[PDF] Hans Dieter Baehr · Karl Stephan Heat and Mass Transfer

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Hans Dieter Baehr · Karl Stephan

Heat and Mass Transfer

Hans Dieter Baehr · Karl Stephan

Heatand

MassTransfer

Second, revised Edition

With 327 Figures

123

Dr.-Ing. E.h. Dr.-Ing. Hans Dieter Baehr

Professor em. of Thermodynamics, University of Hannover, Germany

Dr.-Ing. E.h. mult. Dr.-Ing. Karl Stephan

Professor (em.) Institute of Thermodynamics and Thermal Process Engineering

University of Stuttgart

70550 Stuttgart

Germany

e-mail: stephan@itt.uni-stuttgart.de

Library of Congress Control Number: 2006922796

ISBN-10 3-540-29526-7 Second Edition Springer Berlin Heidelberg New York ISBN-13 978-3-540-29526-6 Second Edition Springer Berlin Heidelberg New York ISBN 3-540-63695-1 First Edition Springer-Verlag Berlin Heidelberg New York

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Preface to the second edition

In this revised edition of our book we retained its concept: The main empha- sis is placed on the fundamental principles of heat and mass transfer and their application to practical problems of process modelling and the apparatus design. Like the first edition, the second edition contains five chapters and several appendices, particularly a compilation of thermophysical property data needed for the solution of problems. Changes are made in those chapters presenting heat and mass transfer correlations based on theoretical results or experimental findings. They were adapted to the most recent state of our knowledge. Some of the worked examples, which should help to deepen the comprehension of the text, were revised or updated as well. The compilation of the thermophysical property data was revised and adapted to the present knowledge. Solving problems is essential for a sound understanding and for relating prin- ciples to real engineering situations. Numerical answers and hints to the solution of problems are given in the final appendix. The new edition also enabled us to correct printing errors and mistakes. In preparing the new edition we were assisted by Jens K

¨orber, who helped

us to submit a printable version of the manuscript to the publisher. We owe him sincere thanks. We also appreciate the efforts of friends and colleagues who provided their good advice with constructive suggestions.

Bochum and Stuttgart,H.D. Baehr

March 2006K. Stephan

Preface to the first edition

This book is the English translation of our German publication, which appeared in

1994 with the title "W

¨arme und Stoff¨ubertragung" (2nd edition Berlin: Springer Verlag 1996). The German version originated from lecture courses in heat and mass transfer which we have held for many years at the Universities of Hannover and Stuttgart, respectively. Our book is intended for students of mechanical and chemical engineering at universities and engineering schools, but will also be of use to students of other subjects such as electrical engineering, physics and chemistry. Firstly our book should be used as a textbook alongside the lecture course. Its intention is to make the student familiar with the fundamentals of heat and mass transfer, and enable him to solve practical problems. On the other hand we placed special emphasis on a systematic development of the theory of heat and mass transfer and gave extensive discussions of the essential solution methods for heat and mass transfer problems. Therefore the book will also serve in the advanced training of practising engineers and scientists and as a reference work for the solution of their tasks. The material is explained with the assistance of a large number of calculated examples, and at the end of each chapter a series of exercises is given. This should also make self study easier. Many heat and mass transfer problems can be solved using the balance equa- tions and the heat and mass transfer coefficients, without requiring too deep a knowledge of the theory of heat and mass transfer. Such problems are dealt with in the first chapter, which contains the basic concepts and fundamental laws of heat and mass transfer. The student obtains an overview of the different modes of heat and mass transfer, and learns at an early stage how to solve practical problems and to design heat and mass transfer apparatus. This increases the mo- tivation to study the theory more closely, which is the object of the subsequent chapters. In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. viii Preface The third chapter covers convective heat and mass transfer. The derivation of the mass, momentum and energy balance equations for pure fluids and multi- component mixtures are treated first, before the material laws are introduced and the partial differential equations for the velocity, temperature and concentration fields are derived. As typical applications we consider heat and mass transfer in flow over bodies and through channels, in packed and fluidised beds as well as free convection and the superposition of free and forced convection. Finally an introduction to heat transfer in compressible fluids is presented. In the fourth chapter the heat and mass transfer in condensation and boil- ing with free and forced flows is dealt with. The presentation follows the book, "Heat Transfer in Condensation and Boiling" (Berlin: Springer-Verlag 1992) by K. Stephan. Here, we consider not only pure substances; condensation and boiling in mixtures of substances are also explained to an adequate extent. Thermal radiation is the subject of the fifth chapter. It differs from many other presentations in so far as the physical quantities needed for the quantita- tive description of the directional and wavelength dependency of radiation are extensively presented first. Only after a strict formulation of Kirchhoff"s law, the ideal radiator, the black body, is introduced. After this follows a discussion of the material laws of real radiators. Solar radiation and heat transfer by radiation are considered as the main applications. An introduction to gas radiation, important technically for combustion chambers and furnaces, is the final part of this chapter. As heat and mass transfer is a subject taught at a level where students have already had courses in calculus, we have presumed a knowledge of this field. Those readers who only wish to understand the basic concepts and become familiar with simple technical applications of heat and mass transfer need only study the first chapter. More extensive knowledge of the subject is expected of graduate mechanical and chemical engineers. The mechanical engineer should be familiar with the fundamentals of heat conduction, convective heat transfer and radiative transfer, as well as having a basic knowledge of mass transfer. Chemical engineers also require, in addition to a sound knowledge of these areas, a good understanding of heat and mass transfer in multiphase flows. The time set aside for lectures is generally insufficient for the treatment of all the material in this book. However, it is important that the student acquires a broad understanding of the fundamentals and methods. Then it is sufficient to deepen this knowledge with selected examples and thereby improve problem solving skills. In the preparation of the manuscript we were assisted by a number of our colleagues, above all by Nicola Jane Park, MEng., University of London, Imperial College of Science, Technology and Medicine. We owe her sincere thanks for the translation of our German publication into English, and for the excellent cooperation.

Hannover and Stuttgart,H.D. Baehr

Spring 1998K. Stephan

Contents

Nomenclaturexvi

1 Introduction. Technical Applications1

1.1 Thedifferenttypesofheattransfer...................... 1

1.1.1 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Steady, one-dimensional conduction of heat . . . . . . . . . . . . . 5

