Heat Transfer

Lessons with Examples Solved by Matlab

First Edition
By Tien-Mo Shih

Included in this preview:

Copyright Page
Table of Contents
Excerpt of Chapter 1

For additional information on adopting this book

for your class, please contact us at 800.200.3908
x501 or via e-mail at info@cognella.com
 Heat Transfer
Lessons with Examples Solved by Matlab

By Tien-Mo Shih
The University of Maryland, College Park
Bassim Hamadeh, Publisher
Christopher Foster, Vice President
Michael Simpson, Vice President of Acquisitions
Jessica Knott, Managing Editor
Stephen Milano, Creative Director
Kevin Fahey, Cognella Marketing Program Manager
Melissa Barcomb, Acquisitions Editor
Sarah Wheeler, Project Editor
Erin Escobar, Licensing Associate

Copyright 2012 by University Readers, Inc. All rights reserved. No part of this pub-
lication may be reprinted, reproduced, transmitted, or utilized in any form or by any
electronic, mechanical, or other means, now known or hereafter invented, including
photocopying, microfilming, and recording, or in any information retrieval system
without the written permission of University Readers, Inc.

First published in the United States of America in 2012 by University Readers, Inc.

Trademark Notice: Product or corporate names may be trademarks or registered trade-

marks, and are used only for identification and explanation without intent to infringe.

15 14 13 12 11 12345

Printed in the United States of America

ISBN: 978-1-60927-544-0
 Dedication
T o my w ife, Ti ngti ng Z hu, whos e l ov e a n d ve g e ta ri a n c o o k i n g a re th e p ri m a ry
dri v i ng force of thi s w ri ti n g e n d e a vo r.
 Contents

Preface xix

Lesson 1: Introduction (What and Why) 1

1. Heat Transfer Is an Old Subject 1
1-1 Partially True 2
1-2 Modifications of the Perception 2

2. What Is the Subject of Heat Transfer? 3

2-1 Definition 3
2-2 Second Income 3

3. Candle Burning for Your Birthday Party 5

3-1 Unknowns and Governing Equations 5
3-2 Solution Procedure 6

4. Why Is the Subject of Heat Transfer Important? 7

4-1. Examples of Heat Transfer Problems in Daily Life 7
4-2 Examples of Heat Transfer Problems in Industrial Applications 8

5. Three Modes of Heat Transfer 8

6. Prerequisites 9
7. Structure of the Textbook 9
8. Summary 10
9. References 10
10. Exercise Problems 11
11. Appendix 11
A-1 Sum Up 1, 2, 3,, 100 11
A-2 Plot f(x) 12
A-3 Solve Three Linear Equations 12
Lesson 2: Introduction (Three Laws) 13
1. Fouriers Law 13
2. Law of Convective Heat Transfer 15
3. Wind-Chill Factor (WCF) 16
4. Stefan-Boltzmann Law of Radiative Emission 17
5. Sheet Energy Balance 18
6. Formation of Ice Layers on Car Windshield and Windows 18
6-1 Clear Sky Overnight 18
6-2 Cloudy Sky Overnight 20
7. Rule of Assume, Draw, and Write (ADW) 20
8. Summary 22
9. References 22
10. Exercise Problems 22
11. Appendix: Taylors Series Expansion 23

Lesson 3: One-Dimensional Steady State Heat Conduction 25

1. Governing Equation for T(x) or T(i) 25
2. A Single-Slab System 26
3. A Two-Slab System 28
4. Three or More Slabs 29
5. Severe Restrictions Imposed by Using Electrical Circuit Analogy 30
6. Other Types of Boundary Conditions 30
7. Thermal Properties of Common Materials Table 1 31
8. Summary 31
9. References 31
10. Exercise Problems 31
11. Appendix: A Matlab Code for One-D Steady-State Conduction 32
Lesson 4: One-Dimensional Slabs with Heat Generation 35
1. Introduction 35
2. Governing Equations 36
3. Heat Conduction Related to Our Bodies 37
3-1 Estimate the Heat Generation of Our Bodies 37
3-2 A Coarse Grid System 39
3-3 Optimization 40

4. Discussions 41
4-1 Radiation Boundary Condition 41
4-2 Sketch the Trends and Check our Speculations
with Running the Matlab Code 42
4-3 Check the Global Energy Balance 43

5. Summary 43
6. References 43
7. Exercise Problems 43
8. Appendix: A Matlab Code for Optimization 44

Lesson 5: One-Dimensional Steady-State Fins 47

1. Introduction 48
1-1 Main Purpose of Fins 48
1-2 Difference between 1-D Slabs and 1-D Fins 49
1-3 A Quick Estimate 49

2. Formal Analyses 51
2-1 Governing Equation 51
2-2 A Standard Matlab Code Solving T(x) and qb for a Fin 51
2-3 Equivalent Governing Differential Equation 52

