flow measurement handbook industrial designs operating principles performance and applications

CHAPTER 1 Q Sensitivity coefficient f{x Function for Normal distribution K K factor in pulses per unit flow quantity q v Volumetric flow rate q vo Volumetric flow rate at calibration poi

FLOW MEASUREMENT HANDBOOK

Flow Measurement Handbook is an information-packed reference for

engineers on flow-measuring techniques and instruments Striking abalance between laboratory ideal and the realities of field experience,

it provides a wealth of practical advice on the design, operation, andperformance of a broad range of flowmeters

The book begins with a brief review of essentials of accuracy andflow, how to select a flowmeter, and various calibration methods Fol-lowing this, each chapter is devoted to a class of flowmeter and in-cludes detailed information on design, application, installation, cali-bration, operation, advantages, and disadvantages

Among the flowmeters discussed are orifice plates, venturi meters,standard nozzles, critical flow venturi nozzles, variable area and otherdevices depending on momentum of the flow, volumetric flowme-ters such as positive displacement, turbine, vortex shedding, swirl,fluidic, electromagnetic and ultrasonic meters, and mass flowmetersincluding thermal and Coriolis More than 80 different types and 250applications are listed in the index There are also chapters coveringprobes, a brief introduction to modern control, and manufacturingimplications

For those readers who want more background information, manychapters conclude with an appendix on the mathematical theory be-hind the techniques discussed The final chapter takes a look at direc-tions in which the technology is likely to go in the future

Engineers will use this practical handbook to solve problems inflowmeter design and application and to improve performance.Roger C Baker is a Visiting Industrial Fellow in the Manufacturing andManagement Division of the Department of Engineering, University

of Cambridge; Visiting Professor, Cranfield University; and Director

of Technical Programmes, the Gatsby Charitable Foundation

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 2RU, UK

Published in the United States of America by Cambridge University Press, New York

www Cambridge org

Information on this title: www.cambridge.org/9780521480109

© Cambridge University Press 2000

This book is in copyright Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 2000

This digitally printed first paperback version 2005

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Baker, R C.

Flow measurement handbook : industrial designs, operating

principles, performance, and applications / Roger C Baker.

To Liz, Sarah and Paul, Mark, John and Rachel

Preface page xix Acknowledgments xxi Nomenclature xxiii

CHAPTER 1 Introduction l

1.1 Initial Considerations 11.2 Do We Need a Flowmeter? 21.3 How Accurate? 41.4 A Brief Review of the Evaluation of Standard Uncertainty 71.5 Sensitivity Coefficients 91.6 What Is a Flowmeter? 91.7 Chapter Conclusions (for those who plan to skip the mathematics!) 131.8 Mathematical Postscript 15

l.A.l Introduction 151.A.2 The Normal Distribution 16

1.A.3 The Student t Distribution 17

1.A.4 Practical Application of Confidence Level 19I.A.5 Types of Error 201.A.6 Combination of Uncertainties 21I.A.7 Uncertainty Range Bars, Transfer Standards,

and Youden Analysis 21

CHAPTER 2 Fluid Mechanics Essentials 24

2.1 Introduction 242.2 Essential Property Values 242.3 Flow in a Circular Cross-Section Pipe 242.4 Flow Straighteners and Conditioners 272.5 Essential Equations 302.6 Unsteady Flow and Pulsation 322.7 Compressible Flow 342.8 Multiphase Flow 362.9 Cavitation, Humidity, Droplets, and Particles 382.10 Gas Entrapment 39

2.11 Steam 392.12 Chapter Conclusions 41

CHAPTER 3 Specification, Selection, and Audit 42

3.1 Introduction 423.2 Specifying the Application 423.3 Notes on the Specification Form 433.4 Flowmeter Selection Summary Tables 463.5 Other Guides to Selection and Specific Applications 533.6 Draft Questionnaire for Flowmeter Audit 553.7 Final Comments 55

APPENDIX 3.A Specification and Audit Questionnaires 56

3.A.1 Specification Questionnaire 563.A.2 Supplementary Audit Questionnaire 58

CHAPTER 4 Calibration 61

4.1 Introduction 614.1.1 Calibration Considerations 614.1.2 Typical Calibration Laboratory Facilities 644.1.3 Calibration from the Manufacturer's Viewpoint 654.2 Approaches to Calibration 664.3 Liquid Calibration Facilities 694.3.1 Flying Start and Stop 694.3.2 Standing Start and Stop 724.3.3 Large Pipe Provers 744.3.4 Compact Provers 74

4.4 Gas Calibration Facilities 77 4.4.1 Volumetric Measurement 77

4.4.2 Mass Measurement 794.4.3 Gas/Liquid Displacement 80

4.4.4 pvT Method 80

4.4.5 Critical Nozzles 814.4.6 Soap Film Burette Method 814.5 Transfer Standards and Master Meters 824.6 In Situ Calibration 844.7 Calibration Uncertainty 914.8 Traceability and Accuracy of Calibration Facilities 924.9 Chapter Conclusions 93

5.1 Introduction 955.2 Essential Background Equations 975.3 Design Details 1005.4 Installation Constraints 1025.5 Other Orifice Plates 106

5.6 Deflection of Orifice Plate at High Pressure 1065.7 Effect of Pulsation 1095.8 Effects of More Than One Flow Component 1135.9 Accuracy Under Normal Operation 1175.10 Industrially Constructed Designs 1185.11 Pressure Connections 1195.12 Pressure Measurement 1225.13 Temperature and Density Measurement 1245.14 Flow Computers 1245.15 Detailed Studies of Flow Through the Orifice Plate, Both

Experimental and Computational 1245.16 Application, Advantages, and Disadvantages 1275.17 Chapter Conclusions 127

APPENDIX 5.A Orifice Discharge Coefficient 128

6.1 Introduction 1306.2 Essential Background Equations 1316.3 Design Details 1346.4 Commercially Available Devices 1356.5 Installation Effects 1356.6 Applications, Advantages, and Disadvantages 1376.7 Chapter Conclusions 138

7.1 Introduction 1407.2 Design Details of a Practical Flowmeter Installation 1417.3 Practical Equations 1437.4 Discharge Coefficient C 1457.5 Critical Flow Function C* 1467.6 Design Considerations 147

7.7 Measurement Uncertainty 148

7.8 Example 1497.9 Industrial and Other Experience 1517.10 Advantages, Disadvantages, and Applications 1527.11 Chapter Conclusions 152

CHAPTER 8 Other Momentum-Sensing Meters 153

8.1 Introduction 1538.2 Variable Area Meter 1538.2.1 Operating Principle and Background 1548.2.2 Design Variations 1548.2.3 Remote Readout Methods 1558.2.4 Design Features 1568.2.5 Calibration and Sources of Error 157

8.2.6 Installation 1578.2.7 Unsteady and Pulsating Flows 1588.2.8 Industrial Types, Ranges, and Performance 1588.2.9 Computational Analysis of the Variable Area Flowmeter 1598.2.10 Applications 1598.3 Spring-Loaded Diaphragm (Variable Area) Meters 1598.4 Target (Drag Plate) Meter 1628.5 Integral Orifice Meters 1638.6 Dall Tubes and Devices that Approximate to Venturis

and Nozzles 1638.7 Wedge and V-Cone Designs 1658.8 Differential Devices with a Flow Measurement Mechanism

in the Bypass 1678.9 Slotted Orifice Plate 1688.10 Pipework Features - Inlets 1688.11 Pipework Features - Bend or Elbow Used as a Meter 1698.12 Averaging Pitot 1708.13 Laminar or Viscous Flowmeters 1738.14 Chapter Conclusions 176

APPENDIX 8.A History, Equations, and Accuracy Classes

for the VA Meter 177

8.A.1 Some History 1778.A.2 Equations 1788.A.3 Accuracy Classes 180

CHAPTER 9 Positive Displacement Flowmeters 182

9.1 Introduction 1829.1.1 Background 1829.1.2 Qualitative Description of Operation 1839.2 Principal Designs of Liquid Meters 1849.2.1 Nutating Disk Meter 1849.2.2 Oscillating Circular Piston Meter 1849.2.3 Multirotor Meters 1859.2.4 Oval Gear Meter 1859.2.5 Sliding Vane Meters 1879.2.6 Helical Rotor Meter 1899.2.7 Reciprocating Piston Meters 1909.2.8 Precision Gear Flowmeters 1909.3 Calibration, Environmental Compensation, and Other Factors

Relating to the Accuracy of Liquid Flowmeters 1919.3.1 Calibration Systems 1929.3.2 Clearances 1949.3.3 Leakage Through the Clearance Gap Between Vane

and Wall 1949.3.4 Slippage Tests 1969.3.5 The Effects of Temperature and Pressure Changes 1979.3.6 The Effects of Gas in Solution 197

