Heat Transfer

S. P. Venkateshan

Heat TransferThird Edition

123

S. P. VenkateshanDepartment of Mechanical EngineeringIndian Institute of Technology MadrasChennai, Tamil Nadu, India

ISBN 978-3-030-58337-8 ISBN 978-3-030-58338-5 (eBook)https://doi.org/10.1007/978-3-030-58338-5

Jointly published with ANE Books Pvt. Ltd. In addition to this printed edition, there is a local printededition of this work available via Ane Books in South Asia (India, Pakistan, Sri Lanka, Bangladesh,Nepal and Bhutan) and Africa (all countries in the African subcontinent).ISBN of the Co-Publisher’s edition: 978-9-385-46207-8

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer NatureSwitzerland AG 2021This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publishers nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publishers remain neutral with regard tojurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To My Parents with Respect

To the Shakkottai Family with Affection

Preface to the Springer Edition

The first edition of the book titled “A First Course in Heat Transfer” was publishedabout 15 years ago. The second edition appeared in 2009. Time was ripe to revisethe book further and the result was this third edition, which was published by AneBooks Pvt. Ltd. in 2017. With corrections, the same is now being brought out asSpringer edition.

At one time it appeared that the field of ‘heat transfer’ had reached saturation andthere was not much new in it. However, things have changed significantly in recenttimes. New applications in fields such as microelectronic devices, nuclear reactors,space propulsion systems, 3D printing made it necessary to move beyond what waspossible only a few years ago. Improvements in computers and measuring instru-ments have made the field interesting once more and there is scope for muchresearch in this area. The author has been involved in several of these developmentsand feels that it is time to look at the subject with renewed interest!

This edition is brought out with a complete overhaul of the book. Many newworked out examples are included in this edition. Also, many new topics have beenadded to bring the book, undoubtedly not only to a higher level, but also to a higherlevel of relevance. I have tried to intersperse the elementary aspects with severaladvanced topics so that the interested reader can explore more recent developmentsin heat transfer with a higher level of preparation.

The aim of the present edition remains the same as the earlier editions, viz., tomove from elementary to advanced in a slow but steady progression. In order tokeep the length of the book under check, I have tried to reformat the entire bookusing more advanced features available in “latex” along with the graphic envi-ronment “tikz”. All plots have been redone using QtiPlot and the line drawings havebeen redone using “tikz”. Problems at the end of each chapter are still the best waythe reader will be challenged to test his learning of the material discussed in thechapters of the book.

vii

I have tried to correct as far as possible all errors of various types in the earliereditions. However, I will be grateful if the reader would bring to my notice anyerrors still found in this edition.

Chennai, India S. P. VenkateshanMay 2020

viii Preface to the Springer Edition

Preface to the Second Edition

The present book is an augmented and fully revised version of my earlier book AFirst Course in Heat Transfer. The book is now out of print and not available.Several typographical and factual errors that had crept into the book have now beencorrected in the present book, which is titled “Heat Transfer”. The change of thetitle seemed reasonable because the contents had been augmented to include manytopics that were omitted in the earlier book. The number of chapters have grown to17 as against 15 in the earlier book. Notable new material is to be found in all topicsof heat transfer. Convection heat transfer and radiation heat transfer portions havebeen significantly improved with additional material that takes these closer to thecurrent literature in these areas. I have also included the most relevant references asfootnotes for the convenience of the reader. The present revised augmented bookhas taken over two years of my time.

I have improved the level of the book with additional materials included inalmost all the topics. In view of the growing importance of numerical methods Ihave expanded the part that deals with numerical methods. Several appendicesdealing with background material not easily accessible to the student are added tomake it possible to deal with more advanced heat transfer topics in the book. Thefirst time reader may want to skip some of these topics, without loss of continuity.

A few years ago I recorded a set of video lectures on Heat Transfer through theEducational Technology Cell of the Indian Institute of Technology, Madras. Theseare available in DVD form from the Educational Technology Cell, IIT Madras.These have subsequently been broadcast periodically over Eklavya, the Technologychannel. As a part of the video effort I prepared notes on the various topics coveredin the lecture series. The notes looked interesting and I felt that it would be worthwhile converting the Notes in to a book form and make it available to a wideraudience. It seemed that, with many of the Regional Colleges of Engineering beingupgraded to National Institutes of Technology, an introductory book directedtowards students joining these institutes would be worth publishing and the presentbook is the result. With the general level of undergraduate programs undergoingqualitative change, the book should be relevant to undergraduate students studyingin any of the many engineering colleges that have started functioning throughout

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the country. I believe the present book can be covered in a semester, say in the thirdyear of the B.Tech. program as is done at the IIT’s including IIT Madras. Someadvanced sections may be omitted for this purpose.

