White fluid mechanics 7th c2011 txtbk3

1. Conversion Factors from BG to SI Units To convert from To Multiply by Acceleration ft/s2 m/s2 0.3048 Area ft2 m2 9.2903 E 2 mi2 m2 2.5900 E 6 acres m2 4.0469 E 3 Density slug/ft3 kg/m3 5.1538 E 2 lbm/ft3 kg/m3 1.6019 E 1 Energy ft-lbf J 1.3558 Btu J 1.0551 E 3 cal J 4.1868 Force lbf N 4.4482 kgf N 9.8067 Length ft m 0.3048 in m 2.5400 E 2 mi (statute) m 1.6093 E 3 nmi (nautical) m 1.8520 E 3 Mass slug kg 1.4594 E 1 lbm kg 4.5359 E 1 Mass ow slug/s kg/s 1.4594 E 1 lbm/s kg/s 4.5359 E 1 Power ftlbf/s W 1.3558 hp W 7.4570 E 2

2. Conversion Factors from BG to SI Units (Continued) To convert from To Multiply by Pressure lbf/ft2 Pa 4.7880 E 1 lbf/in2 Pa 6.8948 E 3 atm Pa 1.0133 E 5 mm Hg Pa 1.3332 E 2 Specic weight lbf/ft3 N/m3 1.5709 E 2 Specic heat ft2 /(s2 R) m2 /(s2 K) 1.6723 E 1 Surface tension lbf/ft N/m 1.4594 E 1 Temperature F C tC (tF 32) R K 0.5556 Velocity ft/s m/s 0.3048 mi/h m/s 4.4704 E 1 knot m/s 5.1444 E 1 Viscosity lbfs/ft2 Ns/m2 4.7880 E 1 g/(cms) Ns/m2 0.1 Volume ft3 m3 2.8317 E 2 L m3 0.001 gal (U.S.) m3 3.7854 E 3 uid ounce (U.S.) m3 2.9574 E 5 Volume ow ft3 /s m3 /s 2.8317 E 2 gal/min m3 /s 6.3090 E 5 5 9

3. EQUATION SHEET Ideal-gas law: p RT, Rair 287 J/kg-K Hydrostatics, constant density: p2 p1 (z2 z1), g Buoyant force: (displaced volume) CV momentum: Steady ow energy: (p/V2 /2gz)out hfriction hpump hturbine Incompressible continuity: Incompressible stream function Bernoulli unsteady irrotational ow: Pipe head loss: where f Moody chart friction factor Laminar flat plate ow: , , Isentropic ow: , , Prandtl-Meyer expansion: K (k1)/(k1), K1/2 tan1 [(Ma2 1)/K]1/2 tan1 (Ma2 1)1/2 Gradually varied channel ow: , Fr V/Vcritdy/dx (S0 S)/(1 Fr2 ) p0/p (T0/T)k(k1) 0/ (T0/T)1/(k1) T0 /T15(k1)/26Ma2 CD Drag/11 2V2 A2; CL Lift/11 2V2 A2 CD 1.328/ReL 1/2 cf 0.664/Rex 1/2 /x 5.0/Rex 1/2 hf f(L/d)V2 /(2g) /t dp/ V2 /2 gz Const u /y; v /x (x,y): V 0 (p/V2 /2gz)in g 3(AV)V4out g 3(AV )V4in gF d/dt1 CV Vd2 FB fluid Surface tension: Hydrostatic panel force: , yCPIxxsin/(hCGA), xCPIxysin/(hCGA) CV mass: CV angular momentum: Acceleration: Navier-Stokes: Velocity potential : Turbulent friction factor: Orice, nozzle, venturi ow: , Turbulent at plate ow: , 2-D potential ow: One-dimensional isentropic area change: A/A*(1/Ma)[1{(k1)/2}Ma2 ](1/2)(k1)/(k1) Uniform ow, Mannings n, SI units: Euler turbine formula: Power Q(u2Vt2 u1Vt1), u r V0(m/s) (1.0/n)3Rh(m)42/3 S0 1/2 2 2 0 cf 0.027/Rex 1/7 , CD 0.031/ReL 1/7 /x 0.16/Rex 1/7 d/DQCdAthroat 32p/5(14 )641/2 2.0 log10 3/(3.7d) 2.51/1Red 1f)4 1/1f u /x; v /y; w /z (x,y,z) (dV/dt)gp2 V u(V/x) v(V/y)w(V/z) dV/dt V/t gAV(r0V)outgAV(r0V)in gM0 d/dt(CV (r0V)d) g(AV)in 0 d/dt(CV d) g(AV)out F hCGA p Y(R1 1 R2 1 )

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5. Fluid Mechanics whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page i ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

6. McGraw-Hill Series in Mechanical Engineering Alciatore/Histand Introduction to Mechatronics and Measurement Systems Anderson Computational Fluid Dynamics: The Basics with Applications Anderson Fundamentals of Aerodynamics Anderson Introduction to Flight Anderson Modern Compressible Flow Beer/Johnston Vector Mechanics for Engineers: Statics and Dynamics Beer/Johnston Mechanics of Materials Budynas Advanced Strength and Applied Stress Analysis Budynas/Nisbett Shigleys Mechanical Engineering Design engel Heat and Mass Transfer: A Practical Approach engel Introduction to Thermodynamics & Heat Transfer engel/Boles Thermodynamics: An Engineering Approach engel/Cimbala Fluid Mechanics: Fundamentals and Applications engel/Turner Fundamentals of Thermal-Fluid Sciences Dieter Engineering Design: A Materials & Processing Approach Dieter Mechanical Metallurgy Dorf/Byers Technology Ventures: From Idea to Enterprise Finnemore/Franzini Fluid Mechanics with Engineering Applications Hamrock/Schmid/Jacobson Fundamentals of Machine Elements Heywood Internal Combustion Engine Fundamentals Holman Experimental Methods for Engineers Holman Heat Transfer Kays/Crawford/Weigand Convective Heat and Mass Transfer Meirovitch Fundamentals of Vibrations Norton Design of Machinery Palm System Dynamics Reddy An Introduction to Finite Element Method Schey Introduction to Manufacturing Processes Smith/Hashemi Foundations of Materials Science and Engineering Turns An Introduction to Combustion: Concepts and Applications Ugural Mechanical Design: An Integrated Approach Ullman The Mechanical Design Process White Fluid Mechanics White Viscous Fluid Flow whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page ii ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

7. Seventh Edition Frank M. White University of Rhode Island Fluid Mechanics whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page iii ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

8. FLUID MECHANICS, SEVENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions 2008, 2003, and 1999. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOC/DOC 1 0 9 8 7 6 5 4 3 2 1 0 www.mhhe.com ISBN 978-0-07-352934-9 MHID 0-07-352934-6 Vice President & Editor-in-Chief: Marty Lange Vice President, EDP/Central Publishing Services: Kimberly Meriwether-David Global Publisher: Raghothaman Srinivasan Senior Sponsoring Editor: Bill Stenquist Director of Development: Kristine Tibbetts Developmental Editor: Lora Neyens Senior Marketing Manager: Curt Reynolds Senior Project Manager: Lisa A. Bruodt Production Supervisor: Nicole Baumgartner Design Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St. Louis, Missouri (USE) Cover Image: Copyright SkySails Senior Photo Research Coordinator: John C. Leland Photo Research: Emily Tietz/Editorial Image, LLC Compositor: Aptara, Inc. Typeface: 10/12 Times Roman Printer: R. R. Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data White, Frank M. Fluid mechanics / Frank M. White. 7th ed. p. cm. (Mcgraw-Hill series in mechanical engineering) Includes bibliographical references and index. ISBN 9780073529349 (alk. paper) 1. Fluid mechanics. I. Title. TA357.W48 2009 620.106dc22 2009047498 whi29346_fm_i-xvi.qxd 12/30/09 1:16PM Page iv ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

9. Frank M. White is Professor Emeritus of Mechanical and Ocean Engineering at the University of Rhode Island. He studied at Georgia Tech and M.I.T. In 1966 he helped found, at URI, the rst department of ocean engineering in the country. Known primarily as a teacher and writer, he has received eight teaching awards and has written four textbooks on uid mechanics and heat transfer. From 1979 to 1990 he was editor-in-chief of the ASME Journal of Fluids Engineering and then served from 1991 to 1997 as chairman of the ASME Board of Editors and of the Publications Committee. He is a Fellow of ASME and in 1991 received the ASME Fluids Engineering Award. He lives with his wife, Jeanne, in Narragansett, Rhode Island. v About the Author whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page v ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

10. To Jeanne whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page vi ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

11. Preface xi Chapter 1 Introduction 3 1.1 Preliminary Remarks 3 1.2 History and Scope of Fluid Mechanics 4 1.3 Problem-Solving Techniques 6 1.4 The Concept of a Fluid 6 1.5 The Fluid as a Continuum 8 1.6 Dimensions and Units 9 1.7 Properties of the Velocity Field 17 1.8 Thermodynamic Properties of a Fluid 18 1.9 Viscosity and Other Secondary Properties 25 1.10 Basic Flow Analysis Techniques 40 1.11 Flow Patterns: Streamlines, Streaklines, and Pathlines 41 1.12 The Engineering Equation Solver 46 1.13 Uncertainty in Experimental Data 46 1.14 The Fundamentals of Engineering (FE) Examination 48 Problems 49 Fundamentals of Engineering Exam Problems 57 Comprehensive Problems 58 References 61 Chapter 2 Pressure Distribution in a Fluid 65 2.1 Pressure and Pressure Gradient 65 2.2 Equilibrium of a Fluid Element 67 2.3 Hydrostatic Pressure Distributions 68 2.4 Application to Manometry 75 2.5 Hydrostatic Forces on Plane Surfaces 78 2.6 Hydrostatic Forces on Curved Surfaces 86 2.7 Hydrostatic Forces in Layered Fluids 89 2.8 Buoyancy and Stability 91 2.9 Pressure Distribution in Rigid-Body Motion 97 2.10 Pressure Measurement 105 Summary 109 Problems 109 Word Problems 132 Fundamentals of Engineering Exam Problems 133 Comprehensive Problems 134 Design Projects 135 References 136 Chapter 3 Integral Relations for a Control Volume 139 3.1 Basic Physical Laws of Fluid Mechanics 139 3.2 The Reynolds Transport Theorem 143 3.3 Conservation of Mass 150 3.4 The Linear Momentum Equation 155 3.5 Frictionless Flow: The Bernoulli Equation 169 3.6 The Angular Momentum Theorem 178 3.7 The Energy Equation 184 Summary 195 Problems 195 Word Problems 224 Fundamentals of Engineering Exam Problems 224 Comprehensive Problems 226 Design Project 227 References 227 Contents vii whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page vii ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

