process heat transfer principles and applications

Process Heat Transfer
Dedication
This book is dedicated to C.C.S.
Process Heat Transfer
Principles and Applications
R.W. Serth
Department of Chemical and Natural Gas Engineering,
Texas A&M University-Kingsville,
Kingsville, Texas, USA
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British Library Cataloguing in Publication Data
Serth, R. W.
Process heat transfer : principles and applications
1. Heat - Transmission 2. Heat exchangers 3. Heat exchangers - Design
4. Heat - Transmission - Computer programs
I. Title
621.4

022
Library of Congress Catalog number: 2006940583
ISBN: 978-0-12-373588-1
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Contents
Preface viii
Conversion Factors x
Physical Constants xi
Acknowledgements xii
1 Heat Conduction 1
1.1 Introduction 2
1.2 Fourier’s Law of Heat Conduction 2
1.3 The Heat Conduction Equation 6
1.4 Thermal Resistance 15
1.5 The Conduction Shape Factor 19

1.6 Unsteady-State Conduction 24
1.7 Mechanisms of Heat Conduction 31
2 Convective Heat Transfer 43
2.1 Introduction 44
2.2 Combined Conduction and Convection 44
2.3 Extended Surfaces 47
2.4 Forced Convection in Pipes and Ducts 53
2.5 Forced Convection in External Flow 62
2.6 Free Convection 65
3 Heat Exchangers 85
3.1 Introduction 86
3.2 Double-Pipe Equipment 86
3.3 Shell-and-Tube Equipment 87
3.4 The Overall Heat-Transfer Coefficient 93
3.5 The LMTD Correction Factor 98
3.6 Analysis of Double-Pipe Exchangers 102
3.7 Preliminary Design of Shell-and-Tube Exchangers 106
3.8 Rating a Shell-and-Tube Exchanger 109
3.9 Heat-Exchanger Effectiveness 114
4 Design of Double-Pipe Heat Exchangers 127
4.1 Introduction 128
4.2 Heat-Transfer Coefficients for Exchangers without Fins 128
4.3 Hydraulic Calculations for Exchangers without Fins 128
4.4 Series/Parallel Configurations of Hairpins 131
4.5 Multi-tube Exchangers 132
4.6 Over-Surface and Over-Design 133
4.7 Finned-Pipe Exchangers 141
4.8 Heat-Transfer Coefficients and Friction Factors for Finned Annuli 143
4.9 Wall Temperature for Finned Pipes 145
4.10 Computer Software 152

vi CONTENTS
5 Design of Shell-and-Tube Heat Exchangers 187
5.1 Introduction 188
5.2 Heat-Transfer Coefficients 188
5.3 Hydraulic Calculations 189
5.4 Finned Tubing 192
5.5 Tube-Count Tables 194
5.6 Factors Affecting Pressure Drop 195
5.7 Design Guidelines 197
5.8 Design Strategy 201
5.9 Computer software 218
6 The Delaware Method 245
6.1 Introduction 246
6.2 Ideal Tube Bank Corr elations 246
6.3 Shell-Side Heat-Transfer Coefficient 248
6.4 Shell-Side Pressure Drop 250
6.5 The Flow Areas 254
6.6 Correlations for the Correction Factors 259
6.7 Estimation of Clearances 260
7 The Stream Analysis Method 277
7.1 Introduction 278
7.2 The Equivalent Hydraulic Network 278
7.3 The Hydraulic Equations 279
7.4 Shell-Side Pressure Drop 281
7.5 Shell-Side Heat-Transfer Coefficient 281
7.6 Temperature Profile Distortion 282
7.7 The Wills–Johnston Method 284
7.8 Computer Software 295
8 Heat-Exchanger Networks 327
8.1 Introduction 328

8.2 An Example: TC3 328
8.3 Design Targets 329
8.4 The Problem Table 329
8.5 Composite Curves 331
8.6 The Grand Composite Curve 334
8.7 Significance of the Pinch 335
8.8 Threshold Problems and Utility Pinches 337
8.9 Feasibility Criteria at the Pinch 337
8.10 Design Strategy 339
8.11 Minimum-Utility Design for TC3 340
8.12 Network Simplification 344
8.13 Number of Shells 347
8.14 Targeting for Number of Shells 348
8.15 Area Targets 353
8.16 The Driving Force Plot 356
8.17 Super Targeting 358
8.18 Targeting by Linear Programming 359
8.19 Computer Software 361
CONTENTS vii
9 Boiling Heat Transfer 385
9.1 Introduction 386
9.2 Pool Boiling 386
9.3 Correlations for Nucleate Boiling on Horizontal Tubes 387
9.4 Two-Phase Flow 402
9.5 Convective Boiling in Tubes 416
9.6 Film Boiling 428
10 Reboilers 443
10.1 Introduction 444
10.2 Types of Reboilers 444
10.3 Design of Kettle Reboilers 449

