Heat Transfer E-Book

Adrian Bejan

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 1026

Veröffentlichungsjahr: 2022

Beschreibung

HEAT TRANSFER Provides authoritative coverage of the fundamentals of heat transfer, written by one of the most cited authors in all of Engineering Heat Transfer presents the fundamentals of the generation, use, conversion, and exchange of heat between physical systems. A pioneer in establishing heat transfer as a pillar of the modern thermal sciences, Professor Adrian Bejan presents the fundamental concepts and problem-solving methods of the discipline, predicts the evolution of heat transfer configurations, the principles of thermodynamics, and more. Building upon his classic 1993 book Heat Transfer, the author maintains his straightforward scientific approach to teaching essential developments such as Fourier conduction, fins, boundary layer theory, duct flow, scale analysis, and the structure of turbulence. In this new volume, Bejan explores topics and research developments that have emerged during the past decade, including the designing of convective flow and heat and mass transfer, the crucial relationship between configuration and performance, and new populations of configurations such as tapered ducts, plates with multi-scale features, and dendritic fins. Heat Transfer: Evolution, Design and Performance: * Covers thermodynamics principles and establishes performance and evolution as fundamental concepts in thermal sciences * Demonstrates how principles of physics predict a future with economies of scale, multi-scale design, vascularization, and hierarchical distribution of many small features * Explores new work on conduction architecture, convection with nanofluids, boiling and condensation on designed surfaces, and resonance of natural circulation in enclosures * Includes numerous examples, problems with solutions, and access to a companion website Heat Transfer: Evolution, Design and Performance is essential reading for undergraduate and graduate students in mechanical and chemical engineering, and for all engineers, physicists, biologists, and earth scientists.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

Notes

About the Author

Acknowledgments

List of Symbols

About the Companion Website

1 Introduction

1.1 Fundamental Concepts

1.2 The Objective of Heat Transfer

1.3 Conduction

1.4 Convection

1.5 Radiation

1.6 Evolutionary Design

References

Problems

Notes

2 Unidirectional Steady Conduction

2.1 Thin Walls

2.2 Cylindrical Shells

2.3 Spherical Shells

2.4 Critical Insulation Radius

2.5 Variable Thermal Conductivity

2.6 Internal Heat Generation

2.7 Evolutionary Design: Extended Surfaces (Fins)

References

Problems

Notes

3 Multidirectional Steady Conduction

3.1 Analytical Solutions

3.2 Integral Method

3.3 Scale Analysis

3.4 Evolutionary Design

References

Problems

Notes

4 Time-Dependent Conduction

4.1 Immersion Cooling or Heating

4.2 Lumped Capacitance Model (The “Late” Regime)

4.3 Semi-infinite Solid Model (The “Early” Regime)

4.4 Unidirectional Conduction

4.5 Multidirectional Conduction

4.6 Concentrated Sources and Sinks

4.7 Melting and Solidification

4.8 Evolutionary Design

References

Problems

Notes

5 External Forced Convection

5.1 Classification of Convection Configurations

5.2 Basic Principles of Convection

5.3 Laminar Boundary Layer

5.4 Turbulent Boundary Layer

5.5 Other External Flows

5.6 Evolutionary Design

References

Problems

Notes

6 Internal Forced Convection

6.1 Laminar Flow Through a Duct

6.2 Heat Transfer in Laminar Flow

6.3 Turbulent Flow

6.4 Total Heat Transfer Rate

6.5 Evolutionary Design

References

Problems

Note

7 Natural Convection

7.1 What Drives Natural Convection?

7.2 Boundary Layer Flow on Vertical Wall

7.3 Other External Flows

7.4 Internal Flows

7.5 Evolutionary Design

References

Problems

Notes

8 Convection with Change of Phase

8.1 Condensation

8.2 Boiling

8.3 Evolutionary Design

References

Problems

Notes

9 Heat Exchangers

9.1 Classification of Heat Exchangers

9.2 Overall Heat Transfer Coefficient

9.3 Log-Mean Temperature Difference Method

9.4 Effectiveness–NTU Method

9.5 Pressure Drop

9.6 Evolutionary Design

References

Problems

10 Radiation

10.1 Introduction

10.2 Blackbody Radiation

10.3 Heat Transfer Between Black Surfaces

10.4 Diffuse-Gray Surfaces

10.5 Participating Media

10.6 Evolutionary Design

References

Problems

Notes

Appendix A: Constants and Conversion Factors

Appendix B: Properties of Solids

References

Appendix C: Properties of Liquids

References

Appendix D: Properties of Gases

References

Appendix E: Mathematical Formulas

Error Function

Leibniz's Formula for Differentiating an Integral

Hyperbolic Functions

Reference

Appendix F: Turbulence Transition

References

Appendix G: Extremum Subject to Constraint

Author Index

Subject Index

End User License Agreement

List of Tables

Chapter 3

Table 3.1 The first six roots of Eq. (3.42).

Table 3.2 Three types of boundary conditions and the corresponding homogeneo...

Table 3.3 Shape factors (

S

) for several configurations with isothermal surfa...

Table 3.4 How, not what: the balancing of high resistivity flow with low res...

Chapter 4

Table 4.1 Constants for the solution for temperature in a cylinder, Eq. (4.6...

Table 4.2 Constants for the solution for temperature in a sphere, Eq. (4.73)...

Chapter 5

Table 5.1 The governing equations for constant-property flow in Cartesian co...

Table 5.2 The governing equations for constant-property flow in cylindrical ...

Table 5.3 The governing equations for constant-property flow in spherical co...

Table 5.4 Heat transfer results for laminar boundary layer flows near walls ...

Chapter 6

Table 6.1 Friction factors (

f

), cross section shape numbers (

B

), and Nusselt...

Table 6.2 Friction factors and Nusselt numbers for heat transfer to laminar ...

Chapter 7

Table 7.1 Constants in Eq. (7.86) for laminar natural convection on immersed...

Table 7.2 Average Nusselt numbers for chimney flow in the narrow-channel lim...

Chapter 8

Table 8.1 Empirical constants for the nucleate pool boiling correlations (8....

Table 8.2 Surface tension and other physical properties needed for calculati...

Chapter 9

Table 9.1 Representative values of the fouling factor

r

s

(m

2

·

K/W).

Table 9.2 Representative orders of magnitude of the overall heat transfer co...

Chapter 10

Table 10.1 Principal values and the asymptotic behavior of the dimensionless...

Table 10.2 Minicatalog of geometric view factors.

Table 10.3 Metallic surfaces: representative values of total hemispherical e...

Table 10.4 Nonmetallic surfaces: representative values of total hemispherica...

Table 10.5 The equivalent length

L

e

for several gas volume shapes.

a)

Appendix F

Table F.1 Traditional critical numbers for transition in several key flows a...

List of Illustrations

Preface

Figure 1 The evolution, spreading and merger of heat transfer with thermodyn...

