Heat and Mass Transfer

Springer-Verlag Berlin Heidelberg GmbH

Hans Dieter Baehr . Karl Stephan

Heat and Mass Transfer

Translated by Nicola Jane Park

With 327 Figures

, Springer

Professor Dr. Hans Dieter Baehr Institut fUr Thermodynamik Universitat Hannover CallinstraBe 36 D-30167 Hannover, Germany

Professor Dr. Karl Stephan Institut fUr Thermodynamik und Thermische Verfahrenstechnik Universitat Stuttgart Pfaffenwaldring 9

D-7oS69 Stuttgart, Germany

Translated from the second German Edition "Wiirme- und Stoff ubertragung" (Springer- Verlag, 1996) by Nicola Jane Park, MEng., University of London, Imperial College of Science, Technology and Medicine.

Library of Congress Cataloging-in-Publication Data

Die Deutsche Bibliothek - Cip-Einheitsaufnahme Baehr, Hans Dieter: Heat and mass transfer I H. D. Baehr; K. Stephan. Trans!. by Nicola JanePark. Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1998

Dt. Ausg. u. d. T.: Baehr, Hans Dieter: Wiirme- und Stoffiibertragung

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights oftranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1998

Originally published by Springer-Verlag Berlin Heidelberg New York in 1998.

Softcover reprint ofthe hardcover I st edition 1998

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Production: ProduServ GmbH Verlagsservice, Berlin Typesetting: Camera-ready by authors SPIN: 10548563 60/3020-543210 - Printed on acid -free paper

ISBN 978-3-662-06661-7 ISBN 978-3-662-03659-4 (eBook)DOI 10.1007/978-3-662-03659-4

Preface

This book is the English translation of our German publication, which appeared in 1994 with the title "Wiirme und Stoffiibertragung" (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 equa­tions 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 mo­tivation 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.

VI Preface

The third chapter covers convective heat and mass transfer. The derivation of the mass, momentum and energy balance equations for pure fluids and multi­component 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 boil­ing with free and forced flows is dealt with. The presentation follows the book, "Heat Transfer in Condensation and Boiling" (Berlin: Springer-Verlag 1988) 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 quantita­tive 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 multi phase 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, Spring 1998

H.D. Baehr K. Stephan

Contents

Nomenclature xiv

1 Introduction. Technical Applications 1

1

2

5

1.1 The different types of heat transfer. . 1.1.1 Heat conduction ....... . 1.1.2 1.1.3 1.1.4 1.1.5

1.1.6

Steady, one-dimensional conduction of heat Convective heat transfer. Heat transfer coefficient Determining heat tramJer coefficients. Dimensionless numbers Thermal radiation . Radiative exchange

10 15 25 27

1.2 Overall heat transfer. . . . 30 1.2.1 The overall heat transf,er coefficient 30 1.2.2 Multi-layer walls . . . . . . . . . . . 32 1.2.3 Overall heat transfer through walls with extended surfaces 33 1.2.4 Heating and cooling of thin walled vessels 37

1.3 Heat exchangers . . . . . . . . . . . . . . . . . . 39 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 cocurrent heat exchangers. . 48 1.3.4 Crossflow heat exchangers. . . . . . . . . . . . . 56 1.3.5 Operating characteristics of further flow configurations. Diagrams 62

1.4 The different types of mass transfer . . 64 1.4.1 Diffusion ............. 66

1.4.1.1 Com position of mixtures 66 1.4.1.2 Diffusive fluxes . . . . . 67 1.4.1.3 Fick's Law. . . . . . . . 70

1.4.2 Diffusion through a semipermeable plane. Equimolar diffusion 72 1.4.3 Convective mass transfer 76

1.5 Mass transfer theories . . . . . 1.5.1 Film theory ...... . 1.5.2 Boundary layer theory. 1.5.3 Penetration and surface renewal theories 1.5.4 Application of film theory to evaporative cooling

79 79 83 85 87

viii Contents

1.6 Overall mass transfer .. 90

1.7 Mass transfer apparatus. 93 1.7.1 Material balances 94 1. 7.2 Concentration profiles and heights of mass transfer columns. 97

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

2 Heat conduction and mass diffusion 105

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 material

properties .................. 109 2.1.3 Boundary conditions. . . . . . . . . . . . . III 2.1.4 Temperature dependent material properties ll4 2.1.5 Similar temperature fields. . . . . . . . . . ll5