9.4 Accuracy and Calibration 1989.5 Principal Designs of Gas Meters 1999.5.1 Wet Gas Meter 1999.5.2 Diaphragm Meter 2009.5.3 Rotary Positive Displacement Gas Meter 2029.6 Positive Displacement Meters for Multiphase Flows 2039.7 Meter Using Liquid Plugs to Measure Low Flows 2059.8 Applications, Advantages, and Disadvantages 2059.9 Chapter Conclusions 206

APPENDIX 9.A Theory for a Sliding Vane Meter 207

9.A.I Flowmeter Equation 2079.A.2 Expansion of the Flowmeter Due to Temperature 2099.A.3 Pressure Effects 2109.A.4 Meter Orientation 2109.A.5 Analysis of Calibrators 2119.A.6 Application of Equations to a Typical Meter 213

10.1 Introduction 21510.1.1 Background 21510.1.2 Qualitative Description of Operation 21510.1.3 Basic Theory 21610.2 Precision Liquid Meters 22110.2.1 Principal Design Components 22110.2.2 Bearing Design Materials 22310.2.3 Strainers 22410.2.4 Materials 22410.2.5 Size Ranges 22510.2.6 Other Mechanical Design Features 22510.2.7 Cavitation 22610.2.8 Sensor Design and Performance 22710.2.9 Characteristics 22810.2.10 Accuracy 22810.2.11 Installation 22910.2.12 Maintenance 23110.2.13 Viscosity, Temperature, and Pressure 23210.2.14 Unsteady Flow 23210.2.15 Multiphase Flow 23210.2.16 Signal Processing 23310.2.17 Applications 23310.2.18 Advantages and Disadvantages 23410.3 Precision Gas Meters 23410.3.1 Principal Design Components 23410.3.2 Bearing Design 23510.3.3 Materials 23610.3.4 Size Range 23610.3.5 Accuracy 236

Special DesignsOther Propeller and Turbine Meters

APPENDIX 10.A Turbine Flowmeter Theory

Installation EffectsEffect of Pulsation and Pipeline VibrationTwo-Phase Flows

Size and Performance Ranges and Materials

in Industrial DesignsComputation of Flow Around Bluff BodiesApplications, Advantages, and DisadvantagesFuture Developments

Swirl Meter - Industrial Design

245245246246

251 253 253 253 254 255 257 259 260 263 264 264 264 267 267 268 269 270

271272272273

11.4.3 Materials 27311.4.4 Installation Effects 27311.4.5 Applications, Advantages, and Disadvantages 27311.5 Fluidic Flowmeter 27411.5.1 Design 27411.5.2 Accuracy 27511.5.3 Installation Effects 27611.5.4 Applications, Advantages, and Disadvantages 27611.6 Other Proposed Designs 27611.7 Chapter Conclusions 276

11.A.I Vortex Shedding from Cylinders 27811.A.2 Order of Magnitude Calculation of Shedding Frequency 279CHAPTER 12 Electromagnetic Flowmeters 28212.1 Introduction 28212.2 Operating Principle 28212.3 Limitations of the Theory 28412.4 Design Details 28612.4.1 Sensor or Primary Element 28612.4.2 Transmitter or Secondary Element 28912.5 Calibration and Operation 29212.6 Industrial and Other Designs 29312.7 Installation Constraints - Environmental 29512.7.1 Surrounding Pipe 29612.7.2 Temperature and Pressure 29612.8 Installation Constraints - Flow Profile Caused by Upstream Pipework 29712.8.1 Introduction 29712.8.2 Theoretical Comparison of Meter Performance Due to

Upstream Flow Distortion 29712.8.3 Experimental Comparison of Meter Performance Due to

Upstream Flow Distortion 29812.8.4 Conclusions on Installation Requirements 29912.9 Installation Constraints - Fluid Effects 30012.9.1 Slurries 30012.9.2 Change of Fluid 30012.9.3 Nonuniform Conductivity 30012.10 Multiphase Flow 30112.11 Accuracy Under Normal Operation 30112.12 Applications, Advantages, and Disadvantages 30212.12.1 Applications 30212.12.2 Advantages 30312.12.3 Disadvantages 30312.13 Chapter Conclusions 304

APPENDIX 12.A Brief Review of Theory 305

12 A.I Introduction 305

12.A.2 Electric Potential Theory 30712.A.3 Development of the Weight Vector Theory 30712.A.4 Rectilinear Weight Function 30812.A.5 Axisymmetric Weight Function 31012.A.6 Performance Prediction 31012.A.7 Further Extensions to the Theory 311

CHAPTER 13 Ultrasonic Flowmeters 312

13.1 Introduction 31213.2 Transit-Time Flowmeters 31513.2.1 Simple Explanation 31513.2.2 Flowmeter Equation and the Measurement of

Sound Speed 31613.2.3 Effect of Flow Profile and Use of Multiple Paths 31913.3 Transducers 32213.4 Size Ranges and Limitations 32513.5 Signal Processing and Transmission 32513.6 Accuracy 32713.6.1 Reported Accuracy - Liquids 32713.6.2 Reported Accuracy - Gases 32713.6.3 Manufacturers' Accuracy Claims 32813.6.4 Special Considerations for Clamp-On Transducers 32813.7 Installation Effects 33013.7.1 Effects of Distorted Profile by Upstream Fittings 33013.7.2 Unsteady and Pulsating Flows 33413.7.3 Multiphase Flows 33513.8 General Published Experience in Transit-Time Meters 33513.8.1 Experience with Liquid Meters 33513.8.2 Gas Meter Developments 33813.9 Applications, Advantages, and Disadvantages 34413.10 Doppler Flowmeter 34513.10.1 Simple Explanation of Operation 34513.10.2 Operational Information 34613.10.3 Applications, Advantages, and Disadvantages 34613.11 Correlation Flowmeter 34613.11.1 Operation of the Correlation Flowmeter 34613.11.2 Installation Effects 34713.11.3 Other Published Work 34813.11.4 Applications, Advantages, and Disadvantages 34913.12 Other Ultrasonic Applications 34913.13 Chapter Conclusions 350

APPENDIX 13.A Simple Mathematical Methods and Weight

Function Analysis Applied to Ultrasonic Flowmeters 351

13.A.1 Simple Path Theory 35113.A.2 Use of Multiple Paths to Integrate Flow Profile 35313.A.3 Weight Vector Analysis 35513.A.4 Doppler Theory 355

CHAPTER 14 Mass Flow Measurement Using Multiple Sensors

for Single- and Multiphase Flows 357

Multiple Differential Pressure Meters

14.2.1 Hydraulic Wheatstone Bridge Method

14.2.2 Theory of Operation

14.2.3 Industrial Experience

14.2.4 Applications

Multiple Sensor Methods

Multiple Sensor Meters for Multiphase Flows

14.4.1 Background

14.4.2 Categorization of Multiphase Flowmeters

14.4.3 Multiphase Metering for Oil Production

Chapter Conclusions

14.5.1 What to Measure If the Flow Is Mixed

14.5.2 Usable Physical Effects for Density Measurement

14.5.3 Separation or Multicomponent Metering

14.5.4 Calibration

14.5.5 Accuracy

357 357 359 359 360 361 361 362 362 363 365 367 367 368 369 369 370 371

15.1 Introduction 37115.2 Capillary Thermal Mass Flowmeter - Gases 37115.2.1 Description of Operation 37115.2.2 Operating Ranges and Materials for Industrial Designs 37415.2.3 Accuracy 37415.2.4 Response Time 37415.2.5 Installation 37515.2.6 Applications 37615.3 Calibration of Very Low Flow Rates 37615.4 Thermal Mass Flowmeter - Liquids 37615.4.1 Operation 37615.4.2 Typical Operating Ranges and Materials for Industrial Designs 37715.4.3 Installation 37815.4.4 Applications 37815.5 Insertion and In-Line Thermal Mass Flowmeters 37815.5.1 Insertion Thermal Mass Flowmeter 37915.5.2 In-Line Thermal Mass Flowmeter 38115.5.3 Range and Accuracy 38115.5.4 Materials 38115.5.5 Installation 38115.5.6 Applications 38215.6 Chapter Conclusions 383

APPENDIX 15.A Mathematical Background to the Thermal

Mass Flowmeters 384

15.A.I Dimensional Analysis Applied to Heat Transfer 38415.A.2 Basic Theory of ITMFs 385

15.A.3 General Vector Equation 38615.A.4 Hastings Flowmeter Theory 38815.A.5 Weight Vector Theory for Thermal Flowmeters 389