Chapter 1

Figure 1.1 The relationships between the Kelvin, Celsius, Rankine, and Fahre...

Figure 1.2 Two heating processes for measuring (a) the specific heat at cons...

Figure 1.3 Unidirectional conduction through a solid body with internal heat...

Figure 1.4 Classification of thermally conducting media in terms of their ho...

Figure 1.5 Dependence of thermal conductivity on temperature (the

k

data are...

Figure 1.6 Three-dimensional Cartesian system of coordinates.

Figure 1.7 Cylindrical system of coordinates.

Figure 1.8 Spherical system of coordinates.

Figure E1.1

Figure 1.9 External flow configuration of convective heat transfer.

Figure 1.10 Internal flow configuration of convective heat transfer.

Figure 1.11 Effect of the fluid type and flow regime on the order of magnitu...

Figure E1.2

Figure E1.3

Figure 1.12 Thermal radiation across an evacuated space.

Figure 1.13 The destruction of useful power. Drawing from 1976 [22] and 1982...

Figure 1.14 (a) Reversible heating from below, (b) reversible heating from a...

Figure P1.4

Figure P1.5

Figure P1.7

Figure P1.8

Figure P1.9

Figure P1.11

Figure P1.15

Figure P1.16

Chapter 2

Figure 2.1 Thermal resistance posed by a sufficiently thin wall.

Figure 2.2 Composite wall and the structure of its thermal resistance.

Figure 2.3 Thin wall sandwiched between two flows: the definition of overall...

Figure E2.1

Figure 2.4 Radial conduction through a cylindrical shell.

Figure 2.5 Composite cylindrical shell with convective heat transfer on both...

Figure 2.6 Radial conduction through a spherical shell.

Figure 2.7 Effect of the outer radius on the overall thermal resistance of a...

Figure E2.2

Figure 2.8 Unidirectional conduction through a solid with temperature-depend...

Figure 2.9 Steady temperature distribution due to uniform internal heat gene...

Figure 2.10 Increase in wall heat flux over the area covered by fins.

Figure 2.11 Longitudinal conduction through a fin with constant cross-sectio...

Figure 2.12 Fin with insulated tip versus fin with finite heat transfer rate...

Figure 2.13 Geometrical reasoning behind the concept of corrected length,

L

c

Figure 2.14 Efficiency of two-dimensional fins with rectangular, triangular,...

Figure 2.15 Longitudinal conduction through a fin with variable cross-sectio...

Figure 2.16 Efficiency of annular fins with constant thickness.

Figure 2.17 Pattern of heat flux lines through a two-dimensional fin with re...

Figure 2.18 Scale drawing of the optimum profile of a plate fin of fixed vol...

Figure 2.19 One-dimensional conduction along a heat tube with fixed length a...

Figure 2.20 Heat spreader with steady line heat source and fins on the upper...

Figure 2.21 (a) The design of Figure 2.20a in the limit where the base thick...

Figure E2.3

Figure P2.5

Figure P2.6

Figure P2.7

Figure P2.8

Figure P2.9

Figure P2.11

Figure P2.14

Figure P2.15

Figure P2.16

Figure P2.19

Figure P2.20

Figure P2.23

Figure P2.24

Figure P2.25

Figure P2.26

Figure P2.27

Figure P2.30

Figure P2.31

Chapter 3

Figure 3.1 Two-dimensional conducting medium with isothermal boundaries (a),...

Figure 3.2 Isotherms and heat flux lines in the two-dimensional rectangular ...

Figure 3.3 Infinitely long plate fin with finite heat transfer coefficient o...

Figure 3.4 The superposition of two known solutions (

θ

1

 + 

θ

2

) as a...

Figure 3.5 Cylindrical object with a temperature difference between one end ...

Figure 3.6 The behavior of the Bessel functions

J

0

,

Y

0

, and

J

1

.

Figure 3.7 Three-dimensional (finite width) generalization of the heat trans...

Figure 3.8 Bar with rectangular cross section and uniform rate of internal h...

Figure E3.2

Figure 3.9 Slender elemental volume with uniform volumetric heat generation ...

Figure 3.10 Elemental conduction volume with progressively greater freedom t...

Figure 3.11 Evolutionary invasion of a conducting tree into a conducting bod...

Figure P3.2

Figure P3.4

Figure P3.5

Figure P3.6

Figure P3.7

Figure P3.8

Figure P3.10

Figure P3.11

Figure P3.12

Figure P3.13

Chapter 4

Figure 4.1 Thermally penetrated layer in a body immersed suddenly in a fluid...

Figure 4.2 The transition from the early regime to the late regime.

Figure 4.3 Penetration of heat conduction into a semi-infinite solid with is...

Figure 4.4 Temperature distribution in an isothermal semi-infinite solid (

T

i

Figure 4.5 The domains of applicability of the lumped capacitance and semi-i...

Figure E4.1

Figure E4.2

Figure 4.6 Plate of thickness 2

L

immersed suddenly in a fluid with convectio...

Figure 4.7 Temperature history in the midplane of a plate immersed suddenly ...

Figure 4.8 Relationship between the temperature in any plane (

x

) and the tem...

Figure 4.9 Total heat transfer between a plate and the surrounding fluid, as...

Figure 4.10 Temperature history on the centerline of a cylinder immersed sud...

Figure 4.11 Relationship between the local temperature (

r

) and the centerlin...

Figure 4.12 Total heat transfer between a cylinder and the surrounding fluid...

Figure 4.13 Temperature history in the center of a sphere (

r

o

 = sphere radiu...

Figure 4.14 Relationship between the temperature at any radius (

r

) and the t...

Figure 4.15 Total heat transfer between a sphere and the surrounding fluid, ...

Figure 4.16 The volume-averaged temperature

and the total heat transfer fr...

Figure 4.17 The time-dependent temperature in a bar immersed in fluid, as th...

Figure 4.18 The time-dependent temperature of a short cylinder immersed in f...

Figure 4.19 Multiplication rules for the temperature distribution in a semi-...

Figure 4.20 The time-dependent temperature of a parallelepiped immersed in f...

Figure 4.21 The temperature distribution in the vicinity of an instantaneous...

Figure 4.22 The thermal wake behind a continuous line source moving through ...

Figure 4.23 The thermal wake left behind a continuous point source in a movi...

Figure 4.24 Melting (a) and solidification (b) into a semi-infinite isotherm...

Figure 4.25 Melting of a semi-infinite solid at the melting point (

T

i

 = 

T

m

) ...

Figure 4.26 History of the melting (or solidification) front position in a s...

Figure 4.27 The deformed shape of the melting front when heating is from the...

Figure 4.28 Conducting finite-size volume with several embedded line heat so...

Figure 4.29 Line-shaped invasion, followed by consolidation by traversal dif...

Figure 4.30 Tree-shaped invasion, showing the narrow regions covered by diff...

Figure P4.2

Figure P4.3

Figure P4.14

Figure P4.24 (a) After 10 days and (b) much later.

