Handbook of Measurement in Science and Engineering, Volume 2

By Myer Kutz

A multidisciplinary reference of engineering measurement tools, techniques, and applications—Volume 2

"When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the stage of science." — Lord Kelvin

Measurement falls at the heart of any engineering discipline and job function. Whether engineers are attempting to state requirements quantitatively and demonstrate compliance; to track progress and predict results; or to analyze costs and benefits, they must use the right tools and techniques to produce meaningful, useful data.

The Handbook of Measurement in Science and Engineering is the most comprehensive, up-to-date reference set on engineering measurements—beyond anything on the market today. Encyclopedic in scope, Volume 2 spans several disciplines—Materials Properties and Testing, Instrumentation, and Measurement Standards—and covers:

• Viscosity Measurement
• Corrosion Monitoring
• Thermal Conductivity of Engineering Materials
• Optical Methods for the Measurement of Thermal Conductivity
• Properties of Metals and Alloys
• Electrical Properties of Polymers
• Testing of Metallic Materials
• Testing and Instrumental Analysis for Plastics Processing
• Analytical Tools for Estimation of ParticulateComposite Material Properties
• Input and Output Characteristics
• Measurement Standards and Accuracy
• Tribology Measurements
• Surface Properties Measurement
• Plastics Testing
• Mechanical Properties of Polymers
• Nondestructive Inspection
• Ceramics Testing
• Instrument Statics
• Signal Processing
• Bridge Transducers
• Units and Standards
• Measurement Uncertainty
• Data Acquisition and Display Systems

Vital for engineers, scientists, and technical managers in industry and government, Handbook of Measurement in Science and Engineering will also prove ideal for members of major engineering associations and academics and researchers at universities and laboratories.

• Language: English
• Publisher: Wiley
• Release date: Dec 3, 2015
• ISBN: 9781118453278

Book preview

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Contributors

Part IV: Materials Properties and Testing

Chapter 31: Viscosity Measurement

31.1 Viscosity Background

31.2 Common Units of Viscosity

31.3 Major Viscosity Measurement Methods

31.4 ASTM Standards for Measuring Viscosity

31.5 Questions to Ask when Selecting a Viscosity Measurement Technique

References

Chapter 32: Tribology Measurements

32.1 Introduction

32.2 Measurement of Surface Roughness

32.3 Measurement of Friction

32.4 Measurement of Wear

32.5 Measurement of Test Environment

32.6 Measurement of Material Characteristics

32.7 Measurement of Lubricant Characteristics

32.8 Wear Particle Analysis

32.9 Industrial Measurements

32.10 Summary

Chapter 33: Corrosion Monitoring

33.1 What is Corrosion Monitoring?

33.2 The Role of Corrosion Monitoring

33.3 Corrosion Monitoring System Considerations

References

Chapter 34: Surface Properties Measurement

34.1 Introduction

34.2 Surface Properties

34.3 Microstructural Analysis

34.4 Compositional Analysis

34.5 Phase Analysis

34.6 Mechanical Testing

34.7 Corrosion Properties

34.8 Standards for Surface Engineering Measurement

References

Chapter 35: Thermal Conductivity of Engineering Materials

35.1 Introduction

35.2 Stationary Methods for Measurement of the Thermal Conductivity

35.3 Transient Methods for the Measurement of the Thermal Conductivity

35.4 Test Results on Various Engineering Materials

References

Chapter 36: Optical Methods for the Measurement of Thermal Conductivity

36.1 Thermal Boundary Resistance May Limit Accuracy in Contact-Based Thermal Conductivity (κ) Measurements

36.2 Optical Measurements of κ May Avoid Contact-Related Issues

36.3 Thermoreflectance (TR)

36.4 Characteristics of Thermoreflectance from Si Thin Films—Modeling and Calibration

36.5 Experimental Procedures

36.6 Results and Discussion

36.7 Summary and Outlook

Acknowledgments

References

Chapter 37: Selection of Metals for Structural Design

37.1 Introduction

37.2 Common Alloy Systems

37.3 What are Alloys and What Affects Their Use?

37.4 What are the Properties of Alloys and How Are Alloys Strengthened?

37.5 Manufacture of Alloy Articles

37.6 Alloy Information

37.7 Metals at Lower Temperatures

37.8 Metals at High Temperatures

37.9 Melting and Casting Practices

37.10 Forging, Forming, Powder Metallurgy, and Joining of Alloys

37.11 Surface Protection of Materials

37.12 Postservice Refurbishment and Repair

37.13 Alloy Selection: A Look at Possibilities

37.14 Level of Property Data

37.15 Thoughts on Alloy Systems

37.16 Selected Alloy Information Sources

Further Readings

Chapter 38: Mechanical Properties of Polymers

38.1 Microstructure and Morphology of Polymers-Amorphous Versus Crystalline

38.2 General Stress-Strain Behavior

38.3 Viscoelasticity

38.4 Mechanical Models of Viscoelasticity

38.5 Time-Temperature Dependence

38.6 Deformation Mechanisms

38.7 Crazing

38.8 Fracture

38.9 Modifying Mechanical Properties

38.10 Load-Bearing Applications: Creep, Fatigue Resistance, and High Strain Rate Behavior

References

Chapter 39: Electrical Properties of Polymers

39.1 Introductory Remarks

39.2 Polarity and Permittivity

39.3 Measurements of Dielectric Permittivity

39.4 Polarization and Dipole Moments in Isotropic Systems

39.5 Thermostimulated Depolarization Currents

39.6 Conductivity in Polyelectrolytes and Polymer-Electrolytes as Separators for Low Temperature Fuel Cells and Electrical Batteries

39.7 Semiconductors and Electronic Conducting Polymers

39.8 Ferroelectricity, Pyroelectricity, and Piezoelectricity in Polymers

39.9 Nonlinear Polarization in Polymers

39.10 Elastomers for Actuators and Sensors

39.11 Electrical Breakdown in Polymers

References

Chapter 40: Nondestructive Inspection*

40.1 Introduction

40.2 Liquid Penetrants

40.3 Radiography

40.4 Ultrasonic Methods

40.5 Magnetic Particle Method

40.6 Thermal Methods

40.7 Eddy Current Methods

References

Chapter 41: Testing of Metallic Materials

41.1 Mechanical Test Laboratory

41.2 Tensile and Compressive Property Testing

41.3 Creep and Stress Relaxation Testing

41.4 Hardness and Impact Testing

41.5 Fracture Toughness Testing

41.6 Fatigue Testing

41.7 Other Mechanical Testing

41.8 Environmental Considerations

Acknowledgments

References

Chapter 42: Ceramics Testing

42.1 Introduction

42.2 Mechanical Testing

42.3 Thermal Testing

42.4 Nondestructive Evaluation Testing

42.5 Electrical Testing

42.6 Summary

References

Chapter 43: Plastics Testing

43.1 Introduction

43.2 Mechanical Properties

43.3 Thermal Properties

43.4 Electrical Properties

43.5 Weathering Properties

43.6 Optical Properties

Further Readings

Chapter 44: Testing and Instrumental Analysis for Plastics Processing: Key Characterization Techniques

