <str<strong>on</strong>g>of</str<strong>on</strong>g>a c<strong>on</strong>centrati<strong>on</strong> camp. My<br />
eyes saw what no pers<strong>on</strong> should witness. Gas<br />
chambers built by learned engineers, children<br />
pois<strong>on</strong>ed by educated physicians, infants killed<br />
lJy trained nurses, woman and babies shot and<br />
killed by high school and college graduates.<br />
So 1 am suspicious <str<strong>on</strong>g>of</str<strong>on</strong>g>educati<strong>on</strong>. My request<br />
is, help your students to be human. Your<br />
efforts must never produce leamed m<strong>on</strong>sters,<br />
skilled psychopaths or educated Eikmans.<br />
Reading and writing and spelling and history<br />
and arithmetic are <strong>on</strong>ly imp<strong>on</strong>ant if<str<strong>on</strong>g>the</str<strong>on</strong>g>y serve<br />
to make our students HUMAN. "<br />
Quotati<strong>on</strong> taken from a book by H Genard<br />
"Each <str<strong>on</strong>g>of</str<strong>on</strong>g>us has a time for life.<br />
There is a misc<strong>on</strong>cepti<strong>on</strong> that we haveforever.<br />
In reality we have such a briefmoment in time<br />
but remember even that briefmoment is yours.<br />
It's yours to celebrate - to misuse it, to loose<br />
it, is to devalue our greatest gift - <str<strong>on</strong>g>the</str<strong>on</strong>g> gift <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
life.<br />
It's our pers<strong>on</strong>al celebrati<strong>on</strong> with so much to<br />
be joyous over, like <str<strong>on</strong>g>the</str<strong>on</strong>g> right to love, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
w<strong>on</strong>der <str<strong>on</strong>g>of</str<strong>on</strong>g> mingling with unique people, <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
smiles, <str<strong>on</strong>g>of</str<strong>on</strong>g>hugging each o<str<strong>on</strong>g>the</str<strong>on</strong>g>r, <str<strong>on</strong>g>of</str<strong>on</strong>g>goodfood, <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
flowers, <str<strong>on</strong>g>of</str<strong>on</strong>g> trees, <str<strong>on</strong>g>of</str<strong>on</strong>g> sunlight, <str<strong>on</strong>g>of</str<strong>on</strong>g> green grass<br />
and cool summer breezes, <str<strong>on</strong>g>of</str<strong>on</strong>g> high flying<br />
ballo<strong>on</strong>s - oh and so much more. This is our<br />
time, <str<strong>on</strong>g>the</str<strong>on</strong>g> time <str<strong>on</strong>g>of</str<strong>on</strong>g> our lives. D<strong>on</strong>'t miss a<br />
minute <str<strong>on</strong>g>of</str<strong>on</strong>g>it, it passes so quickly. The time to<br />
live and celebrate is NOW.•<br />
Leo Buscaglia
Preamble<br />
ACKNOWLEDGEMENrS<br />
Page v<br />
I wish to express my gratitude to <str<strong>on</strong>g>the</str<strong>on</strong>g> following pers<strong>on</strong>s and organisati<strong>on</strong>s for <str<strong>on</strong>g>the</str<strong>on</strong>g>ir assistance<br />
towards <str<strong>on</strong>g>the</str<strong>on</strong>g> completi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this <str<strong>on</strong>g>the</str<strong>on</strong>g>sis:<br />
Pr<str<strong>on</strong>g>of</str<strong>on</strong>g>Paul Slatter for supervising this project, for his help and guidance, and for his assistance<br />
with <str<strong>on</strong>g>the</str<strong>on</strong>g> prepap.ti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> this manuscript.<br />
Pr<str<strong>on</strong>g>of</str<strong>on</strong>g> Francis Petersen for his help and advice.<br />
Pr<str<strong>on</strong>g>of</str<strong>on</strong>g> Mike de Kock for <str<strong>on</strong>g>the</str<strong>on</strong>g> use <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Hydrotransport test facilities at <str<strong>on</strong>g>the</str<strong>on</strong>g> University <str<strong>on</strong>g>of</str<strong>on</strong>g>Cape<br />
Town.<br />
Mr A Siko and Mr D Botha for <str<strong>on</strong>g>the</str<strong>on</strong>g>ir assistance in repairing <str<strong>on</strong>g>the</str<strong>on</strong>g> test facility.<br />
B Tech students, Mr P Ndamage and Mr J Martin, for <str<strong>on</strong>g>the</str<strong>on</strong>g>ir practical assistance.<br />
Murray & Roberts for transporting <str<strong>on</strong>g>the</str<strong>on</strong>g> raw materials.<br />
GeoScience Laboratories for assistance with Particle Size Distributi<strong>on</strong>s.
Preamble<br />
PUBLICATIONS<br />
The following paper was presented by <str<strong>on</strong>g>the</str<strong>on</strong>g> author in support <str<strong>on</strong>g>of</str<strong>on</strong>g> this <str<strong>on</strong>g>the</str<strong>on</strong>g>sis:<br />
Page vi<br />
"Particle Roughness Turbulence", 13 th Internati<strong>on</strong>al C<strong>on</strong>ference <strong>on</strong> Slurry Handling and<br />
Pipeline Transport, Johannesburg, South Africa, September 1996, BHR Group C<strong>on</strong>ference<br />
Series - edited by J F Richards<strong>on</strong>, Mechanical Engineering Publicati<strong>on</strong> Limited, UK, pg 237<br />
257.
Preamble Page vii<br />
CHAPTER 1: INTRODUCTION<br />
CONTENTS<br />
Page<br />
1.1 Statement <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> problem 1.1<br />
1.1.1 Solid Particle Effect 1.2<br />
1.2 Objective 1.4<br />
1.3 Methodology 1.5<br />
1.3.1 Literature Review 1.5<br />
1.3.2 Experimental Work 1.5<br />
1.3.3 Analyses <str<strong>on</strong>g>of</str<strong>on</strong>g> Data 1.5<br />
1.4 Benefits 1.6<br />
CHAPI'ER 2: LITERATURE REVIEW<br />
2.1 Introducti<strong>on</strong> 2.1<br />
2.2 Flow Patterns 2.1<br />
2.2.1 Laminar Flow 2.1<br />
2.2.2 Transiti<strong>on</strong> from Laminar to Turbulent Flow 2.2<br />
2.2.3 Turbulent Flow 2.3<br />
2.3 Reynolds Number 2.3<br />
2.4 Rheology 2.4<br />
2.4.1 Shear Stress 2.4<br />
2.4.2 Yield Stress 2.4<br />
2.4.3 Newt<strong>on</strong>'s Law <str<strong>on</strong>g>of</str<strong>on</strong>g> Viscosity 2.5<br />
2.5 Fluid Classificati<strong>on</strong> 2.6<br />
2.5.1 Newt<strong>on</strong>ian Fluids 2.6<br />
2.5.2 N<strong>on</strong>-Newt<strong>on</strong>ian Fluids 2.7<br />
2.5.3 Characteristics <str<strong>on</strong>g>of</str<strong>on</strong>g> N<strong>on</strong>-Newt<strong>on</strong>ian Behaviour 2.7
Preamble Page ix<br />
2.14 C<strong>on</strong>clusi<strong>on</strong>s 2.44<br />
2.14.1 Laminar Flow 2.44<br />
2.14.2 Transiti<strong>on</strong> from Laminar to Turbulent Flow 2.44<br />
2.14.3 Turbulent Flow 2.44<br />
2.14.4 Objectives <str<strong>on</strong>g>of</str<strong>on</strong>g> Thesis 2.45<br />
CHAPTER 3: EXPERIMENTAL WORK<br />
3.1 Introducti<strong>on</strong>. 3.1<br />
3.2 Testing Facilities 3.1<br />
3.2.1 The East Rig 3.1<br />
3.2.2 The Mini Rig 3.7<br />
3.3 Measured Variables 3.8<br />
3.3.1 Pressure Measurement 3.8<br />
3.3.2 Flow Measurement 3.11<br />
3.4 Calibrati<strong>on</strong> 3.12<br />
3.4.1 Pipeline 3.12<br />
3.4.2 Calibrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Differential Pressure Transducer 3.13<br />
3.4.3 Magnetic Flow Meters 3.16<br />
3.5 O<str<strong>on</strong>g>the</str<strong>on</strong>g>r Measured Variables 3.18<br />
3.5.1 Slurry Density 3.18<br />
3.5.2 Solids Relative Density 3.18<br />
3.5.3 Slurry Temperature 3.18<br />
3.5.4 Particle Size Distributi<strong>on</strong> 3.19<br />
3.6 Material 3.20<br />
3.6.1 Kaolin Clay 3.20<br />
3.6.2 Rock Flour 3.21<br />
3.6.3 Sand 3.21<br />
3.7 Mixtures 3.22<br />
3.7.1 Mixture Kaolin Clay and Rock Flour 3.22<br />
3.7.2 Mixture Kaolin Clay and Sand 3.22
Preamble<br />
3.8 Experimental Procedure<br />
3.9 C<strong>on</strong>clusi<strong>on</strong>s<br />
CHAPfER 4:<br />
4.1 Introducti<strong>on</strong><br />
4.2 Pipeline Tests<br />
3.8.1 Start-up Procedure<br />
Page x<br />
3.23<br />
3.23<br />
3.24<br />
RESULTS AND ANALYSIS 4.1<br />
4.2.1 Pipe Roughness 4.2<br />
4.2.2 Particle Roughness Effect 4.2 .<br />
4.2.3 Kaolin Clay 4.3<br />
4.2.4 11ixtures 4.4<br />
4.2.5 Representative Particle Size 4.4<br />
4.2.6 Roughness Functi<strong>on</strong> Correlati<strong>on</strong> Using Optimum Particle Sizes 4.9<br />
4.2.7 Slurry Temperature 4.9<br />
4.3 Rheological Characterizati<strong>on</strong><br />
4.4 Viscous Sub-Layer<br />
4.5 Influence <str<strong>on</strong>g>of</str<strong>on</strong>g> C<strong>on</strong>centrati<strong>on</strong><br />
4.6 Theoretical 110dels<br />
4.6.1 Turbulent 110del Performance<br />
4.6.2 11aude & Whitmore Correlati<strong>on</strong><br />
4.6.3 Bowen Correlati<strong>on</strong><br />
4.7 Data from <str<strong>on</strong>g>the</str<strong>on</strong>g> Literature<br />
4.8 C<strong>on</strong>clusi<strong>on</strong>s<br />
CHAPTERS:<br />
5.1 Introducti<strong>on</strong><br />
DISCUSSION<br />
5.2 Particle Roughness Effect<br />
5.3 Roughness Functi<strong>on</strong> Correlati<strong>on</strong> Using Optimum Particle Sizes<br />
4.1<br />
4.1<br />
4.11<br />
4.12<br />
4.16<br />
4.18<br />
4.18<br />
4.24<br />
4.24<br />
4.25<br />
4.26<br />
5.1<br />
5.1<br />
5.2
Preamble Page xi<br />
5.4 Temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> Slurry<br />
5.5 Viscous Sub-Layer 5.2<br />
5.6 Influence <str<strong>on</strong>g>of</str<strong>on</strong>g> Particle Size 5.3<br />
5.7 Theoretical Models 5.3<br />
5.8 Reynolds Number vs Fricti<strong>on</strong> Factor Plots 5.6<br />
5.9 C<strong>on</strong>clusi<strong>on</strong>s 5.8<br />
CHAPrER 6:<br />
6.1 Introducti<strong>on</strong><br />
6.2 Summary and C<strong>on</strong>clusi<strong>on</strong>s<br />
SUMMARy & CONCLUSIONS<br />
6.3 Future Research Recommendati<strong>on</strong>s<br />
6.4 Final C<strong>on</strong>clusi<strong>on</strong>s<br />
REFERENCES<br />
APPENDICES<br />
APPENDIX A - EXPERIMENTAL RESULTS<br />
A.I Detailed Pipe Test Results<br />
APPENDIX B - CONFERENCE PAPER<br />
RI Particle Roughness Turbulence<br />
6.1<br />
6.1<br />
6.2<br />
6.3
Preamble Page xii<br />
LIST OF FIGURES<br />
Page<br />
2.1 illustrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>'s law <str<strong>on</strong>g>of</str<strong>on</strong>g> viscosity 2.5<br />
2.2 Rheological models for n<strong>on</strong>-Newt<strong>on</strong>ian fluids 2.8<br />
2.3 Forces acting <strong>on</strong> a fluid element in larninar flow 2.12<br />
2.4 Typical pipeline test data for three pipe diameters 2.14<br />
2.5 Graphical representati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Metzner & Reed approach 2.15<br />
2.6 Magnified view <str<strong>on</strong>g>of</str<strong>on</strong>g> a rough wall pipe showing regi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>turbulent</strong> flow 2.18<br />
2.7 Magnified view <str<strong>on</strong>g>of</str<strong>on</strong>g> a rough wall pipe showing <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer<br />
and <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry <str<strong>on</strong>g>particle</str<strong>on</strong>g>s 2.20<br />
2.8 Graphical comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Blasius, Knudsen & Katz and v<strong>on</strong> Karmen<br />
fricti<strong>on</strong> factor equati<strong>on</strong>s for <strong>turbulent</strong> flow 2.21<br />
2.9 Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> law <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall for smooth and rough pipes 2.25<br />
2.10 Moody diagram 2.28<br />
2.11 The Dodge & Metzner correlati<strong>on</strong> shown <strong>on</strong> a fricti<strong>on</strong> factor-Reynolds<br />
number diagram<br />
2.29<br />
2.12 Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Dodge & Metzner predicti<strong>on</strong> curve and Kemblowski &<br />
Kolodziejski experimental data for <str<strong>on</strong>g>the</str<strong>on</strong>g> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> a 30% aqueous kaolin<br />
suspensi<strong>on</strong> at n'=O,39 (taken from Kemblowski & Kolodziejski, 1973) 2.32<br />
2.13 illustrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> area ratio 2.34<br />
2.14 Unsheared plug geometry 2.35<br />
2.15 Velocity pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile in <strong>turbulent</strong> flow 2.38<br />
2.16 Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>turbulent</strong> velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> between a n<strong>on</strong>-Newt<strong>on</strong>ian<br />
slurry and air at Re=40600<br />
2.17 Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> relative turbulence intensities between a n<strong>on</strong>-Newt<strong>on</strong>ian<br />
slurry and air at Re=40600<br />
2.42<br />
2.43
Preamble Page xiii<br />
2.18 Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> relative turbulence intensities between bent<strong>on</strong>ite clay<br />
suspensi<strong>on</strong>s (C y =4% and Cy =6%) and water 2.43<br />
3.1 Isometric drawing <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> East Rig 3.2<br />
3.2 Solids handling pump and variable speed hydraulic drive 3.3<br />
3.3 Vertical counterflow secti<strong>on</strong> 3.4<br />
3.4 Horiz<strong>on</strong>tli test secti<strong>on</strong> 3.5<br />
3.5 The 80mm, l50mm and 200mm horiz<strong>on</strong>tal return pipelines 3.5<br />
3.6 Steel hopper and weigh tank 3.6<br />
3.7 The Mini Rig 3.7<br />
3.8 Pressure tapping and solids collecting pod 3.8<br />
3.9 Manometer board 3.10<br />
3.10 Layout <str<strong>on</strong>g>of</str<strong>on</strong>g> valves for a manometer for <str<strong>on</strong>g>the</str<strong>on</strong>g> East and Mini Rigs 3.10<br />
3.11 Data logging set-up 3.11<br />
3.12 Typical output obtained from calibrati<strong>on</strong> programme 3.15<br />
3.13 Current signal vs time for <str<strong>on</strong>g>the</str<strong>on</strong>g> l50mm magnetic flow meter for a desired<br />
speed setting 3.17<br />
3.14 Calibrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> l50mm magnetic flow meter, showing laminar and<br />
<strong>turbulent</strong> data over <str<strong>on</strong>g>the</str<strong>on</strong>g> full test range 3.17<br />
3.15 Particle <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s using <str<strong>on</strong>g>the</str<strong>on</strong>g> ASTM and Malvern Particle Sizer<br />
for <str<strong>on</strong>g>the</str<strong>on</strong>g> same mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay, rock flour and sand 3.19<br />
3.16 Solids materials used for testing purposes 3.21<br />
4.1 Kaolin sec<strong>on</strong>d data test set Ty =5,8; K=0,OO676; n=0,645 4.1<br />
4.2 Roughness functi<strong>on</strong> correlati<strong>on</strong> for n<strong>on</strong>-Newt<strong>on</strong>ian slurries 4.3<br />
4.3 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets K_lO 4.5<br />
4.4 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets K_20 4.6
Preamble Page xiv<br />
4.5 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets RF_lO 4.6<br />
4.6 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets RF_20 4.7<br />
4.7 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets RF_30 4.7<br />
4.8 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets S_1O 4.8<br />
4.9 Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets S_20 4.8<br />
4.lO Optimum representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for test sets S_20 4.9<br />
4.11 Roughness functi<strong>on</strong> correlati<strong>on</strong> using optimum <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>s 4.lO<br />
4.12 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> test KMRL20 including data points at <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
extreme temperatures 4.11<br />
4.13 Pseudo-shear diagram showing rheological characterizati<strong>on</strong> for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin<br />
test sets 4.13<br />
4.14 Pseudo-shear diagram showing rheological characterizati<strong>on</strong> for Mixture 1<br />
test sets 4.13<br />
4.15 Pseudo-shear diagram showing rheological characterizati<strong>on</strong> for Mixture 2<br />
test sets 4.14<br />
4.16 Viscous sub-layer thickness predicti<strong>on</strong>s for kaolin clay 4.14<br />
4.17 Viscous sub-layer thickness predicti<strong>on</strong>s for Mixture 1 4.15<br />
4.18 Viscous sub-layer thickness predicti<strong>on</strong>s for Mixture 2 4.15<br />
4.19 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> 150mm pipe showing <str<strong>on</strong>g>the</str<strong>on</strong>g> data for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin<br />
test sets 4.16<br />
4.20 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> 150mm pipe showing <str<strong>on</strong>g>the</str<strong>on</strong>g> data for Mixture 1<br />
test sets 4.17<br />
4.21 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> 150mm pipe showing <str<strong>on</strong>g>the</str<strong>on</strong>g> data for Mixture 2<br />
test sets 4.17<br />
4.22 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> 80mm pipeline: KERSlO: dss =30JLm 4.18<br />
4.23 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> 80mm pipeline: RFERSlO: dss =52JLm 4.19<br />
4.24 Pseudo-shear diagram for <str<strong>on</strong>g>the</str<strong>on</strong>g> 80mm pipeline: SERS!O: dss =137JLm 4.19
Preamble<br />
LIST OF TABLES<br />
Page xvi<br />
Page<br />
3.1 Pipeline diameters 3.12<br />
3.II Pipeline roughness 3.13<br />
3.III d S5 Sizes for <str<strong>on</strong>g>the</str<strong>on</strong>g> various data test sets 3.20<br />
.<br />
4.1 Optimum and experimental representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for Mixture 2 test sets 4.6<br />
4.II Summary <str<strong>on</strong>g>of</str<strong>on</strong>g> slurry properties 4.8<br />
4.III Turbulent model performance - average percentage error 4.17<br />
4.IV Turbulent model performance - log standard error 4.17<br />
5.1 Number c<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in viscous sub-layer 5.5<br />
I
Preamble<br />
NOMENCLATURE<br />
Svrnbol Descripti<strong>on</strong> Unit<br />
a c<strong>on</strong>stant<br />
A c<strong>on</strong>stant<br />
cross secti<strong>on</strong>al area m2 A, area ratio<br />
b c<strong>on</strong>stant<br />
B c<strong>on</strong>stant<br />
roughness functi<strong>on</strong><br />
c c<strong>on</strong>stant<br />
C c<strong>on</strong>centrati<strong>on</strong><br />
CD drag coefficient<br />
d <str<strong>on</strong>g>particle</str<strong>on</strong>g> diameter !Lm<br />
D internal pipe diameter m<br />
E error functi<strong>on</strong><br />
rheological parameter<br />
f Fanning fricti<strong>on</strong> factor<br />
F force N<br />
k c<strong>on</strong>stant<br />
hydraulic roughness !Lm<br />
K fluid c<strong>on</strong>sistency index Pa.s·<br />
K' apparent fluid c<strong>on</strong>sistency index Pa.s·'<br />
I average amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> liquid oscillati<strong>on</strong><br />
:£ PrandtI mixing length m<br />
L pipe length m<br />
m slope<br />
rheological parameter<br />
M mass kg<br />
n flow behaviour index<br />
Page xvii
Preamble<br />
n' apparent flow behaviour index<br />
N number <str<strong>on</strong>g>of</str<strong>on</strong>g> items<br />
p mixing length factor<br />
pressure Pa<br />
q ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> solid to liquid oscillati<strong>on</strong> amplitude<br />
Q volumetric flow rate <str<strong>on</strong>g>of</str<strong>on</strong>g> slurry m 3 /s<br />
r radius at a point in <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe m<br />
correlati<strong>on</strong> coefficient<br />
R radius <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe m<br />
Re Reynolds number<br />
S relative density<br />
u point velocity m/s<br />
u+ dimensi<strong>on</strong>less velocity<br />
V average slurry velocity m/s<br />
V. shear velocity m/s<br />
X unknown quantity<br />
abscissa value<br />
y distance from <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall m<br />
y+ dimensi<strong>on</strong>less wall distance<br />
y ordinate value<br />
et proporti<strong>on</strong>al to<br />
shape factor<br />
shear stress ratio<br />
Preamble<br />
T shear stress Pa<br />
T y yield stress Pa<br />
q, functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
X<br />
rheological parameter<br />
v<strong>on</strong> Kannan c<strong>on</strong>stant<br />
0 velocity functi<strong>on</strong><br />
SUbscripts<br />
o<br />
85<br />
ann<br />
c<br />
calc<br />
m-<br />
max<br />
N<br />
NN<br />
obs<br />
P<br />
plug<br />
r<br />
res<br />
s<br />
shear<br />
v<br />
w<br />
x<br />
at <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall<br />
85 th percentile <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s passing<br />
refers to <str<strong>on</strong>g>the</str<strong>on</strong>g> annulus<br />
critical<br />
calculated<br />
loss, liquid<br />
mixture (slurry)<br />
maximum<br />
Newt<strong>on</strong>ian<br />
n<strong>on</strong>-Newt<strong>on</strong>ian<br />
observed (experimental)<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> plug<br />
roughness<br />
residual<br />
solids<br />
z<strong>on</strong>e over which shear occurs<br />
volumetric<br />
water<br />
representative <str<strong>on</strong>g>size</str<strong>on</strong>g><br />
Page xix
CHAPTER 1
Chapter I<br />
For example:<br />
Introducti<strong>on</strong> Page 1.3<br />
• In experiments carried out by Maude & Whitmore (1956) it was postulated that in<br />
detennining <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> suspensi<strong>on</strong>s, <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> is a significant<br />
factor. Maude & Whitmore (1958) hence performed a study <strong>on</strong> dilute m<strong>on</strong>o-
Chapter 1 Introducti<strong>on</strong> Page lA<br />
flow, which in turn affects <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer.<br />
However, even with findings such as <str<strong>on</strong>g>the</str<strong>on</strong>g>se, most <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models which have been<br />
proposed to predict <strong>turbulent</strong> flow behaviour have not taken into account <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g>s inherent in <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid. In fact Mun (1988) states that <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> is <str<strong>on</strong>g>the</str<strong>on</strong>g> most<br />
frequently overlooked piece <str<strong>on</strong>g>of</str<strong>on</strong>g> data in published literature data. Only as recently as 1994<br />
was a new model been proposed by Slatter (1994) to account for not <strong>on</strong>ly <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g><br />
.<br />
slurry but also <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> solids.<br />
1.2 OBJECTIVE<br />
The aim <str<strong>on</strong>g>of</str<strong>on</strong>g> this study is to model <strong>turbulent</strong> flow using Slatter's model as well as o<str<strong>on</strong>g>the</str<strong>on</strong>g>r<br />
models already well established in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature, to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD <strong>on</strong> n<strong>on</strong><br />
Newt<strong>on</strong>ian flow in pipes and to anaiyze <str<strong>on</strong>g>the</str<strong>on</strong>g> results.<br />
The majority <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> test work c<strong>on</strong>ducted by Slatter (1994) was d<strong>on</strong>e using kaolin clay as <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
solids material with <strong>on</strong>ly 15 % <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> tests having a representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> exceeding<br />
50jLm. The average representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> remaining tests was 29JLm.<br />
This study is essentially a c<strong>on</strong>tinuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> work c<strong>on</strong>ducted by Slatter (1994) and hence tests<br />
were c<strong>on</strong>ducted using kaolin clay to first c<strong>on</strong>firm <str<strong>on</strong>g>the</str<strong>on</strong>g> accuracy <str<strong>on</strong>g>of</str<strong>on</strong>g> Slatter's model. Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r<br />
tests were <str<strong>on</strong>g>the</str<strong>on</strong>g>n carried out using a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay and rock flour (mixture 1) and a<br />
mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay, rock flour and sand (mixture 2) in order to obtain a representative<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> higher than that tested by Slatter (1994) and to observe <str<strong>on</strong>g>the</str<strong>on</strong>g> influence <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> and <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> <strong>on</strong> <strong>turbulent</strong> flow predicti<strong>on</strong>s using <str<strong>on</strong>g>the</str<strong>on</strong>g> various<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models.
Chapter 1 Introducti<strong>on</strong> Page 1.5<br />
1.3 METHODOLOGY<br />
The following steps were undertaken with <str<strong>on</strong>g>the</str<strong>on</strong>g> above objective in mind.<br />
1.3.1 Literature Review<br />
The literature review covers <str<strong>on</strong>g>the</str<strong>on</strong>g> following <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models:<br />
- Newt<strong>on</strong>ian approximati<strong>on</strong><br />
- Dodge and Metzner (1959)<br />
- Torrance (1963)<br />
- Kemb10wski & Kolodziejski (1973)<br />
- Wils<strong>on</strong> & Thomas (1985)<br />
- Slatter (1994)<br />
The correlati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Maude & Whitmore (1956, 1958) is also reviewed as it is <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>ly<br />
correlati<strong>on</strong> previous to Slatter's <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical model to take any account <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> suspensi<strong>on</strong>.<br />
1.3.2 Experimental Work<br />
Experimental work was c<strong>on</strong>ducted at <str<strong>on</strong>g>the</str<strong>on</strong>g> University <str<strong>on</strong>g>of</str<strong>on</strong>g>Cape Town's Hydrotransport Facility<br />
using a pumped recirculating pipe test rig c<strong>on</strong>sisting <str<strong>on</strong>g>of</str<strong>on</strong>g>pipe diameters 25mm, 80mm, 150mm<br />
and 200mm.<br />
1.3.3 Analyses <str<strong>on</strong>g>of</str<strong>on</strong>g> Data<br />
The aforementi<strong>on</strong>ed <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models were used to anaIyze <str<strong>on</strong>g>the</str<strong>on</strong>g> data obtained.