Figure P4.25

Figure P4.26

Figure P4.27

Chapter 5

Figure 5.1 The field of convection heat transfer and the main configurations...

Figure 5.2 The conservation of mass in an infinitesimal control volume in a ...

Figure 5.3 The development of the momentum equation for the

x

direction: the...

Figure 5.4 The four contributions to the statement of energy conservation in...

Figure 5.5 Velocity boundary layer in laminar flow near a plane wall.

Figure 5.6 The similarity velocity profile for laminar boundary layer flow o...

Figure 5.7 The thermal boundary layer in low-

Pr

fluids (a) and high-

Pr

fluid...

Figure 5.8 Laminar, transition, and turbulent sections in the boundary layer...

Figure 5.9 The laminar section and the beginning of transition in the air bo...

Figure 5.10 The behavior of an instantaneous quantity (

u

) in turbulent flow ...

Figure 5.11 The structure of the apparent shear stress

τ

app

and the app...

Figure 5.12 The behavior of the boundary layer thickness and wall shear stre...

Figure 5.13 The behavior of the local heat transfer coefficient in the lamin...

Figure E5.2

Figure 5.14 Single cylinder (or sphere) in cross-flow and the features of th...

Figure 5.15 Drag coefficients of a smooth sphere and a single smooth cylinde...

Figure 5.16 Banks of cylinders in cross-flow: aligned (a) versus staggered (...

Figure 5.17 The effect of the number of rows on the array-averaged Nusselt n...

Figure 5.18 Turbulent, nonbuoyant round jet mixing with a stagnant reservoir...

Figure 5.19 Plate surface with specified heat transfer rate.

Figure 5.20 Large organs belong on large vehicles and animals. Every flow co...

Figure P5.8

Figure P5.9

Figure P5.14

Figure P5.19

Figure P5.31

Figure P5.50

Figure P5.52

Chapter 6

Figure 6.1 Entrance region and fully developed flow region of laminar flow i...

Figure 6.2 Laminar flow through a parallel-plate channel.

Figure 6.3 Force balance over the flow control volume (the upper-left comer)...

Figure 6.4 Thermal entrance region and the thermally fully developed flow in...

Figure 6.5 Fully developed temperature distribution in a tube with uniform h...

Figure 6.6 The Nusselt number for laminar flow through a tube with uniform w...

Figure 6.7 Fully developed temperature distribution in a tube with constant ...

Figure 6.8 The Nusselt number for laminar flow through a tube with constant ...

Figure 6.9 Longitudinal velocity profile and apparent shear stress distribut...

Figure 6.10 Friction factor for fully developed laminar and turbulent flow i...

Figure 6.11 Distribution of temperature along a duct: (a) isothermal wall an...

Figure 6.12 Round tube with specified mass flowrate.

Figure 6.13 T-shaped construct of round tubes.

Figure 6.14 Stack of parallel heat-generating plates cooled by forced convec...

Figure 6.15 Intersection-of-asymptotes method: the recommended spacing as th...

Figure P6.7

Figure P6.10

Figure P6.12

Figure P6.24

Figure P6.26

Figure P6.27

Figure P6.28

Figure P6.29

Figure P6.30

Figure P6.31

Figure P6.32

Figure P6.35

Chapter 7

Figure 7.1 Wall jet driven by buoyancy along a heated wall, and pressure dis...

Figure 7.2 High Prandtl number fluids: thermal and velocity boundary layers ...

Figure 7.3 Low Prandtl number fluids: thermal and velocity boundary layers o...

Figure 7.4 Similarity temperature and velocity profiles for laminar natural ...

Figure 7.5 Laminar, transition, and turbulent sections on a wall with natura...

Figure 7.6 Average Nusselt number for laminar flow along an isothermal wall ...

Figure 7.7 Plane walls inclined relative to the vertical direction.

Figure 7.8 Horizontal surfaces with central plume (a) and without plume flow...

Figure 7.9 Horizontal cylinder or sphere immersed in a fluid at a different ...

Figure 7.10 Vertical cylinders with boundary layer flow on the lateral surfa...

Figure 7.11 Shapes and orientations of isothermal bodies immersed in a fluid...

Figure 7.12 Vertical channel with isothermal walls; the top and bottom ends ...

Figure 7.13 Enclosure filled with fluid and heated and cooled along the two ...

Figure 7.14 Flow regimes for natural convection in enclosures heated from th...

Figure 7.15 Shallow enclosures heated from the side: average Nusselt number ...

Figure 7.16 Horizontal fluid layer between parallel walls and heated from be...

Figure 7.17 Two-dimensional rolls and three-dimensional hexagonal cells in a...

Figure E7.4

Figure E7.5

Figure 7.18 The effect of tilt angle on the heat transfer rate and flow patt...

Figure 7.19 Natural convection in the annular space between horizontal conce...

Figure 7.20 Vertical stack of heat-generating plates cooled by natural conve...

Figure 7.21 Plate-to-plate spacing at the intersection of the small-

D

and th...

Figure 7.22 The evolution of heat transfer density toward higher values, sho...

Figure P7.8

Figure P7.18

Figure P7.21

Figure P7.25

Figure P7.27

Figure P7.29

Chapter 8

Figure 8.1 Flow regimes of the film of condensate on a cooled vertical surfa...

Figure 8.2 Laminar film of condensate in a reservoir of stationary saturated...

Figure 8.3 Effect of Prandtl number on heat transfer from a laminar film of ...

Figure 8.4 Inertia-restrained and friction-restrained film condensation on a...

Figure 8.5 The

L

-averaged heat transfer coefficient for laminar, wavy, and t...

Figure 8.6 Total condensation rate (or

Re

L

) versus condensation driving para...

Figure 8.7 Vertical surfaces with thin films of condensate that can be regar...

Figure 8.8 Film condensation on inclined plane and spherical surfaces.

Figure 8.9 Film condensation on a single horizontal cylinder and on a vertic...

Figure 8.10 Film of condensate on a horizontal strip of width

L

or a horizon...

Figure 8.11 Film condensation on a horizontal cylinder in cross-flow (a) and...

Figure 8.12 Condensation in a vertical tube with cocurrent flow of vapor.

Figure 8.13 Annular film condensation in a tube with

fast

vapor flow (a) and...

Figure 8.14 The surface cleaning effect due to the departure of one large dr...

Figure 8.15 Nucleate pool boiling of a subcooled liquid (a) and a saturated ...

Figure 8.16 The four regimes of pool boiling in water at atmospheric pressur...

Figure 8.17 The pool boiling curve in a temperature-controlled experiment (a...

Figure E8.3

Figure 8.18 The film boiling regime on a sphere or horizontal cylinder.

Figure 8.19 Steady production of power using a one phase-change material and...

Figure 8.20 Cascade of melting and solidification in two materials placed in...

