The CRC Handbook of Mechanical Engineering, Second Edition [2ed.] 8123918135, 9780849308666, 0-8493-0866-6

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The

CRC HANDBOOK

of

MECHANICAL ENGINEERING SECOND EDITION

© 2005 by CRC Press LLC

The Mechanical Engineering Handbook Series Series Editor

Frank Kreith Consulting Engineer

Published Titles Air Pollution Control Technology Handbook Karl B. Schnelle, Jr. and Charles A. Brown Computational Intelligence in Manufacturing Handbook Jun Wang and Andrew Kusiak Fuel Cell Technology Handbook Gregor Hoogers Handbook of Heating, Ventilation, and Air Conditioning Jan F. Kreider Hazardous and Radioactive Waste Treatment Technologies Handbook Chang Ho Oh Inverse Engineering Handbook Keith A. Woodbury Opto-Mechatronic Systems Handbook: Techniques and Applications Hyungsuck Cho The CRC Handbook of Mechanical Engineering, Second Edition Frank Kreith and D. Yogi Goswami The CRC Handbook of Thermal Engineering Frank Kreith The Handbook of Fluid Dynamics Richard W. Johnson The MEMS Handbook Mohamed Gad-el-Hak Biomedical Technology and Devices Handbook James Moore and George Zouridakis

Forthcoming Titles Multi-Phase Flow Handbook Clayton T. Crowe Shock and Vibration Handbook Clarence W. de Silva

© 2005 by CRC Press LLC

The

CRC HANDBOOK

of

MECHANICAL ENGINEERING SECOND EDITION

Edited by

Frank Kreith and

D. Yogi Goswami

CRC PR E S S Boca Raton London New York Washington, D.C. © 2005 by CRC Press LLC

0866_C00.fm Page iv Tuesday, August 10, 2004 5:04 PM

Library of Congress Cataloging-in-Publication Data The CRC handbook of mechanical engineering / edited by Frank Kreith, Yogi Goswami. – 2nd ed. p. cm. — (Mechanical engineering handbook series) Includes bibliographical references and index. ISBN 0-8493-0866-6 (alk. paper) 1. Mechanical engineering—Handbooks, manuals, etc. I. Kreith, Frank. II. Goswami, D. Yogi. III. Series. TJ151.C73 2004 621.3—dc22

2004051970

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 0-8493-0866-6/05/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2005 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0866-6 Library of Congress Card Number 2004051970 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

© 2005 by CRC Press LLC

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Preface

The purpose of the second edition of the CRC Handbook of Mechanical Engineering is to continue providing practicing engineers in industry, government, and academia with up-to-date information on the most important topics of modern mechanical engineering. The book covers traditional topics such as thermodynamics, solid and fluid mechanics, heat and mass transfer, materials, controls, energy conversion, manufacturing and design, robotics, environmental engineering, economics and project management, patent law, and transportation. These topics were covered in the first edition, but they have been updated, new references have been added, and efforts have been made to provide information on computer technology related to the topics at the end of each chapter. But in the 10 years since the first edition of this handbook was published, changes have taken place in engineering and technology and additional topics that were not treated in the first edition have come to the fore. To bring the new edition up-to-date, chapters on topics such as nanotechnology, MEMS, electronic packaging, global climate change, electric and hybrid vehicles, and bioengineering have been added. Moreover, the editorial supervision for the second edition has been broadened by the addition of Yogi Goswami as co-editor. In a work of the size of the second edition of this handbook, it is unavoidable that certain errors or omissions may have occurred. Therefore, the editors appreciate the readers calling any of these shortcomings to their attention and every effort will be made to correct them. We also welcome continuous feedback from readers about topics that may have been omitted and should be considered for inclusion in future editions of this work. The editors would like to thank all the contributors, as well as the CRC staff, especially Helena Redshaw, Jessica Vakili, Cindy Carelli, and Susan Fox, for their assistance in the preparation of this handbook. Frank Kreith [email protected] D. Yogi Goswami [email protected]

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Editors

Dr. Frank Kreith is Professor Emeritus of Engineering at the University of Colorado, Boulder. From 1998 to 2001 he served as the American Society of Mechanical Engineers [ASME] International Legislative Fellow for Energy and Environment at the National Conference of State Legislatures (NCSL) where he provided assistance on energy, transportation, and environmental protection to legislators in all 50 state governments. Prior to joining NCSL in 1988, Dr. Kreith was the Chief of Thermal Research at the Solar Energy Research Institute (SERI), currently the National Renewable Energy Laboratory. During his tenure at SERI, he participated in the Presidential Domestic Energy Review, served as an advisor to the Governor of Colorado, and was the editor of the ASME Journal of Solar Energy Engineering. In 1993, he received the first General Achievement Award from SERI. From 1951 to 1977, Dr. Kreith taught at the University of California, Lehigh University, and the University of Colorado. He is the author of over 100 peer-reviewed articles and the author of textbooks on heat transfer, solar energy, and transportation. He is the recipient of the Charles Greeley Abbot Award from ASES and the Max Jakob Award from ASME-AIChE. In 1992, he received the Ralph Coats Roe Medal from ASME for providing technical information to legislators about energy conservation and environmental protection, and in 1997 the Washington Award for “unselfish and preeminent service in advancing human progress.” In 1998, Dr. Kreith was awarded the ASME medal for research, publications, and public service. Dr. Kreith has served as a consultant and advisor all over the world. His assignments include consultancies to Vice Presidents Rockefeller and Gore, the U.S. Department of Energy, NATO, the U.S. Agency for International Development, and the United Nations. He is a registered professional engineer, a Life Fellow of ASME and a Fellow ASES and the American Association for the Advancement of Science [AAAS]. D. Yogi Goswami, Ph.D., P.E., is a professor of mechanical and aerospace engineering and Director of the Solar Energy and Energy Conversion Laboratory at the University of Florida, Gainesville. Dr. Goswami is internationally known for his research in fundamental and applied areas of renewable energy. He has published as an author or editor 7 books, 12 book chapters, 4 conference proceedings, and more than 100 refereed technical papers. He also holds 5 U.S. patents and 1 worldwide patent. In 2001 he received the University of Florida’s award of Research Foundation Professor. Dr. Goswami is the Editor-in-Chief of Advances in Solar Energy: An Annual Review of the Developments in Solar Energy, published by the American Solar Energy Society (ASES). He is also the Editor-in-Chief of the Solar Energy journal, published by the International Solar Energy Society (ISES) and a past associate editor of the ASME Journal of Solar Engineering. Dr. Goswami has chaired a number of task forces to advise the U.S. Congress and the federal administration on energy policy. He has also made oral presentations on behalf of ISES to the United Nations charrette on world sustainable energy.

© 2005 by CRC Press LLC

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Dr. Goswami is a recognized leader in professional scientific and technical societies. He is the President of ISES and a member of the Board of Governors of the ASME International. He has also served as Senior Vice President of ASME-International (2000–2003), President of the International Association for Solar Energy Education (IASEE, 2000–2002), and a member of the Board of Directors of ISES (2000–2002). He has also served as a Vice President for Energy Resources Board of ASME-International (1989–1993), Vice President of IASEE (1998–2000), and a member of the Board of Directors of ASES (1996–2000). Dr. Goswami is a recipient of the John Yellott Award for Solar Energy from ASME and the Charles Greely Abbott award of the ASES, and more than 20 other awards from engineering and scientific societies. He is a registered professional engineer and a fellow of ASME International and ASES.

© 2005 by CRC Press LLC

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Contributor List

Elsayed M. Afify

Richard Bajura

Kenneth B. Black

North Carolina State University Raleigh, North Carolina

West Virginia University Morgantown, West Virginia

University of Massachusetts Amherst, Massachusetts

Talyan Altan

William W. Bathie

Carl J. Bliem (Deceased)

Ohio State University Columbus, Ohio

William F. Ames Georgia Institute of Technology Atlanta, Georgia

David C. Anderson Purdue University West Lafayette, Indiana

Vincent W. Antonetti Consultant Poughkeepsie, NY

Nagaraj K. Arakere University of Florida Gainesville, Florida

Anthony F. Armor Electric Power Research Institute Palo Alto, California

Iowa State University Ames, Iowa

Kenneth J. Bell Oklahoma State University Stillwater, Oklahoma

Dale E. Berg Sandia National Laboratories Albuquerque, New Mexico

Stanley A. Berger

Robert F. Boehm University of Nevada Las Vegas, Nevada

E. Richard Booser Consultant Vero Beach, Florida

Michael L. Brown Harley-Davidson Milwaukee, Wisconsin

University of California Berkeley, California

Matthew Buczek

Arthur K. Bergles

General Electric R & D Center Schenectady, New York

Rensselaer Polytechnic Institute Troy, New York

Desikan Bharathan

George Cain Georgia Institute of Technology Atlanta, Georgia

National Renewable Energy Laboratory Golden, Colorado

Gonzaga University Spokane, Washington

Bharat Bhushan

Van P. Carey

Ohio State University Columbus, Ohio

University of California Berkeley, California

Roger E. A. Arndt University of Minnesota Minneapolis, Minnesota

CJB Consulting

Massimo Capobianchi

Barbara Atkinson Lawrence Berkeley National Laboratory Berkeley, California

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Stephen M. Birn

Tien-Chien Chang

Moog Inc., Aircraft Group Torrance, California

Purdue University West Lafayette, Indiana

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John C. Chen

Philip C. Crouse

Steven I. Freedman

Lehigh University Bethlehem, Pennsylvania

Philip C. Crouse and Associates Dallas, Texas

Gas Research Institute Deerfield, Illinois

Liang-Yu Chen

Peter S. Curtiss

Ohio Aerospace Institute NASA Glenn Research Center Cleveland, Ohio

Stuart W. Churchill University of Pennsylvania Philadelphia, Pennsylvania

Wendy Clark National Renewable Energy Laboratory Golden, Colorado

Robert Clear Lawrence Berkeley National Laboratory Berkeley, California

Hugh W. Coleman University of Alabama Huntsville, Alabama

Curtiss Engineering Boulder, Colorado

Mark R. Cutkosky Stanford University Stanford, California

Andrea Denver Lawrence Berkeley National Laboratory Berkeley, California

Kenneth R. Diller University of Texas Austin, Texas

Nevis E. Cook, Jr. Colorado School of Mines Golden, Colorado

Jeff R. Crandall University of Virginia Charlottesville, Virginia

Sumit Ghosh Stevens Institute of Technology Hoboken, New Jersey

Bhuvenesh C. Goswami Clemson University Clemson, South Carolina

D. Yogi Goswami University of Florida Gainesville, Florida

John Fildes Northwestern University Evanston, Illinois

Hank Grant University of Oklahoma Norman, Oklahoma

John Firor National Center for Atmospheric Research Boulder, Colorado

Rutgers University East Brunswick, New Jersey

William F. Fischer, III

Gregory W. Hall

H.E. Cook University of Illinois at UrbanaChampaign Urbana, Illinois

Mohamed Gad-el-Hak Virginia Commonwealth University Richmond, Virginia

Lanxide Corporation Newark, Delaware

John M. Fitzgerald University of Texas Fort Worth, Texas

Jean-Pierre Fleurial California Institute of Technology Duarte, California

Victor A. Greenhut

University of Virginia Charlottesville, Virginia

Ronald R. Hewitt Cohen Colorado School of Mines Golden, Colorado

K.G.T. Hollands University of Waterloo Waterloo, Ontario, Canada

Malcolm J. Crocker

Dan M. Frangopol

Trevor D. Howes

Auburn University Auburn, Alabama

University of Colorado Denver, Colorado

University of Connecticut Storrs, Connecticut

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Shou-Heng Huang

Francis E. Kennedy

Kam Lau

Raytheon Appliance Tech Center Coralville, Iowa

Dartmouth College Hanover, New Hampshire

Automated Precision, Inc. Rockville, Maryland

Iqbal Husain

John Kern

Zalman Lavan (Deceased)

University of Akron Akron, Ohio

James D. Idol Rutgers University Piscataway, New Jersey

Tissa Illangasekare Colorado School of Mines Golden, Colorado

Herbert A. Ingley University of Florida Gainesville, Florida

Thomas R. Irvine, Jr. (Deceased) State University of New York, Stony Brook Stony Brook, New York

B. Johan Ivarsson University of Virginia Charlottesville, Virginia

William D. Jackson HMJ Corporation Kensington, Maryland

Chand K. Jotshi AT&TL, Inc. Gainesville, Florida

Richard L. Kautz

Siemens Power Corporation Milwaukee, Wisconsin

Illinois Institute of Technology Evanston, Illinois

Jungho Kim University of Maryland College Park, Maryland

Andrew C. Lee Purdue University West Lafayette, Indiana

Nam Ho Kim University of Florida Gainesville, Florida

Kok-Meng Lee Georgia Institute of Technology Atlanta, Georgia

David E. Klett North Carolina A&T State University Greensboro, North Carolina

Richard L. Lehman Rutgers University Piscataway, New Jersey

Yoram Koren

John Leonard II

University of Michigan Ann Arbor, Michigan

Georgia Institute of Technology Atlanta, Georgia

Steven H. Kosmatka

Frank L. Lewis

Portland Cement Institute Skokie, Oregon

University of Texas at Arlington Arlington, Texas

Jan F. Kreider

Alex Lezuo

Kreider & Associates Boulder, Colorado

Siemens Power Generation Erlangen, Germany

Frank Kreith

Steven Y. Liang

University of Colorado Boulder, Colorado

Georgia Institute of Technology Atlanta, Georgia

National Institute of Standards & Technology Boulder, Colorado

Ajay Kumar

Noam Lior

NASA Langley Research Center Hampton, Virginia

University of Pennsylvania Philadelphia, Pennsylvania

Carl J. Kempf

Ashok V. Kumar

Kai Liu

NSK Ltd. Gunma, Japan

University of Florida Gainesville, Florida

University of Texas Fort Worth, Texas

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Tien-I. Liu

Anthony F. Mills

California State University Sacramento, California

University of California Los Angeles, California

Sergey Edward Lyshevski

Gregory L. Mines

Rochester Institute of Technology Rochester, New York

Roberto Pagano (Deceased) University of Florida Gainesville, Florida

Chan S. Park

Idaho National Engineering Laboratory Idaho Falls, Idaho

Auburn University Auburn, Alabama

Michael F. Modest

John A. Pearce

University of California Irvine, California

The Pennsylvania State University University Park, Pennsylvania

University of Texas Austin, Texas

Roop L. Mahajan

Robert J. Moffat

University of Colorado Boulder, Colorado

Stanford University Stanford, California

Georgia Institute of Technology Atlanta, Georgia

Ioan Marinescu

Michael J. Moran

Kansas State University Manahattan, Kansas

The Ohio State University Columbus, Ohio

Alan T. McDonald

Takeo Nakagawa

Purdue University West Lafayette, Indiana

Fine Tech Corporation Tokyo, Japan

James E. McMahon

Ronald M. Nelson

Lawrence Berkeley National Laboratory Berkeley, California

Iowa State University Ames, Iowa

Marc J. Madou

Paul Norton Mehran Mehregany Case Western Reserve University Cleveland, Ohio

National Renewable Energy Laboratory Golden, Colorado

Michael Merker

Andrew D. Oliver

American Society of Mechanical Engineers New York, New York

Sandia National Laboratories Albuquerque, New Mexico

Ira Pence

George A. Peters Peters & Peters Santa Monica, California

Walter D. Pilkey University of Virginia Charlottesville, Virginia

David W. Plummer Sandia National Laboratories Albuquerque, New Mexico

John W. Priest University of Texas Dallas, Texas

Ari Rabl Ecole des Mines de Paris Paris, France

George Raithby University of Waterloo Waterloo, Ontario, Canada

Ralph P. Overend Michael D. Meyer Georgia Institute of Technology Atlanta, Georgia

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National Renewable Energy Laboratory Golden, Colorado

Srihari Rajgopal Case Western Reserve University Cleveland, Ohio

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K.P. Rajurkar

Robert E. Schafrik

Daniel J. Strange

University of Nebraska–Lincoln Lincoln, Nebraska

National Research Council Washington, D.C.

Alfred University Tucson, Arizona

Mysore L. Ramalingam

Paul Schonfeld

UES, Inc. Dayton, Ohio

University of Maryland College Park, Maryland

Marshall J. Reed

Ramesh K. Shah

U.S. Department of Energy Washington, D.C.

Kitt C. Reinhardt Air Force Research Laboratory Kirtland AFB, New Mexico

Rolf D. Reitz University of Wisconsin Madison, Wisconsin

Joel L. Renner Idaho National Engineering Laboratory Idaho Falls, Idaho

Robert Reuther U.S. Department of Energy Morgantown, West Virginia

Alan Ridilla General Electric R & D Center Schenectady, New York

Giorgio Rizzoni The Ohio State University Columbus, Ohio

Rochester Institute of Technology Rochester, New York

N.V. Suryanarayana

Thomas E. Shannon University of Tennessee Knoxville, Tennessee

Larry W. Swanson

Thomas B. Sheridan Massachusetts Institute of Technology Cambridge, Massachusetts

GE EER Irvine, California

Yashitsuga Taketomi

Sherif A. Sherif

NSK Ltd. Gunma, Japan

University of Florida Gainesville, Florida

J.M.A. Tanchoco

Paul W. Shuldiner

Purdue University West Lafayette, Indiana

University of Massachusetts Amherst, Massachusetts

Donald D. Tippett

University of North Carolina Charlotte, North Carolina

University of Alabama in Huntsville Huntsville, Alabama

W. Glenn Steele, Jr.

Jessica Todd

Scott Smith

Mississippi State University Mississippi State, Mississippi

G.T. Stevens, Jr.

Honeywell, Engines, Systems and Services Tempe, Arizona

Univeristy of Texas, Arlington Arlington, Texas

Bela I. Sandor

California State Polytechnic University Pasadena, California

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Massachusetts Institute of Technology Amherst, Massachusetts

Michigan Technological University Houghton, Michigan

Ryan Roloff

University of Wisconsin Madison, Wisconsin

Nam P. Suh

University of Colorado Boulder, Colorado

M. Tomizuka University of California Berkeley, California

William B. Stine Y.L. Tong Georgia Institute of Technology Atlanta, Georgia

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James S. Tulenko

John Webster

Su-Hsia Yang

University of Florida Gainesville, Florida

University of Connecticut Storrs, Connecticut

Purdue University West Lafayette, Indiana

Blake P. Tullis

Walter T. Welford (Deceased)

Darrin J. Young

Utah State University Logan, Utah

J. Paul Tullis Tullis Engineering Consultants Logan, Utah

Jonathan W. Valvano

Imperial College of London London, England

Case Western Reserve University Cleveland, Ohio

Thomas H. Young Frank M. White University of Rhode Island Narragansett, Rhode Island

Merchant & Gould, P.C. Denver, Colorado

Federica Zangrando Donald F. Wilcock (Deceased) Tribolock, Inc.

National Renewable Energy Laboratory Golden, Colorado

Roland Winston

Paolo Zannetti

University of California Merced, California

The EnviroComp Institute Fremont, California

C. Channy Wong

Chen Zhou

Sandia National Laboratory Albuquerque, New Mexico

Georgia Institute of Technology Atlanta, Georgia

Stephen Wood

Hong Zhou

Florida Institute of Technology Melbourne, Florida

University of California Irvine, California

W.M.Wang

Lynn L. Wright

Christian A. Zorman

University of Nebraska–Lincoln Lincoln, Nebraska

Oak Ridge National Laboratory Oak Ridge, Tennessee

Case Western Reserve University Cleveland, Ohio

Weiping Wang

Toskiaki Yamaguchi

Phillip J. Zulueta

General Electric R & D Center Schenectady, New York

NSK Ltd. Gunma, Japan

Jet Propulsion Laboratory Pasadena, California

University of Texas Austin, Texas

Ian D. Walker Clemson University Clemson, South Carolina

Chris Wang IBM Tampa, Florida

Shan K. Wang Consultant Alhambra, California

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Table of Contents

Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Mechanics of Solids

Bela I. Sandor

Introduction Bela I. Sandor Statics Bela I. Sandor and Ryan Roloff Dynamics Stephen M. Birn and Bela I. Sandor Vibrations Bela I. Sandor and Stephen M. Birn Mechanics of Materials Bela I. Sandor, Stephen M. Birn, and Michael L. Brown Structural Integrity and Durability Bela I. Sandor Comprehensive Example of Using Mechanics of Solids Methods Bela I. Sandor

Engineering Thermodynamics

Michael J. Moran

Fundamentals Michael J. Moran Control Volume Applications Michael J. Moran Property Relations and Data Michael J. Moran Combustion Michael J. Moran Exergy Analysis Michael J. Moran Vapor and Gas Power Cycles Michael J. Moran Guidelines for Improving Thermodynamic Effectiveness Michael J. Moran

Fluid Mechanics

Frank Kreith

Fluid Statics Stanley A. Berger Equations of Motion and Potential Flow Stanley A. Berger Similitude: Dimensional Analysis and Data Correlation Stuart W. Churchill Hydraulics of Pipe Systems J. Paul Tullis and Blake P. Tullis Open Channel Flow Frank M. White External Incompressible Flows Alan T. McDonald Compressible Flow Ajay Kumar and Jessica Todd Multiphase Flow John C. Chen

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3.9 3.10 3.11 3.12 3.13 3.14 3.15

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

New-Newtonian Flows Thomas F. Irvine Jr. and Massimo Capobianchi Tribology, Lubrication, and Bearing Design Francis E. Kennedy, E. Richard Booser, and Donald F. Wilcock Pumps and Fans Robert F. Boehm Liquid Atomization and Spraying Rolf D. Reitz Flow Measurement Jungho Kim, Alan T. McDonald, and Sherif A. Sherif Pressure Measurement Jungho Kim Micro/Nanotribology Bharat Bhushan

Heat and Mass Transfer

Frank Kreith

Conduction Heat Transfer Robert F. Boehm Convection Heat Transfer George D. Raithby, K.G. Terry Hollands, and N.V. Suryanarayana Radiation Michael F. Modest Phase-Change Van P. Carey, John C. Chen, and Noam Lior Heat Exchangers Ramesh K. Shah and Kenneth J. Bell Temperature and Heat Transfer Measurements Robert J. Moffat and Jungho Kim Mass Transfer Anthony F. Mills Applications Arthur E. Bergles, Anthony F. Mills, Larry W. Swanson, and Vincent W. Antonetti Non-Newtonian Fluids — Heat Transfer Thomas F. Irvine, Jr. and Massimo Capobianchi Bioheat Transfer Kenneth R. Diller, Jonathan W. Valvano, and John A. Pearce

Electrical Engineering

Giorgio Rizzoni

Introduction Giorgio Rizzoni Fundamentals of Electric Circuits Giorgio Rizzoni Resistive Network Analysis Giorgio Rizzoni AC Network Analysis Giorgio Rizzoni AC Power Giorgio Rizzoni Frequency Response, Filters, and Transient Analysis Giorgio Rizzoni Electronics Giorgio Rizzoni Power Electronics Giorgio Rizzoni

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5.9 5.10 5.11 5.12

Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6

Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Operational Amplifiers Giorgio Rizzoni Digital Circuits Giorgio Rizzoni Measurements and Instrumentation Giorgio Rizzoni Electromechanical Systems Giorgio Rizzoni

Mechanical System Controls

Jan F. Kreider

Human–Machine Interaction Thomas B. Sheridan The Need for Control of Mechanical Systems Peter S. Curtiss Control System Analysis Peter S. Curtiss Control System Design and Application Peter S. Curtiss Advanced Control Topics Peter S. Curtiss, Jan Kreider, Ronald M. Nelson, and Shou-Heng Huang Control of Distributed Generation Technologies Peter S. Curtiss and Jan F. Kreider

Energy Resouces

D. Yogi Goswami

Introduction D. Yogi Goswami Types of Derived Energy D. Yogi Goswami Fossil Fuels Robert Reuther, Richard Bajura, and Philip C. Crouse Biomass Energy Ralph P. Overend and Lynn L. Wright Nuclear Resources James S. Tulenko Solar Energy Resources D. Yogi Goswami Wind Energy Resources Dale E. Berg Geothermal Energy Joel L. Renner and Marshall J. Reed

Energy Conversion

D. Yogi Goswami

Steam Power Plant John Kern Gas Turbines Steven I. Freedman Internal Combustion Engines David E. Klett and Elsayed M. Afify Hydraulic Turbines Roger E.A. Arndt Stirling Engines William B. Stine Advanced Fossil Fuel Power Systems Anthony F. Armor Energy Storage Chand K. Jotshi and D. Yogi Goswami Nuclear Power Roberto Pagano and James S. Tulenko Nuclear Fusion Thomas E. Shannon

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8.10 8.11 8.12 8.13

8.14 8.15

Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16

Chapter 10 10.1 10.2

Solar Thermal Energy Conversion D. Yogi Goswami Wind Energy Conversion Dale E. Berg Energy Conversion of the Geothermal Resource Carl J. Bliem and Gregory L. Mines Direct Energy Conversion Kitt C. Reinhardt, D. Yogi Goswami, Mysore L. Ramalingam, Jean-Pierre Fleurial, and William D. Jackson Ocean Energy Technology Desikan Bharathan and Federica Zangrando Combined-Cycle Power Plants Alex Lezuo

Air-Conditioning and Refrigeration Herbert A. Ingley and Shan K. Wang Introduction Shan K. Wang Psychrometrics Shan K. Wang Air-Conditioning Processes and Cycles Shan K. Wang Refrigerants and Refrigeration Cycles Shan K. Wang Outdoor Design Conditions and Indoor Design Criteria Shan K. Wang Principles of Load Calculations Ari Rabl and Peter S. Curtiss Air Handling Units and Packaged Units Shan K. Wang Refrigeration Components and Evaporative Coolers Shan K. Wang Water Systems Herbert A. Ingley and Shan K. Wang Heating Systems Shan K. Wang Refrigeration Systems Herbert A. Ingley and Shan K. Wang Thermal Storage Systems Shan K. Wang Air System Basics Shan K. Wang Absorption System Shan K. Wang Air-Conditioning Systems and Selection Shan K. Wang Desiccant Dehumidification and Air-Conditioning Zalman Lavan

Transportation

Frank Kreith

Transportation Planning Michael D. Meyer Design of Transportation Facilities John Leonard II and Michael D. Meyer

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10.3 10.4 10.5 10.6 10.7

Chapter 11 11.1 11.2 11.3 11.4 11.5

Chapter 12 12.1 12.2 12.3 12.4 12.5 12.6

12.7

Chapter 13 13.1 13.2

13.3

Operations and Environmental Impacts Michael D. Meyer, Paul W. Shuldiner, and Kenneth B. Black Transportation Systems Paul Schonfeld Alternative Fuels for Motor Vehicles Paul Norton and Wendy Clark Electric and Hybrid Vehicles Iqbal Husain Intelligent Transportation Systems Sumit Ghosh

Engineering Design

Ashok V. Kumar

Introduction Ashok V. Kumar Elements of the Design Process Nam P. Suh Design Tools Ashok V. Kumar Structural Design Criteria Nagaraj K. Arakere Design Optimization Nam Ho Kim

Materials

Bhuvenesh C. Goswami

Metals Victor A. Greenhut Polymers James D. Idol and Richard L. Lehman Adhesives Richard L. Lehman Wood Daniel J. Strange Portland Cement Concrete Steven H. Kosmatka Composites Bhuvenesh C. Goswami, Weiping Wang, R. Allan Ridilla, Mathew B. Buczek, Richard L. Lehman, and Daniel J. Strange Ceramics and Glass Richard L. Lehman, Daniel J. Strange, and William F. Fischer, III

Modern Manufacturing

Scott Smith

Introduction Scott Smith Unit Manufacturing and Assembly Processes Robert E. Schafrik, Steven Y. Liang, Trevor D. Howes, John Webster, Ioan Marinescu, Scott Smith, K. P. Rajurkar, W. M. Wang, Talyan Altan, Weiping Wang, Alan Ridilla, Matthew Buczek, S. H. Cho, Ira Pence, Toskiaki Yamaguchi, Yashitsugu Taketomi, and Carl J. Kempf Essential Elements in Manufacturing Processes and Equipment John Fildes, Yoram Koren, M. Tomizuka, and Kam Lau

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13.4

13.5 13.6

13.7

Chapter 14

Design and Analysis Tools in Manufacturing David C. Anderson, Tien-Chien Chang, Hank Grant, Tien-I. Liu, J. M. A. Tanchoco, Andrew C. Lee, and Su-Hsia Yang Rapid Prototyping Takeo Nakagawa Underlying Paradigms in Manufacturing Systems and Enterprise Management for the 21st Century H. E. Cook and Chris Wang Electronics Manufacturing: Processes, Optimization, and Control Roop L. Mahajan

Robotics

Frank Lewis

14.1 14.2 14.3 14.4

Introduction Frank Lewis Commercial Robot Manipulators John M. Fitzgerald Robot Configurations Ian D. Walker End Effectors and Tooling Mark R. Cutkosky and Peter McCormick 14.5 Sensors and Actuators Kok-Meng Lee 14.6 Robot Programming Languages — Robot Systems Stephen Wood 14.7 Robot Dynamics and Control Frank L. Lewis 14.8 Planning and Intelligent Control Chen Zhou 14.9 Design of Robotic Systems Kok-Meng Lee 14.10 Robot Manufacturing Applications John W. Priest and G. T. Stevens, Jr. 14.11 Industrial Material Handling and Process Applications of Robots John M. Fitzgerald 14.12 Mobile, Flexible-Link, and Parallel-Link Robots Kai Liu

Chapter 15 15.1 15.2 15.3 15.4 15.5

MEMS Technology

Mohamed Gad-el-Hak

Introduction Mohamed Gad-el-Hak MEMS Technology and Its Applications Mohamed Gad-el-Hak Microscale Manufacturing Processes Marc J. Madou and Hong Zhou MEMS Packaging Liang-Yu Chen and Phillip J. Zulueta Reliability and MEMS Srihari Rajgopal, Christian A. Zorman, Darrin J. Young, and Mehran Mehregany

© 2005 by CRC Press LLC

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15.6 15.7

Chapter 16 16.1 16.2 16.3 16.4 16.5 16.6

16.7

Chapter 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7

Chapter 18 18.1 18.2 18.3

Fluid Flow in Microdevices Mohamed Gad-el-Hak Solid Mechanics of Microdevices Fernando Bitsie, C. Channy Wong, Andrew D. Oliver, and David W. Plummer

Environmental Engineering Ari Rabl and Jan F. Kreider Introduction Ari Rabl and Jan F. Kreider Benchmarks and Reference Conditions Ari Rabl, Nevis Cook, Ronald R. Hewitt Cohen, and Tissa Illangasekare Sources of Pollution and Regulations Jan F. Kreider, Nevis Cook, Tissa Illangasekare, and Ronald R. Hewitt Cohen Regulations and Emission Standards Nevis Cook, Ronald R. Hewitt Cohen, and Jan F. Kreider Mitigation of Water and Air Pollution Jan F. Kreider, Nevis Cook, and Ronald R. Hewitt Cohen Environmental Modeling Paolo Zannetti, Tissa Illangasekare, Ronald R. Hewitt Cohen, Nevis Cook, Ari Rabl, and Peter S. Curtiss Global Climate Change John Firor

Engineering Economics and Project Management Chan S. Park and Donald D. Tippett Engineering Economic Decisions Chan S. Park Establishing Economic Equivalence Chan S. Park Measures of Project Worth Chan S. Park Cash Flow Projections Chan S. Park Sensitivity and Risk Analysis Chan S. Park Design Economics Chan S. Park Project Management Donald D. Tippett

Nanotechnology

Sergey Edward Lyshevski

Introduction Sergey Edward Lyshevski Applications of Engineering Biomimetics in Nanomachines Prototyping Sergey Edward Lyshevski. Nanomachines Synthesis and Classification Sergey Edward Lyshevski

© 2005 by CRC Press LLC

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18.4 18.5 18.6 18.7 18.8

Chapter 19

Synthesis, Design and Analysis of Nanomachines Sergey Edward Lyshevski Synchronous Reluctance Nanomachines Sergey Edward Lyshevski Permanent-Magnet Synchronous Nanomachines Sergey Edward Lyshevski Induction Nanomachines Sergey Edward Lyshevski Conclusions Sergey Edward Lyshevski

Mathematics

William F. Ames and George Cain

19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 19.10 19.11 19.12 19.13

Tables Linear Algebra and Matrices George Cain Vector Algebra and Calculus George Cain Difference Equations William F. Ames Differential Equations William F. Ames Integral Equations William F. Ames Approximation Methods William F. Ames Integral Transforms William F. Ames Calculus of Variations William F. Ames Optimization Methods George Cain Engineering Statistics Y. L. Tong Numerical Methods William F. Ames Experimental Uncertainty Analysis W.G. Steele and H.W. Coleman 19.14 Chaos R. L. Kautz 19.15 Fuzzy Sets and Fuzzy Logic Dan M. Frangopol

Chapter 20 20.1 20.2 20.3 20.4 20.5 20.6

Patent Law and Miscellaneous Topics Patents and Other Intellectual Property Thomas H. Young Product Liability and Safety George A. Peters Biomechanics B. Johan Ivarsson, Jeff R. Crandall, Gregory W. Hall, and Walter D. Pilkey Mechanical Engineering Codes and Standards Michael Merker Optics Roland Winston and Walter T. Welford Water Desalination Noam Lior

© 2005 by CRC Press LLC

Frank Kreith

0866_C00.fm Page xxiii Tuesday, August 10, 2004 5:04 PM

20.7 20.8

Appendices A. B. C. D. E.

Noise Control Malcolm J. Crocker Lighting Technology Barbara Atkinson, Andrea Denver, Robert Clear, and James E. McMahon

Paul Norton Properties of Gases and Vapors Properties of Liquids Properties of Solids Gases and Vapors Miscellaneous

© 2005 by CRC Press LLC

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1 Mechanics of Solids 1.1 1.2

Introduction Statics Vectors. Equilibrium of Particles. Free-Body Diagrams • Forces on Rigid Bodies • Equilibrium of Rigid Bodies Forces and Moments in Beams • Simple Structures and Machines • Distributed Forces • Friction • Work and Potential Energy • Moments of Inertia

1.3

Dynamics Kinematics of Particles • Kinetics of Particles • Kinetics of Systems of Particles • Kinematics of Rigid Bodies • Kinetics of Rigid Bodies in Plane Motion • Energy and Momentum Methods for Rigid Bodies in Plane Motion • Kinetics of Rigid Bodies in Three Dimensions

1.4

Vibrations Undamped Free and Forced Vibrations • Damped Free and Forced Vibrations • Vibration Control • Random Vibrations. Shock Excitation • Multiple-Degree-of-Freedom Systems. Modal Analysis • Vibration-Measuring Instruments

1.5

Stress • Strain • Mechanical Behaviors and Properties of Materials • Uniaxial Elastic Deformations • Stresses in Beams • Deflections of Beams • Torsion • Statically Indeterminate Axially Loaded Members • Buckling • Impact Loading • Combined Stresses • Pressure Vessels • Thick-Walled Cylinders and Interference Fits • Experimental Stress Analysis and Mechanical Testing

Bela I. Sandor University of Wisconsin–Madison

Ryan Roloff Honeywell

1.6

Michael L. Brown Harley-Davidson

Structural Integrity and Durability Finite Element Analysis. Stress Concentrations • Fracture Mechanics • Creep and Stress Relaxation • Fatigue

Stephen M. Birn Moog Inc., Aircraft Group

Mechanics of Materials

1.7

Comprehensive Example of Using Mechanics of Solids Methods The Project • Concepts and Methods

1.1 Introduction Bela I. Sandor Engineers use the concepts and methods of mechanics of solids in designing and evaluating tools, machines, and structures ranging from wrenches to cars to spacecraft. The required educational background for these includes courses in statics, dynamics, mechanics of materials, and related subjects. For example, knowledge of dynamics of rigid bodies is needed in generalizing the spectrum of service loads on a car, which is essential Professor Sandor wishes to acknowledge the contributions made by Richard C. Duveneck, Ian K. Glasgow, David A. Jahnke, Maan H. Jawad, and Christopher J. Watson.

© 2005 by CRC Press LLC

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FIGURE 1.1.1 (See Color Figure 1.) Artist’s concept of a moving stainless steel roadway to drive the suspension system through a spinning, articulated wheel, simulating three-dimensional motions and forces. (MTS Systems Corp., Minneapolis, MN. With permission.) Notes: Flat-Trac® Roadway Simulator, R&D100 Award-winning system in 1993.

in defining the vehicle’s deformations and long-term durability. In regard to structural integrity and durability, the designer should think not only about preventing the catastrophic failures of products, but also of customer satisfaction. For example, a car with gradually loosening bolts (which is difficult to prevent in a corrosive and thermal and mechanical cyclic loading environment) is a poor product because of safety, vibration, and noise problems. Sophisticated methods are available to assure a product’s performance and reliability, as exemplified in Figure 1.1.1. A similar but even more realistic test setup is shown in Color Figure 1.1 It is common experience among engineers that they need to review some old knowledge or learn something new because what they needed at the moment is not at their fingertips. This chapter may help in such a situation. Within the constraints of a single book on mechanical engineering, it provides overviews of topics with modern perspectives; illustrations of typical applications; modeling to solve problems quantitatively with realistic simplifications; equations and procedures; useful hints and reminders of common errors; trends of relevant material and mechanical system behaviors; and references to additional information. This chapter is like an emergency toolbox. It includes a coherent assortment of basic tools, such as vector expressions useful for calculating bending stresses caused by a three-dimensional force system on a shaft, and sophisticated methods, such as life prediction of components using fracture mechanics and modern measurement techniques. In many cases much more information should be considered than is covered in this chapter.

1.2 Statics Bela I. Sandor and Ryan J. Roloff Vectors. Equilibrium of Particles. Free-Body Diagrams Two kinds of quantities are used in engineering mechanics. A scalar quantity has only magnitude (mass, time, temperature, etc.). A vector quantity has magnitude and direction (force, velocity, etc.). Vectors are 1

Color figures follow page 1-104.

© 2005 by CRC Press LLC

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1-3

Mechanics of Solids

FIGURE 1.2.1 Components Fm and Fn of the vector F.

FIGURE 1.2.2 Addition of concurrent vectors F and P.

FIGURE 1.2.3 Addition of concurrent vectors A, B, and C.

represented here by arrows and bold-face symbols and are used in analysis according to universally applicable rules that facilitate calculations in a variety of problems. The vector methods are indispensable in three-dimensional mechanics analyses; however, in simple cases, equivalent scalar calculations are sufficient. Vector Components and Resultants. Parallelogram Law A given vector F may be replaced by two or three other vectors that have the same net effect and representation. This is illustrated for the chosen directions m and n for the components of F in two dimensions (Figure 1.2.1). Conversely, two concurrent vectors F and P of the same units may be combined to get a resultant R (Figure 1.2.2). Any set of components of a vector F must satisfy the parallelogram law. According to Figure 1.2.1, the laws of sines and cosines may be useful: Fn F F = m = sin α sin β sin 180° − (α + β)

[

]

[

(1.2.1)

]

F = F + F − 2 Fn Fm cos 180° − (α + β) 2

2 n

2 m

Any number of concurrent vectors may be summed, mathematically or graphically, and in any order, using the preceding concepts (see Figure 1.2.3). Unit Vectors Mathematical manipulations of vectors are greatly facilitated by the use of unit vectors. A unit vector n has a magnitude of unity and a defined direction. The most useful of these are the unit coordinate vectors i, j, and k as shown in Figure 1.2.4. The three-dimensional components and associated quantities of a vector F are shown in Figure 1.2.5. The unit vector n is collinear with F. The vector F is written in terms of its scalar components and the unit coordinate vectors, F = Fx i + Fy j + Fz k = Fn © 2005 by CRC Press LLC

(1.2.2)

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1-4

Chapter 1

FIGURE 1.2.4 Unit vectors in Cartesian coordinates (the same i, j, and k set applies in a parallel x′y′z′ system of axes).

FIGURE 1.2.5 Three-dimensional components of a vector F.

where Fx = F cos θ x

Fy = F cos θ y

Fz = F cos θ z

F = Fx2 + Fy2 + Fz2 n x = cos θ x

n y = cos θ y

nz = cos θ z

n x2 + n y2 + nz2 = 1 n x n y nz 1 = = = Fx Fy Fz F The unit vector notation is convenient for the summation of concurrent vectors in terms of scalar or vector components: Scalar components of the resultant R: Rx =

∑F

x

Ry =

∑F

Rz =

y

∑F

(1.2.3)

z

Vector components: Rx =

∑F = ∑ F i x

x

Ry =

∑F = ∑ F j y

y

Rz =

∑F = ∑ F k z

z

(1.2.4)

Vector Determination from Scalar Information A force, for example, may be given in terms of its magnitude F, its sense of direction, and its line of action. Such a force can be expressed in vector form using the coordinates of any two points on its line of action. The vector sought is F = Fx i + Fy j + Fz k = Fn © 2005 by CRC Press LLC

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1-5

Mechanics of Solids

The method is to find n on the line of points A(x1, y1, z1) and B(x2, y2, z2): n=

d x i + d y j + dz k vector A to B = distance A to B d x2 + d y2 + d z2

where dx = x2 – x1, dy = y2 – y1, dz = z2 – z1. Scalar Product of Two Vectors. Angles and Projections of Vectors The scalar product, or dot product, of two concurrent vectors A and B is defined by A ⋅ B = ABcosφ

(1.2.5)

where A and B are the magnitudes of the vectors and φ is the angle between them. Some useful expressions are A ⋅ B = B ⋅ A = Ax Bx + Ay By + Az Bz φ = arccos

Ax Bx + Ay By + Az Bz AB

The projection F ′ of a vector F on an arbitrary line of interest is determined by placing a unit vector n on that line of interest, so that F ′ = F ⋅ n = Fx n x + Fy n y + Fz nz Equilibrium of a Particle A particle is in equilibrium when the resultant of all forces acting on it is zero. In such cases, the algebraic summation of rectangular scalar components of forces is valid and convenient:

∑F = 0 ∑F = 0 ∑F = 0 x

y

(1.2.6)

z

Free-Body Diagrams Unknown forces may be determined readily if a body is in equilibrium and can be modeled as a particle. The method involves free-body diagrams, which are simple representations of the actual bodies. The appropriate model is imagined to be isolated from all other bodies, with the significant effects of other bodies shown as force vectors on the free-body diagram. Example 1 A mast has three guy wires. The initial tension in each wire is planned to be 200 lb. Determine whether this is feasible to hold the mast vertical (Figure 1.2.6). Solution. The three tensions of known magnitude (200 lb) must be written as vectors. R = TAB + TAC + TAD

TAB = (tension AB)( unit vector A to B) = 200 lb n AB = 200 lb =

200 lb 5 + 10 + 4 2

© 2005 by CRC Press LLC

2

2

(−5i − 10 j + 4k)

( d i + d j + d k) x

y

z

d

ft = −84.2 lb i − 168.4 lb j + 67.4 lb k ft

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1-6

Chapter 1

FIGURE 1.2.6 A mast with guy wires.

TAC =

200 lb (5i − 10 j + 4k) ft = 84.2 lb i + 168.4 lb j + 67.4 lb k 11.87 ft

TAD =

200 lb (0i − 10 j + 6k) ft = −171.5 lb j − 102.9 lb k 11.66 ft

The resultant of the three tensions is R=

∑ F i + ∑ F j + ∑ F k = (−84.2 + 84.2 + 0) lb i + (−168.4 − 168.4 − 171.5) lb j x

y

z

+ (67.4 + 67.4 − 102.9) lb k = 0 lb i − 508 lb j + 31.9 lb k There is a horizontal resultant of 31.9 lb at A, so the mast would not remain vertical.

Forces on Rigid Bodies All solid materials deform when forces are applied to them, but often it is reasonable to model components and structures as rigid bodies, at least in the early part of the analysis. The forces on a rigid body are generally not concurrent at the center of mass of the body, which cannot be modeled as a particle if the force system tends to cause body rotation. Moment of a Force The turning effect of a force on a body is called the moment of the force, or torque. The moment MA of a force F about a point A is defined as a scalar quantity M A = Fd

(1.2.7)

where d (the moment arm or lever arm) is the nearest distance from A to the line of action of F. This nearest distance may be difficult to determine in a three-dimensional scalar analysis; a vector method is needed in that case. Equivalent Forces Sometimes the equivalence of two forces must be established for simplifying the solution of a problem. The necessary and sufficient conditions for the equivalence of forces F and F′ are that they have the same magnitude, direction, line of action, and moment on a given rigid body in static equilibrium. Thus, F = F ′ and © 2005 by CRC Press LLC

M A = M A′

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1-7

Mechanics of Solids

FIGURE 1.2.7 Schematic of testing a ball joint of a car.

FIGURE 1.2.8 Right-hand rule for vector products.

For example, the ball joint A in Figure 1.2.7 experiences the same moment whether the vertical force is pushing or pulling downward on the yoke pin. Vector Product of Two Vectors A powerful method of vector mechanics is available for solving complex problems such as the moment of a force in three dimensions. The vector product (or cross product) of two concurrent vectors A and B is defined as the vector V = A × B with the following properties: 1. 2. 3. 4. 5.

V is perpendicular to the plane of vectors A and B. The sense of V is given by the right-hand rule (Figure 1.2.8). The magnitude of V is V = AB sinq, where q is the angle between A and B. A × B Σ B × A, but A × B = –(B × A). For three vectors, A × (B + C) = A × B + A × C.

The vector product is calculated using a determinant, i V = Ax Bx

j Ay By

k Az = Ay Bz i + Az Bx j + Ax By k − Ay Bx k − Ax Bz j − Az By i Bz

(1.2.8)

Moment of a Force about a Point The vector product is very useful in determining the moment of a force F about an arbitrary point O. The vector definition of moment is MO = r × F

(1.2.9)

where r is the position vector from point O to any point on the line of action of F. A double arrow is often used to denote a moment vector in graphics. The moment MO may have three scalar components, Mx , My , Mz, which represent the turning effect of the force F about the corresponding coordinate axes. In other words, a single force has only one moment about a given point, but this moment may have up to three components with respect to a coordinate system, © 2005 by CRC Press LLC

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1-8

Chapter 1

M O = M x i + M y j + Mz k Triple Products of Three Vectors Two kinds of products of three vectors are used in engineering mechanics. The mixed triple product (or scalar product) is used in calculating moments. It is the dot product of vector A with the vector product of vectors B and C, Ax A ⋅ (B × C) = Bx Cx

Ay By Cy

Az Bz = Ax By Cz − Bz Cy + Ay ( Bz Cx − Bx Cz ) + Az Bx Cy − By Cx Cz

(

)

(

)

(1.2.10)

The vector triple product (A × B) × C = V × C is easily calculated (for use in dynamics), but note that

(A × B) × C ≠ A × (B × C) Moment of a Force about a Line It is common that a body rotates about an axis. In that case, the moment M
, of a force F about the axis, say line,
, is usefully expressed as nx M
= n ⋅ M O = n ⋅ (r × F) = rx Fx

ny ry Fy

nz rz Fz

(1.2.11)

where n is a unit vector along the line,
, and r is a position vector from point O on
, to a point on the line of action of F. Note that M
is the projection of MO on line,
. Special Cases • The moment about a line
is zero when the line of action of F intersects
(the moment arm is zero). • The moment about a line
is zero when the line of action of F is parallel to
(the projection of MO on
is zero). Moment of a Couple A pair of forces equal in magnitude, parallel in lines of action, and opposite in direction is called a couple. The magnitude of the moment of a couple is M = Fd where d is the distance between the lines of action of the forces of magnitude F. The moment of a couple is a free vector M that can be applied anywhere to a rigid body with the same turning effect, as long as the direction and magnitude of M are the same. In other words, a couple vector can be moved to any other location on a given rigid body if it remains parallel to its original position (equivalent couples). Sometimes a curled arrow in the plane of the two forces is used to denote a couple, instead of the couple vector M, which is perpendicular to the plane of the two forces. Force-Couple Transformations Sometimes it is advantageous to transform a force to a force system acting at another point, or vice versa. This method is illustrated in Figure 1.2.9. © 2005 by CRC Press LLC

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1-9

Mechanics of Solids

FIGURE 1.2.9 Force-couple transformations.

FIGURE 1.2.10 Model of a torque wrench.

1. A force F acting at B on a rigid body can be replaced by the same force F acting at A and a moment MA = r × F about A. 2. A force F and moment MA acting at A can be replaced by a force F acting at B for the same total effect on the rigid body. Simplification of Force Systems Any force system on a rigid body can be reduced to an equivalent system of a resultant force R and a resultant moment MR. The equivalent force-couple system is formally stated as n

R=

∑

n

Fi

and M R =

i =1

∑ i =1

n

Mi =

∑ (r × F ) i

i

(1.2.12)

i =1

where MR depends on the chosen reference point. Common Cases • • • •

The resultant force is zero, but there is a resultant moment: R = 0, MR ≠ 0. Concurrent forces (all forces act at one point): R ≠ 0, MR = 0. Coplanar forces: R ≠ 0, MR ≠ 0. MR is perpendicular to the plane of the forces. Parallel forces: R ≠ 0, MR ≠ 0. MR is perpendicular to R.

Example 2 The torque wrench in Figure 1.2.10 has an arm of constant length L but a variable socket length d = OA because of interchangeable tool sizes. Determine how the moment applied at point O depends on the length d for a constant force F from the hand. Solution. Using MO = r × F with r = Li + dj and F = Fk in Figure 1.2.10, M O = ( Li + dj) × Fk = Fdi − FLj Judgment of the Result According to a visual analysis, the wrench should turn clockwise, so the –j component of the moment is justified. Looking at the wrench from the positive x direction, point A has a tendency to rotate counterclockwise. Thus, the i component is correct using the right-hand rule. © 2005 by CRC Press LLC

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1-10

Chapter 1

Equilibrium of Rigid Bodies The concept of equilibrium is used for determining unknown forces and moments of forces that act on or within a rigid body or system of rigid bodies. The equations of equilibrium are the most useful equations in the area of statics, and they are also important in dynamics and mechanics of materials. The drawing of appropriate free-body diagrams is essential for the application of these equations. Conditions of Equilibrium A rigid body is in static equilibrium when the equivalent force-couple system of the external forces acting on it is zero. In vector notation, this condition is expressed as

∑F = 0 ∑ M = ∑ (r × F) = 0

(1.2.13)

O

where O is an arbitrary point of reference. In practice it is often most convenient to write Equation (1.2.13) in terms of rectangular scalar components,

∑F = 0 ∑M ∑F = 0 ∑M ∑F = 0 ∑M x

x

=0

y

y

=0

z

z

=0

Maximum Number of Independent Equations for One Body 1. One-dimensional problem: ΣF = 0 2. Two-dimensional problem:

∑F = 0 ∑F = 0 ∑M = 0 ∑F = 0 ∑M = 0 ∑M = 0 ∑M = 0 ∑M = 0 ∑M = 0 x

or or

y

x

A

A

A

( x axis not ⊥ AB)

B

B

( AB not
BC)

C

3. Three-dimensional problem:

∑F = 0 ∑F = 0 ∑F = 0 ∑M = 0 ∑M = 0 ∑M = 0 x

y

x

z

y

z

where xyz are orthogonal coordinate axes, and A, B, C are particular points of reference. Calculation of Unknown Forces and Moments In solving for unknown forces and moments, always draw the free-body diagram first. Unknown external forces and moments must be shown at the appropriate places of action on the diagram. The directions of unknowns may be assumed arbitrarily, but should be done consistently for systems of rigid bodies. A © 2005 by CRC Press LLC

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1-11

Mechanics of Solids

FIGURE 1.2.11 Example of two-dimensional modeling.

FIGURE 1.2.12 Example of three-dimensional modeling.

negative answer indicates that the initial assumption of the direction was opposite to the actual direction. Modeling for problem solving is illustrated in Figure 1.2.11 and Figure 1.2.12. Notes on Three-Dimensional Forces and Supports Each case should be analyzed carefully. Sometimes a particular force or moment is possible in a device, but it must be neglected for most practical purposes. For example, a very short sleeve bearing cannot support significant moments. A roller bearing may be designed to carry much larger loads perpendicular to the shaft than along the shaft. Related Free-Body Diagrams When two or more bodies are in contact, separate free-body diagrams may be drawn for each body. The mutual forces and moments between the bodies are related according to Newton’s third law (action and reaction). The directions of unknown forces and moments may be arbitrarily assumed in one diagram, but these initial choices affect the directions of unknowns in all other related diagrams. The numbers of unknowns and of usable equilibrium equations increase with the number of related free-body diagrams. Schematic Example in Two Dimensions (Figure 1.2.13) Given: F1, F2, F3, M Unknowns: P1, P2, P3, and forces and moments at joint A (rigid connection)

FIGURE 1.2.13 Free-body diagram. © 2005 by CRC Press LLC

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1-12

Chapter 1

FIGURE 1.2.14 Related free-body diagrams.

Equilibrium Equations

∑ F = −F + P = 0 ∑F = P + P − F − F = 0 ∑ M = P c + P (c + d + e) + M − F a − F ( a + b ) = 0 x

1

y

1

O

3

2

1

2

3

2

2

3

Three unknowns (P1, P2, P3) are in three equations. Related Free-Body Diagrams (Figure 1.2.14) Dimensions a, b, c, d, and e of Figure 1.2.13 are also valid here. New Set of Equilibrium Equations Left part: (OA)

∑ F = −F + A x

1

x

∑F = P + A y

∑M Right side: ( AB)

1

O

y

=0

− F2 = 0

= P1c + Ay (c + d ) + M A − F2 a = 0

∑ F = −A x

x

+ P3 = 0

∑F = P − A − F = 0 ∑ M = −M + P e + M − F f = 0 y

2

A

y

A

3

2

3

Six unknowns (P1, P2, P3, Ax , Ay , MA ) are in six equations. Note that, in the first diagram (Figure 1.2.13), the couple M may be moved anywhere from O to B. M is not shown in the second diagram (O to A) because it is shown in the third diagram (in which it may be moved anywhere from A to B). Example 3 The arm of a factory robot is modeled as three bars (Figure 1.2.15) with coordinates A: (0.6, –0.3, 0.4) m; B: (1, –0.2, 0) m; and C: (0.9, 0.1, –0.25) m. The weight of the arm is represented by WA = –60 Nj at A, and WB = –40 Nj at B. A moment MC = (100i – 20j + 50k) N ⋅ m is applied to the arm at C. Determine the force and moment reactions at O, assuming that all joints are temporarily fixed. © 2005 by CRC Press LLC

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1-13

Mechanics of Solids

FIGURE 1.2.15 Model of a factory robot.

Solution. The free-body diagram showing the unknown force and moment reactions at O, is drawn in Figure 1.2.15b. From Equation (1.2.13),

∑F = 0 FO + WA + WB = 0 FO − 60 N j − 40 N j = 0 FO = 100 N j

∑M

O

=0

M O + M C + (rOA × WA ) + (rOB × WB ) = 0 M O + (100i − 20 j + 50k) N ⋅ m + (0.6i − 0.3 j + 0.4k) m × (−60 N j) + (i − 0.2 j) m × (−40 N j) = 0 M O + 100 N ⋅ m i − 20 N ⋅ m j + 50 N ⋅ m k − 36 N ⋅ m k + 24 N ⋅ m i − 40 N ⋅ m k = 0 M O = (−124i + 20 j + 26k) N ⋅ m Example 4 A load of 7 kN may be placed anywhere within A and B in the trailer of negligible weight. Determine the reactions at the wheels at D, E, and F, and the force on the hitch H mounted on the car, for the extreme positions A and B of the load. The mass of the car is 1500 kg, and its weight is acting at C (see Figure 1.2.16).

FIGURE 1.2.16 Analysis of a car with trailer. © 2005 by CRC Press LLC

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1-14

Chapter 1

Solution. The scalar method is best here. Put the load at position A first. For the trailer alone, with y as the vertical axis: ΣMF = 7(1) – Hy (3) = 0, Hy = 2.33 kN On the car: Hy = 2.33 kN ↓ Ans. ΣFy = 2.33 – 7 + Fy = 0, Fy = 4.67 kN ↑ Ans. For the car alone: ΣME = –2.33(1.2) – Dy (4) + 14.72(1.8) = 0 Dy = 5.93 kN ↑ Ans. ΣFy = 5.93 + Ey – 14.72 – 2.33 = 0 Ey = 11.12 kN ↑ Ans. Put the load at position B next. For the trailer alone: ΣMF = 0.8(7) – Hy (3) = 0, Hy = –1.87 kN On the car: Hy = 1.87 kN ↓ Ans. ΣFy = –1.87 – 7 + Ey = 0 Ey = 8.87 kN ↑ Ans. For the car alone: ΣME = –(1.87)(1.2) – Dy(4) + 14.72(1.8) = 0 Dy = 7.19 kN ↑Ans. ΣFy = 7.19 + Ey – 14.72 – (–1.87) = 0 Ey = 5.66 kN ↑ Ans.

Forces and Moments in Beams Beams are common structural members whose main function is to resist bending. The geometric changes and safety aspects of beams are analyzed by first assuming that they are rigid. The preceding subsections enable one to determine (1) the external (supporting) reactions acting on a statically determinate beam; and (2) the internal forces and moments at any cross section in a beam. Classification of Supports Common supports and external reactions for two-dimensional loading of beams are shown in Figure 1.2.17.

FIGURE 1.2.17 Common beam supports. © 2005 by CRC Press LLC

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Mechanics of Solids

1-15

FIGURE 1.2.18 Internal forces and moments in a cantilever beam.

FIGURE 1.2.19 Preferred sign conventions.

Internal Forces and Moments The internal force and moment reactions in a beam caused by external loading must be determined for evaluating the strength of the beam. If there is no torsion of the beam, three kinds of internal reactions are possible: a horizontal normal force H on a cross section; vertical (transverse) shear force V; and bending moment M. These reactions are calculated from the equilibrium equations applied to the left or right part of the beam from the cross section considered. The process involves free-body diagrams of the beam and a consistently applied system of signs. The modeling is illustrated for a cantilever beam in Figure 1.2.18. Sign conventions. Consistent sign conventions should be used in any given problem. These can be arbitrarily set up, but the following is slightly advantageous. It makes the signs of the answers to the equilibrium equations correct for the directions of the shear force and bending moment. A moment that makes a beam concave upward is taken as positive. Thus, a clockwise moment is positive on the left side of a section, and a counterclockwise moment is positive on the right side. A shear force that acts upward on the left side of a section, or downward on the right side, is positive (Figure 1.2.19). Shear Force and Bending Moment Diagrams The critical locations in a beam are determined from shear force and bending moment diagrams for the whole length of the beam. The construction of these diagrams is facilitated by following the steps, which are illustrated for a cantilever beam in Figure 1.2.20. 1. Draw the free-body diagram of the whole beam and determine all reactions at the supports. 2. Draw the coordinate axes for the shear force (V) and bending moment (M) diagrams directly below the free-body diagram. 3. Immediately plot those values of V and M that can be determined by inspection (especially when they are zero), observing the sign conventions. 4. Calculate and plot as many additional values of V and M as are necessary for drawing reasonably accurate curves through the plotted points, or do it all by computer. © 2005 by CRC Press LLC

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Chapter 1

FIGURE 1.2.20 Construction of shear force and bending moment diagrams.

FIGURE 1.2.21 Shear force and bending moment diagrams of a component in a structure.

Example 5 A construction crane is modeled as a rigid bar AC that supports the boom by a pin at B and wire CD. The dimensions are AB = 10
, BC = 2
, BD = DE = 4
. Draw the shear force and bending moment diagrams for bar AC (Figure 1.2.21). Solution. From the free-body diagram of the entire crane,

∑F = 0 ∑F = 0

∑M

Ax = 0

− P + Ay = 0

− P(8
) + M A = 0

Ay = P

M A = 8P


x

© 2005 by CRC Press LLC

y

A

=0

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Mechanics of Solids

Now separate bar AC and determine the forces at B and C.

∑F = 0

∑F = 0

∑M

− Bx + TCDx = 0

P − By − TCDy = 0

−

x

(a ) Bx =

y

2 T 5 CD

( b) By = P −

1 T 5 CD

−

A

=0

2 T (12
) + Bx (10
) + M A = 0 5 CD 24
20
T + T = −8P
5 CD 5 CD (c) TCD =

8 5 P = 2 5P 4

From (a) and (c), Bx = 4P and TCDx = 4P. From (b) and (c), By = P – 2P = –P and = TCDy = 2P. Draw the free-body diagram of bar AC horizontally, with the shear force and bending moment diagram axes below it. Measure x from end C for convenience and analyze sections 0 ≤ x ≤ 2
and 2
≤ x ≤ 12
(Figure 1.2.21b to 1.2.21f). 1. 0 ≤ x ≤ 2


∑F = 0

∑M

−4 P + VK1 = 0

M K1 + 4 P( x ) = 0

VK1 = 4 P

M K1 = −4 Px

y

K

=0

2. 2
≤ x ≤ 12


∑F = 0

∑M

4 P − 4 P + VK2 = 0

M K2 − 4 P( x − 2
) + 4 P( x ) = 0

VK2 = 0

M K2 = −8P


y

K

=0

At point B, x = 2
, MK1 = –4P(2
) = –8P
= MK2 = MA. The results for section AB, 2
≤ x ≤ 12
, show that the combined effect of the forces at B and C is to produce a couple of magnitude 8P
on the beam. Thus, the shear force is zero and the moment is constant in this section. These results are plotted on the axes below the free-body diagram of bar A-B-C.

Simple Structures and Machines Equilibrium equations are used to determine forces and moments acting on statically determinate simple structures and machines. A simple structure is composed solely of two-force members. A machine is composed of multiforce members. The method of joints and the method of sections are commonly used in such analysis. Trusses Trusses consist of straight, slender members whose ends are connected at joints. Two-dimensional plane trusses carry loads acting in their planes and are often connected to form three-dimensional space trusses. Two typical trusses are shown in Figure 1.2.22. © 2005 by CRC Press LLC

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Chapter 1

FIGURE 1.2.22 Schematic examples of trusses.

To simplify the analysis of trusses, assume frictionless pin connections at the joints. Thus, all members are two-force members with forces (and no moments) acting at the joints. Members may be assumed weightless or may have their weights evenly divided to the joints. Method of Joints Equilibrium equations based on the entire truss and its joints allow for determination of all internal forces and external reactions at the joints using the following procedure. 1. Determine the support reactions of the truss. This is done using force and moment equilibrium equations and a free-body diagram of the entire truss. 2. Select any arbitrary joint where only one or two unknown forces act. Draw the free-body diagram of the joint assuming unknown forces are tensions (arrows directed away from the joint). 3. Draw free-body diagrams for the other joints to be analyzed, using Newton’s third law consistently with respect to the first diagram. 4. Write the equations of equilibrium, ΣFx = 0 and ΣFy = 0, for the forces acting at the joints and solve them. To simplify calculations, attempt to progress from joint to joint in such a way that each equation contains only one unknown. Positive answers indicate that the assumed directions of unknown forces were correct, and vice versa. Example 6 Use the method of joints to determine the forces acting at A, B, C, H, and I of the truss in Figure 1.2.23a. The angles are α = 56.3°; β = 38.7°; φ = 39.8°; and θ = 36.9°. Solution. First the reactions at the supports are determined and are shown in Figure 1.2.23b. A joint at which only two unknown forces act is the best starting point for the solution. Choosing joint A, the solution is progressively developed, always seeking the next joint with only two unknowns. In each

FIGURE 1.2.23 Method of joints in analyzing a truss.

© 2005 by CRC Press LLC

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Mechanics of Solids

diagram, circles indicate the quantities known from the preceding analysis. Sample calculations show the approach and some of the results. Joint A:

∑F = 0

∑F = 0

FAI = 0

FAB − Ay = 0

x

y

FAB − 50 kips = 0 FAB = 50 kips (tension) Joint H:

∑F = 0

∑F = 0

x

y

FGH sin β − FCH cos α − FBH = 0

FCH sin α + FDH + FGH cos β − FHI = 0

FGH (0.625) + (60.1 kips) (0.555) − 0 = 0

−(60.1 kips) (0.832) + FDH − (53.4 kips) (0.780) + 70 kips = 0

FGH = −53.4 kips (compression)

FDH = 21.7 kips (tension)

Method of Sections The method of sections is useful when only a few forces in truss members need to be determined regardless of the size and complexity of the entire truss structure. This method employs any section of the truss as a free body in equilibrium. The chosen section may have any number of joints and members in it, but the number of unknown forces should not exceed three in most cases. Only three equations of equilibrium can be written for each section of a plane truss. The following procedure is recommended: 1. Determine the support reactions if the section used in the analysis includes the joints supported. 2. Section the truss by making an imaginary cut through the members of interest, preferably through only three members in which the forces are unknowns (assume tensions). The cut need not be a straight line. The sectioning is illustrated by lines l-l, m-m, and n-n in Figure 1.2.24. 3. Write equations of equilibrium. Choose a convenient point of reference for moments to simplify calculations such as the point of intersection of the lines of action for two or more of the unknown forces. If two unknown forces are parallel, sum the forces perpendicular to their lines of action. 4. Solve the equations. If necessary, use more than one cut in the vicinity of interest to allow writing more equilibrium equations. Positive answers indicate assumed directions of unknown forces were correct, and vice versa.

FIGURE 1.2.24 Method of sections in analyzing a truss.

© 2005 by CRC Press LLC

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Chapter 1

Space Trusses A space truss can be analyzed with the method of joints or with the method of sections. For each joint, there are three scalar equilibrium equations, ΣFx = 0, ΣFy = 0, and ΣFz = 0. The analysis must begin at a joint at least one force is known and no more than three forces are unknown. The solution must progress to other joints in a similar fashion. Six scalar equilibrium equations are available when the method of sections is used: ΣFx = 0; ΣFy = 0; ΣFz = 0; ΣMx = 0; ΣMy = 0; and ΣMz = 0. Frames and Machines Multiforce members (with three or more forces acting on each member) are common in structures. In these cases the forces are not directed along the members, so they are a little more complex to analyze than the two-force members in simple trusses. Multiforce members are used in two kinds of structures. Frames are usually stationary and fully constrained. Machines have moving parts, so the forces acting on a member depend on the location and orientation of the member. The analysis of multiforce members is based on the consistent use of related free-body diagrams. The solution is often facilitated by representing forces by their rectangular components. Scalar equilibrium equations are the most convenient for two-dimensional problems, and vector notation is advantageous in three-dimensional situations. Often, an applied force acts at a pin joining two or more members, or a support or connection may exist at a joint between two or more members. In these cases, a choice should be made of a single member at the joint on which to assume the external force to be acting. This decision should be stated in the analysis. The following comprehensive procedure is recommended. Three independent equations of equilibrium are available for each member or combination of members in two-dimensional loading; for example, ΣFx = 0, ΣFy = 0, ΣMA = 0, where A is an arbitrary point of reference. 1. Determine the support reactions if necessary. 2. Determine all two-force members. 3. Draw the free-body diagram of the first member on which the unknown forces act, assuming that the unknown forces are tensions. 4. Draw the free-body diagrams of the other members or groups of members using Newton’s third law (action and reaction) consistently with respect to the first diagram. Proceed until the number of equilibrium equations available is no longer exceeded by the total number of unknowns. 5. Write the equilibrium equations for the members or combinations of members and solve them. Positive answers indicate that the assumed directions for unknown forces were correct, and vice versa.

Distributed Forces The most common distributed forces acting on a body are parallel force systems, such as the force of gravity. These can be represented by one or more concentrated forces to facilitate the required analysis. Several basic cases of distributed forces are presented here. The important topic of stress analysis is covered in mechanics of materials. Center of Gravity The center of gravity of a body is the point where the equivalent resultant force caused by gravity is acting. Its coordinates are defined for an arbitrary set of axes as

x=

© 2005 by CRC Press LLC

∫ x dW W

y=

∫ y dW W

z=

∫ z dW W

(1.2.14)

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1-21

Mechanics of Solids

where x, y, z are the coordinates of an element of weight dW, and W is the total weight of the body. In the general case, dW = γ dV, and W = ∫γ dV, where γ = specific weight of the material and dV = elemental volume. Centroids If γ is a constant, the center of gravity coincides with the centroid, which is a geometrical property of a body. Centroids of lines L, areas A, and volumes V are defined analogously to the coordinates of the center of gravity,

Lines:

x=

∫ x dL

y=

∫ y dL

z=

∫ z dL

(1.2.15)

Areas:

x=

∫ x dA

y=

∫ y dA

z=

∫ z dA

(1.2.16)

Volumes:

x=

∫ x dV

y=

∫ y dV

z=

∫ z dV

(1.2.17)

W

A

V

L

A

V

L

A

V

For example, an area A consists of discrete parts A1, A2, A3, where the centroids x1, x2, x3 of the three parts are located by inspection. The x coordinate of the centroid of the whole area A is x obtained from Ax = A1 x1 + A2 x2 + A3 x3. Surfaces of revolution. The surface areas and volumes of bodies of revolution can be calculated using the concepts of centroids by the theorems of Pappus. Distributed Loads on Beams The distributed load on a member may be its own weight and/or some other loading such as from ice or wind. The external and internal reactions to the loading may be determined using the condition of equilibrium. External reactions. Replace the whole distributed load with a concentrated force equal in magnitude to the area under the load distribution curve and applied at the centroid of that area parallel to the original force system. Internal reactions. For a beam under a distributed load w(x), where x is distance along the beam, the shear force V and bending moment M are related according to Figure 1.2.25 as w( x ) = −

dV dx

V=

dM dx

(1.2.18)

FIGURE 1.2.25 Internal reactions in a beam under distributed loading.

© 2005 by CRC Press LLC

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Chapter 1

Other useful expressions for any two cross sections A and B of a beam are

VA − VB = MB − MA =

∫

xB

∫

xB

xA

w( x ) dx = area under w( x ) (1.2.19)

V dx = area under shear force diagram

xA

Example 7 See Figure 1.2.26. Distributed Loads on Flexible Cables The basic assumptions of simple analyses of cables are that there is no resistance to bending and that the internal force at any point is tangent to the cable at that point. The loading is denoted by w(x), a continuous but possibly variable load, in terms of force per unit length. The differential equation of a cable is d 2 y w( x ) = dx 2 To

(1.2.20)

where To = constant = horizontal component of the tension T in the cable. Two special cases are common: • Parabolic cables. The cable supports a load w that is uniformly distributed horizontally. The shape of the cable is a parabola given by y=

wx 2 2To

( x = 0 at lowest point)

(1.2.21)

In a symmetric cable, the tension is T = To2 + w 2 x 2 . • Catenary cables. When the load w is uniformly distributed along the cable, the cable’s shape is given by y=

To w

 wx   cosh T − 1   o

The tension in the cable is T = To + wy.

FIGURE 1.2.26 Shear force and bending moment diagrams for a cantilever beam. © 2005 by CRC Press LLC

(1.2.22)

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Mechanics of Solids

FIGURE 1.2.27 Models showing friction forces.

Friction A friction force F (or
, in typical other notation) acts between contacting bodies when they slide relative to one another, or when sliding tends to occur. This force is tangential to each body at the point of contact, and its magnitude depends on the normal force N pressing the bodies together and on the material and condition of the contacting surfaces. The material and surface properties are lumped together and represented by the coefficient of friction µ. The friction force opposes the force that tends to cause motion, as illustrated for two simple cases in Figure 1.2.27. The friction forces F may vary from zero to a maximum value, Fmax = µN

(0 ≤ F ≤ F ) max

(1.2.23)

depending on the applied force that tends to cause relative motion of the bodies. The coefficient of kinetic friction µk (during sliding) is lower than the coefficient of static friction m or µs; µk depends on the speed of sliding and is not easily quantified. Angle of Repose The critical angle θc at which motion is impending is the angle of repose, where the friction force is at its maximum for a given block on an incline. tanθ c =

F = µs N

(1.2.24)

Thus, θc is measured to obtain µs. Note that, even in the case of static, dry friction, µs depends on temperature; humidity; dust and other contaminants; oxide films; surface finish; and chemical reactions. The contact area and the normal force affect µs only when significant deformations of one or both bodies occur. Classifications and Procedures for Solving Friction Problems The directions of unknown friction forces are often, but not always, determined by inspection. The magnitude of the friction force is obtained from Fmax = µs N when it is known that motion is impending. Note that F may be less than Fmax. The major steps in solving problems of dry friction are organized in three categories as follows: A. Given: bodies, forces, or coefficients of friction are known. Impending motion is not assured: F ≠ µs N. Procedure: to determine if equilibrium is possible: 1. Construct the free-body diagram. 2. Assume that the system is in equilibrium. 3. Determine the friction and normal forces necessary for equilibrium. 4. Results: (a) F < µs N; the body is at rest; (b) F > µs N; motion is occurring, so static equilibrium is not possible. Because there is motion, F = µk N. Complete solution requires principles of dynamics. © 2005 by CRC Press LLC

0866_book.fm Page 24 Thursday, August 5, 2004 10:16 AM

1-24

Chapter 1

B. Given: bodies, forces, or coefficients of friction are given. Impending motion is specified. F = µs N is valid. Procedure: to determine the unknowns: 1. Construct the free-body diagram. 2. Write F = µsN for all surfaces where motion is impending. 3. Determine µs or the required forces from the equation of equilibrium. C. Given: bodies, forces, and coefficients of friction are known. Impending motion is specified, but the exact motion is not given. The possible motions may be sliding, tipping, or rolling, or relative motion if two or more bodies are involved. Alternatively, the forces or coefficients of friction may need to be determined to produce a particular motion from several possible motions. Procedure: to determine the exact motion that may occur, or unknown quantities required: 1. Construct the free-body diagram. 2. Assume that motion is impending in one of the two or more possible ways. Repeat this for each possible motion and write the equation of equilibrium. 3. Compare the results for the possible motions and select the likely event. Determine the required unknowns for any preferred motion. Wedges and Screws A wedge may be used to raise or lower a body. Thus, two directions of motion must be considered in each situation, with the friction forces always opposing the impending or actual motion. The self-locking aspect of a wedge may be of interest. The analysis is straightforward, using interrelated free-body diagrams and equilibrium equations. Screw threads are special applications of the concept of wedges. Square threads are the easiest to model and analyze. The magnitude M of the moment of a couple required to move a square-threaded screw against an axial load P is M = Pr tan(α + φ)

(1.2.25)

where r = radius of the screw α = tan–1 (L/2πr) = tan–1 (np/2πr) L = lead = advancement per revolution n = multiplicity of threads p = pitch = distance between similar points on adjacent threads φ = tan–1µ The relative values of α and φ control whether a screw is self-locking; φ > α is required for a screw to support an axial load without unwinding. Disk Friction Flat surfaces in relative rotary motion generate a friction moment M opposing the motion. For a hollow member with radii Ro and Ri , under an axial force P, M=

R 3 − Ri3 2 µP o2 3 Ro − Ri2

(1.2.26)

The friction moment tends to decrease (down to about 75% of its original value) as the surfaces wear. Use the appropriate µs or µk value. Axle Friction The friction moment M of a rotating axle in a journal bearing (sliding bearing) is approximated (if µ is low) as © 2005 by CRC Press LLC

0866_book.fm Page 25 Thursday, August 5, 2004 10:16 AM

1-25

Mechanics of Solids

M = Prµ

(1.2.27)

where P = transverse load on the axle r = radius of the axle Use the appropriate µs or µk value. Rolling Resistance Rolling wheels and balls have relatively low resistance to motion compared to sliding. This resistance is caused by internal friction of the materials in contact, and it may be difficult to predict or measure. A coefficient of rolling resistance a is defined with units of length, Fr P

a≅

(1.2.28)

where r = radius of a wheel rolling on a flat surface F = minimum horizontal force to maintain constant speed of rolling P = load on wheel Values of a range upward from a low of about 0.005 mm for hardened steel elements. Belt Friction The tensions T1 and T2 of a belt, rope, or wire on a pulley or drum are related as

(T2 > T1 )

T2 = T1e µβ

(1.2.29)

where β = total angle of belt contact, radians (β = 2πn for a member wrapped around a drum n times). Use µs for impending slipping and µk for slipping. For a V belt of belt angle 2φ, T2 = T1e µβ sin φ

Work and Potential Energy Work is a scalar quantity. It is the product of a force and the corresponding displacement. Potential energy is the capacity of a system to do work on another system. These concepts are advantageous in the analysis of equilibrium of complex systems, in dynamics, and in mechanics of materials. Work of a Force The work U of a constant force F is U = Fs

(1.2.30)

where s = displacement of a body in the direction of the vector F. For a displacement along an arbitrary path from point 1 to 2, with dr tangent to the path, U=

∫

2

1

F ⋅ dr =

∫ (F dx + F dy + F dz) 2

1

x

y

In theory, there is no work when: • A force is acting on a fixed, rigid body (dr = 0, dU = 0). • A force acts perpendicular to the displacement (F · dr = 0). © 2005 by CRC Press LLC

z

0866_book.fm Page 26 Thursday, August 5, 2004 10:16 AM

1-26

Chapter 1

Work of a Couple A couple of magnitude M does work U = Mθ

(1.2.31)

where θ = angular displacement (radians) in the same plane in which the couple is acting. In a rotation from angular position α to β, U=

β

β

α

α

∫ M ⋅ dθ = ∫ ( M

x

d θ x + M y dθ y + M z dθ z

)

Virtual Work The concept of virtual work (through imaginary, infinitesimal displacements within the constraints of a system) is useful to analyze the equilibrium of complex systems. The virtual work of a force F or moment M is expressed as δU = F ⋅ δr δU = M ⋅ δθ There is equilibrium if m

δU =

∑

n

Fi ⋅ δri +

i =1

∑ M ⋅ δθ i

j

=0

(1.2.32)

j =1

where the subscripts refer to individual forces or couples and the corresponding displacements, ignoring frictional effects. Mechanical Efficiency of Real Systems Real mechanical systems operate with frictional losses, so input work = useful work + work of friction ( output work )

The mechanical efficiency η of a machine is η=

output work useful work = input work total work required 0 < η 0. Equilibrium is unstable if (d 2V/dq2) < 0. Equilibrium is neutral only if all derivatives of V are zero. In cases of complex configurations, evaluate derivatives of higher order as well.

Moments of Inertia The topics of inertia are related to the methods of first moments. They are traditionally presented in statics in preparation for application in dynamics or mechanics of materials. © 2005 by CRC Press LLC

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Chapter 1

FIGURE 1.2.28 Mass element dM in xyz coordinates.

FIGURE 1.2.29 Area A in the xy plane.

Moments of Inertia of a Mass The moment of inertia dIx of an elemental mass dM about the x axis (Figure 1.2.28) is defined as

(

)

dI x = r 2 dM = y 2 + z 2 dM where r is the nearest distance from dM to the x axis. The moments of inertia of a body about the three coordinate axes are Ix =

∫r

Iy =

∫ (x

Iz =

∫ (x

∫ (y

dM =

2

2

)

+ z 2 dM

)

2

+ z 2 dM

2

+ y 2 dM

(1.2.34)

)

Radius of Gyration. The radius of gyration rg is defined by rg = I x / M , and similarly for any other axis. It is based on the concept of the body of mass M being replaced by a point mass M (same mass) at a distance rg from a given axis. A thin strip or shell with all mass essentially at a constant distance rg from the axis of reference is equivalent to a point mass for some analyses. Moment of Inertia of an Area The moment of inertia of an elemental area dA about the x axis (Figure 1.2.29) is defined as dI x = y 2 dA where y is the nearest distance from dA to the x axis. The moments of inertia (second moments) of the area A about the x and y axes (because A is in the xy plane) are Ix =

∫y

2

Iy =

dA

∫x

2

dA

(1.2.35)

The radius of gyration of an area is defined the same way as it is for a mass: rg = I x / A , etc. Polar Moment of Inertia of an Area The polar moment of inertia is defined with respect to an axis perpendicular to the area considered. In Figure 1.2.29, this may be the z axis. The polar moment of inertia in this case is JO = © 2005 by CRC Press LLC

∫r

2

dA =

∫ (x

2

)

+ y 2 dA = I x + I y

(1.2.36)

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Mechanics of Solids

Parallel-Axis Transformations of Moments of Inertia It is often convenient first to calculate the moment of inertia about a centroidal axis and then transform this with respect to a parallel axis. The formulas for the transformations are I = IC + Md 2

for a mass M

I = IC + Ad 2

for an area A

J O = J C + Ad 2

for an area A

(1.2.37)

where I or JO = moment of inertia of M or A about any line, IC or JC = moment of inertia of M or A about a line through the mass center or centroid and parallel to
d = nearest distance between the parallel lines Note that one of the two axes in each equation must be a centroidal axis. Products of Inertia The products of inertia for areas and masses and the corresponding parallel-axis formulas are defined in similar patterns. Using notations in accordance with the preceding formulas, products of inertia are I xy =

∫ xy dA

or

∫ xy dM

I yz =

∫ yz dA

or

∫ yz dM

I xz =

∫ xz dA

or

∫ xz dM

for area,

for mass (1.2.38)

Parallel-axis formulas are I xy = I x ′y′ + A d x d y

or

I x ′y′ + M d x d y

I yz = I y′z ′ + A d y d z

or

I y ′z ′ + M d y d z

I xz = I x ′z ′ + A d x d z

or

I x ′z ′ + M d x d z

for area,

for mass (1.2.39)

Note that the moment of inertia is always positive. The product of inertia may be positive, negative, or zero; it is zero if x or y (or both) is an axis of symmetry of the area. Transformations of known moments and product of inertia to axes that are inclined to the original set of axes are possible but not covered here. These transformations are useful for determining the principal (maximum and minimum) moments of inertia and the principal axes when the area or body has no symmetry. The principal moments of inertia for objects of simple shape are available in many texts.

1.3 Dynamics Stephen M. Birn and Bela I. Sandor The two major categories in dynamics are kinematics and kinetics. Kinematics involves the time- and geometry-dependent motion of a particle, rigid body, deformable body, or a fluid without considering the forces that cause the motion. It relates position, velocity, acceleration, and time. Kinetics combines the concepts of kinematics and the forces that cause the motion. © 2005 by CRC Press LLC

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Kinematics of Particles Scalar Method The scalar method of particle kinematics is adequate for one-dimensional analysis. A particle is a body whose dimensions can be neglected (in some analyses, very large bodies are considered particles). The equations described here are easily adapted and applied to two and three dimensions. Average and Instantaneous Velocity The average velocity of a particle is the change in distance divided by the change in time. The instantaneous velocity is the particle’s velocity at a particular instant. vave =

∆x ∆t

vinst = lim

∆t → 0

∆x dx = = x˙ ∆t dt

(1.3.1)

Average and Instantaneous Acceleration The average acceleration is the change in velocity divided by the change in time. The instantaneous acceleration is the particle’s acceleration at a particular instant. aave =

∆v ∆t

ainst = lim

∆t → 0

∆v dv = = v˙ = x˙˙ ∆t dt

(1.3.2)

Displacement, velocity, acceleration, and time are related to one another. For example, if velocity is given as a function of time, the displacement and acceleration can be determined through integration and differentiation, respectively. The following example illustrates this concept. Example 8 A particle moves with a velocity v(t) = 3t 2 – 8t. Determine x(t) and a(t), if x(0) = 5. Solution. 1. Determine x(t) by integration. v=

dx dt

v dt = dx

∫ 3t

2

− 8t dt =

∫ dx

t 3 − 4t 2 + C = x from x(0) = 5

C=5

x (t ) = t 3 − 4 t 2 + 5 2. Determine a(t) by differentiation. a=

(

dv d 3t 2 − 8t = dt dt

)

a(t ) = 6t − 8 Four key points can be seen from these graphs (Figure 1.3.1).

© 2005 by CRC Press LLC

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FIGURE 1.3.1 Plots of a particle’s kinematics.

1. v = 0 at the local maximum or minimum of the x-t curve. 2. a = 0 at the local maximum or minimum of the v-t curve. 3. The area under the v-t curve in a specific time interval is equal to the net displacement change in that interval. 4. The area under the a-t curve in a specific time interval is equal to the net velocity change in that interval. Useful Expressions Based on Acceleration Equations for nonconstant acceleration are a=

dv ⇒ dt

v dv = a dx ⇒

© 2005 by CRC Press LLC

v

t

v0

0

∫ dv = ∫ a dt v

x

v0

x0

∫ v dv = ∫ a dx

(1.3.3)

(1.3.4)

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FIGURE 1.3.2 Relative motion of two particles along a straight line.

Equations for constant acceleration (projectile motion; free fall) are v = at + v0 v 2 = 2 a( x − x 0 ) + v02 x=

(1.3.5)

1 2 at + v0 t + x 0 2

These equations are only to be used when the acceleration is known to be a constant. Other expressions are available, depending on how a variable acceleration is given as a function of time, velocity, or displacement. Scalar Relative Motion Equations The concept of relative motion can be used to determine the displacement, velocity, and acceleration between two particles that travel along the same line. Equation (1.3.6) provides the mathematical basis for this method. These equations can also be used when analyzing two points on the same body that are not attached rigidly to each other (Figure 1.3.2). xB A = xB − xA vB A = vB − vA

(1.3.6)

aB A = aB − aA The notation B/A represents the displacement, velocity, or acceleration of particle B as seen from particle A. Relative motion can be used to analyze many different degrees-of-freedom systems. A degree of freedom of a mechanical system is the number of independent coordinate systems needed to define the position of a particle. Vector Method The vector method facilitates the analysis of two- and three-dimensional problems. In general, curvilinear motion occurs and is analyzed using a convenient coordinate system. Vector Notation in Rectangular (Cartesian) Coordinates Figure 1.3.3 illustrates the vector method. The mathematical method is based on determining v and a as functions of the position vector r. Note that the time derivatives of unit vectors are zero when the xyz coordinate system is fixed. The scalar components ( x˙, y˙, ˙˙ x, …) can be determined from the appropriate scalar equations previously presented that only include the quantities relevant to the coordinate direction considered. r = xi + yj + zk v= a= © 2005 by CRC Press LLC

dr dx dy dz = i+ j + k = x˙i + y˙j + z˙k dt dt dt dt

dv d 2 x d2y d2z = 2 i + 2 j + 2 k = ˙˙ xi + ˙˙ yj + ˙˙ zk dt dt dt dt

(1.3.7)

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Mechanics of Solids

FIGURE 1.3.3 Vector method for a particle.

FIGURE 1.3.4 Tangential and normal components. C is the center of curvature.

A few key points should be remembered when considering curvilinear motion. First, the instantaneous velocity vector is always tangent to the path of the particle. Second, the speed of the particle is the magnitude of the velocity vector. Third, the acceleration vector is not tangent to the path of the particle and not collinear with v in curvilinear motion. Tangential and Normal Components Tangential and normal components are useful in analyzing velocity and acceleration. Figure 1.3.4 illustrates the method and Equation (1.3.8) comprises the governing equations for it.

v = v nt a = at n t + an n n dv dt

at =

ρ=

[

v2 ρ

an =

1 + (dy dx )

]

2 32

d 2 y dx 2

(1.3.8)

ρ = r = constant for a circular path The osculating plane contains the unit vectors nt and nn, thus defining a plane. When using normal and tangential components, it is common to forget to include the component of normal acceleration, especially if the particle travels at a constant speed along a curved path. For a particle that moves in circular motion, v = rθ˙ = rω

© 2005 by CRC Press LLC

at =

dv θ = rα = r˙˙ dt

an =

v2 = rθ˙ 2 = rω 2 r

(1.3.9)

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FIGURE 1.3.5 Motion of a particle in polar coordinates.

FIGURE 1.3.6 Motion of a particle in cylindrical coordinates.

Motion of a Particle in Polar Coordinates Sometimes it may be best to analyze particle motion by using polar coordinates as follows (Figure 1.3.5):

(always tangent to the path)

v = r˙n r + rθ˙ n θ

dθ ˙ = θ = ω, rad s dt

(

)

(

(1.3.10)

)

a = r˙˙ − rθ˙ 2 n r + r˙˙ θ + 2r˙θ˙ n θ For a particle that moves in circular motion the equations simplify to dθ˙ ˙˙ ˙ = θ = ω = α, rad s 2 dt v = rθ˙ n θ

(1.3.11)

θn θ a = − rθ˙ 2 n r + r˙˙ Motion of a Particle in Cylindrical Coordinates Cylindrical coordinates provide a means of describing three-dimensional motion as illustrated in Figure 1.3.6. v = r˙n r + rθ˙ n θ + z˙k

(

)

(

)

θ + 2r˙θ˙ n θ + ˙˙ a = ˙˙ r − rθ˙ 2 n r + r˙˙ zk

(1.3.12)

Motion of a Particle in Spherical Coordinates Spherical coordinates are useful in a few special cases but are difficult to apply to practical problems. The governing equations for them are available in many texts. Relative Motion of Particles in Two and Three Dimensions Figure 1.3.7 shows relative motion in two and three dimensions. This can be used in analyzing the translation of coordinate axes. Note that the unit vectors of the coordinate systems are the same. Subscripts are arbitrary but must be used consistently because rB/A = –rA/B, etc. © 2005 by CRC Press LLC

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Mechanics of Solids

FIGURE 1.3.7 Relative motion using translating coordinates.

rB = rA + rB A vB = vA + vB A

(1.3.13)

aB = aA + aB A

Kinetics of Particles Kinetics combines the methods of kinematics and the forces that cause the motion. Several useful methods of analysis are based on Newton’s second law. Newton’s Second Law The magnitude of the acceleration of a particle is directly proportional to the magnitude of the resultant force acting on it and inversely proportional to its mass. The direction of the acceleration is the same as the direction of the resultant force. F = ma

(1.3.14)

where m is the particle’s mass. Three key points should be remembered when applying this equation. 1. F is the resultant force. 2. a is the acceleration of a single particle (use aC for the center of mass for a system of particles). 3. The motion is in a nonaccelerating reference frame. Equations of Motion The equations of motion for vector and scalar notations in rectangular coordinates are

∑

∑ F = ma ∑ F = ma

Fx = ma x

y

∑

y

(1.3.15) Fz = maz

The equations of motion for tangential and normal components are

∑ F = ma n

∑ © 2005 by CRC Press LLC

n

=m

v2 ρ

dv Ft = mat = mv˙ = mv ds

(1.3.16)

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Chapter 1

The equations of motion in a polar coordinate system (radial and transverse components) are

∑ F = ma = m(r˙˙ − rθ˙ ) ∑ F = ma = m(r˙˙θ − 2r˙θ˙ ) 2

r

r

θ

(1.3.17)

θ

Procedure for Solving Problems 1. Draw a free-body diagram of the particle showing all forces. (The free-body diagram will look unbalanced because the particle is not in static equilibrium.) 2. Choose a convenient nonaccelerating reference frame. 3. Apply the appropriate equations of motion for the reference frame chosen to calculate the forces or accelerations applied to the particle. 4. Use kinematics equations to determine velocities and/or displacements if needed. Work and Energy Methods Newton’s second law is not always the most convenient method for solving a problem. Work and energy methods are useful in problems involving changes in displacement and velocity, if it is not necessary to calculate accelerations. Work of a Force The total work of a force F in displacing a particle P from position 1 to position 2 along any path is U12 =

∫

2

F ⋅ dr =

1

∫ (F dx + F dy + F dz) 2

1

x

y

z

(1.3.18)

Potential and Kinetic Energies • Gravitational potential energy: U12 = elevation difference

∫

x2

2

∫ W dy = Wh = V , where W = weight and h = vertical g

1

1 k ( x 22 − x12 ) = Ve , where k = spring constant 2 • Kinetic energy of a particle: T = 1/2mv2, where m = mass and v = magnitude of velocity.

• Elastic potential energy: U =

kx dx =

x1

Kinetic energy can be related to work by the principle of work and energy, U12 = T2 − T1

(1.3.19)

where U12 is the work of a force on the particle moving it from position 1 to position 2. T1 is the kinetic energy of the particle at position 1 (initial kinetic energy). T2 is the kinetic energy of the particle at position 2 (final kinetic energy). Power Power is defined as work done in a given time. power = where v is velocity. © 2005 by CRC Press LLC

dU F ⋅ dr = = F⋅v dt dt

(1.3.20)

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Mechanics of Solids

Important units and conversions of power are 1 W =1 J s =1N⋅m s 1 hp = 550 ft ⋅ lb s = 33,000 ft ⋅ lb min = 746 W 1 ft ⋅ lb s = 1.356 J s = 1.356 W Advantages and Disadvantages of the Energy Method Using the energy method in engineering problems offers four advantages: 1. 2. 3. 4.

Accelerations do not need to be determined. Modifications of problems are easy to make in the analysis. Scalar quantities are summed, even if the path of motion is complex. Forces that do not do work are ignored.

The main disadvantage of the energy method is that quantities of work or energy cannot be used to determine accelerations or forces that do no work. In these instances, Newton’s second law must be used. Conservative Systems and Potential Functions Sometimes it is useful to assume a conservative system where friction does not oppose the motion of the particle. The work in a conservative system is independent of the path of the particle, and potential energy is defined as U 

12 work of F from 1 to 2

=

−∆ V 

difference of potential energies at 1 and 2

A special case is one in which the particle moves in a closed path. One trip around the path is called a cycle. U=

∫ dU = ∫ F ⋅ dr = ∫ (F dx + F dy + F dz) = 0 x

y

z

(1.3.21)

In advanced analysis, differential changes in the potential energy function (V) are calculated by the use of partial derivatives,  ∂V ∂V ∂V  F = Fx i + Fy j + Fz k = − i+ j+ k ∂y ∂z   ∂x Conservation of Mechanical Energy Conservation of mechanical energy is assumed if kinetic energy (T) and potential energy (V) change back and forth in a conservative system (the dissipation of energy is considered negligible). Equation (1.3.22) formalizes such a situation, in which position 1 is the initial state and position 2 is the final state. The reference (datum) should be chosen to reduce the number of terms in the equation. T1 + V1 = T2 + V2

(1.3.22)

Linear and Angular Momentum Methods The concept of linear momentum is useful in engineering when the accelerations of particles are not known but the velocities are. The linear momentum is derived from Newton’s second law, G = mv © 2005 by CRC Press LLC

(1.3.23)

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Chapter 1

FIGURE 1.3.8 Definition of angular momentum for a particle.

The time rate of change of linear momentum is equal to force. When mv is constant, the conservation of momentum equation results,

∑ F = G˙ = dt (mv) d

∑F = 0

(1.3.24)

(conservation of momentum)

mv = constant

The method of angular momentum is based on the momentum of a particle about a fixed point, using the vector product in the general case (Figure 1.3.8). H O = r × mv

(1.3.25)

The angular momentum equation can be solved using a scalar method if the motion of the particle remains in a plane, H O = mrv sin φ = mrvθ = mr 2 θ˙ If the particle does not remain in a plane, then the general space motion equations apply. They are derived from the cross product r × mv, H O = H x i + H y j + Hz k

(

H x = m yvz − zv y

)

H y = m( zv x − xvz )

(

H z = m xv y − yv x

(1.3.25a)

)

Time Rate of Change of Angular Momentum In general, a force acting on a particle changes its angular momentum: the time rate of change of angular momentum of a particle is equal to the sum of the moments of the forces acting on the particle. Vectors:

˙ = d (r × mv) = r × H O dt

Scalars:

∑M ∑M

O

x

= H˙ x

=0

∑F = ∑H

∑M

y

= H˙ y

(1.3.26)

O

∑M

z

= H˙ z

H O = r × mv = constant

(conservation of angular momentum) © 2005 by CRC Press LLC

(1.3.27)

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Mechanics of Solids

A special case is when the sum of the moments about point O is zero. This is the conservation of angular momentum. In this case (motion under a central force), if the distance r increases, the velocity must decrease, and vice versa. Impulse and Momentum Impulse and momentum are important in considering the motion of particles in impact. The linear impulse and momentum equation is t2

∫ F dt = t1

impulse

mv 2 − mv1  

final momentum

(1.3.28)

initial momentum

Conservation of Total Momentum of Particles Conservation of total momentum occurs when the initial momentum of n particles is equal to the final momentum of those same n particles, n

n

∑ (m v ) i

i 1

 i

total initial momentum at time t1

=

∑ (m v ) i

(1.3.29)

i 2

 i

total final momentum at time t2

When considering the response of two deformable bodies to direct central impact, the coefficient of restitution is used. This coefficient e relates the initial velocities of the particles to the final velocities, e=

v Bf − v Af vA − vB

relative velocity of separation relative velocity of approach

=

(1.3.30)

For real materials, 0 < e < 1. If both bodies are perfectly elastic, e = 1, and if either body is perfectly plastic, e = 0.

Kinetics of Systems of Particles There are three distinct types of systems of particles: discrete particles; continuous particles in fluids; and continuous particles in rigid or deformable bodies. This subsection considers methods for discrete particles that have relevance to the mechanics of solids. Methods involving particles in rigid bodies will be discussed in later sections. Newton’s Second Law Applied to a System of Particles Newton’s second law can be extended to systems of particles, n

n

∑F = ∑m a i

i =1

i

(1.3.31)

i

i =1

Motion of the Center of Mass The center of mass of a system of particles moves under the action of internal and external forces as if the total mass of the system and all the external forces were at the center of mass. Equation (1.3.32) defines the position, velocity, and acceleration of the center of mass of a system of particles. n

mrC =

∑ i =1

© 2005 by CRC Press LLC

n

mi ri

mv C =

∑ i =1

n

mi v i

ma C =

∑m a i

i =1

i

∑ F = ma

C

(1.3.32)

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Chapter 1

Work and Energy Methods for a System of Particles Gravitational Potential Energy. The gravitational potential energy of a system of particles is the sum of the potential energies of the individual particles of the system: n

Vg = g

∑

n

∑ W y = mgy

mi yi =

i i

i =1

C

= WyC

(1.3.33)

i =1

where g = acceleration of gravity yC = vertical position of center of mass with respect to a reference level Kinetic energy. The kinetic energy of a system of particles is the sum of the kinetic energies of the individual particles of the system with respect to a fixed reference frame, 1 2

T=

n

∑m v

2 i i

(1.3.34)

i =1

A translating reference frame located at the mass center C of a system of particles can be used advantageously, with T=

1 2 mvC 2  

motion of total mass imagined to be concentrated at C

n

+

∑

1 m v′2 2 i =1 i i 

(v ′ are with respect to a translating frame)

(1.3.35)

motion of all particles relative to C

Work and Energy The work and energy equation for a system of particles is similar to the equation stated for a single particle: n

∑

n

Ui′ =

i =1

n

∑ ∑T Vi +

i

i =1

(1.3.36)

i =1

U ′ = ∆V + ∆T Momentum Methods for a System of Particles Moments of Forces on a System of Particles. The moments of external forces on a system of particles about a point O are given by n

n

∑ (r × F ) = ∑ M i

i

i =1

i =1

n

iO

+

∑ (r × m a ) i

i

i

(1.3.37)

i =1

Linear and Angular Momenta of a System of Particles. The resultant of the external forces on a system of particles equals the time rate of change of linear momentum of that system: n

G=

∑m v i

i =1

© 2005 by CRC Press LLC

i

∑ F = G˙

(1.3.38)

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Mechanics of Solids

The angular momentum equation for a system of particles about a fixed point O is n

∑ (r × m a )

HO =

i

i

i

i =1

∑M

(1.3.39) n

˙ = =H O

O

∑ (r × m a ) i

i

i

i =1

This last equation means that the resultant of the moments of the external forces on a system of particles equals the time rate of change of angular momentum of that system. Angular Momentum about the Center of Mass The preceding equations work well for stationary reference frames, but sometimes a special approach may be useful, noting that the angular momentum of a system of particles about its center of mass C is the same whether it is observed from a fixed frame at point O or from the centroidal frame, which may be translating but not rotating. In this case H O = H C + rC × mv C

∑M

O

˙ + r × ma =H C C C

(1.3.40)

Conservation of Momentum The conservation of momentum equations for a system of particles is analogous to that for a single particle. G = constant   H O = constant  H C = constant 

not the same constants in general

Impulse and Momentum of a System of Particles The linear impulse momentum for a system of particles is n

∑ ∫ F dt = G i =1

t2

t1

i

2

− G1 = mv C2 − mv C1

(1.3.41)

The angular impulse momentum for a system of particles is n

∑∫ M i =1

t2

t1

iO

dt = H O2 − H O1

(1.3.42)

Kinematics of Rigid Bodies Rigid body kinematics is used when the methods of particle kinematics are inadequate to solve a problem. A rigid body is defined as one in which the particles are rigidly connected. This assumption allows for some similarities to particle kinematics. The two kinds of rigid body motion are translation and rotation. These motions may occur separately or in combination.

© 2005 by CRC Press LLC

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Chapter 1

FIGURE 1.3.9 Translational motion of a rigid body.

FIGURE 1.3.10 Rigid body rotating about a fixed axis.

Translation Figure 1.3.9 models the translational motion of a rigid body. rB = rA + rB A

(r

BA

= constant

v B = r˙B = r˙ A = v A

) (1.3.43)

a B = v˙ B = v˙ A = a A These equations represent an important fact: when a rigid body is in translation, the motion of a single point completely specifies the motion of the whole body. Rotation about a Fixed Axis Figure 1.3.10 models a point P in a rigid body rotating about a fixed axis with an angular velocity ω. The velocity v of point P is determined assuming that the magnitude of r is constant, v=ω×r

(1.3.44)

The acceleration a of point P is determined conveniently by using normal and tangential components,

(

)

aP = α × r + ω × ω × r     at

at = ρα

an

(1.3.45)

an = ρω 2

Note that the angular acceleration a and angular velocity w are valid for any line perpendicular to the axis of rotation of the rigid body at a given instant.

© 2005 by CRC Press LLC

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Mechanics of Solids

Kinematics Equations for Rigid Bodies Rotating in a Plane For rotational motion with or without a fixed axis, if displacement is measured by an angle θ, ω=

Angular speed: α=

Angular acceleration:

dθ dt

dω dω =ω dt dθ

For a constant angular speed ω, Angular displacement:

(θ = θ

θ = θo + ωt

o

at t = 0)

For a constant angular acceleration α,

(ω = ω

ω = ω o + αt

θ = θo + ω ot +

o

at t = 0)

1 2 αt 2

ω 2 = ω o2 + 2α(θ − θ o ) Velocities in General Plane Motion General plane motion of a rigid body is defined by simultaneous translation and rotation in a plane. Figure 1.3.11 illustrates how the velocity of a point A can be determined using Equation (1.3.46), which is based on relative motion of particles. vA =

v 

B translation

+ ω × rA B    

(1.3.46)

rotation

Five important points should be remembered when solving general plane motion problems, including those of interconnected rigid bodies. 1. The angular velocity of a rigid body in plane motion is independent of the reference point. 2. The common point of two or more pin-jointed members must have the same absolute velocity even though the individual members may have different angular velocities.

FIGURE 1.3.11 Analysis of velocities in general plane motion.

© 2005 by CRC Press LLC

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Chapter 1

3. The points of contact in members that are in temporary contact may or may not have the same absolute velocity. If sliding occurs between the members, the points in contact have different absolute velocities. The absolute velocities of the contacting particles are always the same if no sliding takes place. 4. If the angular velocity of a member is not known, but some points of the member move along defined paths (i.e., the end points of a piston rod), these paths define the directions of the velocity vectors and are useful in the solution. 5. The geometric center of a wheel rolling on a flat surface moves in rectilinear motion. If no slipping occurs at the point of contact, the linear distance that the center point travels is equal to the portion of the rim circumference that has rolled along the flat surface. Instantaneous Center of Rotation The method of instantaneous center of rotation is a geometric method of determining the angular velocity when two velocity vectors are known for a given rigid body. Figure 1.3.12 illustrates the method. This procedure can also be used to determine velocities parallel to one of the given velocities, by similar triangles. Velocities vA and vB are given; thus the body is rotating about point I at that instant. Point I has zero velocity at that instant, but generally has an acceleration. This method does not work for the determination of angular accelerations.

FIGURE 1.3.12 Schematic for instantaneous center of rotation.

Acceleration in General Plane Motion Figure 1.3.13 illustrates a method of determining accelerations of points of a rigid body. This is similar to (but more difficult than) the procedure of determining velocities.

(

a B = a A + α × rB A + ω × ω × rB A aB =

( ) ( )

+ aB A + aB A t  n  translation aA 

rotation

FIGURE 1.3.13 Accelerations in general plane motion.

© 2005 by CRC Press LLC

) (1.3.47)

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Mechanics of Solids

Six key points should be considered when solving this kind of a problem. 1. The angular velocity and acceleration of a rigid body in plane motion are independent of the reference point. 2. The common points of pin-jointed members must have the same absolute acceleration even though the individual members may have different angular velocities and angular accelerations. 3. The points of contact in members that are in temporary contact may or may not have the same absolute acceleration. Even when no sliding between the members occurs, only the tangential accelerations of the points in contact are the same; the normal accelerations are frequently different in magnitude and direction. 4. The instantaneous center of zero velocity in general has an acceleration and should not be used as a reference point for accelerations unless its acceleration is known and included in the analysis. 5. If the angular acceleration of a member is not known, but some points of the member move along defined paths, the geometric constraints of motion define the directions of normal and tangential acceleration vectors and are useful in the solution. 6. The geometric center of a wheel rolling on a flat surface moves in rectilinear motion. If no slipping occurs at the point of contact, the linear acceleration of the center point is parallel to the flat surface and equal to ra for a wheel of radius r and angular acceleration α. General Motion of a Rigid Body Figure 1.3.14 illustrates the complex general motion (three-dimensional) of a rigid body. It is important to note that here the angular velocity and angular acceleration vectors are not necessarily in the same direction as they are in general plane motion. Equation (1.3.48) gives the velocity and acceleration of a point on the rigid body. These equations are the same as those presented for plane motion. v B = v A + ω × rB A

(

a B = a A + α × rB A + ω × ω × rB A

)

(1.3.48)

( ) + (a )

aB = aA + aB A

t

BA n

The most difficult part of solving a general motion problem is determining the angular acceleration vector. There are three cases for the determination of the angular acceleration: 1. The direction of ω is constant. This is plane motion and α = ω˙ can be used in scalar solutions of problems. 2. The magnitude of ω is constant but its direction changes. An example of this is a wheel that travels at a constant speed on a curved path. 3. The magnitude and direction of ω change. This is space motion because all or some points of the rigid body have three-dimensional paths. An example of this is a wheel that accelerates on a curved path.

FIGURE 1.3.14 General motion of a rigid body.

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Chapter 1

FIGURE 1.3.15 Rigid body fixed at point O.

A useful expression can be obtained from the second item and Figure 1.3.15. The rigid body is fixed at point O and ω has a constant magnitude. Let ω rotate about the Y axis with angular velocity Ω. The angular acceleration is determined from Equation (1.3.49). α=

dω =Ω×ω dt

(1.3.49)

For space motion, it is essential to combine the results of the first two items, which provide components of α for the change in magnitude and the change in direction. The following example illustrates the procedure. Example 9 The rotor shaft of an alternator in a car is in the horizontal plane. It rotates at a constant angular speed of 1500 rpm while the car travels at v = 60 ft/sec on a horizontal road of 400 ft radius (Figure 1.3.16). Determine the angular acceleration of the rotor shaft if v increases at the rate of 8 ft/sec2. Solution. There are two components of α. One is the change in the direction of the rotor shaft’s ωx, and the other is the change in magnitude from the acceleration of the car. • Component from the change in direction. Determine ωc of the car. Use Equation (1.3.49): v = rω c ω c = 0.15 rad sec k i α = ωc × ω = 0 157.1

j 0 0

k 0.15 = 23.6 j rad sec 2 0

• Component from the acceleration of the car. Use Equation (1.3.9): α C r = at α C = 0.02k rad sec 2

FIGURE 1.3.16 Schematic of shaft’s motion. © 2005 by CRC Press LLC

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Mechanics of Solids

The angular acceleration of the rotor shaft is α = (23.6 j + 0.02k) rad sec 2 This problem could also be solved using the method in the next section. Time Derivative of a Vector Using a Rotating Frame The basis of determining time derivatives of a vector using a rotating frame is illustrated in Figure 1.3.17.

(Q) XYZ

( )

˙ = Q

xyz

+Ω×Q

FIGURE 1.3.17 Time derivative of a vector using a rotating reference frame.

Analysis of Velocities and Accelerations Using Rotating and Translating Frames With the concept of general motion understood, an advantageous method of determining velocities and accelerations is available by the method of rotating reference frames. This method can be used in two cases: 1. For a common origin of XYZ and xyz, with r a position vector to a point P, v P = v xyz + Ω × r ˙ × r + Ω × (Ω × r ) + 2 Ω × v a P = a xyz + Ω xyz

(1.3.50)

2. For the origin A of xyz translating with respect XYZ:

( )

v P = v A + r˙P A

xyz

+ Ω × rP A

(

)

(1.3.51)

˙ ×r +Ω× Ω×r a P = a A + a xyz + Ω P A P A + 2Ω × v xyz where Ω is the angular velocity of the xyz frame with respect to XYZ. 2Ω × vxyz is the Coriolis acceleration.

Kinetics of Rigid Bodies in Plane Motion Equation of Translational Motion The fundamental equation for rigid body translation is based on Newton’s second law. In Equation (1.3.52), a is the acceleration of the center of mass of the rigid body, no matter where the resultant force acts on the body. The sum of the external forces is equal to the mass of the rigid body times the acceleration of the mass center of the rigid body, independent of any rotation of the body.

∑ F = ma © 2005 by CRC Press LLC

C

(1.3.52)

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Chapter 1

Equation of Rotational Motion Equation (1.3.53) states that the sum of the external moments on the rigid body is equal to the moment of inertia about an axis times the angular acceleration of the body about that axis. The angular acceleration α is for the rigid body rotating about an axis. This equation is independent of rigid body translation.

∑M

C

= IC α

(1.3.53)

˙ , H = I ω . An application is illustrated in Color Figure 2. where ΣM C = H C C C Applications of Equations of Motion It is important to use the equations of motion properly. For plane motion, three scalar equations are used to define the motion in a plane:

∑ F = ma x

∑ F = ma

Cx

y

Cy

∑M

C

= IC α

(1.3.54)

=0

(1.3.55)

If a rigid body undergoes only translation,

∑ F = ma x

Cx

∑ F = ma y

Cy

∑M

C

If the rigid body undergoes pure rotation about the center of mass,

∑F = 0 x

∑F = 0

∑M

y

C

= IC α

(1.3.56)

Rigid body motions are categorized according to the constraints of the motion: 1. Unconstrained Motion: Equation (1.3.54) is directly applied, with all three equations independent of one another. 2. Constrained Motion: the equations in Equation (1.3.54) are not independent of one another. Generally, a kinematics analysis must be made to determine how the motion is constrained in the plane. Consider two special cases: a. Point constraint: the body has a fixed axis. b. Line constraint: the body moves along a fixed line or plane. When considering systems of rigid bodies, it is important to remember that, at most, only three equations of motion are available from each free-body diagram for plane motion to solve for three unknowns. The motion of interconnected bodies must be analyzed using related free-body diagrams. Rotation about a Fixed Axis not through the Center of Mass The methods presented previously are essential in analyzing rigid bodies that rotate about a fixed axis, which is common in machines (shafts, wheels, gears, linkages). The mass of the rotating body may be nonuniformly distributed as modeled in Figure 1.3.18. Note that rC is the nearest distance between the fixed axis O and the mass center C. The figure also defines the normal and tangential coordinate system used in Equation (1.3.57), which comprises the scalar equations of motion using normal and tangential components. The sum of the forces must include all reaction forces on the rigid body at the axis of rotation.

∑ F = mr ω n

© 2005 by CRC Press LLC

C

2

∑ F = mr α t

C

∑M

O

= IO α

(1.3.57)

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FIGURE 1.3.18 Rotation of a rigid body about a fixed axis.

General Plane Motion A body that is translating and rotating is in general plane motion. The scalar equations of motion are given by Equation (1.3.54). If an arbitrary axis A is used to find the resultant moment,

∑M

A

= I Aα + r × m a C

(1.3.58)

where C is the center of mass. It is a common error to forget to include the cross-product term in the analysis. General plane motion has two special cases: rolling and sliding. Figure 1.3.19 shows pure rolling of a wheel without slipping with the center of mass C at the geometric center of the wheel. This is called pure rolling of a balanced wheel. From this figure the scalar equation of motion results, aCx = rα

∑M

A

= I Aα

(1.3.59)

For balanced wheels that slide or do not slide, the following schematic is helpful:

If slipping is not certain, assume that no slipping has occurred and check whether
≤ µs N. If
> µs N (not possible; there is sliding), start the solution over using
= µk N but not using aCx = rα, which is not valid here. For the problem involving unbalanced wheels (the mass center and geometric center do not coincide), Equation (1.3.60) results: aCx ≠ rα

a G = rα

( ) + (a )

aC = aG + aC G = aG + aC G

FIGURE 1.3.19 Pure rolling of a wheel. © 2005 by CRC Press LLC

n

CG t

(1.3.60)

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Chapter 1

Energy and Momentum Methods for Rigid Bodies in Plane Motion Newton’s second law in determining kinetics relationships is not always the most efficient, although it always works. When particles are considered, energy and momentum methods are often useful to analyze rigid bodies in plane motion. Work of a Force on a Rigid Body The work of a force acting on a rigid body moving from position 1 to 2 is U12 =

∫

2

F ⋅ dr =

1

2

∫ F ⋅ v dt

(1.3.61)

1

Work of a Moment The work of a moment has a similar form, for angular positions θ, U12 =

∫

θ2

θ1

M ⋅ dθ

(1.3.62)

In the common case in which the moment vector M is perpendicular to the plane of motion, M ⋅ dθ = M dθ. It is important to note those forces that do no work: • Forces that act at fixed points on the body do not do work. For example, the reaction at a fixed, frictionless pin does no work on the body that rotates about that pin. • A force that is always perpendicular to the direction of the motion does no work. • The weight of a body does no work when the body’s center of gravity moves in a horizontal plane. • The friction force
at a point of contact on a body that rolls without slipping does no work because the point of contact is the instantaneous center of zero velocity. Kinetic Energy of a Rigid Body The kinetic energy of a particle only consists of the energy associated with its translational motion. The kinetic energy of a rigid body also includes a term for the rotational energy of the body, T = Ttrans + Trot =

1 2 1 mv + I ω 2 2 C 2 C

(1.3.63)

where C is the center of mass of the rigid body. The kinetic energy of a rigid body rotating about an arbitrary axis at point O is T=

1 I ω2 2 O

Principle of Work and Energy The principle of work and energy for a rigid body is the same as that used for particles with the addition of the rotational energy terms. T2 = T1 + U12

(1.3.64)

where T1 = initial kinetic energy of the body T2 = final kinetic energy of the body U12 = work of all external forces and moments acting on the body moving from position 1 to 2 This method is advantageous when displacements and velocities are the desired quantities. © 2005 by CRC Press LLC

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Mechanics of Solids

FIGURE 1.3.20 Impulse and momentum for rigid bodies.

Conservation of Energy The conservation of energy in a conservative rigid body system is T1 + V1 = T2 + V2

(1.3.65)

where T = kinetic energy V = total potential energy (gravitational and elastic) Power The net power supplied to or required of the system is power = T˙trans + T˙rot + V˙g + V˙e

(1.3.66)

This can be calculated by taking time derivatives of the kinetic and potential energy terms. Each term is considered positive when it represents the power supplied to the system and negative when power is taken from the system. Impulse and Momentum of a Rigid Body Impulse and momentum methods are particularly useful when time and velocities are of interest. Figure 1.3.20 shows how rigid bodies are to be considered for this kind of analysis. Notice that rotational motion of the rigid body must be included in the modeling. The impulse of the external forces in the given interval is

∫ ∑ F dt = m (v t2

C2

t1

C2

− v C1

)

(1.3.67)

where t is time C is the center of mass Σ F includes all external forces The impulse of the external moments in the given interval is

∫ ∑M t2

t1

C

dt = H C2 − H C1

For plane motion, if ΣM is parallel to w, the scalar expressions are © 2005 by CRC Press LLC

(1.3.68)

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Chapter 1

∫ ∑M

C

∫ ∑M

O

t2

t1

(1.3.69)

t2

t1

dt = IC (ω 2 − ω 1 ) dt = IO (ω 2 − ω 1 )

for rotation about a fixed point O

Impulse and Momentum of a System of Rigid Bodies A system of rigid bodies can be analyzed using one of the two following procedures, illustrated in Figure 1.3.21: 1. Apply the principle of impulse and momentum to each rigid member separately. The mutual forces acting between members must be included in the formulation of the solution. 2. Apply the principle of impulse and momentum to the entire system of bodies, ignoring the mutual forces between members. Conservation of Momentum The principle of conservation of linear and angular momentum of particles can be extended to rigid bodies that have no external forces or moments acting on them. The conservation of linear momentum means that the center of mass C moves at a constant speed in a constant direction,

∑ F = 0 ⇒ ∆G = 0

(1.3.70)

v C1 = v C2 Likewise, for conservation of angular momentum of rigid bodies,

∑ M = 0 ⇒ ∆H

C

=0

(1.3.71)

IC ω 1 = IC ω 2 For a system of rigid bodies, use the same fixed reference point O for all parts of the system. Thus, for plane motion, ∆H O = 0

IO ω 1 = IO ω 2

(1.3.72)

Two important points should be remembered when these equations are used. First, ∆HC = 0 does not imply that ∆HO = 0, or vice versa. Second, conservation of momentum does not require the simultaneous conservation of angular and linear momenta (for example, there may be an angular impulse while linear momentum is conserved).

FIGURE 1.3.21 System of rigid bodies.

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Mechanics of Solids

FIGURE 1.3.22 Motion of an inclined, spinning top.

Kinetics of Rigid Bodies in Three Dimensions The concepts of plane rigid body motion can be extended to the more complicated problems in three dimensions, such as those of gyroscopes and jet engines. This section briefly covers some fundamental topics. Many additional topics and useful methods are included in the technical literature. Angular Momentum in Three Dimensions For analyzing three-dimensional angular momentum, three special definitions are used. These can be visualized by considering a spinning top (Figure 1.3.22): • Precession — rotation of the angular velocity vector about the y axis • Space Cone — locus of the absolute positions of the instantaneous axis of rotation • Body Cone — locus of the positions of the instantaneous axis relative to the body; the body cone appears to roll on the space cone (not shown here) Equation (1.3.73) provides the scalar components of the total angular momentum. H x = I x ω x − I xy ω y − I xz ω z H y = − I xy ω x + I y ω y − I yz ω z

(1.3.73)

H z = − I zx ω x − I zy ω y + I z ω z Impulse and Momentum of a Rigid Body in Three-Dimensional Motion The extension of the planar motion equations of impulse and momentum to three dimensions is straightforward: linear momentum of mass center (G) System momenta =  angular momentum about mass center (H C )

(1.3.74)

where G and H have different units. The principle of impulse and momentum is applied for the period of time t1 to t2, G 2 = G1 + (external linear impulses) 1

2

H C2 = H C1 + (external angular impulses) 1

2

© 2005 by CRC Press LLC

(1.3.75)

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Chapter 1

Kinetic Energy of a Rigid Body in Three-Dimensional Motion The total kinetic energy of a rigid body in three dimensions is T=

1 2 1 mvC + ω ⋅ H C 2 2   

(1.3.76)

1 ω ⋅ HO 2

(1.3.77)

translation of mass center

rotation about mass center

For a rigid body that has a fixed point O, T= Equations of Motion in Three Dimensions The equations of motion for a rigid body in three dimensions are extensions of the equations previously stated:

∑ F = ma ∑

( )

˙ = H ˙ MC = H C C

C

(1.3.78)

xyz

+ Ω × HC

where aC = acceleration of mass center HC = angular momentum of the body about its mass center xyz = frame fixed in the body with origin at the mass center Ω = angular velocity of the xyz frame with respect to a fixed XYZ frame Note that an arbitrary fixed point O may be used for reference if done consistently. Euler’s Equations of Motion Euler’s equations of motion results from the simplification of allowing the xyz axes to coincide with the principal axes of inertia of the body.

∑M ∑M ∑M

(

)

x

= I x ω˙ x − I y − I z ω y ω z

y

= I y ω˙ y − ( I z − I x )ω z ω x

z

= I z ω˙ z − I x − I y ω x ω y

(

(1.3.79)

)

where all quantities must be evaluated with respect to the appropriate principal axes. Solution of Problems in Three-Dimensional Motion In order to solve a three-dimensional problem, it is necessary to apply the six independent scalar equations:

∑ F = ma x

© 2005 by CRC Press LLC

Cx

∑ F = ma y

Cy

∑ F = ma z

Cz

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Mechanics of Solids

∑M ∑M ∑M

x

= H˙ x + ω y H z − ω z H y

y

= H˙ y + ω z H x − ω x H z

z

= H˙ z + ω x H y − ω y H x

(1.3.80)

These equations are valid in general. Some common cases are briefly stated: • Unconstrained motion. The six governing equations should be used with xyz axes attached at the center of mass of the body. • Motion of a body about a fixed point. The governing equations are valid for a body rotating about a noncentroidal fixed point O. The reference axes xyz must pass through the fixed point to allow using a set of moment equations that do not involve the unknown reactions at O. • Motion of a body about a fixed axis. This is the generalized form of plane motion of an arbitrary rigid body. The analysis of unbalanced wheels and shafts and corresponding bearing reactions falls in this category.

1.4 Vibrations Bela I. Sandor and Stephen M. Birn Vibrations in machines and structures should be analyzed and controlled if they have undesirable effects such as noise, unpleasant motions, or fatigue damage with potentially catastrophic consequences. Conversely, vibrations are sometimes employed to useful purposes, such as for compacting materials.

Undamped Free and Forced Vibrations The simplest vibrating system has motion of one degree of freedom (DOF) described by the coordinate x in Figure 1.4.1. (An analogous approach is used for torsional vibrations, with similar results.) Assuming that the spring has no mass and that no damping occurs in the system, the equation of motion for free vibration (motion under internal forces only; F = 0) is mx˙˙ + kx = 0

or

˙˙ x + ω2 x = 0

(1.4.1)

where ω = k / m = natural circular frequency in radians per second. The displacement x as a function of time t is x = C1 sin ωt + C2 cos ωt

FIGURE 1.4.1 Model of a simple vibrating system.

© 2005 by CRC Press LLC

(1.4.2)

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Chapter 1

where C1 and C2 are constants depending on the initial conditions of the motion. Alternatively, x = A sin(ωt + φ) where C1 = Acosφ, C2 = Asinφ, and φ is the phase angle, another constant. A complete cycle of the motion occurs in time τ, the period of simple harmonic motion, τ=

2π m = 2π (seconds per cycle) k ω

The frequency in units of cycles per second (cps) or hertz (Hz) is f = 1/τ. The simplest case of forced vibration is modeled in Figure 1.4.1, with the force F included. Using typical simplifying assumptions as done previously, the equation of motion for a harmonic force of forcing frequency Ω, is mx˙˙ + kx = Fo sin Ωt

(1.4.3)

The vibrations of a mass m may also be induced by the displacement d = dosinΩt of a foundation or another mass M to which m is attached by a spring k. Using the same reference point and axis for x and d, the equation of motion for m is mx˙˙ + k ( x − do sin Ωt ) = 0 mx˙˙ + kx = kdo sin Ωt

(1.4.4)

where do is the amplitude of vibration of the moving support M, and Ω is its frequency of motion. The general solution of the forced vibration in the steady state (after the initial transient behavior) is x = A sin Ωt A=

Fo Fo k = k − mΩ 2 1 − (Ω ω ) 2

(1.4.5)

where Ω is the forcing frequency and w is the natural circular frequency of the system of m and k. Resonance. The amplitude of the oscillations in forced vibrations depends on the frequency ratio Ω/ω. Without damping or physical constraints, the amplitude would become infinite at Ω = ω, the condition of resonance. Dangerously large amplitudes may occur at resonance and at other frequency ratios near the resonant frequency. A magnification factor is defined as MF =

1 F A = = Fo k do 1 − (Ω ω ) 2

(1.4.6)

Several special cases of this are noted: 1. 2. 3. 4.

Static loading: Ω = 0, or Ω
ω; MF
1. Resonance: Ω = ω; MF = 8. High-frequency excitation: Ω  ω; MF. 0. Phase relationships: the vibration is in phase for Ω < ω, and it is 180° out of phase for Ω > ω.

© 2005 by CRC Press LLC

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Mechanics of Solids

FIGURE 1.4.2 Model of a damped vibrating system.

Damped Free and Forced Vibrations A vibrating system of one degree of freedom and damping is modeled in Figure 1.4.2. The equation of motion for damped free vibrations (F = 0) is mx˙˙ + cx˙ + kx = 0

(1.4.7)

The displacement x as a function of time t is x = e λt

(1.4.8) 2

λ 1,2 =

−c c  k ±  −  2m  2m m

The value of the coefficient of viscous damping c that makes the radical zero is the critical damping coefficient cc = 2m k / m = 2mω. Three special cases of damped free vibrations are noted: 1. Overdamped system: c > cc; the motion is nonvibratory or aperiodic 2. Critically damped system: c = cc; this motion is also nonvibratory; x decreases at the fastest rate possible without oscillation of the mass 3. Underdamped system: c < cc; the roots λ1,2 are complex numbers; the displacement is x = Ae − ( c 2 m )t sin(ω d t + φ) where A and φ are constants depending on the initial conditions, and the damped natural frequency is  c ωd = ω 1 −    cc 

2

The ratio c/cc is the damping factor ζ. The damping in a system is determined by measuring the rate of decay of free oscillations. This is expressed by the logarithmic decrement δ, involving any two successive amplitudes xi and xi + 1, δ = ln

xi 2 πζ
2 πζ = xi +1 1 − ζ2

The simplifying approximation for δ is valid for up to about 20% damping (ζ
0.2).

© 2005 by CRC Press LLC

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Chapter 1

FIGURE 1.4.3 Magnification factor in damped forced vibration.

The period of the damped vibration is τd = 2π/ωd. It is a constant, but always larger than the period of the same system without damping. In many real systems the damping is relatively small (ζ < 0.2), where τd
τ and ωd
ω can be used. The equation of motion for damped forced vibrations (Figure 1.4.2; F ≠ 0) is mx˙˙ + cx˙ + kx = Fo sin Ωt

(1.4.9)

The solution for steady-state vibration of the system is x = A sin(Ωt − φ)

(1.4.10)

where the amplitude and phase angle are from A=

Fo

( c Ω)

2

tan φ =

(

+ k − mΩ 2

)

2

cΩ k − mΩ 2

The magnification factor for the amplitude of the oscillations is MF =

A A = = Fo k do

1

[2ζ(Ω ω)]

2

[

+ 1 − (Ω ω )

]

2 2

(1.4.11)

This quantity is sketched as a function of the frequency ratio Ω/ω for several damping factors in Figure 1.4.3. Note that the amplitude of vibration is reduced at all values of Ω/ω if the coefficient of damping c is increased in a particular system.

Vibration Control Vibration Isolation It is often desirable to reduce the forces transmitted, or the noise and motions inside or in the neighborhood of vibrating machines and structures. This can be done to some extent within the constraints of space and additional weight and cost by the use of isolators, such as rubber engine mounts and wheel suspension systems in cars. Many kinds of isolating materials and systems are available commercially. © 2005 by CRC Press LLC

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Mechanics of Solids

FIGURE 1.4.4 Transmissibility patterns of a vibration isolator.

The effectiveness of vibration isolation is expressed by the transmissibility TR, the ratio of the force transmitted FT to the disturbing force Fo . A simple isolation system is modeled as a spring and a dashpot in parallel, for which the transmissibility is given by Equation (1.4.12) and sketched in Figure 1.4.4.

TR =

FT = Fo

1 + 4ζ 2 (Ω ω )

[

1 − (Ω ω )

]

2 2

2

+ 4ζ 2 (Ω ω )

(1.4.12) 2

When damping is negligible, TR


1 (Ω ω ) 2 − 1

Note from the figure that • Vibration isolation occurs at Ω/ω > 2 . • Isolation efficiency increases with decreasing stiffness of the isolation mounts. • Damping reduces isolation efficiency. However, some damping is normally required if resonance may occur in a system even for short periods. • The response curves are essentially independent of damping when Ω/ω is large (>3) ~ and damping 2 – 1]. is low (ζ < 0.2). Here TR
1/[(Ω/ω) ~ • For a system with more than one excitation frequency, the lowest excitation frequency is of primary importance. The efficiency of an isolating system is defined by the reduction R in transmissibility, R = 1 – TR If a certain reduction R in transmissibility is desired, the appropriate stiffness k of an isolation system is obtained from ω = k / m and Ω = ω

2−R 1− R

A small magnitude of stiffness k makes the reduction R in transmissibility large. It is difficult to achieve isolation for very low excitation frequencies because of the required large static deflections. To obtain highly efficient isolation at low excitation frequencies, a large supporting mass M may be utilized, with the value of ω = k /( m + M ) © 2005 by CRC Press LLC

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Chapter 1

Vibration Absorption In some cases, a vibratory force is purposely generated in a system by a secondary spring-mass system to oppose a primary disturbing force and thereby reduce or eliminate the undesirable net effect. An interesting example of this is the “tuned-mass damper” in a few skyscrapers, designed to counter the oscillatory motions caused by wind. The secondary spring-mass system has disadvantages of its own, such as extra weight, complexity, and effectiveness limited to a single frequency. Balancing of Rotating Components The conditions of static or dynamic unbalance of rotating bodies have long been recognized. These can be analyzed by the methods of elementary mechanics; simple tests can be performed in many cases and adequate corrections can be made routinely to achieve balance, such as for the wheels of automotive vehicles. Three categories of increasing complexity are distinguished. 1. Static unbalance. The distributed or lumped masses causing unbalance are in a single axial plane and all on the same side of the axis of rotation (Figure 1.4.5). Thin disks are also in this category. Static unbalance is detected in a static test because the center of gravity of the body is not on the axis, and correction is made by adding or removing mass at a convenient radial distance from the axis. 2. Static balance with dynamic unbalance. This may be the case when the masses causing unbalance are in a single axial plane but on opposite sides of the axis of rotation (Figure 1.4.6a). Static balance is achieved if the center of gravity of the body is on the axis, but dynamic unbalance results from the couple of the unbalance forces (mω2r) during rotation, causing a shaking of the axle. 3. Static and dynamic unbalance. This is the general case of unbalance, which can be visualized by not allowing m1 and m2 and the axis of rotation to lie in the same plane (Figure 1.4.6b).

FIGURE 1.4.5 Schematic of static unbalance.

FIGURE 1.4.6 Schematic of two cases of dynamic unbalance. © 2005 by CRC Press LLC

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The magnitude and angular position of a body’s unbalance can be determined using a dynamic balancing machine. Here the shaking forces are measured by electronically sensing the small oscillations of the bearings that can be correlated with the position of the body. Critical Speed of Rotating Shafts A rotating shaft may become dangerously unstable and whirl with large lateral amplitudes of displacement at a critical speed of rotation. The critical speed, in revolutions per second, corresponds with the natural frequency of lateral vibration of the system. Thus, it can be analytically predicted fairly well and can be safely measured in a real but nonrotating machine with high precision. If unavoidable, as at startup, the critical speed should be passed over rapidly. Other ways of minimizing the problems of whirling shafts include the proper balancing of rotors and the replacing of bent shafts and worn bearings.

Random Vibrations. Shock Excitation Many structures are subjected to nonharmonic excitations and respond with transient vibrations rather than steady-state motions. Random vibration is often caused by shock excitation, which implies that the loading occurs suddenly, in a short time with respect to the natural period of vibration of the system. Such a loading, typically caused by impact conditions, may be highly irregular in terms of amplitude, waveform, and repetition (Figure 1.4.7), but normally it is possible to extract practically uniform critical events from the loading history for purposes of future design and life prediction. For most practical purposes, this plot represents aperiodic motion, in which the important quantities are the maximum and average large amplitudes and the projected total repetitions (in this case, at the rate of about 1000 per day) over the design life of the structure. The small-amplitude transient vibrations associated with the large events are likely to be negligible here in terms of dynamic behavior as well as fatigue damage, although the relatively large number of small oscillations may cause concern in some cases. Random vibrations are difficult to deal with analytically. Numerical methods involving computers are advantageous to obtain response (or shock) spectrums of a system, assuming key parameters and simple models of nonharmonic excitations such as impulsive forces and force step functions. Because the

FIGURE 1.4.7 Strain-time history at one strain-gage location on a steel bridge caused by two trucks moving in opposite directions. (A) Garbage truck in the near lane; (B) tractor trailer in the far lane. Weights unknown. (Data courtesy Mark J. Fleming, University of Wisconsin–Madison.) © 2005 by CRC Press LLC

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FIGURE 1.4.8 Simply supported beam in two modes of vibration.

maximum transient response is relatively insensitive to damping, an undamped system is useful in modeling response spectrums. Experimental techniques are needed to verify the analytical predictions, especially when the behavior of a multiple-degree-of-freedom system is determined from the response spectrum of a single-degree-of-freedom system.

Multiple-Degree-of-Freedom Systems. Modal Analysis The analysis of a system with more than one degree of freedom requires an independent coordinate for each degree of freedom to describe the configurations. Thus, an n-degree-of-freedom system has n natural frequencies and n normal modes of vibration. Complex systems can be classified as (1) discrete and lumped-parameter systems with finite numbers of degrees of freedom; or (2) continuous elastic bodies of distributed mass with infinite number of degrees of freedom (in theory). A common example of the latter is a vibrating beam, with the first two modes of vibration shown in Figure 1.4.8. Each nodal point is a point of zero deflection. Usually the fundamental natural frequency (the lowest) is the most important, and only the lowest few frequencies are considered in practice. A system’s harmonic vibrations are its principal modes. The system can vibrate nonharmonically in many ways. Periodic motion of complex wave form can be analyzed as a combination of principal-mode vibrations. The classical method of mathematical solution and the experimental techniques become increasingly cumbersome and sometimes inaccurate for a system of more than a few degrees of freedom. The recent emergence of sophisticated numerical (finite element; Figure 1.4.9) and experimental (electro-optics) techniques has resulted in significant progress in this area. The synergistic aspects of several new methods are especially remarkable. For example, damage caused by vibrations can significantly affect a system’s own modal behavior and, consequently, the rate of damage evolution. Such nonlinear changes of a system can now be investigated and eventually predicted by the hybrid applications of computerized numerical methods; fatigue and fracture mechanics (Section 1.6); and high-speed, noncontacting, full-field vibration and stress imaging (Section 1.4, “Vibration-Measuring Instruments,” and Section 1.5, “Experimental Stress Analysis and Mechanical Testing”). These enhance the already powerful modern methods of modal analysis for accurately describing the response of multiple-degree-of-freedom systems.

Vibration-Measuring Instruments Many kinds of instruments can be used for the experimental investigation of vibrating systems. They range from simple, inexpensive devices to sophisticated electro-optics with lasers or infrared detectors, with the list still expanding in many areas. The basic quantities of interest regarding a vibrating system are the displacement, velocity, acceleration, and frequency. A typical sensor (or pickup or transducer) for determining these is the piezoelectric accelerometer, which is attached to the vibrating machine or structure to be analyzed. The complete setup normally includes amplifiers, frequency analyzer, oscilloscope, and recorders. An instrumented impact hammer may be used to provide well-defined impulse excitation to determine the natural frequencies of structures. The frequency analyzer can display the accelerometer output in the time or the frequency domain. Other kinds of devices used for vibration sensing include seismic spring-mass systems, electrical-resistance strain gages, and electromagnetic transducers. Care must be exercised in matching a transducer to the task at hand because reliable data can be obtained only if the transducer has a “flat-response” frequency region for the measurements of interest. © 2005 by CRC Press LLC

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FIGURE 1.4.9 Modal analysis of a vibrating plate. (Photo courtesy David T. Corr, University of Wisconsin–Madison.)

For example, electromagnetic vibrometers (or seismometers) are low-frequency transducers that have low natural frequency compared to the frequency of the motion to be measured. At the other extreme, piezoelectric accelerometers are designed to have higher natural frequency than the frequency to be measured. It is also important to use transducers of negligible mass compared to the mass of the vibrating system being measured. Very small, light-weight accelerometers are available to satisfy this condition in many cases. In some situations, however, only noncontacting means of motion measurement provide satisfactory results. Optical techniques are prominent in this area, offering several advantages besides the noncontacting measurement capability. They can be full-field techniques, which means that data may be obtained rapidly from many points on a body using one instrument. They have excellent resolution and precision, and some of them are easy to use. Three kinds of optical instruments are distinguished here for vibratory system analysis, depending on the primary quantity measured: 1. Displacement measurement. Holography and speckle pattern imaging have excellent resolution, but they are adversely affected by unstable measuring conditions. They are most useful in laboratory applications. 2. Velocity measurement. Laser Doppler systems provide time-resolved, accelerometer-like measurements. They are relatively unaffected by measuring conditions, and are simple and rugged enough to use in the laboratory or in the field. Several important capabilities of such a vibration pattern imaging system are worth mentioning (Color Figure 3 through Color Figure 7): • Noncontacting; the structure’s response is not affected by the instrumentation; applicable in some hazardous environments (hot structures, etc.), and short or long range (over 200 m) on natural surfaces • Single-point or full-field data acquisition at high resolution from areas of 0.5 × 0.5 mm to 8 × 8 m; up to 500 individual points can be programmed • Wide frequency range; 0 to 100 kHz (for example, Figure 1.4.10) © 2005 by CRC Press LLC

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FIGURE 1.4.10 Laser-based, noncontacting vibration analysis of a point on a car door. (Data courtesy of Ometron Inc., Sterling, VA.)

FIGURE 1.4.11 Schematic of modal analysis of a jet engine turbine blade by thermal imaging of the stress field caused by complex vibration. For sample data, see Color Figure 8.

• • • • • •

Sensitivity to a wide range of vibration velocities; 0.005 to 1000 mm/sec Large depth of focus; ±3 m at 10-m working distance Node spacing down to a few millimeters can be resolved Resolution of small displacements, down to the wavelength of the laser source (typically, ≈1 Å) Safe, class II laser system; 60. Yielding would occur first at smaller values of L/r. Ratios of 200 or higher indicate very slender members that cannot support large compressive loads. Several common end conditions of slender columns are shown schematically in Figure 1.5.29. Secant Formula Real columns are not perfectly straight and homogeneous and are likely to be loaded eccentrically. Such columns first bend and deflect laterally, rather than buckle suddenly. The maximum elastic compressive stress in this case is caused by the axial and bending loads and is calculated for small deflections from the secant formula, σ max =

P  ec  L 1 + sec A  r 2  2r

P   EA  

(1.5.48)

where e is the eccentricity of the load P (distance from the neutral axis of area A) and c is measured from the neutral axis to the outer layer of the column where σmax occurs. The load and stress are nonlinearly related; if several loads are on a column, the loads should be properly combined first before using the secant formula, rather than linearly superposing several individually determined stresses. Similarly, factors of safety should be applied to the resultant load. Inelastic Buckling For columns that may yield before buckling elastically, the generalized Euler equation, also called the Engesser equation, is appropriate. This involves substituting the tangent modulus ET (tangent to the stress–strain curve) for the elastic modulus E in the Euler equation: © 2005 by CRC Press LLC

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σ cr =

π 2 ET

(k L r )2

(1.5.49)

Note that ET must be valid for the stress σcr , but ET is dependent on stress when the deformations are not entirely elastic. Thermal or plastic-strain events may even alter the stress–strain curve of the material, thereby further changing ET . Thus, Equation (1.5.49) should be used with caution in a trial-and-error procedure.

Impact Loading A mass impacting another object causes deformations that depend on the relative velocity between them. The simplest model for such an event is a mass falling on a spring. The maximum dynamic deformation d of a linearly responding spring is related to the static deformation dst (the deformation caused by a weight W applied slowly) by a factor that depends on h, the height of free fall from a static position.  2h  d = dst 1 + 1 + dst  

(1.5.50)

The dynamic and static stresses are related in a similar way:  2h  σ = σ st 1 + 1 + dst  

(1.5.51)

The quantity in parentheses is called the impact factor and shows the magnification of deflection or stress in impacts involving free fall. Note that the real impact factor is somewhat smaller than what is indicated here because some energy is always dissipated by friction during the fall and deceleration of the body. This includes internal friction during plastic flow at the points of contact between the bodies. Other small errors may result from neglecting the mass and possible inelasticity of the spring. A special value of the impact factor is worth remembering. When the load is applied suddenly without a prior free fall, h = 0, and d = 2 dst

and σ = 2σ st

This means that the minimum impact factor is about two; it is likely to be larger than two, causing perhaps a “bottoming out” of the spring, or permanent damage somewhere in the structure or the payload supported by the spring.

Combined Stresses Combinations of different kinds of loads on a member are common. The resultant states of stress at various points of interest can be determined by superposition if the material does not yield. The threedimensional visualization and correct modeling of such a problem are typically the most difficult parts of the solution, followed by routine calculations of the stress components and resultants. No new methods of analysis are needed here. The approach is to sketch an infinitesimal cube at each critical point in the member and determine the individual stresses (magnitudes and signs) acting on that element, generally caused by axial, shear, bending, torsion, and internal pressure loading. This is illustrated for a case of medium complexity in Figure 1.5.30. Consider a solid circular rod of radius R, fixed at z = 0 (in the xy plane), and loaded by two forces at point B of a rigid arm. Set up the stress analysis at point A (–R, 0, 0), assuming that no stress concentration is present at the wall fixture of the rod (Figure 1.5.30a). © 2005 by CRC Press LLC

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FIGURE 1.5.30 Illustration of stress analysis for combined axial, shear, bending, and torsion loading.

First, the equivalent loading at the origin 0 is determined (Figure 1.5.30b). This can be done most accurately in vector form. The individual stresses at point A are set up in the subdiagram (c). Check that each stress (even in symbolic form) has the proper units of force per area. The net normal force in this case is σ1 + σ2, and the net shear stress is τ1 + τ2. The state of stress is different at other points in the member. Note that some of the stresses at a point could have different signs, reducing the resultant stress at that location. Such is the case at a point C diametrically opposite to point A in the present example (R, 0, 0), where the axial load F and My generate normal stresses of opposite signs. This shows the importance of proper modeling and setting up a problem of combined loads before finding the numerical solution.

Pressure Vessels Pressure vessels, which are an important special category of combined stresses, are made in different shapes and sizes (Figure 1.5.31 and Color Figure 10) and are used in diverse applications. The applications range from air receivers in gasoline stations to nuclear reactors in submarines to heat exchangers in refineries. The required thicknesses for some commonly encountered pressure vessel components depend on the geometry as follows. Cylindrical Shells The force per unit length in the hoop (tangential) direction, Nt , required to contain a given pressure p in a cylindrical shell is obtained by taking a free-body diagram (Figure 1.5.32a) of the cross section. Assuming the thickness t to be much smaller than the radius R and summing forces in the vertical direction gives 2 Nt L = 2 RLp or Nt = pR © 2005 by CRC Press LLC

(1.5.52)

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FIGURE 1.5.31 Various pressure vessels. (Photos courtesy Nooter Corp., St. Louis, MO.)

The corresponding hoop stress is σt = pR/t. The longitudinal force per unit length, Nx , in the cylinder due to pressure is obtained by summing forces in the axial direction (Figure 1.5.32b), © 2005 by CRC Press LLC

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FIGURE 1.5.32 Analysis of cylindrical pressure vessels.

2 πRN x = πR 2 p or Nx = p R 2

(1.5.53)

The corresponding axial stress is σx = pR/2t. It is seen that the magnitude of Nt (and σt ) is twice that of Nx (and σx). If S is the allowable stress and t is the required minimum thickness, t = pR S

(1.5.54)

Spherical Shells A free-body diagram of the spherical cross section is shown in Figure 1.5.33. Summation of forces gives t = p R 2S

(1.5.55)

Example 10 Determine the required thickness of the shell and heads of the air receiver shown in Figure 1.5.34 if p = 100 psi and S = 15,000 psi. Solution. From Equation (1.5.54), the required thickness for the cylindrical shell is t = 100 × 18 15, 000 = 0.12 in. The required head thickness from Equation 1.5.55 is t = 100 × 18 2 × 15, 000 = 0.06 in.

FIGURE 1.5.33 Analysis of spherical pressure vessels.

FIGURE 1.5.34 Sketch of a pressure vessel. © 2005 by CRC Press LLC

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FIGURE 1.5.35 Analysis of conical shells.

Conical Shells The governing equations for the longitudinal and circumferential forces in a conical shell (Figure 1.5.35a) due to internal pressure are similar to Equation (1.5.52) and Equation (1.5.53) for cylindrical shells, with the radius taken normal to the surface. Thus, Nt = pr cosα

(1.5.56)

N x = pr 2 cosα

(1.5.57)

where α is half the apex angle of the cone. The junction between a conical and cylindrical shell, Figure 1.5.35b, is subjected to an additional force, H, in the horizontal direction due to internal pressure. The magnitude of this additional force per unit length can be obtained by taking a free-body diagram as shown in Figure 1.5.35b: H = N x sin α

(1.5.58)

A ring is usually provided at the cone-to-cylinder junction to carry the horizontal force H. The required area A of the ring is obtained from Figure 1.5.35c as H 2r = 2 AS or

(

A = Hr S = ( N x sin α ) r S = pr 2 sin α

) (2S cos α)

(1.5.59)

The stress in the ring is compressive at the large end of the cone and tensile at the small end due to internal pressure. This stress may reverse in direction due to other loading conditions such as weight of contents and end loads on the cone due to wind and earthquake loads. Example 11 Determine the required thickness of the two cylindrical shells and cone shown in Figure 1.5.36a due to an internal pressure of 200 psi. Calculate the area of the rings required at the junctions. Assume the allowable stress to be 20 ksi in tension and 10 ksi in compression. Solution. From Equation (1.5.54), the thickness of the large cylinder is t = 200 × 60 20, 000 = 0.60 in. The thickness of the small cylinder is t = 200 × 30 20, 000 = 0.30 in.

© 2005 by CRC Press LLC

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FIGURE 1.5.36 Cylindrical shells with cone connection.

The thickness of the cone is obtained from Equation (1.5.56) as t = 200 × 60 (20, 000 × cos 30°) = 0.69 in. The required area of the ring at the large end of the cone is obtained from Equation (1.5.59) using the allowable compressive stress of 10 ksi: A = 200 × 60 2 × sin 30° (2 × 10, 000 × cos 30°) = 20.78 in.2 The required area of the ring at the small end of the cone is obtained from Equation (1.5.59) using the allowable tensile stress of 20 ksi: A = 200 × 30 2 × sin 30° (2 × 20, 000 × cos 30°) = 2.60 in.2 The rings at the junction are incorporated in a number of ways such as those shown in Figure 1.5.36b. Nozzle Reinforcement Reinforcements around openings in pressure vessels are needed to minimize the local stress in the area of the opening. The calculation for the needed reinforcement around an opening is based on the concept that pressure in a given area of a vessel is contained by the material in the vessel wall surrounding the pressure. Thus, in Figure 1.5.37, if one takes an infinitesimal length dL along the cylinder, the force caused by the pressure within this length is given by the quantity pR dL. The force in the corresponding vessel wall is given by St dL. Equating these two quantities results in the expression t = pR/S that is given earlier as Equation (1.5.54). Similarly for the nozzle in Figure 1.5.37, T = pr/S. The intersection of the nozzle with the cylinder results in an opening where the pressure in area ABCD is not contained by any material. Accordingly, an additional area must be supplied in the vicinity of the opening to prevent overstress of the vessel. The required area A is determined from Figure 1.5.37 as A = p Rr S Substituting Equation (1.5.54) into this expression gives A = tr

(1.5.60)

This equation indicates that the needed additional area is equal to the removed area of the vessel wall.

© 2005 by CRC Press LLC

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FIGURE 1.5.37 Nozzle reinforcement.

Creep–Fatigue of Boilers and Pressure Vessels See Figure 1.6.27 in Section 1.6, “Fatigue.” Composite Materials for Pressure Vessels Some pressure vessels can be made of fibrous composite materials with high strength-to-weight ratios. The advantages of using such a material are remarkable in the case of a tubular vessel, where the hoop stress is twice the longitudinal stress, if the fiber quantities and orientations are optimally designed to resist the applied load caused by internal pressure. Simplistically (because a basic element of a composite is strong along the fibers and weak perpendicular to the fibers), this requires twice as many fibers oriented circumferentially as axially. In practice, fibers are commonly laid at ± (a winding angle) at which the hoop and axial stress components are equal, to efficiently create an optimized configuration. Example 12 Determine the minimum weight of the tube portion of a thin-walled cylindrical pressure vessel of r = 8 in. (20 mm),
= 10 ft (3.05 m), p = 8 ksi (55 MPa); t = ? Assume using a typical graphite/epoxy composite of 60% fibers by volume with allowable tensile stress σy = 300 ksi (207 MPa) at 0.058 lb/in.3 (1600 kg/m3). For comparison, consider a steel of σy = 200 ksi (138 MPa) at 0.285 lb/in.3 (7890 kg/m3). Solution. Composite: σy = pr/t, t = 0.213 in. (5.41 mm) for circumferential plies and σy = pr/2t, t = 0.107 in. (2.72 mm) for axial plies Total minimum wall thickness: 0.32 in. (8.13 mm) Total material in tube: 112 lb (50.8 kg) Steel: σy = pr/t, t = 0.32 in. (8.13 mm) Total material in tube: 550 lb (249 kg) = 4.9 (composite material) Note that additional considerations exist in practice, such as cost and potential problems in making adequate connections to the tube.

Thick-Walled Cylinders and Interference Fits The previous section provided design considerations and analysis of thin-walled pressure vessels. However, some applications require thick-walled pressure vessel analysis, as is the case when r/t > 10, for example, when small diameter tubes are used to transport high pressure fluids. This section covers thickwalled pressure vessel analysis and a special case for interference fits. Referring to Figure 1.5.38, the hoop and radial stress components (dependent on pressure and geometry) are shown without derivation in Equation (1.5.61a) and Equation (1.5.61b). Internal Presssure

External Pressure

Component Component     2 2 pi Ri Ro po Ro2 Ri2 = + + 1 1 Hoop Ro2 Ri2 r2 Ro2 Ri2 r2

© 2005 by CRC Press LLC

(1.5.61a)

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FIGURE 1.5.38 Thick-walled cylinder analysis nomenclature.

σ Radial =

pi Ri2  Ro2  po Ro2  Ri2  − 1 1− − Ro2 − Ri2  r 2  Ro2 − Ri2  r 2 

(1.5.61b)

These equations can be simplified by setting the internal or external pressure to zero. As Ri approaches Ro , with t = Ro – Ri the thin-wall pressure vessel equation, σHoop = pR/t (where σHoop = σt as stated in the pressure vessel section), is apparent. A thick-walled cylinder with end caps exerts an axial force on the caps, thus applying a longitudinal stress component on the cylinder as given by

σz =

pi Ri2 − po Ro2 Ro2 − Ri2

(1.5.62)

The interference fit analysis is a special case that can be used for nesting pressure vessels (for example, to increase a pressure vessel’s capacity) and for use as an assembly method (for example, to join an electric motor core to its output shaft). The interference fit equations are based on Equation (1.5.61a), Equation (1.5.61b), and Figure 1.5.39. An interface pressure p is generated by expanding the outer cylinder’s inside diameter over the larger inner cylinder’s outside diameter by a radial interference δ. The first step in formulating the interference fit analysis is determining the hoop stress equations in the assembled inner and outer cylinders. These hoop stresses are calculated by recognizing that Figure 1.5.38 and Equation (1.5.61a) and Equation (1.5.61b) can be reduced by using p, the interface pressure, as the internal pressure on the outer cylinder and the external pressure on the inner cylinder (the internal and external pressures on the assembly are equal to zero). b2 + a2 b2 − a2

(1.5.63a)

c 2 + b2 c 2 − b2

(1.5.63b)

σ HoopInner = − p

σ HoopOuter = p © 2005 by CRC Press LLC

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FIGURE 1.5.39 Interference fit schematic and analysis notations.

To solve Equation (1.5.63a) and Equation (1.5.63b), the interface pressure p must be determined. Referring to Figure 1.5.38 and using the strain relationships that result from the change in the interface diameters (from the change in circumference), the interferences are expressed as functions of radius,  pb  c 2 + b 2 + ν02   2 2 Eo  c − b 

(1.5.64a)

 − pb  b 2 + a 2 − νi2   2 2 Ei  b − a 

(1.5.64b)

δo =

δi =

Recognizing that δ = δo – δi , the interference fit equation is stated as δ=

  pb  b 2 + a 2 pb  c 2 + b 2 + ν02  −  2 2 − νi2   2 2 Eo  c − b   Ei  b − a

(1.5.65)

Given the geometry and material properties, the interface pressure p is calculated from Equation (1.5.65). This equation is useable for an assembly of two different materials, as is the case of a steel pin pressed into an aluminum housing. However, further simplification of Equation (1.5.65) is possible if both cylinders are of the same material; with Ei = Eo and νi = νo ,

(

)(

 2 2 2 2 Eδ c − b b − a  p= b  2b 2 c 2 − a 2

(

)

)  

(1.5.66)

With the interface pressure p obtained, the hoop stresses at the interface can be found using Equation (1.5.63a) and Equation (1.5.63b). At this point, the design engineer should verify that the outer cylinder does not yield at this stress. Yielding of the outer cylinder leads to a reduced interference fit that may be detrimental to the assembly. If the stress at the interface is deemed too high, the assembly geometries should be iterated and the problem solved again. This process is to be repeated until the margins of safety are acceptable in the design. The stress distributions across the thickness in the hoop and radial directions on the inner and outer cylinders are solved using Equation (1.5.61a) and Equation (1.5.61b), by recognizing the outer cylinder external pressure po = 0 and pi = p, and the inner cylinder internal pressure po = p and pi = 0. The interference © 2005 by CRC Press LLC

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FIGURE 1.5.40 End-stop assembly.

fit equations presented previously are valid under the assumption that the inner and outer cylinders are of the same length. In the case of a hub fit over a shaft, a stress concentration should be employed. It is worthwhile to note that a person must be able to assemble such a fit. This is usually accomplished by heating the outer cylinder and cooling the inner cylinder to develop a suitable clearance for assembly. It is important to understand the limits to which the cylinders can be heated and cooled with the available equipment and, possibly more importantly, without altering their material properties. Neglecting the practicality of the interference fit process often leads to unnecessary assembly problems. Example 13 A mechanical stop is attached to a linear ballscrew assembly by pinning the stop to the screw as shown in Figure 1.5.40. The screw is recessed below the top surface of the end-stop. The minimum wall section between the pin outer edge and the top surface of the screw is 0.011 in. (the result of a complex and partially flawed design process that will not be discussed here). The pin of diameter 0.1250 in. is to be inserted into a reamed hole of 0.1240 in. Calculate the hoop stress in the “outer cylinder” for the screw. Solution. Using Equation (1.5.65), a = 0 in., b = 0.1250/2 in., and c = (0.1240 + 2 × 0.011)/2 in. The maximum interference fit is 0.0010 in. by completion of the tolerance analysis. Assuming both parts are steel (E = 30 × 106 psi), the interface pressure is

(30 ×10 )(0.0005)  (0.0730 − 0.0625 )(0.0625 − 0 )  = 32,000 psi (interface pressure) (0.0625)  2(0.0625 )(0.0730 − 0)  6

p=

© 2005 by CRC Press LLC

2

2

2

2

2

2

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FIGURE 1.5.41 Assembled ballscrew end-stop with fractures. (Photo courtesy Moog Inc., Torrance, CA.)

With the interface pressure (also σRadial at b) solved, the hoop stress in the outer cylinder is calculated by using Equation (1.5.63b): σ Hoop

Outer

= 32, 000

0.07302 + 0.06252 = 207,700 psi 0.07302 − 0.06252

The engineer should recognize that the hoop stress calculated here is above the yield strength and ultimate strength of most steels. The photograph of the assembly in Figure 1.5.41 shows the fractured thin-walled section. This example illustrates that the engineer must check even seemingly benign loadings and geometries with respect to relevant material properties to assure structural integrity.

Experimental Stress Analysis and Mechanical Testing Experimental stress analysis is based mostly on the measurement of strains, which may be transformed into stresses. A variety of techniques is available to measure strains. A few of these are described here. Properties of Strain-Measuring Systems Strain-measuring systems are based on a variety of sensors, including mechanical, optical, and electrical devices. Each has some special advantages but can usually be adapted for other needs as well. No one system is entirely satisfactory for all practical requirements, so it is necessary to optimize the gage system to each problem according to a set of desirable characteristics. Some of the common characteristics used to evaluate the system’s adequacy for a typical application are • The calibration constant for the gage should be stable; it should not vary with time, temperature, or other environmental factors. • The gage should be able to measure strains with an accuracy of ±1 µε over a strain range of ±10%. • The gage size, i.e., the gage length l0 and width w0, should be small so that strain at a point is approximated with small error. • The response of the gage, largely controlled by its inertia, should be sufficient to permit recording of dynamic strains with frequency components exceeding 100 kHz. • The gage system should permit on-location or remote readout. • Both the gage and the associated auxiliary equipment should be inexpensive. • The gage system should be easy to install and operate. • The gage should exhibit a linear response to strain over a wide range. © 2005 by CRC Press LLC

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Three of these basic characteristics deserve further mention here: the gage length l0; gage sensitivity; and range of the strain gage. The gage length is often the most important because in nonlinear strain fields the error will depend on the gage length. Sensitivity is the smallest value of strain that can be read on the scale associated with the strain gage and should not be mistaken for accuracy or precision. The sensitivity chosen should not be higher than necessary because it needlessly increases the complexity of the measuring method and introduces new problems. The range of the strain gage refers to the maximum value of strain that can be recorded. Because the range and sensitivity of the gage are interrelated, it is often necessary to compromise between the two for optimal performance of both. Various compromises have resulted in two main kinds of strain gages: extensometers and electrical strain gages. Although electrical strain gage systems are numerous, only electrical-resistance strain gages will be considered here. Extensometers Various extensometers involving mechanical, electrical, magnetic, or optical devices are used in material test systems. A typical extensometer (Figure 1.5.42) is used in the conventional tensile test in which the stress-strain diagram is recorded. This kind of extensometer is attached to the specimen by knife edges and spring clips. Electrical-resistance strain gages are attached to the cross-flexural member and provide the strain output. The main advantage of extensometers is that they can be reused and recalibrated after each test. The disadvantages are that they are much larger and more expensive than electrical-resistance strain gages.

FIGURE 1.5.42 Extensometer attached to a tensile specimen.

Electrical-Resistance Strain Gages The electrical-resistance strain gage fulfills most of the requirements of an optimum system and is widely used for experimental stress analysis. This gage consists of a metal-foil grid bonded to a polymer backing (Figure 1.5.43). A Wheatstone bridge is often used to enhance its ability to measure changes in resistance. As a specimen is deformed, the strain is transmitted to the grid, which has a current applied to it. The change in resistance of the grid is converted to a voltage signal output of the Wheatstone bridge. The basic equation used with this system is

FIGURE 1.5.43 Model of metal-foil strain gages.

∆R = SA ε R

(1.5.67)

where R is the resistance of the gage; ε is the applied strain; and SA is the sensitivity, or gage factor, of the metallic alloy used in the conductor. The most commonly used alloy is a copper–nickel alloy called Advance, for which the sensitivity is 2.1. Electrical-Resistance Strain Gage Mounting Methods For precision strain measurements, the correct adhesive and proper mounting procedures must be employed. The adhesive serves a vital function in the strain-measuring system: it must transmit the strain from the specimen to the sensing element without distortion. Bonding a strain gage to a specimen is one © 2005 by CRC Press LLC

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of the most critical steps in the entire process of measuring strain with an electric-resistance strain gage. When mounting a strain gage, it is important to prepare the surface of the component where the gage is to be located carefully. This includes sanding, degreasing, etching, cleaning, and, finally, neutralizing the surface where the gage is to be mounted. Next, the surface is marked to allow accurate orientation of the strain gage. The gage is then put in place and held with tape while the adhesive is allowed to dry. Several of the adhesive systems commonly used for this are epoxy cements, cyanoacrylate cement, polyester adhesives, and ceramic adhesives. Once the adhesive has been placed, the drying process becomes vitally important because it can cause residual stresses in the grid work of the gage, which could influence the output. After allowing the adhesive to dry, the cure must be tested to ensure complete drying. Failure to do so will affect the stability of the gage and the accuracy of the output. The cure state of the adhesive can be tested by various resistance tests. Also, the bonded surface is inspected to determine if any voids are present between the gage and the specimen due to bubbling of the adhesive. After the bonding process is complete, the lead wires are attached from the soldering tabs of the gage to an anchor terminal, which is also bonded to the test specimen. This anchoring terminal is used to protect the fragile metal-foil gages. Finally, wires are soldered from this anchor terminal to the instrumentation used to monitor the resistance changes. Gage Sensitivities and Gage Factor The electrical-resistance strain gage has a sensitivity to axial and transverse strain. The magnitude of the transverse strain transmitted to the grid depends on a number of factors, including the thickness and elastic modulus of the adhesive; the carrier material; the grid material; and the width-to-thickness ratio of the axial segments of the grid. Sometimes it is necessary to calculate the true value of strain that includes all contributions from εa =

(∆R R) Sg

1 − v0 K t 1 + K t (ε t ε a )

(1.5.68)

where εa = the normal strain along the axial direction of the gage εt = the normal strain along the transverse direction of the gage v0 = 0.285 is Poisson’s ratio for the calibration beam Kt = the transverse-sensitivity factor of the gage The strain gage sensitivity factor, Sg , is a calibration constant provided by the manufacturer. By using Equation (1.5.67) and Equation (1.5.68), the percent error involved in neglecting the transverse sensitivity can be calculated. These errors can be significant for large values of Kt and εt /εa , so it may be necessary to correct for the transverse sensitivity of the gage (Figure 1.5.44).

FIGURE 1.5.44 Error as a function of transverse-sensitivity factor with the biaxial strain ratio as a parameter.

© 2005 by CRC Press LLC

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FIGURE 1.5.45 Three gage elements placed at arbitrary angles relative to the x and y axes.

Strain Analysis Methods Electrical-resistance strain gages are normally employed on the free surface of a specimen to establish the stress at a particular point on this surface. In general, it is necessary to measure three strains at a point to define the stress or the strain field completely. For this general case in which nothing is known about the stress field or its directions before experimental analysis, three-element rosettes are required to establish the stress field. This is accomplished by using the three-element gage with orientations at arbitrary angles, as shown in Figure 1.5.45. Using this setup, the strains εx , εy , and γxy can be determined. These values can be used to determine the principal strains and principal directions:

(

)

(ε

(

)

(ε

ε1 =

1 1 ε xx + ε yy + 2 2

ε2 =

1 1 ε xx + ε yy − 2 2 tan 2φ =

xx

− ε yy

)

xx

− ε yy

)

2

2

+ γ 2xy + γ 2xy

(1.5.69)

γ xy ε xx − ε yy

where φ is the angle between the principal axis (σ1) and the x axis. The principal stresses can be computed using the principal strains: σ1 =

E (ε + vε 2 ) 1 − v2 1

(1.5.70)

E σ2 = (ε + vε1 ) 1 − v2 2 These expressions give the complete state of stress because the principal directions are known from Equation (1.5.69). Optical Methods of Strain Analysis Moiré Method of Strain Analysis. The moiré technique depends on an optical phenomenon of fringes caused by relative displacement of two sets of arrays of lines. The arrays used to produce the fringes may be a series of straight parallel lines; a series of radial lines emanating from a point; a series of concentric circles; or a pattern of dots. The straight parallel line “grids” are used most often for strain analysis work and consist of equal width lines with opaque spacing of the same width between them. These straight parallel lines are spaced in a “grating” scheme of typically 50 to 1000 lines per inch for moiré work.

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In the cross-grid system of two perpendicular line arrays, the grid placed on the specimen is referred to as the model grid. The second grid is referred to as the reference grid and is overlaid on top of the model grid. Often a thin layer of oil or some other low-friction substance is placed between the model grid and the reference grid to keep them in contact while attempting to minimize the transmission of strains from the model to the reference grid. To obtain a moiré fringe pattern, the grids are first aligned on the unloaded model so that no pattern is present. The model is loaded and light is transmitted through the two grids. Strain displacement is observed in the model grid while the reference grid remains unchanged. A moiré fringe pattern is formed each time the model grating undergoes a deformation in the primary direction equal to the pitch p of the reference grating. For a unit gage length, ∆L = np, where ∆L is the change in length per unit length, p is the pitch of the reference grating, and n is the number of fringes in the unit gage length. In order to calculate εx , εy , and γxy , two sets of gratings must be applied in perpendicular directions. Then displacements u and v (displacements in the x and y directions, respectively) can be established and the Cartesian strain components can be calculated from slopes of the displacement surfaces: εxx = ∂u/∂x, εyy = ∂v/∂y, and γxy = ∂v/∂x + ∂u/∂y. The displacement gradients in the z direction, ∂w/∂x and ∂w/∂y, have been neglected here because they are not considered in moiré analysis of in-plane deformation fields. Photoelasticity. The method of photoelasticity is based on the physical behavior of transparent, noncrystalline, optically isotropic materials that exhibit optically anisotropic characteristics, referred to as temporary double refraction, while they are stressed. To observe and analyze these fringe patterns, a device called a polariscope is used. Two kinds of polariscope are common: the plane polariscope and the circular polariscope. The plane polariscope (Figure 1.5.46) consists of a light source, two polarizing elements, and the model. The axes of the two polarizing elements are oriented at a 90° angle from each other. If the specimen is not stressed, no light passes through the analyzer and a dark field is observed. If the model is stressed, two sets of fringes — isoclinics and isochromatics — will be obtained. Black isoclinic fringe patterns are the loci of points where the principal-stress directions coincide with the axis of the polarizer. These fringe patterns are used to determine the principal stress directions at all points of a photoelastic model. When

FIGURE 1.5.46 Schematic of a stressed photoelastic model in a plane polariscope. © 2005 by CRC Press LLC

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FIGURE 1.5.47 Schematic of a stressed photoelastic model in a circular polariscope.

the principal stress difference is zero (n = 0) or sufficient to produce an integral number of wavelengths of retardation (n = 1, 2, 3, …), the intensity of light emerging from the analyzer is zero. This condition for extinction gives a second fringe pattern, called isochromatics, where the fringes are the loci of points exhibiting the same order of extinction (n = 0, 1, 2, 3, …): n=N=

h (σ − σ 2 ) fσ 1

(1.5.71)

where N is the isochromatic fringe order. The order of extinction n depends on the principal stress difference (σ1 – σ2), the thickness h of the model, and the material fringe value fσ . When monochromatic light is used, the isochromatic fringes appear as dark bands. When white light is used, the isochromatic fringes appear as a series of colored bands. Black fringes appear in this case only where the principal stress difference is zero. A circular polariscope is a plane polariscope with two additional polarizing plates, called quarter-wave plates, added between the model and the original polarizing plates (Figure 1.5.47). The two quarter-wave plates are made of a permanently doubly refracting material. The circular polariscope is used to eliminate the isoclinic fringes while maintaining the isochromatic fringes. To accomplish this, monochromatic light must be used because the quarter-wave plates are designed for a specific wavelength of light. For the dark-field arrangement shown, no light is passed through the polariscope when the model is unstressed. A light-field arrangement is achieved by rotating the analyzer 90°. The advantage of using light- and dark-field analysis is that twice as much data are obtained for the whole-field determination of σ1 – σ2. If a dark-field arrangement is used, n and N still coincide, as in Equation (1.5.71). If a light-field arrangement is used, they are not coincident. In this case, Equation (1.5.71) becomes N= © 2005 by CRC Press LLC

1 h +n= (σ − σ 2 ) 2 fσ 1

n = 0, 1, 2, 3, …

(1.5.72)

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By determining the isoclinic fringes and the isochromatic fringes, the principal-stress directions and the principal-stress difference can be obtained. In order to obtain the individual principal stresses, a stress separation technique would need to be employed. The advantages of the photoelastic method are that it allows a full-field stress analysis and it makes it possible to determine the magnitude and direction of the principal stresses. The disadvantages are that it requires a plastic model of the actual component and it takes a considerable effort to separate the principal stresses. Major advances have been made in this area recently, and a variety of special-purpose equipment is available to solve difficult engineering problems efficiently. Readers are encouraged to visit Internet sites such as www.stressphotonics.com for more information. Thermoelastic Stress Analysis. Modern thermoelastic stress analysis (TSA) employs advanced differential thermography (or AC thermography) methods based on dynamic thermoelasticity and focal-plane-array infrared equipment capable of rapidly measuring small temperature changes (down to 0.001°C) caused by destructive or nondestructive alternating stresses. Stress resolutions comparable to those of strain gages can be achieved in a large variety of materials. The digitally stored data can be processed in nearreal time to determine the gradient stress fields and related important quantities (such as combinedmode stress intensity factors) in complex components and structures, with no upper limit in temperature. The efficient, user-friendly methods can be applied in the laboratory, field, vehicles, and structures such as bicycles, automobiles, aircraft, surgical implants, welded bridges, and microelectronics. Optimum design, rapid prototyping, failure analysis, life prediction, and rationally accelerated testing can be facilitated with the new TSA methods (Color Figure 8 and Color Figure 11 through Color Figure 14). Major advances have been made in this area recently, and a variety of special-purpose equipment is available to solve difficult engineering problems efficiently. Readers are encouraged to visit Internet sites, such as www.stressphotonics.com, for more information. Brittle Coatings. If a coating is applied to a specimen that is thin in comparison with the thickness of the specimen, then the strains developed at the surface of the specimen are transmitted without significant change to the coating. This is the basis of the brittle coating method of stress analysis. The two kinds of coatings available are resin-based and ceramic-based coatings. The ceramic-based coatings are seldom used due to the high application temperatures (950 to 1100°F) required. The coatings are sprayed on the component until a layer approximately 0.003 to 0.010 in. thick has accumulated. It is also necessary to spray calibration bars with the coating at the same time in order to obtain the threshold strain at which the coating will crack. These calibration bars are tested in a cantilever apparatus and the threshold strain is calculated using the flexure formula and Hooke’s law. Once the threshold strain is known and the actual specimen has been tested, the principal stress perpendicular to the crack can be determined by using Hooke’s law. The procedure is to load the component, apply the coating, and then quickly release the loading in steps to observe any cracks. The main advantages of this method are that magnitude and direction of the principal strains can be quickly obtained and that the coating is applied directly to the component. This also allows a quick analysis of where the maximum stress regions are located so that a better method can be used to obtain more accurate results. The main disadvantage is that the coatings are very sensitive to ambient temperature and might not have sufficiently uniform thickness. Mechanical Testing Standards. Many engineering societies have adopted mechanical testing standards; the most widely accepted are the standards published by the American Society for Testing and Materials. Standards for many engineering materials and mechanical tests (tension, compression, fatigue, plane strain fracture toughness, etc.) are available in the Annual Book of ASTM Standards. Open-Loop Testing Machines. In an open-loop mechanical testing system, no feedback is offered to the control mechanism that would allow for continuous adjustment of the controlled parameter. Instead, the chosen parameter is “controlled” by the preset factory adjustments of the control mechanism. It is © 2005 by CRC Press LLC

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not possible for such a machine continually to adjust its operation to achieve a chosen (constant or not constant) displacement rate or loading rate. A human operator can be added to the control loop in some systems in an attempt to maintain some parameter, such as a loading rate, at a constant level. This is a poor means of obtaining improved equipment response and is prone to error. Closed-Loop Testing Machines. In a closed-loop, most commonly electrohydraulic, testing system, a servo controller is used to control the chosen parameter continuously. When the difference between the desired value programmed in and the actual value measured is small, the servo controller adjusts the flow of hydraulic fluid to the actuator to reduce the difference (the error). This correction occurs at a rate much faster than any human operator could achieve. A standard system makes 10,000 adjustments per second automatically. A typical closed-loop system (Color Figure 9, Color Figure 11, and Color Figure 15) allows the operator to control load, strain, or displacement as a function of time and can be adjusted to control other parameters as well. This makes it possible to perform many different kinds of tests, such as tension, compression, torsion, creep, stress relaxation, fatigue, and fracture. Impact Testing. The most common impact testing machines utilize a pendulum hammer or a dropped weight. In the pendulum system, a hammer is released from a known height and strikes a small notched specimen, causing it to fracture. The hammer proceeds to some final height. The difference between the initial and final heights of the hammer is directly proportional to the energy absorbed by the specimen. For the Charpy test, the specimen is mounted horizontally with the ends supported so that the pendulum will strike the specimen in midspan, opposite the notch. In the Izod test, the specimen bottom is mounted in a vertical cantilever support so that the pendulum will strike the specimen at a specific distance above the notch, near the unsupported top end. A large variety of drop-weight tests is also available to investigate the behaviors of materials and packages during impact. Hardness Testing. The major hardness tests are the Brinell, Rockwell, Vickers, and Shore scleroscope tests. The Brinell hardness test uses a hardened steel ball indenter that is pushed into the material under a specified force. The diameter of the indentation left in the surface of the material is measured and a Brinell hardness number is calculated from this diameter. The Rockwell hardness test differs from the Brinell test in that it uses a 120° diamond cone with a spherical tip for hard metals and a 1/16-in. steel ball for soft metals. The Rockwell tester gives a direct readout of the hardness number. The Rockwell scale consists of a number of different letter designators (B, C, etc.) based on the depth of penetration into the test material. The Vickers hardness test uses a small pyramidal diamond indenter and a specified load. The diagonal length of the indentation is measured and used to obtain the Vickers hardness number. The Shore scleroscope uses a weight that is dropped on the specimen to determine the hardness. This hardness number is determined from the rebound height of the weight.

1.6 Structural Integrity and Durability Bela I. Sandor The engineer is often concerned about the long-term behavior and durability of machines and structures. Designs based only on statics, dynamics, and basic mechanics of materials are typically able to satisfy only minimal performance and reliability requirements. For realistic service conditions, numerous degradations may need to be considered. A simple and common approach is to use safety factors based on experience and judgment. The degradations could become severe and require sophisticated analyses if unfavorable interactions occur. For example, fatigue with corrosion or high temperatures is difficult to predict accurately, and much more so when corrosion is occurring at a high temperature. There are many kinds of degradations and interactions between them, and a large (and still growing) technical literature is available in most of these areas. The present coverage cannot possibly do justice to © 2005 by CRC Press LLC

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the magnitude of the most serious problems and the available resources to deal with them. Instead, the material here highlights some common problems and provides fundamental concepts to prepare for more serious efforts. The reader is encouraged to study the technical literature (including that by technical societies such as ASM, ASME, ASNT, ASTM, SAE), attend specialized short courses, and seek consulting advice (ASM, ASTM, Teltech) as necessary.

Finite Element Analysis. Stress Concentrations The most common problem in creating a machine or structure with good strength-to-weight ratio is to identify its critical locations and the corresponding maximum stresses or strains and to adjust the design optimally. This is difficult if a member’s geometry, including the geometry and time dependence of the loading, is complex. The modern analytical tool for addressing such problems is finite element analysis (FEA) or finite element modeling (FEM). Finite Element Analysis The finite element (FE) method was developed by engineers using physical insight. In all applications, the analyst seeks to calculate a field quantity: in stress analysis, it is the displacement field or the stress field; in thermal analysis, it is the temperature field or the heat flux; and so on. Results of the greatest interest are usually peak values of the field quantity or its gradients. The FE method is a way of getting a numerical solution to a specific problem. An FEA does not produce a formula as a solution, nor does it solve a class of problems. Also, the solution is approximate unless the problem is so simple that a convenient exact formula is already available. Furthermore, it is important to validate the numerical solution instead of trusting it blindly. The power of the FE method is its versatility. The structure analyzed may have arbitrary shape, arbitrary supports, and arbitrary loads. Such generality does not exist in classical analytical methods. For example, temperature-induced stresses are usually difficult to analyze with classical methods, even when the structure geometry and the temperature field are simple. The FE method treats thermal stresses as readily as stresses induced by mechanical load, and the temperature distribution can be calculated by FE. However, it is easy to make mistakes in describing a problem to the computer program. Therefore, it is essential that the user have a good understanding of the problem and the modeling so that errors in computed results can be detected by judgment. Stress Concentrations Geometric discontinuities cause localized stress increases above the average or far-field stress. A stress raiser’s effect can be determined quantitatively in several ways, but not always readily. The simplest method, if applicable, is to use a known theoretical stress concentration factor, Kt , to calculate the peak stress from the nominal, or average, value: σ max = K t σ ave

(1.6.1)

This is illustrated in Figure 1.6.1. The area under the true stress distribution always equals the area under the nominal stress level,

∫σ A

true

dA =

∫σ A

ave

dA = σ ave A

(1.6.2)

The factor Kt depends mainly on the geometry of the notch, not on the material, except when the material deforms severely under load. Kt values are normally obtained from plots such as in Figure 1.6.2 and are strictly valid only for ideally elastic, stiff members. Kt values can also be determined by FEA or by several experimental techniques. There are no Kt values readily available for sharp notches and cracks, but one can always assume that such discontinuities produce the highest stress concentrations, sometimes factors of tens. This is the reason why brittle, high-strength materials are extremely sensitive even to © 2005 by CRC Press LLC

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FIGURE 1.6.1 Stress distribution (simplistic) in a notched member under uniaxial load.

FIGURE 1.6.2 Samples of elastic stress concentration factors. (Condensed from Figure 10.1, Figure 10.2 and Figure 10.2, Dowling N.E., Mechanical Behavior of Materials, Prentice Hall, Englewood Cliffs, NJ, 1993. With permission.)

minor scratches. In fatigue, for example, invisible toolmarks may lead to premature, unexpected failures in strong steels. Many other factors may seem similar to Kt , but they should be carefully distinguished. The first is the true stress concentration factor Kσ, defined as Kσ =

σ max σ ave

(1.6.3)

which means that Kσ = Kt (by Equation 1.6.1) for ideally elastic materials. Kσ is most useful in the case of ductile materials that yield at the notch tip and lower the stress level from that indicated by Kt . Similarly, a true strain concentration factor, Kε, is defined as

Kε =

ε max ε ave

(1.6.4)

where εave = σave /E. Furthermore, a large number of stress intensity factors are used in fracture mechanics, and these (such as K, Kc , KI, etc.) are easily confused with Kt and Kσ , but their definitions and uses are different as seen in the next section. © 2005 by CRC Press LLC

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FIGURE 1.6.3 Modes of deformation.

Fracture Mechanics Notches and other geometric discontinuities are common in solid materials, and they tend to facilitate the formation of cracks, which are in turn more severe stress raisers. Sharp cracks and their further growth are seldom simple to analyze and predict because the actual stresses and strains at a crack tip are not known with the required accuracy. In fact, this is the reason that classical failure theories (maximum normal stress, or Rankine, theory; maximum shear stress, or Tresca, theory; distortion energy, or von Mises or octahedral shear stress, theory), elegantly simple as they are, are not sufficiently useful in dealing with notched members. A powerful modern methodology in this area is fracture mechanics, which was originated by A. A. Griffith2 in 1920 and has grown in depth and breadth enormously in recent decades. The space here is not adequate to list all of the significant references in this still expanding area. The purpose here is to raise the engineer’s awareness to a quantitative, practically useful approach in dealing with stress concentrations as they affect structural integrity and durability. Brittle and Ductile Behaviors. Embrittlements Brittleness and ductility are often the first aspects of fracture considerations, but they often require some qualifications. Simplistically, a material that fractures in a tension test with 0% reduction of area (RA) is perfectly brittle (and very susceptible to fracture at stress raisers), while one with 100% RA is perfectly ductile (and quite tolerant of discontinuities). Between these extremes fall most engineering materials, with the added complication that embrittlement is often made possible by several mechanisms or environmental conditions. For example, temperature, microstructure, chemical environment, internal gases, and certain geometries are common factors in embrittlement. A few of these will be discussed later. Linear-Elastic Fracture Mechanics (LEFM) A major special case of fracture mechanics is when little or no plastic deformation occurs at the critical locations of notches and cracks. It is important that even intrinsically ductile materials may satisfy this condition in common circumstances. Modes of Deformation. Three basic modes of deformation (or crack surface displacement) of cracked members are defined as illustrated schematically in Figure 1.6.3. Each of these modes is very common, but Mode I is the easiest to deal with analytically and experimentally, so most data available are for Mode I. Stress Intensity Factors. The stresses on an infinitesimal element near a crack tip under Mode I loading are obtained from the theory of linear elasticity. Referring to Figure 1.6.4, 2

The Griffith criterion of fracture states that a crack may propagate when the decrease in elastic strain energy is at least equal to the energy required to create the new crack surfaces. The available elastic strain energy must also be adequate to convert into other forms of energy associated with the fracture process (heat from plastic deformation, kinetic energy, etc.). The critical nominal stress for fracture according to the Griffith theory is proportional to

1/ crack length . This is significant since crack length, even inside a member, is easier to measure nondestructively than stresses at a crack tip. Modern, practical methods of fracture analysis are sophisticated engineering tools on a common physical and mathematical basis with the Griffith theory. © 2005 by CRC Press LLC

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FIGURE 1.6.4 Coordinates for fracture analysis.

σx =

KI f (θ) + … 2 πr 1

σy =

KI f (θ) + … 2 πr 2

τ xy =

KI f (θ) + … 2 πr 3

(1.6.5)

τ xz = τ zx = 0 There are two special cases of σz : 1. σz = 0 for plane stress (thin members) 2. σz = v(σx + σy ) for plane strain, with εz = 0 (thick members) The factor K in these and similar expressions characterizes the intensity or magnitude of the stress field near the crack tip. It is thus called the stress intensity factor, which represents a very useful concept different from that of the well-known stress concentration factor. KI is a measure of the severity of a crack, and most conveniently it is expressed as K I = σ πa f (geometry)

(1.6.6)

where a is the crack length and f is a function of the geometry of the member and of the loading (typically, f  1 ± 0.25). Sometimes f includes many terms, but all stress intensity factors have the same essential features and units of stress length. In any case, expressions of K for many common situations are available in the literature, and numerical methods are presented for calculating special K values. Differential thermography via dynamic thermoelasticity is a powerful, efficient modern method for the measurement of actual stress intensity factors under a variety of complex conditions (Section 1.6, “Experimental Stress Analysis and Mechanical Testing”; Figure 1.6.12; Color Figure 8; Color Figure 11 through Color Figure 14). Fracture Toughness of Notched Members The stress intensity factor, simply K for now, is analogous to a stress-strain curve, as in Figure 1.6.5. K increases almost linearly from 0 at σ = 0, to a value Kc at a critical (fracture) event. Kc is called the fracture

FIGURE 1.6.5 Kc = fracture toughness of a particular member. © 2005 by CRC Press LLC

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FIGURE 1.6.6 KIc = plane strain fracture toughness of material.

FIGURE 1.6.7 Plane strain fracture toughness ranges (approximate).

toughness of a particular member tested. It depends on the material, but it is not a reliable material property because it depends on the size of the member too much. This is illustrated in Figure 1.6.6 for plates of the same material but different thicknesses. At very small thickness, Kc tends to drop. More significantly, Kc approaches a lower limiting value at large thickness (>A). This worst-case value of Kc is called KIc , the plane strain fracture toughness in Mode I. It may be considered a pseudomaterial property because it is independent of geometry at least over a range of thicknesses. It is important to remember that the thickness effect can be rather severe. An intrinsically ductile metal may fracture in an apparently brittle fashion if it is thick enough and has a notch. Fracture Toughness Data. Certain criteria about crack sharpness and specimen dimensions must be satisfied in order to obtain reliable basic KIc data (see ASTM Standards). These data for many engineering materials are available in the technical literature. A schematic overview of various materials’ KIc values is given in Figure 1.6.7. Note that particular expected values are not necessarily attained in practice. Poor material production or manufacturing shortcomings and errors could result in severely lowered toughness. On the other hand, special treatments or combinations of different but favorably matched materials (as in composites) could substantially raise the toughness. Besides the thickness effect, there are a number of major influences on a given material’s toughness, and they may occur in favorable or unfavorable combinations. Several of these are described here schematically, showing general trends. Note that some of the actual behavior patterns are not necessarily as simple or well defined as indicated. Yield Strength. High yield strength results in a low fracture toughness (Figure 1.6.8); therefore, it should be chosen carefully, understanding the consequences.

FIGURE 1.6.8 Yield strength effect on toughness. © 2005 by CRC Press LLC

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FIGURE 1.6.9 Temperature effect on toughness.

FIGURE 1.6.10 Trends of toughness degradations.

Temperature. Two kinds of temperature effect on toughness should be mentioned here. They may appear, at least for part of the data, as in Figure 1.6.9, with high temperature causing increased toughness. One temperature effect is by the increased ductility at higher temperature. This tends to lower the yield strength (except in low-carbon steels that strain-age at moderately elevated temperatures, about 100 to 500°C); increase the plastic zone at the notch tip; and effectively blunt the stress concentration. Another effect — the distinct temperature-transition behavior in low-carbon steels (BCC metals, in general; easily shown in Charpy tests) — is caused by microstructural changes in the metal and is relatively complex in mechanism. Loading Rate. The higher the rate of loading, the lower the fracture toughness is in most cases. Note that toughness results obtained in notch-impact or explosion tests are most relevant to applications in which the rate of loading is high. Microstructural Aspects. In some cases, apparently negligible variations in chemical composition or manufacturing processes may have a large effect on a material’s fracture toughness. For example, carbon, sulfur, and hydrogen contents may be significant in several embrittling mechanisms. Also, the common mechanical processing of cold or hot working (rolling, extruding, forging) influences the grain structure (grain size and texture) and the corresponding toughness. Neutron radiation also tends to cause microscopic defects, increasing the yield strength and consequently lowering the ductility and toughness of the material. Overview of Toughness Degradations. A multitude of mechanisms and situations must be considered singly and in realistic combinations, as illustrated schematically in Figure 1.6.10 (review Figure 1.6.6 for relevant toughness definitions). Degrading Factors Some chemical compositions Sharper notch Greater thickness Faster loading Lower temperature Higher yield strength Hostile chemical environment Liquid metal embrittlement Tensile residual stress Neutron irradiation Microstructural features Moisture Gases in solid solution Surface hardening Note that the toughness can drop essentially to zero in some cases. © 2005 by CRC Press LLC

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Crack Propagation Crack growth may be classified as stable (subcritical) or unstable (critical). Often stable cracks become unstable in time, although the opposite behavior, cracks decelerating and even stopping, is sometimes possible. Unstable cracks under load control are extremely dangerous because they propagate at speeds nearly 40% of the speed of sound in that particular solid. For example, in steels, this means a crack growth speed of about 1 mi/sec. Thus, warnings and even electronically activated, automated countermeasures during unstable propagation are useless. The only reasonable course is to provide, by design and proper manufacture, preventive measures such as ductile regions in a structure where cracks become stable and slow to grow, thus allowing for inspection and repair. Each of the three kinds of stable crack growth is important in its own right; interactions among the three are possible. Under steady loads, environmentally assisted crack growth (also called stress corrosion cracking) and creep crack growth are commonly found. Under cyclic loading, fatigue crack growth is likely to occur. In each case the rate of crack growth tends to accelerate in time or with progressive cycles of load if the loads are maintained while the cracks reduce the load-bearing cross-sectional area. This common situation, caused by increasing true stresses, is illustrated schematically in Figure 1.6.11, where a0 is an initial flaw’s size, da/dN and da/dt are the fatigue and creep crack growth rates, respectively, and ac is the critical crack size. The rate of stable crack growth is controlled by the stress intensity factor. This will be discussed later.

FIGURE 1.6.11 Crack growth rates under constant load.

Design and Failure Analysis Using Stress Intensity Concepts The concept of stress intensity of cracked members is highly useful and practical. Three major possibilities are outlined here with respect to the essential framework of K ∝ stress crack length

(1.6.7)

Here K may be an operating stress intensity factor or a KIc value, a material property (the units are the same). In design, the idea is to fix one or two quantities by some initial constraints of the case, then work with the results according to Equation (1.6.7). • Operating stress and material (KIc ) are predetermined. This forces one to measure crack length and set the maximum allowable size of cracks. • Operating stress and detectable crack size are predetermined. This forces one to choose an appropriate material with the required KIc value. • The material (KIc value) and the detectable crack size are predetermined. This forces one to limit the operating stress accordingly. Similar thinking can be used in failure analysis and corresponding design iteration. For example, the critical crack size at the end of the stable propagation (and start of the unstable, high-speed growth) can

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FIGURE 1.6.12 Practical fracture mechanics with NDE: nearly instantaneous measurement of crack size and the actual stress intensity factor via advanced thermoelastic stress analysis. The member’s loading (including boundary conditions) need not be known to obtain reliable data using this method.

often be determined by looking at the broken parts. The material property, KIc , can also be estimated from the parts at hand, and thus the stress that caused the failure can be calculated. Whether the stress was within normal bounds or was an overload from misuse of the equipment can be determined. These are powerful, quantitative methods useful in improving designs and manufacturing. Special Methods The many other important and useful methods in fracture mechanics cannot even be listed here. For example, several methods in the area of elastic–plastic fracture mechanics are available. Within this area, mainly applicable to thin members of ductile materials, the J-integral approach alone has been covered in a large number of books and journal articles. Nondestructive Evaluation Because all of fracture mechanics is based on knowing the crack size and its location and orientation, nondestructive evaluation (NDE) is a major part of quantitative, predictive work in this area. Many techniques of NDE are available, and some are still rapidly evolving. Two major categories of NDE methods are defined here: 1. Geometry-based methods. At best, the size, shape, location, and orientation of a flaw are measured. Considerable additional effort is needed to estimate the effect of the flaw on structural integrity and durability. Common methods involve acoustic, magnetic, microwave, optical (including thermal), or x-ray instruments. 2. Stress-based methods. A flaw’s effect on the stress–strain field is directly measured, which is often much more important than just finding that flaw (a flaw of a given geometry may be benign or malignant, depending on the stress field of the neighborhood). Only a few optical methods are readily available for stress-based NDE; the most effective one for laboratory and field applications is thermoelastic stress analysis by infrared means (Figure 1.6.12; Color Figure 8; Color Figure 11 through Color Figure 14; Section 1.5, “Experimental Stress Analysis and Mechanical Testing”).

Creep and Stress Relaxation Creep and stress relaxation are related time- and temperature-dependent phenomena, with creep occurring under load control and stress relaxation under deformation control. In both cases the material’s temperature is a governing factor regarding what happens. Specifically, for most metals, the creep and relaxation regimes are defined as high homologous (relative, dimensionless) temperatures, normally those above half the melting point in absolute temperature for each metal. Thus, solder at room temperature creeps significantly under load, while steel and aluminum do not. However, some creep and relaxation may occur even at low homologous temperatures, and they are not always negligible. For polymers, the creep regime is above the glass transition temperature. This is typically not far from room temperature.

© 2005 by CRC Press LLC

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FIGURE 1.6.13 Creep under constant load. dε/dt = A(σ)n. A and n are material parameters.

FIGURE 1.6.14 Stress relaxation under constant deformation. σ = σ0e–Et/η. E and η are material parameters.

FIGURE 1.6.15 Approximate stress vs. rupture lives of S-590 alloy as functions of temperature. (After Figure 15.8, Dowling, N.E., Mechanical Behavior of Materials. Prentice Hall, Englewood Cliffs, NJ, 1993. With permission.)

Figure 1.6.13 and Figure 1.6.14 show trends of creep and stress relaxation in the large-scale phenomenon region. Stress vs. rupture life curves for creep may be nearly linear when plotted on log–log coordinates (Figure 1.6.15). Mechanical Models of Viscoelastic Behaviors Creep and stress relaxation appear to be combinations of behaviors of viscous liquids and elastic solids. The so-called viscoelastic phenomena are commonly modeled by simple mechanical components, springs and dashpots, as in Figure 1.6.16. The Maxwell model and related others are based on such elements. The Maxwell model for creep under constant stress σ0 is ε = ε1 + ε 2 = © 2005 by CRC Press LLC

σ + E

t

∫σ 0

0

dt =

σ 0 σ 0t + E η

(1.6.8)

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FIGURE 1.6.16 Viscoelastic elements.

For relaxation, ε = constant and σ varies, so dε 1 dσ σ =0= + dt E dt η σ

dσ E =− η σ0 σ

∫

t

∫ dt, 0

(1.6.9)

σ = σ 0 e − Et η

Time–Temperature Parameters. Life Estimation It is often necessary to extrapolate from laboratory creep test data, which are limited in time (from days to years), to real service lives, which tend to be from years to several decades. Time–temperature parameters are useful for this purpose. Three common parameters are outlined here. Note that no such parameter is entirely reliable in all cases; they are best if used consistently in direct comparisons of materials. Sherby–Dorn Parameter (PSD ) 1 PSD = log θ r = log tr − 0.217Q  T

(1.6.10)

where, for steady-state creep, θr = temperature-compensated time to rupture tr = rupture time, hours Q = activation energy = constant T = temperature, K Stress-life data at high T and low tr are needed to plot PSD vs. stress, in order to predict a longer tr at a lower T. Larson–Miller Parameter (PLM ) This approach is analogous to the Sherby–Dorn approach, but is based on different assumptions and equations. PLM = 0.217Q = T (log t r + C)

(1.6.11)

where C = –logθr  20 for steels. For using temperature in degrees Fahrenheit (as in most of the data): PLM © 2005 by CRC Press LLC

°F

= 1.8PLM

K

(1.6.12)

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Manson–Haferd Parameter (PMH ) PMH =

T − Ta log t r − log t a

(1.6.13)

where Ta and ta are temperature and time constants representing a point of convergence for a family of data points. As shown previously, for different temperature scales: PMH

°F

= 1.8PMH

K

(1.6.14)

Overview. The greater the extrapolation using any parameter, the greater the likelihood of error is. A factor of 10 or less extrapolation in life is often reasonable. At very large extrapolations, damage mechanisms may be different from those of the tests and unpredictable service loading and environmental conditions as well.

Fatigue Fatigue is a process of damage evolving in a material due to repeated loads, also called cyclic loads. This is a common degradation that affects virtually all solid materials, and thus it is often the main (or a contributing) factor in the failure of vehicles, machinery, structures, appliances, toys, electronic devices, and surgical implants. Many apparently well-designed and -fabricated items that fail inexplicably have problems rooted in the fatigue area. Nearly two centuries of fatigue studies and engineering efforts have resulted in a huge, and still expanding, technical literature. This brief review can cover only a few major topics, some old but valuable items of wisdom, and practical modern methods. Three important approaches are presented: the stressbased (useful for long lives), strain-based (useful for short lives), and fracture mechanics methods. Definitions Constant-amplitude, stress- or strain-controlled cycling is common in testing and some service situations. Figure 1.6.17 shows the stress (σ) quantities in such cycling. Similar notations are used for strains. In completely reversed stress, σm = 0 and R = –1. Zero-to-tension (a special case of pulsating tension) has σmin = 0 and R = 0.

FIGURE 1.6.17 Notation for constant-amplitude stress cycling.

Material Properties in Cyclic Loading The mechanical properties of some materials are gradually changed by cyclic plastic strains. The changes that occur are largest early in the fatigue life and become negligible beyond about 20 to 50% of the life. The most important material properties that could change significantly this way are the flow properties (yield strength, proportional limit, strain hardening exponent), while the modulus of elasticity is little © 2005 by CRC Press LLC

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affected. For metals, three initial conditions can be defined using the strain hardening exponent n as a key parameter. The concept of a cyclic stress–strain curve, as opposed to that in monotonic (static) loading, is also used to distinguish possible material behaviors in fatigue, as follows. • Stable: 0.15 < n < 0.2 (approx.) The monotonic and cyclic stress–strain curves are the same for most practical purposes (though seldom coincident). Examples: 7075-T6 Al; 4142 steel (550 BHN) • Cycle-dependent softening: n < 0.15 (approx.) (means initially hard, cold-worked material) The cyclic stress-strain curve falls significantly below the monotonic curve, which means a gradually decreasing deformation resistance as cyclic loading progresses. The cyclic yield strength may be less than half the tensile yield strength in some cases. Examples: 4340 steel (350 BHN); 4142 steel (400 BHN) • Cycle-dependent hardening: n > 0.2 (approx.) (means initially soft, annealed material) The cyclic stress–strain curve is significantly above the monotonic curve, which means a gradually increasing deformation resistance as cyclic loading progresses. Examples: 2024-T4 Al; 4142 steel (670 BHN) Note that the hardest steels tend to harden further in cyclic loading. Thus, a given steel (such as 4142) may be stable, softening, or hardening, depending on its initial hardness. In the technical literature, primes are normally used to denote cyclic material properties. For example, σ ′y is the yield strength obtained from a cyclic stress–strain curve. Stress vs. Life (S-N) Curves The most common and historical fatigue life plots present data of stress amplitude (simplistically, S or Sa) on a linear scale vs. cycles to failure (N or Nf ) on a logarithmic scale as in Figure 1.6.18. Many steels (plain carbon or low alloy) appear to have a distinct fatigue limit. For other metals that do not have such a limit (aluminum, for example), an arbitrary fatigue limit is defined as a stress amplitude corresponding to a specified life, typically 107 or 108 cycles.

FIGURE 1.6.18 Schematic of S–N curves.

Trends in S-N Curves There are many influences on the shape and position of a material’s fatigue life curve as briefly discussed below. Ultimate Strength. It is widely believed that, at least for steels, the fatigue limit σe is about one half of the ultimate strength σu. In fact, this is a gross oversimplification; actual values are lower or higher than that in many cases. Mean Stress, Residual Stress. Several main points are worth remembering: residual stresses (also called self-stresses) are common, and they are to be treated as mean stresses (by sign and magnitude) in fatigue; a tensile mean stress lowers the life while a compressive one increases it. Simplistically, a tensile mean stress lowers the allowable cyclic stress amplitude, according to Figure 1.6.19 where © 2005 by CRC Press LLC

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FIGURE 1.6.19 Schematic of tensile mean stress effect.

(if yielding is to be prevented)

σ m + σ a ≤ σ u or σ y

For example, if σm = 0.7σu , then the maximum alternating stress for one cycle is 0.3σu . This kind of graphical relationship is called a Goodman diagram. Several special expressions are used for dealing with the detrimental effects of tensile mean stresses. For example, the modified Goodman equation is σa σm + =1 σe σu

(1.6.15)

where σe is the fatigue limit for fully reversed loading. Sometimes curved lines represent real behavior better than the linear theory shown in Figure 1.6.19. In that case, the Gerber parabola may be appropriate, in the form of 2

σa  σm  + =1 σ e  σ u 

for σ m ≥ 0

(1.6.16)

Another approach worth mentioning is the mechanistically elegant and sensible Morrow expression, which will be presented later. Note that tensile mean stresses are generally detrimental and that many approaches have been proposed to deal with them, although no single method is capable of good predictions in all cases. In practice, it is best to use a particular method that has a good track record for the material and situation at hand. Constant-life diagrams are useful, elaborate derivatives of the Goodman approach, if they include a broad data base (Figure 1.6.20). Notch Effects. Stress raisers can be extremely detrimental in fatigue, except when they help create localized compressive residual stresses in ductile metals, delaying crack formation and growth. These are discussed in connection with the strain-based approach. Microstructure. Large grain size (annealed metals) lowers the fatigue strength, and small grain size (by cold working) increases it, especially at long lives, under load control. Surface Effects. The condition of a material’s surface may influence the fatigue behavior in many ways, typically in combinations. Toolmarks are common detrimental features, especially because they often are aligned perpendicular to the principal tensile stress in axial or bending loading. An example is a shaft cut in a lathe. Note that in the case of high-strength, hard materials even invisible scratches from grinding and buffing may be stress raisers. Machining also tends to create tensile or compressive residual stresses in surface layers. © 2005 by CRC Press LLC

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FIGURE 1.6.20 Constant-life diagram.