44.1 Ftir Spectroscopy

44.2 Chromatography (GC, GC-MSD, GC-FID, and HPLC)

44.3 DSC and Thermogravimetry (TGA)

44.4 Rheometry

References

Chapter 45: Analytical Tools for Estimation of Particulate Composite Material Properties

45.1 Introduction

45.2 Concepts in Statistical Quality Control

45.3 Effective Property Estimates

45.4 Summary

References

Part V: Instrumentation

Chapter 46: Instrument Statics

46.1 Terminology

46.2 Static Calibration

46.3 Statistics in the Measurement Process

References

Chapter 47: Input and Output Characteristics

47.1 Introduction

47.2 Familiar Examples¹ of Input-Output Interactions

47.3 Energy, Power, Impedance

47.4 Operating Point of Static Systems

47.5 Transforming the Operating Point

47.6 Measurement Systems

47.7 Distributed Systems in Brief

47.8 Concluding Remarks

References

Chapter 48: Bridge Transducers

48.1 Terminology

48.2 Flexural Devices in Measurement Systems

48.3 The Resistance Strain Gage

48.4 The Wheatstone Bridge

48.5 Resistance Bridge Balance Methods

48.6 Resistance Bridge Transducer Measurement System Calibration

48.7 Resistance Bridge Transducer Measurement System Considerations

48.8 AC Impedance Bridge Transducers

References

Further Readings

Chapter 49: Signal Processing

49.1 Frequency-Domain Analysis of Linear Systems

49.2 Basic Analog Filters

49.3 Basic Digital Filter

49.4 Stability and Phase Analysis

49.5 Extracting Signal from Noise

References

Chapter 50: Data Acquisition and Display Systems

50.1 Introduction

50.2 Data Acquisition

50.3 Process Data Acquisition

50.4 Data Conditioning

50.5 Data Storage

50.6 Data Display and Reporting

50.7 Data Analysis

50.8 Data Communications

50.9 Other Data Acquisition and Display Topics

50.10 Summary

References

Part VI: Measurement Standards

Chapter 51: Mathematical and Physical Units, Standards, and Tables*

51.1 Symbols and Abbreviations

Bibliography for Letter Symbols

Bibliography for Graphic Symbols

51.2 Mathematical Tables

51.3 Statistical Tables¹

51.4 Units and Standards

Bibliography for Units and Measurements

51.5 Tables of Conversion Factors⁹

51.6 Standard Sizes

51.7 Standard Screws¹³

Chapter 52: Measurement Uncertainty

52.1 Introduction

52.2 Literature

52.3 Evaluation of Uncertainty

52.4 Discussion

Disclaimer

References

Chapter 53: Measurements

53.1 Standards and Accuracy

53.2 Impedance Concepts

53.3 Error Analysis

References

Index

HANDBOOK OF MEASUREMENT IN SCIENCE AND ENGINEERING

Title Page

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.

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Library of Congress Cataloging-in-Publication Data:

Handbook of Measurement in Science and Engineering / Myer Kutz, editor.

volumes cm

  Includes bibliographical references and index.

  ISBN 978-0-470-40477-5 (volume 1) – ISBN 978-1-118-38464-0 (volume 2) – ISBN 978-1-118-38463-3 (set) 1. Structural analysis (Engineering) 2. Dynamic testing. 3. Fault location (Engineering) 4. Strains and stresses-Measurement. I. Kutz, Myer.

  TA645.H367 2012

  620’.0044–dc23

2012011739

To Jayden, Carlos, Rafael, Irena, and Ari. Watch them grow.

PREFACE

The idea for the Handbook of Measurement in Science and Engineering came from a Wiley book first published over 30 years ago. It was Fundamentals of Temperature, Pressure and Flow Measurements, written by a sole author, Robert P. Benedict, who also wrote Wiley books on gas dynamics and pipe flow. Bob was a pleasant, unassuming, and smart man. I was the Wiley editor for professional-level books in mechanical engineering when Bob was writing such books, so I knew him as a colleague. I recall meeting him in the Wiley offices at a time when he seemed to be having some medical problems, which he was reluctant to talk about. Recently, I discovered a book published in 1972 by a London firm, Pickering & Inglis, which specializes in religion. This book was Journey Away From God, an intriguing title. The author’s name was Robert P. Benedict. I do not know whether the two Benedicts are in fact the same person, although Amazon seems to think so. (See the Robert P. Benedict page.) In any case, I do not recall Bob’s mentioning the book when we had an occasion to talk.

The moral of this story, if there is one, is that the men and women who contributed the chapters in this handbook are real people, who have real-world concerns, in addition to the expertise required to write about technology. They have families, jobs, careers, and all manner of cares about the minutia of daily life to deal with. And that they have been able to find the time and energy to write these chapters is remarkable. I salute them.

I have spent a lot of time in my life writing and editing books. I wrote my first Wiley book somewhat earlier than Bob Benedict wrote his. When Wiley published Temperature Control in 1967, I was in my mid-twenties and was a practicing engineer, working on temperature control of the Apollo inertial guidance system at the MIT Instrumentation Lab, where I had done my bachelor’s thesis. One of the coauthors of my book was to have been a Tufts Mechanical Engineering Professor by the name of John Sununu (yes, that John Sununu), but he and the other coauthor dropped out of the project before the contract was signed. So I wrote the short book myself.

Bob Benedict’s measurement book, the third edition of which is still in print, surfaced several years ago, during a discussion I was having with one of my Wiley editors, Bob Argentieri, about possible projects we could collaborate on. It turned out that no one had attempted to update Benedict’s book. I have not been a practicing engineer for some time, so I was not in a position to do an update as a single author—or even with a collaborator or two. Most of my career life has been in scientific and technical publishing, however, and for over a decade I have conceived of, and edited, numerous handbooks for several publishers. (I also write fiction, but that is another story.) So, it was natural for me to think about using Benedict’s book as the kernel of a much larger and broader reference work dealing with engineering measurements. The idea, formed during that discussion, that I might edit a contributed handbook on engineering measurements took hold, and with the affable and expert guidance of my other Wiley editor, George Telecki, the volume you are holding in your hands, or reading on an electronic device, came into being.

Like many such large reference works, this handbook went through several iterations before the final table of contents was set, although the general plan for arrangement of chapters has been the same throughout the project. The initial print version of the handbook is divided into two volumes. The chapters are arranged essentially by engineering discipline. The first volume contains 30 chapters related to five engineering disciplines, which are divided into three parts:

Part I, Civil and Environmental Engineering, which contains seven chapters, all but one of them dealing with measurement and testing techniques for structural health monitoring, GIS and computer mapping, highway bridges, environmental engineering, hydrology, and mobile source emissions (the exception being the chapter on traffic congestion management, which describes the deployment of certain measurements);

Part II, Mechanical and Biomedical Engineering, which contains 16 chapters, all of them dealing with techniques for measuring dimensions, surfaces, mass properties, force, resistive strain, vibration, acoustics, temperature, pressure, velocity, flow, heat flux, heat transfer for non-boiling two-phase flow, solar energy, wind energy, human movement, and physiological flow;

Part III, Industrial Engineering, which contains seven chapters dealing with statistical quality control, evaluating and selecting technology-based projects, manufacturing systems evaluation, measuring performance of chemical process equipment, industrial energy efficiency, industrial waste auditing, and organizational performance measurement.