Figure P8.2

Figure P8.7

Figure P8.8

Figure P8.10

Figure P8.12

Figure P8.13

Figure P8.16

Figure P8.17

Figure P8.18

Figure P8.23

Chapter 9

Figure 9.1 (a) Double-pipe parallel flow and (b) counterflow heat exchangers...

Figure 9.2 Plate fin cross-flow heat exchangers and the use of longitudinal ...

Figure 9.3 Single-pass and multipass shell-and-tube heat exchangers.

Figure 9.4 Shell-and-tube heat exchanger and three examples of baffle design...

Figure 9.5 Parallel-plate channel with interrupted fins (a) and bank of finn...

Figure 9.6 Ordering of heat exchangers according to their degree of compactn...

Figure 9.7 Heat exchanger surface with fins and scale on both sides.

Figure E9.1

Figure 9.8 The temperature distribution in a parallel flow heat exchanger.

Figure 9.9 Flow arrangements to which the heat transfer relationship (9.21) ...

Figure 9.10 Correction factor

F

for cross-flow (single-pass) heat exchangers...

Figure 9.11 Correction factor

F

for cross-flow (single-pass) heat exchangers...

Figure 9.12 Correction factor

F

for cross-flow (single-pass) heat exchangers...

Figure 9.13 Correction factor

F

for shell-and-tube heat exchangers with one ...

Figure 9.14 Correction factor

F

for shell-and-tube heat exchangers with two ...

Figure E9.2

Figure E9.3

Figure 9.15 The effect of the heat transfer conductance

UA

on the temperatur...

Figure 9.16 The effectiveness of a parallel flow heat exchanger.

Figure 9.17 The effectiveness of a counterflow heat exchanger.

Figure 9.18 The effectiveness of a cross-flow (single-pass) heat exchanger i...

Figure 9.19 The effectiveness of a cross-flow (single-pass) heat exchanger i...

Figure 9.20 The effectiveness of a cross-flow (single-pass) heat exchanger i...

Figure 9.21 The effectiveness of shell-and-tube heat exchangers with one she...

Figure 9.22 The effectiveness of shell-and-tube heat exchangers with two she...

Figure E9.4

Figure 9.23 Abrupt contraction in a two-dimensional flow passage.

Figure 9.24 Abrupt contraction and enlargement loss coefficients for a heat ...

Figure 9.25 Abrupt contraction and enlargement loss coefficients for a heat ...

Figure 9.26 Abrupt enlargement in a two-dimensional flow passage and the sep...

Figure 9.27 The cross-sectional contraction and enlargement experienced by a...

Figure 9.28 Arrays of aligned tubes: the coefficients

f

and

χ

for the p...

Figure 9.29 Arrays of staggered tubes: the coefficients

f

and

χ

for the...

Figure 9.30 Pressure drop and heat transfer data for the flow passage throug...

Figure 9.31 First construct containing a large number of stacked elemental v...

Figure 9.32 Three-dimensional flow construct with two dendritic streams in c...

Figure 9.33 Tube-based heat exchanger-architecture illustrating the use of c...

Figure 9.34 A flow system has shape and size. The shape is the trade-off bet...

Figure 9.35 The space occupied by a general system with heat transfer across...

Figure 9.36 (a) The evolution of heat transfer mechanisms and (b) compact co...

Figure P9.2

Figure P9.3

Figure P9.13

Figure P9.14

Figure P9.15

Figure P9.19

Figure P9.23

Figure P9.24

Chapter 10

Figure 10.1 The thermal radiation interaction between two bodies at differen...

Figure 10.2 The definitions of total absorptivity, reflectivity, and transmi...

Figure 10.3 The wavelength and frequency domains of thermal radiation and th...

Figure 10.4 Enclosure with perfectly reflecting internal surfaces and blackb...

Figure 10.5 Pencil of rays aligned with the direction normal to

dA

(a) and t...

Figure 10.6 The infinitesimal solid angle

dA

n

/

r

2

associated with the directi...

Figure 10.7 The effects of wavelength and temperature on the monochromatic h...

Figure 10.8 The dimensionless radiation function

E

b

(0–

λT

)/

σT

4

and ...

Figure 10.9 (a) The definition of the radiation function

E

b

(0–

λT

) and (...

Figure E10.1

Figure 10.10 The geometric parameters needed for calculating the view factor...

Figure 10.11 The geometric view factor between two parallel rectangles.

Figure 10.12 The geometric view factor between two perpendicular rectangles ...

Figure 10.13 Two cases in which the additivity property can be used to calcu...

Figure 10.14 Enclosure formed by

n

surfaces and the dependence of the view f...

Figure 10.15 Examples of enclosures consisting of only two surfaces. (a) Two...

Figure E10.3a

Figure E10.3b

Figure E10.4

Figure 10.16 The emission characteristics of a directional emitter (a) and a...

Figure 10.17 The gray-surface model: the monochromatic emissive power (a) an...

Figure 10.18 The absorption characteristics of a directional absorber (a) an...

Figure 10.19 The reflection characteristics of a directional reflector (a) a...

Figure 10.20 Two-surface enclosure for the derivation of Kirchhoff's law,

....

Figure 10.21 Enclosure defined by two diffuse-gray surfaces, and the thermal...

Figure 10.22 The surface radiosity as the superposition of diffuse emission ...

Figure E10.5

Figure 10.23 Enclosure formed by

n

diffuse-gray surfaces, and the resistance...

Figure E10.6

Figure 10.24 The attenuation of the monochromatic radiation that penetrates ...

Figure 10.25 Hemispherical space filled with gas radiating to a black surfac...

Figure 10.26 The emissivity of carbon dioxide in a mixture with nonparticipa...

Figure 10.27 Correction factor for the emissivity of carbon dioxide in a mix...

Figure 10.28 The emissivity of water vapor in a mixture with nonparticipatin...

Figure 10.29 Correction factor for the emissivity of water vapor in a mixtur...

Figure 10.30 Correction for gas emissivity when carbon dioxide and water vap...

Figure 10.31 Gray medium enclosed by two diffuse-gray surfaces, and the radi...

Figure 10.32 Convective-cooling model for moderate temperature solar collect...

Figure 10.33 Extraterrestrial solar power plant with radiative heat transfer...

Figure 10.34 The whole Earth is an engine + brake system, containing innumer...

Figure P10.1

Figure P10.5

Figure P10.11

Figure P10.12

Figure P10.14

Figure P10.15

Figure P10.16

Figure P10.17

Figure P10.18

Figure P10.19

Figure P10.21

Figure P10.22

Figure P10.28

Figure P10.29

Figure P10.30

Figure P10.31

Figure P10.32

Figure P10.33

Figure P10.34

Appendix B

Figure B.1

Appendix F

Figure F.1

Guide

Cover

Table of Contents

Title Page

Copyright

Preface

About the Author

Acknowledgments

List of Symbols

About the Companion Website

Begin Reading

Appendix A: Constants and Conversion Factors

Appendix B: Properties of Solids

Appendix C: Properties of Liquids

Appendix D: Properties of Gases

Appendix E: Mathematical Formulas

Appendix F: Turbulence Transition

Appendix G: Extremum Subject to Constraint

Author Index

Subject Index

End User License Agreement

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Other Books by Adrian Bejan

Entropy Generation Through Heat and Fluid Flow, Wiley, 1982.