10.4 Design of Horizontal Thermosyphon Reboilers 467
10.5 Design of Vertical Thermosyphon Reboilers 473
10.6 Computer Software 488
11 Condensers 539
11.1 Introduction 540
11.2 Types of Condensers 540
11.3 Condensation on a Vertical Surface: Nusselt Theory 545
11.4 Condensation on Horizontal Tubes 549
11.5 Modifications of Nusselt Theory 552
11.6 Condensation Inside Horizontal Tubes 562
11.7 Condensation on Finned Tubes 568
11.8 Pressure Drop 569
11.9 Mean Temperature Difference 571
11.10 Multi-component Condensation 590
11.11 Computer Software 595
12 Air-Cooled Heat Exchangers 629
12.1 Introduction 630
12.2 Equipment Description 630
12.3 Air-Side Heat-Transfer Coefficient 637
12.4 Air-Side Pressure Drop 638
12.5 Overall Heat-Transfer Coefficient 640
12.6 Fan and Motor Sizing 640
12.7 Mean Temperature Difference 643
12.8 Design Guidelines 643
12.9 Design Strategy 644
12.10 Computer Software 653
Appendix 681
Appendix A Thermophysical Properties of Materials 682
Appendix B Dimensions of Pipe and Tubing 717
Appendix C Tube-Count Tables 729

Appendix D Equivalent Lengths of Pipe Fittings 737
Appendix E Properties of Petroleum Streams 740
Index 743
Preface
This book is based on a course in process heat transfer that I have taught for many years. The course
has been taken by seniors and first-year graduate students who have completed an introductory
course in engineering heat transfer. Although this background is assumed, nearly all students need
some review before proceeding to more advanced material. For this r eason, and also to make the
book self-contained, the first three chapters provide a r eview of essential material normally covered
in an introductory heat transfer course. Furthermore, the book is intended for use by practicing
engineers as well as university students, and it has been written with the aim of facilitating self-study.
Unlike some books in this field, no attempt is made herein to cover the entire panoply of heat trans-
fer equipment. Instead, the book focuses on the types of equipment most widely used in the chemical
process industries, namely, shell-and-tube heat exchangers (including condensers and reboilers),
air-cooled heat exchangers and double-pipe (hairpin) heat exchangers. Within the confines of a sin-
gle volume, this approach allows an in-depth treatment of the material that is most relevant from an
industrial perspective, and provides students with the detailed knowledge needed for engineering
practice. This approach is also consistent with the time available in a one-semester course.
Design of double-pipe exchangers is presented in Chapter 4. Chapters 5–7 comprise a unit dealing
with shell-and-tube exchangers in operations involving single-phase fluids. Design of shell-and-tube
exchangers is covered in Chapter 5 using the Simplified Delaware method for shell-side calcula-
tions. For pedagogical reasons, more sophisticated methods for performing shell-side heat-transfer
and pressure-drop calculations are presented separately in Chapter 6 (full Delaware method) and
Chapter 7 (Stream Analysis method). Heat exchanger networks are covered in Chapter 8. I nor-
mally pr esent this topic at this point in the course to provide a change of pace. However, Chapter
8 is essentially self-contained and can, therefore, be covered at any time. Phase-change operations
are covered in Chapters 9–11. Chapter 9 presents the basics of boiling heat transfer and two-phase
flow. The latter is encountered in both Chapter 10, which deals with the design of reboilers, and
Chapter 11, which covers condensation and condenser design. Design of air-cooled heat exchang-
ers is presented in Chapter 12. The material in this chapter is essentially self-contained and, hence,

it can be covered at any time.
Since the primary goal of both the book and the course is to provide students with the knowl-
edge and skills needed for modern industrial practice, computer applications play an integral role,
and the book is intended for use with one or more commercial software packages. HEXTRAN
(SimSci-Esscor), HTRI Xchanger Suite (Heat Transfer Research, Inc.) and the HTFS Suite (Aspen
Technology, Inc.) are used in the book, along with HX-Net (Aspen Technology, Inc.) for pinch
calculations. HEXTRAN affords the most complete coverage of topics, as it handles all types of heat
exchangers and also performs pinch calculations for design of heat exchanger networks. It does
not perform mechanical design calculations for shell-and-tube exchangers, however, nor does it
generate detailed tube layouts or setting plans. Furthermore, the methodology used by HEXTRAN
is based on publicly available technology and is generally less refined than that of the other software
packages. The HTRI and HTFS packages use proprietary methods developed by their respective
research organizations, and are similar in their level of refinement. HTFS Suite handles all types
of heat exchangers; it also performs mechanical design calculations and develops detailed tube
layouts and setting plans for shell-and-tube exchangers. HTRI Xchanger Suite lacks a mechanical
design feature, and the module for hairpin exchangers is not included with an academic license.
Neither HTRI nor HTFS has the capability to perform pinch calculations.
As of this writing, Aspen Technology is not providing the TASC and ACOL modules of the HTFS
Suite under its university program. Instead, it is offering the HTFS-plus design package. This
package basically consists of the TASC and ACOL computational engines combined with slightly
modified GUI’s from the corresponding BJAC programs (HETRAN and AEROTRAN), and packaged
with the BJAC TEAMS mechanical design program. This package differs greatly in appearance and
to some extent in available features from HTFS Suite. However, most of the results presented in the
text using TASC and ACOL can be generated using the HTFS-plus package.
PREFACE ix
Software companies are continually modifying their products, making differences between the
text and current versions of the software packages unavoidable. However, many modifications
involve only superficial changes in format that have little, if any, effect on results. More substantive
changes occur less frequently, and even then the effects tend to be relatively minor. Nevertheless,
readers should expect some divergence of the software from the versions used herein, and they