16.1 Introduction 39116.2 The Fuel Flow Transmitter 39216.2.1 Qualitative Description of Operation 39416.2.2 Simple Theory 39416.2.3 Calibration Adjustment 39516.2.4 Meter Performance and Range 39616.2.5 Application 39616.3 Chapter Conclusions 397CHAPTER 17 Coriolis Flowmeters 39817.1 Introduction 39817.1.1 Background 39817.1.2 Qualitative Description of Operation 40017.1.3 Experimental Investigations 40217.2 Industrial Designs 40217.2.1 Principal Design Components 40417.2.2 Materials 40717.2.3 Installation Constraints 40717.2.4 Vibration Sensitivity 40817.2.5 Size and Flow Ranges 40817.2.6 Density Range and Accuracy 40917.2.7 Pressure Loss 41017.2.8 Response Time 41017.2.9 Zero Drift 41017.3 Accuracy Under Normal Operation 41217.4 Performance in Two-Component Flows 41317.4.1 Air-Liquid 41417.4.2 Sand in Water 41417.4.3 Pulverized Coal in Nitrogen 41417.4.4 Water-in-Oil Measurement 41417.5 Industrial Experience 41517.6 Calibration 41617.7 Applications, Advantages, Disadvantages, and Cost Considerations 41617.7.1 Applications 41617.7.2 Advantages 41817.7.3 Disadvantages 41917.7.4 Cost Considerations 41917.8 Chapter Conclusions 420

APPENDIX 17.A A Brief Note on the Theory of Coriolis Meters 421

17.A.I Simple Theory 421

17.A.3 Theoretical Developments 424

CHAPTER 18 Probes for Local Velocity Measurement in Liquids

and Gases 427

18.1 Introduction 42718.2 Differential Pressure Probes - Pitot Probes 42818.3 Differential Pressure Probes - Pitot-Venturi Probes 43018.4 Insertion Target Meter 43118.5 Insertion Turbine Meter 43118.5.1 General Description of Industrial Design 43118.5.2 Flow-Induced Oscillation and Pulsating Flow 43318.5.3 Applications 43418.6 Insertion Vortex Probes 43518.7 Insertion Electromagnetic Probes 43518.8 Insertion Ultrasonic Probes 43618.9 Thermal Probes 43718.10 Chapter Conclusions 437

19.1 Introduction 43819.1.1 Analogue Versus Digital 43919.1.2 Present and Future Innovations 43919.1.3 Industrial Implications 44019.1.4 Chapter Outline 44019.2 Instrument 44119.2.1 Types of Signal 44119.2.2 Signal Content 44219.3 Interface Box Between the Instrument and the System 44319.4 Communication Protocol 44419.4.1 Bus Configuration 44419.4.2 Bus Protocols 44519.5 Communication Medium 44619.5.1 Existing Methods of Transmission 44619.5.2 Present and Future Trends 44619.5.3 Options 44719.6 Interface Between Communication Medium and the Computer 44819.7 The Computer 44819.8 Control Room and Work Station 44819.9 Hand-Held Interrogation Device 44919.10 An Industrial Application 44919.11 Future Implications of Information Technology 449

CHAPTER 20 Some Reflections on Flowmeter Manufacture,

Production, and Markets 451.

20.1 Introduction 45120.2 Instrumentation Markets 45120.3 Making Use of the Science Base 45320.4 Implications for Instrument Manufacture 45420.5 The Special Features of the Instrumentation Industry 454

20.6 Manufacturing Considerations 45520.6.1 Production Line or Cell? 45520.6.2 Measures of Production 45620.7 The Effect of Instrument Accuracy on Production Process 45620.7.1 General Examples of the Effect of Precision of Construction

on Instrument Quality 45720.7.2 Theoretical Relationship Between Uncertainty in

Manufacture and Instrument Signal Quality 45720.7.3 Examples of Uncertainty in Manufacture Leading to

Instrument Signal Randomness 45920.8 Calibration of the Finished Flowmeters 46120.9 Actions for a Typical Flowmeter Company 461

CHAPTER 21 Future Developments 463

21.1 Market Developments 46321.2 Existing and New Flow Measurement Challenges 46321.3 New Devices and Methods 46521.3.1 Devices Proposed but Not Exploited 46521.3.2 New Applications for Existing Devices 46721.3.3 Microengineering Devices 46721.4 New Generation of Existing Devices 46921.5 Implications of Information Technology 47021.5.1 Signal Analysis 47021.5.2 Redesign Assuming Microprocessor Technology 47021.5.3 Control 47021.5.4 Records, Maintenance, and Calibration 47121.6 Changing Approaches to Manufacturing and Production 47121.7 The Way Ahead 47121.7.1 For the User 47121.7.2 For the Manufacturer 47121.7.3 For the Incubator Company 47121.7.4 For the R&D Department 47221.7.5 For the Inventor/Researcher 47221.8 Closing Remarks 472

Bibliography 473

A Selection of International Standards 475 Conferences 479 References 483 Index 515 Main Index 515 Flowmeter Index 518 Flowmeter Application Index 521

This is a book about flow measurement and flowmeters written for all in the try who specify and apply, design and manufacture, research and develop, maintainand calibrate flowmeters It provides a source of information on the published re-search, design, and performance of flowmeters as well as on the claims of flowmetermanufacturers It will be of use to engineers, particularly mechanical and processengineers, and also to instrument companies' marketing, manufacturing, and man-agement personnel as they seek to identify future products

indus-I have concentrated on the process, mechanical, and fluid engineering aspectsand have given only as much of the electrical engineering details as are necessaryfor a proper understanding of how and why the meters work I am not an electri-cal engineer and so have not attempted detailed explanations of modern electricalsignal processing I am also aware of the speed with which developments in signalprocessing would render any descriptions which I might give out of date

In the bibliography, other books dealing with flow measurement are listed, and

my intention is not to retread ground covered by them, more than is necessary andunavoidable, but to bring together complementary information I also make theassumption that the flowmeter engineer will automatically turn to the appropriatestandard; therefore, I have tried to avoid reproducing information that should beobtained from those excellent documents I include a brief list that categorizes afew of the standards according to meter or application I also recommend that thoseinvolved in new developments keep a watchful eye on the regular conferences, whichcarry much of the latest developments in the business

I hope, therefore, that this book will provide a signpost to the essential tion required by all involved in the development and use of flowmeters, from thefield engineer to the chief executive of the entrepreneurial company that is devel-oping its product range in this technology

informa-In this book, following introductory chapters on accuracy, flow, selection, andcalibration, I have attempted to provide a clear explanation of each type of flowmeter

so that the reader can easily understand the workings of the various meters I havethen attempted to bring together a significant amount of the published informationthat explains the performance and applications of flowmeters The two sources forthis are the open literature and the manufacturers' brochures I have also introduced,

to a varying extent, the mathematics behind the meter operations, but to avoiddisrupting the text, I have consigned this, in most cases, to the appendices at theend of many chapters This follows the approach that I have used for technical reviewpapers on turbine meters, Coriolis meters, and to a lesser extent earlier papers on

electromagnetic flowmeters, positive displacement flowmeters, and flowmeters inmultiphase flows.

However, when searching the appropriate databases for flowmeter papers, Iquickly realized that including references to all published material was unrealistic

I have attempted to select those references that appeared to be most relevant andavailable to the typical reader of this book However, the reader is referred to thelist of journals and conferences that were especially valuable in writing this book

In particular, the Journal of Flow Measurement and Instrumentation has filled a gap in

the market, judging by the large number and high quality of the papers published

by the journal It is likely that, owing to the problems of obtaining papers, I haveomitted some that should have been included

Topics that I do not consider to be within the subject of bulk flow measurement

of liquids and gases, and that are not covered in this book, are metering pumps,flow switches, flow controllers, flow measurement of solids and granular materials,open channel flow measurement, hot-wire local velocity probes or laser doppleranemometers, and subsidiary instrumentation

In two areas where I know that I am lacking in first-hand knowledge - moderncontrol methods and manufacturing - 1 have included a brief review, which shouldnot be taken as expert information However, I want to provide a source of infor-mation for existing and prospective executives in instrumentation companies whomight need to identify the type of products for their companies' future develop-ments This requires a knowledge of the market for each type of flowmeter andalso an understanding of who is making each type of instrument It requires somethought regarding the necessities of manufacturing and production and the impli-cations for this in any particular design

I have briefly referred to future directions for development in each chapter whereappropriate, and in the final chapter I have drawn these ideas together to provide aforward look at flow metering in general

The techniques for precise measurement of flow are increasingly important day when the fluids being measured, and the energy involved in their movement,may be very expensive If we are to avoid being prodigal in the use of our naturalresources, then the fluids among them should be carefully monitored Flow mea-surement contributes to that monitoring and, therefore, demands high standards ofprecision and integrity