Convection Heat Transfer, 1984, Wiley, Fourth Edition, 2013.

Advanced Engineering Thermodynamics, 1988, Wiley, Fourth Edition, 2016.

Convection in Porous Media, with D. A. Nield, 1992, Springer, Fifth Edition, 2017.

Heat Transfer, Wiley, 1993.

Thermal Design and Optimization, with G. Tsatsaronis and M. Moran, Wiley, 1996.

Entropy Generation Minimization, CRC Press, 1996.

Shape and Structure, from Engineering to Nature, Cambridge University Press, 2000.

Heat Transfer Handbook, with A. D. Kraus, eds., Wiley, 2003.

La loi constructale, with S. Lorente, L'Harmattan, Paris, 2005.

Constructal Theory of Social Dynamics, with G. W. Merkx, eds., Springer, 2007.

Design with Constructal Theory, with S. Lorente, Wiley, 2008.

DESIGN IN NATURE: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organization, with J. P. Zane, Doubleday, 2012.

THE PHYSICS OF LIFE: The Evolution of Everything, St. Martin's Press, 2016.

FREEDOM AND EVOLUTION: Hierarchy in Nature, Society and Science, Springer Nature, 2020.

TIME AND BEAUTY: Why Time Flies and Beauty Never Dies, World Scientific, 2022.

Heat Transfer: Evolution, Design and Performance, Wiley, 2022.

Heat Transfer

Evolution, Design and Performance

Adrian Bejan

Duke University

This edition first published 2022© 2022 John Wiley & Sons, Inc.

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Cover Design: WileyCover Image: © Adrian Bejan

Preface

The discipline of heat transfer grew as a sequence of solved problems. From these roots the doctrine emerged as a predictive method, not as a random collection of solved problems. The first problems were the most fundamental and the simplest, and they bear the names of their creators: Fourier, Prandtl, Nusselt, Reynolds, and their contemporaries. As the field grew, the problems became more ad-hoc and applied (i.e. relevant to this, but not to that), more complicated and disunited, and much more numerous and forgettable.

Hidden in this voluminous stream are the fundamental principles that emerge. Identifying these and building with them the structure of the discipline is the main characteristic of the present book. I teach not only structure but also strategy:

The structure is drawn with sharp lines: heat transfer versus thermodynamics, conduction versus convection, convection versus radiation, external convection versus internal convection, forced convection versus natural convection, combined convection and conduction, phase change, and radiation.

The strategy is to start with the simplest method (scale analysis), and follow with more laborious and exact methods. Scale analysis is powerful because it teaches how to determine (on the back of an envelope) the proper orders of magnitude of all the physical features that matter (temperature difference, heat flux, fluid velocity, boundary layer thickness, lost power). It reveals the correct dimensionless groups, which are the fewest such numbers. With them, we learn how to predict and correlate in the most compact form the results obtained analytically, numerically, and experimentally.

This book is an idea-book that points toward the future of the discipline, in four ways:

The relationship between configuration and performance. Traditionally, the fundamentals of heat transfer are taught by first postulating the configuration (the “boundary conditions”) and then solving the governing equations. The resulting solution describes the flow fields (pressure, temperature, velocity) and the currents that flow on these fields in the assumed configuration. The solution permits the calculation of global features such as pressure drop and heat transfer rate, which are important in practice.

The key word is “describe”. How these features affect the desired performance of the greater installation that uses that configuration is another matter. Addressed even less is how to discover the flow configuration in the first place. This is the new point of view from which this book teaches thermal sciences. It teaches how configuration affects performance, and how to configure the flows such that performance is enhanced. It establishes “performance” and “evolution” (the freedom to change1) as fundamental concepts in thermal sciences.

New configurations are being developed, adopted, and joined by new arrivals. This is the universal phenomenon of evolution as physics, bio and non bio, which includes technology evolution. Chief among the growing population of designs are the tree-shaped vasculatures that bathe entire areas and volumes. They are transforming the field of smart, high-density and multi-functional materials, and bringing them close to animal design. This book shows how principles of physics such as the constructal law

2

predict a future with more degrees of freedom, economies of scale, multi-scale design, vascularization, and hierarchical distribution of many small features among the few large features.

The oneness of natural convection and forced convection is a central feature. Traditionally, natural convection is presented as being “free,” i.e. unlike forced convection configurations for which people must constantly pay with power. In reality, natural convection too is driven by power from invisible “engines” that inhabit the flow field. These engines drive the flow, and because of fluid flow and heat transfer they dissipate their power output in internal “brakes” that account for the irreversibility of the relative motion between fluid layers, and the resistance overcome by heat currents.

Thermodynamics is an essential component in a heat transfer treatise (

Figure 1

). Why, because nothing moves and nothing flows (heat, fluid) unless it is driven. Power is dissipated throughout the flow and temperature fields. Flows do not happen unless differences and gradients (pressure, temperature) are sustained over time, so that power is generated in order to drive flows.

The book rests on material from Heat Transfer (1993), and it features new work (formulas, figures, problems, references) and the history of ideas and terminology. Words have meaning, and a very important word is optimization: the activity of making changes and choosing between the alternatives that emerge. To opt is to make a choice. To be able to choose, one must have freedom to change the existing configuration (design, organization) and to choose from the alternate configurations that emerge after the change. To optimize is a relentless activity of making choices when confronted with the endless number of forks in the road of evolution.

To choose is natural. Every moving thing, animate and inanimate, does it because of freedom. Every river alters its course and bed cross section to flow more easily. Every animal group varies its migration routes to facilitate its movement, which is its life. Every wounded tissue heals itself in order to keep the whole body moving, which means to keep the whole alive.3,4

Figure 1 The evolution, spreading and merger of heat transfer with thermodynamics and evolution during the past two centuries.

To change, choose and change again is the plot of the never ending movie called evolution. To find better choices is so good that it is addictive. The addiction reveals what “good” means. Freedom and design are concepts that belong in physics. They were placed firmly in physics as the constructal law (Figure 1).

With freedom new changes are made, more choices emerge, old bests die, and future bests are born. The best is short lived, precious today and derisory tomorrow. This truth is the mother of all evolution, including technology, and this book teaches it in terms of evolutionary design for heat and fluid flow and power.

Adrian Bejan

January 2021

Durham, NC

Notes

1

   A. Bejan,

Freedom and Evolution: Hierarchy in Nature, Society and Science

, Springer Nature, New York, 2020.

2

   For a finite-size flow system to persist in time (to live) it must evolve freely such that it offers greater access to what flows.

3

   A. Bejan,

The Physics of Life: The Evolution of Everything

, St. Martin's Press, New York, 2016.

