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An Introduction to Fluid Mechanics and Transport Phenomena FLUID MECHANICS and ITS APPLICATIONS Vo L U M E 8 6

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An Introduction to Fluid Mechanics and Transport Phenomena FLUID MECHANICS and ITS APPLICATIONS Vo L U M E 8 6 An Introduction to Fluid Mechanics and Transport Phenomena FLUID MECHANICS AND ITS APPLICATIONS Vo l u m e 8 6 Series Editor:R.MOREAU MADYLAM Ecole Nationale Supérieure d’Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d’Hères Cedex, France Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modeling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advance- ment. Fluids have the ability to transport matter and its properties as well as to transmit force, therefore fluid mechanics is a subject that is particularly open to cross fertilization with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For other titles published in this series, go to www.springer.com/series/5980 G. Hauke An Introduction to Fluid Mechanics and Transport Phenomena G. Hauke Área de Mecánica de Fluidos Centro Politécnico Superior Universidad de Zaragoza C/MariadeLuna3 50018 Zaragoza Spain ISBN-13: 978-1-4020-8536-9 e-ISBN-13: 978-1-4020-8537-6 Library of Congress Control Number: 2008932575 © 2008 Springer Science+Business Media, B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com To my wife, my parents and my children Preface This text is a brief introduction to fundamental concepts of transport phe- nomena within a fluid, namely momentum, heat and mass transfer. The em- phasis of the text is placed upon a basic, systematic approach from the fluid mechanics point of view, in conjunction with a unified treatment of transport phenomena. In order to make the book useful for students, there are numerous ex- amples. Each chapter presents a collection of proposed problems, whose solu- tions can be found in the Problem Solutions Appendix. Also the Self Evalu- ation chapter gathers exercises from exams, so readers and students can test their understanding of the subject. Most of the content can be taught in a course of 45 hours and has been employed in the course Transport Phenomena in Chemical Engineering at the Centro Polit´ecnico Superior of the University of Zaragoza. The text is aimed at beginners in the subject of transport phenomena and fluid mechanics, emphasizing the foundations of the subject. The text is divided into four parts: Fundamentals, Conservation Principles, Dimensional Analysis;Theory and Applications, and Transport Phenomena at Interfaces. In the first part, Fundamentals, basic notions on the subject are intro- duced: definition of a fluid, preliminary hypothesis for its mathematical treat- ment, elementary kinematics, fluid forces, especially the concept of pressure, and fluid statics. In the Conservation Principles part, the conservation equations that gov- ern transport phenomena are presented and explained, both in integral and differential form. Emphasis is placed on practical applications of integral equa- tions. Also, constitutive equations for transport by diffusion are contained in this part. In the third part, Dimensional Analysis;Theory and Applications, the im- portant tool of dimensional analysis and the laws of similitude are explained. Also the dimensionless numbers that govern transport phenomena are derived. VIII Preface The last part, Transport Phenomena at Interfaces, explains how most transport processes originate at interfaces. Some aspects of the concept of boundary layer are presented and the usage of transport coefficients to solve practical problems is introduced. Finally the analogies between transport coef- ficients are explained. There are a great number of people whose help in writing this book I would like to acknowledge. First my parents, for providing an intellectually challen- ging environment and awakening my early interest in engineering and fluid mechanics. Professor C. Dopazo, for his inspiring passion for fluid mechanics. My family, wife and children, for their love and support. C. P´erez-Caseiras for providing ideas to strengthen the text. Many colleagues and friends, who have accompanied me during these years, especially professors T.J.R. Hughes and E. O˜nate, and friends Jorge, Antonio, Connie and Ed. Finally, I would like to acknowledge the encouragement of Nathalie Jacobs. Without them, this project would not have been possible. Zaragoza, Guillermo Hauke May 2008 Contents Nomenclature .................................................XV Introduction ................................................... 1 Part I Fundamentals 1 Basic Concepts in Fluid Mechanics ........................ 5 1.1 TheConceptofaFluid .................................. 5 1.1.1 TheMacroscopicPointofView ..................... 5 1.1.2 TheMicroscopicPointofView...................... 8 1.2 TheFluidasaContinuum................................ 8 1.3 Local Thermodynamic Equilibrium . 10 Problems............................................... 10 2 Elementary Fluid Kinematics .............................. 11 2.1 DescriptionofaFluidField............................... 11 2.1.1 LagrangianDescription ............................ 11 2.1.2 EulerianDescription............................... 13 2.1.3 Arbitrary Lagrangian-Eulerian Description (ALE) . 15 2.2 The Substantial or Material Derivative . 15 2.3 MechanismsofTransportPhenomena...................... 19 2.4 Streamlines,TrajectoriesandStreaklines................... 20 2.4.1 CalculationofStreamlines.......................... 22 2.4.2 CalculationofTrajectories.......................... 24 2.4.3 CalculationofStreaklines .......................... 25 2.5 TheConceptofFlux..................................... 26 Problems............................................... 30 XContents 3 Fluid Forces ............................................... 33 3.1 Introduction............................................ 33 3.2 BodyForces............................................ 34 3.3 SurfaceForces .......................................... 35 3.3.1 TheStressTensor................................. 35 3.3.2 TheConceptofPressure........................... 40 3.4 SurfaceTension......................................... 43 3.5 Summary............................................... 45 Problems............................................... 45 4 Fluid Statics ............................................... 47 4.1 The Fundamental Equation of Fluid Statics . 47 4.2 Applications............................................ 48 4.2.1 Hydrostatics...................................... 49 4.2.2 Manometry....................................... 51 4.2.3 FluidStaticsofanIsothermalPerfectGas............ 51 4.2.4 Forces over Submerged Surfaces . 52 Problems............................................... 62 Part II Conservation Principles 5 Transport Theorems ....................................... 69 5.1 FluidVolumeandControlVolume......................... 69 5.2 TransportTheorems..................................... 70 5.2.1 FirstTransportTheorem........................... 72 5.2.2 SecondTransportTheorem......................... 73 5.2.3 ThirdTransportTheorem.......................... 73 6 Integral Conservation Principles ........................... 75 6.1 MassConservation....................................... 75 6.2 MomentumEquation .................................... 78 6.2.1 DecompositionoftheStressTensor.................. 79 6.3 AngularMomentumEquation............................. 82 6.4 TotalEnergyConservation ............................... 85 6.4.1 BodyForceStemmingfromaPotential .............. 87 6.5 OtherEnergyEquations.................................. 89 6.5.1 MechanicalEnergyEquation........................ 89 6.5.2 InternalEnergyEquation .......................... 95 6.5.3 Energy Transfer Between Mechanical and Internal Energy........................................... 96 6.6 ConservationofChemicalSpeciesEquation................. 96 6.6.1 IntroductoryDefinitions............................ 96 6.6.2 DerivationoftheConservationEquations............. 98 6.6.3 Chemical Species Equations for Molar Concentrations . 102 Contents XI 6.6.4 Equations with Respect to the Molar Average Velocity . 103 6.7 Equation of Volume Conservation for Liquids . 103 6.8 Outline ................................................105 6.9 Initial and Boundary Conditions . 107 6.9.1 InitialConditions .................................107 6.9.2 Boundary Conditions . 107 Problems...............................................111
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