Heat transfer as a discipline has grown over the past two centuries to a maturescience. Rapid developments took place during the Second World War. Later thespace age brought a new emphasis to the study of heat transfer under harsh con-ditions. The energy crisis focused the attention of heat transfer experts on solarenergy applications. Developments in microelectronics have in recent times moti-vated heat transfer research. Heat Transfer in manufacturing processes like lasermachining, electron beam welding, metal casting, to name a few, have also beenmajor areas of recent research. However, it is fortunate that most of these have notrequired any more mathematics than that contained in a book like “AdvancedEngineering Mathematics” by Kreyszig 1993. The knowledge of Physics requiredis more or less that covered in the Plus 2 followed by what is taught in the first yearin most engineering colleges in the country. The background knowledge of FluidMechanics may be obtained from a book like “Introduction to Fluid Mechanics” byFox and McDonald 1995. A good grounding in the fundamentals of thermody-namics, as is covered in the first year of engineering, is all that is needed toundertake a study of heat transfer. The present book assumes that the student hasalready had exposure to the above by the time he decides to use this book.

I do not have any pretensions regarding the originality of the material that formsthe bulk of the book. These have been considered in one form or another by all theprevious authors who have written books on Heat Transfer. I can only claim to acertain way the material has been presented in the present book. I use examples thatare close to reality. A practicing Heat transfer engineer would probably think ofsimilar examples when he is designing thermal systems. The problems are not theplug and play type. They do require some amount of “modeling” effort on the partof the student. Also, exercises at the end of each chapter require a fair amount ofthinking on the part of the reader. I have deliberately avoided giving answers toproblems. This will discourage the student from trying to get the provided answer byhook or by crook. We all make mistakes, in modeling or in calculations, and we learnmore from mistakes than from perfectly executed solutions, the first time. This, Ibelieve, is the best way of attaining some self sufficiency on the part of the student.

The writing of the book has involved support from several people.Dr. N. Ramesh, a former student of mine produced the first hand written versionof the notes. Mrs. Lakshmi Suresh typed the first draft with a lot of care. I made allthe figures and plots using the many software resources available on my PC.Particularly useful was MathCad 7 Professional and Microsoft EXCEL whichhelped in checking and rechecking the many solved examples presented in the book.

I have enjoyed writing the book since it gave me opportunity to learn a lot ofnew things. I hope the book also interests the students and other readers who mayuse it for learning heat transfer.

S. P. Venkateshan

x Preface to the Second Edition

Acknowledgements

Writing a book requires the help and support of many people. I have been fortunatein having a large number of students who have helped me in this effort. All mystudents are involved in heat transfer research and have provided support by carryingout my pet projects with dedication. At several places in the book, I have includeddata and results from theses of my students. The institution where I work—IndianInstitute of Technology Madras——has provided an ambience conducive to aca-demic pursuit and made this book revision a smooth affair. My past student PrasannaSwaminathan wrote a class file “bookspv.cls” which has helped me in improving theaesthetic quality of the book.

Most importantly I thank my wife for the wholehearted support she has extendedby allowing me to work with the computer at home for long hours every day.

May 2016 S. P. Venkateshan

xi

Contents

1 Introduction to the Study of Heat Transfer . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Assumptions in the Study of Heat Transfer . . . . . . . . . . 21.3 Basic Heat Transfer Processes and Examples . . . . . . . . . . . . . 4

1.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Steady Conduction in One Dimension . . . . . . . . . . . . . . . . . . . . . . . 172.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 On Thermal Conductivity Values . . . . . . . . . . . . . . . . 172.1.2 Approaches to the Study of Conduction Heat

Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Steady One Dimensional Conduction . . . . . . . . . . . . . . . . . . . 19

2.2.1 One-Dimensional Conduction in a UniformArea Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Steady One-Dimensional Conduction in CylindricalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.3 Steady Radial Conduction in a Solid Cylinderwith Internal Heat Generation . . . . . . . . . . . . . . . . . . 47

2.2.4 One-Dimensional Radial Conduction in SphericalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 Unsteady Heat Transfer in Lumped Systems . . . . . . . . . . . . . . . . . 653.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Governing Equation and the General Solution . . . . . . . . . . . . 66

3.2.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.2 Electrical Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.3 Characteristic Length Scale . . . . . . . . . . . . . . . . . . . . 68

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3.2.4 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.5 Response of a First-Order System in Particular

Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.3 Second-Order Thermal System: Response to Step Input . . . . . 853.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Heat Transfer from Extended Surfaces . . . . . . . . . . . . . . . . . . . . . . 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Fins of Uniform Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2.2 Solution to the Fin Equation . . . . . . . . . . . . . . . . . . . 1034.2.3 Uniform Area Fin Subject to Third Kind

Boundary Condition at the Tip . . . . . . . . . . . . . . . . . 1104.3 Variable Area Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.3.1 General Analysis of Variable Area Fins . . . . . . . . . . . 1144.3.2 Particular Cases of Variable Area Fins . . . . . . . . . . . . 115

4.4 Fins of Minimum Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.1 Uniform Fin of Optimum Proportions . . . . . . . . . . . . 1284.4.2 Uniform Spine (pin Fin) of Optimum Proportions . . . 131

4.5 Heat Transfer from Fin Arrays . . . . . . . . . . . . . . . . . . . . . . . . 1364.5.1 Overall Surface Efficiency of a fin array . . . . . . . . . . 1364.5.2 Effectiveness of a Fin Array . . . . . . . . . . . . . . . . . . . 1374.5.3 Fin Array Applications . . . . . . . . . . . . . . . . . . . . . . . 138

4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5 Multidimensional Conduction Part I . . . . . . . . . . . . . . . . . . . . . . . . 1515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.1.1 Integral Form of Governing Equation . . . . . . . . . . . . 1515.1.2 Differential Form of Governing Equation . . . . . . . . . . 1535.1.3 Simplified Form of Energy Equation . . . . . . . . . . . . . 1565.1.4 Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.2 One-Dimensional Transient Conduction . . . . . . . . . . . . . . . . . 1575.2.1 Transients in a Semi-infinite Solid . . . . . . . . . . . . . . . 1585.2.2 Approximate Integral Method Due to Goodman . . . . . 1735.2.3 One-Dimensional Transient Problem: Space

Domain Finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.3 Steady Conduction in Two Dimensions . . . . . . . . . . . . . . . . . 183

5.3.1 Steady Conduction in a Rectangle . . . . . . . . . . . . . . . 1835.3.2 Steady Conduction in a Rectangle With Heat

Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.3.3 Steady Two-Dimensional Conduction

in Cylindrical Co-Ordinates . . . . . . . . . . . . . . . . . . . . 1905.3.4 Shape Factors for Some Useful Configurations . . . . . . 201

xiv Contents

5.3.5 Solution to Laplace Equation in a Cylinder . . . . . . . . 2065.3.6 Solution to a Practical Problem . . . . . . . . . . . . . . . . . 2085.3.7 Solution to Laplace Equation in Spherical

Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2125.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6 Multidimensional Conduction Part II . . . . . . . . . . . . . . . . . . . . . . . 2196.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.1.2 Basic Problem in Cartesian Coordinates . . . . . . . . . . . 2206.1.3 Basic Problem in Cylindrical Coordinates . . . . . . . . . 2276.1.4 Basic Problem in Spherical Co-Ordinates . . . . . . . . . . 229

6.2 One-Term Approximation and Heisler Charts . . . . . . . . . . . . . 2356.3 Transient Conduction in More Than One Dimension . . . . . . . . 235

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.3.2 Transient Conduction in an Infinitely Long

Rectangular Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2366.3.3 Transient Heat Conduction in a Rectangular

Block in the form of a brick . . . . . . . . . . . . . . . . . . . 2426.3.4 Transient Heat Conduction in a Circular Cylinder

of Finite Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7 Numerical Solution of Conduction Problems . . . . . . . . . . . . . . . . . . 2537.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.1.1 A Simple Example: One-Dimensional SteadyConduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

7.1.2 Numerical Solution of a Fin Problem . . . . . . . . . . . . . 2577.1.3 Solution of Nodal Equations by TDMA . . . . . . . . . . . 2627.1.4 Steady Radial Conduction in a Cylinder . . . . . . . . . . . 2647.1.5 Steady Radial Conduction in a Spherical Shell . . . . . . 269

7.2 Conduction in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . 2727.2.1 Steady Heat Conduction in Two Dimensions:

Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 2727.2.2 Steady Heat Conduction in Two Dimensions:

Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 2827.2.3 One-Dimensional Transient in a Bar . . . . . . . . . . . . . 2877.2.4 Transient Heat Transfer in a Conducting

Convecting Fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2967.2.5 One-Dimensional Transient in a Solid Cylinder . . . . . 2997.2.6 One-Dimensional Transient in a Solid Sphere . . . . . . . 302

7.3 Transient Conduction in Two and Three Dimensions . . . . . . . 306

Contents xv

7.3.1 Transient Conduction in a Rectangle: ExplicitFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

7.3.2 ADI Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3097.3.3 Modification of the ADI Method for Three

Dimensional Transient Conduction . . . . . . . . . . . . . . 3147.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