3. Fins Losing Radiation to Clear Sky Overnight 52

4. A Seemingly Puzzling Phenomenon 54
5. Fin Efciency 55
6. Optimization 55
6-1 Constraint of Fixed Total Volume 56
6-2 Optimization for Rod Bundles 56
 7. Summary 58
8. References 58
9. Exercise Problems 58
10. Appendix 59
A-1 Explanation of the Seemingly Puzzling Phenomenon 59
A-2 Cubic Temperature Profi le 60
A-3 Constraint of Fixed Fin Mass (or Volume) 62
A-4 Fin Bundles and Optimization 63

Lesson 6: Two-Dimensional Steady-State Conduction 65

1. Governing Equations 66
1-1 Derivation of the General Governing Equation 66
1-2 Three-Interior-Node System 67
1-3 A Special Case 70

2. A Standard Matlab Code Solving 2-D Steady-State Problems 70

3. Maximum Heat Loss from a Cylinder Surrounded by Insulation Materials 70
3-1. Governing Equation 71
3-2 An Optimization Problem 72

4. Summary 72
5. References 73
6. Exercise Problems 73
7. Appendix 74
A-1 Gauss-Seidel Method 74
A-2 A Standard 2-D Steady State Code 75
A-3 Optimization Problem in the Cylindrical Coordinates 77

Lesson 7: Lumped-Capacitance Models (Zero-Dimension

Transient Conduction) 79
1. Introduction 79
1-1 Adjectives 79
1-2 Justifications 80
 1-3 Examples 80
1-4 Objectives and Control Volumes 81
1-5 The Most Important Term and the First Law of Thermodynamics 81

2. Detailed Analyses of a Can-of-Coke Problem 82

2-1 Modeling 82
2-2 Assumptions in the Modeling 83
2-3 A Matlab Code Computing T(t) for the Can of Coke Problem 83
2-4 Discussions 84

3. When Is It Appropriate to Use the Lumped-Capacitance Model? 85

3-1 A Three-Node Example 85
3-2 An Analogy 86

4. A Simple Way to Relax the Bi < 0.002 Constraint 87

5. Why Stirring the Food When We Fry It? 88
6. Summary 90
7. References 90
8. Exercise Problems 90
9. Appendix 91
A-1 Discussions of Adjectives 91
A-2 Comparisons of Three Cases 92

Lesson 8: One-Dimensional Transient Heat Conduction 95

1. Kitchen Is a Good Place to Learn Heat Transfer 95
1-1 Governing Equation of One-D Transient Heat Conduction 95
1-2 The Generic Core of One-D Transient Code 97
1-3 Various Boundary Conditions 97
1-4 A One-D Transient Matlab Code 97
1-5 Global Energy Balance 99

2. Other One-D Transient Heat Conduction Applications 100

2-1. Semi-Infinite Solids 100
2-2. Revisit Fin Problems 100
2-3. Multi-layer Slabs with Heat Generation 102
 3. Differential Governing Equation for One-D Transient Heat Conduction 103
4. Summary 103
5. References 103
6. Exercise Problems 103
7. Appendix 104
A-1 Semi-infinite Solids 104
A-2 Transient 1-D Fins 105
A-3 Example 4-1 Revisited 106

Lesson 9: Two-D Transient Heat Conduction 109

1. Governing Equation for T(i, j) 109
1-1 General Case 109
1-2 Special Cases 110

2. A Standard Matlab Code for Readers to Modify 113

3. Speculation on Steel Melting in Concrete Columns during 9/11 113
4. Possible Numerical Answers 114
5. Use a Two-D code to Solve One-D Transient Heat Conduction Problems 115
6. Exact Solutions for Validation of Codes 116
7. Advanced Heat Conduction Problems 117
7-1 Moving Interface (or called Stefan Problem) 117
7-2. Irregular Geometries 118
7-3. Non-Fourier Law (Hyperbolic-Type Heat Conduction Equation) 118

8. Summary 120
9. References 120
10. Exercise Problems 120
11. Appendix 121
A-1 A Standard Transient 2-D Code for Readers to Modify 121
A-2. Investigation of Transient 1-D Heat Conduction for a Steel Column 122
A-3. Investigation of Transient 2-D Heat Conduction for the Concrete Column 123
Lesson 10: Forced-Convection External Flows (I) 125
1. Soup-Blowing Problem 125
2. Boundary- Layer Flows 128
3. A Cubic Velocity Prole 130
31 Determine the coefficients 130
3-2 Application of Eq. (3) 131