2.2 Steady-state heat conduction . . . . . . . . . . . . ll9 2.2.1 Geometric one-dimensional heat conduction with heat sources ll9 2.2.2 Longitudinal heat conduction in a rod . . . . . 122 2.2.3 The temperature distribution in fins and pins. 127 2.2.4 Fin efficiency . . . . . . . . . . . . . . . . . . . 131 2.2.5 Geometric multi-dimensional heat flow. . . . . 134

2.2.5.1 Superposition of heat sources and heat sinks. 2.2.5.2 Shape factors

2.3 Transient heat conduction .. 2.3.1 Solution methods

135 139

140 141

2.3.2 The Laplace transformation. 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 Periodic temperature variations . . . . . . . . . . . . .. 156

2.3.4 Cooling or heating of simple bodies in one-dimensional heat flow 159 2.3.4.1 Formulation of the problem 159 2.3.4.2 Separating the variables . . . . . . . . 161 2.3.4.3 Results for the plate . . . . . . . . . . 162 2.3.4.4 Results for the cylinder and the sphere 167 2.3.4.5 Approximation for large times: Restriction to the first term

in the series . . . . . . . . . . . . . . . . . . 169 2.3.4.6 A solution for small times . . . . . . . . . . 170

2.3.5 Cooling and heating in multi-dimensional heat flow 172 2.3.5.1 Product solutions. . . . . . . . . . . . 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 Improved approximations 184

2.3.7 Heat sources . . . . . . . . . . . . . . . . 185

Contents

2.3.7.1 Homogeneous heat sources .... . 2.3.7.2 Point and linear heat sources .. .

IX

185 187

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 The finite difference equation 193 2.4.1.2 The stability condition . . . . 195 2.4.1.3 Heat sources. . . . . . . . . . 196

2.4.2 Discretisation of the boundary conditions 197 2.4.3 The implicit difference method from J. Crank and P. Nicolson 202 2.4.4 Noncartesian coordinates. Temperature dependent material prop-

erties .................................. 206 2.4.4.1 The discretisation of the self-adjoint differential operator. 206 2.4.4.2 Constant material properties. Cylindrical coordinates 207 2.4.4.3 Temperature dependent material properties . . . 209

2.4.5 Transient two- and three-dimensional temperature fields. . . . 210 2.4.6 Steady-state temperature fields. . . . . . . . . . . . . . . . . . 213

2.4.6.1 A simple finite difference method for plane, steady-state temperature fields ............. 213

2.4.6.2 Consideration of the boundary conditions 216

2.5 Mass diffusion ........................ 221 2.5.1 Remarks on quiescent systems ... . . . . . . . . 221 2.5.2 Derivation of the differential equation for the concentration field 224 2.5.3 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . 229 2.5.4 Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . .. 230 2.5.5 Steady-state mass diffusion with catalytic surface reaction. . .. 233 2.5.6 Steady-state mass diffusion with homogeneous chemical reaction 237 2.5.7 Transient mass diffusion. . . . . . . . . . . . . . . . . . . . . .. 241

2.5.7.1 Transient mass diffusion in a semi-infinite solid . . . .. 241 2.5.7.2 Transient mass diffusion in bodies of simple geometry with

one-dimensional mass flow 243

2.6 Exercises .. 244

3 Convective heat and mass transfer. Single phase flow 251

3.1 Preliminary remarks: Longitudinal, frictionless flow over a flat plate 251

3.2 The balance equations . . . . . . . . 256 3.2.1 Reynolds' transport theorem 256 3.2.2 The mass balance .. . . . . 258

3.2.2.1 Pure substances. . . 258 3.2.2.2 Multicomponent mixtures

3.2.3 The momentum balance . . . . . . 3.2.3.1 The stress tensor ..... 3.2.3.2 Cauchy's equation of motion. 3.2.3.3 The strain tensor . . . . . . .