4

   A. Bejan and J. P. Zane,

Design in Nature

, Doubleday, New York 2012.

About the Author

Adrian Bejan was awarded the 2018 Benjamin Franklin Medal “for his pioneering interdisciplinary contributions in thermodynamics and convection heat transfer that have improved the performance of engineering systems, and for constructal theory, which predicts natural design and its evolution in engineering, scientific, and social systems.”

He received the 2019 Humboldt Research Award for lifetime achievement for “his pioneering interdisciplinary contributions to thermodynamics and Constructal Law – a law of physics that predicts natural design and its evolution in biology, geophysics, climate change, technology, social organization, evolutionary design and development, wealth and sustainability.”

His degrees are from M.I.T.: BS (1971, Honors Course), MS (1972, Honors Course) and PhD (1975). He was a Fellow in the Miller Institute for Basic Research in Science at the University of California, Berkeley (1976–1978). At Duke University, he is the J. A. Jones Distinguished Professor. His research is in heat transfer, thermodynamics, and the physics that governs organization and evolution in nature.

Professor Bejan is the author of 700 peer-refereed journal articles and 30 books. His books are used worldwide in multiple editions and languages. He received the top international awards for thermal sciences. He is an honorary member of the American Society of Mechanical Engineers and a member of the Academy of Europe and the national academies of Mexico, Turkey, Moldova, and Romania.

He was awarded 18 honorary doctorates from universities in 11 countries, for example, the Swiss Federal Institute of Technology (ETH Zurich), the University of Rome I “La Sapienza,” the National Institute of Applied Sciences (INSA) Lyon, and the University of Pretoria.

Acknowledgments

In this book, I benefited from the continuous creative support offered by Deborah Fraze and my most recent doctoral students and visiting professors: Umit Gunes, Abdulrahman Almerbati, Sinan Gucluer, Hitoshi Matsushima, and Hamad Almahmoud.

My spirits were kept high by Mary and our children Cristina, Teresa, and William. I am grateful for the continuous support from Wiley and my colleagues Sylvie Lorente, Pezhman Mardanpour, Tanmay Basak, Hooman Farzaneh, William Worek, Jose Lage, Jaime Cervantes, Abel Hernandez, Alexandru Morega, Heitor Reis, Bahri Sahin, Yousef Haseli, Adrian Sabau, and Shigeo Kimura.

List of Symbols

a

i

impurity-scattering resistivity coefficient (m·K

2

/W),

Eq. (1.26)

a

n

dimensionless characteristic values,

Eq. (3.43)

and

Table 3.1

a

p

phonon-scattering resistivity coefficient (m/W·K),

Eq. (1.26)

A

area (m

2

)

A

c

cross-sectional area (m

2

)

A

c

minimum free-flow area (m

2

),

Eq. (9.65)

A

exp

fin surface exposed to the fluid (m

2

)

A

f

finned area (m

2

), contributed by the exposed surfaces of the fins,

Chapter 9

A

fr

frontal area (m

2

),

Eq. (9.65)

A

n

normal or projected area (m

2

)

A

u

unfinned area (portions) of the wall (m

2

),

Chapter 9

A

0

bare surface, projected surface (m

2

),

Figure 2.10

A

0,

f

finned portion of bare surface (m

2

),

Figure 2.10

A

0,

u

unfinned portion of bare surface (m

2

).

Figure 2.10

b

thermal stratification parameter,

Eq. (7.71)

b

,

b

T

transversal length scales (m),

Eqs. (5.148)

–

(5.149)

b

n

dimensionless characteristic values,

Table 4.1

B

cross-section shape number,

Eq. (6.30)

B

driving parameter for film condensation,

Eq. (8.26)

Be

Bejan number,

Be

 = Δ

P

·

L

2

/

μα

,

Appendix A

Bi

Biot number,

Eq. (2.58)

Bo

y

Boussinesq number,

Bo

y

 = 

Ra

y

Pr

c

specific heat of incompressible substance (J/kg·K)

c

speed of light in vacuum,

Appendix A

c

v

specific heat at constant volume (J/kg·K)

c

P

specific heat at constant pressure (J/kg·K)

C

capacity rate (W/K), or

ṁc

P

C

c

correction factor,

Figure 10.27

C

D

drag coefficient,

Eq. (5.138)

C

f,x

local skin friction coefficient,

Eq. (5.39)

average skin friction coefficient,

Eq. (5.55)

C

sf

empirical constant for liquid–surface combination,

Table 8.1

C

w

correction factor,

Figure 10.29

d

diameter (m)

D

diameter (m)

D

h

hydraulic diameter (m),

Eq. (6.28)

D

i

inner diameter (m)

D

o

outer diameter (m)

e

specific internal energy (J/kg),

Eq. (5.12)

E

energy (J)

E

modulus of elasticity (N/m

2

)

E

total hemispherical emissive power (W/m

2

)

E

b

total hemispherical blackbody emissive power (W/m

2

)

E

b,λ

monochromatic hemispherical blackbody emissive power (W/m·m

2

)

Ec

Eckert number,

Appendix A

E

λ

monochromatic hemispherical emissive power (W/m m

2

)

f

factor,

Figures 9.28

and

9.29

f

friction factor,

Eq. (6.24)

f

v

vortex shedding frequency (s

–1

),

Eq. (5.137)

F

correction factor,

Figures 9.10

–

9.14

F

force (N)

F

similarity streamfunction profile,

Eq. (7.45)

F

D

drag force (N)

F

n

normal force (N)

Fo

Fourier number (dimensionless time),

Eqs. (4.63)

F

r

,

F

θ

,

F

z

body forces per unit volume (N/m

3

),

Table 5.2

F

r

,

F

φ

,

F

θ

body forces per unit volume (N/m

3

),

Table 5.3

F

t

tangential force (N)

F

12

geometric view factor,

Eq. (10.33)

g

gravitational acceleration (m/s

2

)

G

mass velocity (kg/m

2

·s),

Eq. (9.58)

G

similarity vertical velocity profile,

Eq. (7.43)

G

total irradiation (W/m

2

)

Gr

y

Grashof number based on temperature difference and height

y

,

Gr

y

 = 

gβy

3

Δ

T

/

ν

2

 = 

Ra

y

/

Pr

Gz

Graetz number,

Eq. (6.53)

G

ℒ

constant,

Table 7.1

G

λ

monochromatic irradiation (W/m

2

·m)

h

heat transfer coefficient for external flow (W/m

2

·K),

Eq. (1.50)

h

heat transfer coefficient for internal flow (W/m

2

·K),

Eq. (1.51)

h

Planck's constant (J·s),

Appendix A

h

specific enthalpy (J/kg)

h

e

effective heat transfer coefficient (W/m

2

·K),

Eq. (9.7)

h

f

specific enthalpy of saturated liquid (J/kg)

h

fg

latent heat of condensation (J/kg),

h

g

 − 

h

f

augmented latent heat of condensation (J/kg),

Eqs. (8.10)

and

(8.17)

augmented latent heat of condensation (J/kg),

Eq. (8.41)

h

g

specific enthalpy of saturated vapor (J/kg)

h

s

specific enthalpy of saturated solid (J/kg)

h

sf

latent heat of melting, or of solidification (J/kg),

h

f

 − 

h

s

h

x

local heat transfer coefficient (W/m

2

·K) at position

x

average heat transfer coefficient (W/m

2

·K) averaged over length

x

heat transfer coefficient (W/m

2

·K) averaged over cylinder or sphere of diameter

D

H

enthalpy (J)

