Heat and mass transfer

Hans Dieter Baehr · Karl Stephan
Heat and Mass Transfer


Hans Dieter Baehr · Karl Stephan

Heat and
Mass Transfer
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:

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
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
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Preface to the second edition

In this revised edition of our book we retained its concept: The main emphasis 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 principles 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 Kăorber, 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,
March 2006

H.D. Baehr
K. Stephan


Preface to the first edition

This book is the English translation of our German publication, which appeared in
1994 with the title Wăarme und Stoă
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 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.


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 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 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 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 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,
Spring 1998

H.D. Baehr
K. Stephan


Contents

xvi

Nomenclature

1

1 Introduction. Technical Applications
1.1 The different types of heat transfer . . . . . . . . . . . . . . . . . . . .
1.1.1 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Steady, one-dimensional conduction of heat . . . . . . . . . . .
1.1.3 Convective heat transfer. Heat transfer coefficient . . . . . . .

1.1.4 Determining heat transfer coefficients. Dimensionless numbers
1.1.5 Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.6 Radiative exchange . . . . . . . . . . . . . . . . . . . . . . . .

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1.2 Overall heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 The overall heat transfer coefficient . . . . . . . . . . . . .

1.2.2 Multi-layer walls . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Overall heat transfer through walls with extended surfaces
1.2.4 Heating and cooling of thin walled vessels . . . . . . . . . .

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1.3 Heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Types of heat exchanger and flow configurations . . . . . . . . . .
1.3.2 General design equations. Dimensionless groups . . . . . . . . . .
1.3.3 Countercurrent and cocurrent heat exchangers . . . . . . . . . . .
1.3.4 Crossflow heat exchangers . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 Operating characteristics of further flow configurations. Diagrams

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1.4 The different types of mass transfer . . . . . . .
1.4.1 Diffusion . . . . . . . . . . . . . . . . . .
1.4.1.1 Composition of mixtures . . . . .
1.4.1.2 Diffusive fluxes . . . . . . . . . .
1.4.1.3 Fick’s law . . . . . . . . . . . . .
1.4.2 Diffusion through a semipermeable plane.
1.4.3 Convective mass transfer . . . . . . . . .
1.5 Mass
1.5.1
1.5.2
1.5.3
1.5.4

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transfer theories . . . . . . . . . . . . . . . . . . .
Film theory . . . . . . . . . . . . . . . . . . . . .
Boundary layer theory . . . . . . . . . . . . . . .
Penetration and surface renewal theories . . . .
Application of film theory to evaporative cooling

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x

Contents

1.6 Overall mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.7 Mass transfer apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.1 Material balances . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.2 Concentration profiles and heights of mass transfer columns . . . .

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1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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2 Heat conduction and mass diffusion
2.1 The heat conduction equation . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Derivation of the differential equation for the temperature field
2.1.2 The heat conduction equation for bodies with constant
material properties . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Temperature dependent material properties . . . . . . . . . . .
2.1.5 Similar temperature fields . . . . . . . . . . . . . . . . . . . . .

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2.2 Steady-state heat conduction . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Geometric one-dimensional heat conduction with heat sources
2.2.2 Longitudinal heat conduction in a rod . . . . . . . . . . . . . .
2.2.3 The temperature distribution in fins and pins . . . . . . . . . .
2.2.4 Fin efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Geometric multi-dimensional heat flow . . . . . . . . . . . . . .
2.2.5.1 Superposition of heat sources and heat sinks . . . . . .
2.2.5.2 Shape factors . . . . . . . . . . . . . . . . . . . . . . .

