Heat and Mass Transfer • Hans Dieter Baehr  Karl Stephan

Heat and Mass Transfer

Third, revised edition

With 343 Figures, Many Worked Examples and Exercises

123 Dr.-Ing. E.h. Dr.-Ing. Hans Dieter Baehr Dr.-Ing. E.h. mult. Dr.-Ing. Karl Stephan Professor em. of Thermodynamics Professor em. Institute of Thermodynamics University of Hannover and Thermal Process Engineering Germany University of Stuttgart 70550 Stuttgart Germany [email protected]

ISBN 978-3-642-20020-5 e-ISBN 978-3-642-20021-2 DOI 10.1007/978-3-642-20021-2 Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2011934036

c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, 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. Violations are liable to prosecution under the German Copyright Law. 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.

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Springer is part of Springer Science+Business Media (www.springer.com) Preface to the third edition

In this edition of our book we still retained its concept: The main emphasis is placed on the fundamental principles of heat and mass transfer and their applica- tion to practical problems of process modelling and apparatus design. Like the previous editions, the third edition contains five chapters and several appendices, particularly a compilation of thermophysical property data needed for the solution of problems. In this third edition the text has been expanded to include recent develop- ments. In chapter 2 on “Numerical solution to heat conduction problems” we added now an introduction into the numerical solution of heat conduction problems with the method of Finite Elements. It shall familiarise the student with the principles of the method, so that he will be able to solve basic problems by himself. For the solution of more complicated problems it is advisable to make use of one of the many programs available either commercially or on the Internet. The acquired knowledge will help to do so. The chapter 3 on “Convective heat and mass transfer, Single phase flow” was thoroughly revised. It is now complemented by a chapter on heat and mass transfer in porous bodies. In chapter 4 the subchapter explaining the phenomena of the mechanism of boiling heat transfer was revised. It presents now the recent state of our knowledge in the field. As in the previous editions changes were 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. Also the compilation of the thermophysical properties, needed to solve heat and mass transfer problems, was revised and adapted to our present knowledge. As a con- sequence many of the worked examples and exercises had to be updated with the new thermophysical properties. The list of references was updated as well, including recent publications in the field. The worked examples not only illustrate the application of the theory, but are also relevant for apparatus design. Solving the exercises is essential for a sound understanding and for relating principles to real engineering situations. Numerical answers and hints to the solution of the exercises are given in the final appendix. The new edition also presented an opportunity to correct printing errors and mistakes. vi Preface

In preparing this edition we were assisted by Annika Zeitz and Sebastian Asenbeck, who helped us to submit a printable version of the manuscript to the publisher. We owe them sincere thanks. We also appreciate the efforts of friends and colleagues who provided their good advice with constructive suggestions. Par- ticular thanks are due to Dr. Arno Laesecke at the National Institute of Science and Technology (NIST), Boulder, Colorado, USA.

Bochum and Stuttgart, H.D. Baehr April 2011 K. 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 Stoffubertragung”¨ (7th edition Berlin: Springer- Verlag 2010). 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. Preface ix

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 1998 K. Stephan Contents

Nomenclature xviii

1 Introduction. Technical Applications 1 1.1 The different types of heat transfer ...... 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 Thermal radiation ...... 25 1.1.6 Radiative exchange ...... 27 1.2 Overall heat transfer ...... 31 1.2.1 The overall heat transfer coefficient ...... 31 1.2.2 Multi-layer walls ...... 33 1.2.3 Overall heat transfer through walls with extended surfaces .... 34 1.2.4 Heating and cooling of thin walled vessels ...... 38 1.3 Heat exchangers ...... 40 1.3.1 Types of heat exchanger and flow configurations ...... 41 1.3.2 General design equations. Dimensionless groups ...... 45 1.3.3 Countercurrent and cocurrent heat exchangers ...... 50 1.3.4 Crossflow heat exchangers ...... 57 1.3.5 Operating characteristics of further flow configurations. Diagrams 64 1.4 The different types of mass transfer ...... 65 1.4.1 Diffusion ...... 67 1.4.1.1 Composition of mixtures ...... 67 1.4.1.2 Diffusive fluxes ...... 68 1.4.1.3 Fick’s law ...... 71 1.4.2 Diffusion through a semipermeable plane. Equimolar diffusion .. 73 1.4.3 Convective mass transfer ...... 77 1.5 Mass transfer theories ...... 81 1.5.1 Film theory ...... 81 1.5.2 Boundary layer theory ...... 85 1.5.3 Penetration and surface renewal theories ...... 87 1.5.4 Application of film theory to evaporative cooling ...... 88 Contents xi