8 Basics of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3238.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

8.1.1 Fundamental Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . 3248.1.2 Preliminaries and Definitions . . . . . . . . . . . . . . . . . . . 325

8.2 Cavity or Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . 3358.2.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3358.2.2 Thermodynamics of Black Body Radiation . . . . . . . . 336

8.3 Wavelength Distribution of Black Body Radiation . . . . . . . . . 3388.3.1 About Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.3.2 Number of Degenerates Modes

in a Three-Dimensional Cavity . . . . . . . . . . . . . . . . . 3428.3.3 Planck Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 3438.3.4 Properties of the Planck Distribution Function . . . . . . 345

8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

9 Surface Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3599.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

9.1.1 Surface Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3609.2 Spectral and Hemispherical Surface Properties . . . . . . . . . . . . 361

9.2.1 Spectral Hemispherical Quantities . . . . . . . . . . . . . . . 3619.2.2 Total Hemispherical Quantities . . . . . . . . . . . . . . . . . 3639.2.3 Band Model for a Non-gray Surface . . . . . . . . . . . . . 3649.2.4 Equilibrium Temperature of a Surface . . . . . . . . . . . . 3689.2.5 Selective Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

9.3 Angle-Dependent Surface Properties . . . . . . . . . . . . . . . . . . . . 3759.3.1 Some Results from Electromagnetic Theory . . . . . . . . 3759.3.2 Specular Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3789.3.3 Hemispherical Reflectance . . . . . . . . . . . . . . . . . . . . . 3939.3.4 Real or Engineering Surfaces . . . . . . . . . . . . . . . . . . 398

9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

10 Radiation in Enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40510.2 Evacuated Enclosure with Gray Diffuse Walls . . . . . . . . . . . . 406

10.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40610.2.2 Diffuse Radiation Interchange Between Two

Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

xvi Contents

10.2.3 Angle Factor Algebra and Its Applications . . . . . . . . . 40910.2.4 Three-Dimensional Enclosures . . . . . . . . . . . . . . . . . . 420

10.3 Radiation Heat Transfer in Enclosures with Gray DiffuseWalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42710.3.1 Method of Detailed Balancing . . . . . . . . . . . . . . . . . . 42710.3.2 Radiation Shields . . . . . . . . . . . . . . . . . . . . . . . . . . . 42910.3.3 Radiosity Irradiation Method of Enclosure

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43110.3.4 Electrical Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . 444

10.4 Enclosure Analysis Under Special Circumstances . . . . . . . . . . 45010.4.1 Enclosure Containing Diffuse Non-gray Surfaces . . . . 45010.4.2 Gray Enclosures Containing Diffuse and Specular

Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45310.4.3 Enclosure Analysis with Surfaces of Non-uniform

Radiosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45710.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

11 Radiation in Participating Media . . . . . . . . . . . . . . . . . . . . . . . . . . 47511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47511.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

11.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47611.2.2 Equation of Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 476

11.3 Absorption of Radiation in Different Media . . . . . . . . . . . . . . 47811.3.1 Transmittance of a Solid Slab . . . . . . . . . . . . . . . . . . 47811.3.2 Absorption of Radiation by Liquids . . . . . . . . . . . . . . 48111.3.3 Absorption of Radiation by Gases . . . . . . . . . . . . . . . 48211.3.4 Radiation in an Isothermal Gray Gas Slab

and the Concept of Mean Beam Length . . . . . . . . . . . 48811.4 Modeling of Gas Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 496

11.4.1 Basics of Gas Radiation Modeling . . . . . . . . . . . . . . . 49611.4.2 Band Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

11.5 Radiation in a Non-isothermal Participating Medium . . . . . . . . 50911.5.1 Radiation Transfer in a Gray Slab: . . . . . . . . . . . . . . 50911.5.2 Radiation Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 51111.5.3 Solution of Integral Equation . . . . . . . . . . . . . . . . . . . 51311.5.4 Discrete Ordinate Method . . . . . . . . . . . . . . . . . . . . . 517

11.6 Enclosure Analysis in the Presence of an Absorbingand Emitting Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52611.6.1 Zone Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52711.6.2 Example of Zone Analysis . . . . . . . . . . . . . . . . . . . . 53011.6.3 Application of DOM to Two-Surface Enclosure

with a Non-isothermal Participating Medium . . . . . . . 53411.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Contents xvii

12 Laminar Convection In Internal Flow . . . . . . . . . . . . . . . . . . . . . . 54512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

12.1.1 Classification of Flows . . . . . . . . . . . . . . . . . . . . . . . 54512.1.2 Fluid Properties and Their Variation . . . . . . . . . . . . . 547