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2.3 Transient heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 The Laplace transformation . . . . . . . . . . . . . . . . . . . . .
2.3.3 The semi-infinite solid . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3.1 Heating and cooling with different boundary conditions
2.3.3.2 Two semi-infinite bodies in contact with each other . . .
2.3.3.3 Periodic temperature variations . . . . . . . . . . . . . .
2.3.4 Cooling or heating of simple bodies in one-dimensional heat flow
2.3.4.1 Formulation of the problem . . . . . . . . . . . . . . . .
2.3.4.2 Separating the variables . . . . . . . . . . . . . . . . . .
2.3.4.3 Results for the plate . . . . . . . . . . . . . . . . . . . .
2.3.4.4 Results for the cylinder and the sphere . . . . . . . . . .
2.3.4.5 Approximation for large times: Restriction to the first
term in the series . . . . . . . . . . . . . . . . . . . . . .
2.3.4.6 A solution for small times . . . . . . . . . . . . . . . . .
2.3.5 Cooling and heating in multi-dimensional heat flow . . . . . . .
2.3.5.1 Product solutions . . . . . . . . . . . . . . . . . . . . . .
2.3.5.2 Approximation for small Biot numbers . . . . . . . . . .
2.3.6 Solidification of geometrically simple bodies . . . . . . . . . . . .
2.3.6.1 The solidification of flat layers (Stefan problem) . . . . .
2.3.6.2 The quasi-steady approximation . . . . . . . . . . . . .
2.3.6.3 Improved approximations . . . . . . . . . . . . . . . . .
2.3.7 Heat sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

xi

2.3.7.1 Homogeneous heat sources . . . . . . . . . . . . . . . . . . 186
2.3.7.2 Point and linear heat sources . . . . . . . . . . . . . . . . 187
2.4 Numerical solutions to heat conduction problems . . . . . . . . . . . . . .
2.4.1 The simple, explicit difference method for transient heat conduction
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1.1 The finite difference equation . . . . . . . . . . . . . . . .

2.4.1.2 The stability condition . . . . . . . . . . . . . . . . . . . .
2.4.1.3 Heat sources . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Discretisation of the boundary conditions . . . . . . . . . . . . . .
2.4.3 The implicit difference method from J. Crank and P. Nicolson . .
2.4.4 Noncartesian coordinates. Temperature dependent material
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4.1 The discretisation of the self-adjoint differential operator .
2.4.4.2 Constant material properties. Cylindrical coordinates . .
2.4.4.3 Temperature dependent material properties . . . . . . . .
2.4.5 Transient two- and three-dimensional temperature fields . . . . . .
2.4.6 Steady-state temperature fields . . . . . . . . . . . . . . . . . . . .
2.4.6.1 A simple finite difference method for plane, steady-state
temperature fields . . . . . . . . . . . . . . . . . . . . . .
2.4.6.2 Consideration of the boundary conditions . . . . . . . . .
2.5 Mass
2.5.1
2.5.2
2.5.3
2.5.4
2.5.5
2.5.6
2.5.7

diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Remarks on quiescent systems . . . . . . . . . . . . . . . . . . .
Derivation of the differential equation for the concentration field
Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . .
Steady-state mass diffusion with catalytic surface reaction . . . .
Steady-state mass diffusion with homogeneous chemical reaction

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

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2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
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3 Convective heat and mass transfer. Single phase flow

3.1 Preliminary remarks: Longitudinal, frictionless flow over a flat plate . . . 253
3.2 The balance equations . . . . . . . . . . . . .
3.2.1 Reynolds’ transport theorem . . . . .
3.2.2 The mass balance . . . . . . . . . . .
3.2.2.1 Pure substances . . . . . . . .
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 . . . . . . .

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xii

Contents

3.2.4

3.2.5

3.2.3.4 Constitutive equations for the solution of the
momentum equation . . . . . . . . . . . . . . . . . .
3.2.3.5 The Navier-Stokes equations . . . . . . . . . . . . . .