Chapter 1 Introducti<strong>on</strong> Page 1.6<br />
1.4 BENEFITS<br />
Slatter's model is mainly <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical at this stage and it is envisaged that through applying<br />
practical research to <str<strong>on</strong>g>the</str<strong>on</strong>g> model that it will go some way to establishing a new model for<br />
industry and to make it more acceptable.
CHAPTER 2
2.1 INTRODUCTION<br />
CHAPTER 2<br />
LITERATURE REVIEW<br />
The presentati<strong>on</strong> in this chapter deals with <str<strong>on</strong>g>the</str<strong>on</strong>g> rheological fundamentals relevant to fluid flow<br />
in pipes. These fundamentals include <str<strong>on</strong>g>the</str<strong>on</strong>g> classificati<strong>on</strong>, measurement and interpretati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
n<strong>on</strong>-Newt<strong>on</strong>ian behaviour.<br />
.<br />
2.2 FLOW PATTERNS<br />
The nature <str<strong>on</strong>g>of</str<strong>on</strong>g> flow <str<strong>on</strong>g>of</str<strong>on</strong>g>a fluid depends <strong>on</strong> a number <str<strong>on</strong>g>of</str<strong>on</strong>g> factors. Factors influencing flow are:<br />
• <str<strong>on</strong>g>the</str<strong>on</strong>g> physical properties <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid and its mass flow rate,<br />
• <str<strong>on</strong>g>the</str<strong>on</strong>g> geometry <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>tainer or pipeline.<br />
The flow <str<strong>on</strong>g>of</str<strong>on</strong>g> a fluid can be characterized into <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> three categories namely larninar flow,<br />
transiti<strong>on</strong>al flow or <strong>turbulent</strong> flow.<br />
2.2.1 L4uninar Flow<br />
In larninar flow layers <str<strong>on</strong>g>of</str<strong>on</strong>g> fluid move relative to each o<str<strong>on</strong>g>the</str<strong>on</strong>g>r without any macroscopic<br />
intermixing (Holland, 1973) ie. <str<strong>on</strong>g>the</str<strong>on</strong>g>re are no comp<strong>on</strong>ents <str<strong>on</strong>g>of</str<strong>on</strong>g> velocity normal to <str<strong>on</strong>g>the</str<strong>on</strong>g> directi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> flow (Govier & Aziz, 1972). From <str<strong>on</strong>g>the</str<strong>on</strong>g> above statement it can be understood why laminar<br />
flow is also referred to as viscous or streamline flow in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature.<br />
For <str<strong>on</strong>g>the</str<strong>on</strong>g> purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> this study <str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic model has been used for larninar flow<br />
predicti<strong>on</strong>s (secti<strong>on</strong> 2.6), although o<str<strong>on</strong>g>the</str<strong>on</strong>g>r models in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature may also be used.<br />
2.2.2 Transiti<strong>on</strong> from Laminar to Turbulent Flow<br />
There are several methods in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature that can be used to identify <str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> between<br />
•
Chapter 2 Literature Review Page 2.6<br />
However, even though <str<strong>on</strong>g>the</str<strong>on</strong>g> above statements can be taken to be true for Newt<strong>on</strong>ian fluids<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>re are times when certain fluids deviate from <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian behaviour. From <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical<br />
c<strong>on</strong>siderati<strong>on</strong>s as presented by Grunberg & Nissan (1945) it can be c<strong>on</strong>cluded that at high<br />
rates <str<strong>on</strong>g>of</str<strong>on</strong>g> shear <str<strong>on</strong>g>the</str<strong>on</strong>g> viscosity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>se fluids (eg. n-pentane) may become a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> shear<br />
rate. O<str<strong>on</strong>g>the</str<strong>on</strong>g>r fluids which behave as Newt<strong>on</strong>ian at lower shear stresses also exhibit n<strong>on</strong><br />
Newt<strong>on</strong>ian behaviour above certain shear stresses. In experiments c<strong>on</strong>ducted by We1tmann<br />
(1948) it was shown that several oils have a n<strong>on</strong>-Newt<strong>on</strong>ian behaviour above a limiting shear<br />
stress <str<strong>on</strong>g>of</str<strong>on</strong>g> approximately 958 Pa. This seems to c<strong>on</strong>tradict <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that normally at high<br />
Reynolds numbers n<strong>on</strong>-Newt<strong>on</strong>ian behaviour decreases (see secti<strong>on</strong> 2.5.3.d).<br />
i Examples <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>ian behaviour include:<br />
• all gases,'<br />
• all liquids or soluti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> low molecular weight (Metzner, 1956).<br />
2.5.2 N<strong>on</strong>-Newt<strong>on</strong>ian fluids<br />
Unlike Newt<strong>on</strong>ian fluids, <str<strong>on</strong>g>the</str<strong>on</strong>g> term viscosity has no meaning for a n<strong>on</strong>-Newt<strong>on</strong>ian fluid unless<br />
it is related to a particular shear rate (Holland, 1973). However, an apparent viscosity can<br />
be defiiled as (Holland 1973, Wils<strong>on</strong> 1986):<br />
N<strong>on</strong>-Newt<strong>on</strong>ian fluids are normally divided into <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> two categories (Metzner, 1956):<br />
(a) Fluids with properties independent <str<strong>on</strong>g>of</str<strong>on</strong>g> tome or <str<strong>on</strong>g>the</str<strong>on</strong>g> durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> shear.<br />
(2.5)<br />
(b) More complex fluids where <str<strong>on</strong>g>the</str<strong>on</strong>g> relati<strong>on</strong>ship between shear stress and shear<br />
rate depends <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> durati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> shear.<br />
A graphical representati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> several possible relati<strong>on</strong>ships between shear stress and shear<br />
rate for n<strong>on</strong>-Newt<strong>on</strong>ian fluids, called a rheogram, is showlLin Figure 2.2.
Chapter 2 Literature Review Page 2.8<br />
... Cb) Particle Shape<br />
In a Newt<strong>on</strong>ian liquid, if <str<strong>on</strong>g>the</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are rigid spherical <str<strong>on</strong>g>particle</str<strong>on</strong>g>s, <str<strong>on</strong>g>the</str<strong>on</strong>g>y impart<br />
pseudoplastic behaviour. Rigid ellipsoidal and elastic spherical <str<strong>on</strong>g>particle</str<strong>on</strong>g>s will normally<br />
result in visco-elastic behaviour (Charles & Kenchingt<strong>on</strong>, 1976).<br />
I (c) C<strong>on</strong>centrati<strong>on</strong><br />
The increase <str<strong>on</strong>g>of</str<strong>on</strong>g> solids c<strong>on</strong>centrati<strong>on</strong> increases <str<strong>on</strong>g>the</str<strong>on</strong>g> degree <str<strong>on</strong>g>of</str<strong>on</strong>g>n<strong>on</strong>-Newt<strong>on</strong>ian behaviour<br />
and larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s may form a n<strong>on</strong>-Newt<strong>on</strong>ian mixture at increased c<strong>on</strong>centrati<strong>on</strong><br />
(Lazarus, 1992). Thomas (1963) showed that for <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <str<strong>on</strong>g>the</str<strong>on</strong>g> range 0,35 to 13 Itm<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> yield stress is inversely proporti<strong>on</strong>al to <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> diameter and proporti<strong>on</strong>al to<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> cube <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> volumetric c<strong>on</strong>centrati<strong>on</strong>.<br />
I (d) Reynolds Number<br />
Athigh Reynolds numbers n<strong>on</strong>-Newt<strong>on</strong>ian behaviour decreases (Lazarus, 1992). This<br />
is due to <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian fluids at high Reynolds<br />
numbers is characterized by increased inertial forces compared with <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />
forces. This is in c<strong>on</strong>tradicti<strong>on</strong> to <str<strong>on</strong>g>the</str<strong>on</strong>g> findings <str<strong>on</strong>g>of</str<strong>on</strong>g> Weltman (1948) (see secti<strong>on</strong><br />
2.5.1). In laminar flow and at "low Reynolds numbers <str<strong>on</strong>g>the</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian<br />
characteristics or behaviour becomes more prominent.<br />
2.6 SLURRY CLASSIFICATION<br />
Solid-liquid suspensi<strong>on</strong>s (ie. slurries) can be classified in <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> two regimes according to<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> manner in which <str<strong>on</strong>g>the</str<strong>on</strong>g>y flow.<br />
2.6.1 Homogeneous Slurries<br />
This type <str<strong>on</strong>g>of</str<strong>on</strong>g>slurry <str<strong>on</strong>g>of</str<strong>on</strong>g>ten exhibits n<strong>on</strong>-Newt<strong>on</strong>ian rheology (Thomas 1976, Lazarus 1992) and<br />
although it is characterized by a uniform c<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s about <str<strong>on</strong>g>the</str<strong>on</strong>g> axis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe,<br />
it flows as a single-phase fluid (Thomas 1976). This definiti<strong>on</strong> encompasses n<strong>on</strong>-settling
Chapter 2 Literature Review Page 2.9<br />
slurries as well as those requiring <strong>turbulent</strong> flow to maintain homogeneity. This study will<br />
c<strong>on</strong>centrate <strong>on</strong> homogeneous slurries.<br />
2.6.2 Heterogeneous Slurries<br />
By c<strong>on</strong>trast, heterogeneous slurries do not behave as single-phase fluid ie. <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid and solid<br />
phase retain <str<strong>on</strong>g>the</str<strong>on</strong>g>ir separate identities. Hence, <str<strong>on</strong>g>the</str<strong>on</strong>g>se slurries will flow with a n<strong>on</strong><br />
axisymmetric c<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>. The solids phase tends to have larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s than<br />
homogeneous slurries.<br />
2.7 TIME-INDEPENDENT FLUIDS<br />
2.7.1 Introducti<strong>on</strong><br />
Bingharn plastic, pseudoplastic and dilatant fluids all fall into this category. However, all<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g>se rheological relati<strong>on</strong>ships as well as <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian and yield dilatant relati<strong>on</strong>ships can<br />
be described by <str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic model (sometimes referred to in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature as <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
generalised Bingharn model). The rheological assumpti<strong>on</strong> for <str<strong>on</strong>g>the</str<strong>on</strong>g> yield-pseudoplastic model,<br />
as suggested by Herschel & Bulkley (1926), is as follows:<br />
T = T + K [_ dUJ D<br />
Y dr'<br />
where T y is <str<strong>on</strong>g>the</str<strong>on</strong>g> yield stress;<br />
(2.6)<br />
K is <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid c<strong>on</strong>sistency index which characterizes <str<strong>on</strong>g>the</str<strong>on</strong>g> "thickness" <str<strong>on</strong>g>of</str<strong>on</strong>g> a fluid and<br />
is similar to <str<strong>on</strong>g>the</str<strong>on</strong>g> viscosity <str<strong>on</strong>g>of</str<strong>on</strong>g> a Newt<strong>on</strong>ian fluid,<br />
n is <str<strong>on</strong>g>the</str<strong>on</strong>g> flow behaviour index and characterizes <str<strong>on</strong>g>the</str<strong>on</strong>g> extent <str<strong>on</strong>g>of</str<strong>on</strong>g>deviati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g>a fluid from<br />
Newt<strong>on</strong>ian behaviour (Metzner,1956).<br />
The above menti<strong>on</strong>ed rheological relati<strong>on</strong>ships can be described using <str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic<br />
model depending <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> values <str<strong>on</strong>g>of</str<strong>on</strong>g> T y and n:<br />
(a)<br />
(b)<br />
Bingham plastic<br />
Yield pseudoplastic<br />
{Ty>O and n=l}<br />
{Ty>O and n < 11
Chapter 2 Literature Review Page 2.11<br />
The relati<strong>on</strong>ship between shear rate and <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> is given by<br />
i' = -<br />
dV<br />
dr<br />
Length (L)<br />
"<br />
Fluid element<br />
Figure 2.3: Forces acting <strong>on</strong> a fluid element in laminar flow<br />
(2.9)<br />
(2.10)<br />
A force balance <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> cylindrical element in Figure 2.3 will yield <str<strong>on</strong>g>the</str<strong>on</strong>g> relati<strong>on</strong>ship <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
shear stress at a radial positi<strong>on</strong> r to <str<strong>on</strong>g>the</str<strong>on</strong>g> overall axial pressure drop over <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe <str<strong>on</strong>g>of</str<strong>on</strong>g> length<br />
L:<br />
or,<br />
(2.11)
Chapter 2 Literature Review Page 2.12<br />
The partial integrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> equati<strong>on</strong> 2.9 using equati<strong>on</strong> 2.10 and equati<strong>on</strong> 2.12 leads to:<br />
3<br />
rD '. du<br />
Q = - - • J r- - - dT ,<br />
8 [ TO ] 0 dr<br />
where <str<strong>on</strong>g>the</str<strong>on</strong>g> shear stress at <str<strong>on</strong>g>the</str<strong>on</strong>g> wall is given by:<br />
T = Ddp<br />
o 4 L<br />
(2.12)<br />
(2.13)<br />
(2.14)<br />
The flow rate for <str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic model, <str<strong>on</strong>g>the</str<strong>on</strong>g> Bingham plastic model or <str<strong>on</strong>g>the</str<strong>on</strong>g> power law<br />
model can hence be obtained by substituti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> relevant rheological equati<strong>on</strong> ie. equati<strong>on</strong><br />
2.6, equati<strong>on</strong> 2.7 or equati<strong>on</strong> 2.8 respectively. As menti<strong>on</strong>ed, <str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic model<br />
- - -is<br />
to used for this study <str<strong>on</strong>g>the</str<strong>on</strong>g>refore substituting equati<strong>on</strong> 2.6 into equati<strong>on</strong> 2.13 yields:<br />
4n<br />
1<br />
• n 3<br />
TO<br />
K<br />
2.7.3 Generalized Approach for Laminar Flow<br />
(2.15)<br />
Metzner & Reed (1955) proposed a generalized correlati<strong>on</strong> for any time independent fluid.<br />
This is based up<strong>on</strong> a relati<strong>on</strong>ship developed by Rabinowitsch (1929) and Mo<strong>on</strong>ey (1931) for<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> calculati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> shear rates at <str<strong>on</strong>g>the</str<strong>on</strong>g> wall <str<strong>on</strong>g>of</str<strong>on</strong>g> a pipe, where:<br />
[ _dU] = l. [ 8Y]<br />
dr 0 4 D<br />
I 8Y<br />
+ - - To'<br />
4 D<br />
(2.16)
Chapter 2 Literature Review Page 2.16<br />
At fixed values <str<strong>on</strong>g>of</str<strong>on</strong>g> T y , K and n, a root mean square error <str<strong>on</strong>g>of</str<strong>on</strong>g> fit functi<strong>on</strong> E can be defined for<br />
a series <str<strong>on</strong>g>of</str<strong>on</strong>g> N data points in <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar flow regi<strong>on</strong>,<br />
E =<br />
N - 1<br />
2<br />
(2.23)<br />
The K value for minimum error K.u.. can be found by setting aE/aK=O for a fixed value <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Ty and n,<br />
K. = 11<br />
auo<br />
T<br />
2 y ]<br />
+ l+n<br />
The T y and n values are <str<strong>on</strong>g>the</str<strong>on</strong>g>n optimized to give a global minimum for E.<br />
2.9 VELOCITY DISTRIBUTION FOR TURBULENT FLOW<br />
n<br />
(2.24)<br />
Turbulent flow models for n<strong>on</strong>-Newt<strong>on</strong>ian fluids are normally based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> analytical<br />
methods adopted for Newt<strong>on</strong>ian fluid <strong>turbulent</strong> behaviour (eg. Torrance 1963, Wils<strong>on</strong> &<br />
Thomas 1985, 1987, SIatter 1994). In this secti<strong>on</strong> <strong>turbulent</strong> flow for Newt<strong>on</strong>ian fluids is<br />
COvered although <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s is also c<strong>on</strong>sidered as it is felt that this should<br />
form part <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> basis for <strong>turbulent</strong> flow analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> any n<strong>on</strong>-Newt<strong>on</strong>ian flow model.<br />
2.9.1 Wall Roughness<br />
The type <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>turbulent</strong> flow encountered depends <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall roughness (Govier & Aziz,<br />
1972). Under microscopic investigati<strong>on</strong> it can be seen that <str<strong>on</strong>g>the</str<strong>on</strong>g> inner wall <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe is not<br />
smooth but many protrusi<strong>on</strong>s exist. The term wall, pipe or surface TOughness is used to<br />
describe <str<strong>on</strong>g>the</str<strong>on</strong>g> complex <str<strong>on</strong>g>size</str<strong>on</strong>g>, shape and spacing <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>se protrusi<strong>on</strong>s. The elttent <str<strong>on</strong>g>of</str<strong>on</strong>g> roughness<br />
will depend <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> material used in manufacturing <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe and <str<strong>on</strong>g>the</str<strong>on</strong>g> method <str<strong>on</strong>g>of</str<strong>on</strong>g> its<br />
manufacture. For a given fluid and velocity, roughness in a pipe increases <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure drop
Chapter 2 Literature Review<br />
Viscous Sub-layer Thickness<br />
---------------------._----------._-.-._-----<br />
Page 2.18<br />
Figure 2.6: Magnified view <str<strong>on</strong>g>of</str<strong>on</strong>g> a rough wall pipe showing regi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>turbulent</strong> flow<br />
2.9.2 The Effect <str<strong>on</strong>g>of</str<strong>on</strong>g> Solid Particles<br />
Although a <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <strong>on</strong> energy gradients in <strong>turbulent</strong> flow has been reported by<br />
Maude & Whitmore (1956, 1958), Mun (1988) and Slatter (1994), it is still customary in<br />
homogenous n<strong>on</strong>-Newt<strong>on</strong>ian slurries to ignore <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are present. As<br />
previously menti<strong>on</strong>ed in Chapter 1 it has been found that <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> and <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> do<br />
influence flow behaviour (philipp<str<strong>on</strong>g>of</str<strong>on</strong>g>f 1944, Hedstrom 1952, Orr & Blocker 1955,<br />
Zettlemoyer & Lower 1955, Maude & Whitrnore 1956, Thomas 1983, Mun 1988, Slatter<br />
1994) and yet it is <str<strong>on</strong>g>the</str<strong>on</strong>g> most overlooked piece <str<strong>on</strong>g>of</str<strong>on</strong>g> data in <strong>turbulent</strong> flow analysis (Mun, 1988).<br />
Particles will cause a decrease in <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity gradient similar to <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall<br />
roughness (Slatter, 1994) and should <str<strong>on</strong>g>the</str<strong>on</strong>g>refore be taken into account in <strong>turbulent</strong> flow<br />
analyses. This can be understood if <str<strong>on</strong>g>the</str<strong>on</strong>g> following is c<strong>on</strong>sidered.<br />
The change in velocity as <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall is approached is very rapid. The magnitude <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
change in <str<strong>on</strong>g>the</str<strong>on</strong>g> regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall is in <str<strong>on</strong>g>the</str<strong>on</strong>g> order <str<strong>on</strong>g>of</str<strong>on</strong>g> hn/s (Slatter, 1994) over <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g>a typical <str<strong>on</strong>g>particle</str<strong>on</strong>g>. Ifsolid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are present in <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid <str<strong>on</strong>g>the</str<strong>on</strong>g>y will resist-shear and hence<br />
impede <str<strong>on</strong>g>the</str<strong>on</strong>g> rapid changes in velocity.
Chapter 2 Literature Review Page 2.19<br />
2.9.3 C<strong>on</strong>tinuum Approximati<strong>on</strong><br />
From secti<strong>on</strong> 2.9.1 and secti<strong>on</strong> 2.9.2 it can be seen that wall roughness and <str<strong>on</strong>g>the</str<strong>on</strong>g> presence <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s will affect <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity gradient. This fact should be stressed as virtually all<br />
researchers in <str<strong>on</strong>g>the</str<strong>on</strong>g> field <str<strong>on</strong>g>of</str<strong>on</strong>g> hydrotransport describe homogeneous solid-liquid suspensi<strong>on</strong>s<br />
using c<strong>on</strong>tinuum models. The definiti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>tinuum as given by Parker (1994) is:<br />
"The study <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s <str<strong>on</strong>g>of</str<strong>on</strong>g> energy. matter and o<str<strong>on</strong>g>the</str<strong>on</strong>g>r physical quannnes under<br />
.<br />
circumstances where <str<strong>on</strong>g>the</str<strong>on</strong>g>ir discrete (composed <str<strong>on</strong>g>of</str<strong>on</strong>g> separate and distinct pans) nature is<br />
unimportant and <str<strong>on</strong>g>the</str<strong>on</strong>g>y may be regarded as (in general. complex) c<strong>on</strong>tinuous ftmcti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
positi<strong>on</strong>. "<br />
The problem with using c<strong>on</strong>tinuum models is that homogeneous solid-liquid suspensi<strong>on</strong>s can<br />
never be truly homogeneous. The c<strong>on</strong>tinuum nature <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g>se slurries is an approximati<strong>on</strong> and<br />
is a state to which <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tend asymptotically (Shook & Roco, 1991). Although this<br />
approximati<strong>on</strong> is deemed to hold good, it <strong>on</strong>ly does so as l<strong>on</strong>g as <str<strong>on</strong>g>the</str<strong>on</strong>g> scale <str<strong>on</strong>g>of</str<strong>on</strong>g> fineness<br />
required by <str<strong>on</strong>g>the</str<strong>on</strong>g> subsequent modelling is not surpassed by <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> (Lumley, 1978).<br />
Hence, when c<strong>on</strong>sidering <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer, <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>tinuum approximati<strong>on</strong> MUST be<br />
compromised (Slatter 1994, Slatter et al 1996). This can clearly be seen in Figure 2.7, a<br />
magnified view <str<strong>on</strong>g>of</str<strong>on</strong>g> a rough pipe wall showing <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness and <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g>s, which indicates that <str<strong>on</strong>g>the</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are large compared with <str<strong>on</strong>g>the</str<strong>on</strong>g> scale <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
modelling. It is <str<strong>on</strong>g>the</str<strong>on</strong>g>refore imperative that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical model account for <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <strong>turbulent</strong> flow.<br />
Maude & Whitmore (1956, 1958) and Slatter (1994) are at <str<strong>on</strong>g>the</str<strong>on</strong>g> time <str<strong>on</strong>g>of</str<strong>on</strong>g> writing <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>ly<br />
known researchers to have taken <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> into account in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow analyses <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>ir <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models. Their analyses are presented in secti<strong>on</strong> 2.11.6 and secti<strong>on</strong> 2.11.5<br />
respectively.<br />
2.9.4 Turbulence<br />
Blasius (1913) was <str<strong>on</strong>g>the</str<strong>on</strong>g> first to suggest a standard empirical relati<strong>on</strong>ship between <str<strong>on</strong>g>the</str<strong>on</strong>g> Reynolds<br />
number and <str<strong>on</strong>g>the</str<strong>on</strong>g> fricti<strong>on</strong> factor for fully developed turbulenLNewt<strong>on</strong>ian flow.
Chapter 2 Literature Review<br />
• ..Page<br />
2.20<br />
Figure 2.8: Magnified view <str<strong>on</strong>g>of</str<strong>on</strong>g> a rough pipe wall showing <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer<br />
and <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry <str<strong>on</strong>g>particle</str<strong>on</strong>g>s<br />
The Blasius equati<strong>on</strong> is applicable to <str<strong>on</strong>g>the</str<strong>on</strong>g> range 3000 < Re < 100 000 where,<br />
f = 0,079 Reo. 25 • (2.28)<br />
Knudsen & Katz (1958) proposed a similar equati<strong>on</strong> for fully developed <strong>turbulent</strong> Newt<strong>on</strong>ian<br />
flow supposedly applicable to <str<strong>on</strong>g>the</str<strong>on</strong>g> range 5000. < Re < 200 000 where,<br />
f = 0,046 Reo. 2 •<br />
(2.29)<br />
Equati<strong>on</strong> 2.29 is about 10% below equati<strong>on</strong> 2.28 at low Reynolds numbers but meets it at<br />
higher Reynolds numbers (Bowen, 1961). A logarithmic plot <str<strong>on</strong>g>of</str<strong>on</strong>g>f vs Re (Figure 2.8) for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
above equati<strong>on</strong>s yields a straight line and over <str<strong>on</strong>g>the</str<strong>on</strong>g> years evidence has supported this single<br />
line correlati<strong>on</strong>. For example, Cadwell & Babbitt (1941), who studied <str<strong>on</strong>g>the</str<strong>on</strong>g> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> muds,<br />
Sludges and suspensi<strong>on</strong>s in circular pipes c<strong>on</strong>cluded that head loss in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow<br />
regime can be calculated from <str<strong>on</strong>g>the</str<strong>on</strong>g> Blasius equati<strong>on</strong> if<str<strong>on</strong>g>the</str<strong>on</strong>g> density <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry and a proper<br />
viscosity (defined under larninar flow c<strong>on</strong>diti<strong>on</strong>s) are used.<br />
The v<strong>on</strong> Karmen equati<strong>on</strong> (1930) based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> Prandtl (1926) mixing length model was<br />
proposed to represent moreexactly <str<strong>on</strong>g>the</str<strong>on</strong>g> experimental data for Reynolds numbers for <str<strong>on</strong>g>the</str<strong>on</strong>g> range<br />
3000 < Re < 300 000 where,
Chapter 2 Literature Review Page 2.21<br />
I{ =<br />
4 log(Re If) - 0,4 .<br />
(2.30)<br />
Even though this equati<strong>on</strong> was proposed for a much wider range <str<strong>on</strong>g>of</str<strong>on</strong>g> Reynolds numbers, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Reynolds numbers <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> vast majority <str<strong>on</strong>g>of</str<strong>on</strong>g>n<strong>on</strong>-Newt<strong>on</strong>ian fluids rarely exceeds 100 000 due<br />
to <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous nature <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>se fluids.<br />
However, even though <str<strong>on</strong>g>the</str<strong>on</strong>g> simple Blasius type equati<strong>on</strong>s can be used, most <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> work to<br />
develop semi-<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models for <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>ian fluids in pipes has centred<br />
around <str<strong>on</strong>g>the</str<strong>on</strong>g> mixing length model <str<strong>on</strong>g>of</str<strong>on</strong>g> Prandtl (1926).<br />
0.01<br />
0.00 1-f---r--r-;..-rrTT.,,---.----.T"T"T"TTTT--r-r.....,...,....,.-T"TTi<br />
1000 10000 100000 1000000<br />
Reynolds Number (Re)<br />
Figure 2.9: Graphical comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Blasius, Knudsen & Katz and v<strong>on</strong><br />
Karmen fricti<strong>on</strong> factor equati<strong>on</strong>s for <strong>turbulent</strong> flow<br />
2.9.5 Smooth Wall Turbulence<br />
In <strong>turbulent</strong> flow, <str<strong>on</strong>g>the</str<strong>on</strong>g> interchange <str<strong>on</strong>g>of</str<strong>on</strong>g>momentum between layers, due to eddy formati<strong>on</strong>, sets<br />
up shear stresses, also koown as Reynolds stresses (Schlichting, 1968). The shear stress in<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> core must be related to <str<strong>on</strong>g>the</str<strong>on</strong>g> shear rate in order to obtain <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g><br />
and flow rate (Janna, 1983).