12.2 Dimensional Analysis and Similarity . . . . . . . . . . . . . . . . . . . 55112.2.1 Dimensional Analysis of a Flow Problem . . . . . . . . . 55112.2.2 Notion of “Similarity” . . . . . . . . . . . . . . . . . . . . . . . . 55412.2.3 Dimensional Analysis of Heat Transfer Problem . . . . . 556

12.3 Internal Flow Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 55912.3.1 Fundamentals of Steady Laminar Tube Flow . . . . . . . 56012.3.2 Governing Equation Starting from First

Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56112.3.3 Governing Equation Starting with the NS

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56312.3.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56312.3.5 Fully Developed Flow in a Parallel Plate Channel . . . 56712.3.6 Concept of Fluid Resistance . . . . . . . . . . . . . . . . . . . 570

12.4 Laminar Heat Transfer in Tube Flow . . . . . . . . . . . . . . . . . . . 57212.4.1 Bulk Mean Temperature . . . . . . . . . . . . . . . . . . . . . . 57312.4.2 Variation of the Bulk Mean Temperature . . . . . . . . . . 57412.4.3 Tube Flow with Uniform Wall Heat Flux . . . . . . . . . 57512.4.4 Fully Developed Temperature with Uniform Wall

Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57612.4.5 Tube Flow with Constant Wall Temperature . . . . . . . 58012.4.6 Fully Developed Tube Flow with Constant Wall

Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58112.5 Laminar Fully Developed Flow and Heat Transfer

in Non-circular Tubes and Ducts . . . . . . . . . . . . . . . . . . . . . . 58812.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58812.5.2 Parallel Plate Channel with Asymmetric Heating . . . . 58912.5.3 Parallel Plate Channel with Symmetric Heating . . . . . 59012.5.4 Fully Developed Flow in a Rectangular Duct . . . . . . . 59012.5.5 Fully Developed Heat Transfer in a Rectangular

Duct: Uniform Wall Heat Flux Case . . . . . . . . . . . . . 59312.5.6 Fully Developed Flow and Heat Transfer Results

in Several Important Geometries . . . . . . . . . . . . . . . . 59512.6 Laminar Fully Developed Heat Transfer to Fluid Flowing

in an Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59612.6.1 Fully Developed Flow in an Annulus . . . . . . . . . . . . 59612.6.2 Fully Developed Temperature in an Annulus . . . . . . . 599

xviii Contents

12.7 Flow and Heat Transfer in Laminar Entry Region . . . . . . . . . . 60212.7.1 Heat Transfer in Entry Region of Fully Developed

Tube Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60212.7.2 Mean Nusselt Number and Useful Correlations . . . . . 604

12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

13 Laminar Convection in External Flow . . . . . . . . . . . . . . . . . . . . . . 61113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61113.2 Laminar Boundary Layer Flow Past a Surface . . . . . . . . . . . . 612

13.2.1 Order of Magnitude Analysis and the BoundaryLayer Approximation . . . . . . . . . . . . . . . . . . . . . . . . 613

13.2.2 Laminar Boundary Layer over a Flat Plate:Velocity Boundary Layer . . . . . . . . . . . . . . . . . . . . . 618

13.2.3 Laminar Thermal Boundary Layer over a FlatPlate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

13.3 Boundary Layer Flow in the Presence of Stream-WisePressure Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63713.3.1 Inviscid Flow Past the Wedge . . . . . . . . . . . . . . . . . . 63813.3.2 Flow Within the Boundary Layer . . . . . . . . . . . . . . . 64013.3.3 Temperature Profiles in Falkner–Skan Flows . . . . . . . 644

13.4 Integral Form of Boundary Layer Equations . . . . . . . . . . . . . . 64713.4.1 Momentum and Energy Integral Equations . . . . . . . . . 64813.4.2 Approximate Solution for Boundary Layer Flow

Past a Flat Plate Using a Polynomial Profilefor Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

13.4.3 Approximate Solution for Boundary LayerTemperature Profile for Flow Past a Flat PlateUsing a Polynomial Profile for Temperature . . . . . . . . 658

13.4.4 Integral Method Applied to Boundary LayerFlow with Axial Pressure Gradient . . . . . . . . . . . . . . 661

13.4.5 Thwaites’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . 66313.5 Cylinder in Cross Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

13.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67013.5.2 Laminar Flow Normal to a Cylinder . . . . . . . . . . . . . 67213.5.3 Laminar Boundary Layer Flow Past a Cylinder . . . . . 67313.5.4 Effect of Pressure Gradient on Boundary Layer

Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67613.5.5 Drag Force on a Cylinder in Cross Flow . . . . . . . . . . 678