12. Chapter 4 Differential Relations for Fluid Flow 229 4.1 The Acceleration Field of a Fluid 230 4.2 The Differential Equation of Mass Conservation 232 4.3 The Differential Equation of Linear Momentum 238 4.4 The Differential Equation of Angular Momentum 244 4.5 The Differential Equation of Energy 246 4.6 Boundary Conditions for the Basic Equations 249 4.7 The Stream Function 253 4.8 Vorticity and Irrotationality 261 4.9 Frictionless Irrotational Flows 263 4.10 Some Illustrative Incompressible Viscous Flows 268 Summary 276 Problems 277 Word Problems 288 Fundamentals of Engineering Exam Problems 288 Comprehensive Problems 289 References 290 Chapter 5 Dimensional Analysis and Similarity 293 5.1 Introduction 298 5.2 The Principle of Dimensional Homogeneity 296 5.3 The Pi Theorem 302 5.4 Nondimensionalization of the Basic Equations 312 5.5 Modeling and Its Pitfalls 321 Summary 333 Problems 333 Word Problems 342 Fundamentals of Engineering Exam Problems 342 Comprehensive Problems 343 Design Projects 344 References 344 Chapter 6 Viscous Flow in Ducts 347 6.1 Reynolds Number Regimes 347 6.2 Internal versus External Viscous Flow 352 6.3 Head LossThe Friction Factor 355 6.4 Laminar Fully Developed Pipe Flow 357 6.5 Turbulence Modeling 359 6.6 Turbulent Pipe Flow 365 6.7 Four Types of Pipe Flow Problems 373 6.8 Flow in Noncircular Ducts 379 6.9 Minor or Local Losses in Pipe Systems 388 6.10 Multiple-Pipe Systems 397 6.11 Experimental Duct Flows: Diffuser Performance 403 6.12 Fluid Meters 408 Summary 429 Problems 430 Word Problems 448 Fundamentals of Engineering Exam Problems 449 Comprehensive Problems 450 Design Projects 452 References 453 Chapter 7 Flow Past Immersed Bodies 457 7.1 Reynolds Number and Geometry Effects 457 7.2 Momentum Integral Estimates 461 7.3 The Boundary Layer Equations 464 7.4 The Flat-Plate Boundary Layer 467 7.5 Boundary Layers with Pressure Gradient 476 7.6 Experimental External Flows 482 Summary 509 Problems 510 Word Problems 523 Fundamentals of Engineering Exam Problems 524 Comprehensive Problems 524 Design Project 525 References 526 Chapter 8 Potential Flow and Computational Fluid Dynamics 529 8.1 Introduction and Review 529 8.2 Elementary Plane Flow Solutions 532 8.3 Superposition of Plane Flow Solutions 539 8.4 Plane Flow Past Closed-Body Shapes 545 8.5 Other Plane Potential Flows 555 8.6 Images 559 8.7 Airfoil Theory 562 8.8 Axisymmetric Potential Flow 574 8.9 Numerical Analysis 579 viii Contents whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page viii ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

13. Summary 593 Problems 594 Word Problems 604 Comprehensive Problems 605 Design Projects 606 References 606 Chapter 9 Compressible Flow 609 9.1 Introduction: Review of Thermodynamics 609 9.2 The Speed of Sound 614 9.3 Adiabatic and Isentropic Steady Flow 616 9.4 Isentropic Flow with Area Changes 622 9.5 The Normal Shock Wave 629 9.6 Operation of Converging and Diverging Nozzles 637 9.7 Compressible Duct Flow with Friction 642 9.8 Frictionless Duct Flow with Heat Transfer 654 9.9 Two-Dimensional Supersonic Flow 659 9.10 Prandtl-Meyer Expansion Waves 669 Summary 681 Problems 682 Word Problems 695 Fundamentals of Engineering Exam Problems 696 Comprehensive Problems 696 Design Projects 698 References 698 Chapter 10 Open-Channel Flow 701 10.1 Introduction 701 10.2 Uniform Flow: The Chzy Formula 707 10.3 Efcient Uniform-Flow Channels 712 10.4 Specic Energy: Critical Depth 714 10.5 The Hydraulic Jump 722 10.6 Gradually Varied Flow 726 10.7 Flow Measurement and Control by Weirs 734 Summary 741 Problems 741 Word Problems 754 Fundamentals of Engineering Exam Problems 754 Comprehensive Problems 754 Design Projects 756 References 756 Chapter 11 Turbomachinery 759 11.1 Introduction and Classication 759 11.2 The Centrifugal Pump 762 11.3 Pump Performance Curves and Similarity Rules 768 11.4 Mixed- and Axial-Flow Pumps: The Specic Speed 778 11.5 Matching Pumps to System Characteristics 785 11.6 Turbines 793 Summary 807 Problems 807 Word Problems 820 Comprehensive Problems 820 Design Project 822 References 822 Appendix A Physical Properties of Fluids 824 Appendix B Compressible Flow Tables 829 Appendix C Conversion Factors 836 Appendix D Equations of Motion in Cylindrical Coordinates 838 Answers to Selected Problems 840 Index 847 Contents ix whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page ix ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

15. The seventh edition of Fluid Mechanics sees some additions and deletions but no philosophical change. The basic outline of eleven chapters, plus appendices, remains the same. The triad of integral, differential, and experimental approaches is retained. Many problem exercises, and some fully worked examples, have been changed. The informal, student-oriented style is retained. A number of new photographs and gures have been added. Many new references have been added, for a total of 435. The writer is a rm believer in further reading, especially in the postgraduate years. The total number of problem exercises continues to increase, from 1089 in the rst edition, to 1675 in this seventh edition. There are approximately 20 new problems added to each chapter. Most of these are basic end-of-chapter problems, classied according to topic. There are also Word Problems, multiple-choice Fundamentals of Engineering Problems, Comprehensive Problems, and Design Projects. The appendix lists approximately 700 Answers to Selected Problems. The example problems are structured in the text to follow the sequence of recommended steps outlined in Sect. 1.3, Problem-Solving Techniques. The Engineering Equation Solver (EES) is available with the text and continues its role as an attractive tool for uid mechanics and, indeed, other engineering problems. Not only is it an excellent solver, but it also contains thermophysical properties, publication-quality plotting, units checking, and many mathematical functions, including numerical integration. The author is indebted to Sanford Klein and William Beckman, of the University of Wisconsin, for invaluable and continuous help in preparing and updating EES for use in this text. For newcomers to EES, a brief guide to its use is found on this books website. There are some revisions in each chapter. Chapter 1 has added material on the history of late 20th century uid mechanics, notably the development of Computational Fluid Dynamics. A very brief introduction to the acceleration eld has been added. Boundary conditions for slip ow have been added. There is more discussion of the speed of sound in liquids. The treatment of thermal conductivity has been moved to Chapter 4. Content Changes Learning Tools General Approach xi Preface whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page xi ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

16. Chapter 2 introduces a photo, discussion, and new problems for the deep ocean submersible vehicle, ALVIN. The density distribution in the troposphere is now given explicitly. There are brief remarks on the great Greek mathematician, Archimedes. Chapter 3 has been substantially revised. Reviewers wanted Bernoullis equation moved ahead of angular velocity and energy, to follow linear momentum. I did this and followed their specic improvements, but truly extensive renumbering and rear- ranging was necessary. Pressure and velocity conditions at a tank surface have an improved discussion. A brief history of the control volume has been added. There is a better treatment of the relation between Bernoullis equation and the energy equation. There is a new discussion of stagnation, static and dynamic pressures, and boundary conditions at a jet exit. Chapter 4 has a great new opener: CFD for ow past a spinning soccer ball. The total time derivative of velocity is now written out in full. Fouriers Law, and its application to the differential energy equation, have been moved here from Chapter 1. There are 21 new problems, including several slip-ow analyses. The Chapter 5 introduction expands on the effects of Mach number and Froude number, instead of concentrating only on the Reynolds number. Ipsens method, which the writer admires, is retained as an alternative to the pi theorem. The new opener, a giant disk-band-gap parachute, allows for several new dimensional analysis problems. Chapter 6 has a new formula for entrance length in turbulent duct ow, sent to me by two different researchers. There is a new problem describing the ow in a fuel cell. The new opener, the Trans-Alaska Pipeline, allows for several innovative problems, including a related one on the proposed Alaska-Canada natural gas pipeline. Chapter 7 has an improved description of turbulent ow past a at plate, plus recent reviews of progress in turbulence modeling with CFD. Two new aerodynamic advances are reported: the Finaish-Witherspoon redesign of the Kline-Fogelman airfoil and the increase in stall angle achieved by tubercles modeled after a humpback whale. The new Transition ying car, which had a successful maiden ight in 2009, leads to a number of good problem assignments. Two other photos, Rocket Man over the Alps, and a cargo ship propelled by a kite, also lead to interesting new problems. Chapter 8 is essentially unchanged, except for a bit more discussion of modern CFD software. The Transition autocar, featured in Chapter 7, is attacked here by aerodynamic theory, including induced drag. Chapter 9 beneted from reviewer improvement. Figure 9.7, with its 30-year-old curve-ts for the area ratio, has been replaced with ne-gridded curves for the area- change properties. The curve-ts are gone, and Mach numbers follow nicely from Fig. 9.7 and either Excel or EES. New Trends in Aeronautics presents the X-43 Scram- jet airplane, which generates several new problem assignments. Data for the proposed Alaska-to-Canada natural gas pipeline provides a different look at frictional choking. Chapter 10 is basically the same, except for new photos of both plane and circular hydraulic jumps, plus a tidal bore, with their associated problem assignments. Chapter 11 has added a section on the performance of gas turbines, with application to turbofan aircraft engines. The section on wind turbines has been updated, with new data and photos. A wind-turbine-driven vehicle, which can easily move directly into the wind, has inspired new problem assignments. xii Preface whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page xii ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