260 262 264 268 269

x Contents

3.2.3.4 Constitutive equations for the solution of the momentum equation . . . . . . . . . . . . 271

3.2.3.5 The Navier-Stokes equations. . 272 3.2.4 The energy balance ........... 272

3.2.4.1 Dissipated energy and entropy 278 3.2.4.2 Constitutive equations for the solution of the energy equation279 3.2.4.3 Some other formulations of the energy equation 281

3.2.5 Summary ................ .

3.3 Influence of the Reynolds number on the flow.

283

285

3.4 Simplifications to the Navier-Stokes equations 288 3.4.1 Creeping flows . . . . 288 3.4.2 Frictionless flows . . . 289 3.4.3 Boundary layer flows 289

3.5 The boundary layer equations 291 3.5.1 The velocity boundary layer 291 3.5.2 The thermal boundary layer 294 3.5.3 The concentration boundary layer 298 3.5.4 General comments on the solution of boundary layer equations 298

3.6 Influence of turbulence on heat and mass transfer 302 3.6.1 Turbulent flows near solid walls. 306

3.7 External forced flow . . . . . . . . . . . 309 3.7.1 Parallel flow along a flat plate . 310

3.7.1.1 Laminar boundary layer 311 3.7.1.2 Turbulent flow .. 323

3.7.2 The cylinder in cross flow . . . . 327 3.7.3 Tube bundles in cross flow ... 331 3.7.4 Some empirical equations for heat and mass transfer in external

forced flow . 335

3.8 Internal forced flow 337 3.8.1 Laminar flow in circular tubes 337

3.8.1.1 Hydrodynamic, fully developed, laminar flow 338 3.8.1.2 Thermal, fully developed, laminar flow . . . . 340 3.8.1.3 Heat transfer coefficients in thermally fully developed, lam-

inar flow . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343 3.8.1.4 The thermal entry flow with fully developed velocity profile 346 3.8.1.5 Thermally and hydrodynamically developing flow 350

3.8.2 Turbulent flow in circular tubes 352 3.8.3 Packed beds ........... 353 3.8.4 Fluidized beds ......... . 357 3.8.5 Some empirical equations for heat and mass transfer in flow through

channels, packed and fluidized beds 366

3.9 Free flow .................. . 3.9.1 The momentum equation ..... . 3.9.2 Heat transfer in laminar flow on a vertical wall

369 372 375

Contents

3.9.3 Some empirical equations for heat transfer in free flow. 3.9.4 Mass transfer in free flow ..

3.10 Overlapping of free and forced flow.

3.11 Compressible flows ......... . 3.11.1 The temperature field in a compressible flow 3.11.2 Calculation of heat transfer

3.12 Exercises ............. .

4 Convective heat and mass transfer. Flows with phase change

4.1 Heat transfer in condensation ....... . 4.1.1 The different types of condensation 4.1.2 Nusselt's film condensation theory . 4.1.3 4.1.4 4.1.5 4.1.6 4.1.7 4.1.8

Deviations from Nusselt's film condensation theory. Influence of non-condensable gases .. Film condensation in a turbulent film Condensation of flowing vapours . Dropwise condensation Condensation of vapour mixtures. 4.1.8.1 The temperature at the phase interface. 4.1.8.2 The material and energy balance for the vapour.

4.1.8.3 Calculating the area of a condenser. 4.1.9 Some empirical equations

4.2 Heat transfer in boiling . . . . . 4.2.1 The different types of heat transfer. 4.2.2 The formation of vapour bubbles .. 4.2.3 Bubble frequency and departure diameter

Xl

380 382

383

384 385 392

395

401

401 402

404 408 412 418 421 427 430 434 438 440 441

443 443 448 451

4.2.4 Boiling in free flow. The Nukijama curve 455 4.2.5 Stability during boiling in free flow. . . . 456 4.2.6 Calculation of heat transfer coefficients for boiling in free flow 460 4.2.7 Some empirical equations for heat transfer during nucleate boiling

in free flow . . . . . . . . . . . . . . 463 4.2.8 Two-phase flow. . . . . . . . . . . . 467

4.2.8.1 The different flow patterns. 467 4.2.8.2 Flow maps. . . . . . . . . . 470 4.2.8.3 Some basic terms and definitions 471 4.2.8.4 Pressure drop in two-phase flow. 474

4.2.8.5 The different heat transfer regions in two-phase flow 481

4.2.8.6 Heat transfer in nucleate boiling and convective evaporation483 4.2.8.7 Critical boiling states .................... , 486