1.6 Overall mass transfer ...... 92 1.7 Mass transfer apparatus ...... 94 1.7.1 Material balances ...... 95 1.7.2 Concentration profiles and heights of mass transfer columns .... 98 1.8 Exercises ...... 103

2 Heat conduction and mass diffusion 107 2.1 The heat conduction equation ...... 107 2.1.1 Derivation of the differential equation for the temperature field . . 108 2.1.2 The heat conduction equation for bodies with constant material properties ...... 111 2.1.3 Boundary conditions ...... 113 2.1.4 Temperature dependent material properties ...... 116 2.1.5 Similar temperature fields ...... 117 2.2 Steady-state heat conduction ...... 121 2.2.1 Geometric one-dimensional heat conduction with heat sources . . 121 2.2.2 Longitudinal heat conduction in a rod ...... 124 2.2.3 The temperature distribution in fins and pins ...... 129 2.2.4 Fin efficiency ...... 133 2.2.5 Geometric multi-dimensional heat flow ...... 136 2.2.5.1 Superposition of heat sources and heat sinks ...... 137 2.2.5.2 Shape factors ...... 141 2.3 Transient heat conduction ...... 142 2.3.1 Solution methods ...... 143 2.3.2 The Laplace transformation ...... 144 2.3.3 The semi-infinite solid ...... 151 2.3.3.1 Heating and cooling with different boundary conditions . 151 2.3.3.2 Two semi-infinite bodies in contact with each other .... 156 2.3.3.3 Periodic temperature variations ...... 158 2.3.4 Cooling or heating of simple bodies in one-dimensional heat flow . 161 2.3.4.1 Formulation of the problem ...... 161 2.3.4.2 Separating the variables ...... 163 2.3.4.3 Results for the plate ...... 165 2.3.4.4 Results for the cylinder and the sphere ...... 169 2.3.4.5 Approximation for large times: Restriction to the first term in the series ...... 171 2.3.4.6 A solution for small times ...... 173 2.3.5 Cooling and heating in multi-dimensional heat flow ...... 174 2.3.5.1 Product solutions ...... 175 2.3.5.2 Approximation for small Biot numbers ...... 178 2.3.6 Solidification of geometrically simple bodies ...... 179 2.3.6.1 The solidification of flat layers (Stefan problem) ...... 180 2.3.6.2 The quasi-steady approximation ...... 183 2.3.6.3 Improved approximations ...... 186 2.3.7 Heat sources ...... 187 xii Contents

2.3.7.1 Homogeneous heat sources ...... 188 2.3.7.2 Point and linear heat sources ...... 189 2.4 Numerical solution to heat conduction problems with difference methods 194 2.4.1 The simple, explicit difference method for transient heat conduction problems ...... 195 2.4.1.1 The finite difference equation ...... 195 2.4.1.2 The stability condition ...... 197 2.4.1.3 Heat sources ...... 198 2.4.2 Discretisation of the boundary conditions ...... 199 2.4.3 The implicit difference method from J. Crank and P. Nicolson . . 205 2.4.4 Noncartesian coordinates. Temperature dependent material properties ...... 208 2.4.4.1 The discretisation of the self-adjoint differential operator . 209 2.4.4.2 Constant material properties. Cylindrical coordinates . . 210 2.4.4.3 Temperature dependent material properties ...... 212 2.4.5 Transient two- and three-dimensional temperature fields ...... 213 2.4.6 Steady-state temperature fields ...... 216 2.4.6.1 A simple finite difference method for plane, steady-state temperature fields ...... 216 2.4.6.2 Consideration of the boundary conditions ...... 219 2.5 Numerical solution to heat conduction problems with the method of Finite elements ...... 224 2.5.1 The finite element method applied to geometrical one-dimensional, steady-state temperature fields ...... 225 2.5.2 The finite element method applied to steady-state plane tempera- ture fields ...... 230 2.5.3 The finite element method applied to transient, geometrical one- dimensional heat conduction problems ...... 236 2.5.4 Extension to transient, geometrical two-dimensional heat conduc- tion problems ...... 241 2.6 Mass diffusion ...... 242 2.6.1 Remarks on quiescent systems ...... 242 2.6.2 Derivation of the differential equation for the concentration field . 245 2.6.3 Simplifications ...... 250 2.6.4 Boundary conditions ...... 251 2.6.5 Steady-state mass diffusion with catalytic surface reaction ..... 254 2.6.6 Steady-state mass diffusion with homogeneous chemical reaction . 258 2.6.7 Transient mass diffusion ...... 262 2.6.7.1 Transient mass diffusion in a semi-infinite solid ...... 263 2.6.7.2 Transient mass diffusion in bodies of simple geometry with one-dimensional mass flow ...... 264 2.7 Exercises ...... 266 Contents xiii