13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

14 Convection in Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68514.2 Time-Averaged Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

Contents xix

14.2.1 Turbulent Shear Stress and Turbulent Heat Flux . . . . . 68914.2.2 Turbulent Boundary Layer Equations . . . . . . . . . . . . . 690

14.3 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69014.3.1 Prandtl’s Mixing Length Theory . . . . . . . . . . . . . . . . 69114.3.2 Universal Velocity Distribution . . . . . . . . . . . . . . . . . 69214.3.3 Velocity Profiles in Pipe Flow . . . . . . . . . . . . . . . . . . 693

14.4 Pressure Drop and Heat Transfer in Turbulent Pipe Flow . . . . 69614.4.1 Pressure Drop in Turbulent Pipe Flow . . . . . . . . . . . . 69614.4.2 Heat Transfer in Turbulent Pipe Flow . . . . . . . . . . . . 69914.4.3 Application of Average Heat Transfer Coefficient

Concept to a Practical Application . . . . . . . . . . . . . . . 70114.5 Turbulent Boundary Layer over a Flat Plate . . . . . . . . . . . . . . 704

14.5.1 Approximate Analysis of Turbulent FlowParallel to a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . 704

14.5.2 Heat Transfer in the Turbulent BoundaryLayer over a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . 706

14.5.3 Calculation of Drag with Flow Being PartlyLaminar and Partly Turbulent . . . . . . . . . . . . . . . . . . 712

14.6 Cylinder in Cross Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71514.6.1 Heat Transfer for Flow Normal to a Tube Bank . . . . . 719

14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

15 Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72715.2 Analysis of Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . 729

15.2.1 Thermodynamic Analysis of a Co-Current HeatExchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

15.2.2 Thermal Analysis of a Co-Current HeatExchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

15.2.3 Overall Heat Transfer Coefficient . . . . . . . . . . . . . . . . 73315.2.4 Alternate Approach—�� NTU Relationship

for a Co-Current Heat Exchanger . . . . . . . . . . . . . . . 73615.2.5 Counter-Current Heat Exchanger . . . . . . . . . . . . . . . . 739

15.3 Other Types of Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . 74515.3.1 Analysis of Shell and Tube Heat Exchanger . . . . . . . . 74715.3.2 Analysis of a Cross Flow Heat Exchanger

by the �� NTU Approach . . . . . . . . . . . . . . . . . . . . . 75015.3.3 Analysis of a Cross Flow Heat Exchanger

by LMTD Correction Factor Approach . . . . . . . . . . . . 75515.3.4 General Remarks on Heat Exchangers . . . . . . . . . . . . 758

15.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760

xx Contents

16 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76316.2 Laminar Natural Convection from a Vertical Isothermal

Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77016.2.1 Isothermal Vertical Plate—Integral Solution . . . . . . . . 77116.2.2 Exact Solution of Ostrach . . . . . . . . . . . . . . . . . . . . . 78316.2.3 Comparison with Experimental Results . . . . . . . . . . . 787

16.3 Turbulent Natural Convection from a Vertical IsothermalPlate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78816.3.1 Approximate Integral Analysis . . . . . . . . . . . . . . . . . 78816.3.2 Useful Nusselt Number Correlations . . . . . . . . . . . . . 790

16.4 Natural Convection from Other Geometries . . . . . . . . . . . . . . 79416.4.1 Correlation for Horizontal Plates . . . . . . . . . . . . . . . . 79416.4.2 Correlation for Vertical Cylinders . . . . . . . . . . . . . . . 79716.4.3 Correlation for Horizontal Cylinders . . . . . . . . . . . . . 797

16.5 Heat Transfer Across Fluid Layers . . . . . . . . . . . . . . . . . . . . . 80216.5.1 Horizontal Fluid Layers . . . . . . . . . . . . . . . . . . . . . . 80216.5.2 Vertical Fluid Layers . . . . . . . . . . . . . . . . . . . . . . . . 80316.5.3 Inclined Air Layers . . . . . . . . . . . . . . . . . . . . . . . . . . 808

16.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

17 Special Topics in Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 81517.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81517.2 Multi-mode Problem Involving Radiation . . . . . . . . . . . . . . . . 816

17.2.1 Transient Cooling of a Lumped System . . . . . . . . . . . 81617.2.2 Radiation Error in Thermometry . . . . . . . . . . . . . . . . 81817.2.3 Duct Type Space Radiator . . . . . . . . . . . . . . . . . . . . 82117.2.4 Uniform Area Fin Losing Heat by Convection

and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82717.2.5 Radiating-Conducting-Convecting Fin