The energy balance . . . . . . . . . . . . . . . . . . . . . . .
3.2.4.1 Dissipated energy and entropy . . . . . . . . . . . .
3.2.4.2 Constitutive equations for the solution of the energy
equation . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4.3 Some other formulations of the energy equation . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Influence of the Reynolds number on the flow . . . . . . . . . . . . . . . . 287
3.4 Simplifications to the Navier-Stokes equations
3.4.1 Creeping flows . . . . . . . . . . . . . .
3.4.2 Frictionless flows . . . . . . . . . . . . .
3.4.3 Boundary layer flows . . . . . . . . . .
3.5 The boundary layer equations . . . . . . .
3.5.1 The velocity boundary layer . . . .
3.5.2 The thermal boundary layer . . . .
3.5.3 The concentration boundary layer .
3.5.4 General comments on the solution of

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3.6 Influence of turbulence on heat and mass transfer . . . . . . . . . . . . . 304
3.6.1 Turbulent flows near solid walls . . . . . . . . . . . . . . . . . . . . 308
3.7 External forced flow . . . . . . . . . . . . . .
3.7.1 Parallel flow along a flat plate . . . .
3.7.1.1 Laminar boundary layer . . .
3.7.1.2 Turbulent flow . . . . . . . .
3.7.2 The cylinder in crossflow . . . . . . .
3.7.3 Tube bundles in crossflow . . . . . . .
3.7.4 Some empirical equations for heat and
external forced flow . . . . . . . . . .

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

3.8.1.1 Hydrodynamic, fully developed, laminar flow . . . . .
3.8.1.2 Thermal, fully developed, laminar flow . . . . . . . . .
3.8.1.3 Heat transfer coefficients in thermally fully developed,
laminar flow . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1.4 The thermal entry flow with fully developed velocity
profile . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1.5 Thermally and hydrodynamically developing flow . . .
3.8.2 Turbulent flow in circular tubes . . . . . . . . . . . . . . . . .
3.8.3 Packed beds . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.4 Fluidised beds . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.5 Some empirical equations for heat and mass transfer in flow
through channels, packed and fluidised beds . . . . . . . . . . .

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.

341
341
342
344

. . 346
.

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.

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.

349
354
355
357
361

. . 370

3.9 Free flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373


Contents

3.9.1
3.9.2
3.9.3
3.9.4

The momentum equation . . . . . . . . . . . . . . . .

Heat transfer in laminar flow on a vertical wall . . . .
Some empirical equations for heat transfer in free flow
Mass transfer in free flow . . . . . . . . . . . . . . . .

xiii

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376
379
384
386

3.10 Overlapping of free and forced flow . . . . . . . . . . . . . . . . . . . . . . 387
3.11 Compressible flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
3.11.1 The temperature field in a compressible flow . . . . . . . . . . . . 389
3.11.2 Calculation of heat transfer . . . . . . . . . . . . . . . . . . . . . . 396
3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
405

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 Deviations from Nusselt’s film condensation theory . . .
4.1.4 Influence of non-condensable gases . . . . . . . . . . . .
4.1.5 Film condensation in a turbulent film . . . . . . . . . .

4.1.6 Condensation of flowing vapours . . . . . . . . . . . . .
4.1.7 Dropwise condensation . . . . . . . . . . . . . . . . . .
4.1.8 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 size of a condenser . . . . . . .
4.1.9 Some empirical equations . . . . . . . . . . . . . . . . .

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405
406
408
412
416
422
426
431
435
439
443
445
446

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 . . . . . . . . . . . . .
4.2.4 Boiling in free flow. The Nukijama curve . . . . . . . . . . . . .
4.2.5 Stability during boiling in free flow . . . . . . . . . . . . . . . . .
4.2.6 Calculation of heat transfer coefficients for boiling in free flow .
4.2.7 Some empirical equations for heat transfer during nucleate
boiling in free flow . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.8 Two-phase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.8.1 The different flow patterns . . . . . . . . . . . . . . . . .
4.2.8.2 Flow maps . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.8.3 Some basic terms and definitions . . . . . . . . . . . . .
4.2.8.4 Pressure drop in two-phase flow . . . . . . . . . . . . . .

4.2.8.5 The different heat transfer regions in two-phase flow . .
4.2.8.6 Heat transfer in nucleate boiling and convective
evaporation . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.8.7 Critical boiling states . . . . . . . . . . . . . . . . . . . .
4.2.8.8 Some empirical equations for heat transfer in two-phase
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.9 Heat transfer in boiling mixtures . . . . . . . . . . . . . . . . . .