Surface treatments such as carburizing or nitriding of steels affect the fatigue life by changes in chemical composition, microstructure, hardness, or residual stress. Shot peening, surface rolling, or burnishing is done to introduce compressive residual stresses, which delay cracking in long-life service. Plating (chromium, nickel) tends to create layers of poor fatigue resistance and harmful tensile residual stresses. Shot peening after plating is a beneficial step. Environment. Hostile chemical environments can severely reduce most materials’ fatigue resistance. Common causes of problems are salt water, salt in the air, salt on the road, moisture, and even pollutants in the air. For example, sulfur in the air results in aggressive sulfuric acid on machines and structures. Statistical Scatter. Statistical scatter is always in a material’s fatigue life at any given stress level, especially at long lives. The scatter band may cover several orders of magnitude in life at a single stress level. Because of the scatter, there is no unique fatigue life curve for any material — the curve depends not only on physical factors such as environment, but also on the number of tests done. It is not sufficient to do a handful of tests and draw a curve somewhere through the data points. As a simple rule, to have a high level of confidence (>99%) in a fatigue life curve, at least six identical tests are needed to obtain a mean value at each of several levels of stresses in the general life range of interest. A curve through these mean values is fairly representative of the average life curve (50% probability of failure), but still may not be adequate to deal with scatter. Note that the minimum number of test specimens according to the ASTM Standard E 739 is 6 to 12 for preliminary, exploratory work or for research and development and component testing, and 12 to 24 for design allowables or reliability assessment. Ideally, additional analysis is done, using Gaussian (normal) statistical distribution or some other model, such as the Weibull distribution. The latter is particularly informative in determining the probability of fatigue failure. The practical problem is that engineers may require very low probabilities of failure (less than 1%), but neither the necessary mathematical methods nor the data bases are available for that. A family of fatigue life curves for various probabilities of failure and other relevant considerations for one material are shown schematically in Figure 1.6.21 through Figure 1.6.23. Variable Amplitude Loading Many machines, vehicles, and structures experience random or blockwise changing loading. They can be simplistically modeled for life prediction using the Palmgren–Miner rule, illustrated in Figure 1.6.24. There are two major assumptions for this rule for completely reversed loading: 1. Every cycle at a given level of stress amplitude causes the same amount of damage, whether the cycle is early or late in the life. 2. The percentage of damage caused by a cycle of load at any level of stress is equivalent to the same percentage of damage at any other level of stress. © 2005 by CRC Press LLC

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FIGURE 1.6.21 Schematic S–N curves with various probabilities of failure.

FIGURE 1.6.22 Probability aspects of fatigue depending on stress level.

FIGURE 1.6.23 Probability aspects of fatigue depending on applied stress and product strength.

Thus, because 100% of the life Nfi is exhausted at failure at any single stress amplitude σi , in multilevel loading the life fractions sum to unity, as mathematically formulated here and illustrated in Figure 1.6.24: N1 N + 2 +… = N f1 N f 2

∑N

Ni

=1

(1.6.17)

fi

where Ni is the actual number of cycles at σi and Nfi is the life at σi. In practice, summations of about 0.8 to 1.2 can be accepted, saying that the Palmgren–Miner rule is valid in that case. Gross deviations from summations of one are common, especially when the mean stress is not zero. Modified versions of the basic rule for such cases should be used with caution. © 2005 by CRC Press LLC

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FIGURE 1.6.24 Schematic for Palmgren–Miner rule.

Cycle Counting. Highly irregular loading requires the use of special cycle counting methods, such as level crossing, range counting, or rainflow cycle counting. The latter is the best modern method, lending itself to efficient field data acquisition and computer work (ASTM Standard E1049; SAE Fatigue Design Handbook). Multiaxial Fatigue Complex states of stress are common in engineering components, and in fatigue analysis they may cause serious difficulties. Although many methods are available, none of them is adequate for all cases. The simplest situations that might be handled reasonably well involve fully reversed loading by in-phase or 180° out-of-phase proportional stresses at the same frequency. Multiaxial fatigue testing is difficult and expensive, so it is often desired to use uniaxial test data for predicting the multiaxial behavior. A typical approach for this is based on computing an effective stress amplitude se from the amplitudes of the principal stresses σ1a, σ2a, σ3a. With the concept of the octahedral shear yield criterion, σe =

1 2

(σ

− σ 2 a ) + (σ 2 a − σ 3a ) + (σ 3a − σ1a ) 2

1a

2

2

(1.6.18)

where in-phase stresses are positive and 180° out-of-phase stresses are negative. The life is estimated by entering σe on the appropriate S-N curve. Note that mean stresses, localized or general yielding, creep, and random frequencies of loading further complicate the problem and require more sophisticated methods than outlined here. Strain vs. Life (ε-N) Curves A strain-based approach is necessary in fatigue when measurable inelastic strains occur. In general, total strain consists of elastic, plastic, and creep strains, with the latter two in the category of inelastic strains: εt = εe + ε p + εc

© 2005 by CRC Press LLC

(1.6.19)

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FIGURE 1.6.25 Hysteresis loop.

FIGURE 1.6.26 Schematic of strain vs. life curves.

When εp or/and εc are dominant, the life is relatively short and the situation is called low-cycle fatigue (LCF), as opposed to high-cycle fatigue (HCF), where εe is dominant. The mechanics of LCF can be understood by first considering hysteresis loops of elastic and plastic strains as defined in Figure 1.6.25. Simplistically, HCF means a thin loop (a straight line at very long life) and LCF means a fat loop. Strain-life plots are especially useful in the LCF regime where material properties (εf , σf ) obtained in monotonic tension tests are directly useful in fatigue life prediction as shown in Figure 1.6.26. Most commonly, the total strain amplitude εa is plotted vs. the life 2Nf , with a corresponding equation (called Coffin–Manson equation) for fully reversed loading: εa =

σf

(2 N ) E f

b

( )

+ ε f 2N f

c

(1.6.20)

It is remarkable that all metals are similar to one another in their values of the exponents b (≈ –0.1) and c (≈ –0.6), differing only in fracture strength σf and fracture ductility εf . These allow a simplistic fatigue life prediction if at least σf and εf are known. If there is a mean stress, its effect is equivalent to an altered fracture strength. Using the Morrow approach in a simplified version, εa =

σf  σ  1 − m  2N f  σf  E 

( )

b

( )

+ ε f 2N f

c

where σm is positive for tensile and negative for compressive mean stress. © 2005 by CRC Press LLC

(1.6.21)

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Notch Effects The localized plastic strains of notched members complicate fatigue analysis considerably. It should be noted, first of all, that the theoretical stress concentration factor Kt is not entirely relevant to such members because yielding lowers the actual peak stresses from those predicted. This leads to the definitions of the true stress and strain concentration factors: Kσ =

peak stress ave. stress

Kε =

peak strain ave. strain

(1.6.22)

According to Neuber’s rule: Kt = Kσ Kε

(1.6.23)

which is useful for notch analysis in fatigue. This expression is strictly true for ideally elastic behavior and is qualitatively evident for elastic–plastic deformations. Residual Stresses at Notches. An extremely important, and somewhat surprising, phenomenon can occur in notched members if they yield locally under variable-amplitude loading. If a large load (called an overload) causes yielding at a notch and is followed only by smaller loads, a residual stress of the opposite sign to the overload’s sign is generated at the root of the notch. Thus, a tensile overload (such as at one side of a shaft in a straightening operation) creates a compressive residual stress, and vice versa. These stresses may remain in the member for a long time or be relaxed by other plastic strain events or by annealing. Of course, such stresses are effective mean stresses and can alter the life greatly. Creep–Fatigue Interactions Inelastic strains (plastic and creep strains) are the basic causes of time- and cycle-dependent damage processes. When both kinds of strains occur during the life of a particular component, complex damage interactions may arise. The simplest and most elegant approach in such a case is to sum both of the different damages linearly (as in the Palmgren–Miner summation for pure fatigue), assuming that they are equivalent to one another. In other words, assume that X percentage of creep life exhausted is equivalent to the same X percentage of fatigue life exhausted. Thus, a linear expression involving time and cycle fractions can be stated:

∑t

ti

+

ri

pure creep

∑N

nj fj

pure fatigue

=

1 at failure

(1.6.24)

where ti = actual time spent at stress level i in creep tri = total time to rupture at stress level i nj = actual number of cycles at stress level j Nfj = cycles to failure at stress level j This idealized linear expression is plotted as a dashed line in Figure 1.6.27; in contrast, a more realistic ASME code and possible severe degradations are also plotted. Many other methods (such as damage rate equations; strain-range partitioning) can be used to deal with creep–fatigue problems, but none of them is adequate for all situations. The difficulty is mainly because of the need to account for important, complex details of the loading cycle (frequency, hold times, temperature, and deformation wave shape). Fracture Mechanics Method in Fatigue Cyclic loading can cause crack growth with or without the presence of a hostile chemical environment. The rate of crack growth depends on the stress intensity factor K ∝ σ a . Investigations of this dependence © 2005 by CRC Press LLC

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FIGURE 1.6.27 Schematic of creep–fatigue interactions. The bilinear damage rule is recommended in the ASME Boiler and Pressure Vessel Code, Section III, Code Case N47.

FIGURE 1.6.28 Schematic of fatigue crack propagation data.

have led to the development of powerful techniques in design and failure analysis. The fatigue crack growth behavior is quantified by the Paris equation: da m = C ( ∆K ) dN

(1.6.25)

where da/dN = crack growth rate C, m = material constants ∆K = Kmax – Kmin = stress intensity factor range Kmax ∝ σmax Kmin ∝ σmin Typical data for a wide range of crack growth rates have patterns as in Figure 1.6.28, where ∆Kth is a threshold value akin to a fatigue limit. The linear part of the curve is useful for life prediction and failure analysis. Abridged Example of a Modern Fatigue Analysis Many of the concepts mentioned earlier are applied in Sandia National Laboratories’ “User’s Manual for FAROW: Fatigue and Reliability of Wind Turbine Components,” SAND94-2460, November 1994. FAROW is a computer program for the probabilistic analysis of large wind turbines, using structural reliability techniques to calculate the mean time to failure; probability of failure before a target lifetime; relative importance of each of the random inputs; and sensitivity of the reliability to all input parameters. The method is useful whether extensive data are available or not (showing how much can be gained by reducing the uncertainty in each input). It helps one understand the fatigue reliability of a component and indicates how to improve the reliability. The sample figures (Figure 1.6.29 through Figure 1.6.32) illustrate some of the key data and results for the machines and materials considered. © 2005 by CRC Press LLC

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FIGURE 1.6.29 Relative importance factors as fractions of the total influence on the probability of failure. (Courtesy Sandia National Laboratories, Albuquerque, NM.) 4.6 4.5 Log10 (Eff Alt Stress), psi

Least Square Curve Fit 4.4 Variation of 2 Std Dev 4.3 4.2 4.1

Teledyne Engr. Services Failed Specimen RunOut Specimen

4.0 3.9 3.8 4

Southern University

5

6

7

8

9

Log10 (Cycles to Failure)

FIGURE 1.6.30 Fatigue life data for 6063 Al. (Courtesy Sandia National Laboratories, Albuquerque, NM.)

FIGURE 1.6.31 Fatigue life data for uniaxial fiberglass composite. (Courtesy Sandia National Laboratories, Albuquerque, NM.) © 2005 by CRC Press LLC

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FIGURE 1.6.32 Example FAROW results for probability of premature failure as a function of target lifetime. (Courtesy Sandia National Laboratories, Albuquerque, NM.)

Note especially a large discrepancy between mean lifetime and probability of failure in a few years. A mean lifetime of 600 years was calculated for a critical component, using the median values for all the random variables considered and using the constant values for all the other input parameters. However, the probability of the component failing in less than 5 years is estimated at 7.6% (Figure 1.6.32). This shows the uncertainty — even in sophisticated fatigue life calculations — because of reasonable uncertainty in the inputs and the sensitivity of fatigue life to parameter variation.

1.7 Comprehensive Example of Using Mechanics of Solids Methods Bela I. Sandor A concise overview of an engineering project is presented to illustrate the relevance and coordinated application of several concepts and methods in this chapter. The sketchy outline is limited in breadth and depth, emphasizes modern methods, and is not aiming for completeness in any particular area.

The Project Analyze the currently used A-shaped arm of the suspension system of a small, special-purpose ground vehicle. The goal is to redesign the component to save weight and, more importantly, reduce the cost of manufacturing while assuring the product’s reliability over its expected service life.

Concepts and Methods Statics Vectors Free-body diagrams. Equilibrium Two-force member: shock absorber Frame components Beams. Bending moments Moments of inertia Center of mass Dynamics Velocity, acceleration Rigid-body dynamics General plane motion Relative motion

© 2005 by CRC Press LLC

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Vibrations Natural frequency Damping. Logarithmic decrement Mechanics of Materials Stress and strain. Transformation equations. Principal stresses. Maximum shear stress Material properties. Material selection Bending stresses. Beam optimization Strain gages. Mechanical testing with closed-loop equipment Durability Stress concentrations. Finite element analysis Cumulative fatigue damage. Cycle counting in random loading. Mean stresses. Goodman diagrams. Life prediction Thermoelastic stress analysis Illustrations A few aspects of the project are graphically illustrated in Color Figure 16 and Figure 1.7.1 through Figure 1.7.3.

FIGURE 1.7.1 Accelerometer data from front suspension system of vehicle. Logarithmic decrement ∂ = ln(x1/x2); damping ratio ζ = 0.16.

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FIGURE 1.7.2 Axial stress and force vs. time in shock absorber shaft.

Defining Terms Statics Equilibrium: A concept used to determine unknown forces and moments. A rigid body is in equilibrium when the equivalent force-couple system of the external forces acting on it is zero. The general conditions of equilibrium are expressed in vector form (ΣF = 0; ΣMO = Σ[r × F] = 0) or scalar form (ΣFx = 0; ΣFy = 0; ΣFz = 0; ΣMx = 0; ΣMy = 0; ΣMz = 0). Equivalent force-couple system: Any system of forces and moments acting on a rigid body can be reduced to a resultant force and a resultant moment. Transformations of a force-couple system involving chosen points of reference are easy to make. These are useful for determining unknown forces and moments and the critical locations in structural members. Free-body diagram: A method of modeling and simplifying a problem for the efficient use of the equilibrium equations to determine unknown forces and moments. A body or group of bodies is imagined to be isolated from all other bodies, and all significant external forces and moments (known or unknown) are shown to act on the free-body model. Dynamics Equations of motion: Expressions of the acceleration of a body related to the forces acting on the body. The basic equation of motion for a particle of mass m is ΣF = ma. Many other equations of motion may be stated, depending on the dimensions of the body and its motion (such as twoor three-dimensional motion) and the coordinate system chosen.

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FIGURE 1.7.3 Stresses σx , σy , and τxy measured at one point of the A-arm by strain gages as the vehicle travels over bumps.

Kinematics: The analysis of motion based on geometry and time-dependent aspects. Forces may or may not be associated with the motion, but the analysis does not involve considerations of forces. The parameters of interest in kinematics are position, displacement, velocity, acceleration, and time. Kinetics: The analysis of motion based on kinematics and the effects of forces on masses. Vibrations Forced vibration: Involves an exciting force applied periodically during the motion. A forced vibration may also be described in terms of the displacement of a foundation or primary mass that supports the vibrating system. Free vibration: Occurs when only two kinds of forces are acting on a mass: (1) the elastic restoring force within the system; and (2) the force of gravity or other constant forces that cause no displacement from the equilibrium configuration of the system. Resonance: A critical aspect of forced vibrations that occurs when the forcing frequency equals the system’s natural frequency. In this condition, the amplitude of the displacements becomes infinite in theory or dangerously large in practice when the damping is small. Near-resonance conditions may also be undesirable.

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Mechanics of Materials Flexure formula: Used to calculate the bending stresses in beams. Must be applied with modifications if there are inelastic deformations or unsymmetric bending, or for composite beams and curved beams. Hooke’s law: Applicable for calculating uniaxial or multiaxial stress–strain responses when the material acts entirely elastically. Involves the modulus of elasticity E and Poisson’s ratio v. Principal stresses: The maximum and minimum normal stresses at a point on an infinitesimal element. An important related quantity is the absolute maximum shear stress. These quantities can be determined (given an arbitrary state of applied stress) from stress transformation equations or from their graphical solution, Mohr’s circle. Principal strains are determined in a similar way. Stress–strain diagram: Shows the stress–strain response and many important mechanical properties for a material. These properties depend greatly on the material’s chemical composition and several other factors of fabrication and service conditions. Monotonic (tension or compression) and cyclic loading conditions may result in grossly different mechanical behaviors, even for a given material. Structural Integrity and Durability Rate of crack growth: A measure of damage evolution and remaining life of a member. In fatigue, the crack propagation rate da/dN depends on the stress intensity factor range DK and material properties. This relationship is the basis of the powerful, well-established damage-tolerant design method. Stress concentration factor: The localized stress-raising effect of a geometric discontinuity. Many potentially confusing forms of quantifying this effect exist. The most prominent factors are distinguished concisely: • Theoretical stress concentration factor, Kt = σmax/σave Depends on geometry of notch, not on material Has no engineering units • True stress concentration factor, Kσ = σmax/σave Depends on geometry of notch and material; Kσ = Kt for perfectly elastic material, Kσ < Kt for ductile material Has no engineering units • True strain concentration factor, Kε = εmax/εave, εave = σave/E Depends on geometry of notch and material; Kε = Kt for perfectly elastic material, Kε > Kt for ductile material Has no engineering units Stress intensity factor : A measure of the severity of a crack or the intensity of the stress field near the crack tip. Many potentially confusing forms of this factor exist; they have identical engineering units of stress length, but a variety of definitions and applications. A few are listed concisely: • Opening-mode stress intensity factor, KI Depends on geometry of a crack and applied stress, not on material Units of stress length • Plane strain fracture toughness, KIC Depends on material but not on geometry above a certain thickness, and not on applied stress Units of stress length • Stress intensity factor range, ∆K = Kmax – Kmin Depends on geometry of a crack and applied cyclic stress, not on material Units of stress length

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References Statics and Dynamics Hibbeler, R.C. 2001. Engineering Mechanics: Statics and Dynamics, 9th ed. Prentice Hall, Inc., Upper Saddle River, NJ. Sandor, B.I. 1987. Engineering Mechanics: Statics and Dynamics, 2nd ed. Prentice Hall, Englewood Cliffs, NJ. Vibrations Harris, C.M. and Crede, C.E. 1988. Shock and Vibration Handbook, 3rd ed. McGraw–Hill, New York. James, M.L., Smith, G.M., Wolford, J.C., and Whaley, P.W. 1994. Vibration of Mechanical and Structural Systems, 2nd ed. Harper Collins College Publishers, New York. Wowk, V. 1991. Machinery Vibration: Measurement and Analysis. McGraw–Hill, New York. Wowk, V. 1993. Machinery Vibration: Balancing. McGraw–Hill, New York. Mechanics of Materials Cook, R.D. and Young, W.C. 1985. Advanced Mechanics of Materials. Macmillan, New York. Dally, J.W. and Riley, W.F. 1991. Experimental Stress Analysis, 3rd ed. McGraw–Hill, New York. Hibbeler, R.C. 1997. Mechanics of Materials, 3rd ed. Prentice Hall, Englewood Cliffs, NJ. Jawad, M.H. and Farr, J.R. 1989. Structural Analysis and Design of Process Equipment, 2nd ed. John Wiley & Sons, New York. Kobayashi, A.S. (Ed.). 1993. Handbook on Experimental Mechanics, 2nd ed. Society for Experimental Mechanics, Bethel, CT. Pilkey, W.D., 1997. Peterson’s Stress Concentration Factors, 2nd ed. John Wiley & Sons, New York. Shigley, J.E. and Mischke, C.R. 1989. Mechanical Engineering Design, 5th ed. McGraw–Hill, New York. Young, W.C. 1989. Roark’s Formulas for Stress and Strain, 6th ed. McGraw–Hill, New York. Structural Integrity and Durability Anderson, T.L. 1994. Fracture Mechanics: Fundamentals and Applications, 2nd ed., CRC Press, Boca Raton, FL. Boyer, J.E. 1986. Atlas of Fatigue Curves. American Society for Metals, Metals Park, OH. Cook, R.D. 1995. Finite Element Modeling for Stress Analysis. John Wiley & Sons, New York. Dowling, N.E. 1993. Mechanical Behavior of Materials. Prentice Hall, Englewood Cliffs, NJ. Fuchs, H.O. and Stephens, R.I. 1980. Metal Fatigue in Engineering. John Wiley & Sons, New York. Gallagher, J.P. (Ed). 1983. Damage Tolerant Design Handbook, 4 vols. Metals and Ceramics Information Ctr., Battelle Columbus Labs, Columbus, OH. Murakami, Y. (Ed). 1987. Stress Intensity Factors Handbook, 2 vols. Pergamon Press, Oxford, U.K. Rice, R.C. (Ed). 1988. Fatigue Design Handbook, 2nd ed. SAE Publ. No. AE-10. Society of Automotive Engineers, Warrendale, PA.

Further Information Many technical societies are active in various areas of mechanics of solids, and they are excellent, steady sources of long-accepted and new information, some of which is available within hours. They also organize committee work, conferences, symposia, short courses, and workshops; establish codes and standards; and publish books, papers, journals, and proceedings covering the latest developments in numerous specialties. A short list of societies is given here; note that they tend to have international breadth, regardless of the name. It is wise to belong to several relevant societies and at least scan their announcements.

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ASM International (formerly American Society for Metals) (800-336-5152) ASME — American Society for Mechanical Engineers (800-843-2763) ASNT — American Society for Nondestructive Testing (800-222-2768) ASTM — American Society for Testing and Materials (215-299-5585) SAE — Society of Automotive Engineers (412-776-4841) SEM — Society for Experimental Mechanics (203-790-6373) SES — Standards Engineering Society (513-223-2410) As a hint of the scope and magnitude of what is available from the large technical societies, here are selected offerings of ASTM: • ASTM staff access/tel: 215-299-5585; fax: 215-977-9679; e-mail: [email protected] • ASTM Standardization News, a monthly magazine, regularly presents information on “the development of voluntary full consensus standards for materials, products, systems and services and the promotion of related knowledge… the research, testing and new activities of the ASTM standards-writing committees… the legal, governmental and international events impacting on the standards development process” (quotes from the masthead). • Over 50 volumes of ASTM Standards Samples of standards: Friction, wear, and abrasion (B611 on wear resistance of carbides; G77 on ranking of materials in sliding wear) Fracture mechanics (E399 on fracture toughness testing of metals) Fatigue (E466 on axial fatigue tests of metals; D671 on flexural fatigue of plastics) • Training courses for ASTM Standards (215-299-5480) • ASTM International Directory of Testing Laboratories • ASTM Directory of Scientific & Technical Consultants & Expert Witnesses • ASTM Special Technical Publications (STP) are books of peer-reviewed papers on recent research and developments Samples of STPs: STP 1198 — Nondestructive Testing of Pavements and Backcalculation of Moduli, 2nd Vol.; 1995 STP 1231 — Automation in Fatigue and Fracture: Testing and Analysis; 1995.

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2 Engineering Thermodynamics 2.1

Fundamentals Basic Concepts and Definitions • The First Law of Thermodynamics, Energy • The Second Law of Thermodynamics, Entropy • Entropy and Entropy Generation

2.2

Control Volume Applications Conservation of Mass • Control Volume Energy Balance • Control Volume Entropy Balance • Control Volumes at Steady State

2.3

Property Relations and Data Basic Relations for Pure Substances • P-v-T Relations • Evaluating ∆h, ∆u, and ∆s • Fundamental Thermodynamic Functions • Thermodynamic Data Retrieval • Ideal Gas Model • Generalized Charts for Enthalpy, Entropy, and Fugacity • Multicomponent Systems

2.4

Combustion Reaction Equations • Property Data for Reactive Systems • Reaction Equilibrium

2.5

Exergy Analysis Defining Exergy • Control Volume Exergy Rate Balance • Exergetic Efficiency • Exergy Costing

2.6

Vapor and Gas Power Cycles Rankine and Brayton Cycles • Otto, Diesel, and Dual Cycles • Carnot, Ericsson, and Stirling Cycles

Michael J. Moran The Ohio State University

2.7

Guidelines for Improving Thermodynamic Effectiveness

Although various aspects of what is now known as thermodynamics have been of interest since antiquity, formal study began only in the early 19th century through consideration of the motive power of heat: the capacity of hot bodies to produce work. Today the scope is larger, dealing generally with energy and entropy, and with relationships among the properties of matter. Moreover, in the past 25 years engineering thermodynamics has undergone a revolution, both in terms of the presentation of fundamentals and in the manner that it is applied. In particular, the second law of thermodynamics has emerged as an effective tool for engineering analysis and design.

0-8493-0866-6/05/$0.00+$1.50 © 2005 by CRC Press LLC

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Chapter 2

2.1 Fundamentals Classical thermodynamics is concerned primarily with the macrostructure of matter. It addresses the gross characteristics of large aggregations of molecules and not the behavior of individual molecules. The microstructure of matter is studied in kinetic theory and statistical mechanics (including quantum thermodynamics). In this chapter, the classical approach to thermodynamics is featured.

Basic Concepts and Definitions Thermodynamics is both a branch of physics and an engineering science. The scientist is normally interested in gaining a fundamental understanding of the physical and chemical behavior of fixed, quiescent quantities of matter and uses the principles of thermodynamics to relate the properties of matter. Engineers are generally interested in studying systems and how they interact with their surroundings. To facilitate this, engineers have extended the subject of thermodynamics to the study of systems through which matter flows. System In a thermodynamic analysis, the system is the subject of the investigation. Normally the system is a specified quantity of matter and/or a region that can be separated from everything else by a well-defined surface. The defining surface is known as the control surface or system boundary. The control surface may be movable or fixed. Everything external to the system is the surroundings. A system of fixed mass is referred to as a control mass or as a closed system. When there is flow of mass through the control surface, the system is called a control volume, or open, system. An isolated system is a closed system that does not interact in any way with its surroundings. State, Property The condition of a system at any instant of time is called its state. The state at a given instant of time is described by the properties of the system. A property is any quantity whose numerical value depends on the state but not the history of the system. The value of a property is determined in principle by some type of physical operation or test. Extensive properties depend on the size or extent of the system. Volume, mass, energy, and entropy are examples of extensive properties. An extensive property is additive in the sense that its value for the whole system equals the sum of the values for its parts. Intensive properties are independent of the size or extent of the system. Pressure and temperature are examples of intensive properties. A mole is a quantity of substance having a mass numerically equal to its molecular weight. Designating the molecular weight by M and the number of moles by n, the mass m of the substance is m = nM. One kilogram mole, designated kmol, of oxygen is 32.0 kg and one pound mole (lbmol) is 32.0 lb. When an extensive property is reported on a unit mass or a unit mole basis, it is called a specific property. An overbar is used to distinguish an extensive property written on a per-mole basis from its value expressed per unit mass. For example, the volume per mole is v , whereas the volume per unit mass is v, and the two specific volumes are related by v = Mv. Process, Cycle Two states are identical if, and only if, the properties of the two states are identical. When any property of a system changes in value there is a change in state, and the system is said to undergo a process. When a system in a given initial state goes through a sequence of processes and finally returns to its initial state, it is said to have undergone a cycle. Phase and Pure Substance The term phase refers to a quantity of matter that is homogeneous throughout in both chemical composition and physical structure. Homogeneity in physical structure means that the matter is all solid, or all liquid, or all vapor (or equivalently all gas). A system can contain one or more phases. For example, © 2005 by CRC Press LLC

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2-3

Engineering Thermodynamics

a system of liquid water and water vapor (steam) contains two phases. A pure substance is one that is uniform and invariable in chemical composition. A pure substance can exist in more than one phase, but its chemical composition must be the same in each phase. For example, if liquid water and water vapor form a system with two phases, the system can be regarded as a pure substance because each phase has the same composition. The nature of phases that coexist in equilibrium is addressed by the phase rule (Section 2.3, Multicomponent Systems). Equilibrium Equilibrium means a condition of balance. In thermodynamics the concept includes not only a balance of forces, but also a balance of other influences. Each kind of influence refers to a particular aspect of thermodynamic (complete) equilibrium. Thermal equilibrium refers to an equality of temperature, mechanical equilibrium to an equality of pressure, and phase equilibrium to an equality of chemical potentials (Section 2.3, Multicomponent Systems). Chemical equilibrium is also established in terms of chemical potentials (Section 2.4, Reaction Equilibrium). For complete equilibrium the several types of equilibrium must exist individually. To determine if a system is in thermodynamic equilibrium, one may think of testing it as follows: isolate the system from its surroundings and watch for changes in its observable properties. If there are no changes, it may be concluded that the system was in equilibrium at the moment it was isolated. The system can be said to be at an equilibrium state. When a system is isolated, it cannot interact with its surroundings; however, its state can change as a consequence of spontaneous events occurring internally as its intensive properties, such as temperature and pressure, tend toward uniform values. When all such changes cease, the system is in equilibrium. At equilibrium. temperature and pressure are uniform throughout. If gravity is significant, a pressure variation with height can exist, as in a vertical column of liquid. Temperature A scale of temperature independent of the thermometric substance is called a thermodynamic temperature scale. The Kelvin scale, a thermodynamic scale, can be elicited from the second law of thermodynamics (Section 2.1, The Second Law of Thermodynamics, Entropy). The definition of temperature following from the second law is valid over all temperature ranges and provides an essential connection between the several empirical measures of temperature. In particular, temperatures evaluated using a constantvolume gas thermometer are identical to those of the Kelvin scale over the range of temperatures where gas thermometry can be used. The empirical gas scale is based on the experimental observations that (1) at a given temperature level all gases exhibit the same value of the product pv (p is pressure and v the specific volume on a molar basis) if the pressure is low enough, and (2) the value of the product pv increases with the temperature level. On this basis the gas temperature scale is defined by T=

1 lim( pv ) R p→ 0

where T is temperature and R is the universal gas constant. The absolute temperature at the triple point of water (Section 2.3, P-v-T Relations) is fixed by international agreement to be 273.16 K on the Kelvin temperature scale. R is then evaluated experimentally as R = 8.314 kJ/kmol · K (1545 ft · lbf/lbmol · °R). The Celsius termperature scale (also called the centigrade scale) uses the degree Celsius (°C), which has the same magnitude as the kelvin. Thus, temperature differences are identical on both scales. However, the zero point on the Celsius scale is shifted to 273.15 K, as shown by the following relationship between the Celsius temperature and the Kelvin temperature: T (°C) = T (K) − 273.15 On the Celsius scale, the triple point of water is 0.01°C and 0 K corresponds to –273.15°C. © 2005 by CRC Press LLC

(2.1)

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Chapter 2

Two other temperature scales are commonly used in engineering in the U.S. By definition, the Rankine scale, the unit of which is the degree rankine (°R), is proportional to the Kelvin temperature according to T (°R) = 1.8T (K)

(2.2)

The Rankine scale is also an absolute thermodynamic scale with an absolute zero that coincides with the absolute zero of the Kelvin scale. In thermodynamic relationships, temperature is always in terms of the Kelvin or Rankine scale unless specifically stated otherwise. A degree of the same size as that on the Rankine scale is used in the Fahrenheit scale, but the zero point is shifted according to the relation T (°F) = T (°R) − 459.67

(2.3)

Substituting Equation 2.1 and Equation 2.2 into Equation 2.3 gives T (°F) = 1.8T (°C) + 32

(2.4)

This equation shows that the Fahrenheit temperature of the ice point (0°C) is 32°F and of the steam point (100°C) is 212°F. The 100 Celsius or Kelvin degrees between the ice point and steam point corresponds to 180 Fahrenheit or Rankine degrees. To provide a standard for temperature measurement taking into account both theoretical and practical considerations, the International Temperature Scale of 1990 (ITS-90) is defined in such a way that the temperature measured on it conforms with the thermodynamic temperature, the unit of which is the kelvin, to within the limits of accuracy of measurement obtainable in 1990. Further discussion of ITS-90 is provided by Preston-Thomas (1990).

The First Law of Thermodynamics, Energy Energy is a fundamental concept of thermodynamics and one of the most significant aspects of engineering analysis. Energy can be stored within systems in various macroscopic forms: kinetic energy, gravitational potential energy, and internal energy. Energy can also be transformed from one form to another and transferred between systems. For closed systems, energy can be transferred by work and heat transfer. The total amount of energy is conserved in all transformations and transfers. Work In thermodynamics, the term work denotes a means for transferring energy. Work is an effect of one system on another that is identified and measured as follows: work is done by a system on its surroundings if the sole effect on everything external to the system could have been the raising of a weight. The test of whether a work interaction has taken place is not that the elevation of a weight is actually changed, nor that a force actually acted through a distance, but that the sole effect could be the change in elevation of a mass. The magnitude of the work is measured by the number of standard weights that could have been raised. Since the raising of a weight is in effect a force acting through a distance, the work concept of mechanics is preserved. This definition includes work effects such as is associated with rotating shafts, displacement of the boundary, and the flow of electricity. Work done by a system is considered positive: W > 0. Work done on a system is considered negative: W < 0. The time rate of doing work, or power, is symbolized by W˙ and adheres to the same sign convention. Energy A closed system undergoing a process that involves only work interactions with its surroundings experiences an adiabatic process. On the basis of experimental evidence, it can be postulated that when a closed system is altered adiabatically, the amount of work is fixed by the end states of the system and is © 2005 by CRC Press LLC

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Engineering Thermodynamics

independent of the details of the process. This postulate, which is one way the first law of thermodynamics can be stated, can be made regardless of the type of work interaction involved, the type of process, or the nature of the system. As the work in an adiabatic process of a closed system is fixed by the end states, an extensive property called energy can be defined for the system such that its change between two states is the work in an adiabatic process that has these as the end states. In engineering thermodynamics the change in the energy of a system is considered to be made up of three macroscopic contributions: the change in kinetic energy, KE, associated with the motion of the system as a whole relative to an external coordinate frame, the change in gravitational potential energy, PE, associated with the position of the system as a whole in the Earth’s gravitational field, and the change in internal energy, U, which accounts for all other energy associated with the system. Like kinetic energy and gravitational potential energy, internal energy is an extensive property. In summary, the change in energy between two states of a closed system in terms of the work Wad of an adiabatic process between these states is

( KE2 − KE1 ) + ( PE2 − PE1 ) + (U2 − U1 ) = −Wad

(2.5)

where 1 and 2 denote the initial and final states, respectively, and the minus sign before the work term is in accordance with the previously stated sign convention for work. Since any arbitrary value can be assigned to the energy of a system at a given state 1, no particular significance can be attached to the value of the energy at state 1 or at any other state. Only changes in the energy of a system have significance. The specific energy (energy per unit mass) is the sum of the specific internal energy, u, the specific kinetic energy, v2/2, and the specific gravitational potential energy, gz, such that

specific energy = u +

v2 + gz 2

(2.6)

where the velocity v and the elevation z are each relative to specified datums (often the Earth’s surface) and g is the acceleration of gravity. A property related to internal energy u, pressure p, and specific volume v is enthalpy, defined by h = u + pv

(2.7a)

H = U + pV

(2.7b)

or on an extensive basis

Heat Closed systems can also interact with their surroundings in a way that cannot be categorized as work, as, for example, a gas (or liquid) contained in a closed vessel undergoing a process while in contact with a flame. This type of interaction is called a heat interaction, and the process is referred to as nonadiabatic. A fundamental aspect of the energy concept is that energy is conserved. Thus, since a closed system experiences precisely the same energy change during a nonadiabatic process as during an adiabatic process between the same end states, it can be concluded that the net energy transfer to the system in each of these processes must be the same. It follows that heat interactions also involve energy transfer. Denoting the amount of energy transferred to a closed system in heat interactions by Q, these considerations can be summarized by the closed system energy balance:

(U2 − U1 ) + ( KE2 − KE1 ) + ( PE2 − PE1 ) = Q − W © 2005 by CRC Press LLC

(2.8)

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2-6

Chapter 2

The closed system energy balance expresses the conservation of energy principle for closed systems of all kinds. The quantity denoted by Q in Equation 2.8 accounts for the amount of energy transferred to a closed system during a process by means other than work. On the basis of experiments it is known that such an energy transfer is induced only as a result of a temperature difference between the system and its surroundings and occurs only in the direction of decreasing temperature. This means of energy transfer is called an energy transfer by heat. The following sign convention applies: Q > 0: heat transfer to the system Q < 0: heat transfer from the system The time rate of heat transfer, denoted by Q˙ , adheres to the same sign convention. Methods based on experiment are available for evaluating energy transfer by heat. These methods recognize two basic transfer mechanisms: conduction and thermal radiation. In addition, theoretical and empirical relationships are available for evaluating energy transfer involving combined modes such as convection. Further discussion of heat transfer fundamentals is provided in Chapter 4. The quantities symbolized by W and Q account for transfers of energy. The terms work and heat denote different means whereby energy is transferred and not what is transferred. Work and heat are not properties, and it is improper to speak of work or heat “contained” in a system. However, to achieve economy of expression in subsequent discussions, W and Q are often referred to simply as work and heat transfer, respectively. This less formal approach is commonly used in engineering practice. Power Cycles Since energy is a property, over each cycle there is no net change in energy. Thus, Equation 2.8 reads for any cycle Qcycle = Wcycle That is, for any cycle the net amount of energy received through heat interactions is equal to the net energy transferred out in work interactions. A power cycle, or heat engine, is one for which a net amount of energy is transferred out by work: Wcycle > 0. This equals the net amount of energy transferred in by heat. Power cycles are characterized both by addition of energy by heat transfer, QA, and inevitable rejections of energy by heat transfer, QR : Qcycle = QA − QR Combining the last two equations, Wcycle = QA − QR The thermal efficiency of a heat engine is defined as the ratio of the net work developed to the total energy added by heat transfer: η=

Wcycle QA

= 1−

QR QA

(2.9)

The thermal efficiency is strictly less than 100%. That is, some portion of the energy QA supplied is invariably rejected QR ≠ 0. © 2005 by CRC Press LLC

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Engineering Thermodynamics

The Second Law of Thermodynamics, Entropy Many statements of the second law of thermodynamics have been proposed. Each of these can be called a statement of the second law or a corollary of the second law since, if one is invalid, all are invalid. In every instance where a consequence of the second law has been tested directly or indirectly by experiment it has been verified. Accordingly, the basis of the second law, like every other physical law, is experimental evidence. Kelvin-Planck Statement The Kelvin-Planck statement of the second law of thermodynamics refers to a thermal reservoir. A thermal reservoir is a system that remains at a constant temperature even though energy is added or removed by heat transfer. A reservoir is an idealization, of course, but such a system can be approximated in a number of ways — by the Earth’s atmosphere, large bodies of water (lakes, oceans), and so on. Extensive properties of thermal reservoirs, such as internal energy, can change in interactions with other systems even though the reservoir temperature remains constant, however. The Kelvin-Planck statement of the second law can be given as follows: It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir. In other words, a perpetual-motion machine of the second kind is impossible. Expressed analytically, the Kelvin-Planck statement is Wcycle ≤ 0

(single reservoir)

where the words single reservoir emphasize that the system communicates thermally only with a single reservoir as it executes the cycle. The “less than” sign applies when internal irreversibilities are present as the system of interest undergoes a cycle and the “equal to” sign applies only when no irreversibilities are present. Irreversibilities A process is said to be reversible if it is possible for its effects to be eradicated in the sense that there is some way by which both the system and its surroundings can be exactly restored to their respective initial states. A process is irreversible if there is no way to undo it. That is, there is no means by which the system and its surroundings can be exactly restored to their respective initial states. A system that has undergone an irreversible process is not necessarily precluded from being restored to its initial state. However, were the system restored to its initial state, it would not also be possible to return the surroundings to their initial state. There are many effects whose presence during a process renders it irreversible. These include, but are not limited to, the following: heat transfer through a finite temperature difference; unrestrained expansion of a gas or liquid to a lower pressure; spontaneous chemical reaction; mixing of matter at different compositions or states; friction (sliding friction as well as friction in the flow of fluids); electric current flow through a resistance; magnetization or polarization with hysteresis; and inelastic deformation. The term irreversibility is used to identify effects such as these. Irreversibilities can be divided into two classes, internal and external. Internal irreversibilities are those that occur within the system, while external irreversibilities are those that occur within the surroundings, normally the immediate surroundings. As this division depends on the location of the boundary there is some arbitrariness in the classification (by locating the boundary to take in the immediate surroundings, all irreversibilities are internal). Nonetheless, valuable insights can result when this distinction between irreversibilities is made. When internal irreversibilities are absent during a process, the process is said to be internally reversible. At every intermediate state of an internally reversible process of a closed system, all intensive properties are uniform throughout each phase present: the temperature, pressure, specific volume, and other intensive properties do not vary with position. The discussions to follow compare the actual and internally reversible process concepts for two cases of special interest. © 2005 by CRC Press LLC

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2-8

Chapter 2

For a gas as the system, the work of expansion arises from the force exerted by the system to move the boundary against the resistance offered by the surroundings: W=

2

2

∫ Fdx = ∫ pAdx 1

1

where the force is the product of the moving area and the pressure exerted by the system there. Noting that Adx is the change in total volume of the system, W=

∫

2

pdV

1

This expression for work applies to both actual and internally reversible expansion processes. However, for an internally reversible process p is not only the pressure at the moving boundary but also the pressure of the entire system. Furthermore, for an internally reversible process the volume equals mv, where the specific volume v has a single value throughout the system at a given instant. Accordingly, the work of an internally reversible expansion (or compression) process is W=m

∫

2

pdv

(2.10)

1

When such a process of a closed system is represented by a continuous curve on a plot of pressure vs. specific volume, the area under the curve is the magnitude of the work per unit of system mass (area a-b-c′-d′ of Figure 2.3, for example). Although improved thermodynamic performance can accompany the reduction of irreversibilities, steps in this direction are normally constrained by a number of practical factors often related to costs. For example, consider two bodies able to communicate thermally. With a finite temperature difference between them, a spontaneous heat transfer would take place and, as noted previously, this would be a source of irreversibility. The importance of the heat transfer irreversibility diminishes as the temperature difference narrows; and as the temperature difference between the bodies vanishes, the heat transfer approaches ideality. From the study of heat transfer it is known, however, that the transfer of a finite amount of energy by heat between bodies whose temperatures differ only slightly requires a considerable amount of time, a large heat transfer surface area, or both. To approach ideality, therefore, a heat transfer would require an exceptionally long time and/or an exceptionally large area, each of which has cost implications constraining what can be achieved practically. Carnot Corollaries The two corollaries of the second law known as Carnot corollaries state: (1) the thermal efficiency of an irreversible power cycle is always less than the thermal efficiency of a reversible power cycle when each operates between the same two thermal reservoirs; (2) all reversible power cycles operating between the same two thermal reservoirs have the same thermal efficiency. A cycle is considered reversible when there are no irreversibilities within the system as it undergoes the cycle, and heat transfers between the system and reservoirs occur ideally (that is, with a vanishingly small temperature difference). Kelvin Temperature Scale Carnot corollary 2 suggests that the thermal efficiency of a reversible power cycle operating between two thermal reservoirs depends only on the temperatures of the reservoirs and not on the nature of the substance making up the system executing the cycle or the series of processes. With Equation 2.9 it can be concluded that the ratio of the heat transfers is also related only to the temperatures, and is independent of the substance and processes:

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2-9

Engineering Thermodynamics

 QC   Q  rev = ψ (TC , TH )  H cycle

where QH is the energy transferred to the system by heat transfer from a hot reservoir at temperature TH , and QC is the energy rejected from the system to a cold reservoir at temperature TC . The words rev cycle emphasize that this expression applies only to systems undergoing reversible cycles while operating between the two reservoirs. Alternative temperature scales correspond to alternative specifications for the function ψ in this relation. The Kelvin temperature scale is based on ψ(TC , TH ) = TC /TH . Then  QC  TC  Q  rev = T  H H

(2.11)

cycle

This equation defines only a ratio of temperatures. The specification of the Kelvin scale is completed by assigning a numerical value to one standard reference state. The state selected is the same used to define the gas scale: at the triple point of water the temperature is specified to be 273.16 K. If a reversible cycle is operated between a reservoir at the reference-state temperature and another reservoir at an unknown temperature T, then the latter temperature is related to the value at the reference state by  Q T = 273.16   Q ′  rev

cycle

where Q is the energy received by heat transfer from the reservoir at temperature T, and Q′ is the energy rejected to the reservoir at the reference temperature. Accordingly, a temperature scale is defined that is valid over all ranges of temperature and that is independent of the thermometric substance. Carnot Efficiency For the special case of a reversible power cycle operating between thermal reservoirs at temperatures TH and TC on the Kelvin scale, combination of Equation 2.9 and Equation 2.11 results in ηmax = 1 −

TC TH

(2.12)

called the Carnot efficiency. This is the efficiency of all reversible power cycles operating between thermal reservoirs at TH and TC . Moreover, it is the maximum theoretical efficiency that any power cycle, real or ideal, could have while operating between the same two reservoirs. As temperatures on the Rankine scale differ from Kelvin temperatures only by the factor 1.8, the above equation may be applied with either scale of temperature. The Clausius Inequality The Clausius inequality provides the basis for introducing two ideas instrumental for quantitative evaluations of processes of systems from a second law perspective: entropy and entropy generation. The Clausius inequality states that  δQ 

∫  T 

≤0

(2.13a)

b

where δQ represents the heat transfer at a part of the system boundary during a portion of the cycle, and T is the absolute temperature at that part of the boundary. The symbol δ is used to distinguish the

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2-10

Chapter 2

differentials of nonproperties, such as heat and work, from the differentials of properties, written with the symbol d. The subscript b indicates that the integrand is evaluated at the boundary of the system executing the cycle. The symbol ∫ indicates that the integral is to be performed over all parts of the boundary and over the entire cycle. The Clausius inequality can be demonstrated using the Kelvin-Planck statement of the second law, and the significance of the inequality is the same: the equality applies when there are no internal irreversibilities as the system executes the cycle, and the inequality applies when internal irreversibilities are present. The Clausius inequality can be expressed alternatively as  δQ 

∫  T 

= − Sgen

(2.13b)

b

where Sgen can be viewed as representing the strength of the inequality. The value of Sgen is positive when internal irreversibilities are present, zero when no internal irreversibilities are present, and can never be negative. Accordingly, Sgen is a measure of the irreversibilities present within the system executing the cycle. In the next section, Sgen is identified as the entropy generated (or produced) by internal irreversibilities during the cycle.

Entropy and Entropy Generation Entropy Consider two cycles executed by a closed system. One cycle consists of an internally reversible process A from state 1 to state 2, followed by an internally reversible process C from state 2 to state 1. The other cycle consists of an internally reversible process B from state 1 to state 2, followed by the same process C from state 2 to state 1 as in the first cycle. For these cycles, Equation 2.13b takes the form      

∫

2

∫

2

1

1

δQ    + T A  δQ    + T B 

∫

1

∫

1

2

2

δQ   = − Sgen = 0 T C δQ   = − Sgen = 0 T C

where Sgen has been set to zero since the cycles are composed of internally reversible processes. Subtracting these equations leaves   

∫

2

1

δQ    =  T A 

∫

2

1

δQ   T B

Since A and B are arbitrary, it follows that the integral of δQ/T has the same value for any internally reversible process between the two states: the value of the integral depends on the end states only. It can be concluded, therefore, that the integral defines the change in some property of the system. Selecting the symbol S to denote this property, its change is given by  S2 − S1 =  

∫

2

1

δQ   T  int

(2.14a)

rev

where the subscript int rev indicates that the integration is carried out for any internally reversible process linking the two states. This extensive property is called entropy. Since entropy is a property, the change in entropy of a system in going from one state to another is the same for all processes, both internally reversible and irreversible, between these two states. In other words, once the change in entropy between two states has been evaluated, this is the magnitude of the entropy change for any process of the system between these end states. © 2005 by CRC Press LLC

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2-11

Engineering Thermodynamics

The definition of entropy change expressed on a differential basis is  δQ  dS =    T  int

(2.14b)

rev

Equation 2.14b indicates that when a closed system undergoing an internally reversible process receives energy by heat transfer, the system experiences an increase in entropy. Conversely, when energy is removed from the system by heat transfer, the entropy of the system decreases. This can be interpreted to mean that an entropy transfer is associated with (or accompanies) heat transfer. The direction of the entropy transfer is the same as that of the heat transfer. In an adiabatic internally reversible process of a closed system the entropy would remain constant. A constant entropy process is called an isentropic process. On rearrangement, Equation 2.14b becomes

(δQ)intrev = TdS Then, for an internally reversible process of a closed system between state 1 and state 2,

∫

2

Qint = m Tds

(2.15)

1

rev

When such a process is represented by a continuous curve on a plot of temperature vs. specific entropy, the area under the curve is the magnitude of the heat transfer per unit of system mass. Entropy Balance For a cycle consisting of an actual process from state 1 to state 2, during which internal irreversibilities are present, followed by an internally reversible process from state 2 to state 1, Equation 2.13b takes the form 2

 δQ 

1

 δQ 

∫  T  + ∫  T  1

2

b

int rev

= − Sgen

where the first integral is for the actual process and the second integral is for the internally reversible process. Since no irreversibilities are associated with the internally reversible process, the term Sgen accounting for the effect of irreversibilities during the cycle can be identified with the actual process only. Applying the definition of entropy change, the second integral of the foregoing equation can be expressed as S1 − S2 =

1

 δQ 

∫  T  2

int rev

Introducing this and rearranging the equation, the closed system entropy balance results: S2 − S1 =

© 2005 by CRC Press LLC

2

 δQ 

∫  T  1

+ Sgen b

______ ______

______

entropy entropy change transfer

entropy generation

(2.16)

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2-12

Chapter 2

When the end states are fixed, the entropy change on the left side of Equation 2.16 can be evaluated independently of the details of the process from state 1 to state 2. However, the two terms on the right side depend explicitly on the nature of the process and cannot be determined solely from knowledge of the end states. The first term on the right side is associated with heat transfer to or from the system during the process. This term can be interpreted as the entropy transfer associated with (or accompanying) heat transfer. The direction of entropy transfer is the same as the direction of the heat transfer, and the same sign convention applies as for heat transfer: a positive value means that entropy is transferred into the system, and a negative value means that entropy is transferred out. The entropy change of a system is not accounted for solely by entropy transfer, but is also due to the second term on the right side of Equation 2.16 denoted by Sgen. The term Sgen is positive when internal irreversibilities are present during the process and vanishes when internal irreversibilities are absent. This can be described by saying that entropy is generated (or produced) within the system by the action of irreversibilities. The second law of thermodynamics can be interpreted as specifying that entropy is generated by irreversibilities and conserved only in the limit as irreversibilities are reduced to zero. Since Sgen measures the effect of irreversibilities present within a system during a process, its value depends on the nature of the process and not solely on the end states. Entropy generation is not a property. When applying the entropy balance, the objective is often to evaluate the entropy generation term. However, the value of the entropy generation for a given process of a system usually does not have much significance by itself. The significance is normally determined through comparison. For example, the entropy generation within a given component might be compared to the entropy generation values of the other components included in an overall system formed by these components. By comparing entropy generation values, the components where appreciable irreversibilities occur can be identified and rank ordered. This allows attention to be focused on the components that contribute most heavily to inefficient operation of the overall system. To evaluate the entropy transfer term of the entropy balance requires information regarding both the heat transfer and the temperature on the boundary where the heat transfer occurs. The entropy transfer term is not always subject to direct evaluation, however, because the required information is either unknown or undefined, such as when the system passes through states sufficiently far from equilibrium. In practical applications, it is often convenient, therefore, to enlarge the system to include enough of the immediate surroundings that the temperature on the boundary of the enlarged system corresponds to the ambient temperature, Tamb. The entropy transfer term is then simply Q/Tamb. However, as the irreversibilities present would not be just those for the system of interest but those for the enlarged system, the entropy generation term would account for the effects of internal irreversibilities within the system and external irreversibilities present within that portion of the surroundings included within the enlarged system. A form of the entropy balance convenient for particular analyses is the rate form: dS = dt

Q˙ j

∑T j

+ S˙gen

(2.17)

j

where dS/dt is the time rate of change of entropy of the system. The term Q˙ j / Tj represents the time rate of entropy transfer through the portion of the boundary whose instantaneous temperature is Tj . The term S˙gen accounts for the time rate of entropy generation due to irreversibilities within the system. For a system isolated from its surroundings, the entropy balance is

(S2 − S1 )isol = Sgen

(2.18)

where Sgen is the total amount of entropy generated within the isolated system. Since entropy is generated in all actual processes, the only processes of an isolated system that actually can occur are those for which the entropy of the isolated system increases. This is known as the increase of entropy principle.

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2-13

Engineering Thermodynamics

2.2 Control Volume Applications Since most applications of engineering thermodynamics are conducted on a control volume basis, the control volume formulations of the mass, energy, and entropy balances presented in this section are especially important. These are given here in the form of overall balances. Equations of change for mass, energy, and entropy in the form of differential equations are also available in the literature (see, e.g., Bird et al., 1960).

Conservation of Mass When applied to a control volume, the principle of mass conservation states: The time rate of accumulation of mass within the control volume equals the difference between the total rates of mass flow in and out across the boundary. An important case for engineering practice is one for which inward and outward flows occur, each through one or more ports. For this case the conservation of mass principle takes the form dmcv = dt

∑ m˙ − ∑ m˙ i

i

(2.19)

e

e

The left side of this equation represents the time rate of change of mass contained within the control volume, m˙ i denotes the mass flow rate at an inlet, and m˙ e is the mass flow rate at an outlet. The volumetric flow rate through a portion of the control surface with area dA is the product of the velocity component normal to the area, vn, times the area: vn dA. The mass flow rate through dA is ρ(vn dA). The mass rate of flow through a port of area A is then found by integration over the area m˙ =

∫ ρv A

n

dA

For one-dimensional flow the intensive properties are uniform with position over area A, and the last equation becomes m˙ = ρvA =

vA v

(2.20)

where v denotes the specific volume and the subscript n has been dropped from velocity for simplicity.

Control Volume Energy Balance When applied to a control volume, the principle of energy conservation states: The time rate of accumulation of energy within the control volume equals the difference between the total incoming rate of energy transfer and the total outgoing rate of energy transfer. Energy can enter and exit a control volume by work and heat transfer. Energy also enters and exits with flowing streams of matter. Accordingly, for a control volume with one-dimensional flow at a single inlet and a single outlet,     d (U + KE + PE )cv v2 v2 = Q˙ cv − W˙ + m˙  ui + i + gz i  − m˙  ue + e + gz e  2 2 dt     ___________

(2.21)

___________

where the underlined terms account for the specific energy of the incoming and outgoing streams. The terms Q˙ cv and W˙ account, respectively, for the net rates of energy transfer by heat and work over the boundary (control surface) of the control volume. © 2005 by CRC Press LLC

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2-14

Chapter 2

Because work is always done on or by a control volume where matter flows across the boundary, the quantity W˙ of Equation 2.21 can be expressed in terms of two contributions: one is the work associated with the force of the fluid pressure as mass is introduced at the inlet and removed at the exit. The other, denoted as W˙ cv, includes all other work effects, such as those associated with rotating shafts, displacement of the boundary, and electrical effects. The work rate concept of mechanics allows the first of these contributions to be evaluated in terms of the product of the pressure force, pA, and velocity at the point of application of the force. To summarize, the work term W˙ of Equation 2.21 can be expressed (with Equation 2.20) as W˙ = W˙ cv + ( pe Ae )v e − ( pi Ai )v i

(2.22)

= W˙ cv + m˙ e ( pe ve ) − m˙ i ( pi vi )

The terms m˙ i (pvi) and m˙ e (peve) account for the work associated with the pressure at the inlet and outlet, respectively, and are commonly referred to as flow work. Substituting Equation 2.22 into Equation 2.21, and introducing the specific enthalpy h, the following form of the control volume energy rate balance results:     d (U + KE + PE )cv v2 v2 = Q˙ cv − W˙ cv + m˙ i  hi + i + gz i  − m˙ e  he + e + gz e  2 2 dt    

(2.23)

To allow for applications where there may be several locations on the boundary through which mass enters or exits, the following expression is appropriate: d (U + KE + PE )cv = Q˙ cv − W˙ cv + dt







v i2



v e2

∑ m˙  h + 2 + gz  − ∑ m˙  h + 2 + gz  i

i

i

e

i

e

e

(2.24)

e

Equation 2.24 is an accounting rate balance for the energy of the control volume. It states that the time rate of accumulation of energy within the control volume equals the difference between the total rates of energy transfer in and out across the boundary. The mechanisms of energy transfer are heat and work, as for closed systems, and the energy accompanying the entering and exiting mass.

Control Volume Entropy Balance Like mass and energy, entropy is an extensive property. And like mass and energy, entropy can be transferred into or out of a control volume by streams of matter. As this is the principal difference between the closed system and control volume forms, the control volume entropy rate balance is obtained by modifying Equation 2.17 to account for these entropy transfers. The result is dScv = dt

Q˙ j

∑ T + ∑ m˙ s − ∑ m˙ s i i

j

j

i

e e

+ S˙gen

e

_____ ______________________ _________ rate of entropy change

rate of entropy transfer

(2.25)

rate of entropy generation

where dScv /dt represents the time rate of change of entropy within the control volume. The terms m˙ i si and m˙ e se account, respectively, for rates of entropy transfer into and out of the control volume associated with mass flow. One-dimensional flow is assumed at locations where mass enters and exits. Q˙ j represents © 2005 by CRC Press LLC

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2-15

Engineering Thermodynamics

the time rate of heat transfer at the location on the boundary where the instantaneous temperature is Tj ; and Q˙ j / Tj accounts for the associated rate of entropy transfer. S˙gen denotes the time rate of entropy generation due to irreversibilities within the control volume. When a control volume comprises a number of components, S˙gen is the sum of the rates of entropy generation of the components.

Control Volumes at Steady State Engineering systems are often idealized as being at steady state, meaning that all properties are unchanging in time. For a control volume at steady state, the identity of the matter within the control volume change continuously, but the total amount of mass remains constant. At steady state, Equation 2.19 reduces to

∑ m˙ = ∑ m˙ i

i

(2.26a)

e

e

The energy rate balance of Equation 2.24 becomes, at steady state, 0 = Q˙ cv − W˙ cv +







v i2



v e2

∑ m˙  h + 2 + gz  − ∑ m˙  h + 2 + gz  i

i

i

e

i

e

e

(2.26b)

e

At steady state, the entropy rate balance of Equation 2.25 reads 0=

Q˙ j

∑ T + ∑ m˙ s − ∑ m˙ s i i

j

j

i

e e

+ S˙gen

(2.26c)

e

Mass and energy are conserved quantities, but entropy is not generally conserved. Equation 2.26a indicates that the total rate of mass flow into the control volume equals the total rate of mass flow out of the control volume. Similarly, Equation 2.26b states that the total rate of energy transfer into the control volume equals the total rate of energy transfer out of the control volume. However, Equation 2.26c shows that the rate at which entropy is transferred out exceeds the rate at which entropy enters, the difference being the rate of entropy generation within the control volume owing to irreversibilities. Applications frequently involve control volumes having a single inlet and a single outlet, as, for example, the control volume of Figure 2.1 where heat transfer (if any) occurs at Tb: the temperature, or a suitable average temperature, on the boundary where heat transfer occurs. For this case the mass rate balance, Equation 2.26a, reduces to m˙ i = m˙ e . Denoting the common mass flow rate by m˙ , Equation 2.26b and Equation 2.26c read, respectively,    v 2 − v e2  + g(z i − z e ) 0 = Q˙ cv − W˙ cv + m˙ (hi − he ) +  i   2    0=

Q˙ cv + m˙ (si − se ) + S˙gen Tb

(2.27a)

(2.28a)

When Equation 2.27a and Equation 2.28a are applied to particular cases of interest, additional simplifications are usually made. The heat transfer term Q˙ cv is dropped when it is insignificant relative to other energy transfers across the boundary. This may be the result of one or more of the following: (1) the outer surface of the control volume is insulated; (2) the outer surface area is too small for there to be effective heat transfer; (3) the temperature difference between the control volume and its surroundings is small enough that the heat transfer can be ignored; (4) the gas or liquid passes through the control volume so quickly that there is not enough time for significant heat transfer to occur. The work term © 2005 by CRC Press LLC

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2-16

Chapter 2

FIGURE 2.1 One-inlet, one-outlet control volume at steady state.

W˙ cv drops out of the energy rate balance when there are no rotating shafts, displacements of the boundary, electrical effects, or other work mechanisms associated with the control volume being considered. The changes in kinetic and potential energy of Equation 2.27a are frequently negligible relative to other terms in the equation. The special forms of Equation 2.27a and Equation 2.28a listed in Table 2.1 are obtained as follows: when there is no heat transfer, Equation 2.28a gives

se − si =

S˙gen m˙

≥0

(2.28b)

(no heat transfer) Accordingly, when irreversibilities are present within the control volume, the specific entropy increases as mass flows from inlet to outlet. In the ideal case in which no internal irreversibilities are present, mass passes through the control volume with no change in its entropy — that is, isentropically. For no heat transfer, Equation 2.27a gives    v 2 − v e2  + g(z i − z e ) W˙ cv = m˙ (hi − he ) +  i   2   

(2.27b)

A special form that is applicable, at least approximately, to compressors, pumps, and turbines results from dropping the kinetic and potential energy terms of Equation 2.27b, leaving W˙ cv = m˙ (hi − he )

(compressors, pumps, and turbines)

(2.27c)

In throttling devices a significant reduction in pressure is achieved simply by introducing a restriction into a line through which a gas or liquid flows. For such devices W˙ cv = 0 and Equation 2.27c reduces further to read © 2005 by CRC Press LLC

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2-17

Engineering Thermodynamics

TABLE 2.1 Energy and Entropy Balances for One-Inlet, One-Outlet Control Volumes at Steady State and No Heat Transfer Energy balance    v 2 − v e2  + g(z i − z e ) W˙ cv = m˙ (hi − he ) +  i   2    Compressors, pumps, and turbinesa

(2.27b)

W˙ cv = m˙ (hi − he )

(2.27c)

he ≅ hi

(2.27d)

v e = v i2 + 2(hi − he )

(2.27f)

Throttling Nozzles, diffusersb

Entropy balance se − si =

S˙gen ≥0 m˙

(2.28b)

a For an ideal gas with constant cp, Equation 1′ of Table 2.7 allows Equation 2.27c to be written as

˙ p (Ti − Te ) ‘ W˙ cv = mc

(2.27c′)

The power developed in an isentropic process is obtained with Equation 5′ of Table 2.7 as ( k −1) k  ˙ p Ti 1 − ( pe pi ) W˙ cv = mc  

(s = c)

(2.27c″)

where cp = kR/(k – 1). For an ideal gas with constant cp, Equation 1′ of Table 2.7 allows Equation 2.27f to be written as b

v e = v i2 + 2c p (Ti − Te )

(2.27f′)

The exit velocity for an isentropic process is obtained with Equation 5′ of Table 2.7 as v e = v i2 + 2c p Ti 1 − ( pe pi ) 

( k −1)

k

 

(s = c)

(2.27f″)

where cp = kR/(k – 1).

he ≅ hi

(throttling process)

(2.27d)

That is, upstream and downstream of the throttling device, the specific enthalpies are equal. A nozzle is a flow passage of varying cross-sectional area in which the velocity of a gas or liquid increases in the direction of flow. In a diffuser, the gas or liquid decelerates in the direction of flow. For such devices, W˙ cv = 0. The heat transfer and potential energy change are also generally negligible. Then Equation 2.27b reduces to 0 = hi − he + Solving for the outlet velocity © 2005 by CRC Press LLC

v i2 − v e2 2

(2.27e)

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2-18

Chapter 2

v e = v i2 + 2(hi − he )

(nozzle, diffuser)

(2.27f)

Further discussion of the flow-through nozzles and diffusers is provided in Chapter 3. The mass, energy, and entropy rate balances, Equations 2.26, can be applied to control volumes with multiple inlets and/or outlets, as, for example, cases involving heat-recovery steam generators, feedwater heaters, and counterflow and crossflow heat exchangers. Transient (or unsteady) analyses can be conducted with Equation 2.19, Equation 2.24, and Equation 2.25. Illustrations of all such applications are provided by Moran and Shapiro (1995). Example 1 A turbine receives steam at 7 MPa, 440°C and exhausts at 0.2 MPa for subsequent process heating duty. If heat transfer and kinetic/potential energy effects are negligible, determine the steam mass flow rate, in kg/hr, for a turbine power output of 30 MW when (a) the steam quality at the turbine outlet is 95%, (b) the turbine expansion is internally reversible. Solution. With the indicated idealizations, Equation 2.27c is appropriate. Solving, m˙ = W˙ cv /(hi − he ). Steam table data (Table A.5) at the inlet condition are hi = 3261.7 kJ/kg, si = 6.6022 kJ/kg · K. (a) At 0.2 MPa and x = 0.95, he = 2596.5 kJ/kg. Then m˙ =

 10 3 kJ sec   3600 sec  30 MW (3261.7 − 2596.5) kJ kg  1 MW   1 hr 

= 162, 357 kg hr (b) For an internally reversible expansion, Equation 2.28b reduces to give se = si. For this case, he = 2499.6 kJ/kg (x = 0.906), and m˙ = 141,714 kg/hr. Example 2 Air at 500°F, 150 lbf/in.2, and 10 ft/sec expands adiabatically through a nozzle and exits at 60°F, 15 lbf/in.2. For a mass flow rate of 5 lb/sec determine the exit area, in in.2. Repeat for an isentropic expansion to 15 lbf/in.2. Model the air as an ideal gas (Section 2.3, Ideal Gas Model) with specific heat cp = 0.24 Btu/lb · °R (k = 1.4). Solution. The nozle exit area can be evaluated using Equation 2.20, together with the ideal gas equation, v = RT/p: Ae =

m˙ ν e m˙ ( RTe pe ) = ve ve

The exit velocity required by this expression is obtained using Equation 2.27f′ of Table 2.1, v e = v i2 + 2c p (Ti − Te ) 2  32.174 lb ⋅ ft sec 2  10 ft  Btu   778.17 ft ⋅ lbf  =  + 2 0.24 440°R)  (      s  lb ⋅ R   1 Btu 1 lbf   

= 2299.5 ft sec Finally, with R = R /M = 53.33 ft · lbf/lb · °R, © 2005 by CRC Press LLC

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Engineering Thermodynamics

FIGURE 2.2 Open feedwater heater.

 5 lb   53.3 ft ⋅ lbf  520°R ( )  sec   lb ⋅ °R  = 4.02 in.2 Ae =  2299.5 ft  15 lbf   sec   in.2  Using Equation 2.27f″ in Table 2.1 for the isentropic expansion, ve =



(10)2 + 2(0.24)(778.17)(960)(32.174) 1 −  

15  150 

0.4 1.4

  

= 2358.3 ft sec Then Ae = 3.92 in.2. Example 3 Figure 2.2 provides steady-state operating data for an open feedwater heater. Ignoring heat transfer and kinetic/potential energy effects, determine the ratio of mass flow rates, m˙ 1 / m˙ 2 . Solution. For this case Equation 2.26a and Equation 2.26b reduce to read, respectively, m˙ 1 + m˙ 2 = m˙ 3 0 = m˙ 1h1 + m˙ 2 h2 − m˙ 3 h3 Combining and solving for the ratio m˙ 1 / m˙ 2 , m˙ 1 h2 − h3 = m˙ 2 h3 − h1 Inserting steam table data, in kJ/kg, from Table A.5, m˙ 1 2844.8 − 697.2 = = 4.06 697.2 − 167.6 m˙ 2 Internally Reversible Heat Transfer and Work For one-inlet, one-outlet control volumes at steady state, the following expressions give the heat transfer rate and power in the absence of internal irreversibilities: © 2005 by CRC Press LLC

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Chapter 2

 Q˙ cv   m˙  int =   rev

 W˙ cv   m˙  int = −   rev

∫

2

νdp +

1

2

∫ Tds

(2.29)

1

v12 − v 22 + g( z1 − z2 ) 2

(2.30a)

(see, e.g., Moran and Shapiro, 1995). If there is no significant change in kinetic or potential energy from inlet to outlet, Equation 2.30a reads 2  W˙ cv   m˙  int = − 1 νdp   rev

∫

(∆ke = ∆pe = 0)

(2.30b)

The specific volume remains approximately constant in many applications with liquids. Then Equation 30b becomes  W˙ cv   m˙  int = − v( p2 − p1 )  

(v = constant)

(2.30c)

rev

When the states visited by a unit of mass flowing without irreversibilities from inlet to outlet are described by a continuous curve on a plot of temperature vs. specific entropy, Equation 2.29 implies that the area under the curve is the magnitude of the heat transfer per unit of mass flowing. When such an ideal process is described by a curve on a plot of pressure vs. specific volume, as shown in Figure 2.3, the magnitude of the integral ∫vdp of Equation 2.30a and Equation 2.30b is represented by the area a-b-c-d behind the curve. The area a-b-c′-d′ under the curve is identified with the magnitude of the integral ∫pdv of Equation 2.10.

FIGURE 2.3 Internally reversible process on p–v coordinates.

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Engineering Thermodynamics

2.3 Property Relations and Data Pressure, temperature, volume, and mass can be found experimentally. The relationships between the specific heats cv and cp and temperature at relatively low pressure are also accessible experimentally, as are certain other property data. Specific internal energy, enthalpy, and entropy are among those properties that are not so readily obtained in the laboratory. Values for such properties are calculated using experimental data of properties that are more amenable to measurement, together with appropriate property relations derived using the principles of thermodynamics. In this section property relations and data sources are considered for simple compressible systems, which include a wide range of industrially important substances. Property data are provided in the publications of the National Institute of Standards and Technology (formerly the U.S. Bureau of Standards), of professional groups such as the American Society of Mechanical Engineering (ASME), the American Society of Heating. Refrigerating, and Air Conditioning Engineers (ASHRAE), and the American Chemical Society, and of corporate entities such as Dupont and Dow Chemical. Handbooks and property reference volumes such as included in the list of references for this chapter are readily accessed sources of data. Property data are also retrievable from various commercial online data bases. Computer software is increasingly available for this purpose as well.

Basic Relations for Pure Substances An energy balance in differential form for a closed system undergoing an internally reversible process in the absence of overall system motion and the effect of gravity reads dU = (δQ)int − (δW )int rev

rev

From Equation 2.14b, (δQ)int = TdS. When consideration is limited to simple compressible systems: rev systems for which the only significant work in an internally reversible process is associated with volume change, (δW )int = pdV, the following equation is obtained: rev dU = TdS − pdV

(2.31a)

Introducing enthalpy, H = U + pV, the Helmholtz function, Ψ = U – TS, and the Gibbs function, G = H – TS, three additional expressions are obtained: dH = TdS + Vdp

(2.31b)

dΨ = − pdV − SdT

(2.31c)

dG = Vdp − SdT

(2.31d)

Equations 2.31 can be expressed on a per-unit-mass basis as du = Tds − pdv

(2.32a)

dh = Tds + vdp

(2.32b)

dψ = − pdv − sdT

(2.32c)

dg = vdp − sdT

(2.32d)

Similar expressions can be written on a per-mole basis.

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2-22 TABLE 2.2

Chapter 2 Relations from Exact Differentials

Maxwell Relations Since only properties are involved, each of the four differential expressions given by Equations 2.32 is an exact differential exhibiting the general form dz = M(x, y)dx + N(x, y)dy, where the second mixed partial derivatives are equal: (∂M/∂y) = (∂N/∂x). Underlying these exact differentials are, respectively, functions of the form u(s, v), h(s, p), ψ(v, T), and g(T, p). From such considerations the Maxwell relations given in Table 2.2 can be established. Example 4 Derive the Maxwell relation following from Equation 2.32a. Solution. The differential of the function u = u(s, v) is © 2005 by CRC Press LLC

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Engineering Thermodynamics

 ∂u   ∂u  du =   ds +   dv  ∂s  v  ∂v  s By comparison with Equation 2.32a,  ∂u  T =  ,  ∂s  v

 ∂u  −p=   ∂v  s

In Equation 2.32a, T plays the role of M and –p plays the role of N, so the equality of second mixed partial derivatives gives the Maxwell relation,  ∂T   ∂p    = −   ∂v  s  ∂s  v Since each of the properties T, p, v, and s appears on the right side of two of the eight coefficients of Table 2.2, four additional property relations can be obtained by equating such expressions:  ∂u   ∂h    =  ,  ∂s  v  ∂s  p

 ∂u   ∂ψ     =  ∂v  s  ∂v  T

 ∂g   ∂h    =  ,  ∂p  s  ∂p  T

 ∂ψ   ∂g    =   ∂T  v  ∂T  p

These four relations are identified in Table 2.2 by brackets. As any three of Equations 2.32 can be obtained from the fourth simply by manipulation, the 16 property relations of Table 2.2 also can be regarded as following from this single differential expression. Several additional first-derivative property relations can be derived; see, e.g., Zemansky, 1972. Specific Heats and Other Properties Engineering thermodynamics uses a wide assortment of thermodynamic properties and relations among these properties. Table 2.3 lists several commonly encountered properties. Among the entries of Table 2.3 are the specific heats cv and cp . These intensive properties are often required for thermodynamic analysis, and are defined as partial derivations of the functions u(T, v) and h(T, p), respectively,  ∂u  cv =    ∂T  v

(2.33)

 ∂h  cp =    ∂T  p

(2.34)

Since u and h can be expressed either on a unit mass basis or a per-mole basis, values of the specific heats can be similarly expressed. Table 2.4 summarizes relations involving cv and cp. The property k, the specific heat ratio, is k=

cp cv

(2.35)

Values for cv and cp can be obtained via statistical mechanics using spectroscopic measurements. They can also be determined macroscopically through exacting property measurements. Specific heat data for © 2005 by CRC Press LLC

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2-24

TABLE 2.3

Chapter 2

Symbols and Definitions for Selected Properties

Property

Symbol

Pressure

Definition

Property

Symbol

Definition

p

Specific heat, constant volume

cv

Temperature

T

Specific heat, constant pressure

cp

(∂u ∂T ) v (∂h ∂T ) p

Specific volume

v

Volume expansivity

β

Specific internal energy

u

Isothermal compressivity

κ

−

1 (∂v ∂p) T v

Specific entropy

s

Isentropic compressibility

α

−

1 (∂v ∂p) s v

Specific enthalpy

h

u + pv

Isothermal bulk modulus

B

− v(∂p ∂v) T

Specific Helmholtz function

ψ

u – Ts

Isentropic bulk modulus

Bs

− v(∂p ∂v) s

Specific Gibbs function

g

h – Ts

Joule-Thomson coefficient

µJ

Compressibility factor

Z

pv/RT

Joule coefficient

η

(∂T ∂p) h (∂T ∂v)u

Specific heat ratio

k

cp /cv

Velocity of sound

c

1 (∂v ∂T ) p v

− v 2 (∂p ∂v) s

common gases, liquids, and solids are provided by the handbooks and property reference volumes listed among the Chapter 2 references. Specific heats are also considered in Section 2.3 as a part of the discussions of the incompressible model and the ideal gas model. Figure 2.4 shows how cp for water vapor varies as a function of temperature and pressure. Other gases exhibit similar behavior. The figure also gives the variation of cp with temperature in the limit as pressure tends to zero (the ideal gas limit). In this limit cp increases with increasing temperature, which is a characteristic exhibited by other gases as well. The following two equations are often convenient for establishing relations among properties:  ∂x   ∂y      = 1  ∂y  z ∂x z

(2.36a)

 ∂y   ∂z   ∂x        = −1  ∂z  x  ∂x  y  ∂y  z

(2.36b)

Their use is illustrated in Example 5. Example 5 Obtain Equation 2 and Equation 11 of Table 2.4 from Equation 1. Solution. Identifying x, y, z with s, T, and v, respectively, Equation 2.36b reads  ∂T   ∂v   ∂s        = −1  ∂v  s  ∂s  T  ∂T  v Applying Equation 2.36a to each of (∂T/∂v)s and (∂v/∂s)T , 1  ∂v   ∂s   ∂s  = −      =−  ∂T  v (∂T ∂v)s (∂v ∂s)T  ∂T  s  ∂v  T

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Engineering Thermodynamics

TABLE 2.4

Specific Heat Relationsa

 ∂u   ∂s  cv =   = T    ∂T  v  ∂T  v  ∂p   ∂v  = −T      ∂T  v  ∂T  s  ∂h   ∂s  cp =   = T   ∂T  p  ∂T  p  ∂v   ∂p  = T     ∂T  p  ∂T  s  ∂p   ∂v  c p − cv = T      ∂T  v  ∂T  p

(1) (2)

(3)

(4)

(5)

2

 ∂v   ∂p  = −T      ∂T  p  ∂v  T = cp =

(7)

 1   ∂v  T   − v    µ J  ∂T p 

(8)

 1   ∂p  T   − p η   ∂T  v 

(9)

 ∂v   ∂p  =     ∂p  T  ∂v  s

10)

cv = − k=

Tvβ 2 κ

(6)

cp cv

 ∂2 p   ∂cv    = T 2   ∂v  T  ∂T  v

(11)

 ∂c p   ∂2v   ∂p  = −T  ∂T 2   T p

(12)

a See, for example, Moran, M.J. and Shapiro, H.N. 1995. Fundamentals of Engineering Thermodynamics, 3rd ed. Wiley, New York, chap. 11.

Introducing the Maxwell relation from Table 2.2 corresponding to ψ(T, v),  ∂s   ∂v   ∂p    = −     ∂T  v  ∂T  s  ∂T  v With this, Equation 2 of Table 2.4 is obtained from Equation 1, which in turn is obtained in Example 6. Equation 11 of Table 2.4 can be obtained by differentiating Equation 1 with repect to specific volume at fixed temperature, and again using the Maxwell relation corresponding to ψ.

P-v-T Relations Considerable pressure, specific volume, and temperature data have been accumulated for industrially important gases and liquids. These data can be represented in the form p = f (v, T ), called an equation of state. Equations of state can be expressed in tabular, graphical, and analytical forms.

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Chapter 2

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2-26

FIGURE 2.4 cp of water vapor as a function of temperature and pressure. (Adapted from Keenan, J.H., Keyes, F.G., Hill, P.G., and Moore, J.G. 1969 and 1978. Steam Tables — S.I. Units (English Units). John Wiley & Sons, New York.)

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Engineering Thermodynamics

2-27

FIGURE 2.5 Pressure-specific volume-temperature surface and projections for water (not to scale).

P-v-T Surface The graph of a function p = f (v, T) is a surface in three-dimensional space. Figure 2.5 shows the p-v-T relationship for water. Figure 2.5b shows the projection of the surface onto the pressure-temperature plane, called the phase diagram. The projection onto the p–v plane is shown in Figure 2.5c. Figure 2.5 has three regions labeled solid, liquid, and vapor where the substance exists only in a single phase. Between the single phase regions lie two-phase regions, where two phases coexist in equilibrium. The lines separating the single-phase regions from the two-phase regions are saturation lines. Any state represented by a point on a saturation line is a saturation state. The line separating the liquid phase and the two-phase liquid-vapor region is the saturated liquid line. The state denoted by f is a saturated liquid state. The saturated vapor line separates the vapor region and the two-phase liquid-vapor region. The state denoted by g is a saturated vapor state. The saturated liquid line and the saturated vapor line meet at the critical point. At the critical point, the pressure is the critical pressure pc, and the temperature is the

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Chapter 2

critical temperature Tc . Three phases can coexist in equilibrium along the line labeled triple line. The triple line projects onto a point on the phase diagram. The triple point of water is used in defining the Kelvin temperature scale (Section 2.1, Basic Concepts and Definitions; The Second Law of Thermodynamics, Entropy). When a phase change occurs during constant pressure heating or cooling, the temperature remains constant as long as both phases are present. Accordingly, in the two-phase liquid-vapor region, a line of constant pressure is also a line of constant temperature. For a specified pressure, the corresponding temperature is called the saturation temperature. For a specified temperature, the corresponding pressure is called the saturation pressure. The region to the right of the saturated vapor line is known as the superheated vapor region because the vapor exists at a temperature greater than the saturation temperature for its pressure. The region to the left of the saturated liquid line is known as the compressed liquid region because the liquid is at a pressure higher than the saturation pressure for its temperature. When a mixture of liquid and vapor coexists in equilibrium, the liquid phase is a saturated liquid and the vapor phase is a saturated vapor. The total volume of any such mixture is V = Vf + Vg; or, alternatively, mv = mfvf + mgvg, where m and v denote mass and specific volume, respectively. Dividing by the total mass of the mixture m and letting the mass fraction of the vapor in the mixture, mg /m, be symbolized by x, called the quality, the apparent specific volume v of the mixture is v = (1 − x )vf + xvg (2.37a)

= vf + xvfg

where vfg = vg – vf . Expressions similar in form can be written for internal energy, enthalpy, and entropy: u = (1 − x )uf + xug (2.37b)

= uf + xufg h = (1 − x )hf + xhg

(2.37c)

= hf + xhfg s = (1 − x )sf + xsg

(2.37d)

= sf + xsfg

For the case of water, Figure 2.6 illustrates the phase change from solid to liquid (melting): a-b-c; from solid to vapor (sublimation): a′-b′-c′; and from liquid to vapor (vaporization): a″-b″-c″. During any such phase change the temperature and pressure remain constant and thus are not independent properties. The Clapeyron equation allows the change in enthalpy during a phase change at fixed temperature to be evaluated from p-v-T data pertaining to the phase change. For vaporization, the Clapeyron equation reads  dp  = hg − hf  dT  sat T v − v g f

(

)

(2.38)

where (dp/dT)sat is the slope of the saturation pressure-temperature curve at the point determined by the temperature held constant during the phase change. Expressions similar in form to Equation 2.38 can be written for sublimation and melting.

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Engineering Thermodynamics

FIGURE 2.6 Phase diagram for water (not to scale).

The Clapeyron equation shows that the slope of a saturation line on a phase diagram depends on the signs of the specific volume and enthalpy changes accompanying the phase change. In most cases, when a phase change takes place with an increase in specific enthalpy, the specific volume also increases, and (dp/dT)sat is positive. However, in the case of the melting of ice and a few other substances, the specific volume decreases on melting. The slope of the saturated solid-liquid curve for these few substances is negative, as illustrated for water in Figure 2.6. Graphical Representations The intensive states of a pure, simple compressible system can be represented graphically with any two independent intensive properties as the coordinates, excluding properties associated with motion and gravity. While any such pair may be used, there are several selections that are conventionally employed. These include the p-T and p-v diagrams of Figure 2.5, the T-s diagram of Figure 2.7, the h-s (Mollier) diagram of Figure 2.8, and the p-h diagram of Figure 2.9. The compressibility charts considered next use the compressibility factor as one of the coordinates. Compressibility Charts The p-v-T relation for a wide range of common gases is illustrated by the generalized compressibility chart of Figure 2.10. In this chart, the compressibility factor, Z, is plotted vs. the reduced pressure, pR, reduced temperature, TR, and pseudoreduced specific volume, v R′ , where Z=

pv RT

(2.39)

and pR =

p , pc

TR =

T , Tc

v R′ =

v

( RT

c

pc

)

(2.40)

In these expressions, R is the universal gas constant and pc and Tc denote the critical pressure and temperature, respectively. Values of pc and Tc are given for several substances in Table A.9. The reduced

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Chapter 2

FIGURE 2.7 Temperature-entropy diagram for water. (Source: Jones, J.B. and Dugan, R.E. 1996. Engineering Thermodynamics, Prentice-Hall, Englewood Cliffs, NJ, based on data and formulations from Haar, L., Gallagher, J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere, Washington, D.C.)

isotherms of Figure 2.10 represent the best curves fitted to the data of several gases. For the 30 gases used in developing the chart, the deviation of observed values from those of the chart is at most on the order of 5% and for most ranges is much less.1 Figure 2.10 gives a common value of about 0.27 for the compressibility factor at the critical point. As the critical compressibility factor for different substances actually varies from 0.23 to 0.33, the chart is inaccurate in the vicinity of the critical point. This source of inaccuracy can be removed by restricting the correlation to substances having essentially the same Zc values. which is equivalent to including the critical compressibility factor as an independent variable: Z = f (TR, pR, Zc). To achieve greater accuracy variables other than Zc have been proposed as a third parameter — for example, the acentric factor (see, e.g., Reid and Sherwood, 1966). Generalized compressibility data are also available in tabular form (see, e.g., Reid and Sherwood, 1966) and in equation form (see, e.g., Reynolds, 1979). The use of generalized data in any form (graphical, tabular, or equation) allows p, v, and T for gases to be evaluated simply and with reasonable accuracy. When accuracy is an essential consideration, generalized compressibility data should not be used as a substitute for p-v-T data for a given substance as provided by computer software, a table, or an equation of state. Equations of State Considering the isotherms of Figure 2.10, it is plausible that the variation of the compressibility factor might be expressed as an equation, at least for certain intervals of p and T. Two expressions can be written that enjoy a theoretical basis. One gives the compressibility factor as an infinite series expansion in pressure,

1 To determine Z for hydrogen, helium, and neon above a T of 5, the reduced temperature and pressure should R be calculated using TR = T/(Tc + 8) and PR = p/(pc + 8), where temperatures are in K and pressures are in atm.

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Engineering Thermodynamics

FIGURE 2.8 Enthalpy-entropy (Mollier) diagram for water. (Source: Jones, J.B. and Dugan, R.E. 1996. Engineering Thermodynamics. Prentice-Hall, Englewood Cliffs, NJ, based on data and formulations from Haar, L., Gallagher, J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere, Washington, D.C.)

Z = 1 + Bˆ (T ) p + Cˆ (T ) p 2 + Dˆ (T ) p 3 + … and the other is a series in 1/ v , Z = 1+

B(T ) C(T ) D(T ) + 2 + 3 +… v v v

These expressions are known as virial expansions, and the coefficients Bˆ , Cˆ Dˆ , … and B, C, D … are called virial coefficients. In principle, the virial coefficients can be calculated using expressions from statistical mechanics derived from consideration of the force fields around the molecules. Thus far only the first few coefficients have been calculated and only for gases consisting of relatively simple molecules. The coefficients also can be found, in principle, by fitting p-v-T data in particular realms of interest. Only the first few coefficients can be found accurately this way, however, and the result is a truncated equation valid only at certain states. Over 100 equations of state have been developed in an attempt to portray accurately the p-v-T behavior of substances and yet avoid the complexities inherent in a full virial series. In general, these equations exhibit little in the way of fundamental physical significance and are mainly empirical in character. Most © 2005 by CRC Press LLC

Chapter 2

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2-32

FIGURE 2.9 Pressure-enthalpy diagram for water. (Source: Jones, J.B. and Dugan, R.E. 1996. Engineering Thermodynamics. Prentice-Hall, Englewood Cliffs, NJ, based on data and formulations from Haar, L., Gallagher, J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere, Washington, D.C.)

2-33

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Engineering Thermodynamics

– FIGURE 2.10 Generalized compressibility chart (TR = T/TC, pR = p/pC, vR′ = vp C/RTC) for pR ≤ 10. (Source: Obert, E.F. 1960 Concepts of Thermodynamics. McGrawHill, New York.)

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Chapter 2

are developed for gases, but some describe the p-v-T behavior of the liquid phase, at least qualitatively. Every equation of state is restricted to particular states. The realm of applicability is often indicated by giving an interval of pressure, or density, where the equation can be expected to represent the p-v-T behavior faithfully. When it is not stated, the realm of applicability often may be approximated by expressing the equation in terms of the compressibility factor Z and the reduced properties, and superimposing the result on a generalized compressibility chart or comparing with compressibility data from the literature. Equations of state can be classified by the number of adjustable constants they involve. The RedlichKwong equation is considered by many to be the best of the two-constant equations of state. It gives pressure as a function of temperature and specific volume and thus is explicit in pressure: p=

RT a − v − b v (v + b)T 1 2

(2.41)

This equation is primarily empirical in nature, with no rigorous justification in terms of molecular arguments. Values for the Redlich-Kwong constants for several substances are provided in Table A.9. Modified forms of the equation have been proposed with the aim of achieving better accuracy. Although the two-constant Redlich-Kwong equation performs better than some equations of state having several adjustable constants, two-constant equations tend to be limited in accuracy as pressure (or density) increases. Increased accuracy normally requires a greater number of adjustable constants. For example, the Benedict-Webb-Rubin equation, which involves eight adjustable constants, has been successful in predicting the p-v-T behavior of light hydrocarbons. The Benedict-Webb-Rubin equation is also explicit in pressure, p=

(

)

bRT − a γ γ RT  C 1 aα c + 6 + 3 2 1 + 2  exp − 2  + BRT − A − 2  2 +  v   v T v v3 v v T  v 

(2.42)

Values of the Benedict-Webb-Rubin constants for various gases are provided in the literature (see, e.g., Cooper and Goldfrank, 1967). A modification of the Benedict-Webb-Rubin equation involving 12 constants is discussed by Lee and Kessler, 1975. Many multiconstant equations can be found in the engineering literature, and with the advent of high speed computers, equations having 50 or more constants have been developed for representing the p-v-T behavior of different substances. Gas Mixtures Since an unlimited variety of mixtures can be formed from a given set of pure components by varying the relative amounts present, the properties of mixtures are reported only in special cases such as air. Means are available for predicting the properties of mixtures, however. Most techniques for predicting mixture properties are empirical in character and are not derived from fundamental physical principles. The realm of validity of any particular technique can be established by comparing predicted property values with empirical data. In this section, methods for evaluating the p-v-T relations for pure components are adapted to obtain plausible estimates for gas mixtures. The case of ideal gas mixtures is discussed in Section 2.3, Ideal Gas Model. In Section 2.3, Multicomponent Systems, some general aspects of property evaluation for multicomponent systems are presented. The total number of moles of mixture, n, is the sum of the number of moles of the components, ni: j

n = n1 + n2 + …n j =

∑n

i

(2.43)

i =1

The relative amounts of the components present can be described in terms of mole fractions. The mole fraction yi of component i is yi = ni /n. The sum of the mole fractions of all components present is equal

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Engineering Thermodynamics

to unity. The apparent molecular weight M is the mole fraction average of the component molecular weights, such that j

∑y M

M =

i

(2.44)

i

i =1

The relative amounts of the components present also can be described in terms of mass fractions: mi /m, where mi is the mass of component i and m is the total mass of mixture. The p-v-T relation for a gas mixture can be estimated by applying an equation of state to the overall mixture. The constants appearing in the equation of state are mixture values determined with empirical combining rules developed for the equation. For example, mixture values of the constants a and b for use in the Redlich-Kwong equation are obtained using relations of the form  a= 

j

∑ i =1

2

 ya  , 

j

b=

12 i i

∑y b

i i

(2.45)

i =1

where ai and bi are the values of the constants for component i. Combination rules for obtaining mixture values for the constants in other equations of state are also found in the literature. Another approach is to regard the mixture as if it were a single pure component having critical properties calculated by one of several mixture rules. Kay’s rule is perhaps the simplest of these, requiring only the determination of a mole fraction averaged critical temperature Tc and critical pressure pc : j

Tc =

∑

j

yi Tc,i ,

pc =

i =1

∑y p i

c,i

(2.46)

i =1

where Tc,i and pc,i are the critical temperature and critical pressure of component i, respectively. Using Tc and pc, the mixture compressibility factor Z is obtained as for a single pure component. The unkown quantity from among the pressure p, volume V, temperature T, and total number of moles n of the gas mixture can then be obtained by solving Z = pV/nR T. Additional means for predicting the p-v-T relation of a mixture are provided by empirical mixture rules. Several are found in the engineering literature. According to the additive pressure rule, the pressure of a gas mixture is expressible as a sum of pressures exerted by the individual components:

]

p = p1 + p2 + p3 … T ,V

(2.47a)

where the pressures p1, p2, etc. are evaluated by considering the respective components to be at the volume V and temperature T of the mixture. The additive pressure rule can be expressed alternatively as 

j

Z=

∑ y Z  i

i =1

(2.47b)

i

T ,V

where Z is the compressibility factor of the mixture and the compressibility factors Zi are determined assuming that component i occupies the entire volume of the mixture at the temperature T. The additive volume rule postulates that the volume V of a gas mixture is expressible as the sum of volumes occupied by the individual components:

]

V = V1 + V2 + V3 … © 2005 by CRC Press LLC

p ,T

(2.48a)

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Chapter 2

where the volumes V1, V2, etc. are evaluated by considering the respective components to be at the pressure p and temperature T of the mixture. The additive volume rule can be expressed alternatively as 

j

Z=

∑ y Z  i

i =1

(2.48b)

i

p,T

where the compressibility factors Zi are determined assuming that component i exists at the pressure p and temperature T of the mixture.

Evaluating ∆h, ∆u, and ∆s Using appropriate specific heat and p-v-T data, the changes in specific enthalpy, internal energy, and entropy can be determined between states of single-phase regions. Table 2.5 provides expressions for such property changes in terms of particular choices of the independent variables: temperature and pressure, and temperature and specific volume. Taking Equation 1 of Table 2.5 as a representative case, the change in specific enthalpy between states 1 and 2 can be determined using the three steps shown in the accompanying property diagram. This requires knowledge of the variation of cp with temperature at a fixed pressure p′, and the variation of [v – T(∂v/∂T)p] with pressure at temperatures T1 and T2: 1-a: Since temperature is constant at T1, the first integral of Equation 1 in Table 2.5 vanishes, and ha − h1 =

∫ [v − T (∂v ∂T ) ]dp p′

p

p1

a-b: Since pressure is constant at p′, the second integral of Equation 1 vanishes, and hb − ha =

T2

∫ c (T, p′)dT T1

p

b-2: Since temperature is constant at T2, the first integral of Equation 1 vanishes, and h2 − hb =

∫ [v − T (∂v ∂T ) ]dp p2

p′

p

Adding these expressions, the result is h2 – h1. The required integrals may be performed numerically or analytically. The analytical approach is expedited when an equation of state explicit in specific volume is known. Similar considerations apply to Equation 2 to Equation 4 of Table 2.5. To evaluate u2 – u1 with Equation 3, for example, requires the variation of cv with temperature at a fixed specific volume v′, and the variation of [T(∂p/∂T)v – p] with specific volume at temperatures T1 and T2. An analytical approach to performing the integrals is expedited when an equation of state explicit in pressure is known. As changes in specific enthalpy and internal energy are related through h = u + pv by h2 − h1 = (u2 − u1 ) + ( p2 v2 − p1v1 )

(2.49)

only one of h2 – h1 and u2 – u1 need be found by integration. The other can be evaluated from Equation 2.49. The one found by integration depends on the information available: h2 – h1 would be found when an equation of state explicit in v and cp as a function of temperature at some fixed pressure is known, u2 – u1 would be found when an equation of state explicit in p and cv as a function of temperature at some specific volume is known. © 2005 by CRC Press LLC

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Engineering Thermodynamics

∆h, ∆u, ∆s Expressions TABLE 2.5

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Chapter 2

Example 6 Obtain Equation 1 of Table 2.4 and Equation 3 and Equation 4 of Table 2.5. Solution. With Equation 2.33 and the Maxwell relation corresponding to ψ(T, v) from Table 2.2, Equation 3′ and Equation 4′ of Table 2.5 become, respectively,  ∂u  du = cv dT +   dv  ∂v  T  ∂s   ∂p  ds =   dT +   dv  ∂T  v  ∂T  v Introducing these expressions for du and ds in Equation 2.32a, and collecting terms,   ∂s    ∂u   ∂p   T   − cv  dT =   + p − T    dv ∂ ∂ ∂T v  T v v T    Since T and v are independent, the coefficients of dT and dv must vanish, giving, respectively, c  ∂s    = v  ∂T  v T  ∂u   ∂p    = T  − p  ∂v  T  ∂T  v The first of these corresponds to Equation 1 of Table 2.4 and Equation 4 of Table 2.5. The second of the above expressions establishes Equation 3 of Table 2.5. With similar considerations, Equation 3 of Table 2.4 and Equation 1 and Equation 2 of Table 2.5 may be obtained.

Fundamental Thermodynamic Functions A fundamental thermodynamic function is one that provides a complete description of the thermodynamic state. The functions u(s, v), h(s, p), ψ(T, v), and g(T, p) listed in Table 2.2 are fundamental thermodynamic functions. In principle, all properties of interest can be determined from a fundamental thermodynamic function by differentiation and combination. Taking the function ψ(T, v) as a representative case, the properties v and T, being the independent variables, are specified to fix the state. The pressure p and specific entropy s at this state can be determined by differentiation of ψ(T, v), as shown in Table 2.2. By definition, ψ = u – Ts, so specific internal energy is obtained as u = ψ + Ts with u, p, and v known, the specific enthalpy can be found from the definition h = u + pv. Similarly, the specific Gibbs function is found from the definition g = h – Ts. The specific heat cv can be determined by further differentiation cv = (∂u/∂T)v . The development of a fundamental function requires the selection of a functional form in terms of the appropriate pair of independent properties and a set of adjustable coefficients that may number 50 or more. The functional form is specified on the basis of both theoretical and practical considerations. The coefficients of the fundamental function are determined by requiring that a set of selected property values and/or observed conditions be statisfied in a least-squares sense. This generally involves property data requiring the assumed functional form to be differentiated one or more times, for example p-v-T © 2005 by CRC Press LLC

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Engineering Thermodynamics

and specific heat data. When all coefficients have been evaluated, the function is tested for accuracy by using it to evaluate properties for which accepted values are known such as velocity of sound and JouleThomson data. Once a suitable fundamental function is established, extreme accuracy in and consistency among the thermodynamic properties are possible. The properties of water tabulated by Keenan et al. (1969) and by Haar et al. (1984) have been calculated from representations of the Helmholtz function.

Thermodynamic Data Retrieval Tabular presentations of pressure, specific volume, and temperature are available for practically important gases and liquids. The tables normally include other properties useful for thermodynamic analyses, such as internal energy, enthalpy, and entropy. The various steam tables included in the references of this chapter provide examples. Computer software for retrieving the properties of a wide range of substances is also available, as, for example, the ASME Steam Tables (1993) and Bornakke and Sonntag (1996). Increasingly, textbooks come with computer disks providing thermodynamic property data for water, certain refrigerants, and several gases modeled as ideal gases — see, e.g., Moran and Shapiro (1996). The sample steam table data presented in Table 2.6 are representative of data available for substances commonly encountered in mechanical engineering practice. Table A.5 and Figure 2.7 to Figure 2.9 provide steam table data for a greater range of states. The form of the tables and figures, and how they are used are assumed to be familiar. In particular, the use of linear interpolation with such tables is assumed known. Specific internal energy, enthalpy, and entropy data are determined relative to arbitrary datums and such datums vary from substance to substance. Referring to Table 2.6a, the datum state for the specific internal energy and specific entropy of water is seen to correspond to saturated liquid water at 0.01°C (32.02°F), the triple point temperature. The value of each of these properties is set to zero at this state. If calculations are performed involving only differences in a particular specific property, the datum cancels. When there are changes in chemical composition during the process, special care should be exercised. The approach followed when composition changes due to chemical reaction is considered in Section 2.4. Liquid water data (see Table 2.6d) suggests that at fixed temperature the variation of specific volume, internal energy, and entropy with pressure is slight. The variation of specific enthalpy with pressure at fixed temperature is somewhat greater because pressure is explicit in the definition of enthalpy. This behavior for v, u, s, and h is exhibited generally by liquid data and provides the basis for the following set of equations for estimating property data at liquid states from saturated liquid data: v(T , p) ≈ vf (T )

(2.50a)

u(T , p) ≈ uf (T )

(2.50b)

[

]

h(T , p) ≈ h f (T ) + vf p − psat (T )

(2.50c)

−−−−−−−−− s(T , p) ≈ sf (T )

(2.50d)

As before, the subscript f denotes the saturated liquid state at the temperature T, and psat is the corresponding saturation pressure. The underlined term of Equation 2.50c is often negligible, giving h(T, p) ≈ hf (T), which is used in Example 3 to evaluate h1. In the absence of saturated liquid data, or as an alternative to such data, the incompressible model can be employed: v = constant Incompressible model:  u = u(T ) © 2005 by CRC Press LLC

(2.51)

Sample Steam Table Data (a)

Properties of Saturated Water (Liquid-Vapor): Temperature Table

Specific Volume (m3/kg) Temp (°C)

Pressure (bar)

Saturated Liquid (vf × 103)

.01 4 5 6 8

0.00611 0.00813 0.00872 0.00935 0.01072

1.0002 1.0001 1.0001 1.0001 1.0002

Saturated Vapor (ug)

Saturated Liquid (hf )

206.136 157.232 147.120 137.734 120.917

0.00 16.77 20.97 25.19 33.59

2375.3 2380.9 2382.3 2383.6 2386.4

0.01 16.78 20.98 25.20 33.60

(b)

Pressure (bar)

Temp (°C)

0.04 0.06 0.08 0.10 0.20

28.96 36.16 41.51 45.81 60.06

1.0040 1.0064 1.0084 1.0102 1.0172

Enthalpy (kJ/kg)

Saturated Liquid (uf )

Entropy (kJ/kg · K)

Evap. (hfg)

Saturated Vapor (hg)

Saturated Liquid (sf )

Saturated Vapor (sg)

2501.3 2491.9 2489.6 2487.2 2482.5

2501.4 2508.7 2510.6 2512.4 2516.1

0.0000 0.0610 0.0761 0.0912 0.1212

9.1562 9.0514 9.0257 9.0003 8.9501

Properties of Saturated Water (Liquid-Vapor): Pressure Table

Specific Volume (m3/kg) Saturated Liquid (vf × 103)

Internal Energy (kJ/kg)

Saturated Vapor (vg)

Internal Energy (kJ/kg)

Enthalpy (kJ/kg)

Saturated Vapor (vg)

Saturated Liquid (uf )

Saturated Vapor (ug)

Saturated Liquid (hf )

34.800 23.739 18.103 14.674 7.649

121.45 151.53 173.87 191.82 251.38

2415.2 2425.0 2432.2 2437.9 2456.7

121.46 151.53 173.88 191.83 251.40

Entropy (kJ/kg · K)

Evap. (hfg)

Saturated Vapor (hg)

Saturated Liquid (sf )

Saturated Vapor (sg)

2432.9 2415.9 2403.1 2392.8 2358.3

2554.4 2567.4 2577.0 2584.7 2609.7

0.4226 0.5210 0.5926 0.6493 0.8320

8.4746 8.3304 8.2287 8.1502 7.9085

Chapter 2

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TABLE 2.6

(c) T(°C)

Properties of Superheated Water Vapor

3

u(kJ/kg) h(kJ/kg) s(kJ/kg · K) v(m /kg) p = 0.06 bar = 0.006 MPa (Tsat 36.16°C)

Sat. 80 120 160 200

23.739 27.132 30.219 33.302 36.383

T(°C)

v × 10 (m3/kg)

20 80 140 200 Sat.

1.0006 1.0280 1.0784 1.1555 1.1973

2425.0 2487.3 2544.7 2602.7 2661.4

2567.4 2650.1 2726.0 2802.5 2879.7 (d)

8.3304 8.5804 8.7840 8.9693 9.1398

u(kJ/kg) h(kJ/kg) s(kJ/kg · K) p = 25 bar = 2.5 MPa (Tsat 223.99°C) 86.30 336.86 590.52 852.8 962.1

4.526 4.625 5.163 5.696 6.228

2473.0 2483.7 2542.4 2601.2 2660.4

2631.4 2645.6 2723.1 2800.6 2878.4

7.7158 7.7564 7.9644 8.1519 8.3237

Properties of Compressed Liquid Water

3

83.80 334.29 587.82 849.9 959.1

v(m3/kg) u(kJ/kg) h(kJ/kg) s(kJ/kg · K) p = 0.35 bar = 0.035 MPa (Tsat = 72.69°C)

0.2961 1.0737 1.7369 2.3294 2.5546

v × 103 (m3/kg) u(kJ/kg) h(kJ/kg) s(kJ/kg · K) p = 50 bar = 5.0 MPa (Tsat = 263.99°C) 0.9995 1.0268 1.0768 1.1530 1.2859

83.65 333.72 586.76 848.1 1147.8

88.65 338.85 592.15 853.9 1154.2

0.2956 1.0720 1.7343 2.3255 2.9202

Source: Moran, M.J. and Shapiro, H.N. 1995. Fundamentals of Engineering Thermodynamics, 3rd ed. Wiley, New York, as extracted from Keenan, J. H., Keyes, F.G., Hill, P.G., and Moore, J.G. 1969. Steam Tables. Wiley, New York.

2-41

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Sample Steam Table Data

Engineering Thermodynamics

TABLE 2.6 (continued)

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Chapter 2

This model is also applicable to solids. Since internal energy varies only with temperature, the specific heat cv is also a function of only temperature: cv(T) = du/dT. Although specific volume is constant, enthalpy varies with both temperature and pressure, such that h(T , p) = u(T ) + pv

(2.52)

Differentiation of Equation 2.52 with respect to temperature at fixed pressure gives cp = cv . The common specific heat is often shown simply as c. Specific heat and density data for several liquids and solids are provided in Table B.2, Table C.1, and Table C.2. As the variation of c with temperature is slight, c is frequently taken as constant. When the incompressible model is applied. Equation 2.49 takes the form h2 − h1 =

∫

T2

T1

c(T ) dT + v( p2 − p1 )

(2.53)

= cave (T2 − T1 ) + v( p2 − p1 ) Also, as Equation 2.32a reduces to du = Tds, and du = c(T)dT, the change in specific entropy is ∆s =

∫

T2

T1

c(T ) dT T (2.54)

T = cave ln 2 T1

Ideal Gas Model Inspection of the generalized compressibility chart, Figure 2.10, shows that when pR is small, and for many states when TR is large, the value of the compressibility factor Z is close to 1. In other words, for pressures that are low relative to pc, and for many states with temperatures high relative to Tc , the compressibility factor approaches a value of 1. Within the indicated limits, it may be assumed with reasonable accuracy that Z = 1 — that is, pv = RT or pv = RT

(2.55a)

where R = R /M is the specific gas constant. Other forms of this expression in common use are pV = nRT ,

pV = mRT

(2.55b)

Referring to Equation 3′ of Table 2.5, it can be concluded that (∂u/∂v)T vanishes identically for a gas whose equation of state is exactly given by Equation 2.55, and thus the specific internal energy depends only on temperature. This conclusion is supported by experimental observations beginning with the work of Joule, who showed that the internal energy of air at low density depends primarily on temperature. These considerations allow for an ideal gas model of each real gas: (1) the equation of state is given by Equation 2.55 and (2) the internal energy and enthalpy are functions of temperature alone. The real gas approaches the model in the limit of low reduced pressure. At other states the actual behavior may depart substantially from the predictions of the model. Accordingly, caution should be exercised when invoking the ideal gas model lest significant error is introduced. Specific heat data for gases can be obtained by direct measurement. When extrapolated to zero pressure, ideal gas-specific heats result. Ideal gas-specific heats also can be calculated using molecular models of

© 2005 by CRC Press LLC

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Engineering Thermodynamics

TABLE 2.7

Ideal Gas Expressions for ∆h, ∆u, and ∆s Variable Specific Heats

h(T2 ) − h(T1 ) =

T2

∫ c (T )dT T1

p

s (T2 , p2 ) − s (T1 , p1 ) = u (T2 ) − u (T1 ) =

Constant Specific Heats

∫

T2

T1

∫

T2

c p (T ) T

T1

dT − R ln

p2 p1

cv (T ) dT

s (T2 , v2 ) − s(T1 , v1 ) =

∫

T2

T1

c v (T ) v dT + R ln 2 T v1

(1)

h(T2 ) − h(T1 ) = c p (T2 − T1 )

(2)

s (T2 , p2 ) − s (T1 , p1 ) = c p ln

(3)

u (T2 ) − u (T1 ) = cv (T2 − T1 )

(4)

s (T2 , v2 ) − s (T1 , v1 ) = cv ln

s2 = s 1 pr (T2 ) pr (T1 )

v r (T2 ) v r (T1 )

(1′) T2 p − R ln 2 T1 p1

T2 v + R ln 2 T1 v1

(2′)

(3′) (4′)

s2 = s1 =

p2 p1

(5)

T2  p2  = T1  p1 

=

v2 v1

(6)

T2  v1  = T1  v2 

( k −1)

k

(5′) k −1

(6′)

matter together with data from spectroscopic measurements. Table A.9 provides ideal gas-specific heat data for a number of substances. The following ideal gas-specific heat relations are frequently useful: c p (T ) = cv (T ) + R cp =

kR , k −1

cv =

(2.56a)

R k −1

(2.56b)

where k = cp /cv . With the ideal gas model, Equation 1 to Equation 4 of Table 2.5 give Equation 1 to Equation 4 of Table 2.7, respectively. Equation 2 of Table 2.7 can be expressed alternatively using s°(T) defined by s°(T ) ≡

∫

T

0

c p (T ) T

(2.57)

dT

as s(T2 , p2 ) − s(T1 , p1 ) = s°(T2 ) − s°(T1 ) − R ln

p2 p1

(2.58)

Expressions similar in form to Equation 2.56 to Equation 2.68 can be written on a molar basis. For processes of an ideal gas between states having the same specific entropy, s2 = s1, Equation 2.58 gives

[ [

] ]

p2 exp s°(T2 ) R = p1 exp s°(T1 ) R or with pr = exp[s°(T)/R] p (T ) p2 = r 2 p1 pr (T1 ) © 2005 by CRC Press LLC

(s2 = s1 )

(2.59a)

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Chapter 2

A relation between the specific volume and temperatures for two states of an ideal gas having the same specific entropy can also be developed: v2 vr (T2 ) = v1 vr (T1 )

(s2 = s1 )

(2.59b)

Equations 2.59 are listed in Table 2.7 as Equation 5 and Equation 6, respectively. Table A.8 provides a tabular display of h, u, s°, pr , and vr vs. temperature for air as an ideal gas. Tabulations of h , u , and s° for several other common gases are provided in Table A.2. Property retrieval software also provides such data; see, e.g., Moran and Shapiro (1996). The use of data from Table A.8 for the nozzle of Example 2 is illustrated in Example 7. When the ideal gas-specific heats are assumed constant, Equation 1 to Equation 6 of Table 2.7 become Equation 1′ tο Equation 6′, respectively. The specific heat cp is taken as constant in Example 2. Example 7 Using data from Table A.8, evaluate the exit velocity for the nozzle of Example 2 and compare with the exit velocity for an isentropic expansion to 15 lbf/in.2. Solution. The exit velocity is given by Equation 2.27f v e = v i2 + 2(hi − he ) At 960 and 520°R, Table A.8 gives, respectively, hi = 231.06 Btu/lb and he = 124.27 Btu/lb. Then 2 10 ft  Btu   778.17 ft ⋅ lbf   32.174 lb ⋅ ft sec 2  ve =  + 2(231.06 − 124.27)    s   lb     1 Btu 1 lbf 

= 2312.5 ft sec Using Equation 2.59a and pr data from Table A.8, the specific enthalpy at the exit for an isentropic expansion is found as follows: pr (Te ) = pr (Ti )

pe 15  = 1.061 = 10.61  150  pi

Interpolating with pr data, he = 119.54 Btu/lb. With this, the exit velocity is 2363.1 ft/sec. The actual exit velocity is about 2% less than the velocity for an isentropic expansion, the maximum theoretical value. In this particular application, there is good agreement in each case between velocities calculated using Table A.8 data and, as in Example 2, assuming cp constant. Such agreement cannot be expected generally, however. See, for example, the Brayton cycle data of Table 2.15. Polytropic Processes An internally reversible process described by the expression pvn = constant is called a polytropic process and n is the polytropic exponent. Although this expression can be applied with real gas data, it most generally appears in practice together with the use of the ideal gas model. Table 2.8 provides several expressions applicable to polytropic processes and the special forms they take when the ideal gas model is assumed. The expressions for ∫pdv and ∫vdp have application to work evaluations with Equation 2.10 and Equation 2.30, respectively. In some applications it may be appropriate to determine n by fitting pressure-specific volume data. Example 8 illustrates both the polytropic process and the reduction in the compressor work achievable by cooling a gas as it is compressed. © 2005 by CRC Press LLC

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Engineering Thermodynamics

Polytropic Processes: pvn = Constanta

TABLE 2.8

Ideal Gasb

General p2  v1  = p1  v2 

n

n ( n −1)

n

T  p2  v1  =  = 2 p1  v2   T1 

(1)

(1′)

n = 0: constant pressure n = ±∞: constant specific volume

n = 0: constant pressure n = ±∞: constant specific volume n = 1: constant temperature n = k: constant specific entropy when k is constant

n=1

n=1 2

2

v2

∫ pdv = p v ln v 1

∫

1

1

2

− vdp = − p1 v1 ln 1

v2

∫ pdv = RT ln v

(2)

1 1

p2 p1

∫

2

− vdp = − RT ln

(3)

1

n≠1

(2′)

1

p2 p1

(3′)

n≠1 2

∫ pdv = 1

p2 v2 − p1 v1 1− n

pv  p  = 1 1 1 −  2  n − 1   p1  

∫

2

− vdp = 1

=

2

∫ pdv = 1

( n −1)

n

   

(4)

n ( p v − p1v1 ) 1− n 2 2 ( n −1) n  np1 v1   p2   1 −  n − 1   p1   

R(T2 − T1 ) 1− n

( n −1) n  RT1   p2   1 − =  n − 1   p1   

∫

2

− vdp = 1

(5) =

(4′)

nR (T − T1 ) 1− n 2 nRT1   p2  1 − n − 1   p1  

( n −1)

n

   

(5′)

a For polytropic processes of closed systems where volume change is the only work mode, Equation 2, Equation 4, and Equation 2′, Equation 4′ are applicable with Equation 2.10 to evaluate the work. When each unit of mass passing through a one-inlet, one-exit control volume at steady state undergoes a polytropic process, Equation 3, Equation 5, and Equation 3′, Equation 5′ are applicable with Equation 2.30a and Equation 2.30b to evaluate the power. Also note that generally,

−

2

∫ vdp = n∫ 1

2

pdv.

1

b

Example 8 A compressor operates at steady state with air entering at 1 bar, 20°C and exiting at 5 bar. (a) If the air undergoes a polytropic process with n = 1.3, determine the work and heat transfer, each in kJ/kg of air flowing. Repeat for (b) an isothermal compression and (c) an isentropic compression.

© 2005 by CRC Press LLC

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Chapter 2

Solution. Using Equation 5′ of Table 2.8 together with Equation 2.30b, ( n −1) n  W˙ cv nRT1   p2    = 1− m˙ n − 1   p1    

1.3   8.314 kJ  = (293 K) 1 − (5)0.3 1.3  0.3   28.97 kg ⋅ K 

[

= −163.9

]

kJ kg

(The area behind process 1-2 of Figure 2.11, area 1-2-a-b, represents the magnitude of the work required, per unit mass of air flowing.) Also, Equation 1′ of Table 2.8 gives T2 = 425 K.

FIGURE 2.11 Internally reversible compression processes.

An energy rate balance at steady state and enthalpy data from Table A.8 gives Q˙ cv W˙ cv = + h2 − h1 m˙ m˙ = −163.9 + (426.3 − 293.2) = −30.8 (b) Using Equation 3′ of Table 2.8 together with Equation 2.30b, © 2005 by CRC Press LLC

kJ kg

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Engineering Thermodynamics

W˙ cv p = − RT ln 2 m˙ p1 8.314  = − (293) ln 5  28.97  = −135.3

kJ kg

Area 1-2′-a-b on Figure 2.11 represents the magnitude of the work required, per unit of mass of air flowing. An energy balance reduces to give Q˙ cv / m˙ = W˙ cv / m˙ = –135.3 kJ/kg. (c) For an isentropic compression, Q˙ cv = 0 and an energy rate balance reduces to give W˙ cv / m˙ = –(h2s – h1), where 2s denotes the exit state. With Equation 2.59a and pr data, h2s = 464.8 kJ/kg (T2s = 463K). Then W˙ cv / m˙ = –(464.8 – 293.2) = –171.6 kJ/kg. Area 1-2s-a-b on Figure 2.11 represents the magnitude of the work required, per unit of mass of air flowing. Ideal Gas Mixtures When applied to an ideal gas mixture, the additive pressure rule (Section 2.3, p-v-T Relations) is known as the Dalton model. According to this model, each gas in the mixture acts as if it exists separately at the volume and temperature of the mixture. Applying the ideal gas equation of state to the mixture as a whole and to each component i, pV = nRT, piV = ni RT , where pi, the partial pressure of component i, is the pressure that component i would exert if ni moles occupied the full volume V at the temperature T. Forming a ratio, the partial pressure of component i is pi =

ni p = yi p n

(2.60)

where yi is the mole fraction of component i. The sum of the partial pressures equals the mixture pressure. The internal energy, enthalpy, and entropy of the mixture can be determined as the sum of the respective properties of the component gases, provided that the contribution from each gas is evaluated at the condition at which the gas exists in the mixture. On a molar basis, j

U=

∑

j

∑y u

ni ui or u =

i i

i =1 j

H=

∑

j

ni hi or h =

i =1

∑

∑y h

i i

(2.61b)

i =1

j

S=

(2.61a)

i =1

j

ni si or s =

i =1

∑y s

i i

(2.61c)

i =1

The specific heats cv and c p for an ideal gas mixture in terms of the corresponding specific heats of the components are expressed similarly: j

cv =

∑y c

i vi

(2.61d)

i =1

j

cp =

∑y c

i pi

i =1

© 2005 by CRC Press LLC

(2.61e)

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Chapter 2

When working on a mass basis, expressions similar in form to Equations 2.61 can be written using mass and mass fractions in place of moles and mole fractions, respectively, and using u, h, s, cp, and cv in place of u , h , s , c p , and cV , respectively. The internal energy and enthalpy of an ideal gas depend only on temperature, and thus the ui and hi terms appearing in Equations 2.61 are evaluated at the temperature of the mixture. Since entropy depends on two independent properties, the si terms are evaluated either at the temperature and the partial pressure pi of component i, or at the temperature and volume of the mixture. In the former case j

S=

∑ n s (T , p ) i i

i

i =1

(2.62)

j

=

∑ n s ( T , x p) i i

i

i =1

Inserting the expressions for H and S given by Equation 2.61b and Equation 2.61c into the Gibbs function, G = H – TS, j

G=

∑

j

ni hi (T ) − T

i =1

∑ n s (T , p ) i i

i

i =1

(2.63)

j

=

∑ n g (T , p ) i i

i

i =1

where the molar-specific Gibbs function of component i is gi(T, pi ) = hi (T) – Tsi (T, pi ). The Gibbs function of i can be expressed alternatively as gi (T , pi ) = gi (T , p ′) + RT ln( pi p ′) = gi (T , p ′) + RT ln( xi p p ′)

(2.64)

were p′ is some specified pressure. Equation 2.64 is obtained by integrating Equation 2.32d at fixed temperature T from pressure p′ to pi . Moist Air An ideal gas mixture of particular interest for many practical applications is moist air. Moist air refers to a mixture of dry air and water vapor in which the dry air is treated as if it were a pure component. Ideal gas mixture principles usually apply to moist air. In particular, the Dalton model is applicable, and so the mixture pressure p is the sum of the partial pressures pa and pv of the dry air and water vapor, respectively. Saturated air is a mixture of dry air and saturated water vapor. For saturated air, the partial pressure of the water vapor equals psat (T), which is the saturation pressure of water corresponding to the drybulb (mixture) temperature T. The makeup of moist air can be described in terms of the humidity ratio (specific humidity) and the relative humidity. The bulb of a wet-bulb thermometer is covered with a wick saturated with liquid water, and the wet-bulb temperature of an air-water vapor mixture is the temperature indicated by such a thermometer exposed to the mixture. When a sample of moist air is cooled at constant pressure, the temperature at which the sample becomes saturated is called the dew point temperature. Cooling below the dew point temperature results in the condensation of some of the water vapor initially present. When cooled to a final equilibrium state at a temperature below the dew point temperature, the original sample would consist of a gas phase of dry air and saturated water vapor in equilibrium with a liquid water phase.

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Engineering Thermodynamics

Psychrometric charts are plotted with various moist air parameters, including the dry-bulb and wetbulb temperatures, the humidity ratio, and the relative humidity, usually for a specified value of the mixture pressure such as 1 atm. Further discussion of moist air and related psychrometric principles and applications is provided in Chapter 9.

Generalized Charts for Enthalpy, Entropy, and Fugacity The changes in enthalpy and entropy between two states can be determined in principle by correcting the respective property change determined using the ideal gas model. The corrections can be obtained, at least approximately, by inspection of the generalized enthalpy correction and entropy correction charts, Figure 2.12 and Figure 2.13, respectively. Such data are also available in tabular form (see, e.g., Reid and Sherwood, 1966) and calculable using a generalized equation for the compressibility factor (Reynolds, 1979). Using the superscript * to identify ideal gas property values, the changes in specific enthalpy and specific entropy between states 1 and 2 are  h * − h   h* − h   h2 − h1 = h2* − h1* − RTc  −     RTc  2  RTc  1 

(2.65a)

 s * − s   s* − s   s2 − s1 = s2* − s1* − R   −    R  2  R  1 

(2.65b)

The first underlined term on the right side of each expression represents the respective property change assuming ideal gas behavior. The second underlined term is the correction that must be applied to the ideal gas value to obtain the actual value. The quantities (h * − h ) / RTc and (s * − s ) / R at state 1 would be read from the respective correction chart or table or calculated, using the reduced temperature TR1 and reduced pressure pR1 corresponding to the temperature T1 and pressure p1 at state 1, respectively. Similarly, (h * − h ) / RTc and (s * − s ) / R at state 2 would be obtained using TR2 and pR2. Mixture values for Tc and pc determined by applying Kay’s rule or some other mixture rule also can be used to enter the generalized enthalpy correction and entropy correction charts. Figure 2.14 gives the fugacity coefficient, f/p, as a function of reduced pressure and reduced temperature. The fugacity f plays a similar role in determining the specific Gibbs function for a real gas as pressure plays for the ideal gas. To develop this, consider the variation of the specific Gibbs function with pressure at fixed temperature (from Table 2.2) ∂g   =v ∂p  T For an ideal gas, integration at fixed temperature gives g * = RT ln p + C(T ) where C(T) is a function of integration. To evaluate g for a real gas, fugacity replaces pressure, g = RT ln f + C(T ) In terms of the fugacity coefficient the departure of the real gas value from the ideal gas value at fixed temperature is then

© 2005 by CRC Press LLC

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Chapter 2

FIGURE 2.12 Generalized enthalpy correction chart. (Source: Adapted from Van Wylen, G. J. and Sonntag, R. E. 1986. Fundamentals of Classical Thermodynamics, 3rd ed., English/SI. Wiley, New York.)

g − g * = RT ln

f p

(2.66)

As pressure is reduced at fixed temperature, f/p tends to unity, and the specific Gibbs function is given by the ideal gas value. © 2005 by CRC Press LLC

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Engineering Thermodynamics

2-51

FIGURE 2.13 Generalized entropy correction chart. (Source: Adapted from Van Wylen, G. J. and Sonntag, R. E. 1986. Fundamentals of Classical Thermodynamics, 3rd ed., English/SI. Wiley, New York.)

Multicomponent Systems In this section are presented some general aspects of the properties of multicomponent systems consisting of nonreacting mixtures. For a single phase multicomponent system consisting of j components, an extensive property X may be regarded as a function of temperature, pressure, and the number of moles © 2005 by CRC Press LLC

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Chapter 2

FIGURE 2.14 Generalized fugacity coefficient chart. (Source: Van Wylen, G. J. and Sonntag, R. E. 1986. Fundamentals of Classical Thermodynamics, 3rd ed., English/SI. Wiley, New York.)

of each component present in the mixture: X = X(T, p, n1, n2, … nj). Since X is mathematically homogeneous of degree one in the n’s, the function is expressible as j

X=

∑n X i

i =1

where the partial molar property Xi is by definition

© 2005 by CRC Press LLC

i

(2.67)

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Engineering Thermodynamics

Xi =

∂X  ∂ni  T , p,n

(2.68)




and the subscript n
denotes that all n’s except ni are held fixed during differentiation. As Xi depends in general on temperature, pressure, and mixture composition: Xi (T, p, n1, n2, … nj), the partial molal property Xi is an intensive property of the mixture and not simply a property of the ith component. Selecting the extensive property X to be volume, internal energy, enthalpy, entropy, and the Gibbs function, respectively, gives j

V=

∑

j

U=

ni Vi ,

i =1

∑

i

i

i =1

j

H=

∑n U j

S=

ni Hi ,

i =1

∑n S

i i

(2.69)

i =1

j

G=

∑n G i

i

i =1

where Vi , Ui , Hi , Si , and Gi denote the respective partial molal properties. When pure components, each initially at the same temperature and pressure, are mixed, the changes in volume, internal energy, enthalpy, and entropy on mixing are given by j

∆Vmixing =

∑ n (V − v ) i

i

i

(2.70a)

i =1 j

∆U mixing =

∑ n (U − u ) i

i

i

(2.70b)

i =1 j

∆H mixing =

∑ n (H − h ) i

i

i

(2.70c)

i =1 j

∆Smixing =

∑ n (S − s ) i

i

i

(2.70d)

i =1

where vi , ui , hi , and si denote the molar-specific volume, internal energy, enthalpy, and entropy of pure component i. Chemical Potential The partial molal Gibbs function of the ith component of a multicomponent system is the chemical potential, µi , µ i = Gi =

∂G  ∂ni  T , p,n




Like temperature and pressure, the chemical potential, µi is an intensive property.

© 2005 by CRC Press LLC

(2.71)

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Chapter 2

When written in terms of chemical potentials, Equation 2.67 for the Gibbs function reads j

G=

∑n µ i

(2.72)

i

i =1

For a single component sysrem, Equation 2.72 reduces to G = nµ; that is, the chemical potential equals the molar Gibbs function. For an ideal gas mixture, comparison of Equation 2.63 and Equation 2.72 suggests µi = gi (T, pi); that is, the chemical potential of component i in an ideal gas mixture equals its Gibbs function per mole of gas i evaluated at the mixture temperature and the partial pressure of the ith gas of the mixture. The chemical potential is a measure of the escaping tendency of a substance in a multiphase system: a substance tends to move from the phase having the higher chemical potential for that substance to the phase having a lower chemical potential. A necessary condition for phase equilibrium is that the chemical potential of each component has the same value in every phase. The Gibbs phase rule gives the number F of independent intensive properties that may be arbitrarily specified to fix the intensive state of a system at equilibrium consisting of N nonreacting components present in P phases: F = 2 + N – P. F is called the degrees of freedom (or the variance). For water as a single component, for example, N = 1 and F = 3 – P. • For a single phase, P = 1 and F = 2: two intensive properties can be varied independently, say temperature and pressure, while maintaining a single phase. • For two phases, P = 2 and F = 1: only one intensive property can be varied independently if two phases are maintained — for example, temperature or pressure. • For three phases, P = 3 and F = 0: there are no degrees of freedom; each intensive property of each phase is fixed. For a system consisting of ice, liquid water, and water vapor at equilibrium, there is a unique temperature: 0.01°C (32.02°F) and a unique pressure: 0.6113 kPa (0.006 atm). The phase rule does not address the relative amounts that may be present in the various phases. With G = H – TS and H = U + pV, Equation 2.72 can be expressed as j

U = TS − pV +

∑n µ i

(2.73)

i

i =1

from which can be derived j

dU = TdS − pdV +

∑ µ dn i

i

(2.74)

i =1

When the mixture composition is constant, Equation 2.74 reduces to Equation 2.31a. Ideal Solution The Lewis-Randall rule states that the fugacity fi of each component i in an ideal solution is the product of its mole fraction and the fugacity of the pure component, fi, at the same temperature, pressure, and state of aggregation (gas, liquid, or solid) as the mixture: fi = yi fi

(Lewis-Randall rule)

(2.75)

The following characteristics are exhibited by an ideal solution: Vi = vi , Ui = ui , Hi = hi . With these, Equation 2.70a, Equation 2.70b, and Equation 2.70c show that there is no change in volume, internal energy, or enthalpy on mixing pure components to form an ideal solution. The adiabatic mixing of different pure components would result in an increase in entropy, however, because such a process is irreversible. © 2005 by CRC Press LLC

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Engineering Thermodynamics

The volume of an ideal solution is j

V=

∑

j

ni vi =

i =1

∑V

(ideal solution)

i

(2.76)

i =1

where Vi is the volume that pure component i would occupy when at the temperature and pressure of the mixture. Comparing Equation 2.48a and Equation 2.76, the additive volume rule is seen to be exact for ideal solutions. The internal energy and enthalpy of an ideal solution are j

U=

∑ i =1

j

ni ui ,

H=

∑n h

i i

(ideal solution)

(2.77)

i =1

where ui and hi denote, respectively, the molar internal energy and enthalpy of pure component i at the temperature and pressure of the mixture. Many gaseous mixtures at low to moderate pressures are adequately modeled by the Lewis Randall rule. The ideal gas mixtures considered in Section 2.3, Ideal Gas Model, is an important special case. Some liquid solutions also can be modeled with the LewisRandall rule.

2.4 Combustion The thermodynamic analysis of reactive systems is primarily an extension of principles presented in Sections 2.1 to 2.3. It is necessary, though, to modify the methods used to evaluate specific enthalpy and entropy.

Reaction Equations In combustion reactions, rapid oxidation of combustible elements of the fuel results in energy release as combustion products are formed. The three major combustible chemical elements in most common fuels are carbon, hydrogen, and sulfur. Although sulfur is usually a relatively unimportant contributor to the energy released, it can be a significant cause of pollution and corrosion. The emphasis in this section is on hydrocarbon fuels, which contain hydrogen, carbon, sulfur, and possibly other chemical substances. Hydrocarbon fuels may be liquids, gases, or solids such as coal. Liquid hydrocarbon fuels are commonly derived from crude oil through distillation and cracking processes. Examples are gasoline, diesel fuel, kerosene, and other types of fuel oils. The compositions of liquid fuels are commonly given in terms of mass fractions. For simplicity in combustion calculations, gasoline is often considered to be octane, C8H18, and diesel fuel is considered to be dodecane, C12H26. Gaseous hydrocarbon fuels are obtained from natural gas wells or are produced in certain chemical processes. Natural gas normally consists of several different hydrocarbons, with the major constituent being methane, CH4. The compositions of gaseous fuels are commonly given in terms of mole fractions. Both gaseous and liquid hydrocarbon fuels can be synthesized from coal, oil shale, and tar sands. The composition of coal varies considerably with the location from which it is mined. For combustion calculations, the makeup of coal is usually expressed as an ultimate analysis giving the composition on a mass basis in terms of the relative amounts of chemical elements (carbon, sulfur, hydrogen, nitrogen, oxygen) and ash. Coal combustion is considered further in Chapter 8, Energy Conversion. A fuel is said to have burned completely if all of the carbon present in the fuel is burned to carbon dioxide, all of the hydrogen is burned to water, and all of the sulfur is burned to sulfur dioxide. In practice, these conditions are usually not fulfilled and combustion is incomplete. The presence of carbon monoxide (CO) in the products indicates incomplete combustion. The products of combustion of actual combustion reactions and the relative amounts of the products can be determined with certainty only by experimental means. Among several devices for the experimental determination of the composition of © 2005 by CRC Press LLC

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Chapter 2

products of combustion are the Orsat analyzer, gas chromatograph, infrared analyzer, and flame ionization detector. Data from these devices can be used to determine the makeup of the gaseous products of combustion. Analyses are frequently reported on a “dry” basis: mole fractions are determined for all gaseous products as if no water vapor were present. Some experimental procedures give an analysis including the water vapor, however. Since water is formed when hydrocarbon fuels are burned, the mole fraction of water vapor in the gaseous products of combustion can be significant. If the gaseous products of combustion are cooled at constant mixture pressure, the dew point temperature (Section 2.3, Ideal Gas Model) is reached when water vapor begins to condense. Corrosion of duct work, mufflers, and other metal parts can occur when water vapor in the combustion products condenses. Oxygen is required in every combustion reaction. Pure oxygen is used only in special applications such as cutting and welding. In most combustion applications, air provides the needed oxygen. Idealizations are often used in combustion calculations involving air: (1) all components of air other than oxygen (O2) are lumped with nitrogen (N2). On a molar basis air is then considered to be 21% oxygen and 79% nitrogen. With this idealization the molar ratio of the nitrogen to the oxygen in combustion air is 3.76; (2) the water vapor present in air may be considered in writing the combustion equation or ignored. In the latter case the combustion air is regarded as dry; (3) additional simplicity results by regarding the nitrogen present in the combustion air as inert. However, if high-enough temperatures are attained, nitrogen can form compounds, often termed NOX, such as nitric oxide and nitrogen dioxide. Even trace amounts of oxides of nitrogen appearing in the exhaust of internal combustion engines can be a source of air pollution. The minimum amount of air that supplies sufficient oxygen for the complete combustion of all the combustible chemical elements is the theoretical, or stoichiometic, amount of air. In practice, the amount of air actually supplied may be greater than or less than the theoretical amount, depending on the application. The amount of air is commonly expressed as the percent of theoretical air or the percent excess (or percent deficiency) of air. The air-fuel ratio and its reciprocal the fuel-air ratio, each of which can be expressed on a mass or molar basis, are other ways that fuel-air mixtures are described. Another is the equivalence ratio: the ratio of the actual fuel-air ratio to the fuel-air ratio for complete combustion with the theoretical amount of air. The reactants form a lean mixture when the equivalence ratio is less than unity and a rich mixture when the ratio is greater than unity. Example 9 Methane, CH4, is burned with dry air. The molar analysis of the products on a dry basis is CO2, 9.7%; CO, 0.5%; O2, 2.95%; and N2, 86.85%. Determine (a) the air-fuel ratio on both a molar and a mass basis, (b) the percent of theoretical air, (c) the equivalence ratio, and (d) the dew point temperature of the products, in °F, if the pressure is 1 atm. Solution. (a) The solution is conveniently conducted on the basis of 100 lbmol of dry products. The chemical equation then reads aCH 4 + b(O 2 + 3.76 N 2 ) → 9.7CO 2 + 0.5CO + 2.95O 2 + 86.85N 2 + cH 2 O where N2 is regarded as inert. Water is included in the products together with the assumed 100 lbmol of dry products. Balancing the carbon, hydrogen, and oxygen, the reaction equation is 10.2CH 4 + 23.1(O 2 + 3.76N 2 ) → 9.7CO 2 + 0.5CO + 2.95O 2 + 86.85N 2 + 20.4H 2 O The nitrogen also balances, as can be verified. This checks the accuracy of both the given product analysis and the calculations conducted to determine the unknown coefficients. Exact closure cannot be expected with measured data, however. On a molar basis, the air-fuel ratio is © 2005 by CRC Press LLC

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Engineering Thermodynamics

AF =

lbmol(air ) 23.1(4.76) = 10.78 lbmol(fuel) 10.2

On a mass basis lb(air ) 28.97  = 19.47 AF = (10.78)   16.04  lb(fuel) (b) The balanced chemical equation for the complete combustion of methane with the theoretical amount of air is CH 4 + 2(O 2 + 3.76 N 2 ) → CO 2 + 2H 2 O + 7.52N 2 The theoretical air-fuel ratio on a molar basis is

(AF)

theo

=

lbmol(air ) 2(4.76) = 9.52 lbmol(fuel) 1

The percent theoretical air is then

% theoretical air =

(AF) (AF)

theo

=

10.78 = 1.13(113%) 9.52

(c) Equivalence ratio = ( FA)/( FA) theo = 9.52/10.78 = 0.88. The reactants form a lean mixture. (d) To determine the dew point temperature requires the partial pressure pv of the water vapor. The mole fraction of the water vapor is yv =

20.4 = 0.169 100 + 20.4

Since p = 1 atm, pv = 0.169 atm = 2.48 lbf/in.2. With psat = 2.48 lbf/in.2, the corresponding saturation temperature from the steam tables is 134°F. This is the dew point temperature.

Property Data for Reactive Systems Tables of thermodynamic properties such as the steam tables provide values for the specific enthalpy and entropy relative to some arbitrary datum state where the enthalpy (or alternatively the internal energy) and entropy are set to zero. When a chemical reaction occurs, however, reactants disappear and products are formed, and it is generally no longer possible to evaluate ∆h and ∆s so that these arbitrary datums cancel. Accordingly, special means are required to assign specific enthalpy and entropy for application to reacting systems. Property data suited for the analysis of reactive systems are available from several sources. The encyclopedic JANAF Thermochemical Tables is commonly used. Data for a wide range of substances are retrievable from Knacke et al. (1991), which provides both tabular data and analytical expressions readily programmable for use with personal computers of the specific heat, enthalpy, entropy, and Gibbs function. Textbooks on engineering thermodynamics also provide selected data, as, for example, Moran and Shapiro (1995). © 2005 by CRC Press LLC

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Chapter 2

Enthalpy of Formation An enthalpy datum for reacting systems can be established by assigning arbitrarily a value of zero to the enthalpy of the stable elements at a standard reference state where the temperature is Tref = 298.15 K (25°C) and the pressure is pref , which may be 1 bar or 1 atm depending on the data source. The term stable simply means that the particular element is chemically stable. For example, at the standard state the stable forms of hydrogen, oxygen, and nitrogen are H2, O2, and N2 and not the monatomic H, O, and N. The molar enthalpy of a compound at the standard state equals its enthalpy of formation, symbolized  here by h f . The enthalpy of formation is the energy released or absorbed when the compound is formed from its elements, the compound and elements all being at Tref and pref . The enthalpy of formation may be determined by application of procedures from statistical thermodynamics using observed spectroscopic data. The enthalpy of formation also can be found in principle by measuring the heat transfer in a reaction in which the compound is formed from the elements. In this chapter, the superscript ° is used to denote pref . For the case of the enthalpy of formation, the reference temperature Tref is also intended by this symbol. Table 2.9 gives the values of the enthalpy of formation of various substances at 298 K and 1 atm. The molar enthalpy of a substance at a state other than the standard state is found by adding the molar enthalpy change ∆h between the standard state and the state of interest to the molar enthalpy of formation: 

[

(

h (T , p) = h f + h (T , p) − h Tref , pref 

)] = h

 f

+ ∆h

(2.78)

That is, the enthalpy of a substance is composed of h f , associated with the formation of the substance from its elements, and ∆h, associated with a change of state at constant composition. An arbitrarily chosen datum can be used to determine ∆h, since it is a difference at constant composition. Accordingly, ∆h can be evaluated from sources such as the steam tables and the ideal gas tables. The enthalpy of combustion, hRP , is the difference between the enthalpy of the products and the enthalpy of the reactants, each on a per-mole-of-fuel basis, when complete combustion occurs and both reactants and products are at the same temperature and pressure. For hydrocarbon fuels the enthalpy of combustion is negative in value since chemical internal energy is liberated in the reaction. The heating value of a fuel is a positive number equal to the magnitude of the enthalpy of combustion. Two heating values are recognized: the higher heating value and the lower heating value. The higher heating value is obtained when all the water formed by combustion is a liquid; the lower heating value is obtained when all the water formed by combustion is a vapor. The higher heating value exceeds the lower heating value by the energy that would be required to vaporize the liquid water formed at the specified temperature. Heating values are typically reported at a temperature of 25°C (77°F) and a pressure of 1 bar (or 1 atm). These values also depend on whether the fuel is a liquid or a gas. A sampling is provided on a unit-massof-fuel basis in Table 2.10. In the absence of work W˙ cv and appreciable kinetic and potential energy effects, the energy liberated on combustion is transferred from a reactor at steady state in two ways: the energy accompanying the exiting combustion products and by heat transfer. The temperature that would be achieved by the products in the limit of adiabatic operation is the adiabatic flame or adiabatic combustion temperature. For a specified fuel and specified temperature and pressure of the reactants, the maximum adiabatic flame temperature is realized for complete combustion with the theoretical amount of air. Example 10 provides an illustration. The measured value of the temperature of the combustion products may be several hundred degrees below the calculated maxunum adiabatic flame temperature, however, for several reasons including the following: (1) heat loss can be reduced but not eliminated; (2) once adequate oxygen has been provided to permit complete combustion, bringing in more air dilutes the combustion products, lowering the temperature; (3) incomplete combustion tends to reduce the temperature of the products, and combustion is seldom complete; (4) as result of the high temperatures achieved, some of the combustion products may dissociate. Endothermic dissociation reactions also lower the product temperature. © 2005 by CRC Press LLC

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Engineering Thermodynamics

TABLE 2.9 Enthalpy of Formation, Gibbs Function of Formation, and Absolute Entropy of Various Substances at 298 K and 1 atm 

h f and g f (kJ/kmol), s  (kJ/kmol•K) 

Substance

Formula

hf

g f

s

Carbon Hydrogen Nitrogen Oxygen Carbon monoxide Carbon dioxide Water

C(s) H2(g) N2(g) O2(g) CO(g) CO2(g) H2O(g) H2O(l) H2O2(g) NH3(g) O(g) H(g) N(g) OH(g) CH4(g) C2H2(g) C2H4(g) C2H6(g) C3H6(g) C3H8(g) C4H10(g) C5H12(g) C8H18(g) C8H18(l) C6H6(8) CH3OH(g) CH3OH(I) C2H5OH(g) C2H5OH(I)

0 0 0 0 –110,530 –393,520 –241,820 –285,830 –136,310 –46,190 249,170 218,000 472,680 39,460 –74,850 226,730 52,280 –84,680 20,410 –103,850 –126,150 –146,440 –208,450 –249,910 82,930 –200,890 –238,810 –235,310 –277,690

0 0 0 0 –137,150 –394,380 –228,590 –237,180 –105,600 –16,590 231,770 203,290 455,510 34,280 –50,790 209,170 68,120 –32,890 62,720 –23,490 –15,710 –8,200 17,320 6,610 129,660 –162,140 –166,290 –168,570 174,890

5.74 130.57 191.50 205.03 197.54 213.69 188.72 69.95 232.63 192.33 160.95 114.61 153.19 183.75 186.16 200.85 219.83 229.49 266.94 269.91 310.03 348.40 463.67 360.79 269.20 239.70 126.80 282.59 160.70

Hydrogen peroxide Ammonia Oxygen Hydrogen Nitrogen Hydroxyl Methane Acetylene Ethylene Ethane Propylene Propane Butane Pentane Octane Benzene Methyl alcohol Ethyl alcohol

Source: Adapted from Wark, K. 1983. Thermodynamics, 4th ed. McGraw-Hill, New York, as based on JANAF Thermochemical Tables, NSRDS-NBS-37, 1971; Selected Values of Chemical Thermodynamic Properties, NBS Tech. Note 270-3, 1968; and API Research Project 44, Carnegie Press, 1953.

TABLE 2.10 Heating Values in kJ/kg of Selected Hydrocarbons at 25°C Higher Valuea

Lower Valueb

Hydrocarbon

Formula

Liquid Fuel

Gas. Fuel

Liquid Fuel

Gas. Fuel

Methane Ethane Propane n-Butane n-Octane n-Dodecane Methanol Ethanol

CH4 C2H6 C3H8 C4H10 C8H18 C12H26 CH3OH C3H5OH

— — 49,973 49,130 47,893 47,470 22,657 29,676

55,496 51,875 50,343 49,500 48,256 47,828 23,840 30,596

— — 45,982 45,344 44,425 44,109 19,910 26,811

50,010 47,484 46,352 45,714 44,788 44,467 21,093 27,731

a b

H2O liquid in the products. H2O vapor in the products.

© 2005 by CRC Press LLC

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Chapter 2

Absolute Entropy A common datum for assigning entropy values to substances involved in chemical reactions is realized through the third law of thermodynamics, which is based on experimental observations obtained primarily from studies of chemical reactions at low temperatures and specific heat measurements at temperatures approaching absolute zero. The third law states that the entropy of a pure crystalline substance is zero at the absolute zero of temperature, 0 K or 0°R. Substances not having a pure crystalline structure have a nonzero value of entropy at absolute zero. The third law provides a datum relative to which the entropy of each substance participating in a reaction can be evaluated. The entropy relative to this datum is called the absolute entropy. The change in entropy of a substance between absolute zero and any given state can be determined from measurements of energy transfers and specific heat data or from procedures based on statistical thermodynamics and observed molecular data. Table 2.9, Table A.2 and Table A.8 provide absolute entropy data for various substances. In these tables, pref = 1 atm. When the absolute entropy is known at pressure pref and temperature T, the absolute entropy at the same temperature and any pressure p can be found from

) [

(

(

s (T , p) = s T , pref + s (T , p) − s T , pref

)]

(2.79)

For an ideal gas, the second term on the right side of Equation 2.79 can be evaluated by using Equation 2.58, giving s (T , p) = s  (T ) − R ln

p pref

(ideal gas)

(2.80)

In this expression, s  (T) denotes the absolute entropy at temperature T and pressure pref . The entropy of the ith component of an ideal gas mixture is evaluated at the mixture temperature T and the partial pressure pi: si (T, pi). For the ith component, Equation 2.80 takes the form si (T , pi ) = si (T ) − R ln

pi pref (2.81)

yp = s (T ) − R ln i pref

(ideal gas)

 i

where si (T) is the absolute entropy of component i at temperature T and pref . Example 10 Liquid octane at 25°C, 1 atm enters a well insulated reactor and reacts with dry air entering at the same temperature and pressure. For steady-state operation and negligible effects of kinetic and potential energy, determine the temperature of the combustion products for complete combustion with the theoretical amount of air, and (b) the rates of entropy generation and exergy destruction, each per kmol of fuel. Solution. For combustion of liquid octane with the theoretical amount of air, the chemical equation is C 8 H18 (l) + 12.5O 2 + 47N 2 → 8CO 2 + 9H 2 O(g) + 47N 2 (a) At steady state, the control volume energy rate balance reduces to read 0= © 2005 by CRC Press LLC

0 0 Q˙ cv W˙ − cv + n˙ F n˙ F

∑ n (h i

R

 f

) ∑ n (h

+ ∆h −

e

i

P

 f

+ ∆h

)

e

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Engineering Thermodynamics

where R denotes reactants, P denotes products, and the symbols for enthalpy have the same significance as in Equation 2.78. Since the reactants enter at 25°C, the corresponding ( ∆h )i terms vanish, and the energy rate equation becomes

∑ n ( ∆h ) = ∑ n h − ∑ n h 

e

i

e

P

fi

e

R

 fe

P

Introducing coefficients from the reaction equation, this takes the form

( )

8 ∆h

CO2

( )

+ 9 ∆h

( )

− 8 h f 



H 2O( g )

( )

( )

+ 47 ∆h



CO2

+ 9 hf

( )

 =  h f 

N2

( ) 

H 2O( g )

+ 47 ∆h f

N2

( ) 

C8H18 ( l )

+ 12.5 h f

( ) 

O2

+ 47 h f

N2

 

 

Using data from Table 2.9 to evaluate the right side,

( )

8 ∆h

CO2

( )

+ 9 ∆h

H 2O( g )

( )

+ 47 ∆h

= 5,074,630 kJ kmol (fuel)

N2

Each ∆h term on the left side of this equation depends on the temperature of the products, Tp, which can be solved for iteratively as Tp = 2395 K. (b) The entropy rate balance on a per-mole-of-fuel basis takes the form 0=

∑ j

0 S˙gen Q˙ j Tj + s F + 12.5sO2 + 47sN2 − 8sCO2 + 9sH O( g ) + 47sN 2 + 2 n˙ F n˙ F

) (

(

)

or on rearrangement, S˙gen n˙ F

(

)

(

= 8sCO2 + 9sH O( g ) + 47sN 2 − s F − 12.5sO2 + 47sN 2 2

)

The absolute entropy of liquid octane from Table 2.9 is 360.79 kJ/mol · K. The oxygen and nitrogen in the combustion air enter the reactor as components of an ideal gas mixture at Tref , pref . With Equation 2.81, where p = pref , and absolute entropy data from Table 2.9,

( )

sO2 = sO 2 Tref − R ln yO2 = 205.03 − 8.314 ln 0.21 = 218.01 kJ kmol ⋅ K

( )

sN 2 = sN 2 Tref − R ln yN 2 = 191.5 − 8.314 ln 0.79 = 193.46 kJ kmol ⋅ K The product gas exits as a gas mixture at 1 atm, 2395 K with the following composition: yCO2 = 8/64 = 0.125, yH2O( g ) = 9/64 = 0.1406, yN 2 = 47/64 = 0.7344. With Equation 2.81, where p = pref , and absolute entropy data at 2395 K from Table A.2, sCO2 = 320.173 − 8.314 ln 0.125 = 337.46 kJ kmol ⋅ K sH2O = 273.986 − 8.314 ln 0.1406 = 290.30 kJ kmol ⋅ K sN2 = 258.503 − 8.314 ln 0.7344 = 261.07 kJ kmol ⋅ K © 2005 by CRC Press LLC

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Chapter 2

Inserting values, the rate of entropy generation is S˙gen n˙ F

= 8(337.46) + 9(290.30) + 47(261.07) − 360.79 − 12.5(218.01) − 47(193.46) = 5404 kJ kmol ⋅ K

Using Equation 2.87 and assuming T0 = 298 K, the rate of exergy destruction is E˙ D / n˙ F = 1.61 × 106 kJ/kmol. Gibbs Function of Formation Paralleling the approach used for enthalpy, a value of zero is assigned to the Gibbs function of each stable element at the standard state. The Gibbs function of formation of a compound equals the change in the Gibbs function for the reaction in which the compound is formed from its elements. Table 2.9 provides Gibbs function of formation data of various substances at 298 K and 1 atm. The Gibbs function at a state other than the standard state is found by adding to the Gibbs function of formation the change in the specific Gibbs function ∆g between the standard state and the state of interest:

[

(

g (T , p) = g f + g (T , p) − g Tref , pref

)] = g

 f

+ ∆g

(2.82a)

where

[

(

∆g = h (T , p) − h Tref , pref

)] − [T s (T , p) − T s (T ref

ref

, pref

)]

(2.82b)

The Gibbs function of component i in an ideal gas mixture is evaluated at the partial pressure of component i and the mixture temperature. As an application, the maximum theoretical work that can be developed, per mole of fuel consumed, is evaluated for the control volume of Figure 2.15, where the fuel and oxygen each enter in separate streams and carbon dioxide and water each exit separately. All entering and exiting streams are at the same temperature T and pressure p. The reaction is complete: b b Ca Hb +  a +  O 2 → aCO 2 + H 2 O   4 2 This control volume is similar to idealized devices such as a reversible fuel cell or a van’t Hoff equilibrium box. For steady-state operation, the energy rate balance reduces to give W˙ cv Q˙ cv b b = + hF +  a +  hO2 − ahCO2 − hH2O  4 2 n˙ F n˙ F where n˙ F denotes the molar flow rate of the fuel. Kinetic and potential energy effects are regarded as negligible. If heat transfer occurs only at the temperature T, an entropy balance for the control volume takes the form 0=

© 2005 by CRC Press LLC

S˙gen Q˙ cv n˙ F b b + s F +  a +  sO2 − asCO2 − sH2O +  T 4 2 n˙ F

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Engineering Thermodynamics

FIGURE 2.15 Device for evaluating maximum work.

Eliminating the heat transfer term from these expressions, an expression for the maximum theoretical value of the work developed per mole of fuel is obtained when the entropy generation term is set to zero:  W˙ cv  b b b b        n˙  int = hF +  a + 4  hO2 − ahCO2 − 2 hH2O (T , p) − T s F +  a + 4  sO2 − asCO2 − 2 sH2O (T , p)  F      rev

This can be written alternatively in terms of the enthalpy of combustion as  W˙ cv  b b     n˙  int = − hRP (T , p) − T s F +  a + 4  sO2 − asCO2 − 2 sH2O (T , p)  F   

(2.83a)

rev

or in terms of Gibbs functions as  W˙ cv     n˙  int = gF +  a +  F   rev

b b  gO2 − agCO2 − gH2O (T , p)  4 2 

(2.83b)

Equation 2.83b is used in the solution to Example 11. Example 11 Hydrogen (H2) and oxygen (O2), each at 25°C, 1 atm, enter a fuel cell operating at steady state, and liquid water exits at the same temperature and pressure. The hydrogen flow rate is 2 × 10–4 kmol/sec and the fuel cell operates isothermally at 25°C. Determine the maximum theoretical power the cell can develop, in kW. Solution. The overall cell reaction is H2 + 1/2 O2 → H2O(
), and Equations 2.83 are applicable. Selecting Equation 2.83b, and using Gibbs function data from Table 2.9,  W˙ cv  1    n˙  int =  gH2 + 2 gO2 − gH2O(
)  (25°C, 1 atm)  F  rev

=0+

© 2005 by CRC Press LLC

1 (0) − (−237,180) = 237,180 kJ kmol 2

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Chapter 2

Then

(W˙ )

cv int rev

kJ   kmol   kW  =  237,180 2 × 10 −4   = 47.4 kW  kmol   s   1kJ s 

Reaction Equilibrium Let the objective be to determine the equilibrium composition of a system consisting of five gases A, B, C, D, and E, at a temperature T and pressure p, subject to a chemical reaction of the form v A A + v B B ↔ vC C + v D D where the v’s are stoichiometric coefficients. Component E is assumed to be inert and thus does not appear in the reaction equation. The equation suggests that at equilibrium the tendency of A and B to form C and D is just balanced by the tendency of C and D to form A and B. At equilibrium, the temperature and pressure would be uniform throughout the system. Additionally, the equation of reaction equilibrium must be satisfied: v A µ A + v B µ B = vC µ C + v D µ D

(2.84a)

where the µ’s are the chemical potentials (Section 2.3, Multicomponent Systems) of A, B, C, and D in the equilibrium mixture. In principle, the composition that would be present at equilibrium for a given temperature and pressure can be determined by solving this equation. For ideal gas mixtures, the solution procedure is simplified by using the equilibrium constant K(T ) and the following equation: y vC y vD  p  K (T ) = CvA DvB   y A y B  pref 

vC + vD − v A − vB

v n C n vD  p pref  = CvA DvB   nA nB  n 

(2.84b) vC + vD − v A − vB

where yA, yB , yC , and yD denote the mole fractions of A, B, C, and D in the equilibrium mixture and n = nA + nB + nC + nD + nE, where the n’s denote the molar amounts of the gases in the mixture. Tabulations of K(T ) for each of several reactions of the form Equation 2.84a are provided in Table 2.11. An application of Equation 2.84b is provided in Example 12. Example 12 One kmol of CO reacts with the theoretical amount of dry air to form an equilibrium mixture of CO2, CO, O2, and N2 at 2500 K, 1 atm. Determine the amount of CO in the equilibrium mixture, in kmol. Solution. The reaction of CO with the theoretical amount of dry air to form CO2, CO, O2, and N2 is CO +

1 z O 2 + 1.88N 2 → zCO + O 2 + (1 − z )CO 2 + 1.88N 2 2 2

where z is the amount of CO, in kmol, present in the equilibrium mixture. The total number of moles n is n=z+

© 2005 by CRC Press LLC

z 5.76 + z + (1 − z ) + 1.88 = 2 2

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Engineering Thermodynamics

TABLE 2.11 Logarithms to the Base 10 of the Equilibrium Constant K log10 K /2O2 + 1/2N2 H2O ⇔ H2 + 1/2O2 ⇔NO

1

Temp (K) H2 ⇔ 2H O2 ⇔ 2O N2 ⇔ 2N 298 500 1000 1200 1400 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500

–71.224 –40.316 –17.292 –13.414 –10.630 –8.532 –7.666 –6.896 –6.204 –5.580 –5.016 –4.502 –4.032 –3.600 –3.202 –2.836 –2.494 –2.178 –1.882 –1.606 –1.348 –1.106 –0.878 –0.664 –0.462

–81.208 –45.880 –19.614 –15.208 –12.054 –9.684 –8.706 –7.836 –7.058 –6.356 –5.720 –5.142 –4.614 –4.130 –3.684 –3.272 –2.892 –2.536 –2.206 –1.898 –1.610 –1.340 –1.086 –0.846 –0.620

–159.600 –92.672 –43.056 –34.754 –28.812 –24.350 –22.512 –20.874 –19.410 –18.092 –16.898 –15.810 –14.818 –13.908 –13.070 –12.298 –11.580 –10.914 –10.294 –9.716 –9.174 –8.664 –8.186 –7.736 –7.312

–15.171 –8.783 –4.062 –3.275 –2.712 –2.290 –2.116 –1.962 –1.823 –1.699 –1.586 –1.484 –1.391 –1.305 –1.227 –1.154 –1.087 –1.025 –0.967 –0.913 –0.863 –0.815 –0.771 –0.729 –0.690

–40.048 –22.886 –10.062 –7.899 –6.347 –5.180 –4.699 –4.270 –3.886 –3.540 –3.227 –2.942 –2.682 –2.443 –2.224 –2.021 –1.833 –1.658 –1.495 –1.343 –1.201 –1.067 –0.942 –0.824 –0.712

CO2 +H2 H2Ο ⇔ CO2 ⇔ ⇔ OH + 1/2H2 CO + 1/2 O2 CO +H2O –46.054 –26.130 –11.280 –8.811 –7.021 –5.677 –5.124 –4.613 –4.190 –3.776 –3.434 –3.091 –2.809 –2.520 –2.270 –2.038 –1.823 –1.624 –1.438 –1.265 –1.103 –0.951 –0.809 –0.674 –0.547

–45.066 –25.025 –10.221 –7.764 –6.014 –4.706 –4.169 –3.693 –3.267 –2.884 –2.539 –2.226 –1.940 –1.679 –1.440 –1.219 –1.015 –0.825 –0.649 –0.485 –0.332 –0.189 –0.054 +0.071 +0.190

–5.018 –2.139 –0.159 +0.135 +0.333 +0.474 +0.530 +0.577 +0.619 +0.656 +0.688 +0.716 +0.742 +0.764 +0.784 +0.802 +0.818 +0.833 +0.846 +0.858 +0.869 +0.878 +0.888 +0.895 +0.902

Temp (°R) 537 900 1800 2160 2520 2880 3060 3240 3420 3600 3780 3960 4140 4320 4500 4680 4860 5040 5220 5400 5580 5760 5940 6120 6300

Source: Based on data from the JANAF Thermochemical Tables, NSRDS-NBS-37, 1971.

At equilibrium CO2 ↔ CO + 1/2 O2; and Equation 2.84b takes the form z( z 2 ) 1− z

12

K=

 p pref     (5.76 + z ) 2 

12

where p/pref = 1. At 2500 K, Table 2.11 gives K = 0.0363. Solving iteratively, z = 0.175.

2.5 Exergy Analysis The method of exergy analysis (availability analysis) presented in this section enables the location, cause, and true magnitude of energy resource waste and loss to be determined. Such information can be used in the design of new energy-efficient systems and for improving the performance of existing systems. Exergy analysis also provides insights that elude a purely first-law approach. For example, on the basis of first-law reasoning alone, the condenser of a power plant may be mistakenly identified as the component primarily responsible for the plant’s seemingly low overall performance. An exergy analysis correctly reveals not only that the condenser loss is relatively unimportant (see the last two rows of the Rankine cycle values of Table 2.15), but also that the steam generator is the principal site of thermodynamic inefficiency owing to combustion and heat transfer irreversibilities within it. When exergy concepts are combined with principles of engineering economy, the result is known as thermoeconomics. Thermoeconomics allows the real cost sources at the component level to be identified: capital investment costs, operating and maintenance costs, and the costs associated with the destruction © 2005 by CRC Press LLC

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Chapter 2

and loss of exergy. Optimization of thermal systems can be achieved by a careful consideration of such cost sources. From this perspective thermoeconomics is exergy-aided cost minimization. Discussions of exergy analysis and thermoeconomics are provided by Bejan et al. (1996), Moran (1989), and Moran and Shapiro (1995). In this section salient aspects are presented.

Defining Exergy An opportunity for doing work exists whenever two systems at different states are placed in communication because, in principle, work can be developed as the two are allowed to come into equilibrium. When one of the two systems is a suitably idealized system called an environment and the other is some system of interest, exergy is the maximum theoretical useful work (shaft work or electrical work) obtainable as the systems interact to equilibrium, heat transfer occurring with the environment only. (Alternatively, exergy is the minimum theoretical useful work required to form a quantity of matter from substances present in the environment and to bring the matter to a specified state.) Exergy is a measure of the departure of the state of the system from that of the environment, and is therefore an attribute of the system and environment together. Once the environment is specified, however, a value can be assigned to exergy in terms of property values for the system only, so exergy can be regarded as an extensive property of the system. Exergy can be destroyed and generally is not conserved. A limiting case is when exergy would be completely destroyed, as would occur if a system were to come into equilibrium with the environment spontaneously with no provision to obtain work. The capability to develop work that existed initially would be completely wasted in the spontaneous process. Moreover, since no work needs to be done to effect such a spontaneous change, the value of exergy can never be negative. Environment Models with various levels of specificity are employed for describing the environment used to evaluate exergy. Models of the environment typically refer to some portion of a system’s surroundings, the intensive properties of each phase of which are uniform and do not change significantly as a result of any process under consideration. The environment is regarded as composed of common substances existing in abundance within the Earth’s atmosphere, oceans, and crust. The substances are in their stable forms as they exist naturally, and there is no possibility of developing work from interactions — physical or chemical — between parts of the environment. Although the intensive properties of the environment are assumed to be unchanging, the extensive properties can change as a result of interactions with other systems. Kinetic and potential energies are evaluated relative to coordinates in the environment, all parts of which are considered to be at rest with respect to one another. For computational ease, the temperature T0 and pressure p0 of the environment are often taken as standard-state values, such as 1 atm and 25°C (77°F). However, these properties may be specified differently depending on the application. T0 and p0 might be taken as the average ambient temperature and pressure, respectively, for the location at which the system under consideration operates. Or, if the system uses atmospheric air, T0 might be specified as the average air temperature. If both air and water from the natural surroundings are used, T0 would be specified as the lower of the average temperatures for air and water. Dead States When a system is in equilibrium with the environment, the state of the system is called the dead state. At the dead state, the conditions of mechanical, thermal, and chemical equilibrium between the system and the environment are satisfied: the pressure, temperature, and chemical potentials of the system equal those of the environment, respectively. In addition, the system has no motion or elevation relative to coordinates in the environment. Under these conditions, there is no possibility of a spontaneous change within the system or the environment, nor can there be an interaction between them. The value of exergy is zero. © 2005 by CRC Press LLC

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Engineering Thermodynamics

Another type of equilibrium between the system and environment can be identified. This is a restricted form of equilibrium where only the conditions of mechanical and thermal equilibrium must be satisfied. This state of the system is called the restricted dead state. At the restricted dead state, the fixed quantity of matter under consideration is imagined to be sealed in an envelope impervious to mass flow, at zero velocity and elevation relative to coordinates in the environment, and at the temperature T0 and pressure p0. Exergy Balances Exergy can be transferred by three means: exergy transfer associated with work, exergy transfer associated with heat transfer, and exergy transfer associated with the matter entering and exiting a control volume. All such exergy transfers are evaluated relative to the environment used to define exergy. Exergy is also destroyed by irreversibilities within the system or control volume. Exergy balances can be written in various forms, depending on whether a closed system or control volume is under consideration and whether steady-state or transient operation is of interest. Owing to its importance for a wide range of applications, an exergy rate balance for control volumes at steady state is presented next.

Control Volume Exergy Rate Balance At steady state, the control volume exergy rate balance takes the form 0=

∑ E˙

q, j

− W˙ cv +

j

∑ E˙ − ∑ E˙ i

e

i

− E˙ D

e

________________________ rates of exergy transfer

_________

(2.85a)

rate of exergy destruction

or

0=



T0 

∑ 1 − T  Q˙ − W˙ + ∑ m˙ e − ∑ m˙ e j

j

j

cv

i i

i

e e

− E˙ D

(2.85b)

e

W˙ cv has the same significance as in Equation 2.22: the work rate excluding the flow work. Q˙ j is the time rate of heat transfer at the location on the boundary of the control volume where the instantaneous temperature is Tj. The associated rate of exergy transfer is  T  E˙ q, j = 1 − 0  Q˙ j Tj  

(2.86)

As for other control volume rate balances, the subscripts i and e denote inlets and outlets, respectively. The exergy transfer rates at control volume inlets and outlets are denoted, respectively, as E˙ i = m˙ i ei and E˙ e = m˙ e ee . Finally, E˙ D accounts for the time rate of exergy destruction due to irreversibilities within the control volume. The exergy destruction rate is related to the entropy generation rate by E˙ D = T0 S˙gen © 2005 by CRC Press LLC

(2.87)

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Chapter 2

The specific exergy transfer terms ei and ee are expressible in terms of four components: physical exergy ePH, kinetic exergy eKN, potential exergy ePT, and chemical exergy eCH: e = e PH + e KN + e PT + e CH

(2.88)

The first three components are evaluated as follows: e PH = (h − h0 ) − T0 (s − s0 ) e KN =

(2.89a)

1 2 v 2

(2.89b)

e PT = gz

(2.89c)

In Equation 2.89a, h0 and s0 denote, respectively, the specific enthalpy and specific entropy at the restricted dead state. In Equation 2.89b and Equation 2.89c, v and z denote velocity and elevation relative to coordinates in the environment, respectively. The chemical exergy eCH is considered next. Chemical Exergy To evaluate the chemical exergy, the exergy component associated with the departure of the chemical composition of a system from that of the environment, the substances comprising the system are referred to the properties of a suitably selected set of environmental substances. For this purpose, alternative models of the environment have been developed. For discussion, see, for example, Moran (1989) and Kotas (1995). Exergy analysis is facilitated, however, by employing a standard environment and a corresponding table of standard chemical exergies. Standard chemical exergies are based on standard values of the environmental temperature T0 and pressure p0 — for example, 298.15 K (25°C) and 1 atm, respectively. A standard environment is also regarded as consisting of a set of reference substances with standard concentrations reflecting as closely as possible the chemical makeup of the natural environment. The reference substances generally fall into three groups: gaseous components of the atmosphere, solid substances from the lithosphere, and ionic and noninonic substances from the oceans. The chemical exergy data of Table 2.12 correspond to two alternative standard exergy reference environments, called here model I and model II, that have gained acceptance for engineering evaluations. Although the use of standard chemical exergies greatly facilitates the application of exergy principles, the term standard is somewhat misleading since there is no one specification of the environment that suffices for all applications. Still, chemical exergies calculated relative to alternative specifications of the environment are generally in good agreement. For a broad range of engineering applications the simplicity and ease of use of standard chemical exergies generally outweigh any slight lack of accuracy that might result. In particular, the effect of slight variations in the values of T0 and p0 about the values used to determine the standard chemical exergies reported in Table 2.12 can be neglected. The literature of exergy analysis provides several expressions allowing the chemical exergy to be evaluated in particular cases of interest. The molar chemical exergy of a gas mixture, for example, can be evaluated from j

e CH =

∑ i =1

j

yi eiCH + RT0

∑ y ln y i

i =1

where eiCH is the molar chemical exergy of the ith component.

© 2005 by CRC Press LLC

i

(2.90)

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TABLE 2.12 Standard Molar Chemical Exergy, eCH (kJ/kmol), of Various Substances at 298 K and p0 Substance Nitrogen Oxygen Carbon dioxide Water Carbon (graphite) Hydrogen Sulfur Carbon monoxide Sulfur dioxide Nitrogen monoxide Nitrogen dioxide Hydrogen sulfide Ammonia Methane Ethane Methanol Ethyl alcohol

Formula

Model Ia

Model IIb

N2(g) O2(g) CO2(g) H2O(g) H2O(l) C(s) H2(g) S(s) CO(g) SO2(g) NO(g) NO2(g) H2S(g) NH3(g) CH4(g) C2H6(g) CH3OH(g) CH3OH(l) C2H5OH(g) C2H5OH(l)

640 3,950 14,175 8,635 45 404,590 235,250 598,160 269,410 301,940 88,850 55,565 799,890 336,685 824,350 1,482,035 715,070 710,745 1,348,330 1,342,085

720 3,970 19,870 9,500 900 410,260 236,100 609,600 275,100 313,400 88,900 55,600 812,000 337,900 831,650 1,495,840 722,300 718,000 1,363,900 1,357,700

a Ahrendts, J. 1977. Die Exergie Chemisch Reaktionsfähiger Systeme, VDI-Forschungsheft. VDI-Verlag, Dusseldorf, 579. Also see Reference States, Energy — The International Journal, 5: 667–677, 1980. In Model I, p0 = 1.019 atm. This model attempts to impose a criterion that the reference environment be in equilibrium. The reference substances are determined assuming restricted chemical equilibrium for nitric acid and nitrates and unrestricted thermodynamic equilibrium for all other chemical components of the atmosphere, the oceans, and a portion of the Earth’s crust. The chemical composition of the gas phase of this model approximates the composition of the natural atmosphere. b Szargut, J., Morris, D. R., and Steward, F. R. 1988. Energy Analysis of Thermal, Chemical, and Metallurgical Processes. Hemisphere, New York. In Model II, p0 = 1.0 atm. In developing this model a reference substance is selected for each chemical element from among substances that contain the element being considered and that are abundantly present in the natural environment, even though the substances are not in completely mutual stable equilibrium. An underlying rationale for this approach is that substances found abundantly in nature have little economic value. On an overall basis, the chemical composition of the exergy reference environment of Model II is closer than Model I to the composition of the natural environment, but the equilibrium criterion is not always satisfied.

Example 13 Ignoring the kinetic and potential exergies, determine the exergy rate, in kJ/kg, associated with each of the following streams of matter: (a) Saturated water vapor at 20 bar. (b) Methane at 5 bar, 25°C. Let T0 = 298 K, p0 = 1.013 bar (1 atm).

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Solution. Equation 2.88 reduces to read e = (h − h0 ) − T0 (s − s0 ) + e CH (a) From Table A.5, h = 2799.5 kJ/kg, s = 6.3409 kJ/kg · K. At T0 = 298 K (25°C), water would be a liquid; thus with Equation 2.50c and Equation 2.50d, h0 ≈ 104.9 kJ/kg, s0 ≈ 0.3674 kJ/kg · K. Table 2.12 (model I) gives eCH = 45/18.02 = 2.5 kJ/kg. Then e = (2799.5 − 104.9) − 298(6.3409 − 0.3674) + 2.5 = 914.5 + 2.5 = 917.0 kJ kg Here the specific exergy is determined predominately by the physical component. (b) Assuming the ideal gas model for methane, h – h0 = 0. Also, Equation 2.58 reduces to give s – s0 = –Rlnp/p0. Then, Equation 2.88 reads e = RT0 ln p p0 + e CH With eCH = 824,350/16.04 = 51,393.4 kJ/kg from Table 2.12 (model I),  8.314 kJ  5 kJ e= + 51, 393.4  (298 K) ln ⋅ 16 . 04 kg K 1 . 013 kg   = 246.6 + 51, 393.4 = 51,640 kJ kg Here the specific exergy is determined predominately by the chemical component. The small difference between p0 = 1.013 bar and the value of p0 for model I has been ignored.

Exergetic Efficiency The exergetic efficiency (second law efficiency, effectiveness, or rational efficiency) provides a true measure of the performance of a system from the thermodynamic viewpoint. To define the exergetic efficiency both a product and a fuel for the system being analyzed are identified. The product represents the desired result of the system (power, steam, some combination of power and steam, etc.). Accordingly, the definition of the product must be consistent with the purpose of purchasing and using the system. The fuel represents the resources expended to generate the product and is not necessarily restricted to being an actual fuel such as a natural gas, oil, or coal. Both the product and the fuel are expressed in terms of exergy. For a control volume at steady state whose exergy rate balance reads E˙ F = E˙ P + E˙ D + E˙ L the exergetic efficiency is E˙ E˙ + E˙ ε = ˙P = 1 − D ˙ L EF EF

(2.91)

where the rates at which the fuel is supplied and the product is generated are E˙ F and E˙ P , respectively. E˙ D and E˙ L denote the rates of exergy destruction and exergy loss, respectively. Exergy is destroyed by © 2005 by CRC Press LLC

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irreversibilities within the control volume, and exergy is lost from the control volume via stray heat transfer, material streams vented to the surroundings, and so on. The exergetic efficiency shows the percentage of the fuel exergy provided to a control volume that is found in the product exergy. Moreover, the difference between 100% and the value of the exergetic efficiency, expressed as a percent, is the percentage of the fuel exergy wasted in this control volume as exergy destruction and exergy loss. To apply Equation 2.91, decisions are required concerning what are considered as the fuel and the product. Table 2.13 provides illustrations for several common components. Similar considerations are used to write exergetic efficiencies for systems consisting of several such components, as, for example, a power plant. Exergetic efficiencies can be used to assess the thermodynamic performance of a component, plant, or industry relative to the performance of similar components, plants, or industries. By this means the performance of a gas turbine, for instance, can be gauged relative to the typical present-day performance level of gas turbines. A comparison of exergetic efficiencies for dissimilar devices — gas turbines and heat exchangers, for example — is generally not significant, however. The exergetic efficiency is generally more meaningful, objective, and useful than other efficiencies based on the first or second law of thermodynamics, including the thermal efficiency of a power plant, the isentropic efficiency of a compressor or turbine, and the effectiveness of a heat exchanger. The thermal efficiency of a cogeneration system, for instance, is misleading because it treats both work and heat transfer as having equal thermodynamic value. The isentropic turbine efficiency (Equation 2.95a) does not consider that the working fluid at the outlet of the turbine has a higher temperature (and consequently a higher exergy that may be used in the next component) in the actual process than in the isentropic process. The heat exchanger effectiveness fails, for example, to identify the exergy destruction associated with the pressure drops of the heat exchanger working fluids. Example 14 Evaluate the exergetic efficiency of the turbine in part (a) of Example 1 for T0 = 298 K. Solution. The exergetic efficiency from Table 2.13 is W˙ W˙ ε= ˙ = ˙ E1 − E2 m˙ (e1 − e2 ) Using Equation 2.88 and Equation 2.89a, and noting that the chemical exergy at 1 and 2 cancels, ε=

[

W˙

]

m˙ (h1 − h2 ) − T0 (s1 − s2 )

Since W˙ = m˙ (h1 − h2 ), W˙ ε= ˙ ˙ 0 (s2 − s1 ) W + mT Finally, using data from Example 1 and s2 = 6.8473 kJ/kg · K, ε=

=

30 MW  kJ   1 MW  162, 357 kg  30 MW +  298 K) (6.8473 − 6.6022)  (    3600 s   kg ⋅ K   10 3 kJ sec  30 MW

(30 + 3.29) MW

© 2005 by CRC Press LLC

= 0.9(90%)

Turbine or Expander

Extraction Turbine

Compressor, Pump, or Fan

Heat Exchangerb

Mixing Unit

Gasifier or Combustion Chamber

· EP

· W

· W

· · E2 – E1

· · E2 – E1

· E3

· E3

· EF

· · E1 – E2

· · · E 1 – E2 – E3

· W

· · E3 – E4

· · E1 + E2

· · E 1 + E2

ε

W E1 − E 2

W E1 − E 2 − E3

E 2 − E1 W

E 2 − E1 E3 − E 4

E3 E1 + E 2

E3 E1 + E 2

Boiler

Component

( E − E ) + ( E − E ) ( E + E ) + ( E + E ) ( E − E ) + ( E − E ) ( E + E ) − ( E + E ) 6

5

8

7

1

2

3

4

6

5

8

7

1

2

3

4

a

For discussion, see Bejan et al. (1996). This definition assumes that the purpose of the heat exchanger is to heat the cold stream (T1 ≥ T0). If the purpose of the heat exchanger is to provide cooling (T3 ≥ T0), then the · · · · · · following relations should be used: EP = E4 – E3 and EF = E1 – E2. b

Chapter 2

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TABLE 2.13 The Exergetic Efficiency for Selected Components at Steady Statea

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Engineering Thermodynamics

FIGURE 2.16 Steam turbine/electric generator used to discuss exergy costing.

Exergy Costing Since exergy measures the true thermodynamic values of the work, heat, and other interactions between the system and its surroundings as well as the effect of irreversibilities within the system, exergy is a rational basis for assigning costs. This aspect of thermoeconomics is called exergy costing. Referring to Figure 2.16 showing a steam turbine-electric generator at steady state, the total cost to produce the electricity and exiting steam equals the cost of the entering steam plus the cost of owning and operating the device. This is expressed by the cost rate balance for the turbine-generator: C˙ e + C˙ 2 = C˙1 + Z˙

(2.92a)

where C˙ e is the cost rate associated with the electricity, C˙1 and C˙ 2 are the cost rates associated with the entering steam and exiting steam, respectively, and Z˙ accounts for the cost rate associated with owning and operating the system, each annualized in $ per year. With exergy costing, the cost rates C˙1 , C˙ 2 , and C˙ e are evaluated in terms of the associated rate of exergy transfer and a unit cost. Equation 2.92a then appears as ce W˙ e + c2 E˙ 2 = c1 E˙1 + Z˙

(2.92b)

The coefficients c1, c2, and ce in Equation 2.92b denote the average cost per unit of exergy for the associated exergy rate. The unit cost c1 of the entering steam would be obtained from exergy costing applied to the components upstream of the turbine. Assigning the same unit cost to the exiting steam: c2 = c1 on the basis that the purpose of the turbine-generator is to generate electricity and thus all costs associated with owning and operating the system should be charged to the power, Equation 2.92b becomes

(

)

ce W˙ e = c1 E˙1 − E˙ 2 + Z˙

(2.92c)

The first term on the right side accounts for the cost of the net exergy used and the second term accounts for cost of the system itself. Introducing the exergetic efficiency from Table 2.13, the unit cost of the electricity is ce =

c1 Z˙ + ˙ ε We

(2.93)

This equation shows, for example, that the unit cost of electricity would increase if the exergetic efficiency were to decrease owing to a deterioration of the turbine with use. © 2005 by CRC Press LLC

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Chapter 2

Example 15 A turbine-generator with an exergetic efficiency of 90% develops 7 × 107 kW · hr of electricity annually. The annual cost of owning and operating the system is $2.5 × 105. If the average unit cost of the steam entering the system is $0.0165 per kW · hr of exergy, evaluate the unit cost of the electricity. Solution. Substituting values into Equation 2.93, ce =

$2.5 × 10 5 year $0.0165 kW ⋅ h + 0.9 7 × 10 7 kW ⋅ h year

= 0.0183 + 0.0036 = $0.0219 kW ⋅ h

2.6 Vapor and Gas Power Cycles Vapor and gas power systems develop electrical or mechanical power from energy sources of chemical, solar, or nuclear origin. In vapor power systems the working fluid, normally water, undergoes a phase change from liquid to vapor, and conversely. In gas power systems, the working fluid remains a gas throughout, although the composition normally varies owing to the introduction of a fuel and subsequent combustion. The present section introduces vapor and gas power systems. Further discussion is provided in Chapter 8. Refrigeration systems are considered in Chapter 9. The processes taking place in power systems are sufficiently complicated that idealizations are typically employed to develop tractable thermodynamic models. The air standard analysis of gas power systems considered later in the present section is a noteworthy example. Depending on the degree of idealization, such models may provide only qualitative information about the performance of the corresponding realworld systems. Yet such information is frequently useful in gauging how changes in major operating parameters might affect actual performance. Elementary thermodynamic models can also provide simple settings to assess, at least approximately, the advantages and disadvantages of features proposed to improve thermodynamic performance.

Rankine and Brayton Cycles In their simplest embodiments vapor power and gas turbine power plants are represented conventionally in terms of four components in series, forming, respectively, the Rankine cycle and the Brayton cycle shown schematically in Table 2.14. The thermodynamically ideal counterparts of these cycles are composed of four internally reversible processes in series: two isentropic processes alternated with two constant pressure processes. Table 2.14 provides property diagrams of the actual and corresponding ideal cycles. Each actual cycle is denoted 1-2-3-4-1; the ideal cycle is 1-2s-3-4s-1. For simplicity, pressure drops through the boiler, condenser, and heat exchangers are not shown. Invoking Equation 2.29 for the ideal cycles, the heat added per unit of mass flowing is represented by the area under the isobar from state 2s to state 3: area a-2s-3-b-a. The heat rejected is the area under the isobar from state 4s to state 1: area a-1-4s-b-a. Enclosed area 1-2s-3-4s-1 represents the net heat added per unit of mass flowing. For any power cycle, the net heat added equals the net work done. Expressions for the principal energy transfers shown on the schematics of Table 2.14 are provided by Equation 1 to Equation 4 of the table. They are obtained by reducing Equation 2.27a with the assumptions of negligible heat loss and negligible changes in kinetic and potential energy from the inlet to the outlet of each component. All quantities are positive in the directions of the arrows on the figure. Using these expressions, the thermal efficiency is η=

(h

3

− h4 ) − (h2 − h1 ) h3 − h2

(2.94)

To obtain the thermal efficiency of the ideal cycle, h2s replaces h2 and h4s replaces h4 in Equation 2.94. © 2005 by CRC Press LLC

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Engineering Thermodynamics

TABLE 2.14 Rankine and Brayton Cycles Rankine Cycle

Brayton Cycle

W˙ p   = m˙ (h2 − h1 ) W˙ c  Q˙ = m˙ (h − h )

(> 0)

(1)

(> 0)

(2)

W˙ t = m˙ (h3 − h4 )

(> 0 )

(3)

in

3

2

Q˙ out = m˙ (h1 − h4 )

(> 0 )

(4)

Decisions concerning cycle operating conditions normally recognize that the thermal efficiency tends to increase as the average temperature of heat addition increases and/or the temperature of heat rejection decreases. In the Rankine cycle, a high average temperature of heat addition can be achieved by superheating the vapor prior to entering the turbine, and/or by operating at an elevated steam-generator pressure. In the Brayton cycle an increase in the compressor pressure ratio p2/p1 tends to increase the average temperature of heat addition. Owing to materials limitations at elevated temperatures and pressures, the state of the working fluid at the turbine inlet must observe practical limits, however. The turbine inlet temperature of the Brayton cycle, for example, is controlled by providing air far in excess © 2005 by CRC Press LLC

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Chapter 2

of what is required for combustion. In a Rankine cycle using water as the working fluid, a low temperature of heat rejection is typically achieved by operating the condenser at a pressure below 1 atm. To reduce erosion and wear by liquid droplets on the blades of the Rankine cycle steam turbine, at least 90% quality should be maintained at the turbine exit: x4 > 0.9. The back work ratio, bwr, is the ratio of the work required by the pump or compressor to the work developed by the turbine: bwr =

h2 − h1 h3 − h4

(2.95)

As a relatively high specific volume vapor expands through the turbine of the Rankine cycle and a much lower specific volume liquid is pumped, the back work ratio is characteristically quite low in vapor power plants — in many cases on the order of 1 to 2%. In the Brayton cycle, however, both the turbine and compressor handle a relatively high specific volume gas, and the back ratio is much larger, typically 40% or more. The effect of friction and other irreversibilities for flow-through turbines, compressors, and pumps is commonly accounted for by an appropriate isentropic efficiency. The isentropic turbine efficiency is ηt =

h3 − h4 h3 − h4 s

(2.95a)

ηc =

h2 s − h1 h2 − h1

(2.95b)

The isentropic compressor efficiency is

In the isentropic pump efficiency, ηp , which takes the same form as Equation 2.95b, the numerator is frequently approximated via Equation 2.30c as h2s – h1 ≈ v1∆p, where ∆p is the pressure rise across the pump. Simple gas turbine power plants differ from the Brayton cycle model in significant respects. In actual operation, excess air is continuously drawn into the compressor, where it is compressed to a higher pressure; then fuel is introduced and combustion occurs; finally the mixture of combustion products and air expands through the turbine and is subsequently discharged to the surroundings. Accordingly, the low-temperature heat exchanger shown by a dashed line in the Brayton cycle schematic of Table 2.14 is not an actual component, but included only to account formally for the cooling in the surroundings of the hot gas discharged from the turbine. Another frequently employed idealization used with gas turbine power plants is that of an air-standard analysis. An air-standard analysis involves two major assumptions: (1) as shown by the Brayton cycle schematic of Table 2.14, the temperature rise that would be brought about by combustion is effected instead by a heat transfer from an external source; (2) the working fluid throughout the cycle is air, which behaves as an ideal gas. In a cold air-standard analysis the specific heat ratio k for air is taken as constant. Equation 1 to Equation 6 of Table 2.7 together with data from Table A.8 apply generally to air-standard analyses. Equation 1′ to Equation 6′ of Table 2.7 apply to cold air-standard analyses, as does the following expression for the turbine power obtained from Table 2.1 (Equation 27c″):

[

kRT3 ( k −1) k W˙ t = m˙ 1 − ( p4 p3 ) k −1

]

(2.96)

(Equation 2.96 also corresponds to Equation 5′ of Table 2.8 when n = k.) An expression similar in form can be written for the power required by the compressor. © 2005 by CRC Press LLC

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Engineering Thermodynamics

TABLE 2.15 Sample Calculations for the Rankine and Brayton Cycles of Table 2.14 Rankine Cycle Given data:

p1 = p4 = 8 kPa (saturated liquid at 1) T3 = 480°C (superheated vapor at 3) p2 = p3 = 8 MPa W˙ net = 100 MW Ideal cycle: ηt = ηp = 100% Actual cycle: ηt = 85%, ηp = 70%

Parameter

a b

Ideal Cycle

Actual Cycle

x4 h2 (kJ/ kg)

0.794 181.9a

0.873 185.4

m˙ ( kg/h ) η (%)

2.86 × 105 39.7

3.38 × 105 33.6

Q˙ out (MW)

151.9

197.6

E˙ q,out (MW)b

8.2

10.7

h2s ≈ h1 + v1∆p Equation 2.86 with T0 = 298 K, Tj = Tsat (8 kPa) = 315 K

Brayton Cycle Given data:

p1 = p4 = 1 bar p2 = p3 = 10 bar T3 = 1400 K ηt = ηc = 100%

Air-Standard Analysis

Cold Air-Standard Analysis k = 1.4

T2 (K) T4 (K)

574.1 787.7

579.2 725.1

W˙ net m˙ ( kJ/kg)

427.2

397.5

η (%) bwr

45.7 0.396

48.2 0.414

Parameter

For the simple Rankine and Brayton cycles of Table 2.14 the results of sample calculations are provided in Table 2.15. The Brayton cycle calculations are on an air-standard analysis basis.

Otto, Diesel, and Dual Cycles Although most gas turbines are also internal combustion engines, the name is usually reserved to reciprocating internal combustion engines of the type commonly used in automobiles, trucks, and buses. Two principal types of reciprocating internal combustion engines are the spark-ignition engine and the compression-ignition engine. In a spark-ignition engine a mixture of fuel and air is ignited by a spark plug. In a compression ignition engine air is compressed to a high-enough pressure and temperature that combustion occurs spontaneously when fuel is injected. In a four-stroke internal combustion engine, a piston executes four distinct strokes within a cylinder for every two revolutions of the crankshaft. Figure 2.17 gives a pressure-displacement diagram as it might © 2005 by CRC Press LLC

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Chapter 2

FIGURE 2.17 Pressure-displacement diagram for a reciprocating internal combustion engine.

be displayed electronically. With the intake valve open, the piston makes an intake stroke to draw a fresh charge into the cylinder. Next, with both valves closed, the piston undergoes a compression stroke raising the temperature and pressure of the charge. A combustion process is then initiated, resulting in a highpressure, high-temperature gas mixture. A power stroke follows the compression stroke, during which the gas mixture expands and work is done on the piston. The piston then executes an exhaust stroke in which the burned gases are purged from the cylinder through the open exhaust valve. Smaller engines operate on two-stroke cycles. In two-stroke engines, the intake, compression, expansion, and exhaust operations are accomplished in one revolution of the crankshaft. Although internal combustion engines undergo mechanical cycles, the cylinder contents do not execute a thermodynamic cycle, since matter is introduced with one composition and is later discharged at a different composition. A parameter used to describe the performance of reciprocating piston engines is the mean effective pressure, or mep. The mean effective pressure is the theoretical constant pressure that, if it acted on the piston during the power stroke, would produce the same net work as actually developed in one cycle. That is, mep =

net work for one cycle displacement volume

(2.97)

where the displacement volume is the volume swept out by the piston as it moves from the top dead center to the bottom dead center. For two engines of equal displacement volume, the one with a higher mean effective pressure would produce the greater net work and, if the engines run at the same speed, greater power. Detailed studies of the performance of reciprocating internal combustion engines may take into account many features, including the combustion process occurring within the cylinder and the effects of irreversibilities associated with friction and with pressure and temperature gradients. Heat transfer between the gases in the cylinder and the cylinder walls and the work required to charge the cylinder and exhaust the products of combustion also might be considered. Owing to these complexities, accurate modeling of reciprocating internal combustion engines normally involves computer simulation. To conduct elementary thermodynamic analyses of internal combustion engines, considerable simplification is required. A procedure that allows engines to be studied qualitatively is to employ an airstandard analysis having the following elements: (1) a fixed amount of air modeled as an ideal gas is the © 2005 by CRC Press LLC

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system; (2) the combustion process is replaced by a heat transfer from an external source and generally represented in terms of elementary thermodynamic processes; (3) there are no exhaust and intake processes as in an actual engine: the cycle is completed by a constant-volume heat rejection process; (4) all processes are internally reversible. The processes employed in air-standard analyses of internal combustion engines are selected to represent the events taking place within the engine simply and mimic the appearance of observed pressuredisplacement diagrams. In addition to the constant volume heat rejection noted previously, the compression stroke and at least a portion of the power stroke are conventionally taken as isentropic. The heat addition is normally considered to occur at constant volume, at constant pressure, or at constant volume followed by a constant pressure process, yielding, respectively, the Otto, Diesel, and Dual cycles shown in Table 2.16. Reducing the closed system energy balance, Equation 2.8, gives the following expressions for heat and work applicable in each case shown in Table 2.16: W12 = u1 − u2 m

(< 0)

W34 = u3 − u4 m

(> 0)

Q41 = u1 − u4 m

(< 0)

Table 2.16 provides additional expressions for work, heat transfer, and thermal efficiency identified with each case individually. The thermal efficiency, evaluated from Equation 2.9, takes the form η = 1−

Q41 m QA m

Equation 1 to Equation 6 of Table 2.7 together with data from Table A.8, apply generally to air-standard analyses. In a cold air-standard analysis the specific heat ratio k for air is taken as constant. Equation 1′ to Equation 6′ of Table 2.7 apply to cold air-standard analyses, as does Equation 4′ of Table 2.8, with n = k for the isentropic processes of these cycles. Referring to Table 2.16, the ratio v1/v2 is the compression ratio, r. For the Diesel cycle, the ratio v3 /v2 is the cutoff ratio, rc . Figure 2.18 shows the variation of the thermal efficiency with compression ratio for an Otto cycle and Diesel cycles having cutoff ratios of 2 and 3. The curves are determined on a cold airstandard basis with k = 1.4 using the following expression: η = 1−

k 1  rc − 1    r  k (rc − 1)  k −1

(constant k )

(2.98)

where the Otto cycle corresponds to rc = 1. As all processes are internally reversible, areas on the p-v and T-s diagrams of Table 2.16 can be interpreted, respectively, as work and heat transfer. Invoking Equation 2.10 and referring to the p-v diagrams, the areas under process 3-4 of the Otto cycle, process 2-3-4 of the Diesel cycle, and process x-3-4 of the Dual cycle represent the work done by the gas during the power stroke, per unit of mass. For each cycle, the area under the isentropic process 1-2 represents the work done on the gas during the compression stroke, per unit of mass. The enclosed area of each cycle represents the net work done per unit mass. With Equation 2.15 and referring to the T-s diagrams, the areas under process 2-3 of the Otto and Diesel cycles and under process 2-x-3 of the Dual cycle represent the heat added per unit of mass. For each cycle, the area under the process 4-1 represent the heat rejected per unit of mass. The enclosed area of each cycle represents the net heat added, which equals the net work done, each per unit of mass. © 2005 by CRC Press LLC

(a) Otto Cycle

(b) Diesel Cycle

(

W23 =0 m

W23 = p2 v3 − v2 m

Q23 = u3 − u2 m

Q23 = h3 − h2 m

u4 − u1 u3 − u2

© 2005 by CRC Press LLC

η = 1−

u4 − u1 h3 − h2

W2 x Q2 x = 0, = ux − u2 m m

(

)

Wx 3 = p3 v3 − v2 , m η = 1−

(u

x

Qx 3 = h3 − hx m

u4 − u1 − u2 + h3 − hx

) (

)

Chapter 2

η = 1−

)

(c) Dual Cycle

0866_book.fm Page 80 Friday, August 6, 2004 2:25 PM

2-80

TABLE 2.16 Otto, Diesel, and Dual Cycles

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2-81

FIGURE 2.18 Thermal efficiency of the cold air-standard Otto and Diesel cycles, k = 1.4.

Carnot, Ericsson, and Stirling Cycles Three thermodynamic cycles that exhibit the Carnot efficiency (Equation 2.12) are the Carnot, Ericsson, and Stirling cycles shown in Figure 2.19. Each case represents a reversible power cycle in which heat is added from an external source at a constant temperature TH (process 2-3) and rejected to the surroundings at a constant temperature TC (process 4-1). Carnot cycles can be configured both as vapor power cycles and as cycles executed by a gas in a piston-cylinder assembly (see, e.g., Moran and Shapiro, 1995). Carnot cycles also can be executed in systems where a capacitor is charged and discharged, a paramagnetic substance is magnetized and demagnetized, and in other ways. Regardless of the type of device and the working substance used, the Carnot cycle always has the same four internally reversible processes in series: two isentropic processes alternated with two isothermal processes. The Ericsson and Stirling cycles also consist of four internally reversible processes in series: heating from state 1 to state 2 (at constant pressure in the Ericsson cycle and at constant volume in the Stirling cycle), isothermal heating from state 2 to state 3 at temperature TH, cooling from state 3 to state 4 (at constant pressure in the Ericsson cycle and at constant volume in the Stirling cycle), and isothermal cooling from state 4 to state 1 at temperature TC . An ideal regenerator allows the heat input required for process 1-2 to be obtained from the heat rejected in process 3-4. Accordingly, as in the Carnot cycle all the heat added externally occurs at TH and all of the heat rejected to the surroundings occurs at TC . The Ericsson and Stirling cycles are principally of theoretical interest as examples of cycles that exhibit the same thermal efficiency as the Carnot cycle: Equation 2.12. However, a practical engine of the pistoncylinder type that operates on a closed regenerative cycle having features in common with the Stirling cycle has been under study in recent years. This engine, known as the Stirling engine, offers the opportunity for high efficiency together with reduced emissions from combustion products because the combustion takes place externally and not within the cylinder as in internal combustion engines. In the Stirling engine, energy is transferred to the working fluid from products of combustion, which are kept separate. It is an external combustion engine.

2.7 Guidelines for Improving Thermodynamic Effectiveness Thermal design frequently aims at the most effective system from the cost viewpoint. Still, in the cost optimization process, particularly of complex energy systems, it is often expedient to begin by identifying a design that is nearly optimal thermodynamically; such a design can then be used as a point of departure for cost optimization. Presented in this section are guidelines for improving the use of fuels (natural gas, © 2005 by CRC Press LLC

0866_book.fm Page 82 Friday, August 6, 2004 2:25 PM

2-82

Chapter 2

T 3

TC

1

4

s=c

2

s=c

TH

s

(A)

T TH

3

=

2

c =

p

c

p

s TC

4

1

s

(B)

T

2

3

c υ=

υ=

c

TH

TC 4

1

(C)

s

FIGURE 2.19 (A) Carnot, (B) Ericsson, and (C) Stirling cycles.

oil, and coal) by reducing sources of thermodynamic inefficiency in thermal systems. Further discussion is provided by Bejan et al. (1996). To improve thermodynamic effectiveness it is necessary to deal directly with inefficiencies related to exergy destruction and exergy loss. The primary contributors to exergy destruction are chemical reaction, heat transfer, mixing, and friction, including unrestrained expansions of gases and liquids. To deal with them effectively, the principal sources of inefficiency not only should be understood qualitatively, but also determined quantitatively, at least approximately. Design changes to improve effectiveness must be © 2005 by CRC Press LLC

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2-83

done judiciously, however, for the cost associated with different sources of inefficiency can be different. For example, the unit cost of the electrical or mechanical power required to provide for the exergy destroyed owing to a pressure drop is generally higher than the unit cost of the fuel required for the exergy destruction caused by combustion or heat transfer. Since chemical reaction is a significant source of thermodynamic inefficiency, it is generally good practice to minimize the use of combustion. In many applications the use of combustion equipment such as boilers is unavoidable, however. In these cases a significant reduction in the combustion irreversibility by conventional means simply cannot be expected, for the major part of the exergy destruction introduced by combustion is an inevitable consequence of incorporating such equipment. Still, the exergy destruction in practical combustion systems can be reduced by minimizing the use of excess air and by preheating the reactants. In most cases only a small part of the exergy destruction in a combustion chamber can be avoided by these means. Consequently, after considering such options for reducing the exergy destruction related to combustion, efforts to improve thermodynamic performance should focus on components of the overall system that are more amenable to betterment by cost-effective conventional measures. In other words, some exergy destructions and energy losses can be avoided, others cannot. Efforts should be centered on those that can be avoided. Nonidealities associated with heat transfer also typically contribute heavily to inefficiency. Accordingly, unnecessary or cost-ineffective heat transfer must be avoided. Additional guidelines follow: • The higher the temperature T at which a heat transfer occurs in cases where T > T0, where T0 denotes the temperature of the environment (Section 2.5), the more valuable the heat transfer and, consequently, the greater the need to avoid heat transfer to the ambient, to cooling water, or to a refrigerated stream. Heat transfer across T0 should be avoided. • The lower the temperature T at which a heat transfer occurs in cases where T < T0, the more valuable the heat transfer and, consequently, the greater the need to avoid direct heat transfer with the ambient or a heated stream. • Since exergy destruction associated with heat transfer between streams varies inversely with the temperature level, the lower the temperature level, the greater the need to minimize the streamto-stream temperature difference. • Avoid the use of intermediate heat transfer fluids when exchanging energy by heat transfer between two streams Although irreversibilities related to friction, unrestrained expansion, and mixing are often secondary in importance to those of combustion and heat transfer, they should not be overlooked, and the following guidelines apply: • Relatively more attention should be paid to the design of the lower temperature stages of turbines and compressors (the last stages of turbines and the first stages of compressors) than to the remaining stages of these devices. • For turbines, compressors, and motors, consider the most thermodynamically efficient options. • Minimize the use of throttling; check whether power recovery expanders are a cost-effective alternative for pressure reduction. • Avoid processes using excessively large thermodynamic driving forces (differences in temperature, pressure, and chemical composition). In particular, minimize the mixing of streams differing significantly in temperature, pressure, or chemical composition. • The greater the mass rate of flow, the greater the need to use the exergy of the stream effectively. • The lower the temperature level, the greater the need to minimize friction. Flowsheeting or process simulation software can assist efforts aimed at improving thermodynamic effectiveness by allowing engineers to readily model the behavior of an overall system, or system components, under specified conditions and do the required thermal analysis, sizing, costing, and optimization. Many of the more widely used flowsheeting programs: ASPEN PLUS, PROCESS, and CHEMCAD are of the sequential-modular type. SPEEDUP is a popular program of the equation-solver type. Since © 2005 by CRC Press LLC

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Chapter 2

process simulation is a rapidly evolving field, vendors should be contacted for up-to-date information concerning the features of flowsheeting software, including optimization capabilities (if any). As background for further investigation of suitable software, see Biegler (1989) for a survey of the capabilities of 15 software products.