3 Convective heat and mass transfer. Single phase flow 275 3.1 Preliminary remarks: Longitudinal, frictionless flow over a flat plate ... 275 3.2 The balance equations ...... 280 3.2.1 Reynolds’ transport theorem ...... 280 3.2.2 The mass balance ...... 282 3.2.2.1 Pure substances ...... 282 3.2.2.2 Multicomponent mixtures ...... 284 3.2.3 The momentum balance ...... 286 3.2.3.1 The stress tensor ...... 288 3.2.3.2 Cauchy’s equation of motion ...... 292 3.2.3.3 The strain tensor ...... 293 3.2.3.4 Constitutive equations for the solution of the momentum equation ...... 295 3.2.3.5 The Navier-Stokes equations ...... 296 3.2.4 The energy balance ...... 297 3.2.4.1 Dissipated energy and entropy ...... 302 3.2.4.2 Constitutive equations for the solution of the energy equation ...... 303 3.2.4.3 Some other formulations of the energy equation ...... 305 3.2.5 Summary ...... 308 3.3 Influence of the Reynolds number on the flow ...... 310 3.4 Simplifications to the Navier-Stokes equations ...... 313 3.4.1 Creeping flows ...... 313 3.4.2 Frictionless flows ...... 314 3.4.3 Boundary layer flows ...... 314 3.5 The boundary layer equations ...... 315 3.5.1 The velocity boundary layer ...... 315 3.5.2 The thermal boundary layer ...... 319 3.5.3 The concentration boundary layer ...... 323 3.5.4 General comments on the solution of boundary layer equations . . 323 3.6 Influence of turbulence on heat and mass transfer ...... 327 3.6.1 Turbulent flows near solid walls ...... 331 3.7 External forced flow ...... 335 3.7.1 Parallel flow along a flat plate ...... 336 3.7.1.1 Laminar boundary layer ...... 336 3.7.1.2 Turbulent flow ...... 348 3.7.2 The cylinder in crossflow ...... 353 3.7.3 Tube bundles in crossflow ...... 357 3.7.4 Some empirical equations for heat and mass transfer in external forced flow ...... 361 3.8 Internal forced flow ...... 364 3.8.1 Laminar flow in circular tubes ...... 364 3.8.1.1 Hydrodynamic, fully developed, laminar flow ...... 365 xiv Contents

3.8.1.2 Thermal, fully developed, laminar flow ...... 367 3.8.1.3 Heat transfer coefficients in thermally fully developed, laminar flow ...... 369 3.8.1.4 The thermal entry flow with fully developed velocity profile ...... 372 3.8.1.5 Thermally and hydrodynamically developing flow ..... 377 3.8.2 Turbulent flow in circular tubes ...... 379 3.8.3 Packed beds ...... 380 3.8.4 Porous bodies ...... 384 3.8.4.1 Fluid flow and momentum balance ...... 384 3.8.4.2 The energy balance ...... 387 3.8.4.3 Heat transfer inside channels ...... 392 3.8.5 Fluidised beds ...... 398 3.8.6 Some empirical equations for heat and mass transfer in flow through channels, packed and fluidised beds ...... 407 3.9 Free flow ...... 411 3.9.1 The momentum equation ...... 413 3.9.2 Heat transfer in laminar flow on a vertical wall ...... 417 3.9.3 Some empirical equations for heat transfer in free flow ...... 421 3.9.4 Mass transfer in free flow ...... 423 3.10 Overlapping of free and forced flow ...... 425 3.11 Compressible flows ...... 426 3.11.1 The temperature field in a compressible flow ...... 426 3.11.2 Calculation of heat transfer ...... 433 3.12 Exercises ...... 437