H

enthalpy flowrate per unit length (W/m),

Eq. (8.5)

H

height (m)

i

specific enthalpy (J/kg),

Eq. (5.16)

I

b

total intensity of blackbody radiation (W/m

2

·sr)

I

b,λ

intensity of monochromatic blackbody radiation (W/m

3

·sr)

I

λ

intensity of monochromatic radiation (W/m

3

·sr)

j

H

Colburn

j

H

factor,

Eq. (9.75)

J

electric current density (A/m

2

)

J

radiosity (W/m

2

)

Ja

Jakob number,

Eq. (8.19)

J

0

zeroth-order Bessel function of the first kind,

Figure 3.6

and

Appendix E

J

1

first-order Bessel function of the first kind,

Eq. (3.63)

k

Boltzmann's constant,

Appendix A

k

thermal conductivity (W/m·K)

k

avg

average thermal conductivity (W/m·K),

Eq. (2.50)

k

e

thermal conductivity due to conduction electrons (W/m·K)

k

l

thermal conductivity due to lattice vibrations (W/m·K)

thermal resistivity due to impurity scattering (m·K/W)

thermal resistivity due to phonon scattering (m·K/W)

k

s

sand roughness scale (mm),

Figure 6.10

K

constant coefficient

K

permeability (cm

2

),

Appendix B

K

c

contraction loss coefficient.

Figures 9.23

and

9.24

K

e

enlargement loss coefficient,

Figures 9.23

and

9.24

l

equivalent length (m),

Eq. (7.84)

l

length (m)

l

mixing length (m),

Eq. (5.109)

L

characteristic length (m),

Eq. (7.76)

L

length (m)

ℒ

equivalent length (m),

Eqs. (5.140)

and

(7.85)

L

c

corrected fin length (m),

Eq. (2.95)

L

e

equivalent length (m),

Table 10.5

L

0

Lorentz constant, 2.45 × 10

–8

(V/K)

2

m

fin parameter (m

–1

),

Eq. (2.75)

m

integer

m

mass (kg)

ṁ

mass flowrate (kg/s)

ṁ

′

mass flowrate per unit length (kg/s·m)

ṁ

″

mass flux (kg/m

2

·s)

M

dimensionless factor,

Eq. (6.84)

n

direction normal to the boundary (m),

Table 3.2

n

integer

NTU

number of heat transfer units,

Eq. (9.27)

Nu

x

Nusselt number based on the local heat transfer coefficient

h

x

x

/

k

overall Nusselt number based on the surface-averaged heat transfer coefficient

, where

D

is the diameter

constant,

Table 7.1

overall Nusselt number based on the

x

-averaged heat transfer coefficient

p

perimeter (m)

p

perimeter of contact with fluid (wetted perimeter) (m)

P

dimensionless parameter,

Figures 9.10

–

9.14

P

pressure (

Pa

or N/m

2

)

P

mechanical power (W)

Pe

D

Péclet number based on diameter,

U

∞

D

/

α

,

UD

/

α

Pe

x

Péclet number based on longitudinal length,

U

∞

x

/

α

Pr

Prandtl number,

Pr

 = 

v

/

α

Pr

t

turbulent Prandtl number,

Pr

t

 = 

ε

M

/

ε

H

q

heat transfer rate (W)

q

b

total heat transfer rate through the fin (W)

q

tip

heat transfer through the tip of the fin (W)

q

′

heat transfer rate per unit length (W/m)

q

″

heat flux (W/m

2

)

volumetric rate of internal heat generation (W/m

3

)

local wall heat flux (W/m

2

)

x

-averaged wall heat flux (W/m

2

),

Eq. (5.80)

q

1–2

one-way heat current (W) from 1 to 2

q

1–2

net heat current (W) from 1 to 2

Q

heat transfer (J)

Q

′

heat transfer interaction per unit length (J/m)

Q

″

heat transfer interaction per unit area (J/m

2

)

r

radial coordinate (m),

Figures 1.7

and

1.8

r

i

inner radius (m)

r

o

outer radius (m)

r

o,c

critical outer radius (m),

Eqs. (2.42)

and

(2.43)

r

s

thermal resistance of the scale (m

2

·K/W),

Eq. (9.5)

R

dimensionless parameter,

Figures 9.10

–

9.14

R

radius (m)

R

function of

r

only,

Chapter 3

R

ideal gas constant (kJ/kg·K),

Appendix D

universal ideal gas constant,

Appendix A

Ra

y

Rayleigh number based on temperature difference and height

y

,

Ra

y

 = 

gβy

3

Δ

T

/

αv

Rayleigh number based on heat flux and height

y

,

Re

Reynolds number

V

max

D

h

/

v

,

Eq. (9.70)

Re

D

Reynolds number based on diameter,

U

∞

D

/

v

,

UD

/

v

Re

l

local Reynolds number,

Appendix F

Re

x

Reynolds number based on longitudinal length,

U

∞

x

/

v

Re

y

condensate film Reynolds number, 4Γ(

y

)/

μ

l

,

Eq. (8.22)

R

i

internal radiation resistance (m

–2

),

Eq. (10.78)

R

r

radition thermal resistance (m

–2

),

Eq. (10.45)

R

t

thermal resistance (K/W),

Eq. (2.8)

s

empirical constant,

Table 8.1

s

n

dimensionless characteristic values,

Table 4.2

S

conduction shape factor,

Eq. (3.33)

and the header of

Table 3.3

S

entropy (J/K)

St

x

-independent Stanton number

h

/

ρc

P

U

Ste

Stefan number,

Eq. (4.119)

St

x

local Stanton number

h

x

/

ρc

P

U

∞

t

thickness (m)

t

time (s)

t

c

transition time scale (s),

Eq. (4.9)

T

temperature (K or °C),

Eqs. (1.5)

and

Figure 1.1

T

b

base temperature in fin analysis (K),

Chapter 2

T

b

bulk, or mean temperature (K or °C)

T

c

center temperature (K),

Chapter 4

T

i

initial temperature (K)

T

m

mean, or bulk temperature (K),

Eq. (6.33)

T

m

melting point temperature (K)

T

sat

saturation temperature (K)

T

w

wall temperature (K or °C)

T

0

surface temperature (K),

Chapter 4

T

0

reference temperature (K or °C)

T

∞

free-stream or reservoir temperature (K or °C)

u

specific internal energy (J/kg)

u

velocity component in the

x

direction (m/s)