References Ahrendts, J. 1980. Reference states. Energy Int. J. 5: 667–677. ASHRAE Handbook 1993 Fundamentals. 1993. American Society of Heating, Refrigerating, and Air Conditioning Engineers, Atlanta. ASME Steam Tables, 6th ed. 1993. ASME Press, Fairfield, NJ. Bejan, A., Tsatsaronis, G., and Moran, M. 1996. Thermal Design and Optimization, John Wiley & Sons, New York. Biegler, L.T. 1989. Chemical process simulation. Chem. Eng. Progr. October: 50–61. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. 1960. Transport Phenomena. John Wiley & Sons, New York. Bolz, R.E. and Tuve, G.L. (eds.). 1973. Handbook of Tables for Applied Engineering Science, 2nd ed. CRC Press, Boca Raton, FL. Bornakke, C. and Sonntag, R.E. 1996. Tables of Thermodynamic and Transport Properties. John Wiley & Sons, New York. Cooper, H.W. and Goldfrank, J.C. 1967. B-W-R constants and new correlations. Hydrocarbon Processing. 46(12): 141–146. Gray, D.E. (ed.). 1972. American Institute of Physics Handbook. McGraw-Hill, New York. Haar, L. Gallagher, J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere, New York. Handbook of Chemistry and Physics, annual editions. CRC Press, Boca Raton, FL. JANAF Thermochemical Tables, 3rd ed. 1986. American Chemical Society and the American Institute of Physics for the National Bureau of Standards. Jones, J.B. and Dugan, R.E. 1996. Engineering Thermodynamics. Prentice-Hall, Englewood Cliffs, NJ. Keenan, J.H., Keyes, F.G., Hill, P.G., and Moore, J.G. 1969 and 1978. Steam Tables. John Wiley & Sons, New York (1969, English Units; 1978, SI Units). Keenan, J.H., Chao, J., and Kaye, J. 1980 and 1983. Gas Tables — International Version, 2nd ed. John Wiley & Sons, New York (1980, English Units; 1983, SI Units). Knacke, O., Kubaschewski, O., and Hesselmann, K. 1991. Thermochemical Properties of Inorganic Substances, 2nd ed. Springer-Verlag, Berlin. Kotas, T.J. 1995. The Exergy Method of Thermal Plant Analysis, Krieger, Melbourne, FL. Lee, B.I. and Kessler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 21: 510–527. Liley, P.E. 1987. Thermodynamic properties of substances. In Marks’ Standard Handbook for Mechanical Engineers, E.A. Avallone and T. Baumeister, (eds.). 9th ed. McGraw-Hill, New York, Sec. 4.2. Liley, P.E., Reid, R.C., and Buck, E. 1984. Physical and chemical data. In Perrys’ Chemical Engineers, Handbook, R.H. Perry and D.W. Green, (eds.). 6th ed. McGraw-Hill, New York, Sec. 3. Moran, M.J. 1989. Availability Analysis — A Guide to Efficient Energy Use. ASME Press, New York. Moran, M.J. and Shapiro, H.N. 2004. Fundamentals of Engineering Thermodynamics, 5th ed. John Wiley & Sons, New York. Moran, M.J. and Shapiro, H.N. 2004. IT: Interactive Thermodynamics. Computer software to accompany Fundamentals of Engineering Thermodynamics, 5th ed. developed by Intellipro Inc., John Wiley & Sons, New York. Obert, E.F. 1960. Concepts of Thermodynamics. McGraw-Hill, New York. Preston-Thomas, H. 1990. The International Temperature Scale of 1990 (ITS-90). Metrologia. 27: 3–10. Reid, R.C. and Sherwood, T.K. 1966. The Properties of Gases and Liquids, 2nd ed. McGraw-Hill, New York. © 2005 by CRC Press LLC

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Reid, R.C., Prausnitz, J.M., and Poling, B.E. 1987. The Properties of Gases and Liquids, 4th ed. McGrawHill, New York. Reynolds, W.C. 1979. Thermodynamic Properties in SI — Graphs, Tables and Computational Equations for 40 Substances. Department of Mechanical Engineering, Stanford University, Palo Alto, CA. Stephan, K. 1994. Tables. In Dubbel Handbook of Mechanical Engineering, W. Beitz and K.-H. Kuttner, (eds.). Springer-Verlag, London, Sec. C11. Szargut, J., Morris, D.R., and Steward, F.R. 1988. Exergy Analysis of Thermal, Chemical and Metallurgical Processes. Hemisphere, New York. Van Wylen, G.J., Sonntag, R.E., and Bornakke, C. 1994. Fundamentals of Classical Thermodynamics, 4th ed. John Wiley & Sons, New York. Wark, K. 1983. Thermodynamics, 4th ed. McGraw-Hill, New York Zemansky, M.W. 1972. Thermodynamic symbols, definitions, and equations. In American Institute of Physics Handbook, D.E. Gray, (ed.). McGraw-Hill, New York, Sec. 4b.

© 2005 by CRC Press LLC

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3

Frank Kreith University of Colorado, Boulder

Stanley A. Berger University of California, Berkeley

Stuart W. Churchill

Fluid Mechanics

University of Pennsylvania

J. Paul Tullis Tullis Engineering Consultants

Blake P. Tullis Utah State University

Frank M. White

3.1

Equilibrium of a Fluid Element • Hydrostatic Pressure • Manometry • Hydrostatic Forces on Submerged Objects • Pressure Variation in Rigid-Body Motion of a Fluid

University of Rhode Island

Ajay Kumar NASA Langley Research Center

3.2

University of Colorado, Boulder

John C. Chen Lehigh University

Thomas F. Irvine, Jr. (Deceased) 3.3

Massimo Capobianchi 3.4

Consultant

3.5

Donald F. Wilcock (Deceased)

3.6

Robert F. Boehm

3.7

3.8

Sherif A. Sherif University of Florida

Bharat Bhushan The Ohio State University

© 2005 by CRC Press LLC

Multiphase Flow Introduction • Fundamentals • Gas–Liquid Two-Phase Flow • Gas–Solid, Liquid–Solid Two-Phase Flows

University of Maryland

Alan T. McDonald

Compressible Flow Introduction • One-Dimensional Flow • Normal Shock Wave • One-Dimensional Flow with Heat Addition • Quasi-OneDimensional Flow • Two-Dimensional Supersonic Flow • Further Information

University of Wisconsin

Purdue University

External Incompressible Flows Introduction and Scope • Boundary Layers • Drag • Lift • Boundary Layer Control • Computation vs. Experiment

Rolf D. Reitz Jungho Kim

Open Channel Flow Definition • Uniform Flow • Critical Flow • Hydraulic Jump • Weirs • Gradually Varied Flow

Tribolock, Inc.

University of Nevada, Las Vegas

Hydraulics of Pipe Systems Basic Equations • Fluid Friction • Minor Losses • Pipe Selection • Valve Selection • Centrifugal Pump Selection and Performance • Other Considerations

Dartmouth College

E. Richard Booser

Similitude: Dimensional Analysis and Data Correlation Dimensional Analysis • Correlation of Experimental Data and Theoretical Values

Gonzaga University

Francis E. Kennedy

Equations of Motion and Potential Flow Integral Relations for a Control Volume • Reynolds Transport Theorem • Conservation of Mass • Conservation of Momentum • Conservation of Energy • Differential Relations for Fluid Motion • Mass Conservation — Continuity Equation • Momentum Conservation • The Navier–Stokes Equations • Energy Conservation — Mechanical and Thermal Energy Equations • Boundary Conditions • Vorticity in Incompressible Flow • Stream Function • Inviscid Irrotational Flow: Potential Flow

Jessica Todd

State University of New York, Stony Brook

Fluid Statics

3.9

New-Newtonian Flows Introduction • Classification of Non-Newtonian Fluids • Apparent Viscosity • Constitutive Equations • Rheological Property Measurements • Fully Developed Laminar Pressure Drops for Time-Independent Non-Newtonian Fluids • Fully Developed Turbulent Flow Pressure Drops • Viscoelastic Fluids • Further Information

0866_book.fm Page 2 Thursday, August 5, 2004 10:16 AM

3-2

Chapter 3

3.10 Tribology, Lubrication, and Bearing Design Introduction • Sliding Friction and Its Consequences • Lubricant Properties • Fluid Film Bearings • Thrust Bearings • Oil-Film Bearing Materials • Dry and Semilubricated Bearings • Rolling Element Bearings • Lubricant Supply Methods • Dynamic Seals

3.11 Pumps and Fans Introduction • Pumps • Centrifugal and Other Velocity-Head Pumps • Positive-Displacement Pumps • Selecting a Pump Based upon Flow Considerations • Vacuum Pumps • Fans

3.12 Liquid Atomization and Spraying Spray Characterization • Atomizer Design Considerations • Atomizer Types

3.13 Flow Measurement Introduction • Direct Methods • Restriction Flow Meters for Flow in Ducts • Linear Flow Meters • Traversing Methods • HotWire Anemometry • Laser Doppler Anemometry

3.14 Pressure Measurements Standards • Other Pressure Gages

3.15 Micro/Nanotribology Introduction • Experimental Techniques • Surface Roughness, Adhesion, and Friction • Scratching, Wear, and Indentation • Boundary Lubrication

3.1 Fluid Statics Stanley A. Berger Equilibrium of a Fluid Element If the sum of the external forces acting on a fluid element is zero, the fluid will be at rest or moving as a solid body — in either case, the fluid element is in equilibrium. This section considers fluids in such an equilibrium state. For fluids in equilibrium, the only internal stresses acting will be normal forces because the shear stresses depend on velocity gradients, and all such gradients, by the definition of equilibrium, are zero. If one then carries out a balance between the body forces, assumed proportional to volume or mass — such as gravity — and the normal surface stresses acting on an elementary prismatic fluid volume, the resulting equilibrium equations, after shrinking the volume to zero, show that the normal stresses at a point are the same in all directions; because they are known to be negative, this common value, called the pressure, is denoted by –p.

Hydrostatic Pressure Carrying out an equilibrium of forces on an elementary volume element dxdydz, the forces being pressures acting on the faces of the element and gravity acting in the –z direction, one obtains ∂p ∂p ∂p = = 0, and = −ρg ≡ − γ ∂x ∂y ∂z

(3.1.1)

where γ is the specific weight of the fluid. The first two of these imply that the pressure is the same in all directions at the same vertical height in a gravitational field. The third shows that the pressure increases with depth in a gravitational field, the variation depending on ρ(z). For homogeneous fluids, for which ρ = const., this last equation can be integrated, yielding p2 − p1 = −ρg ( z 2 − z1 ) = −ρg ( h2 − h1 ) © 2005 by CRC Press LLC

(3.1.2)

0866_book.fm Page 3 Thursday, August 5, 2004 10:16 AM

3-3

Fluid Mechanics

open to test region p

open to atmosphere

d h1

patm

liquid, γ h2

Po, at lowest level

FIGURE 3.1.1 U-tube manometer.

or p2 + ρgh2 = p1 + ρgh1 = const.

(3.1.3)

where h denotes the elevation. These are the equations for the hydrostatic pressure variation. When applied to problems in which a liquid, such as the ocean, lies below the atmosphere, with constant pressure patm, h is usually measured from the ocean/atmosphere interface; p at any distance h below this interface differs from patm by an amount p − patm = ρgh

(3.1.4)

Pressures may be given as absolute pressure, which is pressure measured relative to absolute vacuum, or gauge pressure, pressure measured relative to atmospheric pressure.

Manometry The hydrostatic pressure variation may be employed to measure pressure differences in terms of heights of liquid columns, Such devices are called manometers and are commonly used in wind tunnels and in a host of other applications and devices. Consider, for example, the U-tube manometer shown in Figure 3.1.1 filled with liquid of specific weight γ, the left leg open to the atmosphere and the right to the region whose pressure p is to be determined. In terms of the quantities shown in the figure: in left leg:

p0 − ρgh2 = patm

(3.1.5a)

in right leg:

p0 − ρgh1 = p

(3.1.5b)

The difference is p − patm = −ρg ( h1 − h2 ) = −ρgd = − γd

(3.1.6)

and p is determined in terms of the height difference d = h1 – h2 between the levels of the fluid in the two legs of the manometer. © 2005 by CRC Press LLC

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3-4

Chapter 3

p0

free surface

0

y

h dA

dA

0

n dF

x

p θ

y

FIGURE 3.1.2 Hydrostatic force on a plane surface.

Hydrostatic Forces on Submerged Objects Now consider the force acting on a submerged object due to the hydrostatic pressure. This is given by

F=

∫∫ p dA = ∫∫ p ⋅ n dA = ∫∫ ρgh dA + p ∫∫ dA 0

(3.1.7)

where h is the variable vertical depth of the element dA; n is the local (outward) normal to this elemental area, and p0 is the pressure at the surface. In turn, consider plane and nonplanar surfaces. Forces on Plane Surfaces Consider the planar surface A at an angle θ to a free surface shown in Figure 3.1.2. The force on one side of the planar surface, from Equation (3.1.7), is

F = ρgn

∫∫ h dA + p An

(3.1.8)

0

A

but h = y sin θ, so

∫∫ h dA = sin θ∫∫ y dA = y A sin θ = h A c

A

c

(3.1.9)

A

where the subscript c indicates the distance measured to the centroid of the area A. Thus, the total force (on one side) is

F = γhc A + p0 A

(3.1.10)

Thus, the magnitude of the force is independent of the angle θ and is equal to the pressure at the centroid, γ hc + p0, times the area. If gauge pressure is used, the term p0 A in Equation (3.1.10) is dropped. Because p is not evenly distributed over A, but varies with depth, F does not act through the centroid. The point of action of F, called the center of pressure, can be determined by considering moments in Figure 3.1.2. The moment of the hydrostatic force acting on the elementary area dA about the axis perpendicular to the page passing through the point 0 on the free surface is y dF = y(γ y sin θ dA) = γ y 2 sin θ dA so, if ycp denotes the distance to the center of pressure, © 2005 by CRC Press LLC

(3.1.11)

0866_book.fm Page 5 Thursday, August 5, 2004 10:16 AM

3-5

Fluid Mechanics

ycp F = γ sin θ

∫∫ y dA = γ sin θ I 2

x

(3.1.12)

where Ix is the moment of inertia of the plane area with respect to the axis formed by the intersection of the plane containing the planar surface and the free surface (e.g., 0x). Dividing by F = γ hc A = γ yc sin θA gives ycp =

Ix yc A

(3.1.13)

By using the parallel axis theorem I x = I xc + Ay c2 , where Ixc is the moment of inertia with respect to an axis parallel to 0x passing through the centroid, Equation (3.1.13) becomes y cp = y c +

I xc yc A

(3.1.14)

which shows that, in general, the center of pressure lies below the centroid. Similarly, xcp can be found by taking moments about the y axis, specifically x cp F = γ sin θ

∫∫ xy dA = γ sin θ I

xy

(3.1.15)

or x cp =

I xy yc A

(3.1.16)

where Ixy is the product of inertia with respect to the x and y axes. Again, the parallel axis theorem, Ixy = Ixyc + Axc yc, where the subscript c denotes the value at the centroid, allows Equation (3.1.16) to be written x cp = x c +

I xyc yc A

(3.1.17)

This completes the determination of the center of pressure (xcp , ycp). Note that if the submerged area is symmetrical with respect to an axis passing through the centroid and parallel to the x or y axis, Ixyc = 0 and xcp = xc; also, as yc increases, ycp → yc . Centroidal moments of inertia and centroidal coordinates for some common areas are shown in Figure 3.1.3. Forces on Curved Surfaces The most convenient approach to calculating the pressure force on a curved surface is by separating it into its horizontal and vertical components. A free-body diagram of the forces acting on the volume of fluid lying above a curved surface together with the conditions of static equilibrium of such a column shows that: • The horizontal components of force on a curved submerged surface are equal to the forces exerted on the planar areas formed by the projections of the curved surface onto vertical planes normal to these components; the lines of action of these forces are calculated as described earlier for planar surfaces. • The vertical component of force on a curved submerged surface is equal in magnitude to the weight of the entire column of fluid lying above the curved surface and acts through the center of mass of this volume of fluid. © 2005 by CRC Press LLC

0866_book.fm Page 6 Thursday, August 5, 2004 10:16 AM

3-6

Chapter 3

R

b/2 c

c

x

x

b/2

y a 2

y

a 2

A = πR2

A = ab 1xc

1 1xc = ab3 2 1 1yc = ba3 12 1xyc

1xyc

=0

πR4 4

=0

d

c

2b 3

x

R

R

πR2 A= 2 1xc

= 0.11 R4

1yc

= 0.39 R4

c

b 3

y

1xyc

= 1yc =

x

b

y a+d 3

a

ab A= 2 1xc

=

1xyc

=

ab3 36 a(a − 2d)b2 72

=0

FIGURE 3.1.3 Centroidal moments of inertia and coordinates for some common areas.

These three components of force, two horizontal and one vertical, need not meet at a single point, so there is, in general, no single resultant force. They may, however, be combined into a single force at any arbitrary point of application together with a moment about that point. Hydrostatic Forces in Layered Fluids The preceding results employing the linear hydrostatic variation of pressure are valid only for homogeneous fluids. In heterogeneous fluids consisting of individual layers, each of constant density, the pressure varies linearly, with a different slope in each layer. The preceding analyses must be carried out by computing and summing the separate contributions to forces and moments. Buoyancy The principles used to compute hydrostatic forces may be used to determine the net pressure force acting on completely submerged or floating bodies. These laws of buoyancy, the principles of Archimedes, are: • A completely submerged body experiences a vertical upward force equal to the weight of the displaced fluid. • A floating or partially submerged body displaces its own weight in the fluid in which it floats (i.e., the vertical upward force is equal to the body weight). The line of action of the buoyancy force in both these principles passes through the centroid of the displaced volume of fluid. This point, the center of buoyancy, need not correspond to the center of mass © 2005 by CRC Press LLC

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3-7

Fluid Mechanics

of the body (the body may be of nonuniform density). Previously, it has also been assumed that the displaced fluid has a constant γ. If this is not so, e.g., as in a layered fluid, the magnitude of the buoyant force is still equal to the weight of the displaced fluid, but its line of action passes through the center of gravity of the displaced volume, not the centroid. A body whose weight is exactly equal to that of the volume of fluid it displaces is said to be neutrally buoyant and will remain at rest at any point of immersion in a (homogeneous) fluid.

Pressure Variation in Rigid-Body Motion of a Fluid In rigid-body motion of a fluid, all the particles translate and rotate as a whole; there is no relative motion between particles and thus no viscous stresses because these are proportional to velocity gradients. The equation of motion is then a balance among pressure, gravity, and the fluid acceleration, specifically:

(

∇p = ρ g − a

)

(3.1.18)

where a is the uniform acceleration of the body. Equation (3.1.18) shows that the lines of constant pressure, including a free surface if any, are perpendicular to the direction g – a. Two important applications of this are to fluids in uniform linear translation and rigid-body rotation. Although such problems are not, strictly speaking, fluid statics problems, their analysis and the resulting pressure variation are similar to those for static fluids. Uniform Linear Acceleration For a fluid partially filling a large container moving to the right with constant acceleration a = (ax, ay), the geometry of Figure 3.1.4 shows that the magnitude of the pressure gradient in the direction n normal to the accelerating free surface, in the direction g – a, is

(

)

1

2 2 dp  = ρ  ax2 + g + ay  dn  

(3.1.19)

and the angle to the horizontal of the free surface is  a  θ = tan −1  x   g + a y 

(3.1.20)

y

free surface

a

x

θ n g

p = const.

g−a

FIGURE 3.1.4 A fluid with a free surface in uniform linear acceleration. © 2005 by CRC Press LLC

fluid at rest

0866_book.fm Page 8 Thursday, August 5, 2004 10:16 AM

3-8

Chapter 3

axis of rotation z

r

− rΩ2er Ω g

FIGURE 3.1.5 A fluid with a free surface in rigid-body rotation.

Rigid-Body Rotation Consider a fluid-filled circular cylinder rotating uniformly with angular velocity
= Ωez (Figure 3.1.5). The only acceleration is the centripetal acceleration
× (
× r) = –rΩ2er , so Equation 3.1.18 becomes: ∇p =

∂p ∂p er + e z = ρ( g − a ) = ρ rΩ 2 er − ge z ∂r ∂z

(

)

(3.1.21)

or

∂p = ρr Ω2 , ∂r

∂p = −ρg = − γ ∂z

(3.1.22)

Integration of these equations leads to

1 p = p0 − γ z + ρr 2Ω2 2

(3.1.23)

where p0 is a reference pressure (at r = z = 0). Thus, at any fixed r, the pressure varies hydrostatically in the vertical direction, and the constant pressure surfaces, including the free surface, are paraboloids of revolution.

Further Information The reader may find more detail and additional information on the topics in this section in any of the many excellent introductory texts on fluid mechanics, such as White, F.M. 2003. Fluid Mechanics, 5th ed., McGraw–Hill, Boston. Munson, B.R., Young, D.F., and Okiishi, T.H. 2003. Fundamentals of Fluid Mechanics, 4th ed., John Wiley & Sons, New York. © 2005 by CRC Press LLC

0866_book.fm Page 9 Thursday, August 5, 2004 10:16 AM

3-9

Fluid Mechanics

3.2 Equations of Motion and Potential Flow Stanley A. Berger Integral Relations for a Control Volume Like most physical conservation laws, those governing motion of a fluid apply to (moving) material particles or systems of such particles. This so-called Lagrangian viewpoint is generally not as useful in practical fluid flows as an analysis using fixed (deformable) control volumes — the Eulerian viewpoint. The relationship between these two viewpoints can be deduced from the Reynolds transport theorem, from which one also most readily derives the governing integral and differential equations of motion.

Reynolds Transport Theorem An extensive quantity B, which can be a scalar, vector, or tensor, is defined as any physical property of the fluid (e.g., momentum, energy) and b as the corresponding value per unit mass (the intensive value). The Reynolds transport theorem for a moving and arbitrarily deforming control volume CV, with boundary CS (see Figure 3.2.1), states that

(

)

d d B system = dt dt

∫∫∫ ρb dυ + ∫∫ ρb(V ⋅ n) dA r

CV

(3.2.1)

CS

where Bsystem is the total quantity of B in the system (any mass of fixed identity); n is the outward normal to the CS, Vr = V(r,t) – VCS(r,t), the velocity of the fluid particle, V(r,t), relative to that of the CS, VCS(r,t); and d/dt on the left-hand side is the derivative following the fluid particles, i.e., the fluid mass comprising the system. The theorem states that the time rate of change of the total B in the system is equal to the rate of change within the CV plus the net flux of B through the CS. To distinguish between the d/dt that appear on the two sides of Equation (3.2.1) but which have different interpretations, the derivative on the lefthand side, following the system, is denoted by D/Dt and is called the material derivative. This notation is used in what follows. For any function f(x, y, z, t), Df ∂f = + V ⋅∇f Dt ∂t For a CV fixed with respect to the reference frame, Equation (3.2.1) reduces to CS dν

V (r,t)

CV

VCS(r,t) dA

FIGURE 3.2.1 Control volume. © 2005 by CRC Press LLC

n

0866_book.fm Page 10 Thursday, August 5, 2004 10:16 AM

3-10

Chapter 3

(

)

D d Bsystem = Dt dt

∫∫∫ ρb d υ + ∫∫ ρb (V ⋅ n) dA

CV (fixed)

(3.2.2)

CS

(In addition, for this case, the time derivative operator in the first term on the right-hand side may be moved inside the integral, in which case it is then to be interpreted as the partial derivative ∂/∂t.)

Conservation of Mass Applying Equation (3.2.2) for a fixed control volume, with Bsystem the total mass in the system, then, because conservation of mass requires that DBsystem /Dt = 0, it follows, because b = Bsystem /m = 1, that ∂ρ

∫∫∫ ∂t dυ + ∫∫ ρ(V ⋅ n) dA = 0 CV (fixed)

(3.2.3)

CS

This is the integral form of the conservation of mass law for a fixed control volume. For steady flow, Equation (3.2.3) reduces to

∫∫ ρ(V ⋅ n) dA = 0

(3.2.4)

CS

whether compressible or incompressible. For an incompressible flow, ρ = constant, so

∫∫ (V ⋅ n) dA = 0

(3.2.5)

CS

whether the flow is steady or unsteady.

Conservation of Momentum The conservation of (linear) momentum states that Ftotal ≡

∑(external forces acting on the fluid system ) = DDtM ≡ DtD ∫∫∫ ρV dυ

(3.2.6)

system

where M is the total system momentum. For an arbitrarily moving, deformable control volume, it then follows from Equation (3.2.1) with b set to V, that Ftotal =

d dt

∫∫∫ ρV dυ + ∫∫ ρV (V ⋅ n) dA r

CV

(3.2.7)

CS

This expression is only valid in an inertial coordinate frame. For the equivalent expression in a noninertial frame, the relationship between the acceleration aI in an inertial frame and that in a noninertial frame, aR, is used: aI = aR + © 2005 by CRC Press LLC

d2R dΩ + 2Ω Ω ×V + Ω × Ω × r + ×r dt dt 2

(

)

(3.2.8)

0866_book.fm Page 11 Thursday, August 5, 2004 10:16 AM

3-11

Fluid Mechanics

where R is the position vector of the origin of the noninertial frame with respect to that of the inertial frame;
is the angular velocity of the noninertial frame; and r and V the position and velocity vectors in the noninertial frame. The third term on the right-hand side of Equation (3.2.8) is the Coriolis acceleration and the fourth term is the centrifugal acceleration. In a noninertial frame, Equation (3.2.7) is

Ftotal −

 d2R

∫∫∫  dt

2

(

)

+ 2Ω Ω ×V + Ω × Ω × r +

system

d = dt

 dΩ D × r  ρdυ = dt Dt 

∫∫∫ ρV dυ

system

(3.2.9)

∫∫∫ ρV d υ + ∫∫ ρV (V ⋅ n) dA r

CS

CV

where the frame acceleration terms of Equation (3.2.8) have been brought to the left-hand side because, to an observer in the noninertial frame, they act as “apparent” body forces. For a fixed control volume in an inertial frame for steady flow, it follows from the preceding that Ftotal =

∫∫ ρV (V ⋅ n) dA

(3.2.10)

CS

This expression is the basis of many control volume analyses for fluid flow problems. The cross product of r, the position vector with respect to a convenient origin, with the momentum Equation (3.2.6) written for an elementary particle of mass dm, noting that (dr/dt) × V = 0, leads to the integral moment of momentum equation:

∑M−M

I

=

∫∫∫

D ρ ( r × V )dυ Dt system

(3.2.11)

where ΣM is the sum of the moments of all the external forces acting on the system about the origin of r, and MI is the moment of the apparent body forces (see Equation 3.2.9). The right-hand side can be written for a control volume using the appropriate form of the Reynolds transport theorem.

Conservation of Energy The conservation of energy law follows from the first law of thermodynamics for a moving system D Q˙ − W˙ = Dt

∫∫∫ ρe dυ

(3.2.12)

system

where Q˙ is the rate at which heat is added to the system; W˙ the rate at which the system works on its surroundings; and e the total energy per unit mass, the specific energy. For a particle of mass dm, the contributions to e are the internal energy u; the kinetic energy V2/2; and the potential energy, which, in the case of gravity (the only body force to be considered here) is gz, where z is the vertical displacement opposite to the direction of gravity. (No energy transfer due to chemical reaction, as well as no magnetic or electric fields, is assumed.) For a fixed control volume, it follows from Equation (3.2.2) and Equation (3.2.12) (with b = e = u + (V2/2) + gz) that •

•

Q −W =

© 2005 by CRC Press LLC

d dt



∫∫∫ ρu + 2 V CV

1

2

 + gz  dυ + 



∫∫ ρu + 2 V CS

1

2

 + gz  (V ⋅ n ) dA 

(3.2.13)

0866_book.fm Page 12 Thursday, August 5, 2004 10:16 AM

3-12

Chapter 3

V2, Q A2, section 2 Pump z 2 − z1

V1, Q A1, section 1

FIGURE 3.2.2 Pump producing pressure increase.

Problem An incompressible fluid flows through a pump at a volumetric flow rate Qˆ . The (head) loss between sections 1 and 2 (see Figure 3.2.2) is equal to βρV12/2 (V is the average velocity at the section). Calculate the power that must be delivered by the pump to the fluid to produce a given increase in pressure, ∆p = p2 – p1. Solution: The principal equation needed is the energy equation (Equation 3.2.13). The term W˙ , the rate at which the system does work on its surroundings, for such problems has the form: W˙ = − W˙ shaft +

∫∫ pV ⋅ n dA

(P.3.2.1)

CS

where W˙ shaft represents the work done on the fluid by a moving shaft, such as by turbines, propellers, fans, etc., and the second term on the right side represents the rate of working by the normal stress, the pressure, at the boundary. For steady flow in a control volume coincident with the physical system boundaries and bounded by sections 1 and 2, Equation (3.2.13) reduces to (u ≡ 0), Q˙ + W˙ shaft −

1

∫∫ pV ⋅ n dA = ∫∫ 2 ρV CS

2

CS

 + γ z(V ⋅ n ) dA 

(P.3.2.2)

Using average quantities at sections 1 and 2, and the continuity equation (Equation 3.2.5), which reduces in this case to V 1 A 1 = V 2 A 2 = Qˆ ,

(P.3.2.3)

Equation (P.3.2.2) can be written as 1 • • Q + W shaft − p2 − p1 Qˆ =  ρ V22 − V12 + γ z 2 − z1  2

(

)

(

) (



) Qˆ 

(P.3.2.4)

Q˙ , the rate at which heat is added to the system, is here equal to –βρV12/2, the head loss between sections 1 and 2. Equation (P.3.2.4) then can be rewritten •

W shaft = βρ

© 2005 by CRC Press LLC

(

)

(

)

V12 1 + ( ∆p )Qˆ + ρ V22 − V12 Qˆ + γ z 2 − z1 Qˆ 2 2

0866_book.fm Page 13 Thursday, August 5, 2004 10:16 AM

3-13

Fluid Mechanics

or, in terms of the given quantities, 2 ˆ3 βρQˆ 2 ˆ + 1 ρ Q  1 − A 2  + γ (z − z )Qˆ W˙ shaft = + p Q ∆ ( )  2 1 2 A 22  A 12 A 12 

(P.3.2.5)

Thus, for example, if the fluid is water (ρ ≈ 1000 kg/m3, γ = 9.8 kN/m3), Qˆ = 0.5 m3/sec, the heat loss is 0.2ρV12/2, and ∆p = p2 – p1 = 2 × 105 N/m2 = 200 kPa, A1 = 0.1 m2 = A2/2, (z2 – z1) = 2 m, we find, using Equation (P.3.2.5), •

W shaft =

0.2(1000)(0.5)2

(0.1)2

(0.5) (1 − 4) + (9.8 × 103 )(2)(0.5) 1 + ( 2 × 105 )( 0.5 ) + (1000) 2 (0.2)2 3

= 5, 000 + 10, 000 − 4, 688 + 9, 800 = 20,112 Nm/ sec = 20,,112 W =

20,112 hp = 27 hp 745.7

Differential Relations for Fluid Motion In the previous section, the conservation laws were derived in integral form. These forms are useful in calculating, generally using a control volume analysis, gross features of a flow. Such analyses usually require some a priori knowledge or assumptions about the flow. In any case, an approach based on integral conservation laws cannot be used to determine the point-by-point variation of the dependent variables, such as velocity, pressure, temperature, etc. To do this requires the use of the differential forms of the conservation laws, which are presented next.

Mass Conservation — Continuity Equation Applying Gauss’s theorem (the divergence theorem) to Equation (3.2.3) yields  ∂ρ



∫∫∫  ∂t + ∇⋅(ρV ) dυ = 0

(3.2.14)

CV (fixed)

which, because the control volume is arbitrary, immediately yields ∂ρ + ∇ ⋅ (ρV ) = 0 ∂t

(3.2.15)

Dρ + ρ∇ ⋅V = 0 Dt

(3.2.16)

Dρ ∂ρ = + V ⋅∇ρ Dt ∂t

(3.2.17)

This can also be written as

using the fact that

© 2005 by CRC Press LLC

0866_book.fm Page 14 Thursday, August 5, 2004 10:16 AM

3-14

Chapter 3

Special cases: 1. Steady flow [(∂/∂t) ()
0] ∇ ⋅ (ρV ) = 0

(3.2.18)

∇ ⋅V = 0

(3.2.19)

2. Incompressible flow ( D ρ Dt ≡ 0)

Momentum Conservation As a consequence of mass conservation for a system, the right-hand side of Equation (3.2.6) can be written as D Dt

∫∫∫ ρV dυ ≡ ∫∫∫ ρ Dt dυ DV

system

(3.2.20)

system

The total force acting on the system is the sum of the body forces Fb and surface forces Fs . Body forces are often given as forces per unit mass (e.g., gravity) and thus can be written

Fb =

∫∫∫ ρf dυ

(3.2.21)

system

The surface forces are represented in terms of the second-order stress tensor1 σ = {σij}, where σij is defined as the force per unit area in the i direction on a planar element whose normal lies in the j direction.2 From elementary angular momentum considerations for an infinitesimal volume, it can be shown that σij is a symmetric tensor and therefore has only six independent components. The total surface force exerted on the system by its surroundings is Fs =

∫∫ σ ⋅ n dA, with i − component F = ∫∫ σ n dA si

ij

j

(3.2.22)

system surface

The integral momentum conservation law, Equation (3.2.6), can then be written

∫∫∫ ρ Dt dυ = ∫∫∫ ρf dυ + ∫∫ σ ⋅ n dA DV

syystem

system

(3.2.23)

system surface

The application of the divergence theorem to the last term in Equation (3.2.23) leads to

∫∫∫ ρ Dt dυ = ∫∫∫ ρf dυ + ∫∫∫ ∇⋅ σ dυ DV

system

system

(3.2.24)

system

1 It is assumed that the reader is familiar with elementary Cartesian tensor analysis and the associated subscript notation and conventions. The reader for whom this is not true should skip the details and concentrate on the final principal results and equations given at the ends of the next few subsections. 2 This assignment of roles to the first and second subscripts of the stress tensor is a convention that is far from universal. Frequently, their roles are reversed: the first subscript denotes the direction of the normal to the planar element and the second denotes the direction of the force.

© 2005 by CRC Press LLC

0866_book.fm Page 15 Thursday, August 5, 2004 10:16 AM

3-15

Fluid Mechanics

where ∇ ⋅ σ ≡ {∂σij/∂xj}. Because Equation (3.2.24) holds for any material volume, it follows that ρ

DV = ρf + ∇ ⋅σ Dt

(3.2.25)

[With the preceding decomposition of Ftotal, Equation (3.2.10) can be written

∫∫∫ ρf dυ + ∫∫ σ ⋅ n dA = ∫∫ ρV (V ⋅ n) dA CV

CS

(3.2.26)

CS

If ρ is uniform and f is a conservative body force, i.e., f = –∇Ψ, where Ψ is the body force potential, then Equation (3.2.26), after application of the divergence theorem to the body force term, can be written

∫∫ (−ρΨn + σ ⋅ n) dA = ∫∫ ρV (V ⋅ n) dA CS

(3.2.27)

CS

This integral form of the momentum equation, involving integrals only over the surface of the control volume, is commonly used in control volume analyses, particularly when the body force term is absent.] Analysis of Rate of Deformation The principal aim of the following two subsections is to derive a relationship between the stress and the rate of strain to be used in the momentum equation (Equation 3.2.25). The reader less familiar with tensor notation may skip these sections, apart from noting some of the terms and quantities defined therein, and proceed directly to Equation (3.2.38) or Equation (3.2.39). The relative motion of two neighboring points P and Q, separated by a distance η, can be written (u is the local velocity) u(Q) = u(P) + (∇u) or, equivalently, writing ∇u as the sum of antisymmetric and symmetric tensors, u(Q) = u(P) +

(

)

(

)

1 1 ∇u ) − ( ∇u ) *  + ( ∇u ) + ( ∇u ) *  ( 2 2

(3.2.28)

where ∇u = {∂ui /∂xj }, and the superscript * denotes transpose, so (∇u)* = {∂uj /∂xi }. The second term on the right-hand side can be rewritten in terms of the vorticity, ∇ × u, so Equation (3.2.28) becomes

(

)

1 1 u(Q) = u(P) + (∇ × u) ×  + ( ∇u ) + ( ∇u ) *  2 2

(3.2.29)

which shows that the local rate of deformation consists of a rigid-body translation, a rigid-body rotation with angular velocity ½(∇ × u) and a velocity or rate of deformation. The coefficient of η in the last term in Equation (3.2.29) is defined as the rate-of-strain tensor and is denoted by e, in subscript form ∂u j  1  ∂u e ij =  i + 2  ∂x j ∂x i 

(3.2.30)

From e, one can define a rate-of-strain central quadric, along the principal axes of which the deforming motion consists of a rate of pure extension or contraction. © 2005 by CRC Press LLC

0866_book.fm Page 16 Thursday, August 5, 2004 10:16 AM

3-16

Chapter 3

Relationship between Forces and Rate of Deformation Now the required relationship between the stress tensor and the rate of deformation is considered. Assuming that in a static fluid the stress reduces to a (negative) hydrostatic or thermodynamic pressure, equal in all directions, one can write σ = − p I + τ or σ ij = − pδ ij + τ ij

(3.2.31)

where τ is the viscous part of the total stress and is called the deviatoric stress tensor; I is the identity tensor, and δij is the Kronecker delta (δij = 0 if i ≠ j; δij = 1 if i = j). It is further assumed that (1) the fluid exhibits no preferred directions; (2) the stress is independent of any previous history of distortion; and (3) the stress depends only on the local thermodynamic state and the kinematic state of the immediate neighborhood. Specifically, τ is assumed to be linearly proportional to the first spatial derivatives of u, the coefficient of proportionality depending only on the local thermodynamic state. These assumptions, and the relations below that follow from them, are appropriate for a Newtonian fluid. Most common fluids, such as air and water under most conditions, are Newtonian, but many other fluids, including many that arise in industrial applications, exhibit so-called non-Newtonian properties. The study of such non-Newtonian fluids, such as viscoelastic fluids, is the subject of the field of rheology. With the preceding Newtonian fluid assumptions and the symmetry of τ , which follows from the symmetry of σ, one can show that the viscous part τ of the stress can be written as τ = λ(∇ ⋅ u ) I + 2µe

(3.2.32)

Thus, the total stress for a Newtonian fluid is σ = − p I + λ(∇ ⋅ u ) I + 2µe

(3.2.33)

 ∂u ∂u j   ∂u  σ ij = − pδ ij + λ k  δ ij + µ i +   ∂x k   ∂x j ∂x i 

(3.2.34)

or, in subscript notation

(The Einstein summation convention is assumed here: namely, that in any expression containing a repeated subscript, such as in the second term on the right-hand side above, that subscript is to be given all its possible values and the results then summed; note also that ∇ ⋅ u = ∂uk/∂xk = ekk .) The coefficient λ is called the second viscosity and µ the absolute viscosity or, more commonly, the dynamic viscosity, or simply the “viscosity.” For a Newtonian fluid, λ and µ depend only on local thermodynamic state, primarily on the temperature. Note that, from Equation (3.2.34), whereas in a fluid at rest the pressure is an isotropic normal stress (equal to p in all directions), this is not the case for a moving fluid because, in general, σ11 ≠ σ22 ≠ σ33. To have a quantity analogous to p for a moving fluid we define the pressure in a moving fluid as the – negative mean normal stress, denoted, say, by p , thusly: 1 p = − σ ii 3

(3.2.35)

(σii is the trace of σ and an invariant of σ, independent of the orientation of the axes.) From Equation (3.2.34) 1 2   p = − σ ii = p −  λ + µ ∇ ⋅ u  3 3  © 2005 by CRC Press LLC

(3.2.36)

0866_book.fm Page 17 Thursday, August 5, 2004 10:16 AM

3-17

Fluid Mechanics –

For an incompressible fluid, ∇ ⋅ u = 0 and thus p
p. The quantity (λ + ⅔ µ) is called the bulk viscosity. If one assumes that the deviatoric stress tensor τ ij makes no contribution to the mean normal – stress, it follows that λ + ⅔ µ = 0, so again p = p. This condition, λ = –⅔ µ, is called the Stokes assumption or hypothesis. If neither the incompressibility nor the Stokes assumptions are made, the difference between – p and p is usually negligibly small because generally (λ + ⅔ µ)∇ · u 5000) for large values of Pr. Elimination of µ as well as D results in h c p (τ w ρ)

12

© 2005 by CRC Press LLC

=A

0866_book.fm Page 31 Thursday, August 5, 2004 10:16 AM

3-31

Fluid Mechanics

or f Nu = A Re Pr    2

12

which appears to be an approximate asymptote for Re → ∞ and Pr → 0. Elimination of both cp and ρ again yields the appropriate result for laminar flow, indicating that ρ rather than um is the meaningful variable to eliminate in this respect. The numerical value of the coefficient A in the several expressions above depends on the mode of heating, a true variable, but one from which the purely functional expressions are independent. If jw the heat flux density at the wall, and Tw – Tm , the temperature difference between the wall and the bulk of the fluid, were introduced as variables in place of h ≡ jw /(Tw – Tm), another group such as cp(Tw – Tm ) (Dρ/µ)2 or ρcp(Tw – Tm )/τw or c p (Tw − Tm )/ um2 , which represents the effect of viscous dissipation, would be obtained. This effect is usually but not always negligible. (See Chapter 4.) Example 3.3.3: Free Convection from a Vertical Isothermal Plate The behavior for this process may be postulated to be represented by

{

h = φ g, β, Tw − T∞ , x, µ, ρ, c p , k

}

where g is the acceleration due to gravity, β is the volumetric coefficient of expansion with temperature, T∞ is the unperturbed temperature of the fluid, and x is the vertical distance along the plate. The corresponding tabulation is

M L θ T

h

g

β

Tw – T∞

x

µ

ρ

cp

k

1 0 –3 –1

0 1 –2 0

0 0 0 –1

0 0 0 1

0 1 0 0

1 –1 –1 0

1 –3 0 0

0 2 –2 –1

1 1 –3 1

The minimal number of dimensionless groups indicated by both methods is 9 – 4 = 5. A satisfactory set of dimensionless groups, as found by any of the methods illustrated in Example 1 is 2  ρ2 gx 3 c p µ  ρx   hx = φ 2 , , β(Tw − T∞ ), c p (Tw − T∞ )    k k  µ    µ

It may be reasoned that the buoyant force which generates the convective motion must be proportional to ρgβ(Tw – T∞), thus, g in the first term on the right-hand side must be multiplied by β(Tw – T∞), resulting in 2  ρ2 gβ(Tw − T∞ ) x 3 c p µ  ρx   hx = φ , , − , − T T c T T β ( ) ( )   w p w ∞ ∞  k k µ2  µ   

The effect of expansion other than on the buoyancy is now represented by β(Tw – T∞), and the effect of viscous dissipation by cp(Tw – T∞)(ρx/µ)2. Both effects are negligible for all practical circumstances. Hence, this expression may be reduced to  ρ2 gβ(Tw − T∞ ) x 3 c p µ  hx = φ ,  k k  µ2  © 2005 by CRC Press LLC

0866_book.fm Page 32 Thursday, August 5, 2004 10:16 AM

3-32

Chapter 3

or Nu x = φ {Grx , Pr} where Nux = hx/k and Grx = ρ2gβ(Tw – T∞)x3/µ2 is the Grashof number. Elimination of x speculatively now results in 2 3 hx  ρ gβ(Tw − T∞ ) x  =  φ {Pr} k  µ2  13

or Nu x = Grx1 3 φ {Pr} This expression appears to be a valid asymptote for Grx → ∞ and a good approximation for the entire turbulent regime. Eliminating µ speculatively rather than x results in 2 2 3 hx  ρ c p gβ(Tw − T∞ ) x  = φ  k k2  

or

{

Nu x = φ Grx Pr 2

}

The latter expression appears to be a valid asymptote for Pr → 0 for all Grx, that is, for both the laminar and the turbulent regimes. The development of a valid asymptote for large values of Pr requires more subtle reasoning. First cp µ/k is rewritten as µ/ρα where α = k/ρcp . Then ρ is eliminated speculatively except as it occurs in ρgβ(Tw – T∞) and k/ρcp . The result is  c p ρ2 gβ(Tw − T∞ ) x 3  hx = φ  µk k   or Nu x = φ {Ra x } where Ra x =

c p ρ2 gβ(Tw − T∞ ) x 3 µk

= Grx Pr

is the Rayleigh number. The expression appears to be a valid asymptote for Pr → ∞ and a reasonable approximation for even moderate values of Pr for all Grx, that is, for both the laminar and the turbulent regimes. Eliminating x speculatively from the above expressions for small and large values of Pr results in

(

Nu x = A Grx Pr 2

)

13

= A(Ra x Pr )

13

and Nu x = B(Grx Pr )

13

© 2005 by CRC Press LLC

= B(Ra x )

13

0866_book.fm Page 33 Thursday, August 5, 2004 10:16 AM

3-33

Fluid Mechanics

The former appears to be a valid asymptote for Pr → 0 and Grx → ∞ and a reasonable approximation for very small values of Pr in the turbulent regime, while the latter is well confirmed as a valid asymptote for Pr → ∞ and Grx → ∞ and as a good approximation for moderate and large values of Pr over the entire turbulent regime. The expressions in terms of Grx are somewhat more complicated than those in terms of Rax, but are to be preferred since Grx is known to characterize the transition from laminar to turbulent motion in natural convection just as ReD does in forced flow in a channel. The power of speculation combined with dimensional analysis is well demonstrated by this example in which valid asymptotes are thereby attained for several regimes.

Correlation of Experimental Data and Theoretical Values Correlations of experimental data are generally developed in terms of dimensionless groups rather than in terms of the separate dimensional variables in the interests of compactness and in the hope of greater generality. For example, a complete set of graphical correlations for the heat transfer coefficient h of Example 3.3.2 above in terms of each of the six individual independent variables and physical properties might approach book length, whereas the dimensionless groupings both imply that a single plot with one parameter should be sufficient. Furthermore, the reduced expression for the turbulent regime implies that a plot of Nu/Re f 1/2 vs. Pr should demonstrate only a slight parametric dependence on Re or Re f 1/2. Of course, the availability of a separate correlation for f as a function of Re is implied. Theoretical values, that is, ones obtained by numerical solution of a mathematical model in terms of either dimensional variables or dimensionless groups, are presumably free from imprecision. Even so, because of their discrete form, the construction of a correlation or correlations for such values may be essential for the same reasons as for experimental data. Graphical correlations have the merit of revealing general trends, of providing a basis for evaluation of the choice of coordinates, and most of all of displaying visually the scatter of the individual experimental values about a curve representing a correlation or their behavior on the mean. (As mentioned in the previous subsection, the omission of a variable may give the false impression of experimental error in such a plot.) On the other hand, correlating equations are far more convenient as an input to a computer than is a graphical correlation. These two formats thus have distinct and complementary roles; both should generally be utilized. The merits and demerits of various graphical forms of correlations are discussed in detail by Churchill (1979), while the use of logarithmic and arithmetic coordinates, the effects of the appearance of a variable in both coordinates, and the effects of the distribution of error between the dependent and independent variable are further illustrated by Wilkie (1985). Churchill and Usagi (1972; 1974) proposed general usage of the following expression for the formulation of correlating equations: y n {x} = y0n {x} + y∞n {x}

(3.3.6)

where yo{x} and y∞{x} denote asymptotes for small and large values of x, respectively, and n is an arbitrary exponent. For convenience and simplicity, Equation (3.3.6) may be rearranged in either of the following two forms:

(Y ( x ))n = 1 + Z n {x}

(3.3.7)

or n

 Y {x}  1  Z x  = 1+ Zn x {}  { }

(3.3.8)

where Y{x} ≡ y{x}/yo{x} and Z{x} ≡ y∞{x}/yo{x}. Equation (3.3.6), Equation (3.3.7), and Equation (3.3.9) are hereafter denoted collectively as the CUE (Churchill–Usagi equation). The principle merits of the CUE as a canonical expression for correlation are its simple form, generality, and minimal degree of © 2005 by CRC Press LLC

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explicit empiricism, namely, only that of the exponent n, since the asymptotes yo{x} and y∞{x} are ordinarily known in advance from theoretical considerations or well-established correlations. Furthermore, as will be shown, the CUE is quite insensitive to the numerical value of n. Although the CUE is itself very simple in form, it is remarkably successful in representing closely very complex behavior, even including the dependence on secondary variables and parameters, by virtue of the introduction of such dependencies through yo{x} and y∞{x}. In the rare instances in which such dependencies are not represented in the asymptotes, n may be correlated as a function of the secondary variables and/or parameters. Although the CUE usually produces very close representations, it is empirical and not exact. In a few instances, numerical values of n have been derived or rationalized on theoretical grounds, but even then some degree of approximation is involved. Furthermore, the construction of a correlating expression in terms of the CUE is subject to the following severe limitations: 1. The asymptotes yo{x} and y∞{x} must intersect once and only once; 2. The asymptotes yo{x} and y∞{x} must be free of singularities. Even though a singularity occurs beyond the asserted range of the asymptote, it will persist and disrupt the prediction of the CUE, which is intended to encompass all values of the independent variable x; and 3. The asymptotes must both be upper or lower bounds. In order to avoid or counter these limitations it may be necessary to modify or replace the asymptotes with others. Examples of this process are provided below. A different choice for the dependent variable may be an option in this respect. The suitable asymptotes for use in Equation (3.3.6) may not exist in the literature and therefore may need to be devised or constructed. See, for example, Churchill (1988b) for guidance in this respect. Integrals and derivatives of the CUE are generally awkward and inaccurate, and may include singularities not present or troublesome in the CUE itself. It is almost always preferable to develop a separate correlating equation for such quantities using derivatives or integrals of yo{x} and y∞{x}, simplified or modified as appropriate. The Evaluation of n Equation (3.3.6) may be rearranged as   y {x}  n  ln 1 +  ∞     y0 {x}   n=  y{x}  ln    y0 {x} 

(3.3.9)

and solved for n by iteration for any known value of y{x}, presuming that yo{x} and y∞{x} are known. If y{x*} is known, where x* represents the value of x at the point of intersection of the asymptotes, that is, for yo{x} = y∞{x}, Equation (3.3.9) reduces to n=

ln{2}

 y{x *}  ln    y0 {x *} 

(3.3.10)

and iterative determination of n is unnecessary. A graphical and visual method of evaluation of n is illustrated in Figure 3.3.1 in which Y{Z} is plotted vs. Z for 0 ≤ Z ≤ 1 and Y{Z}/Z vs. 1/Z for 0 ≤ 1/Z ≤ 1 in arithmetic coordinates with n as a parameter. Values of y{x} may be plotted in this form and the best overall value of n selected visually (as illustrated in Figure 3.3.2). A logarithmic plot of Y{Z} vs. Z would have less sensitivity relative to the dependence on n. (See, for example, Figure 1 of Churchill and Usagi, 1972.) Figure 3.3.1 explains in part the success of the CUE. Although y and x may both vary from 0 to ∞, the composite variables plotted in Figure 3.3.1 © 2005 by CRC Press LLC

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3-35

FIGURE 3.3.1 Arithmetic, split-coordinate plot of Equation 3.3.10. (From Churchill, S.W. and Usagi, R. AIChE J. 18(6), 1123, 1972. With permission from the American Institute of Chemical Engineers.)

FIGURE 3.3.2 Arithmetic, split-coordinate plot of computed values and experimental data for laminar free convection from an isothermal vertical plate. (From Churchill, S.W. and Usagi, R. AIChE J. 18(6), 1124, 1972. With permission from the American Institute of Chemical Engineers.)

are highly constrained in that the compound independent variables Z and 1/Z vary only between 0 and 1, while for n ≥ 1, the compound dependent variables Y{Z} and Y{Z}/Z vary only from 1 to 2. Because of the relative insensitivity of the CUE to the numerical value of n, an integer or a ratio of two small integers may be chosen in the interest of simplicity and without significant loss of accuracy. For example, the maximum variance in Y (for 0 ≤ Z ≤ 1) occurs at Z = 1 and increases only 100(21/20 – 1) = 3.5% if n is decreased from 5 to 4. If yo{x} and y∞{x} are both lower bounds, n will be positive, and if they are both upper bounds, n will be negative. To avoid extending Figure 3.3.1 for negative values of n, 1/y{x} may simply be interpreted as the dependent variable. © 2005 by CRC Press LLC

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Intermediate Regimes Equation (3.3.6), Equation (3.3.7), and Equation (3.3.8) imply a slow, smooth transition between yo{x} and y∞{x} and, moreover, one that is symmetrical with respect to x*(Z = 1). Many physical systems demonstrate instead a relatively abrupt transition, as for example from laminar to turbulent flow in a channel or along a flat plate. The CUE may be applied serially as follows to represent such behavior if an expression yi{x} is postulated for the intermediate regime. First, the transition from the initial to the intermediate regime is represented by y1n = y0n + yin

(3.3.11)

Then the transition from this combined regime to the final regime by

(

y m = y1m + y∞m = y0n + yin

)

mn

+ y∞m

(3.3.12)

Here, and throughout the balance of this subsection, in the interests of simplicity and clarity, the functional dependence of all the terms on x is implied rather written out explicitly. If yo is a lower bound and yi is implied to be one, y1 and y∞ must be upper bounds. Hence, n will then be positive and m negative. If yo and yi are upper bounds, y1 and y∞ must be lower bounds; then n will be negative and m positive. The reverse formulation starting with y∞ and y1 leads by the same procedure to

(

y n = y0n + yim + y∞m

)

nm

(3.3.13)

If the intersections of yi with yo and y∞ are widely separated with respect to x, essentially the same pair of values for n and m will be determined for Equation (3.3.12) and Equation (3.3.13), and the two representations for y will not differ significantly. On the other hand, if these intersections are close in terms of x, the pair of values of m and n may differ significantly and one representation may be quite superior to the other. In some instances a singularity in yo or y∞ may be tolerable in either Equation (3.3.12) or (3.3.13) because it is overwhelmed by the other terms. Equation (3.3.12) and Equation (3.3.13) have one hidden flaw. For x → 0, Equation (3.3.12) reduces to   y m y → y0 1 +  ∞     y0    

1m

(3.3.14)

If yo is a lower bound, m is necessarily negative, and values of y less than yo are predicted. If yo /y∞ is sufficiently small or if m is sufficiently large in magnitude, this discrepancy may be tolerable. If not, the following alternative expression may be formulated, again starting from Equation (3.3.11):

(y

n

− y0n

)

m

(

= yinm + y∞n − y0n

)

m

(3.3.15)

Equation (3.3.15) is free from the flaw identified by means of Equation (3.3.14) and invokes no additional empiricism, but a singularity may occur at y∞ = yo , depending on the juxtapositions of yo , yi , and y∞ . Similar anomalies occur for Equation (3.3.13) and the corresponding analog of Equation (3.3.14), as well as for behavior for which n < 0 and m > 0. The preferable form among these four is best chosen by trying each of them. One other problem with the application of the CUE for a separate transitional regime is the formulation of an expression for yi{x}, which is ordinarily not known from theoretical considerations. Illustrations of the empirical determination of such expressions for particular cases may be found in Churchill and Usagi (1974), Churchill and Churchill (1975), and Churchill (1976; 1977), as well as in Example 3.3.5 below. © 2005 by CRC Press LLC

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FIGURE 3.3.3 Arithmetic, split-coordinate plot of experimental data for the pressure drop in flow through a packed bed of spheres. (From Churchill, S.W. and Usagi, R. AIChE J. 18(6), 1123, 1972. With permission from the American Institute of Chemical Engineers.)

Example 3.3.4: The Pressure Gradient in Flow through a Packed Bed of Spheres The pressure gradient at asymptotically low rates of flow (the creeping regime) can be represented by the Kozeny–Carman equation, Φ = 150 Rep, and at asymptotically high rates of flow (the inertial regime) by the Burke–Plummer equation, Φ = 1.75 (Rep)2, where Φ = ρε2dp(–dPf /dx)µ2(1 – ε), Rep = dpuoρ/µ(1 – ε), dp = diameter of spherical particles, m, ε = void fraction of bed of spheres, dPf /dx = dynamic pressure gradient (due to friction), Pa/m, and uo = superficial velocity (in absence of the spheres), m/sec. For the origin of these two asymptotic expressions see Churchill (1988a). They both have a theoretical structure, but the numerical coefficients of 150 and 1.75 are basically empirical. These equations are both lower bounds and have one intersection. Experimental data are plotted in Figure 3.3.3, which has the form of Figure 3.3.1 with Y = Φ/150 Rep, Y/Z = Φ/(1.75 Rep)2 and Z = 1.75 Re 2p / 150 Rep = Rep /85.7. A value of n = 1 is seen to represent these data reasonably well on the mean, resulting in

( )

Φ = 150 Re p + 1.75 Re p

2

which was originally proposed as a correlating equation by Ergun (1952) on the conjecture that the volumetric fraction of the bed in “turbulent” flow is proportional to Rep. The success of this expression in conventional coordinates is shown in Figure 3.3.4. The scatter, which is quite evident in the arithmetic split coordinates of Figure 3.3.3, is strongly suppressed in a visual sense in the logarithmic coordinates of Figure 3.3.4. Example 3.3.5: The Friction Factor for Commercial Pipes for All Conditions The serial application of the CUE is illustrated here by the construction of a correlating equation for both smooth and rough pipes in the turbulent regime followed by combination of that expression with ones for the laminar and transitional regimes. The Turbulent Regime The Fanning friction factor, fF , for turbulent flow in a smooth round pipe for asymptotically large rates of flow (say ReD > 5000) may be represented closely by the empirical expression: © 2005 by CRC Press LLC

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FIGURE 3.3.4 Logarithmic correlation of experimental data for the pressure drop in flow through a packed bed of spheres. (From Churchill, S.W. and Usagi, R. AIChE J. 18(6), 1123, 1972. With permission from the American Institute of Chemical Engineers.)

 2 f   F

12

 f  = 0.256 + 2.5 ln  F   2 

12

 Re D  

A corresponding empirical representation for naturally rough pipe is  2 f   F

12

D = 3.26 + 2.5 ln   e

Direct combination of these two expressions in the form of the CUE does not produce a satisfactory correlating equation, but their combination in the following rearranged forms:

e

(1 2.5)( 2

fF )

12

f  = 1.108  F   2

12

Re D

and e

(1 2.5)( 2

fF )

12

D = 3.68    e

with n = –1 results in, after the reverse rearrangement,

 2 f   F

12

12    fF    Re D    2   = 0.256 + 2.5 ln   12 f e 1 + 0.3012   F  Re  D   D  2  

The exact equivalent of this expression in structure but with the slightly modified numerical coefficients of 0.300, 2.46, and 0.304 was postulated by Colebrook (1938–1939) to represent his own experimental data. The coefficients of the expression given here are presumed to be more accurate, but the difference in the predictions of fF with the two sets of coefficients is within the band of uncertainty of the experimental © 2005 by CRC Press LLC

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data. The turbulent regime of the “friction-factor” plot in most current textbooks and handbooks is simply a graphical representation of the Colebrook equation. Experimental values are not included in such plots since e, the effective roughness of commercial pipes, is simply a correlating factor that forces agreement with the Colebrook equation. Values of e for various types of pipe in various services are usually provided in an accompanying table, that thereby constitutes an integral part of the correlation. The Laminar Region The Fanning friction factor in the laminar regime of a round pipe (Red < 1800) is represented exactly by the following theoretical expression known as Poiseuille’s law: fF = 16/ReD . This equation may be rearranged as follows for convenience in combination with that for turbulent flow:  2 f   F

Re D ( f F 2)

12

12

=

8

The Transitional Regime Experimental data as well as semitheoretical computed values for the limiting behavior in the transition may be represented closely by (fF /2) = (ReD /37500)2. This expression may be rewritten, in terms of (2/fF)1/2 and ReD(fF /2)1/2, as follows:  fF     2

12

 37500  =   Re ( f 2)1 2   D F 

12

Overall Correlation The following correlating equation for all ReD(fF /2)1/2 and e/D may now be constructed by the combination of the expressions for the turbulent and transition regimes in the form of the CUE with n = 8, and then that expression and that for the laminar regime with n = –12, both components being chosen on the basis of experimental data and predicted values for the full regime of transition:  2 f   F

12

12   8 =   Re ( f 2)1 2    D F

   4 12   37500    1.108 Re D ( f F 2)  + 2.5 ln  +   12 1 2 e   Re D ( f F 2)  1 + 0.3012  Re D ( f F 2)      a   

8

     

−3 2 −1 12

      

The absolute value signs are only included for aesthetic reasons; the negative values of the logarithmic term for very small values of ReD(fF /2)1/2 do not affect the numerical value of (2/fF)1/2 in the regime in which they occur. This overall expression appears to have a complicated structure, but it may readily be recognized to reduce to its component parts when the corresponding term is large with respect to the other two. It is insensitive to the numerical values of the two arbitrary exponents. For example, doubling their values would have almost no effect on the predictions of (fF /2)1/2. The principal uncertainty is associated with the expression for the transition regime, but the overall effect of the corresponding term is very small. The uncertainties associated with this correlating equation are common to most graphical correlations and algebraic expressions for the friction factor, and are presumed to be fairly limited in magnitude and to be associated primarily with the postulated value of e. Although the overall expression is explicit in ReD(fF /2)1/2 rather than ReD, the latter quantity may readily be obtained simply by multiplying the postulated value of ReD(fF /2)1/2 by the computed values of (2/fF)1/2. © 2005 by CRC Press LLC

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References Buckingham, E. 1914. On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev., Ser. 2, 4(4):345–375. Churchill, S.W. 1976. A comprehensive correlating equation for forced convection from plates. AIChE J. 22(2):264–268. Churchill, S.W. 1977. Comprehensive correlating equation for heat, mass and momentum transfer in fully developed flow in smooth tubes. Ind. Eng. Chem. Fundam. 16(1):109–116. Churchill, S.W. 1979. The Interpretation and Use of Rate Data. The Rate Process Concept, rev. printing, Hemisphere Publishing Corp., Washington, D.C. Churchill, S.W. 1981. The use of speculation and analysis in the development of correlations. Chem. Eng. Commun. 9:19–38. Churchill, S.W. 1988a. Flow through porous media, Chapter 19 in Laminar Flows. The Practical Use of Theory, pp. 501–538, Butterworths, Boston. Churchill, S.W. 1988b. Derivation, selection, evaluation and use of asymptotes. Chem. Eng. Technol. 11:63–72. Churchill, S.W. and Churchill, R.U. 1975. A general model for the effective viscosity of pseudoplastic and dilatant fluids. Rheol. Acta. 14:404–409. Churchill, S.W. and Usagi, R. 1972. A general expression for the correlation of rates of transfer and other phenomena. AIChE J. 18(6):1121–1128. Churchill, S.W. and Usagi, R. 1974. A standardized procedure for the production of correlations in the form of a common empirical equation. Ind. Eng. Chem. Fundam. 13(1):39–44. Colebrook, C.R. 1938–1939. Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng. 11(5024):133–156. Ergun, S. 1952. Fluid flow through packed beds. Chem. Eng. Prog. 48(2):81–96. Hellums, J.D. and Churchill, S.W. 1964. Simplifications of the mathematical description of boundary and initial value problems. AIChE J. 10(1):110–114. Wilkie, D. 1985. The correlation of engineering data reconsidered. Int. J. Heat Fluid Flow. 8(2):99–103. Zlokarnik, M. 1991. Dimensional Analysis and Scale-Up in Chemical Engineering. Springer-Verlag, Berlin.

3.4 Hydraulics of Pipe Systems J. Paul Tullis and Blake P. Tullis Principles involved in the design and operation of pipelines include: • Establish the required flow for present and future demands, determining the route, profile, and elevation differences, as well as need for and benefits of storage capabilities. • Determine whether the flow will be supplied by gravity, pumps, or a combination. • Select the appropriate pipe material and pressure class, considering fluid type; soil conditions; system pressure; pipe size; whether the pipe will be buried; etc. • Consider the possibility of crushing from external loads and collapse due to internal vacuum pressures. • Select the optimum pipe diameter. • Select valves and pumps that have performance characteristics compatible with the system. • Establish operating procedures for the pumps and valves to avoid excessive hydraulic transients. • Analyze the cavitation potential of the valves and pumps. • Select appropriate type, size, and location of air valves and check valves. • Establish procedures for filling, flushing, and draining the system.

© 2005 by CRC Press LLC

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Basic Equations Solving fluid flow problems involves the application of one or more of the three basic equations: continuity, momentum, and energy. These three basic tools are developed from the law of conservation of mass, Newton’s second law of motion, and the first law of thermodynamics. The simplest form of the continuity equation is for one-dimensional incompressible steady flow in a closed conduit. Applying continuity between any two sections gives: A1V1 = A2V2 = Q

(3.4.1)

For a variable density, the equation can be written: •

ρ1 A1V1 = ρ2 A2V2 − m

(3.4.2)

in which A is the cross-sectional area of the pipe; V is the mean velocity at that same location; Q is the flow rate; ρ is the fluid density; and m· is the mass flow rate. The equations are valid for steady flow in any rigid conduit as long as there is no addition or loss of liquid between sections 1 and 2. For steady-state pipe flow, the momentum equation relates the sum of forces in a given direction acting on a control volume (a section of the fluid inside the pipe), to the net momentum flux through the control volume in the same direction. The most common forces are pressure and friction. ΣFx = ρ2 A2V2V2 x − ρ1 A1V1V1x

(3.4.3)

For incompressible flow, this equation can be reduced to:

(

ΣFx = ρQ V2 x − V1x

)

(3.4.4)

These equations can easily be applied to a three-dimensional flow problem by adding equations in the y and z directions. A general form of the energy equation applicable to incompressible pipe flow is: PI V2 P V2 + Z1 + 1 = 2 + Z 2 + 2 − H p + H t + H f γ 2g γ 2g

(3.4.5)

The units are energy per unit weight of liquid: ft-lb/lb or N-m/N, or simply length. The first three terms are pressure head (P/γ); elevation head (Z) (above some datum); and velocity head (V2/2g). The last three terms on the right side of the equation are the total dynamic head added by a pump (Hp), removed by a turbine (Ht ), and the total friction and minor head losses (Hf ). The sum of the first three terms in Equation (3.4.5) is defined as the total head and the sum of the pressure and elevation heads is referred to as the piezometric head.

Fluid Friction This subsection discusses pressure changes resulting from incompressible flow for pipes with circular geometry. However, the results can be generalized for a pipe of noncircular geometry by substituting for the diameter D in any of the equations, the hydraulic diameter, Dh, defined as Dh= A/P. A is the crosssectional area of the conduit and P is the wetted perimeter (πD for a circular pipe flowing full).

© 2005 by CRC Press LLC

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FIGURE 3.4.1 The Moody diagram.

The analysis in this subsection can also be applied to gases and vapors, provided the Mach number in the duct does not exceed 0.3. For greater values of the Mach number, the compressibility effect becomes significant and the reader should see Section 3.7 on compressible flow. Friction loss (Hf ) depends on pipe diameter (D); length (L); pipe roughness (e); fluid density (ρ) or specific weight (γ); viscosity (ν); and flow velocity (V). Dimensional analysis can be used to provide a functional relationship among the friction loss Hf , pipe dimensions, fluid properties, and flow parameters. The resulting equation is called the Darcy-Weisbach equation: Hf =

fLV 2 fLQ 2 = 2 gD 1.23gD5

(3.4.6)

The friction factor f is a measure of pipe roughness and has been evaluated experimentally for numerous pipes. The data were used to create the Moody friction factor chart shown as Figure 3.4.1. The friction factor f and the subsequent friction loss of a flowing liquid depend on whether the flow is laminar or turbulent. Laminar flow exists when viscous forces are large compared to inertial forces. The Reynolds number is the ratio of inertia forces to viscous forces and is a convenient parameter for determining whether a flow condition will be laminar or turbulent. The Reynolds number is defined as: Re =

ρVd Vd = µ ν

(3.4.7a)

V is the mean flow velocity; d is diameter; ρ is fluid density; µ is dynamic viscosity; and ν is kinematic viscosity. For laminar flow (Re < 2000), f is only a function of Re and is calculated by f = 64/Re. Laminar flow in pipes is unusual for practical applications. To illustrate this, the velocity of water flowing in a 1-m diameter pipe at 20°C would need to be less than or equal to 2 mm/sec to be laminar. Most practical pipe flow problems are in the turbulent region. © 2005 by CRC Press LLC

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At Reynolds numbers between about 2000 and 4000, the flow is unstable as a result of the onset of turbulence (critical zone in Figure 3.4.1). In this range, friction loss calculations are difficult because it is impossible to determine a unique value of f. Fortunately, few pipe flow problems involve Reynolds numbers below 4000. For Re > 4000, the flow becomes turbulent and f is a function of Re and relative pipe roughness (e/D), where e is the equivalent roughness height and D is pipe diameter. At high Re, f eventually depends only on the relative roughness height, e/D. This region is defined as fully turbulent flow. The Reynolds number at which this occurs depends on the relative roughness of the pipe Using the Moody chart in Figure 3.4.1 to get f requires that Re and e/D be known. Calculating Re is direct if the water temperature, flow, and pipe diameter are known. The problem then reduces to one of obtaining a good value for e. The values of e listed in Figure 3.4.1 should be considered as approximate and used only if more accurate values cannot be obtained from the pipe supplier. For noncircular pipes, the only change in the friction loss equation is replacing the diameter with the hydraulic radius (R). R is the ratio of the flow area to the wetter perimeter. For a circular pipe, D = 4R, The Moody diagram (Figure 3.4.1) is a graphical representation of the Colebrook and White equation:  2e 18.7 1 = 1.74 − 0.869 ln  +  D Re f f

  

(3.4.7b)

The Colebrook and White equation requires a trial-and-error or iterative solution because f appears on both sides of the equation. Haaland (1983) developed an explicit relationship that approximates the Colebrook and White equation. The relationship provides reasonable accuracy in the range of 4000 < Re < 108 and 0 < e/d < 0.05 1.11   1 6.9  e   = −0.782 ln  +   f  Re  3.7 D  

(3.4.8)

If the flow or pipe diameter is not known, the solution to the Darcy–Weisbach or Haaland equations becomes a trial-and-error or iterative process. For long gravity flow pipelines, the criterion for selecting the pipe diameter is simply finding the smallest pipe that can pass the required flow without the friction and minor losses exceeding the available head. For pumped systems, optimizing the pipe diameter is based on an economic analysis that compares the installed pipe cost with the cost of building and operating the pumping plant. Pipe cost is proportional to D and pumping cost is inversely proportional to D. The optimum pipe diameter is selected as the one that provides the lowest total cost. The pipe roughness usually varies with time due to buildup of solid deposits or organic growths. Manufacturing methods and tolerances also cause variations in the surface roughness. Consequently, the friction factor for any pipe can only be approximated and proper allowance should be made for these uncertainties.

Minor Losses Flow through valves, orifices, elbows, transitions, etc. causes flow separation, which results in the generation and dissipation of turbulent eddies. The resulting head loss, Hm , referred to as a minor loss, is proportional to the velocity head: Hm = K l

(

Q2 2 gAm2

)

(3.4.9)

in which Kl is the minor loss coefficient and Am is the flow area at the inlet to the minor loss element. Although these types of losses are referred to as minor, in short piping systems; long piping systems with © 2005 by CRC Press LLC

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small velocity heads; or other systems in which the friction loss is relatively small, the minor losses can be responsible for the majority of the system head loss. The minor loss coefficient Kl is analogous to fL/D in Equation (3.4.6). The summation of all friction and minor losses in a pipe system can be expressed as: H l = H f + Hm  H l =  

(3.4.10)

 K   l 2   Q 2 = CQ 2  2 gA  m  

∑

   fL  +  2 gDA 2  p 

C=

∑  2gDA  + ∑  2gA

∑

(3.4.11)

in which: 



fL

2 p

 K  l  2  

(3.4.12)

m

It is important to use the correct pipe inside diameter for each pipe section and minor loss. The variation of the actual inside pipe diameters from the nominal is a function of size and pressure class. In an effort to simplify calculations, some have expressed the minor losses as an equivalent pipe length parameter: Lequ = Kl D/f. It represents the length of pipe that produces the same head loss as the local or minor loss. This is a simple, but potentially inaccurate, method of including minor losses. The problem is that the friction coefficient varies from pipe to pipe, so the equivalent length will not have a unique value. When minor losses are truly minor, this problem becomes academic. However, when local losses are an important part of the analysis, it is recommended that the minor loss coefficients Kl be used rather than an equivalent length. The following summarizes the testing programs conducted to evaluate the loss coefficients of pipe fittings. The documents are only available from ASHRAE. • Reducing Ells, and Pipe Reducers (ASHRAE 2002 H-1405) Number: 250 Radius: short and long Sizes: 2 and 4 in. (5 and 10 cm) Materials: threaded iron and welded steel Manufacturers: 11 • Threaded and Forged Weld Pipe Tees (ASHRAE 2002 H-1404) Number: 142 Sizes: 2 and 4 in. (5 and 10 cm) Materials: threaded iron and welded steel Manufacturers: 11 • Large Pipe Ells, Reducers and Expansions (ASHRAE 2002 H-1672a) Number: 32 Radius: long Sizes: 12, 16, 20, and 24 in. (30.5, 40.6, 50.8, and 61 cm) Materials: welded steel Manufacturers: 4 • Large Pipe Tees (ASHRAE 2002 H-1672b) Number: 328 Sizes: 12 and 16 in. (30.5 and 40.6 cm) Materials: welded steel Manufacturers: 4 © 2005 by CRC Press LLC

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• Close Coupled Pipe Ells (ASHRAE 2002 H-1673) Sizes: 2 and 4 in. (5 and 10 cm) Radius: Short Spacing: 0, 1, 2, 3, 4, 5, 10, 20, and 50 pipe dia. Materials: threaded iron and welded steel • PVC Pipe Elbows, Reducers and Expansions, RP-1193 (ASHRAE 2003,TECH-00034-2003) Number: 170 Schedule: 80 Sizes: 2, 4, 6, and 8 in. (5.08, 10.16, 15.25, and 20.32 cm) Materials: injection molded and fabricated PVC Manufacturers: 7 • PVC Pipe Tees, RP-1193 (ASHRAE 2003 TECH-00035-2003) Number: 61 Schedule: 80 Sizes: 2, 4, 6, and 8 in. (5.08, 10.16, 15.25, 20.32 cm) Materials: injection molded and fabricated PVC Manufacturers: 7 Test results show that the loss coefficients for fittings vary with material; manufacturer; method of fabrication; accuracy of installation; and, for tees, with the percent flow distribution between the branches. The loss coefficient consistently reduced as the size of the fitting increased. Currently, such variations in minor loss coefficients are not accounted for in published minor loss data. The ASHRAE study also looked at the effect of closely spaced elbows in series (Rahmeyer, 2002e, Technical Paper H-1673). The testing included two sizes of elbows, four different alignments, and spacing from 0 to 20 pipe diameters. The results showed that the combined loss coefficient for the two elbows was never greater than the sum of the individual loss coefficients. At a spacing of 20 diameters, there was no effect. At 10 diameters, the combined loss coefficients were between 2 and 6% lower than the sum of the individual coefficients. At 3 diameters, the combined coefficient was 10 to15% lower. For closecoupled elbows, the combined loss coefficient was between 5 and 28% lower, depending on the orientation of the elbows. Comparing the magnitude of (Σ(fL/2gAp2) to Σ(Kl /2gAm2)) will determine how much care should be given to the selection of the Kl values. Typical values of Kl are listed in Table 3.4.1 with a range of values from the recent ASHRAE studies. For details on the loss coefficient for a specific fitting and for loss coefficients not found in the table, the reader is referred to Rahmeyer (2002a,b,c; 2003a,b) and Miller (1990).

Pipe Selection Materials commonly used for pressure pipe transporting liquids are ductile iron; concrete; steel; fiberglass; PVC; and polyolefin (polyethylene and polypropylene). For each of these pipe materials, national committees have developed specifications that discuss external loads; internal design pressure; available sizes; quality of materials; corrosive environments; installation practices; and linings. Standards are available from the following organizations: • • • • • • •

American Water Works Association (AWWA) American Society of Mechanical Engineers (ASME) American Society for Testing and Materials (ASTM) American National Standards Institute (ANSI) Canadian Standards Association (CSA) Federal Specifications (FED) Plastic Pipe Institute (PPI)

In addition, manuals and other standards have been published by various manufacturers and manufacturers’ associations. These specifications and standards should be used to guide the selection of pipe © 2005 by CRC Press LLC

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TABLE 3.4.1 Minor Loss Coefficients Item

Kl

Pipe inlets Projecting pipe Sharp corner-flush Slightly rounded Bell mouth Sudden expansionsa (based on inlet velocity, V1) Sudden contractionsb (based on outlet velocity, V2) A2/A1 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.5–0.9 0.50 0.04–0.5 0.03–0.1 (1-A1/A2)2 (1/Cc-1)2 0.8 0.9

Cc 0.624 0.632 0.643 0.659 0.681 0.712 0.755 0.813 0.892 Steel bell reducerse (welded, D1/ D2 = 1.2–1.33) 0.053–0.23 Steel bell expanderse (welded, D2/ D1 = 1.2–1.33) 0.02–0.11 PVC fabricated reducerse (D1/ D2 = 1.33–1.5) 0.12–0.68 PVC fabricated expanderse (D2/ D1 = 1.2–1.33) 0.07–1.19 ASHRAEe

Bends Short radius, r/d = 1 90 45 30 Long radius, r/d = 1.5 90 45 30 Mitered (one miter) 90 60 45 30 PVC injection molded elbowse PVC fabricated type I elbowse PVC fabricated type II elbowse Teese Valves

Typical Valuesc

0.33–0.74

0.3–0.6 0.10 0.06

0.089–0.26

0.07–0.33 0.09 0.06

1.10 0.40–0.59 0.35–0.44 0.11–0.19 0.68–1.00 0.40–0.42 0.73–0.76

Average Values

Range

d

Check valves Swing check Tilt disc Lift Double door Full open gate Full open butterfly Full open globe

1.0 1.2 4.6 1.32 0.15 0.40 4.0

0.29–2.2 0.27–2.62 0.85–9.1 1.0–1.8 0.1–0.3 0.2–0.6 3–10

Sources: Streeter, V.L., and Wylie, E.B. (1975). Fluid Mechanics, 6th ed., McGraw–Hill, New York, p. 304. b Streeter, V.L., and Wylie, E.B. (1975). Fluid Mechanics, 6th ed., McGraw–Hill, New York, p. 305. c Miller, D.S. (1990). Internal Flow Systems — Design and Performance Prediction, 2nd ed., Gulf Publishing Company, Houston, TX. d Kalsi Engineering and Tullis Engineering Consultants (1993). Application Guide for Check Valves in Nuclear Power Plants, Revision 1, NP-5479. Prepared for Nuclear Maintenance Applications Center, Charlotte, NC. e Rahmeyer, W. (2002a). Pressure loss coefficients of threaded and forged weld pipe fittings for ells, reducing ells, and pipe reducers. Technical paper H-1405, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. a

© 2005 by CRC Press LLC

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TABLE 3.4.1 (continued)

Minor Loss Coefficients

Rahmeyer, W. (2002b). Pressure loss coefficients of pipe fittings for threaded and forged weld pipe tees. Technical paper H-1404, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002c). Pressure loss data for large pipe ells, reducers and expansions. Technical paper H-1672a, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002d). Pressure loss data for large pipe tees. Technical paper H-1672b, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002e). Pressure loss coefficients for close coupled pipe ells. Technical paper H-1673, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2003a). Pressure loss data for PVC pipe elbows, reducers and expansions RP-1193. Technical paper TECH-00034-2003, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2003b). Pressure loss data for PVC pipe tees RP-119. Technical paper TECH-00035-2003, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA.

material. ASCE (1992) contains a description of most of these pipe materials and a list of the specifications for the various organizations, which apply to each material. The document also discusses the various pipe-linings available for corrosion protection. The following are relevant publications for selecting the proper type and pressure class of pipe: (ASCE 1992, 1993; AWWA 1995 (M9), 1989 (M11), 2003 (M41); PPI 1980 (M23). For air and low-pressure liquid applications, available pipe materials include unreinforced concrete; corrugated steel; smooth sheet metal; spiral rib (sheet metal); and HDPE (high-density polyethylene). The choice of a material for a given application depends on pipe size, pressure requirements, resistance to collapse from internal vacuums or external loads, resistance to internal and external corrosion, ease of handling and installing, useful life, and economics.

Valve Selection Valves serve a variety of functions; including, isolation; flow control; pressure regulation; preventing reverse flow; limiting maximum pressures; and releasing/admitting air. This section discusses characteristics of and principles for selecting and operating control valves, check valves, air valves, and pressure relief valves. For a description of various types of valves and details regarding their function, see Tullis 2005. Control Valves Valve selection and operation rarely receive the same detailed engineering attention given to other components of piping systems. Their function is often not demanding and detailed attention to valve selection may not be necessary. However, in some applications, the lack of attention to selecting the proper type and size valve can lead to excessive maintenance, poor performance, and other serious consequences. To find the right valve, the user often relies on the valve vendors for engineering advice, and rightly so, because they should be the most familiar with their product. However, when the system requirements are nonstandard, or severe, few vendors have enough engineering experience or performance data on their valves to enable them to provide reliable information. To minimize valve problems, one must identify the valves in the system that need special attention. For such valves, the engineer needs tools for making a correct analysis. The following subsections provide information for selecting and operating valves when special engineering is required to avoid problems. © 2005 by CRC Press LLC

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When analyzing a flow control valve, the following criteria should be considered: • The valve should not produce excessive pressure drop when full open. • The valve should control the flow rate over approximately 50% of its movement. • The maximum operating torque must not exceed the capacity of the operator or valve shaft and connections for any flow condition. • The cavitation intensity should be limited to the appropriate level. • The valve should be operated so that the resulting pressure transients do not exceed the safe limits of the system. Excessive Pressure Drop What constitutes excessive pressure drop for a control valve depends on the overall purpose of the piping system and on the magnitude of the other losses in the system. A valve considered to have an excessive loss in one system might be totally acceptable in another. To determine the acceptability of the full open pressure loss of the control valve, the magnitude of the loss valve coefficient (Kl/2gAm2) should be compared to the total system loss coefficient C (Equation 3.4.12). For example, the loss coefficient for a full-open globe valve can be 10 times higher than a full-open butterfly valve. However, if the loss coefficient for the globe valve is small compared to the total system loss coefficient, the globe valve would be an acceptable choice. Controllability For many flow control applications, it is desirable to select a valve that has linear control characteristics. Some valve vendors advertise that their valves are linear. The problem with such a claim is that the linearity of a control valve depends on the system in which it is installed. In a short pipe, a valve may linearly control the flow. However, the same valve installed in a long pipe will only control the flow near the closed position. The flow in a pipeline can be calculated using the energy equation (Equation 3.4.5) and the system loss equation (Equation 3.4.11). The result is: Q=

∆Z       ∑  fL  + ∑  K l   2 2   2 gDA   2 gAm   p   

(3.4.13)

The ability of the valve to control flow depends on the relative values of the valve’s loss coefficient (Kl /2gAm2) compared to the pipe’s friction coefficient (fL/2gdAp2). For a short system, a valve can almost linearly reduce the flow because the valve loss becomes large compared to the friction loss as the valve closes. In contrast, for a long pipeline Kl /2gAm2 will be small compared to fL/2gdAp2 until the valve is near its closed position. To demonstrate the relationship between a valve and system, consider a butterfly valve that will be used to control the flow between two reservoirs with an elevation difference of ∆Z. System A is a short pipe (0.3 m dia., 100 m long, ∆Z = 10 m, f = 0.0138) where pipe friction is small (fL/2gdAp2 = 46.9). System B is a long pipe (0.3 m dia., 10,000 m long, ∆Z = 200 m, f = 0.0138) with high friction (fL/2gdAp2 = 4690). Initially, assume that the same butterfly valve will be used in both pipes and it will be the same size as the pipe diameter. For the valve, assume that the Kl full open is 0.187 and at 50% open it is 8.4. Correspondingly, Kl /2gAm2 = 1.905 and 85.7. For system A, the flow with the valve full open will be 0.453 m3/sec and at 50% open 0.275 m3/sec, a reduction of 39%. Repeating these calculations over the full range of valve openings would show that the flow for system A reduces almost linearly as the valve closes. For system B, the flow with the valve full open will be 0.206 m3/sec and at 50% open 0.205 m3/sec, a reduction of less than 1%. The valve in system B will not start to control the flow until it has closed more than 50%. A line-size butterfly valve is obviously not a good choice for a control valve in system B. One solution to this problem is to use a smaller valve. If the butterfly valve installed in system B were half the pipe diameter, it would control the flow over most of its stroke. © 2005 by CRC Press LLC

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The range of opening over which the valve controls the flow also has a significant effect on the magnitude of the transient pressures caused by closing the valve. The valve closure time must be long enough to avoid excessive transient pressures. If the valve is oversized and does not start reducing the flow until it is more than 50% closed, over half of the closing time is wasted and the effective valve closure time is less than half the actual closing time. The solution to this problem is to reduce the valve size so it provides control over most of its movement. The size of the control valve, as well as its associated open/close time, can also have an impact on the overall system control logic. The closing time for the valve must be fast enough to match the changing flow requirements for the rest of the system without generating excessive transient pressures. For example, if the valve controls the water level in a storage tank and is actuated by a level sensor, the valve must close fast enough that the tank does not overfill. A case study of a pipeline rupture illustrates the need for selecting the proper size of control valve. The pipeline was 6 ft (2 m) in diameter, about 30 miles (51 km) long, and supplied water to two storage tanks. The pipes to the storage tanks each contained a 5-ft (1.52-m) diameter butterfly valve. The combined area of the two 5-ft (1.52-m) valves is equivalent to one 7-ft (2.13-m) valve. Because the valves were oversized, they only controlled flow over the last 25% of their movement. The pipeline ruptured due to a pressure transient that resulted from closing the valves too fast. If smaller control valves had been installed, the pipeline may not have ruptured during initial flow tests. Torque Selection of the correct operator for a specific quarter-turn valve and application requires knowing the maximum torque that the operator will be required to supply. This requires analyzing the system for the entire range of expected flow conditions so that the maximum flow torque can be identified. The maximum torque is also needed when selecting the valve to be sure that the valve shaft and connections are adequately designed. It is the responsibility of the valve manufacturer to provide the flow and torque characteristics and limits for their valve. It is the responsibility of the system engineers and/or operators to see that the valves are operated within these limits. The four primary sources of torques for quarter-turn valves are seating torque; bearing friction torque; packing friction torque; and hydrodynamic (flow) torque. These torque values are usually determined experimentally and should be available from the valve manufacturer. All four torques should be evaluated to determine the maximum torque for a given valve. Seating friction torque develops when the plug or disk moves in or out of the seat (sealing surface). For small valves with soft seats, the seating torque can be larger than the other three torques combined. Bearing friction torque develops when a load is placed on the bearing surface by the valve shaft when there is a pressure differential across the valve. Because the pressure drop significantly increases as a valve closes, the bearing torque is greatest at small openings. Packing friction torque is caused by the valve shaft rubbing against the packing material. Packing is the material placed between the valve shaft and valve body to prevent leakage. Packing friction torque can be particularly troublesome because the packing conditions can be modified in the field. If a packing leaks, the normal procedure is to tighten the packing until the leak stops. This can significantly increase the packing torque and may prevent the operator from opening the valve. When valve packing is adjusted, the ability of the actuator to close the valve under all operating conditions should be confirmed. Forces induced by the fluid flowing through the valve cause a hydrodynamic torque, which usually acts to close the valve. The magnitude of the torque varies with flow rate, pressure drop, and valve opening. The valve opening, where maximum hydrodynamic torque occurs, depends on the valve design and system characteristics. In short systems, in which friction loss is low and velocities are high, a quarterturn valve will see maximum torques at large openings where the flow rate is high. In long systems with high friction losses and lower velocities, the same valve will see maximum torque at smaller openings where the pressure drop is high. The hydrodynamic forces should be evaluated over the entire valve operating range and corresponding system flow conditions to identify the maximum operating torque. Operating torque is normally greater when the valve is being opened because the hydrodynamic torque usually acts to close © 2005 by CRC Press LLC

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the valve and the bearing and packing torques oppose the direction of valve motion. During closure, the bearing and packing torques act in the direction opposite to that of the hydrodynamic torque. Cavitation Cavitation is a process in which vapor bubbles form, grow, become unstable, and collapse. This causes excessive noise, vibrations, erosion damage; the valve can lose capacity if subjected to the most severe stage of cavitation. Evaluating the intensity of cavitation and limiting it to an acceptable level are important in control valve selection. The first step in a cavitation evaluation is to decide on the acceptable level of cavitation. Three cavitation limits typically used to quantify the intensity of cavitation for a control valve are: critical, incipient damage, and choking cavitation. The cavitation design limit appropriate for a given application varies with valve type, valve function, details of the piping layout and location, and the frequency and duration of operation. Critical cavitation is typically considered a conservative design limit. It corresponds to the onset of light but constant cavitation noise and is appropriate as a design limit for valves that need to operate essentially free of adverse cavitation effects. Incipient damage corresponds to onset of pitting (material removal). It is an appropriate design limit when significant noise and vibrations can be tolerated but no damage is desired. Choking cavitation (sometimes called flashing) represents the condition in which the mean pressure immediately downstream from the valve drops to vapor pressure and the flow rate is at its maximum for a given upstream pressure. Between critical and incipient damage, the noise level can become objectionable. Between incipient damage and choking cavitation, the erosion damage and vibration levels can be severe. Using choking as a design condition may be appropriate for a pressure relief valve, where valve operation is short lived and infrequent; maximum flow rate through the valve is required; and cavitation damage can be tolerated for short periods of time. It should not be used for valves intended for long-term, low-maintenance operation. One of the challenges with cavitation has been a lack of consistent terminology between industries. For example, some industries define the onset of cavitation as the condition in which the performance of the valve begins to drop off due to heavy cavitation. This is a correct definition for onset of choking cavitation, but it is misleading to imply that it represents onset of cavitation. Choking is, in fact, the final stage of cavitation — well beyond the point at which damage begins. Promoting this incorrect definition of “onset of cavitation” has resulted in many valves suffering extensive damage. A similar definition of onset of cavitation exists in the pump industry, where the cavitation index NPSHr is often assumed to be the point at which cavitation begins. In reality, it represents onset of choking cavitation. Pump cavitation is discussed in more detail later in the chapter. The intensity of cavitation and the corresponding noise, vibration, and erosion damage at the valve are at their maximum just before a valve chokes. If the valve operates at a flow condition beyond choking (sometimes referred to as supercavitation), the cavitation events create one large vapor cavity and the mean pressure in the pipe is essentially vapor pressure. The collapse of the large vapor cavity usually occurs at the first significant downstream flow disturbance — such as an elbow, tee, valve, or orifice — or when the frictional forces in the pipe are sufficient to generate the necessary pressure recovery. During supercavitation, damage may not occur inside the valve, but there will be serious vibration and material erosion problems farther downstream where the collapse occurs. The cavitation intensity for valves is quantified with a cavitation index σ, which represents the ratio of forces preventing cavitation (high local pressure) to forces causing cavitation (high pressure differential). σ=

(Pd + Pb − Pv ) (Pu − Pd)

(3.4.14)

Pu and Pd are the gage pressures upstream and downstream from the valve; Pb is the barometric pressure; and Pv is the absolute vapor pressure.

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The smaller the σ value is, the greater the chance that cavitation will occur. To evaluate the potential for cavitation of a valve or other minor loss element at a particular flow condition, the value of the system σ (Equation 3.4.14 evaluated at system pressure conditions) must be calculated and compared with σ values corresponding to the various cavitation intensity levels, i.e., critical, incipient damage, and choking. These cavitation intensity level values vary with valve type and valve opening and must be determined experimentally. Numerous valves have been tested to evaluate critical cavitation and onset of choking. Consequently, many valve manufacturers should be able to provide experimental data for these limits. Only limited data, however, are available identifying onset of cavitation damage because of the difficulty and laborintensive nature of laboratory damage testing. This is unfortunate because onset of damage is the most important cavitation limit. Experimental data for several types of valves are available in the literature (Tullis, 1989, 1993). Information from these sources can be used if information from the valve manufacturer is not available. It is important to note that most valves have unique cavitation characteristics and that valve-specific cavitation data should be used for design purposes. As an example, if an incipient damage level of cavitation intensity were selected as the limiting operating condition, the experimentally determined σ-incipient damage would be compared with the σ-system. If the σ-system is larger than σ-incipient damage, then the level of cavitation for that condition is acceptable. Because some level of uncertainty is associated with the experimentally determined cavitation intensity parameters, designers may wish to include a factor of safety when limiting the cavitation intensity of the valve or other minor loss device. Cavitation characteristics of a valve can be and typically are subject to size and pressure scale effects. A discussion on cavitation scale effects is presented by Tullis (1989, 1993). If the cavitation analysis indicates that the valve, orifice, or other device will be operating at a cavitation level greater than can be tolerated, various techniques can be used to limit the level of cavitation. One is to select a different type of valve. Recent developments have produced several new valves better suited for limiting the levels of cavitation or reducing the potential for cavitation damage. Some of these valves operate on the principle of dropping the pressure in stages. They have multiple paths with numerous sharp turns or sudden expansions in series and are sometimes referred to as “stack valves.” The number of restrictions depends on the total pressure drop required and the system pressure downstream from the last restriction. Because the pressure downstream from each restriction is progressively smaller, for cavitation purposes, each stage is designed so the pressure drop across each subsequent restriction is less than the preceding one (by approximately 50%). Consequently, the first stage (on the high-pressure inlet side) will be able to provide a pressure drop many times the pressure drop at the last stage, with both stages operating near the same level of cavitation intensity. As a result, these types of valves can operate at very large pressure differentials without experiencing cavitation damage. One limitation of stack valves is that they are only usable in clean systems. For untreated water systems, the stack valve is not appropriate because of plugging problems. As an alternative, conventional valves and orifices can be installed in series to reduce the cavitation in stages. Proper spacing of valves and orifices is important. The spacing between valves depends upon the type. For most quarter-turn valves, a spacing of five to eight pipe diameters is needed for pressure recovery and uniform flow re-establishment between each valve. This prevents disk flutter and ensures normal pressure drop characteristics at each valve. Globe valves, however, can be installed with no piping between them and have satisfactory operation. The primary disadvantages of locating conventional valves in series are the space requirement, increased complexity of the control logic, and cost. For details on this type of application, see Tullis (1989, 1993). Another recent advance in the fight against cavitation is the development of the sleeve valve. These valves use a perforated cylindrical sleeve to create multiple jets that discharge radially inward. An external sliding collar controls the number of holes exposed. These have excellent cavitation characteristics because they can operate at fairly heavy cavitation levels without damaging the valve or outlet pipe. The valve’s success is attributed to two things. Small holes discharging into a large chamber is the optimum configuration to suppress cavitation in a single-stage device. The other factor is that the jets from the multiple © 2005 by CRC Press LLC

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holes converge toward the center of the outlet sleeve, keeping the collapsing cavitation events away from the boundary until just before the valve chokes. These valves can operate at σ values below 0.2 without damage. Conventional valves typically experience cavitation damage at σ values greater than 1.0. The cavitation performance of conventional valves varies considerably. For example, skirted cone valves and solid ball valves generate significantly less cavitation than unskirted cone and nonsolid ball valves. This is because, for valves with a solid plug, all the flow goes through two ports in series (one at the inlet side and one at the outlet side of the plug). Similarly to stack valves, reducing the pressure in stages provides superior cavitation performance, compared to a single-stage control valve. The unskirted cone and nonsolid ball valves have two throttling ports in series, but they also allow flow to pass around the plug. The flow passage around the plug experiences only a single-stage pressure drop, resulting in increased cavitation potential. Skirted cone and solid ball valves also have better cavitation performance than butterfly, gate, and segmented ball valves because those valves only have a single stage of pressure drop. See Tullis (1989, 1993, 2003) for descriptions of the different types of valves and their cavitation characteristics. For some applications, cavitation can be avoided by using a free discharge valve. Fixed cone valves are specifically designed for this application. However, these are specially designed valves and are relatively expensive. Some conventional valves can also be used for free discharge, if they can be adequately vented. Injecting air to suppress cavitation is a technique that has been used for many years with varying degrees of success to suppress cavitation damage. If an adequate amount of air is injected into the proper region, noise, vibrations, and erosion damage can be significantly reduced. For systems that can tolerate some air injection, aeration is often the cheapest and most effective remedy for cavitation. If all else fails, cavitation damage can be suppressed by plating critical areas of the pipe and valve with cavitation-resistant materials. The final option is simply to replace damaged valves as needed. For additional information on valve cavitation, see AWWA (2001, M49); Knapp et al. (1970); ISA (2000); and Tullis (1989, 1993). Transients Hydraulic transients refer to the dynamic pressure waves that result from rapid acceleration or deceleration of flow and are common in pipeline operation. They can be generated as a result of pipeline filling and air-flushing techniques; valve operation techniques, pump startup and shutdown, and pipe ruptures. Transient pressures can exceed the safe operating limit of the pipe if adequate design provisions and operational procedures are not established. The larger the incremental velocity change is and the faster the changes occur, the larger the resulting pressure change will be. Transients caused by slow velocity changes, such as the rise and fall of the water level in a tank, are called surges. Surge analysis, or “rigid column theory,” involves the numerical solution of a simple ordinary differential equation (force = mass × acceleration). The compressibility of the fluid and the elasticity of the conduit are ignored and the entire column of fluid is assumed to move as a rigid body. When changes in velocity occur rapidly, the compressibility of the liquid and the elasticity of the pipe become important and must be included in the analysis. This procedure is often called “elastic” or “water hammer” analysis and involves tracking acoustic pressure waves through the pipe. The analysis requires solving partial differential equations. An equation predicting the head rise ∆H caused by a sudden change of velocity ∆V = V2 – V1 can be derived by applying the unsteady momentum equation to a control volume of a section of the pipe at which the change of flow occurred. Consider a partial valve closure, which instantly reduces the velocity by an amount ∆V. Reduction of the velocity can only be accomplished by an increase in the pressure upstream of the valve of magnitude ∆H. The pressure wave of magnitude ∆H travels in the upstream direction of the pipe at the acoustic velocity, a, which is a function of the fluid and pipe material properties. The increased pressure compresses the liquid and slightly expands the pipe. The transient head rise due to an incremental change in velocity is described as ∆H = –a ∆V/g, for a >> ∆V © 2005 by CRC Press LLC

(3.4.15)

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This equation can be used to account for the accumulative effect of multiple incremental changes of velocity until the initial pressure wave returns to the point of origin. The wave speed depends on the pipe material, wall thickness, diameter, type of liquid, and the amount of air in the system. For a steel pipe with no entrained or trapped air, the wave speed is typically about 3000 fps (1000 m/sec). Based on this wave speed, Equation (3.4.15) predicts that an instant velocity change of 3 fps (1 m/sec) causes a transient pressure of about 300 ft (100 m). This illustrates why transient pressures can easily damage pipes. The derivation of Equation (3.4.15) was based on an assumption of an instantaneous incremental velocity change or an instant valve closure. Instant closure actually refers to a finite time. It is the longest time over which a valve can be closed and still cause a pressure rise equal to that of an instant closure. It is the time required for the initial transient pressure wave to travel from the point of origin to the next upstream (or downstream) boundary condition and return to the point of origin. This wave travel time equals 2L/a seconds, where L is the pipe length between the point of origin and the adjacent boundary condition and a is the acoustic wave speed. The maximum transient pressure rise associated with an instant valve closer will occur at the valve if the closure time is less than or equal to 2L/a seconds. Computational techniques for estimating transient pressures are too complex to be done with simple hand calculations. The solution involves solving partial differential equations derived from the equations of motion and continuity. These partial differential equations are normally solved by the method of characteristics — a technique that transforms the equations into ordinary differential equations. After integration, the equations can be solved numerically by finite differences (Tullis, 1989; Wiley and Streeter, 1993). To complete the analysis, equations describing the boundary conditions are required. Typical boundary conditions are the connection of a pipe to a reservoir, a valve, changes in pipe diameter or material, pipe junctions, etc. Friction loss is included in the development of the basic equations and minor losses are handled as boundary conditions. Every pipe system should have at least a cursory transient analysis performed to identify the possibility of serious transients and decide whether a detailed analysis is necessary. If an analysis indicates that transients are a problem, methods of controlling them include: • • • • •

Increasing the closing time of the control valve Using a smaller valve to provide better control Designing special facilities for filling, flushing, and removing air from pipelines Increasing the pressure class of the pipeline Using pressure relief valves, surge tanks, air chambers, etc.

Restricted Valve Opening Many conventional control valves cannot safely and/or accurately regulate flow near the closed and fullopen positions. Near the closed position, two of the potential problems are seat damage due to high velocities and inability to set the valve opening accurately when the connections between the control element and the operator are loose. Near full open, some valves lose control, meaning that flow does not change as the valve position is changed. For globe style valves, this occurs when the stroke is too long. For a butterfly valve, it is due to the shape of the disk and changes in the flow pattern around the disk at large openings. For some disk shapes, the flow can actually decrease at valve openings greater than about 90%. This problem is magnified when the valve is installed in a long system in which the valve loss is small compared to the system friction loss (see the section on “Controllability”). Another reason that some quarter-turn valves should not be operated at large openings is that some valves experience torque reversals, which result in disk flutter that can lead to fatigue of the shaft and/or connections. Torque reversals are a function of the plug or disk design and usually only occur for flow in one direction. Valve vendors usually indicate a preferred flow direction for valve installation to avoid torque reversal problems. © 2005 by CRC Press LLC

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Check Valves Selecting the wrong type or size of check valve can result in poor performance, severe transients, and frequent repairs (Kalsi Engineering and Tullis Engineering Consultants, 1993). Proper check valve selection requires understanding the characteristics of the various types of check valves and analyzing how they will function as a part of the system in which they will be installed. For a description of the characteristics of common types of check valves, see Kalsi Engineering and Tullis Engineering Consultants (1993) and Tullis (2003). A check valve that operates satisfactorily in one system may be totally inadequate in another. Each type has unique characteristics that give it advantages or disadvantages compared with the others. The characteristics of check valves that should be considered in the selection process include: • • • • •

Closure speed of check valves relative to the rate of flow reversal of the system Stability of the disk and its sensitivity to upstream disturbances The flow required to fully open and firmly backseat the disc The pressure drop at maximum flow Sealing effectiveness and ease of maintenance

Disk stability varies with flow rate, disk position, and upstream disturbances and is an important factor in determining the useful life of a check valve. For most applications, it is preferable to size the check valve (especially swing check valve) so that the disk is fully open and firmly back-seated at normal flow rates. It is a mistake to oversize a swing check valve that is located just downstream from a disturbance such as a pump, elbow, or control valve. The disk will not firmly back seat and it will be subjected to severe motion and accelerate wear. Reducing the valve size reduces this problem. The transient pressure rise generated at check valve closure is another important consideration. The pressure rise is a function of how fast the check valve closes compared to the speed of flow reversal (Thorley, 1989). Systems in which rapid flow reversals occur include parallel pumps, where one pump is stopped while the others are still operating; systems that have air chambers or surge tanks close to the check valve; and short systems with high elevation heads. For these systems, a high-energy source downstream from the check valve causes the flow to reverse quickly. As the disk nears its seat, it starts to restrict the reverse flow. This builds the pressure, accelerates the disk, and slams it into the seat. Results of laboratory experiments, field tests, and computer simulations show that dramatic reductions in the transient pressures can be achieved by replacing a slow-closing swing check valve with a fast-acting check valve. For example, in a system containing parallel pumps in which the transient was generated by stopping one of the pumps, the peak transient pressure was reduced from 745 to 76 kPa when the swing check was replaced with a nozzle check valve. The closing speed of a valve is determined by the mass of the disk, the forces closing the disk, and the travel distance from fully open to fully closed. Fast-closing valves have the following properties: the disk (including all moving parts) is lightweight, closure is assisted by springs, and the full stroke of the disk is short. Swing check valves are the slowest closing valves because they violate all three of these criteria: they have heavy disks, no springs, and long disk travel. The nozzle check valve is one of the fastest closing valves because the closing element is light, spring loaded, and has a short stroke. The silent, duo, double door, and lift check valves with springs are similar to nozzle valves in their closing times, mainly because of the closing force of the spring. Air Valves There are three types of automatic air valves: • Air/vacuum valves are designed for releasing large quantities of air while the pipe is being filled and for admitting air when the pipe is being drained. Air/vacuum valves typically contain a float, which rises and closes the orifice as the valve body fills with water. Once the line is pressurized, the float cannot reopen to remove air that may subsequently accumulate. If the pressure becomes negative during a transient or while draining, the float drops and admits air into the line. At least

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one of the air/vacuum valves should be sized for maximum flow from a full pipe break. It must be large enough for its air flow rate under vacuum conditions to equal the maximum drainage rate of the water from a pipe break and at an internal pipe pressure above the pipe collapse pressure. When sized for vacuum service, the air/vacuum valves will actually be oversized for filling the pipe. However, this does not cause any problems because the filling rate must be controlled by the water inflow rate and not by the discharge capacity of the air/vacuum valves. • Air release valves contain a small orifice and are designed to release small quantities of pressurized air not released by the air/vacuum valves and small quantities of air that accumulate after initial filling and pressurization. The small orifice is controlled by a plunger activated by a float at the end of a lever arm. As air accumulates in the valve body, the float drops and opens the orifice. As the air is expelled, the float rises and closes off the orifice. • Combination valves are made up of two valves, a large valve that functions as an air/vacuum valve and a small one that functions as an air release valve. The installation can consist of an air/vacuum valve and an air release valve plumbed in parallel, or the two can be housed in a single valve body. Most air valve installations require combination valves. Guidelines for sizing air valves are available from valve manufacturers and AWWA (2001, M51). The use of large manual air release valves should be avoided because they can cause severe transients. If the pipeline is filled with the manual air valves closed, the trapped air will be pressurized to full system pressure. When the manual air valve is manually opened, the pressurized air escapes at sonic velocity, causing rapid acceleration of the liquid toward the air valve. When the last of the air is released and the water hits the air valve, the velocity of the water is suddenly reduced and high transient pressures can be generated. If manual air valves are installed, they should be small so that the air release rate is properly controlled. Air valves should be placed at prominent high points or at intervals if there are few or no high points. The pipe profile needs to be studied relative to possible pipe ruptures so that the largest air/vacuum valves can be located at the high points where they can protect against pipe collapse. At other high points, smaller air/vacuum valves can be installed. Velocity of the flow during filling is important. A safe way to fill a pipe is to limit the fill rate to an average flow velocity of about 0.3 m/sec (1 ft/sec) until the air/vacuum valves close. Once they are closed, the fill rate can be increased to about 1 m/sec (3 ft/sec), keeping the system pressure as low as possible. This will move the remaining air to the air release valve. When possible, the system should not be pressurized until all of the air has been removed. Trapped air at high pressure can generate severe transients if the air is allowed to pass through a control valve or manual air release valve. Relief Valves Pressure relief valves (PRVs) are installed to limit maximum system pressures. They can automatically open when the system pressure exceeds a set pressure or they can be programmed to open in anticipation of a transient or surge. The two general types of relief valves are: nonpilot activated and pilot activated. The selection of which type is appropriate depends on the size of valve required and if the opening and/or closing rate needs to be controlled. The characteristics of nonpilot-activated valves are: • • • • • •

They are only available in small sizes. The valve is held closed with a spring. The valve opens when line pressure exceeds the spring setting. They are fast acting but the speed of opening and closing is not controlled. They automatically close when the pressure drops below the spring setting. Only small changes can be made in the pressure setting by adjusting the compression of the spring.

Pilot-activated relief valves are opened and closed by system pressure and a restoring spring. The characteristics of pilot-activated valves are:

© 2005 by CRC Press LLC

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• • • • • • • •

The main valve is generally a globe-style valve that is controlled by a small pilot valve. There is no size limitation. The valves are held closed by a spring and fluid pressure. The pilot valve is a miniature PRV and opens the main PRV by bleeding fluid from the hydraulic pressure chamber of the main valve. Opening and closing speeds are controlled by throttling valves in the pilot system. The range of pressures can be adjusted by changing the pilot PRV. They are generally slow acting, so they do not protect against rapid pressure transients. They can be programmed to be “surge anticipators.”

The pressure setting of the PRV can be adjusted over a small range by changing the compression of the spring in the pilot valve. However, it is important that the spring not be overcompressed. It is possible to compress the pilot spring completely if the adjusting rod is screwed in all the way. If this is done, the pilot PRV will remain closed and the main PRV cannot open. It was this type of improper adjustment that caused the failure of a major pipeline. Large pressure range changes require changing the spring and piston size of the pilot valve. For added safety, multiple PRVs should be installed. Typical options include two identical valves, each with 100% capacity, or three valves, each with 50% capacity. PRVs need to be serviced periodically to ensure that they are functional. Foreign matter in the liquid can plug the pilot valves and render them inoperable.

Centrifugal Pump Selection and Performance Optimizing the life of a piping system requires proper selection, operation, and maintenance of the pumps. During the selection process, the designer must be concerned about matching the pump performance to the system requirements and anticipate problems that will be encountered when the pumps are started or stopped and when the pipe is filled and drained. The design should also consider the effect of variations in present and future flow demands. Selection of the pumps should not be based on the least initial cost but rather on the least total cost, considering pump performance and reliability. This subsection deals with pump selection and selected operational problems. Single Pump Selection Selecting a pump for a particular service requires matching the system requirements to the capabilities of the pump. The process consists of developing a system equation by applying the energy equation to evaluate the pumping head required to overcome the elevation difference between reservoirs friction, and minor losses. For a pump supplying water between two reservoirs or tanks, the pump head required to produce a given discharge can be expressed as Hp = ∆Z + Hl or Hp = ∆Z + CQ2

(3.4.16)

in which Q is flow rate, ∆Z is the downstream reservoir elevation minus the upstream reservoir elevation, and the constant C is defined by Equation (3.4.12). The total dynamic head of a pump is defined by: Hp =

(V2 2 − V12 ) (P2 − P1) + + Z 2 − Z1 + H l 2g γ

(3.4.17)

Figure 3.4.2 shows a system curve for a pipe having an elevation lift of 82 ft (25 m) and moderate friction losses. When the elevations of the upstream and/or downstream reservoirs are variable, a family of system curves can be developed using different values of ∆Z. The three pump curves shown in Figure 3.4.2 represent different impeller diameters. The intersections of the system curve with the pump curves identify the flow rate that each impeller would supply if © 2005 by CRC Press LLC

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FIGURE 3.4.2 Pump selection for a single pump.

installed in that system. For this system, A and B impellers would be a good choice because they operate at or near their best efficiency range. Figure 3.4.2 shows the head and flow produced by the B pump when operating in that system are 97 ft (30 m) and 450 gpm (28.4 l/s), respectively. The net positive suction head (NPSH) and brake horsepower (bhp) are obtained as shown in the figure. Multiple Pump Selection The selection process is more complex when the system demand varies due to variations in reservoir elevation or to changing flow requirements. If the system must operate over a range of reservoir elevations, the pump should be selected so that the system curve, based on the mean water level (or that most frequently encountered), intersects the pump curve to the right of the midpoint of the best efficiency range. If the water level variation is not too great, the pump may not be able to operate efficiently over the complete flow range. If the flow range is large, multiple pumps or variable-speed drives may be needed. Selection of multiple pumps and the decision about installing them in parallel or in series depend on the amount of friction in the system. Parallel installations are most effective for low-friction systems. Series pumps work best in high-friction systems. For parallel constant speed pump operation, the combined two-pump curve is constructed by adding the flow of each pump. Such a curve is shown in Figure 3.4.3 (labeled 2 pumps). The intersection of the two-pump curve with the system curve identifies the head and combined flow for the two pumps. The pump efficiency for each pump is determined by projecting horizontally to the left to intersect the singlepump curve. For this example, a C pump, when operating by itself, will have an efficiency of 83%. With two C pumps operating, the efficiency of each will be about 72%. For the pumps to operate most efficiently for a one- or two-pump operation, the pump curve should intersect the system curve to the right of its best efficiency point. Variable-Speed Pumps Recent improvements in reliability and reductions in cost have made variable-speed pumps a viable option for installations in which a wide range of flow is required. Several variable speed technologies are available. One that is increasing in popularity, especially in the water and wastewater industries, is the variable-frequency drive (VFD). These are electronic controllers that adjust the speed of the electric pump motor by modulating the power delivered. VFD units can be retrofitted to existing pumps, as well as being supplied with new units. © 2005 by CRC Press LLC

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FIGURE 3.4.3 Selection of parallel pumps.

For a specific centrifugal pump, head/capacity curves for different motor speeds and/or different diameter impellers (shaved impellers) for noncavitating conditions can be estimated using the following affinity laws (Karassik et al., 1976). (Note: D represents impeller diameter, not pump diameter.)  N  D  Q2 = Q1  2   2   N1   D1 

(3.4.18)

2

N  D  H 2 = H1  2   2   N1   D1  3

N  D  P2 = P1  2   2   N1   D1 

2

(3.4.19)

3

(3.4.20)

To approximate pump curves for a family of geometrically similar pumps for noncavitating conditions, the following set of affinity equations can be used:  N  D  Q2 = Q1  2   2   N1   D1 

3

(3.4.21)

2

N  D  H 2 = H1  2   2   N1   D1  3

N  D  P2 = P1  2   2   N1   D1  © 2005 by CRC Press LLC

2

(3.4.22)

5

(3.4.23)

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The D ratios in Equation (3.4.21) through Equation (3.4.23) can be the pump or the impeller diameter. This is because the equations are only valid for geometrically similar pumps where the ratio of pump diameter to impeller diameter is constant. Either set of equations can be used to compute changes due to motor speed variations. Figure 3.4.4 and Figure 3.4.5 show pump curves for different motor speeds and different systems for a pump with a constant impeller diameter but operating at different motor speeds. The curves for the different speeds are generated by scaling Q and H values from the 1770-rpm curve and using Equation (3.4.18) and Equation (3.4.19) or Equation (3.4.20) and Equation (3.4.22) to obtain the adjusted values of Q and H. The variable-speed drive can operate at any speed from almost zero to full motor speed (and even higher). Assume that a 10:1 flow range is desired. The speed range required by the pump to provide a given flow range is controlled by the system. Figure 3.4.4 shows a system with a significant elevation change and relatively low friction. The pump can provide a 10:1 flow range at motor speeds between 1770 and 1310 rpm. For the high-friction system with a small elevation change shown in Figure 3.4.5, the speed variation will be between 1770 and 800 rpm for a flow range of 10:1. Variable-speed drives offer several advantages. Single-speed drives start the motor abruptly, subjecting the rotating element to high torque and electrical current surges several times the full load current. For a constant speed deep well pump, rapid start of the motor can also cause transient and vibration problems related to compression and release of air trapped in the pump column between the water level and check valve. In contrast, variable-speed drives provide a “soft” start, gradually ramping up the motor to operating speed. This allows the air to be released slowly and reduces electrical and mechanical stress on the rotating elements, reduces maintenance, and extends motor and pump life. Because the speed can be controlled to a fraction of a percent variation, variable-speed units provide fine control of the system operating conditions. One of the primary justifications for using a variable speed drive is cost savings resulting from reduced power demands and reduced maintenance of the motor, pump, and discharge control valves. For constant-speed pumps, reduced flow is achieved by throttling the discharge valve. This can be compared to operating an automobile at full throttle and trying to control speed by braking. The pressure drop across the valve required to reduce the flow results in wasted energy and creates the possibility of cavitation. Cavitation problems are more problematic for high-friction systems with small elevation changes due to the lack of back pressure at small flows and high-pressure differentials across the valve. Figure 3.4.4 and Figure 3.4.5 demonstrate the difference in cavitation potential and energy wasted by the throttled discharge control valve in two systems. In the figures, the lines labeled “throttled” represent conditions in which the discharge control valve is throttled to provide a flow of 400 gpm (25.2 l/s) with the motor at full speed. The pump and system curves show that the head upstream from the valve will be 194 ft (59 m) and the back pressure provided by the system to the downstream side of the valve will be 108 ft (33 m). The head loss across the valve is therefore 194 – 108 = 86 ft (26.2 m). Using Equation (3.4.14) with PV = 32 ft (9.8 m) gives σ = 1.63. For the system shown in Figure 3.4.5, the head upstream from the valve is again 194 ft (59 m) and the back pressure provided by the system is 47 ft (14.3 m), so the head loss across the valve is 147 ft (44.8 m) and σ = 0.54. Because σ for the valve installed in the high-friction system (Figure 3.4.5) is onethird as much as the σ for the valve in the low-friction system, the valve in the high-head system will experience considerably more cavitation. For the high-friction system (Figure 3.4.5), a variable speed pump provides more power savings than the system with low friction. In the former, the variable-speed pump would operate at a head of 47 ft (14.3 m) (compared to 194 ft (59.1 m) for a constant-speed pump), so the power consumption at a flow rate of 400 gpm (25.2 l/s) would be only 24% of the power consumption of a constant-speed pump. Pump Operation Starting a constant-speed pump with the pipeline empty can result in filling at a rapid rate because there is little frictional resistance initially to build back pressure. As a result, the pump will operate at a flow © 2005 by CRC Press LLC

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190 Throttled

170

1770 rpm

Head, feet

150

1600

130 1400

System

110 13100

90 70 50 0

500

1000

1500

2000

1500

2000

Flow, gpm FIGURE 3.4.4 Variable-speed pump operating in a low-friction system.

200 180

Throttled

1770 rpm

160 1600

Head, feet

140 120

1400

System

100 1200

80 60 40

800

20 0 0

500

1000 Flow, gpm

FIGURE 3.4.5 Variable-speed pump operating in a high-friction system.

greater than its design flow. This may cause the pump to cavitate, but the more serious problem is the possibility of high pressures generated by the rapid filling of the pipe. Provisions should be made to control the rate of filling. This can be done by starting the pump against a partially open discharge control valve and bypassing some of the flow around the pump, which allows the system to be filled slowly and safely. If the pipe remains full after a pump is shut down and no air is trapped in the pipe, subsequent start-up of the pumps generally does not create serious transient problems. If the pump has a VFD drive, the pipeline can be filled slowly without throttling the discharge valve and using a bypass. © 2005 by CRC Press LLC

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For some systems, uncontrolled shutdown of the pumps due to power failures can generate high transient pressures. The problem is more severe for pipelines that have a large elevation change and multiple high points. The magnitude of the transient is related to the pipe length, the magnitude of the elevation change, the pump characteristics, and the type of check valve used. Computer simulations are needed to develop solutions to this transient problem. Numerous mechanical devices and techniques have been used to suppress pump shut-down transients. These include increasing the rotational inertia of the pump, using surge tanks or air chambers near the pump, pressure relief valves, vacuum-breaking valves, and surge-anticipating valves. Selection of the proper transient control device will improve reliability, extend the economic life of the system, and reduce maintenance. Pump Cavitation The cavitation potential of a pump is quantified by a term referred to as the net positive suction head (NPSH). The available NPSH (NPSHa) is the total head at the pump suction relative to the liquid vapor pressure. For a wet-pit pump installation, NPSHa can be calculated by: NPSHa = Hb – Hv + Zs – Hl

(3.4.24)

in which, Hb is the barometric pressure at the pump installation; Hv is the absolute vapor pressure head of the liquid; Zs is the elevation of the water level in the wet well; and Hl is the friction loss in the suction piping. Note that Zs is negative if the liquid level in the wet well is below the elevation of the pump. For dry-pit pumps, NPSHa can be calculated by: NPSHa = Hb + H v +

Ps

+

Vs2 2g

(3.4.25)

The NPSH required by the pump (NPSHr), as defined by ANSI/HI (1998), corresponds to the flow condition in which the pump efficiency is reduced by 3% due to heavy cavitation. NPSHr is evaluated experimentally by maintaining a constant flow and reducing the pressure in the suction pipe (or lowering the level in the wet well) until the pump’s efficiency is reduced by 3%. NPSHr varies with flow rate; for some pumps, NPSHr increases with flow rate and, for others, it decreases with flow rate. The intensity of cavitation generally increases when the pump is operated significantly above or below the best efficiency point even if the NPSHr reduces. It is important to understand the intensity of the cavitation when a pump is operating at or near NPSHr. It is common practice to assume that if NPSHa > NPSHr, no cavitation will occur. This is not true. In order for the pump’s efficiency to reduce by 3%, the cavitation must be heavy enough that part of the impeller is engulfed in a large vapor cavity. Research has shown that, for some pumps, cavitation begins when the NPSH is several times higher than NPSHr (Grist, 1974). When cavitation first begins, it is manifest by individual cavitation events that create a light crackling sound. The noise is not objectionable and the cavitation does not cause vibrations or damage to the pump. As NPSH is reduced, the noise and vibrations increase. At some point, the cavitation is heavy enough that it causes erosion damage to the impeller and pump casing. Once the cavitation is in the damaging region, it often sounds like gravel passing through the pump. As the suction pressure reduces and the system NPSH approaches NPSHr, the cavitation no longer consists of individual events. It has entered the most advanced stage in which so many cavitation events occur that they form into large vapor cavities that envelope part of the impeller. This condition is similar to the choked-flow or super-cavitating condition described in the section on control valves, in which a large vapor cavity extends into the downstream pipe. When choking cavitation occurs in a pump, the efficiency reduces because the pump is no longer pumping just a liquid but is pumping a mixture of liquid and vapor. Consequently, even if NPSHa > NPSHr, the pump may not be operating cavitation-damage free. Adding a factor of safety of at least 5 to 10 ft (1.5 to 3 m) to the published NPSHr values is recommended © 2005 by CRC Press LLC

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to reduce the risk of severe vibrations and cavitation damage. There is a relationship between vibrations and cavitation. First, the forces caused by implosion of the cavitation events induce vibrations. Second, once measurable erosion damage occurs on the impeller, it becomes unbalanced. This further increases the intensity of the vibrations and accelerates bearing wear. The cavitation damage is also progressive. Once material is removed from the leading edge or from the impeller surface, the resulting roughened areas become sources of additional cavitation. Suction Conditions Pump installations can be classified as booster, dry pit, or wet pit. Booster pump installations refer to systems in which no wet well is close to the pump and suction conditions are controlled by the suction piping. Dry pit refers to installations in which the liquid is supplied from a wet well or vessel and the pump is connected via a short length of pipe and fittings. Wet pit identifies installations in which the impeller or the entire pump is submerged. Such installations are also commonly referred to as wet wells, pump sumps, or pumping pits. Pump impellers are designed assuming that the approach flow will be relatively steady, uniform, and one-dimensional. Consequently, for acceptable pump operation, the flow needs to enter the pump free of undesirable flow conditions. Standards have been developed that identify the amount of circulation, degree of flow nonuniformity, unsteadiness of the flow, and strength of vortices acceptable for pump installations (IR, 1985; ANSI/HI, 1998). Circulation and nonuniform flow distribution can cause the flow to approach the impeller blades at the wrong velocity and/or the wrong angle, thus causing flow separation on the impeller blades. Organized vortices and high levels of turbulence cause the impeller blades to respond much like the wings of an aircraft flying through a storm. These undesirable flow conditions can result in vibrations, cavitation, accelerated wear, and loss of efficiency. Considerable care should be used to provide good suction flow conditions to the pumps. Hydraulic model studies are routinely performed for large wet-well pump installations to develop modifications to improve the flow approaching the pump (Sweeney and Rockwell, 1982; Tullis, 1979). These studies have shown that designing pumping pits based only on published guidelines does not guarantee acceptable suction conditions. Modifications to the pumping pits are usually required to provide acceptable flow conditions, even if the sump meets the ANSI/HI guidelines. On the other hand, field experience has shown that some pumps with suction conditions that do not meet the ANSI/HI standards still operate without problems. This seems to be a function of the quality of the pump. For example, the authors are aware of one installation in which two variable-speed dry-pit pumps (built by different manufacturers) were installed side by side and connected to the same wet well with similar piping. The larger of the two pumps had higher suction velocities but operated normally. The smaller pump experienced severe cavitation and vibrations at all motor speeds. The difference in performance was attributed to differences in the quality of the two pumps.

Other Considerations External Loads In some situations, the external load is the controlling factor determining whether the pipe will collapse. The magnitude of the external load and the resistance of the pipe to collapse depend on numerous factors, including: • • • • • • •

Pipe diameter Pipe material Pipe deformation (deviation from circular cross section) Trench width Depth of cover Specific weight of the soil Degree of soil saturation

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• • • •

3-63

Type of backfill material Method used to backfill Degree of compaction Live loads

The cumulative effect of all these sources of external loading requires considerable study and analysis, beyond the scope of this chapter. Because no simple guidelines for evaluating external pipe loads are available, the reader is referred to Watkins and Anderson (2000) and Spangler and Handy (1973) for details on how to perform calculations of earth loading. Limiting Velocities There are concerns about maximum and minimum velocity limits. If the velocity is too low, problems may develop due to settling of suspended solids and air being trapped at high points and along the crown of the pipe. The safe lower velocity limit to avoid collecting air and sediment depends on the amount and type of sediment and on the pipe diameter and pipe profile. Generally, velocities greater than about 1 m/sec (3 ft/sec) are sufficient to move trapped air to air release valves and keep the sediment in suspension. Problems associated with high velocities are: • • • • •

Erosion of the pipe wall or liner (especially if coarse, abrasive suspended sediment is present) Cavitation at control valves and other restrictions Increased pumping costs Removal of air at air release valves Increased operator size requirements and concern about valve shaft failures due to excessive flow torques • Increased risk of hydraulic transients

Each of these should be considered before making the final pipe diameter selection. A typical upper velocity for many applications is 20 ft/sec (6 m/sec). However, with proper pipe design and analysis, higher velocities can be tolerated.

Nomenclature m· ρ γ µ υ ∆H ∆V A a Am Ap C Cc D e F f g H Hb

Mass flow rate Density Specific weight Dynamic viscosity Kinematic viscosity Transient head rise Velocity change Area Wave speed Area at inlet to pipe fitting for minor losses Area of pipe Total system loss coefficient, (Equation 3.4.12) Contraction coefficient Diameter Pipe roughness Momentum force Friction factor Acceleration of gravity Pressure head Barometric pressure head

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Hf hf hl Hp Ht Hv k Kl L Lequ N NPSHa NPSHr P Pb Pd Pu Pv Q R r Re V Z Zs ∆Z σ

Chapter 3

Total head loss, friction and local Friction loss Minor loss Pump head Turbine head Absolute vapor pressure head of liquid Relative roughness, e/D Minor loss coefficient Pipe length Equivalent length for minor loss Motor speed Available net positive suction head Required net positive suction head Pressure Barometric pressure Pressure downstream from valve Pressure upstream from valve Absolute vapor pressure Flow rate Hydraulic radius, ratio of the flow area to the wetted perimeter Radius Reynolds number Velocity Elevation Elevation of liquid level in pump suction well (negative when below pump) Elevation change Cavitation index

References ANSI/HI 9.8-1998 (1998). Pump intake design. Hydraulic Institute and American National Standards Institute Inc., Parsippany, NJ. ASCE (1992). Pressure pipeline design for water and wastewater. Prepared by the Committee on Pipeline Planning of the Pipeline Division of the American Society of Civil Engineers. ASCE (1993). Steel penstocks. Prepared by the ASCE Task Committee on Manual of Practice for Steel Penstocks No. 79, Energy Division, American Society of Civil Engineers. AWWA (1989). Steel pipe — a guide for design and installation (M11), ED. 1989. American Water Works Association. AWWA (2003). Ductile iron pipe and fitting (M41), Second ed. 2003. American Water Works Association. AWWA (1995). Concrete pressure pipe (M9), 1995 ED. American Water Works Association. AWWA (2001). Butterfly valves: torque, head loss and cavitation analysis (M49). American Water Works Association. AWWA (2001). Air-release, air/vacuum and combination air valves (M51). American Water Works Association. Grist, E. (1974). Net positive suction head requirements for avoidance of unacceptable cavitation erosion in centrifugal pumps. Proceedings of the Conference on Cavitation, Edinburgh, Scotland, Sept. 3–5, 1974, Institute of Mechanical Engineers, London. Haaland, S. (1983). Simple and explicit formulas for the friction factor in turbulent pipe flow. ASME J. Fluids Eng., 105, March 1983, 89–90.

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IR (1985). Test standards for pump intake models. Engineered Pump Division, Ingersoll–Rand Co., Atlanta GA. ISA-RP75.23 (2000). Considerations for evaluating control valve cavitation. Instrument Society of America, Research Triangle Park, NC. Kalsi Engineering and Tullis Engineering Consultants (1993). Application Guide for Check Valves in Nuclear Power Plants, Revision 1, NP-5479. Prepared for Nuclear Maintenance Applications Center, Charlotte, NC. Karassik, I.J., Krutzsch, W.C., and Fraser, W.H. (1976), Pump Handbook. McGraw–Hill, New York. Knapp, R.T., Daily, J.W., and Hammitt, F.G. (1970). Cavitation. McGraw–Hill, New York. Miller, D.S. (1990). Internal Flow Systems — Design and Performance Prediction, 2nd ed. Gulf Publishing Company, Houston, TX. Morrison, E.B. (1969). Monograph for the design of thrust blocks. Civil Eng., 39, June, 50–51, American Society of Civil Engineers. PPI (1980). PVC pipe design and installation (M23). Plastics Pipe Institute, Inc. Rahmeyer, W. (2002a). Pressure loss coefficients of threaded and forged weld pipe fittings for ells, reducing ells, and pipe reducers. Technical paper H-1405, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002b). Pressure loss coefficients of pipe fittings for threaded and forged weld pipe tees. Technical paper H-1404, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002c). Pressure loss data for large pipe ells, reducers and expansions. Technical paper H-1672a, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002d). Pressure loss data for large pipe tees. Technical paper H-1672b, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2002e). Pressure loss coefficients for close coupled pipe ells. Technical paper H-1673, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2003a). Pressure loss data for PVC pipe elbows, reducers and expansions, RP-1193. Technical paper TECH-00034-2003, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Rahmeyer, W. (2003b). Pressure loss data for PVC pipe tees, RP-119. Technical paper TECH-00035-2003, American Society of Heating, Refrigeration and Air Conditioning Engineering, Atlanta, GA. Spangler, M.G. and Handy, R.L. (1973). Soil Engineering, 3rd ed. Intext Educational Publishers, New York, Chaps. 25 and 26. Streeter, V.L. and Wylie, E.B. (1975). Fluid Mechanics, 6th ed., McGraw–Hill, New York. Sweeney, C.E. and Rockwell, G.E. (1982). Pump sump design acceptance through hydraulic model testing. Proc. IAHR Symp. Operating Probl. Pump Stations Power Plants, Amsterdam, September 1982. Thorley, A.R.D. (1989). Check Valve Behavior under Transient Flow Conditions: A State-of-the-Art Review. vol. 111. ASME, June. Tullis, J.P. (1979). Modeling in design of pumping pits. J. Hydraulics Division, ASCE, 105(HY9), Proc. paper 14812, September, 1979, 1053–1063. Tullis, J.P. (1989). Hydraulics of Pipelines – Pumps, Valves, Cavitation, Transients. John Wiley & Sons, New York. Tullis, J.P. (1993). Cavitation guide for control valves. NUREG/CR-6031, U.S. Nuclear Regulatory Commission, Washington, D.C. Tullis, J.P. (2005). Valves, in The Engineering Handbook, 2nd ed., Dorf, R.C., Ed. CRC Press, Boca Raton, FL. Watkins, R.K. and Anderson, L.R. (2000). Structural Mechanics of Buried Pipes. CRC Press, Boca Raton, FL. Wylie, E.B. and Streeter, V.L. (1993). Fluid Transients in Systems. Prentice Hall, Englewood Cliffs, NJ.

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Chapter 3

FIGURE 3.5.1 Definition sketch for an open channel.

3.5 Open Channel Flow Frank M. White Definition The term open channel flow denotes the gravity-driven flow of a liquid with a free surface. Technically, we may study any flowing liquid and any gas interface. In practice, the vast majority of open channel flows concern water flowing beneath atmospheric air in artificial or natural channels. The geometry of an arbitrary channel is shown in Figure 3.5.1. The area A is for the water cross section only, and b is its top width. The wetted perimeter P covers only the bottom and sides, as shown, not the surface (whose air resistance is neglected). The water depth at any location is y, and the channel slope is θ, often denoted as So = sin θ. All of these parameters may vary with distance x along the channel. In unsteady flow (not discussed here) they may also vary with time.

Uniform Flow A simple reference condition, called uniform flow, occurs in a long straight prismatic channel of constant slope So . There is no acceleration, and the water flows at constant depth with fluid weight exactly balancing the wetted wall shear force: ρgLA sin θ = τw PL, where L is the channel length. Thus, τw = ρgRhSo, where Rh = A/P is called the hydraulic radius of the channel. If we relate wall shear stress to the Darcy friction factor f, τw = (f/8)ρV2, we obtain the basic uniform flow open channel relation: Uniform flow: V =

8g Rh So , where f

8g = C = Chézy coefficient f

(3.5.1)

Antoine Chézy first derived this formula in 1769. It is satisfactory to base f upon the pipe-flow Moody diagram (Figure 3.4.1) using the hydraulic diameter, Dh = 4Rh, as a length scale. That is, f = fcn (VDh /ν, ε/Dh) from the Moody chart. In ordinary practice, however, engineers assume fully rough, high-Reynoldsnumber flow and use Robert Manning’s century-old correlation: C≈

12 ζ 16 ζ R , or Vuniform ≈ Rh2 3 So and Q = VA n h n

(3.5.2)

where ζ is a conversion factor equal to 1.0 in SI units and 1.486 in English units. The quantity n is Manning’s roughness parameter, with typical values, along with the associated roughness heights ε, listed in Table 3.5.1. © 2005 by CRC Press LLC

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TABLE 3.5.1 Average Roughness Parameters for Various Channel Surfaces Average Roughness Height ε

Artificial lined channels Glass Brass Steel; smooth Painted Riveted Cast iron Cement; finished Unfinished Planed wood Clay tile Brickwork Asphalt Corrugated metal Rubble masonry Excavated earth channels Clean Gravelly Weedy Stony; cobbles Natural channels Clean and straight Sluggish, deep pools Major rivers Floodplains Pasture, farmland Light brush Heavy brush Trees

n

ft

mm

0.010 ± 0.002 0.011 ± 0.002 0.012 ± 0.002 0.014 ± 0.003 0.015 ± 0.002 0.013 ± 0.003 0.012 ± 0.002 0.014 ± 0.002 0.012 ± 0.002 0.014 ± 0.003 0.015 ± 0.002 0.016 ± 0.003 0.022 ± 0.005 0.025 ± 0.005

0.0011 0.0019 0.0032 0.0080 0.012 0.0051 0.0032 0.0080 0.0032 0.0080 0.012 0.018 0.12 0.26

0.022 ± 0.004 0.025 ± 0.005 0.030 ± 0.005 0.035 ± 0.010

0.12 0.26 0.8 1.5

37 80 240 500

0.030 ± 0.005 0.040 ± 0.010 0.035 ± 0.010

0.8 3 1.5

240 900 500

0.035 ± 0.010 0.05 ± 0.02 0.075 ± 0.025 0.15 ± 0.05

1.5 6 15

500 2000 5000

0.3 0.6 1.0 2.4 3.7 1.6 1.0 2.4 1.0 2.4 3.7 5.4 37 80

?

?

Critical Flow Since the surface is always atmospheric, pressure head is not important in open channel flows. Total energy E relates only to velocity and elevation: Specific energy E = y +

V2 Q2 = y+ 2g 2 gA 2

At a given volume flow rate Q, the energy passes through a minimum at a condition called critical flow, where dE/dy = 0, or dA/dy = b = gA3/Q2:  bQ 2  Acrit =    g 

13

Vcrit =

Q  gA  =  crit  Acrit  b 

12

(3.5.3)

where b is the top-surface width as in Figure 3.5.1. The velocity Vcrit equals the speed of propagation of a surface wave along the channel. Thus, we may define the Froude number Fr of a channel flow, for any cross section, as Fr = V/Vcrit. The three regimes of channel flow are Fr < 1: subcritical flow;

Fr = 1: critical flow;

Fr > 1: supercritical flow

There are many similarities between Froude number in channel flow and Mach number in variable-area duct flow (see Section 3.6). © 2005 by CRC Press LLC

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Chapter 3

For a rectangular duct, A = by, we obtain the simplified formulas Vcrit = gy

Fr =

V gy

(3.5.4)

independent of the width of the channel. Example 3.5.1 Water (ρ = 998 kg/m3, µ = 0.001 kg/m · sec) flows uniformly down a half-full brick 1-m-diameter circular channel sloping at 1°. Estimate (a) Q; and (b) the Froude number. Solution 3.5.1 (a). First compute the geometric properties of a half-full circular channel: A=

π (1 m)2 = 0.393 m 2 ; 8

P=

π (1 m) = 1.57 m; 2

A 0.393 = = 0.25 m P 1.57

R=

From Table 3.5.1, for brickwork, n ≈ 0.015. Then, Manning’s formula, Equation (3.5.2) predicts V=

1.0 ζ 16 12 m Rh So = (0.25)1 6 (sin1°)1 2 ≈ 3.49 ; n 0.015 sec

Q = 3.49(0.393) ≈ 1.37

m3 Solution 3.5.1(a) sec

The uncertainty in this result is about ±10%. The flow rate is quite large (21,800 gal/min) because 1°, although seemingly small, is a substantial slope for a water channel. One can also use the Moody chart. with V ≈ 3.49 m/sec, compute Re = ρVDh/µ ≈ 3.49 E6 and ε/Dh ≈ 0.0037, then compute f ≈ 0.0278 from the Moody chart. Equation (3.5.1) then predicts 8g RS = f h o

V=

8(9.81) m (0.25) (sin1°) ≈ 3.51 ; 0.0278 sec

Q = VA ≈ 1.38

m3 sec

Solution 3.5.1 (b). With Q known from part (a), compute the critical conditions from Equation (3.5.3):  bQ 2  Acrit =    g 

13

1.0(1.37) 2  =   9.81 

13

= 0.576 m 2 ,

Vcrit =

m 1.37 Q = = 2.38 sec Acrit 0.576

Hence Fr =

V 3.49 = ≈ 1.47 (supercritical) Solution 3.5.1(b) Vcrit 2.38

Again the uncertainty is approximately ±10%, primarily because of the need to estimate the brick roughness.

Hydraulic Jump In gas dynamics (Section 3.6), a supersonic gas flow may pass through a thin normal shock and exit as a subsonic flow at higher pressure and temperature. By analogy, a supercritical open channel flow may pass through a hydraulic jump and exit as a subcritical flow at greater depth, as in Figure 3.5.2. Application of continuity and momentum to a jump in a rectangular channel yields V2 = V1 © 2005 by CRC Press LLC

y1 y2

y2 =

[

y1 −1 + 1 + 8 Fr12 2

] where Fr = 1

V1 >1 gy1

(3.5.5)

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FIGURE 3.5.2 A two-dimensional hydraulic jump.

FIGURE 3.5.3 Geometry and notation for flow over a weir.

Both the normal shock and the hydraulic jump are dissipative processes: the entropy increases and the effective energy decreases. For a rectangular jump,

∆E = E1 − E2

( y − y1 ) = 2 4 y1 y2

3

>0

(3.5.6)

For strong jumps, this loss in energy can be up to 85% of E1. The second law of thermodynamics requires ∆E > 0 and y2 > y1 or, equivalently, Fr1 > 1, Note from Figure 3.5.2 that a hydraulic jump is not thin. Its total length is approximately four times the downstream depth. Jumps also occur in nonrectangular channels, and the theory is much more algebraically laborious.

Weirs If an open channel flow encounters a significant obstruction, it will undergo rapidly varied changes which are difficult to model analytically but can be correlated with experiment. An example is the weir in Figure 3.5.3 (colloquially called a dam), which forces the flow to deflect over the top. If L Sc

yn

S−1

y(0)

Fr > 1

Fr > 1

S−2 S−3

(a)

yn = y c

C−1

Fr < 1 Critical S0 = Sc

Fr > 1

C−3

(b)

yn

Mild S0 < Sc

yc

Fr < 1

M−1

Fr < 1

M−2 Fr > 1

M−3

(c)

yc Horizontal S0 = 0 yn = ∞

Fr < 1

H−2

Fr > 1

H−3

(d)

A−2 Fr < 1

A−3

yc Fr > 1 Adverse S0 < 0 yn = imaginary (e)

FIGURE 3.5.4 Classification of solution curves for gradually varied flow.

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Chapter 3

Example 3.5.2 Water, flowing at 2.5 m3/sec in a rectangular gravelly earth channel 2 m wide, encounters a broad-crested weir 1.5 m high. Using gradually varied theory, estimate the water depth profile back to 1 km upstream of the weir. The bottom slope is 0.10. Solution. We are given Q, Y = 1.5 m, and b = 2 m. We may calculate excess water level H at the weir (see Figure 3.5.3) from Equation (3.5.7) and Equation (3.5.8): Q = 2.5

m3 12 = Cd beff g1 2 H 3 2 = 0.462(2.0 − 0.1H )(9.81) H 3 2 , sec

solve for H ≈ 0.94 m

Since the weir is not too wide, we have subtracted 0.1 H from b as recommended. The weir serves as a “control structure” which sets the water depth just upstream. This is our initial condition for gradually varied theory: y(0) = Y + H = 1.5 + 0.94 ≈ 2.44 m at x = 0. Before solving Equation (3.5.10), we find the normal and critical depths to get a feel for the problem:  2.0 yn  m3 1.0 = 2.0 yn )  (  sec 0.025  2.0 + 2 yn 

Normal depth:

Q = 2.5

Critical depth:

 bQ 2  Ac = 2.0 yc −    g 

13

32

sin 0.1° ,

solve yn ≈ 1.14 m

13

 2.0(2.5) 2  =  ,  9.81 

solve yc ≈ 0.54 m

We have taken n ≈ 0.025 for gravelly earth, from Table 3.5.1. Since y(0) > yn > yc , we are on a mild slope M – 1 “backwater” curve, as in Figure 3.5.4. For our data, Equation (3.5.10) becomes

(

2 2 2 2 43 dy So − n Q ζ A Rh ≈ dx 1 − Q 2 b gA3

( )

)

where Q = 2.5, b = 2, ζ = 1, A = 2y, So = sin 0.1°, Rh = 2y/(2 + 2y), g = 9.81, y(0) = 2.44 at x = 0. Integrate numerically backward, that is, for ∆x < 0, until x = –1 km = –1000 m. The complete solution curve is shown in Figure 3.5.5. The water depth decreases upstream and is approximately y ≈ 1.31 m at x = –1000 m. If slope and channel width remain constant, the water depth asymptotically approaches the normal depth yn far upstream.

Nomenclature English symbols A b C Cd Dh E f Fr g H L n P

= = = = = = = = = = = = =

water cross section area channel upper-surface width Chézy coefficient, Equation (3.5.1) weir discharge coefficient, Equation (3.5.7) hydraulic diameter, = 4Rh specific energy, = y + V2/2g Moody friction factor Froude number, = V/Vcrit acceleration of gravity excess water level above weir, Figure 3.5.3 weir length, Figure 3.5.3 Manning roughness factor, Table 3.5.1 wetted perimeter

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3-73

FIGURE 3.5.5 Backwater solution curve for Example 3.5.2.

Q Rh S So V x y Y

= = = = = = = =

volume flow rate hydraulic radius, = A/P frictional slope, Equation (3.5.10) bottom slope average velocity horizontal distance along the channel water depth weir height, Figure (3.5.3)

Greek Symbols ε ρ µ ν ζ

= = = = =

wall roughness height, Table 3.5.1 fluid density fluid viscosity fluid kinematic viscosity, = µ/ρ conversion factor, = 1.0 (SI) and 1.486 (English)

Subscripts c,crit = critical, at Fr = 1 n = normal, in uniform flow

References Ackers, P. et al. 1978. Weirs and Flumes for Flow Measurement, John Wiley, New York. Bos, M.G. 1985. Long-Throated Flumes and Broad-Crested Weirs, Martinus Nijhoff (Kluwer), Dordrecht, The Netherlands. Bos, M.G., Replogle, J.A., and Clemmens, A.J. 1984. Flow-Measuring Flumes for Open Channel Systems, John Wiley, New York. Brater, E.F. 1976. Handbook of Hydraulics, 6th ed., McGraw-Hill, New York. Chow, V.T. 1959. Open Channel Hydraulics, McGraw-Hill, New York. French, R.H. 1985. Open Channel Hydraulics, McGraw-Hill, New York. Henderson, F.M. 1966. Open Channel Flow, Macmillan, New York. Sellin, R.H.J. 1970. Flow in Channels, Gordon & Breach, London. Spitzer, D.W. (Ed.). 1991. Flow Measurement: Practical Guides for Measurement and Control, Instrument Society of America, Research Triangle Park, NC. © 2005 by CRC Press LLC

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Chapter 3

FIGURE 3.6.1 Viscous flow around an airfoil (boundary layer thickness exaggerated for clarity).

3.6 External Incompressible Flows Alan T. McDonald Introduction and Scope Potential flow theory (Section 3.2) treats an incompressible ideal fluid with zero viscosity. There are no shear stresses; pressure is the only stress acting on a fluid particle. Potential flow theory predicts no drag force when an object moves through a fluid; this is obviously incorrect because all real fluids are viscous and cause drag forces. The objective of this section is to consider the behavior of viscous, incompressible fluids flowing over objects. A number of phenomena that occur in external flow at high Reynolds number over an object are shown in Figure 3.6.1. The freestream flow divides at the stagnation point and flows around the object. Fluid at the object surface takes on the velocity of the body as a result of the no-slip condition. Boundary layers form on the upper and lower surfaces of the body; flow in the boundary layers is initially laminar, then transition to turbulent flow may occur (points “T”). Boundary layers thickening on the surfaces cause only a slight displacement of the streamlines of the external flow (their thickness is greatly exaggerated in the figure). Separation may occur in the region of increasing pressure on the rear of the body (points “S”); after separation, boundary layer fluid no longer remains in contact with the surface. Fluid that was in the boundary layers forms the viscous wake behind the object. The Bernoulli equation is valid for steady, incompressible flow without viscous effects. It may be used to predict pressure variations outside the boundary layer. Stagnation pressure is constant in the uniform inviscid flow far from an object, and the Bernoulli equation reduces to 1 p∞ + ρV 2 = constant 2

(3.6.1)

where p∞ is pressure far upstream, ρ is density, and V is velocity. Therefore, the local pressure can be determined if the local freestream velocity, U, is known.

Boundary

Layers

The Boundary Layer Concept The boundary layer is the thin region near the surface of a body in which viscous effects are important. By recognizing that viscous effects are concentrated near the surface of an object, Prandtl showed that only the Euler equations for inviscid flow need be solved in the region outside the boundary layer. Inside © 2005 by CRC Press LLC

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Fluid Mechanics

FIGURE 3.6.2 Boundary layer on a flat plate (vertical thickness exaggerated for clarity).

the boundary layer, the elliptic Navier–Stokes equations are simplified to boundary layer equations with parabolic form that are easier to solve. The thin boundary layer has negligible pressure variation across it; pressure from the freestream is impressed upon the boundary layer. Basic characteristics of all laminar and turbulent boundary layers are shown in the developing flow over a flat plate in a semi-infinite fluid. Because the boundary layer is thin, disturbance of the inviscid flow outside the boundary layer is negligible, and the pressure gradient along the surface is close to zero. Transition from laminar to turbulent boundary layer flow on a flat plate occurs when Reynolds number based on x exceeds Rex = 500,000. Transition may occur earlier if the surface is rough, pressure increases in the flow direction, or separation occurs. Following transition, the turbulent boundary layer thickens more rapidly than the laminar boundary layer as a result of increased shear stress at the body surface. Boundary Layer Thickness Definitions Boundary layer disturbance thickness, δ, is usually defined as the distance, y, from the surface to the point where the velocity within the boundary layer, u, is within 1% of the local freestream velocity, U. As shown in Figure 3.6.2, the boundary layer velocity profile merges smoothly and asymptotically into the freestream, making δ difficult to measure. For this reason and for their physical significance, two integral measures of boundary layer thickness are defined. Displacement thickness, δ*, is defined as δ* = δ

∫

∞

0

 u   y 1 −  d    U   δ

(3.6.2)

Physically, δ*, is the distance the solid boundary would need to be displaced into the freestream in a frictionless flow to produce the mass flow deficit caused by the viscous boundary layer. Momentum thickness, θ, is defined as θ = δ

∫

∞

0

u  u   y 1− d U  U   δ 

(3.6.3)

Physically, θ is the thickness of a fluid layer with velocity U, for which the momentum flux is the same as the deficit in momentum flux within the boundary layer (momentum flux is momentum per unit time passing a cross section). Because δ* and θ are defined in terms of integrals for which the integrand vanishes in the freestream, they are easier to evaluate experimentally than disturbance thickness δ. Exact Solution of the Laminar Flat-Plate Boundary Layer Blasius obtained an exact solution for laminar boundary layer flow on a flat plate. He assumed a thin boundary layer to simplify the streamwise momentum equation. He also assumed similar velocity profiles © 2005 by CRC Press LLC

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in the boundary layer so that, when written as u/U = f(y/δ), velocity profiles do not vary with x. He used a similarity variable to reduce the partial differential equations of motion and continuity to a single thirdorder ordinary differential equation. Blasius used numerical methods to solve the ordinary differential equation. Unfortunately, the velocity profile must be expressed in tabular form. The principal results of the Blasius solution may be expressed as δ 5 = x Re x

(3.6.4)

and Cf =

τw 0.664 = 1 Re x ρU 2 2

(3.6.5)

These results characterize the laminar boundary layer on a flat plate; they show that laminar boundary layer thickness varies as x 1/2 and wall shear stress varies as 1/x 1/2. Approximate Solutions The Blasius solution cannot be expressed in closed form and is limited to laminar flow. Therefore, approximate methods that give solutions for laminar and turbulent flow in closed form are desirable. One such method is the momentum integral equation (MIE), which may be developed by integrating the boundary layer equation across the boundary layer or by applying the streamwise momentum equation to a differential control volume (Fox et al., 2004). The result is the ordinary differential equation τ  δ *  θ dU dθ = w − + 2  U dx dx ρU 2  θ

(3.6.6)

The first term on the right side of Equation (3.6.6) contains the influence of wall shear stress. Because τw is always positive, it always causes θ to increase. The second term on the right side contains the pressure gradient, which can have either sign. Therefore, the effect of the pressure gradient can be to increase or to decrease the rate of growth of boundary layer thickness. Equation (3.6.6) is an ordinary differential equation that can be solved for θ as a function of x on a flat plate (zero pressure gradient), provided a reasonable shape is assumed for the boundary layer velocity profile and shear stress is expressed in terms of the other variables. Results for laminar and turbulent flat-plate boundary layer flows are discussed next. Laminar Boundary Layers — A reasonable approximation to the laminar boundary layer velocity profile is to express u as a polynomial in y. The resulting solutions for δ and τw have the same dependence on x as the exact Blasius solution. Numerical results are presented in Table 3.6.1. Comparing the approximate and exact solutions shows remarkable agreement in view of the approximations used in the analysis. The trends are predicted correctly and the approximate values are within 10% of the exact values. Turbulent Boundary Layers — The turbulent velocity profile may be expressed well using a power law, u/U = (y/δ)1/n, where n is an integer between 6 and 10 (frequently 7 is chosen). For turbulent flow, it is not possible to express shear stress directly in terms of a simple velocity profile; an empirical correlation is required. Using a pipe flow data correlation gives δ 0.382 = x Re1x 5 © 2005 by CRC Press LLC

(3.6.7)

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TABLE 3.6.1 Exact and Approximate Solutions for Laminar Boundary Layer Flow over a Flat Plate at Zero Incidence Velocity Distribution

θ δ

δ* δ

2/15 39/280 (4 – π)/2 π 0.133

1/3 3/8 (π – 2)/ π 0.344

 y u =f =f η U  δ

( )

f (η) = 2 η – η2 f (η) = 3/2 η – 1/2 η3 f (η) = sin (π/2 η) Exact

a=

δ Re x x

b = Cf

5.48 4.64 4.80 5.00

Re x

0.730 0.647 0.654 0.664

TABLE 3.6.2 Thickness and Skin Friction Coefficient for Laminar and Turbulent Boundary Layers on a Flat Plate Boundary Layer Thickness/x

Skin Friction Coefficient

Reynolds Number

Laminar BL

Turbulent BL

Laminar BL

Turbulent BL

2E + 05 5E + 05 1E + 06 2E + 06 5E + 06 1E + 07 2E + 07 5E + 07

0.0112 0.00707 0.00500 0.00354 0.00224 0.00158 0.00112 0.000707

0.0333 0.0277 0.0241 0.0210 0.0175 0.0152 0.0132 0.0110

0.00148 0.000939 0.000664 0.000470 0.000297 0.000210 0.000148 0.0000939

0.00517 0.00431 0.00375 0.00326 0.00272 0.00236 0.00206 0.00171

Turbulent/Laminar Ratio BL thickness 2.97 3.92 4.82 5.93 7.81 9.62 11.8 15.6

Skin friction 3.48 4.58 5.64 6.95 9.15 11.3 13.9 18.3

Note: BL = boundary layer.

and Cf =

τw 0.0594 = 1 Re1x 5 2 ρU 2

(3.6.8)

These results characterize the turbulent boundary layer on a flat plate. They show that turbulent boundary layer thickness varies as x 4/5 and wall shear stress varies as 1/x 1/5. Approximate results for laminar and turbulent boundary layers are compared in Table 3.6.2. At a Reynolds number of 1 million, wall shear stress for the turbulent boundary layer is nearly six times as large as for the laminar layer. For a turbulent boundary layer, thickness increases five times faster with distance along the surface than for a laminar layer. These approximate results give a physical feel for relative magnitudes in the two cases. The MIE cannot be solved in closed form for flows with nonzero pressure gradients. However, the role of the pressure gradient can be understood qualitatively by studying the MIE. Effect of Pressure Gradient Boundary layer flow with favorable, zero, and adverse pressure gradients is depicted schematically in Figure 3.6.3. (Assume a thin boundary layer, so that flow on the lower surface behaves as external flow on a surface, with the pressure gradient impressed on the boundary layer.) The pressure gradient is favorable when ∂p/∂x < 0; zero when ∂p/∂x = 0; and adverse when ∂p/∂x > 0, as indicated for regions 1, 2, and 3. Viscous shear always causes a net retarding force on any fluid particle within the boundary layer. For zero pressure gradient, shear forces alone can never bring the particle to rest. (Recall that for laminar and turbulent boundary layers, the shear stress varied as 1/x 1/2 and 1/x 1/5, respectively; shear stress never becomes zero for finite x.) Because shear stress is given by τw = µ ∂u/∂y)y = 0 , the velocity gradient cannot © 2005 by CRC Press LLC

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FIGURE 3.6.3 Boundary layer flow with pressure gradient (thickness exaggerated for clarity).

be zero. Therefore, flow cannot separate in a zero pressure gradient; shear stresses alone can never cause flow separation. In the favorable pressure gradient of region 1, pressure forces tend to maintain the motion of the particle, so flow cannot separate. In the adverse pressure gradient of region 3, pressure forces oppose the motion of a fluid particle. An adverse pressure gradient is a necessary condition for flow separation. Velocity profiles for laminar and turbulent boundary layers are shown in Figure 3.6.2. It is easy to see that the turbulent velocity profile has much more momentum than the laminar profile. Therefore, the turbulent velocity profile can resist separation in an adverse pressure gradient better than the laminar profile. The freestream velocity distribution must be known before the MIE can be applied. A first approximation is obtained by applying potential flow theory to calculate the flow field around the object. Much effort has been devoted to calculation of velocity distributions over objects of known shape (the “direct” problem) and to determination of shapes to produce a desired pressure distribution (the “inverse” problem). Detailed discussion of such calculation schemes is beyond the scope of this section; the state of the art continues to progress rapidly.

Drag Any object immersed in a viscous fluid flow experiences a net force from the shear stresses and pressure differences caused by the fluid motion. Drag is the force component parallel to, and lift is the force component perpendicular to, the flow direction. Streamlining is the art of shaping a body to reduce fluid dynamic drag. Airfoils (hydrofoils) are designed to produce lift in air (water); they are streamlined to reduce drag and thus to attain high lift–drag ratios. In general, lift and drag cannot be predicted analytically for flows with separation, but progress continues on computational fluid dynamics methods. For many engineering purposes, drag and lift forces are calculated from experimentally derived coefficients, discussed next. Drag coefficient is defined as CD =

FD 1 ρV 2 A 2

(3.6.9)

where ½ρV2 is dynamic pressure and A is the area upon which the coefficient is based. Common practice is to base drag coefficients on projected frontal area (Fox et al., 2004). © 2005 by CRC Press LLC

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FIGURE 3.6.4 Drag coefficient vs. Reynolds number for a smooth flat plate parallel to the flow.

Similitude was treated in Section 3.3. In general, the drag coefficient may be expressed as a function of Reynolds number; Mach number; Froude number; relative roughness; submergence divided by length; and so forth. This section considers neither high-speed flow nor free-surface effects; only Reynolds number and roughness effects on drag coefficient will be considered. Friction Drag The total friction drag force acting on a plane surface aligned with the flow direction can be found by integrating the shear stress distribution along the surface. The drag coefficient for this case is defined as friction force divided by dynamic pressure and wetted area in contact with the fluid. Because shear stress is a function of Reynolds number, so is drag coefficient (see Figure 3.6.4). In Figure 3.6.4, transition occurs at Rex = 500,000; the dashed line represents the drag coefficient at larger Reynolds numbers. A number of empirical correlations may be used to model the variation in CD shown in Figure 3.6.4 (Schlichting, 1979). Extending the laminar boundary layer line to higher Reynolds numbers shows that it is beneficial to delay transition to the highest possible Reynolds number. Some results are presented in Table 3.6.3; drag is reduced more than 50% by extending laminar boundary layer flow to ReL = 106.

TABLE 3.6.3 Drag Coefficients for Laminar, Turbulent, and Transition Boundary Layers on a Flat Plate Drag Coefficient

Reynolds Number

Laminar BL

Turbulent BL

Transition

Laminar/ Transition

2E + 05 5E + 05 1E + 06 2E + 06 5E + 06 1E + 07 2E + 07 5E + 07

0.00297 0.00188 0.00133 0.000939 0.000594 0.000420 0.000297 0.000188

0.00615 0.00511 0.00447 0.00394 0.00336 0.00300 0.00269 0.00235

— 0.00189 0.00286 0.00314 0.00304 0.00284 0.00261 0.00232

— — 0.464 0.300 0.195 0.148 0.114 0.081

Note: BL = boundary layer. © 2005 by CRC Press LLC

% Drag Reduction — — 53.6 70.0 80.5 85.2 88.6 9.19

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TABLE 3.6.4 Drag Coefficient Data for Selected Objects (Re > 1000) Object

CD(RE ≥ 103)

Diagram

Square prism b/h = ∞

2.05

b/h = 1

1.05

Disk

1.17

Ring

1.20a

Hemisphere (open end facing flow)

1.42

Hemisphere (open end facing downstream)

0.38

C-section (open side facing flow)

2.30

C-section (open side facing downstream)

1.20

a

Based on ring area. Source: Data from Hoerner, S.F. 1965. Fluid-Dynamic Drag, 2nd ed. Published by the author, Midland Park, NJ.

Pressure Drag A thin flat surface normal to the flow has no area parallel to the flow direction. Therefore, there can be no friction force parallel to the flow; all drag is caused by pressure forces. Drag coefficients for objects with sharp edges tend to be independent of Reynolds number (for Re > 1000) because the separation points are fixed by the geometry of the object. Drag coefficients for selected objects are shown in Table 3.6.4. Rounding the edges that face the flow reduces drag markedly. Compare the drag coefficients for the hemisphere and C-section shapes facing into and away from the flow. Also note that the drag coefficient for a two-dimensional object (long square cylinder) is about twice that for the corresponding threedimensional object (square cylinder with b/h = 1). Friction and Pressure Drag: Bluff Bodies Friction and pressure forces contribute to the drag of bluff bodies (see Shapiro, 1960, for a good discussion of the mechanisms of drag). As an example, consider the drag coefficient for a smooth sphere shown in Figure 3.6.5. Transition from laminar to turbulent flow in the boundary layers on the forward portion of the sphere causes a dramatic dip in drag coefficient at the critical Reynolds number (ReD ≈ 2 × 105). The turbulent boundary layer is better able to resist the adverse pressure gradient on the rear of the © 2005 by CRC Press LLC

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FIGURE 3.6.5 Drag coefficient vs. Reynolds number for a smooth sphere. (From Schlichting, H. 1979. Boundary Layer Theory, 7th ed., McGraw–Hill, New York. With permission.)

sphere, so separation is delayed and the wake is smaller, causing less pressure drag. Surface roughness (or freestream disturbances) can reduce the critical Reynolds number. For example, dimples on a golf ball cause the boundary layer to become turbulent and, therefore, lower the drag coefficient in the range of speeds encountered in a drive. Streamlining Streamlining is adding a faired tail section to reduce the extent of separated flow on the downstream portion of an object (at high Reynolds number where pressure forces dominate drag). The adverse pressure gradient is taken over a longer distance, delaying separation. However, adding a faired tail increases surface area, causing skin friction drag to increase. Thus, streamlining must be optimized for each shape. For example, front contours are of principal importance in road vehicle design; the angle of the back glass also is important (in most cases the entire rear end cannot be made long enough to control separation and reduce drag significantly).

Lift Lift coefficient is defined as CL =

FL 1 ρV 2 A 2

(3.6.10)

Note that lift coefficient is based on projected planform area. Airfoils Airfoils are shaped to produce lift efficiently by accelerating flow over the upper surface to produce a low-pressure region. Because the flow must again decelerate, inevitably a region of adverse pressure gradient must be near the rear of the upper surface (pressure distributions are shown clearly in Hazen, 1965). Lift and drag coefficients for airfoil sections depend on Reynolds number and angle of attack between the chord line and the undisturbed flow direction. The chord line is the straight line joining the leading and trailing edges of the airfoil (Abbott and von Doenhoff, 1959). As the angle of attack is increased, the © 2005 by CRC Press LLC

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minimum pressure point moves forward on the upper surface and the minimum pressure becomes lower. This increases the adverse pressure gradient. At some angle of attack, the adverse pressure gradient is strong enough to cause the boundary layer to separate completely from the upper surface, causing the airfoil to stall. The separated flow alters the pressure distribution, reducing lift sharply. Increasing the angle of attack also causes the drag coefficient to increase. At some angle of attack below “stall,” the ratio of lift to drag — the lift–drag ratio — reaches a maximum value. Drag Due to Lift For wings (airfoils of finite span), lift and drag also are functions of aspect ratio. Lift is reduced and drag increased compared with infinite span because end effects cause the lift vector to rotate rearward. For a given geometric angle of attack, this reduces effective angle of attack, reducing lift. The additional component of lift acting in the flow direction increases drag; the increase in drag due to lift is called induced drag. The effective aspect ratio includes the effect of planform shape. When written in terms of effective aspect ratio, the drag of a finite-span wing is C D = C D ,∞ +

C L2 πar

(3.6.11)

where ar is effective aspect ratio and the subscript ∞ refers to the infinite section drag coefficient at CL. For further details consult the references. The lift coefficient must increase to support aircraft weight as speed is reduced. Therefore, induced drag can increase rapidly at low flight speeds. For this reason, minimum allowable flight speeds for commercial aircraft are closely controlled by the FAA.

Boundary Layer Control The major part of the drag on an airfoil or wing is caused by skin friction. Therefore, it is important to maintain laminar flow in the boundary layers as far aft as possible; laminar flow sections are designed to do this. It also is important to prevent flow separation and to achieve high lift to reduce takeoff and landing speeds. These topics fall under the general heading of boundary layer control. Profile Shaping Boundary layer transition on a conventional airfoil section occurs almost immediately after the minimum pressure at about 25% chord aft the leading edge. Transition can be delayed by shaping the profile to maintain a favorable pressure gradient over more of its length. The U.S. National Advisory Committee for Aeronautics (NACA) has developed several series of profiles that delayed transition to 60 or 65% of chord, reducing drag coefficients (in the design range) 60% compared with conventional sections of the same thickness ratio (Abbott and von Doenhoff, 1959). Flaps and Slats Flaps are movable sections near the trailing edge of a wing. They extend and/or deflect to increase wing area and/or increase wing camber (curvature), to provide higher lift than the clean wing. Many aircraft also are fitted with leading edge slats that open to expose a slot from the pressure side of the wing to the upper surface. The open slat increases the effective radius of the leading edge, improving maximum lift coefficient. The slot allows energized air from the pressure surface to flow into the low-pressure region atop the wing, energizing the boundary layers and delaying separation and stall. Suction and Blowing Suction removes low-energy fluid from the boundary layer, reducing the tendency for early separation. Blowing via high-speed jets directed along the surface reenergizes low-speed boundary layer fluid. The objective of both approaches is to delay separation, thus increasing the maximum lift coefficient the wing © 2005 by CRC Press LLC

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can achieve. Powered systems add weight and complexity; they also require bleed air from the engine compressor, reducing thrust or power output. Moving Surfaces Many schemes have been proposed to utilize moving surfaces for boundary layer control. Motion in the direction of flow reduces skin friction and thus the tendency to separate; motion against the flow has the opposite effect. The aerodynamic behavior of sports balls — baseballs, golf balls, and tennis balls — depends significantly on aerodynamic side force (lift, down force, or side force) produced by spin. These effects are discussed at length in Fox et al. (2004) and its references.

Computation vs. Experiment Experiments cannot yet be replaced completely by analysis. Progress in modeling, numerical techniques, and computer power continues to be made, but the role of the experimentalist likely will remain important for the foreseeable future. Computational Fluid Dynamics (CFD) Computation of fluid flow requires accurate mathematical modeling of flow physics and accurate numerical procedures to solve the equations. The basic equations for laminar boundary layer flow are well known. For turbulent boundary layers, it is not generally possible to resolve the solution space into sufficiently small cells to allow direct numerical simulation. Instead, empirical models for the turbulent stresses must be used. Advances in computer memory storage capacity and speed (e.g., through use of massively parallel processing) continue to increase the resolution that can be achieved. A second source of error in CFD work results from the numerical procedures required to solve the equations. Even if the equations are exact, approximations must be made to discretize and solve them using finite-difference or finite-volume methods. Whichever is chosen, the solver must guard against introducing numerical instability, round-off errors, and numerical diffusion (Hoffman, 2001). Role of the Wind Tunnel Traditionally, wind tunnel experiments have been conducted to verify the design and performance of components and complete aircraft. Design verification of a modern aircraft may require expensive scale models, several thousand hours of wind tunnel time at many thousands of dollars an hour, and additional full-scale flight testing. New wind tunnel facilities continue to be built and old ones refurbished. This indicates a need for continued experimental work in developing and optimizing aircraft configurations. Many experiments are designed to produce baseline data to validate computer codes. Such systematic experimental data can help to identify the strengths and weaknesses of computational methods. CFD tends to become only indicative of trends when massive zones of flow separation are present. Takeoff and landing configurations of conventional aircraft, with landing gear, high-lift devices, and flaps extended, tend to need final experimental confirmation and optimization. Many studies of vertical takeoff and vectored thrust aircraft require testing in wind tunnels.

Defining Terms Boundary layer: Thin layer of fluid adjacent to a surface where viscous effects are important; viscous effects are negligible outside the boundary layer. Drag coefficient: Force in the flow direction exerted on an object by the fluid flowing around it, divided by dynamic pressure and area. Lift coefficient: Force perpendicular to the flow direction exerted on an object by the fluid flowing around it, divided by dynamic pressure and area. Pressure gradient: Variation in pressure along the surface of an object. For a favorable pressure gradient, pressure decreases in the flow direction; for an adverse pressure gradient, pressure increases in the flow direction. © 2005 by CRC Press LLC

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Separation: Phenomenon that occurs when fluid layers adjacent to a solid surface are brought to rest and boundary layers depart from the surface contour, forming a low-pressure wake region. Separation can occur only in an adverse pressure gradient. Transition: Change from laminar to turbulent flow within the boundary layer. The location depends on distance over which the boundary layer has developed; pressure gradient; surface roughness; freestream disturbances; and heat transfer.

References Abbott, I.H. and von Doenhoff, A.E. 1959. Theory of Wing Sections, Including a Summary of Airfoil Data. Dover, New York. Fox, R.W., McDonald, A.T. and Pritchard, P.J. 2004. Introduction to Fluid Mechanics, 6th ed. John Wiley & Sons, New York. Hazen, D.C. 1965. Boundary Layer Control, film developed by the National Committee for Fluid Mechanics Films (NCFMF) and available on videotape from Encyclopedia Britannica Educational Corporation, Chicago. Hoerner, S.F. 1965. Fluid-Dynamic Drag, 2nd ed. Published by the author, Midland Park, NJ. Hoffman, J.D. 2001. Numerical Methods for Engineers and Scientists, 2nd ed. Marcel Dekker, New York. Schlichting, H. 1979. Boundary-Layer Theory, 7th ed. McGraw–Hill, New York. Shapiro, A.H. 1960. The Fluid Dynamics of Drag, film developed by the National Committee for Fluid Mechanics Film (NCFMF) and available on videotape from Encyclopedia Britannica Educational Corporation, Chicago.

Further Information A comprehensive source of basic information is the Handbook of Fluid Dynamics, V.L. Streeter, Ed., McGraw–Hill, New York, 1960. Timely reviews of important topics are published in the Annual Review of Fluid Mechanics series (Annual Reviews, Inc., Palo Alto, CA.). Each volume contains a cumulative index. ASME (American Society of Mechanical Engineers, New York, NY) publishes the Journal of Fluids Engineering quarterly, which contains fluid machinery and other engineering applications of fluid mechanics. The monthly AIAA Journal and bimonthly Journal of Aircraft (American Institute for Aeronautics and Astronautics, New York, NY) treat aerospace applications of fluid mechanics.

3.7 Compressible Flow Ajay Kumar and Jessica Todd Introduction This section deals with compressible flow. Only one- or two-dimensional steady, inviscid flows under perfect gas assumption are considered. Readers are referred to other sources of information for unsteady effects, viscous effects, and three-dimensional flows. The term compressible flow is routinely used to define variable density flow which is in contrast to incompressible flow, where the density is assumed to be constant throughout. In many cases, these density variations are principally caused by the pressure changes from one point to another. Physically, the compressibility can be defined as the fractional change in volume of the gas element per unit change in pressure. It is a property of the gas and, in general, can be defined as τ= © 2005 by CRC Press LLC

1 dρ ρ dp

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where τ is the compressibility of the gas, ρ is the density, and p is the pressure being exerted on the gas. A more precise definition of compressibility is obtained if we take into account the thermal and frictional losses. If during the compression the temperature of the gas is held constant, it is called the isothermal compressibility and can be written as τT =

1  ∂ρ    ρ  ∂p  T

However, if the compression process is reversible, it is called the isentropic compressibility and can be written as τs =

1  ∂ρ    ρ  ∂p  s

Gases in general have high compressibility (τT for air is 10–5 m2/N at 1 atm) as compared with liquids (τT for water is 5 × 10–10 m2/N at 1 atm). Compressibility is a very important parameter in the analysis of compressible flow and is closely related to the speed of sound, a, which is the velocity of propagation of small pressure disturbances and is defined as  ∂p  a2 =    ∂ρ  s

 ∂p  a=    ∂ρ  s

or

In an isentropic process of a perfect gas, the pressure and density are related as p = constant ργ Using this relation along with the perfect gas relation p = ρRT, we can show that for a perfect gas a = γRT =

γp ρ

where γ is the ratio of specific heats at constant pressure and constant volume, R is the gas constant, and T is the temperature. For air under normal conditions, γ is 1.4 and R is 287 m2/sec2 K so that the speed of sound for air becomes a = 20.045 T m/sec where T is in kelvin. Another important parameter in compressible flows is the Mach number, M, which is defined as the ratio of the gas velocity to the speed of sound or M=

V a

where V is the velocity of gas. Depending upon the Mach number of the flow, we can define the following flow regimes: M
1 Incompressible flow M < 1 Subsonic flow M ≈ 1 Transonic flow M > 1 Supersonic flow M  1 Hypersonic flow © 2005 by CRC Press LLC

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Subsonic through hypersonic flows are compressible in nature. In these flows, the velocity is appreciable compared with the speed of sound, and the fractional changes in pressure, temperature, and density are all of significant magnitude. We will restrict ourselves in this section to subsonic through flows only. Before we move on to study these flows, let us define one more term. Let us consider a gas with static pressure p and temperature T, traveling at some velocity V and corresponding Mach number M. If this gas is brought isentropically to stagnation or zero velocity, the pressure and temperature which the gas achieves are defined as stagnation pressure p0 and stagnation temperature T0 (also called total pressure and total temperature). The speed of sound at stagnation conditions is called the stagnation speed of sound and is denoted as a0.

One-Dimensional Flow In one-dimensional flow, the flow properties vary only in one coordinate direction. Figure 3.7.1 shows two streamtubes in a flow. In a truly one-dimensional flow illustrated in Figure 3.7.1(a), the flow variables are a function of x only and the area of the stream tube is constant. On the other hand, Figure 3.7.1(b) shows a flow where the area of the stream tube is also a function of x but the flow variables are still a function of x only. This flow is defined as the quasi-one-dimensional flow. We will first discuss the truly one-dimensional flow. In a steady, truly one-dimensional flow, conservation of mass, momentum, and energy leads to the following simple algebraic equations. ρu = constant p + ρu 2 = constant h+

(3.7.1)

u2 + q = constant 2

where q is the heat added per unit mass of the gas. These equations neglect body forces, viscous stresses, and heat transfer due to thermal conduction and diffusion. These relations given by Equation 3.7.1, when applied at points 1 and 2 in a flow with no heat addition, become ρ1u1 = ρ2 u2 p1 + ρ1u12 = p2 + ρ2 u22 h1 +

u12 u2 = h2 + 2 2 2

FIGURE 3.7.1 (a) One-dimensional flow; (b) quasi-one-dimensional flow. © 2005 by CRC Press LLC

(3.7.2)

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The energy equation for a calorically perfect gas, where h = cpT, becomes c p T1 +

u12 u2 = c p T2 + 2 2 2

Using cp = γR/(γ – 1) and a2 = γRT, the above equation can be written as a12 u2 a2 u2 + 1 = 2 + 2 γ −1 2 γ −1 2

(3.7.3)

Since Equation (3.7.3) is written for no heat addition, it holds for an adiabatic flow. If the energy equation is applied to the stagnation conditions, it can be written as cpT +

u2 = c p T0 2

T0 γ −1 2 M = 1+ T 2

(3.7.4)

It is worth mentioning that in arriving at Equation (3.7.4), only adiabatic flow condition is used whereas stagnation conditions are defined as those where the gas is brought to rest isentropically. Therefore, the definition of stagnation temperature is less restrictive than the general definition of stagnation conditions. According to the general definition of isentropic flow, it is a reversible adiabatic flow. This definition is needed for the definition of stagnation pressure and density. For an isentropic flow, γ γ ( γ −1) p0  ρ0  T  =  = 0 T p  ρ

(3.7.5)

From Equation (3.7.4) and Equation (3.7.5), we can write γ γ −1 p0  γ − 1 2 ( ) = 1+ M  2 p 

(3.7.6)

1 γ −1 ρ0  γ −1 2 ( ) M = 1+  2 ρ 

(3.7.7)

Values of stagnation conditions are tabulated in Anderson (1982) as a function of M for γ = 1.4.

Normal Shock Wave A shock wave is a very thin region (of the order of a few molecular mean free paths) across which the static pressure, temperature, and density increase whereas the velocity decreases. If the shock wave is perpendicular to the flow, it is called a normal shock wave. The flow is supersonic ahead of the normal shock wave and subsonic behind it. Figure 3.7.2 shows the flow conditions across a normal shock wave which is treated as a discontinuity. Since there is no heat added or removed, the flow across the shock wave is adiabatic. By using Equations (3.7.2) the normal shock equations can be written as ρ1u1 = ρ2 u2 p1 + ρ1u12 = p2 + ρ2 u22 h1 + © 2005 by CRC Press LLC

u12 u2 = h2 + 2 2 2

(3.7.8)

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FIGURE 3.7.2 Flow conditions across a normal shock.

Equations (3.7.8) are applicable to a general type of flow; however, for a calorically perfect gas, we can use the relations p = ρRT and h = cpT to derive a number of equations relating flow conditions downstream of the normal shock to those at upstream. These equations (also known as Rankine–Hugoniot relations) are p2 2γ M2 − 1 = 1+ p1 γ +1 1

(

)

ρ2 u1 (γ + 1) M12 = = ρ1 u2 2 + ( γ − 1) M12

(3.7.9)

  2 + ( γ − 1) M12  T2 h2  2γ M12 − 1   = = 1 +  2 T1 h1  γ + 1   ( γ + 1) M1 

(

)

γ −1 2 M1 2 M = γ −1 γM12 − 2 1+

2 2

Again, the values of p2 /p1, ρ2 /ρ1, T2 /T1, etc. are tabulated in Anderson (1982) as a function of M1 for γ = 1.4. Let us examine some limiting cases. As M1 → 1, Equations (3.7.9) yield M2 → 1, p2 /p1 → 1, ρ2 /ρ1 → 1, and T2 /T1 → 1. This is the case of an extremely weak normal shock across which no finite changes occur. This is the same as the sound wave. On the other hand, as M1 → ∞, Equations (3.7.9) yield M2 →

ρ2 γ −1 γ +1 = 0.378; → = 6; 2γ ρ1 γ −1

p2 T → ∞; 2 → ∞ p1 T1

However, the calorically perfect gas assumption no longer remains valid as M1 → ∞. Let us now examine why the flow ahead of a normal shock wave must be supersonic even though Equations (3.7.8) hold for M1 < 1 as well as M1 > 1. From the second law of thermodynamics, the entropy change across the normal shock can be written as s2 − s1 = c p ln

T2 p − R ln 2 T1 p1

By using Equations (3.7.9) it becomes    2 + ( γ − 1) M12    2γ 2γ s2 − s1 = c p ln 1 + M12 − 1   M12 − 1   − R ln 1 + 2 γ γ γ + + 1 M 1 + 1 ( )      1 

(

© 2005 by CRC Press LLC

)

(

)

(3.7.10)

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Equation (3.7.10) shows that the entropy change across the normal shock is also a function of M1 only. Using Equation (3.7.10) we see that s2 − s1 = 0 for M1 = 1 < 0 for M1 < 1 > 0 for M1 > 1 Since it is necessary that s2 – s1
0 from the second law, M1 ≥ 1. This, in turn, requires that p2/p1
1, ρ2/ρ1
1, T2/T1
1, and M2  1. We now examine how the stagnation conditions change across a normal shock wave. For a calorically perfect gas, the energy equation in Equations (3.7.9) gives c p T01 = c p T02

or T01 = T02

In other words, the total temperature remains constant across a stationary normal shock wave. Let us now apply the entropy change relation across the shock using the stagnation conditions. s2 − s1 = c p ln

T02 p − R ln 02 T01 p01

Note that entropy at stagnation conditions is the same as at the static conditions since to arrive at stagnation conditions, the gas is brought to rest isentropically. Since T02 = T01, s2 − s1 = − R ln

p02 p01

p02 − s −s = e ( 2 1) p01

R

(3.7.11)

Since s2 > s1 across the normal shockwave, Equation (3.7.11) gives P02 < P01 or, in other words, the total pressure decreases across a shock wave.

One-Dimensional Flow with Heat Addition Consider one-dimensional flow through a control volume as shown in Figure 3.7.3. Flow conditions going into this control volume are designated by 1 and coming out by 2. A specified amount of heat per unit mass, q, is added to the control volume. The governing equations relating conditions 1 and 2 can be written as ρ1u1 = ρ2 u2 p1 + ρ1u12 = p2 + ρ2 u22 h1 +

(3.7.12)

u12 u2 + q = h2 + 2 2 2

The following relations can be derived from Equation (3.7.12) for a calorically perfect gas q = c p (T02 − T01 ) © 2005 by CRC Press LLC

(3.7.13)

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Chapter 3

FIGURE 3.7.3 One-dimensional control volume with heat addition.

p2 1 + γM12 = p1 1 + γM22 2

T2  1 + γM12   M2  = T1  1 + γM22   M1  2

ρ2  1 + γM22   M1  = ρ1  1 + γM12   M2 

(3.7.14)

2

(3.7.15)

2

(3.7.16)

Equation (3.7.13) indicates that the effect of heat addition is to directly change the stagnation temperature T0 of the flow. Table 3.7.1 shows some physical trends which can be obtained with heat addition to subsonic and supersonic flow. With heat extraction the trends in Table 3.7.1 are reversed. Figure 3.7.4 shows a plot between enthalpy and entropy, also known as the Mollier diagram, for onedimensional flow with heat addition. This curve is called the Rayleigh curve and is drawn for a set of TABLE 3.7.1 Effect of Heat Addition on Subsonic and Supersonic Flow M1 < 1 M2 p2 T2 u2 T02 p02

M1 > 1

Increases Decreases Increases for M1 < γ–1/2 and decreases for M1 > γ–1/2 Increases Increases Decreases

Decreases Increases Increases Decreases Increases Decreases

FIGURE 3.7.4 The Rayleigh curve.

© 2005 by CRC Press LLC

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given initial conditions. Each point on this curve corresponds to a different amount of heat added or removed. It is seen from this curve that heat addition always drives the Mach numbers toward 1. For a certain amount of heat addition, the flow will become sonic. For this condition, the flow is said to be choked. Any further increase in heat addition is not possible without adjustment in initial conditions. For example, if more heat is added in region 1, which is initially supersonic, than allowed for attaining Mach 1 in region 2, then a normal shock will form inside the control volume which will suddenly change the conditions in region 1 to subsonic. Similarly, in case of an initially subsonic flow corresponding to region 1′, any heat addition beyond that is needed to attain Mach 1 in region 2, the conditions in region 1′ will adjust to a lower subsonic Mach number through a series of pressure waves. Similar to the preceding heat addition or extraction relationships, we can also develop relationships for one-dimensional steady, adiabatic flow but with frictional effects due to viscosity. In this case, the momentum equation gets modified for frictional shear stress. For details, readers are referred to Anderson (1982).

Quasi-One-Dimensional Flow In quasi-one-dimensional flow, in addition to flow conditions, the area of duct also changes with x. The governing equations for quasi-one-dimensional flow can be written in a differential form as follows using an infinitesimal control volume shown in Figure 3.7.5. d (ρuA) = 0

(3.7.17)

dp + ρu du = 0

(3.7.18)

dh + u du = 0

(3.7.19)

dρ du dA + + =0 ρ u A

(3.7.20)

Equation (3.7.17) can be written as

which can be further written as follows for an isentropic flow:

(

)

dA du = M2 − 1 A u

FIGURE 3.7.5 Control volume for quasi-one-dimensional flow.

© 2005 by CRC Press LLC

(3.7.21)

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Chapter 3

FIGURE 3.7.6 Compressible flow in converging and diverging ducts.

FIGURE 3.7.7 Schematic of a typical supersonic wind tunnel.

Some very useful physical insight can be obtained from this area–velocity relation. • For subsonic flow (0 ≤ M < 1), an increase in area results in decrease in velocity, and vice versa. • For supersonic flow (M > 1), an increase in area results in increase in velocity, and vice versa. • For sonic flow (M = 1), dA/A = 0, which corresponds to a minimum or maximum in the area distribution, but it can be shown that a minimum in area is the only physical solution. Figure 3.7.6 shows the preceding results in a schematic form. It is obvious from this discussion that for a gas to go isentropically from subsonic to supersonic, and vice versa, it must flow through a convergent–divergent nozzle, also known as the de Laval nozzle. The minimum area of the nozzle at which the flow becomes sonic is called the throat. This physical observation forms the basis of designing supersonic wind tunnels shown schematically in Figure 3.7.7. In general, in a supersonic wind tunnel, a stagnant gas is first expanded to the desired supersonic Mach number. The supersonic flow enters the test section where it passes over a model being tested. The flow then is slowed down by compressing it through a second convergent–divergent nozzle, also known as a diffuser, before it is exhausted to the atmosphere. Now, using the equations for quasi-one-dimensional flow and the isentropic flow conditions, we can derive a relation for the area ratio that is needed to accelerate or decelerate the gas to sonic conditions. Denoting the sonic conditions by an asterisk, we can write u* = a*. The area is denoted as A*, and it is obviously the minimum area for the throat of the nozzle. From Equation (3.7.17) we have ρuA = ρ*u * A* A ρ* u * ρ* ρ 0 u * = = A* ρu ρ0 ρ u

(3.7.22)

1 γ −1 ρ0  γ −1 2 ( ) M = 1+  ρ  2

(3.7.23)

Under isentropic conditons,

© 2005 by CRC Press LLC

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FIGURE 3.7.8 Variation of area ratio A/A* as a function of Mach number for a quasi-one-dimensional flow. 1 γ −1 1 γ −1 ρ0  γ − 1 ( )  γ + 1 ( ) = 1 + =  2  ρ*  2 

(3.7.24)

Also, u*/u = a*/u. Let us define a Mach number M* = u/a*. M* is known as the characteristic Mach number and it is related to the local Mach number by the following relation:

M

*2

γ +1 2 M 2 = γ −1 2 1+ M 2

(3.7.25)

Using Equation (3.7.23) through Equation (3.7.25) in Equation (3.7.22) we can write 2  A  = 1  2  1 + γ − 1 M 2     *   A  M 2  γ + 1  2 

( γ +1) ( γ −1)

(3.7.26)

Equation (3.7.26) is called the area Mach number relation. Figure 3.7.8 shows a plot of A/A* against Mach number. A/A* is always ≥ 1 for physically viable solutions. The area Mach number relation says that for a given Mach number, there is only one area ratio A/A*. This is a very useful relation and is frequently used to design convergent–divergent nozzles to produce a desired Mach number. Values of A/A* are tabulated as a function of M in Anderson (1982). Equation (3.7.26) can also be written in terms of pressure as follows: 12

   ( γ −1) γ    1 γ 1 − p   p   p0    p0  A   = A*  γ − 1 1 2  2  ( γ +1) 2( γ −1)  2   γ + 1

(3.7.27)

Nozzle Flow Using the area relations, we can now plot the distributions of Mach number and pressure along a nozzle. Figure 3.7.9 shows pressure and Mach number distributions along a given nozzle and the wave config© 2005 by CRC Press LLC

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Chapter 3

FIGURE 3.7.9 Effect of exit pressure on flow through a nozzle.

urations for several exit pressures. For curves a and b, the flow stays subsonic throughout and the exit pressure controls the flow in the entire nozzle. On curve c, the throat has just become sonic, and so the pressure at the throat, and upstream of it, can decrease no further. There is another exit pressure corresponding to curve j (pj < pc) for which a supersonic isentropic solution exists. But if the pressure lies between pc and pj , there is no isentropic solution possible. For example, for an exit pressure pd , a shock will form in the nozzle at location s which will raise the pressure to pd′ and turn the flow subsonic. The pressure will then rise to pd as the subsonic flow goes through an increasing area nozzle. The location, s, depends on the exit pressure. Various possible situations are shown in Figure 3.7.9. It is clear that if the exit pressure is equal to or below pf , the flow within the nozzle is fully supersonic. This is the principle used in designing supersonic wind tunnels by operating from a high-pressure reservoir or into a vacuum receiver, or both. Diffuser If a nozzle discharges directly into the receiver, the minimum pressure ratio for full supersonic flow in the test section is  p0  p = 0 p   E  min p f where pf is the value of pE at which the normal shock stands right at the nozzle exit. However, by adding an additional diverging section, known as a diffuser, downstream of the test section as shown in Figure 3.7.10 it is possible to operate the tunnel at a lower pressure ratio than p0 /pf . This happens because the diffuser can now decelerate the subsonic flow downstream of the shock isentropically to a stagnation pressure p0′ . The pressure ratio required then is the ratio of stagnation pressures across a normal shock wave at the test section Mach number. In practice, the diffuser gives lower than expected recovery as a result of viscous losses caused by the interaction of shock wave and the boundary layer which are neglected here. © 2005 by CRC Press LLC

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3-95

FIGURE 3.7.10 Normal shock diffuser.

FIGURE 3.7.11 Supersonic flow over a corner.

The operation of supersonic wind tunnels can be made even more efficient; i.e., they can be operated at even lower pressure ratios than p0 / p0′ , by using the approach shown in Figure 3.7.7 where the diffuser has a second throat. It can slow down the flow to subsonic Mach numbers isentropically and, ideally, can provide complete recovery, giving p0′ = p0. However, due to other considerations, such as the starting process of the wind tunnel and viscous effects, it is not realized in real life.

Two-Dimensional Supersonic Flow When supersonic flow goes over a wedge or an expansion corner, it goes through an oblique shock or expansion waves, respectively, to adjust to the change in surface geometry. Figure 3.7.11 shows the two flow situations. In Figure 3.7.11(a) an oblique shock abruptly turns the flow parallel to the wedge surface. The Mach number behind the shock is less than ahead of it, whereas the pressure, temperature, and density increase. In the case of an expansion corner, oblique expansion waves smoothly turn the flow to become parallel to the surface downstream of the expansion corner. In this case, the Mach number increases, but the pressure, temperature, and density decrease as the flow goes through the expansion corner. Oblique shocks and expansion waves occur in two- and three-dimensional supersonic flows. In this section, we will restrict ourselves to steady, two-dimensional supersonic flows only. Oblique Shock Waves The oblique shock can be treated in the same way as the normal shock by accounting for the additional velocity component. If a uniform velocity v is superimposed on the flow field of the normal shock, the resultant velocity ahead of the shock can be adjusted to any flow direction by adjusting the magnitude and direction of v. If v is taken parallel to the shock wave, as shown in Figure 3.7.12, the resultant velocity ahead of the shock is w1 = u12 + v12 and its direction from the shock is given by β = tan–1 (u1/v). On the downstream side of the shock, since u2 is less than u1, the flow always turns toward the shock. The magnitude of u2 can be determined by the normal shock relations corresponding to velocity u1 and the © 2005 by CRC Press LLC

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Chapter 3

FIGURE 3.7.12 Oblique shock on a wedge.

magnitude of v is such that the flow downstream of the shock turns parallel to the surface. Since imposition of a uniform velocity does not affect the pressure, temperature, etc., we can use normal shock relations with Mach number replaced in them to correspond to velocity u1 or u1/a1, which is nothing but M1 sin β. Thus, oblique shock relations become p2 2γ M 2 sin 2 β − 1 = 1+ p1 γ +1 1

(3.7.28)

ρ2 (γ + 1) M12 sin 2 β = ρ1 ( γ − 1) M12 sin 2 β + 2

(3.7.29)

  2 + ( γ − 1) M12 sin 2 β  T2 a22  2γ M12 sin 2 β − 1   = 2 = 1 +  2 2 T1 a1  γ + 1   ( γ + 1) M1 sin β 

(3.7.30)

(

)

(

)

The Mach number M2 (= w2 /a2) can be obtained by using a Mach number corresponding to velocity u2 (= w2 sin(β – θ)) in the normal shock relation for the Mach number. In other words, γ −1 2 2 M1 sin β 2 M sin (β − θ) = γ −1 γM12 sin 2 β − 2 2 2

1+

2

(3.7.31)

To derive a relation between the wedge angle θ and the wave angle β, we have from Figure 3.7.12 tanβ =

u1 v

tan(β − θ) =

and

u2 v

so that tan(β − θ) u2 ρ1 ( γ − 1) M12 sin 2 β + 2 = = = tan β u1 ρ2 (γ + 1) M12 sin 2 β This can be simplified to tan θ = 2 cot β

© 2005 by CRC Press LLC

M12 sin 2 β − 1 M ( γ + cos 2β) + 2 2 1

(3.7.32)

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Fluid Mechanics

FIGURE 3.7.13 Oblique shock characteristics.

Dennard and Spencer (1964) have tabulated oblique shock properties as a function of M1. Let us now make some observations from the preceding relations. From the normal shock relations, M1 sin β
1. This defines a minimum wave angle for a given Mach number. The maximum wave angle, of course, corresponds to the normal shock or β = π/2. Therefore, the wave angle β has the following range sin −1

1 π ≤β≤ M 2

(3.7.33)

Equation (3.7.32) becomes zero at the two limits of β. Figure 3.7.13 shows a plot of θ against β for various values of M1. For each value of M1, there is a maximum value of θ. For θ < θmax, there are two possible solutions having different values of β. The larger value of β gives the stronger shock in which the flow becomes subsonic. A locus of solutions for which M2 = 1 is also shown in the figure. It is seen from the figure that with weak shock solution, the flow remains supersonic except for a small range of θ slightly smaller than θmax. Let us now consider the limiting case of θ going to zero for the weak shock solution. As θ decreases to zero, β decreases to the limiting value µ, given by M12 sin 2 µ − 1 = 0 µ = sin −1

1 M1

(3.7.34)

For this angle, the oblique shock relations show no jump in flow quantities across the wave or, in other words, there is no disturbance generated in the flow. This angle µ is called the Mach angle and the lines at inclination µ are called Mach lines. © 2005 by CRC Press LLC

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Chapter 3

FIGURE 3.7.14 Lifting flat plate.

Thin-Airfoil Theory For a small deflection angle ∆θ, it can be shown that the change in pressure in a flow at Mach M1 is given approximately by ∆p ≈ p1

γM12 M12 − 1

∆θ

(3.7.35)

This expression holds for both compression and expansion. If ∆p is measured with respect to the freestream pressure, p1, and all deflections to the freestream direction, we can write Equation (3.7.35) as γM12

p − p1 = p1

M12 − 1

θ

(3.7.36)

where θ is positive for a compression and negative for expansion. Let us define a pressure coefficient Cp , as Cp =

p − p1 q1

where q1 is the dynamic pressure and is equal to γ p1 M12 / 2. Equation (3.7.36) then gives Cp =

2θ

(3.7.37)

M12 − 1

Equation (3.7.37) states that the pressure coefficient is proportional to the local flow deflection. This relation can be used to develop supersonic thin-airfoil theory. As an example, for a flat plate at angle of attack α0 (shown in Figure 3.7.14), the pressure coefficients on the upper and lower surfaces are Cp = ∓

2α 0 M12 − 1

The lift and drag coefficients can be written as CL =

© 2005 by CRC Press LLC

(p

L

− pU ) c cos α 0 q1c

(

)

= C pL − C pU cos α 0

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Fluid Mechanics

FIGURE 3.7.15 Arbitrary thin airfoil and its components.

CD =

(p

L

− pU ) c sin α 0

(

)

= C pL − C pU sin α 0

q1c

where c is the chord length of the plate. Since α0 is small, we can write 4α 0

CL =

M12 − 1

4α 20

CD =

,

M12 − 1

(3.7.38)

A similar type of expression can be obtained for an arbitrary thin airfoil that has thickness, camber, and angle of attack. Figure 3.7.15 shows such an airfoil. The pressure coefficients on the upper and lower surfaces can be written as C pU =

dyU , M − 1 dx 2

 dy L  −  M − 1  dx  2

C pL =

2 1

2 1

(3.7.39)

For the thin airfoil, the profile may be resolved into three separate components as shown in Figure 3.7.15. The local slope of the airfoil can be obtained by superimposing the local slopes of the three components as dyU dh dh = − (α 0 + α c ( x )) + = − α( x ) + dx dx dx dy L dh dh = − (α 0 + α c ( x )) − = − α( x ) − dx dx dx

(3.7.40)

where α = α0 + αc (x) is the local total angle of attack of the camber line. The lift and drag for the thin airfoil are given by L = q1

D = q1

∫

c

0

∫ (C c

0

pL

)

− C pU dx

  dy L   dyU   C pL  − dx  + C pU  dx   dx  

Let us define an average value of α (x) as α= © 2005 by CRC Press LLC

1 c

c

∫ α( x) dx 0

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Chapter 3

Using Equation (3.7.40) and the fact that α 0 = α and α c = 0 by definition, the lift and drag coefficients for the thin airfoil can be written as CL =

CD =

4α 0 M12 − 1

 dh  2   + α C2 ( x ) + α 20   M12 − 1  dx  4

(3.7.41)

Equations (3.7.41) show that the lift coefficient depends only on the mean angle of attack whereas the drag coefficient is a linear combination of the drag due to thickness, drag due to camber, and drag due to lift (or mean angle of attack).

References Anderson, J.D. 2002. Modern Compressible Flow: With Historical Perspective, McGraw-Hill, New York. Dennard, J.S. and Spencer, P.B. 1964. Ideal-Gas Tables for Oblique-Shock Flow Parameters in Air at Mach Numbers from 1.05 to 12.0. NASA TN D-2221. Liepmann, H.W. and Roshko, A. 2002. Elements of Gas Dynamics, Dover Publications.

Further Information This section discussed only one- or two-dimensional steady, inviscid compressible flows under perfect gas assumption. Further reading on this topic can be found in the following. Kentfield, J.A.C. 1992. Nonsteady, One-Dimensional, Internal, Compressible Flows: Theory and Application, Oxford University Press. Carscallen, W.E. and Oosthuizen, P. H. 1997. Compressible Fluid Flow, McGraw-Hill. Information on advanced topics such as three-dimensional flows, viscous effects, and unsteady flows can be found in the following. Ghrist, R. W., Holmes, P., and Sullivan, M. C. 1997. Knots and Links in Three-Dimensional Flows, Vol. 165, Springer Verlag, New York. Golovachov, Y. P. and Ioffe, A. F. 1995. Numerical Simulation of Viscous Shock Layer Flows: Fluid Mechanics and Its Applications, Vol. 33, Kluwer Academic Publishers, Dordrecht, the Netherlands. Pironneau, O., Rodi, W., Ryhming, I. L. Savill, A. M., and Troung, T. V., (Eds.) 1992. Numerical Simulation of Unsteady Flows and Transition to Turbulence, Cambridge University Press. Spriggs, S. 1976. Three-Dimensional Boundary Layer Flow, 1960–1975, Gec Power Engineering Ltd. Library. A reference for fundamentals of numerical methods with programming code is Chapra, S. C. and Canale, R. P. 2001. Numerical Methods for Engineers: With Software and Programming Application, 4th ed., McGraw-Hill. Numerous sources for programming codes are available on the World Wide Web, most of which is free, and can be found by using a search engine such as www.google.com. Matrix Solvers for C++: www.vector-space.com/free.htm Fortran and MatLab codes: www.unige.ch/math/folks/hairer/software.html

© 2005 by CRC Press LLC

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3-101

3.8 Multiphase Flow John C. Chen Introduction Classic study of fluid mechanics concentrates on the flow of a single homogeneous phase, e.g., water, air, steam. However, many industrially important processes involve simultaneous flow of multiple phases, e.g., gas bubbles in oil, wet steam, dispersed particles in gas or liquid. Examples include vapor–liquid flow in refrigeration systems, steam–water flows in boilers and condensers, vapor–liquid flows in distillation columns, and pneumatic transport of solid particulates. In spite of their importance, multiphase flows are often neglected in standard textbooks. Fundamental understanding and engineering design procedures for multiphase flows are not nearly so well developed as those for single-phase flows. An added complexity is the need to predict the relative concentrations of the different phases in the multiphase flows, a need that doesn’t exist for single-phase flows. Inadequate understanding not withstanding, a significant amount of data have been collected and combinations of theoretical models and empirical correlations are used in engineering calculations. This knowledge base is briefly summarized in this section and references are provided for additional information. While discussions are provided of solid–gas flows and solid–liquid flows, primary emphasis is placed on multiphase flow of gas–liquids since this is the most often encountered class of multiphase flows in industrial applications. A multiphase flow occurs whenever two or more of the following phases occur simultaneously: gas/vapor, solids, single-liquid phase, multiple (immiscible) liquid phases. Every possible combination has been encountered in some industrial process, the most common being the simultaneous flow of vapor/gas and liquid (as encountered in boilers and condensers). All multiphase flow problems have features which are characteristically different from those found in single-phase problems. First, the relative concentration of different phases is usually a dependent parameter of great importance in multiphase flows, while it is a parameter of no consequence in single-phase flows. Second, the spatial distribution of the various phases in the flow channel strongly affects the flow behavior, again a parameter that is of no concern in single-phase flows. Finally, since the density of various phases can differ by orders of magnitude, the influence of gravitational body force on multiphase flows is of much greater importance than in the case of single-phase flows. In any given flow situation, the possibility exists for the various phases to assume different velocities, leading to the phenomena of slip between phases and consequent interfacial momentum transfer. Of course, the complexity of laminar/turbulent characteristics occurs in multiphase flows as in single-phase flows, with the added complexity of interactions between phases altering the laminar/turbulent flow structures. These complexities increase exponentially with the number of phases encountered in the multiphase problem. Fortunately, a large number of applications occur with just two phase flows, or can be treated as pseudo-two-phase flows. Two types of analysis are used to deal with two-phase flows. The simpler approach utilizes homogeneous models which assume that the separate phases flow with the same identical local velocity at all points in the fluid. The second approach recognizes the possibility that the two phases can flow at different velocities throughout the fluid, thereby requiring separate conservation equations for mass and momentum for each phase. Brief descriptions of both classes of models are given below.