should not be unduly concerned if their results differ somewhat from those presented in the text.
Indeed, even the same version of a code, when run on different machines, can produce slightly
different results due to differences in round-off errors. With these caveats, it is hoped that the
detailed computer examples will prove helpful in learning to use the software packages, as well as
in understanding their idiosyncrasies and limitations.
I have made a concerted effort to introduce the complexities of the subject matter gradually
throughout the book in order to avoid overwhelming the reader with a massive amount of detail
at any one time. As a result, information on shell-and-tube exchangers is spread over a number of
chapters, and some of the finer details are introduced in the context of example problems, including
computer examples. Although there is an obvious downside to this strategy, I nevertheless believe
that it represents good pedagogy.
Both English units, which are still widely used by American industry, and SI units are used in this
book. Students in the United States need to be pr oficient in both sets of units, and the same is true
of students in countries that do a large amount of business with U.S. firms. In order to minimize
the need for unit conversion, however, working equations are either given in dimensionless form
or, when this is not practical, they are given in both sets of units.
I would like to take this opportunity to thank the many students who have contributed to this
effort over the years, both directly and indirectly through their participation in my course. I would
also like to express my deep appreciation to my colleagues in the Department of Chemical and
Natural Gas Engineering at TAMUK, Dr. Ali Pilehvari and Mrs. Wanda Pounds. Without their help,
encouragement and friendship, this book would not have been written.
Conversion Factors
Acceleration 1 m/s
2
=4.2520 ×10
7
ft/h
2
Area 1m
2

=10.764 ft
2
Density 1 kg/m
3
=0.06243 lbm/ft
3
Energy 1J =0.239 cal =9.4787 ×10
−4
Btu
Force 1N =0.22481 lbf
Fouling factor 1m
2
·K/W =5.6779 h ·ft
2
·
◦
F/Btu
Heat capacity flow rate 1 kW/K =1 kW/
◦
C
=1895.6 Btu/h ·
◦
F
Heat flux 1W/m
2
=0.3171 Btu/h ·ft
2
Heat generation rate 1 W/m
3
=0.09665 Btu/h ·ft

3
Heat transfer coefficient 1 W/m
2
·K =0.17612 Btu/h ·ft
2
·
◦
F
Heat transfer rate 1W =3.4123 Btu/h
Kinematic viscosity and thermal 1m
2
/s =3.875 ×10
4
ft
2
/h
diffusivity
Latent heat and specific enthalpy 1 kJ/kg =0.42995 Btu/lbm
Length 1m =3.2808 ft
Mass 1kg =2.2046 lbm
Mass flow rate 1 kg/s =7936.6 lbm/h
Mass flux 1 kg/s ·m
2
=737.35 lbm/h ·ft
2
Power 1kW =3412 Btu/h
=1.341 hp
Pressure (stress) 1Pa(1N/m
2
) =0.020886 lbf/ft

2
=1.4504 ×10
−4
psi
=4.015 ×10
−3
in. H
2
O
Pressure 1.01325 ×10
5
Pa =1 atm
=14.696 psi
=760 torr
=406 .8 in. H
2
O
Specific heat 1 kJ/kg ·K =0.2389 Btu/lbm ·
◦
F
Surface tension 1 N/m =1000 dyne/cm
=0.068523 lbf/ft
Temperature K =
◦
C +273.15
=(5/9)(
◦
F +459.67) =(5/9)(
◦
R)

Temperature difference 1K =1
◦
C =1.8
◦
F =1.8
◦
R
Thermal conductivity
1 W/m ·K =0.57782 Btu/h ·ft ·
◦
F
Thermal resistance 1 K/W =0.52750
◦
F ·h/Btu
Viscosity 1 kg/m ·s =1000 cp =2419 lbm/ft ·h
Volume
1m
3
=35.314 ft
3
=264.17 gal
Volumetric flow rate 1m
3
/s =2118.9 ft
3
/min(cfm)
=1.5850 ×10
4
gal/min (gpm)
lbf: pound force and lbm: pound mass.

