fundamentals of heat exchanger design

FUNDAMENTALS OF HEAT
EXCHANGER DESIGN
Fundamentals of Heat Exchanger Design. Ramesh K. Shah and Dušan P. Sekulic
Copyright © 2003 John Wiley & Sons, Inc.
FUNDAMENTALS OF
HEAT EXCHANGER
DESIGN
Ramesh K. Shah
Rochester Institute of Technology, Rochester, New York
Formerly at Delphi Harrison Thermal Systems, Lockport, New York
Dus
ˇ
an P. Sekulic
´
University of Kentucky, Lexington, Kentucky
JOHN WILEY & SONS, INC.
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1
*
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Library of Congress Cataloging-in-Publication Data:
Shah, R. K.
Fundamentals of heat exchanger design / Ramesh K. Shah, Dus
ˇ
an P. Sekulic
´
.
p. cm.
Includes index.
ISBN 0-471-32171-0
1. Heat exchangers–Design and construction. I. Sekulic
´
, Dus
ˇ

an P. II. Title.
TJ263 .S42 2003
621.402
0
5–dc21
2002010161
Printed in the United States of America
10987654321
Contents
Preface xv
Nomenclature xix
1 Classification of Heat Exchangers 1
1.1 Introduction 1
1.2 Classification According to Transfer Processes 3
1.2.1 Indirect-Contact Heat Exchangers 3
1.2.2 Direct-Contact Heat Exchangers 7
1.3 Classification According to Number of Fluids 8
1.4 Classification According to Surface Compactness 8
1.4.1 Gas-to-Fluid Exchangers 11
1.4.2 Liquid-to-Liquid and Phase-Change Exchangers 12
1.5 Classification According to Construction Features 12
1.5.1 Tubular Heat Exchangers 13
1.5.2 Plate-Type Heat Exchangers 22
1.5.3 Extended Surface Heat Exchangers 36
1.5.4 Regenerators 47
1.6 Classification According to Flow Arrangements 56
1.6.1 Single-Pass Exchangers 57
1.6.2 Multipass Exchangers 64
1.7 Classification According to Heat Transfer Mechanisms 73
Summary 73

References 73
Review Questions 74
2 Overview of Heat Exchanger Design Methodology 78
2.1 Heat Exchanger Design Methodology 78
2.1.1 Process and Design Specifications 79
2.1.2 Thermal and Hydraulic Design 83
2.1.3 Mechanical Design 87
2.1.4 Manufacturing Considerations and Cost Estimates 90
2.1.5 Trade-off Factors 92
2.1.6 Optimum Design 93
2.1.7 Other Considerations 93
2.2 Interactions Among Design Considerations 93
Summary 94
References 94
Review Questions 95
Problems 95
3 Basic Thermal Design Theory for Recuperators 97
3.1 Formal Analogy between Thermal and Electrical Entities 98
3.2 Heat Exchanger Variables and Thermal Circuit 100
3.2.1 Assumptions for Heat Transfer Analysis 100
3.2.2 Problem Formulation 102
3.2.3 Basic Definitions 104
3.2.4 Thermal Circuit and UA 107
3.3 The "-NTU Method 114
3.3.1 Heat Exchanger Effectiveness " 114
3.3.2 Heat Capacity Rate Ratio C* 118
3.3.3 Number of Transfer Units NTU 119
3.4 Effectiveness – Number of Transfer Unit Relationships 121
3.4.1 Single-Pass Exchangers 122
3.5 The P-NTU Method 139

3.5.1 Temperature Effectiveness P 140
3.5.2 Number of Transfer Units, NTU 140
3.5.3 Heat Capacity Rate Ratio R 141
3.5.4 General P–NTU Functional Relationship 141
3.6 P–NTU Relationships 142
3.6.1 Parallel Counterflow Exchanger, Shell Fluid Mixed, 1–2
TEMA E Shell 142
3.6.2 Multipass Exchangers 164
3.7 The Mean Temperature Difference Method 186
3.7.1 Log-Mean Temperature Difference, LMTD 186
3.7.2 Log-Mean Temperature Difference Correction Factor F 187
3.8 F Factors for Various Flow Arrangements 190
3.8.1 Counterflow Exchanger 190
3.8.2 Parallelflow Exchanger 191
3.8.3 Other Basic Flow Arrangements 192
3.8.4 Heat Exchanger Arrays and Multipassing 201
3.9 Comparison of the "-NTU, P–NTU, and MTD Methods 207
3.9.1 Solutions to the Sizing and Rating Problems 207
3.9.2 The "-NTU Method 208
3.9.3 The P-NTU Method 209
3.9.4 The MTD Method 209
3.10 The -P and P
1
ÀP
2
Methods 210
3.10.1 The -P Method 210
3.10.2 The P
1
ÀP

2
Method 211
vi CONTENTS
3.11 Solution Methods for Determining Exchanger Effectiveness 212
3.11.1 Exact Analytical Methods 213
3.11.2 Approximate Methods 213
3.11.3 Numerical Methods 213
3.11.4 Matrix Formalism 214
3.11.5 Chain Rule Methodology 214
3.11.6 Flow-Reversal Symmetry 215
3.11.7 Rules for the Determination of Exchanger Effectiveness
with One Fluid Mixed 216
3.12 Heat Exchanger Design Problems 216
Summary 219
References 219
Review Questions 220
Problems 227
4 Additional Considerations for Thermal Design of Recuperators 232
4.1 Longitudinal Wall Heat Conduction Effects 232
4.1.1 Exchangers with C
*
¼ 0 236
4.1.2 Single-Pass Counterflow Exchanger 236
4.1.3 Single-Pass Parallelflow Exchanger 239
4.1.4 Single-Pass Unmixed–Unmixed Crossflow Exchanger 239
4.1.5 Other Single-Pass Exchangers 239
4.1.6 Multipass Exchangers 239
4.2 Nonuniform Overall Heat Transfer Coefficients 244
4.2.1 Temperature Effect 248
4.2.2 Length Effect 249