U

average longitudinal velocity (m/s)

U

internal energy (J)

U

mean velocity (m/s),

Eq. (6.1)

U

overall heat transfer coefficient (W/m

2

·K)

U

∞

free stream velocity (m/s)

v

specific volume (m

3

/kg)

v

velocity in the

y

direction (m/s)

v

n

normal velocity (m/s)

V

mean longitudinal velocity (m/s)

V

volume (m

3

)

풱

volume (m

3

),

Chapter 9

w

mechanical transfer rate, or power (W)

W

width (m)

W

work transfer (J)

work transfer rate, or power (W)

x

Cartesian coordinate (m),

Figure 1.6

x

tr

transition length (m)

X

flow entrance length (m),

Eqs. (6.4′)

and

(6.65)

X

function of

x

only,

Chapter 3

X

l

longitudinal pitch (m)

X

t

transversal pitch (m)

X

T

thermal entrance length,

Eqs. (6.32)

and

(6.65)

dimensionless longitudinal pitch

X

l

/

D

dimensionless transversal pitch

X

t

/

D

X, Y, Z

body forces per unit volume (N/m

3

),

Table 5.1

y

Cartesian coordinate (m),

Figure 1.6

Y

function of

y

only,

Chapter 3

Y

0

zeroth-order Bessel function of the second kind,

Figure 3.6

z

axial position in cylindrical coordinates (m),

Figure 1.7

z

Cartesian coordinate (m),

Figure 1.6

Z

function of

z

only,

Chapter 3

Greek Letters

α

heat transfer area density (m

2

/m

3

),

Eq. (9.64)

α

thermal diffusivity (m

2

/s),

α

 = 

k

/

ρc

P

α

total absorptivity

α

total hemispherical absorptivity

α

0

temperature coefficient of electrical resistivity (°C

–1

),

Appendix B

α

λ

monocromatic hemispherical absorptivity

directional monochromatic absorptivity

β

coefficient of volumetric thermal expansion (K

–1

),

Eq. (5.18)

Γ

condensate mass flowrate per unit length (kg/s·m),

Eq. (8.4)

δ

film thickness (m),

Chapter 8

δ

skin thickness, boundary layer thickness (m)

δ

velocity boundary layer thickness (m),

Eq. (5.25)

δ

*

displacement thickness (m),

Eq. (5.58)

δ

s

thickness of shear layer (m),

Eq. (7.38)

δ

T

thermal boundary layer thickness (m),

Eq. (5.60)

δ

99

velocity boundary layer thickness (m),

Eq. (5.57)

Δ

P

pressure drop (N/m

2

),

Eq. (6.27)

Δ

T

temperature difference (K)

Δ

T

lm

log-mean temperature difference (K),

Eqs. (6.105)

and

(9.22)

Δ

ɛ

correction term,

Figure 10.30

ε

heat exchanger effectiveness,

Eq. (9.29)

ε

overall surface efficiency,

Eq. (9.4)

ε

total hemispherical emissivity

ε

f

fin effectiveness,

Eq. (2.100)

ε

H

thermal eddy diffusivity (m

2

/s),

Eq. (5.100)

ε

M

momentum eddy diffusivity (m

2

/s),

Eq. (5.99)

ε

0

overall projected-surface effectiveness,

Eq. (2.62)

ε

λ

monochromatic hemispherical emissivity

directional monochromatic emissivity

η

fin efficiency,

Eq. (2.98)

η

similarity variable

η

c

compressor isentropic efficiency

η

p

pump isentropic efficiency

θ

angular coordinate (rad),

Figures 1.7

and

1.8

θ

excess temperature (K),

Eq. (2.71)

θ

momentum thickness (m),

Eq. (5.59)

θ

similarity temperature profile,

Eq. (7.46)

θ

thermal potential function (W/m),

Eq. (2.47)

θ

b

excess temperature of fin base (K)

κ

von Karman's constant,

Eq. (5.112)

κ

λ

monochromatic extinction coefficient (m

–1

)

λ

characteristic value,

Chapter 3

λ

dimensionless parameter in the Stefan solution,

Eq. (4.118)

λ

wavelength (m)

μ

characteristic value,

Chapter 3

μ

viscosity (kg/s·m)

ν

frequency (s

–1

)

ν

kinematic viscosity (m

2

/s),

ν

 = 

μ

/

ρ

ρ

density (kg/m

3

)

ρ

total reflectivity

ρ

e

electrical resistivity (W·m/A

2

)

σ

contraction ratio,

Eq. (9.53)

σ

Stefan–Boltzmann constant,

Appendix A

σ

surface tension (N/m),

Table 8.2

σ

xx

normal stress (N/m

2

),

Eq. (5.8)

τ

angle of enclosure inclination (rad),

Figure 7.18

τ

total transmissivity

τ

w,x

local wall shear stress (N/m

2

)

x

-averaged wall shear stress (N/m

2

),

Eq. (5.54)

τ

xy

tangential stress (N/m

2

),

Eq. (5.8)

τ

λ

monochromatic transmissivity

ϕ

angle of wall inclination (rad),

Figure 7.7

ϕ

angular coordinate (rad),

Figure 1.8

ϕ

dimensionless temperature profile,

Eq. (6.45)

φ

porosity,

Appendix B

Φ

viscous dissipation function (s

–2

),

Eq. (5.15)

Φ

i

mass fraction

ρ

i

/

ρ

χ

correction factor,

Figures 9.28

and

9.29

ψ

streamfunction (m

2

/s),

Eq. (5.45)

ω

solid angle (sr)

Subscripts

( )

a

absorbed,

Chapter 10

( )

acc

acceleration

( )

app

apparent

( )

avg

average

( )

b

base of the fin,

Chapter 2

( )

b

black,

Chapter 10

( )

b

bulk, mean

( )

c

carbon dioxide,

Chapter 10

( )

c

centerline, center, midplane

( )

c

cold

( )

c

compressor

( )

eddy

eddy transport

( )

f

fluid

( )

g

gas,

Chapter 10

( )

h

hot

( )

i

initial

( )

i

inner

( )

in

initial

( )

in

inlet

( )

l

liquid

( )

max

maximum

( )

min

minimum

( )

mol

molecular diffusion

( )

o

outer

( )

opt

optimal

( )

out

outlet

( )

p

pump

( )

r

reflected,

Chapter 10

( )

rad

radiation

( )

ref

reference

( )

s

shield,

Chapter 10

( )

s

straight,

Chapter 9

( )

s

surface,

Chapter 10

( )

sat

saturation

( )

v

vapor

( )

v

water vapor

( )

w

wall

( )

w

water vapor,

Chapter 10

( )

0

nozzle

( )

0

reference

( )

0

wall

( )

∞

free stream

Superscripts

(

−

)

time averaged, or volume averaged

( )′

fluctuation,

Eq. (5.88)

( )

+

wall coordinates,

Eq. (5.117)