Fundamentals Consider n phases in concurrent flow through a duct with cross-sectional area Ac. Fundamental quantities that characterize this flow are

© 2005 by CRC Press LLC

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Chapter 3

m˙ i = mass flow rate of ith phase ui = velocity of ith phase α i = volume fraction of ith phase in channel Basic relationships between these and related parameters are Gi = mass flux of ith phase =

(3.8.1)

m˙ i Ac

vi = superficial velocity of ith phase =

(3.8.2)

Gi ρi

ui = actual velocity of ith phase =

(3.8.3)

vi αi

xi = flow quality of ith phase =

m˙ i

=

n

Gi

(3.8.4)

n

∑m ∑G

i

i

i

i =1

α i = volume fraction of ith phase  xi  ρ u    = n i i  xi  ρ u   i i i =1

(3.8.5)

∑

In most engineering calculations, the above parameters are defined as average quantities across the entire flow area, Ac. It should be noted, however, that details of the multiphase flow could involve local variations across the flow area. In the latter situation, Gi , vi , and αi are often defined on a local basis, varying with transverse position across the flow area. Pressure drop along the flow channel is associated with gravitational body force, acceleration forces, and frictional shear at the channel wall. The total pressure gradient along the flow axis can be represented as dP  dP  dP dP = +  +       dz dz g dz a dz  f

(3.8.6)

where  dP  = − g cosθ ⋅  dz  g © 2005 by CRC Press LLC

n

∑α ρ

i i

i =1

(3.8.7)

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θ = angle of channel from vertical and  dP  = −  dz  a

n

∑G

i

i =1

dui dz

(3.8.8)

 dP  = − ρu f  dz  f 2D 2

(3.8.9)

ρ = density of multiphase mixture (3.8.10)

n

=

∑ρ α i

i

i =1

u = an average mixture velocity =

1 ρ

n

(3.8.11)

∑G

i

i =1

f = equivalent Darcy friction factor for the multiphase flow In applications, the usual requirement is to determine pressure gradient (dP/dz) and the volume fractions (αi). The latter quantities are of particular importance since the volume fraction of individual phases affects all three components of the pressure gradient, as indicated in Equation (3.8.7) to Equation (3.8.11). Correlations of various types have been developed for prediction of the volume fractions, all but the simplest of which utilize empirical parameters and functions. The simplest flow model is known as the homogeneous equilibrium model (HEM), wherein all phases are assumed to be in neutral equilibrium. One consequence of this assumption is that individual phase velocities are equal for all phases everywhere in the flow system: ui = u for all i

(3.8.12)

This assumption permits direct calculation of the volume fractions from known mass qualities: αi =

xi  xi  ρi ρ   i i =1 n

∑

(3.8.13)

The uniform velocity for all phases is the same as mixture velocity: 1 u= ρ

n

∑G

i

(3.8.14)

i =1

where 1 = ρ © 2005 by CRC Press LLC

n

 xi 

∑  ρ  i =1

i

(3.8.15)

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This homogeneous model permits direct evaluation of all three components of axial pressure gradient, if flow qualities (xi ) are known:  dP  = − g cosθ n  dz  g  xi  ρ   i i =1

∑

 dP  = −   dz  a 

 du

n

∑ G  ⋅ dz i

(3.8.16)

(3.8.17)

i =1

2  dP  = − ρu ⋅ f  dz  f 2 Df

(3.8.18)

where u and ρ are given by Equation (3.8.14) and Equation (3.8.15). Predicting the coefficient of friction (f to clear) remains a problem, even in the homogeneous model. For cases of fully turbulent flows, experience has shown that a value of 0.02 may be used as a first-order approximation for (f to clear). More-accurate estimates require empirical correlations, specific to particular classes of multiphase flows and subcategories of flow regimes. The following parts of this section consider the more common situations of two-phase flows and describe improved design methodologies specific to individual situations.

Gas–Liquid Two-Phase Flow The most common case of multiphase flow is two-phase flow of gas and liquid, as encountered in steam generators and refrigeration systems. A great deal has been learned about such flows, including delineation of flow patterns in different flow regimes, methods for estimating volume fractions (gas void fractions), and two-phase pressure drops. Flow Regimes A special feature of multiphase flows is their ability to assume different spatial distributions of the phases. These different flow patterns have been classified in flow regimes, which are themselves altered by the direction of flow relative to gravitational acceleration. Figure 3.8.1 and Figure 3.8.2 (Delhaye, 1981) show the flow patterns commonly observed for co-current flow of gas and liquid in vertical and horizontal channels, respectively. For a constant liquid flow rate, the gas phase tends to be distributed as small bubbles at low gas flow rates. Increasing gas flow rate causes agglomeration of bubbles into larger slugs and plugs. Further increasing gas flow rate causes separation of the phases into annular patterns wherein liquid concentrates at the channel wall and gas flows in the central core for vertical ducts. For horizontal ducts, gravitational force tends to drain the liquid annulus toward the bottom of the channel, resulting in stratified and stratified wavy flows. This downward segregation of the liquid phase can be overcome by kinetic forces at high flow rates, causing stratified flows to revert to annular flows. At high gas flow rates, more of the liquid tends to be entrained as dispersed drops; in the limit one obtains completely dispersed mist flow. Flow pattern maps are utilized to predict flow regimes for specific applications. The first generally successful flow map was that of Baker (1954) for horizontal flow, reproduced here in Figure 3.8.3. For vertical flows, the map of Hewitt and Roberts (1969), duplicated in Figure 3.8.4, provides a simple method for determining flow regimes. Parameters used for the axial coordinates of these flow maps are defined as follows:  ρ gρ  λ=   ρaρw  © 2005 by CRC Press LLC

12

(3.8.19)

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3-105

FIGURE 3.8.1 Flow patterns in gas–liquid vertical flow. (From Lahey, R.T., Jr. and Moody, F.I. 1977. The Thermal Hydraulics of a Boiling Water Nuclear Reactor, The American Nuclear Society, LaGrange, IL. With permission.)

FIGURE 3.8.2 Flow patterns in gas–liquid horizontal flow. © 2005 by CRC Press LLC

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FIGURE 3.8.3 Flow pattern map for horizontal flow (Baker, 1954). (From Collier, J.G. 1972. Convective Boiling and Condensation, McGraw-Hill, London. With permission.) 2  σ w   µ    ρ w   ψ=   σ   µ w   ρ     

13

G ρ

j = volumetric flux,

(3.8.20)

(3.8.21)

Void Fractions In applications of gas–liquid flows, the volume fraction of gas (αg) is commonly called “void fraction” and is of particular interest. The simplest method to estimate void fraction is by the HEM. From Equation (3.8.13), the void fraction can be estimated as αg =

xg

(

xg + 1 − xg

(3.8.22)

ρg

)ρ



where αg , xg , ρg , ρ
are cross-sectional averaged quantities. In most instances, the homogenous model tends to overestimate the void fraction. Improved estimates are obtained by using separated-phase models which account for the possibility of slip between gas and liquid velocities. A classic separated-phase model is that of Lockhart and Martinelli (1949). The top portion of Figure 3.8.5 reproduces the Lockhart–Martinelli correlation for void fraction (shown as α) as a function of the parameter X which is defined as  dP dP  X =   ÷       dz  fg   dz f where © 2005 by CRC Press LLC

12

(3.8.23)

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FIGURE 3.8.4 Flow pattern map for vertical flow (Hewitt and Roberts, 1969). (From Collier, J.G. 1972. Convective Boiling and Condensation, McGraw-Hill, London. With permission.)

 dP  = frictional pressure gradient of liquid phase flowing alone in channel  dz  f  dP  = frictional pressure gradient of gas phase flowing alone in channel  dz  fg Often, flow rates are sufficiently high such that each phase if flowing alone in the channel would be turbulent. In this situation the parameter X can be shown to be  1 − xg  Xtt =    xg 

0.9

 ρg  ρ   

0.5

 µ  µ   g

0.1

(3.8.24)

Another type of separated-phase model is the drift-flux formulation of Wallis (1969). This approach focuses attention on relative slip between phases and results in slightly different expressions depending © 2005 by CRC Press LLC

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FIGURE 3.8.5 Correlations for void fraction and frictional pressure drop (Lockhart and Martinelli, 1949). (From Collier, J.G. 1972. Convective Boiling and Condensation, McGraw-Hill, London. With permission.)

on the flow regime. For co-current upflow in two of the more common regimes, the drift-flux model gives the following relationships between void fraction and flow quality: Bubbly flow or churn-turbulent flow: αg =

xg ρg    uo ρ g     + Co  x g + 1 − x g ρ   G  

(

)

(3.8.25)

Dispersed drop (mist) flow: u ρ  1 − 1 − α g  o  α 2g + 1  G  xg =   ρ 1 − 1 − α g 1 −   ρ  g

(

)

(

)

(3.8.26)

where uo = terminal rise velocity of bubble, in bubbly flow, or terminal fall velocity of drop in churnturbulent flow Co = an empirical distribution coefficient  1.2 © 2005 by CRC Press LLC

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Pressure Drop Equation (3.8.16) through Equation (3.8.18) permit calculation of two-phase pressure drop by the homogeneous model, if the friction coefficient (f ) is known. One useful method for estimating (f ) is to treat the entire two-phase flow as if it were all liquid, except flowing at the two-phase mixture velocity. By this approach the frictional component of the two-phase pressure drop becomes    dP  = 1 + x  ρ − 1  ⋅  dP    g  dz  f     dz  fG  ρg 

(3.8.27)

where (dP/dz)f
G = frictional pressure gradient if entire flow (of total mass flux G) flowed as liquid in the channel. The equivalent frictional pressure drop for the entire flow as liquid, (dP/dz)f
g, can be calculated by standard procedures for single-phase flow. In using Equation (3.8.16) through Equation (3.8.18), the void fraction would be calculated with the equivalent homogeneous expression Equation (3.8.13). A more accurate method to calculate two-phase pressure drop is by the separated-phases model of Lockhart and Martinelli (1949). The bottom half of Figure 3.8.5 shows empirical curves for the Lockhart–Martinelli frictional multiplier, φ:  dP dP  φ i =   ÷     dz  f  dz  fi  

12

(3.8.28)

where (i) denotes either the fluid liquid phase (f ) or gas phase (g). The single-phase frictional gradient is based on the ith phase flowing alone in the channel, in either viscous laminar (v) or turbulent (t) modes. The most common case is where each phase flowing alone would be turbulent, whence one could use Figure 3.8.5 to obtain  dP  = frictional pressure gradient for two-phase flow  dz  f (3.8.29) = φ 2gtt

dP ⋅   dz  fg

where (dP/dz)fg is calculated for gas phase flowing alone and X = Xtt as given by Equation (3.8.24). The correlation of Lockhart–Martinelli has been found to be adequate for two-phase flows at low-tomoderate pressures, i.e., with reduced pressures less than 0.3. For applications at higher pressures, the revised models of Martinelli and Nelson (1948) and Thom (1964) are recommended.

Gas–Solid, Liquid–Solid Two-Phase Flows Two-phase flows can occur with solid particles in gas or liquid. Such flows are found in handling of granular materials and heterogeneous reaction processing. Concurrent flow of solid particulates with a fluid phase can occur with various flow patterns, as summarized below. Flow Regimes Consider vertical upflow of a fluid (gas or liquid) with solid particles. Figure 3.8.6 illustrates the major flow regimes that have been identified for such two-phase flows. At low flow rates, the fluid phase percolates between stationary particles; this is termed flow through a fixed bed. At some higher velocity a point is reached when the particles are all suspended by the upward flowing fluid, the drag force between particles and fluid counterbalancing the gravitational force on the particles. This is the point of minimum © 2005 by CRC Press LLC

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Chapter 3

FIGURE 3.8.6 Flow patterns for vertical upflow of solid particles and gas or liquid. (From Chen, J.C. 1994. Proc. Xth Int. Heat Transfer Conf., Brighton, U.K., 1:369–386. With permission.)

fluidization, marking the transition from fixed to fluidized beds. Increase of fluid flow rate beyond minimum fluidization causes instabilities in the two-phase mixture, and macroscopic bubbles or channels of fluid are observed in the case of gaseous fluids. In the case of liquid fluids, the two-phase mixture tends to expand, often without discrete bubbles or channels. Further increase of fluid velocity causes transition to turbulent fluidization wherein discrete regions of separated phases (fluid slugs or channels and disperse suspensions of particles) can coexist. Depending on specific operating conditions (e.g., superficial fluid velocity, particle size, particle density, etc.), net transport of solid particles with the flowing fluid can occur at any velocity equal to or greater than that associated with slug flow and turbulent flow. Further increases in fluid velocity increase the net transport of solid particles. This can occur with large-scale clusters of solid particles (as exemplified by the fast fluidization regime) or with dilute dispersions of solid particles (as often utilized in pneumatic conveying). For engineering application of fluid–solid two-phase flows, the important thresholds between flow regimes are marked by the fluid velocity for minimum fluidization, terminal slip, and saltation threshold. Minimum Fluidization The transition from flow through packed beds to the fluidization regime is marked by the minimum fluidization velocity of the fluid. On a plot pressure drop vs. superficial fluid velocity, the point of minimum fluidization is marked by a transition from a linearly increasing pressure drop to a relatively constant pressure drop as shown in Figure 3.8.7 for typical data, for two-phase flow of gas with sand particles of 280 µm mean diameter (Chen, 1996). The threshold fluid velocity at minimum fluidization is traditionally derived from the Carman–Kozeny equation,

U mf © 2005 by CRC Press LLC

(ρ =

s

)

− ρ f (φ dp) g 150µ f

2

⋅

α 2mf

(1 − α ) mf

(3.8.30)

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FIGURE 3.8.7 Transition at minimum fluidization. (From Chen, J.C. 1996. In Annual Review of Heat Transfer, Vol. VII, Begal House, Washington, D.C. With permission.)

where φ = sphericity of particles (unity for spherical particles) αmf = volumetric fraction of fluid at minimum fluidization Small, light particles have minimum fluidization voidage (αmf ) of the order 0.6, while larger particles such as sand have values closer to 0.4. An alternative correlation for estimating the point of minimum fluidization is that of Wen and Yu (1966): U mf d p ρ f µf

= (33.7 + 0.041Ga)

0.5

− 33.7

(3.8.31)

where Ga = ρ f d p3 (ρ s − ρ f )g / µ 2f . When the fluid velocity exceeds Umf , the two-phase mixture exists in the fluidized state in which the pressure gradient is essentially balanced by the gravitational force on the two-phase mixture:

[

dP = g α sρs + α f ρ f dz

]

(3.8.32)

This fluidized state exists until the fluid velocity reaches a significant fraction of the terminal slip velocity, beyond which significant entrainment and transport of the solid particles occur. Terminal Slip Velocity For an isolated single particle the maximum velocity relative to an upflowing fluid is the terminal slip velocity. At this condition, the interfacial drag of the fluid on the particle exactly balances the gravitational body force on the particle:

(

Ut = U f − U s

)

(

t

where CD = coefficient of drag on the particle. © 2005 by CRC Press LLC

)

 4d ρ − ρ  1  p s f = ⋅  CD  3ρ f  

12

(3.8.33)

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Chapter 3

The coefficient of drag on the particle (CD) depends on the particle Reynolds number:

Re p =

(

ρ f d p U f − Us µf

)

(3.8.34)

The following expressions may be used to estimate CD as appropriate: CD =

32 , Re p

Re p ≤ 1 (3.8.35)

18.5 CD = , Re 0.67 p

1 ≤ Re p ≤ 10

3

Pneumatic Conveying A desirable mode of pneumatic conveying is two-phase flow with solid particles dispersed in the concurrent flowing fluid. Such dispersed flows can be obtained if the fluid velocity is sufficiently high. For both horizontal and vertical flows, there are minimum fluid velocities below which saltation of the solid particles due to gravitational force occurs, leading to settling of the solid particles in horizontal channels and choking of the particles in vertical channels. Figure 3.8.8 and Figure 3.8.9 for Zenz and Othmer (1960) show these different regimes of pneumatic conveying for horizontal and vertical transport, respectively. Figure 3.8.8 shows that for a given rate of solids flow (W) there is a minimum superficial fluid velocity below which solid particles tend to settle into a dense layer at the bottom of the horizontal channels. Above this saltation threshold, fully dispersed two-phase flow is obtained. In the case of vertical

FIGURE 3.8.8 Flow characteristics in horizontal pneumatic conveying. (From Zeng, F.A. and Othmer, D.F. 1960. Fluidization and Fluid-Particle Systems, Reinhold, New York. With permission.)

© 2005 by CRC Press LLC

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3-113

FIGURE 3.8.9 Flow characteristics in vertical pneumatic conveying. (From Zeng, F.A. and Othmer, D.F. 1960. Fluidization and Fluid-Particle Systems, Reinhold, New York. With permission.)

transport illustrated in Figure 3.8.9, there is a minimum fluid velocity below which solid particles tend to detrain from the two-phase suspension. This choking limit varies not only with particle properties but also with the actual rate of particle flow. Well-designed transport systems must operate with superficial fluid velocities greater than these limiting saltation and choking velocities. Zenz and Othmer (1960) recommend the empirical correlations represented in Figure 3.8.10 estimating limiting superficial fluid velocities at incipient saltation or choking, for liquid or gas transport of uniformly sized particles. Note that these correlations are applicable for either horizontal or vertical concurrent flow. Figure 3.8.10 is duplicated from the original source and is based on parameters in engineering units, as noted in the figure. To operate successfully in dispersed pneumatic conveying of solid particles, the superficial fluid velocity must exceed that determined from the empirical correlations of Figure 3.8.10.

Nomenclature Ac Co dp fD G j m· P u v x z

cross-sectional flow area of channel Wallis’ distribution coefficient diameter of solid particles Darcy friction factor mass flow flux, kg/m2 · sec volumetric flow flux, m/sec mass flow rate, kg/sec pressure, N/m2 velocity in axial flow direction, m/sec superficial velocity in axial flow direction, m/sec mass flow quality axial coordinate

© 2005 by CRC Press LLC

Chapter 3

© 2005 by CRC Press LLC

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3-114

FIGURE 3.8.10 Correlations for limiting velocities in pneumatic conveying. (From Zeng, F.A. and Othmer, D.F. 1960. Fluidization and FluidParticle Systems, Reinhold, New York. With permission.)

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3-115

Greek Letters α λ φ φi ψ σ θ

volume fraction parameter in Baker flow map sphericity of solid particles frictional multiphase for pressure drag, Equation (3.8.28) parameter in Baker flow map surface tension angle from vertical

Subscripts a f g l mf p s t w

air fluid phase gas phase liquid phase minimum fluidization particle solid phase terminal slip water

References Baker, O. 1954. Design of pipelines for simultaneous flow of oil and gas, Oil Gas J. Chen, J.C. 1994. Two-phase flow with and without phase changes: suspension flows. Keynote lecture, Proc. Xth Int. Heat Transfer Conf., Brighton, U.K., 1:369–386. Chen, J.C. 1996. Heat transfer to immersed surfaces in bubbling fluidized beds, in Annual Review of Heat Transfer, Vol. VII, Bengel House, Washington, D.C. Collier, J.G. 1972. Convective Boiling and Condensation, McGraw-Hill, London. Delhaye, J.M. 1981. Two-phase flow patterns, in Two-Phase Flow and Heat Transfer, A.E. Bergles, J.G. Collier, J.M. Delhaye, G.F. Newitt, and F. Mayinger, Eds., Hemisphere Publishing, McGraw-Hill, New York. Hewitt, G.F. and Roberts, D.N. 1969. Studies of Two-Phase Flow Patterns by Simultaneous X-Ray and Flash Photography, Report AERE-M 2159. Lahey, R.T., Jr. and Moody, F.I. 1977. The Thermal Hydraulics of a Boiling Water Nuclear Reactor, The American Nuclear Society, La Grange, IL. Lockhart, R.W. and Martinelli, R.C. 1949. Proposed correlation of data for isothermal two-phase twocomponent flow in pipes, Chem. Eng. Progr., 45:39. Martinelli, R.C. and Nelson, D.B. 1984. Prediction of pressure drop during forced-circulation boiling of water, Trans. ASME, 70:695–702. Thom, J.R.S. 1964. Prediction of pressure drop during forced circulation boiling of water, Int. J. Heat Mass Transfer, 7:709–724. Wallis, G.B. 1969. One-Dimensional Two-Phase Flow, McGraw-Hill, New York. Wen, C.Y. and Yu, Y.H. 1966. A generalized method of predicting the minimum fluidization velocity, AIChE J., 12:610–612. Zenz, F.A. and Othmer, D.F. 1960. Fluidization and Fluid-Particle Systems, Reinhold, New York.

© 2005 by CRC Press LLC

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3.9 New-Newtonian Flows Thomas F. Irvine Jr. and Massimo Capobianchi Introduction An important class of fluids exists which differ from Newtonian fluids in that the relationship between the shear stress and the flow field is more complicated. Such fluids are called non-Newtonian or rheological fluids. Examples include various suspensions such as coal–water or coal–oil slurries, food products, inks, glues, soaps, polymer solutions, etc. An interesting characteristic of rheological fluids is their large “apparent viscosities”. This results in laminar flow situations in many applications, and consequently the engineering literature is concentrated on laminar rather than turbulent flows. It should also be mentioned that knowledge of non-Newtonian fluid mechanics and heat transfer is still in an early stage and many aspects of the field remain to be clarified. In the following sections, we will discuss the definition and classification of non-Newtonian fluids, the special problems of thermophysical properties, and the prediction of pressure drops in both laminar and turbulent flow in ducts of various cross-sectional shapes for different classes of non-Newtonian fluids.

Classification of Non-Newtonian Fluids It is useful to first define a Newtonian fluid since all other fluids are non-Newtonian. Newtonian fluids possess a property called viscosity and follow a law analogous to the Hookian relation between the stress applied to a solid and its strain. For a one-dimensional Newtonian fluid flow, the shear stress at a point is proportional to the rate of strain (called in the literature the shear rate) which is the velocity gradient at that point. The constant of proportionality is the dynamic viscosity, i.e., τ y, x = µ

du = µγ˙ dy

(3.9.1)

where x refers to the direction of the shear stress y the direction of the velocity gradient, and γ˙ is the shear rate. The important characteristic of a Newtonian fluid is that the dynamic viscosity is independent of the shear rate. Equation (3.9.1) is called a constitutive equation, and if τx,y is plotted against γ˙ , the result is a linear relation whose slope is the dynamic viscosity. Such a graph is called a flow curve and is a convenient way to illustrate the viscous properties of various types of fluids. Fluids which do not obey Equation (3.9.1) are called non-Newtonian. Their classifications are illustrated in Figure 3.9.1 where they are separated into various categories of purely viscous time-independent or time-dependent fluids and viscoelastic fluids. Viscoelastic fluids, which from their name possess both viscous and elastic properties (as well as memory), have received considerable attention because of their ability to reduce both drag and heat transfer in channel flows. They will be discussed in a later subsection. Purely viscous time-dependent fluids are those in which the shear stress in a function only of the shear rate but in a more complicated manner than that described in Equation (3.9.1). Figure 3.9.2 illustrates the characteristics of purely viscous time-independent fluids. In the figure, (a) and (b) are fluids where the shear stress depends only on the shear rate but in a nonlinear way. Fluid (a) is called pseudoplastic (or shear thinning), and fluid (b) is called dilatant (or shear thickening). Curve (c) is one which has an initial yield stress after which it acts as a Newtonian fluid, called Buckingham plastic, and curve (d), called Hershel-Buckley, also has a yield stress after which it becomes pseudoplastic. Curve (e) depicts a Newtonian fluid. Figure 3.9.3 shows flow curves for two common classes of purely viscous time-dependent nonNewtonian fluids. It is seen that such fluids have a hysteresis loop or memory whose shape depends upon © 2005 by CRC Press LLC

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3-117

FIGURE 3.9.1 Classification of fluids.

FIGURE 3.9.2 Flow curves of purely viscous, time-independent fluids: (a) pseudoplastic; (b) dilatant; (c) Bingham plastic; (d) Hershel–Buckley; (e) Newtonian.

FIGURE 3.9.3 Flow curves for purely viscous, time-dependent fluids: (a) thixotropic; (b) rheopectic. © 2005 by CRC Press LLC

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the time-dependent rate at which the shear stress is applied. Curve (a) illustrates a pseudoplastic timedependent fluid and curve (b) a dilatant time-dependent fluid. They are called, respectively, thixotropic and rheopectic fluids and are complicated by the fact that their flow curves are difficult to characterize for any particular application.

Apparent Viscosity Although non-Newtonian fluids do not have the property of viscosity, in the Newtonian fluid sense, it is convenient to define an apparent viscosity which is the ratio of the local shear stress to the shear rate at that point. µa =

τ γ˙

(3.9.2)

The apparent viscosity is not a true property for non-Newtonian fluids because its value depends upon the flow field, or shear rate. Nevertheless, it is a useful quantity and flow curves are often constructed with the apparent viscosity as the ordinate and shear rate as the abscissa. Such a flow curve will be illustrated in a later subsection.

Constitutive Equations A constitutive equation is one that expresses the relation between the shear stress or apparent viscosity and the shear rate through the rheological properties of the fluid. For example, Equation (3.9.1) is the constitutive equation for a Newtonian fluid. Many constitutive equations have been developed for non-Newtonian fluids with some of them having as many as five rheological properties. For engineering purposes, simpler equations are normally satisfactory and two of the most popular will be considered here. Since many of the non-Newtonian fluids in engineering applications are pseudoplastic, such fluids will be used in the following to illustrate typical flow curves and constitutive equations. Figure 3.9.4 is a qualitative flow curve for a typical pseudoplastic fluid plotted with logarithmic coordinates. It is seen in the figure that at low shear rates, region (a), the fluid is Newtonian with a constant apparent viscosity of µo (called the zero shear rate viscosity). At higher shear rates, region (b), the apparent viscosity begins to decrease until it becomes a straight line, region (c). This region (c) is called the power law region and is an important region in fluid mechanics and heat transfer. At higher shear rates than the power law region, there is another transition region (d) until again the fluid becomes Newtonian in region (e). As discussed below, regions (a), (b), and (c) are where most of the engineering applications occur.

FIGURE 3.9.4 Illustrative flow curve for a pseudoplastic fluid (a) Newtonian region; (b) transition region I; (c) power law region; (d) transition region II; (e) high-shear-rate Newtonian region.

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Power Law Constitutive Equation Region (c) in Figure 3.9.4, which was defined above as the power law region, has a simple constitutive equation: τ = Kγ˙ n

(3.9.3)

µ a = Kγ˙ n−1

(3.9.4)

or, from Equation (3.9.2):

Here, K is called the fluid consistency and n the flow index. Note that if n = 1, the fluid becomes Newtonian and K becomes the dynamic viscosity. Because of its simplicity, the power law constitutive equation has been most often used in rheological studies, but at times it is inappropriate because it has several inherent flaws and anomalies. For example, if one considers the flow of a pseudoplastic fluid (n < 1) through a circular duct, because of symmetry at the center of the duct the shear rate (velocity gradient) becomes zero and thus the apparent viscosity from Equation (3.9.4) becomes infinite. This poses conceptual difficulties especially when performing numerical analyses on such systems. Another difficulty arises when the flow field under consideration is not operating in region (c) of Figure 3.9.4 but may have shear rates in region (a) and (b). In this case, the power law equation is not applicable and a more general constitutive equation is needed. Modified Power Law Constitutive Equation A generalization of the power law equation which extends the shear rate range to regions (a) and (b) is given by µa =

µo µ o 1−n 1+ γ˙ K

(3.9.5)

Examination of Equation (3.9.5) reveals that at low shear rates, the second term in the denominator becomes small compared with unity and the apparent viscosity becomes a constant equal to µO. This represents the Newtonian region in Figure 3.9.4. On the other hand, as the second term in the denominator becomes large compared with unity, Equation (3.9.5) becomes Equation (3.9.4) and represents region (c), the power law region. When both denominator terms must be considered, Equation (3.9.5) represents region (b) in Figure 3.9.4. An important advantage of the modified power law equation is that it retains the rheological properties K and n of the power law model plus the additional property µo. Thus, as will be shown later, in the flow and heat transfer equations, the same dimensionless groups as in the power law model will appear plus an additional dimensionless parameter which describes in which of the regions (a), (b), or (c) a particular system is operating. Also, solutions using the modified power law model will have Newtonian and power law solutions as asymptotes. Equation (3.9.5) describes the flow curve for a pseudoplastic fluid (n < 1). For a dilatant fluid, (n > 1), an appropriate modified power law model is given by  K ˙ n−1  µ a = µ o 1 + γ   µo 

(3.9.6)

Many other constitutive equations have been proposed in the literature (Skelland, 1967; Cho and Hartnett, 1982; Irvine and Karni, 1987), but the ones discussed above are sufficient for a large number of engineering applications and agree well with the experimental determinations of rheological properties.

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TABLE 3.9.1 Rheological Properties Used in the Modified Power Law Equations in Figure 3.9.5 for Three Polymer Solutions of CMC-7H4 CMC 5000 wppm 2500 wppm 1500 wppm

K (N · secn/m2)

n

µo (N · sec/m2)n

2.9040 1.0261 0.5745

0.3896 0.4791 0.5204

0.21488 0.06454 0.03673

Source: Park, S. et al., Proc. Third World Conf. Heat Transfer, Fluid Mechanics, and Thermodynamics, Vol. 1, Elsevier, New York, 1993, 900–908.

Rheological Property Measurements For non-Newtonian fluids, specifying the appropriate rheological properties for a particular fluid is formidable because such fluids are usually not pure substances but various kinds of mixtures. This means that the properties are not available in handbooks or other reference materials but must be measured for each particular application. A discussion of the various instruments for measuring rheological properties is outside the scope of the present section, but a number of sources are available which describe different rheological property measurement techniques and instruments: Skelland (1967), Whorlow (1980), Irvine and Karni (1987), and Darby (1988). Figure 3.9.5 is an illustration of experimental flow curves measured with a falling needle viscometer and a square duct viscometer for polymer solutions of different concentrations. Also known in the figure as solid lines is the modified power law equation used to represent the experimental data. It is seen that Equation (3.9.5) fits the experimental data within ±2%. Table 3.9.1 lists the rheological properties used in the modified power law equations in Figure 3.9.5. It must be emphasized that a proper knowledge of these properties is vital to the prediction of fluid mechanics and heat transfer phenomena in rheological fluids.

Fully Developed Laminar Pressure Drops for Time-Independent Non-Newtonian Fluids Modified Power Law Fluids This important subject will be considered by first discussing modified power law fluids. The reason is that such solutions include both friction factor–Reynolds number relations and a shear rate parameter. The latter allows the designer to determine the shear rate region in which his system is operating and thus the appropriate solution to be used, i.e., regions (a), (b), or (c) in Figure 3.9.4. For laminar fully developed flow of a modified power law fluid in a circular duct, the product of the friction factor and a certain Reynolds number is a constant depending on the flow index, n, and the shear rate parameter, β. f D ⋅ Re m = constant(n,β)

(3.9.7)

where fD is the Darcy friction factor and Rem the modified power law Reynolds number, i.e.,

fD =

Re m =

2

∆p D L H ρu 2

(Darcy friction factor)5

ρuDH µ*

5 It should be noted that the Fanning friction factor is also used in the technical literature. The Fanning friction factor is ¼ of the Darcy friction factor, and will be characterized by the symbol fF .

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FIGURE 3.9.5 Experimental measurements of apparent viscosity vs. shear rate for polymer solutions (CMC-7H4) at different concentrations. (From Park, S. et al., in Proc. Third World Conf. Heat Transfer, Fluid Mechanics, and Thermodynamics, Vol. 1, Elsevier, New York, 1993, 900–908.

µ* =

β=

µo 1+ β

µo  u  K  DH 

1− n

where β is the shear rate parameter mentioned previously which can be calculated by the designer for a certain operating duct (u and d) and a certain pseudoplastic fluid (µo, K, n). The solution for a circular tube has been calculated by Brewster and Irvine (1987) and the results are shown in Figure 3.9.6 and in Table 3.9.2. Referring to 3.9.6, we can see that when the log10 β is less than approximately –2, the duct is operating in region (a) of Figure 3.9.4 which is the Newtonian region and therefore classical Newtonian solutions can be used. Note that in the Newtonian region, Rem reverts to the Newtonian Reynolds number given by Re N =

© 2005 by CRC Press LLC

ρuDH µo

(3.9.8)

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FIGURE 3.9.6 Product of friction factor and modified Reynolds number vs. log10 β for a circular duct. (From Brewster, R.A. and Irvine, T.F., Jr., Wärme und Stoffübertragung, 21, 83–86, 1987. TABLE 3.9.2 Summary of Computed Values of fD · Rem for Various Values of n and β for a Circular Duct fD · Rem for Flow Index: n = b

1.0

0.9

0.8

0.7

0.6

0.5

10–5

64.000

64.000

64.000

64.000

63.999

63.999

–4

64.000

63.999

63.997

63.995

63.993

63.990

10–3

64.000

63.987

63.972

63.953

63.930

63.903

10–2

64.000

63.873

63.720

63.537

63.318

63.055

10–1

64.000

62.851

61.519

59.987

58.237

56.243

0

64.000

58.152

52.377

46.761

41.384

36.299

101

64.000

54.106

45.597

38.308

32.082

26.771

102

64.000

53.371

44.458

36.985

30.716

25.451

103

64.000

53.291

44.336

36.845

30.573

25.314

4

64.000

53.283

44.324

36.831

30.559

25.300

105

64.000

53.282

44.323

36.830

30.557

25.299

Exact solution

64.000

53.282

44.323

36.829

30.557

25.298

10

10

10

Source: Brewster, R.A. and Irvine, T.F., Jr., Wärme und Stoffübertragung, 21, 83–86, 1987. With permission.

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When the value of log10 β is approximately in the range –2 ≤ log10 β ≤ 2, the duct is operating in the transition region (b) of Figure 3.9.4 and the values of fD · Rem must be obtained from Figure 3.9.6 or from Table 3.9.2. When log10 β is greater than approximately 2, the duct is operating in the power law region (c) of Figure 3.9.4 and power law friction factor Reynolds number relations can be used. They are also indicated in Figure 3.9.6 and Table 3.9.2. In this region, Rem becomes the power law Reynolds number given by Re g =

ρu 2−n DHn K

(3.9.9)

For convenience, Brewster and Irvine (1987) have presented a correlation equation which agrees within 0.1% with the results tabulated in Table 3.9.2. f D ⋅ Re m =

1 + 64

1+ β β n 3 n + 1  2 3n +3  4n 

(3.9.10)

Thus, Equation (3.9.10) contains all of the information required to calculate the circular tube laminar fully developed pressure drop for a pseudoplastic fluid depending upon the shear rate region(s) under consideration, i.e., regions (a), (b), or (c) of Figure 3.9.4. Note that in scaling such non-Newtonian systems, both Rem and β must be held constant. Modified power law solutions have been reported for two other duct shapes. Park et al. (1993) have presented the friction factor–Reynolds number relations for rectangular ducts and Capobianchi and Irvine (1992) for concentric annular ducts. Power Law Fluids Since the power law region of modified power law fluids (log10 β ≥ 2) is often encountered, the friction factor–Reynolds number relations will be discussed in detail in this subsection. An analysis of power law fluids which is most useful has been presented by Kozicki et al. (1967). Although the method is approximate, its overall accuracy (±5%) is usually sufficient for many engineering calculations. His expression for the friction factor–Reynolds number product is given by f D ⋅ Re * = 2 6 n

(3.9.11)

where Re * = Kozicki Reynolds number, Re * =

Re g n

 a + bn  8 n−1  n 

(3.9.12)

and a and b are geometric constants which depend on the cross-sectional shape of the duct. For example, for a circular duct, a = 0.25 and b = 0.75. Values of a and b for other duct shapes are tabulated in Table 3.9.3. For additional duct shapes in both closed and open channel flows, Kozicki et al. (1967) may be consulted.

Fully Developed Turbulent Flow Pressure Drops In a number of engineering design calculations for turbulent flow, the shear rate range falls in region (c) of Figure 3.9.4. Thus, power law relations are appropriate for such pressure drop calculations.

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TABLE 3.9.3 Constants a and b for Various Duct Geometrics Used in the Method Due to Kozicki et al. (1967) α*

Geometry

a

b

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0a 0.0 0.25 0.50 0.75 1.00

0.4455 0.4693 0.4817 0.4890 0.4935 0.4965 0.4983 0.4992 0.4997 0.5000 0.5000 0.3212 0.2440 0.2178 0.2121

0.9510 0.9739 0.9847 0.9911 0.9946 0.9972 0.9987 0.9994 1.0000 1.0000 1.0000 0.8482 0.7276 0.6866 0.8766

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00b 2φ (deg)

0.3084 0.3018 0.2907 0.2796 0.2702 0.2629 0.2575 0.2538 0.2515 0.2504 0.2500

0.9253 0.9053 0.8720 0.8389 0.8107 0.7886 0.7725 0.7614 0.7546 0.7510 0.7500

10 20 40 60 80 90 N

0.1547 0.1693 0.1840 0.1875 0.1849 0.1830

0.6278 0.6332 0.6422 0.6462 0.6438 0.6395

4 5 6 8

0.2121 0.2245 0.2316 0.2391

0.6771 0.6966 0.7092 0.7241

a

Parallel plates. Circle. Source: Irvine, T.F., Jr. and Karni, J., in Handbook of Single Phase Convective Heat Transfer, John Wiley and Sons, New York, 1987, pp 20-1–20-57.

b

Hartnett and Kostic (1990) have investigated the various correlations which have appeared in the literature for circular tubes and have concluded that for a circular tube the relation proposed by Dodge and Metzner (1959) is the most reliable for pseudoplastic fluids. It is given by 1 12

fF © 2005 by CRC Press LLC

=

1−(1 2 n ) 4 .0  − 0.40 ⋅ log10 Re ′g ( f F ) 0.75    n1.2 n

(3.9.13)

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FIGURE 3.9.7 Dodge and Metzner relation between Fanning friction factor and Re′g. (From Dodge, D.W. and Metzner, A.B., AIChE J., 5, 189–204, 1959.)

where fF is the Fanning friction factor and    81−n   Re ′g = Re g  n   3n + 1     4n     

(3.9.14)

Figure 3.9.7 is a graphical representation of Equation (3.9.13) which indicates the Dodge and Metzner experimental regions by solid lines, and by dashed lines where the data are extrapolated outside of their experiments. For noncircular ducts in turbulent fully developed flow, only a limited amount of experimental data are available. Kostic and Hartnett (1984) suggest the correlation: 1 12

fF

=

1−(1 2 n ) 4  − 0.40 ⋅ log10 Re * ( f F )   n 0.5 n 0.75

(3.9.15)

where fF is again the Fanning friction factor and Re* is the Kozicki Reynolds number: Re * =

Re g n

 a + bn  8 n−1  n 

(3.9.16)

and a and b are geometric constants given in Table 3.9.3.

Viscoelastic Fluids Fully Developed Turbulent Flow Pressure Drops Viscoelastic fluids are of interest in engineering applications because of reductions of pressure drop and heat transfer which occur in turbulent channel flows. Such fluids can be prepared by dissolving small amounts of high-molecular-weight polymers, e.g., polyacrylamide, polyethylene oxide (Polyox), etc., in © 2005 by CRC Press LLC

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FIGURE 3.9.8 Reduction in friction factors for polyethylene oxide (Polyox) solutions in a small-diameter capillary tube. (From Cho, Y.I. and Harnett, J.P., Adv. Heat Transfer, 15, 59–141, 1982. With permission.)

water. Concentrations as low as 5 parts per million by weight (wppm) result in significant pressure drop reductions. Figure 3.9.8 from Cho and Hartnett (1982) illustrates the reduction in friction factors for Polyox solutions in a small-diameter capillary tube. It is seen that at zero polymer concentration the data agree with the Blasius equation for Newtonian turbulent flow. With the addition of only 7 wppm of Polyox, there is a significant pressure drop reduction and for concentrations of 70 wppm and greater all the data fall on the Virk line which is the maximum drag-reduction asymptote. The correlations for the Blasius and Virk lines as reported by Cho and Hartnett (1982) are fF =

0.079 Re1 4

(Blasius)

f F = 0.20 Re −a0.48

(Virk )

(3.9.17)

(3.9.18)

At the present time, no generally accepted method exists to predict the drag reduction between the Blasius and Virk lines. Kwack and Hartnett (1983) have proposed that the amount of drag reduction between those two correlations is a function of the Weissenberg number, defined as ws =

λu DH

(3.9.19)

where λ = characteristic time of the viscoelastic fluid. They present correlations which allow the friction factor to be estimated at several Reynolds numbers between the Blasius and Virk lines. Fully Developed Laminar Flow Pressure Drops The above discussion on viscoelastic fluids has only considered fully developed turbulent flows. Laminar fully developed flows can be considered as nonviscoelastic but purely viscous non-Newtonian. Therefore, the method of Kozicki et al. (1967) may be applied to such situations once the appropriate rheological properties have been determined.

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Nomenclature a b c di do fD fF h K n N Reg

= = = = = = = = = = = =

duct shape geometric constant duct shape geometric constant duct width (see Table 3.9.3) (m) concentric annuli inner diameter (see Table 3.9.3) (m) concentric annuli outer diameter (see Table 3.9.3) (m) Darcy friction factor Fanning friction factor duct height (see Table 3.9.3) (m) fluid consistency (Nsn/m2) flow index number of sides in polygon (see Table 3.9.3) generalized Reynolds number, Re g =

ρu 2−n DHn K

Rem = modified power law Reynolds number, Re m =

ρuDH µ*

ReN = modified power law Reynolds number Newtonian asymptote, Re N =

ρuDH µo

Rea = apparent Reynolds number Re a =

Re g  3n + 1  4n 

n −1

8 n−1

Re* = Kozicki Reynolds number Re * =

ρu 2−n DHn K  

n

a + bn  n−1 8 n 

Re′g = Metzner Reynolds number    81−n   Re ′g = Re g  n   3n + 1     4n      u t ws x y

= average streamwise velocity (m/sec) = time (sec) = Weissenberg number = direction of shear stress (m) = direction of velocity gradient (m)

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Greek α* = duct aspect ratio in Table 3.9.3 β = shear rate parameter µ β= o K γ˙ ∆P λ µa µo µ∞ µ*

= = = = = = =

 u  D   H

1− n

shear rate (L/sec) presure drop (N/m2) characteristic time of viscoelastic fluid (sec) apparent viscosity (N · sec/m2) zero shear rate viscosity (N · sec/m2) high shear rate viscosity (N · sec/m2) reference viscosity µ* =

µo 1+ β

(N ⋅ sec m ) 2

τo = yield stress (N/m2) τy,x = shear stress (N/m2) φ = half apex angle (see Table 3.9.3) (°)

References Brewster, A.A. and Irvine, T.F. Jr. 1987. Similtude considerations in laminar flow of power law fluids in circular ducts, Wärme und Stoffübertagung, 21:83–86. Capobianchi, M. and Irvine, T.F. Jr. 1992. Predictions of pressure drop and heat transfer in concentric annular ducts with modified power law fluids, Wärme und Stoffübertagung, 27:209–215. Cho, Y.I. and Hartnett, J.P. 1982. Non-Newtonian fluids in circular pipe flow, in Adv. Heat Transfer, 15:59–141. Darby, R. 1988. Laminar and turbulent pipe flows of non-Newtonian fluids, in Encyclopedia of Fluid Mechanics, Vol. 7, Gulf Publishing, Houston, 7:20–53. Dodge, D.W. and Metzner, A.B. 1959. Turbulent flow of non-Newtonian systems, AIChE J., 5:189–204. Harnett, J.P. and Kostic, M. 1990. Turbulent Friction Factor Correlations for Power Law Fluids in Circular and Non-Circular Channels, Int. Comm. Heat and Mass Transfer, 17:59–65. Irvine, T.F. Jr. and Karni, J. 1987. Non-Newtonian fluid flow and heat transfer, in Handbook of Single Phase Convective Heat Transfer, pp. 20-1–20-57, John Wiley and Sons, New York. Kostic, M. and Hartnett, J.P. 1984. Predicting turbulent friction factors of non-Newtonian fluids in noncircular ducts, Int. Comm. Heat and Mass Transfer, 11:345–352. Kozicki, W., Chou, C.H., and Tiu, C. 1967. Non-Newtonian flow in ducts of arbitrary cross-sectional shape, Can. J. Chem. Eng., 45:127–134. Kwack, E.Y. and Hartnett, J.P. 1983. Empirical correlations of turbulent friction factors and heat transfer coefficients for viscoelastic fluids, Int. Comm. Heat and Mass Transfer, 10:451–461. Park, S., Irvine, T.F. Jr., and Capobianchi, M. 1993. Experimental and numerical study of friction factor for a modified power law fluid in a rectangular duct, Proc. Third World Conf. Heat Transfer, Fluid Mechanics, and Thermodynamics, Vol. 1, Elsevier, New York, 1:900–908. Skelland, A.H.P. 1967. Non-Newtonian Flow and Heat Transfer, John Wiley and Sons, New York. Whorlow, R.W. 1980. Rheological Techniques, Halsted Press, New York.

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Further Information It is not possible to include all of the interesting non-Newtonian topics in a section of this scope. Other items which may be of interest and importance are listed below along with appropriate references: hydrodynamic and thermal entrance lengths, Cho and Hartnett (1982); non-Newtonian flow over external surfaces, Irvine and Karni (1987); chemical, solute, and degradation effects in viscoelastic fluids, Cho and Harnett (1982); general references, Skelland (1967), Whorlow (1980), and Darby (1988).

3.10 Tribology, Lubrication, and Bearing Design Francis E. Kennedy, E. Richard Booser, and Donald F. Wilcock Introduction Tribology — the science and technology of contacting surfaces involving friction, wear, and lubrication — is extremely important in nearly all mechanical components. A major focus of the field is on friction, its consequences, especially wear, and its reduction through lubrication and material surface engineering. The improper solution of tribological problems is responsible for huge economic losses in society, including shortened component lives, excessive equipment down time, and large expenditures of energy. It is particularly important that engineers use appropriate means to reduce friction and wear in mechanical systems through the proper selection of bearings, lubricants, and materials for all contacting surfaces. The aim of this chapter is to assist in that endeavor.

Sliding Friction and Its Consequences Coefficient of Friction If two stationary contacting bodies are held together by a normal force W, and if a tangential force is applied to one of them, the tangential force can be increased until it reaches a magnitude sufficient to initiate sliding. The ratio of the friction force at incipient sliding to the normal force is known as the static coefficient of friction, fs. After sliding begins, the friction force always acts in the direction opposing motion and the ratio between that friction force and the applied normal force is the kinetic coefficient of friction, fk. Generally, fk is slightly smaller than fs and both coefficients are independent of the size or shape of the contacting surfaces. Both coefficients are very much dependent on the materials and cleanliness of the two contacting surfaces. For ordinary metallic surfaces, the friction coefficient is not very sensitive to surface roughness. For ultrasmooth or very rough surfaces, however, the friction coefficient can be larger. Typical friction coefficient values are given in Table 3.10.1. Generally, friction coefficients are greatest TABLE 3.10.1

Some Typical Friction Coefficientsa

Material Pair

Static Friction Coefficient, fs In Air

Mild steel vs. mild steel Mild steel vs. copper

0.75 0.53

Copper vs. copper Tungsten carbide vs. copper Tungsten carbide vs. tungsten carbide Mild steel vs. PTFE

1.3 0.35 0.2 0.04

In Vacuo 0.5 (oxidized) 2.0 (clean) 21.0 0.4

Kinetic Friction Coefficient, fk In Air, Dry

Oiled

0.57 0.36

0.16 0.18

0.8 0.4 0.15 0.05

0.1

0.04

a The friction coefficient values listed in this table were compiled from several of the references listed at the end of this section.

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when the two surfaces are identical metals, slightly lower with dissimilar but mutually soluble metals, still lower for metal against nonmetal, and lowest for dissimilar nonmetals. The kinetic coefficient of friction, fk, for metallic or ceramic surfaces is relatively independent of sliding velocity at low and moderate velocities, although often a slight decrease occurs in fk at higher velocities. With polymers and soft metals, the friction coefficient may increase with increasing velocity until a peak is reached, after which the friction may decrease with further increases in velocity or temperature. The decrease in kinetic friction coefficient with increasing velocity, which may become especially pronounced at higher sliding velocities, can be responsible for friction-induced vibrations (stick-slip oscillations) of the sliding systems. Such vibrations are an important design consideration for clutches and braking systems; they can also be important in the accurate control and positioning of robotic mechanisms and precision manufacturing systems. Wear Wear is the unwanted removal of material from solid surfaces by mechanical means; it is one of the leading reasons for the failure and replacement of manufactured products. It has been estimated that the costs of wear, which include repair and replacement along with equipment downtime, constitute up to 6% of the U.S. gross national product (Rabinowicz, 1995). Wear can be classified into four primary types: sliding wear, abrasion, erosion, and corrosive wear. Owing to its importance, wear and its control have been the subject of several handbooks (Peterson and Winer, 1980; Blau, 1992) that the interested reader may consult for further information. Types of Wear Sliding wear occurs to some degree whenever solid surfaces are in sliding contact; the two predominant sliding wear mechanisms are adhesion and surface fatigue. Adhesive wear is caused by strong adhesive forces between the two surfaces within the real area of contact. It results in the removal of small particles from at least one of the surfaces, usually the softer one. These particles can then transfer to the other surface or mix with other material from both surfaces before being expelled as loose wear debris. Adhesive wear can be particularly severe for surfaces that have a strong affinity for each other, such as those made from identical metals. Surface fatigue wear occurs when repeated sliding or rolling/sliding over a wear track results in the initiation of surface or subsurface cracks and the propagation of those cracks produces wear particles in ductile materials by a process called delamination. With brittle materials, sliding wear often occurs by a surface fracture process. After an initial transition or “running-in” period, sliding wear tends to reach a steady-state rate, which is approximated by the following Archard (or Holm–Archard) wear equation: V = K ∗ W ∗ s/H

(3.10.1)

where V = volume of worn material; K = dimensionless wear coefficient; s = sliding distance; W = normal load between the surfaces; and H = hardness of the softer of the two contacting surfaces. The dimensionless wear coefficient gives an indication of the tendency of a given material combination to wear; relative wear coefficient values are given in Figure 3.10.1. In general, wear coefficients are highest for identical metals sliding without lubrication, and wear is decreased by adding a lubricant and by having dissimilar material pairs. Abrasive wear occurs when a hard, rough surface slides against a softer surface (two-body abrasion) or when hard particles slide between softer surfaces (three-body abrasion). This process usually results in material removal by plowing or chip formation, especially when the abraded surface is metallic; surface fracture can occur during abrasion of brittle surfaces. In fact, abrasion mechanisms are similar to those of grinding and lapping, which could be considered intentional abrasion. Consideration of the cutting and plowing processes shows that abrasive wear obeys the same equation (Equation 3.10.1) as sliding wear does (Archard, 1980; Rabinowicz, 1995). Typical wear coefficients for abrasive wear are given in Figure 3.10.1. Because the relative size, hardness, and sharpness of the abrading particles, or surface

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3-131

FIGURE 3.10.1 Typical values of wear coefficient for sliding and abrasive wear. (Modified from Rabinowicz, E., in Wear Control Handbook, M.B. Peterson and W.O. Winder, Eds., 475–506, ASME, New York, 1980, and Rabinowicz, E., Friction and Wear of Materials, 2nd ed., John Wiley & Sons, New York, 1995.)

asperities, also affect abrasive wear rates, the wear coefficients for abrasion must include recognition of those factors (Rabinowicz, 1995). Erosion occurs when solid particles or liquid droplets impinge on a solid surface. When impingement is on a ductile metallic surface, the wear process is similar to that caused by abrasion and is dominated by plastic deformation. Brittle surfaces, on the other hand, tend to erode by surface fracture mechanisms. The material removal rate is dependent on the angle of attack of the particles, with erosion reaching a peak at low angles (about 20°) for ductile surfaces and at high angles (90°) for brittle materials. In either case, the wear rate is proportional to the mass rate of flow of the particles and to their kinetic energy; it is inversely proportional to the hardness of the surface and the energy-absorbing potential (or toughness) of the impinged surface (Schmitt, 1980). Although erosion is usually detrimental, it can be used beneficially in such material removal processes as sand blasting and abrasive water-jet machining. Corrosive wear results from a combination of chemical and mechanical action. It involves the synergistic effects of chemical attack (corrosion) of the surface, followed by removal of the corrosion products by a wear mechanism to expose the metallic surface, and then repetition of those processes. Because many corrosion products act to protect the surfaces from further attack, the removal of those films by wear acts to accelerate the rate of material removal. Corrosive wear can become particularly damaging when it acts in a low-amplitude oscillatory contact, which may be induced by vibration, in which case it is called fretting corrosion. Means for Wear Reduction The following actions can be taken to limit sliding wear: • Ensure that the sliding surfaces are well lubricated. This can best be accomplished by a liquid lubricant; however, grease or solid lubricants such as graphite or molybdenum disulfide can sometimes be effective when liquid lubricants cannot be used. • Choose dissimilar materials for sliding pairs. • Use hardened surfaces. • Add wear-resistant coatings to the contacting surfaces (see the following subsection). • Reduce normal loads acting on the contact. • Reduce surface temperatures (particularly important for polymer surfaces). © 2005 by CRC Press LLC

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Chapter 3

To reduce abrasive wear: • • • •

Use hardened surfaces. Add a hard surface coating. Reduce the roughness of hard surfaces in contact with softer surfaces. Provide for the removal of abrasive particles from contacting surfaces. This can be done by flushing surfaces with liquid and/or filtering liquid coolants and lubricants. • Reduce the size of abrasive particles. To reduce erosion: • • • • •

Modify the angle of impingement of solid particles or liquid droplets. Provide for the removal of solid particles from the impinging stream of fluid. Use hardened surfaces. Use tough materials for surfaces. Add protective coating to surfaces.

Surface Engineering for Friction and Wear Reduction Surface treatments have long been an important remedy for wear problems —an importance that has grown in recent years with the introduction of new techniques to harden surfaces or apply hard surface coatings (Bhushan, 1999). Available processes and characteristics for treating a steel substrate are listed in Table 3.10.2. Thermal transformation hardening processes are used to harden ferrous (primarily steel) surfaces by heating the surface rapidly, transforming it to austenite, and then quenching it to form martensite. The source of heat can be one of the following: an oxyacetylene or oxypropane flame (flame hardening); eddy currents induced by a high-frequency electric field (induction hardening); a beam from a high-power laser (laser hardening); or a focused electron beam (electron beam hardening). The depth and uniformity of the hard layer depend on the rate and method of heating. These processes are characterized by a short process time and all except electron beam hardening (which requires a moderate vacuum) can be done in air. Thermal diffusion processes involve the diffusion of atoms into surfaces to create a hard layer. In the most widely used of these processes — carburizing (or case hardening) — carbon diffuses into a lowcarbon steel surface to produce a hard, carbon-rich “case.” The hardness and thickness of the case depend TABLE 3.10.2

Characteristics of Surface Treatment Processes for Steel Coating or Treated Layer

Process Surface hardening Flame or induction hardening Laser or electron beam hardening Carburizing Carbonitriding Citriding Boronizing Coating Chrome plating Electroless nickel Hardfacing Thermal spraying PVD CVD PACVD Ion implantation

© 2005 by CRC Press LLC

Hardness (HV)

Thickness (µm)

Substrate Temperature (°C)

500–700 500–700

250–6000 200–1000

800–1000 950–1050

650–900 650–900 700–1200 1400–1600

50–1500 25–500 10–200 50–100

800–950 800–900 500–600 900–1100

850–1250 500–700 800–2000 400–2000 100–3000 1000–3000 1000–5000 750–1250

1–500 0.1–500 500–50000 50–1500 0.05–10 0.5–100 0.5–10 0.01–0.25

25–100 25–100 1300–1400 0

Im

Im

Im I

Complex plane sketch

θ=0 I V

θ

V

V

Re

Re

θ

Re I

Explanation

The current is in phase with the voltage.

The current “leads” the voltage.

The current “lags” the voltage.

Power factor

Unity

Leading, < 1

Lagging, < 1

Reactive power

0

Negative

Positive

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5-42

Chapter 5

FIGURE 5.5.3 Circuit for Example 5.5.1.

Example 5.5.1 For the circuit shown in Figure 5.5.3: 1. Calculate the complex power for the load. 2. Correct the power factor by adding a suitable reactance in parallel with the load. Solution. 1. The circuit of Figure 5.5.4 is an inductive load. The total impedance is Z = R + jX L = 50 + j 86.7 Ω = 100∠60° The power factor is then pf = cos θ = cos 60° = 0.5 (lagging) The current drawn from the source by the load is IS =

VS 117∠0° = = 1.17∠ − 60° Z 100∠60°

and the average power is found to be P = V˜S I˜S cos θ = 117 × 1.17 cos 60° = 68.4 W while the reactive power is QL = V˜S I˜S sin θ = 117 × 1.17 sin 60° = 119 VAR Figure 5.5 shows the power triangle for the circuit. 2. The unity power factor for the circuit can be obtained by simply reducing the power factor angle θ to 0°. This can be accomplished by adding a capacitor to the circuit that requires –119 VAR of reactive power. The capacitive power and the inductive power will then cancel each other in the power triangle, resulting in a unity power factor, as shown in Figure 5.5.5. The value of capacitive reactance, Xc, required to cancel the reactive power due to the inductance is found most easily by observing that the total reactive power in the circuit must be the sum of the reactive power due to the capacitance and that due to the inductance. Observing that the capacitor sees the same voltage as the RL load, because of the parallel connection, we can write XC =

© 2005 by CRC Press LLC

V˜S2 117 2 = 115 Ω QC 119

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Electrical Engineering

FIGURE 5.5.4 Power triangle for the circuit.

FIGURE 5.5.5 Power factor correction.

From the expression for the reactance, it is then possible to compute the value of the capacitor that will cancel the reactive power due to the inductor: C=

1 1 = = 23.1 µF ωXC 377 × 115

The reactive component of power needed by the inductance is now balanced by the capacitance, and all the power delivered by the source is real power. The power factor is 1. Example 5.5.2 The instrument used to measure power is called a wattmeter. The external part of a wattmeter consists of four connections and a metering mechanism that displays the amount of real power dissipated by a circuit. The external and internal appearances of a wattmeter are depicted in Figure 5.5.6. Inside the wattmeter are two coils: a current-sensing coil and a voltage-sensing coil. In this example, we assume for simplicity that the impedance of the current-sensing coil, CI, is zero and the impedance of the voltagesensing coil, CV , is infinite. In practice, this will not necessarily be true; some correction mechanism will be required to account for the impedance of the sensing coils. A wattmeter should be connected as shown in Figure 5.5.7, to provide both current and voltage measurements. We see that the current-sensing coil is placed in series with the load and the voltagesensing coil is placed in parallel with the load. In this manner, the wattmeter is seeing the current through and the voltage across the load. Remember that the power dissipated by a circuit element is related to these two quantities. The wattmeter, then, is constructed to provide a readout of the product of the rms values of the load current and the voltage, which is the real power absorbed by the load: P = Re (S) = Re (VI*). 1. For the circuit shown in Figure 5.5.8, show the connections of the wattmeter, and find the power dissipated by the load. 2. Show the connections that will determine the power dissipated by R2. What should the meter read?

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Chapter 5

FIGURE 5.5.6 Wattmeter.

FIGURE 5.5.7 Circuit for Example 5.5.2.

FIGURE 5.5.8 Circuit for Example 5.5.2.

FIGURE 5.5.9 Circuit for Example 5.5.2.

Solution. 1. To measure the power dissipated by the load, we must know the current through and the voltage across the entire load circuit. This means that the wattmeter must be connected as shown in Figure 5.5.9. The wattmeter should read:

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Electrical Engineering

FIGURE 5.5.10 Circuit for Example 5.5.2.

(

P = Re VS I *

)

*   156 ∠0°    156     2 = Re  ∠0°    R1 + R2 + jωL    2       *   110∠0°   = Re 110∠0°     15 + j 7.54   

*    110∠0°   110 2 Re 110∠0°  = Re     16 79 ∠ 26 69 ° 16 79 ∠ − 26 69 . . . . °    

= Re(720.67∠26.69°) = 643.88 W 2. To measure the power dissipated by R2 alone, we must measure the current through R2 and the voltage across R2 alone. The connection is shown in Figure 5.5.10. The meter will read: P = I˜ 2 R2  110 =  152 + 7.54 2 

(

2

 12 × 5  

)

2

  110 2 × 5 = 215 W = 2 2   15 + 7.54 

(

)

The measurement and correction of the power factor for the load are an extremely important aspect of any engineering application in industry that requires the use of substantial quantities of electric power. In particular, industrial plants, construction sites, heavy machinery, and other heavy users of electric power must be aware of the power factor their loads present to the electric utility company. As was already observed, a low power factor results in greater current draw from the electric utility and in greater line losses. Thus, computations related to the power factor of complex loads are of great practical utility to any practicing engineer.

© 2005 by CRC Press LLC

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5-46

Chapter 5

FIGURE 5.5.11 Ideal transformer.

Transformers AC circuits are very commonly connected to each other by means of transformers. A transformer is a device that couples two AC circuits magnetically rather than through any direct conductive connection and permits a “transformation” of the voltage and current between one circuit and the other (for example, by matching a high-voltage, low-current AC output to a circuit requiring a low-voltage, high-current source). Transformers play a major role in electric power engineering and are a necessary part of the electric power distribution network. The objective of this section is to introduce the ideal transformer and the concepts of impedance reflection and impedance matching. The Ideal Transformer The ideal transformer consists of two coils that are coupled to each other by some magnetic medium. There is no electrical connection between the coils. The coil on the input side is termed the primary, and that on the output side the secondary. The primary coil is wound so that it has n1 turns, while the secondary has n2 turns. We define the turns ratio N as N=

n2 n1

(5.5.14)

Figure 5.5.11 illustrates the convention by which voltages and currents are usually assigned at a transformer. The dots in Figure 5.5.11 are related so the polarity of the coil voltage: coil terminals marked with a dot have the same polarity. Since an ideal inductor acts as a short circuit in the presence of DC currents, transformers do not perform any useful function when the primary voltage is DC. However, when a time-varying current flows in the primary winding, a corresponding time-varying voltage is generated in the secondary because of the magnetic coupling between the two coils. This behavior is due to Faraday’s law, as will be explained in Section 5.12. The relationship between primary and secondary current in an ideal transformer is very simply stated as follows: V2 = NV1 I2 =

I1 N

(5.5.15)

An ideal transformer multiples a sinusoidal input voltage by a factor of N and divides a sinusoidal input current by a factor of N. If N is greater than 1, the output voltage is greater than the input voltage and the transformer is called a step-up transformer. If N is less than 1, then the transformer is called a step-down transformer, since V2 is now smaller than V1. An ideal transformer can be used in either direction (i.e., either of its coils may be viewed as the input side or primary). Finally, a transformer with N = 1 is called an isolation transformer and may perform a very useful function if one needs to electrically isolate two circuits from each other; note that any DC currents at the primary will not appear at the secondary coil. An important property of ideal transformers is conservation of power; one can easily verify that an ideal transformer conserves power, since S1 = I1* V1 = NI *2 © 2005 by CRC Press LLC

V2 = I *2 V2 = S2 N

(5.5.16)

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Electrical Engineering

FIGURE 5.5.12 Center-tapped transformer.

That is, the power on the primary side equals that on the secondary. In many practical circuits, the secondary is tapped at two different points, giving rise to two separate output circuits, as shown in Figure 5.5.12. The most common configuration is the center-tapped transformer, which splits the secondary voltage into two equal voltages of half the original amplitude. The most common occurrence of this type of transformer is found at the entry of a power line into a household, where the 240-VAC line is split into two 120-VAC lines (this may help explain why both 240and 120-VAC power are present in your house).

Three-Phase Power The material presented so far in this chapter has dealt exclusively with single-phase AC power, that is, with single sinusoidal sources. In fact, most of the AC power used today is generated and distributed as three-phase power, by means of an arrangement in which three sinusoidal voltages are generated out of phase with each other. The primary reason is efficiency: the weight of the conductors and other components in a three-phase system is much lower than in a single-phase system delivering the same amount of power. Further, while the power produced by a single-phase system has a pulsating nature, a threephase system can deliver a steady, constant supply of power. A three-phase generator producing three balanced voltages — that is, voltages of equal amplitude and frequency displaced in phase by 120° — has the property of delivering constant instantaneous power. Another important advantage of three-phase power is that, as will be explained in Section 5.12, threephase motors have a nonzero starting torque, unlike their single-phase counterpart. The change to three-phase AC power system from the early DC system proposed by Edison was therefore due to a number of reasons: the efficiency resulting from transforming voltages up and down to minimize transmission losses over long distances; the ability to deliver constant power (an ability not shared by singleand two-phase AC systems); a more efficient use of conductors; and the ability to provide starting torque for industrial motors. To begin the discussion of three-phase power, consider a three-phase source connected in the wye (or Y) configuration, as shown in Figure 5.5.13. Each of the three voltages is 120° out of phase with the others, so that, using phasor notation, we may write: Van = V˜an ∠0° Vbn = V˜bn ∠ − 120°

(5.5.17)

Vcn = V˜cn ∠ − 240° = V˜cn ∠120° where the quantities V˜an , V˜bn , and V˜cn are rms values and are equal to each other. To simplify the notation, it will be assumed from here on that V˜an = V˜bn = V˜cn = V˜

(5.5.18)

Section 5.12 will discuss how three-phase AC electric generators may be constructed to provide such balanced voltages. In the circuit of Figure 5.5.13, the resistive loads are also wye-connected and balanced © 2005 by CRC Press LLC

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Chapter 5

FIGURE 5.5.13 Balanced three-phase AC circuit.

FIGURE 5.5.14 Positive, or abc, sequence for balanced three-phase voltages.

(i.e., equal). The three AC sources are all connected together at a node called the neutral node, denoted by n. The voltages V˜an , V˜bn , and V˜cn are called the phase voltages and form a balanced set in the sense that Van + Vbn + Vcn = 0

(5.5.19)

This last statement is easily verified by sketching the phasor diagram. The sequence of phasor voltages shown in Figure 5.5.14 is usually referred to as the positive (or abc) sequence. Consider now the “lines” connecting each source to the load and observe that it is possible to also define line voltages (also called line-to-line voltages) by considering the voltages between the lines aa′ and bb′, aa′ and cc′, and bb′ and cc′. Since the line voltage, say, between aa′ and bb′ is given by Vab = Van + Vnb = Van − Vbn

(5.5.20)

the line voltages may be computed relative to the phase voltages as follows: Vab = V˜∠0° − V˜∠ − 120° = 3V˜∠30° Vbc = V˜∠ − 120° − V˜∠120° = 3V˜∠ − 90°

(5.5.21)

Vca = V˜∠120° − V˜∠0° = 3V˜∠150° It can be seen, then, that the magnitude of the line voltages is equal to 3 times the magnitude of the phase voltages. It is instructive, at least once, to point out that the circuit of Figure 5.5.13 can be redrawn to have the appearance of the circuit of Figure 5.5.15. One of the important features of a balanced three-phase system is that it does not require a fourth wire (the neutral connection), since the current In is identically zero (for balanced load Za = Zb = Zc = Z). Another, more important characteristic of a balanced three-phase power system is that the total power delivered to the balanced load by the three-phase generator is constant. © 2005 by CRC Press LLC

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Electrical Engineering

FIGURE 5.5.15 Balanced three-phase AC circuit (redrawn).

FIGURE 5.5.16 Delta-connected generators.

p(t ) = pa (t ) + pb (t ) + pc (t ) =

3V˜ 2 R

It is also possible to connect the three AC sources in a three-phase system in a so-called delta (or ∆) connection, although in practice this configuration is rarely used. Figure 5.5.16 depicts a set of three delta-connected generators. Balanced Wye Loads Consider again the circuit of Figure 5.5.13, where now the balanced load consists of the three complex impedances Z a = Zb = Zc = Zv = Z y ∠θ

(5.5.22)

From the diagram of Figure 5.5.13, it can be verified that each impedance sees the corresponding phase voltage across itself; thus, since the currents Ia , Ib , and Ic have the same rms value, I˜ , the phase angles of the currents will differ by ±120°. It is therefore possible to compute the power for each phase by considering the phase voltage (equal to the load voltage) for each impedance, and the associated line current. Let us denote the complex power for each phase by S: S = V ⋅ I*

(5.5.23)

so that S = P + jQ ˜˜ cos θ + jVI ˜˜ sin θ = VI

(5.5.24)

where V˜ and I˜ denote, once again, the rms values of each phase voltage and line current. Consequently, the total real power delivered to the balanced wye load is 3P, and the total reactive power is 3Q. Thus, the total complex power, ST , is given by © 2005 by CRC Press LLC

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Chapter 5

FIGURE 5.5.17 Balanced wye generators with balanced delta load.

ST = PT + jQT = 3P + j 3Q =

(3P)2 + (3Q)2 ∠θ

(5.5.25)

and the apparent power is ST = 3 (VI ) cos 2 θ + (VI ) sin 2 θ 2

2

= 3VI and the total real and reactive power may be expressed in terms of the apparent power: PT = ST cos θ (5.5.26) QT = ST sin θ Balanced Delta Loads In addition to a wye connection, it is also possible to connect a balanced load in the delta configuration. A wye-connected generator and a delta-connected load are shown in Figure 5.5.17. It should be noted immediately that now the corresponding line voltage (not phase voltage) appears across each impedance. For example, the voltage across Zc′a′ is Vca. Thus, the three load currents are given by the following expressions:

© 2005 by CRC Press LLC

I ab =

Vab = Z∆

3V∠30° Z ∆ ∠θ

I bc =

Vbc = Z∆

3V∠ − 90° Z ∆ ∠θ

I ca =

Vca = Z∆

3V∠150° Z ∆ ∠θ

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Electrical Engineering

One can readily verify that the two currents (Ia)∆ and (Ia)y will be equal if the magnitude of the deltaconnected impedance is three times larger than Zy: Z ∆ = 3Z y

(5.5.27)

This result also implies that a delta load will necessarily draw three times as much current (and therefore absorb three times as much power) as we wye load with the same branch impedance.

Generation and Distribution of AC Power We now conclude the discussion of power systems with a brief description of the various elements of a power system. Electric power originates from a variety of sources; in Section 5.12, electric generators will be introduced as a means of producing electric power from a variety of energy-conversion processes. In general, electric power may be obtained from hydroelectric, thermoelectric, geothermal, wind, solar, and nuclear sources. The choice of a given source is typically dictated by the power requirement for the given application and by economic and environmental factors. In this section, the structure of an AC power network, from the power-generating station to the residential circuits discussed in the previous section, is briefly outlined. A typical generator will produce electric power at 18 kV, as shown in the diagram of Figure 5.5.18. To minimize losses along the conductors, the output of the generators is processed through a step-up transformer to achieve line voltages of hundreds of kilovolts. Without this transformation, the majority of the power generated would be lost in the transmission lines that carry the electric current from the power station. The local electric company operates a power-generating plant that is capable of supplying several hundred megavolt-amperes (MVA) on a three-phase basis. For this reason, the power company uses a three-phase step-up transformer at the generation plant to increase the line voltage to around 345 kV. One can immediately see that at the rated power of the generator (in MVA) there will be a significant reduction of current beyond the step-up transformer.

FIGURE 5.5.18 Structure of an AC power distribution network. © 2005 by CRC Press LLC

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Chapter 5

FIGURE 5.6.1 A circuit model.

Beyond the generation plant, an electric power network distributes energy to several substations. This network is usually referred to as the power grid. At the substations, the voltage is stepped down to a lower level (10 to 150 kV, typically). Some very large loads (for example, an industrial plant) may be served directly from the power grid, although most loads are supplied by individual substations in the power grid. At the local substations (one of which you may have seen in your own neighborhood), the voltage is stepped down further by a three-phase step-down transformer to 4800 V. These substations distribute the energy to residential and industrial customers. To further reduce the line voltage to levels that are safe for residential use, step-down transformers are mounted on utility poles. These drop the voltage to the 120/240-V three-wire single-phase residential service discussed in the previous section. Industrial and commercial customers receive 460- and/or 208-V three-phase service.

5.6 Frequency Response, Filters, and Transient Analysis The aim of the present section is twofold: first, to exploit AC circuit analysis methods to study the frequency response of electric circuits; and second, to continue the discussion of dynamic circuit equations for the purpose of analyzing the transient response of electrical circuits. The sinusoidal frequency response (or, simply, frequency response) of a circuit provides a measure of how the circuit responds to sinusoidal inputs of arbitrary frequency. In other words, given the input signal amplitude, phase, and frequency, knowledge of the frequency response of a circuit permits the computation of the output signal. The frequency response of a circuit is a measure of the variation of a load-related voltage or current as a function of the amplitude, phase, and frequency of the excitation signal. To express the frequency response of a circuit in terms of variation in output voltage as a function of source voltage, we use the general formula HV ( jω ) =

VL ( jω ) VS ( jω )

(5.6.1)

One method that allows for representation of the load voltage as a function of the source voltage (this is, in effect, what the frequency response of a circuit implies) is to describe the source and attached circuit by means of the Thévenin equivalent circuit. The frequency response of the circuit shown in Figure 5.6.1 is given by the expression Z L Z2 VL ( jω) = HV ( jω) = VS Z L ( Z S + Z1 + Z2 ) + ( Z S + Z1 ) Z2

(5.6.2)

The expression for HV(jω) could be evaluated for any given VS(jω) (i.e., for any given source signal amplitude, phase, and frequency) to determine what the resultant load voltage would be. Note that HV(jω) © 2005 by CRC Press LLC

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Electrical Engineering

is a complex quantity (dimensionless, because it is the ratio of two voltages), and that it therefore follows that VL( jω) is a phase-shifted and amplitude-scaled version of VS(jω): VL ( jω ) = HV ( jω ) ⋅ VS ( jω )

(5.6.3)

VL e jφ L = HV e jφ H ⋅ VS e jφS

(5.6.4)

VL e jφ L = HV VS e j ( φ H +φS )

(5.6.5)

or

where VL = HV ⋅ VS and φL = φH + φS

(5.6.6)

The effect of inserting a linear circuit between a source and a load is best understood by considering that, at any given frequency, ω, the load voltage is a sinusoid at the same frequency as the source voltage, with amplitude given by VL = |HV| · VS and phase equal to φL = φH + φS, where |HV| is the magnitude of the frequency response and φH its phase angle. Both |HV| and φH are functions of frequency. The importance and usefulness of the frequency response concept lies in its ability to summarize the response of a circuit in a single function of frequency, H(jω), which can predict the load voltage or current at any frequency, given the input. Note that the frequency response of a circuit can be defined in four different ways: HV ( jω ) =

VL ( jω ) VS ( jω )

V ( jω ) H Z ( jω ) = L I S ( jω )

H I ( jω ) =

I L ( jω ) I S ( jω )

I ( jω ) HY ( jω ) = L VS ( jω )

(5.6.7)

If HV(jω) and HI(jω) are known, one can directly derive the other two expressions: H Z ( jω ) =

HY ( jω ) =

VL ( jω ) I ( jω ) = Z L ( jω ) L = Z L ( jω ) H I ( jω ) I S ( jω ) I S ( jω )

(5.6.8)

I L ( jω ) VL ( jω ) 1 1 = = H ( jω ) VS ( jω ) Z L ( jω ) VS ( jω ) Z L ( jω ) V

(5.6.9)

With these definitions in hand, it is now possible to introduce one of the central concepts of electrical circuit analysis: filters. The concept of filtering an electrical signal will be discussed in the next section.

Filters There are a host of practical, everyday applications that involve filters of one kind or another. Just to mention two, filtration systems are used to eliminate impurities from drinking water, and sunglasses are used to filter out eye-damaging ultraviolet radiation and to reduce the intensity of sunlight reaching the eyes. An analogous concept applies to electrical circuits: it is possible to attenuate (i.e., reduce in amplitude) or altogether eliminate signals of unwanted frequencies, such as those that may be caused by electrical noise or other forms of interference. This section will be devoted to the analysis of electrical filters. © 2005 by CRC Press LLC

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Chapter 5

FIGURE 5.6.2 A simple RC filter.

Low-Pass Filters Figure 5.6.2 depicts a simple RC filter and denotes its input and output voltages by Vi and Vo. The frequency response for the filter may be obtained by considering the function. H ( jω ) =

Vo ( jω) Vi

(5.6.10)

and noting that the output voltage may be expressed as a function of the input voltage by means of a voltage divider, as follows: Vo ( jω ) = Vi ( jω )

H ( jω ) =

1 jωC 1 = Vi ( jω ) 1 + jωRC R + 1 jωC

(5.6.11)

Vo 1 ( jω) = Vi 1 + jωCR

or H ( jω ) = H ( jω ) e jφ H ( jω )

(5.6.12)

with H ( jω ) =

1 1 + (ωCR)

2

=

1 1 + (ω ω 0 )

2

(5.6.13)

and  ω φ H ( jω ) = − arctan(ωCR) = − arctan   ω0 

(5.6.14)

with ω0 =

© 2005 by CRC Press LLC

1 RC

(5.6.15)

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The simplest way to envision the effect of the filter is to think of the phasor voltage Vi = Vi e jφi scaled by a factor of |H| and shifted by a phase angle φH by the filter at each frequency, so that the resultant output is given by the phasor Vo e jφo , with Vo = H ⋅ Vi φo = φ H + φi

(5.6.16)

and where |H| and φH are functions of frequency. The frequency ω0 is called the cutoff frequency of the filter and, as will presently be shown, gives an indication of the filtering characteristics of the circuit. It is customary to represent H(jω) in two separate plots, representing |H| and φH as functions of ω. These are shown in Figure 5.6.3 in normalized form — that is, with |H| and φH plotted vs. ω/ω0, corresponding to a cutoff frequency ω0 = 1 rad/sec. Note that, in the plot, the frequency axis has been scaled logarithmically. This is a common practice in electrical engineering, because it allows viewing a very broad range of frequencies on the same plot without excessively compressing the low-frequency end of the plot. The frequency response plots of Figure 5.6.3 are commonly employed to describe the frequency response of a circuit, since they can provide a clear idea at a glance of the effect of a filter on an excitation signal. For example, the RC filter of Figure 5.6.2 has the property of “passing” signals at low frequencies (ω
1/RC) and of filtering out signals at high frequencies (ω  1/RC). This type of filter is called a low-pass filter. The cutoff frequency ω = 1/RC has a special significance in that it represents — approximately — the point where the filter begins to filter out the higher-frequency signals. The value of H( jω) at the cutoff frequency is 1/ 2 = 0.707. Note how the cutoff frequency depends exclusively on the values of R and C. Therefore, one can adjust the filter response as desired simply by selecting appropriate values for C and R, and therefore choose the desired filtering characteristics.

FIGURE 5.6.3 Magnitude and phase response plots for RC filter.

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FIGURE 5.6.4 Wheatstone bridge with equivalent circuit and simple capacitive filter.

Example 5.6.1 Wheatstone Bridge Filter The Wheatstone bridge circuit is used in a number of instrumentation applications, including the measurement of force (see Example 5.6.2, describing the strain gauge bridge). Figure 5.6.4 depicts the appearance of the bridge circuit. When undesired noise and interference are present in a measurement, it is often appropriate to use a low-pass filter to reduce the effect of the noise. The capacitor that is connected to the output terminals of the bridge in Figure 5.6.4 constitutes an effective and simple low-pass filter, in conjunction with the bridge resistance. Assume that the average resistance of each leg of the bridge is 350 Ω (a standard value for strain gauges) and that we desire to measure a sinusoidal force at a frequency of 30 Hz. From prior measurements, it has been determined that a filter with a cutoff frequency of 300 Hz is sufficient to reduce the effects of noise. Choose a capacitor that matches this filtering requirement. Solution. By evaluating the Thévenin equivalent circuit for the Wheatstone bridge, calculating the desired value for the filter capacitor becomes relatively simple, as illustrated at the bottom of Figure 5.6.4. The Thévenin resistance for the bridge circuit may be computed by short-circuiting the two voltage sources and removing the capacitor placed across the load terminals: RT = R1 R2 + R3 R4 = 350 350 + 350 350 = 350 Ω Since the required cutoff frequency is 300 Hz, the capacitor value can be computed from the expression ω0 =

1 = 2 π × 300 RT C

or C=

1 1 = = 1.51 µF RT ω 0 350 × 2 π × 300

The frequency response of the bridge circuit is of the form Vout 1 ( jω) = 1 + jωCRT VT This response can be evaluated at the frequency of 30 Hz to verify that the attenuation and phase shift at the desired signal frequency are minimal: Vout 1 ( jω = j 2π × 30) = VT 1 + j 2 π × 30 × 1.51 × 10 −6 × 350 = 0.9951∠ − 5.7° © 2005 by CRC Press LLC

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FIGURE 5.6.5 Unfiltered and filtered bridge output.

Figure 5.6.5 depicts the appearance of a 30-Hz sinusoidal signal before and after the addition of the capacitor to the circuit. High-Pass Filters Just as you can construct a simple filter that preserves low frequencies and attenuates higher frequencies, you can easily construct a high-pass filter that passes mainly those frequencies above a certain cutoff frequency. The analysis of a simple high-pass filter can be conducted by analogy with the preceding discussion of the low-pass filter. Consider the circuit shown in Figure 5.6.6. The frequency response of the filter is Vo jωCR ( jω) = 1 + jωCR Vi or H ( jω ) = H e jφ H with H ( jω ) =

ωCR 1 + (ωCR)

2

φ H ( jω ) = 90° − arctan(ωCR) Amplitude-and-phase response curves for the high-pass filter are shown in Figure 5.6.7. These plots have been normalized to have the filter cutoff frequency ω0 = 1 rad/sec. Note that, once again, it is possible to define a cutoff frequency at ω0 = 1/RC in the same way as was done for the low-pass filter. Band-Pass Filters Building on the principles developed in the preceding sections, we can also construct a circuit that acts as a band-pass filter, passing mainly those frequencies within a certain frequency range. The analysis of

FIGURE 5.6.6 High-pass filter. © 2005 by CRC Press LLC

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FIGURE 5.6.7 Frequency response of a high-pass filter.

FIGURE 5.6.8 RLC band-pass filter.

a simple second-order band-pass filter (i.e., a filter with two energy-storage elements) can be conducted by analogy with the preceding discussions of the low-pass and high-pass filters. Consider the circuit shown in Figure 5.6.8, and the related frequency response function for the filter H(jω) =(Vo /Vi )(jω). We may write the frequency response of the filter as Vo jωCR ( jω) = 2 Vi 1 + jωCR + ( jω ) LC

(5.6.17)

Equation (5.6.17) can often be factored into the following form: Vo jAω ( jω) = Vi ( jω ω1 + 1) ( jω ω 2 + 1)

(5.6.18)

where ω1 and ω2 are the two frequencies that determine the pass-band (or band-width) of the filter — that is, the frequency over which the filter “passes” the input signal. The magnitude and phase plots for the frequency response of the band-pass filter of Figure 5.6.8 are shown in Figure 5.6.9. These plots have been normalized to have the filter pass-band centered at the frequency ω = 1 rad/sec. © 2005 by CRC Press LLC

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FIGURE 5.6.9 Frequency response of RLC band-pass filter.

The expression for the frequency response of a second-order band-pass filter (Equation (5.6.17)) can also be rearranged to illustrate two important features of this circuit: the quality factor, Q, and the resonant frequency, ω0. Let ω0 =

1 LC

and

Q = ω 0 CR =

R ω0 L

(5.6.19)

Then we can write ωCR = ω0CR(ω/ω0) = Q(ω/ω0) and rearrange Equation 5.6.18 as follows: ω jQ ω0 Vo ( jω) = 2 Vi  jω  ω  ω  + jQ ω + 1  0 0

(5.6.20)

In Equation (5.6.20), the resonant frequency, ω0, corresponds to the center frequency of the filter, while Q, the quality factor, indicates the sharpness of the resonance, that is, how narrow or wide the shape of the pass-band of the filter is. The width of the pass-band is also referred to as the bandwidth, and it can easily be shown that the bandwidth of the filter is given by the expression B=

ω0 Q

(5.6.21)

Thus, a high-Q filter has a narrow bandwidth, while a low-Q filter has a large bandwidth and is therefore less selective. The quality factor of a filter provides an immediate indication of the nature of the filter. © 2005 by CRC Press LLC

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FIGURE 5.6.10 Examples of transient response.

FIGURE 5.6.11 Circuit with switched DC excitation.

Transient Analysis In analyzing the frequency response of AC circuits earlier in this chapter, we made the assumption that the particular form of the voltages and currents in the circuit was sinusoidal. There are many signals, however, for which the steady-state sinusoidal representation is not adequate. In particular, the sinusoidal, or AC method of analysis does not apply to transient signals, that is, voltages and currents that vary as a function of time as a consequence of a sudden change in the input. Figure 5.6.10 illustrates the appearance of the voltage across some hypothetical load when a DC and an AC source, respectively, are abruptly switched on at time t = 2 sec. The waveforms in Figure 5.6.10 can be subdivided into three regions: a steady-state region for 0 ≤ t ≤ 0.2 sec; a transient region, for 0.2 sec ≤ t ≤ 2 sec (approximately); and a new steady-state region for t > 2 sec, where the waveform reaches a new steady-state DC or sinusoidal condition. The objective of transient analysis is to describe the behavior of a voltage or current during the transition that takes place between two different steady-state conditions. You already know how to analyze circuits in a sinusoidal steady state by means of phasors. The material presented in the remainder of this chapter will provide the tools necessary to describe the transient response of circuits containing resistors, inductors, and capacitors. A general example of the type of circuit that will be discussed in this section is shown in Figure 5.6.11. The switch indicates that we turn the battery power on at time t = 0. Transient behavior may be expected whenever a source of electrical energy is switched on or off, whether it be AC or DC. A typical example of the transient response to a switched DC voltage would be what occurs when the ignition circuits in an automobile are turned on, so that a © 2005 by CRC Press LLC

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FIGURE 5.6.12 A general model of the transient analysis problem.

FIGURE 5.6.13 Decaying and rising exponential responses.

12-V battery is suddenly connected to a large number of electrical circuits. The degree of complexity in transient analysis depends on the number of energy-stored elements in the circuit; the analysis can become quite involved for high-order circuits. In this chapter, we shall analyze only first- and second-order circuits — that is, circuits containing one or two energy-storage elements, respectively. In electrical engineering practice, we would typically resort to computer-aided analysis for higher-order circuits. A convenient starting point in approaching the transient response of electrical circuits is to consider the general model shown in Figure 5.6.12, where the circuits in the box consist of a combination of resistors connected to a single energy-storage element, either an inductor or a capacitor. Regardless of how many resistors the circuit contains, it is a first-order circuit. In general, the response of a first-order circuit to a switched DC source will appear in one of the two forms shown in Figure 5.6.13 which represent, in order, a decaying exponential and a rising exponential waveform. In the next sections, we will systematically analyze these responses by recognizing that they are exponential in nature and can be computed very easily once we have the proper form of the differential equation describing the circuit. Example 5.6.2 Pulse Response A problem of great practical importance is the transmission of voltage pulses along cables. Short voltage pulses are used to represent the two-level binary signals that are characteristic of digital computers; it is often necessary to transmit such voltage pulses over a long distance through coaxial cables, which are characterized by a finite resistance per unit length and by a certain capacitance per unit length, usually expressed in units of pF/m. A simplified model of a long coaxial cable is shown in Figure 5.6.14. It has the appearance of a low-pass filter. If a 100-m cable has a capacitance of 40 pF/m and a series resistance © 2005 by CRC Press LLC

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FIGURE 5.6.14 Pulse transmission in a coaxial cable.

of 0.2 Ω/m, what will the output pulse look like after traveling the length of the cable? Assume the input pulse has a duration of 0.1 µsec and has an amplitude of 5 V. The load resistance is 150 Ω. Solution. The Thévenin equivalent circuit seen by the capacitor will vary, depending on whether the pulse is “on” or “off ”. In the former case, the equivalent resistance consists of the parallel combination of R1 and RL; in the latter case, since the switch is open, the capacitor is connected only to RL. Thus, the effect of the pulse is to charge C through the parallel combination of R1 and RL during the “on” time (0 ≤ t < 0.1 µsec); the capacitor will then discharge through RL during the “off ” time. This behavior is depicted by the circuit model of Figure 5.6.14 in which the pulse signal is represented by a 5-V battery in series with a switch. The charging time constant of the coaxial cable equivalent circuit when the switch is closed is therefore given by

(

)

τ on = R1 RL C = 17.65 × 4000 × 10 −12 = 0.07 µsec and the transient response of the cable during the “on” time is

(

)

(

v L (t ) = 4.41 1 − e − t τ = 4.41 1 − e −1.42×10

7

t

)

0 ≤ t ≤ 0.1 µsec

where VT = RL /(R1 + RL ) × 5 = 4.41 V. At t = 0.1 µsec we calculate the load voltage to be

(

)

v L (0.1 µsec) = 4.41 1 − e −1.42 = 3.35 V For t ≥ 0.1 µsec, the output will naturally decay to zero, starting from the initial condition, vL (0.1 µsec), with a time constant τoff equal to τ off = RL C = 150 × 4000 × 10 −12 = 0.6 µsec The load voltage will therefore decay according to the following expression:

(

6 −6 v L (t ) = 3.35 e −1.67×10 (t −0.1×10 )

)

t > 0.1 µsec

The appearance of the response is shown in Figure 5.6.15. It should be apparent that as the cable becomes longer, both R1 and C increase, and therefore the output voltage will respond more slowly to the input pulse; according to the simple model of a long coaxial cable given in this example, there is a limit to the maximum distance over which a voltage pulse can be transmitted by cable.

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FIGURE 5.6.15 Pulse response of 100-m-long coaxial cable.

5.7 Electronics This chapter introduces semiconductor-based electronic devices, and in so doing, it provides a transition between the fundamentals of electrical circuit analysis and the study of electronic circuits.

Semiconductors and pn Junctions This section briefly introduces the mechanism of conduction in a class of materials called semiconductors. Semiconductors are typically materials consisting of elements from group IV of the periodic table and having electrical properties falling somewhere between those of conducting and of insulating materials. As an example, consider the conductivity of three common materials. Copper, a good conductor, has a conductivity of 0.59 × 166 S/cm; glass, a common insulator, may range between 10–16 and 10–13 S/cm; while silicon, a semiconductor, has a conductivity that varies from 10–8 to 10–1 S/cm. You see, then, that the name semiconductor is an appropriate one. A conducting material is characterized by a large number of conduction-band electrons, which have a very weak bond with the basic structure of the material. Thus, an electric field easily imparts energy to the outer electrons in a conductor and enables the flow of electric current. The free valence electrons are not the only mechanism of conduction in a semiconductor, however, Whenever a free electron leaves the lattice structure, it creates a corresponding positive charge within the lattice. The vacancy caused by the departure of a free electron is called a hole. Note that whenever a hole is present, we have, in effect, a positive charge. The positive charges also contribute to the conduction process, in the sense that if a valence-band electron “jumps” to fill a neighboring hole, thereby neutralizing a positive charge, it correspondingly creates a new hole at a different location. Thus, the effect is equivalent to that of a positive charge moving. Semiconductor technology rarely employs pure, or intrinsic, semiconductors. To control the number of charge carriers in a semiconductor, the process of doping is usually employed. Doping consists of adding impurities to the crystalline structure of the semiconductor. The amount of these impurities is controlled, and the impurities can be of one of two types. Semiconductors doped with donor elements conduct current predominantly by means of free electrons and are therefore called n-type semiconductors. When an acceptor element is used as the dopant, holes constitute the most common carrier, and the resulting semiconductor is said to be a p-type semiconductor. Doping usually takes place at such levels that the concentration of carriers due to the dopant is significantly greater than the intrinsic concentration of the original semiconductor. If n is the total number of free electrons and p that of holes, then in an n-type doped semiconductor, we have n  ni and p
pi © 2005 by CRC Press LLC

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FIGURE 5.7.1 A pn junction.

FIGURE 5.7.2 Forward- and reverse-biased pn junctions.

Thus, free electrons are the majority carriers in an n-type material, while holes are the minority carriers. In a p-type material, the majority and minority carriers are reversed. A simple section of semiconductor material does not in and of itself possess properties that make it useful for the construction of electronic circuits. However, when a section of p-type material and a section of n-type material are brought in contact to form a pn junction, a number of interesting properties arise. The pn junction forms the basis of the semiconductor diode, a widely used circuit element. Figure 5.7.1 depicts an idealized pn junction, where on the p side, we see a dominance of positive charge carriers, or holes, and on the n side, the free electrons dominate. The charge separation causes a contact potential to exist at the junction. This potential is typically on the order of a few tenths of a volt and depends on the material (about 0.6 to 0.7 V for silicon). The contact potential is also called the offset voltage, Vγ . Consider the diagrams of Figure 5.7.2, where a battery has been connected to a pn junction in the reverse-biased direction (Figure 5.7.2(a)), and in the forward-biased direction (Figure 5.7.2(b)). We assume that some suitable form of contact between the battery wires and the semiconductor material can be established (this is called an ohmic contact). The effect of a reverse bias is to increase the contact potential at the junction. Now, the majority carriers trying to diffuse across the junction need to overcome a greater barrier (a larger potential) and a wider depletion region. Thus, the diffusion current becomes negligible. The only current that flows under reverse bias is the very small reverse saturation current, so that the diode current, iD (defined in the figure), is iD = − I 0 When the pn junction is forward-biased, the contact potential across the junction is lowered (note that VB acts in opposition to the contact potential). Now, the diffusion of majority carriers is aided by the external voltage source; in fact, the diffusion current increases as a function of the applied voltage, according to the expression © 2005 by CRC Press LLC

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FIGURE 5.7.3 Semiconductor diode i-v characteristic.

I d = I0 e qvD

kT

(5.7.1)

where vD is the voltage across the pn junction, k = 1.381 × 10–23 j/K is Boltzmann’s constant, q the charge of one electron, and T the temperature of the material in kelvins (K). The quantity kT/q is constant at a given temperature and is approximately equal to 25 mV at room temperature. The net diode current under forward bias is given by the expression

(

iD = I d − I0 = I0 e qvD

kT

)

−1

(5.7.2)

which is known as the diode equation. Figure 5.7.3 depicts the diode i-v characteristic described by the diode equation for a fairly typical silicon diode for positive diode voltages. Since the reverse saturation current, I0, is typically very small (10–9 to 10–15 A), the expression iD = I0 e qvD

kT

(5.7.3)

is a good approximation if the diode voltage, vD, is greater than a few tenths of a volt. The ability of the pn junction to essentially conduct current in only one direction — that is, to conduct only when the junction is forward-biased — makes it valuable in circuit applications. A device having a single pn junction and ohmic contacts at its terminals, as described in the preceding paragraphs, is called a semiconductor diode, or simply diode. As will be shown later in this chapter, it finds use in many practical circuits. The circuit symbol for the diode is shown in Figure 5.7.4, alongside with a sketch of the pn junction.

Circuit Models for the Semiconductor Diode From the viewpoint of a user of electronic circuits (as opposed to a designer), it is often sufficient to characterize a device in terms of its i-v characteristic, using appropriate circuit models to determine the operating currents and voltages. Ideal Diode Model The large-signal model treats the diode as a simple on–off device (much like a check valve in hydraulic circuits). Figure 5.7.5 illustrates how, on a large scale, the i-v characteristic of a typical diode may be approximated by an open circuit when vD < 0 and by a short circuit when vD ≥ 0 (recall the i-v curves of the ideal short and open circuits presented in Section 5.2). The analysis of a circuit containing a diode may be greatly simplified by using the short-circuit–open-circuit model. From here on, this diode model will be known as the ideal diode model. In spite of its simplicity, the ideal diode model (indicated by the symbol shown in Figure 5.7.5 can be very useful in analyzing diode circuits. © 2005 by CRC Press LLC

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FIGURE 5.7.4 Semiconductor diode circuit symbol.

FIGURE 5.7.5 Large-signal on–off diode model.

One of the important applications of the semiconductor diode is rectification of AC signals, that is, the ability to convert an AC signal with zero average (DC) value to a signal with a nonzero DC value. The application of the semiconductor diode as a rectifier is very useful in obtaining DC voltage supplies from the readily available AC line voltage. Here we illustrate the basic principle of rectification using an ideal diode, for simplicity, and also because the large-signal model is appropriate when the diode is used in applications involving large AC voltage and current levels. Consider the circuit of Figure 5.7.6, where an AC source, vi = 155.56 · sin ωt, is connected to a load by means of a series ideal diode. The diode will conduct only during the positive half-cycle of the sinusoidal voltage — that is, that the condition vD ≥ 0 will be satisfied only when the AC source voltage is positive — and that it will act as an open circuit during the negative half-cycle of the sinusoid (vD < 0). Thus, the appearance of the load voltage will be as shown in Figure 5.7.7 with the negative portion of the sinusoidal waveform cut off. The rectified waveform clearly has a nonzero DC (average) voltage, whereas the average input waveform voltage was zero. When the diode is conducting, or vD ≥ 0, the unknowns vL and iD can be found by using the following equations: iD =

© 2005 by CRC Press LLC

vi RL

when

vi > 0

(5.7.4)

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FIGURE 5.7.6

FIGURE 5.7.7 Ideal diode rectifier input and output voltages.

and v L = iD RL

(5.7.5)

The load voltage, vL and the input voltage, vi , are sketched in Figure 5.7.7. From Equation (5.7.5) it is obvious that the current waveform has the same shape as the load voltage. The average value of the load voltage is obtained by integrating the load voltage over one period and dividing by the period: vload,DC =

ω 2π

∫

πω

0

155.56 sin ωt dt =

155.56 = 49.52 V π

(5.7.6)

The circuit of Figure 5.7.6 is called a half-wave rectifier, since it preserves only half of the waveform. This is not usually a very efficient way of rectifying an AC signal, since half the energy in the AC signal is not recovered. It will be shown in a later section that it is possible to recover also the negative half of the AC waveform by mans of a full-wave rectifier. Offset Diode Model While the ideal diode model is useful in approximating the large-scale characteristics of a physical diode, it does not account for the presence of an offset voltage, which is an unavoidable component in semiconductor diodes. The offset diode model consists of an ideal diode in series with a battery of strength equal to the offset voltage (we shall use the value Vγ = 0.6 V for silicon diodes, unless otherwise indicated). The effect of the battery is to shift the characteristic of the ideal diode to the right on the voltage axis, as shown in Figure 5.93. This model is a better approximation of the large-signal behavior of a semiconductor diode than the ideal diode model. According to the offset diode model, the diode of Figure 5.7.8 acts as an open circuit for vD < 0.6 V, and it behaves like a 0.6-V battery for vD ≥ 0.6 V. The equations describing the offset diode model are as follows: © 2005 by CRC Press LLC

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FIGURE 5.7.8

FIGURE 5.7.9 Offset diode as an extension of ideal diode model.

v D ≥ 0.6 V

Diode → 0.6-V battery

v D < 0.6 V

Diode → Open circuit

(5.7.7)

The diode offset model may be represented by an ideal diode in series with a 0.6-V ideal battery, as shown in Figure 5.7.9.

Practical Diode Circuits The Full-Wave Rectifier The half-wave rectifier discussed earlier is one simple method of converting AC energy to DC energy. The need for converting one form of electrical energy into the other arises frequently in practice. The most readily available form of electric power is AC (the standard 110- or 220-V rms AC line power), but one frequently needs a DC power supply, for applications ranging from the control of certain types of electric motors to the operation of electronic circuits. The half-wave rectifier, however, is not a very efficient AC–DC conversion circuit, because it fails to utilize half the energy available in the AC waveform by not conducting current during the negative halfcycle of the AC waveform. The full-wave rectifier shown in Figure 5.7.10 offers a substantial improvement

FIGURE 5.7.10 Full-wave rectifier.

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in efficiency over the half-wave rectifier. The first section of the full-wave rectifier circuit includes an AC source and a center-tapped transformer (see Section 5.5) with 1:2 N turns ratio. The purpose of the transformer is to obtain the desired voltage amplitude prior to rectification. Thus, if the peak amplitude of the AC source voltage is vs, the amplitude of the voltage across each half of the output side of the transformer will be Nvs; this scheme permits scaling the source voltage up or down (depending on whether N is greater or less than 1), according to the specific requirements of the application. In addition to scaling the source voltage, the transformer also isolates the rectifier circuit from the AC source voltage, since there is no direct electrical connection between the input and output of a transformer. The Bridge Rectifier Another rectifier circuit commonly available “off the shelf ” as a single integrated circuit package is the bridge rectifier, which employs four diodes in a bridge configuration, similar to the Wheatstone bridge already explored in Section 5.2. Figure 5.7.11 depicts the bridge rectifier, along with the associated integrated circuit (IC) package. The analysis of the bridge rectifier is simple to understand by visualizing the operation of the rectifier for the two half-cycles of the AC waveform separately. The key is that, as illustrated in Figure 5.7.12, diodes D1 and D3 conduct during the positive half-cycle, while diodes D2 and D4 conduct during the negative half-cycle. Because of the structure of the bridge, the flow of current through the load resistor is in the same direction (from c to d) during both halves of the cycle; hence, the full-wave rectification of the waveform.

FIGURE 5.7.11 Full-wave bridge rectifier.

FIGURE 5.7.12 Diodes conduct.

© 2005 by CRC Press LLC

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DC Power Supplies and Voltage Regulation The principal application of rectifier circuits is in the conversion of AC to DC power. A circuit that accomplishes this conversion is usually called a DC power supply. In power supply applications, transformers are employed to obtain an AC voltage that is reasonably close to the desired DC supply voltage. DC power supplies are very useful in practice: many familiar electrical and electronic appliances (e.g., radios, personal computers, TVs) require DC power to operate. For most applications, it is desirable that the DC supply be as steady and ripple-free as possible. To ensure that the DC voltage generated by a DC supply is constant, the DC supply is made up of voltage regulators, that is, devices that can hold a DC load voltage relatively constant in spite of possible fluctuations in the DC supply. This section will describe the fundamentals of voltage regulators. A typical DC power supply is made up of the components shown in Figure 5.7.13. In the figure, a transformer is shown connecting the AC source to the rectifier circuit to permit scaling of the AC voltage to the desired level. For example, one might wish to step the 110-V rms line voltage down to 24 V rms by means of a transformer prior to rectification and filtering, to eventually obtain a 12-VDC regulated supply (regulated here means that the output voltage is a DC voltage that is constant and independent of load and supply variations). Following the step-down transformer are a bridge rectifier, a filter capacitor, a voltage regulator, and, finally, the load. The most common device employed in voltage regulation schemes is the Zener diode. Zener diodes function on the basis of the reverse portion of the i-v characteristic of the diode with forward offset voltage Vγ and reverse Zener voltage VZ . The operation of the Zener diode may be analyzed by considering three modes of operation: 1. For vD ≥ Vγ , the device acts as a conventional forward-biased diode (Figure 5.7.14). 2. For VZ < vD < Vγ , the diode is reverse-biased but Zener breakdown has not taken place yet. Thus, it acts as an open circuit. 3. For vD ≤ VZ , Zener breakdown occurs and the device holds a nearly constant voltage, –VZ (Figure 5.7.15). To illustrate the operation of a Zener diode as a voltage regulator, consider the circuit of Figure 5.7.16, where the unregulated DC source, vS, is regulated to the value of the Zener voltage, VZ. Note how the diode must be connected “upside down” to obtain a positive regulated voltage. Note also that if vS is greater than VZ , it follows that the Zener diode is in its reverse-breakdown mode. Thus, one need not worry whether the diode is conducting or not in simple voltage regulator problems, provided that the unregulated supply voltage is guaranteed to stay above VZ (a problem arises, however, if the unregulated

FIGURE 5.7.13 DC power supply.

FIGURE 5.7.14 Zener diode model for forward bias. © 2005 by CRC Press LLC

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FIGURE 5.7.15 Zener diode model for reverse bias.

FIGURE 5.7.16 A Zener diode voltage regulator.

FIGURE 5.7.17 Simplified circuit for Zener regulator.

supply can drop below the Zener voltage). Assuming that the resistance rZ is negligible with respect to RS and RL , we replace the Zener diode with the simplified circuit mode of Figure 5.7.17 consisting of a battery of strength VZ . Three simple observations are sufficient to explain the operation of this voltage regulator: 1. The load voltage must equal VZ , as long as the Zener diode is in the reverse-breakdown mode. Then, iL =

VZ RL

(5.7.10)

2. The load current (which should be constant if the load voltage is to be regulated to sustain VZ ) is the difference between the unregulated supply current, is, and the diode current iz : iL = iS − iZ

(5.7.11)

This second point explains intuitively how a Zener diode operates: any current in excess of that required to keep the load at the constant voltage VZ is “dumped” to ground through the diode. Thus, the Zener diode acts as a sink to the undesired source current. 3. The source current is given by iS

vS − VZ RS

(5.7.12)

The Zener diode is usually rated in terms of its maximum allowable power dissipation. The power dissipated by the diode, PZ, may be computed from © 2005 by CRC Press LLC

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PZ = iZ VZ

(5.7.13)

Thus, one needs to worry about the possibility that iZ will become too large. This may occur either if the supply current is very large (perhaps because of an unexpected upward fluctuation of the unregulated supply), or if the load is suddenly removed and all of the supply current sinks through the diode. The latter case, of an open-circuit load, is an important design consideration. Another significant limitation occurs when the load resistance is small, thus requiring large amounts of current from the unregulated supply. In this case, the Zener diode is hardly taxed at all in terms of power dissipation, but the unregulated supply may not be able to provide the current required to sustain the load voltage. In this case, regulation fails to take place. Thus, in practice, the range of load resistances for which load voltage regulation may be attained is constrained to a finite interval: RL min ≤ RL ≤ RL max

(5.7.14)

where RLmax is typically limited by the Zener diode power dissipation and RLmin by the maximum supply current. Example 5.7.1 This example illustrates the calculation of the range of allowable load resistances for a Zener regulator design. For the Zener regulator shown in Figure 5.7.18, we want to maintain the load voltage at 14 V. Find the range of load resistances for which regulation can be obtained if the Zener diode is rated at 14 V, 5 W. Solution. The minimum load resistance for which a regulated load voltage of 14 V may be attained is found by requiring that the load voltage be 14 V and applying KVL subject to this constraint:  RL min  50  = 14  RL min + 30  RL min =

14 ( R + 30) 50 L min

= 11.7 Ω The maximum current through the Zener diode that does not exceed the diode power rating may be computed by considering the 5-W power rating: iZ max =

5 = 0.357 A 14

The current through the 20-Ω resistor will be 50 − 14 = 1.2 A 30

FIGURE 5.7.18 Circuit for example. © 2005 by CRC Press LLC

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so that the maximum load resistance for which regulation occurs is RL max =

14 1.2 − 0.357

= 16.6 Ω Finally, the range of allowable load resistances is: 16.6 Ω ≥ RL ≥ 11.7 Ω Photodiodes Another property of semiconductor materials that finds common application in measurement systems is their response to light energy. In appropriately fabricated diodes, called photodiodes, when light reaches the depletion region of a pn junction, photons cause hole-electron pairs to be generated by a process called photo-ionization. This effect can be achieved by using a surface material that is transparent to light. As a consequence, the reverse saturation current depends on the light intensity (i.e., on the number of incident photons), in addition to the other factors mentioned earlier. In a photodiode, the reverse current is given by –(I0 + Ip), where Ip is the additional current generated by photo-ionization. The result is depicted in the family of curves of Figure 5.7.19, where the diode characteristic is shifted downward by an amount related to the additional current generated by photo-ionization. Figure 5.7.19 depicts the appearance of the i-v characteristic of a photodiode for various values of Ip, where the i-v curve is shifted to lower values for progressively larger values of Ip. The circuit symbol is depicted in Figure 5.7.20. As displayed in Figure 5.7.19 are three load lines, which depict the three modes of operation of a photodiode. Curve L1 represents normal diode operation, under forward bias. Note that the operating point of the device is in the positive i, positive v (first) quadrant of the i-v plane; thus, the diode dissipates positive power in this mode, and is therefore a passive device, as we already know. On the other hand, load line L2 represents operation of the photodiode as a solar cell; in this mode, the operating point is in the negative i, positive v, or fourth, quadrant, and therefore the power dissipated by the diode is negative. In other words, the photodiode is generating power by converting light energy to electrical

FIGURE 5.7.19 Photodiode i-v curves.

FIGURE 5.7.20 Photodiode circuit symbol.

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FIGURE 5.7.21 Light-emitting diode (LED) circuit symbol.

TABLE 5.7.1 LED Materials and Wavelengths Material GaAs GaAs GaP GaP GaP GaAs0.6P0.4 GaAs0.35P0.65 GaAs0.15P0.85

Dopant

Wavelength (nm)

Color

Zn Si N N Zn,O

900 910–1020 570 590 700 650 632 589

Infrared Infrared Green Yellow Red Red Orange Yellow

N N

energy. Note further that the load line intersects the voltage axis at zero, meaning that no supply voltage is required to bias the photodiode in the solar-cell mode. Finally, load line L3 represents the operation of the diode as a light sensor: when the diode is reverse-biased, the current flowing through the diode is determined by the light intensity; thus, the diode current changes in response to changes in the incident light intensity. The operation of the photodiode can also be reversed by forward-biasing the diode and causing a significant level of recombination to take place in the depletion region. Some of the energy released is converted to light energy by emission of photons. Thus, a diode operating in this mode emits light when forward-biased. Photodiodes used in this way are called light-emitting diodes (LEDs); they exhibit a forward (offset) voltage of 1 to 2 volts. The circuit symbol for the LED is shown in Figure 5.7.21. Gallium arsenide (GaAs) is one of the more popular substrates for creating LEDs; gallium phosphide (GaP) and the alloy GaAs1-x Px are also quite common. Table 5.7.1 lists combinations of materials and dopants used for common LEDs and the colors they emit. The dopants are used to create the necessary pn junction. Example 5.7.2 Opto-Isolators One of the common applications of photodiodes and LEDs is the opto-coupler, or opto-isolator. This device, which is usually enclosed in a sealed package, uses the light-to-current and current-to-light conversion property of photodiodes and LEDs to provide signal connection between two circuits without any need for electrical connections. Figure 5.7.22 depicts the circuit symbol for the opto-isolator. Because diodes are nonlinear services, the opto-isolator is not used in transmitting analog signals: the signals would be distorted because of the nonlinear diode i-v characteristic. However, opto-isolators find a very important application when on–off signals need to be transmitted from high-power machinery to delicate computer control circuitry. The optical interface ensures that potentially damaging large currents cannot reach delicate instrumentation and computer circuits.

FIGURE 5.7.22 Opto isolator. © 2005 by CRC Press LLC

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5-75

FIGURE 5.7.23 Controlled-source models of linear amplifier transistor operation.

Transistors A transistor is a three-terminal semiconductor device that can perform two functions that are fundamental to the design of electronic circuits: amplification and switching. Put simply, amplification consists of magnifying a signal by transferring energy to it from an external source; whereas a transistor switch is a device for controlling a relatively large current between or voltage across two terminals by means of a small control current or voltage applied at a third terminal. In this chapter, we provide an introduction to the two major families of transistors: bipolar junction transistors, or BJTs; and field-effect transistors, or FETs. The operation of the transistor as a linear amplifier can be explained qualitatively by the sketch of Figure 5.7.23 in which the four possible modes of operation of a transistor are illustrated by means of circuit models employing controlled sources. In Figure 5.7.23 controlled voltage and current sources are shown to generate an output proportional to an input current or voltage; the proportionality constant, µ, is called the internal gain of the transistor. As will be shown, the BJT acts essentially as a currentcontrolled device, while the FET behaves as a voltage-controlled device. Transistors can also act in a nonlinear mode, as voltage- or current-controlled switches. When a transistor operates as a switch, a small voltage or current is used to control the flow of current between two of the transistor terminals in an on–off fashion. Figure 5.7.24 depicts the idealized operation of the transistor as a switch, suggesting that the switch is closed (on) whenever a control voltage or current is greater than zero and is open (off) otherwise. It will later become apparent that the conditions for the switch to be on or off need not necessarily be those depicted in Figure 5.7.24.

The Bipolar Junction Transistor (BJT) The pn junction forms the basis of a large number of semiconductor devices. The semiconductor diode, a two-terminal device, is the most direct application of the pn junction. In this section, we introduce the bipolar junction transistor (BJT). As we did in analyzing the diode, we will introduce the physics of transistor devices as intuitively as possible, resorting to an analysis of their i-v characteristics to discover important properties and applications. A BJT is formed by joining three sections of semiconductor material, each with a different doping concentration. The three sections can be either a thin n region sandwiched between p+ and p layers, or a p region between n and n+ layers, where the superscript “plus” indicates more heavily doped material. The resulting BJTs are called pnp and npn transistors, respectively; we shall discuss only the latter in this section. Figure 5.7.25 illustrates the approximate construction, symbols, and nomenclature for the two types of BJTs. © 2005 by CRC Press LLC

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FIGURE 5.7.24 Models of ideal transistor switches.

FIGURE 5.7.25 Bipolar junction transistors.

The most important property of the bipolar transistor is that the small base current controls the amount of the much larger collector current IC = βI B

(5.7.15)

where β is a current amplification factor dependent on the physical properties of the transistor. Typical values of β range from 20 to 200. The detailed operation of bipolar transistors can be explained by resorting to a detailed physical analysis of the npn or pnp structure of these devices. The reader interested in such a discussion of transistors is referred to any one of a number of excellent books on semiconductor electronics. The focus of this section will be on the analysis of the i-v characteristic of the npn BJT, based on the circuit notation defined in Figure 5.7.26. The device i-v characteristics will be presented qualitatively, without deriving the underlying equations, and will be utilized in constructing circuit models for the device. The number of independent variables required to uniquely define the operation of the transistor may be determined by applying KVL and KCL to the circuit of Figure 5.7.26. It should be apparent that two voltages and two currents are sufficient to specify the operation of the device. Note that, since the BJT is a three-terminal device, it will not be sufficient to deal with a single i-v characteristic; it will soon become apparent that two such characteristics are required to explain the operation of this device. One of these characteristics relates the base current, iB, to the base-emitter voltage, vBE; the other relates the collector current, iC, to the collector-emitter voltage, vCE. As will be shown, the latter characteristic actually © 2005 by CRC Press LLC

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FIGURE 5.7.26 Definition of BJT voltages and currents.

consists of a family of curves. To determine these i-v characteristics, consider the i-v curves of Figure 5.7.27 and Figure 5.7.28 using the circuit notation of Figure 5.7.26. In Figure 5.7.27, the collector is open and the BE junction is shown to be very similar to a diode. The ideal current source, IBB, injects a base current, which causes the junction to be forward-biased. By varying IBB, one can obtain the open-collector BE junction i-v curve shown in the figure. If a voltage source were now to be connected to the collector circuit, the voltage vCE and, therefore, the collector current, iC, could be varied, in addition to the base current, iB. The resulting circuit is depicted in Figure 5.7.28(a). By varying both the base current and the collector-emitter voltage, one could then generate a plot of the device collector characteristic. This is also shown in Figure 5.7.28(b).

FIGURE 5.7.27 Determining the BE junction open-collector i-v characteristic.

FIGURE 5.7.28 (a) Ideal test circuit to determine the i-v characteristic of a BJT; (b) the collector-emitter output characteristics of a BJT. © 2005 by CRC Press LLC

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FIGURE 5.7.29 Determination of the operating state of a BJT.

Note that this figure depicts not just a single iC-vCE curve, but an entire family, since for each value of the base current, iB, an iC-vCE curve can be generated. Four regions are identified in the collector characteristic: 1. The cutoff region, where both junctions are reverse-biased, the base current is very small, and essentially no collector current flows 2. The active linear region, in which the transistor can act as a linear amplifier, where the BE junction is forward-biased and the CB junction is reverse-biased 3. The saturation region, in which both junctions are forward-biased 4. The breakdown region, which determines the physical limit of operation of the device. Large-Signal Model of the npn BJT The large-signal model for the BJT recognizes three basic operating modes of the transistor (Figure 5.7.29). When the BE junction is reverse-biased, no base current (and therefore no forward collector current) flows, and the transistor acts (virtually as an open circuit; the transistor is said to be in the cutoff region. In practice, there is always a leakage current flowing through the collector, even when VBE = 0 and IB = 0. This leakage current is denoted by ICEO. When the BE junction becomes forward-biased, the transistor is said to be in the active region, and the base current is amplified by a factor of β at the collector: IC = βI B

(5.7.16)

Since the collector current is controlled by the base current, the controlled-source symbol is used to represent the collector current. Finally, when the base current becomes sufficiently large, the collectoremitter voltage, VCE, reaches its saturation limit, and the collector current is no longer proportional to the base current; this is called the saturation region. The three conditions are described in Figure 5.7.30 in terms of simple circuit models. Example 5.7.3 illustrates the application of this large-signal model in a practical circuit and illustrates how to determine which of the three states is applicable, using relatively simple analysis. Example 5.7.3 LED Driver The circuit shown in Figure 5.7.31 is being used to “drive” (i.e., provide power for) a light-emitting diode (LED) from a desktop computer. The signal available from the computer consists of a 5-V low-current output. The reason for using a BJT (rather than driving the LED directly with the signal available from the computer) is that the LED requires a substantial amount of current (at least 15 mA for the device used in this example) and the computer output is limited to 5mA. Thus, some current amplification is required. The LED has an offset voltage of 1.4 V and a maximum power dissipation rating of 280 mW. The circuit has been designed so that the transistor will be either in cutoff, when the LED is to be turned off, or in saturation, when the LED is to be turned on.

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FIGURE 5.7.30 npn BJT large-signal model.

FIGURE 5.7.31 LED driver circuit.

FIGURE 5.7.32 BE circuit for LED driver.

Assume that the base-emitter junction of the transistor has an offset voltage of 0.7 V and that RB = 1 kΩ and RS = 42.5 Ω. 1. When the computer is supplying 5 V to the circuit, is the transistor in cutoff, in saturation, or in the linear region of operation? 2. Determine the current in the LED when “turned on” and, thus, state whether the LED will exceed its power rating. Solution. The base-emitter circuit is considered first, to determine whether the BE junction is forwardbiased. The equivalent circuit is shown in Figure 5.7.32 Writing KVL around the base circuit, we obtain

(5 − 0.7) = I B (1000) © 2005 by CRC Press LLC

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or I B = 4.3 mA Since this current is greater than zero (i.e., since positive current flows into the base), the transistor is not in cutoff. Next, we need to determine whether the transistor is in the linear active or in the saturation region. One method that can be employed to determine whether the device is in the active region is to assume that it is and to solve for the circuit that results from the assumed condition. If the resulting equations are consistent, then the assumption is justified. Assuming that the transistor is in its active region, the following equations apply: IC = βI B or IC = 95(4.3 mA) = 408.5 mA With reference to the circuit of Figure 5.7.33, KVL may be applied around the collector circuit, to obtain 5 = 1.4 + VCE + IC RS or

(5 − 1.4) − 408.5(42.5) = VCE or VCE = −13.76 V This result is clearly not possible, since the supply voltage is only 5V! It must therefore be concluded that the transistor is not in the linear region and must be in saturation. To test this hypothesis, we can substitute the nominal saturation voltage for the BJT in place of VCE (a typical saturation voltage would be VCE sat = 0.2 V) and verify that in this case we obtain a consistent solution: IC =

(5 − 1.4 − 0.2) 42.5

= 80 mA

This is a reasonable result, stating that the collector current in the saturation region for the given circuit is of the order of 80 mA.

FIGURE 5.7.33 Equivalent collector circuit of LED driver, assuming that the BJT is in the linear active region. © 2005 by CRC Press LLC

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FIGURE 5.7.34 LED driver equivalent collector circuit, assuming that the BJT is in the saturation region.

In the above part, it was determined that the transistor was in saturation. Using the circuit model for saturation, we may draw the equivalent circuit of Figure 5.7.34. Since IC = 80 mA, the power dissipated by the LED may be calculated as follows: PLED = IC VLED = 80 mA × 1.4 V = 112 mW Thus, the power limitation of the LED will not be exceeded.

Field-Effect Transistors The second transistor family discussed in this section operates on the basis of a principle that is quite different from that of the pn junction devices. The concept that forms the basis of the operation of the field-effect transistor, or FET, is that the width of a conducting channel in a semiconductor may be varied by the external application of an electric field. Thus, FETs behave as voltage-controlled resistors. This family of electronic devices can be subdivided into three groups, all of which will be introduced in the remainder of this chapter. Figure 5.7.35 depicts the classification of field-effect transistors, as well as the more commonly used symbols for these devices. These devices can be grouped into three major categories. The first two categories are both types of metal-oxide-semiconductor field-effect transistors, or MOSFETs: enhancement-mode MOSFETs and depletion-mode MOSFETs. The third category consists of junction field-effect transistors, or JFETs. In addition, each of these devices can be fabricated either as an n-channel device or as a p-channel device, where the n or p designation indicates the nature of the doping in the semiconductor channel.

FIGURE 5.7.35 Classification of field-effect transistors. © 2005 by CRC Press LLC

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FIGURE 5.7.36 n-Channel enhancement MOSFET.

The construction of the MOSFET is shown in Figure 5.7.36, along with its circuit symbol. The device consists of a metal gate, insulated from the bulk of the p-type semiconductor material by an oxide layer (thus the terminology metal-oxide-semiconductor). Two n+ regions on either side of the gate are called source and drain, respectively. An electric field can be applied between the gate and the bulk of the semiconductor by connecting the gate to a voltage source. The effect of the electric field, the intensity of which depends on the strength of the gate voltage, is to push positive charge carriers away from the surface of the bulk material. As a consequence of this effect, the p-type material has a lower concentration of positive charge carriers near its surface, and it behaves progressively more like intrinsic semiconductor material and then, finally, like n-type material as the electric field strength increases. Thus, in a narrow layer between n+ regions, the p-type material is inverted and n-type charge carriers become available for conduction. This layer is called the channel. The device of Figure 5.7.36 takes the name n-channel MOSFET. It is also called an enhancement-mode device, because the applied electric field “enhances” the conduction in the channel by attracting n-type charge carriers. There are also depletion-mode devices, in which the application of an electric field “depletes” the channel of charge carriers, reducing the effective channel width. It is useful to think of enhancement-mode devices as being normally off: current cannot flow from drain to source unless a gate voltage is applied. On the other hand, depletion-mode devices are normally on; that is, the gate voltage is used to reduce the conduction of current from drain to source. An analogous discussion could be carried out for p-channel MOSFETs. In a p-channel device, conduction in the channel occurs through positive charge carriers. The correspondence between n-channel and p-channel devices is akin to that between npn and pnp bipolar devices. We are now ready to describe qualitatively the i-v characteristic of this enhancement-mode MOSFET, for small values of drain-to-source voltage, vDS, and for constant vGS, the channel has essentially constant width and therefore acts as a constant resistance. As the gate voltage is changed, this resistance can be varied over a certain range. This mode of operation, for small drain voltages, is called the ohmic state, and, as depicted in Figure 5.7.37, it corresponds to a linear i-v curve for fixed vGS, as would be expected of a resistor. Thus, in the ohmic state the MOSFET acts as a voltage-controlled resistor. As the drain voltage is increased, the gate-to-drain voltage, vGD, decreases, reducing the electric field strength at the drain end of the device. When vGD = vGS – vDS < vT and vDS > vGS – vT , the channel is pinched down, and the electron flow into the drain is physically limited, so that the drain current becomes essentially constant. This phenomenon

FIGURE 5.7.37 MOSFET i-v curve.

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is clearly reflected in the curve of Figure 5.7.37, where it is shown that for drain-to-source voltages above vDS > vGS – vT , the drain current becomes constant, independent of vDS. This mode of operation is the constant-current state. If vDS exceeds a given drain breakdown voltage, BVDS (usually between 20 and 50 V), avalanche breakdown occurs and the drain current increases substantially. Operation in this breakdown state can lead to permanent damage because of the excessive heat caused by the large drain current. Finally, if vGS exceeds the gate breakdown voltage (around 50 V), permanent damage to the oxide layer can occur. It is important to know that it is possible to generate static voltages of magnitude sufficient to exceed this breakdown voltage just by handling the device; thus, some attention must usually be paid to static voltage buildup in handling MOS circuits.

Transistor Gates and Switches Transistor switching circuits form the basis of digital logic circuits. The objective of this section is to discuss the internal operation of these circuits and to provide the reader interested in the internal workings of digital circuits with an adequate understanding of the basic principles. An electronic gate is a device that, on the basis of one or more input signals, produces one of two or more prescribed outputs; as will be seen shortly, one can construct both digital and analog gates. A word of explanation is required, first, regarding the meaning of the words analog and digital. An analog voltage or current — or, more generally, an analog signal — is one that varies in a continuous fashion over time, in analogy (hence the expression analog) with a physical quantity. An example of an analog signal is a sensor voltage corresponding to ambient temperature on any given day, which may fluctuate between, say, 30° and 50°F. A digital signal, on the other hand, is a signal that can take only a finite number of values; in particular, a commonly encountered class of digital signals consists of binary signals, which can take only one of two values (for example, 1 and 0). A typical example of a binary signal would be the control signal for the furnace in a home heating system controlled by a conventional thermostat, where one can think of this signal as being “on” (or 1) if the temperature of the house has dropped below the thermostat setting (desired value), or “off ” (or 0) if the house temperature is greater than or equal to the set temperature (say, 68°F). Figure 5.7.38 illustrates the appearance of the analog and digital signals in this furnace example.

FIGURE 5.7.38 Illustration of analog and digital signals.

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Analog Gates A common form of analog gate — probably the most important, in practice — employs an FET and takes advantage of the fact that current can flow in either direction in an FET biased in the ohmic region. Recall that the drain characteristic of the MOSFET consists of three regions: ohmic, active, and breakdown. A MOSFET amplifier is operated in the active region, where the drain current is nearly constant for any given value of vGS. On the other hand, a MOSFET biased in the ohmic state acts very much as a linear resistor. For example, for an n-channel enhancement MOSFET, the conditions for the transistor to be in the ohmic region are vGS > VT

and

v DS ≤

1 (v − VT ) 4 GS

(5.7.17)

As long as the FET is biased within these conditions, it acts simply as a linear resistor, and it can conduct current in either direction (provided that vDS does not exceed the limits stated in Equation (5.7.17). In particular, the resistance of the channel in the ohmic region is found to be RDS =

VT2 2 I DSS (vGS − VT )

(5.7.18)

so that the drain current is equal to iD ≈

v DS rDS

for

v DS ≤

1 (v − VT ) 4 GS

and

vGS > VT

(5.7.19)

The most important feature of the MOSFET operating in the ohmic region, then, is that it acts as a voltage-controlled resistor, with the gate-source voltage, vGS, controlling the channel resistance, RDS. The use of the MOSFET as a switch in the ohmic region, then, consists of providing a gate-source voltage that can either hold the MOSFET in the cutoff region (vGS < VT) or bring it into the ohmic region. In this fashion, vGS acts as a control voltage for the transistor. Consider the circuit shown in Figure 5.7.39, where we presume that vC can be varied externally and that vin is some analog signal source that we may wish to connect to the load RL at some appropriate time. The operation of the switch is as follows. When vC ≤ VT , the FET is in the cutoff region and acts as an open circuit. When vC > VT (with a value of vGS such that the MOSFET is in the ohmic region), the transistor acts as a linear resistance, RDS. If RDS
RL, then vout ≈ vin. By using a pair of MOSFETs, it is possible to improve the dynamic range of signals one can transmit through this analog gate. MOSFET analog switches are usually produced in integrated circuit (IC) form and denoted by the symbol shown in Figure 5.7.40. Digital Gates BJT Gates. In discussing large-signal models for the BJT, we observed that the i-v characteristic of this family of devices includes a cutoff region, where virtually no current flows through the transistor. On the other hand, when a sufficient amount of current is injected into the base of the transistor, a bipolar transistor will reach saturation, and a substantial amount of collector current will flow. This behavior is quite well suited to the design of electronic gates and switches and can be visualized by superimposing a load line on the collector characteristic, as shown in Figure 5.7.41. The operation of the simple BJT switch is illustrated in Figure 5.7.41 by means of load-line analysis. Writing the load-line equation at the collector circuit, we have vCE = VCC − iC RC © 2005 by CRC Press LLC

(5.7.20

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FIGURE 5.7.39 MOSFET analog switch.

FIGURE 5.7.40 Symbol for bilateral FET analog gate.

FIGURE 5.7.41 BJT switching characteristic. © 2005 by CRC Press LLC

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and vout = vCE

(5.7.21)

Thus, when the input voltage, vin, is low (say, 0 V, for example) the transistor is in the cutoff region and little or no current flows, and vout = vCE = VCC

(5.7.22)

so that the output is “logic high.” When vin is large enough to drive the transistor into the saturation region, a substantial amount of collector current will flow and the collector-emitter voltage will be reduced to the small saturation value, VCE sat, which is typically a fraction of a volt. This corresponds to the point labeled B on the load line. For the input voltage vin to drive the BJT of Figure 5.7.41 into saturation, a base current of approximately 50 mA will be required. Suppose, then, that the voltage vin could take the values 0 or 5 V. Then, if vin = 0 V, vout will be nearly equal to VCC or, again, 5 V. If, on the other hand, vin = 5 V and RB is, say, equal to 85 kΩ [so that the base current required for saturation flows into the base: iB ≈ (5 V – 0.7 V)/RB = 50.6 µA], we have the BJT in saturation, and vout = VCE sat ≈ 0.2 V. Thus, you see that whenever vin corresponds to a logic high (or logic 1), vout takes a value close to 0 V, or logic low (or 0); conversely, vin =“0” (logic “low”) leads to vout = “1”. The values of 5 and 0 V for the two logic levels 1 and 0 are quite common in practice and are the standard values used in a family of logic circuits denoted by the acronym TTL, which stands for transistor-transistor logic. One of the more common TTL blocks is the inverter shown in Figure 5.7.41, so called because it “inverts” the input by providing a low output for a high input, and vice versa. This type of inverting, or “negative”, logic behavior is quite typical of BJT gates (and of transistor gates in general). MOSFET Logic Gates. Having discussed the BJT as a switching element, we might quite naturally suspect that FETs may similarly serve as logic gates. In fact, in some respects, FETs are better suited to be employed as logic stages than BJTs. The n-channel enhancement MOSFET serves as an excellent illustration: because of its physical construction, it is normally off (that is, it is off until a sufficient gate voltage is provided), and therefore it does not require much current from the input signal source. Further, MOS devices offer the additional advantage of easy fabrication into integrated circuit form, making production economical in large volume. On the other hand, MOS devices cannot provide as much current as BJTs, and their switching speeds are not quite as fast — although these last statements may not hold true for long, because great improvements are taking place in MOS technology. Overall, it is certainly true that in recent years it has become increasingly common to design logic circuits based on MOS technology. In particular, a successful family of logic gates called CMOS (for complementary metal-oxide-semiconductor) takes advantage of both p- and n-channel enchancement-mode MOSFETs to exploit the best features of both types of transistors. CMOS logic gates (and many other types of digital circuits constructed using the same technology) consume very little supply power and have become the mainstay in pocket calculators, wristwatches, portable computers, and many other consumer electronics products. Without delving into the details of CMOS technology, we shall briefly illustrate the properties of MOSFET logic gates and of CMOS gates in the remainder of this section. Figure 5.7.42 depicts a MOSFET switch with its drain i-v characteristic. Note the general similarity with the switching characteristic of the BJT shown in the previous section. When the input voltage, vin, is zero, the MOSFET conducts virtually no current and the output voltage, vout, is equal to VDD. When vin is equal to 5 V, the MOSFET Q point moves from point A to point B along with the load line, with vDS = 0.5 V. Thus, the circuit acts as an inverter. Much as in the case of the BJT, the inverter forms the basis of all MOS logic gates. An elementary CMOS inverter is shown in Figure 5.7.43. Note first the simplicity of this configuration, which simply employs two enhancement-mode MOSFETs: p-channel at the top, denoted by the symbol Qp, and n-channel at the bottom, denoted by Qn. Recall from Chapter 8 that when vin is low, transistor © 2005 by CRC Press LLC

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FIGURE 5.7.42 MOSFET switching characteristic.

FIGURE 5.7.43 CMOS inverter.

Qn is off. However, transistor Qp sees a gate-to-source voltage vGS = vin – VDD = – VDD; in a p-channel device, this condition is the counterpart of having vGS = VDD for an n-channel MOSFET. Thus, when Qn is off, Qp is on and acts very much as a small resistance. In summary, when vin is low, the output is vout ≈ VDD. When vin is high, the situation is reversed: Qn is now on and acts nearly as a short circuit while Qp is open (since vGS = 0 for Qp). Thus, vin ≈ 0. The complementary MOS operation is depicted in Figure 5.7.43 in simplified form by showing each transistor as either a short or an open circuit, depending on it state. This simplified analysis is sufficient for the purpose of a qualitative analysis.

5.8 Power Electronics The objective of this section is to present a survey of power electronic devices and systems. Power electronic devices form the “muscle” of many electromechanical systems. For example, one finds such devices in many appliances, in industrial machinery, and virtually wherever an electric motor is found, since one of the foremost applications of power electronic devices is to supply and control the currents and voltages required to power electric machines, such as those introduced in Section 5.12.

Classification of Power Electronic Devices Power semiconductors can be broadly subdivided into five groups: (1) power diodes, (2) thyristors, (3) power bipolar junction transistors (BJTs), (4) insulated-gate bipolar transistors (IGBTs), and (5) static induction transistors (SITs). Figure 5.8.1 depicts the symbols for the most common power electronic devices. © 2005 by CRC Press LLC

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FIGURE 5.8.1 Classification of power electronic devices.

Power diodes are functionally identical to the diodes introduced in Section 5.7, except for their ability to carry much larger currents. A diode conducts in the forward-biased mode when the anode voltage (VA) is higher than the cathode voltage (VK). Three types of power diodes exist: general-purpose, highspeed (fast-recovery), and Schottky. Typical ranges of voltage and current are 3000 V and 3500 A for general-purpose diodes and 3000 V and 1000 A for high-speed devices. The latter have switching times as low as a fraction of a microsecond. Schottky diodes can switch much faster (in the nanosecond range) but are limited to around 100 V and 300 A. The forward voltage drop of power diodes is not much higher than that of low-power diodes, being between 0.5 and 1.2 V. Since power diodes are used with rather large voltages, the forward bias voltage is usually considered negligible relative to other voltages in the circuit, and the switching characteristics of power diodes may be considered near ideal. The principal consideration in choosing power diodes is their power rating. Thyristors function like power diodes with an additional gate terminal that controls the time when the device begins conducting; a thyristor starts to conduct when a small gate current is injected into the

© 2005 by CRC Press LLC

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gate terminal, provided that the anode voltage is greater than the cathode voltage (or VAK > 0 V). The forward voltage drop of a thyristor is of the order of 0.5 to 2 V. Once conduction is initiated, the gate current has no further control. To stop conduction, the device must be reverse-biased; that is, one must ensure that VAK ≤ 0 V. Thyristors can be rated at up to 6000 V and 3500 A. The turn-off time is an important characteristic of thyristors; it represents the time required for the device current to return to zero after external switching of VAK. The fastest turn-off times available are in the range of 10 µsec; however, such turn-off times are achieved only in devices with slightly lower power ratings (1200 V, 1000 A). Thyristors can be subclassified into the following groups: force-commutated and line-commutated thyristors, gate turn-off thyristors (GTOs), reverse-conducting thyristors (RCTs), static induction thyristors (SITHs), gate-assisted turn-off thyristors (GATTs), light-activated silicon-controlled rectifiers (LASCSRs), and MOS-controlled thyristors (MCTs). It is beyond the scope of this chapter to go into a detailed description of each of these types of devices; their operation is typically a slight modification of the basic operation of the thyristor. The reader who wishes to gain greater insight into this topic may refer to one of a number of excellent books specifically devoted to the subject of power electronics. Two types of thyristor-based device deserve some more attention. The triac, as can be seen in Figure 5.8.1, consists of a pair of thyristors connected back to back, with a single gate; this allows for current control in either direction. Thus, a triac may be thought of as bidirectional thyristor. The gate turn-off thyristor (GTO), on the other hand, can be turned on by applying a short positive pulse to the gate, like a thyristor, and can also be turned off by application of a short negative pulse. Thus, GTOs are very convenient in that they do not require separate commutation circuits to be turned on and off. Power BJTs can reach ratings up to 1200 V and 400 A, and they operate in much the same way as a conventional BJT. Power BJTs are used in power converter applications at frequencies up to around 10 kHz. Power MOSFETs can operate at somewhat higher frequencies (a few to several tens of kHz), but are limited in power (typically up to 1000 V, 50 A). Insulated-gate bipolar transistors (IGBTs) are voltage-controlled (because of their insulated gate, reminiscent of insulated-gate FETs) power transistors that offer superior speed with respect to BJTs but are not quite as fast as power MOSFETs.

Classification of Power Electronic Circuits The devices that will be discussed in the present chapter find application in a variety of power electronic circuits. This section will briefly summarize the principal types of power electronic circuits and will qualitatively describe their operation. The following sections will describe the devices and their operation in these circuits in more detail. One possible classification of power electronic circuits is given in Table 5.8.1. Many of the types of circuits are similar to circuits that were introduced in earlier chapters. Voltage regulators were introduced in Section 5.7. Power electronic switches function exactly like the transistor switches described in Section 5.7; their function is to act as voltage- or current-controlled switches to turn AC or DC supplies on and off. Transistor power amplifiers are the high-power version of the BJT and MOSFET amplifiers mentioned in Section 5.7. TABLE 5.8.1 Power Electronic Circuits Circuit Type Voltage regulators Power amplifiers Switches Diode rectifier AC-DC converter (controlled rectifier) AC-AC converter (AC voltage controller) DC-DC converter (chopper) DC-AC converter (inverter)

© 2005 by CRC Press LLC

Essential Features Regulate a DC supply to a fixed voltage output Large-signal amplification of voltages and currents Electronic switches (for example, transistor switches) Converts fixed AC voltage (single- or multiphase) to fixed DC voltage Converts fixed AC voltage (single- or multiphase) to variable DC voltage Converts fixed AC voltage to variable AC voltage (single- or multiphase) Converts fixed DC voltage to variable DC voltage Converts fixed DC voltage to variable AC voltage (single- or multiphase)

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A

B

FIGURE 5.8.2 AC-DC converter circuit and waveform.

Diode rectifiers were discussed in Section 5.7 in their single-phase form; similar rectifiers can also be designed to operate with three-phase sources. The operation of a single-phase full-wave rectifier was summarized in Figure 5.8.2. AC-DC converters are also rectifiers, but they take advantage of the controlled properties of thyristors. The thyristor gate current can be timed to “fire” conduction at variable times, resulting in a variable DC output, as illustrated in Figure 5.8.2, which shows the circuit and behavior of a single-phase AC-DC converter. This type of converter is very commonly used as a supply for DC electric motors. In Figure 5.8.2 α is the firing angle of thyristor T1, where the device starts to conduct. AC-AC converters are used to obtain a variable AC voltage from a fixed AC source. Figure 5.8.3 shows a triac-based AC-AC converter, which takes advantage of the bidirectional capability of triacs to control the rms value of an alternating voltage. Note in particular that the resulting AC waveform is no longer a pure sinusoid even though its fundamental period (frequency) is unchanged. A DC-DC converter, also known as a chopper, or switching regulator, permits conversion of a fixed DC source to a variable DC supply. Figure 5.8.4 shows how such an effect may be obtained by controlling the base-emitter voltage of a bipolar transistor, enabling conduction at the desired time. This results in the conversion of the DC input voltage to a variable-duty-cycle output voltage, whose average value can be controlled by selecting the “on” time of the transistor. DC-DC converters find application as variable voltage supplies for DC electric motors used in electric vehicles.

FIGURE 5.8.3 AC-AC converter circuit and waveform. © 2005 by CRC Press LLC

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FIGURE 5.8.4 DC-DC converter circuit and waveform.

Finally, DC-AC supplies, or inverters, are used to convert a fixed DC supply to a variable AC supply; they find application in AC motor control. The operation of these circuits is rather complex; it is illustrated conceptually in the waveforms of Figure 5.8.5, where it is shown that by appropriately switching two pairs of transistors it is possible to generate an alternating current waveform (square wave). Each of the circuits of Table 5.8.1 will now be analyzed in greater detail.

Rectifiers and Controlled Rectifiers (AC-DC Converters) Thyristors and Controlled Rectifiers In a number of applications, it is useful to be able to externally control the amount of current flowing from an AC source to the load. A family of power semiconductor devices called controlled rectifiers allows for control of the rectifiers state by means of a third input, called the gate. Figure 5.8.6 depicts the appearance of a thyristor, or silicon-controlled rectifier (SCR), illustrating how the physical structure of this device consists of four layers, alternating p-type and n-type material. Note that the circuit symbol for the thyristor suggests that this device acts as a diode, with provision for an additional external control signal. The operation of the thyristor can be explained in an intuitive fashion as follows. When the voltage vAK is negative (i.e., providing reverse bias), the thyristor acts just like a conventional pn junction in the off state. When vAK is forward-biased and a small amount of current is injected into the gate, the thyristor conducts forward current. The thyristor then continues to conduct (even in the absence of gate current), provided that vAK remains positive. Figure 5.8.7 depicts the i-v curve for the thyristor. Note that the thyristor has two stable states, determined by the bias vAK and by the gate current. In summary, the thyristor acts as a diode with a control gate that determines the time when conduction begins. A somewhat more accurate description of thyristor operation may be provided if we realize that the four-layer pnpn device can be modeled as a pnp transistor connected to an npn transistor. Figure 5.8.8 clearly shows that, physically, this is a realistic representation. Note that the anode current, iA, is equal to the emitter current of the pnp transistor (labeled Qp) and the base current of Qp is equal to the collector current of the npn transistor, Qn. Likewise, the base current of Qn is the sum of the gate current and the collector current of Qp. The behavior of this transistor model is explained as follows. Suppose, initially, iG and iBn are both zero. Then it follows that Qn is in cutoff, and therefore iCn = 0. But if iCn = 0, then the base current going into Qp is also zero and Qp is also in cutoff, and iCn = 0, consistent with our initial © 2005 by CRC Press LLC

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FIGURE 5.8.5 DC-AC converter circuit and waveform.

FIGURE 5.8.6 Thyristor structure and circuit symbol.

assumption. Thus, this is a stable state, in the sense that unless an external condition perturbs the thyristor, it will remain off. Now, suppose a small pulse of current is injected at the gate. Then iBn > 0 and Qn starts to conduct, provided, of course, that vAK > 0. At this point, iCn , and therefore iBp , must be greater than zero, so that Qp conducts. It is important to note that once the gate current has turned Qn on, Qp also conducts, so that iCp > 0. Thus, even though iG may cease, once this “on” state is reached, iCp = iBn continues to drive Qn so that the on state is also self-sustaining. The only condition that will cause the thyristor to revert to the off state is the condition in which vAK becomes negative; in this case, both transistors return to the cutoff state. © 2005 by CRC Press LLC

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FIGURE 5.8.7 Thyristor i-v characteristic.

FIGURE 5.8.8 Thyristor two-transistor model.

In a typical controlled rectifier application, the device is used as a half-wave rectifier that conducts only after a trigger pulse is applied to the gate. Without concerning ourselves with how the trigger pulse is generated, we can analyze the general waveforms for the circuit of Figure 5.8.9 as follows. Let the voltage vtrigger be applied to the gate of the thyristor at t = τ. The voltage vtrigger can be a short pulse, provided by a suitable trigger-timing circuit (Section 5.11 will discuss timing and switching circuits). At t = τ, the thyristor begins to conduct, and it continues to do so until the AC source enters its negative cycle. Figure 5.8.10 depicts the relevant waveforms. © 2005 by CRC Press LLC

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FIGURE 5.8.9 Controlled rectifier circuit.

FIGURE 5.8.10 Half-wave controlled rectifier waveforms.

Note how the DC load voltage is controlled by the firing time τ, according to the following expression: v L = VL =

1 T

∫

T 2

τ

vAC (t ) dt

(5.8.1)

where T is the period of vAC(t), Now, if we let vAC (t ) = A sin ωt

(5.8.2)

we can express the average (DC) value of the load voltage VL =

1 T

∫

T 2

τ

A sin ωt dt = (1 + cos ωt )

A 2π

(5.8.3)

in terms of the firing angle, α, defined as α = ωt

(5.8.4)

By evaluating the integral of Equation (5.8.3), we can see that the (DC) load voltage amplitude depends on the firing angle, α: VL = (1 + cosα ) © 2005 by CRC Press LLC

A 2π

(5.8.5)

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FIGURE 5.8.11 The four quadrants of an electric drive.

FIGURE 5.8.12 DC motor.

Electric Motor Drives The advent of high-power semiconductor devices has made it possible to design effective and relatively low-cost electronic supplies that take full advantage of the capabilities of the devices introduced in this chapter. Electronic power supplies for DC and AC motors have become one of the major fields of application of power electronic devices. The last section of this chapter is devoted to an introduction to two families of power supplies, or electric drives: choppers, or DC-DC converters; and inverters, or DC-AC converters. These circuits find widespread application in the control of AC and DC motors in a variety of applications and power ranges. Before we delve into the discussion of the electronic supplies, it will be helpful to introduce the concept of quadrants of operation of a drive. Depending on the direction of current flow and on the polarity of the voltage, an electronic drive can operate in one of four possible modes, as indicated in Figure 5.8.11. Choppers (DC-DC Converters) As the name suggests, a DC-DC converter is capable of converting a fixed DC supply to a variable DC supply. This feature is particularly useful in the control of the speed of a DC motor (described in greater detail in Section 5.12). In a DC motor, shown schematically in Figure 5.8.12, the developed torque, Tm , is proportional to the current supplied to the motor armature, Ia, while the electromotive force (emf), Ea, which is the voltage developed across the armature, is proportional to the speed of rotation of the motor, ωm. A DC motor is an electromechanical energy-conversion system; that is, it converts electrical to mechanical energy (or vice versa if it is used as a generator); if we recall that the product of torque and speed is equal to power in the mechanical domain, and that current times voltage is equal to power in the electrical domain, we conclude that in the ideal case of complete energy conversion, we have Ea × I a = Tm × ω m

(5.8.6)

Naturally, such ideal energy conversion cannot take place; however, we can see that there is a correspondence between the four-electrical quadrants of Figure 5.8.11 and the mechanical power output of the motor: namely, if the voltage and current are both positive or both negative, the electrical power will be positive, and so will the mechanical power. This corresponds to the forward (i, v both positive) and reverse (i,v both negative) motoring operation. Forward motoring corresponds to quadrant I, and reverse motoring to quadrant III in Figure 5.8.12. If the voltage and current are of opposite polarity (quadrants II and IV), electrical energy is flowing back to the electric drive; in mechanical terms this corresponds to a braking condition. Operation in the fourth quadrant can lead to regenerative braking, so called © 2005 by CRC Press LLC

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FIGURE 5.8.13 Step-down chopper (buck converter).

FIGURE 5.8.14 Step-down chopper waveforms.

because power is generated by making current flow back to the source. This mode could be useful, for example, to recharge a battery supply, because the braking energy can be regenerated by returning it to the electric supply. A simple circuit that can accomplish the task of providing a variable DC supply from a fixed DC source is the step-down chopper (buck converter), shown in Figure 5.8.13. The circuit consists of a “chopper” switch, denoted by the symbol S, and a free-wheeling diode. The switch can be any of the power switches described in this chapter, for example, a power BJT or MOSFET, or a thyristor; see, for example, the BJT switch of Figure 5.8.4. The circuit to the right of the diode is a model of a DC motor, including the inductance and resistance of the armature windings and the effect of the back emf Ea. When the switch is turned on (say, at t = 0), the supply Vs is connected to the load and vo = Vs. The load current, io , is determined by the motor parameters. When the switch is turned off, the load current continues to flow through the free-wheeling diode, but the output voltage is now vo = 0. At time T, the switch is turned on again, and the cycle repeats. Figure 5.8.14 depicts the vo and io waveforms. The average value of the output voltage, 〈vo〉, is given by the expression vo =

t1 V = δVS T S

(5.8.7)

where δ is the duty cycle of the chopper. The step-down chopper has a useful range, 0 ≤ vo ≤ VS

(5.8.8)

It is also possible to increase the range of a DC-DC converter to above the supply voltage by making use of the energy-storage properties of an inductor; the resulting circuit is shown in Figure 5.8.15. When the chopper switch, S, is on, the supply current flows through the inductor and the closed switch, storing energy in the inductor; the output voltage, vo, is zero, since the switch is a short circuit. When the switch is open, the supply current will flow through the load via the diode; but the inductor voltage is negative during the transient following the opening of the switch and therefore adds to the source voltage: the energy stored in the inductor while the switch was closed is now released and transferred to the load. This stored energy makes it possible for the output voltage to be higher than the supply voltage for a finite period of time. © 2005 by CRC Press LLC

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FIGURE 5.8.15 Step-up chopper (boost converter).

Let t1 once again be the time during which the chopper conducts; neglecting for the moment the ripple in the supply current, the energy stored in the inductor during this period is Wi = VS I S t1

(5.8.9)

When the chopper is off, the energy released to the load is

(

)

Wi = vo − VS I S (T − t1 )

(5.8.10)

If the system is lossless, the two energy expressions must be equal:

(

)

(5.8.11)

t1 + T − t1 1 T = VS = VS T − t1 1− δ T − t1

(5.8.12)

VS I S t1 = vo − VS I S (T − t1 ) and we can determine the average output voltage to be vo = VS

Since the duty cycle, δ, is always less than 1, the theoretical range of the supply is VS ≤ vo < ∞

(5.8.13)

The waveforms for the boost converter are shown in Figure 5.8.16. A step-up chopper can also be used to provide regenerative braking: if the “supply” voltage is the motor armature voltage and the output voltage is the fixed DC supply (battery) voltage, then power can be made to flow from the motor to the DC supply (i.e., recharging the battery). This configuration is shown in Figure 5.8.17.

FIGURE 5.8.16 Step-up chopper output voltage waveform. © 2005 by CRC Press LLC

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FIGURE 5.8.17 Step-up chopper used for regenerative braking.

FIGURE 5.8.18 Two-quadrant chopper.

Finally, the operation of the step-down and step-up choppers can be combined into a two-quadrant chopper, shown in Figure 5.8.18. The circuit shown schematically in Figure 5.8.18 can provide both regenerative braking and motoring operation in a DC motor. When switch S2 is open and switch S1 serves as a chopper, the circuit operates as a step-down chopper, precisely as was described earlier in this section (convince yourself of this by redrawing the circuit with S2 and D2 replaced by open circuits). Thus, the drive and motor operate in the first quadrant (motoring operation). The output voltage, vo , will switch between VS and zero, as shown in Figure 5.8.14, and the load current will flow in the direction indicated by the arrow in Figure 5.8.18 diode D1 free-wheels whenever S1 is open. Since both output voltage and current are positive, the system operates in the first quadrant. When switch S1 is open and switch S2 serves as a chopper, the circuit resembles a step-up chopper. The source is the motor emf, Ea, and the load is the battery; this is the situation depicted in Figure 5.8.17. The current will now be negative, since the sum of the motor emf and the voltage across the inductor (corresponding to the energy stored during the “on” cycle of S2) is greater than the battery voltage. Thus, the drive operates in the fourth quadrant. Inverters (DC-AC Converters) As will be explained in Section 5.12, variable-speed drives for AC motors require a multiphase variablefrequency, variable-voltage supply. Such drives are called DC-AC converters, or inverters. Inverter circuits can be quite complex, so the objective of this section is to present a brief introduction to the subject, with the aim of illustrating the basic principles. A voltage source inverter (VSI) converts the output of a fixed DC supply (e.g., a battery) to a variable-frequency AC supply. Figure 5.8.19 depicts a half-bridge VSI; once again, the switches can be either bipolar or MOS transistors, or thyristors. The operation of this circuit is as follows. When switch S1 is turned on, the output voltage is in the positive half-cycle, and vo = VS /2. The switching sequence of S1 and S2 is shown in Figure 5.8.20. It is important that each switch be turned off before the other is turned on; otherwise, the DC supply would be short-circuited. Since the load is always going to be inductive in the case of a motor drive, it is important to observe that the load current, io, will lag the voltage waveform, as shown in Figure 5.8.20. As shown in this figure, there will be some portions of the cycle in which the voltage is positive but the current is negative. The function of diodes D1 and D2 is precisely to conduct the load current whenever it is of direction opposite to the © 2005 by CRC Press LLC

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FIGURE 5.8.19 Half-bridge voltage source inverter.

FIGURE 5.8.20 Half-bridge voltage source inverter waveforms.

FIGURE 5.8.21 Full-bridge voltage source inverter.

FIGURE 5.8.22 Half-bridge voltage source inverter output voltage waveform.

polarity of the voltage. Without these diodes, there would be no load current in this case. Figure 5.8.20 also shows which element is conducting in each portion of the cycle. A full-bridge version of the VSI can also be designed as shown in Figure 5.8.21; the associated output voltage waveform is shown in Figure 5.8.22. A 3-phase VSI is shown in Figure 5.8.23.

5.9 Operational Amplifiers In this section we will analyze the properties of a general-purpose amplifier circuit known as the operational amplifier.

The Operational Amplifier An operational amplifier is an integrated circuit, that is, a large collection of individual electrical and electronic circuits integrated on a single silicon wafer. An operational amplifier — or op-amp — can perform a great number of operations, such as addition, filtering, or integration, which are all based on © 2005 by CRC Press LLC

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FIGURE 5.8.23 Three-phase voltage source inverter.

the properties of ideal amplifiers and of ideal circuit elements. The introduction of the operational amplifier in integrated circuit form marked the beginning of a new era in modern electronics. Since the introduction of the first IC op-amp, the trend in electronic instrumentation has been to move away from the discrete (individual-component) design of electronic circuits, toward the use of integrated circuits for a large number of applications. This statement is particularly true for applications of the type the nonelectrical engineer is likely to encounter: op-amps are found in most measurement and instrumentation applications, serving as extremely versatile building blocks for any application that requires the processing of electrical signals. In the following pages, simple circuit models of the op-amp will be introduced. The simplicity of the models will permit the use of the op-amp as a circuit element, or building block, without the need to describe its internal workings in detail. Integrated circuit technology has today reached such an advanced stage of development that it can be safely stated that for the purpose of many instrumentation applications, the op-amp can be treated as an ideal device. Following the introductory material presented in this chapter, more advanced instrumentation applications will be explored in Section 5.11. The Open-Loop Model The ideal operational amplifier behaves very much as an ideal differential amplifier, that is, a device that amplifies the difference between two input voltages. Operational amplifiers are characterized by nearinfinite input resistance and very small output resistance. As shown in Figure 5.9.1 the output of the opamp is an amplified version of the difference between the voltages present at the two inputs:

(

vout = AV (OL ) v + − v −

)

(5.9.1)

The input denoted by a positive sign is called the noninverting input (or terminal), while that represented with a negative sign is termed the inverting input (or terminal). The amplification factor, or gain, AV(OL), is called the open-loop voltage gain and is quite large by design, typically of the order of 105 to 107; it will soon become apparent why a large open-loop gain is a desirable characteristic. Together with the high input resistance and low output resistance, the effect of a large amplifier open-loop voltage gain, AV(OL), is such that op-amp circuits can be designed to perform very nearly as ideal voltage or current amplifiers. In effect, to analyze the performance of an op-amp circuit, only one assumption will be needed: that the current flowing into the input circuit of the amplifier is zero, or iin = 0

(5.9.2)

This assumption is justified by the large input resistance and large open-loop gain of the operational amplifier. The model just introduced will be used to analyze three amplifier circuits in the next part of this section. © 2005 by CRC Press LLC

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FIGURE 5.9.1 Operational amplifier model symbols, and circuit diagram.

The Operational Amplifier in the Closed-Loop Mode The Inverting Amplifier. One of the more popular circuit configurations of the op-amp, because of its simplicity, is the so-called inverting amplifier, shown in Figure 5.9.2.  1 1 1  vS = − vout  + +   RF RS AV (OL ) RF RS AV (OL )  © 2005 by CRC Press LLC

(5.9.3)

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FIGURE 5.9.2 Inverting amplifier.

If the open-loop gain of the amplifier, AV(OL), is sufficiently large, the terms 1/(AV(OL) RF /RS) and 1/AV(OL), are essentially negligible, compared with 1/(RF /RS). As stated earlier, typical values of AV(OL) range from 105 to 107, and thus it is reasonable to conclude that, to a close approximation, the following expression describes the closed-loop gain of the inverting amplifier: vout = −

RF v RS S

Inverting amplifier closed-loop gain

(5.9.4)

Next, we show that by making an additional assumption it is possible to simplify the analysis considerably. Consider that, as was shown for the inverting amplifier, the inverting terminal voltage is given by v− = −

vout AV (OL )

(5.9.5)

Clearly, as AV(OL) approaches infinity, the inverting-terminal voltage is going to be very small (practically, of the order of microvolts). It may then be assumed that in the inverting amplifier, v– is virtually zero: v− ≈ 0

(5.9.6)

This assumption prompts an interesting observation (which may not yet appear obvious at this point): the effect of the feedback connection from output to inverting input is to force the voltage at the inverting input to be equal to that at the noninverting input. This is equivalent to stating that for an op-amp with negative feedback, v− ≈ v+

(5.9.7)

The analysis of the operational amplifier can now be greatly simplified if the following two assumptions are made: 1. iin = 0 2. v − = v +

(5.9.8)

This technique will be tested in the next subsection by analyzing a noninverting amplifier configuration. A useful op-amp circuit that is based on the inverting amplifier is the op-amp summer, or summing amplifier. This circuit, shown in Figure 5.9.3, is used to add signal sources. The primary advantage of using the op-amp as a summer is that the summation occurs independently of load and source impedances, so that sources with different internal impedances will not interact with each other. N

vout = −

∑R n =1

© 2005 by CRC Press LLC

RF Sn

vSn

(5.9.9)

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FIGURE 5.9.3 Summing amplifier.

FIGURE 5.9.4 Noninverting amplifier.

FIGURE 5.9.5 Differential amplifier.

That is, the output consists of the weighted sum of N input signal sources, with the weighting factor for each source equal to the ratio of the feedback resistance to the source resistance. The Noninverting Amplifier. To avoid the negative gain (i.e., phase inversion) introduced by the inverting amplifier, a noninverting amplifier configuration is often employed. A typical noninverting amplifier is shown in Figure 5.9.4; note that the input signal is applied to the noninverting terminal this time. vout R = 1+ F vS RS

Noninverting amplifier closed-loop gain

The Differential Amplifier. The third closed-loop model examined in this chapter is a combination of the inverting and noninverting amplifiers; it finds frequent use in situations where the difference between two signals needs to be amplified. The basic differential amplifier circuit is shown in Figure 5.9.5, where the two sources, v1 and v2, may be independent of each other, or may originate from the same process, as they do in Example 5.9.1. © 2005 by CRC Press LLC

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The following expression for the output voltage is obtained:  −v  R2 1 vout = R2  1 + v2 + v2  R1 + R2 R1 ( R1 + R2 )   R1 =

R2 (v − v1 ) R1 2

In practice, it is often necessary to amplify the difference between two signals that are both corrupted by noise or some other form of interference. In such cases, the differential amplifier provides an invaluable tool in amplifying the desired signal while rejecting the noise. Example 5.9.1 provides a realistic look at a very common application of the differential amplifier. Example 5.9.1 An EKG Amplifier This example illustrates the principle behind a two-lead electrocardiogram (EKG) measurement. The desired cardiac waveform is given by the difference between the potentials measured by two electrodes suitably placed on the patient’s chest, as shown in Figure 5.9.6. A healthy, noise-free EKG waveform, v1 – v2, is shown in Figure 5.9.7. Unfortunately, the presence of electrical equipment powered by the 60-Hz, 110-VAC line current causes undesired interference at the electrode leads: the lead wires act as antennas and pick up some of the 60-Hz signal in addition to the desired EKG voltage. In effect, instead of recording the desired EKG signals, v1 and v2, the two electrodes provide the following inputs to the EKG amplifier, shown in Figure 5.9.8. Lead 1: v1 (t ) + vn (t ) = v1 (t ) + Vn cos(377t + φ n ) Lead 2: v2 (t ) + vn (t ) = v2 (t ) + Vn cos(377t + φ n )

FIGURE 5.9.6 Two-lead electrocardiogram.

FIGURE 5.9.7 EKG waveform.

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FIGURE 5.9.8 EKG amplifier.

The interference signal, Vn cos (377t + φn), is approximately the same at both leads, because the electrodes are chosen to be identical (e.g., they have the same lead lengths) and are in close proximity to each other. Further, the nature of the interference signal is such that it is common to both leads, since it is a property of the environment the EKG instrument is embedded in. On the basis of the analysis presented earlier, then, vout =

[

]

R2 (v + vn (t )) − (v2 + vn (t )) R1 1

or vout =

R2 ( v − v2 ) R1 1

Thus, the differential amplifier nullifies the effect of the 60-Hz interference, while amplifying the desired EKG waveform. The preceding example introduced the concept of so-called common-mode and differential-mode signals. In the EKG example, the desired differential-mode EKG signal was amplified by the op-amp while the common-mode disturbance was canceled. Thus, the differential amplifier provides the ability to reject common-mode signal components (such as noise or undesired DC offsets) while amplifying the differential-mode components. This is a very desirable feature in instrumentation systems. In practice, rejection of the common-mode signal is not always complete: some of the common-mode signal component will always appear in the output. This fact gives rise to a figure of merit called the common-mode rejection ratio, which is discussed later in this section. Often, to provide impedance isolation between bridge transducers and the differential amplifier stage, the signals v1 and v2 are amplified separately. This technique gives rise to the so-called instrumentation amplifier (IA), shown in Figure 5.9.9. Example 5.9.2 illustrates the calculation of the closed-loop gain for a typical instrumentation amplifier. Example 5.9.2 Instrumentation Amplifier In this example, we compute the closed-loop gain of the instrumentation amplifier of Figure 5.9.9. Solution. To carry out the desired analysis as easily as possible, it is helpful to observe that resistor R1 is shared by the two input amplifiers. This corresponds to having each amplifier connected to a resistor equal to R1/2, as shown in Figure 5.9.10(a). Because of the symmetry of the circuit, one can view the shared resistor as being connected to ground in the center, so that the circuit takes the form of a noninverting amplifier, with closed-loop gain given by A = 1+ © 2005 by CRC Press LLC

2 R2 R1

(5.9.10)

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FIGURE 5.9.9 Instrumentation amplifier.

ab

FIGURE 5.9.10 (a) and (b).

Thus, each of the input voltages is amplified by this gain, and the overall gain of the instrumentation amplifier can then be computed by considering that the voltage difference (Av1 – Av2) is then amplified by the differential amplifier stage, with gain RF /R, as shown in Figure 5.9.10(b): vout =

RF R  2R  Av1 − v2 ) = F 1 + 2  (v1 − v2 ) ( R R  R1 

(5.9.11)

Active Filters The class of filters one can obtain by means of op-amp designs is called active filters, because op-amps can provide amplification (gain) in addition to the filtering effects already studied in Section 5.6 for passive circuits (i.e., circuits comprising exclusively resistors, capacitors, and inductors). The easiest way to see how the frequency response of an op-amp can be shaped (almost) arbitrarily is to replace the resistors RF and RS in Figure 5.9.2 and Figure 5.9.4 with impedances ZF and ZS, as shown in Figure 5.9.11. It is a straightforward matter to show that in the case of the inverting amplifier, the expression for the closed loop gain is given by Vout Z ( jω) = − F ZS VS

(5.9.12)

whereas for the noninverting case, the gain is Vout Z ( jω) = 1 + F VS ZS © 2005 by CRC Press LLC

(5.9.13)

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FIGURE 5.9.11 Op-amp circuits employing complex impedances.

FIGURE 5.9.12 Active low-pass filter.

where ZF and ZS can be arbitrarily complex impedance functions and where VS, Vout, IF , and IS are all phasors. Thus, it is possible to shape the frequency response of an ideal op-amp filter simply by selecting suitable ratios of feedback impedance to source impedance. The simplest op-amp low-pass filter is shown in Figure 5.9.12. The closed-loop gain ALP (jω) is then computed to be ALP ( jω ) = −

RF RS ZF =− ZS 1 + jωCF RF

It should be apparent that the response of this op-amp filter is just an amplified version of that of the passive filter. Figure 5.9.13 depicts the amplitude response of the active low-pass filter (in the figure, RF /RS = 10 and 1/RF CF = 1) in two different graphs; the first two plots the amplitude ratio Vout /VS vs. radian frequency, ω, on a logarithmic scale, while the second plots the amplitude ratio 20 log10 (Vout /VS) (in units of dB), also vs. ω on a logarithmic scale. A high-pass active filter can easily be obtained by using the circuit shown in Figure 5.9.14. The following gain function for the op-amp circuit can be derived: AHP ( jω ) = −

ZF RF =− ZS RS + 1 jωCS

jωCS RF =− 1 + jωRS CS

(5.9.14)

The high-pass response is depicted in Figure 5.9.15 in both linear and dB plots (in the figure, RF/RS = 10, 1/RSC = 1). © 2005 by CRC Press LLC

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FIGURE 5.9.13 Normalized response of active low-pass filter.

FIGURE 5.9.14 Active high-pass filter.

FIGURE 5.9.15 Normalized response of active high-pass filter.

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FIGURE 5.9.16 Active band-pass filter.

The band-pass filter of Figure 5.9.16 can be shown to have the following frequency response: ABP ( jω ) = −

jωCS RF ZF =− ZS + j ω C 1 ( F RF ) (1 + jωCS RS )

(5.9.15)

The form of the op-amp response we just obtained should not appear as a surprise. It is very similar (although not identical) to the product of the low-pass and high-pass responses: ALP ( jω ) = −

RF RS 1 + jωCF RF

(5.9.16)

jωCS RF 1 + jωRS CS

(5.9.17)

AHP ( jω ) = −

In particular, the denominator of ABP(jω) is exactly the product of the denominators of ALP(jω) and AHP(jω). It is particularly enlightening to rewrite ALP(jω) in a slightly different form, after making the observation that each RC product corresponds to some “critical” frequency: ω1 =

1 RF CS

ω LP =

1 RF CF

ω HP =

1 RS CS

(5.9.18)

It is easy to verify that for the case where ω HP > ω LP

(5.9.19)

the response of the op-amp filter may be represented as shown in Figure 5.9.17 in both linear and dB plots (in the figure, ω1 = 1, ωHP = 1000, ωHP = 10). The dB plot is very revealing, for its shows that, in effect, the band-pass response is the graphical superposition of the low-pass and high-pass responses shown earlier. The two 3-dB (or cutoff) frequencies are the same as in ALP(jω), 1/RFCF; and in AHP(jω), 1/RSCS. The third frequency, ω1 = 1/RFCS, represents the point where the response of the filter crosses the 0-dB axis (rising slope). Since 0 dB corresponds to a gain of 1, this frequency is called the unity gain frequency. The ideas developed thus far can be employed to construct more complex functions of frequency. In fact, most active filters one encounters in practical applications are based on circuits involving more than one or two energy-storage elements. By constructing suitable functions for ZF and ZS, it is possible to realize filters with greater frequency selectivity (i.e., sharpness of cutoff), as well as flatter band-pass or band-rejection functions (that is, filters that either allow or reject signals in a limited band of frequencies). A few simple applications are investigated in the homework problems. One remark that should be made in passing, though, pertains to the exclusive use of capacitors in the circuits analyzed thus far.

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FIGURE 5.9.17 Normalized response of active band-pass filter.

FIGURE 5.9.18 Op-amp integrator.

Integrator and Differentiator Circuits The Ideal Integrator Consider the circuit of Figure 5.9.18, where vs(t) is an arbitrary function of time (e.g., a pulse train, a triangular wave, or a square wave). The op-amp circuit shown provides an output that is proportional to the integral of vs(t). vout = −

1 RS CF

∫

t

vS (t ′) dt ′

(5.9.20)

−∞

This equation states that the output voltage is the integral of the input voltage. There are numerous applications of the op-amp integrator, most notably the analog computer. The following example illustrates the operation of the op-amp integrator. Example 5.9.3 Charge Amplifiers One of the most common families of transducers for the measurement of force, pressure, and acceleration is that of piezoelectric transducers. These transducers contain a piezoelectric crystal, a crystal that generates an electric charge in response to deformation. Thus, if a force is applied to the crystal (leading to a displacement), a charge is generated within the crystal. If the external force generates a displacement xi , then the transducer will generate a charge q according to the expression © 2005 by CRC Press LLC

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FIGURE 5.9.19 Piezolectric transducer.

FIGURE 5.9.20 Charge amplifier.

q = K p xi Figure 5.9.19 depicts the basic structure of the piezoelectric transducer and a simple circuit model. The model consists of a current source in parallel with a capacitor, where the current source represents the rate of change of the charge generated in response to an external force and the capacitance is a consequence of the structure of the transducer, which consists of a piezoelectric crystal (e.g., quartz or Rochelle salt) sandwiched between conducting electrodes (in effect, this is a parallel-plate capacitor). Although it is possible, in principle, to employ a conventional voltage amplifier to amplify the transducer output voltage, vo, given by vo =

∫

1 1 i dt = C C

∫ dt dt = C = dq

q

K p xi C

it is often advantageous to use a charge amplifier. The charge amplifier is essentially an integrator circuit, as shown in Figure 5.9.20 characterized by an extremely high input impedance. The high impedance is essential; otherwise, the charge generated by the transducer would leak to ground through the input resistance of the amplifier. Because of the high input impedance, the input current into the amplifier is negligible; further, because of the high open-loop gain of the amplifier, the inverting-terminal voltage is essentially at ground potential. Thus, the voltage across the transducer is effectively zero. As a consequence, to satisfy KCL, the feedback current, iF(t) must be equal and opposite to the transducer current, i: i F (t ) = − i and since vout (t ) =

1 i (t ) dt CF F

it follows that the output voltage is proportional to the charge generated by the transducer, and therefore to the displacement: © 2005 by CRC Press LLC

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FIGURE 5.9.21 Op-amp differentiators.

vout (t ) =

1 CF

∫ −i dt = C ∫ − dt dt = − C dq

1

F

q F

=−

K p xi CF

Since the displacement is caused by an external force or pressure, this sensing principle is widely adopted in the measurement of force and pressure. The Ideal Differentiator Using an argument similar to that employed for the integrator, we can derive a result for the ideal differentiator circuit of Figure 5.9.21. The relationship between input and output is obtained by observing that iS (t ) = CS

dvS (t ) dt

(5.9.21)

and i F (t ) =

vout (t ) RF

(5.9.22)

so that the output of the differentiator circuit is proportional to the derivative of the input: vout (t ) = − RF CS

dvS (t ) dt

(5.9.23)

Although mathematically attractive, the differentiation property of this op-amp circuit is seldom used in practice, because differentiation tends to simplify any noise that may be present in a signal.

Physical Limitations of Op-Amps Thus far, the operational amplifier has been treated an ideal device, characterized by infinite input resistance, zero output resistance, and infinite open-loop voltage gain. Although this model is adequate to represent the behavior of the op-amp in a large number of applications, it is important to realize that practical operational amplifiers are not ideal devices, but exhibit a number of limitations that should be considered in the design of instrumentation. In particular, in dealing with relatively large voltages and currents, and in the presence of high-frequency signals, it is important to be aware of the nonideal properties of the opamp. In the present section, we examine the principal limitations of the operational amplifier. Voltage Supply Limits The effect of limiting supply voltages is that amplifiers are capable of amplifying signals only within the range of their supply voltages; VS− < vout < VS+ For most op-amps, the limit is actually approximately 1.5 V less than the supply voltages. © 2005 by CRC Press LLC

(5.9.24)

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FIGURE 5.9.22

Frequency Response Limits Another property of all amplifiers that may pose severe limitations to the op-amp is their finite bandwidth. We have so far assumed, in our ideal op-amp model, that the open-loop gain is a very large constant. In reality, AV(OL) is a function of frequency and is characterized by a low-pass response. For a typical op-amp, AV (OL ) ( jω ) =

A0 1 + jω ω 0

(5.9.25)

The finite bandwidth of the practical op-amp results in a fixed gain-bandwidth product for any given amplifier. The effect of a constant gain-bandwidth product is that as the closed-loop gain of the amplifier is increased, its 3-dB bandwidth is proportionally reduced, until, in the limit, if the amplifier were used in the open-loop mode, its gain would be equal to A0 and its 3-dB bandwidth would be equal to ω0. Thus, the product of gain and bandwidth in any given op-amp is constant. That is, A0 × ω 0 = A1 × ω 1 = A2 × ω 2 = K

(5.9.26)

as is shown in Figure 5.9.22. Input Offset Voltage Another limitation of practical op-amps results because even in the absence of any external inputs, it is possible that an offset voltage will be present at the input of an op-amp. This voltage is usually denoted by ±Vos and it is caused by mismatches in the internal circuitry of the op-amp. The offset voltage appears as a differential input voltage between the inverting and noninverting input terminals. The presence of an additional input voltage will cause a DC bias error in the amplifier output, which can be modeled as shown in Figure 5.9.23.

FIGURE 5.9.23 Op-amp input offset voltage. © 2005 by CRC Press LLC

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FIGURE 5.9.24

FIGURE 5.9.25 Output off-set voltage adjustment.

Input Bias Currents Another nonideal characteristic of op-amps results from the presence of small input bias currents at the inverting and noninverting terminals. Once again, these are due to the internal construction of the input stage of an operational amplifier. Figure 5.9.24 illustrates the presence of nonzero input bias currents (IB) going into an op-amp. Typical values of IB depend on the semiconductor technology employed in the construction of the opamp. Op-amps with bipolar transistor input stages may see input bias currents as large as 1 µA, while for FET input devices, the input bias currents are less than 1 nA. Since these currents depend on the internal design of the op-amp, they are not necessarily equal. One often designates the input offset current Ios as Ios = I B+ − I B−

(5.9.27)

The latter parameter is sometimes more convenient from the standpoint of analysis. Output Offset Adjustment Both the offset voltage and the input offset current contribute to an output offset voltage Vout,os. Some op-amps provide a means for minimizing Vout,os. For example, the µA741 op-amp provides a connection for this procedure. Figure 5.9.25 shows a typical pin configuration for an op-amp in an eight-pin dualin-line package (DIP) and the circuit used for nulling the output offset voltage. The variable resistor is adjusted until vout reaches a minimum (ideally, 0 V). Nulling the output voltage in this manner removes the effect of both input offset voltage and current on the output. Slew Rate Limit Another important restriction in the performance of a practical op-amp is associated with rapid changes in voltage. The op-amp can produce only a finite rate of change at its output. This limit rate is called the slew rate. Consider an ideal step input, where at t = 0 the input voltage is switched from zero to V

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FIGURE 5.9.26 Slew rate limit in op-amps.

FIGURE 5.9.27

volts. Then we would expect the output to switch from 0 to A · V volts, where A is the amplifier gain. However, vout(t) can change at only a finite rate; thus, dvout (t ) = S0 = Slew rate dt max

(5.9.28)

Figure 5.9.26 shows the response of an op-amp to an ideal step change in input voltage, Here S0, the slope of vout(t), represents the slew rate. Short-Circuit Output Current Recall the model for the op-amp, which represented the internal circuits of the op-amp in terms of an equivalent input resistance, Rin, and a controlled voltage source, AV vin. In practice, the internal source is not ideal, because it cannot provide an infinite amount of current (either to the load or to the feedback connection, or both). The immediate consequence of this nonideal op-amp characteristic is that the maximum output current of the amplifier is limited by the so-called short-circuit output current, ISC: Iout < I SC

(5.9.29)

To further explain this point, consider that the op-amp needs to provide current to the feedback path (in order to “zero” the voltage differential at the input) and to whatever load resistance, RL, may be connected to the output. Figure 5.9.27 illustrates this idea for the case of an inverting amplifier, where ISC is the load current that would be provided to a short-circuit load (RL = 0). Common-Mode Rejection Ratio (CMRR) Example 5.9.2 introduced the notion of differential-mode and common-mode signals. If we define Adm as the differential-mode gain and Acm as the common-mode gain of the op-amp, the output of an opamp can then be expressed as follows:  v − v1  vout = Adm (v2 − v1 ) + Acm  2   2 

(5.9.30)

Under ideal conditions, Acm should be exactly zero, since the differential amplifier should completely reject common-mode signals. The departure from this ideal condition is a figure of merit for a differential amplifier and is measured by defining a quantity called the common-mode rejection ratio (CMRR).

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The CMRR is defined as the ratio of the differential-mode gain to the common-mode gain and should ideally be infinite: CMRR =

Adm Acm

The CMRR is often expressed in units of decibels (dB).

5.10 Digital Circuits The objective of this section is to discuss the essential features of digital logic circuits, which are at the heart of digital computers.

Analog and Digital Signals One of the fundamental distinctions in the study of electronic circuits (and in the analysis of any signals derived from physical measurements) is that between analog and digital signals. As discussed in the preceding chapter, an analog signal is an electrical signal whose value varies in analogy with a physical quantity (e.g., temperature, force, or acceleration). For example, a voltage proportional to a measured variable pressure or to a vibration naturally varies in an analog fashion. Figure 5.10.1 depicts an arbitrary analog function of time, f(t). We note immediately that for each value of time, t, f(t) can take one value among any of the values in a given range. For example, in the case of the output voltage of an op-amp, we expect the signal to take any value between +Vsat and –Vsat, where Vsat is the supply-imposed saturation voltage. A digital signal, on the other hand, can take only a finite number of values. This is an extremely important distinction, as will be shown shortly. An example of a digital signal is a signal that allows display of a temperature measurement on a digital readout. Let us hypothesize that the digital readout is three digits long and can display numbers from 0 to 100, and let us assume that the temperature sensor is correctly calibrated to measure temperatures from 0 to 100°F. Further, the output of the sensor ranges from 0 to 5 V, where 0 V corresponds to 0°F and 5 V to 100°F. Therefore, the calibration constant of the sensor is kT = (100° – 0°)/(5 – 0) = 20°/V. Clearly, the output of the sensor is an analog signal; however, the display can show only a finite number of readouts (101, to be precise). Because the display itself can only take a value out of a discrete set of states — the integers from 0 to 100 — we call it a digital display, indicating that the variable displayed is expressed in digital form. Now, each temperature on the display corresponds to a range of voltages: each digit on the display represents one hundredth of the 5-V range of the sensor, or 0.05 V = 50 m V. Thus, the display will read 0 if the sensor voltage is between 0 and 49 mV, 1 if it is between 50 and 99 mV, and so on. Figure 5.10.2 depicts the staircase function relationship between the analog voltage and the digital readout. This quantization of the sensor output voltage is, in effect, an approximation. If one wished to know the temperature with greater precision, a greater number of display digits could be employed.

FIGURE 5.10.1 Analog signal.

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FIGURE 5.10.2 Digital representation of an analog signal.

FIGURE 5.10.3 A binary signal.

The most common digital signals are binary signals. A binary signal is a signal that can take only one of two discrete values and is therefore characterized by transitions between two states. Figure 5.10.3 displays a typical binary signal. In binary arithmetic (which we discuss in the next section), the two discrete values f1 and f0 are represented by the numbers 1 and 0. In binary voltage waveforms, these values are represented by two voltage levels. For example, in the TTL convention, these values are (nominally) 5 and 0 V, respectively; in CMOS circuits, these values can vary substantially. Other conventions are also used, including reversing the assignment — for example, by letting a 0-V level represent a logic 1 and a 5-V level represent a logic 0. Note that in a binary waveform, knowledge of the transition between one stage and another (e.g., from f0 to f1 at t = t2) is equivalent to knowledge of the state. Thus, digital logic circuits can operate by detecting transitions between voltage levels. The transitions are often called edges and can be positive (f0 to f1) or negative (f1 to f0). Virtually all of the signals handled by a computer are binary. From here on, whenever we speak of digital signals, you may assume that the text is referring to signals of the binary type, unless otherwise indicated.

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TABLE 5.10.1 Conversion from Decimal to Binary Decimal Number, n10

Binary Number, n2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000

The Binary Number System The binary number system is a natural choice for representing the behavior of circuits that operate in one of two states (on or off, 1 or 0, or the like). The diode and transistor gates and switches studied in Section 5.7 fall in this category. Table 5.10.1 shows the correspondence between decimal and binary number systems for decimal numbers up to 16. Table 5.10.1 shows that it takes four binary digits, also called bits, to represent the decimal numbers up to 15. Usually, the rightmost bit is called the least significant bit, or LSB, and the leftmost bit is called the most significant bit, or MSB. Since binary numbers clearly require a larger number of digits than decimal numbers, the digits are usually grouped in sets of four, eight, or sixteen. Four bits are usually termed a nibble, eight bits are called a byte, and sixteen bits (or two bytes) form a word. Addition and Subtraction The operations of addition and subtraction are based on the simple rules shown in Table 5.10.2. Figure 5.10.4 provides three examples. The procedure for subtracting binary numbers is based on the rules of Table 5.10.3. A few examples of binary subtraction are given in Figure 5.10.5, with their decimal counterparts. TABLE 5.10.2 Rules for Addition 0+0=0 0+1=1 1+0=1 1 + 1 = 0 (with a carry of 1)

FIGURE 5.10.4 Examples of binary addition.

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TABLE 5.10.3 Rules for Subtraction 0–0=0 1–0=1 1–1=0 0 – 1 = 1 (with a borrow of 1)

FIGURE 5.10.5 Examples of binary subtraction.

Multiplication and Division Whereas in the decimal system the multiplication table consists of 102 = 100 entries, in the binary system we only have 22 = 4 entries. Table 5.10.4 represents the complete multiplication table for the binary number system. TABLE 5.10.4 Rules for Multiplication 0×0=0 0×1=0 1×0=0 1×1=1

Division in the binary system is also based on rules analogous to those of the decimal system, with the two basic laws given in Table 5.10.5. Once again, we need be concerned with only two cases, and just as in the decimal system, division by zero is not contemplated. TABLE 5.10.5 Rules for Division 0÷1=0 1÷1=1

Conversion from Decimal to Binary The conversion of a decimal number to its binary equivalent is performed by successive division of the decimal number by 2, checking for the remainder each time. Figure 5.10.6 illustrates this idea with an example.

FIGURE 5.10.6 Example of conversion from decimal to binary.

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FIGURE 5.10.7 Conversion from decimal to binary.

The same technique can be used for converting decimal fractional numbers to their binary form, provided that the whole number is separated from the fractional part and each is converted to binary form (separately), with the results added at the end. Figure 5.10.7 outlines this procedure by converting the number 37.53 to binary form. Complements and Negative Numbers To simplify the operation of subtraction in digital computers, complements are used almost exclusively. In practice, this corresponds to replacing the operation X – Y with operation X + (– Y). This procedure results in considerable simplification, since the computer hardware need include only adding circuitry. Two types of complements are used with binary numbers: the one’s complement and the two’s complement. The one’s complement of an n-bit binary number is obtained by subtracting the number itself from (2n – 1). Two examples are as follows: a = 0101

(

)

One' s complement of a = 2 4 − 1 − a = (1111) − (0101) = 1010 b = 101101

(

)

One' s complement of b = 2 6 − 1 − b = (111111) − (101101) = 010010 The two’s complement of n-bit binary number is obtained by subtracting the number itself from 2″. Two’s complements of the same numbers a and b used in the preceding illustration are computed as follows: © 2005 by CRC Press LLC

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(a)

(b)

(c)

FIGURE 5.10.8 (a) Eight-bit sign-magnitude binary number; (b) eight-bit one’s complement binary number; (c) eight-bit two’s complement binary number.

a = 0101 Two' s complement of a = 2 4 − a = (10000) − (0101) = 1011 b = 101101 Two' s complement of b = 2 6 − b = (1000000) − (101101) = 010011 Different conventions exist in the binary system to represent whether a number is negative or positive. These are summarized in Figure 5.10.8. The Hexadecimal System It should be apparent by now that representing numbers in base 2 and base 10 systems is purely a matter of convenience, given a specific application. Another base frequently used is the hexadecimal system, a direct derivation of the binary number system. In the hexadecimal (or hex) code, the bits in a binary number are subdivided into groups of four. Since there are 16 possible combinations for a four-bit number, the natural digits in the decimal system (0 through 9) are insufficient to represent a hex digit. To solve this problem, the first six letters of the alphabet are used, as shown in Table 5.10.6. Thus, in hex code, an eight-bit word corresponds to just two digits; for example: 1010 01112 = A716 0010 10012 = 2916 Binary Codes In this subsection, we describe two common binary codes that are often used for practical reasons. The first is a method of representing decimal numbers in digital logic circuits that is referred to as binarycoded decimal, or BCD, representation. In effect, the simplest BCD representation is just a sequence of © 2005 by CRC Press LLC

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TABLE 5.10.6 Hexadecimal Code 0 1 2 3 4 5 6 7 8 9 A B C D E F

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

four-bit binary numbers that stops after the first ten entries, as shown in Table 5.10.7. There are also other BCD codes, all reflecting the same principle: that each decimal digit is represented by a fixed-length binary word. One should realize that although this method is attractive because of its direct correspondence with the decimal system, it is not efficient. Consider, for example, the decimal number 68. Its binary representation by direct conversion is the seven-bit number 1000100. On the other hand, the corresponding BCD representation would require eight bits: 6810 = 01101000 BCD TABLE 5.10.7 BCD Code 0 1 2 3 4 5 6 7 8 9

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

Another code that finds many applications is the Gray code. This is simply a reshuffling of the binary code with the property that any two consecutive numbers differ only by one bit. Table 5.10.8 illustrates the three-bit Gray code. The Gray code can be very useful in practical applications, because in counting up or down according to this code, the binary representation of a number changes only one bit at a time.

Boolean Algebra The mathematics associated with the binary number system (and with the more general field of logic) is called Boolean algebra. The variables in a Boolean, or logic, expression can take only one of two values, usually represented by the numbers 0 and 1. These variables are sometimes referred to as true (1) and false (0). This convention is normally referred to as positive logic. There is also a negative logic convention in which the roles of logic 1 and logic 0 are reversed. In this book we shall employ only positive logic. Analysis of logic functions, that is, functions of logical (Boolean) variables, can be carried out in terms of truth tables. A truth table is a listing of all the possible values each of the Boolean variables can take, © 2005 by CRC Press LLC

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TABLE 5.10.8 Three-Bit Gray Code Binary

Gray

000 001 010 011 100 101 110 111

000 001 011 010 110 111 101 100

and of the corresponding value of the desired function. In the following paragraphs we shall define the basic logic functions upon which Boolean algebra is founded, and we shall describe each in terms of a set of rules and a truth table; in addition, we shall also introduce logic gates. Logic gates are physical devices that can be used to implement logic functions. Elementary logic gates were introduced in Section 5.7. AND and OR Gates The basis of Boolean algebra lies in the operations of logical addition, or the OR operation; and logical multiplication, or the AND operation. Both of these find a correspondence in simple logic gates, as we shall presently illustrate. Logical addition, although represented by the symbol +, differs from conventional algebraic addition, as shown in the last rule listed in Table 5.10.9. Note that this rule also differs from the last rule of binary addition studied in the previous section. Logical addition can be represented by the logic gate called an OR gate, whose symbol and whose inputs and outputs are shown in Figure 5.10.9. The OR gate represents the following logical statement: If either X or Y is true (1), then Z is true (1)

(5.10.1)

This rule is embodied in the electronic gates discussed in Chapter 9, in which a logic 1 corresponds, say, to a 5-V signal and a logic 0 to a 0-V signal. Logical multiplication is denoted by the center dot (·) and is defined by the rules of Table 5.10.10. Figure 5.10.10 depicts the AND gate, which corresponds to this operation. The AND gate corresponds to the following logical statement: If both X and Y are true (1), then Z is true (1)

(5.10.2)

TABLE 5.10.9 Rules for Logical Addition (OR) 0+0=0 0+1=1 1+0=1 1+1=1

FIGURE 5.10.9 Logical addition and the OR gate.

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TABLE 5.10.10 Rules for Logical Multiplication (AND) 0·0=0 0·1=0 1·0=0 1·1=1

FIGURE 5.10.10 Logical multiplication and the AND gate.

One can easily envision logic gates (AND and OR) with an arbitrary number of inputs; three- and fourinput gates are not uncommon. The rules that define a logic function are often represented in tabular form by means of a truth table. Truth tables for the AND and OR gates are shown in Figure 5.10.9 and Figure 5.10.10. A truth table is nothing more than a tabular summary of all of the possible outputs of a logic gate, given all the possible input values. If the number of inputs is 3, the number of possible combinations grows from 4 to 8, but the basic idea is unchanged. Truth tables are very useful in defining logic functions. A typical logic design problem might specify requirements such as “the output Z shall be logic 1 only when the condition (X = 1 AND Y = 1) OR (W = 1) occurs, and shall be logic 0 otherwise.” The truth table for this particular logic function is shown in Figure 5.10.11 as an illustration. The design consists, then, of determining the combination of logic gates that exactly implements the required logic function. Truth tables can greatly simplify this procedure. The AND and OR gates form the basis of all logic design in conjunction with the NOT gate. The NOT gate is essentially an inverter, and it provides the complement of the logic variable connected to its input. The complement of a logic variable X is denoted by X. The NOT gate has only one input, as shown in Figure 5.10.12

FIGURE 5.10.11 Example of logic function implementation with logic gates.

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FIGURE 5.10.12 Complements and the NOT gate.

FIGURE 5.10.13 Solution of a logic problem using logic gates.

To illustrate the use of the NOT gate, or inverter, we return to the design example of Figure 5.10.11, where we required that the output of a logic circuit be Z = 1 only if X = 0 AND Y = 1 OR if W = 1. We recognize that except for the requirement X = 0, this problem would be identical if we stated it as follows: “The output Z shall be logic 1 only when the condition ( X = 1 AND Y = 1) OR (W = 1) occurs, and shall be logic 0 otherwise.” If we use an inverter to convert X to X, we see that the required condition becomes X = 1 AND Y = 1) OR (W = 1). The formal solution to this elementary design exercise is illustrated in Figure 5.10.13. In the course of the discussion of logic gates, extensive use will be made of truth tables to evaluate logic expressions. A set of basic rules will facilitate this task. Table 5.10.11 lists some of the rules of Boolean algebra: TABLE 5.10.11 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

© 2005 by CRC Press LLC

Rules of Boolean Algebra

0+X=X 1+X=1 X+X=X X+ X =1 0·X=0 1·X=X X ·X=X X· X =0 X =X X+Y=Y+X X · Y = Y· X X + (X + Z) = (X + Y) + Z X · (Y · Z) = (X · Y) · Z X · (Y + Z) = X · Y + X · Z X+X·Z=X X · (X + Y) = X (X + Y) · (X + Z) = X + Y · Z X+ X ·Y=X+Y X·Y+Y·Z+ X ·Z=X·Y+ X ·Z

  Commutative law    Associative law  Distributive law Absorption law

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FIGURE 5.10.14 De Morgan’s laws.

To complete the introductory material on Boolean algebra, a few paragraphs need to be devoted to two very important theorems, called De Morgan’s theorems. These are stated here in the form of logic functions:

(X + Y) = X ⋅ Y

(5.10.3)

(X ⋅ Y) = X + Y

(5.10.4)

These two laws state a very important property of logic functions: any logic function can be implemented using only OR and NOT gates, or using only AND and NOT gates. De Morgan’s laws can easily be visualized in term of logic gates, as shown in Figure 5.10.14. The associated truth tables are proof of these theorems. The importance of De Morgan’s laws is in the statement of the duality that exists between AND and OR operations: any function can be realized by just one of the two basic operations, plus the complement operation. This gives rise to two families of logic functions: sums of products and products of sums, as shown in Figure 5.10.15. Any logical expression can be reduced to either one of these two forms. Although the two forms are equivalent, it may well be true that one of the two has a simpler implementation (fewer gates). Example 5.10.1 illustrates this point.

FIGURE 5.10.15 Sun-of-products and product-of-sums logic functions.

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FIGURE 5.10.16 Figure for Example 5.10.1.

Example 5.10.1 Use De Morgan’s theorem to realize the function y = A + (B · C) as a product-of-sums expression, and implement it using AND, OR, and NOT gates. Solution. Knowing that y = y, we can apply the first of De Morgan’s laws to the complement of the function y to obtain the expression

(

)

(

y = A + ( B ⋅ C) = A ⋅ B ⋅ C = A ⋅ B + C

)

Thus,

(

y= y= A⋅ B +C

)

Using logic gates, we can then implement the function as shown in Figure 5.10.16 NAND and NOR Gates In addition to the AND and OR gates we have just analyzed, the complementary forms of these gates, called NAND and NOR, are very commonly used in practice. In fact, NAND and NOR gates form the basis of most practical logic circuits. Figure 5.10.17 depicts these two gates and illustrates how they can be easily interpreted in terms of AND, OR, and NOT gates by virtue of De Morgan’s laws. You can readily verify that the logic function implemented by the NAND and NOR gates corresponds, respectively, to AND and OR gates followed by an inverter. It is very important to note that, by De Morgan’s laws, the NAND gate performs a logical addition on the complements of the inputs. while the NOR gate performs a logical multiplication on the complements of the inputs. Functionally, then, any logic function could be implemented with either NOR or NAND gates only. In the next section we shall learn how to systematically approach the design of logic functions. First, we provide a few examples to illustrate logic design with NAND and NOR gates.

FIGURE 5.10.17 Equivalence of NAND and NOR gates with AND and OR gates.

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FIGURE 5.10.18 Figure for Example 5.10.2.

FIGURE 5.10.19 XOR gate.

FIGURE 5.10.20 Realization of an XOR gate.

Example 5.10.2 Realize the following function using only NAND and NOR gates:

(

)

y = A⋅ B + C Solution. Since the term in parentheses appears as a complement product, it can be obtained by means of a NAND gate. Further, once the function ( A ⋅ B) has been realized, we can see that y is the complemented sum of two terms — that is, it can be obtained directly with a NOR gate. The resulting logic circuit is shown in Figure 5.10.18. Can you find another solution to this problem that employs only two gates? The XOR (Exclusive OR) Gate It is rather common practice for a manufacturer of integrated circuits to provide common combinations of logic circuits in a single integrated circuit package. An example of this idea is provided by the exclusive OR (XOR) gate, which provides a logic function similar, but not identical, to the OR gate we have already studied. The XOR gate acts as an OR gate, except when its inputs are all logic 1s; in this case, the output is a logic 0 (thus the term exclusive). Figure 5.10.19 shows the logic circuit symbol adopted for this gate, and the corresponding truth table. The logic function implemented by the XOR gate is the following: “either X or Y, but not both.” This description can be extended to an arbitrary number of inputs. The symbol adopted for the exclusive OR operation is ⊕, and so we shall write Z=X⊕Y to denote this logic operation. The XOR gate can be obtained by a combination of the basic gates we are already familiar with. For example, if we observe that the XOR function corresponds to Z = X ⊕ Y = (X + Y) · X · ( X + Y ), we can realize the XOR gate by means of the circuit shown in Figure 5.10.20.

Karnaugh Maps and Logic Design In examining the design of logic functions by means of logic gates, we have discovered that more than one solution is usually available for the implementation of a given logic expression. It should also be clear by now that some combinations of gates can implement a given function more efficiently than © 2005 by CRC Press LLC

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FIGURE 5.10.21 Two-, three-, and four-variable Karnaugh maps.

others. How can we be assured of having chosen the most efficient realization? Fortunately, there is a procedure that utilizes a map describing all possible combinations of the variables present in the logic function of interest. This map is called a Karnaugh map, after its inventor. Figure 5.10.21 depicts the appearance of Karnaugh maps for two-, three-, and four-variable expressions in two different forms. As can be seen, the row and column assignments for two or more variables are arranged so that all adjacent terms change by only one bit. For example, in the three- or four-variable map, the columns next to column 01 are columns 00 and 10. Also note that each map consists of 2N cells, where N is the number of logic variables. Each cell in a Karnaugh map contains a minterm, that is, a product of the N variables that appear in our logic expression (in either uncomplemented or complemented form). For example, for the case of three variables (N = 3), there are 23 = 8 such combination, or minterms: X ⋅ Y ⋅ Z , X ⋅ Y ⋅ Z , X ⋅ Y ⋅ Z , X · Y · Z, X · Y ⋅ Z , X · Y · Z, X · Y · Z, and X · Y · Z. The content of each cell — that is, the minterm — is the product of the variables appearing at the corresponding vertical and horizontal coordinates. For example, in the three-variable map, X · Y · Z appears at the intersection of X · Y and Z. The map is filled by placing a value of 1 for any combination of variables for which the desired output is a 1. For example, consider the function of three variables for which we desire to have an output of 1 whenever the variables X, Y, and Z have the following values:

X=0 X=0 X =1 X =1 © 2005 by CRC Press LLC

Y Y Y Y

=1 =1 =1 =1

Z=0 Z =1 Z=0 Z =1

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FIGURE 5.10.22 Truth table and Karnaugh map representations of a logic function.

FIGURE 5.10.23 Karnaugh map for a four-variable expression.

The same truth table is shown in Figure 5.10.22 together with the corresponding Karnaugh map. The Karnaugh map provides an immediate view of the values of the function in graphical form. Further, the arrangement of the cells in the Karnaugh map is such that any two adjacent cells contain minterms that vary in only one variable. This property, as will be verified shortly, is quite useful in the design of logic functions by means of logic gates, especially if we consider the map to be continuously wrapping around itself, as if the top and bottom, and right and left edges were touching each other. For the three-variable map given in Figure 5.10.21, for example, the cell X ⋅ Y ⋅ Z is adjacent to X · Y · Z if we “roll” the map so that the right edge touches the left. Note that these two cells differ only in the variable X, a property we earlier claimed adjacent cells have. Shown in Figure 5.10.23 is a more complex, four-variable logic function which will serve as an example in explaining how Karnaugh maps can be used directly to implement a logic function. First, we define a subcube as a set of 2m adjacent cells, for m = 1, 2, 3, …, N. Thus, a subcube can consist of 1, 2, 4, 8, 16, 32, … cells. All possible subcubes for the four-variable map of Figure 5.10.23 are shown in Figure 5.10.24. Note that there are no four-cell subcubes in this particular case. Note also that there is some overlap between subcubes. Sum-of-Products Realizations Although not explicitly stated, the logic functions of the preceding section were all in sum-of-products form. As you know, it is also possible to realize logic functions in product-of-sums form. This section © 2005 by CRC Press LLC

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FIGURE 5.10.24 One- and two-cell subcubes for the Karnaugh map of Figure 5.10.23.

discusses the implementation of logic functions in sum-of-products form and gives a set of design rules. The next section will do the same for product-of-sums form logical expressions. The following rules are a useful aid in determining the minimal sum-of-products expression: 1. 2. 3. 4.

Begin with isolated cells. These must be used as they are, since no simplification is possible. Find all cells that are adjacent to only one other cell, forming two-cell subcubes. Find cells that form four-cell subcubes, eight-cell subcubes, and so forth. The minimal expression is formed by the collection of the smallest number of maximal subcubes.

The following examples illustrate the application of these principles to a variety of problems. Product-of-Sums Realizations Thus far, we have exclusively worked with sum-of-products expressions, that is, logic functions of the form A · B + C · D. We know, however, that De Morgan’s laws state that there is an equivalent form that appears as a product of sums, for example, (W + Y) · (Y + Z). The two forms are completely equivalent, logically, but one of the two forms may lead to a realization involving a smaller number of gates. When using Karnaugh maps, we may obtain the product-of-sums form very simply by following these rules: 1. Solve for the 0s exactly as for the 1s in sum-of-products expressions. 2. Complement the resulting expression. The same principles stated earlier apply in covering the map with subcubes and determining the minimal expression. The following examples illustrate how one form may result in a more efficient solution than the other. Example 5.10.3 This example illustrates the design of a logic function using both sum-of-products and product-of-sums implementations, thus showing that it may be possible to realize some savings by using one implementation rather than the other. 1. Realize the function f by a Karnaugh map using 0s. 2. Realize the function f by a Karnaugh map using 1s.

© 2005 by CRC Press LLC

x

y

z

f

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

0 1 1 1 1 1 0 0

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FIGURE 5.10.25 Figure for Example 5.10.3.

FIGURE 5.10.26 Figure for Example 5.10.3.

Solution. 1. Using 0s, we obtain the Karnaugh map of Figure 5.10.25, leading to the product-of-sums expression f = ( x + y + z) ⋅ ( x + y ) which requires five gates. 2. If 1s are used, as shown in Figure 5.10.26, a sum-of-products expression is obtained, of the form f = x ⋅y+ x⋅y + y⋅z which requires seven gates. Example 5.10.4 Safety Circuit for Operation of a Stamping Press In this example, the techniques illustrated in the preceding example will be applied to a practical situation. To operate a stamping press, an operator must press two buttons (b1 and b2) 1 m apart from each other and away from the press (this ensures that the operator’s hands cannot be caught in the press). When the buttons are pressed, the logical variables b1 and b2 are equal to 1. Thus, we can define a new variable A = b1 · b2; when A = 1, the operator’s hands are safely away from the press. In addition to the safety requirement, however, other conditions must be satisfied before the operator can activate the press. The press is designed to operate on one of two workpieces, part I and part II, but not both. Thus, acceptable logic states for the press to be operated are “part I is in the press, but not part II” and “part II is in the press, but not part I.” If we denote the presence of part I in the press by the logical variable B = 1 and the presence of part II by the logical variable C = 1, we can then impose additional requirements on the operation of the press. For example, a robot used to place either part in the press could activate a pair of switches (corresponding to logical variables B and C) indicating which part, if any, is in the press. Finally, in order for the press to be operable, it must be “ready”, meaning that it has to have completed any previous stamping operation. Let the logical variable D = 1 represent the ready condition. We have now represented the operation of the press in terms of four logical variables, summarized in the truth table of Table 5.10.12. Note that only two combinations of the logical variables will result in operation of the press: ABCD = 1011 and ABCD = 1101. You should verify that these two conditions correspond © 2005 by CRC Press LLC

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FIGURE 5.10.27 Figure for Example 5.10.4.