4. Summary 133
5. References 133
6. Exercise Problems 133
7. Appendix: Momentum Balance over an Integral Segment 136

Lesson 11: Forced-Convection External Flows (II) 139

1. Nondimensionalization (abbreviated as Ndm) 139
2. Important Dimensionless Parameters in Heat Transfer 144
3. Derivation of Governing Equations 144
4. Categorization 145
5. Summary 146
6. References 146
7. Exercise Problems 146
8. Appendix: Steady-State Governing Equations 148
A-1 Derivation of Governing Equation for u 148
A-2 Governing Equation for T 152
Lesson 12: Forced-Convection External Flows (III) 153
1. Preliminary 153
2. A Classical Approach Reported in the Literature 158
3. Steps to Find Heat Flux at the Wall (from the Similarity Solution) 156
4. The Nu Correlation and Some Discussions 157
5. Derivation of Nu = G (Re, Pr) by Ndm 159
6. Finding Eq. (6b) by Using a Quick and Approximate Method 160
6-1. A Quick and Approximate Method 160
6-2. A Matlab code generating the Nu correlation 161
7. An Example Regarding Convection and Radiation Combined 163
8. Possible Shortcomings of Nu Correlations 164
9. Brief Examination of Two More External Flows 164
10. Summary 165
11. References 165
12. Exercise Problems 166
13. Appendix [to nd f '() and the value of f '' (0)] 168
A-1 A Matlab Code for Solving the Blasius Similarity Equation 168
A-2 Table of , f, f',f '' Distributions 169
A-3 Brief Explanations of the Table 171

Lessons 13: Internal Flows (I)Hydrodynamic Aspect 173

1. Main Differences Between External Flows and Internal Flows 174
2. Two Regimes (or Regions) 174
3. A Coarse Grid to Find u, v, and p in the Developing Regime 176
4. An Analytical Procedure of Finding u(y) in the Fully Developed Regime 178
5. Application of the Results 182
 6. Which Value Should We Use? 183
7. Ndm and Parameter Dependence 184
8. Summary 184
9. References 184
10. Exercise Problems 185
11. Appendix: A Matlab Code for Finding u, v, and p in the Developing Regime 187

Lessons 14: Internal Flows (II)Thermal Aspect 189

1. Denition of Tm 189
2. Denition of Thermally Fully Developed Flows 190
3. Justication of T/x = constant 191
4. A Benecial Logical Exercise of Genetics 193
5. Summary 194
6. References 194
7. Exercise Problems 194

Lessons 15: Internal Flows (III)Thermal Aspect 197

1. Derivation of Nu Value for Uniform qs 197
2. Important Implications of Eq. (5) 199
3. Derivation of Nu value for uniform Ts 200
3-1 Justification 201
3-2 Solution Procedure 202
 4. Let the Faucet Drip Slowly 204
5. Summary 207
6. References 207
7. Exercise Problems 207
8. Appendix: A Matlab Code Computing Nu for the Case of Uniform Ts 208

Lesson 16: Free Convection 211

1. Denition of Free Convection 211

2. Denition of Buoyancy Force 212
2-1 Buoyancy Force on an Object 212
2-2 Buoyancy Force on a Control Volume in the Flow 213
2-3 Density Variations 214

3. The Main Difference Between Free Convection and Forced Convection 214
4. How Does Gr Number Arise? 215
5. T and in Free Convection 216
6. A Four-Cell Buoyancy-Driven Flow in a Square Enclosure 217
6-1 Description 217
6-2 Derivation of Governing Equations 218
6-3 Discussions of the Result 219

7. Does Lighting a Fire in Fireplace Gain Net Energy for the House? 219
7-1 Governing Equations 220
7-2 Nomenclature in the Code 220
7-3 Matlab Code 220

8. Solar Radiation-Ice Turbine 221

8-1 Description of the Machine 221
8-2 Some Analyses 222
 9. Free Convection over a Vertical Plate 223
10. Summary 224
11. References 224
12. Exercise Problems 224
13. Appendix: A Matlab Code Computing Buoyancy-Driven Flows in Four-Cell Enclosures 224

Lesson 17: Turbulent Heat Convection 227

1. Introduction 228
1-1 Speed of Typical Flows 228
1-2 Frequencies of Turbulence and Molecular Collision 228
1-3 Superposition 228

2. A Fundamental Analysis 229

2-1 Governing Equations 229
2-2 Zero-Equation Turbulence Model 230
2-3 Discretized Governing Equation 231