Chapter 2 Literature Review Page 2.26<br />
_1 =-410g [ k].<br />
If 3,7 D<br />
2.10.4 Partially Developed Rough Wall Turbulence<br />
(2.48)<br />
The transiti<strong>on</strong>al flow between smooth and rough pipes as shown in Figure 2.10 was<br />
investigated by Colebrook and White (Colebrook, 1939). The following empirical equati<strong>on</strong><br />
was proposed:<br />
If 1 [ k<br />
= -4 log 3,7 D<br />
2.10.5 Moody Diagram<br />
+ 1,26]<br />
Re If<br />
(2.49)<br />
Moody (1944) was <str<strong>on</strong>g>the</str<strong>on</strong>g> first to present a composite diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> all <str<strong>on</strong>g>the</str<strong>on</strong>g> regi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> interest for<br />
Newt<strong>on</strong>ian flow in pipes. The chart, termed <str<strong>on</strong>g>the</str<strong>on</strong>g> Moody diagram, as shown in Figure 2.10,<br />
represents <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> Re and kiD <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> fricti<strong>on</strong> factor and includes:<br />
2.11<br />
•<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> straight line laminar fricti<strong>on</strong> factor curve<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> smooth pipe <strong>turbulent</strong> fricti<strong>on</strong> factor curve<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> various fully rough <strong>turbulent</strong> fricti<strong>on</strong> factor curves<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> fricti<strong>on</strong> factors<br />
NON-NEWTONIAN TURBULENT FLOW MODELS<br />
Data obtained by Slatter (1994) was analyzed and compared using his new model with <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Torrance (1963) and Wils<strong>on</strong> & Thomas (1985, 1987) models, <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models which have<br />
a str<strong>on</strong>ger analytical background. It was decided to use <str<strong>on</strong>g>the</str<strong>on</strong>g>se two models for analysis and<br />
comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> data for this <str<strong>on</strong>g>the</str<strong>on</strong>g>sis seeing that this is a c<strong>on</strong>tinuati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> work c<strong>on</strong>ducted by<br />
Slatter (1994). It was, however, also decided to incorporate into <str<strong>on</strong>g>the</str<strong>on</strong>g> analysis <str<strong>on</strong>g>the</str<strong>on</strong>g> models <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Dodge & Metzner (1959) and Kemblowski & Kolodziejski (1973), models having a more and<br />
empirical approach.
Chapter 2<br />
2.11.1<br />
Literature Review<br />
The Dodge & Metzner Model<br />
Page 2.28<br />
The n<strong>on</strong>-Newt<strong>on</strong>ian <strong>turbulent</strong> flow model <str<strong>on</strong>g>of</str<strong>on</strong>g> Dodge & Metzner is probably <str<strong>on</strong>g>the</str<strong>on</strong>g> most quoted<br />
and used <str<strong>on</strong>g>of</str<strong>on</strong>g> all flow models due to its simplicity and applicability to a wide range <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong><br />
Newt<strong>on</strong>ian fluids (Mun, 1988). The model was developed from <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar flow work <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
Metzner & Reed (1955) in which sixteen different n<strong>on</strong>-Newt<strong>on</strong>ian materials were studied.<br />
Both Metzner & Reed (1955) and Dodge & Metzner (1958) stated that standard Newt<strong>on</strong>ian<br />
procedures could be used for <strong>turbulent</strong> flow predicti<strong>on</strong>s. In order to obtain a velocity pr<str<strong>on</strong>g>of</str<strong>on</strong>g>Ile<br />
equati<strong>on</strong> Dodge & Metzner a;sumed <str<strong>on</strong>g>the</str<strong>on</strong>g> existence <str<strong>on</strong>g>of</str<strong>on</strong>g>a discrete boundary layer (viscous sub<br />
layer), transiti<strong>on</strong> z<strong>on</strong>e and <strong>turbulent</strong> core in <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe. They also assumed that <str<strong>on</strong>g>the</str<strong>on</strong>g> apparent<br />
flow behaviour index n' influences <str<strong>on</strong>g>the</str<strong>on</strong>g> point velocity in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> core which means that<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow head loss is dependant <strong>on</strong> n'.<br />
Their relati<strong>on</strong>ship for <strong>turbulent</strong> flow is:<br />
1<br />
If = A. log<br />
where<br />
and<br />
c =-<br />
•<br />
0,4<br />
(n ')1.2<br />
[ Re f{1 -.;:l] + c<br />
MR n ,<br />
(2.50)<br />
(2.51)<br />
(2.52)<br />
From <str<strong>on</strong>g>the</str<strong>on</strong>g> assumpti<strong>on</strong>s made by Dodge & Metzner (1959) <str<strong>on</strong>g>the</str<strong>on</strong>g> above relati<strong>on</strong>ship will revert<br />
to <str<strong>on</strong>g>the</str<strong>on</strong>g> smooth wall Newt<strong>on</strong>ian model when n' = I. The above relati<strong>on</strong>ship can also be<br />
represented graphically as shown in Figure 2.11.<br />
2.11.2 The Torrance Model<br />
Torrance developed a relati<strong>on</strong>ship between <str<strong>on</strong>g>the</str<strong>on</strong>g> fanning fricti<strong>on</strong> factor and <str<strong>on</strong>g>the</str<strong>on</strong>g> Reynolds<br />
number for Herschel-BulkIey model fluids ie. <str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic rheological model.
Chapter 2 Literature Review Page 2.30<br />
The two c<strong>on</strong>stants a and b were assumed to be identical to those evaluated by Clapp (1961)<br />
and hence taken to be:<br />
3,8<br />
a =-,<br />
n<br />
and<br />
b = 2,78.<br />
n<br />
(2.54)<br />
(2.55)<br />
For rough walled pipes <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile was modified using Prandtl's assumpti<strong>on</strong> that<br />
u(y) ex y/k and <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>stants re-evaluated using <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian data <str<strong>on</strong>g>of</str<strong>on</strong>g> Nikuradse (1933).<br />
The relati<strong>on</strong>ship for rough wall pipes is:<br />
JT<br />
- -- -2,5 1 og [Re] -<br />
f n k<br />
_ 3,75 + 8,5 .<br />
n<br />
The Reynolds number formulati<strong>on</strong> as given by Torrance is:<br />
(2.56)<br />
(2.57)<br />
The Torrance model is unable to predict <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>set <str<strong>on</strong>g>of</str<strong>on</strong>g> turbulence (as shown by Slatter, 1994)<br />
and thus it was assumed (Mun, 1988) that <str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> occurs at <str<strong>on</strong>g>the</str<strong>on</strong>g> intersecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
laminar and <strong>turbulent</strong> curves.<br />
2.11.3 The Kemblowski & Kolodziejski Model<br />
Kemblowski & Kolodziejski (1973) developed an empirical equati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Blasius type to<br />
predict <str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> power law fluids. Tests undertaken related <str<strong>on</strong>g>the</str<strong>on</strong>g> flow resistances <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
10%,20%,30%,40% and 50% by weight aqueous suspensi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g>kaolin in pipes <str<strong>on</strong>g>of</str<strong>on</strong>g>circular<br />
cross-secti<strong>on</strong>. The test results are given in <str<strong>on</strong>g>the</str<strong>on</strong>g> form <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> dependence <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> coefficient <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
flow resistance <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> generalized Reynolds number
Chapter 2 Literature Review Page 2.31<br />
A = 4f = f (Re) . (2.58)<br />
This forms a relatively flat f-Re line. At higher values <str<strong>on</strong>g>of</str<strong>on</strong>g> Re all lines <str<strong>on</strong>g>of</str<strong>on</strong>g> equati<strong>on</strong> 2.58<br />
approach <str<strong>on</strong>g>the</str<strong>on</strong>g> curve described by <str<strong>on</strong>g>the</str<strong>on</strong>g> equati<strong>on</strong><br />
A = 4f = 0,3164<br />
R 0.25<br />
e MR<br />
(2.59)<br />
Kemblowski & Kolodziejski (1973) compared <str<strong>on</strong>g>the</str<strong>on</strong>g>ir work extensively with <str<strong>on</strong>g>the</str<strong>on</strong>g> Dodge &<br />
Metzner model and found that it did not accurately describe <str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>se aqueous<br />
kaolin suspensi<strong>on</strong>s. In fact <str<strong>on</strong>g>the</str<strong>on</strong>g>y found that <str<strong>on</strong>g>the</str<strong>on</strong>g> data lay <strong>on</strong> a relatively flat f-Re line between<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> lines <str<strong>on</strong>g>of</str<strong>on</strong>g> equati<strong>on</strong> 2.59 and <str<strong>on</strong>g>the</str<strong>on</strong>g> Dodge & Metzner predicti<strong>on</strong> as can be seen in <str<strong>on</strong>g>the</str<strong>on</strong>g> figure<br />
below (Figure 2.12).<br />
Figure 2.12: Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Dodge & Metzner predicti<strong>on</strong> curve and Kemb10wski &<br />
Kolodziejski experimental data for <str<strong>on</strong>g>the</str<strong>on</strong>g> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> a 30% aqueous kaolin<br />
suspensi<strong>on</strong> at n=0,39 (taken from Kemblowski & Kolodziejski, 1973)
Chapter 2 Literature Review Page 2.33<br />
• Yield pseudoplastic fluids (Thomas & Wils<strong>on</strong>, 1987). Their analysis showed<br />
that as <str<strong>on</strong>g>the</str<strong>on</strong>g> value <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> flow behaviour index decreases from unity, <str<strong>on</strong>g>the</str<strong>on</strong>g> plot<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> fricti<strong>on</strong> factor vs Reynolds number c<strong>on</strong>verges towards <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian line.<br />
The plot <str<strong>on</strong>g>the</str<strong>on</strong>g>n parallels <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian line and eventually diverges downwards<br />
from <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian line. It was found that <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>verging behaviour was<br />
exhibited by low c<strong>on</strong>centrati<strong>on</strong> slurries compared to <str<strong>on</strong>g>the</str<strong>on</strong>g> diverging behaviour,<br />
exhibited by high c<strong>on</strong>centrati<strong>on</strong> slurries.<br />
The equati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>ir turbulel)t flow model as applying to pseudoplastic fluids is presented<br />
below.<br />
The thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer increases by a factor called <str<strong>on</strong>g>the</str<strong>on</strong>g> area ratio A. shown<br />
graphically in Figure 2.13. The figure shows that <str<strong>on</strong>g>the</str<strong>on</strong>g> area ratio is defined as <str<strong>on</strong>g>the</str<strong>on</strong>g> ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
integrals <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian and assumed Newt<strong>on</strong>ian rheograms under identical shear<br />
c<strong>on</strong>diti<strong>on</strong>s.<br />
Figure 2.13: Illustrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> area ratio<br />
Shear Rate ---...
Chapter 2 Literature Review Page 2.34<br />
The area ratio A, is thus given by:<br />
1 + :y n]<br />
A<br />
r<br />
= 2 [ 0<br />
1 + n '<br />
and hence <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness is:<br />
(2.64)<br />
(2.65)<br />
where 0" and 0",. are equivalent Newt<strong>on</strong>ian and n<strong>on</strong>-Newt<strong>on</strong>ian viscous sub-layer thicknesses<br />
respectively.<br />
The velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> is given by:<br />
u<br />
-=251n<br />
V. ' [ PV.y]<br />
p.' + 5,5 + 11,6 (A, - 1) - 2,5 In(A,) .<br />
The mean velocity is given by:<br />
V V N<br />
- = - + 11 6 (A - 1) - 2 5 In A - {}<br />
V V' r ' r'<br />
. .<br />
where V N is <str<strong>on</strong>g>the</str<strong>on</strong>g> mean velocity for <str<strong>on</strong>g>the</str<strong>on</strong>g> equivalent Newt<strong>on</strong>ian fluid and n is given by:<br />
{} = -2,5 In [1 _T y ] _ 2,5 T y [1 + 0,5 T y ] •<br />
To TO TO<br />
2.11.5 The Slatter Model<br />
(2.66)<br />
(2.67)<br />
(2.68)<br />
As menti<strong>on</strong>ed in Chapter 1 Slatter (1994) is <strong>on</strong>e <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> few researchers to have taken into<br />
account <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> c<strong>on</strong>tained in <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid for <strong>turbulent</strong> flow analysis.<br />
A new Reynolds number was developed for predicting <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>set <str<strong>on</strong>g>of</str<strong>on</strong>g> turbulence. It was<br />
modelled using <str<strong>on</strong>g>the</str<strong>on</strong>g> assumpti<strong>on</strong> that <str<strong>on</strong>g>the</str<strong>on</strong>g> unsheared plug present due to <str<strong>on</strong>g>the</str<strong>on</strong>g> yield stress acts as<br />
a solid at <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe axis and inhibits turbulence (Figure 2.14).
Chapter 2 Literature Review Page 2.35<br />
I<br />
Pipe wall<br />
Sheared<br />
annulus<br />
The Reyno1ds number is given by<br />
where<br />
and<br />
D ohear = D - D p,ug •<br />
Annular<br />
velocity<br />
Figure 2.14: Unsheared plug geometry<br />
Plug diameter<br />
Pipe diameter<br />
Slatter (1994) used <str<strong>on</strong>g>the</str<strong>on</strong>g> following precepts as <str<strong>on</strong>g>the</str<strong>on</strong>g> basis <str<strong>on</strong>g>of</str<strong>on</strong>g> his <strong>turbulent</strong> flow model:<br />
(2.69)<br />
(2.70)<br />
(2.71)<br />
• The velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> is logarithmic and similar to <str<strong>on</strong>g>the</str<strong>on</strong>g> classic Newt<strong>on</strong>ian<br />
<strong>turbulent</strong> velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> over <str<strong>on</strong>g>the</str<strong>on</strong>g> entire core regi<strong>on</strong>.<br />
• A roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> is caused by <str<strong>on</strong>g>the</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s.
Chapter 2 Literature Review Page 2.36<br />
• Fully developed rough wall <strong>turbulent</strong> flow does exist and <str<strong>on</strong>g>the</str<strong>on</strong>g> partially rough<br />
wall <strong>turbulent</strong> regi<strong>on</strong> is much narrower than for Newt<strong>on</strong>ian fluids.<br />
• Fully developed <strong>turbulent</strong> flow is independent <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> viscous characteristics <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> slurry.<br />
• Plug flow does not occur.<br />
In order to determine whe<str<strong>on</strong>g>the</str<strong>on</strong>g>r smooth wall <strong>turbulent</strong> flow or fully developed rough wall<br />
<strong>turbulent</strong> flow exists a roughness Reynolds number was formulated in terms <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> yield<br />
pseudoplastic model.<br />
The roughness Reynolds number is:<br />
(2.72)<br />
Where cl. is <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>. d.=d85 was found to be good representative<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tested.<br />
If Re, < 3,32 <str<strong>on</strong>g>the</str<strong>on</strong>g>n smooth wall <strong>turbulent</strong> flow exists and <str<strong>on</strong>g>the</str<strong>on</strong>g> mean velocity is given by:<br />
V = 2,5 In [R] -<br />
• dgs<br />
+ 2,5 In Re, + 1,75 .<br />
(2.73)<br />
If Re, > 3,32 <str<strong>on</strong>g>the</str<strong>on</strong>g>n fully developed rough wall <strong>turbulent</strong> flow exists and <str<strong>on</strong>g>the</str<strong>on</strong>g> mean velocity<br />
is given by:<br />
V = 2,5 In [R] -<br />
• d85 Which reduces to<br />
+ 4,75 ,<br />
(2.74)
Chapter 2 Literature Review Page 2.38<br />
(2.77)<br />
Tests c<strong>on</strong>ducted by Slatter (1994) c<strong>on</strong>firmed <str<strong>on</strong>g>the</str<strong>on</strong>g> model to be more accurate than <str<strong>on</strong>g>the</str<strong>on</strong>g> Torrance<br />
(1963) model and Wils<strong>on</strong> & Thomas (1985, 1987) model against which experimental data<br />
was compared. This model was also supported by <str<strong>on</strong>g>the</str<strong>on</strong>g> experimental data <str<strong>on</strong>g>of</str<strong>on</strong>g>Park et al (1989)<br />
and Xu et al (1993).<br />
2.11.6 Maude & Whitmore Correlati<strong>on</strong><br />
Maude & Whitmore (1956, 1958) are two <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> few researchers, if not <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>ly two before<br />
Slatter (1994), to take any account <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s play in <strong>turbulent</strong> flow.<br />
Tests ·were c<strong>on</strong>ducted in thin vertical 2,2mm, 3,5mm and 5mm diameter tubes using emery<br />
slurries at six different c<strong>on</strong>centrati<strong>on</strong>s. The <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> emery slurries<br />
fell within <str<strong>on</strong>g>the</str<strong>on</strong>g> range 20 to 40JLm. Data was presented in <str<strong>on</strong>g>the</str<strong>on</strong>g> form <str<strong>on</strong>g>of</str<strong>on</strong>g>curves <str<strong>on</strong>g>of</str<strong>on</strong>g> fricti<strong>on</strong> factors<br />
against Reynolds numbers.<br />
The main observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Maude & Whitmore (1958) was that <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> fricti<strong>on</strong> factor<br />
(based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall viscosity) was initially higher than <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian fricti<strong>on</strong> factor but fell<br />
below at increased Reynolds numbers. This observati<strong>on</strong> is c<strong>on</strong>tradictory to <str<strong>on</strong>g>the</str<strong>on</strong>g> more widely<br />
publicised finding that <str<strong>on</strong>g>the</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian fricti<strong>on</strong> factor is always less than <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian<br />
fricti<strong>on</strong> factor (eg Wils<strong>on</strong>, 1986).<br />
The phenomena encountered by Maude & Whitmore (1958) were explained in <str<strong>on</strong>g>the</str<strong>on</strong>g> following<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical terms. The mixing length, as developed by Prandtl, is defined as <str<strong>on</strong>g>the</str<strong>on</strong>g> mean<br />
distance which fluid elements move at right angles to <str<strong>on</strong>g>the</str<strong>on</strong>g> directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> flow before <str<strong>on</strong>g>the</str<strong>on</strong>g>y<br />
acquire <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> layer <str<strong>on</strong>g>the</str<strong>on</strong>g>y have entered. Hence, <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid element would pursue<br />
an oscillating passage down <str<strong>on</strong>g>the</str<strong>on</strong>g> tube. However, inertial forces would prevent any high<br />
denSity <str<strong>on</strong>g>particle</str<strong>on</strong>g>s from following exactly <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity changes <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid. The <str<strong>on</strong>g>particle</str<strong>on</strong>g>s<br />
Would instead oscillate with a smaller amplitude and <str<strong>on</strong>g>the</str<strong>on</strong>g> mean mixing length <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
suspensi<strong>on</strong> would decrease with an increase in solids c<strong>on</strong>centrati<strong>on</strong> as compared to that <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> raw fluid. They postulated that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g>ive mixing length is reduced by a <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical
Chapter 2 Literature Review Page 2.39<br />
mixing length factor p<br />
p = P, (I - Cv> + q Pp Cv ,<br />
PI (1 - Cv> + Pp Cv<br />
where Cv = volume <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s per unit volume <str<strong>on</strong>g>of</str<strong>on</strong>g> suspensi<strong>on</strong><br />
(2.78)<br />
q = <str<strong>on</strong>g>the</str<strong>on</strong>g> ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s to <str<strong>on</strong>g>the</str<strong>on</strong>g> amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> fluid, given by;<br />
where I = vot, = average amplitude <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> liquid oscillati<strong>on</strong> and K is given by:<br />
where a = shape factor.<br />
(2.79)<br />
(2.80)<br />
Maude & Whitmore (1958) hence stated that <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure loss and fricti<strong>on</strong> factor should be<br />
reduced in <strong>turbulent</strong> flow. It must be noted however, that while this accounts for flow<br />
behaviour at high ReynoIds numbers, <str<strong>on</strong>g>the</str<strong>on</strong>g> approach does not take into account low Reynolds<br />
number behaviour.<br />
In order to account for low Reynolds number behaviour and to explain why higher-than<br />
Newt<strong>on</strong>ian equivalent fricti<strong>on</strong> factors were being observed, Maude & Whitmore (1958)<br />
c<strong>on</strong>sidered <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspended <str<strong>on</strong>g>particle</str<strong>on</strong>g> <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscosity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> medium forming <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
viscous sub-layer. It is known that as <str<strong>on</strong>g>the</str<strong>on</strong>g> Reynolds number increases <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer<br />
becomes narrower to <str<strong>on</strong>g>the</str<strong>on</strong>g> extent that <str<strong>on</strong>g>the</str<strong>on</strong>g> mean <str<strong>on</strong>g>particle</str<strong>on</strong>g> diameter is greater. Thus <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />
SUb-layer is said to c<strong>on</strong>sist <strong>on</strong>ly <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspending medium. At lower Reynolds numbers, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
viscous sub-layer thickens and <str<strong>on</strong>g>the</str<strong>on</strong>g> whole suspensi<strong>on</strong> is sheared by <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous flow in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
SUb-layer. Hence, <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g>ive viscosity becomes that <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> suspensi<strong>on</strong> and explains <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
higher fricti<strong>on</strong> factors.
Chapter 2 Literature Review Page 2.40<br />
Two equati<strong>on</strong>s were postulated by Maude & Whitrnore (1958) for <strong>turbulent</strong> flow. The<br />
criteri<strong>on</strong> as to which equati<strong>on</strong> to use being when <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> diameter is greater than half <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
thickness <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer. The equati<strong>on</strong> for determining <str<strong>on</strong>g>the</str<strong>on</strong>g> thickness <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />
sub-layer was given by:<br />
o = [11,7 p. - .!.k' Cd] (l - k'cr l .<br />
P, V. 2<br />
The <strong>turbulent</strong> flow equati<strong>on</strong>s for 0 ;:: lhd is given by:<br />
1 _ -0,815 I [66,3 1 _ d k'c ]<br />
If - 0,408p oglO Re If (1 - k'c) D (1 - k'c)<br />
The <strong>turbulent</strong> flow equati<strong>on</strong>s for 0 < lhd is given by:<br />
1<br />
If<br />
2.11.7<br />
-0,815 I [66,3] _ 0,531 + 4 14<br />
oglO ' .<br />
0,408p Re If 0,408p<br />
The Bowen Correlati<strong>on</strong><br />
(2.81)<br />
_ 0,531 + 4,14 .(2.82)<br />
0,408p<br />
(2.83)<br />
Bowen (1961) noted that no universal correlati<strong>on</strong> had been suggested for n<strong>on</strong>-Newt<strong>on</strong>ian<br />
flUids and hence published a method which appeared to be applicable to all fluids. The basis<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> his method was his finding that <strong>on</strong> a log-log pseudo-shear diagram <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> branches<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g>different diameters appeared to describe similar straight lines, with each branch appearing<br />
to be almost parallel to <str<strong>on</strong>g>the</str<strong>on</strong>g> next. He suggested that if <str<strong>on</strong>g>the</str<strong>on</strong>g> shear stress or shear rate were<br />
mUltiplied by a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter, a correlati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> data might be<br />
obtained. Hence, he was able to correlate <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter <str<strong>on</strong>g>effect</str<strong>on</strong>g> by adapting <str<strong>on</strong>g>the</str<strong>on</strong>g> Blasius<br />
equati<strong>on</strong> (equati<strong>on</strong> 2.28) for Newt<strong>on</strong>ian fluids and obtain <str<strong>on</strong>g>the</str<strong>on</strong>g> following correlati<strong>on</strong>:<br />
Where k and b are c<strong>on</strong>stants.<br />
Bowen presented four worked examples to substantiate his method.<br />
(2.84)
Chapter 2 Literature Review Page 2.41<br />
Although this correlati<strong>on</strong> does not provide an explanati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry in<br />
terms <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> physical properties <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry, it does produce a good correlati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong><br />
Newt<strong>on</strong>ian <strong>turbulent</strong> flow pipe data (Harris & Quader, 1971 and Quader & Wilkins<strong>on</strong>, 1980).<br />
2.12 EVIDENCE OF PARTICLE ROUGHNESS<br />
Many researchers report similarity between <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>ian fluids and<br />
n<strong>on</strong>-Newt<strong>on</strong>ian slurries (Caldwell & Babbitt 1941, Hedstrom 1952, Metzner & Reed 1955,<br />
Dodge & Metzner 1959, Tomita 1959, Michiyoshi et al 1966, Edwards & Smith 1980,<br />
Thomas & Wils<strong>on</strong> 1987 and Sive 1988). One <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> more interesting pieces <str<strong>on</strong>g>of</str<strong>on</strong>g> evidence to<br />
date <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> similarity between Newt<strong>on</strong>ian and n<strong>on</strong>-Newt<strong>on</strong>ian slurry <strong>turbulent</strong> flow is<br />
presented by Park et al (1989).<br />
Park et al (1989) investigated <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> structure <str<strong>on</strong>g>of</str<strong>on</strong>g> a n<strong>on</strong>-Newt<strong>on</strong>ian slurry using laser<br />
doppler anemometry and c<strong>on</strong>cluded that <str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> regi<strong>on</strong> is much narrower than for<br />
Newt<strong>on</strong>ian fluids. Their n<strong>on</strong>-Newt<strong>on</strong>ian slurry <strong>turbulent</strong> flow velocity pr<str<strong>on</strong>g>of</str<strong>on</strong>g>ile (Figure 2.16)<br />
agrees well with measurements with air (Newt<strong>on</strong>ian fluid). However, <str<strong>on</strong>g>the</str<strong>on</strong>g>y reported higher<br />
relative turbulence intensities in <str<strong>on</strong>g>the</str<strong>on</strong>g> wall regi<strong>on</strong> for <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry, when compared with air<br />
(Figure 2.17). This would support <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>cept <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness. Pokryvalio &<br />
Grozberg (1995) c<strong>on</strong>fmned <str<strong>on</strong>g>the</str<strong>on</strong>g> findings <str<strong>on</strong>g>of</str<strong>on</strong>g> Park et al (1989) using electro-diffusi<strong>on</strong><br />
techniques for measuring velocity pr<str<strong>on</strong>g>of</str<strong>on</strong>g>iles <str<strong>on</strong>g>of</str<strong>on</strong>g>Bent<strong>on</strong>ite clay suspensi<strong>on</strong>s at c<strong>on</strong>centrati<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
4% and 6% and comparing it with air. They reported a significant increase in turbulence<br />
intensities in <str<strong>on</strong>g>the</str<strong>on</strong>g> wall regi<strong>on</strong> for <str<strong>on</strong>g>the</str<strong>on</strong>g> Bent<strong>on</strong>ite clay suspensi<strong>on</strong> (Figure 2.18), providing<br />
fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r support <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>cept <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness.<br />
2.13 DATA FROM THE LITERATURE<br />
Experimental data obtained by Sive (1988) was used in <str<strong>on</strong>g>the</str<strong>on</strong>g> analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> various models<br />
under c<strong>on</strong>siderati<strong>on</strong>. The tests c<strong>on</strong>ducted by Sive (1988) were d<strong>on</strong>e using a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g>kaolin<br />
clay and a relatively coarse quartz sand, which resulted in a heterogeneous, settling slurry.<br />
The purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> using this data was to see if <str<strong>on</strong>g>the</str<strong>on</strong>g> coarse, settling <str<strong>on</strong>g>particle</str<strong>on</strong>g>s c<strong>on</strong>tributed to <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<strong>turbulent</strong> flow headloss, as proposed by <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model.