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448
449
453
456
460
461
465

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468
472
473
475
476
479
487

. 489
. 492
. 495
. 496


xiv

Contents

4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
503

5 Thermal radiation
5.1 Fundamentals. Physical quantities . . . . . . . . . . . . .
5.1.1 Thermal radiation . . . . . . . . . . . . . . . . . .
5.1.2 Emission of radiation . . . . . . . . . . . . . . . .
5.1.2.1 Emissive power . . . . . . . . . . . . . . .
5.1.2.2 Spectral intensity . . . . . . . . . . . . . .
5.1.2.3 Hemispherical spectral emissive power and

5.1.2.4 Diffuse radiators. Lambert’s cosine law . .
5.1.3 Irradiation . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Absorption of radiation . . . . . . . . . . . . . . .
5.1.5 Reflection of radiation . . . . . . . . . . . . . . . .
5.1.6 Radiation in an enclosure. Kirchhoff’s law . . . .

. . .
. . .
. . .
. . .
. . .
total
. . .
. . .
. . .
. . .
. . .

. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
intensity
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .

503
504
506
506
507
509
513
514
517
522
524

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

. . . . . . .
. . . . . . .
. . . . . . .
wavelength
. . . . . . .

527
527
528

5.3 Radiation properties of real bodies . . . . . . . . . . . . . . .
5.3.1 Emissivities . . . . . . . . . . . . . . . . . . . . . . . .

5.3.2 The relationships between emissivity, absorptivity and
The grey Lambert radiator . . . . . . . . . . . . . . .
5.3.2.1 Conclusions from Kirchhoff’s law . . . . . . .
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 . . . . . . . . . . . . . . . . . . . .

. . . . . . .
. . . . . . .
reflectivity.
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .

537
537

532

540
540
541

542
544
545
548
550

5.4 Solar radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Extraterrestrial solar radiation . . . . . . . . . . . . . . . . .
5.4.2 The attenuation of solar radiation in the earth’s atmosphere
5.4.2.1 Spectral transmissivity . . . . . . . . . . . . . . . . .
5.4.2.2 Molecular and aerosol scattering . . . . . . . . . . .
5.4.2.3 Absorption . . . . . . . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . .

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555
555
558
558
561
562
564
566
568

5.5 Radiative exchange . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 View factors . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Radiative exchange between black bodies . . . . . . . . . . .
5.5.3 Radiative exchange between grey Lambert radiators . . . . .

5.5.3.1 The balance equations according to the net-radiation
method . . . . . . . . . . . . . . . . . . . . . . . . .

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569
570
576
579

. . . 580


Contents

5.5.4

5.5.3.2 Radiative exchange between a radiation source, a radiation
receiver and a reradiating wall . . . . . . . . . . . . . . . .
5.5.3.3 Radiative exchange in a hollow enclosure with two zones .
5.5.3.4 The equation system for the radiative exchange between
any number of zones . . . . . . . . . . . . . . . . . . . . .
Protective radiation shields . . . . . . . . . . . . . . . . . . . . . .

5.6 Gas radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Absorption coefficient and optical thickness . . . . . . . . . . . . .
5.6.2 Absorptivity and emissivity . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Results for the emissivity . . . . . . . . . . . . . . . . . . . . . . .
5.6.4 Emissivities and mean beam lengths of gas spaces . . . . . . . . .
5.6.5 Radiative exchange in a gas filled enclosure . . . . . . . . . . . . .
5.6.5.1 Black, isothermal boundary walls . . . . . . . . . . . . . .
5.6.5.2 Grey isothermal boundary walls . . . . . . . . . . . . . . .
5.6.5.3 Calculation of the radiative exchange in complicated cases

xv

581
585
587
590
594
595
597
600
603
607
607

608
611

5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
Appendix A: Supplements

617

A.1 Introduction to tensor notation . . . . . . . . . . . . . . . . . . . . . . . . 617
A.2 Relationship between mean and thermodynamic pressure . . . . . . . . . 619
A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity
in cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
A.4 Navier-Stokes equations for an incompressible fluid of constant viscosity
in cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
A.5 Entropy balance for mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 622
A.6 Relationship between partial and specific enthalpy . . . . . . . . . . . . . 623
A.7 Calculation of the constants an of a Graetz-Nusselt problem (3.246) . . . 624
Appendix B: Property data