3. Matlab Codes 232

3-1 Laminar Flows in Two-Parallel-Plate Channels 232
3-2 Turbulent Flows 233
3-3 Laminar Flows in Circular Tubes 233

4. Dimples on Golf Balls 234

5. Summary 235
6. References 235
7. Exercise Problems 235
8. Appendix 235
A-1 Laminar Flows in Circular Tubes 235
A-2 Fully Developed Turbulent Flows in Planar Channels 236
Lesson 18: Heat Exchangers and Other Heat Transfer Applications 239
1. Types of Heat Exchangers 239
2. A Fundamental Analysis 240
3. A Traditional Method to Find Heat Exchange 241
4. A Matlab Code 242
5. Comments on the Code 244
6. Other Applications in Heat Transfer 244
6-1 Combustion and Low-Temperature Chemical Reactions 244
6-2 Jets, Plumes, and Wakes 245
6-3 Optimization 245
6-4 Porous Media 245
6-5 Radiation with Participating Gases 245
6-6 Two-Phase Flows 245

7. Summary 246
8. References 246
9. Exercise Problems 247

Lesson 19: Radiation (I) 249

1. Fundamental Concepts 249
1-1 Main Difference between Radiation and Convection 250
1-2 Adjectives Used for Radiative Properties 250

2. Blackbody Radiation 251

2-1. Definition of a Blackbody Surface 251
2-2. Planck Spectral Distribution 252
2-3. Stefan-Boltzmann Law 253
 3. A Coffee Drinking Tip 254
4. Fractions of Blackbody Emission 256
5. Summary 257
6. References 257
7. Exercise Problems 257
8. Appendix 258
A-1 Finding the Value of Stefan-Boltzmann Constant 258
A-2 Table 19-1 Fractions of Blackbody Emission 259

Lesson 20: Radiation (II) 261

1. Emissivity 261
2. Three Other Radiative Properties 263
3. Solar Constant and Effective Temperature of the Sun 264
4. Gray Surfaces 266
5. Kirchhoffs Law 268
6. Energy Balance over a Typical Plate 269
7. Greenhouse Effect (or Global Warming) 269
8. Steady-State Heat Flux Supplied Externally by Us 271
9. Find Steady-State Ts Analytically 272
10. Find Steady-State Ts Numerically 272
11. Find Unsteady Ts Not Involved with the Spectral Emissivity 273
12. Find Unsteady Ts Involved with the Spectral Emissivity 274
13. Find Unsteady Ts with Parameters Being Functions of Wavelength and Time 275
14. Summary 277
15. References 277
16. Exercise Problems 277
Lesson 21: Radiation (III) 281
1. View Factors (or Shape Factors, Conguration Factors) 281
1-1 Definition of the View Factor, F12 282
1-2 Reciprocity Rule 282
1-3 Energy Conservation Rule 282
1-4 View Factor for a Triangle 283

2. Black Triangular Enclosures 283

3. Gray Triangular Enclosures 285
3-1 Definition of Radiosity, J 286

4. Two Parallel Gray Plates with A1 = A2 288

5. Radiation Shield 289
6. Summary 290
7. References 290
8. Exercise Problems 290
 Preface

Heat Transfer: Lessons with Examples Solved by Matlab instructs students in heat
transfer, and cultivates independent and logical thinking ability. The book focuses on
fundamental concepts in heat transfer and can be used in courses in Heat Transfer, Heat
and Mass Transfer, and Transport Processes. It uses numerical examples and equation
solving to clarify complex, abstract concepts such as Kirchhoff s Law in Radiation.

Several features characterize this textbook:

It includes real-world examples encountered in daily life;

Examples are mostly solved in simple Matlab codes, readily for students to run
numerical experiments by cutting and pasting Matlab codes into their PCs;
In parallel to Matlab codes, some examples are solved at only a few nodes, al-
lowing students to understand the physics qualitatively without running Matlab
codes;
It places emphasis on why for engineers, not just how for technicians.

Adopting instructors will receive supplemental exercise problems, as well as access

to a companion website where instructors and students can participate in discussion
forums amongst themselves and with the author.

Heat Transfer is an ideal text for students of mechanical, chemical, and aerospace
engineering. It can also be used in programs for civil and electrical engineering, and
physics. Rather than simply training students to be technicians, Heat Transfer uses
clear examples, structured exercises and application activities that train students to be
engineers. The book encourages independent and logical thinking, and gives students
the skills needed to master complex, technical subject matter.