Chapter 2 Literature Review<br />
2<br />
o Q2 a6 aB . Yjl?<br />
Figure 2.19: Comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> relative <strong>turbulent</strong> intensities between bent<strong>on</strong>ite<br />
clay suspensi<strong>on</strong>s (C v =4% and Cv =6%) and water<br />
(Taken from Pokryvalio & Grozberg, 1995)<br />
2.14 CONCLUSIONS<br />
Page 2.43<br />
The rheo1ogical fundamentals relevant to fluid flow in pipes have been presented and <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models under c<strong>on</strong>siderati<strong>on</strong> have been reviewed<br />
2.14.1 Laminar Flow<br />
The yield pseudoplastic model can be used to model and predict <str<strong>on</strong>g>the</str<strong>on</strong>g> 1aminar flow <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong><br />
Newt<strong>on</strong>ian slurries. The rheological c<strong>on</strong>stants can be accurately detennined using <str<strong>on</strong>g>the</str<strong>on</strong>g> method<br />
adopted by Neill (1988).
Chapter 2<br />
2.14.2<br />
Literature Review<br />
Transiti<strong>on</strong> from Laminar to Turbulent Flow<br />
Page 2.44<br />
There are several methods in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> between larninar and<br />
<strong>turbulent</strong> flow. However, <str<strong>on</strong>g>the</str<strong>on</strong>g> full rheology should be used.<br />
2.14.3 Turbulent flow<br />
The Blasius type equati<strong>on</strong>s can be used to determine <strong>turbulent</strong> flow. However, most <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
work to develop semi-<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models for <strong>turbulent</strong> flow has centred around <str<strong>on</strong>g>the</str<strong>on</strong>g> mixing<br />
length model <str<strong>on</strong>g>of</str<strong>on</strong>g> Prandtl. Hence, <str<strong>on</strong>g>the</str<strong>on</strong>g> logarithmic velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> can be used to model<br />
smooth wall Newt<strong>on</strong>ian <strong>turbulent</strong> flow. A logarithmic velocity <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> with a roughness<br />
functi<strong>on</strong> and a roughness Reynolds number to correlate <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness functi<strong>on</strong> can be used<br />
to model fully developed rough wall Newt<strong>on</strong>ian <strong>turbulent</strong> flow. The Colebrook White<br />
relati<strong>on</strong> can be used to model partially rough wall Newt<strong>on</strong>ian <strong>turbulent</strong> flow.<br />
There are three approaches in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature for modelling n<strong>on</strong>-Newt<strong>on</strong>ian <strong>turbulent</strong> flow data:<br />
• empirical approach eg Bowen (1961),<br />
• approach based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry rheology eg Torrance (1963) or Wils<strong>on</strong> &<br />
Thomas (1985, 1987),<br />
• approach based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> or <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> eg Slatter<br />
(1994).<br />
The c<strong>on</strong>tinuum approximati<strong>on</strong> must be compromised as <str<strong>on</strong>g>the</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s c<strong>on</strong>tained in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
slurry are large compared with <str<strong>on</strong>g>the</str<strong>on</strong>g> scale <str<strong>on</strong>g>of</str<strong>on</strong>g> modelling <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness.<br />
Theoretical models must <str<strong>on</strong>g>the</str<strong>on</strong>g>refore account for <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <strong>turbulent</strong> flow and<br />
hence from <str<strong>on</strong>g>the</str<strong>on</strong>g> three possible approaches that can be adopted, <str<strong>on</strong>g>the</str<strong>on</strong>g> approach using <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> should be used. Although Bowen's method does produce good correlati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian <strong>turbulent</strong> flow pipe data it cannot provide an explanati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry in terms <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> physical properties <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry.<br />
A wall roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>, <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> or <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> has been reported in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
literature (Maude & Whitmore 1956, 1958, Mun 1988, Slatter 1994). This <str<strong>on</strong>g>effect</str<strong>on</strong>g> has been
Chapter 2 Literature Review Page 2.45<br />
supported by data from Park et al (1989) and Pokryvalio & Grozberg (1995).<br />
There are reported differences in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature as to when <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer breaks down.<br />
Although <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> has been reported in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature <str<strong>on</strong>g>the</str<strong>on</strong>g>re is no published<br />
limit <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> validity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g>.<br />
2.14.4 Objectives <str<strong>on</strong>g>of</str<strong>on</strong>g> Thesis<br />
Having reviewed <str<strong>on</strong>g>the</str<strong>on</strong>g> relevant literature <str<strong>on</strong>g>the</str<strong>on</strong>g> following areas for investigati<strong>on</strong> can be noted:<br />
• There is a lack <str<strong>on</strong>g>of</str<strong>on</strong>g> published literature <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>. Slatter (1994)<br />
has developed a new general approach to turbulence flow modelling and c<strong>on</strong>cluded<br />
that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> was valid for <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries that were tested.<br />
However, research is required to determine if Slatter's model is accurate with<br />
increasing <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> and whe<str<strong>on</strong>g>the</str<strong>on</strong>g>r <str<strong>on</strong>g>the</str<strong>on</strong>g> d 85 <str<strong>on</strong>g>size</str<strong>on</strong>g> (as suggested by Slatter) does produce<br />
a good representati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>.<br />
• To investigate if <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD does affect <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry.<br />
• To determine if <str<strong>on</strong>g>the</str<strong>on</strong>g> correlati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Maude & Whitmore (<str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>ly o<str<strong>on</strong>g>the</str<strong>on</strong>g>r model to<br />
incorporate <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> besides <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model) can accurately predict <strong>turbulent</strong><br />
flow behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian homogeneous slurries.<br />
• To determine if increasing <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> has an <str<strong>on</strong>g>effect</str<strong>on</strong>g> <strong>on</strong> <strong>turbulent</strong> flow behaviour.<br />
This can be achieved by using <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models (as described in this chapter) to<br />
model experimental data and to observe and analyze any <str<strong>on</strong>g>effect</str<strong>on</strong>g>s <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> has <strong>on</strong><br />
<strong>turbulent</strong> flow behaviour.
CHAPTER 3
3.1 INTRODUCTION<br />
CHAPTER 3<br />
EXPERIMENTAL WORK<br />
The test work was c<strong>on</strong>ducted at <str<strong>on</strong>g>the</str<strong>on</strong>g> University <str<strong>on</strong>g>of</str<strong>on</strong>g> Cape Town's Hydrotransport Research<br />
Facility. The test facilities used included a pumped recirculating pipe test rig known as <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
East Rig and a pipeline rig known as <str<strong>on</strong>g>the</str<strong>on</strong>g> Mini Rig.<br />
In large scale pipe testing (as used for this study), slurry is pumped in a looped circuit<br />
normally c<strong>on</strong>sisting <str<strong>on</strong>g>of</str<strong>on</strong>g> two pipe diameters, over a wide range <str<strong>on</strong>g>of</str<strong>on</strong>g> velocities. This is d<strong>on</strong>e in<br />
order to measure <str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry (rheology) and <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> pipe flow head loss<br />
which are needed in designing a piping system. Rheological data which is obtained from<br />
laminar flow data can be used to predict <strong>turbulent</strong> pipe flow head loss (Slatter, 1994).<br />
The test facilities which were used c<strong>on</strong>sisted in total <str<strong>on</strong>g>of</str<strong>on</strong>g> four different pipe diameters namely<br />
a 25mm, 80mm, l50mm and 200mm nominal bore. Slurries were tested at mean velocities<br />
ranging froI11 O,lrn/s to 6.2rn/s. Slurries tested included kaolin clay, a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g>kaolin clay<br />
and rock flour (mixture 1) and a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay, rock flour and silica sand (mixture<br />
2) at varying ratios.<br />
3.2 TESTING FACILITIES<br />
3.2.1 The East Rig<br />
Figure 3.1 depicts a schematic diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pumped recirculating pipe test rig.<br />
(a) Pump Specificati<strong>on</strong><br />
Slurry is supplied to <str<strong>on</strong>g>the</str<strong>on</strong>g> East Rig by a four bladed Ma<str<strong>on</strong>g>the</str<strong>on</strong>g>r and Platt 8x6, solids<br />
handling pump, which is driven by a variable speed hydraulic drive (Figure 3.2).
Chapter 3 Experimental Work<br />
Figure 3.2: Solids handling pump & variable speed hydraulic drive<br />
(b) Layout<br />
Page 3.3<br />
Slurry to be used for a test run is collected in a steel hopper which has a capacity 2m 3<br />
from where it is pumped through a looped circuit. Directly after <str<strong>on</strong>g>the</str<strong>on</strong>g> pump <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
l50mm line splits up into an SOmm and l50mm pipeline. These two pipelines have<br />
a vertical counterflow secti<strong>on</strong> (Figure 3.3) and horiz<strong>on</strong>tal secti<strong>on</strong> (Figure 3.4), <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
flow rate being measured in both downcomer secti<strong>on</strong>s by means <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic flux flow<br />
meters. The horiz<strong>on</strong>tal secti<strong>on</strong> is approximately 30m in length, with <str<strong>on</strong>g>the</str<strong>on</strong>g> return<br />
pipelines passing through an in-line heat exchanger and pneumatic diverter valve<br />
before being re-routed back into <str<strong>on</strong>g>the</str<strong>on</strong>g> hopper.
Chapter 3 Experimental Work Page 3.5<br />
Figure 3.4: Horiz<strong>on</strong>tal test secti<strong>on</strong><br />
Figure 3.5: The 8Omm, 150mm and 200mm horiz<strong>on</strong>tal return pipelines
Chapter 3 Experimental Work Page 3.6<br />
3.2.2 Mini Rig<br />
Figure 3.6: Steel hopper and weigh tank<br />
The purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Mini Rig was to obtain accurate measurements <str<strong>on</strong>g>of</str<strong>on</strong>g> velocity and head loss<br />
in laminar flow for <str<strong>on</strong>g>the</str<strong>on</strong>g> determinati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry. Slurry was tapped <str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
from <str<strong>on</strong>g>the</str<strong>on</strong>g> 150rnrn pipeline <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> East Rig through a 25mm secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> PVC piping, which<br />
included a 25mm Altometer magnetic flux flowmeter, and <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry directed into <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh<br />
tank. The weight <str<strong>on</strong>g>of</str<strong>on</strong>g> slurry passing into <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank during a data test point, and <str<strong>on</strong>g>the</str<strong>on</strong>g> time<br />
taken for that specific volume to ftIl <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank, were recorded, in order to ensure that<br />
aCCurate velocity readings were being obtained from <str<strong>on</strong>g>the</str<strong>on</strong>g> flowmeter.<br />
The back pressure from <str<strong>on</strong>g>the</str<strong>on</strong>g> 150mm pipeline was used as <str<strong>on</strong>g>the</str<strong>on</strong>g> driving force for <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry to<br />
pass through <str<strong>on</strong>g>the</str<strong>on</strong>g> Mini Rig.<br />
Using slurry from <str<strong>on</strong>g>the</str<strong>on</strong>g> East Rig ensured that <str<strong>on</strong>g>the</str<strong>on</strong>g> same slurry is tested in both rigs ie. East and<br />
Mini Rigs. Figure 3.7 depicts <str<strong>on</strong>g>the</str<strong>on</strong>g> Mini Rig used for testing purposes.
Chapter 3 Experimental Work Page 3.7<br />
3.3 MEASURED VARIABLES<br />
3.3.1 Pressure Measurement<br />
(a) Pressure Tappings<br />
Figure 3.8: The Mini Rig<br />
Pressure tappings located in <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal test secti<strong>on</strong> in <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall are used for<br />
differential pressure measurements. Figure 3.8 shows a typical pressure tapping<br />
arrangement that was used.<br />
The tapping hole diameter is 3mm and as reported by Hanks (1981) <str<strong>on</strong>g>the</str<strong>on</strong>g> length to<br />
diameter (lJD) ratio (ie. length <str<strong>on</strong>g>of</str<strong>on</strong>g> tapping hole to tapping diameter), which is<br />
c<strong>on</strong>sidered critical, was designed according to <str<strong>on</strong>g>the</str<strong>on</strong>g> ratio<br />
L = 4 (3.1)<br />
D '<br />
to ensure accurate readings. All burrs <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> inside edge <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> tappings were<br />
removed.
Chapter 3 Experimental Work<br />
Figure 3.8: Pressure tapping and solids collecting pod<br />
Page 3.8<br />
Pressure measurement points are located at 45° to <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal to ensure that <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
ingesti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s or air bubbles into <str<strong>on</strong>g>the</str<strong>on</strong>g> solids trap (see Figure 3.9) is kept<br />
to a minimum. There is, however, a valve located <strong>on</strong> each solids trap for flushing<br />
away any unwanted solids which may accumulate.<br />
The manometer and differential pressure transducer (DPT) are c<strong>on</strong>nected by clear<br />
water lines to <str<strong>on</strong>g>the</str<strong>on</strong>g> solids trap.<br />
The test secti<strong>on</strong>s which are 2,995m in length are preceded by unobstructed straight<br />
pipe <str<strong>on</strong>g>of</str<strong>on</strong>g> at least 50 pipe diameters (Govier & Aziz, 1972 and Hanks 1981). The <strong>on</strong>ly
Chapter 3 Experimental Work Page 3.9<br />
excepti<strong>on</strong> is <str<strong>on</strong>g>the</str<strong>on</strong>g> 200mm nominal bore pipe which has a straight entry length <str<strong>on</strong>g>of</str<strong>on</strong>g> 35<br />
pipe diameters.<br />
(b) Manometer Board<br />
The manometer board serves as a centralised point for measuring pressure and flow.<br />
There are six differential water manometers and <str<strong>on</strong>g>the</str<strong>on</strong>g> layout is shown in Figure 3.9 and<br />
Figure 3.10. Electrical signals from <str<strong>on</strong>g>the</str<strong>on</strong>g> DPT and flow meters are output from <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
board to <str<strong>on</strong>g>the</str<strong>on</strong>g> data logging system (Figure 3.11). Head loss measurements can be<br />
measured with manometer menisci visible or masked, although <str<strong>on</strong>g>the</str<strong>on</strong>g> air over water<br />
manometer head (ie. visible) is usually maintained during a test run to provide<br />
c<strong>on</strong>firmati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> head loss measurements. Flushing water is supplied from <str<strong>on</strong>g>the</str<strong>on</strong>g> water<br />
main (400 kPa) and air pressure is supplied at 800 kPa by a compressor.<br />
Figure 3.9: Manometer Board
Chapter 3 Experimental Work Page 3.11<br />
(c) Pressure Transducers<br />
A Gould PD3000 pressure transducer was used for differential pressure measurement.<br />
The transducer employs a strain gauge bridge to c<strong>on</strong>vert differential pressure to an<br />
electrical output which can <str<strong>on</strong>g>the</str<strong>on</strong>g>n be read. The diaphragm is made <str<strong>on</strong>g>of</str<strong>on</strong>g>Hastelloy C with<br />
a 316 stainless steel body.<br />
3.3.2 Flow Measurement<br />
(a) Magnetic Flow Meters<br />
The magnetic flow meters that were used for <str<strong>on</strong>g>the</str<strong>on</strong>g> East Rig were manufactured by Kent<br />
Instruments (80mm pipeline) and Kr<strong>on</strong>e Instruments (150mm & 200mm pipelines).<br />
The principle <str<strong>on</strong>g>of</str<strong>on</strong>g> operati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> both magnetic flow meters used is similar. Firstly, a<br />
magnetic field is set up across <str<strong>on</strong>g>the</str<strong>on</strong>g> bore <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> magnetic flow meter. Liquid flowing<br />
through <str<strong>on</strong>g>the</str<strong>on</strong>g> metering tube will cut <str<strong>on</strong>g>the</str<strong>on</strong>g> magnetic field and thus develop an induced emf<br />
in <str<strong>on</strong>g>the</str<strong>on</strong>g> liquid. This emfis detected by two electrodes in <str<strong>on</strong>g>the</str<strong>on</strong>g> wall <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> metering tube.<br />
The emfis proporti<strong>on</strong>al to <str<strong>on</strong>g>the</str<strong>on</strong>g> flow velocity multiplied by <str<strong>on</strong>g>the</str<strong>on</strong>g> magnetic field strength.<br />
The transducer signal is digitised by a data logger.<br />
3.4 CALIBRATION<br />
The transducers and magnetic flux flow meters are re-ealibrated at regular intervals to ensure<br />
accuracy for collecting research data. The transducers are calibrated at <str<strong>on</strong>g>the</str<strong>on</strong>g> start <str<strong>on</strong>g>of</str<strong>on</strong>g>each day<br />
and <str<strong>on</strong>g>the</str<strong>on</strong>g> flow meters at <str<strong>on</strong>g>the</str<strong>on</strong>g> start <str<strong>on</strong>g>of</str<strong>on</strong>g> each week as well as during a test run.<br />
3.4.1 Pipeline<br />
(a) Pipeline Diameter<br />
The internal pipeline diameter (D) is measured by filling a known length (L) <str<strong>on</strong>g>of</str<strong>on</strong>g> pipe<br />
with water. The amount <str<strong>on</strong>g>of</str<strong>on</strong>g> water required to fill <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe is weighed (Mw) and <str<strong>on</strong>g>the</str<strong>on</strong>g>
Chapter 3 Experimental Work<br />
Table 3.II: Pipeline Roughness<br />
Actual Inside Diameter Pipeline Roughness<br />
(mm) (]Lm)<br />
21,6 4<br />
79,0 7<br />
140,7 9<br />
207,0 130<br />
.<br />
3.4.2 Calibrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Differential Pressure Transducer<br />
The following procedure is used to calibrate <str<strong>on</strong>g>the</str<strong>on</strong>g> DPT <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> East and Mini Rig.<br />
Page 3.13<br />
1. The transducer and manometer that are to be calibrated are flushed with mains<br />
water to ensure that air and any solids have been removed from <str<strong>on</strong>g>the</str<strong>on</strong>g> lines.<br />
2. Air over water manometer head ie. a differential head (H) is set up in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
glass manometer tubes.<br />
3. . This head is measured and <str<strong>on</strong>g>the</str<strong>on</strong>g> DPT output is logged at <str<strong>on</strong>g>the</str<strong>on</strong>g> same time.<br />
4. Steps 2 and 3 are repeated for different manometer heights over <str<strong>on</strong>g>the</str<strong>on</strong>g> full<br />
differential head test range until enough data points have been collected for<br />
calibrati<strong>on</strong>.<br />
A least squares linear regressi<strong>on</strong> is performed <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> measured head and transducer readings<br />
in order to obtain <str<strong>on</strong>g>the</str<strong>on</strong>g> calibrati<strong>on</strong> equati<strong>on</strong>. A set <str<strong>on</strong>g>of</str<strong>on</strong>g> N observed measurements Y <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
corresp<strong>on</strong>ding set <str<strong>on</strong>g>of</str<strong>on</strong>g> N transducer readings X will yield <str<strong>on</strong>g>the</str<strong>on</strong>g> least squares regressi<strong>on</strong> line <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
(Spiegel, 1972)<br />
Y:mX+c<br />
Where<br />
(3.4)
Chapter 3 Experimental Work Page 3.14<br />
N EXY - (EX) (EY)<br />
m =<br />
N E(X 2) - (EX)2<br />
and<br />
EY E(X2) - EX E(XY)<br />
c =<br />
N E(X 2) - (EX)2<br />
(3.5)<br />
(3.6)<br />
An objective measure <str<strong>on</strong>g>of</str<strong>on</strong>g> how well <str<strong>on</strong>g>the</str<strong>on</strong>g> line represents <str<strong>on</strong>g>the</str<strong>on</strong>g> data is given by <str<strong>on</strong>g>the</str<strong>on</strong>g> correlati<strong>on</strong><br />
coefficient (r) equati<strong>on</strong>:<br />
(3.7)<br />
Calibrati<strong>on</strong>s were accepted for r values in <str<strong>on</strong>g>the</str<strong>on</strong>g> range O,99
Chapter 3 Experimental Work Page 3.16<br />
The following procedure was undertaken for calibrating <str<strong>on</strong>g>the</str<strong>on</strong>g> magnetic flow meters.<br />
I. The pump speed was set to <str<strong>on</strong>g>the</str<strong>on</strong>g> desired speed.<br />
2. The weigh tank scale reading was noted (Ml)'<br />
3. The data logger was first started and <str<strong>on</strong>g>the</str<strong>on</strong>g>n <str<strong>on</strong>g>the</str<strong>on</strong>g> diverter valve was opened and <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
stopwatch started at <str<strong>on</strong>g>the</str<strong>on</strong>g> same time.<br />
4. The diverter valve was closed and <str<strong>on</strong>g>the</str<strong>on</strong>g> stopwatch stopped at <str<strong>on</strong>g>the</str<strong>on</strong>g> same time when<br />
sufficient slurry had been collected. The data logger ran for a pre-set time, enough<br />
to cover <str<strong>on</strong>g>the</str<strong>on</strong>g> whole procedure.<br />
5. The weigh tank reading .was noted (M,).<br />
The flow rate for each reading is given by <str<strong>on</strong>g>the</str<strong>on</strong>g> equati<strong>on</strong>:<br />
(3.9)<br />
The average transducer reading cannot be used for calibrati<strong>on</strong> purposes as it is not accurate.<br />
In order to obtain <str<strong>on</strong>g>the</str<strong>on</strong>g> correct transducer reading for each point a graph <str<strong>on</strong>g>of</str<strong>on</strong>g> current signal vs<br />
time was plotted. Figure 3.13 shows a typical output.<br />
As can be seen from <str<strong>on</strong>g>the</str<strong>on</strong>g> graph <str<strong>on</strong>g>the</str<strong>on</strong>g> current signal drops sharply after <str<strong>on</strong>g>the</str<strong>on</strong>g> diverter valve is<br />
opened (point A) before c<strong>on</strong>stantly decreasing (point B to point C). After <str<strong>on</strong>g>the</str<strong>on</strong>g> diverter valve<br />
is closed (point C) <str<strong>on</strong>g>the</str<strong>on</strong>g> current signal reading returns to <str<strong>on</strong>g>the</str<strong>on</strong>g> original value (point D). The<br />
slope <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> line BC remains c<strong>on</strong>stant at <str<strong>on</strong>g>the</str<strong>on</strong>g> varying pump speeds. The transducer reading<br />
is thus read by extending a line EF from <str<strong>on</strong>g>the</str<strong>on</strong>g> slope <str<strong>on</strong>g>of</str<strong>on</strong>g> line BC halfway during <str<strong>on</strong>g>the</str<strong>on</strong>g> time taken<br />
to divert <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry to <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank (point A to point C).<br />
The calibrati<strong>on</strong> equati<strong>on</strong> is <str<strong>on</strong>g>the</str<strong>on</strong>g>n derived by performing a linear regressi<strong>on</strong> <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> flow rate<br />
and <str<strong>on</strong>g>the</str<strong>on</strong>g> transducer readings in each case.<br />
No difference was found between <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar and <strong>turbulent</strong> flow data when calibrated in this<br />
Way, as indicated by Fig 3.14 showing a typical calibrati<strong>on</strong>.
Chapter 3 Experimental Work Page 3.18<br />
3.5 OTHER MEASURED VARIABLES<br />
3.5.1 Slurry Density<br />
Slurry density and relative density are determined by performing <str<strong>on</strong>g>the</str<strong>on</strong>g> following steps.<br />
1. A clean, dry <strong>on</strong>e litre volumetric flask is weighed (M,).<br />
2. The volumetric flask was filled with slurry from a tapping in <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall <strong>on</strong><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> vertical secti<strong>on</strong> above <str<strong>on</strong>g>the</str<strong>on</strong>g> pump before it splits into <str<strong>on</strong>g>the</str<strong>on</strong>g> 80mm and 150mm<br />
pipelines. The volume <str<strong>on</strong>g>of</str<strong>on</strong>g> slurry is approximately 990ml.<br />
3. The flask plus slurry are weighed (Mz).<br />
4. The flask is <str<strong>on</strong>g>the</str<strong>on</strong>g>n filled with water up to <str<strong>on</strong>g>the</str<strong>on</strong>g> graduated mark and weighed<br />
(M3)·<br />
5. The flask is emptied, filled with clear water and weighed again (M4)'<br />
6. Steps 1-5 were repeated except that in step 2 slurry was taken from a tapping<br />
in <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal test secti<strong>on</strong>.<br />
The relative slurry density Sm is defined as:<br />
Sm = __m_as.-s_o_f..."s_lurry--;=---sam_..:,p1,...e__<br />
mass <str<strong>on</strong>g>of</str<strong>on</strong>g> equal volume <str<strong>on</strong>g>of</str<strong>on</strong>g> water<br />
S.. is calculated using <str<strong>on</strong>g>the</str<strong>on</strong>g> above equati<strong>on</strong> and slurry density is calculated from:<br />
p = Sm P w •<br />
(3.10)<br />
(3.11)<br />
Normally <str<strong>on</strong>g>the</str<strong>on</strong>g> use <str<strong>on</strong>g>of</str<strong>on</strong>g>tappings for sampling slurries is not a good procedure to follow however<br />
in this particular case <str<strong>on</strong>g>the</str<strong>on</strong>g> following should be noted:<br />
•<br />
•<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> slurry being tested was homogeneous;<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> point at which slurry was tapped was from an area where <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry had been well<br />
mixed;<br />
Slatter (1994) compared this method against samples taken from o<str<strong>on</strong>g>the</str<strong>on</strong>g>r points in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
pipe system and hopper and found no discrepancies.