4.2.3 Combined Effect 251
4.3 Additional Considerations for Extended Surface Exchangers 258
4.3.1 Thin Fin Analysis 259
4.3.2 Fin Efficiency 272
4.3.3 Fin Effectiveness 288
4.3.4 Extended Surface Efficiency 289
4.4 Additional Considerations for Shell-and-Tube Exchangers 291
4.4.1 Shell Fluid Bypassing and Leakage 291
4.4.2 Unequal Heat Transfer Area in Individual Exchanger Passes 296
4.4.3 Finite Number of Baffles 297
Summary 298
References 298
Review Questions 299
Problems 302
5 Thermal Design Theory for Regenerators 308
5.1 Heat Transfer Analysis 308
5.1.1 Assumptions for Regenerator Heat Transfer Analysis 308
5.1.2 Definitions and Description of Important Parameters 310
5.1.3 Governing Equations 312
CONTENTS vii
5.2 The "-NTU
o
Method 316
5.2.1 Dimensionless Groups 316
5.2.2 Influence of Core Rotation and Valve Switching Frequency 320
5.2.3 Convection Conductance Ratio (hA)* 320
5.2.4 "-NTU
o
Results for a Counterflow Regenerator 321
5.2.5 "-NTU

o
Results for a Parallelflow Regenerator 326
5.3 The ÖŠMethod 337
5.3.1 Comparison of the "-NTU
o
and ÖŠMethods 341
5.3.2 Solutions for a Counterflow Regenerator 344
5.3.3 Solution for a Parallelflow Regenerator 345
5.4 Influence of Longitudinal Wall Heat Conduction 348
5.5 Influence of Transverse Wall Heat Conduction 355
5.5.1 Simplified Theory 355
5.6 Influence of Pressure and Carryover Leakages 360
5.6.1 Modeling of Pressure and Carryover Leakages for a Rotary
Regenerator 360
5.7 Influence of Matrix Material, Size, and Arrangement 366
Summary 371
References 372
Review Questions 373
Problems 376
6 Heat Exchanger Pressure Drop Analysis 378
6.1 Introduction 378
6.1.1 Importance of Pressure Drop 378
6.1.2 Fluid Pumping Devices 380
6.1.3 Major Contributions to the Heat Exchanger Pressure Drop 380
6.1.4 Assumptions for Pressure Drop Analysis 381
6.2 Extended Surface Heat Exchanger Pressure Drop 381
6.2.1 Plate-Fin Heat Exchangers 382
6.2.2 Tube-Fin Heat Exchangers 391
6.3 Regenerator Pressure Drop 392
6.4 Tubular Heat Exchanger Pressure Drop 393

6.4.1 Tube Banks 393
6.4.2 Shell-and-Tube Exchangers 393
6.5 Plate Heat Exchanger Pressure Drop 397
6.6 Pressure Drop Associated with Fluid Distribution Elements 399
6.6.1 Pipe Losses 399
6.6.2 Sudden Expansion and Contraction Losses 399
6.6.3 Bend Losses 403
6.7 Pressure Drop Presentation 412
6.7.1 Nondimensional Presentation of Pressure Drop Data 413
6.7.2 Dimensional Presentation of Pressure Drop Data 414
viii CONTENTS
6.8 Pressure Drop Dependence on Geometry and Fluid Properties 418
Summary 419
References 420
Review Questions 420
Problems 422
7 Surface Basic Heat Transfer and Flow Friction Characteristics 425
7.1 Basic Concepts 426
7.1.1 Boundary Layers 426
7.1.2 Types of Flows 429
7.1.3 Free and Forced Convection 438
7.1.4 Basic Definitions 439
7.2 Dimensionless Groups 441
7.2.1 Fluid Flow 443
7.2.2 Heat Transfer 446
7.2.3 Dimensionless Surface Characteristics as a Function of the
Reynolds Number 449
7.3 Experimental Techniques for Determining Surface Characteristics 450
7.3.1 Steady-State Kays and London Technique 451
7.3.2 Wilson Plot Technique 460

7.3.3 Transient Test Techniques 467
7.3.4 Friction Factor Determination 471
7.4 Analytical and Semiempirical Heat Transfer and Friction Factor
Correlations for Simple Geometries 473
7.4.1 Fully Developed Flows 475
7.4.2 Hydrodynamically Developing Flows 499
7.4.3 Thermally Developing Flows 502
7.4.4 Simultaneously Developing Flows 507
7.4.5 Extended Reynolds Analogy 508
7.4.6 Limitations of j vs. Re Plot 510
7.5 Experimental Heat Transfer and Friction Factor Correlations for
Complex Geometries 511
7.5.1 Tube Bundles 512
7.5.2 Plate Heat Exchanger Surfaces 514
7.5.3 Plate-Fin Extended Surfaces 515
7.5.4 Tube-Fin Extended Surfaces 519
7.5.5 Regenerator Surfaces 523
7.6 Influence of Temperature-Dependent Fluid Properties 529
7.6.1 Correction Schemes for Temperature-Dependent Fluid
Properties 530
7.7 Influence of Superimposed Free Convection 532
7.7.1 Horizontal Circular Tubes 533
7.7.2 Vertical Circular Tubes 535
7.8 Influence of Superimposed Radiation 537
7.8.1 Liquids as Participating Media 538
CONTENTS ix
7.8.2 Gases as Participating Media 538
Summary 542
References 544
Review Questions 548

Problems 553
8 Heat Exchanger Surface Geometrical Characteristics 563
8.1 Tubular Heat Exchangers 563
8.1.1 Inline Arrangement 563
8.1.2 Staggered Arrangement 566
8.2 Tube-Fin Heat Exchangers 569
8.2.1 Circular Fins on Circular Tubes 569
8.2.2 Plain Flat Fins on Circular Tubes 572
8.2.3 General Geometric Relationships for Tube-Fin Exchangers 574
8.3 Plate-Fin Heat Exchangers 574
8.3.1 Offset Strip Fin Exchanger 574
8.3.2 Corrugated Louver Fin Exchanger 580
8.3.3 General Geometric Relationships for Plate-Fin Surfaces 584
8.4 Regenerators with Continuous Cylindrical Passages 585
8.4.1 Triangular Passage Regenerator 585
8.5 Shell-and-Tube Exchangers with Segmental Baffles 587
8.5.1 Tube Count 587
8.5.2 Window and Crossflow Section Geometry 589
8.5.3 Bypass and Leakage Flow Areas 592
8.6 Gasketed Plate Heat Exchangers 597
Summary 598
References 598
Review Questions 599
9 Heat Exchanger Design Procedures 601
9.1 Fluid Mean Temperatures 601
9.1.1 Heat Exchangers with C
*
% 0 603
9.1.2 Counterflow and Crossflow Heat Exchangers 604
9.1.3 Multipass Heat Exchangers 604

9.2 Plate-Fin Heat Exchangers 605
9.2.1 Rating Problem 605
9.2.2 Sizing Problem 617
9.3 Tube-Fin Heat Exchangers 631
9.3.1 Surface Geometries 631
9.3.2 Heat Transfer Calculations 631
9.3.3 Pressure Drop Calculations 632
9.3.4 Core Mass Velocity Equation 632
9.4 Plate Heat Exchangers 632
9.4.1 Limiting Cases for the Design 633
9.4.2 Uniqueness of a PHE for Rating and Sizing 635
x CONTENTS
9.4.3 Rating a PHE 637
9.4.4 Sizing a PHE 645
9.5 Shell-and-Tube Heat Exchangers 646
9.5.1 Heat Transfer and Pressure Drop Calculations 646
9.5.2 Rating Procedure 650
9.5.3 Approximate Design Method 658
9.5.4 More Rigorous Thermal Design Method 663
9.6 Heat Exchanger Optimization 664
Summary 667
References 667
Review Questions 668
Problems 669
10 Selection of Heat Exchangers and Their Components 673
10.1 Selection Criteria Based on Operating Parameters 674
10.1.1 Operating Pressures and Temperatures 674
10.1.2 Cost 675
10.1.3 Fouling and Cleanability 675
10.1.4 Fluid Leakage and Contamination 678