Tien-Mo Shih received his Ph.D. from the University of California, Berkeley, and did
his post-doctoral work at Harvard University. From 1978 until his retirement in 2011 he
was an Associate Professor of Mechanical Engineering at the University of Maryland,
College Park, where he taught courses in thermo-sciences and numerical methods. He
remains active in research in these same areas. His book, Numerical Heat Transfer, was

Preface | xix
translated into Russian and Chinese, and subsequently published by both the Russian
Academy of Sciences and the Chinese Academy of Sciences. He has published numer-
ous research papers, and has been invited regularly to write survey papers for Numerical
Heat Transfer Journal since 1980s.

xx | Heat Transfer: Lessons with Examples Solved by Matlab

 Lesson 1
Introduction (What and Why)

I n the very first lesson of this textbook, we will attempt to describe what the subject
of heat transfer is and why it is important to us. In addition, three modes of heat
transfernamely, conduction, convection, and radiationwill be briefly explained.
The structure of this textbook is outlined.

Nomenclature
cv = specific heat, J/kg-K
Q = heat transfer, J
q = heat flow rate or heat transfer rate, W
q = heat flux, W/m2
T = temperature, in C or K (only when radiation is involved, we use K).
U = internal energy of a control volume, J
V = volume of the piston-cylinder system, m3

1. Heat Transfer Is an Old Subject

On one occasion the author overheard a casual conversation between two students:
How did you like your heat transfer course last semester?
Well, it is OK, I guess; it is kind of old.

Lesson 1 | 1
 1-1 Partially True
The students remark may be partially true. The subject of heat transfer is at least as
old as when Nicolas Carnot designed his Carnot engine in 1824, where Q, the heat
transfer in Joules, was needed for interactions between the piston-cylinder system and
the reservoirs during expansion and compression isothermal processes.
Because it is an old subject, and because the course has been traditional, many
textbooks include solution procedures that were used in old times before computers
were invented or not yet available to the general public. A notable example is Separation
of Variables, a way to solve partial differential equations. Without computers, highly
ingenious mathematical theories, assumptions, and formulations must be introduced.
Eigenvalues and Fouriers series are also included. Students are overwhelmed with such
great amount of math. It is no wonder they feel bored.
Today, with the advent of computers and associated numerical methods, not only
can these equations be solved using simple numerical methods, but also the problems
no longer need to be highly idealized to fit the restrictions imposed by the Separation
of Variables. In the authors opinion, this subject of Separation of Variables may not
belong to textbooks that are supposed to deliver heat transfer knowledge within a short
semester. At most, it should be moved to appendices, or taught in a separate math-
related course.

1-2 Modifications
On the other hand, an old subject can be different from old people, whom young college
students love, but may feel bored to play with. For example, history is an old subject.
It can become either dull or interesting depending on the materials presented. Some
possible modifications can be made for the old subject, like heat transfer, as suggested
below:

(a) Useful, interesting and contemporary topics can be introduced.

(b) The modeling (that converts given problems into equations) can be conducted
over control volumes of finite sizes, not of differential sizes, to avoid the appear-
ance of differential equations.
(c) Simple numerical methods that do not require much math can be introduced
and used to solve these algebraic equations.
(d) Matlab software can be further used to eliminate routine algebraic and arithme-
tic manipulations. In the Matlab environment, results can also be readily plotted
for students to see the trends of heat transfer physics.
(e) Finally, we can try to include as many daily-life problems as possible, in substi-
tution of some industrial-application problems. In college education, the most
important objective is to get students to be motivated and interested. Once they

2 | Heat Transfer: Lessons with Examples Solved by Matlab

 are, and have learned fundamentals, they will have plenty of opportunities to
approach industrial-application problems after graduation.

2. What Is the Subject of Heat Transfer?

2-1 Definition
(a) It consists of three sub-topics: Conduction,
Convection, and Radiation.
(b) It contains information regarding how energy is
transferred from places to places or objects to ob-
jects, how much the quantity of this energy is, the
methods of finding this quantity, and associated
discussions.
(c) In slightly more details, we may state that the
subject is for us to attempt to find temperature
(T) distributions and histories in a system. We can
further imagine that, as soon as we have found
T(x, y, z, t), then we are home. At home, we get to
do things very leisurely and conveniently, such as
lying on the couch, watching TV, and fetching a can of beer from the refrigerator.
Similarly, after having found T(x, y, z, t), we can find other quantities related to
energy (or heat) transfer at our leisure and convenience. Therefore, let us keep in
mind that finding T(x, y, z, t) is the key in the subject of heat transfer.