1.1.3 Convective heat transfer. Heat transfer coefficient . . . . . . . . . 10

1.1.4 Determining heat transfer coefficients. Dimensionless numbers . . 15

1.1.5 Thermalradiation........................... 25

1.1.6 Radiativeexchange .......................... 27

1.2 Overallheattransfer.............................. 30

1.2.1 Theoverallheattransfercoefficient ................. 30

1.2.2 Multi-layerwalls............................ 32

1.2.3 Overall heat transfer through walls with extended surfaces . . . . 33

1.2.4 Heatingandcoolingofthinwalledvessels.............. 37

1.3 Heatexchangers ................................ 40

1.3.1 Types of heat exchanger and flow configurations . . . . . . . . . . 40

1.3.2 General design equations. Dimensionless groups . . . . . . . . . . 44

1.3.3 Countercurrent and cocurrentheatexchangers ........... 49

1.3.4 Crossflowheatexchangers....................... 56

1.3.5 Operating characteristics of further flow configurations. Diagrams 63

1.4 Thedifferenttypesofmasstransfer ..................... 64

1.4.1 Diffusion ................................ 66

1.4.1.1 Compositionofmixtures................... 66

1.4.1.2 Diffusivefluxes ........................ 67

1.4.1.3 Fick"slaw........................... 70

1.4.2 Diffusion through a semipermeable plane. Equimolar diffusion . . 72

1.4.3 Convectivemasstransfer ....................... 76

1.5 Masstransfertheories............................. 80

1.5.1 Filmtheory............................... 80

1.5.2 Boundary layer theory . . . . . . . . . . . . . . . . . . . . . . . . . 84

1.5.3 Penetrationandsurfacerenewaltheories .............. 86

1.5.4 Application of film theory to evaporative cooling . . . . . . . . . . 87

xContents

1.6 Overallmasstransfer ............................. 91

1.7 Masstransferapparatus............................ 93

1.7.1 Materialbalances ........................... 94

1.7.2 Concentration profiles and heights of mass transfer columns . . . . 97

1.8 Exercises .................................... 101

2 Heat conduction and mass diffusion105

2.1 The heat conduction equation . . . . . . . . . . . . . . . . . . . . . . . . 105

2.1.1 Derivation of the differential equation for the temperature field . . 106

2.1.2 The heat conduction equation for bodies with constant

materialproperties........................... 109

2.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 111

2.1.4 Temperaturedependentmaterialproperties............. 114

2.1.5 Similartemperaturefields....................... 115

2.2 Steady-state heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . 119

2.2.1 Geometric one-dimensional heat conduction with heat sources . . 119

2.2.2 Longitudinal heat conduction in a rod . . . . . . . . . . . . . . . . 122

2.2.3 The temperature distribution in fins and pins . . . . . . . . . . . . 127

2.2.4 Finefficiency.............................. 131

2.2.5 Geometricmulti-dimensionalheatflow................ 134

2.2.5.1 Superposition of heat sources and heat sinks . . . . . . . . 135

2.2.5.2 Shapefactors......................... 139

2.3 Transient heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

2.3.1 Solutionmethods ........................... 141

2.3.2 TheLaplacetransformation...................... 142

2.3.3 The semi-infinite solid . . . . . . . . . . . . . . . . . . . . . . . . . 149

2.3.3.1 Heating and cooling with different boundary conditions . 149

2.3.3.2 Two semi-infinite bodies in contact with each other . . . . 154

2.3.3.3 Periodictemperaturevariations............... 156

2.3.4 Cooling or heating of simple bodies in one-dimensional heat flow . 159

2.3.4.1 Formulationoftheproblem ................. 159

2.3.4.2 Separatingthevariables................... 161

2.3.4.3 Resultsfortheplate ..................... 163

2.3.4.4 Results for the cylinder and the sphere . . . . . . . . . . . 167

2.3.4.5 Approximation for large times: Restriction to the first

termintheseries....................... 169

2.3.4.6 Asolutionforsmalltimes.................. 171

2.3.5 Cooling and heating in multi-dimensional heat flow . . . . . . . . 172

2.3.5.1 Productsolutions....................... 172

2.3.5.2 Approximation for small Biot numbers . . . . . . . . . . . 175

2.3.6 Solidification of geometrically simple bodies . . . . . . . . . . . . . 177

2.3.6.1 The solidification of flat layers (Stefan problem) . . . . . . 178

2.3.6.2 The quasi-steady approximation . . . . . . . . . . . . . . 181

2.3.6.3 Improvedapproximations .................. 184

2.3.7 Heatsources .............................. 185

Contents xi

2.3.7.1 Homogeneousheatsources.................. 186

2.3.7.2 Pointandlinearheatsources ................ 187

2.4 Numerical solutions to heat conduction problems . . . . . . . . . . . . . . 192

2.4.1 The simple, explicit difference method for transient heat conduction

problems ................................ 193

2.4.1.1 Thefinitedifferenceequation ................ 193

2.4.1.2 The stability condition . . . . . . . . . . . . . . . . . . . . 195

2.4.1.3 Heatsources.......................... 196

2.4.2 Discretisation of the boundary conditions . . . . . . . . . . . . . . 197

2.4.3 The implicit difference method from J. Crank and P. Nicolson . . 203

2.4.4 Noncartesian coordinates. Temperature dependent material

properties................................ 206

2.4.4.1 The discretisation of the self-adjoint differential operator . 207

2.4.4.2 Constant material properties. Cylindrical coordinates . . 208

2.4.4.3 Temperature dependent material properties . . . . . . . . 209

2.4.5 Transient two- and three-dimensional temperature fields . . . . . . 211

2.4.6 Steady-statetemperaturefields.................... 214

2.4.6.1 A simple finite difference method for plane, steady-state

temperaturefields ...................... 214

2.4.6.2 Consideration of the boundary conditions . . . . . . . . . 217

2.5 Massdiffusion ................................. 222

2.5.1 Remarksonquiescentsystems .................... 222

2.5.2 Derivation of the differential equation for the concentration field . 225

2.5.3 Simplifications............................. 230

2.5.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 231

2.5.5 Steady-state mass diffusion with catalytic surface reaction . . . . . 234

2.5.6 Steady-state mass diffusion with homogeneous chemical reaction . 238

2.5.7 Transientmassdiffusion........................ 242

2.5.7.1 Transient mass diffusion in a semi-infinite solid . . . . . . 243

2.5.7.2 Transient mass diffusion in bodies of simple geometry

withone-dimensionalmassflow............... 244

2.6 Exercises .................................... 246

3 Convective heat and mass transfer. Single phase flow253

3.1 Preliminary remarks: Longitudinal,frictionless flow over a flat plate . . . 253

3.2 Thebalanceequations............................. 258

3.2.1 Reynolds"transporttheorem ..................... 258

3.2.2 Themassbalance ........................... 260

3.2.2.1 Puresubstances........................ 260

3.2.2.2 Multicomponentmixtures.................. 261

3.2.3 Themomentumbalance........................ 264

3.2.3.1 Thestresstensor....................... 266

3.2.3.2 Cauchy"sequationofmotion................. 269

3.2.3.3 Thestraintensor....................... 270

xii Contents

3.2.3.4 Constitutive equations for the solution of the

momentumequation..................... 272