4 Convective heat and mass transfer. Flows with phase change 443 4.1 Heat transfer in condensation ...... 443 4.1.1 The different types of condensation ...... 444 4.1.2 Nusselt’s film condensation theory ...... 446 4.1.3 Deviations from Nusselt’s film condensation theory ...... 450 4.1.4 Influence of non-condensable gases ...... 454 4.1.5 Film condensation in a turbulent film ...... 460 4.1.6 Condensation of flowing vapours ...... 464 4.1.7 Dropwise condensation ...... 469 4.1.8 Condensation of vapour mixtures ...... 473 4.1.8.1 The temperature at the phase interface ...... 477 4.1.8.2 The material and energy balance for the vapour ...... 481 4.1.8.3 Calculating the size of a condenser ...... 483 4.1.9 Some empirical equations ...... 484 4.2 Heat transfer in boiling ...... 486 4.2.1 The different types of heat transfer ...... 486 4.2.2 The formation of vapour bubbles ...... 491 4.2.3 Mechanism of heat transfer in natural convection boiling ..... 494 Contents xv

4.2.4 Bubble frequency and departure diameter ...... 498 4.2.5 Boiling in free flow. The Nukijama curve ...... 500 4.2.6 Stability during boiling in free flow ...... 502 4.2.7 Calculation of heat transfer coefficients for boiling in free flow . . 505 4.2.8 Some empirical equations for heat transfer during nucleate boiling in free flow ...... 509 4.2.9 Two-phase flow ...... 513 4.2.9.1 The different flow patterns ...... 514 4.2.9.2 Flow maps ...... 516 4.2.9.3 Some basic terms and definitions ...... 517 4.2.9.4 Pressure drop in two-phase flow ...... 520 4.2.9.5 The different heat transfer regions in two-phase flow ... 528 4.2.9.6 Heat transfer in nucleate boiling and convective evaporation ...... 530 4.2.9.7 Critical boiling states ...... 533 4.2.9.8 Some empirical equations for heat transfer in two-phase flow ...... 535 4.2.10 Heat transfer in boiling mixtures ...... 536 4.3 Exercises ...... 542

5 Thermal radiation 545 5.1 Fundamentals. Physical quantities ...... 545 5.1.1 Thermal radiation ...... 546 5.1.2 Emission of radiation ...... 548 5.1.2.1 Emissive power ...... 548 5.1.2.2 Spectral intensity ...... 549 5.1.2.3 Hemispherical spectral emissive power and total intensity 551 5.1.2.4 Diffuse radiators. Lambert’s cosine law ...... 555 5.1.3 Irradiation ...... 557 5.1.4 Absorption of radiation ...... 559 5.1.5 Reflection of radiation ...... 564 5.1.6 Radiation in an enclosure. Kirchhoff’s law ...... 566 5.2 Radiation from a black body ...... 569 5.2.1 Definition and realisation of a black body ...... 569 5.2.2 The spectral intensity and the spectral emissive power ...... 571 5.2.3 The emissive power and the emission of radiation in a wavelength interval ...... 574 5.3 Radiation properties of real bodies ...... 580 5.3.1 Emissivities ...... 580 5.3.2 The relationships between emissivity, absorptivity and reflectivity. The grey Lambert radiator ...... 582 5.3.2.1 Conclusions from Kirchhoff’s law ...... 582 5.3.2.2 Calculation of absorptivities from emissivities ...... 583 5.3.2.3 The grey Lambert radiator ...... 585 5.3.3 Emissivities of real bodies ...... 587 xvi Contents

5.3.3.1 Electrical insulators ...... 588 5.3.3.2 Electrical conductors (metals) ...... 590 5.3.4 Transparent bodies ...... 593 5.4 Solar radiation ...... 597 5.4.1 Extraterrestrial solar radiation ...... 598 5.4.2 The attenuation of solar radiation in the earth’s atmosphere ... 600 5.4.2.1 Spectral transmissivity ...... 601 5.4.2.2 Molecular and aerosol scattering ...... 604 5.4.2.3 Absorption ...... 605 5.4.3 Direct solar radiation on the ground ...... 606 5.4.4 Diffuse solar radiation and global radiation ...... 608 5.4.5 Absorptivities for solar radiation ...... 611 5.5 Radiative exchange ...... 612 5.5.1 View factors ...... 613 5.5.2 Radiative exchange between black bodies ...... 619 5.5.3 Radiative exchange between grey Lambert radiators ...... 622 5.5.3.1 The balance equations according to the net-radiation method ...... 623 5.5.3.2 Radiative exchange between a radiation source, a radiation receiver and a reradiating wall ...... 624 5.5.3.3 Radiative exchange in a hollow enclosure with two zones . 628 5.5.3.4 The equation system for the radiative exchange between any number of zones ...... 630 5.5.4 Protective radiation shields ...... 633 5.6 Gas radiation ...... 637 5.6.1 Absorption coefficient and optical thickness ...... 638 5.6.2 Absorptivity and emissivity ...... 640 5.6.3 Results for the emissivity ...... 643 5.6.4 Emissivities and mean beam lengths of gas spaces ...... 646 5.6.5 Radiative exchange in a gas filled enclosure ...... 650 5.6.5.1 Black, isothermal boundary walls ...... 650 5.6.5.2 Grey isothermal boundary walls ...... 651 5.6.5.3 Calculation of the radiative exchange in complicated cases 654 5.7 Exercises ...... 655