With Linearized Radiation . . . . . . . . . . . . . . . . . . . . . 83517.3 Heat Transfer During Melting or Solidification . . . . . . . . . . . . 839

17.3.1 Stefan Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83917.3.2 Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 84317.3.3 Phase Change in a Finite Domain . . . . . . . . . . . . . . . 846

17.4 Heat Transfer During Condensation . . . . . . . . . . . . . . . . . . . . 84817.4.1 Film Condensation Over An Isothermal Vertical

Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84917.4.2 Film Condensation Inside and Outside Tubes . . . . . . . 85517.4.3 Condensation in the Presence of Flowing Vapor . . . . . 857

17.5 Heat Transfer During Boiling . . . . . . . . . . . . . . . . . . . . . . . . 86217.5.1 Pool Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86217.5.2 Some Useful Relations in Pool Boiling . . . . . . . . . . . 865

Contents xxi

17.5.3 Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87017.5.4 Heat Transfer Correlation in Flow Boiling . . . . . . . . . 870

17.6 Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87217.6.1 Laminar Mixed Convection For Flow Over

A Vertical Isothermal Flat Plate . . . . . . . . . . . . . . . . . 87217.6.2 Laminar Mixed Convection in a Parallel Plate

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87817.6.3 Laminar Mixed Convection in a Vertical Parallel

Plate Channel: Fully Developed Solution . . . . . . . . . . 87917.7 Heat Transfer in a Particle Bed . . . . . . . . . . . . . . . . . . . . . . . 884

17.7.1 Flow Characteristics of a Particle Bed . . . . . . . . . . . . 88517.7.2 Heat Transfer Characteristics of a Particle Bed . . . . . . 888

17.8 Heat Transfer in High Speed Flows . . . . . . . . . . . . . . . . . . . . 89917.8.1 Compressible Boundary Layer Flow Parallel

to a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89917.9 Current Topics of Interest in Heat Transfer . . . . . . . . . . . . . . . 90717.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909

Appendix A: Note on Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 915

Appendix B: Note on Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . 933

Appendix C: Basics of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . 939

Appendix D: Heisler Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

Appendix E: Numerical Solution of Algebraic and DifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955

Appendix F: Exponential Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969

Appendix G: Angle Factors and Mean Beam Lengths . . . . . . . . . . . . . . . 973

Appendix H: Basic Equations of Convection Heat Transfer . . . . . . . . . . 991

Appendix I: Useful Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011

xxii Contents

About the Author

Prof. S. P. Venkateshan obtained his Ph.D. from the Indian Institute of Science,Bangalore in 1977. After spending three years at Yale University, he joined IndianInstitute of Technology Madras in 1982. He has been teaching subjects related toThermal Engineering to both UG and PG students for the past 32 years. He haspublished extensively and has more than 100 publications to his credit. The areas ofhis interest are: (a) Interaction of natural convection with radiation, (b) Numericaland experimental heat transfer, (c) Heat transfer in space applications (d) Radiationheat transfer in participating media and (e) Instrumentation. Professor Venkateshanhas been a consultant to ISRO, DRDO, and BHEL in India and NASA in the US.He has three patents to his credit in the area of instrumentation. He has also guidedabout 30 scholars towards the Ph.D. and a similar number of scholars towards theM.S. (by Research) degree at IIT Madras. The present book had its beginnings inthe notes prepared for the course on Heat Transfer taught by him for several years atIIT Madras.

xxiii

Nomenclature

Note:

• Many symbols have more than one meaning. The context will indicate thespecific meaning.

• Symbols in limited use are not given here.

Latin Alphabet Symbols

A Aspect ratio of a cavity, non-dimensional; Area (m2)a Speed of sound (m=sÞB Rotational constant (1=s)Bi Biot number (non-dimensional)Cp or c Specific heat (J=kg � K or J=kg�C)C Thermal capacity (W=°C)c0 Speed of light in vacuum (m=s)Cf Friction coefficient (non-dimensional)d Diameter (m)D Diameter (m)DH Hydraulic diameter (m)E Electric field intensity (N=coul); Emissive power (W=m2) (total) or

(W=m2lm) (spectral)En Exponential integral function of order nEc Eckert number (non-dimensional)Ec Total energy stored (J)erf Error function

xxv

erfc Complementary error functionEu Euler number (non-dimensional)Fij Diffuse view factor between surface i and surface j (non-dimensional)f Frequency (Hz); Friction factor (non-dimensional); Fraction of black

body radiation between 0 and kT (non-dimensional)Fo Fourier number (non-dimensional)g Acceleration due to gravity (m=s2)G Heat generation rate (W=m3); Irradiation (W=m2)Gr Grashof number (non-dimensional)Gz Graetz number (non-dimensional)h Heat transfer coefficient (W=m2�C); Planck’s constant (J� s); Enthalpy