2-2 Second Income

Heat transfer is also intimately related to the subject of thermodynamics. In the latter,
generally, the quantity Q is either given, or calculated from the first law of thermody-
namics. For example, during the isothermal expansion process from state 1 to state 2
for a piston-cylinder system containing air, which is treated as an ideal gas, we have

U = Q + W, (1)

where Q and W denote heat transfer flowing into the system, and work done to the
system, respectively. For an ideal gas inside a piston-cylinder system undergoing an
isothermal process,

U = mcv(T2-T1) = 0,

Lesson 1 | 3
 and
W = -mRT ln(V2/V1).

Therefore,
Q = mRT ln(V2 /V1). (2)

Derivation of Eq. (2) dictates that Q be found based on the first law. In other words,
the first law is already consumed by our desire to find Q. It cannot be re-used to find
other quantities.
The subject of heat transfer is to teach us how to find Q from a different resource.
The relationship between heat transfer and thermodynamics can be better understood
with the following analogy.
John lives with his parents (thermodynamics). The household has been supported
by only one source of income (the first law of thermodynamics), earned by his parents.
John (Q) has been living under the same roof with his parents, being a good student all
the way from a little cute kid in kindergarten to an adult in college. Soon it will be also
his responsibility to seek an independent source of income (heat transfer), so that he
can move out of his parents house and get married.

Example 1-1
How long does it take for a can of Coke, taken out from the refrig-
erator at 5C, to warm up to 15C in a room at 25C? Relevant data
are: m = 0.28 kg, cv = 4180 J/kg-K, and A = 0.01m2.
Sol: Let us attempt to find the answer by first writing down the
first law of thermodynamics given by Eq. (1):

U = Q, (3a)

4 | Heat Transfer: Lessons with Examples Solved by Matlab

 in which W vanishes because there is none. From the subject of thermodynamics,
we have learned

U = mcvT.

Therefore, Eq. (1a) can be changed to

mcvT = Q. (3b)

At this juncture, we see that there are two unknowns in Eq. (3b), T and Q. We can
scratch our heads hard, and we are still unable to find another equation. In the analogy
given above, the first law is the only income of the household. We need the second
income, which is the second equation to solve Eq. (3b). This second equation is to be
provided by the knowledge gained from the subject of heat transfer.
See Problems 1-1 and 1-2.

3. Candle Burning for Your Birthday Party

You are celebrating your last birthday party before you graduate with a glamorous
degree. There are candles lighted on the cake for you to make a wish on and blow
out. Candle fires can also serve as a good example to illustrate heat transfer, with wax
melting, hot air rising, and flame illuminating associated with conduction, convection,
and radiation.

3-1 Unknowns and Governing Equations

You like to buy five boxes of candles. Your parents, being frugal, like to buy only three
boxes of candles. So, the first urgent question is: how long does it take for a candle of a
given size to finish burning? To find out the burning rate of a candle is, by no means,
a trivial question. We need to know the solution of quite a few variables in order to

Lesson 1 | 5
determine the rate. Let us count the number of variables (unknowns) and the number
of governing equations below. These two numbers should exactly be equal, no more,
no fewer.

(a) Three momentum equations govern the flow velocities in x, y, and z directions,
u, v, and w.
(b) Conservation of energy, which is essentially the first law of thermodynamics
mentioned above, governs T.
(c) Conservation of species governs mass fractions of fuel, air, and the combustion
product, including Yfuel, YO2, YN2, YCO2, and YH2O.
(d) Conservation of mass governs the density of unburned air,
(e) The equation of state, will be responsible for p = RT. Note that p and in (d)
and (e) may be switched.

In particular, in category (b), the equation takes the form of

enthalpy_in + conduction_in + radiation_in + combustion_in = (U/t)control-volume. (4)

The control volume can be the entire flame if a crude solution is sought; or it can be
small computational cells, xyz, for us to seek more accurate solutions. Regardless,
the essential point is that, in Eq. (4), all terms should be eventually expressed in terms
of T.
Derivation and solution of governing equations in category (a) should be learned
in your fluid mechanics course. Those in category (c) should be covered in your mass
transfer course if you study chemical engineering. Combustion is a subject that gener-
ally belongs to advanced topics offered in graduate schools.
In a sense, we may state that thermodynamics is only a part of heat transfer.
Nonetheless, it may be the most important part.

3-2 Solution Procedure

(a) Let us assume that the flame is subdivided by a numerical grid of 10,000 nodes.
At each node, there are 3 (u, v, w) + 1 (T) + 5 (5 mass fractions) + 2 (p and ) =
11 variables. Hence, there are exactly 110,000 nonlinear equations.
(b) They can be, in principle, solved simultaneously by using the Newton-Raphson
method. The software program nowadays is probably written in FORTRAN or
C language.
(c) Among 110,000 unknowns, there are a few that are of primary interest to you,
regarding how many candles you should purchase so that they will burn for two
hours (for some reason you like to have candlelight throughout your party).