17. Appendix A has new data on the bulk modulus of various liquids. Appendix B, Compressible Flow Tables, has been shortened by using coarser increments (0.1) in Mach number. Tables with much smaller increments are now on the bookswebsite. Appendix E, Introduction to EES, has been deleted and moved to the website, on the theory that most students are now quite familiar with EES. A number of supplements are available to students and/or instructors at the text website www.mhhe.com/white7e. Students have access to a Student Study Guide developed by Jerry Dunn of Texas A&M University. They are also able to utilize Engineering Equation Solver (EES), uid mechanics videos developed by Gary Set- tles of Pennsylvania State University, and CFD images and animations prepared by Fluent Inc. Also available to students are Fundamentals of Engineering (FE) Exam quizzes, prepared by Edward Anderson of Texas Tech University. Instructors may obtain a series of PowerPoint slides and images, plus the full Solu- tions Manual, in PDF format. The Solutions Manual provides complete and detailed solutions, including problem statements and artwork, to the end-of-chapter problems. It may be photocopied for posting or preparing transparencies for the classroom. Instruc- tors can also obtain access to C.O.S.M.O.S. for the seventh edition. C.O.S.M.O.S. is a Complete Online Solutions Manual Organization System instructors can use to create exams and assignments, create custom content, and edit supplied problems and solutions. Ebooks are an innovative way for students to save money and create a greener environment at the same time. An ebook can save students about half the cost of a traditional textbook and offers unique features like a powerful search engine, high- lighting, and the ability to share notes with classmates using ebooks. McGraw-Hill offers this text as an ebook. To talk about the ebook options, con- tact your McGraw-Hill sales rep or visit the site www.coursesmart.com to learn more. Online Supplements Preface xiii Electronic Textbook Options whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page xiii ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

18. As usual, so many people have helped me that I may fail to list them all. Sheldon Green of the University of British Columbia, Gordon Holloway of the University of New Brunswick, Sukanta K. Dash of The Indian Institute of Technology at Kharagpur, and Pezhman Shirvanian of Ford Motor Co. gave many helpful suggestions. Hubert Chanson of the University of Queensland, Frank J. Cunha of Pratt&Whitney, Samuel Schweighart of Terrafugia Inc., Mark Spear of the Woods Hole Oceanographic Institution, Keith Hanna of ANSYS Inc., Elena Mejia of the Jet Propulsion Laboratory, Anne Staack of SkySails, Inc., and Ellen Emerson White provided great new images. Samuel S. Sih of Walla Walla College, Timothy Singler of SUNY Binghamton, Saeed Moaveni of Minnesota State University, and John Borg of Marquette University were especially helpful with the solutions manual. The following prereviewers gave many excellent suggestions for improving the manuscript: Rolando Bravo of Southern Illinois University; Joshua B. Kollat of Penn State University; Daniel Maynes of Brigham Young University; Joseph Schaefer of Iowa State University; and Xiangchun Xuan of Clemson University. In preparation, the writer got stuck on Chapter 3 but was rescued by the following reviewers: Serhiy Yarusevych of the University of Waterloo; H. Pirouz Kavehpour and Jeff Eldredge of the University of California, Los Angeles; Rayhaneh Akhavan of the University of Michigan; Soyoung Steve Cha of the University of Illinois, Chicago; Georgia Richardson of the University of Alabama; Krishan Bhatia of Rowan Univer- sity; Hugh Coleman of the University of Alabama-Huntsville; D.W. Ostendorf of the University of Massachusetts; and Donna Meyer of the University of Rhode Island. The writer continues to be indebted to many others who have reviewed this book over the various years and editions. Many other reviewers and correspondents gave good suggestions, encouragement, corrections, and materials: Elizabeth J. Kenyon of MathWorks; Juan R. Cruz of NASA Langley Research Center; LiKai Li of University of Science and Technology of China; Tom Robbins of National Instruments; Tapan K. Sengupta of the Indian Institute of Technology at Kanpur; Paulo Vatavuk of Unicamp University; Andris Skattebo of Scand- power A/S; Jeffrey S. Allen of Michigan Technological University; Peter R. Spedding of Queens University, Belfast, Northern Ireland; Iskender Sahin of Western Michigan University; Michael K. Dailey of General Motors; Cristina L. Archer of Stanford University; Paul F. Jacobs of Technology Development Associates; Rebecca Cullion- Webb of the University of Colorado at Colorado Springs; Debendra K. Das of the Uni- versity of Alaska Fairbanks; Kevin OSullivan and Matthew Lutts of the Associated Acknowledgments xiv whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page xiv ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

19. Press; Lennart Luttig and Nina Koliha of REpower Systems AG, Hamburg, Germany; Chi-Yang Cheng of ANSYS Inc.; Debabrata Dasgupta of The Indian Institute of Tech- nology at Kharagpur; Fabian Anselmet of the Institut de Recherche sur les Phenomenes Hors Equilibre, Marseilles; David Chelidze, Richard Lessmann, Donna Meyer, Arun Shukla, Peter Larsen, and Malcolm Spaulding of the University of Rhode Island; Craig Swanson of Applied Science Associates, Inc.; Jim Smay of Oklahoma State University; Deborah Pence of Oregon State University; Eric Braschoss of Philadelphia, PA.; and Dale Hart of Louisiana Tech University. The McGraw-Hill staff was, as usual, enormously helpful. Many thanks are due to Bill Stenquist, Lora Kalb-Neyens, Curt Reynolds, John Leland, Jane Mohr, Brenda Rolwes, as well as all those who worked on previous editions. Finally, the continuing support and encouragement of my wife and family are, as always, much appreciated. Special thanks are due to our dog, Sadie, and our cats, Cole and Kerry. Acknowledgments xv whi29346_fm_i-xvi.qxd 12/14/09 7:09PM Page xv ntt 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

21. Fluid Mechanics whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 1 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

22. Hurricane Rita in the Gulf of Mexico, Sept. 22, 2005. Rita made landfall at the Texas-Louisiana border and caused billions of dollars in wind and ooding damage. Though more dramatic than typical applications in this text, Rita is a true uid ow, strongly inuenced by the earths rota- tion and the ocean temperature. (Photo courtesy of NASA.) 2 whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 2 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

23. 3 Fluid mechanics is the study of uids either in motion (uid dynamics) or at rest (uid statics). Both gases and liquids are classied as uids, and the number of uid engineering applications is enormous: breathing, blood ow, swimming, pumps, fans, turbines, airplanes, ships, rivers, windmills, pipes, missiles, icebergs, engines, lters, jets, and sprinklers, to name a few. When you think about it, almost everything on this planet either is a uid or moves within or near a uid. The essence of the subject of uid ow is a judicious compromise between theory and experiment. Since uid ow is a branch of mechanics, it satises a set of well- documented basic laws, and thus a great deal of theoretical treatment is available. How- ever, the theory is often frustrating because it applies mainly to idealized situations, which may be invalid in practical problems. The two chief obstacles to a workable theory are geometry and viscosity. The basic equations of uid motion (Chap. 4) are too difcult to enable the analyst to attack arbitrary geometric congurations. Thus most textbooks concentrate on at plates, circular pipes, and other easy geometries. It is possible to apply numerical computer techniques to complex geometries, and specialized textbooks are now available to explain the new computational uid dynamics (CFD) approximations and methods [14].1 This book will present many theoretical results while keeping their limitations in mind. The second obstacle to a workable theory is the action of viscosity, which can be neglected only in certain idealized ows (Chap. 8). First, viscosity increases the difculty of the basic equations, although the boundary-layer approximation found by Ludwig Prandtl in 1904 (Chap. 7) has greatly simplied viscous-ow analyses. Sec- ond, viscosity has a destabilizing effect on all uids, giving rise, at frustratingly small velocities, to a disorderly, random phenomenon called turbulence. The theory of turbulent ow is crude and heavily backed up by experiment (Chap. 6), yet it can be quite serviceable as an engineering estimate. This textbook only introduces the standard experimental correlations for turbulent time-mean ow. Meanwhile, there are advanced texts on both time-mean turbulence and turbulence modeling [5, 6] and on the newer, computer-intensive direct numerical simulation (DNS) of uctuating turbulence [7, 8]. 1.1 Preliminary Remarks Chapter 1 Introduction 1 Numbered references appear at the end of each chapter. whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 3 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