3.2.3.5 TheNavier-Stokesequations................. 273

3.2.4 Theenergybalance .......................... 274

3.2.4.1 Dissipatedenergyandentropy ............... 279

3.2.4.2 Constitutive equations for the solution of the energy

equation............................ 281

3.2.4.3 Some other formulations of the energy equation . . . . . . 282

3.2.5 Summary................................ 285

3.3 Influence of the Reynolds number on the flow . . . . . . . . . . . . . . . . 287

3.4 SimplificationstotheNavier-Stokesequations ............... 290

3.4.1 Creepingflows ............................. 290

3.4.2 Frictionlessflows............................ 291

3.4.3 Boundary layer flows . . . . . . . . . . . . . . . . . . . . . . . . . 291

3.5 The boundary layer equations . . . . . . . . . . . . . . . . . . . . . . . . 293

3.5.1 The velocity boundary layer . . . . . . . . . . . . . . . . . . . . . 293

3.5.2 The thermal boundary layer . . . . . . . . . . . . . . . . . . . . . 296

3.5.3 The concentration boundary layer . . . . . . . . . . . . . . . . . . 300

3.5.4 General comments on the solution of boundary layer equations . . 300

3.6 Influence of turbulence on heat and mass transfer . . . . . . . . . . . . . 304

3.6.1 Turbulentflowsnearsolidwalls.................... 308

3.7 Externalforcedflow.............................. 312

3.7.1 Parallelflowalongaflatplate .................... 313

3.7.1.1 Laminar boundary layer . . . . . . . . . . . . . . . . . . . 313

3.7.1.2 Turbulentflow ........................ 325

3.7.2 Thecylinderincrossflow ....................... 330

3.7.3 Tube bundles in crossflow . . . . . . . . . . . . . . . . . . . . . . . 334

3.7.4 Some empirical equations for heat and mass transfer in

externalforcedflow .......................... 338

3.8 Internalforcedflow .............................. 341

3.8.1 Laminarflowincirculartubes .................... 341

3.8.1.1 Hydrodynamic, fully developed, laminar flow . . . . . . . 342

3.8.1.2 Thermal, fully developed, laminar flow . . . . . . . . . . . 344

3.8.1.3 Heat transfer coefficients in thermally fully developed,

laminarflow.......................... 346

3.8.1.4 The thermal entry flow with fully developed velocity

profile ............................. 349

3.8.1.5 Thermally and hydrodynamically developing flow . . . . . 354

3.8.2 Turbulentflowincirculartubes ................... 355

3.8.3 Packedbeds .............................. 357

3.8.4 Fluidisedbeds ............................. 361

3.8.5 Some empirical equations for heat and mass transfer in flow

through channels, packed and fluidised beds . . . . . . . . . . . . . 370

3.9 Freeflow .................................... 373

Contents xiii

3.9.1 Themomentumequation....................... 376

3.9.2 Heat transfer in laminar flow on a vertical wall . . . . . . . . . . . 379

3.9.3 Some empirical equations for heat transfer in free flow . . . . . . . 384

3.9.4 Masstransferinfreeflow....................... 386

3.10Overlappingoffreeandforcedflow...................... 387

3.11Compressibleflows............................... 389

3.11.1Thetemperaturefieldinacompressibleflow ............ 389

3.11.2Calculationofheattransfer...................... 396

3.12Exercises .................................... 399

4 Convective heat and mass transfer. Flows with phase change405

4.1 Heattransferincondensation......................... 405

4.1.1 Thedifferenttypesofcondensation ................. 406

4.1.2 Nusselt"sfilmcondensationtheory.................. 408

4.1.3 Deviations from Nusselt"s film condensation theory . . . . . . . . . 412

4.1.4 Influence of non-condensable gases . . . . . . . . . . . . . . . . . . 416

4.1.5 Filmcondensationinaturbulentfilm ................ 422

4.1.6 Condensationofflowingvapours................... 426

4.1.7 Dropwisecondensation ........................ 431

4.1.8 Condensationofvapourmixtures................... 435

4.1.8.1 The temperature at the phase interface . . . . . . . . . . . 439

4.1.8.2 The material and energy balance for the vapour . . . . . . 443

4.1.8.3 Calculatingthesizeofacondenser............. 445

4.1.9 Someempiricalequations....................... 446

4.2 Heat transfer in boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

4.2.1 Thedifferenttypesofheattransfer.................. 449

4.2.2 The formation of vapour bubbles . . . . . . . . . . . . . . . . . . . 453

4.2.3 Bubble frequency and departure diameter . . . . . . . . . . . . . . 456

4.2.4 Boiling in free flow. The Nukijama curve . . . . . . . . . . . . . . 460

4.2.5 Stability during boiling in free flow . . . . . . . . . . . . . . . . . . 461

4.2.6 Calculation of heat transfer coefficients for boiling in free flow . . 465

4.2.7 Some empirical equations for heat transfer during nucleate

boiling in free flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

4.2.8 Two-phaseflow............................. 472

4.2.8.1 Thedifferentflowpatterns.................. 473

4.2.8.2 Flowmaps........................... 475

4.2.8.3 Some basic terms and definitions . . . . . . . . . . . . . . 476

4.2.8.4 Pressuredropintwo-phaseflow............... 479

4.2.8.5 The different heat transfer regions in two-phase flow . . . 487

4.2.8.6 Heat transfer in nucleate boiling and convective

evaporation.......................... 489

4.2.8.7 Critical boiling states . . . . . . . . . . . . . . . . . . . . . 492

4.2.8.8 Some empirical equations for heat transfer in two-phase

flow .............................. 495

4.2.9 Heat transfer in boiling mixtures . . . . . . . . . . . . . . . . . . . 496

xiv Contents

4.3 Exercises .................................... 501

5 Thermal radiation503

5.1 Fundamentals. Physical quantities . . . . . . . . . . . . . . . . . . . . . . 503

5.1.1 Thermalradiation........................... 504

5.1.2 Emissionofradiation ......................... 506

5.1.2.1 Emissivepower........................ 506

5.1.2.2 Spectralintensity....................... 507

5.1.2.3 Hemispherical spectral emissive power and total intensity 509

5.1.2.4 Diffuse radiators. Lambert"s cosine law . . . . . . . . . . . 513

5.1.3 Irradiation ............................... 514

5.1.4 Absorptionofradiation........................ 517

5.1.5 Reflectionofradiation......................... 522

5.1.6 Radiation in an enclosure. Kirchhoff"s law . . . . . . . . . . . . . 524

5.2 Radiationfromablackbody ......................... 527

5.2.1 Definition and realisation of a black body . . . . . . . . . . . . . . 527

5.2.2 The spectral intensity and the spectral emissive power . . . . . . . 528

5.2.3 The emissive power and the emission of radiation in a wavelength

interval................................. 532

5.3 Radiationpropertiesofrealbodies...................... 537

5.3.1 Emissivities............................... 537

5.3.2 The relationships between emissivity, absorptivity and reflectivity.

ThegreyLambertradiator...................... 540

5.3.2.1 ConclusionsfromKirchhoff"slaw.............. 540

5.3.2.2 Calculation of absorptivitiesfromemissivities....... 541

5.3.2.3 ThegreyLambertradiator ................. 542

5.3.3 Emissivitiesofrealbodies....................... 544

5.3.3.1 Electricalinsulators ..................... 545

5.3.3.2 Electrical conductors (metals) . . . . . . . . . . . . . . . . 548

5.3.4 Transparentbodies........................... 550