The second volume contains 23 chapters divided into three parts:

Part IV, Materials Properties and Testing, which contains 15 chapters dealing with measurement of viscosity, tribology, corrosion, surface properties, and thermal conductivity of engineering materials; properties of metals, alloys, polymers, and particulate composite materials; nondestructive inspection; and testing of metallic materials, ceramics, plastics, and plastics processing;

Part V, Instrumentation, which contains five chapters covering electronic equipment used for measurements;

Part VI, Measurement Standards, which contains three chapters covering units and standards, measurement uncertainty and error analysis.

Major reference works, like this handbook, are generally incomplete when they are first published. Editors cannot wait for tardy contributors, some contributors simply cannot manage to deliver their chapters no matter how much time they are given, and contributors cannot be secured for all the chapters an editor has in mind for a reference work. Among the topics that were either contracted for but were not delivered or for which contributors were not found are surveying engineering, engineering seismology, construction materials properties, turbulence, water quality, wastewater engineering, trace gases in the atmosphere, experimental methods, experimental design, shape and deformation, thermal systems, energy audits, electrical properties of materials, rheology, software engineering, biomedical electronics, physiology, dielectric properties of tissues, productivity, remote sensing, and data analysis.

Of course, such a list, when combined with the chapters being published, does not exhaust the list of possible topics for a measurements handbook. Be that as it may, the usual practice has been to attempt to include additional topics, together with updates of existing chapters, in new editions of a reference work, which tend to appear in 5-to-10-year, or longer, intervals. I have done this successfully in the Mechanical Engineers’ Handbook that I edit for Wiley. That work is now, in its forthcoming fourth edition, a four-volume handbook, in contrast to the single-volume first and second editions.

In the case of this measurements handbook, however, George Telecki has proposed that the online version be dynamic, with 20 or so articles added annually. Furthermore, coverage will be expanded beyond engineering disciplines to include chemistry, life sciences, and physics, thereby justifying the handbook’s title, Handbook of Measurement in Science and Engineering. In addition, existing chapters will be updated as the need arises. I have campaigned for years to get my publishers to adopt this scheme, and I am gratified that Wiley intends to pursue it. I will attempt to get the others to follow suit.

Thanks to Kari Capone for shepherding the manuscript toward production and to the stalwarts Kristen Parrish and Shirley Thomas, for bringing the handbook home. Thanks, also, to my wife Arlene, who helps me with everything else.

MYER KUTZ

Delmar, NY

June, 2012

CONTRIBUTORS

Jacob Egede Andersen, COWI A/S, Lyngby, Denmark

Ann M. Anderson, Union College, Schenectady, NY, USA

Brian E. Anderson, Brigham Young University, Provo, UT, USA

Max S. Aubain, University of California—San Diego, La Jolla, CA, USA

Prabhakar R. Bandaru, University of California—San Diego, La Jolla, CA, USA

Donald E. Beasley, Clemson University, Clemson, SC, USA

Adam C. Bell, Dartmouth, Nova Scotia, Canada

Jonathan D. Blotter, Brigham Young University, Provo, UT, USA

Juergen Blumm, NETZSCH-Geraetebau GmbH, Selb, Germany

Bradford Α. Bruno, Union College, Schenectady, NY, USA

S. Chaudhari, West Virginia University, Morgantown, WV, USA

Peter R.N. Childs, Imperial College London, London, UK

David Clippinger, U.S. Coast Guard Academy, New London, CT, USA

Patrick Collins, MecMesin Ltd., Slinfold, West Sussex, UK

Jim W. Crafton, Innovative Scientific Solutions, Inc., Dayton, OH, USA

Robert L. Crane, Kettering, OH, USA

Alan Cross, Little Neck, NY, USA

Dana Dabiri, University of Washington, Seattle, WA, USA

Rahman Davoodi, University of Southern California, Los Angeles, CA, USA

Maria del Pilar Noriega, ICIPC, Medellin, Antioquia, Columbia

Steven Deutsch, Pennsylvania State University, University Park, PA, USA

Ricardo Díaz-Calleja, ITE (Universidad Politécnica de Valencia), Valencia, Spain

Thomas E. Diller, Virginia Tech, Blacksburg, VA, USA

Matthew J. Donachie, Winchester, NH, USA

Merit Enckell, Royal Institute of Technology (KTH), Stockholm, Sweden; COWI A/S, Lyngby, Denmark

Jennifer A. Farris, Texas Tech, Lubbock, TX, USA

Richard S. Figliola, Clemson University, Clemson, SC, USA

Sergey D. Fonov, Innovative Scientific Solutions, Inc., Dayton, OH, USA

Arnold A. Fontaine, Pennsylvania State University, University Park, PA, USA

Kent L. Gee, Brigham Young University, Provo, UT, USA

Afshin J. Ghajar, Oklahoma State University, Stillwater, OK, USA

Branko Glisic, Princeton University, Princeton, NJ, USA

B. Gopalakrishnan, West Virginia University, Morgantown, WV, USA

Sheryl M. Gracewski, University of Rochester, Rochester, NY, USA

Peter Gregg, GE, Schenectady, NY, USA

Mikell P. Groover, Lehigh University, Bethlehem, PA, USA

D.P. Gupta, West Virginia University, Morgantown, WV, USA

Jerry Lee Hall, Iowa State University, Ames, IA, USA

Sandip P. Harimkar, Oklahoma State University, Stillwater, OK, USA

E.L. Hixson, University of Texas, Austin, TX, USA

Ahmad M. Itani, University of Nevada, Reno, NV, USA

Michael G. Jenkins, University of Washington, Seattle, WA, USA

Gamal E. Khalil, University of Washington, Seattle, WA, USA

Jeremy S. Knopp, Wright Patterson Air Force Base, Dayton, OH, USA

Geert Letens, Royal Military Academy, Brussels, Belgium

Daniel Liu, Exponent, Bowie, MD, USA

Keefe B. Manning, Pennsylvania State University, University Park, PA, USA

Y. Mardikar, West Virginia University, Morgantown, WV, USA

Shawn K. McGuire, Stanford University, Palo Alto, CA, USA

Peter C. McKeighan, Warrenville, IL, USA

Philip C. Milliman, Weyerhaeuser Company, Federal Way, WA, USA

Eric V. Monzon, University of Nevada, Reno, NV, USA

Mrinalini Mulukutla, Oklahoma State University, Stillwater, OK, USA

Tariq Muneer, Edinburgh Napier University, Edinburgh, UK

Mahmood Naim, Union Carbide Corporation, Indianapolis, IN, USA

Walter W. Olson, University of Toledo, Toledo, OH, USA

Gokhan Pekcan, University of Nevada, Reno, NV, USA

Nigel D. Ramoutar, Gleason Works, Rochester, NY, USA

Todd C. Rasmussen, University of Georgia, Athens, GA, USA

Jackie Rehkopf, Exponent, Bowie, MD, USA

Maureen Reitman, Exponent, Bowie, MD, USA

Evaristo Riande*, Instituto de Ciencia and Tecnología de Polímeros (Consejo Superior de Investigaciones Científicas), Madrid, Spain