6 | Heat Transfer: Lessons with Examples Solved by Matlab

 These unknowns are the uprising velocities of melted wax inside the wick. You
multiply them with wax density and the cross-sectional area of the wick to
obtain the burning rate of the candle.

4. Why Is the Subject of Heat Transfer Important?

4-1. Examples of Heat Transfer Problems in Daily Life

Examples related to heat transfer in daily life abound. Let us try to follow you around
in a typical morning in early December, and point out some heat transfer phenomena
you may observe or encounter.

(a) Although many college students are addicted to staying up late (and therefore
getting up late, and sometimes missing classes and exams), you are different. You
are a very self-disciplined student, getting up regularly at 7:00 every morning.
As soon as you get out of bed, you feel a little chilly. Your body loses energy by
conduction from your feet at 37C to the cold floor at 20C, by convection of air
flow surrounding your body, and by radiation from your skin to the wall and
the ceiling. As soon as you open the curtain in your room, you will additionally
lose some radiation from your skin to the outer space (assuming that the sky is
cloudless). Of course, at the same time, you will also receive radiation from the
wall and the ceiling, too.
(b) At 8 am, you walk to your car, which is parked on the driveway of your house near
New York. You may see a layer of ice forming on the car windshield. Overnight
the temperature of the glass dropped to -5C.
(c) Driving down the road, you also see an airplane flying over you. Why are your
car and the airplane capable of moving? Because there is considerable amount
of heat transfer going on inside the combustors. At high temperatures, air is
pressurized and pushes the piston, or exiting the exhaust nozzle of the airplane,
creating a thrust.
(d) On the way to the classroom, you drop by a coffee shop to buy a cup of coffee.
When your hand is holding the coffee cup, it feels warm. There is heat conduc-
tion transferring from the hot coffee to your hand.

There is no need for us to go on further. There is a temperature difference, there is

heat transfer flowing from hot bodies to cold bodies.

Lesson 1 | 7
 4-2 Examples of Heat Transfer Problems in Industrial Applications
(a) How did humans raise fires 4,000 years ago?
(b) Manufacturing bronze marked a significant step for humans to enter civilization.
How did ancient people do that?
(c) The steam engine was first invented 2,000 years ago,
and later improved and commercialized by James
Watt in 1763.
(d) The use of the steam engine induced the industrial
revolution.
(e) We wonder how George Washington spent his sum-
mers in Maryland without air conditioning.
(f) No heat transfer, no laptops that we are using. They
require electronic cooling.
(g) No heat transfer, no electricity that we are enjoying. Turbine blades need to be
cooled, too.
(h) No heat transfer, no cars, no airplanes, etc.

5. Three Modes of Heat Transfer

We mentioned conduction, convection, and radia-

tion in the previous section. They are three distinc-
tive modes in heat transfer.
When our hands touch an ice block, we feel cold
because some energy from our hands transfers to the
block. A child feels warm and loved when hugged by
his mother. This phenomenon is known as conduc-
tion. Usually, heat conduction is associated with
atoms or molecules oscillating in the solid. These
particles do not move away from their fixed lattice
positions.
When solids are replaced with fluids, such as air or
water, then the phenomena become heat convection.
Examples include air circulating in a room, oil flow-
ing in a pipe, and water flowing over a plate. Hence,
we may say that conduction is actually a special case
of convection. When air, oil, and water stop moving,
the convection problems degenerate to heat conduction.
Radiation is a phenomenon of electromagnetic wave propagation. It does not require
any media to be present. A notable example is sunlight, or moonlight, reaching our

8 | Heat Transfer: Lessons with Examples Solved by Matlab

earth through vast outer space, which is a vacuum.
If we talk about romantic stories, the moonlight can
be appropriately mentioned. In terms of heat transfer,
however, we are more interested in the sunlight.

6. Prerequisites

To learn heat transfer well with this textbook, we need to know: (1) the first law of ther-
modynamics for both closed systems and open systems, (2) Taylors Series Expansion,
and (3) some Matlab basics. These three subjects are briefly described in the appendices
of this lesson and next two.