626

Appendix C: Solutions to the exercises

640

Literature

654

Index

671


Nomenclature

Symbol

Meaning

A
Am
Aq
Af
a
a
aλ
a
λ
at
a∗
b
b
C
C
c
c
c
c0
cf
cp

cR
D
Dt
d
dA
dh
E
E0
Eλ
e
F

area
average area
cross sectional area
fin surface area
thermal diffusivity
hemispherical total absorptivity
spectral absorptivity
directional spectral absorptivity
turbulent thermal diffusivity
specific surface area
√
thermal penetration coefficient, b = λc


Laplace constant, b = 2σ/g (L − G )
circumference, perimeter
heat capacity flow ratio
specific heat capacity
concentration

propagation velocity of electromagnetic waves
velocity of light in a vacuum
friction factor
specific heat capacity at constant pressure
resistance factor
binary diffusion coefficient
turbulent diffusion coefficient
diameter
departure diameter of vapour bubbles
hydraulic diameter
irradiance
solar constant
spectral irradiance
unit vector
force

SI units
m2
m2
m2
m2
m2 /s
—
—
—
m2 /s
m2 /m3
1/2
W s /(m2 K)
m

m
—
J/(kg K)
mol/m3
m/s
m/s
—
J/(kg K)
—
m2 /s
m2 /s
m
m
m
W/m2
W/m2
W/(m2 µm)
—
N


Nomenclature

FB
Ff
FR
Fij
F (0, λT )
f
fj

g
H
H
H
H˙
h
h
htot
hi
∆hv
˜v
∆h
I
I
j
j∗
uj
K
Kλ
k
k
k
kG
kH
kj
k1
k1
, k1


kn


L
L

Lλ
L0
LS

buoyancy force
friction force
resistance force
view factor between surfaces i and j
fraction function of black radiation
frequency of vapour bubbles
force per unit volume
acceleration due to gravity
height
radiosity
enthalpy
enthalpy flow
Planck constant
specific enthalpy
specific total enthalpy, htot = h + w2 /2
partial specific enthalpy
specific enthalpy of vaporization
molar enthalpy of vaporization
momentum
directional emissive power
diffusional flux
diffusional flux in a centre of gravity system
diffusional flux in a particle based system
incident intensity
incident spectral intensity
overall heat transfer coefficient

extinction coefficient
Boltzmann constant
spectral absorption coefficient
Henry coefficient
force per unit mass
rate constant for a homogeneous
first order reaction
rate constant for a homogeneous (heterogeneous)
first order reaction
rate constant for a heterogeneous
n-th order reaction
length
total intensity
spectral intensity
reference length
solubility

xvii

N
N
N
—
—
1/s
N/m3
m/s2
m
W/m2
J

J/s
Js
J/kg
J/kg
J/kg
J/kg
J/mol
kg m/s
W/(m2 sr)
mol/(m2 s)
kg/(m2 s)
mol/(m2 s)
W/(m2 sr)
W/(m2 µm)
W/(m2 K)
—
J/K
1/m
N/m2
N/kg
1/s
m/s
mol/(m2 s)
(mol/m3 )n
m
W/(m2 sr)
W/(m2 µm sr)
m
mol/(m3 Pa)

xviii

l
M
M
M
Mλ
M˙
˜
M
m
mr
m
˙
N
Ni
N˙
n
n
n˙
P
Pdiss
p
p+
Q
Q˙
q˙
R
Rcond

Rm
r
r
rλ
rλ

re
r+
r˙
S
S
s
s
s
s
sl

Nomenclature

length, mixing length
mass
modulus, M = a∆t/∆x2
(hemispherical total) emissive power
spectral emissive power
mass flow rate
molecular mass, molar mass
optical mass
relative optical mass
mass flux
amount of substance
dimensionless transfer capability (number of