5.4 Solarradiation................................. 555

5.4.1 Extraterrestrialsolarradiation.................... 555

5.4.2 The attenuation of solar radiation in the earth"s atmosphere . . . 558

5.4.2.1 Spectraltransmissivity.................... 558

5.4.2.2 Molecularandaerosolscattering .............. 561

5.4.2.3 Absorption .......................... 562

5.4.3 Directsolarradiationontheground................. 564

5.4.4 Diffuse solar radiation and global radiation . . . . . . . . . . . . . 566

5.4.5 Absorptivitiesforsolarradiation................... 568

5.5 Radiativeexchange .............................. 569

5.5.1 Viewfactors .............................. 570

5.5.2 Radiative exchange between black bodies . . . . . . . . . . . . . . 576

5.5.3 Radiative exchange between grey Lambert radiators . . . . . . . . 579

5.5.3.1 The balance equations according to the net-radiation

method ............................ 580

Contents xv

5.5.3.2 Radiative exchange between a radiation source, a radiation

receiver and a reradiating wall . . . . . . . . . . . . . . . . 581

5.5.3.3 Radiative exchange in a hollow enclosure with two zones . 585

5.5.3.4 The equation system for the radiative exchange between

anynumberofzones..................... 587

5.5.4 Protectiveradiationshields...................... 590

5.6 Gasradiation.................................. 594

5.6.1 Absorptioncoefficientandopticalthickness............. 595

5.6.2 Absorptivityandemissivity...................... 597

5.6.3 Resultsfortheemissivity....................... 600

5.6.4 Emissivities and mean beam lengths of gas spaces . . . . . . . . . 603

5.6.5 Radiative exchange in a gas filled enclosure . . . . . . . . . . . . . 607

5.6.5.1 Black, isothermal boundary walls . . . . . . . . . . . . . . 607

5.6.5.2 Grey isothermal boundary walls . . . . . . . . . . . . . . . 608

5.6.5.3 Calculation of the radiative exchange in complicated cases 611

5.7 Exercises .................................... 612

Appendix A: Supplements617

A.1Introductiontotensornotation........................ 617 A.2 Relationship between mean and thermodynamic pressure . . . . . . . . . 619 A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity incartesiancoordinates............................ 620 A.4 Navier-Stokes equations for an incompressible fluid of constant viscosity incylindricalcoordinates ........................... 621 A.5Entropybalanceformixtures......................... 622 A.6Relationshipbetweenpartialandspecificenthalpy............. 623

A.7 Calculation of the constantsa

n of a Graetz-Nusselt problem (3.246) . . . 624

Appendix B: Property data626

Appendix C: Solutions to the exercises640

Literature654

Index671

Nomenclature

Symbol Meaning SI units

Aarea m

2 A m average area m 2 A q cross sectional area m 2 A f fin surface area m 2 athermal diffusivity m 2 /s ahemispherical total absorptivity - a λ spectral absorptivity - a ? λ directional spectral absorptivity - a t turbulent thermal diffusivity m 2 /s a ? specific surface area m 2 /m 3 bthermal penetration coefficient,b= ⎷

λc?Ws

1/2 /(m 2 K) bLaplace constant,b= ?

2σ/g(?

L -? G )m

Ccircumference, perimeter m

Cheat capacity flow ratio -

cspecific heat capacity J/(kg K) cconcentration mol/m 3 cpropagation velocity of electromagnetic waves m/s c 0 velocity of light in a vacuum m/s c f friction factor - c p specific heat capacity at constant pressure J/(kg K) c R resistance factor -

Dbinary diffusion coefficient m

2 /s D t turbulent diffusion coefficient m 2 /s ddiameter m d A departure diameter of vapour bubbles m d h hydraulic diameter m

Eirradiance W/m

2 E 0 solar constant W/m 2 E λ spectral irradiance W/(m 2

μm)

eunit vector -

Fforce N

Nomenclature xvii

F B buoyancy force N F f friction force N F R resistance force N F ij view factor between surfacesiandj-

F(0,λT) fraction function of black radiation -

ffrequency of vapour bubbles 1/s f j force per unit volume N/m 3 gacceleration due to gravity m/s 2

Hheight m

Hradiosity W/m

2

Henthalpy J



Henthalpy flow J/s

hPlanck constant J s hspecific enthalpy J/kg h tot specific total enthalpy,h tot =h+w 2 /2J/kg h i partial specific enthalpy J/kg h v specific enthalpy of vaporization J/kg   h v molar enthalpy of vaporization J/mol

Imomentum kg m/s

Idirectional emissive power W/(m

2 sr) jdiffusional flux mol/(m 2 s) j ? diffusional flux in a centre of gravity system kg/(m 2 s) u jdiffusional flux in a particle based system mol/(m 2 s)

Kincident intensity W/(m

2 sr) K λ incident spectral intensity W/(m 2

μm)

koverall heat transfer coefficient W/(m 2 K) kextinction coefficient - kBoltzmann constant J/K k G spectral absorption coefficient 1/m k H

Henry coefficient N/m

2 k j force per unit mass N/kg k 1 rate constant for a homogeneous first order reaction 1/s k ? 1 ,k ?? 1 rate constant for a homogeneous (heterogeneous) first order reaction m/s k ?? n rate constant for a heterogeneous mol/(m 2 s) n-th order reaction (mol/m 3 ) n

Llength m

Ltotal intensity W/(m

2 sr) L λ spectral intensity W/(m 2

μmsr)

L 0 reference length m L S solubility mol/(m 3 Pa) xviii Nomenclature llength, mixing length m

Mmass kg

Mmodulus,M=aΔt/Δx

2 -

M(hemispherical total) emissive power W/m

2 M λ spectral emissive power W/(m 2

μm)

Mmass flow rate kg/s

˜Mmolecular mass, molar mass kg/mol

moptical mass kg/m 2 m r relative optical mass - mmass flux kg/(m 2 s)

Namount of substance mol

N i dimensionless transfer capability (number of transfer units) of the material streami-

Nmolar flow rate mol/s

nrefractive index - nnormal vector - nmolar flux mol/(m 2 s)

Ppower W

P diss dissipated power W ppressure Pa p + dimensionless pressure -

Qheat J

Qheat flow W

qheat flux W/m 2

Rradius m

R cond resistance to thermal conduction K/W R m molar (universal) gas constant J/(mol K) rradial coordinate m rhemispherical total reflectivity - r λ spectral reflectivity - r ?λ directional spectral reflectivity - r e electrical resistivity Ω m r + dimensionless radial coordinate - rreaction rate mol/(m 3 s)

Ssuppression factor in convective boiling -

Sentropy J/K

sspecific entropy J/(kg K) sLaplace transformation parameter 1/s sbeam length m sslip factor,s=w G /w L - s l longitudinal pitch m

Nomenclature xix

s q transverse pitch m

Tthermodynamic temperature K

T e eigentemperature K T St stagnation point temperature K ttime s t + dimensionless time - t k cooling time s t j stress vector N/m 2 t R relaxation time,t R =1/k 1 s t D relaxation time of diffusion,t D =L 2 /Ds

Uinternal energy J

uaverage molar velocity m/s uspecific internal energy J/kg uLaplace transformed temperature K

Vvolume m

3 V A departure volume of a vapour bubble m 3 vspecific volume m 3 /kg

Wwork J



Wpower density W/m

3 W i heat capacity flow rate of a fluidiW/K wvelocity m/s w 0 reference velocity m/s w S velocity of sound m/s w τ shear stress velocity,w τ = ? τ 0 /?m/s w ? fluctuation velocity m/s w + dimensionless velocity - Xmoisture content; Lockhart-Martinelli parameter - 

Xmolar content in the liquid phase -

xcoordinate m xmole fraction in the liquid - x + dimensionlessx-coordinate - x ? quality,x ? =  M G / M L - x ? th thermodynamic quality - 

Ymolar content in the gas phase -

ycoordinate m ymole fraction in the gas phase - y + dimensionlessy-coordinate - znumber - zaxial coordinate m z + dimensionlessz-coordinate - z R number of tube rows - xx Nomenclature