4.2.8.8 Some empirical equations for heat transfer in two-phase flow489

4.2.9 Heat transfer in boiling mixtures 490

4.3 Exercises . . . . . . . . . . . . . . . . . 495

XII Contents

5 Thermal radiation 497

497 498

5.1 Fundamentals. Physical quantities 5.1.1 Thermal radiation ... 5.1.2 Emission of radiation .. 500

5.1.2.1 Emissive power . 500 5.1.2.2 Spectral intensity 501 5.1.2.3 Hemispherical spectral emissive power and total intensity 503 5.1.2.4 Diffuse radiators. Lambert's cosine law. 507

5.1.3 Irradiation ....... 508 5.1.4 Absorption of radiation . . . . . . . . . . . 511 5.1.5 Reflection of radiation . . . . . . . . . . . . 516 5.1.6 Radiation in an enclosure. Kirchhoff's law 518

5.2 Radiation from a black body . . . . . . . . . . . . 521 5.2.1 Definition and realisation of a black body . 521 5.2.2 The spectral intensity and the spectral emissive power . 522 5.2.3 The emissive power and the emission of radiation in a wavelength

interval . . . . . . . . . . . 526

5.3 Radiation properties of real bodies 531 5.3.1 Emissivities......... 531 5.3.2 The relationships between emissivity, absorptivity and reflectivity.

The grey Lambert radiator . . . . . . . . . . . . . . . 534 5.3.2.1 Conclusions from Kirchhoff's law . . . . . . . 534 5.3.2.2 Calculation of absorptivities from emissivities 5.3.2.3 The grey Lambert radiator

5.3.3 Emissivities of real bodies . . . . . . . 5.3.3:1 Electrical insulators 5.3.3.2 Electrical conductors (metals)

5.3.4 Transparent bodies ...... .

534 536 538 539 541 544

5.4 Solar radiation . . . . . . . . . . . . . . . . . 548 5.4.1 Extraterrestrial solar radiation . . . . 549 5.4.2 The attenuation of solar radiation in the earth's atmosphere 551

5.4.2.1 Spectral transmissivity . . . . . . 552 5.4.2.2 Molecular and aerosol scattering 555 5.4.2.3 Absorption ............ 556

5.4.3 Direct solar radiation on the ground ... 5.4.4 Diffuse solar radiation and global radiation 5.4.5 Absorptivities for solar radiation

557 559 562

5.5 Radiative exchange ........... 563 5.5.1 View factors .. . . . . . . . . . 564 5.5.2 Radiative exchange between black bodies 569 5.5.3 Radiative exchange between grey Lambert radiators 572

5.5.3.1 The balance equations according to the net-radiation method573 5.5.3.2 Radiative exchange between a radiation source, a radiation

receiver and a reradiating wall. . . . . . . . . . . . . .. 574 5.5.3.3 Radiative exchange in a hollow enclosure with two zones. 578

Contents

5.5.3.4 The equation system for the radiative exchange between any number of zones

5.5.4 Protective radiation shields

Xlll

580 583

5.6 Gas radiation . . . . . . . . . . . . 587 5.6.1 Absorption coefficient and optical thickness 588 5.6.2 Absorptivity and emissivity . . . . . . . . . 590 5.6.3 Results for the emissivity . . . . . . . . . . 593 5.6.4 Emissivities and mean beam lengths of gas spaces 596 5.6.5 Radiative exchange in a gas filled enclosure 600

5.6.5.1 Black, isothermal boundary walls . . . . . 600 5.6.5.2 Grey isothermal boundary walls. . . . . . 601 5.6.5.3 Calculation of the radiative exchange in complicated cases 604

5.7 Exercises ....... . 605

Appendix A: Supplements 609

609

611

A.l Introduction to tensor notation

A.2 Relationship between mean and thermodynamic pressure

A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity in cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . .. 612

A.4 Navier-Stokes equations for an incompressible fluid of constant viscosity in cylindrical coordinates . . 613