Chapter 3 Experimental Work Page 3.19<br />
3.5.2 Solids Relative Density<br />
The relative density <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> solids (S,) is determined using test method 6B for fine grained<br />
soils from BS 1377 (1975).<br />
3.5.3 Slurry Temperature<br />
A mercury <str<strong>on</strong>g>the</str<strong>on</strong>g>rmometer was used to measure <str<strong>on</strong>g>the</str<strong>on</strong>g> temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry in <str<strong>on</strong>g>the</str<strong>on</strong>g> hopper.<br />
The temperature very rarely went above 20°C, with a temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> between 19°C and<br />
20°C being <str<strong>on</strong>g>the</str<strong>on</strong>g> norm. A rise <str<strong>on</strong>g>of</str<strong>on</strong>g>.approximately 2°C is usually experienced during a test run.<br />
If a temperature rise exceeding 2°C is experienced, data at <str<strong>on</strong>g>the</str<strong>on</strong>g> extreme temperatures is<br />
compared to detect for any temperature <str<strong>on</strong>g>effect</str<strong>on</strong>g>s (Slatter, 1994).<br />
3.5.4. Particle Size Distributi<strong>on</strong><br />
The ASTM (American Standard Testing Method), which is recognized worldwide, was used<br />
to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s and were c<strong>on</strong>ducted at Gibbs Africa laboratories.<br />
In this procedure a sample <str<strong>on</strong>g>of</str<strong>on</strong>g> slurry is thoroughly dried and <str<strong>on</strong>g>the</str<strong>on</strong>g> courser <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are screened<br />
to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>. The <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> finer <str<strong>on</strong>g>particle</str<strong>on</strong>g>s is determined using a<br />
hydrometer. The <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin clay test sets however, were<br />
determined using <str<strong>on</strong>g>the</str<strong>on</strong>g> Malvern 2600/3600 Particle Sizer VF.6 instrument, which if used<br />
correctly produces more accurate results for finer material (Roberts<strong>on</strong>, 1996). Ideally it<br />
would have been preferable to use <strong>on</strong>ly <strong>on</strong>e PSD method however, <str<strong>on</strong>g>the</str<strong>on</strong>g> testing method <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Malvern instrument was not suitable for determining <str<strong>on</strong>g>the</str<strong>on</strong>g> courser <str<strong>on</strong>g>particle</str<strong>on</strong>g>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry and<br />
hence it was decided to use <str<strong>on</strong>g>the</str<strong>on</strong>g> ASTM method for Mixture 1 and Mixture 2 and <str<strong>on</strong>g>the</str<strong>on</strong>g> Malvern<br />
Instrument for <str<strong>on</strong>g>the</str<strong>on</strong>g> finer kaolin clay <str<strong>on</strong>g>particle</str<strong>on</strong>g>s. The <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s are presented<br />
in Appendix A.<br />
Any comparis<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s should be undertaken with cauti<strong>on</strong> as those<br />
prodUced by methods o<str<strong>on</strong>g>the</str<strong>on</strong>g>r than <str<strong>on</strong>g>the</str<strong>on</strong>g> ASTM or <str<strong>on</strong>g>the</str<strong>on</strong>g> Malvern 2600/3600 Particle Sizer VF.6<br />
Instrument may not necessarily agree. An example <str<strong>on</strong>g>of</str<strong>on</strong>g> this can be seen in Figure 3.15 where<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> same slurry (mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin, rock flour and sand) used for <str<strong>on</strong>g>the</str<strong>on</strong>g> ASTM, was used <strong>on</strong><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> Malvern instrument for <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> determjnati<strong>on</strong>s. As can be seen from
Chapter 3 Experimental Work Page 3.22<br />
3.6.2 Rock Flour<br />
Rock Flour was obtained from Hippo Quarries from <str<strong>on</strong>g>the</str<strong>on</strong>g>ir quarry called Peninsula.<br />
3.6.3 Sand<br />
Sand was obtained from C<strong>on</strong>sol in 2Skg bags called No. 2 Foundry Sand. The <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> sand was between 7S and 3S0 /Lm.<br />
3.7 ]',llXTURES<br />
3.7.1 Mixture Kaolin Clay and Rock Flour<br />
The rock flour was found to form a settling slurry when mixed with water. In order to<br />
obtain a homogeneous slurry kaolin clay was used as a suspending agent for <str<strong>on</strong>g>the</str<strong>on</strong>g> rock flour.<br />
Tests were <str<strong>on</strong>g>the</str<strong>on</strong>g>refore c<strong>on</strong>ducted to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> amount <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay needed to suspend <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
rock flour.<br />
Twenty sample bottles each c<strong>on</strong>taining 200g <str<strong>on</strong>g>of</str<strong>on</strong>g> water were used for <str<strong>on</strong>g>the</str<strong>on</strong>g> tests. For Test A,<br />
10 <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> sample bottles c<strong>on</strong>tained rock flour at a relative density <str<strong>on</strong>g>of</str<strong>on</strong>g> 1,1 and for Test B, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
remaining ten sample bottles c<strong>on</strong>tained rockflour at a relative density <str<strong>on</strong>g>of</str<strong>on</strong>g> 1,2.<br />
Kaolin clay was <str<strong>on</strong>g>the</str<strong>on</strong>g>n added to <str<strong>on</strong>g>the</str<strong>on</strong>g> sample bottles in Test A, starting at 5g <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay for<br />
L1e first sample bottle and increasing in Sg amounts per sample bottle until <str<strong>on</strong>g>the</str<strong>on</strong>g> last sample<br />
bottle c<strong>on</strong>tained SOg <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay. The rock flour and kaolin clay were <str<strong>on</strong>g>the</str<strong>on</strong>g>n thoroughly<br />
mixed and allowed to stand. This procedure was repeated for Test B.<br />
It Was found that in both Test A and Test B that 30g <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay was required to suspend<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> rock flour and form a homogeneous slurry. At 25g <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay added, <str<strong>on</strong>g>the</str<strong>on</strong>g> mixture<br />
formed a slow settling slurry and hence <str<strong>on</strong>g>the</str<strong>on</strong>g> critical point lay between <str<strong>on</strong>g>the</str<strong>on</strong>g>se two limits ie. 2Sg<br />
and 30g <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin. It was <str<strong>on</strong>g>the</str<strong>on</strong>g>refore decided to use <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>centrati<strong>on</strong> ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> 30g <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin<br />
clay per 200g water to suspend <str<strong>on</strong>g>the</str<strong>on</strong>g> rock flour for <str<strong>on</strong>g>the</str<strong>on</strong>g> tests in <str<strong>on</strong>g>the</str<strong>on</strong>g> pipelines or a kaolin<br />
c<strong>on</strong>Centrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> at least Cv =S%.
Chapter 3 Experimental Work Page 3.24<br />
is allowed to flow through <str<strong>on</strong>g>the</str<strong>on</strong>g> system for sufficient time so as to allow air in<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> system to escape.<br />
3. The DPT is calibrated and slurry relative density tests performed.<br />
4. The data logging programme is loaded and initialised.<br />
5. The desired flow rate for a test is achieved by c<strong>on</strong>trolling <str<strong>on</strong>g>the</str<strong>on</strong>g> pump speed.<br />
The data logging routine is started and <str<strong>on</strong>g>the</str<strong>on</strong>g> magnetic flow meter and DPT<br />
outputs logged for three to five minutes (Sive, 1988). Check for drift.<br />
The calibrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> magnetic flow meter is performed during a test run using timed weigh<br />
test samples at different time intervals.<br />
A test is completed by repeating <str<strong>on</strong>g>the</str<strong>on</strong>g> run at different pump speeds and <str<strong>on</strong>g>the</str<strong>on</strong>g> data collected is<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>n processed.<br />
When a test is performed <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> Mini Rig, slurry c<strong>on</strong>tinuously empties into <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank<br />
and timed weigh test samples are taken at intervals during <str<strong>on</strong>g>the</str<strong>on</strong>g> test for <str<strong>on</strong>g>the</str<strong>on</strong>g> calibrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
magnetic flow meter.<br />
After a test set has been completed a slurry relative density test is performed again to ensure<br />
that <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry density has remained c<strong>on</strong>stant.<br />
3.9 CONCLUSIONS<br />
The test apparatus used for <str<strong>on</strong>g>the</str<strong>on</strong>g> experimental investigati<strong>on</strong> has been fully described and <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
calibrati<strong>on</strong> and test procedures to enable <str<strong>on</strong>g>the</str<strong>on</strong>g> collecti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> accurate pipeline data have been<br />
presented.<br />
The test results <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian slurries that were tested using <str<strong>on</strong>g>the</str<strong>on</strong>g> apparatus and testing<br />
Jrocedures are presented in Appendix A. The results are reviewed in Chapter 4 and<br />
Jiscussi<strong>on</strong> arising from <str<strong>on</strong>g>the</str<strong>on</strong>g> results is presented in Chapter 5.
CHAPTER 4
Chapter 4 Results and Analysis Page 4.2<br />
Mo<strong>on</strong>ey (1931) (Secti<strong>on</strong> 2.6.2 - Figure 2.5) in that <str<strong>on</strong>g>the</str<strong>on</strong>g> relati<strong>on</strong>ship between <str<strong>on</strong>g>the</str<strong>on</strong>g> shear stress<br />
and shear rate is independent <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter in larninar flow. This was true for all tests<br />
c<strong>on</strong>ducted, where pipediameter had no influence <strong>on</strong> wall shear stress at a given pseudo-shear<br />
rate in <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar regime. This indicates that <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry properties were time independent.<br />
In larninar flow, <strong>on</strong> physical observati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries in <str<strong>on</strong>g>the</str<strong>on</strong>g> transparent secti<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
pipeline, <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry <str<strong>on</strong>g>particle</str<strong>on</strong>g>s near <str<strong>on</strong>g>the</str<strong>on</strong>g> tube wall could be seen to be travelling in straight lines.<br />
As <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity increased <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s took <strong>on</strong> a more random, swirling or eddy moti<strong>on</strong> and<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> from larninar to <strong>turbulent</strong> flow could clearly be seen. The differential<br />
transducer output reading showe,d more variati<strong>on</strong> in <str<strong>on</strong>g>the</str<strong>on</strong>g> transiti<strong>on</strong> regi<strong>on</strong> and c<strong>on</strong>fIrms <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
physical observati<strong>on</strong>. This evidence also supports <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that true turbulence is occurring.<br />
Settling <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s was not observed at any <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>centrati<strong>on</strong>s tested. The intersecti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar and <strong>turbulent</strong> data can be taken as <str<strong>on</strong>g>the</str<strong>on</strong>g> critical point at which turbulence begins<br />
and hence <str<strong>on</strong>g>the</str<strong>on</strong>g> critical velocity can be determined.<br />
The 25mm pipeline <strong>on</strong>ly shows <str<strong>on</strong>g>the</str<strong>on</strong>g> first few data points for <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>set <str<strong>on</strong>g>of</str<strong>on</strong>g> turbulence. This<br />
is due to <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <str<strong>on</strong>g>the</str<strong>on</strong>g> back pressure <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> l50mm pipeline was used as <str<strong>on</strong>g>the</str<strong>on</strong>g> driving force<br />
for <str<strong>on</strong>g>the</str<strong>on</strong>g> 25mm pipe. Hence, <str<strong>on</strong>g>the</str<strong>on</strong>g>re was insufficient driving force to enable <str<strong>on</strong>g>the</str<strong>on</strong>g> recording <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> full <strong>turbulent</strong> range. Low velocities were also recorded in <str<strong>on</strong>g>the</str<strong>on</strong>g> 200mm pipeline due to<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> set-up <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipeline system. In a looped pipeline system <strong>on</strong>e is normally limited to two<br />
pipeline diameters due to <str<strong>on</strong>g>the</str<strong>on</strong>g> available head-capacity. Using <str<strong>on</strong>g>the</str<strong>on</strong>g> looped pipeline system at<br />
UCT, testing was c<strong>on</strong>ducted using as wide a range <str<strong>on</strong>g>of</str<strong>on</strong>g> diameters as was possible <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>on</strong>e<br />
system, but in doing so low velocities in <str<strong>on</strong>g>the</str<strong>on</strong>g> 200mm pipeline were inevitable.<br />
4.2.1 Pipe Roughness<br />
There is quite a marked difference in <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe roughness for <str<strong>on</strong>g>the</str<strong>on</strong>g> three smaller pipelines as<br />
oPposed to <str<strong>on</strong>g>the</str<strong>on</strong>g> 200mm pipeline (Table 3.II). The smaller pipelines (ie. 25mm, 80mm &<br />
lS0mm pipelines) are PVC tubing whereas <str<strong>on</strong>g>the</str<strong>on</strong>g> 200mm pipeline is c<strong>on</strong>structed <str<strong>on</strong>g>of</str<strong>on</strong>g> steel. The<br />
inside <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> 200mm pipe was corroded thus accounting for <str<strong>on</strong>g>the</str<strong>on</strong>g> high pipe roughness value.
Chapter 4 Results and Analysis Page 4.12<br />
25rnm pipeline (as outlined in Chapter 2) and are presented in Table 4.II toge<str<strong>on</strong>g>the</str<strong>on</strong>g>r with <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
pseudo-shear diagram <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> test sets for kaolin clay, mixture 1 and mixture<br />
2 in Figure 4.13 to Figure 4.15.<br />
The rheological parameters obtained were used to analyze <str<strong>on</strong>g>the</str<strong>on</strong>g> test data using <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical<br />
models menti<strong>on</strong>ed in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature review.<br />
* - Sive rheoloo ..y for kaolin <strong>on</strong>l<br />
Table 4.II: Summary <str<strong>on</strong>g>of</str<strong>on</strong>g> Slurry Properties<br />
No Test Set Slurry c,,(%) TycPa) K(Pa.s O ) n S,<br />
I Sivel * Kaolin/Quartz 7,L 4,89 0,2991 0,4840 2,22<br />
3K 10 Kaolin 7,00 9,14 0,0676 0,645E 2,60<br />
4K 20 Kaolin 6,16 5,80 0,0176 0,8154 2,60<br />
5RF 10 Kaolin/Rock Flour 9,6, 3,68 0,0132 0,9474 2,60<br />
6RF_20 Kaolin/Rock Flour 11,00 3,91 0,0105 0,9720 2,60<br />
7RF_30 Kaolin/Rock Flour 13,29 5,53 0,0194 0,964E 2,60<br />
8S 10 Kaolin/Rock Flour/Sand 16,4, 5,82 0,1413 0,557, 2,65<br />
9S 20 Kaolin/Rock Flour/Sand 19,43 5,48 0,1239 0,6363 2,65<br />
10 S_30 Kaolin/Rock Flour/Sand 23,86 8,02 0,1350 0,5911 2,65<br />
4.4 VISCOUS SUB-LAYER<br />
y<br />
The viscous sub-layer thickness can be predicted using <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian approximati<strong>on</strong> and <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Wils<strong>on</strong> & Thomas (1985,1987) and Slatter (1994) models. Figure 4.16 to Figure 4.18 show<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> relati<strong>on</strong>ship between wall shear stress and viscous sub-layer thickness for <str<strong>on</strong>g>the</str<strong>on</strong>g> first test<br />
sets <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay, mixture I and mixture 2 respectively. The Maude & Whitmore (1958)<br />
predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness has been included in <str<strong>on</strong>g>the</str<strong>on</strong>g> figures. Figure 4.18<br />
shows that at <str<strong>on</strong>g>the</str<strong>on</strong>g> higher wall shear stress values, <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness is less than<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> diameter <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s. As discussed in Chapter 2, <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s must <str<strong>on</strong>g>the</str<strong>on</strong>g>refore<br />
have an obstructing <str<strong>on</strong>g>effect</str<strong>on</strong>g> <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thus influencing <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow<br />
behaViour.
Chapter 4 Results and Analysis Page 4.21<br />
Overall <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model (APE - 13.9%, LSE - 0.0325) best modelled <str<strong>on</strong>g>the</str<strong>on</strong>g> test data.<br />
Although <str<strong>on</strong>g>the</str<strong>on</strong>g> Torrance model (19.15%) and <str<strong>on</strong>g>the</str<strong>on</strong>g> Wils<strong>on</strong> & Thomas model (19.66%) are both<br />
below an average percentage error <str<strong>on</strong>g>of</str<strong>on</strong>g> 20 %, which is acceptable in engineering practice<br />
(Cheng, 1970), <str<strong>on</strong>g>the</str<strong>on</strong>g> fact still remains that <str<strong>on</strong>g>the</str<strong>on</strong>g>se models diverge from <str<strong>on</strong>g>the</str<strong>on</strong>g> data at higher shear<br />
stresses, indicating that <str<strong>on</strong>g>the</str<strong>on</strong>g>y do not accurately describe <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
solid-liquid suspensi<strong>on</strong>s.<br />
The <strong>turbulent</strong> model performance can also be seen in Figure 4.25 to Figure 4.29 which<br />
shows a log-log plot <str<strong>on</strong>g>of</str<strong>on</strong>g> TO ob' vs To'""" for <str<strong>on</strong>g>the</str<strong>on</strong>g> five <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models which were c<strong>on</strong>sidered.<br />
A 20 % error line is included in <str<strong>on</strong>g>the</str<strong>on</strong>g> plot. It can be seen from <str<strong>on</strong>g>the</str<strong>on</strong>g> five figures that <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter<br />
model yields <str<strong>on</strong>g>the</str<strong>on</strong>g> best results with virtually all <str<strong>on</strong>g>the</str<strong>on</strong>g> data points within <str<strong>on</strong>g>the</str<strong>on</strong>g> 20% error lines.<br />
Large discrepancies for mixture 1 for all models can be seen.<br />
From <str<strong>on</strong>g>the</str<strong>on</strong>g> performance <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> various <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models (visual appraisal <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> model lines,<br />
average percentage error, log standard error and <str<strong>on</strong>g>the</str<strong>on</strong>g> log-log plots) it would appear that <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Slatter model provides better predicti<strong>on</strong>s than <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r models for <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tested.<br />
1000,.---------------:'71,-----,<br />
o<br />
Kaolin<br />
..<br />
Ntxtu.-. 1<br />
+<br />
Nbcv. 2<br />
Figure 4.25: A log-log plot <str<strong>on</strong>g>of</str<strong>on</strong>g> TO"", vs To""" for <str<strong>on</strong>g>the</str<strong>on</strong>g> Dodge & Metzner<br />
model
Chapter 4 Results and Analysis Page 4.25<br />
5<br />
4.5<br />
4<br />
o 3.5<br />
t-<br />
O><br />
.2 3<br />
2.5<br />
2<br />
ITurbulenl Br<strong>on</strong>ches I<br />
>lE<br />
..<br />
D=207mm<br />
D<br />
D=141mm<br />
....<br />
0= BOmm<br />
-+-<br />
D= 22mm<br />
1.5 +---r--,---c----,---r----,.-----r--,---j<br />
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5<br />
log BY/0<br />
Figure 4.31: Turbulent flow predicti<strong>on</strong>s for <str<strong>on</strong>g>the</str<strong>on</strong>g> Bowen correlati<strong>on</strong> for test<br />
set K 20<br />
The Bowen c<strong>on</strong>stants for test set K_20, as shown in Figure 4.31, are k=lO.13 and b=0.22.<br />
4.8 DATA fROM THE LITERATURE<br />
Experimental data obtained by Sive (1988) were used in <str<strong>on</strong>g>the</str<strong>on</strong>g> analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> various models<br />
under c<strong>on</strong>siderati<strong>on</strong>. The tests c<strong>on</strong>ducted by Sive (1988) were d<strong>on</strong>e using a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g>kaolin<br />
clay and a relatively coarse quartz sand which resulted in a n<strong>on</strong>-homogeneous settling slurry<br />
with a high representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>. The purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> using this data was to see if <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
coarse, settling <str<strong>on</strong>g>particle</str<strong>on</strong>g>s c<strong>on</strong>tributed to <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow headloss, as proposed by <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter<br />
modeL<br />
As can be seen from a typical result in Figure 4.32 taken from Sive (1988), Slatter's model<br />
Was unable to predict <str<strong>on</strong>g>the</str<strong>on</strong>g> test data <str<strong>on</strong>g>of</str<strong>on</strong>g> Sive.
Chapter 4 Results and Analysis Page 4.27<br />
• At <str<strong>on</strong>g>the</str<strong>on</strong>g> higher wall shear stress values for mixture 2 <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness<br />
is less than <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s.<br />
• An increase in <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> caused an increase in <str<strong>on</strong>g>the</str<strong>on</strong>g> wall shear<br />
stress in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow regi<strong>on</strong>.<br />
• The Slatter model best predicted <str<strong>on</strong>g>the</str<strong>on</strong>g> test data and becomes more accurate with<br />
increasing <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>.<br />
• The Maude & Whitmore correlati<strong>on</strong> is unable to accurately predict <strong>turbulent</strong> flow<br />
behaviour.<br />
• The Bowen correlati<strong>on</strong> produces good results but does not provide an explanati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry in terms <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> physical properties <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry.<br />
• Slarter's model is unable to predict <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Sive data for<br />
a n<strong>on</strong>-homogeneous settling slurry.<br />
The results and analysis are discussed in Chapter 5.
CHAPTERS
5.1 INTRODUCTION<br />
CHAPTERS<br />
DISCUSSION<br />
This chapter deals with discussing <str<strong>on</strong>g>the</str<strong>on</strong>g> results and analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> test data as presented in<br />
Chapter 4 and Appendix A.<br />
5.2 PARTICLE ROUGHNESS EFFECT<br />
In Figure 4.2 it is noted that <str<strong>on</strong>g>the</str<strong>on</strong>g> data points for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin tend to lie <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> line for <str<strong>on</strong>g>the</str<strong>on</strong>g> law<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall for smooth pipes, with <str<strong>on</strong>g>the</str<strong>on</strong>g> rock flour data points lying to <str<strong>on</strong>g>the</str<strong>on</strong>g> left. The data<br />
points for <str<strong>on</strong>g>the</str<strong>on</strong>g> sand tend to lie <strong>on</strong> or near <str<strong>on</strong>g>the</str<strong>on</strong>g> curve <str<strong>on</strong>g>of</str<strong>on</strong>g> Nikuradse. This is not in line with<br />
what Slatter's model anticipates. The data points according to <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model should lie<br />
<strong>on</strong> or near <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal asymptote (roughness functi<strong>on</strong> B=8,5) since <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
suSpensi<strong>on</strong>s that were tested were nei<str<strong>on</strong>g>the</str<strong>on</strong>g>r fixed nor uniform in <str<strong>on</strong>g>size</str<strong>on</strong>g>, as <str<strong>on</strong>g>the</str<strong>on</strong>g>y were for<br />
Nikuradse's experiments. And yet it can be seen from Figure 4.2 that <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s for mixture 2 is as great and even greater than for Nikuradse's<br />
experimental data.<br />
For smooth wall <strong>turbulent</strong> flow <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model predicts that <str<strong>on</strong>g>the</str<strong>on</strong>g> data should lie <strong>on</strong> or near<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> oblique asymptote (line for <str<strong>on</strong>g>the</str<strong>on</strong>g> law <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall for smooth pipes) and for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin clay<br />
test sets this is certainly <str<strong>on</strong>g>the</str<strong>on</strong>g> case but yet <str<strong>on</strong>g>the</str<strong>on</strong>g> data points for mixture I lie to <str<strong>on</strong>g>the</str<strong>on</strong>g> left. In fact<br />
most <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> data points lie well outside <str<strong>on</strong>g>the</str<strong>on</strong>g> two asymptotes which describe <str<strong>on</strong>g>the</str<strong>on</strong>g> limits <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
behaViour <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>ian <strong>turbulent</strong> flow. There are many reported instances <str<strong>on</strong>g>of</str<strong>on</strong>g> similarities<br />
between <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>ian and n<strong>on</strong>-Newt<strong>on</strong>ian fluids (secti<strong>on</strong> 2.12) and<br />
Slatter'S findings c<strong>on</strong>firmed this trend. The data <str<strong>on</strong>g>of</str<strong>on</strong>g> Slatter (1994) matched <str<strong>on</strong>g>the</str<strong>on</strong>g> limits <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g>Newt<strong>on</strong>ian <strong>turbulent</strong> flow closely and hence for his correlati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness<br />
functi<strong>on</strong> for his model <str<strong>on</strong>g>the</str<strong>on</strong>g> two asymptotes were chosen. However, <str<strong>on</strong>g>the</str<strong>on</strong>g>se test data points<br />
:<strong>on</strong>tradict <str<strong>on</strong>g>the</str<strong>on</strong>g> findings <str<strong>on</strong>g>of</str<strong>on</strong>g> Slatter and would indicate that d 85 <str<strong>on</strong>g>size</str<strong>on</strong>g> does not give <str<strong>on</strong>g>the</str<strong>on</strong>g> best<br />
representati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>.