transfer units) of the material stream i
molar flow rate
refractive index
normal vector
molar flux
power
dissipated power
pressure
dimensionless pressure
heat
heat flow
heat flux
radius
resistance to thermal conduction
molar (universal) gas constant
radial coordinate
hemispherical total reflectivity
spectral reflectivity
directional spectral reflectivity
electrical resistivity
dimensionless radial coordinate
reaction rate
suppression factor in convective boiling
entropy
specific entropy
Laplace transformation parameter
beam length
slip factor, s = wG /wL
longitudinal pitch

m
kg
—
W/m2
W/(m2 µm)
kg/s
kg/mol
kg/m2
—
kg/(m2 s)
mol
—
mol/s
—
—
mol/(m2 s)
W
W
Pa
—
J
W
W/m2
m
K/W
J/(mol K)
m
—
—
—

Ωm
—
mol/(m3 s)
—
J/K
J/(kg K)
1/s
m
—
m


Nomenclature

sq
T
Te
TSt
t
t+
tk
tj
tR
tD
U
u
u
u
V
VA

v
W
˙
W
˙i
W
w
w0
wS
wτ
w

w+
X
˜
X
x
x
˜
x+
x∗
x∗th
Y˜
y
y˜
y+
z
z
z+
zR

transverse pitch
thermodynamic temperature
eigentemperature
stagnation point temperature
time
dimensionless time
cooling time
stress vector
relaxation time, tR = 1/k1
relaxation time of diffusion, tD = L2 /D
internal energy
average molar velocity
specific internal energy
Laplace transformed temperature
volume
departure volume of a vapour bubble
specific volume
work
power density
heat capacity flow rate of a fluid i
velocity
reference velocity
velocity of sound


shear stress velocity, wτ = τ0 /
fluctuation velocity
dimensionless velocity
moisture content; Lockhart-Martinelli parameter
molar content in the liquid phase
coordinate

mole fraction in the liquid
dimensionless x-coordinate
quality, x∗ = M˙ G /M˙ L
thermodynamic quality
molar content in the gas phase
coordinate
mole fraction in the gas phase
dimensionless y-coordinate
number
axial coordinate
dimensionless z-coordinate
number of tube rows

xix

m
K
K
K
s
—
s
N/m2
s
s
J
m/s
J/kg
K
m3

m3
m3 /kg
J
W/m3
W/K
m/s
m/s
m/s
m/s
m/s
—
—
—
m
—
—
—
—
—
m
—
—
—
m
—
—


xx

Nomenclature

Greek letters
Symbol

Meaning

α
αm
β
βm
β
β
β0
Γ˙

heat transfer coefficient
mean heat transfer coefficient
mass transfer coefficient
mean mass transfer coefficient
thermal expansion coefficient
polar angle, zenith angle
base angle
mass production rate
molar production rate
difference
thickness; boundary layer thickness
Kronecker symbol
volumetric vapour content
volumetric quality

hemispherical total emissivity
hemispherical spectral emissivity
directional spectral emissivity
turbulent diffusion coefficient
dimensionless temperature change of the material stream i
dilatation
strain tensor
void fraction
turbulent viscosity
resistance factor
bulk viscosity
dynamic viscosity
fin efficiency
overtemperature
temperature
dimensionless temperature
isentropic exponent
optical thickness of a gas beam
wave length of an oscillation
wave length
thermal conductivity
turbulent thermal conductivity
diffusion resistance factor
kinematic viscosity