TABLE 5.10.12

Conditions for Operation of Stamping Press

(A) b1 · b2

(B) Part I is in Press

(C) Part II is in Press

(D) Press is Operable

Press Operation 1 = Pressing; 0 = Not Pressing

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0

Note: ↑ Both buttons (b1, b2) must be pressed for this to be a 1.

to the desired operation of the press. Using a Karnaugh map, realize the logic circuitry required to implement the truth table shown. Solution. Table 5.10.12 can be converted to a Karnaugh map, as shown in Figure 5.10.27. Since there are many more 0s than 1s in the table, the use of 0s in covering the map will lead to greater simplification. This will result in a product-of-sums expression. The four subcubes shown in Figure 5.10.27 yield the equation

(

A ⋅ D ⋅ (C + B) ⋅ C + B

)

By De Morgan’s law, this equation is equivalent to A ⋅ D ⋅ (C + B) ⋅ (C ⋅ B) which can be realized by the circuit of Figure 5.10.28. For the purpose of comparison, the corresponding sum-of-products circuit is shown in Figure 5.10.29. Note that this circuit employs a greater number of gates and will therefore lead to a more expensive design. © 2005 by CRC Press LLC

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FIGURE 5.10.28 Figure for Example 5.10.4.

FIGURE 5.10.29 Figure for Example 5.10.4.

Don’t Care Conditions Another simplification technique may be employed whenever the value of the logic function to be implemented can be either a 1 or a 0. This condition may result from the specification of the problem and is not uncommon. Whenever it does not matter whether a position in the map is filled by a 1 or a 0, we use a so-called don’t care entry, denoted by an x. Then the don’t care can be used as either a 1 or a 0, depending on which results in a greater simplification (i.e., helps in forming the smallest number of maximal subcubes).

Combinational Logic Modules The basic logic gates described in the previous section are used to implement more advanced functions and are often combined to form logic modules, which, thanks to modern technology, are available in compact integrated circuit (IC) packages. In this section and the next, we discuss a few of the more common combinational logic modules, illustrating how these can be used to implement advanced logic function. Multiplexers Multiplexers, or data selectors, are combinational logic circuits that permit the selection of one of many inputs. A typical multiplexer (MUX) has 2n data lines, n address lines, and one output. In addition, other control inputs (e.g., enables) may exist. Standard, commercially available MUXs allow for n up to 4; however, two or more MUXs can be combined if a greater range is needed. The MUX allows for one of 2n inputs to be selected as the data output; the selection of which input is to appear at the output is made by way of the address lines. Figure 5.10.30 depicts the block diagram of a four-input MUX. The input data lines are labeled D0, D1, D2, and D3; the data select, or address, lines are labeled I0 and I1; and the output is available in both complemented and uncomplemented form, and is thus labeled F, or F. Finally, an enable input, labeled E, is also provided, as a means of enabling or disabling the MUX: if E = 1, the MUX is disabled; if E = 0, it is enabled. The negative logic (MUX off when E = 1 and on when E = 0) is represented by the small “bubble” at the enable input, which represents a complement operation (just as at the output of NAND and NOR gates). The enable input is useful whenever one is interested in a cascade of MUXs; this would be of interest if we needed to select a line from a large number, say 28 = 256. Then two 4-input MUXs could be used to provide the data selection of 1 of 8. © 2005 by CRC Press LLC

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FIGURE 5.10.30 4:1 MUX.

FIGURE 5.10.31 Internal structure of the 4:1 MUX.

The material described in the previous sections is quite adequate to describe the internal workings of a multiplexer. Figure 5.10.31 shows the internal construction of 4:1 MUX using exclusively NAND gates (inverters are also used, but the reader will recall that a NAND gate can act as an inverter if properly connected). In the design of digital systems (for example, microcomputers), a single line is often required to carry two or more different digital signals. However, only one signal at a time can be placed on the line. A MUX will allow us to select, at different instants, the signal we wish to place on this single line. This property is shown here for a 4:1 MUX. Figure 5.10.32 depicts the functional diagram of a 4:1 MUX, showing four data lines, D0 through D3, and two select lines, I0 and I1. The data selector function of a MUX is best understood in terms of Table 5.10.13. In this truth table, the x’s represent don’t care entries. As can be seen from the truth table, the output selects one of the data lines depending on the values of I1 and I0, assuming that I0 is the least significant bit. As an example, I1I0 = 10 selects D2, which means that the output, F, will select the value of the data line D2. Therefore F = 1 if D2 = 1 and F = 0 if D2 = 0. Read-Only Memory (ROM) Another common technique for implementing logic functions uses a read-only memory, or ROM. As the name implies, a ROM is a logic circuit that holds in storage (“memory”) information — in the form of binary numbers — that cannot be altered but can be “read” by a logic circuit. A ROM is an array of © 2005 by CRC Press LLC

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FIGURE 5.10.32 Functional diagram of four-input MUX. TABLE 5.10.13 I1

I0

D3

D2

D1

D0

F

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

x x x x x x 0 1

x x x x 0 1 x x

x x 0 1 x x x x

0 1 x x x x x x

0 1 0 1 0 1 0 1

memory cells, each of which can store either a 1 or a 0. The array consists of 2m × n cells, where n is the number of bits in each word stored in ROM. To access the information stored in ROM, m address lines are required. When an address is selected, in a fashion similar to the operation of the MUX, the binary word corresponding to the address selected appears at the output, which consists of n bits, that is, the same number of bits as the stored words. In some sense, a ROM can be thought of as a MUX that has an output consisting of a word instead of a single bit. Figure 5.10.33 depicts the conceptual arrangement of a ROM with n = 4 and m = 2. The ROM table has been filled with arbitrary 4-bit words, just for the purpose of illustration. In Figure 5.10.33, if one were to select an enable input of 0 (i.e., on) and values for the address lines of I0 = 0 and I1 = 1, the output word would be W2 = 0110, so that b0 = 0, b1 = 1, b2 = 1, b3 = 0. Depending on the content of the ROM and the number of address and output lines, one could implement an arbitrary logic function. Unfortunately, the data stored in read-only memories must be entered during fabrication and cannot be altered later. A much more convenient type of read-only memory is the erasable programmable readonly memory (EPROM), the content of which can be easily programmed and stored and may be changed if needed. EPROMs find use in many practical applications because of their flexibility in content and

FIGURE 5.10.33 Read-only memory. © 2005 by CRC Press LLC

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ease of programming. The following example illustrates the use of an EPROM to perform the linearization of a nonlinear function. Example 5.10.5 EPROM-Based Lookup Table One of the most common applications of EPROMs is the arithmetic lookup table. A lookup table is similar in concept to the familiar multiplication table and is used to store precomputed values of certain functions, eliminating the need for actually computing the function. A practical application of this concept is present in every automobile manufactured in the U.S. since the early 1980s, as part of the exhaust emission control system. In order for the catalytic converter to minimize the emissions of exhaust gases (especially hydrocarbons, oxides of nitrogen, and carbon monoxide), it is necessary to maintain the air-to-fuel ratio (A/F) as close as possible to the stoichiometric value, that is, 14.7 parts of air for each part of fuel. Most modern-day engines are equipped with fuel injection systems that are capable of delivering accurate amounts of fuel to each individual cylinder; thus, the task of maintaining an accurate A/F amounts to measuring the mass of air that is aspirated into each cylinder and computing the corresponding mass of fuel. Many automobiles are equipped with a mass airflow sensor, capable of measuring the mass of air drawn into each cylinder during each engine cycle. Let the output of the mass airflow sensor be denoted by the variable MA, and let this variable represent the mass of air (in g) actually entering a cylinder during a particular stroke. It is then desired to compute the mass of fuel, MF (also expressed in g), required to achieve an A/F of 14.7. This computation is simply MF =

MA 14.7

Although the above computation is a simple division, its actual calculation in a low-cost digital computer (such as would be used on an automobile) is rather complicated. It would be much simpler to tabulate a number of values of MA, to precompute the variable MF, and then to store the result of this computation into an EPROM. If the EPROM address were made to correspond to the tabulated values of air mass and the content at each address to the corresponding fuel mass (according to the precomputed values of the expression MF = MA/14.7), it would not be necessary to perform the division by 14.7. For each measurement of air mass into one cylinder, an EPROM address is specified and the corresponding content is read. The content at the specific address is the mass of fuel required by the particular cylinder. In practice, the fuel mass needs to be converted into a time interval corresponding to the duration of time during which the fuel injector is open. This final conversion factor can also be accounted for in the table. Suppose, for example, that the fuel injector is capable of injecting KF g of fuel per second; then the time duration, TF , during which the injector should be open in order to inject MF g of fuel into the cylinder is given by TF =

MF s KF

Therefore, the complete expression to be precomputed and stored in the EPROM is TF =

MA s 14.7 × K F

Figure 5.10.34 illustrates this process graphically. To provide a numerical illustration, consider a hypothetical engine capable of aspirating air in the range 0 < MA < 0.51 g and equipped with fuel injectors capable of injecting at the rate of 1.36 g/sec. Thus, the relationship between TF and MA is TF = 50 × M A msec = 0.05 M A sec © 2005 by CRC Press LLC

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FIGURE 5.10.34 Use of EPROM lookup table in automotive fuel injection system.

FIGURE 5.10.35 Lookup table for automotive fuel injection application.

If the digital value of MA is expressed in dg (decigrams, or tenths of g), the lookup table of Figure 5.10.35 can be implemented, illustrating the conversion capabilities provided by the EPROM. Note that in order to represent the quantities of interest in an appropriate binary format compatible with the 8bit EPROM, the units of air mass and of time have been scaled. Decorders and Read and Write Memory Decoders, which are commonly used for applications such as address decoding or memory expansion, are combinational logic circuits as well. Our reason for introducing decoders is to show some of the internal organization of semiconductor memory devices. Figure 5.10.36 shows the truth table for a 2:4 decoder. The decoder has an enable input, G , and select inputs, B and A. It also has four outputs, Y0 through Y3. When the enable input is logic 1, all decoder outputs are forced to logic 1 regardless of the select inputs.

FIGURE 5.10.36 2:4 decoder. © 2005 by CRC Press LLC

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FIGURE 5.10.37 Internal organization of SRAM.

This simple description of decoders permits a brief discussion of the internal organization of an SRAM (static random-access or read and write memory). SRAM is internally organized to provide memory with high speed (i.e., short access time), a large bit capacity, and low cost. The memory array in this memory device has a column length equal to the number of words, W, and a row length equal to the number of bits per word, N. To select a word, an n-to-W decoder is needed. Since the address inputs to the decoder select only one of the decoder’s outputs, the decoder selects one word in the memory array. Figure 5.10.37 shows the internal organization of a typical SRAM. Thus, to choose the desired word from the memory array, the proper address inputs are required. As an example, if the number of words in the memory array is 8, a 3:8 decoder is needed. Data sheets for 2:4 and 3:8 decoders from a CMOS family data book are provided at the end of the chapter.

Sequential Logic Modules Combinational logic circuits provide outputs that are based on a combination of present inputs only. On the other hand, sequential logic circuits depend on present and past input values. Because of this “memory” property, sequential circuits can store information; this capability opens a whole new area of application for digital logic circuits. Latches and Flip-Flops The basic information-storage device in a digital circuit is called a flip-flop. There are many different varieties of flip-flops; however, all flip-flops share the following characteristics: 1. A flip-flop is a bistable device; that is, it can remain in one of two stable states (0 and 1) until appropriate conditions cause it to change state. Thus, a flip-flop can serve as a memory element. 2. A flip-flop has two outputs, one of which is the complement of the other. RS Flip-Flop. It is customary to depict flip-flops by their block diagram and a name — such as Q or X — representing the output variable. Figure 5.10.38 represents the so-called RS flip-flop, which has two inputs, denoted by S and R, and two outputs, Q and Q. The value of Q is called the state of the flipflop. If Q =1, we refer to the device as being in the 1 state. Thus, we need define only one of the two outputs of the flip-flop. The two inputs, R and S, are used to change the state of the flip-flop, according to the following rules: 1. 2. 3. 4.

When R = S = 0, the flip-flop remains in its present state (whether 1 or 0). When S = 1 and R = 0, the flip-flop is set to the 1 state (thus, the letter S, for set). When S = 0 and R = 1, the flip-flop is reset to the 0 state (thus, the letter R, for reset). It is not permitted for both S and R to be equal to 1. (This would correspond to requiring the flip-flop to set and reset at the same time.)

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FIGURE 5.10.38 RS flip-flop.

FIGURE 5.10.39 Timing diagram for the RS flip-flop.

The rules just described are easily remembered by noting that 1s on the S and R inputs correspond to the set and reset commands, respectively. A convenient means of describing the series of transitions that occur as the signals sent to the flipflop inputs change is the timing diagram. A timing diagram is a graph of the inputs and outputs of the RS flip-flop (or any other logic device) depicting the transitions that occur over time. In effect, one could also represent these transitions in tabular form; however, the timing diagram provides a convenient visual representation of the evolution of the state of the flip-flop. Figure 5.10.39 depicts a table of transitions for an RS flip-flop Q, as well as the corresponding timing diagram. It is important to note that the RS flip-flop is level-sensitive. This means that the set and reset operations are completed only after the R and S inputs have reached the appropriate levels. Thus, in Figure 5.10.39 we show the transitions in the Q outputs as occurring with a small delay relative to the transitions in the R and S inputs. It is instructive to illustrate how an RS flip-flop can be constructed using simple logic gates. For example, Figure 5.10.40 depicts a realization of such a circuit consisting of four gates: two inverters and two NAND gates (actually, the same result could be achieved with four NAND gates). Consider the case in which the circuit is in the initial state Q = 0 (and therefore Q = 1). If the input S = 1 is applied, the top NOT gate will see inputs Q = 1 and S = 0, so that Q = ( S ⋅ Q ) = (0 ⋅ 1) = 1 — that is, the flip-flop is set. Note that when Q is set to 1, Q becomes 0. This, however, does not affect the state of the Q output, since replacing Q with 0 in the expression

( )

Q = S ⋅Q does not change the result:

( )

Q = 0⋅0 =1

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FIGURE 5.10.40 Logic gate implementation of the RS flip-flop.

Thus, the cross-coupled feedback from outputs Q and Q to the input of the NAND gates is such that the set condition sustains itself. It is straightforward to show (by symmetry) that a 1 input on the R line causes the device to reset (i.e., causes Q = 0) and that this condition is also self-sustaining. An extension of the RS flip-flop includes an additional enable input that is gated into each of the other two inputs. Figure 5.10.41 depicts an RS flip-flop consisting of two NOR gates. In addition, an enable input is connected through two AND gates to the RS flip-flop, so that an input to the R and S line will be effective only when the enable input is 1. Thus, any transitions will be controlled by the enable input, which acts as a synchronizing signal. The enable signal may consist of a clock, in which case the flipflop is said to be clocked and its operation is said to be synchronous.

FIGURE 5.10.41 RS flip-flop with enable, preset, and clear lines.

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FIGURE 5.10.42 Data latch.

The same circuit of Figure 5.10.41 can be used to illustrate two additional features of flip-flops: the preset and clear functions, denoted by the inputs P and C, respectively. When P and C are 0, they do not affect the operation of the flip-flop. Setting P = 1 corresponds to setting S = 1, and therefore causes the flip-flop to go into the 1 state, thus, the term preset: this function allows the user to preset the flipflop to 1 at any time. When C is 1, the flip-flop is reset, or cleared (i.e., Q is made equal to 0). Note that these direct inputs are, in general, asynchronous; therefore, they allow the user to preset or clear the flipflop at any time. A set of timing waveforms illustrating the function of the enable, preset, and clear inputs is also shown in Figure 5.10.41. Note how transitions occur only when the enable input goes high (unless the preset or clear inputs are used to override the RS inputs). Another extension of the RS flip-flop, called the data latch, is shown in Figure 5.10.42. In this circuit, the R input is always equal to the inverted S input, so that whenever the enable input is high, the flipflop is set. This device has the dual advantage of avoiding the potential conflict that might arise if both R and S were high and reducing the number of input connections by eliminating the reset input. This circuit is called a data latch because once the enable input goes low, the flip-flop is latched to the previous value of the input. Thus this device can serve as a basic memory element. D Flip-Flop. The D flip-flop is an extension of the data latch that utilizes two RS flip-flops, as shown in Figure 5.10.43. In this circuit, a clock is connected to the enable input of each flip-flop. Since Q1 sees an inverted clock signal, the latch is enabled when the clock waveform goes low. However, since Q2 is disabled when the clock is low, the output of the D flip-flop will not switch to the 1 state until the clock goes high, enabling the second latch and transferring the state of Q1 to Q2. It is important to note that the D flip-flop changes state only on the positive edge of the clock waveform: Q1 is set on the negative edge of the clock, and Q2 (and therefore Q) is set on the positive edge of the clock, as shown in the timing diagram of Figure 5.10.43. This type of device is said to be edge-triggered. This feature is indicated by the “knife edge” drawn next to the CLK input in the device symbol. The particular device described here is said to be positive edge-triggered, or leading edge-triggered, since the final output of the flip-flop is set on a positive-going clock transition. On the basis of the rules stated in this section, the state of the D flip-flop can be described by means of the following truth table:

© 2005 by CRC Press LLC

D

CLK

Q

0 1

≠ ≠

0 1

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FIGURE 5.10.43 D flip-flop.

FIGURE 5.10.44 JK flip-flop.

where the symbol ↑ indicates the occurrence of a positive transition. JK Flip-Flop. Another very common type of flip-flop is the JK flip-flop, shown in Figure 5.10.44. The JK flip-flop operates according to the following rules: • • • •

When J and K are both low, no change occurs in the state of the flip-flop. When J = 0 and K = ↓, the flip-flop is reset to 0. When J = ↓ and K = 0, the flip-flop is set to 1. When both J and K are high, the flip-flop will toggle between states at every transition of the clock input.

The symbol ↓ denotes a negative transition. Note that, functionally, the operation of the JK flip-flop can also be explained in terms of two RS flipflops. When the clock waveform goes high, the “master” flip-flop is enabled; the “slave” receives the state of the master upon a negative clock transition. The “bubble” at the clock input signifies that the device is negative or trailing edge-triggered. This behavior is similar to that of an RS flip-flop, except for the J = 1, K = 1 condition, which corresponds to a toggle mode rather than to a disallowed combination of inputs. Figure 5.10.45 depicts the truth table for the JK flip-flop. It is important to note that when both inputs are 0 the flip-flop remains in its previous state at the occurrence of a clock transition; when either input is high and the other is low, the JK flip-flop behaves like the RS flip-flop, whereas if both inputs are high, the output “toggles” between states every time the clock waveform undergoes a negative transition. © 2005 by CRC Press LLC

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FIGURE 5.10.45 Truth table for the JK flip-flop.

Digital Counters One of the more immediate applications of flip-flops is in the design of counters. A counter is a sequential logic device that can take one of N possible states, stepping through these states in a sequential fashion. When the counter has reached its last state, it resets to zero and is ready to start counting again. For example, a three-bit binary up counter would have 23 = 8 possible states, and might appear as shown in the functional block of Figure 5.10.46. The input clock waveform causes the counter to step through the eight states, making one transition for each clock pulse. We shall shortly see that a string of JK flipflops can accomplish this task exactly. The device shown in Figure 5.10.46 also displays a reset input, which forces the counter to equal 0: b2b1b0 = 000. Although binary counters are very useful in many applications, one is often interested in a decade counter, that is, a counter that counts from 0 to 9 and then resets. A four-bit binary counter can easily be configured in principle to provide this function by means of simple logic that resets the counter when it has reached the count 10012 = 910. As shown in Figure 5.10.47, if we connect bits b3 and b1 to a fourinput AND gate, along with b2 and b0 , the output of the AND gate can be used to reset the counter after a count of 10. Additional logic can provide a “carry” bit whenever a reset condition is reached,

FIGURE 5.10.46 Binary up counter.

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FIGURE 5.10.47 Decade counter.

which could be passed along to another decade counter, enabling counts up to 99. Decade counters can be cascaded so as to represent decimal digits in succession. Although the decade counter of Figure 5.10.47 is attractive because of its simplicity, this configuration would never be used in practice because of the presence of propagation delays. These delays are caused by the finite response time of the individual transistors in each logic device and cannot be guaranteed to be identical for each gate and flip-flop. Thus, if the reset signal — which is presumed to be applied at exactly the same time to each of the four JK flip-flops in the four-bit binary counter — does not cause the JK flip-flops to reset at exactly the same time on account of different propagation delays, then the binary word appearing at the output of the counter will change from 1001 to some other number, and the output of the four-input NAND gate will no longer be high. In such a condition, the flip-flops that have not already reset will then not be able to reset, and the counting sequence will be irreparably compromised. What can be done to obviate this problem? The answer is to use a systematic approach to the design of sequential circuits making use of state transition diagrams. This topic is discussed in the references. A simple implementation of the binary counter we have described in terms of its functional behavior is shown in Figure 5.10.48. The figure depicts a three-bit binary ripple counter, which is obtained from a cascade of three JK flip-flops. The transition table shown in the figure illustrates how the Q output of each state becomes the clock input to the next stage, while each flip-flop is held in the toggle mode. The output transitions assume that the clock, CLK, is a simple square wave (all JKs are negative edge-triggered). This 3-bit ripple counter can easily be configured as a divide-by-8 mechanism, simply by adding an AND gate. To divide the input clock rate by 8, one output pulse should be generated for every eight clock pulses. If one were to output a pulse every time a binary 111 combination occurs, a simple AND gate would suffice to generate the required condition. This solution is shown in Figure 5.10.49. Note that the

FIGURE 5.10.48 Ripple counter.

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FIGURE 5.10.49 Divide-by-8 circuit.

FIGURE 5.10.50 Three-bit synchronous counter.

square wave is also included as an input to the AND gate; this ensures that the output is only as wide as the input signal. This application of ripple counters is further illustrated in the following example. A slightly more complex version of the binary counter is the so-called synchronous counter, in which the input clock drives all of the flip-flops simultaneously. Figure 5.10.50 depicts a three-bit synchronous counter. In this figure, we have chosen to represent each flip-flop as a T flip-flop. The clocks to all the flip-flops are incremented simultaneously. The reader should verify that Q0 toggles to 1 first and then Q1 toggles to 1, and that the AND gate ensures that Q2 will toggle only after Q0 and Q1 have both reached the 1 state (Q0 · Q1 = 1). Other common counters are the ring counter and the up-down counter, which has an additional select input that determines whether the counter counts up or down. Example 5.10.6 Measurement of Angular Position One type of angular position encoder is the slotted encoder shown in Figure 5.10.51. This encoder can be used in conjunction with a pair of counters and a high-frequency clock to determine the speed of rotation of the slotted wheel. As shown in Figure 5.10.52, a clock of known frequency is connected to a counter while another counter records the number of slot pulses detected by an optical slot detector as the wheel rotates. Dividing the counter values, one could obtain the speed of the rotating wheel in radians per second. For example, assume a clocking frequency of 1.2 kHz. If both counters are started at zero and at some instant the timer counter reads 2850 and the encoder counter reads 3050, then the speed of the rotating encoder is found to be 1200

cycles 2850 slots slots ⋅ = 1121.3 second 3050 cycles second

and 1121.3 slots sec × 1° per slot × 2 π 360 rad degree = 19.6 rad sec © 2005 by CRC Press LLC

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FIGURE 5.10.51

FIGURE 5.10.52 Calculating the speed of rotation of the slotted wheel.

FIGURE 5.10.53 PMA pulse sequence.

If this encoder is connected to a rotating shaft, it is possible to measure the angular position and velocity of the shaft. Such shaft encoders are used in measuring the speed of rotation of electric motors, machine tools, engines, and other rotating machinery. A typical application of the slotted encoder is to compute the ignition and injection timing in an automotive engine. In an automotive engine, information related to speed is obtained from the camshaft and the flywheel, which have known reference points. The reference points determine the timing for the ignition firing points and fuel injection pulses and are identified by special slot patterns on the camshaft and crankshaft. Two methods are used to detect the special slots (reference points): period measurement with additional transition detection (PMA, and period measurement with missing transition detection (PMM). In the PMA method, an additional slot (reference point) determines a known reference position on the crankshaft or camshaft. In the PMM method, the reference position is determined by the absence of a slot. Figure 5.10.53 illustrates a typical PMA pulse sequence, showing the presence of an additional pulse. The additional slot may be used to determine the timing for the ignition pulses relative to a known position of the crankshaft. Figure 5.10.54 depicts a typical PMM pulse sequence. Because the period of the pulses is known, the additional slot of the missing slot can be easily detected and used as a reference position. How would you implement these pulse sequences using ring counters? Registers A register consists of a cascade of flip-flops that can store binary data, one bit in each flip-flop. The simplest type of register is the parallel input–parallel output register shown in Figure 5.10.55. In this register, the “load” input pulse, which acts on all clocks simultaneously, causes the parallel inputs b0b1b2b3 © 2005 by CRC Press LLC

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FIGURE 5.10.54 PMA pulse sequence.

FIGURE 5.10.55 Four-bit parallel register.

FIGURE 5.10.56 Four-bit shift register.

to be transferred to the respective flip-flops. The D flip-flop employed in this register allows the transfer from bn to Qn to occur very directly. Thus, D flip-flops are very commonly used in this type of application. The binary word b3b2b1b0 is now “stored”, each bit being represented by the state of a flip-flop. Until the “load” input is applied again and a new word appears at the parallel inputs, the register will preserve the stored word. The construction of the parallel register presumes that the N-bit word to be stored is available in parallel form. However, it is often true that a binary word will arrive in serial form, that is, one bit at a time. A register that can accommodate this type of logic signal is called a shift register. Figure 5.10.56 illustrates how the same basic structure of the parallel register applies to the shift register, except that the input is now applied to the first flip-flop and shifted along at each clock pulse. Note that this type of register provides both a serial and a parallel output.

5.11 Measurements and Instrumentation Measurement Systems and Transducers Measurement Systems In virtually every engineering application there is a need for measuring some physical quantities, such as forces, stresses, temperatures, pressures, flows, or displacements. These measurements are performed by physical devices called sensors or transducers, which are capable of converting a physical quantity to a more readily manipulated electrical quantity. Most sensors, therefore, convert the change of a physical © 2005 by CRC Press LLC

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FIGURE 5.11.1 Measurement system.

quantity (e.g., humidity, temperature) to a corresponding (usually proportional) change in an electrical quantity (e.g., voltage or current). Often, the direct output of the sensor requires additional manipulation before the electrical output is available in a useful form. For example, the change in resistance resulting from a change in the surface stresses of a material — the quantity measured by the resistance strain gauges described in Section 5.2 — must first be converted to a change in voltage through a suitable circuit (the Wheatstone bridge) and then amplified from the millivolt to the volt level. The manipulations needed to produce the desired end result are referred to as signal conditioning. The wiring of the sensor to the signal conditioning circuitry requires significant attention to grounding and shielding procedures, to ensure that the resulting signal is as free from noise and interference as possible. Very often, the conditioned sensor signal is then converted to digital form and recorded in a computer for additional manipulation, or is displayed in some form. The apparatus used in manipulating a sensor output to produce a result that can be suitably displayed or stored is called a measurement system. Figure 5.11.1 depicts a typical computer-based measurement system in block diagram form. Sensor Classification There is no standard and universally accepted classification of sensors. Depending on one’s viewpoint, sensors may be grouped according to their physical characteristics (e.g., electronic sensors, resistive sensors), or by the physical variable or quantity measured by the sensor (e.g., temperature, flow rate). Other classifications are also possible. Table 5.11.1 presents a partial classification of sensors grouped according to the variable sensed; we do not claim that the table is complete, but we can safely state that most of the engineering measurements of interest to the reader are likely to fall in the categories listed in Table 5.11.1. Also included in the table are section or example references to sensors described in this chapter. A sensor is usually accompanied by a set of specifications that indicate its overall effectiveness in measuring the desired physical variable. The following definitions will help the reader understand sensor data sheets: Accuracy: Conformity of the measurement to the true value, usually in percent of full-scale reading Error: Difference between measurement and true value, usually in percent of full-scale reading Precision: Number of significant figures of the measurement Resolution: Smallest measurable increment Span: Linear operating range Range: The range of measurable values Linearity: Conformity to an ideal linear calibration curve, usually in percent of reading or of full-scale reading (whichever is greater) Motion and Dimensional Measurements The measurement of motion and dimension is perhaps the most commonly encountered engineering measurement. Measurements of interest include absolute position, relative position (displacement), velocity, acceleration, and jerk (the derivative of acceleration). These can be either translational or rotational measurements; usually, the same principle can be applied to obtain both kinds of measurements. These measurements are often based on changes in elementary properties, such as changes in the resistance of an element (e.g., strain gauges, potentiometers), in an electric field (e.g., capacitive sensors), or in a magnetic field (e.g., inductive, variable-reluctance, or eddy current sensors). Other mechanisms may be based on special materials (e.g., piezoelectric crystals), or on optical signals and imaging systems. © 2005 by CRC Press LLC

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TABLE 5.11.1 Sensor Classification Sensed Variables Motion and dimensional variables

Force, torque, and pressure

Flow

Temperature

Liquid level

Humidity Chemical composition

Sensors Resistive potentiometers Strain gauges Differential transformers (LVDTs) Variable-reluctance sensors Capacitive sensors Piezoelectric sensors Electro-optical sensors Moving-coil transducers Seismic sensors Strain gauges Piezoelectric sensors Capacitive sensors Pitot tube Hot-wire anemometer Differential pressure sensors Turbine meters Vortex shedding meters Ultrasonic sensors Electromagnetic sensors Imaging systems Thermocouples Resistance thermometers (RTDs) Semiconductor thermometers Radiation detectors Motion transducers Force transducers Differential-pressure measurement devices Semiconductor sensors Gas analysis equipment Solid-state gas sensors

Ref. in this Chapter Example 5.2.1 Example 5.12.1 Example 5.12.2 Example 5.4.1 and 5.4.2 Example 5.9.2 Example 5.10.6 Example 5.12.4 Example 5.2.1 Example 5.9.2 Example 5.4.1 and 5.4.2 Section 5.11 Section 5.11 Section 5.11

Section 5.11 Section 5.11

Force, Torque, and Pressure Measurements Another very common class of measurements is that of pressure and force, and the related measurement of torque. Perhaps the single most common family of force and pressure transducers comprises those based on strain gauges (e.g., load cells, diaphragm pressure transducers). Also very common are piezoelectric transducers. Capacitive transducers again find application in the measurement of pressure. Flow Measurementrs In many engineering applications it is desirable to sense the flow rate of a fluid, whether compressible (gas) or incompressible (liquid). The measurement of fluid flow rate is a complex subject; in this section we simply summarize the concepts underlying some of the most common measurement techniques. Shown in Figure 5.11.2 are three different types of flow rate sensors. The sensor in Figure 5.11.2(a) is based on differential pressure measurement and on a calibrated orifice: the relationship between pressure across the orifice, p1 – p2, and flow rate through the orifice, q, is predetermined through the calibration; therefore, measuring the differential pressure is equivalent to measuring flow rate. The sensor in Figure 5.11.2(b) is called a hot-wire anemometer, because it is based on a heated wire that is cooled by the flow of a gas. The resistance of the wire changes with temperature, and a Wheatstone bridge circuit converts this change in resistance to a change in voltage. Also commonly used are hot-film anemometers, where a heated film is used in place of the more delicate wire. A very common application of the latter type of sensor is in automotive engines, where control of the air-to-fuel ratio depends on measurement of the engine intake mass airflow rate. © 2005 by CRC Press LLC

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FIGURE 5.11.2 Devices for the measurement of flow.

TABLE 5.11.2 Thermocouple Data Type E J K R T S

Elements +/– Chromel/constantan Iron/constantan Chromel/alumel Pt(10%)–Rh/Pt Copper/constantan Pt(13%)–Rh/Pt

Seebeck Coefficient (µV/°C)

Range (°C)

Range (mV)

58.70 at 0°C 50.37 at 0°C 39.48 at 0°C 10.19 at 600°C 38.74 at 0°C 11.35 at 600°C

–270 to 1000 –210 to 1200 –270 to 1372 –50 to 1768 –270 to 400 –50 to 1768

–9.835 to 76.358 –8.096 to 69.536 –6.548 to 54.874 –0.236 to 18.698 –6.258 to 20.869 –0.226 to 21.108

Figure 5.11.2(c) depicts a turbine flowmeter, in which the fluid flow causes a turbine to rotate; the velocity of rotation of the turbine (which can be measured by a noncontact sensor — e.g., a magnetic pickup) is related to the flow velocity. Besides the techniques discussed in this chapter, many other techniques exist for measuring fluid flow, some of significant complexity. Temperature Measurements One of the most frequently measured physical quantities is temperature. The need to measure temperature arises in just about every field of engineering. This subsection is devoted to summarizing two common temperature sensors — the thermocouple and the resistance temperature detector (RTD) — and their related signal conditioning needs. Thermocouples. A thermocouple is formed by the junction of two dissimilar metals. This junction results on an open-circuit thermoelectric voltage due to the Seebeck effect, named after Thomas Seebeck, who discovered the phenomenon in 1821. Various types of thermocouples exist; they are usually classified according to the data of Table 5.11.2. The Seebeck coefficient shown in the table is specified at a given temperature because the output voltage of a thermocouple, v, has a nonlinear dependence on temperature. This dependence is typically expressed in terms of a polynomial of the following form: T = a0 + a1v + a2 v 2 + a3 v 3 + … + an v n

(5.11.1)

For example, the coefficients of the J thermocouple in the range –100 to + 1000°C are as follows: © 2005 by CRC Press LLC

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a0 = −0.048868252

a1 = 19, 873.14503

a2 = −128, 614.5353

a3 = 11, 569,199.78

a4 = −264, 917, 531.4

a5 = 2, 018, 441, 314

The use of a thermocouple requires special connections, because the junction of the thermocouple wires with other leads (such as voltmeter leads, for example) creates additional thermoelectric junctions that in effect act as additional thermocouples. For example, in the J thermocouple circuit of Figure 5.11.3, junction J1 is exposed to the temperature to be measured, but junctions J2 and J3 also generate a thermoelectric voltage, which is dependent on the temperature at these junctions, that is, the temperature at the voltmeter connections. One would therefore have to know the voltages at these junctions as well, in order to determine the actual thermoelectric voltage at J1. To obviate this problem, a reference junction at known temperature can be employed; a traditional approach involves the use of a cold junction, so called because it consists of an ice bath, one of the easiest means of obtaining a known reference temperature. Figure 5.11.4 depicts a thermocouple measurement using an ice bath. The voltage measured in Figure 5.11.4 is dependent on the temperature difference T1 – Tref , where Tref = 0°C. The connections to the voltmeter are made at an isothermal block, kept at a constant temperature; note that the same metal is used in both of the connections to the isothermal block. Thus (still assuming a J thermocouple), there is no difference between the thermoelectric voltages at the two copper-iron junctions; these will add to zero at the voltmeter. The voltmeter will therefore read a voltage proportional to T1 – Tref . An ice bath is not always a practical solution. Other cold junction temperature compensation techniques employ an additional temperature sensor to determine the actual temperature of the junctions J2 and J3 of Figure 5.11.3 Resistance Temperature Detectors (RTDs). A resistance temperature detector (RTD) is a variable-resistance device whose resistance is a function of temperature. RTDs can be made with both positive and negative temperature coefficients and offer greater accuracy and stability than thermocouples. Thermistors are part of the RTD family. A characteristic of all RTDs is that they are passive devices, that is,

FIGURE 5.11.3 J thermocouple circuit.

Isothermal block + Voltmeter connections –

T1 Jref

∗∗ ∗

0° C

J1

∗ Ice Tref ∗∗ ∗ bath ∗∗

Iron Constantan Copper

© 2005 by CRC Press LLC

FIGURE 5.11.4 Cold-junction-compensated thermocouple circuit.

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FIGURE 5.11.5 Effect of connection leads on RTD temperature measurement.

they do not provide a useful output unless excited by an external source. The change in resistance in an RTD is usually converted to a change in voltage by forcing a current to flow through the device. An indirect result of this method is a self-heating error, caused by the i2 R heating of the device. Self-heating of an RTD is usually denoted by the amount of power that will raise the RTD temperature by 1°C. Reducing the excitation current can clearly help reduce self-heating, but it also reduces the output voltage. The RTD resistance has a fairly linear dependence on temperature, a common definition of the temperature coefficient of an RTD is related to the change in resistance from 0 to 100°C. Let R0 be the resistance of the device at 0°C and R100 the resistance at 100°C. Then the temperature coefficient, α, is defined to be α=

R100 − R0 Ω 100 − 0 °C

(5.11.2)

A more accurate representation of RTD temperature dependence can be obtained by using a nonlinear (cubic) equation and published tables of coefficients. As an example, a platinum RTD could be described either by the temperature coefficient α = 0.003911, or by the equation

(

RT = R0 1 + AT − BT 2 − CT 3

(

)

= R0 1 + 3.6962 × 10 −3 T − 5.8495 × 10 −7 T 2 − 4.2325 × 10 −12 T 3

)

(5.11.3)

where the coefficient C is equal to zero for temperatures above 0°C. Because RTDs have fairly low resistance, they are sensitive to error introduced by the added resistance of the lead wires connected to them; Figure 5.11.5 depicts the effect of the lead resistances, rL, on the RTD measurement. Note that the measured voltage includes the resistance of the RTD as well as the resistance of the leads. If the leads used are long (greater than 3 m is a good rule of thumb), then the measurement will have to be adjusted for this error. Two possible solutions to the lead problems are the four-wire RTD measurement circuit and the three-wire Wheatstone bridge circuit, shown in Figure 5.11.6(a) and (b), respectively. In the circuit of Figure 5.11.6(a), the resistance of the lead wires from the excitation, rL1 and rL4, may be arbitrarily large, since the measurement is affected by the resistance of only the output lead wires, rL2 and rL3, which can be kept small by making these leads short. The circuit of Figure 5.11.6(b) takes advantage of the properties of the Wheatstone bridge to cancel out the unwanted effect of the lead wires while still producing an output dependent on the change in temperature.

Wiring, Grounding, and Noise The importance of proper circuit connections cannot be overemphasized. Unfortunately, this is a subject that is rarely taught in introductory electrical engineering courses. The present section summarizes some important considerations regarding signal source connections, various types of input configurations, noise sources and coupling mechanisms, and means of minimizing the influence of noise on a measurement.

© 2005 by CRC Press LLC

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FIGURE 5.11.6 Four-wire RTD circuit (a) and three-wire Wheatstone bridge RTD circuit (b).

Signal Sources and Measurement System Configurations Before proper connection and wiring techniques can be presented, we must examine the difference between grounded and floating signal sources. Every sensor can be thought of as some kind of signal source; a general representation of the connection of a sensor to a measurement system is shown in Figure 5.11.7(a). The sensor is modeled as an ideal voltage source in series with a source resistance. Although this representation does not necessarily apply to all sensors, it will be adequate for the purposes of the present section. Figure 5.7.11(b) and Figure 5.7.11(c) show two types of signal sources: grounded and floating. A grounded signal source is one in which a ground reference is established — for example, by connecting the signal low lead to a case or housing. A floating signal source is one in which neither signal lead is connected to ground; since ground potential is arbitrary, the signal source voltage levels (signal low and signal high) are at an unknown potential relative to the case ground. Thus, the signal is said to be floating. Whether a sensor can be characterized as a grounded or a floating signal source ultimately depends on the connection of the sensor to its case, but the choice of connection may depend on the nature of the source. For example, the thermocouple described earlier is intrinsically a floating signal source, since the signal of interest is a difference between voltages. The same thermocouple could become a grounded signal source if one or its two leads were directly connected to ground, but this is usually not a desirable arrangement for this particular sensor. In analogy with a signal source, a measurement system can be either ground-referenced or differential. In a ground-referenced system, the signal low connection is tied to the instrument case ground; in a differential system, neither of the two signal connections is tied to ground. Thus, a differential measurement system is well suited to measuring the difference between two signal levels (such as the output of an ungrounded thermocouple). One of the potential dangers in dealing with grounded signal sources is the introduction of ground loops. A ground loop is an undesired current path caused by the connection of two reference voltages to each other. This is illustrated in Figure 5.11.8, where a grounded signal source is shown connected to

FIGURE 5.11.7 Measurement system and types of signal sources.

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FIGURE 5.11.8 Ground loop in ground-referenced measurement system.

FIGURE 5.11.9 Differential (nonreferenced) measurement system.

a ground-referenced measurement system. Notice that we have purposely denoted the signal source ground and the measurement system ground by two distinct symbols, to emphasize that these are not necessarily at the same potential — as also indicated by the voltage difference ∆V. Now, one might be tempted to tie the two grounds to each other, but this would only result in a current flowing from one ground to the other, through the small (but nonzero) resistance of the wire connecting the two. The net effect of this ground loop would be that the voltage measured by the instrument would include the unknown ground voltage difference ∆V, as shown in Figure 5.11.8. Since this latter voltage is unpredictable, you can see that ground loops can cause substantial errors in measuring systems. In addition, ground loops are the primary cause of conducted noise, as explained later in this section. A differential measurement system is often a way to avoid ground loop problems because the signal source and measurement system grounds are not connected to each other, and especially because the signal low input of the measuring instrument is not connected to either instrument case ground. The connection of a grounded signal source and a differential measurement system is depicted in Figure 5.11.9. If the signal source connected to the differential measurement system is floating, as shown in Figure 5.11.10, it is often a recommended procedure to reference the signal to the instrument ground by means of two identical resistors that can provide a return path to ground for any currents present at the instrument. An example of such input currents could be the input bias currents inevitably present at the input of an operational or instrumentation amplifier.

FIGURE 5.11.10 Measuring signals from a floating source: (a) differential input; (b) single-ended input.

© 2005 by CRC Press LLC

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FIGURE 5.11.11 Noise sources and coupling mechanisms.

The simple concepts illustrated in the preceding paragraphs and figures can assist the user and designer of instrumentation systems in making the best possible wiring connections for a given measurement. Noise Sources and Coupling Mechanisms Noise — meaning any undesirable signal interfering with a measurement — is an unavoidable element of all measurements. Figure 5.11.11 depicts a block diagram of the three essential stages of a noisy measurement: a noise source, a noise coupling mechanism, and a sensor or associated signal-conditioning circuit. Noise sources are always present and are often impossible to eliminate completely; typical sources of noise in practical measurements are the electromagnetic fields caused by fluorescent light fixtures, video monitors, power supplies, switching circuits, and high-voltage (or current) circuits. Many other sources exist, of course, but often the simple sources in our everyday environment are the most difficult to defeat. Figure 5.11.11 also indicates that various coupling mechanisms can exist between a noise source and an instrument. Noise coupling can be conductive; that is, noise currents may actually be conducted from the noise source to the instrument by physical wires. Noise can also be coupled capacitively, inductively and radiatively. Figure 5.11.12 illustrates how interference can be conductively coupled by way of a ground loop. In the figure, a power supply is connected to both a load and a sensor. We shall assume that the load may be switched on and off, and that it carriers substantial currents. The top circuit contains a ground loop: the current i from the supply divides between the load and sensor; since the wire resistance is nonzero, a large current flowing through the load may cause the ground potential at point a to differ from the potential at point b. In this case, the measured sensor output is no longer, vo, but it is now equal to vo + vba, where vba is the potential difference from point b to point a. Now, if the load is switched on and off and its current is therefore subject to large, abrupt changes, these changes will be manifested in the voltage vba and will appear as noise on the sensor output.

FIGURE 5.11.12 Conductive coupling: ground loop and separate ground returns. © 2005 by CRC Press LLC

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FIGURE 5.11.13 Capacitive coupling and equivalent-circuit representation.

FIGURE 5.11.14 Inductive coupling and equalent-circuit representation.

This problem can be cured simply and effectively by providing separate ground returns for the load and sensor, thus eliminating the ground loop. The mechanism of capacitive coupling is rooted in electric fields that may be caused by sources of interference. The detailed electromagnetic analysis can be quite complex, but to understand the principle, refer to Figure 5.11.13(a), where a noise source is shown to generate an electric field. If a noise source conductor is sufficiently close to a conductor that is part of the measurement system, the two conductors (separated by air, a dielectric) will form a capacitor, through which any time-varying currents can flow. Figure 5.11.13(b) depicts an equivalent circuit in which the noise voltage VN couples to the measurement circuit through an imaginary capacitor, representing the actual capacitance of the noise path. The dual of capacitive coupling is inductive coupling. This form of noise coupling is due to the magnetic field generated by current flowing through a conductor. If the current is large, the magnetic fields can be significant, and the mutual inductance between the noise source and the measurement circuit causes the noise to couple to the measurement circuit. Thus, inductive coupling, as shown in Figure 5.11.14 results when undesired (unplanned) magnetic coupling ties the noise source to the measurement circuit. Noise Reduction Various techniques exist for minimizing the effect of undesired interference, in addition to proper wiring and grounding procedures. The two most common methods are shielding and the use of twisted-pair wire. A shielded cable is shown in Figure 5.11.15. The shield is made of a copper braid or of foil and is usually grounded at the source end, but not at the instrument end, because this would result in a ground loop. The shield can protect the signal from a significant amount of electromagnetic interference, especially at lower frequencies. Shielded cables with various numbers of conductors are available commercially. However, shielding cannot prevent inductive coupling. The simplest method for minimizing inductive coupling is the use of twisted-pair wire; the reason for using twisted pair is that untwisted wire can offer

© 2005 by CRC Press LLC

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FIGURE 5.11.15 Shielding.

large loops that can couple a substantial amount of electromagnetic radiation. Twisting drastically reduces the loop area, and with it the interference. Twisted pair is available commercially.

Signal Conditioning A properly wired, grounded, and shielded sensor connection is a necessary first stage of any well-designed measurement system. The next stage consists of any signal conditioning that may be required to manipulate the sensor output into a form appropriate for the intended use. Very often, the sensor output is meant to be fed into a digital computer, as illustrated in Figure 5.11.1. In this case, it is important to condition the signal so that it is compatible with the process of data acquisition. Two of the most important signal-conditioning functions are amplification and filtering. Both are discussed in the present section. Instrumentation Amplifiers An instrumentation amplifier (IA) is a differential amplifier with very high input impedance, low bias current, and programmable gain that finds widespread application when low-level signals with large common-mode components are to be amplified in noisy environments. This situation occurs frequently when a low-level transducer signal needs to be preamplified, prior to further signal conditioning (e.g., filtering). The functional structure of an IC instrumentation amplifier is depicted in Figure 5.11.16. Specifications for a common IC instrumentation amplifier (and a more accurate circuit description) are shown in.

FIGURE 5.11.16 IC instrumentation amplifier. © 2005 by CRC Press LLC

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FIGURE 5.11.17 AD625 instrumentation amplifier data sheet.

Figure 5.11.17 Among the features worth mentioning here are the programmable gains, which the user can set by suitably connecting one or more of the resistors labeled R1 to the appropriate connection. Note that the user may also choose to connect additional resistors to control the amplifier gain, without adversely affecting the amplifier’s performance, since R1 requires no matching. In addition to the pin connection that permits programmable gains, two additional pins are provided, called sense and reference. These additional connections are provided to the user for the purpose of referencing the output voltage to a signal other than ground, by means of the reference terminal, or of further amplifying the output current (e.g., with a transistor stage), by connecting the sense terminal to the output of the current amplifier. Active Filters The need to filter sensor signals that may be corrupted by noise or other interfering or undesired inputs has already been approached in two earlier chapters. In Section 5.6, simple passive filters made of resistors, capacitors, and inductors were analyzed. It was shown that three types of filter frequency response characteristics can be achieved with these simple circuits: low-pass, high-pass, and band-pass. In Section 5.9, the concept of active filters was introduced, to suggest that it may be desirable to exploit the properties of operational amplifiers to simplify filter design, to more easily match source and load impedances, and to eliminate the need for inductors. The aim of this section is to discuss more advanced active filter designs, which find widespread application in instrumentation circuits. Figure 5.11.18 depicts the general characteristics of a low-pass active filter, indicating that within the pass-band of the filter, a certain deviation from the nominal filter gain, A, is accepted, as indicated by the minimum and maximum pass-band gains, A + ε and A – ε. The width of the pass-band is indicated by the cutoff frequency, ωc. On the other hand, the stop-band, starting at the frequency ωs, does not

FIGURE 5.11.18 Prototype low-pass filter response. © 2005 by CRC Press LLC

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FIGURE 5.11.19 Butterworth low-pass filter frequency response. TABLE 5.11.3 Butterworth Polynomials in Quadratic Form Order, n 1 2 3 4 5

Quadratic Factors (s + 1) (s2 + √2s + 1) (s + 1) (s2 + s + 1) (s2 + 0.7654s + 1) (s2 + 1.8478s + 1) (s + 1) (s2 + 0.6180s + 1) (s2 + 1.6180s + 1)

allow a gain greater than Amin. Different types of filter designs achieve different types of frequency responses, which are typically characterized either by having a particularly flat pass-band frequency response (Butterworth filters) or by a very rapid transition between pass-band and stop-band (Chebyshev filters, and Cauer, or elliptical, filters), or by some other characteristic, such as a very linear phase response (Bessel filters). Achieving each of these properties usually involves trade-offs; for example, a very flat pass-band response will usually result in a relatively slow transition from pass-band to stop-band. In addition to selecting a filter from a certain family, it is also possible to select the order of the filter; this is equal to the order of the differential equation that describes the input-output relationship of a given filter. In general, the higher the order, the faster the transition from pass-band to stop-band (at the cost of greater phase shifts and amplitude distortion, however). Although the frequency response of Figure 5.11.18 pertains to a low-pass filter, similar definitions also apply to the other types of filters. Butterworth filters are characterized by a maximally flat pass-band frequency response characteristic; their response is defined by a magnitude-squared function of frequency: H ( jω ) = 2

H02 1 + ε 2ω 2n

(5.11.14)

where ε = 1 for maximally flat response and n is the order of the filter. Figure 5.11.19 depicts the frequency response (normalized to ωc = 1) of first-, second-, third-, and fourth-order Butterworth low-pass filters. The Butterworth polynomials, given in Table 5.11.3 in factored form, permit the design of the filter by specifying the denominator as a polynomial in s. For s = jω, one obtains the frequency response of the filter. Figure 5.11.20 depicts the normalized frequency response of first- to fourth-order low-pass Chebyshev filters (n = 1 to 4), for ε = 1.06. Note that a certain amount of ripple is allowed in the pass-band; the amplitude of the ripple is defined by the parameter ε and is constant throughout the pass-band. Thus, these filters are also called equiripple filters. Cauer, or elliptical, filters are similar to Chebyshev filters, except for being characterized by equiripple both in the pass-band and in the stop-band. Design tables exist to select the appropriate order of Butterworth, Chebyshev, or Cauer filter for a specific application. Three common configurations of second-order active filters, which can be used to implement secondorder (or quadratic) filter sections using a single of op-amp, are shown in Figure 5.11.21. These filters © 2005 by CRC Press LLC

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FIGURE 5.11.20 Chebyshev low-pass filter frequency response.

FIGURE 5.11.21 Sallen and Key active filters.

are called constant-K, or Sallen and Key, filters (after the name of the inventors). The analysis of these active filters, although somewhat more involved than that of the active filters presented in the preceding chapter, is based on the basic properties of the ideal operational amplifier discussed earlier. Consider, for example, the low-pass filter of Figure 5.11.21. The first unusual aspect of the filter is the presence of both negative and positive feedback; that is, feedback connections are provided to both the inverting and the noninverting terminals of the op-amp. The analysis method consists of finding expressions for the input terminal voltages of the op-amp, V + and V –, and using these expressions to derive the input-output relationship for the filter. The frequency response of the low-pass filter is given by H ( jω ) =

K (1 R1 R2 C1C2 )

( jω)

2

  1 1 1 1 + + + (1 − K ) +  R1 R2 C1C2  R1C1 R2 C1 R2 C2

This frequency response can be expressed in more general form as follows: © 2005 by CRC Press LLC

(5.11.5)

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H ( jω ) =

H0 ω C2

( jω)2 + (ω C Q) jω + ω C2

where H0 = K

(5.11.6)

is the DC gain of the filter, and where ωC =

1 R1 R2 C1C2

is the cutoff frequency, and where 1 = Q

R2 C1C2 + R1C1

R1C2 RC + (1 − K ) 1 1 R2 C1 R2 C2

(5.11.17)

is the inverse of the quality factor, Q. The Q of a filter is related to the overshoot in the transient response of the filter; and to the peaking (i.e., sharpness of the resonant peak) of the frequency response, a highQ circuit will display more peaking, or overshoot, than a low-Q circuit.

Analog-to-Digital and Digital-to-Analog Conversion To take advantage of the capabilities of a microcomputer, it is necessary to suitably interface signals to and from external devices with the microcomputer. Such signals may be analog or digital. Depending on the nature of the signal, either an analog or a digital interface circuit will be required. The advantages in memory storage, programming flexibility, and computational power afforded by today’s digital computers are such that the instrumentation designer often chooses to convert an analog signal to an equivalent digital representation, to exploit the capabilities of a microprocessor in processing the signal. In many cases, the data converted from analog to digital form remain in digital form for ease of storage, or for further processing. In some instances it is necessary to convert the data back to analog form. The latter condition arises frequently in the context of control system design, where an analog measurement is converted to digital form and processed by a digital computer to generate a control action (e.g., raising or lowering the temperature of a process, or exerting a force or a torque); in such cases, the output of the digital computer is converted back to analog form, so that a continuous signal becomes available to the actuators. Figure 5.11.22 illustrates the general appearance of a digital measuring instrument and of a digital controller acting on a plant or process. The objective of this section is to describe how the digital-to-analog (D/A) and analog-to-digital (A/D) conversion blocks of Figure 5.11.22 function. After illustrating discrete circuits that can implement simple A/D and D/A converters, we shall emphasize the use of ICs specially made for these tasks. Nowadays, it is uncommon (and impractical) to design such circuits using discrete components: the performance and ease of use of IC packages make them the preferred choice in virtually all applications. Digital-to-Analog Converters We discuss digital-to-analog conversion first because it is a necessary part of analog-to-digital conversion in many A/D conversion schemes. A digital-to-analog converter (DAC) will convert a binary word to an analog output voltage (or current). The binary word is represented in terms of 1s and 0s, where typically (but not necessarily) 1s correspond to a 5-V level and 0s to a 0-V signal. As an example, consider a four-bit binary word: © 2005 by CRC Press LLC

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FIGURE 5.11.22 Block diagrams of a digital measuring instrument and a digital control system.

(

B = (b3 b2 b1b0 ) 2 = b3 ⋅ 2 3 + b2 ⋅ 2 2 + b1 ⋅ 21 + b0 ⋅ 2 0

)

10

(5.11.8)

The analog voltage corresponding to the digital word B would be va = (8b3 + 4b2 + 2b1 + b0 ) δv

(5.11.9)

where δv is the smallest step size by which va can increment. This least step size will occur whenever the least significant bit (LSB), b0, changes from 0 to 1, and is the smallest increment the digital number can make. We shall also shortly see that the analog voltage obtained by the D/A conversion process has a “staircase” appearance because of the discrete nature of the binary signal. The step size is determined on the basis of each given application and is usually determined on the basis of the number of bits in the digital word to be converted to an analog voltage. We can see that, by extending the previous example for an n-bit word, the maximum value va can attain is

(

)

va max = 2 n−1 + 2 n−2 + … + 21 + 2 0 δv

(

)

(5.11.10)

= 2 n − 1 δv It is relatively simple to construct a DAC by taking advantage of the summing amplifier illustrated in Section 5.9. Consider the circuit shown in Figure 5.11.23, where each bit in the word to be converted is represented by means of a 5-V source and a switch. When the switch is closed, the bit takes a value of 1 (5 V); when the switch is not closed, the bit has value 0. Thus, the output of the DAC is proportional to the word bn-1…bn-2…b1b0. If we select Ri = we can obtain weighted gains for each bit so that © 2005 by CRC Press LLC

R0 2i

(5.11.11)

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FIGURE 5.11.23 n-bit digital-to-analog converter (DAC).

va = −

(

)

RF n−1 2 bn−1 + … + 21 b1 + 2 0 b0 ⋅ 5 V R0

(5.11.12)

and so that the analog output voltage is proportional to the decimal representation of the binary word. The practical design of a DAC is generally not carried out in terms of discrete components, because of problems such as the accuracy required of the resistor value. Many of the problems associated with this approach can be solved by designing the complete DAC circuit in integrated circuit (IC) form. The specifications stated by the IC manufacturer include the resolution, that is, the minimum nonzero voltage; the full-scale accuracy; the output range; the output settling time; the power supply requirements; and the power dissipation. The following example illustrates the use of a common integrated circuit DAC. Example 5.11.1 A typical DAC one would use in conjunction with the 8086 microprocessor is the AD558. This is an IC that can be used in a “stand-alone” configuration (without a microprocessor) or with a microprocessor interfaced to it. 1. If one were to set up the AD558 for an output swing of 0 to 10 V, what would be the smallest voltage output increment attainable? 2. On what is the maximum operating frequency (the largest frequency on which the DAC can perform conversion) of the AD558 dependent? Determine the maximum frequency attainable if the converter is to be run at full-scale input. Solution. 1. Since this DAC is an eight-bit device, the total number of digital increments one could expect is 256. Thus, the finest voltage steps one could expect at the output would be 10 = 39.2 mV 255 This means that for every increment of one bit, the output would jump (in a stepwise fashion) by 39.2 mV. 2. The maximum frequency at which a DAC can run is dependent on the settling time. This is defined as the time it takes for the output to settle to within one half of the least significant bit of its final value. Thus, only one transition can be made per settling time. The settling time for the AD558 depends on the voltage range and is defined for a positive-going full-scale step to ±1/2 LSB. The settling time is 1 µsec, and the corresponding maximum conversion frequency is 1 MHz. © 2005 by CRC Press LLC

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FIGURE 5.11.24 A digital voltage representation of an analog voltage.

Analog-to-Digital Converters You should by now have developed an appreciation for the reasons why it is convenient to process data in digital form. The device that makes conversion of analog signals to digital form is the analog-to-digital converter (ADC), and, just like the DAC, it is also available as a single IC package. In addition to discussing analog-to-digital conversion, we shall also introduce the sample-and-hold amplifier. Quantization. The process of converting an analog voltage (or current) to digital form requires that the analog signal be quantized and encoded in binary form. The process of quantization consists of subdividing the range of the signal into a finite number of intervals; usually, one employs 2″ – 1 intervals, where n is the number of bits available for the corresponding binary word. Following this quantization, a binary word is assigned to each interval (i.e., to each range of voltages or currents); the binary word is then the digital representation of any voltage (current) that falls within that interval. You will note that the smaller the interval, the more accurate the digital representation is. However, some error is necessarily always present in the conversion process; this error is usually referred to as quantization error. Let va represent the analog voltage and vd its quantized counterpart, as shown in Figure 5.11.24 for an analog voltage in the range 0 to 0 16 V. In the figure, the analog voltage va takes on a value of vd = 0 whenever it is in the range 0 to 1 V; for 1 ≤ va < 2, the corresponding value is vd = 1; for 2 ≤ va < 3, vd = 2; and so on, until, for 15 ≤ va < 16, we have vd = 15. You see that if we now represent the quantized voltage vd by its binary counterpart, as shown in the table of Figure 5.11.24, each 1-V analog interval corresponds to a unique binary word. In this example, a four-bit word is sufficient to represent the analog voltage, although the representation is not very accurate. As the number of bits increases, the quantized voltage is closer and closer to the original analog signal; however, the number of bits required to represent the quantized value increases. Tracking ADC. Although not the most efficient in all applications, the tracking ADC is an easy starting point to illustrate the operation of an ADC, in that it is based on the DAC presented in the previous section. The tracking ADC, shown in Figure 5.11.25, compares the analog input signal with the output of a DAC; the comparator output determines whether the DAC output is larger or smaller than the analog input to be converted to binary form. If the DAC output is smaller, then the comparator output will cause an up-down counter to count up, until it reaches a level close to the analog signal; if the DAC output is larger than the analog signal, then the counter is forced to count down. Note that the rate at which the up-down counter is incremented is determined by the external clock, and that the binary counter output corresponds to the binary representation of the analog signal. A feature of the tracking ADC is that it follows (“tracks”) the analog signal by changing one bit at a time. Integrating ADC. The integrating ADC operates by charging and discharging a capacitor, according to the following principle: if one can ensure that the capacitor charges (discharges) linearly, then the time it will take for the capacitor to discharge is linearly related to the amplitude of the voltage that has charged © 2005 by CRC Press LLC

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FIGURE 5.11.25 Tracking ADC.

FIGURE 5.11.26 Integrating ADC.

the capacitor. In practice, to limit the time it takes to perform a conversion, the capacitor is not required to charge fully. Rather, a clock is used to allow the input (analog) voltage to charge the capacitor for a short period of time, determined by a fixed number of clock pulses. Then the capacitor is allowed to discharge through a known circuit, and the corresponding clock count is incremented until the capacitor is fully discharged. The latter condition is verified by a comparator, as shown in Figure 5.11.26. The clock count accumulated during the discharge time is proportional to the analog voltage. In the figure, the switch causes the counter to reset when it is connected to the reference voltage, Vref. The reference voltage is used to provide a known, linear discharge characteristic through the capacitor (see the material on the op-amp integrator in Section 5.9. When the comparator detects that the output of the integrator is equal to zero, it switches state and disables the NAND gate, thus stopping the count. The binary counter output is now the digital counterpart of the voltage va. Other common types of ADC are the so-called successive-approximation ADC and the flash ADC. Flash ADC. The flash ADC is fully parallel and is used for high-speed conversion. A resistive divider network of 2″ resistors divides the known voltage range into that many equal increments. A network of 2″ – 1 comparators then compares the unknown voltage with that array of test voltages. All comparators with inputs exceeding the unknown are “on”; all others are “off ”. This comparator code can be converted to conventional binary by a digital priority encoder circuit. For example, assume that the three-bit flash ADC of Figure 5.11.27 is set up with Vref = 8 V. An input of 6.2 V is provided. If we number the comparators from the top of Figure 5.11.27, the state of each of the seven comparators is as given in Table 5.11.4.

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FIGURE 5.11.27 A three-bit flash ADC.

TABLE 5.11.4 State of Comparators in a 3-Bit Flash ADC Comparator

Input on + Line

Input on – Line

Output

1 2 3 4 5 6 7

7V 6V 5V 4V 3V 2V 1V

6.2 V 6.2 V 6.2 V 6.2 V 6.2 V 6.2 V 6.2 V

H L L L L L L

To resolve the uncertainty generated by the finite ADC conversion time of any practical converter, it is necessary to use a sample-and-hold amplifier. The objective of such an amplifier is to “freeze” the value of the analog waveform for a time sufficient for the ADC to complete its task. A typical sample-and-hold amplifier is shown in Figure 5.11.28. It operates as follows. A MOSFET analog switch is used to “sample” the analog waveform. Recall that when a voltage pulse is provided to the sample input of the MOSFET switch (the gate), the MOSFET enters the ohmic region and in effect becomes nearly a short circuit for the duration of the sampling pulse. While the MOSFET conducts, the analog voltage, va, charges the “hold” capacitor, C, at a fast rate through the small “on” resistance of the MOSFET. The duration of the sampling pulse is sufficient to charge C to the voltage va. Because the MOSFET is virtually a short circuit for the duration of the sampling pulse, the charging (RC) time constant is very small, and the capacitor charges very quickly. When the sampling pulse is over, the MOSFET returns to its nonconducting state, and the capacitor holds the sampled voltage without discharging, thanks to the extremely high input impedance of the voltage-follower (buffer) stage. Thus, vSH is the sampled-andheld value of va at any given sampling time.

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FIGURE 5.11.28 Description of the sample-and-hold process.

FIGURE 5.11.29 Sampled data.

The appearance of the output of a typical sample-and-hold circuit is shown in Figure 5.11.29, together with the analog signal to be sampled. The time interval between samples, or sampling interval, tn – tn–1, allows the ADC to perform the conversion and make the digital version of the sampled signal available, say, to a computer or to another data acquisition and storage system. The sampling interval needs to be at least as long as the A/D conversion time, of course, but it is reasonable to ask how frequently one needs to sample a signal to preserve its fundamental properties (e.g., peaks and valleys, “ringing”, fast transition). One might instinctively be tempted to respond that it is best to sample as frequently as possible, within the limitations of the ADC, so as to capture all the features of the analog signal. In fact, this is not necessarily the best strategy. How should we select the appropriate sampling frequency for a given application? Fortunately, an entire body of knowledge exists with regard to sampling theory, which enables the practicing engineer to select the best sampling rate for any given application. Given the scope of this chapter, we have chosen not to delve into the details of sampling theory, but, rather, to provide the student with a statement of the fundamental result: the Nyquist sampling criterion. The Nyquist criterion states that to prevent aliasing2 when sampling a signal, the sample rate should be selected to be at least twice the highest-frequency component present in the signal. Thus, if we were sampling an audio signal (say, music), we would have to sample at a frequency of at least 40 kHz (twice the highest audible frequency, 20 kHz). In practice, it is advisable to select sampling frequencies substantially greater than the Nyquist rate; a good rule of thumb is five to ten times greater. The following example illustrates how the designer might take the Nyquist criterion into account in designing a practical A/D conversion circuit. Example 5.11.2 A typical ADC one would use in conjunction with the 8086 microprocessor is the AD574. This is a successive-approximation converter. 1. What is the accuracy (in volts) the AD574 can provide if VCC = 15.0 V and 0 ≤ Vin ≤ 15.0 V? 2. On the basis of the data sheet, what is the highest-frequency signal you could convert using the AD574? (Assume that VCC = 15.0 V.) 3. If the maximum conversion time available were 40 msec, what would be the highest-frequency signal you could expect to sample on the basis of the Nyquist criterion?

2

Aliasing is a form of signal distortion that occurs when an analog signal is sampled at an insufficient rate.

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Solution. 1. According to the data sheet, the least significant bit (LSB) of this converter limits its accuracy, meaning that the output is accurate within ± 1 bit. For the 0- to 15-V swing, this gives a voltage accuracy of Vmax − Vmin 2n − 1

or

15 × (±1 bit ) = ±3.66 mV 2 −1 12

2. On the basis of the data sheet, the maximum conversion time is 35 µsec. Therefore, the highest frequency of data conversion using the AD574 is fmax =

1 = 28.57 kHz 35 µs

Thus, the highest signal frequency that could be represented, according to the Nyquist principle, is 1 28.57 × 10 3 fmax = = 14.285 kHz 2 2 This is the maximum theoretical signal frequency that can be represented without distortion, according to the Nyquist principle. 3. Following the same procedure discussed in part 2, 1 1   1  = 12.5 kHz f = 2 max  40 × 10 −6   2  Data Acquisition Systems A typical data acquisition system, shown in Figure 5.11.30, often employs an analog multiplexer, to process several different input signals. A bank of bilateral analog MOSFET switches provides a simple and effective means of selecting which of the input signals should be sampled and converted to digital form. Control logic, employing standard gates and counters, is used to select the desired channel (input-signal) and to trigger the sampling circuit and the ADC. When the A/D conversion is completed, the ADC sends an appropriate end of conversion signal to the control logic, thereby enabling the next channel to be sampled.

FIGURE 5.11.30 Data acquisition system. © 2005 by CRC Press LLC

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FIGURE 5.11.31 Multiplexed sampled data.

In the block diagram of Figure 5.11.30, four analog inputs are shown; if these were to be sampled at regular intervals, the sequence of events would appear as depicted in Figure 5.11.31. We notice, from a qualitative analysis of the figure, that the effective sampling rate for each channel is one fourth the actual external clock rate; thus, it is important to ensure that the sampling rate for each individual channel satisfies the Nyquist criterion. Further, although each sample is held for four consecutive cycles of the external clock, we must notice that the ADC can use only one cycle of the external clock to complete the conversion, since its services will be required by the next channel during the next clock cycle. Thus, the internal clock that times the ADC must be sufficiently fast to allow for a complete conversion of any sample within the design range. Timer ICs: the NE555 This section introduces a multipurpose integrated circuit that can perform basic timing functions. The NE555 is a timer circuit capable of producing accurate time delays (pulses) or oscillation. In the timedelay, or monostable, mode, the time delay or pulse duration is controlled by an external RC network. In the astable, or clock generator, mode, the frequency is controlled by two external resistors and one capacitor. Figure 5.11.32 depicts typical circuits for monostable and astable operation of the NE555. Note that the threshold level and the trigger level can also be externally controlled. For the monostable circuit, the pulse width can be computed from the following equation: T = 1.1R1C

(5.11.13)

For the astable circuit, the positive pulse width can be computed from the following equation: T+ = 0.69( R1 + R2 ) C © 2005 by CRC Press LLC

(5.11.14)

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FIGURE 5.11.32 NE555 timer.

and the negative pulse width can be computed from T− = 0.69 R2 C

(5.11.15)

Data Transmission in Digital Instruments One of the necessary aspects of data acquisition and control systems is the ability to transmit and receive data. Often, a microcomputer-based data acquisition system is interfaced to other digital devices, such as digital instruments or other microcomputers. In these cases it is necessary to transfer data directly in digital form. In fact, it is usually preferable to transmit data that are already in digital form, rather than analog voltages or currents. Among the chief reasons for the choice of digital over analog is that digital data are less sensitive to noise and interference than analog signals: in receiving a binary signal transmitted over a data line, the only decision to be made is whether the value of a bit is 0 or 1. Compared with the difficulty in obtaining a precise measurement of an analog voltage or current, either of which could be corrupted by noise or interference in a variety of ways, the probability of making an error in discerning between binary 0s and 1s is very small. Further, as will be shown shortly, digital data are often coded in such a way that many transmission errors may be detected and corrected for. Finally, storage and processing of digital data are much more readily accomplished than would be the case with analog signals. This section explores a few of the methods that are commonly employed in transmitting digital data; both parallel and serial interfaces are considered. Digital signals in a microcomputer are carried by a bus, consisting of a set of parallel wires each carrying one bit of information. In addition to the signal-carrying wires, there are also control lines that determine under what conditions transmission may occur. A typical computer data bus consists of eight parallel wires and therefore enables the transmission of one byte; digital data are encoded in binary according to one of a few standard codes, such as the BCD code described in Section 5.10, or the ASCII code, which is summarized in Table 5.11.5. This bus configuration is usually associated with parallel transmission, whereby all of the bits are transmitted simultaneously, along with some control bits. Figure 5.11.33 depicts the general appearance of a parallel connection. Parallel data transmission can take place in one of two modes: synchronous or asynchronous. In synchronous transmission, a timing clock pulse is transmitted along with the data over a control line. The arrival of the clock pulse indicates that valid data have also arrived. While parallel synchronous transmission can be very fast, it requires the added complexity of a synchronizing clock and is typically employed only for internal computer data transmission. Further, this type of communication can take place only over short distances (approximately 4 m). Asynchronous data transmission, on the other hand, does not take place at a fixed clock rate, but requires a handshake protocol between sending and receiving ends. The handshake protocol consists of the transmission of data ready and acknowledge signals over two separate control wires. Whenever the sending device is ready to transmit data, it sends a pulse over the data ready line. When this signal reaches © 2005 by CRC Press LLC

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TABLE 5.11.5 ASCII Code Graphic or Control

ASCII (hex)

Graphic or Control

ASCII (hex)

NUL SOH STX ETX EOT ENQ ACK BEL BS HT LF VT FF CR SO SI DLE DCl DC2 DC3 DC4 NAK SYN ETB CAN EM SUB ESC FS GS RS US SP ! ” # $ % & ’ ( ) *

00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20 21 22 23 24 25 26 27 28 29 2A

+ , . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U

2B 2C 2D 2E 2F 30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 40 41 42 43 44 45 46 47 48 49 4A 4B 4C 4D 4E 4F 50 51 52 53 54 55

Graphic or Control

ASCII (hex)

V W X Y Z [ \ ] ↑ ← ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ DEL

56 57 58 59 5A 5B 5C 5D 5E 5F 60 61 62 63 64 65 66 67 68 69 6A 6B 6C 6D 6E 6F 70 71 72 73 74 75 76 77 78 79 7A 7B 7C 7D 7E 7F

the receiver, and if the receiver is ready to receive the data, an acknowledge pulse is sent back, indicating that the transmission may occur; at this point, the parallel data are transmitted. Perhaps the most common parallel interface is based on the IEEE 488 standard, leading to the socalled IEEE 488 bus, also referred to as GPIB (for general-purpose instrument bus). The IEEE 488 Bus The IEEE 488 bus, shown in Figure 5.11.34, is an eight-bit parallel asynchronous interface that has found common application in digital instrumentation applications. The physical bus consists of 16 lines, of which 8 are used to carry the data, 3 for the handshaking protocol, and the rest to control the data flow. The bus permits connection of up to 15 instruments and data rates of up to 1 Mbyte/sec. There is a © 2005 by CRC Press LLC

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FIGURE 5.11.33 Parallel data transmission.

FIGURE 5.11.34 IEEE 488 bus.

limitation, however, in the maximum total length of the bus cable, which is 20 m. The signals transmitted are TTL-compatible and employ negative logic, whereby a logic 0 corresponds to a TTL high state (>2 V) and a logic 1 to a TTL low state ( 0 for positive displacements. The converse is true for negative displacements. More formally, if the primary coil has resistance Rp and self-inductance Lp, we can write iRp + L p

di = vex dt

and the voltages induced in the secondary coils are given by v1 = M1

di dt

v2 = M 2

di dt

FIGURE 5.12.6 Linear variable differential transformer. © 2005 by CRC Press LLC

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FIGURE 5.12.7

so that vout = ( M1 − M2 )

di dt

where M1 and M2 are the mutual inductances between the primary and the respective secondary coils, It should be apparent that each of the mutual inductances is dependent on the position of the iron core. For example, with the core at the null position, M1 = M2 and vout = 0. The LVDT is typically designed so that M1 – M2 is linearly related to the displacement of the core, x. Because the excitation is by necessity an AC signal, the output voltage is actually given by the difference of two sinusoidal voltages at the same frequency and is therefore itself a sinusoid, whose amplitude and phase depend on the displacement, x. Thus, vout is an amplitude-modulated (AM) signal. To recover a signal proportional to the actual displacement, it is therefore necessary to use a demodulator circuit. In practical electromagnetic circuits, the self-inductance of a circuit is not necessarily constant; in particular, the inductance parameter, L, is not constant, in general, but depends on the strength of the magnetic field intensity, so that it will not be possible to use such a simple relationship as v = L di/dt, with L constant. If we revisit the definition of the transformer voltage, e=N

dφ dt

(5.12.12)

We see that in an inductor coil, the inductance is given by L=

Nφ λ = i i

(5.12.13)

This expression implies that the relationship between current and flux in a magnetic structure is linear (the inductance being the slope of the line). In fact, the properties of ferromagnetic materials are such that the flux-current relationship is nonlinear, so that the simple linear inductance parameter used in electric circuit analysis is not adequate to represent the behavior of the magnetic circuits of the present chapter. In any practical situation, the relationship between the flux linkage, λ, and the current is nonlinear, and might be described by a curve similar to that shown in Figure 5.12.7. Wherever the i-λ curve is not a straight line, it is more convenient to analyze the magnetic system in terms of energy calculations, since the corresponding circuit equation would be nonlinear. In a magnetic system, the energy stored in the magnetic field is equal to the integral of the instantaneous power, which is the product of voltage and current, just as in a conventional electrical circuit. Wm =

∫ ei dt

However, in this case, the voltage corresponds to the induced emf, according to Faraday’s law: © 2005 by CRC Press LLC

(5.12.14)

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e=

dλ dφ =N dt dt

(5.12.15)

and is therefore related to the rate of change of the magnetic flux. The energy stored in the magnetic field could therefore be expressed in terms of the current by the integral Wm =

dλ

∫ ei dt = ∫ dt i dt = ∫ i dλ

(5.12.16)

It should be straightforward to recognize that this energy is equal to the area above the λ-i curve of Figure 5.12.7. From the same figure, it is also possible to define a fictitious (but sometimes useful) quantity called co-energy, equal to the area under and identified by the symbol W′m . From the figure, it is also possible to see that the co-energy can be expressed in terms of the stored energy by means of the following relationship: Wm′ = iλ − Wm

(5.12.17)

Ampère’s Law As explained in the previous section, Faraday’s law is one of two fundamental laws relating electricity to magnetism. The second relationship, which forms a counterpart to Faraday’s law, is Ampère’s law. Qualitatively, Ampère’s law states that the magnetic field intensity, H, in the vicinity of a conductor is related to the current carried by the conductor; thus Ampère’s law establishes a dual relationship with Faraday’s law. In the previous section, we described the magnetic field in terms of its flux density, B, and flux φ, To explain Ampère’s law and the behavior of magnetic materials, we need to define a relationship between the magnetic field intensity, H, and the flux density, B. These quantities are related by: B = µH = µrµ0H

Wb m 2 or T

(5.12.18)

where the parameter µ is a scalar constant for a particular physical medium (at least, for the applications we consider here) and is called the permeability of the medium. The permeability of a material can be factored as the product of the permeability of free space, µ0 = 4π × 107 H/m, times the relative permeability, µr , which varies greatly according to the medium. For example, for air and for most electrical conductors and insulators, µr is equal to 1. For ferromagnetic materials, the value of µr can take values in the hundreds or thousands. The size of µr represents a measure of the magnetic properties of the material. A consequence of Ampère’s law is that the larger the value of µ, the smaller the current required to produce a large flux density in an electromagnetic structure. Consequently, many electromechanical devices make use of ferromagnetic materials, called iron cores, to enhance their magnetic properties. Table 5.12.1 gives approximate values of µr for some common materials. Conversely, the reason for introducing the magnetic field intensity is that it is dependent of the properties of the materials employed in the construction of magnetic circuits. Thus, a given magnetic field intensity, H, will give rise to different flux densities in different materials. It will therefore be useful to define sources of magnetic energy in terms of the magnetic field intensity, so that different magnetic structures and materials can then be evaluated or compared for a given source. In analogy with electromotive force, this “source” will be termed magnetomotive force (mmf). As stated earlier, both the magnetic flux density and field intensity are vector quantities; however, for ease of analysis, scalar fields will be chosen by appropriately selecting the orientation of the fields, wherever possible. © 2005 by CRC Press LLC

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TABLE 5.12.1 Relative Permeabilities for Common Materials µr

Material Air Permalloy Cast steel Sheet steel Iron

1 100,000 1,000 4,000 5,195

The field generated by a single conducting wire is not very strong; however, if we arrange the wire into a tightly wound coil with many turns, we can greatly increase the strength of the magnetic field. For such a coil with N turns, one can verify visually that the lines of force associated with the magnetic field link all of the turns of the conducting coil, so that we have effectively increased the current linked by the flux lines N-fold. The product N · i is a useful quantity in electromagnetic circuits and is called the magnetomotive force,
(often abbreviated mmf), in analogy with the electromotive force defined earlier:
= Ni

ampere-turns ( A ⋅ t )

(5.12.19)

Typical arrangements are the iron-core inductor and the toroid of Figure 5.12.8. The flux densities for these inductors are given by the expressions B=

µNi l B=

Flux density for tightly wound circular coil

(5.12.20)

µNi 2 πr2

(5.12.21)

Flux density for toroidal coil

Intuitively, the presence of a high-permeability material near a source of magnetic flux causes the flux to preferentially concentrate in the high-µ material, rather than in air, much as a conducting path concentrates the current produced by an electric field in an electric circuit. Figure 5.12.9 depicts an example of a simple electromagnetic structure, which, as we shall see shortly, forms the basis of the practical transformer. Table 5.12.2 summarizes the variables introduced thus far in the discussion of electricity and magnetism.

FIGURE 5.12.8 Practical inductors. © 2005 by CRC Press LLC

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FIGURE 5.12.9 A simple electromagnetic structure.

TABLE 5.12.2 Magnetic Variables and Units Variable

Symbol

Units

I B φ H e
λ

A Wb/m2 = T Wb A/m V A·t Wb · t

Current Magnetic flux density Magnetic flux Magnetic field intensity Electromotive force Magnetomotive force Flux linkage

Magnetic Circuits It is possible to analyze the operation of electromagnetic devices such as the one depicted in Figure 5.12.9 by means of magnetic equivalent circuits, similar in many respects to the equivalent electrical circuits of the earlier chapters. Before we can present this technique, however, we need to make a few simplifying approximations. The first of these approximations assumes that there exists a mean path for the magnetic flux, and that the corresponding mean flux density is approximately constant over the cross-sectional area of the magnetic structure. Thus, a coil wound around a coil with cross-sectional area A will have flux density B=

φ A

(5.12.22)

where A is assumed to be perpendicular to the direction of the flux lines. Figure 5.12.9 illustrates such a mean path and the cross-sectional area, A. Knowing the flux density, we obtain the field intensity: H=

B φ = µ Aµ

(5.12.23)

But then, knowing the field intensity, we can relate the mmf of the coil,
, to the product of the magnetic field intensity, H, and the length of the magnetic (mean) path, l, for one leg of the structure:
= N ⋅ i = H ⋅ l

(5.12.24)

In summary, the mmf is equal to the magnetic flux times the length of the magnetic path, divided by the permeability of the material times the cross-sectional area: © 2005 by CRC Press LLC

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TABLE 5.12.3 Analogy between Electric and Magnetic Circuits Electrical Quantity

Magnetic Quantity

Electrical field intensity, E, V/m Voltage, v, V Current, i, A Current density, J, A/m2 Resistance, R, Ω Conductivity, σ, l/Ω · m

Magnetic field intensity, H, A · t/m Magnetomotive force,
, A · t Magnetic flux, φ, Wb Magnetic flux density, B, Wb/m2 Reluctance,  = l/µA, A · t/Wb Permeability, µ, Wb/A · m


= φ

l µA

(5.12.25)

A review of this formula reveals that the magnetomotive force,
, may be viewed as being analogous to the voltage source in a series electrical circuit, and that the flux, φ, is then equivalent to the electrical current in a series circuit and the term l/µA to the magnetic resistance of one leg of the magnetic circuit. You will note that the term l/µA is very similar to the term describing the resistance of a cylindrical conductor of length l and cross-sectional A, where the permeability, µ, is analogous to the conductivity, σ. The term l/µA occurs frequently enough to be assigned the name of reluctance, and the symbol . In summary, when an N-turn coil carrying a current i is wound around a magnetic core such as the one indicated in Figure 5.12.9, the mmf,
, generated by the coil produces a flux, φ, that is mostly concentrated with the core and is assumed to be uniform across the cross section. Within this simplified picture, then, the analysis of a magnetic circuit is analogous to that of resistive electrical circuits. This analogy is illustrated in Table 5.12.3 and in the examples in this section. The usefulness of the magnetic circuit analogy can be emphasized by analyzing a magnetic core similar to that of Figure 5.12.9, but with a slightly modified geometry. Figure 5.12.10 depicts the magnetic structure and its equivalent circuit analogy. In the figure, we see that the mmf,
= Ni, excites the magnetic circuit, which is composed of four legs: two of mean path length l 1 and cross-sectional area A1 = d1w, and the other two of mean length l 2 and cross section A2 = d2w. Thus, the reluctance encountered by the flux in its path around the magnetic core is given by the quantity series, with series = 21 + 22 and 1 =

l1 µA1

(5.12.26)

l 2 = 2 µA2 It is important at this stage to review the assumptions and simplifications made in analyzing the magnetic structure of Figure 5.12.10: 1. All of the magnetic flux is linked by all of the turns of the coil. 2. The flux is confined exclusively within the magnetic core. 3. The density of the flux is uniform across the cross-sectional area of the core. You can probably see intuitively that the first of these assumptions might not hold true near the ends of the coil, but that it might be more reasonable if the coil is tightly wound. The second assumption is equivalent to stating that the relative permeability of the core is infinitely higher than that of air (presuming that this is the medium surrounding the core): if this were the case, the flux would indeed be © 2005 by CRC Press LLC

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FIGURE 5.12.10 Analogy between magnetic and electric circuits.

confined within the core. It is worthwhile to note that we made a similar assumption when we treat wires in electric circuits as perfect conductors: the conductivity of copper is substantially greater than that of free space, by a factor of approximately 1015. In the case of magnetic materials, however, even for the best alloys, we have a relative permeability only on the order of 103 to 104. Thus, an approximation that is quite appropriate for electric circuits is not nearly as good in the case of magnetic circuits. Some of the flux in a structure such as those of Figure 5.12.9 and Figure 5.12.10 would thus not be confined within the core (this is usually referred to as leakage flux). Finally, the assumption that the flux is uniform across the core cannot hold for a finite-permeability medium, but it is very helpful in giving an approximate mean behavior of the magnetic circuit. The magnetic circuit analogy is therefore far from being exact. However, short of employing the tools of electromagnetic field theory and of vector calculus, or advanced numerical simulation software, it is the most convenient tool at the engineer’s disposal for the analysis of magnetic structures. Example 5.12.2 Magnetic Reluctance Position Sensor A simple magnetic structure, very similar to those examined in the previous examples, finds very common application in the so-called variable-reluctance position sensor, which, in turn, finds widespread application in a variety of configurations for the measurement of linear and angular position and velocity. Figure 5.12.11 depicts one particular configuration that is used in many applications. In this structure, a permanent magnet with a coil of wire wound around it forms the sensor; a steel disk (typically connected to a rotating shaft) has a number of tabs that pass between the pole pieces of the sensor. The area of the tab is assumed equal to the area of the cross section of the pole pieces and is equal to a2. The reason for the name variable-reluctance sensor is that the reluctance of the magnetic structure is variable, depending on whether or not a ferromagnetic tab lies between the pole pieces of the magnet. The principle of operation of the sensor is that an electromotive force, es, is induced across the coil by the change in magnetic flux caused by the passage of the tab between the pole pieces when the disk

FIGURE 5.12.11 Variable-reluctance position sensor. © 2005 by CRC Press LLC

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FIGURE 5.12.12 Variable-reluctance position sensor waveform.

FIGURE 5.12.13 Signal processing for a 60-tooth wheel RPM sensor.

is in motion. As the tab enters the volume between the pole pieces, the flux will increase because of the lower reluctance of the configuration, until it reaches a maximum when the tab is centered between the poles of the magnet. Figure 5.12.12 depicts the approximate shape of the resulting voltage, which, according to Faraday’s law, is given by eS = −

dφ dt

The rate of change of flux is dictated by the geometry of the tab and of the pole pieces, and by the speed of rotation of the disk. It is important to note that, since the flux is changing only if the disk is rotating, this sensor cannot detect the static position of the disk. One common application of this concept is in the measurement of the speed of rotation of rotating machines, including electric motors and internal combustion engines. In these applications, use is made of a 60-tooth wheel, which permits the conversion of the speed rotation directly to units of revolutions per minute. The output of a variable-reluctance position sensor magnetically coupled to a rotating disk equipped with 60 tabs (teeth) is processed through a comparator or Schmitt trigger circuit. The voltage waveform generated by the sensor is nearly sinusoidal when the teeth are closely spaced, and it is characterized by one sinusoidal cycle for each tooth on the disk. If a negative zero-crossing detector is employed, the trigger circuit will generate a pulse corresponding to the passage of each tooth, as shown in Figure 5.12.13. If the time between any two pulses is measured by means of a high-frequency clock, the speed of the engine can be directly determined in units of rev/min by means of a digital counter.

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FIGURE 5.12.14 Permeability and magnetic saturation effects.

Magnetic Materials and B-H Curves In the analysis of magnetic circuits presented in the previous sections, the relative permeability, µr, was treated as a constant. In fact, the relationship between the magnetic flux density, B, and the associated field intensity, H, B = µH

(5.12.27)

is characterized by the fact that the relative permeability of magnetic materials is not a constant, but is a function of the magnetic field intensity. In effect, all magnetic materials exhibit a phenomenon called saturation, whereby the flux density increases in proportion to the field intensity until it cannot do so any longer. Figure 5.12.14 illustrates the general behavior of all magnetic materials. You will note that since the B-H curve shown in the figure is nonlinear, the value of µ (which is the slope of the curve) depends on the intensity of the magnetic field. To understand the reasons for the saturation of a magnetic material, we need to briefly review the mechanism of magnetization. The basic idea behind magnetic materials is that the spin of electrons constitutes motion of charge and therefore leads to magnetic effects, as explained in the introductory section of this chapter. In most materials, the electron spins cancel out, on the whole, and no net effect remains. In ferromagnetic materials, on the other hand, atoms can align so that the electron spins cause a net magnetic effect. In such materials, there exist small regions with strong magnetic properties (called magnetic domains), the effects of which are neutralized in unmagnetized material by other, similar regions that are oriented differently, in a random pattern. When the material is magnetized, the magnetic domains tend to align with each other, to a degree that is determined by the intensity of the applied magnetic field. In effect, the large number of miniature magnets within the material are polarized by the external magnetic field. As the field increases, more and more domains become aligned. When all of the domains have become aligned, any further increase in magnetic field intensity does not yield an increase in flux density beyond the increase that would be caused in a nonmagnetic material. Thus, the relative permeability, µr, approaches 1 in the saturation region. It should be apparent that an exact value of µr cannot be determined; the value of µr used in the earlier examples is to be interpreted as an average permeability, for intermediate values of flux density. As a point of reference, commercial magnetic steels saturate at flux densities around a few teslas. Figure 5.12.17, shown later in this section, will provide some actual B-H curves for common ferromagnetic materials. The phenomenon of saturation carries some interesting implications with regard to the operation of magnetic circuits: the results of the previous section would seem to imply that an increase in the mmf (that is, an increase in the current driving the coil) would lead to a proportional increase in the magnetic flux. This is true in the linear region of Figure 5.12.14; however, as the material reaches saturation, further increases in the driving current (or, equivalently, in the mmf) do not yield further increases in the magnetic flux. There are two more features that cause magnetic materials to further deviate from the ideal model of the linear B-H relationship: eddy currents and hysteresis. The first phenomenon consists of currents that are caused by any time-varying flux in the core material. As you know, a time-varying flux will

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FIGURE 5.12.15 Eddy currents in magnetic structures.

induce a voltage, and therefore a current. When this happens inside the magnetic core, the induced voltage will cause “eddy” currents (the terminology should be self-explanatory) in the core, which depend on the resistivity of the core. Figure 5.12.15 illustrates the phenomenon of eddy currents. The effect of these currents is to dissipate energy in the form of heat. Eddy currents are reduced by selecting highresistivity core materials, or by laminating the core, introducing tiny, discontinuous air gaps between core layers (see Figure 5.12.15). Lamination of the core reduces eddy currents greatly without affecting the magnetic properties of the core. It is beyond the scope of this section to quantify the losses caused by induced eddy currents, but it will be important to be aware of this source of energy loss. Hysteresis is another loss mechanism in magnetic materials; it displays a rather complex behavior, related to the magnetization properties of a material. The curve of Figure 5.12.16 reveals that the B-H curve for a magnetic material during magnetization (as H is increased) is displaced with respect to the curve that is measured when the material is demagnetized. To understand the hysteresis process, consider a core that has been energized for some time, with a field intensity of H1A · t/m. As the current required to sustain the mmf corresponding to H1 is decreased, we follow the hysteresis curve from the point α to the point β. When the mmf is exactly zero, the material displays the remanent (or residual) magnetization Br . To bring the flux density to zero, we must further decrease the mmf (i.e., produce a negative current), until the field intensity reaches the value –H0 (point γ on the curve). As the mmf is made more negative, the curve eventually reaches the point α′. If the excitation current go the coil is now increased, the magnetization curve will follow the path α′ = β′ = γ ′ = α, eventually returning to the original point in the B-H plane, but via a different path. The result of this process, by which an excess magnetomotive force is required to magnetize or demagnetize the material, is a net energy loss. It is difficult to evaluate this loss exactly; however, it can be shown that it is related to the area between the curves of Figure 5.2.16. There are experimental techniques that enable the approximate measurement of these losses.

FIGURE 5.12.16 Hysteresis in magnetization curves.

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(a)

(b)

FIGURE 5.12.17 (a) Magnetization curve for cast iron; (b) magnetization curve for cast steel;

Figure 5.2.17(a) to Figure 5.2.17(c) depict magnetization curves for three very common ferromagnetic materials: cast iron, cast steel, and sheet steel.

Electromechanical Energy Conversion From the material developed thus far, it should be apparent that electromagnetomechanical devices are capable of converting mechanical forces and displacements to electromagnetic energy, and that the converse is also possible. The objective of this section is to formalize the basic principles of energy conversion in electromagnetomechanical systems, and to illustrate its usefulness and potential for application by presenting several examples of energy transducers. A transducer is a device that can convert electrical to mechanical energy (in this case, it is often called an actuator), or vice versa (in which case it is called a sensor).

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(c)

FIGURE 5.12.17 (c) magnetization curve for sheet steel.

Several physical mechanisms permit conversion of electrical to mechanical energy and back, the principal phenomena being the piezoelectric effect, consisting of the generation of a change in electric field in the presence of strain in certain crystals (e.g., quartz), and electrostriction and magnetostriction, in which changes in the dimension of certain materials lead to a change in their electrical (or magnetic) properties. Although these effects lead to some interesting applications, this chapter is concerned only with transducers in which electrical energy is converted to mechanical energy through the coupling of a magnetic field. It is important to note that all rotating machines (motors and generators) fit the basic definition of electromechanical transducers we have just given. Forces in Magnetic Structures It should be apparent by now that it is possible to convert mechanical forces to electrical signals, and vice versa, by means of the coupling provided by energy stored in the magnetic field. In this subsection, we discuss the computation of mechanical forces and of the corresponding electromagnetic quantities of interest; these calculations are of great practical importance in the design and application of electromechanical actuators. For example, a problem of interest is the computation of the current required to generate a given force in an electromechanical structure. This is the kind of application that is likely to be encountered by the engineer in the selection of an electromechanical device for a given task. As already seen in this chapter, an electromechanical system includes an electrical system and a mechanical system, in addition to means through which the two can interact. The principal focus of this chapter has been the coupling that occurs through an electromagnetic field common to both the electrical and the mechanical system; to understand electromechanical energy conversion, it will be important to understand the various energy storage and loss mechanisms in the electromagnetic field. Figure 5.12.18 illustrates the coupling between the electrical and mechanical systems. In the mechanical system, energy loss can occur because of the heat developed as a consequence of friction, while in the electrical system, analogous losses are incurred because of resistance. Loss mechanisms are also present in the magnetic coupling medium, since eddy current losses and hysteresis losses are unavoidable in ferromagnetic materials. Either system can supply energy, and either system can store energy. Thus, the figure depicts the flow of energy from the electrical to the mechanical system, accounting for these various losses. The same flow could be reversed if mechanical energy were converted to electrical form.

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FIGURE 5.12.18 Electromechanical system.

FIGURE 5.12.19 Simple electromagnet.

Moving-Iron Transducers One important class of electromagnetomechanical transducers is that of moving-iron transducers. The aim of this section is to derive an expression for the magnetic forces generated by such transducers and to illustrate the application of these calculations to simple, yet common devices such as electromagnets, solenoids, and relays. The simplest example of a moving-iron transducer is the electromagnet of Figure 5.12.19, in which the U-shaped element is fixed and the bar is movable. In the following paragraphs, we shall derive a relationship between the current applied to the coil, the displacement of the movable bar, and the magnetic force acting in the air gap. The principle that will be applied throughout the section is that in order for a mass to be displaced, some work needs to be done; this work corresponds to a change in the energy stored in the electromagnetic field, which causes the mass to be displaced. With reference to Figure 5.12.19, let fe represent the magnetic force acting on the bar and x the displacement of the bar, in the direction shown. Then the net work into the electromagnetic field, Wm, is equal to the sum of the work done by the electrical circuit plus the work done by the mechanical system. On the basis of a linear approximation, it can be shown that the stored energy in a magnetic structure is given by the expression Wm =

φ
2

(5.12.28)

and since the flux and the mmf are related by the expression φ=

Ni
=  

the stored energy can be related to the reluctance of the structure according to © 2005 by CRC Press LLC

(5.12.29)

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Wm =

φ 2 ( x ) 2

(5.12.30)

where the reluctance has been explicitly shown to be a function of displacement, as is the case in a moving-iron transducer. Finally, then, we shall use the following approximate expression to compute the magnetic force acting on the moving iron: f =−

dWm φ 2 d ( x ) =− dx 2 dx

(5.12.31)

Example 5.12.13 An Electromagnet An electromagnet is used to support a solid piece of steel as shown in Figure 5.12.19. A force of 8900 N is required to support the weight. The cross-sectional area of the magnet core (the fixed part) is 0.01 m2. Determine the minimum current that can keep the weight from falling for x = 1.5 mm. Assume negligible reluctance for the steel parts, and negligible fringing in the air gap. Solution. We have already shown that in magnetic structures with air gaps, the reluctance is mostly due to the air gaps. This explains the assumption that the reluctance of the structure is negligible. For the structure of Figure 5.12.19, the reluctance is therefore given by ( x ) =

l µ0 A

where A = 0.01 m2 and l = 2x, and therefore ( x ) =

2x x = 4 π × 10 −7 × 0.01 1.2566 × 10 −8

The magnitude of the force in the air gap is given by the expression f = =

φ 2 d ( x ) N 2 i 2 d ( x ) = 2 dx 2 2 dx −9 i2 N 2 d i2 2 6.2832 × 10 = 8900 N = (700) 2 2 2  dx 2 x

from which the current can be computed:

i =2× 2

( ) 6.504 A (6.2832 × 10 )

8900 1.5 × 10 −3

(700)2

2

−9

or i = 2.55 A You should recognize the practical importance of these calculations in determining approximate current requirements and force-generation capabilities of electromechanical transducers. Moving-Coil Transducers Another important class of electromagnetomechanical transducers is that of moving-coil transducers. This class of transducers includes a number of common devices, such as microphones, loudspeakers, and © 2005 by CRC Press LLC

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FIGURE 5.12.20 A simple electromechanical motion transducer.

all electric motors and generators. The aim of this section is to explain the relationship between a fixed magnetic field, the emf across the moving coil, and the forces and motions of the moving element of the transducer. Motor Action. A moving-coil transducer can act as a motor when an externally supplied current flowing through the electrically conducting part of the transducer is converted into a force that can cause the moving part of the transducer to be displaced. Such a current would flow, for example, if the support of Figure 5.12.20 were made of conducting material, so that the conductor and the right-hand side of the support “rail” were to form a loop (in effect, a 1-turn coil). The phenomenon we have just described is sometimes referred to as the “Bli law”. Generator Action. The other mode of operation of a moving-coil transducer occurs when an external force causes the coil (i.e., the moving bar, in Figure 5.12.20) to be displaced. This external force is converted to an emf across the coil, as will be explained in the following paragraphs. It is important to observe that since positive and negative charges are forced in opposite directions in the transducer of Figure 5.12.20, a potential difference will appear across the conducting bar; this potential difference is the electromotive force, or emf. The emf must be equal to the force exerted by the magnetic field. In short, the electric force per unit charge (or electric field) e/l must equal the magnetic force per unit charge f/q = Bu. Thus, the relationship (5.12.32)

e = Blu

which holds whenever B, l, and u are mutually perpendicular, as in Figure 5.12.20. It was briefly mentioned that the Blu and Bli laws indicate that, thanks to the coupling action of the magnetic field, a conversion of mechanical to electrical energy — or the converse — is possible. The simple structure of Figures 5.12.20 can, again, serve as an illustration of this energy-conversion process, although we have not yet indicated how these idealized structures can be converted into a practical device. In this section we shall begin to introduce some physical considerations. Before we proceed any further, we should try to compute the power — electrical and mechanical — that is generated (or is required) by our ideal transducer. The electrical power is given by PE = ei = Blui © 2005 by CRC Press LLC

(W)

(5.12.33)

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while the mechanical power required, say, to move the conductor from left to right is given by the product of force and velocity: PM − fext u = Bliu

(W)

(5.12.34)

Example 5.12.4 The Loudspeaker A loudspeaker, shown in Figure 5.12.21, uses a permanent magnet and a moving coil to produce the vibrational motion that generates the pressure waves we perceive as sound. Vibration of the loudspeaker is caused by changes in the input current to a coil; the coil is, in turn, coupled to a magnetic structure that can produce time-varying forces on the speaker diaphragm. A simplified model for the mechanics of the speaker is also shown in Figure 5.12.21. The force exerted on the coil is also exerted on the mass of the speaker diaphragm, as shown in Figure 5.12.22, which depicts a free-body diagram of the forces acting on the loudspeaker diaphragm. The force exerted on the mass, fi , is the magnetic force due to current flow in the coil. The electrical circuit that describes the coil is shown in Figure 5.12.23, where L represents the inductance of the coil, R represents the resistance of the windings, and e is the emf induced by the coil moving through the magnetic field.

FIGURE 5.12.21 Loudspeaker.

FIGURE 5.12.22 Forces acting on loudspeaker diaphragm.

FIGURE 5.12.23 Model of transducer electrical field.

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Determine the frequency response, (U/V)(jω), of the speaker using phasor analysis if the model parameters are L≈0H

R=8Ω

k ≈ 500, 000 N m

m = 0.001 kg N = 47

d = 22.75

B = 1T

Radius of coil = 5 cm

Solution: To determine the frequency response of the loudspeaker, we need to write the fundamental equations that describe the two subsystems that make up the loudspeaker. The electrical subsystem is described by the usual KVL relationship, applied to the circuit of Figure 5.12.23. v=L

di + Ri + e dt

where e is the emf generated by the motion of the coil in the magnetic field. Next, according to Newton’s law, we can write a force balance equation to describe the dynamics of the mechanical subsystem: m

du = fi − fd − fk = fi − du − kx dt

Now, the coupling between the electrical and mechanical systems is expressed in each of the two preceding equations by the terms e and fi : e = Blu fi = Bli Since we desire the frequency response, we use phasor techniques to represent the electrical and mechanical subsystem equations: V( jω ) = jωLI( jω ) + RI( jω ) + Bl U( jω )

( jωm + d ) U( jω) +

K U( jω ) = Bl I( jω ) jω

Electrical equation Mechanical equation

Having assumed that the inductance of the coil is negligible, we are able to simplify the electrical equation and to solve for I(jω): I( jω ) =

V( jω ) − Bl U( jω ) R

Substituting this equivalence into the mechanical equation and accounting for the length of the coil, l = 2πNr, the final expression for the frequency response of the loudspeaker is then given by 2 πNBr U × ( jω) = V Rm

( jω)

or, numerically, © 2005 by CRC Press LLC

2

jω  (2 π ) 2 B 2 N 2 r 2 + d  R + jω m  

  k +  m 

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U 2 π × 47 × 1 × 0.05 × ( jω ) = V 8(0.001)

( jω)2

≈

(

jω 2  (2π) (1)2 (47)2 (0.05)2 + . 22 75  8 + jω . 0 001  

1, 845 jω jω + 13.8 × 10 3 jω + 36.2 × 10 3

)(

  500, 000 + 0.001  

)

jω  0.051  13.88 × 10 3  = jω jω  1 +  1 +  36.2 × 10 3   13.8 × 10 3  This frequency response shows that the speaker has a lower cutoff frequency fct = 13,800/2π ≈ 2200 Hz and an upper cutoff frequency of fch = 36,000/2π ≈ 5800 Hz. In practice, a loudspeaker with such a frequency response would be useful only as a midrange speaker.

Rotating Electric Machines The range of sizes and power ratings and the different physical features of rotating machines are such that the task of explaining the operation of rotating machines in a single chapter may appear formidable at first. Some features of rotating machines, however, are common to all such devices. This introductory section is aimed at explaining the common properties of all rotating electric machines. We begin our discussion with reference to Figure 5.12.24, in which a hypothetical rotating machine is depicted in a cross-sectional view. In the figure, a box with a cross inscribed in it indicates current flowing into the page, while a dot represents current out of the plane of the page. In Figure 5.12.24, we identify a stator, of cylindrical shape, and a rotor, which, as the name indicates, rotates inside the stator, separated from the latter by means of an air gap. The rotor stator each consist of a magnetic core, some electrical insulation, and the windings necessary to establish a magnetic flux (unless this is created by a permanent magnet). The rotor is mounted on a bearing-supported shaft, which can be connected to mechanical loads (if the machine is a motor) or to a prime mover (if the machine is a generator) by means of belts, pulleys, chains, or other mechanical couplings. The windings carry the electric currents that generate the magnetic fields and flow to the electrical loads, and also provide the closed loops in which voltages will be induced (by virtue of Faraday’s law, as discussed in the previous section).

FIGURE 5.12.24 A rotating electric machine.

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TABLE 5.12.4 Configurations of the Three Types of Electric Machines Machine Type DC

Synchronous Induction

Winding

Winding Type

Location

Current

Input and output

Armature

Rotor

Magnetizing Input and output Magnetizing Input Output

Field Armature Field Primary Secondary

Stator Stator Rotor Stator Rotor

AC (winding) DC (at brushes) DC AC DC AC AC

Basic Classification of Electric Machines An immediate distinction can be made between different types of windings characterized by the nature of the current they carry. If the current serves the sole purpose of providing a magnetic field and is independent of the load, it is called a magnetizing, or excitation, current, and the winding is termed a field winding. Field currents are nearly always DC and are of relatively low power, since their only purpose is to magnetize the core (recall the important role of high-permeability cores in generating large magnetic fluxes from relatively small currents). On the other hand, if the winding carries only the load current, it is called an armature. In DC and AC synchronous machines, separate windings exist to carry field and armature currents. In the induction motor, the magnetizing and load currents flow in the same winding, called the input winding, or primary; the output winding is then called the secondary. As we shall see, this terminology, which is reminiscent of transformers, is particularly appropriate for induction motors, which bear a significant analogy to the operation of the transformers. Table 5.12.4 characterizes the principal machines in terms of their field and armature configuration. It is also useful to classify electric machines in terms of their energy-conversion characteristics. A machine acts as a generator if it converts mechanical energy from a prime mover — e.g., an internal combustion engine — to electrical form. Examples of generators are the large machines used in powergenerating plants, or the common automotive alternator. A machine is classified as a motor if it converts electrical energy to mechanical form. The latter class of machines is probably of more direct interest to you, because of its widespread application in engineering practice. Electric motors are used to provide forces and torques to generate motion in countless industrial applications. Machine tools, robots, punches, presses, mills, and propulsion systems for electric vehicles are but a few examples of the application of electric machines in engineering. Note that in Figure 5.12.24 we have explicitly shown the direction of two magnetic fields: that of the rotor, BR, and that of the stator, BS. Although these fields are generated by different means in differential machines (e.g., permanent magnets, AC currents, DC currents), the presence of these fields is what causes a rotating machine to turn and enables the generation of electric power. In particular, we see that in Figure 5.12.24 the north pole of the rotor field will seek to align itself with the south pole of the stator field. It is this magnetic attraction force that permits the generation of torque in an electric motor; conversely, a generator exploits the laws of electromagnetic induction to convert a changing magnetic field to an electric current. To simplify the discussion in later sections, we shall presently introduce some basic concepts that apply to all rotating electric machines. Referring to Figure 5.12.25, we note that all machines the force on a wire is given by the expression f = iw I × B

(5.12.35)

where iw is the current in the wire, I is a vector along the direction of the wire, and × denotes the cross product of two vectors. Then the torque for a multiturn coil becomes T = KBiw sin α © 2005 by CRC Press LLC

(5.12.36)

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FIGURE 5.12.25 Stator and rotor fields and the force acting on a rotating machine.

where B = magnetic flux density caused by the stator field K = constant depending on coil geometry α = angle between B and the normal to the plane of the coil Other Characteristics of Electric Machines As already stated earlier in this chapter, electric machines are energy-conversion devices, and we are therefore interested in their energy-conversion efficiency. Typical applications of electric machines as motors or generators must take into consideration the energy losses associated with these devices. Figure 5.12.26 represents the various loss mechanisms you must consider in analyzing the efficiency of an electric machine for the case of direct-current machines. It is important for you to keep in mind this conceptual flow of energy when analyzing electric machines. The sources of loss in a rotating machine can be separated into three fundamental groups: electrical (I2R) losses, core losses, and mechanical losses. I2R losses are usually computed on the basis of the DC resistance of the windings at 75°C; in practice, these losses vary with operating conditions. The difference between the nominal and actual I2R is usually lumped under the category of stray-load loss. In direct-current machines, it is also necessary to account for the brush contact loss associated with slip rings and commutators. Mechanical losses are due to friction (mostly in the bearings) and windage, that is, the air drag force that opposes the motion of the rotor. In addition, if external devices (e.g., blowers) are required to circulate air through the machine for cooling purposes, the energy expended by these devices is also included in the mechanical losses. Open-circuit core losses consist of hysteresis and eddy current losses, with only the excitation winding energized. Often these losses are summed with friction and windage losses to give rise to the no-load rotational loss. The latter quantity is useful if one simply wishes to compute efficiency. Since open-circuit core losses do not account for the changes in flux density caused by the presence of load currents, an additional magnetic loss is incurred that is not accounted for in this term. Stray-load losses are used to lump the effects of nonideal current distribution in the windings and of the additional core losses just mentioned. Stray-load losses are difficult to determine exactly and are often assumed to be equal to 1.0% of the output power for DC machines; these losses can be determined by experiment in synchronous and induction machines. The performance of an electric machine can be quantified in a number of ways. In the case of an electric motor, it is usually portrayed in the form of a graphical torque-speed characteristic. The torquespeed characteristic of a motor describes how the torque supplied by the machine varies as a function of the speed of rotation of the motor for steady speeds. As we shall see in later sections, the torque-speed © 2005 by CRC Press LLC

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(a)

(b)

FIGURE 5.12.26 (a) Generator losses, direct current; (b) motor losses, direct current.

curves vary in shape with the type of motor (DC, induction, synchronous) and are very useful in determining the performance of the motor when connected to a mechanical load. Figure 5.12.27 depicts the torque-speed curve of a hypothetical motor. We shall presently describe the essential elements of such a graphical representation of motor performance, and we shall later return to analyze the typical performance curve of each type of motor we encounter in our discussion. It is quite likely that in most

FIGURE 5.12.27 Torque-speed curve for an electric motor.

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engineering applications, the engineer is required to make a decision regarding the performance characteristics of the motor best suited to a specified task. In this context, the torque-speed curve of a machine is a very useful piece of information. The first feature we note of the torque-speed characteristic is that it bears a strong resemblance to the i-v characteristics used in earlier chapters to represent the behavior of electrical sources. It should be clear that, according to this torque-speed curve, the motor is not an ideal source of torque (if it were, the curve would appear as a horizontal line across the speed range). One can readily see, for example, that the hypothetical motor represented by the curves of Figure 5.12.27 would produce maximum torque in the range of speeds between approximately 800 and 1400 rev/min. What determines the actual speed of the motor (and therefore its output torque and power) is the torque-speed characteristic of the load connected to it, much as a resistive load determines the current drawn from a voltage source. In the figure, we display the torque-speed curve of a load, represented by the dashed line; the operating point of the motor-load pair is determined by the intersection of the two curves. Another important observation pertains to the fact that the motor of Figure 5.12.27 produces a nonzero torque at zero speed. This fact implies that as soon as electric power is connected to the motor, the latter is capable of supplying a certain amount of torque; this zero-speed torque is called the starting torque. If the load the motor is connected to requires less than the starting torque the motor can provide, then the motor can accelerate the load, until the motor speed and torque settle to a stable value, at the operating point. The motor-load pair of Figure 5.12.27 would behave in the manner just described. However, there may well be circumstances in which a motor might not be able to provide a sufficient starting torque to overcome the static load torque that opposes its motion. Thus, we see that a torque-speed characteristic can offer valuable insight into the operation of a motor. As we proceed to discuss each type of machine in greater detail, we shall devote some time to the discussion of its torque-speed curve. The most common means of conveying information regarding electric machines is the nameplate. Typical information conveyed by the nameplate is 1. 2. 3. 4. 5.

Type of device (e.g., DC motor, alternator) Manufacturer Rated voltage and frequency Rated current and volt-amperes Rated speed and horsepower

The rated voltage is the terminal voltage for which the machine was designed, and which will provide the desired magnetic flux. Operation at higher voltages will increase magnetic core losses, because of excessive core saturation. The rated current and rated volt-amperes are an indication of the typical current and power levels at the terminal that will not cause undue overheating due to copper losses (I2R losses) in the windings. These ratings are not absolutely precise, but they give an indication of the range of excitations for which the motor will perform without overheating. Peak power operation in a motor may exceed rated torque (horsepower) or currents by a substantial factor (up to as much as six or seven times the rated value); however, continuous operation of the motor above the rated performance will cause the machine to overheat, and possibly to sustain damage. Thus, it is important to consider both peak and continuous power requirements when selecting a motor for a specific application. An analogous discussion is valid for the speed rating: while an electric machine may operate above rated speed for limited periods of time, the large centrifugal forces generated at high rotational speeds will eventually cause undesirable mechanical stresses, especially in the rotor windings, leading eventually even to selfdestruction. Another important feature of electric machines is the regulation of the machine speed or voltage, depending on whether it is used as a motor or as a generator, respectively. Regulation is the ability to maintain speed or voltage constant in the face of load variations. The ability to closely regulate speed in a motor or voltage in a generator is an important feature of electric machines; regulation is often improved by means of feedback control mechanisms, some of which will be briefly introduced in this chapter. We shall take the following definitions as being adequate for the intended purpose of this chapter. © 2005 by CRC Press LLC

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FIGURE 5.12.28 Generator and motor action in an electric machine.

Speed regulation = Voltage regulation =

Speed at no load − Speed at rated load Speed at rated load

(5.12.37)

Voltage at no load − Voltage at rated load Voltage at rated load

(5.12.38)

Please note that the rated value is usually taken to be the nameplate value, and that the meaning of load changes depending on whether the machine is a motor, in which case the load is mechanical, or a generator, in which case the load is electrical. Basic Operation of All Rotating Machines We have already seen how the magnetic field in electromechanical devices provides a form of coupling between electrical and mechanical systems. Intuitively, one can identify two aspects of this coupling, both of which play a role in the operation of electric machines: (1) magnetic attraction and repulsion forces generate mechanical torque, and (2) the magnetic field can induce a voltage in the machine windings (coils) by virtue of Faraday’s law. Thus, we may think of the operation of an electric machine in terms of either a motor or a generator, depending on whether the input power is electrical and mechanical power is produced (motor action), or the input power is mechanical and the output power is electrical (generator action). Figure 5.12.28 illustrates the two cases graphically. The coupling magnetic field performs a dual role, which may be explained as follows. When a current i flow through conductors placed in a magnetic field, a force is produced on each conductor, according to Equation (5.12.35). If these conductors are attached to a cylindrical structure, a torque is generated, and if the structure is free to rotate, then it will rotate at an angular velocity ωm. As the conductors rotate, however, they move through a magnetic field and cut through flux lines, thus generating an electromotive force in opposition to the excitation. This emf is also called “counter” emf; it opposes the source of the current i. If, on the other hand, the rotating element of the machine is driven by a prime mover (for example, an internal combustion engine), then an emf is generated across the coil that is rotating in the magnetic field (the armature). If a load is connected to the armature, a current i will flow to the load, and this current flow will in turn cause a reaction torque on the armature that opposes the torque imposed by the prime mover. You see, then, that for energy conversion to take place, two elements are required: (1) a coupling field, B, usually generated in the field winding; and (2) an armature winding that supports the load current, i, and the emf, e. Magnetic Poles in Electric Machines Before discussing the actual construction of a rotating machine, we should spend a few paragraphs to illustrate the significance of magnetic poles in an electric machine. In an electric machine, torque is developed as a consequence of magnetic forces of attraction and repulsion between magnetic poles on the stator and on the rotor; these poles produce a torque that accelerates the rotor and a reaction torque on the stator. Naturally, we would like a construction such that the torque generated as a consequence © 2005 by CRC Press LLC

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FIGURE 5.12.29 A two-pole machine with salient stator poles.

of the magnetic forces is continuous and in a constant direction. This can be accomplished if the number of rotor poles is equal to the number of stator poles. It is also important to observe that the number of poles must be even, since there have to be equal numbers of north and south poles. Figure 5.12.29 depicts a two-pole machine in which the stator poles are constructed in such a way as to project closer to the rotor than to the stator structure. This type of construction is rather common, and poles constructed in this fashion are called salient poles. Note that the rotor could also be constructed to have salient poles. To understand magnetic polarity, we need to consider the direction of the magnetic field in a coil carrying current. Figure 5.12.30 shows how the right-hand rule can be employed to determine the direction of the magnetic flux. If one were to grasp the coil with the right hand, with the fingers curling in the direction of current flow, then the thumb would be pointing in the direction of the magnetic flux. Magnetic flux is by convention viewed as entering the south pole and exiting from the north pole. Thus, to determine whether a magnetic pole is north or south, we must consider the direction of the flux. Figure 5.12.31 shows a cross section of a coil wound around a pair of salient rotor poles. In this case, one can readily identify the direction of the magnetic flux and therefore the magnetic polarity of the poles by applying the right-hand rule, as illustrated in the figure. Often, however, the coil windings are not arranged as simply as in the case of salient poles. In many machines, the windings are embedded in slots cut into the stator or rotor, so that the situation is similar

FIGURE 5.12.30 Right-hand rule.

© 2005 by CRC Press LLC

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FIGURE 5.12.31 Magnetic field in a salient rotor winding.

FIGURE 5.12.32 Magnetic field of stator.

to that of the stator depicted in Figure 5.12.32. This figure is a cross section in which the wire connections between “crosses” and “dots” have been cut away. In Figure 5.12.32, the dashed line indicates the axis of the stator flux according to the right-hand rule, indicating that the slotted stator in effect behaves like a pole pair. The north and south poles indicated in the figure are a consequence of the fact that the flux exits the bottom part of the structure (thus, the north pole indicated in the figure) and enters the top half of the structure (thus, the south pole). In particular, if you consider that the windings are arranged so that the current entering the right-hand side of the stator (to the right of the dashed line) flows through the back end of the stator and then flows outward from the left-hand side of the stator slots (left of the dashed line), you can visualize the windings in the slots as behaving in a manner similar to the coils of Figure 5.12.31, where the flux axis of Figure 5.12.32 corresponds to the flux axis of each of the coils of Figure 5.12.31. The actual circuit that permits current flow is completed by the front and back ends of the stator, where the wires are connected according to the pattern a-a′, b-b′, c-c′, as depicted in the figure. Another important consideration that facilitates understanding the operation of electric machines pertains to the use of AC currents. It should be apparent by now that if the current flowing into the slotted stator is alternating, the direction of the flux will also alternate, so that in effect the two poles will reverse polarity every time the current reverses direction, that is, every half-cycle of the sinusoidal current. Further — since the magnetic flux is approximately proportional to the current in the coil — as the amplitude of the current oscillates in a sinusoidal fashion, so will the flux density in the structure. Thus, the magnetic field developed in the stator changes both spatially and in time. This property is typical of AC machines, where a rotating magnetic field is established by energizing the coil with an alternating current. As we shall see in the next section, the principles underlying the operation of DC and AC machines are quite different: in a direct-current machine, there is no rotating field, but a mechanical switching arrangement (the commutator) makes it possible for the rotor and stator magnetic fields to always align at right angles to each other. © 2005 by CRC Press LLC

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Direct-Current Machines As explained in the introductory section, direct-current (DC) machines are easier to analyze than their AC counterparts, although their actual construction is made rather complex by the need to have a commutator, which reverses the direction of currents and fluxes to produce a net torque. The objective of this section is to describe the major construction features and the operation of direct-current machines, as well as to develop simple circuit models that are useful in analyzing the performance of this class of machines. Physical Structure of DC Machines A representative DC machine was depicted in Figure 5.12.29, with the magnetic poles clearly identified for both the stator and the rotor. Note the salient pole construction of the stator and the slotted rotor. As previously stated, the torque developed by the machine is a consequence of the magnetic forces between stator and rotor poles. This torque is maximum when the angle γ between the rotor and stator poles is 90°. Also, as you can see from the figure, in a DC machine the armature is usually on the rotor, and the field winding is on the stator. To keep this torque angle constant as the rotor spins on its shaft, a mechanical switch, called a commutator, is configured so the current distribution in the rotor winding remains constant and therefore the rotor poles are consistently at 90° with respect to the fixed stator poles. In a DC machine, the magnetizing current is DC, so that there is no spatial alternation of the stator poles due to timevarying currents. To understand the operation of the commutator, consider the simplified diagram of Figure 5.12.33. In the figure, the brushes are fixed, and the rotor revolves at an angular velocity ωm ; the instantaneous position of the rotor is given by the expression θ = ωm t – γ. The commutator is fixed to the rotor and is made up in this example of six segments that are made of electrically conducting material but are insulated from each other. Further, the rotor windings are configured so that they form six coils, connected to the commutator segments as shown in Figure 5.12.33. As the commutator rotates counterclockwise, the rotor magnetic field rotates with it up to θ = 30°. At that point, the direction of the current changes in coils L3 and L6 as the brushes make contact with the next segment. Now the direction of the magnetic field is –30°. As the commutator continues to rotate, the direction of the rotor field will again change from –30° to +30°, and it will switch again when the brushes switch to the next pair of segments. In this machine, then, the torque angle, γ, is not always 90°, but can vary by as much as ±30°; the actual torque produced by the machine would fluctuate by as much as ±14%, since the torque is proportional to sin γ. As the number of segments increases, the torque fluctuation produced by the commutation is greatly reduced. In a practical machine, for example, one might have as many as 60 segments, and the variation of γ from 90° would be only ±3°, with a torque fluctuation of less than 1%. Thus, the DC machine can produce a nearly constant torque (as a motor) or voltage (as a generator).

FIGURE 5.12.33 Rotor winding and commutator. © 2005 by CRC Press LLC

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5-207

FIGURE 5.12.34

Configuration of DC Machines In DC machines, the field excitation that provides the magnetizing current is occasionally provided by an external source, in which case the machine is said to be separately excited (Figure 5.12.34(a)). More often, the field excitation is derived from the armature voltage and the machine is said to be self-excited. The latter configuration does not require the use of a separate source for the field excitation and is therefore frequently preferred. If a machine is in the separately excited configuration, an additional source, Vf , is required. In the self-excited case, one method used to provide the field excitation is to connect the field in parallel with the armature, since the field winding typically has significantly higher resistance than the armature circuit (remember that it is the armature that carries the load current), this will not draw excessive current from the armature. Further, a series resistor can be added to the field circuit to provide the means for adjusting the field current independent of the armature voltage. This configuration is called a shunt-connected machine and is depicted in Figure 5.12.34(b). Another method for selfexciting a DC machine consists of connecting the field in series with the armature, leading to the seriesconnected machine, depicted in Figure 5.12.34(c); in this case, the field winding will support the entire armature current, and thus the field coil must have low resistance (and therefore relatively few turns). This configuration is rarely used for generators, since the generated voltage and the load voltage must always differ by the voltage drop across the field coil, which varies with the load current. Thus, a series generator would have poor (large) regulation. However, series-connected motors are commonly used in certain applications, as will be discussed in a later section. The third type of DC machine is the compound-connected machine, which consists of a combination of the shunt and series configurations. Figure 5.12.34(d) and Figure 5.12.34(e) show the two types of connections, called the short shunt and the long shunt, respectively. Each of these configurations may be connected so that the series part of the field adds to the shunt part (cumulative compounding) or so that it subtracts (differential compounding). DC Machine Models As stated earlier, it is relatively easy to develop a simple model of a DC machine, which is well suited to performance analysis, without the need to resort to the details of the construction of the machine itself. This section will illustrate the development of such models in two steps. First, steady-state models relating © 2005 by CRC Press LLC

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field and armature currents and voltages to speed and torque are introduced; second, the differential equations describing the dynamic behavior or DC machines are derived. When a field excitation is established, a magnetic flux, φ, is generated by the field current, If . From Equation (5.12.36) we know that the torque acting on the rotor is proportional to the product of the magnetic field and the current in the load-carrying wire; the latter current is the armature current, Ia (iw , in Equation 5.12.36). Assuming that, by virtue of the cummutator, the torque angle, γ, is kept very close to 90°, and therefore sin γ = 1, we obtain the following expression for the torque (in units of N-m) in a DC machine: T = kT φI a

for γ = 90°

(5.12.39)

You may recall that this is simply a consequence of the Bli law of Chapter 15. The mechanical power generated (or absorbed) is equal to the product of the machine torque and the mechanical speed of rotation, ωm (in rad/sec), and is therefore given by Pm = ω m T = ω m kT φI a

(5.12.40)

Recall now that the rotation of the armature conductors in the field generated by the field excitation causes a back emf, Eb , in a direction that opposes the rotation of the armature. According to the Blu law then, this back emf is given by the expression. Eb = ka φω m

(5.12.41)

where ka is called the armature constant and is related to the geometry and magnetic properties of the structure. The voltage Eb represents a countervoltage (opposing the DC excitation) in the case of a motor, and the generated voltage in the case of a generator. Thus, the electric power dissipated (or generated) by the machine is given by the product of the back emf and the armature current: Pe = Eb I a

(5.12.42)

The constants kT and ka in Equation (5.12.39) and Equation (5.12.41) are related to geometry factors, such as the dimension of the rotor and the number of turns in the armature winding; and to properties of materials, such as the permeability of the magnetic materials. Note that in the ideal energy-conversion case, Pm = Pe, and therefore ka = kT . We shall, in general, assume such ideal conversion of electrical to mechanical (or vice versa) and will therefore treat the two constants as being identical: ka = kT . The constant ka is given by ka =

pN 2πM

(5.12.43)

where p = number of magnetic poles N = number of conductors per coil M = number of parallel paths in armature winding An important observation concerning the units of angular speed must be made at this point. The equality (under the no-loss assumption) between the constants ka and kT in Equation (5.12.39) and Equation (5.12.41) results from the choice of consistent units, namely, volts and amperes for the electrical quantities, and newton-meters and radians per second for the mechanical quantities. You should be aware that it is fairly common practice to refer to the speed of rotation of an electric machine in units of revolutions per minute (rev/min). In this book, we shall uniformly use the symbol n to denote angular speed in rev/min; the following relationship should be committed to memory: © 2005 by CRC Press LLC

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FIGURE 5.12.35 Electrical circuit model of a separately excited DC machine.

n( rev min) =

60 ω ( rad sec) 2π m

(5.12.44)

If the speed is expressed in rev/min, the armature constant changes as follows: Eb = ka′φn

(5.12.45)

pN 60 M

(5.12.46)

where ka′ =

Having introduced the basic equations relating torque, speed, voltages, and currents in electric machines, we may now consider the interaction of these quantities in a DC machine at steady state, that is, operating at constant speed and field excitation. Figure 5.12.35 depicts the electrical circuit model of a separately excited DC machine, illustrating both motor and generator action. It is very important to note the reference direction of armature current flow and of the developed torque, in order to make a distinction between the two modes of operation. The field excitation is shown as a voltage, Vf , generating the field current, If , that flows through a variable resistor, Rf , and through the field coil, Lf . The variable resistor permits adjustment of the field excitation. The armature circuit, on the other hand, consists of a voltage source representing the back emf, Eb, the armature resistance, Ra, and the armature voltage, Va. This model is appropriate both for motor and for generator action. When Va < Eb, the machine acts as a generator (Ia flows out of the machine). When Va > Eb, the machine acts as a motor (Ia flows into the machine). Thus, according to the circuit model of Figure 5.12.35, the operation of a DC machine at steady state (i.e., with the inductors in the circuit replaced by short circuits) is described by the following equations: If =

Vf Rf

and

Va = Ra I a + Eb

(motor action) (5.12.47)

If =

Vf Rf

and

Va = − Ra I a + Eb

(generator action)

Equation pair (5.12.47) together with Equation (5.12.39) and Equation (5.12.41) may be used to determine the steady-state operating condition of a DC machine. © 2005 by CRC Press LLC

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The circuit model of Figure 5.12.35 permits the derivation of a simple set of differential equations that describe the dynamic analysis of a DC machine. The dynamic equations describing the behavior of a separately excited DC machine are as follows: Va (t ) = I a (t ) Ra + La

dI a (t ) + E b (t ) dt

V f (t ) = I f (t ) R f + L f

(armature circuit)

dI f (t )

(field circuit)

dt

(5.12.48a)

(5.12.48b)

These equations can be related to the operation of the machine in the presence of a load. If we assume that the motor is rigidly connected to an inertial load with moment of inertia J and that the friction losses in the load are represented by a viscous friction coefficient, b, then the torque developed by the machine (in the motor mode of operation) can be written as follows: T (t ) = TL + bω m (t ) + J

dω m (t ) dt

(5.12.49)

where TL is the load torque. TL is typically either constant or some function of speed, ωm, in a motor. In the case of a generator, the load torque is replaced by the torque supplied by a prime mover, and the machine torque, T(t), opposes the motion of the prime mover, as shown in Figure 5.12.35. Since the machine torque is related to the armature and field currents by Equation (5.12.39), Equation (5.12.48) and Equation (5.12.49) are coupled to each other; this coupling may be expressed as follows: T (t ) = ka φI a (t )

(5.12.50)

or ka φI a (t ) = TL + bω m (t ) + J

dω m (t ) dt

(5.12.51)

The dynamic equations described in this section apply to any DC machine. In the case of a separately excited machine, a further simplification is possible, since the flux is established by virtue of a separate field excitation, and therefore φ=

Nf 

If = kf If

(5.12.52)

where Nf is the number of turns in the field coil,  is the reluctance of the structure, and If is the field current.

AC Machines From the previous sections, it should be apparent that it is possible to obtain a wide range of performance characteristics from DC machines, as both motors and generators. A logical question at this point should be, would it not be more convenient in some cases to take advantage of the single- or multiphase AC power that is available virtually everywhere than to expend energy and use additional hardware to rectify and regulate the DC supplies required by direct-current motors? The answer to this very obvious question is certainly a resounding yes. In fact, the AC induction motor is the workhorse of many industrial applications, and synchronous generators are used almost exclusively for the generation of electric power © 2005 by CRC Press LLC

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worldwide. Thus, it is appropriate to devote a significant portion of this chapter to the study of AC machines, and of induction motors in particular. The objective if this section is to explain the basic operation of both synchronous and induction machines, and to outline their performance characteristics. In doing so, we shall also point out the relative advantages and disadvantages of these machines in comparison with direct-current machines. Rotating Magnetic Fields As mentioned in earlier, the fundamental principle of operation o AC machines is the generation of a rotating magnetic field, which causes the rotor to turn at a speed that depends on the speed of rotation of the magnetic field. We shall now explain how a rotating magnetic field can be generated in the stator and air gap of an AC machine by means of AC currents. Consider the stator shown in Figure 5.12.36, which supports windings, a-a′, b-b′, and c-c′. The coils are geometrically spaced 120° apart, and a three-phase voltage is applied to the coils. As you may recall from the discussion of AC power in Section 5.5, the currents generated by a three-phase source are also spaced by 120°, as illustrated in Figure 5.12.37. The phase voltages referencing the neutral terminal would then be given by the expressions va = A cos(ω e t ) 2π  vb = A cos ω e t −  3 2π  vc = A cos ω e t +  3

FIGURE 5.12.36 Two-pole three-phase stator.

FIGURE 5.12.37 Three-phase stator winding currents. © 2005 by CRC Press LLC

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FIGURE 5.12.38 Flux distribution in a three-phase stator winding as a function of angle of rotation.

FIGURE 5.12.39 Rotating flux in a three-phase machine.

FIGURE 5.12.40 Four-pole stator.

where ωe is the frequency of the AC supply, or line frequency. The coils in each winding are arranged in such a way that the flux distribution generated by any one winding is approximately sinusoidal. Such a flux distribution may be obtained by appropriately arranging groups of coils for each winding over the stator surface. Since the coils are spaced 120° apart, the flux distribution resulting from the sum of the contributions of the three windings is the sum of the fluxes due to the separate windings, as shown in Figure 5.12.38. Thus, the flux in a three-phase machine rotates in space according to the vector diagram of Figure 5.12.39, and is constant in amplitude. A stationary observer on the machine’s stator would see a sinusoidally varying flux distribution as shown in Figure 5.12.38. Since the resultant flux of Figure 5.12.38 is generated by the currents of Figure 5.12.37, the speed of rotation of the flux must be related to the frequency of the sinusoidal phase currents. In the case of the stator of Figure 5.12.36, the number of magnetic poles resulting from the winding configuration is two; however, it is also possible to configure the windings so that they have more poles. For example, Figure 5.12.40 depicts a simplified view of a four-pole stator. In general, the speed of the rotating magnetic field is determined by the frequency of the excitation current, f, and by the number of poles present in the stator, p, according to the equation

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ns =

120 f rev min p

or ωs =

2 πns 2 π × 2 f = p 60

(5.12.53)

where ns (or ωs) is usually called the synchronous speed.

The Induction Motor The induction motor is the most widely used electric machine because of its relative simplicity of construction. The stator winding of an induction machine is similar to that of a synchronous machine; thus, the description of the three-phase winding of Figure 5.12.36 also applies to induction machines. The primary advantage of the induction machine, which is almost exclusively used as a motor (its performance as a generator is not very good), is that no separate excitation is required for the rotor. The rotor typically consists of one of two arrangements: a squirrel cage or a wound rotor. The former contains conducting bars short-circuited at the end and embedded within it; the latter consists of a multiphase winding similar to that used for the stator, but electrically short-circuited. In either case, the induction motor operates by virtue of currents induced from the stator field in the rotor. In this respect, its operation is similar to that of a transformer, in that currents in the stator (which acts as a primary coil) induce currents in the rotor (acting as a secondary coil). In most induction motors, no external electrical connection is required for the rotor, thus permitting a simple, rugged construction, without the need for slip rings or brushes. Unlike the synchronous motor, the induction motor does not operate at synchronous speed, but at a somewhat lower speed, which is dependent on the load. Figure 5.12.41 illustrates the appearance of a squirrel-cage induction motor. The following discussion will focus mainly on this very common configuration. You are by now acquainted with the notion of a rotating stator magnetic field. Imagine now that a squirrel-cage rotor is inserted in a stator in which such a rotating magnetic field is present. The stator field will induce voltages in the cage conductors, and if the stator field is generated by a three-phase source, the resulting rotor currents — which circulate in the bars of the squirrel cage, with the conducting path completed by the shorting rings at the end of the cage — are also three-phase, and are determined by the magnitude of the induced voltages and by the impedance of the rotor. Since the rotor currents are induced by the stator field, the number of poles and the speed of rotation of the induced magnetic field are the same as those of the stator field, if the rotor is at rest. Thus, when a stator field is initially applied, the rotor field is synchronous with it, and the fields are stationary with respect to each other. Thus, according to the earlier discussion, a starting torque is generated. If the starting torque is sufficient to cause the rotor to start spinning, the rotor will accelerate up to its operating speed. However, an induction motor can never reach synchronous speed; if it did, the rotor would appear to be stationary with respect to the rotating stator field, since it would be rotating at the same speed. But in the absence of relative motion between the stator and rotor fields, no voltage would be induced in the rotor. Thus, an induction motor is limited to speeds somewhere below the synchronous speed, ns. Let the speed of rotation of the rotor be n; then, the rotor is losing ground with respect to the rotation of the stator field at a speed (ns – n). In effect, this is equivalent to backward motion of the rotor at the slip speed, defined by (ns – n). The slip, s, is usually defined as a fraction of ns: s=

© 2005 by CRC Press LLC

ns − n ns

(5.12.54)

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FIGURE 5.12.41 (a) Squirrel-cage induction motor; (b) conductors in rotor; (c) photo of squirrel-cage induction motor.

which leads to the following expression for the motor speed: n = ns (1 − s)

(5.12.55)

The slip, s, is a function of the load, and the amount of slip in a given motor is dependent on its construction and rotor type (squirrel cage or wound rotor). Since there is a relative motion between the stator and rotor fields, voltages will be induced in the rotor at a frequency called the slip frequency, related to the relative speed of the two fields. This gives rise to an interesting phenomenon: the rotor field travels relative to the rotor at the slip speed sns, but the rotor is mechanically traveling at the speed (1 – s)ns, so that the net effect is that the rotor field travels at the speed sns + (1 − s) ns = ns

(5.12.56)

that is, at synchronous speed. The fact that the rotor field rotates at synchronous speed — although the rotor itself does not — is extremely important, because it means that the stator and rotor fields will continue to be stationary with respect to each other, and therefore a net torque can be produced. © 2005 by CRC Press LLC

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FIGURE 5.12.42 Performance curve for induction motor.

As in the case of DC and synchronous motors, important characteristics of induction motors are the starting torque, the maximum torque, and the torque-speed curve. Performance of Induction Motors The performance of induction motors can be described by torque-speed curves similar to those already used for DC motors. Figure 5.12.42 depicts an induction motor torque-speed curve, with five torque ratings marked a through e. Point a is the starting torque, also called breakaway torque, and is the torque available with the rotor “locked”, that is, in a stationary position. At this condition, the frequency of the voltage induced in the rotor is highest, since it is equal to the frequency of rotation of the stator field; consequently, the inductive reactance of the rotor is greatest. As the rotor accelerates, the torque drops off, reaching a maximum value called the pull-up torque (point b); this typically occurs somewhere between 25 and 40% of synchronous speed. As the rotor speed continues to increase, the rotor reactance decreases further (since the frequency of the induced voltage is determined by the relative speed of rotation of the rotor with respect to the stator field). The torque becomes a maximum when the rotor inductive reactance is equal to the rotor resistance; maximum torque is also called breakdown torque (point c). Beyond this point, the torque drops off, until it is zero at synchronous speed, as discussed earlier. Also marked on the curve are the 150% torque (point d), and the rated torque (point e). A general formula for the computation of the induction motor steady-state torque-speed characteristic is T=

mVS2 RR s 1 2 ω e  RR  2  + ( XS + X R )   RS + s   

(5.12.57)

where m is the number of phases. Different construction arrangements permit the design of induction motors with different torquespeed curves, thus permitting the user to select the motor that best suits a given application. Figure 5.12.43 depicts the four basic classifications, classes A, B, C, and D, as defined by NEMA. The determining features in the classification are the locked-rotor torque and current, the breakdown torque, the pulldown torque, and the percent slip. Class A motors have a higher breakdown torque than class B motors, and a slip of 5% or less. Motors in this class are often designed for a specific application. Class B motors are general-purpose motors; this is the most commonly used type of induction motor, with typical values of slip of 3 to 5%. Class C motors have a high starting torque for a given starting current, and a low slip. These motors are typically used in applications demanding high starting torque but having relatively normal running loads, once running speed has been reached. Class D motors are characterized by high starting torque, high slip, low starting current, and low full-load speed. A typical value of slip is around 13%. © 2005 by CRC Press LLC

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FIGURE 5.12.43 Induction motor classification.

Factors that should be considered in the selection of an AC motor for a given application are the speed range, both minimum and maximum, and the speed variation. For example, it is important to determine whether constant speed is required; what variation might be allowed, either in speed or in torque; or whether variable-speed operation is required, in which case a variable-speed drive will be needed. The torque requirements are obviously important as well. The starting and running torque should be considered; they depend on the type of load. Starting torque can vary from a small percentage of full-load to several times full-load torque. Furthermore, the excess torque available at start-up determines the acceleration characteristics of the motor. Similarly, deceleration characteristics should be considered, to determine whether external braking might be required. Another factor to be considered is the duty cycle of the motor. The duty cycle, which depends on the nature of the application, is an important consideration when the motor is used in repetitive, noncontinuous operation, such as is encountered in some types of machine tools. If the motor operates at zero or reduced load for periods of time, the duty cycle — that is, the percentage of the time the motor is loaded — is an important selection criterion. Last, but by no means least, are the heating properties of a motor. Motor temperature is determined by internal losses and by ventilation; motors operating at a reduced speed may not generate sufficient cooling, and forced ventilation may be required.

Stepping Motors Stepping, or stepper, motors are motors that convert digital information to mechanical motion. The principles of operation of stepping motors have been known since the 1920s; however, their application has seen a dramatic rise with the increased use of digital computers. Stepping motors, as the name suggests, rotate in distinct steps, and their position can be controlled by means of logic signals. Typical applications of stepping motors are line printers, positioning of heads in magnetic disks drives, and any other situation where continuous or stepwise displacements are required. Stepping motors can generally be classified in one of three categories: variable-reluctance, permanentmagnet, and hybrid types. It will soon be shown that the principles of operation of each of these devices bear a definite resemblance to those of devices already encountered in this book. Stepping motors have a number of special features that make them particularly useful in practical applications. Perhaps the most important feature of a stepping motor is that the angle of rotation of the motor is directly proportional to the number of input pulses; further, the angle error per step is very small and does not accumulate. Stepping motors are also capable of rapid response to starting, stopping, and reversing commands, and can be driven directly by digital signals. Another important feature is a self-holding capability that makes it possible for the rotor to be held in the stopped position without the use of brakes. Finally, a wide range of rotating speeds — proportional to the frequency of the pulse signal — may be attained in these motors. Figure 5.12.44 depicts the general appearance of three types of stepping motors. The permanentmagnet-rotor stepping motor, Figure 5.12.44(a), permits a nonzero holding torque when the motor is not energized. Depending on the construction of the motor, it is typically possible to obtain step angles © 2005 by CRC Press LLC

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FIGURE 5.12.44 Stepping motor configurations.

of 7.5, 11.25, 15, 18, 45, or 90°. The angle of rotation is determined by the number of stator poles, as will be illustrated shortly in an example. The variable-reluctance stepping motor, Figure 5.12.44(b), has an iron multipole rotor and a laminated wound stator, and rotates when the teeth on the rotor are attracted to the electromagnetically energized stator teeth. The rotor inertia of a variable-reluctance stepping motor is low, and the response is very quick, but the allowable load inertia is small. When the windings are not energized, the static torque of this type of motor is zero. Generally, the step angle of the variable-reluctance stepping motor is 15°. The hybrid stepping motor, Figure 5.12.44(c), is characterized by multitoothed stator and rotor, the rotor having an axially magnetized concentric magnet around its shaft. It can be seen that this configuration is a mixture of the variable-reluctance and permanent-magnet types. This type of motor generally has high accuracy and high torque and can be configured to provide a step angle as small as 1.8°. For any of these configurations, the principle of operation is essentially the same: when the coils are energized, magnetic poles are generated in the stator, and the rotor will align in accordance with the direction of the magnetic field developed in the stator. By reversing the phase of the currents in the coils or by energizing only some of the coils (this is possible in motors with more than two stator poles), the alignment of the stator magnetic field can take one of a discrete number of positions; if the currents in the coils are pulsed in the appropriate sequence, the rotor will advance in a step-by-step fashion. Thus, this type of motor can be very useful whenever precise incremental motion must be attained. As mentioned earlier, typical applications are printer wheels, computer disk drives, and plotters. Other applications are found in the control of the position of valves (e.g., control of the throttle valve in an engine or of a hydraulic valve in a fluid power-system), and in drug-dispensing apparatus for clinical applications.

The Universal Motor If it were possible to operate a DC motor from a single-phase AC supply, a wide range of simple applications would become readily available. Recall that the direction of the torque produced by a DC machine is determined by the direction of current flow in the armature conductors and by the polarity of the field; torque is developed in a DC machine because the commutator arrangement permits the field and armature currents to remain in phase, thus producing torque in a constant direction. A similar result can be obtained by using an AC supply, and by connecting the armature and field windings in series, as shown in Figure 5.12.45. A series DC motor connected in this configuration can therefore operate on a single-phase AC supply, and is referred to as a universal motor. An additional consideration is that, because of the AC excitation, it is necessary to reduce AC core losses by laminating the stator; thus, the universal motor differs from the series DC motor in its construction features. Typical torque-speed curves for AC and DC operation of a universal motor are shown in Figure 5.12.46. As shown in Figure 5.12.45, © 2005 by CRC Press LLC

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FIGURE 5.12.45 Operation and circuit diagram of a universal motor.

FIGURE 5.12.46 Torque-speed curve of a universal motor.

the load current is sinusoidal and therefore reverses direction each half-cycle; however, the torque generated by the motor is always in the same direction, resulting in a pulsating torque, with nonzero averaging value. As in the case of a DC series motor, the best method for controlling the speed of a universal motor is to change its (rms) input voltage. The higher the rms input voltage, the greater the resulting speed of the motor. Approximate torque-speed characteristics of a universal motor as a function of voltage are shown in Figure 5.12.47.

FIGURE 5.12.47 Torque-speed characteristics of a universal motor.

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FIGURE 5.12.48 Split-phase motor.

FIGURE 5.12.49 Torque-speed curve of split-phase motor.

Single-Phase Induction Motors Thus far, we have not mentioned how the initial starting torque can be provided to a single-phase motor. In practice, single-phase motors are classified by their starting and running characteristics, and several methods exist to provide nonzero starting torque. The aim of this section is to classify single-phase motors by describing their configuration on the basis of the method of starting. For each class of motor, a torquespeed characteristic will also be described. Split-Phase Motors Split-phase motors are constructed with two separate stator windings, called main and auxiliary windings; the axes of the two windings are actually at 90° with respect to each other, as shown in Figure 5.12.48. The auxiliary winding current is designed to be out of phase with the main winding current, as a result of different reactances of the two windings. Different winding reactances can be attained by having a different ratio of resistance to inductance — for example, by increasing the resistance of the auxiliary winding. In particular, the auxiliary winding current, Iaux, leads the main winding current, Imain. The net effect is that the motor sees a two-phase (unbalanced) current that results in a rotating magnetic field, as in any polyphase stator arrangement. Thus, the motor has a nonzero starting torque, as shown in Figure 5.12.49. Once the motor, a centrifugal switch is used to disconnect the auxiliary winding, since a single winding is sufficient to sustain the motion of the rotor. The switching action permits the use of relatively high-resistance windings, since these are not used during normal operation and therefore one need not be concerned with the losses associated with a higher-resistance winding. Figure 5.12.49 also depicts the combined effect of the two modes of operation of the split-phase motor. Split-phase motors have appropriate characteristics (at very low cost) for fans, blowers, centrifugal pumps, and other applications in the range of 1/20 to 1/2 hp. Capacitor-Type Motors Another method for obtaining a phase difference between currents that will give rise to a rotating magnetic field is by the addition of a capacitor. Motors that use this arrangement are termed capacitor-type motors. These motors make different use of capacitors to provide starting or running capabilities, or a combi-

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FIGURE 5.12.50 Capacitor-start motor.

FIGURE 5.12.51 Torque-speed curve for a capacitor-start motor.

FIGURE 5.12.52 Torque-speed curve for a permanent split-capacitor motor.

nation of the two. The capacitor-start motor is essentially identical to the split-phase motor, except for the addition of a capacitor in series with the auxiliary winding, as shown in Figure 5.12.50. The addition of the capacitor changes the reactance of the auxiliary circuit in such a way as to cause the auxiliary current to lead the main current. The advantage of using the capacitor as a means for achieving a phase split is that greater starting torque may be obtained than with the split-phase arrangement. A centrifugal switching arrangement is used to disconnect the auxiliary winding above a certain speed, in the neighborhood of 75% of synchronous speed. Figure 5.12.51 depicts the torque-speed characteristic of a capacitor-start motor. Because of their higher starting torque, these motors are very useful in connection with loads that present a high static torque. Examples of such loads are compressors, pumps, and refrigeration and air-conditioning equipment. It is also possible to use the capacitor-start motor without the centrifugal switch, leading to a simpler design. Motors with this design are called permanent split-capacitor motors; they offer a compromise between running and starting characteristics. A typical torque-speed curve is shown in Figure 5.12.52. A further compromise can be achieved by using two capacitors, one to obtain a permanent phase split and the resulting improvement in running characteristics, the other to improve the starting torque. A small capacitance is sufficient to improve the running performance, while a much larger capacitor provides the temporary improvement in starting torque. A motor with this design is called a capacitorstart capacitor-run motor; its schematic diagram is shown in Figure 5.12.53. Its torque-speed characteristic is similar to that of a capacitor-start motor. Shaded-Pole Motors The last type of single-phase induction motor discussed in this chapter is the shaded-pole motor. This type of motor operates on a different principle from the motors discussed thus far. The stator of a shadedpole motor has a salient pole construction, as shown in Figure 5.12.54, that includes a shading coil © 2005 by CRC Press LLC

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FIGURE 5.12.53 Capacitor-start capacitor-run motor. Main winding

Shading coil

i

FIGURE 5.12.54 Shaded-pole motor.

FIGURE 5.12.55 Torque-speed curve of a shaded-pole motor.

consisting of a copper band wound around part of each pole. The flux in the shaded portion of the pole lags behind the flux in the unshaded part, achieving an effect similar to a rotation of the flux in the direction of the shaded part of the pole. This flux rotation in effect produces a rotating field that enables the motor to have a starting torque. This construction technique is rather inexpensive and is used in motors up to about 1/20 hp. A typical torque-speed characteristic for a shaded-pole motor is given in Figure 5.12.55. Summary of Single-Phase Motor Characteristics For basic classes of single-phase motors are commonly used: 1. Single-phase induction motors are used for the larger home and small business tasks, such as furnace oil burner pumps, or hot water or hot air circulators. Refrigerator compressors, lathes, and bench-mounted circular saws are also powered with induction motors. 2. Shaded-pool motors are used in the smaller sizes for quite, low-cost applications. The size range is for 1/30 hp (24.9 W) to 1/2 hp (373 W), particularly for fans and similar drives in which the starting torque is low. 3. Universal motors will operate on any household AC frequency or on DC without modification or adjustment. They can develop very high speed while loaded, and very high power for their size. Vacuum cleaners, sewing machines, kitchen food mixers, portable electric drills, portable circular saws, and home motion-picture projectors are examples of applications of universal motors. 4. The capacitor-type motor finds its widest field of application at low speeds (below 900 rev/min) and in ratings from 3/4 hp (0.5595 kW) to 3 hp (2.238 kW) at all speeds, especially in fan drives. © 2005 by CRC Press LLC

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References Section 2 Irwin, J.D., 1989. Basic Engineering Circuit Analysis, 3rd ed. Macmillan, New York. Nilsson, J.W., 1989. Electric Circuits, 3rd ed. Addison-Wesley, Reading, MA. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Smith, R.J. and Dorf, R.C., 1992. Circuits, Devices and Systems, 5th ed. John Wiley & Sons, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 3 Irwin, J.D., 1989. Basic Engineering Circuit Analysis, 3rd ed. Macmillan, New York. Nilsson, J.W., 1989. Electric Circuits, 3rd ed. Addison-Wesley, Reading, MA. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Smith, R.J. and Dorf, R.C., 1992. Circuits, Devices and Systems, 5th ed. John Wiley & Sons, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 4 Budak, A., Passive and Active Network Analysis and Synthesis, Houghton Mifflin, Boston. Irwin, J.D., 1989. Basic Engineering Circuit Analysis, 3rd ed. Macmillan, New York. Nilsson, J.W., 1989. Electric Circuits, 3rd ed. Addison-Wesley, Reading, MA. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Smith, R.J. and Dorf, R.C., 1992. Circuits, Devices and Systems, 5th ed. John Wiley & Sons, New York. Van Valkenburg, M.E., 1982, Analog Filter Design, Holt, Rinehart & Winston, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 5 Del Toro, V., 1992. Electric Power Systems, Prentice-Hall, Englewood Cliffs, NJ. Nilsson, J.W., 1989. Electric Circuits, 3rd ed. Addison-Wesley, Reading, MA. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Smith, R.J. and Dorf, R.C., 1992. Circuits, Devices and Systems, 5th ed. John Wiley & Sons, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 6 Irwin, J.D., 1989. Basic Engineering Circuit Analysis, 3rd ed. Macmillan, New York. Nilsson, J.W., 1989. Electric Circuits, 3rd ed. Addison-Wesley, Reading, MA. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Smith, R.J. and Dorf, R.C., 1992. Circuits, Devices and Systems, 5th ed. John Wiley & Sons, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 7 Horowitz, P. and Hill, W., 1989. The Art of Electronics, 2nd ed. Cambridge University Press, New York. Millman, J. and Grabel, A., 1987. Microelectronics. McGraw-Hill, New York. Sedra, A.S. and Smith, K.C., 1991. Microelectronic Circuits, 3rd ed. W.B. Saunders, Philadelphia. © 2005 by CRC Press LLC

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Neamen, D.A., 1994. Semiconductor Physics and Devices. Richard D. Irwin, Burr Ridge, IL. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Streetman, B.G., 1990. Solid State Electronic Devices, 3rd ed. Prentice-Hall, Englewood Cliffs, NJ. Sze, S.M., 1981. Physics of Semiconductor Devices, 2nd ed. Wiley, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 8 Bose, B.K., 1986. Power Electronics and AC Drives, Prentice-Hall, Englewood Cliffs, NJ. Mohan, N., Undeland, T.M., and Robbins, P., Power Electronics, Van Nostrand, New York. Rashid, M.H., 1988. Power Electronics, Prentice-Hall, Englewood Cliffs, NJ. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 9 Horowitz, P. and Hill, W., 1989. The Art of Electronics, 2nd ed. Cambridge University Press, New York. Millman, J. and Grabel, A., 1987. Microelectronics. McGraw-Hill, New York. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Kennedy, E.J., 1988. Operational Amplifier Circuits, Holt, Rinehart & Winston, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 10 Breeding, K.J., 1992. Digital Design Fundamentals, 2nd ed. Prentice-Hall, Englewood Cliffs, NJ. Horowitz, P. and Hill, W., 1989. The Art of Electronics, 2nd ed., Cambridge University Press, New York. Mano, M.M., 1988. Computer Engineering Hardware Design, Prentice-Hall, Englewood Cliffs, NJ. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Sandige, R.S., 1990. Modern Digital Design. McGraw-Hill, New York. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 11 Doebelin, E.O., 1990. Measurement Systems: Application and Design, 4th ed. McGraw-Hill, New York. Annino, R. and Driver, R., 1986. Scientific and Engineering Applications with Personal Computers. Wiley-Interscience, New York. Horowitz, P. and Hill, W., 1989. The Art of Electronics, 2nd ed., Cambridge University Press, New York. Nachitgal, C. (ed.), 1990. Instrumentation and Control: Fundamentals and Applications. John Wiley & Sons, New York. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. Webster, J.G., 1992. Medical Instrumentation: Application and Design, 2nd ed. Houghton Mifflin, Boston. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

Section 12 Del Toro, V., 1990. Basic Electric Machines. Prentice-Hall, Englewood Cliffs, NJ. Fitzgerald, A.E., Kingsley, C., and Umans, S., 1990. Electric Machinery, 5th ed. McGraw-Hill, New York. © 2005 by CRC Press LLC

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Krause, P. and Wasynczuk, O., 1989. Electromechanical Motion Devices. McGraw-Hill, New York. Rizzoni, G., 1966. Principles and Applications of Electrical Engineering, 2nd ed. Richard D. Irwin, Burr Ridge, IL. 1993. The Electrical Engineering Handbook, CRC Press, Boca Raton, FL.

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6 Mechanical System Controls 6.1

Human–Machine Interaction. Direct Manual Control • Supervisory Control • Advanced Control of Commercial Aircraft • Intelligent Highway Vehicles • High-Speed Train Control • Telerobots for Space, under the Sea, and Medicine • Common Criteria for Human Interface Design • Human Workload and Human Error • Trust, Alienation, and How Far to Go with Automation

6.2

The Need for Control of Mechanical Systems

6.3

Control System Analysis

Classical Control System Representation • Examples

Thomas B. Sheridan Massachussetts Institute of Technology

The Linear Process Approximation • Representation of Processes in t, s, and z Domains

6.4

Peter S. Curtiss

Controllers • PID Controllers • Controller Performance Criteria and Stability • Field Commissioning — Installation, Calibration, Maintenance

Curtiss Engineering

Jan F. Kreider Kreider & Associates

6.5

Iowa State University

Shou-Heng Huang

Advanced Control Topics Neural Network-Based Predictive/Adaptive Controllers • Fuzzy Logic Controllers • Fuzzy Logic Controllers for Mechanical Systems

Ronald M. Nelson

Raytheon Co. Appliance Tech Center

Control System Design and Application

6.6

Control of Distributed Generation Technologies Control Techniques

6.1 Human–Machine Interaction Thomas B. Sheridan Over the years, machines of all kinds have improved and become more reliable. However, machines typically operate as components of larger systems, such as transportation, communication, manufacturing, defense, health care, and so on. Although many aspects of such systems can be and have been automated, the human operator is retained in many cases. This may be because of economics, tradition, cost, or (most likely) capabilities of the human to perceive patterns of information and weigh subtle factors in making control decisions, which the machine cannot match. Although the public as well as those responsible for system operation usually demand a human operator, “human error” is a major reason for system failure. Aside from prevention of error, getting the best performance out of the system means that human and machine must be working together effectively — that they be properly “impedance matched.” Therefore, the performance capabilities of the human relative to those of the machine must be taken into account in system design.

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Chapter 6

FIGURE 6.1.1 Direct manual control (a) and supervisory control (b).

Efforts to “optimize” the human–machine interaction are meaningless in the mathematical sense of optimization because most important interactions between human and machine cannot be reduced to a mathematical form, and the objective function (defining what is good) is not easily obtained in any given context. For this reason, engineering the human–machine interaction, much as in management or medicine, remains an art more than a science based on laboratory experiments and practical experience. In the broadest sense, engineering the human–machine interface includes all of ergonomics or human factors engineering, and goes well beyond design of displays and control devices. Ergonomics includes not only questions of sensory physiology (whether the operator can see the displays or hear the auditory warnings), but also questions of biomechanics — how the body moves and whether the operator can reach and apply proper force to the controls. It further includes the fields of operator selection and training; human performance under stress; human factors in maintenance; and many other aspects of the relation of the human to technology. This section focuses primarily on human–machine interaction in control of systems. The human–machine interactions in control are considered in terms of Figure 6.1.1. In Figure 6.1.1a, the human directly controls the machine, i.e., the control loop to the machine is closed through physical sensors; displays; human senses (visual, auditory, tactile); brain; human muscles; control devices; and machine actuators. Figure 6.1.1b illustrates what has come to be called a supervisory control system, wherein the human intermittently instructs a computer as to goals, constraints, and procedures, then turns a task over to the computer to perform automatic control for some period of time. Displays and control devices can be analogic (movement signal directions and extent of control action; isomorphic with the world, such as an automobile steering wheel or computer mouse controls; or a moving needle or pictorial display element). On the other hand, they can be symbolic (dedicated buttons or general purpose keyboard controls, icons or alarm light displays). In normal human discourse, speech (symbolic) and gestures (analogic) are used and alphanumeric text (symbolic) and draw pictures (analogic) are written on paper. The system designer must decide which types of displays or controls best suit a particular application and/or what mix to use. The designer must be aware of important criteria such as whether, for a proposed design, changes in the displays and controls caused by the human operator correspond in a natural and common-sense way to “more” or “less” of some variable as expected by that operator and correspond to cultural norms (such as reading from left to right in western countries), and whether the movement of the display elements correspond geometrically to movements of the controls.

Direct Manual Control In the 1940s, aircraft designers appreciated the need to characterize the transfer function of the human pilot in terms of a differential equation. Indeed, this is necessary for any vehicle or controlled physical process for which the human is the controller (see Figure 6.1.2). In this case, the human operator H and © 2005 by CRC Press LLC

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FIGURE 6.1.2 Direct manual control loop analysis.

the physical process P lie in the closed loop (where H and P are Laplace transforms of the component transfer functions), and the HP combination determines whether the closed loop is inherently stable (i.e., the closed loop characteristic equation 1 + HP = 0 has only negative real roots). In addition to the stability criterion are the criteria of rapid response of process state x to a desired or reference state r with minimum overshoot, zero steady state error between r and output x, and reduction to near zero of the effects of any disturbance input d. (The latter effects are determined by the closed loop transfer functions x = HP/(1 + HP) r + 1/(1 + HP) d, where, if the magnitude of H is large enough, HP/(1 + HP) approaches unity and 1/(1 + HP) approaches 0. Unhappily, some ingredients of H produce delays in combination with magnitude and thereby can cause instability. Therefore, H must be chosen carefully by the human for any given P.) Research to characterize the pilot in these terms resulted in the discovery that the human adapts to a wide variety of physical processes so as to make HP = K(1/s)(e–sT). In other words, the human adjusts H to make HP constant. The term K is an overall amplitude or gain; (1/s) is the Laplace transform of an integrator; and (e–sT) is a delay T long (the latter time delay is an unavoidable property of the nervous system). Parameters K and T vary modestly in a predictable way as a function of the physical process and the input to the control system. This model is now widely accepted and used, not only in engineering aircraft control systems, but also in designing automobiles, ships, nuclear and chemical plants, and a host of other dynamic systems.

Supervisory Control Supervisory control may be defined by the analogy between a supervisor of subordinate staff in an organization of people, and the human overseer of a modern computer-mediated semiautomatic control system. The supervisor gives human subordinates general instructions that they in turn may translate into action. The supervisor of a computer-controlled system does the same. Defined strictly, supervisory control means that one or more human operators are setting initial conditions for, intermittently adjusting, and receiving high-level information from a computer that closes a control loop in a well-defined process through artificial sensors and effectors. For some time period, the computer controls the process automatically. By a less strict definition “supervisory control” is used when a computer transforms human operator commands to generate detailed control actions, or makes significant transformations of measured data to produce integrated summary displays. In this latter case the computer need not have the capability to commit actions based upon new information from the environment, whereas in the first it necessarily must. The two situations may appear similar to the human supervisor because the computer mediates human outputs and inputs, and the supervisor is thus removed from detailed events at the low level. A supervisory control system is represented in Figure 6.1.3. Here the human operator issues commands to a human-interactive computer capable of understanding high-level language and providing integrated summary displays of process state information back to the operator. Typically located in a control room, cockpit, or office near the supervisor, this computer in turn communicates with at least one, and probably many (thus the dotted lines), task-interactive computers located with the equipment they are controlling. The task-interactive computers thus receive subgoal and conditional branching information from the © 2005 by CRC Press LLC

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Chapter 6

FIGURE 6.1.3 Supervisory control.

human-interactive computer. Using such information as reference inputs, task-interactive computers serve to close low-level control loops between artificial sensors and mechanical actuators, i.e., they accomplish the low-level automatic control. The low-level task typically operates at some physical distance from the human operator and his human-friendly, display-control computer. Therefore, the communication channels between computers may be constrained by multiplexing, time delay, or limited bandwidth. The task-interactive computer, of course, sends analog control signals to and receives analog feedback signals from the controlled process, and the latter does the same with the environment as it operates (vehicles moving relative to air, sea, or Earth; robots manipulating objects; process plants modifying products; etc.). Supervisory command and feedback channels for process state information are shown in Figure 6.1.3 to pass through the left side of the human-interactive computer. On the right side are represented decision-aiding functions, with requests of the computer for advice and displayed output of advice (from a data base, expert system, or simulation) to the operator. Many new developments in computer-based decision aids for planning, editing, monitoring, and failure detection are being used as an auxiliary part of operating dynamic systems. Reflection upon the nervous system of higher animals reveals a similar kind of supervisory control wherein commands are sent from the brain to local ganglia, and peripheral motor control loops are then closed locally through receptors in the muscles, tendons, or skin. The brain, presumably, does higher level planning based on its own stored data and “mental models,” an internalized expert system available to provide advice and permit trial responses before commitment to actual response. Theorizing about supervisory control began as aircraft and spacecraft became partially automated. It became evident that the human operator was being replaced by the computer for direct control responsibility and was moving to a new role of monitor and goal-constraint setter. An added incentive was the U.S. space program, which posed the problem of how a human operator on Earth could control a manipulator arm or vehicle on the Moon through a three-second communication round-trip time delay. The only solution that avoided instability was to make the operator a supervisory controller communicating intermittently with a computer on the moon, which in turn closed the control loop there. The rapid development of microcomputers has forced a transition from manual control to supervisory control in a variety of industrial and military applications (Sheridan, 1992). Now, some examples of human–machine interaction will be considered, particularly those illustrating supervisory control in its various forms. First, three forms of vehicle control will be considered, namely, © 2005 by CRC Press LLC

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FMS STATE

AUTOPILOT STATE

AIRCRAFT STATE

AIRCRAFT TRENDS

ENERGY STATE SYSTEM ANOMALIES

ENERGY RESERVES CLR JFK VIA J-84 M370 STATE OF AUTOMATION SYSTEM INTENT

TRAFFIC SYSTEM TRENDS

CLRNC REVISION WAITING

ATC INTENT

ENVIRONMENTAL THREATS

FIGURE 6.1.4 Pilot information requirements. (From Billings, 1991.)

control of modern aircraft, “intelligent” highway vehicles, and high-speed trains — all of which have human operators in the vehicles as well as humans in centralized traffic control centers. Second, telerobots for space, undersea, and medical applications will be discussed.

Advanced Control of Commercial Aircraft Flight Management Systems Aviation has appreciated the importance of human–machine interaction from its beginning, and today exemplifies the most sophisticated forms of such interaction. Although there have been many good examples of display and control design over the years, the current development of the flight management systems (FMSs) is the epitome. It also provides an excellent example of supervisory control in which the pilot flies the aircraft by communicating in high-level language through a computer intermediary. The FMS is a centralized computer that interacts with a great variety of sensors and communication from the ground, as well as many displays and controls within the aircraft. It embodies many functions and mediates most of the pilot information requirements shown in Figure 6.1.4. Gone are the days when each sensor had its own display, operating independently of all other sensor-display circuits. The FMS, for example, brings together all of the various autopilot modes, from long-standing, low-level control modes wherein the aircraft is commanded to go to and hold a commanded altitude, heading, and speed, to more sophisticated modes in which the aircraft is instructed to fly a given course, consisting of a sequence of waypoints (latitudes and longitudes) at various altitudes, and even land automatically at a given airport on a given runway. Figure 6.1.5 illustrates one type of display mediated by the FMS — in this case integrating many formerly separate components of information. Mostly, it is a multicolor plan-view map showing position and orientation of important objects relative to one’s aircraft (the triangle at the bottom): • • • •

Heading (compass arc at top, present heading 175°) Groundspeed plus wind speed and wind direction (upper left) Actual altitude relative to desired altitude (vertical scale on right side) Programmed course connecting various waypoints (OPH and FLT)

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a% 330 336°/15

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FIGURE 6.1.5 Integrated aircraft map display. (From Billings, 1991.)

• Salient VOR radar beacons to the right and left of present position/direction with their codes and frequencies (lower left and right corners) • Location of key VORs along the course (three-cornered symbols) • Location of weather to be avoided (two gray blobs) • Predicted trajectory based on present turn rate, showing that the right turn is appropriately getting back on course Programming the FMS is done through a specialized keyboard and text display unit (Figure 6.1.6) having all the alphanumeric keys plus a number of special function keys. The displays in this case are specialized to the different phases of a flight (taxi, takeoff, departure, en route approach, land, etc.), each phase with up to three levels of pages. The FMS makes clear that designing displays and controls is no longer a matter of what can be built — the computer allows essentially any conceivable display or control to be realized. The computer can also provide a great deal of real-time advice, especially in emergencies, based on its many sensors and stored knowledge about how the aircraft operates. However, pilots are not sure that they need all the information that aircraft designers would like to give them and have an expression “killing us with kindness” to refer to this plethora of available information. The question is what should be designed based on the needs and capabilities of the pilot. Boeing, McDonnell Douglas, and Airbus have different philosophies for designing the FMS. Airbus has been the most aggressive in automating, intending to make piloting easier and safer for pilots from countries with less well-established pilot training. Unfortunately, of the modern commercial jets, these most automated aircraft have had the most accidents — a fact that has precipitated vigorous debate about how far to automate. Air Traffic Control As demands for air travel continue to increase, so do demands for air traffic control. Given what is currently regarded as safe separation criteria, air space over major urban areas is already saturated, so simply adding more airports is not acceptable (in addition to which residents do not want more airports,

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BRT DIR INTC

NAV RAD

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FIGURE 6.1.6 Flight management system control and display unit. (From Billings, 1991.)

with their noise and surface traffic). The need is to reduce separations in the air, as well as to land aircraft closer together or on parallel runways simultaneously. This puts much greater demands on air traffic controllers, particularly at the terminal area radar control centers (TRACONs); here trained operators stare at blips on radar screens and verbally guide pilots entering the terminal airspace from various directions and altitudes into orderly descent and landing patterns with proper separation between aircraft. Currently, many changes being introduced into air traffic control have profound implications for human–machine interaction. Previously, communication between pilots and air traffic controllers was entirely by voice; now digital communication between aircraft and ground (a system called datalink) allows more, as well as more reliable, two-way communication so that weather and runway and wind information, clearances, etc. can be displayed to pilots visually. However, pilots are not so sure that they want this additional technology. They fear the demise of the “partyline” of voice communications with which they are so familiar that permits all pilots in an area to listen in on other pilots’ conversations. New aircraft-borne radar allows pilots to detect air traffic in their own vicinity. Improved groundbased radar detects microbursts or wind shear, which can easily put an aircraft out of control. Both types of radar pose challenges as to how best to warn the pilot and provide guidance as to how to respond. They also pose a cultural change in air traffic control because heretofore pilots have been dependent upon air traffic controllers to advise them of weather conditions and other air traffic. Furthermore, because of the new weather and collision avoidance technology, current plans call for radically altering the rules whereby high-altitude commercial aircraft must stick to well-defined traffic lanes. Instead, pilots will have great flexibility as to altitude (to find the most favorable winds and therefore save fuel) and be able to take great-circle routes straight to their destinations (also saving fuel). However, air traffic controllers are not sure they want to give up the power they have had and become passive observers and monitors who function only in emergencies.

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Intelligent Highway Vehicles Vehicle Guidance and Navigation Systems The combination of GPS (global positioning system) satellites, high-density computer storage of map data, electronic compass, synthetic speech synthesis, and computer-graphic displays allows cars and trucks to know where they are located on the Earth to within 10 m or less, and can guide a driver to a programmed destination by a combination of a map display and speech. Some human factors challenges are in deciding how to configure the map (how much detail to present; whether to make the map north-up with a moving dot representing one’s own vehicle position; or current heading up and rapidly changing with every turn). The computer graphics can also be used to show the turns to anticipate and which lane to get in. Synthetic speech can reinforce these turn anticipations, caution the driver if he is perceived to be headed in the wrong direction or off-course, and even guide him or her back on course. An interesting question is what the computer should say in each situation to get the driver’s attention, to be understood quickly and unambiguously but without being an annoyance. Another question is whether such systems will distract the driver’s attention from the primary tasks, thereby reducing safety. The major vehicle manufacturers have developed such systems, which have been evaluated for reliability and human use and are beginning to be marketed in the U.S., Europe, and Japan. Smart Cruise Control Standard cruise control has a major deficiency in that it knows nothing about vehicles ahead, and one can easily collide with the rear end of another vehicle if not careful. In a smart cruise control system, a microwave or optical radar detects the presence of a vehicle ahead and measures that distance. The question is what to do with this information. Should the driver be warned with some visual or auditory alarm (auditory is better because the driver need not be looking in the right place)? Can a warning be too late to elicit braking, or surprise the driver so that he brakes too suddenly and causes a rear-end accident to his own vehicle? Should the computer automatically apply the brakes by some function of distance to the obstacle ahead, speed, and closing deceleration? If the computer did all the braking, would the driver become complacent and not pay attention, to the point that a serious accident would occur if the radar failed to detect an obstacle, say a pedestrian or bicycle, or the computer failed to brake? Should braking be some combination of human and computer braking, and if so by what algorithm? These are human factors questions currently being researched. It is interesting to note that current developmental systems only decelerate and downshift, mostly because if the vehicle manufacturers sell vehicles that claim to perform braking, they would be open to a new and worrisome area of litigation. The same radar technology that can warn the driver or help control the vehicle can also be applied to cars overtaking from one side or the other. Another set of questions then arises as to how and what to communicate to the driver and whether to trigger some automatic control maneuver in certain cases. Advanced Traffic Management Systems Automobile congestion in major cities has become unacceptable, and advanced traffic management systems are being built in many of these cities to measure traffic flow at intersections (by some combination of magnetic loop detectors, optical sensors, and other means) and regulate stop lights and message signs. These systems can also issue advisories of accidents ahead by means of variable message signs or radio and give advice of alternate routings. In emergencies, they can dispatch fire, police, ambulances, or tow trucks, and in the case of tunnels, can shut down entering traffic completely if necessary. These systems are operated by a combination of computers and humans from centralized control rooms. The operators look at banks of video monitors, which let them see the traffic flow at different locations, and computer-graphic displays of maps, alarm windows, and textual messages. The operators get advice from computer-based “expert systems” that suggest best responses based on measured inputs, and the operator must decide whether to accept the computer’s advice, whether to seek further information, and how to respond. © 2005 by CRC Press LLC

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PipeP

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FIGURE 6.1.7 Prototype of computer-generated display for high-speed trains. (From Askey, 1995.)

High-Speed Train Control With respect to new electronic technology for information sensing, storage, and processing, railroad technology has lagged behind that of aircraft and highway vehicles, but currently is catching up. The role of the human operator in future rail systems is being debated because, for some limited right-of-way trains (e.g., in airports), one can argue that fully automatic control systems now perform safely and efficiently. The train driver’s principal job is speed control (although he must perform many other monitoring duties); in a train this task is much more difficult than in an automobile because of the huge inertia of the train — it takes 2 to 3 km to stop a high-speed train. Speed limits are fixed at reduced levels for curves, bridges, grade-crossings, and densely populated areas; while wayside signals temporarily command lower speeds if maintenance is being performed on the track; there are poor environmental conditions such as rock slides or deep snow are present; or, especially, if another train is ahead. The driver must obey all speed limits and reach the next station on time. Learning to maneuver the train with its long time constants can take months, given that the driver’s only input currently for the speed control task is an indication of current speed. The authors’ laboratory has proposed a new computer-based display that helps the driver anticipate the future effects of current throttle and brake actions. Based on a dynamic model of the train, this approach gives an instantaneous prediction of future train position and speed based on current acceleration, so speed can be plotted on the display, assuming the operator holds to current brake–throttle settings. It also plots trajectories for maximum emergency braking and maximum service braking. In addition, the computer generates a speed trajectory that adheres to all (known) future speed limits, gets the train to the next station on time, and minimizes fuel/energy. Figure 6.1.7 shows the laboratory version of this display, which is currently being evaluated.

Telerobots for Space, under the Sea, and Medicine When nuclear power was first adopted in the late 1940s, engineers began the development of masterslave remote manipulators, by which a human operator at one location could position and orient a device attached to his hand, and a servomechanism-controlled gripper would move in correspondence and © 2005 by CRC Press LLC

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FIGURE 6.1.8 Flight telerobotic servicer prototype design. (Courtesy of NASA.)

handle objects at another location. At about the same time, remotely controlled wheeled vehicles, submarines, and aircraft began to be developed. Such manipulators and vehicles remotely controlled by humans are called teleoperators. Teleoperator technology got a big boost from industrial robot technology, which came in a decade or so later and provided improved vision, force, and touch sensors, actuators, and control software. Large teleoperators were developed for rugged mining and undersea tasks and small teleoperators for delicate tasks such as eye surgery. Eventually, teleoperators were equipped with sensitive force feedback, so that the human operator can see the objects in the remote environment and also feel them in his grasp. Supervisory controlled teleoperators were developed because, during the time of the Apollo flights to the Moon, the desire to control lunar manipulators and vehicles from Earth was stimulated and unavoidable round trip time delays of 3 sec (speed of light from Earth to Moon and back) would not permit simple closed loop control. The human could communicate a subgoal to be reached and a procedure for getting there, and the teleoperator would be turned loose for some short period to perform automatically. Such a teleoperator is called a telerobot. Figure 6.1.8 shows the flight telerobotic servicer (FTS) developed by Martin Marietta for the U.S. Space Station Freedom. It has two seven-degree of freedom (DOF) arms (including gripper) and one five DOF “leg” for stabilizing itself while the arms work. It has two video “eyes” to present a stereo image to its human operator. It can be configured as a master–slave teleoperator (under direct human control) or as a telerobot (able to execute small programmed tasks using its own eyes and force sensors). Unfortunately the FTS project was canceled by Congress. Figure 6.1.9 shows the remotely operated submersible Jason developed by Woods Hole Oceanographic Institution. It is the “big brother” of Jason Junior, which swam into the interior of the ship Titanic and made a widely viewed video record when the latter was first discovered. It has a single manipulator arm, sonar and photosensors, and four thrusters that can be oriented within limited range and enable it to move in any direction. It is designed for depths up to 6000 m — rather severe pressures. It, too, can be operated in direct teleoperator mode or as a telerobot.

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FIGURE 6.1.9 Deep ocean submersible Jason. (Courtesy of Woods Hole Oceanographic Institution.)

Common Criteria for Human Interface Design Design of operator control stations for teleoperators poses the same types of problems as design of controls and displays for aircraft, highway vehicles, and trains. The displays must show the important variables unambiguously to whatever accuracy is required, but more than that must show the variables in relation to one another so as to portray the current “situation” clearly (situation awareness is currently a popular test of the human operator in complex systems). Alarms must get the operator’s attention; indicate what is abnormal by text, symbol, or location on a graphic display; determine where in the system the failure occurred, what the urgency is, and if response is urgent; and even suggest what action to take. (For example, the ground proximity warning in an aircraft gives a loud “Whoop, whoop!” followed by a distinct spoken command, “Pull up, pull up!”) Controls — whether analogic joysticks, master-arms, or knobs — or symbolic special-purpose buttons or general purpose keyboards must be natural and easy to use and require little memory of special procedures (computer icons and windows do well here). The placement of controls and instruments and their mode and direction of operation must correspond to the desired direction and magnitude of system response.

Human Workload and Human Error As noted previously, new technology allows combination, integration, and simplification of displays compared to the intolerable plethora of separate instruments in older aircraft cockpits and plant control rooms. The computer has taken over more and more functions from the human operator. These changes potentially make the operator’s task easier. However, they also allow for much more information to be presented, more extensive advice to be given, etc. These advances have elevated the stature of the human operator from that of providing physical energy and control, to that of providing only continuous control, to finally serving as a supervisor or a robotic vehicle or system. “Expert systems” can now answer the operator’s questions, much as does a human consultant, or whisper suggestions in his ear even if he does not request them. These changes seem to add many cognitive functions that were not present at an earlier time. They make the operator into a monitor of the automation who is supposed to step in when required to set things straight. Unfortunately, people are not always reliable monitors and interveners.

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Mental Workload Under such complexity, it is imperative to know whether the mental workload of the operator is too great for safety. Human–machine systems engineers have sought to develop measures of mental workload, the idea being that as mental load increases the risk of error increases; however, presumably measurable mental load comes before actual lapse into error. Three approaches have been developed for measuring mental workload: 1. Subjective rating scale, typically a 10-level category scale with descriptors for each category from no load to unbearable load 2. Use of physiological indices that correlate with subjective scales, including heart rate and the variability of heart rate; certain changes in the frequency spectrum of the voice; electrical resistance of the skin; diameter of the pupil of the eye; and certain changes in the evoked brain wave response to sudden sound or light stimuli 3. Use of what is called a secondary task, an easily measurable additional task that consumes all of the operator’s attention remaining after the requirements of the primary task are satisfied The latter technique has been used successfully in the laboratory, but has shortcomings in practice in that operators may refuse to cooperate. Such techniques are now routinely applied to critical tasks such as aircraft landing, air traffic control, certain planned tasks for astronauts, and emergency procedures in nuclear power plants. The evidence suggests that supervisory control relieves mental load when things are going normally, but when automation fails, the human operator is subjected to a rapidly increased mental load. Human Error Human error has long been of interest, but only in recent decades has a serious effort been made to understand human error in terms of categories, causation, and remedy. Human errors can be classified in several ways. One is according to whether it is an error of omission (something not done that was supposed to have been done) or commission (something done that was not supposed to have been done). Another is slip (a correct intention for some reason not fulfilled) vs. a mistake (an incorrect intention that was fulfilled). Errors may also be classified according to whether they are in sensing, perceiving, remembering, deciding, or acting. Some special categories of error worth noting are associated with following procedures in operation of systems. One, for example, is called a capture error, wherein the operator, who is very accustomed to a series of steps, say A, B, C, and D, intends at another time to perform E, B, C, F. However, he is “captured” by the familiar sequence B, C and does E, B, C, D. As to effective therapies for human error, proper design to make operation easy and natural and unambiguous is surely the most important. If possible, the system design should allow for error correction before the consequences become serious. Active warnings and alarms are necessary when the system can detect incipient failures in time to take such corrective action. Training is probably next most important after design, but any amount of training cannot compensate for an error-prone design. Preventing exposure to error by guards, locks, or an additional “execute” step can help make sure that the most critical actions are not taken without sufficient forethought. Least effective are written warnings such as posted decals or warning statements in instruction manuals, although many tort lawyers would say the opposite.

Trust, Alienation, and How Far to Go with Automation Trust If an operator does not trust his sensors and displays, expert advisory system, or automatic control system, he will not use them or will avoid using them, if possible. On the other hand, if an operator comes to place too much trust in such systems, he will let down his guard, become complacent, and, when it fails, not be prepared. The question of operator trust in automation is an important current issue in human–machine © 2005 by CRC Press LLC

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interface design. It is desirable that operators trust their systems, but it is also desirable that they be alert, aware of the situation, and ready to take over. Alienation The new human–machine interaction can have a set of broader social effects that can be discussed under the rubric of alienation: 1. People worry that computers can do some tasks, such as memory and calculation, much better than they can. Surely, people should not try to compete in this arena. 2. Supervisory control tends to make people remote from the ultimate operations they are supposed to be overseeing — remote in space, desynchronized in time, and interacting with a computer instead of the end product or service. 3. People lose the perceptual–motor skills that, in many cases, gave them their identity. They become “deskilled” and, if ever called upon to use their previous well-honed skills, they cannot. 4. Increasingly, people who use computers in supervisory control or in other ways, whether intentionally or not, are denied access to the knowledge to understand what is going on inside the computer. 5. Partly as a result of the preceding factor, the computer becomes mysterious, and the untutored user comes to attribute more capability, wisdom, or blame to the computer than is appropriate. 6. Because computer-based systems are growing more complex and people are being “elevated” to roles of supervising larger and larger aggregates of hardware and software, the stakes naturally become higher. A human error before might have gone unnoticed and been easily corrected; now such an error could precipitate a disaster. 7. The last factor in alienation is similar to the first, but all encompassing — namely, the fear that a “race” of machines is becoming more powerful than the human race. The preceding seven factors and the fears that they engender, whether justified or not, must be reckoned with. Computers must be made not only “human friendly” but also not alienating with respect to these broader factors. Operators and users must become computer literate at the level of sophistication with which they can deal. How Far to Go with Automation The trend toward supervisory control is definitely changing the role of the human operator, posing fewer requirements on continuous sensory–motor skill and more on planning, monitoring, and supervising the computer. As computers take over more sensory–motor skill functions, new questions are raised regarding how the interface should be designed to provide the best cooperation between human and machine. Among these questions are: To what degree should the system be automated? How much “help” from the computer is desirable? What are the points of diminishing returns? Table 6.1.1 lists 10 levels of automation, from 0 to 100% computer control. Obviously, few tasks have achieved 100% computer control; however, new technology pushes relentlessly in that direction. It is TABLE 6.1.1 Scale of Degrees of Automation 1 2 3 4 5 6 7 8 9 10

The computer offers no assistance; the human must do it all The computer offers a complete set of action alternatives, and Narrows the selection down to a few, or Suggests one alternative, and Executes that suggestion if the human approves, or Allows the human a restricted time to veto before automatic execution, or Executes automatically, then necessarily informs the human, or Informs the human only if asked, or Informs the human only if it, the computer, decides to do so The computer decides everything and acts autonomously, ignoring the human

Source: From Sheridan, 1987. © 2005 by CRC Press LLC

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instructive to consider the various intermediate levels of Table 6.1.1 not only in terms of how capable and reliable the technology is, but also in terms of what is desirable in terms of the human operators’ and general public’s safety and satisfaction. The current controversy about how much to automate large commercial transport aircraft is often couched in these terms.

6.2 The Need for Control of Mechanical Systems Peter S. Curtiss Process control typically involves some mechanical system that needs to be operated in such a fashion that the output of the system remains within its design operating range. The objective of a process control loop is to maintain the process at the setpoint under the following dynamic conditions: • The setpoint is changed. • The load on the process is changed. • The transfer function of the process is changed or a disturbance is introduced.

Classical Control System Representation Feedback-Loop System A feedback (or closed-loop) system contains a process, a sensor, and a controller. Figure 6.2.1 shows some of the components and terms used when discussing feedback loop systems. • Process. A process is a system that produces a motion, temperature change, flow, pressure, or many other actions as a function of the actuator position and external inputs. The output of the process is called the process value. If a positive action in the actuator causes an increase in the process value, the process is called direct acting. If positive action in the actuator decreases the process value, it is called reverse acting. • Sensor. A sensor is a pneumatic, fluidic, electronic, or other device that produces some kind of signal indicative of the process value. • Setpoint. The setpoint is the desired value for a process output. The difference between the setpoint and the process value is called the process error. • Controller. A controller sends signals to an actuator to effect changes in a process. The controller compares the setpoint and the process value to determine the process error. It then uses this error to adjust the output and bring the process back to the setpoint. The controller gain dictates the amount that the controller adjusts its output for a given error. • Actuator. An actuator is a pneumatic, fluidic, electric, or other device that performs any physical action that will control a process. • External disturbances. An external disturbance is any effect that is unmeasured or unaccounted for by the controller.

FIGURE 6.2.1 Typical feedback control schematic diagram.

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• Time constants. The time constant of a sensor or process is a quantity that describes the dynamic response of the device or system. Often the time constant is related to the mass of an object or other dynamic effect in the process. For example, a temperature sensor may have a protective sheath around it that must first be warmed before the sensor registers a change of temperature. Time constants can range from seconds to hours. • Dead time. The dead time or lag time of a process is the time between the change of a process and the time this change arrives at the sensor. The delay time is not related to the time constant of the sensor, although the effects of the two are similar. Large dead times must be properly treated by the control system to prevent unstable control. • Hysteresis. Hysteresis is a characteristic response of positioning actuators that results in different positions, depending on whether the control signal is increasing or decreasing. • Dead band. The dead band of a process is that range of the process value in which no control action is taken. A dead band is usually used in two-position control to prevent “chattering” or in split-range systems to prevent sequential control loops from fighting each other. • Control point. The control point is the actual, measured value of a process (i.e., the setpoint + steady-state offset + compensation). • Direct/reverse action. A direct-acting process will increase in value as the signal from the controller increases. A reverse-acting process will decrease in value as the signal from the controller increases. • Stability. The stability of a feedback control loop is an indication of how well the process is controlled or, alternatively, how controllable the process is. The stability is determined by any number of criteria, including overshoot, settling time, correction of deviations due to external disturbances, etc. • Electric control. Electric control is a method of using low voltages (typically 24 VAC) or line voltages (110 VAC) to measure values and effect changes in controlled variables. • Electronic control. Electronic controls use solid-state electronic components that are used for measurement and amplification of measured signals and the generation of proportional control signals. • Pneumatic control. Pneumatic controls use compressed air as the medium for measuring and controlling processes. • Open-loop systems. An open-loop system is one in which no feedback occurs, e.g., a whole-house attic fan. It will continue to run even though the house may have already cooled off. Also, timed on/off devices are open loops.

Examples Direct-Acting Feedback Control A classic control example is a reservoir in which the fluid must be maintained at a constant level. Figure 6.2.2 shows this process schematically; the key features of this direct-acting system are labeled. The control action of this system will be referred to shortly after defining some terms are defined: • Cascaded (master–slave) control loops. If a process consists of several subprocesses, each with a relatively different transfer function, it is often useful to use cascaded control loops. For example, consider a building housing a manufacturing line in which 100% outside air is used that must also have very strict control of room air temperature. The room temperature is controlled by changing the position of a valve on a coil at the main air-handling unit that supplies the zone. Typically, the time constant of the coil will be much smaller than the time constant of the room. A single feedback loop would probably result in poor control because both processes involve so much dead time. The solution is to use two controllers: the first (the master) compares the room temperature to the thermostat setting and sends a signal to the second (the slave), which uses that signal as its own setpoint for controlling the coil valve. The slave controller measures the output

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Chapter 6

FIGURE 6.2.2 Example of a controlled process.

FIGURE 6.2.3 Example of the effect of compensation control.

•

•

•

• •

of the coil, not the temperature of the room. The controller gain on the master can be set lower than that of the slave to prevent excessive cycling. Sequential control loops. Sometimes control action is needed at more than one point in a process. An example of this is an air-handling unit that contains heating and cooling coils in order to maintain a fixed outlet air temperature in any season. Typically, a sequential (or split-range) system in an air-handling unit will have three temperature ranges of operation: the first for heating mode; the last for cooling mode; and a middle dead-band region in which neither the cooling nor heating coils are operating. Most sequential loops are simply two different control loops acting from the same sensor. The term sequential refers to the fact that in most of these systems the components are in series in the air or water stream. Combined feed-forward/feedback loops. As pointed out earlier, feed-forward loops can be used when the effects of an external disturbance on a system are known. An example of this is outside air temperature reset control used to modify supply air temperatures. The control loop contains a discharge air temperature sensor (the primary sensor) and an outdoor air temperature sensor (the compensation sensor). The designer should have some idea about the influence of the outside temperature on the heating load and can then assign an authority to the effect of the outside air temperature on the controller setpoint. As the outdoor temperature increases, the control point decreases and vice versa (see Figure 6.2.3). Predictive control. Predictive control uses a model of the process to predict what the process value will be at some point in the future based upon current and past conditions. The controller then specifies a control action to be taken in the present that will reduce future process errors. Adaptive control. Adaptive controllers modify their gains dynamically in order to adapt to current process conditions. Supervisory controllers. Supervisory controllers are used to govern the operation of an entire plant and/or control system. These may be referred to as distributed control systems (DCSs), which can be used to govern the control of individual feedback loops and can also be used to ensure some

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Mechanical System Controls

FIGURE 6.2.4 Typical supervisory controller.

kind of optimal performance of the entire plant. The controller will vary setpoints and operating modes in an attempt to minimize a cost function. Figure 6.2.4 shows a basic diagram of a supervisory controller.

6.3 Control System Analysis Peter S. Curtiss The Linear Process Approximation To design controllers, it is necessary to have a dynamic process as well as control system representation. This section describes the key points of the most common such representation — that of linear processes and their controls. A process is basically a collection of mechanical equipment in which an input is changed or transformed somehow to produce an output. Many processes will be near steady state, while others may be in a more or less constant state of change. Building control systems are used as an illustration. Steady-State Operation The true response of a seemingly simple process can be, in fact, quite complex. It is very difficult to identify and quantify every single input due to the stochastic nature of life. However, practically any process can be approximated by an equation that takes into account the known input variables and produces a reasonable likeness to the actual process output. It is convenient to use differential equations to describe the behavior of processes. For this reason, the “complexity” of the function will be denoted by the number of terms in the corresponding differential equation (i.e., the order or degree of the differential equation). In a linear system analysis, a step change in the control signal is usually considered and the response observed. The following descriptions will assume a step input to the function, as shown in Figure 6.3.1 Note that a step change such as this is usually unlikely in most fields of control outside of electronic systems and even then can only be applied to a digital event such as a power supply being switched on or a relay being energized. Zero-order system output has a one-to-one correspondence to the input: y(t ) = a0 ⋅ u(t )

FIGURE 6.3.1 Step change in control signal. © 2005 by CRC Press LLC

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FIGURE 6.3.2 Effective dead time of a process subjected to a step change in controlled signal.

FIGURE 6.3.3 Connected water containers used for example of dynamic response.

First-order functions will produce a time-varying output with a step-change as input: dy(t ) + a1 ⋅ y(t ) = b1 ⋅ u(t ) dt and higher order functions will produce more complex outputs. The function that relates the process value to the controller input is called the transfer function of the process. The time between the application of the step change, t0, and the time at which the full extent of the change in the process value has been achieved is called the transfer period. A related phenomenon is process dead time. If sufficient physical distance is present between the process output and the sensor assigned to measuring it, then one observes dead time during which the process output is not affected by the control signal (as in Figure 6.3.2). The process gain (or static gain) is the ratio of the percentage change of the process output to the corresponding percentage change of the control signal for a given response. For example, gain can be positive (as in a heating coil) or negative (as in a cooling coil). Dynamic Response In practice, very few processes are controlled in a step-wise fashion. Usually, the control signal is constantly modulating much the way that one makes small changes to the steering wheel of a car when driving down the highway. The dynamic process of level control in buckets filled with water is now considered. (See Figure 6.2.3.) Imagine that the level of water in a bucket on the left is the control signal and the level of water in a bucket on the right is the process value. It is obvious that a step change in the control signal will bring about a first-order response of the process value. Suppose, however, that a periodic signal is applied to the level of the bucket on the left. If the frequency of the signal is small enough, a response in the level in the bucket on the right varies as a function of this driving force, but with a delay and a decrease in the amplitude. Here the dynamic process gain is less than one even though the static process gain is 1. This process has no dead time; as soon as the control signal begins to increase, the process value will also begin to increase. The dynamic process gain, therefore, can be defined similarly to that of the static gain — it is the ratio of the amplitude of the two signals, comparable to the normalized ranges used in the static gain definition. © 2005 by CRC Press LLC

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FIGURE 6.3.4 Generalized feedback loop.

The dynamic gain, as its name suggests, is truly dynamic. It will change according to the transfer function as well as to the frequency of the control signal. As the frequency increases, the output will lag even farther behind the input and the gain will continue to decrease. At one point, the frequency may be exactly right to cancel any past effects of the input signal (i.e., the phase shift is 180°) and the dynamic gain will approach zero. If the frequency rises further, the process output may decrease as the control signal increases (easily the case with a building cooling or heating coil due to the mass effects) and the dynamic gain will be negative. At this point, it is convenient to define a feedback loop mathematically. A general feedback loop is shown in Figure 6.3.4. The controller, actuator, and process have been combined into the forward transfer function (or openloop transfer function) G and the sensor and dead time have been combined into the feedback path transfer function H. The overall closed-loop transfer function is defined as C G = R 1+ G ⋅H The right-hand side of this equation is usually a ratio of two polynomials when using Laplace or z transforms. The roots of the numerator are called the zeros of the transfer function and the roots of the denominator are called the poles (Shinners, 1978). The denominator of the closed loop transfer function, 1 + G·H, is called the characteristic function. When the characteristic function is set equal to zero, it yields the characteristic equation 1 + G·H = 0 The characteristic equation can be used to assess process control stability during system design.

Representation of Processes in t, s, and z Domains How a process truly behaves can never be known. The world is an inherently stochastic place and any model of a system is going to approximate at best. Nonetheless, it is necessary to choose some kind of representation in order to perform any useful analysis. This section will consider three different domains: the continuous time domain; the frequency domain; and the discrete time domain. The frequency domain is useful for certain aspects of controller design and the discrete time domain is used in digital controllers. Continuous Time Domain Representation of a Process In the time domain, a process is represented by a differential equation such as: dn y d n−1 y d n− 2 y dy d mu d m−1u du a a a + + + ... + + = + + ... + bm−1 + bmu a y b b n 1 n − 0 1 2 1 dt dt dt n dt n−1 dt n− 2 dt m dt m−1 This is just a generalization of the first-order system equation described earlier. Frequency Domain Representation of a Process — Laplace Transforms The solution of higher order system models, closed form solution, is difficult in the time domain. For this reason, process transfer functions are often written using Laplace transforms. A Laplace transform is a mapping of a continuous time function to the frequency domain and is defined as © 2005 by CRC Press LLC

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Chapter 6

F (s) =

∫

∞

f (t )e − st dt

0

This formulation allows one to greatly simplify problems involving ordinary differential equations that describe the behavior of systems. A transformed differential equation becomes purely algebraic and can be easily manipulated and solved. These solutions are not of great interest in modern control system design, but the transformed system + controller differential equation is very useful in assessing control stability. This is the single key aspect of Laplace transforms that is of most interest. Of course, it is possible to solve only the governing differential equation for the system directly and explore stability in that fashion. The Laplace transform of the previous differential equation is sn Y(s) + A1 sn–1 Y(s) +… + An–1 sY(s) + AnY(s) = B0 sm U(s) + B1 sn–1 U(s) +… + Bn–1 s U(s) + BnU(s) This equation can be rewritten as Y(s)·(sn + A1 sn–1 +… + An–1 s + An) = U(s)·(B0 sm + B1 sn–1 +… + Bn–1 s + Bn) so that the transfer function is found from Y (s) s m + B1s m−1 + ... + Bm−1s + Am = U (s) s n + A1s n−1 + ... + An−1s + An This is the expression used for stability studies. Discrete Time Domain Representation of a Process A process in the discrete time domain is described (Radke and Isermann, 1989) by y(k) = a1 y(k − 1) + a2 y(k − 2) + a3 y(k − 3) + ... +b1u(k − 1)) + b2u(k − 2) + b3u(k − 3) + ... This representation is of use when one is designing and analyzing the performance of direct digital control (DDC) systems. Note that the vectors a and b are not the same as for the continuous time domain equation. The z-transform uses the backward shift operator and therefore the z-transform of the discrete time equation is given by

(

) (

y 1 − a1z −1 − a2 z −2 − a3 z −3 + ... = u b1z −1 − b2 z −2 − b3 z −3 + ...

)

The transfer function can now be found: y b z −1 − b z −2 − b z −3 + ... = 1 −1 2 −2 3 −3 u 1 − a1z − a2 z − a3 z + ... z-Transform Details Because z-transforms are important in modern control design and are not treated elsewhere in this handbook, some basics of their use are given next. More and more control applications are being turned over to computers and DDC systems. In such systems, the sampling is not continuous as required for a Laplace transform. The control loop schematic is shown in Figure 6.3.5. © 2005 by CRC Press LLC

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Mechanical System Controls

FIGURE 6.3.5 Sampled feedback loop.

It would be prohibitively expensive to include a voltmeter or ohmmeter on each loop; therefore, the controller employs what is called a zero-order hold. This basically means that the value read by the controller is “latched” until the next value is read in. This discrete view of the world precludes the use of Laplace transforms for analyses, thus making it necessary to find some other means of simplifying the simulation of processes and controllers. The following indicates briefly how z-transforms of controlled processes can be derived and how they are used in a controls application. In the design section of this chapter, the z-transform will be used to assess controller stability. Recall that the Laplace transform is given as ∞

{ } ∫ f ( t )e


f (t ) =

− st

dt

0

Now suppose a process that is sampled at a discrete, constant time interval T. The index k will be used to count the intervals At time t = 0, k = 0, At time t = T, k = 1, At time t = 2T, k = 2, At time t = 3T, k = 3, and so forth. The equivalent Laplace transform of a process that is sampled at a constant interval T can be represented as

{

∞

} ∑ f (kT )e


f ∗ (t ) =

− s ·kT

k =0

Substituting the backward-shift operator z for eTs results in the definition of the z-transform: Z { f (t )} =

∞

∑ f (kT )z

−k

k =0

6.4 Control System Design and Application Peter S. Curtiss Controllers Controllers are akin to processes in that they have gains and transfer functions. Generally, there is no dead time in a controller or it is so small as to be negligible. Steady-State Effects of Controller Gain Recall that the process static gain can be viewed as the total change in the process value due to a 100% change in the controller output. A proportional controller acts like a multiplier between an error signal © 2005 by CRC Press LLC

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Chapter 6

and this process gain. Under stable conditions, therefore, there must be some kind of error to yield any controller output. This is called the steady-state or static offset. Dynamic Effects of Controller Gain Ideally, a controller gain value is chosen that compensates for the dynamic gain of the process under normal operating conditions. The total loop dynamic gain can be considered as the product of the process, feedback, and controller gains. If the total dynamic loop gain is one, the process will oscillate continuously at the natural frequency of the loop with no change in amplitude of the process value. If the loop gain is greater than one, the amplitude will increase with each cycle until the limits of the controller or process are reached or until something fails. If the dynamic loop gain is less than one, the process will eventually settle down to stable control. Controller Bias The controller bias is a constant offset applied to the controller output. It is the output of the controller if the error is zero: u = K ⋅e + M where M is the bias. This is useful for processes that become nonlinear at the extremes or for processes in which the normal operating conditions are at a nonzero controller output.