10.1.5 Fluids and Material Compatibility 678
10.1.6 Fluid Type 678
10.2 General Selection Guidelines for Major Exchanger Types 680
10.2.1 Shell-and-Tube Exchangers 680
10.2.2 Plate Heat Exchangers 693
10.2.3 Extended-Surface Exchangers 694
10.2.4 Regenerator Surfaces 699
10.3 Some Quantitative Considerations 699
10.3.1 Screening Methods 700
10.3.2 Performance Evaluation Criteria 713
10.3.3 Evaluation Criteria Based on the Second Law of
Thermodynamics 723
10.3.4 Selection Criterion Based on Cost Evaluation 724
Summary 726
References 726
Review Questions 727
Problems 732
11 Thermodynamic Modeling and Analysis 735
11.1 Introduction 735
11.1.1 Heat Exchanger as a Part of a System 737
11.1.2 Heat Exchanger as a Component 738
11.2 Modeling a Heat Exchanger Based on the First Law of
Thermodynamics 738
11.2.1 Temperature Distributions in Counterflow and Parallelflow
Exchangers 739
11.2.2 True Meaning of the Heat Exchanger Effectiveness 745
CONTENTS xi
11.2.3 Temperature Difference Distributions for Parallelflow and
Counterflow Exchangers 748
11.2.4 Temperature Distributions in Crossflow Exchangers 749

11.3 Irreversibilities in Heat Exchangers 755
11.3.1 Entropy Generation Caused by Finite Temperature Differences 756
11.3.2 Entropy Generation Associated with Fluid Mixing 759
11.3.3 Entropy Generation Caused by Fluid Friction 762
11.4 Thermodynamic Irreversibility and Temperature Cross Phenomena 763
11.4.1 Maximum Entropy Generation 763
11.4.2 External Temperature Cross and Fluid Mixing Analogy 765
11.4.3 Thermodynamic Analysis for 1–2 TEMA J Shell-and-Tube
Heat Exchanger 766
11.5 A Heuristic Approach to an Assessment of Heat Exchanger
Effectiveness 771
11.6 Energy, Exergy, and Cost Balances in the Analysis and Optimization
of Heat Exchangers 775
11.6.1 Temperature–Enthalpy Rate Change Diagram 776
11.6.2 Analysis Based on an Energy Rate Balance 779
11.6.3 Analysis Based on Energy/Enthalpy and Cost Rate Balancing 783
11.6.4 Analysis Based on an Exergy Rate Balance 786
11.6.5 Thermodynamic Figure of Merit for Assessing Heat
Exchanger Performance 787
11.6.6 Accounting for the Costs of Exergy Losses in a Heat
Exchanger 791
11.7 Performance Evaluation Criteria Based on the Second Law of
Thermodynamics 796
Summary 800
References 801
Review Questions 802
Problems 804
12 Flow Maldistribution and Header Design 809
12.1 Geometry-Induced Flow Maldistribution 809
12.1.1 Gross Flow Maldistribution 810

12.1.2 Passage-to-Passage Flow Maldistribution 821
12.1.3 Manifold-Induced Flow Maldistribution 834
12.2 Operating Condition–Induced Flow Maldistribution 837
12.2.1 Viscosity-Induced Flow Maldistribution 837
12.3 Mitigation of Flow Maldistribution 844
12.4 Header and Manifold Design 845
12.4.1 Oblique-Flow Headers 848
12.4.2 Normal-Flow Headers 852
12.4.3 Manifolds 852
Summary 853
References 853
xii CONTENTS
Review Questions 855
Problems 859
13 Fouling and Corrosion 863
13.1 Fouling and its Effect on Exchanger Heat Transfer and Pressure Drop 863
13.2 Phenomenological Considerations of Fouling 866
13.2.1 Fouling Mechanisms 867
13.2.2 Single-Phase Liquid-Side Fouling 870
13.2.3 Single-Phase Gas-Side Fouling 871
13.2.4 Fouling in Compact Exchangers 871
13.2.5 Sequential Events in Fouling 872
13.2.6 Modeling of a Fouling Process 875
13.3 Fouling Resistance Design Approach 881
13.3.1 Fouling Resistance and Overall Heat Transfer Coefficient
Calculation 881
13.3.2 Impact of Fouling on Exchanger Heat Transfer Performance 882
13.3.3 Empirical Data for Fouling Resistances 886
13.4 Prevention and Mitigation of Fouling 890
13.4.1 Prevention and Control of Liquid-Side Fouling 890

13.4.2 Prevention and Reduction of Gas-Side Fouling 891
13.4.3 Cleaning Strategies 892
13.5 Corrosion in Heat Exchangers 893
13.5.1 Corrosion Types 895
13.5.2 Corrosion Locations in Heat Exchangers 895
13.5.3 Corrosion Control 897
Summary 898
References 898
Review Questions 899
Problems 903
Appendix A: Thermophysical Properties 906
Appendix B: "-NTU Relationships for Liquid-Coupled Exchangers 911
Appendix C: Two-Phase Heat Transfer and Pressure Drop Correlations 913
C.1 Two-Phase Pressure Drop Correlations 913
C.2 Heat Transfer Correlations for Condensation 916
C.3 Heat Transfer Correlations for Boiling 917
Appendix D: U and C
UA
Values for Various Heat Exchangers 920
General References on or Related to Heat Exchangers 926
Index 931
CONTENTS xiii
Preface
Over the past quarter century, the importance of heat exchangers has increased immen-
sely from the viewpoint of energy conservation, conversion, recovery, and successful
implementation of new energy sources. Its importance is also increasing from the stand-
point of environmental concerns such as thermal pollution, air pollution, water pollu-
tion, and waste disposal. Heat exchangers are used in the process, power, transportation,
air-conditioning and refrigeration, cryogenic, heat recovery, alternate fuels, and
manufacturing industries, as well as being key components of many industrial products

available in the marketplace. From an educational point of view, heat exchangers
illustrate in one way or another most of the fundamental principles of the thermal
sciences, thus serving as an excellent vehicle for review and application, meeting the
guidelines for university studies in the United States and oversees. Significant advances
have taken place in the development of heat exchanger manufacturing technology as well
as design theory. Many books have been published on the subject, as summarized in
the General References at the end of the book. However, our assessment is that none of
the books available seems to provide an in-depth coverage of the intricacies of heat
exchanger design and theory so as to fully support both a student and a practicing
engineer in the quest for creative mastering of both theory and design. Our book was
motivated by this consideration. Coverage includes the theory and design of exchangers
for many industries (not restricted to, say, the process industry) for a broader, in-depth
foundation.
The objective of this book is to provide in-depth thermal and hydraulic design theory
of two-fluid single-phase heat exchangers for steady-state operation. Three important
goals were borne in mind during the preparation of this book:
1. To introduce and apply concepts learned in first courses in heat transfer, fluid
mechanics, thermodynamics, and calculus, to develop heat exchanger design
theory. Thus, the book will serve as a link between fundamental subjects men-
tioned and thermal engineering design practice in industry.
2. To introduce and apply basic heat exchanger design concepts to the solution of
industrial heat exchanger problems. Primary emphasis is placed on fundamental
concepts and applications. Also, more emphasis is placed on analysis and less on
empiricism.
3. The book is also intended for practicing engineers in addition to students.
Hence, at a number of places in the text, some redundancy is added to make the
concepts clearer, early theory is developed using constant and mean overall heat
transfer coefficients, and more data are added in the text and tables for industrial
use.
To provide comprehensive information for heat exchanger design and analysis in a