My knowledge of this subject has benefited from many others with whom I haveworked and talked over the years These include colleagues from industry andacademia, and students, whether in short courses or longer-term degree courses andresearch I hope that the book does justice to all that they have taught me

In writing this book, I have drawn on the information from many manufacturers,and some have been particularly helpful in agreeing to the use of information anddiagrams I have acknowledged these companies in the captions to the figures Somewent out of their way to provide artwork, and I am particularly grateful to them.Unfortunately, space, in the end, prevented me from using many of the excellentdiagrams and photographs with which I was provided

In the middle of already busy lives, the following people kindly read throughsections of the book, of various lengths, and commented on them: Heinz Bernard(Krohne Ltd.), Reg Cooper (Bailey-Fischer & Porter), Terry Cousins (T&SK Flow Con-sultants), Chris Gimson (Endress & Hauser), Charles Griffiths [Flow Automation(UK) Ltd.], John Hemp (Cranfield University), Yousif Hussain (Krohne Ltd.), Peterlies-Smith (Yokagawa United Kingdom Limited), Alan Johnson (Fisher-Rosemount),David Lomas (ABB Kent-Taylor), Graham Mason (GEC-Marconi Avionics), JohnNapper (formerly with FMA Ltd.), Kyung-Am Park (Korea Research Institute of Stan-dards and Science), Bob Peters (Daniel Europe Ltd.), Roger Porkess (University ofPlymouth), Phil Prestbury (Fisher-Rosemount), David Probert (Cambridge Univer-sity), Karl-Heinz Rackebrandt (Bailey-Fischer & Porter), Bill Pursley (NEL), JaneSattary (NEL), Colin Scott (Krohne Ltd.), John Salusbury (Endress & Hauser), DaveSmith (NEL), Ian Sorbie (Meggitt Controls), Eddie Spearman (Daniel Europe Ltd.),

J D Summers-Smith (formerly with I.C.I), and Ben Weager (Danfoss FlowmeteringLtd.) I am extremely grateful to them for taking time to do this and for the con-structive comments they gave Of course, I bear full responsibility for the final script,although their help and encouragement was greatly valued

I am also grateful to Dr Michael Reader-Harris for his advice on the orifice platedischarge coefficient equation, and to Prof Stan Hutton for his help and encourge-ment when Chapter 10 was essentially in the form of technical papers

I acknowledge with thanks the following organizations that have given sion to use their material:

permis-ASME for agreeing to the reproduction of Figures 5.10(a), 10.11, 10.16, 17.1, 18.2,and 18.4

Elsevier Science Ltd for permission to use Figures 4.18, 5.5, 5.10(b), 5.12, 8.6,11.7,11.11,11.13,11.14,11.17,13.9, 21.1, 21.2 and for agreement to honor my right

to use material from my own papers for Chapters 10 and 17

National Engineering Laboratory (NEL) for permission to reproduce Figures 4.9,4.14-4.16, 4.19, 5.11, 11.4-11.6, and 14.5

Professional Engineering Publishing for permission to draw on material from theIntroductory Guide Series of which I am Editor, and to the Council of the Insti-tution of Mechanical Engineers for permission to reproduce material identified

in the text as being from Proceedings Part C, Journal of Mechanical Engineering Science, Vol 205, pp 217-229, 1991.

Extracts from BS EN ISO 5167-1:1997 are reproduced with the permission of BSIunder licence no PD\ 19980886 Complete editions of the standards can be obtained

by mail from BSI Customer Services, 389 Chiswick High Road, London W4 4AL,United Kingdom

I am also grateful for the help and encouragement given to me by many in thepreparation of this book It would be difficult to name them, but I am grateful for eachcontribution The support of my family must be mentioned In various ways they allcontributed - by offering encouragement, by undertaking some literature searches,

by doing some early typing work, and by helping with some of the diagrams Iparticularly thank my wife whose encouragement and help at every stage, not tomention putting up with a husband glued to the word processor through days,evenings, holidays, etc., ensured that the book was completed

My editor has been patient, first as I overran the agreed delivery date and then

as I overran the agreed length I am grateful to Florence Padgett for being willing tooverlook this lack of precision!

CHAPTER 1

Q Sensitivity coefficient

f{x) Function for Normal distribution

K K factor in pulses per unit flow quantity

q v Volumetric flow rate

q vo Volumetric flow rate at calibration point

u(xi) Standard uncertainty for the zth quantity

u c (y) Combined standard uncertainty

z Normalized coordinate (x — /x)/a

l± Mean value of data for normal curve

v Degrees of freedom

o Standard deviation (a 2 variance)

4>(z) Area under Normal curve [e.g., 4>(0.5) is

the area from z = -oo to z = 0.5]

</>(x) Function for normalized Normal

Local speed of sound

Specific heat at constant pressure

Specific heat at constant volume

APioss Pressure loss across a pipe fitting

q v Volumetric flow rate

q m Mass flow rate

V Velocity in pipe, Volume of pipework

and other vessels between the source

of the pulsation and the flowmeter position

Vb Velocity on pipe axis Kms Fluctuating component of velocity

V Mean velocity in pipe

z Elevation above datum

y Ratio of specific heats

n Bearing rotational speed

a Combined roughness of the two

contacting surfaces of the bearing

CHAPTER 4

Q Concentration of tracer in the main

stream at the downstream sampling

point

Cdmean Mean concentration of tracer measured

downstream during time t

Q Concentration of tracer in the injected

stream

C u Concentration of tracer in the main

stream upstream of injection point

(if the tracer material happens to be

q v Volumetric flow rate in the line

q V i Volumetric flow rate of injected tracer

R Gas constant for a particular gas

T Temperature

t Collection time during calibration,

Integration period for tracer

Part of discharge coefficient which

allows for position of taps

Discharge coefficient for infinite

Reynolds number

Expression in orifice plate bending

formula

Constant

Pipe diameter (ID)

Orifice plate support diameter

Orifice diameter

Velocity of approach factor (1 - £ 4 )~~ 1/2: ,

Thickness of the orifice plate

Total error in the indicated flow rate of a

flowmeter in pulsating flow

Elastic modulus of plate material

Thickness of the orifice

Correction factors used to obtain the

mass flow of a (nearly) dry steam flow

f

H h K

qm qv

Re r

= h/D

= l' 2 /D (The prime signifies that the

measurement is from the downstream face of the plate)

Distance of the upstream tapping from the upstream face of the plate

Distance of the downstream tapping from the downstream face of the plate (The prime signifies that the

measurement is from the downstream face of the plate)

= 2L' 2/(1 - p)

Index Downstream pressure Upstream pressure Differential pressure, pressure drop between pulsation source and meter Mass flow rate

Volumetric flow rate Reynolds number usually based on the pipe ID

Radius of upstream edge of orifice plate Time

Volume of pipework and other vessels between the source of the pulsation and the flowmeter position

Mean velocity in pipe with pulsating flow

Root-mean-square value of unsteady velocity fluctuation in pipe with pulsating flow

Dryness fraction Flow coefficient, CE

Diameter ratio, d/D

Ratio of specific heats

Small changes or errors in q m , etc.

Expansibility (or expansion) factor Expansibility (or expansion) factor for

orifice

Isentropic exponent Density at the upstream pressure tapping cross-section

Yield stress for plate material Ratio of two-phase pressure drop to liquid flow pressure drop

Maximum allowable percentage error in pulsating flow

6

Coefficient of discharge Part of coefficient of discharge affected

by Reynolds number

C t p Coefficient for wet gas flow equation

Coo Discharge coefficient for infinite

q g Gas volumetric flow rate

qi Liquid volumetric flow rate

q m Mass flow rate

q tp Apparent volumetric flow rate when

liquid is present in the gas stream

q v Volume flow rate

Re Reynolds number based on D

Red Reynolds number based on d

Vsg Superficial gas velocity

A2 Outlet cross-sectional area

A* Throat cross-sectional area

c p Specific heat at constant pressure

c v Specific heat at constant volume

Mi Mach number at inlet when stagnation

conditions cannot be assumed

M Molecular weight

n Exponent in Equation (7.12)

po Stagnation pressure

pi Pressure at inlet when stagnation

conditions cannot be assumed

p2i Ideal outlet pressure

p2max Actual maximum outlet pressure

p* Throat pressure in choked conditions

R

Re d

To

% X Z

ZQ p y

K V

PO

Mass flow Universal gas constant Reynolds number based on the throat diameter

Stagnation temperature Throat temperature in choked conditions Mole fraction of each component of a gas mixture