Physical Constants
Quantity Symbol Value
Universal gas constant

R
0.08205 atm ·m
3
/kmol ·K
0.08314 bar ·m
3
/kmol ·K
8314 J/kmol ·K
1.986 cal/mol ·K
1.986 Btu/lb mole ·
◦
R
10.73 psia ·ft
3
/lb mole ·
◦
R
1545 ft ·lbf/lb mole ·
◦
R
Standard gravitational acceleration g 9.8067 m/s
2
32.174 ft/s
2
4.1698 ×10
8

ft/h
2
Stefan-Boltzman constant σ
SB
5.670 ×10
−8
W/ m
2
·K
4
1.714 ×10
−9
Btu/h ·ft
2
·
◦
R
4
Acknowledgements
Item Special Credit Line
Figure 3.1 Reprinted, with permission, from Extended Surface Heat Transfer by D. Q. Kern and
A. D. Kraus. Copyright © 1972 by The McGraw-Hill Companies, Inc.
Table 3.1 Reprinted, with permission, from Perry’s Chemical Engineers’ Handbook, 7th edn.,
R. H. Perry and D. W. Green, eds. Copyright © 1997 by The McGraw-Hill Companies, Inc.
Figure 3.6 Reprinted, with permission, from Extended Surface Heat Transfer by D. Q. Kern and
A. D. Kraus. Copyright © 1972 by The McGraw-Hill Companies, Inc.
Figure 3.7 Reprinted, with permission, from Extended Surface Heat Transfer by D. Q. Kern and
A. D. Kraus. Copyright © 1972 by The McGraw-Hill Companies, Inc.
Table 3.2 Reproduced, with permission, from J. W. Palen and J. Taborek, Solution of shell side flow
pressure drop and heat transfer by stream analysis method, Chem. Eng. Prog. Symposium

Series, 65, No. 92, 53–63, 1969. Copyright © 1969 by AIChE.
Table 3.5 Reprinted, with permission, from Perry’s Chemical Engineers’ Handbook, 7th edn.,
R. H. Perry and D. W. Green, eds. Copyright © 1997 by The McGraw-Hill
Companies, Inc.
Figure 4.1 Copyright © 1998 from Heat Exchangers: Selection, Rating and Thermal Design by S. Kakac
and H. Liu. Reproduced by permission of Taylor & Francis, a division of Informa plc.
Figure 4.2 Copyright © 1998 from Heat Exchangers: Selection, Rating and Thermal Design by S. Kakac
and H. Liu. Reproduced by permission of Taylor & Francis, a division of Informa plc.
Figure 4.4 Reprinted, with permission, from Extended Surface Heat Transfer by D. Q. Kern and
A. D. Kraus. Copyright © 1972 by The McGraw-Hill Companies, Inc.
Figure 4.5 Reprinted, with permission, from Extended Surface Heat Transfer by D. Q. Kern and
A. D. Kraus. Copyright © 1972 by The McGraw-Hill Companies, Inc.
Figure 5.3 Reproduced, with permission, from R. Mukherjee, Effectively design shell-and-tube heat
exchangers, Chem. Eng. Prog., 94, No. 2, 21–37, 1998. Copyright © 1998 by AIChE.
Figure 5.4 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-in-
Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figures 6.1–6.5 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-in-
Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Table 6.1 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-in-
Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 6.10 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-in-
Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 7.1 Reproduced, with permission, from J. W. Palen and J. Taborek, Solution of shell side flow
pressure drop and heat transfer by stream analysis method, Chem. Eng. Prog. Symposium
Series, 65, No. 92, 53–63, 1969. Copyright © 1969 by AIChE.
Table, p. 283 Reproduced, with permission, from R. Mukherjee, Effectively design shell-and-tube heat
exchangers, Chem. Eng. Prog., 94, No. 2, 21–37, 1998. Copyright © 1998 by AIChE.
Figure 8.20 Reprinted from Computers and Chemical Engineering, Vol. 26, X. X. Zhu and X. R. Nie,
Pressure Drop Considerations for Heat Exchanger Network Grassroots Design, pp. 1661–
1676, Copyright © 2002, with permission from Elsevier.