book of reasonable length, we have opted not to include detailed theoretical derivations
of many results, as they can be found in advanced convection heat transfer textbooks.
Instead, we have presented some basic derivations and then presented comprehensive
information through text and concise tables.
An industrial heat exchanger design problem consists of coupling component and
system design considerations to ensure proper functioning. Accordingly, a good design
engineer must be familiar with both system and component design aspects. Based on
industrial experience of over three decades in designing compact heat exchangers for
automobiles and other industrial applications and more than twenty years of teaching,
we have endeavored to demonstrate interrelationships between the component and sys-
tem design aspects, as well as between the needs of industrial and learning environments.
Some of the details of component design presented are also based on our own system
design experience.
Considering the fact that heat exchangers constitute a multibillion-dollar industry in
the United States alone, and there are over 300 companies engaged in the manufacture
of a wide array of heat exchangers, it is difficult to select appropriate material for an
introductory course. We have included more material than is necessary for a one-
semester course, placing equal emphasis on four basic heat exchanger types: shell-and-
tube, plate, extended surface, and regenerator. The choice of the teaching material to
cover in one semester is up to the instructor, depending on his or her desire to focus on
specific exchanger types and specific topics in each chapter. The prerequisites for this
course are first undergraduate courses in fluid mechanics, thermodynamics, and heat
transfer. It is expected that the student is familiar with the basics of forced convection
and the basic concepts of the heat transfer coefficient, heat exchanger effectiveness, and
mean temperature difference.
Starting with a detailed classification of a variety of heat exchangers in Chapter 1, an
overview of heat exchanger design methodology is provided in Chapter 2. The basic
thermal design theory for recuperators is presented in Chapter 3, advanced design theory
for recuperators in Chapter 4, and thermal design theory for regenerators in Chapter 5.
Pressure drop analysis is presented in Chapter 6. The methods and sources for obtaining

heat transfer and flow friction characteristics of exchanger surfaces are presented in
Chapter 7. Surface geometrical properties needed for heat exchanger design are covered
in Chapter 8. The thermal and hydraulic designs of extended-surface (compact
and noncompact plate-fin and tube-fin), plate, and shell-and-tube exchangers are out-
lined in Chapter 9. Guidelines for selecting the exchanger core construction and surface
geometry are presented in Chapter 10. Chapter 11 is devoted to thermodynamic analysis
for heat exchanger design and includes basic studies of temperature distributions in heat
exchangers, a heuristic approach to an assessment of heat exchanger effectiveness, and
advanced topics important for modeling, analysis, and optimization of heat exchangers
as components. All topics covered up to this point are related to thermal–hydraulic
design of heat exchangers in steady-state or periodic-flow operation. Operational
problems for compact and other heat exchangers are covered in Chapters 12 and 13.
They include the problems caused by flow maldistribution and by fouling and corrosion.
Solved examples from industrial experience and classroom practice are presented
throughout the book to illustrate important concepts and applications. Numerous review
questions and problems are also provided at the end of each chapter. If students can
answer the review questions and solve the problems correctly, they can be sure of their
grasp of the basic concepts and material presented in the text. It is hoped that readers will
xvi PREFACE
develop good understanding of the intricacies of heat exchanger design after going
through this material and prior to embarking on specialized work in their areas of
greatest interest.
For the thermal design of a heat exchanger for an application, considerable intellec-
tual effort is needed in selecting heat exchanger type and determining the appropriate
value of the heat transfer coefficients and friction factors; a relatively small effort is
needed for executing sizing and optimizing the exchanger because of the computer-
based calculations. Thus, Chapters 7, 9, and 10 are very important, in addition to
Chapter 3, for basic understanding of theory, design, analysis, and selection of heat
exchangers.
Material presented in Chapters 11 through 13 is significantly more interdisciplinary

than the rest of the book and is presented here in a modified methodological approach. In
Chapter 11 in particular, analytical modeling is used extensively. Readers will participate
actively through a set of examples and problems that extend the breadth and depth of the
material given in the main body of the text. A number of examples and problems in
Chapter 11 require analytical derivations and more elaborate analysis, instead of illus-
trating the topics with examples that favor only utilization of the formulas and comput-
ing numerical values for a problem. The complexity of topics requires a more diverse
approach to terminology, less routine treatment of established conventions, and a more
creative approach to some unresolved dilemmas.
Because of the breadth of the subject, the coverage includes various design aspects and
problems for indirect-contact two-fluid heat exchangers with primarily single-phase
fluids on each side. Heat exchangers with condensing and evaporating fluids on one
side can also be analyzed using the design methods presented as long as the thermal
resistance on the condensing or evaporating side is small or the heat transfer coefficient
on that side can be treated as a constant. Design theory for the following exchangers
is not covered in this book, due to their complexity and space limitations: two-phase
and multiphase heat exchangers (such as condensers and vaporizers), direct-contact
heat exchangers (such as humidifiers, dehumidifiers, cooling towers), and multifluid
and multistream heat exchangers. Coverage of mechanical design, exchanger fabrication
methods, and manufacturing techniques is also deemed beyond the scope of the
book.
Books by M. Jakob, D. Q. Kern, and W. M. Kays and A. L. London were considered
to be the best and most comprehensive texts on heat exchanger design and analysis
following World War II. In the last thirty or so years, a significant number of books
have been published on heat exchangers. These are summarized in the General
References at the end of the book.
This text is an outgrowth of lecture notes prepared by the authors in teaching courses
on heat exchanger design, heat transfer, and design and optimization of thermal systems
to senior and graduate students. These courses were taught at the State University of
New York at Buffalo and the University of Novi Sad, Yugoslavia. Over the past fifteen