A.5 Entropy balance for mixtures 613

A.6 Relationship between partial and specific enthalpy 615

A.7 Calculation of the constants an of a Graetz-Nusselt problem (3.243) 616

Appendix B: Property data 618

Appendix C: Solutions to the exercises 632

Literature 646

Index 663

Nomenclature

Symbol Meaning SI units

A area m 2

Am average area m 2

Aq cross sectional area m 2

Af fin surface area m 2

a thermal diffusivity m 2/s a hemispherical total absorptivity a). spectral absorptivity a' ). directional spectral absorptivity

at turbulent thermal diffusivity m 2/s a* specific surface area m 2/m3

b thermal penetration coefficient, b = v>:Ce W sl/2/(m2 K)

b Laplace constant, b = V2cr / 9 (ilL - IlG) m

C circumference, perimeter m

C heat capacity flow ratio c specific heat capacity J/(kg K) c concentration mol/m3

C propagation velocity of electromagnetic waves m/s Co velocity of light in a vacuum m/s Cf friction factor cp specific heat capacity at constant pressure J/(kg K)

CR resistance factor D binary diffusion coefficient m 2/s D t turbulent diffusion coefficient m 2/s d diameter m

dA departure diameter of vapour bubbles m

dh hydraulic diameter m

E irradiance W/m2

Eo solar constant W/m2

E). spectral irradiance W /(m2 iJ.m)

e unit vector

F force N

Nomenclature xv

FB buoyancy force N

Fr friction force N

FR resistance force N

Fij view factor between surfaces i and j

F(O, AT) fraction function of black radiation

I frequency of vapour bubbles l/s

Ij force per unit volume N/m3

9 acceleration due to gravity m/s2

H height m

H radiosity W/m2

H enthalpy J

if enthalpy flow J/s h Planck constant J s

h specific enthalpy J/kg htot specific total enthalpy, htot = h + w 2 /2 J/kg

hi partial specific enthalpy J/kg

l'ihv specific enthalpy of vaporisation J/kg l'ihv molar enthalpy of vaporisation J/mol I momentum kg m/s I directional emissive power W/(m2 sr)

j diffusional flux mol/(m2 s) j* diffusional flux in a centre of gravity system kg/(m 2 s)

tij diffusional flux in a particle based system mol/(m2 s) f{ incident intensity W/(m2 sr) f{A incident spectral intensity W /(m2 Jim) k overall heat transfer coefficient W/(m2 K) k extinction coefficient

k Boltzmann constant J/K kG spectral absorption coefficient l/m kH Henry coefficient N/m2

kj force per unit mass N/kg kl rate constant for a homogeneous

first order reaction l/s k;,k~ rate constant for a homogeneous (heterogeneous)

first order reaction m/s k" n rate constant for a heterogeneous mol/(m2s)

n-th order reaction (mol/m3 )n L length m L total intensity W/(m2 sr) LA spectral intensity W /(m 2 Jim sr) La reference length m

Ls solubility mol/(m3 Pal

XVI Nomenclature

length, mixing length m

M mass kg

M modulus, M = at::..t/ t::..x 2

M (hemispherical total) emissive power W/m 2

M), spectral emissive power W/(m 2 pm) 1\1 mass flow rate kg/s M molecular mass, molar mass kg/mol m optical mass kg/m2

mr relati ve optical mass

m mass flux kg/(m 2 s) N amount of substance mol N; dimensionless transfer capability (number of

transfer units) of the material stream i

N molar flow rate molls n refractive index

n normal vector

it molar flux mol/(m2 s) p power W

Pdiss dissipated power W p pressure Pa p+ dimensionless pressure

Q heat J Q heat flow W q heat flux W/m2

R radius m

Rcond resistance to thermal conduction K/W Rm molar (universal) gas constant J/(moIK)

" radial coordinate m

" hemispherical total reflectivity 1'). spectral reflectivity

1" A directional spectral reflectivity

" e electrical resistivity nm 1'+ dimensionless radial coordinate

r reaction rate mol/(m3 s) 5 suppression factor in convective boiling

5 entropy J/K S specific entropy J/(kgK) S Laplace transformation parameter lis S beam length m

S slip factor, S = WG/WL

SI longitudinal pitch m

Nomenclature XVll

Sq transverse pitch m

T thermodynamic temperature K

Te eigentemperature K Tst stagnation point temperature K

time t+ dimensionless time

tk cooling time

tj stress vector N/m2

tR relaxation time, tR = 1/ kJ S

tD relaxation time of diffusion, tD = £2/ D s

U internal energy J u average molar velocity m/s u specific internal energy J/kg u Laplace transformed temperature K