ACKNOWLEDGEMENTS xiii
Item Special Credit Line
Figure 9.2 Copyright © 1997 from Boiling Heat Transfer and Two-Phase Flow, 2nd edn., by
L. S. Tong and Y. S. Tang. Reproduced by permission of Taylor & Francis, a division
of Informa plc.
Figures 10.1–10.5 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 10.6 Reproduced, with permission, from A. W. Sloley, Properly design thermosyphon
reboilers, Chem. Eng. Prog., 93, No. 3, 52–64, 1997. Copyright © 1997 by AIChE.
Table 10.1 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Appendix 10.A Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn.,
R. H. Perry and C. H. Chilton, eds. Copyright © 1973 by The McGraw-Hill
Companies, Inc.
Figure 11.1 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 11.3 Copyright © 1998 from Heat Exchangers: Selection, Rating and Thermal Design by
S. Kakac and H. Liu. Reproduced by permission of Taylor & Francis, a division of
Informa plc.
Figure 11.6 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 11.7 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 11.8 Reprinted, with permission, from Distillation Operation by H. Z. Kister. Copyright ©
1990 by The McGraw-Hill Companies, Inc.
Figure 11.11 Reprinted, with permission, from G. Breber, J. W. Palen and J. Taborek, Prediction
of tubeside condensation of pure components using flow regime criteria, J. Heat
Transfer, 102, 471–476, 1980. Originally published by ASME.
Figure 11.12 Copyright © 1998 from Heat Exchangers: Selection, Rating and Thermal Design by
S. Kakac and H. Liu. Reproduced by permission of Taylor & Francis, a division of

Informa plc.
Figures 11.A1–11.A3 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Figure 12.5 Copyright © 1991 from Heat Transfer Design Methods by J. J. McKetta, Editor.
Reproduced by permission of Taylor & Francis, a division of Informa plc.
Figures 12.A1–12.A5 Copyright © 1988 from Heat Exchanger Design Handbook by E. U. Schlünder, Editor-
in-Chief. Reproduced by per mission of Taylor & Francis, a division of Informa plc.
Table A.1 Copyright © 1972 from Handbook of Thermodynamic Tables and Charts by
K. Raznjeviˇc. Reproduced by permission of Taylor & Francis, a division of
Informa plc.
Table A.3 Reprinted, with permission, from Heat Transfer, 7th edn., by J. P. Holman. Copyright
© 1990 by The McGraw-Hill Companies, Inc.
Table A.4 Copyright © 1972 from Handbook of Thermodynamic Tables and Charts by
K. Raznjeviˇc. Reproduced by permission of Taylor & Francis, a division of
Informa plc.
Table A.7 Copyright © 1972 from Handbook of Thermodynamic Tables and Charts by
K. Raznjeviˇc. Reproduced by permission of Taylor & Francis, a division of
Informa plc.
xiv ACKNOWLEDGEMENTS
Item Special Credit Line
Table A.8 Reprinted, with permission, from ASME Steam Tables, American Society of Mechanical
Engineers, New York, 1967. Originally published by ASME.
Table A.9 Reprinted, with permission, from Flow of Fluids Through Valves, Fittings and Pipe, Technical
Paper 410, 1988, Crane Company. All rights reserved.
Table A.11 Copyright © 1975 from Tables of Ther mophysical Properties of Liquids and Gases, 2nd edn., by
N. B. Vargaftik. Reproduced by permission of Taylor & Francis, a division of Informa plc.
Table A.13 Copyright © 1972 from Handbook of Thermodynamic Tables and Charts by K. Raznjeviˇc.
Reproduced by permission of Taylor & Francis, a division of Informa plc.
Table A.15 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R. H. Perry and
C. H. Chilton, eds. Copyright © 1973 by The McGraw-Hill Companies, Inc.

Table A.17 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R. H. Perry and
C. H. Chilton, eds. Copyright © 1973 by The McGraw-Hill Companies, Inc.
Figure A.1 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R. H. Perry and
C. H. Chilton, eds. Copyright © 1973 by The McGraw-Hill Companies, Inc.
Table A.18 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R. H. Perry and
C. H. Chilton, eds. Copyright © 1973 by The McGraw-Hill Companies, Inc.
Figure A.2 Reprinted, with permission, from Chemical Engineers’ Handbook, 5th edn., R. H. Perry and
C. H. Chilton, eds. Copyright © 1973 by The McGraw-Hill Companies, Inc.
1
HEAT
CONDUCTION
Contents
1.1 Introduction 2
1.2 Fourier’s Law of Heat Conduction 2
1.3 The Heat Conduction Equation 6
1.4 Thermal Resistance 15
1.5 The Conduction Shape Factor 19
1.6 Unsteady-State Conduction 24
1.7 Mechanisms of Heat Conduction 31
1/2 HEAT CONDUCTION
1.1 Introduction
Heat conduction is one of the three basic modes of thermal energy transport (convection and
radiation being the other two) and is involved in virtually all process heat-transfer operations. In
commercial heat exchange equipment, for example, heat is conducted through a solid wall (often
a tube wall) that separates two fluids having different temperatures. Furthermore, the concept of
thermal resistance, which follows from the fundamental equations of heat conduction, is widely used
in the analysis of problems arising in the design and operation of industrial equipment. In addition,
many routine pr ocess engineering problems can be solved with acceptable accuracy using simple
solutions of the heat conduction equation for rectangular, cylindrical, and spherical geometries.
This chapter provides an introduction to the macroscopic theory of heat conduction and its engi-