Greek letters

Symbol Meaning SI units

αheat transfer coefficient W/(m

2 K) α m mean heat transfer coefficient W/(m 2 K)

βmass transfer coefficient m/s

β m mean mass transfer coefficient m/s

βthermal expansion coefficient 1/K

βpolar angle, zenith angle rad

β 0 base angle rad

Γ mass production rate kg/(m

3 s)

γmolar production rate mol/(m

3 s)

Δ difference -

δthickness; boundary layer thickness m

δ ij

Kronecker symbol -

εvolumetric vapour content -

ε ? volumetric quality -

εhemispherical total emissivity -

ε λ hemispherical spectral emissivity - ε ?λ directional spectral emissivity - ε D turbulent diffusion coefficient m 2 /s ε i dimensionless temperature change of the material streami- ε ii dilatation 1/s ε ji strain tensor 1/s ε p void fraction - ε t turbulent viscosity m 2 /s

ζresistance factor -

ζbulk viscosity kg/(m s)

ηdynamic viscosity kg/(m s)

η f fin efficiency -

Θ overtemperature K

?temperature K ? + dimensionless temperature -

κisentropic exponent -

κ G optical thickness of a gas beam -

Λ wave length of an oscillation m

λwave length m

λthermal conductivity W/(K m)

λ t turbulent thermal conductivity W/(K m)

μdiffusion resistance factor -

νkinematic viscosity m

2 /s

Nomenclature xxi

νfrequency 1/s

?density kg/m 3

σStefan-Boltzmann constant W/(m

2 K 4 )

σinterfacial tension N/m

ξmass fraction -

τtransmissivity -

τ λ spectral transmissivity -

τshear stress N/m

2 τ ji shear stress tensor N/m 2

Φ radiative power, radiation flow W

Φ viscous dissipation W/m

3 ?angle, circumferential angle rad

Ψ stream function m

2 /s

ωsolid angle sr

ωreference velocity m/s

ωpower density W/m

2

Subscripts

Symbol Meaning

A air, substance A

abs absorbed

B substance B

C condensate, cooling medium

diss dissipated

E excess, product, solidification

e exit, outlet eff effective eq equilibrium

F fluid, feed

f fin, friction G gas g geodetic, base material

I at the phase interface

i inner, inlet id ideal in incident radiation, irradiation

K substance K

L liquid

lam laminar m mean, molar (based on the amount of substance) max maximum min minimum xxii Nomenclature nnormal direction o outer, outside

Pparticle

ref reflected, reference state

S solid, bottom product, sun, surroundings

s black body, saturation tot total trans transmitted turb turbulent uin particle reference system

V boiler

W wall, water

αstart

δat the pointy=δ

λspectral

ωend

0 reference state; at the pointy=0

∞at a great distance; in infinity

Dimensionless numbers

Ar=[(?

S -? F )/? F ]?d 3P g/ν 2 ?Archimedes number

Bi=αL/λBiot number

Bi D =βL/DBiot number for mass transfer

Bo=q/(mΔh

v ) boiling number Da=k ??1 L/DDamk¨ohler number (for 1st order heterogeneous reaction) Ec=w 2 /(c p

Δ?)Eckertnumber

Fo=at/L

2

Fourier number

Fr=w 2 /(gx) Froude number Ga=gL 3 /ν 2

Galilei number

Gr=gβΔ?L

3 /ν 2

Grashof number

Ha=?k 1 L 2 /D? 2

Hatta number

Le=a/DLewis number

Ma=w/w

S

Mach number

Nu=αL/λNusselt number

Pe=wL/aP´eclet number

Ph=h E /[c(? E -? 0 )] phase change number

Pr=ν/aPrandtl number

Ra=GrPrRayleigh number

Re=wL/νReynolds number

Sc=ν/DSchmidt number

Sh=βL/DSherwood number

St=α/(w?c

p ) Stanton number

St=1/PhStefan number

1 Introduction. Technical Applications

In this chapter the basic definitions and physical quantities needed to describe heat and mass transfer will be introduced, along with the fundamental laws of these processes. They can already be used to solve technical problems, such as the transfer of heat between two fluids separated by a wall, or the sizing of appa- ratus used in heat and mass transfer. The calculation methods presented in this introductory chapter will be relatively simple, whilst a more detailed presentation of complex problems will appear in the following chapters.

1.1 The different types of heat transfer

In thermodynamics, heat is defined as the energy that crosses the boundary of a system when this energy transport occurs due to a temperature difference between the system and its surroundings, cf. [1.1], [1.2]. The second law of thermodynamics states that heat always flows over the boundary of the system in the direction of falling temperature. However, thermodynamics does not state how the heat transferred depends on this temperature driving force, or how fast or intensive this irreversible process is. It is the task of the science of heat transfer to clarify the laws of this process. Three modes of heat transfer can be distinguished: conduction, convection, and radiation. The following sections deal with their basic laws, more in depth in- formation is given in chapter 2 for conduction, 3 and 4 for convection and 5 for radiation. We limit ourselves to a phenomenological description of heat transfer processes, using the thermodynamic concepts of temperature, heat, heat flow and heat flux. In contrast to thermodynamics, which mainly deals with homogeneous systems, the so-calledphases, heat transfer is a continuum theory which deals with fields extended in space and also dependent on time. This has consequences for the concept of heat, which in thermodynamics is defined as energy which crosses the system boundary. In heat transfer one speaks of a heat flow also within the body. This contradiction with thermodynamic terminology can be resolved by considering that in a continuum theory the mass and volume elements of the body are taken to be small systems, between which energy can be transferred as heat. Therefore, when one speaks of heat flow within

2 1 Introduction. Technical Applications

a solid body or fluid, or of the heat flux vector field in conjunction with the temperature field, the thermodynamic theory is not violated. As in thermodynamics, the thermodynamic temperatureTis used in heat transfer. However with the exception of radiative heat transfer the zero point of the thermodynamic temperature scale is not needed, usually onlytemperature differencesare important. For this reason a thermodynamic temperature with an adjusted zero point, an example being the Celsius temperature, is used. These thermodynamic temperature differences are indicated by the symbol?, defined as ?:=T-T 0 (1.1) whereT 0 can be chosen arbitrarily and is usually set at a temperature that best fits the problem that requires solving. WhenT 0 = 273.15 K then?will be the

Celsius temperature. The value forT

0 does not normally need to be specified as temperature differences are independent ofT 0 .