24. Thus there is theory available for uid ow problems, but in all cases it should be backed up by experiment. Often the experimental data provide the main source of information about specic ows, such as the drag and lift of immersed bodies (Chap. 7). Fortunately, uid mechanics is a highly visual subject, with good instru- mentation [911], and the use of dimensional analysis and modeling concepts (Chap. 5) is widespread. Thus experimentation provides a natural and easy comple- ment to the theory. You should keep in mind that theory and experiment should go hand in hand in all studies of uid mechanics. Like most scientic disciplines, uid mechanics has a history of erratically occurring early achievements, then an intermediate era of steady fundamental discoveries in the eighteenth and nineteenth centuries, leading to the twenty-rst-century era of modern practice, as we self-centeredly term our limited but up-to-date knowledge. Ancient civilizations had enough knowledge to solve certain ow problems. Sailing ships with oars and irrigation systems were both known in prehistoric times. The Greeks produced quantitative information. Archimedes and Hero of Alexandria both postulated the parallelogram law for addition of vectors in the third century B.C. Archimedes (285212 B.C.) formulated the laws of buoyancy and applied them to oating and sub- merged bodies, actually deriving a form of the differential calculus as part of the analysis. The Romans built extensive aqueduct systems in the fourth century B.C. but left no records showing any quantitative knowledge of design principles. From the birth of Christ to the Renaissance there was a steady improvement in the design of such ow systems as ships and canals and water conduits but no recorded evidence of fundamental improvements in ow analysis. Then Leonardo da Vinci (14521519) stated the equation of conservation of mass in one-dimensional steady ow. Leonardo was an excellent experimentalist, and his notes contain accurate descriptions of waves, jets, hydraulic jumps, eddy formation, and both low-drag (streamlined) and high-drag (parachute) designs. A Frenchman, Edme Mariotte (16201684), built the rst wind tunnel and tested models in it. Problems involving the momentum of uids could nally be analyzed after Isaac Newton (16421727) postulated his laws of motion and the law of viscosity of the linear uids now called newtonian. The theory rst yielded to the assumption of a perfect or frictionless uid, and eighteenth-century mathematicians (Daniel Bernoulli, Leonhard Euler, Jean dAlembert, Joseph-Louis Lagrange, and Pierre-Simon Laplace) produced many beautiful solutions of frictionless-ow problems. Euler, Fig. 1.1, developed both the differential equations of motion and their integrated form, now called the Bernoulli equation. DAlembert used them to show his famous paradox: that a body immersed in a frictionless uid has zero drag. These beautiful results amounted to overkill, since perfect-uid assumptions have very limited application in practice and most engineering ows are dominated by the effects of viscosity. Engineers began to reject what they regarded as a totally unrealistic theory and developed the science of hydraulics, relying almost entirely on experiment. Such experimentalists as Chzy, Pitot, Borda, Weber, Francis, Hagen, Poiseuille, Darcy, Manning, Bazin, and Weisbach produced data on a variety of ows such as open channels, ship resistance, pipe ows, waves, and turbines. All too often the data were used in raw form without regard to the fundamental physics of ow. 1.2 History and Scope of Fluid Mechanics 4 Chapter 1 Introduction Fig. 1.1 Leonhard Euler (1707 1783) was the greatest mathematician of the eighteenth century and used Newtons calculus to develop and solve the equations of motion of inviscid ow. He published over 800 books and papers. [Courtesy of the School of Mathematics and Statistics, University of St Andrew, Scotland.] whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 4 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

25. At the end of the nineteenth century, unication between experimental hydraulics and theoretical hydrodynamics nally began. William Froude (18101879) and his son Robert (18461924) developed laws of model testing; Lord Rayleigh (18421919) proposed the technique of dimensional analysis; and Osborne Reynolds (18421912) published the classic pipe experiment in 1883, which showed the importance of the dimensionless Reynolds number named after him. Meanwhile, viscous-ow theory was available but unexploited, since Navier (17851836) and Stokes (18191903) had successfully added newtonian viscous terms to the equations of motion. The result- ing Navier-Stokes equations were too difcult to analyze for arbitrary ows. Then, in 1904, a German engineer, Ludwig Prandtl (18751953), Fig. 1.2, published perhaps the most important paper ever written on uid mechanics. Prandtl pointed out that uid ows with small viscosity, such as water ows and airows, can be divided into a thin viscous layer, or boundary layer, near solid surfaces and interfaces, patched onto a nearly inviscid outer layer, where the Euler and Bernoulli equations apply. Boundary-layer theory has proved to be a very important tool in modern ow analysis. The twentieth- century foundations for the present state of the art in uid mechanics were laid in a series of broad-based experiments and theories by Prandtl and his two chief friendly competi- tors, Theodore von Krmn (18811963) and Sir Geoffrey I. Taylor (18861975). Many of the results sketched here from a historical point of view will, of course, be discussed in this textbook. More historical details can be found in Refs. 12 to 14. The second half of the twentieth century introduced a new tool: Computational Fluid Dynamics (CFD). The earliest paper on the subject known to this writer was by A. Thom in 1933 [47], a laborious, but accurate, hand calculation of ow past a cylinder at low Reynolds numbers. Commercial digital computers became available in the 1950s, and personal computers in the 1970s, bringing CFD into adulthood. A legendary rst textbook was by Patankar [3]. Presently, with increases in computer speed and memory, almost any laminar ow can be modeled accurately. Turbulent ow is still calculated with empirical models, but Direct Numerical Simulation [7, 8] is possible for low Reynolds numbers. Another ve orders of magnitude in computer speed are needed before general turbulent ows can be calculated. That may not be possible, due to size limits of nano- and pico-elements. But, if general DNS develops, Gad-el-Hak [14] raises the prospect of a shocking future: all of uid mechanics reduced to a black box, with no real need for teachers, researchers, writers, or uids engineers. Since the earth is 75 percent covered with water and 100 percent covered with air, the scope of uid mechanics is vast and touches nearly every human endeavor. The sciences of meteorology, physical oceanography, and hydrology are concerned with naturally occurring uid ows, as are medical studies of breathing and blood circu- lation. All transportation problems involve uid motion, with well-developed spe- cialties in aerodynamics of aircraft and rockets and in naval hydrodynamics of ships and submarines. Almost all our electric energy is developed either from water ow or from steam ow through turbine generators. All combustion problems involve uid motion as do the more classic problems of irrigation, ood control, water supply, sewage disposal, projectile motion, and oil and gas pipelines. The aim of this book is to present enough fundamental concepts and practical applications in uid mechanics to prepare you to move smoothly into any of these specialized elds of the science of owand then be prepared to move out again as new technologies develop. 1.2 History and Scope of Fluid Mechanics 5 Fig. 1.2 Ludwig Prandtl (1875 1953), often called the father of modern uid mechanics [15], developed boundary layer theory and many other innovative analyses. He and his students were pioneers in ow visualization techniques. [Aufnahme von Fr. Struckmeyer, Gottingen, courtesy AIP Emilio Segre Visual Archives, Lande Collection.] whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 5 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

26. Fluid ow analysis is packed with problems to be solved. The present text has more than 1700 problem assignments. Solving a large number of these is a key to learning the subject. One must deal with equations, data, tables, assumptions, unit systems, and solution schemes. The degree of difculty will vary, and we urge you to sample the whole spectrum of assignments, with or without the Answers in the Appendix. Here are the recommended steps for problem solution: 1. Read the problem and restate it with your summary of the results desired. 2. From tables or charts, gather the needed property data: density, viscosity, etc. 3. Make sure you understand what is asked. Students are apt to answer the wrong questionfor example, pressure instead of pressure gradient, lift force instead of drag force, or mass ow instead of volume ow. Read the problem carefully. 4. Make a detailed, labeled sketch of the system or control volume needed. 5. Think carefully and list your assumptions. You must decide if the ow is steady or unsteady, compressible or incompressible, viscous or inviscid, and whether a control volume or partial differential equations are needed. 6. Find an algebraic solution if possible. Then, if a numerical value is needed, use either the SI or BG unit systems, to be reviewed in Sec. 1.6. 7. Report your solution, labeled, with the proper units and the proper number of signicant gures (usually two or three) that the data uncertainty allows. We shall follow these steps, where appropriate, in our example problems. From the point of view of uid mechanics, all matter consists of only two states, uid and solid. The difference between the two is perfectly obvious to the layperson, and it is an interesting exercise to ask a layperson to put this difference into words. The technical distinction lies with the reaction of the two to an applied shear or tangential stress. A solid can resist a shear stress by a static deection; a uid cannot . Any shear stress applied to a uid, no matter how small, will result in motion of that uid. The uid moves and deforms continuously as long as the shear stress is applied. As a corollary, we can say that a uid at rest must be in a state of zero shear stress, a state often called the hydrostatic stress condition in structural analysis. In this condition, Mohrs circle for stress reduces to a point, and there is no shear stress on any plane cut through the element under stress. Given this denition of a uid, every layperson also knows that there are two classes of uids, liquids and gases. Again the distinction is a technical one concern- ing the effect of cohesive forces. A liquid, being composed of relatively close-packed molecules with strong cohesive forces, tends to retain its volume and will form a free surface in a gravitational eld if unconned from above. Free-surface ows are dominated by gravitational effects and are studied in Chaps. 5 and 10. Since gas molecules are widely spaced with negligible cohesive forces, a gas is free to expand until it encounters conning walls. A gas has no denite volume, and when left to itself without connement, a gas forms an atmosphere that is essentially hydrostatic. The hydrostatic behavior of liquids and gases is taken up in Chap. 2. Gases cannot form a free surface, and thus gas ows are rarely concerned with gravitational effects other than buoyancy. 1.4 The Concept of a Fluid 1.3 Problem-Solving Techniques 6 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 6 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

27. Figure 1.3 illustrates a solid block resting on a rigid plane and stressed by its own weight. The solid sags into a static deection, shown as a highly exaggerated dashed line, resisting shear without ow. A free-body diagram of element A on the side of the block shows that there is shear in the block along a plane cut at an angle through A. Since the block sides are unsupported, element A has zero stress on the left and right sides and compression stress p on the top and bottom. Mohrs circle does not reduce to a point, and there is nonzero shear stress in the block. By contrast, the liquid and gas at rest in Fig. 1.3 require the supporting walls in order to eliminate shear stress. The walls exert a compression stress of p and reduce Mohrs circle to a point with zero shear everywherethat is, the hydrostatic condition. The liquid retains its volume and forms a free surface in the container. If the walls are removed, shear develops in the liquid and a big splash results. If the container is tilted, shear again develops, waves form, and the free surface seeks a horizontal con- guration, pouring out over the lip if necessary. Meanwhile, the gas is unrestrained and expands out of the container, lling all available space. Element A in the gas is also hydrostatic and exerts a compression stress p on the walls. 1.4 The Concept of a Fluid 7 Static deflection Free surface Hydrostatic condition LiquidSolid A A A (a) (c) (b) (d) 0 0 A A Gas (1) p p p p p = 0 2 1 = p = p 1 Fig. 1.3 A solid at rest can resist shear. (a) Static deection of the solid; (b) equilibrium and Mohrs circle for solid element A. A uid cannot resist shear. (c) Containing walls are needed; (d) equilibrium and Mohrs circle for uid element A. whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 7 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