E.A. Ripperger, University of Texas, Austin, TX, USA

Pierre R. Roberge, Royal Military College of Canada, Kingston, Ontario, Canada

Nagui M. Rouphail, North Carolina State University, Raleigh, NC, USA

Prasanta Sahoo, Jadavpur University, Kolkata, India

Marvin Sellers, Aerospace Testing Alliance, Arnold Air Force Base, TN, USA

Johan Silfwerbrand, Royal Institute of Technology (KTH), Stockholm, Sweden

Lilla Safford Smith, Union College, Schenectady, NY, USA

Vishu Shah, Consultek Consulting Group, Brea, CA, USA

Scott D. Sommerfeldt, Brigham Young University, Provo, UT, USA

Gary S. Spring, Merrimack College, North Andover, MA, USA

Sriram Sundararajan, Iowa State University, Ames, IA, USA

Clement C. Tang, University of North Dakota, Grand Forks, ND, USA

David Tellet, Society of Allied Weight Engineers, Inc. (SAWE, Inc.)

Yieng Wei Tham, Edinburgh Napier University, Edinburgh, UK

Hans J. Thamhain, Bentley University, Waltham, MA, USA

John Turnbull, Case Western Reserve University, Cleveland, OH, USA

Mark Tuttle, University of Washington, Seattle, WA, USA

Daniel A. Vallero, Duke University, Chapel Hill, NC, USA

Eileen M. Van Aken, Virginia Tech, Blacksburg, VA, USA

Mohan Venigalla, George Mason University, Fairfax, VA, USA

C. Visvanathan, Asian Institute of Technology, Klongluang Pathumthani, Thailand

Patrick L. Walter, Texas Christian University, Fort Worth, TX, USA

Jack H. Westbrook, Ballston Spa, NY, USA

Jesse Yoder, Flow Research, Inc., Wakefield, MA, USA

Magd E. Zohdi, Louisiana State University, Baton Rouge, LA, USA

Tarek I. Zohdi, University of California, Berkeley, CA, USA

*Deceased

PART IV

MATERIALS PROPERTIES AND TESTING

CHAPTER 31

VISCOSITY MEASUREMENT

ANN M. ANDERSON, BRADFORD A. BRUNO, AND LILLA SAFFORD SMITH

31.1 Viscosity background

31.2 Common units of viscosity

31.2.1 Absolute viscosity, μ

31.2.2 Kinematic viscosity, v

31.2.3 Nonstandard units

31.2.4 Distinction between rheology and viscometry

31.2.5 Mathematical formalism

31.2.6 Relation of viscosity to molecular theory

31.2.7 Effect of pressure and temperature on viscosity

31.2.8 Correlations of viscosity with temperature for gases

31.2.9 Correlations of viscosity with temperature for liquids 31.2.10 Effect of pressure on viscosity

31.3 Major viscosity measurement methods

31.3.1 Drag-type viscometers

31.3.2 Bubble (tube) viscometers

31.3.3 Rotational viscometers

31.3.4 Flow-type viscometers

31.3.5 Orifice-type (cup) viscometers

31.3.6 Vibrational (resonant) viscometers

31.4 ASTM standards for measuring viscosity

31.5 Questions to ask when selecting a viscosity measurement technique

References

31.1 VISCOSITY BACKGROUND

The most significant mechanical difference between materials classified as fluids and those classified as solids is in their reaction to shear stresses. (Recall that a shear stress is a distributed force, or force per unit area, whose direction of action is within the plane of application. If you slide your open hand over a desk top the friction between your hand and the desk will create a shear stress on your hand.) As illustrated in Figure 31.1, a solid (within its elastic limit) will deform through some limited angle while a shear stress is applied, and will then return to its original configuration when the stress is removed. A fluid on the other hand will deform continuously as long as a shear stress is applied, and will not return to its original shape when the stress is removed. For a fluid the rate of shear deformation is related to the magnitude of the shear stress applied, while for a solid the amount of shear deformation is related to the magnitude of the shear stress applied. This seemingly innocuous mechanical difference in reaction to shear stresses gives rise to the large difference in character between solids and fluids. In fact it gives rise to all of the behaviors that one thinks of as inherently fluid: the ability to flow, the ability to fill volumes of arbitrary shape, the ability to spread out and wet certain surfaces, etc. It also gives rise to the complex nature of fluid mechanics because it allows for the very large material deformations that in turn give rise to phenomena like turbulence.

FIGURE 31.1 Fluid (viscous) versus Solid (elastic) behavior. (a) Fluid Behavior: A thin layer of fluid held between two parallel plates, the top plate caused to move relative to the bottom plate by application of a shear stress, will continue to deform (experience shear strain) as long as the shear stress is applied. The rate of shear strain is related to the magnitude of the shear stress. When the shear stress is removed the fluid will remain in its deformed state. (b) Solid (Elastic) Behavior: A solid will deform (experience shear strain) through some fixed angle when a shear stress is applied. The amount of shear strain is related to the magnitude of the shear stress. When the shear stress is released the solid will return to its original form.

Viscosity (or more precisely the shear viscosity, defined below) is the material property that defines the quantitative relation between the applied shear stress and the shear deformation rate in a fluid. Qualitatively the viscosity indicates the thickness or resistance to flow of a fluid. Since viscosity is the property that controls and quantifies the shear stress/shear rate behavior that is definitional to fluids, it is in many regards the most important physical property of a fluid.

Unfortunately, as alluded to above, the term viscosity is actually used to denote several related, but different, physical properties. It is important to understand these distinctions in terms from the outset. First, the term viscosity is most commonly used in conjunction with effects arising from shear forces and shear deformations in fluids. When used in this context, the most common one, the property is more precisely called the shear viscosity or the first coefficient of viscosity. However, when used in this sense, it is almost always simply referred to as viscosity. This is contrasted with the bulk viscosity, associated with volume dilatation. Bulk viscosity is rarely an important parameter and hence is not as well known or understood as the more common shear viscosity. Bulk viscosity is discussed briefly in Section 31.1.3. Second, it should be noted that even the shear viscosity described above is often stated in two different forms, the absolute or dynamic viscosity, μ, and the kinematic viscosity or momentum diffusivity, ν, where ν = μ/ρ and ρ is the fluid’s density. Although the dynamic and kinematic viscosities are clearly related properties, they are dimensionally dissimilar and it is critically important to always distinguish between them. More is said on the distinction between dynamic and kinematic viscosity in the following section on common viscosity units.

The remainder of this chapter begins by discussing the units in which viscosity is measured. Then the distinction between the larger field of rheology and its subfield viscometry is made in the context of differentiating between the so-called Newtonian and non-Newtonian fluids. After that the chapter provides a brief theoretical and mathematical overview of viscosity. Finally, the majority of the chapter provides detailed and practical information on methods for measuring viscosity.