Compressibility factor Compressibility factor at stagnation conditions

d/D

Ratio of specific heats Error

Isentropic exponent Kinematic viscosity Density at stagnation conditions

CHAPTER 8

A Cross-sectional area of the pipe, Constant A' Constant

Af Cross-sectional area of float

A x Cross-sectional area of tapering tube at

C c Contraction coefficient, Constants in

curve fit for target meter discharge coefficient

D Pipe diameter

d Throat diameter for pipe inlet

E Full-scale or upper range value of flow

rate used in precision calculation

F Summation error in flow rate

g Gravity

K Loss coefficient, Precision class, Bend or

elbow meter coefficient

L Length of laminar flow tube

M Actual flow rate used in precision

calculation

p Pressure

q v Volumetric flow rate

R Radius of bend or elbow

Re Reynolds number

V Velocity, Volume of float

V Mean velocity in tube

v Specific volume of gas

x Height of float in tube

/x Viscosity

fig Viscosity of calibration gas at

flowing conditions /x std Viscosity of reference gas at

standard conditions

p Density

Pi Density of float material

CHAPTER 9

E Young's modulus of elasticity

F Friction force

g Acceleration due to gravity

L Length of clearance gap in direction

of flow

I Axial length of clearance gap

£ ax Axial length of measuring chamber

M Mass of rotor

N Rotational speed of rotor

N[ Rotational speed of cam disk of calibrator

(clutch system)

NIN Rotational speed of shaft into calibrator

(epicyclic system)

N m ax M a x i m u m rotational speed of rotor

N o Rotational speed of outer ring of

calibrator (clutch system)

NOUT Rotational speed of shaft out of

calibrator (epicyclic system)

N s Slippage in NIN

n Number of teeth o n gear

pa Downstream pressure

p u Upstream pressure

<Zideai A s u s e d i n Equation (9.A.6)

leakage Leakage flow rate

<?BULK Bulk volumetric flow rate

<7siip Volumetric flow rate through meter at no

rotation

q v Volumetric flow rate

R Radius of friction wheel

r Radius of point of friction wheel

contact

r x Inner radius of measuring chamber

r o Outer radius of measuring chamber

r s Shaft radius

T Temperature, Torque

TIN Torque on input shaft to calibrator

TOUT Torque out from calibrator

7b Constant drag torque

T\ Speed-dependent drag torque

t Clearance between stationary and

moving members

t 0 Thickness of outer casing

u Fluid velocity

y Position coordinate across clearance gap

a m Coefficient of linear expansion of metal

«i Coefficient of volume expansion of

liquid

A Change in quantity, Distance between

center of rotation of arm and center of

discs in calibrator

8 Reduction in area of the measuring

chamber due to blade sections

9 Q Angular position of center disk of

calibrator

0[ Angular position of cam disk of calibrator

0 o Angular position of outer ring of

calibrator

Viscosity Angular position of arm of calibrator

OTHER / 1

m

1,2 1-8

SUBSCRIPTS

Dummy suffix for summation Liquid

Metal Different materials Epicyclic gears

K h

L N n

P

poi P02

Cross-sectional area of the effective annular flow passage at the rotor blades Turbine meter aerodynamic torque coefficient

Constant coefficients Axial length of rotor Turbine meter aerodynamic torque coefficient, Bearing length, Frequency response coefficient unless used as an exponent

Proportional to 1/K

Drag coefficient Drag coefficient adjusted to allow forC h

Fluid drag coefficient Constant for a particular design (Behaves like an adjustment to the main

aerodynamic drag term coefficient) Lift coefficient

Chord Drag force, Pipe diameter, Impeller diameter

Rotor response parameter Fluid drag parameter Nonfluid drag parameter Additional geometric variables Meter error

Nonfluid forces Frequency of blade passing Mass moment of inertia of rotor system about rotor axis

Incidence angle

Incidence at radius r Lattice effect coefficient, K factor

(pulses/unit volume) Lattice coefficient at radius r Turbine meter resisting torque coefficients

Constant used in equation for helical blade angle

Lift force, Helical pitch of blades Number of blades

Rotational speed (frequency) Index in error equation Stagnation pressure at inlet Stagnation pressure at outlet

pi Static pressure at inlet

p2 Static pressure at outlet

q Actual average flow rate, Index in error

equation

qo Time average flow rate over period T

q\ Initial flow rate

qi Final flow rate

q b Base flow rate

q\ Indicated average flow rate

<?max Maximum flow rate for which the meter

is designed

<7min Minimum flow rate for which the meter

is designed

q n Normal flow rate

qt Flow rate at time t

^trans Flow rate at change of precision

q v Volumetric flow rate

R Pipe bend radius

T Temperature, Fundamental period of

pulsating flow, Torque

TB Bearing drag torque

T c Temperature at calibration

T d Driving torque

Tpo Mechanical friction torque o n rotor at

zero speed

T h H u b fluid drag torque

T n Nonfluid drag torque

7^> Temperature at operation

T r Retarding torque

T t Blade tip clearance drag torque

T w H u b disk friction drag torque

t Time

£B Blade thickness

£R Relaxation time

V N o n d i m e n s i o n a l fluid velocity

Vo Time average value of V z over period T

V\ Inlet relative flow velocity

V2 Outlet relative flow velocity

^rnax M a x i m u m value of V z

^rnin M i n i m u m value of V z

V z Axial velocity (instantaneous inlet fluid

velocity of pulsating flow)

W Rotor blade velocity, Nondimensional

instantaneous rotor velocity under

X

CHAPTER

A

a B

D

f

H h K L

A/7 Patmos Pgmin

Pv qv

Re S s

Blade angle at radius r measured from

axial direction of meter Relative inlet angle of flow Relative outlet angle of flow Mean of the inlet and outlet flow field Full-flow amplitude relative to average flow

Deflection of flow at blade outlet from blade angle

Flow deviation factor Dynamic viscosity Kinematic viscosity Fluid density

Nondimensional time (t/T), Period of

modulation Rotor coast time to standstill Instantaneous rotor angular velocity

11

Pipe full flow area, Constant of value about 3.0 used in equation for avoiding cavitation in flows past vortex meters Flow area past bluff body

Area when integrating vorticity Constant of value about 1.3 used in equation for avoiding cavitation in flows past vortex meters

Pipe ID Shedding frequency Streamwise length of bluff body Parallel flats on sides of bluff body at leading edge

Calculation factor for shedding frequency (Zanker and Cousins 1975),

K factor = pulses/unit volume

Length of bluff body across pipe between end fittings

Pressure drop across vortex meter (about 1 bar at 10 m/s)

Atmospheric pressure Minimum back pressure 5D downstream

of vortex meter Saturated liquid vapor pressure Volumetric flow rate

Reynolds number Strouhal number Length along curve when integrating velocity around a vortex

Velocity of flow Velocity of flow past bluff body Diameter or width of bluff body Shear layer thickness

Injected swirling flow rate/total flow rate

Vorticity

CHAPTER 12

A Used for area of electrode leads forming

a loop causing quadrature signals in the

electromagnetic flowmeter

a Pipe radius

B Magnetic flux density in tesla

B Magnetic flux density vector

Bo Maximum value of magnetic flux

density

Be 6 component of magnetic flux density

b Inner radius of an annulus of conducting

fluid in two phase annular flow

D Diameter of pipe

E Electric field vector

f Frequency of magnetic field excitation

j Current density vector

/ Length of wire traversing magnetic field

Ap Pressure of the pipeline fluid above

atmospheric

q v Volumetric flow rate

r Radial coordinate

5 Electromagnetic flowmeter sensitivity

t Pipe wall thickness

U Electric potential

AI/EE Voltage between electrodes

AC/p Voltage across a wire P moving through

a magnetic field (similarly for Q and R)

W Weight function vector

W Rectilinear flow weight function

W" Axisymmetric weight function

Frequency of stable quartz oscillator for

ultrasonic measurement system

Frequency for the downstream

sing-around pulse train

Transmitted doppler frequency

Reflected doppler frequency

Frequency for the upstream sing-around

pulse train

y

z

z (3

7

e

X

p Pm

T

x

fit) h I L

PR PI

Re

Ryxir) T t tm At

\/n is the index for turbulent profile

curve fit, Index Ultrasound power reflected Ultrasound power transmitted Mass flow rate

Reynolds number Correlation coefficient Correlation time for integration Pipe wall thickness

Downstream wave transit time Mean wave transit time Upstream wave transit time Difference between these two wave transit times

Beam positions for ultrasonic flowmeter using Gaussian quadrature

Received voltages for ultrasonic flowmeter

Flow velocity in pipe Mean velocity in pipe Flow velocity profile along pipe chord Ultrasonic flowmeter created flow Acoustic field for cases (1) and (2) Relative flow velocity in the direction of the acoustic beam