28. In the previous discussion, clear decisions could be made about solids, liquids, and gases. Most engineering uid mechanics problems deal with these clear casesthat is, the common liquids, such as water, oil, mercury, gasoline, and alcohol, and the common gases, such as air, helium, hydrogen, and steam, in their common temperature and pressure ranges. There are many borderline cases, however, of which you should be aware. Some apparently solid substances such as asphalt and lead resist shear stress for short periods but actually deform slowly and exhibit denite uid behavior over long periods. Other substances, notably colloid and slurry mixtures, resist small shear stresses but yield at large stress and begin to ow as uids do. Specialized textbooks are devoted to this study of more general deformation and flow, a field called rheology [16]. Also, liquids and gases can coexist in two-phase mixtures, such as steamwater mixtures or water with entrapped air bubbles. Specialized textbooks present the analysis of such multiphase ows [17]. Finally, in some situations the distinction between a liquid and a gas blurs. This is the case at temperatures and pressures above the so-called critical point of a substance, where only a single phase exists, primarily resembling a gas. As pressure increases far above the critical point, the gaslike substance becomes so dense that there is some resemblance to a liquid and the usual thermodynamic approximations like the perfect-gas law become inaccurate. The critical temperature and pressure of water are Tc 647 K and pc 219 atm (atmosphere2 ) so that typical problems involving water and steam are below the critical point. Air, being a mixture of gases, has no distinct critical point, but its principal component, nitrogen, has Tc 126 K and pc 34 atm. Thus typical problems involving air are in the range of high temperature and low pressure where air is distinctly and denitely a gas. This text will be concerned solely with clearly identiable liquids and gases, and the borderline cases just discussed will be beyond our scope. We have already used technical terms such as uid pressure and density without a rig- orous discussion of their denition. As far as we know, uids are aggregations of molecules, widely spaced for a gas, closely spaced for a liquid. The distance between molecules is very large compared with the molecular diameter. The molecules are not xed in a lattice but move about freely relative to each other. Thus uid density, or mass per unit volume, has no precise meaning because the number of molecules occupying a given volume continually changes. This effect becomes unimportant if the unit volume is large compared with, say, the cube of the molecular spacing, when the number of molecules within the volume will remain nearly constant in spite of the enormous interchange of particles across the boundaries. If, however, the chosen unit volume is too large, there could be a noticeable variation in the bulk aggregation of the particles. This situation is illustrated in Fig. 1.4, where the density as calculated from molecular mass m within a given volume is plotted versus the size of the unit volume. There is a limiting volume * below which molecular variations may be important and above which aggre- gate variations may be important. The density of a uid is best dened as (1.1) lim S* m 1.5 The Fluid as a Continuum 8 Chapter 1 Introduction 2 One atmosphere equals 2116 lbf/ft2 101,300 Pa. whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 8 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

29. The limiting volume * is about 109 mm3 for all liquids and for gases at atmospheric pressure. For example, 109 mm3 of air at standard conditions contains approximately 3 107 molecules, which is sufcient to dene a nearly constant density according to Eq. (1.1). Most engineering problems are concerned with physical dimensions much larger than this limiting volume, so that density is essentially a point function and uid properties can be thought of as varying continually in space, as sketched in Fig. 1.4a. Such a uid is called a continuum, which simply means that its variation in properties is so smooth that differential calculus can be used to analyze the substance. We shall assume that continuum calculus is valid for all the analyses in this book. Again there are borderline cases for gases at such low pressures that molecular spacing and mean free path3 are comparable to, or larger than, the physical size of the system. This requires that the continuum approximation be dropped in favor of a molecular theory of rareed gas ow [18]. In principle, all uid mechanics problems can be attacked from the molecular viewpoint, but no such attempt will be made here. Note that the use of continuum calculus does not preclude the possibility of discontinuous jumps in uid properties across a free surface or uid interface or across a shock wave in a compressible uid (Chap. 9). Our calculus in analyzing uid ow must be exible enough to handle discontinuous boundary conditions. A dimension is the measure by which a physical variable is expressed quantitatively. A unit is a particular way of attaching a number to the quantitative dimension. Thus length is a dimension associated with such variables as distance, displacement, width, deection, and height, while centimeters and inches are both numerical units for expressing length. Dimension is a powerful concept about which a splendid tool called dimensional analysis has been developed (Chap. 5), while units are the numerical quantity that the customer wants as the nal answer. In 1872 an international meeting in France proposed a treaty called the Metric Con- vention, which was signed in 1875 by 17 countries including the United States. It was an improvement over British systems because its use of base 10 is the foundation of our number system, learned from childhood by all. Problems still remained because 1.6 Dimensions and Units 1.6 Dimensions and Units 9 Microscopic uncertainty Macroscopic uncertainty 0 1200 * 10-9 mm3 Elemental volume Region containing fluid = 1000 kg/m3 = 1100 = 1200 = 1300 (a) (b) Fig. 1.4 The limit denition of continuum uid density: (a) an elemental volume in a uid region of variable continuum density; (b) calculated density versus size of the elemental volume. 3 The mean distance traveled by molecules between collisions (see Prob. P1.5). whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 9 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

30. even the metric countries differed in their use of kiloponds instead of dynes or newtons, kilograms instead of grams, or calories instead of joules. To standardize the metric system, a General Conference of Weights and Measures, attended in 1960 by 40 countries, proposed the International System of Units (SI). We are now undergo- ing a painful period of transition to SI, an adjustment that may take many more years to complete. The professional societies have led the way. Since July 1, 1974, SI units have been required by all papers published by the American Society of Mechanical Engineers, and there is a textbook explaining the SI [19]. The present text will use SI units together with British gravitational (BG) units. In uid mechanics there are only four primary dimensions from which all other dimensions can be derived: mass, length, time, and temperature.4 These dimensions and their units in both systems are given in Table 1.1. Note that the kelvin unit uses no degree symbol. The braces around a symbol like {M} mean the dimension of mass. All other variables in uid mechanics can be expressed in terms of {M}, {L}, {T}, and {}. For example, acceleration has the dimensions {LT2 }. The most crucial of these secondary dimensions is force, which is directly related to mass, length, and time by Newtons second law. Force equals the time rate of change of momentum or, for constant mass, F ma (1.2) From this we see that, dimensionally, {F} {MLT 2 }. The use of a constant of proportionality in Newtons law, Eq. (1.2), is avoided by dening the force unit exactly in terms of the other basic units. In the SI system, the basic units are newtons {F}, kilograms {M}, meters {L}, and seconds {T}. We dene 1 newton of force 1 N 1 kg 1 m/s2 The newton is a relatively small force, about the weight of an apple (0.225 lbf). In addition, the basic unit of temperature { } in the SI system is the degree Kelvin, K. Use of these SI units (N, kg, m, s, K) will require no conversion factors in our equations. In the BG system also, a constant of proportionality in Eq. (1.2) is avoided by dening the force unit exactly in terms of the other basic units. In the BG system, the basic units are pound-force {F}, slugs {M}, feet {L}, and seconds {T}. We dene 1 pound of force 1 lbf 1 slug 1 ft/s2# The British Gravitational (BG) System # The International System (SI) Primary Dimensions 10 Chapter 1 Introduction Table 1.1 Primary Dimensions in SI and BG Systems Primary dimension SI unit BG unit Conversion factor Mass {M} Kilogram (kg) Slug 1 slug 14.5939 kg Length {L} Meter (m) Foot (ft) 1 ft 0.3048 m Time {T} Second (s) Second (s) 1 s 1 s Temperature {} Kelvin (K) Rankine (R) 1 K 1.8R 4 If electromagnetic effects are important, a fth primary dimension must be included, electric current {I}, whose SI unit is the ampere (A). whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 10 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

31. One lbf 4.4482 N and approximates the weight of four apples. We will use the abbreviation lbf for pound-force and lbm for pound-mass. The slug is a rather hefty mass, equal to 32.174 lbm. The basic unit of temperature {} in the BG system is the degree Rankine, R. Recall that a temperature difference 1 K 1.8R. Use of these BG units (lbf, slug, ft, s, R) will require no conversion factors in our equations. There are other unit systems still in use. At least one needs no proportionality constant: the CGS system (dyne, gram, cm, s, K). However, CGS units are too small for most applications (1 dyne 105 N) and will not be used here. In the USA, some still use the English Engineering system, (lbf, lbm, ft, s, R), where the basic mass unit is the pound of mass. Newtons law (1.2) must be rewritten: (1.3) The constant of proportionality, gc, has both dimensions and a numerical value not equal to 1.0. The present text uses only the SI and BG systems and will not solve problems or examples in the English Engineering system. Because Americans still use them, a few problems in the text will be stated in truly awkward units: acres, gallons, ounces, or miles. Your assignment will be to convert these and solve in the SI or BG systems. In engineering and science, all equations must be dimensionally homogeneous, that is, each additive term in an equation must have the same dimensions. For example, take Bernoullis incompressible equation, to be studied and used throughout this text: Each and every term in this equation must have dimensions of pressure {ML1 T2 }. We will examine the dimensional homogeneity of this equation in detail in Ex. 1.3. A list of some important secondary variables in uid mechanics, with dimensions derived as combinations of the four primary dimensions, is given in Table 1.2. A more complete list of conversion factors is given in App. C. p 1 2 V2 gZ constant The Principle of Dimensional Homogeneity F ma gc , where gc 32.174 ft # lbm lbf # s2 Other Unit Systems 1.6 Dimensions and Units 11 Table 1.2 Secondary Dimensions in Fluid Mechanics Secondary dimension SI unit BG unit Conversion factor Area {L2 } m2 ft2 1 m2 10.764 ft2 Volume {L3 } m3 ft3 1 m3 35.315 ft3 Velocity {LT 1 } m/s ft/s 1 ft/s 0.3048 m/s Acceleration {LT2 } m/s2 ft/s2 1 ft/s2 0.3048 m/s2 Pressure or stress {ML1 T2 } Pa N/m2 lbf/ft2 1 lbf/ft2 47.88 Pa Angular velocity {T1 } s1 s1 1 s1 1 s1 Energy, heat, work {ML2 T2 } J N m ft lbf 1 ft lbf 1.3558 J Power {ML2 T3 } W J/s ft lbf/s 1 ft lbf/s 1.3558 W Density {ML3 } kg/m3 slugs/ft3 1 slug/ft3 515.4 kg/m3 Viscosity {ML1 T1 } kg/(m s) slugs/(ft s) 1 slug/(ft s) 47.88 kg/(m s) Specic heat {L2 T2 1 } m2 /(s2 K) ft2 /(s2 R) 1 m2 /(s2 K) 5.980 ft2 /(s2 R) whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 11 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