31.2 COMMON UNITS OF VISCOSITY

There are several systems of units used with viscosity; many of them are archaic and/or closely tied to one specific viscosity measuring technique (e.g., the Saybolt cup and the Saybolt Universal Second, and the Krebs unit) or one particular industry (e.g., SAE oil grade and the automotive industry). It is impossible to capture all of these systems in one document, but an attempt is made below to define and relate the most common and standard units associated with viscosity measurement.

31.2.1 Absolute Viscosity, μ

In terms of the SI (Le Systeme Internationale d’Unites) system of fundamental units the derived units for absolute viscosity, μ, are kg/m × s which is equivalent to Pa · s (Pascal-seconds). This grouping of units has not received a name of its own. In the closely related cgs (centimeter, gram, and second) system of units, the derived unit of g/cm × s or dyne × s/cm² is called a Poise (after Poiseuille). More commonly a centipoise, cP = 1/100th of a Poise is used. In the FPS (foot, pound, and second) system of units, the units of absolute viscosity are IbF · s/in², which is called the Reyn (after Osbourne Reynolds). Refer to Table 31.1 for a collection of units of absolute viscosity.

TABLE 31.1 Units for Absolute/Dynamic Viscosity, μ

31.2.2 Kinematic Viscosity, ν

Recall that the dynamic (kinematic) viscosity, v, is defined as the absolute viscosity divided by the fluid density, ρ. In the SI system of fundamental units the units for kinematic viscosity are meter square per second, which is not a named grouping. It should be noted that the units of kinematic viscosity (m²/s) are identical to the units of thermal diffusivity used in heat transfer, and mass (species) diffusivity used in diffusion. This leads to the kinematic viscosity being referred to as the coefficient of momentum diffusivity by analogy. In the cgs system the unit of kinematic viscosity is the centimeter square per second called the Stokes (after G.G. Stokes). More commonly the kinematic viscosity is given in centistokes (cSt) where 100 cSt = 1 Stokes. In the FPS system kinematic viscosity would be foot square per second or inch square per second, neither of which is a named unit.

31.2.3 Nonstandard Units

Kinematic viscosity is also often given in Saybolt Universal Seconds or SUS (also sometimes SSU Saybolt Seconds Universal or SUV Saybolt Universal Viscosity), which is directly related to the Saybolt viscosity cup measuring system (see Section 31.2.2). Of course, the unit of seconds is not a dimensionally correct unit for the physical quantity of kinematic viscosity, so this system is problematic. The Saybolt measurement system is based on ASTM method D88 and measurements in SUS can be converted into more standard (dimensionally correct) viscosity units using procedures provided in ASTM 2161. There are countless other such legacy" scales of viscosity associated with different industries, and unfortunately there is often no standard method for converting these legacy measures into dimensionally correct viscosity units. A number of online viscosity converters exist (see www.coleparmer.com, www.gardco.com, or www.cannon.com, for example) (Table 31.2).

TABLE 31.2 Units for Kinematic Viscosity, v

31.2.4 Distinction Between Rheology and Viscometry

A simple linear relationship between shear stress and shear strain rate is observed in a wide variety of fluids (Figure 31.2a). The constant slope of the line labeled Newtonian is the (shear) viscosity of the fluid. Fluids demonstrating such a relationship are known as Newtonian fluids. Many common fluids like air, gases in general, water, or simple oils demonstrate Newtonian behavior meaning constant viscosity with respect to strain rate over a very wide range (many orders of magnitude) of strain rates. The measurement of the shear viscosity of Newtonian fluids is referred to as viscometry and is the focus of this chapter.

FIGURE 31.2 Newtonian and non-Newtonian fluid behavior (a) shear stress vs. strain rate (b) shear stress vs. duration of applied strain rate.

Fluids with more complicated molecular structures (e.g., polymers) or fluids with other phases suspended in them (e.g., mixtures, slurries, and colloids) often demonstrate more complicated shear stress to strain rate behaviors (see Figure 31.2a). Fluids exhibiting such behaviors are broadly characterized as non-Newtonian fluids. Non-Newtonian fluids can be further classified according to how they react to changes in shear deformation rates, to the duration of application of the applied loading, and to whether or not they exhibit a threshold elastic (solid-like) shear resistance prior to deforming like a fluid.

Fluids that show increasing apparent viscosity (the apparent viscosity is the local slope of the stress versus strain rate curve, see Figure 31.2a) as the applied strain rate increases are called shear thickening or dilatant fluids. The classic example of a shear thickening fluid is a mixture of cornstarch in water. If one attempts to shear this fluid quickly (e.g., hit it with a hammer) the viscosity will rise to such a level that the fluid seems almost solid, the hammer blow will bounce off the surface. Yet at lower shear rates the mixture will act like a normal fluid (e.g., a hammer set on its surface would sink right into the fluid). Fluids which show the opposite behavior (decrease in apparent shear viscosity with increasing strain rate) are called shear thinning or pseudo-plastic fluids. A common example of a shear thinning fluid would be no drip paint, which behaves as a fairly thick (viscous) fluid while adhering to a paintbrush (a low shear rate circumstance), but which spreads easily (i.e., exhibits lower viscosity) when the paintbrush is dragged along a surface thereby increasing the shear strain rate applied to the fluid (Cengel and Cimbala, 2010).

Some fluids will thin (produce a lower shear stress resisting the motion) or some will thicken (produce a higher shear stress resisting the motion) as the duration for which a constant strain rate is applied increases, (see Figure 31.2b). Fluids exhibiting the former behavior are referred to as thixotropic the latter as rheopectic. Such fluids are also sometimes referred to as time-thinning or time-thickening fluids. Examples of thixotropic fluids include yogurt and some classes of paint. Rheopectic behavior is much rarer. Examples include gypsum paste and printers ink. Newtonian fluids exhibit constant strain rate with regard to loading duration for a constant applied shear stress.

Newtonian fluids will exhibit constant strain rate to shear stress behavior down to very low (theoretically zero) applied shear stresses. However some fluids, called Bingham plastic fluids will initially show solid-like behavior until a threshold shear stress (called the yield stress) is applied; after which they will show fluid-like behavior (continuously deforming while the shear stress is applied) (see Figure 31.2a). A common example of this type of fluid is toothpaste, which will not flow at all until a threshold shear value is exceeded. Broadly this kind of behavior is described as viscoelasticity. Bingham plastic materials can show dilatant, Newtonian, or pseudo-plastic behavior after their yield point. It is also worth noting that these terms are often not consistently applied.

The study and measurement of these more complicated, non-Newtonian, shear stress/shear strain rate behaviors is a subset of the larger science referred to as rheology and is largely beyond the scope of this chapter.

31.2.5 Mathematical Formalism

In this section we develop the mathematical formulations governing viscosity, and explain the roles and relations between shear viscosity and bulk viscosity. The discussion begins with the consideration of a very simple one-dimensional (1D) flow situation, and then introduces the more general 3D form of the equations.