γ˙
∆
δ
δij
ε

ε∗
ε
ελ
ε
λ
εD
εi
ε˙ii
ε˙ji
εp
εt
ζ
ζ
η
ηf
Θ
ϑ
ϑ+
κ
κG
Λ
λ
λ
λt
µ
ν

SI units
W/(m2 K)
W/(m2 K)
m/s

m/s
1/K
rad
rad
kg/(m3 s)
mol/(m3 s)
—
m
—
—
—
—
—
—
m2 /s
—
1/s
1/s
—
m2 /s
—
kg/(m s)
kg/(m s)
—
K
K
—
—
—
m

m
W/(K m)
W/(K m)
—
m2 /s


Nomenclature

ν

σ
σ
ξ
τ
τλ
τ
τji
Φ
Φ
ϕ
Ψ
ω
ω
ω˙

frequency
density
Stefan-Boltzmann constant
interfacial tension

mass fraction
transmissivity
spectral transmissivity
shear stress
shear stress tensor
radiative power, radiation flow
viscous dissipation
angle, circumferential angle
stream function
solid angle
reference velocity
power density

Subscripts
Symbol

Meaning

A
abs
B
C
diss
E
e
eff
eq
F
f
G

g
I
i
id
in
K
L
lam
m
max
min

air, substance A
absorbed
substance B
condensate, cooling medium
dissipated
excess, product, solidification
exit, outlet
effective
equilibrium
fluid, feed
fin, friction
gas
geodetic, base material
at the phase interface
inner, inlet
ideal
incident radiation, irradiation
substance K

liquid
laminar
mean, molar (based on the amount of substance)
maximum
minimum

xxi

1/s
kg/m3
W/(m2 K4 )
N/m
—
—
—
N/m2
N/m2
W
W/m3
rad
m2 /s
sr
m/s
W/m2


xxii

n
o

P
ref
S
s
tot
trans
turb
u
V
W
α
δ
λ
ω
0
∞

Nomenclature

normal direction
outer, outside
particle
reflected, reference state
solid, bottom product, sun, surroundings
black body, saturation
total
transmitted
turbulent
in particle reference system
boiler

wall, water
start
at the point y = δ
spectral
end
reference state; at the point y = 0
at a great distance; in infinity

Dimensionless numbers


Ar = [(S − F )/F ] d3P g/ν 2
Bi = αL/λ
BiD = βL/D
Bo = q/
˙ (m∆h
˙
v)
Da = k1

L/D
Ec = w2 / (cp ∆ϑ)
Fo = at/L2
Fr = w2 / (gx)
Ga = gL3 /ν 2
Gr = gβ∆ϑL3 /ν 2

2
Ha = k1 L2 /D
Le = a/D
Ma = w/wS
N u = αL/λ

P e = wL/a
P h = hE / [c (ϑE − ϑ0 )]
P r = ν/a
Ra = GrP r
Re = wL/ν
Sc = ν/D
Sh = βL/D
St = α/ (wcp )
St = 1/P h

Archimedes number
Biot number
Biot number for mass transfer
boiling number
Damkă
ohler number (for 1st order heterogeneous reaction)
Eckert number
Fourier number
Froude number
Galilei number
Grashof number
Hatta number
Lewis number
Mach number
Nusselt number
P´eclet number
phase change number
Prandtl number
Rayleigh number
Reynolds number

Schmidt number
Sherwood number
Stanton number
Stefan 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 apparatus 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 information 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-called phases, 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 temperature T is used in heat
transfer. However with the exception of radiative heat transfer the zero point
of the thermodynamic temperature scale is not needed, usually only temperature
differences are 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 ϑ, defined as
ϑ := T − T0

(1.1)

where T0 can be chosen arbitrarily and is usually set at a temperature that best
fits the problem that requires solving. When T0 = 273.15 K then ϑ will be the

Celsius temperature. The value for T0 does not normally need to be specified as
temperature differences are independent of T0 .