Vector weight function Axial length between transducers in ultrasonic flowmeter

Length of chord in the plane of the ultrasonic path

Distance along chord used in ultrasonic flowmeter

Acoustic impedance Ultrasonic beam deflection distance on opposite wall

Ultrasonic beam deflection angle Ratio of specific heats for a particular gas Ultrasound beam angle

Ultrasound wave length Density of material Mean density in ultrasonic meter Measurement period for ultrasonic system, Time period between ultrasonic wave peaks Mean ultrasonic correlation time

A/?A Pressure drop to throat of venturi A

A/?B Pressure drop to throat of venturi B

ApAB etc Differential pressure between

venturi throats, across limbs of the

hydraulic Wheatstone bridge, or across

diagonals of the hydraulic Wheatstone

bridge

q m Total mass flow

q vp Metering pump volumetric transfer flow

V Velocity in the meter

7 Ratio of specific heats

f> Density of the fluid

c p Specific heat at constant pressure

c v Specific heat at constant volume

k Thermal conductivity of the fluid

k f Constant allowing for heat transfer and

temperature difference at zero flow

I Finite difference dimension

<j a Rate of heat addition per unit volume

q\y Heat flux

q h Heat flux vector

q m Mass flow rate

q v Volumetric flow rate

R Resistance of heating element

Re Reynolds number

5 Flow signal

So Flow signal at start

St Stanton number

T Absolute temperature of the fluid

Ti,Tz,T c Temperatures used in finite difference

T Time difference between markers on

the angular momentum meter rotating assembly

co Angular velocity of the rotor

CHAPTER

A d

F h

qm

r 8r 8r' T t V

V

v (!)

v (2)w

0

p X

Inertia in plane of normal oscillation Constant, Allows for the fact that the twist of the tube will not form a straight integration

Spring constant of the U-tube in twisting oscillation

Spring constant of the U-tube in normal oscillation

Length of the U-tube

Element of mass equal to pA8r'

Mass flow Radius Elementary length of tube Elementary length of fluid Torque

Time Flow velocity Vector velocity Oscillatory velocity field set up in the stationary fluid by the driving oscillator

Oscillatory velocity field set up in the stationary fluid by the Coriolis forces Vector weight function

Twist angle of U-tube Amplitude of twist angle of U-tube when

in sinusoidal motion Density

Difference in transit time of two halves

of twisted U-tube Phase difference between the total velocities at the two sensing points Angular velocity of the pipe caused by the vibration

cos

COu

CHAPTER

Amplitude of the angular velocity of the

pipe caused by the vibration

p3, p4 Dimensional and other factors

q m Mass flow rate

q w Volumetric flow rate

5 Flow signal

At Time difference between upstream and

downstream waves or pulses

t m Mean time of transit used to obtain

sound speed

0 Ultrasound beam angle

r Difference in transit time of two halves

of twisted U-tube

co Driving frequency

co s Natural frequency of U-tube in

twisting oscillation

COMBINED ALL CHAPTERS

A Cross-section of pipe, Function of p and

Re, Cross-sectional area of the pipe,

Constant, Cross-sectional area of the

effective annular flow passage at the

rotor blades, Constant of value about 3.0

used in equation for avoiding cavitation

in flows past vortex meters, Used for area

of electrode leads forming a loop

causing quadrature signals in the

electromagnetic flowmeter, Weighting

factors for Gaussian quadrature, Area of

the duct, Constant equal to 1 In

Q

C L

CR

Constant Cross-sectional area of float Flow area past bluff body Cross-sectional area of tapering tube

at height x

Outlet cross-sectional area, Annular area around float, Annular area around target Throat cross-sectional area

Constant, Area of target, Turbine meter aerodynamic torque coefficient, Area when integrating vorticity, Pipe radius

Constant coefficients Expression in orifice plate bending formula

Constant Constant Constant Constant, Axial length of rotor, Constant

of value about 1.3 used in equation for avoiding cavitation in flows past vortex meters, Magnetic flux density in tesla, Constant equal to (2/TT) 0 - 5

Magnetic flux density vector Maximum value of magnetic flux density

0 component of magnetic flux density

Inner radius of an annulus of conducting fluid in two-phase annular flow Constant, Turbine meter aerodynamic torque coefficient, Bearing length, Frequency response coefficient unless used as an exponent

Constant Constant Constant Discharge coefficient, Coefficient,

Proportional to 1/K, Function of

temperature, Cost Contraction coefficient, Constants in curve fit for target meter discharge coefficient

Drag coefficient Drag coefficient adjusted to allow for C h

Concentration of tracer in the main stream at the downstream sampling point

Mean concentration of tracer measured

downstream during time t

Fluid drag coefficient Constant for a particular design (Behaves like an adjustment to the main

aerodynamic drag term coefficient) Concentration of tracer in the injected stream

Lift coefficient

CRe Part of discharge coefficient affected by Re

Craps P a r t °f discharge coefficient that allows

for position of taps

Qp Coefficient for wet gas flow equation

C u Concentration of tracer in the main

stream upstream of injection point (if the

tracer material happens to be present)

Coo Discharge coefficient for infinite

Reynolds number

C* Critical flow function

c Sound/ultrasound speed, Chord,

D Pipe diameter (ID), Drag force, Impeller

diameter, Function of temperature

D' Orifice plate support diameter

D\ Rotor response parameter

D2 Fluid drag parameter

D3 Nonfluid drag parameter

d Diameter of tube bundle straightener

tubes, Orifice diameter, Throat diameter,

Heating element diameter, Width of

Coriolis flowmeter U-tube

d t Diameter of tapping

d\ Additional geometric variable

dz Outlet diameter, Additional geometric

variable

E velocity of approach factor (1 - £4 )~ 1 / 2 ,

Thickness of the orifice plate, Full-scale

or upper range value of flow rate used in

precision calculation, Young's modulus

of elasticity, Fluid compressibility

E Electric field vector

Ej Total error in the indicated flow rate of a

flowmeter in pulsating flow

E* Elastic modulus of plate material

e Thickness of the orifice, Meter error

eq Electronics quality

F Correction factors used to obtain the

mass flow of a (nearly) dry steam flow,

Summation error in flow rate, Friction

force, Nonfluid forces, Force due to

Coriolis acceleration

Fr g Superficial gas Froude number

f Frequency of the pulsation, Obtained

from Equation (7.17), Frequency of blade

passing, Shedding frequency, Frequency

of magnetic field excitation, Frequency

of stable quartz oscillator for ultrasonic

measurement system

fy Frequency for the downstream

sing-around pulse train

ft Transmitted doppler frequency

ft Reflected doppler frequency

f u Frequency for the upstream sing-around

pulse train

A f Difference between the sing-around

frequencies, Frequency shift in doppler flowmeter

f{t) Function of t fix) Function for Normal distribution

g Acceleration due to gravity

H Hodgson number, Streamwise length of

bluff body

h Thickness of orifice plate, Parallel flats on

sides of bluff body at leading edge, Displacement of ultrasonic beam from axis of pipe

frmin Lubrication film thickness / Mass moment of inertia of rotor system

about rotor axis, Driving current in ultrasonic flowmeter, Current through

resistance R

I s Inertia in plane of twisting oscillation

J u Inertia in plane of normal oscillation

i Incidence angle

/(r) Incidence at radius r

j Current density vector

K Pressure loss coefficient, Related to the

criterion for Hodgson's number, Precision class, Bend or elbow meter

coefficient, Lattice effect coefficient, K

factor (pulses/unit volume), Calculation factor for shedding frequency (Zanker and Cousins 1975), Constant, Allows for the fact that the twist of the tube will not form a straight integration

Kir) Lattice coefficient at radius r K\,Ki Turbine meter resisting torque

k Coverage factor, Roughness, Thermal

conductivity of the fluid, Coefficient

k! Constant allowing for heat transfer and

temperature difference at zero flow

k s Adiabatic compressibility

L Length of laminar flow tube, Length of

clearance gap in direction of flow, Lift force, Helical pitch of blades, Length of bluff body across pipe between end fittings, Distance along path in transit-time flowmeter, Distance between transducers in ultrasonic correlation flowmeter, Finite difference dimension

V 2 /D (The prime signifies that the

measurement is from the downstream

face of the plate)

Length of wire traversing magnetic field,

Length of the U-tube

Distance of the upstream tapping from

the upstream face of the plate

Distance of the downstream tapping

from the downstream face of the plate

(The prime signifies that the

measurement is from the downstream

face of the plate)