Shear viscosity is defined mathematically by Newton’s Law of Viscosity. Newton’s law defines viscosity as the physical property that relates the shear stress produced as a reaction to an applied strain deformation rate. If one considers the simple flow shown in Figure 31.1a, a thin layer of fluid confined between two infinite parallel flat plates the upper moving within its own plane relative to the lower, the total shear strain rate on any layer in the fluid is given by ∂u/∂y which is the rate of change of x direction velocity in the perpendicular (y) direction. Newton’s law states that the shear stress experienced on the lower face of such a layer is given by:

(31.1) equation

where μ is the (shear) viscosity of the fluid at the applied strain rate. If the shear viscosity is constant with regard to strain rate then the fluid is said to be Newtonian. If the fluid exhibits a more complex relationship between shear stress and strain rate the fluid is defined as non-Newtonian. Distinctions between Newtonian and non-Newtonian behavior is discussed in greater detail in Section 31.1.2.

The viscosity picture becomes more complicated if we allow for more complex (3D) motions of the fluid. Any motion that a particle of fluid can undertake can be constructed from a superposition of the following four simpler types of motion: pure translation (movement without rotation or deformation), pure rotation (rotation without movement or deformation), pure linear strain (deformation without motion that does not disturb angles within the fluid particle), and pure shear strain (deformation without motion which does change angles within the fluid particle). Figure 31.3 illustrates these types of motions in two dimensions. In a fully general flow any combination of these motions can occur in any or all of the three coordinate directions. In such cases independent shear deformations (Figure 31.3d) can occur on any or all of three orthogonal planes. The fluid can also undergo purely extensional deformations (i.e., elongations without shear, Figure 31.3c) in any or all of the three dimensions. These extensional deformations will in some circumstances also contribute to the stress response of the fluid, for example, if they combine in such a way that the volume of the fluid element changes then the bulk viscosity described below will also be important. Thus for general 3D deformations it is necessary to use tensors (a branch of mathematics that describes vectors pointing in several directions) to describe the full relation between the stress at a point in the fluid and the resulting strain.

FIGURE 31.3 Types of fluid motion and deformation illustrated in 2-D.

The stress response of a Newtonian fluid element in response to a fully general deformation is given in White (2006) as

(31.2)

equation

Here μ is the shear viscosity, γ is the Bulk Viscosity coefficient and δij is the Kronecker delta function (i.e., δ = 1 when i = j, and δ = 0 when i ≠ j), and i and j are indices used to refer to the three orthogonal planes. Those interested in the derivation of this relation are directed to White or any other graduate level fluids text.

One important point to note arising from Equation (2) is that the stress response to purely extensional strains is described by the bulk viscosity, γ, or through the closely related second coefficient of viscosity μ’ (the bulk viscosity, γ is equal to the second coefficient of viscosity, μ’, plus 2/3 μ, see Owczarek (1964), for example, for a more thorough discussion). The topic of bulk viscosity is largely beyond the scope of this chapter, but these few comments are made to inform the reader of when it may be safely ignored, and when it may become important. First it is important to note that the bulk viscosity is not as well understood or characterized as the more common shear viscosity. Fortunately, for Newtonian fluids, the bulk viscosity coefficient occurs only in combination with the divergence of the velocity field ; and for incompressible fluids conservation of mass (i.e., continuity) requires that . Hence the bulk viscosity will play no role in a truly incompressible fluid.

Of course no substance is truly 100% incompressible, so if one is concerned with acoustic issues (which are inherently a compressibility phenomenon) or flows at significant Mach number the bulk viscosity may play a role. One way to gain insight on when the bulk viscosity may play a significant role is to examine its role in viscous dissipation. Ultimately the role of viscosity (shear or bulk) is to irreversibly convert the mechanical energy in a flow into thermal energy (heat). This effect is known as viscous dissipation. Hence the magnitude of the dissipation caused by the viscosity can give one indication of the qualitative importance of viscosity in the flow. The dissipation (Φ) caused by the bulk viscosity is given by Shaughnessy et al. (2005) as:

(31.3) equation

where γ is the bulk viscosity and ρ is the density. In order to be significant, either large changes in density are required, or the changes in density must occur over very short time scales. Thus, due to the relative magnitudes of the extensional strains typically involved, the dissipation due to bulk viscosity (and indeed the bulk viscosity itself) may safely be ignored for almost all practical applications except shock waves (which involve large changes in density) and attenuation of high frequency (small dt) sound or ultrasound. Henceforth in this chapter the term viscosity shall refer to shear viscosity only.

31.2.6 Relation of Viscosity to Molecular Theory

As discussed above, ultimately the role of viscosity is to dissipate the ordered kinetic energy associated with the macroscopic motions of a flow to disordered, randomly distributed, microscopic molecular energy (i.e., thermal energy). As such it becomes clear that the quantity that we refer to as viscosity is the macroscopic manifestation of molecular level effects; much in the same way that the macroscopic quantity of pressure represents the average net force per unit area caused by countless individual molecular collisions on a surface. That is to say that viscosity, like pressure, temperature, and density, is a continuum property of a substance. It makes no more sense to talk about the viscosity of a single molecule than it does to talk about the density of a single atom of a gas. The fact that the viscosity is an emergent property arising only for large collections of molecules places an important limitation on the concept of fluid viscosity, namely the continuum limit. Essentially the (continuum) concept of viscosity breaks down when applied on length scales that are comparable to the mean free path of the molecules in the fluid, or when applied on time scales that are comparable to the mean time period between molecular collisions in the fluid. For ordinary macroscopically sized flows at ordinary pressures the continuum limit is not a concern. However in applications such as nanotechnology (where the length scales of concern become very small) or rarefied gas dynamics (where the mean free path of molecules in very low density gases becomes very large) this limit should be kept in mind.

Examining how the macroscopically observed property of viscosity arises from molecular effects can provide insight and physical intuition about viscosity. If we examine how momentum is transported by the thermal (random) motions of molecules within a flow we can come to understand the molecular basis for viscosity.

First, let us consider the physics qualitatively. To do so consider again the simple 1D shear flow shown in Figure 31.1. Specifically consider the molecules near a plane parallel to the top plate and half way between the top plate and the bottom plate. All of these molecules will have velocities that are composed of their individual, random (thermal) velocities (which average out to zero bulk velocity), plus a small extra velocity component which depends on the local value of the bulk velocity. Those molecules slightly above the plane will, on average, have slightly larger values of velocity in the x direction and therefore also have slightly higher values of x direction momentum. Similarly, on average, those below the plane will have slightly lower values of x velocity and momentum. All of the molecules (above and below the plane) will have random thermal velocities with components in all three directions. The y component of these random velocities will occasionally carry molecules across the plane in both directions. But because of the asymmetry in velocity above and below the plane (i.e., because of the gradient of velocity perpendicular to the plane) the net effect of these random cross plane exchanges will be to transport x momentum from above the plane to below the plane. That is, there will be a net flux of x momentum in the negative y direction.

In fact the quantity we call viscosity is precisely defined by this diffusive transport (i.e., transport caused by random molecular motions rather than bulk macroscopic fluid motions) of momentum in the direction opposite to a velocity gradient. A simple dimensional analysis will convince the reader that a stress, such as shear stress (measured in Pa = kg/m s²), is dimensionally identical to a flux of momentum (transport of momentum per second per unit area = kg/m s²). Thus, referring back to Equation (1), we can consider viscosity to be the fluid property that relates the diffusive momentum flux (τ) to the velocity gradient driving it.