PID Controllers Many mechanical systems are controlled by proportional-integral-derivative (PID) controllers. Many permutations of such controllers use only certain portions of the PID controllers or use variations of this kind of controller. This subsection considers this very common type of controller. Proportional Control Proportional control results in action that is linear with the error. The proportional term, Kp ⋅e, has the greatest effect when the process value is far from the desired setpoint. However, very large values of Kp will tend to force the system into oscillatory response. The proportional gain effect of the controller goes to zero as the process approaches setpoint. Purely proportional control should therefore only be used when • The time constant of the process is small and thus a large controller gain can be used. • The process load changes are relatively small so that the steady-state offset is limited. • The steady-state offset is within an acceptable range. Integral Control Integral makes a process adjustment based on the cumulative error, not its current value. The integral term Ki is the reciprocal of the reset time, Tr , of the system. The reset time is the duration of each error summing cycle. Integral control can cancel any steady-state offsets that would occur when using purely proportional control. This is sometimes called reset control. Derivative Control Derivative makes a process adjustment based on the current rate of change of the process control error. Derivative control is typically used in cases in which a large time lag occurs between the controlled device and the sensor used for the feedback. This term has the overall effect of preventing the actuator signal from going too far in one direction or another and can be used to limit excessive overshoot. PID Controller in Time Domain The PID controller can be represented in a variety of ways. In the time domain, the output of the controller is given by

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Mechanical System Controls

t  de(t )   u(t ) = K p e(t ) + K i e(t ) dt + K d dt   0  

∫

PID Controller in the s Domain It is relatively straightforward to derive the Laplace transform of the time domain PID equation. The transfer function of the controller is K pKi  U (s)  = K p + + K pKds  E(s)  s  This controller transfer function can be multiplied by the process transfer function to yield the overall forward transfer function G of an s-domain process model. The criteria described earlier can then be used to assess overall system stability. PID Controller in the z Domain Process data are measured discretely at time intervals ∆t, and the associated PID controller can be represented by  u(k) = K p e(k) + K i ∆t 

k

∑e(i) + K i =0

d

e(k) − e(k − 1)   ∆t 

The change of the output from one time step to the next is given by u(k) – u(k – 1), so the PID difference equation is:  K    K  K  u(k) − u(k − 1) = K p  1 + d  e(k) +  K i ∆t − 1 − 2 d  e(k − 1) +  d  e(k − 2)       t t ∆ ∆ ∆ t   and can be simplified as u(k) − u(k − 1) = q0e(k) + q1e(k − 1) + q2e(k − 2) where  K   K  K  q0 = K p  1 + d  ; q1 = K p  K i ∆t − 1 − 2 d  ; q2 = K p  d   ∆t    ∆t  ∆t  Note that this can be written as

(

) (

u 1 − z −1 = e q0 + q1z −1 + q2 z −2

)

The z-domain transfer function of the PID controller is then given as u(z ) q0 + q1z −1 + q2 z −2 q0 z 2 + q1z + q2 = = e(z ) z2 − z 1 − z −1

© 2005 by CRC Press LLC

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Chapter 6

Controller Performance Criteria and Stability Performance Indices Obviously, in feedback loops one wishes to reduce the process error quickly and stably. The control systems engineer can use different cost functions in the design of a given controller depending on the criteria for the controlled process. Some of these cost functions (or performance indices) are listed here:

ISE

Integral of the square of the error

∫e

ITSE

Integral of the time and the square of the error

∫ te

ISTAE

Integral of the square of the time and the absolute error

∫t

ISTSE

Integral of the square of the time and the square of the error

∫t e

2

2

2

e

2 2

These indices are readily calculated in with DDC systems and can be used to compare the effects of different controller settings, gains, and even control methods. Stability Stability in a feedback loop means that the feedback loop will tend to converge on a value as opposed to exhibiting steady-state oscillations or divergence. Recall that the closed loop transfer function is given by C G = R 1 + GH and that the denominator, 1 + GH, when equated to zero, is called the characteristic equation. Typically, this equation will be a polynomial in s or z depending on the method of analysis of the feedback loop. Two necessary conditions for stability are that (1) all powers of s must be present in the characteristic equation from zero to the highest order; and (2) all coefficients in the characteristic equation must have the same sign. Note that the process may still be unstable even when these conditions are satisfied. Roots of the Characteristic Equation The roots of the characteristic equation play an important role in determining the stability of a process. These roots can be real and/or imaginary and can be plotted as shown in Figure 6.4.1. In the s-domain, if all the roots are in the left half-plane (i.e., to the left of the imaginary axis), then the feedback loop is guaranteed to be asymptotically stable and will converge to a single output value. If one or more roots are in the right half-plane, then the process is unstable. If one or more roots lie on the imaginary axis and none are in the right half-plane, then the process is considered to be marginally stable.

FIGURE 6.4.1 Placement of roots in the imaginary plane (showing unit circle). © 2005 by CRC Press LLC

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Mechanical System Controls

FIGURE 6.4.2 Simple feedback control loop.

FIGURE 6.4.3 Root locus of s2 + (1.25 + K)s + 1.25 = 0.

In the z-domain, if all the roots lie within the unit circle about the origin, then the feedback loop is asymptotically stable and will converge. If one or more roots lie outside the unit circle, then the process is unstable. If one or more roots lie on the unit circle and none are outside the unit circle, then the process is marginally stable. Root locus example: Consider the feedback loop shown in Figure 6.4.2. The characteristic equation is given by 1 + GH = 0, or  s  1  1+ K  =0  s + α   s + β  For different values of K the roots of this equation can be plotted. The graph in Figure 6.4.3 shows an example plot when the characteristic equation is given by s2 + (1.25 + K)s + 1.25 = 0. The plot shows that a system described by this characteristic demonstrates stable response for a process gain of 0.0 ≤ K ≤ 10.0. For gains greater than 10, at least one root exists in the right half-plane and the process is not under stable control. Note that the root locus plot is always symmetric about the real axis and that the number of separate segments of the locus is equal to the number of roots of the characteristic equation (i.e., the number of poles of the closed-loop transfer function). © 2005 by CRC Press LLC

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Routh–Hurwitz Stability Criteria The Routh–Hurwitz method is an tabular manipulation of the characteristic equation in the frequency domain and is used to assess stability. If the characteristic equation is given by a0 s n + a1s n−1 + ... + an−1s + an = 0 then the Routh–Hurwitz method constructs a table from the coefficients as follows: sn s n−1 s n− 2 s n− 3 

a0 a1 X1 Y1 

a2 a3 X2 Y2 








a4 a5 X3 Y3 

where X1 =

a1a2 − a0a3 ; a1

X2 =

Y1 =

X1a3 − a1 X 2 ; X1

Y2 =

a1a4 − a0a5 ; a1

X3 =

a1a6 − a0a7
a1

X1a5 − a1 X 3
X1

and so forth. The number of roots in the right-hand plane of the s-domain is equal to the number of sign changes in the first column, i.e., the column containing a0, a1, X1, Y1, etc. In other words, if all the elements in the first column have the same sign, then no roots are in the right-hand plane and the process is stably controlled. Also, for special cases of the characteristic equation: • If the first element of any row is zero but the remaining elements are not, then use some small value ε and interpret the final results as ε → 0. • If one of the rows before the final row is entirely zeros, then (1) there is at least one pair of real roots of equal magnitude but opposite signs; or (2) there is at least one pair of imaginary roots that lie on the imaginary axis; or (3) there are complex roots symmetric about the origin.

Field Commissioning — Installation, Calibration, Maintenance Tuning of Feedback Loops The tuning of a controller involves finding controller gains that will ensure at least a critically damped response of the process to a change in setpoint or process disturbance. A good starting point for PID constants is that derived during the design phase by the stability assessment approaches described earlier. However, real processes do not necessarily behave as their models would suggest and actual field tuning of controls is needed during the system commissioning process. Pole-Zero Cancellation One method of obtaining the desired critically damped response of a process is to determine the closedloop transfer function in the form C (s + A1)(s + A2 )…(s + Am ) = R (s + B1)(s + B2 )…(s + Bn ) The coefficients A and B will depend on the process characteristics and the controller gains. The objective of pole-zero cancellation is to find values for the controller gains that will set some numerator © 2005 by CRC Press LLC

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Mechanical System Controls

FIGURE 6.4.4 Reaction curve components.

coefficients equal to those in the denominator, effectively canceling terms. As can be imagined, however, this can be a very difficult exercise, particularly when working with complex roots of the equations. This method can only be used with very simple system models. Reaction Curve Techniques Often it is advisable to test a feedback loop in situ. Several techniques have been developed that allow for the derivation of “good” PID constants for a given open-loop response. Consider the process response shown in Figure 6.4.4 where ∆c is the change of process output; ∆u is the change of controller; L is the time between change and intersection; and T is the time between lower intersection and upper intersection. The following variables can be defined: A = ∆u /∆c ; B = T/L, and R = L/T. These values can be used with the equations given in Table 6.4.1 to estimate “decent” control constants. The users of these constants

TABLE 6.4.1 Equations for Finding PID Constants Using Zeigler–Nichols and Cohen and Coon Reaction Curve Tests Zeigler–Nichols Controller Components

Kp

Cohen and Coon

Kp Ki

Kd Kp

—

—

AB  1 +

Kp

 

R

P

AB

P+I

0.9AB

3.3L

—

AB  1.1 +

P+D

—

—

—

AB  1.25 +

0.5L

AB  1.33 +

P+I+D

© 2005 by CRC Press LLC

1.2AB

2L

 



3 R



12 

 

R

 

R

L



Kd Kp

—

—

30 + 3R

—

9 + 20 R —



6

4

Kp Ki

L

32 + 6 R 13 + 8 R

L

L

6 − 2R 22 + 3R 4 11 + 2 R

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Chapter 6

TABLE 6.4.2 Equations for Estimating PID Constants Using the Ultimate Frequency Test Controller Components P P+I P+I+D

Kp

Kp Ki

Kd Kp

0.5 Kp∗ 0.45 Kp∗ 0.6 Kp∗

— 0.8T ∗ 0.5T ∗

— — 0.125T ∗

should be aware, however, that they are based on the typical response of second-order systems and may not provide good values for all processes. Ultimate Frequency The ultimate frequency test involves increasing the proportional gain of a process until it begins steadystate oscillations. Kp∗ is defined as the proportional gain that results in steady oscillations of the controlled system and T ∗ is the period of the oscillations. The desired controller gains are given in Table 6.4.2. Note that the use of the ultimate period test is not always easy to do in practice and may be prohibited in certain cases by a process operations manager.

6.5 Advanced Control Topics Peter S. Curtiss, Jan Kreider, Ronald M. Nelson, and Shou-Heng Huang Neural Network-Based Predictive/Adaptive Controllers Neural networks are powerful modeling tools used for predicting nonlinear behavior of processes and require a minimum of knowledge about the physical system involved. This approach can be used to predict the behavior of a process and can calculate the future value of the process variables. The effects of current modifications to the future value of the controlled process can be easily quantified and used to obtain the desired process response. Overview of Neural Networks The artificial neural network attempts to mimic a few aspects of the behavior of biological neural networks. Inputs to a biological nerve cell are carried along the dendrites of that cell. These inputs come from the positive impulse signals of other cells but may be converted to negative signals by the chemical interactions at the synapse between the cells. All of the inputs are then carried to the soma where they add or subtract from the overall potential difference between the interior of the soma and the surrounding fluid. Once the cell potential rises above a certain level, the cell “fires” and sends signals to other cells along its axon. The artificial cell behaves in much the same way, except that the output signal is analog instead of digital. Signals from sending cells are passed along to a receiving cell through a series of connections. Each connection has an associated weighting factor that acts as a multiplier on the signal from the sending cell. All the inputs to a cell are summed (along with a cell bias, if included) and the resulting value is used to generate the output of the receiving cell. The output of the cell is referred to as the cell activation and the function that uses the net input to generate the cell activation is called the activation function. The latter can theoretically be of any form, although linear and sigmoidal functions are frequently used. Figure 6.5.1 shows a comparison between a biological cell and an artificial cell. When many different cells are combined together into a richly connected network (Figure 6.5.2), the result can behave mathematically as a nonlinear regression capable of mapping inputs to outputs for complex relationships. The trick is to find a series of weights W that allow the network to provide the desired outputs using specific inputs. A number of references to network training are cited in the bibliography at the end of this chapter. © 2005 by CRC Press LLC

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FIGURE 6.5.1 Biological cell vs. artificial cell.

FIGURE 6.5.2 Artificial neural network consisting of several layers of mathematical models of biological neurons.

Using Networks for Controlling Feedback Loop Processes Neural networks offer the potential for and have demonstrated improved control of processes through predictive techniques. The concept is fairly simple: train a network to predict the dynamic behavior of a process and then use these predictions to modify the controller output to place the process at a desired setpoint R(t) at some time in the future. Initial results from computer simulations of such a controller are presented in Curtiss et al. (1993a, b, c). Anderson (1989) described a computer simulation in which a network was trained to recognize the dynamic properties of an inverted pendulum (e.g., a broom balanced on an open palm). A control system was developed in which the angle and position of the pendulum were used to move the supporting base in order to maintain the pendulum upright. A neural network-based predictive controller is outlined in the classic discussion by Nguyen and Widrow (1989) on the “truck backer-upper” problem in which a tractor-trailer is backed into position at a loading dock. Properly tuned fixed-gain controllers will usually work over a relatively wide range of process operation provided that the external perturbations and influences are small or time invariant. With nonlinear processes, however, a conventional control algorithm can lead to unstable control if the gains were chosen for a range different from the current operating conditions. © 2005 by CRC Press LLC

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FIGURE 6.5.3 Network used for process prediction.

Architecture of the Network With the neural network approach, it is possible to overcome these problems by using as many additional salient inputs (the auxiliary inputs) as necessary and by incorporating an inherently nonlinear model to accomplish the control objectives. The network is trained using examples of the time-dependent relationship between a value of the feedback and previous values of the feedback, the controller output, and the auxiliary inputs. An example of the network architecture required for this is shown in Figure 6.5.3. In practice, it is not necessary to limit the number of previous measurements of any of the inputs, although the final size of the network and the corresponding training time and memory requirements need to be taken into consideration. The network, once trained, can predict the future feedback value for any controller output. The trick is to find the controller output that causes the future process value to match the setpoint. This is accomplished by finding the derivative of the future error with respect to the current controller signal. Starting with the current process conditions, the feedback value is predicted at each time step into the future over a preset time window as shown in Figure 6.5.4. During each step of the prediction, the values for the controller output and auxiliary inputs are held constant. This simulation is performed twice: the first time with a small increase in the controller output and the second time with a small decrease. This allows for calculation of the change of the future process value (and thus the change of the future error) as a function of the change in the current controller output. The controller output is then modified by ∆U (t ) = −Gnet ⋅ E f (t )

∂E f (t ) ∂U (t )

where Ef is the future error and Gnet is the network controller gain. For a multiple-output controller, the additional outputs are simply added as more outputs of the network and the future predictions repeated several times to find the correct partial derivatives. Many different variations on this theme are possible — for example, using the sum of the absolute values of all the errors over the prediction window (or the sum of the square of the errors, etc.) instead of simply the future error. Computer simulated results of such tests are provided by Curtiss et al. (1993c). Estimating Size of the Prediction Time Window It is possible to use the network model to determine the size of the time window by estimating the amount of time required for the process to reach some future steady state after a simulated change in the controller output. An example of such an open-loop response is shown in Figure 6.5.5. Here the © 2005 by CRC Press LLC

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FIGURE 6.5.4 Schematic of procedure for determining future process value and error.

FIGURE 6.5.5 Example of computer-simulated process step change (used to determine size of time window.)

network is simulating the response of a reverse-acting process after a decrease in actuator position at time step 0. About 70% (ln2) of total rise time is achieved after 15 time steps. This kind of calculation can be performed during the control sequence and should indicate the proper time window size. Example of PID vs. Network Controller Figure 6.5.6 shows an example of a process under PID control that demonstrates nonlinearity at different ranges of actuator position. Figure 6.5.7 shows the same process under the influence of a predictive neural network controller that had been trained on the process. Note that the network-based controller does not show the same problems of unstable control in certain actuator ranges. The size of the time window (15 time steps) was determined using the method discussed in the previous subsection. Using Networks as Supervisory Controllers The previous section discussed the use of neural networks to minimize a predicted error of a feedbackloop process. It is possible to apply a similar methodology for supervisory plant control to optimize the process according to some cost function. A network is first trained to predict the cost function under a wide range of operating conditions. This network is then used to predict what will happen with different control strategies. Figure 6.5.8 shows a schematic of this technique. The left side of the figure shows the training mode, where the network is attempting to associate the various plant inputs with the cost function output. There can be multiple inputs, including uncontrollable variables (e.g., ambient conditions, plant loads, etc.) and controlled variables (i.e., the various process setpoints.) Once the network is sufficiently trained, it is used to find values for the setpoints under any set of uncontrolled variables. The technique for doing so is similar to the back-propagation training technique of the network. The inputs corresponding to the controlled variables are replaced with virtual nodes © 2005 by CRC Press LLC

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FIGURE 6.5.6 Example of computer simulation using PID controller.

FIGURE 6.5.7 Example of computer simulation using neural network controller.

whose outputs are always unity. These nodes are connected to the predictor network through adjustable weights. The optimization occurs by finding values for these weights that allow the model to predict a desired output. These weights can be found through any number of search methods, including the gradient descent technique used in back-propagation training. In this case, the predictor network is “trained” normally, except that all weights in the network are static except those connected to the virtual nodes. Once weights have been found that produce the desired output, the setpoints can be found from direct interpretation of these weights. Constraints can be imposed on the weights through physical limitations (e.g., freezing points) or from predictions from local-loop neural network controllers.

Fuzzy Logic Controllers Fuzzy logic controllers use conditional relationships to analyze one or more inputs — that is, the inputs are subject to a series of if…then queries to produce some intermediate values. An example would be something like a simple cruise control on an automobile:

© 2005 by CRC Press LLC

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FIGURE 6.5.8 Using network model to optimize process control.

• If vehicle speed = much lower than setpoint, then need to increase speed = large • If vehicle speed = slightly lower than setpoint, then need to increase speed = small These intermediate values are then used to determine the actual change of speed in the car: • If need to increase speed = large, then increase of throttle position = 10% • If need to increase speed = small, then increase of throttle position = 3% In fuzzy control, the satisfaction of a particular if statement may not lead to or be restricted by a true or false response. A range of weighting coefficients is assigned to a particular conditional, with the coefficients generally decreasing as the certainty of a specific condition decreases. In the preceding example, “need to increase speed = large” may be assigned a certainty of 1 when the speed of the vehicle is less than 50% of the desired speed of the car. This certainty will decrease as the speed of the car increases, so that “need to increase speed = large” may be 0 when the speed of the car is greater than 90% of the desired speed, but the certainty of “need to increase speed = small” will be large. It is possible that, for a given speed of the car, two or more conditions may be satisfied with different magnitudes of certainty. These conditions can then be applied to the output rules along with their respective certainties to determine the actual increase (or decrease) in the controller output. When the speed of the car is below the desired speed, the initial rules may yield, for example: • Need to increase speed = large with certainty = 0.3 • Need to increase speed = small with certainty = 0.7 The actual output would then be Increase of throttle position = (0.3 × 10% + 0.7 × 3%)/(0.3 + 0.7) = 5.1% The following subsection formalizes some of these ideas and includes a detailed example. The chapter on mathematics contains the formalism underlying fuzzy set theory and fuzzy logic. The reader is referred to that chapter and one that follows for the technical basis for fuzzy logic controllers.

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Fuzzy Logic Controllers for Mechanical Systems Introduction In the last decade, fuzzy logic controllers (FLCs) have been receiving more attention (Daley and Gill, 1985; Leigh and Wetton, 1983; Xu, 1989; Yasunobu and Miyamoto, 1985), not only in test cases but also in real industrial process control applications, including building mechanical systems (Meijer, 1992; Sakai and Ohkusa, 1985; Ono et al., 1989; Togai and Maski 1991; Huang and Nelson, 1991). The basic idea of this approach is to incorporate the experience of human operators in the design of controllers. From a set of linguistic rules describing operators’ control strategies, a control algorithm can be constructed (Ralston and Ward, 1985). Computer simulations and experiments have shown that FLCs may have better performance than those obtained by conventional controllers. In particular, FLCs appear very useful when the processes are too complex for analysis using conventional control algorithms or when the available information is qualitative, inexact, or uncertain. Thus, fuzzy logic control may be viewed as a compromise between conventional precise mathematical control and human-like decision making, as indicated by Gupta (Gupta and Tsukamoto, 1980). However, fuzzy logic controllers sometimes fail to obtain satisfactory results with the initial rule set drawn from the operators’ experiences. This is because some differences exist between the way a plant is operated by an experienced operator and by a fuzzy logic controller using the rules based directly on his experience. It is often difficult to express human experience exactly using linguistic rules in a simple form. Sometimes no experience is available that can be used to construct control rules for FLCs. In these cases, it is necessary to design, develop, and modify control rules for FLCs to obtain optimal performance. Few discussions have been undertaken about rule development and adjustment strategies for FLCs (Scharf and Mandic, 1985; Ollero and Williams, 1989; Sheridah, 1984; Wakileh and Gill, 1988). Basic Aspects of an FLC An FLC includes three parts: fuzzifier; fuzzy reasoning unit; and defuzzifier. The fuzzifier converts ordinary inputs into their fuzzy counterparts; the fuzzy reasoning unit creates fuzzy control signals based on these fuzzy variables and the defuzzifier converts the fuzzy control signals into the real control outputs. The block diagram of a fuzzy control system is shown in Figure 6.5.9, where e, d, and u are tracking ˜ and u˜ are their fuzzy counterparts, respectively; error, derivative error, and output control action; e˜, d, y is the controlled parameter; and r is the set point for y. Kp is the scale factor for e; Kd is the scale factor for d; and Ko is the output gain. The control rules expressed in natural language are expressed in the following form: IF (e is A) AND (d is B) THEN (u is C) where A, B, and C are fuzzy subsets defined on the universes of discourse of e, d, and u, respectively. Every rule is interpreted into a fuzzy reasoning matrix: Θ

Rk =  Ak (e) ⊗ Bk (d) ⊗ Ck (u) FLC r + −

~ e

e Kp d dt

Kd d

Fuzzifier ~ d

Fuzzy Reasoning Unit

FIGURE 6.5.9 The block diagram of a fuzzy control system. © 2005 by CRC Press LLC

k = (1, N )

~ u Defuzzifier

Ko

u

HVAC System

y

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TABLE 6.5.1 The Membership Function of Input of FLC A(e), B(d)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

PL PM PS ZZ NS NM NL

0.0 0.0 0.0 0.0 0.0 0.5 1.0

0.0 0.0 0.0 0.0 0.1 0.8 0.8

0.0 0.0 0.0 0.0 0.5 1.0 0.5

0.0 0.0 0.0 0.1 0.8 0.8 0.1

0.0 0.0 0.0 0.5 1.0 0.5 0.0

0.0 0.0 0.1 0.8 0.8 0.1 0.0

0.0 0.0 0.5 1.0 0.5 0.0 0.0

0.0 0.1 0.8 0.8 0.1 0.0 0.0

0.0 0.5 1.0 0.5 0.0 0.0 0.0

0.1 0.8 0.8 0.1 0.0 0.0 0.0

0.5 1.0 0.5 0.0 0.0 0.0 0.0

0.8 0.8 0.1 0.0 0.0 0.0 0.0

1.0 0.5 0.0 0.0 0.0 0.0 0.0

where N is the number of rules, the symbol ⊗ denotes aggregation operator, and the symbol Θ denotes an align-turning operator (see Chap. 19). The general fuzzy relation matrix R can be constructed as the union of the individual rules: R=

∪

N k =1

Rk

This matrix represents the relationship between the fuzzy inputs and the fuzzy control output. The fuzzy control output can then be calculated from the known fuzzy input e˜ and d˜ by: Θ u = e ⊗ d  R

where the symbol
denotes the max–min composition operator (see Chapter 19). The input universe of discourse for tracking error e or derivative error d is divided into several degrees connected with a number of fuzzy subsets by membership functions. In this study, e and d can each range from –6 to +6, and 13 degrees are used: –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6. Also, seven fuzzy subsets are defined as: NL, NM, NS, ZZ, PS, PM, PL where the first letters N and P mean negative and positive; the second letters L, M, and S mean large, middle, and small; and ZZ means zero. These degrees and fuzzy subsets are shown in Table 6.5.1, which uses a 1.0-0.8-0.5-0.1 distribution. For example, if e = 3, then its membership in PL is 0.1, its membership in PM is 0.8, etc. A similar analysis is given to the outputs for the control action indicated in Table 6.5.2, which uses a 1.0-0.7-0.2 distribution. The fuzzifier converts ordinary inputs into their fuzzy counterparts. In this study, a fuzzy singleton is ˜ with used as a fuzzification strategy, which interprets an input, e (or d), into a fuzzy value, e˜ (or d), membership function (µ) equal to zero except at the element nearest to the real input, where µ = 1.0. For example, if e = 3.2 and the nearest element is 3, then the fuzzy singleton will be: d˜ =(0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0.) This fuzzy singleton has membership function µ = 1.0 at the point of element e = 3. The defuzzifier converts the fuzzy control output created by the rule-based fuzzy reasoning unit into a real control action. In this study, weighted combination method is used as defuzzification strategy, which can be explained by the following example: © 2005 by CRC Press LLC

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TABLE 6.5.2 The Membership Function of Output of FLC C(u)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

VS (level 7) ST (level 6) SU (level 5) ME (level 4) SS (level 3) SM (level 2) TI (level 1)

0.0 0.0 0.0 0.0 0.0 0.2 1.0

0.0 0.0 0.0 0.0 0.0 0.7 0.7

0.0 0.0 0.0 0.0 0.2 1.0 0.2

0.0 0.0 0.0 0.0 0.7 0.7 0.0

0.0 0.0 0.0 0.2 1.0 0.2 0.0

0.0 0.0 0.0 0.7 0.7 0.0 0.0

0.0 0.0 0.2 1.0 0.2 0.0 0.0

0.0 0.0 0.7 0.7 0.0 0.0 0.0

0.0 0.2 1.0 0.2 0.0 0.0 0.0

0.0 0.7 0.7 0.0 0.0 0.0 0.0

0.2 1.0 0.2 0.0 0.0 0.0 0.0

0.7 0.7 0.0 0.0 0.0 0.0 0.0

1.0 0.2 0.0 0.0 0.0 0.0 0.0

Note: Abbreviations mean that the output control actions are Very Strong; Strong; SUbstrong; Medium; Slightly Small; Small; and TIny.

If u˜ = (0, 0, 0, 0, 0, 0, 0, 0.2, 0.4, 0.8, 0.7, 0.5, 0.1.) then u = [0.2(1) + 0.4(2) + 0.8(3) + 0.7(4) + 0.5(5) + 0.1(6)]/[0.2 + 0.4 + 0.8 + 0.7 + 0.5 + 0.1] = 3.4 Rule Refinement An FLC is characterized by a set of linguistic statements that are usually in the form of “if–then”' rules. The initial set of rules is usually constructed based on the operators’ experiences, or sometimes by analyzing the dynamic process of the controlled plant. Both approaches require modifying the initial set of rules to obtain an optimal rule set. This is called rule refinement. Figure 6.5.10 shows an initial rule set analyzed on a “linguistic plane.” The horizontal axis expresses the fuzzy subsets defined on the universe of discourse for the tracking error (e), and the vertical axis expresses the fuzzy subsets defined on the universe of discourse for the derivative error (d). Both have seven fuzzy “values”: NL, NM, NS, ZZ, PS, PM, and PL. Output control action levels are on the cross points of these fuzzy values; these are also fuzzy subsets having seven values from level 1 (tiny) to level 7 (very strong). For example, the cross point of e = NM and d = PM indicates u = level 5. This corresponds to the rule: IF (e is NM) AND (d is PM) THEN (u is level 5) For example, the initial rule set could be based on the following control strategies. First, it tries to keep a proportional relationship between the control action (u) and the tracking error (e). Note that if the derivative error (d) is ZZ, then the output control action (u) increases from level 1 to level 7 when the tracking error (e) changes from NL to PL. Second, the influence of derivative error (d) is considered such that if it is positive, then the control action (u) is increased a little bit, and if it is negative, then the control action (u) is decreased. For example, if the tracking error (e) keeps PM, the control action (u) increases from level 6 to level 7 when the derivative error (d) is positive, and it decreases from level 6 to level 5 when the derivative error (d) is negative. Consider a second order plant with a transfer function: H (s) =

1.0 s 2 + 0.1s + 1.0

that is controlled using the initial rule set to respond to a step input for computer simulation. The performance trajectory of the FLC is shown by the arrows in Figure 6.5.10 and the dynamic process of the normalized controlled parameter (CP) is shown in Figure 6.5.11, where the horizontal axis indicates the number of sample period (SP). The dynamic process can be divided into two stages. © 2005 by CRC Press LLC

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PL PM PS ZZ NS NM NL

C

NL

NM

NS

1

3

3

d ZZ D 4

1 1 1

3 2 2

3 3 3

1 1 1

1 1 1

2 2 1

PS

PM

PL

7

7

7

4 4 4

6 6 5

7 7 6

7 7 A 7

4

5 5 5

6 5 5

7 7 7

4 4

e

B

FIGURE 6.5.10 The initial rule set and performance trajectory on the linguistic plane. CP

C

2.0

1.0

B D

A 0

10

20

30

SP

FIGURE 6.5.11 The dynamic process corresponding to Figure 6.5.10.

At the first stage, there is a strong oscillation with a higher frequency and, at the second stage, there is a moderate swing with a smaller frequency. Looking at the performance trajectory in the linguistic plane reveals that the stronger oscillation occurs at the out-cycle (points further from the center). As time increases, the state moves to the in-cycle near the center of the plane and becomes moderate. This shows that FLCs have the desirable property of a structure-variable controller. The rules at the out-cycle belong to one kind of structure for the first stage, and the rules at the in-cycle belong to another structure for the second stage. If the initial rule set does not satisfy a good design for a controller, then it can be modified by intuitive reasoning. A rule set is often symmetrically positioned about the central point, which is the desired stable operating point at which the tracking error (e) and the derivative error (d) equal zero and the control action (u) is medium. When a positive step increase is imposed to the set point, the tracking error (e) has the biggest value and the derivative error (d) is zero at the beginning time (point A in the linguistic plane). With the regulating action, the tracking error (e) will decrease, the derivative error (d) will be negative, and the performance trajectory will enter into the right-bottom block in the linguistic plane. Thus, the rules in this area have the most important effect on the behavior of the first stage of the dynamic process. The most important area responsible for the behavior of the second stage is the central block. To avoid strong oscillations, it is apparent that the control actions in the right-bottom block should be decreased. The modified rule set and its simulation of response to a step input are shown in Figure 6.5.12. The performance trajectory expressed in the linguistic plane is just like spiral (Figure 6.5.12). It can be seen that the performance of the control system has been improved, but a small oscillation still exists and a little overshoot is indicated by point C in Figure 6.5.13. Once again, the rule set is modified and the final rule set and its simulation of response to a step input are shown in Figure 6.5.16 and Figure 6.5.17. © 2005 by CRC Press LLC

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NL

NM

NS

PL PM

3 2

4 3

5 4

PS ZZ

1 1

C 1 1

1 1

NS

1

1

NM

1

1

1

NL

1

1

1

G

1

d ZZ PS PM D 6 7 7 7 5 7 D 7 4 7 4 7 7 E 7 7 4 3F 4 5 2

3

PL 7 7 7 A 7 7

e

6

4

5

B

FIGURE 6.5.12 The second rule set on the linguistic plane.

CP 2.0 C

1.0

G

D B

F E

A 0

10

20

30

SP

FIGURE 6.5.13 The dynamic process corresponding to Figure 6.5.12.

d

PL PM PS ZZ NS NM NL

NL

NM

NS

ZZ

PS

PM

PL

3 3 1 1 1 1 1

5 4 1 1 1 1 1

5 4 1 1 1 1 1

6 5 4 4 4

7 7 7 7 7 4 3

7 7 7 7 7 4 3

7 7 7 7 7 5 5

3 2

e

FIGURE 6.5.14 The third rule set on the linguistic plane.

The final rule set gives good performance with a short rise time and a very small overshoot and is considered satisfactory. By analyzing the performance trajectory on the linguistic plane, a rule set is refined. It relies heavily on intuitive reasoning when comparing the dynamic process of the controlled parameter for the present rule set with the desired one. © 2005 by CRC Press LLC

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CP 2.0

1.0

0

10

20

30

SP

FIGURE 6.5.15 The dynamic process corresponding to Figure 6.5.14.

FIGURE 6.5.16 Noncomplete membership function. Membership function NL

−6

NM

NS

−4

−2

1

ZZ

0

PS

PM

2

4

PL

6

e or d

FIGURE 6.5.17 Heavy overlap membership function.

Completeness and Interaction of Rules and Selection of Membership Functions The second significant influence on the behavior of an FLC is the membership functions. They should be chosen carefully in the adjustment process. As mentioned previously, the fuzzy subsets, language variables, NL, NM, NS, ZZ, PS, PM, and PL, are defined on the universe discourse of tracking error (e) or derivative error (d). Some possible membership functions are shown in Figure 6.5.16 through Figure 6.5.18. The membership functions should be chosen to make these language variables have suitable coverage on the universe of discourse. For the case of Figure 6.5.16, the whole range is not covered by these language variables. For some values of e or d, the membership functions of all language variables are zero. In this case, an empty output control action could be created. This means that the control actions are lost for those points not covered by any input fuzzy subset. This is referred to as the noncompleteness of control rules. FLCs should satisfy the condition of completeness for their membership functions; the membership function shown in Figure 6.5.17 cannot be used for an effective fuzzy logic controller. In other words, the union of all fuzzy subsets, Xi , i = [1,7], should be greater than zero for all e ∈ E, i.e., © 2005 by CRC Press LLC

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Membership function NL

−6

NM

NS

−4

1

−2

ZZ

PS

PM

PL

2

4

6

0

e or d

FIGURE 6.5.18 Moderate overlap membership function.

∪

∀ e ∈E

7 i =1

X i (e) > 0

On the other hand, interaction can take place among the rules if the overlap of fuzzy subsets occurs on the range of the universe of discourse. In this case, the membership functions have the forms shown in Figure 6.5.17 and Figure 6.5.18. The interaction tends to smooth out the set of control rules. Consider the single-input–single-output case for simplicity. The rule set is: IF (e is Ai ), THEN (u is Ci ) i = [1,N] where N is the number of rules in the set. These rules are incorporated into a fuzzy relation matrix as follows: R=

∪

N i =1

Ri =

∪

N i =1

( A ⊗C ) i

i

If the fuzzy value of input e is known as e˜, the fuzzy output u˜ then can be calculated as follows: u = e  R If e˜ is Ai , u˜ is expected to be Ci . However, now the interaction of rules due to overlap results in: Ci ⊆ Ai  R The equality is established only when no overlap occurs. This analysis is based on the fuzzy logic scheme including max–min composition operator. A more detailed example of the numeric calculation is given in the next subsection. If the overlap is heavy, as shown in Figure 6.5.17, large deformation will occur and the control rules will lose their original shape. In the limit, as the membership functions become unity for all values, the output of the FLC will always be the same fuzzy quantity. This means that the fuzzy reasoning system conveys no valuable information and the FLC has lost its efficacy. A moderate overlap, shown in Figure 6.5.18, is desirable to allow for reasoning with uncertainty and the need for completeness of the control rules. How does one determine the “size” of overlap? At present, intuitive judgment is used to choose membership functions when adjusting an FLC. There appears to be some latitude in choosing the amount of overlap on which the performance of an FLC does not change significantly. The quantitative analysis will be given after further research. When the control rules are modified in the linguistic plane, the overlapping membership functions let the rules near the performance trajectory have an effect on the output control actions. This is because interactions occur among the neighboring rules.

© 2005 by CRC Press LLC

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CP 2.0 Ko = 8.4 Ko = 1.6 1.0

Ko = 4.2

0

10

20

30

SP

FIGURE 6.5.19 The influence of Ko on the behavior of FLCs.

Scale Factors and Output Gain The scale factors, Kp and Kd , and the output gain, Ko , shown in Figure 6.5.19, also have significant influence on the behavior of an FLC. Their influence is not as complicated as those of rules and membership functions. The adjustment for the scale factors and output gain is comparatively simple. The scale factor Kp relates the actual range of tracking error (e) to the universe of discourse (E) defined in the fuzzy logic system. In this work, E consists of 13 degrees as indicated earlier. Then Kp is determined as the ratio of the range of E to the range of the real variable: Kp =

Emax − Emin emax − emin

For scale factor Kd , similar analysis leads to: Kd =

Dmax − Dmin dmax − dmin

(6.5.1)

where D is the universe of discourse for derivative error (d) defined in the fuzzy logic system. Small Kp or Kd will narrow the control band, while large Kp or Kd will lead to loss of control for large inputs. The output gain Ko is defined as follows: Ko =

umax − umin U max − U min

(6.5.2)

This is the ratio of range of real output control action (u) to the range of its universe of discourse (U) defined in the fuzzy logic system. Ko acts as an amplification factor of the whole FLC. Figure 6.5.19 shows the influence of Ko on the step response simulation of an FLC with the final rule set used in Figure 6.5.14. Increasing Ko results in a shorter rise time. The performance trajectory in the linguistic plane will become steeper for the first stage and oscillation occurs. Decreasing Ko results in a longer rise time and the performance trajectory in the linguistic plane will become moderate during the first stage. However, in this simulation, oscillation still occurred. This is because different Ko, larger or smaller, results in a new route of the performance trajectory, which will activate the different rules that might cause oscillation. Therefore, the influence of output gain, Ko, should be considered together with the change of the activated rules.

© 2005 by CRC Press LLC

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Conclusion A fuzzy logic controller can perform much better than a conventional one, such as a PID controller, if the FLC has been well constructed. The main disadvantage of using FLCs today seems to be the lack of a systematic procedure for the design of FLCs. The general method for designing an FLC is to use trial and observation. No useful mathematical tool has yet been developed for the design of an FLC because of its fuzziness, complexity, and nonparameterization. Three significant elements have notable influence on the behavior of a fuzzy logic controller: • Control rules expressed in linguistic language • Membership functions defined for fuzzy subsets • Scale factors attached to the input and the output gains The control rules play the main role in forming the dynamics of FLCs. The rule set can be analyzed and modified using the performance trajectory technique and evaluated using the dynamic process curve of the controlled parameter. The membership functions define the “shape” of fuzzy subsets. They should have appropriate width to avoid noncompleteness and suitable interaction among the fuzzy control rules. The scale factors (Kp and Kd ) and output gain (Ko) serve as amplification factors. At present, each application must be individually designed. The initial sets of rules are specifically set up for different applications. Work is now under way to develop a self-adaptive fuzzy logic controller, which will choose the initial set of rules automatically according to the process dynamics and refine it on the basis of the global performance evaluation.

6.6 Control of Distributed Generation Technologies Peter S. Curtiss and Jan F. Kreider Distributed power generation (DG) is any small-scale power generation technology that provides electric power at a site closer to customers than central station generation; it is usually interconnected to the distribution system or directly to the customer’s facilities (Curtiss et al., 1999). Distributed generation technologies include small combustion turbine generators; internal combustion; reciprocating engines and generators; photovoltaic panels; wind turbines; and fuel cells. Table 6.6.1 provides an overview of feasible DG technologies using present or near-future technologies. Distributed generation can provide a multitude of services to utilities and consumers, including standby generation; peak shaving capability; baseload generation; or cogeneration. Less well-understood benefits including ancillary services such as voltage/VAR support, network stability, and others may ultimately be of more economic benefit than simple energy-related benefits. Electric restructuring has spurred the consideration of DG power because the buyers and sellers of electricity will need to be more responsive to market forces. Central utilities suffer from the burden of significant stranded costs; however, DG avoids this cost. DG is a priority in parts of the country where the spinning reserve margins are shrinking, where industrial and commercial users and transmission and distribution (T&D) constraints are limiting power flows (DCPA, 1998). TABLE 6.6.1 Summary of Distributed Generation Technologies

Dispatchable Capacity range (MW) Efficiency Capital cost ($/kW) O&M cost (¢/kWh)

IC Engine

Turbine

PVs

Wind Turbine

Fuel Cells

Yes 0.05–5 35% 200–350 ~1

Yes 0.025–25 15–35% 450–850 0.5–0.65

No 0.001–1 6–19% ~6000 0.1–0.4

No 0.01–1 25% ~1000 ~1

Yes 0.2–2 30–55% ~3500 0.1–0.2

Notes: Efficiencies of fossil and renewable DG technologies are not directly comparable. O&M costs do not include fuel. Source: Adapted from Distributed Power Coalition of America (DCPA), 1998. URL: www.dcpa.org. © 2005 by CRC Press LLC

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Control Techniques The control techniques chosen for distributed generation will depend on the type of equipment installed (Curtiss, 2000). In the case of wind or solar power generation, the main goal is to get as much energy out of the system as possible in order to recoup the installation cost. For combustion-based processes, however, the cost of fuel and maintenance must also be taken into account. The goal of the control scheme is to determine whether the on-site generation should be operating during a particular hour. Generally, a simple hour-ahead control method is sufficient if the start-up transients of the generator are not too inefficient. Otherwise, it is necessary to develop methods that perform a certain degree of prediction to determine the effects of scheduled dispatch over the period of several hours or even several days. Threshold Control In threshold control, the turbines run whenever the building electrical load is greater than a predetermined threshold. The number of turbines initially installed is equal to the difference between the annual peak and the threshold, divided by the nominal power output of each installed unit: Number installed =

kWPEAK − kWTHRESHOLD kWUNIT

If the electrical load of the building is greater than the threshold, then the number of turbines operating is equal to that required to reduce the grid load to the threshold limit: Number operating =

kWBUILDING − kWTHRESHOLD kWUNIT

A problem with this control method is deciding where to assign the threshold limit. A high limit means the turbine is used for peak shaving and will reduce the number of operating hours. A low limit forces the generators to run more often and is akin to base loading. A threshold of zero indicates that the generators would try to operate whenever possible. This specific case is referred to as always-on control. Buyback Priority Buyback priority is used in cases in which the building operator wishes to produce electricity and sell any or all of the produced power back to the utility. There are two versions of buyback control; one takes advantage of a simple buyback rate and the other responds to net metering, in which the value of produced power is used to offset the traditional electrical bill. Simple Buyback In the case of simple buyback the generators would normally use the threshold control scheme as described previously. If the buyback cost is greater than the equivalent cost of gas, then all the generators run and the excess is sold to the utility. The number of turbines installed depends on the projected income the building operator expects to make from selling electricity. The control method finds the incremental sum of all fuel used to get the total cost for the hour: Total cost = ∆$kWhGRID + ∆$BtuGRID – ∆$kWhBUYBACK The ∆$ term implies that the non-RTP gas and electric costs are evaluated on a monthly (i.e., billing period) incremental basis. For example, the change of the grid electricity bill is ∆$kWhGRID = M$(kWh1, kWh2, …, kWhN–1) – M$(kWh1, kWh2, …, kWhN) where M$ is the monthly bill amount (including consumption and demand fees, surcharges, and taxes) based on N hourly electricity use values for that billing period. This allows the true bills to be calculated, including any time-of-use and block components. © 2005 by CRC Press LLC

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Unfortunately, these latter components also affect the linearity of the cost function — the cost function is not necessarily linear under these conditions. The algorithm for determining whether to use buyback, therefore, should (1) determine the loads on the building for a given hour; (2) calculate the total cost function for all integral numbers of generators operating, from zero to the number installed; and (3) find the number of generators, which minimizes the total cost function. Net Metering Control In the net metering scenario, the electrical meter “runs backwards” if excess electricity is produced on site. Once the meter reaches zero, buyback rates apply. As with the buyback priority control, the incremental sum of all fuel uses is calculated to get the total cost for the hour: Total cost = ∆$kWhGRID + ∆$BtuGRID – ∆$kWhBUYBACK. The ∆$ terms are the incremental costs as discussed in the buyback priority control. Consequently, the control algorithm is the same as in the buyback priority with the exception that the ∆$kWhGRID term here refers to the adjusted (i.e., rolled back) meter usage and the ∆$kWhBUYBACK amount is decreased by the kWh that go into reducing ∆$kWhGRID. If the monthly sum is positive (i.e., more electricity has been used from the grid than produced on site), then the monthly bill is based on simple effective aggregation of hourly consumption plus demand and fees. Otherwise, the customer is refunded the value of excess electricity produced as dictated by buyback rate. Cooling/Heating Priority Control In some cases, distributed generation will be applied to satisfy a cooling load (through auxiliary absorption cooling or direct electrical connection to conventional cooling equipment) or a heating load (through heat recovery). In this mode of control, the generators operate primarily to satisfy these loads and the satisfaction of the electrical load is a secondary benefit. The number of generators installed is sufficient to meet the annual peak thermal load and the control algorithm has the generators operating as required to meet the thermal load of the building. No consideration is given to the value of electricity. Optimal Control Ideally, distributed generation would be operated using an algorithm that reduces the operating cost over the lifetime of the equipment such that the cost to the building operator is minimized. If the building is subject to a real-time pricing rate schedule then the optimization can be trivial; the costs of grid electricity and locally produced electricity are compared at each hour and, when the former is more expensive, the on-site generators are operated. However, more conventional rate structures such as block rates and timeof-use rates as accumulated over a billing period can make the calculation of instantaneous “next kWh” costs much more difficult. In this case, the electricity bill CELEC at any given hour is

CELEC

 kWhBLDG (1) − kWhGEN (1)    kWhBLDG (2) − kWhGEN (2) = Φ KWH   
   kWhBLDG (k) − kWhGEN (k)   kWBLDG (1) − kWGEN (1)    kWBLDG (2) − kWGEN (2) + Φ KW   
   kWBLDG (k) − kWGEN (k) 

where ΦKWH is the utility function used to calculate the bill based on consumption; ΦKW is the function used for demand; kWhBLDG(1) is the total electric load at hour 1; kWhGEN(1) is the kWh offset from the on-site generation equipment at hour 1; and so forth. © 2005 by CRC Press LLC

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The calculation must be performed for each hour of the billing period to account for the hourly building load, any time-of-use components of the utility rate, and any ambient temperature or solar dependencies of the generation equipment. If the generators use natural gas to produce electricity (an internal combustion engine, microturbine, or fuel cell), then a similar calculation is performed for the gas consumption. Assuming no demand component for gas, the total gas bill up to hour k of the billing period is given as  BtuBLDG (1) + BtuGEN (1)    BtuBLDG (2) + BtuGEN (2) CGAS = ΦGAS   
   BtuBLDG (k) + BtuGEN (k)  where BtuGEN is the incremental gas consumption of the generation equipment at each hour. Note that kWhGEN, kWGEN, and BtuGEN can have zero values at any hour, depending on whether the generation equipment is operating for that hour. To determine if the generators should operate at hour k + 1, the total cost CELEC + CGAS should be evaluated twice: once using values for the terms kWhGEN, kWGEN, and BtuGEN based on the estimated generator performance and then again with these values set to zero. If the former is greater than the latter, then the generators should not be run for that hour. Complete Optimization The procedure just described is sufficient for performing an optimization based on a single type of generation equipment without accounting for any other inputs. To be truly optimal, however, the algorithm should account for any different capacities of generators installed, any utility incentives, and the variable operation and maintenance costs experienced during operation. Such an optimization uses an algorithm similar to that described here. The structure and calculation methods used for the electricity and gas utility rate schedules must be known. The optimization routine must also be able to keep track of all data acquired during a given billing period and provide cost estimates for the current hour. Any utility-sponsored incentives and rebates should be tallied, along with the method of their application (e.g., by kilowatt hours produced, kilowatts installed, etc.) At each hour of the billing period, the optimization routine determines the number of generators that should run for that hour. This requires a prediction of the building load data for that hour, including: • • • • • • •

whole building kWh use whole building Btu use kWh used for domestic water heating Btu used for domestic water heating kWh used for space heating Btu used for space heating kWh used for space cooling

The electrical and thermal output from each generation device in the building must then be determined. This may require monitoring of the ambient temperature, wind speed, and insolation. Parametrics are then run examining the benefit of operating each generator, accounting for any generators that may already be operating and for any part-load characteristics of generators that are not operating at full load. The cost function in the analysis is truly a cost function; the costs of providing on-site electrical and thermal costs are compared with those of grid consumption and the lowest-cost option is chosen. To assess these costs properly, the grid electricity consumption kWhGRID is adjusted by the decrease of grid electricity consumption due to distributed power generation: kWhGRID = kWhBLDG – kWhGEN – kWhCOOL © 2005 by CRC Press LLC

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Chapter 6

where kWhBLDG is the building load and kWhGEN is the amount of electricity from the generators. The term kWhCOOL is nonzero if the generator provides direct cooling through absorption cooling and must be corrected for the nominal efficiency of the conventional cooling equipment:

∑Q = •

kWhCOOL

COOL

COPCOOL

where the summation is taken over all devices that provide supplemental cooling. If the distributed generation equipment includes any gas-fired devices, the incremental cost of natural gas consumption must also be taken into account, BtuGRID = BtuBLDG + BtuGEN – BtuHEAT where BtuBLDG is the building load and BtuGEN is the consumption of gas by the generators: BtuGEN =

∑W

GEN

(PLR)

where the summation is taken over all devices that convert gas to electricity. The work term must also be corrected by the part load efficiency of any generators that are not at full load. The term BtuHEAT represents any credit that can be applied due to heat recovery from the generators that precludes the use of conventional space or water heating sources. As with the cooling term, this credit is adjusted by the nominal efficiency of the conventional sources:

∑Q = •

BtuHEAT

HEAT

ηHEAT

The total operating cost can now be calculated from the incremental rates, incentives, and maintenance costs:  kWhBLDG (1) − kWhGEN (1)    kWhBLDG (2) − kWhGEN (2) CTOTAL = Φ KWH        kWhBLDG (k) − kWhGEN (k)   kWBLDG (1) − kWGEN (1)    kWBLDG (2) − kWGEN (2) + Φ KW        kWBLDG (k) − kWGEN (k)   BtuBLDG (1) + BtuGEN (1)    BtuBLDG (2) + BtuGEN (2) + ΦGAS        BtuBLDG (k) + BtuGEN (k)  − ΦCRED  

∑kWh

GEN

+ ΦO& M kWINST +  © 2005 by CRC Press LLC

 

∑kWh

GEN

 

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6-47

where ΦCRED represents a positive cash flow based on any utility incentives that are provided, including transmission loss credits, wheeling charge credits, voltage support credits, etc. The term ΦO&M is used to account for any operation and maintenance costs that arise from operating the generation equipment. Finally, a cost matrix is compiled that represents all reasonable combinations of generation available to that building. The combination with the lowest cost is chosen and implemented.

References Anderson, C., 1989, Learning to control an inverted pendulum using neural networks, IEEE Control Syst. Mag., April, 31–36. Askey, S.Y. 1995. Design and Evaluation of Decision Aids for Control of High Speed Trains: Experiments and a Model, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, June. Billings, C.E. 1991. Human-Centered Aircraft Automation: A Concept and Guidelines, NASA Ames Research Center, Moffet Field, CA. Curtiss, P., 2000, Control of distributed electrical generation systems, ASHRAE Trans., 106, Pt. 1. Curtiss, P.S., Kreider, J.F., and Brandemuehl, M.J., 1993a, Artificial neural networks proof of concept for local and global control of commercial building HVAC systems, Proc. ASME Int. Solar Energy Conf., Washington, D.C., pp. 429–443. Curtiss, P.S., Kreider, J.F. and Brandemuehl, M.J., 1993b, Energy management in central HVAC plants using neural networks, Proc. ASHRAE Annu. Winter Meeting, Chicago, IL. Curtiss, P.S., Brandemuehl, M.J. and Kreider, J.F., 1993c, Adaptive control of HVAC processes using predictive neural networks, ASHRAE Trans., 99, Pt 1, pp. 496–504. Curtiss, P., Cohen, D., and Kreider, J.F., 1999, A methodology for technical and financial assessment of distributed generation in the U.S., Proc. ASME ISEC April 1999 Conf. Maui, HI. Daley, S. and Gill, K.F., 1985, The fuzzy logic controller: an alternative design scheme? Computers Ind., 6, 3–14. Gupta, M. M. and Tsukamoto, Y., 1980, Fuzzy logic controllers — a perspective, Proc. Joint Automatic Control Conf., FA10-C, August 1980, San Francisco. Huang, S.H. and Nelson, R.M., 1991., A PID-law-combining fuzzy controller for HVAC applications, ASHRAE Trans., 97, Pt. 2, 768–774. Leigh, R. and Wetton, M., 1983, Thinking clearly with fuzzy logic, Process Eng., 64, 36–37. Meijer, G., 1992, Fuzzy logic-controlled A/Cs heat pumps, IEA Heat Pump Centre Newslett., 10(1). Ollero, A. and Williams, J., 1989, Direct digital control, auto-tuning, and supervision using fuzzy logic, Fuzzy Sets Syst., 30, 135–153. Ono, H., Ohnish, T., and Terada, Y. ,1989, Combustion control of refuse incineration plant by fuzzy logic, Fuzzy Sets Syst., 32, 193–206. Nguyen, D.H. and Widrow, B., 1989, The truck backer-upper: an example of self learning in neural networks, Proc. Int. Joint Conf. Neural Networks, 2, 357–363. Radke, F. and Isermann, R., 1987, A parameter-adaptive PID-controller with stepwise parameter optimization, Automatica, 23, 449–457. Ralston, P.A. and Ward, T.L., 1985, Fuzzy control of industrial process, in Appl. Fuzzy Set Methodol. Ind. Eng., 29–45. B.V., North Holland. Elsevier Science Publishers. Sakai, Y. and Ohkusa, K., 1985, A fuzzy controller in turning process automation, in Ind. Appl. Fuzzy Control, 139–151. B.V., North Holland. Elsevier Science Publishers. Scharf, E.M. and Mandic, N.J., 1985, The application of a fuzzy controller to the control of a multidegreeof-freedom robot arm, in Ind. Appl. Fuzzy Control, 1–18. B.V., North Holland. Elsevier Science Publishers. Sheridan, S.E., 1984, Automatic kiln control at Oregon portland cement company’s Durkee plant utilizing fuzzy logic, IEEE Trans. Ind. Appl., 20, 562–568. Sheridan, T.B. 1987. Supervisory control. In G. Salvendy, Ed., Handbook of Human Factors/Ergonomics, Wiley, New York.

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Shinners, S. M., 1978, Modern Control System Theory and Application, Reading, MA: Addison–Wesley Publishing Co. Togai and Maski, 1991, An example of fuzzy logic control, Computer Design, 30, 93–103. Wakileh, B.A. and Gill, K.F., 1988, se of fuzzy logic in robotics, Computers Ind., 10, 35–46. Xu, C.W., 1989, Fuzzy system identification, IEEE Proc., 136(4), Pt. D, 146–150. Yasunobu, S. and Miyamoto, S., 1985, Automatic train operation system by predictive fuzzy control, Ind. Appl. Fuzzy Control, 1–18. B.V., North Holland. Elsevier Science Publishers.

Bibliography Distributed Power Coalition of America (DCPA) (1998). URL: www.dcpa.org. Huang, S.-H. and Nelson, R.M., 1993, Rule development and adjustment strategies of a fuzzy logic controller for an HVAC system: part two — experiment, ASHRAE Trans., (Submitted for review.) MacArthur, J.W., Grald, E.W., and Konar, A.F., 1989, An effective approach fir dynamically compensated adaptive control, ASHRAE Trans., 95(2), 415–423.

Additional Reading The following are suggested reading for those interested in learning more about neural networks and their use in control systems: Helferty, J.J., Collins, J.B., Wong, L.C., and Kam, M., 1989, A learning strategy for the control of a onelegged hopping robot, Proc. 1989 Am. Control Conf., 896–901. Kuperstein, M. and Rubinstein, J., 1989, Implementation of an adaptive neural network controller for sensory-motor coordination, IEEE Control Syst. Mag., April, 25–30. Lan, M., 1989, Adaptive control of unknown dynamical systems via neural network approach, Proc. 1989 Am. Control Conf., 910–915. Liu, H., Iderall, T., and Bekey, G., 1989, Neural network architecture for robot hand control, IEEE Control Syst. Mag., April, 38–41. Miller, R.C. and Seem, J.E., 1991, Comparison of artificial neural networks with traditional methods of predicting return time from night or weekend setback, ASHRAE Trans., 97(2), 500–508. Psaltis, D., Sideris, A., and Yamamura, A.,1988, A multilayered neural network controller, IEEE Control Syst. Mag., April, 17–21. Rumelhart, D.E. and McClelland, J.L., 1986, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Cambridge, MA: MIT Press. Wasserman, P.D., 1989, Neural Computing: Theory and Practice," New York: Van Nostrand Reinhold.

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7 Energy Resouces D. Yogi Goswami University of Florida

Robert Reuther U.S. Department of Energy

Richard Bajura West Virginia University

Philip C. Crouse Philip C. Crouse and Associates, Inc.

7.1 7.2 7.3

Introduction Types of Derived Energy Fossil Fuels

7.4

Biomass Energy

Coal • Environmental Aspects • Oil • Natural Gas Biomass Feedstock Technologies • Biomass Conversion Technologies

Ralph P. Overend National Renewable Energy Laboratory

7.5

Nuclear Resources

Lynn L. Wright

7.6

Solar Energy Resources

The Nuclear Fuel Cycle • Processing of Nuclear Fuel

Oak Ridge National Laboratory

Solar Energy Availability • Earth-Sun Relationships • Solar Time • Solar Radiation on a Surface • Solar Radiation on a Horizontal Surface • Solar Radiation on a Tilted Surface • Solar Radiation Measurements • Solar Radiation Data

James S. Tulenko University of Florida

Dale E. Berg Sandia National Laboratories1

7.7

Wind Origins • Wind Power • Wind Shear • Wind Energy Resource • Wind Characterization • Wind Energy Potential

Joel L. Renner Idaho National Engineering Laboratory

Marshall J. Reed U.S. Department of Energy

Wind Energy Resources

7.8

Geothermal Energy Heat Flow • Types of Geothermal Systems • Geothermal Energy Potential • Geothermal Applications • Environmental Constraints • Operating Conditions

This chapter describes the primary as well as derived energy sources. The objective is to provide information on the extent, availability, measurements and estimation, properties, and limitations of each type of resource. These considerations are important for an engineer to know and understand before attempting selection and design of an energy conversion system. The chapter also includes environmental impacts of energy resources since the environmental aspects are expected to play a major role in the selection of energy resources. In addition, there is a brief discussion of the costs associated with each resource to help in the economic analysis and comparison of the resources. The chapter starts with an introduction and background of a historical perspective on energy use and projections of the future energy needs in the U.S., the industrialized countries, and the world. The primary energy sources described in this chapter include fossil fuels such as coal, natural gas, petroleum (including their synthetic derivatives), biomass (including refuse-derived biomass fuels), nuclear, solar radiation, wind, geothermal, and ocean. In addition there is a brief section on derived energy sources including electricity. So, the terminology and units used for each energy resource and their equivalence are provided. 1 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.

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Chapter 7

Quadrillion Btu 800 History

Projections

623

568 600

517 471 348

400 285

368

404

311

243 207 200

0 1970

1975

1980

1985

1990

1995

2001

2010

2015

2020

2025

FIGURE 7.1.1 Historical and projected energy consumption. (From EIA. International Energy Outlook 2001, U.S. DOE, DOE/EIA-0484 (2001), Washington, D.C.)

7.1 Introduction D. Yogi Goswami Global energy consumption in the last 50 years has increased at a very rapid rate. Present trends in global population growth, rapid industrialization, and urbanization in major population centers of the world suggest that the world energy demand will continue to increase in the next 50 years (U.S. DOE, 2001). Figure 7.1.1 shows the historical and projected world energy consumption compiled by the Energy Information Agency. The energy resources available to furfill the world demand include Fossil fuels (oil, coal, natural gas) Nuclear fuels Geothermal Solar radiation Hydropower Biomass (crops, wood, municipal solid waste) Wind Ocean Out of all the energy resources, fossil fuels have been used the most (88% of total consumption) because of their extremely high energy densities and simplicity of conversion and use. Figure 7.1.2 shows the world energy consumption by resource. Recent concerns about the environment are expected to increase the use of natural gas for power production. Renewable energy resources, such as solar energy, wind, and biomass, are also expected to increase their share of the energy use. There is a strong sentiment in the world in favor of exploiting renewable energy resources, especially because of environmental concerns. How far that sentiment translates into practical use will depend on the development of the renewable energy technologies and prices of the fossil fuels.

© 2005 by CRC Press LLC

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FIGURE 7.1.2 World energy consumption by resource. (From EIA. International Energy Outlook 2001, U.S. DOE, DOE/EIA-0484 (2001), Washington, D.C.)

Defining Terms MTOE: Mega tons of oil equivalent; 1 MTOE = 42.63 × 1012 Btu. Quadrillion Btu: 1015 British thermal units (Btu), also known as Quad; 1 Btu = 1055 joules).

References EIA. International Energy Outlook 2001, International U.S. DOE, DOE/EIA-0484 (2001), Washington, D.C. IEA. 1994. World Energy Outlook, Economic Analysis Division, International Energy Agency, Paris. U.S. DOE. 1991. National Energy Strategy — Powerful Ideas for America, 1991. National Technical Information Service, U.S. Department of Commerce, Springfield, VA.

Further Information Historial and projected energy consumption are published annually by the Energy Information Agency, U.S. Department of Energy, Washington, D.C., and International Energy Agency, Paris.

7.2 Types of Derived Energy D. Yogi Goswami Energy from renewable and nonrenewable fuels can be converted to the derived energy forms — thermal, mechanical, and electrical, which are useful for various end uses such as transportation, buildings (heating, cooling, lighting), agricultural, and industrial. The derived energy forms are easily transformed from one type to the other. Figure 7.2.1 shows the projected U.S. energy use by end-use sector.

© 2005 by CRC Press LLC

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Chapter 7

History

60

Projections

Residential Commercial Industrial Transportation

50 40

30 20

10 0 1970

1980

1990

2000

2010

2020

2025

FIGURE 7.2.1 Projected U.S. energy use by end-use sector. (From EIA. Annual Energy Outlook, 2001, U.S. DOE, DOE/EIA-0383 (2001), Washington, D.C.) Percent of Total

19.7%

20.2%

20.7%

20.6%

16.5%

17.0%

14.6%

12.0%

Renewables Nuclear

30.9% 36.6%

34.1%

33.4%

Coal Natural Gas Oil

26.5%

17.7%

18.8%

22.2%

9.6%

9.9%

9.1%

10.0%

1995

1999

2010

2020

FIGURE 7.2.2 World electricity output. (From EIA. International Energy Outlook, 2001, U.S. DOE, DOE/EIA-0484 (2001), Washington, D.C.)

Transportation is mainly dependent on oil resources. Efforts to reduce urban air pollution are expected to increase the use of electricity as the preferred energy form for urban transportation. For most of the other end uses electricity will continue to be the preferred energy form. Therefore, it is important to understand the activity in the area of electricity production. Figure 7.2.2 shows the world installed electricity generation capacity by primary energy sources. The United States produces 770 GW (gigawatts), representing more than 25% of the world electricity capacity. Other major electricity producers are Russia, Europe, Japan, and China. It is expected that China, India, and Southeast Asian countries will add major electricity capacity in the next 20 years. © 2005 by CRC Press LLC

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Integrated resource planning (IRP), or least-cost planning, is the process used to optimize the resource options and minimize the total consumer costs including environmental and health costs that may be attributed to the resource. IRP examines all of the options, including the demand-side options, to minimize the total costs. There is considerable emphasis on IRP in a number of states in United States for future electric capacity and on demand-side management (DSM) for the current capacity (Kreith and Burmeister, 1993). The IRP process generally includes some combination of the following steps (Kreith and Burmeister, 1993): development of a load forecast; inventory of existing resources; identification of additional electrical capacity needs; demand-side management programs; screening and identification of options that are feasible; uncertainty analysis in view of uncertainty of future load, fuel prices, capital costs, etc; and selection of a resource or a mix of resources. Demand Side Management DSM refers to a mix of electrical utility-sponsored custom incentives and disincentives that influence the amount and timing of customer demand in order to better utilize the available resources. Kreith and Burmeister (1993) and SERI (1991) list a number of DSM strategies.

Defining Terms Demand-side management (DSM): Refers to a mix of incentives and disincentives that influence the amount and timing of energy use in order to better utilize the available resources. Integrated resource planning (IRP): The process to optimize the resource options and minimize the total consumer costs including environmental and health costs that may be all attributed to the resource.

References EIA. International Energy Outlook, 2001, U.S. DOE, DOE/EIA-0484 (2001), Washington, D.C. EIA. Annual Energy Outlook, 2001, U.S. DOE, DOE/EIA-0383 (2001), Washington, D.C. Kreith, F. and Burmeister, G. 1993. Energy Management and Conservation. National Conference of State Legislatures, Denver, CO. SERI. 1991. Demand Side Management Pocket Guide Book, Volume 1: Residential Technologies; and Volume 2: Commercial Technologies. SERI (Now National Renewable Energy Laboratory), Golden, CO.

Further Information Annual reviews published by the EIA, U.S. Department of Energy, and the International Energy Agency (see References) provide a wealth of information on electricity capacity and energy consumption by enduse sectors.

7.3 Fossil Fuels Coal

Robert Reuther Coal Composition and Classification Coal is a sedimentary rock formed by the accumulation and decay of organic substances, derived from plant tissues and exudates, which have been buried over periods of geological time, along with various mineral inclusions. Coal is classified by type and rank. Coal type classifies coal by the plant sources from which it was derived. Coal rank classifies coal by its degree of metamorphosis from the original plant sources and is therefore a measure of the age of the coal. The process of metamorphosis or aging is termed coalification. The study of coal by type is known as coal petrography. Coal type is determined from the examination of polished sections of a coal sample using a reflected-light microscope. The degree of reflectance and the color of a sample are identified with specific residues of the original plant tissues. These various residues © 2005 by CRC Press LLC

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TABLE 7.3.1 Coal Maceral Groups and Macerals Maceral Group Vitrinite

Exinite

Inertinite

Maceral

Derivation

Collinite Telinite Pseudovitrinite Sporinite Cutinite Alginite Micrinite Macrinite Semifusinite Fusinite Sclerotinite

Humic gels Wood, bark, and cortical tissue ? (Some observers place in the inertinite group) Fungal and other spores Leaf cuticles Algal remains Unspecified detrital matter, 650 TWh based on the current >1.0 Gt of cane. Current cane generation is estimated to be 30 to 50 TWh y–1 because only sufficient electricity is generated in most instances to power the mill in season. Electric Power Generation from Biomass Prime Mover Systems and Fuels Power generation takes place in prime movers, a technical term to describe engines of all kinds attached to alternators to generate electricity. Prime movers include the: • Steam engine • Steam turbine engine (the Rankine cycle) • Internal combustion engine (ICE), which comes in two types: the Otto or gasoline-based spark ignition (SI) engine, and the diesel or compression ignition (CI) engine • Gas turbine engine (Brayton cycle) • Stirling engine Each of these requires that the raw biomass be processed to some level and then used in the prime mover. Eventually fuel cells will replace the prime mover and alternator requirement by generating electricity directly from biomass-derived hydrogen fuels. The steam cycle (already discussed) uses combustor and boiler combinations to generate steam, requiring that the fuel be reduced in size (perhaps dried to some level) and have physical contaminants removed.

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The high-pressure steam is then expanded through a steam engine at small scales or through a turbine at larger scales. The efficiency of conversion technologies for combustion steam boiler systems is very scale dependent — a small steam engine or turbine would not exceed 10%. However, typically sufficient biomass is available over a transportation radius of 10 to 80 km to operate a unit in the range of 10 to 50 MW electricity output. Current Rankine cycles, at that scale, operate in the range of 25 to 30% efficiency and, as a consequence, require approximately 0.75 to 1 t of dry biomass to produce 1 MWh of electricity. Industrial and power generation boilers range in size from 100 to 300 MW thermal output. The major types of boilers are: pile burners; grate boilers; suspension fired boilers; fluidized beds; and circulating fluid beds. Recent trends in power generation are to use circulating and bubbling fluidized bed combustors, although the majority of units in current service are stoker-fired moving grate units. Biomass resources can be used in ICEs and gas turbine systems only if they are converted into clean liquid or gaseous fuels. Ethanol, biodiesel from lipids, or Fischer–Tropsch liquids can be used with little alteration to SI or CI engines. Gaseous fuels include the mixture of methane and carbon dioxide (biogas) produced by the action of microorganisms on biomass in anaerobic digestion (AD). AD is conducted at an industrial scale, using sewage or effluent streams containing high levels of soluble sugars, alcohols, and acids, as well as in landfills. The installed landfill power generation capacity in the U.S. is now in excess of 1 GW (Goldstein, 2002). The energy content of biogas is typically 20 to 25 MJ Nm–3 or between 50 and 60% that of natural gas. Fuel gases can also be produced by thermal gasification; when this is carried out at small scales, the gasifying agent is usually air. The product gas, which has a low calorific value (LCV) with a heating value of 12 to 15% that of natural gas, is often called a producer gas. LCV gas at about 5 to 6 MJ Nm–3 has carbon monoxide (CO), hydrogen (H2), and methane (CH4) as the main fuel components, diluted with a lot of nitrogen and carbon dioxide. Larger scale processes can utilize pure oxygen, enriched air, or an indirect heating method to produce medium calorific value (MCV) gas in the range of 15 to 25 MJ Nm–3 heating value with essentially the same fuel gas components but with much less inert diluent. Clean MCV gases can be burnt without much modification of ICEs or gas turbines. Cofiring Biomass with Coal In a biomass/coal cofiring operation, a biomass fuel is used to replace a portion of the coal fed to an existing coal-fired boiler. This has been practiced, tested, or evaluated for a variety of boiler technologies. There are four types of coal boilers: pulverized coal (PC); cyclone; stoker; and fluidized bed. PC boilers are the most common type in the U. S., representing about 92% of the U.S. coal-generating capacity, with cyclone boilers as the next most common, with about 7% representation. Demonstrations have been undertaken at coal plants ranging in size from about 30 MWe through to 700 MWe. Wood residues, hybrid poplar, and switchgrass have all been tested as supplemental fuels, and several utilities have biomass cofiring plants in commercial operation. Solid biomass can be fed to coal boilers by blending biomass on the coal pile or separately injecting biomass into the boiler. Introducing blended feeds to PC boilers can cause operational problems with the pulverizer, so the biomass proportion is limited to no more than 2 to 3% by heat input (4 to 5% by mass). Separate injection allows for the introduction of higher biomass percentages to the PC boiler — typically up to 15% on a heat input basis (about 30% by mass). However, separate injection requires additional fuel handling equipment and increased fuel preparation. Capital costs for the blended feed approach are typically $50/100 kW–1. For the separate feed approach, capital costs are typically higher, in the range of $175/200 kW–1. Cofiring can reduce boiler efficiency to some degree. For example, cofiring in a PC boiler with 10% of the heat input from wood may decrease the boiler efficiency by 0.5 to 1.5%; after “tuning” the boiler’s combustion output and adjusting for any efficiency losses, the combustion efficiency to electricity would be approximately 33%. Because coal plants comprise more than half of U.S. power plant capacity currently in operation, cofiring technology has the advantage of improving environmental performance at existing power plants, while providing fuel source flexibility and using proven and familiar equipment. It reduces air pollution

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TABLE 7.4.5 Heating Values of Fuel Gases

Hydrogen Carbon monoxide Methane Ethane Propane Butane

HHVa LHVb Btu ft –3 c

HHVa LHVb MJ Nm–3 d

325 322 1013 1792 2590 3370

12.75 12.63 39.74 69.63 99.02 128.39

275 322 913 1641 2385 3113

10.79 12.63 35.81 63.74 91.16 118.56

Note: Conversion factors for 1 MJ Nm–3 at 273.15 K and 101.325 kPa.
25.45 Btu ft–3 at 60°F and 14.73 psia. Inverse 1 Btu ft–3 at 60°F and 30 in. Hg.
0.0393 MJ Nm–3. a Higher heating value. b Lower heating value. c Standard temperatures and pressure of dry gas are 60°F and 14.73 psia (NIST, 2004). d S.I. units

emissions, GHG emissions, and the amount of waste ash generated as a by-product of the combustion. In addition, it requires relatively low up-front capital expenses compared to other renewable energy options; this makes it a straightforward and inexpensive way to diversify the fuel supply and divert biomass from landfill disposal. Thermal Gasification Technologies The conversion of biomass into a gaseous fuel opens up modern applications in electric power generation, the manufacture of liquid fuels, and the production of chemicals from biomass. The chemistry of gasification of biomass is best viewed as an extension of the pyrolysis process. Pyrolysis is simply defined as the chemical changes occurring in the solid biomass when heat is applied to a material in the absence of oxygen. The products of biomass pyrolysis include water, charcoal (or more correctly char, a carbonaceous solid), oils or tars, and permanent gases including methane, hydrogen, carbon monoxide, and carbon dioxide. The majority of gasifiers are partial oxidation reactors, in which just sufficient air or oxygen is introduced to burn part of the input biomass to provide the heat for pyrolysis and gasification. If the oxidant is air, the product gas is diluted by the nitrogen present. Although air is 79% nitrogen, the stoichiometry of partial oxidation is such that the final LCV product, gas, has about 50% nitrogen as a diluent. The energy content of the typical gases produced in biomass gasification is shown in Table 7.4.5. The use of pure oxygen as the gasification agent eliminates the nitrogen diluent and can produce medium calorific value (MCV) gases in the range of 10 to 20 MJ Nm–3. An alternative strategy is to carry out the gasification process by means of indirect heat the product stream is even higher in calorific value, because neither nitrogen nor the carbon dioxide produced from the combustion in-situ of the partial oxidation processes is present in the product gas stream. The challenges to achieve a clean and useable fuel gas have been addressed through gasifier design and postgasification processing to remove tar and particulate contaminants from the gas stream. Gasifier Systems The main challenge in gasification is enabling the pyrolysis and gas-reforming reactions to take place, using the minimum amount of energy, in reactors that are economical to construct. During a history dating back to the late 18th century, an extraordinary number of different designs and process configurations have been proposed. Prior to the development of fluidized bed technologies in the 1920s, the majority of the gasifiers were so-called fixed bed units. The flow of gasifying agents, usually air and steam,

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could be cocurrent with the biomass feed, or countercurrent; these are often described, respectively, as downdraft and updraft gasifiers. Downdraft gasification was widely used during the Second World War as an on-board fuel gas generator to offset the lack of gasoline. Millions of units were constructed and then abandoned as soon as petroleum supplies resumed. Units derived from the automotive application are marketed today as stationary generating sets equipped with ICEs, with SI or CI for power production in remote locations and in developing countries without grid systems. The simplest and oldest gasifier is the counterflow moving bed, which consists of an insulated shaft into which the feedstock (typically pieces larger than 3 cm on a side) are fed. The shaft is filled to a depth of 0.5 to 2 times the vessel diameter, and the mass of material is supported on a grate. The oxidant (air or enriched air/oxygen) and sometimes steam are introduced below the grate. The grate material is ignited and the hot gases flow up through the bed, exchanging heat with the down-flowing biomass material and, at the same time, pyrolyzing and drying it. At steady state, the bed level is maintained by continuous additions of the feed. The product gases include the desired fuel gases (methane, carbon monoxide, hydrogen, and C2 hydrocarbons), nitrogen, carbon dioxide, and water vapor, which exit at relatively low temperatures (C3) alcohols. Also, mixed thermochemical syngas and biotechnology routes to ethanol are under development. Ethanol can be synthesized from syngas using an anaerobic bacterium, Clostridium ljungdahlii. The growth conditions of the bacterium are managed so as to maximize the ethanol yield over the production of acetate (as acetic acid) from the key biological pathway intermediate acetyl-CoA, which is derived from the syngas. Although not yet commercialized, many of the development challenges of producing the bacterium and making effective gas–liquid contactors and bioreactors have been overcome. Yields are similar to those from the inorganic high temperature Fischer–Tropsch catalysts.

References Anonymous. (1994). Sweden’s largest biofuel-fired cfb up and running, Tampere, Finland, 16–17. Cannell, M.G.R. (2003). Carbon sequestration and biomass energy offset: theoretical, potential and achievable capacities globally, in Europe and the U.K. Biomass Bioenergy. 24: 97–116. © 2005 by CRC Press LLC

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Cralle, H. T., and Vietor, D. M. (1989). Productivity: solar energy and biomass, in Biomass Handbook, O. Kitani and C. W. Hal, Eds., Gordon and Breach Science Publishers, New York. 11–20. Ezzati, M. and Kammen, D.M. (2001). Quantifying the effects of exposure to indoor air pollution from biomass combustion on acute respiratory infections in developing countries. Environ. Health Perspect. 109(5): 481–488. Gallagher, P., Dikeman, M., Fritz,J., Wailes, E., Gauther, W., and Shapouri, H. (2003). Biomass from crop residues: cost and supply estimates. U.S. Department of Agriculture, Office of the Chief Economist, Office of energy Policy and New Uses. Agricultural Economic Report No. 819. Goldstein, J. (2002). Electric utilities hook up to biogas. Biocycle. March 2002: 36–37. Graham, R.L. et al. (1995). The Effect of location and facility demand on the marginal cost of delivered wood chips from energy crops: a case study of the state of Tennessee. In Proc. 2nd Biomass Conf. Am.: Energy, Environ., Agric., Ind., 1324–1333. Hall, D.O., Rosillo–Calle, F., Williams, R.H., and Woods, J. (1993). Biomass for energy: supply prospects, in Renewable Energy: Sources for Fuels and Electricity, T.B. Johansson, H. Kelly, A.K.N. Reddy, and R.H. Williams, Eds., Island Press, Washington, D.C., 593–651. Jenkins, B.M. (1997). A Comment on the optimal sizing of a biomass utilization facility under constant and variable cost scaling. Biomass Bioenergy. 13(1/2): 1–9. Lynd, L.R., C.E. Wyman, and T.U. Gerngross. (1999). Biocommodity engineering. Biotechnol. Progress. 15(5): 777–793. McKeever, D. (2003). Taking inventory of woody residuals. Biocycle. July 2003: 31–35. McLaughlin, S. and L. Kszos, personal communication. (2003). (These managers of switchgrass research at Oak Ridge National Laboratory have summarized the past 15 years of switchgrass research in a Biomass and Bioenergy 2005 paper in press). NIST (2004). Handbook 44, Specifications, Tolerances, and Other Technical Requirements for Weighing and Measuring Devices. Butcher, T., Crown, L., Suiter, R., and Williams, J., Eds. Prasad, K. (1985). Stove design for improved dissemination, in Wood-Stove Dissemination, Robin Clarke, Ed., Intermediate Technology Publications. London 59–74. Sampson, R.N. et al. (1993). Biomass management and energy. Water, Air, Soil Pollut. 70: 139–159. Stahl, K., M. Neergard, and J. Nieminen. (2000). Final report: Varnamo demonstration programme. In Progress in Thermochemical Biomass Conversion, Ed. A.V. Bridgwater, Blackwell Sciences Ltd., Oxford U.K. 549–563. Walsh, M.E., R.L. Perlack, A. Turhollow, D.G. de la Torre Ugarte, D.A. Becker, R.L. Graham, S.E. Slinsky, and D.E. Ray. (2000). Biomass feedstock availability in the United States: 1999 state level analysis. Report prepared for the U.S. Department of Energy found at: http://bioenergy.ornl.gov/pubs/ resource_data.html. Wright, L.L. (1994). Production technology status of woody and herbaceous crops. Biomass Bioenerg., 6(3): 191–209. Wyman, C.E. (1994). Ethanol from lignocellulosic biomass: technology, economics, and opportunities, BioResource Technol., 50(1): 3–16. Zhang, M., C. Eddy, K. Deanda, M. Finkelstein, and S. Picataggio. (1995). Metabolic engineering of a pentose metabolism pathway in ethanologenic Zymomonas mobilis. Science. 267(5195): 13 January 1995. 240–243.

Further information Combustion is widely described in the literature. The International Energy Agency recently produced an extremely useful reference book on the topic of biomass combustion: van Loo, S. and J. Koppejan, Eds. (2002). Handbook of Biomass Combustion and Cofiring. Enschede, Netherlands, Twente University Press (A multiauthor IEA Bioenergy collaboration Task 32 publication). Power generation is described and analyzed in: EPRI (1997). Renewable Energy Technology Characterizations. Washington, D.C., Electric Power Research Institute. © 2005 by CRC Press LLC

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Gasification in general and synthetic liquid fuels are well described in: Probstein, R. F. and R. E. Hicks (1982). Synthetic Fuels. New York, McGraw–Hill Inc. Anaerobic digestion is the subject of a wonderful handbook that is only available in German at present: Schulz, H. and B. Eder (2001). Bioga Praxis: grundlagen, planung, anlagenbau, beispiele. Freiburg, Germany, Ökobuch Verlag, Staufen bei Freiburg. Ethanol from lignocellulosics as well as a useful section on starch ethanol can be obtained in: Wyman, C. E., Ed. (1996). Handbook on Bioethanol: Production and Utilization. Applied Energy Technology Series. Washington, D.C., Taylor and Francis. Additional information on ethanol can be obtained from the Web sites of the National Corn Growers Association (http://www.ncga.com) and the Renewable Fuels Association (http://www.ethanolrfa.org/). The last three decades of biomass activity in the United States are described in: Chum, H. L. and R. P. Overend (2003). Biomass and Bioenergy in the United States. In Advances in Solar Energy: an Annual Review of Research and Development. Ed. Y. Goswami. American Solar Energy Society, Boulder, CO. USA. 83–148. Additional information on energy efficiency and renewable energy technologies can be obtained from the U.S. Energy Efficiency and Renewable Energy Websites http://www.eere.energy.gov/biomass.html, and http://www.eere.energy.gov/RE/bioenergy.html. Also the Website http://bioenergy.ornl.gov/ provides useful resource information and many links to other bioenergy Websites.

7.5 Nuclear Resources James S. Tulenko The Nuclear Fuel Cycle Sources of Nuclear Fuels and World Reserves Nuclear power can use two naturally occurring elements, uranium and thorium, as the sources of its fissioning energy. Uranium can be a fissionable source (fuel) as mined (Candu Reactors in Canada), while thorium must be converted in a nuclear reactor into a fissionable fuel. Uranium and thorium are relatively plentiful elements ranking about 60th out of 80 naturally occurring elements. All isotopes of uranium and thorium are radioactive. Today, natural uranium contains, in atomic abundance, 99.2175% Uranium-238 (U238); 0.72% Uranium-235 (U235); and 0.0055% Uranium-234 (U234). Uranium has atomic number 92, meaning all uranium atoms contain 92 protons, with the rest of the mass number being composed of neutrons. Uranium-238 has a half-life of 4.5 × 109 years (4.5 billion years), U-235 has a halflife of 7.1 × 108 years (710 million years), and U-234 has a half-life of 2.5 × 105 years (250 thousand years). Since the age of the earth is estimated at 3 billion years, roughly half of the U-238 present at creation has decayed away, while the U-235 has changed by a factor of sixteen. Thus, when the earth was created, the uranium-235 enrichment was on the order of 8%, enough to sustain a natural reactor of (there is evidence of such an occurrence in Africa). The U-234 originally created has long disappeared, and the U-234 currently present occurs as a product of the decay of U-238. Uranium was isolated and identified in 1789 by a German scientist, Martin Heinrich Klaproth, who was working with pitchblend ores. No one could identify this new material he isolated, so in honor of the planet Uranus which had just been discovered, he called his new material Uranium. It wasn’t until 1896, when the French scientist Henri Becquerel accidentally placed some uranium salts near some paperwrapped photographic plates, that radioactivity was discovered. Until 1938, when the German scientists Otto Hahn and Fritz Shassroen succeeded in uranium fission by exposure to neutrons, uranium had no economic significance except in coloring ceramics, where it proved valuable in creating various shades of orange, yellow, brown, and dark green. When a uranium atom is fissioned it releases 200 million electron volts of energy; the burning of a carbon (core) atom releases 4 electron volts. This diffenence of 50 million times in energy release shows the tremendous difference in magnitude between chemical and nuclear energy. © 2005 by CRC Press LLC

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Uranium is present in the earth’s crust to the extent of four parts per million. This concentration makes uranium about as plentiful as beryllium, hafnium, and arsenic; and greater in abundance than tungsten, molybdenum, and tantalum. Uranium is an order of magnitude more plentiful than silver and a hundred times more plentiful than gold. It has been estimated that the amount of uranium in the earth’s crust to a depth of 12 miles is of the order of 100 trillion tons. Thorium, which is composed of only one isotope, Thorium-232, has a half-life of 14 billion years (1.4 × 1010 yr), is more than three times more abundant than uranium, and is in the range of lead and gallium in abundance. Thorium was discovered by Berjelius in 1828 and named after Thor, the Scandinavian god of war. For reference, copper is approximately five times more abundant than thorium and twenty times more abundant than uranium. Uranium is chemically a reactive element; therefore, while it is relatively abundant, it is found chemically combined as an oxide (U3O8 or UO2) and never as a pure metal. Uranium is obtained in three ways, either by underground mining, open pit mining, or in situ leaching. An economic average ore grade is normally viewed as .2% (4 pounds per short ton), though recently ore grades as low as .1% have been exploited. A large quantity of uranium exists in sea-water which has an average concentration of 3 × 10-3 ppm, yielding an estimated uranium quantity available in sea-water of 4000 million tons. A pilot operation was successfully developed by Japan to recover uranium from sea-water, but the cost was about $900/lb, and the effort was shut down as uneconomical. The major countries with reserves of urnaium in order of importance are Australia, United States, Russia, Canada, South Africa, and Nigeria. The countries with major thorium deposits are India, Brazil, and the United States. It is estimated that for a recovery value of $130/kg ($60/lb), the total uranium reserves in these countries are approximately 1.5 million tonnes of uranium in the U.S., 1 million tonnes of uranium in Australia, .7 million tonnes of uranium in Canada, and 1.3 million tonnes of uranium in the former Soviet Union. As mentioned earlier, thorium reserves are approximately four times greater. With the utilization of breeder reactors, there is enough uranium and thorium to provide electrical power for the next thousand years at current rates of usage.

Processing of Nuclear Fuel Once the uranium ore is mined it is sent to a concentrator (mill) where it is ground, treated, and purified. Since the ore is of a grade of .1 to .2% uranium, a ton of ore contains only between 1 to 2 kilograms of uranium per 1000 kilograms of ore. Thus, thousands to tonnes of ore have to be extracted and sent to a mill to produce a relatively small quantity of uranium. In the concentration process approximately 95% of the ore is recovered as U3O8 (yellowcake) to a purity grade of about 80%. Thus, assuming 0.15% uranium ore, the milling and processing of a metric ton (1000 kg) of ore yields a concentrate of 1.781 kg (1.425 kg of uranium and 0.356 kg of impurities). For this reason the mills must be located relatively close to the mine site. The ore tailings (waste) amounts to 998.219 kg and contains quantities of radon and other urnaium decay products and must be disposed of as a radioactive waste. The U3O8 concentrate is then taken to a conversion plant where the concentrate is further purified (the 20% impurities are removed) and the uranium yellowcake is converted to uranium hexafluoried UF6). The uranium hexafluoride is a gas at fairly low temperature and is an ideal material for the U-235 isotope enriching processes of either gaseous diffusion or gaseous centrifuge. The UF6 is shipped in steel cylinders in a solid state, and UF6 is vaporized by putting the cylinder in a steam bath. If the uranium is to be enriched to 4% U235, then 1 kilogram of 4% U235 product will require 7.4 kilograms of natural uranium feed and will produce 6.4 kilograms of waste uranium (tails or depleted uranium) with a U235 isotope content of 0.2%. This material is treated as a radioactive waste. Large quantities of tails (depleted uranium) exist as UF6 in their original shipping containers at the enriching plants. Depleted uranium (a dense material) has been used as shields for radioactive sources, armor piercing shells, balancing of helicopter rotor tips, yacht hold ballast, and balancing of passenger aircraft. The enriched UF6 is then sent to a fabrication plant where it is converted to a uranium dioxide (UO2) powder. The powder is pressed and sintered into cylindrical pellets which are placed in zircaloy tubes (an © 2005 by CRC Press LLC

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URANIUM MINES & MILLS

CONVERSION TO UF6

ENRICHING

CONVERSION TO FUEL

PLUTONIUM RECOVERED URANIUM REACTOR

REPROCESSING

WASTE STORAGE BY PRODUCTS FIGURE 7.5.1 The nuclear fuel cycle.

alloy of zirconium), pressurized with helium, and sealed. The rods are collected in an array (~17 × 17) bound together by spacer grids, with top and bottom end fittings connected by tie rods or guide tubes. Pressurized water reactor fuel assemblies, each containing approximately 500 kilograms of uranium, are placed in a reactor for 3 to 4 years. A single fuel assembly produces 160,000,000 kilowatt hours of electricity and gives 8,000 people their yearly electric needs for its three years of operation. When the fuel assembly is removed from the reactor it must be placed in a storage pond to allow for removal of the decay heat. After approximately five years of wet storage, the fuel assembly can be removed to dry storage in concrete or steel containers. In the United States the current plan is to permanently store the nuclear fuel, with the Department of Energy assuming responsiblility for the “spent” fuel. The money for the government to handle the storage comes from a fee of 1 mill per kilowatt hour paid by consumers of nuclear-generated electricity. A mill is a thousandth of a dollar or a tenth of a penny. Thus, the fuel assembly described above would have collected $160,000 in the waste fund for the Department of Energy to permanently store the fuel. In Europe, when the fuel is taken out of wet storage it is sent to a reprocessing plant where the metal components are collected for waste disposal; and the fuel is chemically recovered as 96% uranium, which is converted to uranium dioxide for recycling to the enrichment plant, 1% plutomium, which is converted to fuel or placed in storage, and 3% fission products which are encased in glass and permanently stored. The important thing to remenber about the fuel cycle is the small quantitiy of radioactive fission products (1.5 kilograms) which are created as radioactive waste in producing power which can serve the yearly eletricity needs of 8,000 people for the three years that it operates. The schematic of the entire fuel cycle showing both the United States system (once-through) and the European (recycle) system is given in Figure 7.51.

7.6 Solar Energy Resources D. Yogi Goswami The sun is a vast nuclear power plant of the fusion variety which generates power in the form of radiant energy at a rate of 3.8 × 1023 kW. An extremely small fraction of this is intercepted by Earth, but even © 2005 by CRC Press LLC

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this small fraction amounts to the huge quantity of 1.8 × 1014 kW. On the average, about 60% of this energy incident at the outer edge of the atmosphere, reaches the surface. To compare these numbers with our energy needs, consider the present electrical-generating capacity in the United States, which is approximately of 7 × 108 kW. This is equivalent to an average solar radiation falling on only 1000 square miles in a cloudless desert area. It must, however, be remembered that solar energy is distributed over the entire surface of Earth facing the sun, and it seldom exceeds 1.0 kW/m2. Compared to other sources, such as fossil fuels or nuclear power plants, solar energy has a very low energy density. However, solar radiation can be concentrated to achieve very high energy densities. Indeed, temperatures as high as 3000 K have been achieved in solar furnaces. Solar energy technology has been developed to a point where it can replace most of the fossil fuels or fossil fuel-derived energy. In many applications it is already economical, and it is a matter of time before it becomes economical for other applications as well. This section deals in the availability of solar radiation, including methods of measurement, calculation, and available data.

Solar Energy Availability Detailed information about solar radiation availability at any location is essential for the design and economic evaluation of a solar energy system. Long-term measured data of solar radiation are available for a large number of locations in the United States and other parts of the world. Where long-term measured data are not available, various models based on available climatic data can be used to estimate the solar energy availability. The solar energy is in the form of electromagnetic radiation with the wavelengths ranging from about 0.3 µm (10–6 m) to over 3 µm, which correspond to ultraviolet (less than 0.4 µm), visible (0.4 and 0.7 µm), and infrared (over 0.7 µm). Most of this energy is concentrated in the visible and the near-infrared wavelength range (see Figure 7.6.1). The incident solar radiation, sometimes called insolation, is measured as irradiance, or the energy per unit time per unit area (or power per unit area). The units most often used are watts per square meter (W/m2), British thermal units per hour per square foot (Btu/hr-ft2), and Langleys (calories per square centimeter per minute, cal/cm2-min).

Spectral irradiance (W/m2 . µm)

2400

Air mass zero solar spectrum, 1353 W/m2 Black-body curve 5762 K (normalized) 1353 W/m2 Air mass two solar spectrum α 0.66, β 0.085, H2O 2 cm, O3 0.34 cm, 691.2 W/m2

1600

Air mass two solar spectrum without molecular absorption

O3

800

H2O UV

O2 H2O 1R

Visible O3 0

0.2

H2O 0.8

1.4

H2O, CO2

2.0

H2O, CO2

2.6

Wavelength λ (µm)

FIGURE 7.6.1 Spectral distribution of solar energy at sea level. (Reprinted by permission from Goswami, D.Y., Kreith, F., and Kreider, J.F., Principles of Solar Engineering, Taylor and Francis, Philadelphia, PA, 2000.) © 2005 by CRC Press LLC

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The amount of solar radiation falling on a surface normal to the rays of the sun outside the atmosphere of the earth (extraterrestrial) at mean Earth-sun distance (D) is called the solar constant, Io. Measurements by NASA indicated the value of solar constant to be 1353 W/m2 (±1.6%). This value was revised upward and the present accepted value of the solar constant is 1377 W/m2 (Quinlan, 1979) or 437.1 Btu/hr-ft2 or 1.974 langleys. The variation in seasonal solar radiation availability at the surface of Earth can be understood from the geometry of the relative movement of Earth around the sun.

Earth-Sun Relationships Figure 7.6.2 shows the annual motion of Earth around the sun. The extraterrestrial solar radiation varies throughout the year because of the variation in the Earth-sun distance (D) as: I = Io ( D Do )

2

(7.6.1)

which may be approximated as (Spencer, 1971)

(D D ) o

2

= 1.00011 + 0.034221cos( x ) + 0.00128 sin( x ) + 0.000719 cos(2 x ) + 0.000077 sin(2 x ) (7.6.2)

where x = 360( N − 1) 365°

(7.6.3)

and N = Day number (starting from January 1 as 1). The axis of the Earth is tilted at an angle of 23.45° to the plane of its elliptic path around the sun. This tilt is the major cause of the seasonal variation of solar radiation available at any location on Earth. The angle between the Earth-sun line and a plane through the equator is called solar declination, δ. The declination varies between –23.45° to +23.45° in 1 year. It may be estimated by the relation: δ = 23.45° sin[360 (284 + N ) 365°]

(7.6.4)

The apparent motion of the sun around the earth is shown in Figure 7.6.3. The solar altitude angle, β, and the solar azimuth angle, Φ, describe the position of the sun at any time. Polar Axis

Ecliptic axis Sep. 21

23.45°

89.83 million miles 1.471 × 1011 m

Sun

95.9 million miles 1.521 × 1011 m

Dec. 21

June 21 March 21

Ecliptic Plane

FIGURE 7.6.2 Annual motion of the Earth around the sun. (Adapted from Goswami, D.Y., Kreith, F., and Kreider, J., Principles of Solar Engineering, Taylor and Francis, Philadelphia, PA, 2000.)

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V

West α Φ South

S

o

North

East

FIGURE 7.6.3 Apparent daily path of the sun across the sky from sunrise to sunset, showing the solar altitude and azimuth angles.

Solar Time The sun angles are found from the knowledge of solar time, which differs from the local time. The relationship between solar time and local standard time (LST) is given by Solar Time = LST + ET + 4( Lst − Lloc )

(7.6.5)

where ET is the equation of time, which is a correction factor in minutes that accounts for the irregularity of the motion of the Earth around the sun. Lst is the standard time meridian and Lloc is the local longitude. ET can be calculated from the following empirical equation: ET(in minutes) = 9.87 sin 2 B − 7.53 cos B − 1.5 sin B

(7.6.6)

where B = 360(N – 81)/365°. The sun angles α (altitude) and Φ (azimuth) can be found from the equations: sinα = cos
cosδcosH + sin
sinδ

(7.6.7)

sinΦ = cosδsinH/cosα

(7.6.8)

Number of minutes from local solar noon 4 min degree

(7.6.9)

where
= latitude angle,

and H = Hour angle =

(At solar noon, H = 0, so α = 90 – |
– δ| and Φ = 0.)

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TABLE 7.6.1 Average Values of Atmospheric Optical Depth (τ) and Sky Diffuse Factor (C) for 21st Day of Each Month Month

1

2

3

4

5

6

7

8

9

10

11

12

τ C

0.142 0.058

0.144 0.060

0.156 0.071

0.180 0.097

0.196 0.121

0.205 0.134

0.207 0.136

0.201 0.122

0.177 0.092

0.160 0.073

0.149 0.063

0.142 0.057

Source: Threlkeld, J.L. and Jordan, R.C., ASHRAE Trans., 64, 45, 1958.

Solar Radiation on a Surface As solar radiation, I, passes through the atmosphere, some of it is absorbed by air and water vapor, while some gets scattered by molecules of air, water vapor, aerosols, and dust particles. The part of solar radiation that reaches the surface of the Earth with essentially no change in direction is called direct or beam normal radiation, IbN. The scattered radiation reaching the surface from the atmosphere is called diffuse radiation, Id. IbN can be calculated from the extraterrestrial solar irradiance, I, and the atmospheric optical depth τ as (Goswami et al., 1981; ASHRAE, 1995) I bN = Ie −τ sec θ z

(7.6.10)

where θz is the solar zenith angle (angle between the sun rays and the vertical). The atmospheric optical depth determines the attenuation of the solar radiation as it passes through the atmosphere. Threlkeld and Jordan (1958) calculated values of τ for average atmospheric conditions at sea level with a moderately dusty atmosphere and amounts of precipitable water vapor equal to the average value for the United States for each month. These values are given in Table 7.6.1. To account for the differences in local conditions from the average sea level conditions Equation (7.6.10) is modified by a parameter called Clearness Number, Cn, introduced by Threlkeld and Jordan (1958): I bN = CnIe −τ sec θ z

(7.6.11)

values of Cn vary between 0.85 and 1.15.

Solar Radiation on a Horizontal Surface Total incident solar radiation on a horizontal surface is given by It ,Horizontal = I bN cos θ z + CI bN = I bN sinβ + CI bN

(7.6.12) (7.6.13)

where θz is called the solar zenith angle and C is called the sky diffuse factor, as given in Table 7.6.1.

Solar Radiation on a Tilted Surface For a surface of any orientation and tilt as shown in Figure 7.6.4, the angle of incidence, θ, of the direct solar radiation is given by cosθ = cosαcosγsinβ + sinαcosβ

(7.6.14)

where γ is the angle between horizontal projections of the rays of the sun and the normal to the surface. β is the tilt angle of the surface from the horizontal.

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FIGURE 7.6.4 Definitions of solar angles for a tilted surface.

For a tilted surface with angle of incidence θ, the total incident solar radiation is given by Ib = I bN cosθ + Idiffuse + I reflected

(7.6.15)

I diffuse = CI bN (1 + cosβ ) 2

(7.6.16)

I reflected = ρI bN (C + sin α )(1 − cos β ) 2

(7.6.17)

where

and

where ρ is the reflectivity of the surroundings. For ordinary ground or grass, ρ is approximately 0.2 while for ground covered with snow it is approximately 0.8.

Solar Radiation Measurements Two basic types of instruments are used in measurements of solar radiation. These are (see Figure 7.6.5): 1. Pyranometer: An instrument used to measure global (direct and diffuse) solar radiation on a surface. This instrument can also be used to measure the diffuse radiation by blocking out the direct radiation with a shadow band. 2. Pyrheliometer: This instrument is used to measure only the direct solar radiation on a surface normal to the incident beam. It is generally used with a tracking mount to keep it aligned with the sun. More-detailed discussions about these and other solar radiation measuring instruments can be found in Zerlaut (1989).

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FIGURE 7.6.5 Two basic instruments for solar radiation: (a) pyranometer; (b) pyrheliometer.

Solar Radiation Data Measured values of solar radiation data for locations in the United States are available from the National Climatic Center in Asheville, NC. A number of states have further presented solar radiation data for locations in those states in readily usable form. Weather services and energy offices in almost all the countries have available some form of solar radiation data or climatic data that can be used to derive solar radiation data for locations in those countries. Table 7.6.2 to Table 7.6.4 give solar radiation data for clear days for south-facing surfaces in the Northern Hemisphere (and northern-facing surfaces in the Southern Hemisphere) tilted at 0°, 15°, 30°, 45°, 60°, 75°, and vertical, for latitudes 0°, 30°, and 60°. The actual average solar radiation data at a location is less than the values given in these tables because of the cloudy and partly cloudy days in addition to the clear days. The actual data can be obtained either from long-term measurements or from modeling based on some climatic parameters, such as percent sunshine. Worldwide solar radiation data is available from the World Radiation Data Center (WRDC). WRDC has been archiving data from over 500 stations and operates a website in collaboration with NREL (wrdc-mgo.nrel.gov). © 2005 by CRC Press LLC

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TABLE 7.6.2 Average Daily Total Solar Radiation on South-Facing Surfaces in Northern Hemisphere; Latitude = 0°N Month

Horiz.

15°

30°

45°

60°

75°

90°

1 2 3 4 5 6 7 8 9 10 11 12

31.11 32.34 32.75 31.69 29.97 28.82 29.22 30.59 31.96 32.18 31.33 30.51

34.13 33.90 32.21 29.13 26.08 24.43 25.08 27.48 30.51 32.82 33.80 33.90

35.13 33.45 29.79 24.93 20.81 18.81 19.66 22.87 27.34 31.54 34.28 35.27

34.02 31.03 25.67 19.39 14.64 12.54 13.48 17.13 22.65 28.44 32.72 34.53

30.90 26.80 20.12 12.97 8.34 6.66 7.45 10.82 16.78 23.73 29.24 31.73

25.96 21.05 13.53 6.59 4.92 5.07 5.17 5.58 10.18 17.72 24.08 27.05

19.55 14.18 6.77 4.97 5.14 5.21 5.31 5.32 5.33 10.84 17.58 20.83

TABLE 7.6.3 Average Daily Total Solar Radiation on South-Facing Surfaces in Northern Hemisphere; Latitude = 30°N Month

Horiz.

15°

30°

45°

60°

75°

90°

1 2 3 4 5 6 7 8 9 10 11 12

17.19 21.47 26.81 31.48 34.49 35.61 35.07 32.60 28.60 23.41 18.50 15.90

22.44 26.14 30.04 32.71 33.96 34.24 34.06 33.00 30.87 27.38 23.48 21.19

26.34 29.25 31.50 32.06 31.56 31.03 31.21 31.54 31.35 29.74 27.05 25.21

28.63 30.59 31.09 29.57 27.49 26.28 26.76 28.35 30.02 30.33 28.98 27.68

29.15 30.06 28.84 25.44 22.08 20.40 21.11 23.68 26.97 29.10 29.14 28.44

27.86 27.70 24.90 19.96 15.82 13.97 14.77 17.89 22.42 26.14 27.51 27.43

24.85 23.68 19.54 13.60 9.49 8.02 8.68 11.57 16.67 21.66 24.20 24.71

TABLE 7.6.4 Average Daily Total Solar Radiation on South-Facing Surfaces in Northern Hemisphere; Latitude = 60°N Month

Horiz.

15°

30°

45°

60°

75°

90°

1 2 3 4 5 6 7 8 9 10 11 12

1.60 5.49 12.82 21.96 30.00 33.99 32.26 25.37 16.49 8.15 2.70 0.82

3.54 9.38 17.74 26.22 32.79 35.82 34.47 28.87 21.02 12.39 5.27 2.06

5.26 12.71 21.60 28.97 33.86 35.93 34.97 30.80 24.34 15.90 7.53 3.16

6.65 15.25 24.16 30.05 33.17 34.29 33.71 31.02 26.22 18.45 9.31 4.07

7.61 16.82 25.22 29.38 30.73 31.00 30.78 29.53 26.54 19.85 10.51 4.71

8.08 17.32 24.73 27.00 26.72 26.26 26.36 26.42 25.27 20.01 11.03 5.05

8.03 16.72 22.71 23.09 21.45 20.46 20.80 21.94 22.51 18.92 10.84 5.07

Note: Values are in megajoules per square meter. Clearness number = 1.0; ground reflection = 0.2.

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Defining Terms Diffuse radiation: Scattered solar radiation coming from the sky. Direct or beam normal radiation: Part of solar radiation coming from the direction of the sun on a surface normal to the sun’s rays. Equation of time: Correction factor in minutes, to account for the irregularity of the Earth’s motion around the sun. Extraterrestrial solar radiation: Solar radiation outside Earth’s atmosphere. Insolation: Incident solar radiation measured as W/m2 or Btu/hr-ft2. Solar altitude angle: Angle between the solar rays and the horizontal plane. Solar azimuth angle: Angle between the true south horizontal line and the horizontal projection of the sun’s rays. Solar constant: Extraterrestrial solar radiation at the mean Earth-sun distance. Solar declination: Angle between the Earth-sun line and a plane through the equator.

References ASHRAE. 1995. 1995 HVAC Applications, ASHRAE, Atlanta, GA. Goswami, D.Y. 1986. Alternative Energy in Agriculture, Vol. 1, CRC Press, Boca Raton, FL. Goswami, D.Y., Klett, D.E., Stefanakos, E.K., and Goswami, T.K. 1981. Seasonal variation of atmospheric clearness numbers for use in solar radiation modelling, AIAA J. Energ., 5(3) 185. Goswami, D.Y., Kreith, F., and Kreider, J. 2000. Principles of Solar Engineering, Taylor and Francis, Philadelphia, PA. Kreith, F. and Kreider, J.F. 1978. Principles of Solar Engineering, Hemisphere Publishing, Washington, D.C. Quinlan, F.T., Ed. 1979. SOLMET Volume 2: Hourly Solar Radiation — Surface Meteorological Observations, National Oceanic and Atmospheric Administration, Asheville, NC. Spencer, J.W. 1971. Fourier series representation of the position of the sun, Search, 2, 172. Threlkeld, J.L. and Jordan, R.C. 1958. Direct radiation available on clear days, ASHRAE Trans., 64, 45. Zerlaut, G. 1989. Solar Radiation Instrumentation, Chapter 5, Solar Resources, R.L. Hulstrom, Ed., The MIT Press, Cambridge, MA.

Further Information Solar Resources, edited by R.H. Hulstrom, MIT Press, Cambridge, MA, 1989. World Radiation Data Center (WRDC), St. Petersburg, Russia: WRDC, operating under the auspices of World Meteorological Organization (WMO), has been archiving data over 500 stations and operates a website in collaboration with NREL (wrdc-mgo.nrel.gov).

7.7 Wind Energy Resources2 Dale E. Berg Wind Origins The primary causes of atmospheric air motion, or wind, are uneven heating of the Earth by solar radiation and the Earth’s rotation. Differences in solar radiation absorption at the surface of the Earth and transference back to the atmosphere create differences in atmospheric temperature, density, and pressure, 2

This work was supported by the United States Department of Energy under Contract DE-AC04-94AL85000.

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which in turn create forces that move air from one place to another. For example, land and water along a coastline absorb radiation differently, and this is the dominant cause of the light winds or breezes normally found along a coast. The Earth’s rotation gives rise to semipermanent global wind patterns such as trade winds, westerlies, easterlies, and subtropical and polar jets.

Wind Power The available power in the wind with air density ρ, passing through an area A, perpendicular to the wind, at a velocity U, is given by Power = ½ρAU3

(7.7.1)

Air density decreases with increasing temperature and increasing altitude above sea level. The effect of temperature on density is relatively weak and is normally ignored because these variations tend to average out over the period of a year. The density difference due to altitude, however, is significant; it does not average out and cannot be ignored. For example, the air density at Denver, Colorado (elevation 1600 m, or 5300 ft, above sea level), is approximately 14% lower than at sea level, so wind at Denver contains 14% less power than wind of the same velocity at sea level. From Equation (7.7.1), it is obvious that the most important factor in the available wind power is the velocity of the wind — an increase in wind velocity of only 20%, e.g., from 5 to 6 m/s (11.2 to 13.4 mph), yields a 73% increase in available wind power.

Wind Shear Wind moving across the Earth’s surface is slowed by trees, buildings, grass, rocks, and other obstructions in its path. The result is a wind velocity that varies with height above the Earth’s surface — a phenomena known as wind shear. For most situations, wind shear is positive (wind speed increases with height), but situations in which the wind shear is negative or inverse are not unusual. In the absence of actual data for a specific site, a commonly used approximation for wind shear in an open area is: U/Uo = (h/ho)α

(7.7.2)

where U = the velocity at a height h Uo = the measured velocity at height ho α = the wind shear exponent The wind shear exponent, α, varies with terrain characteristics, but usually falls between 0.10 and 0.25. Wind over a body of open water is normally well modeled by a value of α of about 0.10; wind over a smooth, level, grass-covered terrain such as the U.S. Great Plains by an α of about 0.14; wind over row crops or low bushes with a few scattered trees by an α of 0.20; and wind over a heavy stand of trees, several buildings, or hilly or mountainous terrain by an α of about 0.25. Short-term shear factors as large as 1.25 have been documented in rare, isolated cases. The available wind power at a site can vary dramatically with height due to wind shear. For example, for α = 0.20, Equation (7.7.1) and Equation (7.7.2) reveal that the available wind power at a height of 50 m is approximately {(50/10)0.2}3 = 2.63 times the available wind power at a height of 10 m.

Wind Energy Resource The amount of energy available in the wind (the wind energy resource) is the average amount of power available in the wind over a specified period of time — commonly 1 year. If the wind speed is 20 m/s,

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Wind Speed Distribution 0.08 Amarillo, TX Airport

0.07

Rayleigh (6.6 m/s)

Probability Density

0.06 0.05 0.04 0.03 0.02 0.01 0.00

0

5

10 Wind Speed, m/s

15

20

FIGURE 7.7.1 Rayleigh and measured wind speed distributions.

the available power is very large at that instant, but if it only blows at that speed for 10 h per year and the rest of the time the wind speed is near zero, the resource for the year is small. Therefore, the site wind speed distribution, or the relative frequency of occurrence for each wind speed, is very important in determining the resource. This distribution is often presented as a probability density function, such as the one shown in Figure 7.7.1. The probability that the wind occurs in any given wind speed range is given by the area under the density function for that wind speed range. If the actual wind speed probability density distribution is not available, it is commonly approximated with the Rayleigh distribution, given by: f (U ) =

 πU 2 πU exp − 2 4U  4U 

(7.7.3)

where ƒ(U) = the frequency of occurrence of wind speed U U = the yearly average wind speed The measured wind speed distribution at the Amarillo, Texas, airport (yearly average wind speed of 6.6 m/s) is plotted in Figure 7.7.1, together with the Rayleigh distribution for that wind speed. It is obvious that the Rayleigh distribution is not a good representation for the Amarillo airport. How large is the wind energy resource? Even though wind energy is very diffuse, the total resource is very, very large. In the U.S. and many other countries around the world, the resource is large enough to supply the entire current energy consumption of the country, potentially. In 1987, scientists at Batelle Pacific Northwest Laboratory (PNL) in the U.S. carefully analyzed and interpreted the available longterm wind data for the U.S. and summarized their estimate of the wind energy resources in the Wind Energy Resource Atlas of the United States (Elliott et al., 1987). Their summary for the entire U.S. is reproduced in Figure 7.7.2. The results are presented in terms of wind power classes based on the annual average power available per square meter of intercepted area (see the legend on Figure 7.7.2). Scientists at Denmark’s Risø National Laboratory have produced a European wind atlas (Troen and Petersen, 1989) that estimates the wind resources of the European Community countries and summarizes the resource available at a 50 m height for five different topographic conditions. A summary of these © 2005 by CRC Press LLC

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Energy Resouces

results is reproduced in Figure 7.7.3. The estimates presented in Figure 7.7.2 and Figure 7.7.3 are quite crude and have been superceded in recent years by much higher resolution maps, made possible by improvements in wind resource computer modeling programs and increases in computer speed. Many countries around the world have recently embarked on high-resolution mapping efforts to quantify their wind resources and identify those areas of highest resource accurately. The resultant resource maps are frequently available to the public, but in some cases a payment is required to obtain them. High-resolution wind resource maps of the individual states in the U.S. may be found on the Web at www.eere.energy.gov/windpoweringamerica/wind_resources.html. Similar maps for some other countries may be found at www.rsvp.nrel.gov/wind_resources.html, and information on where to find maps and/or data for other countries may be found at www.windatlas.dk/index.htm. Remember that even the highest resolution resource estimates are just that — estimates. The actual wind resources in any specific area can vary dramatically from those estimates and should be determined with long-term, site-specific measurements.

Wind Characterization Wind speed, direction, distribution, and shear can vary significantly over fairly short distances in the horizontal or vertical directions, so in order to get the best possible estimate of the wind energy resource at a particular location, it is important to measure the wind resource at the specific site and height of interest. However, a comprehensive site characterization normally requires measuring the wind for at least 12 months, according to meteorologists at PNL (Wegley et al., 1980). This is a very time-consuming and potentially expensive effort. Long-term data from the nearest airport or weather recording station can help determine whether the data obtained at a site are representative of normal winds for the site or of higher or lower than average winds. Wegley et al. (1980) and Gipe (1993) give suggestions on methods of using available data from nearby sites to estimate site wind speed with minimal on-site data. Sites of wind power class 4 or above (at least 200 W/m2 at 10 m height or 400 W/m2 at 50 m height) are often considered economic for utility-scale wind power development with available wind technology. Sites of wind power class 3 (150 to 200 W/m2 at 10 m height or 300 to 400 W/m2 at 50 m height) are not considered economic for utility development today but are likely to become economic with near-term wind technology advances. Sites of wind power class 2 or lower (less than 150 W/m2 at 10 m height or 300 W/m2 at 50 m height) are usually considered economic only for remote or hybrid wind power systems.

Wind Energy Potential With a wind speed distribution and a turbine power curve (the electrical power generated by the turbine at each wind speed) properly adjusted for the local air density, the wind energy potential, or gross annual wind energy production, for a specific site can be estimated as:  Energy = 0.85 8760 

n

∑ i =1

 f (U i ) ∆U i P(U i ) 

(7.7.4)

where 8760 = the number of hours in a year n = the number of wind speeds considered ƒ(Ui)∆Ui = the probability of a wind speed occurring in the wind-speed range ∆Ui P(Ui) = the electrical power produced by the turbine at wind speed Ui, the center of the range ∆Ui The leading 0.85 factor assumes 15% in losses (10% due to power transfer to the grid, control system losses, and decreased performance due to dirty blades; 5% due to operation within an array of wind turbines). If the turbine is not inside an array, replace 0.85 with 0.90. Wind energy potential is typically 20 to 35% of the wind energy resource. © 2005 by CRC Press LLC

Chapter 7

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FIGURE 7.7.2 Map of U.S. wind energy resources. Reproduced from Elliott et al. Wind Energy Resource Atlas of the United States. (Courtesy of National Renewable Energy Laboratory, Golden, Colorado.)

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FIGURE 7.7.3 Map of European wind energy resources. Reproduced from Troen and Petersen, 1989. European Wind Atlas. (Courtesy of Risø National Laboratory, Roskilde, Denmark.)

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Defining Terms Wind energy potential: total amount of energy that can actually be extracted from the wind, taking into account the efficiency of the wind turbine. Wind energy resource: total amount of energy present in the wind. Wind shear: change in wind velocity with increasing height above the ground. Wind speed distribution: probability density of occurrence of each wind speed over the course of a year for the site in question

References Elliott, D.L., Holladay, C.G., Barchet, W.R., Foote, H.P., and Sandusky, W.F. 1987. Wind Energy Resource Atlas of the United States, DOE/CH10094-4, Solar Energy Research Institute, Golden, Colorado. Gipe, P. 1993. Wind Power for Home & Business — Renewable Energy for the 1990s and Beyond, Chelsea Green Publishing Company, Post Mills, VT. Troen, I. and Petersen, E.L. 1989. European Wind Atlas, Risø National Laboratory, Roskilde, Denmark. Wegley, H.L., Ramsdell, J.V., Orgill, M.M., and Drake, R.L. 1980. A Siting Handbook for Small Wind Energy Conversion Systems, PNL-2521, Pacific Northwest Laboratory, Richland, WA.

Further Information Wind Characteristics — An Analysis for the Generation of Wind Power, J.S. Rohatgi and V. Nelson, Alternative Energy Institute, West Texas A&M University, is an excellent source for additional information on the wind resource. Wind Turbine Technology, Fundamental Concepts of Wind Turbine Engineering, D. Spera, Ed., ASME Press, New York, contains a wealth of information on wind energy resources, history, and technology, together with extensive reference lists. Extensive information on wind energy resources and technology may also be found on the World Wide Web. Excellent sites to start with include those of the U.S. National Renewable Energy Laboratory Wind Energy Technology Center at www.nwtc.nrel.gov; the Danish Risø National Laboratory at www.risoe.dk/vea/index.htm; the American Wind Energy Association at www.awea.org; the British Wind Energy Association at www.britishwindenergy.co.uk; and the European Wind Energy Association at www.ewea.org.

7.8 Geothermal Energy Joel L. Renner and Marshall J. Reed The word Geothermal comes from the combination of the Greek words gê, meaning Earth, and thérm, meaning heat. Quite literally, geothermal energy is the heat of the Earth. Geothermal resources are concentrations of the Earth’s heat, or geothermal energy, that can be extracted and used economically now or in the reasonable future. Currently, only concentrations of heat associated with water in permeable rocks can be exploited. Heat, fluid, and permeability are the three necessary components of all exploited geothermal fields. This section of Energy Resources will discuss the mechanisms for concentrating heat near the surface, the types of geothermal systems, and the environmental aspects of geothermal production.

Heat Flow Temperature within the Earth increases with depth at an average of about 25°C/km. Spatial variations of the thermal energy within the deep crust and mantle of the Earth give rise to concentrations of thermal

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energy near the surface of the Earth that can be used as an energy resource. Heat is transferred from the deeper portions of the Earth by conduction of heat through rocks, by the movement of hot, deep rock toward the surface, and by deep circulation of water. Most high-temperature geothermal resources are associated with concentrations of heat caused by the movement of magma (melted rock) to near-surface positions where the heat is stored. In older areas of continents, such as much of North America east of the Rocky Mountains, heat flow is generally 40 to 60 mWm–2 (milliwatts per square meter). This heat flow coupled with the thermal conductivity of rock in the upper 4 km of the crust yields subsurface temperatures of 90 to 110°C at 4 km depth in the Eastern United States. Heat flow within the Basin and Range (west of the Rocky Mountains) is generally 70 to 90 mWm–2, and temperatures are generally greater than 110°C at 4 km. There are large variations in the Western United States, with areas of heat flow greater than 100 mWm–2 and areas which have generally lower heat flow such as the Cascade and Sierra Nevada Mountains and the West Coast. A more detailed discussion of heat flow in the United States is available in Blackwell et al. (1991).

Types of Geothermal Systems Geothermal resources are hydrothermal systems containing water in pores and fractures. Most hydrothermal resources contain liquid water, but higher temperatures or lower pressures can create conditions where steam and water or only steam are the continuous phases (White et al., 1971; Truesdell and White, 1973). All commercial geothermal production is expected to be restricted to hydrothermal systems for many years because of the cost of artificial addition of water. Successful, sustainable geothermal energy usage depends on reinjection of the maximum quantity of produced fluid to augment natural recharge of hydrothermal systems. Other geothermal systems that have been investigated for energy production are (1) geopressuredgeothermal systems containing water with somewhat elevated temperatures (above normal gradient) and with pressures well above hydrostatic for their depth; (2) magmatic systems, with temperature from 600 to 1400°C; and (3) hot dry rock geothermal systems, with temperatures from 200 to 350°C, that are subsurface zones with low initial permeability and little water. These types of geothermal systems cannot be used for economic production of energy at this time.

Geothermal Energy Potential The most recent report (Huttrer, 1995) shows that 6800 MWe (megawatts electric) of geothermal electric generating capacity is on-line in 21 countries (Table 7.8.1). The expected capacity in the year 2000 is 9960 MWe. Table 7.8.2 lists the electrical capacity of U.S. geothermal fields. Additional details of the U.S. generating capacity are available in DiPippo (1995) and McClarty and Reed (1992). Geothermal resources also provide energy for agricultural uses, heating, industrial uses, and bathing. Freeston (1995) reports that 27 countries had a total of 8228 MWt (megawatts thermal) of direct use capacity. The total energy used is estimated to be 105,710 TJ/year (terajoules per year). The thermal energy used by the ten countries using the most geothermal resource for direct use is listed in Table 7.8.3. The U.S. Geological Survey has prepared assessments of the geothermal resources of the U.S. Muffler(1979) estimated that the identified hydrothermal resource, that part of the identified accessible base that could be extracted and utilized at some reasonable future time, is 23,000 MWe for 30 years. This resource would operate power plants with an aggregate capacity of 23,000 MWe for 30 years. The undiscovered U.S. resource (inferred from knowledge of Earth science) is estimated to be 95,000 to 150,000 MWe for 30 years.

Geothermal Applications In 1991, geothermal electrical production in the United States was 15,738 GWh (gigawatt hours), and the largest in the world (McLarty and Reed, 1992). © 2005 by CRC Press LLC

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TABLE 7.8.1 Installed and Projected Geothermal Power Generation Capacity Country Argentina Australia China Costa Rica El Salvador France Greecea Iceland Indonesia Italy Japan Kenya Mexico New Zealand Nicaragua Philippines Portugal (Azores) Russia Thailand Turkey U.S. Totals

1995

2000

0.67 0.17 28.78 55 105 4.2 0 49.4 309.75 631.7 413.705 45 753 286 35 1227 5 11 0.3 20.6 2816.775 6797.975

n/ab n/a 81 170 165 n/a n/a n/a 1080 856 600 n/a 960 440 n/a 1978 n/a 110 n/a 125 3395 9960

a

Greece has closed its 2.0 MWe Milos pilot plant. n/a = information not available. Source: Huttrer, G.W., in Proceedings of the World Geothermal Congress, 1995, International Geothermal Association, Auckland, N.Z., 1995, 3–14. With permission. b

Most geothermal fields are water dominated, where liquid water at high temperature, but also under high (hydrostatic) pressure, is the pressure-controlling medium filling the fractured and porous rocks of the reservoir. In water-dominated geothermal systems used for electricity, water comes into the wells from the reservoir, and the pressure decreases as the water moves toward the surface, allowing part of the water to boil. Since the wells produce a mixture of flashed steam and water, a separator is installed between the wells and the power plant to separate the two phases. The flashed steam goes into the turbine to drive the generator, and the water is injected back into the reservoir. Many water-dominated reservoirs below 175°C used for electricity are pumped to prevent the water from boiling as it is circulated through heat exchangers to heat a secondary liquid that then drives a turbine to produce electricity. Binary geothermal plants have no emissions because the entire amount of produced geothermal water is injected back into the underground reservoir. The identified reserves of lower-temperature geothermal fluids are many times greater than the reserves of high-temperature fluids, providing an economic incentive to develop more-efficient power plants. Warm water, at temperatures above 20°C, can be used directly for a host of processes requiring thermal energy. Thermal energy for swimming pools, space heating, and domestic hot water are the most widespread uses, but industrial processes and agricultural drying are growing applications of geothermal use. In 1995, the United States was using over 500 TJ/year of energy from geothermal sources for direct use (Lienau, et al., 1995). The cities of Boise, ID; Elko, NV; Klamath Falls, OR; and San Bernardino and Susanville, CA have geothermal district-heating systems where a number of commercial and residential buildings are connected to distribution pipelines circulating water at 54 to 93°C from the production wells (Rafferty, 1992).

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TABLE 7.8.2 U.S. Installed Geothermal Electrical Generating Capacity in MWe Rated State/Field California Casa Diablo Coso East Mesa East Mesa Honey Lake Valley Salton Sea The Geysers Hawaii Puna Nevada Beowawe Brady Hot Springs Desert Peak Dixie Valley Empire Soda Lake Steamboat Steamboat Stillwater Wabuska Utah Roosevelt Cove Fort Cove Fort

Plant Capacity

Type

27 240 37 68.4 2.3 440 1797

B 2F 2F B B 2F S

25

H

16 21 8.7 66 3.6 16.6 35.1 14.4 13 1.2

2F 2F 2F 2F B B B 1F B B

20 2 9

1F B S

Note: S = natural dry steam, 1F = single flash, 2F = double flash, B = binary, H = hybrid flash and binary.

TABLE 7.8.3 Geothermal Energy for Direct Use by the Ten Largest Users Worldwide Country China France Georgia Hungary Iceland Italy Japan New Zealand Russia U.S. Total

Flow Rate, kg/sec

Installed Power, MWt

Energy Used, TJ/year

8,628 2,889 1,363 1,714 5,794 1,612 1,670 353 1,240 3.905 37,050

1,915 599 245 340 1,443 307 319 264 210 1,874 8,664

16,981 7,350 7,685 5,861 21,158 3,629 6,942 6,614 2,422 13.890 112,441

Source: Freeston, D.H., in Proceedings of the World Geothermal Congress, 1995, International Geothermal Association, Auckland, N.Z., 1995, 15–26. With permission.

The use of geothermal energy through ground-coupled heat pump technology has almost no impact on the environment and has a beneficial effect in reducing the demand for electricity. Geothermal heat pumps use the reservoir of constant temperature, shallow groundwater and moist soil as the heat source during winter heating and as the heat sink during summer cooling. The energy efficiency of geothermal heat pumps is about 30% better than that of air-coupled heat pumps and 50% better than electricresistance heating. Depending on climate, advanced geothermal heat pump use in the United States reduces energy consumption and, correspondingly, power-plant emissions by 23 to 44% compared to

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advanced air-coupled heat pumps, and by 63 to 72% compared with electric-resistance heating and standard air conditioners (L’Ecuyer et al., 1993).

Environmental Constraints Geothermal energy is one of the cleaner forms of energy now available in commercial quantities. Geothermal energy use avoids the problems of acid rain, and it greatly reduces greenhouse gas emissions and other forms of air pollution. Potentially hazardous elements produced in geothermal brines are removed from the fluid and injected back into the producing reservoir. Land use for geothermal wells, pipelines, and power plants is small compared with land use for other extractive energy sources such as oil, gas, coal, and nuclear. Geothermal development projects often coexist with agricultural land uses, including crop production or grazing. The average geothermal plant occupies only 400 m2 for the production of each gigawatt hour over 30 years (Flavin and Lenssen, 1991). The low life-cycle land use of geothermal energy is many times less than the energy sources based on mining, such as coal and nuclear, which require enormous areas for the ore and processing before fuel reaches the power plant. Low-temperature applications usually are no more intrusive than a normal water well. Geothermal development will serve the growing need for energy sources with low atmospheric emissions and proven environmental safety. All known geothermal systems contain aqueous carbon dioxide species in solution, and when a steam phase separates from boiling water, CO2 is the dominant (over 90% by weight) noncondensible gas. In most geothermal systems, noncondensible gases make up less than 5% by weight of the steam phase. Thus, for each megawatt-hour of electricity produced in 1991, the average emission of carbon dioxide by plant type in the United States was 990 kg from coal, 839 kg from petroleum, 540 kg from natural gas, and 0.48 kg from geothermal flashed-steam (Colligan, 1993). Hydrogen sulfide can reach moderate concentrations of up to 2% by weight in the separated steam phase from some geothermal fields. At The Geysers geothermal field in California, either the Stretford process or the incineration and injection process is used in geothermal power plants to keep H2S emissions below 1 ppb (part per billion). Use of the Stretford process in many of the power plants at The Geysers results in the production and disposal of about 13,600 kg of sulfur per megawatt of electrical generation per year. Figure 7.8.1, shows a typical system used in the Stretford process at The Geysers (Henderson and Dorighi, 1989). The incineration process burns the gas removed from the steam to convert H2S to SO2, the gases are absorbed in water to form SO 3−2 and SO −42 in solution, and iron chelate is used to form S2 O 3−2 (Bedell and Hammond, 1987). Figure 7.8.2 shows an incineration abatement system (Bedell and Hammond, 1987). The major product from the incineration process is a soluble thiosulfate which is injected into the reservoir with the condensed water used for the reservoir pressure-maintenance program. Sulfur emissions for each megawatt-hour of electricity produced in 1991, as SO2 by plant type in the United States was 9.23 kg from coal, 4.95 kg from petroleum, and 0.03 kg from geothermal flashed-steam (Colligan, 1993). Geothermal power plants have none of the nitrogen oxide emissions that are common from fossil fuel plants. The waters in geothermal reservoirs range in composition from 0.1 to over 25 wt% dissolved solutes. The geochemistry of several representative geothermal fields is listed in Table 7.8.4. Temperatures up to 380°C have been recorded in geothermal reservoirs in the United States, and many chemical species have a significant solubility at high temperature. For example, all of the geothermal waters are saturated in silica with respect to quartz. As the water is produced, silica becomes supersaturated, and, if steam is flashed, the silica becomes highly supersaturated. Upon cooling, amorphous silica precipitates from the supersaturated solution. The high flow rates of steam and water from geothermal wells usually prevent silica from precipitating in the wells, but careful control of fluid conditions and residence time is needed to prevent precipitation in surface equipment. Silica precipitation is delayed in the flow stream until the water reaches a crystallizer or settling pond. There the silica is allowed to settle from the water, and the water is then pumped to an injection well.

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FIGURE 7.8.1 Typical equipment used in the Stretford process for hydrogen sulfide abatement at The Geysers geothermal field. (Based on the diagram of Henderson, J.M. and Dorighi, G.P., Geotherm. Resour. Counc. Trans., 13, 593–595, 1989.)

FIGURE 7.8.2 Equipment used in the incineration process for hydrogen sulfide abatement at The Geysers geothermal field. (Based on the diagram of Bedell, S.A. and Hammond, C.A., Geotherm. Resour. Counc. Bull., 16(8), 3–6, 1987.)

Operating Conditions For electrical generation, typical geothermal wells in the United States have production casing pipe in the reservoir with an inside diameter of 29.5 cm, and flow rates usually range between 150,000 and

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Field

T(°C)

Na

K

Li

Ca

Mg

Cl

F

Br

SO4

Reykyavik, Iceland Hveragerdi, Iceland Broadlands, N. Zealand Wairekai, New Zealand Cerro Prieto, Mexico Salton Sea, California Roosevelt, Utahb

100 216 260 250 340 340 Tr . To prevent stratification, with the warm supply air staying at a higher level, (Ts – Tr) > 20°F is not recommended. w Lb/Lb

w Lb/Lb

r ru ru

r

m

m

s sf ch

h ch

sf s

o T, °F

(a)

(b)

T, °F

w, lb/lb

r

sf s aw m

(c)

ru

T, °F

FIGURE 9.3.5 Basic air-conditioning cycle — winter modes: (a) warm air supply without space humidity control, (b) cold air supply without space humidity control, and (c) cold air supply with space humidity control. ch = air leaving heating coil, h = air leaving humidifer, and aw = air leaving air washer. © 2005 by CRC Press LLC

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Warm Air Supply with Space Humidity Control This operating cycle (see Figure 9.3.5[b]) is often used for hospitals, nurseries, etc. or in locations where winter is very cold. The state point of supply air must be determined first by drawing a space conditioning line with known SHRs and then from the calculated supply temperature differential ∆Ts. The difference in humidity ratio (ws – wch) is the water vapor must be added at the humidifier. Humidifying capacity can be calculated from Equation 9.3.8. Cold Air Supply with Space Humidity Control This operating cycle (shown in Figure 9.3.5[c]) is widely used in industrial applications such as textile mills where a cold air supply is needed to remove machine load in winter and maintains the space relative humidity required for the manufacturing process. An outdoor air and recirculating air mixture is often used for the required cold air supply. An air washer is adopted for winter humidification. Air Economizer Mode In the air economizer mode, as shown by the middle dotted line cycle o″-cc-sf-s-r in Figure 9.3.4, all outdoor air or an outdoor air-recirculating air mixture is used to reduce the refrigeration capacity and improve the indoor air quality during spring, fall, or winter. When all outdoor air is admitted, it is an open cycle. Outdoor air is cooled and often dehumidified to point cc. After absorbing fan and duct heat gains, it is supplied to the conditioned space. Space air is exhausted entirely through openings, relief dampers, or relief/exhaust fans to the outside. An all-outdoor air-operating mode before the space is occupied is often called an air purge operation, used to dilute space air contaminants. Cool-Down and Warm-Up Modes In summer, when an air system is shut down during an unoccupied period at night, the space temperature and relative humidity often tend to increase because of infiltration of hot and humid air and heat transfer through the building envelope. The air system is usually started before the space is occupied in cooldown mode to cool the space air until the space temperature falls within predetermined limits. In winter, the air system is also started before the occupied period to warm up the space air to compensate for the nighttime space temperature setback to 55 to 60°F for energy saving or the drop of space temperature due to heat loss and infiltration. If dilution of indoor air contaminants is not necessary, only recirculating space air is used during cooldown or warm-up periods in order to save energy. Care should be taken in suggesting cool-down or warm-up modes as for some applications after-hour conditions could result in damage to interior surfaces or materials.

9.4 Refrigerants and Refrigeration Cycles Shan K. Wang Refrigeration and Refrigeration Systems Refrigeration is the cooling effect of the process of extracting heat from a lower temperature heat source, a substance or cooling medium, and transferring it to a higher temperature heat sink, probably atmospheric air and surface water, to maintain the temperature of the heat source below that of the surroundings. A refrigeration system is a combination of components, equipment, and piping, connected in a sequential order to produce the refrigeration effect. Refrigeration systems that provide cooling for air-conditioning are classified mainly into the following categories: 1. Vapor compression systems. In these systems, a compressor(s) compresses the refrigerant to a higher pressure and temperature from an evaporated vapor at low pressure and temperature. The compressed refrigerant is condensed into liquid form by releasing the latent heat of condensation to the condenser water. Liquid refrigerant is then throttled to a low-pressure, low-temperature vapor, © 2005 by CRC Press LLC

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producing the refrigeration effect during evaporation. Vapor compression is often called mechanical refrigeration, that is, refrigeration by mechanical compression. 2. Absorption systems. In an absorption system, the refrigeration effect is produced by means of thermal energy input. After liquid refrigerant produces refrigeration during evaporation at very low pressure, the vapor is absorbed by an aqueous absorbent. The solution is heated by a directfired gas furnace or waste heat, and the refrigerant is again vaporized and then condensed into liquid form. The liquid refrigerant is throttled to a very low pressure and is ready to produce the refrigeration effect again. 3. Gas expansion systems. In an air or other gas expansion system, air or gas is compressed to a high pressure by compressors. It is then cooled by surface water or atmospheric air and expanded to a low pressure. Because the temperature of air or gas decreases during expansion, a refrigeration effect is produced.

Refrigerants, Cooling Mediums, and Absorbents A refrigerant is a primary working fluid used to produce refrigeration in a refrigeration system. All refrigerants extract heat at low temperature and low pressure during evaporation and reject heat at high temperature and pressure during condensation. A cooling medium is a working fluid cooled by the refrigerant during evaporation to transport refrigeration from a central plant to remote cooling equipment and terminals. In a large, centralized airconditioning system, it is more economical to pump the cooling medium to the remote locations where cooling is required. Chilled water and brine are cooling media. They are often called secondary refrigerants to distinguish them from the primary refrigerants. A liquid absorbent is a working fluid used to absorb the vaporized refrigerant (water) after evaporation in an absorption refrigeration system. The solution that contains the absorbed vapor is then heated. The refrigerant vaporizes, and the solution is restored to its original concentration to absorb water vapor again. A numbering system for refrigerants was developed for hydrocarbons and halocarbons. According to ANSI/ASHRAE Standard 34-1997, the first digit is the number of unsaturated carbon–carbon bonds in the compound. This digit is omitted if the number is zero. The second digit is the number of carbon atoms minus one. This is also omitted if the number is zero. The third digit denotes the number of hydrogen atoms plus one. The last digit indicates the number of fluorine atoms. For example, the chemical formula for refrigerant R-123 is CHCl2CF3. In this compound: No unsaturated carbon–carbon bonds, first digit is 0 There are two carbon atoms, second digit is 2–1=1 There is one hydrogen atom, third digit is 1+1=2 There are three fluorine atoms, last digit is 3 To compare the relative ozone depletion of various refrigerants, an index called the ozone depletion potential (ODP) has been introduced. ODP is defined as the ratio of the rate of ozone depletion of 1 lb of any halocarbon to that of 1 lb of refrigerant R-11. For R-11, ODP = 1. Similar to the ODP, halocarbon global warming potential (HGWP) is an index used to compare the global warming effect of a halocarbon refrigerant with the effect of refrigerant R-11.

Classification of Refrigerants Nontoxic and nonflammable synthetic chemical compounds called halogenated hydrocarbons, or simply halocarbons, were used almost exclusively in vapor compression refrigeration systems for comfort air-conditioning until 1986. Because chlorofluorocarbons (CFCs) cause ozone depletion and global warming, they must be replaced. A classification of refrigerants based on ozone depletion follows (see Table 9.4.1): © 2005 by CRC Press LLC

Chemical Formula

© 2005 by CRC Press LLC

Evaporating Pressure, psia

Condensing Pressure, psia

Compression Ratio

Refrigeration Effect, Btu/lb

135.6 111.9 49.7

340.2 276.2 138.8

2.51 2.47 2.79

37.1 65.2

44.8

124.3

2.77

82.09 5.8 27.9

201.5 20.8 80.92

2.46 3.59 2.90

Molecular Mass

Ozone Depletion Potential (ODP)

52.02 120.03 102.03 84.0 66.05 134.1

0.0 0.0 0.0 0.0 0.0 0.0

0.14 0.84 0.26

0.0

0.98

0.0

0.94

0.0

0.49

0.0

0.70

0.05 0.02 0.02

0.40 0.02

0.02

0.63

0.37

0.22

0.04

0.24

86.48 152.93 136.47

69.0 62.9 5.21

Chapter 9

Hydrofluorocarbons HFCs R-32 Difluoromethane CH2F2 R-125 Pentafluoroethane CHF2CF3 R-134a Tetrafluoroethane CF3CH2F R-143a Trifluoroethane CH3CF3 R-152a Difluoroethane CH3CHF2 R-245ca Pentafluoropropane CF3CF2CH3 HFC’s azeotropic blends R-507 R-125/R-143 (45/55) HFC’s near azeotropic blends R-404A R-125/R-143a (44/52/4) R-407A R-32/R-125/R-134a (20/40/40) R-407C R-32/R-125/R-134a (23/25/52) Hydrochlorofluorocarbons HCFCs and their azeotropic blends R-22 Chlorodifluoromethane CHCIF2 R-123 Dichlorotrifluoroethane CHCl2CF3 R-124 Chlorotetrafluoroethane CHFClCF3 HCFC’s near azeotropic blends R-402A R-22/R-125/R-290 (38/60/2) HCFC’s azeotropic blends R-401A R-22/R-124/R-152a (53/34/13) R-401B R-22/R-124/R-152a (61/28/11)

Global Warming Potential (HGWP)

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TABLE 9.4.1 Properties of Commonly Used Refrigerants 40°F Evaporating and 100°F Condensin

Chemical Formula Inorganic compounds R-717 Ammonia R-718 Water R-729 Air

NH3 H2O

Molecular Mass 17.03 18.02 28.97

Chlorofluorocarbons CFCs, halons BFCs and their azeotropic blends R-11 Trichlorofluoromethane CCl3F 137.38 R-12 Dichlorodifluoromethane CCl2F2 120.93 R-13B1 Bromotrifluoromethane CBrF3 148.93 R-113 Trichlorotrifluoroethane CCl2FCClF2 187.39 R-114 Dichlorotetrafluoroethan CCl2FCF3 170.94 e R-500 R-12/R-152a 99.31 (73.8/26.2) R-502 R-22/R-115 111.63 (48.8/51.2)

Ozone Depletion Potential (ODP) 0 0 0 1.00 1.00 10 0.80 1.00

0.283

Global Warming Potential (HGWP)

Evaporating Pressure, psia

Condensing Pressure, psia

Compression Ratio

Refrigeration Effect, Btu/lb

0

71.95

206.81

2.87

467.4

1.00 3.20

6.92 50.98

23.06 129.19

3.33 2.53

68.5 50.5

1.4 3.9

2.64 14.88

10.21 45.11

3.87 3.03

54.1 42.5

59.87

152.77

2.55

60.5

4.10

9-33

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Air-Conditioning and Refrigeration

TABLE 9.4.1 (continued)

Properties of Commonly Used Refrigerants 40°F Evaporating and 100°F Condensing

Replacement of

Trade Name

Hydrofluorocarbons HFCs R-32 R-125 R134a R-12 R143a R-152a R-245ca HFC’s azeotropic blends R-507 R-502 Genetron AZ-50 HFC’s near azeotropic blends R-404A R-22 SUVA HP-62 R-407A R-22 KLEA 60 R-407C R-22 KLEA 66 Hydrochlorofluorocarbons HCFC’s and their azeotropic blends R-22 R-123 R-11 R-124 HCFC’s near azeotropic blends R-402A R-502 SUVA HP-80 HCFC’s azeotropic blends R-401A R-12 MP 39 R-401B R-12 MP 66 Inorganic compounds R-717 R-718 R-729

Specific Volume of Vapor ft3/lb

Compresssor Displacement cfm/ton

Power Consumption hp/ton

Critical Temperature °F

Discharge Temperature °F

0.63 0.33 0.95

173.1 150.9 213.9

103

1.64

235.9

Flammability

Safety

Nonflammable Nonflammable

A1 A1

Lower flammable

A2

A1/A1a A1/A1a A1/A1a 0.66 5.88 1.30

1.91 18.87 5.06

0.696 0.663 0.698

204.8 362.6 252.5

127

Nonflammable Nonflammable

A1 B1

A1/A1a A1/A1a A1/A1a 3.98

1.70

0.653

271.4

207

Lower flammability Nonflammable Nonflammable

B2

Chapter 9

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TABLE 9.4.1 (continued)

Trade Name

Specific Volume of Vapor ft3/lb

Chlorofluorocarbons CFCs, halons BFCs, and their azeotropic blends R-11 5.43 R-12 5.79 R-13B1 0.21 R-113 10.71 R-114 2.03 R-500 R-12/R-152a (73.8/26.2) 0.79 R-502 R-22/R-115 (48.8/51.2)

Compresssor Displacement cfm/ton

Power Consumption hp/ton

Critical Temperature °F

Discharge Temperature °F

15.86 3.08

0.636 0.689

39.55 9.57 3.62

0.71 0.738 0.692

388.4 233.6 152.6 417.4 294.3 221.9

104 100 103 86 86 105 98

Flammability Nonflammable Nonflammable Nonflammable Nonflammable Nonflammable Nonflammable Nonflammable

Safety A1 A1 A1 A1 A1 A1 A1

Source: Adapted with permission from ASHRAE Handbooks 1993 Fundamentals. Also from refrigerant manufacturers. First classification is that safety classification of the formulated composition. The second is the worst case of fractionation.

9-35

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Properties of Commonly Used Refrigerants 40°F Evaporating and 100°F Condensing

Replacement of

a

Air-Conditioning and Refrigeration

TABLE 9.4.1 (continued)

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Hydrofluorocarbons (HFCs) HFCs contain only hydrogen, fluorine, and carbon atoms and cause no ozone depletion. HFCs group include R-134a, R-32, R-125, and R-245ca. HFC’s Azeotropic Blends or Simply HFC’s Azeotropes An azeotrope is a mixture of multiple components of volatilities (refrigerants) that evaporate and condense as a single substance and do not change in volumetric composition or saturation temperature when they evaporate or condense at constant pressure. HFC’s azeotropes are blends of refrigerant with HFCs. ASHRAE assigned numbers between 500 and 599 for azeotropes. HFC’s azeotrope R-507, a blend of R-125/R-143, is the commonly used refrigerant for low-temperature vapor compression refrigeration systems. HFC’s Near Azeotropic A near azeotrope is a mixture of refrigerants whose characteristics are near those of an azeotrope. Because the change in volumetric composition or saturation temperature is rather small for a near azeotrope, such as, 1 to 2°F, it is thus named. ASHRAE assigned numbers between 400 and 499 for zeotropic. R-404A (R-125/R-134a/R-143a) and R-407B (R-32/R-125/R134a) are HFC’s near azeotrope. R-32 is flammable; therefore, its composition is usually less than 30% in the mixture. HFC’s near azeotropes are widely used for vapor compression refrigeration systems. Zeotropes or nonazeotropes, including near azeotropes, show a change in composition due to the difference between liquid and vapor phases, leaks, and the difference between charge and circulation. A shift in composition causes the change in evaporating and condensing temperature/pressure. The difference in dew point and bubble point during evaporation and condensation is called glide, expressed in °F. Near azeotrope has a smaller glide than zeotropes. The midpoint between the dew point and bubble point is often taken as the evaporating and condensing temperature for refrigerant blends. Hydrochlorofluorocarbons (HCFCs) and Their Zeotropics HCFCs contain hydrogen, chlorine, fluorine, and carbon atoms and are not fully halogenated. HCFCs have a much shorter lifetime in the atmosphere (in decades) than CFCs and cause far less ozone depletion (ODP 0.02 to 0.1). R-22, R-123, R-124, etc. are HCFCs. HCFCs are the most widely used refrigerants today. HCFC’s near azeotropes and HCFC’s zeotropes are blends of HCFCs with HFCs. They are transitional or interim refrigerants and are scheduled for a restriction in production starting in 2004. Inorganic Compounds These compounds include refrigerants used before 1931, like ammonia R-717, water R-718, and air R-729. They are still in use because they do not deplete the ozone layer. Because ammonia is toxic and flammable, it is used in industrial applications. Inorganic compounds are assigned numbers between 700 and 799 by ASHRAE. Chlorofluorocarbons, Halons, and Their Azeotropic Blends CFCs contain only chlorine, fluorine, and carbon atoms. CFCs have an atmospheric lifetime of centuries and cause ozone depletion (ODP from 0.6 to 1). R-11, R-12, R-113, R-114, R-115… are all CFCs. Halons or BFCs contain bromide, fluorine, and carbon atoms. R-13B1 and R-12B1 are BFCs. They cause very high ozone depletion (ODP for R-13B1 = 10). Until 1995, R-13B1 was used for very low temperature vapor compression refrigeration systems. Phaseout of CFCs, BFCs, HCFCs, and Their Blends On September 16, 1987, the European Economic Community and 24 nations, including the United States, signed a document called the Montreal Protocol. It is an agreement to restrict the production and consumption of CFCs and BFCs in the 1990s because of ozone depletion. The Clean Air Act amendments, signed into law in the United States on November 15, 1990, concern two important issues: the phaseout of CFCs and the prohibition of deliberate venting of CFCs and HCFCs. © 2005 by CRC Press LLC

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In February 1992, President Bush called for an accelerated ban of CFCs in the United States. In late November 1992, representatives of 93 nations meeting in Copenhagen agreed to phase out CFCs beginning January 1, 1996. Restriction on the use of HCFCs will start in 2004, with a complete phaseout by 2030. In the earlier 1990s, R-11 was widely used for centrifugal chillers, R-12 for small and medium-size vapor compression systems, R-22 for several other vapor compression systems, and CFC/HCFC blend R-502 for low-temperature vapor compression systems. Because of the phaseout of CFCs and BFCs before 1996 and HCFCs in the early years of the next century, alternative refrigerants have been developed to replace them: • R-123 (an HCFC of ODP = 0.02) to replace R-11 is a short-term replacement that causes a slight reduction in capacity and efficiency. R-245ca (ODP = 0) may be the long-term alternative to R-11. • R-134a (an HFC with ODP = 0) to replace R-12 in broad applications. R-134a is not miscible with mineral oil; therefore, a synthetic lubricant of polyolester is used. • R-404A (R-125/R-134a/143a) and R-407C (R-32/R-125/R-134a) are both HFCs near azeotropic of ODP = 0. They are long-term alternatives to R-22. For R-407C, the composition of R-32 in the mixture is usually less than 30% so that the blend will not be flammable. R-407C has a drop of only 1 to 2% in capacity compared with R-22. • R-507 (R-125/R-143a), an HFC’s azeotropic with ODP = 0, is a long-term alternative to R-502. Synthetic polyolester lubricant oil will be used for R-507. There is no major performance difference between R-507 and R-502. R-402A (R-22/R-125/R-290), an HCFC’s near azeotropic, is a shortterm immediate replacement, and drop-in of R-502 requires minimum change of existing equipment except for reset of a higher condensing pressure.

Required Properties of Refrigerants A refrigerant should not cause ozone depletion. A low global warming potential is required. Additional considerations for refrigerant selection are 1. Safety, including toxicity and flammability. ANSI/ASHRAE Standard 34-1997 classifies the toxicity of refrigerants as Class A and Class B. Class A refrigerants are of low toxicity. No toxicity was identified when their time-weighted average concentration was less than or equal to 400 ppm, to which workers can be exposed for an 8-hr workday and 40-hr work week without adverse effect. Class B refrigerants are of higher toxicity and produce evidence of toxicity. ANSI/ASHRAE Standard 34-1982 classifies the flammability of refrigerants as Class 1, no flame propagation; Class 2, lower flammability; and Class 3, higher flammability. The safety classification of refrigerants is based on the combination of toxicity and flammability: A1, A2, A3, B1, B2, and B3. R-134a and R-22 are in the A1 group, lower toxicity and nonflammable; R-123 in the B1 group, higher toxicity and nonflammable; and R-717 (ammonia) in the B2 group, higher toxicity and lower flammability. 2. Effectiveness of refrigeration cycle. High effectiveness is a desired property. The power consumed per ton of refrigeration produced, hp/ton or kW/ton, is an index for this assessment. Table 9.4.1 gives values for an ideal single-stage vapor compression cycle. 3. Oil miscibility. Refrigerant should be miscible with mineral lubricant oil because a mixture of refrigerant and oil helps to lubricate pistons and discharge valves, bearings, and other moving parts of a compressor. Oil should also be returned from the condenser and evaporator for continuous lubrication. R-22 is partially miscible. R-134a is hardly miscible with mineral oil; therefore, synthetic lubricant of polyolester will be used. 4. Compressor displacement. Compressor displacement per ton of refrigeration produced, in cfm/ton, directly affects the size of the positive displacement compressor and therefore its compactness. Ammonia R-717 requires the lowest compressor displacement (1.70 cfm/ton) and R-22 the second lowest (1.91 cfm/ton). © 2005 by CRC Press LLC

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5. Desired properties: • Evaporating pressure pev should be higher than atmospheric. Then noncondensable gas will not leak into the system. • Lower condensing pressure for lighter construction of compressor, condenser, piping, etc. • A high thermal conductivity and therefore a high heat transfer coefficient in the evaporator and condenser. • Dielectric constant should be compatible with air when the refrigerant is in direct contact with motor windings in hermetic compressors. • An inert refrigerant that does not react chemically with material will avoid corrosion, erosion, or damage to system components. Halocarbons are compatible with all containment materials except magnesium alloys. Ammonia, in the presence of moisture, is corrosive to copper and brass. • Refrigerant leakage can be easily detected. Halide torch, electronic detector, and bubble detection are often used.

Ideal Single-Stage Vapor Compression Cycle Refrigeration Process A refrigeration process shows the change of the thermodynamic properties of the refrigerant and the energy and work transfer between the refrigerant and surroundings. Energy and work transfer is expressed in British thermal units per hour, or Btu/hr. Another unit in wide use is ton of refrigeration, or ton. A ton = 12,000 Btu/hr of heat removed; i.e., 1 ton of ice melting in 24 hr = 12,000 Btu/hr. Refrigeration Cycles When a refrigerant undergoes a series of processes like evaporation, compression, condensation, throttling, and expansion, absorbing heat from a low-temperature source and rejecting it to a higher temperature sink, it is said to have undergone a refrigeration cycle. If its final state is equal to its initial state, it is a closed cycle; if the final state does not equal the initial state, it is an open cycle. Vapor compression refrigeration cycles can be classified as single stage, multistage, compound, and cascade cycles. A pressure-enthalpy diagram or p-h diagram is often used to calculate the energy transfer and to analyze the performance of a refrigeration cycle, as shown in Figure 9.4.1. In a p-h diagram, pressure p, in psia or psig logarithmic scale, is the ordinate, and enthalpy h, in Btu/lb, is the abscissa. The saturated liquid and saturated vapor line encloses a two-phase region in which vapor and liquid coexist. The two-phase region separates the subcooling liquid and superheated vapor regions. The constant-temperature line is nearly vertical in the subcooling region, horizontal in the two-phase region, and curved down sharply in the superheated region. In the two-phase region, a given saturated pressure determines the saturated temperature and vice versa. The constant-entropy line is curved upward to the right-hand side in the superheated region. Each kind of refrigerant has its own p-h diagram. Refrigeration Processes in an Ideal Single-Stage Cycle An ideal cycle has isentropic compression, and pressure losses in the pipeline, valves, and other components are neglected. All refrigeration cycles covered in this section are ideal. Single stage means a single stage of compression. There are four refrigeration processes in an ideal single-stage vapor compression cycle, as shown in Figure 9.4.2(a) and (b): 1. Isothermal evaporation process 4–1 — The refrigerant evaporates completely in the evaporator and produces refrigeration effect qrf, in Btu/lb: qrf = (h1 − h4 ) © 2005 by CRC Press LLC

(9.4.1)

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Constant temperature line

Air-Conditioning and Refrigeration

ine ntr op yl

Satura

Two-phase region

30

Constant temperature line

dry Con stan t

100 80 70 60 50 40

nt e

li q

nes s fr

uid

lin e

200

Sa tur ate d

Pressure p, psig

300

Superheated vapor

Co nst a

Subcooled liquid

400

por lin e

600

ted va

800

acti on

1000

me line

nt volu

Consta

20

60

80

100

120

140 160 Enthalpy h, Btu/lb

180

200

220

FIGURE 9.4.1 Skeleton of pressure-enthalpy diagram for R-22.

where h1,h4 = enthalpy of refrigerant at state points 1 and 4, respectively, Btu/lb. 2. Isentropic compression process 1–2 — Vapor refrigerant is extracted by the compressor and compressed isentropically from point 1 to 2. The work input to the compressor Win, in Btu/lb, is Win = (h2 − h1 )

(9.4.2)

where h2 = enthalpy of refrigerant at state point 2, Btu/lb. The greater the difference in temperature/pressure between the condensing pressure pcon and evaporating pressure pev , the higher will be the work input to the compressor. 3. Constant pressure condensation process 2–3 — Hot gaseous refrigerant discharged from the compressor is condensed in the condenser into liquid, and the latent heat of condensation is rejected to the condenser water or ambient air. The heat rejection during condensation, q2–3, in Btu/lb, is − q2−3 = (h2 − h3 )

(9.4.3)

where h3 = enthalpy of refrigerant at state point 3, Btu/lb. 4. Throttling process 3–4 — Liquid refrigerant flows through a throttling device (e.g., an expansion valve, a capillary tube, or orifices) and its pressure is reduced to the evaporating pressure. A portion of the liquid flashes into vapor and enters the evaporator. This is the only irreversible process in the ideal cycle, usually represented by a dotted line. For a throttling process, assuming that the heat gain from the surroundings is negligible: h3 = h4


The mass flow rate of refrigerant m r , in lb/min, is © 2005 by CRC Press LLC

(9.4.4)

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Chapter 9

Condenser

p, psia 3

liqu

id

ed va p

liquid

Two-phase

rate d

1 pev

Satu

Expansion valve

4

2 or

2

Compressor

Satur at

3

pcon Subcooled

Superheating

4

Evaporator h3=h4 (a) p, psia sc

h2

p, psia

Subcooling

3«

h1 (b)

3

3

2

pcon

2

pcon

Lcon Superheating region

Tsc pev

4«

h4«

pev

1

4

h4

h1

h2 h, Btu/Lb

4

1

h3 = h4

(c)

1« Superheating

h1 h1«

h, Btu/Lb

(d)

FIGURE 9.4.2 A single-stage ideal vapor compression refrigeration cycle: (a) schematic diagram, (b) p-h diagram, (c) subcooling, and (d) superheating.




m r = qrc 60qrf

(9.4.5)

where qrc = refrigeration capacity of the system, Btu/hr. The ideal single-stage vapor compression refrigeration cycle on a p-h diagram is divided into two pressure regions: high pressure (pcon) and low pressure (pev).

Coefficient of Performance of Refrigeration Cycle The coefficient of performance (COP) is a dimensionless index used to indicate the performance of a thermodynamic cycle or thermal system. The magnitude of COP can be greater than 1. • If a refrigerator is used to produce a refrigeration effect, COPref is COPref = qrf Win

(9.4.6)

• If a heat pump is used to produce a useful heating effect, its performance denoted by COPhp is COPhp = q2−3 Win © 2005 by CRC Press LLC

(9.4.7)

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Air-Conditioning and Refrigeration

• For a heat recovery system when both refrigeration and heating effects are produced, the COPhr is denoted by the ratio of the sum of the absolute values of qrf and q2-3 to the work input, or

(

COPhr = qrf + q2 −3

)W

in

(9.4.8)

Subcooling and Superheating Condensed liquid is often subcooled to a temperature lower than the saturated temperature corresponding to the condensing pressure pcon, in psia or psig, as shown in Figure 9.4.2(c). Subcooling increases the refrigeration effect to qrf.sc as shown in Figure 9.4.2(c): qrf.sc = (h4′ − h1 ) > (h4 − h1 )

(9.4.9)

The enthalpy of subcooled liquid refrigerant hsc approximately equals the enthalpy of the saturated liquid refrigerant at subcooled temperature hs.sc, both in Btu/lb: hsc = h3′ = h4′ = h1.con − c pr (Tcon − Tsc ) ≈ hs.sc

(9.4.10)

where h3′, h4′ = enthalpy of liquid refrigerant at state points 3′ and 4′ respectively, Btu/lb hl.con = enthalpy of saturated liquid at condensing temperature, Btu/lb cpr = specific heat of liquid refrigerant at constant pressure, Btu/lb °F Tcon = condensing temperature or saturated temperature of liquid refrigerant at condensing pressure, °F Tsc = temperature of subcooled liquid refrigerant, °F The purpose of superheating is to prevent liquid refrigerant flooding back into the compressor and causing slugging damage as shown in Figure 9.4.2(d). The degree of superheating depends mainly on the types of refrigerant feed, construction of the suction line, and type of compressor. The state point of vapor refrigerant after superheating of an ideal system must be at the evaporating pressure with a specific degree of superheat and can be plotted on a p-h diagram for various refrigerants.

Refrigeration Cycle of Two-Stage Compound Systems with a Flash Cooler A multistage system employs more than one compression stage. Multistage vapor compression systems are classified as compound systems and cascade systems. A compound system consists of two or more compression stages connected in series. It may have one high-stage compressor (higher pressure) and one low-stage compressor (lower pressure), several compressors connected in series, or two or more impellers connected internally in series and driven by the same motor. The compression ratio Rcom is defined as the ratio of discharge pressure pdis to the suction pressure at the compressor inlet psuc: Rcom = pdis psuc

(9.4.11)

Compared to a single-stage system, a multistage has a smaller compression ratio and higher compression efficiency for each stage of compression, greater refrigeration effect, lower discharge temperature at the high-stage compressor, and greater flexibility. At the same time, a multistage system has a higher initial cost and more complicated construction. The pressure between the discharge pressure of the high-stage compressor and the suction pressure of the low-stage compressor of a multistage system is called interstage pressure pi, in psia. Interstage © 2005 by CRC Press LLC

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Chapter 9



 










 


 %

!









%




# $

"!










#  $







 

&'



 






FIGURE 9.4.3 Two-stage compound system with a flash cooler: (a) schematic diagram and (b) refrigeration cycle.

pressure for a two-stage system is usually determined so that the compression ratios are nearly equal between two stages for a higher COP. Then the interstage pressure is pi =

(p

con

pev )

(9.4.12)

where pcon, pev = condensing and evaporating pressures, psia. For a multistage system of n stages, then, the compression ratio of each stage is Rcom = ( pcon psuc )

1n

(9.4.13)

Figure 9.4.3(a) shows a schematic diagram and Figure 9.4.3(b) the refrigeration cycle of a two-stage compound system with a flash cooler. A flash cooler, sometimes called an economizer, is used to subcool the liquid refrigerant to the saturated temperature corresponding to the interstage pressure by vaporizing a portion of the liquid refrigerant in the flash cooler. Based on the principle of heat balance, the fraction of evaporated refrigerant, x, or quality of the mixture in the flash cooler is

(

x = h5′ − h8

) (h

7

− h8 )

(9.4.14)

where h5′, h7, h8 = enthalpy of the refrigerant at state points 5′, 7, and 8, respectively, Btu/lb. The coefficient of performance of the refrigeration cycle of a two-stage compound system with a flash cooler, COPref, is given as COPref = qrf Win = (1 − x ) (h1 − h9 )

[(1 − x)(h

2

]

− h1 ) + (h4 − h3 )

(9.4.15)

where h1, h2, h3, h4, h9 = enthalpy of refrigerant at state points 1, 2, 3, 4, and 9, respectively, Btu/lb. The
mass flow rate of refrigerant flowing through the condenser, m r , in lb/min, can be calculated as


m r = qrc 60qrf

(9.4.16)

Because a portion of liquid refrigerant is flashed into vapor in the flash cooler and goes directly to the second-stage impeller inlet, less refrigerant is compressed in the first-stage impeller. In addition, the © 2005 by CRC Press LLC

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Air-Conditioning and Refrigeration



 




 





(













 













% %



&




'
" 





)  
*





! 

! 

$ #!

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FIGURE 9.4.4 Cascade system: (a) schematic diagram and (b) refrigeration cycle.

liquid refrigerant in the flash cooler is cooled to the saturated temperature corresponding to the interstage temperature before entering the evaporator, which significantly increases the refrigeration effect of this compound system. Two-stage compound systems with flash coolers are widely used in large central airconditioning systems.

Cascade System Characteristics A cascade system consists of two independently operated single-stage refrigeration systems: a lower system that maintains a lower evaporating temperature and produces a refrigeration effect and a higher system that operates at a higher evaporating temperature as shown in Figure 9.4.4(a) and (b). These two separate systems are connected by a cascade condenser in which the heat released by the condenser in the lower system is extracted by the evaporator in the higher system. A heat exchanger is often used between the liquid refrigerant from the condenser and the vapor refrigerant leaving the evaporator of the lower system. When the system is shut down in summer, a relief valve connected to a stored tank should be used to relieve the higher pressure of refrigerant at the higher storage temperature. The main advantages of a cascade system compared with a compound system are that different refrigerants, oils, and equipment can be used for the lower and higher systems. Disadvantages are the overlap of the condensing temperature of the lower system and the evaporating temperature of the higher system because of the heat transfer in the cascade condenser and a more complicated system. The refrigeration effect qrf of the cascade system is qrf = (h1 − h4 )

(9.4.17)

where h1, h4 = enthalpy of the refrigerant leaving and entering the evaporator of the lower system, Btu/lb. The total work input to the compressors in both higher and lower systems Win, in Btu/lb, can be calculated as Win = (h2 − h1′ ) + m h (h6 − h5 ) m1


© 2005 by CRC Press LLC




(9.4.18)

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Chapter 9

where h2 = enthalpy of refrigerant discharged from the compressor of the lower system h1′ = enthalpy of the vapor refrigerant leaving the heat exchanger h6, h5 = enthalpy of the refrigerant leaving and entering the high-stage compressor

m h , m1 = mass flow rate of the refrigerant of the higher and lower systems, respectively The coefficient of performance of a cascade system COPref is


COPref = qrf Win = m1 (h1 − h4 ) m1 (h2 − h1′ ) + m h (h6 − h5 )  

(9.4.19)

9.5 Outdoor Design Conditions and Indoor Design Criteria Shan K. Wang Outdoor Design Conditions In principle, the capacity of air-conditioning equipment should be selected to offset or compensate for the space load so that indoor design criteria can be maintained if the outdoor weather does not exceed the design values. Outdoor and indoor design conditions are used to calculate the design space loads. In energy use calculations, hour-by-hour outdoor climate data of a design day should be adopted instead of summer and winter design values. ASHRAE Handbook 2001 Fundamentals (Chapter 27) and Wang’s Handbook of Air Conditioning and Refrigeration (Chapter 7) both list tables of climate conditions for the U.S. and Canada based on the data from the National Climate Data Center (NCDC), U.S. Air Force, U.S. Navy, and Canadian Atmospheric Environment Service.

Indoor Design Criteria and Thermal Comfort Indoor design criteria, such as space temperature, humidity, and air cleanliness, specify the requirements for the indoor environment as well as the quality of an air-conditioning or HVAC&R project. The human body requires energy for physical and mental activity. This energy comes from the oxidation of food. The rate of heat release from the oxidation process is called the metabolic rate, expressed in met (1 met = 18.46 Btu/h.ft2). The metabolic rate depends mainly on the intensity of the physical activity of the human body. Heat is released from the human body by two means: sensible heat exchange and evaporative heat loss. Experience and experiments all show that there is thermal comfort only under these conditions: • Heat transfer from the human body to the surrounding environment causes a steady state of thermal equilibrium; that is, there is no heat storage in the body core and skin surface. • Evaporative loss or regulatory sweating is maintained at a low level. The physiological and environmental factors that affect the thermal comfort of the occupants in an air-conditioned space are mainly: 1. Metabolic rate M determines the amount of heat that must be released from the human body. 2. Indoor air temperature Tr and mean radiant temperature Trad, both in °F. The operating temperature To is the weighted sum of Tr and Trad. Trad is defined as the temperature of a uniform black enclosure in which the surrounded occupant would have the same radiative heat exchange as in an actual indoor environment. Tr affects both the sensible heat exchange and evaporative losses, and Trad affects only sensible heat exchange. In many indoor environments, Trad ≈ Tr . 3. Relative humidity of the indoor air ϕr , in %, which is the primary factor that influences evaporative heat loss. 4. Air velocity of the indoor air vr , in fpm, which affects the heat transfer coefficients and therefore the sensible heat exchange and evaporative loss. 5. Clothing insulation Rcl, in clo (1 clo = 0.88 h.ft2.°F/Btu), affects the sensible heat loss. Clothing insulation for occupants is typically 0.6 clo in summer and 0.8 to 1.2 clo in winter. © 2005 by CRC Press LLC

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Air-Conditioning and Refrigeration

Indoor Temperature, Relative Humidity, and Air Velocity For comfort air-conditioning systems, according to ANSI/ASHRAE Standard 55-1992, the following indoor design temperatures and air velocities apply for conditioned spaces where the occupant’s activity level is
1.2 met, mean air speed
30 fpm, indoor space relative humidity is 50% (in summer only), and Tr = Trad: Clothing Insulation (clo)

Indoor Temperature (°F)

0.9 0.5

68–75 73–79

Winter Summer

Regarding the indoor humidity: 1. Many comfort air-conditioning systems are not equipped with humidifiers. Winter indoor relative humidity should not be specified in such circumstances. 2. When comfort air-conditioning systems are installed with humidifiers, ASHRAE/IES Standard 90.1-2001 provides specific requirements for economy of operation. 3. Indoor relative humidity should not exceed 75% to avoid increasing bacterial and viral populations. 4. For air-conditioning systems that use flow rate control in a chilled water cooling coil, space indoor relative humidity may be substantially higher in part load than at full load. Therefore, for comfort air-conditioning systems, the recommended indoor relative humidities, in %, are

Summer Winter With humidifier Without humidifier

Tolerable range

Preferred value

30–65

40–50 25–30 Not specified

In surgical rooms or similar health care facilities, the indoor relative humidity is often maintained at 40 to 60% year round.

Indoor Air Quality and Outdoor Ventilation Air Requirements According to the National Institute for Occcupational Safety and Health (NIOSH), 1989, the causes of indoor air quality complaints in buildings are inadequate outdoor ventilation air, 53%; indoor contaminants, 15%; outdoor contaminants, 10%; microbial contaminants, 5%; construction and furnishings, 4%; unknown and others, 13%. For space served by air-conditioning systems using low- and mediumefficiency air filters, according to the U.S. Environmental Protection Agency (EPA) and Consumer Product Safety Commission (CPSC) publication “A Guide to Indoor Air Quality” (1988) and the field investigations reported by Bayer and Black (1988), indoor air contaminants may include some of the following: 1. Total particulate concentration. This concentration comprises particles from building materials, combustion products, mineral fibers, and synthetic fibers. In February 1989, the EPA specified the allowable indoor concentration of particles of 10 µm and less in diameter (which penetrate deeply into lungs) as: 50 µg/m3 (0.000022 grain/ft3), 1 year 150 µg/m3 (0.000066 grain/ft3), 24 hr In these specifications, “1 year” means maximum allowable exposure per day over the course of a year. 2. Formaldehyde and organic gases. Formaldehyde is a colorless, pungent-smelling gas. It comes from pressed wood products, building materials, and combustion. Formaldehyde causes eye, nose, and © 2005 by CRC Press LLC

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3.

4.

5.

6.

Chapter 9

throat irritation as well as coughing, fatigue, and allergic reactions. Formaldehyde may also cause cancer. Other organic gases come from building materials, carpeting, furnishings, cleaning materials, etc. Radon. Radon, a colorless and odorless gas, is released by the decay of uranium from the soil and rock beneath buildings, well water, and building materials. Radon and its decay products travel through pores in soil and rock and infiltrate into buildings along the cracks and other openings in the basement slab and walls. Radon at high levels causes lung cancer. The EPA believes that levels in most homes can be reduced to 4 pCi/l (picocuries per liter) of air. The estimated national average is 1.5 pCi/l, and levels as high as 200 pCi/l have been found in houses. Biologicals. These include bacteria, fungi, mold and mildew, viruses, and pollen. They may come from wet and moist walls, carpet furnishings, and poorly maintained dirty air-conditioning systems and may be transmitted by people. Some biological contaminants cause allergic reactions, and some transmit infectious diseases. Combustion products. These include environmental tobacco smoke, nitrogen dioxide, and carbon monoxide. Environmental tobacco smoke from cigarettes is a discomfort factor to other persons who do not smoke, especially children. Nicotine and other tobacco smoke components cause lung cancer, heart disease, and many other diseases. Nitrogen dioxide and carbon monoxide are both combustion products from unvented kerosene and gas space heaters, woodstoves, and fireplaces. Nitrogen dioxide (NO2) causes eye, nose, and throat irritation; may impair lung function; and increases respiratory infections. Carbon monoxide (CO) causes fatigue at low concentrations; impaired vision, headache, and confusion at higher concentrations; and is fatal at very high concentrations. Houses without gas heaters and gas stoves may have CO levels varying from 0.5 to 5 parts per million (ppm). Human bioeffluents. These include the emissions from breath including carbon dioxide exhaled from the lungs, body odors from sweating, and gases emitted as flatus.

There are three basic means of reducing the concentration of indoor air contaminants and improving indoor air quality: (1) eliminate or reduce the source of air pollution, (2) enhance the efficiency of air filtration, and (3) increase the ventilation (outdoor) air intake. Dilution of the concentrations of indoor contaminants by outdoor ventilation air is often the simple and cheapest way to improve indoor air quality. The amount of outdoor air required for metabolic oxidation is rather small. Outdoor air requirements listed in ANSI/ASHRAE Standard 62-1999 can be used in the ventilation procedure method to satisfy indoor air quality requirements. The minimum outside air ventilation per person for any type of space is 15 cfm. This requirement is based on the analysis of dilution of CO2 as the representative human bioeffluent to an allowable indoor concentration within 0.07% (700 ppm) of the outdoor concentration. Field measurements of daily maximum CO2 levels in office buildings reported by Persily (1993) show that most of them were within the range 400 to 820 ppm. The quality of outdoor air must meet the EPA’s National Primary and Secondary Ambient Air Quality Standards, some of which is listed below: Long-Term Concentration Pollutants

µg/m3

ppm

Particulate SO2 CO

75 80

0.03

NO2 Lead

100 1.5

0.055

Exposure 1 year 1 year

Short-Term Concentration µg/m3

ppm

Exposure

260 365 40,000 10,000

0.14 35 9

24 hr 24 hr 1 hr 8 hr

1 year 3 months

Here exposure means average period of exposure. If unusual contaminants or unusually strong sources of contaminants are introduced into the space, or recirculated air is to replace part of the outdoor air supply for occupants, then acceptable indoor air © 2005 by CRC Press LLC

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quality is achieved by controlling known and specific contaminants. This is called an indoor air quality procedure. Refer to ANSI/ASHRAE Standard 62-1999 for details. Clean Rooms Electronic, pharmaceutical, and aerospace industries and operating rooms in hospitals all need strict control of air cleanliness during manufacturing and operations. According to ASHRAE Handbook 1991 HVAC Applications, clean rooms can be classified as follows based on the particle count per ft3: Particle Size 0.5 µm and Larger

Class

5 µm and Larger

Particle Count per ft3 Not to Exceed 1 10 100 1000 10,000 100,000

1 10 100 1000 10,000 100,000

0 0

65 700

For clean rooms, space temperature is often maintained at 72 ± 2°F and space humidity at 45 ± 5%. Here, ±2°F and ±5% are allowable tolerances. Federal Standard 209B specifies that the ventilation (outdoor air) rate should be 5 to 20% of the supply air. Space Pressure Differential Most air-conditioning systems are designed to maintain a slightly higher pressure than the surroundings, a positive pressure, to prevent or reduce infiltration and untreated air entering the space directly. For laboratories, restrooms, or workshops where toxic, hazardous, or objectional gases or contaminants are produced, a slightly lower pressure than the surroundings, a negative pressure, should be maintained to prevent or reduce the diffusion of these contaminants’ exfiltrate to the surrounding area. For comfort air-conditioning systems, the recommended pressure differential between the indoor and outdoor air is 0.02 to 0.05 in in. WG. WG indicates the pressure at the bottom of a top-opened water column of specific inches of height; 1 in. WG = 0.03612 psig. For clean rooms, Federal Standard No. 209B, Clean Rooms and Work Stations Requirements (1973), specifies that the minimum positive pressure between the clean room and any adjacent area with lower cleanliness requirements should be 0.05 in. WG with all entryways closed. When the entryways are open, an outward flow of air is to be maintained to prevent migration of contaminants into the clean room. In comfort systems, the space pressure differential is usually not specified in the design documents. Sound Levels Noise is any unwanted sound. In air-conditioning systems, noise should be attenuated or masked with another less objectionable sound. Sound power is the capability to radiate power from a sound source exited by an energy input. The intensity of sound power is the output from a sound source expressed in watts (W). Due to the wide variation of sound output at a range of 1020 to 1, it is more convenient to use a logarithmic expression to define a sound power level Lw, in dB:

(

)

Lw = 10 log w 10 −12 W re 1 pW

(9.5.2)

where w = sound source power output, W. The human ear and microphones are sound pressure sensitive. Similarly to the sound power level, the sound pressure level Lp, in dB, is defined as: © 2005 by CRC Press LLC

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(

)

Lp = 20 log p 2 × 10 −5 Pa re 20 µPa

(9.5.3)

where p = sound pressure, Pa. The sound power level of any sound source is a fixed output. It cannot be measured directly; it can only be calculated from the measured sound pressure level. The sound pressure level at any one point is affected by the distance from the source and the characteristics of the surroundings. Human ears can hear frequencies from 20 Hz to 20 kHz. For convenience in analysis, sound frequencies are often subdivided into eight octave bands. An octave is a frequency band in which the frequency of the upper limit of the octave is double the frequency of the lower limit. An octave band is represented by its center frequency, such as 63, 125, 250, 500, 1000, 2000, 4000, and 8000 Hz. On 1000 Hz the octave band has a higher limit of 1400 Hz and a lower limit of 710 Hz. Human ears do not respond in the same way to low frequencies as to high frequencies. The object of noise control in an air conditioned space is to provide background sound low enough that it does not interfere with the acoustical requirements of the occupants. The distribution of background sound should be balanced over a broad range of frequencies, that is, without whistle, hum, rumble, and beats. The most widely used criteria for sound control are the noise criteria NC curves. The shape of NC curves is similar to the equal-loudness contour representing the response of the human ear. NC curves also intend to indicate the permissible sound pressure level of broad-band noise at various octave bands rated by a single NC curve. NC curves are practical and widely used. Other criteria used are room criteria RC curves and A-weighted sound level, dBA. RC curves are similar to NC curves except that the shape of the RC curves is a close approximation to a balanced, blandsounding spectrum. The A-weighted sound level is a single value and simulates the response of the human ear to sound at low sound pressure levels. The following are abridged indoor design criteria, NC or RC range, listed in ASHRAE Handbook 1987 Systems and Applications: Type of Area

Recommended NC or RC Range (dB)

Hotel guest rooms Office Private Conference Open Computer equipment Hospital, private Churches Movie theaters

30–35 30–35 25–30 30–35 40–45 25–30 25–30 30–35

For industrial factories, if the machine noise in a period of 8 hr exceeds a prescribed level given in dBA, Occupational Safety and Health Administration Standard Part 1910.95 requires the occupants to use personal protection equipment. If the period is shorter, the dBA level can be slightly higher. Refer to this standard for details.

9.6 Principles of Load Calculations Ari Rabl and Peter Curtiss Design Conditions Loads are the heat that must be supplied or removed by HVAC equipment to maintain a space at the desired conditions. Loads depend on the indoor conditions that one wants to maintain and on the © 2005 by CRC Press LLC

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weather. The latter is not known in advance. If the HVAC equipment is to guarantee comfort at all times, it must be designed for peak conditions. What are the extremes? For most buildings it would not be practical to aim for total protection by choosing the most extreme weather on record and adding a safety margin. Such oversizing of the HVAC equipment would be excessive, not just in initial cost but also in operating cost; most of the time, the equipment would run with poor part-load efficiency. Therefore, compromise — reducing the cost of the HVAC equipment significantly while accepting the risk of slight discomfort under rare extremes of weather — is necessary. The greater the extreme is, the more rarely it occurs. Wind speed is another weather-dependent variable that has a bearing on loads. Traditionally the ASHRAE (2001) value v win = 15 mi h ( 6.7 m s )

(9.6.1)

has been recommended for heating loads, if extreme conditions (such as an exposed hilltop location) are not implied. For cooling loads, a value half as large is recommended: Vsum = 7.5 mi s ( 3.4 m s )

(9.6.2)

because wind tends to be less strong in summer than in winter. Of particular interest is the surface heat transfer coefficient (radiation plus convection) ho for which ASHRAE (2001) recommends the design values:

(

)

(

)

22.7 W m 2 · K

ho,win = 6.0 Btu h·ft 2 ·°F 34.0 W m 2 · K

(

ho,sum = 4.0 Btu h · ft 2 · °F

(

)

(9.6.3)

)

(9.6.4)

This coefficient is only one of several components of the calculation of thermal loads, and it enters only through the building heat transmission coefficient defined in the next section. The better a building is insulated and tightened, the less its heat transmission coefficient Ktot depends on wind. With current practice for new construction in the United States, typical wind speed variations may change the heat transmission coefficient by about 10% relative to the value at design conditions.

Building Heat Transmission Coefficient One of the most important terms in the heat balance of a building is the heat flow across the envelope. Heat flow can be assumed to be linear in the temperature difference when the range of temperatures is sufficiently small; this is usually a good approximation for heat flow across the envelope. Thus, one can calculate the heat flow through each component of the building envelope as the product of its area A; its conductance U; and the difference Ti – To between the interior and outdoor temperatures. The total conductive heat flow from interior to exterior is Q˙ cond =

∑U

k

A k (Ti − To ),

(9.6.5)

k

with the sum running over all parts of the envelope that have a different composition. It is convenient to define a total conductive heat transmission coefficient Kcond, or UA value, as K cond =

∑U k

© 2005 by CRC Press LLC

k

Ak

(9.6.6)

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so that the conductive heat flow for the typical case of a single interior temperature Ti can be written as Q˙ cond = K cond (Ti − To )

(9.6.7)

In most buildings, the envelope consists of a large number of different parts; the greater the desired accuracy is, the greater the amount of detail to be taken into account. As a simplification, one can consider a few major groups and use effective values for each. The three main groups are glazing, opaque walls, and roof. The reason for distinguishing the wall and the roof lies in the thickness of the insulation: roofs tend to be better insulated because it is easier and less costly to add extra insulation there than in the walls. With these three groups, one can write Kcond = Uglaz Aglaz + Uwall Awall + Uroof Aroof

(9.6.8)

if one takes for each the appropriate effective value. Results for aggregate U values for walls and roofs of typical construction can be found in ASHRAE lookup tables. In the energy balance of a building, one other term is proportional to Ti – To . It is the flow of sensible heat [W (Btu/h)] due to air exchange: Q˙ air = ρc p V˙ (Ti − To )

(9.6.9)

where ρ = density of air cp = specific heat of air V˙ = air exchange rate ft3/h (m3/s) At standard conditions, 14.7 psia (101.3 kPa) and 68°F (20°C), the factor ρcp has the value ρcp = 0.018 Btu/(ft3 · °F) [1.2 kJ/(m3 · K)]

(9.6.10)

In USCS units, if V˙ is in cubic feet per minute, it must be converted to cubic feet per hour by multiplying by 60 (ft3/h)/(ft3/min). It is convenient to combine the terms proportional to Ti – To by defining the total heat transmission coefficient Ktot of the building as the sum of conductive and air change terms: Ktot = Kcond + ρcp V˙

(9.6.11)

Note that V˙ increases with the temperature difference for a number of reasons (see Kreider et al., 2001).

Heat Gains Heat gains affect heating as well as cooling loads. In addition to solar gains, heat gains occur from occupants, lights, and equipment such as appliances, motors, computers, and copiers. Power densities for lights in office buildings are around 20 to 30 W/m2. For lights and for resistive heaters, the nominal power rating (i.e., the rating on the label) is usually close to the power drawn in actual use. However, for office equipment, that would be quite misleading; the actual power has been measured to be much lower, often by a factor of two to four (Norford et al., 1989). Some typical values are indicated in Table 9.6.1. In recent decades, the computer revolution has brought a rapid increase in electronic office equipment, and the impact on loads has become quite important, comparable to that of lighting. The energy consumption for office equipment is uncertain: will the occupants turn off the computers between uses or keep them running nights and weekends? For special equipment such as laboratories or kitchens, it is advisable to estimate the heat gains by taking a close look at the inventory of the equipment to be installed, paying attention to the possibility that much of the heat may be drawn directly to the outside by exhaust fans. © 2005 by CRC Press LLC

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TABLE 9.6.1 Typical Heat Gain Rates for Several Kinds of Equipment Heat Gain Equipment Television set Refrigerator Personal computer (desktop) Impact printer Laser printer Copier

Btu/h

W

Comments

170–340 340–680 170–680 34–100 500 500–1000

50–100 100–200 50–200 10–30 standby 150 standby 150–300 standby

Recent models more efficient Almost independent of use while turned on Increases about twofold during printing Increases about twofold during printing Increases about twofold during printing

Note: Measured values are often less than half of the nameplate rating. Source: Based on ASHRAE, 1989, Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta; Norford, L.K. et al., 1989, in T.B. Johansson et al., Eds. Electricity: Efficient End Use and New Generation Technologies, and Their Planning Implications. Lund University Press, Lund, Sweden, 427–460; and updates.

TABLE 9.6.2 Nominal Heat Gain Values from Occupants Total

Sensible

Latent

Activity

Btu/h

W

Btu/h

W

Btu/h

W

Seated at rest Seated, light office work Standing or walking slowly Light physical work Heavy physical work

340 410 495 850 1600

100 120 145 250 470

240 255 255 310 630

70 75 75 90 185

100 150 240 545 970

30 45 70 160 285

Source: Based on ASHRAE, 1989, Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta.

Heat gain from occupants depends on the level of physical activity. Nominal values are listed in Table 9.6.2. It is instructive to reflect on the origin of this heat gain. The total heat gain must be close to the caloric food intake because most of the energy is dissipated from the body as heat. An average of 100 W corresponds to  1 kcal  100 W = 0.1 kJ/s ×   × (24 × 3600 s/day) = 2064 kcal/day  4.186 kJ 

(9.6.12)

indeed, a reasonable value compared to the typical food intake (note that the dietician’s calorie is really a kilocalorie). The latent heat gain must be equal to the heat of vaporization of water that is exhaled or transpired. Dividing 30 W by the heat of vaporization of water, yields a water quantity of 30 W/(2450 kJ/kg) = 12.2 × 10–6 kg/s, or about 1.1 kg/24 h. That also appears quite reasonable. The latent heat gain due to the air exchange is Q˙ air ,lat = V˙ ρh fg (W o − W i ) where

V˙ = volumetric air exchange rate, ft3/min (m3/s or L/s) ρ = density, lbm /ft3 (kg/m3) ρ hfg = 4840 Btu/(h · ft3/min) [3010 W/(L/s)] at standard conditions Wi , Wo = humidity ratios of indoor and outdoor air

© 2005 by CRC Press LLC

(9.6.13)

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Load (heat supplied or removed by HVAC equipment)

Space to be conditioned

Conduction (roof, walls, glazing) Conduction (ground) Air exchange, sens. + latent Solar radiation People, sens. + latent Equipment, sens. + latent Lights

Heat capacity, sens. + latent

FIGURE 9.6.1 The terms in a load calculation.

Heat Balance Load calculations are like accounting. One considers all the heat that is generated in the space or flows across the envelope; the total energy, including the thermal energy stored in the space, must be conserved according to the first law of thermodynamics. The principal terms are indicated in Figure 9.6.1. Outdoor air, occupants, and possibly certain kinds of equipment contribute sensible and latent heat terms. Load calculations are straightforward in the static limit, i.e., if all input is constant. As discussed in the following subsection, that is usually an acceptable approximation for the calculation of peak heating loads. However, for cooling loads, dynamic effects (i.e., heat storage) must be taken into account because some of the heat gains are absorbed by the mass of the building and do not contribute to the loads until several hours later. Dynamic effects are also important whenever the indoor temperature is allowed to float. Sometimes it is appropriate to distinguish several aspects of the load. If the indoor temperature is not constant, the instantaneous load of the space may differ from the rate at which heat is being supplied or removed by the HVAC equipment. The load for the heating or cooling plant is different from the space load if there are significant losses from the distribution system or if part of the air is exhausted to the outside rather than returned to the heating or cooling coil. It is convenient to classify the terms of the static energy balance according to the following groups. The sensible energy terms are: • Conduction through building envelope other than ground: Q cond = K cond (Ti − To )

(9.6.14)

• Conduction through floor, Q floor • Heat due to air exchange (infiltration and/or ventilation), at rate V˙ : Q air = V ρc p (Ti − To )

(9.6.15)

• Heat gains from solar radiation, lights, equipment (appliances, computers, fans, etc.), and occupants: Q gain = Q sol + Q lit + Q equ + Q occ

(9.6.16)

Combining the heat loss terms and subtracting the heat gains, one obtains the total sensible load: Q = Q cond + Q air + Q floor − Q gain ± Q stor

(9.6.17)

where a term, Q stor , has been added on the right to account for storage of heat in the heat capacity of the building (the terms thermal mass and thermal inertia are also used to designate this effect). A dynamic analysis includes this term; a static analysis neglects it. © 2005 by CRC Press LLC

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Q floor has been kept as a separate item because it should not be taken proportional to Ti – To except in cases like a crawl space, where the floor is in fairly direct contact with outside air. More typical is conduction through massive soil, for which other methods are required. In traditional construction, the floor term has usually been small, and often it has been neglected altogether; however, in superinsulated buildings it can be relatively important. Using the total heat transmission coefficient Ktot, K tot = K cond + V ρc p

(9.6.18)

one can write the sensible load in the form Q = K tot (Ti − To ) + Q floor − Q gain ± Q stor

(9.6.19)

For signs, take the convention that Q is positive when there is a heating load and negative when there is a cooling load. Sometimes, however, it will be preferable to have a positive sign for cooling loads. In that case, subscripts c and h will be added with the understanding that Q c = −Q and Q h = Q

(9.6.20)

The latent heat gains are mainly due to air exchange, equipment (such as in the kitchen and bathroom), and occupants. Their sum is Q lat = Q lat ,air + Q lat ,occ + Q lat ,equ

(9.6.21)

The total load is the sum of the sensible and the latent loads. During the heating season, the latent gain from air exchange is usually negative (with the signs of Equation 9.6.13) because the outdoor air is relatively dry. A negative Qlat implies that the total heating load is greater than the sensible heating load alone, but this is relevant only if humidification can maintain the specified humidity ratio Wi . For buildings without humidification, one has no control over Wi , and it is pointless to calculate the latent contribution to the heating load at a fictitious value of Wi .

Zones So far the interior has been considered as a single zone at uniform temperature — a fair approximation for simple houses, for certain buildings without windows (such as warehouses), or for buildings dominated by ventilation. In large or complex buildings, however, one must usually calculate the loads separately for a number of different zones, for several reasons. An obvious case is a building in which different rooms are maintained at different temperatures, e.g., a house with an attached sunspace. Here, the heat balance equation is written for each zone, but with an additional term: Q j −k = U j −k A j −k (Tj − Tk )

(9.6.22)

for the heat flow between zones j and k. However, even when the entire building is kept at the same temperature, multizone analysis becomes necessary if the spatial distribution of heat gains is too nonuniform. Consider, for example, a building with large windows on the north and south sides, during a sunny winter day when the gains just balance the total heat loss. In that case, neither heating nor cooling would be required, according to a one-zone analysis — but how can the heat from the south get to the north? Heat flow is the product of the heat transfer coefficient and the temperature difference, as in Equation (9.6.22). Temperature differences between occupied zones are small, usually not more than a few Kelvins; © 2005 by CRC Press LLC

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2

3

5 1

4

FIGURE 9.6.2 Example of recommended zoning. Thick lines represent zones, labeled 1 through 5. Dashed lines represent subzones.

otherwise there would be complaints about comfort. The heat transfer coefficients between zones are often not sufficiently large for effective redistribution of heat, especially if walls or partitions are part of the space — thus the thermodynamically perverse fact that many large buildings require simultaneous heating and cooling. The problem of divergent zone loads is one of the prime targets for energy conservation in large buildings. The first step is to reduce the loads through the envelope, by improved insulation and control of solar radiation. The smaller the loads are, the smaller the differences between the loads. Careful attention must be paid to the design of the HVAC system and the choice of its zones. Finally, heat pump may be able to recover and redistribute heat between zones. The basic criterion for zoning is the ability to control the comfort conditions; the control is limited by the number of zones one is willing to consider. To guarantee comfort, the HVAC plant and distribution system must be designed with sufficient capacity to meet the load of each zone. In choosing the zones for a multizone analysis, the designer should try to match the distribution of heat gains and losses. A common and important division is between interior and perimeter zones, because the interior is not exposed to the changing environment. Different facades of the perimeter should be considered separately for cooling load calculations, as suggested in Figure 9.6.2. Corner rooms should be assigned to the facade with which they have the most in common; usually, this will be the facade where a corner room has the largest windows. Corner rooms are often the critical rooms in a zone, requiring more heating or cooling (per unit floor area) than single-facade rooms of the same zone. Actually, a zoning analysis has different levels, corresponding to different levels of the HVAC system. In an air system, major zones correspond to each air handler. Within each air handler zone, the air ducts, air outlets, and heating or cooling coils must have sufficient capacity and controllability to satisfy the loads of each subzone; the design flow rates for each room are scaled according to the design loads of the room. For best comfort (and if cost were no constraint), each zone should have its own air handler and each room its own thermostat. There is a tradeoff between equipment cost and achievable comfort, and the best choice depends on the circumstances. If temperature control is critical, one installs separate air handlers for each of the five zones in Figure 9.6.2 and separate thermostats for each room. To save equipment cost, one often assigns several zones to one air handler and several rooms to one thermostat; however, the more divergent the loads are, the more problematic the control. For the building of Figure 9.6.2, a single air handler and five thermostats may be adequate if the distribution of heat gains is fairly uniform and if the envelope is well insulated, with good control of solar gains. Another example is a house in which air distribution system has a single fan (typical of all but the largest houses). Even though there is only one major zone, the detailed design of the distribution system demands some attention to subzones. Within each room, the peak heating capacity should match the peak heat loss. Also, it is advisable to place heat sources close to points with large heat loss, i.e., under windows (unless they are highly insulating). © 2005 by CRC Press LLC

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9-55

The choice of zones is not always clear-cut, and the design process may be iterative. Depending on the distribution of gains and losses, one may want to assign several rooms to a zone, one room to a zone, or even several zones to a room (if it is very large). With finer zonal detail, one improves the control of comfort, but at the price of greater calculation effort and higher HVAC system cost. In an open office space, no obvious boundary exists between interior and perimeter; here a good rule is to make the perimeter zone as deep as the penetration depth of direct solar radiation, typically a few meters. Spaces connected by open doors, for example, offices and adjacent hallways, can sometimes be treated as a single zone. Separate zones are advisable for rooms with large computers or energy-intensive equipment. In multistory buildings, one may want to treat the top floor apart from the rest. The calculation of peak heating loads and capacities can often be done without defining different perimeter zones because peak heating loads occur when the sun is not present; with uniform internal gains, the corresponding thermal balance is uniform around the perimeter. Although the calculation can be carried out for a single zone, the operation requires multiple zones because the heating system must allow separate control of different facades to compensate for the variability of solar gains during the day. For cooling loads, a multizone analysis is essential, even at the calculation stage, because the loads occur when the sun is shining. Peak cooling loads require a dynamic analysis whereas peak heating loads can be estimated quite well by static models (at least in the absence of thermostat setback). Compared to heating loads, the calculation of cooling loads of large buildings is thus doubly complicated: It requires multiple zones and dynamic analysis if one wants reasonable accuracy. A related issue is the coincidence between peak loads of different zones. To determine the capacity of the central plant, one needs to know the peak load of the totality of zones served by the plant. This is usually less than the simple sum of the individual peak loads because of noncoincidence. The term diversity is used to designate the ratio of the actual system peak to the sum of the individual peak loads. In practice, one often finds diversity around 0.6 to 0.8 for large buildings or groups of buildings (e.g., university campuses); for better estimates at the design stage, computer simulations are recommended.

Heating Loads Because the coldest weather is likely to occur during periods without solar radiation, it is advisable not to rely on the benefit of solar heat gains when calculating peak heating loads (unless the building contains long-term storage). If the indoor temperature Ti is constant, a static analysis is sufficient and the calculation of the peak heating load Q˙ h ,max is very simple: find the design heat loss coefficient Ktot ; multiply by the design temperature difference Ti – To ; and subtract the internal heat gains on which one can count during the coldest weather: Q˙ h ,max = K tot (Ti − To ) − Q˙ gain

(9.6.23)

What would happen if the thermostat were set back at night? For a rough indication, consider that a house of lightweight construction typical in the U.S. requires heat input at the rate of 1.2 kW if its temperature is increased by 2 K in 5 h. For setback recovery after winter nights, one might want rates that are several times faster, say, 4 K in 2.5 h. Assuming that the heat input is proportional to the warmup rate,1 the extra load for setback recovery (also known as the pickup load) would be 4 × 1.2 kW = 4.8 kW, comparable to the static design heat load. In this case, the capacity of the heating system would need to be doubled relative to the case without setback. In a given situation, the required extra capacity depends on the amount of setback Ti – To , the acceptable recovery time, and building construction. For reasonable accuracy, a dynamic analysis is recommended. 1

Actually, at faster rates the effective heat capacity is smaller (the heat pulse takes longer than 1 h to penetrate the entire mass); thus, the real increment for setback recovery is less. One does not know how much without a detailed dynamic analysis. © 2005 by CRC Press LLC

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Optimizing the capacity of the heating system involves a tradeoff between energy savings and capacity savings, with due attention to part-load efficiency. As a general rule for residences, ASHRAE (1989) recommends oversizing by about 40% for a night setback of 10°F (5.6 K), to be increased to 60% oversizing if additional setback occurs during the day. In any case, some flexibility can be provided by adapting the operation of the building. If the capacity is insufficient, one can reduce the depth and duration of the setback during the coldest periods. In commercial buildings with mechanical ventilation, the demand for extra capacity during setback recovery is reduced if the outdoor air intake is closed during unoccupied periods. In winter that should always be done for energy conservation (unless air quality problems demand high air exchange at night).