Chapter 5 Discussi<strong>on</strong> Page 5.2<br />
Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r investigati<strong>on</strong> needs to be c<strong>on</strong>ducted and a large data base established to determine<br />
whe<str<strong>on</strong>g>the</str<strong>on</strong>g>r <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> will:<br />
• follow <str<strong>on</strong>g>the</str<strong>on</strong>g> Nikuradse trend;<br />
• c<strong>on</strong>tinue increasing, following more closely <str<strong>on</strong>g>the</str<strong>on</strong>g> oblique asymptote;<br />
• or if it does follow <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal asymptote as proposed by Slatter.<br />
5.3 ROUGHNESS FUNCTION CORRELATION USING OPTIMUM PARTICLE<br />
SIZES<br />
If, however, <str<strong>on</strong>g>the</str<strong>on</strong>g> optimum <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>s are used to plot <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness functi<strong>on</strong> correlati<strong>on</strong><br />
as shown in Figure 4.11 <str<strong>on</strong>g>the</str<strong>on</strong>g> data points for mixture 2 tend to lie just above <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal<br />
asymptote. On <str<strong>on</strong>g>the</str<strong>on</strong>g> whole <str<strong>on</strong>g>the</str<strong>on</strong>g> data points tend to follow more closely <str<strong>on</strong>g>the</str<strong>on</strong>g> assumpti<strong>on</strong>s <strong>on</strong><br />
which <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model is based. What this does bring into questi<strong>on</strong> is that <str<strong>on</strong>g>the</str<strong>on</strong>g> d S5 <str<strong>on</strong>g>size</str<strong>on</strong>g> does<br />
not necessarily give <str<strong>on</strong>g>the</str<strong>on</strong>g> best representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>. These optimum <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>s<br />
indicate that <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> will vary depending <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry<br />
being transported. In fact it would seem from Table 4.1, looking at <str<strong>on</strong>g>the</str<strong>on</strong>g> optimum <str<strong>on</strong>g>size</str<strong>on</strong>g>s for<br />
kaolin clay and mixture 2, that a slurry with a steep PSD for <str<strong>on</strong>g>the</str<strong>on</strong>g> smaller <str<strong>on</strong>g>particle</str<strong>on</strong>g>s (eg. kaolin<br />
clay test sets) would have a higher representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> than a slurry which has a PSD<br />
shape that is less steep from small to large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s. In Table 4.1 <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin clay test sets<br />
have an average <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> d x =d 91 as opposed to mixture 2 tests sets which have an<br />
average representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> d. =d 71 • Therefore <str<strong>on</strong>g>the</str<strong>on</strong>g> higher fracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> larger<br />
Particles present in <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry <str<strong>on</strong>g>the</str<strong>on</strong>g> lower <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>, as <str<strong>on</strong>g>the</str<strong>on</strong>g> best<br />
representati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> is reached at a lower d x value.<br />
5.4 SLURRY TEMPERATURE<br />
As reported in secti<strong>on</strong> 4.2.7 an increase in temperature <str<strong>on</strong>g>of</str<strong>on</strong>g> between goC to 10°C made no<br />
significant difference and no temperature <str<strong>on</strong>g>effect</str<strong>on</strong>g>s were found. It is surprising to see this trend<br />
since this would cause a decrease in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscosity for water <str<strong>on</strong>g>of</str<strong>on</strong>g> approximately 20%. This<br />
Could influence <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry and might lead <strong>on</strong>e to questi<strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> method <str<strong>on</strong>g>of</str<strong>on</strong>g>
Chapter 5 Discussi<strong>on</strong> Page 5.4<br />
Whatever <str<strong>on</strong>g>the</str<strong>on</strong>g> point is at which <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer breaks down <str<strong>on</strong>g>the</str<strong>on</strong>g> fact still remains that<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer <str<strong>on</strong>g>of</str<strong>on</strong>g> a homogeneous slurry cannot exist if it is smaller than <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s<br />
which comprise <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry. As can be seen from Figure 4.18 (mixture 2) <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub<br />
layer thickness is smaller than <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s which comprise <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry at higher waIl shear<br />
stresses. The thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer for kaolin clay (Figure 4.16) is smaller than<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s which comprise <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry at higher shear stresses but a much smaller fracti<strong>on</strong><br />
than mixture 2. At <str<strong>on</strong>g>the</str<strong>on</strong>g>se higher wall shear stress values <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model best models <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
test data for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin clay and mixture 2 test sets. This could be ascribed to <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that<br />
Slatter's model accounts for <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s affecting <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer.<br />
To c<strong>on</strong>firm that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s will affect <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer and are relevant for <strong>turbulent</strong><br />
flow analysis <str<strong>on</strong>g>the</str<strong>on</strong>g> number c<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer was<br />
calculated. In order to calculate <str<strong>on</strong>g>the</str<strong>on</strong>g> number c<strong>on</strong>centrati<strong>on</strong> a few assumpti<strong>on</strong>s had to made.<br />
The assumpti<strong>on</strong>s were as follows:<br />
• <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s comprising <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry are spherical in shape;<br />
• <str<strong>on</strong>g>the</str<strong>on</strong>g> large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s were taken to be greater than <strong>on</strong>e half <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
viscous sub-layer in <strong>turbulent</strong> flow;<br />
• <strong>on</strong>e metre <str<strong>on</strong>g>of</str<strong>on</strong>g> pipe tubing was c<strong>on</strong>sidered for calculati<strong>on</strong>s;<br />
• an average radius is assumed for <str<strong>on</strong>g>the</str<strong>on</strong>g>" large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s.<br />
From Figure 4.16 to Figure 4.18 we can determine <str<strong>on</strong>g>the</str<strong>on</strong>g> thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer in<br />
<strong>turbulent</strong> flow. The thickness <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer for kaolin clay, mixture 1 and mixture<br />
2 is approximately 40JLm, 300JLm and 55JLm respectively. The percentage <str<strong>on</strong>g>of</str<strong>on</strong>g> large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s<br />
present in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer can <str<strong>on</strong>g>the</str<strong>on</strong>g>refore be determined from <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD (ie. percentage<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> Particles greater than <strong>on</strong>e half <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness). From <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter and<br />
Viscous SUb-layer thickness <str<strong>on</strong>g>the</str<strong>on</strong>g> volume <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer can be determined and<br />
knOWing <str<strong>on</strong>g>the</str<strong>on</strong>g> percentage <str<strong>on</strong>g>of</str<strong>on</strong>g> large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s <str<strong>on</strong>g>the</str<strong>on</strong>g> volume taken up by <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />
Sub-layer can be calculated. The mass <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s can be obtained from <str<strong>on</strong>g>the</str<strong>on</strong>g> volume and<br />
jensityand similarly by c<strong>on</strong>sidering <strong>on</strong>e <str<strong>on</strong>g>particle</str<strong>on</strong>g> present in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer, it's mass<br />
:an be calculated and hence <str<strong>on</strong>g>the</str<strong>on</strong>g> number <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s. The results obtained are shown in
Chapter 5<br />
Table 5.1.<br />
Discussi<strong>on</strong> Page 5.5<br />
Table 5.1: Number C<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Large Particles in Viscous Sub-layer<br />
Test % Large No. Particles<br />
Particles per m 2<br />
KW 22 163<br />
K 20 20 296<br />
.<br />
RFlO 4 24<br />
RF 20 5 36<br />
RF 30 6 57<br />
S 10 47 336<br />
S 20 55 463<br />
S 30 62 540<br />
From Table 5.1 it can be seen that <str<strong>on</strong>g>the</str<strong>on</strong>g> highest number <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>particle</str<strong>on</strong>g>s per square meter was 540.<br />
This indicates that if <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s were evenly spaced that approximately <strong>on</strong>e large <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
would be found every 5cm. This is fairly low for <strong>on</strong>e would expect for a <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness<br />
<str<strong>on</strong>g>effect</str<strong>on</strong>g> for <str<strong>on</strong>g>the</str<strong>on</strong>g>re to be at least <strong>on</strong>e large <str<strong>on</strong>g>particle</str<strong>on</strong>g> per millimetre (Slatter, 1996) and yet even<br />
with a low number c<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g>re is evidence <str<strong>on</strong>g>of</str<strong>on</strong>g> a <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>. In fact,<br />
Slatter (1994) detected a <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> even when testing slurries with a<br />
volumetric c<strong>on</strong>centrati<strong>on</strong> as low as 2 %. It would <str<strong>on</strong>g>the</str<strong>on</strong>g>refore appear from tests c<strong>on</strong>ducted by<br />
Slatter (1994) and test c<strong>on</strong>ducted for this <str<strong>on</strong>g>the</str<strong>on</strong>g>sis tllat <str<strong>on</strong>g>the</str<strong>on</strong>g> large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s have a dominant <str<strong>on</strong>g>effect</str<strong>on</strong>g><br />
<strong>on</strong> tUrbulence even when <str<strong>on</strong>g>the</str<strong>on</strong>g> number c<strong>on</strong>centrati<strong>on</strong> is extremely. This would indicate that<br />
:he Particle roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> is due to <str<strong>on</strong>g>the</str<strong>on</strong>g> mere fact that <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are present in <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry per<br />
,e but it is strange that a small number produce <str<strong>on</strong>g>the</str<strong>on</strong>g> same <str<strong>on</strong>g>effect</str<strong>on</strong>g> as a large number.
Chapter 5 Discussi<strong>on</strong> Page 5.6<br />
The number <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>particle</str<strong>on</strong>g>s per square meter for <str<strong>on</strong>g>the</str<strong>on</strong>g> mixture I test sets were c<strong>on</strong>siderably lower<br />
than <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r test sets. This could be due to <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
mixture I is relatively large compared to <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin and mixture 2 viscous sub-layer thickness<br />
and hence <str<strong>on</strong>g>the</str<strong>on</strong>g> percentage <str<strong>on</strong>g>of</str<strong>on</strong>g> large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s present (due to <str<strong>on</strong>g>the</str<strong>on</strong>g> definiti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a large <str<strong>on</strong>g>particle</str<strong>on</strong>g>)<br />
is low. It is interesting to note that number <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>particle</str<strong>on</strong>g>s per meter square <str<strong>on</strong>g>of</str<strong>on</strong>g> viscous sub-layer<br />
for <str<strong>on</strong>g>the</str<strong>on</strong>g> 25mm, 80mm, 150mm and 200mm pipelines is <str<strong>on</strong>g>the</str<strong>on</strong>g> same per test set due to <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
number c<strong>on</strong>centrati<strong>on</strong> being independent <str<strong>on</strong>g>of</str<strong>on</strong>g> diameter.<br />
What would streng<str<strong>on</strong>g>the</str<strong>on</strong>g>n <str<strong>on</strong>g>the</str<strong>on</strong>g> case bf <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness turbulence is if <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow<br />
headloss in a vertical test secti<strong>on</strong> was found to be <str<strong>on</strong>g>the</str<strong>on</strong>g> same as <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal. There is<br />
evidence to suggest (eg. Maude & Whitmore, 1956, Wils<strong>on</strong>, 1996) that <str<strong>on</strong>g>particle</str<strong>on</strong>g>s have a<br />
tendency in vertical pipes to move inward away from <str<strong>on</strong>g>the</str<strong>on</strong>g> wall (ie. away from <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />
sub-layer). If this is indeed <str<strong>on</strong>g>the</str<strong>on</strong>g> case <str<strong>on</strong>g>the</str<strong>on</strong>g>n it could affect <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g>iveness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>. Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r test should be c<strong>on</strong>ducted to c<strong>on</strong>firm if <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal headloss and<br />
vertical headloss are in agreement.<br />
5.6 INFLUE."iCE OF PARTICLE SIZE<br />
In Chapter 4, it was stated that an increase in wall shear stress with an increase in<br />
c<strong>on</strong>centrati<strong>on</strong> for <str<strong>on</strong>g>the</str<strong>on</strong>g> 150mm pipeline tests" for mixture 2 was due to an increase in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>. The d S5 <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>s for <str<strong>on</strong>g>the</str<strong>on</strong>g> three 150mm pipeline tests were<br />
l37/tm, 158j.Lm and 170j.Lm and <str<strong>on</strong>g>the</str<strong>on</strong>g> increase in wall shear stress can clearly be seen in Figure<br />
4.21. Slatter's model best predicted <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow data for mixture 2 as it is based <strong>on</strong><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> and is able to account for <str<strong>on</strong>g>the</str<strong>on</strong>g> increase in wall shear stress. The<br />
greater <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> a homogeneous slurry <str<strong>on</strong>g>the</str<strong>on</strong>g> greater will be <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Increase in <str<strong>on</strong>g>the</str<strong>on</strong>g> wall shear stress and <str<strong>on</strong>g>the</str<strong>on</strong>g> greater will be <str<strong>on</strong>g>the</str<strong>on</strong>g> possibility <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical<br />
models being unable to predict <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow data.<br />
HOwever, this phenomen<strong>on</strong> could also be ascribed to <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <str<strong>on</strong>g>the</str<strong>on</strong>g>re is an increase in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
rheology and <str<strong>on</strong>g>the</str<strong>on</strong>g> density <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry. To c<strong>on</strong>firm that <str<strong>on</strong>g>the</str<strong>on</strong>g> increase in wall shear stress is<br />
Jue to an increase in <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>, <str<strong>on</strong>g>the</str<strong>on</strong>g> sensitivity to changes in wall shear stress due to
Chapter 5 Discussi<strong>on</strong> Page 5.9<br />
5.7 THEORETICAL MODELS<br />
As stated Slatter's model was c<strong>on</strong>sistently <str<strong>on</strong>g>the</str<strong>on</strong>g> best model for <str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin test sets and for<br />
mixture 2 with increasing representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>. This is attributed to <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that his<br />
model is able to account for <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> whereas <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r models do not. Where<br />
Slarter's model fails to predict <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow data <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> mixture 1 test sets, it could be<br />
for <str<strong>on</strong>g>the</str<strong>on</strong>g> following reas<strong>on</strong>s. Firstly, it should be noted that <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer thickness is<br />
relatively large in <strong>turbulent</strong> flow (± 300JLm) as opposed to <str<strong>on</strong>g>the</str<strong>on</strong>g> possible <str<strong>on</strong>g>effect</str<strong>on</strong>g>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe<br />
and <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>s. In o<str<strong>on</strong>g>the</str<strong>on</strong>g>r words <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer is suppressing <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> and Slatter's model which is over predicting <str<strong>on</strong>g>the</str<strong>on</strong>g> shear stress values is over<br />
compensating for a roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> which is not that significant. It is interesting to note for<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> test data <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> large pipes (ie. 200mm pipes) for mixture 1, when <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g><br />
could be a factor, <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model is fairly good - an average<br />
percentage error <str<strong>on</strong>g>of</str<strong>on</strong>g>2.59%, 2,98% and 15,48% for <str<strong>on</strong>g>the</str<strong>on</strong>g> three tests. Also, from Table 5.1 it<br />
can be seen that <str<strong>on</strong>g>the</str<strong>on</strong>g>re are relatively fewer larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s present in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer<br />
as opposed to <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r test sets due to <str<strong>on</strong>g>the</str<strong>on</strong>g> thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer (see secti<strong>on</strong><br />
5.5). Ano<str<strong>on</strong>g>the</str<strong>on</strong>g>r reas<strong>on</strong> is that <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology is tending towards being a Bingham plastic (ie. <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
value for n is approaching n=l) and under <str<strong>on</strong>g>the</str<strong>on</strong>g>se c<strong>on</strong>diti<strong>on</strong>s <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model tends to over<br />
predict <str<strong>on</strong>g>the</str<strong>on</strong>g> shear stress values. At high shear rate values for a Bingham plastic as opposed<br />
to yield pseudoplastic <str<strong>on</strong>g>the</str<strong>on</strong>g> shear stress values will be greater as <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous forces<br />
predominate in this regi<strong>on</strong> for Bingham plastics.<br />
Although Slatter's model is <str<strong>on</strong>g>the</str<strong>on</strong>g> best model overall. <str<strong>on</strong>g>the</str<strong>on</strong>g> data presented <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
rOughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> (Figure 4.2) does bring into questi<strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> validity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> assumpti<strong>on</strong>s <strong>on</strong><br />
Which <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model is based. However, because Slatter's model is based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> it proves very useful when undertaking for example scale-up from a smooth<br />
Walled small bore (eg. PVC pipe) to a rough wall large-scale pipe (eg. galvanised ir<strong>on</strong>), or<br />
Vice versa. Normally, for example, if <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure drop in <strong>turbulent</strong> flow is predicted by <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
sCale-up method <str<strong>on</strong>g>of</str<strong>on</strong>g> Bowen, a corrected pressure gradient would have to be estimated for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
change in roughness. This could be d<strong>on</strong>e by using <str<strong>on</strong>g>the</str<strong>on</strong>g> Govier & Aziz (1972) approach <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
mUltiplying <str<strong>on</strong>g>the</str<strong>on</strong>g> predicted pressure gradient by a ratio <str<strong>on</strong>g>of</str<strong>on</strong>g> fricti<strong>on</strong> factors obtained from <str<strong>on</strong>g>the</str<strong>on</strong>g>
Chapter 5 Discussi<strong>on</strong> Page 5.10<br />
Moody chart. This is ano<str<strong>on</strong>g>the</str<strong>on</strong>g>r downfall <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> Bowen method. However, if using <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter<br />
method you would simply select <str<strong>on</strong>g>the</str<strong>on</strong>g> appropriate roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> for <str<strong>on</strong>g>the</str<strong>on</strong>g> Roughness Reynolds<br />
Number (ie. <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> or <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe hydraulic roughness, whichever is <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
greater). It is suggested however, that test work is first c<strong>on</strong>ducted in smooth pipes to ensure<br />
that <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness turbulence is indeed occurring. If testing takes place in a pipe with<br />
pipe hydraulic roughness higher than <str<strong>on</strong>g>the</str<strong>on</strong>g> representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness<br />
<str<strong>on</strong>g>effect</str<strong>on</strong>g> will not be detected.<br />
This investigati<strong>on</strong> does highlight <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> is valid for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
slurries tested and must be used in <strong>turbulent</strong> flow modelling. The <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g><br />
should be incorporated in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian slurries, as has been used<br />
in <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>ory for Slatter's analysis, to account for rough wall <strong>turbulent</strong> flow and to predict<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> change from smooth wall to rough wall <strong>turbulent</strong> flow.<br />
5.8 REYNOLDS NUMBER VS FRICTION FACTOR PLOTS<br />
Figure 5.3 to Figure 5.5 shows <str<strong>on</strong>g>the</str<strong>on</strong>g> plot <str<strong>on</strong>g>of</str<strong>on</strong>g> Reynolds number vs fricti<strong>on</strong> factor, also called<br />
Moody plots, for <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian approximati<strong>on</strong> Reynolds Number, Metzner & Reed Reynolds<br />
Number, and a Reynolds number published by Slatter & Lazarus (1993) <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> form<br />
Re = 8 P V 2<br />
Ty"'K D<br />
[ 8V) n<br />
(5.1)<br />
which can be derived by c<strong>on</strong>sidering <str<strong>on</strong>g>the</str<strong>on</strong>g> Newt<strong>on</strong>ian approximati<strong>on</strong> and substituting <str<strong>on</strong>g>the</str<strong>on</strong>g> bulk<br />
shear rate in place <str<strong>on</strong>g>of</str<strong>on</strong>g> true shear rate (Slatter, 1994). Included in <str<strong>on</strong>g>the</str<strong>on</strong>g> plots are <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar<br />
flow line (ie. f=16/Re) and <str<strong>on</strong>g>the</str<strong>on</strong>g> Prandtlline (Newt<strong>on</strong>ian smooth wall).<br />
The Moody chart shows that for <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> Newt<strong>on</strong>ian fluids, pipe roughness can<br />
have an appreciable <str<strong>on</strong>g>effect</str<strong>on</strong>g>. As can be seen from Figure 5.3 <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow data tends to<br />
lie in straight horiz<strong>on</strong>tal lines for <str<strong>on</strong>g>the</str<strong>on</strong>g> various slurries and does not follow <str<strong>on</strong>g>the</str<strong>on</strong>g> oblique Prandtl<br />
line as suggested by Kemblowski & Kolodziejski (1973). This indicates that f is c<strong>on</strong>stant<br />
with increasing Reynolds numbers and hence it is independent <str<strong>on</strong>g>of</str<strong>on</strong>g> rheology. This trend is
Chapter 5 Discussi<strong>on</strong> Page 5.13<br />
5.7 CONCLUSIONS<br />
Experimental work has been c<strong>on</strong>ducted using homogeneous n<strong>on</strong>-Newt<strong>on</strong>ian slurries and <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
analysis and discussi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> results has been presented. As can be seen from <str<strong>on</strong>g>the</str<strong>on</strong>g> results<br />
Slatter's model best predicts <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models which were<br />
c<strong>on</strong>sidered. From <str<strong>on</strong>g>the</str<strong>on</strong>g> evidence it can be c<strong>on</strong>cluded that <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> are justified in adopting <str<strong>on</strong>g>the</str<strong>on</strong>g> approach and in fact <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> which does exist should be c<strong>on</strong>sidered in <strong>turbulent</strong> flow analysis. There is<br />
room for improving <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model and fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r test work should be c<strong>on</strong>ducted to fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r<br />
vindicate or disprove <str<strong>on</strong>g>the</str<strong>on</strong>g> assumpti<strong>on</strong> <strong>on</strong> which <str<strong>on</strong>g>the</str<strong>on</strong>g> model is based but <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness<br />
<str<strong>on</strong>g>effect</str<strong>on</strong>g> is defmitely a viable starting point to c<strong>on</strong>sider in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> homogeneous<br />
n<strong>on</strong>-Newt<strong>on</strong>ian suspensi<strong>on</strong>s. The following c<strong>on</strong>clusi<strong>on</strong>s can be drawn from <str<strong>on</strong>g>the</str<strong>on</strong>g> discussi<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> results and analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> test data:<br />
• A <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> does exist.<br />
• Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r investigati<strong>on</strong> is required and a large data base should be established to<br />
accurately determine <str<strong>on</strong>g>the</str<strong>on</strong>g> limit <str<strong>on</strong>g>of</str<strong>on</strong>g> validity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g><br />
• The d 85 <str<strong>on</strong>g>size</str<strong>on</strong>g> does not necessarily give <str<strong>on</strong>g>the</str<strong>on</strong>g> best representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> but will depend <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> shape <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD.<br />
• The viscous sub-layer cannot exist if it is smaller than <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s which comprise<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> slurry and <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>tinuum approximati<strong>on</strong> must be compromised in <strong>turbulent</strong> flow<br />
analysis.<br />
• The viscous sub-layer breaks down for Re, > 3,32 which is at a viscous sub-layer<br />
thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> approximately" = 3*d 85 for <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter model. For <str<strong>on</strong>g>the</str<strong>on</strong>g> Maude &<br />
Whitmore (1956, 1958) correlati<strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer breaks down when <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> diameter is greater than half <str<strong>on</strong>g>the</str<strong>on</strong>g> thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer. Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r<br />
investigati<strong>on</strong> is required to accurately predict <str<strong>on</strong>g>the</str<strong>on</strong>g> exact point at which <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-
Chapter 5 Discussi<strong>on</strong> Page 5.14<br />
layer does break down.<br />
• There are fewer large <str<strong>on</strong>g>particle</str<strong>on</strong>g>s present in <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous sub-layer per square meter than<br />
would be anticipated for a <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>.<br />
• Slatter's model can be used for scale-up <str<strong>on</strong>g>of</str<strong>on</strong>g>pipes with different roughnesses without<br />
having to use an estimated pressure gradient which a downfall <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Bowen method.<br />
• The fricti<strong>on</strong> factor is c<strong>on</strong>stant with increasing Reynolds Numbers and vindicates <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness approach adopted by Slatter.<br />
• The solids Particle Size Distributi<strong>on</strong> is an important property <str<strong>on</strong>g>of</str<strong>on</strong>g>a slurry for <strong>turbulent</strong><br />
flow behaviour and should be used to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>. The<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> should be incorporated in <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong><br />
Newt<strong>on</strong>ian slurries.
CHAPTER 6
Chapter 6 Summary and C<strong>on</strong>clusi<strong>on</strong>s Page 6.4<br />
• The relevant literature <strong>on</strong> slurry flow has been reviewed, including a new approach<br />
based <strong>on</strong> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness turbulence.<br />
• An experimental investigati<strong>on</strong> covering wide ranges <str<strong>on</strong>g>of</str<strong>on</strong>g> flow c<strong>on</strong>diti<strong>on</strong>s and slurry<br />
properties has been c<strong>on</strong>ducted and <str<strong>on</strong>g>the</str<strong>on</strong>g> results analyzed.<br />
• Analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>se results shows that <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness turbulence approach<br />
adopted by Slatter (1994) bas been validated.<br />
• Recommendati<strong>on</strong>s for fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r research have been made.