31.2.7 Effect of Pressure and Temperature on Viscosity

31.2.7.1 Low Density Gases

We can make this relationship more quantitative by examining the actual molecular interactions occurring in the fluid. The following development parallels that given by Bird et al. (1960), but similar developments can be found in any text covering the kinetic theory of gases. The simplest model of viscosity arises from a consideration of simple kinetic theory of a low density gas where we assume that the gas molecules are rigid spheres with diameter "d that only interact through collisions (i.e., there are no forces causing action at a distance" between molecules). Thus molecules will exchange momentum and come into equilibrium with their surroundings only through collisions. Kinetic theory for rigid sphere molecules also provides these other important results that will be used in this development:

The average random (thermal) velocity of the molecules in the gas will be

(31.4) equation

where k is Boltzman’s Constant, m is the mass of the individual molecules and T is the absolute temperature.

The mean free path of a molecule between collisions, λ, will be

(31.5) equation

where n is the number density (number per unit volume) of the gas molecules.

The average frequency of collision per unit area (from one side) on any plane in the flow

(31.6) equation

If one again considers the situation of a plane between the two plates in Figure 31.1 it can be shown that a molecule will travel an average vertical distance of 2/3 λ between collisions. The flux of x momentum transferred across the plane from below is then Zmvx,y−2/3/λ where vx,y−2/3λ is the average excess (nonthermal) x velocity component at a y location one collision distance below the plane (i.e., at y−2/3λ). Similarly the flux of x momentum transferred across the plane from above is, Zmvx,y-2/3/λ. Thus the net x momentum flux is:

(31.7) equation

And, if the x velocity gradient in the vicinity of y is linear we can replace the vx at locations above and below the plane with a first order Taylor series in terms of the gradient at the plane, yielding:

(31.8)

equation

Simplifying and substituting in the definition of Z gives:

(31.9) equation

Comparing back to Equation (31.1) we can see that:

(31.10) equation

Further substitution for and λ yields:

(31.11) equation

Thus if the molecules are considered as perfectly rigid spheres (the simplest model) then . Note also that the viscosity of such a gas is not expected to be a function of pressure based on this very simple molecular model. This is an important result which is largely borne out by experimental observations of real low-density gases; their dynamic viscosity is observed to depend only very weakly on pressure.

Of course molecules are not perfectly rigid spheres for that would imply no force whatsoever between molecules until they come into contact (when their center to center distance equals d) and then an infinite repulsion force. Clearly the idea of an infinite repulsion force is unphysical. In fact all molecules will show some action at a distance and, more importantly, act as if they have some give or flexibility when they collide which will eliminate the unphysical infinite forces inherent in the simple rigid sphere model. Typically these interactions are modeled as an intermolecular potential function from which the magnitudes of attractive and/or repulsive forces as a function of center to center distance can be calculated; as can an effective molecular diameter. The exact nature of these intermolecular potential functions is complicated, and has received extensive study, but is largely beyond the scope of this chapter. The interested reader is directed to Kogan (1969) or Vicenti and Kruger (1965), for example. Here it is sufficient to make a few points. First, as more complex and precise relations for the potential theory are used, the predictions of macroscopic properties such as viscosity improve markedly. Second, the action at a distance effects associated with these potentials allow for pressure to have an effect on the viscosity of gases. But, as stated above, this is typically found to be a weak effect for common gases. And finally, the functional relation between the viscosity and the temperature in low-density gases depends critically on the details of this potential function.

The next simplest model of intermolecular relations (after the rigid sphere model) is one proposed by Maxwell. In such a model the potential function drops off as 1/s⁴ where s is the center to center distance between the molecules. With such a model it can be shown that the effective molecular diameter will vary as: and so, referring back to Equation 11, μ ~ T. That is, a gas composed of Maxwellian molecules will have a viscosity that will increase linearly with temperature, as opposed to with the square root of temperature predicted from the rigid sphere model.

31.2.8 Correlations of Viscosity with Temperature for Gases

In reality, gases typically exhibit behaviors between the two extremes discussed in the section above (rigid and Maxwellian) and so their viscosity’s dependence on temperature is commonly correlated with a power-type law, as given by White (2006):

(31.12) equation

Where the parameters n, To, and μo are specific to the particular gas. Values for "n" for most simple gas molecules fall between 0.5 and 1, as predicted by the above discussion. Some values for n, To, and μo can be found in White.

Another common correlation technique for gas viscosities is based on the work of Sutherland (1893) (as covered in Vicenti and Kruger, 1965). Here viscosity is correlated as

(31.13) equation

where S is the so-called Sutherland Parameter and To, and μo are reference values. A completely equivalent form of this equation:

(31.14) equation

where

(31.15) equation

is often used as well. Poling et al. (2004) provided a detailed discussions of several much more detailed methods for estimating the variation of viscosity of both pure gases and mixtures of gases at various temperatures.

31.2.9 Correlations of Viscosity with Temperature for Liquids

In real (higher density, more complex molecular structure) gases and especially in liquids intermolecular forces (beyond the collisional forces discussed previously) play a critically important role. Molecules in such substances can exert significant force (and hence transfer significant momentum) at a distance without colliding. Since viscosity as a property arises from transfers of momentum, these actions at a distance must be accounted for in any physical model that hopes to adequately predict a material’s viscosity. The nature and magnitude of these non-collisional interactions are so complex and so large in liquids that currently no one general model exists that will adequately predict the viscosity of all liquids. Instead many specialized empirical and semi-empirical relations are available. Some general trends however are observed for liquids, the most important being that the viscosity of liquids falls off strongly with increasing temperatures. One type of curve fit that is recommended for liquid viscosity, recommended by White (2006) is

(31.16) equation

where a, b, and c are curve fit parameters and To and μo are reference values. Another, slightly simpler, empirical correlation often used is Andrade’s equation (Munson et al., 2009)

(31.17) equation

which is often presented in the alternate form

(31.18) equation

Viswanath et al. (2007) provide a lengthy discussion of such correlation methods and coefficients A and B for a wide variety of liquids.

31.2.10 Effect of Pressure on Viscosity

The effects of pressure on viscosity are not nearly as significant as the effects of temperature. In many practical circumstances it is entirely sufficient to simply neglect the effect of pressure on viscosity. This has lead to pressure effects being much less studied, and to data on viscosity at different pressures being much more sparse.

For low density gases, the molecular dynamics models discussed above (refer to Equation 31.10) indicate that the absolute viscosity, μ, should not depend on pressure at all; due to the competing effects of increasing number density (n) and decreasing mean free path (λ) as pressure is increased. Of course the value of the kinematic viscosity, ν = μ/ρ, of gases will decrease with increasing pressure due to the increase in density, p, of the gas as the pressure increases. These predictions are largely borne out by experimental data for common low-density (ideal) gases. Poling et al. (2004) provide a detailed discussion of several methods for estimating the viscosities of both pure gases and mixtures of gases at higher pressures.