7. Structure of the Textbook

This textbook is divided into seven major parts. Each part consists of three lessons.
Part 1 describes the introduction of the subject, three laws that govern three heat
transfer modes, and one of the simplest 1-D heat conduction steady state systems
without any heat generation.
In Part 2, non-zero heat generation, 1-D fins, and 2-D steady state heat conduction
are analyzed. They are slightly more advanced topics than those in Part 1.
We then consider transient heat conduction problems exclusively in Part 3. Zero-D,
1-D, and 2-D problems are presented, respectively, in Lessons 7, 8, and 9.
Part 4 enters the subject of heat convection. This part should constitute the most
important subject in heat transfer, and can be considered the peak of the heat transfer
course during the semester. In this part, external flows are studied.
In Part 5, we focus our attention on internal flows. Some internal flow problems
yield exact solutions. As in Part 4, the analyses start with hydrodynamics, and then
enter the thermal aspect.
Part 6 describes application-oriented topics, including free convection, turbulent
flows, and heat exchangers. They are also slightly more advanced than those in previ-
ous parts. Instructors may consider skip this part in their classes if time or students
understanding level does not permit.
Finally, Part 7 introduces basic knowledge of thermal radiation. In addition, some
problems combining radiation with conduction and convection are also considered.

Lesson 1 | 9
 Such organization is also made with assign-
ing students exams in mind. According to the
authors teaching experience at UMCP, straight-
forward simple grading systems may be better
than complicated grading systems. The latter is
exemplified by an array of activities, such as two
midterms (30%), homework sets (10%), team
projects (10%), quizzes (15%), presentations
(5%), final exam (25%), and attendance (5%), among others.
Instead, one part is associated with one exam. Seven exams cover a semester of 14
weeks, with one exam taken every two weeks. The attendance rates tend to be high;
students burden is lowered when exams are more frequent and are of smaller scope.
Furthermore, if too much is emphasized on group work, students may have the op-
portunities to hitch a ride with their group mates. Or they end up sharing the work, and
understanding only a small fraction of the entire group work that they are responsible
for. Hence the grades become less accurate measures of their true performances in the
class.
See Problem 1-3.

8. Summary

In this lesson, we have attempted to explain

(a) what the subject of heat transfer is,
(b) how it is related to thermodynamics,
(c) why it is important.
The content of the entire textbook is also outlined. Why the textbook is organized
into 7 parts is justified.

9. References

1. Wikipedia.
2. Claus Borqnakke and Richard E. Sonntag, Fundamentals of Classical
Thermodynamics, Wiley, 7th edition, 2008

10 | Heat Transfer: Lessons with Examples Solved by Matlab

 10. Exercise Problems

10-1 An electrical stove in the kitchen is initially at 25C. Assume that it generates Qg =
1.8 kW. At the same time, it loses energy to the air in the kitchen by a formula given as
Q_out = 3*(T-T) in Watts.
Other relevant data include: m = 1 kg, cv = 1200 J/kg-K, and T = 25C. Take t = 20
sec. For simplicity, use the explicit method during time integration.
Find T (t) of the stove, and plot T(t) vs. time for a period of 20 minutes.

10-2 A heat transfer system dissipates a rate that can be approximated by

Q = c1 T2, (in W)
where c1 = 0.1 W/k2. The system is initially at 200C. Other relevant data include:
m=2, cv = 1000 J/kg-K. Take t = 10 sec, and use the explicit method. What is the
temperature of the system at t = 10 minutes? Compare with the exact solution, too.

10-3 Which of the following states is true?

(a) The primary unknown in the equation governing conservation of energy is heat
flux, q.
(b) During the isothermal expansion process of a Carnot cycle for a piston-cylinder
system, the work is found by using the first law of thermodynamics.
(c) The primary unknown in the equation governing conservation of energy is T.
(d) In an isobaric process for a piston-cylinder system containing air, Q = cvT.

11. Appendix

Matlab Basics Used in this textbook are limited to only simple operations. Frequently
used ones are described below.

(A-1) Sum up 1, 2, 3, , 100

clc; clear
sum=0;
for i=1:100
sum = sum + i;
end
total = sum % = 5050

Lesson 1 | 11
 (A-2) Plot f(x) = x2 + sin (x) versus x between 0 and 2.
clc; clear
L=2; nx=20; dx=L/nx; nxp=nx+1; 6
for i=1:nxp
4
x(i)=(i-1)*dx; f
f(i)=x(i)*x(i)+sin(x(i)); 2
end
plot(x, f); xlabel(x); ylabel(f ); grid on 0
0 0.5 1 1.5 2
x
(A-3) Solve three linear equations simultaneously.
3x + y z = 3
x +4y +2z = 7
-x 3y + 4z = 0

clc; clear
a(1,1)=3; a(1,2)=1; a(1,3)=-1; b(1)=3;
a(2,1)=1; a(2,2)=4; a(2,3)=2; b(2)=7;
a(3,1)=-1; a(3,2)=-3; a(3,3)=4; b(3)=0;
q=a\b; q % = 1 1 1 The superscripted prime means transpose

12 | Heat Transfer: Lessons with Examples Solved by Matlab