1.1.1 Heat conduction

Heat conduction is the transfer of energy between neighbouring molecules in a substance due to a temperature gradient. In metals also the free electrons transfer energy. In solids which do not transmit radiation, heat conduction is the only process for energy transfer. In gases and liquids heat conduction is superimposed by an energy transport due to convection and radiation. The mechanism of heat conduction in solids and fluids is difficult to understand theoretically. We do not need to look closely at this theory; it is principally used in the calculation of thermal conductivity, a material property. We will limit ourselves to the phenomenological discussion of heat conduction, using the thermodynamic quantities of temperature, heat flow and heat flux, which are sufficient to deal with most technically interesting conduction problems. The transport of energy in a conductive material is described by the vector field ofheat flux q=q(x,t).(1.2) In terms of a continuum theory the heat flux vector represents the direction and magnitude of the energy flow at a position indicated by the vectorx. It can also be dependent on timet. The heat fluxqis defined in such a way that the heat flow dQthrough a surface element dAis d

Q=q(x,t)ndA=|q|cosβdA.(1.3)

Herenis the unit vector normal (outwards) to the surface, which withqforms the angleβ, Fig. 1.1. The heat flow dQis greatest whenqis perpendicular to dA makingβ= 0. The dimension of heat flow is energy/time (thermal power), with

1.1 The different types of heat transfer 3

Fig. 1.1: Surface element with normal vectorn

and heat flux vectorq SI unit J/s = W. Heat flux is the heat flow per surface area with units J/s m 2 = W/m 2 . The transport of energy by heat conduction is due to a temperature gradient in the substance. The temperature?changes with both position and time. All temperatures form a temperature field ?=?(x,t). Steady temperature fields are not dependent on the timet. One speaks of unsteady or transient temperature fields when the changes with time are important. All points of a body that are at the same temperature?, at the same moment in time, can be thought of as joined by a surface. This isothermal surface or isotherm separates the parts of the body which have a higher temperature than?,from those with a lower temperature than?. The greatest temperature change occurs normal to the isotherm, and is given by the temperature gradient grad?= ∂? x e x + ∂? y e y + ∂? z e z (1.4) wheree x ,e y ande z represent the unit vectors in the three coordinate directions. The gradient vector is perpendicular to the isotherm which goes through the point being considered and points to the direction of the greatest temperature increase.

Fig. 1.2:PointPon the isotherm

?= const with the temperature gra- dient grad?from (1.4) and the heat "ux vectorqfrom (1.5) Considering the temperature gradients as the cause of heat flow in a conductive material, it suggests that a simple proportionality between cause and e
ect may be assumed, allowing the heat "ux to be written as q=-λgrad?.(1.5)

4 1 Introduction. Technical Applications

This is J. B. Fourier"s

1 basic law for the conduction of heat, from 1822. The minus sign in this equation is accounting for the 2nd law of thermodynamics: heat "ows in the direction offallingtemperature, Fig. 1.2. The constant of proportion in (1.5) is a property of the material, the thermal conductivity

λ=λ(?,p).

It is dependent on both the temperature?and pressurep, and in mixtures on the composition. The thermal conductivityλis a scalar as long as thematerialis isotropic, which means that the ability of the material to conduct heat depends on position within the material, but for a given position not on the direction. All materials will be assumed to be isotropic, apart from a few special examples in Chapter 3, even though several materials do have thermal conductivities that depend on direction. This can be seen in wood, which conducts heat across its fibres significantly better than along them. In suchnon-isotropic mediumλis a tensor of second order, and the vectorsqand grad?form an angle in contrast to Fig. 1.2. In isotropic substances the heat "ux vector is always perpendicular to the isothermal surface. From (1.3) and (1.5) the heat "ow d 

Qthrough a surface

element dAoriented in any direction is d 

Q=-λ(grad?)ndA=-λ


n dA.(1.6) Here∂?/∂nis the derivative of?with respect to the normal (outwards) direction to the surface element. Table 1.1: Thermal conductivity of selected substances at 20 ◦

C and 100 kPa

Substanceλin W/Km

Substanceλin W/Km

Silver 427

Water 0.598

Copper 399

Hydrocarbons 0.10...0.15

Aluminium 99.2 % 209

CO 2

0.0162

Iron 81

Air 0.0257

Steel Alloys 13...48

Hydrogen 0.179

Brickwork 0.5...1.3

Krypton 0.0093

Foam Sheets 0.02...0.09

R 123 0.0090

The thermal conductivity, with SI units of W/Km, is one of the most impor- tant properties in heat transfer. Its pressure dependence must only be considered for gases and liquids. Its temperature dependence is often not very significant and can then be neglected. More extensive tables ofλare available in Appendix B, 1 Jean Baptiste Fourier (1768-1830) was Professor for Analysis at the Ecole Polytechnique in Paris and from 1807 a member of the French Academy of Science. His most important work

Th´eorie analytique de la chaleurŽ appeared in 1822. It is the first comprehensive mathematical

theory of conduction and cointains the Fourier SeriesŽ for solving boundary value problems in transient heat conduction.

1.1 The different types of heat transfer 5

Tables B1 to B8, B10 and B11. As shown in the short Table 1.1, metals have very high thermal conductivities, solids which do not conduct electricity have much lower values. One can also see that liquids and gases have especially small values forλ. The low value for foamed insulating material is because of its structure. It contains numerous small, gas-filled spaces surrounded by a solid that also has low thermal conductivity.

1.1.2 Steady, one-dimensional conduction of heat

As a simple, but practically important application, the conduction of heat inde- pendent of time, so called steady conduction, in a "at plate, in a hollow cylinder and in a hollow sphere will be considered in this section. The assumption is made that heat "ows in only one direction, perpendicular to the plate surface, and ra- dially in the cylinder and sphere, Fig. 1.3. The temperature field is then only dependent on one geometrical coordinate. This is known as one-dimensional heat conduction. Fig. 1.3: Steady, one dimensional conduction.aTemperature profile in a flat plate of thicknessδ=r 2 -r 1 ,bTemperature profile in a hollow cylinder (tube wall) or hollow sphere of inner radiusr 1 and outer radiusr 2 The position coordinate in all three cases is designated byr. The surfaces r= const are isothermal surfaces; and therefore?=?(r). We assume that?has the constant values?=? W1 , whenr=r 1 ,and?=? W2 , whenr=r 2 . These two surface temperatures shall be given. A relationship between theheat flow Qthrough the flat or curved walls, and the temperature difference? W1 -? W2 , must be found. For illustration we assume? W1 >? W2 , without loss of generality. Therefore heat "ows in the direction of increasingr. The heat flow 

Qhas a certain

value, which on the inner and outer surfaces, and on each isothermr=constis the same, as in steady conditions no energy can be stored in the wall. Fouriers law gives the following for the heat "ow 

Q=q(r)A(r)=-λ(?)

d? dr

A(r).(1.7)

In theflat wallAis not dependent onr:A=A

1 =A 2 . If the thermal conductivity is constant, then the temperature gradient d?/drwill also be constant. The steady

6 1 Introduction. Technical Applications

temperature profile in a plane wall with constantλis linear. This is not true in the case of both the cylinder and the sphere, and also ifλchanges with temperature.

In these more general cases (1.7) becomes

-λ(?)d?=  Q dr A(r) and after integrating over the wall thicknessδ=r 2 -r 1 - ? W2? ? W1

λ(?)d?=

 Q r2? r1 dr A(r) .

From the mean value theorem for integration comes

-λ m (? W2 -? W1 )=  Q  A m or  Q= λ m δ A m (? W1 -? W2 ).(1.8) The heat "ow is directly proportional to the di
erence in temperature between the two surfaces. The driving force oftemperature di
erence is analogous to the potential di
erence (voltage) in an electric circuit and soλ m A m /δis thethermal conductanceand its inverse R cond := δ  m A m (1.9) thethermal resistance. In analogy to electric circuits we get  Q=(? W1 -? W2 )/R cond .(1.10) Theaverage thermal conductivitycan easily be calculated using λ m := 1 (? W2 -? W1 ) ? W2? ? W1

λ(?)d?.(1.11)

In many cases the temperature dependence ofλcan be neglected, givingλ m =λ.