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APPENDICES
APPENDIX A
Appendix A Detailed Pipe Test Results<br />
A.I DETAUED PIPE TEST RESULTS<br />
The detailed pipe test results are presented in this secti<strong>on</strong>.<br />
Each test set (ie. 25mm, 8Omm, 150mm and 2oomm) is preceded by <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g><br />
<str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> for that particular test set.<br />
A table <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> data points <str<strong>on</strong>g>of</str<strong>on</strong>g> wall shear stress and velocity precedes each pipeline test sheet.<br />
Each test sheet c<strong>on</strong>tains <str<strong>on</strong>g>the</str<strong>on</strong>g> test apparatus, material, slurry properties, <strong>turbulent</strong> model<br />
performance and <str<strong>on</strong>g>the</str<strong>on</strong>g> test data plotted <strong>on</strong> a pseudo-shear diagram.<br />
The test code (eg. SERSlO) indicates <str<strong>on</strong>g>the</str<strong>on</strong>g> following:<br />
S This describes <str<strong>on</strong>g>the</str<strong>on</strong>g> material ie. sand (mixture 2)<br />
or K=kaolin or RF=rock flour (mixture 1)<br />
ER This describes <str<strong>on</strong>g>the</str<strong>on</strong>g> apparatus ie East Rig or Mini Rig<br />
S This indicates <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe <str<strong>on</strong>g>size</str<strong>on</strong>g> = small, medium or large<br />
IO This indicates <str<strong>on</strong>g>the</str<strong>on</strong>g> test identifier<br />
A.I
Appendix A Detailed Pipe Test Results<br />
TEST: KERSIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRITICAL VELOCITY: 2.32 m/s<br />
EJ Wall Shear Stress<br />
[pa]<br />
Velocity<br />
fm/s]<br />
1. 68.82 5.55<br />
2. 53.50 4.88<br />
3. 42.89 4.34<br />
4. 34.48 3.84<br />
5. 27.30 3.36<br />
6. 20.09 2.84<br />
7. 15.64 2.34<br />
8. 13.01 1.87<br />
9. 12.03 1.52<br />
10. 11.38 1.04<br />
11. 10.99 0.80<br />
12. 10.56 0.54<br />
A.5
Appendix A Detailed Pipe Test Results<br />
TEST: KERMIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRITICAL VELOCITY: 2.94 m/s<br />
EJI Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[m/s]<br />
1. 76.46 6.20<br />
2. 65.48 5.91<br />
3. 54.43 5.38<br />
4. 45.77 4.93<br />
5. 38.04 4.56<br />
6. 32.65 4.14<br />
7. 27.17 3.77<br />
8. 22.99 3.36<br />
9. 18.83 3.11<br />
10. 14.95 2.77<br />
11. 13.41 2.57<br />
12. 11.97 2.18<br />
13. 11.47 1.85<br />
14. 10.79 1.42<br />
IS. 10.16 1.15<br />
I<br />
A.7
Appendix A Detailed Pipe Test Results<br />
TEST: KERLIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRITICAL VELOCITY: 2.21 m/s<br />
No.<br />
1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
8.<br />
9.<br />
10.<br />
11.<br />
12.<br />
13.<br />
14.<br />
22.12<br />
20.03<br />
18.57<br />
16.30<br />
13.82<br />
13.19<br />
12.82<br />
11.88<br />
11.65<br />
10.89<br />
10.79<br />
10.54<br />
10.33<br />
9.33<br />
Velocity<br />
[m/s]<br />
2.98<br />
2.84<br />
2.76<br />
2.60<br />
2.36<br />
2.16<br />
2.00<br />
1.81<br />
1.63<br />
1.45<br />
1.24<br />
1.07<br />
0.91<br />
0.67<br />
A.9
Appendix A Detailed Pipe Test Results<br />
TEST: KMRL20<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRITICAL VELOCITY: 1.67 m1s<br />
EJ Wall Shear Stress<br />
[pa]<br />
Velocity<br />
[m1s]<br />
I. 22.89 2.60<br />
2. 19.92 2.40<br />
3. 17.51 2.22<br />
4. 14.90 2.03<br />
5. 12.21 1.83<br />
6. 11.38 1.69<br />
7. 10.43 1.52<br />
8. 10.37 1.45<br />
9. 9.94 1.25<br />
10. 9.47 1.03<br />
11. 8.83 0.74<br />
12. 8.35 0.58<br />
13. 7.87 0.32<br />
14. 7.15 0.19<br />
A.12
Appendix A Detailed Pipe Test Results<br />
TEST: KERS20<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRITICAL VELOCITY: 1.95 m/s<br />
I No. Wall Shear Stress<br />
[Pa] I<br />
Velocity<br />
[m/s]<br />
1. 84.69 5.42<br />
2. 71.01 5.14<br />
3. 61.12 4.92<br />
4. 51.53 4.49<br />
5. 40.97 4.03<br />
6. 34.38 3.61<br />
7. 26.94 3.14<br />
8. 17.60 2.64<br />
9. 13.58 2.25<br />
10. 8.44 1.69<br />
11. 7.95 1.21<br />
12. 7.00 0.75<br />
I<br />
A.14
Appendix A Detailed Pipe Test Results<br />
TEST:KERM20<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRmCAL VELOCITY: 2.22 m1s<br />
EJI Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[m1s]<br />
1. 73.65 6.19<br />
2. 65.48 5.90<br />
3. 56.23 5.47<br />
4. 45.10 4.83<br />
5. 32.36 4.05<br />
6. 22.57 3.32<br />
7. 17.58 2.88<br />
8. 13.50 2.56<br />
9. 11.33 2.23<br />
10. 9.39 2.00<br />
11. 8.18 1.75<br />
12. 7.82 1.59<br />
13. 7.32 1.42<br />
I<br />
A.16
Appendix A Detailed Pipe Test Results<br />
TEST: KERL20<br />
SLURRY: Water<br />
Kaolin Clay<br />
CRITICAL VELOCITY: 1.55 m1s<br />
No. IWall Shear Stress<br />
[pa] I<br />
Velocity<br />
[m1s]<br />
1. 7.64 2.89<br />
2. 13.76 2.49<br />
3. 12.22 2.32<br />
4. 10.08 2.00<br />
5. 8.82 1.74<br />
6. 7.25 1.37<br />
7. 5.99 1.05<br />
8. 5.44 0.72<br />
I<br />
A.18
Appendix A Detailed Pipe Test Results<br />
TEST: RFMRL10<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.71 mfs<br />
No. Wall Shear Stress Velocity<br />
[pa] [mfs]<br />
1. 25.25 2.48<br />
2. 19.58 2.19<br />
3. 17.00 2.03<br />
4. 14.72 1.91<br />
5. 13.47 1.87<br />
6. 11.24 1.71<br />
7. 10.36 1.60<br />
8. 9.91 1.44<br />
9. 9.47 1.30<br />
10. 8.77 1.11<br />
11. 8.22 0.95<br />
12. 7.60 0.74<br />
13. 6.88 0.56<br />
14. 6.33 0.44<br />
A.22
Appendix A Detailed Pipe Test Results<br />
TEST: RFERSIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.96 mls<br />
EJI Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[mls]<br />
1. 87.50 5.43<br />
2. 78.45 5.13<br />
3. 64.13 4.86<br />
4. 51.25 4.39<br />
5. 44.36 4.02<br />
6. 39.63 3.71<br />
7. 30.61 3.35<br />
8. 24.78 2.94<br />
9. 21.17 2.65<br />
10. 13.84 2.20<br />
11. 9.31 1.78<br />
12. 5.88 1.30<br />
13. 4.91 0.68<br />
I<br />
A.24
Appendix A Detailed Pipe Test Results<br />
TEST: RFERMIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 2.24 rnIs<br />
No. IWaIl Shear Stress<br />
(pa] I<br />
Velocity<br />
[rnIs]<br />
1. 78.79 6.17<br />
2. 68.44 5.66<br />
3. 55.61 5.25<br />
4. 47.58 4.77<br />
5. 41.72 4.50<br />
6. 35.36 4.10<br />
7. 28.87 3.71<br />
8. 22.43 3.31<br />
9. 16.33 2.89<br />
10. 14.72 2.69<br />
11. 12.45 2.42<br />
12. 9.17 2.15<br />
13. 5.01 1.49<br />
14. 4.88 1.21<br />
I<br />
A.26
Appendix A Detailed Pipe Test Results<br />
TEST: RFERLIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.27 rnfs<br />
No.<br />
I<br />
Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[m1s]<br />
1. 21.22 2.79<br />
2. 16.23 2.40<br />
3. 13.40 2.16<br />
4. 10.11 1.83<br />
5. 8.69 1.63<br />
6. 6.47 1.34<br />
7. 4.97 0.91<br />
8. 4.62 0.71<br />
I<br />
A.28
Appendix A Detailed Pipe Test Results<br />
TEST: RFMRL20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.81 rnIs<br />
No.<br />
I<br />
Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[rnIs]<br />
1. 24.49 2.64<br />
2. 19.49 2.36<br />
3. 16.78 2.17<br />
4. 14.98 2.05<br />
5. 11.40 1.82<br />
6. 10.12 1.52<br />
7. 9.05 1.19<br />
8. 8.63 1.02<br />
9. 7.89 0.82<br />
10. 7.35 0.70<br />
11. 6.65 0.46<br />
12. 5.97 0.35<br />
13. 5.39 0.25<br />
I<br />
A.32
Appendix A Detailed Pipe Test Results<br />
TEST: RFERS20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.48 m1s<br />
No. IWaIl Shear Stress Velocity<br />
[pa] [m/s]<br />
1. 90.31 5.73<br />
2. 66.13 4.85<br />
3. 53.70 4.35<br />
4. 45.21 3.98<br />
5. 40.27 3.67<br />
6. 32.40 3.23<br />
7. 24.48 2.81<br />
8. 18.47 2.45<br />
9. 11.58 1.96<br />
10. 6.13 1.35<br />
11. 5.64 1.08<br />
I<br />
I<br />
A.34
Appendix A Detailed Pipe Test Results<br />
TEST:RFERM20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 2.22 m/s<br />
No. IWall Shear Stress<br />
[pa] I<br />
Velocity<br />
[m/s]<br />
1. 77.92 6.14<br />
2. 66.64 5.53<br />
3. 55.30 5.06<br />
4. 42.66 4.51<br />
5. 35.04 4.07<br />
6. 27.36 3.57<br />
7. 19.49 3.07<br />
8. 14.07 2.64<br />
9. 10.45 2.28<br />
10. 8.53 2.01<br />
11. 4.95 1.63<br />
12. 4.76 1.39<br />
I<br />
A.36
Appendix A Detailed Pipe Test Results<br />
TEST: RFERL20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.38 rnIs<br />
No. IWall Shear Stress<br />
[pa] I<br />
Velocity<br />
[rnIs]<br />
1. 22.93 2.76<br />
2. 19.47 2.56<br />
3. 15.61 2.32<br />
4. 13.61 2.13<br />
5. 11.21 1.94<br />
6. 9.03 1.71<br />
7. 7.59 1.51<br />
8. 6.49 1.15<br />
9. 5.68 0.88<br />
I<br />
A.38
Appendix A Detailed Pipe Test Results<br />
TEST: RFMRL30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRlTICAL VELOCITY: 1.69 rnfs<br />
No. Wall Shear Stress Velocity<br />
[pa] [rnfs]<br />
1. 24.60 2.31<br />
2. 22.37 2.15<br />
3. 21.00 2.00<br />
4. 18.8 1.84<br />
5. 16.80 1.64<br />
6. 15.94 1.48<br />
7. 15.30 1.36<br />
8. 13.90 1.13<br />
9. 12.60 0.94<br />
10. 11.32 0.74<br />
11. 9.62 0.46<br />
A.42
Appendix A Detailed Pipe Test Results<br />
TEST: RFERS30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.87 mls<br />
EJI Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[mls]<br />
I. 93.29 5.70<br />
2. 76.66 5.12<br />
3. 66.45 4.79<br />
4. 53.79 4.30<br />
5. 41.64 3.69<br />
6. 34.18 3.28<br />
7. 24.34 2.80<br />
8. 15.82 2.24<br />
9. 9.71 1.64<br />
10. 8.58 1.03<br />
11. 7.83 0.66<br />
I<br />
A.44
Appendix A Detailed Pipe Test Results<br />
TEST:RFERM30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 2.26 rnIs<br />
No. Wall Shear Stress Velocity<br />
[pa] [rnIs]<br />
I. 77.82 6.02<br />
2. 62.69 5.26<br />
3. 45.36 4.46<br />
4. 37.51 4.10<br />
5. 29.93 3.70<br />
6. 23.89 3.33<br />
7. 18.06 2.86<br />
8. 14.09 2.65<br />
9. 11.22 2.38<br />
10. 10.00 2.16<br />
11. 7.81 1.70<br />
A.46
Appendix A Detailed Pipe Test Results<br />
TEST: RFERL30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
CRITICAL VELOCITY: 1.42 m/s<br />
No. Wall Shear Stress Velocity<br />
[pa] [m/s]<br />
1. 26.58 2.71<br />
2. 22.94 2.62<br />
3. 20.00 2.53<br />
4. 17.60 2.36<br />
5. 15.39 2.15<br />
6. 11.97 1.86<br />
7. 10.30 1.59<br />
8. 8.65 1.22<br />
9. 6.89 0.92<br />
A.48
Appendix A Detailed Pipe Test Results<br />
TEST: SMRLIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 1.69 rnIs<br />
No. Wall Shear Stress Velocity<br />
[Pa] [rnIs]<br />
1. 22.78 2.23<br />
2. 19.86 2.10<br />
3. 16.87 1.96<br />
4. 14.99 1.80<br />
5. 13.57 1.70<br />
6. 12.24 1.51<br />
7. 11.85 1.29<br />
8. 11.52 1.13<br />
9. 11.00 0.88<br />
10. 10.41 0.70<br />
11. 9.97 0.56<br />
12. 9.27 0.39<br />
A.52
Appendix A Detailed Pipe Test Results<br />
TEST: SERSIO<br />
SLURRY; Water<br />
Kaolin Clay<br />
RockFIour<br />
Silica Sand<br />
CRITICAL VELOCITY: 2.10 m/s<br />
EJI I<br />
1. 95.43 5.42<br />
Wall Shear Stress Velocity<br />
[pa] fm/s]<br />
2. 86.31 5.15<br />
3. 76.92 4.86<br />
4. 65.80 4.55<br />
5. 55.38 4.19<br />
6. 46.20 3.73<br />
7. 37.72 3.27<br />
8. 26.76 2.83<br />
9. 17.03 2.24<br />
10. 10.42 1.57<br />
11. 9.10 0.99<br />
12. 8.76 0.65<br />
I<br />
A.54
Appendix A Detailed Pipe Test Results<br />
TEST: SERMIO<br />
SLURRY: Water<br />
Kaolin Clay<br />
RockFIour<br />
Silica Sand<br />
CRITICAL VELOCITY: 2.31 m/s<br />
No. Wall Shear Stress Velocity<br />
[pa] [m/s]<br />
1. 79.91 5.40<br />
2. 69.15 5.09<br />
3. 59.87 4.75<br />
4. 48.00 4.26<br />
5. 39.42 3.87<br />
6. 31.47 3.43<br />
7. 23.62 2.99<br />
8. 18.18 2.65<br />
9. 15.26 2.46<br />
10. 12.27 2.29<br />
11. 9.54 1.86<br />
12. 8.26 1.41<br />
A.56
Appendix A Detailed Pipe Test Results<br />
TEST: SERLIO<br />
SLURRY; Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 1.27 mfs<br />
No. Wall Shear Stress Velocity<br />
[pal [mfsl<br />
1. 22.54 2.69<br />
2. 20.30 2.63<br />
3. 17.85 2.47<br />
4. 15.10 2.23<br />
5. 12.59 1.94<br />
6. 10.74 1.63<br />
7. 9.06 1.22<br />
8. 7.75 0.73<br />
A.58
Appendix A Detailed Pipe Test Results<br />
TEST: SMRL20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 1.66 rnIs<br />
I<br />
No.<br />
I<br />
Wall Shear Stress<br />
[pa]<br />
Velocity<br />
[rnIs]<br />
1. 21.65 2.12<br />
2. 19.94 2.00<br />
3. 18.03 1.90<br />
4. 16.21 1.79<br />
5. 15.00 1.69<br />
6. 14.81 1.62<br />
7. 13.98 1.43<br />
8. 13.55 1.28<br />
9. 13.22 1.16<br />
10. 13.04 1.07<br />
11. 12.46 0.97<br />
12. 11.80 0.81<br />
13. 13.39 0.71<br />
I<br />
A.62
Appendix A Detailed Pipe Test Results<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
TEST:SERS20<br />
CRITICAL"VELOCITY: 2.02 m/s<br />
61 Wall Shear Stress<br />
[pa] I<br />
Velocity<br />
[m/5]<br />
I. 98.13 5.37<br />
2. 89.52 5.07<br />
3. 76.39 4.75<br />
4. 67.33 4.37<br />
5. 51.14 3.79<br />
6. 46.64 3.52<br />
7. 32.99 3.01<br />
8. 23.49 2.47<br />
9. 15.33 2.02<br />
10. 11.85 1.46<br />
lI. 10.68 0.96<br />
I<br />
A.64
Appendix A Detailed Pipe Test Results<br />
TEST:SERM20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL"VELOCITY: 2.17 m/s<br />
No. Wall Shear Stress Velocity<br />
[pa] [m/s]<br />
1. 79.66 5.23<br />
2. 70.60 4.92<br />
3. 59.76 4.54<br />
4. 51.51 4.11<br />
5. 43.55 3.83<br />
6. 36.77 3.52<br />
7. 31.59 3.26<br />
8. 25.62 2.92<br />
9. 18.90 2.61<br />
10. 13.11 2.17<br />
11. 8.89 1.76<br />
12. 8.73 1.50<br />
A.66
Appendix A Detailed Pipe Test Results<br />
TEST: SERL20<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 1.55 mls<br />
EJI I<br />
1. 21.20 2.72<br />
Wall Shear Stress Velocity<br />
[pa] [m1s]<br />
2. 18.51 2.50<br />
3. 14.90 2.17<br />
4. 11.50 1.83<br />
5. 8.97 1.29<br />
6. 8.01 0.90<br />
,<br />
A.68
Appendix A Detailed Pipe Test Results<br />
TEST: SMRL30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 1.68 rn/s<br />
No. Wall Shear Stress Velocity<br />
[pa] [rn/s]<br />
1. 22.78 2.02<br />
2. 20.87 1.95<br />
3. 19.09 1.82<br />
4. 17.60 1.73<br />
5. 16.18 1.57<br />
6. 15.48 1.35<br />
7. 14.61 1.05<br />
8. 13.96 0.81<br />
9. 13.18 0.61<br />
10. 12.36 0.44<br />
11. 11.60 0.28<br />
12. 10.47 0.13<br />
A.72
Appendix A Detailed Pipe Test Results<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
TEST: SERS30<br />
CRlTICAL'VELOCITY: 2.10 m1s<br />
No.<br />
I<br />
Wall Shear Stress<br />
[pal I<br />
Velocity<br />
[m/sl<br />
1. 101.92 5.19<br />
2. 77.39 4.53<br />
3. 64.57 4.24<br />
4. 54.48 3.88<br />
5. 41.09 3.33<br />
6. 29.37 2.75<br />
7. 16.10 1.93<br />
8. 12.17 1.01<br />
I<br />
A.74
Appendix A Detailed Pipe Test Results<br />
TEST: SERM30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 2.13 m/s<br />
No. Wall Shear Stress Velocity<br />
[pa} [m/sJ<br />
1. 79.66 5.15<br />
2. 70.44 4.85<br />
3. 60.32 4.49<br />
4. 53.04 4.22<br />
5. 44.58 3.82<br />
6. 36.25 3.40<br />
7. 26.89 - 2.93<br />
8. 21.44 2.60<br />
9. 12.76 1.94<br />
10. 11.98 1.64<br />
A.76
Appendix A Detailed Pipe Test Results<br />
TEST: SERL30<br />
SLURRY: Water<br />
Kaolin Clay<br />
Rock Flour<br />
Silica Sand<br />
CRITICAL VELOCITY: 1.27 m1s<br />
EJ J. 20.94 2.56<br />
Wall Shear Stress Velocity<br />
[pa] fm/sI<br />
2. 17.96 2.29<br />
3. 14.90 1.87<br />
4. 12.36 1.48<br />
5. 11.05 1.19<br />
6. 10.42 0.94<br />
A.78
APPENDIXB
Appendix B C<strong>on</strong>ference Paper<br />
B.1 CONFERENCE PAPER - PARTICLE ROUGHNESS TURBULENCE<br />
The c<strong>on</strong>ference paper that was presented by <str<strong>on</strong>g>the</str<strong>on</strong>g> author at HYDROTRANSPORT 13 in<br />
support <str<strong>on</strong>g>of</str<strong>on</strong>g> this <str<strong>on</strong>g>the</str<strong>on</strong>g>sis is presented in this appendix.<br />
The details <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>ference paper are as follows:<br />
"Particle Roughness Turbulence", 13 th Internati<strong>on</strong>al C<strong>on</strong>ference <strong>on</strong> Slurry Handling and<br />
Pipeline Transport, Johannesburg, South Africa, September 1996, BRR Group C<strong>on</strong>ference<br />
Series - edited by J F Richards<strong>on</strong>, Mechanical Engineering Publicati<strong>on</strong> Limited, UK, pg 237<br />
257.<br />
B.l
Appendix B C<strong>on</strong>ference Paper<br />
ABSTRACT<br />
PARTICLE ROUGHNESS<br />
TURBULENCE<br />
PT SLATTER, G S THORVALDSEN & F W PETERSEN<br />
Cape Technik<strong>on</strong>, Cape Town, South Africa<br />
The <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g>n<strong>on</strong>-Newt<strong>on</strong>ian slurries has remained a problem, despite much<br />
research in this area. The successful resoluti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g>this problem is vitally imp<strong>on</strong>anr not<br />
<strong>on</strong>ly for hydrotransp<strong>on</strong> applicati<strong>on</strong>s involving fine slurries, but also for mixed regime<br />
slurries, where <str<strong>on</strong>g>the</str<strong>on</strong>g> vehicle comp<strong>on</strong>ent is usually a n<strong>on</strong>-Newt<strong>on</strong>ian slurry. Two major<br />
problem areas are that <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g>se slurries appears unrelated to<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>ir laminar behaviour and yet has been found to be strikingly similar to Newt<strong>on</strong>ian<br />
<strong>turbulent</strong> behaviour, in spite <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> obvious difference in rheology.<br />
This paper explores <str<strong>on</strong>g>the</str<strong>on</strong>g>se and o<str<strong>on</strong>g>the</str<strong>on</strong>g>r problem areas in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature and shows how<br />
previous <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models havefailed to address <str<strong>on</strong>g>the</str<strong>on</strong>g>m adequately. A new approach to<br />
turbulence modelling is reviewed which does address <str<strong>on</strong>g>the</str<strong>on</strong>g>se problem areas. This approach<br />
is based <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> panicle roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>, but is as yet relatively untested outside <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
range ojslurries and panicle <str<strong>on</strong>g>size</str<strong>on</strong>g>s <strong>on</strong> which it was originally evaluated.<br />
An experimentalprogramme has been initiated to investigate and accumulate a data base<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g>a wider range <str<strong>on</strong>g>of</str<strong>on</strong>g>n<strong>on</strong>-Newt<strong>on</strong>ian slurries. including slurries with a<br />
bimodal panicle <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>. These new data are analysed and <str<strong>on</strong>g>the</str<strong>on</strong>g> results are<br />
presented and discussed. It is c<strong>on</strong>cluded that <str<strong>on</strong>g>the</str<strong>on</strong>g> new approach to turbulence modelling<br />
using <str<strong>on</strong>g>the</str<strong>on</strong>g> panicle roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> is valid for <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tested.<br />
1. INTRODUCTION<br />
The predicti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g><strong>turbulent</strong> flow or pipe flow energy requirements from <strong>on</strong>ly <str<strong>on</strong>g>the</str<strong>on</strong>g> viscous<br />
properties <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian suspensi<strong>on</strong>s has over <str<strong>on</strong>g>the</str<strong>on</strong>g> years been questi<strong>on</strong>ed by<br />
researchers. It has been found that <str<strong>on</strong>g>the</str<strong>on</strong>g> flow behaviour and rheology <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g>se suspensi<strong>on</strong>s<br />
is influenced by such factors as <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>, shape, weight and <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> (philipp<str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
1944, Hedstrom 1952, Orr & Blocker 1955, Zettlemoyer & Lower 1955, Maude &<br />
Whitmore 1956, Thomas 1963, Thomas 1983, Mun 1988, Slatter 1994).<br />
The <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian suspensi<strong>on</strong>s is complex and is <str<strong>on</strong>g>of</str<strong>on</strong>g>ten c<strong>on</strong>sidered<br />
problematic to design. However, in many situati<strong>on</strong>s it has beneficial characteristics,<br />
B.2
Appendix B C<strong>on</strong>ference Paper B.3<br />
preventing suspensi<strong>on</strong>s settling and enabling higher throughputs (Mun, 1988). Many<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models have <str<strong>on</strong>g>the</str<strong>on</strong>g>refore been proposed to try and explain and predict <strong>turbulent</strong><br />
flow behaviour. However, two major problem areas are that <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> behaviour <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
n<strong>on</strong>-Newt<strong>on</strong>ian slurries appears unrelated to <str<strong>on</strong>g>the</str<strong>on</strong>g>ir laminar behaviour and yet has been<br />
found strikingly similar to Newt<strong>on</strong>ian <strong>turbulent</strong> flow behaviour, in spite <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> obvious<br />
difference in rheology.<br />
This paper explores <str<strong>on</strong>g>the</str<strong>on</strong>g>se and o<str<strong>on</strong>g>the</str<strong>on</strong>g>r problem areas in <str<strong>on</strong>g>the</str<strong>on</strong>g> literature and shows how<br />
previous models have failed to address <str<strong>on</strong>g>the</str<strong>on</strong>g>m adequately. Slatter's (1994) approach to<br />
turbulence modelling, based <strong>on</strong> his findings <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>, is reviewed.<br />
This model addresses <str<strong>on</strong>g>the</str<strong>on</strong>g>se problem areas - however <str<strong>on</strong>g>the</str<strong>on</strong>g> model was initially evaluated<br />
<strong>on</strong> a limited number <str<strong>on</strong>g>of</str<strong>on</strong>g> slurries and <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g>s.<br />
Experimental work was c<strong>on</strong>ducted at <str<strong>on</strong>g>the</str<strong>on</strong>g> University <str<strong>on</strong>g>of</str<strong>on</strong>g> Cape Town's (UCT)<br />
Hydrotransport Department, South Africa, using a pumped recirculating pipe test rig <str<strong>on</strong>g>of</str<strong>on</strong>g><br />
pipe diameters 25mm, 80mm, 150mm and 200mm. N<strong>on</strong>-Newt<strong>on</strong>ian slurries with varying<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g>s (PSD's) were tested and analysed, and <str<strong>on</strong>g>the</str<strong>on</strong>g> results are presented<br />
and discussed.<br />
2. LITERATURE REVIEW<br />
2.1 The Yield Pseudoplastic Model<br />
N<strong>on</strong>-Newt<strong>on</strong>ian slurries can <str<strong>on</strong>g>of</str<strong>on</strong>g>ten be modelled as yield pseudoplastics (Govier & Aziz,<br />
1972 and Hanks, 1979) and <str<strong>on</strong>g>the</str<strong>on</strong>g> larninar flow <str<strong>on</strong>g>of</str<strong>on</strong>g> all <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tested have been<br />
successfully characterized using this model. The c<strong>on</strong>stitutive rheo10gical equati<strong>on</strong> is<br />
where Ty is <str<strong>on</strong>g>the</str<strong>on</strong>g> yield stress, K is fluid c<strong>on</strong>sistency index and n is <str<strong>on</strong>g>the</str<strong>on</strong>g> flow behaviour<br />
index.<br />
2.2 Laminar Pipe Flow<br />
For laminar pipe flow, volumetric discharge, Q, and average velocity, V, can be<br />
determined using <str<strong>on</strong>g>the</str<strong>on</strong>g> equati<strong>on</strong><br />
(1)<br />
(2)
Appendix B C<strong>on</strong>ference Paper<br />
where TO = DJip/4L and V=Q/A.<br />
2.3 Rheological Characterizati<strong>on</strong><br />
Rheological characterizati<strong>on</strong> involves choosing a rheological model which best fits <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
data (<str<strong>on</strong>g>the</str<strong>on</strong>g> yield pseudoplastic model is used for this study) and <str<strong>on</strong>g>the</str<strong>on</strong>g>n determining <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
values <str<strong>on</strong>g>of</str<strong>on</strong>g> T y , K and n for a particular slurry. The viscous flow data in <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar regi<strong>on</strong><br />
is coincident for <str<strong>on</strong>g>the</str<strong>on</strong>g> different tube diameters and <str<strong>on</strong>g>the</str<strong>on</strong>g> rheological c<strong>on</strong>stants (Ty, K and<br />
n) were determined using <str<strong>on</strong>g>the</str<strong>on</strong>g> method proposed by Lazarus & Slatter (1988) and Neill<br />
(1988).<br />
2.4 Turbulent Flow<br />
Turbulence is a natural form <str<strong>on</strong>g>of</str<strong>on</strong>g> fluid moti<strong>on</strong> which is characterized by large, random<br />
swirling or eddy moti<strong>on</strong>s both parallel and transverse to <str<strong>on</strong>g>the</str<strong>on</strong>g> directi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> main flow.<br />
Particle paths cross and velocity (both directi<strong>on</strong> and magnitude) and pressure, fluctuate<br />
<strong>on</strong> a c<strong>on</strong>tinuous random basis. The flow behaviour becomes extremely complex and full<br />
rigorous analysis becomes impossible (Tennekes & Lumley, 1972).<br />
The <strong>turbulent</strong> flow <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian slurries that were tested were modelled using<br />
Slatter's model as well as <str<strong>on</strong>g>the</str<strong>on</strong>g> following models which are already well established in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
literature,<br />
- Newt<strong>on</strong>ian approximati<strong>on</strong><br />
- Dodge and Metzner (1959)<br />
- Torrance (1963)<br />
- Kemblowski & Kolodziejski (1973)<br />
- Wils<strong>on</strong> & Thomas (1985)<br />
2.4.1 Newt<strong>on</strong>ian Approximati<strong>on</strong><br />
In order to make use <str<strong>on</strong>g>of</str<strong>on</strong>g>standard Newt<strong>on</strong>ian <str<strong>on</strong>g>the</str<strong>on</strong>g>ory, a value for <str<strong>on</strong>g>the</str<strong>on</strong>g> viscosity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid<br />
is required. Usually <str<strong>on</strong>g>the</str<strong>on</strong>g> term viscosity is meaningless <strong>on</strong>ce a n<strong>on</strong>-Newt<strong>on</strong>ian approach<br />
has been adopted. However, an apparent or secant viscosity (Holland, 1973 and Wils<strong>on</strong>,<br />
1986) can be defined as<br />
Note that JL' is not a c<strong>on</strong>stant for a given fluid and pipe diameter, but must be evaluated<br />
at a given value for TO.<br />
(3)<br />
B.4
Appendix B C<strong>on</strong>ference Paper B.7<br />
and can never be truly homogeneous. Treating <str<strong>on</strong>g>the</str<strong>on</strong>g>m as a c<strong>on</strong>tinuum is <str<strong>on</strong>g>the</str<strong>on</strong>g>refore an<br />
approximati<strong>on</strong>, which works well in <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar regime. The term homogeneous, when<br />
applied to slurries is taken as meaning uniform, stable, spatial <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>particle</str<strong>on</strong>g>s.<br />
2.4.6.1 The Laminar Sub-layer<br />
Velocity comp<strong>on</strong>ents normal to <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe axis cannot exist at <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall, and<br />
turbulence is suppressed in this regi<strong>on</strong>. Viscous forces are dominant and a laminar sublayer<br />
exists for some finite thickness o. It has been shown (Slatter, 1994) that <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer is <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> same order <str<strong>on</strong>g>of</str<strong>on</strong>g> magnitude as <str<strong>on</strong>g>the</str<strong>on</strong>g> larger<br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <str<strong>on</strong>g>the</str<strong>on</strong>g> sluny over <str<strong>on</strong>g>the</str<strong>on</strong>g> range <str<strong>on</strong>g>of</str<strong>on</strong>g>wall shear stresses found in <strong>turbulent</strong> flow. Since<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s will be larger than <str<strong>on</strong>g>the</str<strong>on</strong>g> thickness <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer at <str<strong>on</strong>g>the</str<strong>on</strong>g> higher<br />
wall shear stress values, <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g>s must <str<strong>on</strong>g>the</str<strong>on</strong>g>refore have an obstructing <str<strong>on</strong>g>effect</str<strong>on</strong>g> <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
shear in <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer. The c<strong>on</strong>tinuum approximati<strong>on</strong> must <str<strong>on</strong>g>the</str<strong>on</strong>g>refore be<br />
compromised in this regi<strong>on</strong>.<br />
2.4.6.2 Wall Roughness and <str<strong>on</strong>g>the</str<strong>on</strong>g> Effect <str<strong>on</strong>g>of</str<strong>on</strong>g> Solid Particles<br />
Under microscopic investigati<strong>on</strong> it can be seen that <str<strong>on</strong>g>the</str<strong>on</strong>g> inner wall <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe is not<br />
smooth but many protrusi<strong>on</strong>s exist. The term wall, pipe or surface roughness is used to<br />
describe <str<strong>on</strong>g>the</str<strong>on</strong>g> complex <str<strong>on</strong>g>size</str<strong>on</strong>g>, shape and spacing <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g>se protrusi<strong>on</strong>s. For a given fluid and<br />
velocity, roughness in a pipe increases <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure drop compared with c<strong>on</strong>diti<strong>on</strong>s<br />
existing in a "smooth" pipe (Bowen 1961, Cheng 1975). At low Reynolds numbers when<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer is sufficiently thick to cover <str<strong>on</strong>g>the</str<strong>on</strong>g> protrusi<strong>on</strong>s <strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
roughness will not be <str<strong>on</strong>g>effect</str<strong>on</strong>g>ive. At high Reynolds numbers when <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer<br />
is not sufficiently thick to cover <str<strong>on</strong>g>the</str<strong>on</strong>g> protrusi<strong>on</strong>s, roughness will be <str<strong>on</strong>g>effect</str<strong>on</strong>g>ive. The<br />
progressive penetrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer by <str<strong>on</strong>g>the</str<strong>on</strong>g> wall roughness will generate a<br />
wake <str<strong>on</strong>g>of</str<strong>on</strong>g> eddies, stimulating turbulence. This physical phenomen<strong>on</strong> is <str<strong>on</strong>g>of</str<strong>on</strong>g> fundamental<br />
importance to <str<strong>on</strong>g>the</str<strong>on</strong>g> understanding <str<strong>on</strong>g>of</str<strong>on</strong>g> rough wall <strong>turbulent</strong> flow.<br />
Particles will also have a similar <str<strong>on</strong>g>effect</str<strong>on</strong>g> to that <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> wall roughness, in causing a<br />
decrease in <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity gradient and should be taken into account in <strong>turbulent</strong> flow<br />
analysis.<br />
The change in velocity as <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe wall is approached is very rapid, <str<strong>on</strong>g>the</str<strong>on</strong>g> velocity gradient<br />
being in <str<strong>on</strong>g>the</str<strong>on</strong>g> order <str<strong>on</strong>g>of</str<strong>on</strong>g> lmls over <str<strong>on</strong>g>the</str<strong>on</strong>g> diameter <str<strong>on</strong>g>of</str<strong>on</strong>g> a typical <str<strong>on</strong>g>particle</str<strong>on</strong>g> in <str<strong>on</strong>g>the</str<strong>on</strong>g> regi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
pipe wall. Ifsolid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s are present in <str<strong>on</strong>g>the</str<strong>on</strong>g> fluid <str<strong>on</strong>g>the</str<strong>on</strong>g>y will resist shear and impede such<br />
rapid changes in velocity.<br />
A roughness Reynolds number can be used to determine various regi<strong>on</strong>s <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>turbulent</strong><br />
flow in a pipe (Schlichting, 1960). The inadequate formulati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness<br />
Reynolds number for n<strong>on</strong>-Newt<strong>on</strong>ian slurries is a problem with previous models eg a<br />
formulati<strong>on</strong> excluding <str<strong>on</strong>g>the</str<strong>on</strong>g> yield stress (Torrance, 1963 and Hanks & Dadia, 1971).