32. EXAMPLE 1.1 A body weighs 1000 lbf when exposed to a standard earth gravity g 32.174 ft/s2 . (a) What is its mass in kg? (b) What will the weight of this body be in N if it is exposed to the moons standard acceleration gmoon 1.62 m/s2 ? (c) How fast will the body accelerate if a net force of 400 lbf is applied to it on the moon or on the earth? Solution We need to nd the (a) mass; (b) weight on the moon; and (c) acceleration of this body. This is a fairly simple example of conversion factors for differing unit systems. No property data is needed. The example is too low-level for a sketch. Newtons law (1.2) holds with known weight and gravitational acceleration. Solve for m: Convert this to kilograms: m 31.08 slugs (31.08 slugs)(14.5939 kg/slug) 454 kg Ans. (a) The mass of the body remains 454 kg regardless of its location. Equation (1.2) applies with a new gravitational acceleration and hence a new weight: F Wmoon mgmoon (454 kg)(1.62 m/s2 ) 735 N Ans. (b) This part does not involve weight or gravity or location. It is simply an application of Newtons law with a known mass and known force: F 400 lbf ma (31.08 slugs) a Solve for Ans. (c) Comment (c): This acceleration would be the same on the earth or moon or anywhere. Many data in the literature are reported in inconvenient or arcane units suitable only to some industry or specialty or country. The engineer should convert these data to the SI or BG system before using them. This requires the systematic application of conversion factors, as in the following example. EXAMPLE 1.2 Industries involved in viscosity measurement [27, 36] continue to use the CGS system of units, since centimeters and grams yield convenient numbers for many uids. The absolute viscosity () unit is the poise, named after J. L. M. Poiseuille, a French physician who in 1840 performed pioneering experiments on water ow in pipes; 1 poise 1 g/(cm-s). The kinematic viscosity () unit is the stokes, named after G. G. Stokes, a British physicist who a 400 lbf 31.08 slugs 12.87 ft s2 a0.3048 m ft b 3.92 m s2 Part (c) Part (b) F W 1000 lbf mg (m)(32.174 ft/s2 ), or m 1000 lbf 32.174 ft/s2 31.08 slugs Part (a) 12 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:57 Page 12 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

33. in 1845 helped develop the basic partial differential equations of uid momentum; 1 stokes 1 cm2 /s. Water at 20C has 0.01 poise and also 0.01 stokes. Express these results in (a) SI and (b) BG units. Solution Approach: Systematically change grams to kg or slugs and change centimeters to meters or feet. Property values: Given 0.01 g/(cm-s) and 0.01 cm2 /s. Solution steps: (a) For conversion to SI units, Ans. (a) For conversion to BG units Ans. (b) Comments: This was a laborious conversion that could have been shortened by using the direct viscosity conversion factors in App. C. For example, BG SI/47.88. We repeat our advice: Faced with data in unusual units, convert them immediately to either SI or BG units because (1) it is more professional and (2) theoretical equations in uid mechanics are dimensionally consistent and require no further conversion factors when these two fundamental unit systems are used, as the following example shows. EXAMPLE 1.3 A useful theoretical equation for computing the relation between pressure, velocity, and altitude in a steady ow of a nearly inviscid, nearly incompressible uid with negligible heat transfer and shaft work5 is the Bernoulli relation, named after Daniel Bernoulli, who published a hydrodynamics textbook in 1738: (1) where p0 stagnation pressure p pressure in moving uid V velocity density Z altitude g gravitational acceleration p0 p 1 2 V2 gZ 0.01 cm2 s 0.01 cm2 (0.01 m/cm)2 (1 ft/0.3048 m)2 s 0.0000108 ft2 s 0.01 g cm # s 0.01 g(1 kg/1000 g)(1 slug/14.5939 kg) (0.01 m/cm)(1 ft/0.3048 m)s 0.0000209 slug ft # s Part (b) 0.01 cm2 s 0.01 cm2 (0.01 m/cm)2 s 0.000001 m2 s 0.01 g cm # s 0.01 g(1 kg/1000g) cm(0.01 m/cm)s 0.001 kg m # s Part (a) 1.6 Dimensions and Units 13 5 Thats an awful lot of assumptions, which need further study in Chap. 3. whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 13 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

34. (a) Show that Eq. (1) satises the principle of dimensional homogeneity, which states that all additive terms in a physical equation must have the same dimensions. (b) Show that consistent units result without additional conversion factors in SI units. (c) Repeat (b) for BG units. Solution We can express Eq. (1) dimensionally, using braces, by entering the dimensions of each term from Table 1.2: {ML1 T2 } {ML1 T2 } {ML3 }{L2 T2 } {ML3 }{LT2 }{L} {ML1 T2 } for all terms Ans. (a) Enter the SI units for each quantity from Table 1.2: {N/m2 } {N/m2 } {kg/m3 }{m2 /s2 } {kg/m3 }{m/s2 }{m} {N/m2 } {kg/(m s2 )} The right-hand side looks bad until we remember from Eq. (1.3) that 1 kg 1 N s2 /m. Ans. (b) Thus all terms in Bernoullis equation will have units of pascals, or newtons per square meter, when SI units are used. No conversion factors are needed, which is true of all theoretical equations in uid mechanics. Introducing BG units for each term, we have {lbf/ft2 } {lbf/ft2 } {slugs/ft3 }{ft2 /s2 } {slugs/ft3 }{ft/s2 }{ft} {lbf/ft2 } {slugs/(ft s2 )} But, from Eq. (1.3), 1 slug 1 lbf s2 /ft, so that Ans. (c) All terms have the unit of pounds-force per square foot. No conversion factors are needed in the BG system either. There is still a tendency in English-speaking countries to use pound-force per square inch as a pressure unit because the numbers are more manageable. For example, standard atmospheric pressure is 14.7 lbf/in2 2116 lbf/ft2 101,300 Pa. The pascal is a small unit because the newton is less than lbf and a square meter is a very large area. Note that not only must all (uid) mechanics equations be dimensionally homogeneous, one must also use consistent units; that is, each additive term must have the same units. There is no trouble doing this with the SI and BG systems, as in Example 1.3, but woe unto those who try to mix colloquial English units. For example, in Chap. 9, we often use the assumption of steady adiabatic compressible gas ow: h 1 2V2 constant Consistent Units 1 4 5slugs/(ft # s2 )6 5lbf # s2 /ft6 5ft # s2 6 5lbf/ft2 6 Part (c) 5kg/(m # s2 )6 5N # s2 /m6 5m # s2 6 5N/m2 6 Part (b) Part (a) 14 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 14 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

35. where h is the uid enthalpy and V2 /2 is its kinetic energy per unit mass. Colloquial thermodynamic tables might list h in units of British thermal units per pound mass (Btu/lb), whereas V is likely used in ft/s. It is completely erroneous to add Btu/lb to ft2 /s2 . The proper unit for h in this case is ft lbf/slug, which is identical to ft2 /s2 . The conversion factor is 1 Btu/lb 25,040 ft2 /s2 25,040 ft lbf/slug. All theoretical equations in mechanics (and in other physical sciences) are dimensionally homogeneous; that is, each additive term in the equation has the same dimensions. However, the reader should be warned that many empirical formulas in the engineering literature, arising primarily from correlations of data, are dimensionally inconsistent. Their units cannot be reconciled simply, and some terms may contain hidden variables. An example is the formula that pipe valve manufacturers cite for liquid volume ow rate Q (m3 /s) through a partially open valve: where p is the pressure drop across the valve and SG is the specic gravity of the liquid (the ratio of its density to that of water). The quantity CV is the valve ow coefcient, which manufacturers tabulate in their valve brochures. Since SG is dimensionless {1}, we see that this formula is totally inconsistent, with one side being a ow rate {L3 /T} and the other being the square root of a pressure drop {M1/2 /L1/2 T}. It follows that CV must have dimensions, and rather odd ones at that: {L7/2 /M1/2 }. Nor is the resolution of this discrepancy clear, although one hint is that the values of CV in the literature increase nearly as the square of the size of the valve. The pres- entation of experimental data in homogeneous form is the subject of dimensional analysis (Chap. 5). There we shall learn that a homogeneous form for the valve ow relation is where is the liquid density and A the area of the valve opening. The discharge coefcient C d is dimensionless and changes only moderately with valve size. Please believeuntil we establish the fact in Chap. 5that this latter is a much better for- mulation of the data. Meanwhile, we conclude that dimensionally inconsistent equations, though they occur in engineering practice, are misleading and vague and even dangerous, in the sense that they are often misused outside their range of applicability. Engineering results often are too small or too large for the common units, with too many zeros one way or the other. For example, to write p 114,000,000 Pa is long and awkward. Using the prex M to mean 106 , we convert this to a concise p 114 MPa (megapascals). Similarly, t 0.000000003 s is a proofreaders night- mare compared to the equivalent t 3 ns (nanoseconds). Such prexes are common and convenient, in both the SI and BG systems. A complete list is given in Table 1.3. Convenient Prexes in Powers of 10 Q Cd Aopeninga p b 1/2 Q CV a p SG b 1/2 Homogeneous versus Dimensionally Inconsistent Equations 1.6 Dimensions and Units 15 Table 1.3 Convenient Prexes for Engineering Units Multiplicative factor Prex Symbol 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 10 deka da 101 deci d 102 centi c 103 milli m 106 micro 109 nano n 1012 pico p 1015 femto f 1018 atto a whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 15 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