neering applications. The key concept of thermal resistance, used throughout the text, is developed
here, and its utility in analyzing and solving problems of practical interest is illustrated.
1.2 Fourier’s Law of Heat Conduction
The mathematical theory of heat conduction was developed early in the nineteenth century by
Joseph Fourier [1]. The theory was based on the results of experiments similar to that illustrated
in Figure 1.1 in which one side of a rectangular solid is held at temperature T
1
, while the opposite
side is held at a lower temperature, T
2
. The other four sides are insulated so that heat can flow
only in the x-direction. For a given material, it is found that the rate, q
x
, at which heat (thermal
energy) is transferred from the hot side to the cold side is proportional to the cross-sectional area,
A, across which the heat flows; the temperature difference, T
1
−T
2
; and inversely proportional to
the thickness, B, of the material. That is:
q
x
∝
A(T
1
− T
2
)
B

Writing this relationship as an equality, we have:
q
x
=
kA(T
1
− T
2
)
B
(1.1)
T
2
q
x
x
Insulated
Insulated
Insulated
B
q
x
T
1
Figure 1.1 One-dimensional heat conduction in a solid.
HEAT CONDUCTION 1/3
The constant of proportionality, k, is called the thermal conductivity. Equation (1.1) is also applicable
to heat conduction in liquids and gases. However, when temperature differences exist in fluids, con-
vection currents tend to be set up, so that heat is generally not transferred solely by the mechanism
of conduction.

The thermal conductivity is a property of the material and, as such, it is not really a constant, but
rather it depends on the thermodynamic state of the material, i.e., on the temperature and pressure
of the material. However, for solids, liquids, and low-pressure gases, the pressure dependence is
usually negligible. The temperature dependence also tends to be fairly weak, so that it is often
acceptable to treat k as a constant, particularly if the temperature difference is moderate. When the
temperature dependence must be taken into account, a linear function is often adequate, particularly
for solids. In this case,
k = a + bT (1.2)
where a and b are constants.
Thermal conductivities of a number of materials are given in Appendices 1.A–1.E. Many other
values may be found in various handbooks and compendiums of physical property data. Process
simulation software is also an excellent source of physical property data. Methods for estimating
thermal conductivities of fluids when data are unavailable can be found in the authoritative book
by Poling et al. [2].
The form of Fourier’s law given by Equation (1.1) is valid only when the thermal conductivity
can be assumed constant. A more general result can be obtained by writing the equation for an
element of differential thickness. Thus, let the thickness be x and let T =T
2
−T
1
. Substituting
in Equation (1.1) gives:
q
x
=−kA
T
x
(1.3)
Now in the limit as x approaches zero,
T

x
→
dT
dx
and Equation (1.3) becomes:
q
x
=−kA
dT
dx
(1.4)
Equation (1.4) is not subject to the restriction of constant k. Furthermore, when k is constant, it can
be integrated to yield Equation (1.1). Hence, Equation (1.4) is the general one-dimensional form of
Fourier’s law. The negative sign is necessary because heat flows in the positive x-direction when
the temperatur e decreases in the x-direction. Thus, according to the standard sign convention that
q
x
is positive when the heat flow is in the positive x-direction, q
x
must be positive when dT /dx is
negative.
It is often convenient to divide Equation (1.4) by the area to give:
ˆ
q
x
≡ q
x
/A =−k
dT
dx

(1.5)
where
ˆ
q
x
is the heat flux. It has units of J/s ·m
2
=W/m
2
or Btu/h ·ft
2
. Thus, the units of k are
W/m ·K or Btu/h ·ft ·
◦
F.
Equations (1.1), (1.4), and (1.5) are restricted to the situation in which heat flows in the x-direction
only. In the general case in which heat flows in all three coordinate directions, the total heat flux is
1/4 HEAT CONDUCTION
obtained by adding vectorially the fluxes in the coordinate directions. Thus,
→
ˆ
q =
ˆ
q
x
→
i
+
ˆ
q

y
→
j
+
ˆ
q
z
→
k
(1.6)
where
→
ˆ
q is the heat flux vector and
→
i
,
→
j
,
→
k
are unit vectors in the x-, y-, z-directions, respectively.
Each of the component fluxes is given by a one-dimensional Fourier expression as follows:
ˆ
q
x
=−k
∂T
∂x

ˆ
q
y
=−k
∂T
∂y
ˆ
q
z
=−k
∂T
∂z
(1.7)
Partial derivatives are used here since the temperature now varies in all three directions. Substituting
the above expressions for the fluxes into Equation (1.6) gives:
→
ˆ
q =−k