years or more, the notes of the first author have been used for teaching purposes at a
number of institutions, including the University of Miami by Professor S. Kakac¸ ,
Rensselaer Polytechnic Institute by Professors A. E. Bergles and R. N. Smith,
Rochester Institute of Technology by Professor S. G. Kandlikar, Rice University by
Professor Y. Bayazitog
ˇ
lu, University of Tennessee Space Center by Dr. R. Schultz,
University of Texas at Arlington by Professor A. Haji-Sheikh, University of
Cincinnati by Professor R. M. Manglik, Northeastern University by Professor Yaman
Yener, North Carolina A&T State University by Professor Lonnie Sharpe, Auburn
PREFACE xvii
University by Dr. Peter Jones, Southern Methodist University by Dr. Donald Price,
University of Tennessee by Professor Edward Keshock, and Gonzaga University by
Professor A. Aziz. In addition, these course notes have been used occasionally at a
number of other U.S. and foreign institutions. The notes of the second author have
also been used for a number of undergraduate and graduate courses at Marquette
University and the University of Kentucky.
The first author would like to express his sincere appreciation to the management
of Harrison Thermal Systems, Delphi Corporation (formerly General Motors
Corporation), for their varied support activities over an extended period of time. The
second author acknowledges with appreciation many years of support by his colleagues
and friends on the faculty of the School of Engineering, University of Novi Sad, and
more recently at Marquette University and the University of Kentucky. We are also
thankful for the support provided by the College of Engineering, University of
Kentucky, for preparation of the first five and final three chapters of the book. A special
word of appreciation is in order for the diligence and care exercised by Messrs. Dale Hall
and Mack Mosley in preparing the manuscript and drawings through Chapter 5.
The first author is grateful to Professor A. L. London of Stanford University for
teaching him the ABCs of heat exchangers and for providing constant inspiration and
encouragement throughout his professional career and particularly during the course of

preparation of this book. The first author would also like to thank Professors Sadik
Kakac¸ of the University of Miami and Ralph Webb of the Pennsylvania State University
for their support, encouragement, and involvement in many professional activities
related to heat exchangers. The second author is grateful to his colleague and friend
Professor B. S. Bac
ˇ
lic
´
, University of Novi Sad, for many years of joint work and teaching
in the fields of heat exchanger design theory. Numerous discussions the second author
have had with Dr. R. Gregory of the University of Kentucky regarding not only what
one has to say about a technical topic, but in particular how to formulate it for a reader,
were of a great help in resolving some dilemmas. Also, the continuous support and
encouragement of Dr. Frederick Edeskuty of Los Alamos National Laboratory, and
Professor Richard Gaggioli of Marquette University were immensely important to the
second author in an effort to exercise his academic experience on both sides of the
Atlantic Ocean. We appreciate Professor P. V. Kadaba of the Georgia Institute of
Technology and James Seebald of ABB Alstom Air Preheater for reviewing the complete
manuscript and providing constructive suggestions, and Dr. M. S. Bhatti of Delphi
Harrison Thermal Systems for reviewing Chapters 1 through 6 and Dr. T. Skiepko of
Bialystok Technical University for reviewing Chapter 5 and providing constructive
suggestions. The constructive feedback over a period of time provided by many students
(too numerous to mention by name) merits a special word of appreciation.
Finally, we must acknowledge the roles played by our wives, Rekha and Gorana, and
our children, Nilay and Nirav Shah and Vis
ˇ
nja and Aleksandar Sekulic
´
, during the
course of preparation of this book. Their loving care, emotional support, assistance,

and understanding provided continuing motivation to compete the book.
We welcome suggestions and comments from readers.
Ramesh K. Shah
Dus
ˇ
an P. Sekulic
´
xviii PREFACE
NOMENCLATURE
The dimensions for each symbol are represented in both the SI and English systems of
units, where applicable. Note that both the hour and second are commonly used as units
for time in the English system of units; hence a conversion factor of 3600 should be
employed at appropriate places in dimensionless groups.
A total heat transfer surface area (both primary and secondary, if any) on one
side of a direct transfer type exchanger (recuperator), total heat transfer
surface area of all matrices of a regenerator,
{
m
2
,ft
2
A
c
total heat transfer area (both primary and secondary, if any) on the cold side
of an exchanger, m
2
,ft
2
A
eff

effective surface area on one side of an extended surface exchanger [defined by
Eq. (4.167)], m
2
,ft
2
A
f
fin or extended surface area on one side of the exchanger, m
2
,ft
2
A
fr
frontal or face area on one side of an exchanger, m
2
,ft
2
A
fr;t
window area occupied by tubes, m
2
,ft
2
A
fr;w
gross (total) window area, m
2
,ft
2
A

h
total heat transfer surface area (both primary and secondary, if any) on the
hot fluid side of an exchanger, m
2
,ft
2
A
k
fin cross-sectional area for heat conduction in Section 4.3 (A
k;o
is A
k
at the
fin base), m
2
,ft
2
A
k
total wall cross-sectional area for longitudinal conduction [additional
subscripts c, h, and t, if present, denote cold side, hot side, and total (hot þ
cold) for a regenerator] in Section 5.4, m
2
,ft
2
A
*
k
ratio of A
k

on the C
min
side to that on the C
max
side [see Eq. (5.117)],
dimensionless
A
o
minimum free-flow (or open) area on one fluid side of an exchanger, heat
transfer surface area on tube outside in a tubular exchanger in Chapter 13
only, m
2
,ft
2
A
o;bp
flow bypass area of one baffle, m
2
,ft
2
A
o;cr
flow area at or near the shell centerline for one crossflow section in a shell-and-
tube exchanger, m
2
,ft
2
A
o;sb
shell-to-baffle leakage flow area, m

2
,ft
2
A
o;tb
tube-to-baffle leakage flow area, m
2
,ft
2
A
o;w
flow area through window zone, m
2
,ft
2
A
p
primary surface area on one side of an exchanger, m
2
,ft
2
A
w
total wall area for heat conduction from the hot fluid to the cold fluid, or total
wall area for transverse heat conduction (in the matrix wall thickness direc-
tion), m
2
,ft
2
a short side (unless specified) of a rectangular cross section, m, ft

a amplitude of chevron plate corrugation (see Fig. 7.28), m, ft
NOMENCLATURE xix
{
Unless clearly specified, a regenerator in the nomenclature means either a rotary or a fixed-matrix regenerator.
B parameter for a thin fin with end leakage allowed, h
e
=mk
f
, dimensionless
Bi Biot number, Bi ¼ hð=2Þ=k
f
for the fin analysis; Bi ¼ hð=2Þ=k
w
for the
regenerator analysis, dimensionless
b distance between two plates in a plate-fin heat exchanger [see Fig. 8.7 for b
1
or b
2
(b on fluid 1 or 2 side)], m, ft
b long side (unless specified) of a rectangular cross section, m, ft
c some arbitrary monetary unit (instead of $, £, etc.), money
C flow stream heat capacity rate with a subscript c or h,
_
mmc
p
,W=K, Btu/hr-8F
C correction factor when used with a subscript different from c, h, min, or max,
dimensionless
C unit cost, c/J(c/Btu), c/kg (c/lbm), c/kW [c/(Btu/hr)], c/kW Áyr(c /Btu on

yearly basis), c/m
2
(c/ft
2
)
C annual cost, c/yr
C* heat capacity rate ratio, C
min
=C
max
, dimensionless
"
CC flow stream heat capacitance, Mc
p
, C
d
,WÁs=K, Btu/8 F
C
D
drag coefficient, Áp=ðu
2
1
=2g
c
Þ, dimensionless
C
max
maximum of C
c
and C

h
,W=K, Btu/hr-8F
C
min
minimum of C
c
and C
h
, W/K, Btu/hr-8F
C
ms
heat capacity rate of the maldistributed stream, W/K, Btu/hr-8F
C
r
heat capacity rate of a regenerator, M
w
c
w
N or M
w
c
w
=P
t
[see Eq. (5.7) for the
hot- and cold-side matrix heat capacity rates C
r;h
and C
r;c
], W/K, Btu/hr-8F