(J=kg)H Magnetic field intensity, Non-dimensional heat generation parameterhR Radiation heat transfer coefficient (W=m2�C)hsf Latent heat of melting/solidification (J=kg)I Moment of inertia (kg �m2)I Radiation intensity (W=m2 � sr) (total); Radiation intensity (W=m2� lm �

sr) (spectral)J Radiosity (W=m2)J Rotational quantum numberk Boltzmann constant (kJ=kmol � K)k Thermal conductivity (W=m�C or W=m � K)L Length (m)LMTD Logarithmic mean temperature difference (�C or K)Lm Mean beam length (m)m Complex index of refraction; Fin parameter for a uniform area fin (m�1);

Mass (kg)M Mach number (non-dimensional); Reduced mass (kg)N Environmental parameter (non-dimensional)n Refractive indexn! Unit normal vectorNRC Radiation conduction interaction parameter (non-dimensional)NTU Number of transfer units (non-dimensional)Nu Nusselt number (non-dimensional)p Fin parameter for a variable area fin (m�1

2)P Perimeter (m); Power (W)p Pressure (bar or Pa)Pe Peclet number (non-dimensional)Pr Prandtl number (non-dimensional)q Heat flux (W=m2)q! Heat flux vector (W=m2)Q Total heat transfer (W)R Electrical resistance (X)r Radial coordinate or radius (m); Recovery factor (non-dimensional)

xxvi Nomenclature

R Radius (m); Ramp rate (°C=s); Capacity ratio (non-dimensional);Thermal resistance (�C=W or m2�C=W)

Ra Rayleigh number (non-dimensional)Re Reynolds number (non-dimensional)Rf Fouling resistance (�C=W or m2�C=W)S Conduction shape factor (non-dimensional); Solar constant (W=m2);

Surface area (m2)SD Diagonal pitch (m)SL Longitudinal pitch (m)St Stanton number (non-dimensional)ST Transverse pitch (m)Ste Stefan number (non-dimensional)T Temperature (�C or K)t Time (s)t Transmittivity or transmittancet� Time lag (s); Charging time (s)Tm Melting temperature (�C or K)Tref Reference temperature (�C or K)U Free stream velocity (m=s); Overall heat transfer coefficient (W=m2�C)u x component of velocity, (m/s)V Potential energy (J)v y component of velocity (m=s)v Vibrational quantum numberV Volume (m3)x x-coordinate (m)y y-coordinate (m)z z-coordinate (m)

Greek Symbols

a Absorptivity (no unit); Thermal diffusivity (m2=s)b Isobaric volumetric expansion coefficient (K�1); Wedge angle (rad or �)c Ratio of specific heats (no unit); Surface tension (N=m)d Boundary layer thickness (m); Condensate layer thickness (m); Depth of

penetration (m); Phase angle (rad)e Eddy viscosity (kg=m � s); Effectiveness of a fin array (no unit); Emissivity

(no unit); Heat exchanger effectiveness (no unit)eH Eddy diffusivity of heat (m2=s)/ Sub-cooling parameter (non-dimensional)g Blasius similarity variable (non-dimensional); Fin efficiency

(non-dimensional); Heat exchanger effectiveness (non-dimensional)

Nomenclature xxvii

gR Radiating fin efficiency (non-dimensional)j Absorption coefficient (m�1)k Vacuum wavelength (m)l Cosine of angle with respect to normal (no unit); Dynamic viscosity

(kg=m � s); Non-dimensional fin parameter (no unit)m Kinematic viscosity (m2=s); Photon frequency (Hz)h Angle (� or rad); Characteristic rotational temperature (K); Non-dimensional

temperature (no unit)href Environmental parameter (non dimensional)q Density (kg=m3); Reflectivity (no unit); Resistivity (X�m)r Stefan–Boltzmann constant (W=m2K4); Surface tension (N=m)¿ Characteristic time (s); Optical thickness (no unit); Shear stress (Pa); Time

constant (s); Transmittivity (no unit)ˆ Stream function (s�1); Film resistance number (non-dimensional); Radiation

number (non-dimensional)x Circular frequency (rad=s)X Solid angle (sr)

Subscripts

amb Pertaining to the ambientc Pertaining to convectionch Based on a characteristic length scalef Liquidfg Liquid-vapori Pertaining to insulation layerk Pertaining to conductionsat Saturation conditionsf Solid-liquidR Pertaining to radiationr Radialw At the wall1 Pertaining to free-stream or ambient1,2, etc. Pertaining to a specific position

xxviii Nomenclature