CLTD/SCL/CLF Method for Cooling Loads Because of thermal inertia, it is advisable to distinguish several heat flow rates. The heat gain is the rate at which heat is transferred to or generated in a space. The cooling load is the rate at which the cooling equipment would have to remove thermal energy from the air in the space in order to maintain constant temperature and humidity. Finally, the heat extraction rate is the rate at which the cooling equipment actually does remove thermal energy from the space. Conductive heat gains and radiative heat gains do not enter the indoor air directly; rather, they pass through the mass of the building, increasing its temperature relative to the air. Only gradually are they transferred to the air. Thus their contribution to the cooling load is delayed, and there is a difference between heat gain and cooling load. Averaged over time, these rates are, of course, equal, by virtue of the first law of thermodynamics. The heat extraction rate is equal to the cooling load only if the temperature of the indoor air is constant (as assumed in this section). Otherwise the heat flow to and from the building mass causes the heat extraction rate to differ from the cooling load. To account for transient effects without resorting to a full-fledged dynamic analysis, a special shorthand method has been developed that uses the cooling load temperature difference (CLTD) and cooling load factor (CLF). To explain the principles, note that the cooling load due to conduction across an envelope element of area A and conductance U would be simply Q˙ cond = U A (To – Ti)

(9.6.24)

under static conditions, i.e., if indoor temperature Ti and outdoor temperature To were both constant. When the temperatures vary, this is no longer the case because of thermal inertia. However, if the temperatures follow a periodic pattern, day after day, Q˙ c ,cond will also follow a periodic pattern. Once Q˙ c ,cond has been calculated, one can define a CLTD as the temperature difference that gives the same cooling load when multiplied by UA. If such temperature differences are tabulated for typical construction and typical temperature patterns, they can be looked up for quick determination of the load. Thus, the conductive cooling load is Q˙ cond ,t = U A CLTDt

(9.6.25)

where the subscript t indicates the hour t of the day. Likewise, if there is a constant radiative heat gain in a zone, the corresponding cooling load is simply equal to that heat gain. If the heat gain follows a periodic pattern, the cooling load also follows a periodic pattern. The cooling load factor (CLF) is defined such that it yields the cooling load at hour t when multiplied by the daily maximum Q˙ max of the heat gain: Q˙ c ,rad ,t = Q˙ max CLFt

(9.6.26)

The CLFs account for the fact that radiative gains (solar, lights, etc.) are first absorbed by the mass of the building, becoming a cooling load only as they are transferred to the air. Only convective gains can © 2005 by CRC Press LLC

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Air-Conditioning and Refrigeration

be counted as cooling load without delay. Some heat gains, for example, from occupants, are partly convective and partly radiative; the corresponding CLFs take care of that. The CLTDs and CLFs of ASHRAE have been calculated by means of the transfer functions discussed in the next section. To keep the bulk of numerical data within reasonable limits, only a limited set of standard construction types and operating conditions has been considered. Some correction factors are available to extend the applicability, however, without escaping the constraint that the indoor temperature Ti be constant. For a list of these factors, see ASHRAE (2001) or Kreider et al. (2001). Roof CLTD Value Selection The CLTD/SCL/CLF method uses 10 types of roofs. The roof types are numbered 1, 2, 3, 4, 5, 8, 9, 10, 13, and 14. The roof type chosen depends on the principal roof material; location of the mass in the roof; overall R value of the roof; and whether the ceiling is suspended. The tables of new roof CLTD values are calculated based on an indoor temperature of 78°F, maximum and mean outdoor temperatures of 95 and 85°F, respectively, and a daily range of 21°F. Once the 24 CLTD values are selected, they are each adjusted by Corrected CLTD = CLTD + (78 – Ti) + (Tom – 85)

(9.6.27)

where Ti is the actual inside design dry bulb temperature and Tom is the mean outside design dry bulb temperature, Tom = Outside design dry bulb temperature −

nge Daily ran 2

(9.6.28)

No adjustments to the CLTD are recommended for color or ventilation. The CLTD charts are usually published for several different latitudes; interpolation between the latitudes for an exact site is acceptable. Wall CLTD Value Selection The CLTD/SCL/CLF uses 15 wall types numbered sequentially 1 through 16 with no wall 8. The wall type is chosen based on the principal wall material; secondary wall material; location of the mass in the wall; and overall wall R value. The tables of wall CLTD values are broken down by latitude. The wall CLTDs were calculated using the same conditions as the roof CLTD values and may require adjustments based on the actual inside and ambient conditions. Interpolation between the tables may be necessary to obtain the correct values for a given site. Once the roof and wall CLTD values have been selected and adjusted as necessary, the conductive heat flow through the roof and walls is calculated for each hour t q(t) = U A CLTD(t)

(9.6.29)

where U = overall heat transfer coefficient for the surface (Btu/h·ft2·°F) A = area of surface CLTD = cooling load temperature difference Glass CLTD Value Selection As with the roof and wall CLTDs, the fenestration CLTD values may need to be corrected. The conductive load calculation from the glass uses the same method as that for the roof and walls. Solar Cooling Load The solar cooling load (SCL) is used to calculate the radiative (solar) heat gain through any glass surface in the building. The radiative solar gains are then given by © 2005 by CRC Press LLC

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Chapter 9

q(t) = A SC SCL(t)

(9.6.30)

where A is the area of the glass surface and SC is the shading coefficient. The shading coefficient is the ratio of the actual solar heat gain to that from the reference window used to calculate the SCL. Using the SCL value tables requires that one know the number of walls, floor covering, and inside shading, as well as a number of other variables for the zone. The tables are also broken down by building type, with different tables for zones in • • • •

Single-story buildings Top floor of multistory buildings Middle floors of multistory buildings First floor of multistory buildings

The zone types used for determining the SCL are not necessarily the same zone type used for the CLF tables. Once the zone type has been determined, the SCL is found from lookup tables. Accounting for Adjacent Zones The CLTD/SCL/CLF method treats the conductive heating load from any adjacent spaces through internal partitions, ceilings, and floors as a simple steady-state energy flow: q(t) = U A (Ta – Tr)

(9.6.31)

where Ta is the temperature in the adjacent space and Tr is the temperature of the room in question. Occupant Loads People within a space add sensible as well as latent loads to the space. The heating load at any given hour due to the occupants is given as q(t) = N Fd [qs CLF(t) + ql]

(9.6.32)

where N = the number of people in the space Fd = the diversity factor CLF = the cooling load factor for occupants on a given schedule As implied by Equation (9.6.32), the latent load is assumed to translate immediately into a cooling load on the system while the sensible load is subject to some time delay as dictated by the mass of the room — i.e., its capability to absorb heat and release it at a later time. The diversity factor Fd takes into account the variability of the actual number of occupants in the space. The CLF values come from tables. To find the CLF, it is first necessary to determine the zone type. This is done in a fashion similar to that for the solar cooling loads. That is, the building type, room location, and floor coverings must be known before the zone type can be found. Note that the zone type for occupants and equipment is not the same as for lighting. The same holds true for the solar cooling load: the zone type for occupants is not the same as the zone type for the SCL. Once the zone type has been determined, the occupant CLF is found from the lookup tables. Zone type A is for light construction and the zones get progressively more massive for types B, C, and D. Figure 9.6.3 shows the cooling load factors for type A and D zones that are occupied for 12 h. Note that the occupant CLF will be 1.0 for all hours in a building with high occupant density (greater than 1 person per 10 ft2) such as auditoriums and theaters. The CLF will also be 1.0 in buildings with occupancy 24 h/day. Lighting Loads At any given hour the load due to the lighting is approximated as q(t) = watts Fd Fsa CLF(t) © 2005 by CRC Press LLC

(9.6.33)

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CLF

Air-Conditioning and Refrigeration

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Type A zone Type D zone

0

3

6

9 12 15 18 Hour after entry into space

21

24

FIGURE 9.6.3 Occupant cooling load factors for type A and type D zones for a space occupied for 12 h.

where Watts = the total lamp wattage in the space Fd = the diversity factor Fsa = a ballast special allowance factor CLF = the cooling load factor for the lights for a given on/off schedule The diversity factor Fd takes into account the variability of the actual wattage of lights on at any given time. The lighting CLF values come from tables and are found in a fashion similar to that for the occupants. The zone types for lighting are not necessarily the same zone types for the solar cooling load or the occupants. Note that the lighting CLF will be 1.0 for buildings in which the lights are on 24 h/day or the cooling system is shut off at night or on the weekends. If the calculations are done in inch–pound units, then the result from Equation (9.6.33) is multiplied by 3.41 to convert watts to British thermal units per hour. Appliance and Equipment Loads Equipment can add heat through resistive heating or from electrical motors operating in the equipment. The CLTD/SCL/CLF method accounts for both types of equipment heat separately. In addition, the equipment loads are further broken down into sensible or latent components. The latent components are assumed to become immediate loads on the cooling system. The latent loads are found in tables devoted to hospital equipment, restaurant equipment, and office equipment. Latent loads are cited only for the hospital and restaurant equipment. The sensible component of the loads is adjusted by q(t) = qsa CLF(t)

(9.6.34)

where qsa is the sensible heat gain per appliance as found from the tables. The cooling load factor is found by first determining the zone type and then looking up the CLF in a table appropriate for that zone type, as was done for the occupants and lighting. Although the zone type is similar for occupants and equipment, it may not be the same as that for lighting. The total cooling load in the space is then found from the sum of the sensible and latent loads. If a cooling load is due to equipment with electrical motors that run equipment in the space, then the space cooling load is incremented by q(t ) = 2545

HP Fl Fu CLF(t ) η

(9.6.35)

where HP = the rated horsepower of the motor η = the efficiency Fl = the load factor average (power used divided by rated horsepower, typically around 12) Fu = the motor use factor (accounting for intermittent use) © 2005 by CRC Press LLC

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The term 2545 converts from horsepower to Britsh thermal units per hour and CLF is the cooling load factor for the equipment on a given schedule. Equation (9.6.32a) assumes that the equipment and the motor are located within the space. If the equipment is in the space but the motor is located outside the space, then this equation is derated by the motor efficiency: q(t ) = 2545 HP Fl Fu CLF(t )

(9.6.36)

Conversely, if the motor is inside the space but it acts on equipment outside the space, the cooling load is incremented by q(t ) = 2545 HP

1− η Fl Fu CLF(t ) η

(9.6.37)

As with the lighting, the CLF is always 1.0 when the cooling system does not operate 24 h/day. Air Infiltration The infiltrating air is assumed to become a load on the cooling system immediately.

Transfer Functions for Dynamic Load Calculations Basis of the Method The load Q˙ can be considered the response of the building or room to the driving terms {Ti, To, Q˙ sol, etc.) that act on it. The transfer function method calculates the response of a system by making the following basic assumptions: • Discrete time steps: all functions of time are represented as series of values at regular time steps (hourly in the present case). • Linearity: the response of a system is a linear function of the driving terms and of the state of the system. • Causality: the response at time t can depend only on the past, not on the future. As an example, suppose a single driving term u(t) and the response y(t). To make the expressions more readable, indicate the time dependence as a subscript, in the form y(t) = yt, u(t) = ut, and so on. Then, according to the transfer function model, the relation between the response and the driving term is of the form: yt = –(a1yt–1∆t + a2yt–2∆t + · · · + anyt–n∆t) + (b0ut + b1ut–1∆t + b2ut–2∆t + · · · + bmut–m∆t) (9.6.38) with time step ∆t = 1 h

(9.6.39)

where a1 to an and b0 to bm are coefficients that characterize the system; they are independent of the driving term or response. Equation (9.6.33) is obviously linear. It satisfies causality because yt depends only on the past values of the response (yt–1∆t to yt–n∆t) and on present and past values of the driving terms (ut to ut–m∆t). The past state of the system enters because of the coefficients a1 to an and b1 to bm ; this is how thermal inertia is taken into account. The response is instantaneous only if these coefficients are zero. The greater their number and magnitude are, the greater the weight of the past. The accuracy of the model increases as the number of coefficients is enlarged and as the time step is reduced. For most load calculations, hourly time resolution and a handful of coefficients per driving term will suffice. The coefficients are called transfer function coefficients. Incidentally, the relation between u and y could be written in symmetric form: a0yt + a1yt–1∆t + a2yt–2∆t + · · · + anyt–n∆t = b0ut + b1ut–1∆t + b2ut–2∆t + · · · + bmut–m∆t © 2005 by CRC Press LLC

(9.6.40)

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which is equivalent because one can divide both sides of the equation by a0. Because the roles of u and y are symmetric, one can use the same model to find, for example, the load (i.e., the heat Q to be supplied or removed) as a function of Ti, or Ti as a function of Q . Equation (9.6.35) can be readily generalized to the case where there are several driving terms. For instance, if the response Ti is determined by two driving terms, heat input Q , and outdoor temperature To, then one can write the transfer function model in the form ai ,0Ti ,t + ai ,1Ti ,t −1∆t +  + ai ,nTi ,t −n∆t = ao,0To,t + ao,1To,t −1∆t +  + ao,mTo,t −m∆t +aQ ,0Q t + aQ ,1Q t =1∆t + aQ ,2Q t − 2 ∆t +  + aQ ,rQ t −r ∆t

(9.6.41)

with three sets of transfer function coefficients: ai,0 to ai,n; ao,0 to ao,m ; and aQ,0 to aQ,r . This equation can be considered an algorithm for calculating Ti,t, hour by hour, given the previous values of Ti and the driving terms To and Q . Likewise, if Ti and To were given as driving terms, one could calculate Q as response. Any set of response and driving terms can be handled in this manner. Thus, loads can be calculated hour by hour, for any driving terms (meteorological data, building occupancy, heat gain schedules, etc.); in fact, this method is used by the computer simulation program DOE2.1 (Birdsall et al., 1990). Once the necessary numerical values of the transfer function coefficients have been obtained, the calculation of peak loads is simple enough for a spreadsheet. One specifies the driving terms for the peak day and iterates an equation like Equation (9.6.36) until the result converges to a steady daily pattern. Transfer function coefficients have been calculated and listed for a wide variety of standard construction types (ASHRAE, 1989). In the remainder of this section, the ASHRAE transfer function method is discussed in detail. The method involves three steps: • Calculate the conductive heat gain (or loss) for each distinct component of the envelope, by Equation (9.6.37). • Calculate the load of the room at constant temperature, based on this conductive heat gain (or loss) as well as any other heat source in the room, by Equation (9.6.42). • Calculate the heat extraction (or addition) rate for the cooling (or heating) device and thermostat setpoints of the room. Conductive Heat Gain The conductive heat gain (or loss) Q cond ,t at time t through the roof and walls is calculated according to the formula: Q cond ,t = −

∑ d Q n

n≥1

cond ,t −n∆t

 + A 

∑b T

n os ,t −n∆t

n≥0

− Ti



∑ c  n

(9.6.42)

n≥0

where A = area of roof or wall, m2 (ft2) ∆t = time step = 1 h To,st = sol–air temperature of outside surface at time t bn, cn, dn = coefficients of conduction transfer function The indoor temperature Ti is multiplied by the sum of the cn values, so the individual cn coefficients are not needed (because Ti is assumed constant at this point). In general, the initial value Q˙ cond,t = O is not known; its value does not matter if the calculation is repeated over a sufficient number of time steps until the resulting pattern becomes periodic within the desired accuracy. Usually a few days to a week will be sufficient. Numerical values of the coefficients of the conduction transfer function are listed in Table 9.6.3: roofs in section (a) of the table and walls in section (b). If the room in question is adjacent to rooms at a different temperature, the heat gain across the partitions is also calculated according to Equation (9.6.37). © 2005 by CRC Press LLC

n=0

n=1 (a) Roofs

© 2005 by CRC Press LLC

bn dn bn dn bn dn bn dn bn dn bn dn bn dn bn dn bn dn

0.00487 1.00000 0.00056 1.00000 0.00000 1.00000 0.00059 1.00000 0.00000 1.00000 0.00001 1.00000 0.00001 1.00000 0.00000 1.00000 0.00000 1.00000

0.03474 –0.35451 0.01202 –0.60064 0.00065 –1.34658 0.00867 –1.11766 0.00024 –1.40605 0.00066 –1.24348 0.00060 –1.39181 0.00000 –2.29459 0.00000 –2.27813

n=3

n=4

n=5

n=6

Σcn

U

0.01365 0.02267 0.01282 0.08602 0.00339 0.59384 0.00688 0.23731 0.00217 0.58814 0.00163 0.28742 0.00197 0.46337 0.00002 1.93694 0.00007 1.82162

0.00036 –0.00005 0.00143 –0.00135 0.00240 –0.09295 0.00037 –0.00008 0.00251 –0.09034 0.00049 –0.01274 0.00086 –0.04714 0.00014 –0.75741 0.00024 –0.60696

0.00000 0.00000 0.00001 0.00000 0.00029 0.00296 0.00000 0.00000 0.00055 0.00444 0.00002 0.00009 0.00005 0.00058 0.00024 0.14252 0.00016 0.07696

0.00000 0.00000 0.00000 0.00000 0.00000 –0.00001 0.00000 0.00000 0.00002 –0.00006 0.00000 0.00000 0.00000 0.00000 0.00011 –0.01251 0.00003 –0.00246

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00002 0.00046 0.00000 0.00001

0.05362

0.080

0.02684

0.055

0.00673

0.043

0.01652

0.138

0.00550

0.057

0.01477

0.090

0.00349

0.140

0.00053

0.034

0.00050

0.045

Chapter 9

Layers E0 A3 B25 E3 E2 A0 Steel deck with 3.33 in. insulation Layers E0 A3 B14 E3 E2 A0 Steel deck with 5 in. insulation Layers E0 E1 B15 E4 B7 A0 Attic roof with 6 in. insulation Layers E0 B22 C12 E3 E2 C12 A0 1.67 in. insulation with 2 in. h.w. concrete RTS Layers E0 E5 E4 B12 C14 E3 E2 A0 3 in. insul. w/4 in. I.w. conc. deck and susp. clg. Layers E0 E5 E4 C5 B6 E3 E2 A0 1 in. insul. w/4 in. h.w. conc. deck and susp. clg. Layers E0 E5 E4 C13 B20 E3 E2 A0 6 in. h.w. deck w/0.76 in. insul. and Susp. clg. Layers E0 E5 E4 B15 C15 E3 E2 A0 6 in. insul. w/6 in. I.w. conc. deck and susp. clg. Layers E0 C13 B15 E3 E2 C12 A0 6 in. h.w. deck w/6 in. ins. and 2 in. h.w. RTS

n=2 a

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9-62

TABLE 9.6.3 Coefficients of Conduction Transfer Function

a

bn dn bn dn bn dn bn dn bn dn bn dn bn dn bn dn

0.00768 1.00000 0.00016 1.00000 0.00411 1.00000 0.00099 1.00000 0.00000 1.00000 0.00000 1.00000 0.00000 1.00000 0.00000 1.00000

0.03498 –0.24072 0.00545 –0.93389 0.03230 –0.76963 0.00836 –0.93970 0.00061 –1.52480 0.00000 –2.00875 0.00005 –1.78165 0.00001 –2.12812

0.00719 0.00168 0.00961 0.27396 0.01474 0.04014 0.00361 0.04664 0.00289 0.67146 0.00013 1.37120 0.00064 0.96017 0.00019 1.53974

0.00006 0.00000 0.00215 –0.02561 0.00047 –0.00042 0.00007 0.00000 0.00183 –0.09844 0.00044 –0.37897 0.00099 –0.16904 0.00045 –0.45512

0.00000 0.00000 0.00005 0.00014 0.00000 0.00000 0.00000 0.00000 0.00018 0.00239 0.00030 0.03962 0.00030 0.00958 0.00022 0.05298

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00005 –0.00165 0.00002 –0.00016 0.00002 –0.00158

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00002 0.00000 0.00000 0.00000 0.00002

0.04990

0.066

0.01743

0.055

0.05162

0.191

0.01303

0.122

0.00552

0.109

0.00093

0.043

0.00200

0.106

0.00089

0.112

Note: U, bn, and cn are in Btu/(h · ft2 · °F); dn and A are dimensionless. Layer sequence left to right = inside to outside. Source: From ASHRAE, 1989, Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta. With permission.

© 2005 by CRC Press LLC

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Layers E0 A3 B1 B13 A3 A0 Steel siding with 4 in. insulation Layers E0 E1 B14 A1 A0 A0 Frame wall with 5 in. insulation Layers E0 C3 B5 A6 A0 A0 4 in. h.w. concrete block with 1 in. insulation Layers E0 A6 C5 B3 A3 A0 4 in. h.w. concrete with 2 in. insulation Layers E0 E1 C8 B6 A1 A0 8 in. h.w. concrete block with 2 in. insulation Layers E0 A2 C2 B15 A0 A0 Face brick and 4 in. I.w. conc. block with 6 in. insul. Layers E0 C9 B6 A6 A0 A0 8 in. common brick with 2 in. insulation Layers E0 C11 B6 A1 A0 A0 12 in. h.w. concrete with 2 in. insulation

Air-Conditioning and Refrigeration

(b) Wallsa

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Chapter 9

It is instructive to establish the connection of the transfer function coefficients with the U value. In the steady-state limit, i.e., when Q cond, Tos, and Ti are all constant, Equation (9.6.37) becomes Q cond





∑ d = A T ∑b − T ∑ c  n

os

n≥1

n

i

n ≥0

n

(9.6.43)

n ≥0

where d0 = 1. Because that limit also contains Q cond = A U (Tos – Ti)

(9.6.44)

the coefficients of Tos and Ti must be equal,

∑b = ∑ c n

n≥0

n

(9.6.45)

n≥0

and the U value is given by

∑c U= ∑d

n

n≥0

(9.6.46)

n

n≥0

The Load at Constant Temperature The preceding calculation of the conductive heat gain (or loss) is to be repeated for each portion of the room envelope that has a distinct composition. The relation between these conductive gains and the total load depends on the construction of the entire room. For example, a concrete floor can store a significant fraction of the heat radiated by lights or by a warm ceiling, thus postponing its contribution to the cooling load of the room. For each heat gain component Q gain , the corresponding cooling load Q c (or reduction of the heating load) at constant Ti is calculated by using another set of coefficients: the coefficients vn and wn, of the room transfer function Q c ,t = v 0Q gain,t + v1Q gain ,t −1∆t + v 2Q gain ,t − 2 ∆t +  −w1Q c ,t −1∆t − w 2Q c ,t − 2 ∆t − 

(9.6.47)

with the subscript t indicating time, as before. The coefficient w0 of Q c ,t is not shown because it is set equal to unity. Equation (9.6.42) must be applied separately to each of the heat gain types and the resulting cooling load components, Q c ,t , are added to obtain the total cooling load of the room at time t. The heat gain types are as follows: • Solar gain (through glass without interior shade) and the radiative component of heat from occupants and equipment • Conduction through envelope and solar radiation absorbed by interior shade • Lights • Convective gains (from air exchange, occupants, equipment) For lights the coefficients depend on the arrangement of the lighting fixture and the ventilation system. © 2005 by CRC Press LLC

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Specific numbers vary a great deal with the circumstances; however, the general pattern is common to all peak cooling loads: thermal inertia attenuates and delays the peak contributions of individual load components. The total peak is usually less than the result of a steady-state calculation, although it could be more if the time delays act in the sense of making the loads coincide. In contrast to peak loads, daily average loads can be determined by a static calculation if the average indoor temperature is known; that follows from the first law of thermodynamics. However, if the thermostat allows floating temperatures, the indoor temperature is, in general, not known without a dynamic analysis. With the transfer functions described so far, one can calculate peak loads when the indoor temperature Ti is constant. That is how the cooling load factors and cooling load temperature differences of the previous section have been determined. Of course, the indoor temperature Ti may vary, not only because of variable thermostat setpoints but also because of limitations of the HVAC equipment (capacity, throttling range, imperfect control). The extension to variable Ti requires one additional transfer function, which is described in ASHRAE (1997) and Kreider et al. (2001).

9.7 Air Handling Units and Packaged Units Shan K. Wang Terminals and Air Handling Units A terminal unit, or terminal, is a device or equipment installed directly in or above the conditioned space to cool, heat, filter, and mix outdoor air with recirculating air. Fan-coil units, VAV boxes, fan-powered VAV boxes, etc. are all terminals. An air handling unit (AHU) handles and conditions the air, controls it to a required state, and provides motive force to transport it. An AHU is the primary equipment of the air system in a central airconditioning system. The basic components of an AHU include a supply fan with a fan motor, a water cooling coil, filters, a mixing box except in a makeup AHU unit, dampers, controls, and an outer casing. A return or relief fan, heating coil(s), and humidifier are optional depending on requirements. The supply volume flow rate of AHUs varies from 2000 to about 60,000 cfm. AHUs are classified into the followings groups according to their structure and location. Horizontal or Vertical Units Horizontal AHUs have their fan, coils, and filters installed at the same level as shown in Figure 9.7.1(a). They need more space and are usually for large units. In vertical units, as shown in Figure 9.7.1(b), the supply fan is installed at a level higher than coils and filters. They are often comparatively smaller than horizontal units. Draw-Through or Blow-Through Units In a draw-through unit, as shown in Figure 9.7.1(a), the supply fan is located downstream of the coils. Air is evenly distributed over the coil section, and the fan discharge can easily be connected to a supply duct of nearly the same air velocity. In a blow-through unit, as shown in Figure 9.7.1(c), the supply fan is located upstream of the coils. It usually has hot and cold decks with discharge dampers connected to warm and cold ducts, respectively. Factory-Fabricated and Field Built-Up Units Factory-fabricated units are standard in construction and layout, low in cost, of higher quality, and fast in installation. Field built-up units or custom-built units are more flexible in construction, layout, and dimensions than factory-built standardized units. Rooftop and Indoor Units A rooftop AHU, sometimes called a penthouse unit, is installed on the roof and will be completely weatherproof. An indoor AHU is usually located in a fan room or ceiling and hung like small AHU units. © 2005 by CRC Press LLC

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Chapter 9

Filters

Coils

Heating coil

Supply fan Filters

Hot deck

Supply fan

Mixing box (a) Mixing box

Supply fan

Cooling (c) coil

Cold deck

Mixing box

Coil

Filters (b)

FIGURE 9.7.1 Type of air handling units: (a) horizontal draw-through unit, (b) vertical draw-through unit, and (c) multizone blow-through unit.

Make-Up Air and Recirculating Units A make-up AHU, also called a primary-air unit, is used to condition outdoor air entirely. It is a oncethrough unit. There is no return air and mixing box. Recirculating units can have 100% outdoor air intake or mixing of outdoor air and recirculating air.

Packaged Units A packaged unit (PU) or Packaged Terminal Air Conditioner (PTAC) is a self-contained air conditioner. It conditions the air and provides it with motive force and is equipped with its own heating and cooling sources. The packaged unit is the primary equipment in a packaged air-conditioning system and is always equipped with a DX coil for cooling, unlike an AHU. R-22, R-134a, and others are used as refrigerants in packaged units. The portion that handles air in a packaged unit is called an air handler to distinguish it from an AHU. Like an AHU, an indoor air handler has an indoor fan, a DX coil (indoor coil), filters, dampers, and controls. Packaged units can be classified according to their place of installation: rooftop, indoor, and split packaged units. Rooftop Packaged Units A rooftop packaged unit is mounted on the roof of the conditioned space as shown in Figure 9.7.2. From the types of heating/cooling sources provided, rooftop units can be subdivided into: • Gas/electric rooftop packaged unit, in which heating is provided by gas furnace and cooling by electric power-driven compressors. • Electric/electric rooftop packaged unit, in which electric heating and electric power-driven compressors provide heating and cooling.

© 2005 by CRC Press LLC

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FIGURE 9.7.2 A cut view of a rooftop package unit. (Source: Mammoth, Inc. Reprinted by permission.)

• Rooftop packaged heat pump, in which both heating and cooling are provided by the same refrigeration system using a four-way reversing valve (heat pump) in which the refrigeration flow changes when cooling mode is changed to heating mode and vice versa. Auxiliary electric heating is provided if necessary. Rooftop packaged units are single packaged units. Their cooling capacity may vary from 3 to 220 tons with a corresponding volume flow rate of 1200 to 80,000 cfm. Rooftop packaged units are the most widely used packaged units. Indoor Packaged Units An indoor packaged unit is also a single packaged and factory-fabricated unit. It is usually installed in a fan room or a machinery room. A small or medium-sized indoor packaged unit could be floor mounted directly inside the conditioned space with or without ductwork. The cooling capacity of an indoor packaged unit may vary from 3 to 100 tons and volume flow rate from 1200 to 40,000 cfm. Indoor packaged units are also subdivided into: • Indoor packaged cooling units • Indoor packaged cooling/heating units, in which heating may be provided from a hot water heating coil, a steam heating coil, and electric heating • Indoor packaged heat pumps Indoor packaged units have either an air-cooled condenser on the rooftop or a shell-and-tube or double-tube water-cooled condenser inside the unit. Split Packaged Units A split packaged unit consists of two separate pieces of equipment: an indoor air handler and an outdoor condensing unit. The indoor air handler is often installed in the fan room. Small air handlers can be ceiling hung. The condensing unit is usually located outdoors, on a rooftop or podium or on the ground. A split packaged unit has its compressors and condenser in its outdoor condensing unit, whereas an indoor packaged unit usually has its compressors indoors. The cooling capacity of split packaged units varies from 3 to 75 tons and the volume flow rate from 1200 to 30,000 cfm.

© 2005 by CRC Press LLC

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Rating Conditions and Minimum Performance Air Conditioning and Refrigeration Institute (ARI) Standards and ASHRAE/IES Standard 90.1-2001 specified the following rating indices: • Energy efficiency ratio (EER) is the ratio of equipment cooling capacity, in Btu/hr, to the electric input, in W, under rating conditions. • SEER is the seasonal EER, or EER during the normal annual usage period. • IPLV is the integrated part-load value. It is the summarized single index of part-load efficiency of PUs based on weighted operations at several load conditions. • HSPF is the heating seasonal performance factor. It is the total heating output of a heat pump during its annual usage period for heating, in Btu, divided by the total electric energy input to the heat pump during the same period, in watt-hours. For water- and evaporatively cooled packaged units including heat pumps, refer to ASHRAE/IES Standard 90.1-2001 and also ARI Standards.

Coils Coils, Fins, and Water Circuits Coils are indirect contact heat exchangers. Heat transfer or heat and mass transfer takes place between conditioned air flowing over the coil and water, refrigerant, steam, or brine inside the coil for cooling, heating, dehumidifying, or cooling/dehumidifying. Chilled water, brine, and refrigerants that are used to cool and dehumidify the air are called coolants. Coils consist of tubes and external fins arranged in rows along the air flow to increase the contact surface area. Tubes are usually made of copper; in steam coils they are sometimes made of steel or even stainless steel. Copper tubes are staggered in 2, 3, 4, 6, 8, or up to 10 rows. Fins are extended surfaces often called secondary surfaces to distinguish them from the primary surfaces, which are the outer surfaces of the tubes. Fins are often made from aluminum, with a thickness Ft = 0.005 to 0.008 in., typically 0.006 in. Copper, steel, or sometimes stainless steel fins are also used. Fins are often in the form of continuous plate fins, corrugated plate fins to increase heat transfer, crimped spiral or smooth spiral fins that may be extruded from the aluminum tubes, and spine pipes, which are shaved from the parent aluminum tubes. Corrugated plate fins are most widely used. Fin spacing Sf is the distance between two fins. Fin density is often expressed in fins per inch and usually varies from 8 to 18 fins/in. In a water cooling coil, water circuits or tube feeds determine the number of water flow passages. The greater the finned width, the higher the number of water circuits and water flow passages. Direct Expansion (DX) Coil In a direct expansion coil, the refrigerant, R-22, R-134a, or others, is evaporated and expanded directly inside the tubes to cool and dehumidify the air as shown in Figure 9.7.3(a). Refrigerant is fed to a distributor and is then evenly distributed to various copper tube circuits typically 0.375 in. in diameter. Fin density is usually 12 to 18 fins/in. and a four-row DX coil is often used. On the inner surface of the copper tubes, microfins, typically at 60 fins/in. and a height of 0.008 in., are widely used to enhance the boiling heat transfer. Air and refrigerant flow is often arranged in a combination of counterflow and cross flow and the discharge header is often located on the air-entering side. Refrigerant distribution and loading in various circuits are critical to the coil’s performance. Vapor refrigerant cleaning the DX coil is superheated 10 to 20°F in order to prevent any liquid refrigerant from flooding back to the reciprocating compressors and damaging them. Finally, the vapor refrigerant is discharged to the suction line through the header. For comfort air-conditioning systems, the evaporating temperature of refrigerant Tev inside the tubes of a DX coil is usually between 37 and 50°F. At such a temperature, the surface temperature of the coil is often lower than the dew point of the entering air. Condensation occurs at the coil’s outside surface, © 2005 by CRC Press LLC

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Air-Conditioning and Refrigeration

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FIGURE 9.7.3 Types of coils: (a) direct expansion coil, (b) water cooling coil, (c) water heating coil, and (d) steam heating coil.

and the coil becomes a wet coil. A condensate drain pan is necessary for each vertically banked DX coil, and a trap should be installed to overcome the negative pressure difference between the air in the coil section and the ambient air. Some units are equipped with IAQ drainpans — drain pans that are sloped to drain connections so that they completely drain. Face velocity of the DX coil va, in fpm, is closely related to the blow-off of the water droplets of the condensate, the heat transfer coefficients, the air-side pressure drop, and the size of the air system. For corrugated fins, the upper limit is 600 fpm, with an air-side pressure drop of 0.20 to 0.30 in. WG/row. A large DX coil is often divided into two refrigerant sections, each with its own expansion valve, distributor, and discharge header. For a packaged unit of a specific model, size, face velocity and condition of entering air and outdoor air, the DX coil’s cooling capacities in nominal tons, number of rows, and fin density are all fixed values. © 2005 by CRC Press LLC

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Water Cooling Coils — Dry–Wet Coils In a water cooling coil, chilled water at a temperature of 40 to 50°F, brine, or glycol-water at a temperature of 34 to 40°F during cold air distribution enters the coil. The temperature of chilled water, brine, or glycol-water is usually raised 12 to 24°F before it leaves the water cooling coil. The water tubes are usually copper tubes of 1/2 to 5/8 in. diameter with a tube wall thickness of 0.01 to 0.02 in. They are spaced at a center-to-center distance of 0.75 to 1.25 in. longitudinally and 1 to 1.5 in. transversely. These tubes may be staggered in 2, 3, 4, 6, 8, or 10 rows. Chilled water coils are often operated at a pressure of 175 to 300 psig. As in a DX coil, the air flow and water flow are in a combination of counterflow and cross flow. The temperature of the outer surface of a chilled water cooling coil at the air entering side Tse is often greater than the dew point of the entering air Tae′′, or Tse > Tae′′. The outer surface temperature of the coil at the air leaving side Tsl may be smaller than Tae′′, or Tsl < Tae′′. Then the water cooling coil becomes a dry–wet coil with part of the dry surface on the air entering side and part of the wet surface on the air leaving side. A dry–wet boundary divides the dry and wet surfaces. At the boundary, the tube outer surface temperature Tsb = Tae′′ as shown in Figure 9.7.3(b). A condensate drain pan is necessary for a dry–wet coil. A water cooling coil is selected from the manufacturer’s selection program or from its catalog at (1) a dry and wet bulb temperature of entering air, such as 80°F dry bulb and 67°F wet bulb; (2) an entering water temperature, such as 44 or 45°F; (3) a water temperature rise between 10 and 24°F; and (4) a coil face velocity between 400 and 600 fpm. The number of rows and fins per inch is varied to meet the required sensible and cooling coil load, in Btu/hr. Water Cooling Coil–Dry Coil When the temperature of chilled water entering the water cooling coil Twe ≥ Tae′′, condensation will not occur on the outer surface of the coil. This coil becomes a sensible cooling–dry coil, and the humidity ratio of the conditioned air wa remains constant during the sensible cooling process. The construction of a sensible cooling–dry coil, such as material, tube diameter, number of rows, fin density, and fin thickness, is similar to that of a dry–wet coil except that a dry coil always has a poorer surface heat transfer coefficient than a wet coil, and therefore a greater coil surface area is needed; the maximum face velocity of a dry coil can be raised to va ≤ 800 fpm; and the coil’s outer surface is less polluted. The effectiveness of a dry coil ∈dry is usually 0.55 to 0.7. Water Heating Coil The construction of a water heating coil is similar to that of a water cooling coil except that in water heating coils hot water is supplied instead of chilled water and there are usually fewer rows, only 2, 3, and 4 rows, than in water cooling coils. Hot water pressure in water heating coils is often rated at 175 to 300 psig at a temperature up to 250°F. Figure 9.7.3(c) shows a water heating coil. Steam Heating Coil In a steam heating coil, latent heat of condensation is released when steam is condensed into liquid to heat the air flowing over the coil, as shown in Figure 9.7.3(d). Steam enters at one end of the coil, and the condensate comes out from the opposite end. For more even distribution, a baffle plate is often installed after the steam inlet. Steam heating coils are usually made of copper, steel, or sometimes stainless steel. For a steam coil, the coil core inside the casing should expand or contract freely. The coil core is also pitched toward the outlet to facilitate condensate drainage. Steam heating coils are generally rated at 100 to 200 psig at 400°F. Coil Accessories and Servicing Coil accessories include air vents, drain valves, isolation valves, pressure relief valves, flow metering valves, balancing valves, thermometers, pressure gauge taps, condensate drain taps, and even distribution baffles. They are employed depending on the size of the system and operating and serving requirements. © 2005 by CRC Press LLC

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Coil cleanliness is important for proper operation. If a medium-efficiency air filter is installed upstream of the coil, dirt accumulation is often not a problem. If a low-efficiency filter is employed, dirt accumulation may block the air passage and significantly increase the pressure drop across the coil. Coils should normally be inspected and cleaned every 3 months in urban areas when low-efficiency filters are used. Drain pans should be cleaned every month to prevent buildup of bacteria and microorganisms. IAQ drain pans can be specified to minimize this cleaning. Coil Freeze-Up Protection Improper mixing of outdoor air and recirculating air in the mixing box of an AHU or PU may cause coil freeze-up when the outdoor air temperature is below 32°F. Outdoor air should be guided by a baffle plate and flow in an opposite direction to the recirculating air stream so that they can be thoroughly mixed without stratification. Preheat coils can also be used to temper outside air before it enters the mixing box. Run the chilled water pump for the idle coil with a water velocity of 2.5 ft/sec, so that the cooling coil will not freeze when the air temperature drops to 32°F. A better method is to drain the water completely. For a hot water coil, it is better to reset the hot water temperature at part-load operation instead of running the system intermittently. A steam heating coil with inner distributor tubes and outer finned heating tubes provides better protection against freeze-up.

Air Filters Air Cleaning and Filtration Air cleaning is the process of removing airborne particles from the air. Air cleaning can be classified as air filtration and industrial air cleaning. Industrial air cleaning involves the removal of dust and gaseous contaminants from manufacturing processes as well as from the space air, exhaust air, and flue gas for air pollution control. In this section, only air filtration is covered. Air filtration involves the removal of airborne particles presented in the conditioned air. Most of the airborne particles removed by air filtration are smaller than 1 µm, and the concentration of these particles in the airstream seldom exceeds 2 mg/m3. The purpose of air filtration is to benefit the health and comfort of the occupants as well as meet the cleanliness requirements of the working area in industrial buildings. An air filter is a kind of air cleaner that is installed in AHUs, PUs, and other equipment to filter the conditioned air by inertial impaction or interception and to diffuse and settle fine dust particles on the fibrous medium. The filter medium is the fabricated material that performs air filtration. Operating performance of air filters is indicated by their: • Efficiency or effectiveness of dust removal • Dust holding capacity mdust, which is the amount of dust held in the air filter, in grains/ft2 • Initial pressure drop when the filter is clean ∆pfi and final pressure drop ∆pff when the filter’s mdust is maximum, both in in. WG • Service life, which is the operating period between ∆pfi and ∆pff Air filters in AHUs and PUs can be classified into low-, medium-, and high-efficiency filters and carbon activated filters. Test Methods The performance of air filters is usually tested in a test unit that consists of a fan, a test duct, the tested filter, two samplers, a vacuum pump, and other instruments. Three test methods with their own test dusts and procedures are used for the testing of low-, medium-, and high-efficiency air filters. The weight arrestance test is used for low-efficiency air filters to assess their ability to remove coarse dusts. Standard synthetic dusts that are considerably coarser than atmospheric dust are fed to the test unit. By measuring the weight of dust fed and the weight gain due to the dust collected on the membrane of the sampler after the tested filter, the arrestance can be calculated. © 2005 by CRC Press LLC

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The atmospheric dust spot efficiency test is used for medium-efficiency air filters to assess their ability to remove atmospheric dusts. Atmospheric dusts are dusts contained in the outdoor air, the outdoor atmosphere. Approximately 99% of atmospheric dusts are dust particles 1 µm that make up 70% of the total weight. Untreated atmospheric dusts are fed to the test unit. Air samples taken before and after the tested filter are drawn through from identical fiber filter-paper targets. By measuring the light transmission of these discolored white filter papers, the efficiency of the filter can be calculated. Similar atmospheric dust spot test procedures have been specified by American Filter Institute (AFI), ASHRAE Standard 52.1, and former National Bureau of Standards (NBS). The DOP penetration and efficiency test or simply DOP test is used to assess high-efficiency filters removing dusts particles of 0.18 µm. According to U.S. Military Standard MIL-STD-282 (1956), a smoke cloud of uniform dioctyl phthalate (DOP) droplets 0.18 µm in diameter, generated from the condensation of the DOP vapor, is fed to the test unit. By measuring the concentration of these particles in the air stream upstream and downstream of the tested filter using an electronic particle counter or laser spectrometer, the penetration and efficiency of the air filter can be calculated. Low-Efficiency Air Filters ASHRAE weight arrestance for low-efficiency filters is between 60 and 95%, and ASHRAE dust spot efficiency for low-efficiency filters is less than 20%. These filters are usually in panels as shown in Figure 9.7.4(a). Their framework is typically 20 × 20 in. or 24 × 24 in. Their thickness varies from 1 to 4 in.

11.5 in

n 24 i

Activated carbon tray

24 in

Air flow

Prefilter

(c)

(d)

Framework

Bag, fibrous mat

Filter media (a)

(b)

FIGURE 9.7.4 Various types of air filters: (a) low efficiency, (b) medium efficiency, (c) HEPA and ULPA filters, and (d) activated carbon filter.

© 2005 by CRC Press LLC

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For low-efficiency filters, the filter media are often made of materials such as • Corrugated wire mesh and screen strips coated with oil, which act as adhesives to enhance dust removal. Detergents may be used to wash off dusts so that the filter media can be cleaned and reused — they are therefore called viscous and reusable. • Synthetic fibers (nylon, terylene) and polyurethane foam can be washed, cleaned, and reused if required — dry and reusable. • Glass fiber mats with fiber diameter greater than 10 µm. The filter medium is discarded when its final pressure drop is reached — dry and disposable. The face velocity of the panel filter is usually between 300 and 600 fpm. The initial pressure drop varies from 0.05 to 0.25 in. WG and the final pressure drop from 0.2 to 0.5 in. WG. Medium-Efficiency Air Filters These air filters have an ASHRAE dust spot efficiency usually between 20 and 95%. Filter media of medium-efficiency filters are usually made of glass fiber mat with a fiber diameter of 10 to 1 µm using nylon fibers to join them together. They are usually dry and disposable. In addition: • As the dust spot efficiency increases, the diameter of glass fibers is reduced, and they are placed closer together. • Extended surfaces, such as pleated mats or bags, are used to increase the surface area of the medium as shown in Figure 9.7.4(b). Air velocity through the medium is 6 to 90 fpm. Face velocity of the air filter is about 500 fpm to match the face velocity of the coil in AHUs and PUs. • Initial pressure drop varies from 0.20 to 0.60 in. WG and final pressure drop from 0.50 to 1.20 in. WG. High-Efficiency Particulate Air (HEPA) Filters and Ultra-Low-Penetration Air (ULPA) Filters HEPA filters have a DOP test efficiency of 99.97% for dust particles ≥0.3 µm in diameter. ULPA filters have a DOP test efficiency of 99.999% for dust particles ≥0.12 µm in diameter. A typical HEPA filter, shown in Figure 9.7.4(d), has dimensions of 24 × 24 × 11.5 in. Its filter media are made of glass fibers of submicrometer diameter in the form of pleated paper mats. The medium is dry and disposable. The surface area of the HEPA filter may be 50 times its face area, and its rated face velocity varies from 190 to 390 fpm, normally at a pressure drop of 0.50 to 1.35 in. WG for clean filters. The final pressure drop is 0.8 to 2 in. WG. Sealing of the filter pack within its frame and sealing between the frame and the gaskets are critical factors that affect the penetration and efficiency of the HEPA filter. An ULPA filter is similar to a HEPA filter in construction and filter media. Both its sealing and filter media are more efficient than those of a HEPA filter. To extend the service life of HEPA filters and ULPA filters, both should be protected by a mediumefficiency filter, or a low-efficiency and a medium-efficiency filter in the sequence low–medium just before the HEPA or ULPA filters. HEPA and ULPA filters are widely used in clean rooms and clean spaces. Often the removal and disposal of HEPA and ULPA filters require special handling due to the concentration of biological species present on the filter. Activated Carbon Filters These filters are widely used to remove objectional odors and irritating gaseous airborne particulates, typically 0.003 to 0.006 µm in size, from the air stream by adsorption. Adsorption is physical condensation of gas or vapor on the surface of an activated substance like activated carbon. Activated substances are extremely porous. One pound of activated carbon contains 5,000,000 ft2 of internal surface. Activated carbon in the form of granules or pellets is made of coal, coconut shells, or petroleum residues and is placed in trays to form activated carbon beds as shown in Figure 9.7.4(d). A typical carbon tray is 23 × 23 × 5/8 in. thick. Low-efficiency prefilters are used for protection. When air flows through the carbon beds at a face velocity of 375 to 500 fpm, the corresponding pressure drop is 0.2 to 0.3 in. WG. © 2005 by CRC Press LLC

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Control valve Jacketed distribution manifold

Dry steam

Dry chamber Silencing substance

Steam

Air

Orifice

Inner steam discharge tubes

Orifice

Separating chamber

(a) Baffles

Spraying nozzle

Elimator

Water filter

Circulating pump (b)

FIGURE 9.7.5 Steam grid humidifier (a) and air washer (b).

Humidifiers A humidifier adds moisture to the air. Air is humidified by: (1) heating the liquid to evaporate it; (2) atomizing the liquid water into minute droplets by mechanical means, compressed air, or ultrasonic vibration to create a larger area for evaporation; (3) forcing air to flow through a wetted element in which water evaporates; and (4) injecting steam into air directly before it is supplied to the conditioned space. For comfort air-conditioning systems, a steam humidifier with a separator as shown in Figure 9.7.5(a) is widely used. Steam is supplied to a jacketed distribution manifold. It enters a separating chamber with its condensate. Steam then flows through a control valve, throttles to a pressure slightly above atmospheric, and enters a dry chamber. Due to the high temperature in the surrounding separating chamber, the steam is superheated. Dry steam is then discharged into the ambient air stream through the orifices on the inner steam discharge tubes. For an air system of cold air supply with humidity control during winter mode operation, an air washer is economical for large-capacity humidification in many industrial applications. An air washer is a humidifier, a cooler, a dehumidifier, and an air cleaner. An air washer usually has an outer casing, two banks of spraying nozzles, one bank of guide baffles at the entrance, one bank of eliminators at the exit, a water tank, a circulating pump, a water filter, and other accessories as shown in Figure 9.7.5(b). Outer casing, baffles, and eliminators are often made of plastics or sometimes stainless steel. Spraying nozzles are usually made of brass or nylon, with an orifice diameter of 1/16 to 3/16 in., a smaller orifice for humidification, and a larger orifice for cooling and dehumidification. An eccentric inlet connected to the discharge chamber of the spraying nozzle gives centrifugal force to the water stream © 2005 by CRC Press LLC

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and atomizes the spraying water. Water is supplied to the spraying nozzle at a pressure of 15 to 30 psig. The distance between two spraying banks is 3 to 4.5 ft, and the total length of the air water from 4 to 7 ft. The air velocity inside an air washer is usually 500 to 800 fpm. Selection of AHUs and PUs • The size of an AHU is usually selected so that the face velocity of its coil is 600 fpm or less in order to prevent entrained condensate droplets. The cooling and heating capacities of an AHU can be varied by using coils of different numbers of rows and fin densities. The size of a PU is determined by its cooling capacity. Normally, the volume flow rate per ton of cooling capacity in PUs is 350 to 400 cfm. In most packaged units whose supply fans have belt drives, the fan speed can be selected so that the volume flow rate is varied and external pressure is met. • To improve the indoor air quality, save energy, and prevent smudging and discoloring building interiors, a medium-efficiency filter of dust spot efficiency ≥50% and an air economizer are preferable for large AHUs and PUs. • See ANSI/ASHRAE/IESNA 90.1-2001 for energy standards on equipment selection.

9.8 Refrigeration Components and Evaporative Coolers Shan K. Wang Refrigeration Compressors A refrigeration compressor is the heart of a vapor compression system. It raises the pressure of refrigerant so that it can be condensed into liquid, throttled, and evaporated into vapor to produce the refrigeration effect. It also provides the motive force to circulate the refrigerant through condenser, expansion valve, and evaporator. According to the compression process, refrigeration compressors can be divided into positive displacement and nonpositive displacement compressors. A positive displacement compressor increases the pressure of the refrigerant by reducing the internal volume of the compression chamber. Reciprocating, scroll, rotary, and screw compressors are all positive displacement compressors. The centrifugal compressor is the only type of nonpositive displacement refrigeration compressor widely used in refrigeration systems today. Based on the sealing of the refrigerant, refrigeration compressors can be classified as • Hermetic compressors, in which the motor and the compressor are sealed or welded in the same housing to minimize leakage of refrigerant and to cool the motor windings by using suction vapor • Semihermetic compressors, in which motor and compressor are enclosed in the same housing but are accessible from the cylinder head for repair and maintenance • Open compressors, in which compressor and motor are enclosed in two separate housings Refrigeration compressors are often driven by a motor directly or by a gear train. Performance Indices Volumetric efficiency ηv of a refrigeration compressor is defined as





ηv = V a.v V p where

(9.8.1)




V a.v = actual induced volume of the suction vapor at suction pressure, cfm


V p = calculated displacement of the compressor, cfm Isentropic efficiency ηisen, compression efficiency ηcp, compressor efficiency ηcom, and mechanical efficiency ηmec are defined as © 2005 by CRC Press LLC

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ηisen = (h2 − h1 ) (h2′ − h1 ) = ηcp ηmec = ηcom ηcp = Wsen Wv

(9.8.2)

ηmec = Wv Wcom where h1 , h2 , h2′ = enthalpy of the suction vapor, ideal discharged hot gas, and actual discharged hot gas, respectively, Btu/lb Wisen, Wv, Wcom = isentropic work = (h2 – h1), work delivered to the vapor refrigerant, and work delivered to the compressor shaft, Btu/lb The actual power input to the compressor Pcom, in hp, can be calculated as Pcom = m r (h2 − h1 ) (42.41ηisen ηmo )








m r = V p ηv ρsuc

(9.8.3)

ηmo = Pcom Pmo where




m r = mass flow rate of refrigerant, lb/min ρsuc = density of suction vapor, lb/ft3 Pmo = power input to the compressor motor, hp Power consumption, kW/ton refrigeration, is an energy index used in the HVAC&R industry in addition to EER and COP. Currently used refrigeration compressors are reciprocating, scroll, screw, rotary, and centrifugal compressors. Reciprocating Compressors In a reciprocating compressor, as shown in Figure 9.8.1(a), a crankshaft connected to the motor shaft drives 2, 3, 4, or 6 single-acting pistons moving reciprocally in the cylinders via a connecting rod. The refrigeration capacity of a reciprocating compressor is a fraction of a ton to about 200 tons. Refrigerants R-22 and R-134a are widely used in comfort and processing systems and sometimes R-717 in industrial applications. The maximum compression ratio Rcom for a single-stage reciprocating compressor is about 7. Volumetric efficiency ηv drops from 0.92 to 0.65 when Rcom is raised from 1 to 6. Capacity control of reciprocating compressor including: on-off and cylinder unloader in which discharge gas is in short cut and return to the suction chamber. Although reciprocating compressors are still widely used today in small and medium-sized refrigeration systems, they have little room for significant improvement and will be gradually replaced by scroll and screw compressors. Scroll Compressors A scroll compressor consists of two identical spiral scrolls assembled opposite to each other, as shown in Figure 9.8.1(b). One of the scrolls is fixed, and the other moves in an orbit around the motor shaft whose amplitude equals the radius of the orbit. The two scrolls are in contact at several points and therefore form a series of pockets. Vapor refrigerant enters the space between two scrolls through lateral openings. The lateral openings are then sealed and the formation of the two trapped vapor pockets indicates the end of the suction process. The vapor is compressed and the discharge process begins when the trapped gaseous pockets open to the discharge port. Compressed hot gas is then discharged through this opening to the discharge © 2005 by CRC Press LLC

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Discharge valve

Fixed scroll

Suction valve

Trapped gas pocket

Discharge opening

Orbiting scroll

Piston Cylinder (a) Discharge valve

(b)

Fixed vane

Suction Cylindrical housing

Rolling piston

Female rotor Dc

Discharge end

Suction end

Dr

Male rotor Eccentric shaft

Roller

(d)

(c) First-stage impeller

Second-stage impeller

Hermetic motor

Inlet vanes

Diffuser Volute (e)

FIGURE 9.8.1 Various types of refrigeration compressors: (a) reciprocating, (b) scroll, (c) rotary, (d) twin-screw, and (e) centrifugal.

© 2005 by CRC Press LLC

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line. In a scroll compressor, the scrolls touch each other with sufficient force to form a seal but not enough to cause wear. The upper limit of the refrigeration capacity of currently manufactured scroll compressors is 60 tons. A scroll compressor has ηv > 95% at Rcom = 4 and ηisen = 80%. A scroll compressor also has only about half as many parts as a reciprocating compressor at the same refrigeration capacity. Few components result in higher reliability and efficiency. Power input to the scroll compressor is about 5 to 10% less than to the reciprocating compressor. A scroll compressor also operates more smoothly and is quieter. Rotary Compressors Small rotary compressors for room air conditioners and refrigerators have a capacity up to 4 tons. There are two types of rotary compressors: rolling piston and rotating vane. A typical rolling piston rotary compressor is shown in Figure 9.8.1(c). A rolling piston mounted on an eccentric shaft is kept in contact with a fixed vane that slides in a slot. Vapor refrigerant enters the compression chamber and is compressed by the eccentric motion of the roller. When the rolling piston contacts the top housing, hot gas is squeezed out from the discharge valve. Screw Compressors These are also called helical rotary compressors. Screw compressors can be classified into single-screw compressors, in which there is a single helical rotor and two star wheels, and twin-screw compressors. Twin-screw compressors are widely used. A typical twin-screw compressor, as shown in Figure 9.8.1(d) consists of a four-lobe male rotor and a six-lobe female rotor, a housing with suction and discharge ports, and a sliding valve to adjust the capacity during part load. Normally, the male rotor is the driver. Twin-screw compressors are often direct driven and of hermetic type. Vapor refrigerant is extracted into the interlobe space when the lobes are separated at the suction port. During the successive rotations of the rotor, the volume of the trapped vapor is compressed. When the interlobe space is in contact with the discharge port, the compressed hot gas discharges through the outlet. Oil injection effectively cools the rotors and results in a lower discharge temperature. Oil also provides a sealing effect and lubrication. A small clearance of 0.0005 in. as well as the oil sealing minimizes leakage of the refrigerant. The refrigeration capacity of twin-screw compressors is 50 to 1500 tons. The compression ratio of a twin-screw compressor can be up to 20:1. R-22 and R-134a are the most widely used refrigerants in comfort systems. In a typical twin-screw compressor, ηv decreases from 0.92 to 0.87 and ηisen drops from 0.82 to 0.67 when Rcom increases from 2 to 10. Continuous and stepless capacity control is provided by moving a sliding valve toward the discharge port, which opens a shortcut recirculating passage to the suction port. Twin-screw compressors are more efficient than reciprocating compressors. The low noise and vibration of the twin-screw compressor together with its positive displacement compression results in more applications today. Centrifugal Compressors A centrifugal compressor is a turbomachine and is similar to a centrifugal fan. A hermetic centrifugal compressor has an outer casing with one, two, or even three impellers internally connected in series and is driven by a motor directly or by a gear train. At the entrance to the first-stage impeller are inlet guide vanes positioned at a specific opening to adjust refrigerant flow and therefore the capacity of the centrifugal compressor. Figure 9.8.1(e) shows a two-stage hermetic centrifugal compressor. The total pressure rise in a centrifugal compressor, often called head lift, in psi, is due to the conversion of the velocity pressure into static pressure. Although the compression ratio Rcom of a single-stage centrifugal compressor using R-123 and R-22 seldom exceeds 4, two or three impellers connected in series satisfy most of the requirements in comfort systems. © 2005 by CRC Press LLC

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Because of the high head lift to raise the evaporating pressure to condensing pressure, the discharge velocity at the exit of the second-stage impeller approaches the acoustic velocity of saturated vapor vac of R-123, 420 ft/sec at atmospheric pressure and a temperature of 80°F. Centrifugal compressors need high peripheral velocity and rotating speeds (up to 50,000 rpm) to produce such a discharge velocity. It is not economical to manufacture small centrifugal compressors. The available refrigeration capacity for centrifugal compressors ranges from 100 to 10,000 tons. Centrifugal compressors have higher volume flow per unit refrigeration capacity output than positive displacement compressors. Centrifugal compressors are efficient and reliable. Their volumetric efficiency almost equals 1. At design conditions, their ηisen may reach 0.83, and it drops to 0.6 during part-load operation. They are the most widely used refrigeration compressors in large air-conditioning systems.

Refrigeration Condensers A refrigeration condenser or simply a condenser is a heat exchanger in which hot gaseous refrigerant is condensed into liquid and the latent heat of condensation is rejected to the atmospheric air, surface water, or well water. In a condenser, hot gas is first desuperheated, then condensed into liquid, and finally subcooled. The capacity of a condenser is rated by its total heat rejection Qrej, in Btu/hr, which is defined as the total heat removed from the condenser during desuperheating, condensation, and subcooling. For a refrigeration system using a hermetic compressor, Qrej can be calculated as Qrej = U con Acon ∆Tm = 60 m r (h2 − h3′ ) = qrl + (2545Pcom ) ηmo


(9.8.4)

where Ucon = overall heat transfer coefficient across the tube wall in the condenser, Btu/hr.ft2.°F Acon = condensing area in the condenser, ft2 ∆Tm = logarithmic temperature difference, °F
m r = mass flow rate of refrigerant, lb/min h2, h3′ = enthalpy of suction vapor refrigerant and hot gas, Btu/lb qrl = refrigeration load at the evaporator, Btu/hr A factor that relates Qrej and qrl is the heat rejection factor Frej, which is defined as the ratio of total heat rejection to the refrigeration load, or Frej = Qrej qrl = 1 + (2545Pcom ) (qrl ηmo )

(9.8.5)

Fouling factor Rf, in hr.ft2.°F/Btu, is defined as the additional resistance caused by a dirty film of scale, rust, or other deposits on the surface of the tube. ARI Standard 550-88 specifies the following for evaporators and condensers: Field fouling allowance New evaporators and condensers

0.00025 hr.ft2.°F/Btu 0

According to the cooling process used during condensation, refrigeration condensers can be classified as air-cooled, water-cooled, and evaporative-cooled condensers. Air-Cooled Condensers In an air-cooled condenser, air is used to absorb the latent heat of condensation released during desuperheating, condensation, and subcooling. An air-cooled condenser consists of a condenser coil, a subcooling coil, condenser fans, dampers, and controls as shown in Figure 9.8.2(a). There are refrigeration circuits in the condensing coil. Condensing coils are usually made of copper tubes and aluminum fins. The diameter of the tubes is 1/4 to 3/4 in., typically 3/8 in., and the fin density is 8 to 20 fins/in. On the inner surface of the copper tubes, microfins, © 2005 by CRC Press LLC

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typically 60 fins/in. with a height of 0.008 in., are used. A condensing coil usually has only two to three rows due to the low pressure drop of the propeller-type condenser fans. A subcooling coil is located at a lower level and is connected to the condensing coil. Hot gas from the compressor enters the condensing coil from the top. When the condensate increases, part of the condensing area can be used as a subcooling area. A receiver is necessary only when the liquid refrigerant cannot all be stored in the condensing and subcooling coils during the shut-down period in winter. Cooling air is drawn through the coils by a condenser fan(s) for even distribution. Condenser fans are often propeller fans for their low pressure and large volume flow rate. A damper(s) may be installed to adjust the volume flow of cooling air.
In air-cooled condensers, the volume flow of cooling air per unit of total heat rejection V ca / Qu.rej is 600 to 1200 cfm/ton of refrigeration capacity at the evaporator, and the optimum value is about 900 cfm/ton. The corresponding cooling air temperature difference — cooling air leaving temperature minus outdoor temperature (Tca.l – To) — is around 13°F. The condenser temperature difference (CTD) for an air-cooled condenser is defined as the difference between the saturated condensing temperature corresponding to the pressure at the inlet and the air intake temperature, or (Tcon.i – To). Air-cooled condensers are rated at a specific CTD, depending on the evaporating temperature of the refrigeration system Tev in which the air-cooled condenser is installed. For a refrigeration system having a lower Tev, it is more economical to equip a larger condenser with a smaller CTD. For a comfort air-conditioning system having a Tev of 45°F, CTD = 20 to 30°F. A higher condensing temperature Tcon, a higher condensing pressure pcon, and a higher compressor
power input may be due to an undersized air-cooled condenser, lack of cooling air or low V ca / Qu.rej value, a high entering cooling air temperature at the roof, a dirty condensing coil, warm air circulation because of insufficient clearance between the condenser and the wall, or a combination of these. The clearance should not be less than the width of the condensing coil. If pcon drops below a certain value because of a lower outdoor temperature, the expansion valve in a reciprocating vapor compression system may not operate properly. At a low ambient temperature To, the following controls are often used: • Duty cycling, turning the condenser fans on and off until all of them are shut down, to reduce cooling air volume flow • Modulating the air dampers to reduce the volume flow • Reducing the fan speed Some manufacturers’ catalogs start low ambient control at To = 65°F and some specify a minimum operating temperature at To = 0°F. Water-Cooled Condensers In a water-cooled condenser, latent heat of condensation released from the refrigerant during condensation is extracted by water. This cooling water, often called condenser water, is taken directly from river, lake, sea, underground well water or a cooling tower. Two types of water-cooled condensers are widely used for air-conditioning and refrigeration: doubletube condensers and horizontal shell-and-tube condensers. A double-tube condenser consists of two tubes, one inside the other. Condenser water is pumped through the inner tube and refrigerant flows within the space between the inner and outer tubes in a counterflow arrangement. Because of its limited condensing area, the double-tube condenser is used only in small refrigeration systems. A horizontal shell-and-tube water-cooled condenser using halocarbon refrigerant usually has an outer shell in which copper tubes typically 5/8 to 3/4 in. in diameter are fixed in position by tube sheets as shown in Figure 9.8.2(b). Integral external fins of 19 to 35 fins/in. and a height of 0.006 in. and spiral internal grooves are used for copper tubes to increase both the external and the inner surface area and their heat transfer coefficients. © 2005 by CRC Press LLC

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Condenser fans

Condenser coil Subcooling coil Hot gas inlet

(a)

Baffle plate

Tube sheet

Condenser water out

Condenser water in Liquid line (b) Fan

Elimnator Water supply Condensing coil Hot gas

Liquid line to evaporator

Subcooling coil Receiver

(c)

FIGURE 9.8.2 Various types of refrigeration condensers: (a) air-cooled, (b) two-pass shell-and-tube water-cooled, and (c) evaporative cooled.

© 2005 by CRC Press LLC

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Hot gas from the compressor enters the top inlet and is distributed along the baffle to fill the shell. Hot gas is then desuperheated, condensed, subcooled into liquid, and discharged into the liquid line at the bottom outlet. Usually one sixth of the volume is filled with subcooled liquid refrigerant. Subcooling depends on the entering temperature of condenser water Tce, in °F, and usually varies between 2 and 8°F. Condenser water enters the condenser from the bottom for effective subcooling. After extracting heat from the gaseous refrigerant, condenser water is discharged at a higher level. Two-pass or three-pass water flow arrangements are usually used in shell-and-tube water-cooled condensers. The two-pass arrangement means that water flows from one end to the opposite end and returns to the original end. Two-pass is the standard setup. In a shell-and-tube water-cooled condenser, the condensing temperature Tcon depends mainly on the entering temperature of condenser water Tce , the condenser area, the fouling factor, and the configuration of the copper tube. Evaporative Condenser An evaporative condenser uses the evaporation of water spray on the outer surface of the condensing tubes to remove the latent heat of condensation of refrigerant during condensation. An evaporative condenser consists of a condensing coil, a subcooling coil, a water spray, an induced draft or sometimes forced draft fan, a circulating water pump, a water eliminator, a water basin, an outer casing, and controls as shown in Figure 9.8.2(c). The condensing coil is usually made of bare copper, steel, or sometimes stainless steel tubing. Water is sprayed over the outside surface of the tubing. The evaporation of a fraction of condenser water from the saturated air film removes the sensible and latent heat rejected by the refrigerant. The wetted outer surface heat transfer coefficient hwet is about four or five times greater than the dry surface heat transfer coefficient ho, in Btu/hr.ft2.°F. The rest of the spray falls and is collected by the basin. Air enters from the inlet just above the basin. It flows through the condensing coil at a face velocity of 400 to 700 fpm, the water spray bank, and the eliminator. After air absorbs the evaporated water vapor, it is extracted by the fan and discharged at the top outlet. The water circulation rate is about 1.6 to 2 gpm/ton, which is far less than that of the cooling tower. An evaporative condenser is actually a combination of a water-cooled condenser and a cooling tower. It is usually located on the rooftop and should be as near the compressor as possible. Clean tube surface and good maintenance are critical factors for evaporative condensers. An evaporative condenser also needs low ambient control similar as in an air-cooled condenser. Comparison of Air-Cooled, Water-Cooled, and Evaporative Condensers An air-cooled condenser has the highest condensing temperature Tcon and therefore the highest compressor power input. For an outdoor dry bulb temperature of 90°F and a wet bulb temperature of 78°F, a typical air-cooled condenser has Tcon = 110°F. An evaporative condenser has the lowest Tcon and is most energy efficient. At the same outdoor dry and wet bulb temperatures, its Tcon may be equal to 95°F, even lower than that of a water-cooled condenser incorporating with a cooling tower, whose Tcon may be equal to 100°F. An evaporative condenser also consumes less water and pump power. The drawback of evaporative condensers is that the rejected heat from the interior zone is difficult to recover and use as winter heating for perimeter zones and more maintenance is required.

Evaporators and Refrigerant Flow Control Devices An evaporator is a heat exchanger in which the liquid refrigerant is vaporized and extracts heat from the surrounding air, chilled water, brine, or other substance to produce a refrigeration effect. Evaporators used in air-conditioning can be classified according to the combination of the medium to be cooled and the type of refrigerant feed, as the following. Direct expansion DX coils are air coolers, and the refrigerant is fed according to its degree of superheat after vaporization. DX coils were covered earlier.

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9-83

Direct expansion ice makers or liquid overfeed ice makers are such that liquid refrigerant is forced through the copper tubes or the hollow inner part of a plate heat exchanger and vaporized. The refrigeration effect freezes the water in the glycol-water that flows over the outside surface of the tubes or the plate heat exchanger. In direct expansion ice makers, liquid refrigerant completely vaporizes inside the copper tubes, and the superheated vapor is extracted by the compressor. In liquid overfeed ice makers, liquid refrigerant floods and wets the inner surface of the copper tubes or the hollow plate heat exchanger. Only part of the liquid refrigerant is vaporized. The rest is returned to a receiver and pumped to the copper tubes or plate heat exchanger again at a circulation rate two to several times greater than the evaporation rate. Flooded shell-and-tube liquid coolers, or simply flooded liquid coolers, are such that refrigerant floods and wets all the boiling surfaces and results in high heat transfer coefficients. A flooded shell-and-tube liquid cooler is similar in construction to a shell-and-tube water-cooled condenser, except that its liquid refrigeration inlet is at the bottom and the vapor outlet is at the top. Water velocity inside the copper tubes is usually between 4 and 12 ft/sec and the water-side pressure normally drops below 10 psi. Flooded liquid coolers can provide larger evaporating surface area and need minimal space. They are widely used in large central air-conditioning systems. Currently used refrigerant flow control devices include thermostatic expansion valves, float valves, multiple orifices, and capillary tubes. A thermostatic expansion valve throttles the refrigerant pressure from condensing to evaporating pressure and at the same time regulates the rate of refrigerant feed according to the degree of superheat of the vapor at the evaporator’s exit. A thermostatic expansion valve is usually installed just prior to the refrigerant distributor in DX coils and direct-expansion ice makers. A thermostatic expansion valve consists of a valve body, a valve pin, a spring, a diaphragm, and a sensing bulb near the outlet of the DX coil, as shown in Figure 9.7.3(a). The sensing bulb is connected to the upper part of the diaphragm by a connecting tube. When the liquid refrigerant passes through the opening of the thermostatic expansion valve, its pressure is reduced to the evaporating pressure. Liquid and a small fraction of vaporized refrigerant then flow through the distributor and enter various refrigerant circuits. If the refrigeration load of the DX coil increases, more liquid refrigerant vaporizers. This increases the degree of superheat of the leaving vapor at the outlet and the temperature of the sensing bulb. A higher bulb temperature exerts a higher saturated pressure on the top of the diaphragm. The valve pin then moves downward and widens the opening. More liquid refrigerant is allowed to enter the DX coil to match the increase of refrigeration load. If the refrigeration load drops, the degree of superheat at the outlet and the temperature of the sensing bulb both drop, and the valve opening is narrower. The refrigeration feed decreases accordingly. The degree of superheat is usually 10 to 20°F. Its value can also be adjusted manually by varying the spring tension. The bulbs of the thermostatic expansion valves can be charged with the same refrigerant as in the system or “cross-charged” by using a different refrigerant to improve response. A float valve is a valve in which a float is used to regulate the valve opening to maintain a specific liquid refrigerant level. A lower liquid level causes a lower valve pin and therefore a wider opening and vice versa. In a centrifugal refrigeration system, two or more orifice plates, multiple orifices, are sometimes installed in the liquid line between the condenser and the flash cooler and between the flash cooler and the flooded liquid cooler to throttle their pressure as well as to regulate the refrigerant feed. A capillary tube, sometimes called a restrictor tube, is a fixed length of small-diameter tubing installed between the condenser and the evaporator to throttle the refrigerant pressure from pcon to pev and to meter the refrigerant flow to the evaporator. Capillary tubes are usually made of copper. The inside diameter Dcap is 0.05 to 0.06 in. and the length Lcap from an inch to several feet. There is a trend to use short capillary tubes of Lcap /Dcap between 3 and 20. Capillary tubes are especially suitable for a heat pump system in which the refrigerant flow may be reversed.

© 2005 by CRC Press LLC

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w, lb/lb

Wetted medium

Fan

80% 70

0.016

r

2

50%

Air 0.012

1 70

0.008 90 T, °F

80

Sump Pump

(a) Wet air ex

w, lb/lb ex 80% 0.018

70

Cooled air

50% 2

70

0.014

r

1 0.010

80

2

1

90 T, °F

r

Plate heat exchange

Sump Pump

(b) w, lb/lb 80%

Indirect evaporative cooler

50%

Wet air ex

Direct evaporative cooler

0.014 0.010

1

r 3

2

3

0.006

60

1

1

2 70

80

90 T, °F

0.002

(c)

FIGURE 9.8.3 Types of evaporative coolers: (a) direct, (b) indirect, and (c) indirect–direct.

Evaporative Coolers An evaporative cooling system is an air-conditioning system in which air is cooled evaporatively. It consists of evaporative coolers, fan(s), filters, dampers, controls, and others. A mixing box is optional. An evaporative cooler could be a stand-alone cooler or installed in an air system as a component. There are three types of evaporative coolers: (1) direct evaporative coolers, (2) indirect evaporative coolers, and (3) indirect–direct evaporative coolers. Direct Evaporative Cooler In a direct evaporative cooler, the air stream to be cooled directly contacts the water spray or wetted medium as shown in Figure 9.8.3(a). Evaporative pads made of wooden fibers with necessary treatment at a thickness of 2 in., rigid and corrugated plastics, impregnated cellulose, or fiber glass all dripping with water are wetted mediums.

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The direct evaporation process 12 takes place along the thermodynamic wet bulb line on the psychrometric chart. Saturation effectiveness εsat is an index that assesses the performance of a direct evaporative cooler:

(

ε sat = (Tae − Tal ) Tae − Tae*

)

(9.8.6)

where T, T * = temperature and thermodynamic wet bulb temperature of air stream, °F. Subscript ae indicates the entering air and al the leaving air. εsat usually varies between 0.75 and 0.95 at a water–air ratio of 0.1 to 0.4. Indirect Evaporative Coolers In an indirect evaporative cooler, the cooled-air stream to be cooled is separated from a wetted surface by a flat plate or tube wall as shown in Figure 9.8.3(b). A wet-air stream flows over the wetted surface so that liquid water is evaporated and extracts heat from the cooled-air stream through the flat plate or tube wall. The cooled-air stream is in contact with the wetted surface indirectly. The core part of an indirect evaporative cooler is a plate heat exchanger. It is made of thin polyvinyl chloride plates 0.01 in. thick and spaced from 0.08 to 0.12 in. apart to form horizontal passages for cooled air and vertical passages for wet air and water. As in a direct evaporative cooler, there are also fan(s), water sprays, circulating pump, air intake, dampers, controls, etc. An indirect evaporative cooling process is represented by a horizontal line on a psychrometric chart, which shows that humidity ratio remains constant. If the space air is extracted and used as the wet air intake, the wet air will be exhausted at point x at nearly saturated state. The performance of an indirect evaporative cooler can be assessed by its performance factor ein, which is calculated as: ein = (Tca.e − Tca.l ) (Tca.e − Ts.a )

(9.8.7)

where Tca.e, Tca.l = temperature of cooled air entering and leaving the indirect evaporative cooler, °F, and Ts.a = temperature of the saturated air film on the wet air side and is about 3°F higher than the wet bulb temperature of the entering air, °F. An indirect evaporative cooler could be so energy efficient as to provide evaporative cooling with an EER up to 50 instead of 9 to 12 for a reciprocating compression refrigeration system. Direct–Indirect Evaporative Cooler. A direct–indirect evaporative cooler is a two-stage evaporating cooler, as shown in Figure 9.15.6(c), in which the first-stage indirect evaporative cooler is connected in series with a second-stage direct evaporative cooler for the purpose of increasing the evaporating effect. Operating Characteristics. The saturation effectiveness εsat and performance factor ein are both closely related to the air velocity flowing through the air passages. For a direct evaporative cooler, face velocity is usually less than 600 fpm to reduce drift carryover. For an indirect evaporative cooler, face velocity vs is usually between 400 to 1000 fpm. A higher vs results at a greater air-side pressure drop. Scofield et al. (1984) reported the performance of an indirect–direct evaporative cooler in Denver, Colorado. Outdoor air enters the indirect cooler at a dry bulb of 93°F and a wet bulb of 67.5° and was evaporatively cooled to 67.5°F dry bulb and 49.8°F wet bulb with an ein = 0.76 as shown in Figure 9.8.3(c). In the direct cooler, conditioned air was further cooled to a dry bulb of 53.5°F and the wet bulb remained at 49.8°F at a saturation effectiveness εsat = 0.8. In locations where outdoor wet bulb To′ ≤ 60°F, a direct evaporative can often provide an indoor environment of 78°F and a relative humidity of 60%. In locations To′ ≤ 68°F, an indirect–direct evaporative cooler can maintain a comfortable indoor environment. In locations To′ ≥ 72°F, an evaporative cooler with a supplementary vapor compression refrigeration may be cost effective. Because the installation cost of an indirect–direct cooler is higher than that of refrigeration, cost analysis is required to select the right choice. Evaporative coolers are not suitable for dehumidification except in locations where To′ ≤ 60°F.

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9.9 Water Systems Herbert A. Ingley and Shan K. Wang Types of Water Systems In central and space conditioning air-conditioning systems, water that links the central plant and the air handling units or terminals, that extracts condensing heat, or that provides evaporative cooling may be classified as • Chilled water system, in which chilled water is first cooled in the centrifugal, screw, and reciprocating chillers in a central plant. Chilled water is then used as a cooling medium to cool the air in the cooling coils in AHUs and terminals. • Evaporative-cooled water system, used to cool air directly or indirectly in evaporative coolers. • Hot water system, in which hot water is heated in the boiler and then used to heat the air through heating coils in AHUs, terminals, or space finned-tube heaters. • Dual-temperature water system, in which chilled water and hot water are supplied to and returned from the coils in AHUs and terminals through separate or common main and branch pipes. Using common main and branch pipes requires a lengthy changeover from chilled water to hot water or vice versa for a period of several hours. • Condenser water system, which is a kind of cooling water system used to extract the latent heat of condensation from the condensing refrigerant in a water-cooled condenser and heat of absorption from the absorber. Water systems can also be classified according to their operating characteristics. Closed System In a closed system, water forms a closed loop for water conservation and energy saving when it flows through the coils, chillers, boilers, heaters, or other heat exchangers and water is not exposed to the atmosphere. Open System In an open system, water is exposed to the atmosphere. A condenser water system with a cooling tower would be an example of an open system. Once-Through System In a once-through system, water flows through a heat exchanger(s) only once without recirculation. A condenser water system using well water that is ultimately rejected to a pond would be an example of a once-through system.

Basics Volume Flow and Temperature Difference The rate of heat transfer between water and air or water and refrigerant when water flows through a heat exchanger qw , in Btu/hr, can be calculated as qw = 500 V gal (Twl − Twe ) = 500 V gal ∆Tw


where







V gal = volume flow rate of water, gpm Twl, Twe = temperature of water leaving and entering the heat exchanger, °F ∆Tw = temperature rise or drop of water when it flows through a heat exchanger, °F 500
8.34 lb/gal ∗ 1 Btu/lb°F ∗ 60 min/hr © 2005 by CRC Press LLC

(9.9.1)

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The temperature of chilled water leaving the water chiller Tel should not be lower than 38°F in order to prevent freezing in the evaporator. Otherwise, brine or glycol-water should be used. The Tel of chilled water entering the coil Twe and the temperature difference of chilled water leaving and entering the coil ∆Tw directly affect the temperature of air leaving the cooling coil Tcc. The lower Twe, the higher will be the compressor power input. The smaller ∆Tw, the greater will be the water volume flow rate, the pipe size, and the pump power. For chilled water in conventional comfort systems, Twe is usually 40 to 45°F and ∆Tw 12 to 24°F. Only in cold air distribution, Twe may drop to 34°F. For a cooling capacity of 1 ton
refrigeration, a ∆Tw of 12°F requires a V gal = 2 gpm. For hot water heating systems in buildings, hot water often leaves the boiler and enters the heating coil or heaters at a temperature Twe of 180 to 200°F. It returns at 150 to 160°F. For dual-temperature systems, hot water is usually supplied at 100 to 150°F and returns at a ∆Tw of 20 to 40°F. Pressure Drop Usually the pressure drop of water in pipes due to friction for HVAC&R systems, Hf, is in the range 1.0 ft/100 ft length of pipe to 4 ft/100 ft. A pressure loss of 2.5 ft/100 ft is most often used. Figure 9.9.1(a), (b), and (c) shows the friction charts for steel, copper, and plastic pipes for closed water systems.

Water Piping The piping materials of various water systems for HVAC&R are as follows: Chilled water Hot water Condenser water

Black and galvanized steel Black steel, hard copper Black steel, galvanized ductile iron, polyvinyl chloride (PVC)

The pipe thickness varies from Schedule 10, a light wall pipe, to Schedule 160, a very heavy wall pipe. Schedule 40 is the standard thickness for a pipe of up to 10 in. diameter. For copper tubing, type K is the heaviest, and type L is generally used as the standard for pressure copper tubes. Steel pipes of small diameter are often joined by threaded cast-iron fittings. Steel pipes of diameter 2 in. and over, welded joints, and bolted flanges are often used. In a water system, the maximum allowable working pressure for steel and copper pipes at 250°F varies from 125 to 400 psig, depending on the pipe wall thickness. Not only pipes, but also their joints and fittings should be considered. During temperature changes, pipes expand and contract. Both operating and shut-down periods should be taken into consideration. Bends like U-, Z-, and L-bends, loops, and sometimes packed expansion joints, bellows, or flexible metal hose mechanical joints are used. ASHRAE/IES Standard 90.1-2001 specifies minimum thickness of pipe insulation for chilled water and heating hot water piping. Corrosion, Impurities, and Water Treatments Corrosion is a destructive process caused by a chemical or electrochemical reaction on metal or alloy. In water systems, dissolved impurities cause corrosion and scale and the growth of microbiologicals like algae, bacteria, and fungi. Scale is the deposit formed on a metal surface by precipitation of the insoluble constituents. In addition to the dissolved solids, unpurified water may contain suspended solids. Currently used chemicals include crystal modifiers to change the crystal formation of scale and sequestering chemicals. Growth of bacteria, algae, and fungi is usually treated by biocides to prevent the formation of an insulating layer resulting in lower heat transfer as well as restricted water flow. Chlorine and its compounds are effective and widely used. Blow-down is an effective process in water treatment and should be considered as important as chemical treatments. Piping Arrangements Main and Branch Pipes. In a piping circuit as shown in Figure 9.9.2(a), chilled water from a chiller or hot water from a boiler is often supplied to a main pipe and then distributed to branch pipes that connect © 2005 by CRC Press LLC

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FIGURE 9.9.1 Friction chart for water in pipes: (a) steel pipe (schedule 40), (b) copper tubing, and (c) plastic pipe (schedule 80). (Souce: ASHRAE Handbook 1993 Fundamentals. Reprinted with permission.)

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 %



%

 


%



%



%



%



   



  

 



















 

  

 !  ""  %

%

 %

 



 

%

 # 



 

 # 

  

 FIGURE 9.9.2 Piping arrangements: (a) two-pipe direct return system, (b) two-pipe reverse system, and (c) fourpipe system.

to coils and heat exchangers. Chilled or hot water from the coils and heat exchangers is accumulated by the return main pipe through return branch pipes and then returned to the chiller or boiler. Constant Flow and Variable Flow. In a constant-flow water system, the volume flow rate at any crosssectional plane of the supply and return mains remains constant during the entire operating period. In a variable-flow water system, the volume flow rate varies when the system load changes during the operating period. Direct Return and Reverse Return. In a direct return system, the water supplies to and returns from various coils through various piping circuits. ABCHJKA, … ABCDEFGHJKA are not equal in length, as shown in Figure 9.9.2(a). Water flow must be adjusted and balanced by using balance valves to provide required design flow rates at design conditions. In a reverse-return system, as shown in Figure 9.9.2(b), the piping lengths for various piping circuits including the branch and coil are almost equal. Water flow rates to various coils are easier to balance. © 2005 by CRC Press LLC

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Two-Pipe or Four-Pipe. In a dual-temperature water system, the piping from the chiller or boiler to the coils can be either a two-pipe system with a supply main and return main as shown in Figure 9.9.2(a) or (b) or a four-pipe system with a chilled water supply main, a hot water supply main, a chilled water return main, and a hot water return main as shown in Figure 9.9.2(c). The two-pipe system needs a changeover from chilled to hot water and vice versa. A four-pipe system is more expensive to install.

9.10 Heating Systems Shan K. Wang Warm Air Furnaces A warm air furnace is a device in which gaseous or liquid fuel is directly fired or electric resistance heaters are used to heat the warm supply air. Natural gas, liquefied petroleum gas (LPG), oil, electric energy, or occasionally wood may be used as the fuel or energy input. Among these, natural gas is most widely used. In a warm air furnace, the warm air flow could be upflow, in which the warm air is discharged at the top, as shown in Figure 9.10.1(a) and (b); downflow, with the warm air discharged at the bottom; or horizontal flow, with the warm air discharged horizontally. Natural Vent Combustion Systems. There are two types of combustion systems in a natural gas-fired warm air furnace: natural vent or power vent combustion systems. In a natural vent or atmospheric vent combustion system, the buoyancy of the combustion products carries the flue gas flowing through the heat exchanger and draft hood, discharging from the chimney or vent pipe. The gas burner is an atmospheric burner. In an atmospheric burner, air is exracted for combustion by the suction effect of the high-velocity discharged gas and the buoyance effect of the combustion air. An atmospheric burner can be either an in-shot or an up-shot burner or multiple ports. Atmospheric burners are simple, require only a minimal draft of air, and need sufficient gas pressure for normal functioning.

Warm air

Vent

Combustion air

Warm air Vent pipe

Clamshell heat exchange

Relief opening

Primary heat exchanger

Burner

Secondary (condensing) heat exchanger

Vent pipe

Burner

Flame

Fan

Filter

Induced draft fan

Recirculating fan

Filter

(a)

(b)

FIGURE 9.10.1 Upflow warm air gas furnace: (a) a natural-vent gas furnace and (b) a power-vent high-efficiency gas furnace. © 2005 by CRC Press LLC

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Two types of ignition have been used in burners: standing pilot and spark ignition. In standing pilot ignition, the small pilot flame is monitored by a sensor and the gas supply is shut off if the flame is extinguished. Spark ignition fires intermittently only when ignition is required. It saves gas fuel if the furnace is not operating. In a natural vent combustion system, the heat exchanger is often made from cold-rolled steel or aluminized steel in the shape of a clamshell or S. A fan or blower is always used to force the recirculating air flowing over the heat exchanger and distribute the heated air to the conditioned space. A low-efficiency disposable air filter is often located upstream of the fan to remove dust from the recirculating air. A draft hood is also installed to connect the flue gas exit at the top of the heat exchanger to a vent pipe or chimney. A relief air opening is employed to guarantee that the pressure at the flue gas exit is atmospheric and operates safely even if the chimney is blocked. The outer casing of the furnace is generally made of heavy-gauge steel with access panels. Power Vent Combustion Systems. In a power vent combustion system, either a forced draft fan is used to supply the combustion air or an induced draft fan is used to induce the flue gas to the vent pipe or chimney. A power vent is often used for a large gas furnace or a high-efficiency gas furnace with condensing heat exchangers. Gas burners in a power vent system are called power burners. The gas supply to the power burner is controlled by a pressure regulator and a gas valve to control the firing rate. Intermittent spark ignition and hot surface ignition that ignites the main burners directly are often used. Usually, there are two heat exchangers in a power vent combustion system: a primary heat exchanger and a secondary or condensing heat exchanger. The primary heat exchanger constitutes the heating surface of the combustion chamber. When the water vapor in the flue gas is condensed by indirect contact with the recirculating air, part of the latent heat of condensation released is absorbed by the air. Thus the furnace efficiency is increased in the secondary or condensing heat exchanger. Both primary and secondary heat exchangers are made from corrosion-resistant steel. A fan is also used to force the recirculating air to flow over the heat exchangers and to distribute the heated air to the conditioned space. Most natural gas furnaces can use LPG. LPG needs a pressure of 10 in. WG at the manifold, compared with 3 to 4 in. for natural gas. It also needs more primary air for gas burners. Oil furnaces are usually forced draft and installed with pressure-atomizing burners. The oil pressure and the orifice size of the injection nozzle control the firing rate. Furnace Performance Indices. The performance of a gas-fired furnace is usually assessed by the following indices: • Thermal efficiency Et, in percent, is the ratio of the energy output of heated air or water to the fuel energy input during specific test periods using the same units: Et = 100(fuel energy output) (fuel energy input)

(9.10.1)

• Annual fuel utilization efficiency (AFUE), in percent, is the ratio of the annual output energy from heated air or water to the annual input energy using the same units: AFUE = (100 annual output energy) (annual input energy)

(9.10.2)

• Steady-state efficiency (SSE) is the efficiency of a given furnace according to an ANSI test procedure, in percent: SSE = 100(fuel input − fuel loss) (fuel input )

(9.10.3)

Jakob et al. (1986) and Locklin et al. (1987), in a report on ASHRAE Special Project SP43, gave the following performance indices based on a nighttime setback period of 8 hr with a setback temperature of 10°F: © 2005 by CRC Press LLC

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Description Natural vent Pilot ignition Intermittent ignition Intermittent ignition plus vent damper Power vent Noncondensing Condensing

AFUE (%)

SSE (%)

64.5 69 78

77 77 77

81.5 92.5

82.5 93

ASHRAE/IES Standard 90.1-2001 specifies AFUEs for both gas-fired and oil-fired furnaces of heating capacity qrl, Tdis drops continually until it reaches 50°F, point A in Figure 9.11.2(b), and the DDC controller shuts down one of the scroll compressors. The operating point immediately shifts to B′ on the three-compressor curve. Because the refrigeration capacity at point B′ qrc is 29 tons, which is less than the required qrl = 35 tons, both Tdis and Tsuc rise. When the operating point moves up to B* and Tdis reaches 56°F, the DDC controller starts all four scroll compressors at operating point A″ with a refrigeration capacity of 42 tons. Since qrc > qrl, the operating point again moves downward along the four-compressor curve and forms an operating cycle A″AB′ and B*. The timing of the operating period on four- or three-compressor performance curves balances any required qrl between 29 and 42 tons. Less evaporation at part load in the DX coil results in a greater superheating region and therefore less refrigeration capacity to balance the reduction of refrigeration capacity of the compressor(s) as well as the condensing unit. The condition © 2005 by CRC Press LLC

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4 scroll

DX-coil 30ft2 A«« A

Cooling capacity qrc, ton

40

A« 3 scroll B««

30

20

B

DX-coil 15ft2

B«

C«« C

B* 2 scroll

C* 1 scroll

10

D««

D D«

30

B*,A«

56 55

Upper limit A«

53 51 50

Control band, 4°F

Set point Lower point

A,B«

C«

D*

40 Tsuc, °F (a)

50

Number of scroll compressors

50

Discharge temperature Tdis, °F

Chapter 9

4 3 2 1 0 (b)

FIGURE 9.11.2 Capacity control of a DX refrigeration system: (a) performance curves and operating points and (b) locus of control point.

will be similar when qrl < 30 tons, only three- or two-compressor, or two- or one-compressor, or even on–off of one compressor forms an operating cycle. Main Problems in DX Systems • Liquid slugging is formed by a mixture of liquid refrigerant and oil. It is formed because of the flooding back of liquid refrigerant from the DX coil to the crankcase of the reciprocating compressor due to insufficient superheating. It also may be caused by migration of liquid refrigerant from the warmer indoor DX coil to the colder outdoor compressor during the shut-down period in a split packaged unit. Liquid slugging dilutes the lubricating oil and causes serious loss of oil in the crankcase of the reciprocating compressor. Liquid slugging is incompressible. When it enters the compression chamber of a reciprocating compressor, it may damage the valve, piston, and other components. Pump-down control and installation of a crankcase heater are effective means of preventing liquid refrigerant migration and flooding back. • Compressor short cycling — For on–off control, too short a cycle, such as less than 3 min, may pump oil away from the compressor or damage system components. It is due mainly to a too close low-pressure control differential or to reduced air flow. • Defrosting — If the surface of a DX coil is 32°F or lower, frost accumulates on it. Frost blocks the air passage and reduces the rate of heat transfer. It should be removed periodically. The process of removing frost is called defrosting. • Air at a temperature above 36°F, hot gas inside the refrigerant tubes and an installed electric heating element can be used for defrosting. The defrosting cycle is often controlled by sensing the temperature or pressure difference of air entering the DX coil during a fixed time interval. • Refrigerant charge — Insufficient refrigerant charge causes lower refrigeration capacity, lower suction temperature, and short on–off cycles. Overcharging refrigerant may cause a higher condensing pressure because part of the condensing surface becomes flooded with liquid refrigerant. Heat Pumps A heat pump in the form of a packaged unit is also a heat pump system. A heat pump can either extract heat from a heat source and reject heat to air and water at a higher temperature for heating, or provide © 2005 by CRC Press LLC

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refrigeration at a lower temperature and reject condensing heat at a higher temperature for cooling. During summer, the heat extraction, or refrigeration effect, is the useful effect for cooling in a heat pump. In winter, the rejected heat and the heat from a supplementary heater provide heating in a heat pump. There are three types of heat pumps: air-source, water-source, and ground-coupled heat pumps. Ground-coupled heat pumps have limited applications. Water-source heat pump systems are covered in detail in a later section. Air-Source Heat Pump. An air-source heat pump, or air-to-air heat pump, is a DX system with an additional four-way reversing valve to change the refrigerant flow from cooling mode in summer to heating mode in winter and vice versa. The variation in connections between four means of refrigerant flow — compressor suction, compressor discharge, DX coil exit, and condenser inlet — causes the function of the indoor and outdoor coils to reverse. In an air-source heat pump, the coil used to cool or to heat the recirculating/outdoor air is called the indoor coil. The coil used to reject heat to or absorb heat from the outside atmosphere is called the outdoor coil. A short capillary or restrict tube is often used instead of a thermostatic expansion valve. Both reciprocating and scroll compressors are used in air-source heat pumps. R-22 is the refrigerant widely used. Currently available air-source heat pumps usually have a cooling capacity of 11/2 to 40 tons. Cooling and Heating Mode Operation. In cooling mode operation, as shown in Figure 9.11.1(b), the solenoid valve is deenergized and drops downward. The high-pressure hot gas pushes the sliding connector to the left end. The compressor discharge connects to the outdoor coil, and the indoor coil connects to the compressor inlet. In heating mode operation, as shown in Figure 9.11.1(c), the solenoid plunger moves upward and pushes the slide connector to the right-hand side. The compressor discharge connects to the indoor coil, and the outdoor coil exit connects to the compressor suction. System Performance. The performance of an air-source heat pump depends on the outdoor air temperature To, in °F as well as the required space heating load qrh. During cooling mode operation, both the refrigeration capacity qrc, in Btu/hr, and EER for the heat pump EERhp, in Btu/hr.W, increase as To drops. During heating mode operation, the heating capacity qhp, in Btu/hr, and COPhp decrease, and qrh increases as the To drops. When qrh > qhp, supplementary heating is required. If COPhp drops below 1, electric heating may be more economical than a heat pump. If on–off is used for compressor capacity control for an air-source heat pump in a split packaged unit, refrigerant tends to migrate from the warmer outdoor coil to the cooler indoor coil in summer and from the warmer indoor coil to the cooler outdoor coil in winter during the off period. When the compressor starts again, 2 to 5 min of reduced capacity is experienced before the heat pump can be operated at full capacity. Such a loss is called a cycling loss. In winter, most air-source heat pumps switch from the heating mode to cooling mode operation and force the hot gas to the outdoor coil to melt frost. After the frost is melted, the heat pump is switched back to heating mode operation. During defrosting, supplementary electric heating is often necessary to prevent a cold air supply from the air-source heat pump. Minimum Performance. ANSI/ASHRAE/IESNA Standard 90.1-2001 specifies a minimum performance for air-cooled DX systems in packaged units as covered in Section 9.7. Centrifugal Chillers A chiller is a refrigeration machine using a liquid cooler as an evaporator to produce chilled water as the cooling medium in a central air-conditioning system. A centrifugal chiller, as shown in Figure 9.11.3(a), is a refrigeration machine using a centrifugal compressor to produce chilled water. It is often a factoryassembled unit with an integrated DDC control system and sometimes may separate into pieces for transportation. A centrifugal chiller is also a centrifugal vapor compression refrigeration system. Refrigerants. As mentioned in Section 9.4, production of CFCs, including R-11 and R-12, ceased at the end of 1995 with limited exception for service. R-123 (HCFC) will replace R-11. The chiller’s efficiency © 2005 by CRC Press LLC

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Chapter 9

Condenser

Cooling water

Motor cooling circuit Inlet vanes

Impeller

Flash cooler

Compressor Hermetic motor

Orifice plates

Vapor refrigerant Liquid refrigerant

Chilled water

Evaporator (liquid-cooler)

Orifice plates

120

0.2

0.4

0.6

Compression

O lin

e

100

P

rge

80

Load ratio LR 1.0

0.8

Surge region

d

Su

Percentage of system design system head, %

(a)

eme

A

tem Sys

hea

Q

R

Sch

h

40

0.25 open 0.5 open

Open 0.75 open

eC

T W

20 0

performance

50% curve 40%

eB em

60 Sc

87% efficiency ηcp 85% 80% 75% 70% Centrifugal 60% compressor

Sc

m he

20 40 60 80 100 120 Percentage of design volume flow, % (b)

FIGURE 9.11.3 A two-stage water-cooled centrifugal chiller: (a) schematic diagram and (b) centrifugal compressor performance map at constant speed.

may drop 2 to 4%, and a capacity reduction of 5% is possible. R-123 has low toxicity. Its allowable exposure limit was raised to 50 ppm in 1997 from 30 ppm in 1993 by its manufacturers. A monitor and alarm device to detect R-123 in air should be installed in plant rooms and places where there may be refrigerant leaks. See ANSI/ASHRAE Standard 15-2001 for other requirements. R-134a (HFC) will replace R-12. According to Lowe and Ares (1995), as a result of the changeout from R-12 to R-134a for a 5000-hp centrifugal chiller in Sears Tower, Chicago, its speed increased from 4878 to 5300 rpm, its cooling capacity is 12 to 24% less, and its efficiency is 12 to 16% worse. System Components. A centrifugal chiller consists of a centrifugal compressor, an evaporator or liquid cooler, a condenser, a flash cooler, throttling devices, piping connections, and controls. A purge unit is optional. © 2005 by CRC Press LLC

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• Centrifugal compressor — According to the number of internally connected impellers, the centrifugal compressor could have a single, two, or more than two stages. A two-stage impeller with a flash cooler is most widely used because of its higher system performance and comparatively simple construction. Centrifugal compressors having a refrigeration capacity less than 1200 tons are often hermetic. Very large centrifugal compressors are of open type. A gear train is often required to raise the speed of the impeller except for very large impellers using direct drive. • Evaporator — Usually a liquid cooler of flooded shell-and-tube type evaporator is adopted because of its compact size and high rate of heat transfer. • Condenser — Water-cooled, horizontal shell-and-tube condensers are widely used. • Flash cooler — For a two-stage centrifugal compressor, a single-stage flash cooler is used. For a three-stage compressor, a two-stage flash cooler is used. • Orifice plates and float valves — Both multiple-orifice plates such as that shown in Figure 9.11.3(a) and float valves are used as throttling devices in centrifugal chillers. • Purge unit — R-123 has an evaporating pressure pev = 5.8 psia at 40°F, which is lower than atmospheric pressure. Air and other noncondensable gases may leak into the evaporator through cracks and gaps and usually accumulate in the upper part of the condenser. These noncondensable gases raise the condensing pressure, reduce the refrigerant flow, and lower the rate of heat transfer. A purge unit uses a cooling coil to separate the refrigerant and water from the noncondensable gases and purge the gases by using a vacuum pump. Performance Ratings. The refrigeration cycle of a typical water-cooled, two-stage centrifugal chiller with a flash cooler was covered in Section 9.4. Centrifugal chillers have the same refrigeration capacity as centrifugal compressors, 100 to 10,000 tons. The integrated part-load value (IPLV) of a centrifugal chiller or other chillers at standard rating conditions can be calculated as: IPLV = 0.1( A + B) 2 + 0.5( B + C) 2 + 0.3(C + D) 2 + 0.1D

(9.11.1)

where A, B, C, and D = kW/ton or COP at 100, 75, 50, and 25% load, respectively. If the operating conditions are different from the standard rating conditions, when Tel is 40 to 50°F, for each °F increase or decrease of Tel, there is roughly a 1.5% difference in refrigeration capacity and energy use; when Tce is between 80 to 90°F, for each °F of increase or decrease of Tce, there is roughly a 1% increase or decrease in refrigeration capacity and 0.6% in energy use. ANSI/ASHRAE/IESNA Standard 90.1-2001 and ARI Standard 550 specify the minimum performance for water-cooled water chillers. Air-cooled centrifugal chillers have COPs from 2.5 to 2.8. Their energy performance is far poorer than that of water-cooled chillers. Their application is limited to locations where city water is not allowed to be used as makeup water for cooling towers. Capacity Control. The refrigeration capacity of a centrifugal chiller is controlled by modulating the refrigerant flow at the centrifugal compressor. There are mainly two types of capacity controls: varying the opening and angle of the inlet vanes, and using an adjustable-frequency AC inverter to vary the rotating speed of the centrifugal compressor. When the opening of the inlet vanes has been reduced, the refrigerant flow is throttled and imparted with a rotation. The result is a new performance curve at lower head and flow. If the rotating speed of a centrifugal compressor is reduced, it also has a new performance curve at lower volume flow and head. Inlet vanes are inexpensive, whereas the AC inverter speed modulation is more energy efficient at partload operation. Centrifugal Compressor Performance Map. Figure 9.11.3(b) shows a single-stage, water-cooled centrifugal compressor performance map for constant speed using inlet vanes for capacity modulation. A performance map consists of the compressor’s performance curves at various operating conditions. The performance © 2005 by CRC Press LLC

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curve of a centrifugal compressor shows the relationship of volume flow of refrigerant V r and its head lift ∆p or compression efficiency ηcp at that volume flow. It is assumed that ηcp for a two-stage compressor is equal to the average of the two single-stage impellers having a head of their sum. On the map, the required system head curve indicates the required system head lift at that volume flow of refrigerant. The intersection of the performance curve and the required system head curve is called the operating point O, P, Q, R, … as shown in Figure 9.11.3(b). One of the important operating characteristics of a centrifugal chiller (a centrifugal vapor compression refrigeration system as well) is that the required system head lift is mainly determined according to the difference in condensing and evaporating pressure ∆pc-e = (pcon – pev). The pressure losses due to the refrigerant piping, fittings, and components are minor.
In Figure 9.11.3(b), the abscissa is the percentage of design volume flow of refrigerant, % V r , or load ratio; the ordinate is the percentage of design system head ∆Hs.d, or percentage of design temperature lift (Tcon – Tev). Here load ratio LR is defined as the ratio of the refrigeration load to the design refrigeration load qrl/qrl.d. There are three schemes of required system head curves: • Scheme A — Tce = constant and Tel = constant • Scheme B — Tel = constant and a drop of 2.5°F of Tce for each 0.1 reduction of load ratio • Scheme C — A reset of Tel of 1°F increase and a drop of 2.5°F of Tce for each 0.1 reduction of load ratio





At design V r and system head Hs.d, ηcp = 0.87. As V r , load ratio, and required system head ∆p decrease, ηcp drops accordingly. Surge is a unstable operation of a centrifugal compressor or fan resulting in vibration and noise. In a centrifugal chiller, surge occurs when the centrifugal compressor is unable to develop a discharge pressure that satisfies the requirement at the condenser. A centrifugal compressor should never be operated in the surge region. Part-Load Operation. During part-load operation, if Tel and Tce remain constant, the evaporating temperature Tev tends to increase from the design load value because there is a greater evaporating surface and a smaller temperature difference (Tel – Tev). Similarly, Tcon tends to decrease. The ratio of actual compressor power input at part load to the power input at design load may be slightly higher or lower than the load ratios, depending on whether the outdoor wet bulb is constant or varying at part load or whether there is a Tel reset; it also depends on the degree of drop of ηcp at part load. Specific Controls. In addition to generic controls, specific controls for a centrifugal chiller include: • • • • •

Chilled water leaving temperature Tel and reset Condenser water temperature Tce control On and off of multiple chillers based on actual measured coil load Air purge control Safety controls like oil pressure, low-temperature freezing protection, high condensing pressure control, motor overheating, and time delaying

Centrifugal Chillers Incorporating Heat Recovery. A HVAC&R heat recovery system converts waste heat or waste cooling from any HVAC&R process into useful heat and cooling. A heat recovery system is often subordinate to a parent system, such as a heat recovery system to a centrifugal chiller. A centrifugal chiller incorporating a heat recovery system often uses a double-bundle condenser in which water tubes are classified as tower bundles and heating bundles. Heat rejected in the condenser may be either discharged to the atmosphere through the tower bundle and cooling tower or used for heating through the heating bundle. A temperature sensor is installed to sense the temperature of return hot water from the heating coils in the perimeter zone. A DDC controller is used to modulate a bypass three-way valve which determines the amount of condenser water supplied to the heating bundle. The tower and heating bundles may be enclosed in the same shell, but baffle sheets are required to guide the water flows. A centrifugal chiller incorporating a heat recovery system provides cooling for the interior zone and heating for the perimeter zone simultaneously in winter with a higher COPhr . However, it needs a higher © 2005 by CRC Press LLC

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condenser water-leaving temperature Tcl of 105 to 110°F, compared with 95°F or even lower in a coolingonly centrifugal chiller. An increase of 10 to 12°F of the difference (Tcon – Tev) requires an additional 10 to 15% power input to the compressor. For a refrigeration plant equipped with multiple chillers, it is more energy efficient and lower in first cost to have only part of them equipped with double-bundle condensers. Screw Chillers A screw chiller or a helical rotary chiller is a refrigeration machine using a screw compressor to produce chilled water. A factory-fabricated and assembled screw chiller itself is also a screw vapor compression refrigeration system. Twin-screw chillers are more widely used than single-screw chillers. A twin-screw chiller consists of mainly a twin-screw compressor, a flooded shell-and-tube liquid cooler as evaporator, a water-cooled condenser, throttling devices, an oil separator, an oil cooler, piping, and controls as shown in Figure 9.11.4(a). The construction of twin-screw compressors has already been covered. For evaporator, condenser, and throttling devices, they are similar to those in centrifugal chillers. Most twin-screw chillers have a refrigeration capacity of 100 to 1000 tons. Following are the systems characteristics of screw chillers. Variable Volume Ratio. The ratio of vapor refrigerant trapped within the interlobe space during the intake process Vin to the volume of trapped hot gas discharged Vdis is called the built-in volume ratio of the twinscrew compressor Vi = Vin/Vdis, or simply volume ratio, all in ft3. There are two types of twin-screw chiller: fixed and variable volume ratio. For a twin-screw chiller of fixed volume ratio, the isentropic efficiency ηisen becomes maximum when the system required compression ratio Rs.com ≈ Vi. Here Rs.com = pcon/pev. If pdis > pcon, overcompression occurs, as shown in Figure 9.11.4(b). The discharged hot gas reexpands to match the condensing pressure. If pdis < pcon, undercompression occurs (Figure 9.11.4[c]). A volume of gas at condensing pressure reenters the trapped volume at the beginning of the discharge process. Both over- and undercompression cause a reduction of ηisen. For a twin-screw chiller of variable volume ratio, there are two slides: a sliding valve is used for capacity control and a second slide. By moving the second slide back and forth, the radial discharge port can be relocated. This allows variation of suction and discharge pressure levels and still maintains maximum efficiency. Economizer. The hermetic motor shell is connected to an intermediate point of the compression process and maintains an intermediate pressure pi between pcon and pev. Liquid refrigerant at condensing pressure pcon is throttled to pi, and a portion of the liquid is flashed into vapor. This causes a drop in the temperature of the remaining liquid refrigerant down to the saturated temperature corresponding to pi. Although the compression in a twin-screw compressor is in continuous progression, the mixing of flashed gas with the compressed gas at the intermediate point actually divides the compression process into two stages. The resulting economizing effect is similar to that of a two-stage compound refrigeration system with a flash cooler: an increase of the refrigeration effect and a saving of the compression power from (pcon – pev) to (pcon – pi). Oil Separation, Oil Cooling, and Oil Injection. Oil entrained in the discharged hot gas enters an oil separator. In the separator, oil impinges on an internal perforated surface and is collected because of its inertia. Oil drops to an oil sump through perforation. It is then cooled by condenser water in a heat exchanger. A heater is often used to vaporize the liquid refrigerant in the oil sump to prevent dilution of the oil. Since the oil sump is on the high-pressure side of the refrigeration system, oil is forced to the rotor bearings and injected to the rotors for lubrication. Oil slugging is not a problem for twin-screw compressors. When suction vapor contains a small amount of liquid refrigerant that carries over from the oil separator, often called wet suction, it often has the benefit of scavenging the oil from the evaporator. Twin-screw compressors are positive displacement compressors. They are critical in oil lubrication, sealing, and cooling. They are also more energy efficient than reciprocating compressors. Twin-screw chillers are gaining more applications, especially for ice-storage systems with cold air distribution. © 2005 by CRC Press LLC

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Chapter 9

Cooling tower

Condenser

Twin-screw compressor

3 Oil/gas separator

Hermatic motor 8

Oil cooler

Refrigerant pipe Chilled water pipe Condenser water pipe Oil pipe

Evaporator 9 Chilled water (a) pdis

pcom

pcom

pdis

psuc

(b)

V, ft3

psuc

(c)

V, ft3

FIGURE 9.11.4 A typical twin-screw chiller: (a) schematic diagram, (b) over-compression, and (c) undercompression.

© 2005 by CRC Press LLC

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9.12 Thermal Storage Systems Shan K. Wang Thermal Storage Systems and Off-Peak Air-Conditioning Systems Many electric utilities in the United States have their on-peak hours between noon and 8 p.m. during summer weekdays, which include the peak-load hours of air-conditioning. Because the capital cost of a new power plant is so high, electric utilities tend to increase their power output by using customers’ thermal energy storage (TES) systems, or simply thermal storage systems, which are much less expensive. A thermal storage system as shown in Figure 9.12.1(a) may have the same refrigeration equipment, like chillers, additional storage tank(s), additional piping, pumps, and controls. The electric-driven compressors are operated during off-peak, partial-peak, and on-peak hours. Off-peak hours are often nighttime hours. Partial-peak hours are hours between on-peak and off-peak hours in a weekday’s 24-hr day-andnight cycle. Chilled water or ice is stored in tanks to provide cooling for buildings during on-peak hours when higher electric demand and electric rates are effective. Although thermal storage systems operate during nighttime when outdoor dry and wet bulbs are lower, they are not necessarily energy saving due to lower evaporating temperature, additional pump power, and energy losses. Thermal storage systems significantly reduce the electric energy cost. Utilities in the United States often use higher electric demand and rates as well as incentive bonus to encourage the shift of electric load from on-peak to off-peak hours by using thermal storage systems and others. Such a shift not only saves expensive capital cost, but also increases the use of base-load highefficiency coal and nuclear plants instead of inefficient diesel and gas turbine plants. The air-conditioning systems that operate during off-peak and partial-peak hours for thermal storage, or those that use mainly natural gas to provide cooling to avoid higher electric demand and rates during on-peak hours, are called off-peak air-conditioning systems. These systems include ice-storage and chilledwater storage systems, desiccant cooling systems, absorption systems, and gas engine chillers. Absorption chillers and desiccant cooling systems are covered in other sections. Gas engine-driven reciprocating chillers are often a cogeneration plant with heat recovery from engine jacket and exhaust gas, and will not be covered here. Full Storage and Partial Storage Ice and chilled-water storage systems are the two most common thermal energy storage systems today. Knebel (1995) estimated that more than 4000 cool storage systems are operated in various commercial buildings. The unit of stored thermal energy for cooling is ton-hour, or ton.hr. One ton.hr is the refrigeration capacity of one refrigeration ton during a 1-hr period, or 12,000 Btu. In order to achieve minimum life-cycle cost, thermal storage systems could be either full storage or partial storage. For a full-storage, or load shift, thermal storage system, all refrigeration compressors cease to operate during on-peak hours. Building refrigeration loads are entirely offset by the chilled water or brine from the thermal storage system within an on-peak period. In a partial storage, or load-leveling, thermal storage system as shown in Figure 9.12.1(b) all or some refrigeration compressor(s) are operated during on-peak hours. Direct cooling is the process in which refrigeration compressors produce refrigeration to cool the building directly. During a specific time interval, if the cost of direct cooling is lower than the stored energy, the operation of a thermal storage system is said to be in chiller priority. On the contrary, if the cost of direct cooling is higher than the cost of stored energy, the operation is said to be at storage priority. The optimum size of a thermal storage system is mainly determined according to the utility’s electric rate structure, especially a time-of-day structure whose electric rates are different between on-peak, partial-peak, and off-peak hours. Not only the design day’s instantaneous building peak cooling load is important, but also an hour-by-hour cooling load profile of the design day is required for thermal storage design. A simple payback or a life-cycle cost analysis is usually necessary. © 2005 by CRC Press LLC

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Coils CV-7

CV-11

Modular storage tanks Return CV-6 4 CV-10

CV-4,5 CV-8

CV-9

Supply

5

Building pumps

CV-3

Chiller pump

CV-1

Chiller pump

CV-2

Chiller 1

Chiller 2

Both on-peak and off-peak

On-peak

Off-peak (a)

Iceburning

Icemaking

Offpeak

Direct cooling

Brine

Direct cooling

Onpeak

Ice

Offpeak

Water (c)

1200 Brine

Ice

800 400 Water 24

4

8

12 Noon Time (b)

16

20

24

Polyethylene tube

(d)

FIGURE 9.12.1 A brine-coil ice-storage system: (a) schematic diagram, (b) partial-storage time schedule, (c) ice making, and (d) ice burning.

Ice-Storage Systems System Characteristics In an ice-thermal-storage system, or simply an ice-storage system, ice is stored in a tank or other containers to provide cooling for buildings in on-peak hours or on- and partial-peak hours. One pound of ice can store [(1 × 144) + (55 – 35)] = 164 Btu instead of (60 – 44) = 16 Btu for chilled water storage. For the same cooling capacity, the storage volume of ice is only about 12% of chilled water. In addition, an air© 2005 by CRC Press LLC

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conditioning system using ice storage often adopts cold air distribution to supply conditioned air at a temperature typically at 44°F. Cold air distribution reduces the size of air-side equipment, ducts, and investment as well as fan energy use. It also improves the indoor air quality of the conditioned space. Since the late 1980s, ice storage has gained more applications than chilled water storage. Brine is a coolant that does not freeze and flash during normal operation. The freezing point of brine, which has a mass fraction of ethylene glycol of 25%, drops to 10°F, and a mass fraction of a propylene glycol brine of 25% drops to 15°F. Glycol-water, when glycol is dissolved in water, is another coolant widely used in ice-storage systems. Ice crystals are formed in glycol-water when its temperature drops below its freezing point during normal operation. In an ice-storage system, ice making or charging is a process in which compressors are operated to produce ice. Ice burning, or discharging, is a process in which ice is melted to cool the brine or glycolwater to offset a refrigeration load. Brine-Coil Ice-Storage Systems. Currently used ice-storage systems include brine-coil, ice-harvester, and ice-on-coil systems. According to Knebel (1995), the brine-coil ice-storage system is most widely used today because of its simplicity, flexibility, and reliability as well as using modular ice-storage tanks. In a typical brine-coil ice-storage system, ice is charged in multiple modular factory-fabricated storage tanks as shown in Figure 9.12.1(a). In each storage tank, closely spaced polyethylene or plastic tubes are surrounded by water. Brine, a mixture of 25 to 30% of ethylene glycol by mass and 70 to 75% water, circulates inside the tubes at an entering temperature of 24°F during the ice-making process. The water surrounding the tubes freezes into ice up to a thickness of about 1/2 in. as shown in Figure 9.12.1(c). Brine leaves the storage tank at about 30°F. Water usually at atmospheric pressure is separated from brine by a plastic tube wall. Plastic tubes occupy about one tenth of the tank volume, and another one tenth remains empty for the expansion of ice. Multiple modular storage tanks are always connected in parallel. During the ice-melting or -burning process, brine returns from the cooling coils in the air-handling units (AHUs) at a temperature of 46°F or higher. It melts the ice on the outer surface of the tubes and is thus cooled to a temperature of 34 to 36°F, as shown in Figure 9.12.1(d). Brine is then pumped to the AHUs to cool and dehumidify the air again. Ice-Harvester Ice-Storage Systems. In an ice-harvester system, glycol-water flows on the outside surface of the evaporator and forms ice sheets with a thickness of 0.25 to 0.40 in. within 20 to 30 min. Ice is harvested in the form of flakes when hot gas is flowing inside the tubes of the evaporator during a time interval of 20 to 30 sec. Ice flakes fall to the glycol-water in the storage tank below. The glycol-water at a temperature of 34°F is then supplied to the cooling coils in AHUs for conditioning. After cooling and dehumidifying the air, glycol-water returns to the storage tank at a temperature of 50 to 60°F and is again cooled to 34°F again. Ice harvesting is an intermittent process. It has a cycle loss due to harvesting of about 15%. In addition, because of its operating complexity and maintenance, its applications are more suitable for large icestorage systems. Ice-on-Coil Ice-Storage Systems. In an ice-on-coil system, refrigerant is evaporated in the coils submerged in water in a storage tank. Ice of a thickness not exceeding 0.25 in. builds up on the outer surface of the coils. The remaining water in the storage tank is used for cooling in AHUs. Ice-on-coil systems need large amounts of refrigerant charge and are less flexible in operation. They are usually used in industrial applications. Ice-in-Containers Ice-Storage Systems. Ice-in-containers ice-storage systems store ice in enclosed containers. Brine circulating over the containers produces the ice inside containers. Complexity in control of the ice inventory inside the containers limits the application of the ice-in-containers systems.

Chilled-Water Storage Systems Basics Chilled-water storage uses the same water chiller and a similar coolant distribution system, except for additional water storage tanks and corresponding piping, additional storage pumps, and controls. The larger the chilled-water storage system, the lower the installation cost per ton.hr storage capacity. © 2005 by CRC Press LLC

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Various types of storage tanks had been used in chilled-water storage systems during the 1970s. A diaphragm tank uses a rubber diaphragm to separate the colder and warmer water. Baffles divide the tank into cells and compartments. Today, stratified tanks have become the most widely used chilledwater storage systems because of their simplicity, low cost, and negligible difference in loss of cooling capacity between stratified tanks and other types of storage tanks. During the storage of chilled water, the loss in cooling capacity includes direct mixing, heat transfer between warmer return chilled water and colder stored chilled water, and also heat transfer between warmer ambient air and colder water inside the tank. An enthalpy-based easily measured index called figure of merit (FOM) is often used to indicate the loss in cooling capacity during chilled-water storage. FOM is defined as: FOM = qdis qch

(9.12.1)

where qdis = cooling capacity available in the discharge process, Btu/hr qch = theoretical cooling capacity available during charging process, Btu/hr Charging is the process of filling the storage tank with colder chilled water from the chiller. At the same time, warmer return chilled water is extracted from the storage tank and pumped to the chiller for cooling. Discharging is the process of discharging the colder stored chilled water from the storage tank to AHUs and terminals. Meanwhile, the warmer return chilled water from the AHUs and terminals fills the tank. Stratified Tanks. Stratified tanks utilize the buoyancy of warmer return chilled water to separate it from the colder stored chilled water during charging and discharging. Colder stored chilled water is always charged and discharged from bottom diffusers, and the warmer return chilled water is introduced to and withdrawn from the top diffusers. Chilled-water storage tanks are usually vertical cylinders and often have a height-to-diameter ratio of 0.25:0.35. Steel is the most commonly used material for above-grade tanks, with a 2-in.-thick spray-on polyurethane foam, a vapor barrier, and a highly reflective coating. Concrete, sometimes precast, prestressed tanks are widely used for underground tanks. A key factor to reduce loss in cooling capacity during chilled water storage is to reduce mixing of colder and warmer water streams at the inlet. If the velocity pressure of the inlet stream is less than the buoyancy pressure, the entering colder stored chilled water at the bottom of tank will stratify. Refer to Wang’s handbook (1993) and Knebel (1995) for details. A thermocline is a stratified region in a chilled-water storage tank of which there is a steep temperature gradient. Water temperature often varies from 42°F to about 60°F. Thermocline separates the bottom colder stored chilled water from the top warmer return chilled water. The thinner the thermocline, the lower the mixing loss. Diffusers and symmetrical connected piping are used to evenly distribute the incoming water streams with sufficient low velocity, usually lower than 0.9 ft/sec. Inlet stream from bottom diffusers should be downward and from the top diffusers should be upward or horizontal. Field measurements indicate that stratified tanks have a FOM between 0.85 to 0.9.

9.13 Air System Basics Shan K. Wang Fan-Duct Systems Flow Resistance Flow resistance is a property of fluid flow which measures the characteristics of a flow passage resisting
the fluid flow with a corresponding total pressure loss ∆p, in in. WG, at a specific volume flow rate V , in cfm: © 2005 by CRC Press LLC

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∆p = RV 2

(9.13.1)

where R = flow resistance (in. WG/(cfm)2). For a duct system that consists of several duct sections connected in series, its flow resistance Rs, in in. WG/(cfm)2, can be calculated as Rs = R1 + R2 + … + Rn

(9.13.2)

where R1, R2, … Rn = flow resistance of duct section 1, 2, … n in the duct system (in. WG/(cfm)2). For a duct system that consists of several duct sections connected in parallel, its flow resistance Rp, in in. WG/(cfm)2, is: 1

Rp = 1

R1 + 1

R2 + … + 1

(9.13.3)

Rn

Fan-Duct System In a fan-duct system, a fan or several fans are connected to ductwork or ductwork and equipment. The volume flow and pressure loss characteristics of a duct system can be described by its performance curve,
called system curve, and is described by ∆p = R V 2 . An air system or an air handling system is a kind of fan-duct system. In addition, an outdoor ventilation air system to supply outdoor ventilation air, an exhaust system to exhaust contaminated air, and a smoke control system to provide fire protection are all air systems, that is, fan-duct systems. Primary, Secondary, and Transfer Air Primary air is the conditioned air or makeup air. Secondary air is often the induced space air, plenum air, or recirculating air. Transfer air is the indoor air that moves to a conditioned space from an adjacent area. System-Operating Point A system-operating point indicates the operating condition of an air system or fan-duct system. Since the operating point of a fan must lie on the fan performance curve, and the operating point of a duct system on the system curve, the system operating point of an air system must be the intersection point Ps of the fan performance curve and system curve as shown in Figure 9.13.1(a). Fan-Laws


For the same air system operated at speeds n1 and n2, both in rpm, their relationship of V volume flow rate, in cfm, system total pressure loss, in in. WG, and fan power input, in hp, can be expressed as





V 2 V 1 = n2 n1 ∆pt2 ∆pt1 = (n2 n1 ) (ρ2 ρ1 ) 2

(9.13.4)

P2 P1 = (n2 n1 ) (ρ2 ρ1 ) 3

where ρ = air density (lb/ft3). Subscripts 1 and 2 indicate the original and the changed operating conditions. For air systems that are geometrically and dynamically similar: V 2 V 1 = ( D2 D1 ) (n2 n1 )





3

∆pt2 ∆pt1 = ( D2 D1 ) (n2 n1 ) (ρ2 ρ1 ) 2

2

P2 P1 = ( D2 D1 ) (n2 n1 ) (ρ2 ρ1 ) 5

where D = diameter of the impeller (ft). © 2005 by CRC Press LLC

3

(9.13.5)

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Fan curve with system effect

∆pt, in WG ∆pts

∆psy

Fan curve

∆pt, in WG

Modulation curve

Ps ∆ps,i Fan curve without system effect

∆ps,o

∆pse A

Pd System curve . ∆pts = RsVs2 including system effect

C D ∆psy

E 50°

System curve without system effect 0

20°

.

Vs

.

Set point of duct static pressure control ∆pfix

V, cfm

(a) System curve damper modulation

Fan curve

∆pt, in WG

Q 4.0

Inlet vanes opening

B

70°

90°

30°

.

V, cfm

(b)

System curve P

Inlet vanes Damper loss Speed modulation

2.0

.

P—V ∆pfix

.

20 P , f hp 15 10

P—V (50%)

5

10,000

. 20,000 V, cfm

(c)


FIGURE 9.13.1 Air system V - ∆pt performance: (a) system operating point and system effect, (b) modulation curve, and (c) damper, inlet vanes, and fan speed modulation.

Geometrically similar means that two systems are similar in shape and construction. For two systems that are dynamically similar, they must be geometrically similar, and in addition, their velocity distribution or profile of fluid flow should also be similar. When fluid flows in the air systems are at high Reynolds number, such as Re > 10,000, their velocity profiles can be considered similar to each other.

System Effect The system effect ∆pse, in in. WG, of an air system is its additional total pressure loss caused by uneven or nonuniform velocity profile at the fan inlet, or at duct fittings after the fan outlet, due to the actual inlet and outlet connections as compared with the total pressure loss of the fan test unit during laboratory ratings. The selected fan total pressure which includes the system effect ∆pts, in in. WG, as shown in Figure 9.13.1(a), can be calculated as ∆pts = ∆psy + ∆pse = ∆psy + ∆ps.i + ∆ps.o = ∆psy + Cs.i (vfi 4005) + Cs.o (vfo 4005) 2

© 2005 by CRC Press LLC

2

(9.13.6)

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where ∆psy = calculated total pressure loss of the air system, in WG ∆ps.i, ∆ps.o = fan inlet and outlet system effect loss, in WG Cs.i, Cs.o = local loss coefficient of inlet and outlet system effect, in WG vfi, vfo = velocity at fan inlet (fan collar) and fan outlet, fpm Both Cs.i and Cs.o are mainly affected by the length of connected duct between the duct fitting and fan inlet or outlet, by the configuration of the duct fitting, and by the air velocity at the inlet or outlet. Because vfi and vfo are often the maximum velocity of the air system, system effect should not be overlooked.

Modulation of Air Systems Air systems can be classified into two categories according to their operating volume flow: constant volume and variable-air-volume systems. The volume flow rate of a constant volume system remains constant during all the operating time. Its supply temperature is raised during part load. For a variableair-volume (VAV) system, its volume flow rate is reduced to match the reduction of space load at partload operation. The system pressure loss of a VAV system can be divided into two parts: variable part ∆pvar and fixed part ∆pfix, which is the set point of the duct static pressure control as shown in Figure 9.13.1(b) and (c). The modulation curve of a VAV air system its its operating curve, or the locus of system operating points when its volume flow rate is modulated at full- and part-load operation. The volume flow and system pressure loss of an air system can be modulated either by changing its fan characteristics or by varying its flow resistance of the duct system. Currently used types of modulation of volume flow rate of VAV air systems are 1. Damper modulation uses an air damper to vary the opening of the air flow passage and therefore its flow resistance. 2. Inlet vanes modulation varies the opening and the angle of inlet vanes at the centrifugal fan inlet and then gives different fan performance curves. 3. Inlet cone modulation varies the peripheral area of the fan impeller and therefore its performance curve. 4. Blade pitch modulation varies the blade angle of the axial fan and its performance curve. 5. Fan speed modulation using adjustable frequency AC drives varies the fan speed by supplying a variable-frequency and variable-voltage power source. There are three types of AC drives: adjustable voltage, adjustable current, and pulse width modulation (PWM). The PWM is universally applicable. Damper modulation wastes energy. Inlet vanes are low in cost and are not so energy efficient compared with AC drives and inlet cones. Inlet cones are not expensive and are suitable for backward curved centrifugal fans. Blade pitch modulation is energy efficient and is mainly used for vane and tubular axial fans. AC or variable frequency drives (VFD) are the most energy-efficient type of modulation; however, they are expensive and often considered cost effective for air systems using larger centrifugal fans. Example 9.13.1 A multizone VAV system equipped with a centrifugal fan has the following characteristics: °

V (cfm)

5,000

10,000

15,000

20,000

25,000

∆pt, in. WG P, hp

4.75

4.85 17.0

4.83 18.6

4.60 20.5

4.20 21.2

At design condition, it has a volume flow rate of 22,500 cfm and a fan total pressure of 4.45 in. WG. The set point of duct static pressure control is 1.20 in. WG. © 2005 by CRC Press LLC

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When this VAV system is modulated by inlet vanes to 50% of design flow, its fan performance curves show the following characteristics: °

V (cfm)

5,000

10,000

11,250

∆pt, in. WG P, hp

3.6

2.5 7.5

2.1 7.8

Determine the fan power input when damper, inlet vanes, or AC drive fan speed modulation is used. Assume that the fan total efficiency remains the same at design condition when the fan speed is reduced. Solutions 1. At 50% design flow, the volume flow of this VAV system is 0.5 × 22,500 = 11,250 cfm. The flow resistance of the variable part of this VAV system is


Rvar = ∆pva = RV 2 = (4.45 − 1.20) (22, 500) = 6.42 × 10 −9 in. WG (cfm) 2

2

When damper modulation is used, the system operating point Q must be the intersection of the
fan curve and the system curve that has a higher flow resistance and a V = 11,250 cfm. From Figure 9.13.1(c), at point Q, the fan power input is 17.0 hp. 2. From the given information, when inlet vane modulation is used, the fan power input is 7.8 hp. 3. The total pressure loss of the variable part of the VAV system at 50% volume flow is


∆pvar = Rvar V 2 = 6.42 × 10 −9 (11, 250) = 0.81 in. WG 2

From Figure 9.13.1(c), since the fan power input at design condition is 21.2 hp, then its fan total efficiency is: ηf = V∆ptf (6356 Pf ) = 22, 500 × 4.45 (6356 × 21.2) = 74.3%


The fan power input at 50% design volume flow is: P = V∆ptf (6356 ηf ) = 11, 250(0.81 + 1.20) (6356 × 0.743) = 4.8 hp


Damper modulation has a energy waste of (17 – 4.8) = 12.2 hp

Fan Combinations in Air-Handling Units and Packaged Units Currently used fan combinations in air-handling units (AHUs) and packaged units (PUs) (except dualduct VAV systems) are shown in Figure 9.13.2(a), (b), and (c): Supply and Exhaust Fan/Barometric Damper Combination An air system equipped with a single supply fan and a constant-volume exhaust fan, or a supply fan and a barometric damper combination as shown in Figure 9.13.2(a), is often used in buildings where there is no return duct or the pressure loss of the return duct is low. An all-outdoor air economizer cycle is usually not adopted due to the extremely high space pressure. A barometric relief damper is often installed in or connected to the conditioned space to prevent excessively high space pressure. When the spacepositive pressure exerted on the barometric damper is greater than the weight of its damper and/or a spring, the damper is opened and the excessive space pressure is relieved.

© 2005 by CRC Press LLC

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Supply fan

Constant volume exhaust fan

Relief fan

ru

o

Supply fan

Exhaust fan

ru

m

o

m

r

r

(a)

(b) Return fan

Supply fan

Exhaust fan

ru

o

m

r

(c)

FIGURE 9.13.2 Fan combinations: (a) supply and exhaust fan, (b) supply and relief fan, and (c) supply and return fan.

Supply and Relief Fan Combination Figure 9.13.2(b) shows the schematic diagrams of an air system of supply fan and relief fan combination. A relief fan is used to relieve undesirable high positive space pressure by extracting space air and relieving it to the outside atmosphere. A relief fan is always installed in the relief flow passage after the junction of return flow, relief flow, and recirculating flow passage, point ru. It is usually energized only when the air system is operated in air economizer mode. A relief fan is often an axial fan. Since the relief fan is not energized during recirculating mode operation, the volume flow and pressure characteristics of a supply fan and relief fan combination are the same as that in a single supply fan and barometric damper combination when they have the same design parameters. Supply Fan and Return Fan Combination A return fan is always installed at the upstream of the junction of return, recirculating, and exhaust flow passage, point ru as shown in Figure 9.13.2(c). A supply and return fan combination has similar pressure and volume flow characteristics as that of a supply and relief fan combination, except a higher total pressure at point ru. If the return fan is improperly selected and has an excessive fan total pressure, total pressure at point m may be positive. There will be no outdoor intake at the PU or AHU, and at the same time there will also be a negative space pressure and an infiltration to the space.

© 2005 by CRC Press LLC

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Chapter 9



 






 

 











&!#'





°*



 

! ""#




 $$

  

%
 $
#$$ 

 



 () "  #"!" #
"   



°*



FIGURE 9.13.3 (a) Year-round operation and (b) discharge air temperature for a VAV reheat system.

Comparison of These Three Fan Combination Systems A supply fan and barometric damper combination is simpler and less expensive. It is suitable for an air system which does not operate in an air economizer mode and has a low pressure drop in the return system. For those air systems whose pressure drop for the return system does not exceed 0.3 in. WG, or there is a considerable pressure drop in relief or exhaust flow passage, a supply and relief fan combination is recommended. For air systems whose return system has a pressure drop exceeding 0.6 in. WG, or those requiring a negative space pressure, a supply and return fan combination seems more appropriate.

Year-Round Operation and Economizers Consider a typical single-duct VAV reheat system to serve an occupied space whose indoor temperature is 75°F with a relative humidity of 50%. During summer, the off-coil temperature is 55°F. The year-round operation of this air system can be divided into four regions on the psychrometric chart, as shown in Figure 9.13.3(a): • Region I — Refrigeration/evaporative cooling. In this region, the enthalpy of the outdoor air ho is higher than the enthalpy of the recirculating air hru , ho > hru. It is more energy efficient to condition the mixture of recirculating air and minimum outdoor air. • Region II — Free cooling and refrigeration. In this region, ho ≤ hru. It is more energy efficient and also provides better indoor air quality to extract 100% outdoor air. © 2005 by CRC Press LLC

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• Region III — Free cooling evaporative cooling, and refrigeration. In this region, extract 100% outdoor air for free cooling because ho ≤ hru. Use evaporative cooling and refrigeration to cool and humidify if necessary. • Region IV — Winter heating. Maintain a 55°F supply temperature by mixing the recirculating air with the outdoor air until the outdoor air is reduced to a minimum value. Provide heating if necessary. An economizer is a device consisting of dampers and control that uses the free cooling capacity of either outdoor air or evaporatively cooled water from the cooling tower instead of mechanical refrigeration. An air economizer uses outdoor air for free cooling. There are two kinds of air economizers: enthalpy-based, in which the enthalpy of outdoor and recirculating air is compared, and temperaturebased, in which temperature is compared. Economizers require routine maintenance and calibration in order to realize maximum savings. A water economizer uses evaporatively cooled water.

Fan Energy Use For an air system, fan energy use for each cfm of conditioned air supplied from the AHUs and PUs to the conditioned space within a certain time period, in W/cfm, can be calculated as W cfm = 0.1175∆psy

( ηf ηm )

(9.13.7)

where ∆psy = mean system total pressure loss during a certain time period, in. WG ηf = fan total efficiency ηm = combined motor and drive (direct drive or belt drive) efficiency For an air system using a separate outdoor ventilation system, its fan energy use, in W/cfm, is then calculated as

[

W cfm = (1 + Ro.s ) 0.1175∆psy

(ηf ηm )]

(9.13.8)

where Ro.s = ratio of outdoor air volume flow rate to supply volume flow rate.

Outdoor Ventilation Air Supply Basics • An adequate amount of outdoor ventilation air supply is the key factor to provide acceptable indoor air quality (IAQ) for a conditioned space. Although an inadequate amount of outdoor ventilation air supply causes poor IAQ, an oversupply of outdoor ventilation air other than in an air economizer cycle is often a waste of energy. • According to local codes and ANSI/ASHRAE Standard 62, the minimum outdoor ventilation rate for each person must be provided at the outdoor air intake of AHU or PU, or by an outdoor air ventilation system. If the minimum outdoor ventilation air rate is reduced by using higher efficiency filters to remove air contaminants in the recirculating air, then indoor air contaminant concentration must be lower than the specified level in ANSI/ASHRAE Standard 62. • For a multizone air system, although the ratio of outdoor ventilation air rate to supply air volume flow rate required may be varied from zone to zone, the excessive outdoor air supply to a specified zone will increase the content of unused outdoor air in the recirculating air in AHU or PU. This helps to solve the problem in any zone that needs more outdoor air. • Since the occupancy in many buildings is often variable and intermittent, a demand-based variable amount of outdoor ventilation air control should be used instead of time-based constant volume outdoor ventilation air control, except during the air economizer cycle. © 2005 by CRC Press LLC

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• Carbon dioxide (CO2) is a gaseous body effluent. CO2 is an indicator of representative odor and an indicator of adequate outdoor ventilation rate at specific outdoor and indoor air concentration in a control zone at steady state. For most of the comfort air-conditioning systems, it is suitable to use CO2 as a key parameter to control the intake volume flow rate of outdoor ventilation air to maintain an indoor CO2 concentration not exceeding 800 to 1000 ppm in a critical or representative zone. As mentioned in Section 9.5, Persily (1993) showed that the actual measured indoor daily maximum CO2 concentration levels in five buildings were all within 400 to 820 ppm. If a field survey finds that a specific indoor air contaminant exceeds a specified indoor concentration, then a gas sensor for this specific contaminant or a mixed gas sensor should be used to control this specific indoor concentration level. Types of Minimum Outdoor Ventilation Air Control. There are four types of minimum outdoor ventilation air control that are currently used: • Type I uses a CO2 sensor or a mixed gas sensor and a DDC controller to control the volume flow rate of outdoor ventilation air for a separate outdoor ventilation air system on the demand-based principle. • Type II uses a CO2 or mixed gas sensor and a DDC controller to control the ratio of the openings between outdoor and recirculating dampers and, therefore, the volume flow rates of outdoor air and recirculating air in AHUs or PUs on the demand-based principle. • Type III uses a flow sensor or a pressure sensor and a DDC controller to control the openings of outdoor and recirculating dampers to provide a nearly constant volume outdoor air intake in VAV AHUs or VAV PUs. • Type IV adjusts the opening of the outdoor damper manually to provide a constant volume of outdoor air in constant-volume AHUs and PUs. If the outdoor intake is mounted on the external wall without a windshield, the volume flow of outdoor ventilation air intake will be affected by wind force and direction. Type I is the best minimum outdoor ventilation air control for the air system. For a VAV system, it is expensive. Type II is a better choice. Type III is more complicated and may cause energy waste. Type IV has the same result as Type III and is mainly used in constant-volume systems. Outdoor intake must be located in a position away from the influence of exhaust outlets. Fans, control dampers, and filters should be properly operated and maintained in order to provide a proper amount of outdoor ventilation air as well as an acceptable IAQ.

9.14 Absorption System Shan K. Wang Absorption systems use heat energy to produce refrigeration as well as heating if it is required. Water is the refrigerant and aqueous lithium bromide (LiBr) is widely used as the carrier to absorb the refrigerant and provide a higher coefficient of performance. The mixture of water and anhydrous LiBr is called solution. The composition of a solution is usually expressed by its mass fraction, or percentage of LiBr, often called concentration. When the water vapor has boiled off from the solution, it is called concentrated solution. If the solution has absorbed the water vapor, it is called diluted solution. Absorption systems can be divided into the following categories: • Absorption chillers use heat energy to produce refrigeration. • Absorption chiller/heaters use direct-fired heat input to provide cooling or heating separately. • Absorption heat pumps extract heat energy from the evaporator, add to the heat input, and release them both to the hot water for heating. • Absorption heat transformers raise the temperature of the waste heat source to a required level. © 2005 by CRC Press LLC

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Most recently installed absorption chillers use direct-fired natural gas as the heat source in many locations in the United States where there are high electric demand and electric rates at on-peak hours. Absorption chillers also are free from CFC and HCFC refrigerants. An energy cost analysis should be done to determine whether an electric chiller or a gas-fired absorption chiller is the suitable choice. Absorption heat pumps have only limited applications in district heating. Most absorption heat transformers need industrial waste heat. Both of them will not be covered here.

Double-Effect Direct-Fired Absorption Chillers Figure 9.14.1(a) shows a double-effect direct-fired absorption chiller. Double effect means that there are two generators. Direct fired means that gas is directly fired at the generator instead of using steam or hot water. A single-effect absorption chiller using steam as the heat input to its single generator has a COP only from 0.7 to 0.8, whereas a double-effect direct-fired absorption chiller has a COP approximately equal to 1 and therefore is the mot widely used absorption chiller in the United States for new and retrofit projects today. The refrigeration capacity of double-effect direct-fired absorption chillers varies from 100 to 1500 tons. A double-effect direct-fired absorption chiller mainly consists of the following components and controls: • Evaporator — An evaporator is comprised of a tube bundle, spray nozzles, a water trough, a refrigerant pump, and an outer shell. Chilled water flows inside the tubes. A refrigerant pump sprays the liquid refrigerant over the outer surface of the tube bundle for a higher rate of evaporation. A water trough is located at the bottom to maintain a water level for recirculation. • Absorber — In an absorber, there are tube bundles in which cooling water flows inside the tubes. Solution is sprayed over the outer surface of the tube bundle to absorb the water vapor. A solution pump is used to pump the diluted solution to the heat exchanger and low-temperature generator. • Heat exchangers — There are two heat exchangers: low-temperature heat exchanger in which the temperature of hot concentrated solution is lower, and high-temperature heat exchanger in which the temperature of hot concentrated solution is higher. In both heat exchangers, heat is transferred from the hot concentrated solution to the cold diluted solution. Shell-and-tube or plate-and-frame heat exchangers are most widely used for their higher effectiveness. • Generators — Generators are also called desorbers. In the direct-fired generator, there are the fire tube, flue tube, vapor/liquid separator, and flue-gas economizer. Heat is supplied from the gas burner or other waste heat source. The low-temperature generator is often of the shell-and-tube type. The water vapor vaporized in the direct-fired generator is condensed inside the tubes. The latent heat of condensation thus released is used to vaporize the dilute solution in the lowtemperature generator. • Condenser — A condenser is usually also of the shell-and-tube type. Cooling water from the absorber flows inside the tubes. • Throttling devices — Orifices and valves are often used as throttling devices to reduce the pressure of refrigerant and solution to the required values. • Air purge unit — Since the pressure inside the absorption chiller is below atmospheric pressure, air and other noncondensable gases will leak into it from the ambient air. An air purge unit is used to remove these noncondensable gases from the chiller. A typical air purge unit is comprised of a pickup tube, a purge chamber, a solution spray, cooling water tubes, a vacuum pump, a solenoid valve, and a manual shut-off valve. When noncondensable gases leak into the system, they tend to migrate to the absorber where pressure is lowest. Noncondensable gases and water vapor are picked from the absorber through the pickup tube. Water vapor is absorbed by the solution spray and returned to the absorber through a liquid trap at the bottom of the purge chamber. Heat of absorption is removed by the cooling water inside the tubes. Noncondensable gases are then evacuated from the chamber periodically by a vacuum pump to the outdoor atmosphere. © 2005 by CRC Press LLC

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To cooling tower

50 mm Hg abs

300°

Condenser

Low temperature generator

195°

85°

390 mm Hg abs

High temp solution pump

60.3%

47.7 mm Hg abs

Exhaust Vapor gas to solution stack separator 350°

From cooling tower

66%

7

3

97°

Evaporator

Exhaust gas economizer

63.0% 91°

44°

306°

Chilled water 54° 42°

5 58.6%

6.8 mm Hg abs

95° 1

121°

5 mm Hg abs

151°

Absorber

2 4

High-temp heat exchanger

8

Fuel input

192°

Low-temp heat exchanger Refrigerant pump

Burner Direct-fired generator

261° 6

Dilute solution pump

(a) 40%

50% 55%

1000

7«

7

6

6«

200

Va p

or

pr

es

su

150 Full-load

100

60 40

2«

2

5

1«

50

ine

l on

20 10

100

3« 4« 3 4 8 8«

Refrigerant temperature, °F

H

g

ab

s

Partload

300

m ,m

65% 200 70%

500 400

re

60%

1 100

ati lliz a t s

50

y

Cr

150

200 Solution temperature, °F

250

300

350

(b)

FIGURE 9.14.1 A double-effect direct-fired reverse-parallel-flow absorption chiller: (a) schematic diagram (reprinted by permission from the Trane catalog) and (b) absorption cycle.

Palladium cells are used to continuously remove a small amount of hydrogen that is produced due to corrosion. Corrosion inhibitors like lithium chromate are needed to protect the machine parts from the corrosive effect of the absorbent when air is present. © 2005 by CRC Press LLC

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Absorption Cycles, Parallel-, Series-, and Reverse-Parallel Flow An absorption cycle shows the properties of the solution and its variation in concentrations, temperature, and pressure during absorbing, heat exchanging, and concentration processes on an equilibrium chart as shown in Figure 9.14.1(b). The ordinate of the equilibrium chart is the saturated temperature and pressure of water vapor, in °F and mm Hg abs. The abscissa is the temperature of the solution, in °F. Concentration lines are incline lines. At the bottom of the concentration lines, there is a crystallization line or saturation line. If the mass of fraction of LiBr in a solution which remains at constant temperature is higher than the saturated condition, that part of LiBr exceeding the saturation condition tends to form solid crystals. Because there are two generators, the flow of solution from the absorber to generators can be in series flow, parallel flow, or reverse-parallel flow. In a series-flow system, the diluted solution from the absorber is first pumped to the direct-fired generator and then to the low-temperature generator. In a parallelflow system, diluted solution is pumped to both direct-fired and low-temperature generators in parallel. In a reverse-parallel-flow system as shown in Figure 9.14.1(a), diluted solution is first pumped to the low-temperature generator. After that, the partly concentrated solution is then sent to the direct-fired generator as well as to the intermediate point 4 between high- and low-temperature heat exchangers in parallel. At point 4, partly concentrated solution mixes with concentrated solution from a direct-fired generator. A reverse-parallel-flow system is more energy efficient. Solution and Refrigerant Flow In a typical double-effect direct-fired reverse-parallel-flow absorption chiller operated at design full load, water is usually evaporated at a temperature of 42°F and a saturated pressure of 6.8 mm Hg abs in the evaporator. Chilled water returns from the AHUs or fan coils at a temperature typically 54°F, cools, and leaves the evaporator at 44°F. A refrigeration effect is produced due to the vaporization of water vapor and the removal of latent heat of vaporization from the chilled water. Water vapor in the evaporator is then extracted to the absorber due to its lower vapor pressure. It is absorbed by the concentrated LiBr solution at a pressure of about 5 mm Hg abs. After absorption, the solution is diluted to a concentration of 58.6% and its temperature increases to 95°F (point 1). Most of the heat of absorption and the sensible heat of the solution is removed by the cooling water inside the tube bundle. Diluted solution is then pumped by a solution pump to the low-temperature generator through a low-temperature heat exchanger. In the low-temperature generator, the dilute solution is partly concentrated to 60.3% at a solution temperature of 180°F (point 3). It then divides into two streams: one of them is pumped to the directfired generator through a high-temperature heat exchanger, and the other stream having a slightly greater mass flow rate is sent to the intermediate point 4. In the direct-fired generator, the concentrated solution leaves at a concentration of 66% and a solution temperature of 306°F (point 7). The mixture of concentrated and partly concentrated solution at point 4 has a concentration of 63% and a temperature of 192°F. It enters the low-temperature heat exchanger. Its temperature drops to 121°F before entering the absorber (point 5). In the direct-fired generator, water is boiled off at a pressure of about 390 mm Hg abs. The boiledoff water vapor flows through the submerged tube in the low-temperature generator. The release of latent heat of condensation causes the evaporation of water from the dilution solution at a vapor pressure of about 50 mm Hg abs. The boiled-off water vapor in the low-temperature generator flows to the condenser through the top passage and is condensed into liquid water at a temperature of about 99°F and a vapor pressure of 47.7 mm Hg abs. This condensed liquid water is combined with the condensed water from the submerged tube at the trough. Both of them return to the evaporator after its pressure is throttled by an orifice plate. Part-Load Operation and Capacity Control During part-load operation, a double-effect direct-fired reverse-parallel-flow absorption chiller adjusts its capacity by reducing the heat input to the direct-fired generator through the burner. Lower heat input © 2005 by CRC Press LLC

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results at less water vapor boiled off from the solution in the generators. This causes the drop in solution concentration, the amount of water vapor extracted, the rate of evaporation, and the refrigeration capacity. Due to less water vapor being extracted, both evaporating pressure and temperature will rise. Since the amount of water vapor to be condensed is greater than that boiled off from the generators, both the condensing pressure and condensing temperature decrease. Coefficient of Performance (COP) The COP of an absorption chiller can be calculated as COP = 12, 000 q1g

(9.14.1)

where qlg = heat input to the direct-fired generator per ton of refrigeration output (Btu/hr.ton).

9.15 Air-Conditioning Systems and Selection Shan K. Wang Basics in Classification The purpose of classifing air-conditioning or HVAC&R systems is to distinguish one type from another so that an optimum air-conditioning system can be selected according to the requirements. Proper classification of air-conditioning systems also will provide a background for using knowledge-based expert systems to help the designer to select an air-conditioning system and its subsystems. Since air system characteristics directly affect the space indoor environmental parameters and the indoor air quality, the characteristics of an air system should be clearly designated in the classification. The system and equipment should be compatible with each other. Each system has its own characteristics which are significantly different from others.

Individual Systems As described in Section 9.1, air conditoning or HVAC&R systems can be classified as individual, space, packaged, and central systems. Individual systems usually have no duct and are installed only in rooms that have external walls and external windows. Individual systems can again be subdivided into the following. Room Air-Conditioner Systems A room air conditioner is the sole factory-fabricated self-contained equipment used in the room airconditioning system. It is often mounted on or under the window sill or on a window frame as shown in Figure 9.1.1. A room air-conditioner consists mainly of an indoor coil, a small forward-curved centrifugal fan for indoor coil, a capillary tube, a low-efficiency dry and reusable filter, grilles, a thermostat or other controls located in the indoor compartment, and a rotary, scroll, or reciprocating compressor, an outdoor coil, and a propeller fan for the outdoor coil located in the outdoor compartment. There is an outdoor ventilation air intake opening and a manually operated damper on the casing that divides the indoor and outdoor compartments. Room air-conditioners have a cooling capacity between 1/2 to 2 tons. The following are system characteristics of a room air-conditioner system: Room heat pump system is a room air-conditioner plus a four-way reversing valve which provides both the summer cooling and winter heating. Air system: single supply fan Fan, motor, and drive combined efficiency: 25% Fan energy use: 0.3 to 0.4 W/cfm Fan speed: HI-LO 2-speed or HI-MED-LO 3-speed Outdoor ventilation air system: type IV © 2005 by CRC Press LLC

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Cooling system: DX system, air-cooled condenser EER 7.5 to 9.5 Btu/hr.W Evaporating temperature Tev at design load: typically 45°F Heating system: electric heating (if any) Part-load: on–off of refrigeration compressor Sound level: indoor NC 45 to 50 Maintenance: More maintenance work is required. Summer and winter mode air-conditioning cycles of a room air-conditioning system are similar to that shown in Figure 9.3.4. Packaged Terminal Air-Conditioner (PTAC) Systems A packaged terminal air-conditioner is the primary equipment in a PTAC system. A PTAC system is similar to a room air-conditioner system. Their main differences are • A PTAC uses a wall sleeve and is intended to be mounted through the wall. • Heating is available from hot water, steam, heat pump, electric heater, and sometimes even directfired gas heaters. PTACs are available in cooling capacity between 1/2 to 1 1/2 tons and a heating capacity of 2500 to 35,000 Btu/hr. Larger PTAC units have capacities in the range of 5 to 7 1/2 tons and may come equipped with heat recovery options.

Space (Space-Conditioning) Systems Most space conditioning air-conditioning systems cool, heat, and filtrate their recirculating space air above or in the conditioned space. Space conditioning systems often incorporate heat recovery by transferring the heat rejected from the interior zone to the perimeter zone through the condenser(s). Space systems often have a separate outdoor ventilation air system to supply the required outdoor ventilation air. Space systems can be subdivided into four-pipe fan-coil systems and water-source heat pump systems.

Four-Pipe Fan-Coil Systems In a four-pipe fan-coil unit system, space recirculating air is cooled and heated at a fan coil by using four pipes: one chilled water supply, one heating hot water supply, one chilled water return, and one heating hot water return. Outdoor ventilation air is conditioned at a make-up AHU or primary AHU. It is then supplied in a neutral thermal state to the fan coil where it mixes with the recirculating air, as shown in Figure 9.15.1(a), or is supplied to the conditioned space directly. A fan-coil unit or a fan coil is a terminal as shown in Figure 9.15.1(b). Fan-coil units are available in standard sizes 02, 03, 04, 06, 08, 10, and 12 which correspond to 200 cfm, 400 cfm, and so on in volume flow. The following are system characteristics of a four-pipe fan-coil system: A water-cooling electric heating fan-coil system uses chilled water for cooling and an electric heater for heating as shown in Figure 9.1.2. This system is often used in a location that has a mild winter. Air system: Fan-coil, space air recirculating Fan, motor, and drive combined efficiency: 25% Fan speed: HI-LO 2-speed and HI-MED-LO 3-speed External pressure for fan coil: 0.06 to 0.2 in. WG System fan(s) energy use: 0.45 to 0.5 W/cfm No return air and return air duct Outdoor ventilation air system: type I An exhaust system to exhaust part of the outdoor ventilation air © 2005 by CRC Press LLC

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Chapter 9

Make-up air AHU O.A.

Exhaust system

Fan coil

Conditioned space

(a) HW supply

HW return

Mixing plenum

o

Centrifugal fan Outdoor ventilation air CW supply

of Drain pan

r

fc

occ

m

os

CW Filter return Recirculating air

r o

DDC controller

(b)

s

m oh

o = Outdoor air occ = Outdoor air having cooling coil of = Outdoor air at fan outlet os = Outdoor air supply

r m s oh

of os

= Space air = Mixture = Supply air = Outdoor air leaving heating coil

(c)

FIGURE 9.15.1 A four-pipe fan-coil system: (a) schematic diagram, (b) fan-coil unit, and (c) air-conditioning cycle.

Cooling system: chilled water from centrifugal or screw chiller Water-cooled chiller energy use: 0.4 to 0.65 kW/ton Heating system: hot water from boiler, electric heater Part load: control the flow rate of chilled and hot water supplied to the coil. Since air leaving coil temperature Tcc rises during summer mode part load, space relative humidity will be higher. Sound level: indoor NC 40 to 45 Maintenance: higher maintenance cost System fan(s) energy use: 0.45 to 0.55 W/cfm (includes all fans in the four-pipe fan-coil system) An air-conditioning cycle for a four-pipe fan-coil system with outdoor ventilation air delivered to the suction side of the fan coil is shown in Figure 9.15.1(c). A part of the space cooling and dehumidifying load is usually taken care by the conditioned outdoor ventilation air from the make-up AHU. A double-bundle © 2005 by CRC Press LLC

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Air-Conditioning and Refrigeration 

( 
 


 



 
 














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FIGURE 9.15.2 A water-source heat pump system: (a) vertical system and (b) system schematic diagram.

condenser is often adopted in a centrifugal chiller to incorporate heat recovery for providing winter heating. Water-Source Heat Pump Systems Water-source heat pumps (WSHPs) are the primary equipment in a water-source heat pump system as shown in Figure 9.15.2(a). A water-source heat pump usually consists of an air coil to cool and heat the air; a water coil to reject and extract heat from the condenser water; a forward-curved centrifugal fan; reciprocating, rotary, or scroll compressor(s); a short capillary tube; a reversing valve; controls; and an outer casing. WSHPs could be either a horizontal or vertical unit. WSHPs usually have cooling capacities between 1/2 to 26 tons. Small-capacity WSHPs of 3 tons or less without ducts are used in perimeter zones, whereas large-capacity WSHPs with ductwork are used only in interior zones. In addition to the WSHPs, a WSHP system usually is also comprised of an evaporative cooler or cooling tower to cool the condenser water; a boiler to provide the supplementary heat for the condenser water if necessary; two circulating pumps, one of them being standby; and controls, as shown in Figure 9.15.2(b). A separate outdoor ventilation air system is required to supply outdoor air to the WSHP or directly to the space. During hot humid weather, when the outdoor wet bulb can reach 78°F, all the WSHPs are operated in the cooling mode. Condenser water leaves the evaporative cooler at a temperature typically 92°F and absorbs condensing heat rejected from the condensers — the water coils in WSHPs. Condenser water is then raised to 104°F and enters the evaporative cooler. In an evaporative cooler, condenser water is evaporatively cooled indirectly by atmospheric air, so that it would not foul the inner surface of water coils in WSHPs. During moderate weather, the WSHPs serving the shady side of a building may be in heating mode, and while serving the sunny side of the building and the interior space in cooling mode. During cold weather, most of the WSHPs serving perimeter zones are in heating mode, while serving interior spaces are in cooling mode except morning warm-up. Cooling WSHPs reject heat to the condenser water loop; meanwhile heating WSHPs absorb heat from the condenser water loop. The condenser water is usually maintained at 60 to 90°F. If its temperature rises above 90°F, the evaporative cooler is energized. If it drops below 60°F, the boiler or electric heater is energized. A WSHP system itself is a combination of © 2005 by CRC Press LLC

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WSHP and a heat recovery system to transfer the heat from the interior space and sunny side of the building to the perimeter zone and shady side of building for heating in winter, spring, and fall. System characteristics of air, cooling, and heating in a WSHP system are similar to a room conditioner heat pump system. In addition: Outdoor ventilating air system: type I and IV Water system: two-pipe, close circuit Centrifugal water circulating pump Water storage tank is optional To prevent freezing in locations where outdoor temperature may drop below 32°F, isolate the outdoor portion of the water loop, outdoor evaporative cooler, and the pipe work from the indoor portion by using a plate-and-frame heat exchanger. Add ethylene or propylene glycol to the outdoor water loop for freezing protection. There is another space system called a panel heating and cooling system. Because of its higher installation cost and dehumidification must be performed in the separate ventilation systems, its applications are very limited. A space conditioning system has the benefit of a separate demand-based outdoor ventilation air system. A WSHP system incorporates heat recovery automatically. However, its indoor sound level is higher; only a low-efficiency air filter is used for recirculating air, and more space maintenance is required than central and packaged systems. Because of the increase of the minimum outdoor ventilation air rate, it may gain more applications in the future.

Packaged Systems In packaged systems, air is cooled directly by a DX coil and heated by direct-fired gas furnace or electric heater in a packaged unit (PU) instead of chilled and hot water from a central plant in a central system. Packaged systems are different from space conditioning systems since variable-air-volume supply and air economizer could be features in a packaged system. Packaged systems are often used to serve two or more rooms with supply and return ducts instead of serving individual rooms only in an individual system. As mentioned in Section 9.7, packaged units are divided according to their locations into rooftop, split, or indoor units. Based on their operating characteristics, packaged systems can be subdivided into the following systems: Single-Zone Constant-Volume (CV) Packaged Systems Although a single-zone CV packaged system may have duct supplies to and returns from two or more rooms, there is only a single zone sensor located in the representative room or space. A CV system has a constant supply volume flow rate during operation except the undesirable reduction of volume flow due to the increase of pressure drop across the filter. A single-zone CV packaged system consists mainly of a centrifugal fan, a DX coil, a direct-fired gas furnace or an electric heater, a low or medium efficiency filter, mixing box, dampers, DDC controls, and an outer casing. A relief or a return fan is equipped for larger systems. A single-zone CV packaged system serving a church is shown in Figure 9.1.3. This system operates on basic air-conditioning cycles as shown in Figure 9.3.4 during cooling and heating modes. The system characteristics of a single-zone CV packaged system are Air system: single supply fan, a relief or return fan for a large system Fan, motor, and drive combined efficiency: 40 to 45% Fan total pressure: 1.5 to 3 in. WG Fan(s) energy use: 0.45 to 0.8 W/cfm Outdoor ventilation air system: type IV and II Enthalpy or temperature air economizer © 2005 by CRC Press LLC

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Cooling systems: DX system, air cooled Compressor: reciprocating or scroll EER: 8.9 to 10.0 Btu/hr.W Heating system: direct-fired gas furnace, air-source heat pump, or electric heating Part load: on–off or step control of the compressor capacity, DX-coil effective area, and the gas flow to the burner Sound level: indoor NC 35 to 45 Maintenance: higher maintenance cost than central systems Single-zone, CV packaged systems are widely used in residences, small retail stores, and other commercial buildings. Constant-Volume Zone-Reheat Packaged Systems System construction and system characteristics of a CV zone-reheat system are similar to the single-zone CV packaged systems except: 1. It serves multizones and has a sensor and a DDC controller for each zone. 2. There is a reheating coil or electric heater in the branch duct for each zone. A CV zone-reheat packaged system cools and heats simultaneously and therefore wastes energy. It is usually used for the manufacturing process and space needs control of temperature and humidity simultaneously. Variable-Air-Volume Packaged Systems A variable-air-volume (VAV) system varies its volume flow rate to match the reduction of space load at part load. A VAV packaged system, also called a VAV cooling packaged system, is a multizone system and uses a VAV box in each zone to control the zone temperature at part load during summer cooling mode operation, as shown in Figure 9.15.3(a). A VAV box is a terminal in which the supply volume flow rate of the conditioned supply air is modulated by varying the opening of the air passage by means of a single blade damper, as shown in Figure 9.15.3(b), or a moving disc against a cone-shaped casing. The following are the system characteristics of a VAV packaged system: Single-zone VAV packaged system which serves a single zone without VAV boxes. A DDC controller modulates the position of the inlet vanes or the fan speed according to the signal of the space temperature sensor. Air system: a supply/relief fan or supply/return fan combination. Space pressurization control by a relief/return fan Fan, motor, and drive combined efficiency: 45% Supply fan total pressure: 3.75 to 4.5 in. WG Fan(s) energy use at design condition: 1 to 1.25 W/cfm VAV box minimum setting: 30% of peak supply volume flow Outdoor ventilation air system: type II and III Economizer: enthalpy air economizer or water economizer Cooling system: DX coil, air-, water-, or evaporative-cooled condenser Compressor: reciprocating, scroll, and screw EER: 8.9 to 12 Btu/hr.W Capacity: 20 to 100 tons Part load: zone volume flow modulation by VAV box; step control of compressor capacity; modulation of gas flow to burner; and discharge air temperature reset Smoke control: exhausts smoke on the fire floor, and supplies air and pressurizes the floors immediately above or below the fire floor Diagnostics: a diagnostic module displays the status and readings of various switches, dampers, sensors, etc. and the operative problems by means of expert system © 2005 by CRC Press LLC

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Packaged unit

1

1

2 Supply fan

Condensing unit

3

4

12

DX coil

5

6

6

Relief fan

5

EO 4

Supply duct

O.A. E.A.

ER

Return duct

3

11

Filter

TD

PD

13

12

11

2

4

TR3

TR2

TR1

CO2

PR

Perimeter zone

Interior zone

(a) 2 30°

Damper

Air flow

Damper closed

Fully open

Reheating coil

8 in.

Actuator Linkages Stroke

1

T 1

(b)

TD = Discharge air temperature ER = Return air enthalpy EO = Outdoor air enthalpy CO2 = Carbon dioxide TR = Space temperature PD = Duct static pressure PR = Space pressure Multiple T = Temperature outlets T

2

1

1

(c)

FIGURE 9.15.3 A variable-air-volume (VAV) package system: (a) schematic diagram, (b) VAV box, (c) reheating box, (d) parallel-flow fan-powered VAV box.

Maintenance: higher than central system Sound level: indoor NC 30 to 45 Heating system characteristics as well as the air-conditioning cycles are similar as that in a single-zone CV packaged system. VAV Reheat Packaged Systems A VAV reheat packaged system has its system construction and characteristics similar to that in a VAV packaged system except in each VAV box there is an additional reheating coil. Such a VAV box is called a reheating VAV box, as shown in Figure 9.15.2(a) and 9.15.3(c). VAV reheat packaged systems are used to serve perimeter zones where winter heating is required. © 2005 by CRC Press LLC

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Air-Conditioning and Refrigeration

4 Backdraft damper

Recirculating plenum air

Heating coil

Cold primary air

Supply air

2

1

3

2

1

(d)

FIGURE 9.15.3d

Fan-Powered VAV Packaged Systems A fan-powered VAV packaged system is similar to that of a VAV packaged system except fan-powered VAV boxes as shown in Figure 9.15.3(d) are used instead of VAV boxes. There are two types of fan-powered VAV boxes: parallel-flow and series-flow boxes. In a parallel-flow fan-powered box, the plenum air flow induced by the fan is parallel with the cold primary air flow through the VAV box. These two air streams are then combined and mixed together. In a series-flow box, cold primarily from the VAV box is mixed with the induced plenum air and then flows through the small fan. The parallel-flow fan-powered VAV box is more widely used. In a fan-powered VAV box, volume flow dropping to minimum setting, extracting of ceiling plenum air, and energizing of reheating coil will actuate in sequence to maintain the space temperature during part-load/heating mode operation. A fan-powered VAV box can also mix the cold primary air from cold air distribution with the ceiling plenum air and provides greater space air movements during minimum space load. Packaged systems are lower in installation cost and occupy less space than central systems. During the past two decades, DDC-controlled packaged systems have evolved into sophisticated equipment and provide many features that only a built-up central system could provide before.

Central Systems Central systems use chilled and hot water that comes from a central plant to cool and heat the air in the air-handling units (AHUs). Central systems are built-up systems. The most clean, most quiet thermalstorage systems, and the systems which offer the most sophisticated features, are always central systems. Central systems can be subdivided into the following. Single-Zone Constant-Volume Central Systems A single-zone CV central system uses a single controller to control the flow of chilled water, hot water, or the opening of dampers to maintain a predetermined indoor temperature, relative humidity, or air contaminants. They are often used in manufacturing factories. The system characteristics of a singlezone CV central system are Single-zone CV air washer central system uses an air washer to control both space relative humidity and temperature. This system is widely used in textile mills. The reason to use constant volume is to dilute the fiber dusts produced during manufacturing. A rotary filter is often used for high dust-collecting capacity.

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Air system: supply and return fan combination Fan, motor, and drive combined efficiency: 45 to 50% Outdoor ventilation air system: type II and IV Economizer: air or water economizer Smoke control: exhaust smoke on the fire floor, and pressurize adjacent floor(s) or area Cooling system: centrifugal or screw chiller, water-cooled condenser Cooling energy use: 0.4 to 0.65 kW/ton Heating system: hot water from boiler or from heat recovery system Part load: modulate the water mass flow to cooling and heating coils in AHUs, and discharge air temperature reset Sound level: indoor NC 30 to 45. Silencers are used both in supply and return air systems if they are required Maintenance: in central plant and fan rooms, lower maintenance cost Single-Zone CV Clean Room Systems This is the central system which controls the air cleanliness, temperature, and relative humidity in Class 1, 10, 100, 1000, and 10,000 clean rooms for electronic, pharmaceutical, and precision manufacturing and other industries. Figure 9.15.4(a) shows a schematic diagram of this system. The recirculating air unit (RAU) uses prefilter, HEPA filters, and a water cooling coil to control the space air cleanliness and required space temperature, whereas a make-up air unit (MAU) supplies conditioned outdoor air, always within narrow dew point limits to the RAU at any outside climate, as shown in Figure 9.15.4(b). A unidirectional air flow of 90 fpm is maintained at the working area. For details, refer to ASHRAE Handbook 1991 HVAC Applications and Wang’s Handbook of Air Conditioning and Refrigeration. CV Zone-Reheat Central Systems These systems have their system construction and characteristics similar to that for a single-zone CV central system, except they serve multizone spaces and there is a reheating coil, hot water, or electric heating in each zone. CV zone-reheat central systems are often used for health care facilities and in industrial applications. VAV Central Systems A VAV central system is used to serve multizone space and is also called VAV cooling central system. Its schematic diagram is similar to that of a VAV packaged system (Figure 9.15.3) except air will be cooled or heated by water cooling or heating coils in the AHUs. The same VAV box shown in Figure 9.15.3(b) will be used in a VAV central system. The system characteristics of VAV central systems are as follows: Single-zone VAV central system differs from a VAV central system only because it serves a single zone, and therefore there is no VAV box in the branch ducts. Supply volume flow is modulated by inlet vanes and AC inverter. Air system: supply and relief/return fan combination Fan, motor, and drive combined efficiency for airfoil centrifugal fan with AC inverter fan speed modulation: 55% Fan(s) energy use: 0.9 to 1.2 W/cfm VAV box: minimum setting 30% of peak supply volume flow Outdoor ventilation air system: type I, II, and III Cooling system: centrifugal, screw, and reciprocating chillers, water-cooled condenser, with energy use 0.4 to 0.65 kW/ton; or sometimes absorption chiller Heating system: hot water from boiler or electric heating at the terminals Economizer: air and water economizer Part load: zone volume flow modulation, discharge air temperature reset, and chilled water temperature reset Smoke control: exhausts smoke from the fire floor and pressurizes the immediate floors above and below © 2005 by CRC Press LLC

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2

12

1

2

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5

Make-up air fan Humidifier

HEPA 99.97% 0.3 µm

Prefilter P

MAU

C ph

H

h

cc

T2

maf

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P

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Sound attenuators C

RAU sc 3

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Recirculating 3 fan

Process exhaust

ULPA filters 99.9997% 0.12 µm 11

P 5

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P1 H1 T1 Class 10 clean room

(a) w, lb/lb r = Space air maf = Make-up air fan m = Mixture h = Humidifier s = Supply air ph = Preheating sc = Sensible cooling coil o = Outdoor cc = Cooling coil 60

60%

90%

70

o 0.016 40%

0.012

20% 50 cc maf

40

h

sc s

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ph maf

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44 46

50

60

70

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90

100

T, °F

(b)

FIGURE 9.15.4 A single-zone CV clean room system: (a) schematic diagram and (b) air-conditioning cycle. (Source: Wang, S. K., Handbook of Air Conditioning and Refrigeration, McGraw-Hill, 1993. Reprinted by permission.)

Sound level: indoor NC 20 to 40. Silencers are often used both in supply and return systems. Maintenance: less space maintenance VAV central systems are widely used for interior zone in buildings. © 2005 by CRC Press LLC

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VAV Reheat Central Systems A VAV reheat system is similar in system construction and characteristics to that in a VAV central system except that reheating boxes are used instead of VAV boxes in a VAV central system. Fan-Powered VAV Central Systems A fan-powered VAV central system is similar in system construction and characteristics to that in a VAV central system except that fan-powered VAV boxes are used instead of VAV boxes. Dual-Duct VAV Central Systems A dual-duct VAV system uses a warm air duct and a cold air duct to supply both warm and cold air to each zone in a multizone space. Warm and cold air are first mixed in a mixing VAV box, and are then supplied to the conditioned space. The warm air duct is only used for perimeter zones. A mixing VAV box consists of two equal air passages, one for warm air and one for cold air, arranged in parallel. Each of them has a single blade damper and its volume flow rate is modulated. Warm and cold air are then combined, mixed, and supplied to the space. A dual-duct VAV system is usually either a single supply fan and a relief/return fan combination, or a warm air supply fan, a cold air supply fan, and a relief/return fan. A separate warm air fan and cold air supply fan are beneficial in discharge air temperature reset and fan energy use. During summer cooling mode operation, the mixture of recirculating air and outdoor air is used as the warm air supply. The heating coil is not energized. During winter heating mode operation, the mixture of outdoor and recirculating air or 100% outdoor air is used as the cold air supply; the cooling coil is not energized. Because there is often air leakage at the dampers in the mixing VAV box, more cold air supply is needed to compensate for the leaked warm air or leaked cold air. Other system characteristics of a dual-duct VAV central system are similar to a VAV central system. Dual-Duct CV Central System This is another version of a dual-duct VAV central system and is similar in construction to a dual-duct VAV system, except that a mixing box is used instead of a mixing VAV box. The supply volume flow rate from a mixing box is nearly constant. Dual-duct CV central systems have only limited applications, like health care facilities, etc. Some of the air-conditioning systems are not listed because they are not effective or are a waste of energy, and therefore rarely used in new and retrofit projects such as: • High-velocity induction space conditioning systems which need a higher pressure drop primary air to induce recirculating air in the induction unit and use more energy than fan-coil systems • Multizone central systems which mix warm and cool air at the fan room and use a supply duct from fan room to each control zone • Air skin central systems which use a warm air heating system to offset transmission loss in the perimeter zone and overlook the effect of the solar radiation from variation building orientations In the future, there will be newly developed systems added to this classification list.

Air-Conditioning System Selection As described in Section 9.1, the goal of an air-conditioning or HVAC&R system is to provide a healthy, comfortable, manufacturable indoor environment at acceptable indoor air quality, keeping the system energy efficient. Various occupancies have their own requirements for their indoor environment. The basic considerations to select an air-conditioning system include: 1. The selection of an air-conditioning system must satisfy the required space temperature, relative humidity, air cleanliness, sound level, and pressurization. For a Class 100 clean room, a singlezone CV clean room system is always selected. A four-pipe fan-coil space conditioning system is © 2005 by CRC Press LLC

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4. 5.

usually considered suitable for guest rooms in hotels for operative convenience, better privacy, and a guaranteed outdoor ventilation air system. A concert hall needs a very quiet single-zone VAV central system for its main hall and balcony. The size of the project has a considerable influence on the selection. For a small-size residential air-conditioning system, a single-zone constant-volume packaged system is often the first choice. Energy-efficient measures are specified by local codes. Comparison of alternatives by annual energy-use computer programs for medium and large projects is often necessary. Selection of energy source includes electricity or gas, and also using electrical energy at off-peak hours, like thermal storage systems is important to achieve minimum energy cost. For a building whose sound level requirement is not critical and conditioned space is comprised of both perimeter and interior zones, a WSHP system incorporating heat recovery is especially suitable for energy saving. First cost or investment is another critical factor that often determines the selection. Selection of an air-conditioning system is the result of synthetical assessment. It is difficult to combine the effect of comfort, reliability, safety, and cost. Experience and detailed computer program comparisons are both important.

The selection procedure usually begins whether an individual, space conditioning, packaged, central system, or CV, VAV, VAV reheat, fan-powered VAV, dual-duct VAV, or thermal storage system is selected. Then the air, refrigeration, heating, and control subsystems will be determined. After that, choose the option, the feature, the construction, etc. in each subsystem.

Comparison of Various Systems The sequential order of system performance — excellent, very good, good, satisfactory — regarding temperature and relative humidity control (T&HC), outdoor ventilation air (OA), sound level, energy use, first cost, and maintenance for individual, space conditioning (SC), packaged, and central systems is as follows: Excellent (low or less) T&HC IAQ Sound Energy use First cost Maintenance

Central Space Central Individual Individual Central

Very Good Packaged Central Packaged Space Packaged Packaged

Good

Satisfactory

Space Packaged Space Packaged Space Space

Individual Individual Individual Central Central Individual

Among the packaged and central systems, VAV cooling systems are used only for interior zones. VAV reheat, fan-powered VAV, and dual-duct VAV central systems are all for perimeter zones. VAV reheat systems are simple and effective, but have a certain degree of simultaneous cooling and heating when their volume flow has been reduced to minimum setting required for proper ventilation. Fan-powered VAV systems have the function of mixing cold primary air with ceiling plenum air. They are widely used in ice-storage systems with cold air distribution. Fan-powered VAV is also helpful to create a greater air movement at minimum cold primary air flow. Dual-duct VAV systems are effective and more flexible in operation. They are also more complicated and expensive.

Subsystems Air Systems The economical size of an air system is often 10,000 to 25,000 cfm. A very large air system always has higher duct pressure loss and is more difficult to balance. For highrise buildings of four stories and higher, © 2005 by CRC Press LLC

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floor-by-floor AHU(s) or PU(s) (one or more AHU or PU per floor) are often adopted. Such an arrangement is beneficial for the balance of supply and return volume flow in VAV systems and also for fire protection. A fan-powered VAV system using a riser to supply less cold primary to the fan-powered VAV box at various floors may have a larger air system. Its risers can be used as supply and exhaust ducts for a smoke-control system during a building fire. In air systems, constant-volume systems are widely used in small systems or to dilute air contaminants in health care facilities and manufacturing applications. VAV systems save fan energy and have better operating characteristics. They are widely used in commercial buildings and in many factories. Refrigeration Systems For comfort air-conditioning systems, the amounts of required cooling capacity and energy saving are dominant factors in the selection of the refrigeration system. For packaged systems having cooling capacity less than 100 tons, reciprocating and scroll vapor compression systems with air-cooled condensers are most widely used. Evaporative-cooled condensers are available in many packaged units manufactured for their lower energy use. Scroll compressors are gradually replacing the reciprocating compressors for their simple construction and energy saving. For chillers of cooling capacity of 100 tons and greater, centrifugal chillers are still most widely used for effective operation, reliability, and energy efficiency. Screw chillers have become more popular in many applications, especially for ice-storage systems. Heating Systems For locations where there is a cold and long winter, a perimeter baseboard hot water heating system or dual-duct VAV systems are often a suitable choice. For perimeter zones in locations where winter is mild, winter heating is often provided by using warm air supply from AHU or PU from terminals with electric or hot water heaters. Direct-fired furnace warm air supply may be used for morning warm-up. For interior or conditioned zones, a cold air supply during occupied periods in winter and a warm air supply from the PUs or AHUs during morning warm-up period is often used. Control Systems Today, DDC microprocessor-based control with open data communication protocol is often the choice for medium- and large-size HVAC&R projects. For each of the air, cooling, and heating systems, carefully select the required generic and specific control systems. If a simple control system and a more complicated control system can provide the same required results, the simple one is always the choice.

Energy Conservation Recommendations 1. Turn off electric lights, personal computers, and office appliances when they are not needed. Shut down AHUs, PUs, fan coils, VAV boxes, compressors, fans, and pumps when the space or zone they serve is not occupied or working. 2. Provide optimum start and stop for the AHUs and PUs and terminals daily. 3. Temperature set point should be at its optimum value. For comfort systems, provide a dead band between summer and winter mode operation. Temperature of discharged air from the AHU or PU and chilled water leaving the chiller should be reset according to space or outdoor temperature or the system load. 4. Reduce air leakages from ducts and dampers. Reduce the number of duct fittings and pipe fittings and their pressure loss along the design path if this does not affect the effectiveness of the duct system. The maximum design velocity in ducts for comfort systems should not exceed 3000 fpm. 5. Adopt first the energy-efficient cooling methods: air and water economizer, evaporative cooler, or ground water instead of refrigeration. 6. Use cost-effective high-efficiency compressors, fans, pumps, and motors as well as evaporativecooled condensers in PUs. Use adjustable-frequency fan speed modulation for large centrifugal fans. Equipment should be properly sized. Over-sized equipment will not be energy efficient.

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7. Use heat recovery systems and waste heat for winter heating or reheating. Use a heat-pump system whenever its COPhp is greater than 1. 8. For medium- and large-size air-conditioning systems, use VAV systems instead of CV systems except for health care or applications where dilution of air contaminant is needed. Use variable flow for building-loop and distribution-loop water systems. 9. Use double- and triple-pane windows with low emissive coatings. Construct low U-value roofs and external walls.

9.16 Desiccant Dehumidification and Air-Conditioning Zalman Lavan Introduction Desiccant air-conditioning is a promising emerging technology to supplement electrically driven vapor compression systems that rely almost exclusively on R22 refrigerant that causes depletion of the ozone layer. To date, this technology has only a limited market, e.g., in supermarkets where the latent heat loads are very high, in specialized manufacturing facilities that require very dry air, and in hospitals where maximum clean air is required. However, recent emphasis on increased air change requirements (see ASHRAE standards, ANSI 62), improved indoor air quality, and restriction on use of CFC refrigerants (see The Montreal Protocol Agreement, as amended in Copenhagen in 1992, United Nations Environmental Programme, 1992) may stimulate wider penetration of desiccant-based air-conditioning which can be used as stand-alone systems or in combination with conventional systems. (See Table 9.4.1 for properties of some refrigerants.)

Sorbents and Desiccants Sorbents are materials which attract and hold certain vapor or liquid substances. The process is referred to absorption if a chemical change takes place and as adsorption if no chemical change occurs. Desiccants, in both liquid and solid forms, are a subset of sorbents that have a high affinity to water molecules. Liquid desiccants absorb water molecules, while solid desiccants adsorb water molecules and hold them on their vast surfaces (specific surface areas are typically hundreds of square meters per gram). While desiccants can sorb water in both liquid and vapor forms, the present discussion is limited to sorption of water vapor from adjacent air streams. The sorption driving force for both liquid and solid desiccants is a vapor pressure gradient. Adsorption (in solid desiccants) and absorption (in liquid desiccants) occur when the water vapor partial pressure of the surrounding air is larger than that at the desiccant surface. When an air stream is brought in contact with a desiccant, water vapor from the air is attracted by the desiccant, the air is dehumidified, and the water content of the desiccant rises. As the water sorbed by the desiccant increases, the sorption rate decreases and finally stops when sorption equilibrium is reached. For dehumidification to be resumed, water must be removed from the desiccant by heating. This process is referred to as desorption, reactivation, or regeneration. The heat of sorption (or desorption) is generally higher than the latent heat of vaporization of water; it approaches the latter as sorption equilibrium is reached. Some typical liquid desiccants are water solutions of calcium chloride (CaCl), lithium chloride (LiCl), lithium bromide (LiBr), and triethylene glycol. The equilibrium water vapor pressure at the solution surface as a function of temperature and water content is shown in Figure 9.16.1 for water-lithium chloride solution. The surface vapor pressure (and dew point) increases with increasing solution temperature and decreases with increasing moisture content. Common solid desiccants are silica gel, molecular sieves (zeolites), activated alumina, and activated carbon. The equilibrium sorption capacity (or moisture content) at a constant temperature, referred to © 2005 by CRC Press LLC

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FIGURE 9.16.1 Surface vapor pressure of water-lithium chloride solutions. (Source: ASHRAE 1993, Fundamentals Handbook, chap. 19. With permission.)

as an isotherm, is usually presented as percent water (mass of water divided by mass of dry desiccant) vs. percent relative humidity (vapor pressure divided by saturation vapor pressure). Sorption capacity decreases with increasing temperature, but the spread of isotherms is relatively small (especially for concave down isotherms). Figure 9.16.2 shows normalized loading (sorption capacity divided by sorption capacity at 100% relative humidity) vs. relative humidity for silica gel, molecular sieve, and a generic desiccant, type 1 (modified) or simply 1-M (Collier et al., 1986).

Dehumidification Dehumidification by vapor compression systems is accomplished by cooling the air below the dew point and then reheating it. The performance is greatly hindered when the desired outlet dew point is below 40°F due to frost formation on the cooling coils (ASHRAE, Systems and Equipment Handbook, 1992). Desiccant dehumidification is accomplished by direct exchange of water vapor between an air stream and a desiccant material due to water vapor pressure difference. Figure 9.16.3 shows the cyclic operation of a desiccant dehumidification system. In sorption (1–2), dry and cold desiccant (point 1) sorbs moisture since the vapor pressure at the surface is lower than that of the air stream. During this process the moisture content (loading or uptake) increases, the surface vapor pressure increases, and the liberated heat of sorption raises the desiccant temperature. During desorption (2–3), the desiccant is subjected to a hot air stream, and moisture is removed and transferred to the surrounding air. The surface vapor pressure is increased and the desiccant temperature rises due to the added heat. The cycle is closed by cooling (3–1). The desiccant is cooled while its moisture content is constant and the surface vapor pressure is lowered. The above cycle of sorption, desorption, and cooling can be modified by combining the sorption process with cooling to approach isothermal rather than adiabatic sorption. © 2005 by CRC Press LLC

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FIGURE 9.16.2 Normalized solid desiccant isotherms.

FIGURE 9.16.3 Cyclic dehumidification processes.

Desirable Characteristics for High-Performance Liquid and Solid Desiccant Dehumidifiers High equilibrium moisture sorption capacity High heat and mass transfer rates Low heat input for regeneration Low pressure drop © 2005 by CRC Press LLC

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Large contact transfer surface area per unit volume Compatible desiccant/contact materials Inexpensive materials and manufacturing techniques Minimum deterioration and maintenance Additional Requirements for Liquid Desiccant Dehumidifiers Small liquid side resistance to moisture diffusion Minimum crystallization Additional Requirements for Solid Desiccant Dehumidifiers The desiccant should not deliquesce even at 100% relative humidity. The airflow channels should be uniform. The desiccant should be bonded well to the matrix. The material should not be carciogenic or combustible.

Liquid Spray Tower Figure 9.16.4 is a schematic of a liquid spray tower. A desiccant solution from the sump is continuously sprayed downward in the absorber, while air, the process stream, moves upward. The air is dehumidified and the desiccant solution absorbs moisture and is weakened. In order to maintain the desired solution concentration, a fraction of the solution from the sump is passed through the regenerator, where it is heated by the heating coil and gives up moisture to the desorbing air stream. The strong, concentrated solution is then returned to the sump. The heat liberated in the absorber during dehumidification is removed by the cooling coil to facilitate continuous absorption (see Figure 9.16.1 and Figure 9.16.3). The process air stream exits at a relatively low temperature. If sufficiently low water temperature is available (an underground well, for example), the process stream could provide both sensible and latent cooling. The heating and cooling coils, shown in Figure 9.16.4, are often eliminated and the liquid solutions are passed through heating and cooling heat exchangers before entering the spray towers. Advantages The system is controlled to deliver the desired level of dry air by adjusting the solution concentration. Uniform exit process stream conditions can be maintained. A concentrated solution can be economically stored for subsequent drying use. The system can serve as a humidifier when required by simply weakening the solution. When used in conjunction with conventional A/C systems, humidity control is improved and energy is conserved.

FIGURE 9.16.4 Liquid desiccant dehumidifier with heating and cooling coils. © 2005 by CRC Press LLC

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Disadvantages Some desiccants are corrosive. Response time is relatively large. Maintenance can be extensive. Crystallization may be a problem.

Solid Packed Tower The dehumidification system, shown in Figure 9.16.5, consists of two side-by-side cylindrical containers filled with solid desiccant and a heat exchanger acting as a desiccant cooler. The air stream to be processed is passed through dry desiccant in one of the containers, while a heated air stream is passed over the moist desiccant in the other. Adsorption (1–2) takes place in the first container, desorption (2–3) in the other container, and cooling (3–1) occurs in the desiccant cooler. The function of the two containers is periodically switched by redirecting the two air streams. Advantages No corrosion or crystallization Low maintenance Very low dew point can be achieved Disadvantages The air flow velocity must be low in order to maintain uniform velocity through the containers and to avoid dusting. Uniform exit process stream dew point cannot be maintained due to changing moisture content in the adsorbing desiccant.

FIGURE 9.16.5 Solid packed tower dehumidification. (From Harriman, L. G., III. 1990. The Dehumidification Handbook, 2nd ed. Munters Cargocaire. With permission.) © 2005 by CRC Press LLC

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FIGURE 9.16.6 Rotary desiccant dehumidification wheel. (Source: ASHRAE 1992, Systems and Equipment Handbook, chap. 22. With permission.)

Rotary Desiccant Dehumidifiers A typical rotary solid desiccant dehumidifier is shown in Figure 9.16.6. Unlike the intermittent operation of packed towers, rotary desiccant dehumidifiers use a wheel (or drum) that rotates continuously and delivers air at constant humidity levels. Desiccant wheels typically consist of very fine desiccant particles dispersed and impregnated with a fibrous or ceramic medium shaped like a honeycomb or fluted corrugated paper. The wheel is divided into two segments. The process stream flows through the channels in one segment, while the regenerating (or reactivating) stream flows through the other segment. Desiccant Material The desired desiccant properties for optimum dehumidification performance are a suitable isotherm shape and a large moisture sorption capacity. The isotherms of silica gel are almost linear. The moisture sorption capacity is high; the desiccant is reactivated at relatively low temperatures and is suitable for moderate dehumidification. Molecular sieves have very steep isotherms at low relative humidity. The desiccant is reactivated at relatively high temperatures and is used for deep dehumidification. The isotherm of the type 1-M yields optimum dehumidification performance (Collier et al., 1986), especially when used in conjunction with high regeneration temperatures. The Desiccant Wheel Some considerations for selection of desiccant wheels are: Appropriate desiccant materials Large desiccant content Wheel depth and flute size (for large contact surface area and low pressure drop) Size and cost The actual performance depends on several additional factors that must be addressed. These include: Inlet process air temperature and humidity Desired exit process air humidity Inlet reactivating air temperature and humidity © 2005 by CRC Press LLC

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Face velocity of the two air streams Size of reactivation segment It should be noted that: Higher inlet process air humidity results in higher exit humidity and temperature (more heat of sorption is released). Lower face velocity of the process stream results in lower exit humidity and higher temperature. Higher regeneration temperatures result in deeper drying, hence lower exit process air humidity and higher temperature. When lower exit air temperature is required, the exit process air should be cooled by a heat exchanger. Final cooling of the exit process air can be achieved by partial humidification (this counteracts in part previous dehumidification). The following is a range of typical parameters for rotary desiccant wheels: Rotation speed: 4 to 10 rpm Desiccant fraction: 70 to 80% Flute size: 1 to 2 mm Reactivation segment: 25 to 30% of wheel Face velocity: 300 to 700 fpm Reactivating temperature: 100 to 300°F

Hybrid Cycles A limited number of hybrid systems consisting of desiccant dehumidifiers and electrically driven vapor compression air-conditioners are presently in use in supermarkets. This application is uniquely suited for this purpose since the latent heat loads are high due to the large number of people and frequent traffic through doors. Also, low relative humidity air is advantageous for open-case displays. Vapor compression systems are inefficient below a dew point of 45 to 50°F. When used in supermarkets, they require high airflow rates, the air must be reheated for comfort, and the evaporator coils must be defrosted frequently. Hybrid systems offer improved performance and lower energy cost in these cases. Figure 9.16.7 shows a typical hybrid air-conditioning system for supermarkets. A mixture of outdoor and recirculated air is first passed through the desiccant and sensible heat exchanger wheels, where it is dehumidified and precooled. It then enters the conventional chiller before it is introduced to the interior of the supermarket. The sensible heat exchanger wheel is cooled by outdoor air and the desiccant wheel is regenerated by air heated with natural gas. Energy cost can be further reduced by preheating the reactivating air stream with waste heat rejected from the condenser of the refrigeration and/or airconditioning systems. The advantages of these hybrid systems are Air-conditioning requirement is reduced by up to 20%. The vapor compression system operates at a higher coefficient of performance (COP) since the evaporator coils are at a higher temperature. Airflow requirements are reduced; electric fan energy is saved and duct sizes are reduced. The refrigeration cases run more efficiently since the frequency of defrost cycles is greatly reduced.

Solid Desiccant Air-Conditioning Several stand-alone desiccant air-conditioning systems were suggested and extensively studied. These systems consist of a desiccant wheel, a sensible heat exchanger wheel, and evaporating pads. Sorption can be adiabatic or cooled (if cooling is combined with sorption). When room air is dehumidified and recirculated, the system is said to operate in the recirculating mode. When 100% outside air is used as the process stream, the system operates in the ventilating mode. © 2005 by CRC Press LLC

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FIGURE 9.16.7 Hybrid air-conditioning system for supermarkets.

Ventilation Mode In the adsorption path the process air stream drawn from the outdoors is passed through the dry section of the desiccant wheel where it is dehumidified and heated by the liberated heat of sorption. It then passes through the sensible heat exchanger wheel and exits as dry but slightly warm air. The hot and dry air leaving the dehumidifier enters the heat exchanger, where it is sensibly cooled down to near room temperature. It is then passed through the evaporative cooler, where it is further cooled and slightly humidified as it enters the conditioned space. In the desorption path, air is drawn from the conditioned space; it is humidified (and thus cooled) in the evaporative cooler. The air stream enters the sensible heat exchanger, where it is preheated, and it is then heated to the desired regeneration temperature by a suitable heat source (natural gas, waste heat, or solar energy), passed through the desiccant wheel (regenerating the desiccant material), and discharged out of doors. Performance. In order to achieve high performance, the maximum moisture content of the desiccant should be high and the isotherm should have the optimum shape (1 M). In addition, Zheng et al. (1993) showed that the optimum performance is very sensitive to the rotational speed of the desiccant wheel. Glav (1966) introduced stage regeneration. He showed that performance is improved when the reactivation segment of the wheel is at a temperature which increases in the direction of rotation. Collier (Collier et al., 1986) showed that well-designed open-cycle desiccant cooling systems can have a thermal COP of 1.3. This, however, would require the use of high-effectiveness sensible heat exchangers, which would be large and expensive. Smaller and more affordable heat exchangers should yield system COPs in the order of unity. An extensive review of the state-of-the-art assessment of desiccant cooling is given by Pesaran et al. (1992).

Conclusions Desiccant-based air-conditioning offers significant advantages over conventional systems. Desiccant systems are already successfully used in some supermarkets. It is expected that these systems will gradually attain wider market penetration due to environmental requirements and potential energy savings. The advantages of desiccant air-conditioning are summarized below: No CFC refrigerants are used. Indoor air quality is improved. Large latent heat loads and dry air requirements are conveniently handled. Individual control of temperature and humidity is possible. The energy source may be natural gas and/or waste heat. Less circulated air is required. Summer electric peak is reduced. © 2005 by CRC Press LLC

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Defining Terms Absorb, absorption: When a chemical change takes place during sorption. Adsorb, adsorption: When no chemical change occurs during sorption. Dehumidification: Process of removing water vapor from air. Desiccant: A subset of sorbents that has a particular affinity to water. Desorb, desorption: Process of removing the sorbed material from the sorbent. Isotherm: Sorbed material vs. relative humidity at a constant temperature. Reactivation: Process of removing the sorbed material from the sorbent. Recirculation: Indoor air only is continuously processed. Regeneration: Process of removing the sorbed material from the sorbent. Sorbent: A material that attracts and holds other gases or liquids. Sorption: Binding of one substance to another. Staged regeneration: When the temperature of the regeneration segment of the desiccant wheel is not uniform. Ventilation mode: 100% of outdoor air is processed.

References AMCA. 1973. Fan and Systems Publication 201. AMCA, Arlington Heights, IL. Amistadi, H. 1993. Design and drawing software review, Eng. Syst. 6:18–29. ANSI/ASHRAE. 1992. ANSI/ASHRAE Standard 34-1992, Numbering Designation and Safety Classification of Refrigerants. ASHRAE, Atlanta, GA. ASHRAE. 1989. Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta. ASHRAE. 1991. ASHRAE Handbook, HVAC Applications. ASHRAE, Atlanta, GA. ASHRAE. 1992. ASHRAE Handbook, HVAC Systems and Equipment. ASHRAE, Atlanta, GA. ASHRAE. 1994. ASHRAE Handbook, Refrigeration. ASHRAE, Atlanta, GA. ASHRAE. 1997. Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta. ASHRAE. 2001. ASHRAE Handbook, Fundamentals. ASHRAE, Atlanta, GA. Bayer, C.W. and Black, M.S. 1988. IAQ evaluations of three office buildings. ASHRAE J. 7:48–52. Birdsall, B., W.F. Buhl, K.L. Ellington, A.E. Erdem, and F.C. Winkelmann. 1990. Overview of the DOE2.1 building energy analysis program. Report LBL-19735, rev. 1. Lawrence Berkeley Laboratory, Berkeley, CA. Bushby, S.T. and Newman, H.M. 1994. BACnet: a technical update, ASHRAE J. 1:S72–84. Carlson, G.F. 1968. Hydronic systems: analysis and evaluation, I. ASHRAE J. 10:2–11. Collier, R.K. 1989. Desiccant properties and their effect on cooling system performance. ASHRAE Trans. 95(1):823–827. Collier, R.K, Cale, T.S., and Lavan, Z. 1986. Advanced Desiccant Materials Assessment, pb-87-172805/XAB. Gas Research Institute, Chicago, IL. DOE. 1981. DOE-2 Reference Material (Version 2.1A). National Technical Information Service, Springfield, VA. Dorgan, C.E. and Elleson, J.S. 1988. Cold air distribution. ASHRAE Trans. I:2008–2025. Durkin, J. 1994. Expert Systems Design and Development. Macmillan, New York. EIA. 1994. Commercial Buildings Characteristics 1992. U.S. Government Printing Office, Washington, D.C. Elyashiv, T. 1994. Beneath the surface: BACnetTM data link and physical layer options. ASHRAE J. 11:32–36. EPA/CPSC. 1988. The Inside Story: A Guide to Indoor Air Quality. Environmental Protection Agency, Washington, D.C. Fanger, P.O., Melikow, A.K., Hanzawa, H., and Ring, J. 1989. Turbulence and draft. ASHRAE J. 4:18–25. © 2005 by CRC Press LLC

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Fiorino, D.P. 1991. Case study of a large, naturally stratified, chilled-water thermal storage system. ASHRAE Trans. II:1161–1169. Gammage, R.B., Hawthorne, A.R., and White, D.A. 1986. Parameters Affecting Air Infiltration and Air Tightness in Thirty-One East Tennessee Homes, Measured Air Leakage in Buildings, ASIM STP 904. American Society of Testing Materials, Philadelphia. Glav, B.O. 1966. Air Conditioning Apparatus, U.S. Patent No. 3251402. Goldschmidt, I.G. 1994. A data communucations introduction to BACnetTM. ASHRAE J. 11:22–29. Gorton, R.L. and Sassi, M.M. 1982. Determination of temperature profiles and loads in a thermally stratified air-conditioning system. I. Model studies. ASHRAE Trans. II:14–32. Grimm, N.R. and Rosaler, R.C. 1990. Handbook of HVAC Design. McGraw-Hill, New York. Harriman, L.G. III. 1990. The Dehumidification Handbook Second Edition. Munters Cargocaire, Amesbury, MA. Hartman, T.B. 1989. TRAV — a new HVAC concept. Heating/Piping/Air Conditioning. 7:69–73. Hayner, A.M. 1994. Engineering in quality. Eng. Syst. 1:28–33. Heyt, H.W. and Diaz, M.J. 1975. Pressure drop in spiral air duct. ASHRAE Trans. II:221–232. Huebscher, R.G. 1948. Friction equivalents for round, square, and rectangular ducts. ASHRAE Trans. 101–144. Hummel, K.E., Nelson, T.P., and Tompson, P.A. 1991. Survey of the use and emissions of chlorofluorocarbons from large chillers. ASHRAE Trans. II:416–421. Jakob, F.E., Locklin, D.W., Fisher, R.D., Flanigan, L.G., and Cudnik, L.A. 1986. SP43 evaluation of system options for residential forced-air heating. ASHRAE Trans. IIB:644–673. Kimura, K. 1977. Scientific Basis of Air Conditioning. Applied Science Publishers, London. Knebel, D.E. 1995. Current trends in thermal storage. Eng. Syst. 1:42–58. Korte, B. 1994. The health of the industry. Heating/Piping/Air Conditioning. 1:111–112. Kreider, J.F., P. Curtiss, and A. Rabl (2001). Heating and Cooling of Buildings: Design for Efficiency, McGraw–Hill, New York. Locklin, D.W., Herold, K.E., Fisher, R.D., Jakob, F.E., and Cudnik, R.A. 1987. Supplemental information from SP43 evaluation of system options for residential forced-air heating. ASHRA Trans. II:1934–1958. Lowe, R. and Ares, R. 1995. From CFC-12 to HFC-134a: an analysis of a refrigerant retrofit project. Heating/Piping/Air Conditioning. 1:81–89. McQuiston, F.C. and Spitler, J.D. 1992. Cooling and Heating Load Calculating Manual, 2nd ed. ASHRAE, Atlanta, GA. Mitalas, G.P. 1972. Transfer function method of calculating cooling loads, heat extraction rate and space temperature, ASHRAE J. 12:52–56. Mitalas, G.P. and Stephenson, D.G. 1967. Room thermal response factors. ASHRAE Trans. 2, III.2.1. Modera, M.P. 1989. Residential duct system leakage: magnitude, impact, and potential for reduction. ASHRAE Trans. II:561–569. Molina, M.J. and Rowland, S. 1974. Stratospheric sink for chloromethanes: chlorine atom catalyzed destruction of ozone. Nature. 249:810–812. NIOSH. 1989. Congressional Testimony of J. Donald Miller, M.D., before the Subcommittee of Superfund, Ocean, and Water Protection, May 26, 1989. NIOSH, Cincinnati, Cleveland. Norford, L.K., A. Rabl, J.P. Harris, and J. Roturier (1989). Electronic office equipment: the impact of market trends and technology on end use demand. In T.B. Johansson et al., Eds. Electricity: Efficient End Use and New Generation Technologies, and Their Planning Implications. Lund University Press, Lund, Sweden, 427–460. Parsons, B.K., Pesaran, A.A., Bharathan, D., and Shelpuk, B. 1989. Improving gas-fired heat pump capacity and performance by adding a desiccant dehumidification subsystem. ASHRAE Trans. I:835–844. Persily, A.K. 1993. Ventilation, carbon dioxide, and ASHRAE Standard 62-1989. ASHRAE J. 7:40–44. Pesaran, A.A., Penny, T.R., and Czanderna. 1992. Desiccant Cooling: State-of-the-Art Assessment. National Renewable Energy Laboratory, Golden, CO. © 2005 by CRC Press LLC

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Reynolds, S. 1994. CFD modeling optimizes contaminant elimination. Eng. Syst. 2:35–37. Rowland, S. 1992. The CFC controversy: issues and answers. ASHRAE J. 12:20–27. Rudoy, W. and Duran, F. 1975. Development of an improved cooling load calculation method. ASHRAE Trans. II:19–69. Scofield, C.M. and DesChamps, N.H. 1984. Indirect evaporative cooling using plate-type heat exchangers. ASHRAE Trans. I B:148–153. Shinn, K.E. 1994. A specifier’s guide to BACnetTM. ASHRAE J. 4:54–58. Sowell, E.F. 1988. Classification of 200,640 parametric zones for cooling load calculations. ASHRAE Trans. II:754–777. Spitler, J.D., McQuiston, F.C., and Lindsey, K.L. 1993. The CLTD/SCL/CLF Cooling Calculation Method. ASHRAE Trans. I:183–192. Straub, H.E. and Cooper, J.G. 1991. Space heating with ceiling diffusers. Heating/Piping/Air Conditioning. May:49–55. Tackett, R.K. 1989. Case study: office building use ice storage, heat recovery, and cold air distribution. ASHRAE Trans. I:1113–1121. Threlkeld, J.L. 1970. Thermal Environmental Engineering. Prentice-Hall, Englewood Cliffs, NJ. The Trane Company. 1992. TRANE TRACE 600, Engineering Manual. The Trane Co., Lacrosse, WI. Tinsley, W.E., Swindler, B., and Huggins, D.R. 1992. Rooftop HVAC system offers optimum energy efficiency. ASHRAE J. 3:24–28. Tsal, R.J., Behls, H.F., and Mangel, R. 1988. T-method duct design. I. Optimizing theory. ASHRAE Trans. II:90–111. Tsal, R.J., Behls, H.F., and Mangel, R. 1988. T-method duct design. II. Calculation procedure and economic analysis. ASHRAE Trans. II:112–150. United Nations Environmental Programme. 1992. Report of the fourth meeting of the parties to the Montreal protocol on substances that deplete the ozone layer, November 23–25, 1992, Copenhagen. Vaculik, F. and Plett, E.G. 1993. Carbon dioxide concentration-based ventilation control. ASHRAE Trans. I:1536–1547. Van Horn, M. 1986. Understanding Expert Systems. Bantam Books, Toronto. Wang, S.K. 1993. Handbook of Air Conditioning and Refrigeration. McGraw-Hill, New York. Wang, S.K., Leung, K.L., and Wong, W.K. 1984. Sizing a rectangular supply duct with transversal slots by using optimum cost and balanced total pressure principle. ASHRAE Trans. II A:414–429. Williams, P.T., Baker, A.J., and Kelso, R.M. 1994. Numerical calculation of room air motion. III. Threedimensional CFD simulation of a full scale experiment. ASHRAE Trans. I:549–564. Wong, S.P.W. and Wang, S.K. 1990. Fundamentals of simultaneous heat and moisture transfer between the building envelope and the conditioned space air. ASHRAE Trans. II:73–83. Wright, D.K. 1945. A new friction chart for round ducts. ASHRA Trans. 303–316. Zheng, W., Worek, W.M., and Novosel, D. 1993. Control and optimization of rotational speeds for rotary dehumidifiers. ASHRAE Trans. 99(1).

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10 Transportation Frank Kreith University of Colorado

Michael D. Meyer Georgia Institute of Technology

John Leonard II Georgia Institute of Technology

Paul W. Shuldiner University of Massachusetts

Kenneth B. Black University of Massachusetts

Paul Schonfeld University of Maryland

Paul Norton National Renewable Energy Laboratory

Wendy Clark National Renewable Energy Laboratory

Iqbal Husain University of Akron

Sumit Ghosh Stevens Institute of Technology

10.1 Transportation Planning Basic Framework of Transportation Planning • Transportation Modeling

10.2 Design of Transportation Facilities Components of the Project Development Process • Basic Concepts of Project Design • Intermodal Transportation Terminals or Transfer Facilities • Advanced Technology Projects

10.3 Operations and Environmental Impacts Fundamental Equations • Flow, Speed, and Density Relationships • Level of Service (LOS) • Highway Capacity • Intersection Capacity • Traffic Control Devices • Coordinated System Operations • Environmental Impacts

10.4 Transportation Systems Transportation System Components • Evaluation Measures • Air Transportation • Railroad Tranportation • Highway Transportation • Water Transportation • Public Transportation

10.5 Alternative Fuels for Motor Vehicles. Overview • Advantages and Disadvantages of Alternative Fuels

10.6 Electric and Hybrid Vehicles EV System • Energy Sources and Stores • Electric Motors • Hybrid Electric Vehicles • Fuel Cell EVs

10.7 Intelligent Transportation Systems Origin of ITS • Current Status of ITS • Promises for the Future

Introduction Frank Kreith An efficient and economically viable transportation system is an essential part of a modern industrial society. This is particularly true in the U.S., where growth of suburbia requires the average American worker to commute a considerable distance daily between home and work. The situation is exacerbated in many locations by a lack of adequate public transportation, which requires commuters to travel by private automobiles. The use of single occupancy vehicles not only causes congestion, delays, and air pollution, but also imposes a severe economic penalty on many Americans. A recent consumer expenditure survey showed that transportation for most Americans is an expense second only to housing. The vast majority of the transportation spending (98%) is for the purchase, operation, and maintenance of automobiles. From a national perspective, the transportation sector presents enormous challenges for the future. At present, more than 97% of the fuel used for ground transportation in the U.S. is petroleum-based and over 50% is imported. The transportation sector accounts for approximately 20% of the gross domestic product, and the cost of gas and oil has become a growing concern to the government. In 1999, the U.S. Department of Energy presented three oil price scenarios with predictions through the year

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2020. In the reference case, prices were predicted to rise slowly to $23 per barrel by 2020, and in the high price scenario, an increase to about $30 per barrel was predicted. Today, the price of oil already exceeds $40 per barrel, and barring a worldwide recession, there is no reason to expect a long-term drop in crude oil prices in the future. The transportation sector is also the nation’s largest single source of air pollution, with personal vehicles producing 26% of volatile organic compounds, 32% of nitrous oxide, and 62% of carbon monoxide. Thus, ways to promote transportation efficiency are one of the most important facets of reducing fossil fuel consumption and improving environmental quality. Transportation engineering is a highly intradisciplinary field, dealing with the planning, design, construction, maintenance, and operation of various transportation modes. This section presents an overview of transportation engineering, emphasizing planning, design, operation, environmental impact, system analysis, and emerging issues, such as alternative fuels, electric and hybrid vehicles, and intelligent highway systems. Emphasis is placed on those facets of transportation that impact the mechanical engineering profession, but the area also needs social, political, and management inputs to arrive at appropriate engineering solutions of specific problems.