∂T
∂x
→
i
+
∂T
∂y
→
j
+
∂T

∂z
→
k

(1.8)
The vector in parenthesis is the temperature gradient vector, and is denoted by
→
∇
T . Hence,
→
ˆ
q =−k
→
∇
T (1.9)
Equation (1.9) is the three-dimensional form of Fourier’s law. It is valid for homogeneous, isotropic
materials for which the thermal conductivity is the same in all directions.
Equation (1.9) states that the heat flux vector is proportional to the negative of the temperature
gradient vector. Since the gradient direction is the direction of greatest temperature increase, the
negative gradient direction is the direction of greatest temperature decrease. Hence, Fourier’s law
states that heat flows in the direction of greatest temperature decrease.
Example 1.1
The block of 304 stainless steel shown below is well insulated on the front and back surfaces, and
the temperature in the block varies linearly in both the x- and y-directions, find:
(a) The heat fluxes and heat flows in the x- and y-directions.
(b) The magnitude and direction of the heat flux vector.
5°C

10°C
x

y
5 cm
10 cm
5 cm
15°C
0°C
HEAT CONDUCTION 1/5
Solution
(a) From Table A.1, the thermal conductivity of 304 stainless steel is 14.4 W/m ·K. The cross-
sectional areas are:
A
x
=10 ×5 = 50 cm
2
= 0.0050 m
2
A
y
=5 × 5 = 25 cm
2
= 0.0025 m
2
Using Equation (1.7) and replacing the partial derivatives with finite differences (since the
temperature variation is linear), the heat fluxes are:
ˆ
q
x
=−k
∂T
∂x

=−k
T
x
=−14.4

−5
0.05

= 1440 W/m
2
ˆ
q
y
=−k
∂T
∂y
=−k
T
y
=−14.4

10
0.1

=−1440 W/m
2
The heat flows are obtained by multiplying the fluxes by the corresponding cross-sectional
areas:
q
x

=
ˆ
q
x
A
x
= 1440 × 0.005 = 7.2W
q
y
=
ˆ
q
y
A
y
=−1440 × 0.0025 =−3.6W
(b) From Equation (1.6):
→
ˆ
q =
ˆ
q
x
→
i
+
ˆ
q
y
→

j
→
ˆ
q =1440
→
i
− 1440
→
j




→
ˆ
q




=[(1440)
2
+ (−1440)
2
]
0.5
= 2036.5 W/m
2
The angle, θ, between the heat flux vector and the x-axis is calculated as follows:
tan θ =

ˆ
q
y
/
ˆ
q
x
=−1440/1440 =−1.0
θ =−45
◦
The direction of the heat flux vector, which is the direction in which heat flows, is indicated in
the sketch below.
1/6 HEAT CONDUCTION
x
y
q
45°
1.3 The Heat Conduction Equation
The solution of problems involving heat conduction in solids can, in principle, be reduced to the
solution of a single differential equation, the heat conduction equation. The equation can be derived
by making a thermal energy balance on a differential volume element in the solid. For the case of
conduction only in the x-direction, such a volume element is illustrated in Figure 1.2. The balance
equation for the volume element is:
{rate of thermal energy in}−{rate of thermal energy out}+{net rate of thermal
energy generation}={rate of accumulation of thermal energy} (1.10)
The generation term appears in the equation because the balance is made on thermal energy, not
total energy. For example, thermal energy may be generated within a solid by an electric current
or by decay of a radioactive material.
The rate at which thermal energy enters the volume element across the face at x is given by the
product of the heat flux and the cross-sectional area,

ˆ
q
x


x
A. Similarly, the rate at which thermal
energy leaves the element across the face at x +x is
ˆ
q
x


x+x
A. For a homogeneous heat source
x
x
x
ϩ ∆x
q
ˆ
xx
q
ˆ
ϩ∆x
∆x
xx
Figure 1.2 Differential volume element used in derivation of conduction equation.
HEAT CONDUCTION 1/7
of strength

˙
q per unit volume, the net rate of generation is
˙
qAx. Finally, the rate of accumulation
is given by the time derivative of the thermal energy content of the volume element, which is
ρc(T − T
ref
)Ax, where T
ref
is an arbitrary reference temperature. Thus, the balance equation
becomes:
(
ˆ
q
x
|
x
−
ˆ
q
x
|
x+x
)A +
˙
qAx = ρc
∂T
∂t
Ax
It has been assumed here that the density, ρ, and heat capacity, c, are constant. Dividing by Ax

and taking the limit as x →0 yields:
ρc
∂T
∂t
=−
∂
ˆ
q
x
∂x
+
˙
q
Using Fourier’s law as given by Equation (1.5), the balance equation becomes:
ρc
∂T
∂t
=
∂
∂x

k ∂T
∂x

+
˙
q
When conduction occurs in all three coordinate directions, the balance equation contains y- and
z-derivatives analogous to the x-derivative. The balance equation then becomes:
ρc