Appendix B C<strong>on</strong>ference Paper<br />
2.4.6.3 Representative Particle Size<br />
The wall roughness and <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> was taken into account in <str<strong>on</strong>g>the</str<strong>on</strong>g> roughness Reynolds<br />
number as given by Slatter (1994).<br />
The <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> roughness <strong>on</strong> turbulence can be thought <str<strong>on</strong>g>of</str<strong>on</strong>g> as an aggravati<strong>on</strong> at <str<strong>on</strong>g>the</str<strong>on</strong>g> wall<br />
which stimulates turbulence. Clearly <str<strong>on</strong>g>the</str<strong>on</strong>g>n <str<strong>on</strong>g>the</str<strong>on</strong>g> larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s will have a more dominant<br />
<str<strong>on</strong>g>effect</str<strong>on</strong>g> <strong>on</strong> turbulence than <str<strong>on</strong>g>the</str<strong>on</strong>g> smaller <str<strong>on</strong>g>particle</str<strong>on</strong>g>s. Also, <str<strong>on</strong>g>the</str<strong>on</strong>g> larger <str<strong>on</strong>g>particle</str<strong>on</strong>g>s will shield <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
smaller <strong>on</strong>es, reducing <str<strong>on</strong>g>the</str<strong>on</strong>g>ir <str<strong>on</strong>g>effect</str<strong>on</strong>g>iveness in stimulating turbulence (Colebrook, 1939).<br />
For all <str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tested to date it was found that <str<strong>on</strong>g>the</str<strong>on</strong>g> d S5 <str<strong>on</strong>g>size</str<strong>on</strong>g> was a good representati<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> roughness <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>effect</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> solid <str<strong>on</strong>g>particle</str<strong>on</strong>g>s in <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry ie
Appendix B C<strong>on</strong>ference Paper<br />
IfRe, < 3,32 <str<strong>on</strong>g>the</str<strong>on</strong>g>n smooth wall <strong>turbulent</strong> flow exists and <str<strong>on</strong>g>the</str<strong>on</strong>g> mean velocity is given by<br />
:.Y.- = 2,5 In [-!.] + 2,5 In Re + 1,75 .<br />
V. dss • '<br />
If Re,. > 3,32 <str<strong>on</strong>g>the</str<strong>on</strong>g>n fully developed rough wall <strong>turbulent</strong> flow exists and <str<strong>on</strong>g>the</str<strong>on</strong>g> mean<br />
velocity is given by<br />
Y.- [-!.] = 2,5 In +<br />
V. dss<br />
4,75 ,<br />
and <str<strong>on</strong>g>the</str<strong>on</strong>g> fricti<strong>on</strong> factor is c<strong>on</strong>stant.<br />
This correlati<strong>on</strong> produces a transiti<strong>on</strong> from <str<strong>on</strong>g>the</str<strong>on</strong>g> smooth to <str<strong>on</strong>g>the</str<strong>on</strong>g> rough flow c<strong>on</strong>diti<strong>on</strong> which<br />
is abrupt.<br />
Tests c<strong>on</strong>ducted by Slatter (1994) c<strong>on</strong>firmed <str<strong>on</strong>g>the</str<strong>on</strong>g> model to be more accurate than <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Torrance (1963) model and Wils<strong>on</strong> & Thomas (1985, 1987) model, when evaluated<br />
against experimental data.<br />
2.4.7 Data from <str<strong>on</strong>g>the</str<strong>on</strong>g> Literature<br />
Experimental data obtained by Sive (1988) was used in <str<strong>on</strong>g>the</str<strong>on</strong>g> analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> various<br />
models under c<strong>on</strong>siderati<strong>on</strong>. The tests c<strong>on</strong>ducted by Sive (1988) were d<strong>on</strong>e using a<br />
mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay and a relatively coarse quartz sand, which resulted in a<br />
heterogeneous, settling slurry. The purpose <str<strong>on</strong>g>of</str<strong>on</strong>g> using this data was to see if <str<strong>on</strong>g>the</str<strong>on</strong>g> coarse,<br />
settling <str<strong>on</strong>g>particle</str<strong>on</strong>g>s c<strong>on</strong>tributed to <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow headloss, as proposed by <str<strong>on</strong>g>the</str<strong>on</strong>g> Slatter<br />
model.<br />
3. EXPERIMENTAL WORK<br />
The test facility at ucr which was used had four different pipe diameters namely<br />
25mm, 80mm, 150mm and 200mm nominal bore and slurries were tested at mean<br />
velocities ranging from 0, Imls to 8m/s. Slurries tested included kaolin clay and mixture<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay and fine sand at varying ratios.<br />
(10)<br />
(11)<br />
(12)<br />
B.9
Appendix B C<strong>on</strong>ference Paper B.IO<br />
A test set c<strong>on</strong>ducted using <str<strong>on</strong>g>the</str<strong>on</strong>g> facility is defined as a set <str<strong>on</strong>g>of</str<strong>on</strong>g> tests using <str<strong>on</strong>g>the</str<strong>on</strong>g> four different<br />
pipe diameters but <str<strong>on</strong>g>the</str<strong>on</strong>g> same slurry.<br />
3.1 Test Facility<br />
Slurry to be used for a test run is collected in a steel hopper which has a capacity 2m 3 •<br />
From <str<strong>on</strong>g>the</str<strong>on</strong>g>re it is pumped by a four bladed Ma<str<strong>on</strong>g>the</str<strong>on</strong>g>r and Plat! 8x6 solids handling pump,<br />
which is driven by a variable speed hydraulic drive, through <str<strong>on</strong>g>the</str<strong>on</strong>g> pipe circuit. Directly<br />
after <str<strong>on</strong>g>the</str<strong>on</strong>g> pump <str<strong>on</strong>g>the</str<strong>on</strong>g> l50mm line splits up into a 90mm and l50mm pipeline. These two<br />
pipelines have a vertical inverted "U" secti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> l7m, <str<strong>on</strong>g>the</str<strong>on</strong>g> flowrate being measured in<br />
both downcomer secti<strong>on</strong>s by means <str<strong>on</strong>g>of</str<strong>on</strong>g> magnetic flux flow meters. A horiz<strong>on</strong>tal secti<strong>on</strong><br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> approximately 30m follows with <str<strong>on</strong>g>the</str<strong>on</strong>g> return pipelines passing through an in-line heat<br />
exchanger and pneumatic diverter valve before being re-routed back into <str<strong>on</strong>g>the</str<strong>on</strong>g> hopper.<br />
The l50mm horiz<strong>on</strong>tal return pipeline splits into a l50mm line and a 200mm pipeline<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> two joining toge<str<strong>on</strong>g>the</str<strong>on</strong>g>r after l2m. The 80mm and 150mm pipelines are PVC with clear<br />
viewing and test secti<strong>on</strong>s located in <str<strong>on</strong>g>the</str<strong>on</strong>g> horiz<strong>on</strong>tal return pipelines. The 200mm line is<br />
steel.<br />
Flowrate can be calibrated by diverting <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry using <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> pneumatic diverter valve<br />
into <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank which is located al<strong>on</strong>gside <str<strong>on</strong>g>the</str<strong>on</strong>g> hopper. The weigh tank, which has<br />
a capacity <str<strong>on</strong>g>of</str<strong>on</strong>g> 1,5m 3 and is placed <strong>on</strong> a 1750kg mass scale, is used to calibrate <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
magnetic flux flow meters. After flowrate determinati<strong>on</strong>s <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry is directed back into<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> looped circuit.<br />
In order to accurately determine <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry a 25mm tube was used. A<br />
tapping is taken from <str<strong>on</strong>g>the</str<strong>on</strong>g> l50mm horiz<strong>on</strong>tal return pipeline and using <str<strong>on</strong>g>the</str<strong>on</strong>g> back pressure,<br />
slurry was circulated through <str<strong>on</strong>g>the</str<strong>on</strong>g> 25mm pipeline into <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank. The flowrate was<br />
measured using a 25mm nominal bore Altometer magnetic flux flowmeter. The flowrate<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry was checked by doing a calibrati<strong>on</strong> using <str<strong>on</strong>g>the</str<strong>on</strong>g> weigh tank during a 25mm<br />
pipe test.<br />
No external agitati<strong>on</strong> was required to maintain solids suspensi<strong>on</strong> in <str<strong>on</strong>g>the</str<strong>on</strong>g> hopper for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
slurries which were tested.<br />
3.2 Measured Variables<br />
Readings <str<strong>on</strong>g>of</str<strong>on</strong>g> head loss were taken at varying velocities and <str<strong>on</strong>g>the</str<strong>on</strong>g> data obtained was used<br />
to plot a pseudo-shear diagram.<br />
3.3 Water Tests<br />
The hydraulic pipe roughness for each <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> four pipe <str<strong>on</strong>g>size</str<strong>on</strong>g>s was first determined
Appendix B C<strong>on</strong>ference Paper B.Il<br />
through c<strong>on</strong>ducting clear water pipeline tests. Mean velocity and wall shear stress were<br />
measured for velocities over <str<strong>on</strong>g>the</str<strong>on</strong>g> test range and roughness was determined using <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
Colebrook / White (1939) equati<strong>on</strong>.<br />
3.3 Material Used<br />
The solids material used for testing purposes was kaolin clay and sand. Tests were<br />
c<strong>on</strong>ducted using kaolin clay and <str<strong>on</strong>g>the</str<strong>on</strong>g>n a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay and No. 2 Foundry Sand<br />
from C<strong>on</strong>sol at varying c<strong>on</strong>centrati<strong>on</strong>s.<br />
- Kaolin Clay<br />
Kaolin clay was obtained from <str<strong>on</strong>g>the</str<strong>on</strong>g> Serina Kaolin (Pty) Ltd which is currently mining a<br />
kaolin deposit at Brakkeklo<str<strong>on</strong>g>of</str<strong>on</strong>g>, Fish Reek, Cape Town, South Africa. The kaolin was<br />
delivered in <str<strong>on</strong>g>the</str<strong>on</strong>g> form <str<strong>on</strong>g>of</str<strong>on</strong>g> pellets and filter cakes. To obtain a homogenous slurry <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
pellets or filter cake was thoroughly mixed with water using a recirculating pipe after<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> pump back into <str<strong>on</strong>g>the</str<strong>on</strong>g> hopper before circulating <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry through <str<strong>on</strong>g>the</str<strong>on</strong>g> pipelines.<br />
- Sand<br />
Sand was obtained from C<strong>on</strong>sol in 25kg bags called No. 2 Foundry Sand. The <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
<str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> sand was between 75 and 350 I'm.<br />
3.3.1 Mixtures <str<strong>on</strong>g>of</str<strong>on</strong>g> Kaolin Clay and Sand<br />
The sand used was found to form a settling slurry with water. In order to obtain a<br />
homogeneous slurry, kaolin clay was used as a suspending agent for <str<strong>on</strong>g>the</str<strong>on</strong>g> sand. Tests<br />
were <str<strong>on</strong>g>the</str<strong>on</strong>g>refore c<strong>on</strong>ducted to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> amount <str<strong>on</strong>g>of</str<strong>on</strong>g> kaolin clay needed to suspend <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
sand. It was found that kaolin clay at a c<strong>on</strong>centrati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g>at least Cv = 5% was required.<br />
3.4 Particle Size Distributi<strong>on</strong><br />
The <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> <str<strong>on</strong>g>distributi<strong>on</strong></str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin clay/sand mixture was obtained by combining<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> PSD <str<strong>on</strong>g>of</str<strong>on</strong>g><str<strong>on</strong>g>the</str<strong>on</strong>g> kaolin clay (determined using a Malvern 2600/3600 Particle Sizer VF.6)<br />
and <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> sand determined through sieving. The dS5 was <str<strong>on</strong>g>the</str<strong>on</strong>g>n obtained from <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
combined PSD's as shown below in Figure 1.<br />
4. ANALYSIS & RESULTS<br />
The measured variables <str<strong>on</strong>g>of</str<strong>on</strong>g> head loss and velocity were used to plot pseudo-shear<br />
diagrams ie wall shear stress vs pseudo-shear rate. A typical pseudo-shear diagram is<br />
shown in Figure 2 and <str<strong>on</strong>g>the</str<strong>on</strong>g> improvement obtained using <str<strong>on</strong>g>the</str<strong>on</strong>g> new model, is clearly visible,<br />
as opposed to <str<strong>on</strong>g>the</str<strong>on</strong>g> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models.
Appendix B C<strong>on</strong>ference Paper<br />
4.3.3 Kaolin Clay<br />
Four sets <str<strong>on</strong>g>of</str<strong>on</strong>g>kaolin clay tests were first c<strong>on</strong>ducted to c<strong>on</strong>firm <str<strong>on</strong>g>the</str<strong>on</strong>g> results already obtained<br />
by Slatter (1994). The results obtained supported <str<strong>on</strong>g>the</str<strong>on</strong>g> existing data set.<br />
4.3.4 Mixture Kaolin Clay & Sand<br />
Tests using a mixture <str<strong>on</strong>g>of</str<strong>on</strong>g>kaolin clay and sand were c<strong>on</strong>ducted in order to obtain a higher<br />
representative <str<strong>on</strong>g>particle</str<strong>on</strong>g> <str<strong>on</strong>g>size</str<strong>on</strong>g> and different PSD to <str<strong>on</strong>g>the</str<strong>on</strong>g> existing data set.<br />
4.3.5 Theoretical Models<br />
The <strong>turbulent</strong> flow performance <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models is tabulated in Table III<br />
showing <str<strong>on</strong>g>the</str<strong>on</strong>g> average percentage error from <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> experimental data. Of <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>the</str<strong>on</strong>g>oretical models under c<strong>on</strong>siderati<strong>on</strong> Slatter's model best predicted <str<strong>on</strong>g>the</str<strong>on</strong>g> test data. The<br />
WJ1s<strong>on</strong> & Thomas and Torrance models under-predicted head loss regularly. The<br />
Kemblowski & Kolodziejski model was also c<strong>on</strong>sidered, however it did not produce<br />
accurate results, significantly over-predicting <str<strong>on</strong>g>the</str<strong>on</strong>g> head loss.<br />
Table III : Turbulent Model Performance: - Average Percentage Error<br />
[;] Test ISlurry I D&M I Torr. I W&T I K&K I Slatter I<br />
1 KSMRLIO Kaolin/Sand 65.68 4.89 21.51 324.83 6.60<br />
2 KSERLlO Kaolin/Sand 26.18 18.05 30.27 232.41 17.78<br />
3 KSERMIO Kaolin/Sand 12.77 21.56 24.89 32.69 10.06<br />
4 KSERSIO Kaolin/Sand 31.85 34.66 38.46 12.99 4.86<br />
5 KSMRL20 Kaolin/Sand 7.22 11.60 12.54 286.07 4.18<br />
6 KSERL20 Kaolin/Sand 10.61 3.54 20.85 265.16 11.93<br />
7 KSERM20 Kaolin/Sand 6.55 12.01 14.95 33.64 6.63<br />
8 KSERS20 Kaolin/Sand 17.50 21.72 28.06 21.53 7.89<br />
9 KSMRL30 Kaolin/Sand 64.79 4.25 18.79 246.03 4.63<br />
10 KSERL30 Kaolin/Sand 23.56 12.54 26.56 245.00 24.69<br />
11 KSERM30 Kaolin/Sand 12.50 19.85 23.21 34.71 5.43<br />
12 KSERS30 Kaolin/Sand 24.94 27.55 35.32 44.25 8.32<br />
B.16
Appendix B C<strong>on</strong>ference Paper B.18<br />
This investigati<strong>on</strong> highlights <str<strong>on</strong>g>the</str<strong>on</strong>g> fact that <str<strong>on</strong>g>the</str<strong>on</strong>g> PSD is a vitally important property <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
slurry and should be used for <str<strong>on</strong>g>the</str<strong>on</strong>g> <strong>turbulent</strong> flow analysis <str<strong>on</strong>g>of</str<strong>on</strong>g> n<strong>on</strong>-Newt<strong>on</strong>ian slurries, as<br />
has been used in <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>ory for <str<strong>on</strong>g>the</str<strong>on</strong>g> new analysis, to account for rough wall <strong>turbulent</strong><br />
flow and to predict <str<strong>on</strong>g>the</str<strong>on</strong>g> change from smooth wall to rough wall <strong>turbulent</strong> flow.<br />
Park et al (1989) and Pokryvalio & Grozberg (1995) used very different slurries and<br />
experimental techniques, yet both reported significantly higher relative turbulence<br />
intensities in <str<strong>on</strong>g>the</str<strong>on</strong>g> wall regi<strong>on</strong> for <str<strong>on</strong>g>the</str<strong>on</strong>g>ir slurries, when compared with air. This empirical<br />
evidence str<strong>on</strong>gly supports <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>cept <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness turbulence.<br />
Where Slarter's model fails to -accurately predict Sive's data it is felt <str<strong>on</strong>g>the</str<strong>on</strong>g> reas<strong>on</strong> is that<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> slurries tested by Sive were settling slurries and not homogenous slurries <strong>on</strong> which<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical model was originally based. Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r investigati<strong>on</strong> is required to determine<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> limit <str<strong>on</strong>g>of</str<strong>on</strong>g> validity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>. In fact, <str<strong>on</strong>g>the</str<strong>on</strong>g> limit <str<strong>on</strong>g>of</str<strong>on</strong>g> validity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> could provide a more logical basis for vehicle/load cut-<str<strong>on</strong>g>of</str<strong>on</strong>g>f in<br />
mixed regime slurries than is used at present.<br />
Future <str<strong>on</strong>g>the</str<strong>on</strong>g>oretical modelling <str<strong>on</strong>g>of</str<strong>on</strong>g>n<strong>on</strong>-Newt<strong>on</strong>ian slurries should take into account both <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
PSD and <str<strong>on</strong>g>the</str<strong>on</strong>g> rheology <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> slurry.<br />
6. CONCLUSION<br />
Particle roughness turbulence does not apply to settling slurries (as tested by Sive,<br />
1988). Fur<str<strong>on</strong>g>the</str<strong>on</strong>g>r investigati<strong>on</strong> is required to determine <str<strong>on</strong>g>the</str<strong>on</strong>g> limit <str<strong>on</strong>g>of</str<strong>on</strong>g>validity <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g><br />
roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g>, and this limit could well serve as <str<strong>on</strong>g>the</str<strong>on</strong>g> criteri<strong>on</strong> for vehic1elload cut-<str<strong>on</strong>g>of</str<strong>on</strong>g>f<br />
in mixed regime slurries.<br />
The <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness turbulence model is <str<strong>on</strong>g>the</str<strong>on</strong>g> first model to incorporate <str<strong>on</strong>g>the</str<strong>on</strong>g> c<strong>on</strong>tinuum<br />
compromise which must arise as <str<strong>on</strong>g>the</str<strong>on</strong>g> laminar sub-layer thickness is limited, and it is <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
<strong>on</strong>ly model to explain <str<strong>on</strong>g>the</str<strong>on</strong>g> empirical evidence <str<strong>on</strong>g>of</str<strong>on</strong>g> increased turbulence intensities in <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
wall regi<strong>on</strong>.<br />
It can be c<strong>on</strong>cluded from <str<strong>on</strong>g>the</str<strong>on</strong>g> test data results that <str<strong>on</strong>g>the</str<strong>on</strong>g> new approach to turbulence<br />
modelling using <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>particle</str<strong>on</strong>g> roughness <str<strong>on</strong>g>effect</str<strong>on</strong>g> is valid and can be successfully adopted for<br />
n<strong>on</strong>-Newt<strong>on</strong>ian slurries.<br />
Acknowledgement<br />
The co-operati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> Pr<str<strong>on</strong>g>of</str<strong>on</strong>g>. Mike de Kock for <str<strong>on</strong>g>the</str<strong>on</strong>g> use <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> Hydrotransport test facilities<br />
at <str<strong>on</strong>g>the</str<strong>on</strong>g> University <str<strong>on</strong>g>of</str<strong>on</strong>g> Cape Town is gratefully acknowledged.
Appendix B C<strong>on</strong>ference Paper<br />
s solids<br />
v volumetric<br />
x representative <str<strong>on</strong>g>size</str<strong>on</strong>g><br />
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