C
*
r
total matrix heat capacity rate ratio, C
r
=C
min
, C
*
r;h
¼ C
r;h
=C
h
, C
*
r;c
¼ C
r;c
=C
c
,
dimensionless
"
CC
r
total matrix wall heat capacitance, M
w
c
w

or C
r
P
t
[see Eq. (5.6) for hot- and
cold-side matrix heat capacitances
"
CC
r;h
and
"
CC
r;c
], W Á s=K, Btu/8F
"
CC
*
r
ratio of
"
CC
r
to
"
CC
min
, dimensionless
C
UA
cost per unit thermal size (see Fig. 10.13 and Appendix D), c/W/K

C
us
heat capacity rate of the uniform stream, W/K, Btu/hr-8F
C
w
matrix heat capacity rate; same as C
r
, W/K, Btu/hr-8F
"
CC
w
total wall heat capacitance for a recuperator, M
w
c
w
,WÁs=K, Btu/8F
"
CC
w
*
ratio of
"
CC
w
to
"
CC
min
, dimensionless
CF cleanliness factor, U

f
=U
c
, dimensionless
c specific heat of solid, J=kg Á K,
{
Btu/lbm-8F
c annual cost of operation percentile, dimensionless
c
p
specific heat of fluid at constant pressure, J=kg Á K, Btu/lbm-8F
c
w
specific heat of wall material, J=kg Á K, Btu/lbm-8F
d exergy destruction rate, W, Btu/hr
D
baffle
baffle diameter, m, ft
D
ctl
diameter of the circle through the centers of the outermost tubes, D
otl
À d
o
,
m, ft
D
h
hydraulic diameter of flow passages, 4r
h

,4A
o
=P,4A
o
L=A,or4=,m,ft
xx NOMENCLATURE
{
J ¼ joule ¼ newton  meter ¼ watt  second; newton ¼ N ¼ kg Á m=s
2
:
D
h;w
hydraulic diameter of the window section, m, ft
D
otl
diameter of the outer tube limit (see Fig. 8.9), m, ft
D
p
port or manifold diameter in a plate heat exchanger, m, ft
D
s
shell inside diameter, m, ft
d differential operator
d
c
collar diameter in a round tube and fin exchanger, d
o
þ 2,m,ft
d
e

fin tip diameter of a disk (radial) fin, m, ft
d
i
tube inside diameter, m, ft
d
o
tube (or pin) outside diameter, tube outside diameter at the fin root for a
finned tube after tube expansion, if any, m, ft
d
w
wire diameter, m, ft
d
1
tube hole diameter in a baffle, m, ft
_
ee exergy rate, W, Btu/hr
E energy, J, Btu
E activation energy in Chapter 13 [see Eq. (13.12)], J=kg Á mol, Btu/lbm-mole
E fluid pumping power per unit surface area,
_
mm Áp=A,W=m
2
, hp/ft
2
Eu row average Euler number per tube row, Áp=ðu
2
m
N
r
=2g

c
Þ or
Áp=ðG
2
N
r
=2g
c
Þ, dimensionless
e surface roughness size, m, ft
e
þ
roughness Reynolds number, eu
*
=, dimensionless
F log-mean temperature difference correction factor [defined by Eq. (3.183)],
dimensionless
f Fanning friction factor, 
w
=ðu
2
m
=2g
c
Þ, Áp g
c
D
h
=ð2LG
2

Þ, dimensionless
f
D
Darcy friction factor, 4f, dimensionless
f
tb
row average Fanning friction factor per tube for crossflow to tubes, used in
Chapter 7, Áp=ð4G
2
N
r
=2g
c
Þ, Eu/4, dimensionless
G fluid mass velocity based on the minimum free area,
_
mm=A
o
(replace A
o
by A
o;c
for the crossflow section of a tube bundle in a shell-and-tube heat exchan-
ger), kg=m
2
Á s, lbm/hr-ft
2
Gr Grashof number [defined by Eq. (7.159)], dimensionless
Gz Graetz number,
_

mmc
p
=kL [see Eqs. (7.39) and (12.53)], dimensionless
Gz
x
local Graetz number,
_
mmc
p
=kx, dimensionless
g gravitational acceleration, m/s
2
, ft/sec
2
g
c
proportionality constant in Newton’s second law of motion, g
c
¼ 1 and
dimensionless in SI units, g
c
¼ 32:174 lbm-ft/lbf-sec
2
H head or velocity head, m, ft
H fluid enthalpy, J, Btu
_
HH enthalpy rate, used in Chapter 11, W, Btu/hr
Hg Hagen number, defined by Eq. (7.23), dimensionless
*
H

thermal boundary condition referring to constant axial as well as peripheral
wall heat flux; also constant peripheral wall temperature; boundary
condition valid only for the circular tube, parallel plates, and concentric
annular ducts when symmetrically heated
NOMENCLATURE xxi
*
H1
thermal boundary condition referring to constant axial wall heat flux with
constant peripheral wall temperature
*
H2
thermal boundary condition referring to constant axial wall heat flux with
constant peripheral wall heat flux
h heat transfer coefficient [defined by Eqs. (7.11) and (7.12)], W=m
2
Á K, Btu/
hr-ft
2
-8F
h specific enthalpy, J/kg, Btu/lbm
h
e
heat transfer coefficient at the fin tip, W=m
2
Á K, Btu/hr-ft
2
-8F
h
‘g
specific enthalpy of phase change, J/kg, Btu/lbm

_
II
irr
irreversibility rate (defined in Table 11.3), W, Btu/hr
I
n
ðÁÞ modified Bessel function of the first kind and nth order
i
j
flow direction indicator, i
j
¼þ1orÀ1, fluid j ¼ 1 or 2, dimensionless
J mechanical to thermal energy conversion factor, J ¼ 1 and dimensionless in SI
units, J ¼ 778:163 lbf-ft/Btu
J
i
correction factors for the shell-side heat transfer coefficient for the Bell–
Delaware method [see Eq. (9.50)]; i ¼ c for baffle cut and spacing; i ¼ ‘ for
baffle leakage effects, including both shell-to-baffle and tube-to-baffle leak-
age; i ¼ b for the bundle bypass flow (C and F streams); i ¼ s for variable
baffle spacing in the inlet and outlet sections; i ¼ r for adverse temperature
gradient buildup in laminar flow, dimensionless
j Colburn factor, St Pr
2/3
, ðh=Gc
p
ÞPr
2=3
, dimensionless
K pressure loss coefficient, Áp=ðu