Ifλchangeslinearlywith?then

λ m = 1 2 ?

λ(?

W1 )+λ(? W2 ) ? .(1.12) This assumption is generally sucient for the region? W1 ≤?≤? W2 asλcan rarely be measured with a relative error smaller than 1 to 2%.

Theaverage areaA

m in (1.8) is defined by 1 A m := 1 r 2 -r 1 r2? r1 dr A(r) (1.13)

1.1 The different types of heat transfer 7

We have

A(r)= ?         A 1 =A 2 foraflatplate

2πLrfor a cylinder of lengthL

4πr

2 for a sphere. (1.14)

From (1.13) we get

A m = ?           A 1 =A 2 = 1 2 (A 1 +A 2 ) flat plate (A 2 -A 1 )/ln(A 2 /A 1 ) cylinder ⎷ A 1 A 2 sphere. (1.15)

The average areaA

m is given by the average of both surface areasA 1 =A(r 1 ) andA 2 =A(r 2 ). One gets the arithmetic mean for the flat plate, the logarithmic mean for the cylinder and the geometric mean for the sphere. It is known that ? A 1 A 2 ≤ A 2 -A 1 ln(A 2 /A 1 ) ≤ 1 2 (A 1 +A 2 ). For the thermal resistance to conduction it follows R cond = ?                 δ  m A flat plate ln(d 2 /d 1 )

2πLλ

m cylinder (d 2 /d 1 )-1

2πd

2 λ m sphere (1.16) The wall thickness for the cylinder (tube wall) and sphere is

δ=r

2 -r 1 = 1 2 (d 2 -d 1 ) so thatR cond can be expressed in terms of both diametersd 1 andd 2 . The temperature profile in each caseshall also be determined. We limit our- selves to the caseλ=const.WithA(r) from (1.14) and integrating -d?=  Q  dr A(r) the dimensionless temperature ratio is ?(r)-? W2 ? W1 -? W2 = ?                   r 2 -r r 2 -r 1 flat plate ln(r 2 /r) ln(r 2 /r 1 ) cylinder

1/r-1/r

2 1/r 1 -1/r 2 sphere (1.17)

8 1 Introduction. Technical Applications

Fig. 1.4: Steady temperature profile from

(1.17) in a "at, cylindrical and spherical wall of the same thicknessδand withr 2 /r 1 =3 As already mentioned the temperature change is linear in the flat plate. The cylinder has a logarithmic, and the sphere a hyperbolic temperature dependence on the radial coordinates. Fig. 1.4 shows the temperature profile according to (1.17) in walls of equal thickness. The largest deviation from the straight line by the logarithmic and hyperbolic temperature profiles appears at the pointr=r m , where the cross sectional areaA(r) assumes the valueA(r m )=A m according to (1.15). Example 1.1:A flat wall of thicknessδ=0.48 m, is made out of fireproof stone whose thermal conductivity changes with temperature. With the Celsius temperature?, between 0 ◦

C and 800

◦

C it holds that

λ(?)=

λ 0 1-b? (1.18) whereλ 0 =0.237 W/Km andb=4.41·10 -4 K -1 . The surface temperatures are? W1 = 750
◦ Cand? W2 = 150 ◦

C. The heat flux q=



Q/Aand the temperature profile in the wall

need to be calculated.

From (1.8) the heat "ux is

q= λ m δ (? W1 -? W2 ) (1.19) with the average thermal conductivity λ m = 1 ? W2 -? W1 ?W2? ?W1

λ(?)d?=

λ 0 b(? W1 -? W2 ) ln 1-b? W2 1-b? W1 .

Putting as an abbreviationλ(?

Wi )=λ i (i=1,2) we get λ m = ln(λ 1 /λ 2 ) 1 λ 2 - 1 λ 1 = ln(λ 1 /λ 2 ) λ 1 -λ 2 λ 1 λ 2 .(1.20)

The average thermal conductivityλ

m can be calculated using theλvalues for both surfaces. It is the square of the geometric mean divided by the logarithmic mean of the two values λ 1 andλ 2 . This yields from (1.18) λ 1 =λ(? W1 )=0.354W/Km,λ 2 =λ(? W2 )=0.254W/Km,

1.1 The different types of heat transfer 9

and from thatλ m =0.298 W/Km. The heat flux follows from (1.19) as q= 373W/m 2 . Under the rather inaccurate assumption thatλvaries linearly with the temperature, it would follow that λ m = 1 2 (λ 1 +λ 2 )=0.304W/Km Although this value is 1.9% too large it is still a useful approximation, as its deviation from the exact value is within the bounds of uncertainty associated with the measurement of thermal conductivity. To calculate the temperature profile in the wall we will use (1.7) as the starting point, -λ(?)d?=qdr, and withx=r-r 1 this gives -λ 0 ? ? ?W1 d? 1-b? = λ 0 b ln 1-b? 1-b? W1 =qx .

With qfrom (1.19) andλ

m from (1.20) it follows that ln 1-b? 1-b? W1 = x  ln 1-b? W2 1-b? W1 or 1-b? 1-b? W1 = ? 1-b? W2 1-b? W1 ? x/δ .

Finally using (1.18) we get

?(x)= 1 b ? 1- λ 0 λ 1 ? λ 1 λ 2 ? x/δ ? (1.21) for the equation to calculate the temperature profile in the wall.

Fig. 1.5: Steady temperature profile

?=?(x/δ) from (1.21) in a flat wall with temperature dependent thermal conductivity according to (1.18). ?is the deviation of the temperature profile from the straight line which is valid for a constant value ofλ, right hand scale. Fig. 1.5 shows?(x) and the deviation Δ?(x) from the linear temperature profile between ? W1 and? W2 . At high temperatures, where the thermal conductivity is large, the temper- ature gradient is smaller than at lower temperatures, whereλ(?) is smaller. At each point

10 1 Introduction. Technical Applications

in the wall the product q=-λ(?) d? dx has to be the same. Smaller values of the thermal conductivity are "compensated" by larger temperature gradients.

1.1.3 Convective heat transfer. Heat transfer coefficient

In a flowing fluid, energy is transferred not only through heat conduction but also by the macroscopic movement of the "uid. When we imagine an area located at a given position within the "uid, heat "ows through this area by conduction due to the temperature gradient and in addition energy as enthalpy and kinetic energy of the "uid which crosses the area. This is known as convective heat transfer which can be described as the superposition of thermal conduction and energy transfer by the "owing "uid. Heat transfer between a solid wall and a "uid, e.g. in a heated tube with a cold gas "owing inside it, is of special technical interest. The "uid layer close to the wall has the greatest e
ect on the amount of heat transferred. It is known as theboundary layerand boundary layer theory founded by L. Prandtl 2 in 1904 is the area of "uid dynamics that is most important for heat and mass transfer. In the boundary layer the velocity component parallel to the wall changes, over a small distance, from zero at the wall to almost the maximum value occurring in the core "uid, Fig. 1.6. The temperature in the boundary layer also changes from that at the wall? W to? F at some distance from the wall. Heat will "ow from the wall into the "uid as a result of the temperature di
erence? W -? F , but if the fluid is hotter than the wall,? F >? W , the fluid will be cooled as heat "ows into the wall. The heat "ux at the wall q W depends on the temperature and velocity fields in the "uid; their evaluation is quite complex and can lead to considerable problems in calculation. One puts therefore q W =α(? W -? F ) (1.22) with a new quantity, the localheat transfer coefficient, defined by

α:=

q W ? W -? F .(1.23)

This definition replaces the unknown heat "ux q

W , with the heat transfer coef- ficient, which is also unknown. This is the reason why many researchers see the introduction ofαas unnecessary and superfluous. Nevertheless the use of heat 2 Ludwig Prandtl (1875-1953) was Professor for Applied Mechanics at the University of G¨ottingen from 1904 until his death. He was also Director of the Kaiser-Wilhelm-Institut for Fluid Mechanics from 1925. His boundary layer theory, and work on turbulent "ow, wing theory and supersonic "ow are fundamental contributions to modern "uid mechanics.