Appendix A: Supplements 660 A.1 Introduction to tensor notation ...... 660 A.2 Relationship between mean and thermodynamic pressure ...... 662 A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity in cartesian coordinates ...... 663 A.4 Navier-Stokes equations for an incompressible fluid of constant viscosity in cylindrical coordinates ...... 664 A.5 Entropy balance for mixtures ...... 665 Contents xvii

A.6 Relationship between partial and specific enthalpy ...... 666

A.7 Calculation of the constants an of the Graetz-Nusselt problem (3.245) . . 667

Appendix B: Property data 669

Appendix C: Solutions to the exercises 683

Literature 701

Index 720 Nomenclature

Symbol Meaning SI units A area m2 2 Am average area m 2 Aq cross sectional area m 2 Af fin surface area m a thermal diffusivity m2/s a hemispherical total absorptivity — aλ spectral absorptivity — aλ directional spectral absorptivity — 2 at turbulent thermal diffusivity m /s ∗ 2 3 a specific surface area√ m /m 1/2 2 b thermal penetration coefficient, b = λc Ws /(m K) b Laplace constant, b = 2σ/g (L − G)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 c0 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 m2/s 2 Dt turbulent diffusion coefficient m /s d diameter m dA departure diameter of vapour bubbles m dh hydraulic diameter m E irradiance W/m2 2 E0 solar constant W/m 2 Eλ spectral irradiance W/(m μm) e unit vector — F force N Nomenclature xix

FB buoyancy force N Ff friction force N FR resistance force N Fij view factor between surfaces i and j — F (0,λT) fraction function of black radiation — f frequency of vapour bubbles 1/s 3 fj force per unit volume N/m g acceleration due to gravity m/s2 H height m H radiosity W/m2 H enthalpy J H˙ enthalpy flow J/s h Planck constant J s h specific enthalpy J/kg 2 htot specific total enthalpy, htot = h + w /2J/kg hi partial specific enthalpy J/kg Δhv specific enthalpy of vaporization J/kg Δh˜v molar enthalpy of vaporization 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/(m2 s) 2 uj diffusional flux in a particle based system mol/(m s) K incident intensity W/(m2 sr) 2 Kλ incident spectral intensity W/(m μm) k overall heat transfer coefficient W/(m2 K) k extinction coefficient — k Boltzmann constant J/K kG spectral absorption coefficient 1/m 2 kH Henry coefficient N/m kj force per unit mass N/kg k1 rate constant for a homogeneous first order reaction 1/s k1,k1 rate constant for a homogeneous (heterogeneous) first order reaction m/s 2 kn rate constant for a heterogeneous mol/(m s) n-th order reaction (mol/m3)n L length m L total intensity W/(m2 sr) 2 Lλ spectral intensity W/(m μmsr) L0 reference length m 3 LS solubility mol/(m Pa) xx Nomenclature

l length, mixing length m M mass kg M modulus, M = aΔt/Δx2 — M (hemispherical total) emissive power W/m2 2 Mλ spectral emissive power W/(m μm) M˙ mass flow rate kg/s M˜ molecular mass, molar mass kg/mol m optical mass kg/m2 mr relative optical mass — m˙ mass flux kg/(m2 s) N amount of substance mol Ni dimensionless transfer capability (number of transfer units) of the material stream i — N˙ molar flow rate mol/s n refractive index — n normal vector — n˙ 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/(mol K) r radial coordinate m r hemispherical total reflectivity — rλ spectral reflectivity — rλ directional spectral reflectivity — re electrical resistivity Ω m r+ dimensionless radial coordinate — r˙ reaction rate mol/(m3 s) S suppression factor in convective boiling — S entropy J/K s specific entropy J/(kg K) s Laplace transformation parameter 1/s s beam length m s slip factor, s = wG/wL — sl longitudinal pitch m Nomenclature xxi