2
m
=2g
c
Þ; subscripts: b for a circular bend, s for
a miter bend, and v for a screwed valve in Chapter 6, and br for branches in
Chapter 12, dimensionless
Kð1Þ incremental pressure drop number for fully developed flow (see Table 7.2 for
the definition), dimensionless
K
c
contraction loss coefficient for flow at heat exchanger entrance, dimensionless
K
e
expansion loss coefficient for flow at heat exchanger exit, dimensionless
K
n
ðÁÞ modified Bessel function of the second kind and nth order
k fluid thermal conductivity for fluid if no subscript, W=m Á K, Btu/hr-ft-8F
k
f
thermal conductivity of the fin material in Chapter 4 and of the foulant
material in Chapter 13, W=m Á K, Btu/hr-ft-8F
k
w
thermal conductivity of the matrix (wall) material, W=m Á K, Btu/hr-ft-8F
L fluid flow (core) length on one side of an exchanger, m, ft
L
f
fin flow length on one side of a heat exchanger, L

f
L,m,ft
L
h
plate length in a PHE for heat transfer (defined in Fig. 7.28), m, ft
L
p
plate length in a PHE for pressure drop (defined in Fig. 7.28), m, ft
L
1
flow (core) length for fluid 1 of a two-fluid heat exchanger, m, ft
L
2
flow (core) length for fluid 2 of a two-fluid heat exchanger, m, ft
L
3
noflow height (stack height) of a two-fluid heat exchanger, m, ft
L
q
Le
´
veˆ que number, defined by Eq. (7.41), dimensionless
xxii NOMENCLATURE
‘ fin height or fin length for heat conduction from primary surface to either fin
tip or midpoint between plates for symmetric heating, ‘ ¼ðd
e
À d
o
Þ=2 for
individually finned tubes, ‘ with this meaning used only in the fin analysis

and in the definition of 
f
,m,ft
‘
c
baffle cut, distance from the baffle tip to the shell inside diameter (see Fig. 8.9),
m, ft
‘
ef
effective flow length between major boundary layer disturbances, distance
between interruptions, m, ft
‘
s
strip length of an offset strip fin, m, ft
‘* flow length between interruptions, ‘
ef
=ðD
h
Á Re Á PrÞ, dimensionless
‘
c
*
baffle cut, ‘
c
=D
s
, dimensionless
m molecular weight (molar mass) of a gas, kg/kmol, lbm/lb mole
M
A

foulant material mass per unit heat transfer surface area in Chapter 13, m/A,
kg/m
2
, lbm/ft
2
M
w
mass of a heat exchanger core or the total mass of all matrices of a regenerator,
kg, lbm
m fin parameter [defined by Eqs. (4.62) and (4.65); see also Table 4.5 for other
definitions], 1/m, 1/ft
m mass of a body or fluid in a control volume, kg, lbm
_
mm fluid mass flow rate, u
m
A
o
, kg/s, 1bm/hr
_
mm
n
fluid mass flow rate for nominal flow passages in Chapter 12, kg/s, 1bm/hr
N number of subexchangers in gross flow maldistributed exchanger or a number
of differently sized/shaped passages in passage-to-passage nonuniformity,
used in Chapter 12
N rotational speed for a rotary regenerator, rev/s, rpm
N
b
number of baffles in a plate-baffled shell-and-tube exchanger
N

c
number of fluid channels in a plate heat exchanger
N
f
number of fins per unit length in the fin pitch direction, l/m, l/ft
N
p
number of fluid 1 passages in a two-fluid heat exchanger
N
p
number of pass divider lanes through the tube field that are parallel to the
crossflow stream in a shell-and-tube exchanger
N
0
p
number of separating plates in a plate-fin exchanger, number of pass divider
lanes in a shell-and-tube exchanger
N
r
number of tube rows in the flow direction
N
r;c
number of effective tube rows crossed during flow through one baffle section,
N
r;cc
þ N
r;cw
N
r;cc
number of effective tube rows crossed during flow through one crossflow

section (between baffle tips)
N
r;cw
number of effective tube rows crossed during flow through one window zone in
a segmental baffled shell-and-tube heat exchanger
N
t
total number of tubes in an exchanger, total number of holes in a tubesheet, or
total number of plates in a plate heat exchanger
N
t;b
total number of tubes associated with one segmental baffle
NOMENCLATURE xxiii
N
t;c
number of tubes at the tube bundle centerline cross section
N
t; p
number of tubes per pass
N
t;w
number of tubes in the window zone
N
0
t
number of tubes in a specified row
NTU number of exchanger heat transfer units, UA=C
min
[defined by Eqs. (3.59)
through (3.64)], it represents the total number of transfer units in a multipass

unit, dimensionless
NTU
1
number of exchanger heat transfer units based on fluid 1 heat capacity rate,
UA=C
1
; similarly, NTU
2
¼ UA=C
2
, dimensionless
NTU
c
number of exchanger heat transfer units based on C
c
, UA=C
c
, dimensionless
NTU
h
number of exchanger heat transfer units based on C
h
, UA=C
h
, dimensionless
NTU
o
modified number of heat transfer units for a regenerator [defined by Eq.
(5.48)], dimensionless
NTU* number of heat transfer units at maximum entropy generation, dimensionless

Nu Nusselt number [defined by Eqs. (7.26) and (7.27)], dimensionless
n, n
p
number of passes in an exchanger
n
c
number of cells of a regenerator matrix per unit of frontal area, 1/m
2
, 1/ft
2
n
f
total number of fins on one fluid side of an extended-surface exchanger
n
t
number of tubes in each pass
ntu
c
number of heat transfer units based on the cold fluid side, ð
o
hAÞ
c
=C
c
,
dimensionless
ntu
*
cost
reduction in ntu [defined by Eq. (12.44)], dimensionless

ntu
h
number of heat transfer units based on the hot fluid side, ð
o
hAÞ
h
=C
h
,
dimensionless
P fluid pumping power,
_
mm Áp=,W,hp
P temperature effectiveness for one fluid stream [defined by Eqs. (3.96) and
(3.97)], dimensionless
P wetted perimeter of exchanger passages on one fluid side, P ¼ A=L ¼
A
fr
,m,ft
} deposition probability function, dimensionless
P
c
cold-gas flow period, duration of the cold-gas stream in the matrix or duration
of matrix in the cold-gas stream, used in Chapter 5, s, sec
P
h
hot-gas flow period, duration of the hot-gas stream in the matrix or duration
of matrix in the hot-gas stream, used in Chapter 5, s, sec
P
r

reversal period for switching from hot- to cold-gas stream, or vice versa, in a
fixed-matrix regenerator, used in Chapter 5, s, sec
P
t
total period between the start of two successive heating (or cooling) periods in
a regenerator, used in Chapter 5, P
t
¼ P
h
þ P
c
þ P
r
% P
h
þ P
c
, s, sec
Pe Pe
´
clet number, Re Á Pr, dimensionless
Pr Prandtl number, c
p
=k, u
m
D
h
=, dimensionless
p fluid static pressure, Pa, lbf/ft
2

(psf ) or lbf/in
2
(psi)
{
xxiv NOMENCLATURE
{
Pa ¼ Pascal ¼ N=m
2
¼ kg=m Á s
2
;N¼ newton ¼ kg Ám=s
2
;psf¼ lbf=ft
3
;psi¼ lbf=in
3
:
p porosity of a matrix, a ratio of void volume to total volume of a matrix, r
h
,
dimensionless
p* ratio of cold-fluid inlet pressure to hot-fluid inlet pressure, p
c;i
=p
h;i
, dimension-
less
p
d
fin pattern depth, peak-to-valley distance, excluding fin thickness (see Fig.