∂T
∂t
=
∂
∂x

k∂T
∂x

+
∂
∂y

k∂T
∂y

+
∂
∂z

k∂T
∂z

+
˙
q (1.11)
Equation (1.11) is listed in Table 1.1 along with the corresponding forms that the equation takes in
cylindrical and spherical coordinates. Also listed in Table 1.1 are the components of the heat flux
vector in the three coordinate systems.
When k is constant, it can be taken outside the derivatives and Equation (1.11) can be

written as:
ρc
k
∂T
∂t
=
∂
2
T
∂x
2
+
∂
2
T
∂y
2
+
∂
2
T
∂z
2
+
˙
q
k
(1.12)
or
1

α
∂T
∂t
=∇
2
T +
˙
q
k
(1.13)
where α ≡k/ρc is the thermal diffusivity and ∇
2
is the Laplacian operator. The thermal diffusivity
has units of m
2
/s or ft
2
/h.
1/8 HEAT CONDUCTION
Table 1.1 The Heat Conduction Equation
A. Cartesian coordinates
ρc
∂T
∂t
=
∂
∂x

k
∂T

∂x

+
∂
∂y

k
∂T
∂y

+
∂
∂z

k
∂T
∂z

+
˙
q
The components of the heat flux vector,
→
ˆ
q ,are:
ˆ
q
x
=−k
∂T

∂x
ˆ
q
y
=−k
∂T
∂y
ˆ
q
z
=−k
∂T
∂z
B. Cylindrical coordinates (r, φ,z)
x
y
z
r
(x, y, z) ϭ (r, φ, z)
φ
ρc
∂T
∂t
=
1
r
∂
∂r

kr

∂T
∂r

+
1
r
2
∂
∂φ

k
∂T
∂φ

+
∂
∂z

k
∂T
∂z

+
˙
q
The components of
→
ˆ
q are:
ˆ

q
r
=−k
∂T
∂r
;
ˆ
q
φ
=
−k
r
∂T
∂φ
;
ˆ
q
z
=−k
∂T
∂z
C. Spherical coordinates (r, θ, φ)
x
y
z
r
(x, y, z) ϭ (r, θ, φ)
φ
θ
HEAT CONDUCTION 1/9

ρ c
∂T
∂ t
=
1
r
2
∂
∂r

kr
2
∂T
∂r

+
1
r
2
sin θ
∂
∂θ

k sin θ
∂T
∂θ

+
1
r

2
sin
2
θ
∂
∂φ

k
∂T
∂φ

+
˙
q
The components of
→
ˆ
q are:
ˆ
q
r
=−k
∂T
∂r
;
ˆ
q
θ
=−
k

r
∂T
∂θ
;
ˆ
q
φ
=−
k
r sin θ
∂T
∂φ
The use of the conduction equation is illustrated in the following examples.
Example 1.2
Apply the conduction equation to the situation illustrated in Figure 1.1.
Solution
In order to make the mathematics conform to the physical situation, the following conditions are
imposed:
(1) Conduction only in x-direction ⇒ T =T(x), so
∂T
∂y
=
∂T
∂z
=0
(2) No heat source ⇒
˙
q =0
(3) Steady state ⇒
∂T

∂t
=0
(4) Constant k
The conduction equation in Cartesian coordinates then becomes:
0 = k
∂
2
T
∂x
2
or
d
2
T
dx
2
= 0
(The partial derivative is replaced by a total derivative because x is the only independent variable
in the equation.) Integrating on both sides of the equation gives:
dT
dx
= C
1
A second integration gives:
T = C
1
x + C
2
Thus, it is seen that the temperature varies linearly across the solid. The constants of integration
can be found by applying the boundary conditions:

(1) At x =0 T =T
1
(2) At x =BT=T
2
The first boundary condition gives T
1
=C
2
and the second then gives:
T
2
= C
1
B + T
1
1/10 HEAT CONDUCTION
Solving for C
1
we find:
C
1
=
T
2
− T
1
B
The heat flux is obtained from Fourier’s law:
ˆ
q

x
=−k
dT
dx
=−kC
1
=−k

T
2
− T
1

B
= k

T
1
− T
2

B
Multiplying by the area gives the heat flow:
q
x
=
ˆ
q
x
A =

kA(T
1
− T
2
)
B
Since this is the same as Equation (1.1), we conclude that the mathematics are consistent with the
experimental results.
Example 1.3
Apply the conduction equation to the situation illustrated in Figure 1.1, but let k =a +bT, where a
and b are constants.
Solution
Conditions 1–3 of the previous example are imposed. The conduction equation then becomes:
0 =
d
dx

k
dT
dx

Integrating once gives:
k
dT
dx
= C
1
The variables can now be separated and a second integration performed. Substituting for k,we
have:
(a + bT )dT = C

1
dx
aT +
bT
2
2
= C
1
x + C
2
It is seen that in this case of variable k, the temperature profile is not linear across the solid.
The constants of integration can be evaluated by applying the same boundary conditions as in the
previous example, although the algebra is a little more tedious. The results are:
C
2
= aT
1
+
bT
2
1
2
C
1
= a
(T
2
− T
1
)

B
+
b
2B
(T
2
2
− T
2
1
)