1.1 The different types of heat transfer 11

Fig. 1.6:Velocityw(left) and temperature?(right) profiles in a fluid as a function of distance from the wally.δandδ t represent the velocity and temperature boundary layer thicknesses. transfer coefficients seems to be reasonable, because whenαis known both the basic questions in convective heat transfer can be easily answered: What is the heat "ux q W for a given temperature difference? W -? F , and what difference in temperature? W -? F causes a given heat flux q W between the wall and the fluid? In order to see how the heat transfer coecient and the temperature field in the "uid are related, the immediate neighbourhood of the wall (y→0) is considered. Here the "uid adheres to the wall, except in the case of very dilute gases. Its velocity is zero, and energy can only be transported by heat conduction. So instead of (1.22) the physically based relationship (Fourier"s law) is valid: q W =-λ ? ∂? y ? W ,(1.24) whereλ,ortobemoreexactλ(? W ), is the thermal conductivity of the fluid at the wall temperature. The heat "ux q W is found from the gradient of the temperature profile of the "uid at the wall, Fig. 1.7. From the definition (1.23), it follows for the heat transfer coecient

α=-λ

? ∂? y ? W ? W -? F .(1.25) From this it is clear thatαis determined by the gradient of the temperature profile at the wall and the di
erence between the wall and "uid temperatures. Therefore,

Fig. 1.7: Fluid temperature?=?(y)

as a function of distance from the wall yand illustration of the ratioλ/αas a subtangent

12 1 Introduction. Technical Applications

to calculate the heat transfer coefficient, knowledge of the temperature field in the fluid is required. This is, in turn, influenced by the velocity field within the fluid. So, in addition to the energy balances from thermodynamics, the equations of fluid motion from fluid mechanics furnish the fundamental relationships in the theory of convective heat transfer. A simple graphical illustration ofαfollows from (1.25). As shown in Fig. 1.7 the ratioλ/αis the distance from the wall at which the tangent to the temperature profile crosses the?=? F line. The length ofλ/αis of the magnitude of the (thermal) boundary layer thickness which will be calculated in sections 3.5 and

3.7.1 and which is normally a bit larger thanλ/α. A thin boundary layer indicates

good heat transfer whilst a thick layer leads to small values ofα.

The temperature of the fluid?

F far away from the wall, appears in (1.23), the definition of the local heat transfer coefficient. If a fluid flows around a body, so called external flow, the temperature? F is taken to be that of the fluid so far away from the surface of the body that it is hardly influenced by heat transfer.? F is called thefree flow temperature, and is often written as? ∞ . However, when a fluid flows in a channel, (internal flow), e.g. in a heated tube, the fluid temperature at each point in a cross-section of the channel will be influenced by the heat transfer from the wall. The temperature profile for this case is shown in Figure 1.8.? F is defined here as across sectional average temperaturein such a way that? F is also a characteristic temperature for energy transport in the fluid along the channel axis. This definition of? F links the heat flow from the wall characterised byα and the energy transported by the flowing fluid.

Fig. 1.8: Temperature profile in a chan-

nel cross section. Wall temperature? W and average fluid temperature? F

To define?

F we will take a small section of the channel, Fig. 1.9. The heat flow from the wall area dAto the fluid is dQ=α(? W -? F )dA.(1.26) From the first law of thermodynamics, neglecting the change in kinetic energy, we have d

Q=?H+dH?

-H=dH.(1.27) The flow of heat causes a change in the enthalpy flow

Hof the fluid. The cross sectional average

fluid temperature? F is now defined such that the enthalpy flow can be written as H=? (Aq) ?wh(?)dA q =Mh(? F ) (1.28)

1.1 The different types of heat transfer 13

Fig. 1.9: Energy balance for a channel section (left); fluid velocitywand temperature ?profiles in channel cross section (right) as the product of the mass "ow rate  M= ? (Aq) ?wdA q and the specific enthalpyh(? F ) at the average temperature? F . ? F is also called theadiabatic mixing temperature. This is the average temperature of the "uid when all elements in a cross section are mixed adiabatically in a container leaving it with the constant temperature? F . According to the first law, the enthalpy flow 

Hwith which the

unmixed "uid enters the adiabatic container must be equal to the enthalpy "ow  Mh(? F )ofthe "uid as it leaves the container. This is implied by (1.28) where? F has been implicitly defined.

To calculate the adiabatic mixing temperature?

F the pressure dependence of the specific enthalpy is neglected. Then setting h(?)=h 0 +[c p ] ? 0 (?-? 0 ) and h(? F )=h 0 +[c p ] ?F ?0 (? F -? 0 ) with [c p ] ? 0 as the average specific heat capacity of the fluid between?and the reference tem- perature? 0 at whichh(? 0 )=h 0 , we get from (1.28) ? F =? 0 + 1  M[c p ] ?F ?0 ? (Aq) ?w[c p ] ? 0 (?-? 0 )dA q .(1.29) For practical calculations a constant specific heat capacityc p is assumed, giving ? F = 1  M ? (Aq) ?w?dA q (1.30) as well as d  H=  Mc p d? F .(1.31) The adiabatic mixing temperature from (1.30) is the link between the local heat transfer coef- ficientαfrom (1.23) and the enthalpy flow for every cross section, because from (1.26), (1.27) and (1.31) follows d 

Q=α(?

W -? F )dA=  Mc p d? F .(1.32)

The adiabatic mixing temperature?

F is different from the integrated average of the cross sec- tional temperature ? m = 1 A q ? (Aq) ?dA q .

14 1 Introduction. Technical Applications

Fig. 1.10: Average fluid temperature?

F ,wall temperature? W and local heat transfer coeffi- cient as functions of the axial distancez,when heating a "uid in a tube of lengthL Both temperatures are only equal if the velocity at each point in the cross section is the same, i.e. in plug "ow withw=const. So far we have considered the local heat transfer coefficient, which can be di
erent at every point of the wall. In practice generally only anaverage heat transfer coecientα m is required in order to evaluate the flow of heat 

Qfrom an

areaAinto the fluid: 

Q=α

m

AΔ?

or α m :=  Q

AΔ?

.(1.33)

In this definition forα

m the temperature difference Δ?can still be chosen at will; a reasonable choice will be discussed later on. If the local heat transfer coecientαis known,α m can be found by integration.

This gives for the "ow of heat transferred

 Q= ? (A) q(A)dA=