7.30), m, ft
p
f
fin pitch, 1=N
f
,m,ft
p
t
tube pitch, center-to-center distance between tubes, m, ft
Áp fluid static pressure drop on one fluid side of a heat exchanger core [see
Eq. (6.28)], Pa, psf (psi)
Áp* ¼ Áp=ðu
2
m
=2g
c
Þ, dimensionless
Áp
b
fluid static pressure drop associated with a pipe bend, Pa, psf (psi)
Áp
b;i
fluid static pressure drop associated with an ideal crossflow section between
two baffles, Pa, psf (psi)
Áp
c
fluid static pressure drop associated with the tube bundle central section
(crossflow zone) between baffle tips, Pa, psf (psi)
Áp
gain

pressure drop reduction due to passage-to-passage nonuniformity [defined by
Eq. (12.36)], Pa, psf (psi)
Áp
s
shell-side pressure drop, Pa, psf (psi)
Áp
w;i
fluid static pressure drop associated with an ideal window section, Pa, psf (psi)
Q heat transfer in a specified period or time, J, Btu
q total or local (whatever appropriate) heat transfer rate in an exchanger, or
heat ‘‘duty,’’ W, Btu/hr
q* normalized heat transfer rate, q=½ð
_
mmc
p
ÞðT
2;i
À T
1;i
Þ, dimensionless
q
0
heat transfer rate per unit length, q=L, W/m, Btu/hr-ft
q
00
heat flux, heat transfer rate per unit surface area, q=A, W/m
2
, Btu/hr-ft
2
q

e
heat transfer rate through the fin tip, W, Btu/hr
q
0
heat transfer rate at the fin base, W, Btu/hr
q
max
thermodynamically maximum possible heat transfer rate in a counterflow heat
exchanger as expressed by Eq. (3.42), and also that through the fin base as
expressed by Eq. (4.130), W, Btu/hr
R universal gas constant, 8.3143 kJ=kmol Á K, 1545.33 1bf-ft/1b mole-8R
R heat capacity rate ratio [defined by Eqs. (3.105) and (3.106)], dimensionless
R thermal resistance based on the surface area A; R ¼ 1=UA ¼ overall thermal
resistance in a two-fluid exchanger, R
h
¼ 1=ðhAÞ
h
¼ hot-side film resistance
(between the fluid and the wall), R
c
¼ cold-side film resistance, R
f
¼ fouling
resistance, and R
w
¼ wall thermal resistance [definitions found after Eq.
(3.24)], K/W, hr-8F/Btu
^
RR unit thermal resistance,
^

RR ¼ RA ¼ 1=U,
^
RR
h
¼ 1=ð
o
hÞ
h
,
^
RR
w
¼ 1=ð
o
hÞ
c
,
^
RR
w
¼ 
w
=A
w
,m
2
Á K=W, hr-ft
2
8F/Btu
R* ratio of thermal resistances on the C

min
to C
max
side, 1=ð
o
hAÞ*; it is also the
same as the ratio of hot to cold reduced periods, Å
h
=Å
c
, Chapter 5, dimen-
sionless
NOMENCLATURE xxv
R* total thermal resistance (wall, fouling, and convective) on the enhanced (or
plain with subscript p) ‘‘outside’’ surface side normalized with respect to the
thermal resistance ½1=ðhA
i; p
Þ of ‘‘inside’’ plain tube/surface (see Table 10.5
for explicit formulas), dimensionless
~
RR gas constant for a particular gas, R/m,J=kg Á K, 1bf-ft=1bm-8R
^
RR
f
fouling factor or unit thermal resistance (‘‘fouling resistance’’), 1=h
f
,
m
2
Á K=W, hr-ft

2
-8F/Btu
R
i
pressure drop correction factor for the Bell–Delaware method, where i ¼ b for
bundle bypass flow effects (C stream), i ¼ ‘ for baffle leakage effects (A and
E streams), i ¼ s for unequal inlet/outlet baffle spacing effects, dimension-
less
Ra Rayleigh number [defined by Eq. (7.160)], dimensionless
Re Reynolds number based on the hydraulic diameter, GD
h
=, dimensionless
Re
d
Reynolds number based on the tube outside diameter and mean velocity,
u
m
d
o
=, dimensionless
Re
dc
Reynolds number based on the collar diameter and mean velocity, u
m
d
c
=,
dimensionless
Re
o

Reynolds number based on the tube outside diameter and free stream
(approach or core upstream) velocity, u
1
d
o
=, dimensionless
r radial coordinate in the cylindrical coordinate system, m, ft
r
c
radius of curvature of a tube bend (see Fig. 6.5), m, ft
r
f
fouling factor or fouling resistance r
f
¼
^
RR
f
¼ 1=h
f
¼ 
f
=k
f
,m
2
Á K=W, hr-ft
2
-
8F/Btu

r
h
hydraulic radius, A
o
L=A or D
h
=4, m, ft
r
i
tube inside radius, m, ft
S entropy, J/K, Btu/8R
S* normalized entropy generation rate,
_
SS
irr
=C
2
or
_
SS
irr
=C
max
, dimensionless
_
SS
irr
entropy generation rate, W/K, Btu/hr-8R
St Stanton number, h=Gc
p

,St
o
¼ U=Gc
p
, dimensionless
s specific entropy in Chapter 11, J=kg Á K, Btu/lbm-8R
s complex Laplace independent variable with Laplace transforms only in
Chapter 11, dimensionless
s spacing between adjacent fins, p
f
À ,m,ft
T fluid static temperature to a specified arbitrary datum, except for Eqs. (7.157)
and (7.158) and in Chapter 11 where it is defined on an absolute temperature
scale, 8C, 8F
*
T
thermal boundary condition referring to constant wall temperature, both
axially and peripherally
T
c;o
flow area average cold-fluid outlet temperature unless otherwise specified, 8C,
8F
T
h;o
flow area average hot-fluid outlet temperature unless otherwise specified, 8C,
8F
T
‘
temperature of the fin tip, 8C, 8F
T

m
fluid bulk mean temperature, 8C, 8F
xxvi NOMENCLATURE