36. EXAMPLE 1.4 In 1890 Robert Manning, an Irish engineer, proposed the following empirical formula for the average velocity V in uniform ow due to gravity down an open channel (BG units): (1) where R hydraulic radius of channel (Chaps. 6 and 10) S channel slope (tangent of angle that bottom makes with horizontal) n Mannings roughness factor (Chap. 10) and n is a constant for a given surface condition for the walls and bottom of the channel. (a) Is Mannings formula dimensionally consistent? (b) Equation (1) is commonly taken to be valid in BG units with n taken as dimensionless. Rewrite it in SI form. Solution Assumption: The channel slope S is the tangent of an angle and is thus a dimensionless ratio with the dimensional notation {1}that is, not containing M, L, or T. Approach (a): Rewrite the dimensions of each term in Mannings equation, using brackets {}: This formula is incompatible unless {1.49/n} {L1/3 /T}. If n is dimensionless (and it is never listed with units in textbooks), the number 1.49 must carry the dimensions of {L1/3 /T}. Ans. (a) Comment (a): Formulas whose numerical coefcients have units can be disastrous for engineers working in a different system or another uid. Mannings formula, though pop- ular, is inconsistent both dimensionally and physically and is valid only for water ow with certain wall roughnesses. The effects of water viscosity and density are hidden in the numerical value 1.49. Approach (b): Part (a) showed that 1.49 has dimensions. If the formula is valid in BG units, then it must equal 1.49 ft1/3 /s. By using the SI conversion for length, we obtain (1.49 ft1/3 /s)(0.3048 m/ft)1/3 1.00 m1/3 /s Therefore Mannings inconsistent formula changes form when converted to the SI system: Ans. (b) with R in meters and V in meters per second. Comment (b): Actually, we misled you: This is the way Manning, a metric user, rst proposed the formula. It was later converted to BG units. Such dimensionally inconsistent formulas are dangerous and should either be reanalyzed or treated as having very limited application. SI units: V 1.0 n R2/3 S1/2 5V6 e 1.49 n f 5R2/3 6 5S1/2 6 or e L T f e 1.49 n f 5L2/3 6 516 V 1.49 n R2/3 S1/2 16 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 16 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

37. In a given ow situation, the determination, by experiment or theory, of the properties of the uid as a function of position and time is considered to be the solution to the problem. In almost all cases, the emphasis is on the spacetime distribution of the uid properties. One rarely keeps track of the actual fate of the specic uid particles.6 This treatment of properties as continuum-eld functions distinguishes uid mechanics from solid mechanics, where we are more likely to be interested in the tra- jectories of individual particles or systems. There are two different points of view in analyzing problems in mechanics. The rst view, appropriate to uid mechanics, is concerned with the eld of ow and is called the eulerian method of description. In the eulerian method we compute the pressure eld p(x, y, z, t) of the ow pattern, not the pressure changes p(t) that a particle expe- riences as it moves through the eld. The second method, which follows an individual particle moving through the ow, is called the lagrangian description. The lagrangian approach, which is more appropriate to solid mechanics, will not be treated in this book. However, certain numerical analyses of sharply bounded uid ows, such as the motion of isolated uid droplets, are very conveniently computed in lagrangian coordinates [1]. Fluid dynamic measurements are also suited to the eulerian system. For example, when a pressure probe is introduced into a laboratory ow, it is xed at a specic position (x, y, z). Its output thus contributes to the description of the eulerian pressure eld p(x, y, z, t). To simulate a lagrangian measurement, the probe would have to move downstream at the uid particle speeds; this is sometimes done in oceanographic measurements, where owmeters drift along with the prevailing currents. The two different descriptions can be contrasted in the analysis of trafc ow along a freeway. A certain length of freeway may be selected for study and called the eld of ow. Obviously, as time passes, various cars will enter and leave the eld, and the identity of the specic cars within the eld will constantly be changing. The trafc engineer ignores specic cars and concentrates on their average velocity as a function of time and position within the eld, plus the ow rate or number of cars per hour pass- ing a given section of the freeway. This engineer is using an eulerian description of the trafc ow. Other investigators, such as the police or social scientists, may be interested in the path or speed or destination of specic cars in the eld. By following a specic car as a function of time, they are using a lagrangian description of the ow. Foremost among the properties of a ow is the velocity eld V(x, y, z, t). In fact, determining the velocity is often tantamount to solving a ow problem, since other properties follow directly from the velocity eld. Chapter 2 is devoted to the calculation of the pressure eld once the velocity eld is known. Books on heat transfer (for example, Ref. 20) are largely devoted to nding the temperature eld from known velocity elds. The Velocity Field Eulerian and Lagrangian Descriptions 1.7 Properties of the Velocity Field 1.7 Properties of the Velocity Field 17 6 One example where uid particle paths are important is in water quality analysis of the fate of con- taminant discharges. whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 17 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

38. In general, velocity is a vector function of position and time and thus has three components u, , and w, each a scalar eld in itself: (1.4) The use of u, , and w instead of the more logical component notation Vx, Vy, and Vz is the result of an almost unbreakable custom in uid mechanics. Much of this textbook, especially Chaps. 4, 7, 8, and 9, is concerned with nding the distribution of the velocity vector V for a variety of practical ows. The acceleration vector, a dV/dt, occurs in Newtons law for a uid and thus is very important. In order to follow a particle in the Eulerian frame of reference, the nal result for acceleration is nonlinear and quite complicated. Here we only give the formula: (1.5) where (u, v, w) are the velocity components from Eq. (1.4). We shall study this formula in detail in Chap. 4. The last three terms in Eq. (1.5) are nonlinear products and greatly complicate the analysis of general fluid motions, especially viscous ows. While the velocity eld V is the most important uid property, it interacts closely with the thermodynamic properties of the uid. We have already introduced into the discussion the three most common such properties: 1. Pressure p 2. Density 3. Temperature T These three are constant companions of the velocity vector in ow analyses. Four other intensive thermodynamic properties become important when work, heat, and energy balances are treated (Chaps. 3 and 4): 4. Internal energy 5. Enthalpy h p/ 6. Entropy s 7. Specic heats cp and cv In addition, friction and heat conduction effects are governed by the two so-called transport properties: 8. Coefcient of viscosity 9. Thermal conductivity k All nine of these quantities are true thermodynamic properties that are determined by the thermodynamic condition or state of the uid. For example, for a single-phase 1.8 Thermodynamic Properties of a Fluid a dV dt V t u V x v V y w V z The Acceleration Field V(x, y, z, t) iu(x, y, z, t) jv(x, y, z, t) kw(x, y, z, t) 18 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 18 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

39. substance such as water or oxygen, two basic properties such as pressure and temperature are sufcient to x the value of all the others: (p, T) h h(p, T) (p, T) and so on for every quantity in the list. Note that the specic volume, so important in thermodynamic analyses, is omitted here in favor of its inverse, the density . Recall that thermodynamic properties describe the state of a systemthat is, a collection of matter of xed identity that interacts with its surroundings. In most cases here the system will be a small uid element, and all properties will be assumed to be continuum properties of the ow eld: (x, y, z, t), and so on. Recall also that thermodynamics is normally concerned with static systems, whereas uids are usually in variable motion with constantly changing properties. Do the properties retain their meaning in a uid ow that is technically not in equilibrium? The answer is yes, from a statistical argument. In gases at normal pressure (and even more so for liquids), an enormous number of molecular collisions occur over a very short distance of the order of 1 m, so that a uid subjected to sudden changes rapidly adjusts itself toward equilibrium. We therefore assume that all the thermodynamic properties just listed exist as point functions in a owing uid and follow all the laws and state relations of ordinary equilibrium thermodynamics. There are, of course, important nonequilibrium effects such as chemical and nuclear reactions in owing uids, which are not treated in this text. Pressure is the (compression) stress at a point in a static uid (Fig. 1.3). Next to velocity, the pressure p is the most dynamic variable in uid mechanics. Differences or gradients in pressure often drive a uid ow, especially in ducts. In low-speed ows, the actual magnitude of the pressure is often not important, unless it drops so low as to cause vapor bubbles to form in a liquid. For convenience, we set many such problem assignments at the level of 1 atm 2116 lbf/ft2 101,300 Pa. High-speed (compressible) gas ows (Chap. 9), however, are indeed sensitive to the magnitude of pressure. Temperature T is related to the internal energy level of a uid. It may vary considerably during high-speed ow of a gas (Chap. 9). Although engineers often use Celsius or Fahrenheit scales for convenience, many applications in this text require absolute (Kelvin or Rankine) temperature scales: If temperature differences are strong, heat transfer may be important [20], but our concern here is mainly with dynamic effects. The density of a uid, denoted by (lowercase Greek rho), is its mass per unit volume. Density is highly variable in gases and increases nearly proportionally to the pressure level. Density in liquids is nearly constant; the density of water (about 1000 kg/m3 ) Density K C 273.16 R F 459.69 Temperature Pressure 1.8 Thermodynamic Properties of a Fluid 19 whi29346_ch01_002-063.qxd 10/14/09 19:58 Page 19 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

40. increases only 1 percent if the pressure is increased by a factor of 220. Thus most liquid ows are treated analytically as nearly incompressible. In general, liquids are about three orders of magnitude more dense than gases at atmospheric pressure. The heaviest common liquid is mercury, and the lightest gas is hydrogen. Compare their densities at 20C and 1 atm: They differ by a factor of 162,000! Thus the physical parameters in various liquid and gas ows might vary considerably. The differences are often resolved by the use of dimensional analysis (Chap. 5). Other uid densities are listed in Tables A.3 and A.4 (in App. A) and in Ref. 21. The specic weight of a uid, denoted by (lowercase Greek gamma), is its weight per unit volume. Just as a mass has a weight W mg, density and specic weight are simply related by gravity: (1.6) The units of are weight per unit volume, in lbf/ft3 or N/m3 . In standard earth gravity, g 32.174 ft/s2 9.807 m/s2 . Thus, for example, the specic weights of air and water at 20C and 1 atm are approximately Specic weight is very useful in the hydrostatic pressure applications of Chap. 2. Specic weights of other uids are given in Tables A.3 and A.4. Specic gravity, denoted by SG, is the ratio of a uid density to a standard reference uid, usually water at 4C (for liquids) and air (for gases): (1.7) For example, the specic gravity of mercury (Hg) is SGHg 13,580/1000 13.6. Engineers nd these dimensionless ratios easier to remember than the actual numerical values of density of a variety of uids. In thermostatics the only energy in a substance is that stored in a system by molecular activity and molecular bonding forces. This is commonly denoted as internal energy . A commonly accepted adjustment to this static situation for uid ow is to add two more energy terms that arise from newtonian mechanics: potential energy and kinetic energy. Potential and Kinetic Energies SGliquid liquid water liquid 1000 kg/m3 SGgas gas air gas 1.205 kg/m3 Specic Gravity water (998 kg/m3 )(9.807 m/s2 ) 9790 N/m3 62.4 lbf/ft3 air (1.205 kg/m3 )(9.807 m/s2 ) 11.8 N/m3 0.0752 lbf/ft3 g Specic Weight Mercury: 13,580 kg/m3 Hydrogen: 0.0838 kg/m3 20 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:59 Page 20 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