V volume m3

VA departure volume of a vapour bubble m3

v specific volume m 3 /kg W work J vir power density W/m3

Wi heat capacity flow rate of a fluid i W/K W velocity m/s Wo reference velocity m/s Ws velocity of sound m/s W T shear stress velocity, W T = vro/e m/s Wi fluctuation velocity m/s w+ dimensionless velocity

X moisture content; Lockhart-Martinelli parameter

X molar content in the liquid phase x coordinate m x mole fraction in the liquid x+ dimensionless x-coordinate x* quality, x* = MG/ ML x~h thermodynamic quality y molar content in the gas phase

y coordinate m

ii mole fraction in the gas phase y+ dimensionless y-coordinate

z number

z axial coordinate m z+ dimensionless z-coordinate

ZR number of tube rows

xv III

Greek letters

Symbol Meaning

Q heat transfer coefficient

Q m mean heat transfer coefficient

f3 mass transfer coefficient

f3m mean mass transfer coefficient f3 thermal expansion coefficient

f3 polar angle, zenith angle

f30 base angle r mass production rate

"r molar production rate

.6, difference

Nomenclature

o thickness; boundary layer thickness Oij Kronecker symbol c volumetric vapour content

c' volumetric quality c hemispherical total emissivity c>, hemispherical spectral emissivity

directional spectral emissivity

turbulent diffusion coefficient

dimensionless temperature change of the material stream i

dilatation strain tensor

cp void fraction

(

(

1)

1)[

e fJ

turbulent viscosity

resistance factor bulk viscosity dynamic viscosity

fin efficiency overtemperature temperature

fJ+ dimensionless temperature

Ji isentropic exponent

JiG optical thickness of a gas beam

wave length of an oscillation

wave length

thermal conductivity

turbulent thermal conductivity

fJ diffusion resistance factor

II kinematic viscosity

SI units

W/(m2 K) W/(m2 K)

m/s m/s l/K

rad rad

kg/(m3 s)

mol/(m3 s)

m

l/s l/s

kg/(m s) kg/(m s)

K

K

m

m

W/(K m) W/(K m)

Nomenclature

v frequency

(! density

(T Stefan-Boltzmann constant (T interfacial tension

~ mass fraction

T transmissivity

T). spectral transmissivity

T shear stress

Tji shear stress tensor

<l> radiative power, radiation flow

<l> viscous dissipation

'P angle, circumferential angle

1J! stream function

w solid angle

w reference velocity

W power density

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

at the phase interface

inner, inlet

id ideal

m incident radiation, irradiation

K substance K

L liquid

lam laminar

m mean, molar (based on the amount of substance)

max maximum

mm mmlmum

l/s

kg/m3

W /(m2 K4)

N/m

N/m2

N/m2

W

W/m3

rad

m2/s

XIX

xx

n

o

normal direction

outer, outside particle

Nomenclature

P ref

S reflected, reference state

s

solid, bottom product, sun, surroundings

black body, saturation

tot

trans turb

total

transmitted

turbulent

u

v w

in particle reference system boiler

J

>. w

wall, water

start

at the point y = J spectral

end

o 00

reference state; at the point y = 0

at a great distance; in infinity

Dimensionless numbers

Ar = [(es - eF)/eF] (4g/v2) Bi = aL/ >. BiD = f3L/ D

Bo = iI/ (m~hv) Da = k~L/D Ec = w 2 / (cp~t?)

Fo=af/L 2

Fr = w 2 / (gx) Ga = gL3 /v 2

Gr = gf3~t?L3 /v 2

Ha = (k 1 L2/D)2 Le = aiD

Ma = w/ws Nu=aL/>.

Pe = wL/a

Ph = hE/ [c (t?E - t?o)] Pr = via Ra = GrPr

Re = wL/v Sc = v/D Sh = f3L/D

Sf = a/ (weep) Sf = 1/ Ph

Archimedes number Biot number

Biot number for mass transfer boiling number Damk6hler number (for 1st order heterogeneous reaction)

Eckert number Fourier number Froude number Galilei number Grashof number

Hatta number Lewis number

Mach number

Nusselt number

Peciet number

phase change number

Prandtl number

Rayleigh number

Reynolds number

Schmidt number

Sherwood number

Stanton number

Stefan number