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 of heat 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 vector x. It can also
be dependent on time t. The heat flux q˙ is defined in such a way that the heat
flow dQ˙ through a surface element dA is
˙
˙ cos β dA .

dQ˙ = q(x,
t) n dA = |q|

(1.3)

Here n is the unit vector normal (outwards) to the surface, which with q˙ forms
the angle β, Fig. 1.1. The heat flow dQ˙ is greatest when q˙ is perpendicular to dA
making β = 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 vector n
and heat flux vector q˙

SI unit J/s = W. Heat flux is the heat flow per surface area with units J/s m2 =
W/m2 .
The transport of energy by heat conduction is due to a temperature gradient
in the substance. The temperature ϑ changes with both position and time. All
temperatures form a temperature field
ϑ = ϑ(x, t) .
Steady temperature fields are not dependent on the time t. 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 ϑ, 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 ϑ, from
those with a lower temperature than ϑ. The greatest temperature change occurs
normal to the isotherm, and is given by the temperature gradient

grad ϑ =

∂ϑ
∂ϑ
∂ϑ
e +
e +
e
∂x x ∂y y ∂z z

(1.4)

where ex , ey and ez 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: Point P on the isotherm
ϑ = const with the temperature gradient grad ϑ from (1.4) and the heat
flux vector q˙ from (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 effect may
be assumed, allowing the heat flux to be written as
q˙ = −λ grad ϑ .

(1.5)


4

1 Introduction. Technical Applications

This is J. B. Fourier’s1 basic law for the conduction of heat, from 1822. The minus
sign in this equation is accounting for the 2nd law of thermodynamics: heat flows
in the direction of falling temperature, Fig. 1.2. The constant of proportion in
(1.5) is a property of the material, the thermal conductivity
λ = λ(ϑ, p) .
It is dependent on both the temperature ϑ and pressure p, and in mixtures on
the composition. The thermal conductivity λ is a scalar as long as the material is
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 such non-isotropic medium λ is a
tensor of second order, and the vectors q˙ and grad ϑ form an angle in contrast to
Fig. 1.2. In isotropic substances the heat flux vector is always perpendicular to
the isothermal surface. From (1.3) and (1.5) the heat flow dQ˙ through a surface
element dA oriented in any direction is
dQ˙ = −λ ( grad ϑ) n dA = −λ

∂ϑ
dA .
∂n

(1.6)

Here ∂ϑ/∂n is the derivative of ϑ 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

Substance
Silver
Copper
Aluminium 99.2 %
Iron
Steel Alloys
Brickwork
Foam Sheets

λ in W/K m
427
399
209
81
13 . . . 48
0.5 . . . 1.3
0.02 . . . 0.09

Substance
Water
Hydrocarbons
CO2
Air
Hydrogen
Krypton
R 123

λ in W/K m
0.598
0.10 . . . 0.15

0.0162
0.0257
0.179
0.0093
0.0090

The thermal conductivity, with SI units of W/K m, is one of the most important 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 of λ are 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 chaleur” appeared in 1822. It is the first comprehensive mathematical
theory of conduction and cointains the “Fourier Series” 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
for λ. 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 independent of time, so called steady conduction, in a flat plate, in a hollow cylinder
and in a hollow sphere will be considered in this section. The assumption is made
that heat flows in only one direction, perpendicular to the plate surface, and radially 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. a Temperature profile in a flat plate of
thickness δ = r2 − r1 , b Temperature profile in a hollow cylinder (tube wall) or hollow
sphere of inner radius r1 and outer radius r2

The position coordinate in all three cases is designated by r. The surfaces
r = const are isothermal surfaces; and therefore ϑ = ϑ(r). We assume that ϑ has
the constant values ϑ = ϑW1 , when r = r1 , and ϑ = ϑW2 , when r = r2 . These
two surface temperatures shall be given. A relationship between the heat flow
Q˙ through the flat or curved walls, and the temperature difference ϑW1 − ϑW2 ,
must be found. For illustration we assume ϑW1 > ϑW2 , without loss of generality.
Therefore heat flows in the direction of increasing r. The heat flow Q˙ has a certain
value, which on the inner and outer surfaces, and on each isotherm r = const is
the same, as in steady conditions no energy can be stored in the wall.
Fourier’s law gives the following for the heat flow
dϑ
A(r) .
(1.7)
Q˙ = q(r)A(r)
˙
= −λ(ϑ)
dr

In the flat wall A is not dependent on r: A = A1 = A2 . If the thermal conductivity
is constant, then the temperature gradient dϑ/dr will also be constant. The steady