Axial length of clearance gap

Axial length of measuring chamber

Mean of a sample of n readings, Mach

number, Actual flow rate used in

precision calculation, Mass of rotor

Molecular weight

Mach number at inlet when stagnation

conditions cannot be assumed

Maximum rotational speed of rotor

Rotational speed of outer ring of

calibrator (clutch system)

Rotational speed of shaft out of calibrator

Number of measurements, Exponent

in equations, Bearing rotational speed,

Number of teeth on gear, Rotational

speed (frequency)

Probability, Pressure, Bearing load,

Exponent in error equation

Ultrasound power reflected

Ultrasound power transmitted

P02

Pi

P2

P2i P2max

P*

ApB ApAB etc

Apioss

Ap

Qh

q q qo q\

pressure at inlet, p\pi dimensional and

other factors Static pressure at outlet Ideal outlet pressure Actual maximum outlet pressure Throat pressure in choked conditions Pressure drop to throat of venturi A Pressure drop to throat of venturi B Differential pressure between venturi throats, across limbs of the hydraulic Wheatstone bridge, or across diagonals of the hydraulic Wheatstone bridge

Pressure loss across a pipe fitting Differential pressure, Pressure drop between pulsation source and meter, Pressure drop across vortex meter (about 1 bar at 10 m/s), Pressure of the pipeline fluid above atmospheric, Dynamic pressure

Heat transfer Actual average flow rate; Exponent in error equation

Mean of n measurements q\

Time average flow rate over period T

Initial flow rate Final flow rate Rate of heat addition per unit volume Base flow rate

Bulk volumetric flow rate Gas volumetric flow rate Heat flux

Heat flux vector

As used in Equation (9.A.6) Indicated average flow rate Test measurement Liquid volumetric flow rate Leakage flow rate

Mass flow rate Maximum flow rate for which the meter

is designed Minimum flow rate for which the meter

is designed Normal flow rate

Flow rate at time t

Apparent volumetric flow rate when liquid is present in the gas stream Flow rate at change of precision Volumetric flow rate through meter at no rotation

Volumetric flow rate (in the main line)

Volumetric flow rate at calibration point

Metering pump volumetric transfer

flow

Volumetric flow rate of injected tracer

Radius of pipe, Gas constant for a

particular gas, Radius of bend or elbow,

Radius of friction wheel, Resistance of

heating element, Radius of the annulus

of angular momentum meter

Universal gas constant

Reynolds number usually based on the

pipe ID

Reynolds number based on the throat

diameter

Correlation coefficient

Radial coordinate (distance from pipe

axis), Radius of upstream edge of orifice

plate, Radius of point of friction wheel

contact, Exponent, Radius

Hub radius

Inner radius of measuring chamber

Journal bearing radius

Tip radius

Outer radius of measuring chamber,

Meter bore radius

Shaft radius

Elementary length of tube

Elementary length of fluid

Slope of no-flow decay curve at

standstill, Strouhal number,

Electromagnetic flowmeter sensitivity,

Flow signal

Flow signal at start

Stanton number

Exponent, Experimental standard

deviation, Blade spacing, Length along

curve when integrating velocity around

a vortex, Spring constant

Experimental standard deviation of

mean of group q^

Experimental standard deviation of q^

Temperature, Torque, Fundamental

period of pulsating flow

Bearing drag torque

Temperature at calibration

Driving torque

Mechanical friction torque on rotor at

zero speed

Hub fluid drag torque

Torque on input shaft to calibrator

Nonfluid drag torque

Torque out from calibrator

Stagnation temperature, Constant drag

torque, Temperature at operation

Retarding torque

Blade tip clearance drag torque

Hub disk friction drag torque

Speed-dependent drag torque

Student's t, Collection time during

calibration, Integration period for tracer measurement, Time, Clearance between stationary and moving members, Pipe wall thickness Blade thickness

Downstream wave time Mean time of transit used to obtain sound speed

Thickness of outer casing Relaxation time

Upstream wave time Time difference between upstream and downstream waves or pulses

Beam positions for ultrasonic flowmeter using Gaussian quadrature

Expanded uncertainty, Electric potential

Voltage between electrodes Voltage across a wire P moving through

a magnetic field (similarly for Q and R) Received voltages for ultrasonic flowmeter

Standard uncertainty for the /th quantity Combined standard uncertainty Fluid velocity, Amount injected in the sudden injection (integration) method, Volume of pipework and other vessels between the source of the pulsation and the flowmeter position, Volume of float, Nondimensional fluid velocity

Mean velocity in pipe with pulsating flow

Root-mean-square value of unsteady velocity fluctuation in pipe with pulsating flow

Velocity vector Oscillatory velocity field set up in the stationary fluid by the driving oscillator Oscillatory velocity field set up in the stationary fluid by the Coriolis forces Mean velocity in the pipe in meters per second

Maximum value of V z , Velocity of flow

past bluff body

Minimum value of V z

Velocity on pipe axis Ultrasonic undisturbed flow Acoustic field for cases (1) and (2) Superficial gas velocity

Axial velocity (instantaneous inlet fluid velocity of pulsating flow)

Vo Time average value of V z over period T

V\ I n l e t relative flow v e l o c i t y

Vz Outlet relative flow velocity

V Mean velocity in pipe

v Specific volume of gas, Relative flow

velocity in the direction of the acoustic

beam

W Rotor blade velocity, Nondimensional

instantaneous rotor velocity under

pulsating flow, Weighting function

W Weight function vector

W Rectilinear flow weight function

W" Axisymmetric weight function

W\i Relative velocity at the hub

W z Axial component of W

w Diameter or width of bluff body

X Axial length between transducers in

ultrasonic flowmeter, Mole fraction of

each component of a gas mixture,

Angular momentum,

Lockhart-Martinelli parameter

x Coordinate, Dryness fraction, Height of

float in tube

Xi Result of a meter measurement

x Mean of n meter measurements

Y Length of chord in the plane of the

ultrasonic path, Tangential force

y Output quantity, Wetness fraction,

Position coordinate across clearance gap,

Distance along chord used in ultrasonic

z Normalized coordinate (x — ii)/o,

Elevation above datum, Ultrasonic beam

deflection distance on opposite wall

a Flow coefficient CE, Wave shape

coefficient, Thermal coefficient of

expan-sion, Angle between inlet flow direction

and far field flow, Void fraction

a m Coefficient of linear expansion of metal

a\ Coefficient of volume expansion of

liquid

0 Diameter ratio d/D, Blade angle at radius

r measured from axial direction of meter,

Ultrasonic beam deflection angle

01 Relative inlet angle of flow

02 Relative outlet angle of flow

0 m Mean of the inlet and outlet flow field

y Ratio of specific heats

r Full-flow amplitude relative to average

flow

A Change in quantity, Distance between

center of rotation of arm and center of

discs in calibrator

8q m etc.

fi g

p Pi

p m

po o

r

r m

Reduction in area of the measuring chamber due to blade sections, Deflection of flow at blade outlet from blade angle, Shear layer thickness

Small changes or errors in q m , etc.

Expansibility (or expansion) factor, Error, Strain

Expansibility (or expansion) factor for orifice

Friction coefficient, Flow deviation factor

Cylindrical coordinate, Ultrasound beam angle, Angular deflection, Twist angle of U-tube

Angular position of center disk of calibrator

Angular position of cam disk of calibrator Angular position of outer ring of calibrator, Amplitude of twist angle of U-tube when in sinusoidal motion Isentropic exponent

Specific film thickness, Ultrasound wave length

Mean value of data for normal curve, Dynamic viscosity, Fluid viscosity, Electric permeability

Viscosity of calibration gas at flowing conditions

Viscosity of reference gas at standard conditions

Degrees of freedom, Kinematic viscosity

Density, Fluid density Density of float material Mean density in ultrasonic meter Density at stagnation conditions

Standard deviation (a 2 variance), Combined roughness of the two contacting surfaces of the bearing, Conductivity

Yield stress for plate material

Nondimensional time (t/T), Period of

modulation, Measurement period for ultrasonic system, Time period between ultrasonic wave peaks, Time constant

of flowmeter, Time difference between markers on the angular momentum meter rotating assembly, Difference in transit time of two halves of twisted U-tube

Mean ultrasonic correlation time Rotor coast time to standstill Angular position of arm of calibrator, Scalar magnetic potential

Area under Normal curve [e.g., 0(0.5) is

the area from z= - oo to z = 0.5]

Q

Ratio of two-phase pressure drop to

liquid flow pressure drop

Maximum allowable percentage error in

pulsating flow

Function for normalized Normal

distribution

Phase difference between the total

velocities at the two sensing points

Injected swirling flow rate/total flow rate,

Angular velocity of the pipe caused by

the vibration

Amplitude of the angular velocity of the

pipe caused by the vibration in

sinusoidal motion

CO

cos cou

OTHER / / m 1,2 1-8

Instantaneous rotor angular velocity, Vorticity, Angular velocity of the rotor, Driving frequency

Natural frequency of U-tube in twisting oscillation

Natural frequency of U-tube in normal oscillation

SUBSCRIPTS Dummy suffix for summation Liquid

Metal Pipe sections, Different materials Epicyclic gears

Introduction

1.1 INITIAL CONSIDERATIONS

Some years ago at Cranfield, where we had set up a flow rig for testing the effect

of upstream pipe fittings on certain flowmeters, a group of senior Frenchmen werebeing shown around and visited this rig The leader of the French party recalled asimilar occasion in France when visiting such a rig The story goes something likethis

A bucket at the end of a pipe seemed particularly out of keeping with the ing high tech rig When someone questioned the bucket's function, it was explainedthat the bucket was used to measure the flow rate Not to give the wrong impression

remain-in the future, the bucket was exchanged for a shremain-iny new high tech flowmeter In duecourse, another party visited the rig and observed the flowmeter with approval "Andhow do you calibrate the flowmeter?" one visitor asked The engineer responsiblefor the rig then produced the old bucket!