Viscosity data for liquids at high pressure is sparse when compared to the quantity of data available at or near atmospheric pressures. In general, as pressure increases, so does the viscosity of most liquids. Both Viswanath et al. (2007) and Poling et al. (2004) recommend a method for estimating the effect of pressure on liquid viscosity attributed to Lucas (1981):

(31.19) equation

where μ is the viscosity of the liquid at pressure Ρ; μSL is the viscosity of the liquid at vapor pressure Pvp; Pvp is the the vapor pressure; ΔPr = (P – Pvp)/Pc; Tr = T/TC; Pc, Tc are the critical pressure and temperature; ω is the acentric factor

equation

In spite of the theory and correlation techniques described above, in many practical situations one must resort to simply measuring viscosity. The following section describes the different measurement techniques available.

31.3 MAJOR VISCOSITY MEASUREMENT METHODS

Viscometers are designed to make use of the theoretical relationship between shear stress and strain rate to measure viscosity. They do this using simple flows (1D, steady, fully developed) in which both the shear stress and strain rate can be measured. There are three primary types of viscometers: flow, drag, and resonant. The flow-type viscometers measure the rate of flow of the fluid in a tube or through an orifice. The shear stress can be calculated from theory (e.g., capillary tube viscometer) or estimated based on theory (e.g., orifice cup viscometers). Use of these types of viscometers yields values for kinematic viscosity. Design parameters for flow-type viscometers include minimizing entrance and exit effects, maintaining a constant pressure head (which drives the flow), minimizing surface tension effects and mitigating effects of temperature variation. Drag-type viscometers measure either the force on an object as it moves at a specified rate in the fluid (rotational viscometers) or measure the time it takes for an object to move a specified distance through the fluid (falling object and bubble tube viscometers). Use of these types of viscometers yields values for absolute viscosity (except for the bubble tube which measures kinematic viscosity). Design parameters for drag-type viscometers include minimizing the effects of turbulence and flow separation through the specification of a flow condition (generally a low relevant Reynolds number), controlling for transients, minimizing surface tension effects and mitigating effects of temperature variation. The third type of viscometer is the resonant or vibrational viscometer which is most commonly used in in-line process applications. These are designed so that changes in the viscous damping bring about significant changes in the resonance behavior of the instrument. Use of these viscometers yields values for kinematic viscosity.

This section presents information on each of the three major viscometer types. It begins with the drag-type viscometers (falling object, bubble tube, and rotational) followed by the flow-type viscometers (capillary and orifice) and concludes with a discussion of vibrational viscometers. Each section includes information on the theory of operation, a description of the types of viscometers available, a list of available manufacturers, and the capabilities and advantages/limitations. This chapter focuses on the use of laboratory-type viscometers; however some information is included on the use of process viscometers. This list of viscometers is not intended to be exhaustive but includes many of those that are most readily available commercially. There are other specialized methods for measuring viscosity and the reader is referred to Viswanath et al. (2007) for more information.

31.3.1 Drag-Type Viscometers

31.3.1.1 Falling Object Viscometers

Theory of Operation Falling object viscometers determine viscosity (μ) by measuring the drag force acting on a falling object under specific flow conditions. Use of the falling object viscometer requires a separate measurement of density to calculate kinematic viscosity. Figure 31.4 illustrates the forces acting on a falling object. This case shows a spherical object (a ball), however there are a variety of falling objects that can be used such as needles, and cylinders. There are three forces acting on the object: FB the buoyancy force and FD the drag force act upwards while FG the gravitation force (weight) acts down. The buoyancy force is calculated using Archimedes principle and is equal to the weight of the fluid displaced by the object:

FIGURE 31.4 Schematic showing the forces acting on a falling object.

(31.20) equation

where is the volume of the object, ρf is the density of the fluid, and g is the gravitational constant. The weight of the object is simply:

(31.21) equation

where ρB is the density of the object. If the object is travelling at terminal speed, the acceleration will be zero and application of Newton’s second law yields an equation relating the three forces:

(31.22) equation

The drag force is composed of a shearing force (due to the fluid) and a pressure force (due to flow separation). Falling object viscometers are generally designed to operate in the Stokes (creeping) flow regime which is characterized by a lack of flow separation and occurs for very low Reynolds numbers (Re < 0.1). In this case the drag force is due only to the shearing force. For example, if the object is a sphere then the Reynolds number is calculated as

(31.23) equation

Where V is the terminal speed of the object, D is the diameter of the sphere, and μ is the viscosity. For the low Reynolds number situation, the drag force is related to Reynolds number by Stokes Law:

(31.24) equation

Combination of these equations yields an equation for viscosity in terms of the speed, diameter, and density difference:

(31.25) equation

This theory applies to balls moving at low Reynolds number in an infinite media (see Brizard et al. 2005 for development of the theory accounting for more realistic conditions). In many commercial applications, the falling object is placed in a tube of specified diameter (see Figure 31.5). The object, typically a ball, will fall or slide down the tube and the user measures the time it takes for the ball to travel between two timing lines. The first timing line is placed sufficiently far from the top of the viscometer to allow the ball to reach terminal velocity. The manufacturer supplies a calibration equation of the sort:

FIGURE 31.5 Falling ball viscometers (a) Gilmont and (b) Hakke type

(Courtesy of Brookfield Engineering Labs).

(31.26) equation

where K is a calibration constant and t is the measured time to fall the specified distance. To obtain the calibration constant the manufacturer measures the fall time of the ball in a series of liquids of known viscosity.

Types of Viscometers/Options Falling object viscometers use a variety of objects including spheres, needles, and cylinders. Falling object viscometers often come with a set of objects, each with different mass/density which allows one to measure viscosity over a range of values. The most readily available commercial falling object viscometers are the relatively inexpensive Gilmont-type falling ball viscometers and the more expensive, more accurate Haake-type falling ball viscometers as shown in Figure 31.5a and b. The Haake viscometers include a mounting mechanism and an outer chamber that can be used for temperature control of the sample during testing. Falling needle viscometers use thin needle like objects which are designed to minimize wall effects and are more stable as they fall (see Davis and Brenner, 2001). They can be used to measure viscosity of non-Newtonian fluids. Falling cylinder viscometers involve a more complex flow field subject to significant end effects; however, they are useful for measuring viscosity at high pressure (Cristescu et al., 2002). Table 31.3 provides further information about suppliers of falling object viscometers.

TABLE 31.3 Sampling of Common Falling Ball Viscometer Types

Summary Although falling object viscometers are relatively inexpensive, the use of one requires some skill and is labor intensive. The tubes must be carefully cleaned before use and when filling the tube with the fluid of interest, care needs to be taken to avoid air bubbles. They cannot be used with opaque liquids. After setup, each individual measurement can take 1–2 min to complete. Gilmont-type viscometers must be mounted or held vertically and care needs to be taken when handling them to avoid heating up the fluid in the viscometer. Haake-type viscometers are pre-mounted at a specified angle and the falling ball tube is located inside an outer glass tube that can be easily connected to a circulating water bath for temperature control. They are not automated and require manual timing. However, it is relatively easy to compare the viscosity of different fluids by using multiple viscometers. The primary sources of error that arise in the use of a falling ball viscometer are related to temperature effects, handling and contamination.

31.3.2 Bubble (Tube) Viscometers

Theory of Operation Although most drag-type viscometers measure absolute viscosity