10.1 Transportation Planning Michael D. Meyer Transportation planning is undertaken for a variety of reasons. With the provision of much of the world’s transportation infrastructure the responsibility of governments, transportation planning is undertaken primarily to support public officials in their choice of most cost-effective investments. Because transportation investment has a strong influence on how a community evolves, transportation planning must necessarily consider a variety of factors when assessing the cost effectiveness of alternative investment options. For example, transportation investment can strongly influence land use patterns, the attractiveness of different parts of a region for economic development, the equitable distribution of mobility benefits among different population groups, and the environmental consequences of both the construction and operation of transportation facilities. Transportation planning must therefore be forwardlooking, as well as give attention to current problems in the transportation system.

Basic Framework of Transportation Planning The basic framework for transportation planning that could be applied at any scale of application is shown in Figure 10.1.1. The steps shown in this framework are discussed in the following sections.

Develop project concepts

System operations Other sources for project ideas

Program

Prosperity Goals and objectives

Vision Social equity/ quality of life

Performance measures

Alternative improvement strategies

Environmental Quality

Plan

Data

Analysis methods

Transportation Systems Planning

FIGURE 10.1.1 Transportation planning process. © 2005 by CRC Press LLC

Evaluation criteria

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Define a Vision The transportation system can impact society in a variety of ways — providing mobility and accessibility, promoting economic development, contributing to quality of life, as well as negatively affecting the natural environment. The first step in transportation planning thus usually consists of defining what it is that the nation, state or region desires in terms of its future characteristics. Identify Goals and Objectives Once a desired vision is articulated, goals can be identified that relate the vision to the ultimate achievement of a transportation plan. Objectives are more specific statements that indicate the means by which these goals will be achieved. Goals and objectives not only provide overall direction to the transportation planning process, but they also help define the criteria, known as measures of effectiveness, that are used later in the process for evaluating alternative courses of action. Identify Performance Measures An important aspect of a continuing transportation planning process is the monitoring of system performance. This monitoring systematically identifies areas where improvements might occur, and, in addition, helps transportation officials assess the effectiveness of previously implemented actions. Performance measures can focus explicitly on transportation system operations, e.g., the level of freeway delay during the morning peak travel hours, or on other issues of importance to transportation officials, e.g., the level of transportation-related air pollutants emitted during specified periods of time. Collect Data Given that transportation investment is usually aimed at upgrading the physical condition of a facility (e.g., repaving a road or building a new bridge) or at improving its performance (e.g., providing new person-carrying capacity by setting aside highway lanes for multi-occupant vehicles or by building a new road), engineers are continually collecting data on the many different components of the transportation system. The base condition or performance of all the different facilities or services that make up a transportation system is called an inventory. Forecasting future demand for transportation requires engineers and planners to characterize the current and likely future states of the factors that influence this demand. Thus, for example, the type of data that is collected includes such things as current land use and socioeconomic characteristics of the traveling population. Current land use is readily attained through land use inventories. The methods of estimating future land use range from trends analysis to large-scale land use models that predict household and employment sites decades into the future. Important socioeconomic characteristics include level of household income, number of members of the household, number of autos in the household, number of children, age of the head of household, and highest level of education achieved. Each of these factors has been shown through research to influence the amount and type of travel associated with a typical household. Use Analysis Tools to Identify System Deficiencies or Opportunities The analysis tools and methods used to identify transportation deficiencies and improvement opportunities can vary widely. In some cases, computer-based transportation network models are used to estimate future traffic volumes and transit ridership, with the results then compared to existing system capacity to handle such volumes. This comparison relies on one of the more popular performance measures used in transportation planning today, the volume-to-capacity (V/C) ratio. However, given the many different goals and objectives that can characterize a transportation planning process, a wide variety of measures are often used for determining system deficiencies. Other types of analysis tools include time–distance diagrams, queuing models, fluid-flow approximation methods, macro- and micro-simulation models, and mathematical programming techniques. Develop and Analyze Alternatives Various types of strategies can result from the planning process: © 2005 by CRC Press LLC

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1. Improving the physical infrastructure of the transportation system — for example, adding new highway lanes or extending an existing subway line 2. Improving system operations — for example, coordinating traffic signals, improving traffic flow through improved geometric design of intersections, or making transit operations more efficient through schedule coordination 3. Reducing travel demand so that the transportation system can handle peak loads more effectively — for example, flexible working hours, increasing average vehicle occupancy through such measures as carpools or transit use, or raising the “price” of travel through the use of tolls In the past 10 years, the application of advanced transportation technologies to the operation of the transportation system, known as intelligent transportation systems (ITS), has become an important type of strategy in many cities. Thus, it is not uncommon for major cities to now have a centralized traffic management center, with a regional surveillance and traveler communication system that permits transportation system managers to communicate to travelers the best times for travel and which routes are least congested. Evaluate Alternatives Evaluation brings together all of the information gathered on individual alternatives/plans and provides a systematic framework to compare the relative worth of each. This evaluation process most often relies on the various measures of effectiveness that link to the goals and objectives defined at the beginning of the process. Different types of evaluation methods include use of benefit/cost ratios, cost-effectiveness indices, goals matrix analysis, and subjective assessment of the merits of individual alternatives. Develop Transportation Plan One of the most important products of the transportation planning process is the transportation plan. The plan outlines the many different strategies and projects that are necessary to meet the challenges and opportunities facing a state or region. In the U.S., federal law requires that every state and every metropolitan area over 50,000 population have a transportation plan. The state department of transportation (DOT) is responsible for preparing the state transportation plan; an agency called the metropolitan planning organization (MPO) is responsible for preparing the metropolitan transportation plan. Implement Plan Another major product of the transportation planning process is a strategy for implementing all of the actions identified in the plan. In the U.S., federal law requires each state and every metropolitan area over 50,000 population to produce a transportation improvement program that lists the projects that will be implemented over the next 3 to 5 years, identifies which agency is responsible for each project, and describes the source of project funding. The implemented projects will affect the performance of the transportation system. Through a continuing monitoring process, linked directly to important performance measures, the performance of individual projects or of the entire transportation system can be fed back into the planning process as a means of identifying new problems.

Transportation Modeling The level of transportation analysis can vary according to the level of complexity and scale of application of potential solution strategies. Thus, for example, the consideration of a new subway system would necessarily have to be examined from a metropolitan level, while the transportation impacts of a new development site would likely be analyzed at a subregional level. In most cases, however, the modeling process consists of four major steps — trip generation, trip distribution, mode split, and trip assignment. Even though recent models combine some of these steps together during the analysis process, the concept of the “trip” consisting of these four stages still holds. Each study area (whether a nation, state, metropolitan

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Ti Number of Trips Produced in Zone i

i

i

i

m1 m2

j

Tij Number of Trips Produced in Zone i Destined for Zone j

j

Tijm Number of Trips From Zone i to Zone j by Mode m

j

Tijmr Number of Trips From Zone i to Zone j by Mode m via Route r

Trip Generation Leads to Trip Distribution Leads to Mode Choice

m3

i

r1 r2

Leads to Trip Assignment

FIGURE 10.1.2 Transportation modeling framework.

area, or community) is divided into zones of homogeneous characteristics (e.g., similar household incomes) that can then be used as the basic foundation for estimating trips from or attracted to that zone. Most planning studies define these zones to be similar to those used in other data collection activities (e.g., the U.S. census tracts) so that data useful to the transportation study collected by others can be readily linked to the transportation analysis. The transportation system is represented in models as a network of links and nodes. Links represent line-haul facilities, such as roads or transit lines, and nodes represent points of connection, such as an intersection or transit terminal. Given the complex nature of transportation systems, the typical transportation network consists of links representing only highly used facilities or other facilities that are critical to the overall performance of the transportation system. The steps in a typical modeling exercise are shown in Figure 10.1.2. Basic to this approach is the concept of derived demand. Derived demand means that a trip is taken to accomplish some activity at a destination, and that the trip itself is simply a means of reaching this activity. There is no intrinsic value of the trip itself. Thus, modeling trip-making requires linking travel behavior to the characteristics of the trip-maker and to the activities at the origin and destination ends of the trip that will influence the way the trips are made. Trip generation is the process of analytically deriving the number of trips that will be generated from a location or zone based on socioeconomic characteristics of the household, or in the case of freight movement, the zonal economic characteristics. Trip generation also includes predicting the number of trips that will be attracted to each zone in the study area. Number of trips produced in a zone = f (Population socio-economic characteristics, land use, transportation mode availability) Number of trips attracted to a zone = f (Attractiveness of the zone) Two approaches are often used to estimate the number of trips generated. The first uses trip rate models that are based on trip-making behavior as compared to important variables. For example, see Table 10.1.1. The other approach is to use regression models that are estimated either from survey data TABLE 10.1.1 Cross-Classification Analysis, Trips per Day, by Household Size and Income Number of People in Households

Low income Medium income High income

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1

2

3+

2.4 3.5 3.9

3.3 3.8 4.2

4.5 4.8 5.4

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collected throughout the study area or from some other data source, such as the U.S. Census. The following regression equations illustrate this approach. Ti = 184.2 + 120.6 (Workersi) + 34.5 (Autosi)

Zone Trip Productions:

Household Trip Productions: Tih = 0.64 + 2.3 + (Employeei) + 1.5 (HHAutoi) Tj = 54.2 + 0.23 (Officej) + 0.43 (Retailj)

Zonal Attractions:

where Ti = total number of trips generated in zone i; Tih = total trips generated per household in zone i; Tj = total trips attracted to zone j; Workersi = number of workers in zone i; Autosi = number of autos in zone i; Employeei = number of employees per household in zone i; HHAutoi = number of autos per household in zone i; Officej = number of office employees in zone j; and Retailj = number of retail employees in zone j. Trip distribution is the process of estimating the number of trips that travel from each zone to every other zone in the study area. The results of the trip distribution process is a matrix called the trip table, which shows the number of trips traveling between each origin–destination (O-D) pair for the time period being examined. A common method for distributing trips in a zonal system is the gravity model, which is of the following form: Tij = Pi ×

A i × Fij × Kij

∑(A

j

× Fij × Kij )

where Tij = total trips originating in zone i and destined to zone j; Pi = number of trips produced in zone i; Aj = level of attractiveness of zone j (e.g., number of retail employees); Fij = friction or impedance factor between zones i and j (a value usually a function of travel time); and Kij = socioeconomic adjustment factors for trips between zones i and j (a value that represents variables that influence trip making not accounted for by other variables). Mode choice is the process of estimating the percentage of travelers who will use one mode of transportation vs. the others available for a given trip. The basic approach in making this estimation is that each mode has associated with it some empirically known characteristics that, when combined with characteristics of the traveler in a mathematical equation, can define that mode’s utility. Variables such as travel time, travel cost, modal reliability, and so on are often incorporated into a mode’s utility function, along with socioeconomic characteristics of the traveler. Freight models use a similar concept in estimating commodity flows by mode. One of the most familiar forms of mode choice models, based on the concept of consumer choice, is the logit model, which predicts mode shares based on the following equation: Pik =

eUk

∑e

U m

for all modes n

where Pik = probability of individual i choosing mode k; Uk = utility of mode k; Um = utility of mode m; n = number of modes available for trip. The utility of each mode is often represented as a linear function of those variables found to influence an individual’s choice of mode. For example, a utility function for the automobile mode might be of the form, Ua = 6.3 – 0.21 (X1) – 0.43 (X2) – 0.005 (X3) where Ua = utility of automobile; X1 = access and egress time when automobile is chosen; X2 = line-haul travel time; and X3 = cost of travel. © 2005 by CRC Press LLC

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Link Travel Time

Capacity

Free-flow Travel time

Traffic Flow

FIGURE 10.1.3 Link performance function.

The utility functions of other modes available for a specific trip would be similarly specified. The respective probabilities would then be multiplied by the total number of trips between an origin and destination to obtain the number of trips made by mode. Trip assignment is the process of estimating the trip paths through a transportation network based on a trip table (which is produced in trip distribution). The basic concept found in all trip assignment methods is that travelers choose modes that will minimize travel time, that is, they will choose the shortest path through a network (once again, the assumption of derived demand influencing the analysis approach). Link performance functions that relate travel time to the number of vehicles or riders on that link are used to iteratively update estimated link travel times so that minimum path travel times reflect the effect of congestion (see Figure 10.1.3). A portion of the total O-D travel demand is assigned to the network, with travel times then updated based on the link performance function, given the volume on each link. An additional portion of the O-D travel is next assigned given the updated travel times, still following the minimum travel time path through the network. This process continues until all estimated trips have been assigned to a link path in the network. Stochastic assignment is also used in many planning studies. This assignment recognizes that, in certain cases, some subset of trip routes will have associated with them some characteristics that attract specific types of travelers, even if the travel time is longer. A probabilistic approach takes these characteristics into account. In order to develop more behaviorally based travel models, researchers in recent years have focused on the fact that travel arises out of the need to participate in out-of-home activities (work, shopping, school, etc.). This directly leads to the conclusion that what one should study in the first instance is not travel per se, but rather the participation in the activities that ultimately generate travel. This approach has been referred to as “activity-based modeling.” Figure 10.1.4 shows the difference in the traditional approach toward modeling and the activity-based approach. Many activity-based models are being implemented within a micro-simulation framework, within which the behavior of each individual is dynamically simulated over time.

Defining Terms Demand management: Reducing the demand for travel during specific time periods by shifting trips to other times, diverting trips to other routes or modes, or reducing the need for trip-making to begin with. Derived demand: An assumption that travelers make a trip to accomplish some objective at the destination and that the trip itself is simply a means of reaching that activity. © 2005 by CRC Press LLC

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walk noon

PB 12:50 pm

W

transit 7:30 am

4:40 pm

H 10 pm

Actual Travel Schedule S

7 pm auto walk midday

walk

PB

W (2 times) Trip-based Model (2 times)

W transit H H

auto (2 times) S

transit

W

O midday

pm peak am peak H night night auto

PB = Personal Business H = Home

O

W = Work

Daily Schedule Model

O = Other S = Shop

FIGURE 10.1.4 Difference between trip-based modeling and activity-based modeling.

Intelligent transportation systems: Application of surveillance, communication, and control technologies to the management of the transportation system, and in some cases, to the control of individual vehicles. Transportation network: A transportation system is represented in models as a network of links and nodes. Links represent line-haul facilities, such as roads and transit lines, and nodes represent points of connection. Utility function: A mathematical formulation that assigns a numerical value to the attractiveness of individual modes of transportation based primarily on that mode’s characteristics. Zonal system: Each study area (whether nation, state, metropolitan region, or community) is divided into zones of homogeneous characteristics that can then be used as the basic foundation for estimating trips from, or attracted to, that zone.

References Goulias, K. (Ed). 2003. Transportation Systems Planning, CRC Press, Boca Raton, FL. Grava, S. 2003. Urban Transportation Systems, McGraw-Hill, New York. Hall, R. (Ed). 2003. Handbook of Transportation Science, 2nd ed. Kluwer, Boston. Institute of Transportation Engineers. 1997. Trip Generation Handbook, 6th ed. ITE, Washington, DC. Meyer, M. and Miller, E. 2001. Urban Transportation Planning: A Decision-Oriented Approach, 2nd ed. McGraw-Hill, New York. Ortuzar, J. and Willumsen. L.G. 1994. Modelling Transport, 2nd ed. John Wiley & Sons, New York. Taylor, M. A. P., Young, W. and Bonsall, P. W. 1996. Understanding Traffic Systems: Data, Analysis and Presentation, Ashgate, Brookfield, VT. Vuchic, V. 1999. Transportation for Livable Cities, Center for Urban Policy Research, Rutgers, The State University of New Jersey, New Brunswick, NJ.

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Further Information American Association of State Highway and Transportation Officials 444 N. Capitol St. NW Suite 225 Washington, DC 20001 Institute of Transportation Engineers 1099 14th St. NW Suite 300W Washington, DC 20005 Transportation Research Board, National Research Council 500 Fifth Street, NW Washington, DC 20001

10.2 Design of Transportation Facilities John Leonard II and Michael D. Meyer The efficient movement of people and goods requires transportation systems and facilities that are designed to provide sufficient capacity for the demands they face in as safe a manner as possible. In addition, in most modern societies, the design of transportation facilities must explicitly minimize harm to the natural and human-made environment while providing for mitigation measures that relate to those impacts that are unavoidable. In many ways the critical challenge to today’s designers of transportation projects is successfully designing a facility that minimally harms the environment. The design of a transportation facility almost always takes place within the context of a much broader project development process. This process can vary in complexity with the type of project under design and with the scale of implementation. The importance of the project development process to the designer is that it: • Establishes the key characteristics of the project that must be considered in the design • Indicates the time frame that will be followed for project design • Establishes which agencies and groups will be involved in the process and when this involvement will likely occur • Links the specific elements of the project design with other tasks that must be accomplished for the project to be constructed • Satisfies legal requirements for a design process that is open for public review and comment • Indicates the specific products that must be produced by the designers to complete the project design process In most cases the project development process consists of a well-defined set of tasks that must be accomplished before the next task can occur. These tasks include both technical activities and public involvement efforts that are necessary for successful project development.

Components of the Project Development Process Identify Project Need A project need can be identified through a formal planning process or from a variety of other sources, including suggestions from elected officials, agency managers, transportation system users, and citizens. Important in this early portion of project development is an indication of what type of improvement is likely to be initiated. For example, a project could relate to one or more of the following types of improvement strategies:

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• New construction. A transportation facility constructed at a new location • Major reconstruction. Addition of new capacity or significant changes to the existing design of a facility, but usually occurring within the area where the current facility is located • Rehabilitation/restoration. Improvements to a facility usually as it is currently designed and focusing on improving the physical condition of the facility or making minor improvements to enhance safety • Resurfacing. Providing new pavement surface to a transportation facility that prolongs its useful life • Spot improvements. Correction of a problem or hazard at an isolated or specific location Establish Project Limits and Context One of the very first steps in the design process is to define the boundaries or limits of the project. This implies establishing how far the project will extend beyond the area being targeted for improvement and the necessary steps to ensure smooth connections to the existing transportation system. Project boundaries also have important influence on the amount of right-of-way that might have to be purchased by an agency to construct a project. Establish Environmental Impact Requirements The design of a project will most likely be influenced by environmental laws or regulations that require design compliance with environmental mandates. These mandates could relate to such things as wetland protection, preservation of historic properties, use of public park lands, maintaining or enhancing water quality, preserving navigable waterways, protecting fish and wildlife, reducing air pollutants and noise levels, and protecting archaeological resources. One of the first steps in project development is to determine whether the likely project impacts are significant enough to require a detailed environmental study. Develop Strategy for Interagency Coordination and Public Involvement Depending on the complexity and potential impact of a project, the project designer could spend a great deal of time interacting with agencies having some role in or jurisdictional control over areas directly related to the project. These agencies could have jurisdiction by law (e.g., wetlands) or have special expertise that is important to project design (e.g., historic preservation). In addition to interagency coordination, transportation project development is often subject to requirements for public outreach and/or public hearings. An important aspect of recent project development efforts is to develop very early in the process a consensus among involved agencies on what environmental impacts will have to be carefully studied and on the definition of the project purpose and need. Initiate Project Design and Preliminary Engineering Topographic data of the study area and forecasted vehicular volumes expected to use the facility in the design year are used as input into the preliminary design of the horizontal and vertical alignment of the facility, that is, the physical space the facility will occupy once finished. This preliminary engineering step also includes the preparation of initial right-of-way (ROW) plans, which indicate the amount of land that must be available to construct the facility. Preliminary engineering is a critical step for environmental analysis in that it provides the first detailed examination of the scope and extent of potential environmental impacts. Project Engineering Once preliminary engineering has provided the basic engineering information for the project, the more detailed project design begins. This entails specific layouts of horizontal and vertical geometry, soils/subsurface examination and design, design of utility location, drainage design, more detailed ROW plans, and initial construction drawings. Concurrent with this design process, the environmental process continues with updated information on project changes that might cause additional environmental harm, the initiation of any permitting process that might be needed to construct the project (e.g., environmental agency permission to affect wetlands), and public hearings/meetings to keep the public involved with project development.

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ROADWAY GEOMETRY

VISUAL FIELD STRUCTURE MEMORY TRAFFIC SENSORY DETECTION

PERCEPTION (Information processing)

ANALYTIC OPERATIONS

DRIVER GOALS

DECISION MAKING

VEHICLE DYNAMICS AND DISPLAY

WEATHER AND LIGHT LEVELS

SURFACE CONDITIONS AND FORCES DUE TO SPEED AND GEOMETRY

CONTROL RESPONSE

VEHICLE RESPONSE

VEHICLE TYPE AND CONDITION DRIVER PHYSIOLOGICAL AND PSYCHOLOGICAL STATE

CONTROL AND DIRECTIONAL MESSAGES

= INFORMATION INPUTS = DRIVER GUIDANCE AND CONTROL PROCESS

FIGURE 10.2.1 Driver-vehicle-roadway interface.

Final Engineering The final engineering step is the culmination of the design process, which completes the previous design plans to the greatest level of detail. This step includes finalizing ROW plans, cost estimates, construction plans, utility relocation plans, and any agreements with other agencies or jurisdictions that might be necessary to complete the project. Environmental permits are received and final project review for environmental impacts is completed. Context-Sensitive Design One of the important characteristics of transportation facility design is the potentially negative impact that new facilities could have on the surrounding community and natural environment. Engineers and planners have begun to consider such impacts earlier in the project development process so that the context within which a facility is constructed is incorporated into the design itself. This process is called context-sensitive design.

Basic Concepts of Project Design Human Factors Human factors have a great deal of influence on the design of transportation facilities in such things as width of facility, length and location of access/egress points, vehicle braking distance, location of information/guidance aids such as signs, and geometric characteristics of the facility’s alignment. The drivervehicle-roadway interface is shown in Figure 10.2.1. Vehicle or User Performance Factors The dynamics of vehicle motion play an important role in determining effective and safe design. The key vehicle characteristics that relate to facility design criteria include:

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PROPORTION OF SERVICE

Mobility

Arterials

Collectors

Land Access

Locals

FIGURE 10.2.2 Relationship of functionally classified systems in relation to traffic mobility and land access. (Source: American Association of State Highway and Transportation Officials. 2001. A Policy on the Geometric Design of Highways and Streets, AASHTO, Washington, DC. Figure 1 through Figure 5.)

• Vehicle size. Influences vertical and horizontal clearances, turning radii, alignment width, and width of vehicle storage berths. • Vehicle weight. Influences strength of material needed to support vehicle operations. • Vehicle or user performance. Influences specifications for horizontal and vertical geometry, braking distances, operational performance and needed capacity to allow passing and successful maneuvering (e.g., assumed walking speed of pedestrians crossing a road that dictates how long a traffic signal must remain red). Classification Schemes The transportation system serves many functions, ranging from providing access to specific locations to providing high-speed, high-capacity movement over longer distances. Classification schemes are used to represent these various roles and influence the design criteria that are associated with the facilities in each classification category. A common functional classification scheme for highways is shown in Figure 10.2.2. Capacity and Level of Service Every design usually begins with some estimation of the demand for the transportation facility that will likely occur if the facility is built. The key design question then becomes, what facility capacity (e.g., number of road lanes, runways, transit lines, or vehicle departures) is necessary if a certain level of performance is desired? These different levels of performance are referred to as level of service (LOS). Level of service is a critical element in establishing important design factors (see Figure 10.2.3). Design Standards Design standards dictate minimum or maximum values of project characteristics that are associated with a particular facility type. Design standards usually result from extensive study of the relationship between various facility characteristics, vehicle performance, and the safe handling of the vehicles by human operators. Design standards often vary by the “design speed” of the facility (and thus the importance of the facility classification) and by the “design vehicle.” Design standards are often the basis for developing typical cross-sections (see Figure 10.2.4 and Figure 10.2.5).

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FREEWAYS

Level of Service

Minimum Speed (mph)

Maximum Density (pc/mi/In)

Max Service Flow Rate (pcphpl)

Maximum v /c Ratio

Free-Flow Speed = 70 mph A B C D E F

10.0 16.0 24.0 32.0 36.7/39.7 var

70.0 70.0 68.5 63.0 60.0/58.0 var

700 1120 1644 2015 2200/2300 var

0.318/0.304 0.509/0.487 0.747/0.715 0.916/0.876 1.000 var

Free-Flow Speed = 65 mph A B C D E F

10.0 16.0 24.0 32.0 39.3/43.4 var

65.0 65.0 64.5 61.0 56.0/53.0 var

650 1040 1548 1952 2200/2300 var

0.295/0.283 0.473/0.452 0.704/0.673 0.887/0.849 1.000 var

Free-Flow Speed = 60 mph A B C D E F

10.0 16.0 24.0 32.0 41.5/46.0 var

A B C D E F

10.0 16.0 24.0 32.0 44.0/47.9 var

60.0 60.0 60.0 57.0 53.0/50.0 var

600 960 1440 1824 2200/2300 var

0.272/0.261 0.436/0.417 0.655/0.626 0.829/0.793 1.000 var

Free-Flow Speed = 55 mph 55.0 55.0 55.0 54.8 50.0/48.0 var

550 880 1320 1760 2200/2300 var

0.250/0.239 0.400/0.383 0.600/0.574 0.800/0.765 1.000 var

Note: In table entries with split values, the first value is for four-lane freeways, and the second is for six- and eight-lane freeways.

PEDESTRIAN WALKWAYS Expected flows and Speeds Level of Service

Space (sq ft/ped)

Ave. Speed, S (ft/min)

Flow Rate, v (ped/min/ft)

Vol/Cap Ratio, v /c

A B C D E

≥130 ≥ 40 ≥ 24 ≥ 15 ≥6

≥260 ≥250 ≥240 ≥225 ≥150

≤2 ≤7 ≤10 ≤15 ≤25

≤0.08 ≤0.28 ≤0.40 ≤0.60 ≤1.000

F

10 T1 = 10; elseif T1 < -10 T1 = -10; end if T2 > 10 T2 = 10; elseif T2 < -10 T2 = -10; end if T3 > 10 T3 = 10; elseif T3 < -10 T3 = -10; end % Calculate hydrodynamic drag forces from the equation % Fd = Cd*Ap*rho*(Vo^2/2), ASSUME: AUV is 28" % tall, 21" dia. cylinder 24/19 % rho (seawater) = 1.99, Cd = 1.0, Ap = 4.083 sq.ft. Fdx = 1.0*4.083*1.99*(xdot^2/2); if xdot > 0 Fdx = -Fdx; end Fdy = 1.0*4.083*1.99*(ydot^2/2); if ydot > 0 Fdy = -Fdy; end % Cd = 0.9 for cylinder moving parallel to flow with % L/D = 1, Ap = 2.405 sq.ft. © 2005 by CRC Press LLC

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Fdz = 0.87*2.405*1.99*(zdot^2/2); if zdot > 0 Fdz = -Fdz; end % Calculate state derivatives sys(1)= xdot + (Yy*psidot); sys(2)= ydot - (Xx*psidot); sys(3)= x(7); sys(4)= x(8); sys(5)= ((T1 + T2 + Fdx + Fxext)/m) + (ydot*psidot); sys(6)= ((Fdy + Fyext)/m) - (xdot*psidot); sys(7)= (T3 + Fdz + Fzext - W)/m; sys(8)= (((T2-T1)*(d/2)) + Mzext)/Iz; end % Return output states if flag == 3 sys = [x(1),x(2),x(3),x(4),x(5),x(6),x(7),x(8)]; end

Method of Information Assimilation Once the vehicle is working, more complex algorithms are used to program the system, since the vehicle must be able to learn and solve complex situations as it accomplishes its mission. To assimilate the different information and help in solving the complex aspects of a mission, fuzzy logic, neural networks, and expert systems are integrated into the software. Typically the code is object oriented and written in C++ to take advantage of code reusability, interchangeability, inheritance, polymorphism, and maintainability. • Fuzzy logic is used to solve problems that are not directly quantifiable, for instance, whether a surface is smooth or rough. Therefore, fuzzy logic is good for dealing with ambiguity, subjectivity, associativity, and rule systems where conclusions can be drawn from only a small amount of information [OWI, 1998]. • Neural networks are used for classifying objects by using SONAR, for combining and analyzing sensor data, and for navigation. Neural networks are able to account for abrupt changes in external environment, system damage, and uncertain or indeterminate data input. • Expert systems use the knowledge from “experts,” where rules of thumb are incorporated for knowledge that cannot be mathematically determined but must be acquired by experience. An example is the maximum shark bite depth, which has been obtained from the actual bites found on deep-sea oceanographic moorings.

Software Modules An AUV system will have a number of software modules that give the vehicle the ability to operate on its own without human assistance: mission planner, mission executor, guidance system, navigation system, SONAR data processor, mission replanner, and vehicle system monitor. Each of these systems obtains the important data from the various subsystems, analyzes the information, and then takes action according to what is necessary from the given information.

Current Software Trends Most companies that originally programmed the AUV’s subsystem microcontrollers and microprocessors in Basic, FORTRAN, Pascal, or Assembly have either already or are in the process of switching to C or C++. For example, Bluefin Robotics Corp. of Massachusetts initially developed the software for its vehicles © 2005 by CRC Press LLC

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in C, but is now “being upgraded to C++ to make the code more flexible and portable.”5 Visual Basic and Java are also used for remotely operated vehicles (ROVs) where human and Internet interfaces are necessary. These changes show that there is a trend toward a standardization of software for future development of advanced robot systems.

14.7 Robot Dynamics and Control Frank L. Lewis This section deals with the real-time motion control of robot manipulators. In Section 14.8 are covered higher-level planning and control functions, including the generation of the prescribed trajectory that is assumed given in this section. Robot manipulators have complex nonlinear dynamics that might make accurate and robust control difficult. Fortunately, robots are in the class of Lagrangian dynamical systems, so that they have several extremely nice physical properties that make their control straightforward. In this section will be discussed several control techniques including computed torque (e.g., feedback linearization), classical joint control, digital control, adaptive control, robust control, learning control, force control, and teleoperation. More information may be found in Lewis et al. (1993). The advances that made possible this modern approach to robot control were made by Craig (1988), Slotine and Li (Slotine, 1988), Spong and Ortega (Spong and Vidyasagar, 1988), and others.

Robot Dynamics and Properties A robot manipulator can have either revolute joints or prismatic joints. The latter are actuators that function like an automobile antenna, extending and contracting on a linear axis. The values of the angles, for revolute joints, and link lengths, for prismatic joints, are called the link variables, and are denoted q1(t), q2(t), …, qn(t) for joints one, two, and so on. The number of links is denoted n; for complete freedom of motion in space, six degrees of freedom are needed, three for positioning and three for orientation. Thus, most commercial robots have six links. We discuss here robots which are rigid, that is which have no flexibility in the links or in the gearing of the joints; flexible robots are discussed in the section on control of flexible-link and flexible-joint robots. The dynamics of robot manipulators with rigid links can be written as M (q) q˙˙ + Vm (q, q˙ ) q˙ + F(q˙ ) + G(q) + τ d = τ

(14.7.1)

where M(q) is the inertia matrix, Vm (q, q˙ ) is the coriolis/centripetal matrix, F(q˙ ) are the friction terms, G(q) is the gravity vector, τd (t) represents disturbances, and τ (t) is the control input torque. The joint variable q(t) is an n-vector containing the joint angles for revolute joints and lengths for prismatic joints. It is often convenient to write the robot dynamics as M (q) q˙˙ + N (q, q˙ ) q˙ + τ d = τ

(14.7.2)

N (q, q˙ ) ≡ Vm (q, q˙ ) q˙ + F(q˙ ) + G(q)

(14.7.3)

where N (q, q˙ ) represents a vector of the nonlinear terms. The objective of robot control is generally to select the control torques τ (t) so that the robot follows a desired prescribed motion trajectory or exerts a desired force. Examples include spray painting, grinding, or manufacturing assembly operations. The position control objective can be achieved by first defining a desired trajectory qd (t), which is a vector containing the desired values vs. time qdi (t ) of each of joint of the manipulator. This desired trajectory vector qd (t) is determined in a higher-level path planner, based on a even higher-level task decomposition, and then fed to the real-time motion control system. This © 2005 by CRC Press LLC

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section discusses the real-time motion control problem assuming that qd (t), or a similar desired force vector, is given. The robot dynamics in Equation (14.7.1) satisfies some important physical properties as a consequence of the fact that they are a Lagrangian system. These properties significantly simplify the robot control problem. The main properties of which one should be aware are the following. Properties of Robot Arm Dynamics P1 The inertia matrix M(q) is symmetric, positive definite, and bounded so that µ1I ≤ M(q) ≤ µ2I for all q(t). For revolute joints, the only occurrences of the joint variables qi are as sin(qi), cos(qi). For arms with no prismatic joints, the bounds µ1, µ2 are constants. P2 The coriolis/centripetal vector Vm (q, q˙ ) q˙ is quadratic in q˙ and bounded so that ||Vm q˙ || ≤ v B ||q˙ ||2 . ˙ (q) – 2V (q, q˙ ) is P3 The coriolis/centripetal matrix can always be selected so that the matrix M m skew symmetric. This is a statement of the fact that the fictitious forces in the robot system do no work. P4 The friction terms have the approximate form F(q˙ ) = Fv q˙ + Fd (q˙ ), with Fv a diagonal matrix of constant coefficients representing the viscous friction, and Fd (·) a vector with entries like K di sgn(q˙i ), with sgn(·) the signum function, and K di the coefficients of dynamic friction. These friction terms are bounded so that || F(q˙ )|| ≤ v B ||q˙ || + kB for constants vB , kB . P5 The gravity vector is bounded so that ||G(q)|| ≤ gB. For revolute joints, the only occurrences of the joint variables qi are as sin(qi), cos(qi). For revolute joint arms the bound gB is a constant P6 The disturbances are bounded so that ||τd (t)|| ≤ d. P7 The nonlinear robot terms are linear in the parameters of mass and friction so that one can write M (q) q˙˙ + Vm (q, q˙ ) q˙ + F(q˙ ) + G(q) = W (q, q˙, q˙˙) φ

(14.7.4)

where W (q, q˙, q˙˙) is a matrix of known robot functions and φ is a vector of mass and friction coefficient parameters, often unknown. The regression matrix W(·) can be computed for any specified robot arm. The last property, P7, is especially useful in adaptive control approaches. The bounding properties are especially useful in robust control approaches. The skew-symmetry property P3 is vital for Lyapunov control proofs, which provide guaranteed tracking motion and often give the structure of the control loops. It essentially allows some very nice linear systems techniques to be used with the time-varying robot dynamics.

State Variable Representations and Computer Simulation The nonlinear state-variable representation x˙ = f (x, u), with x(t) the internal state and u(t) the control input, is very convenient for many applications, including the derivation of suitable control laws and computer simulation. Once the system has been put into state-space form, it can easily be integrated to obtain simulation time plots using, for instance, a Runge-Kutta integrator; many standard software packages have such integration routines, including MATLAB, MATRIXX, and SIMNON. It is supposed for convenience in this subsection that the disturbance τd (t) is equal to zero. There are three convenient state-space formulations for the robot dynamics in Equation (14.7.1). In the position/velocity state-space form, one defines the state as the 2n-vector x
[q T q˙ T ]T and writes q˙   0   x˙ =  −1  +  −1  − M (q) N (q, q˙ )  M (q)

(14.7.5)

which is in state-space form with u(t)
τ (t). For computer simulation purposes, the matrix inversion M–1(q) is required at every integration time step. For arms with simple dynamics, it is often possible to invert the inertia matrix analytically off-line, © 2005 by CRC Press LLC

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reducing the on-line computational burden. Otherwise, it is more suitable to solve Equation (14.7.2) for q˙˙, required by the integration routine, using least-squares techniques to avoid the inversion of M(q). An alternative linear state-space equation in the form x˙ = Ax + Bu can be defined as I 0  x +  u 0 I

0 x˙ =  0

(14.7.6)

with u(t)
–M–1(q) N (q, q˙ ) + M–1(q)τ. This is known as the Brunovsky Canonical Form. The third state-space formulation is the Hamiltonian form, which derives from Hamilton’s equations of motion. Here the state is defined as the 2n-vector x = (qT p T )T, with p(t)
M (q) q˙ the generalized momentum. Then the state-space equation is   M −1 (q) p −1   +  0 u M q ∂ x˙ = 1 ( ) − I n ⊗ p T p  In  ∂q  2 

(

)

(14.7.7)

with the control input defined by u = τ – G(q) and ⊗ the Kronecker product (Lewis et al., 1993).

Cartesian Dynamics and Actuator Dynamics Cartesian Dynamics The dynamics in Equation (14.7.1) are known as the joint-space dynamics; they are expressed in the jointspace coordinates q. Cartesian coordinates referred to some frame, often the base of the robot manipulator, may be used to describe the position of the end effector of the robot arm. Denote the Cartesian coordinates of the end of the arm as Y(t) = h(q), whose first three coordinates represent position and last coordinates represent orientation. The nonlinear function h(q) gives the end effector Cartesian coordinates in termexcepts of the current joint positions q and is called the arm kinematics transformation. The arm Jacobian relates joint and Cartesian velocites and is defined as J(q)
∂h(q)/∂q so that v ˙ ω  ≡ Y = J (q) q˙  

(14.7.8)

where v(t) is the linear velocity and ω(t) the angular velocity of the end effector. Both these velocities are 3-vectors. Differentiating this equation gives the acceleration transformation Y˙˙ = J q˙˙ + J˙ q˙. By differentiating Equation (14.7.1) one discovers that the dynamics may be written in Cartesian form as MY˙˙ + N + fd = F

(14.7.9)

where M
J–TMJ–1, N
J–T(N – MJ-1 J˙ J-1 y˙ ), and the disturbance is fd
J–Tτd . In the Cartesian dynamics the control input is F, which has three components of force and three of torque. The important conclusion of this discussion is that the Cartesian dynamics are of the same form as Equation (14.7.2). Furthermore, it can be shown that the properties of the robot dynamics hold also in Cartesian form. Therefore, all the control techniques to be described in this section can be used for either the joint-space or the Cartesian dynamics. Actuator Dynamics The robot manipulator is driven by actuators which may be electric, hydraulic, pneumatic, and so on. Considering the case of electric motors it is direct to show that if the armature inductance is negligible, the dynamics of the arm plus actuators can be written as © 2005 by CRC Press LLC

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(J

M

) (

)

(

)

+ R 2 M q˙˙ + BM + R 2 Vm q˙ + RFM + R 2 F + R 2 G = RK M v

(14.7.10)

where the robot arm dynamics are described by M(q), Vm (q, q˙ ), F(q˙ ), G(q), and JM is the motor inertia, BM is given by the rotor damping constant and back emf, and R has diagonal elements containing the gear ratios of the motor/joint couplings. The control input is the motor voltage v(t), with KM the diagonal matrix of motor torque constants. The important conclusion is that the dynamics of the arm-plus-actuators has the same form as the dynamics (Equation (14.7.1) and can be shown to enjoy the same properties of boundedness and linearityin-the-parameters. Therefore, the control methods to be described herein apply to this composite system as well. Similar comments hold for other sorts of actuators such as hydraulic. If the armature inductances of the electric motors are not negligible, then the arm-plus-actuators have a coupled form of dynamics such as those discussed in the section on control of flexible-link and flexible-joint robots. Then special control techniques must be used.

Computed-Torque (CT) Control and Feedback Linearization For many years during the 1960s and 1970s the major techniques for robot dynamics control were based on the computed-torque method, which has many variants, including classical independent joint control. Recently, advanced mathematical techniques based on feedback linearization have been derived. For the rigid-link arms, these are equivalent. It is assumed that the desired motion trajectory for the manipulator qd (t), as determined, for instance, by a path planner, is prescribed. Define the tracking error as e(t ) = qd (t ) − q(t )

(14.7.11)

and differentiate twice to see that the Brunovsky canonical form in Equation (14.7.6) can be written in terms of the state x = [eΤ e·T]T as d e = dt e˙

0 0 

I  e 0 u + 0 e˙  I 

(14.7.12)

u ≡ q˙˙d + M −1 (q) ( N (q, q˙ ) − τ)

(14.7.13)

with

A two-step design procedure now suggests itself. First, use linear system design techniques to select a feedback control u(t) that stabilizes the tracking error system in Equation (14.7.12), then compute the required arm torques using the inverse of Equation (14.7.13), namely, τ = M (q) (q˙˙d − u) + N (q, q˙ )

(14.7.14)

This is a nonlinear feedback control law that guarantees tracking of the desired trajectory. It relies on computing the torque τ that makes the nonlinear dynamics of Equation (14.7.1), equivalent to the linear dynamics of Equation (14.7.12), which is termed feedback linearization. Selecting proportional-plus-derivative (PD) feedback for u(t) results in the PD computed-torque controller

(

)

τ = M (q) q˙˙d + K v e˙ + K p e + N (q, q˙ )

(14.7.15)

and yields the tracking error dynamics e˙˙ = − K v e˙ – Kpe, which is stable as long as the derivative gain matrix Kv and the proportional gain matrix Kp are selected positive definite. It is common to select the gain matrices diagonal, so that stability is ensured as long as all gains are selected positive. © 2005 by CRC Press LLC

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Chapter 14

FIGURE 14.7.1 PD computed-torque controller.

The PD computed-torque controller is shown in Figure 14.7.1, which has a multiloop structure, with a nonlinear inner feedback linearization loop and an outer unity-gain tracking loop. Note that there are actually n outer loops, one for each joint. In this figure, q ≡ [qΤ q·T]T, e ≡ [eΤ e·T]T, q d ≡ [qdT q·dT ]T . To improve steady-state tracking errors, n integrators can be added, one to each joint controller, to place an integrator in the outer tracking loop in the figure. In fact, selecting u(t) as a proportional-plusintegral-plus-derivative controller yields the PID computed-torque controller ε˙ = e

(

)

τ = M (q) q˙˙d + K v e˙ + K p e + Ki ε + N (q, q˙ )

(14.7.16)

which has its own dynamics and gives stable tracking as long as the integral gain Ki is not chosen too large. Example 14.7.1 (Performance of PD and PID Computed-Torque Controllers) The sort of performance to be expected from PD and PID CT controllers is illustrated here. It is desired for a 2-link robot arm to follow, in each of its joints, sinusoidal trajectories qd (t) with period of 2 sec. Ideal PD CT Control. Since CT is theoretically an exact cancellation of nonlinearities, under ideal circumstances the PD CT controller yields performance like that shown in Figure 14.7.2, where the initial tracking errors go to zero quickly, so that each joint perfectly tracks its prescribed trajectory. In this figure are shown the plots for joint 1 tracking error e1(t) and joint 2 tracking error e2(t). PD CT Control with Constant Unknown Disturbance. Now a constant unknown disturbance is added to the robot arm. As shown in Figure 14.7.3, the PD controller now exhibits steady-state tracking errors of e1 = –0.01 rad, e2 = 0.035 rad. PID CT Control. If an integral term is now added to the outer loop to achieve PID CT control, even with a constant unknown disturbance, the simulation results look very much like the original plots in Figure 14.7.2; that is, the integral term has reduced the steady-state tracking errors to zero. ▫ A class of computed torque-like controllers is given by selecting ˆ (q˙˙ − u) + Nˆ τ=M d

(14.7.17)

ˆ , Nˆ are approximations, estimates, or simplified expressions for M(q), N (q, q˙ ) . An example is where M the PD-gravity controller τ = K v e˙ + K p e + G(q) © 2005 by CRC Press LLC

(14.7.18)

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Robotics

14-55

FIGURE 14.7.2 Joint tracking errors using PD computed-torque controller under ideal conditions.

FIGURE 14.7.3 Joint tracking errors using PD computed-torque controller with constant unknown disturbance.

ˆ = I and only includes the gravity nonlinear terms, so that it is very easy to implement which selects M compared to full CT control. This has been used with good results in many applications. ˆ , Nˆ are selected, not as the actual inertia matrix and nonlinear terms, but only as approximations If M or simplified values, it is not always possible to guarantee stable tracking. In fact, the error dynamics of Equation (14.7.12) are then driven by modeling mismatch errors, which can degrade or even destabilize the closed-loop system. Another computed torque-like controller is PID classical joint control, where all nonlinearities of the robot arm are neglected and one selects simply © 2005 by CRC Press LLC

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Chapter 14

ε˙ = e τ = K v e˙ + K p e + Ki ε

(14.7.19)

with the gain matrices diagonal, so that all the joints are decoupled. A PD classical joint controller is shown in Figure 14.7.4, which may seem familiar to many readers. The same figure may be drawn for each joint. In this figure, d(t) represents the neglected nonlinear coupling effects from the other joints, and r is the gear ratio. The motor angle is θ(t) and q(t) is the joint angle. The effective joint inertia and damping are J and B, respectively.

FIGURE 14.7.4 PD classical joint controller.

The simplified classical joint controller is very easy to implement, as no digital computations are needed to determine nonlinear terms. It has been found suitable in many applications if the PD gains are selected high enough, particularly when the gear ratio r is small. Unfortunately, if the gains are selected too high, the control may excite vibratory modes of the links and degrade performance. Moreover, practical applications often benefit by including additional terms such as gravity G(q), desired acceleration feedforward q˙˙d (t ), and various additional nonlinear terms. Example 14.7.2 (Performance of PD-Gravity and Classical Joint Controllers) The sort of performance to be expected from PD-gravity and classical joint controllers is shown in this example It is desired for a 2-link robot arm to follow, in each of its joints, sinusoidal trajectories qd (t) with period of 2 sec. PD-Gravity Controller. The joint 1 and 2 tracking errors are shown in Figure 14.7.5. Note that the errors are small but not exactly zero, a reflection of the fact that the nonlinear coriolis/centripetal terms are missing in the controller. However, the DC error is equal to zero, since gravity compensation is used. (The gravity terms are effectively the “DC terms” of the robot dynamics.) Classical PD Controller. The sort of behavior to be expected from classical (independent joint) control is illustrated in Figure 14.7.6. In this figure, the tracking errors are nonzero, but using large-enough PD gains can often make them small enough. Note that the DC error is no longer equal to zero; the offset is due to ignoring the gravity terms. ▫ Another important CT-like controller is the PD digital controller given by

(

)

τ k = M (qk ) q˙˙dk + K v e˙k + K p ek + N (qk , q˙ k )

(14.7.20)

where the control input can only be computed at the sample times, tk = KT, with T the sample period and k taking on integer values. Digital control is usually required in modern applications, as robot control laws are generally implemented using microprocessors or digital signal processors. Unfortunately, the stability of robot digital controllers has not been generally addressed, so that the traditional approach

© 2005 by CRC Press LLC

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Robotics

14-57

FIGURE 14.7.5 Joint tracking errors using PD-gravity controller.

FIGURE 14.7.6 Joint tracking errors using classical independent joint control.

relies on designing continuous-time controllers, meticulously proving stability, then sampling “fast enough” and holding one’s breath. In practice there are many other problems to be faced in robot controller implementation, including actuator saturation, antiwindup compensation, and so on. See Lewis et al., (1993). Example 14.7.3 (Performance of Digital CT Controllers) The performance of digital robot controllers has several idiosyncrasies of which one should be aware. In this example, it is desired for a 2-link robot arm to follow, in each of its joints, sinusoidal trajectories qd (t) with period of 2 sec. © 2005 by CRC Press LLC

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14-58

Chapter 14

FIGURE 14.7.7 Joint tracking errors using digital computed-torque controller, T = 20 msec.

FIGURE 14.7.8 Joint 2 control torque using digital computed-torque controller, T = 20 msec.

Digital CT Controller. Using a sample period of T = 20 msec yields the tracking error plots shown in Figure 14.7.7. There the performance is quite good for the specific choice of PD gains. The associated control input for joint 2 is shown in Figure 14.7.8. Limit Cycle of Digital Robot Controller. Unacceptable behavior in digital robot controllers can be due to integrator windup problems, selecting too large a sample period, selecting too small a sample period (so © 2005 by CRC Press LLC

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Robotics

14-59

FIGURE 14.7.9 Joint tracking errors using digital computed-torque controller, T = 100 msec.

FIGURE 14.7.10 Joint 2 control torque using digital computed-torque controller, T = 100 msec.

that there is not enough time to perform all control calculations in each period), or the ocurrence of limit cycles. If the sample period is selected as T = 100 msec, everything seems acceptable according to Figure 14.7.9, where the tracking errors are somewhat increased but still small. However, Figure 14.7.10 shoves the control torque for link 2, which has now entered a limit cycle-type behavior due to too large a sample period. ▫ © 2005 by CRC Press LLC

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Chapter 14

Adaptive and Robust Control Computed-torque control works very well when all the dynamical terms M(q), Vm (q, q˙ ), F(q˙ ), G(q) are known. In practice, robot manipulator parameters such as friction coefficients are unknown or change with time, and the masses picked up by the arm are often unknown. Moreover, computing nonlinear terms is difficult to do without exotic microprocessor-based hardware. Therefore, in applications simplified CT controllers that do not compute all nonlinear terms are used (e.g., classical joint control). These methods rely on increasing the PD gains to obtain good performance. However, large control signals may result and stability proofs of such controllers are few and far between. Adaptive and robust control techniques are useful in such situations to improve upon the performance of basic PD control techniques, providing good performance that can be mathematically proven and so relied upon in applications. Such advanced techniques also extend directly to more complicated control objectives such as force control for grinding, polishing, and so on where straight PD methods are inadequate. There are many sorts of adaptive and robust controllers (Lewis et al., 1993). A unifying design technique is presented here that extends as well to intelligent control using neural network and fuzzy logic techniques. Thus, given the desired trajectory qd (t), define the tracking error and filtered tracking error r(t) by e = qd − q

(14.7.21a)

r = e˙ + Λe

(14.7.21b)

with Λ a positive definite design parameter matrix. Common usage is to select Λ diagonal with large positive entries. Then Equation (14.7.21b) is a stable system so that e(t) is bounded as long as the controller guarantees that the filtered error r(t) is bounded. Differentiating Equation (14.7.21b) and invoking Equation (14.7.1), it is seen that the robot dynamics are expressed in terms of the filtered error as Mr˙ = −Vm r + f ( x ) + τ d − τ

(14.7.22)

where the nonlinear robot function is defined as f ( x ) + M (q) (q˙˙d + Λe˙) + Vm (q, q˙ ) (q˙ d + Λe) + F(q˙ ) + G(q)

(14.7.23)

Vector x contains all the time signals needed to compute f (·) and may be defined, for instance, as x
[e T e˙ T qdT q˙ dT q˙˙dT ]T . It is important to note that f (x) contains all the potentially unknown robot arm parameters, except for the Vmr term in Equation (14.7.22), which cancels out in the proofs. A general sort of controller is now derived by setting τ = fˆ + K v r − v(t )

(14.7.24)

with fˆ an estimate of f (x), Kvr = Kv e˙ + KvΛe an outer PD tracking loop, and v(t) an auxiliary signal to provide robustness in the face of disturbances and modeling errors. The estimate fˆ and robustifying signal v(t) are defined differently for adaptive control, robust control, neural net control, fuzzy logic control, etc. The multiloop control structure implied by this scheme is shown in Figure 14.7.11. Adaptive Controller Using nonlinear stability proofs based on Lyapunov or passivity techniques, it can be shown that tracking error stability can be guaranteed by selecting one of a variety of specific controllers. One such is the adaptive controller shown in Figure 14.7.11 and described by the equations

© 2005 by CRC Press LLC

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14-61

Robotics

FIGURE 14.7.11 Adaptive filtered error controller.

τ = W (w) φˆ + K v r φˆ = ΓW T (w) r

(14.7.25)

where Γ is a tuning parameter matrix, generally selected diagonal with positive elements. Matrix W(x) is the (known) regression matrix chosen such that f (x) = W(x)φ, with all the unknown parameters placed into the vector φ; W(x) must be computed off-line in a design phase for the specific robot arm to be controlled (Lewis et al., 1993). In the adaptive controller, the second equation represents the internal dynamics of the controller, where the estimate φˆ of the unknown parameter vector is produced by dynamic on-line tuning. The robot control input τ (t) is then given in terms of φˆ by the first equation. Though it need not be computed to produce the control inputs, the estimate of the nonlinear function is given by fˆ ( x ) = W T(x) φˆ . The adaptive controller shown in Figure 14.7.11 has a multiloop structure with an outer PD tracking loop and an inner nonlinear adaptive loop whose function is to estimate the nonlinear function required for feedback linearization of the robot arm. Robust Saturation Controller Another filtered error controller is the robust saturation controller τ = fˆ + K v r − v  F( x ) − r r ,  v=  F( x ) , − r ε 

r ≥ε

(14.7.26)

r 1250 1252 ts, or x = ±l and z < ts: τxx = 0 and τxy = 0 and τxz = 0, at y = ±a and z > ts, or y = ±l and z < ts: τyy = 0 and τyx = 0 and τyz = 0, at z = 0, z = ts and
x
> a or
y
> a, or z = ts + ta + td: τzz = 0, τzx = 0, and τzy = 0. The symmetry conditions are: at x = 0: u = 0, τyx = 0, and τzx = 0, at y = 0: v = 0, τxy = 0, and τzy = 0. The symmetry conditions basically indicate that the transverse shear stress u and the displacements in the x- and y-directions vanish at the transverse central planes of the x- and y-directions correspondingly. © 2005 by CRC Press LLC

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MEMS Technology

At the interfaces between the die and the die-attaching layer and between the die-attaching layer and the substrate, the bonding conditions are usually assumed to be ideal. This means all displacements are continuous at the interfaces: u
Ss+ = u
Ss –, v
Ss+ = v
Ss –, w
Ss+ = w
Ss –, u
Sa+ = u
Sa–, v
Sa+ = v
Sa–, and w
Sa+ = w
Sa–. Ss is the interface between the die and the attaching layer, and Sa is the interface between the attaching layer and the substrate, respectively. The Relaxation Condition At the die-attaching or wafer-bonding temperature, the attaching or bonding material usually either melts or becomes highly diffusive. So at this temperature, the assembly is thermomechanically relaxed, that is, the stress level of the assembly reaches zero or a minimum. This temperature is usually defined as the relaxation temperature in the following thermal stress/strain discussion. This minimum stress level depends on the physical properties of the assembly materials and the physical/chemical process of the attaching or bonding. For thermal compression of thermal-sonic-compression bonding, relaxation conditions (temperature) may not exist. Analytical Methods The thermomechanical boundary value problem of a die-attach with simple geometry is still complicated enough that a closed exact analytical solution has never been derived after decades of efforts. The thermal mechanical boundary value problem of a sandwich type structure is really not new. It has been identified and studied for decades for thermal expansion of bimetal thermostats and electronic packaging. No closed exact analytical solutions have been derived, even in a narrow temperature range in which the material properties, such as thermal expansion and Young’s modulus, can be approximately treated as constants. Therefore, various approximations and assumptions have been used to obtain approximate analytical solutions. Tomoshenko first established a two-dimensional analytic method for the bimetal thermostat using elastic theory. Assuming that the materials are uniform and isotropic, the temperature field is uniform, td,a,s /a = ε
1, td /ts = O(1) (the thickness of die is at least an order lower than that of the substrate), and ta /td,s = O(ε), an asymptotic solution was derived based on the following assumptions: in the die and substrate, horizontal displacements are linear functions of z, the vertical displacements are a second-order function of z, and the displacements in the die-attaching layer are linear functions of z ( Tomoshenko 1925). Extending Tomoshenko’s elastic theory of bi-metal thermostats, Suhir developed a practical onedimensional model for horizontal distributions of lateral normal (tensile) stress, τN, shear stress, τS, and peeling (transverse normal) stress, τP , at the die/attaching layer interface using the condition of horizontal displacement compatibility at the interface (Suhir 1987):  1 − υ 2 1 − υ 2 ( t + t )2  d s s d  τS =  + + 4D  Ed t d  Est s  

−1 2

2  1 − υ 2s 1 − υd2 ( t d + t s )  τ N =  + + 4D  Ed t d  Es t s 

 t t 3E t t 3E τ P = 6  s d d 2 − d s s 2  12 1 − υd 12 1 − υ s

(

)

(

)

© 2005 by CRC Press LLC

−1 2

(α

 Ed t d  1 + t +t  t ( d s ) 4D 1 − υ2 d  d

(

)

s

− α d )(T − TR )

Sin nh ( Ax ) Cosh ( Al )

(

(15.4.9a)

    α − α T − T 1 − Cosh ( Ax )  (15.4.9b) d )( R) ( s  Cosh ( Al )  

−1   3 3 3   2t a + t d + 2t s   Ed t d + Eat a + Es t s   3G 3G 3G   1 − υ 2 2 1 − υa 1 − υ s2 d s   d   a 

(α s − α d )(T − TR )Cosh( Ax) Cosh( Al)

−1

 td 2t a t s   3D + 3G 3D  a S   d

) (

) (

)

   

−1

(15.4.9c)

0866_book.fm Page 52 Thursday, August 5, 2004 3:37 PM

15-52

Chapter 15

where Gd,a,s, Ed,a,s, υd,a,s, and αd,a,s are shear moduli, and tensile elastic moduli, Poisson’s ratios, and coefficients of thermal expansions (CTE) of the die (d), attaching layer (a), and substrate (s) materials, respectively. Also 1/ 2

2 −1/ 2  1 − υ 2s 1 − υd2 ( t d + t s )   t d 2t a ts     A= + + + + Ed t d 4 D   3Gd 3Ga 3Gs   Es t s 

D = D1 + D2 + D3 =

and

 1  Ed t d3 Eat a3 Es t s3 + + 12  1 − υd2 1 − υa2 1− υ 2s 

) (

(

) (

)

(15.4.9d)

   

(15.4.9e)

These results indicate that the normal stress in the die is high at the central area and sharply decreases to zero in a narrow area at the die edges. Both shear and peeling stresses are high at die edge areas and decrease rapidly approaching the die central area. These formulae have been suggested for direct reliability assessment of die-attach for conventional electronic packaging through calculation of maximum normal stress in the die, the maximum shear stress at the interface, and the maximum peeling stress at the die/attaching layer interface to predict the failures of die material, attaching layer material, and the adhesion of die/attaching layer interface (Hu and Pecht 1993). This engineering theory of thermally induced stresses in bimaterial assemblies was extended to a trimaterial assembly for interfacial thermal shear stresses based on the following assumptions: stresses and strains are in the materials’ elastic regions, no singular stresses exist at the assembly edges, and the deflections of the assembly are small. The interfacial shear stresses and assembly deflection are determined without considering the effects of peeling stress (Suhir 2001): τ s1 = −kT1o

sinh(kx) cosh(kl)

(15.4.10a)

τ s 2 = −kT3o

sinh(kx) cosh(kl)

(15.4.10b)

τs1 and τs2 are shear stresses at the first-layer/second-layer (such as die/die-attaching layer) interface and the second-layer/third-layer (such as attaching layer/substrate) interface, respectively, where 1/ 2

2    1/ 2  2k1k2      k12 + k2 2   k= 1 + 1 − γ  2     k1 + k12      2      

T3o = −

(α 3 − α 2 ) λ11 + (α1 − α 2 ) λ13 ∆T , γλ11λ 33

, T1o = −

γλ11λ 33

k1 = ( λ11 k12 )

1/ 2

, k2 = ( λ 33 k23 )

1/ 2

(t1 + t 2 ) + λ + λ , λ = (t 2 + t3 ) + λ + λ , ti (i = 1, 2, 3) , λ11 = 33 1 2 2 3 3Gi 4D 4D 2

ki =

(α 3 − α 2 ) λ13 + (α1 − α 2 ) λ 33 ∆T ,

λ13 =

(t1 + t 2 )(t 2 + t3 ) − λ 4D

© 2005 by CRC Press LLC

2

, λi =

1 Ei t i

2

(i = 1, 2, 3)

, γ = 1−

λ132 , λ11λ13

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15-53

MEMS Technology

Ei is the elastic modulus of the ith component, ti is the thickness of ith component, and ∆T = T – To is the temperature deviation from the relaxing temperature at which the stresses and strains vanish. The corresponding peeling stresses at the interfaces are h kx cosh , j = 1, 2 τ Pj = m j  χ0 (s, k)V0 (sx) + χ 2 (s, k)V2 (sx) + C j cosh kl

(15.4.11)

where s 2 osh kl ) s 2 V0 (sl) − 1 tanh kl + k V3(sl)(1 − 1 c χ0 (s, k) = , k V0 (sl) − 1V1(sl) + V2 (sl)V3(sl) s 2 s 2 [V2 (sl) − 1]tanh kl + k V1(sl)(1 − 1 / cosh kl) , V0 (sx) = cosh sx cos sx , χ 2 (s, k) = k [V0 (sl) − 1]V1(sl) + V2 (sl)V3(sl)

V2 = sinh sx sin sx , s 4 =

δ12 =

 δ s14 + s24  1+ 1− 2  4 s14 + ss4 

(

)

 D1 + D2 D2 + D3  4 4  , s1 = 4 D D δ , s2 = 4D D δ , 1 2 12 2 3 23 

t1 t t t D + 2 , δ 23 = 3* + 2* , and δ = δ12δ 23 D1D2 D3 E1* 2E2* E3 E 2

where j = 1 refers the first-layer/second-layer interface and j = 2 refers to the second-layer/third-layer interface. It is worthwhile to indicate that these interfacial shear, peeling, and normal stresses are derived from engineering theory. They were not mathematically solved from the displacement governing equations; they were derived from interfacial mechanical equilibrium equations of the die-attach assembly (Suhir 2001). Recently, Ru (2002) developed a nonlocal model assuming that the longitudinal displacement at a bimaterial elastic beam interface also depends on both the shear stress and the second gradient of the interfacial shear stress. Ru’s results satisfy both of the requirements of the zero-longitudinal force and zero shear stress at the free edges. The peeling stress calculated from this model is equilibrated at the interface (Ru 2002).

τ S = α∆T

τ P = α∆T

cosh ( ρ1l )sinh ( ρ2l )sinh ( ρ1x )cos ( ρ2 x ) − sinh ( ρ1l )cos ( ρ2l )cosh ( ρ1x )sin ( ρ2 x ) Σ

t1D2 − D1t 2  ρ1 cosh ( ρ1l )sin ( ρ2l ) − ρ2 sinh ( ρ1l )cos ( ρ2l )  cosh ( ρ1x )cos ( ρ2 x ) −  Σ 2( D1 + D2 )  

ρ1 sinh ( ρ1l )cos ( ρ2l ) + ρ2 cosh ( ρ1l )sin ( ρ2l )  sinh ( ρ1x )sin ( ρ2 x )     Σ  where © 2005 by CRC Press LLC

(15.4.12)

(15.4.13)

0866_book.fm Page 54 Thursday, August 5, 2004 3:37 PM

15-54

Chapter 15

 A 1  ρ1 =  + −1 2   4B 2B   2B 1 2 − A  ρ2 =    4B  A=

,

12

,

t1K1 G1 + t 2 K 2 G2 λ1 + λ 2

B=

12

(t1 +

2 + t 2 2) D1 + D2

2

t13d1K1 G1 + t 23d2 K 2 G2 λ1 + λ 2

(t1 +

2 + t 2 2) D1 + D2

2

,

,

K1, K2, d1, and d2 material property related constants (Ru, 2002). The numerical results derived from these formulas are reasonably consistent with those of finite element analysis. Approximate analytical solutions can provide quick and global stress/strains assessment. In the elastic region, limited parametric studies of the effects of materials, sizes, and process condition on stress/strain distributions can be analyzed using analytical results. However, as it has been noted, the analytical method has limitations as indicated in the following paragraph. For conventional electronic packaging, material-level permanent damages are a critical concern for die-attach thermomechanical reliability. However, for MEMS packaging, especially high-temperature MEMS, we are interested in precise and quantitative three-dimensional stress/strain distributions in order to examine the mechanical effects of packaging stress/strain on device mechanical operation, before the stress/strain reaches the level causing permanent material damage. As indicated in the introduction of this section, various MEMS sensors for aerospace engine monitoring and combustion control operate in a wide range of high temperatures and pressures. Therefore, the stress formulae derived using constant material properties and zero external load may no longer be suitable. The temperature dependence of material properties makes the die-attach boundary value problem much more complicated, and elastic/plastic nonlinear material properties make the derivation of a closed analytical solution to this boundary value problem almost impossible. In addition to the complexity caused by temperature dependent nonlinear material properties, the boundary geometry of the MEMS die may generate another obstacle to analytical solutions. As a simple example, a diaphragm-based piezoresistive MEMS sensor may have a round-shaped diaphragm at the center of a square die. Neither Cartesian nor cylindrical coordinates would fit the boundary geometry of this problem. Boundary geometries of MEMS devices can be much more complicated than those of the diaphragm-based sensor. Therefore, it becomes very difficult to derive an analytical solution of a boundary value problem for a high-temperature MEMS die-attach assembly. Numerical analysis is a powerful method for solving thermomechanical boundary value problems, such as the stress/strain distribution of a die-attach assembly. Finite element analysis (FEA) is the most sophisticated numerical method that can deal with all the previously stated complexities, including the external mechanical load. Chang (2002) successfully used FEA to calculate deflection angles of a nickelcoated microplatform in static magnetic fields generated by a microfabricated DC coil (Chang 2002). Another advantage of FEA is its ability to solve coupled thermal and mechanical loads. The next subsection provides an example in which FEA is used to model stress/strain distributions of an Au thick-film based SiC die-attach over a wide temperature range. This example illustrates the application of FEA in MEMS packaging. In order to avoid possible redundancy the fundamentals of FEA are not covered in © 2005 by CRC Press LLC

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2.2 mm 1.3 mm 0.3 mm

50 µm

20 µm 0.69 mm

AIN 4 mm

FIGURE 15.4.5 A schematic diagram of SiC pressure sensor die-attach using Au as attaching material for FEA simulation.

detail in this section. Readers can refer to other chapters of this handbook for both FEA fundamentals and practice.

Numerical Analysis of Die-attach Thermal Stresses and Strains In this section, FEA simulation of stress/strain distributions of a SiC die-attach assembly over a wide temperature range is used to illustrate the application of numerical analysis for thermal stress/strain analysis in MEMS packaging. The SiC die includes a diaphragm structure. Diaphragm structures are used often in MEMS sensors and packaging, such as piezoresistive and capacitive acoustic (Chen et al. 2004) and pressure sensors (Beheim et al. 2001; Okojie et al. 2001). As shown in Figure 15.4.5, a SiC die (2.2 mm × 2.2 mm × 0.30 mm) is attached to an AlN substrate (4 mm × 4 mm × 0.69 mm) with an Au thick-film layer (20 µm). The round diaphragm of the SiC die has a radius of 0.65 mm and a thickness of 50 microns. The die-attach is processed (cured) at 600°C and then cooled down to room temperature. It is assumed that the die-attach assembly is relaxed at 600°C before it is cooled down to room temperature, and that the cooling process is rapid enough such that the stress/strains caused by temperature deviation from the relaxing temperature are accumulated without relaxation/creeping. Because of the symmetry of the die-attach assembly, the stress/strain distributions of only a quarter of the assembly are simulated at room temperature. Within the ANSYS FEA software tool, the Geometric Non-linearity Option was enabled and Automatic Time-step Process was employed with 10% of the full temperature loading (temperature drops from 600 to 25°C) as the initial time-step. A line search was used with the Newton-Raphson iterative algorithm to improve and accelerate convergence. The basic mechanical properties of materials included in the die-attach (Au, SiC, and AlN) and the temperature dependences of these material properties that were used in FEA simulation are listed in Table 15.4.2. The temperature dependence of the Young’s modulus of AlN in a wide temperature range has not been reported, so it is extrapolated as a constant from the room temperature value. The Poisson’s ratios of single-crystal 6H-SiC and AlN have not been published, so they are estimated from those of other carbides and nitrides, respectively. The thermal and mechanical properties of 6H-SiC are assumed to be isotropic. Material CTEs (mismatches) have the most significant effects on the die-attach assembly thermal stresses/strains. Temperature-dependent CTE data of both substrate and die materials (AlN and SiC), as listed in Table 15.4.2, were used for the FEA simulation. The yield strength data of Au reported in the literature covers a wide range due to different heattreatments applied to the materials used for these experimental tests. In this simulation, a low Au yield strength (250 psi) and temperature-dependent plasticity of Au are used. These simulation results should be close to stress/strain configurations of an Au thick-film based die-attach assembly sufficiently annealed (stored and operated) at 600°C. Plasticity of gold thick-film is a primary factor responsible for thermally induced stresses in a die-attach assembly. Only limited data of Au plasticity are available in published literature. Temperature dependent ultimate tensile strength of Au have been reported (King et al. 1988). © 2005 by CRC Press LLC

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TABLE 15.4.2 Materials Properties of Au, AlN, Al2O3, and SiC Used for FEA Temperature (°C)

CTE (xE–6/C)

Au material properties 15 14.04 20 14.24 105 14.71 205 15.23 305 15.75 405 16.29 505 16.89 605 17.58 705 18.38 SiC material properties –15 1.78 20 2.07 105 2.78 205 305 4.44 505 6.11 605 705 7.78 AlN material properties –15 2.89 20 3.15 105 3.79 205 305 5.29 405 505 6.79 605 705 8.29

E (xE6 psi)

υ

11.09 10.99 10.83 10.59 10.28 9.92 9.51 9.03 8.50

0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44

66.72 66.72 66.42 66.08 65.74 65.05 64.71 64.36

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00

0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

Sources: H.K. Charles and G.V. Clatterbaugh 1994; F.P. McCluskey, R.R. Grzybowski, and T. Podlesak 1996.

An elongation of 65% was reported for annealed gold (Suhir 1987). The yield strength of gold is 250 psi (King et al. 1988) at room temperature, and it is assumed to be independent of temperature in the calculation. Based on these material properties, approximations, and assumptions, temperature-dependent plasticity behavior of gold was constructed as shown in Figure 15.4.6 (Lin and Chen 2002). Figure 15.4.7a and Figure 15.4.7b show the von-Mises stress (the second deviatoric stress invariant) contours for the SiC die from an SiC/Au/AlN die-attach assembly at room temperature with an assumed stress relaxing temperature of 600°C. Von-Mises stress is often used to determine if yielding or damage occurs. Since the maximum von-Mises stress of the die is much lower than the yield strength of SiC material, it is anticipated that there is no material level damage to the SiC die. The level of stress around the inner circle (etching wall) of the bottom of the die is high, whereas the stress level is lower towards the die edges and corners of the bottom of the die. The stress level also significantly decreases toward the top of the die. The level of stress in the diaphragm area is significantly higher than those in the areas surrounding the diaphragm. The stress level of the diaphragm area can be used to estimate the effects of die-attach thermal stress on operation and configuration of diaphragm-based MEMS devices. Figure 15.4.7(c) and Figure 15.4.7(d) show the distribution of peeling stress, which is the normal stress perpendicular to the die/attaching-layer interface, at the die/attaching layer interface. Positive peeling stress indicates that the stress is in tension, and the negative stress indicates the stress is compressive. The tension is relatively high around the inner circle of the bottom of the die. At the corner and edge areas, the peeling stress is negative, so it is compressive. Usually, tensile stress causes material damage/cracks. © 2005 by CRC Press LLC

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Plasticity for Gold Thick Film 1800

stress (psi)

-15C 20C 105C 305C 505C 705C

0 0

0.01

0.02

0.03

0.04

0.05

0.06

Plastic Strain (in/in)

FIGURE 15.4.6 Temperature-dependent plasticity of gold thick film; –15°C and 20°C data overlap each other in the figure.

Compared with the mechanical strength of SiC material, the maximum tensile stress of the SiC die is still low, so no die material damage is anticipated. Figure 15.4.7(e) and Figure 15.4.7(f) show the die shear stress (τzx) distribution contour. The shear stress of the bottom of the die is high at the area close to the edge. Both peeling and shear stresses decrease rapidly toward the top of the die. The peeling and shear stresses in the diaphragm area are also higher than those outside the diaphragm. The peeling (tensile) stress at the die/attaching interface can be directly compared with the die-attach adhesion tensile strength data for reliability analysis and failure mode prediction. The shear stress at the die/attachinglayer interface can be directly compared to the shear strength of the die-attach material and the shear adhesion strength of the die/attaching layer interface for failure/reliability analysis. Figure 15.4.7(g) and Figure 15.4.7(h) show von Mises stress contours on the top and bottom of the Au attaching layer. Von Mises stress is high at the corner region, illustrating that the shear component may dominate the von Mises stress at the die/attaching layer interface area. Figure 15.4.7(i–l) show peeling and shear stresses on the top and bottom of the die-attach layer. The peeling and shear stresses on the top of the attaching layer are exactly the same as those on the bottom of the die (the stress scale bars are in different scales). The peeling and shear stress contours on the bottom of the attaching layer are the same as those on the substrate. The strap structure of the shear stress indicates that τzx increases with the distance from the x = 0 neutral point. Because the die-attach assembly has 90o symmetry in the horizontal plane, the shear contour of τzy can be obtained with 90o rotation of that of τzx. Peeling and shear stresses at the attach layer/substrate interface can be compared with the experimental measurements of the adhesion strength of the attach layer/substrate interface for reliability assessment and failuremechanism prediction. Figure 15.4.7(m) and Figure 15.4.7(n) show the equivalent plastic strain (EPS) distributions of the die-attaching layer after cooling to room temperature and heating back to 500°C (one thermal cycle). The thermal stresses/strains on the diaphragm may cause changes in device configuration and operation. For example, thermal stress can directly generate thermal shift of the piezoresistance of diaphragmbased piezoresistive MEMS sensors such as pressure and acoustic sensors. A thermal shift of the capacitance of a diaphragm-based capacitive sensor due to the deformation of the diaphragm under thermal stress may also occur. These die-attach-related thermal stresses may have significant thermal effects on the packaged devices. In addition to the predications of packaging-related reliability and failure analysis, FEA results may also help MEMS packaging designers to simulate/predict packaging effects on devices. Using the FEA results may allow designers to reduce the parasitic packaging effects through materials selection and structure optimization. In order to study the effects of die-attach materials and die-attach geometries, single-parametric dependence of stress/strain on substrate material, die size, die-attaching layer thickness, and environment/processing temperature can be simulated using FEA (Lin and Chen 2002). © 2005 by CRC Press LLC

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(a) von Mises stresses of SiC with AIN substrate.

(b) von Mises stresses of SiC with AIN substrate.

(c) Peeling stresses of SiC die with AIN substrate.

(d) Peeling stresses of SiC with AIN substrate.

(e) Shear stresses of SiC die with AIN substrate.

(f) Shear stresses of SiC with AIN substrate.

FIGURE 15.4.7 Stress and strain contours of SiC diaphragm attached to AlN substrate with Au thick-film. The unit for stresses is MPa. (From P. McCluskey, K. Meyyappan, and L-Y. Chen, Proceedings of 2004 IEEE Aerospace Conferences. With permission.)

In these FEA simulations, the boundary conditions at both SiC/Au-thick-film and Au-thick-film/substrate interfaces are assumed to be ideal. This means that there is no slip at either interface under shear stress. Mathematically, this assumption implies that all three displacement components are continuous at the two interfaces. We made this assumption since limited interfacial thermomechanical properties of this material system are available. Further, it is difficult to either numerically model or experimentally measure the thermal and mechanical properties of interfaces, especially the temperature dependencies of these interfacial thermomechanical properties. This assumption may have limited applicability for those relatively “loosely” bonded interfaces, especially at elevated temperatures. © 2005 by CRC Press LLC

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(g) von Mises stresses of top of die attaching layer.

(h) von Mises stresses of bottom of die attaching layer.

(i) Peeling stresses of top of die attaching layer.

(j) Peeling stresses of bottom of die attaching layer.

(k) Shear stresses of top of die attaching layer.

(l) Shear stresses of bottom of die attaching layer.

FIGURE 15.4.7 (continued).

The experimental measurement of three-dimensional die-attach stress/strain distributions and detection of the thermal mechanical failure of the Au attaching layer sandwiched between an SiC die and a ceramic substrate can be challenging, especially at elevated temperatures. Therefore, it is important and helpful to numerically validate the FEA results. First, the effect of mesh design/density on the numerical results can be examined by changing the three-dimensional FEA mesh. The convergence of the numerical results with increasing mesh density usually is a good indication of valid results. Second, the stability of the stress/strain dependence on the material properties, such as the CTE of the substrate, can also be used to examine the numerical stability with a small virtual perturbation (upward and downward shifts) of the CTE-temperature curve of the substrate material. Third, the numerical results should (approximately) © 2005 by CRC Press LLC

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(m) Equivalent Plastic Strain (AIN based substrate) of the top of the die attaching layer at room temperature.

Chapter 15

(n) Equivalent Plastic Strain (AIN based substrate) of the top of the die attaching layer at 500°C.

FIGURE 15.4.7 (continued).

satisfy all the boundary/interface and symmetry conditions. This is a simple and quick way to physically validate the numerical results. Fourth, the trends of stress/strain can be compared with those predicted by approximate analytical results if they are available.

Low-Stress Die-Attach Advanced low-stress die attach methods for MEMS packaging will be discussed in this section. Low-Stress Die-Attach Materials As discussed earlier, the die-attach stress/strain increases rapidly with the temperature deviation from TR. TR is the temperature at which the die-attach is thermomechanically relaxed (no stress or almost no stress). Usually, the physical/chemical process of die attaching determines TR such that it is close to the die-attach process/curing temperature. Therefore, the die-attach process temperature is an important process parameter determining die-attach stress/strain and thermal reliability. The ideal situation is that TR is at the center of the operating temperature range (Lin and Chen 2002). A process called lowtemperature transient liquid phase (LTTLP) bonding allows die to be bonded at a relatively low temperature, from 60 to 160°C, yet the bonding layer will remelt only at a much higher temperature. The principle of this bonding process is that the metal(s) of the metallization on the substrate and/or the die dissolve into the melted alloy and form a new alloy that isothermally solidifies (Roman and Eagar 1992). In the mid-1960s it was demonstrated that indium, which melts at 156.7°C, forms InxAuy alloy with a melting temperature of ~300°C (Bernstein 1966). The research on LTTLP has been active again since the early 1990s because of the need for low-processing-temperature die-attach materials for reliable largesize high-power devices packaging. Various low-melting-temperature solders of indium, tin, bismuth, and cadmium have been tested for applications with base metals of gold, silver, and copper (Roman and Eagar 1992). Table 15.4.3 shows a recompiled table of melting temperature and tensile strength of these material systems. Low stress die-attach materials are attracting more attention because of the emerging needs of MEMS packaging. Mechanically, most LTTLP bonds are as strong as an equivalent conventional solder bond (Roman and Eagar 1992). Typically, the remelting temperature and the mechanical strength of LTTLP bonds depend on the processing temperature and temperature profile (temperature vs. time). Low-Stress Die-Attach Structures Li and Tseng suggested a “four-dot” low stress die-attach approach for packaging of MEMS accelerometers that reduced die surface strains caused by CTE mismatches (Li 2001). FEA simulation results indicated that a significant stress reduction could be achieved when the area of die-bonding “dots” located at four die corners were small. This die-attach structure can dramatically reduce or eliminate the transverse © 2005 by CRC Press LLC

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TABLE 15.4.3 Melting Temperatures and Mechanical Strength of Various Solder Materials and Remelting Temperatures of New Alloys Formed with Base materials of Ag, Au, and Cu, Listed with Process Temperatures and Times Solder

Melting Point (°C)

Tensile Strength (psi)

Base Metal

Curing Temp (°C)

Curing Time (hrs)

In In In In97Ag3 In97Ag3 In97Ag3 In80Pb15Ag5 In80Pb15Ag5 In80Pb15Ag5 In52Sn48 In52Sn48 In52Sn48 In44Sn42Cd14 In44Sn42Cd14

156.7 156.7 156.7 146 146 146 149 149 149 118 118 118 93 93

575 575 575 800 800 800 2550 2550 2550 1720 1720 1720 2632 2632

Ag Au Cu Ag Au Cu Ag Au Cu Ag Au Cu Ag Au

In44Sn42Cd14 Bi58Sn42 Bi58Sn42

93 138 138

2632 8000 8000

Cu Ag Au

Bi58Sn42 Bi46Sn34Pb20 Bi46Sn34Pb20 Bi46Sn34Pb20 Bi55.5Pb44.5 Bi55.5Pb44.5 Bi55.5Pb44.5 Bi50Pb26.7Sn13.3Cd10 Bi50Pb26.7Sn13.3Cd10 Bi50Pb26.7Sn13.3Cd10 Bi49In21Pb18Sn12

138 100 100 100 124 124 124 70 70 70 58

8000

6400 6400 6400 5990 5990 5990 6300

Cu Ag Au Cu Ag Au Cu Ag Au Cu Ag

165 165 165 155 155 155 155 155 155 130 130 130 110 110 110 110 150 150 150 150 110 110 110 135 130

3 1 3 18 1 3 24 1 24 60 2 1 20 18 72 1 2 11 96 1 26 72 26 25 26

450 500 500 550 500 500 500 400 500 130 400 500 500 250 400 500 500 200 200 500 500 250 500 500 165

Bi49In21Pb18Sn12 Bi49In21Pb18Sn12 Bi44.7Pb22.6In19.1Sn8.3Cd5.3

58 58 47

6300 6300 5400

Au Cu Ag

Bi44.7Pb22.6In19.1Sn8.3Cd5.3 Bi44.7Pb22.6In19.1Sn8.3Cd5.3 Bi67In33 Bi67In33 Bi67In33 In66.3Bi33.7 In66.3Bi33.7 In66.3Bi33.7 Bi60Cd40 Bi60Cd40 Bi60Cd40 In74Cd26 In74Cd26 In74Cd26 Ga Ga Ga

47 47 109 109 109 72 72 72 144 144 144 123 123 123 29.8 29.8 29.8

5400 5400

Au Cu Ag Au Cu Ag Au Cu Ag Au Cu Ag Au Cu Ag Au Cu

80 110 110 80 110 110 110 110 110 110 110 120

72 26 26 72 48 26 26 26 48 26 10 13

500 1 represents increasing failure rates in the wearout region. By appropriately varying the two parameters, the model can be made to fit a wide range of experimental data. It is common to come across linear plots of data based on the appropriate distribution. Such a plot allows easy derivation of reliability parameters from the slopes and intercepts as well as well a quantitative estimate of the fit. In the case of the Weibull distribution, from Equation 15.5.9, F (t ) = 1 − e

( τ)

− t

γ

Rearranging and performing successive logarithmic operations yields ln  − ln {1 − F (t )} = γ ln(t ) − γ ln(τ)

(15.5.13)

Comparing this to the equation y = mx + c results in y = ln  − ln {1 − F (t )} , m = γ and c = −γ ln(τ) The Weibull probability plot is obtained by plotting ln[–ln{1 – F(t)}] vs. ln(t), where F values may be obtained from sample failure data using the following equation: F (t i ) =

i − 0.3 n + 0.4

where γ and τ are obtained from the line fitted to the data.9 © 2005 by CRC Press LLC

(15.5.14)

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Accelerated Stress Testing To reduce the development and test time facilitating a rapid deployment to market, accelerated life testing is commonly used to force products to fail more quickly than they would under normal operating conditions, the assumption being that the failure mechanism remains the same and failures in time under normal and accelerated conditions scale only by time. Thus, for true acceleration Fa (t ) = F ( AF .t ) where Fa(t) and F(t) are the failure CDF’s under accelerated stress and normal operating conditions, respectively, and AF > 1 is the acceleration factor.10 Where an empirical relationship cannot be defined directly from the tests, existing mathematical models governing similar stress acceleration modes or reactions are made to determine the acceleration factor. For example, the Arrhenius model based on the well-known reaction rate equation is used when stress testing involves raising the temperature to hasten the aging process. The Arrhenius rate reaction equation is given by R = Ae

−

Ea K bT

(15.5.15)

where R is the reaction rate, A is a constant, Ea is the activation energy, Kb is Boltzmann’s constant, and T is the absolute temperature.11 To formulate the stress model, from Equation 15.5.12 we define

L = Be

Ea K bT

(15.5.16)

where L is a life measure such as characteristic time, mean time, median life, etc., B is a model parameter, and T is the temperature (stress) in Kelvin.11 Thus the acceleration factor AF, defined as the ratio of the life during use to that at an increased stress, may be given as11 Ea

E  1 a

1 

 −   L Be Tuse K T  T AF = use = = e  b  use accl   Ea L accl Be Taccl

(15.5.17)

Models also exist that represent other stress acceleration modes. Accelerated tests also help define the operational and destruct limits of a device and clarify the effects of prolonged operating and nonoperating times on devices. Figure 15.5.5 shows a schematic plot of stress vs. the number of failures for an arbitrary device.1 The stress limits for a device are also denoted. The operational limits represent the stress needed to cause the device to malfunction. The device, however, must be able to return to its normal mode on the removal of this stress. The destruct limits represent irreversible failure. The curves about each of these limits represent the failure distributions caused by the specific stresses. The Digital Micromirror Device (DMD) The DMD is one of the first commercialized MEMS devices and possesses “typical” microsystem characteristics, namely, moving parts, contact-type motion, and integrated electronics. It is used to digitally redirect light for visual display applications. In comparison to other display technologies, the DMD is more efficient in providing increased brightness, higher contrast ratio, and better reliability in conjunction with simpler optics in a compact form factor.12 The device consists of an array of square, aluminum alloy reflective mirrors with each square mirror being ~14 micron on a side.1 Each mirror is mounted on an aluminum hinge and is capable of tilting © 2005 by CRC Press LLC

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Lower Destruct Limit

Lower Operating Limit

Upper Operating Limit

Product Specs

Operating margin

Upper Destruct Limit

Operating margin

Destruct Margin

Destruct Margin

Stress FIGURE 15.5.5 Accelerated test-to-failure methodology to determine operational and destruct parameters. (Source: Douglass, M.R. 2003. DMD reliability: a MEMS success story, Proc. of Int. Society for Optical Eng. (SPIE), 4980,1–11.) Mirror –10 deg Mirror +10 deg

Hinge Yoke Spring Tip

CMOS Substrate

FIGURE 15.5.6 Schematic diagram of two DMD pixels with the mirrors made transparent in order to highlight the underlying device architecture. (Source: Sontheimer, A.B. 2002. Digital micromirror device (DMD) hinge memory lifetime reliability modeling, IEEE Int. Reliability Physics Symposium Proc., Dallas, TX, April 7–11, 118–121.)

back and forth from +10° to –10° corresponding to the “on” and “off ” positions of the mirror.5 Mirror rotation is accomplished by tilting the yoke (Figure 15.5.6) using electrostatic attraction. The “on-mirrors” reflect light toward a projection lens to create an on-screen image, and the “off-mirrors” reflect light toward an absorber. The frequency of operation of each mirror during each video frame determines the light intensity. Color images are obtained using either a color wheel or a three-chip setup. Each mechanical structure rests on top of a standard static random access memory (SRAM) cell and is thus individually addressable. Integration with electronics is achieved using CMOS-compatible surface micromachining technology. The close packing density obtained by positioning the electronics and mechanical support structures under the mirrors avoids light diffraction and improves the contrast ratio. Highlights of the technology are shown in Table 15.5.5.5 Reliability testing of the DMD technology followed the approach shown schematically in Figure 15.5.7.1 Failure mechanisms were investigated based on the Failure Modes and Effects Analysis method, as well as characterization tests performed on the fabricated devices. Table 15.5.6 summarizes the failure mechanisms investigated specific to the DMD MEMS superstructure and the methods adopted where necessary to mitigate such failures. The primary lifetime-limiting parameter in the DMD was found to be hinge memory. What follows is a review of the still ongoing reliability modeling approach to characterize this failure mechanism.13 Characterization methods and lifetime estimates are also summarized. © 2005 by CRC Press LLC

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TABLE 15.5.5 DMD Technology Details Device Highlights

Moving Parts: >0.5 Million

Motion Process Packaging Testing

Discrete, contact type; ~ 900*109 contacts/moving part Low temperature; dry-etched wafer level removal of sacrificial layer Die separation after sacrificial layer removal; hermetic welded lid package Electro-optical, prior to die separation

Source: Van Kessel, P.F., Hornbeck, L.J., Meier, R.E., and Douglass, M.R. 1998. A MEMS-based projection display, Proc. of the IEEE, 86(8), August, 1687–1704.

FIGURE 15.5.7 Approach to DMD reliability testing. (Based on References 1, 5, and 8.)

TABLE 15.5.6 DMD Failure Mechanisms Investigated During Reliability Testing Failure Mechanism Investigated Hinge fatigue Hinge memorya

Failure Cause

Stuck mirrors

Material High temperature; metal creep; highduty cycle (on-side to off-side ratio) Particles; stiction; UV light exposure

Damage from environment

Shock, vibration, acceleration, temperature shock, cycling, and storage

Effects on Lifetime No Yes Yes

No

Mitigation N/A Improved hinge material; reversed duty cycle actuation Improved packaging and fabrication methods; spring tip design innovation, dry etch release, passivation layer, hermetic sealing UV filter ( 0.3, or if pressure changes due to viscous forces are sufficiently large. Experiments in gaseous microducts confirm the above arguments. For both low- and high-Machnumber flows, pressure gradients in long microchannels are nonconstant, consistent with the compressible flow equations. Such experiments were conducted by, among others, Prud’homme et al. (1986), Pfahler et al. (1991), van den Berg et al. (1993), Liu et al. (1993; 1995), Pong et al. (1994), Harley et al. (1995), Piekos and Breuer (1996), Arkilic (1997), and Arkilic et al. (1995; 1997a; 1997b). Sample results will be presented in the following subsection. There are three additional scenarios in which significant pressure and density changes may take place without inertial, viscous, or thermal effects. First is the case of quasi-static compression/expansion of a gas in, for example, a piston-cylinder arrangement. The resulting compressibility effects are, however, compressibility of the fluid and not of the flow. Two other situations where compressibility effects must also be considered are problems with length-scales comparable to the scale height of the atmosphere and rapidly varying flows as in sound propagation (Lighthill 1963).

Boundary Conditions The continuum equations of motion described earlier require a certain number of initial and boundary conditions for proper mathematical formulation of flow problems. In this subsection, we describe the boundary conditions at a fluid-solid interface. Boundary conditions in the inviscid flow theory pertain only to the velocity component normal to a solid surface. The highest spatial derivative of velocity in the inviscid equations of motion is first-order, and only one velocity boundary condition at the surface is admissible. The normal velocity component at a fluid-solid interface is specified, and no statement can be made regarding the tangential velocity component. The normal-velocity condition simply states that a fluid-particle path cannot go through an impermeable wall. Real fluids are of course viscous, and the corresponding momentum equation has second-order derivatives of velocity, thus requiring an additional boundary condition on the velocity component tangential to a solid surface. Traditionally, the no-slip condition at a fluid-solid interface is enforced in the momentum equation and an analogous no-temperature-jump condition is applied in the energy equation. The notion underlying the no-slip/no-jump condition is that within the fluid there cannot be any finite discontinuities of © 2005 by CRC Press LLC

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velocity/temperature. Those would involve infinite velocity/temperature gradients and so produce infinite viscous stress/heat flux that would destroy the discontinuity in infinitesimal time. The interaction between a fluid particle and a wall is similar to that between neighboring fluid particles, and therefore no discontinuities are allowed at the fluid-solid interface either. In other words, the fluid velocity must be zero relative to the surface and the fluid temperature must equal to that of the surface. But strictly speaking, those two boundary conditions are valid only if the fluid flow adjacent to the surface is in thermodynamic equilibrium. This requires an infinitely high frequency of collisions between the fluid and the solid surface. In practice, the no-slip/no-jump condition leads to fairly accurate predictions as long as Kn < 0.001 (for gases). Beyond that, the collision frequency is simply not high enough to ensure equilibrium and a certain degree of tangential-velocity slip and temperature jump must be allowed. This is a case frequently encountered in MEMS flows, and we develop the appropriate relations in this subsection. For both liquids and gases, the linear Navier boundary condition empirically relates the tangential velocity slip at the wall ∆u w to the local shear: ∆u w = u fluid − uwall = Ls

∂u ∂y w

(15.6.37)

∂u is the strain rate computed at the wall. In most practical ∂y w situations, the slip length is so small that the no-slip condition holds. In MEMS applications, however, that may not be the case. Once again we defer the discussion of liquids to a later section and focus for now on gases. Assuming isothermal conditions prevail, the above slip relation has been rigorously derived by Maxwell (1879) from considerations of the kinetic theory of dilute, monatomic gases. Gas molecules, modeled as rigid spheres, continuously strike and reflect from a solid surface, just as they continuously collide with each other. For an idealized perfectly smooth wall (at the molecular scale), the incident angle exactly equals the reflected angle and the molecules conserve their tangential momentum and thus exert no shear on the wall. This is termed specular reflection and results in perfect slip at the wall. For an extremely rough wall, on the other hand, the molecules reflect at some random angle uncorrelated with their entry angle. This perfectly diffuse reflection results in zero tangential-momentum for the reflected fluid molecules to be balanced by a finite slip velocity in order to account for the shear stress transmitted to the wall. A force balance near the wall leads to the following expression for the slip velocity: where L s is the constant slip length, and

ugas − uwall = L

∂u ∂y w

(15.6.38)

where L is the mean free path. The right-hand side can be considered as the first term in an infinite Taylor series, sufficient if the mean free path is relatively small enough. Equation 15.6.38 states that significant slip occurs only if the mean velocity of the molecules varies appreciably over a distance of one mean free path. This is the case, for example, in vacuum applications and/or flow in microdevices. The number of collisions between the fluid molecules and the solid in those cases is not large enough for even an approximate flow equilibrium to be established. Furthermore, additional (nonlinear) terms in the Taylor series would be needed as L increases and the flow is further removed from the equilibrium state. For real walls some molecules reflect diffusively and some reflect specularly. In other words, a portion of the momentum of the incident molecules is lost to the wall and a (typically smaller) portion is retained by the reflected molecules. The tangential-momentum-accommodation coefficient σv is defined as the © 2005 by CRC Press LLC

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fraction of molecules reflected diffusively. This coefficient depends on the fluid, the solid and the surface finish, and has been determined experimentally to be between 0.2 and 0.8 (Thomas and Lord 1974; Seidl and Steiheil 1974; Porodnov et al. 1974; Arkilic et al. 1997b; Arkilic 1997), the lower limit being for exceptionally smooth surfaces while the upper limit is typical of most practical surfaces. The final expression derived by Maxwell for an isothermal wall reads ugas − uwall =

2 − σv ∂u L σv ∂y w

(15.6.39)

For σv = 0, the slip velocity is unbounded, while for σv = 1, Equation 15.6.39 reverts to 15.6.38. Similar arguments were made for the temperature-jump boundary condition by von Smoluchowski (1898). For an ideal gas flow in the presence of wall-normal and tangential temperature gradients, the complete (first-order) slip-flow and temperature-jump boundary conditions read ugas − uwall =

=

Pr ( γ − 1) 1 τ w + 34 ( −qx )w γ ρR Tgas 2 R Tgas ρ π

2 − σv σv

(15.6.40)

2 − σ v  ∂u  µ  ∂T  L   + 34   ρ T σv ∂ y  w gas  ∂x  w

Tgas − Twall =

=

2 − σT σT 2 − σT σT

 2( γ − 1)     ( γ + 1)    ρR

1 2 R Tgas π

( −q )

y w

(15.6.41)

 2 γ  L  ∂T       ( γ + 1)  Prr  ∂y    w

where x and y are the streamwise and normal coordinates, ρ and µ, are the fluid density and viscosity, respectively; R is the gas constant; Tgas is the temperature of the gas adjacent to the wall; Twall is the wall temperature; τw is the shear stress at the wall; Pr is the Prandtl number; γ is the specific heat ratio; and qx and qy are the tangential and normal heat flux at the wall, respectively. The tangential-momentum-accommodation coefficient σv and the thermal-accommodation coefficient σT are given by, respectively σv =

σT =

τ i − τr τi − τw

(15.6.42)

dEi − dEr dEi − dEw

(15.6.43)

where the subscripts i, r, and w stand for, respectively, incident, reflected, and solid wall conditions; τ is a tangential momentum flux; and dE is an energy flux. The second term in the right-hand side of Equation 15.6.40 is the thermal creep, which generates slip velocity in the fluid opposite to the direction of the tangential heat flux, that is, flow in the direction of increasing temperature. At sufficiently high Knudsen numbers, a streamwise temperature gradient in a conduit leads to a measurable pressure gradient along the tube. This may be the case in vacuum

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applications and MEMS devices. Thermal creep is the basis for the so-called Knudsen pump — a device with no moving parts — in which rarefied gas is hauled from one cold chamber to a hot one.† Clearly, such a pump performs best at high Knudsen numbers, and is typically designed to operate in the freemolecule flow regime. In dimensionless form, Equation 15.6.40 and Equation 15.6.41 read, respectively

* * ugas − uwall =

 ∂u *  ( γ − 1) Kn2 Re  ∂T *  2 − σv Kn  *  + 23π γ Ec  ∂x *  w σv  ∂y  w

* * − Twall = Tgas

2 − σT σT

 2γ  Kn  ∂T *       ( γ + 1)  Pr  ∂y *    w

(15.6.44)

(15.6.45)

where the superscript * indicates dimensionless quantity, Kn is the Knudsen number, Re is the Reynolds number, and Ec is the Eckert number defined by Ec =

v o2 T = ( γ − 1) o Ma 2 c p ∆T ∆T

(15.6.46)

where vo is a reference velocity, ∆T = (Tgas – To), and To is a reference temperature. Note that very low values of σv and σT lead to substantial velocity slip and temperature jump even for flows with small Knudsen number. The first term in the right-hand side of Equation 15.6.44 is first-order in Knudsen number, while the thermal creep term is second-order, meaning that the creep phenomenon is potentially significant at large values of the Knudsen number. Equation 15.6.45 is first-order in Kn. Using Equation 15.6.8 and Equation 15.6.46, the thermal creep term in Equation 15.6.44 can be rewritten in terms of ∆T and the Reynolds number. Thus, * * ugas − uwall =

 ∂u *  2 − σv ∆T T 1  ∂T *  Kn  *  + 34   σv To Re  ∂x *  w  ∂y  w

(15.6.47)

It is clear that large temperature changes along the surface or low Reynolds numbers lead to significant thermal creep. The continuum Navier-Stokes equations with no-slip/no-temperature jump boundary conditions are valid as long as the Knudsen number does not exceed 0.001. First-order slip/temperature-jump boundary conditions should be applied to the Navier-Stokes equations in the range of 0.001 < Kn < 0.1. The transition regime spans the range of 0.1 < Kn < 10, and second-order or higher slip/temperature-jump boundary conditions are applicable there. Note, however, that the Navier-Stokes equations are first-order accurate in Kn, as will be shown later, and are themselves not valid in the transition regime. Either higherorder continuum equations, for example, Burnett equations, should be used there or molecular modeling should be invoked, abandoning the continuum approach altogether. For isothermal walls, Beskok (1994) derived a higher-order slip-velocity condition as follows:

u gas − u wall =

 2 − σ v   ∂u  L 2  ∂ 2u  L 3  ∂ 3u  L   +  3  +   2 + σ v   ∂y  w 2!  ∂y  w 3!  ∂y  w 

(15.6.48)

† The terminology Knudsen pump has been used by, for example, Vargo and Muntz (1996), but according to Loeb (1961), the original experiments demonstrating such pump were carried out by Osborne Reynolds.

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Attempts to implement the previous slip condition in numerical simulations are rather difficult. Second-order and higher derivatives of velocity cannot be computed accurately near the wall. Based on asymptotic analysis, Beskok (1996) and Beskok and Karniadakis (1994; 1999) proposed the following alternative higher-order boundary condition for the tangential velocity, including the thermal creep term: * * ugas − uwall =

2 − σv Kn σ v 1 − b Kn

 ∂u*   *  + 23π  ∂y  w

(γ

− 1) Kn2 Re Ec γ

 ∂T *   *   ∂x 

(15.6.49) w

where b is a high-order slip coefficient determined from the presumably known no-slip solution, thus avoiding the computational difficulties mentioned earlier. If this high-order slip coefficient is chosen as b = uw″/uw′ , where the prime denotes derivative with respect to y and the velocity is computed from the no-slip Navier-Stokes equations, Equation 15.6.49 becomes second-order accurate in Knudsen number. Beskok’s procedure can be extended to third- and higher-orders for both the slip-velocity and thermal creep terms. Similar arguments can be applied to the temperature-jump boundary condition, and the resulting Taylor series reads in dimensionless form (Beskok 1996)

* * Tgas − Twall =

2 − σT σT

 2γ  1   Kn    ( γ + 1)  Pr    

 ∂T *  Kn2  *  +  ∂y  w 2!

  ∂ 2T *     +…   ∂y *2   w

(15.6.50)

Again, the difficulties associated with computing second- and higher-order derivatives of temperature are alleviated using an identical procedure to that utilized for the tangential velocity boundary condition. Several experiments in low-pressure macroducts or in microducts confirm the necessity of applying slip boundary condition at sufficiently large Knudsen numbers. Among them are those conducted by Knudsen (1909), Pfahler at al. (1991), Tison (1993), Liu et al. (1993, 1995), Pong et al. (1994), Arkilic et al. (1995), Harley et al. (1995), and Shih et al. (1995, 1996). The experiments are complemented by the numerical simulations carried out by Beskok (1994, 1996), Beskok and Karniadakis (1994, 1999), and Beskok et al. (1996). Here we present selected examples of the experimental and numerical results. Tison (1993) conducted pipe flow experiments at very low pressures. His pipe has a diameter of 2 mm and a length-to-diameter ratio of 200. Both inlet and outlet pressures were varied to yield Knudsen number in the range of Kn = 0–200. Figure 15.6.3 shows the variation of mass flow rate as a function of (pi2 – po2), where pi is the inlet pressure and po is the outlet pressure.† The pressure drop in this rarefied pipe flow is nonlinear, characteristic of low-Reynolds-number, compressible flows. Three distinct flow regimes are identified: (1) slip flow regime, 0 < Kn < 0.6, (2) transition regime, 0.6 < Kn < 17, where the mass flow rate is almost constant as the pressure changes, and (3) free-molecule flow, Kn > 17. Note that the demarcation between these three regimes is slightly different from that mentioned earlier. As stated, the different Knudsen number regimes are determined empirically and are therefore only approximate for a particular flow geometry. Shih et al. (1995) conducted their experiments in a microchannel using helium as a fluid. The inlet pressure varied but the duct exit was atmospheric. Microsensors where fabricated in-situ along their MEMS channel to measure the pressure. Figure 15.6.4 shows their measured mass flow rate vs. the inlet pressure. The data are compared to the no-slip solution and the slip solution using three different values of the tangential-momentum-accommodation coefficient, 0.8, 0.9, and 1.0. The agreement is reasonable with the case σv = 1, indicating perhaps that the channel used by Shih et al. was quite rough on the molecular scale. In a second experiment (Shih et al. 1996), nitrous oxide was used as the fluid. The square of the pressure distribution along the channel is plotted in Figure 15.6.5 for five different inlet pressures.

†

The original data in this figure were acquired by S.A. Tison and plotted by Beskok et al. (1996).

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600 400

m × 1012 (kg/s)

200 100 80 60 40

17 > Kn > 0.6 200 > Kn > 17

20 0.6 > Kn > 0.0 10 8 6 4 1

0.1

10 100 (p2i – po2) [Pa2]

1000

104

FIGURE 15.6.3 Variation of mass flow rate as a function of (pi2 – po2). Original data acquired by S.A. Tison and plotted by Beskok et al. (1996).

Mass Flow Rate × 1012 [kg/s]

8

Data No-slip solution Slip solution σν = 1.0 Slip solution σν = 0.9 Slip solution σν = 0.8

7 6 5 4 3 2 1 0

0

5

10

15

20

25

30

35

Inlet Pressure [psig] FIGURE 15.6.4 Mass flow rate vs. inlet pressure in a microchannel. From Shih et al. (1995).

The experimental data (symbols) compare well with the theoretical predictions (solid lines). Again, the nonlinear pressure drop shown indicates that the gas flow is compressible. Arkilic (1997) provided an elegant analysis of the compressible, rarefied flow in a microchannel. The results of his theory are compared to the experiments of Pong et al. (1994) in Figure 15.6.6. The dotted line is the incompressible flow solution, where the pressure is predicted to drop linearly with streamwise distance. The dashed line is the compressible flow solution that neglects rarefaction effects (assumes Kn = 0). Finally, the solid line is the theoretical result that takes into account both compressibility and rarefaction via slip-flow boundary condition computed at the exit Knudsen number of Kn = 0.06. That theory compares most favorably with the experimental data. In the compressible flow through the © 2005 by CRC Press LLC

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Inlet Pressure 8.4 psig 12.1 psig 15.5 psig 19.9 psig 23.0 psig

p2 (psi2) 1600 1400 1200 1000 800 600 400 200 0 0

1000

3000

2000

4000

Channel Length (µm)

FIGURE 15.6.5 Pressure distribution of nitrous oxide in a microduct. Solid lines are theoretical predictions. From Shih et al. (1996).

Nondimensional Pressure

2.8

2.4

2

1.6

1.2

Pong et al. (1994) Outlet Knudsen number = 0.0 Outlet Knudsen number = 0.06 Incompressible flow solution

0.8 0

0.2

0.4

0.6

0.8

1

Nondimensional Position (x) FIGURE 15.6.6 Pressure distribution in a long microchannel. The symbols are experimental data while the lines are different theoretical predictions. From Arkilic (1997).

constant-area duct, density decreases and thus velocity increases in the streamwise direction. As a result, the pressure distribution is nonlinear with negative curvature. A moderate Knudsen number (i.e., moderate slip) actually diminishes, albeit rather weakly, this curvature. Thus, compressibility and rarefaction effects lead to opposing trends, as pointed out by Beskok et al. (1996).

Molecular-Based Models In the continuum models discussed thus far, the macroscopic fluid properties are the dependent variables while the independent variables are the three spatial coordinates and time. The molecular models recognize the fluid as a myriad of discrete particles: molecules, atoms, ions, and electrons. The goal here is to determine the position, velocity, and state of all particles at all times. The molecular approach is either deterministic or probabilistic (refer to Figure 15.6.1). Provided that there is a sufficient number © 2005 by CRC Press LLC

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of microscopic particles within the smallest significant volume of a flow, the macroscopic properties at any location in the flow can then be computed from the discrete-particle information by a suitable averaging or weighted averaging process. This subsection discusses molecular-based models and their relation to the continuum models previously considered. The most fundamental of the molecular models is a deterministic one. The motion of the molecules are governed by the laws of classical mechanics, although, at the expense of greatly complicating the problem, the laws of quantum mechanics can also be considered in special circumstances. The modern molecular dynamics computer simulations (MD) have been pioneered by Alder and Wainwright (1957, 1958, 1970) and reviewed by Ciccotti and Hoover (1986), Allen and Tildesley (1987), Haile (1993), and Koplik and Banavar (1995). The simulation begins with a set of N molecules in a region of space, each assigned a random velocity corresponding to a Boltzmann distribution at the temperature of interest. The interaction between the particles is prescribed typically in the form of a two-body potential energy and the time evolution of the molecular positions is determined by integrating Newton’s equations of motion. Because MD is based on the most basic set of equations, it is valid in principle for any flow extent and any range of parameters. The method is straightforward in principle but there are two hurdles: choosing a proper and convenient potential for particular fluid and solid combinations, and gathering the colossal computer resources required to simulate a reasonable flow-field extent. For purists, the former difficulty is a sticky one. There is no totally rational methodology by which a convenient potential can be chosen. Part of the art of MD is to pick an appropriate potential and validate the simulation results with experiments or other analytical/computational results. A commonly used potential between two molecules is the generalized Lennard-Jones 6–12 potential, to be used in the following subsection and further discussed in the subsection following that. The second difficulty, and by far the most serious limitation of molecular dynamics simulations, is the number of molecules N that can realistically be modeled on a digital computer. Since the computation of an element of trajectory for any particular molecule requires consideration of all other molecules as potential collision partners, the amount of computation required by the MD method is proportional to N2. Some saving in computer time can be achieved by cutting off the weak tail of the potential (see Figure 15.6.11) at, say, rc = 2.5σ, and shifting the potential by a linear term in r so that the force goes smoothly to zero at the cutoff. As a result, only nearby molecules are treated as potential collision partners, and the computation time for N molecules no longer scales with N2. The state of the art of molecular dynamics simulations in the early 2000s is such that with a few hours of CPU time, general-purpose supercomputers can handle around 100,000 molecules. At enormous expense, the fastest parallel machine available can simulate around 10 million particles. Because of the extreme diminution of molecular scales, the above translates into regions of liquid flow of about 0.02 µm (200 Angstroms) in linear size, over time intervals of around 0.001 µs, enough for continuum behavior to set in for simple molecules. To simulate 1 s of real time for complex molecular interactions, for example, including vibration modes, reorientation of polymer molecules, and collision of colloidal particles, requires unrealistic CPU time measured in hundreds of years. MD simulations are highly inefficient for dilute gases where the molecular interactions are infrequent. The simulations are more suited for dense gases and liquids. Clearly, molecular dynamics simulations are reserved for situations where the continuum approach or the statistical methods are inadequate to compute from first principles important flow quantities. Slip boundary conditions for liquid flows in extremely small devices is such a case, as will be discussed in the following subsection. An alternative to the deterministic molecular dynamics is the statistical approach where the goal is to compute the probability of finding a molecule at a particular position and state. If the appropriate conservation equation can be solved for the probability distribution, important statistical properties such as the mean number, momentum, or energy of the molecules within an element of volume can be computed from a simple weighted averaging. In a practical problem, it is such average quantities that concern us rather than the detail for every single molecule. Clearly, however, the accuracy of computing average quantities, via the statistical approach, improves as the number of molecules in the sampled volume increases. The kinetic theory of dilute gases is well advanced, but that for dense gases and liquids © 2005 by CRC Press LLC

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is much less so due to the extreme complexity of having to include multiple collisions and intermolecular forces in the theoretical formulation. The statistical approach is well covered in books such as those by Kennard (1938), Hirschfelder et al. (1954), Schaaf and Chambré (1961), Vincenti and Kruger (1965), Kogan (1969), Chapman and Cowling (1970), Cercignani (1988, 2000), and Bird (1994), and review articles such as those by Kogan (1973), Muntz (1989), and Oran et al. (1998). In the statistical approach, the fraction of molecules in a given location and state is the sole dependent variable. The independent variables for monatomic molecules are time, the three spatial coordinates, and the three components of molecular velocity. Those describe a six-dimensional phase space.† For diatomic or polyatomic molecules, the dimension of phase space is increased by the number of internal degrees of freedom. Orientation adds an extra dimension for molecules that are not spherically symmetric. Finally, for mixtures of gases, separate probability distribution functions are required for each species. Clearly, the complexity of the approach increases dramatically as the dimension of phase space increases. The simplest problems are, for example, those for steady, one-dimensional flow of a simple monatomic gas. To simplify the problem we restrict the discussion here to monatomic gases having no internal degrees of freedom. Furthermore, the fluid is restricted to dilute gases and molecular chaos is assumed. The former restriction requires the average distance between molecules δ to be an order of magnitude larger than their diameter σ. That will almost guarantee that all collisions between molecules are binary collisions, avoiding the complexity of modeling multiple encounters.†† The molecular chaos restriction improves the accuracy of computing the macroscopic quantities from the microscopic information. In essence, the volume over which averages are computed has to have sufficient number of molecules to reduce statistical errors. It can be shown that computing macroscopic flow properties by averaging over a number of molecules will result in statistical fluctuations with a standard deviation of approximately 0.1% if one million molecules are used and around 3% if 1000 molecules are used. The molecular chaos limit requires the length-scale L for the averaging process to be at least 100 times the average distance between molecules (i.e., typical averaging over at least one million molecules). Figure 15.6.7, adapted from Bird (1994), shows the limits of validity of the dilute gas approximation (δ/σ > 7), the continuum approach (Kn < 0.1, as discussed previously), and the neglect of statistical fluctuations (L/δ > 100). Using a molecular diameter of σ = 4 × 10–10 m as an example, the three limits are conveniently expressed as functions of the normalized gas density ρ/ρo or number density n/no, where the reference densities ρo and no are computed at standard conditions. All three limits are straight lines in the log-log plot of L vs. ρ/ρo, as depicted in Figure 15.6.7. Note the shaded triangular wedge inside which both the Boltzmann and Navier-Stokes equations are valid. Additionally, the lines describing the three limits very nearly intersect at a single point. As a consequence, the continuum breakdown limit always lies between the dilute gas limit and the limit for molecular chaos. As density or characteristic dimension is reduced in a dilute gas, the Navier-Stokes model breaks down before the level of statistical fluctuations becomes significant. In a dense gas, on the other hand, significant fluctuations may be present even when the Navier-Stokes model is still valid. The starting point in statistical mechanics is the Liouville equation that expresses the conservation of the N-particle distribution function in 6N-dimensional phase space,††† where N is the number of particles under consideration. Considering only external forces that do not depend on the velocity of the molecules,†††† the Liouville equation for a system of N mass points reads ∂F + ∂t

N

∑ k =1

 ∂F ξk ⋅  + ∂x k

N

 ∂F

∑ F ⋅ ∂ξ k

k =1

= 0

(15.6.51)

k

† The evolution equation of the probability distribution is considered; hence time is the seventh independent variable. †† Dissociation and ionization phenomena involve triple collisions and therefore require separate treatment. ††† Three positions and three velocities for each molecule of a monatomic gas with no internal degrees of freedom. †††† This excludes Lorentz forces, for example.

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Chapter 15

1

3

1010

vie Na

108

r–

pic

lid va ns tio ry ua ssa eq ) ce es 0.1 ne ok < ch S t ( kn oa pr ap

co

10-2

10

Dense gas

100

os icr M

Characteristic dimension L (meter)

102

1,000

Dilute gas (δ/σ > 7)

δ/σ 10,000

Ins

ign

10

-4

10-6

L σ 106

ific an (L/δ t fluc Sig nifi > 1 tuat can ion 00) s t st atis tica l flu ctu atio n

104

s

102 10-8

10-8

10-6 10-4 10-2 Density ratio n/n or ρ/ρ ° °

1

102

FIGURE 15.6.7 Effective limits of different flow models. From Bird (1994).

 where F is the probability of finding a molecule at a particular point in phase space, t is time, ξk is the  three-dimensional velocity  vector for the kth molecule, x k is the three-dimensional position vector for the kth molecule, and F is the external force vector. Note that the product  dot  in the previous equation  is carried out over each of the three components of the vectors ξ , x , and F , and that the summation is over all molecules. Obviously such an equation is not tractable for realistic number of particles. A hierarchy of reduced distribution functions may be obtained by repeated integration of the previous Liouville equation. The final equation in the hierarchy is for the single particle distribution that also involves the two-particle distribution function. Assuming molecular chaos, that final equation becomes a closed one (i.e., one equation in one unknown), and is known as the Boltzmann equation, the fundamental relation of the kinetic theory of gases. That final equation in the hierarchy is the only one that carries any hope of obtaining analytical solutions. A simpler direct derivation of the Boltzmann equation is provided by Bird (1994). For monatomic gas molecules in binary collisions, the integro-differential Boltzmann equation reads ∂ (n f ) ∂ (n f ) ∂ (n f ) + ξj + Fj = J f , f * , j = 1,2,3 ∂t ∂x j ∂ξ j

(

)

(15.6.52)

where n f is  the product of the number density and the normalized velocity distribution function ( dn n = f dξ ); xj and ξj are the coordinates and speeds of a molecule, respectively;† Fj is a known external force; and J(f, f *) is the nonlinear collision integral that describes the net effect of populating and †

Constituting, together with time, the seven independent variables of the single-dependent-variable equation.

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depopulating collisions on the distribution function. The collision integral is the source of difficulty in obtaining analytical solutions to the Boltzmann equation, and is given by

(

∞ 4π

) ∫ ∫n ( f

J f,f* =

2

*

  f1* − f f1 ξr σ dΩ dξ

( )

)

−∞ 0

1

(15.6.53)

 where the superscript * indicates postcollision values, f and f1 represent two different molecules, ξr is the  relative speed between two molecules, σ is the molecular cross section, Ω is the solid angle, and dξ = dξ1 dξ 2 dξ3 . Once a solution for f is obtained, macroscopic quantities such as density, velocity, and temperature can be computed from the appropriate weighted integral of the distribution function. For example: ρ = mn = m ui =

3 2

kT =

∫



∫ (n f ) d ξ

(15.6.54)



∫ ξ f dξ

(15.6.55)

i

1 2

 m ξi ξi f dξ

(15.6.56)

If the Boltzmann equation is nondimensionalized with a characteristic length L and characteristic speed [2(k/m)T]1/2, where k is the Boltzmann constant, m, is the molecular mass, and T is temperature, the inverse Knudsen number appears explicitly in the right-hand side of the equation as follows:



∂f
∂f ∂f 1


* J f , f , j = 1,2,3
+ ξ j
+ Fj
= ∂t ∂x j Kn ∂ξ j

(

)

(15.6.57)



where the topping symbol represents a dimensionless variable, and f is nondimensionalized using a reference number density no. The five conservation equations for the transport of mass, momentum, and energy can be derived by multiplying the previous Boltzmann equation by, respectively, the molecular mass, momentum, and energy, and then integrating over all possible molecular velocities. Subject to the restrictions of dilute gas and molecular chaos stated earlier, the Boltzmann equation is valid for all ranges of Knudsen number from 0 to ∞. Analytical solutions to this equation for arbitrary geometries are difficult mostly because of the nonlinearity of the collision integral. Simple models of this integral have been proposed to facilitate analytical solutions; see, for example, Bhatnagar et al. (1954). There are two important asymptotes to Equation 15.6.57. First, as Kn → ∞, molecular collisions become unimportant. This is the free-molecule flow regime depicted in Figure 15.6.2 for Kn > 10, where the only important collision is that between a gas molecule and the solid surface of an obstacle or a conduit. Analytical solutions are then possible for simple geometries, and numerical simulations for complicated geometries are straightforward once the surface-reflection characteristics are accurately modeled. Second, as Kn → 0, collisions become important and the flow approaches the continuum regime of conventional fluid dynamics. The Second Law specifies a tendency for thermodynamic systems to revert to equilibrium state, smoothing out any discontinuities in macroscopic flow quantities. The number of molecular collisions in the limit Kn → 0 is so large that the flow approaches the equilibrium state in a time short compared to the macroscopic time-scale. For example, for air at standard conditions (T = 288 K; p = 1 atm), each molecule experiences, on the average, 10 collisions per nanosecond and travels 1 micron in the same time period. Such a molecule has already forgotten its previous state after 1 ns. In a particular © 2005 by CRC Press LLC

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flowfield, if the macroscopic quantities vary little over a distance of 1 µm or over a time interval of 1 ns, the flow of STP air is near equilibrium. At Kn = 0, the velocity distribution function is everywhere of the local equilibrium or Maxwellian form:

n −3/2
2 f (0) = π exp  − ξ − u  no  

(

)

(15.6.58)



where ξ and u are, respectively, the dimensionless speeds of a molecule and of the flow. In this Knudsen number limit, the velocity distribution of each element of the fluid instantaneously adjusts to the equilibrium thermodynamic state appropriate to the local macroscopic properties as this molecule moves through the flow field. From the continuum viewpoint, the flow is isentropic, and heat conduction and viscous diffusion and dissipation vanish from the continuum conservation relations. The Chapman-Enskog theory attempts to solve the Boltzmann equation by considering a small per
turbation of f from the equilibrium Maxwellian form. For small Knudsen numbers, the distribution function can be expanded in terms of Kn in the form of a power series:



f = f (0) + Kn f (1) + Kn2 f (2) + 

(15.6.59)

By substituting the above series in the Boltzmann equation (Equation 15.6.57) and equating terms of equal order, the following recurrent set of integral equations result:


J f (0) , f (0) = 0 ,

(

)





∂f (0)
∂f (0)


∂f J f (0) , f (1) =
+ ξ j
+ Fj
,  ∂t ∂x j ∂ξ j

(

)

(15.6.60)

The first integral is nonlinear

and its solution is the local Maxwellian distribution, Equation 15.6.58. The distribution functions f (1) , f (2) , etc., each satisfies an inhomogeneous linear equation whose solution leads to the transport terms needed to close the continuum equations appropriate to the particular level of approximation. The continuum stress tensor and heat flux vector can be written in terms of the distribution function, which in turn can be specified in terms of the macroscopic velocity and temperature and their derivatives (Kogan 1973). The zeroth-order equation yields the Euler equations, the first-order equation results in the linear transport terms of the Navier-Stokes equations, the second-order equation gives the nonlinear transport terms of the Burnett equations, and so on. Keep in mind, however, that the Boltzmann equation as developed in this subsection is for a monatomic gas. This excludes the allimportant air that is composed largely of diatomic nitrogen and oxygen. As discussed earlier, the Navier-Stokes equations can and should be used up to a Knudsen number of 0.1. Beyond that, the transition flow regime commences (0.1 < Kn < 10). In this flow regime, the molecular mean free path for a gas becomes significant relative to a characteristic distance for important flowproperty changes to take place. The Burnett equations can be used to obtain analytical/numerical solutions for at least a portion of the transition regime for a monatomic gas, although their complexity have precluded much results for realistic geometries (Agarwal et al. 1999). There is also a certain degree of uncertainty about the proper boundary conditions to use with the continuum Burnett equations, and experimental validation of the results have been very scarce. Additionally, as the gas flow further departs from equilibrium, the bulk viscosity ( = λ + 23 µ , where λ is the second coefficient of viscosity) is no longer zero, and the Stokes’ hypothesis no longer holds (see Gad-el-Hak [1995] for an interesting summary of the issue of bulk viscosity). In the transition regime, the molecularly based Boltzmann equation cannot easily be solved either, unless the nonlinear collision integral is simplified. So, clearly the transition regime is one of dire need © 2005 by CRC Press LLC

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of alternative methods of solution. MD simulations as mentioned earlier are not suited for dilute gases. The best approach for the transition regime right now is the direct simulation Monte Carlo (DSMC) method developed by Bird (1963, 1965, 1976, 1978, 1994) and briefly described below. Some recent reviews of DSMC include those by Muntz (1989), Cheng (1993), Cheng and Emmanuel (1995) and Oran et al. (1998). The mechanics as well as the history of the DSMC approach and its ancestors are well described in the book by Bird (1994). Unlike molecular dynamics simulations, DSMC is a statistical computational approach to solving rarefied gas problems. Both approaches treat the gas as discrete particles. Subject to the dilute gas and molecular chaos assumptions, the direct simulation Monte Carlo method is valid for all ranges of Knudsen number, although it becomes quite expensive for Kn < 0.1. Fortunately, this is the continuum regime where the Navier-Stokes equations can be used analytically or computationally. DSMC is therefore ideal for the transition regime (0.1 < Kn < 10), where the Boltzmann equation is difficult to solve. The Monte Carlo method is, like its namesake, a random number strategy based directly on the physics of the individual molecular interactions. The idea is to track a large number of randomly selected, statistically representative particles, and to use their motions and interactions to modify their positions and states. The primary approximation of the direct simulation Monte Carlo method is to uncouple the molecular motions and the intermolecular collisions over small time intervals. A significant advantage of this approximation is that the amount of computation required is proportional to N, in contrast to N2 for molecular dynamics simulations. In essence, particle motions are modeled deterministically while collisions are treated probabilistically, each simulated molecule representing a large number of actual molecules. Typical computer runs of DSMC in the 1990s involve tens of millions of intermolecular collisions and fluid-solid interactions. The DSMC computation is started from some initial condition and followed in small time steps that can be related to physical time. Colliding pairs of molecules in a small geometric cell in physical space are randomly selected after each computational time step. Complex physics such as radiation, chemical reactions, and species concentrations can be included in the simulations without the necessity of nonequilibrium thermodynamic assumptions that commonly afflict nonequilibrium continuum-flow calculations. DSMC is more computationally intensive than classical continuum simulations, and should therefore be used only when the continuum approach is not feasible. The DSMC technique is explicit and time marching, and therefore always produces unsteady flow simulations. For macroscopically steady flows, Monte Carlo simulation proceeds until a steady flow is established, within a desired accuracy, at sufficiently large time. The macroscopic flow quantities are then the time average of all values calculated after reaching the steady state. For macroscopically unsteady flows, ensemble averaging of many independent Monte Carlo simulations is carried out to obtain the final results within a prescribed statistical accuracy.

Liquid Flows From the continuum point of view, liquids and gases are both fluids obeying the same equations of motion. For incompressible flows, for example, the Reynolds number is the primary dimensionless parameter that determines the nature of the flow field. True, water, for example, has density and viscosity that are, respectively, three and two orders of magnitude higher than those for air, but if the Reynolds number and geometry are matched, liquid and gas flows should be identical.† For MEMS applications, however, we anticipate the possibility of nonequilibrium flow conditions and the consequent invalidity of the Navier-Stokes equations and the no-slip boundary conditions. Such circumstances can best be researched using the molecular approach. This was discussed for gases earlier, and the corresponding arguments for liquids will be given in this subsection. The literature on non-Newtonian fluids in general and polymers in particular is vast (for example, the bibliographic survey by Nadolink and Haigh [1995] †

Barring phenomena unique to liquids such as cavitation, free surface flows, etc.

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cites over 4900 references on polymer drag reduction alone) and provides a rich source of information on the molecular approach for liquid flows. Solids, liquids, and gases are distinguished merely by the degree of proximity and the intensity of motions of their constituent molecules. In solids, the molecules are packed closely and confined, each hemmed in by its neighbors (Chapman and Cowling 1970). Only rarely would one solid molecule slip from its neighbors to join a new set. As the solid is heated, molecular motion becomes more violent and a slight thermal expansion takes place. At a certain temperature that depends on ambient pressure, sufficiently intense motion of the molecules enables them to pass freely from one set of neighbors to another. The molecules are no longer confined but are nevertheless still closely packed, and the substance is now considered a liquid. Further heating of the matter eventually releases the molecules altogether, allowing them to break the bonds of their mutual attractions. Unlike solids and liquids, the resulting gas expands to fill any volume available to it. Unlike solids, both liquids and gases cannot resist finite shear force without continuous deformation; that is the definition of a fluid medium. In contrast to the reversible, elastic, static deformation of a solid, the continuous deformation of a fluid resulting from the application of a shear stress results in an irreversible work that eventually becomes random thermal motion of the molecules; that is viscous dissipation. There are around 25-million molecules of STP air in a one-micron cube. The same cube would contain around 34-billion molecules of water. So, liquid flows are continuum even in extremely small devices through which gas flows would not be continuum. The average distance between molecules in the gas example is one order of magnitude higher than the diameter of its molecules, while that for the liquid phase approaches the molecular diameter. As a result, liquids are almost incompressible. Their isothermal compressibility coefficient α and bulk expansion coefficient β are much smaller compared to those for gases. For water, for example, a hundred-fold increase in pressure leads to less than 0.5% decrease in volume. Sound speeds through liquids are also high relative to those for gases, and as a result most liquid flows are incompressible.† The exception is the propagation of ultra-high-frequency sound waves and cavitation phenomena. The mechanism by which liquids transport mass, momentum, and energy must be very different from that for gases. In dilute gases, intermolecular forces play no role and the molecules spend most of their time in free flight between brief collisions, at which instances the molecules’ direction and speed abruptly change. The random molecular motions are responsible for gaseous transport processes. In liquids, on the other hand, the molecules are closely packed though not fixed in one position. In essence, the liquid molecules are always in a collision state. Applying a shear force must create a velocity gradient so that the molecules move relative to one another, ad infinitum as long as the stress is applied. For liquids, momentum transport due to the random molecular motion is negligible compared to that due to the intermolecular forces. The straining between liquid molecules causes some to separate from their original neighbors, bringing them into the force field of new molecules. Across the plane of the shear stress, the sum of all intermolecular forces must, on the average, balance the imposed shear. Liquids at rest transmit only normal force, but when a velocity gradient occurs, the net intermolecular force would have a tangential component. The incompressible Navier-Stokes equations describe liquid flows under most circumstances. Liquids, however, do not have a well-advanced molecular-based theory as that for dilute gases. The concept of mean free path is not useful for liquids, and the conditions under which a liquid flow fails to be in quasiequilibrium state are not well defined. There is no Knudsen number for liquid flows to guide us through the maze. We do not know, from first principles, the conditions under which the no-slip boundary condition becomes inaccurate, or the point at which the stress–rate of strain relation or the heat flux–temperature gradient relation fails to be linear. Certain empirical observations indicate that those simple

† Note that we distinguish between a fluid and a flow being compressible/incompressible. For example, the flow of the highly compressible air can be either compressible or incompressible.

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relations that we take for granted occasionally fail to accurately model liquid flows. For example, it has been shown in rheological studies (Loose and Hess, 1989) that non-Newtonian behavior commences when the strain rate approximately exceeds twice the molecular frequency-scale: ∂u γ˙ = ≥ 2 T −1 ∂y

(15.6.61)

where the molecular time-scale T is given by 1

m σ2  2 T =    ε 

(15.6.62)

where m is the molecular mass, and σ and ε are the characteristic length- and energy-scale for the molecules, respectively. For ordinary liquids such as water, this time-scale is extremely small, and the threshold shear rate for the onset of non-Newtonian behavior is therefore extraordinarily high. For highmolecular-weight polymers, on the other hand, m and σ are both many orders of magnitude higher than their respective values for water, and the linear stress-strain relation breaks down at realistic values of the shear rate. The moving contact line when a liquid spreads on a solid substrate is an example where slip flow must be allowed to avoid singular or unrealistic behavior in the Navier-Stokes solutions (Dussan and Davis 1974; Dussan 1976, 1979; Thompson and Robbins 1989). Other examples where slip-flow must be admitted include corner flows (Moffatt 1964; Koplik and Banavar 1995) and extrusion of polymer melts from capillary tubes (Pearson and Petrie 1968; Richardson 1973; Den 1990). Existing experimental results of liquid flow in microdevices are contradictory. This is not surprising given the difficulty of such experiments and the lack of a guiding rational theory. Pfahler et al. (1990; 1991), Pfahler (1992), and Bau (1994) summarize the relevant literature. For small-length-scale flows, a phenomenological approach for analyzing the data is to define an apparent viscosity µa calculated so that if it were used in the traditional no-slip Navier-Stokes equations instead of the fluid viscosity µ, the results would be in agreement with experimental observations. Israelachvili (1986) and Gee et al. (1990) found that µa = µ for thin-film flows as long as the film thickness exceeds 10 molecular layers (≈5 nm). For thinner films, µa depends on the number of molecular layers and can be as much as 105 times larger than µ. Chan and Horn’s (1985) results are somewhat different: the apparent viscosity deviates from the fluid viscosity for films thinner than 50 nm. In polar-liquid flows through capillaries, Migun and Prokhorenko (1987) report that µa increases for tubes smaller than 1 micron in diameter. In contrast, Debye and Cleland (1959) report µa smaller than µ for paraffin flow in porous glass with average pore size several times larger than the molecular lengthscale. Experimenting with microchannels ranging in depths from 0.5 micron to 50 microns, Pfahler et al. (1991) found that µa is consistently smaller than µ for both liquid (isopropyl alcohol; silicone oil) and gas (nitrogen; helium) flows in microchannels. For liquids, the apparent viscosity decreases with decreasing channel depth. Other researchers using small capillaries report that µa is about the same as µ (Anderson and Quinn 1972; Tuckermann and Pease 1981, 1982; Tuckermann 1984; Guvenc 1985; Nakagawa et al. 1990). Very recently, Sharp (2001) and Sharp et al. (2002) asserted that, despite the significant inconsistencies in the literature regarding liquid flows in microchannels, such flows are well predicted by macroscale continuum theory. A case can be made to the contrary, however, as will be seen at the end of this subsection, and the final verdict on this controversy is yet to come. The above contradictory results point to the need for replacing phenomenological models by firstprinciples ones. The lack of molecular-based theory of liquids — despite extensive research by the rheology and polymer communities — leaves molecular dynamics simulations as the nearest weapon to first-principles arsenal. MD simulations offer a unique approach to checking the validity of the traditional © 2005 by CRC Press LLC

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continuum assumptions. However, as was pointed out earlier, such simulations are limited to exceedingly minute flow extent. Thompson and Troian (1997) provide molecular dynamics simulations to quantify the slip-flow boundary condition dependence on shear rate. Recall the linear Navier boundary condition introduced earlier: ∆u w = u fluid − u wall = L s

∂u ∂y

(15.6.63) w

where Ls = the constant slip length ∂u = the strain rate computed at the wall. ∂y w The goal of Thompson and Troian’s simulations was to determine the degree of slip at a solid-liquid interface as the interfacial parameters and the shear rate change. In their simulations, a simple liquid underwent planar shear in a Couette cell as shown in Figure 15.6.8. The typical cell measured 12.51 × 7.22 × h, in units of molecular length-scale σ, where the channel depth h varied in the range of 16.71σ to 24.57σ, and the corresponding number of molecules simulated ranged from 1152 to 1728. The liquid is treated as an isothermal ensemble of spherical molecules. A shifted Lennard-Jones 6–12 potential is U

Solid y

Fluid x

h

Solid

u(y)/U

1

0.5 ∈wf σwf 0.6 1.0 0.6 0.75 0.2 0.75

0 0

0.5

nw 1 4 4 1

y/h FIGURE 15.6.8 Velocity profiles in a Couette flow geometry at different interfacial parameters. All three profiles are for U = σT –1, and h = 24.57 σ. The dashed line is the no-slip Couette-flow solution. From Thompson and Troian (1997). © 2005 by CRC Press LLC

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used to model intermolecular interactions, with energy- and length-scales ε and σ, and cutoff distance rc = 2.2σ:  r  −12  r  −6  r  −12  r  −6  V (r ) = 4 ε   −   −  c  +  c    σ  σ  σ   σ  

(15.6.64)

The truncated potential is set to zero for r > rc. The fluid-solid interaction is also modeled with a truncated Lennard-Jones potential, with energyand length-scales εwf and σwf, and cutoff distance rc. The equilibrium state of the fluid is a well-defined liquid phase characterized by number density n = 0.81σ –3 and temperature T = 1.1 ε/k, where k is the Boltzmann constant. The steady state velocity profiles resulting from Thompson and Troian’s (1997) MD simulations are depicted in Figure 15.6.8 for different values of the interfacial parameters εwf, σwf and nw. Those parameters, shown in units of the corresponding fluid parameters ε, σ, and n, characterize, respectively, the strength of the liquid-solid coupling, the thermal roughness of the interface, and the commensurability of wall and liquid densities. The macroscopic velocity profiles recover the expected flow behavior from continuum hydrodynamics with boundary conditions involving varying degrees of slip. Note that when slip exists, the shear rate γ˙ no longer equals U/h. The degree of slip increases (i.e., the amount of momentum transfer at the wall-fluid interface decreases) as the relative wall density nw increases or the strength of the wallfluid coupling σwf decreases; in other words when the relative surface energy corrugation of the wall decreases. Conversely, the corrugation is maximized when the wall and fluid densities are commensurate and the strength of the wall-fluid coupling is large. In this case, the liquid feels the corrugations in the surface energy of the solid owing to the atomic close-packing. Consequently, there is efficient momentum transfer and the no-slip condition applies, or in extreme cases, a “stick” boundary condition takes hold. Variations of the slip length Ls and viscosity µ as functions of shear rate γ˙ are shown in parts (a) and (b) of Figure 15.6.9, for five different sets of interfacial parameters. For Couette flow, the slip length is computed from its definition, ∆u w γ˙ = (U γ˙ − h ) 2 . The slip length, viscosity, and shear rate are normalized in the figure using the respective molecular scales for length σ, viscosity ε T σ –3, and inverse time T–1. The viscosity of the fluid is constant over the entire range of shear rates (Figure 15.6.9b), indicating Newtonian behavior. As indicated earlier, non-Newtonian behavior is expected for γ˙ ≥ 2 T −1, well above the shear rates used in Thompson and Troian’s simulations. At low shear rates, the slip length behavior is consistent with the Navier model, in other words, is independent of the shear rate. Its limiting value L os ranges from 0 to ~17σ for the range of interfacial parameters chosen (Figure 15.6.9a). In general, the amount of slip increases with decreasing surface energy corrugation. Most interestingly, at high shear rates the Navier condition breaks down as the slip length increases rapidly with γ˙ . The critical shear-rate value for the slip length to diverge, γ˙ c , decreases as the surface energy corrugation decreases. Surprisingly, the boundary condition is nonlinear even though the liquid is still Newtonian. In dilute gases, the linear slip condition and the Navier-Stokes equations, with their linear stress-strain relation, are both valid to the same order of approximation in Knudsen number. In other words, deviation from linearity is expected to take place at the same value of Kn = 0.1. In liquids, in contrast, the slip length appears to become nonlinear and to diverge at a critical value of shear rate well below the shear rate at which the linear stress-strain relation fails. Moreover, the boundary condition deviation from linearity is not gradual but is rather catastrophic. The critical value of shear rate γ˙ c signals the point at which the solid can no longer impart momentum to the liquid. This means that the same liquid molecules sheared against different substrates will experience varying amounts of slip, and vice versa. Based on the above results, Thompson and Troian (1997) suggest a universal boundary condition at a solid-liquid interface. Scaling the slip length Ls by its asymptotic limiting value L os and the shear rate γ˙ by its critical value γ˙ c , collapses the data in the single curve shown in Figure 15.6.10. The data points are well described by the relation © 2005 by CRC Press LLC

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Chapter 15

40

Ls / σ

30

∈wf 0.6 0.1 0.6 0.4 0.2

σwf 1.0 1.0 0.75 0.75 0.75

(a)

nw 1 1 4 4 4

20

10

0 µ / ∈ τ σ-3

(b) 3 2 1 0 0.001

0.01

0.1

1.0

γτ

FIGURE 15.6.9 Variation of slip length and viscosity as functions of shear rate. From Thompson and Troian (1997). 5 γcτ

Los/σ 4

L s / Ls

o

3

0.36 0.14 0.10 0.06

1.9 4.5 8.2 16.8

2 1 0 0.01

0.1

1.0

γ / γc

FIGURE 15.6.10 Universal relation of slip length as a function of shear rate. From Thompson and Troian (1997).

 γ˙  L s = L 1 − ˙γ c    o s

− 12

(15.6.65)

The nonlinear behavior close to a critical shear rate suggests that the boundary condition can significantly affect flow behavior at macroscopic distances from the wall. Experiments with polymers confirm this observation (Atwood and Schowalter 1989). The rapid change in the slip length suggests that for © 2005 by CRC Press LLC

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flows in the vicinity of γ˙ c, small changes in surface properties can lead to large fluctuations in the apparent boundary condition. Thompson and Troian (1997) conclude that the Navier slip condition is but the low-shear-rate limit of a more generalized universal relationship that is nonlinear and divergent. Their relation provides a mechanism for relieving the stress singularity in spreading contact lines and corner flows, as it naturally allows for varying degrees of slip on approach to regions of higher rate of strain. To place the above results in physical terms, consider water† at a temperature of T = 288 K. The energyscale in the Lennard-Jones potential is then ε = 3.62 × 10–21 J. For water, m = 2.99 × 10–26 kg, σ = 2.89 × 10–10 m, and at standard temperature n = 3.35 × 1028 molecules/m3. The molecular time-scale can thus be computed, T = [mσ2/ε]1/2 = 8.31 × 10–13 s. For the third case depicted in Figure 15.6.10 (the open squares), γ˙ c T = 0.1, and the critical shear rate at which the slip condition diverges is thus γ˙ c = 1.2 × 10 11 s–1. Such an enormous rate of strain†† may be found in extremely small devices having extremely high speeds. On the other hand, the conditions to achieve a measurable slip of 17σ (the solid circles in Figure 15.6.9) are not difficult to encounter in microdevices: density of solid four times that of liquid, and the energyscale for wall-fluid interaction that is one fifth of the energy-scale for liquid. The limiting value of slip length is independent of the shear rate and can be computed for water as L os = 17σ = 4.91 × 10 −9 m. Consider a water microbearing having a shaft diameter of 100 µm and rotation rate of 20,000 rpm and a minimum gap of h = 1 µm. In this case, U = 0.1 m/s and the no-slip shear rate is U/h = 105 s–1. When slip occurs at the limiting value just computed, the shear rate and the wall slipvelocity are computed as follows: U = 9.90 × 10 4 s–1 h + 2 L os

(15.6.66)

∆u w = γ˙ L s = 4.87 × 10 −4 m/s

(15.6.67)

γ˙ =

As a result of the Navier slip, the shear rate is reduced by 1% from its no-slip value, and the slip velocity at the wall is about 0.5% of U, small but not insignificant.

Surface Phenomena The surface-to-volume ratio for a machine with a characteristic length of 1 m is 1 m–1, while that for a MEMS device having a size of 1 µm is 106 m–1. The million-fold increase in surface area relative to the mass of the minute device substantially affects the transport of mass, momentum, and energy through the surface. Obviously surface effects dominate in small devices. The surface boundary conditions in MEMS flows have already been discussed earlier. In microdevices, it has been shown that it is possible to have measurable slip-velocity and temperature jump at a solid-fluid interface. In this subsection, we illustrate other ramifications of the large surface-to-volume ratio unique to MEMS, and provide a molecular viewpoint to surface forces. In microdevices, both radiative and convective heat loss/gain are enhanced by the huge surface-tovolume ratio. Consider a device having a characteristic length Ls. Use of the lumped capacitance method to compute the rate of convective heat transfer, for example, is justified if the Biot number (≡ hLs /κs, where h is the convective heat transfer coefficient of the fluid and κs is the thermal conductivity of the solid) is less than 0.1. Small Ls implies a small Biot number and a nearly uniform temperature within the solid. Within this approximation, the rate at which heat is lost to the surrounding fluid is given by † Water molecules are complex ones, forming directional, short-range covalent bonds, thus requiring a more complex potential than the Lennard-Jones to describe the intermolecular interactions. For the purpose of the qualitative example described here, however, we use the computational results of Thompson and Troian (1997), who employed the L-J potential. †† Note however that ˙ for high-molecular-weight polymers would be many orders of magnitude smaller than γc the value developed here for water.

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Chapter 15

ρs L3s c s

dT = −h L2s (Ts − T∞ ) dt

(15.6.68)

where ρs and cs are respectively the density and specific heat of the solid, Ts is its (uniform) temperature, and T∞ is the ambient fluid temperature. Solution of the above equation is trivial, and the temperature of a hot surface drops exponentially with time from an initial temperature Ti :  t Ts ( t ) − T∞ = exp  −  Ti − T∞  T 

(15.6.69)

where the time constant T is given by T=

ρs L3s c s h L2s

(15.6.70)

For small devices, the time it takes the solid to cool down is proportionally small. Clearly, the millionfold increase in surface-to-volume ratio implies a proportional increase in the rate at which heat escapes. Identical scaling arguments can be made regarding mass transfer. Another effect of the diminished scale is the increased importance of surface forces and the waning importance of body forces. Based on biological studies, Went (1968) concludes that the demarcation length-scale is around 1 mm. Below that, surface forces dominate over gravitational forces. A 10-mm piece of paper will fall down when gently placed on a smooth, vertical wall, while a 0.1-mm piece will stick. Try it! Stiction is a major problem in MEMS applications. Certain structures such as long, thin polysilicon beams and large, thin comb drives have a propensity to stick to their substrates and thus fail to perform as designed (Mastrangelo and Hsu 1992; Tang et al. 1989). Conventional dry friction between two solids in relative motion is proportional to the normal force, which is usually a component of the moving device weight. The friction is independent of the contactsurface area because the van der Waals cohesive forces are negligible relative to the weight of the macroscopic device. In MEMS applications, the cohesive intermolecular forces between two surfaces are significant and the stiction is independent of the device mass but is proportional to its surface area. The first micromotor did not move — despite large electric current through it — until the contact area between the 100-micron rotor and the substrate was reduced significantly by placing dimples on the rotor’s surface (Fan et al. 1988, 1989; Tai and Muller 1989). One last example of surface effects that to my knowledge has not been investigated for microflows is the adsorbed layer in gaseous wall-bounded flows. It is well known (Brunauer 1944; Lighthill 1963) that when a gas flows in a duct, the gas molecules are attracted to the solid surface by the van der Waals and other forces of cohesion. The potential energy of the gas molecules drops on reaching the surface. The adsorbed layer partakes the thermal vibrations of the solid, and the gas molecules can only escape when their energy exceeds the potential energy minimum. In equilibrium, at least part of the solid would be covered by a monomolecular layer of adsorbed gas molecules. Molecular species with significant partial pressure — relative to their vapor pressure — may locally form layers two or more molecules thick. Consider, for example, the flow of a mixture of dry air and water vapor at STP. The energy of adsorption of water is much larger than that for nitrogen and oxygen, making it more difficult for water molecules to escape the potential energy trap. It follows that the lifetime of water molecules in the adsorbed layer significantly exceeds that for the air molecules (by 60,000 folds, in fact), and as a result, the thin surface layer would be mostly water. For example, if the proportion of water vapor in the ambient air is 1:1000 (i.e., very low humidity level), the ratio of water to air in the adsorbed layer would be 60:1. Microscopic roughness of the solid surface causes partial condensation of the water along portions having sufficiently strong concave curvature. So, surfaces exposed to nondry airflows are mainly liquid water surfaces. In © 2005 by CRC Press LLC

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most applications, this thin adsorbed layer has little effect on the flow dynamics, despite the fact that the density and viscosity of liquid water are far greater than those for air. In MEMS applications, however, the layer thickness may not be an insignificant portion of the characteristic flow dimension and the water layer may have a measurable effect on the gas flow. A hybrid approach of molecular dynamics and continuum flow simulations or MD-Monte Carlo simulations may be used to investigate this issue. It should be noted that recently, Majumdar and Mezic (1998, 1999) have studied the stability and rupture into droplets of thin liquid films on solid surfaces. They point out that the free energy of a liquid film consists of a surface tension component as well as highly nonlinear volumetric intermolecular forces resulting from van der Waals, electrostatic, hydration, and elastic strain interactions. For water films on hydrophilic surfaces such as silica and mica, Majumdar and Mezic (1998) estimate the equilibrium film thickness to be about 0.5 nm (2 monolayers) for a wide range of ambient-air relative humidities. The equilibrium thickness grows very sharply, however, as the relative humidity approaches 100%. Majumdar and Mezic’s (1998, 1999) results open many questions. What are the stability characteristics of their water film in the presence of airflow above it? Would this water film affect the accommodation coefficient for microduct air flow? In a modern Winchester-type hard disk, the drive mechanism has a read/write head that floats 50 nm above the surface of the spinning platter. The head and platter together with the air layer in between form a slider bearing. Would the computer performance be affected adversely by the high relative humidity on a particular day when the adsorbed water film is no longer “thin”? If a microduct hauls liquid water, would the water film adsorbed by the solid walls influence the effective viscosity of the water flow? Electrostatic forces can extend to almost 1 micron (the Debye length), and that length is known to be highly pH-dependent. Would the water flow be influenced by the surface and liquid chemistry? Would this explain the contradictory experimental results of liquid flows in microducts discussed earlier? The few examples above illustrate the importance of surface effects in small devices. From the continuum viewpoint, forces at a solid-fluid interface are the limit of pressure and viscous forces acting on a parallel elementary area displaced into the fluid, when the displacement distance is allowed to tend to zero. From the molecular point of view, all macroscopic surface forces are ultimately traced to intermolecular forces, which subject is extensively covered in the book by Israelachvilli (1991) and references therein. Here we provide a very brief introduction to the molecular viewpoint. The four forces in nature are (1) the strong and (2) weak forces describing the interactions between neutrons, protons, electrons, etc.; (3) the electromagnetic forces between atoms and molecules; and (4) gravitational forces between masses. The range of action of the first two forces is around 10–5 nm, and hence neither concerns us overly in MEMS applications. The electromagnetic forces are effective over a much larger though still small distance on the order of the interatomic separations (0.1–0.2 nm). Effects over longer range — several orders of magnitude longer — can and do arise from the short-range intermolecular forces. For example, the rise of liquid column in capillaries and the action of detergent molecules in removing oily dirt from fabric are the result of intermolecular interactions. Gravitational forces decay with the distance to second power, whereas intermolecular forces decay much quicker, typically with the seventh power. Cohesive forces are therefore negligible once the distance between molecules exceeds few molecular diameters, while massive bodies like stars and planets are still strongly interacting, via gravity, over astronomical distances. Electromagnetic forces are the source of all intermolecular interactions and the cohesive forces holding atoms and molecules together in solids and liquids. They can be classified into (1) purely electrostatic arising from the Coulomb force between charges, interactions between charges, permanent dipoles, quadrupoles, etc.; (2) polarization forces arising from the dipole moments induced in atoms and molecules by the electric field of nearby charges and permanent dipoles; and (3) quantum mechanical forces that give rise to covalent or chemical bonding and to repulsive steric or exchange interactions that balance the attractive forces at very short distances. The Hellman-Feynman theorem of quantum mechanics states that once the spatial distribution of the electron clouds has been determined by solving the appropriate Schrödinger equation, intermolecular forces may be calculated on the basis of classical electrostatics, in © 2005 by CRC Press LLC

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effect reducing all intermolecular forces to Coulombic forces. Note, however, that intermolecular forces exist even when the molecules are totally neutral. Solutions of the Schrödinger equation for general atoms and molecules are not easy, of course, and alternative modeling are sought to represent intermolecular forces. The van der Waals attractive forces are usually represented with a potential that varies as the inverse-sixth power of distance, whereas the repulsive forces are represented with either a power or an exponential potential. A commonly used potential between two molecules is the generalized Lennard-Jones (L-J 6–12) pair potential given by −6    −12 r  r Vij (r ) = 4 ε cij   − dij     σ    σ  

(15.6.71)

where Vij is the potential energy between two particles i and j, r is the distance between the two molecules; ε and σ are characteristic energy- and length-scales, respectively; and cij and dij are parameters to be chosen for the particular fluid and solid combinations under consideration. The first term in the righthand side is the strong repulsive force that is felt when two molecules are at extremely close range comparable to the molecular length-scale. That short-range repulsion prevents overlap of the molecules in physical space. The second term is the weaker, van der Waals attractive force that commences when the molecules are sufficiently close (several times σ). That negative part of the potential represents the attractive polarization interaction of neutral, spherically symmetric particles. The power of 6 associated with this term is derivable from quantum mechanics considerations, while the power of the repulsive part of the potential is found empirically. The Lennard-Jones potential is zero at very large distances, has a weak negative peak at r slightly larger than σ, is zero at r = σ, and is infinite as r → 0. The force field resulting from this potential is given by Fij (r ) = −

∂Vij ∂r

=

−13 −7  dij  r   48 ε   r  c −     ij σ   σ  2  σ    

(15.6.72)

A typical L-J 6–12 potential and force field are shown in Figure 15.6.11, for c = d = 1. The minimum potential Vmin = –ε, corresponds to the equilibrium position (zero force) and occurs at r = 1.12 σ. The attractive van der Waals contribution to the minimum potential is –2ε, while the repulsive energy contribution is +ε. Thus the inverse 12th-power repulsive force term decreases the strength of the binding energy at equilibrium by 50%. V(r) 4∈

F(r)

σ 48∈ Potential energy Force field

0.3

0.2 0.1 1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0.0

–0.1

r σ

–0.2 –0.3

FIGURE 15.6.11 Typical Lennard-Jones 6–12 potential and the intermolecular force field resulting from it. Only a small portion of the potential function is shown for clarity. © 2005 by CRC Press LLC

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The L-J potential is commonly used in molecular dynamics simulations to model intermolecular interactions between dense gas or liquid molecules and between fluid and solid molecules. As mentioned earlier, such potential is not accurate for complex substances, such as water, whose molecules form directional covalent bonds. As a result, MD simulations for water are much more involved.

Parting Remarks The 40-year-old vision of Richard Feynman of building minute machines is now a reality. Microelectromechanical systems have witnessed explosive growth during the last decade and are finding increased applications in a variety of industrial and medical fields. The physics of fluid flows in microdevices has been explored in this subsection. While we now know considerably more than we did just few years ago, much physics remains to be explored so that rational tools can be developed for the design, fabrication, and operation of MEMS devices. The traditional Navier-Stokes model of fluid flows with no-slip boundary conditions works only for a certain range of the governing parameters. This model basically demands two conditions: 1. The fluid is a continuum, which is almost always satisfied as there are usually more than 1 million molecules in the smallest volume in which appreciable macroscopic changes take place. This is the molecular chaos restriction. 2. The flow is not too far from thermodynamic equilibrium, which is satisfied if there is sufficient number of molecular encounters during a time period small compared to the smallest time-scale for flow changes. During this time period the average molecule would have moved a small distance compared to the smallest flow length-scale. For gases, the Knudsen number determines the degree of rarefaction and the applicability of traditional flow models. As Kn → 0, the time- and length-scales of molecular encounters are small compared to those for the flow, and the velocity distribution of each element of the fluid instantaneously adjusts to the equilibrium thermodynamic state appropriate to the local macroscopic properties as this molecule moves through the flow field. From the continuum viewpoint, the flow is isentropic, and heat conduction and viscous diffusion and dissipation vanish from the continuum conservation relations, leading to the Euler equations of motion. At small but finite Kn, the Navier-Stokes equations describe near-equilibrium, continuum flows. Slip flow must be taken into account for Kn > 0.001. The slip boundary condition is at first linear in Knudsen number, then nonlinear effects take over beyond a Knudsen number of 0.1. At the same transition regime, that is, 0.1 < Kn < 10, the linear stress-rate of strain and heat flux–temperature gradient relations — needed to close the Navier-Stokes equations — also break down, and alternative continuum equations (e.g., Burnett or higher-order equations) or molecular-based models must be invoked. In the transition regime, provided that the dilute gas and molecular chaos assumptions hold, solutions to the difficult Boltzmann equation are sought, but physical simulations such as Monte Carlo methods are more readily executed in this range of Knudsen number. In the free-molecule flow regime, that is, Kn > 10, the nonlinear collision integral is negligible and the Boltzmann equation is drastically simplified. Analytical solutions are possible in this case for simple geometries, and numerical integration of the Boltzmann equation is straightforward for arbitrary geometries, provided that the surface-reflection characteristics are accurately modeled. Gaseous flows are often compressible in microdevices even at low Mach numbers. Viscous effects can cause sufficient pressure drop and density changes for the flow to be treated as compressible. In a long, constant-area microduct, all Knudsen number regimes may be encountered and the degree of rarefaction increases along the tube. The pressure drop is nonlinear, and the Mach number increases downstream, limited only by choked-flow condition. Similar deviation and breakdown of the traditional Navier-Stokes equations occur for liquids as well, but there the situation is more murky. Existing experiments are contradictory. There is no kinetic theory of liquids, and first-principles prediction methods are scarce. Molecular dynamics simulations can be © 2005 by CRC Press LLC

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used, but they are limited to extremely small flow extents. Nevertheless, measurable slip is predicted from MD simulations at realistic shear rates in microdevices. Much nontraditional physics is still to be learned and many exciting applications of microdevices are yet to be discovered. The future is bright for this emerging field of science and technology. Richard Feynman was right about the possibility of building mite-size machines, but was somewhat cautious in forecasting that such machines, while they “would be fun to make,” may or may not be useful.

References Agarwal, R., Yun, K., and Balakrishnan, R. (1999) Beyond Navier Stokes: Burnett Equations for Flow Simulations in Continuum–Transition Regime, AIAA Paper No. 99-3580, Reston, VA. Alder, B.J., and Wainwright, T.E. (1957) Studies in molecular dynamics, J. Chemical Phys. 27, 1208–1209. Alder, B.J., and Wainwright, T.E. (1958) Molecular Dynamics by Electronic Computers, in Transport Processes in Statistical Mechanics, I. Prigogine, Ed., 97–131, Interscience, New York. Alder, B.J., and Wainwright, T.E. (1970) Decay of the velocity auto-correlation function, Phy. Rev. A 1, 18–21. Allen, M.P., and Tildesley, D.J. (1987) Computer Simulation of Liquids, Clarendon Press, Oxford. Anderson, J.L., and Quinn, J.A. (1972) Ionic mobility in microcapillaries, J. Chemical Phys. 27, 1208–1209. Arkilic, E.B. (1997) Measurement of the Mass Flow and Tangential Momentum Accommodation Coefficient in Silicon Micromachined Channels, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA. Arkilic, E.B., Schmidt, M.A., and Breuer, K.S. (1995) Slip Flow in Microchannels, in Rarefied Gas Dynamics 19, ed. J. Harvey and G. Lord, Oxford University Press, Oxford. Arkilic, E.B., Schmidt, M.A., and Breuer, K.S. (1997a) Gaseous slip flow in long microchannels, J. MEMS 6, 167–178. Arkilic, E.B., Schmidt, M.A., and Breuer, K.S. (1997b) TMAC Measurement in Silicon Micromachined Channels, in Rarefied Gas Dynamics 20, C. Shen, Ed., 6 pages, Beijing University Press, Beijing, China. Atwood, B.T., and Schowalter, W.R. (1989) Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit, Rheologica Acta 28, 134–146. Bau, H.H. (1994) Transport processes associated with micro-devices, Thermal Sci. Eng. 2, 172–178. Beskok, A. (1994) Simulation of Heat and Momentum Transfer in Complex Micro-Geometries, MSc Thesis, Princeton University, Princeton, NJ. Beskok, A. (1996) Simulations and Models of Gas Flows in Microgeometries, PhD thesis, Princeton University, Princeton, NJ. Beskok, A., and Karniadakis, G.E. (1994) Simulation of heat and momentum transfer in complex microgeometries, J. Thermophys. & Heat Transfer 8, 355–370. Beskok, A., and Karniadakis, G.E. (1999) A model for flows in channels, pipes and ducts at micro and nano scales, Microscale Thermophys. Eng. 3, 43–77. Beskok, A., Karniadakis, G.E., and Trimmer, W. (1996) Rarefaction and compressibility effects in gas microflows, J. Fluids Eng. 118, 448–456. Bhatnagar, P.L., Gross, E.P., and Krook, M. (1954) A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94, 511–524. Bird, G.A. (1963) Approach to translational equilibrium in a rigid sphere gas, Phys. Fluids 6, 1518–1519. Bird, G.A. (1965) The velocity distribution function within a shock wave, J. Fluid Mech. 30, 479–487. Bird, G.A. (1976) Molecular Gas Dynamics, Clarendon Press, Oxford. Bird, G.A. (1978) Monte Carlo simulation of gas flows, Annu. Rev. Fluid Mech. 10, 11–31. Bird, G.A. (1994) Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford. Brunauer, S. (1944) Physical Adsorption of Gases and Vapours, Oxford University Press, Oxford. Cercignani, C. (1988) The Boltzmann Equation and Its Applications, Springer-Verlag, Berlin.

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Cercignani, C. (2000) Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge University Press, London. Chan, D.Y.C., and Horn, R.G. (1985) Drainage of thin liquid films, J. Chemical Phys. 83, 5311–5324. Chapman, S., and Cowling, T.G. (1970) The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge University Press, Cambridge, Great Britain. Cheng, H.K. (1993) Perspectives on hypersonic viscous flow research, Annu. Rev. Fluid Mech. 25, 455–484. Cheng, H.K., and Emmanuel, G. (1995) Perspectives on hypersonic nonequilibrium flow, AIAA J. 33, 385–400. Ciccotti, G., and Hoover, W.G., eds. (1986) Molecular Dynamics Simulation of Statistical Mechanics Systems, North Holland, Amsterdam, the Netherlands. Debye, P., and Cleland, R.L. (1959) Flow of liquid hydrocarbons in porous vycor, J. Appl. Phys. 30, 843–849. Den, L.M. (1990) Issues in viscoelastic fluid mechanics, Annu. Rev. Fluid Mech. 22, 13–34. Dussan, E.B. (1976) The moving contact line: the slip boundary condition, J. Fluid Mech. 77, 665–684. Dussan, E.B. (1979) On the spreading of liquids on solid surfaces: static and dynamic contact lines, Annu. Rev. Fluid Mech. 11, 371–400. Dussan, E.B., and Davis, S.H. (1974) On the motion of fluid-fluid interface along a solid surface, J. Fluid Mech. 65, 71–95. Fan, L.-S., Tai, Y.-C., and Muller, R.S. (1988) Integrated movable micromechanical structures for sensors and actuators, in IEEE Transactions on Electronic Devices, vol. 35, 724–730. Fan, L.-S., Tai, Y.-C., and Muller, R.S. (1989) IC-processed electrostatic micromotors, Sensors & Actuators 20, 41–47. Gad-el-Hak, M. (1995) Questions in fluid mechanics: Stokes’ hypothesis for a Newtonian, isotropic fluid, J. Fluids Eng. 117, 3–5. Gad-el-Hak, M. (1999) The fluid mechanics of microdevices — the Freeman Scholar lecture, J. Fluids Eng. 121, 5–33. Gad-el-Hak, M. (2000) Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press, London. Gad-el-Hak, M., ed. (2002) The MEMS Handbook, CRC Press, Boca Raton, FL. Gee, M.L., McGuiggan, P.M., Israelachvili, J.N., and Homola, A.M. (1990) Liquid to solidlike transitions of molecularly thin films under shear, J. Chemical Phys. 93, 1895–1906. Guvenc, M.G. (1985) V-Groove Capillary for Low Flow Control and Measurement, in Micromachining and Micropackaging of Transducers, C.D. Fung, P.W. Cheung, W.H. Ko, and D.G. Fleming, Eds., 215–223, Elsevier, Amsterdam, the Netherlands. Haile, J.M. (1993) Molecular Dynamics Simulation: Elementary Methods, John Wiley & Sons, New York. Harley, J.C., Huang, Y., Bau, H.H., and Zemel, J.N. (1995) Gas flow in micro-channels, J. Fluid Mech. 284, 257–274. Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B. (1954) Molecular Theory of Gases and Liquids, John Wiley & Sons, New York. Israelachvili, J.N. (1986) Measurement of the viscosity of liquids in very thin films, J. Colloid Interface Sci. 110, 263–271. Israelachvili, J.N. (1991) Intermolecular and Surface Forces, 2nd ed., Academic Press, New York. Karniadakis, G.Em, and Beskok, A. (2002) Micro Flows: Fundamentals and Simulation, Springer-Verlag, New York. Kennard, E.H. (1938) Kinetic Theory of Gases, McGraw-Hill, New York. Knight, J. (1999) Dust mite’s dilemma, New Scientist 162, no. 2180, 29 May, 40–43. Knudsen, M. (1909) Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren, Annalen der Physik 28, 75–130. Kogan, M.N. (1969) Rarefied Gas Dynamics, Nauka, Moscow. Translated from Russian, L. Trilling, Ed., Plenum, New York. Kogan, M.N. (1973) Molecular Gas Dynamics, Annu. Rev. Fluid Mech. 5, 383–404. © 2005 by CRC Press LLC

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Koplik, J., and Banavar, J.R. (1995) Continuum deductions from molecular hydroynamics, Annu. Rev. Fluid Mech. 27, 257–292. Kovacs, G.T.A. (1998) Micromachined Transducers Sourcebook, McGraw-Hill, New York. Lighthill, M.J. (1963) Introduction. Real and Ideal Fluids, in Laminar Boundary Layers, ed. L. Rosenhead, 1–45, Clarendon Press, Oxford. Liu, J., Tai, Y.C., Lee, J., Pong, K.C., Zohar, Y., and Ho, C.M. (1993) In-situ monitoring and universal modeling of sacrificial psg etching using hydrofluoric acid, in Proc. IEEE Micro Electro Mechanical Systems ’93, 71–76, IEEE, New York. Liu, J., Tai, Y.C., Pong, K., and Ho, C.M. (1995) MEMS for pressure distribution studies of gaseous flows in microchannels, in Proc. IEEE Micro Electro Mechanical Systems ’95, 209–215, IEEE, New York. Loeb, L.B. (1961) The Kinetic Theory of Gases, 3rd ed., Dover, New York. Löfdahl, L., and Gad-el-Hak, M. (1999) MEMS applications in turbulence and flow control, Prog. Aero. Sciences 35, 101–203. Loose, W., and Hess, S. (1989) Rheology of dense fluids via nonequilibrium molecular hydrodynamics: shear thinning and ordering transition, Rheologica Acta 28, 91–101. Madou, M. (2002) Fundamentals of Microfabrication, 2nd ed., CRC Press, Boca Raton, FL. Majumdar, A., and Mezic, I. (1998) Stability regimes of thin liquid films, Microscale Thermophys. Eng. 2, 203–213. Majumdar, A., and Mezic, I. (1999) Instability of ultra-thin water films and the mechanism of droplet formation on hydrophilic surfaces, in Proc. ASME–JSME Thermal Engineering and Solar Energy Joint Conference, San Diego, CA, 15–19 March. Also to appear in J. Heat Transfer. Mastrangelo, C., and Hsu, C.H. (1992) A simple experimental technique for the measurement of the work of adhesion of microstructures, in Technical Digest IEEE Solid-State Sensors and Actuators Workshop, 208–212, IE, New York. Maxwell, J.C. (1879) On stresses in rarefied gases arising from inequalities of temperature, Phil. Trans. R. Soc. Part 1 170, 231–256. Migun, N.P., and Prokhorenko, P.P. (1987) Measurement of the viscosity of polar liquids in microcapillaries, Colloid J. of the USSR 49, 894–897. Moffatt, H.K. (1964) Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18, 1–18. Muntz, E.P. (1989) Rarefied gas dynamics, Annu. Rev. Fluid Mech. 21, 387–417. Nadolink, R.H., and Haigh, W.W. (1995) Bibliography on skin friction reduction with polymers and other boundary-layer additives, Appl. Mech. Rev. 48, 351–459. Nakagawa, S., Shoji, S., and Esashi, M. (1990) A micro-chemical analyzing system integrated on silicon chip, in Proc. IEEE: Micro Electro Mechanical Systems, Napa Valley, CA, IEEE 90CH2832-4, IEEE, New York. Oran, E.S., Oh, C.K., and Cybyk, B.Z. (1998) Direct simulation Monte Carlo: recent advances and applications, Annu. Rev. Fluid Mech. 30, 403–441. Panton, R.L. (1996) Incompressible Flow, 2nd ed., Wiley-Interscience, New York. Pearson, J.R.A., and Petrie, C.J.S. (1968) On melt flow instability of extruded polymers, in Polymer Systems: Deformation and Flow, R.E. Wetton and R.W. Whorlow, Eds., 163–187, Macmillian, London. Pfahler, J. (1992) Liquid Transport in Micron and Submicron Size Channels, PhD thesis, University of Pennsylvania, Philadelphia. Pfahler, J., Harley, J., Bau, H., and Zemel, J.N. (1990) Liquid transport in micron and submicron channels, Sensors & Actuators A 21–23, 431–434. Pfahler, J., Harley, J., Bau, H., and Zemel, J.N. (1991) Gas and Liquid Flow in Small Channels, in Symp. on Micromechanical Sensors, Actuators, and Systems, D. Cho et al., Eds., ASME DSC-Vol. 32, 49–60, ASME, New York. Piekos, E.S., and Breuer, K.S. (1996) Numerical modeling of micromechanical devices using the direct simulation Monte Carlo method, J. Fluids Eng. 118, 464–469.

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Pong, K.-C., Ho, C.-M., Liu, J., and Tai, Y.-C. (1994) Non-Linear Pressure Distribution in Uniform Microchannels, in Application of Microfabrication to Fluid Mechanics, P.R. Bandyopadhyay, K.S. Breuer and C.J. Belchinger, Eds., ASME FED-Vol. 197, 47–52, ASME, New York. Porodnov, B.T., Suetin, P.E., Borisov, S.F., and Akinshin, V.D. (1974) Experimental investigation of rarefied gas flow in different channels, J. Fluid Mech. 64, 417–437. Prud’homme, R.K., Chapman, T.W., and Bowen, J.R. (1986) Laminar compressible flow in a tube, Appl. Scientific Res. 43, 67–74. Richardson, S. (1973) On the no-slip boundary condition, J. Fluid Mech. 59, 707–719. Schaaf, S.A., and Chambré, P.L. (1961) Flow of Rarefied Gases, Princeton University Press, Princeton, NJ. Seidl, M., and Steinheil, E. (1974) Measurement of Momentum Accommodation Coefficients on Surfaces Characterized by Auger Spectroscopy, SIMS and LEED, in Rarefied Gas Dynamics 9, M. Becker and M. Fiebig, Eds., E9.1–E9.2, DFVLR-Press, Porz-Wahn, Germany. Sharp, K.V. (2001) Experimental Investigation of Liquid and Particle-Laden Flows in Microtubes, Ph.D. thesis, University of Illinois at Urbana–Champaign, Illinois. Sharp, K.V., Adrian, R.J., Santiago, J.G., and Molho, J.I. (2002) Liquid Flow in Microchannels, in The Handbook of MEMS, M. Gad-el-Hak, Ed., CRC Press, Boca Raton, FL. Shih, J.C., Ho, C.-M., Liu, J., and Tai, Y.-C. (1995) Non-linear pressure distribution in uniform microchannels, ASME AMD-MD-Vol. 238, New York. Shih, J.C., Ho, C.-M., Liu, J., and Tai, Y-.C. (1996) Monatomic and Polyatomic Gas Flow through Uniform Microchannels, in Applications of Microfabrication to Fluid Mechanics, K. Breuer, P. Bandyopadhyay and M. Gad-el-Hak, Eds., ASME DSC-Vol. 59, 197–203, New York. Tai, Y.-C., and Muller, R.S. (1989) IC-processed electrostatic synchronous micromotors, Sensors & Actuators 20, 49–55. Tang, W.C., Nguyen, T.-C., and Howe, R.T. (1989) Laterally driven polysilicon resonant microstructures, Sensors & Actuators 20, 25–32. Thomas, L.B., and Lord, R.G. (1974) Comparative Measurements of Tangential Momentum and Thermal Accommodations on Polished and on Roughened Steel Spheres, in Rarefied Gas Dynamics 8, eds. K. Karamcheti, Academic Press, New York. Thompson, P.A., and Robbins, M.O. (1989) Simulations of contact line motion: slip and the dynamic contact line, Nature 389, 25 September, 360–362. Thompson, P.A., and Troian, S.M. (1997) A general boundary condition for liquid flow at solid surfaces, Phys. Rev. Lett. 63, 766–769. Tison, S.A. (1993) Experimental data and theoretical modeling of gas flows through metal capillary leaks, Vacuum 44, 1171–1175. Tuckermann, D.B. (1984) Heat Transfer Microstructures for Integrated Circuits, PhD thesis, Stanford University, Stanford, CA. Tuckermann, D.B., and Pease, R.F.W. (1981) High-performance heat sinking for VLSI, IEEE Electron Device Lett. EDL-2, no. 5, May. Tuckermann, D.B., and Pease, R.F.W. (1982) Optimized convective cooling using micromachined structures, J. Electrochemical Soc. 129, no. 3, C98, March. Van den Berg, H.R., Seldam, C.A., and Gulik, P.S. (1993) Compressible laminar flow in a capillary, J. Fluid Mech. 246, 1–20. Vargo, S.E., and Muntz, E.P. (1996) A Simple Micromechanical Compressor and Vacuum Pump for Flow Control and Other Distributed Applications, AIAA Paper No. 96-0310, AIAA, Washington, DC. Vincenti, W.G., and Kruger, C.H. Jr. (1965) Introduction to Physical Gas Dynamics, John Wiley & Sons, New York. Von Smoluchowski, M. (1898) Ueber Wärmeleitung in verdünnten Gasen, Annalen der Physik und Chemie 64, 101–30. Went, F.W. (1968) The size of man, American Scientist 56, 400–413.

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15.7 Solid Mechanics of Microdevices C. Channy Wong, Andrew D. Oliver, and David W. Plummer Characteristics of Surface Micromachined Devices Advances in micromachining technology allow many engineering systems and components to be built smaller and more compact, and having less weight. A system with less weight will have a smaller inertia; thus this system can start running and stop quickly. A smaller system can also be more resistant to shock and vibration, because the strength of a system decreases as the square of its dimensions while the mass and inertia decreases as the cube of its dimensions. Another difference at the microscale is that other forces, such as van der Waal forces, electrostatic attraction, and surface tension, can be dominant. Hence, the cause of failure can be very different, for example, (1) large frictional forces leading to more rubbing between surfaces and wearing out faster, (2) stiction an unwanted adhesion between two structural layers, and (3) excessive stress and stress gradient in the thin-film layers. Among existing micromachining processes, surface micromachining (SMM) is very popular because micromachines built by SMM processes can be created and assembled at the same time that the constituent components are being fabricated. This batch fabrication process can minimize the assembly labor costs. However, SMM does have a disadvantage in that there is a limit on the number and types of layers available to designers. In the following discussion, our focus will be on the surface micromachined devices. Wear Similar to conventional machines, micromachines quite often have surfaces touching, rubbing, or impacting on each other. These generate friction and wear (see Figure 15.7.1). However, unlike convention machines, which use dissimilar materials to reduce friction and wear, thin-film layers of micromachines are usually made of the same material. Hence friction and wear become an important issue when addressing reliability of microsystems (Dugger 1999, Tanner 2000). Different lubrication methods have been investigated. Because of the extremely small gap size, liquid lubricants are difficult to apply. This leaves gas films and self-assembly monolayer films as the choices for lubrication. Stiction Stiction is an unintended adhesion between thin-film layers in the surface micromachined devices (see Figure 15.7.2). Two major categories of stiction have been observed: (1) stiction in the final-release process, and (2) in-use stiction. In surface micromachining, a popular final processing step is to apply aqueous chemical solution (e.g., hydrofluoric acid) to remove the sacrificial layers so that the designed

Rubbing Surfaces

Before

10 mm

Pin Hub After

Accumulated Debris

FIGURE 15.7.1 A pin joint and hub joint before and after accelerated wear. (Photograph courtesy of D. Tanner, Sandia National Laboratories.) © 2005 by CRC Press LLC

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FIGURE 15.7.2 SEM picture of cantilever beams with different lengths. Sticting occurs at those beams on the left. (Photograph courtesy of Sandia National Laboratories.)

parts can move freely and perform work. However, during the release and drying process, capillary forces can pull and hold thin-film layers together and prevent them to move freely and functioning as desired. The in-use stiction occurs when two thin-film layers unexpectedly adhere together during the normal operation. This can be caused by many factors. One explanation is that electrostatic charges are trapped and built up in the insulating materials, such as silicon nitride, after a long usage. Other factors, such as electrostatics discharge (ESD), can cause stiction as well. Stiction failure is one of the major concerns when addressing the reliability of the surface micromachined devices. To minimize stiction and wear and to improve performance and reliability of surface micromachined devices, one popular approach is to treat surfaces by applying a hydrophobic coating. Surface treatment is an efficient and effective approach, though it is equally important to incorporate good judgments when developing and designing a microsystem. A few good engineering practices are: (1) to reduce surface area, (2) to add dimples (a small protrusion on the bottom of a structure that act to reduce the area in contact with the structure below), and (3) to increase the stiffness of structure in the out-of-plane direction. In packaging, it is essential to enclose the parts silicon dies in an environment with a dew point well below the minimum storage or operating temperature of the mechanism.

Microsystems Design Considerations General Guidelines In addition to the design considerations listed earlier — reducing surface areas, adding dimples, and increasing stiffness — another good practice is to always ground thin-film layers to prevent any electrostatic charge being trapped and built up in the insulating materials, such as silicon nitride, after a long usage. One approach of grounding is to use a conductive ground plane to cover any exposed insulating layers and to connect every object to ground. Stress in Films and Stress Gradient Stress can be developed in the thin films for many reasons. A typical source of stress is the differences in the thermal expansion coefficient between various thin films or between thin film and substrate, especially with the thin film is deposited at an elevated temperature (see Figure 15.7.3). Another source is from the doping of semiconductor films if the size of the dopant atoms is quite different from the host atoms. A significant stress buildup or a large stress gradient will lead to cracking, delamination, and/or buckling. To relieve stresses in microstructures, a sound design practice is to use folded flexures (see Figure 15.7.4). A folded flexure helps because each flexure is free to expand or contract in the axial direction. This can minimize the stress caused by the fabrication process and by thermal expansion mismatches © 2005 by CRC Press LLC

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FIGURE 15.7.3 Picture showing the surface profile of a micro-optical shutter as a result of the thermal stress generated from laser spot heating. (Photograph courtesy of O.B. Spahn, Sandia National Laboratories.)

FIGURE 15.7.4 Three flexure designs. From left to right they are folded, meandering flexures, and crab leg flexures.

between the flexures and the substrate. Crab leg flexure and meandering flexure (Figure 15.7.4) can also help to relieve the residual stress in flexures. Unavoidable stress concentration can still exist even though if we apply the design practice described earlier to help manage stress. Most unavoidable stress concentration is created by the microfabrication process. For example, etch release holes may serve as stress concentrations in plates. Sharp corners in Manhattan (right-angle) geometries can also create stress concentrations. Hence, the design engineer needs to adjust the stress calculated from the standard equations as follows: σCONCENTRATION = K t σ NOMINAL

(15.7.1)

where Kt is the stress concentration factor. Next, we will examine three typical component structures: cantilever beam springs, fixed-beam springs, and flexures, which are commonly found in the microsystems, and evaluate their mechanical responses. Cantilever Beam Springs For a cantilever beam with a fixed anchor on one end (Figure 15.7.5), the deformation (y) at any arbitrary point (x) is © 2005 by CRC Press LLC

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Y

P X

L

FIGURE 15.7.5 Cantilever beam spring. The deformation of the beam is due to the load “P.”

y=

(

P 3Lx 2 − x 3 6EI

)

(15.7.2)

where L is the length of the beam, E is Young’s modulus of the beam material, and P is the applied load. The maximum deflection will occur at the end of the beam: y MAX =

PL3 3EI

(15.7.3)

The maximum stress is at the fixed end where the internal bending moment is PL. For a rectangular cross section of thickness h and width w, the maximum stress is σ MAX =

6PL h ⋅w2

(15.7.4)

The fundamental resonant frequency in Hertz (Bathe 1982) becomes f1 =

3.52 EI 2π ρAL4

(15.7.5)

where E is Young’s modulus, A is the cross-sectional area (h·w), and ρ is the density. For a microfabricated cantilever beam made of polycrystalline silicon with dimensions of 50 µm wide, 6 µm thick, and 200 µm long, the electrostatic force required to pull down the beam and touch the electrode 2 µm below is as follows:

)(

(

)

2 4 3EIy 3 0.155N µm 900 µm (2 µm) P= 3 = = 105 µN (200 µm)3 L

assuming that the Young’s modulus for polycrystalline silicon is 155 GPa or 0.155 N/µm2 and the density of solid polycrystalline silicon is 2.33 × 10–18 kg/µm3. The area moment of inertia for a rectangular cross section is I = wt3/12 = (50 µm)(6 µm)3/12 = 900 µm4. If the beam is suddenly released, it will vibrate at the following frequency:

f=

(

)(

)

0.155N µm 2 900 µm 4 (1000 µm m ) 3.52 EI 3.52 = = 198 kHz 2π ρAL4 2π [2.33 10−18 kg µm3](50 µm × 6 µm )( 200 µm )4

© 2005 by CRC Press LLC

(

)

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Y P

X

b a L

FIGURE 15.7.6 Deflection of a fixed-beam spring due to a point load. The applied load is “P,” the load’s location is shown by “a” and “b” on the x-axis, and the spring deflects in the direction of the y-axis.

Fixed-Fixed Beam Springs The deflection of a fixed beam (Figure 15.7.6) at any point, 0 < x < a, with a point-applied load (Timoshenko 1959) is y=

Pb 2 x 2 3aL − ( 3a + b ) x  6L3EI

(15.7.6)

The fundamental frequency for a fixed beam (Bathe 1982) is f1 =

22.4 EI 2 π ρAL4

(15.7.7)

The form of this equation is similar Equation 15.7.5 for cantilever beams. Note that fixing the other end of the beam increases the fundamental frequency by more than a factor of six, with the fixed-fixed beam having the same dimension as the cantilever beam. For a beam anchored on both ends with dimensions of 50 µm wide, 6 µm thick, and 200 µm long, the force applied at the center needed to pull it down to the substrate 2 µm below is P=

)(

(

)

2 4 192EIy 192 0.155N µm 900 µm ( 2 µm ) = = 6700 µN L3 ( 200 µm)3

and the resulting frequency of vibration when released is

( (

)(

)

0.155N µm 2 900 µm 4 (1000 µm m ) 22.4 f= = 1.26 MHz 2π  2.33 10−18 kg µm3  ( 6 µm × 50 µm )( 200 µm )4  

)

This illustrates that the fundamental frequency for the beam anchored at both ends is much higher than for the cantilever beam. Flexures Flexures can be found in many microsystems. For instance, they can be used for rotation. When a torque is applied to a flexure, it causes twisting, which creates shear stress in the element. The shear stress in a circular bar arising from an applied torque is τ= © 2005 by CRC Press LLC

Tc J

(15.7.8)

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MEMS Technology

150 µm

20 µm

150 µm

6 µm

4 µm 150 µm

FIGURE 15.7.7 Dimensions of a polysilicon plate supported by torsional flexures.

where τ is the shear stress, T is the applied torque, c is the distance from the central axis to the point where the stress is desired, and J is the polar area moment of inertia. For a rectangular bar, the resulting shear stress can be found as follows: τ=

9T 2wt 2

(15.7.9)

where w is the width (larger dimension) of the flexure and t is the thickness (smaller dimension). The deflection created by the applied torque is θ=

TL JG

(15.7.10)

where L is the length of the flexure, G is the shear modulus, and J is polar moment of inertia. The polar moment of inertia is defined as J = I x + IY

(15.7.11)

Consider a 150-µm square, polysilicon plate (Figure 15.7.7), which is supported by 6 µm wide torsional flexures; the axis of rotation is through the plate’s midpoint. The plate is 4 µm thick and the flexures are 20 µm long. If one edge is pulled down to the substrate 3 µm from the bottom of the plate, neglecting the deformation of the plate, the flexures are required to rotate through an angle of 2.3° for the plate to touch the substrate:   θ = tan −1  3 µm (150 µm 2) = 2.3° = 0.040 rad  The resulting shear stress in the flexure can be found using Equation 15.7.9, which requires us to determine J and T first. Assuming that the shear modulus of rigidity is 0.0772 N/µm2, the polar area moment of inertia for a rectangular section is J=

wt 2 2 (6 µm)(4 µm)  2 2 4 w +t = (6 µm) + (4 µm)  = 104 µm 12 12

)

(

The torque required to rotate one flexure through the angle θ is T=

(

)(

)

4 2 JGθ 104 µm 0.0772N µm ( 0.04 rad ) = = 1.6 × 10−2 µNm L 20 µm

The resulting shear stress in each flexure is τ= © 2005 by CRC Press LLC

9T 9 × 1.6 × 10−2 N ⋅ µm = = 7.5 × 10−4 N µm 2 = 0.75 GPa 2 2wt 2(6 µm)(4 µm)2

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Chapter 15

To predict the vibration frequency, the plate-flexure system can be treated as a rigid body vibrating about a spring. The differential equation of motion for a simple mass-spring system and the associated natural frequency are as follows:  + K ϕ = 0 iϕ

(15.7.12)

1 K 2π i

(15.7.13)

fn =

where i is the mass moment of inertia (different from the area moment of inertia used to calculate stress) and K is the torsional spring rate. The spring rate of the flexures is

K=

)(

(

)

4 2 T JG 104 µm 0.0772N µm = = = 0.40 µNm rad θ L 20 µm

The mass moment of inertia for a thin rectangular plate vibrating about its center is

i=

 2.1 × 10−13 kg  (150 µm )2 1  mw 2 =  = 3.93 × 10−10 kg ⋅ µm 2 12 12

The natural frequency is fn =

1 (0.40N ⋅ µm)(106 )µm m = 5.1MHz 2π 3.93 × 10−10 kg ⋅ µm 2

Application This section will apply the material covered in the previous section and show how to design a microsystem. A microflex mirror is used as an example. Microflex Mirrors The microflex mirror is a device that deforms out of plane (Garcia 1998). It can be configured as a mirror, an optical shutter, a valve, or any structure that requires a plate or beam to move out of the plane of the base substrate. Figure 15.7.8 shows an SEM photograph of the device. The design consists of a long flexible beam connected to a mirror plate that in turn is connected to two anchors via two additional flexible beams. When a force is placed on the long flexible beam in direction of the anchors, the structure is placed under a compressive force. When the force exceeds the critical value, FCR = (π2EI)/(4L2), the structure buckles. Because the long flexible beam is larger in the direction parallel to the plane of the substrate than it is in the direction away from the substrate, it preferentially buckles out of the plane of the substrate. Since the plate and the two anchor beams are wider than the long flexible beam, the majority of the bending occurs in the long flexible beam and not in the plate or the anchor beams. Consider a main beam with dimensions of 300 µm long, 4 µm wide, and 1 µm thick and that has a Young’s modulus of 155 GPa, its moment of inertia will be I=

wt 3 = 3.3 × 10−25 m 4 12

By treating it as a cantilever beam and neglecting the buckling of the anchors and the mirror, the axial force required for a micromachined polysilicon mirror to buckle is as follows: © 2005 by CRC Press LLC

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MEMS Technology

FIGURE 15.7.8 A flexible pop-up mirror that operates via buckling. (Photograph courtesy of E. Garcia, Sandia National Laboratories.)

FIGURE 15.7.9 Hinged polysilicon micromirror fabricated in the Sandia National Laboratories SUMMiT process. (Photograph courtesy of Sandia National Laboratories.)

Fcr =

π 2 EI π 2 × 155 × 109 N m 2 × 3.3 × 10−25 m 4 = = 1.4 µN 2 4 L2 4 × 300 × 10−6 m

(

)

This analysis shows that it is necessary to have a great deal of force in order to buckle the flexible mirror. To achieve and exceed this buckling force, a transmission developed by Garcia was used to increase the force on the mirror (Garcia 1998), as shown in Figure 15.7.9. The transmission is used to trade displacement for force to ensure that the mirror buckled. The magnitude of deflection after buckling is highly nonlinear and is difficult to predict. © 2005 by CRC Press LLC

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Chapter 15

An interesting point about buckling is that it is hard to determine if the beam will initially buckle toward or away from the substrate. If the beam buckles away from the substrate, it will continue to deflect away from the substrate. If it initially buckles toward the substrate, it will contact the substrate and further compression of the beam will result in the structure buckling away from the substrate.

Acknowledgments We would like to thank many of our colleagues in Microsystems Engineering Science Applications–Technology and Operation Prototype (MESA-TOP) facility at Sandia National Laboratories for their help and efforts. Many of the devices and concepts discussed in this section are due to their efforts. Sandia National Laboratories is a multi-program laboratory operated by the Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL8500.

References Bathe, K-J. 1982. Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, NJ. Dugger, M.T., Poulter, G.A., and Ohlhausen, J.A. 1991. Surface passivation for reduced friction and wear in surface micromachined devices,” Proceedings of the Fall MRS Symposium, Boston, December. Garcia, E.J. 1998. Micro-flex mirror and instability actuation technique, Proceedings of the 1998 IEEE Conference on Micro Electro Mechanical Systems, MEMS98, 470–474. Tanner, D.M. 2000. Proceedings of the 22nd International Conference on Microelectronics, invited keynote, Nis, Yugoslavia, 97–104. Timoshenko, S.P., and Woinowsky-Krieger, S. 1959. Theory of Plates and Shells, McGraw-Hill, New York.

Further Information Gardner, J.W., Varadan, V.K., Awadelkarium, O.O. 2002. Microsensors, MEMS, and Smart Devices, John Wiley & Sons, Ltd., Chichester, UK. Hsu, T.-R. 2002. MEMS and Microsystems, Design and Manufacture, McGraw-Hill, New York. Kovacs, G.T.A. 1998. Micromachined Transducers — Source Book, McGraw-Hill, New York. Senturia, S.D. 2001. Microsystem Design, Kluwer Academic Publishers, Boston.

© 2005 by CRC Press LLC

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16 Environmental Engineering Jan F. Kreider Kreider & Associates

Ari Rabl

16.1

Ecole des Mines de Paris

Nevis E. Cook, Jr.

16.2

Colorado School of Mines

Tissa Illangasekare Colorado School of Mines

Paolo Zannetti The EnviroComp Institute

Peter S. Curtiss

16.3

Sources of Pollution and Regulations

16.4

Regulations and Emission Standards

Sources • Pollutant Monitoring • Air Quality Monitoring Water • Air

16.5

Mitigation of Water and Air Pollution Overview • Air Pollution Control • Water Pollution Control

16.6

Curtiss Engineering

Environmental Modeling Air Pollution Dispersion Modeling • Atmospheric Chemistry • Groundwater Pollution Modeling • Surface Water Pollution Transport Modeling • Impact Pathway Methodology

John Firor National Center for Atmospheric Research

Benchmarks and Reference Conditions Natural Environment • Soils and Water Basin Definitions • Acceptable Levels of Pollutants

Ronald R. Hewitt Cohen Colorado School of Mines

Introduction Environmental Engineering and the Role of Mechanical Engineers • Environmental Burdens and Impacts

16.7

Global Climate Change

16.1 Introduction Ari Rabl and Jan F. Kreider Environmental Engineering and the Role of Mechanical Engineers The subject of environmental science and management is vast and interdisciplinary, ranging from highly technical matters, such as the design of emission control equipment, to socioeconomic matters, such as the valuation of the impacts of pollution. The goal is to prevent or reduce undesirable impacts of human activities on the environment. Within this endeavor are several areas where mechanical engineers can make important contributions. One type of contribution concerns the design of equipment, in particular for the control of emissions; an example is an electrostatic precipitator to reduce emissions of particulates from the flue gas of a boiler or furnace. Another type of contribution concerns the modeling of the dispersion of pollutants in the environment. This chapter covers air pollution, surface water pollution, and groundwater pollution. Since space is limited and mechanical engineers are most likely to be involved in air pollution analysis and abatement projects, our emphasis is on air pollution problems.

© 2005 by CRC Press LLC

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Chapter 16

Impacts Burdens Greenhouse gases (CO2, CH4, N2O, ...) Primary air-pollutants Particulates SO2 NOx CO Toxic metals (As, Cd, Pb, Hg, ...) Toxic organic compounds (e.g., dioxins) VOC Secondary air pollutants O3 (from NO + VOC) Acid rain (from NOx, SOx) Nitrates (from NOx) Sulfates (from SOx) Liquid residues Toxic metals (Pb, Hg, Cd, ...) Toxic organic compounds (e.g., dioxins) COD BOD Solid residues Other Thermal Noise, odor

Extent Space Time G P,F

Climate X

Health

Natural

R R R R R

P P P P P, F

X X x x X

X

R

P, F

X

X

R

P

R R R

P P P

X

X

x x

X X

X X x x

L, R

P, F

X

X

L, R

P, F

X

X

L, R L, R L

P, F P, F P, F

x x x

X X x

L L

P P

x

Environment Agricultural Man-Made

x

x

X x x

X X x x

x X x x

x x x

x x

VOC = volatile organic compounds, COD = chemical oxygen demand, BOD = biological oxygen demand; Impacts: X = important; x = may be important; blank = usually not important; Extent: L = local (up to tens of kilometers); P = present generation; R = regional (hundreds to thousands of kilometers); G = global; F = future generations.

FIGURE 16.1.1 Overview of environmental burdens and major impact categories, with approximate indication of typical geographic extent and typical importance of impact.

Environmental Burdens and Impacts

Ari Rabl As general guidance to the field of environmental engineering, it may be helpful to present the most important environmental burdens and impacts in the form of a matrix, as shown in Figure 16.1.1. Burdens, for example, the emission of a pollutant, are listed in the column on the left; impact categories are listed as a row at the top. Each element in this matrix corresponds to a specific impact of a specific burden. An X indicates that the impact from the corresponding burden is likely to be significant. Particulate air pollution, for instance, has been shown to cause a significant increase in mortality. As an added feature we have indicated the spatial and temporal extent to the burdens. The classic air pollutants (particulates, NOx, and SOx) are dispersed over distances on the order of a thousand kilometers, and they affect essentially only the present generation; thus, the second and third columns show the letters R (for regional) and P (for present generation). Global warming from greenhouse gases, on the other hand, affects the entire globe and will persist over decades or centuries, hence the letters G (for global) and P, F (for present and future generations). © 2005 by CRC Press LLC

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Environmental Engineering

The classification in Figure 16.1.1 is not without ambiguities or problems. For example, we have indicated the impact of greenhouse gases as “climate change” only, even though this category includes such effects as deaths from flooding. The relative importance of impacts may change with improved scientific understanding and with the evolution of societal preferences. One should also note that the assignment of effects to causes is in many cases quite uncertain; for instance, the observed mortality from air pollution could be due to particulates or due to SO2. Some impacts, especially thermal pollution and noise, can be highly site dependent. The cooling water from a power plant, for instance, could damage the ecosystem of a river, or it could be used to improve the output of a fishery. Each of the categories in Figure 16.1.1 could be broken down into subcategories: • Health • Mortality • Injury • Cancer • Respiratory illness • Natural environment • Recreational value of land (including forests) • Recreational value of water (including fishing) • Biodiversity • Agricultural environment • Crops • Cattle (milk, meat, fur, …) • Wood production by forests • Commercial fishing • Man-made environment • Functional buildings • Historical buildings • Other objects (bridges, cars, …) • Noise

16.2 Benchmarks and Reference Conditions Ari Rabl, Nevis Cook, Ronald R. Hewitt Cohen, and Tissa Illangasekare Natural Environment Air Basins Unpolluted air is an idealization, but its composition has been defined as indicated in Table 16.2.1. Unfortunately, measurements of truly unpolluted air were not, and can never be, made because measurement techniques and even the interest in measurements did not exist when air was unpolluted. Now even the most remote sites have mildly polluted air. Although measurements of the undisturbed atmosphere were not made, we can gain some insight into trends of air pollutant burden growth by examining emissions. Figure 16.2.1 shows the emissions of the classical air pollutant species in the U.S. since 1940. The emissions of SO2, PM, and VOC have been decreasing since 1970, but not those of NOx. Note that Table 16.2.1 uses two sets of units for gaseous pollutants, one volumetric, the other mass based. To convert from one to the other, the ideal gas law is used with the result (at 1 atm and 25°C): 1 ppm = MW ∗ 40.9 µg/m3 where MW is the molecular weight. © 2005 by CRC Press LLC

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Chapter 16

TABLE 16.2.1 Gaseous Composition of Unpolluted Air (Dry Basis) ppm (vol) Nitrogen Oxygen Water Argon Carbon dioxide Neon Helium Methane Krypton Nitrous oxide Hydrogen Xenon Organic vapors

780,000 209,400 — 9,300 315 18 5.2 1.0–1.2 1.0 0.5 0.5 0.08 ~0.02

µg/m3 8.95 × 108 2.74 × 108 — 1.52 × 107 5.67 × 105 1.49 × 104 8.50 × 102 6.56–7.87 × 102 3.43 × 103 9.00 × 102 4.13 × 101 4.29 × 102 —

Surface Water Basins Human activity has also dramatically altered the distribution, quantity, and quality of Earth’s water, especially since the industrial revolution. Accurate measurement of many water impurities, particularly trace impurities, has only become possible in the latter part of the twentieth century. Given the quantities and wide distribution of human-generated wastes delivered directly or indirectly (via atmospheric deposition or surface runoff and erosion) to water bodies, recent water quality surveys might not be representative of truly “natural” conditions. As an example, a “pristine,” undeveloped alpine lake may show traces of plutonium that are residuals of 1950s through 1960s atmospheric testing of nuclear weapons. Lead from automobile emissions can be detected in the bottom sediments of the Atlantic Ocean, more than 1500 km from the nearest landmass. A tabulation of the averages and ranges of concentrations of many naturally occurring substances detected in minimally impacted waters can serve as a benchmark, admittedly imperfect, against which to compare “polluted” waters, water-quality criteria, and regulatory standards. Choice of Reference Conditions. The authors propose, as reference conditions, the concentrations of measured impurities in the oceans, “average” rainwaters, and “average” river waters. This choice of reference waters was based on one of two distinct criteria: 1. The volume of water was so large (such as the open oceans) that human activity has had little detectable effect on average water quality conditions. 2. The waste input is small and the water body is rapidly and continuously renewed by unpolluted sources (such as a tributary to a major river). The major and minor constituents of water and their quantities are easily presented in tables. The use of the word “major” indicates materials present or required in large quantities. Table 16.2.2 summarizes data on the constituents of selected natural waters. Note that the concentration data for major constituents (Table 16.2.2) is given in milligrams per liter and the concentration data for the minor constituents (Table 16.2.3) is given in micrograms per liter. Inclusion of boron and fluoride as major constituents is somewhat arbitrary and was based on their occurrence at greater than 1 mg/L in seawater. In these tables, individual entries are average values reported by U.S. regional or national surveys. Ranges are derived from the majority of the data from similar surveys, excluding data obtained from waters apparently contaminated by pollution. Some surveys presented results by stating that “very few” samples exceeded certain concentration values or simply reported measurements as “less than” due to analytical detection limits. These results are preceded by the symbol < in the tables. Although the data given are believed to be representative of water found in natural settings, keep in mind that, especially for industrially important trace constituents, the upper limits of the concentration ranges may include some small level of anthropogenic inputs.

© 2005 by CRC Press LLC

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16-5

Environmental Engineering

30

Emissions (million short tons)

25

20

15

10

5

0 1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Year Fuel Combustion

Industrial Processing

On-road

Non-road

Miscellaneous

(a)

35

Emissions (million short tons)

30 25 20 15 10 5 0 1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Year Fuel Combustion

Industrial Processing

On-road

Non-road

Miscellaneous

(b)

FIGURE 16.2.1 Trends in U.S. national emissions of air pollutants, 1940–1995. (a) NOx, (b) SO2, (c) PM10, (d) VOC. (From EPA, 2000.)

© 2005 by CRC Press LLC

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16-6

Chapter 16

18

Emissions (million short tons)

16 14 12 10 8 6 4 2 0 1940

1945

1950

1955

Fuel Combustion

1960

1965

1970 Year

Industrial Processing

1975

1980

On-road

1985

1990

Non-road

1995

Miscellaneous

(c)

35

Emissions (million short tons)

30 25 20 15 10 5 0 1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Year Fuel Combustion Non-road

Industrial Processing

Solvent Utilization

On-road

Miscellaneous (d)

FIGURE 16.2.1

(continued).

Additional categories of natural and contaminant components of water exist. There are complex, difficult-to-characterize, organic humic materials in natural water bodies that represent the endpoint of decay of formerly living organic matter. There are synthetic chemicals by the thousands, produced currently and in the past, by the chemical industry. Some of these compounds and polymers are completely © 2005 by CRC Press LLC

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Environmental Engineering

TABLE 16.2.2 Major Constituents and Characteristics of Natural Waters (constituent concentrations in mg/L, unless otherwise noted) Constituent Cl– Na+ Mg++ Ca++ K+ Br – Sr++ SiO2 B3– F– pH (units) Hardness (total) Ammonia, as N Nitrate, as N BOD

Oceans 19,000 10,500 2,700 1,350 410 390 142 0.67 8 6.4 4.5 1.3 8.2 — — — —

Rivers

Rain

5.8, 7.8 5.3, 6.3 8.3, 11 3.4, 4.1 13.4, 15 1.3, 2.3 52, 58