sq transverse pitch m T thermodynamic temperature K Te eigentemperature K TSt stagnation point temperature K t time s t+ dimensionless time — tk cooling time s 2 tj stress vector N/m tR relaxation time, tR =1/k1 s 2 tD relaxation time of diffusion, tD = L /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 3 VA departure volume of a vapour bubble m v specific volume m3/kg W work J W˙ power density W/m3 W˙ i heat capacity flow rate of a fluid i W/K w velocity m/s w0 reference velocity m/s wS velocity of sound m/s wτ shear stress velocity, wτ = τ0/ m/s w 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 = M˙ G/M˙ L — ∗ xth thermodynamic quality — Y˜ molar content in the gas phase — y coordinate m y˜ 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 — xxii Nomenclature

Greek letters Symbol Meaning SI units α heat transfer coefficient W/(m2 K) 2 αm mean heat transfer coefficient W/(m K) β mass transfer coefficient m/s βm mean mass transfer coefficient m/s β thermal expansion coefficient 1/K β polar angle, zenith angle rad β0 base angle rad Γ˙ mass production rate kg/(m3 s) γ˙ molar production rate mol/(m3 s) Δ difference — δ thickness; boundary layer thickness m δij Kronecker symbol — ε volumetric vapour content — ε∗ volumetric quality — ε hemispherical total emissivity — ελ hemispherical spectral emissivity — ελ directional spectral emissivity — 2 εD turbulent diffusion coefficient m /s εi dimensionless temperature change of the material stream i — ε˙ii dilatation 1/s ε˙ji strain tensor 1/s εp void fraction — 2 εt turbulent viscosity m /s ζ resistance factor — ζ bulk viscosity kg/(m s) η dynamic viscosity kg/(m s) ηf fin efficiency — Θ overtemperature K ϑ temperature K ϑ+ dimensionless temperature — κ isentropic exponent — κG optical thickness of a gas beam — Λ wave length of an oscillation m λ wave length m λ thermal conductivity W/(K m) λt turbulent thermal conductivity W/(K m) μ diffusion resistance factor — ν kinematic viscosity m2/s Nomenclature xxiii

ν frequency 1/s  density kg/m3 σ Stefan-Boltzmann constant W/(m2 K4) σ interfacial tension N/m ξ mass fraction — τ transmissivity — τλ spectral transmissivity — τ shear stress N/m2 2 τji shear stress tensor N/m Φ radiative power, radiation flow W Φ viscous dissipation W/m3 ϕ angle, circumferential angle rad Ψ stream function m2/s ω solid angle sr ω reference velocity m/s ω˙ power density W/m2

Subscripts Symbol Meaning A air, substance A aambient abs absorbed B substance B C condensate, cooling medium diss dissipated E excess, product, solidification e exit, external, eigentemperature 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 xxiv Nomenclature

min minimum n normal direction o outlet, outside Pparticle ref reflected, reference state S solid, bottom product, sun, surroundings s black body, saturation tot total trans transmitted turb turbulent u in particle reference system V boiler W wall, water α start δ at the point y = δ λ spectral ω end 0 reference state; at the point y =0 ∞ at a great distance; in infinity

Dimensionless numbers   3 2 Ar =[(S − F)/F] dPg/ν Archimedes number Bi = αL/λ Biot number BiD = βL/D Biot number for mass transfer Bo =˙q/ (˙mΔhv) boiling number Da = k1 L/D Damk¨ohler number (for 1st order heterogeneous reaction) 2 Ec = w / (cpΔϑ)Eckertnumber Fo = at/L2 Fourier number Fr = w2/ (gx) Froude number Ga = gL3/ν2 Galilei number 3 2 Gr = gβ ΔϑL /ν Grashof number 2 2 Ha = k1L /D Hatta number Le = a/D Lewis number Ma = w/wS Mach number Nu = αL/λ Nusselt number Pe = wL/a P´eclet number Ph = hE/ [c (ϑE − ϑ0)] phase change number Pr = ν/a Prandtl number Ra = GrP r Rayleigh number Re = wL/ν Reynolds number Sc = ν/D Schmidt number Sh = βL/D Sherwood number St = α/ (wcp) Stanton number St =1/P h Stefan number