This book sets out to guide those who need to make decisions about whether

to use a shiny flowmeter, an old bucket, nothing at all, or a combination of these!

It also provides information for those whose business is the design, manufacture,

or marketing of flowmeters I hope it will, therefore, be of value to a wide variety

of people, both in industry and in the science base, who range across the wholespectrum from research and development through manufacturing and marketing

In my earlier book on flow measurement (Baker 1988/9), I provided a brief statement

on each flowmeter to help the uninitiated This book attempts to give a much morethorough review of published literature and industrial practice

This first chapter covers various general points that do not fit comfortably where In particular, it reviews recent guidance on the accuracy of flowmeters (orcalibration facilities)

else-The second chapter reviews briefly some essentials of fluid mechanics necessaryfor reading this book The reader will find a fuller treatment in Baker (1996), whichalso has a list of books for further reading

A discussion of how to select a flowmeter is attempted in Chapter 3, and someindication of the variety of calibration methods is given in Chapter 4, before going

in detail in Chapters 5-17 into the various high (and low) tech meters available.Chapter 18 deals with probes, Chapter 19 gives a brief note on modern controlsystems, and Chapter 20 provides some reflections on manufacturing and markets.Finally, Chapter 21 raises some of the interesting directions in which the technology

is likely to go in the future

In this book, I have tried to give a balance between the laboratory ideal, themanufacturer's claims, the realities of field experience, and the theory behind thepractice I am very conscious that the development and calibration laboratories aresometimes misleading places, which omit the problems encountered in the field(Stobie 1993), and particularly so when that field happens to be the North Sea Inthe same North Sea Flow Measurement Workshop, there was an example of the un-expected problems encountered in precise flow measurement (Kleppe and Danielsen1993), resulting, in this case, from a new well being brought into operation It hadsignificant amounts of barium and strontium ions, which reacted with sulfate ionsfrom injection water and caused a deposit of sulfates from the barium sulfate andstrontium sulfate that were formed.

With that salutary reminder of the real world, we ask an important - and perhapsunexpected - question

1.2 DO WE NEED A FLOWMETER?

Starting with this question is useful It may seem obvious that anyone who looks tothis book for advice on selection is in need of a flowmeter, but for the process engi-neer it is an essential question to ask Many flowmeters and other instruments havebeen installed without careful consideration being given to this question and with-out the necessary actions to ensure proper documentation, maintenance, and cali-bration scheduling being taken They are now useless to the plant operator and mayeven be dangerous components in the plant Thus before a flowmeter is installed,

it is important to ask whether the meter is needed, whether there are proper tenance schedules in place, whether the flowmeter will be regularly calibrated, andwhether the company has allocated to such an installation the funds needed toachieve this ongoing care Such care will need proper documentation

main-The water industry in the United Kingdom has provided examples of the lems associated with unmaintained instruments Most of us who are involved inthe metering business will have sad stories of the incorrect installation or misuse

prob-of meters Reliability-centered maintenance recognizes that the inherent reliabilitydepends on the design and manufacture of an item, and if necessary this will needimproving (Dixey 1993) It also recognizes that reliability is preferable in criticalsituations to extremely sophisticated designs, and it uses failure patterns to selectpreventive maintenance

In some research into water consumption and loss in urban areas, Hopkins et al.(1995) found that obstacles to accurate measurements were

• buried control valves,

• malfunctioning valves,

• valve gland leakage,

• hidden meters that could not be read, and

• locked premises denying access to meters

They commented that "water supply systems are dynamic functions having to

be constantly expanded or amended Consequently continuous monitoring, sions and amendments of networks records is imperative Furthermore, a proper

In this book, I have made no attempt to alert the reader to the industry-specificregulations and legal requirements, although some are mentioned Some regulationsare touched on by the various authors, and Miller (1996) is a source of informa-tion on many documents The main objective of the Organisation International

de Metrologie Legale (OIML) is to prevent any technical barriers to internationaltrade resulting from conflicting regulations for measuring instruments With regard

to flow measurement, it is particularly concerned with the measurement of tic supplies and industrial supplies of water and gas (Athane 1994) This is becausethere are two parties involved, the supplier and the consumer, and the consumer

domes-is unlikely to be able to ascertain the correct operation of the meter In additionthese measurements are not monitored continually by the supplier, the meters mayfail without anyone knowing, the usage is irregular and widely varying in rate, themeasurements are not repeatable, and the commodities have increased in value con-siderably in recent years

In order to reduce discussions and interpretation problems between ers and authorized certifying institutes, the European Commission is mandating theEuropean standardization body (CEN/CENELEC) to develop harmonized standardsthat will give the technical details and implementation of the requirements based

manufactur-on OIML recommendatimanufactur-ons These are such that a measuring instrument complieswith essential requirements, assuming that the manufacturer has complied withthem (Nederlof 1994)

The manufacturer will also be fully aware of the electromagnetic ity (EMC), which relates to electromagnetic interference In particular, the EMC

compatibil-characteristics of a product are that

• the level of electromagnetic disturbance generated by the instrument will notinterfere with other apparatus, and

• the operation of the instrument will not be adversely affected by electromagneticinterference from its environment

In order to facilitate free movement within the European area the CE mark identifiesproducts that conform to the European essential requirements, and all products must

be so marked within the European Economic Area (DTI 1993, Chambers 1994).First, we consider the knotty problem of how accurate the meter should be

1.3 HOW ACCURATE?

There continues to be inconsistency about the use of terms that relate to accuracyand precision This stems from a slight mismatch between the commonly used termsand those that the purists and the standards use Thus we commonly refer to anaccurate measurement, when strictly we should refer to one with a small value of

uncertainty We should reserve the use of the word accurate to refer to the instrument.

A high quality flowmeter, carefully produced with a design and construction totight tolerances and with high quality materials as well as low wear and fatiguecharacteristics, is a precise meter with a quantifiable value of repeatability Also, itwill, with calibration on an accredited facility, be an accurate meter with a smalland quantifiable value of measurement uncertainty In the context of flowmeters,

the word repeatability is preferred to reproducibility The meanings are elaborated on later, and I regret the limited meaning now given to precision, which I have used

more generally in the past and shall slip back into in this book from time to time! Inthe following chapters, I have attempted to be consistent in the use of these words.However, many claims for accuracy may not have been backed by an accreditedfacility, but I have tended to use the phrase "measurement uncertainty" for theclaims made

Hayward (1977) used the story of William Tell to illustrate precision WilliamTell had to use his cross-bow to fire an arrow into an apple on his little son's head.This was a punishment for failing to pay symbolic homage to an oppressive Austrianruler Tell succeeded because he was an archer of great skill and high accuracy

An archer's ability to shoot arrows into a target provides a useful illustration

of some of the words related to precision So Figure 1.1 (a) shows a target with allthe shots in the bull's-eye Let us take the bull's-eye to represent ±1%, within thefirst ring ±3%, and within the second ring ±5% Ten shots out of ten are on target,but how many will the archer fire before one goes outside the bull's-eye? If thearcher, on average, achieves 19 out of 20 shots within the bull's-eye [Figure l.l(b)], wesay that the archer has an uncertainty of ±1% (the bull's-eye) with a 95% confidence

level (19 out of 20 on the bull's-eye: 19 + 20 = 0.95 = 95 -=-100 = 95%).

Suppose that another archer clusters all the arrows, but not in the bull's-eye,Figure l.l(c) This second archer is very consistent (all the shots are within the samesize circle as the bull's-eye), but this archer needs to adjust his aim to correct the