41. The potential energy equals the work required to move the system of mass m from the origin to a position vector r ix jy kz against a gravity eld g. Its value is mg r, or g r per unit mass. The kinetic energy equals the work required to change the speed of the mass from zero to velocity V. Its value is mV2 or V2 per unit mass. Then by common convention the total stored energy e per unit mass in uid mechanics is the sum of three terms: e V2 (g r) (1.8) Also, throughout this book we shall dene z as upward, so that g gk and g r gz. Then Eq. (1.8) becomes e V2 gz (1.9) The molecular internal energy is a function of T and p for the single-phase pure substance, whereas the potential and kinetic energies are kinematic quantities. Thermodynamic properties are found both theoretically and experimentally to be related to each other by state relations that differ for each substance. As mentioned, we shall conne ourselves here to single-phase pure substances, such as water in its liquid phase. The second most common uid, air, is a mixture of gases, but since the mixture ratios remain nearly constant between 160 and 2200 K, in this temperature range air can be considered to be a pure substance. All gases at high temperatures and low pressures (relative to their critical point) are in good agreement with the perfect-gas law (1.10) where the specic heats cp and cv are dened in Eqs. (1.14) and (1.15). Since Eq. (1.10) is dimensionally consistent, R has the same dimensions as specic heat, {L2 T2 1 }, or velocity squared per temperature unit (kelvin or degree Rankine). Each gas has its own constant R, equal to a universal constant divided by the molecular weight (1.11) where 49,700 ft-lbf/(slugmol R) 8314 J/(kmol K). Most applications in this book are for air, whose molecular weight is M 28.97/mol: (1.12) Standard atmospheric pressure is 2116 lbf/ft2 2116 slug/(ft s2 ), and standard temperature is 60F 520R. Thus standard air density is (1.13) This is a nominal value suitable for problems. For other gases, see Table A.4. air 2116 slug/(ft # s2 ) 31716 ft2 /(s2 # R)4(520R) 0.00237 slug/ft3 1.22 kg/m3 Rair 49,700 ft # lbf/(slugmol # R) 28.97/mol 1716 ft # lbf slug # R 1716 ft2 s2 R 287 m2 s2 # K ## Rgas Mgas p RT R cp cv gas constant State Relations for Gases 1 2 1 2 1 2 1 2 1.8 Thermodynamic Properties of a Fluid 21 whi29346_ch01_002-063.qxd 10/14/09 19:59 Page 21 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

42. One proves in thermodynamics that Eq. (1.10) requires that the internal molecular energy of a perfect gas vary only with temperature: (T). Therefore the specic heat cv also varies only with temperature: or (1.14) In like manner h and cp of a perfect gas also vary only with temperature: (1.15) The ratio of specic heats of a perfect gas is an important dimensionless parameter in compressible ow analysis (Chap. 9) (1.16) As a rst approximation in airow analysis we commonly take cp, cv, and k to be constant: (1.17) Actually, for all gases, cp and cv increase gradually with temperature, and k decreases gradually. Experimental values of the specic-heat ratio for eight common gases are shown in Fig. 1.5. Many ow problems involve steam. Typical steam operating conditions are relatively close to the critical point, so that the perfect-gas approximation is inaccurate. Since no simple formulas apply accurately, steam properties are available both in EES (see Sec. 1.12) and on a CD-ROM [23] and even on the Internet, as a MathPad Corp. applet [24]. Meanwhile, the error of using the perfect-gas law can be moderate, as the following example shows. EXAMPLE 1.5 Estimate and cp of steam at 100 lbf/in2 and 400F, in English units, (a) by the perfect-gas approximation and (b) by the ASME Steam Tables [23] or by EES. Solution Approach (a)the perfect-gas law: Although steam is not an ideal gas, we can estimate these properties with moderate accuracy from Eqs. (1.10) and (1.17). First convert pressure cp kR k 1 6010 ft2 /(s2 # R) 1005 m2 /(s2 # K) cv R k 1 4293 ft2 /(s2 # R) 718 m2 /(s2 # K) kair 1.4 k cp cv k(T) 1 dh cp(T)dT cp a h T b p dh dT cp(T) h p RT h(T) d cv(T)dT cv a T b d dT cv(T) 22 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 19:59 Page 22 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

43. from 100 lbf/in2 to 14,400 lbf/ft2 , and use absolute temperature, (400F 460) 860R. Then we need the gas constant for steam, in English units. From Table A.4, the molecular weight of H2O is 18.02, whence Then the density estimate follows from the perfect-gas law, Eq. (1.10): Ans. (a) At 860R, from Fig. 1.5, ksteam cp /cv 1.30. Then, from Eq. (1.17), Ans. (a) Approach (b)tables or software: One can either read the steam tables or program a few lines in EES. In either case, the English units (psi, Btu, lbm) are awkward when applied to uid mechanics formulas. Even so, when using EES, make sure that the cp kR k 1 (1.3)(2758 ft # lbf/(slug R)) (1.3 1) 12,000 ft # lbf slug R p RT 14,400 lbf/ft2 32758 ft # lbf/(slug # R)4(860 R) 0.00607 slug ft3 Rsteam English MH2O 49,700 ft # lbf/(slugmol R) 18.02/mol 2758 ft # lbf slug R 1.8 Thermodynamic Properties of a Fluid 23 Ar Atmospheric pressure H2 CO Air and N2 O2 Steam CO2 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 10000 3000 4000 5000 =k cp c 2000 Temperature, R Fig. 1.5 Specic-heat ratio of eight common gases as a function of temperature. (Data from Ref. 22.) whi29346_ch01_002-063.qxd 10/14/09 20:00 Page 23 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

44. Variable Information menu species English units: psia and F. EES statements for evaluating density and specic heat of steam are, for these conditions, Rho DENSITY(steam, P 100,T 400) Cp SPECHEAT(steam, P 100,T 400) Note that the software is set up for psia and F, without converting. EES returns the curve- t values Rho 0.2027 lbm/ft3 ; Cp 0.5289 Btu/(lbm-F) As just stated, Btu and lbm are extremely unwieldy when applied to mass, momentum, and energy problems in uid mechanics. Therefore, either convert to ft-lbf and slugs using your own resources, or use the Convert function in EES, placing the old and new units in single quote marks: Rho2 Rho*CONVERT(lbm/ft^3,slug/ft^3) Cp2 Cp*CONVERT(Btu/lbm-F,ft^2/s^2-R) Note that (1) you multiply the old Rho and Cp by the CONVERT function; and (2) units to the right of the division sign / in CONVERT are assumed to be in the denominator. EES returns these results: Rho2 0.00630 slug/ft3 Cp2 13,200 ft2 /(s2 -R) Ans. (b) Comments: The steam tables would yield results quite close to EES. The perfect-gas estimate of is 4 percent low, and the estimate of cp is 9 percent low. The chief reason for the discrepancy is that this temperature and pressure are rather close to the critical point and saturation line of steam. At higher temperatures and lower pressures, say, 800F and 50 lbf/in2 , the perfect-gas law yields properties with an accuracy of about 1 percent. Once again let us warn that English units (psia, lbm Btu) are awkward and must be converted in most uid mechanics formulas. EES handles SI units nicely, with no conversion factors needed. The writer knows of no perfect-liquid law comparable to that for gases. Liquids are nearly incompressible and have a single, reasonably constant specic heat. Thus an idealized state relation for a liquid is const cp cv const dh cp dT (1.18) Most of the ow problems in this book can be attacked with these simple assumptions. Water is normally taken to have a density of 998 kg/m3 and a specic heat cp 4210 m2 /(s2 K). The steam tables may be used if more accuracy is required. The density of a liquid usually decreases slightly with temperature and increases moderately with pressure. If we neglect the temperature effect, an empirical pressure density relation for a liquid is (1.19) where B and n are dimensionless parameters that vary slightly with temperature and pa and a are standard atmospheric values. Water can be tted approximately to the values B 3000 and n 7. p pa (B 1)a a b n B State Relations for Liquids 24 Chapter 1 Introduction whi29346_ch01_002-063.qxd 10/14/09 20:00 Page 24 Debd 208:MHDQ176:whi29346:0073529346:whi29346_pagefiles:

45. Seawater is a variable mixture of water and salt and thus requires three thermodynamic properties to dene its state. These are normally taken as pressure, temperature, and the salinity , dened as the weight of the dissolved salt divided by the weight of the mixture. The average salinity of seawater is 0.035, usually written as 35 parts per 1000, or 35 . The average density of seawater is 2.00 slugs/ft3 1030 kg/m3 . Strictly speaking, seawater has three specic heats, all approximately equal to the value for pure water of 25,200 ft2 /(s2 R) 4210 m2 /(s2 K). EXAMPLE 1.6 The pressure at the deepest part of the ocean is approximately 1100 atm. Estimate the density of seawater in slug/ft3 at this pressure. Solution Equation (1.19) holds for either water or seawater. The ratio p/pa is given as 1100: or Assuming an average surface seawater density a 2.00 slugs/ft3 , we compute Ans. Even at these immense pressures, the density increase is less than 5 percent, which justies the treatment of a liquid ow as essentially incompressible. The quantities such as pressure, temperature, and density discussed in the previous section are primary thermodynamic variables characteristic of any system. Certain secondary variables also characterize specic uid mechanical behavior. The most important of these is viscosity, which relates the local stresses in a moving uid to the strain rate of the uid element. Viscosity is a quantitative measure of a uids resistance to ow. More specically, it determines the uid strain rate that is generated by a given applied shear stress. We can easily move through air, which has very low viscosity. Movement is more difcult in water, which has 50 times higher viscosity. Still more resistance is found in SAE 30 oil, which is 300 times more viscous than water. Try to slide your hand through glycerin, which is ve times more viscous than SAE 30 oil, or blackstrap molasses, another factor of ve higher than glyce

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White fluid mechanics 7th c2011 txtbk3